On the dark matter profile in Sculptor: Breaking the $\beta$ degeneracy
  with Virial shape parameters
  We present a new method for studying tracers in gravitational systems where
higher moments of the line-of-sight velocity distribution are introduced via
Virial equations rather than the Jeans equations. Unlike the fourth order Jeans
equations, the fourth order Virial equations can simply be added to the
standard second order Jeans equation without introducing a new anisotropy
parameter $\beta^{\prime}$. We introduce two new global shape parameters
$\zeta_A$ and $\zeta_B$ which replace the kurtosis as a more statistically
robust measure of the shape of the line of sight velocity distribution. We show
that in the case of stars in dwarf spheroidal galaxies these new parameters can
significantly reduce the range of density profiles that are otherwise
consistent with the observed stellar kinematics (a problem sometimes known as
the $\beta$ degeneracy). Specifically, we find that $\zeta_A$ focuses tightly
on a subset of solutions where cusped density profiles are degenerate with more
concentrated cored dark matter halos. If the number density of stars $\nu(r)$
is fixed, then introducing $\zeta_B$ can further reduce the space of solutions
by constraining the outer slope of the dark matter density profile. Assuming a
Plummer profile for $\nu(r)$ we recover the surprising result that the dark
matter in Sculptor may be cuspy after all, in contrast to the conclusions of
other approaches.
