Institute for Theoretical Astronomy, St. Petersburg, Russia
The seventh satellite of Saturn named Hyperion have the strong attraction influence from the nearest to it sixth Saturn's satellite Titan. Titan is also the most massive satellite in the Saturn's satellite system. Hyperion's and Titan's orbits are so close one to another that the ratio of the semi-major axes of the satellites is not lesser than 0.82. These facts lead to the great difficulties in constructing the analytical theories of motion of Hyperion---e.g. according the Sundman's criterion we haven't right to use the usual analytical developments with respect to the powers of eccentrities to represent the perturbation function in this case.
I used a rough orbit obtained by numerical integration as a reference orbit for integrating the motion of Hyperion by the Encke's method. To this end I represented the rough orbit by a Fourier series describing the rectangular saturnocentric coordinates of the satellite. Then by simple differentiation I obtained the velocities and accelerations needed in the Encke's method. This technique allowed me to use greater integration steps than in the case of straight numerical integration at the same accuracy, which is an obvious advantage from the point of view of computer time saving.