4.2 Methods of Inferring the Mass

As pointed out by Richstone, ¹¹⁶ one possibility to infer the presence of black holes is first to determine the central mass, and then to rule out alternatives to a black hole. This is similar to the stellar case within our Galaxy, where the central mass in some cases exceeds the maximal one allowed for a neutron star. ¹¹⁷ Methods of inferring the central mass in AGN include:

I. Stellar kinematics

The black hole hypothesis implies the existence of central black holes in nearly all galaxies. Interestingly, recent observations have indicated the presence of dark, massive concentrations of matter in nearby galactic nuclei. The basis of these investigations is the collisionless Boltzmann equation

(8)

where v is a typical rotation velocity, R the size scale, F a function of the deprojected star density and the three deprojected components of the velocity dispersion _r, and . Using various kinematic stellar models to fit the data, the result is M₈ ~ 0.1 for M31 ¹¹⁸ and ~ 0.6 for M32, ^{116, 119} ~ 10 for NGC 4594 ¹²⁰ and ~ 1-10 for NGC 3115. ^{121, 122} The famous case of M87 seems unsettled at present. ¹²³

It should be stressed that even though this approach may have given support to the first of Richstone's steps, the second one still remains to be done. Indeed, the central mass can, for instance, comprise a non-relativistic, compact cluster of low-mass stars, instead of a black hole. ¹¹⁶ The observed stellar motion could also be influenced by forces from outside the galactic nucleus, and the inherent assumptions of the stellar kinematical models may be irrelevant. Furthermore, the method seems limited to nearby, edge-on and fast rotating spiral galaxies. ¹²²

II. Bolometric luminosity

The basic accretion formula L = c² yields

(9)

where T is the lifetime of the AGN. If we assume (from, e.g., the extension of extragalactic jets) T ~ 10⁸ years, a luminosity ~ 10⁴⁶ ergs s^-1 and ~ 0.1, we obtain M₈ ~ 0.1. In practice, only L is an observable, so statistical arguments (involving radiation density estimates from QSO counts) are invoked. ^{117, 117} The result lies in the range M₈ ~ 0.1-10. The nature of individual central objects may be difficult to determine using this method.

III. Properties of the broad-line region

The basic assumption is that the cloud motion is purely gravitational, which, as discussed above, may be irrelevant. The first term in Eq. (8) is used, where now v is the BLR cloud velocity and R the BLR size. In addition, a multiplying factor ~ 1 is usually added, whose value depends on whether the motion is bound or not. Gaskell ⁸⁵ pointed out that the direction of the motion is also needed, since a pure outflow says very little about the central mass. Some results for nearby Seyferts are: M₈ ~ 0.1 for NGC 5548, ¹⁰⁵ ~ 0.5 for NGC 4151 ⁸⁵ and ~ 8 for F9. ¹⁰⁴ In order to establish the necessary infall of matter, the method uses the cross-correlation technique between line wing variations. The uncertain size and velocity field structure discussed above limit the relevance of this method. Also, the nature of the central object seems difficult to infer, since the BLR effects occur outside the central engine.

IV. Continuum properties

Accretion disk spectra have been rather successfully fitted to UV-optical continua. ^{27, 36} The original models have been extended to include general relativistic, geometrical and opacity effects. Since the central engine is utilized, the reliability of the method may be higher than for previous approaches.

A higher inclination angle implies a harder spectrum, whereas a higher mass has the opposite effect (the maximum disk temperature is inversely proportional to the mass). The result is a correlation between the inclination angle and the mass, which reduces the number of fitting parameters from three to two: [, M(cos i)]. Although these two parameters can be constrained rather tightly, the allowed range in central mass becomes large, since the inclination angle is undetermined. Typical results for a disk without reprocessing are M₈ ~ 0.1-10 for Seyfert 1s and M₈ ~ 1-100 for elliptical galaxies. ³⁶

The cases which show simultaneous variability in UV and optical probably require that reprocessing must also be taken into account. Malkan ¹²⁶ adopted a simple model, in which reprocessing was assumed to dominate outside a critical disk radius r_c, whereas the emission was due to ``normal'' viscous dissipation inside r<sub>c</sub>. Moving the latter inwards obviously increases the relative fraction of reprocessed optical flux, amounting to ~ 27% for r<sub>c</sub> ~ 25 r<sub>g</sub>. Since the reprocessed flux comes out in optical and UV, the ``original'' disk emission in these ranges must be decreased, which is accomplished by a decrease in the central mass. In the case of NGC 5548, consistency requires M₈ to decrease from 2 to 0.55. This in turn implies a hotter disk, which shifts much of the emission into the unobserved EUV region.

If problems concerning the adopted disk model (usually of the inappropriate Shakura-Sunyaev type), the inclination angle, the reprocessing and relative contributions from stars and the BLR can be resolved, spectral fits using accretion disks may provide reliable estimates of the central mass.

V. X-ray variability - NGC 6814

The short (~ 10³ s) variability timescale in X-rays indicates an origin also in the central engine (cf. Eq. (2)). The only AGN which so far has shown clear evidence of quasi-periodic X-ray variability is NGC 6814. ^{127, 128} The peaks in the power spectrum lie in the range 10^-4±1 Hz, and the variability period (~ 10⁴ s) has been shown to be rather constant over timescales of years, indicating ¹²⁹ || 10^-6. The period also seems virtually uncoupled to luminosity (and hence accretion rate) variations.

There are essentially four classes ¹³⁰ of proposed explanations for the variability of NGC 6814, of which the first three involve a single spot orbiting on the accretion disk surface, ^{131, 132} collisions between a star and the accretion disk ^133-135 and an orbiting screen in the outer disk. ^{136, 137} All of these may suffer from being fine tuned, and the reason why the screen should have an approximately stable structure seems obscure. The spot and screen models may have encountered further difficulties due to the change of folded light-curve structure between the EXOSAT and Ginga observations. This change may be explained by Lense-Thirring precession of the orbit of the star in the collisional model, ¹³³ but the indication of > 0 may have imposed major difficulties for this scenario, since the opposite effect is expected. ¹³⁸

The fourth alternative makes use of acoustic mode behaviour in the innermost part of the accretion disk. ¹⁰ The slim disk acoustic instability frequency increases generally with , which partially may explain the horizontal branch oscillations (HBOs) observed in X-ray binaries. ⁹ However, the instability frequency becomes essentially constant (for a specific radius), when the accretion rate is low (Fig. 6). Thus, the only model which at present seems able to explain the stability and the numerical value of the period, its long-term trend (as indicated by > 0) as well as the change of topology of the folded light-curve may be the acoustic one. ¹³⁹ Also, the presence of a supermassive black hole in NGC 6814 seems strongly supported, since a central mass M₈ ~ 10^-2 then would reside within a volume only a few Schwarzschild radii across. Independent global, numerical and time-dependent calculations of acoustic mode behaviour ^{140, 141} are consistent with this conclusion. Provided the acoustic model holds true, the same behaviour should also apply to other sources, if the inner accretion disk can be observed, and if the accretion rate is low. A search for such sources should consequently concentrate on nearby, low-luminosity and face-on AGN, or their stellar galactic counterparts.

Figure 6. The relation between acoustic instability frequency ac and accretion rate in the inner region of a slim disk. The quantity x is the radius in Schwarzschild units. From Ref. 139.