Astron. Astrophys. 319, 413-429 (1997)

5. The spectral energy distribution

5.1. Luminosity - redshift relation

The conversion of the observed flux into a luminosity involves, besides the distance term, a luminosity K-correction (Schmidt & Green 1986) given as

where is the energy index of the assumed power law continuum spectrum, S .

In the soft X-ray band the spectral slope itself is a function of redshift (see Fig. 9). In the optical band there are indications that the continuum slope is a function of the core dominance R (Baker & Hunstead 1995). Further, an additional correction taking into account the fact that the quasar spectrum is not strictly a power law, but is affected by emission lines and by the Lyman forest depleting the continuum to the blue of Ly (VV 93) should be considered.

For a redshift of z = 2, using for X-rays the range of photon indices as discussed in Sect. 4.4 and for the optical band the different values given by Baker & Hunstead (1995), the K-correction term amounts to approximately 0.57 . This means that an inappropriate knowledge of the spectral properties of a source at higher redshifts can lead to uncertainties in the calculated luminosity of a factor of 3.

In the following we will use for the K-correction in the radio band the power law index determined as above for each individual source. In the optical we take a continuum slope of with line corrections as given by Avni & Tananbaum (1986), and in the X-ray range the z-dependent average power law indices as determined in Sect. 4.4.

In Fig. 11 we show the K-corrected (0.1 - 2.4 keV) luminosities of our sample as a function of redshift. The different object classes are indicated by the usual symbols. For illustrative purposes we include (full curve) the typical Survey detection limit for a source with an X-ray flux of erg cm , ignoring the effects of Galactic absorption. The luminosity K-correction for this curve was done as well with a redshift dependent photon index as determined in Sect. 4.4. It can be seen that many sources have luminosities definitely below the Survey flux limit and are thus only detected in pointed observations. At higher redshifts only flat spectrum and GPS objects are found. A relatively high fraction of very luminous objects can be seen around z 2, in accordance with current schemes of quasar luminosity evolution. The two brightest sources near z = 2 are PKS 2149-306 and the - bright quasar S5 0836+710.

Fig. 11. (0.1-2.4 keV) soft X-ray luminosity as a function of the redshift z. Different object classes are indicated by the usual symbols. Plus signs ( + ) indicate sources without known radio slopes. The full curve represents the luminosity of a source at a Survey flux limit of erg cm , K-corrected with a redshift dependent photon index.

5.2. Luminosity - luminosity correlations

Previous studies of radio-loud quasars (Avni & Tananbaum 1986, Browne & Murphy 1987, Kembhavi et al. 1986, Worrall et al. 1987, Wilkes et al. 1994, Paper I,  II) find a non-linear relationship between the X-ray and optical luminosities of the form . A knowledge of the exact form of this relation is required to relate the quasar statistics (evolution, luminosity function) in the two wave-bands and to understand the quasar's broad band emission. The fitted slopes of the regression curves in the individual papers differ markedly and they further seem to depend on the sample under consideration and on the method employed for the determination of the correlation (see the discussion by Padovani 1992, Franceschini et al. 1994, and La Franca et al. 1995).

Most of the data used previously were from Einstein IPC observations with relatively low quality spectral information and a major source of the statistical uncertainties were the generally rather small samples of different object classes. In both respects the  ROSAT   data provide a substantial improvement and considering the importance of obtaining an exact value of the slope of the luminosity correlations we test different mathematical correlation analysis methods to understand their limitations.

For this we used from the total data set a 'test sample' of 278 flat spectrum sources with 29.6 log(l _o) . When errors were considered in the analysis we took mag for the optical data , and the statistical errors from Table 1 for the X-ray fluxes. In Table 2 we present the results of various methods for the determination of the slope of a correlation of the form log(l _x) log(l _o).

Table 2. Regression analysis for test sample of flat spectrum sources

In the first two tests we employed an unweighted ordinary least square regression which can be found in standard analysis packages like IDL or Numerical Recipes. Taking the x-coordinate (optical luminosity) as independent and the X-ray luminosity as dependent variable, both without errors, we obtain a slope similar to that found in Papers I, II and others. However, reversing dependent and independent variables the fitted slope is not the inverse of the previous one. This indicates that there is no 'physical' separation into an 'independent' coordinate and a coordinate dependent 'variable', and as a more 'reasonable slope' the bi-section of these two slope should be used (Feigelson & Babu 1992). Including the errors of the X-ray luminosity into the analysis results in a similar slope (inside the errors) as without errors. This means that our sample is large enough not to be dominated by statistical errors and that the data points are statistically independent. The method of orthogonal distance regression (ODRPACK, Boggs et al. 1990) gives a slope around , if errors in both variables are considered. Finally, we tried a generalized orthogonal regression proposed by Fasano & Vio (1988, FV) which takes into account both measurement errors and intrinsic variances (i.e., scatter around the regression line) usually unaccounted for by the observational uncertainties. Setting the intrinsic variance we obtain a slope consistent with the corresponding ODR result. With free fitted variance the slope gets even steeper than unity with a variance considerably different from zero. This implies that there is already some intrinsic scatter in the correlation of the luminosities, not accounted for in standard regression analysis techniques.

Thus, considering the relatively large uncertainties inherent in the data, an ODR regression analysis, taking into account errors in both variables is most appropriate, at least for luminosity - luminosity correlations. Therefore, we will employ this method in the following sections for the study of these correlations.

5.3. l _x - l _o - correlation

Fig. 12 shows the correlation of the monochromatic 2 keV X-ray luminosity with the monochromatic 2500 Å optical luminosity. An ODR regression analysis as described above gives a linear fit log (l _x) = (log(l _o) - 30.5) + (27.33 0.07) for the flat spectrum sources and for the steep spectrum sources log (l _x) = (log(l _o) - 30.5) + (27.08 0.08). The Spearman rank correlation coefficients are = 0.71 for flat spectrum and = 0.61 for steep spectrum objects, respectively. For both cases the probability levels are . The different values for the regression constant imply that for a given optical luminosity flat spectrum quasars are about 50% X-ray brighter than steep spectrum quasars.

Fig. 12. The monochromatic X-ray luminosity as a function of the optical 2500 Å luminosity. Symbols as in Fig. 6. The linear regression lines are plotted for flat spectrum quasars (dashed line) and steep spectrum quasars (dash-dotted line).

Very similar slopes for the l _x - l _o relation have been found for the optically selected LBQS sample (Green et al. 1995), for an X-ray selected sample of ROSAT and EMSS sources (Boyle et al. 1993), as well as in many earlier studies (e.g. Wilkes et al. 1994). This seems to indicate that the high energy emission in quasars is of similar origin, independent of the quasar's radio properties.

Of particular interest is the sample of 252 flat spectrum objects, which show a definite flattening of the correlation at high optical luminosities. Using a broken slope for the regression line (break luminosity at log(l _o) the two slopes turn out to be and , respectively. This flattening is obviously a luminosity, not a redshift effect, as an inspection of the optically very bright objects shows that they are found over the whole redshift range.

Ignoring the 10 optically brightest and 2 optically faintest objects, i.e., taking a truncated sample of 240 flat spectrum sources (again with 29.6 log(l _o) ) and performing a regression analysis for restricted redshift ranges we find a steepening of the regression slope with increasing redshift (Table 3).

Table 3. Redshift dependent regression analysis for flat spectrum sources

Whether this steepening is a genuine physical effect needs to be investigated as the influence of observational biases and redshift dependent selection effects on the correlations are not fully understood at present. Changes of the luminosity K-correction for the optical luminosities by varying the spectral slopes () lead to slightly different slopes in the various redshift bins but the general trend of steepening with redshift remains. Worrall et al. (1987) claim that the inclusion of a redshift term in the analysis does not change the correlations noticeably. These findings have been criticized by Padovani (1992), noting that the samples used were inhomogeneous and incomplete.

The possible introduction of detection biases through distance (redshift) effects does not justify the search for correlations amongst the corresponding flux densities. If there exist a relationship between the intrinsic luminosities, for example of the form , a correlation between the flux densities might disappear unless , as shown by Feigelson & Berg (1983) and Kembhavi et al. (1986). To assess quantitatively the extent of a spurious correlation introduced by distance effects the partial linear correlation coefficient R (e.g. Hald 1952) can be used. We determined the partial correlation coefficients R with the effect of redshift (i.e. luminosity distance) eliminated for all luminosity - luminosity correlations studied in this paper and found that in all cases the influence of a luminosity - redshift dependence on the correlations can be ruled out with high probability ().

Luminosity correlations can suffer biases if upper limits are not included in the analysis. We used the tools of the survival analysis package ASURV Rev 1.3 (LaValley et al. 1992) and tested the effects of including the upper limits in the regression analysis. Remembering, that no measurement errors are considered in that analysis the regression slope gets only slightly flatter when the upper limits are included. For example, the slope of the l _x - l _o  regression changes from 0.74 (no upper limits) to 0.70 with upper limits. These changes are smaller than the differences found between various regression analysis methods.

Finally, there might be observational biases as X-ray flux limited observations will preferentially select objects with larger values of l _x /l _o  as discussed by Cheng et al. (1984).

Summarizing, we can say that a simple correlation l _x l _o   as favored by some evolutionary considerations cannot be ruled out, but the various subsamples show different slopes, redshift dependencies, changes in the slopes as a function of the sample parameters and, over all, possibly a non negligible intrinsic dispersion of the source properties. These effects can be imposed by selection and detection biases or by intrinsic, physically different source properties not yet accounted for. Thus, although we have at hand the by far largest sample of X-ray detected radio-loud quasars the conclusions we draw might be valid only for the objects we have 'seen'. In particular, there might be a whole population of 'X-ray quiet' radio-loud quasars, as already discussed in Paper I.

5.4. l _x - l _r - correlations

In Fig.13 we plot the monochromatic X-ray luminosity at 2 keV as function of the 5 GHz radio luminosity: in the upper panel the total radio luminosity is taken, in the bottom panel (for objects where these data are available) the core luminosity.

Fig. 13. The monochromatic X-ray luminosity as a function of the total 5 GHz luminosity (upper panel) and as a function of the 5 GHz core luminosity (lower panel). Open circles: steep spectrum quasars, dots: flat spectrum quasars. The linear regression line is for objects with core luminosities higher than log .

An ODR correlation analysis gives for the correlation of the X-ray luminosity with total radio luminosity a slope of for flat spectrum sources and for the steep spectrum sources. For the 5 GHz radio fluxes typical errors of 15% are used. However, the results are not sensitive to the actual magnitude of the errors. Both slopes are consistent with each other within their uncertainties, but flat spectrum sources are on average more luminous.

Separating the radio core and radio lobe components of a sample of 3CR quasars, Tananbaum et al. (1983) concluded that the X-ray flux is correlated with the core but not with the lobe radio flux. Later Kembhavi et al. (1986) confirmed the much tighter correlation between X-ray and radio core flux indicating that both emission processes are related and originate in the central regions of the objects. The correlation analysis for the X-ray versus core luminosity yielded for flat spectrum sources and for the steep spectrum sources , again consistent with a single slope within the mutual errors. We therefore fitted the whole sample with a single line and obtained a slope of . However, the data (Fig. 13, bottom panel) seem to indicate a steepening of the slope towards higher luminosities ( erg s^-1 Hz^-1). A fit of two lines with a break luminosity of log (l _r) 34.2 resulted in a flat slope of at luminosities lower than the break luminosity, and of at higher core luminosities. However, the reduced of this fit is only marginally lower than the fit with a single line. We thus cannot confirm on a statistically sound basis a clear separation of steep spectrum and flat spectrum quasars as claimed by Baker et al. (1995) for a much smaller data set. However, Fig. 13 is indicative for the existence of two X-ray components, which can be interpreted in terms of two distinct components of the X-ray emission, an unbeamed and a beamed one correlated with (Baker et al. 1995, Browne & Murphy 1987).

To test the relationship between radio morphology and X-ray luminosity we separated out the contribution from the flat spectrum cores to the X-ray luminosity, given by the fitted form at high R (see below, Table 4 and the dashed line in Fig. 15) log (l _x) = 0.97 (log () - 34) + 27.17. With this relation we estimated the core X-ray luminosity . In Fig.14 we plot the logarithmic ratio log(l _x / ) as a function of the radio core dominance R for all objects with known radio core luminosity. This corresponds to the the excess values of log(l _x) above the extrapolated line for the core X-ray luminosity . The ratio log(l _x / ) is found to anti-correlate strongly with R. The dashed line shows the predictions by the two component model of Kembhavi (1993) which are slightly different from those of Browne & Murphy (1987) (see Baker et al. 1995). However, both models contain some ad hoc assumptions which do not seem to hold in the light of our data.

Table 4. Regression analysis for various core dominances

Fig. 14. Plot of the logarithmic ratio of total-to-core X-ray luminosities versus R. See text for details.

Fig. 15. The monochromatic X-ray luminosity as a function of the 5 GHz core luminosity. Full dots are objects with core dominance , open circles objects with .

Our sample is large enough to study subgroups characterized by suitably restricted radio parameters. We find that the measured value of the core flux (i.e., the core luminosity alone) is possibly not the physically dominating parameter. The radio core dominance, R, in connection with seems to be more suited. Performing a regression analysis of l _x in various ranges of the core dominance parameter we find that the slope of the regression curve steepens with R. In Table 4 we list the results of the analysis. Given are the core dominance parameters, the number of objects, and the slope of the correlation, determined by an ODR analysis.

There is a clear trend of a steepening of the slope with increasing core dominance. In Fig. 15 we show the obtained correlation for the objects with highest core dominance (, full dots) and for the objects with very low values of , (open circles). It is interesting to note, that the core luminosity spans more than two orders of magnitudes even for the restricted ranges of R, which indicates large intrinsic differences in the power of the jets. And, even at lowest R-values, where the emission should be completely dominated by unbeamed emission, a strict correlation holds between l _x  and , showing that the unbeamed radio and X-ray components are as well correlated over a wide range of the intrinsic power of the cores.

Further, at a given core luminosity there is a clear tendency that objects with lower R - values show higher X-ray luminosities. From the definition of R this means that even the extended radio flux is positively correlated with the X-ray luminosity, however, with a different slope. Therefore, any simple correlation found between luminosities like l _x must be questionable as other physically relevant parameters are neglected.

Finally, it should be noted that a similar analysis for a R - dependent l _x - l _o correlation does not reveal statistically significant deviations between the slopes of the correlations. This seems to imply that the optical and the X-ray emission are directly linked, i.e., the optical emission is partly beamed as well.

5.5. Redshift - independent quantities

Some of the correlations could be introduced by distance dependent luminosity selection through the inherent sensitivity limits of the different catalogues. Nearly distance independent correlation diagrams can be constructed by using flux or luminosity ratios. It must be noted, however, that the applied K-corrections still introduce different z-dependencies into the calculated luminosities and that the samples themselves are possibly subject to Malmquist biases.

In Fig. 16 we plot the ratios l _x /l _r versus l _o /l _r for the sample of both, radio-loud and radio-quiet sources for which 5 GHz fluxes are available. The flat spectrum and steep spectrum quasars populate a well defined, relatively narrow region in the diagram with an obvious correlation between the luminosity ratios. A regression analysis for the logarithmic luminosity ratios for flat and steep spectrum objects gives a slope of with a correlation coefficient of R_sp = 0.58 and a nearly 100% confidence level for the existence of a correlation. This correlation between luminosity ratios for quasars would imply that we have a dependency of the form l _x l _r l _o , as claimed previously by Kembhavi et al. (1986), Worrall et al. (1987), and others. This further argues against a simple correlation of the form l _x l _o  discussed above in Sect. 5.2.

Fig. 16. Luminosity - ratios l x /l r versus l o /l r. The linear regression line for the steep and flat spectrum sources is plotted as dash-dotted line. The symbols have the usual meaning; crosses are the radio detected quasars not fulfilling the 'radio-loud' flux ratio criterion.

The relatively narrow range of X-ray - to - radio flux ratios (less than two orders of magnitude) compared to the wider spread along the horizontal axis in Fig.16 argues for a closer link between these two radiation mechanisms than that for the X-ray and optical emission. The two objects with very large l _x /l _r ratios are PKS 0558-504 and OZ 453.7 which seem to be 'peculiar' in a general sense. They will be discussed in a subsequent paper, together with other extreme objects of the sample ². Interestingly, the objects with very low radio fluxes, not qualifying as radio-loud quasars are close (but slightly above) the extrapolation of the regression line of the radio-loud objects. They are characterized by a diminishing influence of the radio emission properties and might follow a simple l _x l _o  correlation which will be discussed in more detail by Yuan et al. (1996).

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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