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The Rewards of Patience: An 822 Day Time Delay in the Gravitational Lens SDSS J1004+4112

We present 107 new epochs of optical monitoring data for the four brightest images of the gravitational lens SDSS J1004+4112 observed between October 2006 and June 2007. Combining this data with the previously obtained light curves, we determine the time delays between images A, B and C. We confirm our previous measurement finding that A leads B by dt_BA=40.6+-1.8 days, and find that image C leads image A by dt_CA=821.6+-2.1 days. The lower limit on the remaining delay is that image D lags image A by dt_AD>1250 days. Based on the microlensing of images A and B we estimate that the accretion disk size at a rest wavelength of 2300 angstrom is 10^{14.8+-0.3} cm for a disk inclination of cos{i}=1/2, which is consistent with the microlensing disk size-black hole mass correlation function given our estimate of the black hole mass from the MgII line width of logM_BH/M_sun=8.44+-0.14. The long delays allow us to fill in the seasonal gaps and assemble a continuous, densely sampled light curve spanning 5.7 years whose variability implies a structure function with a logarithmic slope of gamma = 0.35+-0.02. As C is the leading image, sharp features in the C light curve can be intensively studied 2.3 years later in the A/B pair, potentially allowing detailed reverberation mapping studies of a quasar at minimal cost.


Introduction
The quasar SDSS J1004+4112 at z s = 1.734 is split into five images by an intervening galaxy cluster at z l = 0.68 (Inada et al. 2003;Inada et al. 2005;Oguri et al. 2004).
With a maximum image separation of 14. ′′ 62, it is a rare example of a quasar gravitationally lensed by a cluster (Wambsganss 2003;Inada et al. 2006). One of the most interesting applications of this system is to use the time delays between the lensed im-ages to study the structure of the cluster. If we assume the Hubble constant is known, then the delays break the primary model degeneracy of lensing studies (the "mass sheet degeneracy"), and the delay ratios constrain the structure even if the Hubble constant is unknown. After its discovery, several groups modeled the expected time delays in SDSS J1004+4112 and their dependence on the mean mass profile of the cluster (Kawano & Oguri 2006;Oguri et al. 2004;Williams 2004). When we measured the shortest delay in the system, between images A and B, we found a longer delay than predicted by the models (Fohlmeister et al. 2007, hereafter Paper I) where the discrepancy probably arose because the models included the cD galaxy and the cluster halo but neglected the significant perturbations from the member galaxies. As we measure the longer delays, where the cluster potential should be relatively more important than for the merging A/B image pair, we would not expect cluster substructures to play as important a role.
We also expect this lens to have a fairly short time scale for microlensing variability created by stars either in the intracluster medium or in galaxies near the images. The internal velocities of a cluster are much higher than in a galaxy (700 km/s versus 200 km/s), and SDSS J1004+4112's position on the sky is almost orthogonal to the CMB dipole (Kogut et al. 1993), giving the observer a projected motion on the lens plane of almost 300 km/s. In Paper I, we detected microlensing of the continuum emission of the A/B images in Paper I and there is also evidence for microlensing of the CIV broad line (Richards et al. 2004;Lamer et al. 2006;Gómez-Álvarez et al. 2006). Once we have measured the time delays we can remove the intrinsic quasar variability and use the microlensing variability to estimate the mean stellar mass and stellar surface density, the transverse velocities, and the structure of the quasar source (Gil-Merino et al. 2005;Mortonson et al. 2005;Poindexter et al. 2007; Morgan et al. 2007).
Finally, we note that SDSS J1004+4112 could be an ideal laboratory for studying correlations in the intrinsic variability of quasars. With, image C leading images A and B by 2.3 years, sharp variations in image C can be used to plan intensive monitoring of images A and B to measure the response times as a function of wavelength (e.g. Kaspi et al. 2007), with the additional advantage that the delay between A and B provides redundancies that protect against weather, the Moon and the Sun. The long delays between the images also mean that seasonal gaps are completely filled, and we can examine the structure function of the variability with a densely-sampled, gap-free light curve (modulo corrections for microlensing). Such data generally do not exist, since most time variability data for quasars (other than nearby reverberation mapping targets, e.g. Peterson et al. 2004) have very sparse sampling (e.g. Hawkins 2007 on long time scales for a small number of objects or Vanden Berk et al. 2004 on shorter time scales for many objects).
In Paper I (Fohlmeister et al. 2007) we presented three years of optical monitoring data for the four brightest images of SDSS J1004+4112 spanning 1000 days from December 2003 to June 2006. The fifth quasar image, E, is too faint to be detected in our observations. We measured the time delay between the A and B image pair to be ∆τ BA = 38.4 ± 2.0 days. While larger separation lenses tend to have longer time delays, for these two images the propagation time difference is small, because they form a close image pair (3. ′′ 8) from the source lying close to a fold caustic. For the more widely separated C and D images we could only estimate lower limits on the delays of 560 and 800 days relative to image B and A. In this paper we present the 107 new optical monitoring epochs for the 2006/2007 season in §2. When combined with our previous data we have light curves spanning 1250 days that allow us to measure the AC delay in §3. In §4 we use the microlensing variability of the A/B images to measure the size of the quasar accretion disk, and in §5 we measure the structure function of the intrinsic variability. We discuss the future prospects for exploiting this system in §6.

Data
We monitored SDSS J1004+4112 in the rband during the 2006-2007 season using the Fred Lawrence Whipple Observatory (FLWO) 1.2m telescope on Mount Hopkins and the MDM 2.4m Hiltner Telescope on Kitt Peak. The FLWO observations were obtained with Keplercam (0. ′′ 672 pixels) and the MDM observations with RETRO-CAM (Morgan et al. 2005) (0. ′′ 259 pixels). The data reduction was carried out as described in Paper I. We continued to use the same five stars to set the PSF model and the flux scale of each epoch and verified that these flux standards continue to show no variability. Table 1 presents the photometry for the four images in the 2006-2007 season.
In Figure 1 we present the resulting light curves for images A to D for the period from December 2003 to June 2007. The average sampling rate during the 2006/2007 season is once every third day. The FLWO data are noisy, so for Figure 1 we show a running average of the data (one point every five days averaged over ± 7 days) to emphasize the long term trends. Images C and D are offset by 0.3 and 1.0 mag, respectively, so that they do not overlap with image B in the third and fourth season. During this season, images A and

The Time Delay
For the determination of the time delay, we use the methods described in Paper I. Our first step with the new data was to remeasure the A/B delay. The fourth season shows a nice feature with maxima in images A and B near days 4120 and 4080 respectively, followed by a roughly 100 day decline to minima at 4220 (A) and 4180 (B) days. With the dispersion method (Pelt et al. 1994;Pelt et al. 1996) we measure the delay between A and B to be ∆t BA = 40.1 ± 3.5 days. For the  polynomial method we used polynomial orders of N src = 20, 40, 60 and 80 for the source and N µ = 1, 2, 3 and 4 for the microlensing variability and derived the final estimate using the Bayesian weighting of these cases described in Poindexter et al. (2007). We found delays of 40.6 ± 1.8, 40.1 ± 1.8 and 39.8 ± 1.8 (68% confidence regions) depending on whether we weighted the changes in the number of parameters using the Bayesian information criterion (which strongly penalizes extra parameters), the Akaike information criterion (which weakly penalizes extra parameters) or no penalty for extra parameters. These are consistent with our result from Paper I of ∆τ BA = 38.4 ± 2.0 days, but are somewhat more conservative in their treatment of the parameterization and the role of microlensing.
In Paper I we derived a lower limit on the BC delay of ∆τ CB > 560 days and suggested, based on some similarities between the third season for A/B with the first season for C, that a delay of order 700 days was plausible but statistically too weak to claim as a measurement. We now see that the feature in the second season for image C strongly matches the feature we observe in the new season for A and B. Using the dispersion spectra method (Pelt et al. 1994(Pelt et al. , 1996, we find ∆τ CA = 822 ± 7 days and ∆τ CB = 780 ± 6 days where the CA delay is slightly less accurate because the CA overlap is slightly less than the CB overlap due to the alignment of the light curves relative to the seasonal gaps. The three delays are mutually consistent since ∆τ CB = ∆τ CA − ∆τ BA = 782 ± 7 days. For the polynomial method analysis we simultaneously fit A, B and C holding the A/B delay fixed to 40.6 days to find CA delays of 821.6 ± 2.1, 823.0 ± 2.1 and 820.2 ± 2.1 days for the three weighting methods, respectively. Image D should lag the other three images, and we see no feature in the light curve of image D that can be matched to the first season of images A/B. The lower limit on the time delay between images A and D is now ∆τ DA > 1250 days (3.4 years).
We modeled the lens using the same approach as in Paper I, where we include the central cD galaxy, and NFW halo for the cluster dark matter and 12 pseudo-Jaffe models corresponding to cluster galaxies (we added an extra component at (x, y) = (31. ′′ 0, 4. ′′ 0) relative to quasar image A in an effort to reduce the overall shear). The fits were carried out using lensmodel (Keeton 2001) and while adequate they are not satisfactory -it is very difficult to find solutions with no additional quasar images created by the galaxies, and checking for the extra images makes the procedure extraordinarily slow. At present we lack the ability to model this system in detail (including uncertainties) at the precision of the constraints, while simplified models that ignore the galaxies are inca-pable of fitting the data at all. The model predicts an AD delay of order 2000 days (5.5 years), which is consistent with our current lower bound.

Microlensing and the Size of the Quasar Accretion Disk
The residuals of the A and B light curves (see Fig. 2) clearly indicate that microlensing is present. After correcting for the time delay, the mean magnitude differences between A and B for the four seasons are 0.460 ± 0.005, 0.283 ± 0.007, 0.339 ± 0.005 and 0.381 ± 0.007 mag. For the two seasons overlapping with C we find mean magnitude differences, seasonal gradients and second derivatives of 0.590 ± 0.010 mag, −0.04 ± 0.02 mag/year and 0.29 ± 0.09 mag/year 2 for C relative to A and 0.368 ± 0.005 mag, 0.05 ± 0.01 mag/year and 0.18 ± 0.04 mag/year 2 for B relative to A. Fig. 2 shows the superposition of the phased A, B and C light curves and the differences between them that are the signature of microlensing.
We modeled the microlensing for images A/B using the Bayesian Monte Carlo method of Kochanek (2004). We used the microlensing parameters of our (adequate) lens model, with convergence κ and shear γ values of κ = 0.48 and γ = 0.57 for A and κ = 0.47 and γ = 0.39 for B. We allowed the surface density in stars κ * to vary from 10% to 100% of κ increments of 10%. We used a microlens mass function with dn/dM ∝ M −1.3 with a dynamic range in mass of a factor of 50 that approximates the Galactic disk mass function of Gould (2000). We generated 4096 × 4096 pixel magnification patterns with an outer scale of 20 R E where R E is the Einstein radius at the mean stellar mass M . We modeled the disk as a face-on, thin disk (Shakura & Syunyaev 1973) neglecting the central temperature depression and relativistic effects. We measure the disk size R λ as the point where the disk temperature matches the restframe energy of our monitoring band, kT λ = hc/λ, where λ ≃ 2300Å for the r-band at the source redshift (see Morgan et al. 2007). The half-light radius R 1/2 = 2.44R λ should be used to compare to any other disk model, since Mortonson et al. (2005) have shown that the half-light radius depends little on the surface brightness profile of the model. We made four realizations of each of the 10 microlensing models and drew 2 × 10 5 trial light curves for each of the 40 cases so that we would have a reasonable statistical sampling of light curves that fit the data well. We found that for a disk inclination angle i, whether or not we use a prior on the mean microlens mass of 0.1M ⊙ < M < M ⊙ . From the MgII emission line width/black hole mass calibration of Kollmeier et al. (2006), the spectrum of image C from Richards et al. (2004), and a magnification-corrected HST I-band magnitude of 20.9 ± 0.4, we estimate a black hole mass of log M BH /M ⊙ = 8.4 ± 0.2. Fig. 3 compares the disk size estimate to the characteristic scales of such a black hole.

The Structure Function
The quasar structure function can be used as a tool to characterize quasar variability independent of short-timescale monitoring gaps and to compare with theoretical models of quasar variability (e.g. Kawaguchi et al. 1998). The structure function is the variance in the magnitude as a function of the time τ = t j − t i between measurements where m(t j ) and N (τ ) is the number of epochs at that time lag. For SDSS J1004+4112 we can determine the structure function over a moderate time range and with a dense sampling rate and no seasonal gaps if we use the time-delay corrected quasar light curves for images B and C. These

Summary and Conclusions
We present a fourth season of monitoring data for the four bright images of the five image gravitational lens system SDSS J1004+4112. We confirm our previous estimate for the time delay between the merging A/B pair, finding that B leads A by 40.6 ± 1.8 days. We measure the delay for image C for the first time, finding that it leads image A by 821.6 ± 2.1 days. We note that this is nearly twice the longest previously measured delay (the 417 day delay in Q0957+561 (Schild & Thomson 1995;Kundic et al. 1997)). We find a lower bound that D lags A by more than approximately 1250 days. Our current mass model predicts that D lags A by approximately 2000 days, which is consistent with the present limit. The fractional uncertainties in the AB delay are still dominated by sampling and microlensing, while the fractional uncertainties in the AC delay are dominated by cosmic variance due to density fluctuations along the line of sight rather than our measurement uncertainties of 0.3% (e.g. Barkana 1996).
A detailed model of this system, including the constraints from the multiply imaged, higher redshift arcs (Sharon et al. 2005), the X-ray measurements (Ota et al. 2006;Lamer et al. 2006) and a detailed understanding of the uncertainties will be a challenge. We lack a completely satisfactory model for the system at present, in the sense that the modeling process is extraordinarily slow due to the ability of the gravitational potentials associated with the cluster member galaxies to generate additional but undetected images of the quasar, making it impossible to carry out a reliable model survey. The record of models for this system is discouraging. As we noted in Paper I, all three model studies (Oguri et al. 2004;Williams & Saha 2004;Kawano & Oguri 2006) generically predicted shorter AB delays than the observed 40 days, and that this could be plausibly explained by the absence of substructure (i.e. galaxies) in the potential models. The longer AB-C and AB-D delays should be less sensitive to substructure. Oguri et al. (2004) do not include an estimate of the AB-C delays and have A-D delays consistent with our present limits. The range of B-C delays in Williams & Saha (2004) is consistent with our measurement of 820 days, but they predict AD delays shorter than our current lower bound of 1250 days. Kawano & Oguri (2006) predict a range for the longer delays over a broad range of mass distributions, none of which match our delays in detail. However, models with sufficiently long C-B delays generally have C-D delays long enough to agree with our present limits.
Based on our present mass model we used the microlensing between the A and B images to make an estimate of the size of the quasar accretion disk at 2300Å in the quasar rest frame. If we convert this to the expected size at 2500Å assuming the R λ ∝ λ 4/3 scaling for a thin disk and assume the mean disk inclination cos(i) = 1/2 the scale on which the disk temperature matches the photon energy is R 2500Å = 10 15.0±0.3 cm. Comparisons to other disk models should use the half-light radius which is 2.44 times larger. Based on the quasar MgII emission line width we estimate that the black hole mass is 10 8.4±0.2 M ⊙ . For this mass, the microlensing accretion disk size-black hole mass correlation found by Morgan et al. (2007) predicts that R 2500Å = 10 15.3 cm, which is in broad agreement with the measurement. Further observations, the inclusion of additional images, and monitoring in multiple bands should improve these measurements and potentially allow us to determine the mean surface density in stars near the images κ * and their average mass M . Similarly, the ability to construct continuous light curves of the intrinsic variability and to use image C to provide early warning of sharp flux changes that can then be intensively monitored in images A and B may make this system a good candidate for applying reverberation mapping techniques to a massive, luminous quasar. At present, we already see that the system has a structure function typical of quasars.
We thank all the participating observers at both the Harvard-Smithsonian Center for Astrophysics and the MDM Observatory for their support of these observations. This work is also based on observations obtained with the MDM 2.4m Hiltner and 1.3m McGraw-Hill telescopes, which are owned and operated by a consortium consisting of Columbia University, Dartmouth College, the University of Michigan, the Ohio State University and Ohio University. We thank N.F. Bate for valuable comments and encouragements. We also acknowledge support by the European Community's Sixth Framework Marie Curie Research Training Network Programme, Contract No. MRTN-CT-2004-505183 "ANGLES". Note.-The Heliocentric Julian Days (HJD) column gives the date of the observation relative to HJD= 2450000. The χ 2 /N dof column indicates how well our photometric model fit the imaging data. When χ 2 > N dof we rescale the photometric errors presented in this Table by (χ 2 /N dof ) 1/2 before carrying out the time delay analysis to reduce the weight of images that were fit poorly. The image magnitudes are relative to the comparison stars (see text). The magnitudes enclosed in parentheses are not used in the time delay estimates.