Editing SSC/Stability/NeutralMode (section)

=Parabolic Density Distribution=
Here, we build upon our [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|separate discussion]] of equilibrium configurations with a parabolic density distribution.

==Equilibrium Structure==

In an article titled, "Radial Oscillations of a Stellar Model," [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P C. Prasad (1949, MNRAS, 109, 103)] investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,
<div align="center">
<math>\rho_0 = \rho_c\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr] \, ,</math>
</div>
where, <math>\rho_c</math> is the central density and, <math>R</math> is the radius of the star.  Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{r_0} 4\pi r_0^2 \rho_0 dr_0</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi\rho_c r_0^3}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
in which case we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_0 \equiv \frac{G M_r }{r_0^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi G \rho_c r_0}{3}  
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2\biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Delta \equiv \frac{M_r }{4\pi r_0^3\rho_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1 }{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1}
\, .</math>
  </td>
</tr>
</table>

Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P Prasad (1949)] determines that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi G\rho_c^2 R^2}{15} 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\, ,</math>
  </td>
</tr>
</table>
</div>
where, it can readily be deduced, as well, that the central pressure is,
<div align="center">
<math>P_c = \frac{4\pi}{15} G\rho_c^2 R^2 \, .</math>
</div>

<table border="1" width="90%" cellpadding="8" align="center"><tr><td align="left">
<div align="center">'''Specific Entropy Distribution'''</div>


For purposes of later discussion, we find from [[Appendix/Ramblings/PatrickMotl#Tying_Expressions_into_H_Book_Context|a separate examination of specific entropy distributions]], <math>s_0(r_0)</math>, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s_0}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{(\gamma_g-1)}\ln \biggl(\frac{\tau_0}{\rho_0}\biggr)^{\gamma_g} 
=
\frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P_0}{(\gamma_g-1)\rho_0^{\gamma_g}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]s_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \frac{P_c}{(\gamma_g-1)\rho_c^{\gamma_g}} \biggr] 
+
\ln \biggl\{ \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
 \biggr\} 
+
\ln \biggl\{ \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-\gamma_g}\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \frac{P_c}{(\gamma_g-1)\rho_c^{\gamma_g}} \biggr] 
+
\ln \biggl\{ 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
 \biggr\} 
+ (2- \gamma_g)
\ln \biggl\{ \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]\biggr\} \, .
</math>
  </td>
</tr>
</table>

Notice that, independent of the value of <math>\gamma_g</math>, the specific entropy varies with <math>r_0</math> throughout the structure.  According to the [[2DStructure/AxisymmetricInstabilities#Schwarzschild_Criterion|Schwarzschild criterion]], spherically symmetric equilibrium configurations will be stable against convection if the specific entropy increases outward, and unstable toward convection if the specific entropy decreases outward.  Let's examine the slope, <math>ds_0/dr_0</math>, throughout configurations that have a parabolic density distribution.    

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

<tr>
  <td align="right">
<math>\biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]\frac{ds_0}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{r_0}{R^2} \biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
- \frac{2(2- \gamma_g)r_0}{R^2}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{R^2}{r_0}
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]
\biggl[ \frac{\gamma_g - 1}{\Re/\bar{\mu}} \biggr]\frac{ds_0}{dr_0}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
- 2(2- \gamma_g)
\biggl[ 1 - \frac{1}{2}\biggl(\frac{r_0}{R} \biggr)^2 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2\gamma_g- 5)
+ (3 - \gamma_g)
\biggl[\biggl(\frac{r_0}{R} \biggr)^2 \biggr] 
</math>
  </td>
</tr>
</table>

[[File:EntropyDistribution245.png|right|400px]]
The slope is zero when,

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

<tr>
  <td align="right">
<math>
\biggl(\frac{r_0}{R} \biggr)^2 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5 - 2\gamma_g}{3 - \gamma_g} \, .
</math>
  </td>
</tr>
</table>
Moving from the center of the configuration to its surface, <math>0 < (r_0/R)^2 < 1</math>, the slope will go to zero &#8212; hence, the slope of the entropy will change sign

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

==Some Relevant Structural Derivatives==

We note for later use that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{P_c} \cdot \frac{dP_0}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
\frac{d}{dr_0}\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
+
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\frac{d}{dr_0}\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{1}{2R^2} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
\frac{d}{dr_0}\biggl[r_0^2\biggr]
+
2\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\frac{d}{dr_0}\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
-\frac{1}{2R^2} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
- \frac{2}{R^2}\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggr\}\frac{d}{dr_0}\biggl[r_0^2\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{r_0}{R^2}\biggl\{
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
+ 4\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{r_0}{R^2}\biggl\{
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
+ 4
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggr\} \biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{5r_0}{R^2}
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
Checking for detailed force-balance, we note that,

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

<tr>
  <td align="right">
<math>- ~\frac{1}{\rho_0} \cdot \frac{dP_0}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{5r_0}{R^2}\biggl[ \frac{4\pi G \rho_c^2 R^2}{15} \biggr]
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] \cdot \frac{1}{\rho_c} \biggl[1 - \biggl(\frac{r_0}{R} \biggr)^2\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi G \rho_c r_0}{3}
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] \, ,
</math>
  </td>
</tr>
</table>
which is exactly the expression that we have just derived for <math>g_0 = GM_r/r_0^2</math>.
</td></tr></table>

Hence,

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

<tr>
  <td align="right">
<math>\frac{d\ln P_0}{d \ln r_0} = \frac{r_0}{P_0/P_c} \biggl[ \frac{1}{P_c}\cdot \frac{dP_0}{dr_0} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{5r_0^2}{R^2}
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl\{\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]\biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-5 \biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\, ;
</math>
  </td>
</tr>
</table>
and, given that,

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

<tr>
  <td align="right">
<math>\Delta^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
3 
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]^{-1} 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]
\, ,</math>
  </td>
</tr>
</table>

we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{\Delta}\cdot \frac{d\ln P_0}{d \ln r_0} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 15 \biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]^{-1} 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 15 \biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
Also,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>3 \cdot \frac{d\Delta}{dr_0} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{dr_0}\biggl\{ 
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]\frac{d}{dr_0} 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1}
+
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1}\frac{d}{dr_0} 
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2}
\frac{2r_0}{R^2}
+
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} 
\biggl[- \frac{6}{5} \cdot \frac{r_0}{R^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{r_0}{R^2} \biggl\{
2\biggl[1 - \frac{3}{5} \biggl( \frac{r_0}{R} \biggr)^2 \biggr] 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2}
- \frac{6}{5}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-1} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{r_0}{5R^2} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2}\biggl\{
\biggl[10 - 6 \biggl( \frac{r_0}{R} \biggr)^2 \biggr] 
- 
\biggl[ 6 - 6\biggl(\frac{r_0}{R} \biggr)^2 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4r_0}{5R^2} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d\Delta}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4r_0}{15R^2} \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2}
\, ;
</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right">
<math>\frac{15R^2}{4} \cdot \frac{d^2\Delta}{dr_0^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2}
-2r_0
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}\biggl[ -\frac{2r_0}{R^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} \biggl\{
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]
+\frac{4r_0^2}{R^2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} 
\biggl[1 + 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggr]
\, .
</math>
  </td>
</tr>
</table>

==Neutral Mode==
Again, adopting the ''trial'' eigenfunction,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>x_t</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
a - b\Delta
\, ,
</math>
</td>
</tr>
</table>
from the,

<div align="center" id="Delta_Highlighted">
<font color="#770000">'''&Delta;-Highlighted LAWE'''</font><br />

{{Math/EQ_RadialPulsation04}}
</div>
we can write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>\mathrm{LAWE}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- b \cdot \frac{d^2\Delta}{dr_0^2} - \frac{b}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} 
+ \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{(a - b\Delta)}{r_0^2} 
- \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \frac{(a - b\Delta)}{r_0^2} 
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} - r_0\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} 
+ \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl( \frac{a}{b} - \Delta \biggr) 
- \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
- r_0\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl( \frac{a}{b} - \Delta \biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} \biggl(\frac{r_0}{R}\biggr)^2 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} 
\biggl[1 + 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggr]
 </math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
- \frac{16}{15}\biggl(\frac{r_0}{R}\biggr)^2  
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} 
- \frac{4}{15}\biggl(\frac{r_0}{R}\biggr)^2 
\biggl\{ -5 \biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
 \biggr\} 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \alpha
\biggl\{ -5 \biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggr\} 
\biggl\{ \frac{a}{b} - \Delta \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^2\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr]
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
 </math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^4 
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ ~ \frac{5\alpha}{3}
\biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl\{
1 - \frac{3a}{b} -\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2
\biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^2\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr]
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
+ ~ \frac{5\alpha}{3}
\biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl(1 - \frac{3a}{b} \biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^4 
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} 
- ~ \alpha
\biggl(\frac{r_0}{R}\biggr)^4 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>
</table>

Continuing &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(\frac{r_0}{R}\biggr)^2 \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}

\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl\{
\frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr)
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2 \biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
 
- \frac{4}{3}\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr]\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl(\frac{4}{3} - \alpha  \biggr)\biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
 </math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-~ \alpha \biggl(\frac{r_0}{R}\biggr)^6\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(\frac{r_0}{R}\biggr)^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
\biggl\{
\frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr)
\biggl[1-\frac{8}{5}\biggl(\frac{r_0}{R}\biggr)^2 + \frac{3}{5} \biggl( \frac{r_0}{R}\biggr)^4\biggr]
- 
~\frac{4}{3}\biggl[ 1 - \frac{7}{5} \biggl(\frac{r_0}{R}\biggr)^2 + \frac{1}{10}\biggl(\frac{r_0}{R}\biggr)^4\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl(\frac{4}{3} - \alpha  \biggr)\biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
-~ \alpha \biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(\frac{r_0}{R}\biggr)^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
\biggl\{ \biggl[\frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) - \frac{4}{3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl(\frac{r_0}{R}\biggr)^2
\biggl[
-\frac{8\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) + \frac{28}{15}
\biggr]
+ \biggl( \frac{r_0}{R}\biggr)^4 \biggl[\alpha\biggl(1 - \frac{3a}{b} \biggr)
- ~\frac{2}{15} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl(\frac{4}{3} - \alpha  \biggr)\biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
-~ \alpha \biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} 
- r_0\biggl[4 \biggr] \frac{d\Delta}{dr_0} 
+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]\biggl[ - r_0\cdot \frac{d\Delta}{dr_0} + \alpha \biggl( \frac{a}{b} - \Delta \biggr)\biggr]
+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) 
\biggl\{1  - \frac{a}{b}\biggl[\frac{1}{\Delta} \biggr] \biggr\} \frac{\rho_c}{\rho_0} 
\, ,
</math>
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>\frac{\rho_0}{\rho_c}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(1 - x^2\biggr) \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Delta</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\frac{1}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1} \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>\frac{d\ln P_0}{d\ln r_0}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
-5x^2  \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1} \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}
=
-15 x^2 \biggl(1 - \frac{1}{2}x^2\biggr)^{-1} \Delta \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>r_0 \cdot \frac{d\Delta}{dr_0}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>r_0^2 \cdot \frac{d^2\Delta}{dr_0^2}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-3}\biggl( 1 + 3x^2\biggr) \, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} 
- r_0\biggl[4 \biggr] \frac{d\Delta}{dr_0} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]\biggl[ - r_0\cdot \frac{d\Delta}{dr_0} + \alpha \biggl( \frac{a}{b} - \Delta \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
1  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \biggr] \biggr\} \frac{\rho_c}{\rho_0} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-3}\biggl( 1 + 3x^2\biggr)  
- \frac{16}{15} x^2 \biggl(1 - x^2\biggr)^{-2} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]
\biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} 
- \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
1  
- \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1} \, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{x^2}\biggl(1 - x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} \biggl(1 - x^2\biggr)^{-2}\biggl( 1 + 3x^2\biggr)  
- \frac{16}{15} \biggl(1 - x^2\biggr)^{-1} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}
\biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} 
- \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
1  
- \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{x^2}\biggl(1 - x^2\biggr)^3
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} \biggl( 1 + 3x^2\biggr)  
- \frac{16}{15} \biggl(1 - x^2\biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}
\biggl[\frac{a\alpha}{b}\biggl(1 - x^2\biggr)^2 - \frac{4}{15} x^2  
- \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl(1 - \frac{3}{5} x^2\biggr)   
- \frac{3a}{b} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>- 2 \biggl(2 - x^2\biggr)\biggl(5 - x^2\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-~ \biggl(5 - 3 x^2\biggr) 
\biggl[\frac{15a\alpha}{b}\biggl(1 - 2x^2 + x^4\biggr) - 4 x^2  
- \alpha \biggl(5 - 8 x^2 + 3x^4 \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-~ \biggl( \frac{5\sigma_c^2}{2\gamma_g}\biggr) \biggl\{
\biggl(5 - 3 x^2\biggr)   
- \frac{15a}{b} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ -~\frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
20 - 14x^2 + 2x^4
+~ \biggl(5 - 3 x^2\biggr) 
\biggl[ 
\biggl(\frac{15a\alpha}{b}-5\alpha \biggr) + x^2\biggl( \frac{30 a\alpha}{b} -4 + 8\alpha \biggr) + \alpha x^4 \biggl(\frac{15a}{b} - 3 \biggr)
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl( \frac{5\sigma_c^2}{2\gamma_g}\biggr) \biggl[
\biggl(5 - \frac{15a}{b}\biggr) + x^2 \biggl(-3 + \frac{15a}{b}  \biggr) \biggr] \biggl(1 - x^2\biggr) 
\, .
</math>
</td>
</tr>
</table>
Now, if we set <math>(15a/b) = 3</math>, this last expression reduces to,

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

<tr>
<td align="right"><math>-~\frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
20 - 14x^2 + 2x^4
-~ 2\biggl(5 - 3 x^2\biggr) 
\biggl[ 
\alpha + x^2 (2 - 7\alpha ) 
\biggr]
+~ \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggl(1 - x^2\biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
20 - 14x^2 + 2x^4
-~ 2
\biggl[ 5\alpha + x^2 (10-38\alpha ) + x^4(21\alpha - 6) \biggr]
+~ \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggl(1 - x^2\biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>x^0 \biggl[20 - 10\alpha +  \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggr]
+ x^2\biggl[76\alpha -34  - \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggr]
+ 14 x^4 \biggl[1 - 3\alpha \biggr]
</math>
</td>
</tr>
</table>