Numerical Stability of a Family of Osipkov‐Merritt Models

Abstract

We have investigated the stability of a set of nonrotating anisotropic spherical models with a phase-space distribution function of the Osipkov-Merritt type. The velocity distribution in these models is isotropic near the center and becomes radially anisotropic at large radii. The models are special members of the family studied by Dehnen and by Tremaine et al. in which the mass density has a power-law cusp ρ ∝ r-γ at small radii and decays as ρ ∝ r-4 at large radii. The radial-orbit instability of models with γ = 0, 1/2, 1, 3/2, and 2 was studied using an N-body code written by one of us and based on the self-consistent field method developed by Hernquist & Ostriker. These simulations have allowed us to delineate a boundary in the (γ, ra)-plane that separates the stable from the unstable models. This boundary is given by 2Tr/Tt = 2.31 ± 0.27 for the ratio of the total radial to tangential kinetic energy. We also found that the stability criterion df/dQ ≤ 0, recently raised by Hjorth, gives lower values compared with our numerical results. The stability to radial modes of some Osipkov-Merritt γ-models that fail to satisfy the Doremus-Feix criterion ∂f/∂E < 0 has been studied using the same N-body code but retaining only the l = 0 terms in the potential expansion. We have found no signs of radial instabilities for these models.