5.10.1. Infall models
Gunn and Gott (1972) suggested that the intracluster medium was primordial intergalactic gas that had fallen into clusters. This intergalactic gas was never associated with stars or galaxies, and thus could be expected to have no heavy elements. They also noted that some of the intracluster medium could come from ram pressure stripping of interstellar gas out of galaxies. Assuming that clusters were immersed in uniform intergalactic medium, they estimated the amount that would fall into clusters. By comparing this to the amount of intracluster gas deduced from the early X-ray observations of clusters, they could give an upper limit on the density of the intergalactic gas. (This is an upper limit because some of the intracluster gas could have come out of galaxies.) These limits are usually given in terms of c, the density of matter necessary to close the universe
where h0 is the Hubble constant. If IG is the density of intergalactic gas, then Gunn and Gott found IG / c 0.01. They used a rather low value for the gas mass in clusters, and more recent calculations (for example, Cowie and Perrenod, 1978) give IG / c 0.2. Gunn and Gott also noted that infall would heat the intracluster gas to about the observed temperatures (Section 5.3.2).
To determine the final configuration and evolution of the intracluster gas in the infall model, hydrodynamic simulations of the infall have been done by a number of authors (Gull and Northover, 1975; Lea, 1976; Takahara et al., 1976; Cowie and Perrenod, 1978). These calculations all assumed that the cluster potential was fixed; the gas was assumed to fall into the cluster after the cluster itself had collapsed. All of these calculations were one-dimensional simulations of spherical clusters, although a number of different techniques were used to solve the hydrodynamic equations. With the exception of Lea's calculations, these simulations have given similar results.
As the gas first collapses into the core, its density increases and a shock propagates outward from the cluster center and heats the gas. This shock passes through the cluster in 109 yr, essentially the sound crossing time for the cluster (equation 5.54). After the passage of the shock, the hot intracluster gas is nearly hydrostatic, and its further evolution is quasistatic. As the shock moves into the outer parts of the cluster, it weakens; less gas is added to the cluster, and the cluster luminosity is nearly constant. On the other hand, Lea found that the shock heating caused the gas pressure to increase until the inflow was reversed and the intracluster gas expanded. This cooled the gas adiabatically, lowered its pressure, and it collapsed again. This process repeated itself, producing a large number of pulsations with a period of about 5 × 109 yr. During these pulsations, the X-ray luminosity oscillated wildly between roughly 1041 and 1048 erg/s. The other calculations of the infall of intracluster gas have failed to find these oscillations (Gull and Northover, 1975; Takahara et al., 1976; Cowie and Perrenod, 1978; Perrenod, 1978b), and they are probably an artifact of Lea's numerical method. Such oscillations are in violent disagreement with the observed X-ray luminosity function of clusters (Schwartz, 1978).
Gull and Northover (1975) found that the shock strength was nearly constant as it propagated outward; they argued that this occurred because the shock speed was always about the free-fall speed in the cluster. They found that the resulting intracluster gas distribution was nearly adiabatic (Section 5.5.2). On the other hand, more detailed calculations by Cowie and Perrenod (1978) and Perrenod (1978b) found that the cluster gas distributions were not well represented by any polytropic distribution, unless thermal conduction was so effective that the gas was isothermal.
In the absence of significant cooling or thermal conduction, Cowie and Perrenod (1978) showed that the infall models with a fixed cluster potential are characterized by a single parameter, which gives the depth of the cluster potential well, K (r / H0rc)2 where r and rc are the cluster velocity dispersion and core radius, respectively. Models with significant cooling are also characterized by B tcool h0, where the cooling time is evaluated at the cluster center. If thermal conduction is important, the cluster evolution is also determined by the value of C Tg / rc g cs3, where is the thermal conductivity, and Tg, g, and cs are the gas temperature, density, and sound speed. When conduction saturates, the models are independent of c. In general, the gas temperatures in these models scale with r2.
Cowie and Perrenod found that models without significant cooling or conduction showed a very small decrease in the X-ray luminosity with time, less than a 40% reduction from z = 1 to the present. This decrease in luminosity resulted from the slow reexpansion of the intracluster gas as the shock weakened. In models with significant cooling, the cluster evolved to a nearly steady-state cooling flow (Section 5.7). In models with conduction, the X-ray luminosity increased slowly with time, by about 40% from a redshift z = 1 to the present. This occurred because conduction lowered the temperature in the cluster core (Section 5.4.2). The core then contracted so that the increasing density could maintain the pressure in the core. Since the X-ray luminosity increases more rapidly with density than does the pressure (equation 5.21), the luminosity went up.
These models all assumed that the cluster potential was static; the cluster was assumed to collapse before any gas fell into it. Of course, there is no reason why intergalactic gas should wait until the cluster has formed before gaseous infall can occur. Perrenod (1978a, b) calculated the evolution of the intracluster gas in infall models in which the cluster potential varied in time. The cluster potential was taken from White's (1976c) N-body calculations of the collapse of a Coma-like cluster (Section 2.9; Figure 5). In White's models, the cluster first collapses with violent relaxation, then contracts slowly due to two-body interactions between galaxies. This contraction causes the cluster potential well to become deeper, and as a result the intracluster gas temperature and density increase with time. In contrast to the static potential models, Perrenod finds that the X-ray luminosity of his model clusters increases by about an order of magnitude from z = 1 to the present. The sizes of the gas distributions also shrink considerably. If thermal conduction is important, the further contraction in the gas distributions it produces makes them smaller than the observed sizes of X-ray clusters.
One interesting aspect of Perrenod's models is that many of the infall models have a temperature inversion (dTg / dr > 0) in the cluster core, even if there is no significant cooling. This occurs because gas in the core has fallen through a shallower gravitational potential than gas further out. If such a temperature inversion were observed, it might be confused with a cooling flow (Section 5.7).
In White's N-body models, the cluster shows very strong subclustering at the beginning of its collapse, and forms two roughly equal subclusters, which merge as the cluster undergoes violent relaxation. Several double X-ray clusters are known (Section 4.4.2; Figure 18) that appear to be in just this stage of evolution (Forman et al., 1981). Obviously, such subclustering cannot be treated in one-dimensional, spherical, hydrodynamic simulations. Gingold and Perrenod (1979) have made simplified three-dimensional hydro simulations of the evolution of clusters. When applied to the cluster potential from White's N-body models, these verified the previous one-dimensional calculations of Perrenod (1978b). They found that there was no significant enhancement of the X-ray emission from merging subclusters, beyond that predicted by single cluster models. Similar calculations were made by Ikeuchi and Hirayama (1979).
One major concern about all the Perrenod varying-potential models is the use of White's (1976c) N-body calculations for the cluster potential. In this particular set of models by White, the total virial mass of the cluster was assumed to reside in the individual galaxies. This gave the galaxies large masses, which increased their two-body interactions (Section 2.9.1), and caused the cluster core to contract rapidly. However, associating the missing mass in clusters with individual galaxies appears to produce more two-body relaxation in clusters than is observed (Sections 2.8 and 2.9.4); in fact, this was one of White's conclusions from his models. Thus it seems likely that Perrenod's calculations may significantly overestimate the increase with time of the X-ray luminosity and gas temperature and the decrease in the gas core size.
Clusters of galaxies are the largest organized structures in the universe, and X-ray emission from them should be recognizable to large redshifts (Chapter 6). They might therefore be useful as probes of the cosmological structure of the universe. Several cosmological tests have been proposed using X-ray clusters (Schwartz, 1976; Silk and White, 1978); although some of these tests are relatively insensitive to X-ray cluster evolution, most are strongly affected. These models suggest that it will be difficult to apply any tests that require that X-ray clusters have remained unchanged since z = 1 (Perrenod, 1978b; Falle and Meszaros, 1980). On the other hand, in Perrenod's models the luminosity and size of X-ray clusters depend strongly on the density of material in the universe, since this determines the speed with which clusters contract. In principle, this dependence of cluster evolution on density might provide useful cosmological information; in practice, the evolution models are too uncertain to be used reliably for this purpose.
The models described so far have dealt with the evolution of single clusters. Perrenod (1980) has attempted to predict the evolution of the luminosity function of X-ray clusters (Section 4.2). He assumed that galaxies formed before clusters, and that clusters were formed by the gravitational attraction of galaxies. He argued that the merging of clusters tends to produce larger clusters with deeper potential wells, and as a result the average X-ray luminosity increases. White (1982) showed that this argument is incorrect; the increase in the depth of cluster potential wells is more than offset by the decrease in their characteristic densities. Perrenod found a very rapid evolution of the luminosity function to higher luminosities; he predicted that there should be few luminous X-ray clusters at redshifts z 1/2. This evolution depends strongly on the average density of matter in the universe, and Perrenod proposed using it as a cosmological test. However, his basic model for clustering is apparently incorrect (White, 1982).