Geometry of transit orbits in the periodically-perturbed restricted three-body problem

Abstract

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincaré map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.

Extracted equations

• H_CR3BP = (1/2)(p_x^2 + p_y^2) - x*p_y + y*p_x - μ₁/r₁ - μ₂/r₂
• dx/dt = ∂H/∂p_x
• dy/dt = ∂H/∂p_y
• dp_x/dt = -∂H/∂x
• dp_y/dt = -∂H/∂y
• r₁ = sqrt((x + μ₂)^2 + y^2)
• r₂ = sqrt((x - μ₁)^2 + y^2)

Paper claims vs. our run

• The Lagrange point L1 is replaced by a periodic orbit under time-periodic perturbation
  not-testable
  no fidelity score recorded
• The periodic orbit corresponds to a hyperbolic-elliptic fixed point of the stroboscopic Poincaré map
  not-testable
  no fidelity score recorded
• Transit and non-transit orbits can be identified via stable and unstable manifolds of the periodic orbit
  not-testable
  no fidelity score recorded
• The linearized map locally conserves a time-independent effective Hamiltonian function
  not-testable
  no fidelity score recorded
• Phase space geometry of transit orbits is preserved from unperturbed to periodically-perturbed case
  not-testable
  no fidelity score recorded
• Linear symplectic map geometry carries over to the full nonlinear system
  not-testable
  no fidelity score recorded

Parameters