supermassive black hole - How can we infer the mass of SMBH in galaxies that are not active anymore?

I can think of two methods.

Both rely on the dynamics of material surrounding the SMBH, which is affected up to a distance of the order of the "sphere of influence". This is the region where the BH dominates the dynamics as compared to the enclosed mass of the galaxy. The sphere of influence is:

$$ R_{mathrm{infl}} equiv frac{G M_{mathrm{SMBH}}}{sigma_{mathrm{bulge}}^2} >> R_{mathrm{horizon}} approx R_{mathrm{Schwartzschild}} equiv frac{G M_{mathrm{SMBH}}}{c^2} $$

While typically $sigma_{mathrm{bulge}} approx 250 km s^{-1}$, it is well known that $c approx 300000 km s^{-1}$. This means that the influence of a BH can be felt much further away than its event horizon, which is where the accretion takes place. In fact the ratio between the two distances is about 1000000.

Exploiting this fact, astronomers have been using two methods to probe extragalactic*, quiescent SMBH:

• The first method is to observe CO lines (radio astronomy) to trace gas circling the BH. The gas does not need to be near the event horizon, which is much smaller than the sphere of influence. In fact CO observations rely on the gas being relatively dense but cold. Essentially the speed at which the CO gas will rotate is a (quadrature) sum of the declining component due to the stellar mass, plus the Keplerian component due to SMBH mass.


• The second method is completely analogous, but relies on measuring the unresolved kinematics of the stars surrounding the SMBH. This can be done in various bands, but most authors use visible line absorptions to measure the velocity and velocity dispersion (and other moments) of the stars in the regions surrounding the BH. If the kinematics cannot be explained without including a point mass in the middle, then you are done.

See this work, for a comparison of the two methods.

*(extragalactic means outside of our own galaxy)