KAM TORI IN THE SPATIAL RESTRICTEDTHREE-BODY PROBLEM

Jesus F. Palacian & Patricia Yanguas

Universidad Publica de Navarra

CARLES SIMO FEST FOR HIS 60TH BIRTHDAYS’Agaro (Barcelona), May 29th - June 3rd, 2006

Outline

1. The problem:

• Formulation in the inertial frame.

• The comet case.

2. Transformations: two averagings.

3. Reduction to the orbit space.

4. Analysis of the reduced system: relative equilibria.

5. Reconstruction of the flow:

• Symmetries and periodic orbits of the truncated system.

• Quasiperiodic orbits and invariant tori of the original system.

Motivation

• Determination of the existence of Earth–like extrasolar planets:

– More than 1/3 of the exoplanets detected up to now have orbits significantlyelliptic.

– About 2/3 may be orbiting their main star approximately at a mean distancesmaller than the one from Mercury to the Sun.

– The detected planets are big, Jovian–like.

• Planetary orbits of binary systems.

• Capture and escape in the Solar System and extrasolar systems.

A Possible Planet Orbiting a Pair of Binary Stars

The Problem in the Inertial Frame

H(q,p, fp; µ, ap, ep) =1

2(p2

1 + p22 + p2

3)

− 1− µ√[q1 + µ ρ cos (fp)]2 + [q2 + µ ρ sin (fp)]2 + q2

3

− µ√[q1 − (1− µ) ρ cos (fp)]2 + [q2 − (1− µ) ρ sin (fp)]2 + q2

3

.

• µ ∈ (0, 1/2] is the quotient of the mass of one of the primaries and the sum of themasses of the two primaries,

• ρ =ap (1− ep)

1 + ep cos(fp),

• ep ∈ [0, 1) is the eccentricity of the orbits of the two primaries,

• fp is the true anomaly of the eccentricity described by one of the two primaries andap is its corresponding semimajor axis.

A Picture

• Primaries m1 and m2 have masses 1− µ and µ.

• The infinitesimal particle is m3.

• The motion occurs in 3D.

q

q

1 2

3

q

m1

2m

3m

Comet Orbits

• We assume that the infinitesimal particle is far away from both primaries (ap = ε).

• We expand H in powers of ε, arriving at:

H = H0 + εH1 +ε2

2H2 +

ε3

6Hr,

where

H0 =1

2(p2

1 + p22 + p2

3)−1√

q21 + q2

2 + q23

,

H1 = 0,

H2 =(−1 + µ) µ (1− e2

p)2

2 [1 + ep cos (fp)]2 (q21 + q2

2 + q23)

5/2

× [q21 + q2

2 − 2 q23 + 3 (q2

1 − q22) cos (2 fp) + 6 q1 q2 sin (2 fp)]

and Hr contains higher–order terms in ε.

• Upper bound for convergence: ε/|q| < 1/((1− µ) (1 + ep)) ≤ 1/2.

Hamiltonian in a Rotating System

2

The Hamiltonian has five equilibria:

• L1, L2, L3 unstable (Euler),

• L4, L5 linearly stable (Lagrange).

Some Closed Trajectories

-0.015 -0.01 -0.005 0.005 0.01

-0.01

-0.005

0.005

0.01

q1

q2

Ellipses corresponding to the points L1 (light blue), L2 (blue), L3 (green), L4 (yellow)and L5 (red). The primary m1 is describing the brown ellipse, whereas m2 is movingalong the pink ellipse. The value of ep has been fixed to 1/6, whereas µ = 1/4 and thesemimajor axis ε = 10−2

Averaging w.r.t. Time

Using Lie transformations we get the following:

K = K0 + εK1 +ε2

2K2 +

ε3

6Kr,

where

K0 =1

2(p′ 21 + p′ 22 + p′ 23 )− 1√

q′ 21 + q′ 22 + q′ 23

,

K1 = 0,

K2 =(−1 + µ) µ [(1 + 9 e2

p) q′ 21 + (1− 6 e2p) q′ 22 − (2 + 3 e2

p) q′ 23 ]

2 (q′ 21 + q′ 22 + q′ 23 )5/2,

by means of

W =ε (−1 + µ) µ

24 e2p [1 + ep cos (fp)] (q2

1 + q22 + q2

3)5/2

×{ − 24 (1− e2p)

2 [1 + 3 ep cos (fp)] q1 q2

+ 3 e3p sin (fp) [(23 + 7 e2

p) q21 − (7 + 8 e2

p) q22 + (−16 + e2

p) q23]

+ 6 e2p sin (2 fp) [3 (−1 + 4 e2

p) q21 + (3− 8 e2

p) q22 − 5 e2

p q23]

+ e3p sin (3 fp) [(−3 + 13 e2

p) q21 + (3− 8 e2

p) q22 − 5 e2

p q23]}.

Delaunay Coordinates

• `: mean anomaly

• g: argument of pericentre

• h: argument of node

• L: square root of the semimajor axis

• G: modulus of the angular momentum vector G

• H : third component of G

x

y

z

r

fg

I

θG

h

H

x'

'

'

Delaunay Normalisation (1)

We get the Hamiltonian: T = T0 + ε T1 + (ε2/2) T2 where

T0 = − 1

2 L′2,

T1 = 0,

T2 =(−1 + µ) µ

8 L′3 G′5 {(2 + 3 e2p) (3 H ′2 −G′2) + 15 e2

p (G′2 −H

′2) cos (2 h′)}.

The generating function responsible of this transformation is computed in closed formas: V = (ε2/2)V2, where V2 is calculated to satisfy:

1

L3

∂ V2

∂ `= K′

2 − T2.

Delaunay Normalisation (2)

We arrive at the (periodic) expression:

V2 =(−1 + µ) µ

16 e2 G3 r3

×{2 e2 [(2 + 3 e2p) (−1 + 3 c2) + 15 e2

p s2 cos (2 h)] r3 ϕ

+ 2 (2 + 3 e2p) e2 G (−1 + 3 c2) r3 R + 30 e2

p e2 s2 G r3 R cos (2 h)

+ 5 e2p (1 + c)2 G r (2 G4 −G2 r − η2 r2) R cos (2 g + 2 h)

+ 5 e2p (1− c)2 G4 (2 G2 − 3 r) sin (2 g − 2 h)

+ 2 (2 + 3 e2p) s2 G4 (2 G2 − 3 r) sin (2 g)

+ 5 e2p (1 + c)2 G4 (2 G2 − 3 r) sin (2 g + 2 h)

+ 2 (2 + 3 e2p) s2 G r (2 G4 −G2 r − η2 r2) R cos (2 g)

+ 5 e2p (1− c)2 G r (2 G4 −G2 r − η2 r2) R cos (2 g − 2 h)}.

We have introduced c = H/G, s =√

1− c2, η as (1 − e2)1/2 and the equation of thecentre ϕ = fp − `. Besides, r corresponds to the radial distance and R is the radialvelocity.

Reduction

R6 × [0, 2π) −→ R6 −→ ΠL,G

(`, g, h, L, G, H, fp) (`, g, h, L, G, H,−) (−,−, h, H,−; L, G)

ΠL,G ={(π1, π2, π3) ∈ R3 | π2

1 + π22 + π2

3 = G2 ≤ L2}.

The expressions of h, L, G, H as functions of the invariants π1, . . . , π4 are:

sin h =π1√

π21 + π2

2

, cos h =−π2√

π21 + π2

2

,

L2 = 2 (π21 + π2

2 + π23)− π4, G2 = π2

1 + π22 + π2

3, H = π3.

Reduced Hamiltonian:

Tπ(π1, π2; ηp) = (4 η2p − 5) π2

1 − η2p π2

2 where ηp = (1− e2p)

1/2.

Relative Equilibria

i) π1 = π2 = 0, π3 = ±G, which correspond to equatorial orbits, i.e. the motionsof the infinitesimal particle body are coplanar with the motion of the primaries;

ii) π1 = π3 = 0, π2 = ±G: polar orbits (H = 0) with sin h = 0, i.e. motions of thethird body lying on the plane O q1 q3;

iii) π2 = π3 = 0, π1 = ±G: polar orbits (H = 0 and cos h = 0: motions on O q2 q3).

Equilibria α1 α2 α3 Stability

π1 = π2 = 0, π3 = ±G 2(4η2p − 5) −2η2

p 0 Stable

π1 = π3 = 0, π2 = ±G −10(1− η2p) 0 2η2

p Unstable

π2 = π3 = 0, π1 = ±G 0 10(1− η2p) 2(5− 4η2

p) Stable

π1

π3

π2

Flow in the Truncated Original System

• Elliptic relative equilibria of the reduced system correspond to invariant 3–tori of theε3-truncation of the original Hamiltonian. They depend on the actions L and G andthe external parameters µ and ep enjoying the same type of orbital stability.

• Periodic orbits around the elliptic equilibria correspond to invariant 4–tori of the ε3-truncation of the original Hamiltonian.

• The angles parametrising the 3-tori are `, g and t (or fp). We have taken other com-binations when the Delaunay angles are not well defined.

• We have approximated the families of 3-tori and 4-tori, inverting the Lie transforma-tions, up to order ε3. The process may be extended to higher orders.

• The tori can be refined using either analytical or numerical techniques.

Periodic Orbits of the Truncated System

• We select the slopes between two angles of the families of tori to close some of thequasiperiodic orbits confined in the tori.

• Using the discrete symmetry

S1 : (q1, q2, q3, p1, p2, p3, fp) → (q1,−q2,−q3,−p1, p2, p3, 2 k π−fp), k ∈ Z

we have determined symmetric periodic orbits.

• For µ = 1/2 we have also

S2 : (q1, q2, q3, p1, p2, p3, fp) → (−q1, q2, q3, p1,−p2,−p3, 2 k π−fp),S3 : (q1, q2, q3, p1, p2, p3, fp) → (−q1, q2, q3,−p1, p2, p3, 2 k π−fp),

k ∈ Z,

we can obtain doubly–symmetric periodic orbits.

KAM Theory (1)

(H. Russmann, 1990; J.X. Xu, J.G. You & Q.J. Qiu, 1997; S.-N. Chow, Y. Li & Y. Yi,2002)Consider an analytic family of real analytic Hamiltonian systems

H(y, x) = N(y) + εP (y, x)

where (x, y) are action-angle variables. The frequency vector associated to the unper-turbed Hamiltonian is given by ω(y) = (∂N/∂y)(y). For ε sufficiently small:

1. If there is a constant C > 0 such that

rank

∂iω

∂yi : i ∈ Zn+, |i| ≤ C

≡ n over G,

then the majority of unperturbed n-tori on G will persist.

2. If the Hessian matrix ∂2N/∂y2 is non-singular on G, then for any given (γ, τ ), with

γ > ε1

8n+12 , τ > n − 1, all unperturbed Diophantine tori whose frequencies ω(y) areof the Diophantine types (γ, τ ) will persist and give rise to analytic, Diophantine,invariant n-tori of the perturbed system with the same frequencies.

KAM Theory (2)

Time-dependent Hamiltonians (Y. Li & Y. Yi, 2006)Consider a non-autonomous Hamiltonian system

H(y, x) = N(y) + εP (y, x, ω t).

Assume ω satisfies the Diophantine condition

| < k, ω > | > γ′

|k|τ ′∀k ∈ Zn′ \ {0},

for fixed γ′ > 0, τ ′ > n′ − 1. For ε sufficiently small:

1. If condition 1. of previous theorem holds, then the majority of unperturbed n+n′-toriwill persist with last n′ toral frequency equals ω.

2. If condition 2. of previous theorem holds, then the majority of unperturbed n+n′-toriwill persist with the same frequencies.

Reconstruction of the Flow (1)

Original system

Around the elliptic points (four families) we apply KAM theory (for this special isoen-ergetic nondegeneracy case), concluding the persistence of invariant 4-tori of the original(non-truncated) Hamiltonian.

T = T0 + ε T1 + (ε2/2) T2 +O(ε3):

T0 = − 1

2 L2,

T1 = 0,

T2 =(−1 + µ) µ

8 L3 G5{(2 + 3 e2

p) (3 H2 −G2) + 15 e2p (G2 −H2) cos (2 h)}.

Reconstruction of the Flow (2)

• Relative equilibria (±G, 0, 0):

(1) Shift T to the equilibrium by doing h = h± π/2, H = H .

(2) Scale h = ε1/3 h∗, H = ε1/3 H∗ and T ∗ = T /ε2/3 and expand it in powers of ε.

(3) Define h′ =√

2 γ h∗, H ′ = H∗/√

2 γ with γ =3 (1− µ) µ ε2 (1 + 4 e2

p)

8 L3 G5.

(4) Introduce action angle variables: h′ =√2 I1/ω sin ϕ1, H ′ =

√2 ω I1 cos ϕ1 with

ω =3 (1− µ) µ ε2 ep

√5 + 20 e2

p

4 L3 G4.

Reconstruction of the Flow (3)

(5) Average T ∗ over ϕ1 to get T ∗ ≡ T # +O(ε8/3) = T #0 +T #

1 +T #2 +O(ε8/3) where

T #0 = − 1

2 ε2/3 L2, T #

1 =ε4/3 (1− µ) µ (1 + 9 e2

p)

8 L3 G3, T #

2 = −ω I1.

(6) The Hessian ∂2T #/∂(L, G, I1) is non-singular, thus the rank of the followingmatrix is 3.

∂T #

∂L

∂T #

∂G

∂T #

∂I1

∂2T #

∂L2

∂2T #

∂L∂G

∂2T #

∂L∂I1

∂2T #

∂G∂L

∂2T #

∂G2

∂2T #

∂G∂I1

∂2T #

∂I1∂L

∂2T #

∂I1∂G

∂2T #

∂I21

.

(7) Taking fp as the time variable, it is easy to check that the corresponding Diophan-tine condition is satisfied.

So, the 4-invariant tori of the unperturbed system T # = T #0 + T #

1 + T #2

survive as slightly deformed Lagrangian tori with the same frequencies.

Reconstruction of the Flow (4)

Relative equilibria (0, 0,±G):

Delaunay elements are not well defined so we resort to another set of action–angle vari-ables:

• For (0, 0, G):

A1 = ` + g + h, A2 = −h, A3 = −g − h,

B1 = L, B2 = G−H, B3 = L−G.

• For (0, 0,−G):

A1 = ` + g − h, A2 = h, A3 = g − h,

B1 = L, B2 = G + H, B3 = −L + G.

We proceed similarly to the polar orbits, arriving at the same results.

The 4-invariant tori of the unperturbed system survive as slightly deformedLagrangian tori.

The Circular Problem: Doubly Averaged Hamiltonian

T =4∑

i=0(εi/i!) Ti where

T0 = − 1

2 L′2,

T1 = 0,

T2 =(1− µ) µ (G′2 − 3 H ′2)

4 L′3 G′5 ,

T3 = 0,

T4 =3 (1− µ) µ

128 L′5 G′11{3 (27− 91 µ + 91 µ2) G′6 + 24 (1− µ) µ L′G′5

+ [− 135 (L′2 + 6 H ′2) + (1− µ) µ (451 L′2 + 2538 H ′2)] G′4

− 144 (1− µ) µ L′H ′2 G′3

+ 3 H ′2 [45 (10 L′2 + 7 H ′2)− µ (1− µ) (1402 L′2 + 955 H ′2)] G′2

+ 216 (1− µ) µ L′H ′4 G′ − 3 (525− 1553 µ + 1553 µ2) L′2 H ′4

+ 6 (L′2 −G′2) (G′2 −H ′2)

×[15 (G′2 − 7 H ′2)− (1− µ) µ (47 G′2 − 345 H ′2)] cos (2 g′)}.

The Circular Problem: Reduced Phase Space

UL,H = {τ ∈ R3 | τ 22 + τ 2

3 = [(L + τ1)2 −H2] [(L− τ1)

2 −H2]}

τ1

τ3

τ2

τ2

τ3

τ1

1τ

c

e

0

τ3

|H|-L L-|H|

L - H2 2

H - L2 2

|H|>0

r

c-p

r-e

0

τ3

1τ

H=0

-L 2

2L

L-L

The Circular Problem: Relative Equilibria

Bifurcation diagram

-2 -1 0 11 H

L

2

2

6

6

1

Γ

Γ

2

11

2

3

4

Γ5Γ

22

2

Γ1

KAM Tori

Averaging T with respect to g we get a normal form Z which is secular up to terms oforder ε4, Z = ∑4

i=0(εi/i!)Zi, where

Z0 = − 1

2 L2,

Z1 = 0,

Z2 =(1− µ) µ (G2 − 3 H2)

4 L3 G5,

Z3 = 0,

Z4 = F (L, G, H).

The Hessian of Z + (ε2/2)Z2 does not vanish provided that 0 < ε � 1, so Russmannconditions are satisfied and the 3-tori survive around the elliptic points of the Hamil-tonian with the same frequencies as the unperturbed tori.

Transition Between the Circular and the Elliptic Problems

• When ep = 0 the dynamics of the problem is richer than in the case ep > 0.

• In the elliptic situation the polar–type of orbits were equilibria whereas if ep = 0, theyare equilibria only if they are also circular.

• Circular problem: circular–type trajectories with any inclination and orbits whoseinclination is closed to 63o.4349... or to 116o.565... but having any eccentricity.

• The bifurcation lines determine the evolution of the flow.