General Results

The derived masses are independent of the assumed form of the potential. That is, the measured temperature profile has such small statistical errors that all potentials that give the observed temperature profile have the same enclosed mass (10). Of course, these models can disagree in the regime in which there are no data. For ASCA, with its angular resolution of ≈2′ and 18′ effective field of view, there are very few systems in which the mass within 100 h50 kpc or beyond 1.5 h50 Mpc are well constrained. The fraction of the mass that is baryonic, fb (that is, the sum of the gas mass and the galaxy mass with an assumed M/Lv≈5), ranges from 0.09 to 0.25 and may be a weak function of the total mass (11), in the sense that most of the low baryonic fraction clusters have relatively low (<4×1014 M0) total masses. In an encouraging agreement of theory with observation, the derived x-ray masses are in excellent agreement with the scaling relation of Evrard, Metzler, and Navarro (7), with a mean ratio of data/theory of 0.87±0.14.

Clearly, this high baryonic fraction disagrees with big bang nucleosynthesis and Ω=1 and with many simulations of cluster formation and evolution. Analysis of the x-ray-derived ratio of baryonic mass to dark matter inside a cluster allows an increase in fb with radius, but at radii <1 Mpc, this is not required. Recent studies of clusters at larger scales (12) show that most clusters have a decreasing temperature at radii >1/4 of the virial radius. Although this is not fully understood, it arises naturally in low Ω simulations and in simulations with large initial injections of thermal energy (see figure 5 in ref. 7).

As first shown by David, Jones, and Forman in 1994 (11), the “average” M/L derived from x-ray imaging mass estimates is M/L≈150 h50 with a fairly wide range. There is a wide range in the ratio of gas mass to stellar mass, as can be seen clearly in Forman and Jones (13). It is not clear how much of the apparent factor of ≈6 range in gas to stellar mass is due to a lack of published accurate optical photometry for low redshift clusters. Whether this ratio is a monotonic function of mass or has an intrinsic wide scatter is not yet clear, but it seems as if more massive clusters tend to have a higher ratio of gas to stellar mass (11). However, there are some objects, such as Abell 1060 and Abell 1204, that have the same optical richness and x-ray temperature but a factor 30 difference in x-ray luminosity. This may indicate that much of the observed range in gas to stellar mass is due to “cosmic” scatter rather than to a trend.

Comparison of Results with Other Methods

Optical Velocity Dispersion. The x-ray temperature agrees extremely well (on average) with the optical velocity dispersion (β=μmρσ2/kT≈1) (14); however, there is a real variance in the distribution (Fig. 1 a and b). This excellent correlation indicates that, to first order, both the gas and the galaxies are in the same potential, that there are not large radial gradients or anisotropies in the galaxies velocity tensor, and that the virial theorem is not a strongly biased estimator of the mass (15). However, detailed comparison of viral mass estimates (16) and x-ray mass estimates shows a range of ≈2 as expected from the use of the viral theorem (15). In a few clusters with published optical velocity dispersion profiles, it is possible to perform a comparison of the predicted velocity dispersion profile with the x-ray-determined mass profile, under the assumption of isotropic orbits and that agreement between the predicted and observed velocity fields is good.

The <M/L> from the x-ray technique agrees on average with that from virial analysis (12, 16). This is essentially a restatement of the agreement of x-ray temperature and optical velocity dispersion, combined with the roughly isothermal nature of the temperature and velocity dispersion profiles. However, there are clearly outliers (e.g., Abell 1689) in which the virial theorem mass is much larger than the x-ray value. This is presumably due to the effects of mergers that can result in “nonvirial” galaxy velocities and deviations from hydrostatic equilibrium in the gas.

Comparison with Gravitational Lensing Results. There is good agreement (better than 30%) in the derived mass from x-ray and strong and weak lensing measurements for ≈1/2 of the sample [e.g., PKS 0745 (17) and A2390, H.Bohringer, Y. Tanaka, R.M., Y.Ikebe, and M.Hahori, unpublished work], but for ≈1/2 of the sample, the lens mass is significantly greater than the x-ray mass. The sign of the disagreement is consistent with simulations (8, 9) that show that the lensing mass is biased high compared with the true mass because of the spatial correlation of mass and that the x-ray estimate is biased slightly low because of incomplete thermalization of shocks and the

FIG. 1. (a) Optical velocity dispersion compared with emission weighted average x-ray temperature for a large sample of low and high (z> 0.14) redshift clusters. The solid line is not a fit to the data but the expectation if the x-ray temperature and the cluster galaxy velocity dispersion were equal. Note the scatter and the appearance of a few objects with much higher velocity dispersion than expected from the x-ray temperature. These two clusters are also objects with strong gravitational arcs, (b) The distribution of β=μmρσ2/kT for a large sample of well measured clusters. The mean value=1, and the variance is real. This indicates that, although on average the cluster gas and galaxies have the same “temperature,” there is a real variation presumably due to cluster mergers, nonvirialization temperature, and velocity dispersion gradients and foreground/ background projection effects.

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