FURTHER EVIDENCE THAT QUASAR X-RAY EMITTING REGIONS ARE COMPACT: X-RAY AND OPTICAL MICROLENSING IN THE LENSED QUASAR Q J0158–4325^*

Quasars can be highly variable. Since this variability arises in the quasar continuum source itself, it appears in all the images of a gravitationally lensed quasar, but at different times in each image due to differences in the gravitational potential of the lens galaxy and the geometric length of the paths of each image (Refsdal 1964; Cooke & Kantowski 1975). As first pointed out by Chang & Refsdal (1979), the stars near each lensed image also cause temporal variations in the magnification of each image, leading to additional variability. This "microlensing" variability is not correlated between the images, so it functions as both a difficult source of noise in time delay measurements and as a probe of quasar structure because the size of the emission region ultimately controls the amplitude of the microlensing variability.

Traditional time delay measurements require that either the microlensing variability is of sufficiently small amplitude so as to be insignificant in the analysis or that its timescale is significantly longer than that of the quasar's intrinsic variability, allowing for some sort of parametric modeling and subtraction (e.g., Ofek & Maoz 2003; Paraficz et al. 2006; Kochanek et al. 2006; Poindexter et al. 2008). Attempting to measure time delays in systems with significant short-timescale microlensing is both difficult and error prone (see Eigenbrod et al. 2005 for quantitative characterizations of the influence of microlensing on time delay estimates). In Morgan et al. (2008), we applied a new approach to analyzing the light curves of two lensed quasars with significant microlensing and time variability, in which we attempted to simultaneously model the microlensing variability using the Bayesian Monte Carlo technique of Kochanek (2004). In our new approach, we marginalized over the physically realistic models of the microlensing signal to yield a measurement of the most likely time delay. We successfully confirmed the time delay of HE 1104–1805 but failed to measure the delay in Q J0158–4325. In the present paper, we supplement our previously published light curves with four new seasons of monitoring data from two observatories and make a second attempt.

In the models of Morgan et al. (2008), we could also marginalize over the time delay uncertainties and use the microlensing to study the structure of the quasar. The amplitudes and the timescales of the variability due to quasar microlensing are a function of a number of physical variables such as the continuum source size, the mean properties of the lens in the vicinity of the lensed images, and the relative velocities of the source, the lens, and the observer. During the last decade a variety of techniques have been developed to analyze the effects of microlensing on lensed quasar images. Single-epoch photometry in which one or more of the quasar image fluxes are anomalous (i.e., deviates significantly from the predictions of the macroscopic lens model or shows significant wavelength dependencies) can provide (prior-dependent) estimates of source sizes and constrain temperature profiles (e.g., Pooley et al. 2006, 2007; Bate et al. 2008; Floyd et al. 2009; Blackburne et al. 2011; Mosquera et al. 2011; Mediavilla et al. 2011). These studies must also assume a mean mass for the microlensing stars.

Stronger and less prior-dependent constraints can be obtained from analyses of quasar microlensing variability. We have employed the technique of Kochanek (2004) to analyze uncorrelated variability in quasar optical light curves to yield a correlation between the sizes of quasar accretion disks and the masses of their black holes (Morgan et al. 2010), measurements of the structure of quasar accretion disks (e.g., Poindexter et al. 2008), and even a constraint on the orientation of a quasar's accretion disk (in Q2237+0305; Poindexter & Kochanek 2010). We have also modified the Kochanek (2004) technique to simultaneously analyze optical and X-ray light curves, yielding measurements of the size of the X-ray emitting regions in four systems, PG1115+080, HE 1104–1805, RXJ 1131–1231, and HE 0435–1223 (Morgan et al. 2008; Chartas et al. 2009; Dai et al. 2010; Blackburne et al. 2012, respectively). In those papers, we found that the X-ray continuum emission emerges from region ∼10r_g in radius, where r_g is the black hole's gravitational radius r_g = GM_BH/c². This is at least an order of magnitude smaller than sizes measured in the UV/optical, consistent with the arguments of Pooley et al. (2007). These results seem to disfavor X-ray continuum emission from an extended disk-corona model (e.g., Haardt & Maraschi 1991; Merloni 2003), but more measurements are needed both to confirm these results and to begin searching for correlations between the X-ray emitting regions and other physical properties of the quasars. In this paper, we add to the sample of quasars studied at X-ray wavelengths by applying our Monte Carlo microlensing analysis technique to the combined optical and X-ray light curves of Q J0158–4325 to yield size estimates for the hard and soft X-ray continuum emission regions. Making use of our new constraints on the time delay, we are also able to significantly refine our estimate of the optical accretion disk size.

In Section 2 we begin by describing our Chandra X-ray observations (Chen et al. 2012), followed by a presentation of our new optical monitoring data and a brief review of our data reduction process. In Section 3, we describe our time delay measurement attempts and in Section 4 we present the results from our Monte Carlo microlensing simulations. In Section 5, we conclude with a discussion of the implications for the physics of the optical and X-ray continuum sources in this quasar. All calculations in this paper assume a flat ΛCDM cosmology with h = 0.7, Ω_M = 0.3, and Ω_Λ = 0.7.

We imaged the doubly lensed quasar Q J0158–4325 (Morgan et al. 1999) in X-rays using the ACIS imaging spectrometer (Garmire et al. 2003) on the Chandra X-Ray Observatory (Chandra; Weisskopf et al. 2002) on six occasions between 2009 November 6 and 2010 October 6. These observations were a component of a larger Chandra Cycle 11 monitoring program, the details of which are published in Chen et al. (2012). We divided the full observed frame X-ray spectral range (0.4–8.0 keV) into soft (0.4–1.3 keV) and hard (1.3–8.0 keV) energy bands. We analyze the soft and hard X-ray variability separately in order to constrain the energy structure of the X-ray continuum source.

We have monitored Q J0158–4325 for eight seasons in the R band using the SMARTS 1.3 m telescope with the ANDICAM optical/infrared camera (DePoy et al. 2003)⁷ and using the 1.2 m Euler Swiss Telescope as a part of the COSMOGRAIL⁸ project. We analyzed the optical images from both observatories simultaneously, employing the deconvolution algorithm of Magain et al. (1998). In this technique, we measure the point-spread function (PSF) in each frame using the same five reference stars in the field. The astrometry of the lens components is held fixed to the values measured in Hubble Space Telescope H-band images from the CfA-Arizona Space Telescope Lens Survey (CASTLES), the details of which were published in Morgan et al. (2008). Using the corresponding PSFs, we simultaneously fit the photometry of the point sources and a regularized image of the lensing galaxy to all the frames. The resulting QSO image fluxes are calibrated relative to the deconvolved field stars. Finally, we apply a small multiplicative and additive correction to minimize the dispersion in the overlapping sections of the Euler and SMARTS light curves. This empirical adjustment is required to compensate for small color terms introduced by differences in the filters and quantum efficiency curves of the CCDs and to correct for PSF variability caused by minor differences in the flatness of each detector. The new photometric data are presented in Table 1, and our composite eight-season light curves are displayed in Figure 1.

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3.1. Light Curve Preparation

3.2. Simultaneous Microlensing and Time Delay Analysis

Using the same technique that we described in detail in Morgan et al. (2008), we applied the Monte Carlo microlensing analysis method of Kochanek (2004) to the light curves from each trial time delay, in essence attempting to reproduce the observed light curves by random combinations of the unknown variables: the lens galaxy stellar mass fraction, the median microlens mass 〈M〉, and the effective velocity between source, lens, and observer ${\bm v}_e$. We used the recently measured lens redshift z_l = 0.317 (Faure et al. 2009) to update our set of 10 lens models, each consisting of concentric NFW (Navarro et al. 1996) and de Vaucouleurs mass profiles. We parameterize the stellar mass fraction 0.1 ⩽ f_M/L ⩽ 1 in our model sequence by comparing the fractional mass of the de Vaucouleurs component to the mass of a constant M/L model (f_M/L = 1), varying f_M/L between 0.1 and 1.0 in uniform steps. For each model, we generate four unique sets of microlensing magnification patterns using the P³M method described in Kochanek (2004). The magnification patterns are 4096 × 4096 pixel arrays with an outer scale radius of 20R_E, where R_E = 3.4 × 10¹⁶(〈M〉/0.3 M_☉)^1/2 is the Einstein radius, yielding a source plane pixel scale of 1.7 × 10¹⁴(〈M〉/0.3 M_☉)^1/2 cm. We performed four separate Monte Carlo simulations, one per set of magnification patterns. In each simulation, we attempted 10⁸ trials per mass model, and in each trial we attempt to fit all 61 trial time delays, for a grand total of 2.4 × 10¹¹ trials. An example of a set of good fits to the observed flux ratios is shown in Figure 2.

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We use Bayes' theorem to marginalize over the results from our Monte Carlo simulation such that the probability of a given time delay Δt

$$\begin{equation} P(\Delta t| D) \propto \int P(D| {\bm p}, \Delta t) P({\bm p}) P(\Delta t) d{\bm p}, \end{equation} \tag{ 1 }$$

where $P(D|{\bm p}, \Delta t)$ is the probability of fitting the observations in a given trial, $P({\bm p})$ sets the priors on the microlensing variables (see Kochanek 2004; Kochanek et al. 2007) and P(Δt) is the (uniform) prior on the time delay. As in Kochanek (2004), we draw the microlensing variables (e.g., trajectory, effective transverse velocity ${\hat{v}_e}$) from their prior distributions. Since there is no known means of efficiently selecting starting points or velocities for the source, most trial attempts generate very poor fits to the observed light curves. We try to minimize this inefficiency by evaluating the χ² of each trial in real time and aborting those trials with χ²/N_dof > 5 since they contribute negligibly to the Bayesian integrals.

The probability density for the time delay is displayed in Figure 3. To first order, our analysis appears to have failed, since it did not converge on a robust, single-valued estimate for the time delay. We did succeed, however, in effectively eliminating the possibility of positive values for the delay Δt_AB, which is an improvement over our earlier results in Morgan et al. (2008). The probability distribution in Figure 3 appears to be noisy, but the structure is real. For example, if we make a series of trials using only one-tenth the trials of our full simulation, we always obtain distributions consistent with Figure 3. We are left to conclude that the present probability density for the delay is (at least) double-valued. We also attempted a traditional polynomial fit time delay analysis (e.g., Kochanek et al. 2006), in which we fit the quasar's intrinsic and microlensing variability using Legendre polynomials. The best-fit results from this attempt, based on a 30th order fit to the intrinsic and 3rd order fit to the microlensing variability, are also displayed in Figure 3. Again, as was the case with the Monte Carlo technique, our polynomial fits did not converge on a single value for the delay, in fact the (inferior) polynomial fitting technique still allows for some significant probability of a positive (B leading A) time delay. Nevertheless, the Monte Carlo procedure allowed us to study the structure of the quasar with proper statistical weighting of the uncertainties created by the time delay, ultimately enabling a successful joint fit to the observed X-ray and optical light curves using our Monte Carlo method.

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For the simultaneous optical and X-ray analysis, we employed the technique detailed in Dai et al. (2010), in which a large number of good fits to the optical light curves are generated first. To enable better resolution at smaller angular scales, we used a set of 8192 × 8192 magnification patterns with an outer scale of 10R_E, providing a source plane pixel scale of 4.2 × 10¹³(〈M〉/0.3 M_☉)^1/2 cm, ∼2 times larger than the black hole's gravitational radius at r_g = 2.5 × 10¹³ cm. We discuss the limitations of this pixel scale in Section 4. We also tested the Monte Carlo simulation using the original 20R_E magnification patterns to confirm that the 10R_E patterns from the low f_M/L mass models did not introduce any significant systematic errors. We saved all physical variables from the good optical fits (e.g., trajectory, effective transverse velocity v_e) and then fit the X-ray light curves using the same path across the magnification pattern but a different range of source sizes. An example of a good fit from this analysis is also displayed in Figure 2. We then analyze the combined X-ray and optical solutions to yield a set of joint probability densities for the variables of interest, namely the X-ray and optical source sizes. Since the time delay was unknown, we performed this analysis for a range of assumed delays, but the X-ray size was insensitive to the resulting minor changes in the optical solutions. We also tested our optical solutions for the possibility of 20%–30% line emission contamination from the quasar's un-microlensed broad-line region, but this led to only modest changes in the size estimates (see Morgan et al. 2010; Dai et al. 2010).

Similar to our analysis in Morgan et al. (2008), the unknown time delay introduces short-timescale noise in the optical difference light curve that the Monte Carlo code can successfully reproduce using non-physical transverse velocities. To eliminate this non-physicality in our models, we applied a prior on the effective transverse velocity $10 \, \langle M/M_\odot \rangle ^{1/2}\,{\rm km \, {s^{-1}}} \le {\hat{v}}_e \le 10^4 \, \langle M/M_\odot \rangle ^{1/2}\,{\rm km \, s^{-1}}$ where ${\hat{v}}_e$ is in Einstein units. To convert all variables from Einstein units of 〈M/M_☉〉^1/2 into physical units, we combined the resulting probability density for the effective transverse velocity $P({\hat{v}}_e)$ with the distribution for the true transverse velocity P(v_e). We generated the physical model for P(v_e) following the method of Kochanek (2004), where we estimated the peculiar velocities of the source and lens using the power-law fits of Mosquera & Kochanek (2011). We found the transverse velocity of the observer by projection onto to the cosmic microwave background (CMB) dipole v_CMB = 328.7 km s⁻¹, and we used σ_{pec, l} = 277 km s⁻¹ and σ_{pec, s} = 248 km s⁻¹ for the peculiar velocities of lens and source, respectively. We estimated the velocity dispersion in the lens galaxy σ_* = 203 km s⁻¹ using the monopole moment of a singular isothermal ellipsoid lens model.

In Figure 4, we display the probability densities for the size of the continuum emission region at optical and X-ray wavelengths. In our microlensing simulations, we assume a simple thin accretion disk model where the disk radiates locally as a blackbody (e.g., Shakura & Sunyaev 1973). Since we expect the optical accretion disk to be many times larger than the black hole's gravitational radius r_g = GM_BH/c², we assume that we can safely ignore the inner edge of the accretion disk and the associated relativistic corrections (see Morgan et al. 2010). With these assumptions, the surface brightness at rest wavelength λ_rest is

$$\begin{equation} f_{\nu }(r) = {2 h_p c \over \lambda _{{\rm rest}}^3} \left[ \exp \left({r \over r_{s}} \right)^{3/4}-1 \right]^{-1} \end{equation} \tag{ 2 }$$

where the scale length

$$\begin{eqnarray} r_{s} &=& \left[ {45 G \lambda _{{\rm rest}}^4 M_{{\rm BH}} \dot{M} \over 16 \pi ^6 h_p c^2} \right]^{1/3}\nonumber\\ &=& 9.7 \times 10^{15} \left({\lambda _{{\rm rest}} \over {\rm \mbox{$\mu $m}}} \right)^{4/3} \left({M_{{\rm BH}} \over 10^9 \,{M_{\odot }}} \right)^{2/3} \left({L \over \eta L_E} \right)^{1/3} \,{\rm cm} \quad\quad \end{eqnarray} \tag{ 3 }$$

is the radius at which the disk temperature matches the wavelength, $k T_{\lambda _{{\rm rest}}} = h_p c/ \lambda _{{\rm rest}}$, h_p is the Planck constant, k is the Boltzmann constant, M_BH is the black hole mass, $\dot{M}$ is the mass accretion rate, L/L_E is the luminosity in units of the Eddington luminosity, and $\eta =L/(\dot{M}c^2)$ is the accretion efficiency. In previous investigations (e.g., Morgan et al. 2010; Poindexter et al. 2008) we found accretion disk temperature profiles consistent with the T(r)∝r^−3/4 predictions of thin disk theory, so we report the size of the optical continuum emission region in terms of this model.

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Microlensing observations can estimate the temperature slope, since for T∝r^−β the disk size should scale as r_λ∝λ^1/β = λ^4/3 for standard thin disk theory. Although there is general agreement that hotter regions (i.e., shorter wavelengths) are more compact, observational estimates of the slope β have significant uncertainties and dispersion between studies (e.g., Anguita et al. 2008; Eigenbrod et al. 2008; Poindexter et al. 2008; Bate et al. 2008; Floyd et al. 2009; Blackburne et al. 2011). Fortunately, Mortonson et al. (2005) determined that microlensing statistics are insensitive to the choice of model surface brightness profile and depend primarily on the half-light radius. Thus, the size we report here can easily be compared to other estimates using the half-light radius r_1/2 = 2.44r_s. Also, since microlensing statistics are actually sensitive to the projected area of the quasar's accretion disk, we must account for the unknown inclination angle i. We assume that the optical emission comes from a geometrically thin disk so that the de-projected estimate of the size is r_s = r_{s, obs}/(cos i)^1/2 for random orientations. For simplicity, we use the same surface brightness distribution in X-rays but assume that the emission is geometrically thick, so we report the half-light radius r_{1/2, X}.

In Morgan et al. (2010) we demonstrated that in the context of the thin accretion disk model, a quasar's specific luminosity L_λ can also be used to estimate the size of its accretion disk at that rest-frame wavelength. In the observed-frame I band, for example, the so-called flux-based thin disk scale radius is

$$\begin{eqnarray} r_{s,I} &=& 2.83 \times 10^{15} {1 \over \sqrt{\cos i}} \left({ D_{{\rm OS}} \over r_H } \right)\nonumber\\ && \times \left({ \lambda _{I,{\rm obs}} \over {\rm \mbox{$\mu $m}} } \right)^{3/2} 10^{-0.2(I-19)} \, h^{-1} \,{\rm cm}. \end{eqnarray} \tag{ 4 }$$

There is now significant evidence in the literature that flux(luminosity)-based size estimates are consistently smaller than microlensing-based measurements (e.g., Poindexter et al. 2008; Hainline et al. 2012; Blackburne et al. 2012), and in Morgan et al. (2010), we found this to be true in 10 out of the 11 quasars in our sample. The one outlier in that sample was Q J0158–4325, whose flux- and microlensing-based sizes were consistent. Integration of our updated r_s probability density and correction for an average inclination of cos i = 0.5 yields an optical accretion disk size estimate of log [(r_s/cm)(cos (i)/0.5)^1/2] = 15.6 ± 0.3 at λ_rest = 0.31 μm, the rest-frame center of the R band. So in our new, more precise measurement, the microlensing-based size is now ∼0.4 dex larger than the flux-based size (log (r_{s, flux}/cm) = 15.2 ± 0.1), consistent with the discrepancies in the other systems.

Probability densities for the half-light radii of the full, hard, and soft X-ray continuum emission regions are also displayed in Figure 4. Since there is no simple spatial model for the X-ray continuum source, we report the X-ray sizes in terms of the half-light radius r_{1/2, X} for X-ray emission at rest-frame energies 0.9–18.4 keV (full band), 0.9–3.0 keV (soft band), and 3.0–18.4 keV (hard band). Integration of the probability density for the combined hard and soft X-ray flux yields an estimate for the full band half-light radius log (r_{1/2, X}/cm) = 14.2^+0.5_{− 0.6}. The soft and hard size estimates are consistent with that of the full band and each other, where the half-light radius for soft X-ray emission is log (r_{1/2, X, soft}/cm) = 14.3^+0.4_{− 0.5} and the upper limit on the half-light radius for hard X-ray emission is log (r_{1/2, X, hard}/cm) ⩽ 14.6. Because the pixel scale in our magnification patterns is 7.6 × 10¹³ cm for a 1 M_☉ star, the lower limit on our X-ray size estimates is not fully resolved. In essence, once a trial source becomes much smaller than the pixel scale, all solutions become equally likely. In the case of the hard X-ray source, this prevented any convergence on a lower limit, so we also compiled a probability density for the ratio between the soft and hard X-ray half-light radii log (r_{1/2, X, soft}/r_{1/2, X, hard}) = 0.2^+0.6_{− 0.7}. This probability density is plotted in Figure 5 along with the ratio of the optical half-light radius to that in the full X-ray band, which peaks at log (r_{1/2, opt}/r_{1/2, X}) = 1.7 ± 0.5.

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In other systems, Morgan et al. (2008), Dai et al. (2010), and Blackburne et al. (2012) have all found X-ray continuum regions that seem to be concentrated in the vicinity of the black hole's last stable orbit. In Q J0158–4325, the black hole mass M_BH = 1.6 × 10⁸ M_☉ (Peng et al. 2006) implies a gravitational radius log (r_g/cm) = 13.4 and last stable orbit log (r_stable/cm) = 14.2 at 6r_g in the Schwarzschild metric. Consistent with our results in HE 0435–1223, RXJ 1131–1231, and PG1115+080, we find that the majority of the X-ray emission in Q J0158–4325 emerges from a region in the vicinity of the last stable orbit. We also find that the soft and hard X-ray continuum emission regions seem to be the same size, consistent with the findings of Blackburne et al. (2012) for HE 0435–1223.

In Figure 6, we display the results of marginalizing the fits from our Monte Carlo simulation over all variables other than the stellar mass fraction in the macroscopic mass model. The resulting constraint on f_M/L favors a dark-matter-dominated halo, which seems to be marginally consistent with our expectations for a time delay of approximately −8 days.

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To demonstrate the improvement in accuracy gained by adding four seasons of new optical monitoring data and simultaneous analysis with X-ray light curves, we also show the size distribution from our first analysis of this system (Morgan et al. 2008) in Figure 4. While our new measurement is statistically consistent with our previous estimate, the new expectation value is nearly 0.6 dex larger than our original analysis and has nearly twice the precision. We attribute this difference to two effects. First, while we failed to measure the time delay, the new data substantially reduced the allowed range to −30 days ⩽Δt_AB ⩽ 0 days, leading to a significant narrowing of the source size probability density. Flux ratios estimated for an incorrect delay have extra power on short timescales which the model tries to fit using a more compact source size, leading to a broadening of the source size distribution toward smaller sizes. Since the time delay signal with our four new seasons of monitoring data has effectively ruled out positive delays, there are fewer cases in which the Monte Carlo code successfully reproduced the noise from an incorrect delay using a small accretion disk size. Second, the simultaneous analysis of the (admittedly) sparse X-ray light curves provides a very strong constraint on the effective velocity distribution v_e, which, in turn, constrains r_s. Blackburne et al. (2012) gained similar statistical leverage from the use of X-ray photometry in their analysis.

We have made significant improvements in our understanding of the Q J0158–4325 continuum source, but we are still unable to measure the time delay. This may be due to inadequacies in our data or techniques, such as uncertainties in our photometric measurements and/or insufficient signal from the time delay compared to the signal from microlensing, or it could possibly be due to our oversimplification of the accretion disk model. For example, Abolmasov & Shakura (2012) demonstrate that general relativistic effects become important during caustic crossing events and can explain high-magnification features in the light curves of SBS 1520+530 and Q 2237+0305. Several such features are present in the light curves of Q J0158–4325 and are not perfectly fit by our present analysis. To be fair, the results of Abolmasov & Shakura (2012) require disks with large inclination angles, and we see no evidence for this in Q J0158–4325. Nevertheless, we are left to conclude that the modeling of high-amplitude microlensing events may be improved by the inclusion of a fully relativistic disk model or a more sophisticated model for the motions of stars in the lens galaxy (e.g., Poindexter & Kochanek 2010).

At X-ray wavelengths, a consistent picture of the structure of the quasar continuum emission region is beginning to emerge. In the four systems with quantitatively measured continuum emission regions, HE 0435–1223 (Blackburne et al. 2012), RXJ 1131–1231 (Dai et al. 2010), PG 1115+080 (Morgan et al. 2008), and now Q J0158–4325, the X-ray continuum emission source is located at ∼10 r_g. Improving on our previous work, we performed the second complete analysis of the hard and soft X-ray variability, leading to independent estimates for the sizes of the soft and hard X-ray sources. While our hard and soft size estimates are formally consistent with each other, there are some indications that the hard X-ray source is smaller than the soft source. As we discussed in Chen et al. (2012), these data were not designed to produce good statistical estimates for the hard and soft band separately, so we expect that adding more X-ray epochs with better photon statistics should lead to markedly improved X-ray size estimates.

This material is based upon work supported by the National Science Foundation under Grant No. AST-0907848. C.W.M. also gratefully acknowledges support from the Research Corporation for Science Advancement. C.W.M., L.J.H., C.S.K., J.A.B., A.M.M., X.D., and G.C. are supported by Chandra X-Ray Center award 11700501. M.T., F.C., and G.M. acknowledge support from the Swiss National Science Foundation. C.S.K., J.A.B., and A.M.M. are supported by NSF grant AST-1009756.

Facilities: CTIO:2MASS (ANDICAM) - 2MASS Telescope at Cerro Tololo Inter-American Observatory, HST (NICMOS, ACS) - Hubble Space Telescope satellite, Euler1.2m - The Swiss Euler 1.2m Telescope, CXO (ACIS) - Chandra X-ray Observatory satellite