On the Secular Evolution of Extrasolar Planetary Systems
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1 On the Secular Evolution of Extrasolar Planetary Systems Marco Sansottera [a] [a] Namur Center for Complex Systems (naxys) Based on a research work in collaboration with Anne-Sophie Libert [a] JISD 2012 Barcelona,
2 Extrasolar Systems 1995: first detection of an extrasolar planet, a planet orbiting around a star other than the Sun. Nowadays more than 700 have been detected, and more than 2000 candidates.
3 Extrasolar Systems vs. Solar System The main difference between the extrasolar systems and the Solar System regards the shape of the orbits. In the extrasolar systems, the majority of the orbits describe true ellipses (high eccentricities) and no more almost circular like in the Solar System. The classical approach uses the circular approximation as a reference. Dealing with systems with high eccentricities we need to compute the expansion at high order to study the long-term evolution of the extrasolar planetary system.
4 Aim of the work Questions: Can we predict the long-term evolution of the extrasolar systems? How to identify systems that are near to a mean motion resonance? Are the discovered extrasolar systems stable? What is the meaning of the word stable?
5 Aim of the work Questions: Can we predict the long-term evolution of the extrasolar systems? How to identify systems that are near to a mean motion resonance? Are the discovered extrasolar systems stable? What is the meaning of the word stable? Effective stability: a) Will the orbits of the planets remain essentially the same for eternity, or at least for a time comparable to the estimated age of the Universe? b) In the distant future, could dramatic changes happen? For example due to collisions, falls on the star or ejection of a planet.
6 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.
7 Orbital Parameters a (semi-major axis), e (eccentricity), i (inclination), ω (periapsis argument), M (mean anomaly), Ω (longitude of the ascending node).
8 Poincaré variables in the plane Λ j = m 0m j m 0 +m j G(m 0 +m j )a j, λ j = M j +ω j, ξ j = 2Λ j 1 1 ej 2 cos(ω j ), η j = 2Λ j 1 1 ej 2 sin(ω j ), 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.
9 Poincaré variables in the plane Λ j = m 0m j G(m 0 +m j )a j, λ j = M j +ω j, m 0 +m 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 ), 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.
10 Poincaré variables in the plane Λ j = m 0m j G(m 0 +m j )a j, λ j = M j +ω j, m 0 +m 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(10000 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.
11 Expansion of the Hamiltonian In order to compute the Taylor expansion of the Hamiltonian around the fixed value Λ, we have to introduce the translated fast action L = Λ Λ. This is the Hamiltonian, H (T ) = n L+ j 1=2 h (Kep) j 1,0 (L)+µ j 1=0j 2=0 h (T ) j 1,j 2 (L,λ,ξ,η).
12 Expansion of the Hamiltonian In order to compute the Taylor expansion of the Hamiltonian around the fixed value Λ, we have to introduce the translated fast action L = Λ Λ. This is the computed Hamiltonian, H (T ) = n L+ 2 j 1=2 h (Kep) j 1,0 (L)+µ 1 SEC h (T ) j 1,j 2 (L,λ,ξ,η), j 1=0j 2=0 where we also truncate all the coefficients with harmonics of degree greater than 2K. We will choose the lowest possible limits in order to include the fundamental features of the system.
13 First Order Averaging In order to get rid of the fast motion, we consider the averaged Hamiltonian H(L,λ,ξ,η) = H(L,λ,ξ,η) λ. This is the so called first order averaging. Actually it means that we remove from the Hamiltonian all the terms that depends on the fast angles λ.
14 Secular Dynamics Doing the averaging over the fast angles (as we are interested in the secular motions of the planets), the system pass from 4 to 2 degrees of freedom, 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), j N. ξ = η = 0 is an elliptic equilibrium point, so we can introduce action-angle variables via Birkhoff normal form. In this way we remove the dependency on the angles from the Hamiltonian and we can easily solve the equations of motion.
15 Sketch of the procedure (1/2) H = H 0 (ξ,η)+h 2 (ξ,η)+h 4 (ξ,η)+.... Introduce the action-angle variables ξ j = 2Φ j cosϕ j, η j = 2Φ j sinϕ j. Compute the Birkhoff normal form up to a finite order, r, H (r) = Z 0 (Φ)+Z 2 (Φ)+...+Z r (Φ)+R r (Φ,ϕ).
16 Sketch of the procedure (2/2) If the remainder R r is small enough, we can neglect it! The equations of the motion are Φ j (0) = 0, ϕ j (0) = H(r) Φ j. The solutions are Φ j (t) = Φ j (0), ϕ j (t) = ϕ j (0)t +ϕ j (0).
17 Analytical Integration ( ) Secular + NF ( ) η(0),ξ(0) Φ(0),ϕ(0) Numerical integration Φ(t)=Φ(0) ϕ(t)= ϕ(0)t+ϕ(0) ( η(t),ξ(t) ) (NF) 1 ( Φ(t),ϕ(t) )
18 First Order Approximation (HD ) HD : the system is secular Eccentricities analy_hd_134987_ord1 "./ecc_0.dat" "./ecc_1.dat" "./orbitals_1.dat" u 1:4 "./orbitals_2.dat" u 1: e+06 2e+06 3e+06 4e+06 5e+06 6e+06
19 But... Near Mean Motion Resonance (HD ) HD : the system is close to the 4 : 1 MMR. First order averaged Hamiltonian failed Eccentricities analy_hd_108874_ord1 "./ecc_0.dat" "./ecc_1.dat" "./orbitals_1.dat" u 1:4 "./orbitals_2.dat" u 1:
20 Mean Motion Resonance A mean motion resonance (MMR) appears when the revolution periods of the two planets are commensurable T 1 = p, with p,q N. T 2 q For example, if we consider the 2 : 1 resonance,
21 Second Order Averaging In order to get rid of the fast motion, instead of simply erasing the terms depending on fast angles λ, we perform a canonical transformation via Lie Series to kill the terms µh (T ) 0,0 λ:k (λ,ξ,η),..., µh (T ) 0,ECC λ:k (λ,ξ,η), where λ:k means the truncation of the harmonics of degree greater than K. The parameter ECC is chosen to include in the secular model the main effects due to the possible proximity to a mean motion resonance. This is the so called second order averaging.
22 The details of the transformation This procedure is essentially a Kolmogorov s like step of normalization, we have to solve the homological equation n χ λ + µh (T ) 0,j 2 λ:k = 0, and find the generating function χ, that is of order O(µ). H (O2) = expl χ H (T ) = j=0 this transformation, by construction, kill the term 1 j! Lj χh (T ), µh (T F) j 1,j 2 λ:k. The transformed Hamiltonian still has a term of the same type, but at least of order O(µ 2 ). This is the so called Hamiltonian at order two in the masses.
23 Analytical Integration ( ) Secular + NF ( ) η(0),ξ(0) Φ(0),ϕ(0) Numerical integration Φ(t)=Φ(0) ϕ(t)= ϕ(0)t+ϕ(0) ( η(t),ξ(t) ) (NF) 1 ( Φ(t),ϕ(t) )
24 First Order Approximation (HD 11506) HD 11506: the system is close to the 7 : 1 MMR (weak MMR) Eccentricities analy_hd_11506_ord1 "./ecc_0.dat" "./ecc_1.dat" "./orbitals_1.dat" u 1:4 "./orbitals_2.dat" u 1:
25 Second Order Approximation (HD 11506) HD 11506: the system is close to the 7 : 1 MMR (weak MMR) Eccentricities analy_hd_11506_ord2 "./ecc_0.dat" "./ecc_1.dat" "./orbitals_1.dat" u 1:4 "./orbitals_2.dat" u 1:
26 First Order Approximation (HD ) HD : the system is close to the 4 : 1 MMR (stronger MMR) Eccentricities analy_hd_108874_ord1 "./ecc_0.dat" "./ecc_1.dat" "./orbitals_1.dat" u 1:4 "./orbitals_2.dat" u 1:
27 Second Order Approximation (HD ) HD : the system is close to the 4 : 1 MMR (stronger MMR) Eccentricities analy_hd_108874_ord2 "./ecc_0.dat" "./ecc_1.dat" "./orbitals_1.dat" u 1:4 "./orbitals_2.dat" u 1:
28 Conclusions We start with this questions: Can we predict the long-term evolution of the extrasolar systems? How to identify systems that are near to a mean motion resonance?
29 Conclusions We start with this questions: Can we predict the long-term evolution of the extrasolar systems? How to identify systems that are near to a mean motion resonance? Answers: 1) If the system is not too close to a mean motion resonance, providing an approximation of the motions of the secular variables up to order two in the masses, the secular evolution is well approximated via Birkhoff normal form. 2) The secular Hamiltonian at order two in the masses is explicitly constructed via Lie Series, so the generating function contains the information about the proximity to a mean motion.
30 Thanks for your attention! Questions? Comments?
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