Properties of the Intracluster Medium Assuming an Einasto Dark Matter Profile
I investigate an analytical model of galaxy clusters based on the assumptions that the intracluster medium plasma is polytropic and is in hydrostatic equilibrium. The Einasto profile is adopted as a model for the spatial-density distribution of dark matter halos. This model has sufficient degrees of freedom to simultaneously fit X-ray surface brightness and temperature profiles, with five parameters to describe the global cluster properties and three additional parameters to describe the cluster’s cool-core feature. The model is tested with Chandra X-ray data for seven galaxy clusters, including three polytropic clusters and four cool-core clusters. It is found that the model accurately reproduces the X-ray data over most of the radial range. For all galaxy clusters, the data allows to show that the model is essentially as good as that of Vikhlinin et al.(2006) and Bulbul et al.(2010), as inferred by the reduced .
Department of Physics, College of Science, Salahaddin
Kirkuk Street, Erbil, 44001, Iraq
Galaxy clusters, the largest known objects with quasi-relaxed structures in the Universe, are essential probes for tracing the growth of cosmic structures and testing cosmological models \citepmantz2010,allen2011,hann2016. Galaxies, ionized intracluster medium (ICM), and dark matter are considered to be the main components of galaxy clusters. The ICM, the diffuse plasma distributed between galaxies in a cluster, accounts for most of the baryonic constituent of galaxy clusters, with electron temperatures in the approximate range of 1 to 10 keV. Studies of the ICM provide insights into the formation and evolution of galaxy clusters \citep[see e.g.]lin2012,barnes2016. Furthermore, the ICM yields valuable information on the structure and mass distribution of galaxy clusters. For example, one can estimate the cluster’s gas and total masses from the ICM gas density and temperature profiles, under the hypothesis of hydrostatic equilibrium, i.e. that the pressure exerted by the gas is equally counterbalanced by the total gravitational potential. From these mass measurements, it is possible to estimate the gas mass fraction. This gas mass fraction can then be used, with a constraint on the baryon matter density from big-bang nucleosynthesis or Cosmic Microwave Background (CMB) measurements, to set a tight constraint on the total matter density \citep[see e.g.]ettori2013,mantz2014,applegate2016.
Throughout the past few decades, various models have been employed to analyze the structure and morphology of the ICM plasma, most notably the standard isothermal -model \citepcavaliere1976. It is found that the -model provides a reasonable fit to the density structure of the ICM plasma within relatively small radii \citep[see e.g.]mohr1999,laroque2006. However, this model is limited in its ability to reproduce all observed features of the density distribution of the ICM plasma. This is the case, for example, in relaxed galaxy clusters with central-cuspy profiles \citep[e.g.]pointecouteau2004. Moreover, deep X-ray observations with Chandra and XMM-Newton found that the mean temperature profile declines on a large scale \citep[see e.g.]vikhlinin2005,leccardi2008.
Accordingly, other approaches have been attempted to circumvent these issues. One such approach is to use the equation of state to relate the pressure (), density (), and temperature () of the ICM plasma in thermodynamic equilibrium, i.e. \citephoredt2004. Under certain physical conditions, one finds that the pressure and density are related by a polytropic equation of state as , where is the polytropic index, with values that are expected to range from 1 for an isothermal gas to 5/3 for an adiabatic gas. With the aid of the polytropic equation of state, several studies introduced a revised version of the -model to describe the observable properties of galaxy clusters \citep[see e.g.]markevitch1999,ettori2000. These studies found that the mass measurements of typical galaxy clusters under the assumption of the polytropic model can vary significantly within the virial regions, when compared to those derived under the isothermal assumption. However, \citetdegrandi2002 concluded that the polytropic -model does not provide a reasonable fit to the temperature measurements over the full cluster-radial range. Instead, the authors found that the temperature measurements can be better modeled by a phenomenological broken power law.
To describe a broader range of the ICM-related features, different modified versions of the -model have been proposed to model the distribution of the ICM plasma, most notably the \citetvikhlinin2006 model. These authors suggested a model with 17 free parameters to reconstruct the observable properties of the ICM gas. According to this model, the three-dimensional profile of the gas density is
while the gas temperature is given by
where is a phenomenological function that can be expressed as \citepallen2001
where is a free parameter, measuring the amount of central cooling, is the shape parameter, and is the cooling radius. Hence, 9 parameters (, , , , , , , , and ) are used to describe the gas density, and 8 parameters (, , , , , , , and ) for the gas temperature.
Although the \citetvikhlinin2006 model provides a good fit to the density and temperature profiles over the whole radial range, the model is no longer considered physically motivated, but rather is an ad-hoc model tailored to the observed data. Moreover, there are many degeneracies between the model’s best-fitting parameters, implying less precise estimates for their values. This could cause a major issue when one deals with relatively few data points or when one attempts to extrapolate outside the range of observed data.
This has motivated more physically-grounded models with a limited number of free parameters. \citetascasibar2008 presented a simple analytical model based on the assumptions that the ICM plasma is spherical symmetry and is in hydrostatic equilibrium. With only five parameters, the authors found that the model can reconstruct the gas density and temperature profiles of galaxy clusters yielded from the best-fitting parameters of \citetvikhlinin2006 with less than 20 discrepancy over most of the cluster-radial range.
More recently, \citetbulbul2010 proposed an analytical model to describe the observable properties of the diffuse ICM in galaxy clusters based on the assumptions that the ICM plasma is polytropic and is in hydrostatic equilibrium. In the polytropic state, the electron number density, , and temperature, , of the ICM gas are related using a simple power law \citepascasibar2003,
where and are the central electron density and temperature, respectively, and is the polytropic index. Under the assumption of hydrostatic equilibrium, one can relate the properties of the ICM gas to the total gravitational potential, ,
where is the electron density, is the mean mass per particle in units of the proton mass , is the electron pressure, and is the Boltzmann constant.
However, \citetbulbul2010 dropped the polytropic assumption to account for the gas cooling in the cluster center. Adopting a generalized form of the \citetnavarro1996 profile (NFW), with density slope in the outer regions controlled by a free parameter , the three-dimensional gas density and temperature profiles derived by \citetbulbul2010 are
respectively, with five free parameters (, , , , and ) to describe global-cluster properties and three additional parameters (, , and ) to describe the cluster’s cool-core feature.
In recent years, however, many observational studies \citep[e.g.]silva2013,umetsu2014 and high-resolution N-body simulations [[, e.g.]]hayashi2008,dhar2010,dutton2014,klypin2016 have indicated that the Einasto profile \citepnavarro2004 provides a better fit to the spatial-density distribution of dark matter halos than does the NFW profile. In this paper, therefore, I adopt the Einasto profile as a model for dark matter halos instead of the generalized NFW model used by \citetbulbul2010. The model presented in this work represents a slight variation in respect to the model introduced by \citetbulbul2010. The Einasto profile, which has three parameters, has a logarithmic slope that decreases inward more gradually than the singular two-parameter profiles. Furthermore, this three-parameter model allows the density profile to be tailored to each individual halo, thereby yielding improved fits \citep[see e.g.]navarro2004. Adopting this profile, I derive analytical expressions for the thermodynamic properties of the ICM gas relevant to X-ray observation. The model is tested with X-ray data of a sample of seven galaxy clusters. All of the clusters have sufficient signal-to-noise to enable accurate analysis for the radial profiles of projected gas density and temperature. Moreover, the model is compared with the \citetvikhlinin2006 model (Equations (1) and (2)) and the \citetbulbul2010 model (Equations (7) and (8)).
This paper is structured as follows: Section 2 describes the model; testing the model with X-ray Chandra data and comparing it with previous analytical models are presented in Section 3. Finally, in Section 4, I discuss and conclude the results. Throughout this paper, I adopt a CDM cosmology with , , and .
2 Modelling of the Properties of the Intracluster Medium
2.1 Einasto Profile
The model developed in the current paper is essentially the same as the one by \citetbulbul2010 explained in Section 1, but now the generalized NFW profile is replaced by the Einasto profile. A generalized form of the Einasto profile for a spherical density distribution at radius is
where and are the radius and density at which the logarithmic slope , and is the shape parameter.
Since dark matter is the dominant component in galaxy clusters, the Einasto profile is a good approximation for the total density profile. This density profile is further combined with the polytropic and hydrostatic equilibrium assumptions to derive the analytical expressions for the total mass, the electron density, and the electron temperature of the ICM gas.
2.2 Total Mass and Potential Profiles
The total mass of the galaxy cluster, which is mainly made up of dark matter, can be determined by integrating the Einasto profile \citepretana-montenegro2012
where is the lower incomplete gamma function, and
Figure 2.2 shows the total mass profiles for various values of the parameter. The total mass profiles are finite since the Einasto density profile cuts off exponentially at large radii.
For a spherically symmetric mass distribution, the cluster’s gravitational potential is related to the total gravitating mass by
where is the Newtonian gravitational constant. Using Equation (10), the gravitational potential can be found as \citepretana-montenegro2012
and is the upper incomplete gamma function, which is related to the lower incomplete gamma function, , through the complete gamma function,
2.3 Temperature and Density Profiles
Taking advantage of the power-law relation between the density and temperature of the ICM plasma (Equation (4)), the polytropic electron density is
2.4 Pressure profile
2.5 Cool-Core Component
Several studies have shown that the polytropic model can also be applied to cool-core clusters, after adopting a cool-core-corrected temperature profile \citep[see e.g.]landry2013. The temperature profile of such clusters that feature a decline in temperature in the central region can be parameterized by
Recent X-ray and SZ effect observations \citep[see e.g.]arnaud2010,sayers2013 have found that the ICM pressure profile, when scaled appropriately, follows a nearly universal shape, suggesting that it is relatively independent of morphology and dynamical state of the ICM gas. Accordingly, following \citetbulbul2010, in this work it is assumed that the pressure distribution is the same for polytropic and cool-core clusters (Equation (19)). The density profile for cool-core clusters, therefore, can be obtained using ,
To keep the hydrostatic equilibrium assumption (Equation (5)), these modified temperature (Equation (20)) and density (Equation (21)) profiles require to introduce a modified version for the total mass enclosed in radius ,
where the term is significant only in the central region (see Figure 2.2).
2.6 Surface Brightness Profile
The X-ray surface brightness, of a galaxy cluster along the line of sight, , is related to the electron density and temperature distributions of the ICM gas by \citepsuto1998
where is the cluster redshift, and is the X-ray spectral emissivity, which depends on temperature and metallicity, and is calculated in units of counts cm s using the average cluster temperature. Substituting Equations (20) and (21) into Equation (23), the radial profiles of the X-ray surface brightness profile can then be obtained by numerical integration.
Figure 2.6 depicts the radial profiles of the X-ray surface brightness with and without the phenomenological function for different values of the parameter. The effect of the phenomenological function is observable only in the cluster’s central region.
3 Testing Model With X-Ray Observations
3.1 Data Sample
To test the model, archival Chandra X-ray data of seven galaxy clusters were analyzed. The source name, the Chandra observation identification number, the exposure time, the redshift, the Galactic absorption, and the redshift reference of the cluster sample are listed in Table 3.2. The selected clusters have X-ray luminosities in the 0.12.4 keV band of erg s, with redshifts . These galaxy clusters are selected since they have a regular X-ray morphology, and show no or only weak signs of dynamical activity, and the images have sufficient signal-to-noise to enable accurate analysis for the radial profiles of the projected temperature and X-ray surface brightness. Many of these clusters have been studied in literature. In Section 3.8, I compare the mass measurements for Abell 1835, estimated from the current work, with those reported in \citetlandry2013.
3.2 Data Reduction
Data reduction was performed using the Chandra Interactive Analysis of Observations (ciao) version 4.9, with the latest calibration database (caldb) version 4.7.3. I reprocessed the Chandra data using the chandra_repro routine to perform the recommended data preparation, such as checking the source coordinate, filtering the event file to good time intervals, removing streak events, and identifying the bad pixels. This script generates an event file and a bad-pixel file. Since all observations were taken in VFAINT mode, events with significant positive pixels at the border of the event island were excluded by further filtering.
As part of the data reduction, background light curves were examined to detect and remove the flaring periods. For the background dataset, the light curve is generated in the 0.312.0 keV energy band, following the recommendations given by \citetmarkevitch2003. The light curve is analysed using the lc_sigma_clip routine provided by the python script lightcurves.py. This routine removes data points that lie outside a certain sigma value from the mean count rate (a clip was used in this work).
3.3 Background Subtraction
An important aspect of analysis of extended objects is the background subtraction. For this purpose, blank-sky backgrounds were extracted for all observations, processed and reprojected onto the sky to match the cluster observation. Although the background spectrum is remarkably stable, there are short-term and secular changes of the background intensity of as much as because of charged-particle events. Following a method similar to that described in \citetvikhlinin2005, small adjustments to the background normalisation were applied to increase the accuracy of the background, based on data in the energy range 9.512.0 keV, where the effective area of the ACIS detector and the source emission are almost zero.
Besides charged-particle events, soft X-ray emission contributes to the blank-sky-background data. This soft background, which is likely to arise from differences in the extragalactic and Galactic foreground emissions between the source and blank-sky observations, was fitted to a thermal-plasma model. The soft background model was then scaled to the cluster-sky area, and was included as a fixed background component in the cluster’s spectral analysis.
3.4 X-Ray Images and Spectra
The X-ray images of the selected clusters were created in the energy range of 0.57.0 keV in order to determine the radial surface brightness profiles. This energy range was selected to minimise the high-energy-particle background, which rises significantly at low and high energies. An exposure-corrected image of the cluster’s selected region was created, and passed to the source-detection tool to detect and remove point sources and extended substructures. Count values in the point-source regions were replaced with those interpolated from their background. The surface brightness profiles were then extracted in concentric annuli centred at the Chandra selected centre.
Like the surface brightness profiles, X-ray spectra were also extracted in the energy range of 0.57.0 keV from concentric annuli centred at the Chandra selected centre, after excluding point sources and extended substructures. The X-ray spectrum of every galaxy cluster was fitted to a MEKAL model \citepmewe1985, modified by local Galactic absorption. The temperature, abundance, and normalisation were left free in all annuli, whereas the cluster redshift, , and the Galactic absorption, , were fixed (see Table 3.2).
3.5 Sources of Systematic Uncertainty
The radial surface brightness and temperature profiles are subject to various sources of systematic error. The choice of the local background region is the major source of error in the calibration of the Chandra observations. I follow \citetbulbul2010 and consider a uncertainty in the imaging- and spectral-data analysis due to the variation of the count rates of different background regions. Contamination on the optical filter is another major source of uncertainty. This affects the spectral data by \citepbulbul2010, and is added in quadrature to each energy bin in the temperature data. Another possible source of uncertainty is the spatially dependent non-uniformity in the effective area of the ACIS detector. The spatial dependence of the detector efficiency scatters at a level of \citepbulbul2010. This uncertainty, also, is added in quadrature to each annulus in the count rates.
3.6 Model Fitting
A Monte Carlo Markov Chain (MCMC) approach is adopted, as illustrated in \citetbonamente2004, for the fitting process. The parameter space in this approach is explored by moving randomly from a set of parameters to another using the Metropolis-Hastings algorithm. This algorithm typically accepts a move to the new point with the likelihood higher than the old one. Hence, the algorithm gradually moves towards the highest likelihood regions, where the parameter values yield the best fit to the data.
The MCMC method is used to independently calculate the likelihood of the spatial and spectral data with the model. After binning the X-ray data, the log likelihood for the spatial data is given by
where , is the number of counts detected in bin , is the number of counts predicted by the model in bin , and is the measured uncertainty on .
For the spectral data, the log likelihood, , is the same as in the spatial case (Equation (24)), except here , where and are the measured and predicted temperatures, respectively, and is the measured uncertainty on . Then, the joint likelihood of the spatial and spectral models () is calculated, and the goodness of fit is tested using the statistic.
Adopting the MCMC approach, radial profiles of temperature and background-subtracted surface brightness are fitted to the model (Equations (20) and (23)). Five parameters are used to model the global-cluster properties, whereas three describe the cluster’s central region. At the beginning, all clusters were fit to the model letting all parameters, including the cool-core parameters, free to vary. For three clusters (Abell 2218, Abell 2050, and MACS J0647.7+7015), however, it is found that the shape parameter , suggesting that there is no need for the cool-core parameters for these clusters. For these three clusters, therefore, the parameter is set to 1, and then the X-ray data fitted to the model, i.e. using only the global parameters. I classified such clusters as polytopic clusters, i.e. do not possess a cool-core component, whereas the remaining clusters are classified as cool-core clusters.
The values of the best-fitting parameters for the gas temperature (Equation (20)) and density (Equation (21)) profiles are listed in Table 3.6 (see Table 3.9 for their associated reduced ). The best-fitting surface brightness and temperature profiles of all clusters are presented in Figures 3.6 and 3.6. For most polytropic and cool-core clusters, the model accurately reproduces the X-ray surface brightness and temperature profiles over the most radial range. For the Abell 1835 and Abell 2050 clusters, however, the model does not fit well the temperature profile, particularly at intermediate to large radii.
3.7 Mass and Pressure Measurements
With the best-fitting parameters for the gas density and temperature profiles in hand, it is straightforward to determine the cluster masses. The gas mass can be obtained by integrating the density profile over a given volume,
where is the mean mass per electron.
The total mass can be obtained using Equation (22), with the parameter is set to 1 for polytropic clusters. Moreover, the pressure profile can also be obtained using Equation (19). In Table 3.7, I present the gas and total cluster masses enclosed within radii of and , corresponding to densities 2500 and 500 times the critical density of the Universe at the redshift of the cluster, respectively. Also listed in Table 3.7 are the pressure values obtained at these radii. Measurements reported in this table take account of the systematic uncertainties discussed in Section 3.5.
3.8 Comparison With Previous Measurements
The mass measurements for Abell 1835, from this study, are compared here with the results presented in \citetlandry2013. I estimate the cluster masses within the same angular radii as reported in \citetlandry2013. To allow a fair comparison of the cluster masses, their uncertainties on and are adopted.
Using the \citetvikhlinin2006 model, \citetlandry2013 estimated the gas mass of and total mass of within arcsec. These values are consistent, at the level, with the gas mass of and total mass of obtained in the current work for Abell 1835. Within arcsec, the predicted gas and total masses of Abell 1835 by \citetlandry2013, using the \citetvikhlinin2006 model, are and , respectively. These masses are about and larger than the corresponding masses of and given by this study within the same region.
Using the \citetbulbul2010 model, the estimates obtained in the \citetlandry2013 work for the gas and total masses are and , respectively, within arcsec. The corresponding masses predicted in this work are and . The former value is consistent well with that reported by \citetlandry2013, whereas the total mass is about larger than that given by these authors. Within arcsec, the estimated gas and total masses by \citetlandry2013 are and , respectively. The gas mass value is larger by about than the gas mass of derived by this study, but the total mass is statistically consistent with the total mass of estimated by the current study.
Overall, the mass measurements predicted from this study within the radii are in agreement with previous measurements. In the cluster’s outer regions, however, some of the mass measurements are statistically inconsistent with previous measurements. Such discrepancies could be attributed to the choice of model for fitting the X-ray data. \citetlandry2013 found that the choice of model may introduce uncertainties of and , respectively, to the measurements of the cluster gas and total masses at . The temperature profile could be another possible source for the discrepancy between these measurements. The temperature profile used in this analysis is slightly different from that used in \citetlandry2013, and I estimated that the mass measurements for Abell 1835 could be affected by adopting the current temperature profile up to within the radius.
In addition to the mass comparison, the radial pressure distribution for Abell 1835 is estimated using Equation (19), with the best-fitting parameters (Table 3.6), and then compared with those predicted by \citetlandry2013 using the \citetvikhlinin2006 and \citetbulbul2010 models, and assuming . Figure 3.8 shows the radial pressure profiles for Abell 1835 predicted from this work and \citetlandry2013 work, associated with their 68.3 confidence interval. This figure suggests that, despite the differences in the temperature profiles adopted by the current work and \citetlandry2013 work, the parameterized pressure profiles for Abell 1835 are more robust.
3.9 Comparison With Previous Models
The model proposed in the current work is compared with the \citetvikhlinin2006 model (Equations (1) and (2)) and the \citetbulbul2010 model (Equations (7) and (8)) in order to test which model provides a better fit to the X-ray data. For this purpose, the statistic is used as a metric to compare these models. Table 3.9 shows the combined per degree of freedom (reduced ) of the surface brightness and temperature associated with each fit for all studied galaxy clusters. As inferred by the reduced , all the models describe the data equally well. Similar to the \citetbulbul2010 model, however, the current model does not reproduce well the temperature profile at intermediate and outer radii for some clusters, such as Abell 1835 (Figure 3.6).
4 Discussion and Conclusion
In this work, I present an analytical model for the density and temperature profiles of galaxy clusters based on the assumption of hydrostatic equilibrium in the cluster’s gravitational potential. The model represents a variation of the model proposed by \citetbulbul2010. Here, the Einasto profile is adopted to model the spatial-density distribution of dark matter halos instead of the generalized NFW model used by \citetbulbul2010. This three-parameter profile is initially combined with a polytropic equation of state. Then, a cool-core correction is applied to the temperature profile and the gas density profile is derived under the assumption that the pressure profile is the same as in the polytropic case.
The model uses five parameters to describe the global properties of the ICM gas, with three additional parameters to describe the cluster’s core region. The main advantage of this model is the limited number of free parameters, which makes it simple and robust. The robustness feature is particularly important when one attempts to fit data that consist of few measurements. This can be helpful, for example, in galaxy clusters, where the X-ray count rate is low, particularly in the outskirts. Therefore, the proposed model represents a practical improvement compared to the model introduced by \citetvikhlinin2006, which has 17 free parameters. From a computational point of view, it is also more convenient to fit the observations to a model characterised by a few parameters. Another feature of this new model is the weak degeneracies between the best-fitting parameters, compared to the \citetvikhlinin2006 model (see Figures 3.6 and 3.6). This implies that more precise estimates are obtained for parameters.
The model is tested observationally with the X-ray data for polytropic and cool-core clusters. For most clusters, it is observed that the model is able to accurately fit the radial distributions of the ICM properties over the cluster’s full radial range. It is also shown that the model is essentially as good as that of \citetvikhlinin2006 and \citetbulbul2010, as indicated by the reduced . Similar to the \citetbulbul2010 model, however, the model does not fit the temperature profile well enough at intermediate and large radii for some clusters.
Besides its application to model X-ray data, the model can be applied to Sunyaev-Zel’dovich effect observations, making it useful for various cosmological studies. The model can be used, for example, to measure cosmic distances \citepbonamente2006, the cluster pressure profiles \citepbonamente2012, and the gas mass fractions \citepplanck2013. Furthermore, the model can be used to set up the initial conditions in cosmological numerical simulations, since it provides a simple and accurate description of the properties of the ICM plasma.
I am grateful to M. Birkinshaw for his useful comments, and I would like to thank the anonymous referee for valuable comments and suggestions which helped to improve the manuscript.
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