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Chapter 2- Radial Velocities
Contents
2.1 Description of orbits
2.2 Measurement principles and accuracies
2.3 Instrument programmes
2.4 Results to date
2.5 Properties of radial velocity planets
2.6 Multiple planet systems
2.7 Planets around binary and multiple stars
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2.1 Description of orbits (1)
 Principle: the motion of a single planet in orbit around a star causes
the star to undergo a reflex motion around the barycenter (center of
mass) defined as
where M and a refers to the mass and semi-major axis, respectively.
(Notation: subscript p for planet and ★ for the host star; arel is used
for a planet orbit relative to the star)
 Reflex motion results in periodic perturbation of: radial velocity,
angular position on the sky, and in the time arrival of some periodic
signal associated with the host star.
 The orbit is an ellipse described in polar coordinates by
(2.1)
The semi-major axis a, semi-major axis b are related to the eccentricity e
(2.3)
by
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2.1 Description of orbits (2)
ν: true anomaly, angle referred to the elliptical (true) orbit.
E: eccentric anomaly, angle referred to auxiliary circle.
M: mean anomaly, angle refering to a fictitious mean motion around
the orbit related to E and ν.
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2.1 Description of orbits (3)
ν, E are related by
(2.6)
Let P be the orbital period. The mean anomaly at time
pericenter passage is defined as
after
(2.9)
and related to E by
(2.10)
 The position of an object along the orbit at any chosen time t is
obtained first by calculating the mean anomaly M, then solving for E
(transcendental equation 2.10) and then using Equation 2.6 to obtain
ν(t) and r(t) from Equation 2.1.
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2.1 Description of orbits (4)
Orbit specification. A Keplerian orbit in three dimension is described
by the following seven parameters:
: orbit inclination (i=0; face on)
: longitude of the ascending node
: argument of pericenter (measured
in true orbital plane
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2.1 Description of orbits (5)
Kepler’s three laws of planetary motion: (1) the orbit of planet is
an ellipse with the Sun at one focus; (2) the line joining the planet
and the Sun sweeps out equal areas in equal intervals of time; (3) the
squares of the orbital periods of the planets are proportional to the
cubes of their semi-major axes.
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2.1 Description of orbits (6)
(a) Relative orbits: the motion of the planet is relative to the star
rather than the barycenter. The 3rd Kepler’s law is
(2.15)
Since M★ >> Mp
(2.16)
(b) Absolute orbits: the motion of the planet is relative the barycenter.
We have
(2.17)
(2.18)
The sizes of the three orbits are in proportion
a★ : ap : arel = Mp : M★ : (M★ + Mp), with arel = a★ + ap.
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2.1 Description of orbits (7)
The radial velocity semi-amplitude
From Figure 2.2, we have
(2.19)
and
(2.20)
leading to
(2.21)
where K is the radial velocity semiamplitude
(2.22)
Figure 2.2
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2.1 Description of orbits (8)
The shape of the radial velocity curve is determined by e and ω (see
Figure 2.4). Together with P, their combination constrains the value
of a★sin i. Neither a★ nor i can be determined seperately.
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2.1 Description of orbits (9)
Other expressions for K
Substituting 2.17 and 2.18 into 2.22 yields
(2.23)
Alternatively,
(2.26)
For a circular orbit and M★ >> Mp
(2.28)
For Jupiter around the Sun (a=5.2 AU, P=11.9 yr), KJ= 12.5 m s-1.
For Earth,
For an Earth within the HZ of a anM5 (P~10
days; M★~0.1 M), K~1 m/s.
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2.1 Description of orbits (10)
Fitting a single planet
Radial velocity can constrain the following 5 observables: e, P, tp, ω
and the combination K=f(a,e,P,i).
Two additional terms are usually taken into account: (1) the
systemic velocity Υ describing the constant component of the radial
velocity of the system’s centre of mass relative to the solar system
barycentre and (2) a linear trend parameter d,which may
accommodate instrumental drifts as well as unidentified
contributions from massive, long-period companions.
The radial velocity signal of a star with an orbiting planet is thus,
from Equation 2.21,
(2.29)
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2.1 Description of orbits (11)
Data analysis


A χ2 minimization is used for determining the orbital parameters.
Complications due to the non-linear nature of the problem.
Algorithms






Lomb-Scargle periodigram
Levenberg-Marquardt (MPFIT in IDL)
Linearisation techniques
Bayesian methods
Markov Chain Monte Carlo (ex: EXOFIT)
For np planets, there are 5np + 1
parameters to fit, Υ included.

Correction for dynamical interaction
often needed for multiple systems.
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Contents
2.1 Description of orbits
2.2 Measurement principles and accuracies
2.3 Instrument programmes
2.4 Results to date
2.5 Properties of radial velocity planets
2.6 Multiple planet systems
2.7 Planets around binary and multiple stars
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2.2 Measurement principles and accuracies (1)
Doppler shift
An instantaneous measurement of the stellar radial velocity about
the star-planet barycenter is given by the small, systematic Doppler
shift in wavelength of the many (thousands) absorption lines that
make the star’s spectrum.
In the observer’s reference frame, the source is receeding with
velocity v at an angle θ relative to the direction from the observer
source, the change in wavelength Δλ=λobs-λem is given by the
relativistic Doppler shift equation
(2.38)
where λobs, λem are observed and emitted wavelengths, β=v/c.
For v<<c and θ<< π/2, the expression reduces to the classical form
(2.39)
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2.2 Measurement principles and accuracies (2)
 Special relativistic terms are significant corresponding to changes in
vr of several m/s.
 Index of refraction of the air at the spectrograph, nair, and its
dependance in wavelengths introduces errors ≤ 1 m/s.
 Nair=1.000277 at standard temperature and pressure
 Measuring
requires long-term
(months to years) radial velocity accuracies at the level of 1 m/s, i.e.,
one part in 108. Not easy!
 Precision radial velocities usually achieved with dedicated crossdispersed échelle spectrographs with high resolving power
(λ/Δλ ~50 000 – 100 000) operating in the optical (450 – 700 nm).
 High instrumental stability and accurate wavelength calibration is required.
 Relatively large (4m-10m) telescopes required to achieve high signal-to-noise.
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2.2 Measurement principles and accuracies (3)
Echelle spectrum of the Sun
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2.2 Measurement principles and accuracies (4)
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2.2 Measurement principles and accuracies (5)
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2.2 Measurement principles and accuracies (6)
Cross-correlation spectroscopy
 The Doppler shift information is contained in many absorption lines.
 Cross-correlation techniques with masks are used to extract the
information
 Initially with physical masks but limited to match various spectral types
 Nowadays, mask is implemented numerically
Let ε be the Doppler shift, S(v) the spectrum and M(v) the mask, both
expressed in velocity space v, the cross-correlation function is defined
as
(2.40)
 The Doppler shift ε is obtained is obtained by minimizing C(ε).
 The precise shape of C(ε) depends on the intrinsic line shapes and on
the template line width.
 Width of C(ε) yields the stellar rotational velocity v sin i.
 Equivalenth witdh provides a metallicity estimate if Teff is known.
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2.2 Measurement principles and accuracies (7)
Cross-correlation spectroscopy – Principle
Eggenberger & Udry (2009)
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2.2 Measurement principles and accuracies (8)
Cross-correlation spectroscopy - Example
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2.2 Measurement principles and accuracies (9)
Deriving radial velocities from Doppler shifts
The following effects must be taken into account for
deriving the stellar Doppler shift.
 Earth motion
 Frame of reference is the Solar System barycenter.
 Earth movement determined from ephemerides provided by JPL.
 Line shift from gravitational redshift
 636 m/s for the Sun, ~500 m/s for an M5V.
 Stellar space motion
 Intrinsic space motion of the star, acceleration included. Data
obtained from Hipparcos
 Zero point
 Very difficult to establish an absolute reference for radial motion
< 50 m/s
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2.2 Measurement principles and accuracies (10)
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2.2 Measurement principles and accuracies (11)
Wavelength calibration
Wavelength calibration is key for achieving accurate Doppler measurements. It is
certainly one of the most important design drivers of modern precision radial velocity
(PRV) instruments.
 Spectrograph slit width effect
 In velocity space, the slit with can be several km/s wide. This means that the
illumination within the slit must be kept uniform at the level of 10-3 to maintain a
radial velocity accuracy less than 1 m/s.
Various observational and instrumental strategies can be used for PRV
calibration.
 Use telluric (atmospheric) lines
 Advantage of common optical path with the target but limited wavelength range.
Lines are intrinsically variables due to winds. Accuracy possible at the level of ~
20 m/s.
 Gas cell in the optical path of the spectrograph
 Provides a dense and accurate wavelength reference, superimposed on the stellar
lines. HF used in the past but toxic; iodine (I2) now commonly used.
 Pros: large number (1000s) of lines, same optical path as target, provides a
simultaneous tracking of the instrumental point spread function.
 Cons: 20-30% transmission loss, lines not uniformly distributed and clustered
between 500 and 620 nm (exclude M dwarfs). Data analysis not trivial.
 In the infrared: NH3 cell for VLT-CRIRES (R=100 000; 0.96-5.2 μm) yields a
precision of 3-5 m/s over weeks of months.
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2.2 Measurement principles and accuracies (12)
Gas cell spectroscopy principle
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NH3 cell with CRIRES
 Non-optimal experiment done
with CRIRES with a NH3 gas cell
suggests that ~3-5 m/s is possible.
 CRIRES not designed for PRV
 Very encouraging result
 SPIROU will cover a wavelength
range 70 times that of CRIRES!
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2.2 Measurement principles and accuracies (13)
Wavelength calibration
 Fiber-fed spectrograph
 The entrance spectrograph slit is replaced by a fiber and the instrument features
a decidated fiber used for calibration.
 Allow the spectrograph to be installed sway from the telescope in a separate
thermally-controlled room to minimize instrumental wavelength shifts.
 Optical scrambling
 Minimizes variable light illumination due to multiple reflection with the fibers.
 Octogonal fibers are particularly effective at scrambling light.
 Calibration lamps
 Thorium-Argon lamps (e.g. HARPS and predecessor: ELODIE)
• Offer a large number of strong emission lines over a wide optical to infrared range
• No throughput loss
 Laser frequency combs
• Ideal calibration source, wide and uniform wavelength coverage,
• Based on single laser pulse maintained on a repetitive path, circulating in a cavity
• Frequency of the comb:
with
, and T is the round-trip travel
time with n~105-106.
is the carrier envelope frequency.
• Both
and
syncronized by reference to atomic clocks
• Enable high stability at the level of ~0.01 m/s
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2.2 Measurement principles and accuracies (14)
Example of frequency comb from HARPS
Spectrum of a star obtained using the HARPS instrument on the ESO 3.6-metre telescope at the La Silla
Observatory in Chile. The lines are the light from the star spread out through various orders. The dark gaps in
the lines are absorption features from different elements in the star. The regularly spaced bright spots just
above the lines are the spectrum of the laser frequency comb that is used for comparison. The very stable
nature and regular spacing of the frequency comb make it an ideal comparison, allowing the detection of
minute shifts in the star’s spectrum that are induced by the motion of orbiting planets.
Source: http://www.eso.org/public/images/ann12037a/
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2.2 Measurement principles and accuracies (15)
 Exposure metering
 A means to take a small fraction (a few %) of the starlight to
monitor its flux during the exposure. Two purposes:
• Calculate precisely the photon-weighted mid-point of the exposure for
barycentric correction.
• Optimize exposure time to reach required signal-to-noise
 Accuracy limits
 Current sensitivity (instrumental only): 0.3-0.5 m/s, a record held
by the Swiss (Geneva team)
• Corresponds to displacement of a few nm on the detector !
• Accuracy must be maintained over several years.
• RV amplitude independant of distance but SNR considerations limit
observations to stars brighter than V < 8-10
• Easiest targets for RV: massive planets, small P (a) and large e.
(e.g. 51 Pegasus)
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2.2 Measurement principles and accuracies (16)
 Accuracy limits
 Error sources imposing limits in on RV accuracy
• Instrumental (mechanical/thermal stability, wavelength calibration)
• Photon noise (fundamental limit)
– HARPS on 3.6m ESO. ~1 m/s on V~7.5 with SNR~100 on G star in 30s.
• Stellar noise (so-called « jitter » noise)
 Jitter noise
• Activity in the stellar atmosphere (spots, plages)
– Often very significant. Spots with a filling factor of a few % can induce jitter of a
few m/s.
» Varies on timescales comparable to stellar rotation period.
» Correlated with chromospheric (magnetic) activity. Core of CaII H/K lines is a
good proxy (S-index).
– Stellar oscillations
» 15-min exposure sufficient to damp residual RV down to 0.2 m/s (HARPS)
– Surface granulation
» Of order 1 m/s on solar-type star. Hour-long exposure needed to damp (see
next slide).
– Unrecognized (long period) planetary companions
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2.2 Measurement principles and accuracies (17)
Simulation of oscillation jitter
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2.2 Measurement principles and accuracies (18)
 Jitter noise is characterized as an excess in the radial velocity
standard error
(2.44)
is the total RV standard error.
includes all sources of jitter noise.
 Astrophysical contributions are dependent of several variables:
rotational velocity, age, activity
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2.2 Measurement principles and accuracies (19)
 Jitter noise is correlated with rotational velocity (v sin i).
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2.2 Measurement principles and accuracies (20)
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Contents
2.1 Description of orbits
2.2 Measurement principles and accuracies
2.3 Instrument programmes
2.4 Results to date
2.5 Properties of radial velocity planets
2.6 Multiple planet systems
2.7 Planets around binary and multiple stars
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2.3 Instrument Programmes (1)
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2.3 Instrument Programmes (2)
Two good examples of productive instruments
 HARPS (High Accuracy Radial Velocity Planet
Searcher)
 3.6m on LaSilla (Chile)
 Fiber-fed échelle, R=115 000
• Two fibers: one science, one for wavelength calibration




Two CCDs 4kx4k, 15 um pixels)
Wavelength range: 378-691 nm
In operation since 2003
Best instrument in the world (~1 m/s)
• 2nd most productive (# of discoveries)
 HARPS-North
• Copy of HARPS for the 3.6m TNG
on LaPalma. In operation since 2012
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2.3 Instrument Programmes (3)
 KECK HIRES
 10m Keck I telescope (Mauna Kea; Hawaii)
 échelle slit spectrograph, R=80 000
• Iodine cell for wavelength calibration
 Wavelength range: 390-620 nm
 First light in 1993
• Not designed for exoplanet detection !
 Sensitivity: 1-2 m/s
 Most productive in the world
• # of discoveries
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2.3 Instrument Programmes (3)
Future developments (small selection)
 ESPESSO
 VLT 8m.
• Could combine the 4 VLT together.
 HARPS-like fiber-fed échelle spectrograph; R=140 000
 Wavelength range: 350-720 nm
 RV accuracy requirement: 10 cm/s
 First light: 2016
• Available to the community in 2017.
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2.3 Instrument Programmes (4)
Future developments (small selection)
 SPIRou (SPectropolarimètre InfraROUge)
 CFHT 3.6m
 HARPS-like fiber-fed échelle spectropolarimeter
• R=75 000
 Spectral resolution
 Wavelength range: 0.95-2.35 μm
 RV accuracy requirement: 1 m/s
 First light: 2017
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An 8m-class instrument on a 4m class
telescope
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SPIrou data simulator
4Kx4k
K
H
J
Y
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SPIROU vs CRIRES (VLT)
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A spectrum is worth a thousand pictures!
M3V spectrum
Telluric lines
Mauna Kea
Calar Alto
Animation credit: E. Artigau
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A roadmap to habitable exoplanets
Act #1: detection
 Build up a catalog (> 100) of rocky planet
candidates in the habitable zone.
• Transit & RV required
• Lensing very useful for statistic (planet frequency)
Act #2: characterization
 Constrain density of the planet
• Requires both transit & RV data.
 Atmospheric characterization. Seek spectral
signature of H2O, CO2, CH4 and O2/O3.
• Large space-based IR telescope required: JWST.
• Best targets: M dwarfs.
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The M dwarf opportunity
There are lots of them! The most typical
star in the Galaxy is a M3V (M~0.3 M).
They span a factor of 5 in mass.
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Orbital period in the HZ is measured in
weeks, not years
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The RV signal is relatively strong
Higher transit probability: Pt=Rp/a
Sun: Pt=0.5%; M3V: Pt=1.5%; M6V: Pt=2.3%
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But…
M dwarfs are faint.
 Observations in the IR absolutely required
especially for late Ms.
M dwarfs are active. They are fully convective
and show significant magnetic activity (stellar
spots)
 Source of jitter noise for the RV signal. Expected to
be 4-5 smaller in the IR compared to the visible.
RV at IR wavelengths is more complicated
 Lots of telluric lines to deal with.
 Instrumentation is more complex (cryogenic) i.e.
expensive.
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SPIRou’s key science questions
What is the prevalence of habitable planets
around low-mass stars?
Determine η for M dwarfs.
Characterize new super-Earths found through
transit searches (e.g. TESS)
Identify suitable/credible targets for transit
spectroscopy follow-up with JWST/ELT
How do stars/planets form and evolve ?
 What is the role of magnetic field, especially in
young embedded stars?
And much much more !
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Fraction of stars with a
terrestrial planet within the
habitable zone?
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Exoplanet frequency – Kepler data
 Relevant sample: FGK stars
 From extrapolation of the short period transiting
population.
Traub 2012
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Exoplanet frequency – RV data
 HARPS survey of 100 M stars (Bonfils et al, 2011)
 10 nights/yr over 6 yrs: 14 detections.
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Exoplanet frequency – lensing
Cassan et al, 2011.
• Super-Earth frequency betwen 0.5 and 10 AUs:
• On average every star has
planets.
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There are lots of planets,
especially small ones.
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Contents
2.1 Description of orbits
2.2 Measurement principles and accuracies
2.3 Instrument programmes
2.4 Results to date
2.5 Properties of radial velocity planets
2.6 Multiple planet systems
2.7 Planets around binary and multiple stars
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2.4 Results to date (1)
Present radial velocity census
 As of January 19 2015 (exoplanet.eu)
 589 planets in 433 systems
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2.4 Results to date (2)
Mass vs semi-major axis
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2.4 Results to date (3)
Planet detection vs spectral type
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2.4 Results to date (4)
Main sequence stars (HD4113; G5V)
 Early surveys focused on F & G stars
 Lots of absorption lines, low jitter.
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2.4 Results to date (5)
Main sequence stars (55 Cnc; G8V)
 5-planet system
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2.4 Results to date (6)
Main sequence stars (HD40307)
 Three super-Earths system
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2.4 Results to date (7)
M dwarfs surveys
 M dwarfs interesting because
 HZ at small semi-major axes with P~1-3 weeks
 Easier to detect small planets around low-mass stars
 Eta_Earth (Bonfils et al. 2011)
 Gas giants occur less frequently around M dwarfs
 Detection rate of Jovian planets at a < 1 AU earlier than
M5V: 1.3%.
 Low-mass planets (super-Earths, Neptune-mass)
appear more common around M dwarfs
 Only 7 Doppler-detected giant planets (M>0.3 MJ)
known aroun 6 M dwarfs.
 GJ581c: first super-Earths (7.7 ME) found close to the
HZ.
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2.4 Results to date (8)
Early type stars
 Not best targets for RV detection because
 Few absorption lines
 High rotational velocities (v sin i = 100-200 km/s)
 Large atmospheric jitter (~50 m/s) due to surface
inhomogeneities and pulsation.
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2.4 Results to date (9)
Other star categories
 Evolved stars (sub-giants and giants)
 Advantage of cooler atmosphere (more metal lines) and lower rotational
broadning. Ideal for RV.
 About 50 detections (~30 for IV and ~20 for III)
 Open clusters
 Difficult to estimate mass of giants stars as evolutinary track of various
mass all converge to the same region of the HRD.
• Open clusters offers the advantage of a known age -> better estimate of the
host star mass.
 Hyades: a few detections around class III but no close-in planets around 94 mainsequence stars.
 Metal-poor stars
 Planet occurrence rate lower for low-metallicity stars.
 Planet detected around a low-metallicity ([Fe/H=-2.0]) Galactic Halo star.
 Very young stars
 A 6.1 MJ planet around a 100 Myr-old G star (HD70573)
 A 9.8 MJ planet around TW Hya (8-10 Myr), a star with a known circumstellar
disk.
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Contents
2.1 Description of orbits
2.2 Measurement principles and accuracies
2.3 Instrument programmes
2.4 Results to date
2.5 Properties of radial velocity planets
2.6 Multiple planet systems
2.7 Planets around binary and multiple stars
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2.5 Properties of radial velocity planets (1)
Frequency of massive planets
 1200 FGKM stars from the California and Carnegie
Planet Search program (e.g. Marcy et al. 2008)
 87% of stars observed for more than a decade show no
Doppler variations at a 3σ limit of 10 m/s.
 At least 6-7% have giants with Mp>0.5 MJ and a < 5 AU.
• 15% of those are hot-Jupiters (P< 10 days)
 Remaining 6% show long-term radial velocity trends
indicating sub-stellar, brown dwarf of planetary companion
with P > a decade or more.
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2.5 Properties of radial velocity planets (2)
Frequency of planets (from Mayor et al. 2011)
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2.5 Properties of radial velocity planets (3)
Mass distribution
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2.5 Properties of radial velocity planets (4)
Mass distribution (histogram)
 Brown dwarf desert
 There is paucity of RV detected sub-stellar objects in the mass range 1080 MJ. Desert confirmedby imaging data (e.g. Lafrenière et al. 2007)
 Suggest a different formation mechanism for BDs and exoplanets.
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2.5 Properties of radial velocity planets (5)
Low-mass planets
 Below 0.1 MJ, despite incompleteness, mass function
is rising toward lower masses (see next slide)
 As of January 16 2015, 57 planets (in 35 systems)
with M < 10 ME detected by RV.
 Many detections close to detection limits over small
periods of time suggest large population of small
planets perhaps as high as 30%.
 Mass distribution bimodal with a gap around 30 ME
 Expected theoretically from core-accretion formation model
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2.5 Properties of radial velocity planets (6)
Low-mass planet distributions (Mayor et al.2011)
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2.5 Properties of radial velocity planets (7)
Orbits (period distribition)
 Pile-up of planets at P~3d (a < 0.05 AU) + a minimum
of in the interval P~10-100d (a ~ 0.1-0.4 AU)
 Hot-Jupiters formerd further out followed by migration with
some mechanism halting he migration before the planets fall
onto their host star.
 Decline of P beyond 1000d may
not be real. Flat distribution
would imply doubling known rates.
 Large population of of MJ objects
may exist between 3 and 20 AU.
 Giants rare beyond 20-30 AU
(c.f. imaging constraints from
Lafrenière et al. 2007)
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2.5 Properties of radial velocity planets (8)
Mass-period relation
 Correlation between mass ans semi-major axis
 Expected theoretically from core-accretion model
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2.5 Properties of radial velocity planets (9)
Eccentricities





Pre-discovery theories predicted low-eccentricities
Correlation between a and e.
Close-in planets are in preferentially low eccentricity orbits
Small fraction of Solar system analogues.
Eccentric orbits arise from several mechanisms: gravitational
interactions (planet-planet, planet-disk, bound ou unbound
mass).
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2.5 Properties of radial velocity planets (10)
Solar system analogues
 Planetary systems with low e and witha giant at a
comparable semi-major axis.
 Systems with large e are unstable and tend to eject
small close-in planets.
 Example of a solar system analogue HD154345
 Host star: 0.88 M
 Planet properties: 0.92 MJ, a=4.3 AU (P=9.2 yr), e=0.16
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2.5 Properties of radial velocity planets (11)
Host star dependencies
 Correlation between presence of gas giants with high-metallicity
 Correlation does not hold for low-mass planets
 Correlation between mass and planet occurrence
3%
2%
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Contents
2.1 Description of orbits
2.2 Measurement principles and accuracies
2.3 Instrument programmes
2.4 Results to date
2.5 Properties of radial velocity planets
2.6 Multiple planet systems
2.7 Planets around binary and multiple stars
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2.6 Multiple planet systems (1)
General considerations
 Key discoveries
 First multiple system: ν And (Buttler et al. 1999)
 Second system: 47 Uma (Fisher et al. 2002)
 First resonant pair: GJ876 (Marcy et al. 2001a)
 Frequency
 10-15% of confirmed system
 Another 10-15% showing evidence of multiplicity through
long-term RV radial velocity trends.
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2.6 Multiple planet systems (2)

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2.6 Multiple planet systems (3)
General considerations
 Multiple systems and theories of formation
 Core accretion is most compelling scenario
• Dust particule agglomeration.
• With no gas around, process stops to form rocky planets.
• With gas around (beyond ‘’snow’’ line: ~3AU for Solar-type stars),
runaway accretion proceeds to produce an ice- or a giant planet.
 Existence of Hot-Jupiters (a<0.1 AU) where little ice exists
suggests that those formed far out and migrated inwards.
 Existence of resonant systems support migration hypothesis.
 Wide distribution of eccentricities could be due to an early era of
strong planet-planet interaction in multiple systems.
 Coplanarity
 In principle not constrained due sin i uncertainty but N-body
simulations suggests that muliple systems may be substantially
non-coplanar. Hypothesis confirmed by HST astrometric
measurements of ν And c and ν And c with Δicd=29.9±1°
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2.6 Multiple planet systems (4)
General considerations
 Statistics of multiple planet systems more uniform
compared to single systems
 No pile up of hot Jupiters between 0.03-0.07 AU
 No discontinuity at ~ 1 AU
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2.6 Multiple planet systems (5)
General considerations
 Dynamical modelling
 RV signal not simply the sum of all reflex motions of all planets.
 Planet-planet interaction important even on short time scales.
• Lead to evolution of orbital parameters over period of years.
 Dynamical analysis done with two approaches
 N-body numerical simulations.
• May yield deprojected planetary masses and, in favorable conditions
based on stability arguments, relative orbital inclinations
 Analytical methods
• Secular theory
– Ignores terms that depend on planet’s mean motion n=2π/P and high-order orbit
terms.
– Describes non-preriodic secular evolution.
– Predicts that most two-planet systems will have their eccentricities oscillate
through angular momentum exchange.
• Resonant theory
– Ignores terms that depend on planet’s mean motion
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2.6 Multiple planet systems (6)
Resonances
 Orbital resonances arise when two orbiting bodies exert a
regular, periodic gravitational influence on each other as a result
of simple relationship between periods.
 Orbital resonances greatly enhance the ability of bodies to alter
or constrain other’s orbits.
 Most resonances lead to unstable interaction but, under some
circumstances, they can be self-correcting so that the bodies
remain in resonance.
 Example of stable resonance: the 1:2:4 Laplace resonance of Jupiter’s moons:
Ganymede, Europa and Io.
 Example of unstable resonance: Saturn’s inner moons give rise to gaps in the
rings of Saturn. Gaps in circumstellar disks.
 Resonances can involve different periods asociated with the
planetary system.
 Planet rotation and orbit period (spin-orbit) coupling, e.g. Earth-Moon
system.
 Orbital periods of two or more bodies (orbit-orbit coupling)
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2.6 Multiple planet systems (7)
Mean motion resonances
 Resonant systems have orbital period related by
where subscript 1, 2 refer to inner and outer planets, and i and j
are small integers.
 Resonance condition cam also be expressed in terms on mean
motion n=2π/P
(2.45)
 If two planets are conjunction (in alignment) at t=0, then the
next conjunction will occur when
(2.46)
longitude
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2.6 Multiple planet systems (7)
Mean motion resonances - conjunction
 Time interval Δt beween successive conjunctions
(2.47)
(2.48)
 For q=1, each planet completes an integer number of
orbits between successive conjunctions
 Every q-th conjunction occurs at the same longitude in
intertial space.
 q defines the resonance order of the mean motion
resonance
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2.6 Multiple planet systems (8)
Mean motion resonances – conjunction with e2≠0
 If outer planet moves in an eccentric orbit with e2≠0 and
the the longitude of the pericenter precesses,
i.e.
, resonance can still occur if
(2.49)
or
(2.50)
 Every q-th conjunction takes place at the same point in
the other planet’s orbit, but no longer at the same
longitude in inertial space.
 Hence the notion of mean motion resonance.
 Shows the near but not exact commensurabilities in orbital
periods.
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2.6 Multiple planet systems (9)
Mean motion resonances – resonant argument
 Let λ be the mean longitude, the corresponding resonant
arguments (resonant angles), Φ, is defined as
(2.51)
 The resonant argument measures the angular displacement of
the two planets at their point of conjunction.
 Resonance dynamics are important if Φ varies slowly relative to
orbital motion.
 If Φ=0 or oscillates (librates), then the planet are in resonance.
 Exact resonance is the condition when Φ=0.
 Deep resonance is used to describe systems with small libration
amplitudes.
 Outside of exact resonance, a, e,
and Φ all evolve with time.
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2.6 Multiple planet systems (10)
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2.6 Multiple planet systems (11)
Mean motion resonances – origin
 Orbital commensurabilities in the solar system attributed to
dissipative processes early on it its formation, or to the slow
differential increase in the semi-major axes of satellite orbits as a
result of tidal transfer of angular momentum.
 Tidal locking which makes the planet always show the same face to the
star (Earth-moon system).
 Present concensus is that observed exoplanet resonances could
not have formed in situ.
 N-body simulations shows that resonances results from
differential convergent migration in which dissipative processes
(e.g. tidal forces) alter their semi-major axes and eccentricities.
 The occurrence of orbital resonances constitutes strong evidence
of past migration.
 Many planetary systems are found to be dynamically full
 No additional companions can survive in between observed planets with
most pairs of planets lying close to dynamical instability.
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2.6 Multiple planet systems (12)
Mean motion resonances – Hill Radius
 The Hill radius is the radius within which the gravity of one object
dominates that of other bodies within he system.
 Approximates the gravitational sphere of influence of a smaller body
in the face of perturbations from a more massive body.
 For a planet in a circular orbit of radius r, its Hill radius depends on
the planet and central star mass as
(2.58)
 Simple criteria to assess the importance of orbit-orbit interaction.
 Hill radius of the Earth: 1.5x106 km (L2 point)
 A body of mass coming within this radius is likely to become a satellite of the
Earth.
 The moon (at 300 000 km) is confortably within the gravitational sphere of
influence of the Earth.
 Hill radius of the Sun within the Galaxy: ~12 000 AU (~2 light-yr)
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2.6 Multiple planet systems (13)
Lagrange points
 Lagrange points mark positions where the combined
gravitational pull of the two large masses provides precisely the
centripetal force required to orbit with them.
 At L-points, the orbital periods around the two bodies are
equal.
 Earth L2 is the best location for space observatories when
thermal stability is required.
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2.6 Multiple planet systems (14)
Systems with three or more giants - ν And
 First multiple system discovered (three planets)
 Inner planet: Mp sin i= 0.6 MJ, P=4.6d, e~0. Exceeds
minimum stability requirement given by Hill radius
 Little interaction expected with other two companions
 Outer two planets: Mp sin i= 2.0 and 4.2 MJ, a of 0.82 and
2.5 AU, and large e of 0.23 and 0.36, repwectively.
 Stability of two planets strongly depends on the planet masses,
and hence their relative orbital inclinations.
 Certain combinations implies chaotic or unstable orbits
 Numerical simulation studies done to assess stability over
long time scale
 Provides constraints on masses and relative inclinations
 A fourth outer planet was probably part of the system and was
ejected through planet-planet interaction to leave.
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2.6 Multiple planet systems (15)
Systems with three or more giants - ν And
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2.6 Multiple planet systems (16)
Systems with three or more giants - ν And
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2.6 Multiple planet systems (17)
Systems with three or more giants – HD37124
 Initially discovered as a two-planet system (P=150-d,
P~6 yr).
 Dynamical studies showed the system to be unstable.
 Additional observations from Keck-HIRES unveiled a
third planet.
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2.6 Multiple planet systems (18)
Systems with three or more giants – 55 Cnc
 First 5-planet system
 All five planets reside in low-eccentricity (e < 0.1)
orbits
 Stability confirmed through N-body simulations
 Similar attributes to Solar system
 Orbits rather circular and one dominant gas giant at ~6 AU
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2.6 Multiple planet systems (19)
Systems in mean motion resonance
 The 2:1/Laplace resonance system GJ876
 First known M-dwarf host
 Pc=30d, Mc=0.7 MJ; Pb=61d, Mc=2.3 MJ
• Planet c is in the habitable zone.
 A third non-interacting super-Earth, Md=6.8 ME in a 1.9d
orbit, is also detected.
 A fourth planet detected; Pe=124.3d, Me=14.6 ME (~Uranus).
N-body fit show that this four-planet system has an
invariable plane with an inclination of 59.5°, stable for more
than 1 Gyr.
 Fourth planet forms a Laplace (1:2:4) resonance with b and c.
• Pc=30d, Pb=61d, Pe=126.6d
• A Laplace resonance occurs when three or more orbiting bodies
have a simple integer ratio between their orbital periods.
 Comes close to triple conjunction once per orbit of the
othermost planet.
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2.6 Multiple planet systems (19)
Systems in mean motion resonance – GJ876
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2.6 Multiple planet systems (20)
Systems in mean motion resonance – GJ876
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2.6 Multiple planet systems (21)
Other mean motion resonances
 Inclination resonance
 Few observational constraints yet, mostly theoretical inference
 Most studies have focussed on coplanar configurations
 Non coplanar orbits relevant for an understanding of
eccentricity excitation and migration.
 Kozai resonance
 Perturbation of the orbit of an inner planet by the
gravity of another body orbiting farther out.
 This causes libration (oscillation about a constant value)
of the orbit’s argument of pericenter.
 As the orbit librates, there is a periodic exchange
between its inclination and its eccentricity.
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2.6 Multiple planet systems (22)
Other mean motion resonances
 1:1 resonance (e.g. Trojans satellite of Jupiters)
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2.6 Multiple planet systems (23)
Other mean motion resonances
 1:1 resonance (e.g. Quasi-satellites of the Earths )
Tadpole orbits
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2.6 Multiple planet systems (24)
Other mean motion resonances
 1:1 resonance
 No such system firmly indentified yet
 Difficult of detect in periodogram because object have the
same masse, hence same period.
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Contents
2.1 Description of orbits
2.2 Measurement principles and accuracies
2.3 Instrument programmes
2.4 Results to date
2.5 Properties of radial velocity planets
2.6 Multiple planet systems
2.7 Planets around binary and multiple stars
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2.7 Planet around binary and multiple stars (1)
 Most stars occur in binary or multiple star systems
 Many young binaries are known to possess disks
either
 Circumstellar (around one of the stars)
 Circumbinary (surrounding both stars)
 Circumbinary disks may provide the accretion
material necessary for planet formation
 Interesting laboratories for investingating planet
formation mechanisms
 Multiple systems were not a high priority for early
searches of exoplanets.
 More and more systems being discovered through various
techniques
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2.7 Planet around binary and multiple stars (1)
Configurations and stability
 A planet may exist in two stable configurations
 S-type: orbiting one component
 P-type: orbiting both components
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2.7 Planet around binary and multiple stars (2)
Kozai resonance of S-type systems
 Angular momentum exchange between the planet and
the secondary (outer) star results in the planet’s
inclination and eccentricity oscillating synchonously,
i.e. increasing at the expense of the other. The
following quantity is conserved:
(2.69)
 Kosai resonance sets in at inclinations greater than a
critical value of ic=arcsin√0.4≈39.2°
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2.7 Planet around binary and multiple stars (3)
Kozai resonance of S-type systems
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2.7 Planet around binary and multiple stars (4)
Present inventory
 As of late 2010, > 50 planets have been found to be
associated with binary or multiple stars
 Most are gas giants orbiting the primary component, with
projected separations in the range 20-12 000 AU.
 Properties somewhat similar to single systems (next slide)
 Specific example: γ Cep: 1.6
 Stellar system of masses 1.6 and 0.4 M, separated by 18.5 AU (P~57 yr,
e=0.36)
 Planet mass: 1.7 MJ
 Planet stable for binary eccentricity 0.2 eb < 0.45, unstable with 1000 yrs for
eb > 0.5 .
 Planet stable for i < 40°. For large inclinations, system may be locked in a
Kozai resonance.
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2.7 Planet around binary and multiple stars (5)
Observational properties
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2.7 Planet around binary and multiple stars (6)
List of circumbinary planets
Source: wikipedia
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2.8 Summary (1)
 Importance. RV: second most productive method for detecting
exoplanets.
 Nearly 600 discovered by RV as of January 2015
 Orbit specification. Seven Keplerian parameters:
 Five of them constrained by RV: a, e, P, tp and ω.
 Radial velocity amplitude equation:
 Instrumentation
 High-resolution echelle spectroscopy + cross-correlation technique
 Two types of wavelength calibration: gas (I2) cell (e.g. KECK HIRES) and
fiber feed (HARPS)
 Wavelength coverage: mostly optical (300-700 nm)
 Accuracy and limitations
 Best performance achieved over long periods: 1 m/s (made in Switzerland)
 Main limitation is ‘’jitter’’ noise (2-20 m/s) associated with stellar activities
(spots, plages, oscillations, rotation)
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2.8 Summary (2)
 Science highlights
Hot Jupiters (e.g. first planet discovered by RV: 51 Peg)
Frequency: ~75% of stars have planets (any mass) with P< 10 yr
Mass distribution:
. Bimodal, increases towards low-mass
Many systems with large eccentricities indicative of dynamical
processeses (planet interactions, migration) at play.
 Planet occurrence correlates with host mass and metallicity.
 Multiple planets very common




• Many systems in resonance also indicative of dynamical processes.
 Future instrument programmes
 Scientific goal to detect small rocky planets in the habitable zone of their
star.
 ESPRESSO: HAPRS-like instrument on VLT (2016).
 Strong motivation for the development of IR velocimetry (e.g. SPIRou @
CFHT). The ‘’M-dwarf/infrared opportunity’’:
• Lots of nearby M dwarfs, easier to detect Earths, HZ of a few weeks, flux in
the IR.
• M dwarfs are best targets for atmosheric characterization with JWST.
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