Editing ParabolicDensity/Axisymmetric/Structure/Try1thru7 (section)

===Hydrostatic Balance (Algebraic Condition)===
Following our [[Apps/MaclaurinSpheroids#Equilibrium_Structure|separate discussion of the equilibrium structure]] of Maclaurin spheroids, and given that our solution of the Poisson equation fixes the expression for <math>\Phi_\mathrm{grav} </math>, the algebraic expression ensuring hydrostatic balance is,
 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>H(\varpi, z)</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
C_B - \biggl[ \Phi_\mathrm{grav}(\varpi, z) + \Psi(\varpi, z) \biggr] \, ,
</math>
  </td>
</tr>
</table>
where, <math>\Psi</math> is the centrifugal potential.  <font color="red">NOTE:</font> &nbsp; Generally when modeling axisymmetric astrophysical systems (see our [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying discussion of ''simple'' rotation profiles]]) it is assumed that <math>\Psi</math> does not functionally depend on <math>z</math>.  Here, our other constraints &#8212; for example, demanding that the configuration have a parabolic density distribution &#8212; may force us to adopt a <math>z</math>-dependent rotation profile.

Here, we know that the adopted parabolic density distribution gives rise to a gravitational potential of the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\mathbf{x})}{(-\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}  
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] 
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ A_{ss} a_\ell^2  \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\Psi(\varpi, z)</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
C_B - \Phi_\mathrm{grav}(\varpi, z) - H(\varpi, z)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}  
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] 
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ A_{ss} a_\ell^2  \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\} 
-
H_c h(\xi_1) \, .
</math>
  </td>
</tr>
</table>


<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
We presume that the enthalpy profile, as well as the density profile, can be rewritten in terms of [[Appendix/Ramblings/T1Coordinates#T1_Coordinates|T1 Coordinates]].  In particular, defining, <math>q \equiv a_\ell/a_s</math> and,

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

<tr>
  <td align="right">
<math>\xi_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ z^2 + \biggl(\frac{\varpi}{q}\biggr)^2\biggr]^{1 / 2}
=
a_s\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ 1 - \biggl(\frac{\xi_1}{a_s}\biggr)^2 \biggr] \, .</math>
  </td>
</tr>
</table>
Because we expect contours of constant enthalpy <math>(H)</math> to coincide with contours of constant density in equilibrium configurations, we should expect to find that,

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

<tr>
  <td align="right">
<math>\frac{H}{H_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>h(\xi_1) \, .</math>
  </td>
</tr>
</table>
If the "radial" enthalpy profile resembles our [[SSC/Structure/OtherAnalyticModels#SphericalEnthalpyProfile|derived spherical enthalpy profile]], we should expect to find that,

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

<tr>
  <td align="right">
<math>h(\xi_1)</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>h_0 \biggl[1 - h_2 \xi_1^2 - h_4 \xi_1^4 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 1 - \frac{h(\xi_1)}{h_0}</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>h_2 \xi_1^2 + h_4 \xi_1^4</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] 
+ 
h_4 \biggl\{ a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] \biggr\}^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] 
+ 
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4 
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2 
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr] 
</math>
  </td>
</tr>
</table>

</td></tr></table>
Adopting this last expression for the enthalpy, we have,

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

<tr>
  <td align="right">
<math>\frac{h(\xi_1)}{h_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{z}{a_s}\biggr)^2 + \biggl(\frac{\varpi}{a_\ell}\biggr)^2\biggr] 
-
h_4 a_s^4 \biggl[ \biggl(\frac{z}{a_s}\biggr)^4 
+ 2\biggl(\frac{z}{a_s}\biggr)^2\biggl(\frac{\varpi}{a_\ell}\biggr)^2 
+ \biggl(\frac{\varpi}{a_\ell}\biggr)^4 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] 
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} 
\biggr] 
\, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\Psi(\varpi, z)</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}  
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] 
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ A_{ss} a_\ell^2  \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] 
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} 
\biggr] 
\biggr\} \, . 
</math>
  </td>
</tr>
</table>

At the pole of the configuration &#8212; that is, when <math>(\varpi, z) = (0, a_s)</math> &#8212; this statement of hydrostatic balance becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Psi(\varpi, z)</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}  
- \biggl[A_\ell \cancelto{0}{\biggl(\frac{\varpi^2}{a_\ell^2}\biggr)} + A_s \biggl( \frac{a_s^2}{a_\ell^2}\biggr) \biggr] 
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2  \cancelto{0}{\biggl(\frac{\varpi^4}{a_\ell^4}\biggr)} 
+ A_{ss} a_\ell^2  \biggl(\frac{a_s^4}{a_\ell^4}\biggr)   
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr)
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^2} + \biggl(\frac{a_s}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] 
-
h_4 a_s^4 \biggl[\cancelto{0}{\biggl(\frac{\varpi}{a_\ell}\biggr)^4} + \biggl(\frac{a_s}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\cancelto{0}{\varpi^2} a_s^2}{a_\ell^4}\biggr) (1-e^2)^{-1} 
\biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
C_B + \pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}  
- A_s (1-e^2)  
+ \frac{1}{2} A_{ss} a_\ell^2  (1-e^2)^2   
\biggr] 
-
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr] 
\, . 
</math>
  </td>
</tr>
</table>

For centrally condensed configurations, it is astrophysically reasonable to assume that <math>\Psi(\varpi, z)</math> is of the form such that the centrifugal potential goes to zero when <math>\varpi \rightarrow 0</math>.  Adopting that assumption here means that the Bernoulli constant has the value,
 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>C_B</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
  <td align="left">
<math>
H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr] 
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}  
- A_s (1-e^2)  
+ \frac{1}{2} A_{ss} a_\ell^2  (1-e^2)^2   
\biggr] 
\, .
</math>
  </td>
</tr>
</table>
Plugging this expression for <math>C_B</math> back into the general statement of hydrostatic balance gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Psi(\varpi, z)</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\frac{1}{2} I_\mathrm{BT}  
- \biggl[A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) + A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] 
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ A_{ss} a_\ell^2  \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
-
\pi G \rho_c a_\ell^2\biggl[
\frac{1}{2} I_\mathrm{BT}  
- A_s (1-e^2)  
+ \frac{1}{2} A_{ss} a_\ell^2  (1-e^2)^2   
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ H_c h_0 \biggl[ 1 - h_2 a_s^2 - h_4 a_s^4 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
-
H_c h_0 \biggl\{
1 -
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1}\biggr] 
-
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} 
\biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
\biggl[A_s (1-e^2)-A_\ell \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) - A_s \biggl( \frac{z^2}{a_\ell^2}\biggr) \biggr] 
+ \frac{1}{2} \biggl[
A_{\ell \ell} a_\ell^2  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ A_{ss} a_\ell^2  \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ 2A_{\ell s}a_\ell^2 \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- A_{ss} a_\ell^2  (1-e^2)^2   
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2 + \biggl(\frac{z}{a_\ell}\biggr)^2(1-e^2)^{-1} - 1\biggr] 
+
h_4 a_s^4 \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4 + \biggl(\frac{z}{a_\ell}\biggr)^4 (1-e^2)^{-2}
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2)^{-1} -1
\biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\pi G \rho_c a_\ell^2\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) -  \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr] 
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}}  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2   
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+
H_c h_0 \biggl\{
h_2 a_s^2(1-e^2)^{-1}\biggl[ \biggl(\frac{\varpi}{a_\ell}\biggr)^2(1-e^2) + \biggl(\frac{z}{a_\ell}\biggr)^2 - (1-e^2)\biggr] 
+
h_4 a_s^4 (1-e^2)^{-2} \biggl[\biggl(\frac{\varpi}{a_\ell}\biggr)^4(1-e^2)^{2} + \biggl(\frac{z}{a_\ell}\biggr)^4 
+ 2\biggl(\frac{\varpi^2 z^2}{a_\ell^4}\biggr) (1-e^2) - (1-e^2)^{2}
\biggr] 
\biggr\} 
</math>
  </td>
</tr>
</table>

Let's set &hellip;
<div align="center">
<math>H_c h_0 = \pi G \rho_c a_\ell^2 \, ;</math> &nbsp; &nbsp; &nbsp;
<math>h_2 = \frac{A_s(1-e^2)}{a_s^2} \, ;</math> &nbsp; &nbsp; &nbsp;
<math>h_4 = - \frac{ A_{ss}a_\ell^2 (1-e^2)^2 }{ 2a_s^4 } \, .</math> 
</div>
This gives,

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

<tr>
  <td align="right">
<math>\frac{\Psi(\varpi, z)}{\pi G \rho_c a_\ell^2}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
A_s \biggl[- \frac{A_\ell}{A_s} \biggl(\frac{\varpi^2}{a_\ell^2}\biggr) -  \biggl( \frac{z^2}{a_\ell^2}\biggr) + (1-e^2)\biggr] 
+ \frac{A_{ss}a_\ell^2}{2} \biggl[
\frac{A_{\ell \ell}}{A_{ss}}  \biggl(\frac{\varpi^4}{a_\ell^4}\biggr) 
+ \biggl(\frac{z^4}{a_\ell^4}\biggr)   
+ \frac{2A_{\ell s}}{A_{ss}} \biggl( \frac{\varpi^2 z^2}{a_\ell^4}\biggr)
- (1-e^2)^2   
\biggr]
\biggr\} 
</math>
  </td>
</tr>

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

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

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

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

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