domains of effective stability near l5 in the r3bp

Domains of Effective Stability near L5 in the RestrictedThree-body Problem

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Carles Simo Priscilla A. Sousa Silva Maisa O. Terra*

MAiA-UB*Instituto Tecnologico de Aeronautica - SP/Brasil

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Seminari de Sistemes Dinamics UB-UPC

25-04-2012

C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 1 / 35

Summary

1 IntroductionThe Restricted Three-body ProblemReview of previous results

2 Initial conditions for the numerical exploration

3 Confined and unconfined orbits

4 The shape of the effective stability region

5 The escape process

6 The central manifold of L3

7 The central manifold of a family of periodic orbits in the central manifold of L5

8 Ongoing work and Future Perspectives


Introduction

Motivation

Domains where trajectories remainbounded for a very long time span.

Planetary rings with shepherds;Some comets and asteroids, par-ticularly, Near-Earth-AsteroidsSharp boundaries of stability canalso be associated to solutionsof N-body problems known asfigure-8 choreographiesThe design of de space missionsfar away from the Earth Trojans and Greeks from the data of

the MPC - IAU(April 18, 2012)


Introduction The Restricted Three-body Problem

The Restricted Three-body Problem

3 D.O.F. HamiltonianParticle P3 of negligible mass moving under thegravitational influence of P1 and P2 of massesm1 and m2.The primaries describe circular coplanar orbitsaround the barycenter of P1-P2 and are fixed inthe synodic reference frame (which rotates w.r.t.an inertial frame).Non-dimensional variables: distance between theprimaries, the sum of their masses and their an-gular velocity around the barycenter are normal-ized to one.µ = m2/(m1 + m2), m1 > m2 is the only param-eter of the model.



Equations of Motion:x − 2y = Ωx ,

y + 2x = Ωy ,

z = Ωz ,

(1)

Ω(x , y , z) =1

2(x2 + y2) +

1− µr1

+µ

r2+µ(1− µ)

2;

r1 =√

(x − µ)2 + y2 + z2 and r2 =√

(x + 1− µ)2 + y2 + z2.

Integral of motion:

J(x , y , z , x , y , z) = 2Ω(x , y , z)− (x2 + y 2 + z2) = C . (2)

C is called the Jacobi constant.5D manifold M(µ,C) =

(x , y , z, x , y , z) ∈ R6|J(x , y , z, x , y , z) = const.

Hill regions:

Projection of M onto the configuration space;Areas accessible to the trajectories for each C ;Bounded by the Zero Velocity Surfaces.



Equilibrium points

Collinear points: L1,2,3, on the x-axiscenter-center-saddle4D central manifold: vertical and horizontal Lyapunov orbits, Halo orbits, invariant tori,other periodic orbits, chaotic regions.

Triangular points: L4,5, at x = µ− 12, y = ∓

√3

2, z = 0

Planar case: nonlinear stability for µ ∈ (0, µ1)\µ2, µ3µ1 = (9−

√69)

18 , µ2 = (45−√

1833)90 , µ3 = (15−

√213)

30 .

Spatial case: Nonlinear stability for µ ∈ (0, µ1)\µ2, µ3, except a set of initial condi-tions of small Lebesgue measure for fixed µ.

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Markeev, A.P., On the stability of the triangular libration points in the circular bounded three-body problem,Applied Math. Mech. 33 (1969), 105-110.

Markeev, A.P., Stability of the triangular Lagrangian solutions of the restricted three-body problem in thethree-dimensional circular case, Soviet Astronomy 15 (1972), 682-686.

Leontovich, A.M., On the stability of the Lagrange periodic solutions of the restricted problem of threebodies, Soviet Math. Dokl., 3 (1962), 425-428.

Deprit, A. and Deprit-Bartholome, A., Stability of the triangular Lagrangian points, The AstronomicalJournal 72, (1967), 173-179.


Introduction Review of previous results

Rough investigation of the planar R3BP for the Earth-Moon system (480 time units).McKenzie, R. and Szebehely, V., Nonlinear stability motion around the triangular libration points, CelestialMechanics 23 (1981), 223-229.

Numerical evidence that the boundary of the stable domain is related to the centralmanifold of L3.

Gomez, G. and Jorba, A. and Simo, C.and Masdemont, J., Dynamics and Mis-sion Design Near Libration Points, Vol-ume IV: Advanced Methods for Trian-gular Points, World Scientific (2001).



Stability domain around the triangular equilibria for the spatial case

Prediction of practical stability domain around the elliptic equilibrium pointbased on Nekhorosev type estimates.Giorgilli, A., Delshams, A., Fontich, E., Galgani, L., Simo, C. Effective Stability for a Hamil-tonian System near an Elliptic Equilibrium Point, with an application to the Restricted ThreeBody Problem, Journal of Differential Equations 77 (1989), 167-370.

Numerical simulations show a stable domain larger than the one find in theplanar case.Simo, C., Effective Computations in Celestial Mechanics and Astrodynamics, CISM Course onModern Methods of Analytical Mechanics and their Applications (1997).Simo, C., Slides of the talk Boundaries of Stability, given at UB (June 3, 2006).


Initial conditions for the numerical exploration

The zero velocity surface

Initial conditions taken in a 3D subspace with null initial velocity:

x = µ+ (1 + %) cos(2πα),

y = (1 + %) sin(2πα),

z = z0 > 0,

x = y = z = 0,

(3)

% ∈ R roughly limited by

% = −0.25z20 +O(z4

0 )±1

2(K(α)µ)1/2 +O(µ)

α ∈ [0, 0.5]L5 corresponds to % = 0, α = 1/3, z0 = 0

Analogous results to L4 due to

S : (x , y , z , x , y , z , t)↔ (x ,−y , z ,−x , y ,−z ,−t).

IC set expected to provide seed to detect invariant objects at the stability boundary.


Confined and unconfined orbits

Confined orbits

z0 = 0.1 z0 = 0.8


Confined and unconfined orbits

Unconfined orbits

z0 = 0.1 z0 = 0.8


The shape of the effective stability region

Extensive detection of stability transitions

µ = 2× 10−4, ∆ρ = 10−4, t = 104 (rev)



ρ, α, t (rev)

z0 = 0.1 z0 = 0.8



µ = 2× 10−4, ∆ρ = 10−7, t = 106 (rev)

(ρ, α, z) (x , y , z)


The escape process

α = 1/3, z = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.70.8.0, 0.85, 0.867 (xy projection)


The escape process

α = 1/3, z = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.70.8.0, 0.85, 0.867 (xy , z = 0, z > 0)


The central manifold of L3

The case of small z0

z0 = 0.1, α = 0.16, C = 2.99024550, tesc = 8.4 rev

(xy , z = 0, z > 0)

Black: 0 6 t 6 0.1tesc

Red: 0 6 t 6 0.1tesc

Red: before escape

Blue: after escape

(xy projection)



Selecting a piece of anescaping orbit that

approximates anhyperbolic torus:

z0 = 0.4α = 0.33

C = 2.84351423



Correspondence between approximated tori and tori continued from Γv3



C = 2.99764118



C = 2.99033255



C = 2.84351423


The central manifold of a family of periodic orbits in the central manifold of L5

The case of large z0

z0 = 0.46α = 0.35

C = 2.79507059



z0 = 0.8, α = 1/3, C = 2.43554203



Continuation of tori varying ρ

(xmax x ρ)



Γv5

C > C∗: center-center-centerC < C∗: center-center-saddle

γ1,2

center-center-saddle

C∗ = 1.94769998



Γv5 : red

γ1: green

C = C∗: blue

C → 3: γ1 and γ2

approach the secondary.

⇐ xz and yz projection of thesmallest orbits [γ1] obtained by

continuation.



Correspondence between tori at the boundary and tori in the centralmanifold of γ1 and γ2



Initial conditions outside the ZVS

µ = 0.00002, z0 = 0.8, α = 1/3, and v 20 = 0.1

with vx = |v0| cos(theta) and vy = |v0| sin(theta)

min(v) for t = 107 rev: 0.25985451

From the unbounded trajectory: T 2UP

C = 2.2693161922RN = 2.72340848× 10−2.

From the bounded trajectory: T 2DOWN

C = 2.2693161901RN = 2.72365953× 10−2.

Tori were continued in C untilC∗ = 2.43553588 of a reference torusobtained from a transition in the ZVS.



Splitting of the manifolds of γ1 and γ2



Splitting of the manifolds of γ1 and γ2

σ ∼ Ahr exp(−c/h)


Ongoing work and Future Perspectives

Final comments

Small µ: relatively large stability domains with sharpboundaries.

Identification of different objects that play a role in theescape process.

Remark: cantorian structure smoothed out by finite timecomputations.

Checks for ICs outside the ZVS for small z0.

Computation of the hyperbolic invariant manifolds of theobjects sitting at the boundary.

Other mass parameters (other objects may be importantdue to higher order resonances).

More realistic models.

The importance of computing codimension 1 invariantmanifolds!


domains of effective stability near l5 in the r3bp

Documents

triangular libration points

eective stability region

triangular points

hyperbolic torus

initial conditions

nonlinear stability

large z0

body problem