Challenges Constructing Ellipsoidal-Like Configurations
First, let's review the three different approaches that we have described for constructing Riemann S-type ellipsoids. Then let's see how these relate to the technique that has been used to construct infinitesimally thin, nonaxisymmetric disks.
Riemann S-type Ellipsoids
2nd-Order TVE Expressions
As we have discussed in detail in an accompanying chapter, the three diagonal elements of the <math>~(3 \times 3)</math> 2nd-order tensor virial equation are sufficient to determine the equilibrium values of <math>~\Pi</math>, <math>~\Omega_3</math>, and <math>~\zeta_3</math>.
| Indices |
2nd-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids |
| <math>~i</math> |
<math>~j</math> |
| <math>~1</math> |
<math>~1</math> |
|
<math>~0</math>
|
<math>~=</math>
|
<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>~2</math> |
<math>~2</math> |
|
<math>~0</math>
|
<math>~=</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>~3</math> |
<math>~3</math> |
|
<math>~0</math>
|
<math>~=</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> element gives <math>~\Pi</math> directly in terms of known parameters. The <math>~(1, 1)</math> and <math>~(2, 2)</math> elements can then be combined in a couple of different ways to obtain a coupled set of expressions that define <math>~\Omega_3</math> and <math>~f \equiv \zeta_3/\Omega_3</math>.
|
<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) ] |
|
Compressible Structures
See Also
