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We present a new method for studying tracers in gravitational systems wherehigher moments of the line-of-sight velocity distribution are introduced viaVirial equations rather than the Jeans equations. Unlike the fourth order Jeansequations, the fourth order Virial equations can simply be added to thestandard second order Jeans equation without introducing a new anisotropyparameter $\beta^{\prime}$. We introduce two new global shape parameters$\zeta_A$ and $\zeta_B$ which replace the kurtosis as a more statisticallyrobust measure of the shape of the line of sight velocity distribution. We showthat in the case of stars in dwarf spheroidal galaxies these new parameters cansignificantly reduce the range of density profiles that are otherwiseconsistent with the observed stellar kinematics (a problem sometimes known asthe $\beta$ degeneracy). Specifically, we find that $\zeta_A$ focuses tightlyon a subset of solutions where cusped density profiles are degenerate with moreconcentrated cored dark matter halos. If the number density of stars $\nu(r)$is fixed, then introducing $\zeta_B$ can further reduce the space of solutionsby constraining the outer slope of the dark matter density profile. Assuming aPlummer profile for $\nu(r)$ we recover the surprising result that the darkmatter in Sculptor may be cuspy after all, in contrast to the conclusions ofother approaches.

On the dark matter profile in Sculptor: Breaking the $\beta$ degeneracy with Virial shape parameters