Proc. Natl. Acad. Sci. USA Vol. 79, pp. 3913-3917, June 1982 Astronomy

Spatial homogeneity and redshift-distance laws (galaxies/cosmology/Hubble law/Lundmark law) J. F. NICOLL* AND 1. E. SEGALt *Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742; and tDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Contributed by I. E. Segal, March 11, 1982 ABSTRACT Spatial homogeneity in the radial direction of the low redshift region and may be subject to various biases. low-redshift galaxies is subjected to Kafka-Schmidt V/Vm tests Here radial homogeneity is treated by the Kafka-Schmidt V/ using well-documented samples. Homogeneity is consistent with Vm test (see ref. 7), modified by the imposition oflimits on red- the assumption of the Lundmark (quadratic redshift-distance) shifts. The lower limit is required because of peculiar motions, law, but large deviations from homogeneity are implied by the and the upper limit because errors ofmeasurement place a red- assumption of the Hubble (linear redshift-distance) law. These shift limit on completeness (see ref. 2). The V/Vm test retains deviations are similar to what would be expected on the basis of its important property ofimmunity to the observational cutoffs the Lundmark law. Luminosity functions are obtained foreach law when such limits are imposed. In addition, it becomes capable by a nonparametric statistically optimal method that removes the ofbeing quite discriminatory between different redshift-distance observational cutoff bias in complete samples. Although the Hub- the without such limits, the V/Vm test ble law correlation ofabsolute magnitude with redshift is reduced laws, which is not case considerably by elimination of the bias, computer simulations then being substantially equivalent to the analysis of the N(m) show that its bias-free value is nevertheless at a statistically quite relationship. Such limits also tend to produce a sample that is significant level, indicating the self-inconsistency of the law. The subject to fewer and smaller errors. corresponding Lundmark law correlations are quite satisfactory The primary sample used here for statistical analysis is that statistically. The regression ofredshift on magnitude also involves of Visvanathan (2). This appears particularly appropriate be- radial spatial homogeneity and, according to R. Soneira, has slope cause of its homogeneity, completeness, whole sky character, determining the redshift-magnitude exponent independently of and relative freedom from corrections that may be model de- the luminosity function. We have, however, rigorously proved the pendent. It consists of -300 elliptical and similar galaxies; for material dependence of the regression on this function and here comparison, a sample ofde Vaucouleurs (3) that consists largely exemplifyour treatment by using the bias-free functions indicated, of spiral galaxies is also treated, including its angular diameter with results consistent with the foregoing argument. relations. As a consequence ofthe completeness in diameter of the Nilson sample (4) and the inclusion of these measurements For small redshifts z that are beyond the range of influence of in the work ofde Vaucouleurs et al. (5), from which the sample peculiar motions, the observed correlation ofapparent magni- (3) of de Vaucouleurs largely derives, it is to be expected that tude with redshift suggests that the redshift z varies approxi- this sample is likewise complete, down to a limiting diameter, mately as a power p ofthe "distance" r, p being a constant. The here taken conservatively large, at least in the northern galactic distance itselfis not observable and the empirically meaningful hemisphere. Subsamples defined by further limitations on mag- content of such a law derives from its combination with laws nitude, redshift, and galactic latitude are also considered. A relating other observable quantities such as luminosities and wide discrepancy from homogeneity, on the assumption of the angular sizes to distance. Elimination ofthe distance then leads Hubble law, is found in virtually all cases, in contrast to the to predicted relations involving only observed quantities. homogeneity implied by the Lundmark law, and has a character Such predictions are predicated on hypothesized uniformi- similar to that found earlier for a sample oflarger-redshift bright ties ofone type or another. For studies ofthe magnitude-redshift cluster galaxies studied by Peterson (6) in an analysis by Segal relation, uniformity in luminosity is involved; i.e., it is assumed (7). Moreover the departures from homogeneity are quite sim- that the absolute luminosities of galaxies in space have a fre- ilar to those predicted by the Lundmark law. quency distribution that is independent ofthe distance, for gal- Another observable relation, predicated in part on radial spa- axies in the class observed. (Observational imitations may have tial homogeneity and in part on luminosity uniformity, is the the consequence that this physical uniformity may not be di- redshift-magnitude regression. Claims ofSoneira (8) to the con- rectly observed as such, of course, appropriate procedures trary notwithstanding, it has been proved by Nicoll and Segal being required to elicit it.) (9) that this regression depends on the luminosity function and A logically quite distinct type ofuniformity is that ofthe spa- not merely, apart from zero point, on the redshift-magnitude tial distribution ofgalaxies. This spatial uniformity may be fur- exponent p. The latter work is here exemplified by a quanti- ther subdivided into angular and radial homogeneity. The sub- tative determination of the Hubble and Lundmark law predic- ject is classical (1) and, from early on, only radial homogeneity tions ofthis regression, for the Visvanathan sample in the same has appeared tenable on the scales here involved, and only this conservative magnitude and redshift ranges as in the V/Vm type is considered here. This is, briefly, the assumption that tests. the number of galaxies up to distance r varies as r3 in the low- The ROBUSTdeterminations ofluminosity functions treated redshift region, say up to redshifts of 0.01. Classical study has by-Nicoll et at (10) required to this end show independently of is nondiscrim- any assumption of radial spatial homogeneity the self-inconsis- treated the N(m) relationship, which, however, tency of the Hubble law by virtue of its improbably large cor- inatory between different redshift-distance exponents p in relation ofabsolute magnitude with redshift after removal ofthe observational cutoff bias by the ROBUST analysis. This sup- The publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. Abbreviations: LF, luminosity function; OCB, observational cutoffbias. IQI .R Downloaded by guest on September 29, 2021 3914 Astronomy: Nicoll and Segal Proc. Natl. Acad. Sci. USA 79 (1982)

plements and confirms the interpretation ofthe nonparametric C. 8>0 B<0 cross-testing of the Hubble and Lundmark laws reported ear- 0 lier. The Lundmark law is itself entirely self-consistent in the cr 1.0 indicated and other respects. 0 lot a0 00 Because of its dependence on two distinct uniformity hy- O. 0.8 _ at potheses, the redshift-magnitude regression is more difficult 0 a6 o D 0.6 0 6 o to analyze probabilistically than either the magnitude-redshift 0 a0 a o0 .0 0 8 u or V/Vm relations. Despite significant departures from both of 0 0g '- OA a 0 these uniformities, due to partial cancellation of their effects, 1[ p zo // the redshift-magnitude regression predicted by the Hubble law happens not to appear very strongly deviant from the empirical r 0.2 10 relationship, but systematic departures from it are suggested 0 0 E 0, , 02 0 8 1. and not found in the case of the closer-fitting prediction of the U 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.0 Lundmark law. V/Vm The V/Vm tests FIG. 2. V/VYi relations for the same sample as in Fig. 1 in the northern (Left) and southern(Right) galactic hemispheres (118 and 65 The sample of Visvanathan (2) is now considered up to its re- galaxies, respectively). Symbols are as in Fig. 1. ported limiting apparent magnitude of 12.4 and in the conser- vative redshift range 500 < cz < 2,250. The results of V/Vm natural question from a systematic methodological standpoint tests on this sample are shown in Fig. 1. There appear to be ofthe extent to which the Lundmark law explains the deviations systematic deviations of the Hubble law values from expecta- from the Hubble law is addressed by comparison of the V/Vi tion, assuming as throughout unless otherwise stated that the distribution actually observed using the Hubble law analysis galaxy distribution is homogeneous in the radial direction, but with the Lundmark law prediction of the results of the V/Vm the Lundmark law values appear to fit reasonably well. Statis- test when predicated on the Hubble law. As shown in Fig. 1, tical analysis confirms this suggestion from the visual appear- such predictions are in excellent agreement with the results ance of the curve. The Kolmogorov-Smirnov statistic D, de- actually observed. In view of these features, an initial pre- fined as the maximum deviation between the empirical cumu- sumption ofreal physical inhomogeneity, assuming the Hubble lative distribution function of the V/Vm value and that for a law, cannot be sustained consistently with general scientific uniform distribution on the unit interval [0,1], has the values principles. D, = 0.2475 and D2 = 0.0925, where the subscripts 1 and 2 From the standpoint ofthe Hubble law, a number of obser- refer, respectively, to the Hubble and Lundmark laws. The vational or statistical features present themselves as possible corresponding probabilities P of deviations as large as these, explications of this unexpected result; the major ones will be assuming the correctness ofthe respective laws, are < 10' and taken up seriatim. 0.09, using the asymptotic (for large samples) expression P - The radial spatial distribution could conceivably differ in the 2-2nD, where n is the sample size (here 183). The average northern and southern galactic hemispheres in such a way as values of V/Vm similarly deviate from and conform to expec- to produce a spurious fit to the Lundmark law. The results of tation, depending on the law, being 0.359 for the Hubble law separate V/Vm tests for the two hemispheres are shown in Fig. and 0.471 for the Lundmark law. 2. They are quite similar to those for the whole sky; in each The Hubble law deviations in themselves could reasonably hemisphere the Hubble law is rejected bya Kolmogorov-Smirnov be presumed to result from a real physical inhomogeneity, in test at probability levels 5 10-4 while the Lundmark law re- view of known galaxy clustering. Such a presumption would, mains quite acceptable. however, fail to explain the homogeneity represented by the The limitation of the sample to elliptical galaxies could con- observations on the assumption of the Lundmark law, which, ceivably, although in an unknown manner, affect the apparent although imperfect as expected, is statistically acceptable. The homogeneity of the distribution. Fig. 3 shows that tests on a homogeneous and probably complete sample ofspirals give sim-

1.0 C U 1.0 c c ap00 / 2° 0.8 4) 0.8 0 0 o 0 l- So 00 0 so 66068 0 oo 0 0 cr 0 0 = 0.6 0.6 ea 0O 0. .0 0 / .0 4- 0 0 0 L- 0 0 Ad 0.4 0 0.4 .0 000 0 0 E 0 0.2 E 0 0 0 C- I 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 V/Vm V/Vm FIG. 1. V/Vi relations for the Visvanathan sample with limits m FIG. 3. V/Vi relations as in Fig. 1 for the sample of de Vaucou- s 12.4, 500 < cz < 2250 (183 E/SO/SB galaxies). o, Hubble law; o, leurs with angular diameter limits m, < 180 and redshift 500 < cz Lundmark law; A, Lundmark law prediction of Hubble law result; < 2000 and confined to morphological types Sa/Sb/Sc (104pgalaxies). , exact spatial uniformity. Symbols are as in Fig; 1. Downloaded by guest on September 29, 2021 Astronomy: Nicoll and Segal Proc. Natl. Acad. Sci. USA 79 (1982) 3915 mark law to be true and deviant from the Hubble law prediction 1.0 (Fig. 4). CT lower redshift limit to further the O) Raising the suppress pos- ° 0.8 sible effect of peculiar velocities, lowering the upper redshift so 0%~~~~~~ j limit for additional conservatism and exactitude ofobservation, deletion from the sample of the galaxies in a 60 cone centered _ 0.6 ox~ ~ ~ ~ on Virgo, and such have no material effects on the results (Table

.0 1). 0 is inconsistent with radial ho- CL 0.4 Thus, the Hubble law spatial 0/° mogeneity in precisely the magnitude and redshift ranges in _o0 which it originated, for major homogeneous classes of galaxies. -, 0.2 Its inconsistency is moreover of precisely the character pre- E O dicted by the Lundmark law, which is itself entirely consistent u with homogeneity in the well-documented region studied.

0.2 0.4 0.6 0.8 1.0 Self-consistency of redshift-distance laws assuming luminosity uniformity FIG. 4. V/Vm relations as in Fig. 1 for the bright subsample m <11.6. Symbols are as in Fig. 1. The observational nontriviality of a redshift-distance law de- pends fundamentally on the assumption ofluminosity uniform- ilar results, even when the apparent magnitude is replaced by ity, since otherwise the association of redshift and apparent the apparent diameter. (The same is also true with the use of magnitude could be a partially or wholly intrinsic effect. In the apparent magnitudes to projected completeness limits, which, very-large-redshift region, this assumption has been dropped however, are not known exactly; see Table 1.) in the expanding universe cosmology under the heading of"lu- Galactic and other obscuration could conceivably affect the minosity evolution." However in the low-redshift region, z s results, although again there is no special reason to expect it to 0.01, under consideration here, it has not been seriously ques- do so in the manner observed. However, limitations to the polar tioned, and will be used here as in the work of Nicoll et aL (10). regions IBI > 400 give similar results. This included nonparametric cross tests of the Hubble and The V/Vm test is fairly sensitive to magnitude completeness Lundmark laws predicated on this assumption; here the results and so could be affected by progressive incompleteness toward of similar self-tests of these laws are noted. the reported limits, although there is no special reason to expect One of the simplest such self-tests compares the observed this within the redshift region treated here. However, even at correlation r ofabsolute magnitude with redshift (more exactly, the very conservative magnitude limit of 11.6, the results are its logarithm) after elimination of the observational cutoff bias quite similar to what would be expected assuming the Lund- (OCB) with what would be expected on the assumption of the

Table 1. V/Vm tests in restricted subsamples Hubble law Lundmark law Sample czO cz m* Constraint n D P V/Vm D P V/Vm I 500 2250 12.4 183 0.25 0.0000 0.36 0.09 0.09 0.47 I 500 2250 11.6 153 0.15 0.0029 0.44 0.06 0.65 0.44 I 600 1800 12.4 133 0.23 0.0000 0.37 0.08 0.32 0.48 I 1250 2250 12.4 109 0.16 0.0069 0.40 0.10 0.25 0.46 I 500 2250 11.8 168 0.15 0.0014 0.43 0.05 -.5 0.50 I 500 2000 11.8 147 0.15 0.0040 0.43 0.07 ;.5 0.51 I 500 2250 12.4 EBI > 400 154 0.26 0.0000 0.35 0.11 0.05 0.46 I 500 2250 11.8 EBI > 400 140 0.16 0.0011 0.42 0.06 0.5 0.49 I 500 2250 12.4 B > 0 118 0.27 0.0000 0.36 0.11 0.10 0.47 I 500 2250 11.6 B >0 97 0.20 0.0008 0.42 0.11 0.19 0.48 I 500 2250 12.4 B <0 65 0.26 0.0003 0.36 0.13 0.19 0.48 I 500 2250 12.4 B > 0; At 87 0.23 0.0003 0.39 0.07 z.5 0.50 11 500 2000 18 104 0.26 0.0000 0.35 0.07 Z.5 0.49 II 500 2000 17.2 71 0.30 0.0000 0.31 0.12 0.23 0.45 II 500 2000 18 B > 0; At 68 0.31 0.0000 0.31 0.13 0.23 0.44 rn-O 500 2000 18 163 0.27 0.0000 0.33 0.09 0.11 0.47 m-e 600 1800 18 140 0.20 0.0000 0.38 0.05 0.5 0.50 mI-e 500 2000 17.5 129 0.31 0.0000 0.30 0.12 0.05 0.44 11-O 500 2000 18 B > 0; At 108 0.33 0.0000 0.29 0.15 0.02 0.43 m11-e 600 1800 18 B > 0; At 93 0.28 0.0000 0.33 0.11 0.20 0.45 rn-m 500 2000 12.5 174 0.17 0.0001 0.40 0.09 0.11 0.49 r-m 500 2000 12.5 B > 0; At 119 0.20 0.0002 0.38 0.11 0.12 0.46 Sample I is that of Visvanathan (2). Sample H is that of de Vaucouleurs (3), limited to types Sa/Sb/Sc and with "geometric magnitudes" m, con- sisting of angular diameters D converted to a Pogson scale by the equation m,, = 25 - 5logD, where log D is as given in ref. 3, in place of the usual (luminosity) magnitudes. Sample HI is also that of de Vaucouleurs but inclusive of all spirals; 3-B is the sample of geometric magnitudes, and 3- m is that of conventional magnitudes. m* denotes the limiting magnitude, whether luminosity or geometric. n, number of galaxies in the sample; D, Kolmogorov-Smirnov statistic; P, probability of a value ofD as large as that observed, estimated as indicated in the text; larger values are correct in order of magnitude but overestimates. V/Vm values are means. t Constraint A, a 60 cone centered on Virgo has been removed from the sample. Downloaded by guest on September 29, 2021 3916 Astronomy: Nicoll and Segal Proc. Natl. Acad. Sci. USA 79 (1982)

A B In the case of the Visvanathan sample, confined to the red- shift region cz > 500 as usual to eliminate possible material 0.4 effects of peculiar velocities on the observed conditional dis- C) tribution of magnitude for a given redshift, the 290 galaxies co ._ brighter than magnitude 12.4 have a correlation of -0.782 be- 4- o 0.2 tween their Hubble law absolute magnitudes M1 = m - 5logz CO + a constant and logz. This large negative correlation can be ascribed in part to the OCB. Removing the OCB from the es- timated LF by ROBUST, these absolute magnitudes can be C r-, ._5 corrected for the OCB by subtraction from M1 of the redshift-

U) dependent trend in M1 arisingfrom the OCB. At agiven redshift 0D C D C 0.4 z, the sample consists only of objects brighter than absolute >- magnitude M1(z) = 12.4 - 5logz, taking the constant equal to 0 0 for convenience. The average absolute value ofthese objects, (Ml(z)), can be determined numerically as a function ofz from LL 0.2 the ROBUST LF. The difference M1 - M1(z) should then have no trend with z, on a statistical basis, and its Pearson correlation coefficient with logz defines the cited statistic r, expected to be small for a correct theory and fair sample. -0.2 -0.1 0 0.1 -0.2 -0.1 0 0.1 In fact, for the sample cited, the bias-free correlation r is Correlation -0.325. A self-test of the Hubble law consists in the determi- nation of the probability of finding a value of r this large, as- FIG. 5. Distributions of predictions of correlations of OCB-cor- rected absolute magnitudes with (logarithm of) redshift for the Vis- suming the law correct. This probability can be estimated by vanathan sample with limits m c 12.4, cz > 500 (290 galaxies) (A and forming a large number ofrandom samples drawn from the bias- B) and for the bright subsample m c 11.6 (205 galaxies) (C and D). free Hubble law LF, placing the objects at the observed red- (A and C) Hubble law. (B and D) Lundmark law. Arrows show actual shifts, and then computing r as for the original sample, for those values. hypothetical samples having the same LF and analyzed on the assumption ofthe Hubble law. This gives an empirical estimate hypothesis in question. Because the ROBUST process inter- of the distribution of r; the actual value of r observed in the venes to determine the unbiased luminosity function (LF), the original sample can then be associated with a well-defined es- probability distribution of the observed r, assuming the hy- timated probability, according to this distribution, ofobserving pothesis, is not necessarily similar to that of the conventional a value this large, on the same assumptions as earlier-i.e., the Pearson correlation coefficient (whose large-sample asymptotic Hubble law, luminosity uniformity, and completeness of the distribution is well known). Indeed, the statistical complexity sample. The results are shown in Fig. 5 A and B, together with induced by the OCB is such that there is currently no practical the analogous ones for the Lundmark law: The probability of means ofdetermining the distribution of r except by computer a bias-corrected correlation between absolute magnitude and simulations. redshift as large as that observed is extremely small in the case ofthe Hubble law and unexceptionable in the case ofthe Lund- Table 2. Raw (formal) and bias-corrected LFs mark law. A similar result is obtained ifthe subsample consisting Hubble law Lundmark law ofgalaxies ofmagnitudes brighter than 11.6 is used, for extreme Absolute conservatism regarding magnitude completeness in the redshift magnitude Raw Corrected Raw Corrected interval in question, as shown in Fig. 5 C and D. Only 1 of the 8.50 0.0000 0.0000 0.0016 0.0014 1,000 random samples from the Hubble law LF gave rise to a 8.75 0.0035 0.0015 0.0019 0.0017 bias-free correlation as large as observed. 9.00 0.0115 0.0048 0.0000 0.0000 The LFs estimated will be used below and are shown to- 9.25 0.0210 0.0101 0.0084 0.0075 gether with related material in Table 2. The Lundmark law 9.50 0.0306 0.0154 0.0052 0.0046 prediction of the V/Vm distribution expected from a Hubble law 9.75 0.0425 0.0231 0.0151 0.0136 analysis is obtained by making Hubble law V/Vm tests on each 10.00 0.0478 0.0265 0.0334 0.0299 a number of random drawn from the ROBUST-es- 10.25 0.0868 0.0540 0.0640 0.0573 of samples 10.50 0.0915 0.0581 0.1117 0.1000 timated Lundmark law LF, placed at the observed redshifts, 10.75 0.0898 0.0644 0.1521 0.1370 as in the computer simulations. The results are then averaged; 11.00 0.0914 0.0699 0.1604 0.1462 for the points plotted as A in Figs. 1-4, 25 random samples were 11.25 0.0955 0.0843 0.1270 0.1190 used. 11.50 0.0811 0.0789 0.1202 0.1188 11.75 0.0649 0.0729 0.0860 0.0941 The redshift-magnitude regression 12.00 0.0535 0.0695 0.0773 0.1005 12.25 0.0478 0.0679 0.0261 0.0408 Assuming both radial spatial homogeneity and luminosity uni- 12.50 0.0593 0.1109 0.0038 0.0081 formity, the conditional distribution of redshift for given mag- 12.75 0.0581 0.1121 0.0038 0.0119 nitude is predictable on the basis of a given redshift law and a 13.00 0.0076 0.0199 0.0020 0.0077 given LF. An explicit expression for (zlm) is given by Nicoll and 13.25 0.0072 0.0206 0.0000 0.0000 Segal (9), where it is shown that an earlier treatment ofthe red- 13.50 0.0057 0.0228 0.0000 0.0000 shift-magnitude regression by Soneira (8) claiming that the 13.75 0.0032 0.0126 0.0000 0.0000 slope ofthe (zim) relationship is simply p/5, where p is the red- to a Absolute magnitude is defined as m - (5/p)log(cz/2,000). Results shift-distance exponent, cannot properly be applied local are differential LF values as fractions of total and were derived from redshift-distance law, but at best only to the case of a euclidean ROBUST analysis of the Visvanathan sample (3), assumed complete universe in which a redshift-distance power law is valid at ar- in magnitude for redshifts cz > 500 up to the limit m c 12.4. bitrarily large distances. In any distance-limited (or equiva- Downloaded by guest on September 29, 2021 Astronomy: Nicoll and Segal Proc. Natd Acad. Sci. USA 79 (1982) 3917 lently, redshift-limited) region, the precise relationship takes for a model having a built-in redshift trend, so that thef(m - the form 5logz) term still has some effect in reducing the contribution from large redshifts. (zjm) = Gj(m)/Go(m);Gj(m) = f f(m - 5p-llogz)z(&P)+Jldz, Similar determinations of (zim) for other conservative sub- Z0 samples of the samples ofrefs. 2 and 3 yield similar results. In assuming that the lower redshift limit zo is taken sufficiently particular, the same is true when luminosities are replaced by large that the effect of peculiar velocities is negligable and z1 geometric magnitudes for subsamples of ref. 3. is sufficiently small that larger redshift corrections (K correc- tion, etc.) are likewise negligible; here, fis the differential lu- Discussion minosity function corresponding to the exponent p in question. The large deviations from both luminosity uniformity and radial By using the ROBUST LFs described above, these expres- spatial homogeneity shown by the Hubble law in the low-red- sions for (zlm) can be evaluated and compared with the empir- shift region z 5 0.01 raise the question ofwhether there exists ical regression; representative results are shown in Fig. 6. The any material spatial evidence for the Hubble law in the classic Hubble law prediction is not as close as the Lundmark law pre- region and categories ofgalaxies in which it arose. The circum- diction but better than might have been expected on the basis stances that both uniformities are quite consistent with the ofits departures from both spatial homogeneity and luminosity Lundmark law and that the deviations in the case ofthe Hubble uniformity. However, inspection of the above expression for law are similar to what would be expected on the assumption (zlm) shows that some cancellation of the two deviations is to of the Lundmark law provide both positive evidence for the be expected on the assumption of the Lundmark law. Under Lundmark law and further negative evidence counter to the this assumption, the Hubble law expression Ho"6ble law. The ROBUST technique together with the V/Vm test thus (zIm)i = f f(m - 5logz)z3dzj f(m - 51ogz)z2dz provides practical independent tests for the two types of uni- formities, readily applicable to samples that are complete in suffers from excessive weighting at the large-z end of the in- magnitude or in both magnitude and redshift, within specified terval, compared with the Lundmark law limits. They appear to be statistically rather efficient and are effective in discriminating between alternative redshift-distance laws. Moreover, they are applicable also in the large-redshift (zIm) = f f(m - 2.5logz)z3/2dz/ f(m - 2.5logz)z"12dz, region, including the case ofquasar samples, as willbe amplified elsewhere. due to the factor z3 in place ofthe factor Z312, even allowing for In connection with possible appeals to statistical results for- the similar difference in the denominators. This would tend to mally derived from the study ofclusters, it should be noted that, increase (zlm) as derived from a Hubble law analysis and, in- quite apart from the frequent lack of statistically objective se- deed, it generally exceeds the empirical values, but the other lection criteria or the introduction of a posteriori corrections, term in the integrand,f(m - 5logz), tends to be smallerfor large both of which naturally vitiate reliable statistical analysis, the z than the corresponding term for the Lundmark law. Roughly hypothesis commonly taken for granted that all the galaxies in speaking, this is simply the analytic counterpart to the obvious a cluster are at essentially the same distance lacks any model- fact that the Hubble law predicts that relatively more galaxies independent demonstration and is biased toward the Hubble will be at large redshifts, in any given redshift interval, but that law. From the Lundmark law standpoint, there is no reason why thegalaxies actually observed at those redshifts will be relatively many of the galaxies in a cluster should not be at or near the absolutely luminous; although the ROBUST analysis corrects distances indicated by their respective redshifts. This would for the OCB for a model that is correct, it cannot do so altogether reduce the anomalous dispersion in their velocities cited by Holmberg (11) to a level that is likely to be entirely consistent 0 with their apparent gravitational binding. Thus, the apparent 0 0 magnitude distribution of the galaxies in a cluster can by no 48 means be identified with the LF, apart from relocation and 0 0 some truncation at the faint end, except by a hypothesis lacking any direct means ofsubstantiation and implicative ofanomalous 3.20 0 24 results. & 32

0 We thank G. de Vaucouleurs for supplying punched cards containing 428 updated data from ref. 3. - 0 0 a40 co 3.10 44 1. Reiz, A. (1941) Ann. Observatory Lund No. 9, Lund, Sweden: A 414 Study of External Galaxies with Special Regard to the Distribu- tion Problem. 0

0 2. Visvanathan, N. (1979) Astrophys. J. 228, 81-94. 0 3. de Vaucouleurs, G. (i979) Astrophys. J. 227, 380-390. 8 43 4. Nilson, P. (1973) Upps. Astron. Obs. Ann. 6. 3.00 5. de Vaucouleurs, G., de Vaucouleurs, A. & Corwin, H. G., Jr. (1976) Second Reference Catalogue of Bright Galaxies (Univer- sity ofTexas Press, Austin). 8 9 10 I 1 1Z 6. Peterson, B. A. (1970) Astronom. J. 75, 695. Apparent magnitude m 7. Segal, I. E. (1976) Mathematical Cosmology and Extragalactic Astronomy (Academic, New York). FIG. 6. Predicted and empirical (zlm) relations for the Visvana- 8. Soneira, R. M. (1979) Astrophys. J. 230, L63-65. than sample with limits m 5 12.4, 500 < cz < 2250, IBI > 400 (153 9. Nicoll, J. F. & Segal, I. E. (1982) Astrophys. J. 258. galaxies). Symbols are as in Fig. 1 except that A denotes the empirical 10. Nicoll, J. F., Johnson, D., Segal, I. E. & Segal, W. (1980) Proc. value of (zjm), the adjoining integer being the number of galaxies in- NatL Acad. Sci. USA 77, 6275-6279. volved in the empirical average. 11. Holmberg, E. (1961) Ark. Astronom. 2, 559. Downloaded by guest on September 29, 2021