Theoretical study of the partial derivatives produced by numerical integration of satellite orbits

For the two-body system Saturn-Mimas and the theoretical three-body non-resonant system Saturn-Mimas-Tethys we present a theoretical analysis of the behaviour of the partial derivatives of the satellites’ coordinates with respect to the parameters of the system, namely the satellites’ initial conditions and their mass-ratios over Saturn. With the use of Floquet theory for the stability of periodic orbitsweprove that all the partial derivatives have amplitudes that increase linearly with time. Theirmotion is a combination of periodicmotions the periods of which can also be accurately predicted by the theory. This theoreticalmodel can be used for checking the accuracy of the results of the different numerical integration methods used on satellite systems with the purpose of fitting the results to observations or analytical theories. On this basis, in the last part of the paper we extend the investigation of Hadjifotinou&Harper (1995) on the stability and efficience of the 10-order Gauss-Jackson backwarddifference and theRunge-Kutta-NyströmRKN12(10)17M methods by now applying them to the above mentioned threebody system.