Annu. Rev. Astron. Astrophys. 1995. 33: 581-624 Copyright © 1995 by Annual Reviews. All rights reserved

4. STELLAR-DYNAMICAL BLACK HOLE SEARCHES

Sections 4 and 5 review stellar- and gas-dynamical BH searches. Detections are listed in Table 1 (Section 2). Except for M33 (Section 4.8), we will not discuss M_BH upper limits; these can be found in Bower et al. (1993) and in van den Bosch & van der Marel (1995).

This subject began with two papers on the stellar dynamics of M87 (Young et al. 1978; Sargent et al. 1978). They showed that M87 contains an M_BH 3 x 10⁹ M MDO if the stellar velocity distribution is isotropic. We now know that isotropy is unlikely, but these papers nevertheless were seminal. Recently Harms et al. (1994) have confirmed the Sargent and Young conclusions using the refurbished HST. This work is based on emission-line spectroscopy, so we discuss it in Section 5.

4.1 Search Technique

Dynamical mass measurements are conceptually simple. We need to deal with projection and atmospheric blurring (``seeing''); this is time-consuming, but it is routine. Mainly, the analysis is complicated because we need to be careful. This subject is dangerous. We enter it with expectations. We need to protect ourselves, lest we convince ourselves prematurely that we have proved what we expect to find.

The search technique is best described in the idealized case of spherical symmetry and a velocity ellipsoid that everywhere points at the center. Then the first velocity moment of the collisionless Boltzmann equation gives the mass M(r) within radius r,

M(r) = V² r / G + _r² r / G [- (d ln ) / (d ln r) - (d ln _r²) / (d ln r) - (1 - ² / _r²) -
(1 - ² / _r²)], (1)

where V is the rotation velocity; _r, , and are the radial and azimuthal components of the velocity dispersion; G is the gravitational constant. The density is not the total mass density , it is the density of the tracer population whose kinematics we measure. We never see , because the stars that contribute most of the light contribute almost none of the mass. In practice, calculations are made assuming (r) volume brightness, i.e., we assume that M/L for the tracer population is independent of radius. This can be (but usually is not) checked using measurements of color or line strength gradients.

All quantities in Equation 1 are unprojected. We observe projected brightnesses, velocities, and velocity dispersions, so we must derive the ranges of unprojected quantities that are consistent with the observations after projection and seeing convolution. This is tricky.

Kormendy (1988a, c, d) and Dressler & Richstone (1988) independently developed a method of deriving unprojected velocities from the data. Beginning with a trial set of unprojected kinematics, they first calculate model spectra projected along each line of sight by adding spectra of appropriate V and weighted by the local volume brightness. They then convolve the two-dimensional array of projected spectra with the point-spread function (PSF). Finally, they ``observe'' the model with their velocity calculator and iterate it until it agrees with the data. Further, Kormendy does not try to prove uniqueness; rather, he constructs fits that bracket the observations in surface brightness I (r) and in projected V (r) and (r). In particular, he derives low-mass ``error bar'' models in which V and are too small near the center. The above procedure is required because of the complicated response of any velocity calculator to population mixes and to rotational line broadening. It guarantees that non-Gaussian line-of-sight velocity distributions (LOSVDs) are measured in the same way in the galaxies and models. (It does not guarantee that the models have the same LOSVDs as the galaxies; more about this shortly.) If the implied M/L rises rapidly as r -> 0, then we have found an MDO.

Some general properties of the mass measurements follow directly from Equation 1. Rotation and random motions contribute similarly to M(r), but the ² r / G term is multiplied by a factor that involves uncertainties and that can be less than 1. So rapid rotation is a more secure indicator of large masses than are large velocity dispersions. Second, Equation 1 shows why velocity anisotropy is so important, especially in nonrotating giant ellipticals. HST shows that these have shallow power-law profiles I r^-0.1±0.1 at r << r_b (Lauer et al. 1992a; Crane et al. 1993b; Stiavelli et al. 1993; Kormendy et al. 1994, Forbes 1994; Forbes et al. 1994; Ferrarese et al. 1994; Lauer et al. 1995). Then -d ln / d ln r +1. The second term cannot be larger than +1. But the third and fourth terms are negative if _r is larger than and . They can be as small as -1 each, so they can largely cancel the first two terms. This does not prove that anisotropic models are realistic. But it illustrates why they have been so successful in explaining the kinematics of giant ellipticals without BHs (e.g., M87: Section 5.1). In contrast, a BH case is more secure if the density gradient is steep, other things being equal. This is one reason why low-luminosity ellipticals like M32 (Section 4.4) and NGC 3377 (Section 4.7) are better BH candidates than giant ellipticals: no core is resolved, so -d ln / d ln r +2 in Equation 1.

Velocity anisotropy has usually been explored by constructing maximum entropy dynamical models (Richstone & Tremaine 1984, 1988). Once anisotropy becomes the default assumption, it is inherently difficult to prove that an MDO is required: the parameter space to be explored is large, and nature knows more distribution functions than we do. Therefore the biggest advantage of the maximum entropy modeling technique is that it can easily be asked to find the most extreme possible models in the most relevant directions in parameter space. In particular, it can be asked to minimize the central mass (it does so by maximizing the anisotropy). If this fails - if M/L still rises toward the center - then we can be almost certain that an MDO is present.

We say ``almost'' because published maximum entropy models have had significant limitations. Flattening corrections were made post hoc using the tensor virial theorem. Also, the velocity dispersion in the rotation direction was not completely free: ² + V² = ². This means, for example, that it was never possible to make isotropic models that rotate. Both limitations will be removed in future papers. Like other analyses in astronomy, the above procedures provide results at some level of approximation. Strong BH cases (especially M31 and NGC 3115) are ones in which the derived M/L is higher than normal by an amount that is substantially larger than the uncertainties.

Progress in the BH search can come from improvements in analysis as well as in observations. In particular, we can exploit LOSVDs. Their calculation and use have been discussed, e.g., by Bender (1990); Rix & White (1992); van der Marel & Franx (1993); Gerhard (1991, 1993a,b); Winsall & Freeman (1993); Kuijken & Merrifield (1993); Dehnen & Gerhard (1993, 1994); Saha & Williams (1994); Evans & de Zeeuw (1994); van der Marel et al. (1994b); Bender et al. (1994); and Statler (1995). It is convenient to measure departures from Gaussian LOSVDs using an expansion in Gauss-Hermite polynomials H_i (van der Marel & Franx 1993; Gerhard 1993a,b; Dehnen & Gerhard 1993):

LOSVD(v) = [ / sqrt(2 ²)] e^{(v - V)2 / -2 2} {1 + ⁿ_i=3 h_i H_i [(v - V) / ]}, (2)

where is the line strength. If h₃ < 0, then the LOSVD has extra power on the systemic-velocity side of V. If h₄ is negative (positive), then the LOSVD is more square (triangular) than a Gaussian. The h₄ coefficient provides direct observational constraints on the velocity anisotropy: tangentially anisotropic models generally have h₄ < 0; radially anisotropic models generally have h₄ > 0 (e.g., Gerhard 1991, 1993a,b; van der Marel & Franx 1993; Dehnen & Gerhard 1993; van der Marel et al. 1994b). Application of LOSVDs to the BH search has been pioneered independently by R. van der Marel and by O. Gerhard and their collaborators. The following sections discuss the added confidence provided by this second iteration in the analysis.

4.2 M31 (M_BH 3 x 10⁷ M)

M31 is the strongest BH case because of the rapid rotation and large velocity dispersion in its nucleus (Dressler 1984; Kormendy 1987a, 1988a, c; Dressler & Richstone 1988; Richstone et al. 1990). Its dynamics have been measured by four independent groups. And it is the first galaxy for which we have made an iteration in improving the spatial resolution.

M31 contains the best known example of a nuclear star cluster (see Johnson 1961; Kinman 1965; Sandage 1971 for reviews) that is dynamically distinct from the bulge (Tremaine & Ostriker 1982). Light et al. (1974) imaged it with Stratoscope II: at _* 0".1, it is asymmetric; its axial ratio is 0.6, and the brightest spot is near the NE end. Lauer et al. (1993) and Crane et al. (1993b) observed it with HST; Lauer and collaborators show that the nucleus is double (Figure 4).

The rapid rotation of the nucleus (discovered by Lallemand et al. 1960) and its steep central dispersion profile are shown in Figure 5. Dressler & Richstone (1988) and Kormendy (1988a,c) carried out analyses like those described in the previous section. These were complementary: Kormendy considered uncertainties in the light distribution (particularly its flattening) in more detail, and Dressler & Richstone showed that the results are independent of velocity anisotropy. The mass-to-light ratio profiles are somewhat model-dependent, but both papers showed that M/L_V rises near the center to values 100 (Figure 5). This implies that M31 contains an MDO of mass M_BH 3 x 10⁷ M.

Figure 4. HST Planetary Camera V-band isophotes of the nucleus and inner bulge of M31 after Lucy deconvolution (Lauer et al. 1993).

Figure 5. Rotation, dispersion, and M/L ratio profiles in M31.

Lauer et al. (1993) discuss possible explanations of the double nucleus; none is very attractive. Extinction is unlikely: there is no color gradient. M31 may have accreted and almost digested a compact elliptical like M32. But timescales are a problem. At 2r = 0".49 = 1.7 pc (the projected separation), a relative velocity of 200 km s^-1 implies a circular orbit period of 50,000y. Dynamical friction should result in a merger within a few orbital times. Are we just lucky to catch the nuclei in flagrante delicto? Two effects may reduce the dynamical friction. If P1 is accreted, it may fortuitously corotate with the stars of P2 (King et al. 1995). Alternatively, both nuclei could be dominated by BHs. This scenario is ad hoc but not implausible. We do not know whether either effect is sufficient.

Finally, Tremaine (1995) has suggested a model in which the brightness enhancement at P1 is caused by an eccentric nuclear disk. This model requires that a 10^7.3-M BH dominate the potential, otherwise the lifetime of P1 against differential precession is short. Accretion is one of several possible ways that an eccentric nuclear disk could have formed. One advantage of Tremaine's model is that it is specific enough to be testable with better kinematic measurements.

Bacon et al. (1994) add new information about the asymmetries by presenting two-dimensional V and maps (Figure 6). They confirm the conclusion of Dressler & Richstone (1988) and of Kormendy (1988c) that the rotation curve of the nucleus is symmetric about its center (``P2'' in Lauer et al. 1993) and not about the brightest point (``P1''). All studies also agree that the center of the nucleus is the center of the bulge. But this is not the point of maximum . Instead, the brightest and hottest points are displaced from the rotation center by similar amounts in opposite directions. Also, (r) drops rapidly to or below the bulge dispersion on the brighter side of the nucleus. It is natural to wonder (Lauer et al. 1993) whether P1 is cold. Then the displacement of the (r) peak from P2 could be due to a contribution from P1 at the center.

Figure 6. Contours of constant velocity (left) and velocity dispersion (right) in the nucleus of M31 (Bacon et al. 1994). The range of velocities (white to black) is -120 km s-1 V 120 km s-1 and 140 km s-1 240 km s-1. These CFHT observations have spatial sampling = 0".39 and resolution * 0".37. The major axis of the nucleus is horizontal; the white dot marks its photometric center.

Figure 7 shows SIS measurements of the M31 nucleus (Kormendy & Bender 1995). The LOSVD has two components. The steep central rise in seen by previous authors is a feature of P2: P2 is much hotter than the bulge near the center. But P1 is colder than the bulge at all radii. The LOSVDs and the V and fits imply that P1 has 85 km s^-1 at the center. This supports the idea that P1 is an accreted low-luminosity stellar system. Since P1 overlaps P2 at the center, it also confirms that the (r) asymmetry is due to the contribution of P1 at P2.

The accretion hypothesis can be tested. King et al. (1995) make a preliminary comparison of the stellar populations of P1 and P2 using a 175 nm image obtained with HST. In the UV, P2 has a higher surface brightness than P1. But the excess flux is small; it is equivalent to one post-asymptotic-branch giant star. King et al. (1995) suggest that it is due to the central radio source (Table 1). They also confirm that P1 is compact (its brightness drops to zero at r 0".63) and faint (V 14.8; M_V -9.7). P1 is no brighter than a large globular cluster, although it is much denser. Most interestingly, King et al. (1995) see no sign that the (UV - optical) color of P1 is different from that of the bulge. This does not favor accretion. On the other hand, P1 nowhere contributes more than 55% of the total surface brightness. It is difficult to measure its metallicity free of contamination from P2. It will be important to see whether HST shows a drop in spectral line strengths at P1.

HST will also be important for the BH case. Meanwhile, the dramatic kinematics seen with SIS (Figure 7) support the BH detection.

Figure 7. Line-of-sight velocity distributions (top) and V (r) and (r) profiles (bottom) along the nucleus major axis of M31 (Kormendy & Bender 1995). Filled circles show the kinematics of the nucleus plus bulge as observed; crosses show the nuclear kinematics after the bulge spectrum is subtracted.

4.3 NGC 3115 (M_BH 1 x 10⁹ M)

NGC 3115 is a edge-on S0 galaxy (Sandage 1961) whose bulge provides 94% of the total light (Capaccioli et al. 1987). At M_B = -19.9, the bulge is almost as bright as a giant elliptical, but it rotates rapidly enough to be nearly isotropic (Illingworth & Schechter 1982). Kormendy & Richstone (1992, hereafter KR92) showed that its steep central kinematic gradients (Figure 8) make NGC 3115 the strongest BH case after M31.

KR92 derived dynamical models that bracket the observations after projection and seeing convolution. Figure 8 (left) illustrates the best-fitting isotropic model at three spatial resolutions. The _* = 0".44 lines are for the resolution of the KR92 observations; by construction, they fit the open circles. One iteration of improved spatial resolution is now available. The filled circles in Figure 8 show CFHT SIS measurements obtained with a 0".3 slit. At _* = 0".244 ± 0".015, V (r) already reaches its asymptotic value at 1", and the apparent central velocity dispersion has risen from 295 ± 9 km s^-1 to 343 ± 19 km s^-1. This is almost exactly the increase predicted by the KR92 models. The middle lines show the best-fitting isotropic model at _* = 0".244. It fits the rotation curve well, and it falls slightly below the dispersion profile at the center. Also, the isotropic models continue to bracket the data at the improved resolution (Figure 8, right). These models imply that M/L_V rises toward the center to values much larger than those of old stellar populations (Figure 9). Note that M/L_V has already risen by a factor of 5 at 1"; contrary to the claim of Rix (1993), this is not a marginal MDO detection. In fact, the MDO dominates gravitationally to r_BH = G M_BH / ² 1".8. We see that r_BH >> _*. If the M/L_V gradient is due to a central dark mass added to stars with constant M/L_V (r), then M_BH 10⁹ M. The MDO mass can be reduced by allowing _r to be larger than and near the center (Equation 1), but KR92 showed that M_BH must be 10⁸ M or the models predict too little rotation at r 1" to 2". We conclude that a significant iteration in spatial resolution confirms the BH case. This increases our confidence in the MDO detection.

Figure 8. (Left) Kinematics of NGC 3115 compared with the best-fitting isotropic model (D3) from KR92 as seen at various spatial resolutions. (Right) SIS data compared with isotropic models D1 - D5 from KR92 as seen at * = 0".25.

Finally, the top curves in each panel of Figure 8 (left) show the best-fitting isotropic model from KR92 at HST resolution (0".25 aperture and the Faint Object Spectrograph). Kormendy et al. (1995a) observed NGC 3115 with HST in 1994 December. They will combine the results with the SIS data and will construct new dynamical models. It is reasonable to expect that these observations will complete step 1 of the BH search for NGC 3115: i.e., they should tell us definitively whether or not an MDO is present.

Figure 9. Mass-to-light ratio interior to radius r for the isotropic models in Figure 8. All models have M/LV 4 at r 4" to 24"; this value is normal for a bulge of MB = -19.9 (Kormendy 1987b, Figure 3). In contrast, M/LV increases at r < 2" by a factor of 10. From KR92.

NUCLEAR BARS Gerhard (1988, 1989) has suggested that nuclear disks are really bars. When these are oriented end-on, we see stars near the pericenters of elongated orbits, so we measure rotation velocities that are larger than circular. This reduces the mass implied by the rotation curve. We do not know whether constant-M/L nuclear bar models can be constructed. However, we know enough about bars to conclude that they are implausible: (a) Noncircular streaming velocities have been measured in the SB0 galaxy NGC 936 (Kormendy 1983). This has an unusually strong bar, but departures from circular motions are only ± 20%. Even if this bar were viewed end-on and even if all of the light came from the bar, the error in M(r) from the V² r / G term in Equation 1 would be only 40%. The ² r / G term would be unaffected: Kormendy saw no dispersion anisotropy in NGC 936. (b) In NGC 3115, the end-on nuclear bar would have a true major-axis radius of 4", since 1" (where M/L_V looks large) is a pericenter radius. Such a bar would easily be seen in more face-on galaxies. Some galaxies have nuclear bars (see Kormendy 1982 for a review). But we have never seen one as elongated as the main bar in NGC 936, and argument (a) suggests that even this is not enough. Nevertheless, nuclear bar models should be investigated (Gerhard 1992).

VELOCITY ANISOTROPY Velocity anisotropy should be explored in more detail. That is, the limitations in maximum entropy models noted in Section 4.1 should be removed. For NGC 3115, this will be done in Kormendy et al. (1995a).

LINE-OF-SIGHT VELOCITY DISTRIBUTIONS LOSVDs provide new constraints on the mass distribution that we are only beginning to use in the BH search (Gerhard 1993b; van der Marel et al. 1994a,b; van der Marel 1994a,b; Dehnen 1995; Qian et al. 1995). Van der Marel et al. (1994a) and Bender et al. (1994) show that the LOSVDs in rotating systems are usually asymmetric, with extra wings on the systemic-velocity side of the mean V. That is, h₃ (r) and V (r) have opposite signs. For NGC 3115, h₃ -0.05 at r 5". The above authors emphasize that V (r) as derived by fitting a Gaussian to the LOSVD then overestimates the true velocity moment in Equation 1. Van der Marel et al. (1994a) worry that this accounts for the failure of constant-M/L models to fit the kinematics. Similarly, the true velocity dispersion is underestimated when the LOSVD has large wings (van der Marel 1994a,b). Bender et al. (1994; see their figure 3) provide the necessary corrections V and . At 0".5 r 5" in NGC 3115, V -14 to -21 km s^-1. The important point is that V is almost independent of radius. The corrected rotation curve then reaches a slightly lower asymptotic value at r 1" to 5". However, model rotation curves are always scaled to the asymptotic V; this is how the global M/L ratio is derived. Thus the main result of the correction is to lower the global (i.e.,bulge) M/L_V. The change is only ~ 8%. It is small partly because less than half of M(r) as derived from Equation 1 is due to V (r); the dispersion profile is almost unchanged by the LOSVD correction. The strength of the BH case is essentially unaffected, because the shape of the rotation curve is almost unchanged. These remarks are reassuring. Nevertheless, as van der Marel et al. (1994a) emphasize, KR92 did not check whether the LOSVDs of their models fit those of the galaxy. Models of NGC 3115 that exploit LOSVDs are planned (Kormendy et al. 1995a).

M/L RATIOS The most serious caveat remains the fact that large M/L ratios do not uniquely imply a BH. Only step 1 of the BH search - the detection of an MDO - is nearing completion. We cannot exclude the possibility that galaxies contain central clusters of stellar remnants or brown dwarf stars. However, old stellar populations have remarkably consistent mass-to-light ratios M/L_V 1 to 10, with M/L_V L^0.2 (Kormendy 1987a,b; Faber et al. 1987; Djorgovski et al. 1988; Bender et al. 1992, 1993). The increase of M/L_V with L is due to three effects, the metallicity-luminosity correlation (Faber 1973; more luminous galaxies are more metal-rich, so stars emit less V-band light per unit mass because of stronger absorption-line blanketing), the anisotropy-luminosity correlation (Davies et al. 1983), and an age-luminosity correlation (Faber et al. 1992). Given that old stellar populations contain stars with very large and very small mass-to-light ratios, it is surprising that the scatter in M/L is so small. But the small scatter is the reason why we interpret large nuclear M/L ratios as implying MDOs. Stellar densities in NGC 3115 are not outside the range observed in galaxies that lack MDOs or in globular clusters. These have normal populations. The spectra and colors of NGC 3115 also are normal. Therefore we have no reason to expect a dramatic population gradient, and we see no evidence of one. An extreme population gradient would be required: inside 1", the dark star cluster would have to have about five times as much mass as the visible stars. Therefore the most plausible conclusion is that NGC 3115 contains a central BH of mass M_BH ~ 10⁹ M.

To summarize: in NGC 3115, as in M31, the detection of an MDO is probably secure. But the arguments that we are detecting a BH are not rigorous.

4.4 M32 (M_BH 2 x 10⁶ M)

The first galaxy with convincing evidence for an MDO was the dwarf elliptical M32; it is now the most thoroughly studied BH candidate. Like M31, it was long ago known to rotate rapidly near the center (Walker 1962). Dressler (1984) and Tonry (1984, 1987) measured its rotation curve with _* 0".5 to 0".85 seeing at the Hale telescope; they demonstrated that the velocity dispersion rises significantly toward the center. Tonry (1984, 1987) fitted his data with isotropic dynamical models; these imply that M/L increases toward the center. He concluded that M32 contains a BH of mass M_BH (3 to 10) x 10⁶ M.

These results were strengthened by Dressler & Richstone (1988), who obtained _* 0".5 observations of M32 with the Hale telescope. Using the techniques of Section 4.1, they showed that velocity anisotropy provides no escape from the conclusion that an MDO is present. More detailed analysis by Richstone et al. (1990) further supported the case.

Figure 10 illustrates the kinematics.

Figure 10. Kinematics of M32 compared with the dynamical models of van der Marel et al. (1994b), Qian et al. (1995), and Dehnen (1995) kindly recalculated by the above authors for * = 0".20.

The conclusions of Tonry, Dressler, and Richstone have now been confirmed by three independent groups. They provide a significant iteration in spatial resolution over the discovery observations, and they strengthen the analysis by fitting LOSVDs. Therefore M32, with M31 and NGC 3115, is among the most secure BH candidates.

Carter & Jenkins (1993) observed M32 with the WHT, a 0".45 slit, and _* = 0".34. At this resolution, V (r) flattens out to 50 km s^-1 at ~ 1". Carter & Jenkins do not use the dispersion profile to measure masses, but they note that a high central M/L does not follow from V (r) alone.

Van der Marel et al. (1994a) present additional WHT observations, including measurements of Gauss-Hermite coefficients. As in other rotating galaxies, h₃ and V have opposite signs; h₃ -0.07 at r 1". Coefficients h₄ - h₆ are nearly zero. Van der Marel and collaborators note that Tonry's (1987) isotropic models do not fit h₃ (r); in fact, they have h₃ 0 at all radii. This demonstrates an interesting result: the LOSVDs are intrinsically asymmetric; they are not Gaussians rendered asymmetric by rotational line broadening, projection, or seeing. This is important for modelers who try to decipher the galaxy's dynamics. But it does not have a large effect on the BH search, contrary to worries expressed in van der Marel's paper. This can again be seen by correcting the Gaussian-fit rotation velocities to velocity moments appropriate for Equation 1 using Bender et al. (1994). Since h₃ constant at r 1", V -7 km s^-1 is nearly independent of radius (see also Figure 2 in Rix 1993). Thus, the main effect of the correction is again to lower the bulge M/L. This is confirmed by detailed analysis, as follows.

Van der Marel et al. (1994b) construct M32 models with two-integral distribution functions f (E, L_z), where E is total energy and L_z is the axial component of angular momentum. Without MDOs, these models fit V (r) and h₃ (r) - h₆ (r), but they fail to fit the central dispersion gradient. In fact, they predict that falls toward the center at r < 1". [Physically: If stars cannot climb out of their own potential well and therefore make a cuspy profile, they must be cold; see, e.g., Binney 1980; Dehnen 1993; Tremaine et al. 1994).] Similar results are obtained with a moment equation analysis that fits V, , and the skewness of the LOSVDs. Good fits to (r) are obtained when an MDO of mass M_BH = (1.8 ± 0.3) x 10⁶ M is added. These results are in good agreement with the conclusions of Tonry, Dressler, and Richstone.

Qian et al. (1995) take the moment equation models one step further by calculating their distribution functions using the contour integral method of Hunter & Qian (1993). They can then derive the complete LOSVDs and not just their skewness. For the M_BH = 1.8 x 10⁶ M model, the comparison to V, , and h₃ - h₆ shows remarkably good agreement along all slit positions measured by van der Marel et al. (1994b).

Dehnen (1995) has made further f (E, L_z) models of the van der Marel et al. (1994a) observations using a slightly different technique. He recovers f (E, 0) from using a Richardson (1972) - Lucy (1974) algorithm. The distribution function is then multiplied by a guessed function of L_z and used to calculate V, , and h₃ - h₆. These are compared to the observations, and the distribution function is iterated until it agrees with all observables. Slit width, pixel size, and seeing are taken into account. Dehnen's results are closely similar to those of van der Marel et al. (1994b). In particular, he confirms that such models do not fit the kinematics unless they contain an MDO of mass M_BH (1.6 to 2) x 10⁶ M. Again, the models with an MDO provide an excellent fit to h₃(r) - h₆ (r) along the major, minor, skew, and offset axes.

Van der Marel et al. (1994b), Qian et al. (1995), and Dehnen (1995) improve on earlier analyses in several important ways. The models are properly flattened, they fit the non-Gaussian LOSVDs, and they fit kinematic observations at a variety of slit positions, not just along the major axis. The latter point is important: as van der Marel et al. (1994b) point out, Tonry's (1984, 1987) models explain the major-axis dispersion gradient as due to rotational line broadening, but without an intrinsic dispersion gradient, they cannot be in dynamical equilibrium along the minor axis. The f (E, L_z) models solve this problem.

These models also have an important shortcoming compared to the models of Dressler & Richstone (1988). The distribution functions are restricted to be functions of two integrals; in many cases, they are required to be analytic. We can show that these models fail to fit the data without a BH, but we cannot prove that it is impossible to find a more general distribution function that succeeds without a BH. In contrast, the maximum entropy models can be forced to have the smallest possible M/L as r -> 0. When this fails - subject to shortcomings discussed in Section 4.1 - an MDO is required. Thus the distribution function and maximum entropy models are complementary: each has strengths that the other lacks. It is reassuring that they agree. But an exploration of extreme models that successfully fit V, , and h_n without the limitations of published maximum entropy models is still needed.

Finally, Figure 10 shows new observations of M32 obtained with the CFHT and SIS (Kormendy et al. 1995a). These provide a further improvement in spatial resolution: The slit width was 0".35, and the seeing _* was 0".20. At this resolution, the central rotation curve is much steeper; it peaks at V 53 ± 2 km s^-1 at r 0".85. The central dispersion profile is slightly steeper; (0) 94 ± 2 km s^-1. Figure 10 compares the SIS data with the models of van der Marel et al. (1994b), Qian et al. (1995), and Dehnen (1995) as seen at the present resolution. The fit is good at large radii. But V (r) and (r) both reach higher maximum values than the models predict. Therefore the BH case gets stronger at SIS resolution. Modeling of the SIS data is in progress.

In summary, five independent groups have observed and modeled M32. Models without MDOs consistently fail to fit the kinematics. These include maximum entropy models that were instructed to minimize the central M/L. Successful models require that M_BH 2 x 10⁶ M. This result has survived improvements in spatial resolution of a factor of three since the MDO discovery. Given expectations based on Figure 10 and on the predictions of Dehnen (1995) and Qian et al. (1995), HST spectroscopy is feasible. Van der Marel et al. (1995) plan to make these observations. Meanwhile, the case for an MDO is already strong.

4.5 NGC 4594, The Sombrero Galaxy (M_BH 5 x 10⁸ M)

The remaining stellar-dynamical BH cases (Sections 4.5, 4.6, 4.7) are weaker than those of M31, M32, and NGC 3115 because the modeling analyses have explored fewer degrees of freedom on M(r).

NGC 4594 is an almost edge-on Sa galaxy illustrated in the Hubble Atlas (Sandage 1961). Its bulge is as luminous as a giant elliptical (M_B -21.21 at 9.2 Mpc distance), but it rotates rapidly enough so that velocity anisotropies are small (Kormendy & Illingworth 1982). This reduces the uncertainties in the mass measurements.

Kormendy (1988d) observed NGC 4594 with the CFHT (slit width = 0".5, scale = 0".435 pixel^-1, seeing _* = 0".40). The central kinematics reveal a nuclear disk: at r 5" the rotation curve has an inner peak, and the dispersion profile has a minimum ( = 181 ± 6 km s^-1) that is significantly lower than the bulge dispersion ( 240 km s^-1). The nuclear disk is also detected photometrically (Burkhead 1986, 1991; Kormendy 1988d; Crane et al. 1993b; Emsellem et al. 1994a). Its presence guarantees that the LOSVDs are asymmetrical. But the disk is well localized and significantly brighter than the bulge. At r 10" to 15", both the photometry and the kinematics suggest that the disk is negligible; here the major-axis dispersion is equal to that along the minor axis. Therefore Kormendy subtracted the bulge spectrum obtained from the minor axis and scaled to the major-axis bulge profile. He then confined his analysis to the kinematics of the nuclear disk. This is a well-defined kinematic subpopulation that can be used in Equation 1 to measure the total mass distribution. Recently, Kormendy (1994) has shown that asymmetries in the LOSVDs due to superposition of the nuclear disk and bulge were removed by the decomposition; the Kormendy (1988d) models are an adequate fit to the residual LOSVDs.

The machinery of Section 4.1 was used to derive unprojected rotation velocity, velocity dispersion, and brightness profiles that bracket the bulge-subtracted data after projection and seeing convolution. The solutions imply that the mass-to-light ratio rises from normal values M/L_V 7.8 at large radii to M/L_V > 100 near the center. (All values in this section have been corrected to the distance scale of Table 1.) Kormendy & Westpfahl (1989) showed that the outer M/L_V remains constant to ~ 180". So a normal old stellar population dominates the bulge over a large radius range. But an MDO is present at r < 1"; if M/L_V (r) = constant for the stellar population, then M_BH 5 x 10⁸ M.

The main shortcoming of the analysis was that velocity anisotropy was not explored. The rapid rotation and the nuclear disk make it unlikely that _r >> _tangential (Kormendy & Illingworth 1982; Jarvis & Freeman 1985). Nevertheless, anisotropic models should be constructed.

The kinematics of NGC 4594 have been confirmed by five independent groups. Jarvis & Dubath (1988) observed the galaxy with the ESO 3.6 m telescope, a 2" slit, 1".17 pixels, and seeing _* 0".51 to 0".64. Their data agree well with Kormendy's. They estimate the total mass within 3".5 of the center to be 7 x 10⁸ M and ``conclude that there is strong evidence that NGC 4594 contains a super-massive object, possibly a black hole or massive star cluster.'' But they calculate no M/L ratios, so they cannot tell how much of the mass is dark. In fact, comparably luminous galaxies typically have core masses > 10⁹ M in stars (e.g., Kormendy 1982, Table 3). In NGC 4594, the luminosity inside 3".5 is L_V 6 x 10⁸ L (Kormendy 1988d, Figure 10), and M/L_V,bulge 7.8. So Jarvis and Dubath underestimate the mass inside 3".5.

Wagner et al. (1989: ESO 2.2 m telescope, 1".87 pixels, 2" slit) were the first to examine LOSVDs. These clearly show a superposition of rapidly rotating, cold and slowly rotating, hot components. The cross-correlation peaks at the center and at ±3".6 agree well with those illustrated in Kormendy (1994). Wagner et al. (1989) derive a seeing-corrected mass of (3 ± 1.2) x 10⁹ M inside 3".8 radius, but like Jarvis & Dubath (1988), they do not derive M/L ratios.

The kinematics are further confirmed by Carter & Jenkins (1993: WHT, 0".34 pixels, 0".55 slit, seeing _* 0".51). They remark that rotation alone is not rapid enough to imply a large mass-to-light ratio. It is true that the central rise in M/L found by Kormendy (1988d) is largely due to the dispersion gradient. Checking anisotropy is probably more important than Kormendy suggested.

Van der Marel et al. (1994a) took additional spectra with a 1".25 slit, 0".6 pixels, and seeing _* = 0".47. They agree with previous results. The main new contribution is the measurement of LOSVDs. The maximum amplitude of h₃ = -0.15 is reached at r = 6"; |h₃| - |h₆| then fall almost to zero at r = 10" - 20". This is in excellent agreement with the Kormendy (1988d) and Burkhead (1991) conclusion that the nuclear disk dominates near the center but is negligible at intermediate radii.

In summary, the kinematics of NGC 4594 are well confirmed. But none of the published measurements improves on the discovery resolution.

Also, there has been little progress on modeling. Emsellem et al. (1994a) model the Kormendy data using multi-Gaussian expansions of the PSF and galaxy light distributions. Again, isotropic models do not fit the dispersion profile unless there is an MDO of mass 5 x 10⁸ M. However, even the MDO model rotates too slowly. Part of the problem is that Emsellem and collaborators fit the composite (not bulge-subtracted) kinematics; this adds to the uncertainties because of the messy LOSVDs.

More definitive results on NGC 4594 should be available soon. Emsellem et al. (1994b) have obtained two-dimensional spectroscopy with 0".39 spatial sampling on the CFHT. Kormendy et al. (1995a) have taken CFHT SIS spectra with _* = 0".27; at this resolution, the apparent central velocity dispersion is 282 ± 8 km s^-1. They are also scheduled to observe NGC 4594 with HST in 1995 February. The issue of whether NGC 4594 contains an MDO should be settled soon.

4.6 The Galaxy (M_BH 2 x 10⁶ M)

The center of our Galaxy is enormously complicated and well studied. Excellent reviews (Genzel & Townes 1987; Morris 1993; Genzel et al. 1994) and conferences (Backer 1987; Morris 1989; Genzel & Harris 1994) discuss the physics in detail. Papers on a possible Galactic BH include Lynden-Bell & Rees (1971); Rees (1987); Phinney (1989); and de Zeeuw (1993). Here we summarize the AGN evidence and gas dynamics briefly and then concentrate on the stellar-dynamical BH search.

The radio source Sgr A* (Genzel et al. 1994, Figure 2.2) is assumed to be the Galactic center; this is not certain. Sgr A* is spectacularly tiny. Lo et al. (1985) measure a radius of < (1.1 ± 0.1) x 10^-3 arcsec = 9 AU at = 1.3 cm wavelength. Also, ² due to interstellar electron scattering. Strong limits require short ; at 3 mm, 0.07 x 10^-3 arcsec = 0.6 AU (Rogers et al. 1994). This is only 15 times the Schwarzschild radius of the 2 x 10⁶ M MDO suggested by the dynamics. It is easy to be impressed by the small size. But as an AGN, Sgr A* is feeble: its radio luminosity is 10³⁴ erg s^-1 ~ 10^0.4 L. The infrared and high-energy luminosity is much higher (Genzel et al. 1994), but there is no compelling need for a 10⁶-M BH. Mini-AGNs caused by stellar-mass engines are known (e.g., Mirabel et al. 1992). So, to find out whether the Galaxy contains a supermassive BH, we need dynamical evidence.

Genzel & Townes (1987) and Genzel et al. (1994) review the dynamics of neutral and ionized gas near the center. Velocities of 100-140 km s^-1 imply masses of several x 10⁶ M inside 1 pc (Figure 11) it if the gas is in circular motion. This assumption is not well motivated: as discussed in Genzel et al. (1994), stellar winds from luminous young stars combine to make a wind blowing out of the central parsec; a few hundred supernovae are thought to have occurred in the central 10² pc in the past 10⁴-10⁵ y; some noncircular motions are seen, including an expanding bubble of hot gas (Eckart et al. 1992). It is surprising that the motions are so close to gravitational. Without confirmation from stellar dynamics, we would not dare to include the Galaxy in Table 1.

Because of optical extinction, stellar velocities are usually measured using K-band CO band heads. From the first papers (Sellgren et al. 1987; Rieke & Rieke 1988), masses of 10⁶-10⁷ M at r 1 pc were deduced. McGinn et al. (1989) and Sellgren et al.(1990) measured spectra of integrated starlight in apertures of 2".7 - 20" diameter. The rotation curve rises in the inner 15" = 0.6 pc and then flattens out at V 36 km s^-1. The projected dispersion rises from ~ 70 km s^-1 at r > 35" to 125 km s^-1 at r < 20". Interestingly, the CO band strength decreases at small radii; the true strength is consistent with zero at r < 15". The authors suggest that the atmospheres of giant stars have been modified (stripped off?) in the dense environment of the nucleus. This limits the resolution to 0.6 pc, not much better than ground-based resolutions in M31 and M32. Neglecting projection and anisotropy and assuming that r_c = 0.4 pc, McGinn et al. (1989) conclude that the mass distribution is inconsistent with the light distribution: it requires M_BH 2.5 x 10⁶ M. Sellgren et al. (1990) derive M_BH (5.5 ± 1.5) x 10⁶ M.

The best dynamical analyses are by Kent (1992) and by Evans & de Zeeuw (1994). Kent notes that the central K-band starlight is a flattened power law, r^-1.85. He constructs a flattened, isotropic Jeans equation model that fits the stellar and gas kinematics along the major and minor axes. This requires an MDO of mass M_BH 3 x 10⁶ M. Evans & de Zeeuw (1994) make f (E, L_z) models for the same power-law density distribution; these fit the kinematics if M_BH 2 x 10⁶ M.

Kinematic measurements of a number of kinds of stars confirm these results (e.g., OH/IR stars: Lindqvist et al. 1992; He I stars: Krabbe & Genzel 1993, quoted in Genzel et al. 1994). The dynamical mass distribution is compared to that derived from the infrared light distribution in Figure 11. Gas and stellar kinematics agree remarkably well. The conclusion that there is (1 to 3) x 10⁶ M of central dark matter even looks robust to the poorly known core radius of the stars.

Haller et al. (1995) have remeasured the kinematics with a 1".3 slit placed at several positions near Sgr A*. From He I lines in hot stars and CO band heads in cool stars, they find masses at r < 1 pc that are somewhat lower than those reviewed by Genzel et al. (1994). Haller and collaborators raise the possibility that some of the dark matter near the center may be extended in radius. Then M_BH (1 to 2) x 10⁶ M.

Most recently, Krabbe et al. (1995) have obtained two-dimensional K-band spectroscopy of the central 8" x 8" at FWHM = 1" resolution. From 35 individual stellar velocities (mostly based on emission lines), they derive = 153 ± 18 km s^-1 at < r > = 0.245 pc and hence M_BH 3 x 10⁶ M. If the velocity distribution is approximately isotropic, the case for a high central mass-to-light ratio is quite strong.

Nevertheless, the Galactic center BH case is fundamentally more uncertain than those of the best candidates. The observations are sensitive to discreteness and population effects. Absorption-line measurements are not luminosity-weighted along the line of sight because the CO bands disappear in the central 0.5 pc. It is hard to be confident that a particular kinematic tracer is distributed in radius like the 2.2 µm light (which is used to determine d ln d ln r and the stellar mass distribution). Given the peculiar (young?, rejuvenated?) stellar population, it is not clear that M/L_K is constant for the stars. Also, no models have explored anisotropy. On the other hand, we can hope for better spatial resolution than in any other galaxy. The case for an massive dark object is strong enough to be taken seriously. But further work is needed.

If the Galaxy contains a BH, it is starving in a blizzard of food. This could be embarrassing (Section 7).

Figure 11. Galactic center mass distribution (Genzel et al. 1994). The points show dynamical masses derived from various kinematic tracer populations (see text). For two possible core radii, the dashed lines show the mass distribution in stars derived from the K-band brightness distribution assuming M/LK = 1.2.

NGC 3377 (M_BH 8 x 10⁷ M)

NGC 3377 is a normal elliptical galaxy illustrated in the Hubble Atlas. At M_B = -18.8, it rotates rapidly enough to be nearly isotropic. Its core is tiny, so small radii have large luminosity weight in projection. The axial ratio is 0.5; since no elliptical is much flatter, NGC 3377 must be nearly edge-on. Finally, the distance is only 9.9 Mpc. Therefore NGC 3377 is an excellent target for a BH search.

Kormendy et al. (1995b, see Kormendy 1992a) observed NGC 3377 with the CFHT and resolution _* 0".48. The galaxy is kinematically similar to M32: V (r) rises rapidly near the center to 100 km s^-1 and then levels off; (r) 90 km s^-1 at r 2" and then rises to 158 ± 11 km s^-1 at the center. The kinematics are confirmed by Bender et al. (1994). Kormendy et al. (1995b) show that isotropic kinematic models require an MDO of mass M_BH 8 x 10⁷ M. They also fail to find anisotropic models that fit without an MDO, but the failure is not large. Therefore this is a weak MDO detection. After M32, NGC 3377 is only the second elliptical galaxy with stellar-dynamical evidence for an MDO.

4.8 M33 (M_BH 5 x 10⁴ M)

One BH upper limit deserves discussion. M33 is a bulgeless Sc galaxy with a nucleus like a giant globular cluster. Kormendy & McClure (1993) observed it with the CFHT and a camera that uses a tip-tilt mirror to partly correct seeing. At resolution _* = 0".18, the limit on the core radius is r_c < 0.39 pc. This is as small as the smallest cores in globular clusters. The central surface brightness is one of the highest known (Figure 2). M33 is an excellent illustration of the fact that an unusually small and dense core is no evidence for a BH unless kinematic measurements show high velocities (Section 3). The central velocity dispersion measured by Kormendy & McClure (1993) is only 21 ± 3 km s^-1. The central mass-to-light ratio is M/L_V 0.4. Since this is required to explain the stellar population, a conservative limit on M_BH is the limit on the core mass in stars: M_BH 5 x 10⁴ M. M33 is the first giant galaxy in which a dead quasar engine can be ruled out.

The central relaxation time, T_r 2 x 10⁷ y, is so short that core collapse has probably occurred. Also, the center is (B - R) = 0.44 mag bluer than the rest of the nucleus. The color gradient, the F-type spectrum (van den Bergh 1991), and the small M/L ratio imply that the nucleus contains young stars concentrated to the center. Kormendy & McClure (1993) discuss the possibility that the stellar population has been affected by dynamical processes (e.g., stellar collisions).