Shock heating of the merging galaxy cluster A521
A521 is an interacting galaxy cluster located at z=0.247, hosting a low frequency radio halo connected to an eastern radio relic. Previous Chandra observations hinted at the presence of an X-ray brightness edge at the position of the relic, which may be a shock front. We analyze a deep observation of A521 recently performed with XMM-Newton in order to probe the cluster structure up to the outermost regions covered by the radio emission. The cluster atmosphere exhibits various brightness and temperature anisotropies. In particular, two cluster cores appear to be separated by two cold fronts. We find two shock fronts, one that was suggested by Chandra and that is propagating to the east, and another to the southwestern cluster outskirt. The two main interacting clusters appear to be separated by a shock heated region, which exhibits a spatial correlation with the radio halo. The outer edge of the radio relic coincides spatially with a shock front, suggesting this shock is responsible for the generation of cosmic ray electrons in the relic. The propagation direction and Mach number of the shock front derived from the gas density jump, , are consistent with expectations from the radio spectral index, under the assumption of Fermi I acceleration mechanism.
Subject headings:galaxies: clusters: general, — galaxies: clusters: individual( (catalog A521)), — galaxies: clusters: intracluster medium, — shock waves
Collisions between massive galaxy clusters are the most energetic events in the present universe. Part of the kinetic energy released during these collisions is dissipated through supersonic shock fronts propagating in the intracluster medium (ICM) and turbulent motions. While heating the thermal component of the ICM, shocks and turbulence may also accelerate (or reaccelerate) relativistic particles (e.g., Cassano & Brunetti, 2005; Hoeft & Brüggen, 2007; Ryu et al., 2003; Pfrommer et al., 2006; Brunetti & Lazarian, 2007; Vazza et al., 2009). Radio observations probe these complex mechanisms through the detection of diffuse synchrotron emission from the ICM, in the form of giant radio halos, Mpc-scale radio emission in the cluster central regions, and radio relics, sharp-edged radio sources in the cluster periphery (e.g., Ferrari et al., 2008; Cassano, 2009; Venturi, 2011; Brunetti, 2011, for recent reviews)
Observable as sharp X-ray brightness and temperature discontinuities, few shock fronts have been detected so far, because they can only be visible in the brightest cluster regions and in particularly favorable projections (Markevitch et al., 2002, 2005; Russell et al., 2010; Finoguenov et al., 2010; Macario et al., 2011). Peripheral radio relics are believed to be shock fronts that propagated far outside the X-ray bright region, while still accelerating (or re-accelerating) electrons, which produce radio emission and quickly cool after the shock passes, resulting in a characteristic narrow feature (e.g., Ensslin et al., 1998; van Weeren et al., 2011). The physics of giant radio halos is probably more complex. Radio halos plausibly result from the (re)-acceleration and transport of relativistic particles in large turbulent regions of the ICM, although many aspects of the mechanisms generating radio-emitting electrons remain unclear (e.g., Brunetti, 2011, for a recent review). Sharp radio edges (and radio relics) are frequently observed at the border of giant radio halos, suggesting a possible link between merger shocks and the generation of turbulence in the ICM (e.g., Markevitch, 2010; Macario et al., 2011).
A521 is a moderately distant (z=0.247)
and X-ray luminous
(, Arnaud et al., 2000)
The present article will focus on the analysis of a deep observation of A521 recently performed with XMM-Newton , with particular goal to probe the ICM structure up to the outermost regions covered by the cluster radio halo, and radio relic. After discussing data preparation and analyzes issues in Sections 2 and 3, we present the various X-ray brightness and temperature features revealed by this observation in Section 4. We comment on the interplay between thermal and non-thermal components of the ICM in Section 5. Unless otherwise noted, any energy distribution is normalized as a probability density function, while confidence ranges on individual parameter estimates are 68 . In the following, intra-cluster distances are computed as angular diameter distances, assuming a -CDM cosmology with , , . Given these assumptions, an angular separation of 1 arcmin corresponds to a projected intra-cluster distance of 232.5 kpc.
|XMM-Newton||Centre coordinates||MOS1 effective||MOS2 effective||PN effective|
|obs. IDs||exposure time (ks)||exposure time (ks)||exposure time (ks)|
|0603890101 (S)||04h54m22.00s -1016’30.”||15.7 (50.8 %)||15.7 (68.9 %)||12.8 (36.1 %)|
|0603890101 (U)||04h54m22.00s -1016’30.”||64.4 (77.3 %)||64.5 (74.6 %)||57.6 (52.5 %)|
|Note. The fraction of the useful exposure time after solar-flare filtering is shown in brackets.|
2. Observations and data preparation
The EPIC-XMM-Newton data set is a dual observation of A521, performed with focal aim point and central EPIC-MOS CCDs located in the SE cluster outskirts (see Figure 1). In order to remove the contribution of soft proton flares, we filtered the histogram of the photon arrival times through a temporal wavelet analysis. A summary of the ‘good’ exposure time remaining on each of the three EPIC cameras is provided in Table 1. The average good exposure time is about 75 ks.
In order to perform imaging and spatially resolved spectroscopy, we binned photons in sky coordinates and energy , matching the angular and spectral resolution of each focal instrument. To map the surface brightness of extended sources, these photon counts may have to be normalized for spatial and spectral variations of the telescope effective area and detector exposure times. We thus associate an ‘effective exposure’ array, ,
to the photon event cube. Expressible e.g. in , is computed as a linear combination of CCD exposure times, , related to individual observations , with local corrections for useful CCD areas, , Reflexion Grating Spectrometer (RGS) transmissions
3. Data analysis
3.1. Background noise modeling
The cluster emissivity must be separated from an additive, spatially extended and mostly stationary background noise including false photon detections due to charged particle-induced and out-of-time events, but also the Cosmic X-ray Background (CXB), and some Galactic foreground components.
The XMM EPIC background is dominated by
the particle component, which is modeled from observations performed in the
Filter Wheel Closed (FWC) position during revolutions 230 to 2027 as for the
EPIC-MOS cameras, and 355 to 1905 as for the EPIC-PN camera. Following an approach proposed in
e.g. Kuntz & Snowden (2008) or Leccardi & Molendi (2008) this model
sums a quiescent continuum to a set of florescence emission lines
convolved with the energy response of each detector.
It is completed with a residual emission associated with
soft protons, presently only detectable in the case
of the EPIC-PN camera and modelled as a power-low
spectrum normalized to 1.4 cts in the
0.5–1. band. To account for two different spectral shapes in the soft and hard bands,
the quiescent continuum is modelled as the product of a power law with an
inverted error function increasing in the soft band. We set the emission line
energies to the values reported in Leccardi & Molendi (2008), while the soft proton residual is
modelled using an additional power law. Presumably due to differences in the collecting areas
of the imaging and readout detector regions, the EPIC-MOS quiescent continuum exhibit a small
emissivity gradient along the RAWY CCD coordinate, which has been measured
and taken into account in the model. Because the fluorescence lines exhibit a more complex
spatial variation (Lumb et al., 2002; Kuntz & Snowden, 2008), we modelled the
emissivity distribution of the most
Secondary background components include the Cosmic X-ray background and Galactic foregrounds. Being associated with real photon detections, these components are corrected for the effective exposure. The Cosmic X-ray background is modelled with an absorbed power law of index (see, e.g., Lumb et al., 2002), while the Galactic foregrounds are modelled by the sum of two absorbed thermal components accounting for the Galactic Ôtransabsorption emissionÕ (TAE; kT1 = 0.099 keV and kT2 = 0.248 keV, see Kuntz & Snowden, 2000). We estimate emissivities of each of these components from a “joint-fit” of all background components in a region of the field of view located beyond the boundary of X-ray emission in the SE cluster outskirt, but covered by the central MOS CCDs (see also Figure 2). This estimates yields 13.4, 28.0 and 29.5 cts. in the 0.5–1. band as for the two transabsorption and CXB components, respectively (/d.o.f. = 1.23). Our background model, , eventually includes a contribution for the EPIC-PN out-of-time count rate, which is estimated in each energy band as 6.3 % of all photon counts registered along the CCD columns.
3.2. Spectroscopic and surface brightness measurements
To estimate average ICM temperatures, , and metal abundances, , along the line of sight and for a given location of the field of view , we add a source emission spectrum to the background model, and fit the spectral shape of the resulting function, , to the photon energy distribution registered in the energy band (0.3–12 keV):
In this modeling, the source emission spectrum assumes a redshifted and absorbed emission modelled from the Astrophysical Plasma Emission Code (APEC, Smith et al., 2001), with the element abundances of Grevesse & Sauval (1998) and neutral hydrogen absorption cross sections of Balucinska-Church & McCammon (1992). The value has been fixed to , from measurements obtained near A521 in the Leiden/Argentine/Bonn Survey of Galactic HI (Kalberla et al., 2005). It is altered by the mirror effective areas, filter transmissions and detector quantum efficiency, and convolved by a local energy response matrix M(k,l,e,e’) computed from response matrixes files (RMF) tabulated in detector coordinates in the XMM-Newton -EPIC calibration data base.
To compute images and radial profiles of the intra-cluster gas distribution, we estimate a cluster surface brightness map, , from photon counts registered in a given energy band, and corrected for effective exposure and additive background. Assuming an average ICM energy distribution, , we define as a function of an effective exposure map, :
All parameters of are practically determined from spectral fitting of the main cluster emission spectrum: , , , while is estimated in a “soft” energy band ([.5–2.5] ) in order to lower the dependence of on .
3.3. Surface brightness, ICM density and temperature profiles
In the following, surface brightness and temperature profiles have been extracted within cluster sectors oriented approximately along the surface brightness gradients. We derived the radial surface brightness by averaging the surface brightness of Equation (3) in each profile annulus composed of N pixels , as follows:
The background contribution being estimated within a much larger area of the field of view than any sector annulus used to derive , we neglected any systematic uncertainty related to its modeling and estimated the variance on from a weighted mean of the local Poisson fluctuations in :
Projected temperatures and associated confidence interval have been computed within each annulus by fitting a uniform emission model to the data set. To do so, we averaged the emission models of Equation (2) associated with each pixel of the annulus, and estimated the model parameters , via a minimization.
These brightness and temperature profiles have been used to model the underlying density and temperature of the ICM, assuming spherical symmetry of the cluster atmosphere in the vicinity of the features of interest. This was undertaken by projecting and fitting parametric distributions of the three-dimensional (3D) emission measure , , and temperature, , to the observed profiles. In this modeling, projected brightness profiles are convolved with the XMM-Newton focus Point Spread Function (PSF), while projected temperatures are computed assuming the ‘spectroscopic-like’ weighting scheme proposed in Mazzotta et al. (2004). In Section 4.3, the ICM emission measure
3.4. Imaging and spectral-imaging
An image of the cluster is presented on the top panels of Figure 3. To obtain this image, , we corrected the EPIC-XMM-Newton raw photon image for spatially variable effective area and background flux, following Equation (3). The point-like sources have also been modelled by means of an isotropic undecimated B3-spline wavelet analysis (see e.g. Starck et al., 2007), and subtracted from the image.
A map of anisotropic details in the ICM structure is shown on the bottom-left panel of Figure 3. To create this image, we subtracted a wavelet filtered map of the photon rate, , from the photon rate itself, then smoothed the residual image with a gaussian function of typical width . The wavelet filtering has been performed by means of a soft 3 thresholding of B3-spline wavelet coefficients, the significance thresholds being directly computed from the raw —Poisson distributed— photon map, following the multiscale variance stabilization scheme introduced in Zhang et al. (2008).
In order to map the ICM temperature in A521, we used the EPIC-XMM-Newton data set and applied the spectral-imaging algorithm detailed in Bourdin et al. (2004) and Bourdin & Mazzotta (2008, , hereafter B08). Following this algorithm, a set of temperature arrays with associated fluctuations are first computed on various analysis scales , then convolved by complementary high-pass and low-pass analysis filters in order to derive wavelet coefficients. The wavelet coefficients are subsequently thresholded according to a given confidence level in order to restore a de-noised temperature map. Here, the signal analysis have been performed over 6 dyadic scales within an angular resolution range of = [1.7 – 110] arcsec. This was undertaken by averaging the emission modelled by Equation (2) within overlapping meta-pixels , and computing the and arrays by means of a likelihood maximization. The resulting ICM temperature map shown in Figure 3 was then obtained from a B2-spline wavelet analysis (see B08 for details) with coefficients thresholded to the 1 confidence level.
|Sector 1||Sector 2|
|Detector||Region 1||Region 2||Region 3||Region 4||Region 1||Region 2||Region 3|
|Note. The fraction of the total counts is shown in brackets.|
4. ICM thermodynamics
4.1. Intra-cluster gas brightness and thermal structure
The X-ray photon image of Figure 3 reveals us the complex morphology of the intra-cluster gas in A521. On large scales, a northern subcluster with comet shape is apparently falling on the main component. The photon image also reveals the strongly irregular morphology of the surface brightness, presenting various edges indicated with dashed lines. Some of these brightness jumps have been enhanced in the bottom-left image of anisotropic details. They are also noticeable on the surface brightness profiles of Figure 4 and 5. At the interface between the two main interacting cluster components, we observe two bow-shaped brightness jumps, and , joining each other to form a low brightness cross-shaped feature. A third brightness jump with higher curvature radius, , is crossing the southern cluster component from SE to NW, while a fourth one, , is visible at the South-east cluster outskirts.
The ICM temperature map of Figure 3 is strongly irregular, and presents various noticeable features. The northern sub-cluster is clearly cool (). The interacting region separating this cool core from the main cluster to the South appears hotter () and strongly disturbed. The cross-shaped brightness depression observable on the photon image seems to coincide with a hot cross (), in particular, along the brightness jump . The southern part of the main cluster is cooler () than the interacting region, in particular, to the South of the brightness jump .
Bringing together the brightness and temperature maps of Figure 3, we observe that the two brightness jumps and are associated with temperature increases as the brightness decreases, while the brightness jumps and are associated with a temperature decrement. The two jumps and are thus likely to be cold fronts separating the densest parts of the two sub-clusters from their interacting region, while and are probably shock fronts propagating outwards from the colliding clusters.
4.2. Cold fronts and shock heating at the interface between two interacting sub-clusters
One of the most striking features seen in our data is a cross-shaped brightness depletion separating the two colliding sub-clusters. This feature also corresponds to a temperature and entropy enhancement, in particular just outside the two cold fronts, and . What we see is probably shocked gas with high entropy being squeezed by the converging cool core remnants and flowing around the densest part of the two interacting clusters, without penetrating the two cold fronts. The projected layer of shocked gas would thus exhibit maximal temperature and entropy near the two cold fronts, where it is tangentially intercepted by the line of sight. This shocked gas layer might also partly overlay in projection the main sub-cluster from its boundary delineated by cold front , to the southern brightness jump, . A possible interpretation for the origin of this hot gas flow is illustrated on Figure 6. Originally located at the cluster boundary (if there is one), the high entropy gas may have been shock heated between the two clusters starting to interact. It would now expand over the cluster atmosphere, following shock fronts presently propagating outside the cluster cores. One of these shock fronts might be observed to the South of the main cluster as the brightness and temperature jump, .
|Density estimates||Temperature estimates|
|Shock front||Jump amplitude||Mach number||Jump amplitude||Mach number|
4.3. Shock fronts propagation to the cluster outskirts
The 2D gas brightness and temperature maps of Figure 3 suggest the brightness jumps and to be shock fronts propagating outwards the cluster center. Located at various distances from the cluster center, these two shock fronts might have been developed during two successive cluster collisions. In order to analyze ICM thermodynamics across these jumps, we extracted the brightness and temperature profiles shown on Figure 5, corresponding to the two sectors of Figure 7. An estimation of the cluster photon counts in each sector region is provided in Table 2. The brightness jumps and exhibit the typical shape of a projected spherical density jump, convolved with the XMM-Newton PSF. We model the underlying gas density and temperature profiles as two step-like functions with common jump location, following equations (6) and (7). A discussion about the validity of the assumption of the ICM spherical symmetry in the vicinity of the shock fronts is provided in the Appendix. The gas density and temperature distributions corresponding to this model are reported under the projected profiles on Figure 5. The 3D density and temperature jumps associated with these distributions are reported in Table 3, with confidence intervals estimated from the 68 % percentiles of a parameter sample matching several random realizations of the data set.
The direction of the temperature jumps is consistent with the shock front interpretation. The cold front hypothesis would instead imply a temperature increase across the jumps (), which is excluded by the data. Assuming two shocks propagating outwards in the main cluster, one should be able to estimate the shock Mach numbers from the Hugoniot-Rankine density, temperature or pressure jump conditions across the fronts. Such Mach number values are reported on Table 3. The Mach numbers independently estimated from the density and temperature jumps are consistent with each other, though estimates from the temperature jumps have larger uncertainties. We will hereafter use Mach number estimates for both shocks and , from their density jumps: and .
5. Non-thermal ICM emission
A521 hosts a radio relic in its southeastern peripheral region, and a rare low-frequency giant radio halo. In order to investigate the interplay between thermal and non-thermal components of the ICM emission, the 240 MHz radio image obtained from observations performed at the Giant Metrewave Radio Telescope (GMRT) has been superimposed on the X-ray photon and ICM temperature maps of Figure 8.
5.1. The A521 radio halo
The A521 radio halo has been discovered from low-frequency observations at the GMRT (240, 325, 610 MHz, Brunetti et al., 2008, see also Figure 8) and then studied in detail through a deep follow-up Very Large Array observation at 1.4 GHz (Dallacasa et al., 2009). Its very steep spectrum, with spectral index between 325 and 1400 MHz, suggests magneto-hydrodynamic turbulence to be responsible for the in-situ re-acceleration of the relativistic electrons (Brunetti et al., 2008). The radio halo is covering the cluster central region, exhibiting an E-W elongation and reaching the radio relic to the Southeast. When excluding the relic region, the halo appears spatially correlated with the cluster X-ray emission. There is an even better correlation between the radio brightness and the hottest regions of the ICM, –in particular, the radio brightness exhibits a quick drop across the shock.
The complex thermodynamics of the ICM in the cluster center hint at the possible origin of the turbulence that may re-accelerate non-thermal particles in the halo. The two cold fronts and may have developed K-H instabilities at large angles from the main cluster collision axis. As suggested by the spatial correlation between shock-heated regions and the radio emission, turbulence may alternatively have been generated behind the two shocks and , now propagating to the cluster outskirts. In addition, the merger disturbance has likely generated turbulence within the two subcluster core remnants.
5.2. Shock front propagation and the radio relic
A521 has been known to host a SE radio relic observed at various frequencies (Ferrari et al., 2006; Giacintucci et al., 2006, 2008, hereafter, G08). As shown in G08, the integrated synchrotron radiation in the relic exhibit a power-law spectrum with spectral index in the frequency range 235-4890 Mhz, with evidences of steepening of the radio spectrum with increasing distance from the eastern edge. As further noted in G08, the outer edge of the radio relic coincides with the X-ray edge , which we have shown in this work to be a shock front propagating to the cluster outskirts. As observed in several peripherical radio sources of galaxy clusters (see e.g. Brüggen et al. (2012) for a recent review), these facts support the shock electron (re)-acceleration to be at least partly responsible for the radio emission from the relic. Assuming diffuse shock acceleration for the origin of the emitting electrons, in the test particle approach the slope of the injection spectrum of cosmic rays is related to the shock Mach number, M, by (Blandford & Eichler, 1987) . This leads to a spectrum of electrons in the downstream region with slope (implying a synchrotron spectral index ) taking into account radiative losses and assuming stationary conditions. G08 thus predicted a shock propagation with Mach number, , from their measurement of . The steepening of the radio spectrum with increasing distance from the eastern edge further allowed them to predict a shock propagation to the cluster outskirts. The propagation direction and Mach number of the shock front , (cf. section 4.3), are fully consistent with this hypothesis.
It is further worth noticing that the X-ray edge corresponding to the shock front seems to extend in North–South direction more than the radio relic. A first interpretation for this limited extent of the relic might be that the shock would re-accelerate pre-existing relativistic electrons in the ICM. In this case the radio relic could reflect the spatial and energy distribution of the pre-existing electrons across the shock front. In line with this hypothesis, recent analyzes (Kang & Jones, 2007; Kang & Ryu, 2011a, b) suggest that the presence of pre-existing particles in addition to the thermal pool can significantly increase the average efficiency of the particle acceleration and the expected synchrotron emission at weak shocks (M 3). Differences in extension between the shock and the radio relic might alternatively indicate some changes in the efficiency of electron acceleration changes along the shock front, possibly due to local variations of the Mach number (e.g., Hoeft et al., 2008). In this respect the radio relic in A521 is located at the extremity of the NW/SE major galaxy alignment evidenced in Ferrari et al. (2003, see also Figure 9), where indeed recent accretion of subcluster material may have produced inhomogeneities in the ICM.
6. Discussion and conclusions
A521 is a complex cluster system where optical analyzes have revealed at least three galaxy groups to the SE, and four groups to the NW including the cluster BCG group coinciding with the X-ray peak (Ferrari et al., 2003). The X-ray morphology of the BCG group suggests an infall along a NNW–SSE direction (projected onto the sky plane), which is slightly offset with respects to the major NW–SE galaxy alignment (Ferrari et al., 2006). The cluster atmosphere exhibits various brightness and temperature edges associated with cold fronts and shock fronts, that our XMM-Newton data revealed.
The main two interacting gas components in the central region of this system are separated by a region of gas with lower density, higher temperature and entropy. We interpret this feature a flow of high-entropy gas being squeezed by two converging subcluster cores that are delimited by cold fronts. We suggest this high entropy gas to have been heated by shocks formerly developed when the two gas components started to interact. One of these shocks is currently observed to the South of the main component, with Mach number . The hot gas region separating the two interacting components appears spatially correlated with the cluster radio halo. The development of turbulence in the hot gas flowing between the two cool cores may be responsible for high energy electron re-acceleration, yielding the radio halo emission. Merger shock propagation and/or cold fronts may have contributed to the development of these instabilities.
A shock front is observed at the Southeast cluster outskirt. An X-ray brightness edge there has been hinted at by Chandra data (G08), though the statistical significance was marginal. The orientation of this shock front and its large distance from the cluster center suggest that it is associated with a cluster collision that has occurred prior to the current two-component interaction. Our Mach number for this shock, , is consistent with that expected from the spectrum of the radio relic in G08 under the assumption of Fermi I acceleration mechanism. As observed in X-ray follow-ups of other radio relics –A3667, Finoguenov et al. (2010); RXCJ1314.4-2515, Mazzotta et al. (2011)–, its detection supports the shock electron (re)-acceleration to be at least partly responsible for the radio emission from the relic. The detection of a polarization of the relic would be an additional support for this process, complementary to the extension of its synchrotron spectrum to very high radio frequencies, and to evidences for spectral steepening downstream to the shock (G08). Delimited by the shock front, the radio relic seems however to subtend only a fraction of the shock front. Differences in the spatial extent of a radio relic and its companion shock front have also been observed in the colliding cluster RXCJ1314.4–2515 (Mazzotta et al., 2011), where a radio relic seems to be confined to a small section of the shock front presumably distorted by a nonuniform gas flow. These differences may thus reflect variations of the efficiency of particle acceleration across the shock that could be driven by local variations of the Mach number and shock velocity. Deeper X-ray or SZ observations may enable us to investigate this hypothesis, though the present XMM-Newton image does not seem to evidence any strong variation in the amplitude of the surface brightness edge, and thus in the shock Mach number. An alternative hypothesis is that the radio relic would reveal us local inhomogeneities in the properties of the pre-existing relativistic electrons, that would be re-accelerated by the shock passage. The observed connection between the radio halo and the relic may suggests that pre-existing relativistic electrons have first been accelerated by turbulent gas motions responsible for the radio halo emission, then re-accelerated at the shock front.
|Shock||ICM asphericity||ICM slope||Shock curvature||Density jump||Temperature jump||Mach number|
|()||()||()||(derived from )|
The ICM density and temperature distributions intercepting the shocks and has been modelled in Section 3.3 as two step-like functions, assuming the shock center of curvature and the ICM centroid coincide to coincide with each other.
The X-ray image of Figure 3 seems however to show that the shock fronts and S2 are less curved than the closest cluster brightness isophotes. To investigate the systematic uncertainties inherent to our spherical symmetry approximation, we alternatively tried to model the shock front and the ICM density as two spherical distributions with distinct centers. Assuming these two centers to be located in the plane of the sky, the ICM emission measure is now expressed per volume unit, as:
where and refers to the norm of each radius vector in the shock and ICM frames, respectively. Introducing and , the projection of these radius vectors onto the sky plane, a surface brightness profile intercepting the shock is obtained from integration of Equation (1) along the line of sight:
where is related to as a function of , the distance separating the shock from the center of the ICM distribution and , the angle separating the projected radius vector to the shock propagation axis (, see also Fig. 10). In addition to an ICM density slope, , and the shock curvature radius, density and temperature jumps, , and , respectively, the ICM emission measure thus depends on an asphericity parameter: .
We tried to invert and its parameters from a minimization of the distance separating (Equation 2) from the X-ray surface brightness profiles extracted across each shock front (see Figure 5). Some of the searched parameters being degenerated with one another, we first performed this inversion by fixing the asphericity parameter to 0 and 0.5, corresponding to shocks located at distances of and from the cluster center, respectively. We subsequently left all parameters free to vary and report the results of our measurements in Table 4, the confidence interval on each parameter being estimated from the 68 % percentiles of a parameter sample matching several random realizations of the data set. As expected, the shock curvature radius, density and temperature jumps obtained when fixing the asphericity to 0 are consistent with their estimates derived from the spherical model of section 3.3. A marginal difference in the amplitude of the density jump is still noticeable, since Equation (1) yields as for shock , while Equation (6) yields . This difference is probably related to the lack of any variation of the ICM density slope at the shock crossing, following Equation (1). Fixing the aspericity to 0.5 instead of 0 also marginally affect the density jump, essentially due to the degeneracy between the ICM asphericity and density slope. This degeneracy is noticeable in the case of , the shock front observed with the highest statistics. Leaving the ICM asphericity free to vary yields estimates of 0.45 and 0.75 in the case of and , respectively, consistent with the shock curvature radii observed on the X-ray image of Figure 3. The shock density, temperature jump and Mach numbers derived from these various assumptions are in any case consistent with one another, and with their estimates obtained from the sperical model of section 3.3. Given the limited statistics available, it is difficult to break the degeneracy between the ICM asphericity, ICM density slope and shock curvature radius in the vicinity of the shocks. For simplicity purposes, we consequently adopted the spherical model of section 3.3 in order to derive the amplitudes of the density jumps and Mach numbers of the two shocks and .
- affiliation: Dipartimento di Fisica, Università degli Studi di Roma ‘Tor Vergata’, via della Ricerca Scientifica, 1, I-00133 Roma, Italy; email@example.com
- affiliation: Dipartimento di Fisica, Università degli Studi di Roma ‘Tor Vergata’, via della Ricerca Scientifica, 1, I-00133 Roma, Italy; firstname.lastname@example.org
- affiliation: Harvard Smithsonian Centre for Astrophysics, 60 Garden Street, Cambridge MA, 02138, USA
- affiliation: NASA Goddard Space Flight Center, Code 662, Greenbelt, MD 20771, USA
- affiliation: Joint Space-Science Institute, University of Maryland, College Park, MD, 20742-2421, USA
- affiliation: Joint Space-Science Institute, University of Maryland, College Park, MD, 20742-2421, USA
- affiliation: Astronomy Department, University of Maryland, College Park, MD 20742, USA
- affiliation: INAF - Istituto di Radioastronomia, via Gobetti 101, I-40129 Bologna, Italy
- ; X-ray luminosity in the 0.1-2.4 keV band has been corrected for luminosity distance assuming km s Mpc, and .
- EPIC-MOS detectors share a common optical path with the RGS
- Information about these instrumental effects have been obtained from the XMM-Newton -EPIC Current Calibration Files (CCFs)
- Namely the Al, Si and Cu, Ni complexes as for the EPIC-MOS and EPIC-PN cameras, respectively.
- EPIC response matrixes are computed from canned RMFs corresponding to the observation period provided by the XMM-Newton Science Operation Centre.
- More precisely, the ICM emission measure per volume unit.
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