2.7. Spatial distribution of galaxies
The most regular clusters show a smooth galaxy distribution with a concentrated core (Figure 4; Table 1). In general, models to describe the galaxy distribution in these clusters will possess at least five parameters, which can be taken to be the position of the cluster center on the sky, the central projected density of galaxies per unit area of the sky _{0}, and two distance scales r_{c} and R_{h}. The core radius r_{c} is a measure of the size of the central core, and is usually defined so that the projected galaxy density at a distance r_{c} from the cluster center is one half of the central density _{0}. The halo radius R_{h} measures the maximum radial extent of the cluster. Of course, the observed value of the central density _{0} must depend on the range of galaxy magnitudes observed, and the values of other parameters may also depend on galaxy magnitude. If the cluster is elongated, at least two more parameters are necessary; these can be taken to be the orientation of the semimajor axis of the cluster on the sky, and the ratio of semimajor to semiminor axes. However, spherically symmetric galaxy distributions will be discussed first, and r_{c} and R_{h} will be assumed to be independent of galaxy mass or luminosity. Then, one can write
(2.6) |
where n(r) is the spatial volume density of galaxies at a distance r from the cluster center, n_{0} is the central (r = 0) density, (b) is the projected surface density at a projected radius b, and f and F are two dimensionless functions. Obviously,
(2.7) |
A number of models have been proposed to fit the distribution of galaxies. Among the simplest are the isothermal models, which assume a Gaussian radial velocity distribution for galaxies (equation 2.5). If one further assumes that the velocity distribution is isotropic and independent of position, that the galaxy distribution is stationary, and that galaxy positions are uncorrelated, then one can write the galaxy phase space density f(r, v) as
(2.8) |
Now, the time-scale for two-body gravitational interactions in a cluster is much longer than the time for a galaxy to cross the cluster (see Section 2.9). Thus the galaxies can be considered to be a collisionless gas, and the phase space density f is conserved along particle trajectories (Liouville's theorem); f is then a function of only the integrals of the motion ('Jeans' theorem'). If the velocity distribution is isotropic, f does not depend on the orbital angular momentum of the galaxies, and can only depend on the energy per unit mass = 1/2 v^{2} + (r), where (r) is the gravitational potential of the cluster. If we measure (r) relative to the cluster center, the galaxy spatial distribution is thus n(r) = n_{0} exp[-(r) / _{r}^{2}]. If the mass in the cluster is distributed in the same way as the galaxies, then Poisson's equation for the gravitational potential becomes
(2.9) |
where m is the mass per galaxy (Chandrasekhar, 1942). It is conventional to make the change of variables / _{r}^{2}, r / , _{r} / (4 Gn_{0} m)^{1/2}. Then the equation for is
(2.10) |
subject to the boundary conditions (assuming the density is regular at the cluster center) of = 0, d / d = 0 at = 0. Equation (2.10) is identical to the equation for an isothermal gas sphere in hydrostatic equilibrium (Chandrasekhar, 1939).
Figure 4. The dots give the projected galaxy number density observed in 12 regular clusters, from Bahcall (1975). The observed number densities are normalized to the central surface number density _{0} and given as a function of the projected radius b divided by the core radius r_{c}. These parameters were determined by fitting equation 2.11 (the solid curve) to the observed distributions. |
The galaxy density distribution in this isothermal sphere model is then n() = n_{0} exp[-()]. The projected galaxy density can be calculated from equation (2.7) and written as (b) = _{0} F_{isot}(b / ). For convenience, the core radius r_{c} is defined as r_{c} 3, because F_{isot}(3) 0.502, which is obviously close to one-half. The central volume density and projected density are related by _{0} 6.06n_{0} = 2.02n_{0} r_{c}. Unfortunately, neither () nor F_{isot} can be represented by simple analytic functions. Relatively inaccurate tables of these functions are given in Zwicky (1957, p. 139), and more accurate economized analytic approximations to and F_{isot} have been given by Flannery and Krook (1978), and Sarazin (1980), respectively.
At large radii >> 1, n() 2n_{0} / ^{2}, and the total number of galaxies and total mass diverge in proportion to r. Thus the isothermal sphere cannot accurately represent the outer regions of a finite cluster. A number of methods to truncate the isothermal distribution have been used. If one is primarily concerned with representing the galaxy distribution near the core, and if one is not concerned with determining dynamically consistent velocity and spatial distributions, one can simply truncate the isothermal distribution at some radius. Zwicky (1957, p. 140) and Bahcall (1973a) have truncated the isothermal distribution with a uniform surface density cutoff C:
(2.11) |
The cluster galaxy density then falls to zero at a radius R_{h} given by F_{isot}(R_{h}, ) = C / 6.06. Bahcall (1973a) also defines a modified central surface density parameter _{0} /(6.06 - c); then, for small C<< 2, the central volume density is just n_{0} _{0}[1 - (C / 2)^{2}] / (6.06) / . The solid curve for the surface density in Figure 4 is given by equation 2.11.
King (1966) has developed self-consistent truncated density distributions for clusters. The phase-space density he assumes is
(2.12) |
where _{r} is the radial velocity dispersion in an untruncated cluster. The velocity distribution is thus truncated at the escape velocity v_{e}: f(r, |v| v_{e}) = 0 where v_{e}^{2} = -2(r), and the potential goes to zero at infinity () = 0. King showed that equation (2.12) gave an approximate solution to the Fokker-Planck equation for a finite cluster subject to two-body gravitational encounters. As shown in Section 2.9, galaxy clusters are nearly collisionless; however, it is possible that equation (2.12) is a reasonable approximation for the truncated phase-space density. The phase-space density f in equation (2.12) is a function of only the energy per unit mass and the parameter _{r}^{2}, and thus satisfies Jeans' theorem. King integrates equation (2.12) over all velocities to give the density n(r) as a function of the potential (r), and then solves Poisson's equation to give a self-consistent potential. The density n(r) in these models falls continuously to zero at a finite radius R_{h}. The models can be scaled in distance and central density as with the unbounded isothermal models described earlier. The only characteristic parameter is _{r}(0) / _{r} or equivalently R_{h} / r_{c} (where r_{c} is again the core radius). King prefers to use the potential difference between cluster center and edge, W_{0} [(R_{h}) - (0)] / _{r}^{2}. These models predict that the velocity dispersion declines in the outer portions of the cluster, as is observed in Coma (Rood et al., 1972).
Unfortunately, none of these bounded or unbounded isothermal models can be represented exactly in terms of simple analytic functions. However, King (1962) showed that the following analytic functions were a reasonable approximation to the inner portions of an isothermal function:
(2.13) |
where _{0} = 2n_{0}r_{c}. At large radii r >> r_{c}, n(r) n_{0}(r_{c} / r)^{3} in the analytic King model; thus the cluster mass and galaxy number diverge as ln(r / r_{c}). Although this is a slower divergence than the unbounded isothermal model, this analytic King model also must be truncated at some finite radius R_{h}.
Another analytic model is that of de Vaucouleurs (1948a), which was proposed to fit the distribution of surface brightness in elliptical galaxies. However, this distribution also fits many regular clusters (de Vaucouleurs, 1948b). The projected density is
(2.14) |
where r_{e} is an effective radius such that one half of the galaxies lie at projected radii b r_{e}. Accurate tables of the three-dimensional density and potential for this model have been given by Young (1976). The de Vaucouleurs form has several advantages over the isothermal function. It has only one distance scale, the effective radius r_{e}. It also converges to a finite total number of galaxies and cluster mass without a cutoff radius. Numerical simulations of the collapse of clusters seem to lead to distributions similar to this form (see Section 2.9.2). Unfortunately, the de Vaucouleurs form has not been widely used to fit galaxy distributions in clusters, and there have been few attempts to determine objectively whether it or the isothermal models give better fits to the actual distributions.
One major difference between the isothermal functions and the de Vaucouleurs law is that the latter has a density cusp at the cluster center; in fact, as can be seen from Figure 4, many clusters show these cusps, which much be removed in order to fit isothermal sphere models to the galaxy distributions. Beers and Tonry (1986) show that the galaxy distribution in clusters is very sensitive to the position chosen for the cluster center, and that many clusters have central number density spikes if the cluster center is assumed to correspond to the position of a cD galaxy (Section 2.10.1) or the maximum of the X-ray surface brightness (Section 4.4.1). They find that the surface density near this cusp varies roughly as (b) b^{-1}, which is consistent with a singular isothermal sphere (one with r_{c} 0), or with an anisotropic galaxy velocity distribution, with an excess of radial orbits. The presence of these cusps is important to understanding the occurrence of multiple nuclei and companions about cD galaxies (Section 2.10.1).
Another useful fitting form is the Hubble (1930) profile, which is sometimes used to fit the light distribution in elliptical galaxies. It is
(2.15) |
which has the same asymptotic distribution as equation (2.13).
These models (equations 2.11-2.15) have been used to fit the projected distribution of galaxies. In most cases, the galaxy distributions have been fit to the truncated isothermal model (equation 2.11). Figure 4 shows the surface number density distributions in 15 regular clusters, from Bahcall (1975), along with the fitting function (equation 2.11). A compilation of the values of the core radii which have been derived for clusters is given in Table III of Sarazin (1986a), which includes values from Abell (1977), Austin and Peach (1974a), Bahcall (1973a, 1974a, 1975), Bahcall and Sargent (1977), Birkinshaw (1979), Bruzual and Spinrad (1978a, b), des Forets et al. (1984), Dressler (1978c), Havlen and Quintana (1978), Johnston et al. (1981), Koo (1981), Materne et al. (1982), Quintana (1979), Sarazin (1980), Sarazin and Quintana (1987), and Zwicky (1957).
Bahcall (1975) has suggested that the core radii of regular clusters are all very similar, with an average value
(2.16) |
Sarazin and Quintana (1987) find that this may be true for the most compact, regular clusters. However, they also find that the core radius and galaxy distribution depend on the morphology of the cluster (Section 2.5).
The statistical uncertainty in the determination of the core radius or central density of a cluster tends to be rather large ( 30%), because even in a rich cluster only a small fraction of the galaxies are within the core. However, the errors in _{0} and r_{c} are highly anticorrelated, so that the product (_{0}r_{c}) is relatively well-determined. The reason for this is that the number of galaxies within a projected radius b such that r_{c} < b << R_{h} is roughly N(b) 2.17(_{0}r_{c})b for an isothermal model. Thus the uncertainty in the product (_{0}r_{c}) tends to be determined by Poisson statistics on the total number of cluster galaxies observed, and not just by the smaller number in the core (Sarazin and Quintana, 1987). Bahcall (1977b, 1981) has defined a related quantity _{0} as the number of 'bright galaxies' with projected positions within 0.5 Mpc of the cluster center. Here, bright galaxies are those no more than two magnitudes fainter than the third brightest galaxy (m m_{3} + 2). Of course, the magnitude of the third brightest galaxy m_{3} itself depends on richness; _{0} is the number corrected for richness assuming a universal luminosity function (Section 2.4). From the discussion above, it is clear that _{0} (_{0}r_{c}), if _{0} is taken at a standard luminosity level (for example, L^{*} (Section 2.4)), and if r_{c} 0.5 Mpc << R_{h}.
Because they are better determined statistically than the core radius r_{c} or center surface density _{0}, (_{0}r_{c}) and _{0} are often more useful as richness parameters when searching for correlations of integral properties of clusters in the optical, radio, and X-ray region. However, when comparing detailed spatial distributions the core radius is needed.
For example, from the arguments given above _{0}r_{c} _{0} n_{0} r_{c}^{2}, if _{0} and n_{0} are taken at a standard luminosity level. From equation (2.9), r_{c}^{2} = 9_{r}^{2} / 4 Gn_{0}m, where m is the average galaxy mass m _{0} / n_{0}, and _{0} is the central mass density. As n_{0} is defined at a fixed luminosity level, this gives (_{0} r_{c}) _{0} (M/L)^{-1} _{r}^{2}, where (M/L) is the mass-to-light ratio of the cluster (Section 2.8). Bahcall (1981) finds the empirical correlation _{0} 21(_{r} /10^{3} km/s)^{2.2}, which suggests that cluster mass-to-light ratios decrease with _{r}. This relationship may be useful for providing quick estimates of the velocity dispersions of clusters.
Several other size scales can be determined for clusters. The halo radius R_{h} gives the outermost limit of the cluster. Unfortunately, this is very poorly determined, because it depends critically on the assumed background. Moreover, clusters often have very extended haloes or are embedded in extended regions of enhanced density (superclusters). For Coma, the main isothermal distribution of galaxies extends to roughly 4h_{50}^{-1} Mpc; there is then a low-density halo extending to 10h_{50}^{-1} Mpc, which blends into the Coma supercluster which extends to a radius of about 35h_{50}^{-1} Mpc (Rood et al., 1972; Rood, 1975; Chincarini and Rood, 1976; Abell, 1977; Gregory and Thompson, 1978; Shectman, 1982). However, studies of the galaxy covariance function (Peebles, 1974) suggest that there are no preferred scales for galaxy clustering and that the outer regions of clusters and superclusters represent a continuous distribution of clustering.
Other size scales for clusters have been measured that are intermediate between the core and halo size; they include the harmonic mean galaxy separation (Hickson, 1977), which is related to the gravitational radius R_{G} of a cluster (Section 2.8), the de Vaucouleurs effective radius r_{e} defined by equation (2.14), the mean projected distance from the cluster center (Noonan, 1974; Capelato et al., 1980), and the Leir and van den Bergh radius (1977).
While the distribution functions for galaxies discussed above are spherically symmetric, most clusters appear to be at least slightly elongated, and some are highly elongated (Sastry, 1968; Rood and Sastry, 1972; Rood et al., 1972; Bahcall, 1974a; MacGillivray et al., 1976; Thompson and Gregory, 1978; Carter and Metcalfe, 1980; Dressler, 1981; Binggeli, 1982). Carter and Metcalfe (1980) and Binggeli (1982) give ellipticities and position angles for samples of Abell clusters. Their results suggest that clusters have average intrinsic ellipticities of 0.5 - 0.7; thus clusters are actually much more elongated on average than elliptical galaxies.
Carter and Metcalfe (1980) and Binggeli (1982) find that the position angles for the long axes of clusters are significantly aligned with the axis of the first-brightest cluster galaxy. In Sections 2.9.3 and 2.10.1, it is shown that such alignments may result if the first-brightest galaxies are produced by the merger of smaller galaxies through dynamical friction. They might also be produced during the collapse of the cluster.
Thompson (1976) has suggested that the axes of many of the elliptical galaxies in clusters may be aligned with the cluster axis; Adams et al. (1980) find a similar effect in two linear (L; see Section 2.5) clusters. Helou and Salpeter (1982) and Salpeter and Dickey (1985) do not find such alignments for the axes of spiral galaxies in the Virgo or Hercules clusters.
Binggeli (1982) finds that the long axes of Abell clusters tend to point at one another, even when the clusters are separated by as much as 30h_{50}^{-1} Mpc. The alignments of nearby clusters were found to show evidence of a correlation even up to distances a factor of three larger.
In the previous discussion, it has been assumed that the galaxy density decreases monotonically with distance from the cluster center. However, Oemler (1974) found a plateau or local minimum in the projected distribution of galaxies in many clusters at a radius of about 0.4R_{G} (where R_{G} is the gravitational radius). These features would imply the existence of significant oscillations in the three-dimensional galaxy density, although the process of deprojecting to counts is rather unstable (Press, 1976). Although these features are not statistically very significant in any one case, they do appear in a large number of clusters (Omer et al., 1965; Bahcall, 1971; Austin and Peach, 1974a).