A Chandra Study of Radial Temperature Profiles of the Intra-Cluster Medium in 50 Galaxy Clusters
In order to investigate the spatial distribution of the ICM temperature in galaxy clusters in a quantitative way and probe the physics behind, we analyze the X-ray spectra of a sample of 50 galaxy clusters, which were observed with the Chandra ACIS instrument in the past 15 years, and measure the radial temperature profiles out to . We construct a physical model that takes into account the effects of gravitational heating, thermal history (such as radiative cooling, AGN feedback, and thermal conduction) and work done via gas compression, and use it to fit the observed temperature profiles by running Bayesian regressions. The results show that in all cases our model provides an acceptable fit at the 68% confidence level. To further validate this model we select nine clusters that have been observed with both Chandra (out to ) and Suzaku (out to ), fit their Chandra spectra with our model, and compare the extrapolation of the best-fits with the Suzaku measurements. We find that the model profiles agree with the Suzaku results very well in seven clusters. In the rest two clusters the difference between the model and observation is possibly caused by local thermal substructures. Our study also implies that for most of the clusters the assumption of hydrostatic equilibrium is safe out to at least , and the non-gravitational interactions between dark matter and its luminous counterpart is consistent with zero.
Galaxy clusters are the most massive virialized systems in our Universe, whose gravitational potential wells are dominated by dark matter that accounts for up to about 90% of the total mass. Except for the gravity, however, the interaction between dark matter and its luminous counterparts is extraordinary small (e.g., Munoz et al. 2015), thus nearly all of the contemporary studies of clusters have been performed via observing the luminous components, i.e., highly ionized intra-cluster medium (ICM), stellar component, as well as various warm and cool gases, with an emphasis in the X-ray band since the ICM overwhelms other luminous components in mass by a fact of a few (e.g., Makishima et al. 2001 and references therein). The temperature of the ICM, which can be directly measured in space-borne observations, is a fundamental quantity that can be used to either characterize the thermodynamic status of the ICM, or calculate the total gravitating mass and various X-ray scaling relations (Xu et al., 2001), or interpret the Sunyaev-Zeldovich (SZ) effect (Lin et al. 2015). It also provides us with valuable information about astrophysical processes such as AGN feedback, thermal conduction, merger, and radiative cooling. Moreover, the knowledge about the ICM temperature also helps constrain the parameters of our cosmological models (e.g., Kravtsov & Borgani 2012).
Apparently, in order to carry out physical study of a cluster, an accurate determination of the ICM temperature and its spatial variation within a significant part of the virial radius is crucial. As a standard procedure, after the best-fit gas temperatures obtained for the spectra extracted from a set of adjacent annular or pie regions, both interpolation within each of these regions and extrapolation to larger radii are necessary to obtain a smooth gas temperature profile due to the limited capacity of today’s instruments. This invokes a relatively simple and, if possible, universal analytic expression for the temperature profile. As one of the first attempts, Allen et al. (2001) analyzed the Chandra pointing observations of seven relaxed clusters, which are located within a redshift range of , and proposed a 4-parameter empirical profile for the radial distribution of gas temperature. Using Chandra and XMM-Newton data, different but similar temperature profiles were introduced by, e.g., Vikhlinin et al. (2006) to calculate the total gravitating mass of 13 low-redshift relaxed clusters, Zhang et al. (2006) to determine the X-ray scaling relations of 14 distant X-ray luminosity, and Ascasibar et al. (2006) to probe the origin of the cold fronts in the ICM. These profiles and their analogs were evaluated and compared with each other by Gastaldello et al. (2007).
Despite the successful applications of the empirical temperature profiles listed above, the following two questions may still be raised: what is the intrinsic physics behind these empirical profiles and how reliable it is when we extrapolate these profiles out to the skirt region of a cluster? In this work, we address this issue by analyzing the X-ray data of a sample of 50 galaxy clusters (), which are drawn from Chandra’s 15-year archive, and fitting the observed radial temperature profiles with a physical model that takes into account the effects of gravitational heating during the halo collapse, thermal history (such as radiative cooling, AGN feedback, and thermal conduction), and work done via gas compression. In §2 and §3, we describe the sample selection criteria and data analysis, respectively. In §4, we introduce the model and use it to fit the observed temperature profiles. In §5, we select nine clusters that have been observed with Suzaku out to or even beyond ( is defined as the radius within which the mean enclosed mass density of the target is 500 times the critical density of the Universe at the target’s redshift), extrapolate the best-fit Chandra temperature profiles obtained with our new model to and compare the results with the Suzaku measurements. We also discuss the applicability of the hydrostatic equilibrium assumption and the constraint on the non-gravitational interaction between dark and luminous matters. In §6, we summarize our results. Throughout the work we adopt a flat CDM cosmology with density parameters and , and Hubble constant . Unless stated otherwise, we used the solar abundance standards of Grevesse & Sauval (1998) and quote errors at 68% confidence level.
2 Sample Selection and Data Preparation
In order to measure and characterize the spatial distribution of gas temperature in a galaxy cluster, both a high signal-to-noise ratio (SNR) and a balance between sufficient angular resolution and complete detector coverage of the target are necessary. To this end, we constructed our Chandra sample by applying the following selection criteria: (1) a large part of the target out to at least is fully or nearly fully covered by either the S3 CCD or the I0-3 CCDs of the Chandra Advanced CCD Imaging Spectrometer (ACIS) instrument, (2) the number of photons contained inside is no less than 12500 cts, and (3) the target exhibits a relatively regular appearance and possesses no significant substructures. As a result we drew 50 clusters from the Chandra archive, whose redshifts, average temperatures (see §3.3), and X-ray luminosities span ranges of , keV, and erg , respectively (Table 1).
For each observation, we followed the standard Chandra data processing procedure to prepare the data by using CIAO v4.4 and CALDB v4.4.8, by starting with the ACIS level 1 event files, which were collected with a frame time of 3.2 s telemetered in the FAINT or VFAINT mode. We removed all bad pixels and columns, as well as events with ASCA grades 1, 5, and 7 and carried out corrections for the gain, charge transfer inefficiency (for the observations performed after January 30, 2000), astrometry, and cosmic ray afterglow. By examining the light curves extracted in keV from source-free regions or regions less contaminated by the sources on the S3 or I0-3 CCDs, we identified and excluded time intervals contaminated by occasional particle background flares during which the count rate rises to of the mean value. When available, data of S1 CCD were also analyzed to cross check the determination of the contaminated intervals. All the point sources detected beyond the threshold in the ACIS images with CIAO tools celldetect and wavedetect have been masked in the analysis.
3 Data Analysis and Results
In order to construct the local background of each observation, we extract the spectrum from one or several separate boundary regions on the S3 CCD or I0-3 CCDs, where the thermal emission of the cluster is relatively weak but usually cannot be neglected, and fit the extracted spectrum in keV (e.g., Sun et al. 2009; Vikhlinin et al. 2005) with a model that consists of an absorbed thermal APEC component (the absorption is fixed to the Galactic value given in Kalberla et al. 2005; Dickey & Lockman 1990, and the abundance is set to if it is not well constrained; e.g., Panagoulia et al. 2014), an absorbed power-law component for the Cosmic X-ray Background (CXB; ; e.g., Mushotzky et al. 2000; Carter & Read 2007), a Galactic emission component (two APEC components with and keV, respectively; e.g., Gu et al. 2012; Humphrey & Buote 2006), and a particle-induced hard component calculated from the corresponding Chandra blanksky templates provided by the Chandra Science Center. The background template employed in this work, i.e., Galactic + CXB + particle components, can be determined after the best-fit is achieved. When available, we also cross checked our background templates with the keV ROSAT All-Sky Survey (RASS) diffuse background maps, where the particle component is minor, and obtained consistent results. In order to approximate the field-to-field variation of the Galactic and CXB background components in each observation, in the imaging and spectral analysis that follows we estimate the model parameter errors by taking into account both statistical and systematic uncertainties (10%; Kushino et al. 2002) in the background.
3.2 Radial Temperature Distributions of the ICM
In order to calculate the azimuthally averaged gas temperature distributions of the sample clusters, for each observation, we extract the Chandra ACIS spectra from concentric annuli, which are all centered on the X-ray peak and cover the . The width of each annulus is determined in such a way that a minimal photon count of 2500 cts in keV is guaranteed, while in the outmost annulus the condition that the photon count is at least twice the background is simultaneously satisfied. We perform the spectral model fittings by employing the X-ray spectral fitting package XSPEC v12.8.2 (Arnaud 1996), and limit the fittings in keV to minimize the effect of the instrumental background at higher energies and the calibration uncertainties at lower energies.
In the spectral analysis we use the XSPEC model PROJCT to evaluate the influence of the outer spherical shells on the inner ones, and fit the deprojected spectra by applying the optically thin thermal plasma model APEC (Smith et al. 2001), which is absorbed by the foreground photoelectric absorption model WABS. For each cluster, the column density of the WABS model is fixed to the corresponding Galactic value (Kalberla et al. 2005; Dickey & Lockman 1990), and the redshift of the APEC component is fixed to the value given in literature, which can be found in the NASA Extragalactic Database (NED). Whenever the metal abundance of the hot gas is not well constrained, we fix it to 0.3 ( e.g., Panagoulia et al. 2014). For the innermost annulus, we also attempt to add an additional absorbed APEC component to represent the possible cooler phase gas, the origin of which is often ascribed to the central dominating galaxy (e.g., Makishima et al. 2001). When the F-test shows that the fitting is improved at the 90% confidence level, we choose to use the two-phase gas model and define the temperature of the intergalactic gas as that of the hot phase. The obtained gas temperature distributions are shown in Figure 1, along with the errors calculated at 68 % confidences level.
3.3 Spatial Distributions of Gas Density, Gas Entropy and Total Gravitational Mass
We extract the X-ray surface brightness profiles ( is the two dimensional radius) in keV from a set of concentric annular bins that are all centered at the X-ray peak of the gas halo, and correct them by applying the exposure maps to remove the effect of vignetting and exposure time fluctuations. The exposure maps are created using the spectral weights calculated for an incident thermal gas spectrum that possesses the same average temperature and metal abundance as the cluster (§). We assume both hydrodynamic equilibrium and spherical symmetry so that the three-dimensional spatial distribution of the electron density follows either the -model,
or the double- model
when a significant central surface brightness excess is detected in the inner regions (e.g., Makishima et al. 2001), where and are defined as the core radius and slope parameter, respectively. Given the gas density distribution profiles and the radial distributions of gas temperature and metal abundance obtained by running cubic spline interpolation to the best-fit temperatures and abundances (§3.2), we model the extracted X-ray surface brightness profile as
where is the diffuse X-ray background, and is the cooling function. The electron density and gas entropy are determined when the best-fit to the observed surface brightness profile is achieved by minimizing the . Note that the entropy is the customary “entropy” used in the X-ray cluster field, in comparison with the classical definition (see Voit 2005a for a review).
The total gravitating mass of the cluster is calculated as
where is the mean molecular weight per hydrogen atom, is the Boltzmann constant, and is the proton mass. In order to extrapolate the obtained mass profile out to the virial radius, we employ the NFW model (Navarro et al., 1996)
where is the density of the total gravitational mass, and are free parameters in the NFW model. The average temperatures of the clusters are calculated by fitting the spectra extracted in , using the same method described in §3.2. We show the physical properties of clusters in Table 2.
4 Modeling and Fitting of the Observed Temperature Profiles
4.1 Effects of Gravity and Non-gravitational Processes
In this section we attempt to introduce an universal profile to describe the observed gas temperature profiles by taking account the effects of both gravity and non-gravitational processes, the latter includes the polytropic compression and thermal history (such as radiative cooling, AGN feedback, and thermal conduction) of the gas. For a small test gas element, whose position, mass, and particle number density are , , and , respectively, we mark the gravitational energy released during the halo collapse as , and the energy transferred to it by the non-gravitational processes as . Thus the total energy available to increase the internal energy of the test gas element is expressed as
Energy Released in the Gravitational Collapse
First let us consider the gravitational energy release of the test gas element during the collapse of the cluster. As in §3.3, we assume that at present time the gravitational potential of the system can be described by the NFW model (Navarro et al., 1996). Thus when the test gas element falls from infinity to its present position , the gravitational energy release is
where is the gravitational constant. This equation denotes the upper limit on the thermal energy increase that can be caused by the gravitational collapse.
Contributions of the Non-gravitational Processes
Next we investigate the energy transferred to the test gas element by non-gravitational processes. As a simple and reasonable assumption, this part of energy is attributed to the work done by gas compression () and the net heating supply determined by the thermal history which typically involves radiative cooling, AGN heating, and thermal conduction (), i.e., . Assuming that ideal gas undergoing polytropic processes that is characterized by the index during the halo collapse, the state of equations can be written as:
where , , and are the pressure, volume and temperature of the test gas element, respectively, is the universal gas constant, and is the mole number as is the Avogadro constant. Thus the work done by the surrounding gas during the compression is calculated by integrating from the initial state 1 to the final state 2
Now by applying the second law of thermodynamics, we use the entropy change to estimate the amount of net non-gravitational heating as . Clearly the change of the classical entropy satisfies , where as defined in §3.3. Thus we obtain
where is the scaling factor (Chaudhuri et al. 2012), is the observed entropy profile, and is the entropy profile predicted in the hydrodynamical simulation in which only the effect of gravitational energy release is considered. In our calculation, the observed entropy profile is derived by fitting the entropy distribution given in §3.3 with an empirical model
where , and are parameters constrained by observation. The model predicted entropy profile, on the other hand, is quoted from Voit et al. (2005b) as
where , , and are the parameters given by Eqs. 9, 10 and Figure 5 in Voit et al.
Fraction of Energy Transferred into the Thermal Form
Not all of the energy supply available in the gravitational collapse and non-gravitational processes have been transferred into the thermal energy of the test gas element, and part of them may have been mainly stored as kinetic energy. To account for this energy loss, we introduce an efficiency factor to represent the ratio of the energy transferred into the thermal form to the total energy supplied in both gravitational and non-gravitational processes, i.e.,
where , , , and as calculated in Battaglia et al. (2012).
Gas Temperature Profile
Substituting Eqs. 6, 7, 9, 10, and 14 into 13 , and noting that is the effective initial temperature defined under the quasi-static assumption, which is estimated to be of the order and is the observed temperature today, we obtain
where is the average initial energy for a single particle in the gas element, which should be approximately zero. Rewriting this immediate yields the temperature profile
where the temperatures are measured in keV. In this formula, is the initial gas temperature and is assumed to be . The parameter , where is a fixed combination of physical constants. is related to the scaling factor (c.f., Eq. 10), which equals in an isochoric process (e.g., Chaudhuri et al. 2012). , where is the polytropic index to be decided in the fitting. The parameters (c.f., Eq. 14), (c.f., Eq. 12), , and as well as their error ranges are constrained by observations, simulations and/or scaling relations in the Bayesian model fittings (§4.2).
4.2 Fitting Method and Results
By assuming that the parameters in our model are independent, which can be considered as a good approximation in practice, we use Eq. 16 to fit the gas temperature distributions observed with Chandra (§3.2) by running Bayesian regressions (e.g., Andreon & Hurn 2013; Andreon 2012). Compared with chi-squared test and maximum-likelihood estimation, this approach has the advantage that it incorporates observation errors in the model meanwhile it can quantify both the intrinsic scatter and the uncertainties of the known model parameters. To be specific, we apply Bayes theorem to express the posterior probability distribution as the product of the prior distribution, which is tightly constrained by the uncertainty ranges of known model parameters (c.f., Eq. 16), and the likelihood function determined according to the error ranges of gas temperature allowed by the observation, i.e.,
and maximize it with Powell’s method (Powell, 1964), which is powerful for calculating the local maximum of a continuous but complex function. When the maximum is found, the best-fit temperature profile is achieved. We plot the best-fit profiles in Figure 1, where the 68% error bands of the model computed by Monte-Carlo simulation are shown in shadow. To evaluate the goodness of the fittings, we introduce the model efficiency (See Nash & Sutcliffe 1970 and Engeland & Gottschalk 2002), which is defined as
where is the bin number of the observed temperature distribution, is the total number of Monte-Carlo simulation, and , and represent the simulated, model-predicted and observed temperatures, respectively. As shown in Table 2, for all of the clusters, the new profile proposed in this work gives an acceptable fit (), although in five cases () one data bin deviates the from model prediction slightly (68 % confidence level). To describe the degree of the violation to the hydrostatic equilibrium, we also plot the best-fit non-thermal energy fraction as a function of radius for all clusters in Figure 2.
5.1 A Comparison with Suzaku’s Results
In order to further validate our temperature profile model we select nine galaxy clusters (Table 3), seven of which are not involved in our sample due to either a small () field coverage or a low signal-to-noise ratio in the Chandra observation, calculate their gas temperature profiles with the ACIS S3 or I0-3 data, extrapolate the best-fit model profiles derived with Eq. 16 to , and compare the results with those measured with the Suzaku satellite (see references in Table 3), as shown in Figure 3. The difference between the temperatures measured with Chandra and Suzaku caused by the energy dependence of the stacked residuals ratios, (i.e., the energy-dependent difference between the effective areas of the two instruments even after careful calibrations; Schellenberger et al. 2015; Kettula et al. 2013; Nevalainen et al. 2010), has been compensated by using the data provided by Schellenberger et al. (2015)111The authors listed the Chandra and XMM-Newton measurements of gas temperature to show their difference, meanwhile it is known that the Suzaku and XMM-Newton measurements of temperature are consistent with each other in keV..
We find that within (roughly ) of seven clusters our model profiles are in good agreement with the Suzaku measurements at the 68% confidence level. For A1835 the model prediction is higher than the Suzaku measurement in , where a break on the observed gas entropy distribution is detected against the theoretical power-law distribution. This is an indication of the appearance of additional cooler gas, which can be associated with contacting regions between the cluster and large-scale structure environment (Ichikawa et al., 2013). For A2029 our model underestimates the Suzaku temperatures in , which might be due to substructures of the cluster.
5.2 Are the Sample Clusters in Hydrodynamic Equilibrium?
Hydrostatic equilibrium has been widely assumed in X-ray imaging spectroscopic analysis of galaxy clusters because, e.g., by adopting this assumption, the projection effect can be well restricted in measuring the X-ray mass, meanwhile the scatter of X-ray mass measurements is about a factor of two smaller than that of the lensing masses. However, both observations and simulations show that the X-ray mass measurements are generally biased low by due to violation of the hydrostatic equilibrium due to mergers and bulk motions of the gas (e.g., Meneghetti et al. 2010), especially in the skirt regions. When this violation occurs, we may employ the efficiency factor introduced in §4.1 to describe the degree of the violation as , which is typical inside and increases to at (Battaglia et al., 2012). In our model fittings of the Chandra temperature profiles of the sample clusters (§4.2), we usually obtain at small and intermediate radii, whereas reaches to at (Figure 2).
If we force throughout the cluster, we will obtain gas temperatures much higher than the observation in in some cases. As an example in Figure 3 we show the model predicted temperature profiles for the nine clusters discussed in §5.1 (light blue solid lines) when is set. In three clusters (A1835, A2142, and PKS 0745-191) the model profiles significantly deviate from the observed data, indicating that in these clusters the hydrostatic equilibrium is not well established in . For the rest six clusters, however, such a deviation is not seen in or even in within the error ranges. These results imply that for these clusters the assumption of hydrostatic equilibrium is safe at small and intermediate radii.
5.3 None Gravitational Interaction between Dark Matter and Baryonic Matter
The non-gravitational interactions between dark matter and luminous matter are expected to be very weak, therefore the kinetic energy of the dark matter can hardly be transferred to the hot gas. To test this we modify our model by adding a new free parameter to describe the percentage of the dark matter’s kinetic energy that has been transfered to the gas. Assuming that the baryonic fraction of the total gravitating mass is 16%, the model temperature profile Eq. 16 is modified to be
We investigate the gas temperature profiles in a sample of 50 clusters observed with Chandra by introducing a new temperature profile model, which takes into account the effects of both the gravitational heating, thermal history (such as radiative cooling, AGN feedback, and thermal conduction) and work done via gas compression, and use it to fit the observed temperature profiles by running Bayesian regressions. In all cases our model can provide an acceptable fit at the 68% confidence level. Also we find that by extrapolating the best-fit Chandra temperature profiles derived with our model to the results agree very well with the Suzaku measurements. With the new model we show that for most clusters assumption of hydrostatic equilibrium is safe out to at least .
This work was supported by the Ministry of Science and Technology of China (grant No. 2013CB837900), the National Science Foundation of China (grant Nos. 11125313, 11203017, 11433002, 61271349, and 61371147), the Chinese Academy of Sciences (grant No. KJZD-EW-T01), Science and Technology Commission of Shanghai Municipality (grant No. 11DZ2260700), and Shanghai Key Lab for Particle Physics and Cosmology (SKLPPC) (Grant No. 11DZ2260700).
|Name||Average temperatureaaAverage temperature is calculated for region.||( keV)|
|(keV)||( )||(kpc)||( erg )|
|Name||Obsid||RA||DEC||Redshift||Referencebb90% errors quoted from the references, except for A2029 and RXJ0747.5-1917, for which the errors are quote at the confidence level. The projection effect are not corrected expect for A2029 and PKS 0745-191.|
|A1413||5003||11:55:18.168||+23:24:24.49||0.1427||Hoshino et al. 2010|
|A1689||6930||13:11:29.472||-01:20:30.12||0.183||Kawaharada et al. 2010|
|A1795||10898||13:48:52.742||+26:35:27.63||0.0625||Bautz et al. 2009|
|A1835||6880||14:01:01.971||+02:52:40.88||0.2532||Ichikawa et al. 2013|
|A2029||4977||15:10:56.195||+05:44:43.36||0.07728||Walker et al. 2012a|
|A2142||5005||15:58:19.972||+27:13:58.50||0.0909||Akamatsu et al. 2011|
|A2204||7940||16:32:46.981||+05:34:31.87||0.1522||Reiprich et al. 2009|
|A780||575||09:18:05.352||-12:05:46.55||0.0539||Sato et al. 2012|
|PKS 0745-191||6103||07:47:31.430||-19:17:42.29||0.1028||Walker et al. 2012b|
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