## 6. Results of the model fittingAs examples, we present in Fig. 4 the best fit spectra of three sample members, #15, #17, and #18 (Table 1), showing both the IUE and ROSAT data points as well as the prediction from our model calculation. Due to the complex nature of the spectrum in the UV range, in some objects the slope of the UV continuum is not well matched by our model fits (see, for example, object # 18 in Fig. 4). This has no large effect on the best-fit values, however, which are dominated by the X-ray data points, while the UV data points mainly help to constrain the total flux from the accretion disk.
The resulting best fit parameters of all sample members, as well as their errors and the corresponding minimum values are given in Table 4. Cases where the upper or lower errors lie beyond the limit of our calculated grid, are denoted by a minus sign. In most cases acceptable fits are achieved. The distribution of model parameters was studied using a maximum likelihood technique (see Avni 1976) which, based on the assumption that both the model parameters and their statistical errors follow the normal distribution, gives the first (mean) and second moment (i.e., the of the normal distribution) as well as their statistical errors. Fig. 5 and 6 show the 68 %, 90 %, and 99 % confidence contours of the distribution of and , respectively. The case that all sample members have the same and/or best-fit parameter values is excluded at a high statistical significance level (the confidence contours do not intersect the line). We find a mean accretion rate of within a relatively narrow parameter range (). The best-fit accretion rates are below the Eddington accretion rate in all sample members (see Fig. 8), thus fulfilling the requirement for the thin disk approximation (Laor & Netzer, 1989). The viscosity parameters are relatively high () and are spread over a wider range ( for most objects), possibly suggesting some diversity of the underlying physical viscosity mechanism in our sample. Note that, according to its definition, should not greatly exceed unity.
The best-fit central masses which roughly span two orders of
magnitude ( ; see Fig. 7) are in broad
agreement with AGN black hole masses derived from variability and from
general luminosity arguments. As the accretion rates are found to be
confined within a relatively narrow range (),
this implies that, in absolute terms, the mass accretion rates also
span about two orders of magnitude while maintaining a rough
proportionality (within a factor ) with the
central masses over the whole dynamic range. We have tested for any
dependencies of and on
The narrow range of observed accretion rates in terms of the Eddington accretion rate also implies that the large luminosity range covered by AGN must predominantly be due to a similarly large variation in central mass. Note that for the object class studied here, i.e. radio-quiet quasars, the emission is considered to be dominated by an unobscured accretion disk and absorbing material on the line of sight is thus not thought to contribute to the large observed luminosity range. Taken together with the known evolution of the quasar luminosity function (e.g., Boyle et al., 1987 and 1993), i.e. the fact that quasars at high redshifts are considerably more luminous than `local' quasars (by up to a factor of 40 at redshift z = 2, depending on which relative contribution of luminosity and/or density evolution is favoured) it follows that quasars at earlier epoches were more massive than present day quasars by similar factors, giving further support to the concept that many local galaxies (including our own; Genzel & Eckart, 1996 ) contain dormant, super-massive black holes in their centers. See the more detailed discussion of this finding in Brunner et al. (1997). We presently do not know which physical processes are responsible for the fact that high accretion rates (0.3 - 1.0 for the total sample; 0.15 - 1.0 for the low mass/low redshift subsample) are not observed. However, since the definition of the Eddington accretion rate is based on the assumption that both radiation and accretion flow are isotropic, suitable unisotropies of both the accretion flow and the resulting radiation may lead to a reduction of the permitted maximum accretion rates. Dynamical processes in the disk not considered in our present modeling may also result in a limit to the possible accretion rates. We believe that this highly interesting point warrants further theoretical attention. Note that the observed lower cutoff of the distribution of accretion rates () may be due to selection effects: At very low accretion rates no appreciable emission is expected in the X-ray range such that most objects will not be detected in the ROSAT band. We find marginal correlations of the accretion disk parameters with
(see Fig. 10). When one anomalous object, #
20 (Table 1), with the lowest , and at the
same time the highest and
) values in the sample is removed, the
probabilities for randomness of the observed correlations of
on
A statistical comparison of the sample properties of AGN from the ROSAT All Sky Survey using a simpler precursor version of the present accretion disk code has been performed by Friedrich et al. (1997 ) which is in broad agreement with the present study. However, using the improved model, considerably smaller mass accretion rates are sufficient to produce the observed X-ray emission. This is mainly because, contrary to the simpler version, our improved model also takes into account the temperature gradient in the vertical direction of the disk. This means that the local spectra differ from the blackbody even in the optically thick case, leading to harder spectra for the same parameter values. © European Southern Observatory (ESO) 1997 Online publication: April 8, 1998 |