Stability of Extra-solar Planetary Systems

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

3. Period ratio of consecutive planets in a system

4. 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

5. Class Ia – Planet pairs in Mean-Motion Resonance

6. GJ 876

7. Solar System with Saturn initialized on a grid of different initial conditions 2/17/3 5/2 8/3 . 50 Myr Collision Chaos Order Grid: 33x251 Ref: Michtchenko (unpub.)

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

9. HD 82943 i=90 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) A, B our solutions M solution of Mayor et al.

10. The orbits of solution M are bound to a catastrophic event in less than 100,000 years. Ref: Ferraz-Mello et al. (2005)

11. HD 82943 same as before with i=30 deg (masses multiplied by 2) 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)

12. (O-C) of solution B of the previous slide (triangles) vs. (O-C) of the Mayor et al. solution (squares). The two solutions fit the observation equally. Ref: Ferraz-Mello et al. 2005

13. Class Ib – Low-eccentricity Near-resonant pairs Outer Solar System

14. Neighborhood of Uranus  Ref: Michtchenko & Ferraz-Mello, 2001 (Solar system with Uranus initializaed on a grid of different initial conditions). The blue spot is the actual position of Uranus.

15. Class Ib – Low-eccentricity Near-resonant pairs Resonant Pulsar Planets

16. Neighborhood of the 3rd planet of pulsar B1257 +12 collision Grid: 21x101 Pulsar system initialized with planet C on a grid of different initial conditions. The actual position of planet C is shown by a cross. (N.B. I=90 degrees)

17. Neighborhood of the 3rd planet of pulsar B1257 +12 CHAOS ORDER X Grid 21x51

18. Class II – Non-resonant planet pairs with a significant secular dynamics

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 Robutel & Laskar (2000) unpublished 1/5 2/11 Black spot: actual position of And D White line: Colision line with planet C chaotic regular

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

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

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

24. Class III – Hierarchical Planet pairs

25. Data from: Ferraz-Mello et al (2005) [HD 82943], Laughlin et al (2005) [GJ 876], California & Carnegie web page [HD12831] California & Carnegie web page [HD11964] McArthur et al.(2004) [55 Cnc b,c,d], Correia et al. (2005) [HD 202206] McCarthy et al. (2004) [mu Ara = HD 160691], Mayor et al. (2004) [HD 169830] , Udry et al (2002) [HD 168443], Naef et al. (2004) [HD 74156], Santos et al. (2004) [HD 160691e], Konacki & Wolszczan (2003) [PSR 1257+12] Fischer et al. (2003) [all others]

26. Class Ia – Planet pairs in Mean-Motion Resonance

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

28. 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)  2 1   2 1 2 1 [arbitrary units]

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

30. Stable stationary solutions with aligned periastra At conjunction both planets are at the perihelion. + [2/1 resonance]  m /m = const. 2 1 N.B. m < m (“internal case”) 1 2 * Point corresponding to the stationary solution around which move the planets of HD 82943. + Actual position of the orbits.

31. Polar representation of the actual motion of planets HD 82943 c,b Polar readius: Eccentricity Polar angle: distance inter periastra () [2/1 resonance] m2/m1=1.06 gray: inner planet black: outer planet + + The stationary solution is somewhere on the x-axis between the maxima and minima of the eccentricities.

32. Stable asymmetric stationary solutions. 2/1 resonance m < m 2 1 0.11 0.22 0.33 0.44 0.56 0.67 (external cases) 0.78 0.89 1.00 Ref: Ferraz-Mello et al. Cel. Mech. Dyn. Astron. (2003) red: mass ratios

33. ASYMMETRIC APSIDAL COROTATIONS

34. External Internal 1.0 Stable Stationary Solutions. 3/1 resonance. 0.11 0.22 0.67 0.78 0.89 0.33 0.44 0.56 1.1 1.2 Asymmetric Anti-alligned Ref: Ferraz-Mello et al. (2003) PROBLEM: 55 Cnc b,c ..... e1=0.02 e2=0.44 m2/m1=0.28(?????)

35. References downloadable from: http://www.astro.iag.usp.br/~dinamica/usp-unc.htm @ArXiv: Astro-ph/0404166 /0402335 /0301252 /0210577