Accurate mass estimates from the proper motions of dispersion-supported galaxies

Starting with the spherical Jeans equation, we show that there exists a radius where the mass enclosed depends only on the projected tangential velocity dispersion, assuming that the anisotropy profile slowly varies. This is well-approximated at the radius where the log-slope of the stellar tracer profile is $-2$: $r_{-2}$. The associated mass is $M(r_{-2}) = 2 G^{-1} \langle \sigma_{\mathcal{T}}^{2}\rangle^{*} r_{-2}$ and the circular velocity is $V^{2}({r_{-2}}) = 2\langle \sigma_{\mathcal{T}}^{2}\rangle^{*}$. For a Plummer profile $r_{-2} \simeq 4R_e/5$. Importantly, $r_{-2}$ is smaller than the characteristic radius for line-of-sight velocities derived by Wolf et al. 2010. Together, the two estimators can constrain the mass profiles of dispersion-supported galaxies. We illustrate its applicability using published proper motion measurements of dwarf galaxies Draco and Sculptor, and find that they are consistent with inhabiting cuspy NFW subhalos of the kind predicted in CDM but we cannot rule out a core. We test our combined mass estimators against previously-published, non-spherical cosmological dwarf galaxy simulations done in both CDM and SIDM. For CDM, the estimates for the dynamic rotation curves are found to be accurate to $10\%$ while SIDM are accurate to $15\%$. Unfortunately, this level of accuracy is not good enough to measure slopes at the level required to distinguish between cusps and cores of the type predicted in viable SIDM models without stronger priors. However, we find that this provides good enough accuracy to distinguish between the normalization differences predicted at small radii ($r \simeq r_{-2} < r_{\rm core}$) for interesting SIDM models. As the number of galaxies with internal proper motions increases, mass estimators of this kind will enable valuable constraints on SIDM and CDM models.