Numerical integration of the variational equations of satellite orbits

A recurrent power series (RPS) method is constructed for the numerical integration of the equations of motion together with the variational equations of N point masses orbiting around an oblate spheroid. By the term “variational equations” we mean the equations of the partial derivatives of the bodies’ position and velocity components with respect to the initial conditions, the relative masses and the spheroidʹs oblateness coefficients J2 and J4. The construction of recursive relations for the partial derivatives involved in the variational equations is based on partial differentiation of the corresponding recursive relations for the integration of the equations of motion. Since the number of the auxiliary variables needed for this complex system becomes tremendously large when N>1, special care must be taken during computer implementation, so as to minimize the amount of computer memory needed as well as the cost in CPU time. The RPS method constructed in this way is tested for N=1,…,4 using real initial conditions of the Saturnian satellite system. For various sets of satellites, we monitor the behaviour of all the corresponding partial derivatives. The results show a prominent difference in the behaviour of the partial derivatives between resonant and non-resonant orbital systems.