Recent Developments of Integrating the Gravitational Problem of N-Bodies

Abstract

This paper discusses the formulation and the numerical integration of large systems of differential equations occurring in the gravitational problem of n-bodies.

Different forms of the pertinent differential equations of motion are presented, and various regularizing and smoothing transformations are compared. Details regarding the effectiveness and the efficiency of the Kustaanheimo-Stiefel and of other methods are discussed. In particular, a method is described in which some of the phase variables are treated in the regularized system and others in the ordinary system. This mixed method of numerical regularization offers some advantages.

Several numerical integration techniques are compared. A high order Runge-Kutta method, Steffensen’s method, and a finite difference method are investigated, especially with regard to their adaptability to regularization.

The role of integrals and integral invariants is displayed in controlling the accuracy of the numerical integration.

Numerical results are described with 5, 25 and 500 bodies participating. These examples compare the various integration techniques, several regularization methods and different logics in treating binaries.