The Mass and Mass‐to‐Light Ratio for Clusters of Galaxies at Large Radii

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Abstract
We construct models describing the velocity field in the infall regions of clusters of galaxies. In all models the velocity field is the superposition of a radial systematic component that is assumed to be spherically symmetric and a "noise" component of random nature. The latter accounts for the combined effects of small-scale substructure and observational errors. The effect of the noise term is to smear out the caustic envelopes searched for by previous groups, resulting in models that resemble the observations quite well. When the systematic component is known, the infall velocity and the mass profile of the infall region of the clusters can be determined. Given a particular model, it is possible to calculate the distribution function f(cz, θ, m) at all points in the observable (cz, θ, m) phase space outside the virialized region. It is demonstrated, on simulated data, that the use of a maximum likelihood estimator enables identification of the correct model, even at realistic noise levels. To minimize systematic effects caused by deviations from spherical infall, observations from a number of clusters should be used when applying the method to a real data set. Combining data from different clusters is a straightforward procedure. In the models the mass-to-light ratio is allowed to vary freely with radius. Furthermore, the models do not require any knowledge of the virial region. Rather, such knowledge can be used to test the models' ability to produce realistic results by comparing the mass estimated by the models at small radii to the mass estimates obtained for the inner region using the virial theorem. As a corollary we show how distance information of the quality obtained using the Tully-Fisher relation enhances the likelihood signal. Such distance information might be used in the future either to strengthen the results for this type of model or to allow more advanced models to be used (e.g., models breaking the assumption of spherical symmetry). The method relies on observations of redshifts of galaxies at large angular separations from the centers of the clusters. Currently, such data exist only for Coma, to which we apply the method as an example. The estimated mass at r = 2.5 h-1 Mpc falls in the range 1.1-2.4 × 1015 h-1 M☉, depending on the model and on the level of the noise term. We find rturn to be of the order of 10 h-1 Mpc, a result that is rather robust against changes in the model and subdivision of the data in various ways. We find 0.6<Ω0.60/b0.75b is the bias parameter, when using the luminosity profile adopted in the models. Interpreting this result, one must recall that the luminosity function of Coma is not well known in the infall region, and that ideally the method should be used on several clusters simultaneously to minimize the effects of deviations from the spherical model. The method provides a promising way of measuring the infall velocities in the outer regions of clusters. However, a robust determination of cluster properties must await future redshift observations of galaxies in the outer regions of a number of clusters.

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