Analysis of frozen conditions and optimal frozen orbit insertion

The frozen conditions of the orbits of Earth satellites are analyzed using a redundant set of elements, called epicycle elements. As well as avoiding the circular orbit singularity, working through the epicycle elements allows the higher geopotential harmonics to be included in the analytic computation up to an arbitrary number of terms until a satisfactory accuracy is achieved. This is important especially for orbits that are close to the critical inclination of 63.4 deg. An optimal single-burn strategy is analyzed for maneuvering a satellite from arbitrary initial conditions toward frozen conditions. It is shown that, while the orbital elements are varying, the minimum delta-V requirement will not vary with time. An optimal strategy is proposed for the frozen orbit transfer by using multiple small impulsive burns, which are preferred for modern autonomous orbit operations on small satellites. It is shown that the total delta-V required by a series of small burns in the strategy is not greater than the minimum value-required by the optimal single-burn strategy. This strategy is examined through the maneuvering of the UoSAT-12, a 300-kg minisatellite, toward frozen conditions by using its low-cost cold-gas thrusters.