4.2. Two Routes to SBH Demographics in Local Quiescent Galaxies
Because of its tightness, the M_{} - relation provides us with a direct and powerful tool to estimate the mass density of SBHs, _{}, in the local universe. One approach is to combine the known mass density of spheroids (e.g. Fukugita et al. 1998) with the mean ratio between the mass of the SBH and that of the host bulge. Merritt & Ferrarese (2001a) used the M_{} - relation to estimate M_{} for a sample of 32 galaxies for which a dynamical measurement of the mass of the hot stellar component was available (from Magorrian et al. 1998). For this sample, the frequency function N[log(M_{} / M_{bulge})] is well approximated by a Gaussian with <log(M_{} / M_{bulge})> ~ -2.90 and standard deviation ~ 0.45. This implies M_{} / M_{bulge} ~ 1.3 × 10^{-3} or, when combined with the mass density in local spheroids from Fukugita et al. (1998), _{} ~ 5 × 10^{5}. This estimate is a factor of five smaller than obtained by Magorrian et al. (1998) using what we now believe to be inflated values for the masses of the central black holes in many galaxies.
Here I will present an independent derivation of _{} which, while not directly leading to <M_{} / M_{bulge}>, has the advantage of producing an analytical representation of the cumulative SBH mass density as a function of M_{}. The idea is simple: if M_{} correlates with the luminosity of the host bulge, the SBH mass density can be calculated once the luminosity function of bulges is known. Black hole masses are related to bulge luminosity directly through the M_{} - M_{B} relation, a representation of which is given by Ferrarese & Merritt (2000) as log M_{} = -0.36M_{B} + 1.2. Unfortunately, the large scatter of the M_{} - M_{B} relation (Fig. 3), combined with the small number of galaxies on which it is based, makes it impossible to establish whether elliptical and spiral galaxies follow a similar relation. Indeed, the observations of McLure & Dunlop (2001) cast doubts on whether spirals and lenticulars follow an M_{} - M_{B} relation at all. This is unfortunate since the galaxy luminosity function does show a dependence on morphology (e.g., Marzke et al. 1998), and it is therefore desirable to conduct the analysis independently for different Hubble types. An alternative approach is to derive a relation between M_{} and bulge luminosity by combining the M_{} - relation (which, given the present sample, seems independent of the morphology of the host galaxy) with the Faber-Jackson relation for ellipticals and its equivalent for spiral bulges. The drawback here is that the Faber-Jackson relation has large scatter and is ill defined, especially for bulges.
The luminosity function for spheroids can be derived from the luminosity function of galaxies, generally represented as a Schechter function, once a ratio between total and bulge luminosity (which depends on the Hubble type of the galaxy considered) is assumed. The latter is adopted from Table 1 of Fukugita et al. (1998). Here, I will use the galaxy luminosity function derived by Marzke et al. (1998) from the Second Southern Sky Redshift Survey (SSRS2), corrected to H_{0} = 75 km s^{-1} Mpc^{-1} and an Einstein-de Sitter universe. Marzke et al. derived luminosity functions separately for E/S0s and spirals, in a photometric band B_{SSRS2}. This band is similar to the Johnson's B-band, where representations of both the M_{} - M_{B} relation and the Faber-Jackson relation exist: B_{SSRS2} = B + 0.26 (Alonso et al. 1993). A Schechter luminosity function,
(3) |
is then easily transformed into a SBH mass density if L = A M_{}^{k},
(4) |
where _{0} = k_{0}, M_{*} = ( L_{*}10^{0.4 × 0.26} / A)^{1/k}, and L/L_{bulge} = 0.23 for spirals and 0.76 for E/S0 galaxies. is the sum of the ratios between bulge to total B-band luminosity for different Hubble types, each weighted by the fraction of the mean luminosity density contributed by each type (from Fukugita et al. 1998).
Fig. 4 shows the cumulative SBH mass function separately for the E/S0 and spiral populations, derived from the M_{} - M_{B} relation (from Ferrarese & Merritt 2000, dotted lines) and the M_{} - relation (from this paper) combined with the Faber-Jackson relations for ellipticals and spirals (from Kormendy & Illingworth 1983, corrected to H_{0} = 75 km s^{-1} Mpc^{-1}). While the two distributions differ in the details, there is little difference in the total mass density, which falls in the range (4 - 5) × 10^{5} M_{} Mpc^{3}. This is in excellent agreement with the estimate of Merritt & Ferrarese (2001a).
Table 1 summarizes the mass density estimates for SBHs discussed in the preceeding three sections. While a detailed comparison of the distribution of masses remains to be carried out (for instance, Fig. 1 suggests a larger fraction of very massive black holes, M > 10^{9}, in high redshift QSOs than have been found in local galaxies), the overall picture is one of agreement: local studies seem to have recovered the overall mass density inferred from high redshift QSOs. It appears that supermassive black holes are a fundamental component of every large galaxy.
Method | _{} (10^{5} M_{} Mpc^{-3}) |
QSO optical counts, 0.3 < z < 5.0 | 2 - 4 |
AGN X-ray counts, z > 0.3 | 0.6 - 9 |
Spectral fit to the X-ray background, z unknown | 2 - 30 |
Local AGNs, z < 0.1 | 0.05 - 0.6 |
Local Quiescent Galaxies, z < 0.0003 | 4 - 5 |