CHANDRA STUDIES OF THE X-RAY GAS PROPERTIES OF GALAXY GROUPS

All observations were performed with ACIS. Standard data analysis was performed which includes the corrections for the slow gain change⁴ and charge transfer inefficiency (for both the FI and BI chips). We investigated the light curve of source-free regions (or regions with a small fraction of the source emission) to identify and exclude time intervals with particle background flares, including weak flares. The relevant information on the Chandra pointings is listed in Table 1. We corrected for the ACIS low-energy quantum efficiency (QE) degradation due to the contamination on ACIS's optical blocking filter,⁵ which increases with time and is positionally dependent. The dead area effect on the FI chips, caused by cosmic rays, has also been corrected. As the background subtraction is important for this project, we present it in detail in the following section and in the Appendix. We do not use any data on the S4 chip, because of the residual streaks often seen at low levels (after running the CIAO tool "DESTREAK") and the lack of the stowed background data. The "readout artifact," seen in groups with very bright cores, is also corrected (see e.g., V06). We used CIAO3.4 for the data analysis. The calibration files used correspond to Chandra calibration database (CALDB) 3.4.3 from the Chandra X-ray Center, released in 2008 March. We are aware of a possible overcorrection of the Chandra effective area beyond the Iridium M-edge (∼ 2 keV) in CALDB 3.4.3 and before (see David's presentation in the 2007 Chandra Calibration Workshop),⁶ which can bias Chandra temperatures to higher values, especially for hot clusters (T > 4 keV). The difference between Chandra and XMM-Newton temperatures is also shown in Snowden et al. (2008). The calibration work to incorporate this into the Chandra CALDB is ongoing. However, as shown in both David's presentation and Snowden et al. (2008), the agreement between Chandra and XMM-Newton temperatures for ≲ 4–5 keV systems is very good. The highest gas temperature at any radii in our sample is ∼ 3.3 keV for several groups at the center, while temperatures at large radii are much lower and mainly determined by the iron L hump (instead of the continuum slope as for hot clusters). Thus, any changes in our results from this correction should be smaller than the current measurement errors and scatter. In the spectral analysis, a lower energy cut of 0.4 keV (for the BI data) and 0.5 keV (for the FI data) is used to minimize the effects of calibration uncertainties at low energy. The solar photospheric abundance table by Anders & Grevesse (1989) is used in the spectral fits. Uncertainties quoted in this paper are 1σ.

Proper background subtraction is important for the analysis in the low-surface brightness regions of the groups. A detailed discussion of the Chandra background and the relevant data set is presented in the Appendix. We determine the local background based on the stowed background data and modeling. The corresponding stowed background of each observation is scaled according to the flux ratio in the 9.5–12 keV band (e.g., V05). We adopt a 3% uncertainty on the particle background (PB) normalization (5% for BI data in period E, see the Appendix) in the error budgets.

The local cosmic X-ray background (CXB) can be modeled. The best fit of the hard CXB component is determined with an absorbed power law with a photon index of 1.5. The absorption is determined from spectral fits to the group spectra, which involves iterations and is present in Section 3.3. The soft CXB component is adequately described by two thermal components at zero redshift, one unabsorbed component with a fixed temperature of 0.1 keV and another absorbed component with a temperature either derived from spectral fits or fixed at 0.25 keV (see the Appendix). Abundances of both components are fixed at Solar. We can compare our CXB model with other work. V05 used the blank sky background and corrected for differences between a local control field and the blank sky background. As the scaling factors of the PB are close to unity in their sample, V05 ignored the correction for the unresolved hard CXB. However, the correction for hard CXB becomes more important after the middle of 2004 when the PB flux is 30%–50% higher than that in the blank sky background (the Appendix). V05 found that generally a single thermal component with a temperature of ∼ 0.2 keV and solar abundance can fit the soft CXB excess or decrement well. In regions with high RASS R45 values, a second thermal component with a temperature of ∼ 0.4 keV and solar abundance is required. Humphrey et al. (2006) and G07 also used two thermal components with solar abundances to describe the soft CXB. The CXB temperatures are fixed at 0.07 keV and 0.2 keV. As discussed in the Appendix, Snowden et al. (2008) used three thermal components to describe the soft CXB. Thus, our model of the soft CXB only has less freedom than the model by Snowden et al. (2008). However, our model balances the requirement of having enough components to fit the local CXB and avoiding parameter degeneracy in generally low-statistics data. First, the two ∼ 0.1 keV components will naturally be mixed in our energy band when the absorption is low. Second, statistics for most of our groups at large radii are not very good, so uncertainties are mainly statistical errors. As long as the background uncertainties are folded into the final error budgets, very detailed modeling of the soft CXB component is not crucial. Third, we test different models of the soft CXB for groups in our sample for which a large group-emission-free region is available. All spectra can be well fit by our two-component model, while the three-component model by Snowden et al. (2008) makes little or no improvement. In fact, the soft CXB can be well fitted by a single thermal component in most cases. However, the abundance usually has to be free and the best-fit value is usually very close to zero. Therefore, we conclude that the two-component soft CXB model is adequate and also necessary for our analysis.

The Chandra observations in our sample can be classified into two categories, one with regions free of group emission in the field of view (FOV), another with group emission detected to the edge of the field. Cool systems at higher redshifts (e.g., z > 0.04) usually have group-emission-free regions in the off-center chips (e.g., S2 for ACIS-I observations, I2/3 and S1 for ACIS-S observations). There are also two groups with adjacent Chandra pointings for unrelated targets in coincidence (ESO 306-017 and A1692). We took the following approach to look for group-emission-free regions. The radial surface brightness profile is derived with the exposure correction and background subtraction using the scaled stowed background. Different pointings for the same group are combined. The obtained surface brightness profile has group emission plus the CXB. The region where CXB is more dominant than the group emission, if present, can be determined from the flattened portion at the outer region of the surface brightness profile. We used a power law plus a constant at large radii to determine whether a significant group-emission-free region exists and the radial range of that region. If such a region is found, we have a control field with only CXB. In this work, the inner radius of the group-emission-free region is >3' larger than the outermost radius for the group temperature profile. This control field may be a single region on the S2 chip (for the ACIS-I data) or two separate regions on the S1 and I2/I3 chips respectively (for the ACIS-S data). Although we could simply fit the spectrum (or spectra) of this control field to determine the local CXB, the statistics are generally not sufficient. Moreover, the hard CXB in this control field may be larger than that in the outermost bin for the group temperature profile, as the flux of the hard CXB depends on point source excision. Thus, we fit the spectra of the control field and the outermost bin for the group temperature profile together to better constrain the local CXB. The normalizations of the soft CXB in different regions are linked by the ratios of their covered solid angles of the sky, while the normalizations of the hard CXB are not linked. Therefore, we can determine the local CXB in the outermost bin for the group temperature profile. In this work, we assume that the soft CXB is constant across the examined group area. Accurate determination of the local CXB is only important for large radii. The covered area at large radii tends to be in the same direction from the group center so the assumption should be reasonable. There is a complication that the hard CXB may be smaller in inner radial bins, but it is a small effect and can be corrected (also discussed later in this section).

However, many groups in our sample have group emission detected to the edge of the field, so we have to fit the CXB components with the group emission together. Since these groups are limited by the Chandra FOV, the group emission is generally still significant near the edge of the field. We fit the spectra from the two outermost radial bins together, with the normalizations of the soft CXB components linked. Generally in this work, we are conservative and do not include the outermost bin in the temperature profile as the uncertainties are generally large. However, there are a few groups that remain bright to the edge of the field (e.g., NGC 1550, A262, and MKW4). They all have S1 data, so the soft CXB component can be easily separated from the group emission (e.g., Figure 1), owing to the good response of the BI chips at < 1 keV. As the temperatures of the cool gas can be well constrained from the iron hump centroid, group emission can be robustly separated from the soft CXB even to the edge of the field. Thus, we derive temperature profiles to the edge of the field for these groups.

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As we separate the PB from the CXB, the absolute CXB flux can be derived in each group field. In Table 2, we list the flux of the local background components, for both the soft CXB and the hard unresolved CXB (in the outermost bin). We expect the derived soft CXB is on average correlated with the RASS R45 flux in the surrounding area (excluding the source region, see Table 2), which is true as shown in Figure 2. One should be aware of the uncertainty in the R45 flux, as the RASS soft X-ray background maps have 0.2 deg pixels and the various uncertainties combined are not small (e.g., variable SWCX emission). There are also uncertainties related to cross-calibration and cosmic variance of both the soft and hard CXB, so a detailed one-to-one comparison is hardly meaningful. The derived 2–8 keV flux of the hard CXB component depends on the limiting flux of the observation. As discussed in the Appendix, there is an empirical relation between the limiting flux for point sources and the average 2–8 keV flux of unresolved hard CXB (K07). We estimated the average limiting flux of the group observations in the outermost spectral bin. The regions we used to control the local background generally have an area of about one ACIS chip. As shown in Hickox & Markevitch (2006, HM06 hereafter) and K07, the cosmic variance in this angular scale (depending on the two-point angular correlation function of point sources) is 20%–30% and the total hard CXB flux has ∼ 10% uncertainty. Thus, the expected hard CXB flux from the empirical relation is only meaningful in an average sense. We use the CIAO tool MKPSF to generate several PSFs (at an energy of 1.4 keV) in the outermost spectral bin. We then derive the 90% enclosed power aperture and measure the 3σ limits at these regions. Their average is taken as the estimate for the limiting flux for point sources. Compared with the average growth curve determined by K07 (their Figure 19), we have an estimate of the unresolved hard CXB, which is also listed in Table 2. We can see from Table 2 that the general agreement is quite good, while the uncertainties for the hard X-ray CXB are much larger than the difference. It is also true that the limiting flux for point sources depends on the off-axis angle. The Chandra limiting flux changes little within the central 6–7' from the aim point, but increases rapidly beyond 10'. This can be corrected from the slope of the limiting flux versus unresolved hard CXB in K07 and the absolute flux in the outermost bin. The correction is small, also because the errors of the hard CXB are not small.

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Besides the 3% uncertainty on the PB mentioned above (5% for BI data in period E), we also included the following error budgets and added them (and the PB uncertainty) to the statistical uncertainties of temperatures, abundances, and surface brightness in quadrature. For the hard CXB, we set the photon index at 1.1 and 1.9 (e.g., HM06) and repeat the analysis to estimate the local background in those situations. For the soft CXB component, if the temperature of the hotter component cannot be derived from the data and is fixed at 0.25 keV, we change it to 0.2 keV and 0.3 keV and repeat the analysis to estimate the local background in those situations. Thus, we are conservative in the uncertainties for the local CXB.

Once the local CXB is determined, we proceed to derive the projected temperature profile. The radial bins for temperature measurement are decided from the outermost bin, by requiring the temperature to be constrained to better than 30% (1σ with all error budgets) and r_out/r_in = 1.3–1.75. After determining the outermost bin, the inner bins are determined progressively with r_out/r_in = 1.25–1.6. Point sources and subclumps are excluded. The absorption column is determined from the MEKAL fit to the integrated spectrum between 0.1r₅₀₀ and 0.4r₅₀₀. In this process, we also tried the VMEKAL model with extra free parameters of O, Ne, Mg, Si, S, Fe, and Ni abundances, as a lower O/Fe ratio may cause excess absorption (Buote et al. 2003). This process and the determination of the local CXB are done in iterations to obtain the final value of the absorption. If the derived absorption is consistent with the Galactic absorption from the HI survey (Table 2) within 1σ, the Galactic value is used. Only when both MEKAL and VMEKAL fits show excess absorption, the X-ray absorption from the MEKAL fit (always consistent with the VMEKAL fit within 1σ) is adopted (Table 2). About 44% of groups show excess absorption relative to the Galactic value, and indeed we find IRAS 100 μm enhancement in many of these groups (see the Appendix for discussions of some groups). We also examined the absorption variation with radius in each group. Beyond the central 20 kpc, no significant absorption variation is found for any group so we used a fixed absorption column for each group. In several cases, we observed an absorption increase within the central 20 kpc, but this analysis is complicated by the possible multiphase gas around the center. Nevertheless, it has little effect on any of our results. We also discuss the systematic error related to absorption in Section 9.

We used the MEKAL model to fit the spectra of the group emission. The free parameters are temperature, abundance, and normalization, once the absorption is determined. For spectra around the center, we also include a component to account for the LMXB emission from the cD galaxy. The LMXB component is represented by a power law with an index of 1.7, within D₂₅ aperture obtained from HyperLeda. The total LMXB luminosity is fixed from the L_X–L_Ks relation derived in Kim & Fabbiano (2004), where L_Ks is the total K_s band luminosity of the cD from 2MASS. We also assume that the LMXB emission follows the K_s band light (see also Gilfanov 2004).

The projected group temperature profiles are shown in Figures 3–5. We used the algorithm determined by Vikhlinin (2006) to derive the deprojected temperature profiles in a parametric way, which was first applied in V06. The required inputs are the three-dimensional or deprojected profiles of gas density and abundance, and the projected temperature profile. V06 simply used the projected abundance profile as their sample is dominated by hot clusters. However, the emissivity of ≲ 2 keV plasma is sensitive to the chemical abundances so the three-dimensional abundance profile is required for our work. We applied the nonparametric geometrical deprojection (summarized in e.g., Pizzolato et al. 2003; G07) to derive deprojected abundances in wider radial bins, generally merging 2–3 adjacent bins for the temperature profile. The deprojected abundance is an emission-weighted average in each bin so an effective radius in each bin is required. We define the effective radius as the emissivity weighted radius in each bin, where the plasma emissivity, ε(T, Z, r), depends on the three-dimensional profiles of temperature, abundance, and density.

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Since the determination of the three-dimensional profiles of temperature, abundance, and density depends on each other, iterations have to be done to derive the best-fit three-dimensional profiles. Usually at most three iterations are required to stabilize the best fits of these profiles. We assumed the following three-dimensional abundance profile:

$$\begin{eqnarray} &&Z(r) = Z_{0} + Z_{1} \exp (-(r/r_{Z})^{\alpha _{\rm Z}}). \end{eqnarray} \tag{ 1 }$$

This simple form can fit the abundance profiles of all groups in the sample. An example is shown in Figure 6 that represents the best-constrained three-dimensional abundance profile in this sample (thus the most difficult to fit, as the errors are the smallest). Most groups in this sample only have three-dimensional abundances constrained in 3–5 bins with larger errors so good fits can always be achieved with this simple function. Once the best fits of all profiles are achieved, we applied 1000 Monte Carlo simulations to address the uncertainties of the three-dimensional abundance profile. The simulation is realized by scattering the three-dimensional abundance profile according to the measurement errors. In this process, to be conservative, we also include an uncertainty of 10% of the bin size on the effective radius of each bin for the abundance profile. Thus, besides the best-fit three-dimensional abundance profile, we have 1000 simulated profiles to cover the error ranges, which will be used to estimate the uncertainties of the three-dimensional temperature and density profiles.

We used the same form of the three-dimensional temperature profile as used in V06:

$$\begin{eqnarray} &&T(r) = \frac{T_{0}(r/r_{\rm cool})^{a_{\rm cool}}+T_{\rm min}}{(r/r_{\rm cool})^{a_{\rm cool}}+1} * \frac{(r/r_{t})^{-a}}{[1+(r/r_{t})^{b}]^{c/b}}. \end{eqnarray} \tag{ 2 }$$

The exceptions are A1139, A1238, and A2092, which are faintest 2–3 keV systems in this sample so the errors on temperatures are large. Their three-dimensional temperature profiles are modeled as: T(r) = T₀[1 + (r/r_t)^a]^b/a, which fits their temperature profiles very well (Figures 4 and 5). The inner boundary of the fit is 5–20 kpc, depending on the quality of the fits at the center. As we focus on properties at large radii, the detailed choice of the inner boundary (5–20 kpc) is not a concern. Because the sizes of bins at large radii are not small, each bin is further divided into six subbins. The temperature modeling is done at the center of each subbin, only added up (with the algorithm by Vikhlinin 2006) at the end to obtain the expected temperature for each original bin of spectral analysis. In this way, the binsize effect is minimized and we don't need to worry about the accurate determination of effective radius for each radial bin, as most previous work had to address. The uncertainty of the three-dimensional temperature profile is derived from 1000 Monte Carlo simulations. In each simulation, the measured projected temperature profile is scattered according to the measurement errors, and a new simulated abundance profile is input. The density profile is fixed at the best-fit value, as the temperature error is the dominant error source. The reconstructed three-dimensional temperature profiles, with 1σ uncertainties from simulations, are also shown in Figures 3–5. There are groups where a central corona exists (e.g., Sun et al. 2007), so naturally the temperature gradient at 5–20 kpc is large in these cases (e.g., A2462, A160, ESO 306-107, HCG 51, and NGC 6269). Sometimes the three-dimensional temperature appears too low within 5–10 kpc radius (e.g., A160 and NGC 6269), which however affects none of our results in this work as core properties are excluded.

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We extract the surface brightness profile in the 0.7–2 keV band, as suggested by V06 to avoid the 0.6 keV hump in the soft CXB. Point sources and chip gaps are excluded. The scaled stowed background is subtracted so only the group emission and the local CXB is left. Some previous analysis on a surface brightness profile involved the correction of a single exposure map. However, the Chandra exposure map is energy and position dependent. It is not accurate to use a single exposure map (even one convolved with the group spectrum) as there is spectral variation in a group. The most accurate approach is to generate response files for each bin of the surface brightness and convert the raw count rate (without corrections on vignetting and other response) to density, from the derived three-dimensional temperature and abundance profiles and response files. To achieve that, we use XSPEC to generate an MEKAL emissivity matrix that depends on temperature, abundance and position of the radial bin for surface brightness (or response files there), once the absorption is determined. The ranges for temperatures and abundances cover the observed ranges for any particular group. This emissivity matrix provides the conversion factor needed to transfer the observed count rate to the emission measure. We assume the following density model (with 11 free parameters):

$$\begin{eqnarray} n_{e}^{2}\!\!\!& =\!\!\!& n_{0}^{2} \frac{(r/r_{c})^{-\alpha }}{[1+(r/r_{c})^{2}]^{3\beta -\alpha /2}} \frac{1}{[1+(r/r_{s})^{\gamma }]^{\delta /\gamma }} \nonumber \\ \!\!\!&\!\!\!& + n_{02}^{2} [1-(r/r_{c2})^{\alpha _{2}}]^{3\beta _{2}/\alpha _{2}}\; (r < r_{c2}) \end{eqnarray} \tag{ 3 }$$

This model combines the profile used in V06 and the profile proposed by Ettori (2000) for cool cores. We find that this model provides very good fits for all groups in our sample. The density profile is then converted to the emissivity profile from the emissivity matrix. The emissivity profile is then projected along line of sight with the formula of geometrical deprojection (e.g., McLaughlin 1999). The local CXB can be added later with the known CXB spectra and the radial set of the response files. The resulting surface brightness profile can be compared with the observed one. To avoid the binsize effect that may especially affect outskirts as wider bins are required there, we further divide bins within 100 kpc to three subbins and bins beyond the central 100 kpc to six subbins. The conversion and deprojection are done in these subbins. They are later merged to compare with the observed profile. The density errors are estimated from 1000 Monte Carlo simulations, with 1000 corresponding simulated abundance and temperature profiles as inputs.

This method is similar to that used in V06, but we use the three-dimensional temperature and abundance profiles to convert count rates to density. Like V06, we also analyzed the ROSAT PSPC pointed observations for the purpose of constraining gas density profiles at large radii. The inclusion of the PSPC data is especially useful to z < 0.04 groups. We used the software developed by Snowden et al. (1994) to produce flat-fielded PSPC images in the 0.7–2 keV band (more exactly, R567 bands, or PI channels of 70–201). The images are further analyzed as described in Vikhlinin et al. (1999). Seventeen groups have sufficiently long PSPC data (listed in Table 1 with effective exposure), and we included the PSPC surface brightness profile in the modeling of the density profile. The probed outermost radius of the PSPC data for each group is listed in Table 2. We only used the PSPC surface brightness profiles outside the central 3' to avoid the point-spread function (PSF) correction in the core. In all cases, there is good agreement between the Chandra and PSPC surface brightness data. One example is shown in Figure 7.

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We first examined the K–T relations at 30 kpc, 0.15 r₅₀₀, r₂₅₀₀, r₁₅₀₀, r₁₀₀₀, and r₅₀₀ (Figure 11). Entropy values at these radii are K_30kpc, K_0.15r500, K₂₅₀₀, K₁₅₀₀, K₁₀₀₀, K₅₀₀, respectively (note the difference between K₅₀₀ and K_500,adi defined earlier). We chose 30 kpc to represent the entropy level around the BCG. Previous studies often discussed entropy at 0.1 r₂₀₀ (e.g., Ponman et al. 2003), which is about 0.15 r₅₀₀. We also performed the Bivariate Correlated Errors and Intrinsic Scatter (BCES) (Y|X) regression (Akritas & Bershady 1996) to these relations, and the results are listed in Table 5. The entropy values at 30 kpc radius hardly show any correlation with the system temperature. The K–T relation is stronger at 0.15 r₅₀₀, but the intrinsic scatter is still substantial. The large scatter of gas entropy around the center has been known for a while (e.g., Ponman et al. 2003). We also examined the connection between the luminosity of the central radio source (see Table 1) and the entropy scatter at 0.15 r₅₀₀. No correlation is found, before or after the temperature dependence of entropy is removed. However, in Section 7.2, we show that the entropy scatter at 0.15 r₅₀₀ is tightly correlated with the scatter of the gas fraction within r₂₅₀₀.

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Despite large entropy scatter from the core to at least 0.15 r₅₀₀, the K–T relations are tighter at r₂₅₀₀ and beyond. We derived the intrinsic scatter in these relations, using the method described in Pratt et al. (2006) and Maughan (2007). As shown in Figure 11, the intrinsic scatter reduces to 10% at r₂₅₀₀ and stays at that level to r₅₀₀. As the derived intrinsic scatter decreases with increasing measurement errors, could this trend be due to the increasing measurement errors with radius? To answer this question, we compared K₂₅₀₀–T₂₅₀₀ and K_0.15r500–T₂₅₀₀ relations. If we use the relative measurement errors of K_0.15r500 for the corresponding K₂₅₀₀, the intrinsic scatter only increases to 12%. Similarly, if we use the relative measurement errors of K₂₅₀₀ for the corresponding K_0.15r500, the intrinsic scatter only decreases to 28%. Therefore, the significant tightening of the K–T relation from 0.15 r₅₀₀ to r₂₅₀₀ is not caused by different measurement errors. Similarly, we find that the intrinsic scatter in entropy at radii from r₂₅₀₀ to r₁₀₀₀ is affected little by switching measurement errors. The K₅₀₀ values have large measurement errors and half of systems are extrapolated from r₁₀₀₀. Clearly more groups with K₅₀₀ robustly determined are required, but the intrinsic scatter in the current K₅₀₀–T₅₀₀ relation is consistent with the level from r₂₅₀₀ to r₁₀₀₀. Thus, groups behave more regularly from r₂₅₀₀ outward than inside 0.15 r₅₀₀.

We also include 14 clusters from V06 and V09 for comparison and to constrain the K–T relations in a wider temperature range (Figure 12 and Table 5). The measured slopes increase from ∼ 0.50 at 0.15 r₅₀₀ to ∼ 1 at r₅₀₀, mainly caused by the excess entropy of groups at their centers. The slope we find at 0.15 r₅₀₀, 0.494 ± 0.047, is consistent with what Pratt et al. (2006) found at 0.1 r₂₀₀ for 10 groups and clusters with the XMM-Newton data, 0.49 ± 0.15, but smaller than what Ponman et al. (2003) found at 0.1 r₂₀₀ with ROSAT and ASCA data, ∼ 0.65. We, however, caution that the V09 cluster sample lacks noncool core clusters. Pratt et al. (2006) also gave the K–T relation at 0.3 r₂₀₀, which is about r₂₅₀₀. The slope they found, 0.64 ± 0.11, is also consistent with our result, 0.740 ± 0.027. At r₅₀₀, the slope we found is consistent with the expected value from the self-similar relation (1.0). Interestingly, the derived K₅₀₀–T₅₀₀ relation agrees very well with the NKV07 simulations with cooling + star-formation (Figure 12), although the agreement is progressively worse with decreasing radius. The tightening of the K–T relation at r₂₅₀₀ and beyond was also reported in the NKV07 simulations, but the predicted K₂₅₀₀–T relation by NKV07 lies below all groups in our sample (Figures 11 and 12). The NKV07 simulations with cooling + star-formation achieve entropy amplification with strong condensation to drop dense materials out of the X-ray phase. The resulting stellar fraction is about twice the observed value as too much material has cooled. If the star formation is suppressed with more efficient feedback, entropy is lower in the NKV07 simulations, between the results from the simulations with cooling + star-formation and the nonradiative simulations, which further disagrees with observations. It seems that the most challenging task is to explain the excess entropy of groups at r₂₅₀₀. As shown in Borgani & Viel (2009), simply increasing entropy floor from preheating produces too large voids in the Lymanα forest. Thus, it is still an open question on how to generate enough entropy in groups from the center to r₅₀₀ and still preserve other relations like the condensed baryon fraction and properties of the Lymanα forest.

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We derived the ratios of the observed entropy to the baseline entropy from VKB05 at r₅₀₀, r₁₀₀₀, and r₂₅₀₀ (Figure 13). For comparison, we also include the entropy ratios for 14 clusters from V06 and V09. The entropy ratios are always larger than or comparable to unity at all radii. The average ratio decreases with radius, for both groups and clusters, although the decrease is more rapid in groups. For groups, the weighted mean decreases from 2.2 at r₂₅₀₀ to 1.57–1.60 at r₅₀₀, while the mean decreases only ∼ 9% for clusters. Thus, there is still a significant entropy excess over the VKB05 entropy baseline at r₅₀₀, in both groups and clusters. The weighted mean entropy of groups at r₅₀₀ is ∼ 18% larger than that of clusters. If we consider only the T₅₀₀ ≲ 1.4 keV groups, the weighted mean for seven groups is 1.68. Similar studies have been done with ASCA and ROSAT data (Finoguenov et al. 2002; Ponman et al. 2003). When the observed entropy values at r₅₀₀ are scaled with M^−2/3₅₀₀ (which is proportional to K⁻¹_500,adi), both works found that groups have on average twice the scaled entropy of clusters. Ponman et al. (2003) further concluded that the excess entropy is observed in the full mass range. Thus, the results from this work, V06 and V09 confirm the previous finding of the excess entropy at r₅₀₀ over the full mass range, but the magnitude of the excess is smaller, and the difference between groups and clusters is also smaller than previous results, ∼ 15%–20% versus ∼ 100% in Finoguenov et al. (2002) and Ponman et al. (2003).

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As pointed out by Pratt et al. (2006), the magnitude of the excess may be affected by the systematics of the M–T relation, or how robust the hydrostatic equilibrium (HSE) mass is. The NKV07 simulations show that the HSE mass is systematically lower than the real mass, and the difference is biggest in low-mass systems. The real mass is 45% higher than the HSE mass for 1 keV groups, while the difference reduces to 13% for 10 keV clusters. The best-fit M₅₀₀–T₅₀₀ relation from this work and V06 (Section 7.1) is close to the relation for the HSE mass in NKV07. Thus, if this bias is real, the actual entropy ratios at r₅₀₀ are smaller. For 1 keV groups, a 45% higher total mass means a 13% larger r₅₀₀. For a typical entropy slope of 0.7 in this sample (Section 6.3), the entropy ratio at r₅₀₀ is reduced by 17%. Similarly for 5 keV clusters, the entropy ratio at r₅₀₀ is reduced by 8% for an average entropy slope of 0.9 at large radii. Thus, this bias can explain only part of the entropy excess observed. The entropy baseline, we adopted in this work, is from the smoothed particle hydrodynamics simulations, while the Adaptive Mesh Refinement (AMR) simulations produce a baseline with 7% higher normalization (VKB05), likely because of its better capability to catch shocks. However, the entropy excess for groups at r₅₀₀ remains significant.

There are several mechanisms to achieve entropy amplification at r₅₀₀. The first idea relies on modification of accretion, without extra nongravitational processes (Ponman et al. 2003; Voit et al. 2003; Voit & Ponman 2003). If preheating or feedback in small subhalos that are being accreted can eject gas out of the halo and thicken the filaments significantly, the accretion may be smoother than the lumpy accretion in hierarchical mergers. Voit (2005) showed that the smooth accretion can generate ∼ 50% more entropy throughout the cluster than would lumpy hierarchical accretion. However, Borgani et al. (2005) argued that this entropy amplification effect is substantially reduced by cooling. The other ideas resort to nongravitational processes. As discussed in the last section, although the NKV07 simulations predict the K₅₀₀–T₅₀₀ relation very well, they produce too many stars. Thus, models with only cooling may not be enough. Borgani et al. (2005) showed that galactic winds from SN explosions are rather localized and cannot boost entropy enough at large radii. Thus, a feedback mechanism that can distribute heat in a very diffuse way is required. As groups are smaller than hot clusters, AGN outflows from the central galaxy can reach a larger scaled radius. One good example in this sample is 3C449 (Section 8 and the Appendix), with radio lobes extending to at least 3.7 r₅₀₀. Thus, it is still an open question to explain the entropy excess at r₅₀₀, especially in groups. But the much reduced entropy ratios from this work largely alleviate the problem. It is also clear that more systems with the entropy ratio constrained at r₅₀₀ are required for better comparison, especially T₅₀₀ ≲ 1.4 keV groups.

We also derived the entropy slopes at 30 kpc–0.15r₅₀₀, 0.15 r₅₀₀–r₂₅₀₀, r₂₅₀₀–r₁₅₀₀ (or r_det,spe) and r₁₅₀₀–r_det,spe (Figure 14). The scatter is large but the average slopes are about the same (∼ 0.7) beyond 0.15 r₅₀₀. The slope is always shallower than that from pure gravitational processes (∼ 1.1). Mahdavi et al. (2005) analyzed the XMM-Newton data of eight nearby groups and found their entropy profiles are best fitted by a broken power law with the break radius of ∼0.1r₅₀₀. Across the break radius, the entropy slope decreases from 0.92 to 0.42. As the break radius is very near the core, this kind of entropy profiles may be caused by a cool core within 0.1 r₅₀₀. From 0.15 r₅₀₀, there are systems in our sample with an entropy slope of around 0.42, but the weighted mean is significantly larger. The measured average slopes in this work are consistent with the result by Finoguenov et al. (2007; 0.6–0.7) for groups.

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One of the most important aspects of cluster science is to use clusters to study cosmology, which often involves derivation of a cluster mass function. Assuming hydrostatic equilibrium, X-ray observations can be used to derive the cluster mass (at least the HSE value), provided the radial coverage of the data is good. Often we estimate cluster mass from another X-ray observable used as a mass proxy. A frequently used mass proxy is gas temperature, so it is important to understand the cluster M–T relation. There had been a lot of work on the cluster M–T relation before the Chandra and XMM-Newton era (e.g., Finoguenov et al. 2001; Sanderson et al. 2003). However, as emphasized in V06, it is crucial to constrain the gas properties (both the temperature gradient and the density gradient) at large radii (e.g., around r₅₀₀). Systematics like the assumption of a polytropic equation of state and the inadequate fit of the density profile at large radii can bias the HSE mass to lower values (Borgani et al. 2004; V06). The M–T relations have also been constrained with Chandra and XMM-Newton (Arnaud et al. 2005 with the XMM-Newton data; V06 and V09 with the Chandra data). However, the number of clusters used to constrain the M–T relation is still small (10 in Arnaud et al. 2005 and 17 in V09). There are only four systems with temperatures of 2.1–3.0 keV in Arnaud et al. (2005) and only three systems with temperatures of 1.6–3.0 keV in V09 (two overlapping with the Arnaud et al. sample). Our paper adds more < 2.7 keV systems (23 tier 1/2 groups).

The total mass values and uncertainties at interesting radii are derived from the 1000 simulated density and temperature profiles. The determination of the best-fit value and the 1σ errors (at two sides) is mentioned in Section 4. One difference from the determination of entropy is that we include only simulations that produce physically meaningful mass density profiles (mass density always larger than zero). The derived M₅₀₀–T₅₀₀ relation with the tiers 1+2 groups is shown in Figure 15. The BCES fits are listed in Table 6. We also included 14 clusters from V09 to constrain the M–T relation in a wider mass range. Our results show that the M₅₀₀–T₅₀₀ relation can be described by a single power law down to at least M₅₀₀ = 2 × 10¹³ h⁻¹ M_☉. At T₅₀₀ < 1 keV, more systems are needed to examine whether the relation steepens or not. Our M₅₀₀–T₅₀₀ relation is steeper than but still consistent with V09's (1.65 ± 0.04 versus 1.53 ± 0.08). Our slope is consistent with the Borgani et al. (2004) simulations with nongravitational processes included (1.59 ± 0.05), especially if two groups at T₅₀₀ < 1 keV are excluded. The derived M₅₀₀–T₅₀₀ relation can also be compared with that by Arnaud et al. (2005), which has a slope of 1.71 ± 0.09. Arnaud et al. (2005) defined the system temperature as the overall spectroscopic temperature of the 0.1 r₂₀₀–0.5r₂₀₀ region, which should be close to T₁₀₀₀ defined in this work (note r₅₀₀ ∼ 0.66r₂₀₀). The data of the tiers 1 and 2 groups give T₅₀₀/T₁₀₀₀ = 0.96 on average. Then we find that the Chandra M₅₀₀–T relation constrained from 23 groups + 14 V09 clusters is 3%–18% higher than that by Arnaud et al. (2005) at 1–10 keV. We also note that at 1 keV, the normalization of the Chandra M₅₀₀–T₅₀₀ relation is 54% higher than that by Finoguenov et al. (2001). This is expected from the generally higher density gradient around r₅₀₀ derived in this work (Figure 16) than the typical values in Finoguenov et al. (2001; also see the Appendix of V06).

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As shown in Figure 15, our M₅₀₀–T₅₀₀ relation is offset from the true M₅₀₀–T₅₀₀ relation in the NKV07 simulations, from 33% lower at 1 keV to 9% lower at 10 keV. The agreement with the Borgani et al. (2004) simulations is better, from 18% lower at 1 keV to 6% lower at 10 keV. Interestingly, the M_500,HSE–T₅₀₀ relation from the NKV07 simulations is almost the same as the Chandra M₅₀₀–T₅₀₀ relation from 23 groups + 14 V09 clusters. NKV07 attribute turbulence as the extra pressure to deviate the HSE mass from the true mass (also see Rasia et al. 2004; Kay et al. 2004). Indeed for the ICM without a magnetic field, the dynamic viscosity is roughly proportional to T^2.5_ICM. Thus, it may not be surprising that cool groups can develop stronger turbulence. However, the ICM is magnetized and the real magnitude of the ICM turbulence is unknown. The NKV07 simulations only have numerical viscosity that is small. Simulations with viscosity at different strengths are required to better determine this bias term. High resolution X-ray spectra of the ICM may be ultimately required to constrain the turbulence pressure in the ICM.

Kravtsov et al. (2006) suggested a new mass proxy, the Y_X parameter (product of the gas temperature and the gas mass derived from the X-ray image, or M_gas,500T₅₀₀ in this work), which in simulated clusters has a remarkably low scatter of only 5%–7%, regardless of whether the clusters are relaxed or not. The agreement between simulations and observations is also better for the M–Y_X relation than the M–T relation (NKV07). We examined the M₅₀₀–Y_X,500 relation for 23 groups + 14 V09 clusters (Figure 15 and Table 7). Our results indicate that a single power law relation can fit the data very well. Our best fit (0.57 ± 0.01) is the same as the V09 best fit at Y_X > 2 × 10¹³M_☉ keV (0.57 ± 0.05), implying the groups aligned well with clusters. Our best fit is also consistent with the XMM-Newton result by Arnaud et al. (2007) (10 clusters at Y_X > 10¹³M_☉ keV), on both the slope (0.548 ± 0.027) and the normalization (within ∼ 3%). Maughan (2007) assembled 12 clusters at z = 0.14–0.6 from their work and literature (Y_X > 8 × 10¹³M_☉ keV) and found that the slope of the M₅₀₀–Y_X,500 relation is consistent with the fit to the V06 clusters (0.564 ± 0.009). Intrinsic scatter in both the M–T and M–Y_X relations is consistent with zero as the measurement errors for groups are large. If we simply move the best-fit lines up or down to estimate the range of the scatter from the best-fit mass values, the scatter in the M–Y_X relation is about the half that in the M–T relation. The slope is very close to the self-similar value of 0.6, especially if only tier 1 groups are included (0.588 ± 0.012). Our best fits lie between the true mass and the HSE mass from the NKV07 simulations, but the offset is much smaller than that in the M₅₀₀–T₅₀₀ relation. Thus, the Y_X parameter appears to be a robust mass proxy down to at least 2 × 10¹³ h⁻¹ M_☉. With the derived M–T and M–Y_X relations in this work, the slope of the M_gas,500–T₅₀₀ relation is 1.89 ± 0.05 (for tier 1+2 groups + V09 clusters), or 1.86 ± 0.06 (for tier 1 groups + V09 clusters). This value of the slope is consistent with the result by Mohr et al. (1999), 1.98 ± 0.18 (90% confidence).

We derived the enclosed gas fraction profile for each group. The enclosed gas fraction generally increases with radius, and this trend continues to the outermost radius in our analysis, as generally found in V06. The enclosed gas fractions for groups at r₂₅₀₀ have a large scatter (Figure 17). For groups with similar T₅₀₀, f_gas,2500 can be different by a factor as large as 2.5. Both the weighted average and the median of f_gas,2500 in our sample is 0.043, much smaller than the typical value of ∼ 0.09 for V09 clusters. This mean can be compared with the average f_gas,2500 by G07 (0.050 ± 0.011), and we note that the 16 groups in G07 are on average brighter than our systems in a similar redshift range. The intrinsic scatter of f_gas,2500 is tightly correlated with the intrinsic scatter of K_0.15r500, after the temperature dependence of both variables are removed from their relations with temperature (Figure 18). Groups with low f_gas,2500 have high K_0.15r500 relative to the average relations. Thus, the large scatter of the gas fraction within r₂₅₀₀ for groups is tightly correlated with the large scatter of entropy at 0.15 r₅₀₀, and likely also the large scatter of X-ray luminosities for groups. Group properties (e.g., luminosity and central entropy) have large scatter because of the large scatter of the gas fraction around the center (e.g., r₂₅₀₀). Groups are on average fainter than what is expected from the self-similar L–T relation, because groups are generally "gas-poor" within r₂₅₀₀, compared with clusters.

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The enclosed gas fraction within r₅₀₀ is also derived for tiers 1 + 2 groups (Figure 19). We added 14 clusters from V06 and V09 to constrain the f_gas–T₅₀₀ relations. The f_gas,500–T₅₀₀ relation has a slope of ∼ 0.16–0.22, depending on whether both tiers 1 and 2 are included (Figure 19). We also give the f_gas,500–M₅₀₀ relation from the BCES orthogonal fit. For tiers 1 + 2 groups + clusters:

$$\begin{equation} f_{\rm gas, 500} = (0.0616\pm 0.0060) \; h_{73}^{-1.5} \left(\frac{M_{500}}{10^{13}\; h_{73}^{-1}\; M_{\odot }}\right)^{0.135\pm 0.030}. \end{equation} \tag{ 8 }$$

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For tier 1 groups + clusters

$$\begin{equation} f_{\rm gas, 500} = (0.0724\pm 0.0078) h_{73}^{-1.5}\left(\frac{M_{500}}{10^{13}\; h_{73}^{-1}\; M_{\odot }}\right)^{0.093\pm 0.031}. \end{equation} \tag{ 9 }$$

We note that the NKV07 simulations predict f_gas,500 ∝ T^0.152₅₀₀ (or M^0.10₅₀₀), from their best-fit M₅₀₀–T₅₀₀ and M₅₀₀–Y_X,500 relations. The gas fraction predicted in simulations depends on the modeling of cooling (e.g., Kravtsov et al. 2005) and is often tangled with the problem of predicting the right stellar mass fraction in clusters. We also derived the enclosed gas fraction between r₂₅₀₀ and r₅₀₀ for 23 groups in tiers 1 and 2 (Figure 19). Combined with the V06 and V09 results for clusters, the average f_gas–T₅₀₀ has little or no temperature dependence with an average value of ∼ 0.12, although the measurement errors are not small. f_gas,2500-500 can also be derived as

$$\begin{eqnarray} &&f_{\rm gas, 2500-500} = f_{\rm gas, 500} \left(\frac{1 - a \frac{f_{\rm gas, 2500}}{f_{\rm gas, 500}}}{1 - a}\right) \\ &&(a = M_{2500} / M_{500} = 5 (r_{2500}/r_{500})^3). \nonumber \end{eqnarray} \tag{ 10 }$$

We use f_gas,2500 = 0.0347T^0.509₅₀₀ (from the BCES fit to 43 groups and 14 clusters, Figure 17) and f_gas,500 = 0.0708T^0.220₅₀₀ (from the BCES fit to 23 groups and 14 clusters, Figure 19). Combining our results on c₅₀₀ (Section 7.3) with V06's for clusters, roughly we have c₅₀₀ = 5.0(M₅₀₀/10¹³M_☉)^−0.09. Assuming an NFW profile, the r₂₅₀₀/r₅₀₀ ratio can be well approximated as r₂₅₀₀/r₅₀₀≈ 0.322 + 0.178 lg(1.523c₅₀₀) at c₅₀₀ = 1.3–5.0. Thus, combined with the M₅₀₀–T₅₀₀ relation derived in this work, we can estimate the f_gas,2500-500–T₅₀₀ relation. Indeed as shown in Figure 19, f_gas,2500-500 is nearly constant at 1–10 keV. The average f_gas,2500-500 is still ∼ 27% lower than the universal baryon fraction (0.1669 ± 0.0063 from Komatsu et al. 2009). However, one should be aware that the enclosed gas fraction still rises beyond r₅₀₀, as generally found in V06 and this work. We also note that the observed f_gas,2500-500 is consistent with what was found in the simulations of Kravtsov et al. (2005) with cooling and star formation. We conclude that the low gas fraction generally observed in groups is mainly due to the low gas fraction of groups within r₂₅₀₀, or the generally weak ability of the group gas to stay within r₂₅₀₀. Beyond r₂₅₀₀, the groups are more regular and more similar to hot clusters, as also shown by the smaller scatter of their entropy values at r ≳ r₂₅₀₀.

A natural question motivated by these results on entropy and gas fraction is: what is the fraction of the group luminosities within 0.15 r₅₀₀ or r₂₅₀₀? Detailed work on the group L_X–T relation is beyond the scope of this paper and will be presented in a subsequent paper with an extended sample (including nonrelaxed groups). Here we give the curves of the enclosed count fluxes for 17 groups that r₅₀₀ is reached by Chandra or (and) PSPC (Figure 20). As the previous L_X–T relations only used the global spectrum to convert the count rate to the group flux, Figure 20 can be regarded as the growth curves of the enclosed group luminosities. At 0.15 r₅₀₀, the fraction ranges from 13% to 69%. At r₂₅₀₀, the fraction ranges from 51% to 94%. Two extreme cases defining the boundaries are A1238 and AS1101, also two systems with similar T₅₀₀ but very different f_gas,2500 (Table 3). A system with a bright cool core (like AS1101) has a large fraction of the X-ray luminosity within 0.15 r₅₀₀. Its system temperature without excluding the central core (e.g., 0.15 r₅₀₀), therefore, is biased to a lower value. A system without a bright cool core (like A1238) has a small fraction of the X-ray luminosity within 0.15 r₅₀₀, and its system temperature without excluding the central core (e.g., 0.15 r₅₀₀) is usually biased to a higher value (Figures 3–5). All these factors contribute to the large scatter of the group L_X–T relation (e.g., Osmond & Ponman 2004). Thus, the group L_X–T relation can be significantly contaminated by the large difference in the cores (e.g., within 0.15 r₅₀₀).

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We also fitted the total mass density profile with the NFW profile and derived the concentration parameter, c₅₀₀ = r₅₀₀/r_s, where r_s is the characteristic radius of the NFW profile. V06 used an inner radius of 0.05 r₅₀₀, since the stellar mass of the cD is dominant in the center. The V06 sample is mainly composed of clusters. The groups in our sample have r₅₀₀ of 440–800 kpc, so the contribution of the stellar mass at 0.05 r₅₀₀ is still significant for low-temperature systems (see e.g., G07). Thus, we use a fixed inner radius of 40 kpc. The outer radius is the outermost radius for the spectral analysis (r_det,spe in Table 2). We derived the total mass with 1σ uncertainties at radii corresponding to the boundaries of radial bins for spectral analysis between 40 kpc and r_det,spe (Figures 3–5). The resulting mass density profile (at 4–10 radial points in this work) is fitted with an NFW profile and the uncertainty is estimated from 1000 Monte Carlo simulations. The results for 33 groups are present in Table 3 and are plotted with the system mass in Figure 21. For the other 10 groups, c₅₀₀ is very poorly constrained. Our errors on c₅₀₀ are larger than those in V06 and G07 as our errors on mass are larger.

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We compare our results with the best-fit c–M relation from G07. G07 gave the best-fit c_vir–M_vir relation. For their range of c_vir, c₅₀₀ ∼ 0.51c_vir (for c_vir = 10.35). For Δ = 101, we convert G07's best-fit relation to c₅₀₀(1 + z) = 3.96(M₅₀₀/10¹⁴M_☉)^−0.226 (adjusted to our cosmology). As shown in Figure 21, at M₅₀₀ > 4.5 × 10¹³M_☉, the G07 fit describes our results very well. But our results do not show significant mass dependence, so at M₅₀₀ < 4.5 × 10¹³M_☉, our results are systematically below the G07 fit, although the errors are not small. However, the difference mainly comes from three groups (NGC 1550, NGC 533, and NGC 5129) for which G07 found r_s = 41–46 kpc (adjusted to our cosmology), and our inner radius cut at 40 kpc prevents us from measuring such a small value of r_s. In fact, the overdensity radii of these three groups agree better between G07 and this work: r₂₅₀₀ = 206 ± 2 kpc (G07) versus 222 ± 6 from this work for NGC 1550, r₁₂₅₀ = 251 ± 2 kpc (G07) versus 275 ± 30 for NGC 533, r₁₂₅₀ = 217 ± 7 kpc (G07) versus 236 ± 13 for NGC 5129. Excluding these three groups, our results agree well with G07's. Thus, the difference mainly hinges on the determination of r_s that is related to the subtraction of stellar mass, while the results at large radii agree better. We also compare our results with the simulations of Bullock et al. (2001) and further work.⁷ As c₅₀₀ is sensitive to the halo formation time, smaller σ₈, Ω_M, and tilt drive c₅₀₀ smaller. We used the parameters derived in Komatsu et al. (2009; see the caption of Figure 21). As shown in Figure 21, our results are generally consistent with the Bullock et al. simulations. Detailed discussions on the difference of the observed concentration parameter from the prediction can be found in Buote et al. (2007). We should point out that both our and the V06 analyses do not subtract the stellar mass and the X-ray gas mass, while G07 subtracted both components. G07 also included the XMM-Newton data for most of their groups. Future work on c₅₀₀ may need very good measurement of the gas properties to >r₅₀₀ and careful modeling of the stellar and gas components (e.g., G07). The group dark matter mass profile may also need to be examined first to see whether a single NFW profile is the best fit.

We can estimate the enclosed baryon fraction from the cluster stellar fraction estimated before. Lin et al. (2003) gave M_*,500 = 7.30 × 10¹¹M_☉(T_X/1keV)^1.169 (adjusted to our cosmology) from the 2MASS data of nearby groups and clusters, which includes the stellar mass in cluster galaxies. With the M–T relation derived in this work from 23 groups and 14 V09 clusters, f_*,500,Lin = 0.0263(T_X/1keV)^−0.481. Gonzalez et al. (2007) included the intracluster stellar mass and found f_{*,500,Gonzalez} = 0.0380(M₅₀₀/10¹⁴M_☉)^−0.64 (adjusted to our cosmology). With our M₅₀₀–T₅₀₀ relation, f_{*,500,Gonzalez} = 0.0864(T₅₀₀/1keV)^−1.056. Adding the relation for f_gas,500, the total baryon fraction from groups to clusters can be estimated. As shown in Figure 19, there is substantial difference for groups. We also examined the stellar mass of the cD and its relation with other group properties. As shown in Figure 22, the stellar mass of the group cD (which is proportional to its K_s band luminosity) is weakly correlated with the system mass. Low-mass cDs generally reside in low-mass groups. We also examined the relation between L_Ks of the cD and f_gas,2500 but found no correlation. However, in low-mass systems with low f_gas,2500, the stellar mass of the cD can be comparable to the gas mass within r₂₅₀₀.

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We also searched for fossil groups in this sample. From Jones et al. (2003), fossil groups are defined as a bound system of galaxies with the R-band magnitude difference of the two brightest galaxies within half the virial radius larger than 2 mag. In this work, because we do not have homogeneous R-band magnitudes for group galaxies in this sample, we used the 2MASS K_s band magnitude, which is a good indicator of the stellar mass. To ensure large and blue spirals are not left out, we also checked NED and HyperLeda to examine the B-band magnitude difference. Jones et al. (2003) used r₂₀₀ as the virial radius, while we use the exact definition of the virial radius for our cosmology, r_vir = r_Δ (Δ∼  101). For the typical mass concentration of groups in this sample, 0.5 r_vir ∼ r₅₀₀. Thus, we examined the K_s- and B-band magnitude difference for group galaxies within r₅₀₀. Six fossil groups are selected: NGC 741, ESO 306-017, RXJ 1159+5531, NGC 3402, ESO 552-020, and ESO 351-021. RXJ 1159+5531 and ESO 306-017 are known fossil groups (Vikhlinin et al. 1999; Sun et al. 2004). We note that NGC 1132 was considered a fossil group (Mulchaey & Zabludoff 1999). However, NGC 1126 is 8.4' from NGC 1132 with a velocity difference of 438 km s⁻¹ (r₅₀₀ = 16.2' or 440 kpc for the NGC 1132 group). The 2MASS K_s magnitude difference is 1.49 mag, and the B magnitude difference is 2.07 mag. Thus, we do not consider NGC 1132 a fossil group based on the Jones et al. (2003) definition.

Although the sample is small and not representative, we examined whether these fossil groups populate a different position of the scaling relations than nonfossil groups. No significant difference in K–T, f_gas–T, and M–T relations is found. As shown in Figure 22, these fossil groups indeed have massive cDs. The c₅₀₀ of these fossil groups (four listed in Table 3, others poorly constrained) are ∼ 2.4–4.3, on average smaller than the average of this sample, which differs from the claim by Khosroshahi et al. (2007) that fossil groups have higher mass concentration than nonfossil systems.

Besides preheating (from SNe and AGN) and cooling, impulsive heating from the central AGN is often required to explain the observed scaling relations like L–T and K–T (e.g., Lapi et al. 2005). Strong shocks driven by the central AGN may boost the ICM entropy and create an entropy bump. This transient anomaly in the entropy profile may be detected in a large sample. We have searched for such entropy features in our sample and find two promising cases: UGC 2755 and 3C 449 (Figure 23). There are also three groups with a significant break observed in their surface brightness profiles (3C442A, IC1262, and A2462) that may be related to AGN heating. We briefly discuss them in the Appendix. Both UGC 2755 and 3C 449 host a strong FR I radio source with two-sided radio lobes. UGC 2755 has a central corona with a radius of ∼ 3 kpc (see Sun et al. 2007 for the connection of thermal coronae with strong radio sources). From 10 kpc to 80 kpc in radius, the surface brightness profile is very flat. Then there is a sharp break at ∼ 90 kpc. UGC 2755's radio lobes extend to ∼ 100 kpc from the nucleus in the NVSS image, which may naturally explain the entropy bump within ∼ 90 kpc. 3C449 also has a central corona with a radius of ∼ 3 kpc. Its entropy bump is at ∼ 40 kpc–100 kpc and less significant as that in UGC 2755. 3C449's radio outflow is spectacular and can be traced to at least 1.6 Mpc in radius from the NVSS image (compared to its r₅₀₀ of 433 kpc estimated from r–T relation). Nevertheless, the brighter inner part of the radio jets/lobes ends at ∼ 100 kpc from the nucleus. Thus, the entropy bump we observed in 3C 449 may represent the most recent heating event.

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Strong shocks are required to effectively boost entropy. Using the standard Rankine–Hugoniot conditions, the entropy increase after shocks is a sensitive function of the shock Mach number, and weak shocks have little effect on amplifying entropy. A shock with a Mach number of 1.2 only increases entropy by 0.44%. A Mach 2 shock increases entropy by 20%, while Mach 3.3 and 4.4 shocks increase entropy by 100% and 200% respectively. The entropy bump in 3C449 is only at the level of 20%, which can be produced by a single Mach 2 shock. The entropy boost in UGC 2755 is 1.5–2 times, which can be produced by a single Mach 3 shock. The adiabatic sound speed in groups is not high, 540–630 km s⁻¹ in 1.1–1.5 keV ICM for these two groups. In these low density groups, the ambient pressure is much lower than that in the dense cores of hot clusters. Shock deceleration may also be slower. The velocity of the outflow-driven shock is ∼f_P(P_kin/ρr²)^1/3, where ρ is the ICM density and f_P is a structure factor of order unity that depends on the outflow geometry and the preshock density profile (e.g., Ostriker & McKee 1988). n_e ∼ 10⁻³ cm⁻³ around the entropy bumps and P_kin ∼ 10⁴⁴ ergs s⁻¹ from the radio luminosities of two radio AGN and the relation derived by Bîrzan et al. (2004). The estimated velocity is then ∼1300f_P km s⁻¹, which is comparable to the requirement of the entropy boost. Therefore, radio outflows in these two groups are capable of driving Mach 2–3 shocks to produce the observed entropy bumps.

There are ten groups hosting a central radio source that is more luminous than L_1.4GHz = 10²⁴ W Hz⁻¹: 3C 31, 3C 449, UGC 2755, 3C 442A, A160, A2717, AS1101, A3880, A1238 and A2462. What is the typical X-ray gas environment around these radio sources? Six of them lack large cool cores (e.g., ≳ 30 kpc radius), 3C31, 3C 449, UGC 2755, A160, A1238 and A2462. However, all of them host a central corona with a radius of ∼ 3–8 kpc, typical for massive cluster and group galaxies as discussed in Sun et al. (2007). This component is reflected in their temperature profile, except for A1238 as it is faint. The spectrum of A1238's central source can be described by a ∼ 0.8 keV thermal component. Its X-ray luminosity and K_s band luminosity fall on the typical region for coronae and its properties are similar to ESO 137-006 in A3627 (a nearby bright corona, Sun et al. 2007). Four other groups (3C442A, A2717, A3880 and AS1101) host larger cool cores with a radius of ∼ 30 kpc or larger. The cool cores in 3C442A and A2717 are clearly disrupted, likely by the radio sources. Thus, all these strong radio sources have low-entropy ICM (<30 keV cm²) at the center.