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Dressing, Courtney Danielle. 2015. The Prevalence and
Compositions of Small Planets. Doctoral dissertation, Harvard
University, Graduate School of Arts & Sciences.
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June 15, 2017 6:03:20 PM EDT
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http://nrs.harvard.edu/urn-3:HUL.InstRepos:17467474
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(Article begins on next page)
The Prevalence and Compositions
of Small Planets
A dissertation presented
by
Courtney Danielle Dressing
to
The Department of Astronomy
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Astronomy & Astrophysics
Harvard University
Cambridge, Massachusetts
April 2015
c 2015 — Courtney Danielle Dressing
!
All rights reserved.
Dissertation Advisor: Professor David Charbonneau
Courtney Danielle Dressing
The Prevalence and Compositions
of Small Planets
Abstract
This thesis describes three investigations of the galactic abundance and properties of
small planets.
First, I revised the properties of the smallest Kepler target stars and searched
their light curves for transits using a custom transit detection pipeline. Combining
the detected population of 156 planet candidates (including one previously undetected
candidate) with an empirical estimate of the search completeness based on transit
injection and recovery simulations, I found occurrence rates of 0.24+0.18
−0.08 Earth-size
planets (1 − 1.5 R⊕ ) and 0.21+0.11
−0.06 super-Earths (1.5 − 2 R⊕ ) per M dwarf habitable zone.
Consequently, the most probable distances to the nearest non-transiting and transiting
potentially habitable planets are 2.6 ± 0.4 pc and 10.6+1.6
−1.8 pc, respectively.
Second, I conducted an adaptive optics imaging survey of 87 bright Kepler target
stars with ARIES at the MMT to search for nearby stars that might be diluting the
depths of the planetary transits. I identified visual companions within 1## for 5 stars,
between 1## and 2## for 7 stars, and between 2## and 4## for 15 stars. For all stars observed,
I placed limits (typically ∆Ks = 5.3 at 1## and ∆Ks = 5.7 at 2## ) on the presence of
undetected nearby stars.
Third, I investigated the composition of Kepler-93b, a 1.478 ± 0.019 R⊕ planet
with a 4.7-day orbit around a bright (V = 10.2) asteroseismically-characterized host
iii
star with a mass of 0.911 ± 0.033 M$ and a radius of 0.919 ± 0.011 R$ . Based on two
seasons of observations with HARPS-N at the Telescopio Nazionale Galileo and archival
observations from Keck/HIRES, I found a mass of 4.02 ± 0.68 M⊕ and a density of
6.88 ± 1.18 g cm−3 . Comparing Kepler-93b to the other nine exoplanets smaller than
2.7 R⊕ with well-constrained parameters, I found that all dense exoplanets with masses
of approximately 1 − 6 M⊕ are consistent with the same fixed ratio of iron to rock as the
Earth and Venus. There are currently no such planets with masses greater than 7 M⊕ .
Future measurements of the masses and radii of a larger sample of planets receiving a
wider range of stellar insolations will reveal whether the fixed compositional model found
for these seven highly-irradiated dense exoplanets extends to the full population of dense
1 − 6 M⊕ planets.
iv
Contents
Abstract
iii
Acknowledgments
x
Dedication
xiv
1 Introduction
1
1.1
The Small Star Advantage . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
The Small Star Challenge: Stellar Parameters . . . . . . . . . . . . . . . .
5
1.3
Living with a Star: Application of Solar Physics to Exoplanet Studies . . . 10
1.3.1
The Influence of Stellar Activity on Planet Detectability . . . . . . 10
1.3.2
The Influence of Stellar Activity on Planet Habitability . . . . . . . 17
1.4
Distinguishing Planets from Astrophysical False Positives . . . . . . . . . . 21
1.5
Expectations from Planet Formation Theory . . . . . . . . . . . . . . . . . 24
1.6
1.5.1
The Demographics of M Dwarf Systems
. . . . . . . . . . . . . . . 24
1.5.2
The Formation of Terrestrial Planets . . . . . . . . . . . . . . . . . 27
1.5.3
The Role of Photoevaporation . . . . . . . . . . . . . . . . . . . . . 31
The Interior Structure and Composition of Small Planets . . . . . . . . . . 34
1.6.1
The Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6.2
Other Terrestrial Worlds in the Solar System . . . . . . . . . . . . . 37
1.6.3
A Framework for Modeling Terrestrial Exoplanets . . . . . . . . . . 44
v
CONTENTS
1.6.4
The Abundance Ratios of Planet Host Stars . . . . . . . . . . . . . 48
1.6.5
Measuring Planetary Masses from Dynamical Interactions . . . . . 51
1.6.6
Radial Velocity Observations of Small Transiting Planets . . . . . . 54
1.6.7
Comparing the Observations to Models . . . . . . . . . . . . . . . . 56
1.7
Assessing Planetary Habitability . . . . . . . . . . . . . . . . . . . . . . . . 60
1.8
Planet Occurrence Across the HR Diagram . . . . . . . . . . . . . . . . . . 63
1.9
1.8.1
Evolved & High-Mass Stars . . . . . . . . . . . . . . . . . . . . . . 63
1.8.2
Sun-like Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.8.3
Low-Mass Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.8.4
The Role of Metallicity on the Frequency of Low-Mass Planets . . . 84
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2 Revised Properties for Low-Mass Kepler Target Stars and an Initial
Estimate of the Planet Occurrence Rate for Early M Dwarfs
87
2.1
2.2
2.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.1.1
The Small Star Advantage . . . . . . . . . . . . . . . . . . . . . . . 90
2.1.2
Previous Analyses of the Cool Target Stars . . . . . . . . . . . . . . 93
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.2.1
Stellar Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.2.2
Revising Stellar Parameters . . . . . . . . . . . . . . . . . . . . . . 98
2.2.3
Assessing Covariance Between Fitted Parameters . . . . . . . . . . 100
2.2.4
Validating Methodology . . . . . . . . . . . . . . . . . . . . . . . . 102
Revised Stellar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.3.1
2.4
Revised Planet Candidate Properties . . . . . . . . . . . . . . . . . . . . . 118
2.4.1
2.5
Comparison to Previous Work . . . . . . . . . . . . . . . . . . . . . 111
Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Planet Occurrence Around Small Stars . . . . . . . . . . . . . . . . . . . . 128
vi
CONTENTS
2.6
2.5.1
Correcting for Incomplete Phase Coverage . . . . . . . . . . . . . . 130
2.5.2
Calculating the Occurrence Rate . . . . . . . . . . . . . . . . . . . 131
2.5.3
Dependence on Planet Size . . . . . . . . . . . . . . . . . . . . . . . 132
2.5.4
Dependence on Stellar Temperature . . . . . . . . . . . . . . . . . . 138
2.5.5
The Habitable Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.5.6
Planet Candidates in the Habitable Zone . . . . . . . . . . . . . . . 141
2.5.7
Planet Occurrence in the Habitable Zone . . . . . . . . . . . . . . . 145
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3 The Occurrence of Potentially Habitable Planets Orbiting M Dwarfs
Estimated from the Full Kepler Dataset and an Empirical Measurement
of the Detection Sensitivity
155
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
3.2
Stellar Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.3
Planet Detection Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
3.4
3.3.1
Preparing the light curves . . . . . . . . . . . . . . . . . . . . . . . 169
3.3.2
Searching for Transiting Planets . . . . . . . . . . . . . . . . . . . . 171
Vetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.4.1
New Planet Candidate . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.4.2
Accounting for Transit Depth Dilution . . . . . . . . . . . . . . . . 178
3.4.3
False Positive Correction . . . . . . . . . . . . . . . . . . . . . . . . 180
3.4.4
Known Planet Candidates Missed by Our Pipeline . . . . . . . . . . 180
3.5
Planet Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.6
Planet Injection Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.7
3.6.1
Predicting Transit Detectability . . . . . . . . . . . . . . . . . . . . 186
3.6.2
Assessing Pipeline Performance . . . . . . . . . . . . . . . . . . . . 188
3.6.3
Calculating Search Completeness . . . . . . . . . . . . . . . . . . . 193
The Planet Occurrence Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 194
vii
CONTENTS
3.8
3.7.1
Dependence on Planet Radius & Period
. . . . . . . . . . . . . . . 198
3.7.2
Dependence on Planet Radius & Insolation . . . . . . . . . . . . . . 205
3.7.3
The Occurrence of Potentially Habitable Planets . . . . . . . . . . . 205
3.7.4
Implications of Systematic Biases in Modeled Stellar Radii . . . . . 211
Summary & Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4 Adaptive Optics Images III: 87 Kepler Objects of Interest
222
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
4.2
Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
4.3
Target Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.4
Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.5
Visual Companions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
4.6
Detection Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
4.7
Comparison to Previous Surveys . . . . . . . . . . . . . . . . . . . . . . . . 257
4.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
5 The Mass of Kepler-93b and the Composition of Terrestrial Planets
268
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.2
Observations & Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . 272
5.3
Analysis of the Radial Velocity Data . . . . . . . . . . . . . . . . . . . . . 273
5.3.1
5.4
Limits on the Properties of Kepler-93c . . . . . . . . . . . . . . . . 284
Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 285
6 Future Directions
6.1
289
Prospects for Detecting Small Planets Orbiting Nearby Bright Stars . . . . 290
6.1.1
Kepler & K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
6.1.2
TESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
6.1.3
CHEOPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
viii
CONTENTS
6.1.4
JWST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.1.5
PLATO 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
6.1.6
WFIRST-AFTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
6.1.7
Exo-C & Exo-S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.1.8
Current & Upcoming Ground-based Transit Surveys . . . . . . . . . 302
6.1.9
Current & Upcoming RV Projects . . . . . . . . . . . . . . . . . . . 305
6.1.10 Exoplanet Investigations in the Era of ELTs . . . . . . . . . . . . . 313
6.2
The Scope & Precision of Mass Measurement . . . . . . . . . . . . . . . . . 317
6.3
Initial Atmospheric Characterization . . . . . . . . . . . . . . . . . . . . . 320
6.3.1
6.4
Identifying Cloud- and Haze-Free Worlds . . . . . . . . . . . . . . . 322
Detecting & Interpreting Potential Biosignatures . . . . . . . . . . . . . . . 323
References
329
ix
Acknowledgments
First and foremost, I would like to thank David Charbonneau for serving as a
phenomenal thesis advisor for the last five years. His thoughtful comments, steadfast
encouragement, and pioneering spirit helped me grow as a scientist and I will always
be grateful that I had the opportunity to be part of such a wonderful research group.
Thank you, Dave, for providing me with the incredible opportunity to work on exciting
projects with world-class telescopes and for sharing your enthusiasm for scientific
discovery. I would also like to thank the full Charbonneau family for hosting highly
enjoyable group events and for sharing your home with me during my visit to the
Geneva Observatory. Thanks also to current and former Charbonneau group members
Sarah Ballard, Jacob Bean, Zach Berta-Thompson, Jayne Birkby, Chris Burke, Jessie
Christiansen, Francesca DeMeo, Jean-Michel Desert, Jason Dittmann, Francois Fressin,
Jonathan Irwin, Elisabeth Newton, and Sukrit Ranjan for your support and advice.
Thank you to David Latham for chairing my research exam committee, welcoming
me into the Kepler, HARPS-N, and TESS collaborations, writing postdoc reference
letters, and for sharing valuable insight. In addition, thank you for fostering a
collaborative spirit at the CfA by hosting dozens of meetings, wine tastings, and
dinner parties. Thank you to Andrea Dupree for serving on my thesis and research
exam committees and for providing me with the opportunity to observe at the MMT.
In addition, thank you to Elisabeth Adams for teaching me how to reduce ARIES
observations. I would also like to thank Ruth Murray-Clay for serving on my research
exam committee, John Johnson for chairing my thesis committee, and Greg Laughlin for
agreeing to travel to Boston to serve as the external examiner on my thesis committee.
x
CHAPTER 0. ACKNOWLEDGMENTS
Special thanks to Andrew Howard for writing dozens of postdoc reference letters and
sharing insight into planet occurrence rates.
Thank you to the members of the Kepler Team, the HARPS-N Consortium, and
the TESS Team for allowing me to join you on a voyage of scientific discovery. Lars
Buchhave, Xavier Dumusque, Sara Gettel, Mercedes Lopez-Morales, Dave Phillips,
Dimitar Sasselov, and Andrew Vanderburg, thank you for sharing your insight during
our CfA HARPS-N meetings and for teaching me more about instrumentation, stellar
activity, radial velocity data reduction, and planet formation. Natalie Batalha, thank
you for encouraging my participation in the Kepler team and in exoplanet science in
general. Josh Winn and Peter Sullivan, thank you for inviting me to participate in the
TESS simulations group. David Aguilar and Christine Pulliam, thank you for helping us
publicize our results and for running the fabulous Public Observatory Nights series.
Thank you to the full CfA community for creating a wonderful environment to
pursue scientific research. In particular, thank you to the members of the Solar, Stellar,
and Planetary Division, the broader CfA exoplanet community, the observatory night
docents, and the graduate student body. My experience at Harvard wouldn’t have
been nearly as much fun without you! In addition, many thanks to Peg Herlihy, Robb
Scholten, Donna Adams, Geri Barney, Lisa Bastille, Elke Blackstone, Kathy Campbell,
and Nayla Rathle for keeping track of everything and everyone. Thank you also to the
members of the Harvard Origin of Life Initiative for enriching my graduate experience
with fascinating discussions regarding the origin of life on the Earth and the quest for
life on other worlds.
I would also like to thank the Department of Astrophysical Sciences at Princeton
xi
CHAPTER 0. ACKNOWLEDGMENTS
University for integrating undergraduates so fully into the program and for providing
me with a nurturing and highly educational community during college. Thank you in
particular to Jill Knapp, Dave Spiegel, Ed Turner, and Mike McElwain for advising me
on my junior papers and senior theses. Thank you also to Neta Bahcall, Cullen Blake,
Jim Gunn, David Spergel, and Michael Strauss for your advice and support.
I was fortunate to have dozens of fantastic teachers and professors, but I am
particularly indebted to my sixth grade teacher Rocky Curtis and my high school
astronomy teacher Lee Ann Hennig. Mr. Curtis, thank you for going beyond the regular
curriculum and teaching us about cutting edge scientific discoveries. Mrs. Hennig, thank
you very much for providing me with my first introduction to “real” research and for
encouraging me to pursue a PhD in astrophysics. I would also like to thank the Thomas
Jefferson High School for Science & Technology community as a whole for fostering an
environment in which scientific curiosity and enthusiasm for learning were celebrated.
To my friends both at and beyond the CfA, thank you for supporting me and
reminding me to take breaks from research. Thank you to Rue Wilson for helping me
find balance and become more confident in myself.
Finally, thank you to my family for encouraging my early interest in science and
always supporting me. Thank you to my father, Steven Dressing, for teaching me about
the scientific method via dozens of elementary school science projects and for introducing
me to backyard astronomy. Thank you to my mother, Julie Dressing, for proofreading
far too many school essays and for teaching me how to construct effective arguments. I
am grateful to my siblings, James and Kelsey Dressing, for believing in me even when
I doubted myself and for preventing me from taking myself too seriously! Thank you
xii
CHAPTER 0. ACKNOWLEDGMENTS
to my grandparents, June Stumpe, Russell Stumpe, and Florence Hojnacki, and to my
extended family for your undying support. In particular, thank you to Anne and Pat
for telling me about your experiences working in the space industry and for sending me
numerous space-related care packages.
My graduate research was supported by the National Science Foundation through
the Graduate Research Fellowship Program, the Kepler Participating Scientist Program
via grants NNX09AB53G and NNX12AC77G awarded to David Charbonneau, the
NASA Exoplanets Research Program under grant NNX15AC90G awarded to David
Charbonneau, NASA via grant NNX10AK54A awarded to Andrea Dupree, and the John
Templeton Foundation.
xiii
For my amazing family
xiv
Chapter 1
Introduction
One of the questions that has fascinated humanity for millennia is whether there is
intelligent life beyond the Earth. The answer to that question is still unknown, but
the prospects for detecting life elsewhere in the universe have changed considerably
over the last twenty years. Prior to the first detections of planets orbiting other stars,
astronomers could only speculate whether any of the multitude of stars in the night
sky were accompanied by their own pale blue dots. The detections of Latham’s world
(hereafter HD114762b, Latham et al. 1989), the pulsar planets (Wolszczan & Frail
1992), and 51 Peg b (Mayor & Queloz 1995) heralded in the era of the planet detection
gold rush and transformed the quest for other worlds from science fiction into an entire
subfield of astronomy.
The subsequent chapters of this thesis focus on recent work to estimate the frequency
of small, potentially habitable planets orbiting small stars (Chapters 2 & 3), identify
possible astrophysical false positives masquerading as planets (Chapter 4), and constrain
the compositions of small planets (Chapter 5). The purpose of this chapter is to provide
1
CHAPTER 1. INTRODUCTION
a brief overview of the necessary background material. In Section 1.1, I discuss the
motivation for targeting M dwarfs when searching for small planets and potentially
habitable planets in particular. These stars are smaller, less massive, and cooler than
Sun-like stars, thereby increasing the detectability of any associated small planets.
However, the “Small Star Advantage” of enhanced planet detectability is partially
offset by the “Small Star Challenge” of determining accurate stellar parameters for
low-mass stars. In Section 1.2, I discuss the current discrepancies between theoretical
models of low-mass stars and empirical observations. I also describe empirical relations
that provide a convenient way to reduce the reliance on theoretical models. Section 1.3
is devoted to a description of solar physics and the implications of stellar activity on the
detectability and habitability of exoplanets.
Even if the parameters of the host star are well-determined, putative planetary
candidates may actually be false positives. Section 1.4 presents several tests to distinguish
between bona fide transiting planets and astrophysical false positives. I later apply these
tests in Chapters 3 and 4.
The penultimate chapter of this thesis discuss the compositions of small planets.
Sections 1.5 and 1.6 provide context for those chapters by describing theoretical models
of planet formation and the resulting expectations for the compositions of small planets.
Finally, Sections 1.7 discusses important considerations for the habitability of exoplanets,
Section 1.8 reviews current estimates of the planet occurrence rate for various types of
stars, and Section 1.9 provides a brief introduction to the remainder of this thesis.
2
CHAPTER 1. INTRODUCTION
1.1
The Small Star Advantage
Detecting small planets orbiting distant stars is challenging, but the difficulty of the
problem can be reduced by targeting smaller stars (Charbonneau & Deming 2007).
There are several key reasons why potentially habitable planets are easier to detect if
they orbit M dwarfs than if they orbit Sun-like stars:
Deeper Transit Depth: The decrease in brightness δ due to the transit of a planet
across the disk of its host star is given by
πRP2
δ=
πR!2
(1.1)
where RP is the radius of the planet and R! is the radius of the host star. For the
transit of an Earth-size planet across the disk of a Sun-like star, δ = 84 ppm. In
contrast, the transit depth for an Earth-size planet orbiting an early M dwarf is
250 ppm.
Increased Likelihood of Transit: The geometric likelihood PT that a planet will
appear to transit its host star depends on the ratio of the stellar radius R! to the
semimajor axis a of the planetary orbit. For a planet in an eccentric orbit with
eccentricity e and argument of periapsis ω, the full formula can be expressed as in
Kipping (2014):
R! + Rp
PT =
a
!
1 + e sin ω
1 − e2
"
(1.2)
For the purpose of determining the yield of a transit survey, one could then
marginalize over ω to obtain (Barnes 2007):
R! + Rp
PT =
a
3
!
1
1 − e2
"
(1.3)
CHAPTER 1. INTRODUCTION
Due to the cooler temperatures of M dwarfs, potentially habitable planets orbiting
M dwarfs have much smaller orbital semimajor axes and therefore are more likely
to transit. Specifically, the geometric probabilities of transit are 0.5% and 0.9%
for potentially habitable planets orbiting a Sun-like star and an early M dwarf,
respectively. Importantly, Equation 1.3 reveals that planets in eccentric orbits
are more likely to transit than planets in circular orbits. The corollary is also
true: a planet that is observed to transit is more likely to have an eccentric orbit.
As discussed in Chapter 3, this effect should be considered when estimating the
likelihood of planetary transit in order to compute the planet occurrence rate.
More Frequent Transits: A transiting planet in the habitable zone of a Sun-like star
would have an orbital semimajor axis of roughly 1 AU and transit only once per
Earth year. In contrast, the habitable zones of early M dwarfs are much closer to
the star. A transiting planet in the habitable zone of an early M dwarf would have
an orbital semimajor axis of approximately 0.3 AU, corresponding to an orbital
period of approximately 80 days and roughly four transits per Earth year.
Larger Radial Velocity Amplitude: A planet with mass MP in orbit around a star
with mass M! induces a radial velocity signal with semiamplitude K given by:
#
G
K=
MP sin i (M! + MP )−1/2 a−1/2
(1.4)
(1 − e2 )
where G is the gravitational constant and a is the orbital semimajor axis (Lovis &
Fischer 2010). Alternatively, we can employ Kepler’s third law to write K in terms
of the orbital period P :
K=
!
2πG
P
"1/3
4
MP sin i
2/3
M!
1
$
(1 − e2 )
(1.5)
CHAPTER 1. INTRODUCTION
where we have also assumed that the mass of the planet, MP , is much less than
the mass of the star, M! . Due to the lower mass of the host star and the proximity
of the planet to the star, potentially habitable planets orbiting M dwarfs induce
larger radial velocity semi-amplitudes than planets orbiting Sun-like stars (e.g.,
21 cm s−1 for an early M dwarf host star versus 9 cm s−1 for a Sun-like host star).
Galactic Demographics: The peak of the initial mass function occurs within the
M dwarf spectral class (Salpeter 1955; Chabrier 2003) and approximately threequarters of the stars in the solar neighborhood are M dwarfs (Henry et al. 2006;
Winters et al. 2015). Accordingly, including M dwarfs in target lists significantly
expands survey samples.
For reference, Table 1.1 displays the predicted transit depth, transit probability, and
radial velocity semiamplitude for several varieties of planetary systems. The examples
most relevant to this thesis are the various combinations involving early M dwarfs
(Chapters 2 and 3) and the Kepler-93 system (Chapter 5).
1.2
The Small Star Challenge: Stellar Parameters
As demonstrated by the equations in Section 1.1, planetary properties are typically
determined relative to the properties of their host stars. Accordingly, careful host
star characterization is essential for accurately and precisely determining planet radii
and masses. Determining the properties of small stars is notoriously challenging, but
fortunately there are two pathways toward empirical characterization of low-mass stars:
5
CHAPTER 1. INTRODUCTION
Table 1.1. Detectability of Various Planetary Systems
Planet
Star
Sun
Earth in Habitable Zone
K0
M0
M3 M4.5 M6.5
M8
Jupiter
Sun
Close-in Super-Earth
Sun K-93
M0
Stellar Properties
M! ( M! )
1 0.85
0.58
0.41
0.16
0.10
0.09
1
1
0.91
0.58
R! ( R! )
1
0.80
0.58
0.41
0.21
0.16
0.10
1
1
0.92
0.58
5777
5347
3907
3412
3026
2800
2600
5777
5777
5669
3907
Planetary Properties
Mp ( M⊕ )
1
1
1
1
1
1
1
318
4
4
4
Rp ( R⊕ )
1
1
1
1
1
1
1
318
1.5
1.5
1.5
P (Days)
365
225
80
40
17
11
4.8
4333
4.7
4.7
4.7
1
0.69
0.30
0.17
0.07
0.04
0.02
5.2
0.06
0.05
0.05
Fp ( F⊕ )
1
0.99
0.76
0.71
0.69
0.70
0.69
0.98
0.99
1.01
0.04
Teq (K)b
255
254
238
234
232
233
232
112
1085
1037
611
RV & Transit Detectability
K (m s−1 ) 0.09 0.12 0.21
0.34
0.85
1.3
1.9
12.5
1.5
1.6
2.4
δ (ppm)
Teff (K)
a (AU)
a
PT (%)
84
133
252
501
1890
3290
8080
10600
184
218
551
0.46
0.54
0.89
1.12
1.41
1.66
1.90
0.09
8.43
8.00
5.84
Note. — For the K0, M0, and M3 dwarfs, the assumed stellar radii and effective temperatures are from
Table 12 of Boyajian et al. (2012). The masses were estimated by consulting the catalog of low-mass stars
in their Table 6. For Kepler-93 (listed as K-93), the stellar properties are from Ballard et al. (2014). The
properties for the M4.5, M6.5, and M8 were adopted from estimates for GJ 1214 (Charbonneau et al.
2009), Gl 406 (Doyle & Butler 1990; Pavlenko et al. 2006), and VB 10 (Linsky et al. 1995), respectively.
a
The insolation flux boundaries of the habitable zone depend on the spectral type of the host star. See
Section 1.7 for details.
b
Calculated assuming that the planet has an albedo of 0.3 and re-radiates heat from the entire surface.
This ignores the role of gravitational contraction and the greenhouse effect.
6
CHAPTER 1. INTRODUCTION
1. Detached, double-lined eclipsing binaries: Eclipsing binary systems provide
excellent physical laboratories for precisely determining the masses1 and radii of
stars (Andersen 1991; Torres et al. 2010). For systems that have not undergone
mass exchange or significant tidal interactions, the properties of stars in binaries
may be representative of single stars. Figure 1.1 displays the masses and radii for
low-mass stars in the Detached Eclipsing Binary Catalog (DEBCat2 , Southworth
2014).
2. Interferometric Measurements: In special cases, the disk of the star can be
resolved on the sky, enabling a direct measurement of the angular diameter of the
star (e.g., Boyajian et al. 2012; von Braun et al. 2014). If the distance to the star
is accurately known from trigonometric parallax, then the angular measurement
can be converted into a physical radius. Due to the large number of optics required
to combine the light from multiple telescopes, current interferometry projects are
restricted to very bright stars, but empirical relations derived from inferometric
observations can be applied to fainter stars (e.g., Boyajian et al. 2012; Mann
et al. 2013a; Newton et al. 2015). Figure 1.1 also displays the radii and masses of
interferometrically-characterized stars.
1
The masses M1,2 (in units of solar masses) and orbital semimajor axis a (in units of solar radii) for
stars in eclipsing binaries may be estimated from the radial velocity semi-amplitudes K1,2 (in units of
km s−1 ) using the following equations:
%
&3/2
M1,2 sin3 i =1.036149 × 10−7 1 − e2
(K1 + K2 )2 K2,1 P
%
&1/2
a sin i =1.976682 × 10−2 1 − e2
(K1 + K2 ) P
(1.6)
(1.7)
where P is the orbital period in days, e is the eccentricity, and i is the inclination (Torres et al. 2010).
2
http://www.astro.keele.ac.uk/~jkt/debdata/debs.html
7
CHAPTER 1. INTRODUCTION
The combined sample of interferometry targets and eclipsing binary stars has enabled
direct comparison between empirical measurements and stellar models. Compared to
interferometric measurements, stellar models overpredict the temperatures of M dwarfs
by roughly 3% and underpredict the radii by approximately 5% (Boyajian et al. 2012).
In addition, the influence of metallicity on stellar radii is overstated in stellar models;
the relationship between temperature and radius varies less as a function of metallicity
than would be predicted by stellar models (Newton et al. 2015).
The discrepancies between stellar models and empirical observations can significantly
change the assumed properties of any associated planets. For example, Ballard et al.
(2013) revised the classification of the late K dwarf Kepler-61 to reveal that the planet
Kepler-61b is likely too hot to be habitable. Newton et al. (2015) later refit the radii of
Kepler M dwarfs using empirical relations rather than stellar models. They found that
the radii of planet candidates orbiting M dwarfs were typically underestimated by 15%
in the Huber et al. (2014) stellar catalog.
In the future, more accurate tables of molecular opacities and more sophisticated
models of stellar convection will likely reduce the disagreement between stellar models
and empirical observations. In addition, parallaxes from Gaia (Perryman et al. 2001) will
increase the sample of low-mass stars with well-determined distances and technological
advances will allow interferometric surveys to observe fainter targets.
8
CHAPTER 1. INTRODUCTION
Radius (RSun)
1.0
BCAH98
PARSEC
Dartmouth
DEBCat Primaries
DEBCat Secondaries
Interferometry Boyajian+2012
Interferometry von Braun+2014
0.1
0.1
1.0
Mass (MSun)
Figure 1.1: Empirically estimated masses and radii of low-mass stars (points) versus
theoretical stellar isochrones (lines). The blue (crimson) points are primaries (secondaries)
in eclipsing binaries from Southworth (2014). Note that some of the binaries contain
evolved stars. The purple and orange points are stars with interferometrically constrained
radii from Boyajian et al. (2012) and von Braun et al. (2014), respectively. The Boyajian
et al. (2012) sample also includes stars with radii determined by previous interferometric
studies (Lane et al. 2001; Ségransan et al. 2003; Berger et al. 2006; di Folco et al. 2007;
Boyajian et al. 2008; Kervella et al. 2008; Demory et al. 2009; van Belle & von Braun
2009; von Braun et al. 2011, 2012). The gray, navy, and green lines are 1 Gyr solar
metallicity isochrones from BCAH98 (Baraffe et al. 1998), the PAdova and tRieste Stellar
Evolution Code (PARSEC, Bressan et al. 2012), and the 2012 update to the Dartmouth
Stellar Evolutionary Database (Dotter et al. 2008; Feiden et al. 2011).
9
CHAPTER 1. INTRODUCTION
1.3
Living with a Star: Application of Solar Physics
to Exoplanet Studies
In contrast to low-mass stars, theoretical models of Sun-like stars are quite advanced. The
more mature state of models for Sun-like stars is partially due to the reduced complexity
of working at hotter temperatures at which the primary opacity sources are atomic
rather than molecular, but the largest advantage is that the nearest and best studied
star is a G2 dwarf. Unsurprisingly, models of Sun-like stars have benefited tremendously
from a rich legacy of solar physics. The combination of centuries of high-cadence,
spatially-resolved solar observations and modern sophisticated magnetohydrodynamical
models have allowed solar physicists to study phenomena at a level of detail that is
current unreachable for the vast majority of stars. The most important implication for
this thesis is that a star is a dynamic environment whose behavior can influence both the
detectability and habitability of planets.
1.3.1
The Influence of Stellar Activity on Planet
Detectability
From a planet detectability standpoint, the most challenging obstacles are starspots3
and the somewhat ambiguously defined umbrella term of “stellar jitter.” Starspots are
regions where the magnetic field lines extend through the stellar surface. The presence
of the field lines inhibits convection and causes the area around the starspot to be cooler
3
Interestingly, the first observations of sunspots predate this thesis by at least 2180 years. See Clark
& Stephenson (1978) and Wittmann & Xu (1987) for a review of sunspot sightings in ancient China.
10
CHAPTER 1. INTRODUCTION
than the surroundings. In photometric observations, the cooler temperatures of starspots
cause them to appear fainter at a given wavelength than the unspotted surface.
As starspots rotate into and out of view, the brightness of the star will therefore
appear to change in a quasi-predictable fashion. In general, stars have multiple starspots,
so the net brightness variations are a combination of multiple sinusoids with amplitudes
set by the relative sizes and brightnesses of starspots compared to the full stellar disk and
periods determined by the rotational period of the star at the latitude of the starspot.
The overall morphology of the light curve of a spotted star will change gradually over
time as spots appear, grow, shrink, merge, and disappear.
On the Sun, typical spots have lifetimes of days to weeks and have sizes
! 2 square degrees (Schrijver 2002). Individual spots are often formed in “active nests”
with lifetimes of roughly 4–6 solar rotation periods (e.g., Becker 1955; Gaizauskas et al.
1983; Castenmiller et al. 1986; Brouwer & Zwaan 1990). Similar “active site” lifetimes
of weeks to months have been observed for other stars. The central regions of sunspots
(within the umbra) are typically 1800 K cooler than the unspotted photosphere whereas
the average temperature difference across the full spot including the penumbra is 600 K
(Schrijver 2002).
At solar minimum, starspots are formed at moderate latitudes of 20–30◦ and are
relatively rare. For example, the typical number of spots visible in 2008 during the last
solar minimum was only 1–5.4,5 As the 11-year solar cycle progresses from solar minimum
to solar maximum, the active latitude of sunspots gradually shifts towards the equator,
4
http://www.ips.gov.au/Solar/1/6
56
11
CHAPTER 1. INTRODUCTION
obeying “Spörer’s Law of Zones” (Carrington 1858; Maunder 1903) and yielding a figure
reminiscent of a butterfly when active sunspot latitude is plotted as a function of time
(Maunder 1904). By solar maximum, the Sun usually features roughly 80–200 spots
covering 0.1 − 0.5% of the visible hemisphere (Hathaway 2015).
The fraction of the surface covered by spots is known as the “filling fraction” and
is generally believed to increase with decreasing stellar mass. For active stars, filling
fractions as high as 50% are suggested based on fitting TiO absorption features with
two-temperature models (e.g., O’Neal et al. 1996, 1998, 2004). Using Doppler imaging,
Barnes & Collier Cameron (2001) investigated the star spot distribution on the M dwarfs
HK Aqr and RE 1816+541. The reconstructed images of both stars displayed spotted
surfaces, but the spots on HK Aqr were concentrated at low latitudes whereas the spots
on RE 1816+541 appeared latitudinally dispersed.
Starspots can complicate transit surveys by leading to incorrect assumptions about
the fraction of the surface blocked by a transiting planet and causing systematic offsets
in planet radius estimates. There are two main cases to consider:
1. The star is heavily spotted, but the chord of planetary transit does not
cross any starspots. In this case, the transiting planet will block a brighter
region of the stellar surface as it transits. The bright, unspotted region will be
contributing a large fraction of the stellar flux than simple geometry and limb
darkening would suggest. In this case, the radius of the transiting planet will be
overestimated because the planet will block a larger fraction of the flux than would
be expected for the transit of an unspotted star.
2. The transit chord usually intersects starspots, but the spots cannot
12
CHAPTER 1. INTRODUCTION
be resolved in individual transits. Because starspots are fainter than their
surroundings, the passage of a transiting planet over a starspot will cause a slight
bump in brightness relative to a typical transit profile. If the signal-to-noise level
or cadence of the observations is too low to discern the presence of the starspot
in individual transit events, then it is likely that the observer might not realize
that the depths of some transits are diluted. In contrast to the previous case, if
the transiting planet crosses a region of the star that is typically more spotted
than the rest of the star, then the radius of the planet will be underestimated
because the regions of the star along the transit chord will be contributing a
lower-than-expected fraction of the stellar flux.
Both of these challenges are discussed at length by Pont et al. (2008) for the case of
HD 189733b. In addition, Oshagh et al. (2013) addressed the possibilities that poor
modeling of star spots could also lead to erroneous transit duration estimates, incorrect
stellar limb darkening coefficients, or spurious transit timing variations.
In precise data sets acquired at high cadence, starspots can sometimes be resolved
in single transits. This greatly simplifies the determination of the planet radius in Case
2 because the observer is able to confirm that the light curve morphology is consistent
with a planetary transit containing a spot-induced brightening event. Occasionally,
the planetary orbital period, orbit inclination, and starspot lifetime are such that the
planet encounters the same starspots (or the same active latitude of starspots) multiple
times. For systems in these configurations such as WASP-4b (Sanchis-Ojeda et al.
2011), CoRoT-2b (Nutzman et al. 2011), Kepler-17b (Désert et al. 2011b), Kepler-63b
(Sanchis-Ojeda et al. 2013b), and Qatar-2b (Mancini et al. 2014), the repeated transit
13
CHAPTER 1. INTRODUCTION
of starspots can be used to probe the angle between the planetary orbit and the stellar
spin axis as explained by Sanchis-Ojeda et al. (2013a). Starspots can also be used to
constrain the planetary obliquity for misaligned planets if the host star has spots at
particular active latitudes (e.g., the case of HAT-P-11, Sanchis-Ojeda & Winn 2011).
For radial velocity observations, the general phrase “stellar jitter” is often use to
encompass a wide variety of radial velocity variations caused by stellar physics. Although
these effects are typically viewed as a noise source, it is important to remember that
they are the manifestation of real stellar phenomena. An optimistic astronomer might
hope that we will be able to understand these signals more accurately in the future and
interpret them rather than treating them as an insurmountable noise floor. As outlined
by Dumusque et al. (2011), the dominant sources of stellar jitter are:
Oscillation: On short timescales, solar-type stars vibrate due to pressure waves. These
oscillations have a timescale of 5–15 minutes (Schrijver & Zwaan 2000; Broomhall
et al. 2009) and yield radial velocity changes of 10–400 cm s−1 (Schrijver & Zwaan
2000). According to theoretical predictions, the oscillation timescales and RV
amplitudes should decrease with decreasing stellar mass. Christensen-Dalsgaard
(2004) argued that the oscillation frequency scales with the square root of the
mean stellar density and that the amplitude is directly proportional to stellar
luminosity and inversely proportional to stellar mass. Due to the relatively short
oscillation timescales for the FGK stars that are the favored targets of most RV
surveys, the conventional approach to combating radial velocity variations due to
oscillation is to average out the oscillation signature by using integration times that
are at least as long as the oscillation period (Dumusque et al. 2011). For example,
14
CHAPTER 1. INTRODUCTION
the HARPS-N survey has adopted a set integration time of 15 minutes for bright
targets and 30 minutes for fainter targets. (In practice, the observations are often
divided into a series of co-adds in order to avoid saturation.)
Granulation Phenomena: Stellar convection introduces radial velocity signatures on
several different timescales based on the size of the region under consideration.
“Granulation” refers to the smallest convective patterns, regions with diameters
of ≤ 2000 km and lifetimes shorter than 25 minutes (Title et al. 1989; Del Moro
2004). Similarly, “mesogranulation” refers to patterns at intermediate scales
(2000–15000 km) with lifetimes of several hours (Harvey 1984; Palle et al. 1995;
Schrijver & Zwaan 2000). At the largest scales, “supergranulation” produces
convective patterns with diameters of 15000–40000 km and lifetimes as long as
33 hours (Del Moro et al. 2004). Granulation noise can be reduced by acquiring
longer RV observations (roughly 30 minutes in total) so that the total integration
time exceeds the typical granulation timescale. Integrating for multiple hours to
combat mesogranulation noise is unrealistic, but noise due to both mesogranulation
and supergranulation can be reduced by taking several observations per night
and spacing the observations as far apart as possible (Dumusque et al. 2011).
In practice, these observations are usually separated by 2–3 hours given the
observational constraints of changing airmass and contamination from a bright,
nearby G2 star.
Activity: In an unspotted region of the star, the upwelling of material in stellar
convection cells results in a net convective blueshift (Beckers & Nelson 1978). The
amplitude of the shift depends on the details of convection within the particular
star, but the convective blueshift for Sun-like stars is likely similar to the roughly
15
CHAPTER 1. INTRODUCTION
300 m s−1 convective blueshift observed for the Sun (e.g., Dravins et al. 1981).
However, because the strong magnetic field lines that cause starspots and plages
also hinder convection, regions of the stellar surface near starspots or plages will
not exhibit convective blueshift. Instead, these regions will appear to be redshifted
relative to the unspotted regions of the star. Accordingly, the overall RV of the
star will appear to vary as starspots and plages rotate in and out of view. The
amplitude of this RV variation is expected to be similar to the 40–140 cm s−1 range
observed for the Sun near solar minimum and maximum, respectively (Meunier
et al. 2010). At a lesser level, starspots and plages also cause a flux-dependent shift
in the radial velocity. The bulk radial velocity signature is the sum of the signal
from the blue-shifted and red-shifted hemispheres. If one hemisphere has fewer
starspots and more plages, then that hemisphere will appear brighter and dominate
the RV signal. The amplitude of this effect is predicted to be up to 40 cm s−1 for
the Sun during solar maximum and is likely comparable for other FGK stars. For
late K and M dwarfs, however, the magnitude is expected to be larger for (e.g.,
Reiners et al. 2010; Barnes et al. 2011; Andersen & Korhonen 2015). In general,
assuming that the rotational period of the star is sufficiently different from the
orbital periods of known planets, the RV contribution from longer-term variations
in stellar activity could be removed using a filtering procedure such as determining
offsets for chunks of data (e.g., Hatzes et al. 2010; Dumusque et al. 2014).
16
CHAPTER 1. INTRODUCTION
1.3.2
The Influence of Stellar Activity on Planet Habitability
As far as we know, all life on Earth is either directly or indirectly dependent on solar
radiation. Electromagnetic radiation from a host star is therefore frequently considered
an essential requirement for habitability,7 but stellar radiation can also be a potential
danger to living organisms.
From the perspective of life on Earth, solar activity (often described as “space
weather” in this context) can be divided into three main categories: (1) coronal holes
causing fast streams in the solar wind, (2) solar flares, and (3) coronal mass ejections.
Speed increases in the solar wind due to coronal holes can lead to enhanced aurorae and
weak or intermediate geomagnetic storms (Tsurutani et al. 1995, 2006). Solar flares can
also instigate geomagnetic storms, although they are significantly less important than
coronal mass ejections (Gosling 1993).
Coronal mass ejections (CMEs) are the rather violent expulsion of plasma from the
Sun into interplanetary space (Webb & Howard 2012). The ejected plasma is bound to
a magnetic field that is typically wound up into a “flux rope.” Ejected CMEs can be
comparable in size to the full stellar disk and have typical masses of 1.6 × 1012 kg (Webb
& Howard 2012). CMEs are often responsible for triggering geomagnetic storms in the
Earth’s magnetosphere (Gosling 1993) and can cause significant damage to electronics.
In general, CMEs occur roughly 1–5 times per day, with higher rates observed closer
to solar maximum (St. Cyr et al. 2000; Gopalswamy et al. 2005; Gopalswamy et al.
7
A notable exception is a free-floating planet that is dependent on geothermal decay and residual heat
from accretion as energy sources (e.g., Abbot & Switzer 2011).
17
CHAPTER 1. INTRODUCTION
2006), but with the CME cycle lagging behind the sunspot cycle by several months
(Cliver & Webb 1998; Gopalswamy et al. 2003). The speeds of individual CMEs vary
significantly over two orders of magnitude (from 20 km s−1 to faster than 2500 km s−1 ),
with the average CME speed increasing from roughly 150 km s−1 near solar minimum to
approximately 475 km s−1 near solar maximum (Webb & Howard 2012).
On planets orbiting M dwarfs, the danger to alien lifeforms from stellar activity may
be more pronounced due to longer stellar active lifetimes and the closer proximity of the
habitable zone. Based on Sloan Digital Sky Survey spectra of > 38, 000 low mass stars,
West et al. (2008) determined the fraction of active M dwarfs (defined as those displaying
Hα emission) as a function of spectral type. They observed that the activity lifetime
has a strong dependence on spectral type, with early M dwarfs (M0–M3) displaying
typical active lifetimes of 0.5–2 Gyr whereas M5–M7 dwarfs have activity lifetimes of
7–8 Gyr. The rapid increase in activity lifetime between M3 and M5 coincides with the
transition between partially convective and fully convective stellar interiors, suggesting
that the processes governing Hα emission and magnetic activity in general depend on
interior stellar physics, such as the possible transition from a solar-like dynamo to an α2
(Chabrier & Küker 2006) or turbulent (Durney et al. 1993) dynamo.
A planet orbiting an early M dwarf will therefore experience a longer era of intense
stellar flares and coronal mass ejections (CMEs) than a planet orbiting a Sun-like star. If
both planets are in the habitable zones of their respective stars, then the M dwarf planet
will be much closer to its host star and will be more likely to lie within the path of a
given CME. For that reason, several authors have expressed concerns that the habitable
zones of M dwarfs might not be very hospitable environments. When discussing the
effects of high energy radiation on possible lifeforms, it is useful to remember that our
18
CHAPTER 1. INTRODUCTION
own understanding of life in the universe is limited to a single example: life on Earth.
Similarly, we have only one example of biogenesis. To confound matters further, the
exact timing of biogenesis on Earth is uncertain because the early rock record is sparse.
Knowledge of the UV environment is vital for correctly interpreting possible
biosignatures in planetary atmospheres because UV radiation may facilitate rapid
atmospheric loss by increasing the temperature of the upper atmosphere (Tian et al.
2008) and has a significant effect on atmospheric chemistry. For instance, far-UV (FUV;
λ = 912–1700 Å) and near-UV (NUV; λ = 1700–4000 Å) photons can dissociate CO2 and
H2 O to form O2 (Tian et al. 2014). The mere presence of O2 in an exoplanet atmosphere
should therefore not be interpreted as a biosignature because the O2 could easily be
created abiotically, particularly in the atmospheres of planets receiving high levels FUV
and NUV radiation.
The caution against interpreting O2 alone as a biosignature may seem obvious
to modern astrobiologists, but the importance of considering the UV when assessing
planetary habitability was not fully appreciated until the first FUV observations of
M dwarfs. The data revealed that even optically “quiet” M dwarfs without detected
Hα emission have FUV emission far exceeding that predicted by typical quiet M dwarf
models considering only photospheric flux (France et al. 2013). Furthermore, many
M dwarfs display strong Lyα emission lines that contribute approximately as much flux
as the rest of the full FUV+NUV bandpass combined (France et al. 2012).
One program focused on characterizing the UV activity of M dwarfs is the
Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanet host
Stars (MUSCLES) program led by France et al. (2013). The MUSCLES collaboration
19
CHAPTER 1. INTRODUCTION
has published results from Hubble Space Telescope Cosmic Origins Spectrograph (HST
COS) and Space Telescope Imaging Spectrograph (STIS) observations of six M dwarfs
known to host planets. The selected stars (GJ 581, GJ 876, GJ 436, GJ 832, GJ 667C,
and GJ 1214) span a significant range of the M dwarf spectral sequence from M1–M6.
Although all of the MUSCLES stars would be classified as merely “weakly active”
(Walkowicz & Hawley 2009) due to the appearance of Hα in absorption and weak
Ca II H&K emission, all of the stars display chromospheric and transition region
emission lines. In addition, all but GJ 1214 display detectable Lyα flux. Based on the
subset of UV spectra with the highest S/N, France et al. (2013) remarked that the UV
activity of M dwarfs can be highly variable (variations of 50–500%) on short timescales
of 100–1000 seconds.
The spectra provided by the MUSCLES project are very useful for characterizing the
wavelength-dependent UV activity of M dwarfs, but the sample is quite small. In order
to advance our understanding of M dwarf UV activity in general, several researchers
(Browne et al. 2009; Rodriguez et al. 2011, 2013; Shkolnik et al. 2011; Stelzer et al.
2013; Ansdell et al. 2015) cross-correlated catalogs of known M dwarfs with the Galaxy
Evolution Explorer (GALEX, Martin et al. 2005) catalog of NUV sources to check
for serendipitous UV observations of M dwarfs. Most recently, Ansdell et al. (2015)
identified GALEX matches for 4805 early M dwarfs from the Lépine & Gaidos (2011)
catalog. Roughly 20% of the stars were classified as NUV-luminous, meaning that their
NUV − Ks colors were 2.5σ bluer than the value expected for an inactive star.
Ansdell et al. (2015) also cross-matched their M dwarf catalog to the ROSAT
All-Sky Survey Bright Source Catalog (Voges et al. 1999) and the Faint Source Catalog
20
CHAPTER 1. INTRODUCTION
(Voges et al. 2000) to check for X-ray emission. They then rechecked the GALEX catalog
to determine whether any of the stars displayed FUV emission as well as NUV emission.
They discovered that roughly 8% of the full sample (including 40% of the NUV-luminous
stars) displayed emission at NUV, FUV, and X-ray wavelengths.
After correcting for false positives, Ansdell et al. (2015) used a synthetic galactic
population model to investigate the relation between activity and age. Their results
suggest that early M dwarfs experience a 100–200 Myr phase (perhaps as long as 300 Myr;
see Shkolnik & Barman 2014) of saturated NUV emission during which the atmospheres
of associated planets could be affected by photodissociation. During this early phase,
M dwarfs are also likely to have high levels of FUV emission (FFUV /FNUV ≥ 0.1 for
70% of NUV saturated stars), further influencing planetary atmospheric chemistry. In
general, the high levels of UV flux observed for M dwarfs present a compelling case that
stellar models incorporating both photospheric fluxes and chromospheric UV activity
(such as those developed by Grenfell et al. 2014; Rugheimer 2015) are essential for
accurately modeling the atmospheres of planets orbiting M dwarfs.
1.4
Distinguishing Planets from Astrophysical False
Positives
Although many of the putative planets revealed by transit surveys are bona fide planets,
some astrophysical effects can mimic planetary transits. The most common culprits are
background eclipsing binaries (BEBs), hierarchical eclipsing binaries (HEBs), background
stars with transiting planets (BTPs), and companion stars with transiting planets
21
CHAPTER 1. INTRODUCTION
(CTPs). In all of these cases, the depth of the stellar eclipse or planetary transit is
diluted by light from (an) additional star(s) in the aperture. The resulting transit depth
is then shallow enough that the system might be misidentified as a transiting planet
orbiting the (purportedly single) target star. Adaptive optics observations such as those
described in Chapter 4 may sometimes unveil the presence of additional stars in the
photometric aperture and often provide valuable limits on the likelihood that a putative
transit is due to an astrophysical false positive.
Conveniently, there are several tests that can be applied to expose astrophysical false
positives even in cases for which the multi-star systems cannot be visually resolved. One
valuable indicator is the motion of the photocenter during transit. If the only light source
in the photometric aperture is the host star of a transiting planet, then the photocenter
will not shift during transit. In contrast, if there are additional light sources in the
aperture, then the photocenter will shift away from the transit host star during transit.
Accordingly, careful measurements of the position of the photocenter in and out of
planetary transit can reveal whether the target star is indeed the transit source (Bryson
et al. 2013). This test is most effective for revealing BEBs and BTPs; astrophysical false
positives involving physically associated systems do not exhibit noticeable centroid shifts.
The data validation (DV) process of creating a catalog of Kepler planet candidates
from a list of possible “threshold crossing events” (TCEs) is detailed by Batalha et al.
(2010a) and also incorporates a comparison of the depths of odd and even transits. One
might imagine a scenario in which the primary and secondary eclipses of an eclipsing
binary have similar transit depths. If the system is configured such that secondary
eclipse occurs nearly half an orbital phase after planetary transit, the system might
successfully masquerade as a transiting planet with half the true orbital period of the
22
CHAPTER 1. INTRODUCTION
eclipsing binary. Close inspection of the depths and durations of odd and even transits
might distinguish such a system from a true transiting planet. Similarly, the DV process
includes a search for secondary eclipses (which should be undetectable for all but the
largest, most highly irradiated planets) and a search for ellipsoidal variations. Blended
systems that survive the DV process may be subsequently revealed by photometric
observations taken other wavelengths (e.g., Désert et al. 2015, and references therein);
planetary transits are achromatic, but blends comprised of stars with different spectral
types are not.
Importantly, larger transiting planets can be misidentified as smaller transiting
planets if they orbit stars in multi-star systems physically associated with target stars
(CTPs) or if they orbit stars in the backgrounds of target stars (BTPs). Using a
hierarchical model considering larger planets as potential false positives for smaller
planets, Fressin et al. (2013) found that BTPs are the dominant source of false positives
for Earth-size planet candidates surviving the DV process. Accordingly, accurate
knowledge of the occurrence rate of gas giants and Neptunes is required to correctly
estimate the frequency of Earth-size planets in the galaxy.
A correct characterization of stellar multiplicity is also necessary for accurately
estimating planetary occurrence rates. Due to transit depth dilution, the radii of planet
candidates in multi-star systems are often underestimated because the target star is
believed to be single. Accounting for the observed frequency of binary and triple star
systems (Raghavan et al. 2010), Ciardi et al. (2015) calculated that the radius of a
typical Kepler planet candidate for which no follow-up vetting has been performed is
likely underestimated by 60% for planets orbiting A or F stars and by 20% for planets
orbiting K and M dwarfs. Neglecting this effect could lead to an overestimate of the
23
CHAPTER 1. INTRODUCTION
inferred occurrence rate of Earth-size planets by 15–20% in the absence of follow-up
observations or by 5–7% if reconnaissance RV and AO observations are obtained for each
candidate (Ciardi et al. 2015). Parallaxes from Gaia should also improve the accuracy
of the estimate by revealing multiple star systems for which the previously assumed
distance is inconsistent with the distance calculated from the measured parallax.
1.5
Expectations from Planet Formation Theory
The wealth of planets detected by Kepler and ground-based surveys provides a test for
theories of planet formation and migration. Two key predictions that are addressed
in this thesis are the properties of planetary systems orbiting low-mass stars and the
compositions of small planets.
1.5.1
The Demographics of M Dwarf Systems
Long before the launch of Kepler, Laughlin et al. (2004) and Adams et al. (2005) made
three key predictions about planet formation in M dwarf systems:
1. Jovian planets should be rare.
2. Neptunes and rocky planets should be common.
3. The small planets orbiting M dwarfs with higher metallicities should be more
massive than the small planets orbiting M dwarfs with lower metallicities.
The primary explanation for these three predictions is that protoplanetary disks orbiting
M dwarfs are less massive, initially believed to be shorter-lived (but see Pascucci et al.
24
CHAPTER 1. INTRODUCTION
2009), and more easily disrupted than protoplanetary disks orbiting Sun-like stars. In
addition, disks orbiting less massive stars have longer orbital timescales at a given orbital
distance, further increasing the difficulty of forming large planets.
According to the core accretion model (e.g., Mizuno 1980; Hayashi et al. 1985;
Pollack et al. 1996; Ida & Lin 2004), the first stage in the formation of both terrestrial
and gaseous planets is the collision of planetesimals. Some of these planetesimals stick
together to form larger bodies, which may eventually grow to become rocky planets or
the cores of giant planets. In the latter case, the planet accumulates mass quickly enough
to reach the “critical core mass” required to initiate run-away gas accretion before the
protoplanetary disk dissipates. Due to the conspiring factors of slower planet growth,
reduced disk surface density, and previously assumed shorter disk lifetimes, few M dwarf
planets were predicted to be able to accrete enough mass to become gas giants.
The prediction of few M dwarf planetary systems containing gas giants on
short-period orbits has been borne out in reality (Butler et al. 2004, 2006; Endl et al.
2006; Johnson et al. 2007a, see also Section 1.8.3). Although the transit of a Jupiter-sized
planet across the face of an M dwarf produces a deep and relatively easily detected
transit, a query of the Exoplanet Orbit Database8 (Wright et al. 2011; Han et al. 2014)
on 26 March 2015 revealed only twelve planets with minimum mass estimates larger than
94 M⊕ (comparable to Saturn’s mass of 95.2 M⊕ ) orbiting stars less massive than 0.6 M$ .
This list includes two Jovian planets in the same system: a 0.7141 ± 0.0039MJ and a
2.2756 ± 0.0045MJ planet orbit the metal-rich star GJ 876 every 30.1 and 61.1 days,
respectively (GJ 876b and GJ 876c, Marcy et al. 2001). Interestingly, the GJ 876 system
8
www.exoplanets.org
25
CHAPTER 1. INTRODUCTION
also harbors a 6.83 ± 0.40 M⊕ planet with an orbital period of 1.9 days (Rivera et al.
2005) and a 14.6 ± 1.7 M⊕ planet with a period of 124.3 days (Rivera et al. 2010).
Furthermore, no hot Jupiters had been detected in M dwarf systems until the Kepler
era. The archetypal example of a hot Jupiter orbiting an M dwarf, the 0.96RJ planet
KOI-254b orbits a 0.55 R$ host star every 2.45 days (Johnson et al. 2012). The host star
has a higher metallicity than the Sun, adding credence to the theory that low-mass stars
must be enriched in metals in order to have protoplanetary disks that are sufficiently
massive to produce giant planets.
In contrast, Kornet et al. (2006) argued that the surface density of solids should be
higher in debris disks orbiting less massive stars and that giant planets should therefore
be more commonly formed orbiting lower mass stars. Due to the closer proximity of
the snow line to the star in the protoplanetary disks of low-mass stars, their model also
suggested that giant planets should preferentially be formed at closer semimajor axes
with decreasing stellar mass. Specifically, they predicted that gas giants orbiting early
M dwarfs would be formed 25–45% closer to the star than gas giants orbiting Sun-like
stars. However, Kornet et al. (2006) also noted that the minimum metallicity required
to form gas giants at separations less than 5 AU is higher for less massive stars ([Fe/H]
" 0.6 for 0.5 M$ versus [Fe/H] " 0.2 for 4 M$ ) so the influence of metallicity might
be responsible for the observed decline in the occurrence of close-in giant planets with
decreasing stellar mass. (See Section 1.8.4 for a discussion of the influence of metallicity
on planet occurrence.)
The current Kepler planet candidate catalog includes 14 planets larger than 4 R⊕
orbiting M dwarfs, but few of those systems have been examined in detail. For example,
26
CHAPTER 1. INTRODUCTION
KOIs 2842.01 and 2842.02 (now Kepler-446b and 446d) were originally listed with radii
of 25 ± 15 R⊕ and 26 ± 15 R⊕ . The recent revision of their radii to 1.50 ± 0.25 R⊕
and 1.11 ± 0.18 R⊕ , respectively, by Muirhead et al. (2015) demonstrated that some
purportedly large planet candidates orbiting M dwarfs may be significantly smaller than
the corresponding entries in the planet candidate list would suggest. The reason for the
large discrepancies between the revised sizes and the catalog listings is uncertain, but is
likely linked to poor estimates of the impact parameters.
As discussed in detail in Section 1.8.3, the second prediction that small planets
should be common in M dwarf systems also seems to be accurate (e.g., Dressing &
Charbonneau 2013, 2015; Gaidos et al. 2014; Morton & Swift 2014). The accuracy of the
third prediction requires a larger sample of small planets with well-constrained masses,
but there is active discussion regarding the possibility of a correlation between the
metallicity of low-mass stars and the presence of 1.7 − 3.9 R⊕ planets (Buchhave et al.
2014; Schlaufman 2015, see Section 1.8.4).
1.5.2
The Formation of Terrestrial Planets
Below a threshold mass, planets orbiting both M dwarfs and Sun-like stars are expected
to have rocky compositions with abundance ratios comparable to that of the refractory
elements in the original protoplanetary disk. Observationally, the threshold mass below
which planetary compositions are consistent with an Earth-like mixture of rock and iron
appears to be roughly 6 M⊕ , resulting in a maximum radius of approximately 1.6 R⊕
for rocky planets (see Chapter 5, Rogers 2015, and Dressing et al. 2015). Obtaining a
6 M⊕ planet in a close-in orbit requires either delivery of additional planetesimals from
27
CHAPTER 1. INTRODUCTION
the outer regions of the protoplanetary disk (e.g., Hansen & Murray 2012) or an initial
protoplanetary disk density that is much higher than that proposed for the minimum
mass Solar nebula (MMSN, Chiang & Laughlin 2013). Alternatively, protoplanets might
form farther out in the disk in regions where the isolation mass is higher and then
migrate inward (e.g., Terquem & Papaloizou 2007) under Type I Migration (Goldreich
& Tremaine 1980; Ward 1986). Protoplanets might form from collisions between
planetesimals with radii between approximately 10 m and 100 km (oligarchic growth,
Kokubo & Ida 1998, 2000, 2002; Thommes et al. 2003) or from the gradual accretion
of numerous mm- and cm-sized pebbles onto larger cores with diameters of 1–10 km
(pebble accretion, Lambrechts & Johansen 2012).
In theory, measurements of planetary masses and radii like those described in
Chapter 5 may be able to distinguish among the possible pathways of super-Earth
formation. Raymond et al. (2008) suggested that close-in small planets that formed
farther out in the disk and subsequently migrated inward should be composed of
higher fractions of low-density ices than small planets that formed in situ from drier
planetestimals. However, it is likely that the process of forming small planets includes
both migration and in situ formation. Additionally, distinguishing between super-Earths
formed in situ and those formed via migration is possible only if rocky planets that
form in situ cannot retain gaseous envelopes (Raymond et al. 2013). Current theoretical
models suggest that this caveat is true. Calculations by Hansen & Murray (2012) and
Chiang & Laughlin (2013) have demonstrated that small planets that form in situ can
initially accumulate thick atmospheres that would result in low bulk densities, but those
atmospheres typically dissipate when the protoplanetary disk disperses (Ikoma & Hori
2012).
28
CHAPTER 1. INTRODUCTION
A detailed discussion of the multitude of conjectures made by various planet
formation theories is beyond the scope of this thesis, but there are several interesting
theoretical predictions regarding the initial protoplanetary disk properties and the
presence of giant planets. For instance, Kokubo et al. (2006) argued that protoplanetary
disks with higher local disk surface densities Σ0 are expected to result in more massive
average planet masses MP with the scaling Mp ∝ Σ1.1
0 . In addition, more massive disks
are expected to produced a lower total number of planets because embryos forming in
more massive disks can be more easily excited to higher eccentricities, thereby increasing
the likelihood that their larger feeding zones will inhibit the growth of neighboring
embryos (Kokubo et al. 2006; Raymond et al. 2007b).
The presence of massive and/or eccentric outer gas giants also increases the typical
mean eccentricity of growing embryos. In such systems, more embryos and planetesimals
will be excited to unstable orbits and ejected from the system. As a result, any terrestrial
planets will be more massive and less numerous than in systems without massive or
eccentric gas giants (Chambers & Cassen 2002; Levison & Agnor 2003; Raymond et al.
2004).
Furthermore, systems with outer giant planets are expected to harbor drier
terrestrial planets than systems without giant planets. The rationale is that the majority
of water-rich embryos influence by giant planets are scattered outward and ejected
from the system rather than scattered inward toward the growing terrestrial planets.
Accordingly, less water is delivered to terrestrial planets in systems with giant planets
(Chambers & Cassen 2002; Raymond et al. 2004, 2006, 2007a, 2009; O’Brien et al.
2006). The putative anti-correlation between the presence of outer gas giants and
water-rich inner planets could be tested by measuring the masses of inner planets via RV
29
CHAPTER 1. INTRODUCTION
observations or possibly TTVs and constraining their radii and atmospheric compositions
via transmission spectroscopy (see Chapter 6). The presence of giant planets could then
be constrained using a combination of RV observations, astrometric investigations, and
possibly even direct imaging observations (for particularly young systems in which giant
planets are still cooling).
Finally, there may also be an anti-correlation between the presence of cool gas giants
and the presence of highly-irradiated super-Earths. Izidoro et al. (2015) conducted a
series of dynamical simulations indicating that gas giants serve as “dynamical barriers”
that prevent the inward migration of more distant protoplanetary cores. For the case of
our solar system, their model would predict that the early growth of Jupiter prevented
the growing cores Uranus and Neptune from migrating inward, possibly losing their
atmospheres (see Section 1.5.3), and becoming highly irradiated super-Earths.
In contrast, if the observed population of highly irradiated super-Earths formed in
situ (e.g., Hansen & Murray 2012, 2013) then the presence of hot super-Earths and more
distant gas giants should not be anti-correlated. Even if the migration explanation is
correct, hot super-Earths could still be observed in systems with a distant gas giant as
long as the migrating super-Earth formed interior to the gas giant. Nonetheless there
may still be an observable difference between the frequency of super-Earths in systems
with and without outer gas giants because the presence of hot super-Earths in any given
system would depend on whether a growing gas giant planet formed interior to the more
slowly growing embryos and prevented them from migrating inward.
30
CHAPTER 1. INTRODUCTION
1.5.3
The Role of Photoevaporation
Planets in close proximity to their host stars run the risk of losing their outer envelopes
or possibly their entire atmospheres to photoevaporation, hydrodynamic mass loss driven
by short-wavelength stellar radiation. Owen & Jackson (2012) investigated the relative
importance of X-ray and extreme UV radiation as a function of time. They found that
X-ray driven evaporation is most important for planets with relatively large masses
and low densities orbiting in close proximity to stars with high X-ray luminosities.
Because stellar X-ray activity declines as stars age (Ribas et al. 2005), photoevaporation
that begins in the X-ray driven mode may eventually transition to extreme UV driven
photoevaporation. The transition stage will occur earlier for planets that are farther
away from their host stars or for planets with lower initial masses.
Many studies of atmospheric loss from small planets in the extreme UV driven
regime adopt the energy-limited assumption (Watson et al. 1981) that the amount of
energy available to drive mass loss is set by the efficiency factor ' at which high-energy
flux from the star heats the atmosphere. For planets receiving UV fluxes below
approximately 104 erg cm−2 s−1 , Murray-Clay et al. (2009) found that the energy-limited
assumption is reasonable and that the mass loss rate scales approximately linearly with
the extreme UV flux.
Specifically, in the energy-limited regime, the mass loss rate Ṁ for a planet with
mass Mp receiving the flux FXUV , where XUV refers the wavelength range 1 − 1200Å,
can be expressed as in Lopez et al. (2012):
Ṁ ≈
3
π'FXUV RXUV
GMp Ktide
(1.8)
where G is the gravitational constant, RXUV is the size of the planet measured at XUV
31
CHAPTER 1. INTRODUCTION
wavelengths (likely 10–20% bigger than at optical wavelengths), and the scaling factor
Ktide corrects for the fact that mass escapes from the Hill radius rather than RXUV
(Erkaev et al. 2007). The scaling factor can be as large as two for young systems, but it
will decrease as the planet cool and the stellar extreme UV flux decreases. For reference,
Ribas et al. (2005) found that young Sun-like stars display X-ray and XUV emission
enhanced by factors of 100–1000 compared to the current solar levels.
Lopez et al. (2012) found that the relative lack of known low-mass, low density
(LMLD) planets in high insolation flux environments could be explained by a model in
which there is a critical mass-loss threshold above which planets cannot retain H/He
envelopes. A similar explanation was previously developed by Lecavelier Des Etangs
(2007), but Lopez et al. (2012) took advantage of the large population of small transiting
planets discovered between 2007 and 2012 to confirm that the proposed mass-loss
threshold could explain planets with masses as low as 2 M⊕ . To aid observers conducting
RV follow-up observations, Lopez et al. (2012) provided the following prescription for
estimating the masses Mp of highly irradiated small planets (Fp > 500 F⊕ ):
#
π'FXUV,E100 Fp
Mp ≥
tloss,critRp3/2
G
F⊕
(1.9)
where FXUV,E100 = 504 erg s−1 cm−2 is the estimated XUV flux received by the Earth
when the Sun was 100 Myr old and tloss,crit is the critical timescale for mass loss (roughly
12 Gyr not accounting for planetary contraction).
In a follow-up paper, Lopez & Fortney (2013) investigated the dependence of
atmospheric mass loss on planetary core mass and the efficiency of mass loss. They found
that the core mass has a significant influence on the mass-loss timescale, with planets
possessing larger cores more resistant to photoevaporation. Specifically, the threshold
32
CHAPTER 1. INTRODUCTION
flux Fth required to remove half of a planet’s initial reservoir of H and He is:
Fth = 0.5 F⊕
!
Mcore
M⊕
"2.4±0.4 '
' (−0.7±0.1
0.1
(1.10)
Owen & Wu (2013) also studied the role of photoevaporation on the radii of
highly-irradiated low mass planets. In agreement with Lopez et al. (2012), they observed
that the cumulative photoevaporative history of a planet is dominated by mass loss
within the first 100 Myr. Unlike Lopez et al. (2012), Owen & Wu (2013) employed an
adaptive mass loss efficiency ' that varied as the planet mass, planet radius, and X-ray
flux evolved. Owen & Wu (2013) demonstrated that modeling the time-dependence of '
can lead to an estimate of the cumulative mass loss 10× lower than the value predicted
if ' is set to the median efficiency
Although models of photoevaporation are still evolving, there are several key results
with important implications for the compositions of small planets. First, Neptune-mass
planets are far more susceptible to mass loss than are Jupiter-mass planets (Owen & Wu
2013). Second, planets with very small H/He envelopes (< 1% by mass) are unlikely
to retain them if they are highly-irradiated (Lopez & Fortney 2013, 2014; Owen & Wu
2013). Instead, observers might expect to see an “evaporation valley” (Lopez & Fortney
2014) separating highly irradiated massive planets that have retained substantial H/He
envelopes from highly irradiated less massive planets that have lost their atmospheres.
At high fluxes (Fp ≈ 1000 F⊕ ), Lopez & Fortney (2013) predicted that planets with
core masses ≤ 10 M⊕ and 1% H/He envelopes could have lose their envelopes and end up
as 2 − 2.5 R⊕ stripped cores. For context, assuming that Neptune is composed of roughly
10% H/He, 25% rock, and 65% ice (Hubbard et al. 1991, 1995; Podolak et al. 1995),
stripping the entire H/He envelope would leave approximately 11 M⊕ of ice and 4 M⊕
33
CHAPTER 1. INTRODUCTION
of rock. Utilizing the two-component water-rock models developed by Zeng & Sasselov
(2014), such a planet would have a radius of roughly 2.7 R⊕ .
At lower fluxes (Fp ≈ F⊕ ), they estimated that low-mass planets with > 0.1% H/He
envelopes would be able to retain their atmospheres. Nonetheless, Lopez & Fortney
(2013) found that their model predicted that some (possibly quite rare) planets would
fall within the nominal boundaries of the evaporation valley, suggesting that variation
in the initial planet core mass and envelope fraction could blur sharp changes in the
occurrence rate. Additionally, the presence of water-rich planets with varying water mass
fractions would smear out any clear distinctions in the mass-radius diagram of highly
irradiated planets (Lopez & Fortney 2013).
1.6
The Interior Structure and Composition of Small
Planets
The Earth is not a “water world:” the total abundance of water both on the surface and
within the mantle is estimated to be only 1–3 ppt (Marty 2012, and references therein).
The Earth consists of an iron-dominated core and a silicate-rich mantle capped by an
exquisitely thin silicate crust, hydrosphere, and atmosphere. Seismological monitoring
has allowed geophysicists to probe the interior structure of the Earth and Moon. The
composition of the mantle is also constrained by mineralogical analyses of xenoliths,
inclusions of rock fragments (which are occasionally mantle material) within other rocks
(e.g., Nixon 1987).
34
CHAPTER 1. INTRODUCTION
1.6.1
The Earth
The upper mantle extends from the Mohorovicic discontinuity (depths of 30–50 km below
continental crust or 5–10 km beneath oceanic crust) to depths of roughly 300 km (Wenk
& Bulakh 2006). The composition of the upper mantle is predominantly a mixture of
olivine, (Mg,Fe)2 SiO4 , and pyroxene, (Mg,Fe)2 Si2 O6 (Sotin et al. 2010). Overall, the
magnesium-bearing species are more common, resulting in a bulk Mg/(Mg+Fe) ratio
of 89 for the upper mantle. The majority of the upper mantle is solid, but there are
regions of partial melting in subduction zones and near upwelling material. Although
the mantle is largely solid, at least the uppermost portion is plastic enough to experience
convection. The slow convection of the mantle drives the motion of roughly fifteen
oceanic and continental plates comprising the lithosphere (de Pater & Lissauer 2010).
Typical plate velocities are a few cm yr−1 and the relationship between the strength
(and overall existence) of plate tectonics and planetary properties such as surface gravity
and mantle water content is uncertain (Wenk & Bulakh 2006; Baraffe et al. 2014, and
references therein).
Below the upper mantle, there is a transition zone between 300–660 km during
which several pressure-dependent phase changes occur in mineral structures. The
general pattern governing the mantle phase transitions is that mineral structure typically
becomes simpler, more compact, and more symmetric as pressure increases. Specifically,
olivine undergoes a phase transition to become first wadsleyite and then ringwoodite
(a form of spinel) at even higher pressures whereas pyroxene (Mg2 Si2 O6 ) transforms to
majorite (a type of garnet with the composition Mg3 (MgSi)Si3 O12 ). Aluminum-bearing
kyanite (Al2 SiO5 ) transforms into corundum (Al2 O3 ) and stishovite (SiO2 ). Interestingly,
35
CHAPTER 1. INTRODUCTION
roughly 0.1% of the mass of the lower transition zone (roughly 400–660 km) consists
of water (entrapped in magnesium silicates), but this contribution is often neglected in
theoretical models of exoplanet interiors (Wenk & Bulakh 2006).
At depths between 660 km, additional phase changes cause the transformation
of ringwoodite Mg2 SiO4 into MgO (as periclase or magnesiowüstite) and MgSiO3 (as
perovskite). This transition marks the upper extent of the lower mantle, which extends
to the core-mantle boundary (CMB) at a depth of approximately 2900 km (Wenk &
Bulakh 2006). At the intense pressures experienced in the lower mantle (Ringwood
1975),9 the magnesium silicates are expected to be in the form of perovskite (MgSiO3 ;
70% by volume), magnesiowüstite ((Mg,Fe)O; 25% by volume), and ferrite (NaAlSiO4
and (Mg,Fe)(Al,Cr,Fe)2 O4 ; 5% by volume)
Seismological observations have revealed that the Earth’s core consists of a solid
inner core surrounded by a liquid outer core (Wenk & Bulakh 2006). The liquid outer
core is believed to be the source of the Earth’s magnetic field, which has a significant
influence on the Earth’s biosphere by shielding the surface from stellar magnetic activity
(see Section 1.3.2). The dividing line between liquid outer core and the solid inner
core occurs at depth of approximately 5200 km (de Pater & Lissauer 2010). Both the
9
Be wary of placing too much weight on these statements regarding the interior composition of the
Earth. In the words of noted geophysicist Francis Birch, “unwary readers should take warning that
ordinary language undergoes modification to a high pressure form when applied to the interior of the
Earth.” Birch then suggested that the phrases “certain,” “undoubtedly,” “positive proof,” and “pure
iron” should be read as “dubious,” “perhaps,” “vague suggestion,” and “uncertain mixture of all the
elements” in the context of the deep Earth. The same translation may be useful for discussions of
exoplanet interiors.
36
CHAPTER 1. INTRODUCTION
inner and outer layers are composed primarily of iron and (to a lesser extent) nickel,
but the core density estimates from seismology also require incorporation of a lighter
element such as sulfur, oxygen, silicate, carbon, or hydrogen (Wenk & Bulakh 2006; de
Pater & Lissauer 2010). The inclusion of sulfur in the core could easily be explained
by fractionation during the differentiation of the Earth. Prior to the formation of the
Earth, iron sulfide (FeS) and iron oxide (FeO) formed from the interaction of iron with
H2 S and H2 O when the protoplanetary disk cooled to below approximately 700 K and
500K, respectively. Later in the planet formation process, the iron oxide would have
been integrated into olivine and pyroxene in the mantle whereas the iron sulfide would
follow iron to form the core of the planet (de Pater & Lissauer 2010).
1.6.2
Other Terrestrial Worlds in the Solar System
Although our solar system lacks super-Earths, we can at least inspect the properties
of three other terrestrial planets and several large moons in order to test theories of
planetary structure. The purpose of this short tour through the solar system is to
remind the reader of the remarkable diversity of worlds present in our own astronomical
backyard. Each of these worlds likely conceals valuable clues to aid our interpretation of
planets that are most likely far too distant for the in-depth analyses described in this
section.
The nearest test case is the Moon, which is the only other solar system body that
has been visited by humans. The Moon’s dimensionless moment of inertia factor is
I/(MR2 ) = 0.3932 ± 0.0002, which argues that the mass of the Moon is significantly less
centrally concentrated than that of the Earth. For comparison, the factor for the Earth
37
CHAPTER 1. INTRODUCTION
is 0.33 and the factor for a sphere of uniform density is 0.4 (de Pater & Lissauer 2010).
Based on seismological observations of moonquakes using probes left by Apollo
astronauts, the current model of the interior structure of the Moon is a small metallic
core with a radius smaller than 300–400 km covered by a three-layer mantle and a crust
of variable thickness ranging from ≤ 20 km beneath the basaltic maria and over 100 km
beneath the lunar highlands. In general, the crust is thinner on the near side (roughly
60 km) than on the far side (roughly 68 km). The top layer of the mantle (down to
depths of approximately 500 km) is composed primarily of olivine whereas the middle
layer (depths of 500–1000 km) is a mixture of olivine and pyroxene. The lowest layer
of the mantle (depths below roughly 1000 km) may be partially molten. The Moon
currently lacks a magnetic field, but close inspection of the lunar samples retrieved by
Apollo astronauts suggests that the Moon had a strong magnetic field 3–4 Gyr ago (de
Pater & Lissauer 2010).
Mercury was visited by Mariner 10 during three flybys in 1974-1975, but over
thirty years elapsed before MESSENGER10 began orbiting Mercury on 18 March 2011.
The planned European Space Agency mission BepiColombo11 is expected to arrive at
Mercury in 2024. Ground-based radar and optical measurements and the Mariner 10
flybys of Mercury revealed that the planet has a high bulk density (ρ = 5.43 g cm−3 , Ash
et al. 1967; Howard et al. 1974), suggesting that Mercury has a much higher iron fraction
than the Earth. Mercury’s interior structure is estimated to consist of a large, iron-rich
core comprising 75% of the planet’s radius covered by a relatively thin (roughly 600 km)
10
http://messenger.jhuapl.edu/
11
http://sci.esa.int/bepicolombo/
38
CHAPTER 1. INTRODUCTION
rocky mantle. The dimensionless moment of a inertia factor for this configuration is
0.325, demonstrating that Mercury is more centrally concentrated than the Earth (de
Pater & Lissauer 2010).
The assumed high iron fraction of Mercury is commonly attributed to a violent
impact near the end of the planet formation process (e.g., Benz et al. 1988). During
the impact, the majority of Mercury’s mantle would have been ejected from the
planet, causing the remainder of the planet to become significantly more iron-rich.
Alternative explanations for Mercury’s high iron content are equilibrium condensation
and evaporation of Mercury’s crust along with some mantle material (Fegley & Cameron
1987; Cameron et al. 1988).
Intriguingly, Mercury’s large core is likely still partially molten despite the small
size of the planet. The evidence for a partially molten core is that Mercury’s spin rate
changes subtly during the course of its 88-day orbit around the Sun. The changes are
induced by solar torques and the resulting libration is large enough to indicate that
Mercury’s mantle must be able to spin at a different rate from the core. In turn, the
apparent decoupling between core and mantle suggests that the outer core must be
liquid, which is physically plausible if the outer core includes a small amount of sulfur in
addition to iron (de Pater & Lissauer 2010).
Of all of the terrestrial planets, Venus likely has an interior structure most similar
to that of the Earth. The Soviet Venera landers discovered that Venus has a basaltic
surface (Surkov et al. 1984), and the uncompressed density of the planet is 4.3 g cm−3 ,
similar to the 4.4 g cm−3 uncompressed density of the Earth (de Pater & Lissauer 2010).
The lack of a global magnetic field on Venus suggests that the core of the planet is either
39
CHAPTER 1. INTRODUCTION
solid or non-convecting liquid. A solid Venusian core would be expected if Venus formed
with less sulfur than the Earth (as predicted by some models of abundance gradients in
the protoplanetary disk) and ended up with a lower ratio of iron sulfide to iron in the
core.12 Convection in a liquid core could be prevented if either the entire core is liquid
so that the phase transition from liquid core to solid core is absent (this transition is a
major energy source for convection in the Earth’s core) or if the core is cooler than the
mantle (de Pater & Lissauer 2010).
Unlike the Earth, Venus does not have active plate tectonics. Instead, Venus might
currently be in a “stagnant lid” convection regime in between infrequent “catatrosphic
resurfacing” events in which the entire crust of the planet subducts (de Pater & Lissauer
2010). The lack of plate tectonics on Venus is intriguing from a habitability perspective
because the temperature-regulating carbon-silicate cycle relies on crustal subduction to
release the reservoir of calcium carbonate trapped in the crust into CO2 gas that can be
vented by volcanoes (Walker et al. 1981).
The existence of an (approximately) Earth-size planet in our solar system without
active plate tectonics raises the question of whether plate tectonics exist on terrestrial
exoplanets. Of course, the necessity of plate tectonics for life is uncertain, but there
is a chance that the lack of plate tectonics on Venus is linked to the extremely low
water content in the Venusian lithosphere (de Pater & Lissauer 2010). The dry nature
of the crust is likely due to the high surface temperature, which is in turn a result of
the runaway greenhouse effect. One might therefore hypothesize that a hypothetical
12
Iron sulfide has a lower melting temperature than pure iron, so reducing the sulfur content in the
core would be more likely to yield a solid core.
40
CHAPTER 1. INTRODUCTION
Earth-size planet receiving Earth-like levels of insolation from another star might indeed
have active plate tectonics. However, the likelihood of plate tectonics on super Earths
is more uncertain (Sotin et al. 2010, and references therein). Valencia et al. (2007a)
found that more massive planets should have higher shear stresses and thinner plates,
both of which enhance the likelihood of plate tectonics. In contrast, O’Neill & Lenardic
(2007) argued that super Earths are more likely to experience episodic plate tectonics or
stagnant lid convection.
The final nearby terrestrial planet is Mars, the small size of which necessitated the
genesis of the Grand Tack model of early solar system history (Walsh et al. 2011). Mars
has a mass of only 10% of the Earth, but it has an unexpectedly dense mantle with an
uncompressed density of 3.55 g cm−3 . This density is even higher than the 3.34 g cm−3
uncompressed density of the Earth’s mantle. The disparity indicates that the martian
mantle is more iron-rich than the Earth’s mantle, consistent with analyses of martian
meteorites and surface observations by generations of intrepid landers and rovers. The
iron-rich mantle is sandwiched between a thick lithosphere and a partially or fully liquid
core with a radius of 1520–1840 km (de Pater & Lissauer 2010).
Like the Moon, Mars displays clear hemispheric asymmetries. The northern
hemisphere is significantly lower in elevation (≈ 5 km) than the southern hemisphere
and may have been covered by an ancient ocean (e.g., Perron et al. 2007). Although
the polar ocean is disputed, there is clear evidence that parts of Mars were covered by
standing water for a significant amount of time (e.g. Squyres et al. 2004; Perron et al.
2007; di Achille & Hynek 2010; Andrews-Hanna & Lewis 2011, and references therein).
In the outer Solar System, the icy moons of Jupiter and Saturn may resemble smaller
41
CHAPTER 1. INTRODUCTION
versions of ocean planets orbiting distant stars. For instance, the interior structure of
the Jovian moon Europa is expected to be a thick water crust (80–170 km) covering
a rocky mantle/core comprising 90% of the world’s mass (de Pater & Lissauer 2010).
The outermost layer of the water crust is frozen, but Galileo observations hint that
the ice shell rotates non-synchronously, indicating that there is a liquid water interface
between the ice shell and the rocky mantle (Geissler et al. 1998; Pappalardo et al. 1999).
The presence of a liquid layer is further supported by observations of magnetic field
disturbances similar to those expected from a salty subsurface ocean (Khurana et al.
1998; Kivelson et al. 2000; Zimmer et al. 2000).
Like Europa, Ganymede also has an icy surface and a rocky mantle. Ganymede is
more centrally concentrated than Europa and possesses a magnetic field indicating the
presence of a liquid iron core (possibly surrounding a solid inner core). The current set
of observations is inconclusive, but it is possible that Ganymede features a subsurface
liquid water ocean in the middle of the icy surface layer (Anderson et al. 1996; de Pater
& Lissauer 2010).
In the Saturnian system, the large moons Titan and Enceladus are both imaginable
abodes for life. Titan features an intriguingly dense (1.44 bar) nitrogen-dominated
atmosphere covering an ice-rich mantle and a core comprised of a mixture of rock and
iron. Titan may also harbor a subsurface ocean of water and liquid ammonia between
the icy crust and the rocky core (de Pater & Lissauer 2010).
At visible wavelengths, Titan’s surface is concealed by a photochemical haze, but
observations at longer wavelengths have revealed that the albedo of the surface varies
significantly across the planet. The low density of impact craters detected in Cassini
42
CHAPTER 1. INTRODUCTION
radar data suggests that the surface of Titan is geologically young (Elachi et al. 2005;
Porco et al. 2005). Although the low number of small craters is easily explained by
atmospheric shielding (e.g., Ivanov et al. 1997), the paucity of larger craters (20–120 km
across) necessitates recent resurfacing (Lorenz et al. 2007; Wood et al. 2010).
In 2005, the Huygens probe descended through the rich haze in Titan’s atmosphere
to land on the surface. Images from the descent and brief interval of surface observations
displayed fluvial morphology and suggested that Titan may have an active hydrological
cycle in which liquid hydrocarbons substitute for liquid water (Lebreton et al. 2005;
Tomasko et al. 2005). An important open question is the source of the methane
comprising approximately 1.4% of Titan’s atmosphere. The presence of methane could
be explained by either a hydocarbon-fueled hydrological cycle or cryovolcanism, but it
must be actively replenished by some means (Niemann et al. 2005; Maltagliati et al.
2015).
One of the most surprising discoveries of the Cassini mission to Saturn was that the
small moon Enceladus is also geologically active (Porco et al. 2006). Cassini observed
significant outflows of vapor, dust, and ice in geyser plumes originating from cracks
nicknamed“tiger stripes” near the south pole (Hansen et al. 2006; Spencer et al. 2006;
Waite et al. 2006, 2009; Matson et al. 2007; Spitale & Porco 2007). Underneath the icy
surface, Enceladus is expected to have a liquid ocean or sea and a rocky core (Schubert
et al. 2007). Future missions to Enceladus (such as the proposed Enceladus Life Mission)
could fly through the plumes to determine the composition of the jets.
43
CHAPTER 1. INTRODUCTION
1.6.3
A Framework for Modeling Terrestrial Exoplanets
If planetary composition were completely unconstrained, then determining the properties
of all but the densest planets would be futile. Below a maximum density at which only
a pure iron composition could explain a measured mass and radius, there are multiple
combinations of compositions and interior structures that could explain a given set of
measured masses and radii.
For instance, the mini-Neptune GJ 1214b has a mass of 6.55 M⊕ and a radius of
2.68 R⊕ (Charbonneau et al. 2009). As explained by Rogers & Seager (2010), the inferred
bulk density could be explained by three distinct categories of planet models: (1) a water
world with a steam atmosphere, (2) a rocky planet with an iron core, a silicate mantle,
and a puffy H/He envelope, or (3) an ice world with an iron core, silicate mantle, water
shell, and a H/He envelope. If a small planet has a clear atmosphere free of clouds and
hazes, the degeneracy between possible compositions can be partially broken by using
transmission spectroscopy to probe the atmospheric composition (see Section 6.3), but
another way to mitigate the degeneracy is to adopt general assumptions about the likely
building blocks of small planets.
As an example, the composition and interior structure of the Earth can be neatly
summarized by tracking the simple set of four elements (oxygen, iron, silicon, and
magnesium) that together comprise 95% of the total mass of the Earth (Sotin et al.
2007). Incorporating the next most significant set of elements (nickel, sulfur, aluminum,
and calcium) yields 99.9% of the total mass (Javoy 1995; Sotin et al. 2007, 2010).
Although the presence of the latter set of elements has a significant influence on the
oxidation state (Frost et al. 2004) and melting temperature (Chen et al. 2008) of iron,
44
CHAPTER 1. INTRODUCTION
the bulk properties of the Earth (e.g., the masses, radii, and compositions of the core
and mantle) can be understood using a model that includes only the first set of elements
(Sotin et al. 2007).
In this framework, the Earth consists of an iron core and a magnesium-silicate
mantle. Nickel and sulfur are primarily incorporated into the core with the iron,
aluminum is split between magnesium and silicon to preserve charge balance, and
calcium is associated with magnesium. As a result, the interior of a rocky planet can be
(almost) fully explained by tracking the behavior of oxygen, iron, silicon, and magnesium
(Sotin et al. 2007).
More precisely, the interior structure and composition of a terrestrial planet can be
specified by four numbers: (1) the fraction of the total planetary mass in water; (2) the
abundance of magnesium relative to silicon; (3) the abundance of iron relative to silicon;
(4) the ratio Mg# of the number of magnesium atoms in the silicates to the total number
of magnesium and iron atoms in the silicates. Once these four parameters are known,
the compositions and thicknesses of various layers within rocky and ocean planets can
be determined by making the basic assumption that the general structure of the planet
(from the inside out) is an iron-rich core; lower and upper silicate mantles with two
distinct pressure-dependent mineralogies but a single composition; a high-pressure ice
layer; and a thin hydrosphere (i.e., a global ocean or ice shell).
Terrestrial planets like those in our solar system lack the high-pressure ice layer
whereas ocean planets (for instance, an exoplanet resembling a larger version of Europa
or Enceladus) might lack an upper mantle if the pressure at the base of the ice layer is
sufficiently high. Sotin et al. (2007) further assumed that the mantle and core of the
45
CHAPTER 1. INTRODUCTION
planet are completely dry (i.e., all of the water is in the high-pressure ice layer or the
hydrosphere) and that the core contains a fixed ratio of iron and sulfur. The assumption
that the mantle is dry does not hold on the Earth because the silicate mantle does
include some water, but the water mass fraction in the mantle is negligible compared to
the mass fraction of silicates. The set of models (Zeng & Sasselov 2013) that we adopt in
Chapter 5 to investigate the composition of highly irradiated small planets also assumes
a dry mantle and a fixed core composition (pure iron in that case).
The model presented by Sotin et al. (2007) provides a tantalizing recipe for predicting
the compositions and structures of terrestrial exoplanets, but directly measuring the
planetary water mass fraction, Fe/Si ratio, Mg/Si ratio, and Mg# for an exoplanet
appears to be an insurmountable challenge. Fortunately, two degrees of freedom can be
removed if the abundance ratios of the host stars can be used as proxies for the planetary
abundance ratios. This substitution is generally considered reasonable (but see Gaidos
2015, which is discussed below) because the refractory elements comprising the bulk of
terrestrial exoplanets have similar condensation temperatures and therefore condense
out at similar locations in the protoplanetary disk. Consequently, terrestrial planets are
expected to form with Fe/Si and Mg/Si ratios very similar to that of their host stars
(Baraffe et al. 2014, and references therein).
The Mg# is more challenging to constrain because the value depends on the fraction
of iron that remained in the mantle instead of settling in the core. In our Solar System,
Mg# has been constrained to roughly 0.9 for the Earth and roughly 0.7 for Mars, but
the values for other small solar system bodies and all exoplanets are unknown. However,
theoretical models of planet formation suggest that Mg# should generally increase with
planet mass because Mg# is a measure of the degree of differentiation. Differentiation
46
CHAPTER 1. INTRODUCTION
becomes energetically easier for more massive planets because there is a larger reservoir
of gravitational potential energy to heat the planet and partially melt the iron so it sinks
down into the core.
Part of the evidence supporting the theory that the Fe/Si and Mg/Si abundance
ratios of terrestrial exoplanets can be predicted from photospheric abundances is that
the abundance ratios measured in the solar photosphere are very similar to the ratios
measured for carbonaceous chondrites (e.g., Lodders 2003). However, the abundance
ratios for the Earth may be better described by the compositions of enstatite chondrites
(Javoy 1995; Mattern et al. 2005), which have Mg/Si and Fe/Si ratios of 0.734 and
0.878, respectively, compared to the ratios of 1.05 and 0.86 observed for carbonaceous
chondrites (Hoppe 2009).
Although many studies have advocated that the abundance ratios of refractory
elements in terrestrial exoplanets can be constrained by measuring the appropriate
abundance ratios in exoplanet host stars, Gaidos (2015) outlined a series of reasons
why this relationship might not be universal. As described by Gaidos (2015), the
compositions of carbonaceous chondrites indicate formation under oxidizing conditions
in regions with high dust-to-gas ratios, suggesting that they may have formed in the
mid-plane of the protoplanetary disk from dust grains inherited from the molecular
cloud. In contrast, enstatite chondrites have compositions consistent with formation
under reduced conditions. The specific composition of any given terrestrial planet would
therefore depend on the relative fractions of reduced and oxidized planetesimals acquired
during formation. The reprocessing and mixing of material within the protoplanetary
disk would influence the resulting planet composition and might lead to a composition
different from the composition that would have resulted from collapse of a gas with the
47
CHAPTER 1. INTRODUCTION
same composition as the host star.
1.6.4
The Abundance Ratios of Planet Host Stars
If we assume that the compositions of planets are governed by the compositions of
their host stars, then it is important to consider variations in stellar abundances.
Within the solar neighborhood, measured photospheric abundances extend from Fe/Si
ratios of 0.6–1.7 and Mg/Si ratios of 0.8–2 (Sotin et al. 2010). On average, a typical
solar neighborhood star can be expected to have abundance ratios of Fe/Si=1.1 and
Mg/Si=1.3. These values are slightly higher than the values of 0.986 and 1.131 measured
for the Sun and would yield planet core mass fractions between 20–40%.
Due to the likely link between stellar and planetary abundance ratios, several
researchers (e.g., Bodaghee et al. 2003; Beirão et al. 2005; Gilli et al. 2006; Adibekyan
et al. 2012; Teske et al. 2015) have conducted detailed investigations of the compositions
of planet host stars and stars without detected planets. For a typical star, the errors
on the individual elemental abundances are such that the ratios Fe/Si and Mg/Si are
uncertain to the level of 0.3–0.5, which is comparable to the difference between the solar
composition and the composition of enstatite chondrites (Sotin et al. 2010).
In general, abundance ratio investigations begin with the acquisition of high
resolution spectra. For instance, Gilli et al. (2006) used spectra acquired at a variety of
telescopes as part of the CORALIE planet search program. After reducing the spectra,
the next step is to either perform a synthesis model in which model spectra are directly
compared to the reduced spectra in a chi-squared sense or to measure equivalent widths
(EWs) directly and then use a curve of growth analysis to arrive at the underlying stellar
48
CHAPTER 1. INTRODUCTION
parameters. For both approaches, sophisticated atmospheric models and an accurate
and comprehensive list of atomic and molecular features are a necessity.
In 2015, most stellar abundance analyses seem to adopt the latter approach of
measuring EWs. Some astronomers measure EWs “by hand” using a tool such as “splot”
in IRAF while others perform the process automatically using a script like ARES (Sousa
et al. 2007) or DAOSPEC (Stetson & Pancino 2008). The user then adopts a trial
atmospheric model based on the initial estimates of the temperature, metallicity, and
surface gravity of the star. The adopted models are often selected from the ATLAS9
(Kurucz 1993) or MARCS (Gustafsson et al. 2008) suites of atmospheric models. In
most cases, the stars in question are modeled using plane-parallel atmospheres that are
assumed to be in local thermodynamic equilibrium (LTE).
Under LTE conditions, the ratio of number of atoms in a given species in a particular
excitation state Na versus state Nb is determined by the Boltzmann equation (e.g.,
Carroll & Ostlie 2007):
Nb
gb
= e−(Eb −Ea )/kT
Na
ga
(1.11)
in which k is the Boltzmann constant, T is the temperature, and ga and gb are the
statistical weights and Ea and Eb are the energies in states a and b, respectively.
Similarly, the number of atoms in a particular ionization state Ni+1 compared to
state Ni is governed by the Saha equation (e.g., Carroll & Ostlie 2007):
!
"3/2
Ni+1
2Zi+1 2πme kT
=
e−χi /kT
Ni
ne Zi
h2
(1.12)
where me is the electron mass, ne is the electron number density, h is the Planck constant,
)
−(Ej −E1 )/kT
χi is the ionization potential from state i to state i + 1, and Zi = ∞
is
j=1 gj e
the partition function for state i.
49
CHAPTER 1. INTRODUCTION
The numbers of atoms in various ionization and excitation states can therefore be
predicted if the total number of atoms (i.e., the abundance) of a given species and the
atmospheric temperature profile are known. Spectral abundance codes such as MOOG
(Sneden 1973) solve the Boltzmann, Saha, and radiative transfer equations in an iterative
fashion to derive abundances from equivalent widths and an assumed model atmosphere.
In many cases, the initial model atmosphere selected by the user may not be the best
possible description of the star. The user can attempt to refine the choice of atmosphere
by fine-tuning the stellar parameters (e.g., as explained by Sousa 2014). For instance,
the presence of a trend between the inferred Fe I abundance for individual lines and
the measured EW is an indication that the microturbulence velocity is incorrect. The
microturbulence parameter accounts for the presence of non-thermal gas velocities over
small scales on the stellar surface.
Furthermore, if the assumed temperature is incorrect, then different Fe I lines will
yield different iron abundances. The user would therefore adjust the selected temperature
until the measured EWs for all of the Fe I lines are consistent with a single assumed
temperature.
Similarly, the surface gravity can be determined by ensuring that the iron abundance
inferred from the Fe I lines is consistent with the value inferred from the Fe II lines.
Once the microturbulence velocity, effective temperature, and surface gravity have been
constrained from the Fe I and Fe II lines, the user can estimate the abundances of other
elements by measuring the EWs of corresponding spectral lines and conducting a curve
of growth analysis.
For an individual star, the abundance of a particular element is quoted as the
50
CHAPTER 1. INTRODUCTION
logarithm of the ratio of the number of atoms of the element in question compared to
the number of hydrogen atoms (Sotin et al. 2010, and references therein). The number
is normalized by subtracting the same logarithmic abundance ratio computed using the
solar abundances. Mathematically, the abundance [X] of element X is:
! "
! "
X
X
[X] = log
− log
H !
H $
where the subscripts
!
and
$
indicate the stellar and solar values, respectively. By
extension, the abundance ratios Fe/Si and Mg/Si are given by:
! "
! "
Fe
Fe
=
10[Fe]−[Si]
Si !
Si $
!
"
!
"
Mg
Mg
=
10[Mg]−[Si]
Si !
Si $
1.6.5
(1.13)
(1.14)
(1.15)
Measuring Planetary Masses from Dynamical
Interactions
Not all systems are well-suited for radial velocity observations. Fortunately, in special
circumstances, the masses of planets can be inferred from dynamical interactions. In
multi-planet systems, planets can exchange angular momentum. If any of the planets
involved in the angular momentum exchange appear to transit, then transit times will
no longer be strictly periodic. The observed transit time variations (TTVs) can be fit
by a model in which the pattern and amplitude of the TTVs depend on the properties
of the perturber (Agol et al. 2005; Holman & Murray 2005). In some cases (like in the
Kepler-19 system, Ballard et al. 2011), TTVs may reveal the presence of a previously
unknown substellar object.
The TTV method has currently been employed to estimate the masses of dozens
51
CHAPTER 1. INTRODUCTION
of planets, many of which have radii smaller than 2 R⊕ . In many cases, the observed
TTVs yield mass upper limits but are insufficient to fully constrain the planet properties.
Unlike the population of small planets with precisely-measured masses from radial
velocity observations, which follow a tight relation between radius and mass suggestive
of an Earth-like composition (see Section 1.6.7, Chapter 5, and Dressing et al. 2015), the
small TTV planets appear to have a wide range of densities. Two systems in particular
are at odds with the prediction that small planets should have rocky compositions:
Kepler-11 (Lissauer et al. 2011, 2013) and KOI-314 (Kipping et al. 2014).
The Kepler-11 system consists of six transiting planets with radii of 1.8 − 4.2 R⊕
and orbital periods between 10 − 118 days. All six planets exhibit TTVs. The inner
five planets are closely packed between 0.091–0.250 AU and the outermost planet is
slightly more distant at 0.466 AU. Three of the planets in the Kepler-11 system have
inferred masses below 6 M⊕ and might therefore be expected to have rocky compositions.
However, Lissauer et al. (2013) presented the following system properties:
+0.03
+1.25
+0.068
−3
Kepler-11b: 1.9+1.4
−1.0 M⊕ , 1.80−0.05 R⊕ , ρ = 1.72−0.91 g cm , e = 0.045−0.042
+0.05
+0.66
+0.063
−3
Kepler-11c: 2.9+2.9
−1.6 M⊕ , 2.87−0.06 R⊕ , ρ = 0.66−0.35 g cm , e = 0.026−0.013
+0.06
+0.14
+0.007
−3
Kepler-11d: 7.3+0.8
−1.5 M⊕ , 3.12−0.07 R⊕ , ρ = 1.28−0.27 g cm , e = 0.004−0.002
+0.07
+0.11
+0.006
−3
Kepler-11e: 8.0+1.5
−2.1 M⊕ , 4.19−0.09 R⊕ , ρ = 0.58−0.16 g cm , e = 0.012−0.006
+0.04
+0.29
+0.011
−3
Kepler-11f: 2.0+0.8
−0.9 M⊕ , 2.49−0.07 R⊕ , ρ = 0.69−0.32 g cm , e = 0.013−0.009
−3
Kepler-11g: < 25 M⊕ , 3.33+0.06
−0.08 R⊕ , ρ < 4 g cm , e < 0.15
All six planets in the Kepler-11 system (including the three planets with MP < 6 M⊕ )
52
CHAPTER 1. INTRODUCTION
therefore appear to have low densities. Similarly, Kipping et al. (2014) estimated a mass
+0.82
−3
of 1.0+0.4
for the 1.61+0.16
−0.3 M⊕ and a density of 1.31−0.54 g cm
−0.15 R⊕ planet KOI-314c. The
density is inconsistent with a rocky composition, suggesting instead that KOI-314c has a
volatile-rich composition despite its small radius. The KOI-314 system also contains two
additional transiting planets with orbital periods shorter than the 23-day period of KOI314c. KOI-314b (P = 13.8 days) has an estimated radius identical to that of KOI-314c,
but is significantly more dense with an estimated mass of MP = 3.83+1.51
−1.26 M⊕ and an
−3
estimated density of 5.0+3.0
−2.0 g cm . The innermost planet, KOI-314.03 (P = 10.3 days)
has an estimated radius of 0.446+0.062
−0.050 R⊕ , but does not participate in the observed the
TTVs, thwarting attempts at photodynamical mass measurement.
The disagreement between the mass-radius relation proposed in Dressing et al.
(2015) and the estimated densities of the low-mass planets in the Kepler-11 and KOI-314
systems raises two important sets of questions:
1. Are TTV mass estimates unique? Might there be an alternative system
configuration, perhaps with masses more consistent with rocky compositions, that
also reproduces the observed TTVs?
2. Are there actually two distinct populations of small planets? Do rocky worlds like
Kepler-93b coexist with lower density planets like KOI-314c? If so, what factors
determine whether a given small planet will have a rocky composition?
The first point was addressed in part by Lithwick et al. (2012), who developed
analytic formulae to explore the behavior of TTVs for planets in near-resonant orbits.
They discovered a degeneracy between planet masses and free eccentricities that can
hinder attempts to determine the masses of planets from TTVs. As described in detail
53
CHAPTER 1. INTRODUCTION
by Lithwick et al. (2012), this degeneracy can be (partially) broken by considering the
phase of the TTVs. Lithwick et al. (2012) also demonstrated that the typical N-body
simulations used to fit TTVs (e.g., Cochran et al. 2011; Ford et al. 2012; Steffen et al.
2012a; Fabrycky et al. 2012) do not always explore the full parameter space. For some
systems, the masses found using analytic expressions differ from the masses predicted by
the N-body simulations by factors of 2–5.
Wu & Lithwick (2013) applied analytic TTV fits to a larger sample of near-resonant
planets and observed a general trend of decreasing planet density with increasing planet
radius. Separating their sample of planets into “mid-sized” (RP > 3 R⊕ ) and “compact”
(RP < 3 R⊕ ), they noted that the inferred densities of compact planets increased with
increasing insolation flux, providing additional evidence that photoevaporation might
play a role in shaping the mass-radius distribution of close-in planets (see Section 1.5.3).
A subsequent study by Hadden & Lithwick (2014) also noted that close-in planets may
be slightly denser than more distant planets.
1.6.6
Radial Velocity Observations of Small Transiting Planets
As described in Chapter 5, there are currently only ten planets smaller than 2.7 R⊕
with masses and radii determined to better than 20% precision.13 In order of increasing
radius, these planets are:
13
Restricting the sample to planets with mass and radius errors < 20% is useful for discriminating
among the various planet interior models presented in Section 1.6.7.
54
CHAPTER 1. INTRODUCTION
+3.02
−3
1. Kepler-78b: a 1.173+0.159
−0.089 R⊕ planet with an Earth-like density of 5.57−1.31 g cm
and a mass of 1.86+0.38
−0.25 M⊕ in a 0.355 day orbit around a 0.758 M$ star (Howard
et al. 2013; Pepe et al. 2013).
2. Kepler-10b: a 1.47+0.03
−0.02 R⊕ planet with a mass of 3.33 ± 0.49 and a density of
5.8 ± 0.8 g cm−3 in a 0.84 day orbit around a 0.91 M$ star (Dumusque et al. 2014).
3. Kepler-93b: a 1.478 ± 0.019 R⊕ planet with a mass of 4.02 ± 0.68 and a density
of 6.88 ± 1.18 g cm−3 in a 4.73 day orbit around a 0.91 M$ star (Dressing et al.
2015).
4. Kepler-36b: a 1.486 ± 0.035 R⊕ planet with a mass of 4.45+0.33
−0.27 and a density of
−3
7.46+0.74
in a 13.84 day orbit around a 1.07 M$ star (Carter et al. 2012).
−0.59 ± g cm
5. CoRoT-7b: a 1.585 ± 0.064 R⊕ planet with a mass of 4.73 ± 0.95 and a density of
6.59 ± 1.5 g cm−3 in a 0.85 day orbit around a 0.91 M$ star (Barros et al. 2014;
Haywood et al. 2014).
6. 55 Cnc e: a 2.21+0.15
−0.16 R⊕ planet with a mass of 8.09 ± 0.26 M⊕ and a density of
−3
5.51+1.32
in a 0.74 day orbit around a 0.91 M$ star (Gillon et al. 2012;
−1.00 g cm
Nelson et al. 2014).
7. HD 97658b: a 2.34+0.18
−0.15 R⊕ planet with a mass of 7.86 ± 0.73 M⊕ and a density of
−3
3.44+0.91
in a 9.5 day orbit around a 0.75 M$ star (Dragomir et al. 2013).
−0.82 g cm
8. Kepler-10c: a 2.35+0.09
−0.04 R⊕ planet with a mass of 17.2 ± 1.9 M⊕ and a density
of 7.1 ± 1.0 g cm−3 in a 45.29 day orbit around a 0.91 M$ star (Dumusque et al.
2014).
55
CHAPTER 1. INTRODUCTION
9. HIP 116454b: a 2.37 ± 0.13 R⊕ planet with a mass of 10.66 ± 1.85 and a density
of 3.36 ± 0.95 g cm−3 in a 9.1 day orbit around a 0.78 M$ star (Vanderburg et al.
2015).
10. GJ 1214b: a 2.678 ± 0.13 R⊕ planet with a mass of 6.55 ± 0.98 M⊕ and a density
of 1.87 ± 0.40 g cm−3 in a 1.58 day orbit around a 0.16 M$ star (Charbonneau
et al. 2009).
In addition to this modest sample of planets with tight mass and radius constraints,
there are dozens of small planets with less precisely determined masses and radii. On
behalf of the Kepler Science Team, Marcy et al. (2014) conducted an intensive campaign
to measure the masses of small planets with Keck/HIRES. They provided moderate
mass constraints (2σ or better) for 16 transiting planets and coarse mass constraints
or upper limits for 26 additional planet candidates. In general, the ranges of allowed
planet mass are broad and often extend to negative values. The Marcy et al. (2014)
sample is therefore more useful for studying the statistical properties of the mass-radius
distribution (e.g., Weiss & Marcy 2014; Rogers 2015) than for detailed investigations of
individual planets.
1.6.7
Comparing the Observations to Models
The cadre of small planets with measured masses or mass upper limits is compared to
theoretical compositional models in Figure 1.2. The specific set of models is from Zeng &
Sasselov (2013), but several groups have modeled the compositions of small planets (e.g.,
Fortney et al. 2007; Seager et al. 2007; Sotin et al. 2007; Valencia et al. 2006, 2007c,b,
2010; Adams et al. 2008; Baraffe et al. 2008; Grasset et al. 2009; Rogers & Seager 2010;
56
CHAPTER 1. INTRODUCTION
-10
Planet Radius (REarth)
4
-5
0
5
10
Weiss+Marcy (Err>20%)
Weiss+Marcy (Err<20%)
Dressing+ (Err<20%)
15
U
20
N
4
3
3
2
2
VE
1
All Water Planet
50/50 Rock/Water Planet
Earth-like Composition
Maximum Collisional Stripping
Ma
Me
0
-10
-5
0
5
10
Planet Mass (MEarth)
15
1
0
20
Figure 1.2: Mass and radius estimates for the sample of transiting planets with measured
masses or mass upper limits. The planets in orange are the subset of planets listed in
Section 1.6.6 and discussed by Dressing et al. (2015) for which the masses and radii have
been estimated to better than 20%. The remaining points are the collection of planets
analyzed by Weiss & Marcy (2014) and primarily have mass estimates from Marcy et al.
(2014). Planets from the Weiss & Marcy (2014) sample with mass and radius errors below
20% are highlighted in red whereas the remaining planets are shown in gray. The vertical
violet line indicates zero planet mass; planets lying within the shaded lilac region to the
left of the violet line have non-physical masses. The gray shaded region indicates planets
with requiring iron fractions higher than the maximum value predicted from simulations
of collisional stripping (Marcus et al. 2010). The navy dashed, teal dash-dotted, and
light blue solid lines are fully-differentiated, two-component models by Zeng & Sasselov
(2013) of planets with 100% water, 50% water–50% magnesium silicate, and Earth-like
compositions, respectively. For reference, the letters mark the locations of the terrestrial
planets and ice giants in the solar system.
57
CHAPTER 1. INTRODUCTION
Rogers et al. 2011; Wagner et al. 2011; Swift et al. 2012; Alibert 2014). The primary
differences among the different sets of models are variations in the adopted equations of
state, the treatment of radioactive heating, and the compositions of specific layers (e.g.,
a pure iron core versus an iron-nickel alloy with added sulfur).
One of the most noticeable features of the plot is the absence of massive planets
that could be explained by a volatile-free composition consisting solely of rock and iron.
Focusing exclusively on the subset of planets with mass and radius errors below 20%,
there are no planets with Mp " 6 M⊕ for which a rocky composition is consistent with
the data; all of those planets must be comprised of some fraction of volatiles. The specific
fraction of volatiles appears to vary considerably. Some planets have radii so large that
they must possess H/He envelopes, while other planets could be explained by a various
combinations of iron, rock, water, H/He, and other volatiles. As described in detail in
Chapter 6, atmospheric investigations via transmission or emission spectroscopy have
the potential to significantly reduce the allowed range of compositions.
For less massive planets (Mp ! 6 M⊕ ), the observed range of radii extends from
smaller than Mercury to nearly Neptune-sized. Many of the planets in this mass range
have considerable mass uncertainties and current mass estimates numerically consistent
with physically implausible compositions such as negative masses or iron fractions
exceeding the maximum value predicted from models of collisional stripping (Marcus
et al. 2010). Ideally, future RV observations will lead to revised mass estimates with
smaller errors. However, as described in Section 1.6.5 there is indeed a population of
low-mass planets with robustly measured low densities.
Supplementing the Marcy et al. (2014) sample with other transiting planets with
58
CHAPTER 1. INTRODUCTION
mass estimates from either RV or TTV analyses, Weiss & Marcy (2014) investigated
the relationship between mass and radius for planets with radii smaller than 4 R⊕
and orbital periods shorter than 100 days. They proposed the following relations:
ρp = 2.43 + 3.39(Rp / R⊕ )g cm−3 for Rp < 1.5 R⊕ and (Mp / M⊕ ) = 2.69(Rp / R⊕ )0.93
for 1.5 R⊕ ≤ Rp ≤ 4 R⊕ . These relations indicate that planets smaller than 1.5 R⊕
become more dense as their radii increase but that larger planets become less dense with
increasing radius, suggesting that they accumulate larger fractions of volatiles.
In a second study, Rogers (2015) investigated how the fraction of planets that are
consistent with rocky compositions depends on planet size. The paper presented a
hierarchical Bayesian framework considering the full range of allowable masses and radii
for small planets with RV-based mass estimates. Specifically, Rogers (2015) modeled the
planet masses by the posterior distributions resulting from joint MCMC fits to RV and
transit data. For the planet radii, Rogers (2015) constructed gaussian distributions with
means and widths set by the reported planet parameters. Working directly with the
MCMC posteriors instead of adopting the stated mass estimate for each plane allowed
Rogers (2015) to account for the degeneracies among the masses of planets in multiplanet
systems and obtain a more precise picture of the bulk compositions of small planets.
The paper concluded that the majority of planets with radii larger than 1.62 R⊕ have
densities that are too low to be explained by volatile-free mixtures of rock and iron.
Rogers (2015) used a simple step-function to model the transition between rocky
planets and planets requiring volatiles, but the paper also considered the possibility that
the division between mostly rocky and mostly gaseous planets occurs over a gradual
range of radii. As of 2015, the current data set was insufficient to distinguish between an
abrupt transition (i.e., a step-function model) and a gradual transition (i.e., a linear or
59
CHAPTER 1. INTRODUCTION
logistic model). For the given sample size and measurement uncertainties, the increase in
model complexity caused by adding additional terms outweighed possible improvements
in the fit. Similarly, Rogers (2015) found that incorporating a dependence on insolation
flux did not improve the fit enough to warrant the added complexity.
1.7
Assessing Planetary Habitability
At present, the sole example of an inhabited planet is the Earth. Other worlds within
our solar system (e.g., Mars, Titan, Enceladus, Europa) may be capable of harboring life,
but missions to Mars and the outer solar system have yet to reveal definitive evidence
of life. Accordingly, we must look to our own planet when considering which planetary
features might be signposts of extraterrestrial life. For astronomical purposes, these
“biosignatures” must be detectably remotely from great distances rather than requiring
in situ exploration like that conducted by the NASA Mars rover Curiosity.
Possible hallmarks of extraterrestrial life might be disequilibrium chemistry
(Lederberg 1965) or specific combinations of oxidizing and reducing gases (Lovelock
1965). However, some forms of life might utilize available thermodynamic gradients, thus
pushing an atmosphere further towards equilibrium instead of away from equilibrium
(Seager et al. 2012, 2013; Kasting et al. 2014). Additionally, disequilibrium chemistry
might be the result of impacts (Kasting 1990) or photolysis (Zahnle et al. 2008) rather
than evidence for life. The combination of oxidizing and reducing gases is a stronger
biosignature, but the usefulness of certain gases as biosignatures depends on the
properties of the planet in question and the incoming stellar flux (Rugheimer 2015, and
references therein).
60
CHAPTER 1. INTRODUCTION
In preparation for future atmospheric characterization of potentially habitable
planets using the James Webb Space Telescope (JWST ) and the next generation of
extremely large ground-based telescopes (see Chapter 6), astronomers are considering
potential targets. An attractive target would have a radius small enough to suggest a
rocky composition (likely 1.6 R⊕ or smaller, Rogers 2015; Weiss & Marcy 2014; Dressing
et al. 2015) and receive enough stellar insolation that the surface of the planet is warm
enough that the liquid water oceans could exist on the surface of the planet but not so
much insolation that the water would evaporate.
The potential habitability of an exoplanet is often assessed by estimating the
planetary equilibrium temperature based on the properties of the host star and the
reported planet period. Adopting the “traditional” assumptions that the star emits as
a blackbody with temperature T! and that the planet radiates uniformly over its entire
surface, the surface temperature of an exoplanet is:
Teq = T! (1 − a)
1/4
*
R!
2D
(1.16)
where T! is the effective temperature of the star, a is the albedo of the planet, R! is the
radius of the star, and D is the distance of the planet from the star (Carroll & Ostlie
2007).
Although this method provides a decent initial conjecture as to the habitability of
the system, it is not advisable for serious considerations of habitability. The reason is
that the greenhouse effect and the spectral energy distribution of the stellar radiation
are fundamentally important when contemplating planetary habitability. For example,
the predicted equilibrium temperature of the Earth based on Equation 1.16 in the
absence of the greenhouse effect is 255K, but the actual mean surface temperature of the
61
CHAPTER 1. INTRODUCTION
Earth in 2014 was 286K (and rising).14 For an exoplanet within unknown atmospheric
composition, surface topography, and albedo, a better indicator of habitability is the
incident flux upon the planet. Using units of effective solar flux Seff,$ , Kopparapu et al.
(2013b) suggest conservative habitable zone limits of 0.34 − 1.01Seff,$ and optimistic
limits of 0.32 − 1.78Seff,$ . In the conservative case, the outer limit is the “maximum
greenhouse” limit beyond which addicting additional CO2 to the atmosphere of the
planet will cease to warm the planet and the inner limit is the “moist greenhouse” limit
at which there is so much water in the stratosphere of the planet that hydrogen is quickly
lost.
In contrast, the optimistic case has empirical boundaries are based on the assumption
that Mars (outer boundary) and Venus (inner boundary) could have supported liquid
water in the past. The evidence in favor of past water on Mars is relatively clearcut:
orbital and ground-based investigations have revealed fluvial morphology and the
presence of minerals that typically form in aqueous environments (e.g., Malin & Edgett
2000; Squyres et al. 2004; Bibring et al. 2006). These features indicate that water could
have persisted on the surface of Mars as recently as 3.8 Gyr ago. Accounting for the
fainter luminosity of the Sun at that time, this suggests that planets receiving as little
as 0.32Seff,$ might be habitable. In the case of Venus, Solomon & Head (1991) argue
that the surface of Venus has lacked liquid water for at least the last 1 Gyr. The absence
of water on current Venus therefore yields an upper empirical limit of 1.78Seff,$ for the
insolation received by a potentially habitable planet.
14
NOAA National Climatic Data Center, State of the Climate: Global Analysis for December 2014,
published online January 2015, retrieved on March 10, 2015 from http://www.ncdc.noaa.gov/sotc/
global/2014/12
62
CHAPTER 1. INTRODUCTION
1.8
Planet Occurrence Across the HR Diagram
There are many open questions regarding the frequency and characteristics of exoplanets,
but the past quarter century of exoplanet searches has enabled an initial census of the
galactic exoplanet population.
1.8.1
Evolved & High-Mass Stars
At advanced stages of stellar evolution, the detection of planets orbiting pulsars
(Wolszczan & Frail 1992; Backer et al. 1993; Thorsett et al. 1999; Wolszczan et al. 2000)
and subdwarf B stars (Silvotti et al. 2007; Lee et al. 2009) has demonstrated that stellar
evolution does not preclude the existence of close-in planets orbiting evolved stars. The
pulsar planets either migrated inward from larger orbital separations or formed as second
generation planets.15 Due to the small number of known millisecond pulsars for which
searching for planets using timing variations is feasible, the occurrence rate of pulsar
planets is highly uncertain (Wolszczan & Kuchner 2010).
At the high mass end of the main sequence, the detection of planets orbiting A stars
is notoriously difficult due to the paucity of spectral features and fast rotation rates,
both of which lead to poor Doppler precision. Furthermore, the relatively large radii of
the high-mass stars increase the difficulty of detecting transiting planets. In addition,
the hotter temperature of high-mass stars push the habitable zone to much larger orbital
semimajor axes, resulting in a lower geometric likelihood of transit and a longer interval
15
A possible example of second generation planet formation in action is the presence of a cool disk
around the young pulsar 4U 0142+61. The estimated disk mass is 10 M⊕ (Wang et al. 2006).
63
CHAPTER 1. INTRODUCTION
between consecutive transits.
One approach to tackling the first challenge of poor Doppler precision is to look for
planets around “retired” A stars, i.e., stars that would have been classified as A stars
while on the main sequence but have since evolved to become giants (Johnson et al.
2007b, 2008, 2010, 2011a,b). The identification of some of these stars as retired A stars
has been disputed (based on inferred distribution of stellar masses, e.g, Lloyd 2011,
2013), but others have parallaxes, spectra, and interferometric radii consistent with a
past career as A stars (e.g., Johnson et al. 2014).
Based on the first five years of results from the “Retired A Star” project, Bowler
et al. (2010) found that Jovian planets with orbital semimajor axes smaller than 3 AU
are significantly more frequent in high-mass stellar systems than in FGK stellar systems.
Specifically, Bowler et al. (2010) estimated that 26+9
−8 % of retired A stars harbor such a
planet.
Reffert et al. (2015) also investigated the frequency of planets around high-mass
stars. Based on twelve years of observation of 373 G and K giants at Lick Observatory,
Reffert et al. (2015) found that the frequency of giant planets increases with increasing
stellar mass for 1 − 1.9 M$ stars (consistent with previous results, e.g., Lovis & Mayor
2007; Johnson et al. 2010), but that giant planets orbiting more massive stars are rare
(< 0.016 planets per 2.7 − 5 M$ star).
1.8.2
Sun-like Stars
Compared to high-mass stars, Sun-like stars have smaller radii, slower rotation rates,
more spectral features, and lower masses. Accordingly, detecting an Earth-size planet
64
CHAPTER 1. INTRODUCTION
or a Super-Earth orbiting a Sun-like star is significantly easier than detecting a small
planet orbiting a high-mass star. Even so, the detection of an Earth-size planet in the
habitable zone of Sun-like star is still beyond our current capabilities. As explained in
Section 1.1, a true Earth twin would yield a Doppler semi-amplitude of 9 cm s−1 and a
transit depth of 84 ppm. For context, the smallest RV semi-amplitude and transit depth
measured for a confirmed planet are 51 ± 4 cm s−1 (α Centauri Bb, Dumusque et al. 2012)
and 11.9+2.6
−3.1 ppm (Kepler-37b, Barclay et al. 2013), respectively. Note that the orbital
periods of the two planets are 3.2 and 13.4 days, respectively, and that the detection
of such small signals caused by a planet in a one-year orbital would be considerably
more challenging. Furthermore, both detections are barely above the detection threshold
and might be viewed as controversial (for instance, see the reanalysis of the α Centauri
B dataset by Hatzes 2013).
Although Earth twins still elude detection, we now have a much better understanding
of the frequency at which FGK stars host planets with shorter orbital periods or larger
radii. Based on nearly twenty years of radial velocity observations of 475 FGK stars
with the HIRES spectrograph on Keck I (Vogt et al. 1994, 2000), Cumming et al.
(2008) detected 46 planets and investigated the dependence of the planet occurrence
rate on planet mass and orbital period. They found that the occurrence of giant planets
(MP ≥ 0.3MJ ) with periods less than 2000 days is explained by a power law distribution
dN = CM α P β d ln Md ln P with the scalings α = −0.31 ± 0.2 and β = 0.26 ± 0.1.
The best-fit value of the normalization factor C implies that 10.5% of Sun-like stars
have planets with masses between 0.3–10MJ and orbital periods between 2–2000 days.
Concentrating on the orbital period distribution, the Keck/HIRES data revealed a
pile-up of planets with periods near 3 days. This surplus of planets with approximately
65
CHAPTER 1. INTRODUCTION
3-day orbits was also observed in previous studies (e.g., Gaudi et al. 2005).
Considering less massive planets, Howard et al. (2010) estimated the planet
occurrence rate of super-Earths, Neptunes, and Jupiters from five years of Keck/HIRES
observations of 166 GK stars under the NASA–University of California Eta-Earth Survey.
Their sample of stars includes 22 planet host stars with 33 detected planets, roughly half
of which have orbital periods < 50 days. Correcting for search incompleteness, Howard
et al. (2010) found that the increase in planet occurrence with decreasing planet mass
can be described by the power-law df/d log M = 0.39M−0.48 . All of the detected planets
have minimum masses larger than 4 M⊕ , but extending the power law to lower masses
suggests that 23+16
−10 % of GK stars harbor Earth-mass planets (0.5 − 2 M⊕ ) with periods
shorter than 50 days.
The planet occurrence rates derived by Cumming et al. (2008) and Howard et al.
(2010) based Keck/HIRES observations can be compared to the rates estimated from the
European CORALIE and HARPS surveys. Mayor et al. (2011) derived their estimates
via a combined analysis of 13 years of observations with the CORALIE spectrograph at
the 1.2-m EULER Swiss telescope (Udry et al. 2000) and eight years of observations
with the HARPS spectrograph at the ESO 3.6-m telescope. The CORALIE survey
included observations of roughly 1650 stars with a typical single-measurement precision
of 5 m s−1 between 1998–2008 and an improved precision between 2008–2011 after an
instrument upgrade. The HARPS sample was significantly smaller than the CORALIE
sample and consisted of 376 apparently quiet stars with spectral types between late F
and late K. In total, there are 155 detected planets orbiting 102 stars in the HARPS and
CORALIE samples. The majority (> 2/3) of these planets were detected by the HARPS
or CORALIE surveys, but some were discovered by other teams.
66
CHAPTER 1. INTRODUCTION
For a mass and period range comparable to that considered by Cumming et al.
(2008) (MP > 100 M⊕ , P < 10 years for HARPS+CORALIE versus MP ≥ 0.3MJ ,
P < 2000 days for HIRES), Mayor et al. (2011) presented a very similar occurrence
estimate of 9.7 ± 1.3% gas giants per Sun-like star. Including less massive planets,
they reported that 57.1 ± 8% of Sun-like stars host at least one planet with an orbital
period shorter than 100 days. Concentrating on super-Earths and Neptunes, they
found that 47.9 ± 8.5% of FGK stars harbor planets with masses < 30 M⊕ and orbital
periods less than 100 days. Restricting the sample even further to periods shorter than
50 days, Mayor et al. (2011) estimated that 38.8 ± 7.1% of FGK stars host short-period
super-Earths and Neptunes. That estimate is consistent with the rate of 23+16
−10 % of GK
stars determined by Howard et al. (2010).
The four-year Kepler mission inspired several transit-based analyses of the planet
occurrence rate for FGK stars. Using the early Q0-Q2 KOI list, Howard et al. (2012)
determined occurrence rates of 0.130 ± 0.008, 0.023 ± 0.003, and 0.0013 ± 0.002 planets
per star with periods shorter than 50 days and radii of 2 − 4 R⊕ , 4 − 8 R⊕ , and 8 − 32 R⊕ ,
respectively. Fitting a power law of the form df /d log R = kR (R/ R⊕ )α to the planet
occurrence rate as a function of orbital period, they observed that the observations were
well-explained by a model with kR = 2.9+0.5
−0.4 and α = −1.92 ± 0.11.
Howard et al. (2012) also remarked that the planet occurrence rate as a function
of orbital period was better fit by a broken power law model in which the position of
the break depended on planet radius (extending from 1.7 days for 8 − 32 R⊕ planets out
to 7.0 days for 2 − 4 R⊕ planets) than by a simple power law model. The position and
planet radius-dependence of the break point may be a residual signature of planetary
migration. Youdin (2011) further noted that the inferred population of planets with
67
CHAPTER 1. INTRODUCTION
orbital periods shorter than seven days contains significantly fewer large planets than
does the population of longer period planets. The pronounced shortage of planets with
radii of roughly 3 R⊕ at short periods likely provides further evidence of migration and
possible photoevaporation (Youdin 2011).
In a combined analysis of the false positive rate (see Section 1.4) and the frequency
of planets based on the Q1–Q6 KOI catalog, Fressin et al. (2013) estimated that
16.5 ± 3.6% of FGK stars harbor at least one small planet (0.8 − 1.25 R⊕ ) with an orbital
period shorter than 85 days. For shorter orbital periods of 0.8 − 50 days, Fressin et al.
(2013) estimated occurrence rates of 0.15 ± 0.024 Earths (0.8 − 1.25 R⊕), 0.19 ± 0.02
super-Earths (1.25 − 2 R⊕ ), 0.18 ± 0.01 small Neptunes (2 − 4 R⊕ ), 0.013 ± 0.002
large Neptunes (4 − 6 R⊕ ), and 0.013 ± 0.001 giant planets (6 − 22 R⊕ ) per FGK star.
Compared to the earlier estimates by Howard et al. (2012), these results suggest that
small Neptunes are slightly more frequent than previously assumed.
Incorporating significantly more Kepler data and using a custom transit detection
pipeline with well-determined search completeness, Petigura et al. (2013b) estimated that
15.1+1.8
−2.7 % of FGK stars harbor 1 − 2 R⊕ planets with orbital periods between 5–50 days.
This result is similar to the previous estimate by Fressin et al. (2013), which was based
on Kepler observations made over a much shorter temporal baseline. Inspecting the
behavior of the planet occurrence as a function of radius, Petigura et al. (2013b) argued
that the planet occurrence rate reaches a plateau for planets smaller than 2 R⊕ rather
than increasing further with decreasing planet radius.
In a subsequent study, Petigura et al. (2013a) estimated the occurrence rate of
potentially habitable planets orbiting FGK stars. The four-year Kepler data set has
68
CHAPTER 1. INTRODUCTION
extremely low search completeness within the relevant region of radius-period space,
so Petigura et al. (2013a) constrained the behavior of the planet occurrence rate at
shorter periods and then extrapolated outward to their region of interest. Within their
comparison region, they estimated that 20.4% and 26.2% of FGK stars host 1 − 2 R⊕
planets with P < 50 days and P < 100 days, respectively. They then assumed that
planet occurrence is constant in log P and extrapolated that 5.7+1.7
−2.2 % of FGK stars host
1 − 2 R⊕ planets with periods of 200 − 400 days and that 11 ± 4% host 1 − 2 R⊕ planets
receiving 1–4 times as much insolation flux as the Earth. Note that at the time this
thesis was written, a true Earth analog (a 1 R⊕ planet receiving 1 F⊕ of insolation flux
from a Sun-like star) had not been detected.
As discussed in Section 1.7, planets receiving such high values of insolation are most
likely too hot to be habitable, but Petigura et al. (2013a) also provided an estimate of
the expected number of more temperature planets. They estimated that roughly 8.6% of
FGK stars harbor 1 − 2 R⊕ planets within the boundaries of habitable zone as defined
by Kopparapu et al. (2013b). Foreman-Mackey et al. (2014) later conducted a follow-up
analysis in which they used gaussian processes and made less restrictive assumptions
about the functional form of the planet occurrence rate. The specific methodology of
their analysis is described in detail in Chapter 3.
Foreman-Mackey et al. (2014) found a significantly lower frequency of small planets
within the habitable zones of FGK stars. For the same choice of period and radius
boundaries, Foreman-Mackey et al. (2014) estimated 0.019+0.010
−0.008 planets per star, roughly
a factor of three less than the value from Petigura et al. (2013a). Foreman-Mackey
et al. (2014) argued that the primary explanation for the difference is that Petigura
et al. (2013a) adopted a fixed relation for the behavior of the planet occurrence rate
69
CHAPTER 1. INTRODUCTION
versus orbital period rather than considering more flexible models and that they did not
adequately consider the errors in the planet radius estimates.
Silburt et al. (2015) also investigated the planet occurrence rate for Sun-like stars.
They considered a sample of 76,711 Sun-like (R! = 0.8 − 1.2 R$ ) Kepler target stars and
estimated the search completeness using the reported Combined Differential Photometric
Precision (Christiansen et al. 2012, see Chapter 2). Focusing on the distribution of
planet candidates with periods of 20–200 days, they reported that the planet occurrence
rate is higher for planets with radii of 2 − 2.8 R⊕ than for smaller or larger planets. In
total, they estimated that a typical Sun-like star hosts 0.46 ± 0.03 planets with periods
of 20–200 days and radii of 1 − 4 R⊕ . In agreement with Petigura et al. (2013a), Silburt
et al. (2015) noted that the planet occurrence rate is flat in log(P ). Within a broad
habitable zone extending from 0.99–1.7 AU, Silburt et al. (2015) estimated an occurrence
rate of 0.064+0.034
−0.011 small (1 − 2 R⊕ ) planets per star.
1.8.3
Low-Mass Stars
The Keck Planet Search sample described in Cumming et al. (2008) also included 110 M
dwarfs with masses below 0.5 M$ . At the time of the Cumming et al. (2008) paper, the
Keck data permitted the detection of two planets: GJ 876b and GJ 436b. A sample of
two planets was insufficient for a detailed investigation of the dependence of the M dwarf
planet occurrence rate on mass and orbital period, but Cumming et al. (2008) tested
whether the relationship they derived for FGK stars might also explain the M dwarf
population if the normalization factor C were allowed to vary.
Their resulting best-fit normalization factor was ten times lower than the
70
CHAPTER 1. INTRODUCTION
normalization factor fit to the FGK star population, suggesting that giant planets
orbiting M dwarfs are indeed quite rare. Specifically, they estimated that only 2% of
M dwarfs harbor 0.3–10MJ planets with orbital periods between 2–2000 days.16 The low
occurrence rate of giant planets orbiting M dwarfs reported by Cumming et al. (2008) is
consistent with expectations from core accretion models (see Section 1.5.1 and Laughlin
et al. 2004; Ida & Lin 2005; Kennedy & Kenyon 2008).
Revisiting the California Planet Survey sample, Johnson et al. (2010) investigated
the dependence of giant planet occurrence on metallicity and stellar mass. In agreement
with Cumming et al. (2008), they observed that giant planets are significantly more rare
in M dwarf systems (3.4+2.2
−0.9 % of M dwarfs host planets more massive than Saturn with
semimajor axes smaller than 2.5 AU) than in higher mass systems (14% for A stars) even
after the influence of metallicity is removed. However, a later analysis of evolved planet
host stars by Mortier et al. (2013) did not reveal a correlation between the frequency of
giant planets and stellar mass.
Using direct imaging observations to constrain the likelihood that observed RV
accelerations are due to stellar companions rather than long-period planets, Montet
et al. (2014) later reported that 6.5% ± 3.0± of M dwarfs harbor massive (1 − 13MJ )
planets with semimajor axes as large as 20 AU. They also suggested that the occurrence
rate of giant planets f scales with stellar mass M! and metallicity F=[Fe/H] as
f (M! , F ) = CM!a 10bF where the overall normalization C = 0.039+0.056
−0.028 , the mass power
law index a = 0.8+1.1
−0.9 , and the metallicity scaling b = 3.8 ± 1.2. The metallicity scaling b
16
The occurrence rate quoted here includes a correction factor made by Cumming et al. (2008) to
account for the fact that the sample of M dwarf planets with periods < 2000 days planet should also
include GJ 849b (Butler et al. 2006).
71
CHAPTER 1. INTRODUCTION
is steeper than the previously proposed scalings of b = 1.2 and b = 1.26 − 2.94 (Johnson
et al. 2010 and Neves et al. 2013, respectively), suggesting the role of metallicity is even
stronger than previously estimated.
In an analysis of six years of ESO/HARPS RVs of 102 southern M dwarfs, Bonfils
et al. (2013) presented further evidence that giant planets are rarely found in M dwarf
systems. For periods between 1 and 10 days, Bonfils et al. (2013) estimated occurrence
rates of ≤ 0.01 giant planets (100–1000 M⊕ ), 0.03+0.04
−0.01 Neptunes (10-100 M⊕ ), and
0.36+0.24
−0.10 super-Earths (1–10 M⊕ ) per M dwarf. At longer periods of 10–100 days,
+0.50
they reported 0.02+0.03
−0.01 giant planets, < 0.02 Neptunes (10-100 M⊕ ), and 0.52−0.16
super-Earths per M dwarf.
Bonfils et al. (2013) also provided an estimate of the occurrence rate of potentially
habitable planets orbiting M dwarfs. They defined “potentially habitable” planets as
worlds with semimajor axes within the habitable zone boundaries computed by Selsis
et al. (2007) and minimum masses between 1 M⊕ and 10 M⊕ . Their sample of detected
planets included two potentially habitable worlds (Gl 581d and Gl 667Cc) and they
estimated that they would have been able to detect potentially habitable planets around
4.84 “effective” stars from their full sample of 102 M dwarfs. They therefore inferred an
occurrence rate of η⊕ = 0.41+0.54
−0.13 potentially habitable planets per M dwarf.
However, Robertson et al. (2014) later revealed that the signal attributed to the
planet Gl 581d was actually a manifestation of stellar activity. Baluev (2013) had
previously found a reduced significance for Gl 581d by accounting for correlated noise
when fitting the data and Robertson et al. (2014) determined that the putative planetary
signal disappeared entirely when a correlation between Hα activity and measured RV
72
CHAPTER 1. INTRODUCTION
was considered. Excluding Gl 581d from the sample of potentially habitable planets
discovered by HARPS decreased the Bonfils et al. (2013) estimate to 0.33 potentially
habitable planets per M dwarf (Robertson et al. 2014). The revised value is consistent
with the Kepler-based estimate of 0.24+0.18
−0.08 potentially habitable Earth-size planets per
M dwarf (see Chapter 3 and Dressing & Charbonneau 2015).
Radial velocity surveys and transit surveys are both most sensitive to close-in
planets due to the dependence of the transit probability and RV semiamplitude on
period (Equations 1.3 and 1.4, respectively). In contrast, microlensing surveys are most
sensitive to planets near or beyond the snow line. More precisely, microlensing is most
likely to detect planets near the “Einstein ring” at which magnification is maximized
(Gaudi 2012, and references therein). The angular position θE of the Einstein ring is
given by:
θE ≡
!
4GM
Drel c2
"1/2
(1.17)
−1
where M is the mass of the lens star (which is the planet host star), Drel
≡ Dl−1 − Ds−1 ,
Dl is the distance to the lens, and Ds is the distance to the source. In typical cases,
the distances of the source and lens are such that the physical distance rE = θE Dl
corresponding to the Einstein ring is 2–4 AU times a stellar-mass-dependent corrective
factor of (M/ M$ )1/2 .
For reference, Ida & Lin (2005) assumed that the position of the snow line depends
on the mass of the host star as asnow = 2.7M! / M$ AU. Other studies argued that
the decline in the stellar accretion rate has a stronger influence on the position of the
snow line than does the stellar mass. Kennedy & Kenyon (2008) found that between
104.8 years and 106.8 years the snow line moves inward from roughly 5 AU to1 AU for
73
CHAPTER 1. INTRODUCTION
solar mass stars and from roughly 3.5 AU to 0.8 AU for early M dwarfs. The region of
maximum microlensing search sensitivity therefore roughly coincides with the snow line
for Sun-like stars and is likely slightly beyond the snow line for M dwarfs.
Due to the different regions of maximum search sensitivity, the combination of
microlensing and RV surveys therefore provides a highly complementary picture of planet
occurrence. The two methods overlap slightly for planets more massive than 100 M⊕
with orbital periods between roughly 3–10 years, but in general microlensing surveys
are sensitive to much larger orbital separations than are RV surveys (Clanton & Gaudi
2014a). The most common planets detected by microlensing have estimated orbital
periods of 3–24 years and anticipated RV semiamplitudes of 0.09–1.33 m s−1 , rendering
them largely undetectable by current RV surveys (Clanton & Gaudi 2014a).17
Accounting for the different completeness biases in RV and microlensing surveys,
Clanton & Gaudi (2014b) found that the M dwarf planet occurrence rates derived from
the two methods are consistent. Combining the results of both techniques, Clanton &
Gaudi (2014b) reported that the occurrence rate for orbital periods of 1–104 days is
0.029+0.013
−0.015 (super-)Jupiters (1–13MJ ) per M dwarf, roughly a factor of four lower than
for FGK stars. Including less massive gas giants, they estimated an occurrence rate of
4
0.15+0.06
−0.07 planets with masses of 30–10 M⊕ per M dwarf within the same period range.
The occurrence rate for the broader mass range is also lower than the corresponding
17
Note that the individual stars targeted by RV surveys and the population of microlensing host stars
have very little overlap. RV surveys favor bright stars with spectral types between late F and early M
whereas microlensing surveys typically target the galactic bulge. The properties of the host stars of many
planets detected via microlensing are unknown, but they are primarily expected to be M dwarfs simply
because M dwarfs comprise the majority of stars in the galaxy.
74
CHAPTER 1. INTRODUCTION
rate for FGK stars, but the discrepancy is reduced to a factor of 2.2 (Clanton & Gaudi
2014b).
As discussed in more detail in Chapters 2 and 3, the M dwarf planet occurrence
rates inferred from RV surveys are also consistent with those derived from transit
surveys. However, there is very little overlap between the search sensitivities of transit
and microlensing surveys. Comparing the results of transit surveys to RV surveys
requires an assumption about the relationship between planet radius and mass. In this
section, I employ relations defined by the masses and radii of the planets in our own
Solar System and those of the subset of exoplanets with highly precise estimates of both
parameters. In reality, this relationship is likely a smeared distribution rather than a
strict functional one-to-one relation (Wolfgang & Lopez 2014). The conversion between
mass and radius also requires careful consideration of the role of host star radiation in
the inflation of the radii of close-in giant planets (e.g., Burrows et al. 2007) and the
possible photoevaporation of the envelopes of highly irradiated mini-Neptunes (e.g.,
Lopez et al. 2012).
For orbital periods shorter than 50 days, Dressing & Charbonneau (2015, see
Chapter 3) found an occurrence rate of 0.56+0.06
−0.05 Earth-size planets (1 − 1.5 R⊕ ) and
0.46+0.07
−0.05 super-Earths (1.5 − 2 R⊕ ) per M dwarf. Dressing & Charbonneau (2015)
reported much lower rates of occurrence for larger 3 − 4 R⊕ planets (0.06+0.03
−0.02 ) within
the same period range. Extending to longer orbital periods of 0.5 − 100 days, Dressing
+0.08
+0.05
& Charbonneau (2015) estimated occurrence rates of 0.65+0.07
−0.05 , 0.57−0.06 and 0.09−0.03
Earths, super Earths, and Neptunes, respectively. Integrating over radius from
0.5 − 4 R⊕ and orbital period from 0.5–180 days, Dressing & Charbonneau (2015) found
a cumulative planet occurrence rate of 3.02 ± 0.25 planets per star.
75
CHAPTER 1. INTRODUCTION
The resulting cumulative planet occurrence rate is higher than the result from
Gaidos et al. (2014) of 2.01 ± 0.36 planets per M dwarf with radii of 0.5 − 6 R⊕ and
orbital periods shorter than 180 days. Gaidos et al. (2014) based their estimate on
the Q1–Q16 KOI Catalog, revised stellar parameters (Gaidos 2013), and a theoretical
model of planet detectability using a photometric noise estimate based on the measured
Combined Differential Photometric Precision (Christiansen et al. 2012) for each star.
They modeled the detected planet population in an iterative fashion (Gelman & Rubin
1992) by varying the assumed intrinsic distribution of planet occurrence as a function
of radius and period until the population of simulated detected planets mirrored the
population of Kepler planet candidates.
Figure 1.3 compares the planet occurrence distributions inferred by Dressing
& Charbonneau (2015) and Gaidos et al. (2014) in more detail. Both distributions
feature peaks at smaller planet radii and significantly lower planet occurrence rates
for mini-Neptunes than for Earth-size planets, but the shape of the decline in planet
occurrence with increasing planet occurrence is different. In comparison to the Gaidos
et al. (2014) distribution, the Dressing & Charbonneau (2015) distribution features a
broader “shelf” in planet occurrence between roughly 1.25 − 2.25 R⊕ and an elevated
planet occurrence rate for planets with radii of approximately 2.25 − 3.25 R⊕ . The
Dressing & Charbonneau (2015) distribution also appears shifted to larger planet radii.
One explanation for this difference might be that Dressing & Charbonneau (2015)
empirically assessed search completeness using a series of transit injection and recovery
tests whereas Gaidos et al. (2014) assumed that planet detectability could be predicted
from the Combined Differential Photometric Precision (CDPP; a measurement of
the noise in the light curve on the timescale of a planetary transit). We tested this
76
CHAPTER 1. INTRODUCTION
Occurrence per Unit Radius
2.0
Dressing & Charbonneau 2015
DC15 Threshold Model
DC15 Ramp Model
Gaidos et al. 2014
1.5
1.0
0.5
0
1
2
Planet Radius (REarth)
3
4
Figure 1.3: Inferred planet occurrence rate per unit radius versus planet radius for
planets with orbital periods shorter than 180 days. The red histogram was reproduced
from Gaidos et al. (2014) using the ADS Dexter tool (Demleitner et al. 2001) and the
navy histogram displays the occurrence rate from Dressing & Charbonneau (2015) with
the same choice of binning. The other two histograms display how the occurrence rates
reported by Dressing & Charbonneau (2015) would change if the sensitivity were modeled
by an abrupt detection threshold at SNR=12 (teal histogram) or by a smooth ramp
between 0% detection efficiency at SNR=6 and 100% detection efficiency at SNR=16
(purple histogram).
77
CHAPTER 1. INTRODUCTION
Difference (DC15 - SNR12)
4.0
Planet Radius (REarth)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
1
10
Period (Days)
Detection Fraction
-0.32
-0.24
-0.16
-0.08
100
0.00
Difference (DC15 - Ramp)
4.0
Planet Radius (REarth)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
1
10
Period (Days)
Detection Fraction
-0.37
-0.30
-0.22
-0.14
100
-0.06
Figure 1.4: Difference in inferred search completeness between an empirical estimate
based on injection and recovery tests (Dressing & Charbonneau 2015) and an abrupt
threshold at SNR=12 (Top) or a smooth ramp between 0% detection efficiency at SNR=6
and 100% detection efficiency at SNR=12 (Bottom).
78
CHAPTER 1. INTRODUCTION
assumption by using the reported CDPP for each star in our sample to generate models
of the expected sensitivity as a function of planet radius and period for two simple models
of the detection efficiency: (1) a step-function with 0% efficiency below SNR=12 and
100% efficiency above and (2) a ramp from 0% efficiency at SNR=6 to 100% efficiency at
SNR=16. The first model matches the detection threshold used by Gaidos et al. (2014)
and the second model was based on the results of Fressin et al. (2013). Both of these
models differ significantly from the completeness map Dressing & Charbonneau (2015)
derived based on transit injection and recovery simulations.
As shown in Figure 1.4, the completeness inferred by Dressing & Charbonneau
(2015) is lower than the value predicted by either a threshold or ramp model. In both
cases, the difference is most pronounced for small planets (Rp < 1 R⊕ ) in short-period
orbits (P< 10 days) and for 1.5 − 2.5 R⊕ planets with orbital periods of 100 − 200 days.
Adopting the threshold model or the ramp model results in a cumulative planet
occurrence rate of 2.60 ± 0.22 or 2.46 ± 0.21, respectively, for 0.5 − 4 R⊕ planets with
orbital periods of 0.5 − 200 days.
However, consulting Figure 1.3 reveals that adopting either of these completeness
models decreases the overall occurrence rate but does not resolve the overall discrepancy
between the Dressing & Charbonneau (2015) and Gaidos et al. (2014) results. In
contrast, altering the assumed completeness generates a discrepancy for small planet
radii without significantly improving the agreement for larger planets. Accordingly,
the difference between the Dressing & Charbonneau (2015) and Gaidos et al. (2014)
rates cannot be attributed solely (or even mostly) to different assumptions about search
completeness. Alternative explanations include differences in the selection of the stellar
sample, the specific properties assigned to each of the stars considered, the method of
79
CHAPTER 1. INTRODUCTION
smearing the planet candidate distributions, and the overall approach used to calculate
the underlying occurrence rate. Further investigations of the relative influence of each of
these choices would be useful for determining the true dependence of the occurrence rate
on planet radius.
In an independent study of the Q1–Q12 KOI catalog, Morton & Swift (2014)
investigated the radius distribution of M dwarf planets using a weighted kernel density
estimator and a CDPP-based assumption of search sensitivity. They estimated a
cumulative planet occurrence rate of 2.00 ± 0.45 planets per M dwarf with orbital
periods < 150 days and radii 0.5 − 4 R⊕ . This result is consistent with the estimate from
Gaidos et al. (2014) and slightly lower than the estimate from Dressing & Charbonneau
(2015), who found an occurrence rate of 2.85 ± 0.24 planets per star within the same
radius and period boundaries. Morton & Swift (2014) did not detect a turnover in the
planet occurrence rate at 1 R⊕ , although they suggest that there may be a decline below
roughly 0.8 R⊕ . That possible decline is also observed in the occurrence distribution from
Dressing & Charbonneau (2015) shown in Figure 1.3, but the data are also consistent
with a flat occurrence rate between 0.5 R⊕ and 1.25 R⊕ .
Although the exact estimates differ slightly, the occurrence rate of small planets
orbiting M dwarfs appears to be higher than the rate calculated for Sun-like stars (see
Section 1.8.2). Mulders et al. (2015) confirmed this result, stating that the cumulative
occurrence rate of small (1 − 4 R⊕ ) planets with periods shorter than 50 days is roughly
twice and three times higher for M dwarfs than for G and F stars. They also noted that
the planet occurrence rates inferred for F, G, K, and M stars increase with increasing
semimajor axis up to a critical distance and then remain roughly constant as a function
of logarithmic semimajor axis.
80
CHAPTER 1. INTRODUCTION
Intriguingly, if the occurrence rates are normalized and the semimajor axis is
1/3
rescaled by a factor that depends on M!
such that the plateaus occur at the same
scaled semimajor axis, the resulting planet occurrence rates are nearly identical for
FGKM stars. The required scaling factors (1.2, 1.4, and 1.6 for G, K, and M stars,
respectively) are similar to the scaling expected if the radial distribution of planets is
set by the orbital distance at which the estimated rotation speed of the protoplanetary
disk would have been equal to the expected spin rate of the pre-main-sequence star. For
FGKM stars, the semimajor axis of the co-rotation radius is predicted to scale with the
cube root of the stellar mass. Accordingly, if the orbital distance at which the planet
occurrence rate transitions from increasing with increasing log a to remaining flat with
increasing log a is determined by the position of the co-rotation radius within the disk,
then the semimajor axis at which the planet occurrence rate flattens should also scale
with the cube root of the stellar mass. One might then expect that the planet occurrence
rate as a function of the ratio of semimajor axis to the co-rotation radius would appear
self-similar for different types of stars.
Within a conservative habitable zone with boundaries set by the moist greenhouse
inner limit and the maximum greenhouse outer limit defined by Kopparapu et al.
(2013b), Dressing & Charbonneau (2015, see also Chapter 3) estimated occurrences of
+0.10
0.16+0.17
−0.07 Earths and 0.12−0.05 super-Earths per M dwarf. Relaxing the habitable zone
limits to the “optimistic” case of adopting boundaries set by the flux received by Venus
and Mars when they were last expected to have liquid water (see Section 1.7), we found
+0.11
0.24+0.18
−0.08 Earths and 0.21−0.06 super-Earths per M dwarf. As mentioned earlier, these
rates are comparable to the RV-based estimate of 0.33 potentially habitable planets per
M dwarf (Bonfils et al. 2013; Robertson et al. 2014). The RV-based estimate considered
81
CHAPTER 1. INTRODUCTION
planets with masses of 1 − 10 M⊕ , which would roughly correspond to planet radii of
1 − 2 R⊕ if the planets have Earth-like compositions or 1 − 4 R⊕ if the more massive
planets have substantial envelopes of H, He, water, or other volatiles.
Converting planet occurrence rates estimated in terms of the number of planets per
star into the fraction of stars with planets requires an assumption about the mutual
inclinations of planets within a single system. Ballard & Johnson (2014) addressed
the coplanarity of planets orbiting M dwarfs by generating synthetic planetary systems
and determining the multiplicities and mutual inclinations required to reproduce the
detected population of Kepler M dwarf planet candidates. They concluded that M dwarf
planetary systems are bimodal: 55+23
−12 % of systems have only a single planet or highly
misaligned multiple planets while the remaining systems contain 6.1 ± 1.9 planets with
◦
mutual inclinations of 2.0◦ +4.0
−2.0◦ .
A prime example of the second category of M dwarf systems is Kepler-32, a 0.5 M$
star orbited by five transiting planets with radii Rp = 0.81 − 2.7 R⊕ and orbital periods
P = 0.74 − 22.8 days Swift et al. (2013). In order to test whether closely packed
M dwarf planetary systems like Kepler-32 are common, Swift et al. (2013) generated
a mock catalog of Kepler-32 planetary systems with random orientations and mutual
inclinations selected from a Rayleigh distribution as suggested by Lissauer et al. (2011).
The observed numbers of systems with one, two, or more transiting planets were best
explained by a mutual inclination distribution of 1.2◦ ± 0.2◦ , in agreement with the
previous estimate of 1.0 − 2.3◦ for Kepler planet candidates orbiting either Sun-like stars
or low-mass stars (Fabrycky et al. 2014) and with the subsequent result by Ballard &
Johnson (2014) using the updated M dwarf planet candidate list.
82
CHAPTER 1. INTRODUCTION
Muirhead et al. (2015) provided further evidence that compact, coplanar systems
are common around M dwarfs. Concentrating specifically on planets with orbital periods
shorter than 10 days orbiting mid-M dwarfs, they calculated that 21+7
−5 % of mid-M dwarfs
harbor systems of tightly packed short period planets with low mutual inclinations.
Making a few assumptions about the masses of the planets in compact systems and the
initial protoplanetary disk masses, Muirhead et al. (2015) suggested that the planet
formation process may be highly efficient for the subset of mid-M dwarfs hosting compact
multiplanet systems. They also provided the alternative explanations that the planets
are less rocky than they assumed or the protoplanetary disk masses may be higher than
predicted, reducing the required planet formation efficiency. An increased sample of
small planets with well-measured masses and more thorough studies of protoplanetary
disk masses and lifetimes in M dwarf systems will help elucidate which explanation
or combinations thereof are responsible for the commonality of M dwarf multiplanet
systems with low mutual inclinations.
Moving farther down the main sequence, Berta et al. (2013) conducted an analysis
of the four-year MEarth Project data set in order to determine whether the resulting
constraints on the planet occurrence rate for mid- and late-M dwarfs were consistent
with the rates derived for early M dwarfs based on Kepler and RV data. The MEarth
sample size of one detected planet (GJ 1214b, Charbonneau et al. 2009) is small, but the
planet yield is consistent with the assumption that later M dwarfs and earlier M dwarfs
have similar planet occurrence rates. As discussed in Chapter 6, future observations by
MEarth, MEarth-South, K2, TESS, and other surveys will allow further studies of the
dependence of planet occurrence on stellar mass across the M dwarf spectral sequence.
83
CHAPTER 1. INTRODUCTION
1.8.4
The Role of Metallicity on the Frequency of Low-Mass
Planets
The theory that higher metallicity stars are more likely to host giant planets is now
well-established (e.g., Gonzalez 1997; Fischer & Valenti 2005; Johnson et al. 2010; Ghezzi
et al. 2010; Schlaufman & Laughlin 2011; Buchhave et al. 2012; Everett et al. 2013),
but the effect of stellar metallicity on the occurrence rate of smaller planets is less well
understood. Using a large sample of > 400 exoplanet host stars, Buchhave et al. (2014)
investigated the dependence of the planet occurrence rate on both stellar metallicity
and planet radius. They observed that the average metallicities of stars harboring gas
giants (Rp > 3.9 R⊕ ), “gas dwarfs” (1.7 R⊕ < Rp < 3.9 R⊕ ), and terrestrial planets
(Rp < 1.7 R⊕ ) are 0.18 ± 0.02, 0.05 ± 0.01, and −0.02 ± 0.02 dex, respectively. Calculating
the likelihood that the separate groups of planet host stars were actually drawn from the
same overall population, they reported evidence for significant variation in the underlying
metallicity distribution of planet host stars as a function of planet radius. According to
Buchhave et al. (2014), the transitions occur for planets with radii of 3.52+0.74
−0.28 R⊕ (the
giant planet/gas dwarf boundary) at a significance of 4.7+0.6
−0.4 σ and for planets with radii
+0.5
of 1.55+0.88
−0.04 R⊕ (the gas dwarf/terrestrial planet boundary) at a significance of 4.2−0.4 σ.
In a follow-up study, Schlaufman (2015) disputed the existence of the proposed
gas dwarf/terrestrial planet metallicity boundary at 1.7 R⊕ . Using the same stellar
population as Buchhave et al. (2014), Schlaufman (2015) tested whether the observed
distribution of planet radii and host star metallicities was better explained by a linear
relation between planet radius and metallicity or by a mixture of subpopulations with
characteristic radii and metallicities. Schlaufman (2015) found a strong preference for
84
CHAPTER 1. INTRODUCTION
a first-order linear model, but remarked that the best mixture model contained two
populations rather than three. In the best two-population mixture model, planets with
radii larger than 4 R⊕ are more prevalent around metal-rich stars whereas planets smaller
than 4 R⊕ occur at a broader range of metallicities.
Wang & Fischer (2015) conducted an independent study of the planet-metallicity
correlation using a sample of 406 Kepler planet candidates orbiting spectroscopically
characterized host stars with Teff = 4800 − 6500K and surface gravities log g ≥ 4.2.
They found that the planet-metallicity correlation affects small planets as well as large
planets, but that the strength of the metallicity dependence decreases with decreasing
planet radius. Adopting the same planet size categories as Buchhave et al. (2014), they
reported that the occurrence rates of terrestrial, gas dwarf, and gas giant planets were
+0.29
+5.34
enhanced by factors of 1.72+0.19
−0.17 , 2.03−0.26 , and 9.30−3.04 for stars with [Fe/H]> 0.05
compared to stars with [Fe/H]< −0.05.
1.9
Summary
This introduction has provided an overview of the advantages of searching for small
planets around smaller stars (Section 1.1), the challenges of characterizing low-mass
stars (Section 1.2) and the influence of stellar phenomena on planet detectability and
habitability (Section 1.3). Section 1.4 described techniques to distinguish astrophysical
false positives from transiting planets, Section 1.5 provided a brief introduction to planet
formation theory, and Section 1.6 summarized current knowledge of the compositions of
small planets within and beyond the solar system. In addition, Section 1.7 considered the
possible habitability of exoplanets and Section 1.8 summarized estimates of the planet
85
CHAPTER 1. INTRODUCTION
occurrence rate for a variety of spectral types.
The next two chapters of this thesis consider low-mass star characterization and the
planet occurrence rate for M dwarfs in considerably more detail. Chapter 2 revises the
parameters of the smallest Kepler target stars and provides an estimate of the M dwarf
planet occurrence rate based on the first 1.5 yr of Kepler observations. Chapter 3 then
refines that estimate using the full four-year Kepler data set and an empirical estimate
of pipeline sensitivity. Next, Chapter 4 discusses the possibility that the radii of some
Kepler planet candidates might be underestimated due to the presence of additional light
within the target aperture. The chapter focuses on an observing campaign conducted
with ARIES at the MMT to search for additional stars near Kepler target stars.
Chapter 5 then returns to the question of the composition of small planets by presenting
a mass estimate for the 1.478 R⊕ planet Kepler-93b. The chapter also considers the
compositions of small dense planets in general and hypothesizes that all dense planets
with masses of 1 − 6 M⊕ can be explained by an Earth-like mixture of rock and iron.
Finally, Chapter 6 provides a glimpse towards the future of exoplanet discovery and
characterization.
86
Chapter 2
Revised Properties for Low-Mass
Kepler Target Stars and an Initial
Estimate of the Planet Occurrence
Rate for Early M Dwarfs
This thesis chapter originally appeared in the literature as
C. D. Dressing & D. Charbonneau, The Astrophysical Journal,
767, 95, 2013
87
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Abstract
We use the optical and near-infrared photometry from the Kepler Input Catalog to
provide improved estimates of the stellar characteristics of the smallest stars in the
Kepler target list. We find 3897 dwarfs with temperatures below 4000K, including
64 planet candidate host stars orbited by 95 transiting planet candidates. We refit the
transit events in the Kepler light curves for these planet candidates and combine the
revised planet/star radius ratios with our improved stellar radii to revise the radii of
the planet candidates orbiting the cool target stars. We then compare the number of
observed planet candidates to the number of stars around which such planets could have
been detected in order to estimate the planet occurrence rate around cool stars. We
find that the occurrence rate of 0.5 − 4 R⊕ planets with orbital periods shorter than
50 days is 0.90+0.04
−0.03 planets per star. The occurrence rate of Earth-size (0.5 − 1.4 R⊕ )
planets is constant across the temperature range of our sample at 0.51+0.06
−0.05 Earth-size
planets per star, but the occurrence of 1.4 − 4 R⊕ planets decreases significantly at cooler
temperatures. Our sample includes 2 Earth-size planet candidates in the habitable zone,
allowing us to estimate that the mean number of Earth-size planets in the habitable zone
is 0.15+0.13
−0.06 planets per cool star. Our 95% confidence lower limit on the occurrence rate
of Earth-size planets in the habitable zones of cool stars is 0.04 planets per star. With
95% confidence, the nearest transiting Earth-size planet in the habitable zone of a cool
star is within 21 pc. Moreover, the nearest non-transiting planet in the habitable zone is
within 5 pc with 95% confidence.
88
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
2.1
Introduction
The Kepler mission has revolutionized exoplanet statistics by increasing the number
of known extrasolar planets and planet candidates by a factor of five and discovering
systems with longer orbital periods and smaller planet radii than prior exoplanet surveys
(Batalha et al. 2011; Borucki et al. 2012; Fressin et al. 2012; Gautier et al. 2012). Kepler
is a Discovery-class space-based mission designed to detect transiting exoplanets by
monitoring the brightness of over 100,000 stars (Tenenbaum et al. 2012). The majority
of Kepler’s target stars are solar-like F GK dwarfs and accordingly most of the work
on the planet occurrence rate from Kepler has been focused on planets orbiting that
sample of stars (e.g., Borucki et al. 2011b; Catanzarite & Shao 2011; Youdin 2011;
Howard et al. 2012; Traub 2012). Those studies revealed that the planet occurrence rate
increases toward smaller planet radii and longer orbital periods. Howard et al. (2012)
also found evidence for an increasing planet occurrence rate with decreasing stellar
effective temperature, but the trend was not significant below 5100K.
Howard et al. (2012) conducted their analysis using the 1235 planet candidates
presented in Borucki et al. (2011b). The subsequent list of candidates published in
February 2012 (Batalha et al. 2013) includes an additional 1091 planet candidates and
provides a better sample for estimating the occurrence rate. The new candidates are
primarily small objects (196 with Rp < 1.25 R⊕ , 416 with 1.25 R⊕ < Rp < 2 R⊕ , and
421 with 2 R⊕ < Rp < 6 R⊕ ), but the list also includes 41 larger candidates with radii
6 R⊕ < Rp < 15 R⊕ . The inclusion of larger candidates in the Batalha et al. (2013)
sample is an indication that the original Borucki et al. (2011b) list was not complete
at large planet radii and that continued improvements to the detection algorithm may
89
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
result in further announcements of planet candidates with a range of radii and orbital
periods.
In addition to nearly doubling the number of planet candidates, Batalha et al. (2013)
also improved the stellar parameters for many target stars by comparing the estimated
temperatures, radii, and surface gravities in the Kepler Input Catalog (KIC; Batalha
et al. 2010b; Brown et al. 2011) to the values expected from Yonsei-Yale evolutionary
models (Demarque et al. 2004). Rather than refer back to the original photometry,
Batalha et al. (2013) adopted the stellar parameters of the closest Yonsei-Yale model
to the original KIC values in the three-dimensional space of temperature, radius, and
surface gravity. This approach did not correctly characterize the coolest target stars
because the starting points were too far removed from the actual temperatures, radii,
and surface gravities of the stars. In addition, the Yonsei-Yale models overestimate the
observed radii and luminosity of cool stars at a given effective temperature (Boyajian
et al. 2012).
2.1.1
The Small Star Advantage
Although early work (Dole 1964; Kasting et al. 1993) suggested that a hypothetical
planet in the habitable zone (the range of distances at which liquid water could exist on
the surface of the planet) of an M dwarf would be inhospitable because the planet would
be tidally-locked and the atmosphere would freeze out on the dark side of the planet,
more recent studies have been more optimistic. For instance, Haberle et al. (1996)
and Joshi et al. (1997) demonstrated that sufficient quantities of carbon dioxide could
prevent the atmosphere from freezing. In addition, Pierrehumbert (2011) reported that a
90
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
tidally-locked planet could be in a partially habitable “Eyeball Earth” state in which the
planet is mostly frozen but has a liquid water ocean at the substellar point. Moreover,
planets orbiting M dwarfs might become trapped in spin-orbit resonances like Mercury
instead of becoming spin-synchronized.
A second concern for the habitability of planets orbiting M dwarfs is the possibility
of strong flares and high UV emission in quiescence (France et al. 2012). Although
a planet without a magnetic field could require years to rebuild its ozone layer after
experiencing strong flare, the majority of the UV flux would never reach the surface of
the planet. Accordingly, flares do not present a significant obstacle to the habitability
of planets orbiting M dwarfs (Segura et al. 2010). Furthermore, the specific role of UV
radiation in the evolution of life on Earth is uncertain. A baseline level of UV flux might
be necessary to spur biogenesis (Buccino et al. 2006), yet UV radiation is also capable of
destroying biomolecules.
Having established that planets in the habitable zones of M dwarfs could be
habitable despite the initial concern of the potential hazards of tidal-locking and stellar
flares, the motivation for studying the coolest target stars is three-fold. First, several
more years of Kepler observations will be required to detect Earth-size planets in the
habitable zones of G dwarfs due to the higher-than-expected photometric noise due to
stellar variability (Gilliland et al. 2011), but Kepler is already sensitive to the presence
of Earth-size planets in the habitable zones of M dwarfs. Although a transiting planet
in the habitable zone of a G star transits only once per year, a transiting planet in the
habitable zone of a 3800K M star transits five times per year. Additionally, the geometric
probability that a planet in the habitable zone transits the star is 1.8 times greater.
Furthermore, the transit signal of an Earth-size planet orbiting a 3800K M star is
91
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
3.3 times deeper than the transit of an Earth-size planet across a G star because the star
is 45% smaller than the Sun. The combination of a shorter orbital period, an increased
transit probability, and a deeper transit depth greatly reduces the difficulty of detecting
a habitable planet and has motivated numerous planet surveys to target M dwarfs
(Delfosse et al. 1999; Endl et al. 2003; Nutzman & Charbonneau 2008; Zechmeister
et al. 2009; Apps et al. 2010; Barnes et al. 2012; Berta et al. 2012a; Bowler et al. 2012;
Giacobbe et al. 2012; Law et al. 2012).
Second, as predicted by Salpeter (1955) and Chabrier (2003), studies of the solar
neighborhood have revealed that M dwarfs are twelve times more abundant than
G dwarfs. The abundance of M dwarfs, combined with growing evidence for an increase
in the planet occurrence rate at decreasing stellar temperatures (Howard et al. 2012),
implies that the majority of small planets may be located around the coolest stars.
Although M dwarfs are intrinsically fainter than solar-type stars, 75% percent of the
stars within 10 pc are M dwarfs1 (Henry et al. 2006). These stars would be among the
best targets for future spectroscopic investigations of potentially-habitable rocky planets
due to the small radii and apparent brightness of the stars.
Third, confirming the planetary nature and measuring the mass of an Earth-size
planet orbiting within the habitable zone of an M dwarf is easier than confirming and
measuring the mass of an Earth-size planet orbiting within the habitable zone of a
G dwarf. The radial velocity signal induced by a 1 M⊕ planet in the middle of the
habitable zone (a = 0.28AU) of a 3800K, 0.55 M$ M dwarf is 23 cm/s. In comparison,
the RV signal caused by a 1 M⊕ planet in the habitable zone of a G dwarf is 9 cm/s.
1
http://www.recons.org/census.posted.htm
92
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
The prospects for RV confirmation are even better for planets around mid-to-late
M dwarfs: an Earth-size planet in the habitable zone of a 3200K M dwarf would
produce an RV signal of 1 m/s, which is achievable with the current precision of modern
spectrographs (Dumusque et al. 2012). Prior to investing a significant amount of
resources in investigations of the atmosphere of a potentially habitable planet, it would
be wise to first guarantee that the candidate object is indeed a high-density planet and
not a low-density mini-Neptune.
Finally, upcoming facilities such as JWST and GMT will have the capability to
take spectra of Earth-size planets in the habitable zones of M dwarfs, but not Earth-size
planets in the habitable zones of more massive stars. In order to find a sample of
habitable zone Earth-size planets for which astronomers could measure atmospheric
properties with the next generation of telescopes, astronomers need to look for planets
around small dwarfs.
2.1.2
Previous Analyses of the Cool Target Stars
In light of the advantages of searching for habitable planets around small stars, several
authors have worked on refining the parameters of the smallest Kepler target stars.
Muirhead et al. (2012a) collected medium-resolution, K-band spectra of the cool
planet candidate host stars listed in Borucki et al. (2011b) and presented revised
stellar parameters for those host stars. Their sample included 69 host stars with
KIC temperatures below 4400K as well as an additional 13 host stars with higher
KIC temperatures but with red colors that hint that their KIC temperatures were
overestimated. Muirhead et al. (2012a) determined effective temperature and metallicity
93
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
directly from their spectra using the H2 O-K2 index (Rojas-Ayala et al. 2012) and then
constrain stellar radii and masses using Dartmouth stellar evolutionary models (Dotter
et al. 2008; Feiden et al. 2011). We adopt the same set of stellar models in this paper.
Muirhead et al. (2012a) found that one of the 82 targets (Kepler Object of Interest
(KOI) 977) is a giant star and that three small KOIs (463.01, 812.03, 854.01) lie within
the habitable zone.
Johnson et al. (2012) announced the discovery of KOI 254.01, the first short-period
gas giant orbiting an M dwarf. The planet has a radius of 0.96RJup and orbits its host star
KIC 5794240 once every 2.455239 days. In addition to discussing KOI 254.01, Johnson
et al. (2012) also calibrated a relation for determining the masses and metallicities of
M dwarfs from broad-band photometry. They found that J − K color is a reasonable
(±0.15 dex) indicator of metallicity for stars with metallicities between −0.5 and 0.5 dex
and J − K colors within 0.1 magnitudes of the main sequence J − K at the V − K color
of the star in question. The relationship between infrared colors and metallicities was
first proposed by Mould & Hyland (1976) and subsequently confirmed by Leggett (1992)
and Lépine et al. (2007).
Mann et al. (2012) took the first steps toward a global reanalysis of the cool Kepler
target stars. They acquired medium-resolution, visible spectra of 382 target stars and
classified all of the cool stars in the target list as dwarfs or giants using “training sets”
constructed from their spectra and literature spectra. Mann et al. (2012) found that the
majority of bright, cool target stars are giants in disguise and that the temperatures
of the cool dwarf stars are systematically overestimated by 110 K in the KIC. Mann
et al. (2012) reported that correctly classifying and removing giant stars removes the
correlation between cool star metallicity and planet occurrence observed by Schlaufman
94
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
& Laughlin (2011). After removing giant stars from the target list, Mann et al. (2012)
calculated a planet occurrence rate of 0.37 ± 0.08 planets per cool star with radii between
2 and 32 R⊕ and orbital periods less than 50 days. Their result is higher than the
occurrence rate we report in Section 2.5.3, most likely because of our revisions to the
stellar radii.
In this paper, we characterize the coolest Kepler target stars by revisiting the
approach used to create the Kepler Input Catalog (Brown et al. 2011) and tailoring
that method for application to cool stars. Specifically, we extract grizJHK photometry
from the KIC for the 51813 planet search target stars with KIC temperature estimates
≤ 5050K and for the 13402 planet search target stars without KIC temperature estimates
and compare the observed colors to the colors of model stars from the Dartmouth Stellar
Evolutionary Database (Dotter et al. 2008; Feiden et al. 2011). We discuss the features
of the Dartmouth stellar models in Section 2.2.1 and explain our procedure for assigning
revised stellar parameters in Section 2.2.2. We present revised stellar characterizations
in Section 2.3 and improved planetary parameters for the associated planet candidates in
Section 2.4. We address the implications of these results on the planet occurrence rate
in Section 2.5 and conclude in Section 2.6.
2.2
2.2.1
Methods
Stellar Models
The Dartmouth models incorporate both an internal stellar structure code and a
model atmosphere code. Unlike the ATLAS9 models (Castelli & Kurucz 2004) used
95
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
in development of the Kepler Input Catalog, the Dartmouth models perform well for
low-mass stars because the package uses PHOENIX atmospheres to model stars cooler
than 10,000K. The PHOENIX models include low-temperature chemistry and are
therefore well-suited for use with low-mass dwarfs (Hauschildt et al. 1999a,b).
The Dartmouth models include evolutionary tracks and isochrones for a range of
stellar parameters. The tracks and isochrones are available electronically2 and provide
the mass, luminosity, temperature, surface gravity, metallicity, helium fraction, and
α-element enrichment at each evolutionary time step. We consider the full range of
Dartmouth model metallicities (−2.5 ≤ [Fe/H] ≤ 0.5), but we restrict our set of models
to stars with solar α-element enhancement, masses below 1 M$ , and temperatures below
7000K. We exclude models of more massive stars because solar-like stars are well-fit by
the ATLAS9 models used in the construction of the KIC and it is unlikely that a star as
massive as the Sun would have been assigned a temperature lower than our selection cut
TKIC ≤ 5050K.
The Dartmouth team supplies synthetic photometry for a range of photometric
systems by integrating the spectrum of each star over the relevant bandpass. We
downloaded the synthetic photometry for the 2MASS and Sloan Digital Sky Survey
Systems (SDSS) and used relations 1–4 from Pinsonneault et al. (2012) to convert the
observed KIC magnitudes for each Kepler target star to the equivalent magnitudes in
the SDSS system. For cool stars, the correction due to the filter differences is typically
much smaller than the assumed errors in the photometry (0.01 mag in gri and 0.03
mag in zJHK, similar to the assumptions in Pinsonneault et al. 2012). All stars have
2
http://stellar.dartmouth.edu/models/grid.html
96
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
full 2MASS photometry, but 21% of the target stars are missing photometry in one or
more visible KIC bands. For those stars, we correct for the linear offset in all bands and
apply the median correction found for the whole sample of stars for the color-dependent
term. In our final cool dwarf sample, 70 stars lack g-band photometry and 29 stars lack
z-band. We exclude all stars with more than one missing band.
Our final sample of model stars is drawn from a set of isochrones with ages 1–13 Gyr
and spans a temperature range 2708–6998K. The stars have masses 0.01–1.00 M$,
radii 0.102–223 R$, and metallicities −2.5 < [Fe/H] < 0.5. All model stars have solar
α/Fe ratios. There is a deficit of Dartmouth model stars with radii 0.32 − 0.42 R$ ; we
cope with this gap by fitting polynomials to the relationships between temperature,
radius, mass, luminosity, and colors at fixed age and metallicity. We then interpolate
those relationships over a grid with uniform (0.01 R$ ) spacing between 0.17 R$ and
0.8 R$ to derive the parameters for stars that would have fallen in the gap in the original
model grid. We compute the surface gravities for the resulting interpolated models
from their masses and radii. When fitting stars, we use the original grid of model stars
supplemented by the interpolated models. Our fitted parameters may be unreliable for
stars younger than 0.5 Gyr because those stars are still undergoing Kelvin-Helmholtz
contraction.
Distinguishing Dwarfs and Giants
We specifically include giant stars in our model set so that we have the capability to
identify red giants that have been misclassified as red dwarfs (and vice versa). Muirhead
et al. (2012a) discovered one such masquerading giant (KOI 977) in their spectroscopic
97
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
analysis of the cool planet candidate host stars and Mann et al. (2012) have argued that
giant stars comprise 96% ± 1% of the population of bright (Kepmag < 14) and 7% ± 3%
of the population of dim (Kepmag > 14) cool target stars. We are confident in the ability
of our photometric analysis to correctly identify the luminosity class of cool stars because
the infrared colors of dwarfs and giant stars are well-separated at low temperatures. For
instance, our photometric analysis classifies KOI 977 (KIC 11192141) as a cool giant with
+3
+28
an effective temperature of 3894+50
−54 K, radius R! = 36−2 R$ , luminosity L! = 260−25 L$ ,
+0.01
and surface gravity log g = 1.3+0.06
−0.05 . The reported mass (0.99−0.05 M$ ) is near the edge of
our model grid, so refitting the star with a more massive model grid may yield different
results for the stellar parameters.
2.2.2
Revising Stellar Parameters
We assign revised stellar parameters by comparing the observed optical and near infrared
colors of all 51813 cool (TKIC ≤ 5050K) and all 13402 unclassified Kepler planet search
target stars to the colors of model stars. We account for interstellar reddening by
determining the distance at which the apparent J-band magnitude of the model star
would match the observed apparent J-band magnitude of each target star. We then
apply a band-dependent correction assuming 1 magnitude of extinction per 1000 pc in
V -band in the plane of the galaxy (Koppen & Vergely 1998; Brown et al. 2011). We find
the best-fit model for each target star by computing the difference in the colors of a given
target star and all of the model stars. We then scale the differences by the photometric
errors in each band and add them in quadrature to determine the χ2 for a match to each
model star.
98
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
As explained in Section 2.2.2, we incorporate priors on the stellar metallicity
and the height of stars above the plane of the galaxy. We rescale the errors so that
the minimum χ2 is equal to the number of colors (generally 6) minus the number
of fitted parameters (3 for radius, temperature, and metallicity). We then adopt
the stellar parameters corresponding to the best-fit model and set the error bars
to encompass the parameters of all model stars falling within the 68.3% confidence
interval. For example, for KOI 2626 (KID 11768142), we find 68.3% confidence intervals
+120
+0.1
R! = 0.35 R$ +0.11
−0.05 , T! = 3482−57 K, and [Fe/H] = −0.1−0.1 . We find a best-fit mass
+0.02
+63
0.36+0.12
−0.06 M$ and luminosity 0.016−0.005 L$ , resulting in a distance estimate of 159−27 pc.
The corresponding surface gravity is therefore log g = 4.91+0.08
−0.12 .
Priors on Stellar Parameters
We find that fitting the target stars without assuming prior knowledge of the metallicity
distribution leads to an overabundance of low-metallicity stars, so we adopt priors on the
underlying distributions of metallicity and height above the plane. We then determine
the best-fit model by minimizing the equation
χ2i = χ2i,color − 2 ln Pmetallicity,i − 2 ln Pheight,i
(2.1)
where χ2i,color is the total color difference between a target star and model star i,
Pmetallicity,i is the probability that a star has the metallicity of model star i, and Pheight,i
is the probability that a star would be found at the height at which model star i would
have the same apparent J-band magnitude as the target star. We weight the priors so
that each prior has the same weight as a single color.
We set the metallicity prior by assuming that the metallicity distribution of the
99
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
M dwarfs in the Kepler target list is similar to the metallicity distribution of the
343 nearby M dwarfs studied by Casagrande et al. (2008). Following Brown et al. (2011),
we produce a histogram of the logarithm of the number of stars in each logarithmic
metallicity bin and then fit a polynomial to the distribution. We extrapolate the
polynomial down to [Fe/H] = −2.5 and up to [Fe/H] = 0.5 to cover the full range
of allowed stellar models. Our final metallicity prior and the histogram of M dwarf
metallicities from Casagrande et al. (2008) are shown in Figure 2.1. The distribution
peaks at [Fe/H]= −0.1 and has a long tail extending down toward lower metallicities.
We adopt the same height prior as Brown et al. (2011): the number of stars falls off
exponentially with increasing height above the plane of the galaxy and the scale height
of the disk is 300 pc (Cox 2000). Our photometric distance estimates for 77% of our cool
dwarfs are within 300 pc, so adopting this prior has little effect on the chosen stellar
parameters and the resulting planet occurrence rate.
2.2.3
Assessing Covariance Between Fitted Parameters
Our procedure for estimating stellar parameters expressly considers the covariance
between fitted parameters by simultaneously determining the likelihood of each of the
models and determining the range of temperatures, metallicities, and radii that would
encompass the full 68.3% confidence interval. The provided error bars therefore account
for the fact that high-metallicity warm M dwarfs and low-metallicity cool M dwarfs have
similar colors.
We confirm that the quoted errors on the stellar parameters are large enough to
account for the errors in the photometry by conducting a perturbation analysis in which
100
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
log(Counts)
2.0
1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
[Fe/H]
Figure 2.1: Logarithmic number of stars versus logarithmic metallicity bin. The black
histogram displays the distribution of metallicities in the Casagrande et al. (2008) sample
and the green line is our adopted metallicity prior.
101
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
we create 100 copies of each of the Kepler M dwarfs and add Gaussian distributed
noise to the photometry based on the reported uncertainty in each band. We then run
our stellar parameter determination pipeline and compare the distribution of best-fit
parameters for each star to our original estimates. We find that there is a correlation
between higher temperatures and higher metallicities, but that our reported error bars
are larger than the standard deviation of the best-fit parameters.
2.2.4
Validating Methodology
We confirm that we are able to recover accurate parameters for low mass stars from
photometry by running our stellar parameter determination pipeline on a sample of stars
with known distances. We obtained a list of 438 M dwarfs with measured parallaxes,
JHK photometry from 2MASS, and g #r # i# photometry from the AAVSO Photometric
All-Sky Survey3 (APASS) from Jonathan Irwin (personal communication, January 2,
2013) and performed a series of quality cuts on the sample. We removed stars with
parallax errors above 5% and and stars with fewer than two measurements in the
APASS database. We then visually inspected the 2MASS photometry of the remaining
230 stars to ensure that none of them belonged to multiple systems that could have been
unresolved in APASS and resolved in 2MASS. We removed 203 stars with other stars or
quasars within 1’, resulting in a final sample of 26 stars.
We estimate the masses of the 26 stars by running our stellar parameter
determination pipeline to match the observed colors to the colors of Dartmouth model
stars. The APASS g # r # i# photometry was acquired using filters matching the original
3
http://www.aavso.org/apass
102
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
SDSS g #r # i# bands; we convert the APASS photometry to the unprimed SDSS 2.5m gri
bands using the transformation equations provided on the SDSS Photometry White
Paper.4 We then compare the masses assigned by our pipeline to the masses predicted
from the empirical relation between mass and absolute Ks magnitude (Delfosse et al.
2000). As shown in Figure 2.2, our mass estimates are consistent with the mass predicted
by the Delfosse relation. The masses predicted by the pipeline are typically 5% lower
than the mass predicted by the Delfosse relation, but none of these stars have reported
z-band photometry whereas 96% of our final sample of Kepler M dwarfs have full
grizJHK photometry. Accordingly, we do not fit for a correction term because the
uncertainty introduced by adding a scaling term based on fits made to stars with only
five colors would be comparable to the offset between our predicted masses and the
masses predicted by the Delfosse relation.
2.3
Revised Stellar Properties
Our final sample of cool Kepler target stars includes 3897 stars with temperatures below
4000K and surface gravities above log g = 3.6. The sample consists primarily of late-K
and early-M dwarfs, but 201 stars have revised temperatures between 3122 − 3300K. The
revised parameters for all of the cool dwarfs are provided in Table 2.1. We exclude 4420
stars from the final sample because their photometry is consistent with classification
as evolved stars (log g < 3.6) and 608 stars because their photometry is insufficient to
discriminate between dwarf and giant models. We refer to the stars that could be fit
by either dwarf or giant models as “ambiguous” stars. The majority (80%) of the stars
4
http://www.sdss.org/dr5/algorithms/jeg_photometric_eq_dr1.html
103
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Fractional Parallax Error
Estimated Mass (Solar Masses)
0.7
0.6
0.00
0.01
0.02
0.03
0.03
0.04
0.05
0.5
0.4
0.3
0.2
1:1
Fit to the Data
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Mass from Parallax & Delfosse Relation (Solar Masses)
Figure 2.2: Mass estimated by our photometric stellar parameter determination pipeline
versus mass predicted by the Delfosse relation. The dashed red line indicates a 1:1 relation
and the solid blue line is fit to the data. The points are color-coded by the reported
fractional error in the parallax measurement.
104
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
classified as “ambiguous” were not assigned temperatures in the KIC. We find that
96 − 98% of cool bright (Teff < 4000K, Kepmag < 14) stars and 5 − 6% of cool faint
(Teff < 4000K, Kepmag > 14) stars are giants, which is consistent with Mann et al.
(2012). (The precise fractions of giant stars depend on whether the ambiguous stars are
counted as giant stars.) One of the excluded ambiguous stars is KID 8561063 (KOI 961),
which was confirmed by Muirhead et al. (2012b) as a 0.17 ± 0.04 R$ , 3200 ± 65K star
hosting sub-Earth-size three planet candidates. The KIC does not include z-band
photometry for KOI 961 and we were unable to rule out matches with giant stars using
only griJHK photometry.
The distributions of temperature, radius, metallicity, and surface gravity for the
stars in our sample are shown in Figure 2.3. For comparison, we display both fits made
without using priors (left panels) and fits including priors on the stellar metallicity
distribution and the height of stars above the plane of the galaxy (right panels). In both
cases the radii of the majority of stars are significantly smaller than the values given in
the KIC and the surface gravities are much higher. As discussed in Section 2.2.2, the
primary difference between the two model fits is that setting a prior on the underlying
metallicity distribution reduces the number of stars with revised metallicities below
[Fe/H]= −0.6. Since such stars should be relatively uncommon, we choose to adopt the
stellar parameters given by fitting the stars assuming priors on metallicity and height
above the plane.
Incorporating priors, the median temperature of a star in the sample is 3723K and
the median radius is 0.45 R$ . Most of the stars in the sample are slightly less metal-rich
than the Sun (median [Fe/H]=−0.1), but 21% have metallicities 0.0 ≤[Fe/H]< 0.5.
Although nearly all of the stars in the sample (96%) had KIC surface gravities below
105
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.1. Revised Cool Star Properties
KID
Teff (K)
R∗ ( R$ )
M∗ ( M$ )
log g
[Fe/H]
Dist (pc)
1162635 3759+50
−50
0.494+0.05
−0.05
0.505+0.05
−0.05
4.754+0.06
−0.06
-0.10+0.1
−0.1
261.3+17
−12
1292688 3774+77
−50
0.530+0.07
−0.05
0.539+0.06
−0.05
4.722+0.06
−0.07
0.00+0.1
−0.1
282.1+43
−10
1293177 3385+50
−50
0.216+0.05
−0.05
0.204+0.05
−0.05
5.077+0.06
−0.06
-0.40+0.1
−0.1
101.7+15
−10
1293393 3953+137
−54
0.536+0.13
−0.05
0.555+0.12
−0.05
4.725+0.06
−0.13
-0.20+0.4
−0.1
454.2+130
−31
1429729 3903+76
−60
0.523+0.07
−0.05
0.541+0.07
−0.05
4.735+0.06
−0.07
-0.20+0.2
−0.1
380.0+61
−32
1430893 3929+98
−58
0.541+0.07
−0.05
0.564+0.07
−0.05
4.724+0.06
−0.06
-0.10+0.2
−0.1
269.8+42
−20
1433760 3296+50
−50
0.213+0.05
−0.05
0.196+0.05
−0.05
5.072+0.06
−0.06
-0.10+0.1
−0.1
109.8+13
−13
1569682 3860+93
−78
0.514+0.06
−0.07
0.544+0.07
−0.05
4.752+0.07
−0.06
-0.10+0.2
−0.1
262.8+41
−44
1569863 3591+50
−53
0.360+0.05
−0.06
0.384+0.07
−0.06
4.910+0.07
−0.06
-0.30+0.1
−0.1
157.0+25
−29
1572802 3878+53
−88
0.535+0.05
−0.06
0.545+0.05
−0.06
4.719+0.06
−0.06
-0.10+0.1
−0.1
246.4+25
−36
Note. — Table 2.1 is published in its entirety in the electronic edition. A
portion is shown here for guidance regarding its form and content.
106
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
log g = 4.7, our reanalysis indicates that 95% actually have surface gravities above
log g = 4.7. As shown by the purple histograms in each of the panels, the distribution of
stellar parameters for the planet candidate host stars matches the overall distribution of
stellar parameters for the cool star sample.
The two-dimensional distribution of radii and temperatures for our chosen model
fit is shown in Figure 2.4. The spread in the radii of the model points at a given
temperature is due to the range of metallicities allowed in the model suite. At a given
temperature, the majority of the original radii from the KIC lie above the model grid in
a region of radius–temperature space unoccupied by low-mass stars. The discrepancy
between the model radii and the KIC radii is partially due to the errors in the assumed
surface gravities. As shown in Figure 2.3, the surface gravities assumed in the KIC peak
at log(g) = 4.5 with a long tail extending to lower surface gravities whereas the minimum
expected surface gravity for cool stars is closer to log(g) = 4.7.
For a typical cool star, we find that the revised radius is only 69% of the original
radius listed in the KIC and that the revised temperature is 130K cooler than the original
temperature estimate. The majority (96%) of the stars have revised radii smaller than
the radii listed in the KIC and 98% of the stars are cooler than their KIC temperatures.
The revised radius and temperature distribution of planet candidate host stars is similar
to the underlying distribution of cool target stars. The median changes in radius and
temperature for a cool planet candidate host star are −0.19 R$ (−29%) and −102K,
respectively.
We compare the revised and initial parameters for the host stars in more detail in
Figure 2.5. For all host stars except for KOI 1078 (KID 10166274), the revised radii are
107
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Without Priors
With Priors
1200
1000
Number of Stars
Revised Host Stars (x40)
Revised Values
KIC Values
800
600
400
200
0
2950
3338 3725 4112 4500
Effective Temperature (K)
1200
1000
Number of Stars
Number of Stars
Number of Stars
1200
800
600
400
200
0
1000
800
600
400
200
0
2950
1200
1000
800
600
400
200
0
0.2 0.4 0.6 0.8 1.0 1.2
Radius (Solar Radii)
500
0
-2.0
Number of Stars
1500
Revised Host Stars (x40)
Revised Values
KIC Values
Number of Stars
1000
0.2 0.4 0.6 0.8 1.0 1.2
Radius (Solar Radii)
-1.5
-1.0 -0.5 0.0
Metallicity
1000
800
600
400
200
0
4.0
4.3
4.6 4.9
log(g)
5.2
1000
500
0
-2.0
0.5
Number of Stars
Number of Stars
1500
3338 3725 4112 4500
Effective Temperature (K)
5.5
-1.5
-1.0 -0.5 0.0
Metallicity
0.5
1000
800
600
400
200
0
4.0
4.3
4.6 4.9
log(g)
5.2
5.5
Figure 2.3: Histograms of the resulting temperature (top), radius (second from top),
metallicity (third from top), and surface gravity (bottom) distributions for the target
stars with revised temperatures below 4000K. The panels on the left show the distributions
resulting from fitting the stars without setting priors while the stellar parameters in the
right panels were fit assuming priors on metallicity and height above the plane. In all
panels, a histogram of the original KIC values is shown in blue and a histogram of the
revised values is plotted in red. The distribution of cool host stars (multiplied by forty)
is shown in purple in all plots.
108
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Radius (Solar Radii)
1.5
KIC Values
Revised Values
Revised Host Stars
Model Grid
Median Errors
1.0
0.5
0.0
3000
3500
4000
Temperature (K)
4500
Figure 2.4: Revised (red) and original (blue) temperatures and radii of the cool target
stars. The revised values were determined by comparing the observed colors of stars to
the expected colors of Dartmouth model stars (gray) and incorporating priors on the
metallicity and height above the galactic plane. The revised stellar parameters for cool
planet candidate host stars are highlighted in purple. The position of the KIC radii well
above the model grid indicates that many of the combinations of radius and temperature
found in the KIC are nonphysical.
109
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
smaller than the radii listed in the KIC and the revised temperatures for all of the stars
are cooler than the KIC temperatures. Unlike the original values given in the KIC, the
revised temperatures and radii of the cool stars align to trace out a main sequence in
which smaller stars have cooler temperatures by construction.
1.2
Stellar Radius (RSun)
1.0
0.8
0.6
0.4
0.2
3200
KIC
This Work
3400
3600
3800
4000
Stellar Effective Temperature (K)
4200
Figure 2.5: Revised (red circles) and original (blue squares) radii and temperatures for
the planet candidate host stars with revised temperatures below 4000K. The gray lines
connect the initial and final values for each host star.
110
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
2.3.1
Comparison to Previous Work
We validate our revised parameters by comparing our photometric effective temperatures
for a subset of the cool target stars to the spectroscopic effective temperatures from
Muirhead et al. (2012a) and Mann et al. (2012). We exclude the stars KIC 5855851
and KIC 8149616 from the comparison due to concerns that their spectra may have
been contaminated by light from another star (Andrew Mann, personal communication,
January 15, 2013). As shown in Figure 2.6, our revised temperatures are consistent
with the literature results for stars with revised temperatures below 4000K, which is the
temperature limit for our final sample.
At higher temperatures, we find that our temperatures are systematically hotter
than the literature values reported by Muirhead et al. (2012a). The temperatures given
in Muirhead et al. (2012a) are determined from the H2 O-K2 index (Rojas-Ayala et al.
2012), which measures the shape of the spectrum in K-band. Although the H2 O-K2
index is an excellent temperature indicator for cool stars, the index saturates around
4000K, accounting for the disagreement between our temperature estimates and the
Muirhead et al. (2012a) estimates for the hotter stars in our sample.
We also compare our photometric metallicity estimates to the spectroscopic
metallicity estimates from Muirhead et al. (2012a). Given the disagreement between our
temperature estimates and Muirhead et al. (2012a) at higher temperatures, we choose
to plot only the 32 stars with revised temperatures below 4000K and spectroscopic
metallicities from Muirhead et al. (2012a). The top panel of Figure 2.7 compares our
revised metallicities to the spectroscopic metallicities from Muirhead et al. (2012a). We
observe a systematic offset in metallicity with our values typically 0.17 dex lower than
111
Literature Temperature (K)
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
4200
Muirhead+ 2012
Mann+ 2012
4000
3800
3600
3400
3200
3200
3440 3680 3920 4160
Revised Temperature (K)
4400
Figure 2.6: Spectroscopic effective temperatures from Muirhead et al. (2012a) (red circles) and Mann et al. (2012) (blue squares) versus our revised photometric effective temperature estimates. The dashed black line indicates a 1:1 relation. The disagreement for
the hotter stars is attributed to the saturation of the H2 O-K2 index used by Muirhead
et al. (2012a) at temperatures above 4000K.
112
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
the metallicities reported in Muirhead et al. (2012a).
The metallicity difference is dependent on the spectroscopic metallicity of the star,
as depicted in the lower panel of Figure 2.7, which shows the metallicity difference
as a function of the metallicity reported in Muirhead et al. (2012a). For stars with
Muirhead et al. (2012a) metallicities between -0.2 and -0.1 dex, our revised metallicities
are 0.05 dex lower, but for stars with Muirhead et al. (2012a) metallicities above 0.1 dex,
our revised metallicities are 0.3 dex lower.
113
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Temperature Difference (K)
Metallicity Difference Revised [Fe/H]
-156
-73
11
95
178
262
0.0
-0.2
-0.4
-0.6
0.0
-0.2
-0.4
-0.6
-0.5
-0.3
-0.1
0.1
0.3
0.5
Spectroscopically Determined [M/H] from Muirhead et al. (2012)
Figure 2.7: Comparison of our photometric metallicity estimates to the spectroscopic
metallicities from Muirhead et al. (2012a) for stars with revised T < 4000K. The colorcoding indicates our revised stellar temperatures and the dashed red lines mark a 1:1
relation between photometric and spectroscopic metallicities. Top: Revised photometric
metallicity estimates versus spectroscopic metallicity. Bottom: Metallicity difference
(photometric - spectroscopic) versus spectroscopic metallicity.
114
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.2. Revised Properties for Planet Candidates Orbiting Small Stars
t0
P
(Days)
(Days)
1.525
13.815
47.536
0.030
0.4
1.41+0.26
−0.29
248.01b 5364071
4.593
7.028
17.897
0.032
0.6
248.02c
6.158
10.913
21.948
0.047
KOI
KID
247.01
11852982
248.03
5364071
5364071
2.076
2.577
a/R∗ a
10.121
Rp /R∗
0.032
RP
FP
Teff
R∗
( R⊕ )
(F⊕ )
(K)
( R$ )
4.41+5.61
−2.95
3725
0.437
1.83+0.18
−0.26
16.90+3.57
−1.88
3903
0.523
0.8
2.69+0.26
−0.38
9.40+9.86
−6.67
3903
0.523
0.5
1.83+0.18
−0.26
64.39+5.83
−3.94
3903
0.523
4.62+3.62
−2.14
3903
0.523
b
248.04
5364071
11.080
18.596
51.184
0.034
0.5
1.96+0.19
−0.27
249.01
9390653
3.871
9.549
44.353
0.040
0.3
1.60+0.22
−0.22
4.65+7.56
−6.22
3514
0.370
0.3
2.73+0.63
−0.54
6.06+4.21
−3.46
3853
0.447
0.5
2.73+0.63
−0.54
3.85+28.82
−23.70
3853
0.447
31.79+2.07
−1.70
3853
0.447
250.01d
250.02e
9757613
9757613
10.720
11.877
12.283
17.251
34.265
62.567
0.056
0.056
250.03
9757613
1.594
3.544
11.511
0.020
0.5
0.98+0.23
−0.19
250.04
9757613
43.087
46.828
157.259
0.039
0.6
1.92+0.44
−0.38
1.02+2.18
−1.68
3853
0.447
0.7
2.63+0.27
−0.34
26.00+29.45
−15.49
3743
0.488
0.5
0.76+0.08
−0.10
16.81+0.94
−0.50
3743
0.488
3.82+9.79
−8.78
3770
0.479
251.01
251.02
10489206
10489206
0.347
0.157
4.164
5.775
12.214
18.612
0.049
0.014
252.01
11187837
12.059
17.605
33.315
0.045
0.5
2.37+0.25
−0.30
253.01
11752906
4.643
6.383
17.910
0.049
0.8
3.05+0.27
−0.47
21.99+6.33
−5.68
3919
0.574
0.5
10.74+0.98
−1.44
68.37+1.67
−1.30
3837
0.550
0.3
2.77+0.24
−0.34
3.07+8.92
−8.91
3907
0.570
49.78+24.30
−24.90
3410
0.346
254.01f
255.01
5794240
7021681
1.410
24.694
2.455
27.522
11.223
51.142
0.179
0.045
256.01m 11548140
0.200
1.379
4.825
0.454
1.2
17.12+2.48
−2.48
463.01
0.491
18.478
69.231
0.049
0.5
1.80+0.33
−0.38
1.70+1.02
−1.05
3504
0.340
0.9
4.77+0.63
−0.69
36.55+27.40
−18.38
3898
0.490
13.98+1.36
−0.83
3820
0.500
531.01
8845205
10395543
1.255
3.687
14.775
0.089
571.01
8120608
7.166
7.267
22.444
0.025
0.4
1.37+0.14
−0.21
571.02
8120608
3.440
13.343
25.894
0.031
0.7
1.68+0.17
−0.25
6.22+19.62
−13.83
3820
0.500
571.03
8120608
1.184
3.887
12.801
0.023
0.6
1.24+0.12
−0.19
32.20+6.16
−5.33
3820
0.500
0.5
1.39+0.14
−0.21
3.12+2.74
−2.37
3820
0.500
63.67+14.19
−12.27
3626
0.430
571.04
8120608
19.360
22.407
46.240
0.025
596.01
10388286
0.496
1.683
8.565
0.025
0.4
1.17+0.14
−0.16
739.01
10386984
1.214
1.287
6.023
0.026
0.5
1.58+0.19
−0.14
187.26+1.37
−1.19
3994
0.554
781.01
11923270
6.418
11.598
29.230
0.055
0.7
2.54+0.30
−0.30
+28.30
4.65−22.44
3603
0.423
0.5
1.69+0.20
−0.20
2.45+87.00
−50.98
3758
0.474
10.09+1.99
−1.47
3758
0.474
6.67+1.20
−0.78
3564
0.401
817.01
4725681
18.439
23.968
42.743
0.033
817.02
4725681
3.063
8.296
45.449
0.029
0.5
1.49+0.18
−0.17
818.01
4913852
5.940
8.114
25.959
0.038
0.4
1.65+0.21
−0.21
115
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.2—Continued
t0
P
RP
FP
Teff
R∗
b
( R⊕ )
(F⊕ )
(K)
( R$ )
KOI
KID
(Days)
(Days)
a/R∗ a
Rp /R∗
854.01
6435936
33.001
56.055
90.045
0.039
0.4
1.69+0.33
−0.21
0.50+4.96
−3.22
3562
0.400
886.01g
7455287
1.978
8.011
6.286
0.038
1.1
1.38+0.30
−0.27
5.30+3.12
−2.19
3579
0.330
1.3
0.81+0.18
−0.16
3.07+0.35
−0.18
3579
0.330
0.8
1.14+0.25
−0.22
1.47+4.93
−2.48
3579
0.330
12.33+2.85
−1.44
3989
0.544
886.02h
886.03m
7455287
7455287
10.709
5.355
12.072
20.995
6.370
39.246
0.023
0.032
898.01i
7870390
9.615
9.770
27.672
0.042
0.4
2.49+0.23
−0.23
898.02
7870390
2.032
5.170
16.115
0.033
0.5
1.96+0.18
−0.18
28.81+1.36
−0.69
3989
0.544
0.4
2.14+0.20
−0.20
4.71+4.32
−3.18
3989
0.544
0.5
1.27+0.15
−0.25
8.74+10.10
−7.43
3587
0.410
24.26+1.65
−1.22
3587
0.410
898.03j
899.01
7870390
7907423
7.354
3.596
20.090
7.114
41.819
23.515
0.036
0.028
899.02
7907423
2.114
3.307
12.885
0.021
0.4
0.95+0.12
−0.19
899.03
7907423
9.085
15.368
31.920
0.028
0.8
1.24+0.15
−0.24
3.13+4.74
−4.23
3587
0.410
0.4
1.79+0.24
−0.26
4.88+13.16
−11.74
3518
0.370
113.74+1.70
−1.51
3518
0.370
936.01
9388479
7.990
9.468
27.967
0.044
936.02
9388479
0.580
0.893
5.775
0.025
0.4
1.03+0.14
−0.15
947.01
9710326
18.333
28.599
46.796
0.039
0.7
1.84+0.35
−0.26
1.61+2.31
−1.93
3717
0.430
952.01k
9787239
0.274
5.901
19.376
0.039
0.4
2.15+0.28
−0.28
18.06+53.82
−44.92
3787
0.506
0.7
1.94+0.25
−0.26
10.68+1.16
−0.61
3787
0.506
2.98+1.63
−1.10
3787
0.506
952.02l
9787239
4.351
8.752
19.985
0.035
952.03
9787239
18.525
22.780
48.891
0.047
0.4
2.58+0.33
−0.34
952.04
9787239
0.400
2.896
13.641
0.026
0.5
1.43+0.18
−0.19
46.65+25.47
−17.22
3787
0.506
1078.01
10166274
0.720
3.354
16.129
0.035
0.4
1.97+0.24
−0.25
+21.56
43.04−14.85
3878
0.523
0.9
2.50+0.30
−0.31
16.52+8.28
−5.70
3878
0.523
2.49+1.25
−0.86
3878
0.523
1078.02
10166274
1.417
6.877
20.830
0.044
1078.03
10166274
15.729
28.463
71.424
0.039
0.5
2.22+0.27
−0.28
1085.01
10118816
0.219
7.718
26.930
0.018
0.6
1.02+0.11
−0.10
14.89+6.29
−4.05
3878
0.535
1141.01
8346392
3.424
5.728
17.940
0.024
0.5
1.44+0.16
−0.14
+10.78
24.99−7.43
3976
0.550
0.4
0.99+0.13
−0.10
12.32+5.98
−3.67
3778
0.470
178.11+68.87
−52.00
3711
0.475
1146.01
8351704
1.504
7.097
23.314
0.019
1164.01
10341831
0.780
0.934
1.768
0.014
0.3
0.74+0.08
−0.08
1201.01
4061149
0.690
2.758
18.588
0.023
0.4
1.19+0.18
−0.16
43.54+27.86
−16.58
3728
0.482
0.4
2.24+0.20
−0.30
120.22+41.73
−43.59
3872
0.563
0.4
2.14+0.20
−0.20
21.87+8.08
−5.95
3957
0.542
4.20+3.80
−2.16
3424
0.220
0.82+0.74
−0.42
3424
0.220
1393.01
1397.01
9202151
9427402
1.164
0.829
1.695
6.247
7.709
30.153
0.037
0.036
1422.01
11497958
1.568
5.842
22.474
0.035
0.4
0.84+0.19
−0.19
1422.02
11497958
14.559
19.850
51.985
0.038
0.4
0.92+0.21
−0.21
116
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.2—Continued
t0
P
(Days)
(Days)
a/R∗ a
Teff
R∗
( R⊕ )
(F⊕ )
(K)
( R$ )
KID
1422.03
11497958
0.743
3.622
7.933
0.020
0.9
0.47+0.11
−0.11
7.95+7.18
−4.08
3424
0.220
1427.01
11129738
2.463
2.613
9.757
0.023
0.5
1.29+0.12
−0.16
+25.15
67.13−21.75
3979
0.523
0.9
1.02+0.11
−0.15
27.15+11.74
−10.05
3767
0.479
0.8
1.18+0.18
−0.15
8.63+4.77
−2.94
3608
0.400
0.30+0.19
−0.12
3414
0.300
1681.01
11337141
5531953
2.239
6.486
4.044
6.939
7.983
15.493
0.019
0.027
b
FP
KOI
1649.01
Rp /R∗
RP
1686.01
6149553
43.529
56.867
102.482
0.029
0.5
0.95+0.16
−0.16
1702.01
7304449
1.082
1.538
9.008
0.028
0.6
0.80+0.15
−0.15
27.41+20.88
−12.57
3304
0.260
0.4
1.26+0.14
−0.22
19.30+9.46
−8.02
3584
0.450
0.5
0.86+0.10
−0.15
11.09+5.43
−4.61
3584
0.450
53.87+24.11
−16.96
3799
0.492
1843.01
1843.02
5080636
5080636
4.103
4.025
4.195
6.356
19.152
38.543
0.026
0.018
1867.01
8167996
0.033
2.550
9.819
0.022
0.5
1.20+0.12
−0.13
1867.02
8167996
6.446
13.969
26.759
0.045
1.0
2.42+0.25
−0.27
5.58+2.50
−1.76
3799
0.492
0.5
1.07+0.11
−0.12
20.76+9.29
−6.54
3799
0.492
5.68+2.16
−1.65
3950
0.560
1867.03
8167996
2.404
5.212
15.672
0.020
1868.01
6773862
13.183
17.761
76.082
0.034
0.4
2.10+0.19
−0.20
1879.01
8367644
2.731
22.085
69.891
0.053
0.5
2.37+0.38
−0.35
1.96+1.21
−0.75
3635
0.410
1880.01
10332883
0.847
1.151
5.801
0.024
0.7
1.38+0.13
−0.22
182.97+69.88
−72.47
3855
0.530
0.5
1.96+0.19
−0.18
9.30+3.90
−2.52
3901
0.542
35.00+23.72
−13.95
3809
0.455
1907.01
7094486
9.197
11.350
32.483
0.033
2006.01
10525027
0.233
3.273
12.574
0.015
0.5
0.76+0.14
−0.11
2036.01
6382217
7.635
8.411
27.409
0.028
0.4
1.60+0.15
−0.30
13.30+6.10
−5.94
3903
0.523
2036.02
6382217
3.489
5.795
19.205
0.019
0.6
1.07+0.10
−0.20
21.85+10.02
−9.76
3903
0.523
0.5
1.11+0.10
−0.14
21.67+7.87
−7.63
3900
0.537
+47.95
133.13−46.43
3900
0.537
2057.01
9573685
3.200
5.945
18.668
0.019
2058.01
10329835
0.575
1.524
8.046
0.018
0.5
1.05+0.10
−0.13
2090.01
11348997
3.845
5.132
23.462
0.027
0.4
1.44+0.15
−0.22
18.90+8.02
−7.35
3688
0.497
2130.01
2161536
3.445
16.855
50.586
0.031
0.4
1.88+0.17
−0.25
6.27+2.21
−2.17
3972
0.565
1.2
11.32+1.22
−1.29
+14.90
37.83−11.55
3694
0.464
3.20+1.44
−1.21
3591
0.410
2156.01
2556650
0.835
2.852
9.813
0.223
2179.01
10670119
3.851
14.871
43.731
0.027
0.4
1.23+0.15
−0.18
2179.02
10670119
1.564
2.733
21.112
0.026
0.5
1.18+0.14
−0.17
30.65+13.80
−11.59
3591
0.410
0.6
0.96+0.11
−0.13
8.61+4.10
−3.00
3724
0.460
0.5
0.95+0.09
−0.09
120.04+38.46
−34.30
3900
0.537
538.37+219.33
−193.75
3878
0.520
+41.70
102.58−29.72
3815
0.498
2191.01
2238.01
5601258
8229458
7.441
0.313
8.848
1.647
29.479
8.090
0.019
0.016
2306.01
6666233
0.040
0.512
3.419
0.018
0.5
1.04+0.10
−0.15
2329.01
11192235
0.326
1.615
9.104
0.021
0.6
1.16+0.12
−0.12
117
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
2.4
Revised Planet Candidate Properties
Our sample of cool stars includes 64 host stars with 95 planet candidates. As part of
our analysis, we downloaded the Kepler photometry for the 95 planet candidates and
inspected the agreement between the planet candidate parameters provided by Batalha
et al. (2013) and the Kepler data. We used long cadence data from Quarters 1 − 6 for
all KOIs except KOI 531.01, for which we utilized short cadence data from Quarters 9
and 10 due to the range of apparent transit depths observed in the long cadence data.
The long cadence data provide measurements of the brightness of the target stars every
29.4 minutes and the short cadence data provide measurements every 58.9 seconds.
We detrended the data by dividing each data point by the median value of the data
points within the surrounding 1000 minute interval and masked transits of additional
planets in multi-planet systems. We found that the distribution of impact parameters
reported by Batalha et al. (2013) for these planet candidates was biased towards high
values (median b = 0.75) and that the published parameters for several candidates
did not match the observed depth or shape. Accordingly, we used the IDL AMOEBA
minimization algorithm based on Press et al. (2002) to determine the best-fit period and
ephemeris for each planet candidate. We then ran a Markov Chain Monte Carlo analysis
using Mandel & Agol (2002) transit models to revise the planet radius/star radius ratio,
stellar radius/semimajor axis ratio, and inclination for each of the candidates. For
each star, we determined the limb darkening coefficients by interpolating the quadratic
coefficients provided by Claret & Bloemen (2011) for the Kepler bandpass at the effective
temperature and surface gravity found in Section 2.2.2. We adopt the median values
of the resulting parameter distributions as our best-fit values and provide the resulting
118
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.2—Continued
t0
P
(Days)
(Days)
a/R∗ a
FP
Teff
R∗
b
( R⊕ )
(F⊕ )
(K)
( R$ )
KOI
KID
2347.01
8235924
0.352
0.588
3.717
0.016
0.4
0.97+0.09
−0.09
550.32+187.35
−145.42
3972
0.565
2418.01
10027247
15.600
86.830
116.837
0.028
0.5
1.27+0.24
−0.17
0.35+0.24
−0.13
3724
0.414
2453.01
8631751
0.235
1.531
14.100
0.024
0.5
1.03+0.23
−0.18
62.60+57.58
−28.77
3565
0.400
0.4
0.63+0.11
−0.17
87.24+74.20
−48.09
3339
0.288
0.66+0.78
−0.30
3482
0.350
2542.01
6183511
0.000
0.727
4.643
Rp /R∗
RP
0.020
2626.01
11768142
25.703
38.098
36.283
0.036
0.9
1.37+0.43
−0.21
2650.01
8890150
4.280
34.987
54.052
0.027
0.5
1.18+0.40
−0.15
1.15+1.53
−0.47
3735
0.400
2650.02
8890150
2.155
7.054
30.813
0.019
0.5
0.84+0.29
−0.11
9.73+12.94
−3.98
3735
0.400
0.5
0.55+0.08
−0.08
+16.17
28.22−10.41
3410
0.345
2662.01
a This
3426367
0.742
2.104
13.578
0.015
column lists the ratio estimated from the fit to the light curve. We compute the geometric probability of transit
using the semimajor axis determined from the planet orbital period and the host star mass listed in Table 2.1.
b Kepler-49b
(Steffen et al. 2012b; Xie 2013)
c Kepler-49c
(Steffen et al. 2012b; Xie 2013)
d Kepler-26b
(Steffen et al. 2012a)
e Kepler-26c
(Steffen et al. 2012a)
f Confirmed
by Johnson et al. (2012)
g Kepler-54b
(Steffen et al. 2012b)
i Confirmed
by Xie (2013)
j Confirmed
by Xie (2013)
k Kepler-32b
l Kepler-32c
m Transits
(Fabrycky et al. 2012)
(Fabrycky et al. 2012)
noted as “v”-shaped by Batalha et al. (2013)
119
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
planet candidate parameters in the Appendix in Table 2.2. Figures 2.8-2.11 display
detrended and fitted light curves for the three habitable zone planet candidates in our
sample and for one additional candidate at short cadence.
Ten of the planet candidates in our sample have reported transit timing variations
(TTVs), but our fitting procedure assumed a linear ephemeris. Due to the smearing of
ingress and egress caused by fitting a planet candidate exhibiting TTVs with a linear
ephemeris, our simple fitting routine experienced difficulty determining the transit
parameters for those candidates. Rather than use our poorly constrained fits for the
candidates with TTVs, we choose instead to adopt the literature values for KOIs 248.01,
248.02, 886.01, and 886.02 (Kepler-49b, 49c, 54b, and 54c) from Steffen et al. (2012b),
KOIs 250.01 and 250.02 (Kepler-26b and 26c) from Steffen et al. (2012a), KOIs 952.01
and 952.02 (Kepler-32b and 32c) from Fabrycky et al. (2012), and KOIs 898.01 and
898.03 from Xie (2013).
We also adopt the transit parameters for KOIs 248.03, 248.04, and 886.03 from
Steffen et al. (2012b), KOI 250.03 from Steffen et al. (2012a), KOIs 952.03 and 952.04
from Fabrycky et al. (2012), and KOI 254.01 from Johnson et al. (2012) because the
authors completed extensive modeling of their light curves. We cannot adopt values
from Steffen et al. (2012a) for KOI 250.04 because that planet candidate was announced
after publication of Steffen et al. (2012a). Fabrycky et al. (2012) also present transit
parameters for a fifth planet candidate in the KOI 952 system, but we choose not to
add KOI 952.05 to our sample because that planet candidate was not included in the
February 2012 planet candidate list (Batalha et al. 2013) and including KOI 952.05
would necessitate including any other planet candidates that were not included in the
February 2012 KOI list.
120
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Detrended Flux
KOI 854.01
1.002
1.000
0.998
0.996
0.994
0.992
200
300
Flux
rrstar: 0.040, arstar: 73.140, Inc: 89.483
1.002
1.001
1.000
0.999
0.998
0.997
400
500
600
Time (Days) rrstar: 0.039, arstar: 90.045, Inc: 89.759
Batalha
Revised
-4
-2
0
Time (Hours Since Transit)
2
-4
-2
0
Time (Hours Since Transit)
2
4
Residuals
0.004
0.002
0.000
-0.002
MAST Period: 56.056286 Days
AMOEBA Period: 56.054762 Days
4
MAST t0: 33.000000 Days
AMOEBA t0: 33.001157 Days
Figure 2.8: Light curve for KOI 854.01. Top: Detrended light curve with transit times
marked by red dots. Middle: Light curve phased to the best-fit period. The blue
curve indicates the original transit model and the red curve marks our revised fit. The
parameters for the fit are indicated above the middle panel and the period and ephemeris
are marked at the bottom of the figure. The “MAST” values indicate the original period
and ephemeris listed in the planet candidate list at MAST and the “AMOEBA” values
indicate the revised period and ephemeris. Bottom: Residuals for the original transit
model (blue) and our revised model (red).
121
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Detrended Flux
KOI 1422.02
1.002
1.000
0.998
0.996
200
300
Residuals
Flux
rrstar: 0.040, arstar: 34.830, Inc: 88.700
1.002
1.001
1.000
0.999
0.998
0.997
400
500
600
Time (Days) rrstar: 0.038, arstar: 51.985, Inc: 89.553
Batalha
Revised
-2
-1
0
1
Time (Hours Since Transit)
-2
-1
0
1
Time (Hours Since Transit)
2
0.002
0.001
0.000
-0.001
MAST Period: 19.850214 Days
AMOEBA Period: 19.849853 Days
2
MAST t0: 14.560000 Days
AMOEBA t0: 14.559054 Days
Figure 2.9: Light curve for KOI 1422.02 in the same format as Figure 2.8.
For 31 of the remaining 78 planet candidates without revised fits from the literature,
the planet radius/star radius ratios from Batalha et al. (2013) lie within the 1σ error
bars of our revised values. The median changes to the transit parameters for the refit
planet candidates are that the planet radius/star radius ratio decreases by 3%, the star
radius/semimajor axis ratio increases by 18%, and the inclination increases by 0.7◦ .
Combining our improved stellar radii with the revised planet radius/star radius ratios
for all of the planet candidates, we find that the radius of a typical planet candidate
is 29% smaller than the value found by computing the radius from the transit depth
given in Batalha et al. (2013) and the stellar radii listed in the KIC as shown in
122
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Detrended Flux
KOI 2626.01
1.002
1.000
0.998
0.996
200
300
Residuals
Flux
rrstar: 0.025, arstar: 91.029, Inc: 89.815
1.002
1.001
1.000
0.999
0.998
0.997
400
500
600
Time (Days) rrstar: 0.036, arstar: 36.283, Inc: 88.538
Batalha
Revised
-3
-2
-1
0
1
Time (Hours Since Transit)
-3
-2
-1
0
1
Time (Hours Since Transit)
2
3
2
3
0.003
0.002
0.001
0.000
-0.001
-0.002
MAST Period: 38.098240 Days
AMOEBA Period: 38.098428 Days
MAST t0: 25.700000 Days
AMOEBA t0: 25.702688 Days
Figure 2.10: Light curve for KOI 2626.01 in the same format as Figure 2.8.
123
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Figure 2.11: Light curve for KOI 531.01. Top: Light curve phased to best-fit period.
The blue curve indicates the original transit model and the red curve marks our revised fit.
For clarity, only 50% of the data are plotted. The gray point in the lower right indicates
representative error bars. The parameters for the fits are indicated between the panels.
Bottom: Residuals for the original transit model (blue) and our revised model (red).
124
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Figure 2.12. The improvements in the stellar radii account for most of the changes in
the planet candidate radii, but the contributions from the revised transit parameters are
non-negligible for a few planet candidates, most notably KOIs 531.01 and 1843.02.
We computed error bars on the planet candidate radii by computing the fractional
error in the planet radius/star radius ratio and the stellar radius and adding those
differences in quadrature to determine separate upper and lower 1σ error bounds for
each candidate. For a typical candidate in the sample, the 68% confidence region extends
from 86 − 112% of the best-fit planet radius. The best-fit radii and 1σ error bars for the
smallest planet candidates are plotted in Figure 2.13 as a function of orbital period.
2.4.1
Multiplicity
Half (48 out of 95) of our cool planet candidates are located in multi-candidate systems.
We mark the multiplicity of each system in Figure 2.14. As shown in the figure, the
largest planet candidates (KOIs 254.01, 256.01, 531.01, and 2156.01) are in systems with
only one known planet and 93% of the 14 candidates with orbital periods shorter than
2 days belong to single-candidate systems. The one exception is KOI 936.02, which has
an orbital period of 0.89 days and shares the system with KOI 936.01, a 1.8 R⊕ planet
in a 9.47 day orbit. At orbital periods longer than 2 days, 59% of the candidates belong
to systems with at least one additional planet candidate. Our sample contains 47 single
systems, 7 double systems, 6 triple systems, and 4 quadruple5 systems. The fraction
5
Fabrycky et al. (2012) report that the KOI 952 system has five planet candidates, but we count this
system as a quadruple planet system because KOI 952.05 was not included in the February 2012 planet
candidate list.
125
Planet Radius (REarth)
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
All Candidates
Original
Revised
10
1
Planet Radius (REarth)
3400 3600 3800 4000
Stellar Effective Temperature (K)
6
5
Small Candidates
Original
Revised
4
3
2
1
3400 3600 3800 4000
Stellar Effective Temperature (K)
Figure 2.12: Revised (red circles) and original (blue squares) planet radii and stellar
effective temperatures for the 95 planet candidates. The gray lines connect the initial and
final values for each planet candidate. Top: Full planet candidate population. Bottom:
Zoomed-in view of the small planet candidate population.
126
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Stellar Temperature (K)
Radius (Earth Radii)
3200
3360
3520
3680
3840
4000
4
3
2
1
1
10
Orbital Period (Days)
100
Figure 2.13: Revised planet candidate radius versus orbital period for the smallest planet
candidates. The points are color-coded according to the temperature of the host star.
127
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
of single planet systems (73%) is slightly lower than the 79% single system fraction for
the planet candidates around all stars (Fabrycky et al. 2014), but this difference is not
significant.
2.5
Planet Occurrence Around Small Stars
We estimate the planet occurrence rate around small stars by comparing the number of
detected planet candidates with the number of stars searched. Our analysis assumes that
all 64 of the planet candidates are bona fide planet candidates and not false positives.
This assumption is reasonable because previous studies have demonstrated that the false
positive rate is low for the planet candidates identified by the Kepler team (Morton &
Johnson 2011; Fressin et al. 2013).
For a planet with a given radius and orbital period, we calculate the number of stars
searched by determining the depth δ and duration of a transit in front of each of the cool
stars. We then calculate the signal-to-noise ratio for a single transit of each of the stars
by comparing the predicted transit depth to the expected noise level:
SNR1 transit =
δ
σCDPP
(2.2)
where σCDPP is a measure of the expected noise on the timescale of the predicted transit
duration and the depth δ for a central transit is the square of the planet/star radius
ratio.
We determine σCDPP by fitting a curve to the observed Combined Differential
Photometric Precision (CDPP; Christiansen et al. 2012) measured for each star over
3-hr, 6-hr, and 12-hr time periods and then interpolating to find the expected CDPP
128
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Radius (Earth Radii)
All Candidates
Single
Double
Triple
Quad
10
1
1
10
Orbital Period (Days)
Radius (Earth Radii)
3.0
Small Candidates
2.5
Single
Double
Triple
Quad
2.0
1.5
1.0
0.5
1
10
Orbital Period (Days)
Figure 2.14: Revised planet candidate radii versus orbital period for candidates in single
(cross), double (circle), triple (triangle), and quadruple (square) systems. Each multicandidate system is plotted in a different color. Top: Full planet candidate population.
Bottom: Zoomed-in view of the smallest planet candidates.
129
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
for the predicted transit duration. CDPP is available from the data search form on the
Kepler MAST.6
Although the CDPP varies on a quarter-by-quarter basis, we choose to interpolate
the median CDPP value at a given time period for each star across all quarters. We also
repeat our analysis using the minimum and maximum CDPP for each time interval to
quantify the dependence of the planet occurrence rate on our estimate of the noise in the
light curve on the timescale of a transit.
We then estimate the number of transits n that would have been observed by
dividing the number of days the star was observed by the orbital period of the planet.
We assume that the total signal-to-noise scales with the number of transits so that the
total signal-to-noise for a planet with radius Rp orbiting a star with radius R∗ is:
√
SNRtotal = SNR1 transit n =
!
Rp
R∗
"2 √
n
σCDPP
(2.3)
where n is the number of transits. We adopt the 7.1σ detection threshold used by the
Kepler team and require that the total SNR is above 7.1σ in order for a planet to be
detected. We apply this cut both to our detected sample of planet candidates and to the
sample of stars searched.
2.5.1
Correcting for Incomplete Phase Coverage
Previous research on the occurrence rate of planets around Kepler target stars has
assumed that all stars were observed continuously during all quarters. This assumption
is reasonable for the objects in the 2011 planet candidate list, but the failure of Module 3
6
http://archive.stsci.edu/kepler/data_search/search.php
130
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
on January 9, 2010 (Batalha et al. 2013) means that 20% of Kepler’s targets fall on a
failed module every fourth quarter. In addition, some targets fall in the gaps between
the modules and are observed only 1-3 quarters per year even though they never fall on
Module 3.
We account for the missing phase coverage by determining the modules on which
each of the stars fall during each quarter and calculating the fraction of Q1–Q6 that
each star spent within the field-of-view of the detectors. For a star that spends x days of
the 486.5 day Q1–Q6 observation period in the field-of-view of the detectors, we assume
that x/486.5 of transits would be present in the data. Note that our approach does
not account for gaps in phase coverage during each quarter due to planned events and
spacecraft anomalies. We also ignore the temporal spacing of transits relative to the
gaps in phase coverage. This effect is negligible for transits that occur multiple times per
quarter (i.e., durations < 90 days), but the timing becomes important for transits that
occur with periods equal to or longer than the duration of a quarter.
2.5.2
Calculating the Occurrence Rate
Following Howard et al. (2012), we estimate the planet occurrence rate f as a function of
planet radius and orbital period by dividing the number of planet candidates found with
a given radius and period by the number of stars around which those candidates could
have been detected. We account for non-transiting geometries by multiplying the number
of planet candidates found by the inverse of the geometric likelihood ptransit = R∗ /a that
a planet with semi-major axis a would appear to transit a star with radius R∗ . The
131
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
planet occurrence rate over a given period and planet radius range is therefore:
Np (Rp ,P )
f (Rp , P ) =
+
i=1
ai
R∗,i N∗,i
(2.4)
where Np (Rp , P ) is the number of planets with the radius Rp and orbital period P within
the desired intervals, ai is the semimajor axis of planet i , R∗,i is the radius of the host
star of planet i, and N∗,i is the number of stars around which planet i could have been
detected. Like Howard et al. (2012), we estimate the error on the planet occurrence
rate f (R, p) by computing the binomial probability distribution of finding Np (Rp , P )
planets in a given radius and period range when searching Np (Rp , P )/f (Rp, P ) stars.
We determine the 15.9 and 84.1 percentiles of the cumulative binomial distribution and
adopt those values as the 1σ statistical errors on the occurrence rate f (Rp , P ) within the
desired radius and period range.
2.5.3
Dependence on Planet Size
Our final sample of planet candidates orbiting dwarf stars with revised temperatures
below 4000K consists of 47 candidates with radii between 0.5 − 1.4 R⊕ , 43 candidates
with radii between 1.4 − 4 R⊕ , 4 candidates with radii above 4 R⊕ , and 1 candidate
smaller than 0.5 R⊕ . Using Equation 2.4, we find the occurrence rate of planets with
periods shorter than 50 days peaks at 0.29 planets per star for planets with radii between
1.0 − 1.4 R⊕ and decreases for smaller and larger planets. We summarize our findings
for the occurrence rate as a function of planet radius and orbital period in Table 2.3
and in Figure 2.15. Our estimate for the occurrence rate of planets with radii between
0.5 − 4 R⊕ and orbital periods shorter than 50 days is 0.90+0.04
−0.03 planets per star, which
agrees well with the estimate of 1.0+0.1
−0.1 planets per star calculated by Swift et al. (2013).
132
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.3. Planet Occurrence Rate for Late K and Early M Dwarfs
Orbital Period (Days)
Rp ( R⊕ )
0.68 − 10
10 − 50
0.68 − 50
0.5 − 0.7
0.014+0.0129
−0.006 (2)
0.014+0.0129
−0.006 (2)
0.7 − 1.0
+0.0977
0.109+0.0344
−0.025 (12) 0.103−0.046 (2)
0.212+0.0590
−0.044 (14)
1.0 − 1.4
+0.0735
0.108+0.0251
−0.020 (21) 0.177−0.048 (7)
0.285+0.0509
−0.041 (28)
1.4 − 2.0
+0.0490
0.080+0.0245
−0.018 (13) 0.123−0.034 (8)
0.202+0.0443
−0.035 (21)
2.0 − 2.8
0.038+0.0168
−0.011 (7)
+0.0440
0.148+0.0456
−0.033 (12) 0.186−0.034 (19)
2.8 − 4.0
0.005+0.0081
−0.003 (1)
—
0.005+0.0081
−0.003 (1)
4.0 − 5.7
0.004+0.0062
−0.002 (1)
—
0.004+0.0062
−0.002 (1)
5.7 − 8.0
—
—
—
8.0 − 11.3
0.003+0.0044
−0.001 (1)
—
0.003+0.0044
−0.001 (1)
11.3 − 16.0 0.004+0.0055
−0.002 (1)
—
0.004+0.0055
−0.002 (1)
16.0 − 22.6 0.003+0.0041
−0.001 (1)
—
0.003+0.0041
−0.001 (1)
22.6 − 32.0 —
—
—
—
Note. — The number of planets in each bin is given in parentheses.
133
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
We find that the planet occurrence rate per logarithmic bin increases with increasing
orbital period and that the occurrence rate of small (RP < 2.8 R⊕ ) candidates with
periods less than 50 days is higher than the occurrence rate of larger candidates. The
sample includes only three candidates smaller than 0.7 R⊕ , but the low number of planet
candidates smaller than 0.7 R⊕ is likely due to incompleteness in the planet candidate
list and the inherent difficulty of detecting small planets. In contrast, the scarcity of
planet candidates larger than 2.8 R⊕ indicates that large planets rarely orbit small stars
at periods shorter than 50 days.
In order to more closely investigate the dependence of the planet occurrence rate
on orbital period and planet radius, we plot the occurrence rate as a function of planet
radius for planet candidates in three different period groups in Figure 2.16. For the
population of candidates with periods shorter than 50 days, we find that the occurrence
rate is highest for planets with radii between 1 − 1.4 R⊕ and decreases at smaller and
larger radii. The occurrence rate falls to nearly zero for planets larger than 2.8 R⊕ and
to 0.014 planets per star for planets with radii between 0.5 − 0.7 R⊕ . The occurrence
rate of planets smaller than 0.7 R⊕ might be underestimated due to incompleteness in
the Kepler pipeline or there might be a real turnover in the underlying planet radius
distribution at small radii.
Breaking down the sample by orbital period, we find a slight indication that the
planet radius distribution of short-period planets (P < 10 days) is more peaked toward
smaller planet radii than the distribution of longer-period planets (10 < P < 50 days),
but the difference in the occurrence rate is significant only for 2.0 − 2.8 R⊕ planets.
Our result for the occurrence rate of 2 − 4 R⊕ planets within 50 days is 19+5
−4 %, which
is consistent with the 26+8
−9 % occurrence rate for 2 − 4 R⊕ planets found by Howard
134
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Planet Occurrence - d2f/dlogP/dLogRp
0.001
0.002
0.004
0.0079
0.016
0.032
0.000035 0.00007 0.00014 0.00028 0.00056 0.0011
22.6
0.25
0.50
1.0
0.0022
0.0044
0.0088
0.018
0.035
0.078
0.0027
1 (10)
3897
16.0
0.10
0.0037
1 (14)
3897
0.084
0.0029
11.3
Planet Radius, Rp (RE)
0.13
Planet Occurrence - fcell
32.0
1 (11)
3897
8.0
5.7
0.12
0.0042
1 (16)
3895
4.0
2.8
0.065
0.0023
1 (8)
3892
0.055
0.0019
2.0
1.4
0.063
0.11
0.0039
0.13
0.0046
0.15
0.0054
1 (21)
3892
0.89
0.031
1.1
0.039
3.1
0.11
1 (17)
5 (120) 4 (149) 8 (417)
3885
3881
3865
3819
0.30
0.45
1.5
0.85
1.9
0.011
0.016
0.051
0.030
0.066
0.78
0.027
1.1
0.037
1 (7)
3882
3 (40)
3 (59)
6 (189) 3 (110) 4 (235) 1 (94)
1 (115)
3865
3823
3745
3645
3504
3287
2991
0.30
0.42
0.81
1.4
1.3
1.4
2.4
0.011
0.015
0.028
0.050
0.045
0.048
0.084
1.0
2 (14)
4 (38)
4 (53)
5 (94)
6 (164) 3 (137) 2 (121) 2 (180)
3831
3764
3658
3505
3260
2866
2336
1822
0.058
0.19
0.36
1.0
1.4
1.1
1.8
0.0020 0.0068
0.013
0.037
0.051
0.038
0.065
0.7
1 (6)
2 (22)
3546
3307
0.073
0.0026
2 (32)
3 (75)
2918
2361
0.32
0.011
4 (110)
1797
1 (47)
1307
1 (83)
872
5.9
0.21
3.0
0.10
1 (150)
994
1 (138)
353
(7)
1 (14)
1374
0.5 12484
0.68 1.2 2.0 3.4 5.9 10 17 29 50 85 146
Orbital Period, P (days)
Figure 2.15: Planet occurrence rate as a function of planet radius and orbital period in
the style of Figure 4 from Howard et al. (2012). The color-coding of each cell indicates
the planet occurrence within the cell as shown in the legend and the circles mark the radii
and periods of the 95 planet candidates in our sample. Planets marked in blue orbit stars
hotter than 3723K and planets marked in black orbit stars cooler than 3723K. Cells shaded
in white do not contain any planet candidates. The planet candidate list is less complete at
long periods and our estimates of the planet occurrence rate are likely underestimated at
periods longer than 50 days (hatched region). The four numbers within each cell describe
the planet occurrence in that region of parameter space: Top Left: number of detected
planet candidates with signal to noise ratios above 7.1σ and, in parentheses, the number
of non-transiting planets in the same period and radius bin computed by correcting for
the geometric probability of transit; Bottom Left: the number of stars around which a
planet from the center of the grid cell would have been detected with a signal to noise
ratio above 7.1σ; Bottom Right: the planet occurrence rate within the cell; Top Right:
planet occurrence per logarithmic area unit.
135
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
0.35
<50 Days
<10 Days
10-50 Days
Number of Planets per Star
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.5 0.7 1.0 1.4 2.0 2.8 4.0 5.7 8.0 11.3 16.0 22.6 32.0
Planet Radius (RE)
Figure 2.16: Planet occurrence rate as a function of planet radius for all candidates
(black) and candidates with orbital periods shorter than < 10 days (green) or between
10 − 50 days (purple). The error bars indicate the errors from binomial statistics and do
not include errors from the stellar and planetary radius estimates.
136
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
et al. (2012) for target stars with 3600 ≤ Teff ≤ 4100K. Our result is slightly below the
37 ± 8% occurrence rate for 2 − 32 R⊕ planets orbiting 3400 ≤ Teff ≤ 4100 stars found by
Mann et al. (2012) and the 30% occurrence rate for Rp ≥ 2 R⊕ and 3660 ≤ Teff ≤ 4660K
found by Gaidos et al. (2012). Howard et al. (2012) and Gaidos et al. (2012) adopt the
KIC parameters for the target stars, so they overestimate both the stellar radii and
planetary radii for the coolest stars in their sample. Accordingly, many of the planets
that we classify as Earth-size would have ended up with radii above 2 R⊕ in the Howard
et al. (2012) and Gaidos et al. (2012), and studies, therefore increasing the apparent
occurrence rate of 2 − 4 R⊕ planets in those studies.
Additionally, Gaidos et al. (2012) arrive at their occurrence rate by comparing
the number of planet candidates with radii between 2 − 32 R⊕ to the number of stars
around which such planets could have been detected, but they use the noise relation and
distribution from Koch et al. (2010) to predict the expected noise of each star based
on Kepler magnitude rather than using the observed noise. Given that the stellar noise
displays variation even at constant Kepler magnitude, this assumption could contribute
to the slight difference between our occurrence rate and the value reported by Gaidos
et al. (2012).
Despite the sensitivity of Kepler to giant planets orbiting small stars, we find only
four planets with radii > 4 R⊕ in our sample (KOIs 254.01, 256.01, 531.01, and 2156.01).
The implied low occurrence rate of giant planets is consistent with previous estimates of
the giant planet occurrence rate around cool stars (Butler et al. 2004; Bonfils et al. 2006;
Butler et al. 2006; Endl et al. 2006; Johnson et al. 2007a; Cumming et al. 2008; Bonfils
et al. 2013; Howard et al. 2012). The paucity of giant planets orbiting M dwarfs is in
line with expectations from theoretical studies of planet formation (Laughlin et al. 2004;
137
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Adams et al. 2005; Ida & Lin 2005; Kennedy & Kenyon 2008). The formation of a giant
planet via core accretion requires a considerable amount of material and the combination
of longer orbital timescales and lower disk surface density decreases the likelihood that a
protoplanet will accrete enough material to become a gas giant before the disk dissipates.
As an alternative to determining the mean number of planets per star, we also
compute the fraction of stars with planets. The latter number is more relevant when
determining the required number of targets to survey in a planet finding mission. To
compute the fraction of stars that host planets, we repeat the analysis described in
Section 2.5.2 using only one planet per system. We pick the planet used for each system
by determining which of the planets would be easiest to detect. We find that 25% of
cool dwarfs host planets with radii 0.5 − 1.4 R⊕ and orbital periods shorter than 50 days
and that 25% of cool dwarfs host 1.4 − 4 R⊕ planets with periods shorter than 50 days.
These estimates for the fraction of stars with planets are slightly lower than the mean
number of planets per star due to the prevalence of multiplanet systems.
2.5.4
Dependence on Stellar Temperature
The coolest planet host star (KOI 1702) in our sample has a temperature of 3305K and
the hottest planet host star (KOI 739) has a temperature of 3995K. The temperature
range for the entire small star sample spans 3122-4000K, with a median temperature of
3723K. Splitting the cool star population into a cool group (3122K< Teff < 3723K) and
a hot group (3723K≤ Teff ≤ 4000K), we find that the cool star group includes 34 KOIs
orbiting 25 host stars and the hot star group includes 61 KOIs orbiting 39 host stars.
The cool group contains 1957 stars total and the hot group contains 1940 stars total.
138
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
The multiplicity rates for the two groups are similar: 1.4 planets per host star for the
cooler group and 1.6 planets per host star for the hotter group.
In order to investigate the dependence of the planet occurrence rate on host star
temperature, we repeat the analysis described in Section 2.5 for each group separately.
We find that the occurrence rates of Earth-size planets (0.5 − 1.4 R⊕ ) are consistent
with a flat occurrence rate across the temperature range of our sample, but that the
occurrence rate of 1.4 − 4 R⊕ planets is higher for the hot group than for the cool group
or for the full sample. The mean numbers of Earth-size planets (0.5 − 1.4 R⊕ ) and
+0.08
1.4 − 4 R⊕ planets per star with periods shorter than 50 days are 0.57+0.09
−0.06 and 0.61−0.06
+0.07
for the hot group and 0.46+0.09
−0.06 and 0.19−0.05 for the cool group.
The lower occurrence rate of 1.4 − 4 R⊕ planets for the cool group indicates that
cooler M dwarfs have fewer 1.4 − 4 R⊕ planets than hotter M dwarfs, but the planet
occurrence rate for mid-M dwarfs is not well constrained by the Kepler data. Since
Kepler is observing few mid-M dwarfs, the median temperature for the cool star group
is 3520K and only 26% of the stars in the cool group have temperatures below 3400K.
The estimated occurrence rate for the cool star group is therefore most indicative of the
occurrence rate for stars with effective temperatures between 3400K and 3723K. Further
observations of a larger sample of M dwarfs with effective temperatures below 3300K are
required to constrain the planet occurrence rate around mid- and late-M dwarfs.
2.5.5
The Habitable Zone
The concept of a “habitable zone” within which life could exist is fraught with
complications due to the influence of the spectrum of the stellar flux and the composition
139
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
of the planetary atmosphere on the equilibrium temperature of a planet as well as our
complete lack of knowledge about alien forms of life. Regardless, for this paper we adopt
the conventional and naı̈ve assumption that a planet is within the “habitable zone” if
liquid water would be stable on the surface of the planet. For the 64 host stars in our
sample, we determine the position of the liquid water habitable zone by finding the
orbital separation at which the insolation received at the top of a planet’s atmosphere is
within the insolation limits determined by Kasting et al. (1993) for M0 dwarfs. Kasting
et al. (1993) included several choices for the inner and outer boundaries of the habitable
zone. For this paper we adopt the most conservative assumption that the inner edge
of the habitable zone is the distance at which water loss occurs due to photolysis and
hydrogen escape (0.95 AU for the Sun) and the outer edge as the distance at which CO2
begins to condense (1.37 AU for the Sun).
For M0 dwarfs, these transitions occur when the insolation at the orbit of the
planet is Finner = 1.00F⊕ and Fouter = 0.46F⊕ , respectively, where F⊕ is the level of
insolation received at the top of the Earth’s atmosphere. These insolation levels are 9%
and 13% lower than the insolation at the boundaries of the G2 dwarf habitable zone
because the albedo of a habitable planet is lower at infrared wavelengths compared to
visible wavelengths due to the wavelength dependence of Rayleigh scattering and the
strong water and CO2 absorption features in the near-infrared. Additionally, habitable
planets around M dwarfs are more robust against global snowball events in which the
entire surface of the planet becomes covered in ice because increasing the fraction of the
planet covered by ice decreases the albedo of the planet at near-infrared wavelengths and
therefore causes the planet to absorb more radiation, heat up, and melt the ice. This is
not the case for planets orbiting Sun-like stars because ice is highly reflective at visible
140
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
wavelengths and because the stellar radiation peaks in the visible.
We contemplated using the analytic relations derived by Selsis et al. (2007) for the
dependence of the boundaries of the habitable zone on stellar effective temperature, but
the coefficients for their outer boundary equation were fit to the shape of the maximum
greenhouse limit. The analytic relations derived by Selsis et al. (2007) therefore
overestimate the position of the edge of the habitable zone for our chosen limit of the
first condensation of CO2 clouds. Additionally, the equations provided in Selsis et al.
(2007) are valid only for 3700K≤ Teff ≤ 7200K because Kasting et al. (1993) calculated
the boundaries of the habitable zone for stars with temperatures of 3700K, 5700K, and
7200K. Selsis et al. (2007) deals with the lower temperature limit by assuming that
the albedo of a habitable planet orbiting a star with a temperature below 3700K is
sufficiently similar to the albedo of a habitable planet orbiting a 3700K star that the
insolation limits of the habitable zone are unchanged. In this paper, we extend the Selsis
et al. (2007) approximation to use constant insolation limits for all of the stars in our
sample. Given the uncertainties inherent in defining a habitable planet and determining
the temperatures of low-mass stars, our assumption of constant insolation boundaries
should not have a significant effect on our final result for the occurrence rate of rocky
planets in the habitable zones of M dwarfs.
2.5.6
Planet Candidates in the Habitable Zone
As shown in Figure 2.17, the habitable zones for the 64 host stars in our final sample of
dwarfs cooler than 4000K fall between 0.08 and 0.4 AU, corresponding to orbital periods
of 17 − 148 days. Figure 2.17 displays the semimajor axes of all of the planet candidates
141
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Stellar Effective Temperature (K)
4000
250.04
3800
2650.01
2418.01
947.01
1879.01
3600
886.03
854.01
463.01
2626.01
1422.02
3400
0.01
1686.01
Planet Radius (RE)
[Fe/H]
0.4 1.0 1.7 2.3 3.0
-0.6 -0.4 -0.1 0.1
0.10
Planet Semimajor Axis (AU)
HZ
3.0
Host Star Teff (K)
Planet Radius (REarth)
3304
3633
3994
2.5
2.0
1.5
1.0
0.5
0.1
1.0
10.0
100.0
Flux Received by Planet (FEarth)
1000.0
Figure 2.17: Top: Stellar effective temperature and planet semimajor axes for the
95 planet candidates orbiting stars with revised temperatures below 4000K. The points
are color-coded according to the radius of each planet candidate as indicated in the left
legend. The lines indicate the calculated position of the habitable zone (HZ) for each
star and are color-coded accorded to the metallicity of the star as indicated in the right
legend. The three candidates within the HZ (KOIs 854.01, 1422.02, and 2626.01) are
identified by name and highlighted in red. Bottom: Planet radii versus flux for the
planet candidates around stars with revised temperatures below 4000K. The color-coding
indicates the effective temperature of the host star. The green box indicates the habitable
zone as defined in Section 2.5.5.
142
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
and the positions of the habitable zones around their host stars. Nearly all of the planet
candidates orbit closer to their host stars than the inner boundary of the habitable
zone, but two candidates (KOIs 1686.01 and 2418.01) orbit beyond the habitable zone
and two candidates (KOI 250.04 and 2650.01) orbit just inside the inner edge of the
habitable zone. Three candidates fall within our adopted limits: KOIs 854.01, 1422.02,
and 2626.01. These candidates are identified by name in Figure 2.17 and have radii of
1.69, 0.92, and 1.37 R⊕ , respectively. A full list of the stellar and planetary parameters
for the three candidates in the habitable zone and the candidates near the habitable zone
is provided in Table 2.4.
The lateral variation in the position of the habitable zone at a given stellar effective
temperature is due to the range of metallicities found for the host stars. At a given stellar
effective temperature, stars with lower metallicities are less luminous and therefore the
habitable zone is located closer to the star. Adopting a different metallicity prior would
change the metallicities of the host stars and shift the habitable zones slightly inward or
outward. The metallicities and temperatures of the cool stars and planet candidate host
stars are plotted in Figure 2.18. As shown in Figure 2.18, 98% of the cool stars and all of
the planet candidate host stars have metallicities −0.5 ≤ [Fe/H] ≤ 0. There are 17 cool
stars (0.4%) with super-solar metallicities and 75 cool stars (2%) with metallicities below
[Fe/H]= −0.5.
All of the habitable zone candidates orbit stars fit by models with sub-solar metallicity
(KOI 854: [Fe/H]= −0.1, KOI 1422: [Fe/H]= −0.5, KOI 2626: [Fe/H]= −0.1). If we
restrict all of the stars to solar metallicity and redetermine the stellar parameters and
habitable zone boundaries for each planet candidate, then we find that the number of
candidates in the habitable zone remains constant, but that identity of the habitable
143
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
0.5
0.0
[Fe/H]
-0.5
-1.0
-1.5
-2.0
-2.5
3000
Target Star
Host Star
HZ Host Star
3200
3400
3600
3800
Stellar Effective Temperature (K)
4000
Figure 2.18: Revised metallicities versus stellar effective temperature for all stars with
revised temperatures below 4000K (black crosses) and planet candidate host stars (circles).
The three stars hosting planet candidates within the habitable zone are highlighted in
blue; all other planet host stars are marked in red.
144
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
zone candidates changes. KOIs 854.01 and 2626.01 remain in the habitable zone, but
KOI 1422.02 does not. We find that the habitable zones of KOIs 1422 and 2418 move
outward so that KOI 1422.02 is now too close to the star to be within the habitable zone
and that KOI 2418.01 is now within the boundaries of the habitable zone. Because the
number of candidates in the habitable zone is unchanged, our estimate of the occurrence
rate within the habitable zone is not affected by adopting a different metallicity prior.
2.5.7
Planet Occurrence in the Habitable Zone
Our final sample contains three planet candidates in the habitable zone, which is
sufficient to allow us to place a lower limit on the occurrence rate in the habitable zone
of late K and early M dwarfs. We find that planets with the same radii and insolation
as KOIs 854.01, 1422.02, and 2626.01 could have been detected around 2853 (73%), 813
(21%), and 2131 (55%) of the cool dwarfs, respectively. Accordingly, the occurrence rate
of Earth-size (0.5 − 1.4 R⊕ ) planets in the habitable zone is 0.15+0.13
−0.06 planets per star
and the occurrence rate of larger (1.4 − 4 R⊕ ) planets is 0.04+0.06
−0.02 planets per star. We
find lower limits of 0.04 Earth-size planets and 0.008 1.4 − 4 R⊕ planets per cool dwarf
habitable zone with 95% confidence. These occurrence rate estimates are most applicable
for stars with temperatures between 3400K and 4000K because 80% of the stars in our
cool dwarf sample have temperatures above 3400K.
As shown in Figure 2.19, the occurrence rate of 1.4 − 4 R⊕ planets peaks for
insolation levels 2.2 − 4.7 times higher than that received by the Earth (F⊕ ) and falls
off at higher and lower insolation levels. The occurrence rate of Earth-size planets is
roughly constant per logarithmic insolation bin for insolation levels between 0.2 − 50F⊕
145
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.4. Properties of Candidates In or Near the Habitable Zone
KOI
R∗ ( R$ ) [Fe/H]
P (Days)
RP ( R⊕ ) FP (F⊕ )
6149553 3414
0.30
-0.1
56.87
0.95
0.30
2418.01 10027247 3724
0.41
-0.4
86.83
1.27
0.35
6435936 3562
0.40
-0.1
56.05
1.69
0.50
2626.01 11768142 3482
0.35
-0.1
38.10
1.37
0.66
1422.02 11497958 3424
0.22
-0.5
19.85
0.92
0.82
1686.01
854.01
KID
Teff (K)
250.04
9757613 3853
0.45
-0.5
46.83
1.92
1.02
2650.01
8890150 3735
0.40
-0.5
34.99
1.18
1.15
886.03
7455287 3579
0.33
-0.4
21.00
1.14
1.47
947.01
9710326 3717
0.43
-0.3
28.60
1.84
1.61
463.01
8845205 3504
0.34
-0.2
18.48
1.80
1.70
1879.01
8367644 3635
0.41
-0.2
22.08
2.37
1.96
146
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
and decreases for higher levels of insolation. The large error bars at low insolation levels
should shrink as the Kepler mission continues and becomes more sensitive to small
planets in longer-period planets.
Number of Planets per Star
0.4
HZ
0.5-1.4 RE
1.4-4 RE
0.3
0.2
0.1
0.0
0.1
1.0
10.0
100.0
1000.0
Flux at Planet Relative to Flux at Earth
Figure 2.19: Planet occurrence rate versus insolation for Earth-size planets (0.5−1.4 R⊕ ,
blue) and 1.4 − 4 R⊕ planets (red). The green box marks the habitable zone. The error
bars indicate the errors from binomial statistics and do not include errors from the stellar and planetary radius estimates although we do consider those errors as discussed in
Section 2.5.7.
Our result for the occurrence rate of 1.4 − 4 R⊕ planets within the habitable zones
of late K and early M dwarfs is lower than the 42+54
−13 % occurrence rate reported by
Bonfils et al. (2013) from an analysis of the HARPS radial velocity data. The difference
between our results may be due in part to the difficulty of converting measured minimum
147
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
masses into planetary radii and the definition of a “Super Earth” for both surveys.
Small number statistics may also factor into the difference. Bonfils et al. (2013) surveyed
102 M dwarfs and found two Super Earths within the habitable zone: Gl 581c (Selsis
et al. 2007; von Bloh et al. 2007) and Gl 667Cc (Anglada-Escudé et al. 2012; Delfosse
et al. 2013). Their 42% estimate of the occurrence rate of Super Earths in the habitable
zone includes a large correction for incompleteness. In comparison, the Kepler sample
contains 3897 M dwarfs with three small habitable zone planets.
Due to the small sample size and the need to account for uncertainties in the
stellar parameters, we also conduct a perturbation analysis in which we generate 10,000
realizations of each of the 3897 cool dwarfs and recalculate the occurrence rate within
the habitable zone for each realization. We generate the population of cool dwarfs by
drawing 10,000 model fits for each cool dwarf from the Dartmouth Stellar Models. We
weight the probability that a particular model is selected by the likelihoods computed
in Section 2.2 so that the population of models for each star represents the probability
density function for the stellar parameters. For the planet host stars, we then compute
the radii, semimajor axes, and insolation levels of the associated planet candidates. The
full population of perturbed planet candidates is plotted in Figure 2.20. The realization
“ellipses” are diagonally elongated due to the correlation between stellar temperature
and radius.
For each realization of perturbed stars and associated planet candidates, we calculate
the number of cool dwarfs for which each perturbed planet could have been detected.
We report the median occurrence rates and the 68% confidence intervals in Table 2.5 as
a function of planet radius and insolation. The estimated occurrence rates resulting from
the perturbation analysis are consistent with the occurrence rates plotted in Figure 2.19
148
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Planet Radius (REarth)
100
10
1
100
102
104
106
Flux at Planet relative to Flux at Earth
108
Figure 2.20: Planet radii versus insolation for the population of planet candidates generated in the perturbation analysis. The best-fit parameters for each planet candidate are
indicated by red circles and the perturbed realizations are marked by black points. The
green lines mark the boundaries of the habitable zone as defined in Section 2.5.5.
149
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
for the best-fit model parameters.
In addition to refining our estimate of the mean number of planets in the habitable
zone, the perturbation analysis also allows us to estimate the likelihood that each of the
planet candidates lies within the habitable zone. We find that the most likely habitable
planet is KOI 2626.01, which lies within the habitable zone in 4,907 of the 10,000
realizations. KOIs 2650.01, 1422.02, 250.04, and 947.01 are also promising candidates
and are within the habitable zone in 47%, 46%, 28%, and 22% of the realizations,
respectively. KOIs 886.03, 463.01, 1686.01, 1078.03, 1879.01, 817.01, and 571.04 have
much lower habitability fractions (11%, 8%, 7%, 5%, 5%, 3%, and 2%) but still contribute
to the overall estimate of the occurrence rate of planets within the habitable zone of cool
dwarfs.
2.6
Summary and Conclusions
We update the stellar parameters for the coolest stars in the Kepler target list by
comparing the observed colors of the stars to the colors of model stars from the
Dartmouth Stellar Evolutionary Program. Our final sample contains 3897 dwarf stars
with revised temperatures cooler than 4000K. In agreement with previous research,
we find that the temperatures and radii of the coolest stars listed in the KIC are
overestimated. For a typical star, our revised estimates are 130K cooler and 31% smaller.
We also refit the light curves of the associated planet candidates to better constrain the
planet radius/star radius ratios and combine the revised radius ratios with the improved
stellar radii of the 64 host stars to determine the radii of the 95 planet candidates in our
sample.
150
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
Table 2.5. Results of Perturbation Analysis: Planet Occurrence Rate as a Function of
Flux for Late K and Early M Dwarfs
Planet Radius
Flux (FEarth )
0.5 − 1.4 R⊕
1.4 − 4 R⊕
0.10 − 0.21
—
—
0.21 − 0.46
0.256+0.210
−0.142
—
0.46 − 1.00
0.155+0.138
−0.098
0.039+0.038
−0.039
1.00 − 2.17
0.153+0.089
−0.064
0.084+0.033
−0.026
2.17 − 4.73
0.133+0.055
−0.043
0.120+0.031
−0.030
4.73 − 10.27
0.131+0.049
−0.042
0.069+0.023
−0.021
10.27 − 22.33
0.100+0.025
−0.023
0.043+0.013
−0.011
22.33 − 48.55
0.047+0.012
−0.012
0.013+0.006
−0.008
48.55 − 105.55
0.017+0.006
−0.006
0.004+0.004
−0.001
105.55 − 229.45
0.007+0.003
−0.003
0.002+0.001
−0.002
229.45 − 498.81
0.002+0.001
−0.002
—
498.81 − 1084.37
—
—
151
CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
In the next stage of our analysis, we compute the planet occurrence rate by
comparing the number of planet candidates to the number of stars around which Kepler
could have detected planets with the same radius and orbital period or insolation. We
find that the mean number of Earth-size (0.5 − 1.4 R⊕ ) planets and 1.4 − 4 R⊕ planets
+0.05
with orbital periods shorter than 50 days are 0.51+0.06
−0.05 and 0.39−0.04 planets per star,
respectively. Our occurrence rate for 2 − 4 R⊕ planets is consistent with the value
reported by Howard et al. (2012) and our occurrence rate for 2 − 32 R⊕ planets is slightly
lower than the occurrence rate found by Gaidos et al. (2012).
The calculated occurrence rate of Earth-size (0.5 − 1.4 R⊕ ) planets with orbital
periods shorter than 50 days is consistent with a flat occurrence rate for temperatures
below 4000K, but the temperature dependence of the occurrence rate of 1.4−4 R⊕ planets
is significantly different. We estimate an occurrence rate of 0.61+0.08
−0.06 1.4 − 4 R⊕ planets
per hotter star (3723K≤ Teff ≤ 4000K) and 0.19+0.07
−0.05 per cooler star (3122K≤ Teff <
3723K), noting that 74% of the stars in the cool group have temperatures between 3400K
and 3701K. The apparent decline in the 1.4 − 4 R⊕ planet occurrence rate at cooler
temperatures might be due to the decreased surface density in the circumstellar disks
of very low-mass stars and the longer orbital timescales at a given separation (Laughlin
et al. 2004; Adams et al. 2005; Ida & Lin 2005; Kennedy & Kenyon 2008).
We also estimate the occurrence rate of potentially habitable planets around cool
stars. We find that the occurrence rate of small (0.5 –1.4 R⊕ ) planets within the
habitable zone is 0.15+0.13
−0.06 planets per cool dwarf. This result is lower than the M dwarf
planet occurrence rates found by radial velocity surveys (Bonfils et al. 2013), but higher
than some estimates of the occurrence rate for Sunlike stars (e.g., Catanzarite & Shao
2011). The relatively high occurrence rate of potentially habitable planets around cool
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CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
stars bodes well for future missions to characterize habitable planets because the majority
of the stars in the solar neighborhood are M dwarfs. Given that there are 248 early M
dwarfs within 10 parsecs,7 we estimate that there are at least 9 Earth-size planets in the
habitable zones of nearby M dwarfs that could be discovered by future missions to find
nearby Earth-like planets. Applying a geometric correction for the transit probability
and assuming that the space density of M dwarfs is uniform, we find that the nearest
transiting Earth-size planet in the habitable zone of an M dwarf is less than 21 pc away
with 95% confidence. Removing the requirement that the planet transits, we find that
the nearest non-transiting Earth-size planet in the habitable zone is within 5 pc with 95%
confidence. The most probable distances to the nearest transiting and non-transiting
Earth-size planets in the habitable zone are 13 pc and 3 pc, respectively.
Acknowledgments
C.D. is supported by a National Science Foundation Graduate Research Fellowship.
We acknowledge support from the Kepler Participating Scientist Program via grant
NNX12AC77G. We thank the referee, Philip Muirhead, for providing comments that
improved the paper. We acknowledge helpful conversations with S. Ballard, Z. Berta, J.
Carter, R. Dawson, J.-M. Desert, A. Dupree, F. Fressin, A. Howard, J. Irwin, D. Latham,
R. Murray-Clay, and G. Torres. We thank R. Kopparapu for correspondence that led
to a correction. This paper includes data collected by the Kepler mission. Funding for
the Kepler mission is provided by the NASA Science Mission directorate. We thank the
Kepler team for acquiring, reducing, and sharing their data. This publication makes use
7
http://www.chara.gsu.edu/RECONS/census.posted.htm
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CHAPTER 2. SMALL PLANETS AROUND SMALL STARS
of data products from the Two Micron All Sky Survey, which is a joint project of the
University of Massachusetts and the Infrared Processing and Analysis Center/California
Institute of Technology, funded by the National Aeronautics and Space Administration
and the National Science Foundation. All of the data presented in this paper were
obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by
the Association of Universities for Research in Astronomy, Inc., under NASA contract
NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of
Space Science via grant NNX09AF08G and by other grants and contracts.
154
Chapter 3
The Occurrence of Potentially
Habitable Planets Orbiting M
Dwarfs Estimated from the Full
Kepler Dataset and an Empirical
Measurement of the Detection
Sensitivity
This thesis chapter originally appeared in the literature as
C. D. Dressing & D. Charbonneau, submitted to The
Astrophysical Journal, arXiv:150101623
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Abstract
We present an improved estimate of the occurrence rate of small planets orbiting small
stars by searching the full four-year Kepler data set for transiting planets using our own
planet detection pipeline and conducting transit injection and recovery simulations to
empirically measure the search completeness of our pipeline. We identified 156 planet
candidates, including one object that was not previously identified as a Kepler Object
of Interest. We inspected all publicly available follow-up images, observing notes, and
centroid analyses, and corrected for the likelihood of false positives. We evaluated the
sensitivity of our detection pipeline on a star-by-star basis by injecting 2000 transit
signals into the light curve of each target star. For periods shorter than 50 days, we
+0.07
find 0.56+0.06
−0.05 Earth-size planets (1 − 1.5 R⊕ ) and 0.46−0.05 super-Earths (1.5 − 2 R⊕ )
per M dwarf. In total, we estimate a cumulative planet occurrence rate of 2.5 ± 0.2
planets per M dwarf with radii 1 − 4 R⊕ and periods shorter than 200 days. Within a
conservatively defined habitable zone based on the moist greenhouse inner limit and
maximum greenhouse outer limit, we estimate an occurrence rate of 0.16+0.17
−0.07 Earth-size
planets and 0.12+0.10
−0.05 super-Earths per M dwarf habitable zone. Adopting the broader
insolation boundaries of the recent Venus and early Mars limits yields a higher estimate of
+0.11
0.24+0.18
−0.08 Earth-size planets and 0.21−0.06 super-Earths per M dwarf habitable zone. This
suggests that the nearest potentially habitable non-transiting and transiting Earth-size
planets are 2.6 ± 0.4 pc and 10.6+1.6
−1.8 pc away, respectively. If we include super-Earths,
these distances diminish to 2.1 ± 0.2 pc and 8.6+0.7
−0.8 pc.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
3.1
Introduction
In this paper, we focus on the population of planets orbiting small stars. Such planets
are important for constraining the galactic census of exoplanets because the majority
of stars in the galaxy are low-mass stars (Henry et al. 2006; Winters et al. 2015). In
addition to understanding the overall occurrence rate of planets orbiting low-mass stars,
we would like to know how planet occurrence depends on factors such as planet radius,
orbital period, stellar insolation, and host star properties. Furthermore, small stars
afford the best near-future opportunities for detailed characterization studies of small
planets and their atmospheres (Charbonneau & Deming 2007). In order to prepare for
these observations, we would like to know the likely distance to the closest such targets.
Computing the occurrence rate of small planets around small stars is complicated
by the fact that the parameters of low-mass stars are more difficult to measure than
the parameters of Sun-like stars. The main emphasis of the Kepler mission was the
detection of planets around Sun-like stars, so the assumptions made in the construction
of the Kepler Input Catalog (KIC) were tailored to be appropriate for Sun-like stars.
Accordingly, Brown et al. (2011) cautioned against relying on KIC classifications for
stars cooler than 3750K.
Several previous studies have attempted to improve the KIC parameters for the
coolest Kepler target stars. In our previous paper (Dressing & Charbonneau 2013), we
used the photometry provided in the KIC to reclassify all stars cooler than 4000K using
models (Dotter et al. 2008; Feiden et al. 2011) and assumptions more appropriate for
low-mass stars. Gaidos (2013) conducted a similar analysis for the population of planet
candidate host stars. Other authors further constrained the properties of particular
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
low-mass stars and associated planet candidates by acquiring follow-up spectroscopic
and high resolution imaging observations (Johnson et al. 2012; Muirhead et al. 2012a,b;
Ballard et al. 2013; Muirhead et al. 2013; Swift et al. 2013). Recognizing the importance
of characterizing the full target sample as well as the planet host stars in order to
constrain the planet occurrence rate, Mann et al. (2012) acquired spectra for a subset of
non-planet host stars. They found that the majority of bright (Kp < 14) Kepler target
stars are giant stars (several hundred of which were classified as dwarfs in the KIC) but
that 93% of fainter stars are correctly classified as dwarfs.
In this paper, we combine the current best estimates of the properties of small
Kepler target stars in order to estimate the frequency of small planets around small
stars. Our analysis was preceded by several studies of the planet occurrence rate based
on Kepler data and we adopt some techniques from the earlier studies. In particular,
we draw upon the framework established by Howard et al. (2012), Fressin et al. (2013),
Dressing & Charbonneau (2013), Petigura et al. (2013b), and Petigura et al. (2013a).
Working with the first three quarters of Kepler data, Howard et al. (2012) estimated
the frequency of planets around main-sequence GK stars. They found that the
occurrence rate of planets increased sharply with decreasing planet size and moderately
with increasing orbital period. They also found evidence for a cutoff period below which
the planet occurrence rate falls off more quickly with decreasing period. The position
of the cutoff period appeared to move outward from near 2 days for larger planets
(Rp > 8 R⊕ ) to roughly 7 days for 2 − 4 R⊕ planets.
Youdin (2011) used the search completeness estimates from Howard et al. (2012) to
model the planet occurrence rate around Sun-like stars by a joint powerlaw in orbital
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period and planet radius. He found an occurrence rate of 0.19 planets per star with
periods shorter than 50 days and radii larger than 2 R⊕ . He also extrapolated outward
to predict an occurrence rate of roughly three Earth-like planets per star with periods
shorter than a year.
Due to the low false positive rate expected for the Kepler planet candidate sample
(Morton & Johnson 2011), the Howard et al. (2012) and Youdin (2011) analyses assumed
that all of the candidates were bona fide transiting planets. They further assumed that
Kepler would have been able to detect all transiting planets with cumulative SNR above a
set threshold of 10σ. The first assumption biased their occurrence rate estimates toward
higher values while the second assumption would have resulted in an underestimate if
the actual search completeness were lower.
Fressin et al. (2013) conducted a follow-up study of the Kepler planet occurrence rate
incorporating both contamination from false positives and a more sophisticated model of
pipeline sensitivity. In particular, they used a hierarchical approach in which they first
estimated the population of Jupiter-size planet candidates that might be astrophysical
false positives. They then iteratively determined the occurrence rate of small planets
by modeling the fraction of larger planet candidates that might be masquerading as
smaller planet candidates in diluted transit events. Fressin et al. (2013) found a global
false positive rate of 9.4 ± 0.9% and noted that considering false positives is particular
important when calculating the occurrence rate of giant planets (6 − 22 R⊕ , FP rate =
17.7% ± 2.9) and Earth-size planets (0.8 − 1.25 R⊕ , FP rate = 12.3% ± 3.0).
Fressin et al. (2013) also used their hierarchical model to estimate the completeness
threshold of the Kepler pipeline. They found that a linear ramp model in which 0% of
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
signals with SNR < 6 and 100% of signals with SNR > 16 were detected provided a better
fit to the observed planet candidate population than an abrupt step function. Accounting
for both a non-zero false positive rate and a ramp sensitivity model, Fressin et al. (2013)
estimated that 14.9 ± 2.4% of FGK stars host an Earth-size planet (0.8 − 1.25 R⊕ ) with
a period between 0.8 and 50 days. In a similar study of Kepler data, Dong & Zhu (2013)
found that roughly 20% of main sequence stars with 5000K < Teff < 6500K host 1 − 2 R⊕
planets in periods less than 50 days.
More recently, Petigura et al. (2013b) developed their own planet search pipeline
in order to search for additional planet candidates around Kepler stars. Their TERRA
pipeline uses a custom light curve detrending algorithm based on principal component
analysis (Petigura & Marcy 2012). After searching for planets around 42,557 relatively
quiet GK stars, Petigura et al. (2013a) found that 7.7 ± 1.3% of GK stars host small
planets (1 − 2 R⊕ ) in periods between 25 and 50 days. They also extrapolated to predict
that 22% ± 8% of GK stars host 1 − 2 R⊕ planets receiving between 1/4 and 4 times
the insolation received by the Earth. Their calculation incorporated a 10% correction
for false positives. As a benefit of writing their own pipeline, Petigura et al. (2013a)
were able to explicitly measure the completeness of their planet sample by injecting and
attempting to recover transiting planets.
In a follow-up study, Foreman-Mackey et al. (2014) used the reported search
completeness and planet candidates from Petigura et al. (2013a) to rederive the planet
occurrence rate using a hierarchical Bayesian model. The Foreman-Mackey et al. (2014)
analysis differed from the Petigura et al. (2013a) analysis in two key aspects: (1)
Foreman-Mackey et al. (2014) considered measurement errors in the stellar and transit
parameters and (2) they did not assume that the planet occurrence rate was flat in log
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
period, instead using a flexible Gaussian process to model the occurrence rate assuming a
smooth functional form. As a result, Foreman-Mackey et al. (2014) found an occurrence
rate of potentially habitable Earth-size planets three times lower than the Petigura et al.
(2013a) estimate.
Silburt et al. (2015) considered a sample of 76,711 Kepler target stars with radii
of 0.8 − 1.2 R$ and estimated the search completeness using the reported Combined
Differential Photometric Precision (CDPP, Christiansen et al. 2012). They employed
an iterative simulation to investigate the dependence of the planet occurrence rate on
planet radius without subdividing the data into bins and accounted for errors in the
planet radii. For planets with periods of 20–200 days, Silburt et al. (2015) reported that
the occurrence rate is higher for planets with radii of 2 − 2.8 R⊕ than for smaller or
larger planets. In total, they estimated that a typical Sun-like star hosts 0.46 ± 0.03
planets with periods of 20–200 days and radii of 1 − 4 R⊕ . In agreement with Petigura
et al. (2013a), Silburt et al. (2015) noted that the planet occurrence rate is flat in log
period. Within a broad habitable zone extending from 0.99–1.7 AU, Silburt et al. (2015)
estimated an occurrence rate of 0.064+0.034
−0.011 small (1 − 2 R⊕ ) planets per Sun-like star.
Focusing specifically on Kepler’s smallest target stars, we (Dressing & Charbonneau
2013) estimated an occurrence rate of 0.90+0.04
−0.03 planets per star for 0.5 − 4 R⊕ planets
with periods shorter than 50 days. We based our previous analysis on Q1–Q6 Kepler
planet candidate list and assumed that Kepler detected all planets with cumulative
SNR> 7.1σ. Using conservative habitable zone limits from Kasting et al. (1993), we
estimated an occurrence rate of 0.15+0.13
−0.06 potentially habitable Earth-size (0.5 − 1.4 R⊕ )
planets per small star. Kopparapu (2013) then revised this estimate to 0.48+0.12
−0.24 planets
per star using the broader updated habitable zone boundaries from Kopparapu et al.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
(2013b). His result agreed well with the estimate of 0.46+0.18
−0.15 potentially habitable
0.8 − 2 R⊕ planets per star from Gaidos (2013). Unlike Kopparapu (2013), Gaidos (2013)
adopted habitable zone boundaries corresponding to the 50% cloud cover case from Selsis
et al. (2007).
Morton & Swift (2014) adopted a slightly different technique to estimate the
frequency of small planets around small stars. They assumed that the planet radius
distribution is independent of orbital period and modeled each planet using a weighted
kernel density estimator when computing the occurrence rate. They found that the
occurrence rate estimates from Dressing & Charbonneau (2013), Kopparapu (2013),
and Gaidos (2013) for planets smaller than 1.4 R⊕ should be increased by an additional
incompleteness factor of 1.6 if the assumption made by Morton & Swift (2014) about the
period-independence of the planet radius distribution is correct.
Like Silburt et al. (2015), Gaidos et al. (2014) used an iterative simulation to
estimate the planet occurrence rate, but they elected to focus on stars cooler than
4200K. For orbital periods of 1 − 180 days and radii of 0.5 − 6 R⊕ , Gaidos et al. (2014)
calculated a cumulative occurrence rate of 2.01 ± 0.36 planets per M dwarf. Gaidos et al.
(2014) also remarked that the planet occurrence rate is highest for planets with radii of
approximately 1 R⊕ and lower for larger and smaller planets.
The frequency of potentially habitable planets around small stars has also been
estimated from radial velocity surveys. Based on six years of observations with the
HARPS spectrograph, Bonfils et al. (2013) estimated an occurrence rate of 0.41+0.54
−0.13
potentially habitable planets per M dwarf. Their definition of “potentially habitable”
encompassed planets with 1 ≤ M sin i ≤ 10 within the “early Mars” and “recent Venus”
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
boundaries of the habitable zone presented in Selsis et al. (2007). A subsequent study by
Robertson et al. (2014) revealed that GJ 581d, one of the two planets upon which Bonfils
et al. (2013) based their occurrence rate estimate, is likely a manifestation of stellar
activity. Robertson et al. (2014) reported a revised occurrence rate of 0.33 potentially
habitable planets per M dwarf. The updated RV-based estimate is more similar to the
estimates based on Kepler data, but accurately determining an occurrence rate with only
a single planet (GJ 667Cc, Anglada-Escudé et al. 2012; Bonfils et al. 2013; Delfosse et al.
2013) is challenging. Additionally, direct comparison of planet occurrence estimates from
RV and transit surveys is complicated by the need to employ a compositional model to
translate planet masses into radii.
In this paper, we implement the following improvements to refine our 2013 estimate
of the frequency of small planets around small stars:
• We use the full Q0-Q17 Kepler data set.
• We utilize archival spectroscopic and photometric observations to refine the stellar
sample.
• We explicitly measure the pipeline completeness.
• We inspect follow-up observations of planet host stars to properly account for
transit depth dilution due to light from nearby stars.
• We apply a correction for false positives in the planet candidate sample.
• We incorporate a more sophisticated treatment of the habitable zone.
In Section 3.2 we describe the selection of our stellar sample, which includes some stars
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whose parameters have been characterized spectroscopically via follow-up observations.
We explain our planet detection pipeline in Section 3.3 and our procedure for vetting
candidates in Section 3.4. We present light curve fits for the accepted planet candidates
in Section 3.5. In Section 3.6, we assess the completeness of our pipeline. We then
estimate and discuss the planet occurrence rate in Section 3.7 before concluding in
Section 3.8.
3.2
Stellar Sample Selection
We selected our stellar sample by first downloading a table of all 4915 stars with
Teff < 4000 and log g > 3 from the Q1–16 Kepler Stellar Catalog on the NASA Exoplanet
Archive1 (Akeson et al. 2013). This catalog is described in Huber et al. (2014) and
combines the best estimates available for each star from a variety of photometric,
spectroscopic, and asteroseismic analyses. The properties for the stars in the downloaded
sample were primarily determined from photometry (Brown et al. 2011; Dressing
& Charbonneau 2013; Gaidos 2013; Huber et al. 2014), but 2% of the sample had
spectroscopically-derived parameters (Mann et al. 2012; Muirhead et al. 2012a; Mann
et al. 2013b; Martı́n et al. 2013). For the majority of the stars in the sample (79%), the
stellar parameters were drawn from our 2013 analysis (Dressing & Charbonneau 2013).
Some of the stars in the downloaded subset had light curves indicative of binary
stars, variable stars, or enhanced spot activity. In addition, some of the stars were
observed only for a small number of days. In order to accurately estimate the planet
1
http://exoplanetarchive.ipac.caltech.edu
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
occurrence rate for small stars, we wanted to select the subset of stars with the highest
search completeness. We therefore performed the following series of cuts on the sample.
First, we counted the number of timestamps for which each star had “good”
data (i.e., not flagged). We rejected all 2101 stars with fewer than 48940 unflagged
long cadence data points. Since Kepler obtained long cadence data using 29.4 minute
integration times, this cut requires 1000 days of data. One of the main goals of this
paper is to measure the occurrence rate of potentially habitable planets and we wanted
to ensure that Kepler would have been able to observe multiple transits of planets within
the habitable zones (HZ; see Section 3.7.3) of the stars in our final sample. For reference,
the median orbital period at the outer edge of the Kopparapu et al. (2013b) HZ for the
stars in our final sample is 131 days and the longest period at the outer HZ is 207 days.
Second, we removed 63 stars that McQuillan et al. (2013) categorized as likely giants
based on their stochastic photometric variability. The affected stars have red colors
(median J − H = 0.83) consistent with their revised classification as giants. Although
two of the stars had revised classifications from Dressing & Charbonneau (2013), the
remaining 61 had parameters from the Kepler Input Catalog.
For reference, we checked whether any of our target stars were known eclipsing
binaries by consulting the Kepler Eclipsing Binary Catalog. We examined both the
published Version 2 (Slawson et al. 2011) and the online beta vewidth]chapter3/fsion of
the Third Revision.2 Eleven of the target stars were listed in both versions of the catalog,
six stars were listed in Version 2 only and three stars were listed in Version 3 only.
Six of the twenty targets with matches in the Eclipsing Binary Catalogs were listed
2
keplerebs.villanova.edu
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
in the NASA Exoplanet Archive as false positive systems (KID 5820218 = KOI 1048,
KID 6620003 = KOI 1225, KID 8823426 = KOI 1259, KID 9761199 = KOI 1459,
KID 9772531 = KOI 950, and KID 10002261 = KOI 959). Two were listed as planet
candidate host stars (KID 5384713 = KOI 3444 and KID 11853130 = KOI 3263).
Confirmed giant planet KOI 254.01 (KID 5794240, Johnson et al. 2012) was also
included as an EB match because of the very large transit depth. As evidenced by the
presence of KOI 254.01 in the EB catalogs, the catalogs contain both actual EBs and
likely planets. Accordingly, we did not remove the twenty targets with matches in the
EB catalogs from our target sample.
We then detrended all of the light curves as described in detail in Section 3.3.1
using smoothing lengths of 500, 1000, and 2000 minutes. We constructed a histogram
of the flux distributions for each of the detrended light curves and measured the χ2 of
a fit to a Gaussian flux distribution. The flux distribution of a well-behaved single star
should be Gaussian after detrending, but the flux distribution of an eclipsing binary
can appear bimodal. We therefore flagged for visual inspection all 353 stars for which
reduced χ2 > 3 for any of the detrended light curves. We also measured the standard
deviations σ500 , σ1000 , σ2000 of the three detrended light curves for each star and took the
ratios of the standard deviations of light curves detrended using different median filters.
We flagged 42 stars for which any of the ratios σ500 /σ1000 , σ500 /σ2000 , or σ1000 /σ2000 were
below 0.8. This cut was designed to pick out light curves for which the detrending
algorithm failed to remove longer timescale variability.
Finally, we visually inspected the detrended and PDC-MAP photometry (Smith
et al. 2012; Stumpe et al. 2012) for the 395 flagged stars with > 1000 days of data
and large χ2 or standard deviation ratios. We rejected 207 stars with highly variable
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
0.8
Stellar Radius (RSun)
Possible Stars: 4915
Selected Stars: 2543
0.6
0.4
0.2
0.0
2500
3000
3500
Stellar Effective Temperature (K)
4000
Figure 3.1: Radii and stellar effective temperatures of the stars in our final selected subsample (blue) compared to the full sample initially downloaded from the NASA Exoplanet
Archive (black). The red line is an empirical relation between effective temperature and
radius (Mann et al. 2013a, see Section 3.7.4).
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
detrended light curves or strong indications of classification as an eclipsing binary. We
also rejected KID 12207013 because of the unusual light curve morphology displayed
in certain quarters. We accepted the remaining 187 flagged stars, increasing our final
selected sample to 2543 stars. We note that the 208 stars that were rejected during
the visual inspection stage are unlikely to harbor detectable planet candidates exactly
because their light curves are highly variable. Similarly, our injection tests would likely
recover only a small fraction of any planets injected into their light curves. Accordingly,
the exclusion of these stars from our stellar sample has negligible effect on our estimated
rates of planet occurrence.
Our final selected sample of 2543 stars is compared to the initial downloaded sample
in Figure 3.1. The temperature range of the sample extends from 2661K to 3999K, with
a median stellar effective temperature of 3746K. The median stellar radius is 0.47 R$ and
the stars have radii spanning from 0.10 R$ to 0.64 R$ . The metallicity range is [Fe/H]
= -2.5 to [Fe/H] = 0.56, with a slightly sub-solar median metallicity of [Fe/H] = −0.1.
However, most of the metallicity estimates were derived from photometry (Dressing &
Charbonneau 2013) and are not well-constrained. The brightest star in the sample has
a Kepler magnitude Kp = 10.07, but the median brightness is Kp = 15.5. The faintest
star has Kp = 16.3. The sample contains 100 known planet (candidate) host stars with
83 planet candidates and 80 confirmed planets.
3.3
Planet Detection Pipeline
The first step in our planet detection pipeline was to clean the light curves to prepare
them for the transit search. Next, we searched each light curve sequentially for planets,
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
allowing the code to detect multiple planets per star when warranted by the data. We
then accepted or rejected putative detections using the vetting procedure described in
Section 3.4 and checked that the automatically accepted transits were not ephemeris
matches with other KOIs. Finally, we visually inspected all surviving candidates
and reviewed all available follow-up analyses produced by the Kepler team and the
community. We discuss each step of the planet detection pipeline in more detail in the
following sections.
3.3.1
Preparing the light curves
We obtained all available long cadence data for each target via anonymous ftp from the
MAST.3 We then excluded all data points flagged as low quality and detrended each
quarter of data independently. We produced three detrended versions of each light curve
using a running sigma-clipped mean filter with widths of 500, 1000, or 2000 minutes.
When calculating the mean, we excluded all points more than 3σ away from the median
value of the light curve within the filtered region. We then divided the flux data by the
smoothed light curve to obtain a detrended, normalized light curve for that quarter.
Figure 3.2 provides an illustration of our light curve detrending process.
Next, we searched for data gaps and anomalies within the detrended light curves.
We defined a data gap as ≥ 0.75 days of missing photometry. Data gaps are frequently
accompanied by sharp increases or decreases in flux that can confound searches for
planets. We removed these events by excising all data points within 1 day of the start of
a data gap or less than 3 days after the end of a data gap.
3
http://archive.stsci.edu/kepler/publiclightcurves.html
169
Detrended
Flux
Normalized
Flux
PDC-MAP
Flux
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
6200
6000
5800
5600
5400
5200
5000
1.010
1.005
1.000
0.995
0.990
1.004
1.002
1.000
0.998
0.996
200
250
300
350
400
Time (Days)
450
500
Figure 3.2: Illustration of the detrending process using a section of the light curve of
KID 5531953 (KOI 1681). Top: PDC-MAP flux versus time. Middle: Normalized flux
versus time. Bottom: Flux detrended using a 2000 minute filter versus time.
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3.3.2
Searching for Transiting Planets
We designed our planet search algorithm to take advantage of the known stellar
properties for each target star. First, we predicted the expected transit duration as a
function of orbital period based on the mass and radius of the target star (Winn 2010).
We initially computed the transit duration for a central transit of a planet in a circular
orbit, but we reduced the minimum duration considered by a factor of 15 to account for
grazing transits and eccentric orbits. We then constructed a box-fitting least squares
(BLS) periodogram for each of twelve logarithmically spaced intervals between 0.5 and
200 days. Our main planet search program was written in IDL, but the BLS portion was
implemented using the Fortran package and IDL wrapper provided by Scott Fleming.4
We used different boundaries for the transit “duty cycle” (the ratio of the transit
duration to the orbital period) for each period range based on our predicted transit
durations. For each search, we used a light curve detrended with a smoothing filter of
500, 1000, or 2000 minutes. The choice of light curve was set by the expected transit
duration. For predicted transit durations shorter than 200 minutes, we selected the
shortest smoothing filter such that the expected transit duration was less than one tenth
of the smoothing window. For transit durations longer than 200 minutes we used the
2000 minute filter.
We then determined the signal detection efficiency (SDE) for each possible signal
in the composite periodogram using Equation 6 in Kovács et al. (2002). We checked
whether any peaks had SDE > 6 and stopped searching if no peaks were above the
threshold. If peaks were detected, we ranked the peaks in order of decreasing SDE.
4
http://www.personal.psu.edu/users/s/w/swf13/SGE/clio.html
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Starting with the most significant peak, we re-ran the BLS algorithm considering
only periods close to the period of the identified peak. In these high-resolution runs, we
considered transit durations (τdur ) as short as 1/30th of the expected transit duration of
a planet in a circular orbit to increase the chance that our code would be able to recover
planets in grazing or eccentric orbits. We used these higher resolution BLS periodograms
to determine the epochs of the putative transit events. We then ran a Monte Carlo
analysis to find the preliminary transit model (Mandel & Agol 2002) that best described
the candidate event.
In our Monte Carlo analysis, we allowed the transit center to shift by 2 hours
(up to a maximum of 1/200th of the orbital period for short-period events). For each
choice of transit center we generated a new version of the detrended light curve by
dividing the raw PDC-MAP Kepler photometry by a straight line fit to the photometry
immediately preceding and following each putative transit. Specifically, we considered
data points more than one and less than 3.5 expected full transit durations away from
the putative transit center. We assumed circular orbits and estimated quadratic limb
darkening parameters from the Teff and log g of the target star by interpolating between
the coefficients determined by Claret & Bloemen (2011). We considered a/R∗ between
50% and 200% of the expected value for the trigger orbital period, impact parameters
between 0 and 1, and RP /R∗ as large as the square root of the depth of the trigger event.
In some cases, the highest peak in the periodogram was actually at a harmonic of
the true planet period, so we repeated the Monte Carlo transit fitting analysis at
1
n
and
n times the trigger period for n = 2 − 7. We then selected the period for which the
∆χ2 compared to a straight line fit was maximized. We rejected all putative transit
events for which none of the models were preferred at 5σ and recorded the parameters of
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
the best-fit model in other cases. We further refined the planet parameters during the
vetting stage as described in Section 3.4.
We then repeated the process described in the previous three paragraphs to fit the
next highest peak in the periodogram. If the preceding transit fit had been accepted,
then we excised the data near transit prior to fitting the next peak. When all peaks
above the threshold level were exhausted, we generated a new periodogram using only
the out-of-transit data and reran the peak identification and transit model fitting process
with the new periodogram. The code automatically stopped searching for planets when
no peaks with SDE > 6 were found, when none of the transit models for the identified
peaks were accepted, or when the code had completed three iterations of searching for
planets. (Note that multiple planets could be detected in a single round of searching.)
3.4
Vetting
Our transit detection pipeline identified 3111 putative transit events associated with
534 stars. Some of those signals might have been systematics or astrophysical false
positives instead of bona fide transiting planet candidates. Accordingly, we performed a
series of cuts to select the events consistent with transiting planets. First, we visually
inspected the candidate transit events to identify signals that were not clearly associated
with spacecraft systematics or stellar activity. Of the 3111 candidate events, 511 events
survived initial visual inspection. The 2600 events rejected at the visual inspection
stage displayed morphologies consistent with classification as spacecraft systematics or
sinusoidal brightness variations indicative of starspots rather than transiting planets.
The 511 signals surviving visual inspection were associated with 246 unique host stars.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Some of the candidate events were harmonics of signals detected at integer multiples
of the true period. We then ranked the accepted signals for each host star in order of
decreasing ∆χ2 as calculated during the detection phase and iteratively fit and excised
the transits of each signal in order to ensure that the lower ∆χ2 signals were not simply
harmonics of the strongest signals. We rejected all signals with resulting ∆χ2 below 5σ.
This “sequential vetting” step reduced the number of candidate events to 323 possible
transits for 246 unique stars.
Next, we conducted a second, more intensive round of visual examination for the
remaining candidate events. We compared the shapes and depths of odd and even
transits, checked for the appearance of secondary eclipses, considered the depth of
the putative transit relative to other possible features at the same orbital period, and
investigated whether the putative transit events were dominated by a small number of
deep events. After visually vetting the candidates, we checked whether putative signals
had been previously identified as false positives and excluded all such events. We rejected
180 signals (including 9 known false positives) and accepted 143 signals associated with
97 unique stars.
During the vetting stage, we noticed that the phase-folded light curve for
KOI 2283.01 exhibited two transit-like events with markedly different depths when folded
to the 17.402 day orbital period listed in the Q1–16 KOI catalog. We therefore rejected
KOI 2283.01 as a blend containing an eclipsing binary. This interpretation is consistent
with the large observed centroid source offset shift of 5.3σ.
We then performed a final search for additional planets in the systems in which a
previously detected planet had survived the vetting process. We executed this search by
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
phase-folding the detrended light curve on the orbital periods of all accepted signals and
removing all data points within plus or minus one best-fit duration of transit center prior
to re-running the search process detailed in Section 3.3.2. In all cases we used the light
curve that had been detrended using a 2000 minute filter. The motivation for repeating
this search after the first round of vetting was that uncertainties in the initial periods,
durations, and transit centers of the accepted signals might have limited the effectiveness
of the clipping performed in the initial search.
Our second round search revealed 104 candidate signals for 33 stars. We vetted
the signals using the same vetting pipeline as in the first round search and accepted
15 additional candidate transiting planets associated with 12 stars. Next, we excised
the transits of the signals accepted in the second round and performed a third round of
transit searches. No additional signals were accepted during the third round. The full
sample of 156 planet candidates included 143 signals associated with 97 stars revealed in
the first round of searching and 15 signals associated with 12 stars revealed in the second
round.
We compared the periods, P , and epochs, t0 , of the accepted planet candidates to
the catalog of known eclipsing binaries, periodic variable stars, and KOIs compiled by
Coughlin et al. (2014). We excluded the host star from the match process in order to
avoid matching a signal to itself. We did not find any corresponding signals within our
specified match tolerances of
|Pmatch − P | ≤ min (2 hr, 0.001 × P)
and
(3.1)
|t0,match − t0 | ≤ min (4 hr, 0.001 × P)
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
3.4.1
New Planet Candidate
The majority of the accepted planet candidates corresponded to signals previously
identified as KOIs. We found that 155 putative planet candidates had periods and
epochs matching those of known planet candidates or confirmed planets (Borucki et al.
2010, 2011a,b; Batalha et al. 2013; Burke et al. 2014). We accepted one new signal in a
system with previously known KOIs.
The KOI 1681 (KID 5531953) system contains three known planet candidates with
periods of 6.51, 1.99, and 3.53 days. Our pipeline detected a planet candidate in the
system with a radius of 1 R⊕ and a period of 21.9 days, roughly 11 times the 1.99 day
orbital period of KOI 1681.02. We describe additional planet properties in Table 3.1. As
shown in Figure 3.3, the transit signal is still visible in the light curve after the transits
of the other three planets have been removed, suggesting that this is a new planet
candidate rather than an alias of KOI 1681.02. The signal was later identified in the
Q1-17 DR 24 Kepler pipeline run5 and is now listed as planet candidate 1681.04.
5
http://exoplanetarchive.ipac.caltech.edu/cgi-bin/TblView/nph-tblView?app=
ExoTbls&config=q1_q17_dr24_koi
Table 3.1. New Candidate Accepted By Our Pipeline
KID
KOI
P
(days)
5531953
1681
21.913843
t0
(days)
Rp
( R⊕ )
a
(AU)
∆χ2
17.036402
1.03
0.16
41.4
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
15
Planet Number
t0 (Days)
20
Accepted Signals: 4
Known KOIs: 3
10
1681.01
5
0
0
1681.02
1681.03
5
10
15
20
4
3
2
1
0
0
25
50
100
Time (Days)
Detrended Flux
Detrended Flux
P (Days)
1.0000
0.9998
0.9996
0.9994
2
0.9999
0.9998
1.0
Folded Time
1.0000
0.9998
0.9996
0.9994
0.9992
0.9999
0.9998
0.9997
0.9999
0.9998
0.9997
1.5
2.0
Folded Time
2.5
3.0
0.9998
15
20
0.20
Folded Time
0.25
0.30
ΔΧ2: 56.90
P: 3.53
Odd
Even
0.05
Detrended Flux
10
Folded Time
0.15
1.0000
0.10
0.15
Folded Time
0.20
0.25
1.0005
1.0000
0.9995
0.9990
5
0.10
1.0002
3.5
1.0004
1.0002
1.0000
0.9998
0.9996
0.9994
ΔΧ2: 114.87
P: 1.99
Odd
Even
1.0004
0.9996
1.0
6.6
1.0000
0.05
1.0000
6.5
Folded Time
1.0001
1.5
1.0001
0.5
ΔΧ2: 157.72
P: 6.94
Odd
Even
6.4
1.0000
200
1.0002
6
Detrended Flux
Detrended Flux
5
1.0001
0.5
Detrended Flux
3
4
Folded Time
Detrended Flux
Detrended Flux
1
150
ΔΧ2: 41.37
P: 21.91
Odd
Even
16.90
16.95
17.00
17.05
Folded Time
17.10
17.15
17.20
Figure 3.3: Transit signals detected in the KID 5531953 system. Top left: Period
and epochs of all four signals identified by our pipeline (circles) and the known planet
candidates in the system (marked by text). Top right: Transit times for each of the four
planet candidates. Second row from top: Detrended flux versus time folded to the 6.9 day
period of KOI 1681.01. The left panel displays the full binned phase-folded light curve
and the right panel shows a zoomed-in view near transit center. The orange line marks
the transit center. The light gray and dark gray lines show the binned phase-folded light
curve for only the odd and even transits, respectively. We excised data points between
the vertical gray lines before folding the data to the period of the next planet. Third row
from top: Same as second row but for the 1.99 day period of KOI 1681.02. The transits
of KOI 1681.01 are not included. Fourth row from top: Same as previous row but for
the 3.53 day period of KOI 1681.03. The transits of KOI 1681.01 and KOI 1681.02 are
not included. Bottom row: Same as second row but for the 21.9 day period of the new
signal detected by our pipeline. The transits of KOI 1681.01, 1681.02, and 1681.03 are
not included.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
3.4.2
Accounting for Transit Depth Dilution
For the 155 known KOIs in our planet candidate sample, we inspected the DV reports
prepared by the Kepler team and all publicly available follow-up data to check for signs
that the KOIs were false positives. As described below, we learned that several of
the planet host stars in our sample have stellar companions at separations within 1## .
Accordingly, the measured transit depths for those planet candidates would have been
diluted by the additional light in the aperture.
Two of those systems (KOI 1422 and KOI 2626) were well characterized by Cartier
et al. (2014). In their analysis, Cartier et al. (2014) determined stellar parameters
for the double star system KOI 1422 and the triple star system KOI 2626 using HST
WFC3/UVIS photometry. They were unable to constrain which of the stars hosted the
associated planet candidates, but they were able to provide revised estimates for the
radii and orbital parameters of the associated planet candidates for each choice of host
star. In our analysis, we therefore chose to represent the planet candidates in these
systems using “fractional planets” orbiting each of the possible host stars rather than
assuming that the planet candidates orbit the system primaries or excluding them from
the analysis.
For the remaining systems with close stellar companions we did not have sufficient
information about the companion star to model planet candidates orbiting each star
in the system. Instead, we corrected for the transit depth dilution by multiplying the
√
estimated planet radius by the correction factor c = 2.512−∆K + 1, where ∆K is the
difference in Kepler magnitudes between the apparent magnitudes of the target star and
companion star. In the worse case scenario of an equal-brightness binary, the correction
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
increases the estimated radius of the planet candidate by roughly 40%.
We applied correction factors for three systems: KOI 605 (41%), KOI 3010 (41%),
and KOI 3284 (2%). In the KOI-605 system, which contains two candidates, D. Ciardi’s
Keck/NIRC2 adaptive optics images6 revealed that the system consists of two stars
separated by less than 0.## 1 with nearly equal brightness in the Kepler bandpass.
D. Ciardi also acquired Keck/NIRC2 observations7 of KOI 3010 showing that the
system is a close binary with a separation of 0.## 3. The two stars appear to have nearly
equal brightnesses. For KOI 3284, Keck/NIRC2 and Gemini/DSSI images8 revealed a
companion 3.56 magnitudes fainter than the target star at a separation of 0.## 4.
Our correction procedure explicitly assumed that any associated planet candidates
orbit the target star, but they might actually orbit the companion star. If the planet
candidates do indeed orbit the companion star, then the radii of the planet candidates
will need to be reevaluated once the properties of the companion star are established (see
Ciardi et al. 2015 for a detailed discussion). In most cases, the available photometry was
insufficient to determine whether the nearby companion is physically associated with the
target star or to constrain the properties of the companion star.
6
https://cfop.ipac.caltech.edu/edit_obsnotes.php?id=605
7
https://cfop.ipac.caltech.edu/edit_obsnotes.php?id=3010
8
https://cfop.ipac.caltech.edu/edit_obsnotes.php?id=3284
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
3.4.3
False Positive Correction
In addition to correcting for transit depth dilution in systems with nearby companions,
we incorporated a general false positive correction to account for the possibility that
some of the smaller transiting planets in our sample might be diluted eclipses or transits
of larger planets. We therefore consulted Table 1 of Fressin et al. (2013) to determine
the false positive probability (F P P ) for a planet with a given radius. We apply this
radius-dependent correction to the planet occurrence map derived in Section 3.7.
3.4.4
Known Planet Candidates Missed by Our Pipeline
Our sample of accepted candidate events included all but 7 of the 161 known planet
candidates and confirmed planets meeting our sample cuts of planet radii larger than
0.5 R⊕ and orbital periods shorter than 200 days. We list the missed candidates in
Table 3.2. We note that Swift et al. (2015) rejected one of the missed candidates
(KOI 1686.01) as a possible planet because the phase folded light curve did not display
a convincing transit event and that the reported disposition in the NASA Exoplanet
Database was later changed to False Positive. Three of the missed candidates are in the
same system (KOIs 3444.01, 3444.03, and 3444.04) and none of the 7 missed candidates
produced accepted peaks in the BLS periodograms. Although we were reassured that
our pipeline recovered most of the previously detected planet candidates, our goal was
not to reproduce the Kepler planet candidate list but to design a single pipeline that
could be used to both search for planets and characterize pipeline completeness. Thus
we do not consider these additional 7 KOIs in our analysis below.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
10
Petigura+
Short Cadence (57) Rowe+ 2014 (19)
Long Cadence (65) Swift+ 2015 (2)
Cartier+ 2014 (6)
NEA (5)
Ioannidis+ 2014 (2)
Kopparapu+ Moist/Max Greenhouse
Planet Radius (REarth)
Planet Radius (REarth)
10
1
1
10
Period (Days)
100
Kopparapu+ Venus/Mars
1
100.0
10.0
Insolation Flux (FEarth)
1.0
0.1
Figure 3.4: Radii of the planet candidates detected by our pipeline versus orbital period (Left) or insolation flux (Right) with 1σ errors. In most cases, we refit the planet
parameters by conducting an MCMC analysis fitting transit models to the short cadence
(crimson points) or long cadence (teal points) Kepler data. For the remaining 34 planet
candidates, we adopted transit parameters from Cartier et al. (2014, purple points), Ioannidis et al. (2014, brown points), Rowe et al. (2014, black points), Swift et al. (2015, blue
points), or the NASA Exoplanet Archive (gray points). In all cases, the errors on the
planet properties incorporate uncertainties in both stellar and transit parameters. The
green boxes indicate the boundaries within which we report planet occurrence rates and
the arrows in the right panel mark several variations of the habitable zone as explained in
the legend. The errors for KOIs 1422.01, 1422.02, 1422.03, 1422.04, 1422.05, and 2626.01
appear particularly large because we accounted for the possibility that any of the three
stars in the KOI 2626 system or either of the two stars in the KOI 1422 system harbor
the transiting planets (see Section 3.4.2 for details). KOIs 254.01 has an estimated radii
of 11 R⊕ and therefore does not appear on these plots.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
3.5
Planet Properties
Accurate planet radius estimates are a key ingredient in the planet occurrence
calculation. We therefore refined the preliminary transit parameters found in
Section 3.3.2 by conducting a Bayesian Markov Chain Monte Carlo (MCMC) analysis
with a Metropolis-Hastings acceptance criterion (Metropolis et al. 1953). We varied the
orbital period P , epoch of transit center t0 , planet-to-star radius ratio Rp /R∗ , semimajor
axis to stellar radius ratio a/R∗ and impact parameter b. We assumed that all of
the orbits were circular and fixed quadratic limb darkening parameters to the values
predicted from the stellar temperatures and surface gravities (Claret & Bloemen 2011).
We conducted some of the planet fits using short cadence Kepler data to better constrain
the shape of transit during ingress and egress.
For each planet candidate, we ran N chains starting at initial positions set by
perturbing the initial solution found during the detection and validation process by up to
5σ in each parameter. We manually adjusted the step sizes for each parameter such that
the acceptance fractions were between 10–30%. We ran each chain for at least 104 steps
before initiating periodic convergence tests by calculating the Gelman-Rubin potential
scale reduction factor R̂ for each parameter (Gelman et al. 2004). We terminated the
chains when R̂ < 1.05 for all parameters and then accounted for “burn-in” by removing
all steps taken prior to the point at which the likelihood first became higher than the
median likelihood of the chain. After merging the chains, we adopted the median values
of each parameter as the best-fit value and assigned errors encompassing the 68% of
values nearest to the chosen best-fit value. We provide the best-fit parameters for each
detected planet candidate in Table 3.9 and display them in Figure 3.4.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Several of the planet candidates in our sample exhibit large transit timing variations
and were poorly fit by the method described above. In those cases, we adopted the
transit parameters found in previous studies using combined fits to the individual transit
times and the planet properties. For the candidates displaying TTVs, we incorporated
fits from Rowe et al. (2014) for KOIs 248.01, 248.02, 248.03, 248.04, 314.01, 314.02,
314.03, 448.01, 448.02, 886.01, 886.02, and 886.03 and from Ioannidis et al. (2014) for
KOIs 676.01 and 676.02.
In addition, we adopted the light curve parameters from the 2 January 2015 version
of the NASA Exoplanet Archive for 5 candidates: 961.01, 961.03, 1681.01, 2329.01, and
3263.01. We also adopted the fits from Rowe et al. (2014) for KOIs 254.01, 430.01,
430.02, 438.01, 775.02, 868.01, and 961.02, from Swift et al. (2015) for KOIs 1902.01
and 3444.02 and from Cartier et al. (2014) for KOIs 1422.01, 1422.02, 1422.03, 1422.04,
1422.05, and 2626.01. The KOIs with parameters from Cartier et al. (2014) were detected
in multi-star systems in which the identities of the host stars are unknown. As described
in Section 3.4.2, we accounted for all possible system configurations by using fractional
planets distributed around each of the possible host stars.
As depicted in Figure 3.4, we found that 149 of the accepted planet candidates had
revised radii 0.5 < RP < 4 R⊕ and orbital periods 0.5 < P < 200 days. Of the remaining
candidates, one had a shorter period (KOI 961.02, P = 0.45 days), one was too small
(KOI 5692.01), and six were too large (KOI 254.01, 868.01, 901.01, 902.01, 1176.01, and
3263.01). KOI 868.01 also has an orbital period longer than 200 days.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
3.6
Planet Injection Pipeline
In order to accurately measure the planet occurrence rate based on the results of our
planet search, we needed to know the completeness of our planet candidate list. We
measured the completeness of our planet detection pipeline by injecting transiting planets
into the PDC-MAP light curves, detrending them, and running the detrended light
curves through our detection algorithm. We did not introduce the signals at the pixel
level and we are therefore unable to comment on how the initial light curve extraction
process affects transiting planets. We refer instead to Christiansen et al. (2013) for a
discussion of pixel-level effects. They found that transits injected at the pixel level are
usually recovered with high fidelity (final SNR = 96% − 98% expected SNR).
The transit detection process as modeled in this paper consists of two distinct
stages: (1) the putative event is identified as a peak in the BLS periodogram and (2) the
signal is accepted because a transit model provides a 5σ improvement to a straight-line
fit. We took advantage of the two-step nature of the search process when determining
the search completeness for each star in our survey.
For each star, we generated a set of 2000 trial planets with orbital periods drawn
from a log uniform distribution extending from 0.5 to 200 days and uniformly distributed
epochs of transit, radii (0.5 − 4 R⊕), and impact parameters (0 − 1). We then constructed
transit models (Mandel & Agol 2002) for the trial planets using the assigned planetary
parameters and limb darkening parameters estimated from the coefficients in Claret &
Bloemen (2011) based on the stellar temperatures and surface gravities. We resampled
the transit models to the 29.4 minute long cadence integration time.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
The first question we wished to address was whether the transits of the trial planets
would be accepted as 5σ detections in the ideal scenario in which the light curve was
perfectly detrended and the orbital parameters were determined exactly. Accordingly,
we first multiplied the detrended light curves by the transit models (Mandel & Agol
2002) generated using the assigned trial planet parameters. We then checked whether
the difference in the χ2 between the best-fit transit model (which we guessed perfectly)
and a no-transit model exceeded the 5σ detection threshold of ∆χ2 = 30.863. If the
transit model was not preferred, then we recorded the trial planet as a non-detection.
For a typical star, 8% of the trial planets were rejected at this stage.
For the trial planets that would be accepted in the ideal scenario, we then conducted
a more realistic test by multiplying the transit model by the raw PDC-MAP photometry
before detrending using the straight line fit method described in Section 3.3.2. We then
re-checked whether the transit model was preferred at 5σ. Trial planets that were not
accepted at this stage were also recorded as non-detections. Overall, 4% of the trial
planets that were accepted in the ideal case of a perfectly detrended light curve were not
accepted in this more realistic test.
Finally, we tested whether the remaining trial planets would have been identified
as peaks in the BLS periodograms by running a full test for at least 25 trial planets for
each star. We selected the trial planets for the full test by ranking the signals detected
in the second round of testing in order of increasing ∆χ2 , where ∆χ2 was the value at
which they were preferred to a non-transiting model. We chose the first 25 trial planets
in the ranked list for which a random number draw yielded a result greater than 0.5. In
other words, we thinned the sample of trial planets by 50% and selected those closest
to the expected sensitivity threshold. If fewer than ten trial planets were recovered,
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
we conducted up to 25 additional runs (for a total of 50 simulations) until at least ten
planets were recovered.
For the selected trial planets, we multiplied the corresponding transit model by the
raw PDC-MAP photometry for the assigned host star and detrended the light curve
using a 2000 minute median filter as explained in Section 3.3.1. We then fed the injected
light curves into the detection pipeline described in Section 3.3. Although we considered
the full range of planet periods (0.5–200 days), we halted the search process as soon as
the injected signal was detected. If the signal was not detected, we terminated the search
using the usual conditions discussed in Section 3.3.2.
3.6.1
Predicting Transit Detectability
In total, we ran 83699 complete BLS injection simulations for the 2543 stars in our
sample. We also injected 604278 planets that had ∆χ2 below our 5σ detection threshold.
The remaining 4398023 injected planets had ∆χ2 above the detection threshold but
were not tested in the full BLS simulation. We predicted the detectability of these trial
planets by finding the fraction of BLS trial planets recovered as a function of the ∆χ2
computed in the second round of transit model tests. We ranked the BLS trial planets
by ∆χ2 and computed the recovery fraction for each consecutive group of 500 planets.
Next, we smoothed the resulting histogram and predicted the likelihood of detection for
the 4398023 non-BLS runs using a cubic spline interpolation based on the smoothed
histogram. We limited the maximum detection likelihood to 91.2%, which was the
maximum value of the cubic spline in the histogram of the recovery rate for the full BLS
runs. Figure 3.5 displays the histogram of the recovery rate for the BLS trial planets and
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
1.0
Detection Fraction
0.9
0.8
0.7
0.6
0.5
83699 BLS Simulations
4398023 Extrapolations
0.4
50
100
Δχ2
150
200
Figure 3.5: Empirical detection sensitivity versus the ∆χ2 between fitting the detrended
light curve with the injected transit model or with a no-transit model. The light blue histogram depicts the recovery fraction for the 83699 BLS trial planets in bins of 1000 planets.
The solid red line marks the estimated likelihood of detection for the 4398023 non-BLS
injected planets predicted from the smoothed histogram of BLS results.
187
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
the extrapolated likelihoods of detection for the non-BLS runs.
3.6.2
Assessing Pipeline Performance
In total, we injected 5086000 transiting planets into the light curves of the 2543 stars in
our sample. We provide a catalog of injected planet parameters and recovery results in
Table 3.3. Our pipeline successfully recovered 86% of signals injected with an expected
SNR between 15 and 20. For lower SNR, the pipeline performance decreased roughly
linearly with anticipated SNR until reaching 52% recovery for signals with anticipated
SNR between 5 and 7. For the purpose of assessing pipeline performance as a function
of SNR, we modeled the anticipated SNR of a transiting planet as:
SNR =
√
δ
ntransit
CDPPtransit
(3.2)
where δ is the median decrease in brightness during the injected transit, CDPPtransit
is the Combined Differential Photometric Precision (CDPP) on the timescale of a full
transit of a planet on a circular orbit, and ntransit is the number of transits expected
given the orbital period of the planet and the number of days the star was observed.
As in Dressing & Charbonneau (2013), we estimated the CDPP on the timescale of
a planetary transit by interpolating over the provided CDPP measured on 3-, 6-, and
12-hour timescales.
As a benefit of injecting multiple trial planets per star, we generated unique transit
detectability maps for each star in our sample. For example, Figure 3.6 displays the
transit detectability maps for KID 7104554, a Kp = 15.3, Teff = 3957K star with
lower search completeness than the larger sample. We created the star-by-star transit
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
detectability maps by gridding the injected planets in radius/period and radius/insolation
space and calculating the fraction of detectable planets within each grid cell. For the
subset of trial planets that passed both ∆χ2 tests yet were not selected for the full BLS
search, we estimated the recovery fraction as explained in Section 3.6.1.
KID7104554
KID7104554
4.0
3.5 0.911
0.912
0.912
0.912
0.911
0.907
0.904
0.899
0.868
0.836
3.0 0.912
0.912
0.912
0.911
0.903
0.911
0.912
0.849
0.901
0.842
2.5 0.900
0.887
0.912
0.909
0.912
0.899
0.888
0.805
0.839
0.777
2.0 0.907
0.873
0.912
0.909
0.833
0.866
0.806
0.861
0.757
0.542
1.5 0.911
0.808
0.870
0.867
0.844
0.792
0.698
0.485
0.239
0.248
1.0 0.796
0.790
0.706
0.615
0.489
0.223
0.151
0.068
0.000
0.5 0.299
0.211
0.095
0.067
0.085
0.000
0.000
0.000
0.000
1
10
Period (Days)
3.5 0.912
0.912
0.912
0.912
0.910
0.907
0.895
0.907
0.876
0.819
3.0 0.912
0.912
0.912
0.905
0.908
0.910
0.912
0.853
0.900
0.837
2.5 0.906
0.886
0.912
0.910
0.912
0.898
0.863
0.822
0.840
0.770
2.0 0.912
0.881
0.912
0.831
0.867
0.897
0.819
0.845
0.743
0.519
1.5 0.822
0.876
0.869
0.848
0.783
0.837
0.626
0.464
0.259
0.175
0.000
1.0 0.793
0.756
0.731
0.479
0.453
0.255
0.101
0.000
0.000
0.000
0.000
0.5 0.230
0.161
0.094
0.122
0.000
0.000
0.000
0.000
0.000
0.000
Planet Radius (REarth)
Planet Radius (REarth)
4.0
100
100
10
Insolation (FEarth)
1
Figure 3.6: Transit detectability maps for KID 7104554 as a function of planet radius
and orbital period (Left) or insolation flux (Right) based on the results of our injection
simulation. The small red plus symbols mark the 498 injected planets with ∆χ2 below
the 5σ detection threshold. The 25 large circles indicate injected planets with ∆χ2 above
the detection threshold that were recovered (yellow, 17 planets) or undetected (red, 8
planets) during the full BLS test phase. The small gray plus symbols are the remaining
1477 injected planets with ∆χ2 above the detection threshold that were not selected for
the full BLS test. The numbers within each cell denote the recovery fraction within the
cell boundaries and the cells are color-coded so that darker colors correspond to lower
detectability. The green dashed lines mark the maximum greenhouse (Max GH) and
moist greenhouse (Moist GH) insolation limits from Kopparapu et al. (2013b) and the
magenta dot-dashed lines mark the less conservative Recent Venus and Early Mars limits,
also from Kopparapu et al. (2013b).
After generating transit detectability maps in radius-period and radius-insolation
space for each star independently, we created transit detectability maps for the full
sample by summing the individual maps. Although the combined maps displayed in
Figure 3.7 are useful for comparing the sensitivity of any individual star to the sensitivity
189
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.2. Known KOIs Missed By Our Pipeline
KID
KOI
Kepler
name
mag
(Kp)
P
(days)
Rp
( R⊕ )
6149553
8890150
9605552
5384713
5384713
5384713
2986833
1686.01b
2650.02
3102.01
3444.01
3444.03
3444.04
4875.01
...
395b
...
...
...
...
...
15.89
15.99
15.98
13.69
13.69
13.69
15.78
56.87
7.05
9.32
12.67
2.64
14.15
0.91
1.3
1.1
1.0
1.0
0.6
0.8
1.0
a
Kepler
SNRa
5.2c
12.4
6.1
14.7
12.2
9.3
10.2
As reported in the Cumulative KOI table at the NASA Exoplanet Archive on 22 November 2014.
b
As of 22 March 2015, the reported disposition of KOI 1686.01 at the NASA Exoplanet Archive has
been changed to False Positive.
c
The transit SNR for KOI 1686.01 was not reported in the Cumulative KOI table, the Q1–Q16 table,
or the Q1–Q12 table. This is the value from the Q1–Q8 table. The value in the Q1–Q6 table was 7.6.
Sensitivity (2543 Stars)
Sensitivity (2543 Stars)
4.0
3.5 0.912
0.912
0.912
0.911
0.911
0.911
0.910
0.909
0.906
0.900
3.0 0.912
0.911
0.911
0.911
0.911
0.910
0.908
0.904
0.898
0.889
2.5 0.911
0.909
0.909
0.909
0.908
0.906
0.901
0.896
0.887
0.869
2.0 0.909
0.906
0.901
0.902
0.900
0.896
0.887
0.876
0.857
0.819
1.5 0.902
0.896
0.890
0.886
0.881
0.869
0.850
0.819
0.764
0.678
1.0 0.873
0.861
0.847
0.829
0.798
0.750
0.685
0.585
0.471
0.5 0.702
0.630
0.551
0.472
0.390
0.308
0.230
0.160
0.104
1
10
Period (Days)
3.5 0.925
0.914
0.913
0.912
0.911
0.911
0.910
0.909
0.906
0.902
3.0 0.925
0.914
0.912
0.911
0.910
0.909
0.908
0.904
0.900
0.892
2.5 0.924
0.912
0.910
0.909
0.908
0.906
0.902
0.897
0.889
0.875
2.0 0.922
0.908
0.904
0.902
0.900
0.896
0.889
0.879
0.861
0.830
1.5 0.914
0.899
0.892
0.887
0.880
0.869
0.852
0.822
0.775
0.706
0.356
1.0 0.887
0.864
0.848
0.828
0.796
0.754
0.691
0.609
0.512
0.416
0.067
0.5 0.717
0.626
0.556
0.482
0.409
0.337
0.267
0.205
0.151
0.106
Planet Radius (REarth)
Planet Radius (REarth)
4.0
100
100
10
Insolation (FEarth)
1
Figure 3.7: Combined transit detectability maps for the full stellar sample as a function
of planet radius and orbital period (Left) or insolation flux (Right) based on the results
of our injection simulation. The numbers within each cell denote the recovery fraction
within the cell boundaries and the cells are color-coded so that darker colors correspond
to lower detectability. These figures were produced by combining individual completeness
maps for each star such as those displayed in Figure 3.6. As in Figure 3.6, the vertical
lines in the right panel mark two definitions of the habitable zone.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.3. Injected Planets
KID
Period
(Days)
Radius
( R⊕ )
Insolation
( F⊕ )
Recovery
Statusa
003835071
005693298
011560326
008229458
004931385
009691776
010032631
002441562
011013096
002692704
...
0.686
0.876
1.900
2.700
20.420
55.352
85.665
112.865
155.207
184.735
...
0.712
0.531
2.700
0.788
1.717
2.555
1.978
0.925
1.795
1.181
...
409.813
250.636
48.871
61.212
3.056
0.568
0.664
0.195
0.123
0.225
...
1.000
1.000
0.912
0.520
0.650
0.904
0.791
0.000
0.908
0.000
...
Note. — Table 3.3 is published in its entirety in the electronic edition of the Astrophysical Journal. A
portion is shown here for guidance regarding its form and content.
a
For the 83699 injected planets that were tested in the complete pipeline and the 604278 planets with
∆χ2 below our 5σ detection threshold, the recovery status indicates whether the planet was detected (1
for recovered planets, 0 for unrecovered planets). For the remaining 4398023 injected planets that had
∆χ2 above the detection threshold but were not tested in the full BLS simulation, the recovery status
indicates the estimated likelihood of detection (see Section 3.6).
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
of the larger sample, the binning is rather coarse. We therefore generated a second set of
combined sensitivity maps by sorting the full set of 5086000 injected planets into smaller
grid cells in radius/period and radius/insolation space. We then calculated the recovery
fractions within each of the cells to produce the smoother sensitivity maps displayed in
Figure 3.8.
As shown in Figure 3.8, we found that our pipeline is very sensitive to injected
planets with radii larger than 2.5 R⊕ . Such planets were detected with nearly 90%
efficiency out to the maximum injected orbital period of 200 days. Our pipeline had
a significantly harder time detecting 1.0 − 1.5 R⊕ planets with periods longer than
100 days (recovery fraction = 36%) and 0.5 − 1.0 R⊕ planets with periods longer than
5 days (recovery fraction = 22%). Planets smaller than 1.0 R⊕ were nearly undetectable
(recovery rate approximately 6%) at orbital periods longer than 150 days.
Inspecting the transit recovery map as a function of insolation revealed that the
transit detectability changes sharply across the habitable zone (HZ). At the inner edge
of the HZ (median orbital period of 50 days for the stars in our sample), we recovered
84% of 2.0 R⊕ planets and 34% of 1.0 R⊕ planets. At the outer edge of the habitable
zone (median orbital period of 130 days), the sensitivity decreased to 80% for 2.0 R⊕
planets and 25% for 1.0 R⊕ planets. This large change in sensitivity in a very interesting
region of planet radius and insolation space reveals that the pipeline sensitivity within
the habitable zone is not well described by a single number.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Sensitivity (2543 Stars)
4.0
3.5
3.5
Planet Radius (REarth)
Planet Radius (REarth)
Sensitivity (2543 Stars)
4.0
3.0
2.5
2.0
1.5
1.0
2.5
2.0
1.5
1.0
0.5
0.5
1
10
Period (Days)
Detection Fraction
0.00
3.0
0.23
0.46
0.68
100
100
10
Insolation (FEarth)
Detection Fraction
0.91
0.00
0.23
0.46
0.68
1
0.91
Figure 3.8: Smoothed maps of the fraction of injected planets that were detected by
our pipeline. As indicated in the color bars, darker points correspond to lower detection
fractions. Left: Planet radius versus period. Right: Planet radius versus insolation. As
in Figure 3.6, the vertical lines mark two definitions of the habitable zone.
3.6.3
Calculating Search Completeness
The overall planet search completeness depends both on the detectability of a particular
transiting planet and the likelihood that a particular planet will be observed to transit.
We accounted for the latter factor by determining the mean geometric probability of
transit for planets orbiting the stars in our sample at particular periods or insolation
levels. For a given orbital period, we computed the corresponding semimajor axis for a
planet orbiting each of the stars in our sample. Next, we divided the stellar radii by the
calculated semimajor axes to find the transit probability for a planet in a circular orbit.
We then multiplied the transit probability by a correction factor to account for the
fact that the planets in our sample are more likely to be eccentric and were accordingly
more likely to transit (Barnes 2007; Kipping 2014). We adopted a correction factor of
1.08 based on a beta distribution fit by Kipping (2013) to transiting planets with periods
shorter than 382.3 days. Neglecting this correction factor would lead to an underestimate
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
of search completeness and an overestimate of the planet occurrence rate by roughly 8%
(Kipping 2014). The resulting search completeness plots are displayed in Figure 5.4.
Completeness
4.0
3.5
3.5
Planet Radius (REarth)
Planet Radius (REarth)
Completeness
4.0
3.0
2.5
2.0
1.5
1.0
2.5
2.0
1.5
1.0
0.5
0.5
1
10
Period (Days)
Log10 Completeness
-4.33
3.0
-3.42
-2.51
-1.61
100
100
10
Insolation (FEarth)
Log10 Completeness
-0.70
-3.69
-2.94
-2.18
-1.42
1
-0.67
Figure 3.9: Smoothed maps of the search sensitivity accounting for both pipeline sensitivity and the geometric probability of transit. As indicated in the color bars, darker
points correspond to lower detection fractions. Left: Planet radius versus period. Right:
Planet radius versus insolation. As in Figure 3.6, the vertical lines mark two definitions
of the habitable zone.
3.7
The Planet Occurrence Rate
In order to estimate the planet occurrence rate, we first generated smoothed maps of
the detected planet population. For each planet candidate, we counted the number of
links from the MCMC posteriors that fell within each grid cell in radius/period and
radius/insolation space.9 When converting each link of the chains from light curve
parameters to physical values, we accounted for uncertainties in the stellar parameters
9
We directly incorporated the posteriors for the KOIs fit by Rowe et al. (2014). For the other 15 KOIs
with parameters drawn from previously published papers, we modeled the radii and periods by constructing Gaussian distributions using the reported values and errors.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
by drawing new stellar parameters from Gaussians centered at the reported values with
widths set by the reported errors. (In cases where the reported errors were asymmetric,
we adopted the larger value.) We weighted each link so that the total weight equaled one
minus the false positive correction (see Section 3.4.3) for a planet with the given radius.
The errors on the planet radii and insolation flux were large enough that the
posteriors from multiple candidates overlapped to produce smoothed distributions. For
the orbital periods, however, the errors were small enough that each planet appeared
isolated. For the purpose of calculating the planet occurrence rate, we artificially inflated
the spread of the period values so that the standard deviation of log10 P distribution was
equal to 0.1. We then constructed smoothed distributions of the insolation flux received
by each planet by converting the periods into semimajor axes using Kepler’s third law
and stellar masses drawn from gaussian distributions centered on the reported value with
widths set by the reported errors.
As shown in Figure 3.10, the detected planet population has peaks within the region
P = 1 − 20 days and Rp = 0.7 − 2.5 R⊕ . There is also a noticeable lack of large planets
(Rp ≥ 1.7 R⊕ ) and shorter periods (P ≤ 2 days). In radius-insolation space (right panel
of Figure 3.10), the highest peaks of the smoothed candidate distribution are located at
insolations of 20 − 50 and 2.5 − 10 times the insolation received by the Earth.
We estimated the planet occurrence rate by dividing the smoothed maps of
the detected planet population in Figure 3.10 by the smoothed maps of the search
completeness in Figure 5.4. The resulting maps of the planet occurrence rate are
displayed in Figure 3.11. The division reveals that the lack of detected planets in the
upper left corner of the left panel of Figure 3.10 is quite meaningful. That region has
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
MCMC Results
4.0
3.5
3.5
Planet Radius (REarth)
Planet Radius (REarth)
MCMC Results
4.0
3.0
2.5
2.0
1.5
1.0
3.0
2.5
2.0
1.5
1.0
0.5
0.5
1
10
Period (Days)
100
100
10
Insolation (FEarth)
1
Figure 3.10: Smoothed distribution of planet candidates detected by our pipeline. The
color scale is linear with lighter colors indicating a higher number of planets. Left: Planet
radius versus period. Right: Planet radius versus insolation. The orange points with
error bars are the planet candidates detected by our pipeline. As in Figure 3.6, the vertical
lines mark two definitions of the habitable zone.
Planet Occurrence (%)
4.0
3.5
3.5
Planet Radius (REarth)
Planet Radius (REarth)
Planet Occurrence (%)
4.0
3.0
2.5
2.0
1.5
1.0
2.5
2.0
1.5
1.0
0.5
0.5
Recovery < 15%
1
10
Period (Days)
Log10 Occurrence
-7.00
3.0
-5.71
-4.41
-3.12
Recovery < 15%
100
100
Log10 Occurrence
-1.83
-7.00
-5.67
-4.33
-3.00
10
Insolation (FEarth)
1
-1.66
Figure 3.11: Smoothed plot of the derived planet occurrence rate as a function of planet
radius versus orbital period (Left) or insolation (Right). Lighter colors indicate higher
planet occurrence per grid cell and regions in which our pipeline detected < 15% of
injected signals are marked in gray. As in Figure 3.6, the vertical lines in the right panel
mark two definitions of the habitable zone.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Planet Occurrence (%)
4.0
0.000
(91%)
0.008
(91%)
0.18
(91%)
0.18
(91%)
0.36
(91%)
0.51
(91%)
0.32
(91%)
Planet Occurrence (%)
0.21
(90%)
0.42
(90%)
4.0
0.080
(91%)
0.006
(91%)
0.17
(91%)
0.42
(91%)
1.1
(91%)
1.4
(91%)
0.81
(90%)
1.6
(90%)
1.7
(89%)
0.16
(88%)
Planet Radius (REarth)
0.000
Planet Radius (REarth)
0.003
0.030
0.002
0.063
0.001
0.034
0.17
(91%)
(91%)
0.12
(91%)
0.32
(91%)
0.22
(91%)
0.32
(91%)
0.47
(90%)
0.14
(90%)
0.25
(90%)
3.5
3.0
0.000
(91%)
0.004
(90%)
0.23
(90%)
0.96
(90%)
2.7
(90%)
3.8
(90%)
4.6
(90%)
5.8
(89%)
4.2
(88%)
1.1
(86%)
2.5
0.002
(90%)
0.009
(90%)
0.42
(90%)
1.8
(90%)
6.4
(89%)
9.3
(89%)
10
(88%)
12
(87%)
9.6
(86%)
4.5
(82%)
2.0
0.061
(90%)
0.27
(89%)
1.2
(89%)
2.5
(88%)
6.7
(88%)
13
(87%)
14
(85%)
12
(82%)
8.3
(77%)
10
(68%)
1.5
0.46
(87%)
1.4
(86%)
3.5
(85%)
5.7
(83%)
10
(80%)
13
(76%)
16
(70%)
6.4
(59%)
10
(48%)
19
(91%)
0.20
(91%)
(91%)
0.45
(91%)
0.84
(91%)
0.85
(90%)
0.64
(90%)
1.7
(90%)
0.73
(89%)
0.48
(89%)
3.0
(90%)
0.23
(90%)
(90%)
0.90
(90%)
2.1
(90%)
3.2
(90%)
3.3
(90%)
4.7
(89%)
2.6
(88%)
1.5
(87%)
2.5
(90%)
0.50
(90%)
(90%)
1.8
(90%)
5.5
(89%)
7.0
(89%)
8.0
(88%)
10
(87%)
8.3
(86%)
3.6
(83%)
2.0
0.066
(89%)
0.32
0.75
(89%)
(89%)
2.6
(88%)
4.4
(88%)
11
(87%)
10
(85%)
12
(83%)
7.0
(79%)
8.6
(71%)
1.5
0.22
(35%)
(86%)
1.0
1.3
3.1
(86%)
(84%)
5.6
(83%)
6.2
(80%)
9.1
(76%)
17
(70%)
8.7
(62%)
4.4
(51%)
18
(41%)
1.0
0.40
(75%)
1.5
(69%)
4.4
(59%)
1
-4.00
-3.00
-2.00
5.5
(50%)
10
(39%)
12
(29%)
10
Period (Days)
Log10 Occurrence
-5.00
0.011
(91%)
(89%)
3.5
0.5
0.001
11
(19%)
0.21
0.5
Recovery < 15%
(69%)
0.89
1.8
(65%)
100
(57%)
100
Log10 Occurrence
-1.00
-5.00
-4.00
-3.00
-2.00
7.2
(49%)
10
(40%)
13
(33%)
5.5
(25%)
10
Insolation (FEarth)
12
(20%)
Recovery < 15%
1
-1.00
Figure 3.12: Binned planet occurrence rate in period/planet radius space (Left) and
insolation/planet radius space (Right). The numbers within each grid cell indicate the
planet occurrence rate as a percentage (top) and the percentage of injected planets that
were recovered by our pipeline (bottom). The gray regions have injected planet recovery
rates below 15%. Some boxes have large Poisson errors; please see Tables 3.4 –3.8. As in
Figure 3.6, the vertical lines in the right panel mark two definitions of the habitable zone.
very high search completeness, so the lack of detected planets in that region of parameter
space implies that hot mini-Neptunes and Neptunes are rare around low-mass stars. At
the opposite corner of the diagram in the small-planet, long-period regime, the relatively
small number of detected planets does not indicate a low occurrence rate. On the
contrary, those planets were detected despite relatively low search completeness, so the
underlying occurrence rate of such planets is predicted to be high.
Consulting the right panel of Figure 3.11, the estimated occurrence rate of small
planets is highest at insolations below roughly 0.5 F⊕ , but that is a region of low search
completeness and the occurrence rate for such planets is not well constrained. In order
to more readily see trends in the planet occurrence rate as a function of planetary
properties, we binned the smoothed occurrence distributions shown in Figure 3.11
to produce the gridded diagrams shown in Figure 3.12. The gridded version of the
197
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
radius-period diagram (left panel of Figure 3.12) clearly demonstrates that planet
occurrence increases with decreasing planet radius and increasing log10 P .
3.7.1
Dependence on Planet Radius & Period
Figure 3.13, Figure 3.14, and Table 3.4 display the planet occurrence rate as a function of
planet radius and orbital period. As in previous studies (Dressing & Charbonneau 2013;
Morton & Swift 2014), we found that planets with radii > 3 R⊕ are rare around small
stars (at least out to orbital periods of 200 days). Concentrating on planets with periods
shorter than 50 days, we observed a general trend of decreasing planet occurrence with
increasing planet radius between 1 R⊕ and 4 R⊕ . There is an indication in Figure 3.14
that the planet occurrence rate may become flat in log10 P for periods longer than
10 days. However, the errors on the longest orbital bins are large enough that we cannot
distinguish between a brief flattening between 10 − 100 days and a plateau extending out
to much longer orbital periods.
For orbital periods shorter than 50 days, we measure an occurrence rate of 0.56+0.06
−0.05
Earth-size (1 − 1.5 R⊕ ) planets and 0.46+0.07
−0.05 super-Earths (1.5 − 2 R⊕ ) per small star.
+0.08
Extending the period range to 100 days, we estimate 0.65+0.07
−0.05 Earths and 0.57−0.06
super-Earths per star. Overall, we find 2.5 ± 0.2 planets per M dwarf with radii of
1 − 4 R⊕ and periods shorter than 200 days. We provide cumulative planet occurrence
rates for several additional choices of period and radius boundaries in Table 3.5.
198
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Planets Per Star
0.1000
0.0100
0.0010
0.0001
0.5
P = 0.5-1.7 Days
P = 1.7-5.5 Days
P = 5.5-18.2 Days
P = 18.2-60.3 Days
P = 60.3-200.0 Days
1.0
1.5
2.0
2.5
3.0
Planet Radius (REarth)
3.5
4.0
Cumulative Planets Per Star
1.000
0.100
0.010
P = 0.5-1.7 Days
P = 1.7-5.5 Days
P = 5.5-18.2 Days
P = 18.2-60.3 Days
P = 60.3-200.0 Days
0.001
0.5
1.0
1.5
2.0
2.5
3.0
Planet Radius (REarth)
3.5
4.0
Figure 3.13: Planet occurrence (top) and cumulative planet occurrence (bottom) versus
planet radius for planets with periods of 0.5 − 1.7 days (dark green), 1.7 − 5.5 days (teal),
5.5 − 18.2 days (light blue), 18.2 − 60.3 days (navy), and 60.3 − 200 days (purple). The
error bars are based on binomial statistics and the assumed smoothing of the planet
population. In this figure and Figures 3.14–3.15 we do not present occurrence rates for
regions with pipeline sensitivity below 15%.
199
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Planets Per Star
0.1000
0.0100
0.0010
Rp = 0.5-1.0 REarth
Rp = 1.0-1.5 REarth
Rp = 1.5-2.0 REarth
Rp = 2.0-3.0 REarth
Rp = 3.0-4.0 REarth
0.0001
1
Cumulative Planets Per Star
1.000
10
Period (Days)
100
10
Period (Days)
100
Rp = 0.5-1.0 REarth
Rp = 1.0-1.5 REarth
Rp = 1.5-2.0 REarth
Rp = 2.0-3.0 REarth
Rp = 3.0-4.0 REarth
0.100
0.010
0.001
1
Figure 3.14: Planet occurrence (top) and cumulative planet occurrence (bottom) versus
orbital period for planets with radii of 0.5−1 R⊕ (black), 1−1.5 R⊕ (dark gray), 1.5−2.0 R⊕
(brown), 2 − 3 R⊕ (orange), and 3 − 4 R⊕ (red).
200
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.4. Number of Planets Per Star Versus Orbital Period (In Percentage)
Rp ( R⊕ )
0.5 − 1.7 Days
1.7 − 5.5 Days
5.5 − 18.2 Days
18.2 − 60.3 Days
60.3 − 200 Days
0.5 − 1.0
1.92+1.01
−0.64
(70%)
1.83+0.87
−0.58
(87%)
0.33+0.46
−0.16
(90%)
< 0.13
(90%)
< 0.11
(91%)
< 0.11
(91%)
< 0.11
(91%)
9.88+3.45
−2.46
(54%)
9.18+2.62
−1.98
(84%)
3.70+1.82
−1.18
(88%)
2.25+1.57
−0.88
(90%)
1.19+1.21
−0.54
(90%)
0.58+0.91
−0.29
(91%)
0.36+0.72
−0.18
(91%)
23.06+8.29
−5.57
(33%)
< 26.54
(79%)a
20.06+5.45
−4.04
(87%)
15.69+4.91
−3.55
(89%)
6.54+3.51
−2.18
(90%)
2.55+2.41
−1.13
(91%)
0.87+1.65
−0.43
(91%)
17.75+13.34
−6.54
(17%)
23.08+9.38
−6.02
(64%)
26.73+8.99
−6.08
(84%)
23.65+8.81
−5.83
(88%)
10.42+6.68
−3.74
(89%)
2.44+4.14
−1.19
(90%)
0.53+2.31
−0.17
(90%)
—
(6%)
30.70+26.67
−10.54
(41%)
18.90+14.67
−6.99
(74%)
14.12+11.55
−5.51
(84%)
5.30+7.85
−2.52
(88%)
1.83+5.32
−0.80
(89%)
< 2.71
(90%)
1.38+0.93
−0.53
(66%)
1.95+0.93
−0.61
(86%)
0.41+0.51
−0.20
(89%)
< 0.13
(90%)
< 0.11
(90%)
< 0.12
(91%)
< 0.11
(91%)
8.42+3.53
−2.39
(44%)
9.94+2.82
−2.13
(83%)
4.15+1.94
−1.28
(88%)
2.72+1.73
−1.01
(89%)
1.59+1.39
−0.69
(90%)
0.65+1.00
−0.32
(91%)
0.38+0.77
−0.19
(91%)
20.59+8.70
−5.57
(26%)
< 26.63
(75%)
< 23.58
(86%)
18.73+5.39
−3.95
(89%)
8.29+3.96
−2.55
(90%)
3.25+2.72
−1.37
(90%)
1.05+1.82
−0.52
(91%)
—
(11%)
26.85+10.70
−6.79
(58%)
24.59+9.08
−6.00
(82%)
27.58+9.42
−6.31
(87%)
14.51+7.71
−4.62
(89%)
3.37+4.62
−1.62
(90%)
0.56+2.32
−0.19
(90%)
—
(3%)
28.85+28.66
−10.34
(32%)
19.98+16.07
−7.42
(67%)
18.08+13.21
−6.57
(82%)
8.61+9.78
−3.84
(87%)
1.97+5.87
−0.85
(89%)
< 2.34
(89%)
1.0 − 1.5
1.5 − 2.0
2.0 − 2.5
2.5 − 3.0
3.0 − 3.5
3.5 − 4.0
0.5 − 1.0
1.0 − 1.5
1.5 − 2.0
2.0 − 2.5
2.5 − 3.0
3.0 − 3.5
3.5 − 4.0
Note. — In this table and all subsequent occurrence rate tables, the numbers in parentheses are the
fraction of injected planets that were recovered within the given intervals. In addition, the first set of
entries are our estimates of the planet occurrence rate when using the stellar properties in the Huber et al.
(2014) catalog. The second set of entries (below the double line) are alternative estimates constructed by
revising the stellar radii to lie along an empirical temperature/radius relation from Mann et al. (2013b,
see Section 3.7.4 for details).
a
We provide one-sigma upper limits instead of two-sided errors for grid cells with large Poisson errors
and very low occurrence rates.
201
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.5. Cumulative Number of Planets Per Star Versus Orbital Period
(In Percentage)
Rp ( R⊕ )
0.5 − 10 Days
0.5 − 50 Days
0.5 − 100 Days
0.5 − 150 Days
0.5 − 200 Days
0.5 − 1.0
22.08+4.88
−3.79
(54%)
51.43+7.99
−6.02
(44%)
54.73+8.21
−6.13
(40%)
59.71+8.46
−6.21
(37%)
59.71+8.44
−6.21
(36%)
1.0 − 1.5
21.18+3.84
−3.11
(84%)
56.25+6.31
−5.01
(79%)
65.09+6.52
−5.05
(75%)
82.17+6.07
−4.24
(72%)
88.74+5.38
−3.42
(70%)
1.5 − 2.0
10.76+3.23
−2.40
(89%)
45.95+7.05
−5.47
(87%)
57.35+7.59
−5.76
(86%)
66.58+7.78
−5.72
(85%)
69.72+7.77
−5.63
(84%)
2.0 − 2.5
8.68+3.21
−2.26
(90%)
35.62+7.02
−5.35
(89%)
49.81+8.18
−6.13
(89%)
54.59+8.46
−6.26
(88%)
55.72+8.52
−6.29
(88%)
2.5 − 3.0
3.89+2.30
−1.38
(90%)
15.46+5.41
−3.79
(90%)
21.97+6.87
−4.86
(90%)
23.19+7.14
−5.05
(90%)
23.45+7.20
−5.09
(90%)
3.0 − 3.5
1.73+1.64
−0.77
(91%)
4.65+3.10
−1.75
(90%)
7.15+4.32
−2.53
(90%)
7.38+4.44
−2.60
(90%)
7.41+4.45
−2.61
(90%)
3.5 − 4.0
0.72+1.08
−0.36
(91%)
1.63+1.86
−0.77
(91%)
2.11+2.31
−0.98
(91%)
2.27+2.45
−1.05
(90%)
2.27+2.46
−1.05
(90%)
0.5 − 1.0
18.39+5.02
−3.74
(49%)
39.93+8.22
−6.08
(38%)
40.60+8.31
−6.14
(34%)
40.60+8.28
−6.12
(31%)
40.60+8.27
−6.12
(30%)
1.0 − 1.5
22.85+4.08
−3.31
(82%)
60.76+6.57
−5.13
(75%)
68.38+6.65
−5.06
(71%)
85.73+5.85
−3.89
(68%)
91.47 ± 5.04
(66%)
1.5 − 2.0
11.96+3.42
−2.57
(88%)
45.87+7.07
−5.48
(86%)
57.26+7.65
−5.80
(85%)
66.74+7.86
−5.76
(83%)
70.23+7.85
−5.66
(82%)
2.0 − 2.5
9.98+3.46
−2.48
(90%)
42.48+7.42
−5.67
(89%)
58.49+8.27
−6.13
(88%)
65.45+8.39
−6.07
(88%)
67.12+8.40
−6.03
(87%)
2.5 − 3.0
4.94+2.58
−1.63
(90%)
20.60+6.17
−4.44
(90%)
29.85+7.79
−5.63
(90%)
32.40+8.21
−5.92
(89%)
33.01+8.31
−5.99
(89%)
3.0 − 3.5
2.09+1.84
−0.90
(91%)
6.32+3.73
−2.22
(90%)
8.67+4.74
−2.88
(90%)
9.13+4.94
−3.02
(90%)
9.25+4.99
−3.05
(90%)
3.5 − 4.0
3.5 − 4.0
0.83+1.19
−0.41
(91%)
1.92+2.08
−0.89
(91%)
2.14+2.27
−0.98
(90%)
2.20+2.33
−1.01
(90%)
2.21+2.34
−1.02
(90%)
Note. — As in Table 3.4, the entries below the double horizontal line are the estimates based on the
revised stellar radii (see Section 3.7.4).
202
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.6. Number of Planets Per Star Versus Insolation (In Percentage)
Rp ( R⊕ )
0.2 − 1.1 F⊕
1.1 − 6.3 F⊕
6.3 − 35.6 F⊕
35.6 − 200 F⊕
0.5 − 1.0
—
(11%)
22.63+11.24
−6.65
(24%)
23.48+6.45
−4.71
(44%)
3.52+1.67
−1.10
(60%)
1.0 − 1.5
24.02+19.77
−8.69
(47%)
< 35.71
(69%)
13.45+3.68
−2.78
(81%)
5.36+1.65
−1.24
(85%)
1.5 − 2.0
17.70+11.70
−6.18
(75%)
< 31.17
(85%)
10.59+3.41
−2.49
(88%)
1.38+0.93
−0.53
(89%)
2.0 − 2.5
13.97+9.54
−5.07
(85%)
20.16+6.80
−4.73
(88%)
9.84+3.34
−2.40
(89%)
0.75+0.81
−0.35
(90%)
2.5 − 3.0
5.47+6.67
−2.53
(88%)
8.90+5.04
−3.02
(90%)
3.94+2.28
−1.38
(90%)
0.42+0.62
−0.21
(90%)
3.0 − 3.5
1.71+4.48
−0.78
(89%)
2.18+3.15
−1.06
(90%)
1.71+1.65
−0.77
(91%)
0.27+0.53
−0.13
(91%)
3.5 − 4.0
< 2.33
(90%)
0.80+2.28
−0.36
(91%)
0.57+1.14
−0.28
(91%)
0.19+0.44
−0.09
(91%)
0.5 − 1.0
—
(6%)
10.46+9.36
−4.32
(16%)
17.22+6.10
−4.22
(32%)
3.23+1.75
−1.09
(52%)
1.0 − 1.5
26.87+23.57
−9.63
(37%)
29.27+9.12
−6.26
(63%)
13.19+3.79
−2.83
(78%)
5.84+1.75
−1.32
(84%)
1.5 − 2.0
18.99+13.05
−6.69
(71%)
24.54+7.34
−5.21
(83%)
11.35+3.63
−2.64
(87%)
1.79+1.07
−0.64
(88%)
2.0 − 2.5
15.29+10.41
−5.49
(83%)
22.53+7.21
−5.05
(88%)
10.93+3.59
−2.60
(89%)
1.05+0.94
−0.46
(90%)
2.5 − 3.0
7.68+7.80
−3.36
(87%)
11.83+5.75
−3.63
(89%)
4.40+2.46
−1.51
(90%)
0.47+0.65
−0.23
(90%)
3.0 − 3.5
1.90+4.89
−0.87
(89%)
2.70+3.34
−1.28
(90%)
2.11+1.83
−0.91
(90%)
0.20+0.47
−0.10
(91%)
3.5 − 4.0
< 1.81
(90%)
0.76+2.22
−0.33
(90%)
0.75+1.27
−0.37
(91%)
0.19+0.45
−0.09
(91%)
Note. — As in Table 3.4, the entries below the double horizontal line are the estimates based on the
revised stellar radii (see Section 3.7.4).
203
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.7. Cumulative Number of Planets Per Star Versus Insolation
(In Percentage)
Rp ( R⊕ )
0.2 − 200 F⊕
1.0 − 200 F⊕
10.0 − 200 F⊕
50.0 − 200 F⊕
100.0 − 200 F⊕
150.0 − 200 F⊕
0.5 − 1.0
63.36+8.82
−6.34
(34%)
49.63+8.11
−6.09
(41%)
17.79+4.57
−3.46
(51%)
2.09+1.20
−0.74
(58%)
0.61+0.61
−0.28
(61%)
0.06+0.25
−0.02
(62%)
1.0 − 1.5
74.67+6.59
−4.85
(70%)
50.65+6.18
−4.94
(77%)
15.00+3.06
−2.46
(83%)
3.50+1.30
−0.93
(85%)
1.00+0.67
−0.39
(85%)
0.16+0.31
−0.08
(86%)
1.5 − 2.0
57.61+7.57
−5.75
(84%)
39.92+6.49
−5.09
(87%)
6.60+2.31
−1.66
(88%)
0.90+0.74
−0.38
(89%)
0.23+0.38
−0.11
(89%)
0.04+0.20
−0.01
(89%)
2.0 − 2.5
44.72+7.55
−5.76
(88%)
30.74+6.17
−4.77
(89%)
6.41+2.50
−1.74
(90%)
0.33+0.56
−0.17
(90%)
0.02+0.19
−0.00
(90%)
0.00+0.09
−0.00
(90%)
2.5 − 3.0
18.72+5.92
−4.22
(90%)
13.26+4.62
−3.26
(90%)
2.85+1.74
−1.03
(90%)
0.23+0.46
−0.11
(90%)
0.04+0.22
−0.01
(90%)
—
(90%)
3.0 − 3.5
5.87+3.57
−2.10
(90%)
4.16+2.72
−1.56
(90%)
1.28+1.23
−0.57
(91%)
0.19+0.45
−0.09
(91%)
0.02+0.19
−0.00
(91%)
—
(91%)
3.5 − 4.0
2.03+2.21
−0.94
(90%)
1.56+1.78
−0.73
(91%)
0.49+0.79
−0.24
(91%)
0.18+0.43
−0.09
(91%)
—
(91%)
—
(91%)
0.5 − 1.0
34.08+7.85
−5.78
(28%)
30.91+7.35
−5.43
(34%)
15.13+4.79
−3.46
(44%)
2.03+1.32
−0.77
(52%)
0.54+0.61
−0.25
(55%)
0.05+0.23
−0.01
(57%)
1.0 − 1.5
75.17+6.91
−5.00
(64%)
48.30+6.31
−5.02
(73%)
15.61+3.18
−2.55
(81%)
3.95+1.39
−1.01
(83%)
1.30+0.77
−0.47
(84%)
0.22+0.36
−0.11
(84%)
1.5 − 2.0
56.66+7.72
−5.84
(81%)
37.67+6.39
−5.00
(85%)
7.26+2.43
−1.77
(88%)
1.14+0.84
−0.46
(88%)
0.28+0.43
−0.14
(88%)
0.04+0.21
−0.01
(88%)
2.0 − 2.5
49.80+7.75
−5.89
(87%)
34.51+6.47
−5.01
(89%)
7.31+2.68
−1.90
(89%)
0.54+0.69
−0.26
(90%)
0.02+0.21
−0.00
(90%)
—
(90%)
2.5 − 3.0
24.38+6.81
−4.94
(89%)
16.71+5.27
−3.79
(90%)
3.17+1.86
−1.13
(90%)
0.35+0.56
−0.17
(90%)
0.05+0.25
−0.01
(90%)
—
(90%)
3.0 − 3.5
6.91+3.91
−2.36
(90%)
5.01+3.03
−1.79
(90%)
1.51+1.38
−0.67
(91%)
0.17+0.43
−0.08
(91%)
0.03+0.22
−0.00
(91%)
—
(91%)
3.5 − 4.0
1.93+2.03
−0.89
(90%)
1.69+1.82
−0.78
(91%)
0.62+0.92
−0.31
(91%)
0.18+0.44
−0.09
(91%)
< 0.01
(91%)
0.00+0.09
−0.00
(91%)
Note. — As in Table 3.4, the entries below the double horizontal line are the estimates based on the revised stellar radii
(see Section 3.7.4).
204
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
3.7.2
Dependence on Planet Radius & Insolation
In Figure 3.15 and Tables 3.6-3.7, we present the planet occurrence rate as a function of
stellar insolation and planet radius. In general, we find that planet occurrence increases
both with decreasing planet radius (as discussed in Section 3.7.1) and with decreasing
log10 FP . Intriguingly, we observed that the occurrence rates of Earths, Super-Earths,
and mini-Neptunes are comparable for insolations below roughly 30 F⊕ but that planets
larger than 1.5 R⊕ were less common than smaller planets at insolations above 30 F⊕.
The error bars on the coolest insolation bin are rather large, but the divergence of the
Earth and Neptune occurrence relations might be due to photo-evaporation at short
orbital periods.
Across the size range we considered, the planet occurrence rate versus log10 FP rises
with decreasing insolation between 100 − 10 F⊕ and appears roughly flat in log10 FP
between 10 − 0.2 F⊕ . Figure 3.15 hints that the planet occurrence rate might increase
again at cooler insolations, but a larger sample of long period planets will be required to
test that hypothesis.
3.7.3
The Occurrence of Potentially Habitable Planets
The shaded regions in Figure 3.15 display one choice of habitable zone boundaries,
but the definition of the “habitable zone” (HZ) is still rather uncertain. Traditionally,
astronomers have used the term to refer to the distance from the star at which liquid
water could be present on the surface of a planet (Dole 1964; Hart 1979; Kasting et al.
1993). In theory, there could also be water-based life on worlds with subsurface oceans
or non-water-based life on worlds like Titan, but any associated biosignatures would
205
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
be difficult to interpret remotely. Accordingly, astronomers have concentrated thus far
on the search for life as we know it, meaning surface-based lifeforms that depend on
liquid water and might generate biosignatures that could alter the composition of their
homeworld’s atmosphere.
Even within that rather narrow definition, there are many assumptions that can
affect the choice of habitable zone boundaries. In particular, the assumed mass and
composition of the planet’s atmosphere affects the surface pressure and therefore the
temperature range at which water would be liquid (e.g., Vladilo et al. 2013). For instance,
Pierrehumbert & Gaidos (2011) showed that planets with thick hydrogen atmospheres
would have sufficient surface pressure to retain surface liquid water out to distances of
2.4 AU. It is uncertain whether biosignatures would be detectable in such atmospheres
(Seager et al. 2013; Hu et al. 2013), but those worlds could still be habitable.
The presence of clouds adds an additional complication by both cooling and heating
the planet. Clouds are particularly important in the case of tidally-locked planets, which
might be a common fate for planets orbiting within the habitable zones of M dwarfs.
Yang et al. (2013) demonstrated that a tidally-locked M dwarf planet might develop a
persistent cloud patch above the sub-stellar point. That cloud patch would have a higher
albedo than the planetary surface and would allow the planet to be much cooler at a
given separation than a cloud-free model would predict. As a result, the habitable zone
for a cloudy, tidally-locked planet could extend to insolations as high as FP = 1.76 F⊕
for the moist greenhouse limit rather than the limit of FP < 0.88 F⊕ calculated by
Kopparapu et al. (2013a) for a cloud-free model. Even for non-tidally-locked planets,
the presence of clouds can expand the distances corresponding to the boundaries of the
habitable zone by roughly 40% depending on the degree of cloud cover (Selsis et al.
206
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
2007).
The orbital geometry of exoplanets is also important when assessing planetary
habitability. For instance, planets with high obliquities or eccentric orbits might be
partially habitable at certain latitudes or during certain times of year (Williams &
Kasting 1997; Williams & Pollard 2002; Spiegel et al. 2008, 2009; Dressing et al. 2010;
Cowan et al. 2012; Dobrovolskis 2013; Armstrong et al. 2014; Linsenmeier et al. 2015).
Depending on the timescale for the temperature of the planet to change (which depends
on factors such as the fraction of surface covered by ocean), such planets may undergo
periodic global glaciations punctuated by short-lived epochs during which the surface is
warm enough for liquid water (Pierrehumbert 2005; Spiegel et al. 2010). In addition, the
primordial obliquities of close-in planets orbiting M dwarfs may be significantly eroded
by tides (Heller et al. 2011).
Constructing a multi-dimensional habitable zone model for planets orbiting M dwarfs
is beyond the scope of this paper, but we aspire to provide enough information so that
other researchers can assess the abundance of planets within their chosen habitable zone
boundaries. We therefore provide occurrence rates for a few possible choices of habitable
zone boundaries in Table 3.8.
[tbp]
In the most conservative case, we adopt the maximum greenhouse (Max GH) and
moist greenhouse (Moist GH) insolation limits from Kopparapu et al. (2013b). The Max
GH limit is the insolation at which adding additional CO2 can no longer heat the surface
of the planet because Rayleigh scattering begins to dominate over the greenhouse effect.
At the inner edge, the Moist GH limit corresponds to the insolation at which the planet’s
207
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.8. Habitable Zone Occurrence Rates (In Percentage)
FP ( F⊕ )
Outer HZ:
Inner HZ:
0.25 − 0.88
Max GHa
Moist GHa
0.23 − 1.54
Early Marsa
Recent Venusa
0.25 − 4.00
Fixedb
Fixedb
0.5 − 1.0 R⊕
—
(13%)
—
(14%)
25.28+14.90
−7.96
(18%)
15.75+15.34
−6.41
(15%)
20.61+15.15
−7.37
(17%)
27.56+13.70
−7.86
(20%)
0.8 − 1.0 R⊕
4.63+15.19
−1.78
(21%)
13.09+16.88
−5.73
(22%)
20.95+15.07
−7.41
(28%)
13.30+15.31
−5.72
(24%)
17.23+14.71
−6.66
(26%)
20.05+13.68
−6.99
(30%)
1.0 − 1.5 R⊕
15.82+16.60
−6.54
(48%)
24.28+17.58
−8.39
(50%)
46.77+12.33
−8.12
(56%)
21.82+15.03
−7.54
(52%)
31.65+13.20
−8.00
(55%)
46.90+11.00
−7.54
(59%)
1.5 − 2.0 R⊕
11.54+9.97
−4.67
(75%)
20.69+10.80
−6.32
(76%)
36.07+10.00
−6.90
(79%)
21.48+10.25
−6.22
(77%)
28.04+10.10
−6.62
(78%)
39.35+9.32
−6.64
(80%)
2.0 − 2.5 R⊕
10.25+8.58
−4.14
17.09+9.48
−5.50
29.17+9.30
−6.33
18.23+9.27
−5.56
23.99+9.31
−6.06
30.97+8.93
−6.23
+6.66
5.03−2.37
+7.73
10.58−4.04
+7.78
16.61−4.87
+7.62
11.09−4.10
+7.71
13.79−4.53
(89%)
(89%)
(90%)
(89%)
(89%)
17.96−4.89
(90%)
1.0 − 2.0 R⊕
+15.77
27.36−8.40
(61%)
+15.52
44.97−9.29
(63%)
82.84+8.99
−5.33
(68%)
43.31+14.02
−8.74
(64%)
59.68+12.18
−7.96
(67%)
86.25+7.51
−4.45
(69%)
2.0 − 3.0 R⊕
13.88−5.03
(86%)
24.44−6.53
(87%)
+10.44
41.17−6.96
(87%)
25.97−6.55
(87%)
33.87−6.88
(87%)
+10.11
44.16−6.76
(88%)
3.0 − 4.0 R⊕
1.40+4.46
−0.58
(90%)
3.23+5.08
−1.57
(90%)
4.61+5.04
−2.10
(90%)
3.34+5.00
−1.62
(90%)
3.91+5.02
−1.85
(90%)
4.77+4.87
−2.13
(90%)
2.0 − 4.0 R⊕
15.28+9.78
−5.33
(88%)
+10.74
27.67−6.86
(88%)
+10.02
45.78−7.07
(89%)
+10.51
29.32−6.86
(88%)
+10.32
37.78−7.09
(89%)
48.93+9.50
−6.83
(89%)
0.5 − 1.4 R⊕
19.85+18.42
−7.70
(28%)
37.34+20.29
−10.30
(30%)
67.77+13.06
−7.97
(35%)
35.60+18.05
−9.67
(31%)
49.48+15.53
−9.34
(33%)
69.67+11.57
−7.32
(37%)
0.5 − 1.0 R⊕
—
(8%)
—
(9%)
—
(12%)
—
(10%)
—
(11%)
—
(14%)
0.8 − 1.0 R⊕
—
(14%)
4.49+10.81
−2.03
(15%)
8.55+11.74
−3.93
(20%)
5.54+11.98
−2.54
(16%)
7.76+12.30
−3.61
(18%)
9.12+11.70
−4.14
(22%)
1.0 − 1.5 R⊕
19.97+20.84
−7.94
(38%)
28.20+21.29
−9.56
(41%)
49.46+14.16
−8.85
(48%)
26.25+18.49
−8.84
(43%)
35.71+15.79
−9.06
(46%)
48.68+12.74
−8.30
(51%)
1.5 − 2.0 R⊕
13.17+11.23
−5.25
(69%)
+11.89
20.28−6.59
(71%)
+10.60
34.10−7.11
(75%)
+11.08
20.32−6.36
(72%)
+10.73
26.12−6.76
(74%)
36.79+9.76
−6.80
(76%)
2.0 − 2.5 R⊕
11.62+9.53
−4.61
(83%)
18.38+10.29
−5.90
(83%)
32.50+9.85
−6.71
(85%)
19.41+9.97
−5.92
(84%)
26.23+9.91
−6.44
(84%)
34.48+9.41
−6.58
(85%)
2.5 − 4.0 R⊕
7.07+7.75
−3.17
(88%)
12.65+8.64
−4.62
(89%)
21.53+8.61
−5.61
(89%)
13.42+8.52
−4.72
(89%)
17.56+8.59
−5.26
(89%)
23.29+8.34
−5.61
(89%)
1.0 − 2.0 R⊕
33.14+18.53
−9.60
(54%)
48.48+17.89
−10.06
(56%)
83.55+9.92
−5.51
(61%)
46.57+16.20
−9.54
(57%)
61.83+13.69
−8.47
(60%)
85.47+8.53
−4.86 )
(63%)
2.0 − 3.0 R⊕
17.14+10.68
−5.84
(85%)
28.28+11.42
−7.16
(85%)
49.13+10.35
−7.24
(86%)
29.93+11.10
−7.13
(86%)
39.87+10.78
−7.35
(86%)
52.58+9.75
−6.95
(87%)
3.0 − 4.0 R⊕
1.54+4.72
−0.66
(89%)
2.75+5.11
−1.33
(89%)
4.89+5.08
−2.20
(90%)
2.89+5.02
−1.41
(89%)
3.92+5.05
−1.86
(90%)
5.18+4.95
−2.26
(90%)
2.0 − 4.0 R⊕
+10.95
18.69−6.13
(87%)
31.03+11.64
−7.40
(87%)
54.03+10.29
−7.20
(88%)
32.83+11.32
−7.35
(88%)
43.79+10.88
−7.46
(88%)
57.77+9.65
−6.85
(88%)
0.5 − 1.4 R⊕
18.48+20.46
−7.54
(21%)
29.91+21.56
−9.83
(22%)
53.10+14.77
−9.06
(27%)
28.96+19.24
−9.33
(24%)
39.97+16.65
−9.52
(26%)
53.79+13.47
−8.57
(30%)
10.25+8.58
−4.14
2.5 − 4.0 R⊕
+9.47
17.09+9.48
−5.50
0.25 − 1.76
Max GHa
Cloudy Moist GHc
29.17+9.30
−6.33
+9.91
0.25 − 2.78
Max GHa
Desert (a=0.2)d
18.23+9.27
−5.56
+10.21
23.99+9.31
−6.06
0.25 − 5.85
Max GHa
Desert (a=0.8)d
30.97+8.93
−6.23
+7.52
+9.42
Note. — As in Table 3.4, the entries below the double horizontal line are the estimates based on the revised stellar radii (see Section 3.7.4).
a
These habitable zone limits are from Kopparapu et al. (2013b).
b
These are the “simple” habitable zone boundaries adopted by Petigura et al. (2013a).
c
This limit approximates the effect of clouds (Yang et al. 2014) by increasing the flux at the inner edge of the habitable zone a factor of
two compared to the baseline calculation by Kopparapu et al. (2013b).
d
For these calculations, the inner edge of the habitable zone was set to the values predicted by Zsom et al. (2013) for hot “desert” worlds
with low relative humidity.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
stratosphere becomes dominated by water vapor. At that point, the planet’s reservoir of
hydrogen quickly escapes to space.
Table 3.8 also provides estimates based on the assumption that Venus and Mars
were habitable at earlier times in their histories. For a Sun-like star, those constraints
correspond to insolation limits of 1.776 F⊕ and 0.321 F⊕ , respectively (Kopparapu et al.
2013b,a). The insolation boundaries are lower for planets orbiting M dwarfs because
the incoming radiation is redder. For a typical star in our sample (Teff = 3748K), the
boundaries are 1.543 F⊕ and 0.228 F⊕, respectively.
Even more optimistically, Table 3.8 includes HZ occurrence rates using the cloudy
inner HZ from Yang et al. (2014), two choices of desert world albedos from Zsom et al.
(2013), and the convenient limits of 0.25 − 4 F⊕ used by Petigura et al. (2013a). We
do not provide an estimate based on the hydrogen atmosphere HZ of Pierrehumbert &
Gaidos (2011) because our search completeness is very low at the maximum allowed
separation of 2.4 AU.
The appropriate radius range to consider for a potentially habitable planet is more
clearly defined than the appropriate insolation range. Based on radial velocity follow-up
observations of Kepler planet candidates, Rogers (2015) argued that the majority of
planets larger than 1.6 R⊕ contain too many volatiles to be rocky. This result agrees
with previous fits to measured exoplanet masses and radii by Weiss & Marcy (2014) and
simulations by Lopez & Fortney (2014). Furthermore, Dressing et al. (2015) found that
all five exoplanets smaller than 1.6 R⊕ with masses and radii measured to a precision
better than 20% have densities consistent with an Earth-like mixture of iron and silicates.
Like Rogers (2015), they noted that planets larger than 1.6 R⊕ radii have densities
209
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
inconsistent with rocky compositions. Due to observational constraints, the population
of planets with well-constrained densities is strongly biased towards highly irradiated
planets. We therefore include a broader range of radius choices in Table 3.8 to account
for the possibility that the transition between rocky and gaseous planets might occur at
a slightly different radius for less irradiated planets. For instance, the 2.35 R⊕ exoplanet
Kepler-10c has a measured mass of 17.2 ± 1.9 M⊕ and a bulk density of 7.1 ± 1g cm−3 ,
higher than the densities of most 2 − 3 R⊕ planets (Dumusque et al. 2014).
Adopting the most conservative assumptions (1.0 R⊕ < RP < 1.5 R⊕ , outer HZ =
Max GH, inner HZ = Moist GH), we estimate an occurrence rate of 0.16+0.17
−0.07 potentially
habitable 1 − 1.5 R⊕ planets per M dwarf. The predicted occurrence rates of super-Earths
(1.5 − 2.0 R⊕ ) and larger planets (2 R⊕ < RP < 4 R⊕ ) within the same habitable
+0.10
zone boundaries are 0.12+0.10
−0.05 super-Earths and 0.15−0.05 larger planets per M dwarf.
Expanding the radius range to 1 − 2 R⊕ or increasing the habitable zone boundaries
to the limits for recent Venus and early Mars increases the assumed occurrence rate
+0.18
to 0.27+0.16
−0.08 and 0.24−0.08 , respectively. In the most optimistic case, we estimate an
occurrence rate of 0.86+0.08
−0.04 desert worlds with albedos of 0.8 and radii of 1 − 2 R⊕
receiving insolations between the Zsom et al. (2013) inner limit and the Max GH outer
limit. The assumed occurrence rate of potentially habitable M dwarf planets therefore
varies by a factor of four depending on the specific choice of radius and insolation
boundaries.
The range of HZ possibilities in Table 3.8 is particularly useful for comparing our
results to those of previous studies. For instance, Petigura et al. (2013a) estimated
that 22% of FGK stars host 1 − 2 R⊕ planets receiving 0.25 − 4 F⊕ . Within the same
boundaries, we find an occurrence rate of 0.83+0.09
−0.05 small planets per M dwarf HZ.
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CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
The difference in our estimates might suggest that habitable zone planets are more
common around lower-mass stars, but the Petigura et al. (2013a) prediction is based on
an extrapolation of the occurrence rate for shorter period planets to longer period orbits
assuming that planet occurrence is flat in log P . If the planet occurrence rate actually
increases with log P at longer periods, then perhaps the occurrence rates of potentially
habitable planets orbiting FGK and M dwarfs are more similar. Such a change in the
slope of the FGK star planet occurrence rate at longer periods could be explained by the
radial- and temperature-dependence of the physics governing planet formation.
3.7.4
Implications of Systematic Biases in Modeled Stellar
Radii
The stellar parameters for the majority of the stars in our sample were estimated by
fitting Dartmouth stellar models to photometric (Dressing & Charbonneau 2013; Gaidos
2013; Huber et al. 2014) or spectroscopic (Mann et al. 2012; Muirhead et al. 2012a)
observations. The exceptions are one star with parameters from Mann et al. (2013b) and
two very-low mass stars with parameters from Martı́n et al. (2013). Both Mann et al.
(2013b) and Martı́n et al. (2013) estimated stellar radii using empirical relations based
on interferometric observations of low-mass stars (Boyajian et al. 2012).
Several recent studies (e.g., Boyajian et al. 2012; Mann et al. 2013a; Newton et al.
2015) have demonstrated that theoretical stellar models do not accurately reproduce the
observed radii of low-mass stars. As explained in Newton et al. (2015), there are two
main issues:
211
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Planets Per Star
0.1000
0.0100
0.0010
Rp = 0.5-1.0 REarth
Rp = 1.0-1.5 REarth
Rp = 1.5-2.0 REarth
Rp = 2.0-3.0 REarth
Rp = 3.0-4.0 REarth
0.0001
100
Cumulative Planets Per Star
1.000
10
Insolation (FEarth)
1
10
Insolation (FEarth)
1
Rp = 0.5-1.0 REarth
Rp = 1.0-1.5 REarth
Rp = 1.5-2.0 REarth
Rp = 2.0-3.0 REarth
Rp = 3.0-4.0 REarth
0.100
0.010
0.001
100
Figure 3.15: Planet occurrence (top) and cumulative planet occurrence (bottom) versus
insolation for planets with radii of 0.5 − 1 R⊕ (crimson), 1 − 1.5 R⊕ (orange), 1.5 − 2.0 R⊕
(brown), 2 − 3 R⊕ (dark gray), and 3 − 4 R⊕ (black). As in Figure 3.6, the vertical lines
mark two definitions of the habitable zone.
212
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
1. The radii of model stars with Teff < 4000K are smaller than the interferometricallymeasured radii by approximately 0.04 − 0.09 R$ .
2. Variations in metallicity produce significant changes in the modeled radii of
low-mass stars whereas observations reveal that metallicity actually has little
influence on the radii of low-mass stars.
Although the Dartmouth stellar models perform better than many alternative models,
both of these effects may have caused the radii of the stars in our sample to be
systematically underestimated. In order to gauge the magnitude of this effect, we
recalculated the radii for our stellar sample using an empirical temperature/radius
relation for main sequence stars with 3300K < Teff (Equation 6 in Mann et al. 2013a
with the additional significant figures reported by Newton et al. 2015). We did not
consider changes in the stellar temperatures, but Newton et al. (2015) demonstrated
that the temperatures we estimated in Dressing & Charbonneau (2013) were consistent
with predictions based on empirical observations (our values were lower by 40 ± 110K).
Using the empirical temperature/radius relation to revise the radii of the 2437 stars in
our sample with Teff > 3300, we found that the median change in radius (∆R∗ ) was an
increase of 0.026 R$ (6%). The change was highly dependent on the assumed metallicity;
stars with assigned [Fe/H]≤ −0.5 displayed a median size increase of 0.05 R$ (11%)
while the estimated radii of stars with assigned [Fe/H] ≥ 0 shrank by 0.016 R$ (5%).
For the planet host stars in our sample, the increase in the stellar radii leads to
larger predicted radii and increased insolation fluxes for the associated planet candidates.
The median planet radius increase was 6.6%, but the amplitude of the change varied
considerably. The systems most strongly affected by the revision of the stellar radii
213
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
were: KOI 3102 (+48%), KOI 2650 (+26%), KOI 2418 (+20%), KOI 2006 (+17%), and
KOI 812 (+16%). Two of the KOIs in these systems (KOI 3102.01 and KOI 2650.02)
were missed by our planet detection pipeline so they did not enter into our calculation of
the planet occurrence rate.
In addition to altering the radius estimates for the detected planet candidates, the
changes in the stellar radii affect the estimated survey completeness and, in turn, the
derived occurrence rate. If the stellar radii are typically 6% larger, then the search
completeness we displayed in Figure 3.7 for planets between 0.50 − 4.00 R⊕ actually
corresponds to 0.53 − 4.24 R⊕ planets. We accounted for this effect by generating new
search completeness maps following the procedure outlined in Section 3.6 but correcting
the radii and insolation flux environments of the injected planets to reflect the new
radius estimates for each star. We then recalculated the planet occurrence rate using the
updated search completeness maps and the revised planet properties. We present the
corresponding planet occurrence maps in Figure 3.16 and include the resulting planet
occurrence rates below the double horizontal lines in Tables 3.4–3.8.
As expected, the most noticeable difference between the occurrence maps displayed
in Figures 3.12 and 3.16 is that the ridge of high planet occurrence has moved upward
to large radii. Similarly, the region of low search completeness now encompasses a
slightly larger portion of our chosen parameter space. Using the revised stellar radii, we
+0.07
calculated occurrence rates of 0.61+0.07
−0.05 Earth-size planets and 0.46−0.05 super-Earths per
low-mass star with periods shorter than 50 days. These rates are nearly identical to the
estimates presented in Section 3.7. Within the habitable zone, we estimate a frequency
+0.11
of 0.20+0.21
−0.08 Earths and 0.13−0.05 super-Earths per star when adopting the moist GH
inner limit and the maximum GH outer limit from Kopparapu et al. (2013b). These
214
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Planet Occurrence (%, Mann R*)
4.0
0.000
(91%)
0.007
(91%)
0.18
(91%)
0.20
(91%)
0.45
(91%)
0.61
(91%)
0.41
(90%)
0.15
(90%)
0.16
(90%)
Planet Occurrence (%, Mann R*)
4.0
0.057
(91%)
0.14
(91%)
0.51
(91%)
1.4
(90%)
1.8
(90%)
1.4
(90%)
2.0
(90%)
1.5
(89%)
0.45
(88%)
Planet Radius (REarth)
Planet Radius (REarth)
(91%)
0.004
3.0
0.000
(91%)
0.004
(90%)
0.30
(90%)
1.3
(90%)
3.3
(90%)
4.9
(90%)
6.4
(89%)
8.1
(89%)
6.2
(88%)
2.4
(85%)
2.5
0.003
(90%)
0.012
(90%)
0.53
(89%)
2.2
(89%)
7.2
(89%)
11
(89%)
12
(88%)
14
(86%)
11
(84%)
6.7
(79%)
2.0
0.081
(89%)
0.33
(89%)
1.5
(88%)
2.6
(88%)
7.4
(87%)
13
(86%)
13
(83%)
10
(79%)
8.9
(73%)
11
(62%)
1.5
0.48
(86%)
1.5
(85%)
3.7
(83%)
6.2
(82%)
10
(78%)
12
(73%)
18
(65%)
7.9
(50%)
9.6
(39%)
(91%)
0.17
(91%)
0.17
(91%)
0.42
(91%)
0.26
(90%)
0.37
(90%)
0.33
(90%)
0.12
0.004
0.045
0.14
0.49
1.1
1.1
1.2
1.3
0.91
0.002
0.082
0.001
0.046
0.072
(89%)
19
(91%)
(91%)
(91%)
(91%)
(90%)
(90%)
(90%)
(90%)
(89%)
0.60
(88%)
3.0
(90%)
0.33
(90%)
(90%)
0.81
(90%)
2.5
(90%)
3.8
(90%)
4.7
(89%)
6.0
(89%)
3.9
(88%)
2.2
(86%)
2.5
(90%)
0.77
(90%)
(90%)
1.7
(89%)
6.3
(89%)
7.8
(89%)
9.8
(88%)
10.0
(86%)
8.5
(84%)
4.8
(80%)
2.0
0.074
(89%)
0.40
0.99
(89%)
(88%)
2.8
(88%)
4.7
(87%)
11
(85%)
9.6
(83%)
9.2
(80%)
9.0
(73%)
8.7
(65%)
1.5
0.21
(27%)
(85%)
1.0
1.6
3.2
(84%)
(83%)
5.9
(81%)
5.9
(76%)
7.3
(72%)
16
(63%)
9.0
(53%)
5.7
(41%)
20
(34%)
1.0
0.30
(71%)
1.1
(61%)
3.7
(50%)
1
-3.00
-2.00
4.7
(40%)
8.6
(31%)
11
(21%)
10
Period (Days)
Log10 Occurrence
-4.00
(91%)
3.5
0.000
-5.00
0.015
(91%)
(89%)
3.5
0.5
0.001
0.043
0.5
Recovery < 15%
(63%)
0.77
1.8
(59%)
100
(49%)
100
Log10 Occurrence
-1.00
-5.00
-4.00
-3.00
-2.00
6.0
(39%)
8.5
(31%)
7.1
(24%)
2.2
(17%)
10
Insolation (FEarth)
Recovery < 15%
1
-1.00
Figure 3.16: Alternative calculation of the planet occurrence rate in period/planet radius
space (Left) and insolation/planet radius space (Right) using the revised stellar radii (see
Section 3.7.4). The annotations are the same as in Figure 3.12. As in Figure 3.6, the
vertical lines in the right panel mark two definitions of the habitable zone.
estimates are 26% and 14% higher, respectively, than the rates of 0.16+0.17
−0.07 Earths and
0.12+0.10
−0.05 super-Earths per HZ presented in Section 3.7. Although increasing the assumed
stellar radii alters the inferred occurrence rates, the dominant source of error is the
relatively small number of potentially habitable small planets.
3.8
Summary & Conclusions
In this paper, we presented an updated estimate of the planet occurrence rate for
early M dwarfs based on the full four-year Kepler data set. We developed our own
planet detection pipeline to search for transiting planets in the Kepler light curves. We
then characterized the completeness of our pipeline by injecting simulated transiting
planets into the Kepler light curves and attempting to recover them. Our search of the
light curves of 2543 small stars with at least 1000 days of Kepler photometry revealed
3215 possible planetary transits. We thoroughly inspected all available follow-up
215
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
observations for these objects and accounted for transit depth dilution for systems with
close stellar companions. We accepted 156 planet candidates, one of which was not a
previously known Kepler planet candidate.
We then measured the occurrence rate of small planets around small stars by
dividing smoothed maps of the detected planet population by maps of our pipeline
search completeness in radius-period space and radius-insolation space. We found that
Earth-sized planets (1.0 − 1.5 R⊕ ) are common and calculated an occurrence rate of
0.56+0.06
−0.05 Earth-sized planets with periods shorter than 50 days per early M dwarf. We
also found an occurrence of 0.46+0.07
−0.05 super-Earths (1.5 − 2 R⊕ ) with periods shorter than
50 days per early M dwarf. For orbital periods shorter than 200 days and planet radii of
1 − 4 R⊕ , we estimated a cumulative planet occurrence rate of 2.5 ± 0.2 planets per M
dwarf.
Within a conservatively defined habitable zone based on the moist greenhouse and
maximum greenhouse limits (Kopparapu et al. 2013b,a) we estimated occurrence rates of
+0.10
0.16+0.17
−0.07 Earth-size (1.0 − 1.5 R⊕ ) planets and 0.12−0.05 (1.5 − 2.0 R⊕ ) super-Earths per
small star. Adopting a wider planet size range of 1 − 2 R⊕ and considering the effects of
clouds (Yang et al. 2013) increased our estimate to 0.43+0.14
−0.09 potentially habitable planets
per star. Considering desert worlds (Zsom et al. 2013) would increase the measured
occurrence rate to nearly one potentially habitable planet per M dwarf. These estimates
span the range of previous estimates of the occurrence rate of potentially habitable
M dwarf planets.
An order of magnitude calculation multiplying the occurrence rate of potentially
habitable 1 − 1.5 R⊕ planets between the empirical early Mars outer boundary and the
216
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
recent Venus inner boundary by an estimate of the number density of small stars in
the galaxy from the RECONS survey (Henry et al. 2006; Winters et al. 2015) therefore
suggests that the nearest potentially habitable planet is most likely 2.6 ± 0.4 pc away
and is within 3.5 pc with 95% confidence. This estimate assumes that the occurrence
rate of potentially habitable planets orbiting later M dwarfs is identical to that for early
M dwarfs, which is consistent with the results of Berta et al. (2013).
Correcting for the geometric probability of transit (assuming that 1.4% of potentially
habitable M dwarf planets transit), the nearest transiting potentially habitable planet
10
is likely to be 10.6+1.6
Early
−1.8 pc away and is within 14.6 pc with 95% confidence.
M dwarfs at distances of 3.5 pc and 14.6 pc would have apparent K band magnitudes
of 2.9 and 6.0, respectively, well within the magnitude range probed by current and
upcoming planet surveys of nearby, bright stars such as CARMENES (Quirrenbach et al.
2010), CHEOPS (Broeg et al. 2013), ExoplanetSat (Smith et al. 2010), ExTrA (Bonfils
et al. 2014), HPF (Mahadevan et al. 2010), MEarth (Nutzman & Charbonneau 2008;
Berta et al. 2012a), PLATO (Rauer et al. 2014), K2 (Howell et al. 2014), SPECULOOS
(Gillon et al. 2013a), SPIRou (Thibault et al. 2012), and TESS (Ricker et al. 2014).
10
These distance estimates are based on the mean number of planets per star rather than the fraction
of stars with planets. If potentially habitable planets are clustered such that M dwarfs hosting potentially
habitable planets typically feature more than one potentially habitable planet, then the distance estimates
will need to be increased to account for the relatively flat nature of multiplanet systems orbiting M dwarfs
(Ballard & Johnson 2014).
217
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.9. Candidates Accepted By Our Pipeline
Period (Days)
- Err
KID
KOI
Value
2161536
2556650
2715135
2973386
3426367
3642335
3749365
4061149
4139816
4139816
4139816
4139816
4172805
4832837
4832837
4913852
5364071
5364071
5364071
5364071
5384713
5531953
5531953
5531953
5531953
5617854
5640085
5640085
5794240
5809954
6382217
6382217
6435936
6497146
6666233
6679295
6773862
6867155
7021681
7021681
7094486
7135852
7287995
7287995
7287995
7304449
7447200
7447200
7455287
7455287
7455287
7603200
7603200
7603200
7870390
7870390
7870390
7871954
7871954
7907423
7907423
7907423
8013419
8018547
8120608
8120608
8120608
8120608
8120608
2130.01
2156.01
1024.01
3034.01
2662.01
3010.01
1176.01
1201.01
812.01
812.02
812.03
812.04
4427.01
605.01
605.02
818.01
248.01
248.02
248.03
248.04
3444.02
1681.00
1681.01
1681.02
1681.03
1588.01
448.01
448.02
254.01
1902.01
2036.01
2036.02
854.01
3284.01
2306.01
2862.01
1868.01
868.01
255.01
255.02
1907.01
875.01
877.01
877.02
877.03
1702.01
676.01
676.02
886.01
886.02
886.03
314.01
314.02
314.03
898.01
898.02
898.03
1515.01
1515.02
899.01
899.02
899.03
901.01
902.01
571.01
571.02
571.03
571.04
571.05
16.85595526
2.85234932
5.74773353
31.02089195
2.10434036
60.86661711
1.97376228
2.75759481
3.34022436
20.05993080
46.18428829
7.82527952
147.66174063
2.62811645
5.06549822
8.11438395
7.20387100
10.91273200
2.57654900
18.59610800
60.32665084
21.91384343
6.93911381
1.99281275
3.53105829
3.51749675
10.13962300
43.59579200
2.45524062
137.86453830
8.41102527
5.79529807
56.05318853
35.23301880
0.51240811
24.57535492
17.76080479
235.99802060
27.52199799
13.60335797
11.35011141
4.22097130
5.95489377
12.03993346
20.83776773
1.53818130
7.97251347
2.45323590
8.01026000
12.07238900
20.99569400
13.78113900
23.08871300
10.31236400
9.77042372
5.16981333
20.09010000
1.93703537
7.06117534
7.11369666
3.30656751
15.36834908
12.73263426
83.94017967
7.26737224
13.34295116
3.88677934
22.40788397
129.94188177
6.07E-05
3.29E-06
8.53E-06
2.46E-04
4.05E-06
4.91E-04
4.23E-07
6.97E-06
8.45E-06
2.82E-04
6.06E-04
1.24E-04
1.51E-03
2.03E-06
4.22E-05
1.25E-05
8.00E-06
2.10E-05
3.00E-06
7.90E-05
4.47E-05
1.83E-04
2.60E-05
7.47E-06
2.39E-05
3.96E-06
2.20E-05
1.25E-04
1.00E-07
2.87E-04
2.59E-05
3.24E-05
1.31E-03
2.24E-04
5.59E-07
1.37E-04
2.70E-05
3.78E-04
4.98E-05
2.02E-04
2.30E-05
3.02E-06
8.57E-06
2.85E-05
1.97E-04
3.09E-06
1.82E-06
4.75E-07
3.00E-05
1.00E-04
1.43E-04
1.10E-05
3.10E-05
3.60E-05
3.02E-05
2.29E-05
1.33E-04
2.77E-06
1.34E-05
2.66E-05
1.39E-05
1.08E-04
5.97E-06
1.49E-03
1.93E-05
4.25E-05
9.86E-06
1.20E-04
2.32E-03
+ Err
Value
5.96E-05
3.38E-06
8.58E-06
2.30E-04
4.03E-06
5.80E-04
4.24E-07
6.65E-06
8.40E-06
3.03E-04
6.48E-04
1.37E-04
2.16E-03
2.04E-06
4.37E-05
1.24E-05
8.00E-06
2.10E-05
3.00E-06
7.90E-05
4.47E-05
1.93E-04
2.60E-05
7.21E-06
2.40E-05
3.94E-06
2.20E-05
1.25E-04
1.00E-07
2.87E-04
2.61E-05
3.52E-05
1.26E-03
2.27E-04
5.53E-07
1.37E-04
2.62E-05
3.78E-04
4.96E-05
2.01E-04
2.33E-05
3.04E-06
8.63E-06
2.92E-05
1.94E-04
3.12E-06
1.82E-06
4.75E-07
3.00E-05
1.00E-04
1.43E-04
1.10E-05
3.10E-05
3.60E-05
3.20E-05
2.19E-05
1.40E-04
2.71E-06
1.38E-05
2.68E-05
1.42E-05
1.12E-04
6.01E-06
1.38E-03
1.98E-05
4.45E-05
9.88E-06
1.27E-04
2.34E-03
1.93
1.88
1.49
1.62
0.56
2.37
9.11
1.21
2.11
2.14
1.90
1.09
1.56
2.52
0.97
2.14
2.25
2.74
1.55
1.31
2.98
1.03
0.99
0.72
0.68
1.21
1.77
2.48
11.00
1.99
1.51
1.00
2.09
1.01
0.94
1.60
2.13
8.88
2.56
0.75
1.98
2.58
2.06
1.87
1.05
0.84
2.88
3.67
2.39
1.28
1.40
1.34
1.30
0.59
2.38
1.78
2.04
0.95
1.16
1.25
0.99
1.17
5.02
5.23
1.26
1.28
1.06
1.18
1.02
218
R p ( R⊕ )
- Err
+ Err
0.26
0.22
0.19
0.33
0.09
0.27
2.01
0.18
0.32
0.33
0.29
0.18
0.23
0.41
0.17
0.26
0.27
0.36
0.18
0.17
0.34
0.15
0.13
0.10
0.10
0.15
0.22
0.31
0.56
0.40
0.30
0.20
0.29
0.16
0.13
0.18
0.21
1.18
0.31
0.11
0.20
0.45
0.29
0.26
0.16
0.17
0.33
0.41
0.35
0.17
0.18
0.16
0.16
0.07
0.29
0.22
0.25
0.12
0.15
0.18
0.15
0.17
0.86
0.76
0.17
0.17
0.14
0.16
0.14
0.30
0.27
0.21
0.37
0.09
0.33
2.01
0.20
0.33
0.36
0.31
0.22
0.25
0.43
0.19
0.27
0.27
0.45
0.18
0.17
0.36
0.17
0.13
0.11
0.12
0.16
0.23
0.32
0.56
0.44
0.31
0.21
0.32
0.17
0.14
0.20
0.24
1.17
0.31
0.12
0.22
0.46
0.30
0.26
0.19
0.18
0.32
0.42
0.43
0.17
0.19
0.16
0.18
0.08
0.31
0.24
0.27
0.13
0.17
0.20
0.16
0.18
0.87
0.80
0.17
0.18
0.15
0.18
0.16
Value
Fp ( F⊕ )
- Err
+ Err
Sourcea
6.27
37.94
22.53
2.31
28.25
0.94
74.26
39.76
38.49
3.53
1.16
12.37
0.17
69.25
28.88
11.23
14.00
8.04
55.11
3.94
1.08
1.85
8.55
45.27
21.12
40.95
9.50
1.36
67.54
0.23
13.17
21.68
0.65
1.31
541.12
2.51
5.69
0.14
2.27
5.79
9.32
29.90
20.25
7.92
3.81
27.77
14.41
69.37
10.04
5.81
2.77
5.98
3.01
8.81
10.52
24.57
4.02
94.54
16.86
8.49
23.55
3.04
7.98
0.62
11.52
5.12
26.51
2.57
0.25
1.56
8.21
5.97
0.85
8.02
0.18
36.13
10.43
12.17
1.12
0.37
3.91
0.05
28.58
11.92
2.81
3.64
2.08
14.21
1.01
0.18
0.47
2.07
11.46
5.34
9.81
2.35
0.33
9.34
0.06
4.79
7.86
0.16
0.38
143.41
0.56
1.12
0.04
0.54
1.39
1.90
18.13
6.52
2.56
1.23
10.07
3.22
15.43
2.56
1.49
0.70
1.42
0.71
2.10
2.56
5.99
0.98
22.56
4.04
2.37
6.58
0.85
5.02
0.18
3.00
1.34
6.90
0.67
0.06
1.86
9.58
7.60
1.12
9.82
0.21
60.43
12.72
16.10
1.48
0.49
5.19
0.06
43.81
18.35
3.47
4.55
2.58
17.62
1.28
0.21
0.57
2.38
13.98
6.52
11.76
2.84
0.41
10.62
0.08
6.30
10.38
0.20
0.47
173.03
0.67
1.31
0.05
0.65
1.65
2.25
36.70
8.89
3.48
1.67
13.20
3.70
18.00
3.09
1.78
0.85
1.68
0.84
2.47
3.13
7.28
1.20
26.94
4.81
2.90
8.03
1.04
10.44
0.22
3.63
1.61
8.35
0.81
0.08
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
3
3
3
3
4
0
2
0
0
0
3
3
3
4
0
0
1
0
0
0
0
3
1
1
0
0
1
1
1
0
6
6
3
3
3
3
3
3
1
1
1
0
0
1
1
1
0
1
1
1
1
1
0
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.9—Continued
Period (Days)
- Err
KID
KOI
Value
8167996
8167996
8167996
8189801
8229458
8235924
8351704
8367644
8509442
8547140
8561063
8561063
8561063
8631751
8845205
8874090
8874090
8890150
9214942
9388479
9388479
9390653
9427402
9573685
9575728
9710326
9757613
9757613
9757613
9757613
9787239
9787239
9787239
9787239
9787239
10027247
10027323
10027323
10073672
10118816
10166274
10166274
10166274
10329835
10332883
10340423
10340423
10386984
10388286
10489206
10489206
10525027
10525049
10591855
10670119
10670119
10717241
10717241
10925104
10925104
10925104
11129738
11187837
11192235
11348997
11497958
11497958
11497958
11497958
1867.01
1867.02
1867.03
2480.01
2238.01
2347.01
1146.01
1879.01
2992.01
1266.01
961.01
961.02
961.03
2453.01
463.01
1404.01
1404.02
2650.01
1403.01
936.01
936.02
249.01
1397.01
2057.01
5692.01
947.01
250.01
250.02
250.03
250.04
952.01
952.02
952.03
952.04
952.05
2418.01
1596.01
1596.02
2764.01
1085.01
1078.01
1078.02
1078.03
2058.01
1880.01
736.01
736.02
739.01
596.01
251.01
251.02
2006.01
4252.01
2845.01
2179.01
2179.02
430.01
430.02
156.01
156.02
156.03
1427.01
252.01
2329.01
2090.01
1422.01
1422.02
1422.03
1422.04
2.54954597
13.96935372
5.21231974
0.66682583
1.64680052
0.58800191
7.09712741
22.08557431
82.65995306
11.41929554
1.21377022
0.45328734
1.86511477
1.53051487
18.47717612
13.32391953
18.90623146
34.98870981
18.75471931
9.46787010
0.89303356
9.54930070
6.24703152
5.94565639
2.64180062
28.59885823
12.28293091
17.25116937
3.54391419
46.82763492
5.90129534
8.75208025
22.78068566
2.89601166
0.74295972
86.82818564
5.92367766
105.35789935
2.25297009
7.71790094
3.35372553
6.87749137
28.46416009
1.52372638
1.15116708
18.79420980
6.73899074
1.28707954
1.68269527
4.16438060
5.77446840
3.27346499
15.57135826
1.57408931
14.87155876
2.73277006
12.37646610
9.34052690
8.04133834
5.18856137
11.77614238
2.61301749
17.60462615
1.61535973
5.13248495
5.84164124
19.85028393
10.86443187
63.33666060
1.25E-05
9.90E-05
5.15E-05
9.30E-07
2.72E-06
7.34E-07
2.98E-05
5.05E-05
6.70E-04
2.42E-05
3.76E-07
8.60E-08
1.12E-06
1.96E-06
1.92E-04
5.44E-05
2.81E-04
2.25E-03
5.57E-05
6.62E-05
4.90E-06
1.50E-05
1.02E-05
1.53E-05
1.99E-05
2.34E-04
1.43E-05
3.21E-05
1.15E-05
1.60E-04
2.03E-05
2.04E-05
1.28E-04
8.92E-06
3.06E-06
1.57E-03
1.81E-05
5.05E-04
7.20E-06
5.17E-05
1.04E-05
2.35E-05
2.41E-04
2.40E-06
6.89E-07
5.75E-05
3.10E-05
4.42E-06
5.88E-06
2.78E-06
6.66E-05
5.99E-06
9.44E-05
5.05E-06
5.40E-05
5.46E-06
1.30E-05
8.42E-05
9.01E-06
9.66E-06
7.55E-06
5.76E-06
3.24E-05
2.53E-06
9.13E-06
1.02E-05
5.85E-05
4.97E-05
5.57E-04
+ Err
5.91E-06
1.07E-04
5.22E-05
9.58E-07
2.81E-06
7.18E-07
3.27E-05
4.98E-05
7.23E-04
2.45E-05
3.76E-07
8.60E-08
1.12E-06
1.88E-06
2.26E-04
5.62E-05
3.04E-04
3.62E-03
5.57E-05
6.29E-05
4.81E-06
1.42E-05
1.03E-05
1.52E-05
2.22E-05
2.18E-04
1.41E-05
3.18E-05
1.15E-05
1.62E-04
1.94E-05
2.09E-05
1.31E-04
9.34E-06
3.65E-06
1.27E-03
1.89E-05
5.36E-04
7.26E-06
4.78E-05
1.06E-05
2.40E-05
2.27E-04
2.46E-06
6.97E-07
5.95E-05
3.13E-05
4.48E-06
5.95E-06
2.77E-06
6.90E-05
6.07E-06
9.36E-05
5.08E-06
5.01E-05
5.49E-06
1.30E-05
8.42E-05
8.98E-06
9.67E-06
7.52E-06
5.70E-06
3.21E-05
2.53E-06
9.09E-06
1.02E-05
5.85E-05
4.97E-05
5.57E-04
Value
1.12
1.81
1.16
1.31
0.93
1.02
0.82
2.42
2.23
1.54
0.87
0.90
0.69
1.13
1.67
1.35
0.96
0.96
1.78
2.18
1.26
1.62
2.23
1.19
0.46
1.92
2.89
2.51
1.06
2.10
2.13
1.98
2.30
1.09
0.91
1.24
1.04
1.90
1.30
0.94
1.86
2.10
1.93
1.05
1.30
1.82
1.17
1.45
1.38
2.61
0.90
0.74
0.73
0.89
1.27
1.08
2.04
0.73
1.39
1.04
2.02
1.36
2.36
1.17
1.52
3.49
3.10
2.34
2.30
219
R p ( R⊕ )
- Err
+ Err
0.12
0.21
0.13
0.23
0.10
0.10
0.13
0.37
0.35
0.26
0.21
0.21
0.16
0.21
0.28
0.27
0.20
0.15
0.28
0.30
0.18
0.22
0.23
0.17
0.05
0.26
0.44
0.40
0.17
0.33
0.27
0.27
0.29
0.14
0.12
0.17
0.16
0.28
0.22
0.14
0.25
0.28
0.26
0.14
0.20
0.30
0.19
0.18
0.19
0.31
0.12
0.10
0.14
0.10
0.19
0.16
0.26
0.10
0.18
0.13
0.23
0.20
0.29
0.22
0.25
0.30
0.51
0.34
0.34
0.14
0.26
0.15
0.25
0.12
0.13
0.14
0.39
0.37
0.27
0.21
0.21
0.16
0.23
0.28
0.28
0.21
0.17
0.29
0.31
0.18
0.22
0.26
0.19
0.06
0.27
0.45
0.45
0.18
0.36
0.29
0.34
0.29
0.14
0.13
0.23
0.16
0.30
0.24
0.16
0.26
0.31
0.29
0.16
0.21
0.31
0.20
0.19
0.22
0.31
0.15
0.11
0.15
0.13
0.20
0.18
0.27
0.10
0.20
0.15
0.23
0.22
0.29
0.24
0.26
0.50
1.11
0.87
0.66
Value
Fp ( F⊕ )
- Err
+ Err
Sourcea
53.87
5.58
20.77
466.69
118.41
550.07
7.97
1.96
0.70
7.27
17.83
67.88
10.00
61.55
1.27
4.79
3.01
1.17
5.03
6.26
145.89
4.54
21.87
21.39
64.99
1.81
7.83
4.98
41.08
1.31
16.39
9.70
2.71
42.36
260.04
0.35
18.61
0.40
85.43
15.00
34.92
13.39
2.02
131.36
181.87
2.81
11.03
129.24
76.32
30.10
19.47
35.43
5.54
135.82
3.15
30.19
6.15
8.95
15.72
28.20
9.45
58.27
3.79
101.54
18.98
14.89
2.94
5.09
0.61
11.96
1.24
4.61
145.16
22.88
101.56
2.35
0.57
0.20
2.55
7.89
31.62
4.38
21.28
0.41
2.11
1.32
0.38
1.59
1.68
39.21
1.20
4.24
5.60
13.21
0.47
2.76
1.75
14.46
0.46
4.14
2.46
0.69
10.69
66.04
0.09
5.52
0.12
27.60
4.82
9.20
3.51
0.53
34.28
53.73
0.89
3.48
30.84
19.27
6.98
4.52
9.80
2.00
26.43
0.90
8.59
1.57
2.30
3.58
6.40
2.15
17.40
0.93
20.26
5.92
5.58
1.30
1.96
0.37
14.43
1.49
5.56
180.18
26.30
116.51
2.94
0.71
0.25
3.52
11.04
54.63
6.24
28.63
0.51
3.29
2.06
0.62
2.11
2.04
47.53
1.44
4.89
6.80
15.76
0.57
3.88
2.46
20.34
0.65
5.11
3.02
0.84
13.21
80.88
0.11
7.08
0.15
35.63
6.54
11.35
4.35
0.65
41.60
66.69
1.14
4.48
36.80
23.05
8.25
5.33
12.56
2.61
30.77
1.10
10.50
1.92
2.78
4.24
7.60
2.55
22.67
1.10
23.51
7.38
8.77
1.78
7.23
0.53
1
1
1
0
0
0
0
0
0
0
2
3
2
0
1
0
0
1
0
1
1
1
0
0
0
1
1
1
0
1
1
0
1
0
1
0
0
0
0
0
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
0
3
3
1
1
0
0
1
2
0
5
5
5
5
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Acknowledgments
C.D. is supported by a National Science Foundation Graduate Research Fellowship.
Support for this work was provided through the NASA Kepler Mission Participating
Scientist Program grants NNX09AB53G and NNX12AC77G awarded to D.C. This
publication was made possible through the support of a grant from the John Templeton
Foundation. The opinions expressed in this publication are those of the authors and
do not necessarily reflect the views of the John Templeton Foundation. We thank the
anonymous referee for providing feedback that improved the quality of this paper and
the Kepler team for providing the community with a fantastic collection of data. We
are grateful to Jonathan Irwin for sharing a fast implementation of the transit model
(Mandel & Agol 2002) and for providing valuable advice. We thank Jessie Christiansen
for answering questions about the Kepler pipeline completeness and providing helpful
suggestions. Funding for the Kepler mission is provided by the NASA Science Mission
directorate. This publication made use of the Kepler Community Follow-Up Observing
Program website (https://cfop.ipac.caltech.edu), the NASA Exoplanet Archive,
and the Mikulski Archive for Space Telescopes (MAST). The NASA Exoplanet Archive
is operated by the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.
STScI is operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS5-26555. Support for MAST for non-HST data is provided
by the NASA Office of Space Science via grant NNX09AF08G and by other grants and
contracts.
220
CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE
Table 3.9—Continued
Period (Days)
- Err
KID
KOI
Value
11497958
11752906
11752906
11754553
11754553
11754553
11768142
11852982
11853130
11853255
11923270
12066335
12066335
12066569
12302530
12302530
12352520
12506770
1422.05
253.01
253.02
775.01
775.02
775.03
2626.01
247.01
3263.01
778.01
781.01
784.01
784.02
3282.01
438.01
438.02
3094.01
1577.01
34.14189000
6.38316009
20.61727169
16.38481298
7.87740709
36.44516433
38.09723780
13.81496561
76.87935073
2.24336795
11.59822478
19.27103179
10.06525147
49.27623343
5.93119294
52.66160633
4.57700369
2.80624351
2.64E-04
9.25E-06
2.66E-04
7.05E-05
1.12E-05
2.60E-04
2.86E-04
6.45E-05
4.79E-05
1.13E-05
1.50E-05
3.52E-04
2.48E-05
4.37E-04
5.02E-06
1.53E-04
1.33E-05
8.14E-06
+ Err
2.64E-04
9.25E-06
2.85E-04
7.02E-05
1.12E-05
2.91E-04
2.86E-04
6.47E-05
4.79E-05
1.11E-05
1.49E-05
3.51E-04
2.46E-05
5.12E-04
5.02E-06
1.53E-04
1.34E-05
8.33E-06
Value
2.12
2.68
1.48
1.90
2.10
1.96
2.36
1.61
6.83
1.37
2.82
1.81
1.54
2.07
1.80
1.80
1.39
1.40
R p ( R⊕ )
- Err
+ Err
0.34
0.39
0.23
0.26
0.28
0.27
0.53
0.21
1.18
0.19
0.37
0.25
0.20
0.29
0.24
0.25
0.17
0.23
0.79
0.40
0.25
0.27
0.31
0.29
0.44
0.22
1.27
0.20
0.39
0.28
0.22
0.31
0.25
0.27
0.20
0.25
Value
1.42
17.61
3.69
5.56
14.75
1.91
0.92
5.22
0.31
52.26
6.10
3.29
7.83
1.30
23.47
1.28
24.39
59.53
Fp ( F⊕ )
- Err
+ Err
0.72
6.43
1.35
1.61
4.23
0.55
0.46
1.27
0.08
14.75
1.55
0.88
2.10
0.34
7.53
0.41
5.04
18.67
0.93
9.30
1.94
2.08
5.53
0.72
0.57
1.50
0.09
18.85
1.90
1.10
2.63
0.41
10.42
0.56
5.86
24.43
Sourcea
5
1
1
1
3
1
5
1
2
1
0
1
0
0
3
0
0
0
a
The planet parameters are from our fits to the long cadence data (source = 0), our fits to the short cadence data (source = 1), the NASA
Exoplanet Archive (source = 2), Rowe et al. (2014, source = 3), Swift et al. (2015, source = 4), Cartier et al. (2014, source = 5), and Ioannidis
et al. (2014, source = 6).
221
Chapter 4
Adaptive Optics Images III: 87
Kepler Objects of Interest
This thesis chapter originally appeared in the literature as
C. D. Dressing, E. R. Adams, A. K. Dupree, C. Kulesa, & D.
McCarthy, The Astronomical Journal, 148, 78, 2014
Abstract
The Kepler mission has revolutionized our understanding of exoplanets, but some of
the planet candidates identified by Kepler may actually be astrophysical false positives
or planets whose transit depths are diluted by the presence of another star. Adaptive
optics images made with ARIES at the MMT of 87 Kepler Objects of Interest place
limits on the presence of fainter stars in or near the Kepler aperture. We detected visual
companions within 1## for five stars, between 1## and 2## for seven stars, and between 2##
222
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
and 4## for 15 stars. For those systems, we estimate the brightness of companion stars
in the Kepler bandpass and provide approximate corrections to the radii of associated
planet candidates due to the extra light in the aperture. For all stars observed, we
report detection limits on the presence of nearby stars. ARIES is typically sensitive
to stars approximately 5.3 Ks magnitudes fainter than the target star within 1## and
approximately 5.7 Ks magnitudes fainter within 2## , but can detect stars as faint as
∆Ks = 7.5 under ideal conditions.
4.1
Introduction
Since launch in 2009, the Kepler mission has discovered 4234 planet candidates and
confirmed or validated 977 planets (Borucki et al. 2010, 2011a,b; Batalha et al. 2013;
Burke et al. 2014; Rowe et al. 2014). Many of the planet candidates are expected to be
bona fide planets (Borucki et al. 2011a; Morton & Johnson 2011; Fressin et al. 2013),
but a small fraction may actually be astrophysical false positives (Brown 2003) in which
the apparent transit signal is produced by a pair of eclipsing stars physically associated
with the target star (a hierarchical triple) or in the background of the target star (a
background eclipsing binary). Close-in giant planet candidates (Santerne et al. 2012) and
candidate planets around giant stars (Sliski & Kipping 2014) are particularly likely to be
false positives. In other cases, transit signals may be diluted due to the presence of other
stars (physically associated or not) in the target aperture. This would cause the radius
of the planet to be underestimated. Because Kepler has a relatively large plate scale of
nearly 4## per pixel and many target apertures consist of multiple pixels, acquiring higher
resolution follow-up imagery near planet host stars is crucial for untangling potentially
223
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
blended systems.
In order to reduce the odds of classifying blended systems as planet candidates,
the Kepler team performs a series of tests on Kepler Objects of Interest (KOIs) before
nominating them to planet candidate status. The tests include comparing the depths of
odd and even transits in order to identify stellar eclipses that have been misidentified
as planetary transits, checking for ellipsoidal variations, and searching for secondary
eclipses (Batalha et al. 2010a). Background eclipsing binaries can also be identified by
examining the direction and magnitude of the shift in the photocenter during transit
(Bryson et al. 2013). Even systems in which the observed dip is due to a transit on
the target star can have significant centroid motion in crowded fields. However, some
background eclipsing binaries can be identified by computing the “source offset” between
the target star and the transit source (Bryson et al. 2013). A dip due to a transit on the
target star should have a negligible source offset while a dip due to a transit or eclipse of
another star can result in a significant source offset depending on the angular separation
and relative brightnesses of the true source and the target star.
An additional false positive check that can be performed using Kepler data alone
consists of a comparison of the ephemeris of an identified transit signal to the ephemerides
of known eclipsing binaries, variable stars, and other planet candidates. Using this
method and supplementing Kepler data with additional catalogs of eclipsing binaries and
variable stars, Coughlin et al. (2014) found that 12% of the KOIs they inspected were
false positives due to contamination from other known sources. However, because Kepler
does not downlink data for all stars in the field, some contaminated KOIs will not be
revealed via ephemeris matching because the contaminating star will not be downloaded.
Correcting for this effect, Coughlin et al. (2014) caution that 35% of KOIs may be false
224
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
positives due to contamination.
In an ideal case, all planet candidates could be confirmed by obtaining radial
velocity observations and measuring a mass for the transiting planet. However, this
plan is logistically impossible due to the large number of planet candidates, the faint
magnitudes of most Kepler host stars, and the small RV signature expected for most
small planets. In many cases, we must therefore attempt to “validate” planet candidates
by demonstrating that the odds that the transit signal is due to a bona fide transiting
planet are much higher than the odds of a false positive (e.g., Ballard et al. 2011;
Cochran et al. 2011; Torres et al. 2011; Fressin et al. 2011; Borucki et al. 2012; Ford
et al. 2012; Fressin et al. 2012; Lissauer et al. 2012; Morton 2012; Ballard et al. 2013;
Lissauer et al. 2014; Rowe et al. 2014; Wang et al. 2014b).
In order to validate planets and properly correct for diluted transits, we need to
place limits on the presence of other stars close to the target. The Kepler-14 system is
a prime example of the importance of high-resolution imaging: Buchhave et al. (2011)
report that the planetary radius and mass would have been underestimated by 10%
and 60%, respectively, without high-resolution follow-up images. Their analysis of
ground-based follow-up images revealed that the target star is in a close binary system
with a nearby star only 0.5 magnitudes fainter and 0.## 3 away. The Kepler team and
community have used speckle imaging (Howell et al. 2011; Horch et al. 2012; Kane et al.
2014), lucky imaging (Lillo-Box et al. 2012), high-resolution adaptive optics imaging
(Adams et al. 2012, 2013a,b; Law et al. 2014), and Hubble Space Telescope snapshots
(SNAP Program 12893; PI: R. Gilliland) to accomplish this objective.
In this paper, we present adaptive optics images of 87 Kepler planet candidates
225
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
acquired in September 2012, October 2012, and September 2013 in order to investigate
whether any of the target stars are diluted due to nearby stars and to place limits on
the presence of additional stars in the Kepler target aperture. We explain our observing
strategy in Section 4.2, the target sample in Section 4.3, and our data reduction process
in Section 4.4. We then discuss the detected visual companions in Section 4.5 and place
limits on undetected stars in Section 4.6. We compare our findings to the results of
previous surveys in Section 4.7 and conclude in Section 4.8.
4.2
Observations
All observations were taken using the Arizona Infrared Imager and Echelle Spectrograph
(ARIES) behind the adaptive optics system on the 6.5m Multiple-Mirror Telescope
(MMT). We used the target star as a natural guide star and ran the AO system at speeds
between 10 and 550 Hz depending on the brightness of the target star and the current
observing conditions. The resulting full-width at half-maximum of the target star point
spread functions (PSFs) varied between 0.## 1 and 0.## 58, with a median value of 0.## 25. The
airmass of our targets ranged from 1.01 to 2.01, with a median value of 1.16.
We observed all targets using a four-point dither pattern in f /30 mode with a plate
scale of 0## .02085 pixel−1 and a field of view of 20## × 20## . We also observed KOI 886 in
f /15 mode with a plate scale of 0## .04 pixel−1 and a field of view of 40## × 40## , but we
opted to use the images taken in f /30 mode for the final reduction. The field rotator
was turned on for the observations acquired in 2012 but not for the observations taken
in 2013. Accordingly, more distant stars are smeared by field rotation in the images
acquired in 2013.
226
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Under ideal conditions, ARIES is diffraction-limited in J, H, Ks in f /30 mode down
to a limiting magnitude of Ks = 21, and diffraction-limited in Ks in f /15 mode down to
a limiting magnitude of Ks = 22. The measured Strehl ratios were 0.3 in Ks and 0.05
in J for ARIES observations acquired in May 2010 under favorable observing conditions
with uncorrected seeing of 0.## 5 in Ks (Adams et al. 2012).
We varied the integration times for individual exposures between 0.8 seconds
and 89.9 seconds depending on the stellar magnitude. Our observing strategy was
intentionally more sensitive to fainter companions around fainter target stars because
the amount of transit depth dilution is governed by the brightness ratio of the target
star and the contaminating star. For our shortest exposure times of 0.8 seconds, we
were sensitive to companions as faint as Ks = 15.4 − 17.2 depending on the observing
conditions.
We typically repeated the four-point dither pattern four times for a total of
16 images in Ks band per star. For most targets, the dither pattern had a throw of
2## , but we increased the throw to 3## when we noticed nearby stars in the acquisition
image. In nine cases, we also imaged objects with close companions in J band in order
to determine the color of the companion and better estimate the relative contribution of
each star to the flux measured by Kepler. Table 4.1 provides a list of target stars with
detected companions.
227
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
4.3
Target Sample
We conducted our observations as part of the Kepler team follow-up effort. We selected
our targets from the lists of Kepler planet candidates available at the time of our
observing runs in Fall 2012 and Fall 2013. Some of the planet candidates associated with
our target stars were later reclassified as false positives and additional planet candidates
were detected in several systems. When selecting our sample, we prioritized relatively
bright (Kp ! 14) stars with small planet candidates. The Ks magnitude of our selected
sample extends from Ks = 8.6 to Ks = 12.7 with a median magnitude of Ks = 11.6.
4.4
Data Analysis
We reduced the ARIES observations of each star using the IRAF and python pipeline
described in Adams et al. (2012, 2013a,b). We calibrated each set of dithered images
using standard IRAF procedures1 and then used the xmosaic function in the xdimsum
package to combine and sky-subtract the images. For targets with detected companions,
we determined the approximate orientation of the field from the dither pattern. Our
field orientations are therefore approximate and should be treated as general guidelines
with an accuracy of a few degrees.
We searched for visual companions to our target stars by looking for bright objects
in the reduced images using the IRAF routine daophot and by visually inspecting each
image. The automated IRAF routines sometimes triggered on residual PSF speckles and
1
http://iraf.noao.edu
228
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
image artifacts near the edges of the CCD and in a square pattern one-half CCD width
away from bright stars, but those artifacts were easier to identify visually. Due to the
quasi-static nature of speckles, we saw similarities in the speckle pattern throughout
the course of the night. We could therefore distinguish between speckles and visual
companions by whether the objects reappeared in images of multiple target stars or
whether they were unique to a particular target star.
For the companions that passed visual inspection, we measured the magnitude
difference relative to the target star using the IRAF routine phot. We adopted a 5 pixel
aperture in order to sample most of the PSF of the target star without contaminating
the measurement with light from nearby stars. We tested the effect of using larger
apertures for stars observed in poor seeing conditions and found only slight changes
(0.001 - 0.03 magnitudes) in the differential photometry.
For the closest companions (stars within 0.## 5 of the target star), we instead
determined the relative magnitude by simultaneously fitting the PSFs using the same
Mathematica routines as in Adams et al. (2012, 2013a,b). Our PSF fitting routine fits a
Bessel-Lorentzian-Fourier model to each star using six Bessel and four Fourier terms.
We followed the procedure outlined in Adams et al. (2012, 2013a,b) to determine
the approximate Kepler magnitude, Kp, of identified companions. We first measured
the brightness differential between the target stars and companions in Ks (and J when
available) and converted those to apparent magnitudes for the companions using the
target star Ks and J magnitudes reported in the Two Micron All Sky Survey (2MASS)
catalog (Skrutskie et al. 2006) as absolute references. For systems with detected
companions within 2## we assumed that the stars would have been blended in 2MASS. In
229
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
those cases, we recomputed the magnitudes of each component so that the total system
magnitudes were equal to the catalog values. We then estimated the Kp magnitude of
companion stars using the relations provided in Appendix A of Howell et al. (2012).
4.5
Visual Companions
Out of 87 targets, we detected close visual companions for 27 stars: five stars have
detected companions within 1## , seven have detected companions within 2## , and 15 have
detected companions within 4## . We present ARIES images of the stars with companions
within 1## in Figure 4.1, companions within 1 − 2## in Figure 4.2, and companions within
2 − 4## in Figure 4.3. The ARIES field of view extends to 20## × 20## , but objects within 4## ,
the size of a Kepler pixel, are most likely to dilute planetary transits without revealing
their presence by inducing a significant centroid shift.
For stars with detected companions, the properties of the associated planet
candidates will need to be reevaluated to account for the contaminating light in the
aperture. We provide rough dilution corrections to the reported planet radii for stars
with companions at separations < 2## . This dilution correction will increase the radii of
associated planets by a given percentage. Stars at larger separations also contribute to
the background flux due to the large size of Kepler’s target apertures, but the fraction
of companion star flux collected depends on the Kepler pixel response function, which
varies across the focal plane (Bryson et al. 2010), and the specific aperture selected for
the target each quarter as the spacecraft is rotated.
A thorough analysis of the quarter-by-quarter dilution correction for each KOI
230
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
is beyond the scope of this paper, so we restrict our dilution corrections to a simple
order-of-magnitude estimate for the closest companions. In our simple model, we assume
that the transit source orbits the target star and that all of the light from both the target
and the close companion is captured in the target aperture. In some cases, the transit
source might actually be the fainter star detected via adaptive optics imaging rather
than the target star. If the planet candidate actually orbits the fainter star, then the
planet properties must be completely reevaluated based on the properties of the fainter
star. This can result in significant changes to the assumed planet radius, particularly if
the fainter star is a background star and not physically associated with the target star.
We discuss individual target stars with identified companions in the following sections
and list all detected stars within 10## in Table 4.1. We caution that this list may be
incomplete at larger angular separations because some stars may have been off the edge
of the ARIES detector.
231
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.1. Observed Stars with Visual Companions within 10##
(## )
(## )
(◦ )
(Ks)
Star
Dist
Dist Err
P.A.f
∆magg
Kp
(Ks)
KOI
KID
Kp
K00266
7375348
K00364
a
b
c
d
FP
e
2MASS
CP
PC
ND
11.472
10.379
0
2
0
0
1
3.621
0.0036
325.6
6.32
19.5j
7296438
10.087
8.645
0
1
0
0
1
6.019
0.0015
130.6
7.43
18.7j
K00720
9963524
13.749
11.900
4
0
0
0
1
...
...
...
...
...
...
...
...
2
3.861
0.0042
210.7
5.13
19.9
9.047
0.0019
76.3
3.91
18.3
K01279
8628758
13.749
12.246
0
2
0
0
1
K01344
4136466
13.446
12.001
0
1
0
0
1
2.625
0.002
137.8
3.74
18.6
4.022
0.0038
144.1
4.85
K01677
5526717
14.279
12.687
0
2
0
0
19.7
1
0.592
0.001
160.5
2.48
17.6i
K01977
9412760
14.028
11.551
2
0
0
0
1
9.786
0.0016
336.3
5.12
19.5
K02158
5211199
13.052
11.279
0
2
0
0
1
2.268
0.0024
13.5
4.45
18.2
...
...
...
...
...
...
...
...
2
6.484
0.0018
334.3
4.24
17.9
K02159
8804455
13.482
11.982
0
1
0
1
1
1.952
0.0012
323.4
2.52
16.7
...
...
...
...
...
...
...
...
2
6.872
0.0008
323.4
1.1
15.0
K02298
9334893
13.831
11.735
0
1
0
1
1
1.469
0.0004
195.0
1.3
15.2
K02331
12401863
13.467
12.065
0
1
0
0
1
3.884
0.0012
321.3
3.79
18.4
K02399
11461433
14.100
12.187
0
1
0
0
1
4.147
0.002
355.1
4.92
20.0
K02421
8838950
14.363
12.264
0
1
0
0
1
1.118
0.0006
290.3
0.42
15.1
...
...
...
...
...
...
...
...
2
4.002
0.0013
130.9
2.62
17.8
...
...
...
...
...
...
...
...
3
7.772
0.0022
45.6
4.8
20.7
K02426
8081899
13.889
12.199
0
1
0
0
1
8.757
0.0012
150.3
2.58
16.9
K02516
7294743
13.388
11.582
0
1
0
0
1
3.306
0.0014
86.9
4.23
18.3
K02527
7879433
14.131
11.562
0
1
0
0
1
7.909
0.0017
75.8
3.55
17.4
K02623
10916600
13.383
12.075
0
1
0
0
1
5.712
0.0026
116.1
5.39
20.5
K02672
11253827
11.921
10.285
2
0
0
0
1
4.541
0.0018
308.0
5.88
18.8
K02678
6779260
11.799
10.088
0
1
0
0
1
8.060
0.002
144.9
7.49
20.7
K02693
5185897
13.256
10.794
2
1
0
0
1
4.579
0.0015
118.4
5.11
18.4
K02706
9697131
10.268
9.109
0
1
0
0
1
1.618
0.0012
163.6
5.2
16.3
K02722
7673192
13.268
11.993
4
1
0
0
1
3.151
0.0025
280.5
4.14
18.7
...
...
...
...
...
...
...
...
2
7.178
0.002
112.4
4.87
19.7
K02732
9886361
12.805
11.537
2
2
0
0
1
7.697
0.0018
102.8
6.73
21.6
...
...
...
...
...
...
...
...
2
9.846
0.002
155.7
6.2
20.9
K02754
10905911
12.299
10.627
0
1
0
0
1
0.763
0.0001
261.5
1.65
15.3i,h
K02771
11456382
11.751
10.462
0
0
0
1
1
3.574
0.0012
312.4
5.73
18.8
K02790
5652893
13.380
11.486
0
1
0
0
1
0.254
0.0001
130.6
0.62
14.5i
...
...
...
...
...
...
...
...
2
5.662
0.003
240.8
5.39
20.4
...
...
...
...
...
...
...
...
3
5.303
0.0016
6.8
4.84
19.7
...
...
...
...
...
...
...
...
4
8.381
0.0017
63.4
5.28
20.2
K02803
9898447
12.258
10.642
0
1
0
0
1
3.650
0.0005
63.1
2.64
15.1
...
...
...
...
...
...
...
...
2
4.245
0.002
65.6
5.13
18.3
...
...
...
...
...
...
3
8.606
0.0016
205.4
5.18
18.3
13.586
11.514
0
1
0
0
1
1.038
0.0011
263.6
1.82
14.8h
...
...
K02813
11197853
232
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.1—Continued
(## )
(## )
(◦ )
(Ks)
Star
Dist
Dist Err
P.A.f
∆magg
Kp
5.76
20.2
(Ks)
KOI
KID
Kp
K02829
6197215
K02833
9109857
...
...
K02838
6607357
...
...
K02840
6467363
...
...
...
...
K02879
7051984
...
...
K02904
3969687
...
...
K02913
a
b
c
d
FP
e
2MASS
CP
PC
ND
12.824
11.444
12.599
11.123
0
1
0
0
1
9.922
0.0016
120.6
0
1
0
0
1
8.674
0.0006
39.1
4.4
...
...
17.9
...
...
...
...
2
7.216
0.0013
135.5
5.42
19.3
13.421
11.857
0
1
0
1
1
1.748
0.0053
197.2
4.04
18.5
...
...
...
...
...
...
2
7.739
0.0018
137.3
2.83
16.8
13.884
12.261
0
2
0
0
1
4.033
0.0021
297.0
4.16
19.1
...
...
...
...
...
...
2
7.029
0.0033
258.1
5.05
20.3
...
...
...
...
...
...
3
8.221
0.0018
185.9
4.44
19.5
12.771
11.099
0
0
0
1
1
0.423
0.0001
110.9
0.27
13.8i,h
...
...
...
...
...
...
2
5.449
0.0004
223.6
1.41
15.0
12.683
11.359
0
1
0
0
1
0.684
0.001
226.0
2.58
15.8i,h
...
...
...
...
...
...
...
2
5.274
0.0005
49.5
2.84
16.3
...
...
...
...
...
...
...
3
8.488
0.0006
21.3
3.36
17.0
9693006
12.858
11.361
0
1
0
0
1
7.141
0.0014
15.7
4.42
18.3
K02914
6837283
12.199
11.006
0
1
0
0
1
3.740
0.0012
230.4
5.28
17.8h
K02915
5613821
13.346
11.956
0
1
0
0
1
5.226
0.0014
167.3
5.36
20.3
...
...
...
...
...
...
...
...
2
9.276
0.0007
305.6
3.82
18.3
K02939
5473556
13.545
12.006
0
0
0
0
1
2.780
0.0009
131.4
1.84
15.8
...
...
...
...
...
...
...
...
2
4.293
0.0034
202.7
5.96
21.2
...
...
...
...
...
...
...
...
3
4.820
0.0036
300.5
5.68
20.8
...
...
...
...
...
...
...
...
4
9.768
0.0026
355.9
4.16
18.8
...
...
...
...
...
...
...
...
5
8.174
0.0023
84.3
4.5
19.2
K02961
10471515
12.581
11.290
0
1
0
0
1
1.954
0.0021
260.6
6.94
21.5
...
...
...
...
...
...
...
...
2
5.210
0.0009
59.9
4.86
18.8
K02970
5450893
12.861
11.566
0
1
0
0
1
4.393
0.0005
354.8
3.16
16.8
...
...
...
...
...
...
...
...
2
5.802
0.0024
326.3
6.98
22.0
20.7
...
...
K02971
4770174
...
...
...
...
...
...
3
6.254
0.0013
216.2
6.03
12.742
11.438
0
2
0
0
1
3.477
0.0019
36.8
6.69
...
21.4
...
...
...
...
...
...
...
2
4.736
0.0017
352.1
6.81
21.6
...
...
...
...
...
...
...
...
3
7.990
0.0016
352.5
6.08
20.6
...
...
...
...
...
...
...
...
4
6.806
0.0015
147.8
6.98
21.8
...
...
...
...
...
...
...
...
5
7.680
0.0014
216.6
6.31
20.9
K02984
7918652
13.066
11.637
0
1
0
0
1
3.262
0.001
31.9
3.81
17.8
...
...
...
...
...
...
...
...
2
6.620
0.0022
57.8
5.68
20.3
...
...
...
...
...
...
...
...
3
5.848
0.002
328.6
6.84
21.9
...
...
...
...
...
...
...
...
4
5.679
0.0009
200.8
6.46
21.4
K03015
11403530
13.219
11.774
0
1
0
0
1
4.763
0.0014
223.8
5.22
19.9
K03075
3328080
12.994
11.532
0
1
0
0
1
4.289
0.0017
48.8
K03111
8581240
12.863
11.353
0
2
0
0
1
3.334
0.001
...
...
...
...
...
...
...
...
2
5.390
0.0012
233
7.0
21.9
235.1
5.25
19.4
154.7
4.86
18.9
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Figure 4.1: Target stars with detected companions within 1## . Each box is 2## by 2## .
The color scaling is logarithmic for K01677 and linear for the other stars. Only a subset
of stars were imaged in both J and Ks. Here and in Figures 4.2 and 4.3 we include all
available J-band images.
KOI 266
This system contains a 1.6 R⊕ planet candidate with a 25.3 day period and a second
1.8 R⊕ planet candidate with a 47.7 day period (Burke et al. 2014). Our ARIES
observations revealed a star roughly 6.3 Ks magnitudes fainter than KOI 266 at a
234
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.1—Continued
(## )
(## )
(◦ )
(Ks)
Star
Dist
Dist Err
P.A.f
∆magg
Kp
21.7
(Ks)
KOI
KID
...
...
K03117
6523351
...
...
...
...
K03122
12416661
Kp
a
2MASS
CP
b
c
PC
ND
d
FP
e
...
...
...
...
...
...
3
4.178
0.0015
135.4
6.97
13.163
11.508
0
1
0
0
1
2.612
0.0018
286.5
6.1
20.7
...
...
...
...
...
...
2
5.294
0.0035
343.4
5.66
20.1
...
...
...
...
...
...
3
5.182
0.0053
229.0
6.42
21.1
12.086
10.819
0
1
0
0
1
4.437
0.0017
55.8
6.53
20.4
20.0
...
...
...
...
...
...
...
...
2
8.771
0.0015
114.9
6.22
K03128
7609674
13.371
11.900
0
1
0
0
1
6.274
0.0015
106.3
5.14
20.0
K03242
6928906
12.374
11.520
0
1
0
0
1
3.998
0.0012
238.2
8.4
23.8j
a
Apparent magnitude in the Kepler bandpass.
b
Number of confirmed planets associated with the target.
c Number of planet candidates associated with the target.
d
Number of not dispositioned KOIs associated with the target.
e
Number of false positive KOIs associated with the target.
f
Angle measured eastward from north. We caution that the position angles were estimated from the dither pattern and therefore might
differ from the true angle by a few degrees. We note that some stars in Adams et al. (2012) were reported with PA incorrectly listed as 360
minus PA. The affected KOIs (and star numbers) were K00010 (1), K00018 (3), K00068 (1,2,3), K00102 (1,2), K00106 (1), K00113 (2,4),
K00118 (1), K00121 (2), K00122 (1), K00123 (2), K00124 (1), K00126 (1,3), K00137 (1,3), K00148 (1,2,4), K00153 (1), K00251 (2), K00283
(1), and K00306 (1,2).
g
Error on ∆Ks is roughly 0.02 mag.
h
Estimated Kp for a dwarf companion based on both J and Ks photometry.
i
Brightness contrast, separation, and companion Kp determined using PSF fitting.
j
These companions to K00266, K00364, and K03242 were smeared by field rotation and their magnitudes are likely underestimated. More
distant smeared companions to these stars were omitted from this list.
235
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Figure 4.2: Target stars with detected companions between 1## and 2## . Each box is 4##
by 4## . The color scaling is linear for K02298 and K02421 and logarithmic for all other
stars.
236
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Figure 4.3: Target stars with detected companions between 2## and 4## . Each box is 12##
by 12## . KOI 266 also has a companion within 4## , but it is not pictured here because
the companion is smeared due to field rotation. The color scaling is linear for K02939
and logarithmic for all other stars. Some stars also have more distant companions at
separations between 4## and 12”. We provide a list of companions within 10## in Table 4.1.
237
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
distance of 3.## 62. KOI 266 was previously inspected2 using speckle imagery with DSSI on
WIYN, but the nearby star was beyond the 3.## 2 × 3.## 2 speckle field of view. This star
was previously imaged by Adams et al. (2012, 2013b), who reported that the companion
is 6.6 J magnitudes fainter and 6.1 Ks magnitudes fainter than KOI 266, resulting in
an estimated Kepler magnitude of Kp = 19.3. For the rest of this paper, we adopt the
magnitude estimates from Adams et al. (2012, 2013b) because the visual companion is
slightly smeared in our image due to field rotation. The nearby star is also listed in
UKIRT (Lawrence et al. 2007).
KOI 266 was classified by Slawson et al. (2011) as a detached eclipsing binary with
a period of 25.3 days, suggesting that the 1.6 R⊕ planet candidate with the same period
might not actually be a planet. Instead, the observed decrease in flux every 25.3 days
might be an eclipsing binary diluted by the light of a nearby star. The centroid source
offset during transits of KOI 266.01 is 0.## 574 (2.66σ).
KOI 720 (Kepler-221)
This system has four confirmed planets with radii of 2.96, 2.81, 3.05, and 1.56 R⊕
(Borucki et al. 2011b; Rowe et al. 2014). We detected another star 3.## 86 from the target
star. The nearby star is 5.13 Ks magnitudes fainter than KOI 720 and is predicted
to have Kp = 19.9 (∆Kp = 6.2). KOI 720 has been observed with the Differential
Speckle Survey Instrument (DSSI) on WIYN and at low quality with Robo-AO on the
Palomar 1.5-m (Law et al. 2014). The companion we detected is visible in UKIRT
2
The archival observations discussed in this paper were reported on the Kepler Community Follow-up
Observing Program (CFOP) website: http://cfop.ipac.caltech.edu.
238
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
and has a reported J magnitude of 17.67. The maximum dilution correction for a star
5.13 Ks magnitudes fainter than the target star is a 0.2% correction to the planet radii,
so the dilution correction is unlikely to be significant given the nearly 4## separation
between KOI 720 and the companion and the large brightness contrast between the stars.
KOI 1279
This system contains two short-period planet candidates with radii of 1.6 R⊕ and 0.9 R⊕
(Borucki et al. 2011b; Batalha et al. 2013). We detected a star 3.74 Ks magnitudes
fainter than KOI 1279 at a distance of 2.## 62. The star is predicted to have Kp = 18.6
(∆Kp = 4.8). KOI 1279 has also been observed using speckle imaging with DSSI on
WIYN and at low quality with Robo-AO on the Palomar 1.5-m (Law et al. 2014). The
companion we detected was visible in the UKIRT image of the field and has a reported J
magnitude of J = 16.54. KOI 1279 does not exhibit a large source offset during transits,
which supports the interpretation that the planet candidates orbit the target star.
KOI 1677
KOI 1677 hosts a 2.2 R⊕ planet candidate with a 52.1 day orbit and a 0.8 R⊕ candidate
with a 8.5 day orbit (Batalha et al. 2013). We detected a companion 2.48 Ks magnitudes
fainter than KOI 1677 at a distance of 0.## 6. Using the relation from Howell et al. (2012),
the predicted Kp magnitude for the companion is Kp = 17.6 (∆Kp = 3.3). This
object was also detected in a medium-quality Robo-AO image of KOI 1677 and has an
estimated magnitude of i = 18.83 ± 0.44 (Law et al. 2014). Assuming that all of the
flux from the target star and the companion is captured in the Kepler aperture and
239
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
that the planet orbits the target star, the planet radius estimate should be increased by
roughly 2% to account for the contamination from the nearby star. KOI 1677 does not
display a significant offset during the transits of KOI 1677.01, but the centroid analysis
for KOI 1677.02 is not yet available.
KOI 2158
This system contains two planet candidates with periods of 4.6 and 6.7 days and
radii of 1.6 and 1.0 R⊕ , respectively (Batalha et al. 2013). We detected a companion
4.45 Ks magnitudes fainter than KOI 2158 at a distance of 2.## 27. The companion is
predicted to have Kp = 18.2 (∆Kp = 5.2). KOI 2158 has also been observed with DSSI
on WIYN and at medium quality with Robo-AO on the Palomar 1.5-m. The companion
was detected in UKIRT and has a reported J magnitude of 17.15. KOI 2158 does not
exhibit a large source offset during the transits of either planet candidate.
KOI 2159
KOI 2159 hosts one candidate planet with a period of 7.6 days and a radius of 1.1 R⊕
(Batalha et al. 2013). The NASA Exoplanet Archive entry for KOI 2159 also includes a
1 R⊕ false positive at a period of 2.4 days. Our ARIES observations revealed a companion
2.52 Ks magnitudes fainter than KOI 2159 at a distance of 1.## 95. The estimated Kp
magnitude for the companion is 16.7 (∆Kp = 3.2). This companion was also listed as
a likely detection with Robo-AO in Law et al. (2014) based on a medium-quality image
and has an estimated magnitude of i = 17.28 ± 0.53. The star was also detected in
UKIRT and has a J band magnitude of 15.57. The estimated dilution correction due to
240
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
the extra light from the companion is a 3% increase to the planet radius. KOI 2159 does
not display a significant source offset during transit.
KOI 2298
This system contains a 1 R⊕ planet candidate with a 16.7 day orbit (Batalha et al.
2013). NEXSci also reports a 0.8 R⊕ false positive with a 31.8 day period. We
detected a companion 1.3 Ks magnitudes fainter than KOI 2298 at a distance of 1.## 47.
The companion is expected to have Kp = 15.2 (∆Kp = 1.4), indicating that the
contamination from this companion star may lead to a significant underestimate of the
planet radius. In the simple approximation that all light from the companion star is
captured in the Kepler aperture, the radius estimated for the planet should be increased
by 13% to account for the dilution if indeed the planet orbits the target star and the
companion has Kp = 14.9. However, the companion may be the same object identified
roughly 1## away from KOI 2298 in a HIRES guider image3 . The estimated brightness
contrast from the HIRES image is three magnitudes, which implies the companion is
red enough that the dilution correction might be only a 3% change to the radius of the
planet candidate. The false positive KOI 2998.02 failed the centroid test during data
validation, but KOI 2298 does not exhibit a significant source offset during the transit of
KOI 2298.01.
3
https://cfop.ipac.caltech.edu/edit_obsnotes.php?id=2298
241
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
KOI 2331
KOI 2331 hosts a single 1.4 R⊕ planet candidate with a 2.8 day period (Batalha et al.
2013). Our ARIES observations revealed a companion 3.79 Ks magnitudes fainter than
KOI 2331 at a separation of 3.## 88. The predicted Kp magnitude for the companion
is Kp = 18.4 (∆Kp = 4.9). KOI 2331 has also been observed at medium quality
with Robo-AO on the Palomar 1.5-m Law et al. (2014). KOI 2331 does not display a
significant source offset during transit.
KOI 2421
KOI 2421 hosts a 0.7 R⊕ planet candidate with a 2.3 day orbit (Batalha et al. 2013).
We detected two companions 0.42 and 2.62 Ks magnitudes fainter than KOI 2421 at
separations of 1.## 12 and 4.## 0, respectively. The closer companion is predicted to be
Kp = 15.1 (∆Kp = 0.7) and the farther companion is predicted to be Kp = 17.8
(∆Kp = 3.5). KOI 2421 has also been observed using NIRC2 on Keck with a laser guide
star.
Due to the similar brightness of the innermost companion and KOI 2421, this system
will require a significant dilution correction. If all light from the innermost companion
is captured in the Kepler aperture and the planet orbits the target star, then the planet
radius measurement will need to be increased by 23% to account for dilution. KOI 2421
does not exhibit a large source offset during transit, which lends support to the theory
that the planet candidate orbits the target star, but this system should be inspected
closely to confirm that the planet candidate does indeed orbit KOI 2421.
242
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
KOI 2516
KOI 2516 hosts a 1.2 R⊕ planet candidate with a 2.8 day orbit (Batalha et al. 2013).
We detected a companion 4.23 Ks magnitudes fainter than KOI 2516 at a distance of
3.## 31. The estimated Kp magnitude of the companion star is Kp = 18.3 (∆Kp = 4.9).
The companion was previously identified in UKIRT and has a J magnitude of 16.45.
KOI 2516 does not display a significant source offset during transit.
KOI 2706
KOI 2706 hosts a 1.5 R⊕ planet candidate with a 3.1 day orbit (Burke et al. 2014). We
detected a companion 5.2 Ks magnitudes fainter than KOI 2706 at a separation of 1.## 62.
The predicted Kp magnitude for the companion is Kp = 16.3 (∆Kp = 6.0). KOI 2706
has also been observed with DSSI on WIYN and PHARO on the Palomar-5m. The
detected companion is visible in the PHARO observations, but was undetected in the
WIYN speckle imaging. The estimated dilution correction for this system is a 0.2%
increase in the radius of the planet candidate. KOI 2706 exhibits a 0.## 83 (3.7σ) source
offset during transit.4
KOI 2722 (Kepler-402)
This system contains four confirmed planets with radii of 1.4, 1.4, 1.1, and 1.3 R⊕
and one candidate planet with a radius of 1.3 R⊕ (Burke et al. 2014). We detected
4
Data Validation reports containing centroid analyses are available at http://exoplanetarchive.
ipac.caltech.edu for all KOIs.
243
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
a companion 4.14 Ks magnitudes fainter than KOI 2722 at a distance of 3.## 15. The
estimated Kp magnitude of the companion star is Kp = 18.7 (∆Kp = 5.5).
KOI 2754
KOI 2754 hosts a 0.7 R⊕ planet candidate with a 1.3 day period (Burke et al. 2014). Our
ARIES observations revealed a companion 2.11 J magnitudes and 1.65 Ks magnitudes
fainter than KOI 2754 at a separation of 0.## 763. The predicted Kp magnitude of the
companion star is Kp = 15.3 (∆Kp = 3.0) if the star is a dwarf and Kp = 15.2
(∆Kp = 2.9) if the star is a giant. The companion star was also detected in the WIYN
speckle imaging of K02754 acquired with DSSI by Mark Everett5 . The companion
is 3.12 magnitudes fainter than KOI 2754 at 692nm and 2.56 magnitudes fainter at
880nm. Assuming that the planet orbits the target star and that all of the light from
the companion is captured in the Kepler aperture, then the planet radius should be
increased by 3% to account for dilution. KOI 2754 does not exhibit a significant source
offset during transit.
KOI 2771
KOI 2771 was reported to have a 1.7 R⊕ planet with a 0.8 day period, but this signal
has been found to be a false positive (Burke et al. 2014). We detected a companion
5.73 Ks magnitudes fainter than KOI 2771 at a separation of 3.## 57. The estimated
Kp magnitude of the companion is Kp = 18.8 (∆Kp = 7.1). KOI 2771 has also
been observed with DSSI on WIYN and PHARO on the Palomar-5m. The detected
5
https://cfop.ipac.caltech.edu/edit_target.php?id=2754
244
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
companion is visible in both the PHARO observation and in the UKIRT data.
KOI 2790
KOI 2790 hosts a 0.9 R⊕ planet candidate with a 14.0 day period (Burke et al. 2014). Our
ARIES observations revealed a companion 0.62 Ks magnitudes fainter than KOI 2790
at a separation of 0.## 21. The predicted Kp magnitude of the companion is Kp = 14.5
(∆Kp = 1.1). The companion is clearly identifiable in the more recent higher resolution
image of KOI 2790 acquired with NIRC2 on Keck. The approximate increase to the
planet radius is 17% assuming that all of the light from the companion is captured in the
Kepler aperture and that the planet orbits the target star. KOI 2790 does not exhibit a
significant source offset during transit.
KOI 2803
KOI 2803 hosts a 0.5 R⊕ planet candidate with a 2.4 day period (Burke et al. 2014). We
detected a companion 2.64 Ks magnitudes fainter than KOI 2803 at a distance of 3.## 65.
The estimated Kp magnitude of the companion is Kp = 15.1 (∆Kp = 2.9). KOI 2803
has also been observed with speckle imaging using DSSI on WIYN, but the companion
was too far from the star to be detected. The companion we identified was found in
UKIRT at a separation of 3.## 4 and has a reported J magnitude of 18.33. KOI 2803 does
not display a significant source offset during transit.
245
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
KOI 2813
KOI 2813 hosts a 1.2 R⊕ planet candidate with a 0.7 day period (Burke et al. 2014). We
detected a companion 1.43 J magnitudes and 1.82 Ks magnitudes fainter than KOI 2813
at a separation of 1.## 04. The predicted Kp magnitude of the companion is Kp = 14.8
(∆Kp = 1.3). In the simple approximation in which all of the light from the companion
star is captured in the Kepler aperture and the companion orbits the target star, then
the estimated planet radius should be increased by 15% to correct for the extra light in
the aperture. KOI 2813 does not exhibit a significant source offset during transit.
KOI 2838
KOI 2838 hosts a 0.7 R⊕ planet candidate with a 4.8 day period. The Kepler data
also revealed a 7.7 day false positive (Burke et al. 2014). We detected a companion
4.04 Ks magnitudes fainter than KOI 2838 at a distance of 1.## 75. The estimated
Kp magnitude of the companion is Kp = 18.5 (∆Kp = 5.0) and the approximate dilution
correction is a 0.5% increase to the radius of the planet candidate. KOI 2838 does not
display a significant source offset during the transits of KOI 2838.02.
KOI 2879
KOI 2879 was reported to have a 1.4 R⊕ planet with a 0.3 day period (Burke et al. 2014),
but this signal has been found to be a false positive. Our ARIES observations revealed
a companion 0.37 J magnitudes and 0.27 Ks magnitudes fainter than KOI 2879 at a
distance of 0.## 423. Using the J − Ks to Kp − Ks color-color conversion from Howell et al.
(2012), we predict that the Kp magnitude of the companion is Kp = 13.8 (∆Kp = 1.1)
246
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
if the star is a dwarf or Kp = 13.9 (∆Kp = 1.2) if the star is a giant. If any additional
planet candidates are detected around KOI 2879, their radii will need to be increased by
roughly 17% to account for the additional light in the aperture.
KOI 2904
KOI 2904 hosts a 1.2 R⊕ planet with a 16.4 day period (Burke et al. 2014). We detected
a companion 2.74 J magnitudes and 2.58 Ks magnitudes fainter than KOI 2904 at a
separation of 0.## 68. Using the J − Ks to Kp − Ks color-color conversion from Howell et al.
(2012), we predict that the Kp magnitude of the companion is Kp = 15.8 (∆Kp = 3.1)
if the star is a dwarf or Kp = 15.9 (∆Kp = 3.2) if the star is a giant. KOI 2904 has
also been observed with speckle imaging using DSSI on WIYN and the companion was
detected with magnitude differences of 2.83 mags at 692nm and 2.77 mags at 880nm.
The estimated dilution correction for this system is 3% assuming that the planet orbits
the target star and that all of the light from the companion is captured in the Kepler
aperture. KOI 2904 does not display a significant source offset during transit.
KOI 2914
KOI 2914 hosts a 2.0 R⊕ planet with a 21.1 day period (Burke et al. 2014). Our ARIES
observations revealed a companion 5.42 J magnitudes and 5.28 Ks magnitudes fainter
than KOI 2914 at a distance of 3.## 74. The predicted Kp magnitude of the companion
is Kp = 17.8 (∆Kp = 5.6) if the star is a dwarf. KOI 2914 has also been observed
with DSSI on WIYN, but the companion was outside the image area. The detected
companion is likely to be the J = 16.64 source found in UKIRT at a separation of 3.## 95.
247
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
KOI 2914 does not exhibit a significant source offset during transit.
KID 5473556 (formerly KOI 2939)
KOI 2939 (KID 5473556) is an eclipsing binary with a single observed planetary transit
(Welsh et al. 2012) and is no longer listed in the KOI catalog. We detected a companion
1.84 Ks magnitudes fainter than KOI 2939 at a distance of 2.## 78. The estimated
Kp magnitude of the companion is Kp = 15.8 (∆Kp = 2.2).
KOI 2961
KOI 2961 hosts a single planet candidate with a radius of 1.2 R⊕ and an orbital period
of 3.78 days (Burke et al. 2014). Our ARIES observations revealed a companion
6.94 Ks magnitudes fainter than KOI 2961 at a distance of 1.## 95. The predicted
Kp magnitude of the companion is Kp = 21.5 (∆Kp = 8.9) and the estimated dilution
correction is only 0.01% due to the large brightness contrast between KOI 2961 and the
companion.
KOI 2961 has also been observed using speckle imaging with DSSI on WIYN at
692nm and 880nm. The companion we report in this paper was not detected in the
3.## 2x3.## 2 speckle image. At the distance of the companion, the 3-sigma detection limits
for the speckle image were 4.04 magnitudes at 692nm and 3.953 magnitudes at 880nm.
The lack of a detection in the speckle image is therefore unsurprising given the predicted
faintness of the companion at bluer wavelengths. KOI 2961 does not display a significant
source offset during transit.
248
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
KOI 2971
This system contains a 0.8 R⊕ planet candidate with a 6.1 day period and a second
1.1 R⊕ planet candidate with a 31.9 day period (Burke et al. 2014). We detected a
companion 6.69 Ks magnitudes fainter than KOI 2971 at a distance of 3.## 48. The
predicted Kp magnitude of the companion is Kp = 21.4 (∆Kp = 8.7). KOI 2971 has also
been observed using speckle imaging with DSSI on WIYN. The detected companion was
identified in UKIRT and has a reported J magnitude of 20.76. The detection limit near
3.## 5 in our J band ARIES image is J = 15.3, so we are not able to estimate the J band
magnitude of the companion from our data. KOI 2971 does not exhibit a significant
source offset during the transits of either KOI 2971.01 or 2971.02.
KOI 2984
KOI 2984 hosts a 1.1 R⊕ planet candidate with a 11.5 day orbit (Burke et al. 2014). Our
ARIES observations revealed a companion 3.81 Ks magnitudes fainter than KOI 2984
at a separation of 3.## 26. The estimated Kp magnitude of the companion is Kp = 17.8
(∆Kp = 4.8). KOI 2984 has also been observed using DSSI on WIYN and NIRC2 on
Keck. The companion we detected was identified in UKIRT with a J band magnitude of
16.0. No closer companions were detected in the WIYN and NIRC2 images. KOI 2984
does not display a significant source offset during transit.
KOI 3111
This system hosts a 2.1 R⊕ planet candidate with a 10.8 day period and a 1.5 R⊕
planet candidate with a 4.3 day period (Burke et al. 2014). We detected a companion
249
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
5.25 Ks magnitudes fainter than KOI 3111 at a distance of 3.## 33. The predicted
Kp magnitude of the companion is Kp = 19.4 (∆Kp = 6.5). KOI 3111 has also been
observed with speckle imaging using DSSI at WIYN. The companion we identified was
visible in UKIRT and has a J band magnitude of 17.98. KOI 3111 exhibits a 2.## 847
(3.09σ) source offset during the transits of KOI 3111.01, but only a 1.## 89 (1.64σ) source
offset during the transits of KOI 3111.02.
KOI 3117
KOI 3117 hosts a 1.5 R⊕ planet candidate with a 6.1 day period (Burke et al. 2014).
We detected a companion 6.1 Ks magnitudes fainter than KOI 3117 at a separation
of 2.## 61. The estimated Kp magnitude of the companion is Kp = 20.7 (∆Kp = 7.5).
KOI 3117 has also been observed with speckle imaging using DSSI at WIYN, but the
source was not detected in the 3.## 2 by 3.## 2 field of view. The reported 3σ speckle detection
limits for an annulus extending from 1.## 7 - 1.## 9 (the farthest reported separation) are
4.095 magnitudes at 692nm and 3.381 magnitudes at 880nm. The lack of a speckle
detection is not surprising given the large Ks magnitude contrast between the target
and the companion and the likelihood that the companion would be even fainter in the
bluer 692nm and 880nm filters used in the speckle imaging. KOI 3117 does not display a
significant source offset during transit.
250
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
4.6
Detection Limits
In addition to measuring the brightness of companions, we calculated detection limits by
measuring the total amount of flux in annuli centered on the target star. The widths of
the non-overlapping annuli were 0.## 05 for separations within 0.## 2, 0.## 1 between 0.## 2 and
1.## 0, and 1## at separations beyond 1## . We estimated the contribution from background
stars by measuring the mean flux in an annulus with a radius of 10## and subtracted that
background value from the total within each annulus to measure the flux due to the star
at that distance. We then measured the standard deviation within each annulus and
calculated the detection limit for each annulus as 5 standard deviations above the mean
flux. For most targets we found a full width at half-maximum (FWHM) of 0.## 25 and a
limiting magnitude of ∆Ks = 5.3 at 1## . However, under good conditions we are sensitive
to companions as faint as ∆Ks = 7.5 and as close as 0.## 1 (see Figure 4.4). We provide
detection limits for each target in Table 4.2 and plot detection limits as a function of
angular separation for three stars in Figure 4.4.
251
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.2. Limits on the Presence of Nearby Stars for All Observed Stars
FWHM
2MASS
Companion
Limiting ∆Ks for Annulus Centered At
(## )
(Ks)
within 10##
0.## 1
0.## 2
0.## 5
2.## 0
4.## 0
K00159 0.15
11.970
No
–
2.3
4.26 6.15 6.38
6.33
K00266 0.23
10.379
Yes
–
–
3.0
5.54 6.36
6.68
K00330 0.32
12.384
No
–
–
3.07 4.82 4.91
4.92
K00351 0.28
12.482
No
–
–
3.24 4.85 4.82
4.86
K00364 0.1
8.645
Yes
2.45 3.68 5.73 8.03 8.49
8.6
K00392 0.26
12.416
No
–
–
3.26 4.85 5.01
5.03
K00664 0.25
12.001
No
–
–
2.98 4.98 5.23
5.38
K00720 0.25
11.900
Yes
–
–
3.35 5.12 5.25
5.22
K00886 0.49
12.648
No
–
–
1.51
3.26
2.95
K00947 0.25
12.097
No
–
–
3.48 5.26 5.42
5.41
K01219 0.26
12.469
No
–
–
3.44 5.34 5.49
5.51
K01279 0.19
12.246
Yes
–
1.84 3.76 5.16 5.31
5.33
K01344 0.36
12.001
Yes
–
–
2.27 4.37 4.73
4.75
K01677 0.29a
12.687
Yes
–
–
3.5
4.99 5.18
5.17
K01913 0.29
11.664
No
–
–
3.01 4.98 5.19
5.2
K01977 0.23
11.551
Yes
–
–
3.79 5.79
6.1
6.13
K02002 0.32
11.730
No
–
–
2.91 5.12 5.35
5.43
Object
252
1.## 0
3.0
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.2—Continued
FWHM
2MASS
Companion
Limiting ∆Ks for Annulus Centered At
(## )
(Ks)
within 10##
0.## 1
0.## 2
K02072 0.28
11.970
No
–
K02158 0.31
11.279
Yes
K02159 0.3
11.982
K02298 0.19
1.## 0
2.## 0
4.## 0
–
3.27 5.28
5.51
5.51
–
–
3.24 5.56
5.78
5.91
Yes
–
–
2.71 4.73
5.02
4.92
11.735
Yes
–
1.86 4.04 5.57
5.8
5.95
K02331 0.17
12.065
Yes
–
2.12 4.36 6.05
6.21
6.18
K02372 0.3
11.988
No
–
–
2.4
4.12
4.54
4.51
K02399 0.22
12.187
Yes
–
–
3.74 5.27
5.35
5.32
K02421 0.24
12.264
Yes
–
–
3.59 5.31
5.55
5.55
K02426 0.3
12.199
Yes
–
–
2.9
4.53
4.67
4.79
K02516 0.19
11.582
Yes
–
1.74 3.58 5.55
5.86
5.85
K02527 0.29
11.562
Yes
–
–
2.96 5.03
5.4
5.49
K02581 0.22
11.808
No
–
–
3.26 5.23
5.59
5.57
K02585 0.26
12.121
No
–
–
2.97 4.84
5.03
5.07
K02623 0.18
12.075
Yes
–
1.81 3.48 5.39
5.76
5.77
K02672 0.35
10.285
Yes
–
–
6.57
6.91
K02675 0.58
10.907
No
–
–
3.5
4.82
5.5
K02678 0.16
10.088
Yes
–
2.52 4.24 6.77
7.38
7.64
Object
253
0.## 5
2.22 4.75
–
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.2—Continued
Object
FWHM
2MASS
Companion
Limiting ∆Ks for Annulus Centered At
(## )
(Ks)
within 10##
0.## 1
0.## 2
0.## 5
1.## 0
2.## 0
4.## 0
3.0
K02684 0.2
9.564
No
–
1.49
5.25
6.59
7.09
K02687 0.14
8.693
No
–
2.11 3.78 5.97
6.85
6.8
K02693 0.33
10.794
Yes
–
–
2.61 5.32
6.21
6.7
K02706 0.16
9.109
Yes
–
1.9
3.46 5.91
7.67
8.06
K02720 0.19
8.996
No
–
1.52 3.07 5.34
7.01
7.33
K02722 0.23
11.993
Yes
–
–
3.51 5.43
5.61
5.62
K02732 0.23
11.537
Yes
–
–
3.59 6.37
6.81
7.07
K02754 0.08a
10.627
Yes
–
2.21 3.95 6.31
7.05
7.21
K02755 0.13
10.706
No
–
2.41 4.26 6.73
7.42
7.61
K02771 0.38
10.462
Yes
–
–
1.99 4.51
5.91
6.59
K02790 0.27a
11.486
Yes
–
–
3.28 5.56
6.05
5.98
K02792 0.36
9.761
No
–
–
2.03 4.38
5.37
5.81
K02798 0.27
11.470
No
–
–
3.33 5.44
5.68
5.77
K02801 0.18
9.472
No
–
1.66 3.25
5.5
6.04
6.52
K02803 0.2
10.642
Yes
–
1.77 3.43 5.85
6.53
6.55
K02805 0.39
11.919
No
–
–
2.27 4.85
5.43
5.71
K02813 0.34
11.514
Yes
–
–
2.51 4.56
5.36
5.39
254
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.2—Continued
FWHM
2MASS
Companion
Limiting ∆Ks for Annulus Centered At
(## )
(Ks)
within 10##
0.## 1
K02829 0.19
11.444
Yes
K02833 0.15
11.123
K02838 0.44
1.## 0
2.## 0
4.## 0
–
1.92 4.01 6.63
7.14
7.25
Yes
–
2.52 4.64 7.14
7.52
7.54
11.857
Yes
–
–
1.93 4.28
4.68
4.85
K02840 0.29
12.261
Yes
–
–
3.32
5.2
5.4
5.36
K02859 0.25
12.052
No
–
–
3.23 5.32
5.72
5.76
K02867 0.49
10.475
No
–
–
1.64 3.99
5.3
6.09
K02879 0.16a
11.099
Yes
–
2.11
4.0
5.83
6.5
6.8
K02904 0.18a
11.359
Yes
–
2.27
4.1
6.46
7.61
7.63
K02913 0.23
11.361
Yes
–
–
3.43 5.83
6.34
6.4
K02914 0.14
11.006
Yes
–
2.84 4.86 7.32
7.74
7.92
K02915 0.5
11.956
Yes
–
–
1.67 4.22
4.72
5.05
K02916 0.28
12.402
No
–
–
3.33 4.73
4.91
4.89
K02936 0.2
12.764
No
–
1.99 3.92 4.91
4.94
4.93
K02939 0.25
12.006
Yes
–
–
3.4
5.17
5.28
5.32
K02948 0.38
10.322
No
–
–
2.01
4.4
5.74
6.42
K02951 0.32
11.816
No
–
–
2.96 4.82
4.95
4.94
K02961 0.16
11.290
Yes
–
2.47 4.52 6.87
7.3
7.3
Object
255
0.## 2
0.## 5
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.2—Continued
FWHM
2MASS
Companion
Limiting ∆Ks for Annulus Centered At
(## )
(Ks)
within 10##
0.## 1
0.## 2
K02968 0.42
10.293
No
–
–
K02970 0.19
11.566
Yes
K02971 0.13
11.438
K02977 0.31
1.## 0
2.## 0
4.## 0
1.69 3.91
5.44
5.97
–
2.01 4.09 6.76
7.38
7.29
Yes
–
2.67 4.79 6.87
7.3
7.34
12.355
No
–
–
2.75 4.61
4.75
4.82
K02984 0.3
11.637
Yes
–
–
2.9
5.6
6.24
6.42
K03008 0.14
10.694
No
–
2.58 4.45 6.93
7.54
7.76
K03015 0.23
11.774
Yes
–
–
3.48 6.12
6.58
6.71
K03017 0.22
11.757
No
–
–
3.8
5.75
6.02
6.02
K03038 0.29
12.537
No
–
–
0.14
0.3
0.63
0.69
K03060 0.26
11.554
No
–
–
3.28 6.12
6.61
6.7
K03075 0.23
11.532
Yes
–
–
3.7
6.16
6.75
6.87
K03083 0.28
11.401
No
–
–
2.77 5.31
6.09
6.43
K03085 0.31
12.706
No
–
–
2.93 4.68
4.85
4.7
K03097 0.5
10.649
No
–
–
1.51 3.79
5.03
5.78
K03111 0.18
11.353
Yes
–
2.16 4.26 6.62
6.99
7.05
K03117 0.23
11.508
Yes
–
–
3.65 5.59
5.87
6.0
K03122 0.14
10.819
Yes
–
2.8
4.8
7.88
7.88
Object
256
0.## 5
7.13
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Figure 4.4: Detected nearby stars (black crosses or stars) and detection limits (lines)
on the presence of additional stars. We highlight the detected companions and detection
limits for three systems: K01677 (blue), K02904 (magenta) and K02961 (green). K01677
is fainter than K02904 and K02961 by approximately 1.5 Ks magnitudes. The stars in the
vicinity of K00266, K00364, and K03242 were smeared by field rotation and are excluded
from this plot because their magnitudes were underestimated. All of the stars detected
around K00266, K00364, and K03242 are at separations of at least 3.## 5 and were identified
in UKIRT.
4.7
Comparison to Previous Surveys
As discussed in Section 4.5, we detected visual companions within 2## around 11 of the
81 targets that host planet candidates or confirmed planets. The overall companion rate
of 13% for planet (candidate) host stars is slightly lower than the rates of 20% and 17%
257
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Excludes Known False Positives
8
L13 low (338)
L13 med (598)
L13 high (299)
A12 (139)
A13 (18)
This Paper (126)
Kepmag
10
12
14
16
1
10
KOI Radius (REarth)
Figure 4.5: Kepler magnitudes of host stars versus the radii of associated planet candidates for the KOIs observed by Law et al. (2014) with Robo-AO (gray), Adams et al.
(2012) with ARIES and PHARO (teal diamonds), Adams et al. (2013b) with ARIES (purple triangles), and in this paper with ARIES (orange stars). The symbols for the Law
et al. (2014) targets indicate the photometric quality of the observations as described
in their Table 5. The KOI radii were obtained from the cumulative planet candidate
list at the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu/
cgi-bin/ExoTables/nph-exotbls?dataset=cumulative) and have not been corrected
for possible dilution due to the presence of nearby stars.
258
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
All KOIs (3286)
AO Targets (81)
AO, No Comp < 4" (56)
AO, Comp < 4" (25)
AO, Comp < 2" (11)
Cumulative Fraction
0.8
0.6
0.4
0.2
Excludes Known False Positives
0.0
6
8
10
12
14
16
Galactic Latitude (Deg)
18
20
Figure 4.6: Galactic latitude distribution of AO targets (black solid line), AO targets
without identified companions (blue dotted line), AO targets with companions identified
within 4## (orange dashed line), and AO targets with companions identified within 2## (red
dot-dash line) compared to the galactic latitude distribution of all KOIs (gray). We have
excluded all KOIs and AO targets that have been identified as false positives.
259
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Planet Radius (REarth)
10
Candidate (92)
Confirmed (34)
False Positive (9)
1
No Comp < 4" (93)
Comp < 4" (26)
Comp < 2" (10)
Comp < 1" (6)
1
10
Period (Days)
100
Figure 4.7: Radii and periods for the planet candidates (crosses), confirmed planets
(stars), and false positive KOIs (diamonds) orbiting the stars imaged in this study. Stars
for which we did not detect a visual companion within 4## are shown in teal and stars with
visual companions within 1## , between 1 − 2## , and between 2 − 4## are displayed in purple,
orange, and navy, respectively.
260
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
found in Adams et al. (2013b) and Adams et al. (2012), respectively and slightly higher
than the rate of 7.4% found by Law et al. (2014) using Robo-AO on the robotic Palomar
60-inch telescope. Within 3## we found companions for 14 (17%) of the 81 targets hosting
planet candidates or confirmed planets. This rate agrees well with the rate of 17% found
by Lillo-Box et al. (2012) using lucky imaging.
Due to the efficiency of Robo-AO observations, the Robo-AO sample of 715 KOIs
is much larger than the samples of 90, 12, 98, and 87 KOIs observed in Adams et al.
(2012), Adams et al. (2013b), Lillo-Box et al. (2012), and this paper, respectively. The
Robo-AO team was therefore able to divide their sample into different categories and
search for variations in the stellar multiplicity rate as a function of stellar or planetary
properties. They found a slight (1.6σ) discrepancy between the stellar multiplicity of
single KOI systems and multiple KOI systems, but the difference was not statistically
significant. We also found a higher companion fraction for the 56 single KOI systems
(18%) compared to the 26 multiple KOI systems (4%), which lends additional support
to the theory that single KOI systems are more likely to be false positives than multiple
KOI systems (Lissauer et al. 2012, 2014).
Comparing the companion rates from different studies is not straightforward due
to the small sample sizes of most of the studies and the differences in target sample
selection, observing strategy, sensitivity, and weather conditions. The targets discussed
in Law et al. (2014) were selected randomly with the express goal of reproducing the
general features of the full planet candidate population. In contrast, our observations and
those of Adams et al. (2012, 2013b) were prioritized to target small planet candidates
around bright or moderately faint (Kp ! 14) stars. As shown in Figure 4.5, the stars in
the Adams et al. (2012) sample are typically brighter than the stars observed by Law
261
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
et al. (2014) and the stars discussed in this paper. The median Kp magnitude of the
Adams et al. (2012) sample is Kp = 12.2 whereas the median magnitude of our sample
is Kp = 13.3. The Law et al. (2014) sample extends to even fainter magnitudes and has
a median magnitude of Kp = 13.7.
In addition, the surveys reached different detection limits and operated in different
bandpasses. In this paper and in Adams et al. (2013b), KOIs were observed in J or
Ks using ARIES on the MMT. Adams et al. (2012) presented J and Ks observations
acquired with both ARIES on the MMT and PHARO on the Palomar Hale 200 inch
telescope. Lillo-Box et al. (2012) conducted their observations in SDSS i and z bands
using AstraLux on the 2.2 m telescope at Calar Alto Observatory. Finally, Law et al.
(2014) observed their targets at visible wavelengths using an SDSS-i’ filter and a
long-pass filter (LP600) that selects wavelengths redder than 600 nm and cuts off near
1000 nm. The shape of the LP600 filter matches the red end of the Kepler bandpass,
so the contrast ratios measured in LP600 are more similar to the contrast ratios that
the stars would have in the Kepler bandpass than the contrast ratios measured at
near-infrared wavelengths.
In contrast, near-infrared observations are more sensitive to faint, red companions
that may be below the detection limit at visible wavelengths. For example, the faint
(∆Ks = 2.5, estimated ∆Kp = 3.1) companion to KOI 2159 that we discuss in
Section 3.2 was classified as a “likely” Robo-AO detection rather than a “secure”
detection because the detection significance was below their formal 5σ limit. Depending
on weather conditions and the magnitude of the host star, observations with ARIES or
PHARO may also have smaller inner working angles than Robo-AO observations. For
instance, the close-in (0.## 13) companion to KOI 1537 reported by Adams et al. (2012)
262
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
was too close to the target star to be resolved with Robo-AO.
As shown in Figure 4.6, we find that the stars with visual companions are slightly
more likely to be located at lower galactic latitudes than stars without identified
companions. This indicates that some of the visual companions identified within 4## are
likely to be background objects because the background density of stars is higher near
the galactic plane. Concentrating on the stars with companions identified within 2## ,
we see that the bias towards lower galactic latitudes is slightly reduced, as would be
expected if many of the companions identified within 4## are not physically bound to the
target star. Interestingly, none of our target stars have two detected companions within
4## whereas Adams et al. (2012) detected multiple stars within 4## near eight of their
90 targets and Lillo-Box et al. (2012) found that 3% of their targets had at least two
companions within 3## .
4.8
Conclusions
Our sample of target stars hosts 34 confirmed planets, 92 planet candidates, and 9 false
positive KOIs. In Figure 4.7 we display the radii and periods of these KOIs and denote
which objects orbit stars with detected visual companions. Four of the stars with
visual companions within 1## and all of the stars with companions within 2## host planet
candidates smaller than 1.5 R⊕ . In most cases, the estimated dilution corrections for
these systems are small enough that the planet radii would change by only a few percent
after accounting for the extra light in the aperture. In the extreme cases of KOI 2421 and
KOI 2790, however, the approximate dilution corrections of 23% and 17%, respectively,
would increase the radii of the associated planet candidates by over 0.15 R⊕ . The change
263
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
in the planet properties might be even larger if the planet candidates orbit the visual
companions instead of the target stars.
In addition to planet (candidate) host stars, our sample also includes 9 stars
that were previously identified as planet host stars but have been revealed to be false
positives. Our ARIES observations revealed visual companions within 4## of five of those
stars (K02159, K02298, K02771, K02838, and K02879). Although this paper focuses on
the search for companions to planet host stars, knowledge about the contamination of
the light curves of stars without detected planets is equally important for computing
planet occurrence rates. Most calculations of the frequency of planets (e.g., Catanzarite
& Shao 2011; Youdin 2011; Howard et al. 2012; Mann et al. 2012; Traub 2012; Dressing
& Charbonneau 2013; Gaidos 2013; Kopparapu 2013; Petigura et al. 2013a,b; Swift et al.
2013; Morton & Swift 2014; but see Fressin et al. 2013) neglect flux contamination from
nearby stars when estimating the smallest planet that could have been seen around a
particular star, but additional light from a companion star could dilute the transit signals
of small planets and render them undetectable. Failing to account for this dilution could
therefore lead to an overestimate of the search completeness and an underestimate of
the planet occurrence rate. In addition, stars with nearby visual companions of different
spectral types might be misclassified due to their unusual colors, further complicating
estimates of the search completeness.
For stars with companions closer than 2.## 0, we estimated the appropriate dilution
corrections for the radii of associated planet candidates. Depending on the magnitude
differences and angular separations between the target stars and the identified
companions, the approximate corrections to the planet radii varied from 0.2% to 23%.
Given that radial velocity observations (Weiss et al. 2013; Marcy et al. 2014; Weiss &
264
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Marcy 2014) and planet formation models (Lopez & Fortney 2013) have revealed that
the transition between rocky and gaseous planets occurs at roughly 1.5 R⊕ , this change
has important implications for frequency of rocky planets in the galaxy. If the radii of
many Kepler planet candidates are indeed underestimated, then the true frequency of
rocky planets may be lower than previously estimated. However, we must caution that
dilution from background stars will also make the detection of truly tiny planets more
challenging. Accordingly, the Kepler census of rocky planets may be less complete than
previously estimated.
In the next few years, observations of Kepler target stars with Gaia (Perryman et al.
2001) will help disentangle blended systems by providing distance estimates for the host
stars. In the case of blended systems in which the stars have nearly equal brightnesses,
the distance reported by Gaia will be roughly 1.4 times that estimated from photometry
alone. In that case, we will be able to infer that the system is a blend and that the radii
of any planet candidates within the system are underestimated. Until we receive the
Gaia data, however, we can inspect systems individually using ground-based observations
like those presented in this paper and serendipitous space-based observations from the
HST SNAP program (SNAP Program 12893; PI: R. Gilliland).
Acknowledgments
The authors gratefully acknowledge partial support from NASA grant NNX10AK54A.
CD is supported by a National Science Foundation Graduate Research Fellowship.
We thank David Ciardi for coordinating the Kepler Follow-up Observing Program.
We are grateful to Adam Kraus and the anonymous referee for providing helpful
265
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
suggestions to improve the paper. This research has made use of the Kepler Community
Follow-Up Observing Program website (https://cfop.ipac.caltech.edu) and the
NASA Exoplanet Archive, which is operated by the California Institute of Technology,
under contract with the National Aeronautics and Space Administration under the
Exoplanet Exploration Program.
Facilities: MMT (ARIES), Kepler
266
CHAPTER 4. AO IMAGES OF KEPLER TARGET STARS
Table 4.2—Continued
FWHM
2MASS
Companion
Limiting ∆Ks for Annulus Centered At
(## )
(Ks)
within 10##
0.## 1
0.## 2
0.## 5
1.## 0
2.## 0
4.## 0
K03128 0.51
11.900
Yes
–
–
–
4.29
4.88
5.14
K03242 0.1
11.520
Yes
5.7
7.96
8.57
8.6
Object
a
2.48 3.59
FWHM determined using PSF fitting for stars with close companions.
267
Chapter 5
The Mass of Kepler-93b and the
Composition of Terrestrial Planets
This thesis chapter originally appeared in the literature as
C. D. Dressing, D. Charbonneau, X. Dumusque, S. Gettel,
F. Pepe, A. Collier Cameron, D. W. Latham, E. Molinari, S.
Udry, L. Affer, A. S. Bonomo, L. A. Buchhave, R. Cosentino,
P. Figueira, A. F. M. Fiorenzano, A. Harutyunyan, R. D.
Haywood, J. A. Johnson, M. Lopez-Morales, C. Lovis, L.
Malavolta, M. Mayor, G. Micela, F. Motalebi, V. Nascimbeni,
D. F. Phillips, G. Piotto, D. Pollacco, D. Queloz, K. Rice, D.
Sasselov, D. Ségransan, A. Sozzetti, A. Szentgyorgyi, C. Watson,
The Astrophysical Journal, 800, 135, 2015
268
CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
Abstract
Kepler-93b is a 1.478 ± 0.019 R⊕ planet with a 4.7 day period around a bright
(V = 10.2), asteroseismically-characterized host star with a mass of 0.911 ± 0.033 M$
and a radius of 0.919 ± 0.011 R$. Based on 86 radial velocity observations obtained
with the HARPS-N spectrograph on the Telescopio Nazionale Galileo and 32 archival
Keck/HIRES observations, we present a precise mass estimate of 4.02 ± 0.68 M⊕ . The
corresponding high density of 6.88 ± 1.18 g/cc is consistent with a rocky composition of
primarily iron and magnesium silicate. We compare Kepler-93b to other dense planets
with well-constrained parameters and find that between 1 − 6 M⊕ , all dense planets
including the Earth and Venus are well-described by the same fixed ratio of iron to
magnesium silicate. There are as of yet no examples of such planets with masses > 6 M⊕ :
All known planets in this mass regime have lower densities requiring significant fractions
of volatiles or H/He gas. We also constrain the mass and period of the outer companion
in the Kepler-93 system from the long-term radial velocity trend and archival adaptive
optics images. As the sample of dense planets with well-constrained masses and radii
continues to grow, we will be able to test whether the fixed compositional model found
for the seven dense planets considered in this paper extends to the full population of
1 − 6 M⊕ planets.
5.1
Introduction
Small planets are abundant in the galaxy, but the compositional diversity of small planets
is not well understood. Theoretical models of planet formation predict that planets
269
CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
intermediate in size between Earth and Neptune could be gaseous “mini-Neptunes,”
water worlds, or rocky “Super-Earths” (Kuchner 2003; Léger et al. 2004; Valencia et al.
2006; Seager et al. 2007; Fortney et al. 2007; Rogers et al. 2011; Lopez et al. 2012; Zeng &
Sasselov 2013). Recent studies have explored the compositional diversity of small planets
using hierarchical Bayesian modeling of the observed planet radii and measured planet
masses (Rogers 2015) or theoretical models (Wolfgang & Lopez 2014), but a thorough
investigation of planet densities is hindered by the small number of small planets with
well-measured masses and radii. There are currently only nine planets smaller than
2.7 R⊕ with masses measured to 20% precision: 55 Cnc e (Gillon et al. 2012; Nelson et al.
2014), CoRoT-7b (Barros et al. 2014; Haywood et al. 2014), GJ1214b (Charbonneau
et al. 2009), HD97658b (Dragomir et al. 2013), HIP116454b (Vanderburg et al. 2015),
Kepler-36b (Carter et al. 2012), Kepler-78b (Pepe et al. 2013; Howard et al. 2013), and
Kepler-10b and 10c (Dumusque et al. 2014).
The host star Kepler-93 (KIC 3544595, KOI 69) is one of the brightest stars observed
by Kepler (V = 10.2, Kp = 9.93), enabling very high precision photometry of 17 ppm on
six-hour timescales (Christiansen et al. 2012). Kepler observed Kepler-93 throughout the
baseline mission (Quarters 0–17) and conducted observations at short cadence (exposure
time of 58.5 s) beginning in Quarter 2 and extending until the end of the mission. Due to
the high photometric precision of the Kepler-93 observations, the planet was detected in
the first four months of Kepler data (Borucki et al. 2011b). Marcy et al. (2014) acquired
32 Keck HIRES radial velocity observations of Kepler-93 from July 2009 - September
2012 and provided an estimate of 2.6 ± 2.0 M⊕ for the mass of Kepler-93b. Marcy et al.
(2014) also noted a large linear RV trend of 11.2 ± 1.5 m s−1 yr−1 and calculated lower
limits on the mass and period of the perturbing companion of M > 3MJup and P > 5 yr.
270
CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
Incorporating an additional 14 spectra from the 2013 observing season, the HIRES mass
estimate for Kepler-93b increased to 3.8 ± 1.5 M⊕ (Ballard et al. 2014). Nonetheless,
the 40% error on the mass measurement allows a wide range of planetary compositions
including a rocky body, an ice world, and even a substantial primordial envelope of
hydrogen and helium (Ballard et al. 2014).
In contrast, the properties of the host star Kepler-93 are well-constrained. Using
37 months of Kepler short cadence data, Ballard et al. (2014) conducted an asteroseismic
investigation to characterize Kepler-93 in exquisite detail. They estimated an average
stellar density of 1.652 ± 0.006 g cm−3 , a stellar mass of 0.911 ± 0.033 M$ , and a stellar
radius of 0.919 ± 0.011 R$. Adopting priors from their asteroseismic investigation, they
fit the Kepler photometry to obtain a precise radius estimate of 1.478 ± 0.019 R⊕ for
Kepler-93b.
In addition to characterizing the host star, Ballard et al. (2014) present a variety
of evidence that Kepler-93b is a bona fide planet rather than an astrophysical false
positive. First, they report that the steep shape of the ingress and egress portions of the
Kepler-93b light curve cannot be reproduced by a non-planetary companion. Second,
they note that the infrared transit depth they measured with the Spitzer Space Telescope
is consistent with the planetary interpretation of Kepler-93b. Third, they place stringent
limits on the presence of nearby stars based on Keck AO images (Marcy et al. 2014).
Fourth, they state that the stellar density derived from the transit duration (Seager &
Mallén-Ornelas 2003; Nutzman et al. 2011) is consistent with the asteroseismic stellar
density constraint, indicating that the planet likely orbits the target star rather than the
companion causing the large RV trend.
271
CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
In this paper we refine the mass measurement of Kepler-93b from 2.5σ to 6σ
by analyzing two seasons of HARPS-N radial velocities in addition to the publicly
available HIRES data. We discuss these observations and our data reduction methods
in Section 5.2. In Section 5.3, we develop a model to fit the observed radial velocities.
Finally, we discuss the implications of the resulting planet mass and present our
conclusions in Section 5.4.
5.2
Observations & Data Reduction
We obtained 86 spectra of Kepler-93 using the HARPS-N spectrograph on the 3.57-m
Telescopio Nazionale Galileo (TNG) at the Observatorio del Roque de los Muchachos.
HARPS-N is a high-precision, vacuum-stabilized, high-resolution (R * 115, 000) echelle
spectrograph. The design is very similar to the design of the original HARPS instrument
at the ESO 3.6-m (Mayor et al. 2003). The main differences are that HARPS-N is fed
by octagonal fibers rather than circular fibers to improve the scrambling of the light and
features a monolithic 4096 x 4096 CCD instead of the dual CCD configuration used for
the HARPS focal plane (Cosentino et al. 2012).
We acquired 38 and 49 HARPS-N observations of Kepler-93 during the 2013 and
2014 observing seasons, respectively. In most cases, we used an exposure time of
30 minutes and achieved a mean S/N per extracted pixel of 103 at 550nm. (Four of
the spectra had an exposure time of 15 minutes and one had 27 minutes; these were
all gathered in July 2013.) One of the observations collected in 2013 was contaminated
by light from a mercury lamp and was therefore removed from the analysis. The final
HARPS-N dataset analyzed in this paper consists of 86 spectra. In most cases (75 of
272
CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
86 spectra), we observed Kepler-93 using simultaneous thorium argon (observing mode
HARPN ech obs thosimult). The remaining eleven observations were obtained without
simultaneous thorium argon in observing mode HARPN ech obs objAB.
We reduced the data with the standard HARPS-N pipeline by cross-correlating the
observed spectra with a numerical mask based on the spectrum of a G2V star (Baranne
et al. 1996; Pepe et al. 2002). We provide the resulting RVs and their 1σ errors in
Table 5.1 along with the observation BJDs, exposure times, bisector spans, and stellar
#
activity levels as measured by the Ca II log(RHK
) activity indicator (Noyes et al. 1984).
The BJDs in Table 5.1 are provided in UTC, but we converted the times to TDB (the
units used by the Kepler mission) using the IDL routine utc2bjd.pro1 prior to fitting
the RVs. We did not find evidence for a correlation between RV and bisector span or
#
log(RHK
).
5.3
Analysis of the Radial Velocity Data
Our full data set included RVs from four seasons of HIRES observations (2009 July –
2012 September) and two seasons of HARPS-N observations (2013 June – 2014 October).
We fit the combined HARPS-N and HIRES data set by incorporating a single offset
RVoff between the HIRES and HARPS-N data. We used the following general model:
M(ti ) = γ + RVoff + β(ti )
+ K [cos(θ(ti , TC , P, e) + ω) + e cos ω]
1
http://astroutils.astronomy.ohio-state.edu/time/pro/utc2bjd.pro
273
(5.1)
CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
Table 5.1. HARPS-N Radial Velocity Observations of Kepler-93
BJDUTC
-2450000
RV (m/s)
#
log(RHK
) (dex)
texp
Error
(m/s)
Value
Error
(s)
2456462.686262 27335.24
1.02
-28.39
-5.01
0.01
1800
2456463.584483 27337.75
0.94
-31.63
-5.00
0.01
1800
2456464.609617 27331.57
1.75
-27.24
-5.04
0.02
1800
2456465.606438 27342.34
1.00
-32.74
-5.02
0.01
1800
2456466.608850 27337.58
0.86
-29.26
-5.00
0.01
1800
...
...
...
...
...
...
Value
Bisector
...
Note. — (This table is available in its entirety in a machine-readable form in the online
journal. A portion is shown here for guidance regarding its form and content.)
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
where γ is the systemic velocity of Kepler-93, RVoff = RVHARPS−N − RVHIRES is the
offset between the HIRES and HARPS-N RVs, β(ti ) is a long-term RV trend due to a
third component in the system, K is the semi-amplitude due to Kepler-93b, and ω is
the argument of periastron. The function θ is the true anomaly of Kepler-93b at time ti
and depends on the period P , epoch of transit TC , eccentricity e. When fitting eccentric
orbits, we used IDL routine keplereq.pro written by Brian Jackson2 to solve Kepler’s
equation for the eccentric anomaly. The routine uses the method suggested by Mikkola
(1987) as an initial guess.
We considered linear and quadratic parameterizations of the long-term trend β(ti )
and circular and eccentric orbits for Kepler-93b. For all models, we determined an initial
solution using the Levenberg-Marquardt minimization algorithm as implemented by
lmfit in IDL. We then explored the region of parameter-space near the best-fit solution
using a Bayesian Markov Chain Monte Carlo analysis with a Metropolis-Hastings
acceptance criterion (Metropolis et al. 1953). We initialized N chains, where N was twice
the number of free parameters in the chosen model. We selected different initial positions
for each chain by perturbing each free parameter of the best-fit solution by a random
number drawn from a distribution with a width of five times the step size. We tuned
the step sizes such that the acceptance fractions for each parameter were 10–30%. For
the MCMC analysis, we set uniform priors for all parameters except the orbital period
and epoch of transit. We allowed only non-negative values for the RV semi-amplitude K
and the separate stellar jitter terms σsj for the HIRES and HARPS-N observations (see
below).
2
http://www.lpl.arizona.edu/~bjackson/idl_code/keplereq.pro
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
The Kepler photometry places tight constraints on the period and epoch of transit
(Ballard et al. 2014). We incorporated this knowledge into our MCMC analysis by
including Gaussian priors on period and transit epoch in the likelihood calculation. As
shown in Dumusque et al. (2014), using the tight prior from Kepler photometry when
fitting a circular model to RV observations of a planet in a circular orbit yields a result
very similar to that from a combined photometric and spectroscopic fit. We also tested
fitting the data while allowing the epoch of transit to float and find the epoch of transit
at BJD = 2454944.29514. This epoch differs from the value determined by Ballard
et al. (2014) by 4 minutes (0.3σ). The possible shift in the transit center is therefore
insignificant. Accordingly, we adopt the photometric ephemeris determined by Ballard
et al. (2014).
In our calculations, we shifted the epoch of transit close to the start of the HARPS-N
RVs to reduce error propagation. We increased the efficiency of our model fits by
√
√
parameterizing eccentric models using e cos(ω) and e sin(ω) rather than varying e
and ω directly (Ford 2006; Eastman et al. 2013). As in Dumusque et al. (2014), we
accounted for stellar activity by incorporating a stellar jitter term σsj in our adopted
likelihood L:
L=
N
,
i=1

/
1
2π(σi2
+
2
σsj
)
0
exp −
(RV (ti ) − M(ti ))
2
2(σi2 + σsj
)
2
1


(5.2)
where RV (ti ) is the measured RV at each time ti in the set of N observations, M is the
model, σi is the instrumental noise listed in Table 5.1, and the stellar jitter noise σsj is
allowed to adopt a different constant value for the HARPS-N and HIRES data.
We ran each chain for a minimum of 104 steps and checked for convergence by
computing the Gelman-Rubin potential scale reduction factor R̂ for each parameter
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
(Gelman et al. 2004). We stopped the MCMC analysis when R̂ < 1.03 for all parameters.
Next, we accounted for “burn-in” by identifying the point in each chain at which the
likelihood first became higher than the median likelihood of the chain and removing all
earlier steps. After combining all of the chains, we selected the median values of each
parameter as the best-fit value and assigned symmetric errors encompassing 68% of
values closest to the adopted best-fit value.
We then used Bayesian statistics to determine which of the models considered best
describes the data. We followed the method of Chib & Jeliazkov (2001) as described in the
Appendix of Haywood et al. (2014) to calculate the Bayes factor between pairs of models
using the posterior distributions and acceptance probabilities from our MCMC analyses.
This method was previously used by Dumusque et al. (2014) to compare RV models of
the Kepler-10 system. We found that penalties incurred by the additional complexity of
fitting the orbit of Kepler-93b with an eccentric model or fitting the long-term trend with
a quadratic model outweighed the improvement in the likelihood. We also compared
the models by holding the stellar jitter terms fixed to σsj,HARPS−N = 1.56 m s−1 and
σsj,HIRES = 2.03 m s−1 and computing the Bayesian Information Criterion (BIC, Schwarz
1978) and finite sample Akaike Information Criterion (AICC , Hurvich & Tsai 1989).
When considering only the HARPS-N data, we found that the model with a quadratic
trend and a circular orbit for Kepler-93b was preferred over the models with a linear
trend and circular orbit (∆BIC = 6.1, ∆AICC = 8.3), linear trend and eccentric orbit
(∆BIC = 12.4, ∆AICC = 10.4), or quadratic trend and eccentric orbit (∆BIC = 6.6,
∆AICC = 2.5).
Nonetheless, when we included the HIRES data we found that the simplest model
(linear trend and circular orbit) was preferred over the models with a linear trend
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
and eccentric orbit (∆BIC = 5.6, ∆AICC = 0.7), quadratic trend and circular orbit
(∆BIC = 4.7, ∆AICC = 2.1), or quadratic trend and eccentric orbit (∆BIC = 10.5,
∆AICC = 3.0). We therefore treat the perturbation from Kepler-93c as a linear trend
and model the orbit of Kepler-93b as circular. (For the eccentric fits, we found a median
eccentricity of 0.15 and an upper limit of e < 0.31 with 95% confidence.) We present the
resulting system properties including a mass estimate for Kepler-93b of 4.02 ± 0.68 M⊕
in Table 5.2 and display the measured RVs and the best-fit model in Figure 5.1. As
highlighted in Figure 5.2, the HARPS-N residuals are gaussian with a distribution
centered on zero and containing 68% of the data within a half width 1.6 m s−1 . For the
HIRES residuals the region encompassing 68% of the data has a half width of 3.4 m s−1 .
The expected circularization timescale for Kepler-93b is significantly shorter than
the 6.6 ± 0.9 Gyr age of the star (Ballard et al. 2014). Following Goldreich & Soter
(1966), we calculated a tidal circularization timescale of 75 Myr for a 4.02 M⊕ , 1.48 R⊕
planet in an orbit with a = 0.053 AU around a 0.91 M$ star. We assumed Q = 100 based
on the tidal quality factors estimated for terrestrial planets in the Solar System (Yoder
1995; Henning et al. 2009). Obtaining a tidal circularization timescale similar to the age
of the system would require Q = 9000, comparable to the estimate for Neptune (Zhang
& Hamilton 2008). Although the tidal circularization argument is consistent with the
preference for a circular orbit, we caution that the tidal quality factors for exoplanets are
largely unknown.
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
Table 5.2. Parameters for the Kepler-93 System
Value and
1σ Errors
Parameter
Ref.
Kepler-93 (star) = KIC 3544595 = KOI 69
19h 25m 40.s 39
+38d 40m 20.s 45
9.931
8.370
5669 ± 75
0.919 ± 0.011
0.911 ± 0.033
−0.18 ± 0.10
4.470 ± 0.004
6.6 ± 0.9
27337.89 ± 0.51
27304.1 ± 1.5
1.58 ± 0.19
2.09 ± 0.71
1,2
1,2
3
3
4
4
4
4
4
4
5
5
5
5
4.72673978 ± 9.7 × 10−7
2454944.29227 ± 0.00013
0.014751 ± 0.000059
12.496 ± 0.015
89.183 ± 0.044
0.1765 ± 0.0095
0 (fixed)
1.63 ± 0.27
4
4
4
4
4
4
5
5
1.478 ± 0.019
4.02 ± 0.68
6.88 ± 1.18
3.26 ± 0.07
0.053 ± 0.002
1037 ± 13
4
5
5
5
5
4
12.0 ± 0.4
5
> 8.5
> 10
5
5
Right ascension
Declination
Kepler magnitude
2MASS K
Teff (K)
R∗ (solar radii)
M∗ (solar masses)
[F e/H]
log g
Age (Gyr)
Systemic Velocitya(m s−1 )
HIRES Offset (m s−1 )
RV Jitter (HARPS-N)
RV Jitter (HIRES)
Kepler-93b (planet) = KOI 69.01
Transit and orbital parameters
Orbital period P (days)
Transit epoch TC (BJD)
Rp /R∗
a/R∗
Inc (deg)
Impact parameter
Orbital eccentricity e
RV semi-amplitude K (m s−1 )
Planetary Parameters
Rp ( R⊕ )
Mp ( M ⊕ )
ρp (g cm−3 )
log gp (cgs)
a (AU)
Teq (K)b
Kepler-93c (companion)
Fit Parameters
Acceleration (m s−1 yr−1 )
Companion Limits
Mass (MJ )
Orbital period P (yr)
Note. — References: (1) Høg et al. (1998), (2) Høg et al. (2000), (3) Brown et al. (2011), (4) Ballard et al. (2014),
(5) This Paper.
a Systemic
velocity at BJD 2456461.57573945.
b Assuming
a Bond albeo of 0.3.
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
10
HIRES (32)
HARPS−N (86)
5
RV − 27337.88 (m/s)
ΔRV (m/s)
0
0
−5
−20
−10
10
−40
O−C (m/s)
5
10
0
O−C (m/s)
5
−5
0
−5
−10
0
−10
500
1000
BJD − 2455000
1500
2000
−0.2
0.0
0.2
0.4
0.6
Phase
0.8
1.0
1.2
Figure 5.1: Best-fit model (black) for the Kepler-93 system and measured HIRES (light
blue) and HARPS-N (dark blue) RVs after correcting for the offset between HIRES and
HARPS-N. The errors include contributions from both instrumental noise and stellar
jitter. Top Left: Measured RVs versus time after removing the signal of the planet
Kepler-93b. Bottom Left: RV residuals versus time after removing the full planet+trend
fit. Top Right: Phase-folded signal of Kepler-93b after removing the long-term trend
due to Kepler-93c. The large red circles with error bars show the weighted mean and
corresponding uncertainties of the measured RVs, conveniently binned to equal arbitrary
intervals in phase. The points shown in gray are repeated to better reveal the behavior
of the data near phase=0. Bottom Right: RV residuals versus phase after removing the
full planet+trend fit. The red circles are the binned data.
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
25
Median
68%
HIRES (32)
HARPS−N (86)
Count
20
15
10
5
0
−5
0
Residuals (m/s)
5
Figure 5.2: Histogram of the residuals of the HIRES (light blue) and HARPS-N (dark
blue) observations. The dashed lines mark the median of each distribution (-0.16 m/s
for HIRES, 0.002 m/s for HARPS-N) and the dot-dash lines encompass 68% of the measurements. The half width of the 68% interval is 1.6 m s−1 for the HARPS-N data and
3.4 m s−1 for the HIRES data.
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
10
1000
Semimajor Axis (AU)
100
100
RV Baseline
Mass (MJ)
Keck AO
10
RV Trend
0.1
1.0
Separation (")
Figure 5.3: Limits on the mass and separation of the companion Kepler-93c. The Keck
AO observations exclude a companion within the blue region. The combined HIRES and
HARPS-N RVs exclude the teal region due to the amplitude of the trend and the maroon
region due to the baseline of the observations. Kepler-93c is therefore constrained to lie
within the white region. The dashed purple line divides substellar and stellar companions.
These limits assume that the companion has an orbit with i = 90◦ and e = 0.
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
Marcus+ 2010
Zeng + Sasselov 2013
This Paper
Planet Radius (REarth)
2.5
GJ 1214b
2.0
CoRoT-7b
1.5
1.0
O
H2O
%
2
5
O
H
2 37
% HgSiO3 50%
0
0
1
M SiO
25% Mg
50%
Earth
Venus
0.7
1
K10b
HD97658b
K10c
HIP 116454b
3
SiO
g
M
0%
0
3
55Cnc e 1
SiO
Mg
%
50
Fe
%
50
e
%F
0
0
1
K93b K36b
K78b
2
3
4 5
7
Planet Mass (MEarth)
10
20
Figure 5.4: Mass-radius diagram for planets smaller than 2.7 R⊕ with masses measured
to better than 20% precision. The shaded gray region in the lower right indicates planets
with iron content exceeding the maximum value predicted from models of collisional
stripping (Marcus et al. 2010). The solid lines are theoretical mass-radius curves (Zeng &
Sasselov 2013) for planets with compositions of 100% H2 O (blue), 25% MgSiO3 – 75% H2 O
(purple), 50% MgSiO3 – 50% H2 O (green), 100% MgSiO3 (black), 50% Fe – 50% MgSiO3
(red), and 100% Fe (orange). Our best-fit relation based on the Zeng & Sasselov (2013)
models is the dashed light blue line representing an Earth-like composition (modeled as
17% iron and 83% magnesium silicate using a fully-differentiated, two-component model).
The shaded region surrounding the line indicates the 2% dispersion in radius expected
from variation in Mg/Si and Fe/Si ratios (Grasset et al. 2009).
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
5.3.1
Limits on the Properties of Kepler-93c
The baseline of our RV data is too short to measure the period and minimum mass of
the perturber responsible for the long-term trend, but we can place lower limits on the
companion properties. Wang et al. (2014a) conducted a similar analysis of the properties
of Kepler-93c based on AO observations and Keck/HIRES RVs. They found a linear RV
trend of 12.2 ± 0.2 m s−1 yr−1 and argued that Kepler-93c is most likely to have a mass
below 101 MJ and a semi major axis a = 15.5 − 33 AU if it is a stellar companion. For
the substellar case, they found limits of a = 5.5 − 27.6 AU and M = 10 − 80MJ .
Our additional two years of HARPS-N observations have allowed us to further
restrict the allowed parameter space for Kepler-93c. We measured a linear trend
of 12.0 ± 0.4 m s−1 yr−1 for 5 years, implying that Kepler-93c has P > 10 yr and
M > 8.5 MJ . Assuming the 100 pc distance to Kepler-93 estimated by Ballard et al.
(2014), the resulting semimajor axis a > 4.5 AU corresponds to an angular separation
of 0.## 045. At this separation, the detection limit from Keck AO imaging is 1.7 Ks
magnitudes fainter than Kepler-93. We can therefore place an upper limit of Ks > 10.1
on Kepler-93c unless Kepler-93c happened to have an orbital geometry precluding
detection at the epoch of the Keck observations. Converting the Ks upper limit into a
mass limit via the Delfosse et al. (2000) relation3 and the distance, we found a mass
upper limit of 0.64 M$ for angular separations beyond 0.## 045. We display the combined
3
The Delfosse relation predicts stellar mass from KsCIT whereas the Keck observations were ac-
quired in Ks2MASS . We converted between the two systems assuming a color of J − K = 1 and
using the color-dependent conversions provided at http://www.astro.caltech.edu/~jmc/2mass/v3/
transformations/.
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
limits from the AO and RV data in Figure 5.3. In the future, astrometric measurements
from Gaia (Perryman et al. 2001) will likely provide additional constraints on the
properties of the Kepler-93 system. We will then be able to investigate the dynamical
history of the system and test whether Kepler-93c might be responsible for scattering
Kepler-93b inward onto a short-period orbit.
5.4
Discussion and Conclusions
Combining our estimate of 4.02 ± 0.68 M⊕ for the mass of Kepler-93 with the radius
estimate of Rp = 1.478 ± 0.019 R⊕ from Ballard et al. (2014), we find a density of
6.88 ± 1.18 g/cc. In Figure 5.4, we show Kepler-93b on the mass-radius diagram. In
this diagram we plot only those planets smaller than 2.7 R⊕ and with masses determined
to a precision better than 20%. In addition to Venus and the Earth, there are ten
such planets. We observe that Kepler-93b falls in a cluster of planets with radii 50%
larger than that of the Earth, all of which have extremely similar densities: Kepler-10b
−3
(ρ = 5.8 ± 0.8 g cm−3 ; Dumusque et al. 2014), Kepler-36b (ρ = 7.46+0.74
−0.59 g cm ; Carter
et al. 2012), and CoRoT-7b (ρ = 6.56 ± 1.40 g cm−3 ; Barros et al. 2014; Haywood et al.
2014). This cluster falls upon a relation that includes Earth, Venus, and Kepler-78b
(Howard et al. 2013; Pepe et al. 2013), which is itself only 20% larger than the Earth. To
investigate this further, we used the two-component iron-magnesium silicate models of
Zeng & Sasselov (2013) to see if we could find a single composition that explained these
seven worlds.
For the solar system planets, we artificially include mass and radius errors equal
to the mean fractional errors for the exoplanets considered so that they do not have
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
undue influence on the resulting fit. We find the lowest χ2 for a model composition of
83% MgSiO3 and 17% Fe. We arrive at the same best-fit relation when we exclude Earth
and Venus. We caution that the two-component models used in this analysis make two
simplifying approximations about the interior structure of planets that cause the core
mass fraction to be underestimated: (1) the core contains only iron and the mantle
contains only magnesium silicate and (2) the planet is completely dry with no water
content. Accordingly, we expect the actual core mass fraction to be slightly higher by
5-8% to account for incorporation of lighter elements like oxygen, sulfur, and silicon in
the core and the inclusion of water in the mantle. In addition, there could be a change of
roughly 2% towards higher or lower core fractions due to uncertainties in the equations
of state used in the model calculations. Our purpose in this exercise is to test whether
we can find one composition that successfully explains all seven planets, not to place
stringent constraints on the abundance of magnesium silicate or iron.
Intriguingly, all of these planets, which are smaller than 1.6 R⊕ , have a tight
dispersion around this best-fit compositional curve, suggesting that the distribution of
small planet compositions has low intrinsic scatter. In the solar system, the strong
agreement between abundance ratios of elements in meteorites and those of the solar
photosphere (Lodders 2003) is a key constraint by which we deduce the composition of
the interior of the Earth. Therefore, we might look to the bulk abundances of exoplanet
host stars for similar constraints on the interior compositions of their terrestrial planets.
Grasset et al. (2009) use a set of planetary models to investigate the dependence of planet
radii on elemental abundances. Varying the ratios of iron to silicate and magnesium to
silicate within the range observed for the photospheric abundances of nearby exoplanet
host stars (Beirão et al. 2005; Gilli et al. 2006), Grasset et al. (2009) predicted that the
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
radii of terrestrial planets would vary by roughly 2% at a given mass. Our findings are
in agreement with this picture: We measure a mean absolute deviation of 1.9% between
the estimated planet radii and the values predicted by a 83% MgSiO3 /17% Fe model
for planets less massive than 6 M⊕ . Indeed, rocky planets very close to their host stars
seem to obey a well-defined relationship between radius and mass, although with only
5 such examples outside the Solar system, the immediate task is to characterize other
terrestrial exoplanets with similar precision. Increasing the sample of small planets with
well-constrained masses and radii will allow us to learn whether additional rocky planets
could also be explained by a single mass-radius relation and investigate whether the
relation found for close-in planets extends to planets in more distant orbits.
Our mass-radius diagram also includes five planets more massive than 6 M⊕ : 55 Cnc
e, GJ1214b, HD97658b, HIP 116454b, and Kepler-10c. In contrast, none of these more
massive planets have a high density consistent with the best-fit magnesium silicate/iron
composition described above. In agreement with Rogers (2015), we find that planets
larger than approximately 1.6 R⊕ (e.g., more massive than approximately 6 M⊕ ) contain
significant fractions of volatiles or H/He gas. These planets appear to have a diversity
of compositions that is not well-explained by a single mass-radius relation (Wolfgang &
Lopez 2014).
The discussion above focused exclusively on planets smaller than 2.7 R⊕ with masses
measured to better than 20%. Some low-mass worlds with very low densities are known,
notably the Kepler-11 system (Lissauer et al. 2013) and KOI-314c (Kipping et al.
2014). Thus we are not proposing that all planets less massive than 6 M⊕ obey a single
mass-radius relation; rather, we suggest that the rocky analogs of the Earth might do so.
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CHAPTER 5. THE COMPOSITION OF TERRESTRIAL PLANETS
Acknowledgments
The HARPS-N project was funded by the Prodex Program of the Swiss Space
Office (SSO), the Harvard University Origin of Life Initiative (HUOLI), the Scottish
Universities Physics Alliance (SUPA), the University of Geneva, the Smithsonian
Astrophysical Observatory (SAO), and the Italian National Astrophysical Institute
(INAF), University of St. Andrews, Queen’s University Belfast and University of
Edinburgh. The research leading to these results has received funding from the European
Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No.
313014 (ETAEARTH). C. D. is supported by a National Science Foundation Graduate
Research Fellowship. X. D. would like to thank the Swiss National Science Foundation
(SNSF) for its support through an Early Postdoc Mobility fellowship. P. F. acknowledges
support by Fundação para a Ciência e a Tecnologia (FCT) through Investigador FCT
contracts of reference IF/01037/2013 and POPH/FSE (EC) by FEDER funding through
the program “Programa Operacional de Factores de Competitividade - COMPETE”.
This publication was made possible through the support of a grant from the John
Templeton Foundation. The opinions expressed in this publication are those of the
authors and do not necessarily reflect the views of the John Templeton Foundation.
288
Chapter 6
Future Directions
Following the introduction to planets in Chapter 1, the remaining sections of this thesis
have addressed the frequency of small planets orbiting small stars (Chapters 2 and 3),
described a search for nearby stars that may be diluting the measured transit depths of
planet candidates or producing astrophysical false positives (Chapter 4), and explored
the composition of small planets (Chapter 5). Building on the previous material, this
section provides a glimpse of the future of exoplanet studies.
In Section 6.1, I describe the landscape of current and upcoming projects devoted
to the discovery and characterization of exoplanets. I pay particular attention to
space-based flagship missions and the next generation of extremely large ground-based
telescopes. Section 6.2 then provides context for the surveys described in Section 6.1 by
explaining how the expected survey precision will enable us to investigate new regions
of the planetary mass-radius diagram. In Section 6.3, I describe how the technique
of transmission spectroscopy can be used to explore the composition of planetary
atmospheres and partially resolve degeneracies in the bulk compositions of exoplanets.
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CHAPTER 6. FUTURE DIRECTIONS
Finally, Section 6.4 addresses the detectability of biosignatures in the atmospheres of
alien worlds and outlines a possible pathway towards the characterization of a potentially
habitable exoplanet.
6.1
Prospects for Detecting Small Planets Orbiting
Nearby Bright Stars
My graduate experience at Harvard coincided nearly perfectly with the Kepler era, but
Kepler is only one example of the remarkable array of projects that has contributed,
is currently contributing, or will contribute to improving our knowledge of planetary
systems. This section provides a brief introduction to several current and future planet
surveys. Like this thesis as a whole, the emphasis is on studying Earth-size planets and
super-Earths orbiting nearby, bright, and/or low-mass stars. Accordingly, this section
does not discuss the slate of current and future projects focused on direct imaging or
astrometry. The list of projects is restricted to missions and surveys for which first light
occurred or is expected to occur after 2000.
6.1.1
Kepler & K2
The underlying goal of the Kepler mission was to estimate the frequency of Earth-like
planets orbiting Sun-like stars and probe the frequency of planets across a wide range
of stellar and planetary properties. The transits of Earth-size planets orbiting Sun-like
stars are shallow, rare, and unlikely to occur (see Table 1.1), so the Kepler mission was
designed as a “pencil-beam” survey in which Kepler stared at a single 105 square degree
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CHAPTER 6. FUTURE DIRECTIONS
field of view nearly continuously for 3.5 years from spring 2009 until November 2012.1
The selected patch of sky was between the constellations of Cygnus and Lyra (centered
on RA=19h22m40s, Dec=+44◦ 30#00## ). The field was positioned slightly above the plane
of the galaxy (galactic coordinates l = 76.32◦ , b = +13.5◦ ) to reduce confusion from
background stars and far enough from the ecliptic plane to avoid flux contamination
from the Sun, Moon, or solar system planets.
Although a subset of the Kepler target stars were bright stars better suited for
detailed characterization and follow-up observations, 82% of the stars targeted by Kepler
during the main mission were fainter than Kp = 13.2 The relative faintness of the
Kepler target stars compared to the mean brightness of ground-based transit surveys
was a consequence of the adopted mission design. Bright stars are relatively scarce on
the celestial sphere, so a survey targeting bright stars must cover a vast area of sky or
obtain images covering widely separated small regions of the sky near selected targets.
In contrast, the pencil-beam approach adopted by Kepler led to a target list containing a
small number of very bright stars (2212 stars with Kp ≤ 10), a set of 13782 moderately
bright stars well-suited for RV follow-up (Kp = 10 − 12), a large sample of 110,634
moderately faint stars (Kp = 12 − 15), and a multitude of faint stars (80,594 with
Kp > 15) primarily useful for conducting statistical analyses of stellar and planetary
1
Kepler was later awarded a four-year extended mission; the selected field was actually observed for
over four years from 13 May 2009 until 11 May 2013.
2
The Kepler bandpass is a broadband filter extending from 420 nm to 900 nm that is roughly equivalent
to merging the V and R bandpasses (Gilliland et al. 2011). For sun-like stars, the brightness Kp in the
Kepler bandpass is roughly the same as the brightness in R. See http://keplergo.arc.nasa.gov/
CalibrationResponse.shtml for further details.
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CHAPTER 6. FUTURE DIRECTIONS
populations.
In May 2013, the NASA Kepler mission ended when the second of four reaction
wheels malfunctioned. At least three reaction wheels were required to allow the spacecraft
to point stably at the original field of view above the plane of the ecliptic, but the Kepler
team realized that the spacecraft could still point (slightly less stably) along the ecliptic
by using solar radiation pressure in place of a third working reaction wheel. The Kepler
spacecraft in its new mode of operation is now called K2. The K2 mission consists of a
series of approximately 75-day stares searching for planets around 10,000–20,000 stars
per field for a series of pointings along the ecliptic plane (Howell et al. 2014).
The pointing stability of K2 is lower than that of the original Kepler mission, but
custom K2 data reduction pipelines have enabled observers to obtain a precision roughly
a factor of two worse than during the main Kepler mission (Vanderburg & Johnson
2014). The K2 data have already yielded several planet detections, most notably the
first K2 planet (HIP 116454b, Vanderburg et al. 2015) and a system of three transiting
super Earths orbiting the nearby M dwarf EPIC 201367065 (Crossfield et al. 2015). The
three planet system is particularly intriguing because the outermost planet lies near the
“recent Venus” boundary of the habitable zone (Kopparapu et al. 2013b).
As the loss of the second reaction wheel brought the era of detecting small transiting
planets in long-period orbits to a close, the dawn of the K2 era opened up a new
realm of search space. Whereas Kepler focused primarily on stars too faint for detailed
ground-based follow-up observations, many K2 observing proposals feature brighter stars.
In theory, each K2 field should yield approximately the same number of short-period
super Earths that Kepler detected during the first few months of the main mission.
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These targets (the analogs to Kepler-10 and Kepler-93 in the K2 fields) are the ones
most amenable for radial velocity follow-up observations. Accordingly, the combination
of K2 light curves and ground-based follow-up observations with instruments such as
HARPS-N should significantly increase the number of small planets with well-constrained
masses and radii.
6.1.2
TESS
The upcoming Transiting Exoplanet Survey Satellite (TESS3 ) mission is a NASA
Explorer-class mission to search for planets orbiting bright stars across nearly the full
sky. TESS is scheduled to launch in August 2017 aboard a SpaceX Falcon 9 v1.1 rocket
and enter a highly elliptical orbit in a 2:1 resonance with the Moon. The four TESS
cameras will then begin to map the sky by staring at a series of 24◦ by 96◦ sectors for
27 days each. The sectors overlap at the ecliptic poles where a subset of targets will be
observed for nearly a full year. TESS will spend the first year mapping the Northern
sky and then flip to map the Southern sky in year two. The mapping configuration for a
possible extended mission is not yet finalized, but TESS might repeat the initial survey,
concentrate only on one hemisphere for two full years, or rotate the cameras horizontally
to search for planets orbiting stars in the ecliptic plane.
Whereas Kepler was a statistics mission designed to constrain the frequency of
planets in the galaxy, the main goal of TESS (and many K2 guest observer proposals)
is to identify small planets well-suited for mass measurement with ground-based
spectrographs (see Section 6.1.9) and possible atmospheric characterization with Spitzer,
3
http://tess.gsfc.nasa.gov/
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HST, or JWST (see Section 6.3). Due to the 27-day coverage that will be obtained for
most TESS target stars, the vast majority of TESS planet candidates will have orbital
periods shorter than 10 days.
For the full two-year TESS primary mission, Sullivan et al. (2015) predicted a yield
of 39 ± 4 Earth-size planets (< 1.25 R⊕ ), 265 ± 8 super-Earths (1.25 − 2 R⊕ ), 648 ± 17
mini-Neptunes (2 − 4 R⊕), and 133 ± 14 giant planets (> 4 R⊕ ). A small subset of planets
(17 ± 2) are predicted to be both small (Rp < 2 R⊕ ) and relatively cool, receiving between
half and twice the insolation received by the Earth. Almost all of these potentially
habitable planets are expected to orbit stars cooler than 4000K due to the shorter orbital
periods, deeper transit depths, and increased likelihood of transit for planets orbiting
low-mass stars compared to Sun-like stars (see Section 1.1).
6.1.3
CHEOPS
CHEOPS (CHaracterising ExOPlanet Satellite,4 Broeg et al. 2013; Fortier et al.
2014) is an approved ESA S(mall)-Class mission (ESA cost < 50M Euros; total cost
< 150M Euros)5 with a target launch date in 2017. In contrast to survey missions
like Kepler and TESS, CHEOPS is a 3.5-yr targeted mission focused on follow-up
observations of roughly 500 previously known planets and planet candidates with an
emphasis on super-Earths and Neptunes. The 32-cm CHEOPS telescope will conduct
4
http://cheops.unibe.ch/
5
This budget must also cover the cost of the launch vehicle. The nominal plan is to launch the relatively
lightweight CHEOPS spacecraft (expected mass < 250 kg) using a shared launch vehicle (CHEOPS Study
Team 2013).
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photometric measurements of bright (V < 12.5) planet host stars using a red bandpass
(0.4 − 1.1µm) and is expected to reach a precision of 150 ppm/min for a 9th magnitude
star. CHEOPS will both search for transits of planets detected by RV surveys and garner
additional transit observations for known transiting planets to improve the precision of
radius estimates. The planned sun-synchronous orbit will enable long, uninterrupted
observations, enabling CHEOPS to monitor a full transit event without breaks. The
goals of the CHEOPS mission are (1) probe the mass-radius relation for small planets
by refining planet radius estimates; (2) obtain phase curves to study heat redistribution
on Hot Jupiters; (3) identify attractive targets for further ground- and space-based
characterization with the ELTs and JWST.
6.1.4
JWST
The James Webb Space Telescope (JWST,6 Gardner et al. 2006) will be launched to L2
in 2018. As the successor to the Great Observatories, JWST has many scientific goals
and capabilities. For a detailed review of the exoplanet science that will be possible with
JWST, see Belu et al. (2011) and the recent white paper by Beichman et al. (2014).
In brief, JWST will feature four instruments with combined wavelength coverage
extending from 0.6–28µm. One of the main challenge for scheduling JWST observations
will be the necessity of observing multiple transits (2-4 depending on the chosen
instrument settings) in order to obtain information across the full wavelength range of
JWST (Beichman et al. 2014).
6
http://www.jwst.nasa.gov/
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The near-infrared (0.6–5µm) spectrograph NIRSpec7 (Ferruit et al. 2012) will
feature several resolution options ranging from an R=100 prism to an R=2700 grating
and will be useful for transit spectroscopy. NIRISS (the Near-InfraRed Imager and
Slitless Spectrograph, Doyon et al. 2012) will be well-suited for transmission spectroscopy
of brighter stars (down to a bright limit of J=7–8 mag) when used in the grism mode
(R=300–800, λ = 0.6 − 2.6µm). NIRCam (the Near Infrared Camera, Horner & Rieke
2004) will provide photometry at wavelengths between 0.7 − 5µm and grism spectroscopy
(R=1700) for stars as bright as K = 4 between 2.4 − 5µm. MIRI (the Mid-InfraRed
Instrument, Wright et al. 2004) will feature imaging for stars as bright as K = 6 at
8µm and medium resolution spectroscopy (R=3000) at 5 − 28.3µm for stars as bright as
K = 3 (Beichman et al. 2014).
Phase Curves with JWST
For highly irradiated small planets, long-duration phase curve observations by JWST will
probe heat redistribution (e.g., Knutson et al. 2007). On Mercury-like worlds without
signifiant atmospheres, longitudinal heat transport should be negligible. Accordingly, a
phase curve displaying significant longitudinal heat transfer (e.g., hot spot offsets) will
be an indication that the observed planet possess an atmosphere (Seager & Deming
2009). Although Spitzer phase curve measurements for super-Earths have not yet
yielded definite results, the six-fold increase in S/N expected for JWST should result in
meaningful phase curve measurements for planets as small as 4 R⊕ .
7
http://jwst.nasa.gov/nirspec.html
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Transmission Spectroscopy with JWST
For transmission spectroscopy, Batalha et al. (2014, see also the corresponding White
Paper8 ) conducted a series of simulations to assess the detectability of atmospheric
features in the atmospheres of super-Earths orbiting hypothetical nearby M dwarfs with
the same properties as GJ 1214. They considered planets with masses between 1 − 10 M⊕
and temperatures of 400K, 700K, and 1000K. Batalha et al. (2014) also considered two
types of atmospheres: (1) a hydrogen-rich atmosphere with a metal abundance three
times higher than the Sun and (2) a pure water atmosphere.
For planets with hydrogen-rich atmospheres, masses > 1 M⊕ , and temperatures
above 400K, they found that NIRSpec observations of 25 transit events would be sufficient
to obtain 15σ detections of water and methane at 2.7µm and 4.5µm, respectively. In
the case of pure water atmospheres with smaller scale heights, Batalha et al. (2014)
demonstrated that detecting water at the same 15σ level by obtaining 25 transits would
require a surface temperature above 700K.
Batalha et al. (2014) also considered the maximum distance out to which JWST
will provide high S/N detections of water and methane. In the hydrogen-rich case, they
found that the allowable distance limit extended from 11 pc for the coolest, lowest mass
planets (400K, 1 M⊕ ) to 50 pc for hotter, more massive super-Earths (1000K, 10 M⊕ ).
The distances are considerably reduced for planets with pure water atmospheres; water
should be detectable at 15σ in 25 transits out to distances of 26 pc for hot, massive
super-Earths (1000K, 10 M⊕ ), 9.2 pc for slightly cooler super-Earths (700K, 10 M⊕ ), and
7.7 pc for less massive super-Earths (1000K, 4 M⊕ ). See Section 6.4 for a more detailed
8
http://www.stsci.edu/jwst/doc-archive/white-papers/JWSTNirspec_BatalhaFinal.pdf
297
CHAPTER 6. FUTURE DIRECTIONS
discussion of the detection of potential biosignatures in exoplanet atmospheres.
In an independent study, Shabram et al. (2011) simulated the likelihood of
constraining the atmospheric composition of the transiting hot Neptune GJ 436b
(Butler et al. 2004; Gillon et al. 2007; Maness et al. 2007) using JWST transmission
spectroscopy. They found that NIRSpec observations of the 0.7 − 5µm region would be
useful for estimating the relative abundances of HCN and C2 H2 . Similarly, low-resolution
spectroscopy with MIRI at 5 − 10µm would probe the HCN and C2 H2 features near
7µm and further constrain atmospheric mixing. Previous studies have suggested that
the atmosphere of GJ 436b has a low CH4 abundance compared to that expected from
equilibrium chemstry(Stevenson et al. 2010; Madhusudhan & Seager 2011), so MIRI
observations spanning the 3 − 4µm CH4 feature would be particularly interesting.
Secondary Eclipses with JWST
In general, the detection of the secondary eclipse of a potentially habitable planet is a
daunting proposition. Potentially habitable planets are significantly smaller and cooler
than their host stars, so their luminosities are much lower. However, as explained in
Section 1.1, selecting smaller target stars can mitigate the observational challenge of
detecting a small planet. Furthermore, conducting secondary eclipse measurements at
longer wavelengths reduces the luminosity difference between the star and planet from
1010 to 107 (e.g., Seager & Deming 2010).
The secondary eclipse of a cool (300–350K) Earth-size planet should be detectable
by JWST/MIRI if the planet orbits a nearby late M dwarf. Specifically, the expected
S/N at 15µm during an individual 45-minute transit is 0.5 − 1 and the combination of
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25 secondary eclipse measurements should yield a 3 − 5σ detection. That simulation
considers both stellar and zodiacal noise and assumes a somewhat conservative noise
floor of 50 ppm. In an ideal case of an extremely nearby system (< 10 pc), a smaller
target star, or a higher intrinsic precision, JWST/MIRI observations may even reveal
CO2 features at 15µm (Beichman et al. 2014).
6.1.5
PLATO 2.0
PLATO 2.0 (PLAnetary Transits and Oscillations of stars,9 Rauer et al. 2014) is an
approved ESA M(medium)-Class mission (cost < 500M Euros) planned for launch by
2024. The design of PLATO 2.0 combines aspects of both Kepler and TESS. PLATO 2.0
will feature 34 small (120 mm) telescopes mounted on a single platform and will be
stationed in orbit around the L2 Lagrange point. Two of the cameras will be “Fast”
Cameras monitoring bright (V=4–8) stars at a cadence of 2.5 s and the remaining 32 will
be “Normal” Cameras observing fainter stars (V=8–16) at a longer 25 s cadence. The
Normal Cameras will be divided into 4 groups of eight cameras, each of which will look
at the same 1100 square degree field of view. The fields of view of each group of cameras
will partially overlap yielding a central regions with higher photometric precision.
The PLATO 2.0 mission will feature two “Long Stares” (three-year pointings at
single field for 2–3 years; one of these will include the Kepler field of view) and a “Step
and Stare” phase targeting different patches of the sky for 2–5 months each. In total,
PLATO 2.0 is expected to yield asteroseimic observations of roughly 85,000 stars, over
one million stellar light curves, and thousands of detected planets, a few hundred of
9
http://www.oact.inaf.it/plato/PPLC/Home.html
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which might be small planets in the habitable zones of GKM stars.
6.1.6
WFIRST-AFTA
The original strategy of the NASA WFIRST (Wide-Field Infrared Survey Telescope,10
Green et al. 2012) mission was to study dark energy and eoxplanets using a space-based
telescope with a diameter of 1.3–1.5 m. However, the National Reconnaissance Office
unexpectedly gifted NASA with two 2.4-m telescopes in 2013. NASA then convened a
science definition team (SDT) to study the feasibility of conducting the WFIRST mission
using the “Astrophysics Focused Telescope Assets” (AFTA). The SDT estimated that the
WFIRST-AFTA mission would require 82 months to progress from preliminary design to
launch (not including a 1-year Phase A). The nominal launch date for WFIRST-AFTA
is mid-2024.
The newly designed WFIRST-AFTA mission would last six years and feature a
chronograph, slit less grim, IFU, and an infrared imager with a 0.28◦ × 0.28◦ field of view
and a pixel scale of 0.## 11 per pixel. The exoplanet portion of WFIRST-AFTA science will
feature a microlensing survey for both bound and free-floating planets and direct imaging
observations of giant planets and debris disks. Spergel et al. (2015) predicted that the
WFIRST-AFTA microlensing survey will detect 2600 gravitationally bound exoplanets
with masses of 0.03 − 1000 M⊕ . The detected population was estimated to include
approximately 50 Mars-mass planets, 370 Earth-mass planets, and 1030 super-Earths.
While conducting the microlensing survey, WFIRST-AFTA is also expected to detect
roughly 20,000 transiting planets. These planets will be in short-period orbits with
10
http://wfirst.gsfc.nasa.gov/
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semimajor axes of a few 0.1 AU and could be as small as Neptune-radius. For direct
imaging observations, Spergel et al. (2015) calculated that WFIRST-AFTA could achieve
contrast ratios of 10−9 (and possibly better) for separations > 0.## 2.
6.1.7
Exo-C & Exo-S
In 2013, NASA convened two Science and Technology Definition Teams (STDTs) to
consider possible designs for an upcoming probe-scale (cost cap <$1B) mission to directly
image planets orbiting nearby stars11 . The two STDTs were charged with the task of
investigating concepts employing either an internal coronograph (the Exo-C concept) or
an external starshade (the Exo-S concept) to block out light from the planet host star.
Exo-C would be a three-year mission using a 1.5-m telescope in an Earth-trailing
orbit. The optical design featuring an internal coronograph would enable raw contrast
ratios of 10−9 within a 2 − 20λ/D field of view at wavelengths of 450–1000 nm. The
telescope would also feature an integral field spectrometer with spectral resolution of
R=25–70 (Exo-C Interim Report12 ).
Exo-S would be also be a three-year mission, but it would employ a 1.1-m
telescope and a 34-m starshade comprised of 28 7-m petals extending from a 20-m
inner circle. After launching on a shared vehicle, the starshade and telescope would
assume Earth-trailing orbits. Exo-S would offer three possible bands: (1) the blue band
(400–630 nm) with an inner working angle (IWA) of 75 mas, and a starshade/telescope
11
http://exep.jpl.nasa.gov/stdt/
12
http://exep.jpl.nasa.gov/stdt/Exo-C_InterimReport.pdf
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separation of 47,000 km, (2) the green band (510–825 nm) with an IWA of 95 mas,
and a separation of 37,000 km, and (3) the red band (600–1000 nm) with an IWA of
115 mas, and a separation of 30,000 km. Regardless of the choice of band, Exo-S will
offer three resolution options (full-band, three color, or high resolution) and optional use
of a polarizer when in the “full-band” mode. The planet camera and spectrometer will
have fields of view of 1’ and 0.## 2 × 0.## 2, respectively. Exo-S is expected to achieve contrast
ratios of 10−10 (Exo-S Interim Report13 ).
6.1.8
Current & Upcoming Ground-based Transit Surveys
APACHE (A PAthway to the Characterization of Habitable Earths
14
Sozzetti et al.
2013) is a transit survey for small planets orbiting nearby early- and mid-dwarfs
(M0–M5). Like MEarth, APACHE is an array of telescopes based at a single observatory.
The five 40-cm Ritchey-Chrétien APACHE telescopes are fitted with Johnson-Cousins R
& I filters and survey over 3000 M dwarfs from the vantage point of the Astronomical
Observatory of the Autonomous Region of Aosta Valley in the Italian Alps. The
APACHE survey began in 2012.
KELT (the Kilodegree Extremely Little Telescope, Siverd et al. 2009) is a
collaboration to search for transiting hot Jupiters orbiting bright stars (V=8–12) using
small (42 mm), wide-field (26◦ × 26◦ ) robotic telescopes. The twin KELT-North15 and
13
http://exep.jpl.nasa.gov/stdt/Exo-S_InterimReport.pdf
14
http://apacheproject.altervista.org/,http://www.oact.inaf.it/exoit/EXO-IT/Projects/
Entries/2011/12/31_APACHE.html
15
http://www.astronomy.ohio-state.edu/keltnorth/Home.html
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KELT-South16 telescopes began science operations at Winer Observatory in Arizona
in 2005 and at the South African Astronomical Observatory in Sutherland in 2009,
respectively.
HATNet (the Hungarian-made Automated Telescope Network,17 Bakos et al. 2004)
Exoplanet Survey is a search for transiting planets using a set of seven roboticallycontrolled 200-mm cameras attached to telescope mounts. Five of the telescopes are
located at the Fred Lawrence Whipple Observatory at Mt. Hopkins in Arizona and the
remaining two are stationed at Mauna Kea in Hawaii.
HATSouth18 (Bakos et al. 2013) is the southern interpretation of HATNet and
consists of six robotically-controlled stations of four co-mounted 180-mm telescopes. The
stations are distributed in pairs at Las Campanas Observatory in Chile, the High Energy
Stereoscopic System site in Namibia, and the Siding Spring Observatory in Australia.
HATSouth began operations in 2009, six years after the start of HATNet operations in
2003.
MASCARA (the Multi-site All-Sky CAmeRA,19 Snellen et al. 2012; Lesage et al.
2014) is a geographically dispersed set of 5–6 small telescopes that search for transiting
planets orbiting very bright (V=4–8) stars. Due to the relative scarcity of very bright
stars on the night sky, each MASCARA camera observes nearly the full visible sky. The
photometric precision goal is 0.1 − 0.3% for the brightest stars and < 1% variability
16
https://my.vanderbilt.edu/keltsouth/
17
http://hatnet.org/
18
http://hatsouth.org/
19
http://mascara.strw.leidenuniv.nl/
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for “faint stars” over one hour so that Jovian planets transiting Sun-like stars can
be detected in a single transit. Smaller planets should be detectable in phase-folded
MASCARA photometry. The first MASCARA station was scheduled to be installed at
La Palma in summer 2014 following testing in the Netherlands.
MEarth (Nutzman & Charbonneau 2008; Berta et al. 2012a; Irwin et al. 2015) is a
survey for transiting planets orbiting nearby mid- to late-M dwarfs. The initial MEarth
array (now called MEarth-North) is a suite of eight 0.4-m robotic telescopes at the Fred
Lawrence Whipple Observatory in Arizona. MEarth-North began science operations in
2008 and found the mini-Neptune GJ 1214b shortly thereafter (Charbonneau et al. 2009).
The southern counterpart MEarth-South is a nearly identical array of telescopes at
Cerro Tololo Inter-American Observatory in Chile. Science operations at MEarth-South
commenced in January 2014.
NGTS (the Next Generation Transit Survey,20 Wheatley et al. 2013) is a search
for tra=nsiting super-Earths and Neptunes orbiting stars brighter than V = 13. The
survey employs an array of twelve 20 cm telescopes at the ESO Paranal Observatory in
Chile and uses back-illuminated deep-depleted CCDs optimized for operation in the red
optical because the primary targets are K dwarfs and early M dwarfs. NGTS saw first
light in January 2015.
SPECULOOS (Search for habitable Planets ECclipsing ULtra-cOOl Stars,21
Gillon et al. 2013a) is a project to look for transiting planets orbiting late M dwarfs (M6
and cooler) in the southern sky. SPECULOOS will consist of an array of small telescopes
20
http://www.ngtransits.org/index/shtml
21
http://www.mpia.de/homes/ppvi/posters/2K066.pdf
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CHAPTER 6. FUTURE DIRECTIONS
(80 cm–1 m) at the ESO Paranal Observatory in Chile.
In preparation for SPECULOOS, Gillon and collaborators conducted the Ultra-Cool
Dwarfs Transit Survey (UCDTS) in 2011 using the 60-cm TRAPPIST telescope (Jehin
et al. 2011) at La Silla. TRAPPIST/UCDTS consisted of observations of approximately
30 late M dwarfs and indicated that the expected level of stellar variability would not
impede the detection of Earth-size planets (Gillon et al. 2013b). SPECULOOS recently
received a 2M Euro grant from the European Research Council that will fund the
installation of two telescopes and cover project operations until through 2018.
WASP (the Wide Angle Search for Planets,22 Pollacco et al. 2006) is a survey for
planets orbiting bright stars. As of March 2015, the WASP Consortium operates two
sets of camera arrays: the SuperWASP array in La Palma and the WASP-South array
in Sutherland. Each array contains eight cameras with a combined field of view of 480◦ .
The WASP Consortium has been highly successful; the two arrays had detected planets
in 134 systems as of 17 March 2015.
6.1.9
Current & Upcoming RV Projects
APOGEE (Deshpande et al. 2013) is a multi-object, moderate resolution (R=22,500)
spectrograph on the 2.5-m Sloan Foundation Telescope. The instrument operates in the
NIR (1.51 − 1.70µm) and has a precision of approximately 10 m s−1 .
APF (the Automated Planet Finder,23 Radovan et al. 2010; Vogt et al. 2014) is a
22
http://wasp-planets.net/about/
23
http://www.ucolick.org/public/telescopes/apf.html
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2.4-m telescope at Lick Observatory coupled to a high-precision (1 m s−1 ), high-resolution
(R=120,000) spectrograph. APF has wavelength coverage of 490–600 nm and uses an
iodine cell for wavelength calibration. APF operations began in 2013 and the telescope
is now fully automated.
CARMENES (Calar Alto high-Resolution search for M dwarfs with Exoearths
with Near-infrared and optical Echelle Spectrographs,24 Quirrenbach et al. 2010, 2012)
will search for low-mass planets orbiting within the habitable zones of approximately
300 low-mass stars from the vantage point of the 3.5-m telescope at the Calar Alto
Observatory. CARMENES consists of a bluer spectrograph (0.5 − 1.0µm) and a redder
spectrograph (1.0 − 1.7µm). Each channel has high spectral resolution (R = 82,000) and
is expected to obtain a precision of approximately 1 m s−1 . First light is scheduled for
2015 and the construction of a target list of bright, single M dwarfs in the northern sky
is underway (Alonso-Floriano et al. 2015).
CHIRON (CTIO High Resolution spectrometer,25 Schwab et al. 2010; Tokovinin
et al. 2013) is a high-precision, high-resolution (R=90,000 or R=130,000), fiber-fed
spectrograph on the 1.5-m telescope in Chile. The long-term stability is 2 m s−1
(measured over two years) and the stability over shorter intervals (10 days) is 0.5 m s−1
(Plavchan et al. 2015). CHIRON was commissioned in 2012.
CRIRES26 (Kaeufl et al. 2004; Bean et al. 2010) is a K-band high-resolution
(R=100,000) spectrograph on the VLT. The instrument is currently offline for an upgrade
24
https://carmenes.caha.es/
25
http://ftp.ctio.noao.edu/noao/content/chiron
26
http://www.eso.org/sci/facilities/paranal/instruments/crires.html
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CHAPTER 6. FUTURE DIRECTIONS
to CRIRES+, but the previous version achieved a radial velocity precision of 5 m s−1 .
ExTrA (Exoplanets in Transit and their Atmospheres,27 Bonfils et al. 2014) is an
ERC-funded projected devoted to devoted to (spectra)photometry of transiting planets
orbiting mid- to late-M dwarfs. The facility will consist of three 60-cm telescopes feeding
a single R > 200 fiber-fed multi-object spectrograph operating at NIR wavelengths
(0.8 − 1.6µm). ExTrA will both survey 800 M dwarfs for transiting planets and
investigate exoplanet atmospheres using differential spectrophotometry.
ESPaDOnS (an Echelle SpectroPolarimetric Device for the Observation of Stars
at CFHT,28 Donati 2003) is a fiber-fed echelle spectrograph/spectropolarimeter on the
Canada France Hawaii Telescope. The instrument covers a broad wavelength range from
0.3 − 1µm and has a resolving power of 68,000–81,000 depending on the observing mode.
ESPRESSO (the Echelle Spectrograph for Rocky Exoplanet and Stable
Spectroscopic Observations,29 Spanò et al. 2008, 2012; Pepe et al. 2010) will be an
extremely stable (< 10 cm s−1 with a goal of a few cm s−1 ) fiber-fed spectrograph at
the Very Large Telescope. The planned location of ESPRESSO at the coudé focus will
allow the instrument to be used with a single unit telescope or with light from all four
unit telescopes simultaneously. The wavelength range will be 380-686 nm, identical
to that of HARPS and HARPS-N, and appropriate for late F to early M dwarfs. In
addition to providing mass measurements for small planets discovered by transit surveys,
27
http://www.eso.org/sci/meetings/2014/exoelt2014/presentations/Bonfils.pdf,http:
//erc.europa.eu/exoplanets-transit-and-their-atmosphere
28
http://www.ast.obs-mip.fr/projets/espadons/espadons.html
29
http://www.eso.org/sci/facilities/develop/instruments/espresso.html
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CHAPTER 6. FUTURE DIRECTIONS
ESPRESSO will enable detailed investigations of the compositions of stars outside of the
Milky Way galaxy and more precise estimates of fundamental physical constants. First
light for ESPRESSO at the VLT is scheduled for 2016.
HARPS (the High Accuracy Radial velocity Planet Searcher,30 Pepe et al. 2000,
2003; Rupprecht et al. 2004; Lovis et al. 2006) is a high-resolution (R=110,000) fiber-fed
spectrograph on the 3.6-m ESO telescope in La Silla. The wavelength coverage extends
from 380–680 nm and the precision is 1 m s−1 .
HARPS-N (the High Accuracy Radial velocity Planet Searcher North31 Cosentino
et al. 2012, 2014; Langellier et al. 2014) is a near-clone of the original HARPS
spectrograph with a few key improvements. The most important changes are that
HARPS-N employs octagonal fibers and a single monolithic CCD.
HIRES (HIgh Resolution Echelle Spectrometer,32 Vogt et al. 1994) is a high
resolution (R=25,000-85,000) cross-dispersed echelle spectrograph covering the
wavelength range 0.3 − 1µm. The instrument is mounted on the right Nasmyth port
of the 10-m Keck I telescope and has an estimated precision of 1 m s−1 (Howard et al.
2009). Science operations with HIRES began in 1996.
HRS (High-Resolution Spectrograph,33 Tull 1998) is an R=15,000–120,000
spectrograph on the Hobby-Eberly Telescope. The instrument was commissioned in
30
http://www.eso.org/sci/facilities/lasilla/instruments/harps.html
31
https://plone.unige.ch/HARPS-N/
32
http://www2.keck.hawaii.edu/inst/hires/
33
http://hydra.as.utexas.edu/?a=help&h=34
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spring 2001 and has a broad wavelength range of 380–1100 nm (two exposures are
required to obtain the full range). The long-term stability of HRS is < 2.5 m s−1 .
HZPF (the Habitable Zone Planet Finder,34 Mahadevan et al. 2010) is a moderate
resolution (R = 50,000) fiber-fed cross-dispersed Echelle spectrograph currently being
built for future operation at the 10-m Hobby-Eberly Telescope at McDonald Observatory
in Texas. As one would infer from the name, the scientific driver for HZPF is to find
low-mass planets in the habitable zones of their host stars and measure the masses of
potentially habitable planets found by transit surveys. HZPF will concentrate on bright
(J < 10) mid- to late-M dwarfs (M4–M9) and will consequently have a red bandpass
covering 0.9 − 1.65µm.
For typical target stars, HZPF is expected to obtain an RV precision of better than
3 m s−1 . For the brightest target stars, the realized precision may even be better than
1 m s−1 . Obtaining such a high precision requires a wavelength calibration, but the
thorium-argon lamps used for highly precise RV work in the optical are less suitable for
the NIR because the argon lines are too bright relative to the thorium lines (Mahadevan
et al. 2010). HZPF will employ both thorium-argon lamp and a uranium-neon lamp as
secondary calibration sources, but the primary calibrators will be a laser frequency comb
and a Fabry-Pérot etalon (Halverson et al. 2014). A prototype of HZPF has already been
tested at the Hobby-Eberly telescope and the instrument should begin science operations
before the launch of TESS.
iSHELL35 (Rayner et al. 2012) is a high-resolution (R=75,000) near-infrared
34
http://hpf.psu.edu/
35
http://irtfweb.ifa.hawaii.edu/~ishell/
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CHAPTER 6. FUTURE DIRECTIONS
spectrograph under construction for the IRTF. The instrument will cover the H and K
bands and is expected to achieve a radial velocity precision of 2–3 m s−1 . Operations are
scheduled to begin in 2016.
IGRINS (Immersion GRating INfrared Spectrograph,
36
Yuk et al. 2010) is a
near-infrared (H and K band) spectrograph that was commissioned in 2014. The
instrument has a resolution is R=40,000 and is located at the 2.7-m Harlan J. Smith
telescope at McDonald Observatory.
IRD (the Infrared Doppler Instrument,37 Tamura et al. 2012; Kotani et al. 2014)
will be a high-resolution (R = 70,000) fiber-fed echelle spectrograph at the 8.2-m Subaru
telescope with a wavelength range of 0.98 − 1.75µm. The primary scientific objective
for IRD is to detect planets with masses as low as 1 M⊕ orbiting mid- to late-M dwarfs
(M4–M9) and the precision goal is 1 m s−1 . This precision will be realized by using a
laser frequency comb with lines spanning 970–1750 nm as a wavelength standard and
designing the majority of the components from ceramics with low thermal expansion
coefficients.
LCOGT NRES (Las Cumbres Observatory Global Telescope Network of Robotic
Echelle Spectrographs,38 Eastman et al. 2014) is a proposed network of six moderate
resolution (R=50,000) spectrographs. The spectrographs will be dispersed around the
globe and have wavelength coverage extending from 390–860 nm. The anticipated
36
http://www.as.utexas.edu/astronomy/research/people/jaffe/igrins.html,https:
//wikis.utexas.edu/display/IGRINS/IGRINS+Home
37
http://seeds.mtk.nao.ac.jp/ird_pub/Overview.html
38
http://lcogt.net/network/instrumentation/nres
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precision is 1–3 m s−1 .
Maroon-X39 is a proposed high-resolution (R=85,000), fiber-fed spectrograph
planned for the 6.5-m Magellan telescopes. The scientific objective is to measure the
masses of potentially habitable small planets orbiting mid- to late-M dwarfs. Some of
the small planets will be previously known transiting planets, but others may be new
Maroon-X discoveries. Due to the red colors of the target stars, the spectrograph will be
optimized for observations at 700–900 nm. Maroon-X passed PDR in June 2014.
MINERVA (MINiature Exoplanet Radial Velocity Array,40 Swift et al. 2013) is
an array of small robotically controlled telescopes that will search for and characterize
exoplanets orbiting nearby stars using both the radial velocity and transit techniques.
The main objectives of MINERVA are: (1) detect Earth-size planets in short period
orbits (P < 50 days); (2) discover potentially habitable super-Earths; (3) refine the
radius and mass estimates of small planets to probe their interior compositions. Most
of the MINERVA observations will be conducted at optical wavelengths, but the
supplemental MINERVA-Red project will empty a cross-dispersed echelle spectrograph
with wavelength coverage of 800–900 nm (Blake et al. 2015). Project MINERVA was
recently relocated to the Whipple Observatory on Mt. Hopkins in Arizona and will begin
science operations shortly.
PFS (the Planet Finder Spectrograph,41 Crane et al. 2010) is a high resolution
(R=38,000–190,000) echelle spectrograph on the 6.5-m Magellan Clay Telescope at Las
39
http://astro.uchicago.edu/~jbean/spectrograph.html
40
https://www.cfa.harvard.edu/minerva/
41
http://users.obs.carnegiescience.edu/crane/pfs/
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Campanas Observatory in Chile. The instrument uses an iodine cell for wavelength
calibration and covers the wavelength range 388–668 nm. PFS began science operations
on January 1, 2010 and has an estimated precision of 1 m s−1 .
SHREK (the Stable High Resolution Echelle for Keck42 is a proposed replacement
for HIRES. Unlike HIRES, SHREK will be fiber-fed. The spectrograph will have a
resolution of R=85,000 and wavelength coverage from 440–590 nm. The expected
precision is 1 m s−1 , roughly a factor of two improvement over Keck/HIRES (Plavchan
et al. 2015).
SOPHIE (Spectrographe pour l’Observation des PHénomènes des Intérieurs
stellaires et des Exoplanètes,43 Perruchot et al. 2008) is a high-resolution (R=75,000)
spectrograph on the 1.93-m Haute-Provence telescope. The wavelength range of
387–694 nm is similar to that of HARPS and HARPS-N, but SOPHIE is less precise
(approximately 3 m s−1 ).
SPIRou (Spectro-Polarimetre Infra-Rouge,44 Thibault et al. 2012; Artigau et al.
2014) is a fiber-fed, cross-dispersed near-infrared spectropolarimeter that will be installed
at the 3.6-m Canada France Hawaii Telescope on Mauna Kea. First light is scheduled
for 2017. SPIRou will provide high-resolution (R = 75,000) spectra in a red bandpass
(0.98 − 2.35µm) well-suited for observations of mid-M dwarfs. SPIRou is expected to
have an RV stability of approximately 1 m s−1 . The SPIRou science goals are (1) search
for previously unknown planets orbiting nearby low-mass stars, (2) obtain follow-up
42
http://nexsci.caltech.edu/keck_strategic_planning_Sep2014.pdf
43
http://www.obs-hp.fr/guide/sophie/sophie-eng.shtml
44
http://exoplanets.ch/projects/spirou/
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CHAPTER 6. FUTURE DIRECTIONS
mass measurements of planets detected by transit survey like K2 and TESS, and (3)
investigate the magnetic fields of young stellar objects (with an emphasis on embedded
protostars) in order to determine the role of magnetic fields in the formation of stars and
planets.
TRES (Tillinghast Reflector Echelle Spectrograph,45 Szentgyorgyi & Furész 2007)
is a spectrograph on the 1.5-m telescope at Fred Lawrence Whipple Observatory. The
instrument has broad wavelength coverage (380–900 nm) and moderate resolution
(R=44,000). The 15 m s−1 precision is well-suited for initial reconnaissance observations
to vet planet candidates for astrophysical false positives.
6.1.10
Exoplanet Investigations in the Era of ELTs
There are currently three major efforts underway to construct extremely large telescopes
(ELTs). The proposed Giant Magellan Telescope (GMT, http://www.gmto.org/),
Thirty Meter Telescope (TMT, www.tmt.org), and European Extremely Large
Telescope (E-ELT, http://www.eso.org/sci/facilities/eelt/) are expected to
begin operations within the next 5–10 years.
The tremendous light collecting power of these large telescope will enable novel
observing strategies such as the combination of high-dispersion spectroscopy with
high-contrast imaging (e.g., Kawahara & Hirano 2014; Snellen et al. 2015). The two
techniques combined (HDS+HCI) should be sensitive to contrast ratios of approximately
10−7 . Compared to high-contrast imaging alone, HDS+HCI will probe smaller orbital
45
http://tdc-www.harvard.edu/instruments/tres/
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separations and reach higher contrast ratios.
In a set of simulations, Snellen et al. (2015) predicted that E-ELT/METIS (see
below) could achieve a 5σ detection of a cool (Teq = 300K) 1.5 R⊕ planet orbiting α Cen A
in a single night of observations. They further demonstrated that the same strategy
could be employed at optical wavelengths to yield a 10σ detection of a potentially
habitable planet orbiting Proxima Centauri. Their simulations assumed a field of view
of 60 × 60 mas, large enough to investigate planets orbiting within the habitable zones
of nearby M dwarfs.46 In some cases, extant radial velocity or transit observations may
constrain the expected position of the planet within the three-dimensional data cube. In
other cases, the position of the planet will need to be determined via cross correlation
with model spectra of planetary atmospheric features or identified in the residuals after
the stellar signal is removed.
GMT
The GMT will be a segmented mirror telescope with a diameter of 24.5 m and a total
collecting area of 368 m2 . The telescope will be located at Las Campanas Observatory
in Chile and is expected to begin science operations by 2021. The first-light instruments
will include G-CLEF (the GMT-CfA Large Earth Finder Szentgyorgyi et al. 2012, 2014),
a fiber-fed visible (350–950 nm) echelle spectrograph with R = 19,000–105,000 depending
on the mode of operation. In the highest resolution mode, feeding the light through
46
For example, Snellen et al. (2015) assumed that a potentially habitable planet orbiting Proxima
Centauri would have a semimajor axis of 0.032 AU. Given the 1.3 pc distance to Proxima Centauri, the
projected angular separation of the planet would as large as 25 mas for a planet in a circular orbit.
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a scrambler is expected to yield an RV precision high enough to detect the 10 cm s−1
Doppler amplitude induced by an Earth-mass planet in the habitable zone of a Sun-like
star. In addition to measuring the masses of small planets detected by TESS and other
surveys, G-CLEF will enable precise investigations of stellar abundances. The other
first-light instruments will be the visible multi-object spectrograph GMACS, the NIR
IFU and AO imager GMTIFS, and the IR echelle spectrograph GMTNIRS.
TMT
The 30-m primary mirror of the TMT will be composed of 492 hexagonal segments, each
of which will be 1.44 m across and separated from the adjacent segments by only 2.5 mm.
The TMT will be be built on Mauna Kea and feature three first-light instruments:
the Wide Field Optical Spectrometer (WFOS), the Infrared Imaging Spectrometer
(IRIS), and the Infrared Multi-object Spectrometer (IRMS). The slate of “First Decade”
instruments also includes a High Resolution Optical Spectrometer (HROS, Crampton
et al. 2008). HROS will have a resolution of R = 100, 000 and wavelength coverage of
0.31-1.1µm, with a possible redward extension to 1.3µm.
E-ELT
Construction of the E-ELT was authorized in December 2014 and science operations are
expected to begin in 2024. The E-ELT will consist of an enormous segmented primary
mirror (39-m in diameter), a 4.2-m convex secondary mirror, a 3.8-m aspheric concave
tertiary mirror, a small adaptive quaternary mirror (2380 × 2340 mm) with as many
as 8000 actuators, and a flat quinary mirror that will direct the beam to the Nasmyth
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focus. The final image should be very close to diffraction-limited over the full 10# field
of view. The two first-light instruments, the NIR imager MICADO (designed for the
ELT-CAM47 opportunity) and the NIR integral field spectrograph HARMONI (designed
for the ELT-IFU48 opportunity), should generate fascinating results, but the second
generation instrument concepts (a high-resolution spectrograph ELT-HIRES,49 a mid-IR
imager and spectrograph ELT-MIDIR,50 and a multi-object spectrograph ELT-MOS51 )
are more exciting from the perspective of studying small exoplanets.
The ELT-HIRES concept is expected to be fulfilled by a combination of
the previously proposed instruments CODEX (COsmic Dynamics and EXo-earth
experiment,52 Pasquini et al. 2008, 2010a,b; Delabre & Manescau 2010) and SIMPLE53
(Oliva & Origlia 2008; Origlia et al. 2010). Both designs were precise high-resolution
spectrographs, but the planned wavelength range of CODEX was bluer than that of
47
http://www.eso.org/sci/facilities/eelt/docs/ESO-193104_1_Top_Level_Requirements_
for_ELT-CAM.pdf
48
http://www.eso.org/sci/facilities/eelt/docs/ESO-191883_1_Top_Level_Requirements_
for_ELT-IFU.pdf
49
http://www.eso.org/sci/facilities/eelt/docs/ESO-204697_1_Top_Level_Requirements_
for_ELT-HIRES.pdf
50
http://www.eso.org/sci/facilities/eelt/docs/ESO-204965_1_Top_Level_Requirements_
for_ELT-MIDIR.pdf
51
http://www.eso.org/sci/facilities/eelt/docs/ESO-204696_1_Top_Level_Requirements_
for_ELT-MOS.pdf
52
http://www.iac.es/proyecto/codex/
53
http://simple.bo.astro.it
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CHAPTER 6. FUTURE DIRECTIONS
SIMPLE (370–710 nm and 0.8-2.5µm, respectively). Following the Phase A study, the
CODEX and SIMPLE teams merged and plan to build a high-resolution spectrograph
covering both visible and NIR wavelengths (Maiolino et al. 2013).
The ELT-MIDIR opportunity will likely be filled by METIS (the Mid-infrared
E-elT Imager and Spectrograph,54 Brandl et al. 2010), a high-resolution (R=100,000)
IFU and diffraction-limited imager. METIS will provide high-resolution spectroscopy in
L and M bands and high-contrast imaging in L, M, and N bands within a 18 × 18## field
of view. Negotiations between ESO and the METIS Consortium were in progress as of
spring 2015.55
6.2
The Scope & Precision of Mass Measurement
Table 1.1 detailed the radial velocity precision necessary to detect several benchmark
planets. From an astrobiological perspective, the critical numbers are the 9 cm s−1 ,
21 cm s−1 , and 2 m s−1 signatures of Earth-mass planets in the habitable zones of
Sun-like stars, early M dwarfs, and late M dwarfs, respectively. In comparison, the
realized and expected precision of the cadre of radial velocity surveys discussed in
Section 6.1 ranged from 10 cm s−1 for CODEX, ESPRES, ESPRESSO, and G-CLEF to
> 10 m s−1 for APOGEE and TRES. Even if the ambitious goal of obtaining precision
better than 10 cm s−1 is realized, a significant investment of observing time and a careful
selection of targets will be required to measure the masses of true Earth twins.
54
http://metis.strw.leidenuniv.nl/
55
http://www.eso.org/sci/facilities/eelt/instrumentation/index.html
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CHAPTER 6. FUTURE DIRECTIONS
As discussed in Chapter 5, there are currently five exoplanets smaller than 2 R⊕ for
which both the mass and radius have been measured to better than 20% accuracy. All
of these planets orbit relatively bright stars and are close enough to their host stars that
they are highly irradiated. Accordingly, the present-day radii may be more reflective of
the efficiency of photoevaporation rather than a reflection of the underlying distribution
of small planet compositions at the time of formation. Estimates of the masses of small
planets that receive less stellar insolation will be useful for investigating the influence of
photoevaporation.
Future RV campaigns with instruments like those described in Section 6.1 will allow
investigations of the universality and possible dependences of small planet compositions.
In particular, a thesis written in ten years from now will likely be able to provide answers
to the following questions:
1. As a function of planet mass and orbital period, how common are various classes
of planets? For instance, what fraction of planets with masses of 1.5 − 2.5 M⊕
and orbital periods of 10–30 days have rocky compositions, water-dominated
compositions, or puffy H/He atmospheres?
2. Are there regions of parameter space (e.g., planets smaller than a threshold size or
receiving more than a set amount of insolation) for which a one-to-one mass-radius
relation is nearly always upheld?
3. To what extent are the observed radii of super-Earths and mini-Neptunes
determined by photoevaporation rather than planetary formation?
4. Do small planets primarily form in situ or do they more often form beyond the
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snow line and migrate inward?
5. Are there notable differences between the compositions, sizes, and overall number
of small planets in systems with distant giant planets and small planets in systems
without giant planets?
6. Are there detectable compositional gradients within exoplanetary systems? If so,
how similar are they to the compositional gradients observed in protoplanetary
disks and debris disks?
7. How tightly correlated are the elemental abundances of planet host stars and their
terrestrial planets?
While we await answers to these questions, we can make predictions about the
number of planets accessible to such investigations. Using planet occurrence rates derived
from Kepler data (Fressin et al. 2013; Dressing & Charbonneau 2015), I simulated the
population of small planets orbiting bright stars that are likely to be detected by K2. I
found that each K2 campaign should yield roughly 16 ± 4 small, short-period planets
(1 − 3 R⊕ , P < 10 d) orbiting bright stars. I then accounted for the possibility that some
of the detected planets will be poorly suited for HPRV follow-up due to stellar variability,
fast stellar rotation rates, confusion with the stellar rotation period, the presence of
other planets, the presence of nearby stars, or other observational factors.
Overall, I estimated that there should be 6 ± 2 short-period small planets per K2
campaign for which a 6σ mass measurement could be obtained with < 40 hr of observing
time with HARPS-N or an instrument with comparable precision (e.g., HARPS, PFS).
Considering all ten K2 campaigns, this estimate implies that the number of small planets
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with well-measured masses and radii could increase by an order of magnitude due to the
K2 mission alone. Adding in the population of roughly 300 small planets (R < 2 R⊕ )
that are expected to be detected by TESS (Sullivan et al. 2015) will further increase the
number of small planets with well-determined bulk densities.
6.3
Initial Atmospheric Characterization
As discussed in Section 1.6, there is considerable model degeneracy in translating
measured masses and radii into planetary compositions. Fortunately, the allowed range
of compositions can be reduced by measuring the wavelength-dependence of the transit
depth. When a transit event is viewed at different wavelengths, the apparent size of the
planet (and by extension the transit depth) will vary as a function of the abundance of
species in the planet’s atmosphere that absorb at the particular wavelength.
When assessing the detectability of wavelength-dependent transit depth variations,
the most important quantity to consider is the atmospheric scale height, H:
H=
kB T
µg
(6.1)
where kB is the Boltzmann constant, T is the temperature of the planet, µ is the mean
molecular weight of the planetary atmosphere, and g is the gravitational acceleration
(e.g., Meadows & Seager 2010).
Assuming that the thickness of the atmosphere is much smaller than the radius of
the planet Rp neglecting the atmosphere, then the depth δ(λ) of the transit measured at
wavelength λ is given by
δ(λ) =
πRp2 + 2πRp h(λ)
πR!2
320
(6.2)
CHAPTER 6. FUTURE DIRECTIONS
where h(λ) is the effective height of the atmosphere at wavelength λ and is related to
both the overall composition of the atmosphere and the distribution of absorbers within
the atmosphere (Kaltenegger & Traub 2009). Planets with larger atmospheric scale
heights H are capable of displaying a broader diversity of effective heights h(λ).
A planet possessing an atmosphere with high mean molecular weight (e.g., primarily
water vapor) will have a higher µ and a smaller H than a planet with a light atmosphere
(e.g., a roughly solar composition blend dominated by H2 ). In the absence of confounding
clouds or hazes (see below), the two extremes (atmospheres with high and low mean
molecular weights) could therefore be distinguished because the measured transit depth
of the planet possessing the atmosphere with the higher mean molecular weight would
display considerably less variation with wavelength due to the smaller atmospheric scale
height.
As one might expect, the small planet for which the atmosphere has been
best characterized is GJ 1214b. Thorough investigations using HST, Spitzer, and
ground-based instruments have revealed that the transit depth displays very little
wavelength-dependence, demonstrating that the planet does not have the large
atmospheric scale height indicative of a hydrogen-dominated atmosphere (Bean et al.
2011; Carter et al. 2011; Crossfield et al. 2011; Désert et al. 2011a; Berta et al. 2012b;
de Mooij et al. 2012; Fraine et al. 2013; Narita et al. 2013a,b; Teske et al. 2013, but see
Croll et al. 2011). Most recently, Kreidberg et al. (2014) presented HST observations
of 15 transits and concluded that the atmosphere of GJ 1214b must be concealed by
high-altitude clouds or hazes.
The atmosphere of Neptune-mass planet GJ 436b may also be cloudy. Knutson et al.
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CHAPTER 6. FUTURE DIRECTIONS
(2014a) analyzed HST observations of four transits and found that a hydrogen-dominated
atmosphere is inconsistent with the data at the 48σ level. Unlike GJ 1214b, which cannot
be explained be a cloud-free atmosphere, GJ 436b may either have a high-metallicity
atmosphere or possess a lower metallicity atmosphere cloaked by high-level clouds.
Similarly, the 7.9 ± 0.7 M⊕ , 2.3 ± 0.2 R⊕ planet HD 97658b (Dragomir et al. 2013) must
also have a high-metallicity atmosphere or high-level clouds; the measured NIR transit
depths from HST/WFC3 are 10σ discrepant with a cloud-free solar-metallicity model
(Knutson et al. 2014b).
6.3.1
Identifying Cloud- and Haze-Free Worlds
If clouds are a common feature on small exoplanets, then searching for biosignatures
in the atmospheres of potentally habitable planets may be harder than anticipated.
Accordingly, it would be advantageous if astronomers could prioritize follow-up
observations in order to spend a larger fraction of follow-up time on planets with less
shrouded atmospheres. One possible approach would be to prescreen potential candidates
with HST to ensure that their transmission spectra displayed intriguing features.
Alternatively, as proposed by Misra & Meadows (2014), one could look for the
signature of refracted light in the out-of-transit light curve. In a clear exoplanet
atmosphere, there should be a slight increase in flux just before transit ingress and
just after transit egress. The refraction signature should be detectable in < 10 hr of
JWST time or < 5 hr of ELT time for Jovian planets or super-Earths, respectively.
While non-negligible, this time investment is low enough that astronomers may decide
to first verify that a planet displays a refraction signature before investing a significant
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amount of observing time with JWST, WFIRST-AFTA, or Exo-C/S on atmospheric
characterization. Conveniently, the refractive signature should be most identifiable for
potentially habitable planets and for hotter planets with equilibrium temperatures as
high as 500 K (Misra & Meadows 2014).
The potential variability of exoplanet clouds is also important to consider. Although
many fits to transmission spectra consider either clear atmospheres or fully cloudy/hazy
atmospheres in which the entire surface is hidden, real exoplanet clouds are most likely
spatially heterogeneous. As the quality of the data improve, accurately describing
super-Earth transmission spectra will likely require “patchy” cloud models (e.g., Marley
et al. 2010; Morley et al. 2014) or sophisticated general circulation models in which
clouds can dynamically form and dissipate in different regions(e.g., Joshi 2003; Edson
et al. 2011; Yang et al. 2013). These models could then be combined with high quality
super-Earth phase curves and secondary eclipse observations to yield planetary maps
like those generated for hot Jupiters (e.g., Knutson et al. 2007, 2009, 2012; de Wit et al.
2012; Majeau et al. 2012; Demory et al. 2013; Stevenson et al. 2014) and for the Earth
using EPOXI observations (e.g. Cowan et al. 2009, 2011; Fujii et al. 2010, 2013; Robinson
et al. 2011).
6.4
Detecting & Interpreting Potential
Biosignatures
Section 1.7 reviewed the challenges of selecting potential biosignatures; this section
discusses the likelihood of detecting such signals. For example, Kaltenegger & Traub
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(2009) simulated the detectability of spectral features in the atmosphere of an Earth-like
planet orbiting either a Sun-like star or an M dwarf. Their simulations accounted for
the effects of refraction within the atmosphere and suggested that a 6.5-m space-based
telescope like JWST could detect CO2 , H2 O, and maybe even CH4 using transmission
spectroscopy at NIR wavelengths. Observations at longer wavelength could be even more
fruitful, yielding CO2 , H2 O, O3 , CH4 , and HNO3 .
In general, these features will be insignificant (0.2 − 1.7σ) in observations of a single
transit event, so observations of multiple transits will be required to robustly detect
biosignatures. For a total observation time of 200 hr, Kaltenegger & Traub (2009)
predicted that 6.5-m space-based telescope could detect the O3 feature at 0.6 µm at
significance of 16.9σ, 9.1σ, or 9.6σ in the atmosphere of an Earth-like planet in the
habitable zone of a G2V, M0V, or M9V star, respectively, at a distance of 10 pc. NIRISS
or NIRSpec could be used to obtain the required observations. For CH4 at 7.7µm, the
expected significance is 2.0σ, 2.5σ, and 13.8σ, respectively.
Simulating both the expected TESS planet yield and the expected performance of
NIRSpec and MIRI, Deming et al. (2009) investigated the number of super-Earths for
which JWST might be able to provide atmospheric characterization. They estimated that
TESS should yield roughly 330 hot super-Earths for which JWST /MIRI observations
of secondary eclipses would reveal CO2 absorption. Deming et al. (2009) also predicted
that TESS would detect five nearby, potentially habitable super-Earths for which
JWST /NIRSpec observations of primary eclipses could detect water observation at
S/N ≥ 8.
At longer wavelengths, Belu et al. (2011) predicted that JWST would be capable of
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detecting the 9.6 µm O3 feature. Using the current MIRI specifications and performance
expectations as of 2011, Belu et al. (2011) estimated that JWST/MIRI could detect O3
at S/N ≥ 3 in the atmospheres of warm super-Earths orbiting mid- and late-M dwarfs
within 6.7 pc. The feature should be detectable in both transmission and secondary
eclipse spectra for host stars with spectral types M5V or later and in transmission
spectra for planets orbiting M4 dwarfs. The total time investment in either case would
be 2% of the full 5-yr baseline mission.
Another biomarker highlighted in Section 1.7 was O2 . Although insufficient in
isolation, the mutual presence of O2 and a reducing gas such as CH4 in an exoplanet
atmosphere would likely be a signature of alien life. The presence of O2 in our
atmosphere would typically confound efforts to observe O2 in exoplanet atmospheres,
but the high-resolution spectrographs that will be available on ELTs will resolve the O2
bands into well-separated lines that will be Doppler shifted away from the lines produced
in our own atmosphere (e.g. Lowell 1905; Webb & Wormleaton 2001). Simulations by
Snellen et al. (2013) suggested that for an Earth-like planet within the habitable zone of
a GJ 1214-like mid-M dwarf, observations of thirty transits with the E-ELT would result
in a detection of O2 at 3.8σ significance.
However, a follow-up study by Rodler & López-Morales (2014) suggested that
the Snellen et al. (2013) estimate of 30 transits might be overly optimistic. Rodler &
López-Morales (2014) considered the effects of refraction in the planet’s atmosphere
and incorporated both random and correlated noise, whereas the initial study did not
consider refraction and assumed only photon noise. Assuming that the contribution
from red noise is equal to 20% of the white noise contribution, Rodler & López-Morales
(2014) found that a 3σ detection of O2 in the atmosphere of an Earth-like planet orbiting
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CHAPTER 6. FUTURE DIRECTIONS
an I = 10 M4V dwarf or an I = 11.8 M6V dwarf would require observing 60 or 84
transits, respectively, with the E-ELT. Since transits are by nature periodic events, this
implies that a potentially habitable Earth-like planet orbiting a nearby M dwarf would
need to be monitored for 12–25 years to enable the detection of a possible biosignature.
Increasing the significance of the detection from 3σ to 6σ, as will likely be required for
such a noteworthy scientific claim, could require a full century of observing every possible
transit. Fortunately, the required observation time could be reduced by employing an
image slicer like that described by Dekker et al. (2003) to increase the spectral resolution
(Rodler & López-Morales 2014).
The required investment of observing time is less daunting for planets orbiting closer
stars. For a planet within the habitable zone of an M dwarf 5 pc from the Sun, the
requisite numbers of years to obtain a 3σ detection of oxygen (neglecting the influence of
red noise) are 2–30 for E-ELT, 1–33 for SIMPLE/E-ELT, 7–35 for GMT/G-CLEF, and
5–26 for TMT/HROS. The wide ranges are due to the difference in stellar properties from
M1 to M9 and to the two different spectrograph designs used for the visible wavelength
E-ELT simulations.
In general, the required number of transits is larger for later M dwarfs than for
earlier M dwarfs, but the shorter orbital periods of planets in the habitable zones of late
M dwarfs results in lower total time investments. Given the choice between detecting
oxygen at visible wavelengths using the 0.75µm O2 A band or at NIR wavelengths using
the 1.26µm band, the Rodler & López-Morales (2014) study demonstrated that visible
wavelengths should be preferred for M0V–M8V stars, but that NIR observations are
advantageous for M9V stars.
326
CHAPTER 6. FUTURE DIRECTIONS
Rodler & López-Morales (2014) concluded their study by stating that ground-based
detections of O2 in exoplanetary atmospheres will likely be restricted to worlds < 8 pc
from Earth orbiting stars with spectral types later than M3V. For reference, the
CONCH-SHELL database (Gaidos et al. 2014) lists 93 stars within 8 pc and the
(partially overlapping) southern RECONS sample includes 50 systems within 8 pc
(Winters et al. 2015). Naively surmising that the planet occurrence rates for midand late-M dwarfs are identical to those for early M dwarfs, there should be roughly
25 potentially habitable Earth-size planets within 8 pc. Even though the geometric
probability of transit for a planet orbiting an M3V star is relatively high (1%; see
Table 1.1), the likelihood that any of the 25 hypothetical habitable planets will transit
is 22%. Unless there are nearby late M dwarfs with undetermined parallaxes (which
is plausible) or the planet occurrence rate is higher for mid- and late-M dwarfs than
for early M dwarfs (also plausible), the most likely distance to the nearest transiting
Earth-like planet is 10.6+1.6
−1.8 pc (Dressing & Charbonneau 2015, see also Chapter 3).
Even though the 2.6 pc discrepancy between the planet distance recommended
by Rodler & López-Morales (2014) and the actual distance calculated by Dressing &
Charbonneau (2015) might appear discouraging, the history of exoplanet observations
includes numerous stories of instruments outperforming their design specifications or
operating in novel configurations. Studying heat redistribution on hot Jupiters with
Spitzer (Knutson et al. 2007), detecting an Earth-mass planet with HARPS (Dumusque
et al. 2012), and implementing spatial scan mode to obtain 23 ppm transmission
spectroscopy with HST (Knutson et al. 2014b; Kreidberg et al. 2014) are three examples.
Accordingly, it seems reasonable to predict that astronomers will also develop specialized
strategies to conduct previously unanticipated investigations with the E-ELTs and
327
CHAPTER 6. FUTURE DIRECTIONS
JWST. Most likely, the resulting exoplanet discoveries will surpass even our most
ambitious expectations.
328
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