Stability of Extra-solar Planetary Systems C. Beaugé (UNC) S. Ferraz-Mello (USP) T. A. Michtchenko (USP) USP-UNC team on Exoplanets:

(Sequence Number) III II I 28 planets in 12 multiplanet systems (*) (*)  Eri not included since too uncertain.

3 (4) classes Ia – Planets in mean-motion resonances Ib – Low-eccentricity Non-resonant Planet Pairs II – Non-resonant Planets with a Significant Secular Dynamcis III – Weakly interacting Planet Pairs

CLASS Ia – Planet Pairs in Mean Motion Resonance

GJ 876

Solar System without Saturn 50 Myr Collision Chaos Order Grid: 33x251 Ref: Michtchenko (unpub.) 2/1 7/3 5/2 8/3.

M0=1.15 Msun m1=1.7 Mjup/sin i m2=1.8 Mjup/sin i

HD i=90 deg Ref: Ferraz-Mello et al. (2005) Axes: x = e2.cos  y = e2.sin  RED = collision in t<260,000 yrs GRAY=very chaotic WHITE=mild or almost no chaos

Ref: Ferraz-Mello et al. (2005)

HD i=30 deg Axes: x = e2.cos  y = e2.sin  RED = collision in t<260,000 yrs GRAY=very chaotic WHITE=mild or almost no chaos Ref: Ferraz-Mello et al. (2005)

(O-C) of solution B vs. (O-C) of the Mayor et al. solution (squares) Ref: Ferraz-Mello et al. 2005

Solar System without Saturn 50 Myr Collision Chaos Order Grid: 33x251 Ref: Michtchenko (unpub.) 2/1 7/3 5/2 8/3.

CLASS Ib – Low Eccentricity Near-Resonant Planet Pairs

Neighborhood of Uranus  Ref: Michtchenko & Ferraz-Mello, 2001

CLASS Ib – Low-Eccentricity Near-Resonant Planet Pairs

Grid: 21x101 Neighborhood of the 3 rd planet of pulsar B

Grid 21x51 CHAOS ORDER Neighborhood of the 3 rd planet of pulsar B X

Class II – NR Pairs with significant Secular Dynamics 

Dynamical map of the neighborhood of planet  And D cf. Robutel & Laskar (2000) unpublished White line: Colision line with planet C Black spot: actual position of  And D chaotic regular 1/5 2/11

Consequences Poisson-Laplace invariance of semi-major axes in the Solar System (at order m ) In Extra-solar systems (with large eccentricities) =0 as far as a close approximation of the two planets does not occur. i 2

Consequences In a system formed by two coplanar planets the eccentricities vary in anti-phase because of N.B. cos i = 0

Eccentricity variation of upsilon Andromeda planets B(green), C(red), D(blue) in a 100,000-yr simulations.

 Class III – Weakly Interacting Planet Pairs ????

CLASS Ia – Planet Pairs in Mean Motion Resonance

Ref: Papaloizou, Cel. Mech. Dyn. Astron 87 (2003). Model: 2 planets & 1 gap a=0.8 and a=1.4 Fig. scale 2x2

Evolution of a 2-planet system under [2/1 resonance] non-conservative forces (mass ratio 0.54) Ref. Ferraz-Mello et al. Cel. Mech. Dyn. Astron. (2003)     [arbitrary units]

SYMMETRIC APSIDAL COROTATIONS  (0,0)

Stable stationary solutions with aligned periastra At conjunction both planets are at the perihelion. N.B. m < m (“internal case”) m /m = const. Ex: HD82943 m2/m1=1.06  [2/1 resonance]

Stable asymmetric stationary solutions. 2/1 resonance m < m 2 1 (external cases) Ref: Ferraz-Mello et al. Cel. Mech. Dyn. Astron. (2003)

ASYMMETRIC APSIDAL COROTATIONS

Ref: Ferraz-Mello et al. (2003) Stable Stationary Solutions. 3/1 resonance. Asymmetric Anti-alligned External Internal PROBLEM: 55 Cnc b,c..... e1=0.02 e2=0.44 m2/m1=0.28 (?????)

Astro-ph/ / / / References downloadable from:

THE PLANETS OF THE STAR GJ 876 (Gliese 876) MASSES [unit: Jupiter mass] (B) 1.89 /sin i (C) /sin i ~

THE PLANETS OF UPSILON ANDROMEDA MASSES [unit: Jupiter mass] (B) 0.69 /sin i (C) 1.89 (D) 3.75

mutual inclination HD Mayor et al. (2004) planar elements and masses and initial mutual inclination 10. o o planar case

Radial Velocity of a star with 2 planets – HD Roughly 50 percent of the stars for which one planet was detected show evidence of additional companions. Ref. Marcy et al Astroph.J. 556 (2001) Mayor et al. A&A, 2004

 And F8 V L=3L_sun d=13.47 pc M=1.3 M_sun agee 2-3 Gyrs Absence of astrometric variation means sin i > 0,4 (i > 25deg) Absence of transits means sin i < (i < 83 deg)

Corotations as functions of the eccentricities

Intersecting orbits of an asymmetric stationary solution Eccentricities Inner 0.17 Outer 0.38 Mass ratio 0.44 (ext.)

Asymmetric solutions bifurcating from solutions with aligned periastra Asymmetric solutions bifurcating from solutions with anti-aligned periastra

Long-term evolution of the eccentricity (same solution)

Evolution of the eccentricities in the same solution