Proc. Nati. Acad. Sci. USA Vol. 83, pp. 3056-3063, May 1986 Astronomy The galaxy luminosity function and the redshift-distance controversy (A Review) (cosmology/clusters of galaxies) E. E. SALPETER AND G. L. HOFFMAN, JR. Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853; and Lafayette College, Easton, PA 18042 Contributed by E. E. Salpeter, December 20, 1985

ABSTRACT The mean relation between distance and law" or "linear law," p = 1, remarkably early-partly redshift for galaxies is reviewed as an observational question. for a theoretical reason: A simple extrapolation of this The luminosity function for galaxies is an important ingredient expansion law (the same for all observers) leads back to and is given explicitly. We discuss various observational a unique "starting time" for the expansion. Different selection effects that are important for comparison ofthe linear subclasses of the "standard cosmological models" make and quadratic distance-redshift laws. Several lines of evidence different predictions for large redshifts but all reduce to are reviewed, including the distribution of galaxy luminosities the linear law, p = 1, for z << 1. On the other hand, Segal in various redshift ranges, the luminosities ofbrightest galaxies in 1972 proposed a different kind of cosmological model, in groups and clusters at various redshifts, and the Tully-Fish- one that predicts instead the quadratic law, p = 2 (again for er correlation between neutral hydrogen velocity widths and Z << 1). luminosity. All of these strongly favor the linear law over the Much of the literature on the controversy of p = 1 versus quadratic. p = 2 is interlaced with theoretical discussions, but one can consider the redshift-distance relation as an observational question. Segal and his collaborators have raised a number of Section 1. Introduction questions on observational procedures, many in this journal, as well as on theory and statistical analysis. The present In 1920 a "cosmological controversy" was sparked by a paper is meant as a review in a very limited sense: to famous debate between Curtis and Shapley: Are "spiral emphasize the direct observational nature of the question we nebulae" distant galaxies, like our own Milky Way galaxy, or shall not discuss or quote any theoretical papers (not even merely small nebulae inside our galaxy? In 1925 Hubble (1) Newton or Einstein) and avoid sophisticated statistical settled the debate by discovering and analyzing Cepheid tests-presenting observational data in "almost raw form," variable stars in a few such "spirals;" by calibrating against i.e., using only the most naive and transparent statistical Cepheids in our galaxy, he found the distances to these analysis. This omits much of the relevant literature on the spirals to be much larger than the diameter ofour galaxy (=30 controversy and it may seem surprising that we have any- kpc; 1 pc = 3.086 x 1016 m), so they are definitely external thing left to review. Fortunately, there has been such a galaxies. Nevertheless, these few galaxies were all closer remarkable "data explosion" in extragalactic observational than 1 Mpc (= 3.086 x 1024 cm) and all members of the astronomy over the last ten years or so, that the controversy so-called "Local Group" of galaxies (see Section 3). The can now be settled from "almost raw data" alone. The two Cepheid method (with certain refinements) is still the most most important individual developments are (i) the magni- direct way to measure distances, up to a few megaparsecs but tude-limited "CfA survey" of more than 2400 galaxies (3) not beyond. This range includes a handful of galaxy groups with optical redshifts for all and (ii) the development of the similar to our local group, typically containing two or three "Tully-Fisher method" (4) for measuring distances r to spiral galaxies like our galaxy and =20 or 30 smaller ones (some galaxies even when r is very large. Recent extensive work on other fairly direct optical methods extend this distance range the brightest galaxies in clusters is particularly suitable for slightly). the present discussion. In 1929 Hubble (2) published values for the radial velocities Statistical discussions have been given by a number of relative to the sun for various galaxies outside of the local authors but have not yet been applied to the recent obser- group and found that most of them are redshifted, indicating vational data. We give here (without critique) a few refer- that they are receding from us. The sun's orbital velocity in ences (which themselves quote earlier work). First, some our galaxy is =230 km's'l and the sun's total velocity relative early contributions by Segal and his colleagues (5, 6); a to the centroid ofthe Local Group ofgalaxies is =300 km s'l. nonparametric procedure (7) was given in 1983 and a recent It is convenient to express all external velocities relative to paper (8) gives many references. Two pairs of papers (refs. 9 the Local Group centroid, with an uncertainty of =50 km s'1, and 10, 11 and 12) address controversies on statistical and we shall quote all observed redshifts with this correction. techniques. Here we concentrate on direct observational We define redshift z and "velocity" V in terms of the data, especially data leading to an explicit luminosity function observed wavelength Xobs in the simplest formulation, that we feel is essential for the reader to draw his or her own conclusions. We also have to deal with our slightly unusual z = V/c = (Xobs/Xrest)-1, [1] position-we live in the outskirts of a concentration of galaxies, the "Local Supercluster." appropriate for the low-redshift range (z < 0.1) to which we shall restrict ourselves. Section 2. Some definitions and procedures For galaxies outside of the Local Group, an empirical relation between redshift z and distance r of the form z a rP For the sake of readers who are not astronomers we compile (with p of order 1 or 2) was already apparent in Hubble's afew definitions and numerical values (13). We start with (the time. In spite of the small number of observations, most derived) absolute magnitude M, a logarithmic measure ofthe of the astronomical community accepted the "Hubble absolute luminosity L of an object, restrict ourselves to the 3056 Downloaded by guest on September 30, 2021 Astronomy: Salpeter and Hoffman Proc. Natl. Acad. Sci. USA 83 (1986) 3057 "UBV blue magnitude MB" and drop the subscript. In solar the "distance modulus" (m - M) and the distance r. Various units the absolute magnitude M and the apparent magnitude slight refinements of this method have been proposed (e.g., m (measuring the directly observed flux L/r2) read in refs. 21 and 22) and are summarized in a recent paper (23). The scatter in the Tully-Fisher relation is smallest for Sb and M = 5.48 - 2.5 loglo(L/L), Sc galaxies, but it still holds reasonably well for other types m = M + 25 + 5 of spirals. The method works fairly well out to z 0.04. loglo(r/l Mpc), For statistical work on galaxies without measured dis- where r is the distance to the object. We shall test the two tances, two magnitude-limited optical catalogs with galaxy rival for M from m and redshifts are particularly useful. One is the so-called RSA hypotheses deriving V, catalog (24) with a rough magnitude cutoffof 13.2, listing 1246 V galaxies (including southern galaxies); another, the CfA r (1 V \r (1 12 3 survey (3), with a magnitude limit of about 14.5, listing 2401 1 Mpc (Ho km s') oKo km s-) * [3] galaxies. Since different magnitude definitions have been used by different sources in the past there are some ambi- Unlike stars, galaxies do not have a sharply defined "outer guities in the cutoffs but not by more than a few tenths of a edge" and the assigned apparent magnitude m depends on the magnitude. Furthermore, the catalogs are not complete right angular area over which one sums the optical surface bright- up to the cutoff but have a (reasonably well determined) ness CL. For spiral galaxies, like our own, CTL decreases "completeness function." Excluding dwarf galaxies (which exponentially with distance from the center and there is little have a low central surface brightness) it is safe to consider ambiguity in the integral of aL. For our own galaxy the these catalogs complete to apparent magnitude m, about 12.2 exponential scalelength is =6 kpc and the absolute magnitude and 13.8, respectively. An important property ofthe catalogs is M -20.2 (see, e.g., ref. 14), with an uncertainty of order is their redshift completeness; i.e., every galaxy on each list ±1.0. For giant elliptical galaxies, on the other hand, the has a measured velocity. In particular, the catalogs could surface brightness decreases less sharply than exponential have missed only very few galaxies of large redshift but and, for intermediate values of angular distance 6 from the brighter than mc. The "data-explosion" in this field is center, the integrated flux increases approximately as (ln 6 + considerable-a catalog only seven years old (e.g., ref. 25) is constant). For the relatively small redshifts considered here already "old-fashioned." The slightly larger random magni- (z < 0.1) these "optical halos" need not be a problem in tude errors in the CfA catalog are not very troublesome, but principle: The central optical surface brightness crL(O) is the extension to larger redshifts is very important because of typically much brighter than that ofthe night sky (discounting the clustering to be discussed below. dwarf galaxies, which are of no interest here); the angular radius Osb where aL drops to some predetermined surface brightness is large enough (>10 arcsec) so it can in principle Section 3. Groups, dusters, and superclusters: Peculiar be measured accurately without smearingfrom "atmospheric velocities seeing." One can then give a distance-independent measur- ing procedure for apparent total magnitude m in terms of the Most theoretical cosmological models assume that, after flux in an aperture of radius Osb. A similar procedure is used averaging over a "sufficiently large" volume, (i) the density in the "RC2 catalog" (15), and the magnitudes in the CfA distribution of galaxies is uniform and (ii) the differential survey can be adjusted to this system (16). expansion velocity V between two regions depends only on As pointed out before (17), some papers on bright galaxies their separation r. We summarize the evidence for (i) clump- in clusters present magnitude data that are referred to angular ing and (ii) peculiar velocity perturbations superimposed on apertures 6H that correspond to a constant absolute radius if a smooth V(r) relation. the linear redshift-distance law holds but to a varying (i) We have already seen that we live in the Local Group absolute radius for the quadratic law. In principle this could of galaxies with two moderately bright galaxies, our own (M invalidate a fair comparison between the two laws (17), but -20.2) and M31, Andromeda (M -21.6) (24). Compi- in practice recent advances in optical procedures have lations of other galaxy groups, with similar or larger galaxy eliminated this flaw: For many bright galaxies in clusters, content, are now available, including redshifts (26). For our detailed surface photometry and accurate values for various purposes the nearest few are most important, because they distance-independent definitions of angular radius were ob- contain many of the "primary distance calibrator galaxies" tained by Sandage (18) and have been independently con- for which direct optical distance measurements are available firmed (with less tedium) by more recent efforts (19, 20). (e.g., ref. 27). For instance, there are five groups with mean These radii, which are independent of distance and of the recession velocities V between 200 and 400 km s'1 for which assumed law, are found to be compatible with the "Hubble- biased" radii OH. The scatter in individual radii is too large to (r) 5.5 Mpc,' [4] consider this compatibility as a verification of the linear law, =5.8 for = 270 km but at least no great errors are introduced if magnitude rrms Mpc (V) s1', measurements are based on OH. where the linear distance average (r) is appropriate for the The gas-rich "late spirals," class Sb and Sc (which bracket linear law in 3 and the rms our galaxy and Andromeda), are typically regular flat disks Eq. value rrn. is appropriate for with circular rotation, and the maximum rotation velocity is the quadratic law. There persists some controversy over the to be absolute distance scale (some distance estimates are almost empirically known correlated with L. The Tully-Fisher a factor of two smaller). That controversy affects only the method (4) for determining distances to galaxies consists of multiplicative factor in the redshift law, however, not the measuring accurately the internal velocity width AV of a power-law dependence. Some of these groups are rather spiral galaxy from the 21-cm line radiation emitted by the loose and there is some controversy on dynamic group neutral hydrogen (termed by astronomers Hi) in the disk. membership. However, here we are using groups only as a Optical data are needed to apply an inclination correction to convenient way to collate galaxies with roughly similar obtain the "edge-on value" AVO (essentially twice the max- locations and redshifts, so membership is unimportant. imum rotation velocity). Nearby galaxies give an absolute Unfortunately, (V) for these five (and other nearby) groups calibration for the Tully-Fisher relation L(AV0), and the could have an appreciable contribution from peculiar veloc- observed apparent magnitude m of a distant spiral then gives ities (see below). Downloaded by guest on September 30, 2021 3058 Astronomy: Salpeter and Hoffman Proc. Natl. Acad. Sci. USA 83 (1986) Galaxy clusters typically contain 10 to 100 times more distribution on the sky (Fig. 1) is highly concentrated around bright galaxies than a typical group and are correspondingly the Virgo Cluster (all are within 0.1 radians). Since distances rarer. Procedures for defining and classifying the richer types can be neither negative nor imaginary, this cannot be of clusters, the "Abell clusters," have been given and more achieved from pure cosmological flow in either model and than 2000 are known (28). In principle, there is an unavoid- shows that peculiar velocities, IV - 1100 km s1I, can be quite able difficulty inherent in a quantitative definition of a appreciable near cluster cores. (ii) Many clusters are found to cluster, which increases with increasing distance, namely, have hot x-ray-emitting gas bound in their cores, with a the contamination by foreground and background galaxies temperature equivalent to the cluster velocity dispersion (37). and groups (29, 30). Fortunately, this effect is fairly small for For regions further out in a supercluster (in contrast to the our restricted redshift range z < 0.1. Furthermore, as for cluster core) the majority and minority views agree that groups, we shall use clusters mainly as a convenient collec- galaxies are not gravitationally bound and measured veloci- tion ofgalaxies with roughly similar redshifts and location, so ties mainly come from the smooth cosmological flow, the dynamical membership is less important here. Elliptical model V(r). However, there must be smoothly varying small galaxies are more common in dense clusters than in the field deviations from "isotropic Hubble flow" due to the gravita- (31), and rich, dense clusters often have "special" ("cD") tional pull of the cluster. In principle this can be observed, galaxies in their centers that are about half a magnitude using the Tully-Fisher method (and others) to select equally brighter than the brightest normal galaxies in other (poorer) distant galaxies in the direction toward the Virgo Cluster, and clusters at comparable redshift (19). opposite, and then comparing their velocities. Further, this The nearest few Abell clusters are at distances correspond- can be generalized to equal-distance shells at larger dis- ing to almost 5000 km s-1, but we live in a somewhat special tances. This has been done by a large number of observers location since one galaxy cluster that is "almost rich enough" (38-42) with slightly different methods and each with only to be an Abell cluster, the Virgo Cluster, has a recession modest accuracy. There is at least qualitative agreement that velocity of only 1100 km s-1 (distance, 15-25 Mpc). The our deviation 8V from isotropic flow relative to the Virgo Virgo Cluster has comparable numbers of spiral and elliptical center ("radius" 1000 kms-1) is of order 200 kms-' galaxies and a somewhat irregularly shaped "core" or inward and that WV fluctuates but generally increases slowly "cluster proper," of radius about 5 or 60. Many galaxy with increasing shell radius to an asymptote of somewhat less clusters are surrounded by a huge halo of galaxies, of radius than 1000 kms- relative to the microwave background about 10 core radii, forming a kind of "supercluster" (32, 33). radiation a shell of "radius" To The Virgo Cluster is surrounded by such a halo or cloud, the (effectively c) (43). "Virgo Supercluster" or "Local Supercluster" (34, 35), and summarize: the fractional error 8V/V in measured average we live in the outskirts of this supercluster! This fact leads to recession velocity (V), as an indicator of the smooth cosmo- a considerable inhomogeneity (and anisotropy) in the number logical velocity field V(r), is =0.2 at V =1000 km sl and then density of galaxies in our vicinity. fluctuates but generally decreases with increasing distance r. (ii) We are interested in two kinds of "peculiar velocities" Unfortunately, because of the "deviations from Hubble for individual galaxies, both related to clusters (or superclus- flow," group velocities are particularly uncertain for the five ters). The first has to do with the velocity dispersion oa of the groups with velocity between 200 and 400 km's' in Eq. 4, systemic velocities of individual galaxies in a cluster relative where the measurement ofdistance r is particularly direct and to the mean velocity V of the cluster. According to majority accurate. To establish the power law dependence in V(r) the opinion, a cluster (but not its associated supercluster) is a scale factors Ho or Ko in Eq. 3 are not needed, but in the spirit gravitationally bound system and a represents only "internal of concreteness we quote some values: If we use Eq. 4, in motion" of the system. This point of view requires a large spite of the uncertainty, we get Ho = 49 or Ko = 8.0, where amount of "dark" gravitational mass, but it is argued that both the Hubble parameter and the quadratic scale factor Ko dark matter is already required by circular rotation curves in represent what the recession velocity would be in km s'1 at spiral galaxies. According to a minority view (29), a cluster a distance of 1 Mpc. These values are uncertain by easily a represents an extended density enhancement with less grav- factor of two and most workers prefer to determine the scale itational effect (avoiding the necessity ofdark matter), so that factors (even though less directly) from the Virgo Cluster oa derives largely from the differences in recession velocity V(r) for different distances. For the gravitation-less limit, coupled with the linear V(r) law (equivalent to unacpelerated 28.0 expansion from a point since the instant of the "Big Bang"), one can make definite predictions: The cluster radius along 23.0 the line of sight divided by distance is a/Vand, if clusters are roughly spherical, this should be of order the apparent angular radius 6ad in radians. In fact, or/V6>d is typically of 8018. order 10 for rich clusters, which would imply cigar-shaped clusters all pointed along the line of sight (or "fingers of 13.0 god"). For instance, for the Coma Cluster, the_nearest well-studied (36) very-rich cluster, ar- 1000 km-s-, V = 7000 kms-1, and 6rd is of order 0.05 radians. A naive interpreta- 8.G tion of the quadratic law would suggest that cr/2V6md be of order unity, since d log rid log z is half as large as in the linear 3.0 model, but apparently the model dependence can be even stronger (29). -2.01 Fortunately, there are two "model-independent" obser- 13.00 12.50 12.00 vational facts suggesting that there are indeed considerable peculiar velocities for individual galaxies in a bound cluster. Right ascension, hr (i) The most convincing evidence comes from galaxies with FIG. 1. Distribution on the sky of all negative velocity galaxies negative (approaching) velocities in the RSA and CfA cata- outside the local group. The large x marks the center of the Virgo logs (always excluding Local Group galaxies and using cluster; the circle is of 60 radius and demarcates the canonical velocities V relative to the Local Group center). Their "cluster proper." Downloaded by guest on September 30, 2021 Astronomy: Salpeter and Hoffman Proc. Natl. Acad. Sci. USA 83 (1986) 3059

(see, e.g., refs. 41 and 42). With r 14-22 Mpc and V(r) important, but at high V (far) the fainter end ofthe luminosity 1000-1400 km s-l,this gives Ho 50-100 and Ko 2.1-7.1. function is suppressed. Ifone were to evaluate mean absolute magnitudes as a function of V or to compare properties of Section 4. The luminosity function, Malmquist biases, and near-far pairs, the Malmquist effect would then become a the magnitude-redshift relation "Malmquist bias" (45). One can already test the two V(r) laws in Eq. 3 in a qualitative way: If the luminosity function In the earliest days of observational cosmology it was hoped 0(L) had a sharp upper cutoff at some fixed Lmax and 4i were that galaxies would provide "standard candles"-i.e., all very largejust below Lmax, (neither ofwhich is true), the upper have the same luminosity L. It soon became clear that this is asymptote ofthe scatter diagram in Fig. 2 would be horizontal not the case and that one has to study the luminosity function for the linear law, parallel to the heavy dashed line for the 4(L), the probability distribution for different luminosities. quadratic law. Neither is the case, but we must discuss a kind For well-developed spirals (Hubble type Sc) with detailed of "generalized Malmquist effect." optical photographs one can assign a second kind of purely As we go to larger velocities V (with bins of constant A log morphological label, "luminosity class I, II, III, or IV" that V), the volume, and the potential number of galaxies, empirically has some correlation with absolute luminosity. increases (as V3 on the linear, V1.5 on the quadratic law). There was a hope that, for a given class, the luminosity Hence, ifthe upper end ofthe luminosity function 4i does not function might be sufficiently narrow for an "almost standard have a sharp cutoffbut only a moderately sharp decrease, the candle." For instance, it has been suggested (8) that com- absolute magnitude (largest luminosity Lmax) should increase parison at different redshifts, plus the hypothesis of a narrow with increasing V-the Scott effect (46, 47). For particularly luminosity function, might distinguish between the different rare types of bright galaxies, there will even be a first V-bin expansion laws. Fortunately, we now have enough observa- containing any; e.g., the cD galaxy with the lowest V is in the tional data to obtain the luminosity function for a single Abell 2199 cluster with V = 9360 kmts-1. For the linear law luminosity class and unfortunately it is not narrow! An we then should expect to find more galaxies above the thin illustration from the analysis (44) of the RSA catalog for the dashed horizontal line as V increases; this is certainly the most favorable (and brightest) class Spiral I is shown in Fig. case but we cannot say whether or not by the right amount. 2. The Virgo Cluster proper (core) has been omitted from the Similarly, for the quadratic law more galaxies should lie sample, but the Local Supercluster is represented and is quite above the heavy dashed sloping line in Fig. 2 as V increases rich. Consequently there are an appreciable number of (but the increase should be less severe because ofthe weaker galaxies within V 800-1600 kmts'1, say, that should all be V-dependence of volume). This is definitely not the case; in at comparable distances irrespective ofthe V(r) relation. Fig. fact there is a void in the upper right corner even below the 2 shows quite a large spread in luminosity. The luminosity heavy dashed line. This is a serious, qualitative blow to the function 4(L) is not narrow but, on any cosmological model, quadratic hypothesis, but we need more data to make it one assumes that 4' (and also the number of galaxies per unit quantitative. volume) should be independent of distance. Since the individual luminosity classes do not provide The absolute magnitude M derived from the observed "standard candles" anyway, we now lump all morphological apparent magnitude m and Eq. 2, assuming the linear law, is types together and display combined luminosity functions. plotted in Fig. 2. The lower solid line represents the cutoff in We use data from the CfA survey (fainter limit than the RSA apparent magnitude of the catalog, and the thin dashed by more than one magnitude), omitting only galaxies within horizontal line represents a constant (large) luminosity on the 60 ofthe Virgo Cluster center with velocities outside the range linear law, the dashed sloping line on the quadratic law. This 500-1700 kmnsrl (to decrease peculiar velocity effects). We figure illustrates the so-called "Malmquist effect:" At low lump together all galaxies within this velocity range, both in velocities V (near) the apparent magnitude cutoff is not too the Virgo Cluster and out (we verified before combining that the differences between the "in" and "out" samples are not noticeable in Fig. 3 B and B'). Fig. 3 displays data "without -24 -O prejudice to either V(r) law," as Segal has suggested for a long time: Using as our distance unit the distance correspond- ing to V = 1100 km s-1 instead of 1 Mpc as in Eq. 2, we define

-23 - le ml = m - 5 loglo(V/1100 km s-1), [5] M2 = m - 2.5 log1o(V/1100 km sr1). -22S W/d~~~~~~~~~~~* , 4 ml and m2 are then (except for a common additive constant) the absolute magnitudes (-2.5 log L) on the linear and quadratic hypotheses, respectively. We plot the function L24(L) in Fig. 3. -21- Fig. 3 A and A', galaxies appreciably closer than the Virgo Cluster, is unreliable because of peculiar velocities. Fig. 3 B and B' uses the same galaxies with ml M2. The luminosity function here is therefore almost the same for the linear and -20- quadratic models, is based on a large number ofgalaxies with Vnear 1100 kms -' (much ofthe Local Supercluster), and can be used to calibrate 4(L). The smooth curve in Fig. 3 B and 0.3 0.6 1 2 3 6 10 B' is an empirical fitting function, the so-called "Schechter V X 103, km-sY1 function" (48), FIG. 2. Distribution in absolute magnitude and velocity of the Spiral I and I-LI galaxies in the RSA catalog (modified from figure 1 4(L) X L-exp(-L/L*), [61 in ref. 44). Constant luminosity is represented by the horizontal line if the linear law is correct, by the dashed sloping line if the quadratic with /3 1.25 and L* corresponding to a critical absolute law holds. magnitude mn = m2 = 10.5. This choice agrees reasonably Downloaded by guest on September 30, 2021 3060 Astronomy: Salpeter and Hoffman Proc. Natl. Acad. Sci. USA 83 (1986)

8 10 12 14

FIG. 3. The luminosity function (times luminosity) in various redshift ranges with an "absolute" magnitude ml determined according to the linear law or m2 determined according to the quadratic law (see Eq. 5). (A and A') Velocities from 250 to 500 km s1. (B and B') Velocities from 500 to 1700 km s-1. (For each pair, the same galaxies are used although ml and m2 differ slightly for individual galaxies.) (C-F and C'-E') Each pair is based on the same ranges of distance, and numbers in the upper left-hand corners indicate velocities corresponding to the (common) midpoint distance for each pair. The velocity ranges are as follows: C, 1395-2416 kmns-1; C', 1768-5305 km s-'; D, 2416-4184 km s-1; D', 5305-15916 km's'1; E, 4184-7247 km s-1; E', 15916-47749 km s-1; F, 7247-12553 km s-1. The smooth curves (identical in B-F and B'-E' except for normalization) represent the Schechter function that best fits the 1100-km s-1 bin. The histograms are L2q/ computed from the data in the CfA catalog. The breaks from solid lines to dashed lines indicate the points at which the magnitude bin ceases to be complete at the far end ofthe velocity range, and the tic marks on the Schechter curves indicate the points at which the center ofthe distance range ceases to be complete.

well with more careful analysis for the Virgo Cluster (49) and velocity-distance hypotheses: For each pair r is larger than (since the velocities are nearly the same) is independent ofthe for the pair above by \/-; with the redshift distance at V = velocity-distance law. 1100 km s'1 as the common unit of distance, the equivalent Fig. 3 pairs C and C', D and D', and E and E' correspond mean velocities V (V a r and V a r2, respectively) separate to identical ranges of distance r, on each of the two more and more as the distance increases. The labeling of Downloaded by guest on September 30, 2021 Astronomy: Salpeter and Hoffman Proc. Natl. Acad. Sci. USA 83 (1986) 3061 absolute magnitude (ml and M2, respectively, but each -2.5 smaller collection. Because rich clusters are rarer than poor log L) and the normalization per unit spatial volume are the clusters, we have a tendency toward a generalized Malmquist same for each pair. The smooth curves in Fig. 3 C-F are the bias; i.e., as the velocity V increases we tend to pick richer same as the fitted curve in Fig. 3B. The observed histograms collections with a brighter brightest galaxy (45, 46). The should then agree with the smooth curve for the "correct" occurrence ofthe even brighter cD galaxies only for V > 9000 redshift law, if a constant luminosity function and a uniform km s-1 is another small example in this direction. There is, distribution of galaxies is assumed (because of the density however, one anti-Malmquist bias (29) (discussed in Section enhancement in the Local Supercluster the histograms in Fig. 3) if the redshift is measured for only a few galaxies in a 3 C-F should actually lie slightly lower than the smooth "concentration in the sky:" As the redshift increases there is curve). The linear law fits this criterion quite well, even in an increasing probability that the measured galaxy is the Fig. 3F at 9 times the unit distance. The quadratic law, absolute brightest in only a smaller, nearer cluster with a however, violates the criterion more and more with increas- larger, unrelated, background cluster making the measured ing distance. The discrepancy, a progressive "shift to the cluster appear richer than it really is. This effect has been right" of the histogram, is particularly disastrous in Fig. 3E' modeled numerically (30); it can be appreciable for z > 0.2 but at 5.2 times the unit distance (and beyond) where there are is completely unimportant for z < 0.04 (47) and weak for 0.04 simply no galaxies with M2 < 11. Furthermore, the discrep- < z < 0.1. In Fig. 4 we have strongly overcompensated this ancy must be due to an incorrect law, not merely due to a anti-Malmquist bias by excluding the poorest (richness class "hole" at this distance, since there are plenty of galaxy 0) Abell clusters for z > 0.04 but including them for z < 0.04. clusters with V 30000 km s-1 but with all galaxies fainter With the sign of the Malmquist bias in Fig. 4 firmly than M2 11. established to be positive, we now have a single criterion for whichever of the two redshift-distance models V(r) is cor- Section 5. Bright cluster galaxies and Tully-Fisher results rect: The observed scatter diagram in Fig. 4 for clusters (circles) should have a slope close to, but slightly smaller Although galaxy catalogs are even now complete only to a than, the model line. There can be little doubt that the linear magnitude cutoff of m 14.0, modem optical techniques law fits this criterion and that the quadratic law does not. The allow accurate velocity and magnitude measurements for a brightest galaxies in small groups at lower redshifts are also small number of individual galaxies three or four magnitudes shown in Fig. 4 (triangles). If one combined all triangles and fainter. A convenient way to select a small fraction of distant circles, one would obtain a "best-fit" slope intermediate galaxies is to pick the brightest galaxy (or the brightest few) between the two power laws, but that would introduce too in each of a large number of galaxy groups or clusters. This large a generalized Malmquist bias-as shown by the overlap has now been done both for groups (26) and for clusters near V 5000 kms-1, clusters have brighter brightest (50-52) up to z 0.3. We display some of this data in Fig. 4, galaxies than the (very much smaller) groups and a separate again restricting ourselves to redshifts z < 0.1. fitting line would have to be drawn for the triangles. We also give in Fig. 4 the predicted slopes for the linear and Since the Tully-Fisher method gives a direct distance quadratic models, but we must consider any possible dis- estimate for each galaxy used, membership in a group or tance-dependent biases. As we saw in Section 2, detailed cluster is not essential for this method. In Fig. 5 we display surface photometry (18-20) has eliminated Hubble-biased the data in two ways: Fig. SA shows data for 383 galaxies, aperture selection as a source of serious error. Because the every spiral galaxy in the CfA catalog having m < 13.0 and end the a published Hi profile with a signal/noise ratio of >7 (80% of bright of luminosity function (Section 4) is not the total sample of spirals having m < 13.0), and Fig. SB infinitely steep, the brightest of a larger collection ofgalaxies shows data averaged over all spirals with measured Hi widths will typically be slightly brighter than the brightest in a in each of several well-studied groups and clusters: nearby groups from Richter and Huchtmeier (23), nine large clusters from Giovanelli and Haynes (53). Although neither the group sample nor the cluster sample are statistically complete in any sense, neither were they preselected for our present purpose: Each is the largest homogeneous collection of data of its class available. The calibration of the Tully-Fisher correlation we have adopted here is that of Richter and Huchtmeier (23). On each plot we have drawn two lines, one for each ofthe two redshift laws, that should parallel the ridge line of the data under each respective hypothesis. From either plot it is clear that the linear law is consistent with the data while the quadratic law is not. Section 6. Discussion We have seen above that the luminosity function for galaxies, 4X(L), can be obtained from the Local Supercluster indepen- dent of the redshift-distance law. 4(L) is fairly broad and its high-luminosity tail is particularly troublesome-very bright galaxies are rare, but in a large enough collection some will be found. This leads to the so-called generalized Malmquist V, km-s- effect, but we saw that modem observational data are sufficiently redundant that the "effect" does not lead to a FIG.4. Apparent magnitude m versus velocity V for the brightest "bias." We saw then that each of several lines of evidence galaxies in groups from the CfA catalog (v) and Abell clusters (v), including richness class R = 0 for z <0.04, and excluding R = 0 for favors the linear redshift-distance law over the quadratic law: z > 0.04. The sloping lines are parallel to the ones expected if all The luminosity function for various ranges of redshift, the brightest galaxies had the same luminosity, according to the two m-z diagram for brightest galaxies in groups and clusters, and hypotheses (line 1, linear; line 2, quadratic). the Tully-Fisher results. Downloaded by guest on September 30, 2021 3062 Astronomy: Salpeter and Hoffman Proc. Natl. Acad. Sci. USA 83 (1986) the data in Fig. 4 by professional observational cosmologists (47, 50-52), including corrections for a remaining generalized Malmquist bias, gives a much better fit to the linear law than our Fig. 4 and has been extended to higher redshifts, but such analysis requires the input of some theory that we wish to avoid here. Similarly a naive "best-fit-by-eye" to the Tully-Fisher plots in Fig. 5 would give (for individual galaxies)p 1.0 or (for groups and clusters) p 1.1 and again "professionals" could do better. However, the "almost raw data" in Figs. 3-5 extend over a fairly large velocity range and clearly exclude p = 2 with high confidence. We thank R. Giovanelli for data in advance of publication; Drs. G. de Vaucouleurs, W. Huchtmeier, S. Kent, P. Lax, B. M. Lewis, A. Sandage, I. E. Segal, and G. Tammann for interesting suggestions; and K. Zadorozny and B. Boetscher for drawing the figures. This work was supported in part by the National Science Foundation Grants AST 84-15162 at Cornell University and AST 84-06392 at Lafayette College.

I Xf - /1 1 1 1 l l 1. Hubble, E. P. (1925) Astrophys. J. 62, 409-433. 1.0 10.0 100 2. Hubble, E. P. (1929) Proc. Natl. Acad. Sci. USA 15, 168-173. 3. Huchra, J., Davis, M., Latham, D. & Tonry, J. (1983) Astrophys. J. Suppl. 52, 89-120. 4. Tully, R. B. & Fisher, J. R. (1977) Astron. Astrophys. 54, 10000 B 661-673. 5. Segal, I. E. (1975) Proc. Natl. Acad. Sci. USA 72, 2473-2477. 6. Nicoll, J. F. & Segal, I. E. (1975) Proc. Natl. Acad. Sci. USA 72, 4691-4695. 7. Nicoll, J. F. & Segal, I. E. (1983) Astron. Astrophys. 115, 180-188. 8. Segal, I. E. (1985) Publ. Astron. Soc. Jpn. 37, 499-506. 9. Soneira, R. M. (1979) Astrophys. J. Lett. 230, L63-L65. 1000 A z - 10. Nicoll, J. F. & Segal, I. E. (1982) Astrophys. J. 258, 457-466. 11. Tammann, G., Yahil, A. & Sandage, A. (1980) Physical Cosmology (North-Holland, Amsterdam). 12. de Vaucouleurs, G. (1983) Mon. Not. R. Astr. Soc. 202, 367-378. 1000 13. Allen, C. W. (1973) Astrophysical Quantities (University of London Press, London). 14. de Vaucouleurs, G. (1983) Astrophys. J. 268, 451-467. 15. de Vaucouleurs, G., de Vaucouleurs, A. & Corwin, H. (1976) Second Reference Catalogue ofBright Galaxies (University of Texas Press, Austin, TX). 16. Auman, J. R., Hickson, P. & Fahlman, G. G. (1982) Publ. 1.0 10.0 100 Astron. Soc. Pac. 94, 19-25. Roe Mpc 17. Segal, I. E. (1983) Astron. Astrophys. 123, 151-158. 18. Sandage, A. (1972) Astrophys. J. 173, 485-499. 19. Hoessel, J. G. (1980) Astrophys. J. 241, 493-506. FIG. 5. Correlation of distance R obtained from the Tully-Fisher 20. Schneider, D. P., Gunn, J. E. & Hoessel, J. G. (1983) method and velocity V for individual spiral galaxies with m -- 13.0 in Astrophys. J. 268, 476-494. the CfA catalog (A) and averages for well-studied groups and clusters 21. Visvanathan, N. (1983) Astrophys. J. 275, 430-444. (B). The calibration is by the method ofRichter and Huchtmeier (23). 22. Bottinelli, L., Gouguenheim, L., Paturel, G. & de Vaucou- (A) All spiral galaxy types Sa through Sm are included. No selection leurs, G. (1984) Astrophys. J. 280, 34-40. by inclination angle has been made; to correct V to VO we have used 23. Richter, 0. G. & Huchtmeier, W. K. (1984) Astron. sin i,, = max(0.4, sin i). All galaxies within 6° of the Virgo Cluster Astrophys. 132, 253-264. center (and with V < 2500 km s-1) have been plotted at the cluster 24. Sandage, A. & Tammann, G. A. (1981) A Revised Shap- mean velocity (977 km s-1). No corrections have been made for ley-Ames Catalog of Bright Galaxies (Carnegie Institute, Virgocentric deviations from the velocity law, which may increase Washington, DC). the scatter at intermediate velocities but are not biased in favor of 25. Visvanathan, N. (1979) Astrophys. J. 228, 81-94. either law. (B) Mean distances and velocities for nearby groups (a) 26. Geller, M. J. & Huchra, J. P. (1983) Astrophys. J. Suppl. 52, from Richter and Huchtmeier (23) and for nine large clusters (0), 61-88. including Virgo, studied by Giovanelli and Haynes (53). Two circles 27. Sandage, A. & Tammann, G. A. (1982) Astrophys. J. 256, joined by a bar are shown for Virgo; the lower V assumes no 339-345. Virgocentric motion ofthe Local Group while the upper one assumes 28. Abell, G. 0. (1958) Astrophys. J. Suppl. 3, 211-288. a (large) Virgocentric infall, of 450 km s-1. The sloping lines should 29. Segal, I. E. (1983) Astron. Astrophys. 123, 151-158. parallel the ridge line of the data in either plot under each respective 30. Lucey, J. R. (1983) Mon. Not. R. Astr. Soc. 204, 33-43. hypothesis (line 1, linear; line 2, quadratic). 31. Dressler, A. (1980) Astrophys. J. 236, 351-365. 32. Oort, J. H. (1983) Annu. Rev. Astron. Astrophys. 21, 373-428. It is clear that if one were to insist that V xc with an 33. Einasto, J., Joeveer, M., Kivila, A. & Tago, E. (1975) Astr. rP, p Circ. USSR, no. 895, 2. integer, we must choose p = 1 rather than p = 2. We do not 34. de Vaucouleurs, G. (1956) Vistas Astron. 2, 1584-1606. derive a formal best-fit noninteger value for p nor a formal 35. de Vaucouleurs, G. (1978) in The Large Scale Structure of the "standard error in the exponent." If we nevertheless com- Universe: IAU Symposium Number 79, eds. Longair, M. S. & pute a "best-fit" value ofp from the m-z diagram (Fig. 4) for Einasto, J. (Reidel, Dordrecht, The Netherlands), pp. 205-212. brightest galaxies in "Abell clusters" alone (omitting the 36. Kent, S. M. & Gunn, J. E. (1982) Astron. J. 87, 945-971. smaller "groups"), we obtain p ==1.1. A careful analysis of 37. Jones, C. & Forman, W. (1984) Astrophys. J. 276, 38-55. Downloaded by guest on September 30, 2021 Astronomy: Salpeter and Hoffman Proc. Nati. Acad. Sci. USA 83 (1986) 3063

38. Tully, R. B. & Shaya, E. J. (1984) Astrophys. J. 281, 31-55. 45. Malmquist, K. G. (1922) Ark. Mat. Astron. Fys. 16 (23). 39. Aaronson, M., Huchra, J., Mould, J., Schechter, P. & Tully, 46. Scott, E. L. (1957) Astron. J. 62, 248-265. B. (1982) Astrophys. J. 258, 64-76. 47. Postman, M., Huchra, J. P., Geller, M. J. & Henry, J. P. 40. Hart, L. & Davies, R. D. (1982) Nature (London) 297, (1985) Astron. J. 90, 1400-1406. 191-196. 48. Schechter, P. (1976) Astrophys. J. 203, 297-306. 41. de Vaucouleurs, G. & Peters, W. L. (1984) Astrophys. J. 287, 49. Kraan-Kortweg, R. C. (1981) Astron. Astrophys. 104, 280-287. 1-16. 50. Sandage, A. & Hardy, E. (1973) Astrophys. J. 183, 743-757. 42. Tammann, G. A. & Sandage, A. (1985) Astrophys. J. 294, 51. Hoessel, J., Gunn, J. & Thuan, T. (1980) Astrophys. J. 241, 81-95. 486-492. 43. Yahil, A., Walker, D. & Rowan-Robinson, M. (1986) 52. Schneider, D., Gunn, J. & Hoessel, J. (1983) Astrophys. J. 264, Astrophys. J. Lett. 301, L1-L6. 337-355. 44. Tammann, G., Yahil, A. & Sandage, A. (1979) Astrophys. J. 53. Giovanelli, R. & Haynes, M. P. (1985) Astrophys. J. 292, 234, 775-784. 404-425. Downloaded by guest on September 30, 2021