Cluster Masses and Composition 2321 Mass Determination

The precise knowledge of the masses of the most massive, gravitationally confined objects is not only interesting as such, but it is also a prerequisite to many of the astrophysical and cosmological studies with clusters. The X-ray method of mass determination is based on the assumption that the ICM constitutes an atmosphere, which is approximately in hydrostatic equilibrium in the cluster potential described by the equation p VP = -G1ML(r). In most cases a spherically symmetric approximation is used and justified. Then the equation can be reformulated to yield the cluster mass profile

Gjmp \dlogr dlogr J

where mp is the proton mass and J is the mean molecular particle weight (~0.6 for a nearly fully ionized ICM plasma). The observables required for this equation are the absolute temperature profile TX (r) and the shape of the density profile p(r), which are both obtained from X-ray observations.

Since the X-ray emission is proportional to the squared plasma density (with a weak dependence on temperature), the ICM density can be reconstructed from the observed X-ray images if we make some presumptions on the three-dimensional geometry of the clusters that allows us the deprojection of the surface brightness distribution. Also the temperature of the ICM can be determined by X-ray spectroscopy. The thermal emission (mostly bremsstrahlung with some line emission and recombination radiation) leads to optically thin radiation in which every electron-ion collision results in an emitted photon. The shape of the spectrum depends on the temperature and chemical composition of the plasma, but it is independent from the density that determines its normalization. Therefore, temperatures and abundances can be reliably determined from the spectral appearance. The only remaining ambiguities are again that the observed spectrum is a projection of thermal emission along the line of sight and the existence of a range of different temperatures can not easily and unambiguously unfolded. Thus we observe what is often called an "emission measure weighted temperature" and further assumptions play a role in the unfolding and deprojection of the measurement (see e.g. [97]). However, the best observed spectra, e.g., in the X-ray halo of M87 [94] provide good support for an uncomplicated scenario with ionization equilibrium as a valid approximation, local isothermality, and smooth temperature variations as a function of radius, which allows us a good assessment of the structure of the ICM. With the known density and temperature distribution (1) can be solved for the integral mass profile.

The new X-ray satellite observatories XMM-Newton (ESA) and Chandra (NASA) now provide advanced observational capabilities to derive spatially resolved spectroscopic information, allowing us to reconstruct the density and temperature distribution. The Chandra Observatory provides a superior angular resolution of less than 1 arcsec showing many important details in the ICM distribution like cold fronts [152], shock waves [92], and X-ray cavities blown by AGN radio lobes [7,55]. The large X-ray collecting power of XMM-Newton provides good photon statistics for the construction of detailed X-ray spectra from different regions and high resolution spectra from the reflection grating spectrometer (RGS), yielding a good overview on the temperature structure of the clusters.

Figure 23.3 gives an example of one of the best X-ray studies of a galaxy cluster observed with XMM-Newton, of Abell 1413 [119]. The figure illustrates the accuracy of the determination of the cluster surface brightness and temperature

Radius (arcmin) Radius (kpc) Radius R (kpc)

Fig. 23.3 Left: Surface brightness profile (the dotted line shows a ^-model fit to the outer parts of the profile; the central excess is a signature of a cluster cooling core). Middle: temperature profile, and Right: gravitational mass profile of the cluster Abell 1413 determined from XMM-Newton observations by Pratt and Arnaud [119]. The right panel illustrates how well the profile can be fitted by the proposed, theoretically derived mass models by Navarro et al. [108] (NFW) and Moore et al. [103], (MQGSL)

Radius (arcmin) Radius (kpc) Radius R (kpc)

Fig. 23.3 Left: Surface brightness profile (the dotted line shows a ^-model fit to the outer parts of the profile; the central excess is a signature of a cluster cooling core). Middle: temperature profile, and Right: gravitational mass profile of the cluster Abell 1413 determined from XMM-Newton observations by Pratt and Arnaud [119]. The right panel illustrates how well the profile can be fitted by the proposed, theoretically derived mass models by Navarro et al. [108] (NFW) and Moore et al. [103], (MQGSL)

profiles. The temperature profile extends to about a radius of an overdensity of 500 over the critical density of the Universe, r500. This radius is often taken as a conservative measure to separate the well virialized part of the cluster from the infall and intermediate transition region (e.g. [50,156]). The mass profile resulting from the application of (1) is shown in the right panel of Fig. 23.3 as derived by Pratt and Arnaud [119]. The profile can be reasonably well fit with a model profile by Navarro et al. (NFW) [108]. This has also been found in a series of detailed inspections of mass profiles from XMM-Newton and Chandra (e.g. [28,116]), with the conclusion that most well-relaxed appearing clusters follow this description. Clusters showing some distortion, most probably due to recent merger activity, have cores that are often too flat to be fit by NFW profiles, however.

A possibility to test the mass measurement with an independent method is the comparison to the implications from the gravitational lensing effect of clusters. As the deepest gravitational potentials on large scale, galaxy clusters produce the largest observed angular deflections of light rays in our Universe. The more massive and compact galaxy clusters reach the critical projected mass density in the center necessary for the strong gravitational lensing effect, which gives rise to spectacular arcs (distorted images of background galaxies, e.g. [100]). It has been pointed out in some publications that the masses inferred from strong gravitational lensing tend to be larger than those determined from X-ray observations (e.g. [160]). For the comparison with the weak lensing effect which probes cluster masses on larger scales most often good agreement is found, however. Consistent measurements involving both lensing effects are in general also obtained in more regular cluster. The cluster A2390 that shows a prominent tangential and radial arc [114] provides a good example for the latter case as shown in Fig. 23.4 [1,10].

100 200 500 1000 2000 Radius (kpc)

radius (h50 1 Mpc)

Fig. 23.4 Comparison of different methods of mass determinations for the galaxy cluster A2390. Left: X-ray studies withROSAT and ASCA ([10], light shaded region), strong lensing mass ([114] green dot), weak lensing mass ([141] diamonds), and dynamical mass from detailed spectroscopic studies in the optical ([32], dashed line). Right: More recent mass determination based on Chandra data [1] with the same comparison to strong and weak lensing (data points)

100 200 500 1000 2000 Radius (kpc)

radius (h50 1 Mpc)

Fig. 23.4 Comparison of different methods of mass determinations for the galaxy cluster A2390. Left: X-ray studies withROSAT and ASCA ([10], light shaded region), strong lensing mass ([114] green dot), weak lensing mass ([141] diamonds), and dynamical mass from detailed spectroscopic studies in the optical ([32], dashed line). Right: More recent mass determination based on Chandra data [1] with the same comparison to strong and weak lensing (data points)

Prominent lensing clusters for which no good agreement has been obtained comprise, for example, Abell 2218 and CL0024+17, which are famous for their very spectacular Hubble Space Telecope images. CL0024+17, for example, shows several images of the same background galaxy. The X-ray studies with XMM-Newton show smaller masses than the lensing results and indications of a disturbed cluster structure in the central regions ([87,163] for CL0024+17, [120] for A2218). The clue to this mystery comes from the study of the galaxy velocity distributions in these clusters [39,69], which clearly indicate that both clusters are configurations of two major subclusters merging together in the line-of-sight. Most probably due to the projection of the mass of a whole filament, in which the merging clusters are embedded along the line-of-sight, the two-dimensional mass density for lensing is greatly enhanced in these configurations, giving rise to such spectacular lensing objects. While the X-ray mass measurement provides a result on the mass of the collapsed and partly relaxed core of the structure, the lensing measurement comprises a larger fraction of the large-scale structure, which explains the observed discrepancies.

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