X-Ray Variability of SDSS Quasars Based on the SRG/eROSITA All-Sky Survey

We examine the long-term (rest-frame time scales from a few months to $\sim 20$ years) X-ray variability of a sample of 2344 X-ray bright quasars from the SDSS DR14Q Catalogue, based on the data of the SRG/eROSITA All-Sky Survey complemented for $\sim 7$% of the sample by archival data from the XMM-Newton Serendipitous Source Catalogue. We characterise variability by a structure function, $SF^2(\Delta t)$. We confirm the previously known anti-correlation of the X-ray variability amplitude with luminosity. We also study the dependence of X-ray variability on black hole mass, $M_{\rm BH}$, and on an X-ray based proxy of the Eddington ratio, $\lambda_{\rm X}$. Less massive black holes prove to be more variable for given Eddington ratio and time scale. X-ray variability also grows with decreasing Eddington ratio and becomes particularly strong at $\lambda_{\rm X}$ of less than a few per cent. We confirm that the X-ray variability amplitude increases with increasing time scale. The $SF^2(\Delta t)$ dependence can be satisfactorily described by a power law, with the slope ranging from $\sim 0$ to $\sim 0.4$ for different ($M_{\rm BH}$, $\lambda_{\rm X}$) subsamples (except for the subsample with the lowest black hole mass and lowest Eddington ratio, where it is equal to $1.1\pm 0.4$)


INTRODUCTION
The X-ray emission due to accretion of matter onto a black hole (BH) is expected to be variable (e.g.Syunyaev 1973;Lightman & Eardley 1974;Shakura & Sunyaev 1976;Pringle 1981).By studying the variability, together with the spectral properties and, since recently, polarization (Weisskopf et al. 2022) of the X-ray emission, one can get insight into the physics of the accretion disc and its hot corona and obtain constraints on the BH properties such as mass, spin and accretion rate.
Observations indeed reveal that virtually all BH stellar X-ray binaries and active galactic nuclei (AGN) are variable X-ray sources.Thus far, X-ray variability has been studied in greater detail for X-ray binaries (see McClintock & Remillard 2006;Gilfanov 2010;Belloni & Stella 2014 for reviews) than for AGN.This is partly due to the fact that BH X-ray binaries are among the brightest X-ray sources in the sky (their fluxes typically being ∼ 0.1 Crab), whereas even the brightest, nearby Seyfert galaxies usually have fluxes of a few mCrab, while quasars are usually yet fainter, which, of course, narrows the possibilities of their investigation.More important is that all characteristic time scales of accretion discs are 4-9 orders of magnitude ★ E-mail: sprokhorenko@cosmos.ru longer for AGN than for X-ray binaries due to the huge difference in BH mass (∼ 10 5 -10 10  ⊙ vs. ∼ 10 ⊙ ).In particular, the orbital, thermal and viscous times scale with the BH mass  BH and radius  in the disc as follows (Shakura & Sunyaev 1973): (2) where   = 2  BH / 2 is the Schwarzschild radius of the BH,  is the gravitational constant,  is the speed of light,  is the viscosity parameter and  is the scale height of the disc.Therefore, in AGN, it is usually possible to study variability on the dynamical and thermal time scales, but the viscous time scale (in the inner region of the accretion disc) can be probed for relatively light SMBHs only, given the limited duration of monitoring campaigns.X-ray variability of AGN is interesting not only from the point of view of studying the physics of accretion onto supermassive black holes (SMBHs), but also in the cosmological context due to the existence of correlation between the X-ray and UV luminosities of AGN (Tananbaum et al. 1979).If this relation is non-linear, as many studies suggest (e.g.Strateva et al. 2005;Lusso & Risaliti 2016;Salvestrini et al. 2019), then there is an exciting possibility of using quasars as "standard" candles for determination of cosmological parameters (Risaliti & Lusso 2015;Risaliti & Lusso 2019).However, there is a serious obstacle on this route, namely that the UV-X-ray correlation is characterised by a significant scatter.It is thus necessary to understand the origin(s) of this scatter and take it into account when applying this relation to cosmological measurements.Variability of quasars in X-rays and in the optical-UV bands is definitely one of the major causes of the observed scatter (Ulrich et al. 1997;Berk et al. 2004;Kelly et al. 2009;Vagnetti et al. 2013;Caplar et al. 2017;Chiaraluce et al. 2018;Burke et al. 2021;Arévalo et al. 2023).
X-ray variability of AGN can be studied in different ways.One approach consists of conducting extensive observational campaigns of individual objects aimed at obtaining their detailed X-ray light curves and estimating their power spectral density (PSD), covering time scales from minutes to years.This approach is quite demanding in terms of observational time and sensitivity, and has thus far been implemented mostly for bright Seyfert galaxies (e.g.Mushotzky et al. 1993;Nandra et al. 1997;Uttley et al. 2002;Markowitz & Edelson 2004;González-Martín & Vaughan 2012).These studies have shown that AGN PSDs can usually be described by a bending power law, with a slope of ∼ 2 at high frequencies and a flatter slope of ∼ 0-1 at lower frequencies, with the characteristic time scale varying between a few minutes and a few years from one Seyfert galaxy to another (Uttley & McHardy 2005;Vaughan et al. 2005;McHardy et al. 2006).When compared to the X-ray variability properties of BH X-ray binaries, these results support the idea that AGN are scaled-up analogs of the latter.
In studying the X-ray variability of high-luminosity, distant AGN (quasars), one usually encounters the problem that the sampling and quality of the X-ray light curves is insufficient for constructing powerdensity spectra on a source-by-source basis.Nevertheless, given a large enough sample of objects, one can use another approach that consists of estimating some integral characteristics of X-ray variability, e.g.normalized excess variance, for each object and examining how these quantities depend upon quasar properties such as redshift, X-ray luminosity or BH mass as well as on the rest-frame time scale.Within this approach, it is often possible and desirable to bin the objects and/or measurements according to these physical properties, i.e. employ an ensemble-averaged approach, based on the implicit assumption that all AGN are intrinsically similar objects.
This method has been successfully implemented by a number of authors, using quasar samples selected from deep extragalactic surveys by such X-ray observatories as ROSAT, XMM-Newton and Chandra (Almaini et al. 2000;Papadakis et al. 2008;Yang et al. 2016;Paolillo et al. 2017;Paolillo et al. 2023).These studies indicated that the X-ray variability properties of quasars indeed depend on their physical characteristics such as BH mass and accretion rate (see a discussion in Section 5 below).However, the available statistics was usually not sufficient for constraining the dependencies on these parameters simultaneously, so that there is a need to continue this research using larger samples, with a better coverage of temporal frequencies, redshifts, BH masses and accretion rates.
The all-sky X-ray survey that has been conducted since December 2019 by the eROSITA telescope (Predehl et al. 2021) on board the SRG orbital observatory (Sunyaev et al. 2021) for the first time allows us to study the X-ray variability of quasars on a massive basis (using thousands of objects).In this paper, we report the results of such a variability analysis for a sample of X-ray bright quasars detected during the SRG/eROSITA all-sky survey and selected from the SDSS catalogue of optical quasars (Pâris et al. 2018).

SRG/EROSITA-SDSS SAMPLE OF X-RAY BRIGHT QUASARS
We use a sample of X-ray bright quasars found by cross-correlating the catalogue of X-ray sources detected by SRG/eROSITA during the all-sky survey in the 0 • <  < 180 • celestial hemisphere with the optical SDSS DR14 quasar catalogue (SDSS DR14Q, Pâris et al. 2018).The latter covers 9376 square degrees on the sky, 8548 of which are at 0 • <  < 180 • .Specifically, the X-ray sources were selected from the catalogue of point X-ray sources detected on the summed map of five eROSITA all-sky surveys1 in the 0.3-2.3keV energy band and were required to have an average (i.e.determined from the summed map) observed flux of at least 2 × 10 −13 erg s −1 cm −2 in this band.Our analysis of X-ray variability is based on the source fluxes measured in individual eROSITA sky surveys.These fluxes were evaluated by forced X-ray photometry, using the source positions determined from the summed eROSITA map (see Medvedev et al. 2022).
The observed fluxes provided in the eROSITA catalogue are estimated from the measured count rates assuming a universal absorbed power-law spectral model with a photon index of Γ = 2.0 and a Galactic column density of  H = 3 × 10 20 cm −2 .We applied an approximate absorption correction to the observed fluxes adopting this column density, i.e. multiplied the fluxes by 1.18.Hereafter, we denote the unabsorbed fluxes (in the observed 0.3-2.3keV energy band) in individual eROSITA sky surveys by  X and the average eROSITA fluxes by ⟨ X ⟩.The actual Galactic absorption,  H,Gal , in the direction of the different quasars varies significantly, with 90% of the objects having  H,Gal between 10 20 and 5 × 10 20 cm −2 (the median column density is 2×10 20 cm −2 ).This corresponds to ±10% variations in the flux absorption correction.We have not taken this minor factor into account in view of other possible, unaccounted for, uncertainties of the same order: namely, there may be sourceto-source variations in the intrinsic slope of the X-ray continuum and additional absorption intrinsic to the quasars.This is justifiable, since a systematic uncertainty of 10% in flux (and hence luminosity) is much smaller than the width of the luminosity bins (∼ 1 dex) that we use below to study the dependence of X-ray variability on quasar luminosity.We also emphasise that absorption correction has no impact on the determination of the variability amplitude of each given source.
The SDSS DR14Q catalogue comprises spectroscopically studied objects that have been confirmed as quasars via an automated procedure combined with a partial visual inspection of spectra, have luminosities   [ = 2] < −20.5, and either display at least one emission line with a full width at half maximum larger than 500 km s −1 or have interesting/complex absorption features.We conducted the search within the eROSITA 98% positional uncertainty, defined by radius 98.For X-ray bright sources with ⟨ X ⟩ > 2 × 10 −13 erg s −1 cm −2 , this radius is typically ∼ 5 ′′ and does not exceed 6.5 ′′ for the SRG-SDSS X-ray bright quasar sample.We excluded a number of unreliable associations.First, we removed 10 objects that are either not present in the SDSS DR16Qprop catalogue (Wu & Shen 2022) or have redshift estimates in that catalogue that significantly differ from those in SDSS DR14Q2 .Second, we excluded two pairs of quasars (four objects) that are separated from each other by less than 20 ′′ .The expected number of spurious eROSITA-SDSS matches is ∼ 1 (it is less than 2 with a probability of 90%), which is tolerable for our purposes.We obtained this estimate by assigning small (but much larger than 98) angular offsets to all SDSS DR14Q quasars and then counting the number of cross-matches with our eROSITA sample.
To minimise the impact of exceptionally large (and thus significantly non-Gaussian) flux uncertainties on our variability analysis, we excluded measurements (151 in total) with the vignetting corrected eROSITA exposure time of less than 70 seconds (while the typical exposure time of a single observation is ∼ 200 seconds).In addition, we excluded 11  X estimates for which the forced photometry yielded zero values.Such cases occasionally arise because the likelihood function that is used in the forced photometry is not defined for negative fluxes, and although this procedure also provides flux upper limits their consideration would unnecessarily complicate our X-ray variability analysis.Together, these two steps have removed just 162 (i.e. less than 2%) of all flux measurements, which cannot affect the results of this study in any significant way.
Blazars (BL Lac objects and flat-spectrum radio quasars) are a special class of AGN, which, due to the presence of a highly collimated emission component, can have substantially different variability properties from normal (i.e.unbeamed) AGN.We thus tried to carefully clean our sample from blazars.First, we cross-correlated our sample with the 5th edition of the Roma-BZCAT Multifrequency Catalogue of Blazars (Massaro et al. 2015).This yielded 184 counterparts.Second, we cross-correlated our sample with the fourth catalogue of AGN detected by the Fermi Large Area Telescope, Data Release 3 (Ajello et al. 2022), namely with objects classified as 'bll' (BL Lac-type objects), 'bcu' (blazar candidates of unknown types) or 'fsqr' (Flat Spectrum Radio Quasars).This resulted in an additional 10 counterparts.We then cross-matched our sample with the Blazar Radio and Optical Survey (BROS) (Itoh et al. 2020), excluding Gigahertz peaked-spectrum sources and compact steep-spectrum radio sources (based on the corresponding flag in the catalogue) from the search.This provided an additional 67 sources.Finally, we crosscorrelated our sample with the Combined Radio All-Sky Targeted Eight GHz Survey (Healey et al. 2007), which yielded another 18 counterparts.In all these cases, the search was done within 10 ′′ of the SDSS DR14Q optical positions of our objects to take into account that some of the positions given in the blazar catalogues are not very precise (in particular, when only a radio position is available).The expected total number of spurious cross-matches with the blazar catalogues is less than one.
In total, we found 279 blazars or blazar candidates associated with our sample of X-ray bright quasars (see Table 1 for details) and removed them from the sample.It is possible that some of the discarded objects are actually not blazars, but that does not present a problem since our preference is to achieve maximum purity of the quasar sample.
The resulting clean sample consists of 2344 quasars.Of these, 1224 have five eROSITA flux measurements, 1074 have four, 45 have three and one object has only two fluxes.
Our choice of the relatively high X-ray flux limit (2 × 10 −13 erg s −1 cm −2 before absorption correction) for this study is primarily driven by the desire to better control the impact of flux measurement uncertainties on the variability analysis.With this threshold, 93.4% of the individual flux measurements in our sample are based on more than 20 net source counts, so that the corresponding flux uncertainties are expected to be close to Gaussian. 3econdly, at this X-ray flux limit, our sample is characterised by high optical completeness.Figure 1 shows the distribution of the apparent magnitudes of our X-ray selected quasars in the SDSS r band.The maximum of the distribution lies at  ∼ 18, while the effective threshold of the SDSS DR14Q catalogue is significantly deeper.Namely, SDSS DR14Q is based on spectroscopy carried out during the SDSS I/II and SDSS III/IV surveys, which had different quasar selection criteria.Based on fig.6 in Pâris et al. (2018), we can estimate the effective completeness limit of SDSS I/II at  ∼ 19 and that of SDSS III/IV at  ∼ 20.5 (see the corresponding vertical lines in Fig. 1).Therefore, it is unlikely that we are missing a substantial fraction of X-ray bright quasars in the SDSS DR14Q footprint due to their faintness in the optical.Estimating this incompleteness in quantitative terms goes beyond the scope of this study and is not necessary here, since we are not exploring space densities of quasars.
Figure 2 illustrates the main X-ray properties of our quasar sample.Namely, we show the distributions of: (i) average X-ray fluxes, (ii) individual flux measurements, (iii) ratios of the maximum to minimum flux, and (iv) relative flux uncertainties, all based on eROSITA data.For most of the objects, the X-ray flux varies by less than a factor of two between the SRG sky surveys but some quasars demonstrate stronger variability.In particular, the flux varies more than tenfold for 27 sources.The relative flux uncertainty ( X / X ) usually does not exceed 20%.Hereafter  X denotes the 1 flux error (see Medvedev et al. 2022).
Figure 3 shows examples of eROSITA light curves of quasars from our sample.During the SRG all-sky survey, any source is visited approximately every 6 months, with the duration of a single visit depending on the ecliptic latitude (Sunyaev et al. 2021).Namely at low latitudes, visits typically last between one and two days, while for the highest-latitude objects in our sample this duration can reach 20 days.As can be seen from Fig. 3, for such sources, the flux uncertainties are significantly smaller due to the longer exposure time.

XMM-Newton subsample
The SRG/eROSITA data allows us to study the X-ray variability of quasars over time spans of at most 2 years.To extend our analysis to longer time scales (up to ∼ 20 years), we cross-correlated our sample with the XMM-Newton Serendipitous Source Catalogue 4XMM-DR12 (Webb et al. 2020), which contains X-ray flux measurements taken since the year 2000 and covers 3.1% of the sky (for net exposure time ⩾ 1 ks) with a typical sensitivity of a few ×10 −15 erg s −1 cm −2 .We conducted a search using the optical (SDSS DR14Q) positions of the quasars and the 98% localisation regions of the XMM-Newton sources, defined by radius    98.We evaluated the latter as    98 = 1.98 ×  , where   is provided in the 4XMM-DR12 catalogue and describes the 63% localisation region (the factor 1.98 in the formula above corresponds to a twodimensional Gaussian distribution).
The cross-correlation yielded 157 matches, which constitutes 6.7% of the SRG/eROSITA-SDSS X-ray bright quasar sample.The expected number of spurious 4XMM-DR12 counterparts is 0.3 (it is less than 1 with a probability of 95%).This value was found by assigning small angular offsets to the 4XMM-DR12 sources and then counting the cross-matches with our quasar sample.We excluded observations with the  _   value greater than two, as suggested by the XMM-Newton team.This resulted in 156 quasars with at least one good-quality XMM-Newton flux.
Although the typical sensitivity of the 4XMM-DR12 catalogue is some two orders of magnitude better than the flux limit of our sample of X-ray bright quasars, it is in principle possible that some of our objects, due to their variability, were observed but not detected by XMM-Newton.There is no information on such non-detections in the 4XMM-DR12 catalogue but it is available via the HILIGT tool4 (König et al. 2022).We thus used this service to search for XMM-Newton flux upper limits for our quasar sample and did not find any.Therefore, all of our quasars were sufficiently bright to be detected by XMM-Newton whenever they fell into the field of view of its instruments.
We used XMM-Newton fluxes in the soft (0.2-2.0 keV) energy band, which are the sum of the EPIC fluxes listed in the 4XMM-DR12 catalogue in bands 1, 2 and 3 (0.2-0.5, 0.5-1 and 1-2 keV, respectively).These observed fluxes had been obtained from the count rates assuming an absorbed power-law spectral model with a photon index Γ = 1.7 and a Galactic absorption column density of  H = 3×10 20 cm −2 .However, because these fluxes were determined in narrow energy ranges, they are only weakly dependent on the assumed spectral shape, e.g. the differences between the fluxes for Γ = 1.7 and Γ = 2 will not exceed one per cent5 .Hence, this difference will not exceed 1% for the observed flux in the 0.2-2 keV band either.
To enable direct comparison with the eROSITA fluxes, we converted the observed 0.2-2 keV XMM-Newton fluxes to unabsorbed fluxes in the (observed) 0.3-2.3keV energy band assuming an absorbed power-law model with Γ = 2 and  H = 3 × 10 20 cm −2 .This corresponds to a multiplication factor of 1.15.This coefficient would change just slightly, to 1.12, if we adopted Γ = 1.7.Similarly to the SRG/eROSITA data, variations in the Galactic absorption towards the different quasars in the XMM-Newton sample are expected to induce ∼ 10% variations in flux absorption correction, which we neglect.It is worth noting again absorption correction does not affect the inferred variability characteristics of sources.
The 4XMM-DR12 catalogue, composed of serendipitous source detections, is characterised by irregular timing structure, in contrast to the SRG all-sky survey data.To approximately mimic the SRG/eROSITA observation strategy, we performed exposure weighted averaging of the individual fluxes measured by XMM-Newton in observations conducted within 5 days of each other.We also found the corresponding mean dates,  XMM , of such sets of observations.Hereafter, we refer to these merged observations as 'isolated' XMM-Newton observations, or simply as XMM-Newton observations.We chose the maximum interval of 5 days because series of observations of a given target by XMM-Newton usually do not span longer periods, and indeed very few consecutive XMM-Newton flux measurements in our sample are separated by times between 5 and 25 days (rather than by much longer times of several years).
There are 109 of the 156 quasars that have only one isolated XMM-Newton observation, 32 have 2, 8 have 3, and 7 have 4 isolated observations.Figure 4 illustrates the main properties of the XMM-Newton subsample of the SRG/eROSITA-SDSS X-ray bright quasar sample.Specifically, we show the distributions of: (i) fluxes in XMM-Newton observations, (ii) time lags between the latest eROSITA visit and all XMM-Newton observations of a given source, (iii) ratios of the fluxes in XMM-Newton observations and in the latest eROSITA observation, and (iv) the relative uncertainties of XMM-Newton fluxes.We see that the XMM-Newton and eROSITA fluxes, despite the scatter caused by variability, are generally consistent with each other.The median value of the logarithm of their ratio is 0.017 ± 0.024 (the uncertainty was estimated by a bootstrap method), i.e. consistent with zero.We also note that the vast majority of the XMM-Newton fluxes are measured very precisely, to better than 10%.Figure 5 shows examples of long-term X-ray light curves of our quasars as measured by XMM-Newton and SRG/eROSITA.

Physical properties of the quasars
The SDSS DR14Q catalogue (Pâris et al. 2018) provides the spectroscopic redshifts of the quasars under consideration.Ninety per cent of our objects have redshifts between 0.15 and 0.97, and the median redshift is  median = 0.38.Having this information, we can determine the X-ray luminosities of the quasars.To facilitate comparison with previous studies, we define luminosities in the rest-frame 2-10 keV energy band,  X .To this end, we use the unabsorbed fluxes in the observed 0.3-2.3keV band and calculate the -correction for a power-law spectrum with Γ = 1.8, which is a typical slope of (type 1) AGN spectra in the standard X-ray band (e.g.Brightman et al. 2013;Trakhtenbrot et al. 2017).
Due to the range of redshifts in our sample, we actually examine the X-ray variability of different quasars in somewhat different rest-frame energy ranges.Namely, the observed X-ray range of 0.3-2.3keV corresponds to 0.4-3.2keV in the rest-frame at  median = 0.38, whereas for 90% of our objects the probed energies vary between 0.35-2.65 and 0.6-4.5 keV.Therefore, our results might be affected by intrinsic dependence of X-ray variability on energy (due to spectral shape variability or variable intrinsic absorption), if there is any.Some previous studies suggested that the energy dependence is not substantial.In particular, Ponti et al. (2012) found that variability in the soft (0.7-2 keV) and hard (2-10 keV) energy bands was tightly correlated and of similar amplitude on time scales up to 80 ks (where is was probed) for their XMM-Newton sample of AGN.
Hereafter, we denote by ⟨ X ⟩ the luminosity of a given quasar determined from its mean eROSITA flux ⟨ X ⟩.The median value of ⟨ X ⟩ for the sample is 10 44.3 erg s −1 .The sample average statistical uncertainty of log⟨ X ⟩ is 0.03, which directly stems from the sample average uncertainty of log⟨ X ⟩ (since we neglect any uncertainties in the spectroscopic redshift measurements and have adopted the same -correction for all the objects, see above).Figure 6 shows the luminosity-redshift diagram as well as the distributions of  and ⟨ X ⟩ for the whole SRG/eROSITA-SDSS X-ray bright quasar sample and for the XMM-Newton subsample.We see that we effectively probe ∼ 2.5 dex in X-ray luminosity (from ∼ 10 43 to ∼ 10 45.5 erg s −1 ) and that the XMM-Newton subsample has similar redshift and luminosity properties as the entire sample.
We also made use of the catalogue of spectral properties of quasars from SDSS DR14Q (Rakshit et al. 2020), which, in particular, provides estimates of the BH masses,  BH .These are based on the optical continuum luminosity and line width measurements from single-epoch SDSS spectroscopy.The authors used parameters of strong emission lines such as H 4861, Mg II 2798 and C IV 1549 wherever available.It is important to note that although the formal statistical uncertainties of the resulting  BH estimates are small, the systematic uncertainty associated with the underlying empirical relations is significant, ∼ 0.4 dex (e.g.Collin et al. 2006;Shen 2013), and is unknown for a given object.Only one of the 2344 quasars in our sample has no  BH estimate in the Rakshit et al. (2020) catalogue.In addition, the estimates for 108 quasars are of bad quality, according to the QUALITY_MBH flag in that catalogue.We thus exclude these 109 quasars from those considerations below where the value of the black hole mass is required.
To characterise the regime of accretion for a given quasar, we are interested in knowing its Eddington ratio,  Edd =  bol / Edd .
Determining  bol is an untrivial task, because it involves applying a bolometric correction for a given spectral range and can be significantly affected by variability.In particular, although we could adopt the optically-based  bol estimates for our objects provided by Rakshit et al. (2020), we decided to refrain from that because the SDSS spectral observations were typically carried out ∼ 10 years before the SRG survey and may thus have caught a given quasar in a significantly different luminosity state compared to the period of time over which we study its X-ray variability.Instead, we estimated the bolometric luminosities directly from the eROSITA X-ray measurements as  bol =  X  X , using a constant bolometric correction  X = 10 for the 2-10 keV energy band, based on Sazonov et al. (2012).This allows us to define an X-ray based Eddington ratio  X =  X  X / Edd and the corresponding average quantity ⟨ X ⟩ =  X ⟨ X ⟩/ Edd .We emphasize that the so-derived  X is a crude proxy of the true Eddington ratio, since the  X / bol ratio can actually depend on the BH mass and the Eddington ratio, as is actively discussed in the literature (e.g.Vasudevan & Fabian 2007;Vasudevan et al. 2009;Lusso et al. 2012;Bongiorno et al. 2016;Duras et al. 2020).We do not try to take these dependencies into account to avoid circular reasoning.
Figure 7 shows the distributions of  BH and ⟨ X ⟩ for the whole sample and for the XMM-Newton subsample.Again, the XMM-Newton subsample is similar in terms of BH masses and Eddington ratios to the rest of the sample.It is worth noting that the observed distributions of X-ray luminosities, BH masses and Eddington ratios shown in Figs. 6 and 7 are, of course affected by selection effects.In particular, there is a paucity of low-luminosity quasars because we are dealing with a flux limited sample.This implies that for lower-luminosity quasars, mostly located at lower redshifts, we are probing on average somewhat longer rest-frame time scales than for higher-luminosity quasars.This should affect our estimates of the uncertainties of the variability characteristics.However, we do not expect any significant impact of the selection effects on our conclusions regarding the variability trends with physical parameters, since our analysis is based on examining variability properties within fairly narrow  BH ,  X and  X bins, without any cross-talk between these subsamples.

Radio properties
It is also interesting to investigate if the X-ray variability of quasars depends on their radio-loudness, which in turn may be related to BH spin (e.g.Sikora et al. 2007;Tchekhovskoy et al. 2010) or to the magnetic flux threading the BH (Sikora & Begelman 2013).To this end, we cross-correlated our sample with the LOw-Frequency ARray (LOFAR) Two-metre Sky Survey DR2 (LoTSS) (Shimwell et al. 2022), which has covered 27% of the northern sky with good angular resolution (6 ′′ ) and high sensitivity in the 120-168 MHz band.The point-source completeness of the catalogue is ∼ 90% at an integrated (over the solid angle) flux density of 0.45 mJy.Taking into account that some of the radio counterparts of quasars may be extended, we used a radius of 10 ′′ for cross-matching between the SRG/eROSITA-SDSS X-ray bright quasar sample (using the optical coordinates) and LoTSS.
A total of 1542 quasars from the SRG/eROSITA-SDSS X-ray bright quasar sample are located in the footprint of LoTSS.Of these, 1035 have at least one LOFAR counterpart, while 13 quasars have two LOFAR counterparts.We assume that the latter are physically related pairs (e.g.double-lobed radio sources) and thus simply add up their fluxes.In what follows, we compute the radio flux from the LoTSS integrated flux density,   , as  r =   ×48 MHz, since 48 MHz is the width of the LoTSS frequency band.If no radio counterpart is found for a quasar located in the area covered by LoTSS, we assign an upper limit to its radio flux of 0.45 mJy × 48 MHz, since this is the effective sensitivity of LoTSS (see above).The expected number of spurious LOFAR counterparts is 36±6.This value was estimated by assigning small angular offsets to all LoTSS sources and counting the number of cross-matches within 10 ′′ of our quasars.Although the number of spurious radio counterparts is fairly large (which is caused by the high density of LoTSS objects on the sky), it is nevertheless just ∼ 2% of the total number of counterparts and thus cannot significantly affect our results.
Figure 8 shows the distributions of radio fluxes and radio to Xray flux ratios for the quasars located in the LoTSS footprint.We see a bimodal shape, well known from previous studies (e.g.Sikora et al. 2007).The minimum of the distribution between the two peaks approximately lies at  r /⟨ X ⟩ = 0.0001, which is convenient to define as the boundary between radio-quiet and radio-loud quasars.With this definition, 128 quasars (8.3%) turn out to be radio-loud and the remaining 1414 quasars are radio-quiet.
Figure 9 shows the BH mass and Eddington ratio distributions for the radio-quiet and radio-loud samples of quasars located in the LoTSS footprint.We see that the radio-loud sample is dominated by heavy ( BH ∼ 10 9  ⊙ ) BHs, whereas the ⟨ X ⟩ distributions are similar for the radio-quiet and radio-loud samples.

VARIABILITY ANALYSIS
Our goal is to investigate the dependence of X-ray variability on different rest-frame timescales from ∼ half a year to ∼ 20 years and on various physical parameters, namely  X ,  BH ,  X and radioloudness.Since we have very few X-ray data points for each quasar but a large sample of such objects, we necessarily base our analysis on ensemble averaging.As a measure of X-ray variability, we utilize the structure function (SF).It is based on pairs of flux measurements taken at different moments in time and is well suited for studying large samples of objects with poorly sampled light curves.The SF has often been used in astronomy, with the exact definition slightly varying from one work to another (e.g.Simonetti et al. 1985;Di Clemente et al. 1996;Berk et al. 2004;Vagnetti et al. 2011).Here we define it as follows (see e.g.Press et al. 1992;Kozłowski 2016;Vagnetti et al. 2016): where the angle brackets indicate the ensemble average,  X,true is the true (i.e. that would be measured in the absence of noise) X-ray flux of a given object at a given moment,  true ≡ log  X,true , and the rest-frame and observed-frame (hereafter denoted with a subscript "obs") time intervals are related as follows: We thus characterise the variability by the squared difference between the logarithms of the fluxes, or equivalently by the square of the logarithm of the X-ray flux (i.e.luminosity) ratio, on a given restframe time scale.
To ensure the mutual statistical independence of  2 estimates in different Δ bins, we should exclude from the averaging in equation (4) those pairs of flux measurements that can be algebraically expressed through other pairs of measurements for the same object that are already used in the calculation (see also Emmanoulopoulos et al. 2010).To this end, we build each pair from the latest eROSITA observation and some previous observation by eROSITA or XMM-Newton.Hence, for each quasar, we use all the available eROSITA and XMM-Newton observations only once, expect for the latest eROSITA observation, which is used in every pair of flux measurements involved in the averaging.We have adopted the latest eROSITA observation as a reference one because this allows us to exploit the maximum available time scale for each object.
Due to the Poisson statistics of photons, the detector actually measures a flux  X that is different from the true flux  X,true .In logarithmic terms this corresponds to  ≡ log  X =  true +   , where   is a random fluctuation.We thus need to subtract the contribution of photon statistical noise to the SF.To this end, it is reasonable to assume that flux fluctuations (  ) are uncorrelated with the corresponding fluxes (  true ) and with themselves6 .Then, we may compute a one-point estimate of  2 for each (i'th) pair of flux measurements, corrected for the noise (see Press et al. 1992): where  2  ( obs, ) and  2  ( obs, + Δ obs, ) are calculated from the corresponding measured X-ray flux uncertainties as follows: As noted in Section 2, the vast majority of sources in the SRG-SDSS X-ray bright quasar sample have at least 20 net source counts, so that their flux errors should be Gaussian to a first approximation.In order to check the error propagation expressed by equation ( 7), we performed a simple simulation.We assumed a Poisson distribution of counts and computed the standard deviation of log  X .We found that the true standard deviation of log  X is 4% (2%) higher than the estimate given by equation ( 7) for 20 (30) counts.We then repeated this simulation for a normal distribution based on individual flux measurements and the corresponding errors from our data and obtained deviations of 7.5% and 4%, respectively.This minor inaccuracy in the rms calculation is unlikely to significantly affect the results of this work.
We then compute the SF for a given ensemble of flux measurement pairs as follows: where  pairs is the number of independent measurement pairs and ŵ is defined as follows.If a particular quasar has just one pair of measurements that falls into the Δ bin under consideration, then ŵ = w  .If several pairs of measurements for the same quasar fall into the same time difference bin, we calculate ŵ as the mean w  of those pairs.Thereby we intend to avoid attributing too much weight to some quasars compared to others.However, we have also tried to treat all pairs of measurements of the same quasar as independent estimates of w  , which led to negligible changes in the results.

RESULTS
In what follows, we study the dependence of  2 on the quasar restframe time difference Δ for different subsamples of SRG/eROSITA-SDSS X-ray bright quasars.We first construct these dependencies in binned form.To this end, we divide a given subsample into Δ bins of 0.3 dex width in the range from −0.9 to 1.2 dex.We then count the number of independent flux measurement pairs (or equivalently quasars) in each bin.If there are less than 10 pairs in some bin, we merge it with the adjacent bins until the number of pairs in the merged bin reaches 10 or the width of the merged bin reaches 1.2 dex.If upon re-binning there still remain bins that contain less than 10 pairs, we exclude them from the subsequent analysis.This is aimed at increasing the reliability of ensemble-averaging in equation ( 8) and leads to just a minor loss of data.We finally use bootstrapping to estimate the sampling distribution of the mean value of ŵ in each Δ bin, i.e. the sampling distribution of  2 (Δ), and evaluate the corresponding 68% confidence intervals for  2 (Δ).Specifically, we use 10,000 re-samples in each bin.In Appendix A, we provide further details of this procedure and present examples of actually measured ŵ distributions.
We then seek to parameterise the obtained  2 (Δ) dependencies, using ML estimation.The bootstrap-derived  2 sampling distribution around the actually measured  2 value in a given bin can be regarded as the probability density of the uncertainty in measuring  2 in that bin: ( 2 ).We can then assume that, once we have specified some model  2 model (Δ), the probability of measuring a value  2 in the experiment is determined by ( 2 −  2 model ).We can then define the likelihood function as follows: where  2  2  2 =  2  is the data set,    are the parameters of the model  2 model , and the product is taken over all Δ  bins.

Models
A number of previous studies of quasar variability, in particular, in the optical band (MacLeod et al. 2010;Zu et al. 2013;Kozłowski 2016), used 'decorrelation' models to describe the  2 (Δ) dependence.A particular case of this class of models corresponds to the damped random walk (DRW) process.On the other hand, numerous studies of Seyfert galaxies (e.g.Uttley et al. 2002;Uttley & McHardy 2005;McHardy et al. 2004;Vaughan et al. 2005; González-Martín & Vaughan 2012) usually found a bending power law, to satisfactorily describe the observed X-ray PSDs of AGN.In this model, the PSD follows a power law (PSD() ∝  −  ) with a slope of  ≈ 1 at low frequencies and a steeper power law ( ≈ 2) at high frequencies.
The characteristic time scale varies between a few minutes and a few years from one Seyfert galaxy to another and appears to be proportional to BH mass (see Paolillo et al. 2023 and references therein).A bending power-law model might also be suitable for describing the  2 (Δ) dependence of quasar X-ray variability.However, the relationship between PSD and SF is not-trivial (e.g.Emmanoulopoulos et al. 2010).
Both the decorrelation model and the bending power-law model are characterized by three or more parameters.Upon multiple trials, we realized that the statistical quality of our data is not sufficient for constraining them simultaneously.Moreover, we found that  2 (Δ) for different studied susamples can be satisfactorily described by a simple power law: Here, Δ is measured in years so that  characterises the amplitude of the variability on a time scale of 1 year.In more specific terms, the aforementioned models involving a characteristic time scale do not provide a statistically significant improvement in the fit quality (e.g. based on the Akaike information criterion).Therefore, in what follows we discuss  2 (Δ) dependencies in terms of the power-law model given by equation (10).

Luminosity dependence
We first investigate the dependence of variability on X-ray luminosity.
To this end, we divide our sample into three subsamples: 10 42 <  X < 10 44 , 10 44 <  X < 10 45 , and 10 45 <  X < 10 47 erg s −1 .Hereafter, by  X for a flux measurement pair we mean the arithmetic mean of the X-ray luminosities of the two measurements involved.
The results are presented in Fig. 10.We see that X-ray variability tends to increase with increasing time scale and, for a given time scale, with decreasing luminosity.The binned  2 (Δ) dependencies can be satisfactorily described by a power law, whose best-fit parameters are given in Table 2.

Dependence on the BH mass and Eddington ratio
We next investigate the dependence of X-ray variability on BH mass and Eddington ratio.To this end, we divide our sample into nine subsamples: 10 −2.5 <  X < 10 −1.4 , 10 −1.4 <  X < 10 −0.9 and 10 −0.9 <  X < 10 0.5 for 10 6.8 <  BH < 107.8  ⊙ ; 10 −3 <  X < 10 −1.5 , 10 −1.5 <  X < 10 −1 and 10 −1 <  X < 10 0 for 10 7.8 <  BH < 10 8.8  ⊙ ; and 10 −3 <  X < 10 −1.7 , 10 −1.7 <  X < 10 −1.3 and 10 −1.3 <  X < 10 0 for 10 8.8 <  BH < 10 9.8  ⊙ .These subsamples have been selected so (see Fig. 11) that: (i) they contain sufficiently many data points, and in particular SRG/eROSITA-XMM-Newton pairs (corresponding to long time scales), to enable an adequate statistical analysis, (ii) they are well separated from each other in terms of their median  BH and  X values (as indicated by black crosses in Fig. 11), and (iii) there are several  X bins per nearly the same median  BH , so that it is possible to set apart the dependencies of X-ray variability on  X and  BH .Note that a small fraction of the quasars, in particular those with very low ( BH ∼ 10 6.5  ⊙ ) or very high (∼ 10 10  ⊙ ) BH masses, have remained outside of the adopted ( BH ,  X ) regions.Also note that  X for a given flux measurement pair is computed from the mean of the corresponding X-ray luminosities.
The results are presented in Fig. 12 and Table 3.There is a clear trend of increasing variability with increasing time scale for low  X .This trend also persists for medium and high  X , but becomes less pronounced (the power-law slope  ≲ 0.3) and is in fact just marginally detected.Also, for a given time scale, variability tends to increase with decreasing  BH and  X and becomes especially strong for the lowest  X .
To further verify that these results are not significantly affected by our particular ( BH ,  X ) binning, we calculated the first-order partial correlation coefficient (see e.g.Vagnetti et al. 2011) between individual variability estimates ŵ and the corresponding  X values, taking the dependence on  BH into account, for our whole sample (excluding the few quasars without  BH estimates).We obtained a value of −0.223 for this coefficient and the corresponding -value of ∼ 10 −88 .Similarly, we studied the dependence of ŵ on  BH , taking into account the influence of the accretion rate.We obtained the corresponding partial correlation coefficient value of −0.187 with the -value of 7 ∼ 10 −61 .We can thus conclude that the anticorrelation between X-ray variability and accretion rate as well as the anti-correlation between X-ray variability and  BH are robustly established.

Dependence on radio-loudness
We finally investigate the dependence of X-ray variability on radioloudness, based on the samples of radio-quiet and radio-loud quasars defined in Section 2.2.1.As shown in the left panel of Fig. 13, the variability amplitude increases with increasing time scale for radio-quiet quasars.The  2 (Δ) dependence for radio-loud quasars is measured only on times scales shorter ∼ 1.5 years, where it is consistent with a constant value.
The radio-loud subsample is dominated by heavy BHs (see Fig. 9), which may be the cause of their apparently higher variability.To examine whether radio-quiet quasars are intrinsically more variable than radio-loud ones, or this is just another manifestation of the anticorrelation between BH mass and the amplitude of X-ray variability, we constructed a subsample of radio-quiet quasars that have  BH and  X distributions similar to those of the radio-loud sample.We achieved this by collecting as many radio-quiet quasars in each ( BH ,  X ) bin (0.2 dex in  BH and 0.25 dex in  X ) as there are radio-loud ones.Here, we use only quasars that have reliable  BH estimates.We show the resulting  2 (Δ) dependence for the selected radioquiet subsample along with the  2 (Δ) dependence for the radioloud sample in the right panel of Fig. 13.They are not significantly different from each other.This implies that  2 does not depend on radio-loudness once  BH and  X selection effects are removed.
In addition, we have computed the second-order partial correlation coefficient (see Vagnetti et al. 2011) between ŵ and radio-loudness of quasars, which accounts for the  BH and  X dependencies, and found it to be equal to 0.027 with the corresponding p-value of 0.06.This further indicates that there is no significant intrinsic dependence of X-ray variability on radio-loudness.

DISCUSSION
We now discuss our results in the broad context of AGN variability developed in previous studies.As we already discussed in Sections 1 and 4, one common approach to studying X-ray variability of AGN, applicable when well-sampled light curves are available, consists of measuring the PSD as a function of frequency.In the past, it has mostly been applied to nearby and relatively low-luminosity AGN such as Seyfert galaxies.These studies have revealed, in particular, that higher luminosity sources tend to be less variable on short time scales compared to lower luminosity ones but this difference becomes barely noticeable at the longest time scales probed (e.g.Markowitz & Edelson 2004).
Typically, it is difficult to obtain high-quality X-ray light curves, and hence PSDs, for more distant and luminous AGN.Therefore, Xray variability studies of quasars have usually focused on examining the dependence of integral characteristics of X-ray variability, such as fractional root mean square (rms) variability (or, equivalently, normalized excess variance) on SMBH physical parameters, and mostly employed an ensemble-averaging approach.In one of the first detailed studies of X-ray variability of quasars, Papadakis et al. (2008), using a sample of 66 objects at  ∼ 1 observed by XMM-Newton in the Lockman Hole, confirmed the trend (albeit with a large scatter) of decreasing variability amplitude with increasing Xray luminosity on an observed time scale of ∼ 2 months.Yang et al. (2016) and Paolillo et al. (2017) performed similar studies on longer observed time scales of ∼ 15 years, using samples of distant quasars from the Chandra Deep Field-South Survey, and solidified the existence of anti-correlation between X-ray luminosity and variability.In our study, based on SF, we have reaffirmed the anticorrelation of X-ray variability with quasar luminosity on rest-frame time scales from ∼ 0.2 to ∼ 20 years.
It is not the first time that AGN X-ray variability has been studied in terms of SF.In particular, Vagnetti et al. (2011) andVagnetti et al. (2016) previously used XMM-Newton data to investigate the dependence of SF on luminosity and BH mass on rest-frame time scales from 0.1 day to 4 years, whereas Middei et al. (2017) studied the behaviour of SF on time scales up to 20 years.These authors, similar to our work, approximated  (Δ) by a simple power-law model and obtained slopes in the range ∼ 0-0.2 for different AGN subsamples, which are in good agreement with the slopes ∼ 0-0.4 for  2 (Δ) that we infer for our different subsamples (except for the subsample with the lowest BH mass and lowest Eddington ratio, where  = 1.1 ± 0.4) 8 .10 2 10 1 10 0 SF 2 6.8 < log(M BH /M ) < 7.8 2.5<log X <1.4 N pairs = 56 1.4<log X <0.9 N pairs = 356 0.9<log X <0.5 N pairs = 594 10 2 10 1 10 0 SF 2 7.8 < log(M BH /M ) < 8.8 3.0<log X <1.5 N pairs = 918 1.5<log X <1.0 N pairs = 1705 1.0<log X <0.0 N pairs = 1037 10 1 10 0 10 1 Restframe t, years  3.
The shape of the  (Δ) function is expected to reflect (see e.g.MacLeod et al. 2010;Zu et al. 2013;Kozłowski 2016) that of the underlying PSD() function, although the relationship between these two properties is non-trivial (e.g.Emmanoulopoulos et al. 2010).Specifically, if the slope of PSD() changes gradually from  1 to  2 (PSD() ∝  −  ) around some frequency  b , then  (Δ) is also expected to bend at Δ ∼ 1/ b .Our study has not revealed statistically significant evidence for a change of the SF slope across the probed range of time scales.

Key role of the Eddington ratio
From a fundamental point of view, variability properties are expected to be determined by the combination of BH mass and Eddington ratio, as well as possibly BH spin.The interplay between BH mass and Eddington ratio in shaping the long-term X-ray variability of quasars has remained poorly understood so far, largely due the lack of sufficiently large samples of objects covering broad ranges in BH mass and luminosity, and with sufficient sampling on long (∼ 1-10 year) time scales.
For nearby Seyfert galaxies and for time scales shorter than 80 ks, an anti-correlation between variability and BH mass has been established (e.g.O'Neill et al. 2005;Ponti et al. 2012).However, the dependence of variability on the accretion rate remained uncertain.2. Right panel: The same  2 ( ) dependence for the radio-loud sample vs. the  2 ( ) dependence for a subsample of radio-quiet quasars that is similar in terms of  BH and  X distributions to the radio-loud sample.Note that in the right panel, we only use quasars that have good  BH estimates for both the radio-loud and radio-quiet subsamples to check that their  BH and  X distributions are similar.This is the reason for the slightly different number of pairs for the radio-loud samples in the left and right panels.Arguably, the most interesting finding of this study is that Xray variability is substantially stronger at low Eddington ratios,  X ≲ 10 −2 , compared to higher accretion rates, at least over the ∼ 0.2-20 year rest-frame time scales probed by this study.This behaviour pertains to SMBHs of various mass.The trend of decreasing variability amplitude with increasing accretion rate appears to persist, but becomes less pronounced, also when we compare moderateand high-accretion rate objects ( X ∼ 10 −1.25 vs.  X ∼ 10 −0.75 ).There is thus an indication that once the Eddington ratio drops below a few per cent (with a significant uncertainty in this value due to the fact that  X is a crude proxy of the true Eddington ratio), the X-ray emission becomes substantially more variable.Considering the Xray variability amplitude at a given time scale and a given Eddington ratio as a function of BH mass, we find that lighter BHs tend to be more variable on relatively short time scales (less than ∼ 2 years) but we cannot draw a conclusion about the dependence on longer time scales with this sample.
In Appendix B, we provide additional information on the 10 quasars with the lowest  X in our sample.Specifically, we present their X-ray light curves (see Fig. B1) and discuss their optical spectral properties based on the literature.This information can be useful for follow-up studies of these potentially interesting objects.

Comparison with optical variability of quasars
Recently, a study similar to ours, but on the optical variability of quasars, has been conducted by Arévalo et al. (2023), who also similarly used the catalogue of spectral properties of quasars from SDSS DR14Q (Rakshit et al. 2020).Dividing their sample into subsamples by BH mass, Arévalo et al. (2023) found the Eddington ratio and the variability amplitude to be anti-correlated, confirming previous indications (MacLeod et al. 2010;Sánchez-Sáez et al. 2018;Li et al. 2018).They also found that at a given time scale, the amplitude of optical variability decreases with BH mass, with this trend being stronger on short rest-frame time scales of 30-150 days than on longer scales of ∼ 300 days.
These new results on optical variability are similar to our findings on X-ray variability, which suggests that the temporal behaviours of optical and X-ray emission are largely driven by the same physical processes near the SMBH.It is important to note, however, that the results of both studies are not directly comparable to each other, since the variability analysis in Arévalo et al. (2023) is restricted to times scales shorter than 1 year, while our work is mostly focused on longer time scales.

SUMMARY
We have studied the medium-and long-term (rest-frame time scales between a few months and ∼ 20 years) X-ray variability of a large, uniform sample of X-ray bright quasars from the SDSS DR14Q catalogue, based on the data of the SRG/eROSITA all-sky survey complemented for ∼ 7% of the sample by archival data from the 4XMM-DR12 catalogue.The results of this work can be summarized as follows.B1.Only eROSITA data are shown, because none of these objects have XMM-Newton measurements.

Figure 1 .
Figure 1.Distribution of the SDSS r-band apparent magnitudes, corrected for the Galactic extinction, of the SRG-SDSS sample of X-ray bright quasars studied in this work.The whole sample is shown in green.The other histograms show the following subsamples: blue, quasars observed in SDSS I/II only; orange, quasars observed in SDSS III/IV only; red, quasars observed both in SDSS I/II and SDSS III/IV.The vertical blue and orange lines indicate the effective spectroscopic depths of SDSS I/II and SDSS III/IV.

Figure 2 .Figure 3 .
Figure 2. Differential distributions of various X-ray properties of the SRG/eROSITA-SDSS X-ray bright quasar sample, based on eROSITA data.Upper left panel: time-averaged X-ray fluxes.Upper right panel: individual (per SRG sky survey) X-ray flux measurements.Lower left panel: ratio of the maximum to minimum X-ray fluxes.Lower right panel: relative uncertainties of individual flux measurements.Hereafter, log means log 10 .

Figure 4 .Figure 5 .
Figure 4. Differential distributions of various X-ray properties of the XMM-Newton subsample.Upper left panel: flux in the XMM-Newton isolated (see the main text) observations.Upper right panel: time gap between the latest eROSITA observation and all available XMM-Newton observations.Lower left panel: Ratio of the fluxes in the XMM-Newton observations and in the latest eROSITA observation.Lower right panel: Relative flux measurement uncertainty in the XMM-Newton observations.

Figure 6 .Figure 7 .
Figure 6.Average X-ray (2-10 keV) luminosity during the SRG/eROSITA survey vs. redshift (left), as well as the corresponding distributions of the redshifts (middle) and X-ray luminosities (right) of the studied quasars.The whole sample is shown in blue and the XMM-Newton subsample in orange.

Figure 8 .Figure 9 .
Figure 8. Left panel: Distribution of the radio flux densities of the studied quasars, as measured by LOFAR in the 120-168 MHz band.The orange histogram shows actual measurements, while the blue column (on top of the histogram) indicates the number of non-detections with an estimated flux density upper limit of 0.45 mJy.Right panel: The corresponding distribution of the ratios of the LOFAR to eROSITA fluxes (the upper limits are shown on top of the actual measurements).The black vertical line is drawn at  r /⟨ X ⟩ = 0.0001, which we define as a dividing line between radio-quiet and radio-loud quasars.

Figure 10 .
Figure 10. 2 as a function of the rest-frame time scale for different X-ray luminosity subsamples.Error bars indicate 68%-confidence intervals.The number of flux measurement pairs ( pairs ) used in the calculation for each subsample is quoted in the legend.The different dependencies are slightly shifted along the Δ axis with respect to each other for better visibility.The dotted lines show fits power laws (equation 10).The best-fitting parameter values are given inTable 2.

Figure 11 .
Figure 11.Scatter plot of  BH vs.  X for the independent flux measurement pairs used in our analysis.Orange dots denote SRG/eROSITA-XMM-Newton pairs.The black lines divide the ( BH ,  X ) space into 9 regions that have been used for constructing the  (Δ ) dependencies shown in Fig. 12.The black crosses indicate the median  BH and  X values for each bin.

Figure 12 .
Figure 12.  2 as a function of the rest-frame time scale for different ( BH ,  X ) subsamples.Downward arrows indicate 84%-confidence upper limits.The dotted lines show fits by power laws.The best-fitting parameter values are given in Table3.

Figure 13 .
Figure13.Left panel:  2 as a function of the rest-frame time scale for the radio-loud and radio-quiet quasar samples.The dotted lines show fits by power laws.The best-fitting parameter values are given in Table2.Right panel: The same  2 ( ) dependence for the radio-loud sample vs. the  2 ( ) dependence for a subsample of radio-quiet quasars that is similar in terms of  BH and  X distributions to the radio-loud sample.Note that in the right panel, we only use quasars that have good  BH estimates for both the radio-loud and radio-quiet subsamples to check that their  BH and  X distributions are similar.This is the reason for the slightly different number of pairs for the radio-loud samples in the left and right panels.

FFigure
Figure B1.X-ray light curves of the 10 AGN with the lowest  X , listed in TableB1.Only eROSITA data are shown, because none of these objects have XMM-Newton measurements.

Table 2 .
Parameters of the best-fitting  2 (Δ ) functions by power laws in different X-ray luminosity bins.The parameter uncertainties are given at the 1 confidence level.

Table 3 .
Parameters of the best-fitting  2 (Δ ) functions by power laws in different ( BH ,  X ) bins.

Table 4 .
Parameters of the best-fitting  2 (Δ ) functions by power laws for the radio-loud and radio-quiet quasar samples.