Proc. Natl. Acad. Sci. USAVol. 90, pp. 4793-4797, June 1993Colloquium Paper

This paper was presented at a colloquium entitled "Physical Cosmology," organized by a committee chaired by DavidN. Schramm, held March 27 and 28, 1992, at the National Academy of Sciences, Irvine, CA.

Is the velocity-distance relation for galaxies linear?SIDNEY VAN DEN BERGHDominion Astrophysical Observatory, National Research Council, 5071 West Saanich Road, Victoria, BC, V8X 4M6, Canada

ABSTRACT Diameters of ScI galaxies, the luminosities ofsupernovae of type Ia at maximum light, and the brightness ofcentral galaxies in rich clusters are examined as potentialyardsticks or standard candles for study of the velocity-distance relationship. Both supergiant Scl galaxies and super-novae Ia (which have luminosities that differ by up to a factorof -10) are found to be unsuitable for such a study. Theremarkably small luminosity dispersion of first-ranked clustergalaxies (which is not yet understood physically) suggests thatdeviations from a linear velocity-distance relationship areG20% out to red shifts of -40,000 km-si .

In one of the earliest discussions of the radial velocities ofextragalactic nebulae, Lundmark (1) assumed that their mo-tions might be represented in a statistical sense by a relationof the form

V = A + HD +BE, [1]

in which V is the velocity of recession, D is the distance, His the Hubble parameter, in which A and B are fittingparameters. As more velocity data became available (2, 3), itwas found that the velocity-distance relationship could ac-tually be quite well represented by the simple relation

V = HD. [21

This linear version of the extragalactic velocity-distancerelation remained virtually unchallenged until nonlinear re-lations were again introduced by Hawkins (4) and by Segal (5,6). The suggestion that the velocity versus distance relation-ship might be nonlinear has not been received well by themajority of the astronomical community (e.g., Sandage,Tammann, and Yahil, ref. 7). It was generally felt that theapparent nonlinearity of some observed relationships wasdue to various observational selection effects.

"Standard Candles" and Deviations from a Smooth HubbleFlow

Attempts to establish the true nature of the velocity-distancerelation have to rely on very luminous standard candles andmeet with a number of formidable practical difficulties,including: (i) evidence for large-scale deviations from asmooth Hubble flow, (ii) mergers of galaxies, (iii) star for-mation resulting from cooling flows, and (iv) evolution ofstellar populations in galaxies. The discovery of the anisot-ropy of the cosmic microwave background radiation byGorenstein and Smoot (8) showed that the Galaxy is movingwith a velocity of a few hundred km-s-1 relative to theabsolute reference frame provided by this background radi-ation field. Recent work by Dressler et al. (9), Bertschinger

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement"in accordance with 18 U.S.C. §1734 solely to indicate this fact.

et al. (10), and many others has shown that deviations froma smooth Hubble flow occur over redshift intervals as largeas a few thousand km s-1. Very recently Mathewson, Ford,and Buchorn (11) have even suggested that such large sys-tematic streaming motions may occur over redshift ranges ofup to about 104 kms-1. Furthermore, Turner, Cen, andOstriker (12) have shown that, in fact, such large-scaledeviations from a smooth Hubble flow might be expected ontheoretical grounds.

Since "local" deviations from a smooth Hubble flow mayextend over ranges as large as -10,000 km s-s (Az = 0.03),only the most luminous standard candles can be used toestablish the true nature of the velocity-distance relation.The two currently most popular standard candles are thefirst-ranked (cD) galaxies in rich cluster (13-15) (Scd galaxies)and supernovae of type Ia (16) (SN la). The form of theluminosity function of galaxies in rich clusters (17) has alsobeen suggested (but not yet seriously exploited) as a possiblemeans of establishing distances to clusters at large redshifts.Furthermore, attempts have been made recently to use thediameters of ScI galaxies to estimate galaxy distances. How-ever, Block (18) has shown convincingly that such objectsactually exhibit an enormous range in diameters and aretherefore not precision yardsticks (see Fig. 1).

SN Ia.

Three types of supernovae are currently recognized: (i) SNIa, which we believed to be produced by the deflagration ordetonation of white dwarfs that are pushed over the Chan-drasekhar limit; (ii) supernovae of type Ib or Ic (SN Ib or Ic),which are thought to have massive single or binary progen-itors; and (iii) supernovae of type II (SN II), which also haveyoung massive progenitors. At maximum light SN II have arange of a few hundred in luminosity, so that they are notsuitable as standard candles. Not enough observational dataare yet available to make a meaningful estimate of theluminosity dispersion among SN Ib (or SN Ic) at maximumlight.

Since all SN Ia are widely believed (20) to burn the sameamount of 56Ni-+ 56Co-. 56Fe, it seemed reasonable toassume (16, 21) that such objects might turn out to be goodstandard candles. The first hint that all SN Ia might not all beidentical was provided by the observation (22) that suchobjects exhibit a wide range of expansion velocities and thatsome of them, such as SN1986G (23), show nonstandardcolor evolution. Very recently SN1991T (24) and SN199lbghave demonstrated that SN Ia can exhibit very peculiarspectroscopic characteristics at maximum light. Finally,SN199lbg (in the almost dust-free Virgo cluster ellipticalM84) was found to be enormously subluminous at maximumlight. These observations show that not all SN Ia are standard

Abbreviations: SN Ia, supernovae of type Ia, etc; mag, magnitude.

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U.

I

4 4'

v f vt, §. . .... ,. ,,. , ju ,..... _.4

,.,. ..... ,, . ', .0k >; + v

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FIG. 1. Comparison between the Scl galaxies NGC309 in Cetus and NGC4321 (=M100) in the Virgo cluster. Both galaxies are shown to

the same linear scale. The composite figure demonstrates that galaxy diameters are not suitable yardsticks for distance determinations. Both

photographs were obtained with the KPNO 4-m reflector. I am indebted to David Block (Witwatersrand University, Johannesburg) and TomKinman (Kitt Peak National Observatory) for permission to reproduce these images. Reprinted from ref. 19 with permission from the

Astronomical Society of the Pacific (copyright 1992).

candles. It is not yet clear if the class of SN Ia contains a

subclass that is composed of good standard candles.Fig. 2 shows a plot of B(max), the maximum brightness in

blue light, versus distance for all data (complete throughFebruary 1992) on SN Ia for which B(max) is estimated tohave an observational error s 0.5 mag. Distances were

calculated by assuming a smooth Hubble flow with Ho = 75km*s-l Mpc-1. Fig. 2 shows that the dispersion in the Hubblediagram forSN Ta is large. This dispersion is due to (i) internalabsorption of the light of SN Ia that occurs in dusty spiralgalaxies, (ii) deviations from a smooth Hubble flow, (iii)observational errors, and (iv) the intrinsic luminosity disper-sion of SN Ia at maximum light.

Correcting the observed magnitudes of supernovae by AB= 4 E(B - V) where E(B - V) is the reddening in the B - Vcolor and adopting the canonical intrinsic color (B - V)mO -

-0.25 (25) paradoxically increases the dispersion in theHubble diagram for SN Ia! The explanation for this appears

to be (26) that the intrinsic colors of SN Ia are significantlyredder than those previously derived from (mainly photo-graphic) multicolor light curves. The luminosities calculatedfor reddened supernovae in dusty spiral galaxies are de-creased by adopting redder intrinsic colors. On the other

hand, the luminosities of SN la in dust-free ellipticals (whichare assumed to be unreddened) will not be affected byadopting a different intrinsic color.

Fig. 3 shows a Hubble diagram in which absorption cor-

rections were applied by assuming that (B - V)O = 0.0 andAB = 4 E(B V). This figure exhibits significantly less

scatter than Fig. 2. However, the dispersion still is seen to bedisappointingly large. This clearly makes it very difficult touse SN Ta to determine the true nature of the velocity-distance relation. An additional complication is introducedby the observation that SN199lbg was not only subluminousbut also unusually red. This suggests that both the luminos-ities and the colors of SN Ia at maximum light may exhibit asignificant intrinsic dispersion.

Part of the scatter seen in Figs. 2 and 3 might be due todeviations from a smooth Hubble flow. Therefore, the dis-tances to individual supernovae have been recalculated for aVirgocentric inflow model (27) in which the Galaxy is re-

tarded by 300 kms-1. The observed and the absorption-corrected Hubble diagrams for such a model are plotted inFigs. 4 and 5, respectively.

It is disappointing to see that the dispersion in the Hubblediagram for distances derived from a Virgocentric inflow

i

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FIG. 2. Observed Hubble diagram for all SN Ia for which theerror ofB(max) was judged to be s0.5 mag. o, Spirals; *, ellipticals.Uniform Hubble flow with Ho = 75 km s-1'Mpc-1 is assumed. Amean red shift of 1300 km s-1 was adopted for the Virgo cluster. Theregression line shows the locus of points with absolute blue magni-tude MB = -18.8.

model is, at best, only marginally smaller than that for asmooth Hubble flow. This implies that either (i) the true flowmodel for nearby galaxies is not well represented by aVirgocentric infall model, or (ii) the dispersion in the absorp-tion-corrected Hubble diagrams for SN la is dominated bythe intrinsic luminosity dispersion ofSN Ia at maximum light.Therefore, it is reluctantly concluded that it is not yet certainthat SN la are standard candles that can be used to determinethe exact nature of the velocity-distance relation. There isthe hope that it might eventually prove to be possible to select

14k

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Co

E

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FIG. 3. Hubble diagram for a smooth Hubble flow with absorp-tion computed from AB = 4 E(B - V) and (B - V)max = 0.O. o,

Spirals; *, ellipticals.

FIG. 4. Observed Hubble diagram for SN la with the distancecalculated from a Virgocentric infall model. o, Spirals; *, ellipticals.

spectroscopically homogenous subsamples ofSN Ia that are,in fact, standard candles.The effects ofdeviations from a smooth Hubble flow on the

apparent dispersion in the luminosities of SN la at maximumlight can be minimized by considering only those SN la thatoccurred in Virgo cluster galaxies. A listing ofB(max) for theseven accurately observed SN Ia in (or close to) the Virgocluster is given in Table 1. Taken at face value, the data in thistable suggest that well-observed SN Ta have a luminosityrange of =3.8 mag at maximum light. Note in particular thatSN1971G and SN199lbg might constitute a subclass ofSN lathat are subluminous by -3 mag.

14k

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-

co

10 _

5 10 20 40Mpc

FIG. 5. Hubble diagram for SN Ia with distances calculated froma Virgocentric infall model and absorption corrections derived byassuming AB = 4 E(B - V) and (B - V)o = 0.0. Note that thedispersion (Virgocentric infall) is, at best, only marginally smallerthan that in Fig. 3 (smooth Hubble flow). o, Spirals; e, ellipticals.

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Table 1. SN Ia in Virgo cluster galaxies

B(max), Bo(max),SN NGC Type mag mag Remarks*

1957B 4374 El 12.0 12.0 a1961H 4564 E6 11.5 11.5 a1971G 4165 Sa 14.0 14.0 b1984A 4419 SBab: 12.7 12.2 c1990N 4639 SBb II 12.7 12.6 c1991T 4527 Sb 11.6 11.1 c, d1991bg 4374 El -14.9 o14.9 amag, Magnitude.

*Remarks: a, assumed to be unreddened; b, spectrum of type I (i.e.,type Ib or Ic) is not excluded; c, (B - V)Wl = 0.0 is assumed; d,NGC4527 lies just beyond the formal R = 100 limit of the Virgocluster region adopted by Sandage and Tammann (28).

Luminosity Functions of Rich Clusters

Dressler (29, 30) has published photometry of a large numberof galaxies in rich Abell clusters. These data have been usedby van den Bergh (17) to obtain integral luminosity functionsfor 10 rich clusters ofgalaxies. Fig. 6 shows that these integralluminosity functions can be represented by a long linearsection plus a luminous "toe" that differs in strength fromcluster to cluster. Extrapolation ofthis linear section to n(MF)= 0 (where MF is the luminosity in the F passband) yields aparameter MF(A), which exhibits a dispersion of only 0.19mag about a mean value MF(O) = -22.66 + 0.06 + 5 log(Ho/50). Data for individual clusters are listed in Table 2. Itwould clearly be worthwhile to obtain additional MF(O)values for more rich clusters of galaxies to see if MF(O) isindeed a good standard candle, which might place strongconstraints on the linearity of the velocity-distance relation.

Brightest Cluster Galaxies

A number of investigators (13-15, 31) have shown that thebrightest galaxies in clusters are good standard candles. Atgreat distances (and hence at large look-back times), theluminosities ofthe first-ranked galaxies in rich clusters will beaffected by the evolution of stellar populations. The effect ofsuch evolution has been modeled (32, 33). However, these

5oF

MF

FIG. 6. Integral luminosity functions of rich clusters are found toconsist of a linear domain and a high-luminosity "toe." This toe isseen to be quite well-developed in A2256 (Left) but almost absent inA2670 (Right).

Table 2. Data on rich clustersBM MF(0)*,

Cluster type z mag

A98 II-III 0.1034 -22.6:A154 II 0.0652 -22.7A401 I 0.0750 -22.8A665 III 0.180 -22.8A1413 I 0.1427 -22.6:A1940 III 0.1387 -22.4A2029 I 0.0774 -22.3A2218 II 0.1641 -22.8A2256 II-III 0.0594 -22.7A2670 I-II 0.0774 -22.9

BM, Bautz-Morgan; colons denote uncertain valves.*Ho = 50 km s-l Mpc-1 is assumed; MF(O) is corrected for K-dim-ming with qo = + 1 assumed.

calculations do not take into account the luminosity increasesresulting from mergers and from possible star formation incooling flows.

Butcher and Oemler (34) have shown that evolutionaryeffects become important in clusters with z > 0.5. For fieldgalaxies (35) such effects are already important at z ; 0.2. Ittherefore seems prudent to omit clusters with large red shiftsfrom discussions of the velocity-distance relationship. By thesame token, clusters with very small redshifts should be ex-cluded because the relation between redshift and distance fornearby objects is likely to be affected by deviations from asmooth Hubble flow.An almost ideal data base for study ofthe velocity-distance

relation is provided by Hoessel, Gunn, and Thuan (36) whogive colors and redshifts for a complete sample offirst-rankedgalaxies in Abell clusters of richness class 2 1 and in distanceclasses c 4. The redshifts for the objects in the sample fall inthe range 0.02 S z S 0.15. A Hubble diagram for thefirst-ranked galaxies in these Abell clusters is shown in Fig.7. This figure shows that (i) the luminosity dispersion offirst-ranked galaxies is small and (ii) the cD galaxies inclusters of Bautz-Morgan (37) types I and I-II are system-

V (corrected)

FIG. 7. Hubble diagram for first-ranked galaxies in all Abellclusters of richness class 2 1 and distance class - 4 (36). *, Clustersof Bautz-Morgan classes I and I-II. Apparent visual magnitudeshave been corrected for galactic absorption and K-dimming. Theplotted line is the locus of points with absolute visual magnitude Mv= -21.73 + 5 log (Ho/100).

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atically more luminous than the first-ranked galaxies inclusters of Bautz-Morgan types II, II-III, and III. For thebrightest galaxies in clusters of Bautz-Morgan types I andI-II, it is found that <Mv> = -23.24 ± 0.06 + 5 log (Ho/SO),with an rms scatter of 0.31 mag. The small size of thisdispersion is remarkable because observational errors, merg-ers, and bursts of star formation will increase the luminositydispersion of the central galaxies in rich clusters. This resultmight indicate that mergers are not as common as previouslybelieved (38) and that bursts of star formation associated withcooling flows (such as that observed in NGC1275) are rare infirst-ranked galaxies at low redshifts.

It was first thought (39) that, "to a first rough approxima-tion, [first-ranked] galaxy luminosities are drawn at randomfrom a universal distribution." However, more recent work(40) suggests "that the luminosity of the brightest clustergalaxy is determined by some physical process, rather thanby a statistical sampling of a luminosity function." In otherwords, it appears that first-ranked galaxies are in some wayunique (15) and that they seem to have been cast from a"standard mould" that depends only weakly on the richnessof its parent cluster (14). Furthermore, the small present-dayluminosity dispersion offirst-ranked cluster galaxies suggeststhat their initial unique luminosity has not been greatlymodified by subsequent mergers. It has been suggested thatcentral galaxies in rich clusters may become bloated bymergers. If so, their luminosities within a metric radius of=15 kpc might remain more-or-less unchanged, even thoughtheir isophotal luminosity is increased by mergers.

Returning to Fig. 7, it is concluded that presently availabledata on first-ranked galaxies suggest that the velocity-distance relation is linear to :20% out to redshifts of -40,000kms-'.

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