ON THE ORIGIN OF CUSPS IN STELLAR-SYSTEMS

An origin is sought for the ubiquity of cusps, both in computer simula tions of halo formation in hierarchical clustering cosmogonies and in observations of galactic nuclei by the Hubble Space Telescope. The enc ounters of merging clumps that built the galaxies can be described by the collisional Boltzmann equation. Using insights gained by studying the simpler Fokker-Planck equation, we show that there is a steady-sta te, self-consistent, cusped solution of the collisional Boltzmann equa tion corresponding to rho similar to r(-4/3). This equilibrium is both stable and an attractor. It is the natural endpoint of the diffusive encounters of an ensemble of equal mass clumps. The introduction of a mass spectrum weakens the mass density cusp. The spike in the luminosi ty density can be accentuated or softened, depending on the form of th e mass-luminosity relation. Possible applications to the cusped nuclei of early-type galaxies are discussed.