3.3. The Luminosity Function and Colors of Cluster Galaxies
Hubble (1936b) investigated the distribution of absolute magnitudes of 134 spiral and 11 irregular galaxies (most of them not in conspicuous clusters) and represented their luminosity function with a Gaussian curve having a mean absolute photographic magnitude of 14.19 and a standard deviation of 0.85 mag (the mean should be several magnitudes brighter to correspond to the modern distance scale). Hubble's luminosity function applies at best to the type of galaxies represented in his sample (late spirals and irregulars were chosen because in them he could identify what appeared to be individual stars, and hence could estimate distances), and there is no justification for assuming that it holds for elliptical galaxies, which are the major constituents of rich clusters, or, therefore, for cluster galaxies in general. In particular, the great prevalence of dwarf ellipticals was unknown to Hubble at the time of his pioneering investigation. We should, in fact, expect the luminosity functions to vary even among clusters of different morphological type and richness. Considerable confusion has resulted from the failure of some investigators to recognize the inapplicability of Hubble's luminosity function to clusters of galaxies.
Zwicky (1942a) may have been the first to summarize the observations favoring the existence of large numbers of galaxies of low luminosity. He also advanced quasistatisticalmechanical arguments that the gal increasing magnitude. His argument is based on the assumption of a stationary universe in statistical equilibrium, and on the application of the Boltzmann principle to determine the relative numbers of galaxies in clusters and in the field, as well as the relative numbers of stars inside and outside galaxies. The analysis leads to the result that most stars (and other matter) should be in the intergalactic space; moreover, he believed that collisions should result in the fragmentation of some galaxies into smaller parts. Zwicky concludes (italics are his): "...individual stars, multiple stars, open and compact star clusters and stellar systems of increasing population will be found in numbers presumably decreasing in frequency as the stellar content of the systems in question increases."
Few modern investigators would accept the assumption on which Zwicky's conclusions are based; however, his supposition that the galaxian luminosity function rises at faint absolute magnitudes is almost certainly correct  at least for galaxies in rich clusters to the magnitude limits observed. Later, from a rather ingenious analysis of the mean numbers of galaxies within clusters of various angular diameters, Zwicky (1957) derived the following expression for the integrated luminosity function of cluster galaxies:
(1) 
where N(m) is the number of galaxies in the range m between magnitude m and the magnitude of the brightest cluster galaxy. Unfortunately, as Zwicky defines his cluster angular diameters, they are not related to distance in the manner he assumes in the derivation of equation (1), as has been shown by the writer (Abell 1962) and by Scott (1962). Nevertheless, Zwicky's formula may be qualitatively correct, at least at the fainter magnitudes. He states (Zwicky 1957) that equation (1) is consistent with counts of galaxies in several clusters to two or more different magnitude limits (on plates with different exposure times or taken with different telescopes), and also with the distribution of magnitudes obtained with schraffier photometry among the brightest galaxies in several clusters (e.g., Zwicky and Humason 1964a, b).
The published data have been reexamined to determine the galaxian luminosity function by Kiang (1961) and by van den Bergh (1961b). Kiang, from magnitudes of galaxies both in clusters and in the field, adopts Zwicky's form of the differential luminosity function (M) (derivative of eq. [1]) for faint magnitudes, but finds that the function rises more sharply for bright magnitudes; he adopts a cubic law for (M) over the interval of the brightest 2.5 mag. Van den Bergh analyzes the absolute magnitudes of 240 bright field galaxies, of which 48 are ellipticals and 192 are spirals and irregulars. Although the magnitude range considered by van den Bergh is small (4.5 and 6.5 mag for ellipticals and spirals, respectively), his luminosity functions (found separately for ellipticals and for spirals and irregulars) are both in qualitative agreement with Zwicky's formula at faint magnitudes; however, they rise more rapidly at their bright ends.
Derived luminosity functions are, unfortunately, sensitive to photometric procedures, the difficulty of which are often not appreciated. The surface brightness of an elliptical galaxy drops off very slowly at large distances from the center of its projected image; Hubble's (1930) representation of the brightness is, in fact, an inversesquare function of the radial distance. Hubble's interpolation formula, if extended to infinite radius, leads to an infinite luminosity for a galaxy; thus at some distance the galaxian surface brightness must begin to drop more rapidly. Some investigators (Dennison 1954; Liller 1960; de Vaucouleurs 1948) have found finite luminosities (or "total" magnitudes) for elliptical galaxies, but their measures of surface brightness deviate from Hubble's law only at very large radial distances, where the measurements are extremely difficult. For a few elliptical galaxies, convergence of the total luminosity has not been ascertained observationally; photoelectric measures of M87 by Baum (1955), for example, show no evidence of deviation from Hubble's inversesquare law even to very great distances from the center. Even in spiral galaxies, where the spiral arms seem more sharply defined, the contribution to the total luminosity of the Population II coronal components, which may extend far beyond the spiral arms, is not accurately known.
"Total" magnitudes of galaxies are thus not easy to define unambiguously, and investigators should exercise considerable caution in interpreting published magnitudes. Consistent results have been obtained by Sandage (Humason, Mayall, and Sandage 1956), who defines the magnitude of a galaxy according to the total light contained within a given standard isophote, and by Holmberg (1950, 1958), who integrates the light over a galaxian image from several microphotometer tracings across it. Sandage's magnitudes, however, do not correspond to the same fraction of light in galaxies of similar size but different surface brightness (Abell 1962), and with Holmberg's procedure the contribution to the total light of a galaxy from very faint outer extensions that are below the photographic threshold remains unknown.
On the other hand, if the surface brightnesses of galaxies can be represented by a single model, "total" magnitudes for them can be operationally defined. It appears possible to find a satisfactory representation of the distribution of surface brightness in bright elliptical galaxies. Hubble's formula for the brightness I at a distance r from the center along the major axis of the projected image of an elliptical galaxy,
(2) 
(where I_{0} and a are parameters for a particular galaxy), describes the surface brightness over most of its observable image very well. De Vaucouleurs's (1948) formula is
(3) 
where r_{e} is the distance from the center along the major axis to the isophote within which onehalf of the total light is emitted, and I_{e} is the surface brightness at that isophote. It agrees well with the Hubble law for small r, and has the advantage of leading to finite total magnitudes that are in satisfactory agreement with those of Sandage and Holmberg.
The writer has found that a convenient interpolation formula for the distribution of light in an elliptical galaxy is a modification of the Hubble law:
(4) 
Equation (4) agrees with de Vaucouleurs's formula to the precision of presentday photometry, and leads to "total" magnitudes that are (statistically) close to those of Sandage and Holmberg. The writer and Mihalas (Abell and Mihalas 1966) have developed a technique of determining the parameters I_{0} and a, and hence the "total" magnitude, of an elliptical galaxy from measures of the brightness of its extrafocal image on each of two or more calibrated photographs taken different amounts out of focus. The total magnitude obtained is actually the magnitude of a fictitious galaxy with surface brightness given by equation (4), and that has extrafocal images of the same brightness as the measured galaxy. The precise form of the surface brightness law is not important except to establish the zero point of the magnitude scale, for in practice it is rare that contributions to an extrafocal image come from parts of a galaxy for which r/a exceeds 21.4. The magnitudes obtained may not always be accurate for individual galaxies, for all elliptical galaxies may not be built on the same model; but they are statistically selfconsistent, and are free from systematic effects that depend on the distance of a galaxy measured; in particular, they are suited to the study of luminosity functions of elliptical cluster galaxies, and to the comparison of luminosity functions of different clusters.
To date, the writer has applied the procedure to determine the luminosity functions of six clusters, of which four are regular clusters of known redshift. The luminosity functions of these clusters, which consist predominantly of elliptical galaxies, are all similar. If the logarithmic integrated luminosity functions, log N(m), are plotted, they can all be fitted together very satisfactorily with horizontal and vertical shifts that depend only on the relative cluster distances and richnesses. Such a combined integrated luminosity function is shown for four rich clusters in figure 3; they are cluster numbers 1656 (Coma), 2199 (surrounding NGC 6166), 151, and 2065 (Corona Borealis). The coordinates of figure 3 have arbitrary zero points. The smooth curve labeled "Abell" is adopted as the mean function, log N(m), for the four clusters. The dashed curve labeled "Zwicky" corresponds to Zwicky's integrated luminosity function, defined by equation (1).
Figure 3. The logarithmic integrated luminosity functions of four rich clusters. The symbols and curves are explained in the text. The zero points of the ordinates and abscissae are arbitrary. 
Curves of log N(m) compiled from published data of several other observers are plotted in the same manner in figure 4. The points for cluster 1377 (Ursa Major cluster No. 1) are from very old photometry by Baade (1928). The "Zwicky" data for cluster 1656 and the Virgo cluster are from the first two volumes of the Catalogue of Galaxies and Clusters of Galaxies. Only the central part of the Virgo cluster is used, and the Catalogue data covers only the interval of the brightest 2.7 mag in cluster 1656. The "Holmberg" data for the Virgo cluster are from Holmberg (1958). For comparison, there is also shown the logarithmic integrated luminosity function for 48 elliptical field galaxies, taken from van den Bergh (1961b). It must be emphasized that the photometric procedures differed among the various observers; large zeropoint differences between the different magnitude systems are expected, and scale errors may exist as well. Zeropoint and scale errors are especially likely in the early photographic photometry of Baade. It should also be noted that the Virgo data include many spirals, and the apparently excellent agreement between the Virgo luminosity function and that of the other clusters may be fortuitous. The "Abell" and "Zwicky" curves in figure 4 are the same ones as in figure 3.
Figure 4. Cluster logarithmic integrated luminosity functions adapted from the published data several observers. See text for explanation. The scale of the abscissae and ordinates is as in fig. 3, and the zero points are arbitrary. 
The differential luminosity function (M) corresponding to the "Abell" curve in figures 3 and 4 is shown in figure 5. The ordinates are arbitrary; the abscissae are M_{pv}, determined by fitting van den Bergh's luminosity function to the Abell cluster luminosity functions (van den Bergh's absolute magnitudes have been adjusted to correspond to a Hubble constant of 75 km s^{1} Mpc^{1}). It appears that (M) for elliptical galaxies can not be represented by a single exponential function. The function log N(m) shows a very definite change of slope after the interval of the brightest 2 to 3 mag. This abrupt change of slope corresponds to the maximum near the bright end of (M). There is some evidence that at least in the Coma cluster this peak is contributed mainly by galaxies near the central core of the cluster (Rood 1969; Rood and Abell 1973).
Figure 5. The differential cluster luminosity function, derived from figs. 3 and 4. The scale of the ordinates is arbitrary. 
The composite luminosity function for the clusters in figure 3 can be represented rather closely by two intersecting straight lines (dashed lines in fig. 3), which represent simple exponential relations over two different magnitude intervals. These lines are merely interpolation formulae, and the approximately exponential character of the integrated luminosity functions should not be taken too literally. For example, a simple exponential function can hardly be expected to match the very brightest galaxies in different clusters of different BautzMorgan types (some of which have supergiant cD galaxies and some of which do not). Moreover, if the two straight lines were to apply rigorously the differential luminosity function would have to be discontinuous (dashed lines in fig. 5), which is, of course, physically unrealistic. Nevertheless, the magnitude at which the two lines intersect in figure 3, which we define as m^{*}, is an operationally defined point on the luminosity function. If all galaxies in all clusters are selected from the same population with a parent luminosity function like that in figure 3, m^{*} should be a useful "standard candle" for determination of relative distances of clusters.
The luminosity functions of all clusters investigated so far show the same characteristic shape (i.e., change of slope as in fig. 3). The best fit of the luminosity function of a cluster to the one in figure 3 determines m^{*} for that cluster, even if the photometric data are not very complete and if its luminosity function cannot be fitted unambiguously by the two exponential relations. For those clusters of known redshift and for which m^{*} has been so determined, the plot of log z versus m^{*} has a scatter of about 0.1 mag. (s.d.) (Abell 1962; Bautz and Abell 1973), significantly less than in the corresponding Hubble diagram in which the first brightest cluster galaxies are used as standard candles.
The luminosity function data used to construct figures 3 and 4 are summarized in table 2. The cluster numbers are those in the Abell (1958) catalog. The slopes, s_{1} and s_{2}, of the straight lines used to approximate log N(m) are defined by:
(5) 
Log N( m^{*}) is the ordinate of the intersect of the two lines. Note that zeropoint and scale differences may exist between the magnitude systems of the different observers, and that m^{*} is given in photovisual magnitudes for the clusters observed by the writer, and in photographic magnitudes for those observed by others. The slope s_{2} is sensitive (particularly for distant clusters) to the corrections applied to the cluster photometry to take account of foreground and background field galaxies. The writer has not yet analyzed the field in a final way, and the values of s_{2} given for the clusters investigated by him should be considered provisional. For the purpose of discussion he adopts for the mean luminosity function the values, s_{1} = 0.75, and s_{2} = 0.25. (In the limit of faint magnitudes, Zwicky's curve has a slope, s_{2} = 0.20  in reasonable agreement with the value adopted here.)
Cluster  Mean Radial Velocity (km s^{1})  Reference for Velocity  s_{1}  s_{2}  m*  log N(m*)  Observer, or Reference for Photometry 
Virgo  1136  Humason et al. (1956)  0.72  0.20  m_{pg} = 11.9  1.15  Zwicky, Herzog, and Wild (1961) 
Virgo  0.78  0.33  m_{pg} = 10.9  1.25  Holmberg (1958)  
1656  6866  Lovasich et al. (1961)  0.75  m_{pg} 15.6  2.25  Zwicky and Herzog (1963)  
1656  0.78  0.25  m_{pv} = 14.7  2.25  Abell  
2199  8736  Minkowski (1961)  0.75  0.26  m_{pv} = 15.4  2.00  Abell 
1377  15269  Humason et al. (1956)  1.20  0.35  m_{pg} = 16.9  1.25  Baade(1928) 
151  15781  Humason et al. (1956)  0.73  0.33  m_{pv} = 16.4  1.95  Abell 
2065  21651  Humason et al. (1956)  0.72  0.27  m_{pv} = 17.2  2.23  Abell 
Field ellipticals  0.95  0.27  M_{pg} = 19.5†  ‡  van den Bergh (1961b)  
NOTE.  According to
de Vaucouleurs (1961b), the Virgo cluster is at least two clusters seen
in projection.
One component, consisting mostly of elliptical galaxies, has a mean
radial velocity of 950 km s^{1};
the other, consisting mostly of spirals, has a mean radial velocity of
1450 km s^{1}.

Since figures 3  5 and table 2 were prepared for this review, considerable new data on cluster luminosity functions have become available. Gudehus (1971, 1973) has determined luminosity functions for clusters A754 and A1367, and has confirmed that of Abell for A2065. Abell and Mottmann (in preparation; see Bautz and Abell 1973) have obtained the luminosity function for cluster A2670. Eastmond (Abell and Eastmond 1968; Abell 1972) has new photometry for the elliptical galaxies in the Virgo cluster. Noonan (1971) presents counts in clusters A1656 and A2065 to different magnitude limits that are consistent with the luminosity functions for those clusters shown in figure 3. Rood and Baum (1967, 1968), while not discussing the luminosity function per se, do give very useful photometric and other data for 315 Coma cluster galaxies. Huchra and Sargent (1973) have derived a luminosity function for field galaxies from the data in the Reference Catalogue of Bright Galaxies (de Vaucouleurs and de Vaucouleurs 1964) that is also in excellent agreement with the cluster luminosity function in figure 3. Because the more recent data are all consistent with those presented here, the figures and table have not been updated.
The cluster luminosity function has not yet been determined observationally at magnitudes fainter than about m^{*} + 4. Galaxies at this observational limit have absolute magnitudes near M_{v}  16.5; these are still relatively luminous objects, brighter, in fact, than all but four Local Group galaxies. Reaves (1966) searched for dwarf galaxies in the central core of the Coma cluster and found 32 probable dwarfs in the range 14 > M_{v} >  16. If the total content of dwarf galaxies in nearby groups and in the Virgo cluster is typical, we would expect the luminosity function to continue to increase at fainter magnitudes, but the precise form of (M)  for example, whether it increases monotonically  is not known. In particular, there is no justification for extrapolating the interpolation formula (eq. [5]) for the integrated luminosity function beyond the observed magnitude range. Such an extrapolation of equation (5) would predict that the number of Sculptortype systems (with M_{v} in the range 9 to 14) in the Coma cluster is of the order 10^{5}. Nevertheless, even this large number of dwarfs would contribute only slightly to the total luminosity (and presumably also to the total mass) of the cluster; extrapolation of equation (5) to infinite magnitude leads to a finite luminosity for the cluster of 1.2 × 10^{13} times the solar luminosity, and only 13 percent of this light would be contributed by galaxies fainter than m_{pv} = 18.3, the magnitude limit for which the luminosity function is observed.
Colors of galaxies, at least of their inner portions, can be measured far more easily than total magnitudes, and fairly accurate data are available. Published results of photoelectric photometry by de Vaucouleurs (1961a) and photographic photometry by Holmberg (1958) include galaxies both in the field and in the Virgo cluster. Colors of the brighter cluster galaxies appear to be the same as those of galaxies of similar morphological type in the field. For giant ellipticals, B  V color indices are about 0.95 to 1.00; B  V decreases through the sequence of spirals from about 0.8 for Sa's to about 0.5 for Sc's. Both de Vaucouleurs (1961a) and Baum (1959), however, report that colors of elliptical galaxies in the Virgo cluster that are successively fainter than m_{pv} 11.5 are successively bluer than those of giant ellipticals, and at m_{pv} 14, B  V is from 0.6 to 0.8. Sandage (1972) has obtained threecolor observations of six dwarf ellipticals in the Virgo cluster, and of 25 elliptical galaxies in the Coma cluster, and confirms the correlation between B  V and magnitude. He also shows that there is an even stronger dependence of U  B on V, in the sense of fainter galaxies being bluer. On the other hand, both de Vaucouleurs's and Holmberg's data include color measures of several intrinsically fainter Local Group ellipticals of intermediate luminosity, which appear to have colors that are about the same as those of giant ellipticals. Holmberg even includes two dwarf ellipticals (the Leo systems in the Local Group), and they also have approximately "normal" colors.