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

Since before
Baade (1944)
resolved the brightest stars in the bulge of
M31, it has been recognized that spheroidal
components consist of swarms of stars that move about in the
gravitational field that is generated by themselves and any
other components that have an appreciable density in the same volume of
space. Observations show that typical stellar speeds exceed 200 km
s^{-1}, so that
a star completes a half-orbit of semimajor axis *r* kpc in

(1) |

The effective smoothness of the gravitational field in which the stars of a
galaxy move may be measured by comparing the two-body relaxation time
*T*_{2B} to *T*_{cr}.
*T*_{2B} is the time typically required for a star
in the system to be deflected
from the path it would follow in the smooth potential by encounters with
other stars. If the system is only modestly concentrated toward its
center and is made up of *N* equal point masses, a simple
application of the virial theorem and the expression for
*T*_{2B} given by
Spitzer & Härm
(1958)
shows that

(2) |

If the constituent masses are unequal, *N* in this formula should
be set equal to the total mass of the system divided by the mass of the
heaviest point masses. If we take
*N* > 10^{10}, we may conclude that in the main body of a
spheroidal component
*T*_{2B} > 10^{8} × *T*_{cr}
> 10^{14} y, so that the trajectories of stars may be
computed using a smooth mean field for times longer than the Hubble time
*T*_{H}.
However, one should note that there are two circumstances in which two-body
effects can play a role in the dynamics of spheroidal components. The
first is when the effective value of *N* is smaller than
10^{5}. For example, a massive globular cluster may contain
10^{7}
*M*_{},
i.e. 10^{-3} of the mass of a typical
spheroidal component, with the result that a system of globular clusters
orbiting through a spheroidal component corresponds to
*N*
10^{3} and may suffer appreciable evolution within a Hubble time
(Tremaine et al. 1975).
The second case is when a spheroidal component is so centrally concentrated
that its nucleus may become dynamically separate from the main body of the
galaxy. The crossing time
*T*_{cr} may be very small for stars in the nucleus (e.g.
*T*_{cr} < 10^{5} y in the nucleus of M31) and
*N* may be less than or of order 10^{7}, so
that *T*_{2B} < *T*_{H} there.
Spitzer & Saslaw
(1966)
have discussed the significance
of this situation for the formation of active galactic nuclei.

If one confines oneself to consideration of the main body of the galaxy,
the unimportance of two-body evolution enables one to describe the
dynamics of the system with the single-particle distribution function
*f*(**x**, **v**) that gives the
mass density of stars at the point (**x**, **v**) in phase
space. If *f* is a differentiable
function of **x** and **v**, then it evolves in time according
to the collisionless Boltzmann or Vlasov equation:

(3) |

One is usually interested in the steady states of galaxies, when
*f* /
*t* might
be supposed to vanish, at least when *f* is referred to some suitable
rotating coordinate system. Unfortunately, a distribution function *f*
that is initially
smooth may become with the passage of time less and less smooth in certain
parts of the space, with the result that for many potentials differentiable
time-independent solutions of Equation (3) do not exist. When the
*f* that
satisfies Equation (3) becomes rough in this way, one must distinguish
between the fine-grained distribution function *f* and the
coarse-grained distribution function *f*_{c}, obtained by
averaging *f* over macroscopic regions in phase
space. When *f* does not become rough, and the right-hand side of
Equation (3) vanishes for some differentiable *f*, Equation (3) then
states that *f* is constant
along the curves in phase space that are followed by the representative
points of stars as the stars orbit in the potential. That is, the
time-independent Vlasov
equation states that *f* is an integral of the equations of stellar
motion. If *I*_{1}(**x**, **v**), ...,
*I*_{n}(**x**, **v**) is a complete set of integrals,
then

(4) |

which is known as Jeans' theorem. Conversely, any integral or function of any integrals that happen to be known gives solutions to the time-independent Vlasov equation.

Jeans' theorem raises the thorny question of how many integrals there are in a complete set. It is important to distinguish between isolating integrals and nonisolating integrals. An isolating integral is one for which the equation

(5) |

where *C* is a constant, defines a smooth five-dimensional
hypersurface in phase space. It is clear that the minimum number of
isolating integrals is one, because the energy
*E*(**x**, **v**) is always such an integral, and the
maximum
number is five, because the intersection of the *N* five-dimensional
hypersurfaces associated with each isolating integral must contain the
one-dimensional space of an individual orbit. However, very few
potentials admit as many as five isolating integrals. The usual number
of isolating integrals is three, one for each pair of canonical coordinates.

When there are three or more isolating integrals, the orbit of a star is quasi-periodic; that is, the evolution of any phase space coordinate may be expressed as a Fourier series

(6) |

where _{1},
_{2} and
_{3} are
frequencies characteristic of the orbit and nearly all
the power is concentrated in the terms involving small *n*,
*m*, and
(Binney & Spergel
1982).
Four or five isolating integrals arise when two or more of the
frequencies _{i}
are commensurable. An example of this phenomenon is given
by motion in the Kepler potential
~ 1 / *r*. Then
the orbital frequency is the
only independent frequency, and there are five isolating integrals.

In a general potential, some (often the great majority) of the orbits are
quasi-periodic and admit three isolating integrals. These are the
regular orbits.
But there are usually some orbits, called irregular or stochastic
orbits, that are not quasi-periodic. These orbits cannot admit three
isolating integrals because their representative points in phase space
occupy a volume that has more than three dimensions, and therefore
cannot be described as the intersection of three
five-dimensional hypersurfaces. However, these orbits do not pass close to
every point in the five-dimensional hypersurface of their energy
integral, i.e. they are not ergodic in the sense of classical statistical
mechanics. The so-called KAM theorem (e.g.
Moser 1977)
assures us that there are always
regions of the energy hypersurface that are forbidden to them because that
space is occupied by regular orbits. And one finds that even in parts of
the phase space where irregular orbits are in the majority, a given
irregular orbit will spend the majority of its time in certain areas
(Ichimura & Saito
1978,
Goodman &
Schwarzschild 1981,
Binney 1982b).
This complex orbital structure prevents the fine-grained distribution
function *f*, for which the Vlasov equation (3) holds, from
reaching a steady state.

It is uncertain what role the irregular orbits play in the structure and
evolution of spheroidal components. In particular, we do not know what
features of the potential
determine what
proportion of all orbits are irregular. But in
any galaxy in which an appreciable fraction of the stars are on
irregular orbits, the coarse-grained distribution function
*f*_{c}(**x**, **v**), which
is the quantity of astronomical interest, will not obey the
time-independent Vlasov equation and hence will not be a function of
(**x**, **v**) only through the integrals of stellar
motion. Therefore Jeans' theorem should be used more cautiously than
perhaps it has been in the past. Furthermore, irregular orbits may
evolve in a systematic way on time scales that are long compared to the
dynamical time of the system, but not longer than the Hubble time. If
this is so, they may drive galactic evolution in a way that has yet to
be fully explored (Sanders, in preparation).

The ideal basis for the interpretation of observations would be an array of
stellar dynamical models deriving from exact solutions of the Vlasov
equation (3). Unfortunately we are far from possessing such a hoard of
treasure and we have to confine ourselves, for the most part, to
statements about the properties such models would have if we were able
to construct them. Constraints of this
type are obtained by integrating multiples of the Vlasov equation over all
velocities (or over both velocities and space) to obtain moment equations.
Thus when one multiplies Equation (3) by 1 or **v** and integrates
over all
velocities, one obtains the equations that have been called the
equations of stellar hydrodynamics by reason of their similarity to the
equations of fluid flow. [A more convenient and less misleading
designation might be the "Jeans equations" in honor of
Jeans' (1922)
investigation of them.] For future reference note that the equation
obtained on multiplication of (3) by **v** is for a
steady-state system of spherical symmetry

(7) |

where

(8a) (8b) (8c) (8d) |

When the Vlasov equation (3) is multiplied by *x*_{i}
_{j} and integrated
over all velocities and spatial coordinates, the equations of the tensor
virial theorem are obtained
(Chandrasekhar 1964,
Binney 1978a).
This theorem is valuable because it states that the ratios
<
*v*_{z}^{2}> /
<
*v*_{x}^{2}> , etc., of the kinetic energies
associated with the components of motion parallel to the three body-axes of
the system depend on the shape of the system and the speed with which the
figure rotates with respect to inertial space, but not on its radial
density profile. In particular, if one knows the figure of a flattened
axisymmetric galaxy, one
immediately knows how much more kinetic energy is associated with motion
parallel to the equatorial plane than perpendicular to this plane. Some
of this additional kinetic energy will be associated with rotation, and
the rest with anisotropy of the velocity dispersion tensor.

The tensor virial theorem has been used in recent years as the standard framework within which to analyze observations of the kinematics of spheroidal components. However, it should be realized that the popularity of the virial theorem arises not so much from its own merits, but because we lack realistic models of flattened spheroidal components. Consequently, we are obliged to reduce the wealth of information contained in the best recent observations of spheroidal components to the pair of numbers that can be accommodated by the tensor virial theorem. Section 4 discusses the origin of this unfortunate situation.