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Starting with the spherical Jeans equation, we show that there exists aradius where the mass enclosed depends only on the projected tangentialvelocity dispersion, assuming that the anisotropy profile slowly varies. Thisis well-approximated at the radius where the log-slope of the stellar tracerprofile 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 Plummerprofile $r_{-2} \simeq 4R_e/5$. Importantly, $r_{-2}$ is smaller than thecharacteristic radius for line-of-sight velocities derived by Wolf et al. 2010.Together, the two estimators can constrain the mass profiles ofdispersion-supported galaxies. We illustrate its applicability using publishedproper motion measurements of dwarf galaxies Draco and Sculptor, and find thatthey are consistent with inhabiting cuspy NFW subhalos of the kind predicted inCDM but we cannot rule out a core. We test our combined mass estimators againstpreviously-published, non-spherical cosmological dwarf galaxy simulations donein both CDM and SIDM. For CDM, the estimates for the dynamic rotation curvesare found to be accurate to $10\%$ while SIDM are accurate to $15\%$.Unfortunately, this level of accuracy is not good enough to measure slopes atthe level required to distinguish between cusps and cores of the type predictedin viable SIDM models without stronger priors. However, we find that thisprovides good enough accuracy to distinguish between the normalizationdifferences predicted at small radii ($r \simeq r_{-2} < r_{\rm core}$) forinteresting SIDM models. As the number of galaxies with internal proper motionsincreases, mass estimators of this kind will enable valuable constraints onSIDM and CDM models.

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