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.