Rotational locks for gravity gradient satellites

Abstract

Locked-in planar rotational motion for satellites moving in a gravity gradient field is examined using both analytical and numerical techniques. It is shown that rotational locks at spin rates of n/2 (where n is an integer) satellite rotations per orbit revolution exist for specific combinations of satellite inertia properties and orbital eccentricity. For nearly axial symmetric satellites, the maximum and minimum instantaneous rates which permit the satellite to remain in a particular rotational lock are found analytically by applying the averaging techniques of Kryloff and Bogolinboff and that of Symon. For these cases, it is found that the strength of the higher rotational locks (n > 3) are greater than the strength of the n = 2 or 1/1 rotational lock for proper combinations of lock number, n, and orbital eccentricity. Comparison of the results for the case of the planet Mercury are shown to be in good agreement with both observations of the planet and the 2 numerical calculations of Liu. Numerical results were obtained for representative values throughout the range of satellite inertia properties. Periodic solutions of periods 2π and 4 π are found and their variational stability investigated by Floquet analysis. The results which are presented on stability charts show that for satellites that deviate appreciably from axial symmetry, the stable periodic solutions occur at eccentricities which tend to increase as the absolute value of the lock number |n| increases. Estimates of the strength of these rotational locks are found by applying the phase space technique of Brereton and Modi. For nearly axially symmetric satellites, the results of this technique agreed favorably with the analytical results. Rotational locks for satellites that are not nearly axial symmetric were found in general to be considerably weaker than the more frequently investigated 1/1 lock.