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The Prevalence and Compositions of Small Planets The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Dressing, Courtney Danielle. 2015. The Prevalence and Compositions of Small Planets. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Accessed June 15, 2017 6:03:20 PM EDT Citable Link http://nrs.harvard.edu/urn-3:HUL.InstRepos:17467474 Terms of Use This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-ofuse#LAA (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 152 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 153 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 155 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. 156 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 157 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 158 CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE 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 159 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 160 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. 161 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” 162 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 163 CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE 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 164 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 165 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 166 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). 167 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, 168 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. 170 CHAPTER 3. A REFINED ESTIMATE OF THE OCCURRENCE RATE 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 171 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 172 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. 173 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 174 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) 175 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 41.4 176 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. 177 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 178 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 179 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. 180 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. 181 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. 182 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. 183 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. 184 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, 185 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 186 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 188 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. 190 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). 191 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. 192 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 193 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. 194 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 195 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. 196 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. 208 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. 210 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.) 274 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 275 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 276 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 277 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. 278 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. 279 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. 280 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. 281 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. 282 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). 283 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/. 284 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 285 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 286 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. 287 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. 289 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 290 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. 291 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. 292 CHAPTER 6. FUTURE DIRECTIONS 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/ 293 CHAPTER 6. FUTURE DIRECTIONS 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). 294 CHAPTER 6. FUTURE DIRECTIONS 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/ 295 CHAPTER 6. FUTURE DIRECTIONS 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 296 CHAPTER 6. FUTURE DIRECTIONS 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 298 CHAPTER 6. FUTURE DIRECTIONS 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 299 CHAPTER 6. FUTURE DIRECTIONS 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/ 300 CHAPTER 6. FUTURE DIRECTIONS 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 301 CHAPTER 6. FUTURE DIRECTIONS 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 302 CHAPTER 6. FUTURE DIRECTIONS 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/ 303 CHAPTER 6. FUTURE DIRECTIONS 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 304 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 305 CHAPTER 6. FUTURE DIRECTIONS 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 306 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 307 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=34 308 CHAPTER 6. FUTURE DIRECTIONS 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/ 309 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 310 CHAPTER 6. FUTURE DIRECTIONS 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/ 311 CHAPTER 6. FUTURE DIRECTIONS 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/ 312 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/ 313 CHAPTER 6. FUTURE DIRECTIONS 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. 314 CHAPTER 6. FUTURE DIRECTIONS 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 315 CHAPTER 6. FUTURE DIRECTIONS 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 316 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 317 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 318 CHAPTER 6. FUTURE DIRECTIONS 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 319 CHAPTER 6. FUTURE DIRECTIONS 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. 321 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 322 CHAPTER 6. FUTURE DIRECTIONS 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 323 CHAPTER 6. FUTURE DIRECTIONS (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 324 CHAPTER 6. FUTURE DIRECTIONS 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 325 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 References Abbot, D. 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