The X-ray continuum time-lags and intrinsic coherence in AGN
We present the results from a systematic analysis of the X-ray continuum (‘hard’) time-lags and intrinsic coherence between the and various energy bands in the range, for ten X-ray bright and highly variable active galactic nuclei (AGN). We used all available archival XMM-Newton data, and estimated the time-lags following Epitropakis & Papadakis (2016). By performing extensive numerical simulations, we arrived at useful guidelines for computing intrinsic coherence estimates that are minimally biased, have known errors, and are (approximately) Gaussian distributed. Owing to the way we estimated the time-lags and intrinsic coherence, we were able to do a proper model fitting to the data. Regarding the continuum time-lags, we are able to demonstrate that they have a power-law dependence on frequency, with a slope of , and that their amplitude scales with the logarithm of the light-curve mean-energy ratio. We also find that their amplitude increases with the square root of the X-ray Eddington ratio. Regarding the intrinsic coherence, we found that it is approximately constant at low frequencies. It then decreases exponentially at frequencies higher than a characteristic ‘break frequency.’ Both the low-frequency constant intrinsic-coherence value and the break frequency have a logarithmic dependence on the light-curve mean-energy ratio. Neither the low-frequency constant intrinsic-coherence value, nor the break frequency exhibit a universal scaling with either the central black hole mass, or the the X-ray Eddington ratio. Our results could constrain various theoretical models of AGN X-ray variability.
keywords:galaxies: active – X-rays: galaxies – accretion, accretion discs – galaxies: Seyfert – relativistic processes
According to the currently accepted paradigm, active galactic nuclei (AGN) contain a central, super-massive () black hole (BH), onto which matter accretes in a disc-like configuration. A fraction of the low-energy photons emitted by the disc is assumed to be Compton up-scattered by a population of high-energy () electrons, which is often referred to as the X-ray corona. The Compton up-scattered disc photons form a power-law spectrum that is observed in the X-ray spectra of AGN. We will henceforth refer to this source as the X-ray source, and to its emission as the X-ray continuum emission.
In addition to spectral studies, X-ray variability studies can also provide valuable information that can be used to understand the nature of the X-ray source in AGN, which remains largely unknown. One particular, and rather powerful, variability analysis tool is the estimation of ‘time-lags’ (delays) between temporal variations of the X-ray continuum emission in different energy bands (we will henceforth refer to these time-lags as the continuum time-lags). Such studies were first performed for X-ray binaries (XRBs; e.g. Miyamoto & Kitamoto, 1989; Nowak & Vaughan, 1996; Nowak et al., 1999; Pottschmidt et al., 2000), which are also thought to be compact accreting systems, where the central BH has a mass of . Several characteristics of continuum time-lags in XRBs have since been established: a) variations in hard energy-bands are delayed with respect to variations in softer energy-bands, b) the time-lags have an approximately power-law dependence on temporal frequency (with their magnitude decreasing with increasing frequency), and c) the magnitude of the continuum time-lags at a given frequency has an approximately log-linear dependence on the energy separation of the light curves. Continuum time-lags with similar characteristics were later reported in several AGN as well (e.g. Papadakis et al., 2001; McHardy et al., 2004; Arévalo et al., 2006, 2008; Sriram et al., 2009).
Apart from the time-lags, an additional (potentially useful) tool in understanding the nature of the X-ray variability in compact accreting objects is the so-called coherence function, which is a measure of the degree of correlation between variations in two light curves as a function of temporal frequency (Vaughan & Nowak, 1997, henceforth VN97). When correcting for the effect of Poisson noise, the intrinsic coherence in XRBs is generally observed to be frequency- and energy-dependent, remaining close to unity for a wide range of frequencies and for light curves with a small energy separation (VN97). This behaviour is observed in AGN as well, although, contrary to time-lags, quantitative studies of the energy- and frequency-dependence of the intrinsic coherence are limited. This is partly because the methods of intrinsic coherence estimation have not been established as well as the time-lag estimation methods, and because its interpretation is less straight-forward.
The main aim of our work is to perform a systematic study of the energy- and frequency-dependence of the continuum time-lags and of the intrinsic coherence in AGN. To this end, we chose a sample of ten X-ray bright and variable AGN that have been observed many times by XMM-Newton. We relied on the work of Epitropakis & Papadakis (2016, EP16 hereafter) to calculate time-lags that are minimally biased, have known errors, and are approximately Gaussian distributed. Following their work, in this paper we also present the results from an extensive study of the statistical properties of the traditional, Fourier-based intrinsic coherence estimator. We provide practical guidelines that can be used to compute intrinsic coherence estimates that are minimally biased, have known errors, and are approximately Gaussian distributed.
We used all the existing XMM-Newton archival data for these objects to estimate the time-lags and intrinsic coherence between light curves in various energy bands. Our results provide a quantitative description of the dependence of the time-lags and intrinsic coherence on frequency and energy in AGN. We also provide results regarding their scaling with BH mass and (X-ray) Eddington ratio. Our results could be used to constrain theoretical models for the X-ray variability in AGN.
2 Observations and data reduction
|Source||Obs. ID||Exp.||Source||Obs. ID||Exp.|
|1H 0707–495||Mrk 766|
|Pan et al. (2016)||0506200201||38.6||Bentz et al. (2009)||0304030301||98.4|
|Bentz et al. (2016)||0029740801||124.1||IRAS 13224–3809|
|0693781201||131.6||Zhou & Wang (2005)||0673580201||86.7|
|Denney et al. (2010)||0606320101||45.2||0302850201||110.7|
|0606320201||44.0||Oliva et al. (1995)||0727960101||127.5|
|0117710601||15.9||Grier et al. (2012)||0510010701||16.7|
|Gliozzi et al. (2010)||0555170201||113.7||0600540601||114.1|
Table 1 lists the details of the XMM-Newton observations we used. Column 1 lists the source name, redshift, (taken from the NASA/IPAC Extragalactic Database (NED)), central BH mass, , estimate in units of (along with the respective reference below the listed value), and the mean luminosity, , in units of . The luminosity was determined using the mean fluxes listed in the RXTE AGN Timing & Spectral Database, and the respective luminosity distance values listed in the NED (assuming a -CDM cosmology with , , and ). The only exception is IRAS 13324–3809, which is not listed in the former database, for which we used the mean flux reported by Dewangan et al. (2002). In the same column we also list the ratio of the luminosity over the Eddington luminosity (henceforth, the X-ray Eddington ratio, ). Columns 2 and 3 of the same figure show the identification number (ID) of each observation and net exposure in units of , respectively.
We processed data from the XMM-Newton satellite using the Scientific Analysis System (SAS, v. 14.0.0; Gabriel et al., 2004). We only used EPIC-pn (Strüder et al., 2001) data. Source and background light curves were extracted from circular regions on the CCD. The source regions had a fixed radius of 800 pixels () centred on the source coordinates listed on the NASA/IPAC Extragalactic Database. The positions and radii of the background regions were determined by placing them sufficiently far from the location of the source, but within the boundaries of the same CCD chip.
Source and background light curves with a bin size of were extracted, using SAS command evselect, in the following energy bands: , , , , , , , , and . We included the criteria PATTERN==0–4 and FLAG==0 in the extraction process, which select only single- and double-pixel events and reject ‘bad’ pixels from the edges of the detector CCD chips. Periods of high flaring background activity owing to solar activity were determined by observing the light curves (which contain very few source photons) extracted from the whole surface of the detector, and subsequently excluded during the source and background light curve extraction process.
We checked all source light curves for pile-up using the SAS task epatplot, and found that only observations 0670130201, 0670130501, and 0670130901 of Ark 564 are affected. For those observations we used annular instead of circular source regions with inner radii of 280, 200, and 250 pixels (the outer radii were held at 800 pixels), respectively, which we found to adequately reduce the effects of pile-up.
The background light curves were then subtracted from the corresponding source light curves using the SAS command epiclccorr. Most of the resulting light curves were continuously sampled, except for a few cases that contained a small ( per cent of the total number of points in the light curve) number of missing points. These were either randomly distributed throughout the duration of an observation, or appeared in groups of points. We replaced the missing points by linear interpolation, with the addition of the appropriate Poisson noise.
3 Time-lag estimation
|Source||()||Mean c.r.||Source||()||Mean c.r.||/|
|1H 0707–495||(1.50)||0.528||22.0/7.5||Ark 564||(1.50)||9.149||19.6/6.6|
|PKS 0558–504||(1.50)||3.980||6.8/3.1||NGC 7314||(1.50)||2.075||15.5/9.1|
|Mrk 766||(1.50)||3.322||13.8/7.9||Mrk 335||(1.50)||2.689||6.0/3.0|
We calculated the time-lag estimates between light curves in seven energy bands, and the light curves in the energy band (henceforth, the reference band; the reason for this particular reference band choice is explained in Section 4). The energy bands, along with their mean energy, , are listed in column 2 of Table 2. In the case of MCG–5-23-16, which has a very low count rate at energies (owing to the fact that it is an absorbed AGN), we used light curves in the entire energy band. We chose the energy bands to be as narrow as possible to maximise energy resolution, while at the same time maintaining a relatively high mean count rate to minimise Poisson noise effects. We also considered the vs. band time-lags to determine the frequency range over which we fitted the observed time-lags at low frequencies (see Section 3.1). We used light curves with a bin size of . The time-lags were estimated following the prescription of EP16 to ensure that they (approximately) follow a Gaussian distribution with know errors. We provide below a short description of our methodology.
We first partitioned all available light curves in each energy band into segments of duration (the number of segments is listed in column 1 of Table 2). For a given pair of segments we calculated the so-called cross-periodogram at the frequencies , where ( is the total number of points in each segment). The cross-periodogram is an estimator of the cross-spectrum (CS), which, in turn, is a measure of the correlation between two random signals (Priestley, 1981, henceforth P81). Our final estimate for the CS, , was obtained by averaging the individual cross-periodograms at each frequency. We did not average over neighbouring frequencies, as this can introduce a bias at low frequencies (EP16). We only considered frequencies ( in our case) to minimise the effects of light-curve binning on the time-lag estimates.
The time-lag at each frequency is defined as the argument of the CS, divided by the angular frequency (P81). Following standard practice, we thus used
as our estimates of the time-lags and their corresponding error, respectively. The quantity is the so-called coherence estimate, which is defined as (P81; VN97)
and are the traditional periodograms of the two light curves, which are also calculated by averaging over segments. The coherence function between two random processes is a measure of the degree of linear correlation between their corresponding sinusoidal components at each frequency. As we explain in detail in Appendix A, the coherence estimate defined by equation 3 is a biased estimator of the intrinsic coherence of the measured processes. Nevertheless, its estimation plays a crucial role in the determination of reliable time-lag estimates, as demonstrated by EP16.
Figure 1 shows the vs. coherence and time-lag estimates (top and bottom panels, respectively) of Ark 564 and NGC 4051. Both sources are X-ray bright and highly variable. Figure 2 shows the same results for 1H 0707–495 and Mrk 335 (two sources that are fainter, and, in the case of Mrk 355, less variable). They were calculated using equations 3 and 1, respectively. The coherence estimates decrease with increasing frequency in all cases. EP16 showed that, in the presence of measurement errors, the coherence estimates converge to the constant at frequencies where the amplitude of the experimental noise dominates over the amplitude of the intrinsic variations. In fact, EP16 showed that, if the measured processes are intrinsically coherent (i.e. the intrinsic coherence function is equal to unity at all frequencies), is well-fitted by a function of the form
The error of the time-lag estimates increases as the coherence decreases. Therefore, we expect that Poisson noise will severely affect the reliability of the time-lag estimates above a certain critical frequency, . According to EP16, is the frequency at which the mean sample coherence function becomes equal to . At higher frequencies the analytic error prescription (equation 2) underestimates the true scatter of the time-lag estimates around their mean, their distribution becomes uniform, and their mean value converges to zero, irrespective of the intrinsic time-lag spectrum. At frequencies lower than , and as long as , the time-lag estimates are unbiased, equation 2 provides a reliable estimate of their true scatter around the mean, and their distribution is approximately Gaussian.
The horizontal blue dotted-dashed lines in the upper panels of Figs. 1 and 2 indicate the constant value of , and the vertical red lines indicate , i.e. the frequency at which the best-fitting coherence model is equal to this value. EP16 showed that, for a given intrinsic PSD, decreases with decreasing S/N of the light curves (in particular, the one with the smaller mean count rate). As the S/N decreases, the frequency range over which we can obtain realiable time-lag estimates decreases. Therefore, it is not surprising that the critical frequency is highest (lowest) in the case of NGC 4051 (Mrk 355), respectively. However, S/N is not the only parameter that determines . For example, despite the fact that the mean count rate of the and light curves is significantly higher in the case of Mrk 335, . This is because 1H 0707–495 is much more variable. Consequently, the amplitude of the intrinsic variations is higher than the amplitude of the Poisson noise variations in the case of 1H 0707–495, even at frequencies that are four times higher than .
We fitted the coherence estimates of each source (at all energy bands) to the exponential function given by equation 4. We then equated the best-fitting model to the constant to estimate . These values are listed in column 4 of Table 2. The observed time-lag spectra, for all the sources in our sample, are shown in Figs. 23, 25, 27, 29, 31, 33, 35, 37, 39, and 41 in Appendix B. The time-lag estimates in these figures are plotted at frequencies in each case. The time-lags were estimated such that a positive time-lag value indicates that variations in the reference band are delayed with respect to variations in the other energy band (and vice-versa).
The low frequency time-lags between the reference band and those at lower (higher) energies are positive (negative). This shows that X-ray continuum variations in hard energy-bands are always delayed with respect to variations in softer energy-bands. In all cases, the low frequency time-lag amplitude increases with increasing energy separation (the limits in the vertical axis are the same for all sample time-lag spectra in each figure). The frequency range of the vs. time-lags is the smallest among all sample time-lag spectra. This is because the count rate of the light curves is very small. We could not estimate the soft band time-lags seperately in the case of MCG–5-23-16, because the count rate of the corresponding light curves is almost zero (because of absorption). For this source we hence utilised the entire energy band, and estimated the corresponding vs. time-lags. The vs. time-lags of NGC 7314 are poorly determined for the same reason. On the other hand, the hard band time-lags are poorly determined in IRAS 13224–3809, because the source is not particularly bright and has a very soft energy spectrum, hence the count rate at energies is very small.
|1H 0707–495||0.28||Ark 564||0.28|
|PKS 0558–504||0.28||NGC 7314||0.28|
|Mrk 766||0.28||Mrk 335||0.28|
3.1 Modelling the sample time-lag spectra
The statistical properties of the sample time-lag spectra plotted in the relevant Appendix B figures are appropriate for model fitting, using traditional minimisation techniques. We fitted the sample time-lag spectra with a power-law function of the form
where is the mean energy of each band (listed in column 2 of Table 2), is the mean energy of the reference band, is the power-law slope, and is the (energy dependent) amplitude at .
We fitted the model in a limited frequency range between and . These values are listed in column 3 of Table 3, and are indicated by the vertical blue dotted-dashed lines in the vs. time-lag panel of the same figures. The points plotted with filled circles in all panels of the same figures indicate the time-lag estimates used for the model-fitting procedure. The low frequency limit, , is the lowest sampled frequency () in all sources except 1H 0707–495 and MCG–6-30-15. For these sources, we observe a low frequency turn-over in the sample time-lag spectra (see Figs. 23 and 25). This turn-over is more pronounced in the soft band time-lags. We decided to ignore the time-lags at these low frequencies, because the best-fitting results change significantly depending on whether we keep them or not. At high frequencies the sample time-lag spetra may change sign, most probably because of the presence of so-called X-ray reverberation time-lags. Since we are interested in studying the continuum time-lags, we decided to fit the sample time-lag spectra only at frequencies where the time-lags are predominately positive or negative (at energies lower or higher than the reference band, respectively). We defined as the frequency above which the probability that all vs. time-lag estimates in the range are positive is smaller than 0.01. This probability was calculated by assuming that the time-lag estimates have a Gaussian distribution (with a mean and standard deviation given by equation 1 and 2, respectively), and are independent at each frequency. In this case, the aforementioned probability is equal to the product of the integrated (Gaussian) probability distribution functions over the interval of all the time-lag estimates in the range .
For each source we fitted all available sample time-lag spectra simultaneously. We left as a free parameter, and kept the slope, , fixed at the same value for all time-lag spectra. We determined the best-fitting parameter values by locating the minimum of the function, , using the Levenberg-Marquardt method. The 68 per cent (95 per cent) confidence intervals of the best-fitting model parameters were determined by the standard () method for one independent parameter. Unless otherwise mentioned, best-fitting parameters will henceforth be quoted at the 68 per cent confidence level.
3.2 The best-fitting results
The best-fitting results are listed in Table 3, and the best-fitting models are shown as red dashed lines in the relevant Appendix B figures. The best-fitting models describe well the overall shape of the low-frequency sample time-lag spectra. The values in some cases (Ark 564, NGC 7314, and Mrk 335) imply that the power-law model does not fit the data well (the null hypothesis probability, , is smaller than 1 per cent). However, it is not easy to judge the quality of the fits in our case. Although the time-lag estimates should be uncorrelated at each frequency, the fact that the light curves in the various energy bands are correlated implies that (within each source) the time-lags between the reference band and different energy bands should be also be correlated to some extent. In this case, the actual number of degrees of freedom should be smaller than the numbers listed in Table 3. This would imply that the model fit may not be acceptable even in other sources as well, however, as we argue below, we do not believe this is the case.
We fitted the individual sample time-lag spectra of each source with the model defined by equation 5. The fit was acceptable in all cases (). The best-fitting slope values were consistent with the corresponding weighted-mean value for each source, hence the hypothesis of a constant (i.e. energy independent) slope is likely to be true. We could consider the best-fitting results from these fits, however the best-fitting amplitudes were poorly determined in that case. In fact, it was for this reason that we decided to fit all sample time-lag spectra simultaneously for each source: The best-fitting parameter values are consistent (within the errors) in both cases, but the errors are smaller when we fit all time-lag spectra simultaneously. We conclude that a power-law time-lag model, with the same slope at all energies, fits the sample time-lag spectra well.
Figure 3 shows the power-law amplitude, , plotted as a function of the light curve mean-energy ratio, . The logarithm of this ratio is a measure of the energy separation between the light curves. The amplitude’s sign ‘flips’ from positive to negative when and , respectively. This behaviour is the result of the fact that hard energy-band variations are delayed with respect to variations in softer energy-bands. The plots in Fig. 3 show that, in all sources, the power-law time-lag model amplitude increases with increasing energy separation. To quantify this trend we fitted the data plotted in the panels of Fig. 3 with the following model:
Equation 6 describes a function that becomes zero when , increases in magnitude with increasing , and whose sign shifts from positive to negative when and , respectively (as seen in the sample time-lag spectra). The amplitude corresponds to the power-law time-lag amplitude (at ) between the reference band and an energy band with (or ).
Our best-fitting results are listed in Table 4, and the best-fitting models are shown as red dashed lines in Fig. 3. The model fits the data well, except for PKS 0558–504, where the vs. power law time-lag amplitude appears to be significantly higher than for other energy bands. Perhaps the more significant discrepancy between the model and the data appears in Ark 564: a log-linear relation between the time-lag amplitude and energy may be just a first-order approximation in this case. Just like in PKS 0558–504, Mrk 766, and Mrk 335, the ‘amplitude vs. energy’ plot of Ark 564 suggests that the energy dependence is less (more) steep than what the model defined by equation 6 predicts when (), respectively (although the errors of the time-lag amplitudes are larger for the former sources compared to Ark 564).
4 Intrinsic coherence estimation
We discuss in detail the estimation of the intrinsic coherence between two light curves in Appendix A. We followed the prescription described in Section A.5, and estimated the sample intrinsic coherence function between the same light curves that we used to estimate the time-lag spectra. The results are plotted in Figs. 24, 26, 28, 30, 32, 34, 36, 38, 40, and 42 in Appendix B. We first calculated the intrinsic coherence estimates up to . The vertical, blue dotted-dashed lines in the panels of the same figures indicate (estimated as explained in Section A.2; these values are listed in column 4 of Table 1). The intrinsic coherence estimate (as defined by equation 11) at frequencies below should be an unbiased estimator of the intrinsic coherence. Their distribution should (roughly) follow a Gaussian distribution, and their error (as defined by equation A) should be representative of their intrinsic scatter around the mean, provided they are corrected as explained in Section A.3. In addition to those cases where we could not reliably estimate time-lags, turned out to be smaller than the lowest sampled frequency in a few other cases, owing to the very low count rate of the respective light curves (e.g. the vs. sample intrinsic coherence function of 1H 0707–495 and Mrk 335).
In many sources, the sample intrinsic coherence function is not equal to one, even at the lowest sampled frequencies, and they decrease rapidly with increasing frequency. We stress that, in this case, the loss of coherence at high frequencies is not due to the presence of experimental noise in the light curves. The intrinsic coherence amplitude appears to be energy dependent. For example, the vs. sample intrinsic coherence function of MCG–6-30-15 (see Fig. 26) is almost equal to one at all sampled frequencies but, clearly, the vs. sample intrinsic coherence is not equal to one, even at the lowest sampled frequency, and it decreases rapidly with increasing frequency. In fact, the vs. sample intrinsic coherence function (which we do not show here) is even smaller in amplitude.
Since time-lag estimation is less accurate when the coherence is low, we decided to choose the band as our reference band (as opposed to the lowest energy band, which is the usual choice) to estimate both the time-lags and the intrinsic coherence. This band has a mean energy that is around the middle of the total available XMM-Newton EPIC-pn energy range, and therefore the energy separation between and the lowest/highest energy bands we considered is somewhat balanced. In addition, the band is more representative of the X-ray continuum emission, as it is expected to be less affected by components originating from X-ray reflection, or the presence of a warm absorber, compared to other bands.
4.1 Modelling the sample intrinsic coherence
|Mrk 766||0.50||(5.2)||6.3/13||Mrk 335||0.50||(3.2)||4.2/3|
Based on the shape of the sample intrinsic coherence of most sources, we fitted the data with the following model:
Equation 7 describes a function that is constant at low frequencies (equal to ), and then decreases exponentially at frequencies above a ‘break’ frequency, . We determined the best-fitting model parameters using standard minimisation techniques (similar to the modelling of the sample time-lag spectra). The best-fitting results are listed in Table 5, and the best-fitting models are shown as red dashed lines in the relevant Appendix B figures. In some cases we did not detect a significant break frequency, and we list the 68 per cent lower limit on the corresponding best-fitting values in column 4 of Table 5 (we also show the 95 per cent lower limits in parentheses). Furthermore, in some cases, the best-fitting value was equal to one, and we list the respective 68 per cent lower limit in the same column.
In general, the model fits the data well in almost all cases. In most sources, decreases with increasing energy separation between the light curves. The loss of coherence is reinforced by the simultaneous decrease of with increasing energy separation (e.g. MGC–6-30-15, and NGC 4051). In some cases (e.g. IRAS 13224–3809) the intrinsic coherence is equal to one at low frequencies, for all energy separation values we considered. The loss of coherence in this case is because decreases strongly with increasing energy separation between the light curves. We investigate below these issues in more detail.
4.2 The energy dependence of the intrinsic coherence
Figures 4 and 5 show the best-fitting and values as a function . Each panel in these figures corresponds to a different source. The sources are divided into three groups (corresponding to the three rows in each figure) according to a common phenomenological behaviour of as a function of energy.
The first group (first row in Figs. 4 and 5) consists of IRAS 13224–3809, MCG–5-23-16, and NGC 7314 (henceforth, Group A). The best-fitting values of the Group A sources are consistent with one at all energies. The second group (second row in the same figures) consists of 1H 0707–495, MCG–6-30-15, NGC 4051, and Ark 564 (henceforth, Group B). The best-fitting values of Group B show a moderate (up to 10 per cent) decrease from the value of one as the energy separation increases. Arguably, the uncertainty of the best-fitting values of the Group A sources is larger than that those of the Group B sources, hence a meaningful quantitative comparison these two Groups cannot be determined very accurately. The third group (third row in the same figures) consists of PKS 0558–504, Mrk 766, and Mrk 335 (henceforth, Group C). The best-fitting values of Group C show a stronger (up to 30 per cent) decrease from the value of one as the energy separation increases.
To further investigate the dependence of on energy separation, we fitted the data plotted in the panels of Fig. 4 to a function of the form
We did not fit the Group A data because there either are few estimates, or the they are consistent with one. We only fitted the model to the values at soft energies (), as their error is smaller than at hard energies (). The best-fitting results are listed in Table 6. The model provides a statistically acceptable fit to the data of all sources. The Group B and Group C sources are characterised by significantly different best-fitting values. The weighted-mean value of the Group B and C sources is and , respectively. The values of the individual sources within the two Groups are consistent, within the errors, with the Group’s weighted-mean value.
Column 4 of Table 6 lists , where is the energy at which becomes equal to one. According to equation 8, . The value of cannot exceed , since this is the mean energy of the reference band111The best-fitting and values of NGC 4051 were such that ; for that reason we fitted the NGC 4051 data by setting during the fitting procedure, to force an amplitude of 1 for .. The best-fitting models are shown as red dashed lines in Fig. 4. The extension of the best-fitting lines at energies was done assuming that at energies between and , and that the vs. model is symmetric around , whereby (indicated by the vertical, blue dotted-dashed lines in the same figure). This assumption appears to be consistent with the MCG–6-30-15 and NGC 4051 data, where the hard-energy values are as accurately determined as the corresponding soft-energy values. The weighted-mean value is (which corresponds to a weighted-mean value of