Editing ParabolicDensity/Axisymmetric/Structure (section)

====Known Relations====

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

<tr>
  <td align="left"><font color="orange"><b>Density:</b></font></td>
  <td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}  
- A_\ell \chi^2  - A_s \zeta^2  
 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2)  \chi^4 \biggr]
\, .
</math>
  </td>
</tr>

<tr>
  <td align="left">&nbsp;</td>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\partial}{\partial\zeta} \biggl[\frac{ \Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(A_{\ell s}a_\ell^2 )\chi^2 \zeta - 2A_s \zeta  + 2(A_{s s} a_\ell^2) \zeta^3
\, .
</math>
  </td>
</tr>

<tr>
  <td align="left">&nbsp;</td>
  <td align="right">
and, &nbsp; &nbsp; <math>\frac{\partial}{\partial\chi} \biggl[\frac{ \Phi_\mathrm{grav}}{(-\pi G\rho_c a_\ell^2)} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(A_{\ell s}a_\ell^2 )\chi \zeta^2
- 2A_\ell \chi  
+ 2(A_{\ell \ell} a_\ell^2)  \chi^3
\, .
</math>
  </td>
</tr>
</table>

where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>, and the relevant index symbol expressions are:

<table align="center" border=0 cellpadding="3">

<tr>
  <td align="right"><math>I_\mathrm{BT}</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
2A_\ell + A_s (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ;
</math>
  </td>
  <td align="right">[1.7160030]</td>
</tr>

<tr>
  <td align="right">
<math>
A_\ell
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ;
</math>
  </td>
  <td align="right">[0.6055597]</td>
</tr>

<tr>
  <td align="right"><math>A_s</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ;
</math>
  </td>
  <td align="right">[0.7888807]</td>
</tr>

<tr>
  <td align="right">
<math>
a_\ell^2 A_{\ell \ell}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{4e^4}\biggl\{- (3 + 2e^2) (1-e^2)+3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
=
\biggl[\frac{1}{2}-\frac{(A_s - A_\ell)}{4e^2}\biggr]
\, ;
</math>&nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">[0.3726937]</td>
</tr>

<tr>
  <td align="right">
<math>a_\ell^2 A_{ss} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{3}\biggl\{
\frac{( 4e^2 - 3 )}{e^4(1-e^2)} 
+
\frac{3 (1-e^2)^{1 / 2}}{e^4} \biggl[\frac{\sin^{-1}e}{e}\biggr] \biggr\}
=
\frac{2}{3}\biggl[ (1-e^2)^{-1} - \frac{(A_s-A_\ell)}{e^2} \biggr]
\, ;
</math>&nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">[0.7021833]</td>
</tr>

<tr>
  <td align="right">
<math>
a_\ell^2 A_{\ell s}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{ e^4} \biggl\{
(3-e^2) 
-
3 (1-e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] 
\biggr\} 
=
\frac{(A_s - A_\ell)}{e^2}
\, ,
</math>
  </td>
  <td align="right">[0.5092250]</td>
</tr>
</table>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_s}{a_\ell}\biggr)^2  \biggr]^{1 / 2} \, .
</math>
</div>

<font color="red">NOTE: &nbsp; The posted numerical evaluations (inside square brackets) assume that the configuration's eccentricity is</font> <math>e = 0.6 \Rightarrow a_s/a_\ell = 0.8</math>.

Drawing from our separate "[[ParabolicDensity/Axisymmetric/Structure/Try8thru10#6th_Try|6<sup>th</sup> Try]]" discussion &#8212; and as has been highlighted [[AxisymmetricConfigurations/PGE#RelevantCylindricalComponents|here]] for example &#8212; for the axisymmetric configurations under consideration, the <math>\hat{e}_z</math> and <math>\hat{e}_\varpi</math> components of the Euler equation become, respectively,</span>
<table border="1" align="center" cellpadding="10"><tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>{\hat{e}}_z</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
0 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{j^2}{\varpi^3}  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr]  
</math>
  </td>
</tr>

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

Multiplying the <math>\hat{e}_z</math> component through by length <math>(a_\ell)</math> and dividing through by the square of the velocity <math>(\pi G \rho_c a_\ell^2)</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
0 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]\frac{a_\ell}{(\pi G\rho_c a_\ell^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\frac{\rho_c}{\rho}\cdot \frac{\partial }{\partial \zeta}\biggl[ \frac{P}{(\pi G\rho_c^2 a_\ell^2)} \biggr] 
- \frac{\partial }{\partial \zeta}\biggl[ \frac{\Phi}{(-~\pi G\rho_c a_\ell^2)} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\partial }{\partial \zeta}\biggl[ \frac{P}{(\pi G\rho_c^2 a_\ell^2)} \biggr] </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\frac{\rho}{\rho_c}\cdot \frac{\partial }{\partial \zeta}\biggl[ \frac{\Phi}{(-~\pi G\rho_c a_\ell^2)} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\frac{\rho}{\rho_c}\cdot \biggl[ 
2(A_{\ell s}a_\ell^2 )\chi^2 \zeta - 2A_s \zeta  + 2(A_{s s} a_\ell^2) \zeta^3
\biggr] 
</math>
  </td>
</tr>
</table>

Multiplying the <math>\hat{e}_\varpi</math> component through by length <math>(a_\ell)</math> and dividing through by the square of the velocity <math>(\pi G \rho_c a_\ell^2)</math>, we have,

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

<tr>
  <td align="right"><math>{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{j^2}{\varpi^3} \cdot \frac{a_\ell}{(\pi G\rho_c a_\ell^2)} 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi_\mathrm{grav}}{\partial\varpi}\biggr] \frac{a_\ell}{(\pi G\rho_c a_\ell^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="right">
<math>\Rightarrow ~~~
\frac{1}{\chi^3} \cdot \frac{j^2}{(\pi G\rho_c a_\ell^4)} 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\frac{\rho_c}{\rho}\cdot\frac{\partial }{\partial \chi}\biggl[ \frac{P}{(\pi G\rho_c^2 a_\ell^2)} \biggr] 
- \frac{\partial }{\partial \chi}\biggl[ \frac{\Phi_\mathrm{grav}}{(-~\pi G\rho_c a_\ell^2)} \biggr]
</math> 
  </td>
</tr>
</table>