User:Tohline/ThreeDimensionalConfigurations/Challenges

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

2^nd-Order TVE Expressions

As we have discussed in detail in an accompanying chapter, the three diagonal elements of the <math>~(3 \times 3)</math> 2^nd-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) ]

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation