Secular dynamics of extrasolar-systems

Transcription

1 Secular dynamics of extrasolar-systems Marco Sansottera [1] [1] Università degli Studi di Milano [2] Namur Center for Complex Systems (naxys) Based on a research project in collaboration with A.-S. Libert [2] and A. Giorgilli [1] & L. Grassi [1] Complex Planetary Systems (IAU Symposium) Namur,

2 Extrasolar systems vs. Solar System Main differences Shape of the orbits: true ellipses. Giant planets close to the star. Analytic study of the long-term evolution Classical approach: circular approximation as a reference. High eccentricities: high order expansions. Giant planets close to the star: relativistic corrections.

3 Main points High eccentricities of the orbits. Generalize the Lagrange-Laplace secular theory including high-order terms in the eccentricities. (Libert & Henrard, Beaugé et al.,...) Mean-motion resonance effects. Replace the Lagrange-Laplace circular approximation with an invariant torus up to order two in the masses. (Laskar & Robutel, Giorgilli, Locatelli & S.,...) Relativistic effects. Include the main secular terms due to the relativistic corrections extending the Lagrange-Laplace theory. (Laskar, Adams & Laughlin,...)

4 Hamiltonian of the three-body problem F (r, r) = T (0) ( r) + U (0) (r) + T (1) ( r) + U (1) (r), where r are the heliocentric coordinates and r the conjugated momenta. T (0) ( r) = 1 2 U (0) (r) = G 2 ( 1 r j ), m 0 m j j=1 2 j=1 T (1) ( r) = r 1 r 2 m 0, m 0 m j r j, U (1) (r) = G m 1m 2 r 1 r 2.

5 Poincaré variables in the plane Λ j = m 0m j G(m 0 + m j )a j, m 0 + m j λ j = M j + ω j, } {{ } Fast variables, O(1 year). ξ j = 2Λ j 1 1 ej 2 cos(ω j ), η j = 2Λ j 1 1 ej 2 sin(ω j ), }{{} Secular variables, O(1000 years). where a j, e j, M j and ω j are the semi-major axis, the eccentricity, the mean anomaly and periapsis argument of the j-th planet, respectively.

6 Expansion of the Hamiltonian Taylor expansion around a fixed value Λ (translated fast actions), The computed Hamiltonian reads, where h (Kep) j 1,0 H = n L + 2 j 1=2 L = Λ Λ. h (Kep) j 1,0 (L) + µ 1 12 j 1=0 j 2=0 is a hom. pol. of degree j 1 in L and h j1,j 2 is a h j1,j 2 (L, λ, ξ, η), hom. pol. of degree j 1 in L, hom. pol. of degree j 2 in ξ, η, with coeff. that are trig. pol. in λ. Lowest possible limits so as to include the fundamental features.

7 First order averaging Averaged Hamiltonian (2 D.O.F.), H(L, λ, ξ, η) = H(L, λ, ξ, η) λ, namely we remove all terms that depend on the fast angles λ. This is the so called first order averaging. H (sec) = H 0 (ξ, η) + H 2 (ξ, η) + H 4 (ξ, η) +..., where H 2j is a hom. pol. of degree (2j + 2) in (ξ, η), for all j N.

8 Analytic integration Action-angle variables, ξ j = 2Φ j cos ϕ j, η j = 2Φ j sin ϕ j. Birkhoff normal form up to a finite order, r, H (r) = Z 0 (Φ) + Z 2 (Φ) Z r (Φ) + R r (Φ, ϕ). If the remainder is negligible, we get Φ j (0) = 0, ϕ j (0) = H(r) Φ j. (Φ(0),ϕ(0))

9 Analytic integration ( ) Secular + NF (r) ( η(0), ξ(0) Φ (r) (0), ϕ (r) (0) ) Numerical integration Φ (r) (t)=φ (r) (0) ϕ (r) (t)= ϕ (r) (0)t+ϕ (r) (0) ( η(t), ξ(t) ) (NF (r) ) 1 ( Φ (r) (t), ϕ (r) (t) )

10 First order approximation (HD ) HD : the system is secular ECCENTRICITY EVOLUTION e1 (numer.) e1 (ord.1) e2 (numer.) e2 (ord.1) e+06 2e+06 3e+06 4e+06 5e+06 6e+06

11 First order approximation (HD 11964) HD 11964: the system is secular ECCENTRICITY EVOLUTION e1 (numer.) e1 (ord.1) e2 (numer.) e2 (ord.1) e+06 2e+06 3e+06 4e+06 5e+06

12 But... near mean-motion resonance (HD ) HD : the system is close to the 4:1 mean-motion resonance. First order averaged Hamiltonian failed ECCENTRICITY EVOLUTION e1 (numer.) e1 (ord.1) e2 (numer.) e2 (ord.1)

13 Second order averaging Coming back to the original Hamiltonian, H = n L + j 1 2 h j1,0(l) + µ j 1 0 Taking the point L(0) = 0, we have h j1,j 2 (L, λ, ξ, η). j 2 0 L j = µ h 0,j2 (λ, ξ, η). λ j j 2 0 We perform a canonical transformation via Lie Series to kill dependence on fast angles λ in h 0,0 (λ, ξ, η), h 0,1 (λ, ξ, η), h 0,2 (λ, ξ, η),....

14 Kolmogorov-like normalization step We determine the generating function, χ, by solving the equation n KS χ λ + h 0,j2 j 2=0 λ:k F = 0, K S and K F are chosen so as to include the main effects due to the possible proximity to a mean-motion resonance. H (O2) = exp L µχh = j=0 1 j! Lj µχh. This is our Hamiltonian at order two in the masses, namely a torus that is invariant up to order two in the masses.

15 First order approximation (HD ) HD : the system is close to the 9:1 MMR (weak MMR) ECCENTRICITY EVOLUTION e1 (numer.) e1 (ord.1) e2 (numer.) e2 (ord.2)

16 First order approximation (HD ) HD : the system is close to the 9:1 MMR (weak MMR) ECCENTRICITY EVOLUTION e1 (numer.) e1 (ord.1) e2 (numer.) e2 (ord.2)

17 First order approximation (HD ) HD : the system is close to the 4:1 MMR (strong MMR) ECCENTRICITY EVOLUTION e1 (numer.) e1 (ord.1) e2 (numer.) e2 (ord.1)

18 Second order approximation (HD ) HD : the system is close to the 4:1 MMR (strong MMR) ECCENTRICITY EVOLUTION e1 (numer.) e1 (ord.1) e2 (numer.) e2 (ord.1)

19 Proximity to a mean-motion resonance Rate the proximity by looking at the canonical change of coordinates induced by the approximation at order two in the masses. ξ j = ξ j µ ( χ = ξ j 1 µ ) χ, η j ξ j η j η j = η j µ ( χ = η j 1 µ ) χ. ξ j η j ξ j We focus on the norms of the terms δξ j = µ ξ j χ η j and δη j = µ η j χ ξ j, and introduce δ = max(δξ j, δη j ).

20 Mean-motion resonance effects For systems not too close to a mean-motion resonance the secular evolution is well approximated via Birkhoff normal form. We introduce an heuristic and quite rough criterion that we think it is useful to discriminate between the different behaviors: (i) δ : secular; (ii) < δ : near mean-motion resonance; (iii) δ > : in mean-motion resonance. Libert & S., CeMDA (2013).

21 Relativistic effects in the Solar System Le Verrier: the orbital precession of Mercury is slightly faster than what it should be Einstein: general relativity 1938 Einstein-Infeld-Hoffman: relativistic Celestial Mechanics. Relativistic effects are negligible in the study of the secular dynamics of the giant planets: Sun-Jupiter-Saturn system.

22 ...and what about extrasolar systems? There are many giant planets surprisingly close to the star. Relativistic effects are much more intense than in the Solar System. Relativistic corrections can play a significant role in the secular dynamics of extrasolar systems. Laskar, A.&A. (1986) Adams & Laughlin, Astrophys.J. (2006)...

23 Relativistic Hamiltonian H = H 0 + εh c 2 H 2, (1)

24 Relativistic Hamiltonian where H 2 = with n j = 2 i=1 3G 2m 0 2 i=1 H = H 0 + εh c 2 H 2, (1) ( r 4 i 1 8 m ) mi 3 1 ( 4m0 3 r 2 1 r ( r1 r2) i=1 m ( ) i r 2 1 r i + 2 r1 r2 + r 2 2 7G 2 r i r i ( r 1 + r 2) + 2 i=1 3G r 2 i 2m i 2 7G 2 2 r2 2 r 1 r 2 r1 i=1 j=1 r 2 i ( r 1 r 2) ) + i=1 ( ) m0 r i + m 3 i + r 1 r 2 G 2 r i ( r i n i )( r j n i )+ G + 2 r 2 r 1 ( r1 n12)( r2 n12) + G2 m 0m 1m 2 2 r 1 r 2 + G 2 m 0m 1m 2 r i=1 i r 1 r 2 + ( 2 ) + G2 m 0m i 2 r i=1 i 2 (m0 + m m 1m 2 i ) + (m1 + m2), r 2 r 1 2 r j r j and n12 = r 2 r 1 r 2 r 1.

25 Simplified relativistic Hamiltonian Just consider the interactions between the star and the planets. where with H 2 = 2 [ i=1 H = H 0 + εh c 2 H 2, (2) γ 1,i µ 3 Pi 4 γ 2,i Pi 2 i µ i r i γ 3,i (n i P i ) γ 4,i µ i µ i r i r i 2 µ i = m 0m i m 0 m i, β i = G(m 0 + m i ), υ i = m 0 + m i (m 0 + m i ) 2, γ 1,i = 1 3υ i 8, γ 2,i = β i(3 + υ i ), γ 3,i = β iυ i 2 2, γ 4,i = β2 i 2, ], P i = µ i ṙ i (approximation of order O(c 2 )).

26 Secular evolution The averaged Hamiltonian H = H 0 + ε H c 2 H 2 has the same structure as the Newtonian one: action-angle variables via Birkhoff normal form; equations of motion can be easily solved; impact of the relativistic corrections by looking at the quadratic approximations of the Newtonian and relativistic Hamiltonians.

27 Newtonian vs. relativistic evolution (HD ) HD (1): relativistic effects are negligible ECCENTRICITY EVOLUTION Newtonian (numer.) Newtonian (analy) Relativistic (numer.) Relativistic (numer.)

28 Newtonian vs. relativistic evolution (HD ) HD (2): relativistic effects are negligible ECCENTRICITY EVOLUTION Newtonian (numer.) Newtonian (analy) Relativistic (numer.) Relativistic (numer.)

29 Newtonian vs. relativistic evolution (HD 11964) HD (1): relativistic effects weaken the eccentricity variations ECCENTRICITY EVOLUTION Newtonian (numer.) Newtonian (analy) Relativistic (numer.) Relativistic (numer.)

30 Newtonian vs. relativistic evolution (HD 11964) HD (1): relativistic effects weaken the eccentricity variations ECCENTRICITY EVOLUTION Newtonian (numer.) Newtonian (analy) Relativistic (numer.) Relativistic (numer.)

31 When we need to include relativistic corrections? We look at the quadratic parts of the secular Hamiltonians, namely H (New) 0 (η, ξ) = η Aη + ξ Aξ, H (Rel) 0 (η, ξ) = η Bη + ξ Bξ, where A and B are real symmetric 2 2 with B = A 3 G 3/2 2 c 2 0 (m 0+m 1) 3/2 (a 1 )5/2 0 (m 0+m 2) 3/2 (a 2 )5/2. Criteria: relative difference between A and B; analytic computation of the secular frequencies;

32 Impact of relativistic corrections HD coefficient of ξ 2 1, η2 1 : coefficient of ξ 2 2, η2 2 : HD coefficient of ξ 2 1, η2 1 : coefficient of ξ 2 2, η2 2 :

33 Future work... Resonant normal forms. Kolmogorov normal form. Add the mutual inclination. Effective stability time. Improve the determination of the orbital parameters.

34