I know it is possible to infer the mass of a supermassive black hole (SMBH) by many methods, i.e., stellar orbits for out Galaxy, Iron line profile from the accretion disk, and probably other methods (perhaps from the spectrum of the radiation disk itself, that can be related with the central mass, if supposed thermal in origin). What I don't know is: 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)

  • $\begingroup$ Equations can be written with $\LaTeX$ enclosed in single dollar signs, or double dollar signs to put them in an extra line. $\endgroup$
    – Gerald
    Feb 17 '14 at 12:16
  • $\begingroup$ How far are the gas and/or the stars surronding the SMBH? If they are far enough from the central engine (that means out of the Schwartzschild radius in first approximation), we don't need to invoke a SMBH. $\endgroup$
    – Py-ser
    Feb 27 '14 at 5:46
  • 1
    $\begingroup$ In fact you do not need a SMBH to explain measurements taken at e.g. 100 pc (SMBH mass gives a density of 0.25 $M_\odot / pc^3$), but gas discs at radii as small as 5 pc give a density of $10^6 M_\odot / pc^3$, which is difficult to explain without a BH. You have to compare this with the Schwartzschild radius, which is $\approx 20 AU$ or 50000 smaller. $\endgroup$
    – astabada
    Feb 27 '14 at 11:29

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