  We use dwarf spheroidal galaxies as a tool to attempt to put precise lower
limits on the mass of the dark matter particle, assuming it is a sterile
neutrino. We begin by making cored dark halo fits to the line of sight velocity
dispersions as a function of projected radius (taken from Walker et al. 2007)
for six of the Milky Way's dwarf spheroidal galaxies. We test Osipkov-Merritt
velocity anisotropy profiles, but find that no benefit is gained over constant
velocity anisotropy. In contrast to previous attempts, we do not assume any
relation between the stellar velocity dispersions and the dark matter ones, but
instead we solve directly for the sterile neutrino velocity dispersion at all
radii by using the equation of state for a partially degenerate neutrino gas
(which ensures hydrostatic equilibrium of the sterile neutrino halo). This
yields a 1:1 relation between the sterile neutrino density and velocity
dispersion, and therefore gives us an accurate estimate of the Tremaine-Gunn
limit at all radii. By varying the sterile neutrino particle mass, we locate
the minimum mass for all six dwarf spheroidals such that the Tremaine-Gunn
limit is not exceeded at any radius (in particular at the centre). We find
sizeable differences between the ranges of feasible sterile neutrino particle
mass for each dwarf, but interestingly there exists a small range 270-280eV
which is consistent with all dSphs at the 1-$\sigma$ level.
