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Astron. Astrophys. 363, 440-450 (2000)

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5. Discussions

For the above simple models, we have shown that the X-ray deduced mass is consistent with that from the optical data over the scale of 1-3 Mpc under the assumptions of dynamic equilibrium. In a recent paper by Lewis et al. (1999), they also found the X-ray and dynamically deduced mass were consistent for a sample of CNOC clusters at [FORMULA].

On the other hand, in a study of a sample of clusters with giant arcs, Allen (1998) found that the X-ray deduced mass was consistent with the position of the giant arcs for cooling flow clusters but 2-3 times smaller than the lensing mass for non-cooling flow clusters. This was then explained as a direct consequence of the theory that cooling flow clusters were dynamically more relaxed than non-cooling flow clusters since cluster mergers would certainly disrupt a cooling flow. The cooling time for Abell 2104 is [FORMULA] yr at the centre, thus there is no evidence for a cooling flow in this cluster. Pierre et al. (1994) found a red tangential arc [FORMULA] from the centre of the cD galaxy (see Fig. 2). They found that the projected mass within the arc to be [FORMULA]. Here we examine if the arc feature is consistent with the simple cluster potential deduced from the X-ray data. Since the projected density must reach the critical value at [FORMULA], it requires [FORMULA] km s-1 for an arc redshift in the range [FORMULA] assuming the potential is spherically symmetric. However, the X-ray data gave [FORMULA] km s-1 apparently inconsistent with the lensing deduced value, indicating that in this very simplistic model the X-ray mass within the arc radius appears to be [FORMULA] times smaller than needed to produce the giant arc. Our result appears to be consistent with the results of Allen (1998). However, since the model we have adopted so far is very simple and the arc radius is relatively small ([FORMULA]), it is premature at this stage to suggest that the lensing results are inconsistent with the X-ray data under the assumptions of hydrostatic equilibrium and isothermal gas. As it was pointed out in Pierre et al. (1994), the small arc radius is an indication that the local cD potential is probably as important as the global cluster potential in forming the arc feature. Indeed for most clusters with giant arcs, the arc radii are barely larger than the PSPC resolution and probably a few times larger than the HRI resolution, hence an inconsistency between X-ray deduced mass from simple models and that of the strong lensing deduced mass are not sufficient to prove that the cluster is not in dynamic equilibrium. An alternative explanation for the results of Allen (1998) could be that the cooling flow clusters are well modelled by a cluster potential similar to the type given by Eq. 4, but non-cooling flow clusters have a different shape of gravitational potential, e.g. a mass profile that has a broad component in the outer parts of the cluster (e.g. Gioia et al. 1998). It would be difficult for the HRI to reject a model of this kind since it has a high background level and it would be easy to "hide" faint diffuse emission at large radii. In our study of Abell 2104, the current optical image does not extend to the extent of the X-ray emission, thus we need wide-field imaging to find out the true extent of the cluster.

The X-ray emission in the centre of the cluster shows strong ellipticity, the effect such asphericity has on the mass estimates needs to be addressed since the mass estimates given above were calculated under the assumption of spherical symmetry. Neumann & Böhringer (1997), estimated the effects of asphericity on mass estimates of CL0016+16, and found that the total mass was only changed by [FORMULA] when the ellipticity was taken into account. The ellipticity demonstrated in the X-ray image of Abell 2104 is no stronger than that of CL0016+16.

So far we have only considered the isothermal gas models, but the total mass given by Eq. 2 is more sensitive to [FORMULA] than [FORMULA]. It is necessary to explore models with a temperature gradient. Markevitch et al. (1998) found an almost universal decrease in temperature in the outer regions over a radius of 0.3 to 1.8 Mpc in a sample of 30 nearby clusters ([FORMULA]). They found that for a typical 7 keV cluster, the observed temperature profile can be approximated by a polytropic equation of state with [FORMULA]. If we assume that Abell 2104 has a similar large scale temperature profile, then we can quantify the mass ratio between the polytropic and isothermal models as


Since the X-ray emissivity has only a weak dependence on [FORMULA] over the 1-10 keV range (only a 10% change), the X-ray surface brightness varies insignificantly with [FORMULA]. We can then safely take the gas distribution as determined from the isothermal case (i.e. Eq. 3). Thus at [FORMULA] radius (1.46 Mpc), a model with such a temperature gradient would give a mass that is [FORMULA] times smaller than the isothermal case. This would cause the X-ray deduced mass to be strongly inconsistent with the dynamically deduced mass unless [FORMULA] increases with radius in a similar manner as [FORMULA]. Note that a temperature profile that decreases with the radius would also increase the total mass in the inner cluster regions compared to the isothermal model, and thus alleviate the discrepancy between the X-ray mass within the arc radius and the position of the giant arc. Fig. 10 shows the range of mass profiles deduced from the various methods and models discussed in the paper.

[FIGURE] Fig. 10. A comparison of 3D total mass distribution derived by the various methods. The solid curves give the range of X-ray deduced mass for isothermal gas; the dotted curves give the range of mass for the polytropic model. The dashed curves give the range of dynamic mass derived from the galaxy density distribution and velocity dispersion. The curves are plotted only for regions were data is available. The two stars show the range of mass estimates deduced from the position of the lensing arc.

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Online publication: December 11, 2000