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The starting point for estimating the mass of a stellar or galactic system in dynamcal equilibrium is the time-independent Jeans equation:

Equation 1 (1)

Here n(r) is the number density of some "tracer" population (stars, galaxies, etc.), bar{v}(r) is their mean velocity, Phi(r) is the total potential, and sigma2 is the velocity dispersion tensor:

Equation 2 (2)

The Jeans equation is a moment (over velocity space) of the collisionless Boltzmann equation, which describes the "flow" of a set of particles through phase-space in response to the acceleration induced by some gravitational potential Phi(r). Equation (1) is more useful than the full Boltzmann equation because the right hand side depends only on the lowest moments of the phasespace distribution function f (r, v), which are generally far more accessible to observation than the distribution function itself. The Jeans equation is however incomplete in the sense that the distribution function of a collisionless system is not uniquely specified by its lowest moments. Thus it is possible to find solutions to equation (1) which appear reasonable, but are not consistent with any completely nonnegative phase-space density. (This fact can sometimes be used to rule out certain potentials Phi(r); see below.) Furthermore, reduction of the kinematical data to its lowest moments (e.g. n(r), sigma2(r)) can easily conceal the fact that a particular system is far from equilibrium, and therefore not describable by a time-independent equation. This point is particularly important given the recent evidence that many galaxy clusters contain statistically significant substructure (see review by M. Fitchett in this volume).

Galaxy clusters are generically fairly round and slowly-rotating. Assuming spherical symmetry and setting the mean-motion terms to zero, the Jeans equation becomes

Equation 3 (3)

Here M(r) is the total mass contained within r, and sigmar and sigmat are the galaxy velocity dispersions along and tangential to any radius vector. Equation (3) allows us to estimate the mass distribution in a cluster given knowledge of the three functions {n(r), sigmar(r), sigmat(r)}. Until recently, equation (3) was rarely used for this purpose, primarily because of the difficulty of obtaining a usefully-large sample of galaxy radial velocities. This is no longer the case; new techniques, such as multi-object fiber spectrometry, allow one to measure dozens of radial velocities in a single observing run. There are now several clusters for which more than 100 radial velocities have been measured, and for the Coma and Virgo clusters, this number exceeds 300. A more fundamental problem with equation (3) arises from the nature of the information required to evaluate its right hand side. By measuring the radial velocity of a large sample of galaxies, we can in principle determine sigmalos(R), the dependence of the line-of-sight velocity dispersion on (projected) radius from the cluster center. But there is no way to deconvolve a single function of radius sigmalos(R) to obtain the two desired functions {sigmar(r), sigmat(r)}. Physically, this indeterminacy reflects the fact that spatial variations in either velocity anisotropy or cluster mass-to-light ratio may be responsible for the observed variation of sigmalos with R.

Given this indeterminacy, there are several possible ways of proceeding, none completely satisfactory. By far the most common is to make an a priori assumption about the form of the mass distribution - e.g., that the matter is distributed like the observed galaxies - and then to derive a total mass using the virial theorem. If we multiply equation (3) by 4pir3 and integrate from zero to infinity, the result is

Equation 4 (4)

The brackets indicate spatial averages over the observed sample of galaxies. Writing dPhi / dr = GM(r) / r2 = GMinfty F(r) / r2, where F(r) leq 1 is the mass traction within r, equation (4) becomes

Equation 5 (5)

Equation (5), which is a form of the virial theorem, relates the total cluster mass Minfty to the velocity dispersion of the observed sample, and a quantity <r-1F>, which depends on the (generally unknown) form of the matter distribution. Note that, by taking the proper moment of the Jeans equation, we obtained an expression for the total mass that depends only on the total mean square velocity <v2> = 3<vlos2>. It is this lack of dependence on velocity anisotropy (at least in the context of spherical systems) that makes the virial theorem so useful. However it is clear from equation (5) that the inferred total mass will depend strongly on its assumed distribution. If the mass is distributed like the observed galaxies - in the form of "heavy halos", for instance - then the appropriate form of equation (5) is easily shown to be

Equation 6 (6)

(Limber and Mathews 1960), where rij is the projected distance between any pair of galaxies. An equation similar to (6) was used by Zwicky (1933) to infer the mass of the Coma cluster from a sample of seven galaxies. He found a mass-to-light ratio consistent with the best modern estimates, i.e.

Equation 7 (7)

(e.g. Kent and Gunn 1982), where h = H0 in units of 100 km s-1. Applying this mass-to-light ratio to the local universe gives a cosmological density parameter Omega0 approx 0.15.

If the dark matter has a different distribution than the galaxies-as it must if, for instance, Omega0 = 1 - then the virial theorem is not very useful. One can easily derive a lower limit on the mass required to bind a cluster by setting F(r) = M(r) / Minfty = 1, i.e. by putting all of the dark matter at the cluster center. For a cluster like Coma, this assumption reduces the required mass below Zwicky's value by a factor of about five. A more reasonable model (at least in the eyes of most cosmologists) would have a dark matter distribution that is more extended than the galaxies, in which case F(r) is small and the total mass can be arbitrarily large. Thus, although the virial theorem does not constrain the total mass very well, it does imply a relation between the total mass and its distribution: the more centrally concentrated the matter, the less is required to bind the cluster, and vice versa.

One can reduce (though not eliminate) this indeterminacy by making use of the additional information contained within the line-of-sight velocity dispersion profile sigmalos(R) . For any assumed dark matter potential Phi(r), there is a unique set of functions {sigmar2(r), sigmat2(r)} which satisfy the Jeans equation (3) and give the correct projected profile sigmalos(R). But for certain Phi(r), the derived sigmar2 or sigmat2 may be negative at some radii, corresponding to models with negative numbers of galaxies on some orbits. Such models can be ruled out as unphysical. Figure 1 illustrates this technique with the Kent and Gunn (1982) Coma data, and an assumed potential

Equation 8 (8)

corresponding to a mass density that falls off as r-2 at large radii (more slowly than the galaxies). Large values of r0 - i.e., nearly uniform dark matter distributions - require very radial orbits, and for r0 gtapprox 5h-1 Mpc, no solution is possible.

Figure 1

Figure 1. Dependence of Coma galaxy velocity anisotropy on radius, assuming the dark-matter potential of eqn. (8). Curves are labelled by the dark-matter core radius r0 in Mpc. Anisotropies greater than unity are unphysical.

Testing a large number of assumed potentials Phi(r) in this way gives approximate upper and lower bounds on the Coma mass-to-light ratio:

Equation 9 (9)

(The and White 1986; Merritt 1987). The low-mass models are perhaps doubly unphysical, in that they require both a high central concentration of dark matter, as well as galaxy orbits that are close to circular at large radii (gtapprox 1h-1 Mpc). Most theories of the early universe predict that the galaxies would be, if anything, more clustered than the dark matter. Furthermore, to the extent that clusters form via gravitational collapse, the galaxy orbits should be predominantly radial, not circular. However, even excluding the tangentially anisotropic models, the mass of the Coma cluster is still uncertain by a factor of about three.

If the number of galaxy radial velocities available in the Coma cluster were much larger - of order 103 or 104 - one could make further progress at constraining its mass by investigating the form of the line-of-sight velocity distribution as a function of projected radius. In effect, this technique would require any model of Coma to be consistent not only with the Jeans equation, but also with the more detailed Boltzmann equation from which the Jeans equation is derived. No one has yet described the best way to carry out this task (nor is there any cluster that is both sufficiently well observed, and convincingly close to equilibrium, to justify such an analysis. In the case of Coma, one way to make use of the extra information contained within the full velocity distribution function is illustrated in Figure 2. The overall velocity histogram appears marginally most consistent with a high-mass, radial-orbit model; a model with low mass and circular orbits appears strongly inconsistent. There are, however, a number of reasons to be cautious about this comparison. The shape of the velocity histogram can be strongly affected by processes such as rotation and infall which we have so far neglected. In fact, inspection of Figure 2 reveals a possibly significant (~ 97% confidence) degree of skewness in the observed distribution. Cluster rotation by itself would tend to broaden the velocity distribution rather than make it skew; furthermore, Coma does not exhibit significant rotation (Rood et al. 1972). However there is good reason to believe that contamination by foreground galaxies might explain the low-velocity "tail". De Lapparent, Geller and Huchra (1986) show that the Coma cluster appears to sit at the intersection of a number of large-scale galaxy "shells", one of which lies nearly along the line of sight to Coma. These foreground galaxies could significantly affect the form of the overall velocity histogram, even if they have little effect on the inferred dynamics of the central regions.

Figure 2

Figure 2. Velocity histogram for galaxies in Coma. The three curves are derived from models in which the galaxy orbits are isotropic, radial, and circular; the dark matter distributions have been adjusted to give the same line-of-sight velocity dispersion profile in each case. (From Merritt, Ap. J., 313, 121.)

A number of other techniques have been discussed for constraining the orbital kinematics of galaxies in clusters. Pryor and Geller (1984) attempted to use the observed tidal radii and gas content of galaxies in Coma to put limits on their orbital pericenters, and hence on the degree of velocity anisotropy. Their result (that the Coma cluster is close to isotropic within 1h-1 Mpc) is strongly dependent on the uncertain physics of tidal truncation and gas dynamcal ablation; furthermore those authors only considered models in which the dark matter is distributed like the galaxies, while in fact the available velocity data imply a fairly tight relation between the mass distribution and the galaxy orbits, as discussed above. O'Dea, Sarazin and Owen (1987) used the orientation of "narrow angle tail" radio sources in clusters to constrain the' distribution of galaxy orbits, under the assumption that the radio-luminous plasma ejected by a moving galaxy is bent into a tail which marks the path taken by the galaxy through the cluster. Since most clusters contain only a few such radio sources (Coma, for instance, contains only one), those authors were forced to superpose data from many clusters. They obtained the surprising result that galaxy orbits in the inner ~ 0.5h-1 Mpc of their clusters are strongly radial; at large radii the distribution of tail orientations appears to be random. The correct interpretation of this result will probably have to await a better understanding of the gas ablation process. It may be, for instance, that the probability of observing a galaxy as a narrow angle tail depends strongly on its velocity with respect to the intracluster gas, in which case the observed sample could be kinematically biased.

For a long time it was hoped that X-ray observations of hot intracluster gas would resolve the indeterminacy of cluster masses. The equation of hydrostatic equilibrium, in spherical symmetry, states

Equation 10 (10)

where rhog and Tg are the gas density and temperature. Equation (10) is simpler than the Jeans equation (3) since gas is a collisional fluid with an isotropic pressure; thus the two functions {sigmar(r), sigmat(r)} are replaced by one, Tg(r). Furthermore, the statistical accuracy of a mass determination based on the Xray emitting gas can always be increased by lengthening the integration time, whereas the number of bright galaxies in a cluster is limited. Unfortunately, the spatial resolution of the spectral instruments on past X-ray satellites has not been very good, and at present there is no cluster (with the possible exception of Virgo) for which we have an accurate determination of Tg(r). This problem is capable of solution; future satellites, such as AXAF, should yield accurate temperature profiles and hence accurate masses for nearby clusters. However we have so far learned little about cluster masses from X-ray studies that we could not have learned from the kinematics of relatively modest samples of cluster galaxies.

This point is made quantitatively in Figure 3, which presents limits on the mass distribution of the Coma cluster obtained from the kinematical technique described above, as well as the limits from the most recent analysis of the Coma X-ray data (Hughes 1988). In the vicinity of ~ 1h-1 Mpc, both techniques give similar results for the enclosed mass, with error bars that span about a factor of two in both cases. Neither technique places interesting constraints on the central mass density; at large radii, the kinematical mass determination is superior, because currently available X-ray emissivity data for Coma do not extend beyond ~ 50 arcmin approx 1h-1 Mpc.

Figure 3

Figure 3. Constraints on the mass distribution in Coma derived from kinematical and X-ray techniques (h = 1).

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