Steady-State 2^nd-Order Tensor Virial Equations

By satisfying all six — not necessarily unique — components of the Second-Order Tensor Virial Equation, the entire set of Riemann Ellipsoids can be determined.

Summary

Here we employ the 2^nd-order tensor virial equation (TVE),

<math>~ 2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi + \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx \, , </math>

to determine the equilibrium conditions of uniform-density <math>~(\rho)</math> ellipsoids that have semi-axes, <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c),</math> and an internal velocity field, <math>~\vec{u}</math> (as prescribed below), that preserves this specified ellipsoidal shape, as viewed from a frame of reference that is rotating with angular velocity, <math>~\vec\Omega</math>. Because each of the indices, <math>~i</math> and <math>~j</math>, run from 1 to 3, inclusive, this TVE appears to provide nine equilibrium constraints; and once the values of the density and the three semi-axes are specified, there appear to be seven unknowns: <math>~\Pi</math> and the three pairs of velocity-field components <math>~(\Omega_1, \zeta_1)</math>, <math>~(\Omega_2, \zeta_2)</math>, <math>~(\Omega_3, \zeta_3).</math> In practice, however, only five constraints are relevant/independent because, as is encapsulated in …

Riemann's Fundamental Theorem

… non-trivial solutions are obtained only if no more than two of the three pairs of velocity-field components are different from zero.

Following EFE, we will set <math>~\Omega_1 = \zeta_1 = 0</math>, in which case the only applicable TVE constraint relations are the five identified in the following table of equations.

Indices

Each Associated 2^nd-Order TVE Expression

<math>~ \biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi +\biggl\{ ( \Omega_2^2 + \Omega_3^2) + 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 + 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 ~-~(2\pi G\rho) A_1 \biggr\} a^2 + \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2 + \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2 </math>

<math>~ \biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi + \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2 + \biggl\{ \Omega_3^2 + 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~-~( 2\pi G \rho) A_2 \biggr\}b^2 </math>

<math>~ \biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi + \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2 + \biggl\{ \Omega_2^2 + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 - (2\pi G \rho)A_3 \biggr\}c^2 </math>

<math>~\biggl\{ 1 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr] \biggr\} \Omega_2\Omega_3c^2 </math>

<math>~\biggl\{ 1 + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr] \biggr\} \Omega_2 \Omega_3b^2 </math>

General Coefficient Expressions

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, <math>~(a_1,a_2,a_3)</math>, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:

<math> ~A_1 </math>	<math> ~= </math>	<math>~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr) \biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, , </math>
<math> ~A_3 </math>	<math> ~= </math>	<math> ~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[ \frac{(a_2/a_1) \sin\theta - (a_3/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, , </math>
<math> ~A_2 </math>	<math> ~= </math>	<math>~2 - (A_1+A_3) \, ,</math>

where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)</math>	and	<math>~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .</math>
[ EFE, Chapter 3, §17, Eq. (32) ]

Adopted (Internal) Velocity Field

EFE (p. 130) states that the … kinematical requirement, that the motion <math>~(\vec{u})</math>, associated with <math>~\vec{\zeta}</math>, preserves the ellipsoidal boundary, leads to the following expressions for its components:

<math>~u_1</math>	<math>~=</math>	<math>~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,</math>
<math>~u_2</math>	<math>~=</math>	<math>~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,</math>
<math>~u_3</math>	<math>~=</math>	<math>~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .</math>
[ EFE, Chapter 7, §47, Eq. (1) ]

Equilibrium Expressions

[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,

<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>

<math>~- \delta_{ij}\Pi \, .</math>

[This] provides six integral relations which must obtain whenever the conditions are stationary.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2^nd-order TVE takes on the more general form:

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi + \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx \, . </math>
[ EFE, Chapter 2, §12, Eq. (64) ]

EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by

<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>	<math>~=</math>	<math>~-2A_i I_{ij} \, ,</math>
[ EFE, Chapter 3, §22, Eq. (128) ]

where

<math>~I_{ij}</math>	<math>~=</math>	<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
[ EFE, Chapter 3, §22, Eq. (129) ]

is the moment of inertia tensor. Expressions for all nine components of the kinetic energy tensor, <math>~\mathfrak{T}_{ij}</math> are derived in Appendix E, below; and expressions for each of the six Coriolis components can be found in Appendices B, C, & D.

The Three Diagonal Elements

For <math>~i = j = 1</math>, we have,

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11} - \Omega_1\Omega_k I_{k1} + 2\epsilon_{1lm}\Omega_m \int_V \rho u_lx_1 d^3x </math>
	<math>~=</math>	<math>~ 2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11} - \Omega_1^2I_{11} + 2 \Omega_3 \int_V \rho u_2x_1 ~d^3x - 2\Omega_2 \int_V \rho u_3x_1 ~d^3x </math>
	<math>~=</math>	<math>~ 2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi +( \Omega_2^2 + \Omega_3^2) I_{11} + 2 \Omega_3\rho \int_V u_2x ~d^3x - 2\Omega_2\rho \int_V u_3 x~ d^3x </math>
	<math>~=</math>	<math>~ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22} + \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33} ~-~(2\pi G\rho) A_1 I_{11} + \Pi +( \Omega_2^2 + \Omega_3^2) I_{11} + 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 I_{11} + 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 I_{11} </math>
	<math>~=</math>	<math>~ \Pi + \biggl\{ ( \Omega_2^2 + \Omega_3^2) + 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 + 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 ~-~(2\pi G\rho) A_1 \biggr\} I_{11} + \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22} + \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33} </math>
<math>~\Rightarrow~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>	<math>~=</math>	<math>~ \biggl\{ ( \Omega_2^2 + \Omega_3^2) + 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 + 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 ~-~(2\pi G\rho) A_1 \biggr\} a^2 + \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2 + \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2 \, . </math>

Once we choose the values of the (semi) axis lengths <math>~(a, b, c)</math> of an ellipsoid — from which the value of <math>~A_1</math> can be immediately determined — along with a specification of <math>~\rho</math>, this equation has the following five unknowns: <math>~\Pi, \Omega_2, \Omega_3, \zeta_2, \zeta_3</math>. Similarly, for <math>~i = j = 2</math>,

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi + \Omega^2 I_{22} - \Omega_2\Omega_k I_{k2} + 2\epsilon_{2lm}\Omega_m \int_V \rho u_lx_2 d^3x </math>
	<math>~=</math>	<math>~ 2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi + (\Omega_1^2 + \Omega_3^2) I_{22} + 2\Omega_1 \rho \int_V u_3 y ~d^3x - 2\Omega_3 \rho \int_V u_1 y ~d^3x </math>
	<math>~=</math>	<math>~ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33} + \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11} ~-~( 2\pi G \rho) A_2 {I}_{22} + \Pi + (\Omega_1^2 + \Omega_3^2) I_{22} + 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1 I_{22} + 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 I_{22} </math>
	<math>~=</math>	<math>~ \Pi + \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11} + \biggl\{ (\Omega_1^2 + \Omega_3^2) + 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1 + 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~-~( 2\pi G \rho) A_2 \biggr\}{I}_{22} + \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33} </math>
<math>~\Rightarrow~~~-\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>	<math>~=</math>	<math>~ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2 + \biggl\{ (\Omega_1^2 + \Omega_3^2) + 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1 + 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~-~( 2\pi G \rho) A_2 \biggr\}b^2 + \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 c^2 \, . </math>

This gives us a second equation, but an additional pair of (for a total of seven) unknowns: <math>~\Omega_1, \zeta_1</math>. For the third diagonal element — that is, for <math>~i=j=3</math> — we have,

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi + \Omega^2 I_{33} - \Omega_3\Omega_k I_{k3} + 2\epsilon_{3lm}\Omega_m \int_V \rho u_lx_3 ~d^3x </math>
	<math>~=</math>	<math>~ 2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi + (\Omega_1^2 + \Omega_2^2) I_{33} + 2\Omega_2 \rho \int_V u_1 z ~d^3x - 2\Omega_1 \rho \int_V u_2 z ~d^3x </math>
	<math>~=</math>	<math>~\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11} + \biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22} - (2\pi G \rho)A_3 I_{33} + \Pi + (\Omega_1^2 + \Omega_2^2) I_{33} + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 I_{33} + 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1 I_{33} </math>
	<math>~=</math>	<math>~ \Pi + \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11} + \biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22} + \biggl\{ (\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 + 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1 - (2\pi G \rho)A_3 \biggr\}I_{33} </math>
<math>~\Rightarrow ~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>	<math>~=</math>	<math>~ \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2 + \biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 b^2 + \biggl\{ (\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 + 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1 - (2\pi G \rho)A_3 \biggr\}c^2 \, . </math>

This gives us three equations vs. seven unknowns.

Off-Diagonal Elements

Notice that the off-diagonal components of both <math>~I_{ij}</math> and <math>~\mathfrak{W}_{ij}</math> are zero. Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2^nd-order TVE is,

<math>~ 2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j d^3x \, . </math>

For example — as is explicitly illustrated on p. 130 of EFE — for <math>~i=2</math> and <math>~j=3</math>,

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 d^3x} - 2\Omega_3 \int_V \rho u_1x_3 d^3x \, , </math>
[ EFE, Chapter 7, §47, Eq. (3) ]

whereas for <math>~i=3</math> and <math>~j=2</math>,

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 d^3x - 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 d^3x} \, . </math>
[ EFE, Chapter 7, §47, Eq. (4) ]

Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, we recognize that,

<math>~\int_V \rho u_i x_j d^3x</math>	<math>~=</math>	<math>~0</math>	if <math>~i = j \, .</math>
[ EFE, Chapter 7, §47, Eq. (5) ]

This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,

<math>~\mathfrak{T}_{32} </math>

<math>~\mathfrak{T}_{23}</math>

Our first off-diagonal element is, then,

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} - 2\Omega_3 \rho \int_V u_1 z d^3x </math>
	<math>~=</math>	<math>~ - ~ \biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2 - \Omega_2\Omega_3 c^2 - 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_3 \zeta_2 c^2 </math>
	<math>~=</math>	<math>~\biggl\{ \Omega_2\Omega_3 + \biggl[ \frac{\zeta_2 a^2}{a^2 + c^2 }\biggr] \biggl[ 2\Omega_3 + \frac{\zeta_3 b^2}{b^2+a^2}\biggr] \biggr\} c^2 </math>
	<math>~=</math>	<math>~\biggl\{ 1 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr] \biggr\} \Omega_2\Omega_3c^2 \, . </math>

The second is,

<math>~0</math>	<math>~=</math>	<math>~ 2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \rho \int_V u_1 y d^3x </math>
	<math>~=</math>	<math>~ - ~ \biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2 - \Omega_3 \Omega_2 b^2 - 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\Omega_2 \zeta_3 b^2 </math>
	<math>~=</math>	<math>~\biggl\{ \Omega_2 \Omega_3 + \biggl[ \frac{\zeta_3 a^2}{a^2+b^2}\biggr] \biggl[2\Omega_2 + \frac{\zeta_2 c^2}{c^2 + a^2}\biggr] \biggr\} b^2 </math>
	<math>~=</math>	<math>~\biggl\{ 1 + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr] \biggr\} \Omega_2 \Omega_3b^2 \, . </math>

Check against §47 (pp. 130-131) of EFE. Subtracting these first two off-diagonal elements gives,

<math>~\biggl\{ 1 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr] \biggr\} \Omega_2\Omega_3c^2 </math>	<math>~=</math>	<math>~\biggl\{ 1 + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr] \biggr\} \Omega_2 \Omega_3b^2 </math>
<math>~\Rightarrow ~~~ c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 1 + \frac{\zeta_3}{2 \Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr] </math>	<math>~=</math>	<math>~ b^2 + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \biggl[1 + \frac{\zeta_2}{2\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr] </math>
<math>~\Rightarrow ~~~ c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] + \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ \frac{b^2}{b^2+a^2} \biggr] </math>	<math>~=</math>	<math>~ b^2 + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] + \frac{\zeta_3}{\Omega_3} \cdot \frac{\zeta_2}{\Omega_2} \biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2} \biggr] </math>
<math>~\Rightarrow ~~~ c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] </math>	<math>~=</math>	<math>~ b^2 + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \, . </math>
[ EFE, Chapter 7, §47, Eq. (11) ]

Adding the two instead gives,

<math>~ 0</math>	<math>~=</math>	<math>~ \biggl\{ 1 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr] \biggr\} \Omega_2\Omega_3c^2 + \biggl\{ 1 + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr] \biggr\} \Omega_2 \Omega_3b^2 </math>
	<math>~=</math>	<math>~ b^2 + c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr] + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr] </math>
	<math>~=</math>	<math>~ b^2 + c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr] + \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr] + \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 c^2}{(a^2 + c^2)( b^2+a^2 ) }\biggr] \, . </math>
[ EFE, Chapter 7, §47, Eq. (10) ]

How Solution is Obtained

Adding this pair of governing expressions we obtain,

<math>~0</math>	<math>~=</math>	<math>~ \biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} - 2\Omega_3 \int_V \rho u_1x_3 dx \biggr] + \biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx \biggr] </math>
	<math>~=</math>	<math>~4 \mathfrak{T}_{23} - \Omega_2\Omega_3(I_{22}+ I_{33} ) + 2 \int_V \rho u_1 (\Omega_2 x_2 - \Omega_3 x_3) dx \, ; </math>
[ EFE, Chapter 7, §47, Eq. (6) ]

and subtracting the pair gives,

<math>~0</math>	<math>~=</math>	<math>~ \biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} - 2\Omega_3 \int_V \rho u_1x_3 dx \biggr] - \biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx \biggr] </math>
	<math>~=</math>	<math>~ \Omega_2\Omega_3 (I_{22} - I_{33} ) - 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, . </math>
[ EFE, Chapter 7, §47, Eq. (7) ]

Various Degrees of Simplification

Riemann Ellipsoids of Types I, II, & III

In this, most general, case, the two vectors <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>~y-z</math>-plane — that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>. For a given specified density <math>~(\rho)</math> and choice of the three semi-axes <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in our above Summary Table must be used in order to determine the equilibrium configuration's associated values of the five unknowns: <math>~\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>.

Riemann S-Type Ellipsoids

In this case, we assume that <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are aligned with each other and, as well, are aligned with the <math>~z</math>-axis; that is to say, in addition to setting <math>~(\Omega_1, \zeta_1) = (0, 0)</math> we also set <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. So, there are only three unknowns — <math>~\Pi, (\Omega_3, \zeta_3)</math> — and they can be determined by ignoring off-axis expressions and simultaneously solving the diagonal element expressions displayed in our above Summary Table. Furthermore, two of the three diagonal-element expressions can be simplified because we are setting <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. The three relevant equilibrium constraints are:

Indices

2^nd-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids

<math>~ \biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi +\biggl\{ \Omega_3^2 + 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 ~-~(2\pi G\rho) A_1 \biggr\} a^2 + \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2 </math>

<math>~ \biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi + \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2 + \biggl\{ \Omega_3^2 + 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~-~( 2\pi G \rho) A_2 \biggr\}b^2 </math>

<math>~ \biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi - (2\pi G \rho)A_3 c^2 </math>

The <math>~(i, j) = (3, 3)</math> component expression immediately identifies the value of one of the unknowns, namely,

<math>~ \biggl( \frac{2^3\pi^2}{3\cdot 5} \biggr) G \rho^2A_3 a b c^3 \, . </math>

From the remaining pair of diagonal-element expressions, we therefore have,

<math>~ a^2 \Omega_3^2 + 2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 + \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2 ~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \, , </math>

and,

<math>~ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2 + b^2 \Omega_3^2 + 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \, . </math>

Multiplying the first of these two expressions through by <math>~b^2</math> and the second through by <math>~a^2</math>, then subtracting the second from the first gives,

<math>~0</math>	<math>~=</math>	<math>~ b^2\biggl\{ 2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 + \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2 ~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \biggr\} </math>
		<math>~ -~ a^2\biggl\{ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2 + 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \biggr\} </math>
	<math>~=</math>	<math>~ \biggl\{ 2 \biggl[ \frac{b^4 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 ~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )b^2 \biggr\} ~-~ \biggl\{ 2 \biggl[ \frac{a^4 b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) a^2 \biggr\} </math>
<math>~\Rightarrow ~~~ \biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 </math>	<math>~=</math>	<math>~ \pi G\rho \biggl[ \frac{(A_3 c^2 - A_2 b^2) a^2 ~-~(A_3 c^2 - A_1 a^2 )b^2}{ b^2 - a^2} \biggr] </math>
	<math>~=</math>	<math>~ \pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, . </math>
[ EFE, Chapter 7, §48, Eq. (30) ]

Note that — as EFE has done and as we have recorded in a related discussion — the first term on the right-hand-side of this last expression can be expressed more compactly in terms of the coefficient, <math>~A_{12}</math>.

Alternatively, dividing the first expression through by <math>~a^2</math> and the second by <math>~b^2</math>, then adding the pair of expressions gives,

<math>~ 0 </math>	<math>~=</math>	<math>~ \Omega_3^2 + 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 + \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2 ~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )\frac{1}{a^2} </math>
		<math>~+~ \biggl[ \frac{a^2 b^2}{(b^2+a^2)^2}\biggr] \zeta_3^2 + \Omega_3^2 + 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \frac{1}{b^2} </math>
	<math>~=</math>	<math>~ 2\Omega_3^2 + 2 \Omega_3 \zeta_3 + 2\biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2 ~+~2\pi G\rho \biggl[ \frac{A_3 c^2 - A_1 a^2 }{a^2} + \frac{A_3c^2 - A_2 b^2}{b^2}\biggr] \, . </math>

If we divide through by 2, then replace the product, <math>~\Omega_3\zeta_3</math>, in this expression by the relation derived immediately above, we have,

<math>~ \Omega_3^2 + \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2 </math>	<math>~=</math>	<math>~ ~-~\pi G\rho \biggl[ \frac{b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 ) }{a^2b^2} \biggr] ~-~ \pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2)}{ b^2 - a^2} \biggr]\biggl[ \frac{b^2+a^2}{b^2 a^2}\biggr] </math>
	<math>~=</math>	<math>~ \frac{\pi G\rho}{ a^2b^2(a^2-b^2) } \biggl\{ [ b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 )](b^2-a^2) ~+~ [ (A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2) ](b^2+a^2) \biggr\} </math>
	<math>~=</math>	<math>~ \frac{\pi G\rho}{ a^2b^2(a^2-b^2) } \biggl\{ [ - A_1 a^2 b^2 - A_2 a^2 b^2 ](b^2-a^2) ~+~ (A_1 - A_2)a^2b^2 (b^2+a^2) \biggr\} </math>
	<math>~=</math>	<math>~ \frac{2\pi G\rho}{ (a^2-b^2) } \biggl[ A_1 a^2 - A_2 b^2 \biggr] \, . </math>
[ EFE, Chapter 7, §48, Eq. (29) ]

It has become customary to characterize each Riemann S-Type ellipsoid by the value of its equilibrium frequency ratio,

<math>~\equiv</math>

<math>~\frac{\zeta_3}{\Omega_3} \, ,</math>

in which case the relevant pair of constraint equations becomes,

<math>~\biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] f \Omega_3^2 </math>	<math>~=</math>	<math>~ \pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, ; </math>
[ EFE, Chapter 7, §48, Eq. (34) ]

and,

<math>~ \Omega_3^2 \biggl\{1 + \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] f^2 \biggr\} </math>	<math>~=</math>	<math>~ \frac{2\pi G\rho}{ (a^2-b^2) } \biggl[ A_1 a^2 - A_2 b^2 \biggr] \, . </math>
[ EFE, Chapter 7, §48, Eq. (33) ]

These two equations can straightforwardly be combined to generate a quadratic equation for the frequency ratio, <math>~f</math>. Then, once the value of <math>~f</math> has been determined, either expression can be used to determine the corresponding equilibrium value for <math>~\Omega_3^2</math> in the unit of <math>~\pi G \rho</math>. The fact that the value of <math>~f</math> is determined from the solution of a quadratic equation underscores the realization that, for a given specification of the ellipsoidal geometry <math>~(a, b, c)</math>, if an equilibrium exists — i.e., if the solution for <math>~f</math> is real rather than imaginary — a second (adjoint) flow will be realized as well.

Jacobi and Dedekind Ellipsoids

Describe …

Maclaurin Spheroids

Describe …

Appendices: Various Integrals Over Ellipsoid Volume

Throughout this set of appendices, we work with a uniform-density ellipsoid whose surface is defined by the expression,

Appendix A: Volume

Here we seek to find the volume of the ellipsoid via the Cartesian integral expression,

<math>~ \iiint dx ~dy ~dz \, . </math>

Preliminaries

First, we will integrate over <math>~x</math> and specify the integration limits via the expression,

<math>~\equiv</math>

<math>~ a\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \, ; </math>

second, we will integrate over <math>~z</math> and specify the integration limits via the expression,

<math>~\equiv</math>

<math>~ c\biggl[ 1 - \frac{y^2}{b^2} \biggr]^{1 / 2} \, ; </math>

third, we will integrate over <math>~y</math> and set the limits of integration as <math>~\pm b</math>.

Carry Out the Integration

Following thestrategy that has just been outlined, we have,

<math>~V</math>	<math>~=</math>	<math>~ \iint dy ~dz \int_{-x_\ell}^{+x_\ell} dx = \iint dy ~dz \biggl[ x \biggr]_{-x_\ell}^{+x_\ell} = 2\int dy \int x_\ell ~dz </math>
	<math>~=</math>	<math>~ 2a\int dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz = \frac{2a}{c} \int dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz </math>
	<math>~=</math>	<math>~ \frac{2a}{c} \int \frac{dy}{2} \biggl[ z\sqrt{ z_\ell^2- z^2 } + z_\ell^2 \sin^{-1} \biggl( \frac{z}{\|z_\ell \|} \biggr) \biggr]_{-z_\ell}^{+z_\ell} </math>
	<math>~=</math>	<math>~ \frac{2a}{c} \int \biggl[ z_\ell \cancelto{0}{\sqrt{ z_\ell^2- z_\ell^2 }} + z_\ell^2 \sin^{-1} \biggl(1\biggr) \biggr] dy = \frac{2a}{c} \int \biggl[ \frac{\pi}{2} z_\ell^2 \biggr] dy </math>
	<math>~=</math>	<math>~ \pi a c \int_{-b}^{+b} \biggl( 1 - \frac{y^2}{b^2} \biggr) dy = \pi a c \biggl[ y - \frac{y^3}{3b^2} \biggr]_{-b}^{+b} </math>
	<math>~=</math>	<math>~ \frac{4\pi}{3} \cdot a b c\, . </math>

Appendix B: Coriolis Component u₁x₂

<math>~\iiint [u_1 y] ~dx ~dy ~dz</math>	<math>~=</math>	<math>~ \iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} y ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \iiint y^2 ~dx ~dy ~dz + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint yz ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int dz \int_{-x_\ell}^{+x_\ell} dx + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z ~dz \int_{-x_\ell}^{+x_\ell} dx </math>
	<math>~=</math>	<math>~ - \biggl[ \frac{2a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int x_\ell dz + \biggl[ \frac{2a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~x_\ell ~dz </math>
	<math>~=</math>	<math>~ - \biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz + \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} ~dz </math>
	<math>~=</math>	<math>~ - \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz ~+~ \frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int_{-z_\ell}^{+z_\ell} z~\biggl[ z_\ell^2 - z^2 \biggr]^{1 / 2} ~dz </math>
	<math>~=</math>	<math>~ - \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z \sqrt{z_\ell^2 - z^2} + z_\ell^2 \sin^{-1}\biggl(\frac{z}{\|z_\ell \|}\biggr) \biggr\}_{-z_\ell}^{+z_\ell} ~-~ \frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \cdot \frac{1}{3} \biggl\{ \biggl[ z_\ell^2 - z^2 \biggr]^{3 / 2} \biggr\}_{-z_\ell}^{+z_\ell} </math>
	<math>~=</math>	<math>~ - \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z_\ell^2 \sin^{-1}\biggl(\frac{z}{\|z_\ell \|}\biggr) \biggr\}_{-z_\ell}^{+z_\ell} = - \pi a~c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int_{-b}^b y^2 \biggl[1 - \frac{y^2}{b^2} \biggr] dy </math>
	<math>~=</math>	<math>~ - \pi ac\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{y^3}{3} - \frac{y^5}{5b^2} \biggr]_{-b}^{+b} = - 2\pi a b^3 c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{2}{15} \biggr] = - \frac{4\pi abc}{3} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{b^2}{5} \biggr] </math>
	<math>~=</math>	<math>~ - \frac{I_{22}}{\rho} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \, . </math>
[ EFE, Chapter 7, §47, p. 130, Eq. (9a) ]

Appendix C: Coriolis Component u₁x₃

Here we will additionally make use of the integration limits,

<math>~\equiv</math>

<math>~b^2 \biggl(1 - \frac{z^2}{c^2}\biggr) \, .</math>

Integration over the relevant Coriolis component gives,

<math>~\iiint [u_1 z] ~dx ~dy ~dz</math>	<math>~=</math>	<math>~ \iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} z ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~- \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3\iiint \cancelto{0}{y z ~dx ~dy ~dz} + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint z^2 ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~ \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \int_{-x_\ell}^{+x_\ell} dx </math>
	<math>~=</math>	<math>~ 2a\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \biggl\{ \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \biggr\} </math>
	<math>~=</math>	<math>~ \frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int_{-y_\ell}^{+y_\ell} \biggl[ y_\ell^2 - y^2 \biggr]^{1 / 2} dy </math>
	<math>~=</math>	<math>~ \frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \cdot \frac{1}{2}\biggl\{ y \sqrt{y_\ell^2 - y^2} + y_\ell^2 \sin^{-1}\biggr( \frac{y}{\|y_\ell \|} \biggr)\biggr\}_{-y_\ell}^{+y_\ell} </math>
	<math>~=</math>	<math>~ \frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ \frac{\pi}{2} y_\ell^2 \biggr\} dz = \pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ 1 - \frac{z^2}{c^2} \biggr\} dz </math>
	<math>~=</math>	<math>~ \pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{z^3}{3} - \frac{z^5}{5c^2} \biggr\}_{-c}^{+c} = \pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{1}{3} - \frac{1}{5} \biggr\}2c^3 = \frac{4 \pi a b c}{3}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{c^2}{5} \biggr\} </math>
	<math>~=</math>	<math>~+ ~\frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \, . </math>
[ EFE, Chapter 7, §47, p. 130, Eq. (9b) ]

Appendix D: The Other Four Coriolis Components

It follows that,

<math>~\iiint [u_2 x] ~dx ~dy ~dz</math>	<math>~=</math>	<math>~ \iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z} + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x \biggr\} x ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~ +~\frac{I_{11}}{\rho}\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 \, ; </math>
<math>~\iiint [u_2 z] ~dx ~dy ~dz</math>	<math>~=</math>	<math>~ \iiint \biggl\{ - \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z + \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x} \biggr\} z ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~ -~\frac{I_{33}}{\rho} \biggl[\frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \, ; </math>
<math>~\iiint [u_3 x] ~dx ~dy ~dz</math>	<math>~=</math>	<math>~ \iiint \biggl\{ - \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x + \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y} \biggr\} x ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~ -~\frac{I_{11}}{\rho} \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \, ; </math>
<math>~\iiint [u_3 y] ~dx ~dy ~dz</math>	<math>~=</math>	<math>~ \iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x} + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y \biggr\} y ~dx ~dy ~dz </math>
	<math>~=</math>	<math>~ +~\frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 \, . </math>

Appendix E: Kinetic Energy Components

Diagonal Elements

<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{11} = \int_V u_1 u_1 d^3x </math>	<math>~=</math>	<math>~\iiint [u_1^2] ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 y^2 - 2\cancelto{0}{\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \zeta_3} yz + \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 z^2 \biggr\} ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \iiint y^2 ~dx ~dy ~dz + \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2\iiint z^2 ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{22}}{\rho} \biggr] + \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{33}}{\rho} \biggr] \, .</math>

Similarly,

<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{22} = \int_V u_2 u_2 d^3x </math>	<math>~=</math>	<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\}^2 ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{33}}{\rho} \biggr] + \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{11}}{\rho} \biggr] \, ;</math>
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{33} = \int_V u_3 u_3 d^3x </math>	<math>~=</math>	<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}^2 ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~ \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{11}}{\rho} \biggr] + \biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{22}}{\rho} \biggr] \, .</math>

Off-Diagonal Elements

<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{23} = \int_V u_2 u_3 d^3x </math>	<math>~=</math>	<math>~\iiint [u_2 u_3] ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\} \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z \biggr\} \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} ~dx ~dy ~dz</math>
		<math>~+ \iiint \biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\} \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~-~\iiint \biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 \biggr\} x^2~dx ~dy ~dz</math>
	<math>~=</math>	<math>~- ~\frac{I_{11}}{\rho} \biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 </math>
[ EFE, Chapter 7, §47, p. 130, Eq. (8) ]

Similarly,

<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{12} = \int_V u_1 u_2 d^3x </math>	<math>~=</math>	<math>~\iiint [u_1 u_2] ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\} ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~-~ \biggl[ \frac{a^2}{a^2+c^2}\biggr] \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2 \iiint z^2~dx ~dy ~dz</math>
	<math>~=</math>	<math>~-~ \frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2 \, ;</math>
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{31} = \int_V u_3 u_1 d^3x </math>	<math>~=</math>	<math>~\iiint [u_3 u_1] ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~\iiint \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\} \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} ~dx ~dy ~dz</math>
	<math>~=</math>	<math>~ -~ \biggl[ \frac{c^2}{c^2+b^2}\biggr] \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3 \iiint y^2~dx ~dy ~dz</math>
	<math>~=</math>	<math>~ -~ \frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3 \, . </math>

And, finally,

<math>~\mathfrak{T}_{32}</math>

<math>~\mathfrak{T}_{23} \, ;</math>

<math>~\mathfrak{T}_{21}</math>

<math>~\mathfrak{T}_{12} \, ;</math>

and,

<math>~\mathfrak{T}_{13}</math>

<math>~\mathfrak{T}_{31} \, .</math>

User:Tohline/VE/RiemannEllipsoids

Contents

Steady-State 2^nd-Order Tensor Virial Equations

Summary

General Coefficient Expressions

Adopted (Internal) Velocity Field

Equilibrium Expressions

The Three Diagonal Elements

Off-Diagonal Elements

How Solution is Obtained

Various Degrees of Simplification

Riemann Ellipsoids of Types I, II, & III

Riemann S-Type Ellipsoids

Jacobi and Dedekind Ellipsoids

Maclaurin Spheroids

Appendices: Various Integrals Over Ellipsoid Volume

Appendix A: Volume

Preliminaries

Carry Out the Integration

Appendix B: Coriolis Component u₁x₂

Appendix C: Coriolis Component u₁x₃

Appendix D: The Other Four Coriolis Components

Appendix E: Kinetic Energy Components

Diagonal Elements

Off-Diagonal Elements

See Also

Navigation menu

Search

User:Tohline/VE/RiemannEllipsoids

Steady-State 2nd-Order Tensor Virial Equations

Summary

General Coefficient Expressions

Adopted (Internal) Velocity Field

Equilibrium Expressions

The Three Diagonal Elements

Off-Diagonal Elements

How Solution is Obtained

Various Degrees of Simplification

Riemann Ellipsoids of Types I, II, & III

Riemann S-Type Ellipsoids

Jacobi and Dedekind Ellipsoids

Maclaurin Spheroids

Appendices: Various Integrals Over Ellipsoid Volume

Appendix A: Volume

Preliminaries

Carry Out the Integration

Appendix B: Coriolis Component u1x2

Appendix C: Coriolis Component u1x3

Appendix D: The Other Four Coriolis Components

Appendix E: Kinetic Energy Components

Diagonal Elements

Off-Diagonal Elements

See Also

Navigation menu

Search

Steady-State 2^nd-Order Tensor Virial Equations

Appendix B: Coriolis Component u₁x₂

Appendix C: Coriolis Component u₁x₃