A theorem on central velocity dispersions

TL;DR

This paper presents a theorem relating the central velocity dispersion and anisotropy of tracer populations within spherical dark halos, highlighting conditions for finite dispersions and implications for astrophysical modeling.

Contribution

It provides a precise mathematical relation between velocity anisotropy and density cusp slope in spherical dark matter halos, clarifying limitations of common simplifying assumptions.

Findings

For halos with shallow cusps, the tracer cusp slope equals twice the anisotropy parameter.

In halos with steep cusps, the tracer velocity dispersion diverges at the center.

The theorem warns against over-reliance on isotropy and spherical symmetry assumptions.

Abstract

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 'g' of the tracers must be given exactly by g=2b where b 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.