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Astron. Astrophys. 345, 439-447 (1999)

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1. Introduction

Clusters of galaxies, probably the largest known virialized systems, provide an important tool to study the large-scale structure of the universe (e.g. Peebles 1980); for example, they provide an estimate for the dynamical mass density parameter on scales of [FORMULA] 1 Mpc. Clusters can also distort the images of faint background galaxies by weak lensing providing a mean to study the cluster mass distribution and thus placing constraints on large-scale structure formation models (e.g. Miralda-Escudé 1995).

Several methods are used to determine the mass of clusters of galaxies: e.g. velocity dispersions (Bahcall 1977); X-rays of the intracluster gas (Jones & Forman 1984); and gravitational lensing (Tyson et al. 1990). In general, although the three previous methods yield similar results for the total mass of rich clusters, [FORMULA], they provide discordant values for the mass at different radii. This provides a motivation to have well characterized and tested methods for total mass and mass profile determination, owing to their importance e.g. in cosmology (White et al. 1993). We will deal here only with the first method mentioned.

Analysis of the structural parameters in clusters of galaxies show that they are not really spherically symmetric structures, but their shape appears to be somewhat triaxial (e.g. Plionis et al. 1991). Moreover, since some clusters still appear to be on the process of collapse, e.g. judging from the presence of substructures (West 1994), their velocity distribution is probably not isotropic. This may be particularly true at their outer parts where galaxies or material can be still infalling with predominantly radial orbits (White 1992).

Nevertheless, observational limitations and simplicity makes us to assume virial equilibrium and sphericity for clusters when estimating their mass or mass profile. But even in such ideal scenario, methods that use observed radial velocities and projected positions are still a matter of some discussion. On the one hand, for example, Heisler et al. (1985, hereafter HTB) concluded that the projected mass estimator (PME) is probably the best option for estimating the mass of clusters of galaxies from galaxy motions. On the other hand, Perea et al. (1990) reached the conclusion that the best estimator is the virial mass estimator (VME) and that, for example, anisotropy, a mass-spectrum, or the presence of substructure can lead to an overestimation of the mass if they are not properly taken into account.

Furthermore, it seems that the above mass estimators do not even provide adequate masses when applied to N-body simulations; this is more critical, since in these systems we have complete information on the positions and velocities of particles. For example, Thomas & Couchman (1992, hereafter TC) using N-body simulations of the formation of clusters of galaxies report that mass estimates based on the virial theorem underestimate the total mass by a factor [FORMULA], and that the PME yields a factor [FORMULA] too small. TC conclude that the PME is probably the best option to determine the mass using velocity dispersions.

More recently Carlberg et al. (1997a, hereafter CYE) and Carlberg et al. (1996) have raised concerns regarding the consistency of the usual form of the VME when the total system is not entirely sampled. In their investigation of the average mass of galaxy cluster they establish that the VME overestimates their total mass by [FORMULA]%, attributing this discrepancy to the neglect of the surface pressure ([FORMULA]) term in the continuous form of the virial theorem. A correction to virial mass estimates based on this [FORMULA] term is starting to be applied to clusters of galaxies by the community (e.g. Girardi et al. 1998). Note that this finding of CYE is contrary to the trend observed by TC for the virial mass estimate.

From our investigation of the VME and the PME presented below, we will show that although the [FORMULA] term can account for the mass overestimation when the VME is applied to a subsample of an equilibrium gravitational system, the correct application of the VME yields also a correct mass at different radii. The physical reason for the overestimation of mass by the VME in the previous situation lies in an incomplete consideration of the potential energy of the subsystem. We will show that when the potential energy is fully accounted for no discrepancy exists at any radius between the VME and the true mass for an N-body system. In an astronomical situation, however, the previous result relies, of course, on having knowledge of the total extent of the system which is a somewhat ambiguous matter; this relates to the problem raised by CYE. In practical terms, this also affects the use of the [FORMULA] term as a correction factor since one needs a fair knowledge of the system's equilibrium extent, in addition to the velocity dispersion profile of a cluster. On other hand, the VME has the virtue of not depending on the form of the orbital distribution of galaxies in a near-equilibrium cluster, hence being straightforward to apply.

Haller & Melia (1996) have considered also the problem of calculating the mass profile by use of the PME. One of their conclusions is that a boundary term, arising from the finite sampling of a gravitational system, may make appreciable contributions to the mass estimate at the inner regions of stellar systems. Their conclusion is similar to the one we reach here in an independent manner, but this boundary term is studied here to a larger extent using different `cluster' models and the physical reason for the overestimate of the mass is elucidated.

In this work we revisit and test the PME and VME for an isolated spherical system of identical particles with an isotropic velocity distribution. We will quantify in particular the effect of a boundary term usually neglected in the PME and provide bounds to the errors that one may have when using the VME on a subset of an equilibrium system with different profiles. In a future paper we will apply our results to obtain mass profiles of nearby clusters and compare them with e.g. the profiles derived from X-rays.

The plan of the paper is as follows. In Sect. 2, the standard PME formalism is revisited and tested using different spherical models considered appropriate for the description of clusters of galaxies; e.g. those having a `cusp' and a `core'. A correction term is explicitly obtained and tested with N-body experiments. In Sect. 3, the virial theorem, both in its discrete and continuous form, and the effect of the [FORMULA] term on the mass determination are investigated. Finally, in Sect. 4, our main conclusions are presented.

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999