$LastChangedDate: 27-Juil-2011 $

# CL_3b_manifolds

Manifolds (divergent and convergent) from Halo and Lissajous

## Calling Sequence

       manifolds = CL_3b_manifolds(env,lissajous_orb,epsilon,tint,pars)
       [manifold1,...,manifoldN] = CL_3b_manifolds(env,orb,t_orb,epsilon,tint,pars)

## Description

- 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).

## Parameters

env:

  Lagrangian point structure . (see CL_3b_environment)

orb:

  Lissajous or Halo orbit [6xN] (see CL_3b_halo or CL_3b_lissajous)

t_orb:

  Adimensional times of lissajous or halo orbit [1xN]

epsilon:

  Epsilon. In the literature it is said to be about ~1e-9, but as the method is accurate enough, we recommended 1e-5

period:

  Period used to estimate the monodromy. It corresponds to omegahalo for the halo orbits and nu for the Lissajous orbits

tint:

  Integration time (adimensional: use env.OMEGA) (See CL_3b_environment))

pars:

  (string) Name(s) of manifolds to compute. Possible values are 'div','-div','conv' or '-conv'.

manifolds:

  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)

## Bibliography

- 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

## See also

- CL_3b_environment

- CL_3b_lissajous

- CL_3b_halo

## Authors

- CNES - DCT/SB

## Examples

    // 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']);