Annu. Rev. Astron. Astrophys. 1984. 22: 185-222
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3.2 Simulations of Cluster Evolution and cD Models

The long-standing debate over whether the brightest members of galaxy clusters are simply the tail of the bright end of the luminosity function (Peebles 1968, Geller & Peebles 1976) or the products of special formation and/or evolution (Peach 1969, Sandage 1976, Ostriker & Tremaine 1975, Tremaine & Richstone 1977, Dressler 1978a) is not fully resolved (Godwin & Peach 1979), but it seems certain that the most luminous cD galaxies are indeed the special products of dense environments. This can be argued statistically, since the luminosity function for cluster galaxies does not differ significantly from the field (Schechter 1976, Tammann et al. 1979), and yet no luminous cD galaxies are found in the low-density environment of the field. Furthermore, such galaxies seem to occur almost exclusively at local density maxima, even inside the already dense environments of rich clusters (Beers & Geller 1983; cf. N4389, Oemler 1976). The argument for their uniqueness can also be made on a case-by-case basis in clusters like A2029 or A2218 (Dressler 1978b), where the cD is much too luminous to be a continuation of the cluster luminosity function. The clinching piece of evidence that cD galaxies are unique is the high incidence (about one half) of multiple nuclei, compared with the order of magnitude lower frequency for the second- and third-ranked ellipticals in the clusters (Schneider et al. 1983).

Variations of three basic models have been proposed to explain the properties of cD galaxies. (a) Mergers of the brightest cluster members and/or accretion of smaller galaxies are facilitated by dynamical friction between the galaxies and the stellar envelopes of their neighbors (Lecar 1975). This so-called cannibalism model can build a large, luminous, centrally located cD galaxy that will experience runaway growth until its supply of victims is depleted (Ostriker & Tremaine 1975, White 1976, Ostriker & Hausman 1977, Hausman & Ostriker 1978). (b) Many giant ellipticals form in the central parts of rich clusters, but only a centrally located one can retain its extensive stellar envelope against disruption by the mean tidal field of the cluster (Merritt 1984). Tidal debris (stellar envelopes stripped from cluster galaxies), moving in the potential well of the cluster, enhances the appearance of the central galaxy (Richstone 1976). (c) Vast amounts of cooling gas (~ 100 Msun yr-1) from the intracluster medium (ICM) form a cD ``in situ'' at the cluster center (Mushotzky et al. 1981) perhaps precipitated by a large concentration of ``dark'' matter (Blandford & Smarr 1984).

The first two schemes can be studied by comparing the results of computer simulations with observed parameters for cD galaxies and clusters. Early attempts to model cluster evolution (Richstone 1976, Roos & Norman 1979, Aarseth & Fall 1980) demonstrated that tidal interactions could whittle down cluster galaxies. releasing mass and luminosity into the ICM, while mergers could build a significant number of more massive galaxies.

The work of Merritt (1983, 1984), Miller (1983), and Malumuth & Richstone (1984) has significantly expanded the predictive ability of computer simulations of clusters. The Monte Carlo simulation of Malumuth & Richstone (1984) includes the effects of tidal stripping of galaxies by their neighbors, dynamical friction of galaxies with the ICM, and merging. The model galaxies. which initially contain all of the cluster mass are distributed with an isotropic velocity dispersion and follow a King model with Rcore = 500 kpc. After a simulation time of 1010 yr, about 20% of these model clusters develop (by mergers and accretion) central massive galaxies with extensive stellar envelopes composed of the material stripped from cluster galaxies. Malumuth & Richstone use the Bautz-Morgan (1970) type to characterize the prominence of the brightest cluster member, and they find that the development of such cD galaxies is independent of cluster richness, in agreement with Leir & van den Bergh's (1977) classification of the Abell catalog. On the other hand, the luminosity of the envelope of tidal debris is a strong function of cluster richness, again in agreement with observations of poor and rich cD clusters by Thuan & Romanishin (1981), who find that cDs in poor clusters lack the extensive envelopes present in cDs in rich clusters. Another successful prediction of this simulation is that little or no luminosity segregation develops in the galaxy distribution (Oemler 1974, Chincarini & Rood 1977, Dressler 1978c, Sarazin 1980; cf. Quintana 1979, Capelato et al. 1980), even when the galaxies initially carry a significant amount of the cluster mass (cf. Rood 1969, White 1977).

In spite of these successes, however, the simulations by Malumuth & Richstone (1984) fail to produce a high enough frequency of cD galaxies. Although their value of 20% is close to the Leir & van den Berg figure, most clusters in the Abell catalog have not been relaxed for 1010 yr. Probably only ~ 10% of present-epoch clusters are as evolved as the clusters modeled in the simulations, which suggests that the predicted cD frequency is actually an order of magnitude too low.

Merritt's (1984) simulations are even less efficient in producing cD galaxies from dynamical interactions. This appears to be the result of his adopted setup, in which a strong tidal field produced by the cluster mass distribution pares down the galaxies to a maximum size of only 30 kpc. With cross sections this small, mergers between galaxies become very rare, and thus they are neglected in Merritt's simulations. Merritt therefore concludes that if mergers build cD galaxies, they must occur at cluster collapse and only for galaxies on bound orbits (Aarseth & Fall 1980). Little evolution is expected after cluster collapse, as first suggested by Roos & Norman (1979) (cf. Wielen 1979, Cooper & Miller 1982).

The strong tidal limits imposed in Merritt's simulations may be too severe. Early work by Bahcall (1975) suggesting that the core radii of concentrated clusters are constant at about 250 kpc may have been unrepresentative, since subsequent studies by Dressler (1978c), Chincarini (1979), Sarazin (1980), Kent & Gunn (1982), Semeniuk (1982) [see also Baier (1978), and references therein] find larger values of the core radius, typically between 300 and 600 kpc. Dressler (unpublished) has used Sarazin's maximum likelihood technique to obtain similar large values for the most concentrated of his sample of 55 rich clusters (Dressler 1980a). An X-ray-emitting gas should give a good indication of the distribution of the binding mass in the cluster, which is an important agent of tidal limitation. Such observations (Forman & Jones 1982) usually imply core radii of ~ 500 kpc, except when massive central galaxies are present, and in these cases the potential of the central galaxy itself may be responsible for the small apparent size. Of course, the mean tidal fields will be weaker if core radii are larger. Similarly, Merritt's assumption that each galaxy receives the maximum tidal force (i.e. comes to perihelion near one core radius) for an essentially infinite time may also be too stringent. Miller (1983) also points out that if the outer stars have large tangential velocity components, they should be much more difficult to remove.

If one or more of these factors result in an easing of the tidal cutoff, then merging of cluster galaxies cannot be ignored, and significant evolution afer virialization can be expected. If Merritt's parameters are correct, however, the effects of merging and accretion are sufficiently weakened that they will only be important in the context of special ``initial'' conditions (for example, bound orbits and low velocity dispersions). Consequently, Merritt suggests an alternative interpretation of a cD as the chance survivor of many large, massive galaxies that were ripped apart by the mean tidal field. If one of these giant ellipticals comes to rest in the center of the cluster, it will not be torn apart, because of its symmetrical placement within the gravitational field. Inside the core of the cluster, it will retain its envelope and, in addition, will be identified with the more extensive tidal debris belonging to the cluster as a whole (Richstone 1976).

Miller's (1983) simulations are unique in their use of ``collision rules'' taken from N-body simulations of galaxy encounters (see references in Miller 1983). He models a variety of initial conditions, including collapsing clusters from isothermal perturbations (galaxies form before clusters and have initially radial orbits) and pancakelike models (galaxies form late and have an isotropic velocity distribution). A number of runs covering a range of tidal-stripping efficiency and initial conditions show that as little as 10% to as much as 70% of the initial mass in galaxies is released into the cluster potential. Though mergers are important in Miller's simulations, particularly in those where tidal stripping is difficult, he agrees with Merritt that most of the action takes place during cluster collapse, when densities are high and velocity dispersions are still relatively low. Miller emphasizes the importance of measuring the intracluster light that is predicted by all three studies for both cD and non-cD clusters. The intracluster light has been measured in Coma (Thuan & Kormendy 1977, Mattila 1977, Melnick et al. 1977), in Abell 801 and 1132 (Baum 1973), and in Abell 2670 (Oemler 1973) at a typical level of 10-40%, but the measurements are difficult and the results obtained are very uncertain. These types of measurements are much needed and should be facilitated by the new generation of linear area detectors.

Reviewing these cluster simulations, it is apparent that various techniques agree quite well on the qualitative and even quantitative aspects of cluster evolution, but that the results are very sensitive to the orbits and velocities of the member galaxies and the degree to which they are tidally limited. All of the simulations discussed here are relatively inefficient in producing cD galaxies, even given a longer time than is available to most clusters. Unless tidal stripping is much less efficient than thought, the high frequency of cD occurrence must be telling us that the primary evolution of such systems occurs in subcondensations (small groups) before virialization, when tidal stripping is less effective and velocity dispersions are significantly lower. The idea that most of the evolution took place in this ``small group'' phase may also be more compatible with the morphology-density relation for clusters (see Section 4).

A completely different mechanism for forming cD galaxies has followed from the X-ray observations that show large amounts of gas flowing from the ICM into certain centrally located cluster galaxies (Fabian & Nulsen 1977, Canizares et al. 1979, Mushotzky et al. 1981, Fabian et al. 1981a, b). In the context of simple models, the cooling flows that are detected imply an accretion of tens or even hundreds of solar masses of material per year. Continuation of this process for anything approaching a Hubble time will obviously result in the accumulation of mass comparable to a cD galaxy. Since the insides of cD galaxies are much like normal ellipticals (Thuan & Romanishin 1981), one could conclude that either (a) the accretion is taking place onto a preexisting galaxy and has not significantly altered its structure; or (b) the accretion is directed to the center of dark matter binding the cluster (Rcore ~ 500 kpc), but dissipation in the gas causes it to form a much more condensed structure: or (c) a condensed structure in the dark matter, such as the ``black pit'' suggested by Blandford & Smarr (1984), forms a skeleton over which the cD is built from inflowing gas.

Blandford & Smarr's model attempts to explain the large number of fast-moving companion ellipticals found within ~ 10 kpc of many cD galaxies. They propose that the dark binding material of the cluster actually has a much smaller radius (~ 10 kpc) than the radius value Rcore ~ 500 kpc of the galaxy distribution. Binding these companion galaxies with such high orbital velocities would make the time scale for accretion by the cD quite long and would therefore explain why such multiple systems are common. It is, of course, known from observations of the velocity dispersion as a function of radius (Dressler 1979, Carter et al. 1981) that some cD galaxies do not contain such black pits, since the velocity dispersion at 10 kpc would rise to ~ 700 km s-1. Recent observations by Tonry (1984a), however, also seem to rule out black pits with core radii less than 12 kpc in A2199 (NGC 6166) and A2634 (NGC 7720), which are Blandford & Smarr's best cases. The velocity dispersions in these cDs do not rise as is predicted by the model. Tonry's numerical simulations (1984b) show that the orbits of galaxies remain highly elliptical as they are captured. Thus, these simulations are able to reproduce the proper frequency of such companions without an extremely concentrated distribution of dark matter.

Further evidence against the black pit model is that the X-ray core structures of A2199 and A2634 are quite different, as A2634 has a much larger core than is typical for the class (Forman & Jones 1982). It is tantalizing, however, that one prediction made by Blandford & Smarr - that there would be a strong X-ray source at the position of the binary system NGC 6041 a,b in A2151 (Hercules) - is dramatically confirmed (Dressler et al. 1984b).

Let us return to the cases of accretion flows onto a preexisting galaxy or into a dark-matter distribution with a large core (~ 500 kpc). It is well established that the interiors of cD galaxies have properties similar to normal ellipticals (Thuan & Romanishin 1981), including velocity dispersions, mass-to-light ratios, stellar populations (optical spectra and colors), and densities (Faber et al. 1977, Dressler 1979, Malumuth & Kirshner 1981) Valentijn's (1983) report to the contrary of strong color gradients and extremely red nuclei in cDs is in strong disagreement with all previously published data on the color and color gradients of cD galaxies (e.g. Sandage 1972, Schild & Davis 1979, Gallagher et al. 1980, Wirth & Shaw 1983, Lugger 1984) which imply that cD galaxies have the colors of normal ellipticals. These colors are compatible with a burst of star formation some 1010 yr old, as is usually assumed for an elliptical galaxy. The color of a galaxy with a constant rate of star formation over 1010 yr, such as a cD built steadily over a Hubble time, should be markedly different (Searle et al. 1973, Sarazin & O'Connell 1983) unless the initial stellar mass function is truncated at high mass. If the gas is instead forming low-mass, largely undetectable stars (Fabian et al. 1982, Sarazin & O'Connell 1983, Valentijn 1983), then the mass-to-light ratio of cD galaxies should be significantly higher than normal ellipticals. The evidence, however, is to the contrary, since measured mass-to-light ratios for cD galaxies do not differ from those of ellipticals (Malumuth & Kirshner 1981, Thuan & Romanishin 1981).

Because direct evidence for the cooling flow is observed in the form of temperature inversions and optical filaments (Heckman 1981, Fabian et al. 1981b), it is reasonable to conclude that gas accretion rates of order 10 Msun yr-1 are occurring in the centers of some clusters. However, there are no optical observations that support the cases for accretion of hundreds of solar masses per year, a rate that should be capable of building a giant galaxy in a Hubble time. This suggests either that the models of such huge accretion flows are incorrect or that accretion on such a scale has not persisted for such long periods of time. It is worth pointing out, however, that if cD galaxies were built primarily from inflow of this type, then it is probable that all galaxies formed in a similar way, since the properties of cDs and other ellipticals are so similar.

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