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5The Structure of Cold Dark Matter HalosJulio F. Navarro1Steward Observatory, The University of Arizona, Tucson, AZ, 85721, USA.Carlos S. FrenkPhysics Department, University of Durham, Durham DH1 3LE, England.Simon D.M. WhiteMax Planck Institut f�ur Astrophysik, Karl-Schwarzschild Strasse 1, D-85740, Garching, Germany.ABSTRACTWe use N-body simulations to investigate the structure of dark halos in the standardCold Dark Matter cosmogony. Halos are excised from simulations of cosmologically rep-resentative regions and are resimulated individually at high resolution. We study objectswith masses ranging from those of dwarf galaxy halos to those of rich galaxy clusters.The spherically averaged density pro�les of all our halos can be �t over two decades inradius by scaling a simple \universal" pro�le. The characteristic overdensity of a halo, orequivalently its concentration, correlates strongly with halo mass in a way which re ectsthe mass dependence of the epoch of halo formation. Halo pro�les are approximatelyisothermal over a large range in radii, but are signi�cantly shallower than r�2 near thecenter and steeper than r�2 near the virial radius. Matching the observed rotationcurves of disk galaxies requires disk mass-to-light ratios to increase systematically withluminosity. Further, it suggests that the halos of bright galaxies depend only weakly ongalaxy luminosity and have circular velocities signi�cantly lower than the disk rotationspeed. This may explain why luminosity and dynamics are uncorrelated in observedsamples of binary galaxies and of satellite/spiral systems. For galaxy clusters, our halomodels are consistent both with the presence of giant arcs and with the observed struc-ture of the intracluster medium, and they suggest a simple explanation for the disparateestimates of cluster core radii found by previous authors. Our results also highlight twoshortcomings of the CDM model. CDM halos are too concentrated to be consistent withthe halo parameters inferred for dwarf irregulars, and the predicted abundance of galaxyhalos is larger than the observed abundance of galaxies. The �rst problem may implythat the core structure of dwarf galaxies was altered by the galaxy formation process,the second that galaxies failed to form (or remain undetected) in many dark halos.1Bart J. Bok Fellow. 1

1. IntroductionAnalytic calculations and numerical simulationsboth suggest that the density pro�les of dark mat-ter halos may contain useful information regardingthe cosmological parameters of the Universe and thepower spectrum of initial density uctuations. Forexample, the secondary infall models of Gunn & Gott(1972) established that gravitational collapse couldlead to the formation of virialized systems with al-most isothermal density pro�les. This result, fur-ther re�ned by Fillmore & Goldreich (1984) andBertschinger (1985), suggested a cosmological signi�-cance for the observation that the rotation curves ofgalactic disks are at.This analytic work was extended by Ho�man &Shaham (1985) and Ho�man (1988), who pointed outthat the structure of halos should also depend on thevalue of the density parameter and the power spec-trum of initial density uctuations. Studying scale-free spectra, P (k) / kn (�3 < n < 4), these authorsconcluded that halo density pro�les should steepenfor larger values of n and lower values of . Approxi-mately at circular velocity curves were predicted for�2 < n < �1 and � 1. These conclusions werecon�rmed by N-body experiments (Frenk et al. 1985,1988, Quinn et al. 1986, Efstathiou et al. 1988a,Zurek et al. 1988) and supported the then emergingCold Dark Matter scenario, since in this model the ef-fective slope of the power spectrum on galactic massscales is neff � �1:5 (Blumenthal et al. 1984, Daviset al. 1985).It has become customary to model virialized halosby isothermal spheres characterized by two param-eters; a velocity dispersion and a core radius. Forgalaxies, the halo velocity dispersion (or its circularvelocity, de�ned by V 2c = GM=r) is often assumed tobe directly proportional to the characteristic velocityof the observed galaxy. This assumption allows ob-servations to be compared directly to the results ofcosmological N-body simulations or of analytic mod-els for galaxy formation (see e.g. Frenk et al. 1988,White & Frenk 1991, Cole 1991, Lacey et al. 1993,Kau�mann et al. 1993, Cole et al. 1994). How-ever, this simple hypothesis does not seem to be sup-ported by observation. If more massive halos were in-deed associated with faster rotating disks and so withbrighter galaxies, a correlation would be expected be-tween the luminosity of binary galaxies and the rel-ative velocity of their components. Similarly, there

should be a correlation between the velocity of a satel-lite galaxy relative to its primary and the rotationvelocity of the primary's disk. No such correlationsare apparent in existing data (see, e.g. White et al.1983, Zaritsky et al. 1993). A possible explanationmay come from the work of Persic & Salucci (1991),who argue that halo circular velocity is only weaklyrelated to disk rotation speed.Observational estimates of the core radii of darkhalos have also led to con icting results. Large coreradii have been advocated in order to accommodatethe contribution of the luminous component to therotation curves of disk galaxies (Blumenthal et al.1986, Flores et al. 1993); to account for the shapeof the rotation curves of dwarf galaxies (Flores & Pri-mack 1994, Moore 1994); and to reconcile X-ray clus-ter cooling ow models with observations (Fabian etal. 1991). On the other hand, the giant arcs pro-duced by gravitational lensing of background galax-ies by galaxy clusters require cluster core radii to besmall (see e.g. the review by Soucail & Mellier 1994).The very existence of a \core", i.e. a central re-gion where the density is approximately constant, hasbeen challenged by recent high resolution numericalwork. N-body simulations of the formation of galac-tic halos and galaxy cluster halos provide no �rm ev-idence for the existence of a core, beyond that im-posed by numerical limitations (Dubinski & Carlberg1991, Warren et al. 1992, Crone, Evrard & Rich-stone 1994, Carlberg 1994, Navarro, Frenk & White1995b). These studies indicate that dark matter halosare not well approximated by isothermal spheres, butrather have gently changing logarithmic slopes as inthe model proposed by Hernquist (1990) for ellipticalgalaxies, �(r) / 1r(1 + r=rs)3 ; (1)or that proposed by Navarro, Frenk & White (1995b)for X-ray cluster halos,�(r) / 1r(1 + r=rs)2 : (2)These pro�les are singular (although the potentialand mass converge near the center), and possess awell de�ned scale where the pro�le changes shape,the scale radius rs. (We will refer to rs as a \scale"radius and reserve the term \core" to refer to a cen-tral region of constant density.) Near the scale radiusthe pro�les are almost isothermal, so these results are2

quite consistent with the lower resolution simulationsmentioned earlier.The possibility that dark matter density pro�lesmay diverge like r�1 near the center has importantconsequences for the observational issues discussedabove. In this paper, we present the results of a largeset of high resolution numerical simulations aimed atunderstanding the density pro�les of dark halos inthe standard CDM cosmogony. Although this partic-ular cosmogony has fallen into some disrepute sincethe COBE measurement of temperature uctuationsin the microwave background radiation, it is still theprime example of the general class of hierarchical clus-tering theories. Many of the conclusions reached fromstudying this model are directly applicable to othermodels of structure formation. Indeed, preliminaryresults of a study of halos formed in scale-free uni-verses corroborate the conclusions presented below(Navarro et al. in preparation).The study of the structure of dark matter halos isparticularly well suited to the N-body methods de-veloped over the past decade. Care is needed, how-ever, to model their central regions reliably since theseregions are the most vulnerable to systematic errorsintroduced by �nite resolution e�ects. It is particu-larly important to ensure that all halos, irrespective ofmass, are simulated with comparable numerical reso-lution. Halos identi�ed in a single cosmological simu-lation are inevitably resolved to varying degrees sincemassive systems contain more particles than less mas-sive ones. We circumvent this problem by selectinghalos with a wide range of masses from cosmologi-cal simulations of large regions and then resimulatingthem individually at higher resolution. We presentdetails of our numerical procedures in x2, and in x3we test how their limitations a�ect the structure ofour halos. Section 4 presents results for an ensembleof simulations spanning four orders of magnitude inhalo mass. Section 5 discusses some implications ofour results, and x6 summarizes our conclusions.2. The Numerical ExperimentsWe simulate the formation of 19 di�erent systemswith scales ranging from those of dwarf galaxies tothose of rich clusters. The systems were identi�ed intwo large cosmological N-body simulations of a stan-dard = 1 CDM universe carried out with a P3Mcode (Efstathiou et al 1985). These simulations fol-lowed the evolution of 262,144 particles in periodic

boxes of 360 and 30 Mpc on a side, respectively, andwere stopped when the rms uctuation in spheres ofradius 16 Mpc was �8 � 1=b = 0:63, where b de-notes the usual \bias" parameter. (All physical quan-tities quoted in this paper assume a Hubble constantof 50 km/s/Mpc.) At this epoch, which we identifywith the present, we found collapsed systems of vary-ing mass using a friends-of-friends group �nding al-gorithm with the linking parameter set to 10% of themean interparticle separation. The centers of mass ofthese clumps were used as halo centers and the radiusof the sphere encompassing a mean overdensity of 200was calculated in each case. Hereafter, we shall referto this as the \virial" radius of the system and de-note it by r200. After sorting the clumps by the mass,M200, contained within these spheres, we picked 19 ofthem covering a range of four orders of magnitude inmass (� 3� 1011 < M200=M� <� 3� 1015). We didnot select randomly from this mass range, but choseinstead halos grouped roughly into three bins of cir-cular velocity, � 100 km s�1 , 250 km s�1 and > 450km s�1 . This gives some indication of the \cosmicscatter" in the properties of halos of a given mass.The particles in each of the 19 selected systemswere then traced back to the initial time, where abox containing all of them was drawn. This box wasloaded with 323 particles on a cubic grid which wasperturbed using the same waves as in the original sim-ulation, together with additional waves to �ll out thepower spectrum to the Nyquist frequency of the newparticle grid. The size, Lbox, of this \high-resolution"box was chosen so that all systems had, at z = 0,approximately the same number of particles withinthe virial radius, typically 5; 000-10; 000. The gravi-tational softening was chosen to be one percent of thevirial radius, and was kept �xed in physical coordi-nates. (Note that some of the tests described in x3have fewer particles; for these runs the gravitationalsoftening was slightly larger.) The initial redshift ofeach run was chosen so that the median displace-ment of particles within the \high-resolution" boxwas smaller than the initial particle gridsize. Thesechoices ensure that all halos were resolved to a similardegree at z = 0. A summary of the numerical param-eters of the simulations is given in Table 1. The pa-rameters listed in this Table are de�ned throughoutthe text.Tidal e�ects due to distant material are repre-sented by using several thousand massive particlesto coarse-sample the region surrounding the \high-3

resolution" box. This procedure for setting up initialconditions is identical to that described in Navarro,Frenk & White (1995a,b), where further details maybe found. All simulations were run with the N-body code described by Navarro & White (1993); asecond-order accurate, nearest-neighbor binary-treecode, with individual particle timesteps. Typically,the minimum timestep is between 10�4 and 10�5of a Hubble time, low mass systems requiring moretimesteps than more massive ones due to their highercentral densities.3. E�ects of numerical limitationsAs mentioned in x1, numerical limitations can in- uence the structure of model halos. The number ofparticles, the gravitational softening, the initial red-shift, and the timestep can all, in principle, have asigni�cant e�ect. It is important to check explicitlythat our results are insensitive to the particular choiceof numerical parameters we have made. For each ofthe tests described below we chose one of the leastmassive and one of the most massive halos in our se-ries, and we resimulated them varying the numericalparameters in a systematic fashion. Our conclusion isthat, when chosen carefully, numerical parameters donot a�ect the structure of halos in well resolved re-gions, i.e. at radii which are larger than the gravita-tional softening and which enclose a su�ciently largenumber of particles (�>50).3.1. The initial redshiftFigures 1a and 1b show the e�ect of varying the ini-tial redshift, zi, of the simulation for two of our halos,a large halo of mass M200 � 1015M�, correspondingto a rich galaxy cluster, and a small halo of massM200 � 1011M�, corresponding to a dwarf galaxy.These models had only 223 particles within the \high-resolution" box for CPU economy reasons. The pan-els show the density pro�le and the circular velocityas a function of radius. The pro�les are sphericallyaveraged over bins containing about 30 particles each.Three di�erent choices of zi were tried for the smallhalo, 1 + zi = 5:5, 11:0, and 22:0, and two for thelarge system, 1 + zi = 5:5 and 11:0. The densitypro�les show little change, although for 1 + zi = 5:5the central density of the small halo appears to besubstantially underestimated. Since it is a cumulativequantity, the circular velocity pro�le is more sensitiveto variations near the center. Figure 1b shows that

starting at 1 + zi = 5:5 results in a reduced circularvelocity near the center but, as the starting redshiftis increased, the pro�les converge to a unique shapeso that for 1 + zi = 11:0 and 22:0 the two curves areindistinguishable. In the case of the massive system,starting at 1 + zi = 5:5 or at 11:0 had no signi�cante�ect.The importance of the initial redshift may be un-derstood by considering the initial displacement �eldof the particles given by the Zel'dovich approxima-tion. At 1 + zi = 5:5, the median particle displace-ment in the small halo simulation was about 4 timesthe mean interparticle separation. Thus, the uc-tuations are not linear on small scales in the initialconditions and the Zel'dovich approximation breaksdown. The excess kinetic energy imparted to the par-ticles prevents the formation of dense clumps in theearly stages of the simulation, reducing the maximumdensity of the �nal system. The median displace-ment is less than the mean interparticle separationat 1 + zi = 22:0, and less than twice the mean inter-particle separation at 1 + zi = 11:0. In these cases,use of the Ze'ldovich approximation is justi�ed, andthe results converge to a unique solution.In the case of the massive halo, even at 1+zi = 5:5the median displacement was only about twice themean interparticle separation. As a result startingthis \late" had a negligible e�ect on the �nal pro�le.We conclude that as long as the initial ratio betweenthe median particle displacement and the mean in-terparticle separation is of order unity (or less) theresults are insensitive to the particular choice of zi.The initial redshifts of all runs (quoted in Table 1)satisfy this condition.3.2. The gravitational softeningThe gravitational softening, hg, is included in theN-body equations of motion in order to suppress re-laxation e�ects due to two-body encounters. Thisis accomplished if hg is set signi�cantly larger thanthe impact parameter for a large angle de ectionin a typical 2-body encounter, � Gm=�2, where mis the mass of a particle and � the velocity dis-persion of the system (White 1979). Since �2 �GM200=2 r200 = GmN200=2 r200, then hg should belarger than 2 r200=N200. (Here N200 is the total num-ber of particles within the virial radius r200 and wehave assumed that all particles have the same mass.)In our simulations N200 is typically of order a fewthousand, so choosing hg of the order 10�2 � r2004

should be safe.Figures 2a and 2b show that varying the softeninglength by up to a factor of 5 (within the constraintmentioned above) has little impact on the �nal struc-ture of the halos outside about one softening length.Three di�erent values of hg were tried for the largehalo and two for the small system. The �gures dis-play the density and circular velocity pro�les of thesame two halos used in the previous subsection. Thenumber of particles within the \high-resolution" box,however, was increased to 323, and the initial redshiftwas also �xed at 1 + zi = 22:0 in both cases.Note that although hg < 10�2�r200 could be used,too small a value of hg would be counterproductive.Smaller softenings require smaller timesteps for thesame accuracy, with a consequent increase in CPUtime consumption. For the test runs in Figure 2, thenumber of timesteps increased roughly as h1=2g . Thus,the experiments illustrated in this �gure also showthat the �nal halo structures are independent of thenumber of timesteps in the simulation, typically over10; 000 timesteps per run.Finally, comparison of Figures 1 and 2 shows thatincreasing the number of particles by a factor of threehas no noticeable e�ect on the structure of the halos,except that larger values of N allow us to determinethe structure closer to the halo's center. At radii con-taining more than � 50 particles, our results are in-sensitive to the number of particles. For the parti-cle numbers in these runs, we can reliably probe thestructure of halos over approximately two decades inradius, between r200 and 10�2 � r200.4. Results4.1. Density Pro�lesFigure 3 shows the density pro�les of four dark ha-los of di�erent mass. Halo mass increases from leftto right, from � 3 � 1011M� for the smallest sys-tem, to � 3 � 1015M� for the largest. The arrowsindicate the gravitational softening in each simula-tion. In all cases, the virial radius is about two ordersof magnitude larger than the softening length. Thesmooth curves represent �ts to the simulationdata us-ing a model of the form proposed by Navarro, Frenk& White (1995b),�(r)�crit = �c(r=rs)(1 + r=rs)2 ; (3)

where rs = r200=c is a characteristic radius an �crit =3H2=8�G is the critical density (H is the currentvalue of Hubble's constant); �c and c are two dimen-sionless parameters. Note that r200 determines themass of the halo,M200 = 200�crit(4�=3)r3200, and that�c and c are linked by the requirement that the meandensity within r200 should be 200� �crit. That is,�c = 2003 c3�ln(1 + c)� c=(1 + c)� : (4)We will refer to �c as the characteristic overdensityof the halo, to rs as its scale radius, and to c as itsconcentration.A striking feature of Figure 3 is that the same pro-�le shape provides a very good �t to all the halos eventhough they span nearly four orders of magnitude inmass. The agreement holds for radii ranging fromthe gravitational softening length to the virial radius.Halo concentration decreases systematically with in-creasing mass. This may be clearly seen in Figure 4,where we show two of the pro�les of Figure 3 (corre-sponding to the smallest and largest halos), scaled sothat the density is given in units of the critical densityand the radius in units of the virial radius. Althoughwe show pro�les only for a few halos in Figures 3 and4, these are by no means special. Eq. (3) �ts all ourhalos almost equally well, irrespective of mass.4.2. Circular Velocity Pro�lesThe circular velocity pro�le, Vc(r) = (GM (r)=r)1=2,contains the same information as the density pro�lebut is less noisy because it is a cumulative measureof the radial structure of a halo. Circular velocitycurves for all 19 systems are shown in Figure 5. Theyall look rather similar, as expected given the simi-larity in the shapes of the density pro�les. The cir-cular velocity rises near the center, then remains al-most constant over an extended region, and �nallydeclines near the virial radius. There is a notice-able trend with mass. Larger systems have circu-lar velocities that continue to rise to a larger frac-tion of the virial radius than do smaller halos. Thisis more easily seen in Figure 6, where for two ha-los we scale circular velocity by the value at r200,V200 = (GM200=r200)1=2 � (1=2)(r200=kpc) km/s,and radii by r200 itself. Again, we choose the least andmost massive systems to illustrate the trend. Note(i) that the maximum circular velocity of each halo,Vmax, is larger than V200 (by up to 40 per cent); (ii)5

that the ratio Vmax=V200 is larger for low mass halos;and (iii) that the radius rmax at which the peak occursis a larger fraction of the virial radius for more mas-sive systems. This is an alternative way of expressingthe fact that low mass systems are signi�cantly moreconcentrated than high mass ones (see Figure 4).The dashed lines in Figure 6 are �ts to the circularvelocity curve predicted from eq.(3);�Vc(r)V200 �2 = 1x ln(1 + cx) � (cx)=(1 + cx)ln(1 + c) � c=(1 + c) ; (5)where x = r=r200 is the radius in units of the virialradius. Note that for this model the circular velocitypeaks at rmax � 2rs = 2r200=c.The dotted line in Figure 6 is a �t to the circularvelocity curve of the small halo using the Hernquist(1990) model. The circular velocity in this model isgiven by �VH (r)Vmax �2 = 4(r=rmax)(1 + r=rmax)2 : (6)Note that although in the inner regions (for overden-sities larger than � 1000) this model provides as gooda �t to the data as our adopted pro�le (eq. 3), it pre-dicts circular velocities that are too low near the virialradius.4.3. Mass dependence of halo propertiesAs shown in Figures 4 and 6, low mass CDM halosare more centrally concentrated than high mass ones.This is not surprising because lower mass systemsgenerally collapse at higher redshift, when the meandensity of the universe was higher. In the sphericaltop-hat model, the postcollapse density is a constantmultiple of the mean cosmic density at the time of col-lapse. To understand the relation between halo massand characteristic density requires a formal de�nitionof the time of formation. This is not straightforwardbecause systems are continually evolving, accretingmass through mergers with satellites and neighbor-ing clumps. One possibility is to de�ne the formationtime of a halo as the �rst time when half of its �nalmassM was in progenitors with individual masses ex-ceeding some fraction f of M . With this de�nition,the typical formation redshifts of halos of di�eringmass can be predicted analytically.Lacey & Cole (1993) show that a randomly chosenmass element from a halo of massM identi�ed at red-shift z0 was part of a progenitor with mass exceeding

fM at the earlier redshift z with probabilityP (> fM; zjM; z0) = erfc� �0(z � z0)(1 + z0)p2(�20(fM ) ��20(M ))�;(7)where �20(M ) is the variance of the linear power spec-trum at z = 0 when smoothed with a top hat �lter en-closing massM (see, for example, eqs. (6) and (7) ofWhite & Frenk (1991) for a de�nition), and �0 = 1:69is the usual critical linear overdensity for top-hat col-lapse. This equation is valid for z > z0 and f < 1.The formation redshift zform(M; f; z0) is then de�nedby setting P = 1=2 in eq. (7). Lacey & Cole (1994)tested this formula against N-body simulations for thecase f = 0:5 and found excellent agreement.Having assigned a typical formation redshift to ha-los of a given mass, the mean density of the universeat that redshift provides a natural scaling parameter.Let us assume that the characteristic overdensity of ahalo scales as�c(M; f; z0) = C(f)(1 + zform(M; f; z0))3: (8)Figure 7 shows the characteristic density, �c (ob-tained by �tting eq. 3), as a function of halo massexpressed in units of the non-linear mass, M?(z0).This characteristic mass is de�ned by the condition�(M?(z)) = 1:69=(1+ z); for our adopted normaliza-tion of the CDM power spectrum,M? ' 3:3�1013M�at z = 0. The curves in Figure 7 show the predictionsof eq. (8) for various choices of f . The normaliza-tions have been chosen so that the curves all cross atM200 =M?; this implies C(f) = 1:5� 104, 5:4� 103,and 2:0� 103 for f = 0:5, 0:1, and 0:01, respectively.Note that the shapes of these curves are almost inde-pendent of f for f � 1. The agreement between themass-density relation predicted by eqn. (8) and ourN-body results is fair for f = 0:5, and improves forsmaller values of f . Thus the characteristic density ofa halo does indeed seem to re ect its formation time.Figure 8 shows the correlation between concentra-tion, c, and the mass of the halo. Although this �gurecontains the same information as Figure 7 (since �cand c are related by eqn. 4), it shows explicitly thatlarge halos are signi�cantly less concentrated thansmall ones. Note that the dependence of concentra-tion on mass is quite weak; c changes by less than afactor of four while M varies by four orders of magni-tude. Thus, halos di�ering in mass by less than, say,one order of magnitude, have density pro�les that arealmost indistinguishable in the scaled variables used6

in Figure 4. This con�rms the results we presentedin Navarro, Frenk & White (1995b), where we foundthat galaxy cluster halos in the CDM cosmogony arewell approximated by eq. (3) with c � 5. The presenthigher resolution simulations show that such a weakconcentration is appropriate only for the largest halos,M200=M? � 100. With hindsight, the weak depen-dence of c on mass conspired with our use in Navarro,Frenk & White (1995b) of a �xed softening length ofabout 100 kpc for all systems and of a rather \late"starting redshift, zi = 3:74, to give concentration es-timates which are slightly lower than those shown inFigure 8.Figure 9 shows that the maximum circular veloc-ity, Vmax, can di�er by up to 40 percent from thecircular velocity at the virial radius, V200. The ratioVmax=V200 is an interesting quantity to consider be-cause many analytic and numerical studies use V200to characterize the statistical properties of the halopopulation and to compare them with the observedproperties of the galaxy population. For example, it isoften assumed that the rotation velocity of a galacticdisk is identical to V200, an assumption that clearlybreaks down if the circular velocity of a halo varieswith radius as shown in Figures 5 and 6.Finally, Figure 10 shows the correlation betweenthe maximum circular velocity, Vmax, and the radius,rmax, at which the circular velocity attains that max-imum. Note that for Vmax = 220 km/s, the rotationvelocity of the MilkyWay, rmax is about 50 kpc, largerthan the typical optical size of galaxies like the MilkyWay. Thus, if the dark halo of the Milky Way resem-bles our model halos, the observed at rotation curvemust be the result of the dissipational collapse of theluminous component rather than a direct re ection ofthe structure of the dark halo. We discuss further theimplications of these results in the following section.5. Discussion5.1. The shapes of the rotation curves of diskgalaxies.The shapes of the rotation curves of disk galaxieshave been the subject of numerous studies (Burstein& Rubin 1985, Whitmore et al. 1988, Frenk &Salucci 1989, Persic & Salucci 1991, Casertano &van Gorkom 1991). To �rst approximation, the ro-tation velocity is observed to be constant within theluminous radius, but there is good evidence now thatthe rotation curves of less luminous galaxies tend to

be rising whilst those of their brighter counterpartstend to be gently declining. These systematic trendshave been con�rmed in a recent analysis of the largedataset of Matthewson et al. (1992) by Persic &Salucci (1995). They �nd that the optical radiusof a galaxy, ropt, and the rotation velocity at ropt,Vopt, are strongly correlated, ropt � 20(Vopt=200 kms�1)1:16 kpc. (The optical radius is de�ned as theradius that encloses 83% of the total B-band lumi-nosity of the galaxy and corresponds to 3:2 expo-nential disk scalelengths. For a Freeman disk, roptcorresponds to the 25 magB/arcsec2 isophotal ra-dius.) The total B-band luminosity of a galaxy is alsostrongly correlated with Vopt via the Tully-Fisher re-lation, LB � 3:3�109(Vopt=100 km s�1)2:7L� (Pierce& Tully 1988). All these correlations exhibit rathersmall scatter.We now consider the implications of our results onthe structure of dark matter halos for these obser-vations. To compute the rotation curve of a model\disk galaxy" forming within a CDM halo, we needto specify three parameters: the mass of the disk,Mdisk, its exponential scalelength, rdisk, and the massof the halo,M200. (We ignore the contribution of thegalaxy's spheroid in this simpli�ed model.) Accord-ing to the observed correlations mentioned above, Voptuniquely determines the luminosity and scalelength ofthe disk. Requiring that our model should also exhibitthese correlations then reduces the number of param-eters to two and these may be taken to be the diskmass-to-light ratio, (M=L)disk, and the halo mass.These two parameters may be �xed by matching theamplitude of the observed rotation curve, Vopt, andits slope at the optical radius.To carry this program through, we need to allow forthe fact that the halo will respond to disk formationby adjusting slightly according to the mass and radiusof the disk. We assume that the disk is assembledslowly and that the halo is adiabatically compressedduring this process (Blumenthal et al. 1986, Floreset al. 1993). In this approximation, the radius, r, ofeach halo mass shell after the assembly of the disk isrelated to its initial radius, ri, byr [Mdisk(r) +Mhalo(r)] = riMi(ri): (9)Here Mi(ri) is the mass within radius ri before diskformation (found by integrating eqn. 3), Mdisk(r) isthe �nal disk mass within r (assumed to be expo-nential) and Mhalo(r) is the �nal dark matter dis-tribution we wish to calculate. We assume that7

halo mass shells do not cross during compression, sothat Mhalo(r) = Mhalo(ri) = (1 � b)Mi(ri), whereb = 0:06 is the initial baryon fraction and is chosento agree with primordial nucleosynthesis calculations(Walker et al. 1991).The results of this procedure are illustrated in Fig-ure 11, where we plot the circular velocity of theadiabatically compressed halo (dashed lines) and thedisk+halo or \galaxy" rotation curve (solid lines) forgalaxies with Vopt equal to 100, 200, and 300 km/s.Each curve is labeled by the B-band disk mass-to-light ratio required by our model. The \typical"slopes of observed rotation curves near ropt, as givenby Persic, Salucci & Stel (1995), are shown as dottedlines. These and the model �ts may be seen to bedeclining for rapidly rotating disks and gently risingfor slowly rotating disks.Note that in order to match the observed rota-tion curves, our models require that disk mass-to-light ratios increase with luminosity, approximatelyas (M=L)disk / L1=2. This is because even after adi-abatic compression, the circular velocity of the halo iseither rising or at near the optical radius. Thus, toproduce a declining rotation curve, the disk (whoseown rotation curve is declining in this region) mustbe relatively more important in brighter galaxies. As-suming a constant (M=L)disk � 1:2 for all galaxieswould result in nearly at rotation curves at ropt forall models irrespective of Vopt. The required increasein (M=L)disk with rotation velocity is consistent withthe results of Broeils (1992a) and Salucci, Ashman& Persic (1991), who also required a systematicallyvarying (M=L)disk to �t the rotation curves of spiralgalaxies.An important implication of our models is that asVopt for a galaxy increases from 200 to 300 km/s, V200for the surrounding halo increases only from 150 to170 km/s. This kind of behaviour was previouslynoted by Persic and Salucci (1991) and has a num-ber of consequences. It a�ects the interpretation ofdynamical data on binary and satellite galaxies, anissue which we discuss in more detail in x5:2. Itmay also help with an apparent di�culty in theo-ries where galaxies form by the cooling of baryonswithin dark matter halos (White & Rees 1978). As-suming galaxy/halo systems to have singular isother-mal potentials with circular velocity Vc equal to therotation velocity of the central disk, White & Frenk(1991) estimated X-ray luminosities of order 1042erg/s for bright spiral galaxies, well in excess of the

observed upper limits on di�use emission from brightspiral halos. They noted that their model predicteda typical emission temperature of TX = 0:19(Vc=250km/s)2 keV, suggesting that the emission might besoft enough to have been missed. The halo structureimplied by our current models has Vc signi�cantlysmaller than Vopt over the emitting regions and thusimplies even softer emission from the cooling gas.In our models, the dark matter fraction within roptincreases sharply with decreasing galaxy luminosity,from less than � 70% for galaxies with Vopt = 300km/s, to more than � 90% for galaxies with Vopt =100 km/s. The fraction of the total mass within thevirial radius of the halo represented by the disk alsodepends strongly on Vopt and varies from � 0:07 forVopt = 300 km/s, to less than � 0:01 for Vopt = 100km/s. The value for large galaxies is close to our as-sumed value for the universal baryon fraction. Thus,if pregalactic material had a uniform baryonic massfraction to begin with, our models require that thetransformation of baryons into stars should have beenextremely ine�cient in halos with circular velocity be-low about 100 km/s.Such a trend of increasing global mass-to-light ra-tio (M200=LB) with decreasing halo mass is also re-quired in hierarchical clustering models of galaxy for-mation in order to reconcile the steep mass functionof dark halos predicted in these models with the shal-low faint-end slope of the observed galaxy luminosityfunction (see, e.g. White & Rees 1978, Frenk et al.1988, White & Frenk 1991, Lacey et al. 1993, Kau�-mann et al. 1993, Ashman, Salucci and Persic 1993,Cole et al. 1994). Thus a detailed theory for galaxyformation within halos of this type is likely to be moresuccessful than previous semi-analytic models in pro-ducing realistic galaxy luminosity functions.In summary, the observed rotation curves of spiralsare consistent with the structure of CDM halos pro-vided that the assembly of the luminous componentof galaxies was ine�cient in low mass halos. The ro-tation velocity of bright galaxies is largely determinedby the contribution of the disk, and is therefore nota good indicator of the circular velocity of the sur-rounding halo.5.2. The dynamics of binary galaxies andsatellite companions.A major unresolved issue in studies of the dynam-ics of binary galaxies and of satellites orbiting around8

bright spirals is the puzzling lack of correlation be-tween the luminosity of the system and the observedrelative velocity di�erence (White et al. 1983, Zarit-sky et al. 1993, Zaritsky & White 1994). The puz-zle arises because it is commonly assumed that therotation velocity of galactic disks is a good indica-tor of the mean circular velocity of their surroundinghalos. Since brighter galaxies have more rapidly ro-tating disks, a strong correlation is expected betweenthe luminosity of a pair and their velocity di�erence,as well as between the rotation velocity of a primarygalaxy and the orbital velocity of its satellites. Obser-vational datasets rule out such correlations with highsigni�cance. Indeed, the data seem to favor modelswhere galaxies have extended halos with a mass whichdepends only very weakly on their luminosity.The discussion in x5:1 provides an interesting clueto this puzzle. As can be seen in Figure 11, the meancircular velocity of halos surrounding galaxies withrotation velocities larger than about 200 km/s is al-most uncorrelated with Vopt and, consequently, withthe luminosity of the galaxy. (Bright galaxies withVopt of the order or larger than 200 km/s typicallyconstitute the bulk of the systems considered in ob-servational surveys of binaries and satellite/primarysystems.) According to the analysis presented above,bright galaxies are typically surrounded by a halowith mean circular velocity V200 � 160 km/s. Themass of such halo isM200 � 1:9� 1012M� within thevirial radius r200 � 320 kpc. This agrees with the es-timates of Zaritsky & White (1994), who found fromtheir study of the dynamics of satellites surroundingbright spirals that the average circular velocity is be-tween 180 and 200 km/s at 300 kpc. This agreementis encouraging, since the halo properties we infer hereare chosen to match the shape of the disk rotationcurves, and do not use any information about dy-namics at larger radii. The structure of the dark ha-los which surround bright spiral galaxies seems to besimilar to that implied by our models.5.3. The cores of dwarf galaxies.The luminous component typically constitutes onlya small fraction of the total mass within the lumi-nous radius of a dwarf galaxy. Therefore, measure-ments of the internal dynamics of these dark mat-ter dominated systems are a direct probe of the in-ner regions of dark matter halos. High-quality ro-tation curves of several dwarf galaxies indicate thatthe dark halo circular velocity rises almost linearly

with radius over the luminous regions of these galax-ies (Carignan & Freeman 1988, Carignan & Beaulieu1989, Broeils 1992a,b, Jobin & Carignan 1990, Lakeet al. 1990). For spherical symmetry, this impliesthat the halo has a well de�ned core within which thedark matter density is approximately constant. Aspointed out by Moore (1994) and Flores & Primack(1994), this result is inconsistent with the singularhalo models favoured by N-body simulations such asthose presented in this paper.Figure 12 shows the contribution of the dark haloto the observed rotation curve of four dwarf galax-ies (solid lines). The two curves are meant to en-compass the halo contributions allowed by the ob-servations, and correspond to the results of assum-ing a \maximal disk" or a \maximal halo". These�ts assume that the halo structure is of the form�(r) = �0=(1 + (r=rcore)2), and are plotted only inthe radial range where the rotation curve is mea-sured. The parameters for each galaxy have beentaken from Carignan & Freeman (1988, DDO154),Broeils (1992a, DDO168 and NGC3109), and Lake etal. (1990, DDO170). The dotted lines in these plotsare the expected contribution of a CDM dark halo,constrained to match the circular velocity at the out-ermost radius for which the rotation velocity has beenmeasured. The two dotted lines are meant to repre-sent the most and least concentrated halo compatiblewith this constraint and with the scatter in the cor-relations shown in Figure 10 (a factor of � 2 in rmaxfor a given choice of Vmax).Figure 12 shows that, except perhaps for DDO170,where observations constrain the halo parametersvery poorly, the CDM halos appear too concentratedto �t the observations. We thus agree with the conclu-sions of Moore (1994) and Flores & Primack (1994).The discrepancy, however, is less dramatic than foundby these authors (e.g. compare Figure 12 with Figure1 of Moore 1994). The reason for this di�erence isthat CDM halos are actually less concentrated thanassumed by Moore (1994). Moore's �gure is basedon the simulation of a single CDM halo by Dubinski& Carlberg (1991), who found rmax = 26 kpc for anobject with Vmax = 280 km/s. This contrasts withthe mean value of � 60 kpc found in our simulationsfor the same circular velocity (see Figure 10). Weattribute this di�erence to the fact that Dubinski &Carlberg stopped their simulations at z = 1; their runcannot therefore be considered representative of darkhalos identi�ed at z = 0.9

We conclude that, although the cores of dwarfgalaxies pose a signi�cant problem for CDM, theproblem is not as bad as previously thought. Pertur-bations to the central regions of dwarf galaxy halos,resulting perhaps from the sudden loss of a large frac-tion of the baryonic material after a vigorous bout ofstar formation (Dekel and Silk 1986), can in principlereconcile the observations of dwarfs with the structureof CDM halos (Navarro, Eke & Frenk 1995).5.4. The abundance of galactic dark halos.A problem which a�icts all current models ofgalaxy formation based on the standard CDM cos-mogony is related to the high abundance of galacticdark halos (White & Frenk 1991, Kau�mann et al.1993, Cole et al. 1994). Indeed, dark halos of galac-tic size are so numerous in this scenario that it isimpossible to �t simultaneously the observed Tully-Fisher relation and the galaxy luminosity function.This can be shown following the argument of Kau�-mann et al. (1993). Let us assume that there is onlyone galaxy per halo, a conservative assumption sincewe know that the halos of galaxy groups and clusterscontain several galaxies. We can then use the Press-Schechter (1974) theory to compute the number den-sity of galaxies as a function of the circular velocity oftheir surrounding halos, V200 (see, for example, eq. 5of White & Frenk 1991). Assuming that we can inferthe rotation velocity of the disk (Vopt) from V200, theTully-Fisher relation allows us to compute the con-tribution of galaxies with a given rotation velocity tothe mean luminosity density of the universe.Using the B-band Tully-Fisher relation presentedin x5:1, and assuming that the mean luminosity den-sity of the universe is LB = 9:7 � 107L� Mpc�3(Efstathiou et al. 1988b, Loveday et al. 1992), weshow in Figure 13 the cumulative fraction of the totalluminosity density contributed by galaxies with ve-locities larger than Vopt, under the assumption thatV200 = Vopt (solid line). In order to be conserva-tive, we have assumed in this plot that halos withVopt > 300 km/s do not contribute to the luminositydensity at all, thus excluding groups and clusters ofgalaxies from the count. As noted by Kau�mann etal. (1993), even with this restriction the cumulativeluminosity is too large for Vopt � 100 km/s. This im-plies that it is impossible to match the Tully-Fisherrelation and the galaxy luminosity density simultane-ously unless our assumption that each halo containsone galaxy is incorrect.

According to the discussion in x5:1, the mean cir-cular velocity of CDM halos, V200, should actually belower than Vopt in order to account for the shape ofthe rotation curves of disk galaxies. A simple approx-imation to the relationship between halo and galaxycircular velocity isV200 = F (Vopt) = Vopt1 + (Vopt=300 km=s) (10)(see Figure 11). The result of this identi�cation isshown as a dotted line in Figure 13. Clearly, assigninglower circular velocities to the halos of galaxies of agiven luminosity only exacerbates the problem. Thereis now too much luminosity in the standard CDMcosmogony for Vopt � 200 km/s.We conclude that this remains a serious problem.Although it can in principle be alleviated by adoptinga di�erent cosmogony (for example, an open universeor a universe with cold and hot dark matter compo-nents), the most popular forms of these alternativemodels do not readily overcome the di�culty (Heylet al. 1995). Another alternative would be to as-sume that a large number of halos have failed to formgalaxies at all, or that the galaxies they host have sofar escaped detection. The large numbers of systemsbeing detected in systematic surveys for low surfacebrightness galaxies suggest that the existence of sucha population of galaxies cannot be ruled out (see, e.g.Sprayberry et al. 1993, Ferguson & McGaugh 1995).5.5. Core properties of galaxy clusters.We now examine the consequences of the structureof CDM halos discussed in x4 for X-ray and gravita-tional lensing observations of galaxy clusters. An on-going debate concerns the di�ering estimates of coreradius obtained from models of the X-ray emittinggas and from attempts to reproduce the giant arcsobserved in many clusters.The X-ray emitting intracluster medium is oftenapproximated by the hydrostatic, isothermal �-model(Cavaliere & Fusco-Femiano 1976). The density pro-�le of the X-ray emitting gas is then of the form,�ICM / (1+ (r=rcore)2)�3�=2. The core radius, rcore,is usually a sizeable fraction of the total extent ofthe emission, and � is typically found to be � 0:6-0:8 (see Jones & Forman 1984). The halo mass pro-�le can then be derived by assuming that the gas isisothermal and in hydrostatic equilibrium; it followsthe same law as the gas, with the same core radius10

and with outer slope parameter �DM = �=�T , where�T measures the ratio of the \temperatures" of thedark matter and of the gas, �T = �mp�2DM=kTgas(�mp is the mean molecular weight of the gas andk is Boltzmann's constant). This model is frequentlyused to interpret X-ray observations and to constrainthe dark matter distribution near the center of clus-ters (Jones & Forman 1984, Edge & Stewart 1991).Cooling ow models with a �-model mass pro�le havebeen successful at explaining a number of observa-tions, including the central X-ray surface brightness\excess" and the drop in central temperature seen insystems with strong cooling ows. The core radiusof the dark matter is typically inferred to be in therange rc � 100-200 kpc (Fabian et al. 1991).On the other hand, such large cores are ruled outby observations of giant arcs, which require core radiiof order 20-60 kpc when a �-model is used to describethe lensing cluster (Grossman & Narayan 1988). Thisdiscrepancy is resolved if we assume instead that clus-ter halos follow the density pro�le of eq.(3). For theparameters suggested by Figures 7 and 8, halos aresu�ciently concentrated to agree with the gravita-tional lensing constraints. A CDM cluster with meanvelocity dispersion � 1000 km/s placed at z = 0:3can produce giant arcs similar to those seen (Wax-man & Miralda-Escud�e 1995) yet it can also be con-sistent with a large core radius in the X-ray emit-ting gas. Requiring that the ICM be isothermaland in hydrostatic equilibrium in the dark matterpotential results in a well de�ned core. This is il-lustrated in Figure 14, where we plot density pro-�les for an isothermal gas (dotted line) and for thedark matter (solid line). Radii are given in units ofrmax and the density units are arbitrary. We assumethat the gas and dark matter temperatures are equal,kTgas=�mp = �2DM = GM200=2r200.The di�erence in shape between the gas and thedark matter is due to our assumption that the gas isisothermal. Since the dark matter velocity dispersiondrops at radii larger and smaller than rmax (see Fig-ures 5 and 6), the gas structure deviates strongly fromthat of the dark matter at small and large radii. Atsmall radii an isothermal gas develops a well de�nedcore, and at large radii its density drops less rapidlythan the dark matter. Such leveling o� of the gas pro-�le is not observed in real clusters, indicating that thegas temperature must decrease in the outer regions.Hydrodynamical simulations con�rm that this is thecase, and indicate that the gas and dark matter distri-

butions in the outer regions are very similar (Navarro,Frenk & White 1995b). The dashed line shows theresult of �tting the gas pro�le with the � model men-tioned above. In this case, values of rcore = 0:1�rmaxand � = 0:7 give a very good �t to the structure of thegas. For a cluster with a velocity dispersion of 1000km/s, this implies a gas core radius of about 120 kpc.Detailed cooling ow models that use the poten-tial corresponding to eq.(3) rather than the �-modelcan agree well with observation. A recent analysis byWaxman& Miralda-Escud�e (1995) shows that the ob-servational signatures of cooling ows in CDM halosare essentially indistinguishable from those occurringin halos with a true constant density core. We con-clude that the apparent discrepancy between X-rayand gravitational lensing estimates of the core radiusis a direct result of force-�tting a �-model to systemswhose structure is better described by a pro�le moresimilar to that of our CDM halos.6. ConclusionsThe main conclusions of this paper can be summa-rized as follows.1) The density pro�les of CDM halos of all masses canbe well �t by an appropriate scaling of a \univer-sal" pro�le with no free shape parameters. Thispro�le is shallower than isothermal near the centerof a halo, and steeper than isothermal in its outerregions.2) The characteristic overdensities of halos, or equiv-alently their concentrations, correlate with halomass in a way which can be interpreted as re- ecting the di�erent formation redshifts of halosof di�ering mass.3) The observed rotation curves of disk galaxies arecompatible with this halo structure provided thatthe mass-to-light ratio of the disk increases withluminosity. This implies that the halos of brightspirals have masses which correlate only weaklywith their luminosity, and may explain why lumi-nosity and dynamics appear uncorrelated in sam-ples of binary galaxies and of satellite/spiral pairs.Disks with rotation velocity in the range 200 to300 km/s are predicted to have halos with a typ-ical mass of 1:8 � 1012M� within 300 kpc. Thisagrees well with the masses inferred from the dy-namics of observed satellite galaxy samples.4) CDM halos seem too centrally concentrated to beconsistent with observations of the rotation curves11

of dwarf irregulars (Moore 1994, Flores & Primack1994). This may imply that the central regionsof dwarf galaxy halos were substantially alteredduring galaxy formation, for example by suddenbaryonic out ows occurring after a burst of starformation (Navarro, Eke & Frenk 1995.)5) The fact that bright galaxies are surrounded by ha-los with mean circular velocity lower than the ob-served disk rotation velocity exacerbates the dis-crepancy between the number of \galaxy" halospredicted in an = 1 universe and the observednumber of galaxies.6) The predicted structure of galaxy clusters is consis-tent both with X-ray observations of the ICM andwith the presence of giant gravitationally lensedarcs. Previous discrepant estimates of the core ra-dius based on these two kinds of data probablyresult from force-�tting of an inappropriate po-tential structure. A singular pro�le such as thosefavoured by our simulations can be consistent withall current data.We would like to thank the hospitality of the In-stitute for Theoretical Physics of the University ofCalifornia at Santa Barbara, where most of the workpresented here was carried out. JFN would also liketo thank the hospitality of the Max Planck Insti-tut f�ur Astrophysik in Garching, where this projectwas started, as well as to acknowledge useful discus-sions with C. Lacey, N. Katz, and B. Moore. Thiswork has been supported in part by the U.K. PPARCand the National Science Foundation under grant No.PHY94-07194 to the Institute for Theoretical Physicsof the University of California at Santa Barbara.12

Label Lbox 1 + zi hg M200 r200 V200 N200 rs=r200[Mpc] [kpc] [1012M�] [kpc] [km/s]1 3.0 44.0 1.5 0.319 177 88.0 5637 0.0522 3.0 44.0 1.5 0.293 172 85.6 5178 0.0573 3.0 44.0 1.5 0.414 193 96.1 7316 0.0824 3.0 44.0 1.5 0.525 209 103.9 9278 0.0465 6.0 22.0 3.0 2.425 348 173.1 5357 0.0606 6.0 22.0 3.0 2.301 342 170.1 5083 0.0717 6.0 22.0 3.0 3.519 394 196.0 7774 0.0688 6.0 22.0 3.0 2.552 354 176.1 5637 0.1249 12.0 11.0 6.0 28.15 788 392.0 7773 0.12410 12.0 11.0 6.0 20.59 710 353.2 5686 0.07811 12.0 11.0 6.0 29.67 802 398.9 8193 0.13112 12.0 11.0 6.0 22.01 726 361.1 6078 0.08813 12.0 11.0 6.0 26.16 769 382.5 7224 0.07714 12.0 11.0 6.0 22.65 733 364.6 6254 0.06515 18.0 5.5 9.0 102.68 1213 603.4 8401 0.11016 24.0 5.5 12.0 224.35 1574 783.0 7744 0.12117 40.0 5.5 20.0 1109.9 2682 1334 8274 0.15118 48.0 5.5 24.0 1931.5 3226 1605 8334 0.18819 58.0 5.5 30.0 3009.7 3740 1861 7360 0.143Table 1: Parameters of the numerical experiments.13

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This 2-column preprint was prepared with the AAS LATEXmacros v4.0.15

Fig. 1a.| The e�ects of varying the initial redshift onthe density pro�le of simulated halos. The masses ofthe two halos shown are � 1011 and � 1015M�. Thegravitational softening and the virial radius are in-dicated with upward and downward pointing arrows,respectively. Solid, dotted, and dashed lines corre-spond to 1 + zi = 22:0, 11:0, and 5:5, respectively. Fig. 1b.| Circular velocity pro�les for the halosshown in Fig. 1a. Arrows are also as in Fig. 1a.For 1 + zi = 5:5, the circular velocity near the centerof the small halo is substantially underestimated. Forlarger zi, the models converge to a unique pro�le.16

Fig. 2a.| Density pro�les of the same halos shownin Figure 1 for di�erent choices of the gravitationalsoftening, hg. Upward-pointing arrows indicate thesoftening values. Downward pointing arrows indicatethe virial radius. Varying the softening by a factor of�ve has no signi�cant e�ect on the structure of thehalo beyond one gravitational softening length.Fig. 2b.| Circular velocity pro�les for the halosshown in Figure 2a. Arrows are also as in Figure 2a.Di�ering gravitational softening lengths have no sig-ni�cant e�ect on the structure of the halo on regionslarger than hg.Fig. 3.| Density pro�les of four halos spanning fourorders of magnitude in mass. The arrows indicate thegravitational softening, hg, of each simulation. Alsoshown are �ts from eq.3. The �ts are good over twodecades in radius, approximately from hg out to thevirial radius of each system.17

Fig. 4.| Scaled density pro�les of the most and leastmassive halos shown in Figure 3. The large halo is lesscentrally concentrated than the less massive system.Fig. 5.| Circular velocity pro�les of all 19 halos.The pro�les are truncated at the virial radius, r200.The gravitational softening is about 10�2�r200. Notethat all pro�les have the same shape.

Fig. 6.| Scaled circular velocity pro�les of two ha-los, one of the largest and one of the smallest in oursample (solid curves). The dashed lines are �ts witheq.(5). The dotted line is a �t to the low mass sys-tem using a Hernquist model (see eq.6). Note thatthe Hernquist model underestimates the halo circularvelocity at r200.18

Fig. 7.| The characteristic overdensity �c as a func-tion of the mass of the halo. The curves show themass-overdensity relation predicted from the forma-tion times of halos (see text). All curves are normal-ized so that they cross at M200 = M?.Fig. 8.| The concentration c as a function ofthe mass of the halo. The curves show the mass-concentration relation predicted from the formationtimes of halos. All curves are as in Figure 7, and havebeen normalized so that they cross at M200 = M?.

Fig. 9.| The ratio between the maximum circularvelocity of a halo and the value at r200 as a func-tion of the circular velocity at the virial radius. Lowmass systems are more concentrated than more mas-sive ones. The maximum velocity can be up to twicethe value at r200 for a small galaxy halo.19

Fig. 10.| The maximum circular velocity as a func-tion of the radius at which it is attained in halos ofdi�erent mass. Note that a halo with Vmax = 220km/s has a rising rotation curve that extends out toabout 50 kpc, well beyond the luminous radius of agalaxy like the Milky Way. Fig. 11.| Rotation curves of disk+halo systems(solid lines) with parameters chosen to match the ob-servational data of Persic & Salucci (1995) (dottedlines). The dashed lines indicate the contribution ofthe dark halo. Note that the disk mass-to-light ra-tio increases as a function of mass, and that the halocontribution is less important in bright galaxies.20

Fig. 12.| The circular velocities of CDM halos (dot-ted lines) compared with the halo contribution to therotation curve of four dwarf galaxies (solid lines). Thesolid lines encompass the likely contribution of thehalo, and correspond to the \maximal" and \mini-mal" disk hypotheses. CDM halos seem to be signif-icantly more concentrated than allowed by observa-tions. Fig. 13.| The cumulative B-band luminosity den-sity in galaxies with optical rotation velocities largerthan Vopt as predicted by our standard CDM model.We use the Tully-Fisher relation, and assume thatthe abundance of galaxies with rotation velocity Voptmatches that of halos with V200 = Vopt (solid line), orV200 = F (Vopt) as given in the text (dotted line).21

Fig. 14.| The density pro�le of an isothermal gas(dotted line) in hydrostatic equilibriumwithin a CDMhalo whose structure is given by eq.(3) (solid line).The dashed line indicates a �t using the � model.The parameters rc = 0:1 rmax and � = 0:7 give anexcellent �-model �t to the gas pro�le.22