Optimal Smoothing for N-Body Codes

In any collisionless N-body code, there is an optimal choice for the smoothing parameter that minimizes the average error in the force evaluations. We show how to compute the optimal softening length in a direct-summation code and demonstrate that it varies roughly as 1/N^(1/3).Comment: 6 TEX pages, 3 PostScript figures, uses AASTEX, epsf.sty. Submitted to The Astronomical Journal, November 1995; revised January 199

The Dynamical Inverse Problem for Axisymmetric Stellar Systems

The standard method of modelling axisymmetric stellar systems begins from the assumption that mass follows light. The gravitational potential is then derived from the luminosity distribution, and a unique two-integral distribution function f(E,Lz) that generates the stellar density in this potential is found. We show that the gravitational potential can instead be generated directly from the velocity data in a two-integral galaxy, thus allowing one to drop the assumption that mass follows light. The rotational velocity field can also be recovered in a model-independent way. We present regularized algorithms for carrying out the inversions and test them by application to pseudo-data from a family of oblate models.Comment: 23 LATEX pages, 5 Postscript figures, uses AASTEX, epsf.sty. To appear in The Astronomical Journal, Vol. 112, September 199

Interaction of Supermassive Black Holes with their Stellar and Dark Matter Environments

A review of recent theoretical work on the interactions of supermassive single and binary black holes with their nuclear environments, highlighting ways in which the observed structure of nuclei can be used to constrain the formation history of black holes.Comment: 15 pages, to appear in "Growing Black Holes: Accretion in Cosmological Context

Gravitational Encounters and the Evolution of Galactic Nuclei. II. Classical and Resonant Relaxation

Direct numerical integrations of the Fokker-Planck equation in energy-angular momentum space are carried out for stars orbiting a supermassive black hole (SBH) at the center of a galaxy. The algorithm, which was described in detail in an earlier paper, includes diffusion coefficients that describe the effects of both random ("classical") and correlated ("resonant") encounters. Steady-state solutions are similar to the Bahcall-Wolf solution but are modified at small radii due to the higher rate of diffusion in angular momentum, which results in a low-density core. The core radius is a few percent of the influence radius of the SBH. The corresponding phase-space density f(E,L) drops nearly to zero at low energies, implying almost no stars on tightly-bound orbits about the SBH. Steady-state rates of stellar disruption are presented, and a simple analytic expression is found that reproduces the numerical feeding rates with good accuracy. The distribution of periapsides of disrupted stars is also computed. Time-dependent solutions are also computed, starting from initial conditions similar to those produced by a binary SBH. In these models, feeding rates evolve on two timescales: rapid evolution during which the region evacuated by the massive binary is refilled by angular-momentum diffusion; and slower evolution as diffusion in energy causes the density profile at large radii to attain the Bahcall-Wolf form.Comment: 19 pages, 10 figure