Studying the relationship between X-ray emission and accretion in AGN using the XMM–Newton Bright Serendipitous Survey

Abstract

INTRODUCTION

The engine of active galactic nuclei (AGN) is powered by the accretion of matter on to the supermassive black hole (SMBH), placed in the centre of the host galaxy: the matter is heated (10⁵–10⁶ K) through viscous and magnetic processes and forms an accretion disc around the SMBH emitting in the ultraviolet (UV)–optical region. A fraction of energy is also emitted in the X-ray band with a spectrum that can be represented, at first order, by a power law from 0.1 to 100 keV at rest frame. In the now accepted disc–corona model (Haardt & Maraschi 1991), the X-rays are produced in a hot (T = 10⁸–10⁹ K) corona, reprocessing the primary UV–optical emission of the disc via inverse-Compton mechanism. X-rays are a probe of accretion since they are produced in the very inner part of the nucleus and carry direct information on the physics very close to the SMBH. For instance, the hard X-ray spectral index (Γ) gives direct information about the energy distribution of the electrons in the corona, while the intensity of the X-ray emission with respect to the UV–optical emission quantifies the relative importance between disc and corona. This latter quantity is often parametrized with the X-ray bolometric correction K_bol (e.g. Vasudevan & Fabian 2009), defined as the ratio between bolometric luminosity and 2–10 keV luminosity, or with the two-point spectral index α_OX (e.g. Vignali, Brandt & Schneider 2003), defined between 2500 Å and 2 keV. The different values of X-ray spectral index and of the disc/corona luminosity ratio observed from source to source are likely a consequence of fundamental differences in the physical parameters of the central engine.

First studies, essentially based on ROSAT data, suggested correlations between the ‘soft’ spectral index Γ_{(0.5–2.4) keV} and the full width at half-maximum (FWHM) of Hβ emission line coming from the broad line region (BLR; Wang, Brinkmann & Bergeron 1996; Laor et al. 1997; Sulentic, Marziani & Dultzin-Hacyan 2000; Grupe et al. 2004). Assuming that BLR dynamics is directly dependent on the black hole (BH) mass, this correlation was suggesting a direct link between Γ_{(0.5–2.4) keV} and some physical parameters like the BH mass or accretion rate. In particular, it was suggested that the main physical driver of this correlation is the accretion rate normalized to the Eddington luminosity¹ (Eddington ratio): sources accreting close to the Eddington limit produce the steepest values of Γ_{(0.5–2.4) keV} (Laor et al. 1997; Sulentic et al. 2000; Grupe et al. 2004; Wang, Watarai & Mineshige 2004). However, since the measured value of Γ_{0.5–2.4 keV} can be significantly contaminated by the presence of a spectral component called ‘soft excess’,² it was difficult to establish on a firm ground whether it was the slope of the primary emission that correlates with the accretion rate or, instead, it was the intensity of the soft excess.

Using ASCA observations, Brandt, Mathur & Elvis (1997) and Wang et al. (2004) have found that also the ‘hard’ spectral slope (Γ_{(2–10) keV}) has a strong dependence with the FWHM(Hβ). Since the 2–10 keV energy range is not affected by the ‘soft excess’, this result was considered as a compelling indication that the slope of the primary component of the X-ray emission actually correlates with FWHM(Hβ). First studies made with XMM–Newton, Chandra and Swift-X-Ray Telescope (XRT) have further suggested the possible presence of a second trend, i.e. an anticorrelation between Γ_{(2–10) keV} and the BH mass M_BH (Porquet et al. 2004; Piconcelli et al. 2005). The availability of hard X-ray data from XMM–Newton and Chandra and of statistical relations that allow the systematic computation of M_BH on large numbers of AGN have produced in the very recent years a big leap forward on this kind of study, extending the analysis on significantly larger samples, including up to a few hundreds of sources (Kelly et al. 2008; Shemmer et al. 2008; Gu & Cao 2009; Risaliti, Young & Elvis 2009; Grupe et al. 2010; Zhou & Zhang 2010). These studies seem to confirm the presence of a correlation between the hard Γ and the Eddington ratio (Risaliti et al. 2009; Grupe et al. 2010) with some exceptions (Bianchi et al. 2009). Shemmer et al. (2008) have also demonstrated that the observed strong anticorrelation usually observed between Γ and FWHM(Hβ) is a secondary correlation induced by the dependence between Γ and the Eddington ratio.

Also the bolometric correction is expected to be related to the physical parameters that regulate the accretion mechanism. A possible dependence of the K_bol with the luminosity has been suggested (Marconi et al. 2004; Hopkins, Richards & Herquist 2007), but more recent observations seem to point out that the principal dependence is between K_bol and the Eddington ratio (Vasudevan & Fabian 2007, 2009; Kelly et al. 2008; Lusso et al. 2012). An alternative way to study the relative intensity between disc and corona is through the α_OX, defined as the slope between 2500 Å and 2 keV. Past studies generally found a strong correlation between α_OX and L_UV (e.g. Vignali et al. 2003; Marchese et al. 2012) or L_bol (Kelly et al. 2008; Shemmer et al. 2008) while a dependence of α_OX with the Eddington ratio is usually weak or absent (Young, Elvis & Risaliti 2010), contrary to what has been found for K_bol. This is quite surprising since K_bol and α_OX are both supposed to be proxies of the disc/corona relative intensity and, therefore, they are somehow expected to behave in a similar way.

In this paper we investigate the link between X-ray properties and the accretion rate by analysing a well-defined sample of type 1 AGN selected from the XMM–Newton Bright Serendipitous Survey (XBS). In particular, we study the spectral index Γ estimated in the energy range 0.5–10 and 2–10 keV and three different parameters that quantify the ‘X-ray loudness’, i.e. the bolometric correction K_bol, the α_OX and the disc/corona luminosity ratio (i.e. the ratio between the accretion disc luminosity and the 0.1–100 keV X-ray luminosity). The approach followed in this study is to search for statistically significant correlations between these parameters and the value of accretion rate, both absolute and normalized to Eddington luminosity.

The structure of the paper is the following. In Section 2 we describe the survey, the sample selection and the parameters used for our work. In Section 3 we describe the statistical analysis used to find the correlations between the parameters, taking into account a number of potential biases. In Section 4 we present our results. Finally, in Section 5 we report the summary and conclusions.

We assume here a flat Λ cold dark matter (ΛCDM) cosmology with H₀ = 65 km s⁻¹ Mpc⁻¹, |$\Omega _\Lambda = 0.7$| and Ω_M = 0.3.

XMM–NEWTON BRIGHT SERENDIPITOUS SURVEY

The XBS is a wide-angle (∼28 deg²) high Galactic latitude (|b| > 20°) survey based on the XMM–Newton archival data. It is composed of two flux-limited samples: the XMM Bright Source Sample (BSS; 0.5–4.5 keV band, 389 sources) and the XMM Hard Bright Source Sample (HBSS; 4.5–7.5 keV band, 67 sources, with 56 sources in common with the BSS sample), having a flux limit of ∼7 × 10⁻¹⁴ erg cm⁻² s⁻¹ in both energy selection bands. Selection criteria and properties of these samples are described in Della Ceca et al. (2004). The XBS is composed of sources that are detected serendipitously in the field-of-view of the XMM–Newton pointing, thus excluding the targets of the observations. For this reason the XBS can be considered as representative of the X-ray sky down to its flux limit.

To date, the spectroscopic identification level has reached 98 and 100 per cent in the BSS and the HBSS samples, respectively. Most of the spectroscopic identifications are presented and discussed in Caccianiga et al. (2007, 2008).

The availability of good XMM–Newton data for the sources in the XBS sample, spanning the energy range between ∼0.3 and ∼10 keV, allowed us to perform a reliable X-ray spectral analysis for almost every AGN of the sample (Corral et al. 2011).

The sample

Since the goal of this paper is the study of the possible dependence of Γ, K_bol, α_OX and the disc/corona luminosity ratio on the accretion rate, we restrict the analysis to the subsample of radio-quiet 154 type 1 AGN for which all these parameters have been already derived by fitting the UV–optical spectral energy distribution (SEDs) of the sources (Marchese et al. 2012) and by studying the X-ray and optical spectra. The radio-loud AGN of the sample (see Galbiati et al. 2005) were not considered to avoid possible contamination from the relativistic jet to the SED. The analysis of the SEDs was carried out on a subset of objects for which optical and UV data are available (either a detection or an upper limit) from existing catalogues [Sloan Digital Sky Survey (SDSS) and Galaxy Evolution Explorer (GALEX)]. Since the availability of these data depends mainly on the position of the source in the sky and not on its intrinsic properties, this subset can be confidently considered as a representative subsample of the original one (see Marchese et al. 2012). In addition, in order to minimize the uncertainties on the values of L_bol, we have further restricted the analysis on a subsample of objects for which the possible effects of absorption are negligible, i.e. type 1 AGN with an intrinsic absorbing column density, measured from the X-ray spectra, below 5 × 10²⁰ cm⁻². Finally, we have excluded from the analysis the small fraction (∼8 per cent) of ‘elusive’ type 1 AGN, i.e. those sources whose optical spectrum is dominated by the host galaxy (see Severgnini 2003; Caccianiga et al. 2007), due to the impossibility of computing the BH mass through the single epoch (SE) spectral method (e.g. see Peterson 2010; Marziani & Sulentic 2012). In total, the final sample contains 71 objects. A Kolmogorov–Smirvov (K–S) test indicates that this subsample is not statistically different (at 95 per cent confidence level) from the original one from what concerns the Eddington ratio (Fig. 1) and the redshift (Fig. 2) distributions. We have also evaluated the possible impact of the exclusion of ‘elusive’ AGN from the analysis (see Section 3). The final sample used in this work consists of type 1 AGN with rest frame 2–10 keV luminosities ranging from 6 × 10⁴¹ to 9 × 10⁴⁶ erg s⁻¹ and redshift from 0.04 to 2.

Figure 1.

Eddington ratio distribution for the total sample presented in Marchese et al. (2012) (solid black line, 154 AGN) and for the subsample used here (dashed blue line, 71 AGN). The K–S test gives a probability for the null hypothesis (i.e. the two distributions are drawn from the same parent population) of 0.12.

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Figure 2.

Redshift distribution for the total sample presented in Marchese et al. (2012) (solid black line, 154 AGN) and for the subsample used here (dashed blue line, 71 AGN). The K–S test gives a probability for the null hypothesis (i.e. the two distributions are drawn from the same parent population) of 0.35.

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Parameters

In this section, we describe the methods adopted to determine the parameters of interest (all reported in Table C1).

• L_bol and |$\dot{M}$|⁠. Bolometric luminosities were obtained as the sum of the accretion disc luminosity (⁠|$L_{\rm {\rm disc}}$|⁠) and the 0.1–100 keV X-ray luminosity. |$L_{\rm {\rm disc}}$| was obtained by fitting the optical–UV data with a disc model (Marchese et al. 2012), while L_X was obtained by extrapolating the results obtained in the 2–10 keV energy range analysing the XMM–Newton data (Corral et al. 2011). As described in Marchese et al. (2012), the uncertainties on the bolometric luminosities take into account both the statistical errors on photometry and additional sources of error due to the correction for the intrinsic extinction and the long-term variability (since the used photometric data are not simultaneous).

   From bolometric luminosities we estimate the absolute accretion rate, defined as

  \begin{equation} \dot{M} = \frac{L_{\rm bol}}{\eta c^2}, \end{equation}

  (1)

  where η is the efficiency of the mass to energy conversion, assumed to be 0.1. The uncertainties associated with the values of |$\dot{M}$| in Table C1 are those related to the bolometric luminosity, i.e. we do not assume any error on η. The uncertainty on this value is difficult to assess. Marconi et al. (2004) estimate a range of values for η between 0.04 and 0.16 and, therefore, an additional uncertainty on |$\dot{M}$| up to a factor of ∼2 could be expected, besides that reported in Table C1. We note that, as explained above, the bolometric luminosities include the X-ray emission. Therefore, by using these bolometric luminosities to compute |$\dot{M}$| we are implicitly assuming that the energy budget carried by the X-ray emission is directly related to the accretion process.
• M_BH and Eddington ratio λ. Black hole masses of the XBS type 1 AGN are computed in Caccianiga et al. (2013) using the SE method (Peterson 2010; Marziani & Sulentic 2012). This method assumes that the BLR is gravitationally influenced by the SMBH, so the virial theorem can be applied. The velocity dispersion is derived from the broad emission line widths while the BLR size is estimated from the continuum luminosity. The choice of emission lines used for M_BH estimate depends on the redshift of the source. In this sample we used Hβ (for 0 < z ≤ 0.8) and Mg ii at 2798 Å lines (for 0.8 < z ≤ 2). In particular, we adopted the relation discussed in Vestergaard & Peterson (2006) for the Hβ:

  \begin{eqnarray} \mathrm{Log} M_{\rm BH} &=& 6.91 + 2\,\mathrm{Log} \frac{\mathrm{FWHM (H\beta )}}{1000\,\mathrm{km\,s}^{-1}}\nonumber \\ &&+\, 0 .50\,\mathrm{Log} \frac{\lambda L_{\rm 5100\,{A\!\!\!\!\!^{^\circ}}}}{10^{44}\,\mathrm{erg\,s}^{-1}}, \end{eqnarray}

  (2)

  and the relation presented in Shen et al. (2011) for the Mg ii λ2798 Å line:

  \begin{eqnarray} \mathrm{Log} M_{\rm BH} &=& 6.74 + 2\,\mathrm{Log} \frac{\mathrm{FWHM (Mg\,\small {II})}}{1000\,\mathrm{km\,s}^{-1}}\nonumber \\ && +\, 0.62 \,\mathrm{Log} \frac{\lambda L_{\rm 3000\,{A\!\!\!\!\!^{^\circ}}}}{10^{44}\,\mathrm{erg\,s}^{-1}}, \end{eqnarray}

  (3)

  the latter equation has been obtained by Shen et al. (2011) in such a way that the zero-order point (the virial factor) is the same as in the Hβ relation presented above so that the masses are consistently derived from these two equations (see the discussion in Shen et al. 2011). In both relations, the line widths refer to the broad component, and it is assumed that a narrow component has been subtracted during the fitting procedure and that the iron emission has been taken into account. All the details on how the FWHM of the emission lines have been computed are given in Caccianiga et al. (2013). The monochromatic luminosities at 5100 Å (L_5100 Å) and 3000 Å (L_3000 Å), respectively, are derived from the SED fitting presented in Marchese et al. (2012).

   The SE method is intrinsically affected by a large uncertainty, usually estimated between 0.35 and 0.46 dex (Park et al. 2012), essentially due to the unknown geometry of the BLR. Since the presence of large uncertainties can reduce significantly the strength of the correlations involving BH masses (and the derived quantities) we have estimated the impact of these errors on the analysis presented here (see Section 3.2).

   From the BH masses we can estimate the accretion rate normalized to Eddington luminosity, defined as

  \begin{equation} \lambda = \frac{L_{\rm bol}}{L_{\rm Edd}}, \end{equation}

  (4)

  where L_Edd is the Eddington luminosity:

  \begin{equation} L_{\rm Edd} = \frac{4 \pi G c M_{\rm BH} m_{\rm p}}{\sigma _{\rm e}} = 1.26 \times 10^{38} \left(\frac{M_{\rm BH}}{\mathrm{M}_{{\odot }}}\right)\,{\rm erg\,s}^{-1}. \end{equation}

  (5)
• Γ, L_{(2–10) keV}, K_bol, α_OX and disc/corona luminosity ratio. The values of Γ_{(0.5–10) keV} and L_{(2–10) keV} are taken from the spectral X-ray analysis presented in Corral et al. (2011). The bolometric corrections and the values of α_OX are available from Marchese et al. (2012). In particular, the bolometric correction is defined as

  \begin{equation} K_{\rm bol}=\frac{L_{\rm bol}}{L_{\rm (2-10)\,keV}}, \end{equation}

  (6)

  while α_OX is defined as

  \begin{equation} \alpha _{\rm OX} = \frac{\mathrm{Log}(f_{\rm o}/f_{\rm x})}{\mathrm{Log}(\nu _{\rm o}/\nu _{\rm x})}, \end{equation}

  (7)

  where f_o and f_x are, respectively, the rest-frame monochromatic fluxes at ν_o = 1.20 × 10¹⁵ Hz (corresponding to λ_o = 2500 Å) and ν_x = 4.84 × 10¹⁷ Hz (corresponding to E = 2 keV).

   Finally, the disc/corona luminosity ratios, defined as the ratio between the accretion disc luminosity, L_disc, and the 0.1–100 keV X-ray luminosity (L_X), are computed on the basis of the luminosities presented, again, in the Marchese et al. (2012) work.

STATISTICAL ANALYSIS

We perform a non parametric Spearman rank test on each correlation between X-ray properties (spectral index Γ, K_bol, α_OX, disc/corona luminosity ratio) and accretion rate (absolute |$\dot{M}$| and normalized to Eddington luminosity, λ). When the correlation is statistically significant, we perform a fit to the data [using both the ordinary least-squares (OLS) and the bisector methods; Isobe et al. 1990 to derive the functional dependence. We define a very significant correlation if the probability of null hypothesis (the two quantities are not correlated) is P ≤ 0.10 per cent, a significant correlation if P ≤ 1.00 per cent and a marginal correlation if P ≤ 5.00 per cent. For convenience, the main correlation coefficients and probabilities computed in this paper are summarized in Table 1. During the analysis, we evaluate the impact of some possible biases that we detail in the following subsections.

Flux limited nature of the sample

The XBS is a flux-limited sample. The strong L–z correlation, induced by the presence of a flux limit, may create spurious correlations or cancel real ones. This is not a problem for the correlations involving the X-ray loudness (K_bol, α_OX and disc/corona luminosity ratio) since we find that these parameters are not dependent on z (see Table 1). On the contrary, the values of Γ turned out to be marginally dependent on z (see Section 4.1) and, therefore, the correlations involving this quantity are potentially affected by the aforementioned problem. To exclude this possible effect, we use the partial correlation analysis (Kendall & Stuart 1979, see also Appendix) which allows us to evaluate the correlation between two parameters excluding a third variable on which both parameters depend (in this case, the redshift). As further check of the effect of z on the correlations, we analyse the correlations involving Γ in a relatively narrow bin of z (0 ≤ z < 0.4).

Error impact on correlation coefficient

The relation (8) can be derived from linear correlation coefficient, assuming independent errors on variables. Using Monte Carlo simulations we have verified that it can be also applied to Spearman coefficients in the case of a non-linear relation (Appendix A).

The correction presented above is particularly important for the correlations involving the Eddington ratio, since its computation is based on the highly uncertain BH mass estimate. In this work we assume an intrinsic uncertainty on the BH mass of 0.40 dex which corresponds to a correction factor for the Eddington ratio of about ∼1.57.

We note that the correction discussed above can be used only to have an estimate of the intrinsic strength of the correlation under study. The probability associated with the correlation coefficient (to assess the actual presence of a correlation), instead, is still the one associated with the value of r_obs. Therefore, we will apply this correction only to the correlations that have been established to be statistically significant on the basis of the probabilities associated with the values of r_obs.

Induced correlations

|$\dot{M}$| and λ are interrelated quantities since they both depend on bolometric luminosity. A possible correlation, e.g. between Γ and λ, can create an unreal correlation between Γ and |$\dot{M}$|⁠. To verify this situation, we use partial correlation analysis which allows us to calculate the correlation degree between the parameters of X-ray emission and λ, excluding the dependence on |$\dot{M}$| and vice versa. If the correlation disappears by excluding the dependence on the other variable, it is possible that the observed correlation is just induced by the other variable. Conversely, if the correlation remains, then both the observed correlations are likely to be real and not induced by the other variable.

Elusive AGN

RESULTS

Spectral index Γ

The spectral index Γ is found to marginally correlate with the Eddington ratio (⁠|${\boldsymbol r}_{\rm obs} = 0.27$|⁠, P = 1.64 per cent, Fig. 3) while the correlation between Γ and |$\dot{M}$| is not significant (⁠|${\boldsymbol r}_{\rm obs} = 0.17$|⁠, P = 15.86 per cent). Since Γ marginally depends also on z (r_obs = −0.27, P = 1.64 per cent) it is important to verify whether the observed Γ – λ correlation is in some way influenced by the luminosity–z correlation induced by the flux-limited nature of the sample (see discussion in Section 3.1). In Fig. 4 we present the Γ – λ correlation for sources in the range 0 ≤ z < 0.4. This is the range that contains the greatest number of object and offers the widest coverage of Γ – λ plane at the same time. The correlation in this bin of z is highly significant (⁠|${\boldsymbol r}_{\rm obs} = 0.71$|⁠, P < 0.10 per cent).

Figure 3.

Plot of Γ against λ. A typical error is shown in the upper left-hand corner: the green solid error bar is the statistical error, the red dashed one corresponds to the total error on λ (which includes the uncertainty related to the virial method used to estimate the BH masses). The filled points (blue in the colour version) are sources with 0 ≤ z < 0.4, triangles (magenta in the colour version) are sources with 0.4 ≤ z < 0.8 and the stars (red in the colour version) are sources with 0.8 ≤ z < 2.

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Figure 4.

Plot of Γ against λ in the range 0 ≤ z < 0.4. A typical error is shown in the upper left-hand corner: the green solid error bar is the statistical error, the red dashed one corresponds to the total error on λ (which includes the uncertainty related to the virial method used to estimate the BH masses). The solid line represents the OLS best-fitting relation. Blue triangles are the binned data.

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To further check this correlation, we have used the partial correlation method to exclude the dependence on z from the analysis on the total sample of 71 AGN. Again, we find a significant correlation with |${\boldsymbol r}_{\rm obs} = 0.36$| (P = 0.10 per cent). We conclude that the Γ – λ correlation is not induced by z. Rather, the effect of z is to weaken the correlation (see Fig. 3).

It is interesting to establish the origin of the Γ – z dependence. The spectral index Γ was computed using data in the range between 0.5 and 10 keV at rest frame. In this energy range the X-ray spectrum could be contaminated by the presence of the soft excess component. The origin of this component is still unclear. The classical interpretation of the soft excess is represented by the high-energy tail of blackbody emission of the disc accretion (Czerny & Elvis 1987; Grupe et al. 2010). However, this interpretation was questioned when several studies showed that the observed temperature of resulting blackbody is remarkably constant across orders of magnitude of luminosities and BH masses (Gierliński & Done 2004; Crummy et al. 2006). In the spectral analysis discussed in Corral et al. (2011) the soft excess component has been included in the model only if statistically required by the fit. This means that, if the number of counts is not large enough, the presence of the soft excess could be undetected and, thus, not included as additional component in the fitting procedure. In these cases the fit is expected to produce a steeper value of Γ. Notably, the influence of this component depends on z: with increasing z, the soft excess is confined to lower energies and it becomes negligible for z > 1 – 2 (Mateos et al. 2010; Scott et al. 2011). Therefore, the presence of the soft excess can produce a spurious anticorrelation between Γ and z making steeper values of Γ at low redshifts. In order to test whether the soft excess is at the origin of the observed Γ – z dependence, we have re-computed the values of Γ by restricting the data to energies above 2 keV (rest frame) in order to exclude the possible contamination due to the soft excess. The resulting values of Γ_{(2–10) keV} are poorly determined due to the low statistics in the hard part of the spectrum. Nevertheless, they can be used as an independent test of our conclusions. We find that the values of Γ_{(2–10) keV} do not depend on z (⁠|${\boldsymbol r}_{\rm obs} = -0.13$|⁠, P = 28.92 per cent), while they depend on λ, although with a lower significance (⁠|${\boldsymbol r}_{\rm obs} = 0.24$|⁠, P = 4.14 per cent) when compared to Γ. In principle, given the larger errors on Γ_{(2–10) keV} if compared to Γ, we do expect any correlation to be weaker when considering this parameter. Using equation (8) discussed in Section 3.2, it is possible to have an estimate of the impact of the larger errors on the correlations. Since the average error on Γ_{(2–10) keV} (ϵ ∼ 0.20) is a factor ∼2.5 larger than the average error on Γ (ϵ ∼ 0.08) we expect a decrease by a factor of ∼1.3 of the correlation coefficient just due to the increased errors. Thus, if Γ_{(2–10) keV} had the same dependence on z and λ as Γ (r_obs = −0.27 and 0.36, respectively) we should expect to observe correlation coefficients reduced by a factor of 1.3, i.e. r_obs = −0.21 and 0.28, respectively. While the observed coefficient for the Γ_{(2–10) keV} – λ correlation (0.24) is quite close to the expected one (0.28), the Γ_{(2–10) keV} – z correlation coefficient (−0.13) is nearly half than the expected one (−0.21). We consider this as an indication that the Γ_{(0.5–10 keV)} – λ and Γ – λ correlation has probably a similar strength while the dependence of the hard spectral index with redshift is much weaker (if any). These results support both the idea that the dependence between Γ and z is (mainly) induced by the presence of the soft excess and the idea that it is the spectral index of the primary X-ray component, and not the soft excess intensity, that correlates with the Eddington ratio. Clearly, better quality spectra, in particular at energies above 2 keV, are required to put these conclusions on a firmer ground.

Both Γ and, in particular, λ are characterized by uncertainties that are on average large with respect to the variance of the parameters. As explained in Section 3.2, the presence of such large errors reduces significantly the measured strength of the correlation, i.e. the value of r. In order to have a better estimate of the actual level of correlation between Γ and λ, we have thus applied the corrections described in Section 3.2 finding a corrected value of |${\boldsymbol r}_{\rm i}$| of 0.6. In the case of linear correlation, the square of |${\boldsymbol r}_{\rm i}$| gives an indication of how much of the observed variance on Γ is regulated by the value of λ. We thus conclude that about 40 per cent of the variance on the spectral index is explained by λ. This is the strongest correlation found in the sample. We have evaluated the impact of the elusive AGN (Section 3.4) by adding these objects to the sample. We find that their addition improves the Γ – λ correlation while the |$\Gamma -\dot{M}$| correlation remains not significant. We conclude that the observed Γ – λ correlation is not due to the exclusion of the elusive AGN.

Bolometric correction |$\boldsymbol {K_{\rm bol}}$|

We find a significant correlation between K_bol and λ (⁠|${\boldsymbol r}_{\rm obs} = 0.33$|⁠, P = 0.42 per cent, Fig. 5), while the correlation between K_bol and |$\dot{M}$| is only marginally significant (⁠|${\boldsymbol r}_{\rm obs} = 0.27$|⁠, P = 2.14 per cent).

Figure 5.

Plot of K_bol against λ. A typical error is shown in the upper left-hand corner: the green solid error bar is the statistical error, the red dashed one corresponds to the total error on λ (which includes the uncertainty related to the virial method used to estimate the BH masses). The solid line represents the OLS best-fitting relation. Blue triangles are the binned data.

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Figure 6.

SEDs obtained using the results of the Γ – λ and K_bol – λ best fits. The SED on the left represents the case of low accretion (λ ∼ 10⁻³): the K_bol value is low and Γ is flat. The SED on the right represents instead the case of high accretion rate (λ ∼ 1): in this case K_bol is high and Γ is steep.

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The slope obtained using the bisector method (0.72 ± 0.05) is in good agreement with that presented in Lusso et al. (2012) (0.75 ± 0.04) while the OLS slope is significantly (∼2.5σ) flatter (0.18 versus 0.39). The discrepancy is slightly reduced if we fit the data on the same range of K_bol observed in Lusso et al. (2012) (we find 0.24 ± 0.11). Again, we have verified that the observed correlations are not due to the exclusion of the elusive AGN.

In conclusion, the results show that both the spectral index Γ and the bolometric correction K_bol depend significantly on λ: steep Γ (∼2.5) and high K_bol (∼30–60) values correspond to higher λ (∼1), flat Γ (∼1.7) and low K_bol values (∼10) correspond to lower λ (∼10⁻²). Since K_bol depends also on Γ it is possible that the K_bol – λ correlation is induced by the (stronger) Γ – λ correlation. Again, we have verified this hypothesis using the partial correlation analysis and found that the dependence between K_bol and λ can indeed be explained as induced to the Γ – λ correlation.

In order to visualize these dependences we show in Fig. 6 two theoretical SEDs representing two extreme cases of low (λ ∼ 10⁻³, left-hand panel) and high (λ ∼ 1, right-hand panel) accretion rate. We have built these SEDs using a Shakura–Sunyaev disc model with a maximum temperature of 3 eV (corresponding to the average temperature of the sample sources) and a power law in the range between ∼0.01 and 100 keV with a cut-off at 0.1 keV. The values of the spectral index of the X-ray power law and the relative normalizations between the disc and the X-ray component are obtained from our Γ – λ and K_bol – λ fits, i.e. from (10) and (12). In this way the two SEDs of Fig. 6 can be considered as a visual representation of the correlation analysis discussed in the previous sections. To simplify the comparison between the two SEDs, we assumed the same disc emission in both cases. It is clear from the comparison of the two SEDs that the variation of K_bol with λ can be simply explained as due to a change of Γ, as suggested by the partial correlation analysis. We stress that the point where the disc emission intersects the corona emission is not fixed ‘a priori’ but it comes from the values of Γ and K_bol obtained from the fits.

|$\boldsymbol {\alpha _{\rm OX}}$|

Contrary to what is observed for the K_bol, we find a marginally significant anticorrelation between |$\alpha \rm _{\rm OX}$| and λ (⁠|${\boldsymbol r}_{\rm obs} = -0.25$|⁠, P = 3.32 per cent) while we find a significant anticorrelation between α_OX and |$\dot{M}$| (⁠|${\boldsymbol r}_{\rm obs} = -0.41$|⁠, P < 0.10 per cent, Fig. 7). Even if we weight the correlation coefficients for the errors the dependence between α_OX and |$\dot{M}$| remains the strongest one (⁠|${\boldsymbol r}_{\rm i} = -0.41$| versus −0.39). This result confirms what is usually found in the literature, i.e. that the value of α_OX anticorrelates with the bolometric/UV luminosity while it has weaker dependence with the Eddington ratio. The inclusion of the elusive AGN improves the significance of both α_OX – λ and |$\alpha _{\rm OX}\,-\,\dot{M}$| correlations.

Figure 7.

Plot of α_OX against |$\dot{M}$|⁠. A typical error is shown in the upper left-hand corner and it is the average statistical error on α_OX and |$\dot{M}$|⁠. The solid line represents the OLS best-fitting relation. Blue triangles are the binned data.

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Since both K_bol and α_OX are expected to be in some way proxies of the disc/corona relative intensity, the fact of finding two different dependences for these two quantities, one (K_bol) on the relative accretion rate and the other (α_OX) on the absolute accretion, seems difficult to reconcile. However, these two observational parameters are clearly related but not identical. The major difference is the fact that α_OX is defined at given monochromatic frequencies while K_bol is the ratio of two integrated quantities. For a fixed value of K_bol we can measure different values of α_OX depending on the actual spectral shape and vice versa. In particular, the value of α_OX is less sensitive to the slope of the X-ray emission if compared to K_bol (⁠|${\boldsymbol r}_{\rm obs} = -0.24$|⁠, P = 4.04 per cent for α_OX – Γ, and |${\boldsymbol r}_{\rm obs} = 0.53$|⁠, P < 0.1 per cent for K_bol – Γ). As shown in the previous section, the dependence of K_bol to the Eddington ratio is probably induced by a change of Γ so it is probable that the weaker dependence of α_OX on λ is a consequence of the weaker dependence of α_OX on Γ.

On the other hand, the significant dependence of α_OX with |$\dot{M}$| suggests that the disc/corona relative intensity depends also on the absolute accretion rate. We test this hypothesis in the next section by studying directly the disc/corona luminosity ratio.

Disc–corona luminosity ratio

The dependences of K_bol and α_OX discussed in the previous sections seem to suggest a complex relationship between the disc/corona luminosity ratio and the accretion. From the one hand, there is a significant dependence on the Eddington ratio, probably related to a change of X-ray slope with λ. On the other hand, there could be also a dependence of the disc/corona luminosity ratio on the absolute level of accretion rate. We now want to study directly the dependence of the disc/corona luminosity ratio with accretion. As expected, the situation in this case is more complex than the K_bol and α_OX case. We find significant correlation with |$\dot{M}$| (⁠|${\boldsymbol r}_{\rm obs} = 0.37$|⁠, P < 0.10 per cent, Fig. 8) and a marginally significant correlation with λ (⁠|${\boldsymbol r}_{\rm obs} = 0.28$|⁠, P = 1.64 per cent, Fig. 9). We find a similar result if we add the elusive AGN into the analysis. The strength of the two correlations, once corrected for the errors, is quite similar (⁠|${\boldsymbol r}_{\rm i} \sim 0.4$|⁠) so it is difficult to establish if there is a dominant correlation that explains also the other one. It is thus possible that both correlations are in fact present, i.e. that the disc/corona relative intensity depends both on λ and |$\dot{M}$|⁠, as expected from the combination of the results obtained for K_bol and α_OX.

Figure 8.

Plot of disc–corona luminosity ratio against |$\dot{M}$|⁠. A typical error is shown in the upper left-hand corner and it is the average statistical error on disc–corona luminosity ratio and |$\dot{M}$|⁠. The solid line represents the OLS best-fitting relation. Blue triangles are the binned data.

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Figure 9.

Plot of disc–corona ratio against λ. A typical error is shown in the upper left-hand corner: the green solid error bar is the statistical error, the red dashed one corresponds to the total error on λ (which includes the uncertainty related to the virial method used to estimate the BH masses). The solid line represents the OLS best-fitting relation. Blue triangles are the binned data.

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DISCUSSION AND CONCLUSIONS

In this paper we studied the link between X-ray emission and accretion rate in a statistically well-defined and complete sample of 71 type 1 AGN extracted from the XBS. The X-ray properties analysed here are the spectral index Γ in the range 0.5–10 and 2–10 keV band and the X-ray ‘loudness’ parametrized with both the bolometric correction K_bol (defined as the ratio between bolometric luminosity and 2–10 keV luminosity) and the two-points spectral index α_OX. We have also directly analysed the disc/corona luminosity ratio. The spectral index gives direct information about the energy distribution of the electrons in the corona, while the other three parameters quantify, in different ways, the relative importance between disc and corona.

We have considered different possible biases which can influence final results, such as

• soft excess contamination;

• redshift-induced correlations (important in flux-limited samples);

• impact of errors on correlation coefficients (especially on M_BH estimate);

• interconnected dependences due to the fact that the parameters considered in the analysis are not all independent;

• the impact of the exclusion of ‘elusive’ AGN from the analysis on the final results.

The results can be summarized as follows.

• The spectral index Γ depends significantly on accretion rate normalized to Eddington luminosity; in particular, ∼40 per cent of Γ variance could be explained by λ. This correlation is not due to the soft excess contamination, but it probably reflects a true dependence of the slope of the primary X-emission with λ. The Γ – λ dependence can be speculatively attributed to the effect of cooling of the electrons in the corona: for high values of λ, a large number of photons comes from the accretion disc and cools corona electrons rapidly, thus producing steep X-ray spectra while for low values of λ, less photons are available and this makes electron cooling inefficient, thus producing flat X-ray spectra (see for instance Cao 2009).

• The ‘X-ray loudness’ depends both on λ and |$\dot{M}$| but the dependence with λ is probably just the consequence of the (stronger) Γ – λ dependence.

• The strength of the dependence between the ‘X-ray loudness’ and λ or |$\dot{M}$| is different depending on whether we parametrize the X-ray loudness using the K_bol or the α_OX: while K_bol seems to depend mainly on λ, the values of α_OX show a stronger dependence with |$\dot{M}$|⁠. The explanation is likely connected to the different sensitivity of these two parameters to the X-ray spectral index.

We thank the referee for useful comments that improved the paper. We acknowledge Massimo Dotti, Francesco Haardt, Monica Colpi and Valentina Braito for useful discussions and Laura Maraschi and Tommaso Maccacaro for the precious comments. The authors acknowledge financial support from ASI (grant n. I/088/06/0), from the Italian Ministry of Education, Universities and Research (PRIN2010-2011, grant n. 2010NHBSBE) and from the Spanish Ministry of Economy and Competitiveness through grant AYA2012-31447.

REFERENCES