Restricted several bodies-problem asteroid + Sun + 8 planets - PowerPoint PPT Presentation

A numerical study of the Trojan dynamics Philippe Robutel ASD/IMCCE, Observatoire de Paris Collaborations with: J. Bodossian (Paris) F. Gabern (Barcelona) A. Jorba (Barcelona) J. Laskar (Paris) 1

Restricted “several” bodies-problem asteroid + Sun + 8 planets Different models: asteroid + Sun + 4 giant planets asteroid + Sun + Jup + Sat From 3+8*3 = 24 to 3+2*3 = 9 d. f. But even 9 D.F. imply numerical studies: num. integrations of the trajectories + Analysis of the Traj.: Lypunov exponents, Fourier analysis, Frequency Analysis... 2

Test-particles in the Solar system : restricted 10 bodies-problem e e M a E a M V e e a a J Robutel & Laskar (2001) e U a S e a N e a e a Unstability Stable (Q.P.) Unstable Diffusion index 3

Test-particles in the Solar system : restricted 10 bodies-problem e e Sun & planets are given M a E a M V e e a a J Robutel & Laskar (2001) e U a S e a N e a e a Unstability Stable (Q.P.) Unstable Diffusion index 3

Test-particles in the Solar system : restricted 10 bodies-problem e e Sun & planets are given M a E a M a ∈ [0 . 38 : 90] A.U. V e e e ∈ [0 : 0 . 9] small body: a a J I = 0 Robutel & Laskar (2001) e ( λ , ̟ , Ω ) Fixed U a S e a N e a e a Unstability Stable (Q.P.) Unstable Diffusion index 3

Test-particles in the Solar system : restricted 10 bodies-problem e e Sun & planets are given M a E a M a ∈ [0 . 38 : 90] A.U. V e e e ∈ [0 : 0 . 9] small body: a a J I = 0 Robutel & Laskar (2001) e ( λ , ̟ , Ω ) Fixed U a S e Overlap of MMR above coll. lines: a N Global chaos e a e a Unstability Stable (Q.P.) Unstable Diffusion index 3

Test-particles in the Solar system : restricted 10 bodies-problem e e Sun & planets are given M a E a M a ∈ [0 . 38 : 90] A.U. V e e e ∈ [0 : 0 . 9] small body: a a J I = 0 Robutel & Laskar (2001) e ( λ , ̟ , Ω ) Fixed U a S e Overlap of MMR above coll. lines: a N Global chaos e a e a Unstability Stable (Q.P.) Unstable Diffusion index 3

Projection of the observed inner solar system’s objects on the ecliptic Main asteroids belt (~400000) Terrestrial planet’s crossers (~5000) Comets (~200) Jupiter’s trojans (~2000) Trojans can orbit far from L4 or L5 (2 to 2.5 A.U.) L5 L4 J 4

λ = M + ̟ ̟ = ω + Ω 5

fundamental Frequencies (proper frequencies) 3 frequencies for the Trojan: ( n, g, s ) 3 months for Mercury 12 years for Jupiter Orbital motions: periods 164 years for Neptune 1000 years at 100 A.U. Secular motions: periods > 25000 years 6

fundamental Frequencies (proper frequencies) If Q.P. solution ( evolves on a KAM torus) 3 frequencies for the Trojan: ( n, g, s ) 3 months for Mercury 12 years for Jupiter Orbital motions: periods 164 years for Neptune 1000 years at 100 A.U. Secular motions: periods > 25000 years 6

fundamental Frequencies (proper frequencies) If Q.P. solution ( evolves on a KAM torus) s j = 0 ( n j , g j , s j ) 3n-1 planetary frequencies: one of the 3 frequencies for the Trojan: ( n, g, s ) 3 months for Mercury 12 years for Jupiter Orbital motions: periods 164 years for Neptune 1000 years at 100 A.U. Secular motions: periods > 25000 years 6

The Frequency Map J.Laskar (1999) H real analytic for ( I, θ ) ∈ B n × T n H ( I, θ ) = H 0 ( I ) + εH 1 ( I, θ ) 7

The Frequency Map J.Laskar (1999) H real analytic for ( I, θ ) ∈ B n × T n H ( I, θ ) = H 0 ( I ) + εH 1 ( I, θ ) If ε = 0 F : B n �− → Ω ⊂ R n → ν ( I ) = ∇ H 0 ( I ) I �− � � ∂ 2 H 0 ( I ) if det � = 0 ∂I 2 F is a diffeo. (loc.) 7

The Frequency Map J.Laskar (1999) H real analytic for ( I, θ ) ∈ B n × T n H ( I, θ ) = H 0 ( I ) + εH 1 ( I, θ ) 8

The Frequency Map J.Laskar (1999) H real analytic for ( I, θ ) ∈ B n × T n H ( I, θ ) = H 0 ( I ) + εH 1 ( I, θ ) If ε � = 0 but small enough There exists Ω ε set of diophanine frequiencies ↔ KAM tori 8

The Frequency Map J.Laskar (1999) H real analytic for ( I, θ ) ∈ B n × T n H ( I, θ ) = H 0 ( I ) + εH 1 ( I, θ ) If ε � = 0 but small enough There exists Ω ε set of diophanine frequiencies ↔ KAM tori P¨ oschel (1982),: There exists a diffeo. Ψ and a coord. syst. ( ϕ, ν ) such that Ψ is analytical/ ϕ and C ∞ / ν Ψ : T n × Ω �− → T n × B n The flow is linear on: T n × Ω ε : ( ϕ, ν ) �− → ( θ, I ) ν = 0 , ˙ ˙ ϕ = ν 8

The Frequency Map J.Laskar (1999) H real analytic for ( I, θ ) ∈ B n × T n H ( I, θ ) = H 0 ( I ) + εH 1 ( I, θ ) If ε � = 0 but small enough There exists Ω ε set of diophanine frequiencies ↔ KAM tori P¨ oschel (1982),: There exists a diffeo. Ψ and a coord. syst. ( ϕ, ν ) such that Ψ is analytical/ ϕ and C ∞ / ν Ψ : T n × Ω �− → T n × B n The flow is linear on: T n × Ω ε : ( ϕ, ν ) �− → ( θ, I ) ν = 0 , ˙ ˙ ϕ = ν For fix θ ∈ T n : θ = θ 0 F θ 0 : B n − → p 2 (Ψ − 1 ( θ 0 , I )) → Ω ; I − The frequency map F θ 0 is a smooth diffeo. from, the actions space to the frequencies space 8

Goal to obtain numerically a frequency map: defined on B n which coincide with F θ 0 , up to numerical accuracy, on the set of KAM tori numerical tool: Frequency analysis (J.Laskar, 1988,1990) 9

� ae i λ = α k e if k Quasi-periodic decomposition of z 5 = e 5 exp i ̟ 5 k | α j | combinations f j (AU) rad/yr 46.183882 .02005033 n .259757 .52968580 n 5 .058931 .21330868 n 6 .049411 .02004870 n − g 8 + g .040885 .02005196 n + g 8 − g .038045 .01808704 − n + n 8 + g .031431 .02201360 3 n − n 8 − g . . . . . . 10

Quasi-periodic decomposition of z 5 = e 5 exp i ̟ 5 N � z 5 ( t ) ≈ α j exp ( if j t ) j =1 f j = k 5 n 5 + k 6 n 6 p 5 g 5 + p 6 g 6 + q 6 s 6 11

Quasi-periodic decomposition of z 5 = e 5 exp i ̟ 5 | α j | f j (” /yr) k 5 k 6 p 5 p 6 4 . 41 × 10 − 2 +4 . 027603 × 10 0 +0 +0 +1 +0 1 . 59 × 10 − 2 +2 . 800657 × 10 1 +0 +0 +0 +1 6 . 44 × 10 − 4 − 2 . 126393 × 10 4 N − 1 +2 +0 +0 � z 5 ( t ) ≈ α j exp ( if j t ) 6 . 28 × 10 − 4 +5 . 198554 × 10 1 +0 +0 − 1 +2 3 . 86 × 10 − 4 +1 . 411472 × 10 3 − 2 +5 +0 − 2 j =1 1 . 31 × 10 − 4 +2 . 270341 × 10 4 − 1 +3 +0 − 1 f j = k 5 n 5 + k 6 n 6 1 . 05 × 10 − 4 − 8 . 652321 × 10 4 − 2 +3 +0 +0 p 5 g 5 + p 6 g 6 + q 6 s 6 9 . 92 × 10 − 5 +1 . 387493 × 10 3 − 2 +5 +1 − 3 8 . 06 × 10 − 5 +4 . 399535 × 10 4 +0 +1 +0 +0 6 . 45 × 10 − 5 − 4 . 255587 × 10 4 − 2 +4 +0 − 1 4 . 60 × 10 − 5 − 2 . 123995 × 10 4 − 1 +2 − 1 +1 4 . 28 × 10 − 5 − 2 . 128791 × 10 4 − 1 +2 +1 − 1 3 . 66 × 10 − 5 − 1 . 517825 × 10 5 − 3 +4 +0 +0 3 . 49 × 10 − 5 +7 . 596451 × 10 1 +0 +0 − 2 +3 3 . 45 × 10 − 5 +1 . 092546 × 10 5 +1 +0 +0 +0 2 . 54 × 10 − 5 +1 . 435452 × 10 3 − 2 +5 − 1 − 1 2 . 01 × 10 − 5 − 1 . 078152 × 10 5 − 3 +5 +0 − 1 1 . 93 × 10 − 5 − 1 . 995139 × 10 1 +0 +0 +2 − 1 1 . 85 × 10 − 5 +2 . 267943 × 10 4 − 1 +3 +1 − 2 1 . 82 × 10 − 5 +1 . 363514 × 10 3 − 2 +5 +2 − 4 11

Dynamical Maps and Frequency Analysis C. I. planets Fixed ( a j , e j , I j , λ j , ϖ j , Ω j ) ( n j , g j , s j ) given F I (0) ( n, g, s ) ( a ( 0 ) , e ( 0 ) ) ( λ ( 0 ) , ϖ ( 0 ) , Ω ( 0 ) ) 2-d surface in R 3 I (0) fixed 12

[ 0 , T ] [ 0 , T ] [ T , 2 T ] [ T , 2 T ] Frequency Analysis Frequency Analysis (Laskar) (Laskar) + + f 1 f 1 f 2 f 2 Diffusion Diffusion σ = log 10 ( || f 1 − f 2 || / || f 1 || ) σ = log 10 ( || f 1 − f 2 || / || f 1 || ) Dynamical map in the Dynamical map in the Dynamical map in the Dynamical map in the “action space” “action space” “frequency space” “frequency space” (a,e) (a,e) correspondance correspondance 13

e e M a E a M V e e a a J e U a S g6 10:3 2:1 7:3 4:1 3:1 8:3 7:2 5:2 e a N e e a a e a Unstability Stable (Q.P.) Unstable Diffusion index 14

g6 10:3 2:1 7:3 3:1 4:1 8:3 5:2 7:2 e a 4:1 7:2 3:1 8:3 5:2 7:3 I (deg) a (A.U.) 15

Frequency Map (secular) -10 I (0) -20 a (0) -30 s (” /yr ) -40 -50 3:1 8:3 -60 5:2 -70 7:3 -80 -40 -20 0 20 40 60 80 g (” /yr ) 16

Main secular resonances in the asteroid belt Label -10 p q r 1 r 2 r 3 1 1 0 0 − 1 0 -20 8 2 1 0 − 1 0 0 3 -30 3 0 1 0 0 − 1 s (” /yr ) 6 13 4 1 1 − 1 0 − 1 17 19 16 -40 5 1 1 0 − 1 − 1 15 6 1 0 1 − 2 0 -50 7 7 0 1 − 1 1 − 1 18 11 14 -60 8 2 − 2 0 0 0 12 9 2 9 1 − 1 − 1 0 1 4 -70 10 10 1 − 1 0 − 1 1 5 1 11 2 1 0 − 2 1 -80 -40 -20 0 20 40 60 80 12 1 − 2 0 − 1 2 g (” /yr ) 13 1 0 2 − 3 0 14 1 − 1 1 − 2 1 15 1 − 3 0 − 1 3 pg + qs + r 1 g 5 + r 2 g 6 + r 3 s 6 = 0 16 1 − 3 − 1 0 3 17 1 − 4 0 − 1 4 18 2 − 3 0 − 2 3 19 1 − 4 − 1 0 4 17

-10 -20 -30 s (” /yr ) -40 -50 3:1 8:3 -60 5:2 -70 7:3 -80 -40 -20 0 20 40 60 80 g (” /yr ) 18

-10 -20 8 3 -30 6 13 17 19 16 s (” /yr ) -40 15 -50 3:1 7 18 11 14 8:3 -60 12 9 2 4 5:2 -70 10 5 7:3 1 -80 -40 -20 0 20 40 60 80 g (” /yr ) 18