The MEXSAS2 Sample and the Ensemble X-ray Variability of Quasars

We present the second Multi-Epoch X-ray Serendipitous AGN Sample (MEXSAS2), extracted from the 6th release of the XMM Serendipitous Source Catalogue (XMMSSC-DR6), cross-matched with Sloan Digital Sky Survey quasar catalogues DR7Q and DR12Q. Our sample also includes the available measurements for masses, bolometric luminosities, and Eddington ratios. Analyses of the ensemble structure function and spectral variability are presented, together with their dependences on such parameters. We confirm a decrease of the structure function with the X-ray luminosity, and find a weak dependence on the black hole mass. We introduce a new spectral variability estimator, taking errors on both fluxes and spectral indices into account. We confirm an ensemble softer when brighter trend, with no dependence of such estimator on black hole mass, Eddington ratio, redshift, X-ray and bolometric luminosity.

criteria adopted by Kozłowski (2017) are instead to prefer the Mg II black hole mass estimate when available, rather than that from C IV, which is biased from various effects, and to compute bolometric luminosity as a weighted average of those derived from two different available continua. However, Kozłowski (2017) also find that the DR7Q line widths (which are derived from detailed fits of spectral components) are generally more reliable than the DR12Q ones (which are derived from principal component analysis), thus they are to be preferred, when available. To apply the same criteria to the mass and luminosity data from both DR7Q and DR12Q, we adopt the following choice: (i) for quasars included only in DR12Q and not in DR7Q, we take the estimates by Kozłowski (2017) (2178 objects); (ii) for quasars included in both catalogues or only in DR7Q, and in redshift intervals with only one broad line and one continuum luminosity available (z 0.35, 0.9 z 1.5, z 2.25), we take the estimates by Shen et al. (2011) (442 objects); (iii) for quasars included in both catalogues or only in DR7Q, in redshift intervals with two broad lines and continua (0.35 z 0.9, 1.5 z 2.25), we apply the Kozłowski (2017) criteria to the DR7Q data, deriving new estimates (518 objects). The catalogue, including X-ray measurements from XMMSSC and quasar data from SDSS and from our elaboration, will be published elsewhere (Vagnetti et al, in preparation). Here we provide preliminary results of our ensemble analyses of the X-ray variability.

FLUX VARIABILITY AND STRUCTURE FUNCTION
We compute the structure function according to Vagnetti et al. (2016), as a r.m.s. difference of the X-ray flux measured at two epochs differing by τ in the rest-frame, and corrected for the noise contribution, where σ 2 noise,SF = σ 2 n (t) + σ 2 n (t + τ ) is the quadratic contribution of the photometric noise to the observed variations, σ n being the error on log f X at each given time. We use mainly EPIC X-ray fluxes in the XMM-Newton band 9 (0.5-4.5 KeV), as in our previous papers (Vagnetti et al., 2011(Vagnetti et al., , 2016. The SF can be fitted by a power-law SF ∝ τ b . For the whole sample we find a slope b = 0.11 ± 0.01. The SFs in bins of X-ray luminosity and black hole mass are shown in Fig. 1. We confirm a strong anti-correlation with X-ray luminosity, approximately as L −0.22 X , with correlation coefficient r = −0.92 and null probability p(> r) = 0.04, and no dependence on redshift, similarly to Vagnetti et al. (2016). There is an apparent decrease of the SF with black hole mass, approximately as M −0.15 BH , with r = −0.97 and p(> r) = 0.03, but partial correlation analysis suggests that this is due to the strong correlation of mass with X-ray luminosity. Limiting the analysis to the luminosity interval 10 44 erg/s < L X < 10 45 erg/s, the mass-luminosity correlation reduces (r = −0.83, p(> r) = 0.06), and the dependence of the SF on black hole mass is weaker, ∝ M −0.06 BH . We find neither dependence on bolometric luminosity nor on Eddington ratio.

SPECTRAL VARIABILITY
We update the analysis of the spectral variability parameter, initially introduced by Trevese and Vagnetti (2002) and recently adapted to the X-ray band by Serafinelli et al. (2017) relating variations of the photon index Γ with those of the X-ray flux in a given band. The spectra of most Seyfert galaxies with Eddington ratios above 0.01 typically become steeper in their brighter phases (e.g. Markowitz et al., 2003;Sobolewska and Papadakis, 2009;Connolly et al., 2016), as well as for many galactic black hole binary systems (e.g. Remillard and McClintock, 2006;Done et al., 2007;Dong et al., 2014). This is known as 'softer when brighter' behaviour and translates to a negative β value according to Eq. 2. We also found this 'softer when brighter' behaviour, obtaining β = −0.69 ± 0.03 in our ensemble analysis, for the previous version of the MEXSAS sample, using fluxes in the soft X-ray band 0.5-2 keV, and computing variations with respect to the mean values of each source (Serafinelli et al., 2017). Since the ensemble correlation between ∆Γ and ∆ log f X contains a large scatter, also due to the large measurement errors for the fainter sources, we fit a linear relation ∆Γ ∝ ∆ log f X to the MEXSAS2 data set taking the uncertainties in both variables into account. We run a high number of linear fits replacing original data with Gaussian distributed values within the associated error box. Moreover, following Isobe et al. (1990), we computed both ordinary least squares regressions OLS(Y |X) of the dependent variable Y on the independent variable X (which result we call for brevity β xy ), the inverse regression OLS(X|Y ) (β yx ), and the bisector: For the whole sample, we find ensemble values β xy = −0.22 ± 0.04 and β bis = −1.22 ± 0.05, confirming the 'softer when brighter' behaviour for both spectral variability parameters (see Fig. 2).
We then divide our sample in bins of soft X-ray flux, to show that the resulting β does not change within the errors (see Fig. 3). In fact this was expected, because measurement errors do affect the estimate of β, but this must reflect an intrinsic relation between the flux and slope variations which should not change with an observational parameter like the average flux of the sources.
We also divide our sample in bins of black hole mass, M BH , and show the result in Fig. 3. Both β xy and β bis are independent of M BH within the errors, and always compatible with their overall ensemble values.
We similarly analyse possible dependencies on Eddington ratio, redshift, X-ray and bolometric luminosities, finding no evidence of change. This analysis will be reported in an upcoming paper (Vagnetti et al., in prep.).

SUMMARY
-We have updated our X-ray quasar catalogue MEXSAS to MEXSAS2.
-We have confirmed our previous results for the structure function dependence on X-ray luminosity, and no dependence on redshift; we find a weak dependence on black hole mass, and no dependence on bolometric luminosity and Eddington ratio.
-We have developed new spectral variability estimators, by computing ordinary least squares and bisector fits with errors in both variables.
-We obtain 'softer when brighter' trends for the new MEXSAS2 sample using both estimators.
-Our spectral variability estimates, that also take errors on both fluxes and spectral indices into account, do not present a dependence on the flux.