Domains of Effective Stability near L5 in the Restricted Three-body Problem - Carles Sim´ o Priscilla A. Sousa Silva Maisa O. Terra* MAiA-UB *Instituto Tecnol´ ogico de Aeron´ autica - SP/Brasil - - - Seminari de Sistemes Din` amics UB-UPC 25-04-2012 C. Sim´o, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 1 / 35
37
Embed
Domains of Effective Stability near L5 in the R3BP
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Domains of Effective Stability near L5 in the RestrictedThree-body Problem
-
Carles Simo Priscilla A. Sousa Silva Maisa O. Terra*
MAiA-UB*Instituto Tecnologico de Aeronautica - SP/Brasil
---
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
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 2 / 35
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)
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 3 / 35
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.
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 4 / 35
Introduction The Restricted Three-body Problem
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.
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 5 / 35
Introduction The Restricted Three-body Problem
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.
Spatial case: Nonlinear stability for µ ∈ (0, µ1)\µ2, µ3, except a set of initial condi-tions of small Lebesgue measure for fixed µ.
——————————
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.
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 6 / 35
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).
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 7 / 35
Introduction Review of previous results
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).
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 8 / 35
Introduction Review of previous results
C. Simo, P.A. Sousa Silva, M.O. Terra (MAiA-UB) Domains of Effective Stability near L5 in the R3BP 25-04-12 9 / 35
Initial conditions for the numerical exploration
The zero velocity surface
Initial conditions taken in a 3D subspace with null initial velocity: