0004-637X/879/1/35

We present an analysis of Chandra spectra of five gravitationally lensed active galactic nuclei. We confirm the previous detections of FeK α emission lines in most images of these objects with high significance. The line energies range from 5.8 to 6.8 keV, with widths from unresolved to 0.6 keV, consistent with emission close to spinning black holes viewed at different inclination angles. We also confirm the positive offset from the Iwasawa–Taniguchi effect, the inverse correlation between the FeK α equivalent width (EW) and the X-ray luminosity in active galactic nuclei, where our measured EWs are larger in lensed quasars. We attribute this effect to microlensing, and perform a microlensing likelihood analysis to constrain the emission size of the relativistic reflection region and the spin of supermassive black holes, assuming that the X-ray corona and the reflection region, responsible for the iron emission line, both follow power-law emissivity profiles. The microlensing analysis yields strong constraints on the spin and emissivity index of the reflection component for Q 2237+0305, with a > 0.92 and n > 5.4. For the remaining four targets, we jointly constrain the two parameters, yielding a = 0.8 ± 0.16 and an emissivity index of n = 4.0 ± 0.8, suggesting that the relativistic X-ray reflection region is ultracompact and very close to the innermost stable circular orbits of black holes, which are spinning at close to the maximal value. We successfully constrain the half-light radius of the emission region to <2.4 r g ( r g = GM / c 2 ) for Q 2237+0305 and in the range 5.9–7.4 r g for the joint sample.

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1. Introduction

The X-ray spectra of active galactic nuclei (AGNs) are characterized by a continuum emission that is well modeled by a power law (e.g., Guilbert & Rees 1988; Reynolds & Nowak 2003; Brenneman 2013). The UV emission of the accretion disk provides the seed photons and these photons are then inverse Compton scattered by relativistic electrons in the corona to produce the continuum. A portion of these photons are scattered back to the accretion disk and can can create a reprocessed or reflected emission component including fluorescent emission lines, most notably, the Fe Kα emission line at 6.4 keV in the source rest frame (Guilbert & Rees 1988; Fabian et al. 1995; Reynolds & Nowak 2003; Brenneman 2013). The exact locations of the reflection are not well constrained and the process can occur at multiple locations, from the (inner) accretion disk, broad-line regions, to the torus. Measuring the spins of supermassive black holes (SMBHs) at the centers of AGN is important because it is related to the growth history of the black holes, their interaction with the environment, the launching of relativistic jets, and the size of the innermost stable circular orbit (ISCO; e.g., Thorne 1974; Blandford & Znajek 1977; Fabian 2012; Brenneman 2013; Chartas et al. 2017). For example, as an SMBH grows it can provide matter and energy to its surrounding environment through outflows (Fabian 2012). One important method to estimate spins models the general relativistic (GR) and special relativistic distortions of the Fe Kα emission line (e.g., Brenneman 2013; Reynolds 2014). This method has been applied to many nearby Seyferts with most estimates being close to the maximal spin (Reynolds 2014). Another approach is to model the UV-optical SEDs of high redshift quasars (e.g., Capellupo et al. 2015, 2017). These studies again find high spins for high redshift quasars. Quasar microlensing has significantly improved our understanding of the accretion disks (e.g., Dai et al. 2010; Morgan et al. 2010; Mosquera et al. 2013; Blackburne et al. 2014, 2015; MacLeod et al. 2015) and nonthermal emission regions (e.g., Pooley et al. 2006, 2007; Morgan et al. 2008; Chartas et al. 2009, 2016, 2017; Dai et al. 2010; Chen et al. 2011, 2012; Guerras et al. 2016) of quasars, and the demographics of microlenses in the lens galaxy (e.g., Blackburne et al. 2014; Dai & Guerras 2018; Guerras et al. 2018). Since the magnification diverges on the caustics produced by the lensing stars, quasar microlensing can constrain arbitrarily small emission regions if they can be isolated from other emissions, in position, velocity, or energy. In particular, microlensing can be used to constrain the spin of black holes by measuring the ISCO size. In this paper, we will utilize the excess equivalent width (EW) difference between lensed and unlensed quasars, first summarized by Chen et al. (2012), to constrain the size of the reflection region and the spin of quasars. This paper is organized as follows. We present the Chandra observations and the data reduction in Section 2 and the spectral analysis in Section 3. In Section 3 we discuss the significance of the iron line detections and we confirm the offset of Fe Kα EWs of lensed quasars. In Section 4, we carry out a microlensing analysis to estimate the size of the Fe Kα emission region and the spin of the black hole. We discuss the results in Section 5. We assume cosmological parameters of Ω M = 0.27, Ω Λ = 0.73, and H 0 = 70 km s−1 Mpc−1 throughout the paper.

2. Observations and Data Reduction

The observations used in this paper were made with the Advanced CCD Imaging Spectrometer (Garmire et al. 2003) on board the Chandra X-ray Observatory (Weisskopf et al. 2002). Chandra has a point-spread function (PSF) of 0 5 and is therefore able to resolve most of the multiple images in lensed systems, as they have a typical image separation of 1''–2''. We used two sets of data in this analysis. The first, which we call Data Set 1, mainly comes from the Chandra Cycle 11 program and the second, which we call Data Set 2, comes from Chandra Cycles 14–15. We analyzed five lenses: Q J0158−4325, HE 0435−1223, SDSS J1004+4112, HE 1104−1805, and Q 2237+0305. The lens properties are summarized in Table 1. All the data were reprocessed using the Chandra X-ray Center CIAO 4.7 software tool chandra _ repro , which takes data that have already passed through the Chandra X-ray Center Standard Data Processing and filters the event file on the good time intervals, grades, cosmic ray rejection, transforms to celestial coordinates, and removes any observation-specific bad pixel files. Table 1. Gravitational Lenses Analyzed in this Paper Object z s z l R.A. (J2000) Decl. (J2000) Galactic N H Epochs Chandra Expo. Time Data Set I Data Set II Combined (1020 cm−2) (103 s) QJ 0158−4325 1.29 0.317 01:58:41.44 −43:25:04.20 1.95 12 29.9 111.7 141.6 HE 0435−1223 1.689 0.46 04:38:14.9 −12:17:14.4 5.11 10 48.4 217.5 265.9 SDSS J1004+4112 1.734 0.68 10:04:34.91 +41:12:42.8 1.11 11 103.8 145.6 249.4 HE 1104−1805 2.32 0.73 11:06:33.45 −18:21:24.2 4.62 15 110.0 80.5 191.5 Q 2237+0305 1.69 0.04 22:40:30.34 +03:21:28.8 5.51 30 292.4 175.4 467.8 Download table as: ASCIITypeset image

3. Spectral Analysis

4. Microlensing Analysis

We performed a microlensing analysis to interpret the positive EW offset measured in lensed quasars. In magnitude units, we have where f is the differential magnification magnitude between the magnification of the reflection region and the corona. Here, we have included a modest evolution effect, assuming that the rest-frame EW of high redshift quasars are higher than the local ones, at EW fit = 0.2 keV (Iwasawa et al. 2015). We generated magnification maps of these five lenses using the inverse polygon mapping algorithm (Mediavilla et al. 2006, 2011). These magnification maps are 16,0002 pixels and each pixel has a length scale of 0.685 r g for the five lenses. The rest of the pertinent magnification map properties are listed in Table 5 and further details on the magnification maps can be found in Guerras et al. (2016). We then generated images of "average" AGN corona and the reflection region that are modified by the relativistic effects caused by the black hole using the software KERTAP (Chen et al. 2013a, 2013b, 2015). Table 5. Lensing Parameters of the Sample Object Image R E a κ γ Macro Black Hole Massb (lt-day) Magnification (109 M ⊙ ) 0158 A 13.2 0.348 0.428 4.13 0.16 (Mg ii ) 0158 B 13.2 0.693 0.774 1.98 0.16 (Mg ii ) 0435 A 11.4 0.445 0.383 6.20 0.50 (C iv ) 0435 B 11.4 0.539 0.602 6.67 0.50 (C iv ) 0435 C 11.4 0.444 0.396 6.57 0.50 (C iv ) 0435 D 11.4 0.587 0.648 4.01 0.50 (C iv ) J1004 A 9.1 0.763 0.300 29.6 0.39 (Mg ii ) J1004 B 9.1 0.696 0.204 19.7 0.39 (Mg ii ) J1004 C 9.1 0.635 0.218 11.7 0.39 (Mg ii ) J1004 D 9.1 0.943 0.421 5.75 0.39 (Mg ii ) 1104 A 9.1 0.610 0.512 9.09 0.59 (H β ) 1104 B 9.1 0.321 0.217 2.42 0.59 (H β ) 2237 A 38.5 0.390 0.400 4.71 1.20 (H β ) 2237 B 38.5 0.380 0.390 4.30 1.20 (H β ) 2237 C 38.5 0.740 0.730 2.15 1.20 (H β ) 2237 D 38.5 0.640 0.620 3.92 1.20 (H β ) Notes. aEinstein radius size on the source plane for a 0.3 M ⊙ star. bThe black hole mass and the emission line used for the estimates by Morgan et al. ( Einstein radius size on the source plane for a 0.3 Mstar. The black hole mass and the emission line used for the estimates by Morgan et al. ( 2010 ) (Q J0158−4325, HE 0435−1223, SDSS J1004+4112) and Assef et al. ( 2011 ) (HE 1104−1805, Q 2237+0305). Download table as: ASCIITypeset image We assumed that the X-ray corona and reflection region are located very close to the disk in Keplerian motion, following power-law emissivity profiles, I ∝ r−n, but with different emissivity indices. The models have three parameters: the Kerr spin parameter a of the black hole, the power-law index for the emissivity profile of the reflection region n, and the inclination angle of the accretion disk. To simplify the analysis, we set the inclination angle to 40°, a typical inclination angle for a Type I AGN. The emissivity index n for the reflection region was varied from 3.0 to 6.2 in steps of 0.4, and the spin a was varied between 0 and 0.998 in steps of 0.1. Some example Kerr images of the emissivity profiles are shown in Figure 11. These resulting images were then convolved with the magnification maps of each lens image to estimate the amount of microlensing that the corona and reflection region would experience at different locations of the magnification maps. We performed these convolutions in flux units and then converted to magnitude scales. We also tested to see if the orientation of the corona and reflection region with respect to the magnification map matters by rotating the images by 90°, and in general this will not invoke a significant change in the parameter estimations discussed below. The half-light radius for the X-ray continuum emission is expected to be ~10 r g (gravitational radii; Dai et al. 2010; Mosquera et al. 2013), corresponding to emissivity indices of n = 2.2–3.4 for different spins. We then subtracted the two convolved magnification maps from the continuum and reflection models to estimate the differential microlensing between the two emission regions, f j = μ con − μ ref in magnitude units, as a function of source position on the magnification map for a lensed image and common values of a and n. We obtained distributions of these subtracted convolutions by making histograms of the values for randomly selected points in the subtracted convolved images. Figure 11. Example Kerr images of the Fe Kα emissivity profiles from KERTAP (Chen et al. 2015) with a = 0, n = 3.8 (top, left), a = 0, n = 5.8 (top, right), a = 0.9, n = 3.8 (bottom, left), and a = 0.9, n = 5.8 (bottom, right), where the inclination angle is fixed at 40°. Download figure: Standard image High-resolution image Export PowerPoint slide We then performed a likelihood analysis using the microlensing magnification distributions and the EWs we measured in the data. For a given image and fixed n and a, the likelihood of the data given the model is where A is a normalization constant and b j is the bin height of the jth value from the convolved histograms discussed previously. The chi-square is calculated from where EW data and EW err are, respectively, the EW and the EW uncertainty from the spectra analysis done in Section 3, and f j is the amount of differential microlensing between the continuum and reflection regions in magnitudes. Using Equations (2)–(5), we then have a likelihood as a function of n and a, the index of the emissivity profile and spin parameter, for each image of each object. Examples for Q 2237+0305 A and Q J0158−4325 A are shown in Figure 12, where we plot the model likelihood ratios compared to the a = 0.9 and n = 5.8 model. We then combined the likelihoods of all the images from a target where B is the new normalization constant and the multiplication applies to all the images of the target. After calculating L total (n, a) for a grid of (n, a) combinations, we can then marginalize over either a or n to obtain the posterior probability for the emissivity index or spin, separately. We have discarded HE 1104−1805B, the nondetection case, in the microlensing analysis, and since it contributes to less than a tenth of the sample, we do not expect that our results will change significantly. Figure 12. Sample model likelihood ratios as a function of the differential microlensing magnitude for Q 2237+0305 A (left) and Q J0158−4325 A (right). The logarithm likelihood ratios are between the model of (n = 5.8, a = 0.9) and the models of (n = 3.8, a = 0, black), (n = 3.8, a = 0.9, red), and (n = 5.8, a = 0.0, green). The vertical dashed lines indicate the observed differential microlensing magnitude range including uncertainties. Download figure: Standard image High-resolution image Export PowerPoint slide Figures 13 and 14 show the marginalized probabilities for the spin and emissivity index parameters. For Q 2237+0305, we obtain tight constraints on both the spin and emissivity index parameters with well established probability peaks, and the 68% and 90% confidence limits for the spin parameter are a > 0.92 and a > 0.83, respectively, where we linearly interpolate the probabilities to match the designated limits. The corresponding 68% and 90% confidence limits for the emissivity index are n > 5.4 and n > 4.9 for Q 2237+0305. Compared to other targets, Q 2237+0305 has the longest exposure among the sample, the line EWs have small relative uncertainties, and the EW deviations from the Iwasawa–Taniguchi relation are large, and because of these factors, the constraints for Q 2237+0305 are strong. For the remaining four targets Q J0158−4325, HE 0435−1223, SDSS J1004+4112, and HE 1104−1805, the individual constraints are weak. However, since the shapes of the probability distributions are similar (Figure 13 right), we jointly constrain the remaining targets by multiplying the probability functions, yielding 68% and 90% limits of a = 0.8 ± 0.16 and a > 0.41 for the spin parameter and n = 4.0 ± 0.8 and n = 4.2 ± 1.2 for the emissivity index. We need to remove Q 2237+0305 from the joint sample, otherwise the probabilities for the joint sample will be dominated by a single object. We plot the two-dimensional confidence contours of the two parameters for Q 2237+0305 and the remaining sample in Figure 15. We can also bin the two-dimensional parameter space by the half-light radius after the Kerr lensing effect and calculate the corresponding probabilities in each bin. Figure 16 shows the normalized probabilities as a function of the Fe Kα emission radius for Q 2237+0305 and the remaining joint sample in the logarithm scale. The half-light radius are constrained to be <2.4 r g and <2.9 r g (68% and 90% confidence) for Q 2237+0305, and in the range of 5.9–7.4 r g (68% confidence) and 4.4–7.4 r g (90% confidence) for the joint sample. Figure 13. (Left) Logarithm likelihood as a function of black hole spin for Q 2237+0305 and the joint sample excluding Q 2237+0305. (Right) Linear likelihood as a function of black hole spin for Q J0158−4325, HE 0435−1223, SDSS J1004+4112, HE 1104−1805, and the joint sample of the four targets. The 68% and 90% confidence limits for Q 2237+0305 are a > 0.92 and a > 0.83, and the corresponding limits for the joint sample are a = 0.8 ± 0.16 and a > 0.41. Download figure: Standard image High-resolution image Export PowerPoint slide Figure 14. (Left) Logarithm likelihood as a function of emissivity index for Q 2237+0305 and the joint sample excluding Q 2237+0305. (Right) Linear likelihood for Q J0158−4325, HE 0435−1223, SDSS J1004+4112, HE 1104−1805, and the joint sample of the four targets. The 68% and 90% confidence limits for Q 2237+0305 are n > 5.4 and n > 4.9, and the corresponding limits for the joint sample are n = 4.0 ± 0.8 and n = 4.2 ± 1.2. Download figure: Standard image High-resolution image Export PowerPoint slide Figure 15. 68%, 90%, and 99% confidence contours in the two-dimensional parameter space of a and n for Q 2237+0305 (left) and the joint sample of Q J0158−4325, HE 0435−1223, SDSS J1004+4112, and HE 1104−1805 (right). Download figure: Standard image High-resolution image Export PowerPoint slide Figure 16. Logarithm likelihood as a function Fe Kα emission half-light radius for Q 2237+0305 and the joint sample of Q J0158−4325, HE 0435−1223, SDSS J1004+4112, and HE 1104−1805. The 68% confidence ranges of the half-light radius are <2.4 r g for Q 2237+0305 and 5.9–7.4 r g for the joint sample. Download figure: Standard image High-resolution image Export PowerPoint slide

5. Discussion

Under the hypothesis that the higher average EW of lensed quasars for a monitoring sequence of observations is a microlensing effect, we explain the offset using a set of GR corona, reflection, and microlensing models. We perform a microlensing analysis to obtain the likelihood as a function of the index of the emissivity profile for the reflection component and spins of black holes, in which we have included a modest redshift evolution effect on the rest-frame EW of Fe Kα lines, such that the spin values obtained are more conservative. For the joint constraint from a sample of four targets, our analysis showed that the relativistic reflection region is more likely to have an emissivity index of n = 4.0 ± 0.8 and a half-light radius of 5.9–7.4 r g (1σ), and therefore originates from a more compact region relative to the continuum emission region. This result confirms the previous qualitative microlensing argument that points towards the reflection region belonging to a more compact region (e.g., Chen et al. 2012). The result also shows that the X-ray continuum cannot be a simple point source "lamppost" model, confirming the earlier analysis result of Popović et al. (2006). The spin value of the joint sample is constrained to be a = 0.8 ± 0.16. This is in agreement with previous studies reporting high spin measurements (e.g., Reis et al. 2014; Reynolds 2014; Reynolds et al. 2014; Capellupo et al. 2015, 2017) either in the local or high redshift samples. For Q 2237+0305, both the spin and emissivity index parameters are well constrained individually with a > 0.92 and n > 5.4 corresponding to 2.25–3 r g for spins between 0.9 and the maximal value. Overall, our spin measurements favor the "spin-up" black hole growth model, where most of the accretion occurs in a coherent phase with modest anisotropies, especially for z > 1 quasars (e.g., Dotti et al. 2013; Volonteri et al. 2013). Since this paper uses the relative microlensing magnification as a signal to constrain the emissivity profiles of the reflection region, the technique only probes the Fe Kα emission region comparable or smaller than the X-ray continuum emission regions. Emission lines originating from this compact region are theoretically predicted to have a broad line profile and with the peak energy varying with the inclination angle. The broad emission line widths, especially for Q 2237+0305 of ~0.5 keV and 4–8σ broad, and the range of line energies between 5.8 and 6.8 keV, provide the confirmation of this theoretical expectation. For the reflections that occur at much larger distances, at the outer portion of the accretion disk, disk wind, broad line region, or torus, they will result in a narrow Fe Kα line that is not sensitive to this technique. It is also quite possible that our sample is biased because we selected our targets based on their strong microlensing signals at optical wavelengths. However, since being microlensing active and having large spin values are independent, we do not see that this bias will significantly affect our results. Furthermore, HE 1104−1805B was the only image with no detectable Fe Kα features and was discarded in the microlensing analysis, suggesting that our somewhat limited exposure times were sufficiently large as to not introduce any nondetection bias. This will not affect the microlensing constraints for Q 2237+0305, and will only have a limited effect on the joint sample results because it contributes less than a tenth of the sample. Reis et al. (2014) and Reynolds et al. (2014) fit a broad relativistic Fe Kα line to the stacked spectra of gravitationally lensed quasars. This technique assumes that the stacked Fe Kα line profile resembles the unlensed line profile; however, the Fe Kα line peak is observed to be a variable between observations (Chartas et al. 2017). Although this technique has a different set of systematic uncertainties, the resulting constraints are quite similar to the analysis results from this paper. For Q 2237+0305, the line fitting method has yielded a = (90% confidence, Reynolds et al. 2014), and the constraint in this paper is a > 0.83 (90% confidence). Both studies show that Q 2237+0305 has large spin values, while the analysis here points more to a maximal value. The steep emissivity profiles measured in this paper are also broadly consistent with those measurements from local AGN, such as MCG-6-30-15 (Wilms et al. 2001; Vaughan & Fabian 2004; Miniutti et al. 2007), 1H0707−495 (Zoghbi et al. 2010; Dauser et al. 2012), and IRAS 13224−3809 (Ponti et al. 2010). These steep emission profiles can be resulted by combining the light bending, vertical Doppler boost, or ionization effects, to produce slopes as steep as n ~ 7 (Wilms et al. 2001; Vaughan & Fabian 2004; Fukumura & Kazanas 2007; Svoboda et al. 2012). Unfortunately, the spin measurement technique presented in this paper can only be used to analyze the small sample of targets whose X-ray spectra can be measured with sufficient S/N using the current generation of X-ray telescopes. The next generation of X-ray telescopes with an order of magnitude increase in the effective area will allow these measurements in a much larger sample. Ideally, we need sub-arcsec angular resolutions to resolve the lensed images to increase the constraining power for the size and spin measurements. However, a similar analysis can be applied to the total image of the lensed quasars, where the requirement for the angular resolution is less crucial, because the analysis relies on the time-averaged relative microlensing signals between the X-ray continuum and Fe Kα emission regions. In addition, quasar microlensing can induce variability in the polarization signals, especially the polarization angle (Chen 2015), which can be detected by future X-ray polarization missions and put constraints on quasar black hole spins independently.