Proc. Nati. Acad. Sci. USAVol. 83, pp. 3056-3063, May 1986Astronomy

The galaxy luminosity function and the redshift-distancecontroversy (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 andredshift for galaxies is reviewed as an observational question.The luminosity function for galaxies is an important ingredientand is given explicitly. We discuss various observationalselection effects that are important for comparison of the linearand quadratic distance-redshift laws. Several lines of evidenceare reviewed, including the distribution of galaxy luminositiesin various redshift ranges, the luminosities of brightest galaxiesin groups and clusters at various redshifts, and the Tully-Fish-er correlation between neutral hydrogen velocity widths andluminosity. All of these strongly favor the linear law over thequadratic.

Section 1. Introduction

In 1920 a "cosmological controversy" was sparked by afamous debate between Curtis and Shapley: Are "spiralnebulae" distant galaxies, like our own Milky Way galaxy, ormerely small nebulae inside our galaxy? In 1925 Hubble (1)settled the debate by discovering and analyzing Cepheidvariable stars in a few such "spirals;" by calibrating againstCepheids in our galaxy, he found the distances to thesespirals to be much larger than the diameter ofour galaxy (=30kpc; 1 pc = 3.086 x 1016 m), so they are definitely externalgalaxies. Nevertheless, these few galaxies were all closerthan 1 Mpc (= 3.086 x 1024 cm) and all members of theso-called "Local Group" of galaxies (see Section 3). TheCepheid method (with certain refinements) is still the mostdirect way to measure distances, up to a few megaparsecs butnot beyond. This range includes a handful of galaxy groupssimilar to our local group, typically containing two or threegalaxies like our galaxy and =20 or 30 smaller ones (someother fairly direct optical methods extend this distance rangeslightly).

In 1929 Hubble (2) published values for the radial velocitiesrelative to the sun for various galaxies outside of the localgroup and found that most of them are redshifted, indicatingthat they are receding from us. The sun's orbital velocity inour galaxy is =230 km's'l and the sun's total velocity relativeto the centroid ofthe Local Group ofgalaxies is =300 km s'l.It is convenient to express all external velocities relative tothe Local Group centroid, with an uncertainty of =50 km s'1,and we shall quote all observed redshifts with this correction.We define redshift z and "velocity" V in terms of theobserved wavelength Xobs in the simplest formulation,

z = V/c = (Xobs/Xrest)-1, [1]

appropriate for the low-redshift range (z < 0.1) to which weshall restrict ourselves.For galaxies outside of the Local Group, an empirical

relation between redshift z and distance r of the form z a rP(with p of order 1 or 2) was already apparent in Hubble'stime. In spite of the small number of observations, mostof the astronomical community accepted the "Hubble

law" or "linear law," p = 1, remarkably early-partlyfor a theoretical reason: A simple extrapolation of thisexpansion law (the same for all observers) leads back toa unique "starting time" for the expansion. Differentsubclasses of the "standard cosmological models" makedifferent predictions for large redshifts but all reduce tothe linear law, p = 1, for z << 1. On the other hand, Segalin 1972 proposed a different kind of cosmological model,one that predicts instead the quadratic law, p = 2 (again forZ << 1).Much of the literature on the controversy of p = 1 versus

p = 2 is interlaced with theoretical discussions, but one canconsider the redshift-distance relation as an observationalquestion. Segal and his collaborators have raised a number ofquestions on observational procedures, many in this journal,as well as on theory and statistical analysis. The presentpaper is meant as a review in a very limited sense: toemphasize the direct observational nature of the question weshall not discuss or quote any theoretical papers (not evenNewton or Einstein) and avoid sophisticated statisticaltests-presenting observational data in "almost raw form,"i.e., using only the most naive and transparent statisticalanalysis. This omits much of the relevant literature on thecontroversy and it may seem surprising that we have any-thing left to review. Fortunately, there has been such aremarkable "data explosion" in extragalactic observationalastronomy over the last ten years or so, that the controversycan now be settled from "almost raw data" alone. The twomost important individual developments are (i) the magni-tude-limited "CfA survey" of more than 2400 galaxies (3)with optical redshifts for all and (ii) the development of the"Tully-Fisher method" (4) for measuring distances r to spiralgalaxies even when r is very large. Recent extensive work onthe brightest galaxies in clusters is particularly suitable forthe present discussion.

Statistical discussions have been given by a number ofauthors but have not yet been applied to the recent obser-vational data. We give here (without critique) a few refer-ences (which themselves quote earlier work). First, someearly contributions by Segal and his colleagues (5, 6); anonparametric procedure (7) was given in 1983 and a recentpaper (8) gives many references. Two pairs of papers (refs. 9and 10, 11 and 12) address controversies on statisticaltechniques. Here we concentrate on direct observationaldata, especially data leading to an explicit luminosity functionthat we feel is essential for the reader to draw his or her ownconclusions. We also have to deal with our slightly unusualposition-we live in the outskirts of a concentration ofgalaxies, the "Local Supercluster."

Section 2. Some definitions and procedures

For the sake of readers who are not astronomers we compileafew definitions and numerical values (13). We start with (thederived) absolute magnitude M, a logarithmic measure of theabsolute luminosity L of an object, restrict ourselves to the

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"UBV blue magnitude MB" and drop the subscript. In solarunits the absolute magnitude M and the apparent magnitudem (measuring the directly observed flux L/r2) read

M = 5.48 - 2.5 loglo(L/L),m = M + 25 + 5 loglo(r/l Mpc),

where r is the distance to the object. We shall test the tworival hypotheses for deriving M from m and V,

r (1 V \r (1 V 12 31 Mpc (Ho km s') oKo km s-) * [3]

Unlike stars, galaxies do not have a sharply defined "outeredge" and the assigned apparent magnitude m depends on theangular area over which one sums the optical surface bright-ness CL. For spiral galaxies, like our own, CTL decreasesexponentially with distance from the center and there is littleambiguity in the integral of aL. For our own galaxy theexponential scalelength is =6 kpc and the absolute magnitudeisM -20.2 (see, e.g., ref. 14), with an uncertainty of order±1.0. For giant elliptical galaxies, on the other hand, thesurface brightness decreases less sharply than exponentialand, for intermediate values of angular distance 6 from thecenter, the integrated flux increases approximately as (ln 6+constant). For the relatively small redshifts considered here(z < 0.1) these "optical halos" need not be a problem inprinciple: The central optical surface brightness crL(O) istypically much brighter than that of the night sky (discountingdwarf galaxies, which are of no interest here); the angularradius Osb where aL drops to some predetermined surfacebrightness is large enough (>10 arcsec) so it can in principlebe measured accurately without smearing from "atmosphericseeing." One can then give a distance-independent measur-ing procedure for apparent total magnitude m in terms of theflux in an aperture of radius Osb. A similar procedure is usedin the "RC2 catalog" (15), and the magnitudes in the CfAsurvey can be adjusted to this system (16).As pointed out before (17), some papers on bright galaxies

in clusters present magnitude data that are referred to angularapertures 6H that correspond to a constant absolute radius ifthe linear redshift-distance law holds but to a varyingabsolute radius for the quadratic law. In principle this couldinvalidate a fair comparison between the two laws (17), butin practice recent advances in optical procedures haveeliminated this flaw: For many bright galaxies in clusters,detailed surface photometry and accurate values for variousdistance-independent definitions of angular radius were ob-tained by Sandage (18) and have been independently con-firmed (with less tedium) by more recent efforts (19, 20).These radii, which are independent of distance and of theassumed law, are found to be compatible with the "Hubble-biased" radii OH. The scatter in individual radii is too large toconsider this compatibility as a verification of the linear law,but at least no great errors are introduced if magnitudemeasurements are based on OH.The gas-rich "late spirals," class Sb and Sc (which bracket

our galaxy and Andromeda), are typically regular flat diskswith circular rotation, and the maximum rotation velocity isempirically known to be correlated with L. The Tully-Fishermethod (4) for determining distances to galaxies consists ofmeasuring accurately the internal velocity width AV of aspiral galaxy from the 21-cm line radiation emitted by theneutral hydrogen (termed by astronomers Hi) in the disk.Optical data are needed to apply an inclination correction toobtain the "edge-on value" AVO (essentially twice the max-imum rotation velocity). Nearby galaxies give an absolutecalibration for the Tully-Fisher relation L(AV0), and theobserved apparent magnitude m of a distant spiral then gives

the "distance modulus" (m - M) and the distance r. Variousslight refinements of this method have been proposed (e.g.,in refs. 21 and 22) and are summarized in a recent paper (23).The scatter in the Tully-Fisher relation is smallest for Sb andSc galaxies, but it still holds reasonably well for other typesof spirals. The method works fairly well out to z 0.04.For statistical work on galaxies without measured dis-

tances, two magnitude-limited optical catalogs with galaxyredshifts are particularly useful. One is the so-called RSAcatalog (24) with a rough magnitude cutoffof 13.2, listing 1246galaxies (including southern galaxies); another, the CfAsurvey (3), with a magnitude limit of about 14.5, listing 2401galaxies. Since different magnitude definitions have beenused by different sources in the past there are some ambi-guities in the cutoffs but not by more than a few tenths of amagnitude. Furthermore, the catalogs are not complete rightup to the cutoff but have a (reasonably well determined)"completeness function." Excluding dwarf galaxies (whichhave a low central surface brightness) it is safe to considerthese catalogs complete to apparent magnitude m, about 12.2and 13.8, respectively. An important property ofthe catalogsis their redshift completeness; i.e., every galaxy on each listhas a measured velocity. In particular, the catalogs couldhave missed only very few galaxies of large redshift butbrighter than mc. The "data-explosion" in this field isconsiderable-a catalog only seven years old (e.g., ref. 25) isalready "old-fashioned." The slightly larger random magni-tude errors in the CfA catalog are not very troublesome, butthe extension to larger redshifts is very important because ofthe clustering to be discussed below.

Section 3. Groups, dusters, and superclusters: Peculiarvelocities

Most theoretical cosmological models assume that, afteraveraging over a "sufficiently large" volume, (i) the densitydistribution of galaxies is uniform and (ii) the differentialexpansion velocity V between two regions depends only ontheir separation r. We summarize the evidence for (i) clump-ing and (ii) peculiar velocity perturbations superimposed ona smooth V(r) relation.

(i) We have already seen that we live in the Local Groupof galaxies with two moderately bright galaxies, our own (M

-20.2) and M31, Andromeda (M -21.6) (24). Compi-lations of other galaxy groups, with similar or larger galaxycontent, are now available, including redshifts (26). For ourpurposes the nearest few are most important, because theycontain many of the "primary distance calibrator galaxies"for which direct optical distance measurements are available(e.g., ref. 27). For instance, there are five groups with meanrecession velocities V between 200 and 400 km s'1 for which

(r) 5.5 Mpc,'rrms=5.8 Mpc for (V) = 270 km s1', [4]

where the linear distance average (r) is appropriate for thelinear law in Eq. 3 and the rms value rrn. is appropriate forthe quadratic law. There persists some controversy over theabsolute distance scale (some distance estimates are almosta factor of two smaller). That controversy affects only themultiplicative factor in the redshift law, however, not thepower-law dependence. Some of these groups are ratherloose and there is some controversy on dynamic groupmembership. However, here we are using groups only as aconvenient way to collate galaxies with roughly similarlocations and redshifts, so membership is unimportant.Unfortunately, (V) for these five (and other nearby) groupscould have an appreciable contribution from peculiar veloc-ities (see below).

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Galaxy clusters typically contain 10 to 100 times morebright galaxies than a typical group and are correspondinglyrarer. Procedures for defining and classifying the richer typesof clusters, the "Abell clusters," have been given and morethan 2000 are known (28). In principle, there is an unavoid-able difficulty inherent in a quantitative definition of acluster, which increases with increasing distance, namely,the contamination by foreground and background galaxiesand groups (29, 30). Fortunately, this effect is fairly small forour restricted redshift range z < 0.1. Furthermore, as forgroups, we shall use clusters mainly as a convenient collec-tion of galaxies with roughly similar redshifts and location, sodynamical membership is less important here. Ellipticalgalaxies are more common in dense clusters than in the field(31), and rich, dense clusters often have "special" ("cD")galaxies in their centers that are about half a magnitudebrighter than the brightest normal galaxies in other (poorer)clusters at comparable redshift (19).The nearest few Abell clusters are at distances correspond-

ing to almost 5000 km s-1, but we live in a somewhat speciallocation since one galaxy cluster that is "almost rich enough"to be an Abell cluster, the Virgo Cluster, has a recessionvelocity of only 1100 km s-1 (distance, 15-25 Mpc). TheVirgo Cluster has comparable numbers of spiral and ellipticalgalaxies and a somewhat irregularly shaped "core" or"cluster proper," of radius about 5 or 60. Many galaxyclusters are surrounded by a huge halo of galaxies, of radiusabout 10 core radii, forming a kind of "supercluster" (32, 33).The Virgo Cluster is surrounded by such a halo or cloud, the"Virgo Supercluster" or "Local Supercluster" (34, 35), andwe live in the outskirts of this supercluster! This fact leads toa considerable inhomogeneity (and anisotropy) in the numberdensity of galaxies in our vicinity.

(ii) We are interested in two kinds of "peculiar velocities"for individual galaxies, both related to clusters (or superclus-ters). The first has to do with the velocity dispersion oa of thesystemic velocities of individual galaxies in a cluster relativeto the mean velocity V of the cluster. According to majorityopinion, a cluster (but not its associated supercluster) is agravitationally bound system and a represents only "internalmotion" of the system. This point of view requires a largeamount of "dark" gravitational mass, but it is argued thatdark matter is already required by circular rotation curves inspiral galaxies. According to a minority view (29), a clusterrepresents an extended density enhancement with less grav-itational effect (avoiding the necessity of dark matter), so thatoa derives largely from the differences in recession velocityV(r) for different distances. For the gravitation-less limit,coupled with the linear V(r) law (equivalent to unacpeleratedexpansion from a point since the instant of the "Big Bang"),one can make definite predictions: The cluster radius alongthe line of sight divided by distance is a/Vand, if clusters areroughly spherical, this should be of order the apparentangular radius 6ad in radians. In fact, or/V6>d is typically oforder 10 for rich clusters, which would imply cigar-shapedclusters all pointed along the line of sight (or "fingers ofgod"). For instance, for the Coma Cluster, the_nearestwell-studied (36) very-rich cluster, ar- 1000 km-s-, V = 7000kms-1, and 6rd is of order 0.05 radians. A naive interpreta-tion of the quadratic law would suggest that cr/2V6md be oforder unity, since d log rid log z is half as large as in the linearmodel, but apparently the model dependence can be evenstronger (29).

Fortunately, there are two "model-independent" obser-vational facts suggesting that there are indeed considerablepeculiar velocities for individual galaxies in a bound cluster.(i) The most convincing evidence comes from galaxies withnegative (approaching) velocities in the RSA and CfA cata-logs (always excluding Local Group galaxies and usingvelocities V relative to the Local Group center). Their

distribution on the sky (Fig. 1) is highly concentrated aroundthe Virgo Cluster (all are within 0.1 radians). Since distancescan be neither negative nor imaginary, this cannot beachieved from pure cosmological flow in either model andshows that peculiar velocities, IV - 1100 km s1I, can be quiteappreciable near cluster cores. (ii) Many clusters are found tohave hot x-ray-emitting gas bound in their cores, with atemperature equivalent to the cluster velocity dispersion (37).For regions further out in a supercluster (in contrast to the

cluster core) the majority and minority views agree thatgalaxies are not gravitationally bound and measured veloci-ties mainly come from the smooth cosmological flow, themodel V(r). However, there must be smoothly varying smalldeviations from "isotropic Hubble flow" due to the gravita-tional pull of the cluster. In principle this can be observed,using the Tully-Fisher method (and others) to select equallydistant galaxies in the direction toward the Virgo Cluster, andopposite, and then comparing their velocities. Further, thiscan be generalized to equal-distance shells at larger dis-tances. This has been done by a large number of observers(38-42) with slightly different methods and each with onlymodest accuracy. There is at least qualitative agreement thatour deviation 8V from isotropic flow relative to the Virgocenter ("radius" 1000 kms-1) is of order 200 kms-'inward and that WV fluctuates but generally increases slowlywith increasing shell radius to an asymptote of somewhat lessthan 1000 kms- relative to the microwave backgroundradiation (effectively a shell of "radius" c) (43). Tosummarize: the fractional error 8V/V in measured averagerecession velocity (V), as an indicator of the smooth cosmo-logical velocity field V(r), is =0.2 at V =1000 km sl and thenfluctuates but generally decreases with increasing distance r.

Unfortunately, because of the "deviations from Hubbleflow," group velocities are particularly uncertain for the fivegroups with velocity between 200 and 400 km's' in Eq. 4,where the measurement of distance r is particularly direct andaccurate. To establish the power law dependence in V(r) thescale factors Ho or Ko in Eq. 3 are not needed, but in the spiritof concreteness we quote some values: If we use Eq. 4, inspite of the uncertainty, we get Ho = 49 or Ko = 8.0, whereboth the Hubble parameter and the quadratic scale factor Korepresent what the recession velocity would be in km s'1 ata distance of 1 Mpc. These values are uncertain by easily afactor of two and most workers prefer to determine the scalefactors (even though less directly) from the Virgo Cluster

28.0

23.0

8018.

13.0

8.G

3.0

-2.0113.00 12.50 12.00

Right ascension, hr

FIG. 1. Distribution on the sky of all negative velocity galaxiesoutside the local group. The large x marks the center of the Virgocluster; the circle is of 60 radius and demarcates the canonical"cluster proper."

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(see, e.g., refs. 41 and 42). With r 14-22 Mpc and V(r)1000-1400 km s-l,this gives Ho 50-100 and Ko 2.1-7.1.

Section 4. The luminosity function, Malmquist biases, andthe magnitude-redshift relation

In the earliest days of observational cosmology it was hopedthat galaxies would provide "standard candles"-i.e., allhave the same luminosity L. It soon became clear that this isnot the case and that one has to study the luminosity function4(L), the probability distribution for different luminosities.For well-developed spirals (Hubble type Sc) with detailedoptical photographs one can assign a second kind of purelymorphological label, "luminosity class I, II, III, or IV" thatempirically has some correlation with absolute luminosity.There was a hope that, for a given class, the luminosityfunction might be sufficiently narrow for an "almost standardcandle." For instance, it has been suggested (8) that com-parison at different redshifts, plus the hypothesis of a narrowluminosity function, might distinguish between the differentexpansion laws. Fortunately, we now have enough observa-tional data to obtain the luminosity function for a singleluminosity class and unfortunately it is not narrow! Anillustration from the analysis (44) of the RSA catalog for themost favorable (and brightest) class Spiral I is shown in Fig.2. The Virgo Cluster proper (core) has been omitted from thesample, but the Local Supercluster is represented and is quiterich. Consequently there are an appreciable number ofgalaxies within V 800-1600 kmts'1, say, that should all beat comparable distances irrespective of the V(r) relation. Fig.2 shows quite a large spread in luminosity. The luminosityfunction 4(L) is not narrow but, on any cosmological model,one assumes that 4' (and also the number of galaxies per unitvolume) should be independent of distance.The absolute magnitude M derived from the observed

apparent magnitude m and Eq. 2, assuming the linear law, isplotted in Fig. 2. The lower solid line represents the cutoff inapparent magnitude of the catalog, and the thin dashedhorizontal line represents a constant (large) luminosity on thelinear law, the dashed sloping line on the quadratic law. Thisfigure illustrates the so-called "Malmquist effect:" At lowvelocities V (near) the apparent magnitude cutoff is not too

-24 -OS W/d~~~~~~~~~~~-23 -

le

-22 * , 4

-21-

-20-

0.3 0.6 1 2 3 6 10V X 103, km-sY1

FIG. 2. Distribution in absolute magnitude and velocity of theSpiral I and I-LI galaxies in the RSA catalog (modified from figure 1in ref. 44). Constant luminosity is represented by the horizontal lineif the linear law is correct, by the dashed sloping line if the quadraticlaw holds.

important, but at high V (far) the fainter end of the luminosityfunction is suppressed. If one were to evaluate mean absolutemagnitudes as a function of V or to compare properties ofnear-far pairs, the Malmquist effect would then become a"Malmquist bias" (45). One can already test the two V(r)laws in Eq. 3 in a qualitative way: If the luminosity function0(L) had a sharp upper cutoff at some fixed Lmax and 4i werevery largejust below Lmax, (neither ofwhich is true), the upperasymptote ofthe scatter diagram in Fig. 2 would be horizontalfor the linear law, parallel to the heavy dashed line for thequadratic law. Neither is the case, but we must discuss a kindof "generalized Malmquist effect."As we go to larger velocities V (with bins of constant A log

V), the volume, and the potential number of galaxies,increases (as V3 on the linear, V1.5 on the quadratic law).Hence, if the upper end of the luminosity function 4i does nothave a sharp cutoffbut only a moderately sharp decrease, theabsolute magnitude (largest luminosity Lmax) should increasewith increasing V-the Scott effect (46, 47). For particularlyrare types of bright galaxies, there will even be a first V-bincontaining any; e.g., the cD galaxy with the lowest V is in theAbell 2199 cluster with V = 9360 kmts-1. For the linear lawwe then should expect to find more galaxies above the thindashed horizontal line as V increases; this is certainly thecase but we cannot say whether or not by the right amount.Similarly, for the quadratic law more galaxies should lieabove the heavy dashed sloping line in Fig. 2 as V increases(but the increase should be less severe because of the weakerV-dependence of volume). This is definitely not the case; infact there is a void in the upper right corner even below theheavy dashed line. This is a serious, qualitative blow to thequadratic hypothesis, but we need more data to make itquantitative.

Since the individual luminosity classes do not provide"standard candles" anyway, we now lump all morphologicaltypes together and display combined luminosity functions.We use data from the CfA survey (fainter limit than the RSAby more than one magnitude), omitting only galaxies within60 ofthe Virgo Cluster center with velocities outside the range500-1700 kmnsrl (to decrease peculiar velocity effects). Welump together all galaxies within this velocity range, both inthe Virgo Cluster and out (we verified before combining thatthe differences between the "in" and "out" samples are notnoticeable in Fig. 3 B and B'). Fig. 3 displays data "withoutprejudice to either V(r) law," as Segal has suggested for along 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

ml = m - 5 loglo(V/1100 km s-1),M2 = m - 2.5 log1o(V/1100 km sr1). [5]

ml and m2 are then (except for a common additive constant)the absolute magnitudes (-2.5 log L) on the linear andquadratic hypotheses, respectively. We plot the functionL24(L) in Fig. 3.

Fig. 3 A and A', galaxies appreciably closer than the VirgoCluster, is unreliable because of peculiar velocities. Fig. 3 Band B' uses the same galaxies with ml M2. The luminosityfunction here is therefore almost the same for the linear andquadratic models, is based on a large number of galaxies withV near 1100 kms -' (much ofthe Local Supercluster), and canbe used to calibrate 4(L). The smooth curve in Fig. 3 B andB' is an empirical fitting function, the so-called "Schechterfunction" (48),

4(L) X L-exp(-L/L*), [61

with /3 1.25 and L* corresponding to a critical absolutemagnitude mn = m2 = 10.5. This choice agrees reasonably

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8 10 12 14

FIG. 3. The luminosity function (times luminosity) in various redshift ranges with an "absolute" magnitude ml determined according to thelinear 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 from500 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') Eachpair 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' exceptfor normalization) represent the Schechter function that best fits the 1100-km s-1 bin. The histograms are L2q/ computed from the data in theCfA catalog. The breaks from solid lines to dashed lines indicate the points at which the magnitude bin ceases to be complete at the far endofthe 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(since the velocities are nearly the same) is independent ofthevelocity-distance law.

Fig. 3 pairs C and C', D and D', and E and E' correspondto identical ranges of distance r, on each of the two

velocity-distance hypotheses: For each pair r is larger thanfor the pair above by \/-; with the redshift distance at V =1100 km s'1 as the common unit of distance, the equivalentmean velocities V (V a r and V a r2, respectively) separatemore and more as the distance increases. The labeling of

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absolute magnitude (ml and M2, respectively, but each -2.5log L) and the normalization per unit spatial volume are thesame for each pair. The smooth curves in Fig. 3 C-F are thesame as the fitted curve in Fig. 3B. The observed histogramsshould then agree with the smooth curve for the "correct"redshift law, if a constant luminosity function and a uniformdistribution of galaxies is assumed (because of the densityenhancement in the Local Supercluster the histograms in Fig.3 C-F should actually lie slightly lower than the smoothcurve). The linear law fits this criterion quite well, even inFig. 3F at 9 times the unit distance. The quadratic law,however, violates the criterion more and more with increas-ing distance. The discrepancy, a progressive "shift to theright" of the histogram, is particularly disastrous in Fig. 3E'at 5.2 times the unit distance (and beyond) where there aresimply no galaxies with M2 < 11. Furthermore, the discrep-ancy must be due to an incorrect law, not merely due to a"hole" at this distance, since there are plenty of galaxyclusters with V 30000 km s-1 but with all galaxies fainterthan M2 11.

Section 5. Bright cluster galaxies and Tully-Fisher results

Although galaxy catalogs are even now complete only to amagnitude cutoff of m 14.0, modem optical techniquesallow accurate velocity and magnitude measurements for asmall number of individual galaxies three or four magnitudesfainter. A convenient way to select a small fraction of distantgalaxies is to pick the brightest galaxy (or the brightest few)in each of a large number of galaxy groups or clusters. Thishas now been done both for groups (26) and for clusters(50-52) up to z 0.3. We display some of this data in Fig. 4,again restricting ourselves to redshifts z < 0.1.We also give in Fig. 4 the predicted slopes for the linear and

quadratic models, but we must consider any possible dis-tance-dependent biases. As we saw in Section 2, detailedsurface photometry (18-20) has eliminated Hubble-biasedaperture selection as a source of serious error. Because thebright end of the luminosity function (Section 4) is notinfinitely steep, the brightest of a larger collection of galaxieswill typically be slightly brighter than the brightest in a

V, km-s-

FIG.4. Apparent magnitude m versus velocity V for the brightestgalaxies in groups from the CfA catalog (v) and Abell clusters (v),including richness class R = 0 for z <0.04, and excluding R = 0 forz> 0.04. The sloping lines are parallel to the ones expected if allbrightest galaxies had the same luminosity, according to the twohypotheses (line 1, linear; line 2, quadratic).

smaller collection. Because rich clusters are rarer than poorclusters, we have a tendency toward a generalized Malmquistbias; i.e., as the velocity V increases we tend to pick richercollections with a brighter brightest galaxy (45, 46). Theoccurrence of the even brighter cD galaxies only for V > 9000km s-1 is another small example in this direction. There is,however, one anti-Malmquist bias (29) (discussed in Section3) if the redshift is measured for only a few galaxies in a"concentration in the sky:" As the redshift increases there isan increasing probability that the measured galaxy is theabsolute brightest in only a smaller, nearer cluster with alarger, unrelated, background cluster making the measuredcluster appear richer than it really is. This effect has beenmodeled numerically (30); it can be appreciable for z > 0.2 butis completely unimportant for z < 0.04 (47) and weak for 0.04< z < 0.1. In Fig. 4 we have strongly overcompensated thisanti-Malmquist bias by excluding the poorest (richness class0) Abell clusters for z > 0.04 but including them for z < 0.04.With the sign of the Malmquist bias in Fig. 4 firmly

established to be positive, we now have a single criterion forwhichever of the two redshift-distance models V(r) is cor-rect: The observed scatter diagram in Fig. 4 for clusters(circles) should have a slope close to, but slightly smallerthan, the model line. There can be little doubt that the linearlaw fits this criterion and that the quadratic law does not. Thebrightest galaxies in small groups at lower redshifts are alsoshown in Fig. 4 (triangles). If one combined all triangles andcircles, one would obtain a "best-fit" slope intermediatebetween the two power laws, but that would introduce toolarge a generalized Malmquist bias-as shown by the overlapnear V 5000 kms-1, clusters have brighter brightestgalaxies than the (very much smaller) groups and a separatefitting line would have to be drawn for the triangles.

Since the Tully-Fisher method gives a direct distanceestimate for each galaxy used, membership in a group orcluster is not essential for this method. In Fig. 5 we displaythe data in two ways: Fig. SA shows data for 383 galaxies,every spiral galaxy in the CfA catalog having m < 13.0 anda published Hi profile with a signal/noise ratio of >7 (80% ofthe total sample of spirals having m < 13.0), and Fig. SBshows data averaged over all spirals with measured Hi widthsin each of several well-studied groups and clusters: nearbygroups from Richter and Huchtmeier (23), nine large clustersfrom Giovanelli and Haynes (53). Although neither the groupsample nor the cluster sample are statistically complete inany sense, neither were they preselected for our presentpurpose: Each is the largest homogeneous collection of dataof its class available. The calibration of the Tully-Fishercorrelation we have adopted here is that of Richter andHuchtmeier (23). On each plot we have drawn two lines, onefor each ofthe two redshift laws, that should parallel the ridgeline of the data under each respective hypothesis. Fromeither plot it is clear that the linear law is consistent with thedata 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 itshigh-luminosity tail is particularly troublesome-very brightgalaxies are rare, but in a large enough collection some willbe found. This leads to the so-called generalized Malmquisteffect, but we saw that modem observational data aresufficiently redundant that the "effect" does not lead to a"bias." We saw then that each of several lines of evidencefavors the linear redshift-distance law over the quadratic law:The luminosity function for various ranges of redshift, them-z diagram for brightest galaxies in groups and clusters, andthe Tully-Fisher results.

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3062 Astronomy: Salpeter and Hoffman

I Xf - /1 1 1 1 l l1.0 10.0 100

10000 B

1000 A z -

1000

1.0 10.0 100Roe Mpc

FIG. 5. Correlation of distance R obtained from the Tully-Fishermethod and velocity V for individual spiral galaxies with m -- 13.0 inthe CfA catalog (A) and averages for well-studied groups and clusters(B). The calibration is by the method of Richter and Huchtmeier (23).(A) All spiral galaxy types Sa through Sm are included. No selectionby inclination angle has been made; to correct V to VO we have usedsin i,, = max(0.4, sin i). All galaxies within 6° of the Virgo Clustercenter (and with V < 2500 km s-1) have been plotted at the clustermean velocity (977 km s-1). No corrections have been made forVirgocentric deviations from the velocity law, which may increasethe scatter at intermediate velocities but are not biased in favor ofeither law. (B) Mean distances and velocities for nearby groups (a)from Richter and Huchtmeier (23) and for nine large clusters (0),including Virgo, studied by Giovanelli and Haynes (53). Two circlesjoined by a bar are shown for Virgo; the lower V assumes noVirgocentric motion ofthe Local Group while the upper one assumesa (large) Virgocentric infall, of 450 km s-1. The sloping lines shouldparallel the ridge line of the data in either plot under each respectivehypothesis (line 1, linear; line 2, quadratic).

It is clear that if one were to insist that V xc rP, with p aninteger, we must choose p = 1 rather than p = 2. We do notderive a formal best-fit noninteger value for p nor a formal" standard error in the exponent." If we nevertheless com-pute a "best-fit" value ofp from the m-z diagram (Fig. 4) forbrightest galaxies in "Abell clusters" alone (omitting thesmaller "groups"), we obtain p ==1.1. A careful analysis of

the data in Fig. 4 by professional observational cosmologists(47, 50-52), including corrections for a remaining generalizedMalmquist bias, gives a much better fit to the linear law thanour Fig. 4 and has been extended to higher redshifts, but suchanalysis requires the input of some theory that we wish toavoid here. Similarly a naive "best-fit-by-eye" to theTully-Fisher plots in Fig. 5 would give (for individualgalaxies)p 1.0 or (for groups and clusters) p 1.1 and again"professionals" could do better. However, the "almost rawdata" in Figs. 3-5 extend over a fairly large velocity rangeand 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. Thiswork was supported in part by the National Science FoundationGrants AST 84-15162 at Cornell University and AST 84-06392 atLafayette College.

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