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Are planetary systems flat? collaborators: Subo Dong (IAS) Dan Fabrycky (UC Santa Cruz) Boaz Katz (IAS) Aristotle Socrates (IAS) Thursday, April 26, 2012 Why do astronomers think that planetary systems are flat? Thursday, April 26, 2012 Why do astronomers think that planetary systems are flat? • the solar system is flat (maximum inclination 7 deg) • the inner satellite systems of Jupiter, Saturn, Uranus are flat Thursday, April 26, 2012 Why do astronomers think that planetary systems are flat? Thursday, April 26, 2012 Why do astronomers think that planetary systems are flat? • many other astrophysical systems are flat Thursday, April 26, 2012 Why do astronomers think that planetary systems are flat? •disks minimize energy at constant angular momentum Laplace, Exposition du système du monde (1796): – it is astonishing to see all the planets move around the Sun from west to east, and almost in the same plane; all the satellites move around their planets in the same direction and nearly in the same plane as the planets; finally, the Sun, the planets, and all the satellites that have been observed rotate in the direction and nearly in the plane of their orbits...another equally remarkable phenomenon is the small eccentricity of the orbits of the planets and the satellites...we are forced to acknowledge the effect of some regular cause since chance alone could not give a nearly circular form to the orbits of all the planets Thursday, April 26, 2012 Kepler Thursday, April 26, 2012 • launch March 6 2009 • 0.95 meter mirror, 100 megapixel camera, 12 degree diameter field of view • monitor ~105 stars continuously over ~6 yr mission • 20 parts per million photometric precision 1769 Transit of Venus next June 6, 2012, sunrise to 06:55 in Zurich Thursday, April 26, 2012 Thursday, April 26, 2012 Thursday, April 26, 2012 fractional brightness dip: Jupiter 0.01 = 1% Earth 0.0001=0.01% 1% Thursday, April 26, 2012 transit: planet moves in front of the star; U-shape because of limb darkening in star occultation: planet moves behind the star; square shape Thursday, April 26, 2012 transit: planet moves in front of the star; U-shape because of limb darkening in star occultation: planet moves behind the star; square shape Thursday, April 26, 2012 transit: planet moves in front of the star; U-shape because of limb darkening in star occultation: planet moves behind the star; square shape currently 230 planets by this method from the ground and ~2000 candidates from Kepler Thursday, April 26, 2012 van Kerkwijk et al. (2010) • orbital period P = 5.2 days • two curious features: • sinusoidal brightness variations at fundamental and first harmonic • transit (U shape) is shallower than occultation (square well) Thursday, April 26, 2012 van Kerkwijk et al. (2010) • orbital period P = 5.2 days • two curious features: • sinusoidal brightness variations at fundamental and first harmonic • transit (U shape) is shallower than occultation (square well) • both can be explained if the companion is a white dwarf rather than a planet: • occultation is deeper because the white dwarf is hotter than the primary (T=13,000 K vs. 9,400 K) • first harmonic due to tidal distortion of the primary by the white dwarf • fundamental due to Doppler boosting (gives velocity curve to 1 km/s) • white dwarf has mass 0.22±0.03 M⊙; radius 0.043±0.004 R⊙ Thursday, April 26, 2012 Why do astronomers think that planetary systems are flat? • mutual inclination of planets around PSR B1257+12 is < 13 degrees • modeling of radial velocities, astrometry, and dynamics implies that mutual inclination of GJ 876b and c is • < 2 degrees (Correia et al. 2010) • 3-9 degrees (Bean & Seifahrt 2009) • 5-15 degrees (Baluev 2011) • dynamical modeling of transit timing variations in Kepler-9 system implies that mutual inclination is < 10 degrees (Holman et al. 2010) • absence of transit timing variations in Kepler-11 implies inclination of Kepler-11e < 2 degrees (Lissauer et al. 2011) Thursday, April 26, 2012 Why do astronomers think that planetary systems are flat? • mutual inclination of planets around PSR B1257+12 is < 13 degrees • modeling of radial velocities, astrometry, and dynamics implies that mutual inclination of GJ 876b and c is • < 2 degrees (Correia et al. 2010) • 3-9 degrees (Bean & Seifahrt 2009) • 5-15 degrees (Baluev 2011) • dynamical modeling of transit timing variations in Kepler-9 system implies that mutual inclination is < 10 degrees (Holman et al. 2010) • absence of transit timing variations in Kepler-11 implies inclination of Kepler-11e < 2 degrees (Lissauer et al. 2011) Thursday, April 26, 2012 ✘ Why do astronomers think that planetary systems are flat? • mutual inclination of planets around PSR B1257+12 is < 13 degrees • modeling of radial velocities, astrometry, and dynamics implies that mutual inclination of GJ 876b and c is • < 2 degrees (Correia et al. 2010) • 3-9 degrees (Bean & Seifahrt 2009) • 5-15 degrees (Baluev 2011) • dynamical modeling of transit timing variations in Kepler-9 system implies that mutual inclination is < 10 degrees (Holman et al. 2010) • absence of transit timing variations in Kepler-11 implies inclination of Kepler-11e < 2 degrees (Lissauer et al. 2011) Thursday, April 26, 2012 ✘ ✘ Why do astronomers think that planetary systems are flat? • mutual inclination of planets around PSR B1257+12 is < 13 degrees • modeling of radial velocities, astrometry, and dynamics implies that mutual inclination of GJ 876b and c is • < 2 degrees (Correia et al. 2010) • 3-9 degrees (Bean & Seifahrt 2009) • 5-15 degrees (Baluev 2011) • dynamical modeling of transit timing variations in Kepler-9 system implies that mutual inclination is < 10 degrees (Holman et al. 2010) • absence of transit timing variations in Kepler-11 implies inclination of Kepler-11e < 2 degrees (Lissauer et al. 2011) Thursday, April 26, 2012 ✘ ✘ ✘ ✘ Why might we believe that planetary systems are not flat? • in most disks, rms eccentricity is correlated with rms inclination (e.g., asteroids have <i> ≈ <e>) • rms eccentricities in extrasolar planetary systems are much larger than in the solar system • <e> 0.25 implies <i> 15o Thursday, April 26, 2012 Why might we believe that planetary systems are not flat? • in most disks, rms eccentricity is correlated with rms inclination, and rms eccentricities in extrasolar planetary systems are much larger than in the solar system • mutual inclination of upsilon And c and d is 30±1 degrees (McArthur et al. 2010) • extrasolar planetary systems look quite different from the solar system (e.g., hot Jupiters) so there is no reason to expect that they formed in the same way • if planet formation were similar to star formation (collapse of dense cores in a cloud) we would expect an isotropic distribution – distribution of eccentricities of extrasolar planets is quite similar to distribution of eccentricities of binary stars (Black 1997) Thursday, April 26, 2012 Thursday, April 26, 2012 Jupiter Thursday, April 26, 2012 hot Jupiters Jupiter Thursday, April 26, 2012 Why might we believe that planetary systems are not flat? • in most disks, rms eccentricity is correlated with rms inclination, and rms eccentricities in extrasolar planetary systems are much larger than in the solar system • mutual inclination of upsilon And c and d is 30±1 degrees (McArthur et al. 2010) • extrasolar planetary systems look quite different from the solar system (e.g., hot Jupiters) so there is no reason to expect that they formed in the same way • if planet formation were similar to star formation (collapse of dense cores in a cloud) we would expect an isotropic distribution – distribution of eccentricities of extrasolar planets is quite similar to distribution of eccentricities of binary stars (Black 1997) Thursday, April 26, 2012 Ribas & Miralda-Escudé (2007) Thursday, April 26, 2012 Why might we believe that planetary systems are not flat? • in most disks, rms eccentricity is correlated with rms inclination, and rms eccentricities in extrasolar planetary systems are much larger than in the solar system • mutual inclination of υ And c and d is 30±1 degrees (McArthur et al. 2010) • extrasolar planetary systems look quite different from the solar system (e.g., hot Jupiters) so there is no reason to expect that they formed in the same way • if planet formation were similar to star formation (collapse of dense cores in a cloud) we would expect an isotropic distribution • current theoretical models of planet formation in a disk face serious obstacles Thursday, April 26, 2012 Why might we believe that planetary systems are not flat? • rms eccentricities in extrasolar planetary systems are much larger than in the solar system • mutual inclination of υ And c and d • extrasolar planetary systems are different from the solar system • if planet formation were similar to star formation (collapse of dense cores in a cloud) we would expect an isotropic distribution • current theoretical models of planet formation in a disk face serious obstacles • stellar spin axis and planet orbital axis are often mis-aligned (by 7 degrees for the Sun and much more for some other systems) Thursday, April 26, 2012 (from J. Winn) Thursday, April 26, 2012 λ obliquity = angle between spin angular momentum of star and orbital angular momentum of planet Rossiter-McLaughlin measures projected obliquity (λ) = angle between projection of spin angular momentum of star and orbital angular momentum of planet on the sky plane (from J. Winn) Thursday, April 26, 2012 Thursday, April 26, 2012 HAT-P-30b 20 0 −20 v sin i [km s−1] RV [m s−1] HAT P-7b λ = 183±9° Winn et al. (2009) HAT P-30b λ = 74 ± 9° Winn et al. (2009) 4.0 40 −40 O−C [m s−1] 7 10 0 3.5 HAT P-14b λ = 189 ± 5° Winn et al. (2011) 3.0 2.5 −10 −2 0 2 Time from midtransit [hr] 50 60 70 80 ! [deg] 90 100 Fig. 4.— Rossiter-McLaughlin effect for HAT-P-30 Left.—Apparent radial velocity variation on the night of 2011 Feb 21, spanning Thursday, April 2012 a transit. The top 26, panel shows the observed RVs. The bottom panel shows the residuals between the data and the best-fitting model. projected obliquity λ (degrees) Brown et al. (2012) effective temperature (K) Thursday, April 26, 2012 projected obliquity λ (degrees) Brown et al. (2012) observations imply that either: (a) the planetary system is flat, but the stellar spin is tilted relative to the planetary orbital plane, or (b) planetary system is not flat, but the stellar spin is aligned with the mean planetary orbital angular momentum effective temperature (K) Thursday, April 26, 2012 (a) the planetary system is flat, but the stellar spin is tilted relative to the planetary orbital plane • collision with a giant planet: • spin angular momentum of the Sun = 2 X 1048 gm cm2/s • angular momentum of Jupiter-mass object on a parabolic orbit with perihelion of 0.5R⊙ = 6 X 1048 gm cm2/s Thursday, April 26, 2012 (a) the planetary system is flat, but the stellar spin is tilted relative to the planetary orbital plane • collision with a giant planet • tilt the plane of the planetary orbits by external torques • planets stay coplanar so long as tilting time longer than precession times due to their mutual gravitational interactions (104 to 3 X 105 yr) • stellar spin does not follow the tilt if tilting time is shorter than precession time of stellar spin due to planets (~3 X 1010 yr) fffff Thursday, April 26, 2012 (b) planetary system is not flat, but the stellar spin is aligned with the mean planetary orbital angular momentum •form planets in collapsing clouds around the host star, as in binary stars • form planets in a disk, and excite their inclinations later. For example: • migration and resonance capture Thursday, April 26, 2012 Pluto’s peculiar orbit Pluto has: • the highest eccentricity of any “planet” (e = 0.250 ) • the highest inclination of any “planet” ( i = 17o ) • perihelion distance q = a(1 – e) = 29.6 AU that is smaller than Neptune’s semi-major axis ( a = 30.1 AU ) How do they avoid colliding? Thursday, April 26, 2012 Pluto’s peculiar orbit Orbital period of Pluto = 247.7 y Orbital period of Neptune = 164.8 y 247.7/164.8 = 1.50 = 3/2 Resonance ensures that when Pluto is at perihelion it is approximately 90o away from Neptune (Cohen & Hubbard 1965) Thursday, April 26, 2012 Pluto’s peculiar orbit PPluto/PNeptune Pluto’s semi-major axis Pluto’s eccentricity Pluto’s inclination time •early in the history of the solar system there was debris left over between the planets •ejection of this debris by Neptune caused its orbit to migrate outwards •if Pluto were initially in a low-eccentricity, low-inclination orbit outside Neptune it is inevitably captured into 3:2 resonance with Neptune •once Pluto is captured its eccentricity and inclination grow as Neptune continues to migrate outwards •other objects may be captured in the resonance as well Malhotra (1993) Thursday, April 26, 2012 objects in 2:3 resonance Kuiper belt objects comets Thursday, April 26, 2012 t Period ratios in Kepler’s multi-planet systems biggest second biggest third biggest fourth biggest Fabrycky et al. (2012) Thursday, April 26, 2012 Period ratios in Kepler’s multi-planet systems Fabrycky et al. (2012) Thursday, April 26, 2012 planetary system is not flat, but the stellar spin is aligned with the mean planetary orbital angular momentum •form planets in collapsing clouds around the host star, as in binary stars • form planets in a disk, and excite their inclinations later. For example: • migration and resonance capture • Kozai-Lidov oscillations Thursday, April 26, 2012 Kozai-Lidov oscillations • Kozai (1962); Lidov (1962) • arise most simply in restricted three-body problem (two massive bodies on a Kepler orbit + a test particle, e.g., binary star + planet orbiting one member of the binary) •in Kepler potential Φ = -GM/r, eccentric orbits have a fixed orientation Thursday, April 26, 2012 Kozai-Lidov oscillations • now subject the Kepler orbit to a weak, time-independent external force F • because the orbit orientation is fixed even weak external forces act for a long time in a fixed direction relative to the orbit and therefore change the angular momentum or eccentricity • if F ~ ε then timescale for evolution ~ 1/ε" but nature of evolution is independent of ε F Thursday, April 26, 2012 Kozai-Lidov oscillations planet Consider a planet orbiting one member of a binary star system: pericenter • can average over both planetary and binary star orbits • both energy and z-component of angular momentum of planet are conserved • thus the orbit-averaged problem has only one degree of freedom (e.g., eccentricity e and angle in orbit plane from the equator to the pericenter, ω), so potential Φ=Φ(e,ω) • motion is along level surfaces of Φ(e,ω) Thursday, April 26, 2012 binary star orbital plane Kozai-Lidov oscillations • initially circular orbits remain circular if and only if the initial inclination is < 39o = cos-1(3/5)1/2 circular • for larger initial inclinations the phase plane contains a separatrix • circular orbits cannot remain circular, and are excited to high inclination and eccentricity •circular orbits are chaotic radial Thursday, April 26, 2012 Kozai-Lidov oscillations • as initial inclination approaches 90 , maximum eccentricity approaches unity ° circular collision or tidal dissipation • mass and separation of companion affect period of Kozai-Lidov oscillations, but not the amplitude radial Thursday, April 26, 2012 eccentricity oscillations of a planet in a binary star system M3=0.9M⊙ M3=0.08 M⊙ • aplanet = 2.5 AU • companion has inclination 75°, semimajor axis 750 AU, mass 0.08 M⊙ (solid) or 0.9 M⊙ (dotted) (Takeda & Rasio 2005) Thursday, April 26, 2012 Kozai-Lidov oscillations circular • may excite inclinations of planets in some binary star systems, but probably not all planet inclinations: • not all systems have stellar companions • oscillations are suppressed by additional coplanar planets • oscillations can be suppressed by general relativity (!) • also excites eccentricity and high eccentricity is not well-correlated with presence of a companion star Thursday, April 26, 2012 radial Kozai-Lidov oscillations binary single Thursday, April 26, 2012 Kozai-Lidov oscillations and the formation of hot Jupiters Giant planets at small semi-major axes cannot be formed in situ. There are two mechanisms for placing giant planets in orbits of size < 0.1 AU: 1. migration due to gravitational torques from the protoplanetary disk 2. “high-eccentricity migration” is a multi-step process: • planet-planet scattering at 5-10 AU kicks a planet into an eccentric, inclined orbit (e.g., Rasio & Ford 1996) • torques from other planets drive KL oscillations • KL oscillations drive the pericenter of the orbit to < 0.05 AU where tidal friction from the host star drains energy and eccentricity from the orbit • eventually the planet settles on a circular orbit near the star Thursday, April 26, 2012 theory (Kozai-Lidov oscillations) Fabrycky & Tremaine 2007 observations (from R-M effect) Brown et al. 2012 misaligned projected obliquity Thursday, April 26, 2012 theory (Kozai-Lidov oscillations) Fabrycky & Tremaine 2007 observations (from R-M effect) Brown et al. 2012 misaligned • high-eccentricity migration models predicted large obliquities seen in Rossiter-McLaughlin observations • predict that a population of “super-eccentric” planets (e > 0.9) is present in the Kepler planet sample (Socrates et al. 2011 ) projected obliquity Thursday, April 26, 2012 first Kepler data release (Borucki et al. 2011) • • • 155,000 stars • comparison of numbers of multi-planet systems between Kepler and radial-velocity surveys should constrain inclination distribution (Dong & Tremaine 2011) 1235 planetary candidates, ~90-95% real (Morton & Johnson 2011) 115 two-planet systems, 45 three-planet systems, 8 four-planet systems, one 5-planet and one 6-planet system Good news: • Kepler provides a large, homogeneous database with well-defined selection criteria Bad news: • radial-velocity surveys are heterogeneous and have poorly defined selection effects • Kepler and RV surveys cover different range of mass and semi-major axis Thursday, April 26, 2012 radial velocities transits Thursday, April 26, 2012 radial velocities there are reliable, modelindependent ways to correct the exoplanet multiplicity distribution for different survey properties transits Thursday, April 26, 2012 Kepler survey gives: n0 = 123,700 n1 = 737 n2 = 104 n3 = 37 n4 = 7 n5 = 1 n6 = 1 n7,8,9,… = 0 All RV surveys give n0 = ?? n1 = 162 n2 = 24 n3 = 7 n4 = 1 n5 = 1 n6,7,8,… = 0, Note that n0 is not known! Fit using free parameters: • N1, N2, N3, …, NK - number of k-planet systems in parent population • <sin2 i>½ - rms inclination in multi-planet systems Thursday, April 26, 2012 Kepler survey gives: n0 = 123,700 n1 = 737 n2 = 104 n3 = 37 n4 = 7 n5 = 1 n6 = 1 n7,8,9,… = 0 All RV surveys give n0 = ?? n1 = 162 n2 = 24 n3 = 7 n4 = 1 n5 = 1 n6,7,8,… = 0, Note that n0 is not known! Fit using free parameters: • N1, N2, N3, …, NK - number of k-planet systems in parent population • <sin2 i>½ - rms inclination in multi-planet systems Thursday, April 26, 2012 flat Thursday, April 26, 2012 isotropic Estimating the effective number of target stars in RV surveys: 1) Cumming et al. (2008) derive percentage of FGK stars with a planet having M > MJupiter and P < 1 yr, p = 0.019 ± 0.007. All major RV surveys should be complete in this range, and our sample has 46 such planets. Thus ntotal = 46/p = 2400 ± 900 2) Convert Kepler planet radii to mass using empirical mass-radius relation, then compute number of target stars needed to produce observed number of RV planets as a function of maximum period and minimum velocity semiamplitude Thursday, April 26, 2012 ntot=3000±1000 Thursday, April 26, 2012 Dong & Tremaine (2011) Thursday, April 26, 2012 <sin2i>1/2 < 0.1 or <i> < 6o Dong & Tremaine (2011) Thursday, April 26, 2012 • if two planets have circular, coplanar orbits around the same star orbit the same star their impact parameters b and semi-major axes a are related by b2/b1 = a2/a1 • b by Kepler’s law semi-major axis and orbital period are related by Rs a2/a1 = (P2/P1)2/3 • transit duration is 2(Rs2-b2)1/2/v where v ∝ P-1/3 ➨ transit durations in multi-planet systems constrain the inclination distribution (Fabrycky et al. 2012) -- find rms inclination between 1o and 2.5o • advantage: doesn’t require comparison of Kepler and RV samples • disadvantage: only works if the rms inclinations are small compared to Rs/a ~ 0.05 or 3o Thursday, April 26, 2012 Summary • for the first time we have a strong constraint on the inclinations in a large sample of extrasolar planets • typical multi-planet systems in the Kepler sample of planets are flat, with mean inclination < 6o and probably <3o • nevertheless – RV planets have <e> ~ 0.25 so if <i>=<e> we expect much larger <i> ~ 15o • eccentricities have declined ~20% as observations improve in number and quality • in some formation models <i> is as small as 0.5<e> • <e> seems to decline as planet mass shrinks – there is strong observational evidence that many exoplanet systems have large misalignments between the host star and planet angular momenta • these are clues to resolving one of the basic questions of exoplanet formation: do giant planets at small radii get there via – disk migration – high-eccentricity migration – some other process Thursday, April 26, 2012 Summary • for the first time we have a strong constraint on the inclinations in a large sample of extrasolar planets • typical multi-planet systems in the Kepler sample of planets are flat, with mean inclination < 6o and probably <3o • nevertheless – RV planets have <e> ~ 0.25 so if <i>=<e> we expect much larger <i> ~ 15o • eccentricities have declined ~20% as observations improve in number and quality • in some formation models <i> is as small as 0.5<e> • <e> seems to decline as planet mass shrinks – there is strong observational evidence that many exoplanet systems have large misalignments between the host star and planet angular momenta • these are clues to resolving one of the basic questions of exoplanet formation: do giant planets at small radii get there via – disk migration – high-eccentricity migration – some other process Thursday, April 26, 2012 Summary • for the first time we have a strong constraint on the inclinations in a large sample of extrasolar planets • typical multi-planet systems in the Kepler sample of planets are flat, with mean inclination < 6o and probably <3o • nevertheless – RV planets have <e> ~ 0.25 so if <i>=<e> we expect much larger <i> ~ 15o • eccentricities have declined ~20% as observations improve in number and quality • in some formation models <i> is as small as 0.5<e> • <e> seems to decline as planet mass shrinks – there is strong observational evidence that many exoplanet systems have large misalignments between the host star and planet angular momenta • these are clues to resolving one of the basic questions of exoplanet formation: do giant planets at small radii get there via – disk migration – high-eccentricity migration – some other process Thursday, April 26, 2012 Summary • for the first time we have a strong constraint on the inclinations in a large sample of extrasolar planets • typical multi-planet systems in the Kepler sample of planets are flat, with mean inclination < 6o and probably <3o • nevertheless – RV planets have <e> ~ 0.25 so if <i>=<e> we expect much larger <i> ~ 15o • eccentricities have declined ~20% as observations improve in number and quality • in some formation models <i> is as small as 0.5<e> • <e> seems to decline as planet mass shrinks – there is strong observational evidence that many exoplanet systems have large misalignments between the host star and planet angular momenta • these are clues to resolving one of the basic questions of exoplanet formation: do giant planets at small radii get there via – disk migration – high-eccentricity migration – some other process Thursday, April 26, 2012 Summary • for the first time we have a strong constraint on the inclinations in a large sample of extrasolar planets • typical multi-planet systems in the Kepler sample of planets are flat, with mean inclination < 6o and probably <3o • nevertheless – RV planets have <e> ~ 0.25 so if <i>=<e> we expect much larger <i> ~ 15o • eccentricities have declined ~20% as observations improve in number and quality • in some formation models <i> is as small as 0.5<e> • <e> seems to decline as planet mass shrinks – there is strong observational evidence that many exoplanet systems have large misalignments between the host star and planet angular momenta • these are clues to resolving one of the basic questions of exoplanet formation: do giant planets at small radii get there via – disk migration – high-eccentricity migration – some other process Thursday, April 26, 2012