DETECTION OF IRON Kα EMISSION FROM A COMPLETE SAMPLE OF SUBMILLIMETER GALAXIES

Our SMG sample consists of 38 of the 41 significant detections in the 1.2 mm map (Lindner et al. 2011) of the LHN made using the Max Planck Millimeter Bolometer (MAMBO; Kreysa et al. 1998) array on the Institut de Radioastronomie Millimétrique 30 m telescope. We exclude one source that lacks a plausible 20 cm radio counterpart (L20), one that has a likely X-ray counterpart (L26), and one nearby galaxy at z = 0.044 (L29) from the stacking sample. Of our final sample of 38 galaxies, 37 (97%) have robust 20 cm radio counterparts with a chance of spurious association (P; Downes et al. 1986) of P < 0.05; the remaining galaxy, L32, has P = 0.056. Stacking is performed with the coordinates of the SMGs' radio counterparts, which have a mean offset of 24 with respect to the SMG centroids. Five of our stacking targets have positions that are not listed in the 20 cm catalog of Owen & Morrison (2008) because they had S/N < 5.0 (L9, L28, and L36), or they were blended together with nearby radio sources (L17 and L39) during extraction (Owen & Morrison 2008; Lindner et al. 2011). The sample has a mean redshift of 〈z〉 = 2.6 (see Tables 1 and 2).

Our X-ray data are from the 3 × 3 pointing raster mosaic of the LHN obtained with the Advanced CCD Imaging Spectrometer (ACIS-I; Weisskopf et al. 1996) on the Chandra X-ray telescope by Polletta et al. (2006). The final mosaic comprises nine 70 ks pointings arranged with ∼2' overlap (see Figure 1). It covers a total area of ≃ 0.7 deg² and has a limiting conventional broadband (B_C; 0.5–8.0 keV) sensitivity of ∼4 × 10⁻¹⁶ erg s⁻¹ cm⁻² (Polletta et al. 2006). Fiolet et al. (2009) used these same data to search for a stacked X-ray signal among 33 Spitzer 24 μm selected starburst galaxies and found no significant 0.3–8 keV band emission.

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Within the sample of 41 MAMBO detections in the LHN, only L26 has a likely X-ray counterpart (CXOSWJ104523.6+585601) in the catalog of Wilkes et al. (2009). This X-ray source has conventional broad band (B_C; 0.5–8.0 keV), soft band (S_C; 0.5–2.0 keV), and hard band (H_C; 2.0–8.0 keV) X-ray fluxes of $f_{B_C}=2.53\times 10^{-15}\,\rm erg\,s^{-1}\,cm^{-2}$, $f_{S_C}=1.21\times 10^{-15}\,\rm erg\,s^{-1}\,cm^{-2}$, and $f_{H_C}=1.57\times 10^{-15}\rm \, erg\,s^{-1}\,cm^{-2}$, respectively, and a hardness ratio of HR = −0.32^+0.31_{− 0.34}. The hardness ratio is defined by HR = (H_C − S_C)/(H_C + S_C), where H_C and S_c are the counts in the Chandra conventional hard and soft bands, respectively.

We generate 1'' × 1'' pixel gridded maps of the total counts and effective exposure time in the B_C, S_C, and H_C energy bands using the Chandra Interactive Analysis of Observations (CIAO; Fruscione et al. 2006) script fluximage. The characteristic energies input to fluximage to compute effective areas were 4.00 keV, 1.25 keV, and 5.00 keV for the B_C, S_C, and H_C bands, respectively. We then used the CIAO scripts reproject_image to merge the maps of each observation into one mosaic, and dmimgcalc to produce an exposure-corrected flux image in units of photons  cm⁻² s⁻¹.

Figure 2 shows the resulting stacked image in each energy band. The 40'' × 40'' S/N postage stamp images are shown with a color stretch from S/N = −4 to +4. The peak signal-to-noise ratio (S/N) is 3.2, 4.8, and 2.0 in the B_C, S_C, and H_C bands, respectively. The peak of the strong stacked detection in the S_C band has an offset from the mean radio counterpart centroid position of ≲ 1''.

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Our second stacking technique does not use a binned X-ray map. Instead, we compute the stacked count rate and flux by directly counting photons at the stacking positions. The photometric aperture at each stacking location is derived using a technique similar to the optimized stacking algorithm presented in Treister et al. (2011, supplementary information).

The size and shape of the aperture at each stacking position is chosen to maximize the point-source S/N at that position on the ACIS-I chips. The apertures are constructed as follows. For each stacking position (shown in Figure 1), we (1) use the CIAO script mkpsf to generate a two-dimensional image of the local Chandra point spread function (PSF), (2) convolve this PSF with a Gaussian smoothing kernel (see below), and (3) find the enclosed-energy fraction (EEF) contour C_EEF that maximizes the S/N of the flux within the aperture. The area enclosed by this contour defines the aperture. Because Poisson noise from the X-ray background is stronger than the flux at each position and the average exposure time does not change rapidly with increasing aperture size, the S/N within the aperture can be parameterized by C_EEF as

$$\begin{equation} {\rm S/N} \propto \frac{\rm EEF}{\sqrt{A(C_{\rm EEF})}}, \end{equation} \tag{ 1 }$$

where A(C_EEF) is the total area enclosed by the contour C_EEF. This expression is the same as that derived by Treister et al. (2011), except that instead of using only circular apertures, we allow for non-axisymmetric apertures that follow the local shape of the Chandra PSF.

We compensate for the change in shape of the Chandra PSF with photon energy by generating two optimal apertures at each stacking position, one for each energy band (i.e., for characteristic energies of 1.25 keV and 5 keV). Parts of the LHN Chandra mosaic were imaged multiple times due to the overlapping edges of the individual exposures (see Figure 1). For these positions, we find the total effective aperture by maximizing Equation (1) using a linear combination of PSFs, one for each overlapping observation.

We broaden the local Chandra PSFs to accommodate photons that do not lie at the stacking centers due to intrinsic wavelength offsets in the galaxies and astrometric errors. Previous X-ray stacking analyses find the optimal aperture radius to be σ = 125–30 (Lehmer et al. 2005; Georgantopoulos et al. 2011). We find similarly that our broadband stacked S/N is maximized with a smoothing kernel radius σ_kernel = 13 (see Figure 3), so we adopt this value for our subsequent broadband photometry. The smallest angular separation of any pair of stacking targets (15'' for L17 and L39) is larger than the maximum radial extent of the largest X-ray stacking aperture, so we can ignore the effects of X-ray blending within our sample.

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The photons used for background subtraction are collected from arrays of large circular apertures positioned next to each stacking-target position. The apertures are manually positioned to exclude any bright nearby X-ray sources that could contaminate the background estimate. To avoid possible systematic uncertainties associated with the background subtraction (see, e.g., Treister et al. 2011; Willott 2011), we do not impose any S/N-based clipping or additional filtering in the background regions.

Table 3 shows the average stacked count rate and energy flux per galaxy in the three broad energy bands. We find a significant stacked detection in the soft band and no significant detection in the hard band. To convert the stacked count rate into energy flux, we used the web-based CIAO Portable Interactive Multi-Mission Simulator⁵ (PIMMS- version 4.4; Cycle 5). The fluxes are corrected for Galactic absorption using the column density in the direction of the LHN center, N_H = 6.6 × 10¹⁹ cm⁻² (Stark et al. 1992). Our non-detection in the hard band leaves our calculation of the hardness ratio relatively unconstrained, HR = −0.68^+0.51_{− 0.32} (setting a limit on the photon index Γ > 1.2), although it is clear that our sample has a steeply declining photon spectrum characteristic of star-forming galaxies (e.g., Ranalli et al. 2012). This estimate of HR was made after subtracting out the count rate in the soft band that is due to the strong Fe Kα line (see Section 3.3), which is a ∼20% contribution for 13 broadened photometric apertures.

A high photon index of Γ = 1.6 (HR ≃ -0.37) is found by Laird et al. (2010), who stack on SCUBA-detected SMGs in the Chandra Deep Field North (CDF-N). An even steeper photon index of Γ = 1.9 (HR ≃ -0.49) is measured by Georgantopoulos et al. (2011), who stack on LABOCA-detected SMGs in the Extended Chandra Deep Field South (ECDF-S). We have used values of Γ = 1.6 and Γ = 1.9 to compute the stacked flux in each energy band (e.g., see Table 3), although the difference between the two estimates is less than the Poisson uncertainty (see Table 3).

In addition to stacked broadband fluxes, we also calculate the observed-frame and rest-frame stacked count-rate spectra for our sample.

For each stacking target, we use redshift information in the following order of priority, subject to availability: (1) spectroscopic (Polletta et al. 2006; Owen & Morrison 2009; Fiolet et al. 2010), (2) Herschel-based photometric (Magdis et al. 2010; Roseboom et al. 2012), (3) AA-quality optical-based photometric (Strazzullo et al. 2010), and (4) millimeter/radio photometric estimated using the Carilli & Yun (1999) spectral index $\alpha ^{20\,\rm cm}_{850\rm \,\mu m}$ technique (Lindner et al. 2011). The redshift distribution of our sample is shown in Figure 4.

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We use a flat sum of the observed counts at each stacking-target position with no weighting factors. Although this technique gives more weight to the brightest members of the stack, it is necessary given that none of our stacking targets are individually detected, and therefore S/N-based weights (used in, e.g., Treister et al. 2011) cannot be reliably assigned. For the rest-frame data, we separately co-add, blueshift, and bin the background photons to avoid creating artificial spectral features (see, e.g., Yaqoob 2006).

The uncertainty in the rest-frame energy of the photons ΔE_rest as a function of the observed photon energies E_obs due to the typical redshift error Δz is estimated by the equation

$$\begin{equation} \Delta E_{\rm rest} = E_{\rm obs}\,\frac{\langle \Delta z \rangle }{1 + \langle z \rangle }. \end{equation} \tag{ 2 }$$

This uncertainty is always larger than the energy resolution of the ACIS-I chips, so we set the rest-frame energy bin widths to match ΔE_rest (Equation (2)) using our sample's average redshift 〈z〉 = 2.6 and redshift uncertainty 〈Δz〉 = 0.4 (see Table 2).

Figures 5 and 6 show the net observed and rest-frame count-rate spectra for our SMG sample, respectively. The rest-frame spectrum contains a 4σ emission feature with a centroid near 6.7 keV, which we attribute to Fe Kα line emission from a mixture of Fe ionization states including Fe xxv (see Section 5.1.3). It is apparent from this rest-frame spectrum that a significant fraction of the observed soft-band flux is due to the Fe Kα emission line. If strong unresolved Fe Kα emission is a common feature in the X-ray spectra of other SMG samples, then it may artificially lower their measured HR values by inflating their observed soft-band fluxes.

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To ensure our stacking signal is not the result of contamination from a few strong targets, we performed a bootstrapping Monte Carlo analysis to recover the probability distribution for the single 6.7 keV energy bin (bin_6.7). The mean number of on-target counts in bin_6.7 is 16, while the mean number of background counts in the bin is 8. Figure 7 shows that the resulting distribution closely matches that of an ideal Poisson distribution with a mean of 16, confirming that our stacking signal is characteristic of the entire sample, not a few outliers.

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The mean stacked rest-frame hard-band X-ray luminosity, $L_{H_C}$, of our sample is given by

$$\begin{equation} \left< L_{H_C}\right> = 4\pi f_{H_C}\left< d_L^2\right>, \end{equation} \tag{ 3 }$$

where $f_{H_C}$ is the stacked energy flux per galaxy inside the observed 0.56–2.22 keV energy band (the 2.0–8 keV energy band, redshifted by 〈z〉 = 2.6) and d_L is the luminosity distance at the redshift of each stacking target. We convert the observed count rate to an energy flux using PIMMS. The resulting rest-frame X-ray luminosity is $\langle L_{H_C}\rangle = (3.0\pm 1.1) \times 10^{42}\,\rm erg\,s^{-1}$.

To estimate the equivalent width and line flux of the Fe Kα feature, we assume that the line emission is contained only within the single elevated bin at 6.7 keV (see Figure 6) and estimate the local count-rate continuum around the Fe Kα feature by averaging together the eight bins between 3–9 keV (excluding the bin containing the line). This results in an equivalent width of $\rm EW=3.9\pm 2.5\,\rm keV$. Although the EW is relatively unconstrained, it is >1 keV with 90% confidence. Using the nominal equivalent width and Equation (A7), we find a mean stacked Fe Kα line flux of <f_Fe Kα > ≃2.1 × 10⁻¹⁷ erg cm⁻² s⁻¹ and a mean line luminosity <L_Fe Kα > = (1.3 ± 0.4) × 10⁴² erg s⁻¹.

Figure 3 also shows that the S/N of the Fe Kα signal only drops at a larger radius than the broadband signal. If we interpret the X-ray continuum as originating from the galaxies' nuclear regions, then this relative offset between the Fe Kα emission and the X-ray continuum indicates that the Fe Kα photons in our sample are systematically offset from the galaxies' centers. By measuring the distance between the peaks of the two curves in Figure 3, we estimate the radial offset to be ∼1''. For our computation of the Fe Kα line luminosity, we adopt an aperture broadening kernel suited to maximize the S/N of the Fe Kα emission line, σ_kernel = 24.

We use a two-sample Kolmogorov–Smirnov (K-S) test to determine the significance of the apparent angular extension of the Fe Kα emission relative to the continuum emission. First, we compute the Chandra PSF at the location of each stacking target, then sample the PSFs at the positions of their respective collections of optimally selected photons (see Section 3.3). The PSFs are peak normalized and smoothed by an amount σ_smooth to reflect intrinsic wavelength offsets and astrometric errors in the photon positions. We then compare the cumulative distributions of the observed soft-band (0.5–2.0 keV) continuum photons (excluding those in the rest-frame Fe Kα bin) and the rest-frame Fe Kα photons using the two-sample K-S test to determine with what confidence 1 − p (p is the K-S test significance) we can rule out the null hypothesis that the two samples are drawn from a common distribution. When using all 38 stacking positions, we find a maximum confidence of 1 − p = 0.71 at σ_smooth = 05 (63 continuum counts and 15 Fe Kα counts). When we use only the stacking positions that have ⩾1 Fe Kα photon, the maximum confidence occurs at the same value of σ_smooth but has a reduced 1 − p = 0.45 (29 continuum counts and 15 Fe Kα counts). Therefore, the extension in the Fe Kα emission relative to the continuum emission indicated by Figure 3 is a ≳ 1σ (71% confidence) effect, whose significance is limited primarily by the small number of Fe Kα photons.

In this section, we compare our results to those of previous X-ray analyses of SMG samples from the CDF-N (Alexander et al. 2005a; Laird et al. 2010) and the (E)CDF-S (Georgantopoulos et al. 2011).

5.1.1. Detection Rate

5.1.2. L_Fe Kα versus L_IR versus L_20 cm

Figure 8 shows our sample's average X-ray (corrected only for Galactic absorption), radio, and IR luminosities compared to those of other stacked SMG samples (Laird et al. 2010; Georgantopoulos et al. 2011), individually X-ray-detected SMGs (Alexander et al. 2005a; Laird et al. 2010; Georgantopoulos et al. 2011), and nearby LIRGs and ULIRGs (sample drawn from Iwasawa et al. 2009). Where available, we use the L_IR value from Table A2 of Pope et al. (2006) for the SMGs from the CDF-N. For the 12 SMGs in Alexander et al. (2005a) that are not in the catalog of Pope et al. (2006), we scale $L_{\rm \rm FIR}\longrightarrow L_{\rm IR}$ using the average conversion factor f for the eight SMGs common between the two samples, f = 1.42. The 870 μm detected SMGs from the (E)CDF-S are plotted with L_IR = 10–1000 μm. The local LIRGs and ULIRGs from Iwasawa et al. (2009) also have their L_FIR(40–400 μm) estimates scaled by f = 1.42. We also show the total sample luminosity average for Laird et al. (2010), including the contribution from their stacked SMGs that were not individually detected in the X-ray. The average properties of our stacking sample are in agreement with the total luminosities of Laird et al. (2010).

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Figures 8 and 9 also indicate the AGN classification of each galaxy. Galaxies whose mid-IR or X-ray spectral properties are consistent with emission produced entirely by star formation are plotted in red, while those requiring the presence of an AGN are shown in blue. Georgantopoulos et al. (2011) divide their sample by using a probabilistic approach; those galaxies requiring the presence of a torus-dust component in their mid-IR SED according to an F-test are categorized as AGNs. Laird et al. (2010) and Alexander et al. (2005a) separate out the AGN based on the most favored model of their X-ray spectra according to the Cash (1979) statistic. The sample of Iwasawa et al. (2009) is divided based on hardness ratio.

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The division between AGN and non-AGN systems can be roughly determined based on the X-ray scaling relations of purely star-forming galaxies in the local universe (Ranalli et al. 2003; Vattakunnel et al. 2012) shown as solid black lines. The average properties of our stacking sample lie very near the Ranalli et al. (2003) relation. Considering the substantial intrinsic scatter in the spectral classifications of the Laird et al. (2010) sample, our stacking sample also probably contains a substantial fraction of both star-formation-only and AGN-required systems.

5.1.3. Fe Kα Emission Properties

The Fe Kα photons in our stacking sample may be more spatially extended than the continuum photons by ∼1'' (see Section 3.4). Extended and misaligned Fe Kα emission has been observed in Arp 220 (Iwasawa et al. 2005) and NGC 1068 (Young et al. 2001). We may also be blending together the emission from multiple components of merging systems of which only one component has strong Fe Kα emission (like, e.g., Arp 299; Ballo et al. 2004).

The bin width in our stacked rest-frame X-ray spectrum, which is set by the redshift uncertainties of our SMG sample, is larger than the rest-energy separation between Fe Kα emission from neutral and highly ionized iron (0.3 keV); therefore, it is difficult to determine the average Fe ionization fraction in our sample. Close inspection of the photons near the rest-frame Fe Kα line (see Figure 10) reveals a range of values between 6.4 keV and 7.2 keV, with a local maximum at 6.7 keV. Given that the Fe line photons are contributed fairly evenly by the 38 targets in our stacking sample and have been assigned to their bins based on a wide variety of redshift estimation techniques (spectroscopic, optical-photometric, Herschel-photometric, and millimeter/radio-photometric), they are unlikely to all be systematically biased high or low. Therefore, a significant fraction of the detected Fe Kα photons likely originate from the highly ionized species of Fe xxv or Fe xxvi. However, the ∼10% rest-frame uncertainty in the energy of each photon implies an uncertainty in the centroid of the line profile of $\sigma _{\rm centroid}= \rm (FWHM)/({\rm S/N}) \simeq 398\,\rm eV$, insufficient to determine the relative fractions of each ionization state with certainty.

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Strong emission (EW = 1.8 ± 0.9) from highly ionized Fe Kα has been observed in the nearby ULIRG Arp 220 by Iwasawa et al. (2005) using XMM-Newton. Iwasawa et al. (2009) also find strong 6.7 keV emission ($\rm EW=0.9\pm 0.3\,\rm keV$) from the stacked spectrum of nearby ULIRGs (including Arp 220) that have no evidence of AGN emission (termed X-ray-quiet ULIRGs). Alexander et al. (2005a) detected strong ($\rm EW\simeq 1\,\rm keV$) Fe K α emission in the stacked SMG spectrum of the six SMGs in their sample with N_H > 5 × 10²³ cm⁻² and find that the line centroid is between 6.7 keV and 6.4 keV, indicating a substantial contribution from highly ionized gas.

In Figure 11, we compare the relation between L_Kα and L_IR in our sample with those for other individual systems and stacked samples with measured Fe Kα line luminosities and bolometrically dominant energy sources that are well understood. The dashed line represents a linear slope between L_Kα and L_IR and has been normalized to NGC 1068, a nearby prototypical Seyfert II LIRG. Red symbols represent systems that do not have significant observed AGN bolometric contributions, like SMGs and local X-ray-quiet ULIRGs; the blue symbols represent systems that have significant bolometric AGN contributions. Figure 11 shows that the relative Fe Kα/infrared luminosity fraction, L_Kα/L_IR, increases with increasing L_IR. If the Fe Kα emission is due to AGN activity, then this result may be in agreement with the observed trend that LIRGs/ULIRGs tend to be increasingly AGN-dominated with increasing L_IR (e.g., Tran et al. 2001).

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This section discusses three possible physical origins for the Fe Kα emission detected in our stacked SMG sample: supernovae, galactic-scale winds, and AGNs. Because a significant fraction of our sample's Fe Kα emission likely originates from the highly ionized species Fe xxv (see, e.g., Figure 10) and because evidence for highly ionized Fe Kα emission from other (U)LIRGs exists at both high (Alexander et al. 2005a) and low (e.g., Iwasawa et al. 2005) redshifts, the following sections focus on the origin of this high-ionization component.

5.2.1. Supernovae

5.2.2. Galactic-scale Winds

As discussed in Iwasawa et al. (2005), who consider the 6.7 keV emission line in Arp 220, a starburst-driven galactic-scale superwind of hot gas is energetically plausible as the source of the Fe Kα emission. Large outflows could also explain why the Fe Kα line emission appears more extended than the X-ray continuum emission in our stacking sample. To explore this scenario, we used the X-ray spectral-fitting package XSPEC (Arnaud 1996) to model an absorbed diffuse thermal X-ray (zphabs * mekal) spectrum and estimate the gas metallicity required to produce the strong high-ionization Fe Kα emission detected in our SMG sample. We computed the model EW values using the spectral window 6.35–7.05 keV, the same energy width as the bin containing the Fe Kα emission in our stacked rest-frame spectrum (Figure 6), which includes all Fe Kα ionization states. We fixed the gas temperature to Arp 220's best-fit value kT = 7.4 keV (Iwasawa et al. 2005), the gas density to n = 1 cm⁻³, the redshift to our sample's average 〈z〉 = 2.6, and the obscuring hydrogen column density to 2.3 × 10²³ cm⁻² (Section 4). Both the Fe Kα line luminosity and the continuum intensity vary linearly with Z, allowing us to express the relation between EW and Z as

$$\begin{equation} {\rm EW} = \frac{6.67}{1+\frac{Z^{\prime }}{Z}}\,\rm keV, \end{equation} \tag{ 8 }$$

where Z' = 5.29 Z_☉. EW is approximately proportional to Z for Z ≪ Z' and approaches the constant value 6.67 keV for Z ≫ Z'. Because of this nonlinear behavior, an abundance of 0.94 Z_☉ can produce $\rm EW= \,\rm 1{\rm }keV$ (90% confidence lower-limit) while a significantly greater abundance Z ≃ 7.5 Z_☉ is needed to explain our nominal value $\rm EW\simeq 3.9\,\rm keV$. If a significant amount of our rest-frame 2–10 keV luminosity is from X-ray binaries, incapable of generating the observed line emission, then the required metallicity would be even higher. While the lower limit on our measured EW can be explained by thermal emission from a diffuse ionized plasma, especially considering the extreme enrichment taking place in systems like SMGs, generating an EW with a value close to our nominal measurement would require an unrealistic degree of high-z enrichment.

5.2.3. AGN Activity

We begin by labeling all the photons within the optimized apertures (see Section 3.1) of all of the N = 38 stacking targets with the index i, and those within the background regions for the N targets i'. E_i is the energy of the ith photon and T_i is the total effective exposure time in the mosaic at the position of i. The notation i ∈ j refers to all the photons that have energies located in the jth energy bin. The stacked mean count rate in the jth energy bin, R_j, is then

$$\begin{equation} R_j = \frac{1}{N} \sum _{i \in j} \frac{1}{T_i\,\kappa _i}, \end{equation} \tag{ A1 }$$

where κ_i is the aperture correction for the optimal aperture of the energy band of the ith photon. The background mean count rate in the jth bin is

$$\begin{equation} R^{\prime }_j = \frac{1}{N} \sum _{i^{\prime } \in j} \frac{1}{T_{i^{\prime }}\,\kappa _{i^{\prime }}\,c_{i^{\prime }}}, \end{equation} \tag{ A2 }$$

where $c_{i^{\prime }}$ is the ratio of the areas of the background region of the stacking position of the i'th photon and of the optimal aperture of that stacking position. It follows that the expected number of background counts in the jth bin, $\tilde{N}_j$, is

$$\begin{equation} \eta _j=\left< T_i\right>_{i \in j} \times R^{\prime }_j. \end{equation} \tag{ A3 }$$

We use the double-sided 68% confidence upper and lower limits, η^high_j, and η^low_j (Gehrels 1986), to compute the 1σ count-rate deviations in the jth bin due to the background, σ^hi/low:

$$\begin{equation} \sigma {j}^{\rm hi/low} = \frac{\big|\eta _j^{\rm hi/low} - \eta _j \big|}{\left< T_{i\in j}\right>_i }. \end{equation} \tag{ A4 }$$

Therefore, the net count-rate density per galaxy in the jth bin, $\mathbf {R_{j}}$, is

$$\begin{equation} \mathbf {R_j} = \frac{(R_j - \tilde{R_j})}{\Delta E_j} ^{+\sigma (R)_{j}^{\rm hi}/\Delta E_j}_ {-\sigma (R)_{j}^{\rm low}/\Delta E_j} \,\,\, [\,\rm s^{-1}\,keV^{-1}\,], \end{equation} \tag{ A5 }$$

where ΔE_j is the width of the jth energy bin. We calculate the corresponding rest-frame spectrum $\mathbf {R^{\rm rest}_j}$ by binning the photons according to their rest-frame energies, E^rest_i = E_i (1 + z_i), where z_i is the redshift of the stacking target associated with the photon.

We use the CIAO script eff2evt to tabulate the local effective area A_i ($A_{i^{\prime }}$), and quantum efficiency Q_i ($Q_{i^{\prime }}$), for each photon in the on-target (background) apertures.

The mean stacked on-target and background photon fluxes in the jth bin, F_j, and F'_j, are then

$$\begin{equation} F_j = \frac{1}{N} \sum _{i \in j} \frac{1}{T_i\,A_i\,Q_i\,\kappa _i}, \end{equation} \tag{ A6 }$$

and

$$\begin{equation} F^{\prime }_j = \frac{1}{N} \sum _{i^{\prime } \in j} \frac{1}{T_{i^{\prime }}\,A_{i^{\prime }}\,Q_{i^{\prime }}\,\kappa _{i^{\prime }}\,c_{i^{\prime }}}, \end{equation} \tag{ A7 }$$

respectively.