On inclination resonances in Artificial Satellite Theory

• The critical inclination is presented as the result of a resonance.
• The involved motions are the draconitic and anomalistic frequencies.
• The formulation is given in closed form without relying on averaging.

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.