Virial Sequences for Thick Discs and Haloes: Flattening and Global Anisotropy

Abstract

The virial theorem prescribes the ratio of the globally-averaged equatorial to vertical velocity dispersion of a tracer population in spherical and flattened dark haloes. This gives sequences of physical models in the plane of global anisotropy and flattening. The tracer may have any density, though there are particularly simple results for power-laws and exponentials. We prove the flattening theorem: for a spheroidally stratified tracer density with axis ratio q in a dark density potential with axis ratio g, the ratio of globally averaged equatorial to vertical velocity dispersion depends only on q/g. As the stellar halo density and velocity dispersion of the Milky Way are accessible to observations, this provides a new method for measuring the flattening of the dark matter. If the kinematics of the local halo subdwarfs are representative, then the Milky Way's dark halo is oblate with a flattening in the potential of g ~ 0.85, corresponding to a flattening in the dark matter density of ~ 0.7. The fractional pressure excess for power-law populations is roughly proportional to both the ellipticity and the fall-off exponent. Given the same pressure excess, if the density profile of one stellar population declines more quickly than that of another, then it must be rounder. This implies that the dual halo structure claimed by Carollo et al. (2007) for the Galaxy, a flatter inner halo and a rounder outer halo, is inconsistent with the virial theorem. For the thick disc, we provide formulae for the virial sequences of double-exponential discs in logarithmic and Navarro-Frenk-White (NFW) haloes. There are good matches to the observational data on the flattening and anisotropy of the thick disc if the thin disc is exponential with a short scalelength ~ 2.6 kpc and normalisation of 56 solar masses per square parsec, together with a logarithmic dark halo.