Studies in the application of recurrence relations to special perturbation methods

Abstract

Using the rectangular equations of motion for the restricted three-body problem a comparison is made of the Encke and Cowell methods of integration. Each set of differential equations is integrated using Taylor series expansions where the coefficients of the powers of time are determined by recurrence relationships. It is shown that for fairly highly eccentric orbits in which the perturbing force is less than one thousandth of the two-body force the Encke method achieves a considerable saving in machine time. This is also true for almost circular orbits when low or moderate accuracy is required. When very high accuracy is required, however, the Cowell method is faster unless the perturbing force is less than 10⁻⁶ of the two-body force. There is little difference in the accuracy of the two methods, the Cowell method being slightly more accurate when a low or moderate accuracy criterion is imposed.