Aerodynamic Stability of Satellites in Elliptic Low Earth Orbits

TL;DR

This paper analyzes the aerodynamic stability of satellites in elliptic low Earth orbits by modeling their motion with elliptic functions and Hill's equation, aiming to identify stable parameter ranges for low altitude satellite operations.

Contribution

It introduces a novel stability analysis method for satellites in elliptic low Earth orbits using Jacobi elliptic functions and Floquet theory, extending to potential computational tools.

Findings

Identifies parameter ranges for stable satellite motion

Develops a mathematical framework for stability analysis

Proposes a computational approach for satellite stability evaluation

Abstract

Topical observations of the thermosphere at altitudes below $200 \, km$ are of great benefit in advancing the understanding of the global distribution of mass, composition, and dynamical responses to geomagnetic forcing, and momentum transfer via waves. The perceived risks associated with such low altitude and short duration orbits has prohibited the launch of Discovery-class missions. Miniaturization of instruments such as mass spectrometers and advances in the nano-satellite technology, associated with relatively low cost of nano-satellite manufacturing and operation, open an avenue for performing low altitude missions. The time dependent coefficients of a second order non-homogeneous ODE which describes the motion have a double periodic shape. Hence, they will be approximated using Jacobi elliptic functions. Through a change of variables the original ODE will be converted into Hill's ODE for stability analysis using Floquet theory. We are interested in how changes in the coefficients of the ODE affect the stability of the solution. The expected result will be an allowable range of parameters for which the motion is dynamically stable. A possible extension of the application is a computational tool for the rapid evaluation of the stability of entry or re-entry vehicles in rarefied flow regimes and of satellites flying in relatively low orbits.