THE AVERAGE SIZE AND TEMPERATURE PROFILE OF QUASAR ACCRETION DISKS

The standard model used to describe the generation of energy in the nucleus of quasars and active galactic nuclei is the thin accretion disk (Shakura & Sunyaev 1973; Novikov & Thorne 1973). This model not only explains the luminosities of quasars, but also predicts observables such as the disk size and its dependence on wavelength. In particular, the scaling of the disk size with wavelength, or equivalently, the temperature profile of the disk, is a fundamental test for any disk theory. Well away from the inner edge, a temperature profile T∝r^−1/p translates into a size profile r_λ∝λ^p, where the characteristic size can be thought of as the radius where the rest wavelength matches the disk temperature. The simple Shakura & Sunyaev (1973) disk has T∝r^−3/4, and hence p = 4/3.

One of the most effective means of measuring accretion disk sizes is gravitational microlensing of lensed quasars (Wambsganss 2006). There are now many quasar size measurements based on microlensing either of individual objects (Kochanek 2004; Anguita et al. 2008; Eigenbrod et al. 2008; Poindexter et al. 2008; Morgan et al. 2008; Bate et al. 2008; Mosquera et al. 2009; Agol et al. 2009; Floyd et al. 2009; Poindexter & Kochanek 2010; Dai et al. 2010; Mosquera et al. 2011; Mediavilla et al. 2011b; Muñoz et al. 2011; Blackburne et al. 2011a; Hainline et al. 2012; Motta et al. 2012; Mosquera et al. 2013; Blackburne et al. 2013) or of samples of lenses (Pooley et al. 2007; Morgan et al. 2010; Blackburne et al. 2011b; Jiménez-Vicente et al. 2012). A generic outcome of these studies is that the disks are larger than predicted. One possible solution is to modify the disk temperature profiles, which can be tested using microlensing by measuring the dependence of the disk size on wavelength.

There are fewer estimates of the wavelength dependence of the size (see Blackburne et al. 2011b and Morgan et al. 2010, and references therein). Most of the measurements (Anguita et al. 2008; Poindexter et al. 2008; Bate et al. 2008; Eigenbrod et al. 2008; Floyd et al. 2009; Blackburne et al. 2011b; Blackburne et al. 2013) are based on broad band photometry that can be affected by line contamination. These studies generally find that the shorter wavelength emission regions are more compact, but the estimates of the scaling exponent p have both large individual uncertainties and a broad scatter between studies. Only Blackburne et al. (2011b) have found little or no dependence of size on wavelength.

Single-epoch broadband photometry is affected by several potential systematic problems: (1) broad lines, which are known to be less affected by microlensing both observationally (e.g., Guerras et al. 2013a) and theoretically (e.g., Abajas et al. 2002); (2) differential extinction of the images by the lens galaxy (e.g., Falco et al. 1999; Muñoz et al. 2004); and (3) uncertainties in the macro model lens magnification. The effects of extinction and modeling uncertainties can be eliminated by analyzing the time variability produced by microlensing, but the effects of line contamination must still be modeled (e.g., see the Dai et al. 2010 analysis of RXJ1131−1231). We can remove the effect of broad line contamination by using spectroscopy (Mediavilla et al. 2011b; Motta et al. 2012) or narrowband photometry (Mosquera et al. 2009; Mosquera et al. 2011) to measure the continuum between the lines. If the continuum and line flux ratios can be separately measured, then the line flux ratios can also be used to eliminate the effects of extinction and model uncertainties on the continuum flux ratios to the extent that the lines are little affected by microlensing.

Even with spectra, there are residual issues coming from line contributions to the apparent continuum. The most important of these are the emission from Fe ii in the wavelength range from 1800 to 3500 Å and the Balmer continuum in the range from 2700 to 3800 Å (Wills et al. 1985). The contribution of these pseudo-continua to the measured continuum flux varies from object to object, and its effect on the estimate of the continuum microlensing magnifications is difficult to assess. If the UV-emitting Fe ii comes from a region similar in size to the continuum source, as suggested by Guerras et al. (2013b), then it would be similarly affected by microlensing. On the other hand, the Balmer continuum likely forms on larger scales and would be less affected by microlensing (see Maoz et al. 1993). Similar to including a broad line, it would dilute the microlensing of the continuum. Since the Balmer continuum peaks near the Balmer limit at 3646 Å, it primarily affects the redder portion of the wavelength ranges we consider here and so could slightly bias our results toward steeper slopes (higher p). The scale of the effect is difficult to estimate, but it should be modest.

Previous results based on spectroscopic or narrowband studies (SBS0909+532 in Mediavilla et al. 2011b; SDSSJ1004+4112 in Motta et al. 2012; HE1104−1805 in Motta et al. 2012 and in Muñoz et al. 2011; HE0435−1223 in Mosquera et al. 2011) have found a slightly shallower wavelength dependence than the predictions of the standard thin disk with p ≈ 1 rather than p = 4/3. The main objective of the present work is to extend the studies of the exponent of the size–wavelength scaling, p (r_s∝λ^p), using spectroscopic and narrowband data that are less affected by the presence of emission lines or extinction (Section 2). In Section 3, we will calculate an estimate for the average value of the exponent p based on this less line-contaminated data. The main conclusions are presented in Section 4.

In the present study, we use estimates of microlensing magnifications measured either from optical/infrared spectra or narrowband photometry available in the literature. We collected data for the eight lens systems in Table 1. Spectroscopic data are available for HE0512−3329 (Wucknitz et al. 2003), SBS0909+532 (Mediavilla et al. 2011b), QSO0957+561, SDSSJ1004+4112, and HE1104−1805 (Motta et al. 2012). In the latter case, there were two epochs of data showing significant changes in the microlensing that were examined in Motta et al. (2012). We also selected objects with narrowband photometry for which (most of) the observed filters do not contain contamination from strong emission lines and where there is little extinction or where we can make an extinction correction. This adds two more objects: HE0435−1223 (Mosquera et al. 2011) and Q2237+0305 (J. A. Muñoz et al. 2014, in preparation).

For spectroscopic data, the microlensing magnification between two images, 1 and 2, at a given wavelength λ is calculated as Δm = (m₂ − m₁)_micro = (m₂ − m₁)_cont − (m₂ − m₁)_line, where (m₂ − m₁)_line is the flux ratio in an emission line and (m₂ − m₁)_cont is the flux ratio of the continuum adjacent to that emission line. Using this method, the microlensing magnification is isolated from extinction and from the mean lensing magnification, as these affect the line and the adjacent continuum equally (Mediavilla et al. 2009). With narrowband photometry we can also isolate the microlensing in the continuum from the contamination of strong emission lines. In this case, however, additional information from IR or spectroscopic observations may be needed to safely remove extinction (see Mosquera et al. 2011; Muñoz et al. 2011) and to provide an unmicrolensed baseline.

For each object (except Q2237+0305 for which J. A. Muñoz et al. (2014, in preparation) carried out a detailed analysis of multi-epoch data in five different bands), we used the measured microlensing magnifications at three different wavelengths, trying to keep the wavelength baseline as large as possible in order to maximize our sensitivity to chromatic microlensing. Therefore, we used the bluest and reddest ends of the observed ranges, with a third point at an intermediate wavelength, using data from the literature as summarized in Table 1. In some cases, the microlensing magnification at the three wavelengths is directly obtained from the original references. In other cases, the values where calculated from the published line and continuum magnitude differences. For HE0435−1223, for which we have narrowband photometry, the continuum magnifications were taken from the observations and the H-band magnifications were used as an unmicrolensed baseline (Mosquera et al. 2011). Similarly, mid-infrared flux ratios from Minezaki et al. (2009) were used as an unmicrolensed baseline for Q2237+0305. There may be some residual differential extinction in Q2237+0305, but it is very small compared to the measured microlensing chromaticity and can be ignored (see J. A. Muñoz et al. 2014, in preparation). The values of the differential microlensing magnification at each wavelength and their estimated errors are shown in Table 1 along with the definition of the unmicrolensed baseline and the source of the data.

We used a Monte Carlo statistical analysis to estimate the average size and wavelength dependence for this sample of objects. The procedure is very similar to that described in Jiménez-Vicente et al. (2012), but in this case we apply it to the estimates of the microlensing magnification at three different wavelengths instead of two. The structure of the accretion disk is described by a Gaussian profile $I(R)\propto \exp (-R^2/2r_s^2)$, where the characteristic size r_s is a wavelength-dependent parameter (related to the half-light radius by R_1/2 = 1.18r_s). The wavelength dependence of the source size is parameterized by a power law such that r_s(λ)∝λ^p, and as noted earlier, the standard Shakura & Sunyaev (1973) accretion disk model has p = 4/3.

Magnification maps for each image are generated using the inverse polygon mapping algorithm (Mediavilla et al. 2006, 2011a). The surface densities near the lensed images are generally dark matter dominated, so we put 5% of the surface mass density in the form of stars following the estimates of Mediavilla et al. (2009). The one exception was Q2237+0305, where 100% of the mass is in the form of stars because the lensing is dominated by the bulge of a low redshift spiral galaxy. We used a fixed microlens mass of M = M_☉. All linear scales can be rescaled with the square root of the microlens mass (r_s(λ)∝(M/M_☉)^1/2). The values of the convergence κ and shear γ associated with the macrolens models were taken from Mediavilla et al. (2009). For SBS0909+532, the improved values in Mediavilla et al. (2011b) were used. The maps are 2000×2000 pixels in size, with a pixel size of 0.5 lt-days. For the objects in our sample this size corresponds to between 43 and 80 Einstein radii (with the exception of Q2237+0305, for which the map size is 14 Einstein radii). We explored a grid in the parameter space of the logarithmic slope p and the disk size at a reference wavelength r_s = r_s(λ₀). The reference wavelength is taken to be the bluest wavelength for which we have a measured microlensing magnification, which is 1026 Å (rest frame). The grid runs over p = 0.25 × i for i = 0⋅⋅⋅10. We use a natural logarithmic grid in r_s such that ln r_s = 0.3 × j for j = 0⋅⋅⋅11.

For every pair of values (p, r_s), the magnification maps are convolved with Gaussians of the relevant sizes at the three wavelengths r_s(λ_k) = r_s(λ_k/λ₀)^p with k = 1, 2, 3 from which we can compare the microlensing magnification statistics with the observed values at those wavelengths. The likelihood of observing the three microlensing magnifications $\Delta m^{{\rm obs}}_k$ for lens l at these three wavelengths λ_k (k = 1, 2, 3) given the parameters p and ln (r_s) is calculated as

$$\begin{equation} P_l\left(\Delta m^{{\rm obs}}_k | p_i,\ln (r_{sj})\right) \propto \int \Delta m_1 \int \Delta m_2 \int \Delta m_3 N_{ij} e^{-\frac{1}{2} \chi ^2}, \end{equation} \tag{ 1 }$$

where N_ij is the number of trials with Δm₁, Δm₂, and Δm₃ for the case with parameters p_i, ln (r_sj), and

$$\begin{equation} \chi ^2=\sum _{k=1}^{3} \frac{\left(\Delta m^{{\rm obs}}_k-\Delta m_k\right)^2}{\sigma _k^2}. \end{equation} \tag{ 2 }$$

The estimated errors σ_k at the different wavelengths are given in Table 1. The likelihood given by Equation 1 is calculated for each of the eight lenses shown in Table 1.¹⁰ For each value of the pair (p, ln (r_s)), the likelihood is calculated using 10⁸ trials by sampling each of the two magnification maps at 10⁴ points distributed on a regular grid. We construct a joint likelihood function for the parameters p and ln (r_s) by multiplying the individual likelihood functions for the eight lenses:

$$\begin{equation} P\left(\Delta m^{{\rm obs}}_{k,l} | p_i,\ln (r_{sj})\right)\propto \prod _{l=1}^{8} P_l\left(\Delta m^{{\rm obs}}_k | p_i,\ln (r_{sj})\right). \end{equation} \tag{ 3 }$$

These likelihoods are shown in Figure 1 for the individual objects, and Figures 2 and 3 for the whole sample. They can be directly used to estimate values for the parameters p and ln (r_s). In this context, we have used the maximum likelihood estimator. We can also take a Bayesian approach and convert these likelihoods into posterior probabilities by multiplying them by suitable priors. We have used uniform priors for ln (r_s) (equivalent to a logarithmic prior for r_s) and the exponent p. With this choice for the priors, Figures 1–3 also represent the Bayesian posterior probabilities (normalization constants aside). In this approach, the mean value of the posterior probabilities and its dispersion are used as estimators of the parameters.

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The individual likelihood distributions for each of the eight objects are shown in Figure 1. For the case of HE1104−1805, where there is data at two different epochs, the distribution is the product of the probability distributions for the two epochs. Q2237+0305 combines the results for the three independent pairs at six different epochs (see J. A. Muñoz et al. 2014, in preparation). In Figure 1, we also show the results from previous studies. In order to make this comparison, the sizes from the literature have, as necessary, been converted to a Gaussian r_s at our nominal wavelength of 1026 Å using the reported values of p, a microlens mass of M = 1 M_☉, and assuming a fixed half-light radius, because estimates of the half-light radius are largely model independent (Mortonson et al. 2005). The general agreement is very good, with our results being compatible with previous estimates given the uncertainties. The maximum likelihood and Bayesian estimates for the size of the accretion disk r_s at the reference wavelength (1026 Å) and for the chromaticity exponent p are given in Table 2. Figure 2 shows the resulting joint likelihood. Previous estimates are also shown, although only for estimates using logarithmic priors on size to avoid overcrowding. Again, the general agreement is very good. The maximum likelihood estimates are $p=0.75^{+0.2}_{-0.2}$ and $\ln (r_s/\mathrm{lt-day})=1.5^{+0.3}_{-0.3}$, where these are 1σ confidence intervals for 1 parameter. This corresponds to a size of $r_s(1026 \mathrm{ \;\mathring{\rm{A}}})=4.5^{+1.5}_{-1.2}$ lt-days for a microlens mass of 1 M_☉. We repeated the calculations using only the objects with spectroscopic data (i.e., excluding HE0435−1223 and Q2237+0305) to check whether including narrowband photometry introduces any bias, but the results are nearly identical. The most noticeable effect is that without these objects (mostly due to the effect of Q2237+0305), the distribution is slightly more extended toward larger values of p and smaller sizes. The upper limit on p increases by a small amount, resulting in an estimate of $p=0.75^{+0.3}_{-0.2}$. In order to compare the present result for the size with recent estimates, we convert it to a reference wavelength of 1736 Å (using p = 0.75) and for a mass of the microlenses of 0.3 M_☉. The resulting (average) accretion disk size is $3.7^{+1.2}_{-1.0}$ lt-days, which is in good agreement with the value of 4.0 lt-days estimated by Jiménez-Vicente et al. (2012) from microlensing of (basically) this lens sample without chromatic information.

Table 2. Disk Parameter Estimates

	Maximum Likelihood		Bayesian
Object	rs (lt-day)	p	rs (lt-day)	p	$\frac{M_{{\rm BH}}}{10^9 M_\odot }$	Line	Source
HE0435−1223	$11.0^{+5.4}_{-6.5}$	$0.75^{+0.8}_{-0.4}$	$4.8^{+6.2}_{-2.7}$	1.3 ± 0.6	0.50	C iv	Peng et al. (2006)
HE0512−3329	$3.3^{+2.7}_{-1.5}$	$1.25^{+0.6}_{-0.7}$	$2.6^{+1.9}_{-1.1}$	1.4 ± 0.6	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
SBS0909+532	$4.5^{+3.6}_{-3.3}$	$0.75^{+0.5}_{-0.4}$	$2.7^{+2.6}_{-1.3}$	0.9 ± 0.4	1.95	Hβ	Assef et al. (2011)
QSO0957+561	$11.0^{+11}_{-11}$	$0.25^{+0.9}_{-0.2}$	$3.5^{+5.1}_{-2.1}$	0.9 ± 0.7	0.72	Hβ	Assef et al. (2011)
SDSSJ1004+4112	$4.5^{+2.9}_{-4.5}$	$1.00^{+1.3}_{-0.5}$	$2.5^{+2.1}_{-1.1}$	1.3 ± 0.6	2.02	C iv	Peng et al. (2006)
SDSSJ1029+2623	$6.0^{+7.4}_{-6.0}$	0.00+2.5	$2.6^{+3.0}_{-1.4}$	0.9 ± 0.7	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
HE1104−1805	$3.3^{+2.7}_{-1.7}$	$0.75^{+0.3}_{-0.3}$	$2.6^{+2.0}_{-1.1}$	0.9 ± 0.4	0.59	Hβ	Assef et al. (2011)
Q2237+0305	$20.0^{+7.0}_{-10}$	$0.50^{+0.3}_{-0.4}$	$10.0^{+12}_{-5.5}$	0.6 ± 0.4	1.20	Hβ	Assef et al. (2011)

Notes. Maximum likelihood (Columns 2 and 3) and Bayesian (Columns 4 and 5) estimates of sizes and scaling exponents p (r_s∝λ^p) for individual objects. Confidence intervals are at the 1σ level for 1 d.o.f. for the maximum likelihood estimates and marginalized for the second variable for the Bayesian estimates. Mass of the central black hole, line used in the estimate and source (Columns 6–8). Sizes were estimated at 1026 Å (rest frame) for microlenses of mass M = 1M_☉.

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The result for the size is, however, an average value for objects which, in principle, must have a spread in their sizes due to differences in their black hole masses. In order to minimize this effect when combining the likelihoods of different objects, we repeated the calculations scaling the reference size of the disk as

$$\begin{equation} r_s(M_{{\rm BH}})=r_0\left(\frac{M_{{\rm BH}}}{10^9 \;M_\odot }\right)^{2/3}. \end{equation} \tag{ 4 }$$

The 2/3 exponent is expected for Shakura & Sunyaev disks radiating close to the Eddington limit, and Morgan et al. (2010) used microlensing to estimate this scaling with mass and found results consistent with the theoretical prediction. The black hole mass estimates are based on source luminosities and emission line widths. We use estimates from the literature as reported in Table 2, using the Hβ estimates where available. Generally, these black hole mass estimates are viewed as logarithmically reliable with uncertainties of factors of two to three. The resulting joint likelihood for the parameters ln (r₀) and p is shown in Figure 3. The maximum likelihood value is $\ln (r_0)=1.5^{+0.15}_{-0.45}$, corresponding to $r_0=4.5^{+0.7}_{-1.6}$ lt-days, which is very similar to our previous result. This was to be expected, as the average black hole mass for this sample is 1.2 × 10⁹ M_☉, which is very close to the nominal mass used in the scaling. If we assume a typical value of 0.3 M_☉ for the mass of the microlenses, then our estimate for the size of the accretion disk for a nominal black hole mass of 10⁹ M_☉ is very similar to the estimate by Morgan et al. (2010). The estimate of the logarithmic slope p did not change, remaining at $p=0.75^{+0.2}_{-0.2}$.

On scales of r₀, we would not expect a strong object dependence of the value of p for objects sharing the same emission mechanism. In spite of this, there is no general agreement on the value of p from previous studies. Most earlier determinations for individual objects seemed to point toward the canonical value of p = 4/3 (e.g., Eigenbrod et al. 2008; Poindexter et al. 2008; Floyd et al. 2009; Mosquera et al. 2011), with the notable exception of the sample analyzed by Blackburne et al. (2011b), which produced an unexpectedly low value of p = 0.17 ± 0.15 that is essentially compatible with a disk size that is independent of wavelength. Nevertheless, several recent estimates based on different techniques (Muñoz et al. 2011; Mediavilla et al. 2011b; Motta et al. 2012; Blackburne et al. 2013) have been finding values close to p = 1, although with uncertainties large enough to be consistent with the canonical value of p = 4/3. Figure 4 shows the Bayesian estimate of p found for our sample after marginalizing over ln (r₀). We find p = 0.8 ± 0.2, with the value of the standard thin disk model (p = 4/3) being excluded at the 98% confidence level. The significance drops to 90% when objects with only narrowband data are excluded.

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We have used microlensing measurements for 10 image pairs from 8 lensed quasars to study the structure of their accretion disks. By using spectroscopic and narrowband photometric data, we can eliminate the effects of differential extinction, which can be mistaken with chromatic microlensing and minimize any contamination from the weakly microlensed emission lines that can reduce the microlensing amplitude and potentially alter its dependence on wavelength. We find a best estimate of the wavelength dependence of the accretion disk size of $p=0.75^{+0.2}_{-0.2}$ for a maximum likelihood analysis, and p = 0.8 ± 0.2 for a Bayesian analysis (r_s∝λ^p). This estimate is significantly smaller than the p = 4/3 value predicted by the Shakura & Sunyaev (1973) thin disk model with a temperature profile of T∝r^−3/4∝r^−1/p. Thus, our result seems to indicate a steeper temperature profile for the accretion disk.

Agol & Krolik (2000) proposed a model in which the magnetic connection between the black hole and the disk is able to spin the disk up leading to a steeper temperature profile with p = 8/7, which is more consistent with our results. Another alternative is that there is some other structure on top of the accretion disk. Along this line, Abolmasov & Shakura (2012) proposed that super Eddington accretion in the disks of quasars would generate an optically thick spherical envelope that could explain the lower values of the exponent p found by Blackburne et al. (2011b), although they need an ad hoc mechanism to explain the small sizes of the X-ray-emitting regions of microlensed quasars. Note, however, that Dai et al. (2010) find that the scattered light fraction has to be very high (≳ 50%) before microlensing size estimates are significantly shifted by scattering. Abolmasov & Shakura (2012) proposed that this mechanism would only work for the quasars with lower black hole masses, although we do not find any trend of the parameter p with the mass of the black hole. Along this line of thought, Fukue & Iino (2010) found that a fully thermalized relativistic spherical wind around the black hole would show a profile of the type T(R)∝R⁻¹, which is very similar to that produced by our analysis. Note that such steeper temperature profiles make it more difficult to reconcile the larger disk sizes found by microlensing. Solutions to this problem would favor shallower profiles closer to a strongly irradiated disk with p = 2 (see the discussion in Morgan et al. 2010). Our estimate for the size of the accretion disk of r_s = 4.5 lt-days (at 1026 Å in the rest frame) is in agreement with other recent determinations and considerably larger than the expected value from a thin disk radiating as a black body.

The large slope of the temperature profile of quasar accretion disks estimated in this work makes the already serious problem posed by the large sizes determined by microlensing for these objects even worse. As scattering does not seem to be the answer to this discrepancy and other suitable theoretical models are lacking, evidence seems to indicate that some key ingredient in the emission mechanism of accretion disks in these objects may be missing in our present understanding. On the observational side, optical/infrared spectroscopy (or, even better, spectroscopic monitoring) for a larger sample of lensed quasars would be desirable to unambiguously confirm the large sizes and slopes of the temperature profile of quasar accretion disks. In the meantime, on the theoretical front, a thorough revision of the standard model of accretion disks in quasars may also be necessary.

We gratefully acknowledge the anonymous referee for useful suggestions that improved the presentation of this work. This research was supported by the Spanish Ministerio de Educación y Ciencia with the grants AYA2011-24728, AYA2007-67342-C03-01/03, and AYA2010-21741-C03/02. J.J.V. is also supported by the Junta de Andalucía through the FQM-108 project. J.A.M. is also supported by the Generalitat Valenciana with the grant PROMETEO/2009/64. C.S.K. is supported by NSF grant AST-1009756. V.M. gratefully acknowledges support from FONDECYT 1120741.