State Transition Matrix (non-spherical Earth)

Computation of state transition matrix for desired degree and order of gravity field of the Earth.

Even though the state transition matrix of the two-body problem is a reasonable approximation of the actual transition matrix, it is sometimes desirable to take into account at least the major perturbations in the computation. As with the treatment of the perturbed satellite motion, one may not, however, obtain an analytical solution anymore in this case, but has to solve a special set of differential equations - the variational equations - by numerical methods. Aside from the increased accuracy that may be obtained by accounting for perturbations, the concept of the variational equations offers the advantage that it is not limited to the computation of the state transition matrix, but may also be extended to the treatment of partial derivatives with respect to force model parameters.
References:
Cunningham L. E.; On the Computation of the Spherical Harmonic Terms needed during the Numerical1ntegration of the Orbital Motion of an Artificial Satellite; Celestial Mechanics 2, 207-216 (1970).
Montenbruck, O., Gill, E.; Satellite Orbits - Models, Methods, and Applications; Springer-Verlag, Berlin-Heidelberg (2005).