2.4. Luminosity function of galaxies
The luminosity function of galaxies in a cluster gives the number distribution of the luminosities of the galaxies. The integrated luminosity function N(L) is the number of galaxies with luminosities greater than L, while the differential luminosity function n(L)dL is the number of galaxies with luminosities in the range L to L + dL. Obviously, n(L) -dN(L) / dL. Luminosity functions are often defined in terms of galaxy magnitudes m -2.5 log _{10}(L); N( m) is the number of galaxies in a cluster brighter than magnitude m. Observational studies of the luminosity functions of clusters include Zwicky (1957), Kiang (1961), van den Bergh (1961a), Abell (1962, 1975, 1977), Rood (1969), Gudehus (1973), Bautz and Abell (1973), Austin and Peach (1974b), Oemler (1974), Krupp (1974), Austin et al. (1975), Godwin and Peach (1977), Mottmann and Abell (1977), Dressler (1978b), Bucknell et al. (1979), Carter and Godwin (1979), Thompson and Gregory (1980), and Kraan-Korteweg (1981). Figure 2 gives the observed integrated luminosity function for a composite of 13 rich clusters as derived by Schechter (1976) from Oemler's (1974) data.
Figure 2. The luminosity function of galaxies in clusters. N( m_{V}) is the number of galaxies brighter than absolute magnitude m_{V}. The circles are the composite observed luminosity function derived by Schechter (1976). The solid circles exclude cD galaxies, while the open circles show the changes when they are included. The solid, dashed, and dash-dotted curves are the fitting functions of Schechter, Abell, and Zwicky, as discussed in the text. |
Three types of functions are commonly used for fitting the luminosity function. Zwicky (1957) proposed the form
(2.1) |
where K is a constant and m_{1} is the magnitude of the first brightest galaxy. In general, the Zwicky function fits the faint end of the luminosity function adequately, but does not fall off rapidly enough for brighter galaxies (see Figure 2). Clearly, equation (2.1) implies that N(L) = K[(L_{1} / L)^{1/2} - 1], where L_{1} is the luminosity of the first brightest galaxy, and K is the expected number of galaxies in the range 1/4 L_{1} L L_{1}.
Abell (1975) has suggested that the luminosity function N(L) be fit by two intersecting power laws, N(L) = N^{*}(L / L^{*})^{-}, where 5/8 for L < L^{*}, and 15/8 for L > L^{*}. L^{*} is the luminosity at which the two power laws cross, and N^{*} is the expected number of galaxies with L L^{*}. Of course, this form is intended only as a simple and practical fit to the data; the real luminosity function certainly has a continuous derivative n(L), unlike Abell's function. The magnitude luminosity function corresponding to Abell's form is often written as
(2.2) |
where K_{1} and K_{2} are constants. The slopes are approximately s_{1} 0.75 and s_{2} 0.25, and the power laws cross at m = m^{*}, so that K_{1} + s_{1} m^{*} = K_{2} + s_{2} m^{*}. As shown in Figure 2, the Abell form fits both the bright and faint ends of the luminosity function adequately.
Schechter (1976) proposed an analytic approximation for the differential luminosity function
(2.3) |
where L^{*} is a characteristic luminosity, N^{*} (1 - , 1) is the number of galaxies with L > L^{*}, and (a, x) is the incomplete gamma function; Schechter derives a value for the faint end slope of = 5/4. The integrated luminosity function corresponding to equation (2.3) is N(L) = N^{*} (1 - , L/L^{*}).
The advantages of the Schechter function are that it is analytic and continuous, unlike Abell's form, and that it is a real statistical distribution function, unlike Zwicky's form, which requires that the magnitude of the first brightest galaxy (which ought to be a random variable) be specified. An expression similar to the Schechter form was predicted by a simple analytical model for galaxy formation (Press and Schechter, 1974). The Schechter function is steeper than Abell's function at the bright end because it contains an exponential. The Schechter differential luminosity function decreases monotonically with luminosity, while the Abell function has a local maximum at L = L^{*} (Abell, 1975); there is some weak evidence for such a peak, especially near the centers of rich clusters (Rood and Abell, 1973).
The Schechter function fits the observed distribution in many clusters reasonably well from the faint to the bright end (Figure 2), as long as the very brightest galaxies, the cD galaxies, are excluded (Schechter, 1976). These can have luminosities as large as 10L^{*}, and thus are extremely improbable if equation (2.3) holds exactly. However, cD galaxies have a number of distinctive morphological characteristics which suggest that they were formed by special processes which occur primarily in the centers of rich clusters (see Section 2.10.1). In any case, they apparently can be excluded by morphological (as opposed to luminosity) criteria, and equation (2.3) can be taken to be the non cD luminosity function.
The parameter N^{*} in the Abell or Schechter functions is a useful measure of the richness of the cluster. If these luminosity functions are adequate approximations, then fitting the luminosity function to determine N^{*} ought to give a more accurate measure of richness than counting galaxies in magnitude ranges. Note that while the total number of galaxies predicted by Abell's, Zwicky's, or Schechter's functions diverges at the faint end, the total luminosity is finite; for example, the total cluster optical luminosity is L_{opt} = N^{*} (2 - ) L^{*} for the Schechter function, and N^{*} thus measures the total cluster luminosity.
Because of the break in the luminosity function near L^{*}, this parameter represents a characteristic luminosity of cluster galaxies. Moreover, L^{*} is nearly the same for many clusters; it corresponds to an absolute visual magnitude M_{V}^{*} -21.2 + 5 log _{10} h_{50} for Abell's form (Austin et al., 1975), and M_{V}^{*} -21.9 + 5 log _{10}h_{50} for Schechter's form (Schechter, 1976). By comparing these values with the apparent magnitude m^{*} derived from the observed luminosity function of a cluster, one can derive a distance estimate for the cluster (Bautz and Abell, 1973; Schechter and Press, 1976). Unfortunately, this requires that magnitudes be available for a large number of galaxies.
The magnitude of the brightest galaxy in a cluster has often been used as a distance indicator in cosmological studies (see, for example, Hoessel, Gunn, and Thuan, 1980). It is obviously easier to determine observationally than the luminosity function. Moreover, the luminosities of the brightest cluster galaxies show a very small variation from cluster to cluster (Sandage, 1976) and depend only weakly on richness. There has been a controversy concerning whether the luminosities of these brightest galaxies are determined statistically by the cluster luminosity function, or whether the brightest galaxies are produced by some special physical processes operating in clusters (see Dressler, 1984, for a review). Under the statistical hypothesis, all of the galaxy luminosities are random variables drawn from the luminosity function (Geller and Peebles, 1976). Then, the observed number of galaxies in any luminosity range will have a Poisson distribution about the expectation given by the luminosity function, and the luminosity L_{1} of the brightest galaxy will thus be distributed as exp[-N(L_{1})] n(L_{1})dL_{1}. As N(L_{1}) is proportional to richness, the statistical model predicts that L_{1} increases with cluster richness, which is not really observed (Sandage, 1976). However, Schechter and Peebles (1976) have argued that the near constancy of L_{1} results from a selection effect (that is, the observed sample is biased), and that the statistical hypothesis may still be valid.
Alternatively, the brightest cluster galaxies may be affected by special physical processes, such as the tidal interactions or mergers of galaxies (Peach, 1969; Ostriker and Tremaine, 1975; Hausman and Ostriker, 1978; Richstone, 1975). Evidence that this may indeed be the case is given in Section 2.10.1.
While the luminosity functions of many clusters are reasonably well represented by the Schechter or Abell form with a universal value of M^{*}, significant departures exist in a number of clusters (Oemler, 1974; Mottmann and Abell, 1977; Dressler, 1978b). These departures include variations in the value of M^{*}, variations in the slope of the faint end of the luminosity function ( in Schechter's form, equation (2.3)), and variations in the steepness of the bright end of the luminosity function (Dressler, 1978b). These variations are, in many cases, correlated with cluster morphology (Section 2.5). The variations in M^{*} and probably reflect variations in the conditions in the cluster at its formation, while the variations in the bright end slope may result from evolutionary changes, such as the tidal interaction or merging of massive galaxies (Richstone, 1975; Hausman and Ostriker, 1978). In particular, the clusters with the steepest luminosity functions at the bright end often contain cD galaxies (Dressler, 1978b); this may indicate that the brighter galaxies were either eliminated by mergers to form the cD or diminished in brightness through tidal stripping (Section 2.10.1).
Turner and Gott (1976a) have shown that the luminosity function of galaxies in small groups is well represented by equation (2.3). In fact, Bahcall (1979a) has suggested that the luminosity function of all galaxian systems - from single galaxies (in or out of clusters) to the groups and clusters themselves - can be fit in a single function similar to the Schechter form (equation 2.3).