Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.stsci.edu/science/preprints/prep1533/prep1533.ps Äàòà èçìåíåíèÿ: Tue Nov 26 23:57:57 2002 Äàòà èíäåêñèðîâàíèÿ: Sat Dec 22 14:17:32 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: galactic collision

The Dynamics, Structure & History of Galaxies
ASP Conference Series, Vol. nnn, 2002
G. S. Da Costa & E. M. Sadler, eds
Formation of Galactic Disks
S. Michael Fall
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD 21218, USA
Abstract. We review progress in understanding the formation of galac­
tic disks in the standard cosmogonic scenario involving gravitational clus­
tering of baryons and dark matter and dissipative collapse of the baryons.
This scenario accounts remarkably well for the observed properties of
galactic disks if they have retained most of the specific angular momen­
tum they acquired by tidal torques. Early simulations, which included
cooling of the gas but not star formation and the associated feedback, in­
dicated instead that most of the angular momentum of the baryons would
be transferred to the dark matter. Recent simulations indicate that this
angular­momentum problem can be solved partially, and in some cases
entirely, by feedback and other e#ects.
1. Introduction
Two key ingredients in the formation of galactic disks are dissipation and rota­
tion. Dissipation by radiative cooling causes the gas in a protogalaxy to collapse
inward; rotation then halts the collapse in the directions perpendicular but not
parallel to the overall angular­momentum vector, resulting in a thin centrifu­
gally supported disk. Other processes likely to play some role in the formation
and subsequent evolution of galactic disks include dynamical friction, internal
torques, kinematic viscosity, mergers, and star formation and the associated
heating and stirring of the gas (feedback). In this article, we consider the general
cosmogonic scenario in which the extended halos of galaxies form hierarchically
by the gravitational clustering of non­dissipative dark matter, and the luminous
components of galaxies form by a combination of the gravitational clustering
and dissipative collapse of baryons, as proposed by White & Rees (1978). The
formation of galactic disks and the origin of their rotation in this scenario were
first studied by Fall & Efstathiou (1980). Most of the currently popular models
of galaxy formation, including all variants of the cold and warm dark matter
(CDM and WDM) models, are specific versions of this general scenario.
2. Tidal Torques
The formation of objects by gravitational clustering automatically endows them
with some angular momentum (Peebles 1969). This is because the (proto)objects
initially have irregular shapes and hence non­zero quadrupole moments. They
therefore exert tidal torques on each other, with random strengths and direc­
1

2 Fall
tions. The objects acquire most of their angular momentum in the translinear
regime, when their density contrasts are appreciable (##/# # 1), but before they
reach their maximum sizes and begin to contract. The rotation induced by tidal
torques is usually quantified in terms of the dimensionless spin parameter
# # J |E| 1/2 G -1 M -5/2 , (1)
where J , E, and M are the total angular momentum, energy, and mass of the
object, and G is the gravitational constant. Rapidly rotating objects, such as
disks, have # # 1, whereas slowly rotating objects, such as spheroids, have
# # 1.
Cosmological N­body simulations have revealed that the distribution of
spins induced by gravitational clustering alone (i.e., without dissipation) is ap­
proximately lognormal:
p(#)d# # exp # -
1
2 # ln(#/# m )
#(ln #)
# 2
# d ln #, (2)
with
#m # 0.05 and #(ln #) # 0.5. (3)
Most objects rotate slowly although there is a wide range of spins. This dis­
tribution appears to be nearly universal in the sense that it has little or no
dependence on the cosmological parameters, the initial spectrum of density per­
turbation, or the masses and densities of the objects (Barnes & Efstathiou 1987;
Zurek, Quinn, & Salmon 1988; Warren et al. 1992; Cole & Lacey 1996). These
results are very useful, even if they are not yet fully understood theoretically.
3. Analytical Collapse Model
We can relate the factor by which the baryons collapse in the radial direction
before they reach centrifugal balance in a disk to the initial rotation of a pro­
togalaxy as follows (Fall 1983). For simplicity, we consider a galaxy with a
luminous disk (D) and a dark halo (H) but not a luminous spheroid. Further­
more, we assume that the baryons destined to become the disk receive the same
tidal torques as the dark matter before much dissipation occurs and that during
and after the collapse the total specific angular momentum of each component
is conserved:
JD /MD = JH /MH . (4)
In other words, we assume for now that angular momentum is not transferred
between the disk and the halo, or if it is, that this is accompanied by the transfer
of enough mass (outflow or inflow) that equation (4) is still satisfied.
We approximate the disk by an exponential model with a scale radius # -1
(sometimes denoted by R d ) and the halo by a singular isothermal sphere with a
circular velocity v c and a truncation radius r t . Then, neglecting the self­gravity
of the disk, we have
JD /MD = 2v c # -1 , (5)
JH /MH = # 2#v c r t . (6)

Formation of Galactic Disks 3
Equating these gives a very simple relation between the collapse factor #r t of
the baryons in the disk and the spin parameter # of the halo:
#r t = # 2/#. (7)
This implies #r t # 30 for # # 0.05 and hence r t # 100 kpc for a typical spiral
galaxy like the Milky Way (with # -1 # 3 kpc). The collapse factor is larger
in halos with smaller spin parameters and smaller in halos with larger spin
parameters. The collapse factor would also be larger if the baryons were to lose
some of their specific angular momentum.
Equation (7) is an excellent approximation even when the halo has a finite
core radius and when the self­gravity of the disk is included (Fall & Efstathiou
1980; see their Fig. 3). It is also a good approximation if the halo has an NFW
profile (Navarro, Frenk, & White 1996, 1997) that is later deformed by the
(adiabatic) contraction of the disk within it (Mo, Mao, & White 1998). The
radius r t of the halo in equation (7) should be the one within which the baryons
have collapsed onto the disk. This may be compared and contrasted with the
radius r 200 within which the mean density of the protogalaxy is 200 times the
critical (closure) density. The latter is near the transition between the virialized
and infalling parts of the halo (Cole & Lacey 1996). In recent work in this field,
it has been customary to identify r t with r 200 . This assumption, however, does
not have much physical justification beyond the constraint r t
<
# r 200 . It is likely
that r t depends on cooling, heating, and other non­gravitational processes at
least as much as it depends on the gravitational clustering that determines r 200 .
Thus, we have no guarantee that r t and r 200 will be equal or even proportional
to each other. (Anyone who doubts this should compare the cooling and virial
radii in clusters of galaxies.)
4. Scaling Relations
The properties of the halos that form in a hierarchy by gravitational clustering
obey some simple scaling relations. We can combine these with the relation
between the collapse factor and the spin parameter to derive the corresponding
scaling relations for the properties of galactic disks as follows (Fall 1983). We
approximate the relation between the typical masses MH of the halos (within r t )
and their circular velocities v c by a power law, MH # v k
c . The index k depends
in general on the initial spectrum of density perturbations, the cosmological
parameters, the range of masses considered, and the relation between r t and r 200 .
Simple arguments based on the di#erent formation times of objects with di#erent
mean densities give k = 12/(1 - n) for (##/#) rms # M -(3+n)/6 (White & Rees
1978). This implies k # 4 for the e#ective index n # -2 of the CDM spectrum
on galactic scales (Blumenthal et al. 1984). N­body simulations indicate instead
k # 3 for the NFW halos in CDM cosmogonies (Navarro et al. 1997). This index
is appropriate for r t = r 200 and hence for perfectly synchronous formation. In
view of the unknown initial spectrum (index n) and unknown relation between
r t and r 200 , we regard k as a parameter with some theoretical uncertainty.
The scaling relation for the halos can be reexpressed in terms of the typical
luminosities and circular velocities of the disks as
LD # (f D /#D )v k
c , (8)

4 Fall
where f D # MD /MH is the ratio of masses of the disks and halos, and #D #
MD/LD is the mass­to­light ratio in the disks. This may be compared with the
observed Tully­Fisher (1977) relation, LD # v l
c , the index of which varies from
l # 3 in the B band (0.44 µm) to l # 4 in the K band (2.2 µm) (Verheijen 2001).
Most of this variation can be explained by a dependence of #D on v c , indicated
by the observed correlation between the colors and the luminosities of galaxies.
When this e#ect, including the mass of interstellar gas, is taken into account,
the index of the baryonic Tully­Fisher relation is found to be k = 3.5 ± 0.4 for
f D = const (Bell & de Jong 2001).
The central surface brightness of an exponential disk is given by I 0 =
# 2 LD /2#. Using equations (7) and (8) and the relation v 2
c = GMH /r t , we
can rewrite this in the form
I 0 # # -2 v l-2k+4
c # v l-2k+4
c , (9)
where all of the dependence on f D and #D is contained in the factor v l-k
c .
The second proportionality holds because the spin parameters of the halos are
statistically independent of their other properties. For k # 3.5, we expect I 0 #
const in the B band (l # 3) and I 0 # v c in the K band (l # 4). The first of these
is the same as the original Freeman (1970) relation in the B band. It would be
interesting to search for the predicted correlation in the K band.
The scaling relations above indicate how the typical luminosities and surface
brightnesses of galactic disks depend on their circular velocities. The dispersions
about these relations are determined in part by the dispersion in the spin param­
eter. We expect the LD ­v c correlation to have relatively little scatter because it
is independent of # [equation (8)], while we expect the I 0 ­v c correlation to have
a great deal of scatter because it includes the factor # -2 [equation (9)]. These
trends are qualitatively consistent with the observed Tully­Fisher and Freeman
relations. The distribution of I 0 at fixed v c can be derived from equation (2) for
p(#) and equation (7) for #r t (Dalcanton, Spergel, & Summers 1997; Mo et al.
1998; Weil, Eke, & Efstathiou 1998). These results can then be combined with
the luminosity function of galaxies to derive the joint distributions of I 0 and
# -1 (Dalcanton et al. 1997) and LD and # -1 (de Jong & Lacey 2000). In all
these studies, the observed distributions are reproduced quite well provided the
coe#cient in equation (7) is reasonably close to # 2, i.e., provided the disks have
retained most of the specific angular momentum they acquired by tidal torques.
5. Origins of Exponential Disks
The results above are based on the assumption that the total specific angular
momenta in the disks are the same as those in their halos [equation (4)]. We
may refer to this as the weak form of the assumption of angular­momentum
conservation. In addition, it is sometimes supposed that the distributions of
specific angular momentum in the disks are the same as those in their halos.
This is usually expressed in terms of the fractions of mass with specific angular
momentum below h = Rv # in the two components:
MD (h)/MD = MH (h)/MH . (10)

Formation of Galactic Disks 5
We may refer to this as the strong form of the assumption of angular­momentum
conservation. It is analogous to Mestel's (1963) hypothesis that MD (h) would
be conserved in the collapse of a one­component protogalaxy (made before dark
halos were discovered). Note that equation (10) implies equation (4), but the
converse is not true in general. It is possible to change M(h)/M in either
component without a#ecting the corresponding J/M .
Several authors have pointed out that the distribution MD (h)/MD for an
exponential disk with a flat rotation curve is similar to that for a uniform sphere
with a constant angular velocity (Gunn 1982; van der Kruit 1987; Dalcanton et
al. 1997). This in turn resembles the distribution MH (h)/MH caused by tidal
torques (Barnes & Efstathiou 1987; Quinn & Zurek 1988), although there has
been a recent tendency to emphasize the di#erences rather than the similarities
(Firmani & Avila­Reese 2000; Bullock et al. 2001; van den Bosch 2001). Thus,
the strong form of the assumption of angular­momentum conservation provides
a possible explanation for the fact that the radial profiles of most galactic disks
are approximately exponential. The main concern here is that several processes
could alter MD (h)/MD and hence the radial profiles of the disks over their
lifetimes. These include the torques exerted by non­axisymmetric features in
the disks (bars and spiral arms), the non­conservation of angular momentum as
gas flows though shocks in spiral arms, and the exchange of angular momentum
in collisions between clouds, not to mention galactic outflows, fountains, and
mergers. It seems likely that these processes play some role in determining the
present distributions of specific angular momentum in galactic disks.
Lin & Pringle (1987) showed that the radial distributions of the stars in
galactic disks would evolve toward exponential­like profiles if two conditions
were met. (1) The net e#ect of the processes that redistribute angular momen­
tum in the interstellar gas can be described by an e#ective viscosity # e# . (2)
The associated timescale, t # = R 2 /# e# , is about the same as the timescale for
star formation, t # = # g / —
# s . This is plausible because both the redistribution of
angular momentum and the formation of stars may be regulated by instabilities
in the disks, including bars and spiral arms, although we lack a full understand­
ing of just how or even whether this would actually happen. The Lin­Pringle
model has been explored further, including its chemical evolution, by several
authors (Clarke 1989; Yoshii & Sommer­Larsen 1989; Sommer­Larsen & Yoshii
1989, 1990). This model raises the possibility that the ubiquitous exponential
profile is the result of viscous processes operating in the disks after they formed
rather than tidal torques acting on the galaxies while they formed. More likely,
both are involved: the exponential profile is first established by external torques
and is then reinforced by internal viscosity.
6. The Angular­Momentum Problem
About a decade ago, it became possible to simulate the hierarchical formation
of galaxies with both dark matter and baryons by a combination of N­body and
hydrodynamical models. The early simulations included the radiative cooling of
the gas but not the formation of stars and the associated feedback (heating and
stirring of the gas). The results were very di#erent from the analytical models
described above (Navarro & Benz 1991; Navarro & White 1994). The baryonic

6 Fall
objects that formed in these simulations were ellipsoidal in shape and an order
of magnitude smaller than galactic disks of the same scaled mass. Thus, they
resembled the spheroids more than the disks of real galaxies. The reason for this
is that the gas cooled quickly and collapsed into dense subunits within the halos.
A combination of dynamical friction and gravitational torques within the halos
then transferred most of the orbital angular momentum of the baryons to the
dark matter, causing the subunits to sink toward the centers of the protogalaxies,
where they then merged. (For a prescient discussion of how these processes
might determine the di#erent properties of galactic spheroids and disks, see
Zurek et al. 1988.) The discrepancy between the baryonic objects produced
in such simulations and real galactic disks has become known as the angular­
momentum problem of galaxy formation. It is so severe that it is sometimes
referred to as a crisis or catastrophe.
Steinmetz & Navarro (1999) have explored this and related issues with a
series of high­resolution simulations. They have tried to alleviate the angular­
momentum problem by including some feedback in their simulations but find
this makes little di#erence to the outcome. This conclusion, however, depends
on their particular prescription for feedback. Steinmetz & Navarro assume that
the gas is heated only in the immediate vicinity of ongoing star formation. Then,
since the gas is very dense at these locations, the energy input is quickly radiated
away, before it can influence the motions or the thermodynamic state of the gas
at other locations. Steinmetz & Navarro (1999) have also found that the zero­
point of the relation between luminosity and circular velocity in their simulations
di#ers significantly from that of the observed Tully­Fisher relation, especially
in the Einstein­de Sitter cosmological
model(# M =
1,# # = 0) with CDM.
More recently, Eke, Navarro, & Steinmetz (2001) have shown that this problem
is reduced substantially or solved completely in the concordance cosmological
model(# M =
0.3,# # = 0.7) with CDM or WDM.
7. Possible Solutions
It is now widely believed that the solution of the angular­momentum problem
must be found in some extreme form of feedback that prevents the baryons
from collapsing until after the violent relaxation in their halos is complete.
The baryons would then collapse within relatively smooth halos, with little
quadrupole coupling, and hence would retain most of their specific angular
momentum, as was originally, although perhaps naively, envisaged (Fall & Ef­
stathiou 1980). That this idea goes a long way toward solving the angular­
momentum problem has been demonstrated explicitly in a number of recent
simulations (Weil et al. 1998; Sommer­Larsen, Gelato, & Vedel 1999; Eke, Ef­
stathiou, & Wright 2000; Thacker & Couchman 2001). In the Weil et al. and
Eke et al. simulations, the gas evolves adiabatically until the redshift z = 1 and
is then allowed to cool radiatively and collapse. In the Sommer­Larsen et al.
and Thacker & Couchman simulations, the feedback is driven by local star for­
mation but is extensive in both time and position (in contrast to the Steinmetz
& Navarro prescription). The common result of these studies is that the disks
retain much more of their specific angular momentum.

Formation of Galactic Disks 7
Several other e#ects also help. The transfer of angular momentum is less
severe in simulations with warm dark matter than in simulations with cold
dark matter (Sommer­Larsen & Dolgov 2001). The reason for this is that the
halos in WDM simulations have much less substructure and hence less dynamical
friction and internal torques than the halos in CDM simulations. The transfer of
angular momentum is also reduced in simulations with a cosmological constant
(Eke et al. 2000). This happens because protogalaxies acquire their angular
momenta earlier, before the gas cools, and then su#er fewer late mergers in the
#­dominated simulations. It is also likely that some of the baryons in the inner
parts of the protogalaxies, where the specific angular momentum is lowest, either
formed the luminous spheroids or were expelled entirely in galactic winds (Eke
et al. 2000).
The specific angular momenta of real galactic disks can be explained by
a combination of extreme feedback and one or more of the other e#ects men­
tioned above. Does this mean the angular­momentum problem has been solved?
Yes and no. Yes, because we now know that some mechanisms are capable
of reconciling the simulations with observations. No, because we do not yet
know whether these mechanisms operate in real galaxies. The issue of feedback
is especially vexing because it probably depends on features of the interstellar
medium on the scales of individual star­forming clouds, of order parsecs. (For
a promising model of feedback, see Efstathiou 2000.) The energy released by
young stars---in ionizing radiation, stellar winds, and supernova ejecta---is more
than su#cient to alter radically the collapse of the baryons in protogalaxies.
The key issues are where, when, and how this energy is deposited, in particular,
whether it is absorbed within the clouds themselves or is distributed more widely
within the protogalaxies. This, in turn, depends on the sizes and masses of the
clouds, the locations of the stars within them, whether the clouds are porous,
and so forth.
8. Perspective
The goal of the research reviewed here has been to find a physically consistent
model that accounts for the observed properties of galactic disks. Two decades
ago, it seemed as if we might be close to achieving this goal. At that time,
the observed properties of galactic disks were explained remarkably well by an
idealized model in which the baryons and dark matter in protogalaxies acquired
the same specific angular momenta by tidal torques and then conserved them
during and after the collapse of the baryons. A decade later we recognized that,
if the gas in protogalaxies were to cool rapidly, most of the angular momentum
of the baryons would be transferred to the dark matter. We now know that
this angular­momentum problem can be solved by stellar feedback and other
e#ects in principle, but we do not yet know how e#ective these processes are in
practice. In another decade, with some luck, NGST will be in operation, and
we should be able to observe the formation of galactic disks directly. It will be
interesting to see how much theoretical and numerical progress has been made
by then. The simulations will undoubtly improve, but it will be a formidable
challenge to include all the relevant physics on all the relevant scales.

8 Fall
Appreciation. This article is dedicated to Ken Freeman at the celebration of
his sixtieth birthday on Dunk Island. I am grateful to Ken for his friendship
and collaboration over many years.
References
Barnes, J., & Efstathiou, G. 1987. ApJ, 319, 575
Bell, E. F., & de Jong, R. S. 2001, ApJ, 550, 212
Blumenthal, G. R., Faber, S. M., Primack, J. R., & Rees, M. J. 1984, Nature,
311, 517
Bullock, J. S., Dekel, A., Kolatt, T. S., Kravtsov, A. V., Klypin, A. A., Porciani,
C., & Primack, J. R. 2001, ApJ, 555, 240
Clarke, C. J. 1989, MNRAS, 238, 283
Cole, S., & Lacey, C. 1996, MNRAS, 281, 716
Dalcanton, J. J., Spergel, D. N., & Summers, F. J. 1997, ApJ, 482, 659
de Jong, R. S., & Lacey, C. 2000, ApJ, 545, 781
Efstathiou, G. 2000, MNRAS, 317, 697
Eke, V., Efstathiou, G., & Wright, L. 2000, MNRAS, 315, L18
Eke, V. R., Navarro, J. F., & Steinmetz, M. 2001, ApJ, 554, 114
Fall, S. M. 1983, in IAU Symp. 100, Internal Kinematics and Dynamics of Galax­
ies, ed. E. Athanassoula (Dordrecht: Reidel), 391
Fall, S. M., & Efstathiou, G. 1980, MNRAS, 193, 189
Firmani, C., & Avila­Reese, V. 2000, MNRAS, 315, 457
Freeman, K. C. 1970, ApJ, 160, 811
Gunn, J. E. 1982, in Astrophysical Cosmology, ed. H. A. Br˜uck, G. V. Coyne,
& M. S. Longair (Vatican: Pontificia Academia Scientiarum), 233
Lin, D. N. C., & Pringle, J. E. 1987, ApJ, 320, L87
Mestel, L. 1963, MNRAS, 126, 553
Mo, H. J., Mao, S., & White, S. D. M. 1998, MNRAS, 295, 319
Navarro, J. F., & Benz, W. 1991, ApJ, 380, 320
Navarro, J. F., & White, S. D. M. 1994, MNRAS, 267, 401
Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563
Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493
Peebles, P. J. E. 1969, ApJ, 155, 393
Quinn, P. J., & Zurek, W. H. 1988, ApJ, 331, 1
Sommer­Larsen, J., & Dolgov, A. 2001, ApJ, 551, 608
Sommer­Larsen, J., & Yoshii, Y. 1989, MNRAS, 238, 133
Sommer­Larsen, J., & Yoshii, Y. 1990, MNRAS, 243, 468
Sommer­Larsen, J., Gelato, S., & Vedel, H. 1999, ApJ, 519, 501
Steinmetz, M., & Navarro, J. F. 1999, ApJ, 513, 555
Thacker, R. J., & Couchman, H. M. P. 2001, ApJ, 555, L17
Tully, R. B., & Fisher, J. R. 1977, A&A, 54, 661

Formation of Galactic Disks 9
van den Bosch, F. C. 2001, MNRAS, 327, 1334
van der Kruit, P. C. 1987, A&A, 173, 59
Verheijen, M. A. W. 2001, ApJ, 563, 694
Warren, M. S., Quinn, P. J., Salmon, J. K., & Zurek, W. H. 1992, ApJ, 399, 405
Weil, M. L., Eke, V. R., & Efstathiou, G. 1998, MNRAS, 300, 773
White, S. D. M., & Rees, M. J. 1978, MNRAS, 183, 341
Yoshii, Y., & Sommer­Larsen, J. 1989, MNRAS, 236, 779
Zurek, W. H., Quinn, P. J., & Salmon, J. K. 1988, ApJ, 330, 519
Discussion
Quinn: With respect to the angular­momentum catastrophe, I think we all knew
the simulations had problems since the cooling mechanisms (prescriptions) were
not well defined physically and the heating was totally absent. Secondly, the loss
of J/M by a factor # 10 is a great success in making ellipticals and spheroids,
as pointed out by Zurek, Quinn, & Salmon (1988).
Fall: I tend to agree with both of your points. Thanks for reminding me of
your paper with Zurek and Salmon. My impression is that the simulations
without stellar feedback produce mainly spheroid­like objects, whereas those
with extreme feedback produce mainly disk­like objects. Eventually, we should
aim to produce both types of objects in the observed proportions in the same
simulations. It would be nice to find a simple mechanism to do this.
Silk: The feedback models in which the gas is kept hot until z # 1 are going
to have a problem in accounting for the observations of relatively massive disks
at z # 1 and also for the apparent lack of evolution in the Tully­Fisher relation
back to this redshift.
Fall: The kinds of observations you mention are potentially valuable constraints
on, or input to, the models. However, my impression is that such comparisons
are still very tentative. The samples of galaxies at high redshifts have not been
selected in the same way as the nearby samples, and the photometric evolution
of the disks remains quite uncertain.
Illingworth: Physically, why is it that #­dominated cosmologies help with the
angular­momentum problem?
Fall: In #­dominated models, protogalaxies acquire their angular momenta ear­
lier, before the gas cools, and are then disrupted by fewer late mergers than in
matter­dominated models (see Eke et al. 2000).
Bosma: Are the angular­momentum and cuspy­halo problems related?
Fall: One might think so. Both problems seem to be aggravated by substructure
within galactic halos. And for this reason, they should both be alleviated in
models with warm dark matter. Recent simulations indicate, however, that
WDM has more impact on the angular­momentum problem than it does on the
cuspy­halo problem.