Evaluating galaxy dynamical masses from kinematics and jeans equilibrium in simulations

We provide prescriptions to evaluate the dynamical mass ($M_{\rm dyn}$) of galaxies from kinematic measurements of stars or gas using analytic considerations and the VELA suite of cosmological zoom-in simulations at $z=1-5$. We find that Jeans or hydrostatic equilibrium is approximately valid for galaxies of stellar masses above $M_\star \!\sim\! 10^{9.5}M_\odot$ out to $5$ effective radii ($R_e$). When both measurements of the rotation velocity $v_\phi$ and of the radial velocity dispersion $\sigma_r$ are available, the dynamical mass $M_{\rm dyn} \!\simeq\! G^{-1} V_c^2 r$ can be evaluated from the Jeans equation $V_c^2= v_\phi^2 + \alpha \sigma_r^2$ assuming cylindrical symmetry and a constant, isotropic $\sigma_r$. For spheroids, $\alpha$ is inversely proportional to the S\'ersic index $n$ and $\alpha \simeq 2.5$ within $R_e$ for the simulated galaxies. The prediction for a self-gravitating exponential disc, $\alpha = 3.36(r/R_e)$, is invalid in the simulations, where the dominant spheroid causes a weaker gradient from $\alpha \!\simeq\! 1$ at $R_e$ to 4 at $5R_e$. The correction in $\alpha$ for the stars due to the gradient in $\sigma_r(r)$ is roughly balanced by the effect of the aspherical potential, while the effect of anisotropy is negligible. When only the effective projected velocity dispersion $\sigma_l$ is available, the dynamical mass can be evaluated as $M_{\rm dyn} = K G^{-1} R_e \sigma_l^2$, where the virial factor $K$ is derived from $\alpha$ given the inclination and $v_\phi/\sigma_r$. We find that the standard value $K=5$ is approximately valid only when averaged over inclinations and for compact and thick discs, as it ranges from 4.5 to above 10 between edge-on and face-on projections.