On the Sensitivity of the NBody Problem to Small Changes in Initial Conditions
Abstract
This paper summarizes the results of a considerable number of numerical simulations which investigate the instability of the gravitational Nbody problem towards small, "random" changes in the initial positions of the particles. The specific aims are: (1) to follow the growths of the perturbations of individual particles in the simulation, analyzing their statistical properties; (2) to investigate how these statistical properties vary as the number of particles N in the simulation is changed from 16 to 340; and (3) to determine how these properties vary if one changes the 'softening parameter'. Principal conclusions obtained for the simulations considered include the following: (1) The mean efolding time tasterisk for the perturbation in the location of some particle is comparable to the dynamical, or crossing, time t(cr) for all N. (2) There is, however, a weak Ndependence, tasterisk t(cr) decreasing as N increases but tending toward an asymptotic value for large N. (3) The distribution of efolding times for a given N has a distinctive and comparatively reproducible shape, the reproducibility improving as N grows. (4) Increasing the 'softening parameter' leads to a systematic increase in tasterisk, but this time scale remains comparable to t(cr) even when the 'softening parameter' becomes so large as to be comparable to the interparticle spacing.
 Publication:

The Astrophysical Journal
 Pub Date:
 June 1991
 DOI:
 10.1086/170114
 Bibcode:
 1991ApJ...374..255K
 Keywords:

 Computerized Simulation;
 Many Body Problem;
 Numerical Stability;
 Robustness (Mathematics);
 Systems Stability;
 Body Kinematics;
 Computational Astrophysics;
 Numerical Analysis;
 Astrophysics;
 NUMERICAL METHODS