**6.1. Dissipationless collapse and violent
relaxation**

Several authors
(Sandage, Freeman and
Stokes, 1970;
Gott and Thuan, 1976)
have discussed the possibility that the key process in
determining whether a protogalactic cloud becomes an elliptical or
spiral galaxy is the amount of binding energy that is dissipated
during the collapse of the cloud. In the uniform model of spherical
collapse (e.g.
Gunn and Gott, 1972),
a spherical inhomogeneity reaches
its maximum radius
*r*_{max} at redshift *z*_{m} when it achieves an
overdensity (*z*_{m}) =
9^{2} /
16(*z*_{m}).

Hence, it is argued that if star formation is essentially complete at
redshift *z*_{m}, the subsequent collapse will be purely
stellar dynamical
and might lead to the formation of spheroidal gas-free systems
resembling elliptical galaxies
(Gott, 1973,
Gott, 1975).
On the other hand, if the system is predominantly gaseous at
*z*_{m}, then the subsequent
collapse will be highly dissipative, leading to disc-like systems
resembling spiral galaxies. The appeal of the dissipationless model is
obvious. It avoids the complexities of gas dynamical calculations and
models can, in principle, be studied in detail using *N*-body
simulations.

Using the spherical model, we may make a rough estimate of the
redshift at maximum expansion for a typical giant elliptical. At
*z*_{m} the total energy of the protogalaxy is,

(6.1a) |

and

(6.1b) |

After the collapse and subsequent relaxation of the system, the virial
theorem is satisfied, so
*T* = 1/2| *W*|. We now assume that the density
profile is adequately approximated by a de Vaucouleurs
*r*^{1/4} profile
(Section 2.1) with
*R*_{e}
6*h*^{-1} kpc. Then
*W* = - 0.33*GM*^{2} / *R*_{e} and the mass
of the system may be roughly computed using the virial theorem in the
form,

(6.2) |

where is the average 1-dimensional velocity dispersion, 300 km/sec for a giant elliptical galaxy. Since the collapse is dissipationless, total energy is conserved, so Eqs. (6.1) and (6.2) fix the epoch of galaxy formation (e.g. Binney, 1977).

(6.3) |

Consequently, if ellipticals formed by dissipationless collapse, they must have formed rather early, and with correspondingly short collapse times,

(6.4) |

One might also expect that the mass-radius (or mass-velocity dispersion) relationship of ellipticals is related to the initial spectrum of fluctuations. Several authors (e.g. Fall, 1980a; Faber, 1982) have used the hierarchical model (Section 5.3) to derive the relation,

(6.5) |

where *n* is the index of the power-law fluctuation spectrum
[Eq. (5.9)]. The observations (e.g.
Davies et al., 1983)
yield *M*
*R*_{e}^{1.6 ± 0.3}, hence Eq. (6.5) implies
-0.4 < *n* < - 1.6. This application of
Eq. (6.5) is extremely doubtful, however, since the formula ignores
the possibility that lumps forming at early times might later
agglomerate into larger systems and that there could be a substantial
dispersion in the masses of lumps forming at any fixed time. Such a
relation could, in principle, be checked using cosmological *N*-body
simulations.

Much of the work on dissipationless galaxy formation has been
concerned with predicting the surface brightness profiles of relaxed
systems. Elliptical galaxies have remarkably similar density profiles
(Section 2.1), an indication of some common
relaxation phenomenon. For
stellar systems containing only a small number of particles,
relaxation may proceed via two-body encounters. Let *E* be the total
energy of any individual star (*E* = 1/2*mv*^{2} +
); fluctuations in the
gravitational potential result in energy changes
*E* / *E*
~ 1 on a timescale,

(6.6) |

where *N* is the total number of stars in the system and the
crossing-time *t*_{cr} is defined as
*t*_{cr} ~ *R* / *v*, where
*v* is a characteristic
random stellar velocity and *R* is the half-mass radius of the
system. For a giant elliptical galaxy, with
*N*
10^{11}, *v*
250 km/sec, and
*R* 10 kpc, we
obtain
*t*_{R}
10^{16} yrs, i.e. very much longer than
the age of the Universe. Thus two-body encounters are entirely
negligible in typical galaxies. It is useful to consider the
distribution function for the system,
*f* (**x**, **v**, *t*) defined so that the
mass density of stars at any given time in a phase space element
*d*^{3} **x** *d*^{3} **v**
is *f* (**x**, **v**, *t*) *d*^{3}**x**
*d*^{3}**v**. For a collisionless
system, the distribution
function *f* obeys the collisionless Boltzmann equation

(6.7) |

Jeans' theorem states that in a collisionless system in a
steady-state, the distribution function *f* will be a function only of
the integrals of the equations of motion of any individual star. For
example, for an axisymmetric potential there are at least two
isolating integrals, the total energy of a star *E*, and the component
of the star's angular momentum around the axis of symmetry
*J*_{z}. Hence
as an ansatz for *f* we might write *f* (*E*,
*J*_{z}). There exists a wealth of
literature on the existence of additional "third-integrals" for given
forms of axisymmetric potentials (ie. integrals which are not
obviously related to a fundamental symmetry of the Lagrangian). Some
aspects of this problem will be discussed below (see the review by
Binney, 1982,
for a good introduction).

King (1966) showed that elliptical galaxy profiles could be well fitted using the distribution function,

(6.8) |

where the constants
*A*, ,
*E*_{esc} are treated as free parameters. The
distribution function (6.8) is an approximation to the exact
steady-state solution of the Fokker-Planck equation
^{(7)} and the model was
designed primarily for non-rotating spherical globular clusters for
which two-body relaxation is important
(*t*_{R}
10^{8} yrs). The good fit to
elliptical galaxies was, therefore, somewhat puzzling. Since two-body
relaxation is unimportant in giant ellipticals it had been suggested
that some sort of mean field relaxation mechanism operating during the
collapse phase might be responsible for the distribution function
(6.8). The mechanism was first discussed in detail by
Lynden-Bell (1967)
[see also
Shu (1978)
for a recent discussion]. Lynden-Bell
argued that during the collapse of the galaxy one would expect large
fluctuations in the potential
/
~ 1. Since the
potential is changing, individual stars do not follow energy conserving
orbits, hence
*E* / *E*
~ 1 and the stars undergo a kind of violent relaxation. The
subsequent oscillations of the galaxy would be damped within a few
collapse times by Landau damping and phase mixing. Lynden-Bell
calculated the most probable state to which such violent relaxation
would lead, with the result,
^{(8)}

(6.9) |

This distribution function corresponds to the isothermal sphere (e.g. Chandrasekhar, 1960), which is unacceptable as a model for a galaxy since it leads to a divergent total mass. However, it is clear that high energy orbits with periods longer than the collapse time could not be fully populated. Lynden-Bell argues that such incomplete violent relaxation results in a distribution function of the form (6.8). Since Lynden-Bell's classic paper, a great deal of numerical work has been done to test the theory. Several relevant questions are: (i) Does the theory really lead to density profiles characteristic of elliptical galaxies? (ii) To what extent does the final equilibrium configuration depend on initial conditions such as asphericity? (iii) What happens if the initial state is rotating? We discuss these questions next.

^{7} The
Fokker-Planck equation allows an approximate treatment of
close-encounters (e.g.
Michie, 1963).
Back.

^{8} This is the non-degenerate limit
of Lynden-Bell's most probable
state. Degeneracy effects are unimportant for collisionless systems
such as galaxies
(Shu, 1978).
Back.