Editing ParabolicDensity/Axisymmetric/Structure/Try1thru7 (section)

===5<sup>th</sup> Try===

We should leave untouched the ''form'' of the expression for the centrifugal potential, but let its coefficient values remain unspecified.  The enthalpy function will therefore remain flexible, and, in tern, so will the components of the pressure gradient.  We should adjust these new coefficients in such a way that the gradient of the pressure is everywhere perpendicular to the surface of a constant-density contour; this means that the P-constant contours will be identical to the density-constant contours.

====Modifiable 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"><font color="purple"><b>Specific Angular Momentum:</b></font></td>
  <td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2j_1 \chi  - 2 j_3 \chi^3 
\, .
</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="purple"><b>Centrifugal Potential:</b></font></td>
  <td align="right">
<math>
\frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}\biggl[j_3 \chi^4   -2j_1 \chi^2  \biggr]\, .
</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="purple"><b>Enthalpy:</b></font></td>
  <td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr] - \frac{1}{2} I_\mathrm{BT}   
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 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]
- \frac{1}{2}\biggl[j_3 \chi^4   -2j_1 \chi^2  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot  
\biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="purple"><b>Radial Pressure Gradient:</b></font></td>
  <td align="right">
<math> 
\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[ 2j_1 - 2A_\ell  +
2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi 
+
\biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3
\biggr\}
</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:

====Desired Slopes of Normal Vectors====

A vector that is normal to the surface of a constant-density (oblate-spheroidal) contour has the following components:
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>\frac{\partial}{\partial \zeta}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial}{\partial \zeta}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] 
=
-2\zeta (1-e^2)^{-1}
\, .</math>
  </td>
</tr>
</table>
Hence, the slope, <math>m</math>, of this normal vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>m = \biggl\{\frac{\partial}{\partial \zeta}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]\biggr\}
\biggl\{\frac{\partial}{\partial \chi}\biggl[\frac{\rho(\varpi, z)}{\rho_c} \biggr]\biggr\}^{-1}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ -2\zeta (1-e^2)^{-1}}{-2\chi}
=
\frac{\zeta}{\chi(1-e^2)} \, .
</math>
  </td>
</tr>
</table>
Now, if the constant-pressure contours are to lie precisely on top of our constant-density contours, the normals have to have the same slopes.  This means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\partial P}{\partial \zeta}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta}{\chi(1-e^2)} \biggl[\frac{\partial P}{\partial \chi}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \chi(1-e^2)\biggl\{
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta 
\biggl\{
\biggl[ 2j_1 - 2A_\ell  +
2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi 
+
\biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
2A_{\ell s}a_\ell^2 (1-e^2) \chi^3\zeta - 2A_s(1-e^2) \chi\zeta  
+  2A_{ss} a_\ell^2 (1-e^2)\chi \zeta^3   
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
( 2j_1 - 2A_\ell ) \chi \zeta  + 2A_{\ell s}a_\ell^2  \chi \zeta^3
+
(2A_{\ell \ell} a_\ell^2 - 2j_3 )\chi^3 \zeta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
\biggl[ 2A_{\ell s}a_\ell^2 (1-e^2)  
- 
(2A_{\ell \ell} a_\ell^2 - 2j_3 ) \biggr] \chi^3 \zeta
+ \biggl[- 2A_s(1-e^2) - ( 2j_1 - 2A_\ell ) \biggr]\chi\zeta   
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ 2A_{\ell s}a_\ell^2 -  2A_{ss} a_\ell^2 (1-e^2) \biggr] \chi \zeta^3
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
Note &hellip;

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

<tr>
  <td align="right">
<math>3A_{ss}  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_s^2} - 2A_{\ell s}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_\ell^2(1-e^2)} - 2A_{\ell s}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ 3(1-e^2) (A_{ss} a_\ell^2)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2 - 2(1-e^2) (A_{\ell s}a_\ell^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \mathrm{RHS}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2A_{\ell s}a_\ell^2 -  2A_{ss} a_\ell^2 (1-e^2) \biggr\} \chi \zeta^3
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3}\biggl[ (5-2e^2) (A_{\ell s}a_\ell^2) - 2 
\biggr] \chi \zeta^3
\, .
</math>
  </td>
</tr>
</table>

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

In order for the <math>\chi^3\zeta</math> term on the LHS to be zero, we should set &hellip;

<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>
\biggl[ 2A_{\ell s}a_\ell^2 (1-e^2)  
- 
2A_{\ell \ell} a_\ell^2 + 2j_3  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ j_3  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
A_{\ell \ell} a_\ell^2 -  A_{\ell s}a_\ell^2 (1-e^2) 
\, ; 
</math>
  </td>
</tr>

</table>
and in order for the <math>\chi\zeta</math> term on the LHS to be zero, we should set &hellip;

<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>
\biggl[- 2A_s(1-e^2) - ( 2j_1 - 2A_\ell ) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ j_1  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
A_\ell - A_s(1-e^2)
\, .
</math>
  </td>
</tr>
</table>

====Desired Slopes of Tangent Vectors====
Alternatively, if the constant-pressure contours are to lie precisely on top of our constant-density contours, the tangent vectors have to have slopes given by <math>-1/m</math>.  This means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\partial P}{\partial \zeta}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{1}{m}\biggl[\frac{\partial P}{\partial \chi}\biggr]
=
-\frac{\chi(1-e^2)}{\zeta} \biggl[\frac{\partial P}{\partial \chi}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \zeta \biggl\{
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\chi(1-e^2) 
\biggl\{
\biggl[ 2j_1 - 2A_\ell  +
2A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi 
+
\biggl[ 2A_{\ell \ell} a_\ell^2 - 2j_3 \biggr]\chi^3
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl\{
\biggl[ A_{\ell s}a_\ell^2 \chi^2 - A_s \biggr]\zeta^2  
+  A_{ss} a_\ell^2  \zeta^4   
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- (1-e^2)
\biggl\{
\biggl[ j_1 - A_\ell  +
A_{\ell s}a_\ell^2 \zeta^2 \biggr] \chi^2 
+
\biggl[ A_{\ell \ell} a_\ell^2 - j_3 \biggr]\chi^4
\biggr\}
</math>
  </td>
</tr>
</table>