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

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

[ Download PDF file dated ]

Properties of Homogeneous Ellipsoids

Gravitational Potential

Earlier we demonstrated that the acceleration due to the gravitational attraction of a distribution of mass r can be derived from the gradient of a scalar potential F defined as follows:

F(x) º - ò | x' - x |-1 G r(x') d3x'.

[Equation I.H.2]

As has been demonstrated explicitly in Chapter 3 of Chandrasekhar (1987) and summarized in Table 2-2 (p. 57) of Binney & Tremaine (1987), for an homogeneous ellipsoid this volume integral can be evaluated analytically in closed form. Specifically, at an internal point or on the surface of an homogeneous ellipsoid with semi-axes (x,y,z) = (a1,a2,a3),

F(x) = - p Gr { IBTa12 - [ A1x2 + A2y2 + A3z2 ] },

[Equation III.R.1]
=
EFE, Chapter 3, Eq. (40)1,3
BT87, Chapter 2, Table 2-2

where

Ai º a1a2a3 ò { D (ai2 + u) }-1 du [Equation III.R.2]
IBT º a2a3a1-1 ò du/D = A1 + A2(a2/a1)2 + A3(a3/a1)2 [Equation III.R.3]
D º { (a12 + u)(a22 + u)(a32 + u) }1/2 [Equation III.R.4]
=
EFE, Chapter 3, Eqs. (18), (15 & 22)2, & (8), respectively
BT87, Chapter 2, p. 57, Table 2-2

and the integrals defining IBT and Ai are performed over the interval 0 ® ¥. As detailed in the Table III.R.1, below, the integrals defining IBT and Ai can be expressed in terms of standard incomplete elliptic integrals (or more elementary functions if either a2 = a1 or a3 = a2).

Potential Energy Tensor

Particularly when discussing the global properties of equilibrium structures as mandated by the tensor-virial equations, we will find it useful to adopt the following

Definition of the
Chandrasekhar Potential Energy Tensor

Wij º ò r(x) xi aj d3x,

[Equation III.R.5]
=
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]

As has been demonstrated explicitly in both BT87 and EFE, the components of the tensor W 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.

As has been derived in Chapter 3 of Chandrasekhar (1987) and summarized in Table 2-2 (p. 57) of Binney & Tremaine (1987), for an homogeneous ellipsoid with semi-axes (x,y,z) = (a1,a2,a3), the various components of the Chandrasekhar Potential Energy Tensor are given by the expression,

Wij = - 2 p Gr Ai Iij = - (8/15) p2 Gr2 a1a2a3 [Ai ai2 dij ] ,

[Equation III.R.7]
=
EFE, Chapter 3, Eq. (128)
BT87, Chapter 2, p. 57, Table 2-2

where, for homogeneous ellipsoids, the moment of inertia tensor,

Iij = (1/5) M ai2 dij, (M = mass = 4p a1a2a3 r / 3).
[Equation III.R.8]
=
EFE, Chapter 3, Eqs. (129)

Total Potential Energy

For any self-gravitating configuration, the total gravitational potential energy Egrav can be obtained from the trace of the potential energy tensor, that is,

Egrav = W11 + W22 + W33.

[Equation III.R.9]
=
BT87, § 2.5, Eq. (2-127)
EFE, Chapter 2, § 10, Eq. (13)

In particular, for any homogeneous ellipsoid this implies,

Egrav = - (8/15) p2 Gr2 a1a2a3 IBTa12.

[Equation III.R.10]
=
BT87, Chapter 2, p. 57, Table 2-2

Evaluating the Coefficients Ai and IBT

In order to evaluate the functions F(x), Wij, and Egrav for an homogeneous ellipsoid of a given size, shape and density, one need only determine the values of the coefficients Ai and IBT. Clicking on the appropriate box in the following mosaic will take you to an accompanying page that specifies the analytical function for these coefficients in the case of oblate spheroids, prolate spheroids, or general triaxial figures.

Table III.R.1
Spherical
a1 = a2 = a3		 Oblate
a1 = a2 > a3		 Prolate
a1 > a2 = a3		 Triaxial
a1 > a2 > a3
A1 2/3
A2
A3
IBT 2

Show that for a uniform-density sphere of radius R and mass M, equations [III.R.1], [III.R.8], and [III.R.10], reduce to the proper expressions. That is, show that they give the following expressions for the gravitational potential, moment of inertia, and potential energy, respectively, of a

Uniform-Density Sphere:

F(r) = - (3/2) G M R-1 [ 1 - ( 1/3 ) ( r/R )2 ] ;

[Equation III.R.11]

I = Trace[ I ] = (3/5) M R2 ;

[Equation III.R.12]

Egrav = Trace[ W ] = - (3/5) G M2 R-1.

[Equation III.R.13]

Footnotes

1,2In reference EFE, you'll find these equations written in terms of a variable "I" instead of "IBT" as defined here. The two variables are related to one another straightforwardly through the expression, I = IBTa12.

3 Note that throughout his book entitled, "Ellipsoidal Figures of Equilibrium," Chandrasekhar adopts a sign convention for the scalar gravitational potential that is opposite to the sign convention used herein.

Home Page | Preface | Context | Applications | Appendices | Search Index