Regularization of the restricted elliptic three-body problem in the sun-earth L1-centered rotating system

The singular differential equations of motion of the restricted elliptic three-body problem in the sun-earth L-1-centered rotating system are regularized in order to remove the singularity near the earth center, thereby allowing the use of unconstrained optimization software in solving through an iterative scheme the transfer problem from a low circular parking orbit to any location in the vicinity of L-1. The trajectory is integrated backwards from an arbitrarily selected point near L-1 and the maneuver velocity change components are searched on in order to achieve certain target parameters at closest approach to the earth. Unlike the restricted circular problem where use is made of the Jacobi constant in order to reduce the order of the differential system, the present restricted elliptic problem requires the addition of a differential equation for the energy variable resulting in a system of ten first order equations cast in terms of the Levi-Civita-Kustaaheimo-Stiefel regularized u-variables. It is shown that the transfer solution obtained with the circular assumption fails to reach the desired halo-insertion location near L-1 when the trajectory is run within the framework of the more accurate elliptic model justifying thereby the use of this latter model for higher fidelity trajectory generation.