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