Annu. Rev. Astron. Astrophys. 1982. 20:
399-429
Copyright © 1982 by Annual Reviews. All rights reserved |

The variation of mass density, luminosity density, and velocity dispersion as a function of radius are most easily discussed in terms of spherical models. These are much easier to construct than nonspherical models, and neither observation nor theory indicate that the radial profiles of nonspherical galaxies are affected in important ways by the shapes of the systems. In particular, the works of Saaf (1968) and Richstone (1981) indicate that the total angular momentum of a star is approximately conserved when it orbits in a mildly nonspherical potential.

If a galaxy is spherical, stars orbiting in its potential are
constrained by four
independent isolating integrals - the three components of the angular
momentum vector **J** and the energy - and we may invoke Jeans'
theorem to construct models by taking *f* to be an arbitrary positive
function of these four integrals.
If the galaxy is not only spherical, but also spherically symmetric in
all its properties, *f* can depend on
*J*_{x}, *J*_{y} and
*J*_{z} only through the combination
*J*^{2} = *J*_{x}^{2} +
*J*_{y}^{2} + *J*_{z}^{2} so
*f* is then of the form *f* = *f* (*E*, *J*).

SYSTEMS HAVING *f* (*E*) For many years
elliptical galaxies have been discussed in terms of models whose
distribution functions depend only on *E*. These
models are interesting, but it is important to recognize that they
constitute a narrowly restricted class of possible, spherically
symmetric galaxies, and it is
unlikely that Nature confines herself to models of this type. The
majority of these models are modifications of the isothermal sphere,
whose distribution function is simply

(9) |

where (0) is the
potential energy at the center of the system. Integrating
*f*_{I} over all velocities yields the density at radius
*r* as

(10) |

If the system is self-gravitating, one obtains on solving Poisson's equation with replaced by (10),

(11) |

where

(12) |

is the "core radius" at which the projected density falls to very nearly
1/2 of its value at the center. Substituting Equation (11) into (10),
one sees that at large radii,
~
*r*^{-2}. If the system is not self-gravitating, but
sits in the potential of
another isothermal population whose velocity dispersion
_{h}
differs from that of the first system, one has from equations (10) and
(11) that at large radii the density of the first population falls off as
~
*r*^{-2(h /
)2}.
Gunn (1977)
has argued that the brightness profiles of galaxies may fall off more
steeply than as *r*^{-2} because the velocity dispersion
of the luminous stars
is less than that
(_{h})
of the mass-bearing halo population by a factor of order
/
_{h} = 0.82.

An alternative strategy for obtaining from *f*_{I} a system
that looks like an elliptical galaxy is to truncate *f*_{I}
at some maximum energy *E*_{t}.
Woolley (1954)
simply set *f* equal to *f*_{I} for *E* less than
*E*_{t}, and to zero otherwise.
King (1966),
in parallel with
Michie (1963),
eliminated the discontinuity in Woolley's
distribution function at *E*_{t}, by defining for
*E* < *E*_{t}.

(13) |

Wilson (1975)
subsequently eliminated the discontinuity in the gradient
of *f*_{K} at *E*_{t} by defining for
*E* < *E*_{t},

(14) |

Proceeding in this way one may generate a sequence of models, all of which
are effectively isothermal near their centers, where
*E* << *E*_{t}
but which have nonisothermal envelopes.

Hunter (1977) has shown that the structure of the envelopes of these models depends sensitively and in a paradoxical way on the detailed form of the distribution function near the tidal cutoff. In particular, Wilson spheres, which have more heavily truncated distribution functions than King models, have much more extensive envelopes. It follows from this state of affairs that one cannot say a priori whether tidal encounters between galaxies lead to tidal truncation of the galaxies or to distension of their envelopes. The observations discussed below suggest the latter (Kormendy 1977; but see Strom & Strom 1978d).

As is discussed in Section 3.2, the
brightness profiles at the centers of
elliptical galaxies tend to be more peaky than the projected density of an
isothermal sphere. Therefore it is interesting to study model galaxies
whose projected density profiles have a cusp at the center.
Eddington (1916)
showed how to find the *f*(*E*) that generates a galaxy of any
given radial density profile, and one may apply this apparatus
(Binney 1982a)
to find the distribution function
*f*_{1/4}(*E*) that generates the galaxy whose
projected surface density obeys
de Vaucouleurs' (1948)
*r*^{1/4} law of surface brightness. One finds that
*f*_{1/4}(*E*) rises steeply at energies that
correspond to stars confined to the center of the galaxy. It is this
abundance of tightly bound stars that gives rise to the
central density peak and velocity dispersion depression
(Bailey & MacDonald
1981)
that are characteristic of the *r*^{1/4} model.
Binney (1982a)
has proposed a theoretical interpretation of this model.

SYSTEMS HAVING *f*(*E*, *J*)
Eddington (1914; see also
Shiveshwarkar 1936)
considered simple models, based on
*f* = *f*(*E*, *J*), that can probably not
be used to describe any real system, but which do illustrate the way in
which velocity
dispersion anisotropy affects the structure of spherical systems. The
distribution functions of these Eddington models are of the form

(15) |

The part of Eddington's distribution function (15) that depends on *J*
causes the density at radius *r* to drop from the value [Equation (10)]
associated with the isothermal sphere to

(16) |

and causes each tangential component of squared velocity dispersion to diminish by a fraction of the radial component of , where

(17) |

Thus the velocity dispersion tensor in an Eddington model is isotropic at the center and wholly anisotropic at large radii. The radial component of velocity dispersion equals the constant at all radii.

The outermost part of most Eddington models is an envelope in which
~
*r*^{-2} . This envelope bears a superficial resemblance to
the outermost portion of the isothermal sphere, but it is actually of an
entirely different nature because the circular velocity
*v*_{c}(*r*) in an Eddington model
tends to zero as *r*
increases, rather than to a finite constant as in the isothermal sphere
(*v*_{c} = 2^{1/2}
) . On the other hand,
the radial component of velocity dispersion
in an Eddington model is always equal to
. Therefore the
gravitational attraction of stars interior to *r* slows or deflects the
motion of a star with typical
speed less and less as
*r* increases, and stars far from the core of an Eddington
model execute giant oscillations in radius. The system becomes, in fact,
a kind of stellar traffic jam in which each star moves with more or less
uniform velocity on a radial path. The
~
*r*^{-2} increase of density toward the center
has less to do with gravity and dynamics than with congestion of these
trajectories. In terms of Equation (7), one may say that at the outside
of an Eddington model the term on the right-hand side of this equation
has dropped out, leaving the structure to be determined by a balance
between the two terms on the left.

Color gradients (de Vaucouleurs 1961, Strom & Strom 1978a, b, c) in principle offer a way of constraining the degree of radial velocity anisotropy in a galaxy. If all the stars of a galaxy were on circular orbits, so that the galaxy could be considered to be constructed of infinitesimally thin spherical shells, color changes could be perfectly sharp in three-dimensional space and moderately sharp when projected onto the sky. If, on the other hand, the system resembled the asymptotic portion of an Eddington model, all stars would contribute equally to the light at each radius, and no color gradient would be possible.

Michie (1963) studied the models that are related to King models in the same way that Eddington models are related to the isothermal sphere. The distribution function of a Michie model is

(18) |

For sufficiently large anisotropy radius *r*_{a}, a Michie
model behaves like a King
model in that it has a "tidal radius" at which the density goes to
zero. When *r*_{a} is small, the Michie model has an infinite
envelope that resembles the
envelope of an Eddington model. The Michie models that are of the. greatest
interest from the point of view of modeling real galaxies are those that
have finite tidal radii. In these models the anisotropy parameter
never comes
close to unity.