Synthetic Theory for Earth-Moon L2 Halo Orbits

The Earth-Moon Halo L2 orbits are interesting for their ability to be continually visible from Earth, while being in sight of the far side of the Moon. Because of their unstable dynamical behavior, these orbits require numerical integrations coupled with an optimization algorithm to properly compute their evolution.

In this communication, we investigate the performance of “synthetic” models, reconstructed from Fourier series whose coefficients are fitted over the range of the Halo family of orbits. The goal is to obtain accurate explicit models with simple representations. The possibilities given by such models are two-fold: facilitating the computation and representation of the orbits, and enabling analytic operations on the orbits, such as integration and derivation.

First, we numerically compute the orbits in the framework of the Restricted Three-Body Problem, and compute the Fourier series of the evolution of the barycentric rectangular coordinates x(t), y(t), and z(t). Since each Halo orbit can be indexed by a single parameter - the maximum x value, hereafter x0 - we fit the Fourier coefficients of x, y, z and the orbital frequency with respect to x0. Compared with simple polynomials and Chebyshev polynomials, fits with rational functions seem to offer the best performance.

We then compare the accuracy of several synthetic models and compute their maximum error with respect to the true orbits. We found that a model with 5 Fourier coefficients, using rational function fits with 6 coefficients (a total of 114 coefficients) guarantees an accuracy < 100 km for the farthest 8% fraction of the Halo family. A more complete model with 10 Fourier coefficients, using rational function fits with 11 coefficients (a total of 429 coefficients) guarantees an accuracy < 10 km for the farthest 21% fraction of the Halo family (accuracy < 1 km for the farthest 8% fraction). Note that a 10 km offset at this distance corresponds to 5 arcsec from Earth.

We also compare these synthetic models with published analytical models based on the Lindstedt-Poincare perturbative approach.