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</math>  </math>  
</td>  </td>  
+  </tr>  
+  </table>  
+  
+  For example — as is explicitly illustrated on p. 130 of EFE — for <math>~i=2</math> and <math>~j=3</math>,  
+  
+  <table border="0" cellpadding="5" align="center">  
+  <tr>  
+  <td align="right">  
+  <math>~0</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  2 \mathfrak{T}_{23}  \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 dx}  
+   2\Omega_3 \int_V \rho u_1x_3 dx \, ,  
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  whereas for <math>~i=3</math> and <math>~j=2</math>,  
+  
+  <table border="0" cellpadding="5" align="center">  
+  <tr>  
+  <td align="right">  
+  <math>~0</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  2 \mathfrak{T}_{ij}  \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx  
+  \, ,  
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  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">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\int_V \rho u_i x_j dx</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~0</math>  
+  </td>  
+  <td align="right"> if <math>~i = j \, .</math>  
</tr>  </tr>  
</table>  </table> 
Revision as of 13:34, 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.
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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 ,



whereas for and ,



where, in both cases, we have acknowledged that, given the above specific prescription for the internal velocity field, ,



if 
Various Degrees of Simplification
Riemann SType Ellipsoids
Describe …
Jacobi and Dedekind Ellipsoids
Describe …
Maclaurin Spheroids
Describe …
See Also
© 2014  2020 by Joel E. Tohline 