# Construction of J2-Invariant Periodic Relative Motion in Highly Elliptical Orbits

## TL;DR

This paper derives analytical formulas for constructing J2-invariant periodic relative orbits in highly elliptical orbits, enabling diverse geometries despite limited degrees of freedom.

## Contribution

It introduces new analytical methods to design various relative orbit geometries under J2 perturbation with limited orbital element differences.

## Key findings

- Derived formulas for relative orbit differences based on desired geometries.
- Validated geometries include in-track-centered, quadrant-time-centered, and others.
- Demonstrated the feasibility of diverse orbit configurations with limited orbital element variations.

## Abstract

Two satellites with mean orbital elements which differ only in terms of right ascension of the ascending node, argument of perigee, and mean anomaly are notable for having the same mean orbital element secular drift rates due to the J2 perturbation. The relative orbits which result from this configuration are discounted in the literature for not providing sufficiently many degrees of freedom with which to design relative orbit geometries suitable for real world missions. However, this paper will explore a variety of useful geometries which result from this limited design space, and provide analytical formulas for the differences between chief and deputy mean orbital elements as a function of the desired orbit geometry and inertial chief orbit. These include in-track-centered, quadrant-time-centered, offset-circular, rectilinear, boomerang-shaped, and cross-track only geometries.