Mon. Not. R. Astron. Soc. 308, 551–556 (1999)

Collisional baryonic dark matter haloes

Mark A. WalkerSpecial Research Centre for Theoretical Astrophysics, School of Physics, University of Sydney, NSW 2006, Australia

Accepted 1999 April 12. Received 1999 April 7; in original form 1998 July 30

A B S T R A C TIf dark haloes are composed of dense gas clouds, as has recently been inferred, thencollisions between clouds lead to galaxy evolution. Collisions introduce a core in an initiallysingular dark matter distribution, and can thus help to reconcile scale-free initial conditions– such as are found in simulations – with observed haloes, which have cores. A pseudo-Tully–Fisher relation, between halo circular speed and visible mass (not luminosity),emerges naturally from the model: Mvis / V7=2.

Published data conform astonishingly well to this theoretical prediction. For our sample ofgalaxies, the mass–velocity relationship has much less scatter than the Tully–Fisherrelation, and holds as well for dwarf galaxies (where diffuse gas makes a sizeablecontribution to the total visible mass) as it does for giants. It seems very likely that thisvisible-mass/velocity relationship is the underlying physical basis for the Tully–Fisherrelation, and this discovery in turn suggests that the dark matter is both baryonic andcollisional.

Key words: galaxies: evolution – galaxies: haloes – dark matter.

1 I N T RO D U C T I O N

A great variety of dark matter candidates exist, motivated bydiverse pieces of evidence, typically indirect (see, for example, thereview by Trimble 1987). One such piece of evidence comes fromthe ‘Extreme Scattering Events’ (ESEs: Fiedler et al. 1987); theseare radio-wave lensing events caused by dense blobs of plasmacrossing the line of sight. Walker & Wardle (1998a) presented amodel in which these events are caused by ionized materialassociated with planetary-mass, molecular gas clouds in theGalactic halo. This model is good at explaining the ESEphenomenon, but carries with it the implication that most of themass of the Galaxy is in this cold, dense form. If the Galactic darkmatter is really in cold gas clouds – as, in fact, has been proposedpreviously by a number of authors (Pfenniger, Combes & Martinet1994; de Paolis et al. 1995; Gerhard & Silk 1996) – thenconsistency with a variety of data requires that these clouds satisfyseveral constraints (Gerhard & Silk 1996). Foremost amongst theseis the requirement that collisions between clouds should not entirelydeplete the halo of its dark content. In this paper we use the simplesthalo model, an isothermal sphere, to show how cloud–cloudcollisions lead directly to structural changes in the halo (Section2), and how this process leads to predictable visible galaxy masses(Section 3); however, no attempt is made to predict the formwhich the visible mass should assume (stars versus diffuse gas).

2 I S OT H E R M A L H A L O E S

In a halo with one-dimensional velocity dispersion s the typical

relative speed of a pair of clouds is���6p

s. Essentially all collisionsare highly supersonic and even glancing impacts – i.e. those withimpact parameter roughly equal to twice the cloud radius – areexpected to unbind the clouds, with the result that the collisionproducts become visible as diffuse gas (which may subsequentlybe transformed into stars). In this circumstance the cross-sectionfor disruptive collisions between clouds is just four times thegeometric cross-section of a single cloud. Recognizing that eachcollision disrupts two clouds, we see that the collision rate can bewritten as

2d loge r

dt� 8

���6p rs

S; �1�

where r is the halo density, and S is the mean surface density ofthe individual clouds. A more rigourous treatment, in which oneintegrates the collision rate over the velocity distribution of theclouds, yields a numerical factor �32=

����pp � which differs by less

than 10 per cent from that in equation (1).Collisions occur preferentially between pairs of clouds having

antiparallel velocities, so that the collision products have a muchsmaller velocity dispersion than the parent clouds (by a factor of���

8p

), and subsequently undergo infall in the gravitationalpotential. This infall modifies the potential, which in turnleads to evolution of the dark halo density beyond thatdescribed by equation (1). (This is ‘halo compression’ –seeBlumenthal et al. 1986.) We shall not attempt a self-consistenttreatment but, rather, we neglect the evolution of the gravitationalpotential. This allows us to estimate the dark halo densityevolution in a straightforward manner by integrating equation (1);

q 1999 RAS

552 M. A. Walker

the result is

r�r; t� � r�r; 0�1� t=tc�r� ; �2�

where the characteristic time tc�r� � S=8���6p

sr�r; 0�. If we nowmodel the initial distribution of dark matter as a singularisothermal sphere, i.e. r�r; 0� � s2=2pGr2, for radius r, thenequation (2) implies

r�r; t� � s2

2pG�r2 � r2c�; �3�

where

r2c �

4���6pp

s3tGS

: �4�

That is, at t . 0 the halo is a non-singular isothermal sphere withcore radius rc. We can recast equation (4) in terms of the nativecircular speed, V, for the halo, using V � ���

2p

s. Taking theproposed mean column for individual ESE clouds (Walker &Wardle 1998a; see also Section 4) of N , 1025 cm22, i.e.S , 30 g cm22, together with t � 10 Gyr, we obtain

rc , 4:2V1:5100 kpc; �5�

where V � 100V100 km s21. In consequence, at the present epochwe should expect galaxy haloes to possess modest cores, if thosehaloes virialized at redshifts z * 1 (i.e. , 10 Gyr ago). These coreradii ought, in principle, to be measurable from rotation curves ofspirals.

In practice the task of measuring a core in the density profile ofa dark halo is made difficult by the visible galaxy, which itselfcontributes to the rotation curve, and one is obliged to perform adisc–halo decomposition – a process which engenders a numberof uncertainties. Furthermore, the visible galaxy can alter thegravitational potential sufficiently that the halo density distribu-tion changes in response: ‘halo compression’; see Blumenthal et al.(1986). Notice that this effect is not accounted for in equation (5).These difficulties are mitigated to a large extent if the visiblegalaxy is so puny that it acts only as a kinematic tracer, allowingthe rotation curve to be measured but not playing any significantrole in determining its form. Such galaxies exist and have alreadybeen used to demonstrate that, at least in these cases, haloes dopossess cores (Moore 1994; Flores & Primack 1994). This is animportant result because it appears to conflict with simulations ofhalo formation using dissipationless dark matter: these exhibitsingular density profiles (Dubinski & Carlberg 1991; Navarro,Frenk & White 1996; for an alternative view see Kravtsov et al.1998).

Kormendy (1990) has derived scaling laws for real (late-typespiral) galaxy haloes; his relationship between core radius andhalo circular speed is (for H0 � 75 km s21 Mpc21)

rc � 3:6V1:6100 kpc; �6�

encouragingly close to equation (5). This scaling is less securethan the simple fact that cores exist in galaxy haloes, and oneshould beware of undue emphasis on the numerical agreementbetween equations (5) and (6), as the mean cloud column density(S) is really only known to within a factor of a few (see also thecomments in Section 4). Indeed, in the following section we treatS as a free parameter, and the resulting best fit indicates a largervalue of S than used here.

3 A P H Y S I C A L B A S I S F O R T H E T U L LY –F I S H E R R E L AT I O N

The Tully–Fisher (TF) relation is an empirical result that connectsthe width of 21-cm line emission, DV, from a spiral galaxy withthe expected luminosity of the galaxy, L:

L / DVa; �7�

where a . 4 (Strauss & Willick 1995); the scatter in this result isquite small. Unfortunately there is no sound theoretical basis forthe TF relation. Attempts to derive it on the basis of the virialtheorem, plus the assumption of fixed surface brightness for allgalaxies (‘Freeman’s law’), fall foul of the fact that TF holds alsofor very low surface-brightness galaxies (Zwaan et al. 1995;McGaugh & de Blok 1998). Put another way, we expectDV , 2V , so that the kinematics reflect the properties of thedark halo, but L is a manifestation of the stellar component of thegalaxy, and no direct coupling between these two is expected apriori. Indeed, as total mass-to-light ratios can vary by more thanan order of magnitude among the population of observed galaxies,one tends to think of the visible and dark components as almostindependent. However, the collisional process that we described inSection 2 converts dark matter into visible forms, and thus createsa close coupling between the dark halo and the visible galaxy.Such a coupling holds promise for explaining the Tully–Fisherrelation; this appealing attribute of baryonic models has beenrecognized previously (Pfenniger, Combes & Martinet 1994;Gerhard & Silk 1996).

We can compute the total visible mass, Mvis, in the form of starsand gas (without actually saying anything about their relativeproportions), from

Mvis�t� ��∞

0dr 4pr2�r�r; 0�2 r�r; t��: �8�

With the evolving isothermal halo described by equation (3), thisgives

Mvis � ps2

Grc: �9�

Notice that Mvis is perfectly well defined, despite the total mass ofthe dark halo diverging at large radii. If we adopt the reasonablesupposition of a roughly similar stellar mass-to-light ratio for allspirals then, provided that diffuse gas makes a negligiblecontribution to Mvis, we expect L / V3:5, which is close toequation (7). In the usual form, however, TF is a relation betweenglobal 21-cm linewidth, DV, and luminosity. Moreover, variousexponents a . 4 are observed, so it is not immediately obviousthat our theory is in agreement with the data.

Spectral-line imaging (notably 21-cm imaging) gives detailedinformation on the velocity field of a galaxy, allowing accuratedetermination of the rotation curve (hence V), rather than just theglobal property DV which can be determined with a single dishradio telescope. Broeils (1992, chapter 4, appendix A) hasinvestigated the L[DV] and L[V] relationships for a sample of 21galaxies with well-determined rotation curves. He finds that blueluminosity is more tightly correlated with V than DV; the formerdisplays a scatter of 0.22, and the latter 0.28, in log10L. (Moreprecisely, Broeils used the circular speed in the flat part of therotation curve, which we take to be a good estimator for V.)

q 1999 RAS, MNRAS 308, 551–556

Baryonic haloes 553

Furthermore, the relation he derives is L / V3:4, which is verysimilar to the form we predict.

We could go further and compare the normalization of thisresult with that of our own theory, but in doing so we would beforced to introduce an additional quantity, the value of the stellarmass-to-light ratio, which is a priori unknown. This can beavoided if we compare Mvis directly with mass estimates derivedfrom rotation curve decompositions which, in effect, measure thestellar mass-to-light ratio for each galaxy. Furthermore, byutilizing mass, rather than luminosity, we expect better agreementwith our theory because we can include the diffuse gas content.Note that diffuse gas is often a substantial fraction of the totalvisible mass of dwarf galaxies; in the (extreme) case of DDO154 itamounts to 80 per cent of the visible mass (Carignan & Freeman1988). While Broeils (1992) did not investigate the possibility ofany correlation between Mvis and V, he did give maximum discdecompositions of the rotation curves for 15 of the 21 galaxiesstudied; his results are summarized in Table 1, and plotted in Fig.1. Also shown in Fig. 1 is the relation given by equation (9), withthe cloud surface density (S) treated as a free parameter; the fit ismanifestly good.

The very first point that must be made about this result is that itis not simply an artefact of (maximum-disc) rotation-curvedecomposition. This point is clear because rotation speedmeasures M/R, not M, so even if the visible mass were thedominant component we would still be obliged to explain theexistence of a characteristic length-scale having a form similar tothat of equation (4). However, in the flat part of the rotation curve– which has been used to estimate V – the visible mattercontributes less than the dark matter in every case in Table 1.On average the dark matter contributes three quarters of thetotal, for this sample, and for some galaxies the visiblecontribution is negligible. The hypothesis that Fig. 1 simplyreflects the rotation-curve decomposition procedure is thereforeuntenable.

The explicit numerical form of our fit is

Mvis � 7:0 � 109V7=2100 M(; �10�

corresponding to S . 140 g cm22 (for t � 10 Gyr), and

rc � 1:9V3=2100 kpc: �11�

We will not attempt to give a figure of merit for the fit quality, asthe uncertainties associated with the data involve systematicuncertainties in rotation-curve decomposition, and these are hardto quantify. We can, however, measure the scatter of the data aboutthe theoretical prediction: the root-mean-square deviation is 0.084in log10Mvis. By comparison the scatter in the Tully–Fisherrelation for this sample is almost a factor of 3 larger (0.24 inlog10LB – slightly bigger than the scatter of 0.22 for the full

q 1999 RAS, MNRAS 308, 551–556

Table 1. Galaxies with well-deter-mined rotation curves. Circularspeeds are taken from table A.1(column 9), p. 94 of Broeils(1992), while masses are obtainedfrom ‘maximum-disc’ rotation-curvedecompositions – the sum of col-umns 7 (diffuse gas), 9 (disc stars)and 11 (bulge stars) of table 2, p.244, in Broeils (1992). Only 15galaxies are common to both tables.

Name V Mvis

(km s21) (1010 M()

NGC55 86 0.42UGC2259 90 0.40NGC2841 294 35.0NGC3198 149 3.80DDO170 60 0.09NGC247 108 1.05NGC1560 74 0.22NGC2903 180 4.82DDO154 45 0.05NGC6503 115 1.05NGC300 90 0.58NGC2403 134 1.82NGC3109 64 0.10NGC5033 220 10.8NGC7331 238 18.0

Figure 2. As Fig. 1 but for the stellar mass only, i.e. excluding thecontribution of diffuse gas to the total visible mass; the theoretical line isidentical to that in Fig. 1. It is clear that the dwarf galaxies, which have ahigh mass fraction in diffuse gas, systematically depart from therelationship defined by the giants, if only stellar mass is included in theaccounting. Compare this with Fig. 1, where a single relation is valid forthe whole sample.

Figure 1. Data for a sample of 15 galaxies with well-determined rotationcurves (see Table 1), showing the total visible mass as a function of galaxycircular speed (i.e. Mvis[V]). Contributions to Mvis come from stars – bothbulge and disc (determined from ‘maximum disc’ decompositions) –anddiffuse gas. The plotted symbol size is arbitrarily chosen. The line showsthe prediction of equation (9), with cloud surface density (S) treated as afree parameter.

554 M. A. Walker

sample of 21 galaxies studied by Broeils). Notice that only a partof this difference can be accounted for by uncertainties in galaxydistances, as the stellar mass inferred from rotation-curvedecomposition is proportional to distance, while luminosity andthe measured gas mass are both proportional to (distance)2. Wealso note that if the mass contributed by diffuse gas isneglected then the dwarf galaxies systematically depart from therelation defined by the giants; this point is graphically illustratedin Fig. 2, where stellar mass is plotted as a function of halocircular speed. These facts oblige us to conclude that thefundamental connection is between total visible mass and halocircular speed, with the Tully–Fisher relation emerging as anapproximation which is valid when most of the visible mass is instellar form.

To add a little more weight to this conclusion, we emphasizethat our sample of galaxies is very heterogeneous: it spans theentire size spectrum from dwarfs to giants; it includes low surface-brightness objects; most importantly, perhaps, it includes cases inwhich the visible galaxy makes a negligible contribution to therotation curve. This last point is crucial as it requires that theobserved Mvis[V] relation be interpreted as Mvis being determinedby V, and not the other way around. That is, purely from anobservational perspective we can assert that the visible masscontent of a galaxy is determined by the velocity dispersion of thedark matter halo. Our theory shows why this ought to be so, and itfollows that these data support the model of a baryonic dark halo,with clouds of typical surface density S . 140 g cm22.

4 D I S C U S S I O N

The evolution implicit in equation (9) could, in principle, be usedto further test the theory we have presented, but in practice thiswould be difficult. In particular the requisite sensitivity andangular resolution, for determining 21-cm rotation curves ofnormal galaxies at z , 1, are both beyond the reach of currentinstrumentation. Some studies have already been made of Tully–Fisher-type correlations of galaxies at z , 1, based on optical dataalone (e.g. Vogt et al. 1997). These show mild evolution in a senseopposite to that expected in our theory (if a constant mass-to-lightratio is assumed), and we must suppose that this is a result ofstellar populations being younger at earlier epochs. Notice thatthere are serious consequences for galaxy distance estimates basedon a local Tully–Fisher relation if the actual TF relation evolveswith look-back time.

It is well known that the infrared TF relations show less scatterthan their visible counterparts (Aaronson, Huchra & Mould 1979),because infrared photometry is less sensitive to the young, hotstars that tend to dominate the luminosity (but not the total stellarmass) of the galaxy. Extinction corrections are also smaller in theinfrared. This argues that rotation-curve decompositions based oninfrared photometry ought to be better than those based on visiblephotometry. In addition, there is no particular reason to supposethat ‘maximum-disc’ rotation-curve decompositions give the bestestimates of the stellar mass-to-light ratio. Thus, with a studythat is specifically aimed at testing equation (9), it may bepossible to reduce the scatter seen in Fig. 1 below its alreadysmall value. Ideally one would like to work with a sample ofgalaxies from a single cluster, thereby reducing the dispersioncontributed by distance uncertainties; distance errors probablydominate the currently observed scatter of 0.084 dex in the Mvis[V]relation.

Any cluster of galaxies that is well approximated by anisothermal distribution of dark matter should fit into the schemewe have outlined, simply by an appropriate choice of velocitydispersion. [We estimate from the evaporation constraints given byGerhard & Silk (1996) that the clouds can survive for roughly aHubble time in the environment of a rich cluster of galaxies.] Forexample, taking s � 103 km s21 equation (10) predicts a visiblemass of Mvis � 7 � 1013 M(, broadly consistent with the data(Jones & Forman 1984), and equation (11) gives a dark mattercore radius of rc � 100 kpc. This core radius is rather larger thanindicated by analysis of cluster lensing data (Miralda-Escude1993; Flores & Primack 1994) but we must bear in mind thatlensing measures the total surface density, not just the darkcomponent. For galaxies, the greatest visible contributiontypically comes from stars, the presence of which is relativelystraightforward to quantify, but for rich clusters it is usually thehot intracluster medium that dominates, and here the inferred massdistribution is much more model dependent. (Notice, again, thatthe theory we have presented is only partially predictive in that itgives the total visible mass, but does not tell us whether this oughtto be in stars or diffuse gas.) In particular we note (i) it is thoughtthat in ‘cooling flow’ clusters a large amount of gas accumulatesin some (unknown) form at the centre of the cluster (Fabian 1994),and (ii) there are tentative detections of huge amounts of warm(EUV emitting) gas in some clusters (Mittaz, Lieu & Lockman1998).

In the foregoing discussion we have given no consideration tothe material properties of the cloud-collision products, beingcontent with the notion that this stuff becomes part of the visiblepool. A basic analysis of the effects of the shock (Walker &Wardle 1998b) indicates that, for the Galactic halo, the result of acollision will typically be atomic gas, even though the clouds areinitially molecular. This suggests a possible connection with the‘High Velocity Clouds’ (HVCs) (Wakker & van Woerden 1997)which are seen (almost exclusively) in 21-cm emission: someHVCs may simply be material from recent dark cloud collisions.This gas is expected to have such a low column density that it willbe stopped by the diffuse interstellar medium, at the firstencounter with the Galactic plane, thereby contributing to theassembly of the gaseous disc and, subsequently, the Milky Way.

As well as being relevant to the Tully–Fisher relation, themodel we have presented is germane to the ‘Disc-HaloConspiracy’ (Sancisi & van Albada 1987). The conspiracy is so-called because it seems puzzling that (giant) spiral galaxy rotationcurves should be as flat as they are observed to be, given that theacceleration is predominantly because of stars at small galacto-centric radii and dark matter at large radii. In the model we havepresented, however, the conspiracy is no surprise; rather it is aninnate feature, as dark matter is converted to visible forms bycloud–cloud collisions. With the model of Section 2 we cannotsensibly compute rotation curves – because we have neglectedevolution of the gravitational potential – so these statements arenecessarily qualitative. A useful quantitative treatment wouldrequire a self-consistent description of the evolution; more carefulconsideration of the initial conditions (dark matter density profile)would also be appropriate.

The principal difficulty with our model is that the fit shown inFig. 1 requires S . 140 g cm22 (for t � 10 Gyr�; whereasKormendy (1990) measures halo core radii that suggest a smallercloud surface density, S , 40 g cm22. Both of these values arehigher than the original estimate (Walker & Wardle 1998a) basedon scintillation data, and incompatible with each other. Kormendy

q 1999 RAS, MNRAS 308, 551–556

Baryonic haloes 555

(1990) cautions that his derived halo scaling relationships arequite uncertain, and it is evident from his fig. 3 that the scatter inthe measurements is much larger than in our Fig. 1, so the valueS � 140 g cm22 is to be preferred. One possible resolution is thatour theory is simply too crude. For example, in the case of spiralgalaxies the fact that the visible matter has angular momentumrequires that the dark matter should also have some, yet we aremodelling haloes as non-rotating. Perhaps a more refined theorywould yield consistent estimates of halo core radius and visiblemass? We have made a preliminary investigation into thispossibility, using Toomre’s (1982) analytic models of rotatingisothermal haloes. These models span the entire range fromisothermal spheres to cold, rotationally supported discs. We findthat that the prediction Mvis / V7=2 holds for this whole set ofmodels, with a coefficient of proportionality that is only weaklydependent on the degree of rotational support that the halopossesses. Similarly, the inferred core radius is only weaklydependent on rotation, so it seems unlikely that the discrepancycan be explained in this way. The fact that the predicted Mvis isonly a weak function of the degree of halo rotation, for a fixedvalue of V does, however, lend a certain robustness to the modelwe have presented.

Finally we note that the preferred cloud column densitydeduced in this paper, i.e. S . 140 g cm22, is substantially largerthan the value S , 30 g cm22 anticipated on the basis of the ESEdata (Walker & Wardle 1998a). The latter value was derived fromthe sky covering fraction, t , of the ESE clouds, together with therequirement that the cold clouds do not contribute more than thedynamically determined mass for the Galactic dark halo. Usingthe density distribution of equation (3), we find that at highGalactic latitude,

t .2:4 � 1024���������

S100

p ; �12�

at the Solar circle, where S � 100S100 g cm22. Evidently ifS100 � 1:4, then the dynamical constraints limit the coveringfraction of the clouds to be t & 2 � 1024. The observed ESEcloud covering fraction is highly uncertain because what actuallyconstitutes an ESE is only very loosely defined. Selecting allperiods of ‘unusual variability’, Fiedler et al. (1994) reportedtobs , 5 � 1023. On the other hand, if we consider events whoselight curves are reminiscent of the calculations of Walker &Wardle (1998a), then we would include only the events for thesources 09541658 and 17491096, leading to an estimate oftobs , 5 � 1024; this is still 2.5 times larger than we expect forS � 140 g cm22.

To understand how this discrepancy might arise we need torecognize that the size of an ESE cloud relates to the innerboundary of the ionized wind, as it is the free electrons thatcomprise the radio-wave ‘lens’. By contrast the measure of S wehave given in this paper is a characteristic surface density for theunderlying hydrostatic cloud; the wind is irrelevant here as itcontains a negligible fraction of the mass of the cloud. Walker &Wardle (1998a) explicitly assumed that the inner boundary of theionized wind coincided with the surface of the hydrostatic cloud.However, if the inner boundary of the ionised wind is actually theouter boundary of a neutral wind, then the ESE covering fractionwill be substantially larger than the estimate given by equation(12), as is observed to be the case. It is plausible that a neutralwind should underly the ionized wind, but this possibility requirescareful consideration in its own right and will not be pursued here.

Further constraints on the column density of the cold clouds areconsidered by Draine (1998), based on optical refraction by theneutral gas (‘gas lensing’). These constraints are compatible withS . 140 g cm22 provided that the internal density profile of theclouds is not strongly centrally concentrated; convective poly-tropes, for example, are acceptable.

5 C O N C L U S I O N S

Modelling galaxy haloes as isothermal spheres composed ofcollisional, baryonic dark matter leads us to expect non-singulardark halo density distributions with a predictable visible masscontent. Both of these expectations are borne out in the data, withthe latter result very likely being the fundamental basis for theTully–Fisher relation. Although simple, the theory is remarkablygood at predicting (spiral) galaxy masses across the whole sizespectrum from dwarfs to giants. The evident success of thisdescription of galaxy evolution gives strong support to the notionthat galaxy haloes are composed of a vast number of cold, dense,planetary-mass gas clouds.

AC K N OW L E D G M E N T S

Thanks to Mark Wardle, Jeremy Mould, Bohdan Paczynski, KenFreeman and James Binney for their helpful comments. TheSpecial Research Centre for Theoretical Astrophysics is funded bythe Australian Research Council under its Special ResearchCentres Program.

R E F E R E N C E S

Aaronson M., Huchra J., Mould J., 1979, ApJ, 229, 1Blumenthal G. R., Faber S. M., Flores R., Primack J. R., 1986, ApJ, 301,

27Broeils A., 1992, PhD thesis, Univ. GroningenCarignan C., Freeman K. C., 1988, ApJ, 332, L33de Paolis F., Ingrosso G., Jetzer Ph., Roncadelli M., 1995, Phys. Rev. Lett.,

74, 14Draine B. T., 1998, ApJ, 509, L41Dubinski J., Carlberg R. G., ApJ, 378, 496Fabian A. C., 1994, ARA&A, 32, 277Fiedler R. L., Dennison B., Johnston K. J., Hewish A., 1987, Nat, 326, 675Fiedler R. L., Johnston K. J., Waltman E. B., Simon R. S., 1994, ApJ, 430,

581Flores R. A., Primack J. R., 1994, ApJ, 427, L1Gerhard O., Silk J., 1996, ApJ, 472, 34Jones C., Forman W., 1984, ApJ, 276, 38Kormendy J., 1990, in Kron R. G., ed., ASP Conf. Ser. Vol. 10, Evolution

of the Universe of Galaxies. Astron. Soc. Pac., San Francisco, p. 33Kravtsov A. V., Klypin A. A., Bullock J. S., Primack J. R., 1998, ApJ, 502,

48McGaugh S. S., de Blok W. J. G., 1998, ApJ, 499, 41Miralda-Escude J., 1993, ApJ, 403, 497Mittaz J. P. D., Lieu R., Lockman F. J., 1998, ApJ, 498, L17Moore B., 1994, Nat, 370, 629Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563Pfenniger D., Combes F., Martinet L., 1994, A&A, 285, 79Sancisi R., van Albada T. S., 1987, in Kormendy J., Knapp G. R., eds,

Proc. IAU Symp. 117, Dark Matter in the Universe. Kluwer, Dordrecht,p. 67

Strauss M. A., Willick J. A., 1995, Phys. Rep., 261, 271Toomre A., 1982, ApJ, 259, 535

q 1999 RAS, MNRAS 308, 551–556

556 M. A. Walker

Trimble V., 1987, ARA&A, 25, 425Vogt N. et al., 1997, ApJ, 479, L121Wakker B., van Woerden H., 1997, ARA&A, 35, 217Walker M., Wardle M., 1998a, ApJL, 498, L125Walker M., Wardle M., 1999, in Gibson B. K., Putman M. E., eds, ASP

Conf. Ser. Vol. 166, Stromlo Workshop on High Velocity Clouds.Astron. Soc. Pac., San Francisco, p. 269

Zwaan M. A., van der Hulst J. M., de Blok W. J. G., McGaugh S. S., 1995,MNRAS, 273, L35

This paper has been typeset from a TEX/LATEX file prepared by the author.

q 1999 RAS, MNRAS 308, 551–556