Editing ParabolicDensity/Axisymmetric/Structure/Try1thru7 (section)

===3<sup>rd</sup> Try===
From the [[#Radial_Component|above, "2<sup>nd</sup> Try" discussion of the radial component]], we can write the following "EXACT!" relation,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">EXACT!</font></div>

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
- \frac{\rho}{\rho_c} \cdot \frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
+ 
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)}  \biggr]\frac{\partial P}{\partial \chi}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2  \chi^3 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)}  \biggr]\frac{\partial P}{\partial \chi}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot \biggl\{
\biggl[2A_{\ell s}a_\ell^2 \zeta^2 - 2A_\ell \biggr] \chi
+ 2 A_{\ell \ell} a_\ell^2  \chi^3 
+ 
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>
Now, our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math> suggests that the left-hand-side of this expression should be of the form,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
LHS &nbsp;
<math>
\equiv \biggl[\frac{1}{(\pi G \rho_c^2 a_\ell^2)}  \biggr]\frac{\partial P}{\partial \chi}
</math>
  </td>
  <td align="center"><math>\sim</math></td>
  <td align="left">
<math>
c_2\zeta^2 + c_4\zeta^4 \, ,
</math>
  </td>
</tr>
</table>
where it is understood that the coefficients, <math>c_2</math> and <math>c_4</math>, are both functions of <math>\chi</math>.  This should be compared with the "EXACT!" expression for the RHS after multiplying through by the expression for the dimensionless density, that is,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
RHS
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[
1 - \chi^2 - \zeta^2(1-e^2)^{-1}
\biggr] \cdot \biggl\{
\biggl[ 2 A_{\ell \ell} a_\ell^2  \chi^3 
- 2A_\ell \chi  + 
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi \zeta^2 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(1 - \chi^2)\biggl[ 2 A_{\ell \ell} a_\ell^2  \chi^3 
- 2A_\ell \chi  + 
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
+ 2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- \biggl[ 2 A_{\ell \ell} a_\ell^2  \chi^3 
- 2A_\ell \chi  + 
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr](1-e^2)^{-1}\zeta^2
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4 
\, .
</math>
  </td>
</tr>
</table>

Because we are not expecting to see a term that is independent of <math>\zeta</math>, this suggests that the term inside the large square brackets must be zero.  This leads to an expression for the distribution of specific angular momentum of the form,

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<table border="0" align="center" cellpadding="8">

<tr><td align="center" colspan="3"><font color="red">EXCELLENT !!</font></td></tr>

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 2 A_{\ell \ell} a_\ell^2  \chi^3 
- 2A_\ell \chi  + 
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{j^2 }{(\pi G \rho_c a_\ell^4)}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2A_\ell \chi^4  - 2 A_{\ell \ell} a_\ell^2  \chi^6 
\, .
</math>
  </td>
</tr>
</table>

According to our [[AxisymmetricConfigurations/SolutionStrategies#Specifying_Radial_Rotation_Profile_in_the_Equilibrium_Configuration|accompanying discussion of ''Simple'' rotation profiles]], the corresponding centrifugal potential is given by the expression,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\Psi</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \int \frac{j^2(\varpi)}{\varpi^3} d\varpi
=
- (\pi G \rho_c a_\ell^2) \int \frac{1}{\chi^3} \biggl[2A_\ell \chi^4  - 2 A_{\ell \ell} a_\ell^2  \chi^6\biggr]d\chi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \int \biggl[2A_\ell \chi  - 2 A_{\ell \ell} a_\ell^2  \chi^3\biggr]d\chi
=
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2  \biggr]\, .
</math>
  </td>
</tr>
</table>
(Here, we ignore the integration constant because it will be folded in with the Bernoulli constant.)

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

It also means that the RHS expression simplifies to the form,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
RHS
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2A_{\ell s}a_\ell^2 \chi (1 - \chi^2) \zeta^2  
- 2A_{\ell s}a_\ell^2 (1-e^2)^{-1}\chi \zeta^4 \, . 
</math>
  </td>
</tr>
</table>

This should be compared to our [[#RadialDerivative|earlier examination of the radial derivative of]] <math>P_\mathrm{vert}</math>, namely, 
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[ \frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr]\frac{\partial P_\mathrm{vert}}{\partial \chi} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2  +  2A_s )\zeta^2 -  A_{ss} a_\ell^2 \zeta^4   - A_{\ell s}a_\ell^2 (1-e^2)^{-1}\zeta^4 \biggr] \chi 
- 4A_{\ell s}a_\ell^2 \zeta^2\chi^3    
</math>
  </td>
</tr>

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