Dynamics of Extra-solar Planetary Systems

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

2. Period ratio of consecutive planets in a system III II I

3. 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

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

5. GJ 876 (0,0) apsidal corotation resonance

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

8. ASYMMETRIC APSIDAL COROTATIONS

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

11. Polar representation of the motion of planets HD 82943 c,b Polar radius: 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.

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

14. The planets of 47 UMa M = 2.9 M M = 1.1 M 1 Jup 2 Jup

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

16. 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.)

17. 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.

18. 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)

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

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

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

22. Class II – Non-resonant planet pairs with a significant secular dynamics (cont.)

23. Mu Arae = HD 160691 (b,c) ● ● Gozdziewski et al. (ApJ. 2005)

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 Robutel & Laskar (2000) unpublished ● 1/5 2/11 Black spot: actual position of And D White line: Colision line with planet C chaotic regular

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

27. 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

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

29. From Callegari et al, with pulsar planets data Polar Coordinates: Radius vector: e_1 Polar angle: Dv (See Pauwels, 1983) Dynamical Spectrum near the higher frequency

30. Cf. Michtchenko & Malhotra, 2004 unstable Crossing orbits u And

32. Angular Momentum in the case of mutually inclined orbits w.r.t invariable plane

33. Inclination (deg) Ex: HD 82943 Ferraz-Mello et al. ApJ, 2005 26 6

34. TAM &c 2005

35. Red/Blue = periodic solutions (Oscillation of Dv/ Libration of w resp.) Ref: Michtchenko et al. 2005 (submitted) 1

36. Red/Blue = periodic solutions (Oscillation of Dv/ Libration of w resp.) Michtchenko et al. 2005 1

37. With an increased AMD Note the large domain of the Lidov-Kozai resonance

38. References downloadable from: http://www.astro.iag.usp.br/~dinamica/usp-unc.htm @ArXiv: Astro-ph/0509xxx /0505169 /0404166 /0402335 /0301252 /0210577 Planet data used in this lecture (+ updates): See: http:// www.astro.iag.usp.br/~dinamica/exosys.htm

39. Data from: Ferraz-Mello et al (2005) [HD 82943], Laughlin et al (2005) [GJ 876], Vogt et al. (2005) [HD12831, HD 108871 and HD 37124], McArthur et al.(2004) [55 Cnc ], Correia et al. (2005) [HD 202206], Gozdziewski et al. (2005) [mu Ara = HD 160691], Santos et al. (2004) [HD 160691e], Mayor et al. (2004) [HD 169830], Fischer et al (2002) [HD 12661], Ford et al. (2005) [upsilon Andromedae], Konacki & Wolszczan (2003) [PSR 1257+12].

40. 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]

41. 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 Least squares solution (2004).

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

43. (O-C) of solution B of the previous slide (triangles) vs. (O-C) of the least-squares solution (squares). The two solutions fit the observation equally. Ref: Ferraz-Mello et al. 2005 Mayor et al. 2004

44. 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)