The equations of motion of a secularly precessing elliptical orbit

TL;DR

This paper derives a complete set of equations of motion for a secularly precessing elliptical orbit using time as the independent variable, facilitating numerical integration of perturbations in orbital mechanics.

Contribution

It presents a new, fully time-based formulation of the equations of motion for precessing elliptical orbits, improving upon previous models that used anomaly-based variables.

Findings

Provides a complete set of six equations in time for precessing elliptical orbits.

Reformulates existing equations for simplicity and completeness.

Enhances numerical integration of orbital perturbations.

Abstract

The equations of motion of a secularly precessing ellipse are developed using time as the independent variable. The equations are useful when integrating numerically the perturbations about a reference trajectory which is subject to secular perturbations in the node, the argument of pericenter and the mean motion. Usually this is done in connection with Encke's method to ensure minimal rectification frequency. Similar equations are already available in the literature, but they are either given based on the true anomaly as the independent variable, or in mixed mode with respect to the time through the use of a supporting equation to track the anomaly. The equations developed here form a complete and independent set of six equations in the time. Reformulations both of Escobal's and Kyner and Bennett's equations are also provided which lead to a more concise form.