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

Understand

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 $r_{-2} \simeq 4R_e/5$.