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

Understand

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).