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

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

Origin of Virial Equations

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

Standard Lagrangian Representation
of Euler's Equation,

Dv = - (1/r) ÑP - ÑF.

[Equation I.A.1]

 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),

rDv = - ÑP - rÑF .

[Equation I.A.3]

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

rDvj = - ÑjP - rÑjF ,

[Equation I.Z.1]
=

EFE, Chapter 2, § 11, Eq. (38)2

 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:

I II III
òV [ xirDvj ] d3x = - òV [ xi ÑjP ] d3x - òV [ xirÑjF ] d3x .

[Equation I.Z.2]

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

Divergence Theorem:

The integral of the divergence of any vector F over a volume V is equal to the surface integral of the normal component of the vector over the surface S that is bounding V. That is,

 òV Ñ× F d3x = òSF ×n da .
[Equation I.Z.3a]
=

Arfken, § 1.11, Eq. (1.94a)

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,

Form B
of
Gauss's Theorem

 òV ÑG d3x = òSG n da .
[Equation I.Z.3b]
=

Arfken, § 1.11, Eq. (1.102)

Furthermore, it will be useful to notice that, for any fluid variable "q",

 rDq = D( rq ) - q Dr = ¶t( rq ) + v × Ñ( rq ) + ( rq ) Ñ×v = ¶t( rq ) + Ñ× [ ( rq )v ] ,

 where in order to derive the second line we have utilized both expression [VI.M.14], which relates the Lagrangian time-derivative to the Eulerian time-derivative, and the standard Lagrangian representation of the continuity equation [I.B.1]. Hence, via the Divergence Theorem,

 òV ( rDq ) d3x = òV ¶t( rq ) d3x + òS( rq v )×n da = ¶t òV ( rq ) d3x + òS( rq v )×n da .

[Equation I.Z.4]

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,

 òVL Q d3x ,
is the total mass enclosed by the Lagrangian volume VL and its time-derivative,3

 D òVL Q d3x ,

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

Reynolds Theorem
which states:

 D òVL Q d3x = ¶t òV Q d3x + òS( Q v )×n da ,

[Equation I.Z.5]
=

Tassoul '78, § 3.2, Eq. (17)

 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,

 òVL ( rq ) d3x .
 So, by combining equations [I.Z.4] and [I.Z.5] we recognize the following relationship, which has been presented by Chandrasekhar (1987) in the form of a

Lemma:

If q(x,t) is any attribute of a fluid element, then

 D òVL ( rq ) d3x = òV ( rDq ) d3x .

[Equation I.Z.6]
=

EFE, Chapter 2, § 11, Eq. (40)
Tassoul '78, § 3.3, Eq. (23)3

 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:

 The expression under the integral sign on the left-hand-side of equation [I.Z.2] may be manipulated into a variety of different forms. Notice, first, that by bringing the component of the position vector inside the time-derivative and realizing that vi = Dxi , we may write,

xirDvj = r [D( xivj ) - vivj ] .

[Equation I.Z.7]

 Hence, Term I in equation [I.Z.2] may be rewritten as,

 òV (xirDvj ) d3x = òV r D( xivj ) d3x - òV (rvivj ) d3x = D òVL ( rxivj) d3x - 2 Tij ,

[Equation I.Z.8]
=

Tassoul '78, § 3.7, Eq. (136)

 where, in deriving the second line of this relation, we have employed Lemma [I.Z.6] and have adopted the following standard

Definition of the
Kinetic Energy Tensor

 Tij º (1/2) òV rvivj d3x .

[Equation I.Z.9]
=

EFE, Chapter 2, § 9, Eq. (9)
BT87, Chapter 4, Eq. (4-74b)
Tassoul '78, § 3.7, Eq. (137)

 Now we note that both terms on the right-hand-side of equation [I.Z.2] are symmetric in the indices i and j. (This is not immediately obvious by looking at equation [I.Z.2], but is apparent following the discussions, below, of both of these terms.) Hence, the sub-expressions that contribute to Term I on the left-hand-side of the equation must be symmetric in these indices as well. We may therefore rewrite the integrand of the first term on the right-hand-side of equation [I.Z.8] as,

 rxivj = (1/2) r ( xivj + xjvi ) = (1/2) r ( xi Dxj + xj Dxi ) = (1/2) r D( xixj ) .

[Equation I.Z.10]

 Performing a volume integral over this last expression and, once again employing Lemma [I.Z.6], we conclude that,

 òV ( r xivj ) d3x = (1/2) òV r D( xixj ) d3x = (1/2) D òVL ( rxixj) d3x = (1/2) D Iij .

[Equation I.Z.11]

where, in deriving this last line, we have adopted the following standard

Definition of the
Moment of Inertia Tensor

 Iij º òV rxixj d3x .

[Equation I.Z.12]
=

EFE, Chapter 2, § 9, Eq. (4)
BT87, Chapter 4, Eq. (4-76)
Tassoul '78, § 3.7, Eq. (146)

 Finally, then, by combining relation [I.Z.8] with relation [I.Z.11] we see that Term I may be written simply as,

 òV (xirDvj ) d3x = (1/2) D2Iij - 2 Tij ,

[Equation I.Z.13]
=

Tassoul '78, § 3.7, Eq. (136)

Term II:

 Integrating the first term on the right-hand-side of equation [I.Z.2] by parts (and assuming that at all locations in space the gas pressure is isotropic4), we obtain,

 - òV [ xi ÑjP ] d3x = - òV [ Ñj( Pxi ) ] d3x + dij òV P d3x .

[Equation I.Z.14]

Definition of the
Total Internal Energy

 U º òV ( er ) d3x ,

[Equation I.Z.15]

 using Form B [II.A.4] of the ideal gas equation of state to relate the product er to the gas pressure P, and applying Form B of Gauss's Theorem to the first term on the right-hand-side of this equation [I.Z.14], we further conclude that,

 - òV [ xi ÑjP ] d3x = - òS [ Pxi ] nj da + dij ( g - 1 ) U ,

[Equation I.Z.16]

where nj is the jth component of unit vector that is normal to the surface. Finally, then, if we demand that the volume over which the virial is evaluated always be constructed such that the gas pressure goes to zero everywhere on its surface S, we can write Term II simply as,

 - òV [ xi ÑjP ] d3x = dij ( g - 1 ) U .

[Equation I.Z.17]

Term III:

Definition of the
Chandrasekhar Potential Energy Tensor

 Wij º ò r(x) xi aj d3x,

[Equation I.Z.17]
=

BT87, Chapter 2, Eq. (2-123)
EFE, Chapter 2, § 10, Eq. (18)
Tassoul '78, Chapter 3, Eq. (140)

where aj is the jth component of the (vector) gravitational acceleration defined earlier3 in terms of the gravitational potential as,

a = - ÑF ,

[Equation I.H.4]

 then Term III in equation [I.Z.2] becomes,

 - òV [ xir ÑjF ] d3x = + Wij .

[Equation I.Z.18]

As has been demonstrated explicitly in both BT87 and EFE, the components of the tensor Wij may equivalently be expressed in terms of the following double volume integral,

Wij = - (1/2) G ò ò r(x) r(x') [ (xi' - xi)(xj' - xj) ] | x' - x |-3 d3x d3x',

[Equation III.R.6]
=

BT87, Chapter 2, Eq. (2-126)
EFE, Chapter 2, § 10, Eqs. (14) & (15)
Tassoul '78, Chapter 3, Eq. (139)

demonstrating the "manifestly symmetric" nature of the tensor.

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

The 2nd-Order Tensor Virial Equation(Tensor Virial Theorem)

(1/2) D2Iij = 2 Tij + dij (g - 1) U + Wij.

[Equation I.E.1]
=
EFE, p.23, Eq. (51)
BT87, p.213, Eq. (4-78)

Footnotes

1As Chandrasekhar (1987) points out (see, specifically, Chapter 2, § 11(a), p. 21 of EFE), the related 1st-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 Pij that is defined in terms of a nonisotropic velocity dispersion sij as follows: