CL_3b_manifolds

Description

• This function is deprecated.

  Replacement function: CL_3b_manifold

• Computes manifolds for Halo or Lissajous orbits.

  It can compute all four branches depending on the value of pars:

  - pars = "conv": convergent, inward

  - pars = "-conv": convergent, outward

  - pars = "div": divergent, inward

  - pars = "-div": divergent, outward

  Manifolds in output 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 can be computed using CL_3b_halo or CL_3b_lissajous.

  - For the definition of adimensional quantities, see CL_3b_environment.

  - In the literature it is said epsilon should be ~1.e-9, but as the method is accurate enough, we recommend 1.e-5.

Parameters

env:

(struct) Lagrangian point structure

orb:

Lissajous or Halo orbit (adimensional position and velocity) (6xN)

t_orb:

Time instants at which the orbit is given (adimensional) (1xN)

epsilon:

Tolerance. Recommended value: 1.e-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)

pars:

(string) Type of manifold to be computed: "conv", "-conv", "div", "-div" (1xP)

manifolds:

Generated manifolds (6 x n x nb_points, 6 corresponds to position and velocity,

n is the size of each trajectory making up the manifold, nb_points is the number of trajectories)

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 structure for L2 and "Sun-EarthMoon" system: env = CL_3b_environment("S-EM","L2"); // Generate a Halo orbit: Az = 150.e6 / env.D; // adimensional direction = 0; t_orb = linspace(0,180*86400,50) * env.OMEGA; // adimensional [orb,omega] = CL_3b_halo(env, Az, direction, t_orb); // Compute manifolds: epsilon = 1.e-5; tint = 150*86400 * env.OMEGA // adimensional [conv_out, div_out] = .. CL_3b_manifolds(env, orb, t_orb, omega, epsilon, tint, ["-conv","-div"]); // Plot orbit and manifolds (conv_out only) scf(); for i = 1 : size(conv_out,3) param3d(conv_out(1,:,i), conv_out(2,:,i), conv_out(3,:,i)); end param3d(orb(1,:), orb(2,:), orb(3,:)); h = CL_g_select(gce()); h.foreground = 5;