Satellite Orbits and Relative Motion in Levi-Civita Coordinates

TL;DR

This paper develops a unified formalism for satellite trajectories in central force fields with quadratic drag using Levi-Civita coordinates, and derives equations for relative motion and orbit characterization.

Contribution

It introduces a novel unified coordinate system combining polar and Levi-Civita coordinates, enabling better analysis of satellite orbits and relative motion.

Findings

Derived a new orbit equation with energy and angular momentum as invariants

Established a linearized relative motion equation in Levi-Civita coordinates

Numerically verified the derived equations

Abstract

In this paper we consider satellite trajectories in central force field with quadratic drag using two formalisms. The first using polar coordinates in which the angular momentum plays a dominant role. The second is in Levi-Civita coordinates in which the energy plays a central role. We then unify these two formalisms by introducing polar coordinates in Levi-Civita space and derive a new equation for satellite orbits in which energy and and angular momentum are on equal footing {and thus characterize the orbit by its two invariants}. In the second part of the paper we derive in Levi-Civita coordinates a linearized equation for the relative motion of two satellites whose trajectories are in the same plane. We carry out also a numerical verification of these equations.