SCALING RELATIONS AND X-RAY PROPERTIES OF MODERATE-LUMINOSITY GALAXY CLUSTERS FROM 0.3 < z < 0.6 WITH XMM-NEWTON^*

One of our aims was to investigate how improved XMM-Newton imaging would affect the measurements of these faint clusters. Along with improved spectral response and calibrations, the improved resolution allowed us to identify and mask out contaminating point sources. To this end, we compare our measured fluxes to those reported in the initial 160SD paper of Vikhlinin et al. (1998, V98 hereafter).

In the original work, V98 were unable to use a wide aperture to integrate flux due to the large statistical uncertainty introduced by the ROSAT background. Instead, they estimated the flux from the normalization of a β-model (Cavaliere & Fusco-Femiano 1976),

$$\begin{equation} I(r,r_c) = I_0 \left(1 + r^2/r_c^2\right)^{-3\beta + 0.5}. \end{equation} \tag{ 1 }$$

They estimated core radii by fitting a β = 0.67 model to their surface brightness profiles; then, they extrapolated to obtain the flux based on the normalization and shape of the best-fit β-model. Their final reported flux was actually (f_0.6 + f_0.7)/2, where f_0.6 and f_0.7 are the fluxes obtained assuming β = 0.6 and β = 0.7, respectively.

For direct comparison with these results, we integrated counts inside a fixed aperture. In order to avoid biasing these results by our somewhat uncertain estimation of r₂₅₀₀, we adopted a metric aperture of radius 300 $h_{70}^{-1}$ kpc. For the equivalent flux, we used the β −model parameters from V98 to infer the estimated ROSAT fluxes inside 300 $h_{70}^{-1}$ kpc. The errors on these fluxes were kept at the same percent as the originally reported values. Details of this procedure are given in Appendix A. The comparison between our results and V98 is shown in Figure 2.

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Our measured fluxes agree to within 1σ with the modified fluxes of V98 in all but six cases. For RXJ0847.1+3449, including an XMM-Newton point source blended with the cluster causes the measured fluxes to agree within their combined 1σ errors. To match our flux measurement of RXJ0056.9−2740 with that of V98, we only needed to center our aperture on the same position. RXJ2146.0+0423, which we find to be slightly lower in flux than allowed by V98's uncertainty, matches perfectly when we shift to the V98 coordinates and expand the aperture to include a nearby XMM-Newton point source. To account for our expanded aperture, we rederived a new, corrected V98 flux to compare in this case. Similarly, repositioning our aperture around RXJ0522.2−3625 and using a larger aperture brings the two measurements into agreement. Finally, RXJ0826.1+2625 and RXJ2328.8+1453 were originally measured at a significant positional offset from V98 (≈37'' and 45'', respectively). In both cases, it appears as if the ROSAT images blended in nearby point sources. By recentering our aperture around the V98 coordinates and expanding the region to include the neighboring objects, we find agreement between the two sets of flux measurements.

We have reproduced the ROSAT X-ray flux estimates from V98 and demonstrated that blended point sources and off-center apertures affected the flux estimates of these clusters over and above the uncertainty based on counting statistics and background subtraction alone.

In Table 1 we list coordinates for each cluster twice. The coordinates from Hoekstra et al. (2011) are their best estimate of the position of each cluster's BCG. The new coordinates are of an X-ray centroid performed on data from the MOS1 camera around each cluster's X-ray emission. Centroids were computed in five iterations of centroiding an aperture with radius 16'', of images binned to 16 pixel⁻¹. Twelve of the X-ray-determined positions are within 5'' of the BCG position, and the only position more than 12'' from the BCG is for RXJ0826.1+2625, where the BCG identification may be questionable. We plot our results, along with the offsets using X-ray positions from ROSAT, in Figure 3. These XMM-Newton observations provide a significant improvement in the ability to properly detect the cluster position over the original ROSAT detection positions.

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We fit our measurements of bolometric luminosity, temperature, and mass inside r₂₅₀₀ to the relation

$$\begin{equation} \log \left(\frac{{Y}}{{Y}_0}\right) = \alpha \log \left(\frac{{X}}{{X}_0}\right) + {C}_X. \end{equation} \tag{ 2 }$$

X₀ and Y₀ are pivot values, which were 10⁴⁴ erg s⁻¹, 4 keV, and 10¹⁴ M_☉ for luminosity, temperature, and mass, respectively. Luminosity and mass were corrected for redshift evolution by including the factor E(z); fits were therefore of L/E(z) and ME(z). To extend the dynamic range of our sample and to compare our low-mass sample with a higher-mass sample at similar redshift, we also included data from the Canadian Cluster Comparison Project (CCCP; Hoekstra et al. 2012; Mahdavi et al. 2013). This sample of 50 galaxy clusters spans redshifts 0.15 < z < 0.55, and all clusters were required to have a temperature k_BT_X > 3 keV. CCCP data was acquired through the online database.¹¹ In an erratum (Mahdavi et al. 2014) these data have been updated since original publication to fix an error in the bolometric luminosity correction factor. We therefore present all of the cluster properties used for fitting in Table 5; a full table is provided in the online edition.

Table 5. CCCP Cluster Properties within r₂₅₀₀

Name	Mass × E(z)	L/E(z)	kT	Redshift
	($h_{70}^{-1} 10^{14}\,{M}_\odot$)	($h_{70}^{-2} 10^{45}\,\mathrm{erg}\,\mathrm{s}^{-1}$)	(keV)
3C295	3.30 ± 0.86	0.78 ± 0.01	6.43 ± 0.35	0.464
Abell0068	2.87 ± 0.645	0.96 ± 0.02	7.25 ± 0.34	0.255
Abell0115N	0.68 ± 0.46	0.47 ± 0.01	4.84 ± 0.10	0.197
Abell0115S	0.84 ± 0.53	0.32 ± 0.01	5.60 ± 0.24	0.197
Abell0209	2.05 ± 0.43	0.97 ± 0.01	7.14 ± 0.34	0.206

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Individual fits are discussed below, but the results are given in Table 6. Fits including data in this work are labeled "160SD," while those including CCCP data are marked as such. Except where noted, uncertainties in fit values were derived through 50,000 bootstrap resamplings. Fits were performed using the weighted least squares (WLS) and BCES methods described by Akritas & Bershady (1996). Where luminosity was serving as the X variable, we used the WLS and BCES (Y|X) methods, which minimized the residuals in the other parameter. Conversely, when luminosity was the Y variable, we used the BCES (X|Y) method. When fitting the mass-temperature relation, we used the BCES Bisector and Orthogonal methods, which considers the residuals in both variables. To account for asymmetric error bars, we estimated a single, logarithmic error for a value $X^{+\mathrm{u}}_{-\mathrm{d}}$ to be

$$\begin{equation} \sigma = 0.4343 \frac{0.5(u + d)}{X}. \end{equation} \tag{ 3 }$$

For clarity, when describing a relation fit by Equation (2), we call it the Y-X relation, where X is the independent variable.

Our first fit was of the luminosity-mass relation within r₂₅₀₀. When fitting this relation, we did not include RXJ0826.1+2625, as its mass was not well determined (as discussed in Section 2). We first fit this relationship without assuming intrinsic scatter; the resulting best-fit slope was α = 0.435 ± 0.047. This result shows no significant difference from the result for the 50 CCCP clusters alone, but it does not agree with the result for a fit only of the low-mass sample presented here. We caution that this discrepancy is not necessarily indicative of a break in the scaling relation, for reasons we will discuss in Section 5.

When allowing for intrinsic scatter, the best-fit value of α is 0.305 ± 0.042, with an intrinsic scatter of σ_log (M|L) = 0.100. Figure 4 shows both fits along with the cluster properties for both samples. For a direct comparison of the reduced scatter, we fit the M–L relation using luminosities from the original work by Hoekstra et al. (2011). With these, the intrinsic scatter was σ_log (M|L) = 0.262.

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Next we fit the temperature–luminosity relation within r₂₅₀₀, this time using all 15 clusters studied here. We found that the best fit for the entire sample was α = 0.229 ± 0.016 with an intrinsic scatter of σ_log (T|L) = 0.073  ±  0.009, consistent with the fits for the two individual samples. This fit is shown along with the data in Figure 5. When we did not allow for intrinsic scatter, we found the best-fit slope was relatively unchanged, becoming α = 0.225 ± 0.016.

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We also investigated the scaling between mass and temperature within r₂₅₀₀. Again, RXJ0826.1+2625 was excluded from this fit. For the combined sample, the best-fit with the BCES Bisector was α = 1.88 ± 0.21, which was consistent with fits for the sub-samples alone. This fit is shown in Figure 6. In addition, we include data taken from Sun et al. (2009). Masses from that study are not based on weak-lensing measurements, but were instead derived from an assumption of hydrostatic equilibrium. These data were not included in our fits, however.

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In order to more easily compare our work to other studies, we also fit the inverse of these three relations. Using BCES(X|Y), the L–M relation fit for the CCCP+160SD sample is α = 2.33 ± 0.27. In contrast, for BCES(Y|X), the inverse of the M–L relation is α⁻¹ = 2.30. Our BCES(Y|X) fit of L–T is α = 4.47 ± 0.33, while the corresponding fit from the T–L relation is α⁻¹ = 4.36. When fitting T–M, the best fit from BCES Bisector was α = 0.537 ± 0.059, which agrees with the BCES Bisector of M-T, α⁻¹ = 0.532.

We compared our results to previously published individual XMM-Newton results for four clusters (RXJ0110.3+1938, RXJ0847.1+3449, RXJ1117.4+0743, and RXJ1354.2−0221). To investigate the differences, we replicated the analysis of previous observations, including their aperture sizes and cosmology. We were able to reasonably reproduce previous results. The discrepancies arising from systematics such as differences in background choices or particle background screening criteria are smaller than the statistical uncertainty. We find that any apparent differences between our results for these clusters with previous results arise because of differences in aperture sizes, and rarely, choice of aperture centers. The details of this comparison are reported in Appendix B.

Our most obvious source of possible discrepancy with previous works is our choice of apertures, which have a radius r₂₅₀₀ motivated by weak-lensing estimates from Hoekstra et al. (2011) that were unavailable to most of the other studies. Another potential source of X-ray temperature discrepancy is the choice of binning spectral data. Some previous works binned spectral data to as few as 12 counts per spectral bin. We leave our spectra unbinned and fit with the C-statistic. As this work is focusing on faint clusters, we are limited by low photon counts. If data are binned such that only a few counts are in each bin, each bin will have non-Gaussian behavior. Since the χ² statistic is defined for Gaussian-distributed data, it will not be a valid fitting statistic in this case. Alternatively, data can be binned, but doing so potentially degrades spectral resolution. Along with producing better fits for low counts (Nousek & Shue 1989; Tozzi et al. 2006), use of the C-statistic can also avoid biases in the high-count regime (Humphrey et al. 2009). Use of different thermal models for fitting spectra did not cause major deviations in our results. As we were able to reproduce the earlier results while still using an APEC model, this should not therefore bias our results significantly (see also Belsole et al. 2005; Matsushita et al. 2007).

We demonstrated in detail (see Appendix B) that we can recover results of previous works, which verifies their results and ours. However, we caution that the choice of aperture and center affect the estimate of L, T, and M for any cluster, and that results from different analyses cannot be blindly combined.

We have measured scaling relations between weak-lensing mass, X-ray luminosity, and temperature for a sample of clusters with mass and luminosity around the cluster/group boundary and at redshifts 0.3 < z < 0.6. As we used weak-lensing masses and bolometric luminosities and because we only investigated X-ray properties within r₂₅₀₀, no exact comparisons are available for our results. However, we can compare our results to other similar studies, both those focused on local groups and those that include clusters at redshifts similar to what was studied here but more massive than our sample.

Our best fit of the M–L relation within r₂₅₀₀ was, when neglecting intrinsic scatter, α = 0.435 ± 0.047. Hoekstra et al. (2011) fit this relation using almost the same clusters studied here, with ROSAT luminosities taken from the ROSAT measurements, and a higher mass sample, reporting α = 0.68 ± 0.07. Other works (Maughan 2007; Rykoff et al. 2008; Eckmiller et al. 2011; Reichert et al. 2011) find values in the range 0.5 ≲ α ≲ 0.75, consistent with but somewhat steeper than ours.

For the L–T relation within r₂₅₀₀, we found a best fit of α = 4.47 ± 0.33, although the 160SD groups and the CCCP clusters each had shallower slopes when fit independently. Previous results (Maughan 2007; Pratt et al. 2009; Bruch et al. 2010; Eckmiller et al. 2011; Reichert et al. 2011; Nastasi et al. 2014) have reported slopes from 2.5 ≲ α ≲ 4.5. Our results, particularly for the two sub-samples fit individually, are consistent with this range, albeit on the high end.

In this work, we reported the best fit slope of the M–T relation within r₂₅₀₀ was α = 1.88 ± 0.21. Previous works (Sun et al. 2009; Eckmiller et al. 2011; Reichert et al. 2011; Kettula et al. 2013) have reported slopes in the range 1.45 ≲ α ≲ 1.85. As was the case with the T–L relation, our slope for the combined sample is slightly higher than this range, but the group and cluster samples, when fit independently, are both in agreement with these studies.

While comparing our scaling relationships to others is worthwhile, we caution that there are a handful of issues that make direct comparison problematic. As mentioned earlier in this discussion, other works used different radii within which to measure cluster properties. Our choice of r₂₅₀₀ was motivated by the requirements imposed from our weak-lensing masses, but it means we are analyzing X-ray properties in different apertures from other studies.

Another issue that arises when comparing to other studies is the definition of luminosity. In this work, we used bolometric luminosities. However, in the three works that looked at groups that we discuss in this section (Rykoff et al. 2008; Hoekstra et al. 2011; Eckmiller et al. 2011), all fit scaling relations with a luminosity only within the energy band of 0.1–2.4 keV. The importance of energy bands was shown by Markevitch (1998), who found that when switching from luminosities within 0.1–2.4 keV to bolometric luminosities the measured slope of the L–T relation steepened from α = 2.10 ± 0.24 to α = 2.64 ± 0.27. Such a large change in the fit means that we should be careful comparing scaling relations for luminosities derived from different energy bands. As a test of this effect, we fit the M–L and T–L relations using luminosities measured in the 0.1–2.4 keV energy band for the 160SD clusters. The power law indices increased when using the energy limited luminosities from 1.02 ± 0.17 to α = 1.19 ± 0.22 and from 0.293 ± 0.064 to α = 0.334 ± 0.101 for M–L and T–L, respectively. The 160SD sample here has too small a dynamical range to be seriously considered for a scaling relation, but the effect of choosing to fit bolometric luminosities over band-limited luminosities is clear.

Also, while our sample is a subset of a randomly selected survey, it is originally based on X-ray selected clusters. Hicks et al. (2013) suggest that X-ray selection preferentially picks centrally concentrated systems; these systems populate the high L_X side of the T–L relation. Similarly, since our data were drawn from the faint end of a flux-limited sample, we would expect preferentially over-luminous clusters for their mass to be selected.

One of the issues with comparing the difference between the groups examined in this paper and those at low redshift is the ubiquity of masses derived from hydrostatic equilibrium. Hydrostatic masses may somewhat underestimate the mass of galaxy clusters when compared to weak-lensing measurements (Arnaud et al. 2007; Mahdavi et al. 2008, 2013). However, in order to allow a comparison with work on low redshift groups and poor clusters using hydrostatic masses, we make the assumption that both mass estimates are identical.

Looking at Figure 6, we can see that our moderate redshift clusters ($\overline{z} = 0.444$) are almost all hotter and/or less massive than what would be predicted by the lower redshift scaling relations for groups presented in Sun et al. (2009; $\overline{z} = 0.042$), although five low-temperature clusters agree very well. If we test the hypothesis whether our data are fit by the Sun et al. (2009) relationship between mass and temperature, we find a χ² value of 19.09 for 14 clusters (p = 0.089). Therefore, to within 2σ we see no difference between our sample and the low-redshift sample. If we limit this analysis to only those clusters with temperature kT <4 keV, our value of χ² is 3.15 for 9 clusters (p = 0.87). If we do not scale the mass by E(z), χ² becomes 29.53 for 14 clusters (p = 0.0033). This significance is just below 3σ, constituting very weak evidence for the expected self-similar evolution in the temperature-mass relation for groups.

A direct comparison between the CCCP sample and our sample, which affords a comparison between low-mass and high-mass clusters at a similar redshift range, is difficult due to the limited number of clusters in both sets. So to provide some quantification of whether the two populations differ, we utilize Fisher's exact test, which looks at how two properties are distributed in two populations. In this case, we look at how our sample and the CCCP sample compare to the scaling relations. We choose to use Fisher's exact test due to how few objects we have; in this domain, Fisher's exact test is the best, if not the only, test to use (Wall & Jenkins 2012).

For both samples, we count how many clusters lie above the lines of best fit for each scaling relation and how many lie below. Our null hypothesis is that the samples are similar and so the number of clusters above the relation should equal the number below, statistically. We compute p-values for the T–L, M–L, and M–T relations of p = 0.13, p = 0.19, and p = 0.19, respectively. We therefore cannot reject the hypothesis that groups and clusters at intermediate redshift behave identically with respect to the scaling relations derived in this work, so our measurements are consistent with the hypothesis that z ∼ 0.3–0.5 X-ray selected clusters and groups/poor clusters follow similar X-ray scaling laws.

Bruch et al. (2010) first analyzed this cluster with the same observation used in this paper. While their analysis followed a similar path to our own, their reported results are not the same as ours. Our reported bolometric luminosity is similar to theirs ($2.19_{-0.14}^{+0.12}$ and $2.08^{+0.22}_{-0.22}$ × 10⁴³ erg s⁻¹, respectively), but their reported temperature is noticeably lower than our own ($1.46^{+0.26}_{-0.19}$ keV compared to $2.95_{-0.62}^{+0.72}$ keV). The difference in the result may arise from their less stringent cut for selecting good time intervals, their grouping of their data into energy bins, their use of a smaller aperture, and their lack of pn observations, which supply around 50% of the counts but were often problematic to calibrate 5 yr ago. If we also make these choices, we measure a new temperature of $1.27_{-0.11}^{+0.06}$ keV, which agrees with the earlier result.

However, when we reduce our aperture size and bin the spectral data, we find an even lower luminosity; our new bolometric luminosity is $0.79_{-0.05}^{+0.04} \times$ 10⁴³ erg s⁻¹. After private communication with S. Bruch (2008), we discovered that the same spectral fitting results were obtained but not published for an aperture of 0.5 Mpc. Using the 4.647 kpc arcsec⁻¹ scale provided in the refereed paper, we extract spectra from a 10760 aperture. When letting the abundance vary, we find ${T}_{\rm X} = 1.50_{-0.32}^{+0.45}$ keV and ${L}_\mathrm{bolo} = 1.83_{-0.19}^{+0.10} \times$ 10⁴³ erg s⁻¹. In addition, we find 252 and 219 net counts for MOS1 and MOS2, respectively. These results are in agreement with the earlier result, which found 231 and 205 counts for the two cameras. We therefore conclude that their reported X-ray aperture radius of 32'' is incorrectly reported, and the actual aperture used was 0.5 Mpc. Using this aperture, we obtain similar results.

Lumb et al. (2004) originally looked at RXJ0847.1+3449 using XMM-Newton observation 0107860501. They reported higher values for flux and bolometric luminosity, but a cooler temperature. One source of this difference may be the larger spectral extraction area they used—it was 120'', while ours was ≈70''. Therefore, we attempted to reproduce their results by using the same aperture and masks, as that work included images of where point sources were excluded.

Bolometric luminosities reported by Lumb et al. (2004) are not for the 120'' apertures. Rather, they are for apertures scaled to the entire virial radius, as found by using the fitted temperatures and the T–r_v relation of Evrard et al. (1996). In addition, they increased the estimated photon count rate to account for lack of spatial coverage due to chip gaps or masked point sources. We find a comparable luminosity by fitting a MEKAL model to the parameters specified in Table 5 of Lumb et al. (2004). Unlike the reported luminosity, these parameters are for the best fit of the spectrum within 120'' and are the best measure of what a similar aperture luminosity would be from that work. In order to allow for changes in MEKAL over the past ten years, we let the abundance vary but match the flux reported in the original work.

When fitting to data from the larger aperture, our temperature estimate changes from $4.16_{-0.39}^{+0.58}$ keV to $3.72_{-0.41}^{+0.51}$ keV, which agrees with the reported value of $3.62^{+0.58}_{-0.51}$ keV. Similarly, our flux estimate changes from $5.20_{-0.14}^{+0.12} \times$ 10⁻¹⁴ erg s⁻¹ cm⁻² to $6.77_{-0.12}^{+0.14} \times$ 10⁻¹⁴ erg s⁻¹ cm⁻², in agreement with the predicted 7.04 ± 0.3 × 10⁻¹⁴ erg s⁻¹ cm⁻². For bolometric luminosity, our value within r₂₅₀₀ is $1.31_{-0.03 }^{+0.04 } \times 10^{44}\,h^{-2}_{70} \,\mathrm{erg} \, \mathrm{s}^{-1}$, while inside a 120'' aperture it is $1.70_{-0.05}^{+0.06}\times 10^{44}\,h^{-2}_{70}$ erg s⁻¹. The expected luminosity inside that aperture is $1.75\times 10^{44}\,h^{-2}_{70}$ erg s⁻¹.

RXJ1354.2−0221 was also originally investigated by Lumb et al. (2004), and, as before, they find a higher flux, higher luminosity, and a lower temperature than we do. As with RXJ0847.1+3449, their technique deviated in aperture size, binning, and definition of luminosity. Additionally, we filtered this data for intervals of flaring differently than they did, which we adjust for in our reanalysis.

We again find a drop in temperature, which changes from $7.60_{-1.22}^{+1.92}$ keV to $3.88_{-0.59}^{+0.93}$ keV when expanding the aperture, in comparison to the originally reported value of $3.66^{+0.6}_{-0.5}$ keV. Likewise, the flux increases from $6.90_{-0.19}^{+0.15} \times$ 10⁻¹⁴ erg s⁻¹ cm⁻² to $10.17_{-0.22}^{+0.18} \times$ 10⁻¹⁴ erg s⁻¹ cm⁻², which matches the earlier result of 9.8 ± 0.5 × 10⁻¹⁴ erg s⁻¹ cm⁻². Finally, our luminosity rises from $2.11_{-0.12}^{+0.10} \times 10^{44} \,h^{-2}_{70}$ erg s⁻¹ to $2.47_{-0.06}^{+0.09}\times 10^{44} \,h^{-2}_{70}$ erg s⁻¹, which agrees with the predicted expectation of $2.41\times 10^{44} \,h^{-2}_{70}$ erg s⁻¹. As before, we are able to reproduce the earlier results.

Carrasco et al. (2007) used the same observations analyzed here to look at RXJ1117.4+0743. Their reported temperature ($3.3^{+0.7}_{-0.6}$ keV) is slightly lower than our own ($4.30_{-0.38}^{+0.70}$ keV), but they find larger luminosities from 0.5–2.0 keV (4.19 ± 0.35 to our $1.84_{-0.03}^{+0.03}$, in units of 10⁴³ erg s⁻¹) and in a bolometric band (11.8 ± 0.9 to our $5.27_{-0.16}^{+0.08}$ in units of 10⁴³ erg s⁻¹). There are a few differences in our analysis that can bring those results into closer alignment. Along with using a larger aperture—66'' to our choice of 47''—the previous work binned its data to a minimum of 12 counts per energy bin. Making those adjustments is not enough to match the previous work, however, without also using a different background. In the initial paper, the background was described only as "a larger extraction region near the detector border without any visible sources." To that end, we used a background centered around α₂₀₀₀ = 11^h17^m40^s, δ₂₀₀₀ = +07°55^m10^s that was 72'' in size. With this background, we recover similar results to the original reporting: ${T}_{\rm X} = 3.13_{-0.29}^{+0.30}$ keV, ${F}_{[0.5\hbox{--}2.0 \,\mathrm{keV}]} = 5.29_{-0.13}^{+0.12}\times 10^{-14}$ erg s⁻¹ cm⁻², ${L}_{[0.5\hbox{--}2.0 \,\mathrm{keV}]}= 3.90_{-0.14}^{+0.18}\times 10^{43}$ erg s⁻¹, and ${L}_{\mathrm{bolo}} = 8.90_{-0.37}^{+0.47} \times 10^{43}$ erg s⁻¹. Even without knowing their exact background region, we reproduce the results of Carrasco et al. (2007).