Stellar kinematics

The previous method was not perfect, since we did not take into account that $\sigma$ may vary. Now close enough to the MDO where it dominates the mass, we should be able to detect the Keplerian rise in velocity dispersion $\sigma\propto r^{-1/2}$. So people tried to determine $\sigma(r)$ close to the centre.

However again there is a loop hole: $\sigma(r)$ can grow as well in case the velocity distribution becomes anisotropic, e.g. in spherical coordinates $\sigma^2_r\ne\sigma^2_\theta\ne\sigma^2_\phi$. Again using the Jeans' equations, the radial variation in mass can be related to the stellar velocity dispersion as^11.11

$$M(<r) = {V^2\,r\over G} + {\sigma^2_r\,r\over G}$$

$$\times \left[-{d\ln\nu\sigma^2_r\over dr}+({\sigma^2_\theta+\sigma^2_\phi\over\sigma^2_r}-2)\right]\,.$$ (11.2)

$V$ is the circular velocity. Clearly, if we were allowed to choose the three components of $\sigma^2$, we could balance a large increase in density with the velocity tensor becoming more anisotropic, even in the absence of a MDO.

Observed galaxies do indeed show a large increase in $\sigma$ close to the centre, but we cannot rule out that this is due to anisotropic velocities, and not to the presence of a MDO. So once more, the evidence is a bit circumstantial.