TESTING X-RAY MEASUREMENTS OF GALAXY CLUSTER OUTSKIRTS WITH COSMOLOGICAL SIMULATIONS

We use a sample of high-resolution cosmological simulations of cluster-sized systems from Nagai et al. (2007a, 2007b, hereafter N07) that assumes a flat ΛCDM universe with $\Omega _m=1-\Omega _\Lambda =0.3$, Ω_b = 0.04286, h = 0.7, and σ₈ = 0.9. The Hubble constant is defined as 100h km s⁻¹ Mpc⁻¹, and σ₈ is the mass variance within spheres of radius 8 h⁻¹ Mpc. We simulate these clusters with the Adaptive Refinement Tree N-body+gas-dynamics code (Kravtsov 1999; Kravtsov et al. 2002; Rudd et al. 2008), an Eulerian code with spatial and temporal adaptive refinement, and non-adaptive mass refinement (Klypin et al. 2001).

Our simulations include radiative cooling, star formation, metal enrichment, stellar feedback, and energy feedback from supermassive black holes (see C. Avestruz et al., in preparation for more details). Briefly, we seed black hole particles with a seed mass of 10⁵ h⁻¹M_☉ in dark matter halos with M_500c > 2 × 10¹¹h⁻¹M_☉, and allow the black holes to accrete gas according to a modified Bondi accretion model with a density dependent boost factor (Booth & Schaye 2009). Black holes feed back a fraction of the accreted rest mass energy in the form of thermal energy,

$$\begin{equation} \Delta E_{{\rm fb}}=\epsilon _r\epsilon _f\Delta M_{{\rm BH}}, \end{equation} \tag{ 1 }$$

where _r = 0.1 is the radiative efficiency for a Schwarzschild black hole undergoing radiatively efficient Shakura & Sunyaev accretion (Shakura & Sunyaev 1973), and _f = 0.2 is the feedback efficiency parameter tuned to regulate the z = 0 black hole mass.

We focus on two X-ray-luminous clusters from the N07 sample to study the effects of metallicity, line of sight, and dynamical state on X-ray measurements in cluster outskirts. The core-excised X-ray temperatures of CL6 and CL7 are T_X = 4.18 and T_X = 4.11, respectively. The cluster mass and radii are M_500c = 1.43 × 10¹⁴ h⁻¹M_☉ and R_500c = 626.5 h⁻¹ kpc for CL6, and M_500c = 1.37 × 10¹⁴ h⁻¹M_☉ and R_500c = 618.6 h⁻¹ kpc for CL7. Each cluster is selected from a simulation box with size L_box = 80 h⁻¹ Mpc, run on a 128³ uniform grid with eight levels of refinement, and has peak spatial resolution of ≈2.4 h⁻¹ kpc. The corresponding dark matter particle mass in our regions of interest is 2.7 × 10⁸ h⁻¹M_☉. CL6 is an unrelaxed cluster that experienced a 1:1 major merger at z ≈ 0.6, and CL7 is a typical relaxed cluster which has not undergone a major merger for z < 1. We refer the reader to N07 for further details.

Using our simulated cluster sample, we create realistic mock X-ray maps of simulated clusters by (1) generating an X-ray flux map and (2) convolving the flux map with the Chandra response files to create photon maps. Below we outline our mock Chandra pipeline; details can be found in Nagai et al. (2007b).

2.2.1. X-Ray Flux Maps

We generate X-ray flux maps for each of the simulated clusters along three orthogonal line-of-sight projections. To calculate the flux map, we project X-ray emission from hydrodynamical cells along each line of sight within 3 × R_vir (≈4 × R_200c for the clusters in this work) of the cluster center. The X-ray emission as a function of energy from a single hydrodynamical cell of volume ΔV_i is given as

$$\begin{equation} j_{E,i}= n_{e,i}n_{p,i}\Lambda _{E}(T_i,Z_i,z)\Delta V_i, \end{equation} \tag{ 2 }$$

where n_{e, i}, n_{p, i}, T_i, and Z_i are, respectively, the electron number density, proton number density, temperature, and metallicity in that cell volume. We compute the X-ray emissivity, Λ_E(T_i, Z_i, z), using the MEKAL plasma code (Mewe et al. 1985; Kaastra & Jansen 1993; Liedahl et al. 1995) and the solar abundance table from Anders & Grevesse (1989). We multiply the plasma spectrum with the photoelectric absorption corresponding to a hydrogen column density of N_H = 2 × 10²⁰ cm⁻².

We include results from four different flux maps: (1) Z_sim, (2) Z_0.5,(3) Z_0.05, and (4) Z_0.5T_{X, 1 keV}. We generate the Z_sim flux map from X-ray spectra using all hydrodynamic values directly output from our simulation runs. To isolate the effects of metallicity on our simulated spectra, we use Z_i = 0.5 Z_☉ for all ith cell elements in our Z_0.5 flux maps and a constant Z_i = 0.05 Z_☉ for all ith cell elements in our Z_0.05 flux maps. Current observations do not yet probe metal abundance profiles in cluster outskirts, and often extrapolate the abundance at radii beyond R_500c. Given the typical range of metallicity that observers have found near R_500c (Z ≈ 0.3 Z_☉) (e.g., Matsushita et al. 2007; Leccardi & Molendi 2008; Komiyama et al. 2009; Werner et al. 2013), we choose these two values for Z_i to bracket a potential range of metallicities in the outskirts. The order of magnitude difference probes how X-ray measurements of the ICM and inferred quantities might be affected by an incorrect extrapolation or mis-measurement of both a metal-rich and metal-poor environment.

We remove the effects of a multi-temperature medium by fixing both the metal abundance to Z_i = 0.5 Z_☉, and the temperature to T_X = 1 keV in the Z_0.5T_{X, 1 keV} map, which is a characteristic temperature in the outskirts of a galaxy cluster in our probed mass range.

2.2.2. Mock Chandra Photon Maps and Spectra

To simulate mock Chandra data, we convolve the emission spectrum with the response of the Chandra front-illuminated CCDs ACIS (both the Chandra auxiliary response file (ARF) and redistribution matrix file, RMF) and draw a number of photons at each position and spectral channel from the corresponding Poisson distribution. The photon maps for our mock Chandra data has an exposure time of 2 Msec, comparable to the 2.4 Msec deep Chandra observations of A133. Since we are interested in probing physical causes for biased X-ray measurements, we do not include any simulated backgrounds.

From our mock photon maps, we generate images in the 0.7–2 keV band, and use them to identify and mask out clumps from our analysis. We use the wavelet decomposition algorithm described in Vikhlinin et al. (1998). This procedure is consistent with the small-scale clump removal performed by observers.

Detected clumps are within a limited flux of ∼3 × 10⁻¹⁵ erg s⁻¹ cm⁻² in the 0.7–2 keV energy band, corresponding to a luminosity of ∼1.5 × 10⁴² erg s⁻¹ for clusters at z_obs = 0.06, the observational redshift of our clusters. Any remaining small-scale structures fall below our clump detection limit. The total gas mass in excluded clumps is no more than a few percent of the total enclosed gas mass.

X-ray emission in cluster outskirts also receive azimuthally asymmetric contributions from filamentary structures. Similar to the analysis of X-ray emission in A133 outskirts, after clump removal, we identify filamentary regions as annular sections with an average flux that exceeds the average flux within the entire annulus. We exclude the filamentary contribution from our final analysis. We include a sample image map with identified clumps and filaments in Figure 1.

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From the mock Chandra photon map, we extract a grouped spectrum from concentric annular bins centered on the main X-ray peak. Annular regions are defined by r_out/r_in = 1.25, where r_out and r_in are the outer and inner edges of the annular bin, respectively. The spectra exclude photons from regions corresponding to clumps and filaments. For each spectrum, we perform a single-temperature fit in the 0.1–10 keV band using XSPEC version 12.8 with the MEKAL model plasma code and a C-statistic fit (Cash 1979) to extract the projected X-ray temperature T_{X, proj}(r) and abundance Z_proj(r). We use the same Chandra ARF and RMF files to fit a model spectrum to the mock photon spectra. Temperature, metal abundance, and spectral normalization are the only free parameters in the fit.

2.2.3. Analysis of the ICM

To compare with observed Chandra data, we reconstruct the emission measure profile and repeat the deprojection analysis by Vikhlinin et al. (2006). This analysis reconstructs the spherically averaged density and temperature profiles of the ICM with three-dimensional (3D) analytic models. We also estimate the total cluster mass assuming hydrostatic equilibrium from the reconstructed 3D density and temperature profiles.

To calculate the projected emission measure profile EMM = ∫n_en_pdl, we use two main input ingredients: (1) the azimuthally averaged X-ray surface brightness profile XSB and (2) the projected two-dimensional X-ray temperature T_{X, proj} and metallicity profiles Z_proj described in Section 2.2.2. We can relate the deprojected EMM to the two ingredients as follows,

$$\begin{equation} {\rm XSB}=\int dV\;n_en_p\Lambda (T,Z). \end{equation} \tag{ 3 }$$

We show the emission measure profile for CL7 along the y axis, where filamentary features are most prominent in Figure 2, and the corresponding effect of clump and filament removal from the analysis. The solid blue line corresponds to the emission measure profile that include clumps and filamentary substructure, and dotted line corresponds to the emission measure profile that excludes the substructure. The bottom panel shows the ratio between the profile with substructure to the profile without, and illustrates the extent of the excess emission in cluster outskirts. For this cluster, the excess of the best-fit profile can be as much as a factor of two at R_200c to a factor of three at 3 × R_200c. The error bars correspond to the binned data, and the excess in individual bins are as high as a factor of six in the filamentary regions. Excess emission measure from a substructure is visible at R ≈ 2 kpc.

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We extract the X-ray surface brightness profiles from photons in the 0.7–2 keV band in concentric annuli of r_out/r_in = 1.1, giving us a count rate in each annular bin. We use the projected X-ray temperature and metallicity to calculate the MEKAL model plasma emissivity, Λ(T_{X, proj}(r_proj), Z_proj(r_proj), z). With the relation given by Equation (3), we use interpolated values of the model plasma emissivity to convert the count rate into a projected emission measure.

The deprojection process is fully described in Vikhlinin et al. (2006), and can be summarized as follows. The analytic form of our 3D gas density model is given by,

$$\begin{eqnarray} n_pn_e = & \frac{n^2_0\left(r/r_c\right)^{-\alpha }}{\left(1+r^2/r^2_c\right)^{3\beta -\alpha /2}} \frac{1}{\left(1+r^\gamma /r_s^\gamma \right)^{\epsilon /\gamma }}+\frac{n^2_{02}}{\left(1+r^2/r^2_{c2}\right)^{3\beta _2}}.\nonumber\\ \end{eqnarray} \tag{ 4 }$$

This is a modification of the β-model that allows for separate fitting of the gas density slope at small, intermediate, and large radii.

The analytic form of our 3D temperature profile is a product of two terms that describe key features of observed projected temperature profiles (Vikhlinin et al. 2005),

$$\begin{equation} {T}_\mathrm{3D}(r) = t(r)\times t_{\mathrm{cool}}(r). \end{equation} \tag{ 5 }$$

The first term takes into account the temperature decline at large radii with a broken power law with transition radius, r_t,

$$\begin{equation} t(r) = {T}_0\frac{(r/r_t)^{-a}}{(1+(r/r_t)^b)^{c/b}}, \end{equation} \tag{ 6 }$$

with a and b as free parameters. The second term models the temperature decline in the central region due to radiative cooling,

$$\begin{equation} t_{\mathrm{cool}}=\frac{\left(r/r_{\mathrm{cool}}\right)^{a_{\mathrm{cool}}} +{T}_{\mathrm{min}}/{T}_0}{\left(r/r_{\mathrm{cool}}\right)^{a_{\mathrm{cool}}+1}}. \end{equation} \tag{ 7 }$$

We project the 3D model of the temperature profile, weighting multiple temperature components using the algorithm from Vikhlinin et al. (2006). We then fit the projected model to the projected X-ray temperature from our mock data, fitting in the radial range between 0.1 ⩽ r/R_200c ⩽ 3.

The 3D models for gas density and temperature profiles allow us to estimate the total mass of the cluster, assuming hydrostatic equilibrium,

$$\begin{equation} M_\mathrm{HSE}({<}r)=-\frac{rT_\mathrm{3D}(r)}{\mu m_p G}\left(\frac{d\ln\rho _g}{{d\ln}r} +\frac{{d\ln}T_\mathrm{3D}}{{d\ln}r}\right), \end{equation} \tag{ 8 }$$

where μ ≈ 0.59 is the mean molecular weight for the fully ionized ICM, m_p is the proton mass, and we have set the Boltzmann constant to be unity such that the temperature T_3D(r) is in units of energy.

In this section, we check for consistency in gas density measurements from spectral fitting with the XSPEC fit performed using only two free parameters: projected X-ray temperature and normalization of fit. We use the Z_0.5 and Z_0.05 photon maps for this experiment, and fix the abundance parameter to the true abundance of the map.

Figure 3 shows the best-fit value for the projected X-ray temperature for each map. In the cluster outskirts, the best-fit X-ray temperature for the Z_0.05 maps is systematically higher than that of the Z_0.5 map, and the gas density differs by ∼20%. Note that while the true temperature of the two maps in each radial bin is identical, their measurements differ by up to ∼15%. The discrepancy points to the degeneracies between fitting parameters. While the fitting procedure assumes a single-temperature fit, the true ICM is a multi-temperature medium and thus requires more degrees of freedom to capture the inhomogeneities. Both the metal abundance and the normalization of the fit offer additional degrees of freedom to minimize the residuals, so it is not surprising that statistically significant differences can arise in measured X-ray temperatures if the metal line contributions are different. We plot the spectra of photons from the radial bin with the largest deviation in measured X-ray temperature (R ≈ 2 × R_200c) in Figure 4. Compared to the spectrum from the Z_0.5 photon map (red points), the spectrum from the Z_0.05 map (blue points) has a much lower emission from metal line cooling, and therefore it has a suppressed peak in the energy spectrum. The solid lines show the best-fit model spectrum for each map.

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With a metal abundance parameter fixed to the true value of Z = 0.5 Z_☉, the model spectrum overshoots the peak of the mock photon spectrum at the Fe-L complex between 0.8 and 1.4 keV, and the best-fit X-ray temperature is T_X = 0.81 ± 0.01 keV. If we allow the metal abundance to vary as an additional free parameter, both the best-fit metal abundance (Z = 0.21 ± 0.03 Z_☉) and the X-ray temperature (T_X = 0.76 ± 0.02 keV) are lower and the residual difference between the mock photon counts and the model spectrum improves.

In Figure 5, we show how the systematic bias in metal abundance goes away in the absence of a multi-temperature ICM. The axes show the best-fit metal abundance as a function of radius along three different lines of sight for our Z_0.5T_{X, 1 keV} map. This test isolates biases due to density inhomogeneities, lower photon counts, and instrumental response. For a map with uniform temperature, the fitting procedure well recovers the input constant metallicity and does not exhibit a systematic decline. The best-fit projected X-ray temperature is also well recovered to the fixed input value.

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The suppressed peak in the mock photon spectrum in Figure 4 can be attributed to a high temperature phase of gas in the outskirts. For gas with an average temperature of T_X ≈ 1 keV, the Fe–L line would be sensitive to a higher temperature component of gas, even if the metallicity of that component is the same (e.g., see Figure 3 in Mazzotta et al. (2004) and Figure 23 in Böhringer & Werner 2010).

Note that if the abundance parameter were not fixed (magenta dotted line in Figure 4, the best-fit abundance parameter would have to be lower than the true value Z = 0.5 Z_☉ to better minimize the residuals in the fitting procedure. If we were to increase the X-ray temperature in the model spectrum, the residual at the peak would improve, but at the expense of increasing the residual values at other energies.

On the other hand, for the Z_0.05 spectra in Figure 4, there are fewer photons from metal line cooling. Hence, most of the spectral features are hidden in Poisson noise. Because of the suppressed spectral features, the fitting procedure produces a best-fit X-ray temperature and normalization that will better minimize the residuals near the peak of the spectrum as opposed to those at energies with much fewer counts. To minimize the residuals near the peak, the fitted X-ray temperature must then be higher to harden the model spectrum, which is why the best-fit projected X-ray temperature for the Z_0.05 map is systematically higher in the outskirts, where photon counts are low.

The robustness of metal abundance measurements in the low density outskirts is not well understood. In this section, we compare the abundance measurements from spectral fitting to the true abundance from our photon maps. We also discuss the effects on the projected X-ray temperature, projected emission measure and the reconstructed three dimensional gas density profile. We begin by comparing variations in projected X-ray temperature T_{X, proj} and abundance measurements Z_proj from our Z_0.5 and Z_0.05 photon maps, where we have set the input metal abundance to constant values.

Figure 6 compares ICM profiles from our Z_0.5 and Z_0.05 mock X-ray flux maps of CL7, projected along the z axis. Results along other axis projections are qualitatively similar. The top-left panel shows the metal abundance measurements retrieved from our mock spectra, fitting the mock spectra to MEKAL model spectra using XSPEC. We define $f_{Z_\mathrm{fit}/Z_\mathrm{fix}}$ as the ratio between the best-fit abundance parameter to the true abundance of our photon map. The $f_{Z_\mathrm{fit}/Z_\mathrm{fix}}$ profile shows that the while the XSPEC fitting procedure recovers a Z_fit value that is within a factor of two of the true input abundance of the photon map, the abundance recovery is significantly worse at larger radii.

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For the Z_0.5 map, the abundance measurements have a systematic decline from the true value at large radii R ≳ 0.7R_200c. As discussed in Section 3.1, the metal abundance parameter adjusts to compensate for the multi-temperature medium when we perform a single-temperture spectral fit. Thus, the best-fit abundance parameter will be smaller than the true abundance in order to minimize the residual around the suppressed emission lines. However, if there are too few photons from metal line cooling, the poisson errors become large, and therefore this systematic effect is less apparent since the statistical errors dominate. This is why the last two radial bins do not show underestimation in abundance measurements.

For the Z_0.05 map, the photon counts around the peak of the spectrum (0.7–1 keV) is a factor of three lower than that of the Z_0.5 spectrum (see Figure 4). In addition to the intrinsically less pronounced metal lines, the low photon count makes it more difficult for the fitting routine to discern between contributions from metal line cooling and thermal bremsstrahlung emissions, leading to more scatter in the abundance measurements shown in Figure 6. Note that similar scatter is seen in the last two radial bins from the Z_0.5 map, which also has a very low photon count.

The top right panel of Figure 6 shows the projected X-ray temperature profile (red and blue points for the respective abundances) from spectral fitting, and the corresponding best-fit projected 3D temperature model (thin lines of the same color). We measure the relative bias in T_{X, proj} as the ratio of the projected X-ray temperature measured with abundance as a free parameter (the profile in Figure 6 compared with the X-ray temperature measured with the abundance parameter fixed to the true abundance of the map; see Figure 3). The X-ray temperature measurements from fits with abundance as a free parameter (Z = Z_fit) vary up to ∼10% of the measurements with a fixed abundance parameter (Z = Z_true). Note that the relative bias in projected X-ray temperature from spectral fitting is in the same direction as that of the abundance measurement.

The cause of the correlation in the relative bias of both the abundance and X-ray temperature becomes evident when we compare model spectra (see Figure 7). When the best fit abundance is higher (lower) than the true abundance, the peak of the model spectrum will sit above (below) the mock spectrum, and a higher (lower) X-ray temperature is necessary to harden (soften) the spectrum to minimize the residuals. Note that while fixing the metal abundance parameter to a value that is five times the true value significantly biases the value of the best-fit X-ray temperature parameter, it does not significantly increase the error in the X-ray temperature parameter. The X-ray temperature fit is mostly sensitive to the shape of the continuum, and most of the residual contributions to the error in X-ray temperature come from energy bins outside of the Fe–L complex. Here, the model spectra are still within the error bars of the mock data.

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An interesting feature to note in the projected X-ray temperature profile is that the projected 3D temperature model fits the data points well only out to R ≲ R_200c. While the best-fit model profile is within the error bars of the mock data, it does not capture the emerging trend of the decreasing slope of the projected X-ray temperature at larger radii. The departure from the smooth temperature model profile is indicative of the complicated, inhomogeneous temperature structure in cluster outskirts. The smooth temperature model does not have enough degrees of freedom to account for the bumpier features of the T_{X, proj} data points, indicating that perhaps a modified fitting formula would be appropriate for total mass measurements outside of R_200c.

Note that the error bars at large radii for the metal abundance measurements are not as good as those of the X-ray temperature measurements because the abundance fitting requires an order of magnitude more photons to have the same statistical uncertainty as the X-ray temperature. The metal abundance is only affected by line emission contributions as opposed to the X-ray temperature, which also has contributions from thermal bremsstrahlung.

From Equation (3), we can see that for the same X-ray surface brightness XSB at a given radius, a higher (lower) gas density is necessary to compensate for a lower (higher) X-ray emissivity Λ. The bottom two panels in Figure 6 show the corresponding projected emission measure and gas density for the same photon maps as in the top two panels. The projected emission measure varies to 20% at all radii, and the gas density to a level of 15%.

We note that the apparent enhanced bias in projected emission measure at intermediate radii for the Z_0.5 map, shown in red, is due to effects from integrating over the line of sight. The density profile measured with a fixed abundance parameter is steeper at intermediate radii than the density profile measured with the best-fit values for the abundance parameter. Hence, along an equal line of sight, there is a relatively lower emission measure in the fixed abundance case than in the best-fit abundance case.

In summary, the relative bias in abundance and projected X-ray temperature are correlated, and vary up to 50% and 10%, respectively. Both the resulting projected emission measure and gas density profiles vary up to 10% at low and intermediate radii, but deviate by as much as 40% and 20% in emission measure and gas density, respectively.

As described in Section 3.2, a high (low) bias of the abundance and X-ray temperature measurements would lead to a suppressed (enhanced) density measurement in that radial bin.

In this section, we examine the systematic effects on gas profile measurements with a factor of two and five bias in the abundance fitting at all radii. For this test, we use the Z_0.05 map where we take the metallicity to be constant at Z = 0.05 Z_☉ throughout the simulation. We choose this value since it is closer to the metal abundance at large radii R ≳ R_200c in our self-consistent cluster simulation with metal enrichment and advection. We run XSPEC fitting by fixing the abundance parameter at metallicities a factor of two and five from the true value from the map: Z_fix/Z_☉ = 0.01, 0.025, 0.1, 0.25, and we compare the results to those fitted with the true value Z = 0.05 Z_☉. To isolate the effects from all other quantities, we only allow the projected X-ray temperature and the spectral normalization to be free parameters in the fitting.

In Figure 8, we plot the X-ray temperature, projected emission measure, and gas density for each case of Z_fix and compare it to the Z_fix = Z_true = 0.05 Z_☉ case (black solid line). Consistent with what we found in Section 3.2, an over (under) estimate of the abundance will lead to suppressed (enhanced) emission measurements and subsequently gas density measurements.

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If the abundance is measured to within a factor of two of the true metal abundance, the resulting density measurements in the outskirts will be within 15%. However, for the most extreme case of an abundance measurement biased by a factor of five, the density measurements will have up to a 35% bias.

The emission measure profile on the right panel of Figure 8 shows that an abundance measurement that is higher than the true value can steepen the profile in cluster outskirts. For observational measurements that extrapolate the abundance at large radii, this could introduce an even larger bias in the emission measure, density, and temperature. Our tests that fix the abundance measurement to be a factor of two and five times the true abundance show how severely this extrapolation can propagate to inferred quantities, even at R_200c. Most notably, for an assumed abundance that is a factor of five higher than the true abundance, the gas density is biased low by ≈15%.

From Equation (8), we can see that the X-ray measurement of the hydrostatic mass of the cluster is sensitive to three quantities: the best fit 3D model of the temperature profile, the logarithmic slope of the 3D temperature profile, and the logarithmic slope of the gas density profile. The left panel of Figure 9 illustrates how each of these quantities is affected by a systematic over (under) fit abundance.

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For the case where the metal abundance is recovered to within a typical factor of two (Z_fix = 0.025 Z_☉ and Z_fix = 0.1 Z_☉), all three quantities are within <10% at intermediate and large radii. At large radii, the bias in the 3D temperature and logarithmic density slope dominates to drive the hydrostatic mass bias in the same direction. A lower temperature fit in the outskirts corresponds to a shallower logarithmic density slope, and therefore a smaller hydrostatic mass (see dotted blue and dot-dashed green thick lines in the right panel of Figure 9). However, the lower fitted temperature and shallower logarithmic slope correspond to a larger enclosed gas mass (see corresponding thin lines in Figure 9), as underestimates in both abundance and temperature lead to overestimates in emission measure, and therefore bias high the gas density.

Figure 10 shows that the combined bias in enclosed gas mass and the hydrostatic mass drives up the bias in the gas fraction for the low fixed abundance values, and down for the high fixed abundance values. This shows that a poor fit to the X-ray temperature and metallicity in cluster outskirts is potentially causing biases in the Suzaku gas mass fraction measurements at large radii. The same magnitude of error can come from an incorrectly extrapolated metal abundance in cluster outskirts.

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For a standard factor of two error in the abundance measurement, the gas mass fraction is only affected by 10%, but any underestimate of the 3D temperature at large radii due to an overly steep poor fit in the deprojection procedure could lead to an even higher gas mass fraction by driving the hydrostatic mass low.

For an abundance that is measured five times lower than the true abundance, the measured gas mass fraction could be biased high by 70% given a true abundance of Z = 0.05 Z_☉.

While the bias in enclosed gas mass and hydrostatic mass at small radii is rather large, we do not consider this to be representative of biases present in observations; the error in abundance measurement is primarily an issue in the low density outskirts where photon statistics are poor. Additionally, at smaller radii, the gas is much hotter, so the metal abundance and X-ray temperature parameters are less degenerate with each other in the spectral fit.

We test two potential issues for density slope measurements in cluster outskirts to compare with biases in measurements from spectral fitting and metal abundance extrapolation: projection bias from triaxiality and mass accretion history.

To test the bias due to projection effects from traxiality, we compute the inertia tensors of the gas mass at R_500c for our clusters. For CL7, the semi-minor and semi-major axes of the inertia tensor lie almost parallel to the x- and y-axes, respectively, allowing us to use the projections along these axes to make a preliminary assessment of the effects of cluster shape.

To test the effect of accretion history, we compare two clusters, CL6 and CL7, that have similar masses but differ in dynamical states. CL6 has a recent major merger and CL7 has been quiescent since z ∼ 1. We compare these two clusters to identify the extent of steepening, which is mainly due to recent accretion history.

Figure 11 shows the logarithmic gas density slope for our two test cases along various projections, CL6 and CL7. In our relaxed cluster, CL7, the density slope projected along each axis varies by no more than 15%. We would naively expect a significant difference between a line of sight along the semi-major axis (y axis) and along the semi-minor axis (x axis). However, the difference in density slope between these two line-of-sight projections amounts to only a few percent. Effects due to line-of-sight projection are not likely to significantly steepen observed density profiles.

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CL6 is a dynamically disturbed cluster that experienced a 1:1 major merger at z ≈ 0.6 and has a smaller infalling structure at R ≳ R_200c at z = 0. The x- and y-projections correspond to lines of sight that are perpendicular to the plane of accreting substructure. The effects of a merging substructure are visible in the z-projection (dotted green), where the density slope is no longer monotonically increasing at larger radii. The outskirts of CL6 cluster along the x- and y-projections have an observed density profile that exceeds the density slope along the x-projection of CL7 by ≈30%, confirming that X-ray observations should be able to see a difference in density slope between clusters of varying accretion histories.

These two mechanisms have the potential to contribute additional bias to the observed density slope than the bias introduced from spectral fitting (≈5%).