On Inclination Resonances in Artificial Satellite Theory

TL;DR

This paper investigates the mathematical origins of the critical inclination in artificial satellite orbits, revealing how orbital parameters and Earth's oblateness influence this resonance phenomenon.

Contribution

It derives a closed-form expression linking inclination and frequency ratios, clarifying the resonance mechanism at the core of the critical inclination.

Findings

Critical inclination results from frequency commensurability.

Oblateness and orbit size/shaping slightly modify the inclination.

The resonance persists even with short-period effects.

Abstract

The frozen-perigee behavior of elliptic orbits at the critical inclination is usually displayed after an averaging procedure. However, this singularity in Artificial Satellite Theory manifests also in the presence of short-period effects. Indeed, a closed form expression relating orbital inclination and the ratio anomalistic draconitic frequencies is derived for the main problem, which demonstrates that the critical inclination results from commensurability between the periods with which the radial and polar variables evolve in the instantaneous plane of motion. This relation also shows that the critical inclination value is slightly modified by the degree of oblateness of the attracting body, as well as by the orbit's size and shape.