Editing Appendix/Ramblings/HybridSchemeImplications (section)

====Compare====

Here we consider which formalism is best suited for modeling a fully three-dimensional, nonaxisymmetric configuration that is spinning about (usually) its shortest axis with a uniform and time-independent frequency and which, when viewed from a frame that is rotating with that frequency, exhibits a nontrivial but nevertheless steady-state internal flow.  Examples are Riemann S-type ellipsoids, and binary stars in circular orbits. 

It is most desirable to choose a formalism that recognizes the steady-state nature of the flow.  In the vast majority of cases being considered here, this rules out using any scheme that is designed around an inertial-frame coordinate base.   (As a counterexample, Dedekind ellipsoids can be constructed in the inertial frame because <math>~\Omega_f = 0</math> for all models along the Dedekind equilibrium sequence.)  It is quite reasonable, however, to adopt a rotating, cylindrical-coordinate base as has been described above and as is summarized immediately below. 

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
<div align="center">'''Traditional Rotating, Cylindrical-Coordinate Summary'''</div>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~\bold{\hat{k}:}</math></td>
  <td align="right">
<math>~
\frac{\partial (\rho u_z)}{\partial t} + \bold\nabla \cdot (\rho u_z \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{k}}\cdot (\rho \bold{a}) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\bold{\hat{e}_\varpi:}</math></td>
  <td align="right" colspan="1">
<math>~
\frac{\partial ( \rho u_\varpi )}{\partial t} 
+ \bold\nabla \cdot (\rho u_\varpi \bold{u})</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varpi \cdot (\rho \bold{a}) 
+ \biggl[\frac{(\rho \varpi u_\varphi)^2}{\rho \varpi^3} + \rho\Omega_f^2 \varpi + \frac{ 2\Omega_f (\rho \varpi u_\varphi )}{\varpi} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\bold{\hat{e}_\varphi:}</math></td>
  <td align="right" colspan="1">
<math>~
\frac{\partial (\rho \varpi u_\varphi )}{\partial t} 
+ \bold\nabla \cdot (\rho \varpi u_\varphi \bold{u})
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varphi \cdot (\rho \varpi \bold{a})
- 2\Omega_f \varpi \rho u_\varpi  \, .  
</math>
  </td>
</tr>
</table>

</td></tr></table>

In this scheme, all of the velocities and associated momentum densities in all three components of the Euler equation are expressed in terms of the rotating-frame velocity vector, <math>~\bold{u}</math>, or its cylindrical-coordinate-based components, <math>~(u_\varpi, v_\varphi, v_z)</math>.  When the configuration's distorted (nonaxisymmetric) shape is largely supported by rapid rotation, this scheme provides an advantage over other &#8212; for example, inertial-frame-based &#8212; schemes because the fraction of the fluid's total momentum that is being advected across the grid is often quite small. There is a penalty to be paid, however.  Additional "source" terms appear on the right-hand-side of the radial- and azimuthal-component expressions; they are nonlinear in the velocity and introduce cross-talk between the component expressions.

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
<div align="center">'''Hybrid Scheme Summary'''</div>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~\bold{\hat{k}:}</math></td>
  <td align="right">
<math>~
\frac{\partial (\rho v_z)}{\partial t} + \bold\nabla \cdot (\rho v_z \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{k}} \cdot (\rho \bold{a}) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\bold{\hat{e}_\varpi:}</math></td>
  <td align="right">
<math>~
\frac{\partial (\rho v_\varpi)}{\partial t} + \bold\nabla \cdot (\rho v_\varpi \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varpi \cdot (\rho \bold{a}) + \frac{v_\varphi^2}{\varpi} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\bold{\hat{e}_\varphi:}</math></td>
  <td align="right">
<math>~ 
\frac{\partial(\rho \varpi v_\varphi )}{\partial t} 
+ \bold\nabla \cdot (\rho \varpi v_\varphi \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varphi \cdot (\rho \varpi \bold{a}) \, .</math>
  </td>
</tr>
</table>

</td></tr></table>

For Riemann S-type ellipsoids,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u_x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda \biggl(\frac{a}{b}\biggr) y
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~u_y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \lambda \biggl(\frac{b}{a}\biggr) x
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda \biggl[\frac{a}{b} - \frac{b}{a}  \biggr] x y (x^2 + y^2)^{-1 / 2}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\varpi v_\varphi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-
\biggl[ \lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]x^2
-
\biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]y^2
\, .
</math>
  </td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold\nabla \cdot (v_\varpi \bold{u})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial x}\biggl\{ v_\varpi u_x \biggr\} + \frac{\partial }{\partial y}\biggl\{ v_\varpi u_y \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda^2 \biggl(\frac{a}{b}\biggr) \biggl[\frac{a}{b} - \frac{b}{a}  \biggr]
\frac{\partial }{\partial x}\biggl\{  x y^2 (x^2 + y^2)^{-1 / 2}  \biggr\} 
- 
\lambda^2 \biggl(\frac{b}{a}\biggr) \biggl[\frac{a}{b} - \frac{b}{a}  \biggr]
\frac{\partial }{\partial y}\biggl\{ x^2 y (x^2 + y^2)^{-1 / 2}  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y^2 \lambda^2 \biggl(\frac{a}{b}\biggr) \biggl[\frac{a}{b} - \frac{b}{a}  \biggr]
\biggl\{  (x^2 + y^2)^{-1 / 2}  -  x^2  (x^2 + y^2)^{-3 / 2}\biggr\} 
- 
x^2 \lambda^2 \biggl(\frac{b}{a}\biggr) \biggl[\frac{a}{b} - \frac{b}{a}  \biggr]
\biggl\{  (x^2 + y^2)^{-1 / 2} - y^2 (x^2 + y^2)^{-3 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda^2}{\varpi^3} \biggl[\frac{a}{b} - \frac{b}{a}  \biggr] \biggl\{
y^4  \biggl(\frac{a}{b}\biggr) 
- 
x^4 \biggl(\frac{b}{a}\biggr) 
\biggr\}
</math>
  </td>
</tr>
</table>

And,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold\nabla \cdot (\varpi v_\varphi \bold{u})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial}{\partial x} \biggl\{-
\lambda\biggl(\frac{a}{b}\biggr)y\biggl[ \lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]x^2
\biggr\}
+ \frac{\partial}{\partial y} \biggl\{
\lambda\biggl(\frac{b}{a}\biggr)x \biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]y^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{-
2 \lambda\biggl(\frac{a}{b}\biggr)\biggl[ \lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]x y
\biggr\}
+ \biggl\{
2 \lambda\biggl(\frac{b}{a}\biggr)\biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]x y
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ -\biggl(\frac{a}{b}\biggr)\biggl[ \lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]
+ 
\biggl(\frac{b}{a}\biggr)\biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]
\biggr\} 2 \lambda x y
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{-\biggl[ \lambda  -  \biggl(\frac{a}{b}\biggr)\Omega_f\biggr]
+ 
\biggl[ \lambda  - \biggl(\frac{b}{a}\biggr)\Omega_f\biggr]
\biggr\} 2 \lambda x y
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \lambda xy \Omega_f \biggl[\frac{a}{b} - \frac{b}{a} \biggr] \, .
</math>
  </td>
</tr>
</table>
To within an additive constant &#8212; see, for example, our [[Apps/MaclaurinSpheroids#Equilibrium_Structure|associated discussion of Maclaurin spheroids]] &#8212; the gravitational potential and the enthalpy are, respectively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi G \rho \biggl[ A_1 x^2 + A_2 y^2 + A_3 z^2 \biggr] 
= \pi G \rho \biggl[ A_1 \varpi^2 \cos^2\varphi + A_2 \varpi^2 \sin^2\varphi + A_3 z^2 \biggr]
\, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
H_0 \biggl[1 - \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr] 
=
H_0 \biggl[1 - \frac{\varpi^2\cos^2\varphi}{a^2} - \frac{\varpi^2 \sin^2\varphi}{b^2} - \frac{z^2}{c^2} \biggr]
\, .</math>
  </td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{a} = - \bold\nabla (H + \Phi_\mathrm{grav} )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl[ 
\bold{\hat{e}}_\varpi \frac{\partial}{\partial \varpi} \biggl( H + \Phi_\mathrm{grav}\biggr) 
+ \frac{\bold{\hat{e}}_\varphi}{\varpi} \frac{\partial}{\partial \varphi} \biggl( H + \Phi_\mathrm{grav}\biggr) 
+ \bold{\hat{k}} \frac{\partial}{\partial z} \biggl( H + \Phi_\mathrm{grav}\biggr) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
- \bold{\hat{e}}_\varpi \biggl[
-2H_0 \varpi \biggl( \frac{\cos^2\varphi}{a^2} + \frac{\sin^2\varphi}{b^2}\biggr) + 2\pi G \rho \varpi \biggl(A_1\cos^2\varphi + A_2\sin^2\varphi \biggr) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
- \frac{\bold{\hat{e}}_\varphi}{\varpi} 
\biggl[ -2H_0\varpi^2 \sin\varphi \cos\varphi\biggl( \frac{1}{b^2} - \frac{1}{a^2}\biggr) 
+ 2\pi G \rho \varpi^2 \sin\varphi \cos\varphi \biggl( A_2 - A_1 \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
- \bold{\hat{k}} \biggl[ - \frac{2H_0 z}{c^2} + 2\pi G \rho A_3 z \biggr] \, .
</math>
  </td>
</tr>
</table>

----


<font color="red">'''Vertical Component:'''</font> &nbsp;  Because <math>~v_z = 0</math>, it must be true that <math>~\bold{\hat{k}}\cdot \bold{a} = 0</math>.  This, in turn means,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H_0  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi G \rho c^2 A_3  \, . </math>
  </td>
</tr>
</table>

<font color="red">'''Azimuthal Component:'''</font> &nbsp; In steady-state, the partial time-derivative must be zero, so we require,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
\bold\nabla \cdot (\varpi v_\varphi \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi \bold{\hat{e}}_\varphi \cdot \bold{a}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow~~~
2 \lambda xy \Omega_f \biggl[\frac{a}{b} - \frac{b}{a} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \biggl[ 2H_0\varpi^2 \sin\varphi \cos\varphi\biggl( \frac{1}{b^2} - \frac{1}{a^2}\biggr) 
- 2\pi G \rho \varpi^2 \sin\varphi \cos\varphi \biggl( A_2 - A_1 \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2xy \biggl[ H_0\biggl( \frac{1}{b^2} - \frac{1}{a^2}\biggr) 
- \pi G \rho  \biggl( A_2 - A_1 \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi G \rho xy \biggl[ c^2 A_3\biggl( \frac{1}{b^2} - \frac{1}{a^2}\biggr) 
- \biggl( A_2 - A_1 \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow~~~
\lambda \Omega_f \biggl[\frac{a^2 - b^2}{ab}  \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G \rho \biggl[ c^2 A_3\biggl( \frac{a^2 - b^2}{a^2 b^2} \biggr) 
+ \biggl( A_1 - A_2 \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G \rho \biggl[ ( A_1 - A_2 )  - c^2 A_3\biggl( \frac{b^2 - a^2}{a^2 b^2} \biggr) 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow~~~
- a b \lambda \Omega_f 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G \rho \biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2}  - c^2 A_3 
\biggr] \, .
</math>
  </td>
</tr>
</table>
This expression can be used either (a) to give <math>~\Omega_f</math> in terms of a set of known quantities and the unknown parameter, <math>~\lambda</math>; or (b) to give <math>~\lambda</math> in terms of a set of known quantities and the unknown parameter, <math>~\Omega_f</math>.

Recognizing from  [[VE/RiemannEllipsoids#fDefined|here]] that, <math>~f \equiv \zeta/\Omega_f</math> and
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda \biggl( \frac{a}{b} + \frac{b}{a} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \zeta  = - f\Omega_f
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \lambda </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl( \frac{ab}{a^2 + b^2} \biggr)f \Omega_f   \, ,
</math>
  </td>
</tr>
</table>
this last expression can be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
\biggl( \frac{a^2 b^2}{a^2 + b^2} \biggr)f \Omega_f^2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G \rho \biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2}  - c^2 A_3 
\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48, Eq. (34)</font> ]</td></tr>
</table>

<font color="red">'''Radial Component:'''</font> &nbsp; In steady-state, this partial time-derivative also must be zero, so we require,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\bold\nabla \cdot (v_\varpi \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varpi \cdot \bold{a} + \frac{v_\varphi^2}{\varpi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\frac{\lambda^2}{\varpi^3} \biggl[\frac{a}{b} - \frac{b}{a}  \biggr] \biggl\{
y^4  \biggl(\frac{a}{b}\biggr) 
- 
x^4 \biggl(\frac{b}{a}\biggr) 
\biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi G \rho }{\varpi}\biggl[
c^2 A_3 \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) - \biggl(A_1 x^2 + A_2y^2 \biggr) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\frac{1}{\varpi^3} \biggl\{
-\biggl[\lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]x^2
-
\biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]y^2
\biggr\}^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\lambda^2 \biggl\{ a^2 y^4  - b^2 x^4  \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi G \rho b^2 \varpi^2}{(a^2 - b^2)} 
\biggl[ c^2 A_3 x^2  - A_1 a^2 x^2 \biggr]
+  
\frac{2\pi G \rho a^2 \varpi^2}{(a^2 - b^2)} 
\biggl[ c^2 A_3 y^2 - A_2 b^2 y^2  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[\frac{a^2 b^2}{a^2 - b^2} \biggr]  
\biggl\{
\biggl[\lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]x^2
+
\biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]y^2
\biggr\}^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\lambda^2 (a^2 - b^2) \biggl\{ a^2 y^4  - b^2 x^4  \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi G \rho b^2 \varpi^2 x^2 
\biggl[ c^2 A_3  - A_1 a^2 \biggr]
+  
2\pi G \rho a^2 \varpi^2 y^2 
\biggl[ c^2 A_3  - A_2 b^2   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  
\biggl\{
\biggl[\lambda b^2 - a b \Omega_f \biggr]x^2
+
\biggl[ \lambda a^2 - a b \Omega_f\biggr]y^2
\biggr\}^2 \, .
</math>
  </td>
</tr>
</table>

Also,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
- a b \lambda \Omega_f 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G \rho \biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2}  - c^2 A_3 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
c^2 A_3
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2}   + \frac{a b \lambda \Omega_f }{\pi G \rho} 
\, .
</math>
  </td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\lambda^2 (a^2 - b^2) \biggl\{ a^2 y^4  - b^2 x^4  \biggr\}
-  
\biggl\{
\biggl[\lambda b^2 - a b \Omega_f \biggr]x^2
+
\biggl[ \lambda a^2 - a b \Omega_f\biggr]y^2
\biggr\}^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi G \rho b^2 \varpi^2 x^2 
\biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2}   + \frac{a b \lambda \Omega_f }{\pi G \rho}  - A_1 a^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  
2\pi G \rho a^2 \varpi^2 y^2 
\biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2}   + \frac{a b \lambda \Omega_f }{\pi G \rho}  - A_2 b^2   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi G \rho b^2 \varpi^2 x^2 }{(b^2 - a^2) }
\biggl[ ( A_1 - A_2 )a^2b^2   - A_1 a^2 (b^2 - a^2) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  
\frac{ 2\pi G \rho a^2 \varpi^2 y^2 }{ (b^2 - a^2)}
\biggl[ ( A_1 - A_2 )a^2b^2 - A_2 b^2(b^2 - a^2)   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi G \rho a^2 b^2 \varpi^4 }{(b^2 - a^2) }
\biggl[  A_1 a^2 - A_2 b^2  \biggr]
+
2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)
</math>
  </td>
</tr>
</table>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\frac{2\pi G \rho }{(a^2 - b^2) }
\biggl[  A_1 a^2 - A_2 b^2  \biggr]a^2 b^2 \varpi^4 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 a b \lambda \Omega_f (x^2 + y^2) (a^2 y^2 + b^2 x^2)
- \lambda^2 (a^2 - b^2) [ a^2 y^4  - b^2 x^4 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  
\biggl\{
\biggl[\lambda b^2 - a b \Omega_f \biggr]^2 x^4
+
2\biggl[\lambda b^2 - a b \Omega_f \biggr]\biggl[ \lambda a^2 - a b \Omega_f\biggr]y^2x^2
+
\biggl[ \lambda a^2 - a b \Omega_f\biggr]^2 y^4
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 a b \lambda \Omega_f ( a^2 x^2 y^2 + b^2 x^4 + a^2 y^4 + b^2x^2y^2  )
- \lambda^2 (a^2 - b^2) [ a^2 y^4  - b^2 x^4 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  
\biggl[\lambda^2 b^4 -  2a b \Omega_f \lambda b^2 + a^2 b^2 \Omega_f^2 \biggr] x^4
+
2\biggl[\lambda^2 a^2 b^2  - \lambda a b^3 \Omega_f - \lambda a^3 b\Omega_f + a^2 b^2 \Omega_f^2 \biggr]y^2x^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[ \lambda^2 a^4 - 2a b \Omega_f  \lambda a^2 + a^2 b^2 \Omega_f^2 \biggr] y^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^2 y^2 \biggl\{ 2 \biggl[ \lambda^2 a^2 b^2  - \lambda a b^3 \Omega_f - \lambda a^3 b\Omega_f + a^2 b^2 \Omega_f^2 \biggr] + 2ab\lambda \Omega_f (a^2 + b^2) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  
\biggl[\lambda^2 b^4 -  2a b \Omega_f \lambda b^2 + a^2 b^2 \Omega_f^2 + 2ab^3 \lambda \Omega_f + \lambda^2(a^2-b^2)b^2\biggr] x^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[ \lambda^2 a^4 - 2a b \Omega_f  \lambda a^2 + a^2 b^2 \Omega_f^2 + 2ab\lambda \Omega_f a^2 - \lambda^2(a^2-b^2)a^2\biggr] y^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2a^2 b^2 x^2 y^2 \biggl[ \lambda^2 + \Omega_f^2  \biggr]
+  
\biggl[  \Omega_f^2  + \lambda^2 \biggr] a^2 b^2 x^4
+
\biggl[  \Omega_f^2 + \lambda^2 \biggr] a^2 b^2y^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \lambda^2 + \Omega_f^2\biggr]a^2 b^2 \biggl[ x^4 +2x^2y^2 +  y^4 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\frac{2\pi G \rho }{(a^2 - b^2) }
\biggl[  A_1 a^2 - A_2 b^2  \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \lambda^2 + \Omega_f^2\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48, Eq. (33)</font> ]</td></tr>
</table>

{{ SGFfooter }}