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 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... Different models: asteroid + Sun + 8 planets asteroid + Sun + 4 giant planets asteroid + Sun + Jup + Sat

2

e e e e e e e e a a a a a a a a M N U S J M E V Diffusion index Unstability Stable (Q.P.) Unstable

Test-particles in the Solar system :

restricted 10 bodies-problem

Robutel & Laskar (2001)

3

e e e e e e e e a a a a a a a a M N U S J M E V Diffusion index Unstability Stable (Q.P.) Unstable

Test-particles in the Solar system :

restricted 10 bodies-problem

Robutel & Laskar (2001)

Sun & planets are given

3

e e e e e e e e a a a a a a a a M N U S J M E V Diffusion index Unstability Stable (Q.P.) Unstable

Test-particles in the Solar system :

restricted 10 bodies-problem

Robutel & Laskar (2001)

Sun & planets are given small body:

a ∈ [0.38 : 90] A.U. e ∈ [0 : 0.9] (λ, ̟, Ω) Fixed I = 0

3

e e e e e e e e a a a a a a a a M N U S J M E V Diffusion index Unstability Stable (Q.P.) Unstable

Test-particles in the Solar system :

restricted 10 bodies-problem

Robutel & Laskar (2001)

Sun & planets are given small body:

a ∈ [0.38 : 90] A.U. e ∈ [0 : 0.9] (λ, ̟, Ω) Fixed I = 0

Overlap of MMR above coll. lines: Global chaos

3

e e e e e e e e a a a a a a a a M N U S J M E V Diffusion index Unstability Stable (Q.P.) Unstable

Test-particles in the Solar system :

restricted 10 bodies-problem

Robutel & Laskar (2001)

Sun & planets are given small body:

a ∈ [0.38 : 90] A.U. e ∈ [0 : 0.9] (λ, ̟, Ω) Fixed I = 0

Overlap of MMR above coll. lines: Global chaos

3

Projection of the observed inner solar system’s objects on the ecliptic

Main asteroids belt (~400000) Terrestrial planet’s crossers (~5000)

J L4 L5

Jupiter’s trojans (~2000) Comets (~200) Trojans can orbit far from L4 or L5 (2 to 2.5 A.U.)

4

λ = M + ̟ ̟ = ω + Ω

5

fundamental Frequencies

(proper frequencies)

3 frequencies for the Trojan:

(n, g, s)

Orbital motions: periods Secular motions: periods > 25000 years 3 months for Mercury 12 years for Jupiter 164 years for Neptune 1000 years at 100 A.U.

6

fundamental Frequencies

(proper frequencies) If Q.P. solution ( evolves on a KAM torus)

3 frequencies for the Trojan:

(n, g, s)

Orbital motions: periods Secular motions: periods > 25000 years 3 months for Mercury 12 years for Jupiter 164 years for Neptune 1000 years at 100 A.U.

6

fundamental Frequencies

(proper frequencies) If Q.P. solution ( evolves on a KAM torus)

(nj,g j,sj) sj = 0

3n-1 planetary frequencies:

• ne of the

3 frequencies for the Trojan:

(n, g, s)

Orbital motions: periods Secular motions: periods > 25000 years 3 months for Mercury 12 years for Jupiter 164 years for Neptune 1000 years at 100 A.U.

6

H(I, θ) = H0(I) + εH1(I, θ) H real analytic for (I, θ) ∈ Bn × Tn

The Frequency Map

J.Laskar (1999)

7

H(I, θ) = H0(I) + εH1(I, θ) H real analytic for (I, θ) ∈ Bn × Tn If ε = 0 I − → ν(I) = ∇H0(I) if det

• ∂2H0(I)

∂I2

• = 0

F is a diffeo. (loc.) F : Bn − → Ω ⊂ Rn

The Frequency Map

J.Laskar (1999)

7

H(I, θ) = H0(I) + εH1(I, θ) H real analytic for (I, θ) ∈ Bn × Tn

The Frequency Map

J.Laskar (1999)

8

H(I, θ) = H0(I) + εH1(I, θ) H real analytic for (I, θ) ∈ Bn × Tn

The Frequency Map

If ε = 0 but small enough There exists Ωε set of diophanine frequiencies ↔ KAM tori

J.Laskar (1999)

8

H(I, θ) = H0(I) + εH1(I, θ) H real analytic for (I, θ) ∈ Bn × Tn

The Frequency Map

If ε = 0 but small enough There exists Ωε set of diophanine frequiencies ↔ KAM tori Ψ : Tn × Ω − → Tn × Bn (ϕ, ν) − → (θ, I) P¨

• schel (1982),: There exists a diffeo. Ψ and a coord. syst. (ϕ, ν) such that

Ψ is analytical/ϕ and C∞ /ν The flow is linear on: Tn × Ωε: ˙ ν = 0 , ˙ ϕ = ν

J.Laskar (1999)

8

H(I, θ) = H0(I) + εH1(I, θ) H real analytic for (I, θ) ∈ Bn × Tn

The Frequency Map

If ε = 0 but small enough There exists Ωε set of diophanine frequiencies ↔ KAM tori Ψ : Tn × Ω − → Tn × Bn (ϕ, ν) − → (θ, I) P¨

• schel (1982),: There exists a diffeo. Ψ and a coord. syst. (ϕ, ν) such that

Ψ is analytical/ϕ and C∞ /ν The flow is linear on: Tn × Ωε: ˙ ν = 0 , ˙ ϕ = ν Fθ0 : Bn − → Ω ; I − → p2(Ψ−1(θ0, I)) The frequency map Fθ0 is a smooth diffeo. from, the actions space to the frequencies space For fix θ ∈ Tn: θ = θ0

J.Laskar (1999)

8

Goal to obtain numerically a frequency map: defined on Bn which coincide with Fθ0 , up to numerical accuracy,

• n the set of KAM tori

numerical tool: Frequency analysis (J.Laskar, 1988,1990)

9

|αj| fj combinations (AU) rad/yr 46.183882 .02005033 n .259757 .52968580 n5 .058931 .21330868 n6 .049411 .02004870 n − g8 + g .040885 .02005196 n + g8 − g .038045 .01808704 −n + n8 + g .031431 .02201360 3n − n8 − g . . . . . .

Quasi-periodic decomposition of z5 = e5 exp i̟5

aeiλ =

• k

αkeifk

10

Quasi-periodic decomposition of z5 = e5 exp i̟5 fj = k5n5 + k6n6 p5g5 + p6g6 + q6s6 z5(t) ≈

N

• j=1

αj exp (ifjt)

11

Quasi-periodic decomposition of z5 = e5 exp i̟5 fj = k5n5 + k6n6 p5g5 + p6g6 + q6s6

|αj| fj (” /yr) k5 k6 p5 p6 4.41 × 10−2 +4.027603 × 100 +0 +0 +1 +0 1.59 × 10−2 +2.800657 × 101 +0 +0 +0 +1 6.44 × 10−4 −2.126393 × 104 −1 +2 +0 +0 6.28 × 10−4 +5.198554 × 101 +0 +0 −1 +2 3.86 × 10−4 +1.411472 × 103 −2 +5 +0 −2 1.31 × 10−4 +2.270341 × 104 −1 +3 +0 −1 1.05 × 10−4 −8.652321 × 104 −2 +3 +0 +0 9.92 × 10−5 +1.387493 × 103 −2 +5 +1 −3 8.06 × 10−5 +4.399535 × 104 +0 +1 +0 +0 6.45 × 10−5 −4.255587 × 104 −2 +4 +0 −1 4.60 × 10−5 −2.123995 × 104 −1 +2 −1 +1 4.28 × 10−5 −2.128791 × 104 −1 +2 +1 −1 3.66 × 10−5 −1.517825 × 105 −3 +4 +0 +0 3.49 × 10−5 +7.596451 × 101 +0 +0 −2 +3 3.45 × 10−5 +1.092546 × 105 +1 +0 +0 +0 2.54 × 10−5 +1.435452 × 103 −2 +5 −1 −1 2.01 × 10−5 −1.078152 × 105 −3 +5 +0 −1 1.93 × 10−5 −1.995139 × 101 +0 +0 +2 −1 1.85 × 10−5 +2.267943 × 104 −1 +3 +1 −2 1.82 × 10−5 +1.363514 × 103 −2 +5 +2 −4

z5(t) ≈

N

• j=1

αj exp (ifjt)

11

Dynamical Maps and Frequency Analysis

(a j,ej,Ij,λ j,ϖj,Ω j)

• C. I. planets

(nj, gj, sj)

(a(0),e(0))

(λ(0),ϖ(0),Ω(0))

I(0) fixed 2-d surface in R3

(n, g, s)

FI(0)

Fixed given

12

[0,T] [T,2T] σ = log10(||f1 − f2||/||f1||) f1 f2

Frequency Analysis (Laskar)

+

Diffusion Dynamical map in the “frequency space” Dynamical map in the “action space” (a,e) correspondance

[0,T] [T,2T] σ = log10(||f1 − f2||/||f1||) f1 f2

Frequency Analysis (Laskar)

+

Diffusion Dynamical map in the “frequency space” Dynamical map in the “action space” (a,e) correspondance

13

e e e e e e e e a a a a a a a a M N U S J M E V Diffusion index Unstability Stable (Q.P.) Unstable

4:1 3:1 5:2 2:1 7:3 8:3 7:2 10:3 g6

a e 14

3:1 4:1 7:2 8:3 5:2 7:3

a (A.U.) I (deg)

4:1 3:1 5:2 2:1 7:3 8:3 7:2 10:3 g6

a e

15

• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10
• 40
• 20

20 40 60 80

3:1 8:3 5:2 7:3

s (”/yr) g (”/yr)

Frequency Map (secular)

I(0)

a(0)

16

Label p q r1 r2 r3 1 1 −1 2 1 −1 3 1 −1 4 1 1 −1 −1 5 1 1 −1 −1 6 1 1 −2 7 1 −1 1 −1 8 2 −2 9 1 −1 −1 1 10 1 −1 −1 1 11 2 1 −2 1 12 1 −2 −1 2 13 1 2 −3 14 1 −1 1 −2 1 15 1 −3 −1 3 16 1 −3 −1 3 17 1 −4 −1 4 18 2 −3 −2 3 19 1 −4 −1 4

pg + qs + r1g5 + r2g6 + r3s6 = 0

• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10
• 40
• 20

20 40 60 80

1 2 3 4 5 6 7 8 9 10 12 15 16 17 18 19 11 13 14

g (”/yr) s (”/yr)

Main secular resonances in the asteroid belt

17

• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10
• 40
• 20

20 40 60 80

3:1 8:3 5:2 7:3

g (”/yr)

s (”/yr)

18

• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10
• 40
• 20

20 40 60 80

3:1 8:3 5:2 7:3 1 2 3 4 5 6 7 8 9 10 12 15 16 17 18 19 11 13 14

g (”/yr)

s (”/yr)

18

Jovian T rojans

J L4 L5

Jupiter’s trojans (~2000)

19

Robutel, Gabern & Jorba (2005, 2006)

L4 substitute

a e

20

Robutel, Gabern & Jorba (2005, 2006)

L4 substitute

a e

20

R.4.B.P. (Sun+J+Sat+T)

Comparison of different models

21

E.R.T.B.P. (Sun+J+T) R.4.B.P. (Sun+J+Sat+T)

Comparison of different models

21

E.R.T.B.P. (Sun+J+T) R.4.B.P. (Sun+J+Sat+T) R.6.B.P. (Sun+J+Sat+U+N+T)

Comparison of different models

21

fundamental Frequencies

(proper frequencies)

5 planetary frequencies : Trojan :

(n5, n6, g5, g6, s6)

3 for a test-particle :

(n, g, s)

1:1 orbital resonance

n5 = n

(ν, g, s)

22

fundamental Frequencies

(proper frequencies)

5 planetary frequencies : Trojan :

ν ∈ [7500,9200] arcsec/year Tν ∈ [140,155] years

(n5, n6, g5, g6, s6)

3 for a test-particle :

(n, g, s)

1:1 orbital resonance

n5 = n

(ν, g, s)

22

fundamental Frequencies

(proper frequencies)

5 planetary frequencies : Trojan :

ν ∈ [7500,9200] arcsec/year Tν ∈ [140,155] years g ∈ [230,450] arcsec/year Tg ∈ [2880,5634] years

(n5, n6, g5, g6, s6)

3 for a test-particle :

(n, g, s)

1:1 orbital resonance

n5 = n

(ν, g, s)

22

fundamental Frequencies

(proper frequencies)

5 planetary frequencies : Trojan :

s ∈ [−50,5] arcsec/year Ts > 25000 years ν ∈ [7500,9200] arcsec/year Tν ∈ [140,155] years g ∈ [230,450] arcsec/year Tg ∈ [2880,5634] years

(n5, n6, g5, g6, s6)

3 for a test-particle :

(n, g, s)

1:1 orbital resonance

n5 = n

(ν, g, s)

22

2 obvious families of resonances

Family I

E.R.T.B.P. (Sun+J+T)

pν − n5 + qg = 0

pν = n5

23

2 obvious families of resonances

Family I

E.R.T.B.P. (Sun+J+T)

pν − n5 + qg = 0

pν = n5

Family III: Secular resonances

kg + ls + k5g5 + k6g6 + l6s6 = 0

23

Quasi-periodic decomposition of z5 = e5 exp i̟5

|αj| fj (” /yr) k5 k6 p5 p6 4.41 × 10−2 +4.027603 × 100 +0 +0 +1 +0 1.59 × 10−2 +2.800657 × 101 +0 +0 +0 +1 6.44 × 10−4 −2.126393 × 104 −1 +2 +0 +0 6.28 × 10−4 +5.198554 × 101 +0 +0 −1 +2 3.86 × 10−4 +1.411472 × 103 −2 +5 +0 −2 1.31 × 10−4 +2.270341 × 104 −1 +3 +0 −1 1.05 × 10−4 −8.652321 × 104 −2 +3 +0 +0 9.92 × 10−5 +1.387493 × 103 −2 +5 +1 −3 8.06 × 10−5 +4.399535 × 104 +0 +1 +0 +0 6.45 × 10−5 −4.255587 × 104 −2 +4 +0 −1 4.60 × 10−5 −2.123995 × 104 −1 +2 −1 +1 4.28 × 10−5 −2.128791 × 104 −1 +2 +1 −1 3.66 × 10−5 −1.517825 × 105 −3 +4 +0 +0 3.49 × 10−5 +7.596451 × 101 +0 +0 −2 +3 3.45 × 10−5 +1.092546 × 105 +1 +0 +0 +0 2.54 × 10−5 +1.435452 × 103 −2 +5 −1 −1 2.01 × 10−5 −1.078152 × 105 −3 +5 +0 −1 1.93 × 10−5 −1.995139 × 101 +0 +0 +2 −1 1.85 × 10−5 +2.267943 × 104 −1 +3 +1 −2 1.82 × 10−5 +1.363514 × 103 −2 +5 +2 −4

z5(t) ≈

N

• j=1

αj exp (ifjt)

24

n5 − 2n6

2 5(n5 − 2n6) ≈ 8500”/yr

ν ∈ [7500, 9200] ”/yr

Family II

Quasi-periodic decomposition of z5 = e5 exp i̟5

|αj| fj (” /yr) k5 k6 p5 p6 4.41 × 10−2 +4.027603 × 100 +0 +0 +1 +0 1.59 × 10−2 +2.800657 × 101 +0 +0 +0 +1 6.44 × 10−4 −2.126393 × 104 −1 +2 +0 +0 6.28 × 10−4 +5.198554 × 101 +0 +0 −1 +2 3.86 × 10−4 +1.411472 × 103 −2 +5 +0 −2 1.31 × 10−4 +2.270341 × 104 −1 +3 +0 −1 1.05 × 10−4 −8.652321 × 104 −2 +3 +0 +0 9.92 × 10−5 +1.387493 × 103 −2 +5 +1 −3 8.06 × 10−5 +4.399535 × 104 +0 +1 +0 +0 6.45 × 10−5 −4.255587 × 104 −2 +4 +0 −1 4.60 × 10−5 −2.123995 × 104 −1 +2 −1 +1 4.28 × 10−5 −2.128791 × 104 −1 +2 +1 −1 3.66 × 10−5 −1.517825 × 105 −3 +4 +0 +0 3.49 × 10−5 +7.596451 × 101 +0 +0 −2 +3 3.45 × 10−5 +1.092546 × 105 +1 +0 +0 +0 2.54 × 10−5 +1.435452 × 103 −2 +5 −1 −1 2.01 × 10−5 −1.078152 × 105 −3 +5 +0 −1 1.93 × 10−5 −1.995139 × 101 +0 +0 +2 −1 1.85 × 10−5 +2.267943 × 104 −1 +3 +1 −2 1.82 × 10−5 +1.363514 × 103 −2 +5 +2 −4

z5(t) ≈

N

• j=1

αj exp (ifjt)

24

2n5 − 5n6 4 ≈ 350”/yr

g ∈ [230, 450]”/yr

Family IV

(+2g6)

2n5 − 5n6

n5 − 2n6

2 5(n5 − 2n6) ≈ 8500”/yr

ν ∈ [7500, 9200] ”/yr

Family II

Quasi-periodic decomposition of z5 = e5 exp i̟5

|αj| fj (” /yr) k5 k6 p5 p6 4.41 × 10−2 +4.027603 × 100 +0 +0 +1 +0 1.59 × 10−2 +2.800657 × 101 +0 +0 +0 +1 6.44 × 10−4 −2.126393 × 104 −1 +2 +0 +0 6.28 × 10−4 +5.198554 × 101 +0 +0 −1 +2 3.86 × 10−4 +1.411472 × 103 −2 +5 +0 −2 1.31 × 10−4 +2.270341 × 104 −1 +3 +0 −1 1.05 × 10−4 −8.652321 × 104 −2 +3 +0 +0 9.92 × 10−5 +1.387493 × 103 −2 +5 +1 −3 8.06 × 10−5 +4.399535 × 104 +0 +1 +0 +0 6.45 × 10−5 −4.255587 × 104 −2 +4 +0 −1 4.60 × 10−5 −2.123995 × 104 −1 +2 −1 +1 4.28 × 10−5 −2.128791 × 104 −1 +2 +1 −1 3.66 × 10−5 −1.517825 × 105 −3 +4 +0 +0 3.49 × 10−5 +7.596451 × 101 +0 +0 −2 +3 3.45 × 10−5 +1.092546 × 105 +1 +0 +0 +0 2.54 × 10−5 +1.435452 × 103 −2 +5 −1 −1 2.01 × 10−5 −1.078152 × 105 −3 +5 +0 −1 1.93 × 10−5 −1.995139 × 101 +0 +0 +2 −1 1.85 × 10−5 +2.267943 × 104 −1 +3 +1 −2 1.82 × 10−5 +1.363514 × 103 −2 +5 +2 −4

z5(t) ≈

N

• j=1

αj exp (ifjt)

24

4 different families of resonances 4 different families of resonances

25

4 different families of resonances

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

4 different families of resonances

25

4 different families of resonances

Famille III: q s + q6s6 + p5g5 + p6g6 = 0

Secular resonances

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

4 different families of resonances

25

4 different families of resonances

Famille III: q s + q6s6 + p5g5 + p6g6 = 0

Secular resonances

Famille IV: pg - (2n5 −5n6)+ p5g5 + p6g6 = 0

• G. I. + secular frequencies

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

4 different families of resonances

25

4 different families of resonances

Famille III: q s + q6s6 + p5g5 + p6g6 = 0

Secular resonances

Famille IV: pg - (2n5 −5n6)+ p5g5 + p6g6 = 0

• G. I. + secular frequencies

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

4 different families of resonances

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

25

4 different families of resonances

Famille III: q s + q6s6 + p5g5 + p6g6 = 0

Secular resonances

Famille IV: pg - (2n5 −5n6)+ p5g5 + p6g6 = 0

• G. I. + secular frequencies

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

4 different families of resonances

Famille III: q s + q6s6 + p5g5 + p6g6 = 0

Secular resonances

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

25

4 different families of resonances

Famille III: q s + q6s6 + p5g5 + p6g6 = 0

Secular resonances

Famille IV: pg - (2n5 −5n6)+ p5g5 + p6g6 = 0

• G. I. + secular frequencies

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

4 different families of resonances

Famille III: q s + q6s6 + p5g5 + p6g6 = 0

Secular resonances

Famille IV: pg - (2n5 −5n6)+ p5g5 + p6g6 = 0

• G. I. + secular frequencies

Famille I: pν−n5 +qg+q5g5 +q6g6 = 0 Famille II: 5ν−2(n5 −2n6)+ pg+ p5g5 + p6g6 = 0

Secondary resonances

25

E.R.T.B.P (S+J+T)

26

E.R.T.B.P (S+J+T)

(ν,g,s)+n5

26

E.R.T.B.P (S+J+T)

(ν,g,s)+n5

pν−n5 +qg = 0 12ν−n5 +qg = 0 avec q ∈ {8,··· ,13} 13ν−n5 +qg = 0 avec q ∈ {−4,··· ,8} 14ν−n5 +qg = 0 avec q ∈ {−3,··· ,3} ν g

26

a e a e

L4 L4

27

a e a e

L4 L4

4g+(2n5 −5n6)−g5 = 0 4g+(2n5 −5n6)−g5 = 0

27

a e a e

L4 L4

4g+(2n5 −5n6)−2g5 +g6 = 0 4g+(2n5 −5n6)−2g5 +g6 = 0 4g+(2n5 −5n6)−g5 = 0 4g+(2n5 −5n6)−g5 = 0

27

a e a e

L4 L4

s = s6

4g+(2n5 −5n6)−2g5 +g6 = 0 4g+(2n5 −5n6)−2g5 +g6 = 0 4g+(2n5 −5n6)−g5 = 0 4g+(2n5 −5n6)−g5 = 0

27

28

13ν − n5 + qg + q5g5 + q6g6 = 0 14ν − n5 + qg + q5g5 + q6g6 = 0

28

13ν − n5 + qg + q5g5 + q6g6 = 0 14ν − n5 + qg + q5g5 + q6g6 = 0 5ν − 2(n5 − 2n6) − 0g + p5g5 + p6g6 = 0

28

13ν − n5 + qg + q5g5 + q6g6 = 0 14ν − n5 + qg + q5g5 + q6g6 = 0 5ν − 2(n5 − 2n6) − 0g + p5g5 + p6g6 = 0 5ν − 2(n5 − 2n6) − 1g + p5g5 + p6g6 = 0

28

13ν − n5 + qg + q5g5 + q6g6 = 0 14ν − n5 + qg + q5g5 + q6g6 = 0 5ν − 2(n5 − 2n6) − 0g + p5g5 + p6g6 = 0 5ν − 2(n5 − 2n6) − 2g + p5g5 + p6g6 = 0 5ν − 2(n5 − 2n6) − 1g + p5g5 + p6g6 = 0

28

Long-term stability

29

30

31

32

33

34

35

36

37

38

• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀
• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀

Ejection in the neighborhood and above

s = s6

39

• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀
• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀

Ejection in the neighborhood and above

s = s6

Regions where (orange, red) are cleared in 1Gy except 2

σ > −3

39

• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀
• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀

Ejection in the neighborhood and above

s = s6

Regions where (orange, red) are cleared in 1Gy except 2

σ > −3

4g+(2n5 −5n6)−g5 = 0 Gap along

4g + (2n5 − 5n6) − g6 = 0

39

• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀

Ejection in the neighborhood and above

s = s6

• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀

Ejection in the neighborhood and above

s = s6

Regions where (orange, red) are cleared in 1Gy except 2

σ > −3

4g+(2n5 −5n6)−g5 = 0 Gap along

4g + (2n5 − 5n6) − g6 = 0

39

• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀

Ejection in the neighborhood and above

s = s6

Regions where (orange, red) are cleared in 1Gy except 2

σ > −3

• ฀

฀ ฀ ฀ ฀

• ฀

฀ ฀

• ฀

Ejection in the neighborhood and above

s = s6

Regions where (orange, red) are cleared in 1Gy except 2

σ > −3

4g+(2n5 −5n6)−g5 = 0 Gap along

4g + (2n5 − 5n6) − g6 = 0

39

Slow diffusion along

4g+(2n5 −5n6)−g5 = 0

Slow diffusion along

4g+(2n5 −5n6)−g5 = 0

40

41

Slow diffusion along

4g+(2n5 −5n6)−g5 = 0

and ejection at 800 My during 600 My then wandering for 200 Ma Slow diffusion along

4g+(2n5 −5n6)−g5 = 0

and ejection at 800 My during 600 My then wandering for 200 Ma

42

43

8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5 8490 8500 8510 8520 8530 395 395.5 396 396.5 397 397.5 398 398.5 399 399.5

• 5ν−2(n5 −2n6)−0g+qg5 −(q+2)g6 = 0

Overlapping in family II bounded diffusion

44

45

Diffusion along resonances

45

Diffusion along resonances Diffusion transversal to the resonances

45