The Rewards of Patience: An 822 Day Time Delay in the Gravitational Lens SDSS J1004+4112
J. Fohlmeister, C. S. Kochanek, E. E. Falco, C. W. Morgan, J. Wambsganss
aa r X i v : . [ a s t r o - ph ] O c t The Rewards of Patience: An 822 Day Time Delay in theGravitational Lens SDSS J1004+4112
J. Fohlmeister , C. S. Kochanek , E. E. Falco , C. W. Morgan , ,andJ. Wambsganss ABSTRACT
We present 107 new epochs of optical monitoring data for the four brightest images of thegravitational lens SDSS J1004+4112 observed between October 2006 and June 2007. Combiningthis data with the previously obtained light curves, we determine the time delays between imagesA, B and C. We confirm our previous measurement finding that A leads B by ∆ t BA = 40 . ± . τ CA = 821 . ± . τ AD > . ± . cm for a disk inclination of cos i = 1 /
2, which is consistent with the microlensing disksize-black hole mass correlation function given our estimate of the black hole mass from the MgIIline width of log M BH /M ⊙ = 8 . ± .
14. The long delays allow us to fill in the seasonal gaps andassemble a continuous, densely sampled light curve spanning 5.7 years whose variability impliesa structure function with a logarithmic slope of γ = 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.
Subject headings: cosmology: observations – gravitational lensing – quasars: individual: (SDSSJ1004+4112)
1. Introduction
The quasar SDSS J1004+4112 at z s = 1 .
734 issplit into five images by an intervening galaxy clus-ter at z l = 0 .
68 (Inada et al. 2003; Inada et al. 2005;Oguri et al. 2004). With a maximum imageseparation of 14 . ′′
62, it is a rare example ofa quasar gravitationally lensed by a cluster(Wambsganss 2003; Inada et al. 2006). One ofthe most interesting applications of this system isto use the time delays between the lensed im- Astronomisches Rechen-Institut, Zentrum f¨ur As-tronomie der Universit¨at Heidelberg, M¨onchhofstr. 12-14,69120 Heidelberg, Germany Department of Astronomy, Ohio State University, 140West 18th Avenue, Columbus, OH 43210 Smithsonian Astrophysical Observatory, FLWO, P.O.Box 97, Amado, AZ 85645 Department of Physics, United States Naval Academy,572C Holloway Road, Annapolis, MD 21402 ages to study the structure of the cluster. Ifwe assume the Hubble constant is known, thenthe delays break the primary model degener-acy of lensing studies (the “mass sheet degener-acy”), and the delay ratios constrain the struc-ture even if the Hubble constant is unknown.After its discovery, several groups modeled theexpected time delays in SDSS J1004+4112 andtheir dependence on the mean mass profile of thecluster (Kawano & Oguri 2006; Oguri et al. 2004;Williams 2004). When we measured the shortestdelay in the system, between images A and B,we found a longer delay than predicted by themodels (Fohlmeister et al. 2007, hereafter PaperI) where the discrepancy probably arose becausethe models included the cD galaxy and the clusterhalo but neglected the significant perturbationsfrom the member galaxies. As we measure thelonger delays, where the cluster potential should1e relatively more important than for the merg-ing A/B image pair, we would not expect clustersubstructures to play as important a role.We also expect this lens to have a fairly shorttime scale for microlensing variability created bystars either in the intracluster medium or in galax-ies near the images. The internal velocities of acluster are much higher than in a galaxy (700 km/sversus 200 km/s), and SDSS J1004+4112’s posi-tion on the sky is almost orthogonal to the CMBdipole (Kogut et al. 1993), giving the observera projected motion on the lens plane of almost300 km/s. In Paper I, we detected microlensingof the continuum emission of the A/B images inPaper I and there is also evidence for microlens-ing of the CIV broad line (Richards et al. 2004;Lamer et al. 2006; G´omez- ´Alvarez et al. 2006).Once we have measured the time delays we canremove the intrinsic quasar variability and use themicrolensing variability to estimate the mean stel-lar mass and stellar surface density, the transversevelocities, 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 couldbe an ideal laboratory for studying correlationsin the intrinsic variability of quasars. With, im-age C leading images A and B by 2.3 years, sharpvariations in image C can be used to plan inten-sive monitoring of images A and B to measure theresponse times as a function of wavelength (e.g.Kaspi et al. 2007), with the additional advantagethat the delay between A and B provides redun-dancies that protect against weather, the Moonand the Sun. The long delays between the im-ages also mean that seasonal gaps are completelyfilled, and we can examine the structure functionof the variability with a densely-sampled, gap-freelight curve (modulo corrections for microlensing).Such data generally do not exist, since most timevariability data for quasars (other than nearby re-verberation mapping targets, e.g. Peterson et al.2004) have very sparse sampling (e.g. Hawkins2007 on long time scales for a small number of ob-jects or Vanden Berk et al. 2004 on shorter timescales for many objects).In Paper I (Fohlmeister et al. 2007) we pre-sented three years of optical monitoring data forthe four brightest images of SDSS J1004+4112spanning 1000 days from December 2003 to June 2006. The fifth quasar image, E, is too faint tobe detected in our observations. We measured thetime delay between the A and B image pair tobe ∆ τ BA = 38 . ± . . ′′ §
2. When com-bined with our previous data we have light curvesspanning 1250 days that allow us to measure theAC delay in §
3. In § § §
2. Data
We monitored SDSS J1004+4112 in the r-band during the 2006-2007 season using the FredLawrence Whipple Observatory (FLWO) 1.2mtelescope on Mount Hopkins and the MDM 2.4mHiltner Telescope on Kitt Peak. The FLWO ob-servations were obtained with Keplercam (0 . ′′ . ′′
259 pixels). Thedata reduction was carried out as described inPaper I. We continued to use the same five starsto set the PSF model and the flux scale of eachepoch and verified that these flux standards con-tinue to show no variability. Table 1 presents thephotometry for the four images in the 2006-2007season.In Figure 1 we present the resulting light curvesfor images A to D for the period from December2003 to June 2007. The average sampling rateduring the 2006/2007 season is once every thirdday. The FLWO data are noisy, so for Figure 1we show a running average of the data (one pointevery five days averaged over ± ±
3. The Time Delay
For the determination of the time delay, weuse the methods described in Paper I. Our firststep with the new data was to remeasure theA/B delay. The fourth season shows a nicefeature with maxima in images A and B near days 4120 and 4080 respectively, followed by aroughly 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 measurethe delay between A and B to be ∆ t BA =40 . ± . N src = 20, 40, 60 and 80 for the source and N µ = 1, 2, 3 and 4 for the microlensing variabilityand derived the final estimate using the Bayesianweighting of these cases described in Poindexteret al. (2007). We found delays of 40 . ± . . ± . . ± . τ BA = 38 . ± . τ CB >
560 days and suggested, based onsome similarities between the third season for A/Bwith the first season for C, that a delay of order700 days was plausible but statistically too weakto claim as a measurement. We now see that thefeature in the second season for image C stronglymatches the feature we observe in the new seasonfor A and B. Using the dispersion spectra method(Pelt et al. 1994, 1996), we find ∆ τ CA = 822 ± τ CB = 780 ± τ CB = ∆ τ CA − ∆ τ BA = 782 ± . ± . . ± . . ± . τ DA > x, y ) = (31 . ′′ , . ′′
0) relative to quasar image A inan effort to reduce the overall shear). The fitswere carried out using lensmodel (Keeton 2001)and while adequate they are not satisfactory – itis very difficult to find solutions with no additionalquasar images created by the galaxies, and check-ing for the extra images makes the procedure ex-traordinarily slow. At present we lack the abilityto model this system in detail (including uncer-tainties) at the precision of the constraints, whilesimplified models that ignore the galaxies are inca- pable of fitting the data at all. The model predictsan AD delay of order 2000 days (5.5 years), whichis consistent with our current lower bound.
4. Microlensing and the Size of the QuasarAccretion Disk
The residuals of the A and B light curves(see Fig. 2) clearly indicate that microlensing ispresent. After correcting for the time delay, themean magnitude differences between A and B forthe four seasons are 0 . ± . . ± . . ± .
005 and 0 . ± .
007 mag. For thetwo seasons overlapping with C we find meanmagnitude differences, seasonal gradients and sec-ond derivatives of 0 . ± .
010 mag, − . ± .
02 mag/year and 0 . ± .
09 mag/year forC relative to A and 0 . ± .
005 mag, 0 . ± .
01 mag/year and 0 . ± .
04 mag/year for Brelative to A. Fig. 2 shows the superposition ofthe phased A, B and C light curves and the dif-ferences between them that are the signature ofmicrolensing.We modeled the microlensing for images A/Busing the Bayesian Monte Carlo method ofKochanek (2004). We used the microlensing pa-rameters of our (adequate) lens model, with con-vergence κ and shear γ values of κ = 0 .
48 and γ = 0 .
57 for A and κ = 0 .
47 and γ = 0 .
39 forB. We allowed the surface density in stars κ ∗ to vary from 10% to 100% of κ increments of10%. We used a microlens mass function with dn/dM ∝ M − . with a dynamic range in massof a factor of 50 that approximates the Galac-tic disk mass function of Gould (2000). Wegenerated 4096 × h R E i where h R E i is the Einstein radius at the mean stellar mass h M i . We modeled the disk as a face-on, thindisk (Shakura & Syunyaev 1973) neglecting thecentral temperature depression and relativistic ef-fects. We measure the disk size R λ as the pointwhere the disk temperature matches the rest-frame energy of our monitoring band, kT λ = hc/λ ,where λ ≃ R / = 2 . R λ should be used to compareto any other disk model, since Mortonson et al.(2005) have shown that the half-light radius de-pends little on the surface brightness profile of4he model. We made four realizations of each ofthe 10 microlensing models and drew 2 × triallight curves for each of the 40 cases so that wewould have a reasonable statistical sampling oflight curves that fit the data well. We found that R ˚ A = 10 . ± . cm h √ cosi (1)for a disk inclination angle i , whether or not we usea prior on the mean microlens mass of 0 . M ⊙ < h M i < M ⊙ .From the MgII emission line width/black holemass calibration of Kollmeier et al. (2006), thespectrum of image C from Richards et al. (2004),and a magnification-corrected HST I -band mag-nitude of 20 . ± .
4, we estimate a black hole massof log M BH /M ⊙ = 8 . ± .
2. Fig. 3 compares thedisk size estimate to the characteristic scales ofsuch a black hole.
5. The Structure Function
The quasar structure function can be used as atool to characterize quasar variability independentof short-timescale monitoring gaps and to comparewith theoretical models of quasar variability (e.g.Kawaguchi et al. 1998). The structure function S ( τ ) = s N ( τ ) X i 734 of 755 days.For the very different behavior of the image Dlight curve, which could not yet be time-delay con-nected to the other images, we compute the struc-ture function independently for rest frame timelags up to 470 days. As in Vanden Berk et al.2004 we fit the form of the structure function witha power law. The value for the power law index γ = 0 . ± . 02 for the combined image B andC light curves is consistent with that derived for Fig. 2.— The image A, B and C light curvesin their overlap region after shifting by the timedelays. The data are binned in one week inter-vals. The lower box shows the residual magnitudesshifted by the offset between the images, revealingmicrolensing variability of order 0.15 mag. Thelight curve of image B was chosen to have con-stant flux because it has the most overlap withthe over two.the SDSS quasar sample. For image D we finda similar slope of γ = 0 . ± . 03, as expectedfrom the light curve. Time-delay connecting theimage A, B and C lightcurves by subtracting theestimated microlensing variability in the overlapregion of the lightcurves gives a restframe recordof the intrinisic quasar variability over 500 days.The slope of the structure function for the sourcelight curve γ s = 0 . ± . 03 is steeper than for theobserved non-microlensing corrected curves.5 Fig. 3.— Probability distribution for the accre-tion disk size at 2300˚A assuming the mean diskinclination (cos( i ) = 1 / L/L E = 1) with efficiency η = 0 . L = η ˙ M c . 6. Summary and Conclusions We present a fourth season of monitoring datafor the four bright images of the five image gravita-tional lens system SDSS J1004+4112. We confirmour previous estimate for the time delay betweenthe merging A/B pair, finding that B leads Aby 40 . ± . . ± . R λ ∝ λ / scaling for a thin disk and assume themean disk inclination cos( i ) = 1 / R ˚ A = 10 . ± . cm. Comparisons toother disk models should use the half-light radiuswhich is 2 . 44 times larger. Based on the quasar6gII emission line width we estimate that theblack hole mass is 10 . ± . M ⊙ . For this mass, themicrolensing accretion disk size-black hole masscorrelation found by Morgan et al. (2007) pre-dicts that R ˚ A = 10 . cm, which is in broadagreement with the measurement. Further obser-vations, the inclusion of additional images, andmonitoring in multiple bands should improve thesemeasurements and potentially allow us to deter-mine the mean surface density in stars near theimages κ ∗ and their average mass h M i . Similarly,the ability to construct continuous light curves ofthe intrinsic variability and to use image C to pro-vide early warning of sharp flux changes that canthen be intensively monitored in images A and Bmay make this system a good candidate for apply-ing reverberation mapping techniques to a mas-sive, luminous quasar. At present, we already seethat the system has a structure function typical ofquasars.We thank all the participating observers at boththe Harvard-Smithsonian Center for Astrophysicsand the MDM Observatory for their support ofthese observations. This work is also based onobservations obtained with the MDM 2.4m Hilt-ner and 1.3m McGraw-Hill telescopes, which areowned and operated by a consortium consistingof Columbia University, Dartmouth College, theUniversity of Michigan, the Ohio State Universityand Ohio University. We thank N.F. Bate for valu-able comments and encouragements. 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R. & Saha, P. 2004 , AJ, 128, 2631 This 2-column preprint was prepared with the AAS L A TEXmacros v5.2. able 1Light Curves for SDSS J1004+4112 ∗ HJD χ /N dof Image A Image B Image C Image D Observatory Detector4019.006 1.15 3.451 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± able 1— Continued HJD χ /N dof Image A Image B Image C Image D Observatory Detector4082.865 1.23 3.521 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± able 1— Continued HJD χ /N dof Image A Image B Image C Image D Observatory Detector4169.811 1.40 3.632 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± able 1— Continued HJD χ /N dof Image A Image B Image C Image D Observatory Detector4254.660 0.55 3.652 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Note.— The Heliocentric Julian Days (HJD) column gives the date of the observation relative to HJD=2450000. The χ /N dof column indicates how well our photometric model fit the imaging data. When χ > N dof we rescale the photometric errors presented in this Table by ( χ /N dof ) / before carrying out the time delayanalysis to reduce the weight of images that were fit poorly. The image magnitudes are relative to the comparisonstars (see text). The magnitudes enclosed in parentheses are not used in the time delay estimates.before carrying out the time delayanalysis to reduce the weight of images that were fit poorly. The image magnitudes are relative to the comparisonstars (see text). The magnitudes enclosed in parentheses are not used in the time delay estimates.