The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

[ Download PDF file dated ]

The 2^nd-order tensor virial equation (as opposed to the 1^st-order virial equation¹) is derived by taking the "first moment" of the Euler equation. Here we begin the derivation by referring specifically to the

Multiplying this equation [I.A.1] through by the mass density r produces the relation (shown earlier in the context of our discussion of the numerous different forms of Euler's equation),

In terms of its individual Cartesian vector components, this equation [I.A.3] takes the form,

where Ñj refers to the jth component of the gradient operator. The first moment of Euler's equation is then obtained by multiplying equation [I.Z.1] through by each of the components xi of the Cartesian position vector x, then integrating the resulting equations over the entire volume of the physical system under consideration. Performing these steps initially produces a set of nine integral equations (one each for i = 1,2,3 and j = 1,2,3), each containing the following three terms:

Before we attempt to manipulate each one of the identified terms in this equation into a more recognizable form, we must remind the reader of Gauss's Theorem which, in its most familiar form is also known as the

We shall also need to utilize a similar relation involving the volume integral over the gradient of any scalar function G(x), which we will refer to as,

This last step -- in which we have moved the partial time-derivative outside of the volume integration -- is justified as long as the (Eulerian) volume over which we are integrating does not change with time.

Now, the volume V over which we have performed this integration [I.Z.4] has been understood to be an Eulerian volume, that is, a volume whose surface definition S does not change with time. (It is also understood that the fluid may flow through this surface with time.) Conceptually, there is also a Lagrangian volume VL (which, at time "t" may overlap precisely with the Eulerian volume V) whose surface definition may change with time in order to ensure that the volume VL always contains precisely the same constituent particles. It is over this Lagrangian volume that we would like to integrate a variety of different fluid variables Q(x,t) in order to determine their instantaneous global values and ascertain whether or not these global values change with time. For example, if we let Q = r, then the global variable defined by the integral,

should be zero. Well, via a very careful proof (which we will not reproduce here) of the

Tassoul (1978; see specifically § 3.2, pp. 46-47), has demonstrated that the right-hand-side of equation [I.Z.4] is precisely the Lagrangian time-derivative3 of the global variable defined by the volume integral,

We will find this Lemma and Gauss's Theorem to be very useful as we examine, in turn, each of the identified terms in equation [I.Z.2].

Term I:

Term II:

Term III:

So, replacing each of the terms in equation [I.Z.2] by the expressions given, respectively, by equations [I.Z.13], [I.Z.17], and [I.Z.18], we get the following expression for

¹As Chandrasekhar (1987) points out (see, specifically, Chapter 2, § 11(a), p. 21 of EFE), the related 1^st-order equation is obtained by simply (multiplying through by "1" then) integrating the Euler equation over the volume occupied by the fluid. This produces "an equation which simply expresses the uniform motion of the center of mass; it provides no essentially new information."

²Note that throughout his book, Chandrasekhar (1987) adopts a sign convention for the scalar gravitational potential that is opposite to the sign convention used herein.

³By using the color yellow for the operator D, we are following Tassoul's (1978) lead and drawing a distinction between a standard Lagrangian time-derivative and the time differentiation that is being carried out in this expression. Here, the time derivative is being taken of a (global) variable that exhibits no spatial variation and, hence, depends only on time (see footnote 3 on p. 45 of Tassoul '78). Neither Chandrasekhar (1987) nor Binney and Tremaine (1987) have adopted a notation that clearly draws a distinction between the operators D and D, but for added clarity we will do so throughout this H_Book.

⁴In discussing primarily stellar (rather than gas) dynamical systems, Binney and Tremaine (1987) include in the virial equations a more general pressure tensor P_ij that is defined in terms of a nonisotropic velocity dispersion s_ij as follows: