AN ANALYTIC MODEL OF THE PHYSICAL PROPERTIES OF GALAXY CLUSTERS

The cluster gravitational potential is dominated by dark matter, with the intergalactic medium and stars contributing less than approximately ∼20% of the mass (Allen et al. 2004, 2008; Ettori et al. 2009; Vikhlinin et al. 2009). We therefore start with a total mass density distribution that is obtained as a generalization of the Navarro et al. (1996) profile:

$$\begin{equation} \rho _{\rm tot}(r)= \frac{\rho _{i}}{(r/r_{s})(1+r/r_{s})^{\beta }}, \end{equation} \tag{ 1 }$$

where ρ_i is the normalization constant, r_s is a characteristic scale radius, and β+1 is the slope of the density distribution at large radii. Equation (1) is a simplified version of the density distribution introduced by Suto et al. (1998).

The total mass enclosed within radius r can be found by taking the volume integral of the density (Equation (1)):

$$\begin{equation} M(r)=\frac{4\pi \rho _{i}r_{s}^{3}}{(\beta -2)}\left(\frac{1}{\beta -1} +\frac{1/(1-\beta) - r/r_s}{(1+r/r_{s})^{\beta -1}}\right). \end{equation} \tag{ 2 }$$

Equation (2) is indeterminate at β = 2; the limit at β = 2 can be determined using L'Hospital's rule:

$$\begin{equation} M(r) = 4\pi \rho _{i}r_{s}^{3} \left(\ln (1+r/r_s) - \frac{r/r_s}{1+r/r_s}\right)\ \ [\beta =2], \end{equation} \tag{ 3 }$$

and therefore the mass is a continuous function of β with no discontinuity at β = 2.

The gravitational potential at a distance r is found by

$$\begin{equation} d\phi (r) = G M(r)/r^2 dr \end{equation} \tag{ 4 }$$

using the boundary condition ϕ(∞) = 0. Equation (4) can be integrated analytically

$$\begin{equation} \phi (r) = \phi _{0} \left(\frac{1}{(\beta -2)}\frac{(1+r/r_{s})^{\beta -2}-1}{r/r_{s}(1+r/r_{s})^{\beta -2}} \right), \end{equation} \tag{ 5 }$$

where

$$\begin{equation} \phi _{0}= -\frac{4\pi G \rho _{i}r_{s}^{2}}{(\beta -1)}. \end{equation} \tag{ 6 }$$

The potential at β = 2 is also found by using L'Hospital's rule,

$$\begin{equation} \phi (r) = \phi _0 \left(\frac{\ln (1+r/r_s)}{r/r_s} \right) \ \ [\beta =2]. \end{equation} \tag{ 7 }$$

Therefore, the gravitational potential is a continuous function of β with no discontinuity at β = 2. Figure 1 shows the radial distribution of the gravitational potential for 1.0 ⩽ β ⩽ 3.0. The limiting value of β = 1 is shown in Figure 1 corresponding to a constant potential.

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The diffuse gas is assumed to be in hydrostatic equilibrium with the gravitational potential. Assuming spherical symmetry,

$$\begin{equation} \frac{1}{\mu m_p n_{e}(r)}\frac{dP_{e}}{dr} = -\frac{d\phi (r)}{dr}, \end{equation} \tag{ 8 }$$

where P_e(r) = n_e(r)kT(r) is the electron pressure, G denotes the gravitational constant, m_p is the proton mass, μ is the mean molecular weight of the plasma, k is the Boltzmann constant, and n_e(r) is the electron number density. In order to solve Equation (8), we assume that the gas follows a polytropic equation of state,

$$\begin{equation} \frac{n_{e,\rm poly}(r)}{n_{e0}}=\left[\frac{T_{\rm poly}(r)}{T_{0}} \right]^{n}, \end{equation} \tag{ 9 }$$

where n is the polytropic index, and n_e0 and T₀ are the values of the number density and temperature, respectively, at r = 0. The polytropic index n>0 is a free parameter of the model, with the limit n → ∞ describing an isothermal distribution of gas (Eddington 1926).

The temperature profile is obtained as a function of the gravitational potential from Equations (8) and (9),

$$\begin{equation} T_{\rm poly}(r) = -\frac{1}{(n+1)}\frac{\mu m_{p}}{k} \phi (r), \end{equation} \tag{ 10 }$$

and therefore, using Equation (5),

$$\begin{equation} T_{\rm poly}(r)=T_{0}\left(\frac{1}{(\beta -2)}\frac{(1+r/r_{s})^{\beta -2}-1}{r/r_{s}(1+r/r_{s})^{\beta -2}} \right), \end{equation} \tag{ 11 }$$

where the normalization constant T₀ is obtained from Equations (6) and (10):

$$\begin{equation} T_{0}=\frac{4\pi G \mu m_{p}}{k(n+1)}\frac{r_{s}^{2}\rho _{i}}{(\beta -1)}. \end{equation} \tag{ 12 }$$

Equation (10) shows that T_poly(r) ∝ ϕ(r), and therefore Figure 1 also describes T_poly(r) as a function of radius. Equation (12) links the gas temperature to the normalization of the matter density ρ_i, and therefore, the depth of the gravitational potential can be determined from the observed temperature profile.

Using the relation between temperature and gas density provided by the polytropic relation (Equation (9)), the polytropic gas density profile is

$$\begin{equation} n_{e,\rm poly}(r)= n_{e0} \left(\frac{1}{(\beta -2)}\frac{(1+r/r_{s})^{\beta -2}-1}{r/r_{s}(1+r/r_{s})^{\beta -2}} \right)^{n}. \end{equation} \tag{ 13 }$$

The gas pressure is obtained using the ideal gas law $P_{e}(r)= n_{e,\rm poly}(r) k T_{\rm poly}(r)$,

$$\begin{equation} P_{e}(r)= P_{e0} \left(\frac{1}{(\beta -2)}\frac{(1+r/r_{s})^{\beta -2}-1}{r/r_{s}(1+r/r_{s})^{\beta -2}} \right)^{n+1}. \end{equation} \tag{ 14 }$$

In the limit β → 2, the pressure is analytically described by

$$\begin{equation} P_{e}(r)= P_{e0} \left(\frac{\ln (1+r/r_s)}{r/r_s}\right)^{n+1} \hspace{0.5cm} [\beta =2]. \end{equation} \tag{ 15 }$$

The electron pressure for this model has only four free parameters, and it is suitable for the analysis of SZE observations of galaxy clusters (N. Hasler et al. 2010, in preparation).

Although the temperature profile predicted by the polytropic model provides a good description at intermediate to large radii, cool core clusters feature a significant temperature drop in the central region which cannot be approximated by a polytropic equation of state (see, for example, Vikhlinin et al. 2005, 2006; Sanderson et al. 2006; Baldi et al. 2007). For cool core clusters, we introduce a modified temperature profile

$$\begin{equation} T(r)= T_{\rm poly}(r) \tau _{\rm poly}(r), \end{equation} \tag{ 16 }$$

where T_poly(r) is the temperature profile according to the polytropic equation of state (Equation (11)) and τ_cool(r) is a phenomenological core taper function used by Vikhlinin et al. (2006):

$$\begin{equation} \tau _{\rm cool}(r)=\frac{\alpha +(r/r_{\rm cool})^{\gamma }}{1+(r/r_{\rm cool})^{\gamma }}, \end{equation} \tag{ 17 }$$

where 0 < α < 1 is a free parameter that measures the amount of central cooling, and r_cool is a characteristic cooling radius. The temperature profile modified by the core taper function is shown in Figure 2 for representative values of parameters r_cool, γ, and α.

Therefore, the explicit temperature profile for cool core clusters is given by

$$\begin{equation} T(r)=T_{0}\left(\frac{1}{(\beta -2)}\frac{(1+r/r_{s})^{\beta -2}-1}{r/r_{s}(1+r/r_{s})^{\beta -2}} \right)\tau _{\rm cool}(r). \end{equation} \tag{ 18 }$$

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In order to calculate the density distribution for cool core clusters, we assume that the pressure distribution is the same as in the polytropic case (Equation (14)). Therefore, the electron density is given by

$$\begin{eqnarray} n_{e}(r)&=&\frac{P_{e}(r)}{k T(r)}\nonumber \\ &=& n_{e0} \left(\frac{1}{(\beta -2)}\frac{(1+r/r_{s}) ^{\beta -2}-1}{r/r_{s}(1+r/r_{s})^{\beta -2}} \right)^{n}\tau _{\rm cool}^{-1}(r). \end{eqnarray} \tag{ 19 }$$

The behavior of the gas density for various core taper parameters is shown in Figure 2.

For hydrostatic equilibrium to be satisfied, these modified density and temperature distributions require a modified total mass distribution:

$$\begin{equation} M(r)=\frac{4\pi \rho _{i}r_{s}^{3}}{(\beta -2)}\left(\frac{1}{\beta -1} +\frac{1/(1-\beta) - r/r_s}{(1+r/r_{s})^{\beta -1}}\right)\tau _{\rm cool}(r). \end{equation} \tag{ 20 }$$

The only difference between the cool core total mass distribution (Equation (20)) and the polytropic total mass distribution (Equation (2)) is the term τ_cool(r), which is significant only at small radii. At large radii, the effect of the core taper vanishes, and the thermodynamics of the gas is described by the polytropic equation of state.

We use deep Chandra ACIS-I observations of four galaxy clusters to validate our models: two clusters which do not have a cool core component, MS 1137.5+6625 and CL J1226.9+3332, and two cool core clusters, A2204 and A1835. The observations are summarized in Table 1. As part of the data reduction procedure, we applied afterglow, bad pixel, and charge transfer inefficiency corrections to the level 1 event files using CIAO 4.1 and CALDB 4.1.1. Flares in the background due to solar activity are eliminated using light curve filtering as described in Markevitch et al. (2003). Filtered exposure times are also given in Table 1.

For the purpose of background subtraction, we use blank-sky observations. Given that the background is obtained from regions of the sky that may have different soft X-ray fluxes than at the cluster position, we use a peripheral region of the ACIS-I detector to model the difference between the blank-sky and the cluster soft fluxes. This step in the analysis is particularly important for A2204, which lies in a region of significantly higher soft X-ray emission than the average blank-sky region. Spectra and images used in this paper are extracted in the energy band 0.7–7.0 keV, chosen to minimize the effect of calibration uncertainties at the lowest energies and the effect of the detector background at high energy.

Spectra are extracted in concentric annuli surrounding the centroid of X-ray emission after all point sources were removed. An optically thin plasma emission model (APEC in XSPEC) is used, with temperature, abundance, and normalization as free parameters. The redshift and Galactic N_H of the four clusters are shown in Table 1.

We consider possible sources of systematic uncertainty in the Chandra data. The blank-sky background used in our analysis is normalized to the high-energy background level of each cluster observation, determined from peripheral regions of the ACIS detector that are free of cluster emission (following Markevitch et al. 2003). The primary source of uncertainty in the background subtraction is the choice of a peripheral region as a representative of the background at the cluster location. Due to the scatter in the count rate of various peripheral regions in each cluster observation, we estimate a ∼5% uncertainty in the determination of the background level from these Chandra observations. We use this uncertainty in the spectral and imaging data analysis.

Calibration of the ACIS effective area is another significant source of systematic uncertainty in our analysis. For the spectral data used for measuring the gas temperature, the primary source of uncertainty is the low-energy calibration of the effective area and the presence of a contaminant on the optical filter of the ACIS detector. We use the Chandra calibration available in CALDB 4.1.1, which includes a significant change in the effective area calibration which improves the agreement between cluster temperatures obtained with ACIS-I and also with other instruments (such as XMM-Newton's EPIC). With this calibration of the Chandra efficiency, we estimate that any residual systematic error in the measurement of cluster temperatures is of the order ∼5% and add this error to the temperature measured in each bin.

For the imaging data, spatially dependent non-uniformities in the ACIS efficiency are a relevant source of possible systematic error because of the extended nature of the sources. The absolute calibration of the ACIS efficiency is currently at the level of 3%, with possible spatial variations on arcmin scales at the level of ∼1%; we use a 1% error as an additional uncertainty in the count rates for each annulus.

Table 2 provides a summary of the uncertainties included in our analysis of the Chandra data and references to the Chandra calibration information.

The radial profiles of the X-ray surface brightness and temperature observed from the Chandra data are used to determine the best-fit parameters and the goodness of fit for MS 1137.5+6625, CL J1226.9+3332, A1835, and A2204. The X-ray surface brightness is

$$\begin{equation} S_{x}=\frac{1}{4\pi (1+z)^{3}}\int n_{e}^{2} \Lambda _{ee}(T) \, dl, \end{equation} \tag{ 21 }$$

where S_x is in detector units (counts cm⁻² arcmin⁻² s⁻¹), z is the cluster redshift, Λ_ee(T) is the plasma emissivity in detector units (counts cm³ s⁻¹), which we calculate using the APEC code (Smith et al. 2001), and l is the distance along the line of sight.

We validate the model using two polytropic clusters which do not have a cool core component, MS 1137.5+6625 and CL J1226.9+3332, and two cool core clusters, A1835 and A2204. We use the Monte Carlo Markov Chain code described in Bonamente et al. (2004; see Figure 3) for the fit. The model described in Section 2 has eight free parameters, of which five parameters describe the global cluster properties (n_e0,  T_e0,  r_s, β, and n), and three additional parameters (r_cool,  α, and γ) are used to model the central region of the cool core clusters.

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3.3.1. Polytropic Clusters MS 1137.5+6625 and CL J1226.9+3332

3.3.2. Cool Core Clusters, A2204 and A1835

The clusters A2204 and A1835 have a clear cool core component (see Figure 4), which requires the use of the cooling equations described in Section 2.4. The best-fit model parameters are listed in Table 3. We also calculate the gas mass by taking the volume integral of Equation (19) and the total mass using Equation (20) and report the results in Table 4.

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