Manifolds (divergent and convergent) from Halo and Lissajous
manifolds = CL_3b_manifolds(env,lissajous_orb,epsilon,tint,pars) [manifold1,...,manifoldN] = CL_3b_manifolds(env,orb,t_orb,epsilon,tint,pars)
Computes manifolds for Halo and Lissajous orbits.
It can compute all four branches depending on the value of pars:
convergent in (pars="conv"), convergent out (pars="-conv") divergent in (pars="div"), divergent out (pars="-div")
Output manifolds are then given in the same order as pars.
Notes:
- Before using this function, it is needed to create an "environment" (env) for the chosen libration point and the chosen system (see CL_3b_environment).
- A Halo or Lissajous orbit must have been computed before using this function (see CL_3b_halo or CL_3b_lissajous).
Lagrangian point structure . (see CL_3b_environment)
Lissajous or Halo orbit [6xN] (see CL_3b_halo or CL_3b_lissajous)
Adimensional times of lissajous or halo orbit [1xN]
Epsilon. In the literature it is said to be about ~1e-9, but as the method is accurate enough, we recommended 1e-5
Period used to estimate the monodromy. It corresponds to omegahalo for the halo orbits and nu for the Lissajous orbits
Integration time (adimensional: use env.OMEGA) (See CL_3b_environment))
(string) Name(s) of manifolds to compute. Possible values are 'div','-div','conv' or '-conv'.
Generated manifolds (dimensions: 6,n,nb_points). Where first is position and velocities (6), n is extrapolation in time of each point of the original orbit, en nb_points depends on the discretisation of the original orbit( halo or lissajous)
1) Introduction au probleme a trois corps et dynamique linearisee autour des points de Lagrange, G. Collange, Note Technique CCT Mecanique Orbitale num.7, CNES 2006
2) Estimation numerique des varietes stables et instables des orbites quasi-periodiques de Lissajous autour des points d'Euler (Lagrange L1, L2, L3), R. Alacevich, CNES septembre 2006
3) Exploration numerique d'orbites homoclines et heteroclines autour de L1 et L2 dans le probleme restreint a trois corps, rapport de stage, A. Martinez Maida, DCT/SB/MO 2007.0029301, CNES septembre 2007
CNES - DCT/SB
// Example with an Halo orbit: // Build environement around L2 point of system Sun-EarthMoon: env = CL_3b_environment('S-EM','L2'); // Halo orbit : Az = 150e6/env.D; sens = 0; t_orb=linspace(0,180,50)*24*3600*env.OMEGA; //180 days [orb,omega] = CL_3b_halo(env,Az,sens,t_orb); graph1 =figure();xtitle("Halo''s manifold" ,'X','Y');plot2d(orb(1,:),orb(2,:),5); // 4 branches manifolds : epsilon=1e-5; tint =120*24*3600*env.OMEGA//120 days; [div_in,conv_in,conv_out,div_out] = .. CL_3b_manifolds(env,orb,t_orb,omega,epsilon,tint,['div','conv','-conv','-div']); for i=1:size(conv_out,3) plot2d(conv_out(1,:,i),conv_out(2,:,i),2); end plot2d(orb(1,:),orb(2,:),5) // Just one branch : [div] = CL_3b_manifolds(env,orb,t_orb,omega,epsilon,tint,['div']); | ![]() | ![]() |