Rich clusters of galaxies are certainly not due to chance associations or random encounters between galaxies. They are systems in which galaxies are dynamically interacting with each other and with any intracluster gas. In terms of the spatial distribution of member galaxies the richest clusters especially appear stable and gravitationally bound. Unlike the Solar System for example, which matches all the requirements for a stable, gravitationally-bound system, mass estimates made for galaxy clusters do not agree with their being bound. We now examine the dynamics and motions of galaxies within a cluster, which are somewhat more complicated than those of the planets.
A typical cluster galaxy does not describe a simple orbit about the centre of mass of the cluster. It is continually being deviated from its path by the gravitational pulls of all the other galaxies. Close encounters and collisions must be rare, especially in the less dense regions of the cluster, but the cumulative effect of all the distant galaxies is significant. In a region uniformly populated with galaxies, the gravitational attraction from any individual galaxy falls off as its distance squared, but the number of such galaxies increases as the distance squared. Consequently equal shells around any specified galaxy contribute the same total pull independent of distance. Of course if everything is completely symmetrical the pulls cancel, but the distribution of galaxies and gas in a real cluster gives rise to important effects. The overall tendency is for kinetic energy to be shared among the individual galaxies. The time taken after a disturbance for the system to restore equilibrium, in which all galaxies have on average the same energy as any other, is known as the RELAXATION TIME.

The relaxation times for the dense cores of the richest clusters are of the order of the age of the Universe and so these regions may be considered relaxed and therefore stable. It is possible that conditions in the initial collapse of these cores were such that somewhat shorter relaxation times occurred in the past. If clusters were un bound, then they would tend to evaporate away in a timescale similar to that which a galaxy would take to cross the whole cluster, if its present velocity were unchecked. In the case of the Coma cluster core, for example, this time (usually referred to as the CROSSING TIME) is approximately 3 x 108yr, which is much shorter than the age of the Universe. The dispersal of cluster cores would moreover introduce large numbers of elliptical galaxies into the regions between clusters of galaxies. Only subsequent rapid changes in the form of elliptical galaxies could produce the observed lack of ellipticals in these regions. There is no known mechanism for such changes nor observational basis for its occurrence. These arguments strongly suggest that rich clusters are stable and gravitationally bound.

The mass of a cluster of galaxies can be measured basically in two ways. The first method is to study the spread in motion of the duster galaxies. In the same way that stellar masses and galaxy masses can be estimated from binary pairs, masses of large collections of objects can be deduced from their average motion. The greater the mass of the cluster, the greater will be the gravitational pull that the galaxies exert on each other. These forces hold the cluster together, but it is prevented from collapsing completely by orbital motions, as with a binary pair. A relaxed self-gravitating system left to itself and examined at various later stages should on average be found to be in a state such that twice the sum of the kinetic energy of the components should equal the total gravitational potential energy. This is known as the VIRIAL THEOREM and the masses derived from it VIRIAL MASSES. Simply the mass M— V2R/G where V is a weighted mean of the galaxy velocities (obtained from the velocity dispersion of the cluster), R is a characteristic radius (the harmonic average of the separations of galaxies) and G the gravitational constant.

The VELOCITY DISPERSION is a measure of the spread of galaxy velocities (determined from redshift measurements), and, as ob¬served, relates to motions along the line of sight (a geometrical factor (v 3) must be included to change this into the true velocity dispersion). For the Coma cluster the characteristic radius is about 3Mpc. The virial mass of the Coma cluster is then approximate 5*1015Mo.

Another method for determining the mass of a cluster of galaxies is to add up the individual masses of the member galaxies. Analyses of galaxy rotation curves, luminosity distributions, and studies of binary galaxies have been used to obtain the masses of galaxies themselves. These give mass-to-luminosity ratios (M/L ratios) of 30-70 M0/Le -for ellipticals and between 1 and 10MO/L0 for spirals. The estimated total luminosity for the Coma cluster, found by summing the individual galaxies; is about 3 X 1013L0. A mass-to-luminosity ratio of nearly 200 M0/L0 is therefore required to explain the virial mass. The discrepancy between the mass-to-luminosity ratios for the cluster as a whole and its individual members is at least a factor of 5 and perhaps more.

Similar procedures have been followed for many other clusters and the mass discrepancy always seems to occur, with factors of 10 or more being common, and seeming to increase with the size of the system. This suggests that it cannot all be a statistical fluctuation or a chance projection of galaxies occurring in the Coma cluster. The mass discrepancy is independent of the value of the Hubble constant. Drastic approaches such as new laws of physics, involving, for example, a slight change in the inverse square law of gravitation, have been suggested.

We shall take the view here that there is a mass discrepancy look at the various explanations in terms of MISSING MASS The optical luminosity of the cluster is an estimate of the luminosity of detected objects. There may be much more hidden or invisible mass, and if so it must be distributed in space roughly in the same way as the galaxies. If it were much more extended it could not hope to bind the core, and if it were much more compact it would tidally distort the central regions more than is observed. We dis¬cuss in detail the Coma cluster since this is well studied, but most of the arguments apply to other clusters.

One obvious candidate for the missing mass is INTRACLUSTER GAS. This has probably been detected by its X-ray emission, and it has an implied density, if distributed fairly smoothly, of about 103 ions per cubic metre. The radio trail galaxies also point to its presence. One further test would be the detection of a dip in the intensity of the microwave background as seen through a cluster. This would result because the thermal electrons collide microwave background photons, changing their direction and energy. Preliminary results of searches for this effect suggest that it does occur. Unfortunately the total mass of hot gas is only about 10 per cent of the virial mass. Thus it cannot overcome the mass discrepancy by itself. It is possible that denser clouds of cooler gas are embedded in this hot gas. Under certain circumstances fairly dense clouds of, say, molecular hydrogen might be virtually undetectable. Searches for cluster gas have been carried out at many wavelengths. These include looking for redshifted Lyman-a emission in quasar spectra, H? emission, bremsstrahlung emission at radio and optical frequencies and 21-cm emission and absorption. The results for the Coma cluster are summarized. It appears that the cluster cannot be bound by ionized gas.

Other possibilities centre on more condensed matter. Studies of obscuration and colour changes in the centre of the cluster indicate that dust similar to that in our Galaxy is not responsible. However, larger dust particles ranging in size up to football-sized lumps could easily bind the cluster and remain undetectable, but there is no obvious means for producing such objects. Intergalactic black holes might work, as might a population of ‘brown’ dwarfs – stars with masses less than about one-tenth that of the Sun Presumably most of such objects (if the black holes are of stellar mass) would he in the haloes of galaxies, and in fact there are tentative indications that massive galaxies are surrounded by extensive haloes.

Increasing the mass-to-luminosity ratio for galaxies to about 170 is possible but it is not generally considered likely. Such massive galaxies would tend to distort each other tidally far more than is observed to be the case. They might also tend to interact with each other in such a way as to gobble each other up. This might well result in more mass segregation of galaxies than is observed.

The missing mass problem may be resolved by pushing to their limits many of the suggestions listed above. The virial theorem ought to apply in a fairly precise way to a rich and apparently stable cluster such as the Coma cluster. This intriguing problem has sparked off many interesting investigations and taught us that not everything in the Universe is visible – at least not to present-day instruments. Perhaps some new physics, such as the spontaneous creation of galaxies, or non-velocity redshifts, lies behind the frustrating results so far. Few astronomers believe this, but extra-galactic astronomy is bedevilled by problems related to the interpretation of redshifts arid the measurement of extended objects. Several groups of galaxies, such as Stephan’s Quintet, appear to have mass discrepancies large enough to make the Coma problem pale into insignificance. Such groups could, and probably do, contain chance associations, making it difficult to be at all rigorous.

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