Active Galaxies and Supermassive Black Holes
Stellar speeds weigh whole galaxies, and one star on a 16 year orbit weighs the four million solar mass black hole at the center of the Milky Way. The unit then explains how accreting black holes power Seyfert galaxies, quasars and jets, how the Event Horizon Telescope imaged two black hole shadows, and how black holes and clusters shape the galaxies around them.
1 The dynamics of galaxies
Nobody can watch a galaxy rotate: the Sun needs 230 million years for one lap around the Milky Way. A spectrum does better than an image, because the Doppler shift of Unit 3 turns it into a speedometer. Gravity is the only thing that can hold together matter moving at hundreds of kilometers per second, so a speed and a size give a mass. The same argument, applied at smaller and smaller radii, ends at a black hole.
1.1 Rotation curves and the Tully-Fisher relation
Stars and gas in the disk of a spiral galaxy move on nearly circular orbits. Gravity supplies the centripetal acceleration, \(v^2/r = GM/r^2\), and the mass inside the radius \(r\) is \begin{equation} M(<r) = \frac{v^2 r}{G}. \label{eq:mrot} \end{equation} Unit 8 followed \(v(r)\) outward and found the flat rotation curves that are the classic evidence for dark matter. For galaxies it is handy to write \(G = 4.30\times10^{-6}\unit{kpc\,(km\,s^{-1})^2}\,\Msun^{-1}\).
In 1977 Brent Tully and Richard Fisher found that the rotation speed also predicts the luminosity of a spiral, \(L \propto v^4\), with an exponent between 3 and 4 that depends on the wavelength band. The speed comes cheaply from the width of the 21 cm line of hydrogen, which is broadened because one side of the disk approaches while the other recedes. A rough argument shows where the steep power comes from. Equation \eqref{eq:mrot} at the edge of the disk, radius \(R\), gives \(M \propto v^2R\). If all disks had the same ratio of mass to light and the same surface brightness \(L/R^2\), then \(R \propto L^{1/2}\) and \(L \propto v^2L^{1/2}\), which is \(L \propto v^4\). Astronomers use the Tully-Fisher relation to measure distances: a line width looks the same from any distance, it gives \(L\), and \(L\) compared with the measured flux gives the distance (Units 8 and 13).
1.2 Velocity dispersion and virial masses
Elliptical galaxies and the bulges of spirals barely rotate. Their stars move on orbits of all orientations, like bees in a swarm, and the spectrum of the galaxy is the sum of billions of stellar spectra with different Doppler shifts. Every absorption line comes out smeared. The width of the smearing is the velocity dispersion \(\sigma\), the standard deviation of the velocities along the line of sight. Small ellipticals have about \(70\unit{km\,s^{-1}}\) and the giants reach \(350\unit{km\,s^{-1}}\). The virial theorem of Unit 9, \(2K + U = 0\) for the kinetic energy \(K\) and the potential energy \(U\) of a system in equilibrium, turns \(\sigma\) into a mass.
A giant elliptical with \(\sigma = 300\unit{km\,s^{-1}}\) and \(R_e = 8\unit{kpc}\) has \(M \approx 5\times300^2\times8/(4.30\times10^{-6}) = 8.4\times10^{11}\,\Msun\). In the Coma cluster the galaxies move with \(\sigma \approx 1000\unit{km\,s^{-1}}\) inside \(R \approx 1.5\unit{Mpc}\), which gives \(2\times10^{15}\,\Msun\), far more than its stars can supply (Fritz Zwicky’s argument for dark matter in 1933). Ellipticals also have their own version of the Tully-Fisher relation, the Faber-Jackson relation \(L \propto \sigma^4\) of 1976. In short, \(\sigma\) measures how deep the gravitational well of a galaxy is, and Section 3 shows that the central black hole knows this number.