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

2. BASIC CONSIDERATIONS

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 > 10¹⁰, we may conclude that in the main body of a spheroidal component T_2B > 10⁸ × T_cr > 10¹⁴ 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 10⁵. For example, a massive globular cluster may contain 10⁷ 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 10³ 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 < 10⁵ y in the nucleus of M31) and N may be less than or of order 10⁷, 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 I₁(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_z²> / < v_x²> , 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.