Super-Eddington accretion in the Q2237+0305 quasar?
aa r X i v : . [ a s t r o - ph . H E ] D ec Astronomy & Astrophysicsmanuscript no. berdina2020 © ESO 2020December 23, 2020
Super-Eddington accretion in the Q2237+0305 quasar?
L.A.Berdina , , V.S.Tsvetkova , , and V.M.Shulga , , Institute of Radio Astronomy of the National Academy of Sciences of Ukraine, 4 Mystetstv, 61002 Kharkov, Ukrainee-mail: [email protected]; [email protected]; [email protected] Institute of Astronomy of Kharkov National University, Sumskaya 35, 61022 Kharkov, Ukraine V.N.Karazin Kharkov National University, Svobody sq. 4, 61070 Kharkov, Ukraine College of Physics, International Center of Future Science, Jilin University, 2699 Qianjin St., 130012 Changchun, ChinaReceived .. / Accepted ..
ABSTRACT
The interband time lags between the flux variations of the Q2237 + V , R , and I spectral bands. The values of the time lags for filter pairs R − V , I − R , and I − V are significantly higher thanthose predicted by the standard accretion disk model by Shakura and Sunyaev. To explain the discrepancy, the idea of a supercriticalaccretion regime in quasars considered in 1973 by Shakura and Sunyaev is applied. This regime has been shown by them to cause anextended scattering envelope around the accretion disk. The envelope e ffi ciently scatters and re-emits the radiation from the accretiondisk and thus increases the apparent disk size. We made use of analytical expressions for the envelope radius and temperature derivedby Shakura and Sunyaev in their analysis of super-Eddington accretion and show that our results are consistent with the existence ofsuch an envelope. The corresponding parameters of the accretion regime were calculated. They provide the radii of the envelope inthe V , R , and I spectral bands consistent with the inter-band time lags determined in our work. Key words. accretion, accretion disks – quasars: individual: Q2237 +
1. Introduction
Flux variability inherent in active galactic nuclei (AGNs) andquasars is an important source of information about their physi-cal properties and therefore has been closely investigated by as-tronomers for decades (see, e.g., Cristiani et al. 1997; Giveon etal. 1999; Vanden Berk et al. 2004; Wilhite et al. 2005; Magdis &Papadakis 2006; Gopal-Krishna et al. 2013; Kumar et al. 2015;Grier et al. 2019; Yi et al. 2019; Kokubo & Minezaki 2020; Luoet al. 2020). The variability is observed in all spectral regions,from the X-rays to radio wavelengths, and at timescales from afew days to several years (Giveon et al. 1999; Webb & Malkan2000; Wilhite et al. 2005; Schmidt et al. 2012). The characteris-tic amplitudes grow toward longer time lags and shorter wave-lengths. The flux variations in AGNs and quasars in di ff erentspectral bands are often observed with certain time lags betweenthem. This may suggest that these time lags measure the lighttravel times between the quasar regions that radiate in di ff er-ent spectral bands. It may therefore be a useful instrument forstudying spatial structure and physical parameters of AGNs andquasars. This instrument, called ”reverberation mapping” (RM),was initially proposed for measuring the distance between thecentral region of a quasar that is responsible for the hard con-tinuum radiation and a broad emission line region (Blandford &McKee 1982). Initially, RM implied spectroscopic observations.The so-called photometric RM has been widely used recently(e.g. Bachev 2009; Edri et al 2012; Jiang et al. 2016, Mudd et al.2018; Kokubo 2018 ). The photometric RM implied photometryin two or more spectral bands. Some of them contained emissionlines and others did not.In 1997, Wanders et al. (1997) reported reverberation timedelays measured in their spectroscopic RM campaign with theInternational Ultraviolet Explorer (IUE). The authors noted that the most remarkable result is the detection of apparent time shiftsbetween the brightness variations in di ff erent regions of the UVcontinuum. Collier et al. (1999) later realized that the opticalcontinuum variations lag the UV variations in NGC 7469. Fur-ther photometric RM projects provided much evidence of lagsbetween flux variations observed in continua of di ff erent spec-tral regions (e.g., Collier 2001; Sergeev et al. 2005; Cackett etal. 2007; Bachev 2009; Fausnaugh et al. 2016; Fausnaugh etal. 2018; Mudd et al. 2018). Currently, several major interna-tional projects are dedicated to RM of quasars and AGNs, suchas the Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM), the Space Telescope and Optical Reverberation Mapping(STORM), the Lick AGN Monitoring Project (LAMP), the DarkEnergy Survey (DES), and others (Grier et al. 2017; Fausnaughet al. 2018; Mudd et al. 2018; Homayouni et al. 2019; Kinemuchiet al. 2020; Yu et al. 2020).The UV / optical radiation from quasars is generally believedto come from a geometrically thin optically thick accretion diskwith a supermassive black hole in its center. According to thismodel (Shakura & Sunyaev 1973), the accretion disk temper-ature varies along the disk radius as T ∝ r − / , therefore theshorter-wavelength radiation must originate closer to the accre-tion disk center. According to the continuum thermal reprocess-ing scenario proposed by Krolik et al. (1991), its variations musttherefore precede those observed at longer wavelengths. The ex-pected trend of time lags with wavelength must therefore followthe relationship τ ∝ λ / , provided the accretion disk radiates asa blackbody.The paper contains photometric RM results and their anal-ysis applied to the V , R , and I light curves of Q2237 + Article number, page 1 of 11 & Aproofs: manuscript no. berdina2020 tems with a radio-quiet quasar at z q = .
695 that is quadruplylensed by a z g = .
039 Sab galaxy. Four individual macroimageslabeled A through D are arranged in a cross-like pattern aroundthe galaxy nucleus within a circle of approximately 1.8 arcsec indiameter. Until recently, there were two determinations of the in-terband time delays for a quasar of the Q2237 + V obtained by the OGLE collaboration (Udalski et al. 2006) andthe R light curves built from observations at the 1.5-meter tele-scope of the Maidanak observatory. The time delays between thevariations in these two spectral bands were reported in Koptelovaet al. (2006) to be 9 days for image A and 16.2 days for imageC. Koptelova et al. (2010) reported time lags of 5.1-5.6 days and5.1-5.2 days for images A and C, respectively. In addition to dif-ferent sampling rates and time coverage, the two datasets usedin the two works were processed with di ff erent photometry al-gorithms, therefore we decided to repeat the work with morehomogeneous data. To do this, we turned to the monitoring dataof the Q2237 + V , R , and I in 2001-2008,presented by Dudinov et al. (2011). Preliminary processing ofthe 2004 - 2005 light curves in filters V and R has definitelyshown that the variations in filter R lag those in V by about 5days (Berdina et al. 2018, 2019). We show the results of pro-cessing the data of two seasons in all the three filters, V , R , and I , with a more careful analysis of errors and a subsequent dis-cussion of possible mechanisms leading to the particular valuesof the interband time delays and their behavior in wavelengths.In the next section, we describe the initial data we used anddiscuss their suitability for determining the interband time delaysin Q2237 +
2. Initial data and time lags
Of all the existing monitoring data for Q2237 + V obtained in the framework of the OGLEprogram (Udalski et al. 2006) are the most famous. Their highphotometric accuracy, rather good sampling and time coveragemade the OGLE light curves a basis for various microlensingstudies, which provided many estimates of the Q2237 + A similar dataset exists that is much less well known becauseit was published in a poorly accessible journal (Dudinov et al.2011). While the OGLE data are more accurate and cover alonger observing seasons, the data by Dudinov et al. (2011) havethe advantage that they have been obtained through three filters,– V , R , and I of the Johnson-Cousins photometric system. Thee ff ective wavelengths are λ e f f ( V ) = . λ e f f ( R ) = . λ e f f ( I ) = . λ rest ( V ) = . λ rest ( R ) = . λ rest ( I ) = . + z q = . // / databases / index.html). In Fig. 1we reproduce the light curves of Q2237 + V , R , and I light curves for the 2004and 2005 seasons is given in Table 1.The light curves by Dudinov et al. (2011) and those in fil-ter R used by Koptelova et al. (2006; 2010) were obtained fromthe same monitoring data taken with the 1.5-meter telescope of t J =JD - 2450000, days I m a g AA BB CC DD t J =JD - 2450000,days V m a g AA BB CC DD t J =JD - 2450000,days R m a g AA BB CC DD
Fig. 1.
Light curves of Q2237 + V , R , and I from obser-vations by Dudinov et al. 2011. The data for the seasons in 2004 and2005 are presented. the Maidanak Observatory, but they were processed with di ff er-ent photometry algorithms, as described in Vakulik et al. (2004),Dudinov et al. (2011), and Koptelova et al. (2005).Measurements of the interband time delays in gravitationallylensed quasars encounter almost the same di ffi culties as thosethat are inherent in measuring the single-band time delays be-tween the quasar macroimages. These are listed and analyzedin detail by Tewes et al. (2013), for instance. In addition to the Article number, page 2 of 11erdina L.A. et al.: Q2237 + Table 1.
General information about the data of observations of Q2237 + Filter Season duration Number of data Photometry error FWHM(days) points R , season 2004 192 75 0.0142 1.0475 R , season 2005 202 98 0.0141 – V , season 2004 176 48 0.0169 1.0658 V , season 2005 178 38 0.0170 1.0945 I , season 2004 177 52 0.0149 0.9842 I , season 2005 178 39 0.0128 0.9938known di ffi culties, which are common for all time-delay lenses(finite photometric accuracy, poor and irregular sampling, gapsbetween the seasons, variable microlensing), the microlensingevents in Q2237 + + + ff ects is there-fore vital to obtain the unbiased estimates of the time delaysbetween macroimages. A pair of light curves in two di ff erentfilters of the same macroimage can be anticipated to be lessmutually distorted by microlensing: observations show only asmooth growth of microlensing amplitudes toward the shorterwavelengths, but the e ff ects of microlensing still have to be elim-inated. To measure the interband time delays, we therefore ap-plied the method developed by us earlier for determining thedi ff erential time delays in gravitationally lensed quasars in thepresence of microlensing. In short, the method uses some use-ful properties of representing the data of observations by the or-thogonal polynomials, which provide certain simplicity and con-venience in calculations. In particular, it allows any term of thepolynomial that approximates a particular light curve to be elimi-nated or added again without requiring to recalculate the remain-ing expansion coe ffi cients. This property is useful, in particular,to determine the time delays in the presence of microlensing. Adetailed description of the method can be found in Tsvetkovaet al. (2016), where its application is demonstrated in determin-ing the single-band time delays between macroimages of the PG1115 +
080 and HE 0435-1223 gravitationally lensed quasars. InFig. 2 we show the polynomial approximations provided by ourmethod for the light curves in Fig. 1: the curves are reduced tothe same magnitude level, and the first-order terms are elimi-nated.An additional factor restricting the accuracy of determiningtime delays is the character of the intrinsic quasar variability. Inorder to be suitable for determining the time delays, the intrinsicquasar variability curves must contain features with a charac-teristic timescale shorter than (or at least on the order of) theexpected value of the time delay, and with the amplitudes ex-ceeding the photometry errors inherent in observations. Unfor-tunately, this is not always the case with Q2237 + R have a rather long time coverage and are sampled densely enough, with an almost daily cadence that is sometimes inter-rupted by short weather gaps. Although these curves do not fullymeet the criteria indicated above, we decided to use them todetermine the interband time delays. The V and I light curvesshown in Fig. 1 are represented by a smaller number of datapoints than those in R , but not so few as to abandon the attemptof measuring the interband time delays. The time delays for three pairs of filters, τ RV , τ IR , and τ IV calcu-lated from the data of the 2004 and 2005 seasons are presentedin Table 2 for each of the A, B, and C macroimages separately(columns 2, 4, and 6), as well as averaged over the components(columns 3, 5, and 7). We did not use the light curves of thefaintest D component because the photometric accuracy is low.The positive values of the time delays mean that the light curvescorresponding to the first letter in a subscript lag those corre-sponding to the second symbol.To estimate the errors of the time delays, we used the ini-tial light curves modified in the following way. Up to 30% ofthe data points were excluded from an initial pair of light curvessequentially in a random manner, then the procedure of deter-mining the time delay was applied for each such new realizationof the light-curve pairs, and the corresponding new estimate ofthe time delay was obtained. A number of such trials reached20, and then the average time-delay values were calculated for aset of such estimates. The value of the RMS deviation from theaverage was taken as an error of our time-delay estimates. Thisapproach to estimating the errors of measuring the time delaysdi ff ers from the generally accepted approach. A pair of artificialmodel signals is usually synthesized, with the subsequent sig-nal imposing various random noise realizations that imitate thescatter of the initial data points. Our approach is also valid. Atleast, it has the advantage that we need not worry whether theexcepted random noise parameters (e.g., variance and probabil-ity density distribution) are adequate for the real characteristicsof the photometry errors of the processes under comparison.In Fig. 3 we show the histograms, which demonstrate theprobability density distributions for the estimates of the time de-lays inherent in our measurements presented in Table 2. The his-tograms, far enough from being Gaussian, are nevertheless sym-metric enough and have fairly clear maxima, with the exceptionof measurements for τ RV in 2004 and, perhaps, in 2005.Our measurements thus show quite definitely that the quasarintrinsic flux variations in longer wavelengths lag those inshorter wavelengths for all the three pairs of spectral bands, inaccordance with the thermal reprocessing scenario proposed byKrolik et al. (1991). For further consideration, we require that Article number, page 3 of 11 & Aproofs: manuscript no. berdina2020
Season 2004 ∆ m A R t J , JD - 2450000, days ∆ m AV t J , JD - 2450000, days Season 2004 ∆ m B R t J , JD - 2450000, days ∆ m BV t J , JD - 2450000, days Season 2004 ∆ m CR t J , JD - 2450000, days ∆ m C V t J , JD - 2450000, days ∆ m A I t J , JD - 2450000, days ∆ m B I t J , JD - 2450000, days ∆ m C I t J , JD - 2450000, days Season 2005 ∆ m A R t J , JD - 2450000, days ∆ m AV t J , JD - 2450000, days Season 2005 ∆ m B R t J , JD - 2450000, days ∆ m BV t J , JD - 2450000, days Season 2005 ∆ m CR t J , JD - 2450000, days ∆ m C V t J , JD - 2450000, days ∆ m A I t J , JD - 2450000, days ∆ m B I t J , JD - 2450000, days ∆ m C I t J , JD - 2450000, days Fig. 2.
Polynomial approximations for the light curves in Fig. 1: the curves are reduced to the same magnitude level, and the first-order terms areeliminated ( ∆ m is the brightness variations in magnitude relative to the average level for a given season). the time delays are reduced to the source coordinate system,˜ τ = τ/ (1 + z q ). This is shown in Table 3, where the estimatesof ˜ τ RV , ˜ τ IR , and ˜ τ IV averaged over three macroimages are pre-sented for two seasons separately. The 80% confidence intervals( CI ) for the estimated time delays are also presented in this table. We consider the time delays for the 2004 and 2005 seasonsrepresented in Table 3 separately. We further used their valuesaveraged over the seasons, which are (in days)˜ τ RV = . ± .
92; ˜ τ IR = . ± .
69; ˜ τ IV = . ± . . (1) Article number, page 4 of 11erdina L.A. et al.: Q2237 + Table 2.
Relative interband time lags τ (in days) of the Q2237 + RV , IR , and IV pairs of light curves from the monitoring databy Dudinov et al. (2011) (in the observers’s coordinate system). Season 2004 τ RV τ ABCRV τ IR τ ABCIR τ IV τ ABCIV
Comp A 9 . ± .
51 2 . ± .
07 5 . ± . . ± .
31 7 . ± .
81 3 . ± .
08 3 . ± .
37 6 . ± .
91 6 . ± . . ± .
38 4 . ± .
53 7 . ± . τ RV τ ABCRV τ IR τ ABCIR τ IV τ ABCIV
Comp A 7 . ± .
06 0 . ± .
83 7 . ± . . ± .
35 6 . ± .
97 0 . ± .
81 0 . ± .
36 5 . ± .
48 6 . ± . . ± .
62 0 . ± .
19 6 . ± . τ ABCRV τ ABCIR τ ABCIV
Comp ABC 6 . ± .
46 1 . ± .
86 6 . ± . RV and IV pairs of light curves, although the wave-length bases for the filter pairs I − V and R − V di ff er by fourtimes. The least value of the interband time delay was obtainedfor the intermediate wavelength base (and for the longest wave-length range of the filter pairs), namely, between the light curvesin filters I and R . This result is inconsistent with the expectedpower-law dependence of the disk e ff ective radius on wave-length, r λ = r λ ( λ /λ ) ζ , with ζ = / ff erent wavelengths, which for the disk zonesemitting at the e ff ective wavelengths λ and λ can be written as τ = r λ c λ λ ! ζ − , (2)where c it is the velocity of light.We used here a somewhat simplified scheme for emergingreverberation responses, which admits that they are formed foreach of the filters in some annular zones of an accretion disk.The zones are located at the distances from the central sourcewhere the temperatures match the passbands of the correspond-ing filters. An initial signal (fluctuations of the hard radiationfrom the central source region) in its propagation toward thedisk periphery is reprocessed into the longer-wavelength signalswith the time lags determined by the proper light travel times.At the same time, the initial signal undergoes distortions, whichare due to sizes, shapes, and positions of the re-emitting regions.According to our simplified scheme, the observed distortion ofthe initial signal in time (transfer function, or, more exactly, re-sponse function) is determined in particular by the width andbrightness profile of a specific emitting zone, azimuthal distribu-tion of the zone surface brightness, and finally, by the inclinationof the disk plane with respect to the plane of the sky.In each specific case, all these factors are unknown, or areknown with a high degree of uncertainty. To construct the re-sponse function, model representations are therefore commonlyused, which are simplified as a rule. In our case, the simplifica-tion implies that the widths and brightness of the annular zonesdo not change in azimuth. The width of the response functionis then determined by a temperature profile of the annular zone,which matches the corresponding filter passband, while the ef-fect of the disk inclination is reduced to additional broadening Table 3.
Rest-frame time delays ˜ τ of the Q2237 + RV , IR , and IV pairs of light curves averaged over the three compo-nents, and the corresponding confidence intervals CI for the 80% con-fidence level. Season 2004 ˜ τ RV ˜ τ IR ˜ τ IV ˜ τ ABC (days) 2 . ± .
04 1 . ± .
51 2 . ± . CI (1.33; 4.0) (0.53; 1.84) (1.33; 3.56)Season 2005 ˜ τ RV ˜ τ IR ˜ τ IV ˜ τ ABC (days) 2 . ± .
73 0 . ± . . ± . CI (1.38; 3.26) (-0.41; 0.88) (1.41; 3.58)Averaged ˜ τ ABCRV ˜ τ ABCIR ˜ τ ABCIV ˜ τ ABC (days) 2 . ± .
92 0 . ± .
69 2 . ± . ± ( r sin i ) / c , where i is the inclina-tion angle, and c is the velocity of light. It is important that thebroadening is symmetric with respect to the initial signal arrivaltime at a distance r from the source.Poindexter & Kochanek (2010) have concluded from theanalysis of microlensing events in Q2237 + i = r = · cm, will be approximately ± ff ect of a responsefunction with such a width on the initial signal with a charac-teristic variability timescale of a few dozen days and longer willresult only in its insignificant smoothing, while the symmetricnature of the response function will ensure the absence of biasesin estimates of the interband time delays.
3. Discussion
For the standard thin accretion disk by Shakura & Sunyaev(1973), the radius r λ (the disk scale length) where the disk tem-perature reaches the photon energy, kT = hc / ˜ λ , is given by theexpression (Poindexter & Kochanek 2010; Frank et al. 2002) r ˜ λ = G π hc ! / ˜ λ / ( M BH ˙ M ) / , (3)where G is the gravitational constant, h is the Planck constant, c is the light speed, ˜ λ is the rest-frame wavelength, and M BH and˙ M are the black hole mass and accretion rate, respectively. Article number, page 5 of 11 & Aproofs: manuscript no. berdina2020
Filter pairs R - V, season 2004 RV , days P Filter pairs I - R, season 2004 -3 -2 -1 0 1 2 3 4 5 6 7 8 IR , days P Filter pairs I - V, season 2004 IV , days P Filter pairs R - V, season 2005 RV , days P Filter pairs I - R, season 2005 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 IR , days P Filter pairs I - V, season 2005 IV , days P Fig. 3.
Histograms of the probability density distribution P for the estimates of the interband time delays τ RV , τ IR , and τ IV presented in Table 2. Poindexter & Kochanek (2010) and Morgan et al. (2010) pro-posed the following form of expression (3): r λ = . · ˜ λ ! / M BH M ⊙ ! / L η L Edd ! / , (4)where dimensionless quantities M BH / M ⊙ and ˜ λ/ µ m areintroduced, L and L Edd are the disk luminosity and the Eddingtonluminosity limit, L Edd = · ( M BH / M ⊙ ) erg / s, and η is theaccretion e ffi ciency, which varies from 0.06 for the Schwarzshildblack hole to 0.4 for the Kerr black hole. Frank et al. (2002)proposed considering η = . L is also often assumed to be equal to L Edd ,thus providing the upper limit to the estimate of r λ .For Q2237 + M BH determinations are availablein the literature. We have collected most of them in Table 4 andindicate the method used for the determination. The estimatesvary almost by an order of magnitude depending on the method.For example, adopting M BH = · M ⊙ by Morgan et al. (2010),we obtain from Eq. (4) the following values of radii (in centime-ters) for the rest-frame wavelengths corresponding to filters V , R , and I : r V = (2 . ± . · , r R = (2 . ± . · , r I = (4 . ± . · . (5)These values are generally consistent with the determina-tions of the e ff ective accretion disk size in Q2237 + r / is the so-called half-light radiusequal to 2 . r λ (Poindexter & Kochanek 2010). Accordingly, we may expect the following quantities for thedistances between the disk zones that radiate in the correspond-ing spectral bands: r R − r V = (0 . ± . · , r I − r R = (1 . ± . · , r I − r V = (2 . ± . · . (6)Using a simplified model for the occurrence of the reverber-ation responses described above, we therefore obtain the follow-ing values of the interband time delays (in days) predicted by thestandard thin disk model:˜ τ RV = . ± . , ˜ τ IR = . ± . , ˜ τ IV = . ± . . (7)Uncertainties in quantities (5), (6), and (7) were calculatedadopting the formal uncertainty in the M BH determination fromthe line widths to be equal to approximately 0.1 dex (Morganet al. 2010). It should be noted that some authors have reportedless optimistic errors for the broadline and reverberation map-ping estimates of M BH (e.g., Kollmeier et al. 2006; Kelly et al.2008).The interband time delays calculated with the use of Eq.(4) are noticeably shorter than those determined in the presentwork (Table 3 and the quantities in Eq. (1) averaged over thetwo seasons). The discrepancy is large enough and needs to bediscussed. Article number, page 6 of 11erdina L.A. et al.: Q2237 + Table 4.
Estimates of the black hole mass in the Q2237 + Authors M BH · M ⊙ Methods and detailsAgol et al. (2000) 20 From the flux ratios in the mid-IRKochanek (2004) 3.3 From M BH - L relationship assuming η = . L = L Edd
Pooley et al. (2007) 10 From bolometric luminosity assuming η = . L = L Edd
Morgan et al. (2010) 9 From C IV velocity width by Yee & De Robertis (1991)Assef et al. (2011) 6.17 From C IV , H α and H β velocity widthsSluse et al. (2011) 2.0 From C IV velocity width from Assef et al. (2011)Sluse et al. (2011) 17.7 From the disk radius - M BH relationshipMediavilla et al. (2015a, 2015b) 12 From a central depression in microlensing light curve We are not the first to encounter this discrepancy betweenthe measured reverberation time lags and those predicted bythe standard accretion disk model. In particular, Morgan et al.(2010), Edelson et al. (2015), Kokubo (2018), and the DESand STORM project participants (Fausnaugh et al. 2016, 2018;Mudd et al. 2018) pointed out that the measured time delays be-tween the quasar flux variations in di ff erent spectral regions areoften noticeably longer than the expected values derived withinthe standard model of a thin accretion disk. The evident expla-nation of this discrepancy might imply that the observed rever-beration responses arise in some extended disk regions locatedsomewhere at the disk periphery.The idea of the existence of such a structure has repeat-edly been expressed in a number of earlier works dedicated,for instance, to the analysis of flux ratio anomalies, or inter-preting the results of microlensing studies in some gravitation-ally lensed quasars (Witt & Mao 1994; Pooley et al. 2007;Vakulik et al. 2007; Shulga et al. 2014). The similar discrep-ancy between the measured dimensions of quasars and those de-rived from their luminosities has also been discussed by Dai etal. (2003), Poindexter et al. (2008), Morgan et al. (2010), andPoindexter & Kochanek (2010). Jaroszynski et al. (1992) werethe first to admit the existence of an extended outer structurein the Q2237 + B and R . Theysuggested that this additional light could be the radiation fromthe disk reprocessed in the outer regions of the quasar. This ex-tended structure does not experience microlensing, but dampsthe microlensing peaks produced by the compact inner parts ofthe disk. Schild & Vakulik (2003) and Vakulik et al. (2007) haveshown that a two-component quasar structure model consistingof a central compact source and an extended outer feature is ca-pable of better reproducing in simulations the observed ampli-tudes of microlensing light curves than the central source alone.The accretion disk models predict a power-law dependenceof the disk e ff ective radius on wavelength, r λ ∼ λ ζ , with ζ = / + These discrepancies between the results of observations and pre-dictions made from a standard disk model are important indi-cations that the real accretion disk may noticeably di ff er fromthe model. A possible scenario for the emergence of an ex-tended structure around the accretion disk was considered sev-eral decades ago in the classical work by Shakura & Sunyaev(1973). They analyzed a supercritical accretion mode, whichis characterized by the luminosity fixed at the Eddington limit L cr = M / M ⊙ erg / s and by a high accretion rate, ˙ M > ˙ M cr ,where ˙ M cr = L cr /η c , as well as by a low value of the e ffi ciency α of the angular momentum transport in the accreting matter, α ≪
1. According to the classical accretion disk model, the diskthickness increases with distance to the black hole, providing thepossibility that the outer regions of the disk intercept a portion(from 0.1 to 10%) of the hard radiation of the central regionsand re-emit it in the UV and optical continuum, as well as inemission lines of various elements. Shakura & Sunyaev (1973)showed that in the supercritical regime, when ˙ M ≫ ˙ M cr and α ≪
1, an optically thick scattering envelope is formed, whichincreases the apparent disk size. They gave analytical expres-sions for the e ff ective temperature T e f f and radius r e f f of suchan envelope, T e f f ≃ · ˙ m − / m − / α / A − / ( ◦ K ) (8) r e f f ≃ · ˙ m / m / α − / A / ( cm ) . (9)Here, dimensionless parameters are introduced: m = M / M ⊙ ,˙ m = ˙ M / ˙ M cr . Parameter A characterizes a ratio of energy lossesin the Compton processes to those in the free-free transitions. Forphysical conditions of interest in this particular considerations, A is noted to vary from 10 to 300 (Shakura & Sunyaev 1973). Theyalso pointed out that the temperature T e f f is virtually constantin the regions with r > r e f f . The authors further point out thatin the optical wavelengths which correspond to low-frequencyspectral range at the discussed temperature T e f f the radius nearwhich the envelope becomes opaque exceeds r e f f considerably.Setting the value of the optical opacity equal to unity, the authorsobtained the following expression for the optical envelope radius(in centimeters): r opt ≃ α − / ◦ KT ! / Hz ν ! / ˙ m / m / , (10)where (10 ◦ K ) / T and (10 Hz ) /ν are dimensionless temperatureand frequency, respectively. Article number, page 7 of 11 & Aproofs: manuscript no. berdina2020
Table 5.
Half-light radius r / of the Q2237 + ff erent works from the analysis of microlensing events.To compare this with the scale length r λ , the relation r / = . r λ should be used. Authors Spectral range r / (cm)Kochanek (2004) Johnson-Cousins V filter 3 · Wayth et al. (2005) C
III] and Mg II emission lines ≤ . · Vakulik et al. (2007) Johnson-Cousins V filter 2 . · Anguita et al. (2008) SDSS g ‘ and r ‘ + Johnson-Cousins V . · Morgan et al. (2010) Johnson-Cousins V filter 4 . · Poindexter & Kochanek (2010) Johnson-Cousins V filter 1 . · Mosquera et al. (2013) Johnson-Cousins V filter 2 . · Mosquera et al. (2013) Soft X-rays 5 . · Mosquera et al. (2013) Hard X-rays 2 . · Mediavilla et al. (2015a) Johnson-Cousins V filter 7 . · Mediavilla et al. (2015b) Johnson-Cousins V filter 1 . · Munoz et al. (2016) 7 narrow filters + Bessel I , 3510 - 8130 A 2 . · Vives-Arias et al. (2016) Mid-IR (10.36 µ m ) 8 . · In Table 6 we show the values of T e f f , r e f f , and r opt thatwere calculated for two pairs of parameters α and A selected ar-bitrarily within the range of their permissible values indicatedabove, namely, for α = .
05 and A = α = . A =
50. The black hole mass was taken from Table 4 to equal M BH = · M ⊙ , and the dimensionless accretion rate wasadopted to be ˙ m =
17, according to Morgan et al. (2010) andAbolmasov & Shakura (2012). Table 6 shows that the supercrit-ical regime provides values of r opt that are steadily higher thanthose predicted by the standard thin-disk model. The reverber-ation signals at these radii can therefore be expected to exhibitlonger time lags.To reveal the regions of parameters α and A that providethe best fit between the time delays determined in our work andthose calculated for the radii r opt according to expression (10),we plot the di ff erence maps between the measured and predictedvalues of the interband time lags. The maps built in coordinates α and A for each of the three filter pairs separately are shown inFig. 4. The di ff erence values are reproduced in grades of grayshown at the right edge of each row. The brightest regions corre-spond to the smallest di ff erence between the reverberation timedelays calculated according to Eq. (10) and those measured inthe present work. Rather vast areas in coordinates α and A pro-vide the values of the interband time delays predicted with theuse of Eq. 2 and Eq. 10 consistent with our measurements. Theseareas are clearly seen to shift toward the higher values of α forthe longer wavelength range. This is consistent with the analysisof the supercritical regime by Shakura & Sunyaev (1973), whonoted that parameter α can (and must) depend on the disk radius.In particular, for the turbulent mechanism of the angular momen-tum transport, α can be about 1 at the disk periphery, while closerto the black hole, where the accretion picture becomes spherical,parameter α ∼ − .Another quantity that may serve as an indicator of the ac-cretion regime in a particular object is the Eddington ratio, L bol / L Edd . We followed Agol et al. (2009) and adopted L bol = · erg · s − for the bolometric luminosity of Q2237 + L Edd = . · ( M BH / M ⊙ ) erg · s − for the blackhole mass M BH = · M ⊙ (Morgan et al. 2010, see also Table4 ). We obtained L bol / L Edd ≈ . L bol / L Edd and black hole masses for 34 objects from the Kaspi et al. (2000) sample of the PG quasarsfor which the black hole masses are known. They reported that11 objects have super-Eddington luminosities, and 7 have lumi-nosities higher than 0 . · L Edd , that is, more than a half of theobjects are inconsistent with the geometrically thin-disk model.Similar studies have been carried out by Kollmeier et al.(2006) using 407 AGNs from the AGN and Galaxy EvolutionSurvey (AGES). Their L bol / L Edd – M BH plot exhibits condensa-tion of the data points (a ridge of a sort) at L bol / L Edd ≈ / L bol / L Edd built both at fixed BH mass andat fixed luminosity are log-normal and have rather sharp peaksnear L bol / L Edd ≈ /
4, thus confirming that the sample consistsmostly of AGNs radiating near the Eddington limit. With thesetwo works taken into account, we conclude that the Q2237 +
4. Conclusions
The scenario that an extended optically thick envelope forms inthe supercritical accretion regime considered by Shakura & Sun-yaev (1973) therefore is in principle capable of explaining thehigh values of the interband time delays obtained in our work.The analysis of the supercritical regime and of its possible ob-servational manifestations developed further in Abolmasov &Shakura (2012) and Shakura (2018) has also been considered insimulations (e.g., Ohsuga et al. 2005; Okuda et al. 2005; Volon-teri et al. 2015; Sakurai et al 2016; Jiang et al. 2019; Pinto etal. 2020, and others). The results of some observations have alsomade their authors consider the suggestion that some quasarsmay accrete in the super-Eddington regime (Collin et al. 2002;Kollmeier et al. (2006); Lauzuisi et al. 2016; Jin et al. 2017; Liuet al. 2019).As we noted in Sec. 2, the behavior of our estimates of the in-terband time delays in wavelengths is not only inconsistent withthe theoretic expectations, but is also very strange: almost equal τ RV and τ IV and very short τ IR . Although the uncertainties of de-termining the time lags presented in Table 3 are large enough, Article number, page 8 of 11erdina L.A. et al.: Q2237 + Filter pairs R - V A Filter pairs I - V A Filter pairs I - R A Filter pairs R - V A Filter pairs I - V A Filter pairs I - R A Filter pairs R - V A Filter pairs I - V A Filter pairs I - R A Fig. 4. Di ff erence maps between the interband time delays determined in the present work and calculated according to equations (2) and (10). Thebrightest regions correspond to the smallest di ff erence between these values. The maps are built for three black hole masses, from top to bottom: M BH = · M ⊙ , M BH = · M ⊙ , and M BH = · M ⊙ . we would like to draw attention to the low τ IR value. Distractingfrom a possible physical reason, we note that if the values of τ RV and τ IV are really close to each other, the values of τ IR must below indeed, that is, the low τ IR estimate can be regarded as anargument that τ RV and τ IV are measured correctly. Table 6. T ef f , r ef f , and r opt (in centimeters) calculated from Eqs. (8),(9), and (10) for two sets of parameters α and A . The black hole masswas adopted to be M BH = · M ⊙ (see Eq. 4) and the dimension-less accretion rate was ˙ m =
17 according to Morgan et al. (2010) andAbolmasov & Shakura (2012).
Parameters α , A T e f f ( ◦ K ) r e f f (cm) r opt (cm) α = . A =
50 1 . · . · . · α = . A =
100 0 . · . · . · The time delays τ RV , τ IR , and τ IV measured in our worktherefore appear to indicate that the disk radius ceased to dependon the wavelength after the zone in which the blackbody temper-ature corresponds to the passband of filter R . Theoretic analysesmade by Shakura & Sunyaev (1973), as well as presented later inAbolmasov & Shakura (2012) and Shakura (2018) showed that scattering envelopes not only provide larger apparent disk sizes,but make their size less dependent on wavelength when scatter-ing by free electrons is the main source of the envelope opacity.Numerical simulations also predict a constant temperature forthe outer layers of the envelope, namely, for an optical depthless than 5 (Hubeny & Hubeny 1998). Abolmasov & Shakura(2012) also noted that an extended scattering envelope can beexpected to spatially mix the radiation from di ff erent disk zones.These peculiarities of the envelope may qualitatively explain theabnormal behavior of our estimates of the interband time lags inwavelengths.Before we summarise, we add one more comment aboutthe high values of the reverberation time lags. As we indicatedabove, our measurements are inconsistent not only with thethin-disk model predictions, but also with the results of manymicrolensing observations of Q2237 + ff erent spatial scales of the accretion disks,with the smaller of them referring to microlensing. As numericalcalculations by Jaroszynski et al. (1992) have shown, the causticcrossing events are most sensitive to the structure of the innerregions of the disk, while the extended outer regions may only Article number, page 9 of 11 & Aproofs: manuscript no. berdina2020 contribute to the observed amplitudes of microlensing events.We summarise our results below.We have measured the interband time delays between thelight curves of the Q2237 + V , R , and I to be ˜ τ RV = . ± .
92; ˜ τ IR = . ± . , and˜ τ IV = . ± .
86 days (in the source coordinate frame),with flux variations in longer wavelengths lagging those inthe harder range of the spectrum, in accordance with the re-processing scenario.These delays indicate that the observed reverberation re-sponses arise in structures located at distances from the cen-tral black hole that exceed the radii of the corresponding re-gions of the accretion disk predicted by the standard thin-disk model.An explanation for this discrepancy is sought within theassumption about a super-Eddington accretion regime inquasar Q2237 + α and A (angular momentum transport e ffi ciency and the ratio ofenergy losses in the Compton processes to those in the free-free transitions), the super-Eddington regime provides val-ues of the envelope radius that are noticeably higher thanthose predicted by the standard thin-disk model. The rever-beration signals at these radii can therefore be expected toexhibit longer time lags, consistent with our measurements.To reveal the regions of parameters α and A that provide thebest fit between the interband time delays determined in ourwork and those based on the analysis of the super-Eddingtonaccretion regime by Shakura & Sunyaev (1973), we plottedthe di ff erence maps between these values in coordinates α and A (Fig. 4). The areas of the best fit of our results to thoseexpected from the Shakura & Sunyaev (1973) analysis ofthe super-Eddington regime are clearly seen to shift towardsthe higher values of α for the longer wavelength range. Thisis consistent with their indication that parameter α can (andmust) depend on the disk radius.The Eddington ratio, L bol / L Edd , is also an important quantitythat may serve as an indicator of the accretion regime in aparticular object. When we adopt, according to Agol et al.(2009), L Edd = · erg · s − for the Q2237 + M BH = · M ⊙ by Morgan et al. (2010) as known with an un-certainty of 0.3 dex, we obtain 0 . < L bol / L Edd < . + + Acknowledgements.
The authors would like to thank the colleagues from IRAof the NAS of Ukraine and IA of the KhNU for the fruitful discussions of the problems raised in the present work. L. Berdina is grateful to the EAS / Springer(2018) and EPS (2019) grants which have provided an opportunity to report pre-liminary results of the present work at conferences. V. Tsvetkova would like toacknowledge the role of R.E. Schild in giving rise to the RM studies in Kharkov,who tried to draw attention to the problem more than a decade ago. V. Shulgaacknowledges the College of Physics, International Center of Future Science,Jilin University, China, for hosting during the time period of preparing the paperfor publication. Also, the authors are thankful to the anonymous referee for thevaluable comments and suggestions.
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