Solutions to the Variational Equations for Relative Motion of Satellites

The use of integrals of certain reference satellite motions has played an important role in many classical investigations. As a principal example, the solution to the two-body problem is the foundation of Lagrange's planetary equations. The two-body solution has also been used to find solutions that satisfy the associated linear differential "variational" equations that form the basis of certain guidance schemes without additional integration by expanding the solution about a nominal trajectory. These guidance equations have traditionally been written in a nonrotating coordinate system. Recently, solutions to the two-body (and the perturbed two-body) variational equations written in a rotating coordinate system have been found by linearizing solutions to nonlinear equations of relative motion obtained from the two two-body problem solutions. In those solutions, classical and modified orbital elements are varied. Here, the general problem of finding solutions to the variational equations of dynamical systems that are completely integrable is first considered. Then, an alternate solution for linear relative motion of a satellite with respect to another satellite moving in an elliptic orbit is obtained in the form of an analytical state transition matrix by varying the initial polar coordinates, the inclination, and the right ascension of the ascending node of the reference orbit in the analytical solution to the nonlinear equations. Only the inverse of a relatively simple matrix is required at initial time and the new transition matrix is valid for arbitrary elliptic orbits. Examples of relative motion, obtained by evaluating the analytical solution, are presented and are compared with numerical results.