SUZAKU OBSERVATIONS OF THE OUTSKIRTS OF A1835: DEVIATION FROM HYDROSTATIC EQUILIBRIUM

We performed four-pointing Suzaku observations of A1835, named east, south, west, and north, in 2010 July with exposure of ∼50 ks for each pointing. The observation log is summarized in Table 1. Figure 1 shows the XIS image of A1835. The pointings were coordinated so that the X-ray emission centroid of A1835 was located at one corner of each pointing. The XIS mosaic covered the ICM emission out to the virial radius (∼2.9 Mpc or 120) and beyond.

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We used only XIS data in this study. Three out of the four CCD chips were available in these observations: XIS0, XIS1, and XIS3. The XIS1 is a back-illuminated chip with high sensitivity in the soft X-ray energy range, while the XIS0 and XIS3 are front-illuminated. The instrument was operated in the normal clocking mode. We included the data formats of both 5 × 5 and 3 × 3 editing modes in our analysis using xselect (Ver.2.4b). We used version 2.5.16.28 of the processed data screened with the standard filtering criteria.¹³ In order not to reduce the exposure time, event screening with the cutoff rigidity was not performed in our data because we can estimate the non-X-ray background (NXB) reasonably using data outside r_vir. The analysis was performed with HEAsoft ver 6.11 and CALDB 2011-09-06.

For each pointing (four azimuthal directions), we divided the FoV into six concentric annular regions centered on the X-ray emission centroid, (α, δ) = (14^h01^m01865, +02°52'3548) in J2000 coordinates (Zhang et al. 2007), to obtain the temperature and electron density profiles. The inner and outer radii of the annular regions are 00–20, 20–40, 40–60, 60–90, 90–120, and 120–200 (Figure 1). The circular regions around 32 point sources were excluded from the analysis. Furthermore, we subtracted the contribution of luminous point sources outside the excluded regions. The details are described in Appendix A. In the 00–20 region, all the spectra of the east, south, and north directions are unavailable because of calibration sources.

Redistribution matrix files of the XIS were produced by xisrmfgen version 2009-02-28. We generated two ancillary response files (ARFs) using xissimarfgen version 2010-11-05 (Ishisaki et al. 2007), assuming uniform sky (circular region of 20' radius, hereafter UNI-ARF) and surface brightness profile of A1835, where we used a β-model image of 486 × 486 with β = 0.55 and r_c = 0192 based on the ROSAT HRI result as the input X-ray image (Ota & Mitsuda 2004, hereafter β-ARF). Using these ARFs, the normalization of the ICM component derived from the spectral fit for a given spatial region is that for the entire input region to calculate the ARFs. To derive the normalizations of the ICM component for each spatial region, we multiplied the SOURCE_RATIO_REG parameter from the xissimarfgen tool (see also, e.g., Ishisaki et al. 2007; Sato et al. 2007) by the normalizations for the entire input region. We also included the effect of contaminations on the optical blocking filter of the XISs in the ARFs. The NXB were estimated from the database of Suzaku night-earth observations using xisnxbgen version 2010-08-22.

We used the XSPEC v12.7.0 package and ATOMDB v2.0.1 for all spectral fitting. The NXB components were subtracted before the fit. To avoid systematic uncertainties in the background, we used energy ranges of 0.6–7.0 keV for the XIS0, 0.5–5.0 keV for the XIS1, and 0.6–7.0 keV for the XIS3 in all the regions. In addition, we excluded energy band around the Si-K edge (1.82–1.84 keV), because its response was not modeled correctly. We simultaneously fitted all the spectra of the three detectors for the six annular regions toward four azimuthal directions by minimizing the total χ² value. In this fit, relative normalizations between the three sensors were left free to compensate for the cross-calibration errors. The model for the spectral fit was an absorbed thin-thermal emission model represented by phabs × apec for the ICM emission of the cluster, added to the X-ray background (XRB) model. We employed the β-ARF for the ICM component (see Section 2.2). The phabs component models the photoelectric absorption by the Milky Way, parameterized by the hydrogen column density that we fixed to the Galactic value of 2.04 × 10²⁰ cm⁻² (Kalberla et al. 2005). Even if we allowed the hydrogen column density for the phabs model to vary or employ the wabs model fixed at the Galactic column density in the direction of A1835 in the spectral analysis, the resultant temperatures and electron densities of the ICM component are almost the same within ∼3%. The apec is a thermal plasma model by Smith et al. (2001). For a given annulus, each parameter of the ICM component for four azimuthal directions was assumed to have the same value. In the central regions, metal abundance of the ICM component was allowed to vary, while at r > 40, we fixed the metal abundance of the ICM at 0.2. The redshift of the ICM component was fixed to 0.2532.

In order to study the faint X-ray emission from the cluster outskirts, an accurate estimation of the XRB is vitally important. We fitted the spectra in the outermost annulus (120–200 region, which is outside r_vir) for the following three cases. Case-GAL: The XRB model includes three components of the cosmic X-ray background (CXB), unabsorbed 0.1 keV Galactic emission (LHB; representing the local hot bubble and the solar wind charge exchange), and absorbed 0.3 keV Galactic emission (MWH1; representing the Milky Way halo; Yoshino et al. 2009). The normalization for the ICM flux is fixed to zero. Case-GAL+ICM: The XRB model was the same as Case-GAL, but the temperature and normalization for the ICM component were left free. Case-GAL2: In addition to Case-GAL, we added an absorbed 0.6 keV Galactic emission (MWH2; representing the Milky Way halo; Yoshino et al. 2009). This is because several blank fields observed with Suzaku contain the emission with 0.6–0.8 keV (Yoshino et al. 2009). The normalization for the ICM flux was fixed to zero. In all cases, we assumed a power-law spectrum for the CXB with Γ = 1.4. In addition, we modeled the LHB, MWH1, and MWH2 with the apec model, where the redshift and abundance were fixed at 0 and unity, respectively. The temperatures of the LHB, MWH1, and MWH2 were fixed at 0.1 keV, 0.3 keV, and 0.6 keV, respectively. We used UNI-ARF for the XRB components, assuming that the XRB components have flat surface brightness (see Section 2.2). Normalizations of the XRB components were also left free. For the XRB components, we adopted the model formula, phabs × (powerlaw + apec_MWH1) + apec_LHB for Case-GAL and Case-GAL+ICM, and phabs × (powerlaw + apec_MWH1 + apec_MWH2) + apec_LHB for Case-GAL2. Results of the spectral fit are shown in Sections 3.2 and 3.4.

Figure 2 shows the results of the spectral fit for the outermost annulus of 120–200 toward the east and west (opposite azimuthal directions). The best-fit parameters for the XRB components and χ² values are listed in Table 2. The χ² values for Case-GAL and Case-GAL2 are worse than that for Case-GAL+ICM. We estimate F-test probabilities for Case-GAL and Case-GAL+ICM of ∼1 × 10⁻⁷ and for Case-GAL and Case-GAL2 of ∼1 × 10⁻⁴, respectively. Case-GAL is therefore not supported. The intensity of the Galactic emissions (LHB, MWH1) is somewhat higher than that of the typical Galactic emissions (Yoshino et al. 2009). The plausible cause of the higher intensity would be the fact that A1835 is located near the North Polar Spur. For Case-GAL2, we refitted spectra with the temperature for 0.6 keV Galactic emission (MWH2) allowed to be a free parameter. We found that the χ² (2869) becomes slightly better and the resultant MWH2 temperature increases to $0.92_{-0.09}^{+0.11}$ keV. In the r < 120 region, the best-fit ICM parameters were almost the same as those from Case-GAL+ICM. For example, the ICM temperature and electron density in the 90–120 region increased by 8% and 10%, respectively, compared to those for Case-GAL+ICM.

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Table 2. Best-fit Parameters of the X-Ray Background Components

Case	CXB	LHB (0.1 keV)	MWH1 (0.3 keV)	MWH2 (0.6 keV)	12'–20' Region	Reduced-χ2
	SCXBa	NormLHBb	NormMWH1b	NormMHW2b	χ2/bin	(χ2/d.o.f.)
GAL	$5.78_{-0.17}^{+0.17}$	$7.32_{-0.63}^{+0.63}$	$1.05_{-0.05}^{+0.05}$	...	694/420	1.363 (2927/2148)
GAL+ICM	$5.17_{-0.23}^{+0.26}$	$7.54_{-0.63}^{+0.66}$	$0.96_{-0.07}^{+0.06}$	...	655/420	1.344 (2884/2146)
GAL2	$5.66_{-0.18}^{+0.18}$	$8.13_{-0.71}^{+0.72}$	$0.77_{-0.12}^{+0.12}$	$0.16_{-0.06}^{+0.06}$	667/420	1.354 (2907/2147)

Notes. ^aEstimated surface brightness of the CXB after the point-source excision in units of 10⁻⁸ erg cm⁻² s⁻¹ sr⁻¹ (2.0–10.0 keV). ^bNormalization of the apec component scaled by a factor 1/400π assumed in the uniform-sky ARF calculation (circle radius r = 20'). ${\rm Norm} = ({1}/{400\pi}) \int n_{\rm e} n_{\rm H} dV \,/ \,[4\pi \,(1+z)^2 D_{\rm A}^{2}] \times 10^{-20}$ cm⁻⁵ arcmin⁻², where D_A is the angular distance to the source.

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We investigated the validity of the CXB intensity obtained from the spectral fit. To estimate the amplitude of the CXB fluctuations, we scaled the fluctuations measured from Ginga (Hayashida et al. 1989) to our flux limit and FoV area using the method of Hoshino et al. (2010). The fluctuation width is given by the following relation:

$$\begin{eqnarray} &&\frac{\sigma _{\rm Suzaku}}{I_{\rm CXB}} = \frac{\sigma _{\rm Ginga}}{I_{\rm CXB}} \left(\frac{\Omega _{\rm {e,Suzaku}}}{\Omega _{\rm {e,Ginga}}}\right)^{-0.5} \left(\frac{S_{\rm {c,Suzaku}}}{S_{\rm {c,Ginga}}}\right)^{0.25}, \end{eqnarray} \tag{ 1 }$$

where (σ_Suzaku/I_CXB) means the fractional CXB fluctuation width due to the statistical fluctuation of the discrete source number in the FoV. Here, we adopt σ_Ginga/I_CXB = 5%, with S_c (Ginga: 6 × 10⁻¹² erg cm⁻² s⁻¹) representing the upper cutoff of the source flux, and Ω_e (Ginga: 1.2 deg²) representing the effective solid angle of the detector. We show the result, σ/I_CXB, for each annular region in Table 3, where σ is the standard deviation of the CXB intensity, I_CXB. Table 3 shows Ω_e (the solid angle of observed areas), Coverage (the coverage fraction of each annulus, which is the ratio of Ω_e to the total solid angle of the annulus), SOURCE_RATIO_REG (the fraction of the simulated cluster photons that fall in the region compared with the total photons generated in the entire simulated cluster), and σ/I_CXB (the CXB fluctuation due to unresolved point sources). For all directions, σ/I_CXB values in the 90–120 and 120–200 regions are about 6.1% and 3.3%, respectively. The σ/I_CXB value for each direction is higher by a factor of ∼2. These ranges are consistent with those obtained by the method of Bautz et al. (2009) using parameters derived by Moretti et al. (2003), who studied ROSAT, Chandra, and XMM-Newton observations.

The best-fit parameter of the CXB surface brightness (after subtraction of point sources brighter than 2 × 10⁻¹⁴ erg cm⁻² s⁻¹ in 2–10 keV band) is $5.17_{-0.23}^{+0.26}\times 10^{-8}$ erg cm⁻² s⁻¹ sr⁻¹ for Case-GAL+ICM which agrees with that of Hoshino et al. (2010), $4.73_{-0.22}^{+0.13}\times 10^{-8}$ erg cm⁻² s⁻¹ sr⁻¹, within statistical errors taking into account the CXB fluctuation. Using the same threshold, S_CXB derived with previous Suzaku observations were 4–6 × 10⁻⁸ erg cm⁻² s⁻¹ sr⁻¹ (e.g., Moretti et al. 2009; Kawaharada et al. 2010; Hoshino et al. 2010; Sato et al. 2012). Case-GAL and Case-GAL2 gave higher CXB surface brightness by 10%. If point sources brighter than 1.0 × 10⁻¹³ erg cm⁻² s⁻¹ are subtracted, S_CXB were measured to be 6–7 × 10⁻⁸ erg cm⁻² s⁻¹ sr⁻¹ (e.g., Bautz et al. 2009; Simionescu et al. 2011; Akamatsu et al. 2011; Walker et al. 2012a, 2012c). Using this threshold, the CXB surface brightness around A1835 for Case-GAL+ICM is 6.8 × 10⁻⁸ erg cm⁻² s⁻¹ sr⁻¹ which agrees well with the previous studies.

In order to see how far the ICM emission of A1835 is detected, we derived a surface brightness profile in the energy band of 1–2 keV from the XIS mosaic image (XIS0, XIS1, and XIS3 images) excluding the circular regions around 32 point sources (Figure 1). The left panel of Figure 3 shows the raw surface brightness profile in the 1–2 keV band (black crosses) where the background is included and the vignetting effect is not corrected. The 1–2 keV NXB profile, derived from an NXB mosaic image with xisnxbgen, is shown in red in the left panel of Figure 3. We then obtained an XRB mosaic image. Based on the CXB and Galactic (LHB+MWH1) models of A1835 obtained from fitting for the three background cases, we simulated CXB and Galactic images of the four offset observations using xissim with exposures 10 times longer than those of actual observations. The 1–2 keV CXB and Galactic (LHB+MWH1) profiles for Case-GAL+ICM are shown in green and blue in the left panel of Figure 3, respectively. Since there is a contribution of luminous point sources outside the excluded regions, we simulated this residual-point-source signals. Details of the simulation are described in Appendix A. The 1–2 keV residual-point-source profile is shown in orange in the left panel of Figure 3. In the end, we obtained background (CXB+Galactic+NXB+residual-point-source) and background-subtracted (raw−background) profiles for Case-GAL+ICM, as shown in cyan and magenta in the left panel of Figure 3. Here, ±10% systematic error for the CXB intensity is included.

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Since these images were not corrected for the vignetting, we calculated the ratio of the background-subtracted surface brightness to the CXB brightness in the right panel of Figure 3 for the three background cases. As a result, this ratio decreases with radius out to the virial radius (r_vir ∼ 120) and becomes flatter beyond. Beyond r_vir, the background-subtracted signal in the brightness for Case-GAL+ICM accounts for 29% and 26% of the CXB and CXB+LHB+MWH1, respectively. It cannot be explained by the fluctuations of the CXB intensity (∼3.3% in the 120–200 region in Table 3). An additional emission component is required to explain the observed flux beyond r_vir. However, it is uncertain whether this emission comes from the ICM (Case-GAL+ICM) or from the relatively hot (∼0.9 keV) Galactic emission. Since A1835 is located close to the North Polar Spur, it is clear that the background-subtracted signal suffers from Galactic emission uncertainties. The ratios of the remaining signal to the background for Case-GAL and Case-GAL2 are a factor of two smaller than those for Case-GAL+ICM. From χ², Case-GAL+ICM reproduces the spectra beyond r_vir better. We will quantify the systematic errors of the ICM temperature associated with the background subtraction in Section 3.6 and Table 8.

Figure 4 shows results of the spectral fit in the particularly important regions (40–60, 60–90, and 90–120) toward the east and west (opposite azimuthal directions) for Case-GAL+ICM. The best-fit parameters of the ICM components and χ² values in each region, when parameters are tied between the four offset pointings, are listed in Table 4.

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Table 4. Best-fit Parameters of the ICM Components in All Directions

Region	Case-GAL			Case-GAL+ICM			Case-GAL2
	kTAlla	NormAlla,b	χ2/binc	kTAlla	NormAlla,b	χ2/binc	kTAlla	NormAlla,b	χ2/binc
	(keV)			(keV)			(keV)
2'–4'	$7.55_{-0.64}^{+0.65}$	$6.12_{-0.20}^{+0.20} \times 10^{1}$	682/420	$7.55_{-0.64}^{+0.64}$	$6.17_{-0.20}^{+0.20} \times 10^{1}$	680/420	$7.59_{-0.64}^{+0.65}$	$6.11_{-0.20}^{+0.20} \times 10^{1}$	681/420
4'–6'	$4.91_{-0.81}^{+0.99}$	$8.66_{-0.80}^{+0.81}$	457/420	$5.00_{-0.79}^{+0.95}$	$9.14_{-0.81}^{+0.82}$	457/420	$5.13_{-0.86}^{+1.07}$	$8.54_{-0.79}^{+0.81}$	458/420
6'–9'	$2.33_{-0.32}^{+0.52}$	$3.72_{-0.47}^{+0.48}$	481/420	$2.56_{-0.42}^{+0.57}$	$4.14_{-0.48}^{+0.49}$	482/420	$2.60_{-0.46}^{+0.65}$	$3.51_{-0.47}^{+0.49}$	486/420
9'–12'	$1.72_{-0.15}^{+0.53}$	$1.74_{-0.37}^{+0.38}$	489/420	$2.00_{-0.35}^{+0.54}$	$2.23_{-0.40}^{+0.40}$	486/420	$2.07_{-0.44}^{+0.68}$	$1.51_{-0.38}^{+0.40}$	490/420
12'–20'	...	...	694/420	$1.72_{-0.36}^{+0.49}$	$1.03_{-0.27}^{+0.26}$	655/420	...	...	667/420

Notes. ^akT_All represent the temperatures of the sum of all azimuthal regions. Similarly, Norm_All represent normalization. ^bNormalization of the apec component scaled with a factor of ${SOURCE\_RATIO\_REG}/\Omega _{\rm e}\ {\rm Norm} = ({{SOURCE\_RATIO\_REG}}/{\Omega _{\rm e}}) \int n_{\rm e} n_{\rm H} dV \,/ \,[4\pi \,(1+z)^2 D_{\rm A}^{2}]\times 10^{-20}$ cm⁻⁵ arcmin⁻², where D_A is the angular distance to the source. ^cχ² of the fit when parameters are tied between the four offset pointings.

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3.4.1. Temperature Profile

Figure 5 shows radial profiles of projected temperature in all directions, observed with XMM-Newton (Zhang et al. 2007; Snowden et al. 2008), Chandra (Bonamente et al. 2013), and Suzaku (this work). The temperature is ∼8 keV within 20 (∼480 kpc), and it gradually decreases toward the outskirts down to ∼2 keV around the virial radius (r_vir ∼120). At a given radius, the temperatures for the three background cases with Suzaku agree well with each other. For Case-GAL+ICM, the temperature in the 120–200 region joins smoothly with those of inner regions. We shall quantify the systematic error of the Suzaku ICM temperature in Section 3.6. The temperature profile derived from Suzaku agrees well with those from XMM-Newton within 40 (∼0.95 Mpc) and from Chandra (Bonamente et al. 2013) within 75 (∼1.8 Mpc). Bonamente et al. (2013) measured temperature profiles for A1835 out to 100 with Chandra. As their outermost temperature is in the 75–100 regions, we shall compare this with our result at the outskirts in the west in Section 3.5.

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3.4.2. Electron Density Profile

The electron number density profile was calculated from the normalization parameter of the apec model, defined as

$$\begin{eqnarray} &&{\rm Norm} = \frac{10^{-14}}{4\pi (D_{\rm A} (1 + z))^2}\int n_{\rm e} n_{\rm H} dV, \end{eqnarray} \tag{ 2 }$$

where D_A is the angular-size distance to the source in units of cm, n_e is the electron density in units of cm⁻³, and n_H is the hydrogen density in units of cm⁻³. We note that the resultant normalization using an ARF generated by xissimarfgen needs a correction by a factor of SOURCE_RATIO_REG/Ω_e (see details in Section 5.3 of Ishisaki et al. 2007). The left panel of Figure 6 shows the radial profiles of normalization of the apec model for the ICM component scaled with this factor.

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Each annular region, projected in the sky, includes emission from different densities due to integration along the line of sight. Assuming spherical symmetry, we de-convolved the normalization and calculated n_e for each annular region from the outermost region using the method described in Kriss et al. (1983). The resultant radial profiles of n_e in all directions with XMM-Newton (Zhang et al. 2007) and Suzaku (this work) are shown in the right panel of Figure 6 and listed in Table 5. The deprojected electron density profile derived from Suzaku agrees well with that from XMM-Newton within 50 (∼1.2 Mpc). The n_e out to 90 for the three background cases are consistent within statistical errors with each other. However, n_e for Case-GAL and Case-GAL2 in the 90–120 region are 26% and 17% higher, respectively, than that for Case-GAL+ICM because of the projection of the ICM emission outside r_vir.

Table 5. Deprojected Electron Number Densities in All the Directions

Region	Case-GAL	Case-GAL+ICM	Case-GAL2
	ne, Alla	ne, Alla	ne, Alla
2'–4'	$6.64_{-0.13}^{+0.12} \times 10^{-4}$	$6.64_{-0.13}^{+0.12} \times 10^{-4}$	$6.64_{-0.13}^{+0.12} \times 10^{-4}$
4'–6'	$1.86_{-0.14}^{+0.14} \times 10^{-4}$	$1.85_{-0.15}^{+0.14} \times 10^{-4}$	$1.86_{-0.14}^{+0.14} \times 10^{-4}$
6'–9'	$9.57_{-1.11}^{+1.01} \times 10^{-5}$	$9.80_{-1.11}^{+1.02} \times 10^{-5}$	$9.56_{-1.11}^{+1.03} \times 10^{-5}$
9'–12'	$7.91_{-0.89}^{+0.83} \times 10^{-5}$	$6.30_{-1.27}^{+1.06} \times 10^{-5}$	$7.38_{-0.99}^{+0.91} \times 10^{-5}$
12'–20'	...	$4.29_{-0.60}^{+0.51} \times 10^{-5}$	...

Note. ^an_{e, All} represent for electron number densities of the sum of all the azimuthal regions in units of cm⁻³.

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The electron density profile of A1835 for Case-GAL+ICM from 60 (∼0.66r₂₀₀) to 120 (∼1.3r₂₀₀) is well represented by a power-law model with β = 0.88. This value agrees very well with the β = 0.89 derived for electron density profiles in 0.65–1.2r₂₀₀ of 31 clusters observed with ROSAT (Eckert et al. 2012). For Case-GAL and Case-GAL2, the β values derived in the same way are 0.38 and 0.51, respectively, which are significantly smaller than those derived by Eckert et al. (2012). Here, r₂₀₀ is calculated using the average ICM temperature, 〈kT〉, $r_{200} = 2.47 h_{70}^{-1} \sqrt{\langle kT \rangle /\rm {10\, keV}}$ Mpc. This relation was expected from numerical simulations for our cosmology (Henry et al. 2009). The average temperature of A1835 integrated over the radial range of 70 kpc to r₅₀₀ (= 1.39 Mpc $h_{70}^{-1}$ or 585) with XMM-Newton is 7.67 ± 0.21 keV (Zhang et al. 2007), where r₅₀₀ is defined by weak-lensing analysis (Okabe et al. 2010a). Thus, from the average temperature, r₂₀₀ = 2.16 Mpc $h_{70}^{-1}$ or 908, which is close to 929 for r₂₀₀ defined by the weak-lensing analysis.

3.4.3. Entropy Profile

Figure 7 shows the entropy profiles with XMM-Newton and Suzaku calculated as

$$\begin{eqnarray} &&K = \frac{kT}{n_{\rm e}^{2/3}}, \end{eqnarray} \tag{ 3 }$$

where T and n_e are the temperature and deprojected electron density obtained above, respectively. At a given radius, the entropies for the three background cases are consistent within statistical errors with each other. Within r₅₀₀, the derived entropy profiles with XMM-Newton and Suzaku follow a power-law model with a fixed index of 1.1, which was predicted from the accretion shock heating model (Tozzi & Norman 2001; Ponman et al. 2003; Voit et al. 2005). In contrast, beyond r₅₀₀ (∼60), the entropy profiles for the three cases become flatter, in disagreement with the r^1.1 relationship. For Case-GAL+ICM, the entropy profile is flat out to 200 (∼1.7r_vir).

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We investigate here the azimuthal variation of X-ray observables. We fitted all the spectra simultaneously in the same way as for Case-GAL+ICM (Section 3.1), but the temperatures and normalizations for the ICM component within 12' for the four directions were independently determined. The derived χ², 2669 for 2119 degrees of freedom, is smaller than the 2884 for 2146 degrees of freedom of the previous fit for all directions. Table 6 shows the derived temperatures, normalizations, and χ² values for each region. Assuming spherical symmetry we calculated the deprojected electron number density and entropy profiles. The resultant radial profiles of n_e in each direction are listed in Table 7. Figure 8 shows the radial profiles of the ICM temperature, scaled normalization, electron density, and entropy in each direction.

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Table 6. Best-fit Parameters of ICM Components in Each Direction (Case-GAL+ICM)

Region	kTeasta	Normeasta,b	kTsoutha	Normsoutha,b	kTwesta	Normwesta,b	kTnortha	Normnortha,b	χ2/binc
	(keV)		(keV)		(keV)		(keV)
0'–2'	...	...	...	...	$8.25_{-0.53}^{+0.63}$	$1.35_{-0.04}^{+0.04} \times 10^{3}$	...	...	125/105
2'–4'	$7.74_{-1.33}^{+1.80}$	$4.83_{-0.64}^{+0.62} \times 10^{1}$	$7.24_{-1.33}^{+2.10}$	$4.55_{-0.63}^{+0.73} \times 10^{1}$	$6.65_{-0.78}^{+0.99}$	$5.29_{-0.31}^{+0.30} \times 10^{1}$	$8.60_{-0.99}^{+1.42}$	$8.18_{-0.33}^{+0.34} \times 10^{1}$	495/420
4'–6'	$6.40_{-1.85}^{+3.09}$	$9.13_{-1.41}^{+1.48}$	$3.95_{-0.94}^{+1.74}$	$9.32_{-1.69}^{+1.75}$	$5.50_{-1.32}^{+2.55}$	$9.05_{-1.30}^{+1.38}$	$3.98_{-1.11}^{+1.77}$	$9.33_{-1.84}^{+1.94}$	452/420
6'–9'	$3.43_{-1.17}^{+2.24}$	$3.32_{-0.81}^{+0.89}$	$2.76_{-0.64}^{+1.11}$	$5.11_{-0.94}^{+0.98}$	$2.09_{-0.47}^{+0.68}$	$3.74_{-0.80}^{+0.85}$	$2.43_{-0.58}^{+0.78}$	$4.54_{-0.91}^{+0.96}$	471/420
9'–12'	4.01−2.34d	$1.18_{-0.65}^{+0.77}$	$2.81_{-0.91}^{+2.11}$	$3.02_{-0.85}^{+0.92}$	$1.51_{-0.27}^{+0.49}$	$2.31_{-0.67}^{+0.69}$	$1.84_{-0.32}^{+0.77}$	$2.46_{-0.66}^{+0.69}$	471/420

Notes. ^akT_east, kT_south, kT_west, and kT_north represent the temperatures in the east, south, west, and north, respectively. Similarly, Norm_east, Norm_south, Norm_west, and Norm_north represent normalization. ^bNormalization of the apec component scaled with a factor of ${SOURCE\_RATIO\_REG}/\Omega _{\rm e}\ {\rm Norm} = ({{SOURCE\_RATIO\_REG}})/({\Omega _{\rm e}}) \int n_{\rm e} n_{\rm H} dV \,/ \,[4\pi \,(1+z)^2 D_{\rm A}^{2}]\times 10^{-20}$ cm⁻⁵ arcmin⁻², where D_A is the angular distance to the source. ^cχ² of the fit when parameters are separately determined for the four offset pointings. ^dWe could not constrain the 90% confidence level upper limit of the temperature because of the influence of bright point sources (see Appendix B).

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Table 7. Deprojected Electron Number Densities in Each Direction (Case-GAL+ICM)

Region	ne, easta	ne, southa	ne, westa	ne, northa
0'–2'	...	...	$4.29_{-0.07}^{+0.07} \times 10^{-3}$	...
2'–4'	$5.73_{-0.48}^{+0.43} \times 10^{-4}$	$5.54_{-0.49}^{+0.52} \times 10^{-4}$	$6.07_{-0.21}^{+0.20} \times 10^{-4}$	$7.78_{-0.18}^{+0.18} \times 10^{-4}$
4'–6'	$2.12_{-0.22}^{+0.21} \times 10^{-4}$	$1.75_{-0.34}^{+0.30} \times 10^{-4}$	$1.91_{-0.23}^{+0.22} \times 10^{-4}$	$1.82_{-0.36}^{+0.31} \times 10^{-4}$
6'–9'	...	$1.03_{-0.22}^{+0.19} \times 10^{-4}$	$8.70_{-2.26}^{+1.87} \times 10^{-5}$	$1.02_{-0.21}^{+0.18} \times 10^{-4}$
9'–12'	...	$8.26_{-2.11}^{+1.81} \times 10^{-5}$	$6.54_{-2.20}^{+1.68} \times 10^{-5}$	$6.93_{-1.99}^{+1.61} \times 10^{-5}$
6'–12'	$5.58_{-1.39}^{+1.15} \times 10^{-5}$	...	...	...

Note. ^an_{e, east}, n_{e, south}, n_{e, west}, and n_{e, north} represent the electron number densities in the east, south, west, and north, respectively, in units of cm⁻³.

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There is an azimuthal variation of the projected temperatures in the outskirts of r₅₀₀ ≲ r ≲ r_vir. The best-fit temperature in the eastern region is highest, about twice of those in the western and northern regions. The temperature in the southern region is intermediate between them. The differences between individual values are larger than measurement uncertainties, but is not significant. Bonamente et al. (2013) reported with Chandra a temperature of kT = 1.26 ± 0.26 keV in the 75–100 region. We note that they observed the western region of A1835. Their temperature is marginally consistent with our measurements at 60–90 ($kT = 2.09_{-0.47}^{+0.68}$ keV) and at 90–120 ($kT = 1.51_{-0.27}^{+0.49}$ keV) in the west (Figure 8 and Table 6). The electron number density and entropy profiles in the south, west, and north are consistent with one another within the statistical errors in any annulus region. The electron number density and entropy in the outskirts (r₅₀₀ ≲ r ≲ r_vir) in the east are lower and higher than those in other directions, respectively.

We examined the effect of systematic errors on the derived spectral parameters. The level of the CXB fluctuation was scaled from the Ginga result (Hayashida et al. 1989) as shown in Table 3. The CXB fluctuation in the cluster outskirts (60 ≲ r), where correct estimations of the CXB intensity are of utmost importance, is less than 10% in all the directions from Table 3. In the 90–120 region, for example, the CXB fluctuation value in all directions is about 6.1%. Although in the cluster center (r ≲ 60) the CXB fluctuation is higher, the systematic uncertainties caused by the fluctuation became smaller due to much brighter ICM emission. Therefore, we assume that the upper and lower limits of the CXB systematic changes in all directions are ±10%, even when considering uncertainty other than the systematic error due to the spatial variation. For Case-GAL+ICM, we repeated the spectral fit for all directions in the same way but fixed the CXB intensity at the upper and lower 10% from their nominal levels. Table 8 shows the changes of ICM properties in the 90–120 region and the reduced χ² in each variation. In the 90–120 region, the effects of ±10% error for the CXB intensity, on the temperature, electron number density, and entropy are 20%–30%, less than 5%, and 20%–30%, respectively. In each direction, the CXB fluctuation for each annular region should be a factor of two higher than that of for all directions. In the 90–120 region, for example, the CXB fluctuation values in the east, south, west, and north are about 11.5%, 12.6%, 11.2%, and 10.6%, respectively. Therefore, we assume that the upper and lower limits of the CXB systematic changes in each direction are ±15%. In each direction, the change of entropy in the 90–120 region is 20%–40% by ±15% error for the CXB intensity, except for the CXB+15% in the west. This systematic error is comparable to the statistical error. For the CXB+15% in the west, the temperature and entropy are lower by a factor of ∼2 and ∼3, respectively.

The changes of the intensities of the Galactic emissions (LHB, MWH1), by systematic changes in the CXB levels to ±10% and ±15% from their nominal levels, are less than 10%. Then, we repeated the spectral fit by fixing the Galactic (LHB, MWH1) intensity at the upper and lower 10% from their nominal levels, for Case-GAL+ICM, and by fixing the metal abundance of the ICM at 0.1 and 0.3, instead of 0.2. We also show these results in Table 8. The systematic errors due to these effects are less than that for the CXB in the 90–120 region.

Since the spectra with azimuthal variations in temperature are focused to fit with a single temperature model, we obtain a relatively large reduced χ², over 1.3 (see Table 2 and Table 8). A part of the large χ² is due to the azimuthal variation in the temperature and normalization at the 20–40 region: when we set these parameters in the four azimuthal directions free, the χ² for this annular region improved to 495 (Table 6) from 680 (Table 4) for all directions. Furthermore, we estimate changes of χ² by adding an 8% systematic error in the spectral data points for uncertainty analysis. The minimum reduced χ² (χ²/d.o.f.) improved to 1.081 (2321/2146) in all the directions for Case-GAL+ICM. There was no significant effect on the results of the 60–90 and 90–120 regions. In particular, the minimum χ² for the 90–120 region significantly improved from 486 to 412 for 420 bins. Accordingly, the ICM temperature and normalization of this region slightly changed from $2.00_{-0.35}^{+0.54}$ keV to 2.03$_{-0.40}^{+0.66}$ keV and from 2.23 ± 0.40 × 10⁻²⁰ cm⁻⁵ arcmin⁻² to $2.19_{-0.43}^{+0.45} \times 10^{-20}$ cm⁻⁵ arcmin⁻², respectively.

We next discuss the influence of Suzaku's point spread function (PSF). We examined how many of the photons that accumulated in the six annular regions actually came from somewhere else on the sky because of the extended telescope PSF following a procedure described by Sato et al. (2007). Table 9 shows the contribution from each sky region for the 0.5–2 keV energy range. Here, we averaged the values of the three detectors. Although the effect of leakage from the center < 20 to the 20–40 and 40–60 regions is severe, the temperature and normalization derived in this region agree well with those derived from XMM-Newton (Zhang et al. 2007; Snowden et al. 2008) and Chandra (Bonamente et al. 2013). In the cluster outskirts, outside 60, the contributions of the leakage from the central 40 region are less than 20%. We fitted the spectra for the 90–120 region, including the stray light component from the bright central region. Then, we obtained an 8% lower ICM temperature and a 9% smaller normalization. Even if the actual effect of the stray light was twice the current calibrations, the changes would be −14% and −16% for the temperature and normalization, respectively. These differences are significantly smaller than the present statistical errors. Therefore, the temperature changes due to the PSF correction should be mostly small as in previous Suzaku observations of cluster outskirts (George et al. 2009; Reiprich et al. 2009).

Although the temperature profile is a projected one obtained by the two-dimensional spectral analysis, the results of the deprojection fitting do not show any significant difference from the non-deprojection fitting (Bautz et al. 2009; Akamatsu et al. 2011; Walker et al. 2012c).

4.1.1. Mass Estimation

We here estimate the H.E. masses for three background models using azimuthally averaged temperature and electron density profiles and compare with weak-lensing mass (Okabe et al. 2010a). We first fitted the temperature profiles from the XMM-Newton (from 05 to 50; Zhang et al. 2007) and Suzaku (from 20 to 200) results with the β model with a constant, and the electron density profiles from the XMM-Newton (from 00 to 30) and Suzaku (from 20 to 200) results with the double-β + power-law model. The results for Case-GAL+ICM are plotted in Figure 9. The hydrostatic mass, M_H.E.(< r), within the three-dimensional radius r, assuming a spherically symmetric H.E., is calculated from the parametric temperature and electron density profiles with the following formula (Fabricant et al. 1980):

$$\begin{eqnarray} &&M_{\rm {H.E.}}(< r) = -\frac{kT(r)r}{\mu m_p G} \left(\frac{d \ln \rho _{\rm g}(r)}{d \ln r} + \frac{d \ln T(r)}{d \ln r}\right), \end{eqnarray} \tag{ 4 }$$

where G is the gravitational constant, k is the Boltzmann constant, ρ_g is the gas density, kT is the temperature, and μm_p is the mean particle mass of the gas (the mean molecular weight is μ = 0.62). We also calculated gas mass M_gas by integrating the electron density profile.

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Weak-lensing analysis with the Subaru/Suprime-Cam is described in detail in Okabe et al. (2010a). Weak-lensing mass, M_lens, is derived by fitting the Navarro–Frenk–White (NFW) model (Navarro et al. 1997) to the tangential lensing distortion profile in the range 10–180. The NFW profile is a well-defined form for the spherically averaged density profile of dark matter halos with high-resolution numerical simulations. They are parameterized by the virial mass (M_vir) and the halo concentration parameter (c_vir).

Figure 10 shows radial profiles for the H.E. masses (M_H.E.) and gas masses (M_gas) using the three background cases (Case-GAL+ICM, Case-GAL2, and Case-GAL), and weak-lensing mass. The mass (M_H.E. and M_gas) profiles out to 90 for the three cases are consistent within statistical errors with each other. M_H.E. outside 80 unphysically decreases with increasing radius. This means that the H.E. we assumed is inadequate to describe the ICM in the outskirts. The M_H.E. agrees with M_lens within 10–50 from the cluster center, but there is a significant difference in the regions r < 10 and r > r₅₀₀ ≃ 585. The H.E. mass is lower than the weak-lensing one outside r₅₀₀ because of the breakdown of the H.E. assumption. The weak-lensing mass inside 10, extrapolated from the best-fit model obtained by fitting in the region 10 < r < 180, is significantly lower than the H.E. mass. As the weak limit of lensing distortion breaks down in this region, the strong lensing method plays an important role in the reconstruction of the mass distribution. Further study of a strong- and weak-lensing joint analysis, as demonstrated by Broadhurst et al. (2005), would be powerful in obtaining the lensing mass distribution for the entire radial range.

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4.1.2. Gas Mass Fraction

We derived the cumulative gas mass fraction within the three-dimensional radius r, defined as

$$\begin{eqnarray} &&f_{\rm gas}({<} r) = \frac{M_{\rm {gas}}({<} r)}{M_{\rm {total}}({<} r)}, \end{eqnarray} \tag{ 5 }$$

where M_gas(< r) and M_total(< r) are the gas mass and the gravitational mass (hydrostatic mass M_H.E. or lensing mass M_lens), respectively.

The gas mass fraction, $f_{\rm {gas}}^{\rm ({lens})}({<} r) = M_{\rm {gas}}/M_{\rm {lens}}$, combined with complementary Suzaku X-ray and Subaru/Suprime-Cam weak-lensing data set, is shown in Figure 11. As lensing mass does not require an assumption of H.E., a comparison of results derived solely by joint analysis of X-ray data allows us to understand the ICM states and the systematic measurement bias. We found no significant difference between the three background cases. The gas mass fraction in the range r₂₅₀₀ ≲ r ≲ r_vir is approximately constant, accounting for ∼90% of the cosmic mean baryon fraction, Ω_b/Ω_m, derived from seven-year data of the Wilkinson Microwave Anisotropy Probe (WMAP 7; Komatsu et al. 2011). It is in good agreement with recent numerical simulations (Kravtsov et al. 2005; Nagai et al. 2007). It is also consistent with Ω_b/Ω_m at r_vir within a large error. There is no significant radial dependence of gas mass fraction, contrary to recent statistical studies using lensing and X-ray data set (Mahdavi et al. 2008; Zhang et al. 2010). The best-fit value reaches Ω_b/Ω_m at r ∼ 1.1r_vir (r = 130). The gas mass fraction in the central region (r < 10) increases toward the cluster center, conflicting with numerical simulations (Kravtsov et al. 2005; Nagai et al. 2007) and showing that the cooling process increases the stellar fraction and, correspondingly, decreases the gas mass fraction. It implies that the lensing mass is underestimated in the central region in which strong lensing data are essential to reconstruct mass distribution. Future study of joint strong- and weak-lensing analysis will allow a detailed examination of the cluster center.

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The gas mass fraction derived from X-ray data, $f_{\rm {gas}}^{\rm ({H.E.})}(< r) = M_{\rm {gas}}/M_{\rm {H.E.}}$, increases monotonically with increasing radius. The $f_{\rm {gas}}^{\rm ({H.E.})}$ out to the intermediate radius (r < 0.5r₅₀₀) agrees with numerical simulations (Kravtsov et al. 2005; Nagai et al. 2007) and $f_{\rm {gas}}^{\rm ({lens})}$. It, however, significantly exceeds Ω_b/Ω_m beyond r₅₀₀, as found by the Suzaku observations of the northwest region of the Perseus cluster (Simionescu et al. 2011). They interpreted the significant excess as the presence of gas clumping, mainly in the cluster outskirts (Nagai & Lau 2011). However, our gas mass fraction estimated by gas and lensing masses in the range r₅₀₀ ≲ r ≲ r_vir does not exceed the cosmic mean baryon fraction from WMAP, which indicates that the gas-clumsiness effect mentioned by Simionescu et al. (2011) is less significant. The gas mass fraction, along with the low temperature and entropy in the cluster outskirts, supports the fact that the underestimation in H.E. mass is due to the breakdown of H.E. in the cluster outskirts, rather than due to the gas-clumpiness effect. To balance fully the gravity of the lensing mass, we need additional pressure supports such as turbulences and bulk motions caused by infalling matter at the cluster outskirts. The other possibility is deviations in the electron and ion temperatures at the cluster outskirts (e.g., Takizawa 1999; Hoshino et al. 2010; Akamatsu et al. 2011). This possibility is discussed in Section 4.5.

The temperature was measured out to the virial radius r_vir ∼2.9 Mpc, which shows that it decreases from about 8 keV around the center to about 2 keV at r_vir. Figure 12 compares azimuthally averaged temperature for Case-GAL+ICM with the fitting function for the outskirts temperature profile derived from numerical simulations (Burns et al. 2010). The fitting function,

$$\begin{eqnarray} &&\frac{T}{T_{\rm {avg}}} = A \left[ 1+ B \left(\frac{r}{r_{200}} \right) \right] ^{\beta }, \end{eqnarray} \tag{ 6 }$$

is scaled with average temperature and r₂₀₀, and the best-fit values of A = 1.74 ± 0.03, B = 0.64 ± 0.10, and β = −3.2 ± 0.4 are obtained. The temperatures outside 0.3r₂₀₀ agree with those of Burns et al. (2010) within a 1σ error range (dashed lines) and with those of other clusters observed by Suzaku (Burns et al. 2010; Akamatsu et al. 2011). It implies that low temperature (∼0.2〈kT〉) in cluster outskirts is a common feature. Burns et al. (2010) have shown that the kinetic energy of bulk and turbulent motions is as high as 1.5 times the thermal energy at the virial radius. As reported by recent numerical simulations (e.g., Nagai et al. 2007; Piffaretti & Valdarnini 2008; Jeltema et al. 2008; Lau et al. 2009; Fang et al. 2009; Vazza et al. 2009; Burns et al. 2010), the kinetic pressure in the ICM would contribute to balance the total mass.

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Sato et al. (2012) compared the entropy profiles with several clusters of galaxies observed with Suzaku. When scaled with average ICM temperature, the entropy profiles for clusters with ICM temperatures above 3 keV are universal irrespective of the ICM temperature. Walker et al. (2012b) confirmed that the function well represents the ICM entropy profiles of several clusters of clusters, in agreement with the entropy profiles obtained by combining Plank, ROSAT, and XMM-Newton. The entropy profile of A1835 also becomes flat beyond 0.5r₂₀₀, contrary to the r^1.1 relationship expected from the accretion shock heating model (Tozzi & Norman 2001; Ponman et al. 2003; Voit et al. 2005). One possible explanation for the low entropy profiles at cluster outskirts is that kinetic energy accounts for some fraction of energy budget to balance the gravity fully (Bautz et al. 2009; George et al. 2009; Kawaharada et al. 2010; Sato et al. 2012). The flattening of the entropy profile supports the fact that the discrepancy between the M_H.E. and M_lens beyond 0.5r_vir is caused by the deviation from the H.E.

X-ray emission is proportional to the square of the gas density integrated along the line of sight, and thus it is powerful in a denser region of hot gas. The thermal SZE (Sunyaev & Zeldovich 1972) is proportional to the thermal gas pressure integrated along the line of sight. The SZ observation therefore makes a powerful diagnostic of the less dense gas, as in cluster outskirts. As the sensitivity of the SZE to gas clumping (Nagai & Lau 2011) is a function of the pressure differential of the clumps with the surroundings, X-ray and SZE observables are thus complementary and allow us to further constrain the physics of the ICM. Planck is the only SZ experiment with full sky coverage, able to map even nearby clusters to their outermost radii and offering the possibility of an in-depth statistical study through the combination of many observations (Planck Collaboration et al. 2013). We compare the pressure profile, P = kTn_e, using Suzaku observed projected temperature and deprojected electron density for Case-GAL+ICM with that derived from stacked SZ flux for 62 nearby massive clusters from the Planck survey (Planck Collaboration et al. 2013). The results are plotted in Figure 13. Following Planck Collaboration et al. (2013), we normalized the pressure at r₅₀₀ = 1.39 Mpc $h_{70}^{-1}$ measured by weak-lensing analysis (Okabe et al. 2010a). The thermal pressure at the outskirts measured by Planck agrees with our Suzaku X-ray measurement from 20 (∼0.3r₅₀₀) to 120 (∼2r₅₀₀ ∼r_vir), albeit with different sensitivities of gas density, which indicates the reliability of Suzaku measurements. Walker et al. (2012b) have shown this agreement for other clusters observed by Suzaku. The consistency between independent observables implies that there is no strong need to invoke the gas-clumping effect discussed by Simionescu et al. (2011).

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Gravitational lensing masses are available for two clusters, A1835 (Okabe et al. 2010a) and A1689 (Umetsu & Broadhurst 2008), which are among those clusters detected with Suzaku whose X-ray emissions are entirely within the virial radius. Gravitational lensing on background galaxies enables us to directly reconstruct the mass distribution without using any assumptions on the relation between dark matter and baryon distributions. It is important to compare X-ray observables based on lensing properties in order to understand gas properties. Lensing distortion profiles for the two clusters are well expressed by the universal NFW model (Navarro et al. 1997) but not well expressed by a singular isothermal sphere (SIS) model. The virial masses and halo concentrations for the NFW model are $M_{\rm vir}=1.37^{+0.37}_{-0.29}\times 10^{15}M_\odot \,h^{-1}$ and $c_{\rm vir}=3.35^{+0.99}_{-0.79}$ for A1835 (Okabe et al. 2010a) and $M_{\rm vir}=1.47^{+0.59}_{-0.33}\times 10^{15}M_\odot \,h^{-1}$ and c_vir = 12.7 ± 2.9 for A1689 (Umetsu & Broadhurst 2008; Kawaharada et al. 2010), respectively. The virial masses for two clusters are similar, whereas the concentration parameter for A1835 is lower than A1689. Indeed, the Einstein radius determined by strong-lensing analysis (Richard et al. 2010) for A1835 (305) is smaller than that for A1689 (471) in the case when the source redshift is at z_s = 2. It is well established from CDM numerical simulations that the halo concentration is correlated with the halo formation epoch (e.g., Bullock et al. 2001). This is because the central mass density for clusters, corresponding to the concentration, is correlated with those for their progenitors. Therefore, among clusters with similar mass, the age of clusters with higher concentration is likely to be longer (Fujita & Takahara 1999).

If the low thermal pressure and entropy in cluster outskirts discovered by Suzaku (e.g., Kawaharada et al. 2010; Sato et al. 2012; Walker et al. 2012b) are explained by a difference between electron and ion temperatures (e.g., Takizawa 1999), it is statistically expected that the thermalization between electrons and ions through the Coulomb interaction goes on for clusters with high concentration. This is because the thermal equilibration time (<1 Gyr) in cluster outskirts is shorter than their ages, although Wong & Sarazin (2009) have shown that electron and ion temperatures differ by less than a percent within the r_vir. As both the virial mass and redshift for the two clusters are similar, they are a good sample for comparing the ICM properties and investigating a dependence of halo concentration.

Figure 14 shows a comparison of temperature, electron number density, and entropy profiles in all directions for A1835 and A1689, where the temperature and entropy are scaled with the average ICM temperature, 〈kT〉. The radius is also normalized by the r_vir determined by lensing analyses. They are all consistent within statistical errors. It implies that the temperature difference is not critical in explaining the observed ICM properties in the cluster outskirts. As we do not know the details of the formation history, a statistical approach is essential to derive it.

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We discuss here the correlation between temperature anisotropy in cluster outskirts and large-scale structure surrounding the cluster. Kawaharada et al. (2010) have discovered in A1689 that the anisotropic distributions of gas temperature and entropy in the outskirts (r₅₀₀ ≲ r ≲ r_vir) are clearly correlated with the large-scale structure. Similarly, we found anisotropic distributions of temperature and entropy in the outskirts, though their significance levels are low.

We first make a visualization of galaxy distribution using photometric redshifts from the SDSS DR7 catalog (Abazajian et al. 2009),

$$\begin{eqnarray} &&N ({\boldsymbol {\theta }})=\sum _i \int _{z_{\rm min}}^{z_{\rm max}} d {\boldsymbol {\theta ^{\prime }}} p_i(z,{\boldsymbol {\theta ^{\prime }}}) W(|{\boldsymbol {\theta }}-{\boldsymbol {\theta ^{\prime }}}|), \end{eqnarray} \tag{ 7 }$$

where $p_i(z,{\boldsymbol {\theta ^{\prime }}})$ is a photometric redshift probability distribution for the ith galaxy and W(θ) is a weight function. The probability function is computed from the best-fit z_phot and error σ_z, $p(z,{\boldsymbol {\theta ^{\prime }}})=A\exp [-(x-z_{\rm phot})^2/2/\sigma _z^2]$, ignoring a secondary solution of photometric redshift, where A is the normalization by making the integration from z = 0 to z = ∞ equal to unity. We apply the Gaussian smoothing function $W({\bf \theta })=1/(\pi \theta _g^2)\exp (-|{\bf \theta }|^2/\theta _g^2)$ with angular smoothing scale θ_g. We used half the virial radius θ_g = 100 (FWHM = 167) to reconstruct the two-dimensional map of galaxies. We select bright galaxies with magnitude r' < 22 in a photometric redshift slice of |z − z_c| < δz = σ_{v, max}(1 + z_c)/c ≃ 0.0125, where z_c is the cluster redshift, z is a photometric redshift, σ_{v, max} = 3000 km s⁻¹, and c is the velocity of light. The resultant map is shown in Figure 15. The white circle and boxes represent the virial radius obtained by weak-lensing analysis and the FoVs of the XIS pointings, respectively. A filamentary overdensity region outside the virial radius is apparently found in the eastern and southern regions of the XIS pointings. The broad filamentary structure is elongated out to ∼4r_vir in the direction of the southern region. The galaxy number around the virial radius in the eastern region is higher than that in the southern region. The best-fit temperatures in the outermost region for these two regions are higher than those for the other region in Table 6. The northern and western regions with low temperature are in contact with low-density void environments. The correlation between the temperature in cluster outskirts and the large-scale structure outside A1835 is consistent with the result of A1689 (Kawaharada et al. 2010).

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In order to investigate more precisely this correlation, we compare the outskirt temperatures for these two clusters with the number density contrast, δ = n/〈n〉 − 1, of the large-scale structure, without any smoothing procedures applied in making the map. The number densities n in the azimuthal angles of the FoVs of Suzaku are computed using Equation (7) with W(θ) = 1 and the area normalization. Here, 〈n〉 is the azimuthally averaged number density. We measure the density contrast in the annulus regions (1–2)r_vir and (2–4)r_vir centered on the brightest cluster galaxy. The photometric redshift slices are calculated using σ_{v, max} = 3000 km s⁻¹. The standard errors are estimated using the bootstrap method. We measure the deviation of temperatures, T/T_All − 1, with T_all the temperature in all directions. For consistency with the X-ray analysis of A1689 (Kawaharada et al. 2010), we use for A1835 the temperatures measured in 60 < r < 120 which is close to r₅₀₀ ≲ r ≲ r_vir. There is an apparent correlation between the two quantities (Figure 16). In order to quantify this, we computed Spearman's rank correlation coefficient, r_s. The errors are estimated using 10⁴ Monte Carlo redistributions, taking into account the uncertainties. The full data set for two clusters gives r_s = 0.762  ±  0.133 for (1–2)r_vir (Table 10), and the probability of a null hypothesis, $P=0.028_{-0.025}^{+0.067}$, that there is no relationship between the two data sets. It indicates that the correlation does not occur by chance with more than 90% confidence. The result does not change by choosing a luminosity-weighted center or number-weighted center within the virial radius. We also confirmed the same results using the temperature and entropy in the outermost regions (Table 6). Spearman's coefficient using the regions of (2–4)r_vir is smaller and the probability of a null hypothesis is more than the significance level of 10%. The correlation with galaxy distribution in the large-scale structure at a distance of 10 Mpc is not statistically strong. We also investigated Spearman's coefficient for each cluster and could not rule out a null hypothesis, as expected from the low significance level of the temperature anisotropic distribution.

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Table 10. Spearman's Rank Correlation Coefficient between Temperatures in Cluster Outskirts and Density of Large-scale Structure

Name	(1–2)rvir		(2–4)rvir
	rsa	Pb	rsa	Pb
A1835	0.800 ± 0.122	0.200 ± 0.122	0.600 ± 0.171	0.400 ± 0.171
A1689	$0.800_{-0.323}^{+0.200}$	$0.200_{-0.200}^{+0.323}$	$0.800_{-0.247}^{+0.200}$	$0.200_{-0.200}^{+0.247}$
Two clusters	0.762 ± 0.133	$0.028_{-0.025}^{+0.067}$	0.571 ± 0.177	$0.139_{-0.106}^{+0.195}$

Notes. ^aSpearman's rank correlation coefficient. ^bThe probability of a null hypothesis.

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Combining the results of the two clusters, we found that the anisotropic temperature distribution in the cluster outskirts is significantly associated with contacting regions of large-scale structure environment. Further study with a larger sample is of vital importance to obtain more robust result.