5.1. Search techniques for supermassive black holes
The actual searches for supermassive black holes rely primarily on stellar- and gas-dynamical evidence (see, e.g., Kormendy and Richstone 1995, for a review). In particular, the idea is to unambiguously show that the mass-to-light ratio, M/L, increases toward the galactic center to values that are difficult to accomodate with other types of stellar populations. Ideally, one would want to follow this up with the detection of relativistic speeds, but at present even HST cannot resolve orbits at a few gravitational (Schwarzschild) radii (RS = 2GMBH / c2; where MBH is the black hole mass, G is the gravitational constant and c is the speed of light).
Generally, the basic principle behind the early stellar-dyamical search techniques can be explained by the following, simplified picture. Taking the first velocity moment of the collisionless Boltzmann equation gives
where M(r) is the mass enclosed within radius r, Vrot is the rotational velocity, r, , are the components of the velocity dispersion, and t is the density of the tracer stars being observed (usually assumed to be proportional to the volume brightness). A direct measurement of rot and , therefore can determine the central mass. To actually use eq. 15, however, the ranges of unprojected quantities (e.g., rot, ) need to be derived, and various techniques to achieve that have been developed (see for example, Kormendy 1988, Dressler and Richstone 1988, van der Marel 1994, Gerhard 1993).
The more recent search techniques fit axisymmetric, three-integral dynamical models of the galaxy (using the line-of-sight velocity distribution), to the observed light distribution. Basically, the orbits in the (R, z) plane in realistic galactic potentials are often found to have in addition to the two integrals of motion E (the energy) and Lz (the z-component of the angular momentum, where z is the symmetry axis), a third integral, I3, that can be associated with the approximately conserved total angular momentum, L.
The dynamical model assumes axisymmetry and an inclination for the galaxy and first determines the optimal density distributrion that is consistent withthe surface brightness distribution (making certain assumptions about M/LV). Then, a central point mass is added, and the potential calculated. Orbits that sample the phase space of the three integrals of motion are then calculated, and the data of the full line-of-sight velocity distribution are fitted to the models.
Since galactic gas is affected also by forces which are non-gravitational (e.g., radiation pressure), stellar kinematics are considered more secure than gas dynamics in black hole searches. Nevertheless, in a few cases, the presence of the black hole may be revealed by gas-dynamical searches. The prototype of this technique is provided by the radio galaxy M87. Early stellar-dynamical observations revealed that the velocity dispersion continues to rise inward, to r 1".5 (e.g., Sargent et al 1978, Lauer et al. 1992). However, due to the expected anisotropy in the velocity dispersion, models without a black hole could also fit the observations (e.g., Binney and Mamon 1982, Dressler and Richstone 1990, van der Marel 1994). In particular, the last author found 400 km s-1 at r 0".5, which could nevertheless be fitted with anisotropic models that do not include a black hole.
A more definitive answer, however, came in this case from the gas. High-resolution HST observations of the nucleus revealed the presence of a small gas disk (about 20 pc in radius), with a major axis perpendicular to the optical jet (Fig. 20). Spectra taken at two opposite points along the major axis (at a luminosity-weighted mean radius of 16 pc) found emission lines separated by 2V = 916 km s-1 (Ford et al. (1994), Harms et al. (1994)). For an inclination angle of the disk of ~ 42° (implied by the observed axis ratio), the observed velocity amplitude (if interpreted as a circular Keplerian motion) corresponds to a dark mass of MBH 3 × 109 M at the center of M87. These results have been further confirmed by Macchetto et al. (1997), making the black hole in M87 the most massive observed so far (with a relatively secure mass determination). I should caution that an airtight case for circular Keplerian motion is still to be made for M87. Incidentally, the best case for a black hole based on gas dynamics is for the modest active galactic nucleus NGC 4258. In that case, Very Long Baseline Array (VLBA) observations of high-velocity masers show a rotation curve that can be fitted remarkably well with Vrot(r) (832 ± 2)(r / 0.25 pc)-1/2 km s-1, implying a black hole mass (mass interior to 0.18 pc) of MBH 4.1 × 107 M.