Editing ParabolicDensity/Axisymmetric/Structure/Try1thru7 (section)

===4<sup>th</sup> Try===

In our [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|accompanying discussion of Ferrers Potential]], we have derived the expression for the gravitational potential inside (and on the surface of) a triaxial ellipsoid with a parabolic density distribution.  Specifically, for
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho(\mathbf{x})</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr] 
\, ,</math>
  </td>
</tr>
</table>
[[ThreeDimensionalConfigurations/FerrersPotential#GravFor1|we find]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\pi G\rho_c)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT} a_1^2 
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2}  \biggl(A_{11}x^4 +  A_{22}y^4 + A_{33}z^4  \biggr) 
\, .
</math>
  </td>
</tr>
</table>
In this [[ThreeDimensionalConfigurations/FerrersPotential#The_Case_Where_n_=_1|same accompanying discussion]], we plugged this expression for the gravitational potential into the Poisson equation and demonstrated that it properly generates the expression for the parabolic density distribution.  For the axisymmetric configuration being considered here &#8212; with the short axis aligned with <math>c = a_3 = a_s</math> &#8212; these two relations become,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <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 \frac{\varpi^2}{a_\ell^2}  - A_s \frac{z^2}{a_\ell^2}  
+ (A_{\ell s}a_\ell^2 )\frac{ \varpi^2z^2 }{a_\ell^4} + \frac{1}{2}(A_{s s} a_\ell^2) \frac{z^4}{a_\ell^4}
+ \frac{A_{\ell \ell}a_\ell^2}{2}  \biggl[ \frac{(x^4 + 2 x^2y^2 +  y^4 )}{a_\ell^4} \biggr]
\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>
</table>
where, <math>\chi \equiv \varpi/a_\ell</math> and <math>\zeta \equiv z/a_\ell</math>.  (This matches the [[#Gravitational_Potential|expression derived above]].)


----

Discuss scalar relationship between the enthalpy <math>(H)</math> and the effective potential.

As has been detailed in [[AxisymmetricConfigurations/SolutionStrategies#Technique|an accompanying discussion of solution techniques]], a configuration will be in dynamic equilibrium if,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\biggl[ H + \Phi_\mathrm{grav} + \Psi \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ H + \Phi_\mathrm{grav} + \Psi  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
constant
<math>
= C_B
</math>
  </td>
</tr>
</table>

Given that, in our particular case, we have analytic expressions for <math>\Phi_\mathrm{grav}(\chi,\zeta)</math> and for <math>\Psi(\chi,\zeta)</math>, we deduce that, to within a constant, the enthalpy distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ \frac{H(\chi, \zeta) - C_B}{(\pi G\rho_c a_\ell^2)} \biggr]  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\Phi_\mathrm{grav}}{{(\pi G\rho_c a_\ell^2)}} - \frac{\Psi}{{(\pi G\rho_c a_\ell^2)}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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]
- 
\frac{1}{2}\biggl[ A_{\ell \ell}a_\ell^2 \chi^4 - 2A_\ell \chi^2  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}  
- 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
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}  
- A_s \zeta^2  
 + \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(A_{\ell s}a_\ell^2 )\chi^2 \biggr]
</math>
  </td>
</tr>
</table>
Now, according to our [[ParabolicDensity/GravPot#Parabolic_Density_Distribution_2|related discussion of index symbols]],

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

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

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

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

<tr>
  <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_s \zeta^2  
 + \frac{\zeta^2}{2}
\biggl[(A_{s s} a_\ell^2) \zeta^2 + 2(1-e^2)^{-1}\chi^2 - 3(A_{s s}a_\ell^2) \chi^2 \biggr]
</math>
  </td>
</tr>

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

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Examining the radial derivative &hellip;
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(A_{\ell s} a_\ell^2)\zeta^2\chi
\, .
</math>
  </td>
</tr>
</table>
<font color="red">YES !!!</font>  This matches the "radial" pressure-gradient, below.

Now, examining the vertical derivative &hellip;
<table border="0" cellpadding="5" align="center">

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 2A_s \zeta  
+ 
\biggl[2(A_{s s} a_\ell^2) \zeta^3 
+ 2(A_{\ell s} a_\ell^2) \chi^2\zeta \biggr]
</math>
  </td>
</tr>
</table>
<font color="red">HURRAY !!!</font>  This matches the "vertical" pressure-gradient, below.

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


----


<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}{\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="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[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>
Plug in &hellip;

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

<tr>
  <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>
2A_\ell \chi  - 2 A_{\ell \ell} a_\ell^2  \chi^3 
\, .
</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 
+ 
2A_\ell \chi  - 2 A_{\ell \ell} a_\ell^2  \chi^3\biggr\}
</math>
  </td>
</tr>

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

<!-- TEMPORARY PRESSURE (BEGIN)
The result appears to be something like &hellip;

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

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

Hence, examination of the radial component leads to the following suggested expression for the pressure:

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

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

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

While examination of the vertical component leads to the following suggested expression for the pressure:

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