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5
The Structure of Cold Dark Matter Halos
Julio F. Navarro
1
Steward Observatory, The University of Arizona, Tucson, AZ, 85721, USA.
Carlos S. Frenk
Physics Department, University of Durham, Durham DH1 3LE, England.
Simon D.M. White
Max Planck Institut fur Astrophysik, Karl-Schwarzschild Strasse 1, D-85740, Garching, Germany.
ABSTRACT
We use N-body simulations to investigate the structure of dark halos in the standard
Cold Dark Matter cosmogony. Halos are excised from simulations of cosmologically rep-
resentative regions and are resimulated individually at high resolution. We study objects
with 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 in
radius by scaling a simple \universal" prole. The characteristic overdensity of a halo, or
equivalently its concentration, correlates strongly with halo mass in a way which re
ects
the mass dependence of the epoch of halo formation. Halo proles are approximately
isothermal over a large range in radii, but are signicantly shallower than r
2
near the
center and steeper than r
2
near the virial radius. Matching the observed rotation
curves of disk galaxies requires disk mass-to-light ratios to increase systematically with
luminosity. Further, it suggests that the halos of bright galaxies depend only weakly on
galaxy luminosity and have circular velocities signicantly lower than the disk rotation
speed. This may explain why luminosity and dynamics are uncorrelated in observed
samples of binary galaxies and of satellite/spiral systems. For galaxy clusters, our halo
models 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 disparate
estimates of cluster core radii found by previous authors. Our results also highlight two
shortcomings of the CDM model. CDM halos are too concentrated to be consistent with
the halo parameters inferred for dwarf irregulars, and the predicted abundance of galaxy
halos is larger than the observed abundance of galaxies. The rst problem may imply
that 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.
1
Bart J. Bok Fellow.
1

1. Introduction
Analytic calculations and numerical simulations
both suggest that the density proles of dark mat-
ter halos may contain useful information regarding
the cosmological parameters of the Universe and the
power spectrum of initial density
uctuations. For
example, the secondary infall models of Gunn & Gott
(1972) established that gravitational collapse could
lead to the formation of virialized systems with al-
most isothermal density proles. This result, fur-
ther rened by Fillmore & Goldreich (1984) and
Bertschinger (1985), suggested a cosmological signi-
cance for the observation that the rotation curves of
galactic disks are
at.
This analytic work was extended by Homan &
Shaham (1985) and Homan (1988), who pointed out
that the structure of halos should also depend on the
value of the density parameter
and the power spec-
trum of initial density
uctuations. Studying scale-
free spectra, P (k) / k
n
(3 < n < 4), these authors
concluded that halo density proles should steepen
for larger values of n and lower values of
. Approxi-
mately
at circular velocity curves were predicted for
2 < n < 1 and
 1. These conclusions were
con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 emerging
Cold Dark Matter scenario, since in this model the ef-
fective slope of the power spectrum on galactic mass
scales is n
eff
 1:5 (Blumenthal et al. 1984, Davis
et al. 1985).
It has become customary to model virialized halos
by isothermal spheres characterized by two param-
eters; a velocity dispersion and a core radius. For
galaxies, the halo velocity dispersion (or its circular
velocity, dened by V
2
c
= GM=r) is often assumed to
be directly proportional to the characteristic velocity
of the observed galaxy. This assumption allows ob-
servations to be compared directly to the results of
cosmological 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,
Kaumann 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 with
brighter 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 rotation
velocity of the primary's disk. No such correlations
are apparent in existing data (see, e.g. White et al.
1983, Zaritsky et al. 1993). A possible explanation
may come from the work of Persic & Salucci (1991),
who argue that halo circular velocity is only weakly
related to disk rotation speed.
Observational estimates of the core radii of dark
halos have also led to con
icting results. Large core
radii have been advocated in order to accommodate
the contribution of the luminous component to the
rotation curves of disk galaxies (Blumenthal et al.
1986, Flores et al. 1993); to account for the shape
of 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 et
al. 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 be
small (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, has
been challenged by recent high resolution numerical
work. 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 & Carlberg
1991, Warren et al. 1992, Crone, Evrard & Rich-
stone 1994, Carlberg 1994, Navarro, Frenk & White
1995b). These studies indicate that dark matter halos
are not well approximated by isothermal spheres, but
rather have gently changing logarithmic slopes as in
the model proposed by Hernquist (1990) for elliptical
galaxies,
(r) /
1
r(1 + r=r
s
)
3
; (1)
or that proposed by Navarro, Frenk & White (1995b)
for X-ray cluster halos,
(r) /
1
r(1 + r=r
s
)
2
: (2)
These proles are singular (although the potential
and mass converge near the center), and possess a
well dened scale where the prole changes shape,
the scale radius r
s
. (We will refer to r
s
as a \scale"
radius and reserve the term \core" to refer to a cen-
tral region of constant density.) Near the scale radius
the proles are almost isothermal, so these results are
2

quite consistent with the lower resolution simulations
mentioned earlier.
The possibility that dark matter density proles
may diverge like r
1
near the center has important
consequences for the observational issues discussed
above. In this paper, we present the results of a large
set of high resolution numerical simulations aimed at
understanding the density proles of dark halos in
the standard CDM cosmogony. Although this partic-
ular cosmogony has fallen into some disrepute since
the COBE measurement of temperature
uctuations
in the microwave background radiation, it is still the
prime example of the general class of hierarchical clus-
tering theories. Many of the conclusions reached from
studying this model are directly applicable to other
models of structure formation. Indeed, preliminary
results 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 is
particularly 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 these
regions are the most vulnerable to systematic errors
introduced by nite resolution eects. It is particu-
larly important to ensure that all halos, irrespective of
mass, are simulated with comparable numerical reso-
lution. Halos identied in a single cosmological simu-
lation are inevitably resolved to varying degrees since
massive systems contain more particles than less mas-
sive ones. We circumvent this problem by selecting
halos with a wide range of masses from cosmologi-
cal simulations of large regions and then resimulating
them individually at higher resolution. We present
details of our numerical procedures in x2, and in x3
we test how their limitations aect the structure of
our halos. Section 4 presents results for an ensemble
of simulations spanning four orders of magnitude in
halo mass. Section 5 discusses some implications of
our results, and x6 summarizes our conclusions.
2. The Numerical Experiments
We simulate the formation of 19 dierent systems
with scales ranging from those of dwarf galaxies to
those of rich clusters. The systems were identied in
two large cosmological N-body simulations of a stan-
dard
= 1 CDM universe carried out with a P
3
M
code (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, and
were stopped when the rms
uctuation in spheres of
radius 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 constant
of 50 km/s/Mpc.) At this epoch, which we identify
with 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 the
mean interparticle separation. The centers of mass of
these clumps were used as halo centers and the radius
of the sphere encompassing a mean overdensity of 200
was calculated in each case. Hereafter, we shall refer
to this as the \virial" radius of the system and de-
note it by r
200
. After sorting the clumps by the mass,
M
200
, contained within these spheres, we picked 19 of
them covering a range of four orders of magnitude in
mass ( 3 10
11
< M
200
=M

< 3 10
15
). We did
not select randomly from this mass range, but chose
instead halos grouped roughly into three bins of cir-
cular velocity,  100 km s
1
, 250 km s
1
and > 450
km s
1
. This gives some indication of the \cosmic
scatter" in the properties of halos of a given mass.
The particles in each of the 19 selected systems
were then traced back to the initial time, where a
box containing all of them was drawn. This box was
loaded with 32
3
particles on a cubic grid which was
perturbed using the same waves as in the original sim-
ulation, together with additional waves to ll out the
power spectrum to the Nyquist frequency of the new
particle grid. The size, L
box
, of this \high-resolution"
box was chosen so that all systems had, at z = 0,
approximately the same number of particles within
the virial radius, typically 5; 000-10; 000. The gravi-
tational softening was chosen to be one percent of the
virial radius, and was kept xed in physical coordi-
nates. (Note that some of the tests described in x3
have fewer particles; for these runs the gravitational
softening was slightly larger.) The initial redshift of
each run was chosen so that the median displace-
ment of particles within the \high-resolution" box
was smaller than the initial particle gridsize. These
choices ensure that all halos were resolved to a similar
degree 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 throughout
the text.
Tidal eects due to distant material are repre-
sented by using several thousand massive particles
to coarse-sample the region surrounding the \high-
3

resolution" box. This procedure for setting up initial
conditions is identical to that described in Navarro,
Frenk & White (1995a,b), where further details may
be found. All simulations were run with the N-
body code described by Navarro & White (1993); a
second-order accurate, nearest-neighbor binary-tree
code, with individual particle timesteps. Typically,
the minimum timestep is between 10
4
and 10
5
of a Hubble time, low mass systems requiring more
timesteps than more massive ones due to their higher
central densities.
3. Eects of numerical limitations
As mentioned in x1, numerical limitations can in-
uence the structure of model halos. The number of
particles, the gravitational softening, the initial red-
shift, and the timestep can all, in principle, have a
signicant eect. It is important to check explicitly
that our results are insensitive to the particular choice
of numerical parameters we have made. For each of
the tests described below we chose one of the least
massive and one of the most massive halos in our se-
ries, and we resimulated them varying the numerical
parameters in a systematic fashion. Our conclusion is
that, when chosen carefully, numerical parameters do
not aect 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 suciently large
number of particles (

>
50).
3.1. The initial redshift
Figures 1a and 1b show the eect of varying the ini-
tial redshift, z
i
, of the simulation for two of our halos,
a large halo of mass M
200
 10
15
M

, corresponding
to a rich galaxy cluster, and a small halo of mass
M
200
 10
11
M

, corresponding to a dwarf galaxy.
These models had only 22
3
particles within the \high-
resolution" box for CPU economy reasons. The pan-
els show the density prole and the circular velocity
as a function of radius. The proles are spherically
averaged over bins containing about 30 particles each.
Three dierent choices of z
i
were tried for the small
halo, 1 + z
i
= 5:5, 11:0, and 22:0, and two for the
large system, 1 + z
i
= 5:5 and 11:0. The density
proles show little change, although for 1 + z
i
= 5:5
the central density of the small halo appears to be
substantially underestimated. Since it is a cumulative
quantity, the circular velocity prole is more sensitive
to variations near the center. Figure 1b shows that
starting at 1 + z
i
= 5:5 results in a reduced circular
velocity near the center but, as the starting redshift
is increased, the proles converge to a unique shape
so that for 1 + z
i
= 11:0 and 22:0 the two curves are
indistinguishable. In the case of the massive system,
starting at 1 + z
i
= 5:5 or at 11:0 had no signicant
eect.
The importance of the initial redshift may be un-
derstood by considering the initial displacement eld
of the particles given by the Zel'dovich approxima-
tion. At 1 + z
i
= 5:5, the median particle displace-
ment in the small halo simulation was about 4 times
the mean interparticle separation. Thus, the
uc-
tuations are not linear on small scales in the initial
conditions and the Zel'dovich approximation breaks
down. The excess kinetic energy imparted to the par-
ticles prevents the formation of dense clumps in the
early stages of the simulation, reducing the maximum
density of the nal system. The median displace-
ment is less than the mean interparticle separation
at 1 + z
i
= 22:0, and less than twice the mean inter-
particle separation at 1 + z
i
= 11:0. In these cases,
use of the Ze'ldovich approximation is justied, and
the results converge to a unique solution.
In the case of the massive halo, even at 1+z
i
= 5:5
the median displacement was only about twice the
mean interparticle separation. As a result starting
this \late" had a negligible eect on the nal prole.
We conclude that as long as the initial ratio between
the median particle displacement and the mean in-
terparticle separation is of order unity (or less) the
results are insensitive to the particular choice of z
i
.
The initial redshifts of all runs (quoted in Table 1)
satisfy this condition.
3.2. The gravitational softening
The gravitational softening, h
g
, is included in the
N-body equations of motion in order to suppress re-
laxation eects due to two-body encounters. This
is accomplished if h
g
is set signicantly larger than
the impact parameter for a large angle de
ection
in a typical 2-body encounter,  Gm=
2
, where m
is the mass of a particle and  the velocity dis-
persion of the system (White 1979). Since 
2

GM
200
=2 r
200
= GmN
200
=2 r
200
, then h
g
should be
larger than 2 r
200
=N
200
. (Here N
200
is the total num-
ber of particles within the virial radius r
200
and we
have assumed that all particles have the same mass.)
In our simulations N
200
is typically of order a few
thousand, so choosing h
g
of the order 10
2
 r
200
4

should be safe.
Figures 2a and 2b show that varying the softening
length by up to a factor of 5 (within the constraint
mentioned above) has little impact on the nal struc-
ture of the halos outside about one softening length.
Three dierent values of h
g
were tried for the large
halo and two for the small system. The gures dis-
play the density and circular velocity proles of the
same two halos used in the previous subsection. The
number of particles within the \high-resolution" box,
however, was increased to 32
3
, and the initial redshift
was also xed at 1 + z
i
= 22:0 in both cases.
Note that although h
g
< 10
2
r
200
could be used,
too small a value of h
g
would be counterproductive.
Smaller softenings require smaller timesteps for the
same accuracy, with a consequent increase in CPU
time consumption. For the test runs in Figure 2, the
number of timesteps increased roughly as h
1=2
g
. Thus,
the experiments illustrated in this gure also show
that the nal halo structures are independent of the
number of timesteps in the simulation, typically over
10; 000 timesteps per run.
Finally, comparison of Figures 1 and 2 shows that
increasing the number of particles by a factor of three
has no noticeable eect on the structure of the halos,
except that larger values of N allow us to determine
the 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 the
structure of halos over approximately two decades in
radius, between r
200
and 10
2
 r
200
.
4. Results
4.1. Density Proles
Figure 3 shows the density proles of four dark ha-
los of dierent mass. Halo mass increases from left
to right, from  3  10
11
M

for the smallest sys-
tem, to  3  10
15
M

for the largest. The arrows
indicate the gravitational softening in each simula-
tion. In all cases, the virial radius is about two orders
of magnitude larger than the softening length. The
smooth curves represent ts to the simulationdata us-
ing a model of the form proposed by Navarro, Frenk
& White (1995b),
(r)

crit
=

c
(r=r
s
)(1 + r=r
s
)
2
; (3)
where r
s
= r
200
=c is a characteristic radius an 
crit
=
3H
2
=8G is the critical density (H is the current
value of Hubble's constant); 
c
and c are two dimen-
sionless parameters. Note that r
200
determines the
mass of the halo, M
200
= 200
crit
(4=3)r
3
200
, and that

c
and c are linked by the requirement that the mean
density within r
200
should be 200 
crit
. That is,

c
=
200
3
c
3

ln(1 + c) c=(1 + c)


: (4)
We will refer to 
c
as the characteristic overdensity
of the halo, to r
s
as its scale radius, and to c as its
concentration.
A striking feature of Figure 3 is that the same pro-
le shape provides a very good t to all the halos even
though they span nearly four orders of magnitude in
mass. The agreement holds for radii ranging from
the 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 so
that the density is given in units of the critical density
and the radius in units of the virial radius. Although
we show proles only for a few halos in Figures 3 and
4, these are by no means special. Eq. (3) ts all our
halos almost equally well, irrespective of mass.
4.2. Circular Velocity Proles
The circular velocity prole, V
c
(r) = (GM (r)=r)
1=2
,
contains the same information as the density prole
but is less noisy because it is a cumulative measure
of the radial structure of a halo. Circular velocity
curves for all 19 systems are shown in Figure 5. They
all 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 nally
declines 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. This
is more easily seen in Figure 6, where for two ha-
los we scale circular velocity by the value at r
200
,
V
200
= (GM
200
=r
200
)
1=2
 (1=2)(r
200
=kpc) km/s,
and radii by r
200
itself. Again, we choose the least and
most massive systems to illustrate the trend. Note
(i) that the maximum circular velocity of each halo,
V
max
, is larger than V
200
(by up to 40 per cent); (ii)
5

that the ratio V
max
=V
200
is larger for low mass halos;
and (iii) that the radius r
max
at which the peak occurs
is a larger fraction of the virial radius for more mas-
sive systems. This is an alternative way of expressing
the fact that low mass systems are signicantly more
concentrated than high mass ones (see Figure 4).
The dashed lines in Figure 6 are ts to the circular
velocity curve predicted from eq.(3);

V
c
(r)
V
200

2
=
1
x
ln(1 + cx) (cx)=(1 + cx)
ln(1 + c) c=(1 + c)
; (5)
where x = r=r
200
is the radius in units of the virial
radius. Note that for this model the circular velocity
peaks at r
max
 2r
s
= 2r
200
=c.
The dotted line in Figure 6 is a t to the circular
velocity curve of the small halo using the Hernquist
(1990) model. The circular velocity in this model is
given by

V
H
(r)
V
max

2
=
4(r=r
max
)
(1 + r=r
max
)
2
: (6)
Note that although in the inner regions (for overden-
sities larger than  1000) this model provides as good
a t to the data as our adopted prole (eq. 3), it pre-
dicts circular velocities that are too low near the virial
radius.
4.3. Mass dependence of halo properties
As shown in Figures 4 and 6, low mass CDM halos
are more centrally concentrated than high mass ones.
This is not surprising because lower mass systems
generally collapse at higher redshift, when the mean
density of the universe was higher. In the spherical
top-hat model, the postcollapse density is a constant
multiple of the mean cosmic density at the time of col-
lapse. To understand the relation between halo mass
and characteristic density requires a formal denition
of the time of formation. This is not straightforward
because systems are continually evolving, accreting
mass through mergers with satellites and neighbor-
ing clumps. One possibility is to dene the formation
time of a halo as the rst time when half of its nal
mass M was in progenitors with individual masses ex-
ceeding some fraction f of M . With this denition,
the typical formation redshifts of halos of diering
mass can be predicted analytically.
Lacey & Cole (1993) show that a randomly chosen
mass element from a halo of mass M identied at red-
shift z
0
was part of a progenitor with mass exceeding
fM at the earlier redshift z with probability
P (> fM; zjM; z
0
) = erfc


0
(z z
0
)(1 + z
0
)
p
2(

2
0
(fM )

2
0
(M ))

;
(7)
where

2
0
(M ) is the variance of the linear power spec-
trum at z = 0 when smoothed with a top hat lter en-
closing mass M (see, for example, eqs. (6) and (7) of
White & Frenk (1991) for a denition), and 
0
= 1:69
is the usual critical linear overdensity for top-hat col-
lapse. This equation is valid for z > z
0
and f < 1.
The formation redshift z
form
(M; f; z
0
) is then dened
by setting P = 1=2 in eq. (7). Lacey & Cole (1994)
tested this formula against N-body simulations for the
case 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 universe
at that redshift provides a natural scaling parameter.
Let us assume that the characteristic overdensity of a
halo scales as

c
(M; f; z
0
) = C(f)(1 + z
form
(M; f; z
0
))
3
: (8)
Figure 7 shows the characteristic density, 
c
(ob-
tained by tting eq. 3), as a function of halo mass
expressed in units of the non-linear mass, M
?
(z
0
).
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
?
' 3:310
13
M

at z = 0. The curves in Figure 7 show the predictions
of eq. (8) for various choices of f . The normaliza-
tions have been chosen so that the curves all cross at
M
200
= M
?
; this implies C(f) = 1:5 10
4
, 5:4 10
3
,
and 2:0 10
3
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 the
mass-density relation predicted by eqn. (8) and our
N-body results is fair for f = 0:5, and improves for
smaller values of f . Thus the characteristic density of
a 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 gure
contains the same information as Figure 7 (since 
c
and c are related by eqn. 4), it shows explicitly that
large halos are signicantly less concentrated than
small ones. Note that the dependence of concentra-
tion on mass is quite weak; c changes by less than a
factor of four while M varies by four orders of magni-
tude. Thus, halos diering in mass by less than, say,
one order of magnitude, have density proles that are
almost indistinguishable in the scaled variables used
6

in Figure 4. This conrms the results we presented
in Navarro, Frenk & White (1995b), where we found
that galaxy cluster halos in the CDM cosmogony are
well approximated by eq. (3) with c  5. The present
higher resolution simulations show that such a weak
concentration is appropriate only for the largest halos,
M
200
=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 of
about 100 kpc for all systems and of a rather \late"
starting redshift, z
i
= 3:74, to give concentration es-
timates which are slightly lower than those shown in
Figure 8.
Figure 9 shows that the maximum circular veloc-
ity, V
max
, can dier by up to 40 percent from the
circular velocity at the virial radius, V
200
. The ratio
V
max
=V
200
is an interesting quantity to consider be-
cause many analytic and numerical studies use V
200
to characterize the statistical properties of the halo
population and to compare them with the observed
properties of the galaxy population. For example, it is
often assumed that the rotation velocity of a galactic
disk is identical to V
200
, an assumption that clearly
breaks down if the circular velocity of a halo varies
with radius as shown in Figures 5 and 6.
Finally, Figure 10 shows the correlation between
the maximum circular velocity, V
max
, and the radius,
r
max
, at which the circular velocity attains that max-
imum. Note that for V
max
= 220 km/s, the rotation
velocity of the MilkyWay, r
max
is about 50 kpc, larger
than the typical optical size of galaxies like the Milky
Way. Thus, if the dark halo of the Milky Way resem-
bles our model halos, the observed
at rotation curve
must be the result of the dissipational collapse of the
luminous component rather than a direct re
ection of
the structure of the dark halo. We discuss further the
implications of these results in the following section.
5. Discussion
5.1. The shapes of the rotation curves of disk
galaxies.
The shapes of the rotation curves of disk galaxies
have 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 the
luminous radius, but there is good evidence now that
the rotation curves of less luminous galaxies tend to
be rising whilst those of their brighter counterparts
tend to be gently declining. These systematic trends
have been conrmed in a recent analysis of the large
dataset of Matthewson et al. (1992) by Persic &
Salucci (1995). They nd that the optical radius
of a galaxy, r
opt
, and the rotation velocity at r
opt
,
V
opt
, are strongly correlated, r
opt
 20(V
opt
=200 km
s
1
)
1:16
kpc. (The optical radius is dened as the
radius 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, r
opt
corresponds to the 25 mag
B
/arcsec
2
isophotal ra-
dius.) The total B-band luminosity of a galaxy is also
strongly correlated with V
opt
via the Tully-Fisher re-
lation, L
B
 3:310
9
(V
opt
=100 km s
1
)
2:7
L

(Pierce
& Tully 1988). All these correlations exhibit rather
small scatter.
We now consider the implications of our results on
the 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 need
to specify three parameters: the mass of the disk,
M
disk
, its exponential scalelength, r
disk
, and the mass
of the halo, M
200
. (We ignore the contribution of the
galaxy's spheroid in this simplied model.) Accord-
ing to the observed correlations mentioned above, V
opt
uniquely determines the luminosity and scalelength of
the disk. Requiring that our model should also exhibit
these correlations then reduces the number of param-
eters to two and these may be taken to be the disk
mass-to-light ratio, (M=L)
disk
, and the halo mass.
These two parameters may be xed by matching the
amplitude of the observed rotation curve, V
opt
, and
its slope at the optical radius.
To carry this program through, we need to allow for
the fact that the halo will respond to disk formation
by adjusting slightly according to the mass and radius
of the disk. We assume that the disk is assembled
slowly and that the halo is adiabatically compressed
during this process (Blumenthal et al. 1986, Flores
et al. 1993). In this approximation, the radius, r, of
each halo mass shell after the assembly of the disk is
related to its initial radius, r
i
, by
r [M
disk
(r) +M
halo
(r)] = r
i
M
i
(r
i
): (9)
Here M
i
(r
i
) is the mass within radius r
i
before disk
formation (found by integrating eqn. 3), M
disk
(r) is
the nal disk mass within r (assumed to be expo-
nential) and M
halo
(r) is the nal dark matter dis-
tribution we wish to calculate. We assume that
7

halo mass shells do not cross during compression, so
that M
halo
(r) = M
halo
(r
i
) = (1

b
)M
i
(r
i
), where


b
= 0:06 is the initial baryon fraction and is chosen
to 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 the
adiabatically compressed halo (dashed lines) and the
disk+halo or \galaxy" rotation curve (solid lines) for
galaxies with V
opt
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 r
opt
, as given
by Persic, Salucci & Stel (1995), are shown as dotted
lines. These and the model ts may be seen to be
declining for rapidly rotating disks and gently rising
for 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, approximately
as (M=L)
disk
/ L
1=2
. This is because even after adi-
abatic compression, the circular velocity of the halo is
either rising or
at near the optical radius. Thus, to
produce a declining rotation curve, the disk (whose
own rotation curve is declining in this region) must
be relatively more important in brighter galaxies. As-
suming a constant (M=L)
disk
 1:2 for all galaxies
would result in nearly
at rotation curves at r
opt
for
all models irrespective of V
opt
. The required increase
in (M=L)
disk
with rotation velocity is consistent with
the results of Broeils (1992a) and Salucci, Ashman
& Persic (1991), who also required a systematically
varying (M=L)
disk
to t the rotation curves of spiral
galaxies.
An important implication of our models is that as
V
opt
for a galaxy increases from 200 to 300 km/s, V
200
for the surrounding halo increases only from 150 to
170 km/s. This kind of behaviour was previously
noted by Persic and Salucci (1991) and has a num-
ber of consequences. It aects the interpretation of
dynamical data on binary and satellite galaxies, an
issue which we discuss in more detail in x5:2. It
may also help with an apparent diculty in theo-
ries where galaxies form by the cooling of baryons
within dark matter halos (White & Rees 1978). As-
suming galaxy/halo systems to have singular isother-
mal potentials with circular velocity V
c
equal to the
rotation velocity of the central disk, White & Frenk
(1991) estimated X-ray luminosities of order 10
42
erg/s for bright spiral galaxies, well in excess of the
observed upper limits on diuse emission from bright
spiral halos. They noted that their model predicted
a typical emission temperature of T
X
= 0:19(V
c
=250
km/s)
2
keV, suggesting that the emission might be
soft enough to have been missed. The halo structure
implied by our current models has V
c
signicantly
smaller than V
opt
over the emitting regions and thus
implies even softer emission from the cooling gas.
In our models, the dark matter fraction within r
opt
increases sharply with decreasing galaxy luminosity,
from less than  70% for galaxies with V
opt
= 300
km/s, to more than  90% for galaxies with V
opt
=
100 km/s. The fraction of the total mass within the
virial radius of the halo represented by the disk also
depends strongly on V
opt
and varies from  0:07 for
V
opt
= 300 km/s, to less than  0:01 for V
opt
= 100
km/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 mass
fraction to begin with, our models require that the
transformation of baryons into stars should have been
extremely inecient in halos with circular velocity be-
low about 100 km/s.
Such a trend of increasing global mass-to-light ra-
tio (M
200
=L
B
) with decreasing halo mass is also re-
quired in hierarchical clustering models of galaxy for-
mation in order to reconcile the steep mass function
of dark halos predicted in these models with the shal-
low faint-end slope of the observed galaxy luminosity
function (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 galaxy
formation within halos of this type is likely to be more
successful than previous semi-analytic models in pro-
ducing realistic galaxy luminosity functions.
In summary, the observed rotation curves of spirals
are consistent with the structure of CDM halos pro-
vided that the assembly of the luminous component
of galaxies was inecient in low mass halos. The ro-
tation velocity of bright galaxies is largely determined
by the contribution of the disk, and is therefore not
a good indicator of the circular velocity of the sur-
rounding halo.
5.2. The dynamics of binary galaxies and
satellite companions.
A major unresolved issue in studies of the dynam-
ics of binary galaxies and of satellites orbiting around
8

bright spirals is the puzzling lack of correlation be-
tween the luminosity of the system and the observed
relative velocity dierence (White et al. 1983, Zarit-
sky et al. 1993, Zaritsky & White 1994). The puz-
zle arises because it is commonly assumed that the
rotation velocity of galactic disks is a good indica-
tor of the mean circular velocity of their surrounding
halos. Since brighter galaxies have more rapidly ro-
tating disks, a strong correlation is expected between
the luminosity of a pair and their velocity dierence,
as well as between the rotation velocity of a primary
galaxy and the orbital velocity of its satellites. Obser-
vational datasets rule out such correlations with high
signicance. Indeed, the data seem to favor models
where galaxies have extended halos with a mass which
depends only very weakly on their luminosity.
The discussion in x5:1 provides an interesting clue
to this puzzle. As can be seen in Figure 11, the mean
circular velocity of halos surrounding galaxies with
rotation velocities larger than about 200 km/s is al-
most uncorrelated with V
opt
and, consequently, with
the luminosity of the galaxy. (Bright galaxies with
V
opt
of the order or larger than 200 km/s typically
constitute the bulk of the systems considered in ob-
servational surveys of binaries and satellite/primary
systems.) According to the analysis presented above,
bright galaxies are typically surrounded by a halo
with mean circular velocity V
200
 160 km/s. The
mass of such halo is M
200
 1:9 10
12
M

within the
virial radius r
200
 320 kpc. This agrees with the es-
timates of Zaritsky & White (1994), who found from
their study of the dynamics of satellites surrounding
bright spirals that the average circular velocity is be-
tween 180 and 200 km/s at 300 kpc. This agreement
is encouraging, since the halo properties we infer here
are chosen to match the shape of the disk rotation
curves, 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 be
similar to that implied by our models.
5.3. The cores of dwarf galaxies.
The luminous component typically constitutes only
a 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 that
the dark halo circular velocity rises almost linearly
with radius over the luminous regions of these galax-
ies (Carignan & Freeman 1988, Carignan & Beaulieu
1989, Broeils 1992a,b, Jobin & Carignan 1990, Lake
et al. 1990). For spherical symmetry, this implies
that the halo has a well dened core within which the
dark matter density is approximately constant. As
pointed out by Moore (1994) and Flores & Primack
(1994), this result is inconsistent with the singular
halo models favoured by N-body simulations such as
those presented in this paper.
Figure 12 shows the contribution of the dark halo
to 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=r
core
)
2
), and are plotted only in
the radial range where the rotation curve is mea-
sured. The parameters for each galaxy have been
taken from Carignan & Freeman (1988, DDO154),
Broeils (1992a, DDO168 and NGC3109), and Lake et
al. (1990, DDO170). The dotted lines in these plots
are 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 been
measured. The two dotted lines are meant to repre-
sent the most and least concentrated halo compatible
with this constraint and with the scatter in the cor-
relations shown in Figure 10 (a factor of  2 in r
max
for a given choice of V
max
).
Figure 12 shows that, except perhaps for DDO170,
where observations constrain the halo parameters
very poorly, the CDM halos appear too concentrated
to t the observations. We thus agree with the conclu-
sions of Moore (1994) and Flores & Primack (1994).
The discrepancy, however, is less dramatic than found
by these authors (e.g. compare Figure 12 with Figure
1 of Moore 1994). The reason for this dierence is
that CDM halos are actually less concentrated than
assumed by Moore (1994). Moore's gure is based
on the simulation of a single CDM halo by Dubinski
& Carlberg (1991), who found r
max
= 26 kpc for an
object with V
max
= 280 km/s. This contrasts with
the mean value of  60 kpc found in our simulations
for the same circular velocity (see Figure 10). We
attribute this dierence to the fact that Dubinski &
Carlberg stopped their simulations at z = 1; their run
cannot therefore be considered representative of dark
halos identied at z = 0.
9

We conclude that, although the cores of dwarf
galaxies pose a signicant problem for CDM, the
problem 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 of
star formation (Dekel and Silk 1986), can in principle
reconcile the observations of dwarfs with the structure
of CDM halos (Navarro, Eke & Frenk 1995).
5.4. The abundance of galactic dark halos.
A problem which aicts all current models of
galaxy formation based on the standard CDM cos-
mogony is related to the high abundance of galactic
dark halos (White & Frenk 1991, Kaumann et al.
1993, Cole et al. 1994). Indeed, dark halos of galac-
tic size are so numerous in this scenario that it is
impossible 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 only
one galaxy per halo, a conservative assumption since
we know that the halos of galaxy groups and clusters
contain 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 of
their surrounding halos, V
200
(see, for example, eq. 5
of White & Frenk 1991). Assuming that we can infer
the rotation velocity of the disk (V
opt
) from V
200
, the
Tully-Fisher relation allows us to compute the con-
tribution of galaxies with a given rotation velocity to
the mean luminosity density of the universe.
Using the B-band Tully-Fisher relation presented
in x5:1, and assuming that the mean luminosity den-
sity of the universe is L
B
= 9:7  10
7
L

Mpc
3
(Efstathiou et al. 1988b, Loveday et al. 1992), we
show in Figure 13 the cumulative fraction of the total
luminosity density contributed by galaxies with ve-
locities larger than V
opt
, under the assumption that
V
200
= V
opt
(solid line). In order to be conserva-
tive, we have assumed in this plot that halos with
V
opt
> 300 km/s do not contribute to the luminosity
density at all, thus excluding groups and clusters of
galaxies from the count. As noted by Kaumann et
al. (1993), even with this restriction the cumulative
luminosity is too large for V
opt
 100 km/s. This im-
plies that it is impossible to match the Tully-Fisher
relation and the galaxy luminosity density simultane-
ously unless our assumption that each halo contains
one galaxy is incorrect.
According to the discussion in x5:1, the mean cir-
cular velocity of CDM halos, V
200
, should actually be
lower than V
opt
in order to account for the shape of
the rotation curves of disk galaxies. A simple approx-
imation to the relationship between halo and galaxy
circular velocity is
V
200
= F (V
opt
) =
V
opt
1 + (V
opt
=300 km=s)
(10)
(see Figure 11). The result of this identication is
shown as a dotted line in Figure 13. Clearly, assigning
lower circular velocities to the halos of galaxies of a
given luminosity only exacerbates the problem. There
is now too much luminosity in the standard CDM
cosmogony for V
opt
 200 km/s.
We conclude that this remains a serious problem.
Although it can in principle be alleviated by adopting
a dierent cosmogony (for example, an open universe
or a universe with cold and hot dark matter compo-
nents), the most popular forms of these alternative
models do not readily overcome the diculty (Heyl
et al. 1995). Another alternative would be to as-
sume that a large number of halos have failed to form
galaxies at all, or that the galaxies they host have so
far escaped detection. The large numbers of systems
being detected in systematic surveys for low surface
brightness galaxies suggest that the existence of such
a 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 structure
of CDM halos discussed in x4 for X-ray and gravita-
tional lensing observations of galaxy clusters. An on-
going debate concerns the diering estimates of core
radius obtained from models of the X-ray emitting
gas and from attempts to reproduce the giant arcs
observed in many clusters.
The X-ray emitting intracluster medium is often
approximated 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=r
core
)
2
)
3=2
. The core radius, r
core
,
is usually a sizeable fraction of the total extent of
the 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 is
isothermal and in hydrostatic equilibrium; it follows
the same law as the gas, with the same core radius
10

and with outer slope parameter 
DM
= =
T
, where

T
measures the ratio of the \temperatures" of the
dark matter and of the gas, 
T
= m
p

2
DM
=kT
gas
(m
p
is the mean molecular weight of the gas and
k is Boltzmann's constant). This model is frequently
used to interpret X-ray observations and to constrain
the dark matter distribution near the center of clus-
ters (Jones & Forman 1984, Edge & Stewart 1991).
Cooling
ow models with a -model mass prole have
been successful at explaining a number of observa-
tions, including the central X-ray surface brightness
\excess" and the drop in central temperature seen in
systems with strong cooling
ows. The core radius
of the dark matter is typically inferred to be in the
range r
c
 100-200 kpc (Fabian et al. 1991).
On the other hand, such large cores are ruled out
by observations of giant arcs, which require core radii
of order 20-60 kpc when a -model is used to describe
the lensing cluster (Grossman & Narayan 1988). This
discrepancy is resolved if we assume instead that clus-
ter halos follow the density prole of eq.(3). For the
parameters suggested by Figures 7 and 8, halos are
suciently concentrated to agree with the gravita-
tional lensing constraints. A CDM cluster with mean
velocity dispersion  1000 km/s placed at z = 0:3
can produce giant arcs similar to those seen (Wax-
man & Miralda-Escude 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 isothermal
and in hydrostatic equilibrium in the dark matter
potential 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 the
dark matter (solid line). Radii are given in units of
r
max
and the density units are arbitrary. We assume
that the gas and dark matter temperatures are equal,
kT
gas
=m
p
= 
2
DM
= GM
200
=2r
200
.
The dierence in shape between the gas and the
dark matter is due to our assumption that the gas is
isothermal. Since the dark matter velocity dispersion
drops at radii larger and smaller than r
max
(see Fig-
ures 5 and 6), the gas structure deviates strongly from
that of the dark matter at small and large radii. At
small radii an isothermal gas develops a well dened
core, and at large radii its density drops less rapidly
than the dark matter. Such leveling o of the gas pro-
le is not observed in real clusters, indicating that the
gas temperature must decrease in the outer regions.
Hydrodynamical simulations conrm that this is the
case, 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 the
result of tting the gas prole with the  model men-
tioned above. In this case, values of r
core
= 0:1r
max
and  = 0:7 give a very good t to the structure of the
gas. For a cluster with a velocity dispersion of 1000
km/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 -model
can agree well with observation. A recent analysis by
Waxman& Miralda-Escude (1995) shows that the ob-
servational signatures of cooling
ows in CDM halos
are essentially indistinguishable from those occurring
in halos with a true constant density core. We con-
clude that the apparent discrepancy between X-ray
and gravitational lensing estimates of the core radius
is a direct result of force-tting a -model to systems
whose structure is better described by a prole more
similar to that of our CDM halos.
6. Conclusions
The main conclusions of this paper can be summa-
rized as follows.
1) The density proles of CDM halos of all masses can
be well t by an appropriate scaling of a \univer-
sal" prole with no free shape parameters. This
prole is shallower than isothermal near the center
of a halo, and steeper than isothermal in its outer
regions.
2) The characteristic overdensities of halos, or equiv-
alently their concentrations, correlate with halo
mass in a way which can be interpreted as re-
ecting the dierent formation redshifts of halos
of diering mass.
3) The observed rotation curves of disk galaxies are
compatible with this halo structure provided that
the mass-to-light ratio of the disk increases with
luminosity. This implies that the halos of bright
spirals have masses which correlate only weakly
with 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 to
300 km/s are predicted to have halos with a typ-
ical mass of 1:8  10
12
M

within 300 kpc. This
agrees well with the masses inferred from the dy-
namics of observed satellite galaxy samples.
4) CDM halos seem too centrally concentrated to be
consistent with observations of the rotation curves
11

of dwarf irregulars (Moore 1994, Flores & Primack
1994). This may imply that the central regions
of dwarf galaxy halos were substantially altered
during galaxy formation, for example by sudden
baryonic out
ows occurring after a burst of star
formation (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" halos
predicted in an
= 1 universe and the observed
number of galaxies.
6) The predicted structure of galaxy clusters is consis-
tent both with X-ray observations of the ICM and
with the presence of giant gravitationally lensed
arcs. Previous discrepant estimates of the core ra-
dius based on these two kinds of data probably
result from force-tting of an inappropriate po-
tential structure. A singular prole such as those
favoured by our simulations can be consistent with
all current data.
We would like to thank the hospitality of the In-
stitute for Theoretical Physics of the University of
California at Santa Barbara, where most of the work
presented here was carried out. JFN would also like
to thank the hospitality of the Max Planck Insti-
tut fur Astrophysik in Garching, where this project
was started, as well as to acknowledge useful discus-
sions with C. Lacey, N. Katz, and B. Moore. This
work has been supported in part by the U.K. PPARC
and the National Science Foundation under grant No.
PHY94-07194 to the Institute for Theoretical Physics
of the University of California at Santa Barbara.
12

Label L
box
1 + z
i
h
g
M
200
r
200
V
200
N
200
r
s
=r
200
[Mpc] [kpc] [10
12
M

] [kpc] [km/s]
1 3.0 44.0 1.5 0.319 177 88.0 5637 0.052
2 3.0 44.0 1.5 0.293 172 85.6 5178 0.057
3 3.0 44.0 1.5 0.414 193 96.1 7316 0.082
4 3.0 44.0 1.5 0.525 209 103.9 9278 0.046
5 6.0 22.0 3.0 2.425 348 173.1 5357 0.060
6 6.0 22.0 3.0 2.301 342 170.1 5083 0.071
7 6.0 22.0 3.0 3.519 394 196.0 7774 0.068
8 6.0 22.0 3.0 2.552 354 176.1 5637 0.124
9 12.0 11.0 6.0 28.15 788 392.0 7773 0.124
10 12.0 11.0 6.0 20.59 710 353.2 5686 0.078
11 12.0 11.0 6.0 29.67 802 398.9 8193 0.131
12 12.0 11.0 6.0 22.01 726 361.1 6078 0.088
13 12.0 11.0 6.0 26.16 769 382.5 7224 0.077
14 12.0 11.0 6.0 22.65 733 364.6 6254 0.065
15 18.0 5.5 9.0 102.68 1213 603.4 8401 0.110
16 24.0 5.5 12.0 224.35 1574 783.0 7744 0.121
17 40.0 5.5 20.0 1109.9 2682 1334 8274 0.151
18 48.0 5.5 24.0 1931.5 3226 1605 8334 0.188
19 58.0 5.5 30.0 3009.7 3740 1861 7360 0.143
Table 1: Parameters of the numerical experiments.
13

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15

Fig. 1a.| The eects of varying the initial redshift on
the density prole of simulated halos. The masses of
the two halos shown are  10
11
and  10
15
M

. The
gravitational softening and the virial radius are in-
dicated with upward and downward pointing arrows,
respectively. Solid, dotted, and dashed lines corre-
spond to 1 + z
i
= 22:0, 11:0, and 5:5, respectively.
Fig. 1b.| Circular velocity proles for the halos
shown in Fig. 1a. Arrows are also as in Fig. 1a.
For 1 + z
i
= 5:5, the circular velocity near the center
of the small halo is substantially underestimated. For
larger z
i
, the models converge to a unique prole.
16

Fig. 2a.| Density proles of the same halos shown
in Figure 1 for dierent choices of the gravitational
softening, h
g
. Upward-pointing arrows indicate the
softening values. Downward pointing arrows indicate
the virial radius. Varying the softening by a factor of
ve has no signicant eect on the structure of the
halo beyond one gravitational softening length.
Fig. 2b.| Circular velocity proles for the halos
shown in Figure 2a. Arrows are also as in Figure 2a.
Diering gravitational softening lengths have no sig-
nicant eect on the structure of the halo on regions
larger than h
g
.
Fig. 3.| Density proles of four halos spanning four
orders of magnitude in mass. The arrows indicate the
gravitational softening, h
g
, of each simulation. Also
shown are ts from eq.3. The ts are good over two
decades in radius, approximately from h
g
out to the
virial radius of each system.
17

Fig. 4.| Scaled density proles of the most and least
massive halos shown in Figure 3. The large halo is less
centrally 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, r
200
.
The gravitational softening is about 10
2
r
200
. Note
that 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 our
sample (solid curves). The dashed lines are ts with
eq.(5). The dotted line is a t to the low mass sys-
tem using a Hernquist model (see eq.6). Note that
the Hernquist model underestimates the halo circular
velocity at r
200
.
18

Fig. 7.| The characteristic overdensity 
c
as a func-
tion of the mass of the halo. The curves show the
mass-overdensity relation predicted from the forma-
tion times of halos (see text). All curves are normal-
ized so that they cross at M
200
= M
?
.
Fig. 8.| The concentration c as a function of
the mass of the halo. The curves show the mass-
concentration relation predicted from the formation
times of halos. All curves are as in Figure 7, and have
been normalized so that they cross at M
200
= M
?
.
Fig. 9.| The ratio between the maximum circular
velocity of a halo and the value at r
200
as a func-
tion of the circular velocity at the virial radius. Low
mass systems are more concentrated than more mas-
sive ones. The maximum velocity can be up to twice
the value at r
200
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 of
dierent mass. Note that a halo with V
max
= 220
km/s has a rising rotation curve that extends out to
about 50 kpc, well beyond the luminous radius of a
galaxy 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) (dotted
lines). The dashed lines indicate the contribution of
the dark halo. Note that the disk mass-to-light ra-
tio increases as a function of mass, and that the halo
contribution is less important in bright galaxies.
20

Fig. 12.| The circular velocities of CDM halos (dot-
ted lines) compared with the halo contribution to the
rotation curve of four dwarf galaxies (solid lines). The
solid lines encompass the likely contribution of the
halo, 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 larger
than V
opt
as predicted by our standard CDM model.
We use the Tully-Fisher relation, and assume that
the abundance of galaxies with rotation velocity V
opt
matches that of halos with V
200
= V
opt
(solid line), or
V
200
= F (V
opt
) as given in the text (dotted line).
21

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