A Theorem on Central Velocity Dispersions

It is shown that, if the tracer population is supported by a spherical dark halo with a core or a cusp diverging more slowly than that of a singular isothermal sphere, the logarithmic cusp slope γ of the tracers must be given exactly by γ = 2β where β is their velocity anisotropy parameter at the center unless the same tracers are dynamically cold at the center. If the halo cusp diverges faster than that of the singular isothermal sphere, the velocity dispersion of the tracers must diverge at the center too. In particular, if the logarithmic halo cusp slope is larger than two, the diverging velocity dispersion also traces the behavior of the potential. The implication of our theorem on projected quantities is also discussed. We argue that our theorem should be understood as a warning against interpreting results based on simplifying assumptions such as isotropy and spherical symmetry. Subject headings: stellar dynamics — galaxies: halos — galaxies: kinematics and dynamics — dark matter