User:Tohline/VE/RiemannEllipsoids
From VisTrailsWiki
(→The Six OffDiagonal Elements) 
(→The Six OffDiagonal Elements) 

Line 362:  Line 362:  
</td>  </td>  
</tr>  </tr>  
+  <tr><td align="center" colspan="4">[ [[User:Tohline/Appendix/References#EFEEFE]], <font color="#00CC00">Chapter 7, §47, Eq. (3)</font> ]</td></tr>  
</table>  </table>  
whereas for <math>~i=3</math> and <math>~j=2</math>,  whereas for <math>~i=3</math> and <math>~j=2</math>,  
Line 375:  Line 376:  
<td align="left">  <td align="left">  
<math>~  <math>~  
  2 \mathfrak{T}_{  +  2 \mathfrak{T}_{32}  \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx 
+   2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 dx}  
\, ,  \, ,  
</math>  </math>  
</td>  </td>  
</tr>  </tr>  
+  <tr><td align="center" colspan="4">[ [[User:Tohline/Appendix/References#EFEEFE]], <font color="#00CC00">Chapter 7, §47, Eq. (4)</font> ]</td></tr>  
</table>  </table>  
  where  +  where in both cases we have acknowledged that, given the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, 
<table border="0" cellpadding="5" align="center">  <table border="0" cellpadding="5" align="center">  
Line 396:  Line 399:  
<td align="right"> if <math>~i = j \, .</math>  <td align="right"> if <math>~i = j \, .</math>  
</tr>  </tr>  
+  <tr><td align="center" colspan="4">[ [[User:Tohline/Appendix/References#EFEEFE]], <font color="#00CC00">Chapter 7, §47, Eq. (5)</font> ]</td></tr>  
</table>  </table>  
Revision as of 13:42, 4 August 2020
Contents 
SteadyState 2^{nd}Order Tensor Virial Equations
By satisfying all six — not necessarily unique — components of the SecondOrder Tensor Virial Equation, the entire set of Riemann Ellipsoids can be determined.
 Tiled Menu  Tables of Content  Banner Video  Tohline Home Page  
Here we are only interested in determining the equilibrium conditions of uniformdensity ellipsoids that have semiaxes, .
General Coefficient Expressions
As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, , of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:









where, and are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

and 

[ EFE, Chapter 3, §17, Eq. (32) ] 
Adopted (Internal) Velocity Field
EFE (p. 130) states that the … kinematical requirement, that the motion , associated with , preserves the ellipsoidal boundary, leads to the following expressions for its components:









[ EFE, Chapter 7, §47, Eq. (1) ] 
Equilibrium Expressions
[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,



[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:



[ EFE, Chapter 2, §12, Eq. (64) ] 
EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by



[ EFE, Chapter 3, §22, Eq. (128) ] 
where



[ EFE, Chapter 3, §22, Eq. (129) ] 
is the moment of inertia tensor.
The Three Diagonal Elements
For , we have,









Similarly, for ,






and, for ,






The Six OffDiagonal Elements
Notice that the offdiagonal components of both and are zero. Hence, the equilibrium expression that is dictated by each offdiagonal component of the 2^{nd}order TVE is,



For example — as is explicitly illustrated on p. 130 of EFE — for and ,




[ EFE, Chapter 7, §47, Eq. (3) ] 
whereas for and ,




[ EFE, Chapter 7, §47, Eq. (4) ] 
where in both cases we have acknowledged that, given the above specific prescription for the internal velocity field, ,



if 
[ EFE, Chapter 7, §47, Eq. (5) ] 
Various Degrees of Simplification
Riemann SType Ellipsoids
Describe …
Jacobi and Dedekind Ellipsoids
Describe …
Maclaurin Spheroids
Describe …
See Also
© 2014  2020 by Joel E. Tohline 