Whether galaxies form via the collapse and fragmentation of massive adiabatic fluctuations or by the clustering and merging of low mass isothermal fluctuations, dissipative processes may provide the key to understanding many of their observed properties. These include the luminosity function, metallicity gradients, and correlations between global parameters. In this Section, we review the role and implications of dissipation in hierarchical clustering theories (Sections 8.1 and 8.2) and in the pancake theory of galaxy formation (Section 8.3). In Section 8.4 we describe a theory in which galaxies form by the coagulation of small clouds which may be applicable in both hierarchical and pancake theories. The role of pregalactic stars is considered in Section 8.5 where we briefly review a new class of galaxy formation schemes in which density fluctuations on the scales of galaxies and clusters are generated by astrophysical processes operating after recombination.
It is first useful to establish a general criterion for dissipation to
play a general role in galaxy formation. Simple comparisons of cooling
and gravitational time-scales lead to some interesting indications of
characteristic protogalactic scales
(Hoyle, 1953;
Binney, 1976b;
Rees and Ostriker, 1977;
Silk, 1977a).
We assume a primordial abundance
mixture of hydrogen and helium. The cooling rate coefficient in a
diffuse ionized gas at
T 2 ×
104 K may then be approximated by
![]() |
(8.1) |
where Abf is the bound-free cooling coefficient (and is weakly temperature dependent) and Aff is the free-free cooling coefficient. There are separate contributions to Abf from H and He, although only He cooling is important at T > 5 × 104 K. Detailed computations of the coefficients Abf and Aff may be found in Cox and Tucker (1969). The cooling time-scale is written
![]() |
(8.2) |
and is approximately proportional to
T3/2 n-1 for
T
106 K and
T1/2 n-1 for
T
106 K.
We compare tc with the collapse time of a uniform pressure-free sphere
![]() |
(8.3) |
and note that we can write
![]() |
(8.4a) (8.4b) |
Here Rj = (
kT / µ G
)1/2
is the Jeans length and
MJ =
RJ3 / 6. We
conclude that an initially uniform and spherically symmetric
collapsing cloud will first be able to cool once it has collapsed
within a radius of 100 kpc if T > 106 K or
if its mass is less than 1012
M
if
T < 106 K.
Silk (1977b) and
Rees and Ostriker (1977)
have made the interesting point that the characteristic scales given in
Eqs. (8.4) may be expressed, to order of magnitude, in terms of
fundamental physical constants,
![]() |
(8.5a) (8.5b) |
where is the fine
structure constant
=
2
e2 /
(hp c)]. For comparison, the mass of a star is
[hpc /
(Gmp2)]3/2 mp.
These equations neatly
illustrate how it may be possible to interpret the characteristic
masses and radii of galaxies in terms of astrophysical processes that
are independent of the details of the cosmological model (e.g.
H0,
or the photon entropy per baryon). This contrasts with other
characteristic mass scales that we have mentioned, such as the damping
mass for adiabatic fluctuations (Eq. 4.12), the neutrino Jeans mass
(Eq. 5.21) or the Jeans mass just after recombination (Eq. 5.26).
To decide which of these two limits is appropriate, we must examine
the role of pressure. Since the initial collapse is adiabatic
(tc > tgrav), we have
T
2/3.
Now a reasonable guess
for the initial temperature is Ti
104
K. Thus to attain T > 106 K at R > 50 kpc
requires collapse from > 500 kpc where the density is only
< 10-4(M/1012
M
)
cm-3 compared with the mean density in the universe
n
10-5(1 + z)3
h2 cm-3. We conclude
that the mass limit is the more realistic
one. Thus only mass-scales below ~ 1012
M
can cool
effectively. This
compares rather well with the luminous masses of the brightest
galaxies, an L* galaxy having a luminous mass
of ~ 3 × 1011
M
.
The preceding argument applies to uniform spherical collapse. A straightforward generalization suggests that inhomogeneous collapse can be similarly constrained if we compute the critical surface density for a shock between two colliding clouds to be radiative. This quantity depends only on the shock strength, and effectively only on the relative velocity dispersion of the colliding clouds. Again at low velocities, it is equivalent to an upper limit on mass and at high velocities to a bound on radius. This approach will be discussed in detail in Section 8.4.
At large redshifts Compton scattering of electrons by the background radiation provides an important cooling mechanism. The Compton cooling time-scale is
![]() |
(8.6) |
Clearly Compton cooling is effective within a collapse time if tcomp < tgrav, which yields
![]() |
(8.7) |
where ni is the mean density of a cloud that just
begins to recollapse
at redshift z. Hence only after the density increases by a factor 100
at z = 10 or 1010 at z = 100 will atomic
cooling processes
dominate. The necessary collapse factor is so large at z = 100 that
rotational forces are likely to become important: recall that
vrot /
n1/6
for a self-gravitating collapsing cloud that conserves its
angular momentum. At low redshift
(z
10),
Compton cooling will not
affect the arguments which led to Eq. (8.4). However, at high redshift
(z >> 10), the predominantly Compton cooling is
scale-independent: all
mass-scales collapse in free-fall until star formation or rotational
forces intervene. These arguments are summarized in
Figure 8.1.
![]() |
Figure 8.1. The lines separating the shaded
regions show
tc / tgrav = 1. Gas clouds in the
shaded region cannot cool on a free-fall
timescale. Compton cooling, which is scale independent, is dominant
at z |
An important extension of these ideas has been proposed by
White and Rees (1978).
They point out that any acceptable model for galaxy
formation must take into account the large amounts of "missing mass"
known to exist in rich clusters of galaxies and, probably, in dark
halos around spiral galaxies. Any theory which attempts to explain
galaxies by dissipationless collapse would seem to be unable to
explain why the mass-to-light ratios of rich clusters
[M/L ~ 400h(M/L)] are so much larger than
the mass-to-light ratios deduced for
the material within the optical radii of galaxies
[M/L ~ 10h(M/L)
].
Fall (1981)
has expressed this problem in an interesting
way. A typical L* galaxy has a half-light radius
r1/2 ~ 3 kpc and a luminosity overdensity with respect
to the background of
![]() |
(8.8a) |
(using Eqs. 2.19 and 2.22). If, however, galaxies formed without dissipating any binding energy, the mean luminosity overdensity should coincide with the mean mass overdensity inferred from the galaxy two-point correlation function,
![]() |
(8.8b) |
The discrepancy between (8.8a) and (8.8b) is roughly a factor of 103 indicating that typical galaxies have collapsed by a factor of ~ 10 in radius.
White and Rees propose the following model. Most of the material in the universe is assumed to be dark in order to account for the high M/L values of rich clusters. The exact nature of the dark material is unimportant. Any of the usual candidates such as low mass stars, the remnants of supermassive stars, or weakly interacting particles (e.g. gravitinos, photinos, etc., see Section 9) would be acceptable. The important point is that the dark material must be in a form which does not dissipate energy. Primordial gas, destined to form the luminous parts of galaxies, accounts for only a small fraction of the mean mass density of the universe (~ 1 - 10%). The dark material, therefore, provides the dominant contribution to the gravitational potential and can cluster hierarchically in the manner described in Section 5.3. The gas, on the other hand, can radiatively cool and dissipate binding energy to form the dense luminous components of galaxies. The requirement that this could have happened on a time-scale shorter than the Hubble time, using a slight modification of the cooling arguments described above, again leads to a characteristic mass scale similar to Eq. (8.4b). Under some fairly specific assumptions, White and Rees show how the cooling arguments can lead to a galaxy luminosity function that is in reasonable agreement with observations [Eqs. (2.18) and (2.19)].
In summary, the White and Rees theory predicts that each galaxy
should have formed within the potential well of a dark halo and that
the largest galaxies should have masses ~ 1012
M. The high
over-densities and low mass-to-light ratios of galaxies are accounted
for if the gas collapses dissipatively by a factor of ~ 10 in radius
before fragmenting into stars. Systems with masses > 1012
M
would produce groups and clusters, rather than individual galaxies and
should have high mass-to-light ratios. This is because cooling
processes cannot lead to segregation of dark and luminous material on
these scales. On scales larger than individual galaxies, therefore,
the galaxy correlation functions should accurately trace the mass
distribution. In the next section we consider the role of angular
momentum in the White and Rees theory.