Change of Inclination in Passing through Resonance

Abstract

The Earth’s gravity anomalies can give rise to significant changes in inclination for a close satellite when its motion is commensurable with the Earth’s rotation. For a decaying satellite passing through resonance, the change in inclination depends on the value of a resonant variable at exact commensurability (an essentially random quantity). Many different gravity coefficients may contribute significantly, with relative amplitudes which are highly dependent on inclination.

An approximate solution can be found in terms of Fresnel integrals, which is valid in the usual situation when the drag greatly exceeds the resonance forces. This shows that the inclination is almost equally likely to increase or decrease. and that the total change is proportional to (drag)^−1/2, i.e. to the time taken to pass through resonance.

The effect offers a way of deriving gravity coefficients of medium order (e.g. m=15) from the observed magnitude and shape of the variation in inclination. The magnitudes of even higher order gravity coefficients obtained from some resonance with α=2 (e.g. 29/2) or even with α=3 (e.g. 44/3) might yield information on the depth of the sources of the high order gravity field.