On the astrodynamics applications of Weierstrass elliptic and related functions

TL;DR

This paper reviews how Weierstrass elliptic functions provide explicit solutions to classical astrodynamics problems and discusses their applications in trajectory design and satellite motion.

Contribution

It introduces a technique using Weierstrass elliptic functions for solving key astrodynamics problems and explores their practical applications.

Findings

Explicit solutions for classical problems using elliptic functions

Application to low-thrust planetary fly-bys

Modeling satellite motion with gravitational harmonics

Abstract

Weierstrass elliptic and related functions have been recently shown to enable analytical explicit solutions to classical problems in astrodynamics. These include the constant radial acceleration problem, the Stark problem and the two-fixed center (or Euler's) problem. In this paper we review the basic technique that allows for these results and we discuss the limits and merits of the approach. Applications to interplanetary trajectory design are then discussed including low-thrust planetary fly-bys and the motion of an artificial satellite under the influence of an oblate primary including $J_2$ and $J_3$ harmonics.