Resonant Orbital Dynamics in Extrasolar Planetary Systems and the Pluto Satellite System Man Hoi Lee (UCSB)
Introduction: Extrasolar Planetary Systems Extrasolar planet searches have yielded ~ 150 planetary systems to date. Vast majority discovered by detecting the reflex motion of the host star. Transit searches are beginning to yield planets. ~ 18 systems with multiple planets.
~ 1/3 of the known multiple planet systems have a pair of planets known or suspected to be in mean motion resonances. It is now well established that the pair of planets with orbital periods P 30 and 60 days about the star GJ 876 is deep in 2:1 orbital resonance. HD 82943 was the second star discovered to host a pair of planets with orbital periods nearly in the ratio 2:1. Analysis of the HD 82943 planetary system based on a radial velocity data set that combines new measurements obtained with the Keck telecope and the existing CORALIE measurements. (Lee, Butler, Fischer, Marcy, & Vogt 2006, astro ph/ 0512551)
It is likely that mean motion resonances were established during planet formation by the convergent migration of planets due to interactions with the protoplanetary disk. What are the conditions required to assemble the observed 2:1 resonance in the HD 82943 system? (Kley & Lee)
Introduction: the Pluto System
Weaver et al. (2006) discovered two small satellites of Pluto, S/2005 P1 and S/2005 P2, from two images taken with the HST in May 2005. First new satellites of Pluto since discovery of Charon in 1978. Diameter of Charon ~ 1200 km. Size estimates from brightness: If albedo is high and Charon like (0.35), diameter of P1 ~ 60 km. If albedo is low and comet like (0.04), diameter of P1 ~ 170 km. P2 is ~ 20% smaller.
Buie et al. (2006) detected Charon, P2 and P1 in a set of images taken with HST between 6/2002 and 6/2003. Assuming Keplerian orbits, fit by Buie et al. shows that the orbits of P2 and P1 are nearly circular and nearly coplanar with that of Pluto Charon. Orbital periods P 25 days for P2 P 38 days for P1
Unperturbed Keplerian orbits are not good assumptions for orbits of P2 and P1. Mass ratio of Charon Pluto m c /m p = 0.12. Orbital periods of Charon, P2 and P1 nearly in the ratio 1:4:6. Strongest effects from proximity of P2 and P1 to 3:2 commensurability. Prospects for detecting non Keplerian behaviors and constraining the masses of P2 and P1 with existing and future observations. (Lee & Peale 2006)
Keplerian Orbital Elements Semimajor axis a Eccentricity e Longitude of Periapse ϖ Mean Motion n = 2π/P Mean Longitude λ = n (t T peri ) + ϖ = ϖ at periapse = ϖ + 180 o at apoapse
Resonance Variables At the 2:1 commensurability n 1 2 n 2 λ 1 2 λ 2 const But since the periapses precess due to interactions, there are two dynamically significant combinations of frequencies for coplanar orbits: n 1 2 n 2 + dϖ 1,2 /dt 0 Eccentricity type mean motion resonance variables θ 1 = λ 1 2 λ 2 + ϖ 1 const, θ 2 = λ 1 2 λ 2 + ϖ 2 const
θ i is in resonance if it librates about a fixed value.
GJ 876 (Marcy et al. 2001) Two planets with P 30 and 60 days. Because of the short orbital periods and large planetary masses relative to the stellar mass, a dynamical fit to radial velocity data that accounts for the gravitational interaction between the planets is essential (Laughlin & Chambers 2001; Rivera & Lissauer 2001; Laughlin et al. 2005; Rivera et al. 2005; Lee & Peale 2002).
Retrograde apsidal precession of orbits induced by the 2:1 resonances has now been observed for more than one full period (3200 days or only 50 outer planet orbits).
Both θ 1 = λ 1 2λ 2 + ϖ 1 and θ 2 = λ 1 2λ 2 + ϖ 2 librate about 0 with small amplitudes.
The librations of θ 1 = λ 1 2 λ 2 + ϖ 1 and θ 2 = λ 1 2 λ 2 + ϖ 2 about 0 ο mean that Periapses are nearly aligned Conjunctions occur when both planets are near periapse.
Two planets with P 220 and 440 days (Mayor et al. 2004). The HD 82943 System Longer orbital periods and smaller relative planetary masses (compared to GJ 876) mean that a double Keplerian fit is likely adequate.
All 3 body integrations of the best fit double Keplerian model of Mayor et al. for different starting epochs and coplanar sin i are unstable. Mayor et al. noted that there are fits with very different argument of periapse of the outer orbit ω 2 (= ϖ 2 ) that have nearly the same RMS as the best fit, but it was unknown whether such fits would be stable or in 2:1 resonance.
Double Keplerian Fits for HD 82943 Radial velocity data set Keck observations by the California & Carnegie group + CORALIE data from Figure 6 of Mayor et al. (2004) 165 points spanning 6.1 years. Search for best fit double Keplerian model as a function of the poorly constrained e 2 and ω 2.
Parameters of best fit double Keplerian models for HD 82943
Dynamical Analysis of Double Keplerian Fits 3 body integrations of the best fit models up to 5 104 yr Use dynamical stability to narrow the range of reasonable fits Examine dynamical properties of the stable fits.
sin i = 1: Stability requires at least θ 1 to be librating about 0
Fit I: Unstable fit at minimum of χ ν 2
Fit II: Stable fit with the smallest libration amplitudes of both θ 1 and θ 2 about 0
Fit III: Stable fit with large amplitude librations of both θ 1 and θ 2
Fit IV: Stable fit with only θ 1 librating
sin i = 0.5: Stability requires both θ 1 and θ 2 to be librating about 0
Fit II: Stable fit with the smallest libration amplitudes of both θ 1 and θ 2
Origin of the Resonances Mean motion resonances can be easily established during planet formation by convergent migration of planets due to interactions with the protoplanetary disk. BUT continued migration while trapped in resonances can lead to rapid growth of eccentricities, unless there is significant ecc. damping from planet disk interactions.
GJ 876 The observed resonance geometry (with both θ 1 and θ 2 about 0 ) can be easily established by convergent migration. But the small observational upper limits of the eccentricities are a puzzle. (Lee & Peale 2002; Papaloizou 2003; Kley, Lee, Murray, & Peale 2005)
Lee & Peale (2002) updated
Kley, Lee, Murray, & Peale (2005) K = (e 1 de/dt)/(a 1 da/dt)
Hydrodynamical Simulations m 1 /m 0 = 0.00175 and m 2 /m 0 = 0.0059 without disk dispersal with disk dispersal Kley, Lee, Murray, & Peale (2005)
Hydrodynamical simulations of the GJ 876 system do not show significant eccentricity damping from planet disk interaction and the eccentricities quickly exceed the observational upper limits, unless the disk is dispersed shortly after resonance capture. HD 82943 Fit II has e 1 0.42 and e 2 0.14.
Without ecc. damping, observed ecc. not exceeded until orbital radii shrink by 20% (compared to 10% for GJ 876). Observed ecc. would persist during migration if ratio of ecc. damping rate to migration rate K 8 (compared to K > 40 for GJ 876).
Ecc. damping is not as large as K 8, but it is sufficient to delay ecc. growth so that the observed ecc. are not exceeded until orbits shrink by 35% after resonance capture.
Conclusions: HD 82943 The two planets about HD 82943 are almost certainly in 2:1 mean motion resonance. Dynamical stability requires at least θ 1 to be librating about 0 if sin i 1.0. both θ 1 and θ 2 to be librating about 0 if sin i 0.5. The HD 82943 system is consistent with small amplitude librations (as small as 6 for θ 1 and 10 for θ 2 ) of both θ 1 and θ 2 about 0.
The 2:1 resonances in HD 82943 can be established during planet formation by convergent migration due to planet disk interactions without requiring excessive eccentricity damping or the near coincidence of time of resonance capture and disk dispersal time.
The Pluto Satellite System Orbits of S/2005 P2 and S/2005 P1 nearly circular and nearly coplanar with that of Pluto Charon. Orbital periods Charon: P 6.4 days P2: P 25 days P1: P 38 days Mass ratio of Charon Pluto m c /m p = 0.12.
Analytic Theory for Orbits of P2 and P1 P2 and P1 test particles Orbit of Charon relative to Pluto: Keplerian and circular semimajor axis a pc mean motion n pc = [G(m p +m c )/a pc3 ] 1/2 r c = (a pc m p /(m p +m c ), φ c ) r p = (a pc m c /(m p +m c ), φ c +π) φ c = n pc t + pc
The gravitational potential at r = (R, φ) where The axisymmetric k = 0 component identical to that due to two rings: one of mass m p and radius a pc m c /(m p +m c ), and another of mass m c and radius a pc m p /(m p +m c ).
Deviations from point mass potential of mass (m p +m c ) at the origin are of order (m c /m p )(a pc /R) 2 and higher (~0.02 for P2 and 0.01 for P1). Represent orbits as small deviations from circular motion of a guiding center at R 0 : At zeroth order where the mean motion n 0 is given by
At the first order where and epicyclic frequency κ 0 is given by
The solution is Motion described by Circular motion of guiding center at R 0 at mean motion n 0 Epicyclic motion with eccentricity e at frequency κ 0 Forced oscillations of fractional amplitude C k at frequencies k (n 0 n pc ).
With n 0 > n K > κ 0, where n K = [G(m p +m c )/R 03 ] 1/2 is the Keplerian mean motion at R 0 Deviation from Kepler s third law Azimuthal period P 0 = 2π/n 0 is shorter than the Keplerian orbital period P K = 2π/n K. Prograde precession of periapse with period 2π/ n 0 κ 0 1700 days for P2 and 5300 days for P1.
Deviation from Kepler s third law
Numerical Orbit Integrations Size estimates from brightness: If albedo is high and Charon like (0.35), diameter of P1 ~ 60 km. If albedo is low and comet like (0.04), diameter of P1 ~ 170 km. P2 is ~ 20% smaller.
Vairation in orbital radius of P2 is dominated by forced oscillations. No evidence that P2 has any significant epicyclic ecc. Orbit of P1 has significant epicyclic ecc. Small periodic variations in e 2 (400 days) and e 1 (450 days).
Long term prograde precession of periapse in agreement with analytic result. Short term periodic variations on the same timescales as ecc. Timescales are the circulation periods of θ 2 = 2 λ 2 3 λ 1 + ϖ 2 and θ 1 = 2 λ 2 3 λ 1 + ϖ 1.
If masses do not exceed about 1/2 the low albedo ones, amplitudes of short term variations increase for both P2 and P1 with increasing masses. Long term prograde precessions of both ϖ 2 and ϖ 1.
For low albedo masses, smaller variation in e 1 but retrograde precession of ϖ 1 with a period of only 500 days.
θ 2 = 2 λ 2 3 λ 1 + ϖ 2 circulates, but θ 1 = 2 λ 2 3 λ 1 + ϖ 1 is in resonance and librates about 180.
Since existing data can be reasonably fitted by unperturbed Keplerian orbits, they may already be inconsistent with masses of P2 and P1 near the upper end of the ranges allowed by albedo uncertainties. Also produce variation in Charon s ecc. significantly larger than the best fit 0.0 ± 7.0 10 5.
Conclusions: Pluto Satellites Orbits of P2 and P1 significantly non Keplerian due to large mass ratio of Charon Pluto proximity of P2 and P1 to 3:2 commensurability Deviation from Kepler s third law already detected. Effects of proximity to 3:2 commensurability increase with increasing masses for masses within the ranges allowed by the albedo uncertainties. Observations that sample the orbits of P2 and P1 well on the 400 500 day timescales should provide strong constraints on masses of P2 and P1 in the near future.