Abstract Thin accretion disks are too small to explain optical quasar microlensing measurements, and they cannot produce the high levels of observed UV emission. Quasars exhibit % variability, and local fluctuations may be significantly larger since the disk is unlikely to vary coherently (total variance ~ N -1 for N independent zones). We postulate that the disk temperature structure is significantly inhomogeneous, which can explain the discrepancies between observations and thin disk theory: The spectrum can be broad enough to explain the observed UV emission due to the range of temperatures in each annulus. Local spectra have smaller peak values than a single temperature blackbody, requiring a larger emitting area to produce the same flux. Spectral and microlensing constraints can be satisfied by disks with local temperature fluctuations of 0.4 dex without exceeding the observed levels of optical variability. These large fluctuations may be caused by magnetic turbulence in radiation-dominated disks, or may be due to other instabilities. Inhomogeneous Disks The thin disk (‘alpha’) model of Shakura & Sunyaev (1973) has met with mixed success in explaining optical/UV quasar spectra and light curves. Nearly simultaneous variability across optical bands implies a traveling speed of 0.1c; inconsistent with disk instability mechanisms. Quasar spectra are too broad to be well fit by thin disks. Models invoking global Compton scattering atmospheres are disfavored due to the compact X-ray emission region found by microlensing measurements. Optical microlensing observations robustly measure accretion disk sizes that are four times larger on average than thin disks at the same luminosity. Optical quasar variability is well described as a damped random walk with amplitudes of 10-20%. It is unlikely that the observed quasar variability is caused by a coherently varying accretion disk. To produce this variability amplitude with multiple, independent regions in the disk requires larger local variations. The accretion disk temperature T is then no longer a single valued function of the radius r, and the temperature producing the most emission at a given wavelength no longer corresponds to a single radius, r max. The flux from r > r max, exceeds that produced by a thin disk, causing the inhomogeneous disk (ID) to appear larger at that wavelength. Examples We consider a numerical model with evenly spaced zones in log r and φ in the equatorial plane. In each zone, log T follows a different damped random walk with amplitude σ T. The other parameter is n, the total number of zones between r and 2r. The surface brightness is computed from Planck's Law for face-on viewing ignoring all relativistic effects. A sample temperature map from this model is shown in the left panel of Figure 1. Figure 2 shows the ratio of half-light radii between the damped random walk disk and a standard thin disk at the same flux as a function of the fractional variability. For a large number of zones, the disk can be large enough to explain microlensing observations without producing too much optical variability. For few (large) zones, the overall variability becomes unrealistically large before the accretion disk becomes large enough to explain the microlensing measurements. We also consider a model with random Gaussian temperature “flares” set off in the disk at each time step (see the right panel of Figure 1). The resulting size increases and variability amplitudes are similar to those from the damped random walk. Both models produce power spectra and structure functions that are consistent with observations (Kelly et al. 2009; MacLeod et al. 2010). Figure 2. Median size increase vs. fractional variability for various zone sizes for the damped random walk model. Each sequence has values of σ T from Figure 3. Sample microlensing magnification light curves, with intrinsic variability divided out. Best fit power law temperature disk model light curves are shown as dotted lines. DM (SM) indicates a damped random walk (random flare) model. Microlensing Accretion disk sizes can be inferred from microlensing observations by fitting magnification light curves with power-law thin disk (T ~ r -3/4 ) surface brightness maps parameterized by the radius, r s, where hν=kT(r s ). For the power-law disk, r s can also be calculated from the observed L ν in the absence of all lensing effects. The discrepancy between the values of r s inferred from microlensing versus r s inferred from L ν is 0.6 ± 0.3 dex (Morgan et al. 2010). Using the code described by Wambsganss (1990), we simulate magnification patterns for QSO , the Einstein Cross, using lens galaxy parameters found by Kochanek (2004). Choosing random orientations and starting positions, many magnification light curves are produced from the time-varying toy models. Sample light curves are shown in Figure 3 along with best fit power-law disk light curves. The time-steady, power-law disk model produces excellent fits to the microlensing light curves, and the inferred values of r s agree with the disk half-light radii. Spectra Model spectra are compared to the HST composite in the top panel of Figure 4, and can explain high levels of observed UV emission. The bottom panel of Figure 4 shows local spectra at r=20r ms, where r ms is the inner disk edge. The local inhomogeneous spectra peak at a smaller value of λF λ than the single temperature blackbody. To produce the same total flux at that wavelength, the emission must arise from a larger area. Thus, the disk appears larger at that wavelength. Figure 5 summarizes our results. The independent spectral and microlensing size constraints require similar amplitudes of temperature inhomogeneity. Producing the observed variability requires larger σ T at larger n. Allowed parameter values are 0.35 ≤ σ T ≤ 0.5 and 100 ≤ n ≤ The models predict 1% (5%) average (maximum) deviations between inhomogeneous and power law magnification light curves. Figure 4. Top: Model spectra compared to the composite from Zheng et al (1997). The open diamonds show a thin disk spectrum. Bottom: Model spectra (lines) for an annulus compared to a blackbody (diamonds). Acknowledgments This work was partially supported by NASA Earth Space Science Fellowship NNX08AX59H (JD). Physical Mechanisms Quasar accretion disks are expected to be inhomogeneous due to the magnetorotational instability (MRI). MHD simulations of a patch of disk (Hirose et al. 2009a) and global GRMHD simulations (Fragile et al. 2007) show factor of ~2-3 temperature fluctuations. These distributions correspond to σ T = , less than the required σ T =0.35. Hirose et al. (2009b) find that radiation- dominated disks are highly variable, but there is no evidence for the predicted thermal instability. Radiation-dominated disks are also subject to an inflow instability (Lightman & Eardley 1974), which could create large local temperature gradients. Inhomogeneous disks can simultaneously explain discrepancies between observed accretion disk sizes and spectra and those predicted by thin disk theory without exceeding the observed optical variability. Proper modeling of an inhomogeneous disk will require global MHD simulations of radiation-dominated accretion disks. The MRI may be sufficient to produce the required temperature fluctuations, but additional disk instabilities may also be important. References Dexter, J., & Agol, E. 2010, ApJL, submitted Fragile, P. C., et al. 2007, ApJ, 668, 417 Hirose, S., et al. 2009a, ApJ, 704, 781 Hirose, S., et al. 2009b, ApJ, 691, 16 Kelly, B. C., et al ApJ, 698, 895 Kochanek, C. S. 2004, ApJ, 605, 58 Lightman, A. P. & Eardley, D. M. 1974, ApJ, 187, L1 MacLeod, C. L., et al. 2010, ApJ, 721, 1014 Morgan, C. W., et al. 2010, ApJ, 712, 1129 Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 24, 337 Wambsganss, J. 1990, PhD Thesis Zheng, W., et al. 1997, ApJ, 475, 469 Jason Dexter 1 and Eric Agol 2 1 Department of Physics, University of Washington, Seattle, WA, Department of Astronomy, University of Washington, Seattle, WA, Quasar Accretion Disks Are Strongly Inhomogeneous Figure 1 False color temperature maps from the damped random walk (left) and random flare (right) inhomogeneous accretion disk models. The color scale increases logarithmically from blue to red to yellow to white, with a dynamic range of 100. Figure 5. Allowed 68% regions of the σ T vs. n parameter space. Contours of 1% (solid) and 2% (dotted) rms deviation between model light curves and best fit power-law temperature disks are overplotted. Microlensing light curves from IDs are well fit by power law temperature disks IDs explain microlensing sizes without producing too much variability Allowed region All observations require similar parameter values ID spectra can explain the HST composite Pre-print Online! PDF version of the submitted paper and poster: ragol2010_submit.pdf rIOBH.pdf Size Spectrum Variability All Inhomogeneous disk (ID) temperature maps