Structure function of the UV variability of Q0957+561
aa r X i v : . [ a s t r o - ph ] O c t Astronomy&Astrophysicsmanuscript no. astroph c (cid:13)
ESO 2018October 29, 2018
Structure function of the UV variability of Q0957+561
L. J. Goicoechea , V. N. Shalyapin , R. Gil-Merino and A. Ull´an Departamento de F´ısica Moderna, Universidad de Cantabria, Avda. de Los Castros s / n, 39005 Santander, Spaine-mail: [email protected] Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, 12 Proskura St., Kharkov 61085, Ukrainee-mail: [email protected] Instituto de F´ısica de Cantabria (CSIC-UC), Avda. de Los Castros s / n, 39005 Santander, Spaine-mail: [email protected] Robotic Telescopes Group, Centro de Astrobiolog´ıa (CSIC-INTA), associated to the NASA Astrobiology Institute, Ctra de Ajalvir,km 4, 28850 Torrej´on de Ardoz, Madrid, Spaine-mail: [email protected]
Preprint online version: October 29, 2018
ABSTRACT
We present a detailed structure function analysis of the UV variability of Q0957 + − λ ∼ λ ∼ − λ ∼ ∼ ∼ / radio jet, reverberation is probably themain mechanism of variability. Thus, two types of EUV / X-ray fluctuations would be generated within or close to the jet and laterreprocessed by the disc gas in the two emission rings, at λ ∼ λ ∼ ∼ ∼ ∼ Key words. gravitational lensing – black hole physics – quasars: general – quasars: individual: Q0957 +
1. Introduction
Structure functions for optical / UV variability of quasars al-low investigation of processes that cause intrinsic variationsand the properties of intrinsic flares (e.g. Kawaguchi et al.1998; Cid Fernandes et al. 2000; Vanden Berk et al. 2004;de Vries et al. 2005; Wilhite et al. 2008), the composition ofintervening systems (e.g. Lewis & Irwin 1996; Schechter et al.2003), and the nature of the intergalactic medium (e.g. Hawkins2002). The most recent and comprehensive study of ensemblestructure functions of non-lensed quasars has been presented byWilhite et al. (2008). In this work, about 2500 quasars at a me-dian redshift of ∼ ff erent groups ac-cording to their black hole mass and continuum luminosity. Foreach group, Wilhite et al. (2008) derived the square root of thenoise-less structure function for the ugriz bands. They detectedthe well-known anticorrelation between luminosity and variabil-ity, as well as a correlation between variability and black holemass (see also Wold et al. 2007). Taking these findings into ac-count, most of the variations at restframe time lags ≤ ff erences in accretion rate. The analysis by Li & Cao (2008)also supports this last claim. However, Wold et al. (2007) andWilhite et al. (2008) point out the absence of a clear correlation between variability and black hole mass for restframe time lags ≤
100 d. Therefore, short timescale variations could not be re-lated to changes in accretion rate.The Wilhite et al. dataset did not allow for a measure-ment of the variability of very bright and massive quasars.Moreover, the mechanism of variability might di ff er from lo-cal to distant quasars, and other physical properties (apart fromluminosity and black hole mass) may play a role (e.g., cir-cumnuclear activity, presence of a jet, X-ray activity, etc.).Thus, detailed structure functions of well-characterised indi-vidual quasars are important tools for understanding di ff erentquasar populations. With respect to the individual sources, forexample, Cid Fernandes et al. (2000) analysed the members ofa sample of optically selected nearby quasars ( z < Goicoechea et al.: Structure function of the UV variability of Q0957 + ric flares led to poor fits. Collier & Peterson (2001) also stud-ied optical / UV structure functions of 13 local AGN (Seyfert 1galaxies) with very good time coverage and resolution. The flarelifetimes (using symmetric triangular flares and certain lag inter-vals) were τ ∼ −
100 days, with the higher mass AGN havinglarger variability timescales.Many previous studies focused on the initial logarithmicslope of the structure function. For example, using the squareroot of the structure function, a measured initial slope can becompared to predictions of possible physical mechanisms ofquasar variability (e.g. Kawaguchi et al. 1998; Hawkins 2002).The initial logarithmic slope β should be less than or equal to0.5 for square and exponentially decaying flares (Poissonianphenomenological models), whereas symmetric triangular flaresproduce a steep slope β > / X-ray irradiation. The cellular-automaton modelis able to reproduce a slope β ∼ − ff ects. These might be basic in-gredients in the reverberation scenario. The one-dimensionalhydrodynamical simulations by Manmoto et al. (1996) led totime-symmetric flares, i.e., flares having symmetric rise and de-cay. These time-symmetric flares can account for steep growthswith slopes above 0.5. The starburst model (supernova explo-sions) produces an initial slope β ≥ β ∼ − ≥
100 d is roughly 0.5 (e.g. Cid Fernandes et al. 2000;Wilhite et al. 2008). On the other hand, local AGN show avariety of initial slopes over restframe lags ≤
100 d: β ∼ − ≤
60 d,Collier & Peterson (2001) also found that optical and UV slopesare consistent with each other. This supports a common variabil-ity mechanism.Over the last 10 −
15 years, several gravitationally lensedquasars have been monitored more or less regularly. Some ofthem o ff er a unique opportunity to study the origin of the in-trinsic signal of quasars, since their intrinsic fluctuations are re-peated in two or more lensed components with certain time de-lays and magnitude o ff sets (e.g. Schneider et al. 1992). Opticalframes ( g and r bands) of the double quasar Q0957 + +
561 using theAPO g -band light curves (photometric magnitudes), finding ashallow logarithmic slope of ∼ ≤
200 d.Unfortunately, these authors did not subtract the observationalnoise from the photometric measurements (e.g., Simonetti et al.1985; Cid Fernandes et al. 2000; Collier & Peterson 2001),which has a significant e ff ect on the measured variations at theshortest lags. Gil-Merino et al. (2001) found an initial slopeof the square-root noise-less structure function of Q0957 + R band β > <
100 d). Very recently,Fohlmeister et al. (2008) separated intrinsic from extrinsic vari-ations in Q1004 + ∼ ≤ r band).From r -band data of Q0909 + β ∼ ≤
100 d.No simple model was able to accurately reproduce the shape ofthe structure function of the intrinsic luminosity of Q0909 + +
561 (Walsh et al. 1979) isprobably the best-studied lens system. This has been inves-tigated in several spectral regions, including radio, IR, opti-cal, UV and X-ray wavelengths. The radio maps of both com-ponents showed the presence of radio cores and ∼ . ′′ ∼ . ′′
3, which is as-sociated with a circumnuclear environment. Taking into accountthe redshift of the quasar ( z = g and r bands correspond to middle ultraviolet (MUV) emis-sion ( ∼ − +
561 is a very bright and massive object, withboth λ L λ (1350 Å) and λ L λ (3000 Å) exceeding 10 erg s − , and ablack hole mass of 2 − × M ⊙ (Peng et al. 2006). The doublequasar Q0957 +
561 is also a X-ray bright source (e.g. Chartas2000), but a possible X-ray jet has not yet been resolved.Liverpool Quasar Lens Monitoring (LQLM) is a long-termproject to follow the variability of gravitationally lensed quasarswith the Liverpool robotic telescope (Goicoechea et al. 2008).The first phase of this project (LQLM I) included a monitor-ing programme of Q0957 +
561 in the gr optical bands. Thenew LQLM I light curves of Q0957 +
561 (2005 − +
561 are very probablydue to reverberation in the accretion disc around the supermas-sive black hole (see also Collier 2001). In Section 2, the newLQLM I dataset is used to accurately trace the shapes of thesquare-root noise-less structure functions of the quasar UV lu-minosity. These shapes are closely linked to the nature of theUV fluctuations (see here above), so we investigate the mech-anism(s) of variability at two di ff erent restframe wavelengths: λ ∼ g band) and λ ∼ r band). By compar-ing the new shape with the old (APO dataset) at λ ∼
2. Structure function analysis
The APO and LQLM I light curves of the double quasarQ0957 +
561 do not show evidence of extrinsic variability(Kundi´c et al. 1997; Shalyapin et al. 2008). Thus, we directlyobtain the noise-less structure function of the intrinsic luminos-ity at a given restframe wavelength (instead of one structurefunction for each component, a combined record and the cor-responding structure function are made, e.g., Kawaguchi et al.1998). The combined records that we use in this paper are de-picted in Fig. 1. There is no standard form of the structurefunction, but di ff erent approaches to the problem. For example,while magnitudes, S F ( m ), are often used in optical astronomy(e.g. Wilhite et al. 2008, and references therein), monochro-matic fluxes or luminosities, S F ( F ) or S F ( L ), are more rele- oicoechea et al.: Structure function of the UV variability of Q0957 +
561 3
Fig. 1.
APO and LQLM I combined light curves. While thelight curves of Q0957 + + ff sets (producing optimaloverlaps). Top: APO g -band. Middle: LQLM I g -band. Bottom:LQLM I r -band. We note that all magnitudes are expressed inthe SDSS photometric system.vant in radio astronomy or to accurately compare with modelsof variability (e.g. Simonetti et al. 1985; Cid Fernandes et al.2000; Collier & Peterson 2001). The structure function S F ( L )at restframe lag ∆ τ is estimated through the averaged sum S F ( L ) = (1 / N ) X i , j [(10 − . m j − − . m i ) − σ i − σ j ] , (1)where m are magnitudes, σ = . × − . m σ , σ are photomet-ric uncertainties, and the sum includes N pairs verifying τ j − τ i ∼ ∆ τ . Here, L = − . m are monochromatic luminosities in conve-nient units and τ are restframe times (e.g., Cid Fernandes et al.2000; Goicoechea et al. 2008). This S F ( L ) describes typical lu-minosity variabilities at di ff erent restframe lags. We normalizethe original structure function to the luminosity variance andthen take the square-root for convenience, i.e., we analyse thenormalized structure function f = [ S F ( L )] / /σ ( L ).Restframe lags substantially below the restframe duration ofthe records are considered in the analysis. In the initial selection,we take ∆ τ ≤ P / + z ), where P is the duration of each wholecombined record (see Fig. 1). Later, only lags before reachingthe asymptotic behaviours ( f ≤
1) are taken into account. In
Fig. 2.
Normalized structure functions of Q0957 +
561 fromLiverpool telescope (LQLM I) records. These structure func-tions of the UV luminosities at ∼ g band; filled cir-cles) and ∼ r band; open circles) are accurately de-scribed from very short restframe lags ( ∼ −
10 days) to lagswhen [
S F ( L )] / /σ ( L ) ∼ β ∼ − λ ∼ g band; filled circles) is compared to f at λ ∼ r band; open circles). Both growths seem tobe consistent with each other. With respect to the initial loga-rithmic slopes, in Fig. 2 we also show two fits f = A ( ∆ τ ) β withˆ χ = χ / N do f values close to 1 ( N do f is the number of degrees offreedom). The fits over time intervals 3 −
14 d ( g band) and 6 −
50d ( r band) lead to β = ± β = ± σ inter-vals), respectively. These initial slopes disagree with the predic-tion of the cellular-automaton disc-instability model, but roughlyagree with the time-symmetric flares that appear in the one-dimensional hydrodynamical simulations (see Introduction). Inprinciple, the starburst model could also account for the mea-sured slope at ∼ g band). The microlensing slope β ∼ − + +
561 at ∼ ff erent initial growths. This may be a consequenceof evolution in the variability scenario, since both experiments(APO and LQLM I) are separated by ∼ χ ∼ β = ± ∆ τ ≤ β = ± ∆ τ ≤
11 d,and β = ± ∆ τ ≤
16 d (1 σ intervals).In order to discuss the mechanism(s) for (M)UV variability,and to quantify the spectral and time evolution of that mecha-nism(s), we compare the observed shapes of the structure func-tions (Figs. 2 −
3) with predictions of Poissonian models. In afirst level of complexity (simplest models), we use three phe-nomenological models: square flares (SQF), exponentially de-caying flares (EDF), and symmetric triangular flares (STF), aswell as starburst flares (SBF) produced by supernova explosions
Goicoechea et al.: Structure function of the UV variability of Q0957 + Table 1.
Solutions for the main Poissonian models.
Model a Observed structure function b ˆ χ w c τ c (d) τ c (d)SQF(1) + STF(2) f LQLM I (2100 Å) 1.21 0.266 + . − . + . − . + . − . f LQLM I (2600 Å) 0.65 0.170 + . − . + . − . + . − . f APO (2100 Å) 1.44 0.266 + . − . + . − . + . − . SQF(1) + SBF(2) f LQLM I (2100 Å) 1.19 0.706 + . − . + . − . + − f LQLM I (2600 Å) 0.99 0.630 + . − . + . − . + − f APO (2100 Å) 1.78 0.758 + . − . + . − . + − EDF(1) + STF(2) f LQLM I (2100 Å) 1.25 0.902 + . − . + . − . + . − . f LQLM I (2600 Å) 0.73 0.418 + . − . + . − . + . − . f APO (2100 Å) 1.80 0.856 + . − . + . − . + . − . STF(1) + STF(2) f LQLM I (2100 Å) 1.55 0.204 + . − . + . − . + . − . f LQLM I (2600 Å) 0.80 0.182 + . − . + . − . + . − . f APO (2100 Å) 1.31 0.270 + . − . + . − . + . − . STF(1) + SBF(2) f LQLM I (2100 Å) 1.18 0.052 + . − . + . − . + − f LQLM I (2600 Å) 0.72 0.608 + . − . + . − . + − f APO (2100 Å) 1.64 0.646 + . − . + . − . + − a SQF = square flares, EDF = exponentially decaying flares, STF = symmetric triangular flares, and SBF = starburst flares. b Normalized structure functions (see main text) from Liverpool tele-scope (LQLM I) and Apache Point Observatory (APO) data. c All measurements are 1 σ intervals. Fig. 3.
Normalized structure functions of Q0957 +
561 fromApache Point Observatory (APO) and Liverpool telescope(LQLM I) records. The two structure functions of the luminos-ity at ∼ = open triangles, LQLM I = filled circles)correspond to two experiments separated by ∼
10 years, i.e., ∼ β ∼ ≤ β ∼ ≤
16 d.(Aretxaga et al. 1997; Cid Fernandes et al. 2000). For exam-ple, EDF is a simple model to describe asymmetric flares, i.e.,rapid rises and slow declines, or slow rises and rapid declines(the structure function cannot distinguish between both vari-ants). STF is a rough description of the hydrodynamical simula-tions by Manmoto et al. (1996) and a relatively good approachto the X-ray shots found by Negoro et al. (1994). Each model ischaracterised by a shape function s ( τ ), f = [ s ( τ )] / , and theseshape functions appear in Appendix B of Cid Fernandes et al.(2000) and Eq. (17) of Aretxaga et al. (1997). The 4 level-1models of f only include one free parameter: flare lifetime τ (e.g., τ = T for a square flare of duration T and τ = t sg for a supernova explosion, where t sg is the time when the super-nova remnant reaches the maximum of its radiative phase). In thePoissonian framework, we can consider more complex schemes.Thus, in a second level of complexity, the luminosity is assumedto be due to the superposition of a constant background and twoindependent variable components. It can easily be shown that f = [ w s ( τ ) + (1 − w ) s ( τ )] / , where w is the ratio betweenthe variance of the first fluctuating component and the total vari-ance, and s and s are the shape functions of the two indepen-dent components.The possibility that AGN variability is caused by severalindependent processes was suggested in previous work (e.g.Kawaguchi et al. 1998; Collier & Peterson 2001). Here, us-ing 10 level-2 models of f (SQF + SQF, SQF + EDF, SQF + STF,SQF + SBF, EDF + EDF, EDF + STF, EDF + SBF, STF + STF,STF + SBF, and SBF + SBF), we also explore this possibility.Each of these 10 possible combinations is characterised by threefree parameters: w , and two lifetimes τ and τ (see above). Toevaluate the quality of the fits obtained with the full set of 14models, we analyse the ˆ χ (reduced chi-square) values. For anacceptable fit, χ is expected to be in the range N do f ± N do f ) / (allowing χ − N do f di ff erences of up to two standard deviationsof the χ distribution), which implies 0.6 ≤ ˆ χ ≤ N do f ∼ χ are associated with (level-2) models incorporatingeither STF, or SBF, or both of them. However, only one out ofour 42 fits to f LQLM I (2100 Å), f LQLM I (2600 Å), and f APO (2100Å) (3 observed shapes ×
14 models) gives ˆ χ ≤ ff erent time lag bins. Although we use a very popularestimator of uncertainties, Collier & Peterson (2001) noted thatnot all pairs of data in a given bin are independent. To addressthis problem, they multiplied their errors by 2 / . We adopt theCollier & Peterson’s perspective and fit again the observed struc-ture functions. oicoechea et al.: Structure function of the UV variability of Q0957 +
561 5
After slightly enlarging the error bars, we concentrate onlevel-2 models consisting of STF / SBF and anything else. Amongthese 7 models of f , we select those giving ˆ χ < f LQLM I (2100 Å)and f LQLM I (2600 Å), i.e., they fit both LQLM I shapes ac-ceptably well (0.6 ≤ ˆ χ ≤ + STF) indicate that symmetric triangular flares at ∼ ∼ τ (2100 Å) < τ (2600 Å) in a disc local-instability scenario, e.g., the thermal timescale is proportional to( R / GM ) / , where M is the mass of the central black hole and R ∝ λ / is the emission radius (e.g., Goicoechea et al. 2008,and references therein). The solutions for the second model(SQF + SBF) incorporate supernova explosions with t sg (2100 Å) < t sg (2600 Å). However, the timescale t sg is exlusively relatedto the circumstellar density and the total energy released in eachexplosion (e.g., Aretxaga et al. 1997), so a chromatic timescaleis not the expected result.The solutions for the EDF + STF and STF + SBF models arethe most interesting ones (LQLM I data). There is a clear de-generacy between EDF and SBF, which seem to work in a sim-ilar way. Hence, the solutions for the third and fifth modelsare interpreted as evidence in favour of the coexistence of ∼ / X-ray varia-tion in the vicinity of the accretion disc axis that are reprocessedby two annuli of the disc gas. Measurements of time delays(Kundi´c et al. 1997; Collier 2001; Shalyapin et al. 2008), andthe existence of an EUV jet (Hutchings 2003) and a bright X-ray source (Chartas 2000), support this physical scenario. FromTable 1, τ asym =
168 d and τ sym =
105 d is a good compro-mise between the results at the two wavelengths and for the twoflare asymmetric profiles (exponentially decaying and starburstprofile). Although the relative variances depend on wavelength,there is no need to invoke some mechanism other than rever-beration. While the ∼ w asym = ∼ w asym = ff erence betweentime coverages, gaps and artifacts in the g ( ∼ r ( ∼ g -band combined record lasts longer than the LQLM Icombined curve in the r band, so it contains a prominent declinethat is not present in the shorter record. Besides the longer timecoverage, the g -band curve fills a gap corresponding to the peakof an important event. The presence of a few artifacts could alsoplay a role. In Fig. 4, the LQLM I shapes are compared with theadopted solutions: w asym = τ asym =
168 d, and τ sym =
105 dfor f LQLM I (2100 Å), and w asym = τ asym =
168 d, and τ sym =
105 d for f LQLM I (2600 Å).In spite of the enlargement of error bars and the use ofrelatively sophisticated modelling, no model leads to an ac-ceptable global solution at both wavelengths and both epochs.The APO data are only consistent, in terms of ˆ χ , with theSTF + STF model. The corresponding solution includes ∼ Fig. 4.
Observed structure functions and adopted solutions. Forthe LQLM I experiment, we show EDF + STF laws with param-eters: w = τ =
168 d, and τ =
105 d ( g band; solid line),and w = τ =
168 d, and τ =
105 d ( r band; dashed line).SBF + STF laws with the above parameters are also able to re-produce the LQLM I shapes. For the APO data in the g band, wedisplay a STF + STF law with parameters: w = τ =
11 d,and τ =
105 d (dashed-dotted line).d time-symmetric flares and other family of very short-lifetime( τ ∼
10 d) symmetric flares. This last kind of fluctuations is re-sponsible for about 1 / ∼ / X-ray fluctua-tions (reverberation in the accretion disc; see above), we wouldbe dealing with intermittency in the generation of high-energyasymmetric fluctuations (within or close to the jet). As we com-mented here above, the structure function f APO (2100 Å) agreeswith the presence of very short-timescale shots, which are notdetected from the new LQLM I records. These fluctuations maybe caused by some kind of observational (systematic) noise oralternatively, they might be related to an episode of very short-timescale activity inside the disc, the jet or other region. In Fig. 4,the adopted solution (STF + STF model with parameters w = τ =
11 d, and τ =
105 d) fits f APO (2100 Å) reasonablywell.
3. Conclusions and discussion
We present a novel and rigorous analysis of the structure func-tion of the UV variability of the gravitationally lensed quasarQ0957 + − λ ∼ g band data) and λ ∼ r band data). Old Apache Point Observatory records in the g band(1995 − ∼ Goicoechea et al.: Structure function of the UV variability of Q0957 + one variable component (Cid Fernandes et al. 2000, and refer-ences therein), as well as hybrid models incorporating two inde-pendent variable components.Several hybrid (or level-2) models are able to account forboth Liverpool telescope structure functions (see Table 1). Someof them contain flares with unrealistic profile (square flares) andlead to solutions that are di ffi cult to interpret. Fortunately, wealso find reasonable solutions in which ∼ ∼ gr light curves of Q0957 +
561 led to time delays be-tween quasar components and between optical bands that mainlysupport a reverberation scenario (Collier 2001; Shalyapin et al.2008). Thus, reverberation would be the main mechanism ofvariability. The presence of an EUV / radio jet (e.g., Garrett et al.1994; Hutchings 2003) and a bright X-ray source (Chartas2000) also suggests the viability of this mechanism: two typesof EUV / X-ray fluctuations that are generated within or close tothe jet, and later reprocessed by two rings of the disc (each ringcorresponds to a di ff erent restframe wavelength). On the otherhand, one can also justify both kinds of flare profile. For ex-ample, the cellular-automaton model produces asymmetric shots(e.g., Kawaguchi et al. 1998), and hydrodynamical simulationslead to symmetric flares (Manmoto et al. 1996).The ∼ ∼ ∼
10 d) symmetric flares. This kind of flare mightbe caused by observational systematic noise, or perhaps, repre-sent additional evidence for time evolution. Our results do notsupport a previous claim for the possible starburst origin of someevents in the old g -band light curves (Ull´an et al. 2003). Despitethe presence of two twin events (one in each quasar compo-nent) with an anomalous delay (Goicoechea 2002), the asso-ciated shot probably occurred at the base of the jet or in thecircumnuclear region, but it was not originated by a supernovaexplosion.Very recently, several studies have showed evidence thatoptical / UV variability of quasars on restframe timescales >
100 d is mainly driven by variations in accretion rate(e.g., Wold et al. 2007; Ar´evalo et al. 2008; Li & Cao 2008;Wilhite et al. 2008). Here we are discussing the UV variabilityof Q0957 +
561 at restframe lags ≤
100 d. Q0957 +
561 is a verybright and massive object, and this population could not be stud-ied by Wilhite et al. (2008). The first lensed quasar has also anEUV / radio jet (and important X-ray activity; see above), so high-energy variations in the surroundings of the disc axis and theirreverberation are possible. For example, on timescales below100 days, the optical variations of the local Seyfert galaxy NGC5548 are related to its X-ray variations (Czerny et al. 1999).Hence, part of the optical variability of this AGN (timescales <
100 days) could be explained by X-ray reprocessing. Chromaticdelays for the local Seyfert galaxy NGC 7469 also reveal a re- verberation scenario (Collier et al. 1999). Ar´evalo et al. (2008)reported on an illustrative example of mixed variability. The lo-cal quasar MR 2251-178 has been monitored simultaneouslyin X-rays and optical bands. All spectral regions were signif-icantly variable, and the fluctuations were clearly correlated.Ar´evalo et al. (2008) indicated that pure reprocessing of X-rayscannot account for both ∼ ∼ T ∝ r − / (Shakura & Sunyaev 1973). The reverberation hypothesis as-sumes that the optical / UV disc regions are irradiated by EUV / X-ray photons from the vicinity of the disc axis. If the high-energysource is placed on the axis and at a height H X above the thin disc(disc thickness << H X ), the non-standard temperature profile is T ( r ) = " GM ˙ M πσ r + (1 − A ) L X H X πσ ( H x + r ) / / , (2)where G is the gravitation constant, σ is the Stefan constant,˙ M is the mass accretion rate, A is the disc albedo, i.e., the ra-tio of reflection to incident high-energy radiation, and L X is theluminosity of the irradiating source (e.g., Cackett et al. 2007,and references therein). Moreover, considering r >> H X (andthus, a standard temperature profile), the typical radius of the in-tensity distribution at a given restframe wavelength λ should begreater than the standard value (for standard structure, see, e.g.,Shalyapin et al. 2002). Eq. (2) leads to R = " GM ˙ M πσ + (1 − A ) L X H X πσ / " k λ hc / , (3)where k is the Boltzmann constant, h is the Planck constant,and c is the speed of light. This non-standard typical radius isproduced by both the heating due to irradiation and the viscousheating in the disc. Finally, we point out that shallow tempera-ture profiles (from reverberation) could be consistent with mi-crolensing data of some lensed quasars (e.g., Poindexter et al.2008). Moreover, the non-standard sizes of several lensed andmicrolensed quasars (Morgan et al. 2007; Pooley et al. 2007)might be related to relatively high irradiation-to-viscosity ratios IVR = − A ) L X H X / GM ˙ M . Acknowledgements.
We thank an anonymous referee for several comments thatimproved the presentation of our results. Liverpool Quasar Lens Monitoringis a long-term project to follow the optical variability of lensed quasarswith the Liverpool robotic telescope. This paper is partially based on theresults of the first phase of this project (LQLM I; see the Web sitehttp: // grupos.unican.es / glendama / index.htm). We acknowledge the continuingsupport of the Liverpool telescope team. We also thank T. Kundi´c and othermembers of the APO collaboration for providing light curves to us. This researchhas been supported by the Spanish Department of Education and Science grantAYA2007-67342-C03-02 and University of Cantabria funds. RGM holds a grantof the ESP2006-13608-C02-01 project financed by the Spanish Department ofScience and Innovation. References
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