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Line 1,541: Line 1,541:
   <td align="left">
   <td align="left">
<math>~
<math>~
-
\frac{\pi G\rho}{ (a^2-b^2) }  
+
\frac{2\pi G\rho}{ (a^2-b^2) }  
\biggl[  
\biggl[  
-
2A_1   a^2  
+
A_1   a^2  
-
- 2A_2 (b^2)
+
- A_2 b^2  
\biggr] \, .
\biggr] \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
 +
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48, Eq. (29)</font> ]</td></tr>
</table>
</table>

Revision as of 15:16, 8 August 2020


Contents

Steady-State 2nd-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.

Whitworth's (1981) Isothermal Free-Energy Surface
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Summary

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

~0

~=

~
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 \, ,

to determine the equilibrium conditions of uniform-density ~(\rho) ellipsoids that have semi-axes, ~(a_1, a_2, a_3) \leftrightarrow (a, b, c), and an internal velocity field, ~\vec{u} (as prescribed below), that preserves this specified ellipsoidal shape, as viewed from a frame of reference that is rotating with angular velocity, ~\vec\Omega. Because each of the indices, ~i and ~j, 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: ~\Pi and the three pairs of velocity-field components ~(\Omega_1, \zeta_1), ~(\Omega_2, \zeta_2), ~(\Omega_3, \zeta_3). 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 ~\Omega_1 = \zeta_1 = 0, in which case the only applicable TVE constraint relations are the five identified in the following table of equations.


Indices Each Associated 2nd-Order TVE Expression
~i ~j
~1 ~1

~0

~=

~
\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

~2 ~2

~0

~=

~
\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

~3 ~3

~0

~=

~
\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

~2 ~3

~0

~=

~\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

~3 ~2

~0

~=

~\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

General Coefficient Expressions

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


~A_1


~=

~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] \, ,


~A_3


~=


~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] \, ,


~A_2


~=

~2 - (A_1+A_3) \, ,

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

~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)

      and      

~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .

[ EFE, Chapter 3, §17, Eq. (32) ]

Adopted (Internal) Velocity Field

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

~u_1

~=

~- \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 \, ,

~u_2

~=

~- \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 \, ,

~u_3

~=

~- \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 \, .

[ EFE, Chapter 7, §47, Eq. (1) ]

Equilibrium Expressions

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

~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij}

~=

~- \delta_{ij}\Pi \, .

[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 2nd-order TVE takes on the more general form:

~0

~=

~
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
\, .

[ EFE, Chapter 2, §12, Eq. (64) ]

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

~\frac{\mathfrak{W}_{ij}}{\pi G\rho}

~=

~-2A_i I_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (128) ]

where

~I_{ij}

~=

~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (129) ]

is the moment of inertia tensor. Expressions for all nine components of the kinetic energy tensor, ~\mathfrak{T}_{ij} 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 ~i = j = 1, we have,

~0

~=

~
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

 

~=

~
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

 

~=

~
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

 

~=

~
\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}

 

~=

~
\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}

~\Rightarrow~~~ -\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  \, .

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

~0

~=

~
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

 

~=

~
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

 

~=

~
\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}

 

~=

~
\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}

~\Rightarrow~~~-\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_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 \, .

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

~0

~=

~
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

 

~=

~
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

 

~=

~\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}

 

~=

~
\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}

~\Rightarrow ~~~ -\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[ \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 \, .

This gives us three equations vs. seven unknowns.

Off-Diagonal Elements

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

~0

~=

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

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

~0

~=

~
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 \, ,

[ EFE, Chapter 7, §47, Eq. (3) ]

whereas for ~i=3 and ~j=2,

~0

~=

~
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}
\, .

[ 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, ~\vec{u}, we recognize that,

~\int_V \rho u_i x_j d^3x

~=

~0

      if    ~i = j \, .
[ 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,

~\mathfrak{T}_{32}

~=

~\mathfrak{T}_{23}

~=

~\frac{1}{2} \int_V \rho v_2 v_3 d^3x \, .

Our first off-diagonal element is, then,

~0

~=

~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} 
- 2\Omega_3 \rho \int_V u_1 z d^3x

 

~=

~
- ~
\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

 

~=

~\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

 

~=

~\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 \, .

The second is,

~0

~=

~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \rho \int_V u_1 y d^3x

 

~=

~
- ~
\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

 

~=

~\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

 

~=

~\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 \, .

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

~\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

~\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]

~=

~
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]

~\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]

~=

~
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]

~\Rightarrow ~~~ 
c^2  
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]

~=

~
b^2 
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]    \, .

[ EFE, Chapter 7, §47, Eq. (11) ]

Adding the two instead gives,

~
0

~=

~
\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

 

~=

~
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]

 

~=

~
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]      \, .

[ EFE, Chapter 7, §47, Eq. (10) ]

How Solution is Obtained

Adding this pair of governing expressions we obtain,

~0

~=

~
\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]

 

~=

~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 \, ;

[ EFE, Chapter 7, §47, Eq. (6) ]

and subtracting the pair gives,

~0

~=

~
\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]

 

~=

~
\Omega_2\Omega_3 (I_{22} - I_{33} )
- 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, .

[ 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 ~\vec{\Omega} and ~\vec\zeta 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 ~y-z-plane — that is to say, ~(\Omega_1, \zeta_1) = (0, 0). For a given specified density ~(\rho) and choice of the three semi-axes ~(a_1, a_2, a_3) \leftrightarrow (a, b, c), 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: ~\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3).

Riemann S-Type Ellipsoids

In this case, we assume that ~\vec{\Omega} and ~\vec\zeta are aligned with each other and, as well, are aligned with the ~z-axis; that is to say, in addition to setting ~(\Omega_1, \zeta_1) = (0, 0) we also set ~(\Omega_2, \zeta_2) = (0, 0). So, there are only three unknowns — ~\Pi, (\Omega_3, \zeta_3) — 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 ~(\Omega_2, \zeta_2) = (0, 0). The three relevant equilibrium constraints are:


Indices 2nd-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids
~i ~j
~1 ~1

~0

~=

~
\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

~2 ~2

~0

~=

~
\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

~3 ~3

~0

~=

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


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

~\Pi

~=

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

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

~
0

~=

~
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 ) \, ,

and,

~
0

~=

~
\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) \, .

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

~0

~=

~
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\}

 

 

~
-~
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\}

 

~=

~
\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\}

~\Rightarrow ~~~ \biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3

~=

~
\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]

 

~=

~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, .

[ 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, ~A_{12}.

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

~
0

~=

~
\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}

 

 

~+~
\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}

 

~=

~
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] \, .

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

~
\Omega_3^2  
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2

~=

~
~-~\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]

 

~=

~
\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\}

 

~=

~
\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\}

 

~=

~
\frac{2\pi G\rho}{ (a^2-b^2) } 
\biggl[ 
A_1   a^2 
- A_2  b^2 
\biggr] \, .

[ EFE, Chapter 7, §48, Eq. (29) ]

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,

~1

~=

~
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \, .

Appendix A:  Volume

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

~V

~=

~
\iiint  dx ~dy ~dz \, .

Preliminaries

First, we will integrate over ~x and specify the integration limits via the expression,

~x_\ell

~\equiv

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

second, we will integrate over ~z and specify the integration limits via the expression,

~z_\ell

~\equiv

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

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

Carry Out the Integration

Following thestrategy that has just been outlined, we have,

~V

~=

~
\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

 

~=

~
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

 

~=

~
\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}

 

~=

~
\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

 

~=

~
\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}

 

~=

~
\frac{4\pi}{3} \cdot a b c\, .

Appendix B:  Coriolis Component u1x2

~\iiint  [u_1 y] ~dx ~dy ~dz

~=

~
\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

 

~=

~
- \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

 

~=

~
- \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

 

~=

~
- \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

 

~=

~
- \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

 

~=

~
- \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

 

~=

~
- \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}

 

~=

~
- \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

 

~=

~
- \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]

 

~=

~
- \frac{I_{22}}{\rho} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3   \, .

[ EFE, Chapter 7, §47, p. 130, Eq. (9a) ]

Appendix C:  Coriolis Component u1x3

Here we will additionally make use of the integration limits,

~y_\ell^2

~\equiv

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

Integration over the relevant Coriolis component gives,

~\iiint  [u_1 z] ~dx ~dy ~dz

~=

~
\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

 

~=

~-
\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

 

~=

~
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \int_{-x_\ell}^{+x_\ell} dx

 

~=

~
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\}

 

~=

~
\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

 

~=

~
\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}

 

~=

~
\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

 

~=

~
\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\}

 

~=

~+
~\frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \, .

[ EFE, Chapter 7, §47, p. 130, Eq. (9b) ]


Appendix D:   The Other Four Coriolis Components

It follows that,

~\iiint  [u_2 x] ~dx ~dy ~dz

~=

~
\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

 

~=

~
+~\frac{I_{11}}{\rho}\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 \, ;

~\iiint  [u_2 z] ~dx ~dy ~dz

~=

~
\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

 

~=

~
-~\frac{I_{33}}{\rho} \biggl[\frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \, ;

~\iiint  [u_3 x] ~dx ~dy ~dz

~=

~
\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

 

~=

~
-~\frac{I_{11}}{\rho} \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \, ;

~\iiint  [u_3 y] ~dx ~dy ~dz

~=

~
\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

 

~=

~
+~\frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1  \, .

Appendix E:   Kinetic Energy Components

Diagonal Elements

~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{11} = \int_V  u_1 u_1 d^3x

~=

~\iiint  [u_1^2] ~dx ~dy ~dz

 

~=

~\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

 

~=

~\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

 

~=

~
\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

 

~=

~
\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] \, .

Similarly,

~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{22} = \int_V  u_2 u_2 d^3x

~=

~\iiint  [u_2^2] ~dx ~dy ~dz

 

~=

~\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

 

~=

~
\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] \, ;

~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{33} = \int_V  u_3 u_3 d^3x

~=

~\iiint  [u_2^2] ~dx ~dy ~dz

 

~=

~\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

 

~=

~
\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] \, .

Off-Diagonal Elements

~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{23} = \int_V  u_2 u_3 d^3x

~=

~\iiint  [u_2 u_3] ~dx ~dy ~dz

 

~=

~\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

 

~=

~\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

 

 

~+ \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

 

~=

~-~\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

 

~=

~- ~\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

[ EFE, Chapter 7, §47, p. 130, Eq. (8) ]

Similarly,

~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{12} = \int_V  u_1 u_2 d^3x

~=

~\iiint  [u_1 u_2] ~dx ~dy ~dz

 

~=

~\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

 

~=

~-~  
\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

 

~=

~-~  \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  
\, ;

~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{31} = \int_V  u_3 u_1 d^3x

~=

~\iiint  [u_3 u_1] ~dx ~dy ~dz

 

~=

~\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

 

~=

~ -~ 
\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

 

~=

~ -~ \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  \, .

And, finally,

~\mathfrak{T}_{32}

~=

~\mathfrak{T}_{23} \, ;

     

~\mathfrak{T}_{21}

~=

~\mathfrak{T}_{12} \, ;

      and,     

~\mathfrak{T}_{13}

~=

~\mathfrak{T}_{31} \, .

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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