ORBITAL DEFLECTIONS IN N-BODY SYSTEMS

We discuss two types of orbital deflection in N-body systems with a softened field: two-body relaxation and orbital divergence. We study two-body relaxation using tree code N-body simulations of King models with N ranging from 10(2) to 4 x 10(5), and a softening length, epsilon, ranging from 2 x 10(-4) to 7.5 x 10(-2) in units of the half-mass radius. The relaxation time, T(E), depends linearly on N for a fixed softening length, i.e., T(E) is-proportional-to N in contrast to the usually quoted N/ln N dependence. The relaxation time also depends on epsilon as, T(E) is-proportional-to 1/ln (R/epsilon), where R is the radius of the system. At the half-mass radius, T(E) is 0.6-1.0 times the local orbit-averaged Fokker-Planck estimate. The energy evolution of particles is a pure random walk or diffusion process as shown by the constant rate of change of the squared energy fluctuation, SIGMA (DELTAE(i)2)/DELTAt, and the particles' energy power spectrum follows a power law with spectral index -2. The diffusion time is nearly equal to the relaxation time. We study orbital divergence using N-body simulations with N = 2.5 x 10(2) to 4 x 10(3) and epsilon = 0.0125 to 0.1. The orbital divergence follows the exponential law, DELTA(t) = DELTA(0) exp (t/T(e)), where T(e) is the e-folding time and, DELTA(0) is the initial phase separation. The averaged e-folding time is comparable with the crossing time, T(c), and varies as N(alpha)epsilon(beta), where alpha and beta measured from numerical simulations are 0.26 and 0.45, respectively. The deviation of alpha and beta from Goodman, Heggie, and Hut's values of 1/3 and 2/3 is discussed.