HSTPROMO Internal Proper Motion Kinematics of Dwarf Spheroidal Galaxies: II. Velocity Anisotropy and Dark Matter Cusp Slope of Sculptor

We analyze three epochs of HST imaging over 20 years for the Sculptor dwarf spheroidal galaxy, measuring precise proper motions for 119 stars and combining them with 1760 existing line-of-sight velocities. This catalog yields the first radially-resolved 3D velocity dispersion profiles for Sculptor. We confirm mild oblate rotation, with major-axis velocities reaching $\sim 2$ km s$^{-1}$ beyond 20.0 arcmin. Using a methodology similar to that in the first paper in this series, we solve the Jeans equations in oblate axisymmetric geometry to infer the galaxy's mass profile. Our modeling reveals a significant degeneracy due to the unknown galaxy inclination, which is overlooked under spherical symmetry assumptions. This degeneracy allows acceptable fits across a range of dark matter profiles, from cuspy to cored. While we do not directly constrain the inclination with our Jeans models, higher-order line-of-sight velocity moments provide useful additional constraints: comparisons with scalefree models from de Bruijne et al. (1996) favor highly flattened (more face-on) configurations. Adopting an inclination well consistent with these comparisons ($i = 57.1$ degrees), we find, alongside radial velocity anisotropy, a dark matter density slope of $Γ_{\rm dark} = 0.29^{+0.31}_{-0.41}$ within the radial extent of the 3D velocity data, ruling out a cusp with $Γ_{\rm dark} \leq -1$ at 99.8% confidence. This confidence increases for lower inclinations and decreases drastically for nearly edge-on configurations. The results qualitatively agree with $Λ$CDM, SIDM, and Fuzzy DM scenarios that predict core formation, while our specific measurements provide quantitative constraints on the prescriptions of feedback, cross sections, or particle masses required by these models, respectively.