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Astron. Astrophys. 326, 885-896 (1997)

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6. Results of the model fitting

As 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 [FORMULA] 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.

[FIGURE] Fig. 4a-c. UV to X-ray spectra of sample members #15, #17, and #18, showing UV and ROSAT data points (crosses represent 1 [FORMULA] error bars) and model prediction (solid line). The accretion disk component (dot-dashed) and hard X-ray (long dashes) and IR (short dashes) power laws are displayed separately. The spectrum is plotted in the source frame. Residuals are given in units of [FORMULA].

The resulting best fit parameters of all sample members, as well as their [FORMULA] errors and the corresponding minimum [FORMULA] 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 [FORMULA] 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 [FORMULA] and [FORMULA], respectively. The case that all sample members have the same [FORMULA] and/or [FORMULA] best-fit parameter values is excluded at a high statistical significance level (the confidence contours do not intersect the [FORMULA] line). We find a mean accretion rate of [FORMULA] within a relatively narrow parameter range ([FORMULA]). The best-fit accretion rates [FORMULA] are below [FORMULA] 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 ([FORMULA]) and are spread over a wider range ([FORMULA] for most objects), possibly suggesting some diversity of the underlying physical viscosity mechanism in our sample. Note that, according to its definition, [FORMULA] should not greatly exceed unity.

[FIGURE] Fig. 5. Distribution of best-fit accretion rates [FORMULA] in units of the Eddington accretion rate [FORMULA]. Confidence contours (68 %, 90 %, 99 %) of the mean and width (Gaussian [FORMULA]) of the distribution are plotted.

[FIGURE] Fig. 6. Distribution of best-fit viscosity parameters [FORMULA]. Confidence contours (68 %, 90 %, 99 %) of the mean and width (Gaussian [FORMULA]) of the distribution are plotted.

The best-fit central masses which roughly span two orders of magnitude ([FORMULA] ; 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 ([FORMULA]), 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 [FORMULA]) with the central masses over the whole dynamic range. We have tested for any dependencies of [FORMULA] and [FORMULA] on M and find that low central masses also seem to be associated with accretion at a lower fraction of the Eddington accretion rate, [FORMULA]: When M is increased by two orders of magnitude (from [FORMULA] to [FORMULA]) a moderate increase of [FORMULA] by roughly a factor of three (from 0.1 to 0.3) is observed. No such dependence of the viscosity parameter [FORMULA] on central mass is observed. Scatter plots of [FORMULA] and [FORMULA] plotted over M and redshift are shown in Fig. 9. High central masses also imply higher luminosities and, on average, larger distances. Any dependence on central mass thus is also expected to result in a similar dependence on redshift, as is observed (Fig. 9, lower panel). Note that the observed dependencies of [FORMULA] on central mass and redshift can not be attributed to selection effects, alone: Objects with, e.g., central masses of [FORMULA] at a redshift of [FORMULA] would be well above the respective X-ray and UV sensitivity limits if their mass accretion rates were higher than the observed values [FORMULA].

[FIGURE] Fig. 7. Histogram of best-fit central masses, M.

[FIGURE] Fig. 8. Histogram of best-fit mass accretion rates, [FORMULA], in units of the Eddington accretion rate, [FORMULA].

[FIGURE] Fig. 9. Best-fit accretion rates in units of the Eddington accretion rate and viscosity parameters plotted over best-fit central masses and redshifts.

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 [FORMULA] for the total sample; 0.15 - 1.0 [FORMULA] 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 ([FORMULA]) 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 [FORMULA] (see Fig. 10). When one anomalous object, # 20 (Table 1), with the lowest [FORMULA], and at the same time the highest [FORMULA] and [FORMULA]) values in the sample is removed, the probabilities for randomness of the observed correlations of [FORMULA] on M, [FORMULA], and [FORMULA] are 0.16, 0.04, and 0.02, respectively (Spearman rank correlation). Note that both M and [FORMULA] contribute to the total luminosity of the disk ([FORMULA]). Since in the optical spectral range the emission is dominated by the accretion disk while at X-ray energies a large fraction of the emission is supplied by the hard power law component, an increase of either M or [FORMULA] predominantly affects the optical emission and thus results in an increase of the broad-band spectral index [FORMULA], in agreement with the observed correlations. By increasing the viscosity parameter [FORMULA] a larger part of the disk emission is radiated in the X-ray range, thus resulting in a hardening of the broad-band spectral index [FORMULA], again in agreement with observations.

[FIGURE] Fig. 10. [FORMULA] plotted over accretion disk parameters, central mass M, accretion rate [FORMULA], and viscosity parameter [FORMULA]. One anomalous object with the lowest [FORMULA] value in the sample is plotted as an open dimond.

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.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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