A Very Simple Cusped Halo Model

TL;DR

This paper introduces a simple dark halo model based on a transformation of the logarithmic potential, providing analytical formulas for kinematic quantities and applying it to stellar haloes and elliptical galaxies.

Contribution

It presents a new, simplified dark halo model with analytical kinematic formulas and demonstrates its application to galaxy and stellar halo dynamics.

Findings

Radial velocity dispersion tends to a constant in flat rotation curve galaxies.

Milky Way stellar halo is well-fit by an Einasto profile with index ~2.

The model explains the decline of velocity dispersion with radius in the Milky Way.

Abstract

We introduce a very simple model of a dark halo. It is a close relative of Hernquist's model, being generated by the same transformation but this time applied to the logarithmic potential rather than the point mass. The density is proportional to (distance)$^{-1}$ at small radii, whilst the rotation curve is flat at large radii. Isotropic and radially anisotropic distributions functions are readily found, and the intrinsic and line of sight kinematical quantities are available as simple formulae. We also provide an analytical approximation to the Hamiltonian as a function of the actions. As an application, we study the kinematic properties of stellar haloes and tracers in elliptical galaxies. We show that the radial velocity dispersion of a power-law population in a galaxy with a flat rotation curve always tends to the constant value. This holds true irrespective of the anisotropy or the lengthscales of the dark or luminous matter. An analogous result holds for the line of sight or projected velocity dispersion of a power-law surface brightness profile. The radial velocity dispersion of Population II stars in the Milky Way is a strongly declining function of Galactocentric radius. So, if the rotation curve is flat, we conclude that the stellar halo density cannot follow a power-law at large radii, but must decrease more sharply (like an Einasto profile) or be abruptly truncated at large radii. Both the starcount and kinematic data of the Milky Way stellar halo are well-represented by an Einasto profile with index $m\approx 2$ and effective radius $\approx 20$ kpc.