Editing SSC/Stability/GammaVariation

__FORCETOC__ <!-- will force the creation of a Table of Contents -->
<!-- __NOTOC__ will force TOC off -->
=How Does Stability Change with &gamma;<sub>g</sub>?=

==Isolated Uniform-Density Configuration==

===Our Setup===
From our [[SSC/Stability/UniformDensity#Our_Setup|separate discussion]], the relevant LAWE is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{1}{(1 - \chi_0^2)}  \biggl\{ (1 - \chi_0^2) \frac{d^2x}{d\chi_0^2} 
+ \frac{4}{\chi_0}\biggl[1 -  \frac{3}{2}\chi_0^2 \biggr] \frac{dx}{d\chi_0} 
+  \mathfrak{F} x \biggr\}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>0 \, ,</math></td>
</tr>
</table>
where, <math>\chi_0\equiv r_0/R</math>, <math>\alpha \equiv (3-4/\gamma_\mathrm{g})</math>, and 
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\mathfrak{F}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="right">
<math>
\biggl[\frac{3\omega^2}{2\pi \gamma_g G\bar\rho}  - 2 \biggl( 3 - \frac{4}{\gamma_\mathrm{g}}\biggr) \biggr] 
~~~~~\Rightarrow ~~~~~\frac{\gamma_\mathrm{g}\mathfrak{F}}{2} = \biggl[\frac{3\omega^2}{4\pi G\bar\rho} + 4 - 3\gamma_\mathrm{g}  \biggr] 
</math>
  </td>
</tr>
</table>

Also, the two relevant boundary conditions are,
<div align="center">
<math>~\frac{dx}{d\chi_0} = 0</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~\chi_0 = 0 \, ;</math>
</div>

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\chi_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{3\omega^2}{4\pi G \bar\rho}\biggr) </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~\chi_0 = 1 \, .</math>
  </td>
</tr>
</table>
Alternatively, this last expression may be written as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\chi_0}\biggr|_{\chi_0=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\mathfrak{F}}{2} \, .</math>
  </td>
</tr>
</table>

===The Sterne37 Solution===
From the [[SSC/Stability/UniformDensity#Sterne's_General_Solution|general solution]] derived by {{ Sterne37full }}, we have &hellip;
<table border="1" cellpadding="2" align="center">
<tr>
  <td colspan="1" align="center">
[[File:Sterne1937SolutionPlot1.png|350px|center|Sterne (1937)]]
  </td>
  <td colspan="1" align="center">
[[File:Sterne1937CritGamma1.png|350px|center|Sterne (1937)]]
  </td>
</tr>
</table>

The first few solutions are displayed in the following boxed-in image that has been extracted directly from &sect;2 (p. 587) of {{ Sterne37 }}; to the right of his table, we have added a column that expressly records the value of the square of the normalized eigenfrequency that corresponds to each of the solutions presented by {{ Sterne37hereafter }}.

<div align="center">
<table border="1" cellpadding="5">
<tr>
  <td align="center" colspan="1">
Based on exact eigenvector expressions extracted from &sect;2 (p. 587) of &hellip;<br />
{{ Sterne37figure }}
  </td>
  <td align="center" colspan="1">
<math>\frac{\omega^2}{4\pi G \bar\rho}</math>
  </td>
  <td align="center" rowspan="5">
[[File:Sterne37OmegaVsGammaLabeled.png|300px|Sterne's Omega vs. Gamma]]
  </td>
</tr>
<tr>
  <td colspan="1" rowspan="1">
<!-- [[File:Sterne1937SolutionTable1.png|600px|center|Sterne (1937)]] -->
    <table border="0" align="left">
    <tr>
    <td align="right"><math>j=0 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}=0 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right">&nbsp;<math>x = 1</math></td>
    </tr>
    </table>
  </td>
  <td align="center"><math>\gamma - 4/3</math></td>
</tr>

<tr>
  <td colspan="1" rowspan="1">
    <table border="0" align="left">
    <tr>
    <td align="right"><math>j=1 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}= 14 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>x = 1 - (7/5)\chi_0^2</math></td>
    </tr>
    </table>
  </td>
  <td align="center"><math>2(5\gamma - 2)/3</math></td>
</tr>

<tr>
  <td colspan="1" rowspan="1">
    <table border="0" align="left">
    <tr>
    <td align="right"><math>j=2 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}= 36 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>x = 1 - (18/5)\chi_0^2 + (99/35)\chi_0^4</math></td>
    </tr>
    </table>
  </td>
  <td align="center"><math>7\gamma - 4/3</math></td>
</tr>

<tr>
  <td colspan="1" rowspan="1">
    <table border="0" align="left">
    <tr>
    <td align="right"><math>j=3 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}=66 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>x = 1 - (33/5)\chi_0^2 + (429/35)\chi_0^4 - (143/21)\chi_0^6</math></td>
    </tr>
    </table>
  </td>
  <td align="center"><math>12\gamma - 4/3</math></td>
</tr>
</table>
</div>

===Cross-Check===
<b><font color="red">Check j = 0:</font></b>&nbsp; &nbsp;
The eigenvector is <math>x = 1</math>, that is, homologous contraction/expansion, in which case both the first and the second derivative of <math>x</math> are zero.  Hence, this eigenvector is a solution to the LAWE only if <math>\mathfrak{F} = 0</math>.  What about the two boundary conditions?  Well, the central boundary condition is immediately satisfied; for the outer boundary condition, we see that the logarithmic derivative of <math>x</math> is supposed to be zero, which it is because it equals <math>\mathfrak{F}/2</math>.  Finally, since <math>\mathfrak{F} = 0</math>, we see that the oscillation frequency is given by the expression,
<div align="center">
<math>\frac{\omega^2}{4\pi G\bar\rho} = \gamma_\mathrm{g} - 4/3 \, .</math>
</div>

<b><font color="red">Check j = 1:</font></b>&nbsp; &nbsp;
The eigenvector is <math>x = 1 - \tfrac{7}{5} \chi_0^2</math>, hence, <math>dx/d\chi_0 = -\tfrac{14}{5}\chi_0</math>, and, <math>d^2x/d\chi_0^2 = - \tfrac{14}{5} \, .</math>  This means that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">LAWE</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
- \frac{14}{5}(1-\chi_0^2) - \biggl[1 - \frac{3}{2}\chi_0^2\biggr]\frac{56}{5} + \mathfrak{F}\biggl[1 - \frac{7}{5}\chi_0^2\biggr]
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\chi_0^2 \biggl[\frac{14}{5} + \frac{168}{10} - \frac{7}{5}\mathfrak{F} \biggr] + \biggl[-\frac{14}{5} -\frac{56}{5} + \mathfrak{F}\biggr]
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{7}{5}\chi_0^2 \biggl[14 - \mathfrak{F} \biggr] 
+ \biggl[\mathfrak{F} - 14\biggr] \, ,
  </math></td>
</tr>
</table>
which goes to zero if <math>\mathfrak{F} = 14</math>, in which case,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{\omega^2}{4\pi G\bar\rho}  
</math></td>
  <td align="center"><math>=</math></td>
  <td align="right">
<math>
\frac{1}{3}\biggl[ \frac{\gamma_\mathrm{g}\mathfrak{F}}{2} - 4 + 3\gamma_\mathrm{g} \biggr]
=
\frac{2}{3} \biggl[5\gamma_\mathrm{g} - 2 \biggr] \, .
</math>
  </td>
</tr>
</table>
Is the surface boundary condition satisfied?  Well &hellip;
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{d\ln x}{d\chi_0}\biggr|_{\chi_0=1} = \biggl[\frac{1}{x} \cdot \frac{dx}{d\chi_0}\biggr]_{\chi_0=1}  
</math></td>
  <td align="center"><math>=</math></td>
  <td align="right">
<math>
\biggl[\biggl( 1 - \tfrac{7}{5}\chi_0^2\biggr)^{-1} \biggl(-\frac{14}{5}\biggr)\chi_0\biggr]_{\chi_0=1}
=
\biggl[\biggl( - \frac{2}{5} \biggr)^{-1} \biggl(-\frac{14}{5}\biggr)\biggr]
=
+7 \, ,
</math>
  </td>
</tr>
</table>
which matches the desired logarithmic slope, <math>\mathfrak{F}/2</math>.

===Entropy Distribution===

According to our [[Appendix/Ramblings/PatrickMotl#Tying_Expressions_into_H_Book_Context|discussions with P. Motl]], to within an additive constant, the entropy distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P/P_c}{(\gamma_g-1)(\rho/\rho_c)^{\gamma_g}} \biggr] \, .
</math>
  </td>
</tr>
</table>
Now, from the [[SSC/Structure/UniformDensity#Summary|derived properties of a uniform-density sphere]], we know that, <math>\rho/\rho_c = 1</math>, and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1 - \chi_0^2 ) \, .
</math>
  </td>
</tr>
</table>
Hence, again to within an additive constant,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\gamma_g-1)}\biggl\{
\ln \biggl[ \frac{P}{P_c} \biggr] 
\biggr\} 
=
\ln \biggl[(1 - \chi_0^2)^{1/(\gamma_\mathrm{g}-1)} \biggr] \, .
</math>
  </td>
</tr>
</table>
Notice that, if <math>\gamma_\mathrm{g} < 1</math>, the entropy is an ''increasing'' function of the fractional radius, <math>\chi_0</math>, and is therefore ''stable'' against convection according to the [[2DStructure/AxisymmetricInstabilities#Schwarzschild_Criterion|Schwarzschild criterion]].

===Comments on Uniform-Density Configurations===

According to [[#The_Sterne37_Solution|Sterne's stability analysis]], the square of the oscillation frequency, <math>\omega^2/(4\pi G \rho_c)</math>, of the fundamental mode is negative for all values of <math>\gamma_\mathrm{g} < \tfrac{4}{3}</math>.  All models with <math>\gamma_\mathrm{g} < \tfrac{4}{3}</math> are therefore dynamically unstable toward collapse with a radial-displacement eigenfunction given by that of the fundamental mode.  We appreciate as well that all models with <math>\gamma_\mathrm{g}  < \tfrac{2}{5}</math> are (also) dynamically unstable toward collapse with a radial-displacement eigenfunction given by the 1<sup>st</sup> overtone mode.

At the same time, an examination of each model's [[#Entropy_Distribution|entropy distribution]] indicates that models with <math>\gamma_\mathrm{g} > 1</math> are unstable toward convection throughout their entire volume.  Hence, we identify the following model regimes:

<table border="1" align="center" cellpadding="8" width="80%">

<tr>
  <td align="center"><math>\infty \ge \gamma_\mathrm{g} > \tfrac{4}{3}</math></td>
  <td align="left">Dynamically stable against collapse, but unstable toward convection throughout.</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{4}{3} > \gamma_\mathrm{g} > 1</math></td>
  <td align="left">Unstable toward convection throughout and, simultaneously dynamically unstable toward collapse with the eigenfunction provided by the fundamental mode. (All other radial ''overtone'' modes are dynamically stable against collapse.)</td>
</tr>

<tr>
  <td align="center"><math>1 > \gamma_\mathrm{g} > \tfrac{2}{5}</math></td>
  <td align="left">Stable against convection, but dynamically unstable toward collapse with the eigenfunction provided by the fundamental mode. (All other radial ''overtone'' modes are dynamically stable against collapse.)</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{2}{5} > \gamma_\mathrm{g} </math></td>
  <td align="left">Stable against convection, but dynamically unstable ''simultaneously'' toward collapse due to the fundamental and 1<sup>st</sup> overtone modes.</td>
</tr>
</table>

==Lane-Emden in Terms of Various Physical Quantities==
In a [[SSC/Structure/Polytropes#Lane-Emden_Equation|separate discussion]] we derived the,
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
which governs the hydrostatic structure of spherically symmetric polytropes. In this differential equation, 
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\Theta_H</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\frac{H}{H_c} 
= 
\biggl(\frac{\rho}{\rho_c}\biggr)^{1 / n}
=
\biggl( \frac{P}{P_c} \biggr)^{1/(n+1)}
  </math></td>
</tr>
</table>
so the Lane-Emden equation readily can be rewritten in terms of the dimensionless density or the dimensionless pressure.

===For n = 1, in Terms of Pressure===
In terms of the dimensionless pressure, <math>p \equiv P/P_c</math>, the Lane-Emden equation becomes,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>-p^{n/(n+1)}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{1}{\xi^2} \frac{d}{d\xi} \biggl[\xi^2 \frac{dp^{1/(n+1)}}{d\xi} \biggr]</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[\frac{1}{(n+1)^2}\biggr] p^{(-n)/(n+1)} \biggl\{ 
\frac{2(n+1)}{\xi}\cdot \frac{dp}{d\xi} 
- np^{-1} \biggl(\frac{dp}{d\xi}\biggr)^2 
+
(n+1)\frac{d^2p}{d\xi^2} 
\biggr\}\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ -(n+1)^2 p^{2n/(n+1)}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ 
\frac{2(n+1)}{\xi}\cdot \frac{dp}{d\xi} 
- np^{-1} \biggl(\frac{dp}{d\xi}\biggr)^2 
+
(n+1)\frac{d^2p}{d\xi^2} 
\biggr\}\, .
</math>
  </td>
</tr>
</table>
Let's not set n = 1 yet.  Instead, let's first insert the functional behavior of <math>p(\xi)</math> that we know is the proper function for an isolated n = 1 polytrope, namely,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>p</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl(\frac{\sin\xi}{\xi}\biggr)^2 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{dp}{d\xi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{2\sin\xi \cos\xi}{\xi^2} - \frac{2\sin^2\xi}{\xi^3} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{d^2p}{d\xi^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{2 \cos^2\xi}{\xi^2} - \frac{2\sin^2\xi }{\xi^2} - \frac{8\sin\xi \cos\xi}{\xi^3} + \frac{6\sin^2\xi}{\xi^4} 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">and, &nbsp; &nbsp; &nbsp;<math>\biggl[\frac{dp}{d\xi}\biggr]^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[\frac{2\sin\xi \cos\xi}{\xi^2} - \frac{2\sin^2\xi}{\xi^3}\biggr]^2
=
\frac{4\sin^2\xi \cos^2\xi}{\xi^4} - \frac{8\sin^3\xi \cos\xi}{\xi^5} + \frac{4\sin^4\xi}{\xi^6} \, .
</math>
  </td>
</tr>
</table>
Inside the curly braces on the RHS of the Lane-Emden equation, we therefore have,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\biggl\{~~~\biggr\}_\mathrm{RHS}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{2(n+1)}{\xi}\cdot \frac{dp}{d\xi} 
+
(n+1)\frac{d^2p}{d\xi^2} 
- \frac{n\xi^2}{\sin^2\xi} \biggl(\frac{dp}{d\xi}\biggr)^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(n+1) \biggl[ \frac{4\sin\xi \cos\xi}{\xi^3} - \frac{4\sin^2\xi}{\xi^4}  \biggr]
+
(n+1)\biggl[ \frac{2 \cos^2\xi}{\xi^2} - \frac{2\sin^2\xi }{\xi^2} - \frac{8\sin\xi \cos\xi}{\xi^3} + \frac{6\sin^2\xi}{\xi^4} \biggr]
- \frac{n\xi^2}{\sin^2\xi} \biggl[ \frac{4\sin^2\xi \cos^2\xi}{\xi^4} - \frac{8\sin^3\xi \cos\xi}{\xi^5} + \frac{4\sin^4\xi}{\xi^6} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(n+1)\biggl[ \frac{2 \cos^2\xi}{\xi^2} - \frac{2\sin^2\xi }{\xi^2} - \frac{4\sin\xi \cos\xi}{\xi^3} + \frac{2\sin^2\xi}{\xi^4} \biggr]
- n\biggl[ \frac{4\cos^2\xi}{\xi^2} - \frac{8\sin\xi \cos\xi}{\xi^3} + \frac{4\sin^2\xi}{\xi^4} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{2 \cos^2\xi}{\xi^2} - \frac{2\sin^2\xi }{\xi^2} - \frac{4\sin\xi \cos\xi}{\xi^3} + \frac{2\sin^2\xi}{\xi^4} \biggr]
+ n\biggl[ -\frac{2 \cos^2\xi}{\xi^2} - \frac{2\sin^2\xi }{\xi^2} + \frac{4\sin\xi \cos\xi}{\xi^3} - \frac{2\sin^2\xi}{\xi^4} \biggr]
\, .
</math>
  </td>
</tr>
</table>
Now, if we set n = 1, this expression collapses substantially to give,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\biggl\{~~~\biggr\}_\mathrm{RHS}\biggr|_{n=1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{4\sin^2\xi }{\xi^2} 
\, .
</math>
  </td>
</tr>
</table>
Simultaneously, the LHS of the Lane-Emden expression becomes,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math> - \biggl[(n+1)^2 p^{2n/(n+1)}\biggr]_{n=1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> - \biggl[(n+1)^2 \biggl(\frac{\sin\xi}{\xi}\biggr)^{4n/(n+1)}\biggr]_{n=1} = - \frac{4\sin^2\xi }{\xi^2} \, .</math>  </td>
</tr>
</table>
So, the two sides of the expression prove to be identical.

===What About in Terms of Entropy?===

====First Try====
What about, in terms of the entropy?  Well, [[#Entropy_Distribution|from above]], once the value of <math>\gamma_\mathrm{g}</math> has been specified, to within an additive constant, the dimensionless entropy, <math>\Sigma</math>, is given by the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Sigma \equiv \frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\gamma_g-1)}\ln \biggl[ \biggl(\frac{P}{P_c}\biggr) \biggl(\frac{\rho}{\rho_c}\biggr)^{-\gamma_\mathrm{g}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\gamma_g-1)}\ln \biggl[ \Theta_H^{n+1} \biggl(\Theta_H^n\biggr)^{-\gamma_\mathrm{g}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\ln \biggl[ \Theta_H^{n+1 - n\gamma_\mathrm{g}} \biggr]^{1/(\gamma_\mathrm{g}-1)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ e^\Sigma</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \Theta_H^{(n+1 - n\gamma_\mathrm{g})/(\gamma_\mathrm{g}-1)} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \Theta_\mathrm{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red"><b>NOTE:</b></font> &nbsp; If we set <math>\gamma_\mathrm{g} = (1 + 1/n)</math>, then the exponent,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{n + 1 - n\gamma_\mathrm{g}}{\gamma_\mathrm{g}-1} \biggr]_{\gamma_\mathrm{g} = (1 + 1/n)}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
n\biggl[n+1 - (n+1) \biggr] = 0 \, ,
</math>
  </td>
</tr>
</table>
which means that, independent of the functional behavior of the dimensionless enthalpy,

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

<tr>
  <td align="right">
<math>e^\Sigma</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Theta_H^{0} = \mathrm{constant} \, ,
</math>
  </td>
</tr>
</table>
that is, the entropy is uniform throughout the equilibrium configuration.
</td></tr></table>

Generally, then, in terms of the dimensionless entropy, the Lane-Emden equation may be rewritten as,

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

<tr>
  <td align="right">
<math>\frac{1}{\xi^2} \frac{d}{d\xi}\biggl[\xi^2 \frac{d}{d\xi}\biggl( e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} \biggr)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl[e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})}\biggr]^n \, .
</math>
  </td>
</tr>
</table>
That is,

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

<tr>
  <td align="right">
<math>
-e^{n\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]
\frac{1}{\xi^2} \frac{d}{d\xi}\biggl[\xi^2 \biggl( e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} \biggr)\frac{d\Sigma}{d\xi}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]
\frac{1}{\xi^2} \biggl\{
2\xi \biggl( e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} \biggr)\frac{d\Sigma}{d\xi}
+
\xi^2 \biggl( e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} \biggr) \frac{d^2\Sigma}{d\xi^2}
+
\xi^2 \biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]
\biggl( e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} \biggr) \biggl(\frac{d\Sigma}{d\xi}\biggr)^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]
\biggl\{
\frac{2}{\xi} \cdot \frac{d\Sigma}{d\xi}
+
\frac{d^2\Sigma}{d\xi^2}
+
\biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]
\biggl(\frac{d\Sigma}{d\xi}\biggr)^2
\biggr\}\biggl( e^{\Sigma(\gamma_\mathrm{g}-1)/(n+1 - n\gamma_\mathrm{g})} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
-e^{n} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]
\biggl\{
\frac{2}{\xi} \cdot \frac{d\Sigma}{d\xi}
+
\frac{d^2\Sigma}{d\xi^2}
+
\biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]
\biggl(\frac{d\Sigma}{d\xi}\biggr)^2
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Setting,
<div align="center">
<math>\Upsilon \equiv \biggl[ \frac{\gamma_\mathrm{g} - 1}{n+1-n\gamma_\mathrm{g}} \biggr]\Sigma \, ,</math>
</div>
the statement of hydrostatic balance becomes,

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

<tr>
  <td align="right">
<math>
\frac{d^2\Upsilon}{d\xi^2}
+
\biggl(\frac{d\Upsilon}{d\xi}\biggr)^2
+
\frac{2}{\xi} \cdot \frac{d\Upsilon}{d\xi}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-e^{n} 
\, .
</math>
  </td>
</tr>
</table>

<font color="red"><b>What do I do with this???</b></font>


In our [[#Entropy_Distribution|above discussion of uniform-density configurations]], we found that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Sigma = \frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\gamma_\mathrm{g} - 1)^{-1}\ln \biggl[(1 - \xi^2/6) \biggr] \, ,
</math>
  </td>
</tr>
</table>
where we have made the substitution, <math>\chi_0^2 = \xi^2/6</math>.  For this situation, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Upsilon</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m^{-1}\ln \biggl[1 - \frac{\xi^2}{6} \biggr] \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>m</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
n+1-n\gamma_\mathrm{g} \, .
</math>
  </td>
</tr>
</table>
In this case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\Upsilon}{d\xi}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m^{-1} \biggl[1 - \frac{\xi^2}{6} \biggr]^{-1} \cdot \biggl(-\frac{\xi}{3}\biggr) \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl(\frac{d\Upsilon}{d\xi}\biggr)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ \xi^2 }{3^2m^2  } \biggl[1 - \frac{\xi^2}{6} \biggr]^{-2}  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d^2\Upsilon}{d\xi^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\xi}{3m} \biggl[1 - \frac{\xi^2}{6} \biggr]^{-2} \cdot \biggl(-\frac{\xi}{3}\biggr) 
-
\frac{1}{3m} \biggl[1 - \frac{\xi^2}{6} \biggr]^{-1}  \, ;
</math>
  </td>
</tr>
</table>
that is to say,

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

<tr>
  <td align="right">
<math>
-e^{n} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2\Upsilon}{d\xi^2}
+
\biggl(\frac{d\Upsilon}{d\xi}\biggr)^2
+
\frac{2}{\xi} \cdot \frac{d\Upsilon}{d\xi}
</math>
  </td>
</tr>

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \frac{\xi^2}{6} \biggr]^{-2} \biggl\{
- m 
+ \frac{\xi^2}{2\cdot 3^2}\biggl[ m  + 2 \biggr] 
\biggr\}\frac{1}{m^2} \, .
</math>
  </td>
</tr>
</table>
That is,

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

<tr>
  <td align="right">
<math>
- m 
+ \frac{\xi^2}{2\cdot 3^2}\biggl[ m  + 2 \biggr]  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-m^2 e^{n} \biggl[1 - \frac{\xi^2}{6} \biggr]^{2} \, .
</math>
  </td>
</tr>
</table>

<font color="red"><b>What do I do with this???</b></font>

====Second Try====
From [[#Entropy_Distribution|above]] &#8212; and, for simplicity, removing the subscript <math>(\mathrm{g})</math> on <math>\gamma_\mathrm{g}</math> &#8212; we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Sigma \equiv \frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\gamma-1)}\ln \biggl[ \frac{P/P_c}{(\gamma-1)(\rho/\rho_c)^{\gamma_g}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ (\gamma - 1)e^{(\gamma -1)\Sigma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_c}\biggr)^{-\gamma} \frac{P}{P_c}
=
\Theta^{-n\gamma} \cdot \Theta^{(n+1)} = \Theta^{(n+1-n\gamma)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \Theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[(\gamma - 1)e^{(\gamma -1)\Sigma}\biggr]^{1/(n+1 - n\gamma)} 
=
(\gamma - 1)^{1/(n+1 - n\gamma)}e^{(\gamma -1)\Sigma/(n+1 - n\gamma)}
= Ae^{a\Sigma} 
\, ,</math>  
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
(\gamma - 1)^{1/(n+1 - n\gamma)}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>a</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{(\gamma -1)}{(n+1 - n\gamma)} \, .
</math>
  </td>
</tr>
</table>

Hence, in terms of the configuration's entropy profile, the Lane-Emden equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-~A^n e^{an\Sigma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\xi^2}\cdot \frac{d}{d\xi}\biggl[ \xi^2 \cdot \frac{d(Ae^{a\Sigma})}{d\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{aA}{\xi^2}\cdot \frac{d}{d\xi}\biggl[ \xi^2 \cdot (e^{a\Sigma}) \cdot \frac{d\Sigma}{d\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{aA}{\xi^2}\biggl[ 
2\xi \cdot (e^{a\Sigma}) \cdot \frac{d\Sigma}{d\xi} 
+
a\xi^2 \cdot (e^{a\Sigma}) \cdot \biggl(\frac{d\Sigma}{d\xi} \biggr)^2
+
\xi^2 \cdot (e^{a\Sigma}) \cdot \frac{d^2\Sigma}{d\xi^2} 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ - \biggl[ \frac{A^{(n-1)}}{a}\biggr] e^{(n-1)a\Sigma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 
\frac{2}{\xi} \cdot \frac{d\Sigma}{d\xi} 
+
a  \cdot \biggl(\frac{d\Sigma}{d\xi} \biggr)^2
+
\frac{d^2\Sigma}{d\xi^2} 
\biggr] \, .
</math>
  </td>
</tr>
</table>
Again, defining, <math>\Upsilon \equiv a\Sigma</math>, this becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>- (Ae^{\Upsilon})^{(n-1)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{2}{\xi} \cdot \frac{d\Upsilon}{d\xi} 
+
\biggl(\frac{d\Upsilon}{d\xi} \biggr)^2
+
\frac{d^2\Upsilon}{d\xi^2} 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Now, adopting the equilibrium profiles for an n = 1 polytrope &#8212; but without yet setting n = 1 &#8212; we see that the entropy distribution must be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Upsilon = a\Sigma </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{a}{(\gamma-1)}\biggl\{ 
\ln \biggl[ \frac{1}{(\gamma-1)}\biggr] 
+
\ln \biggl[ \frac{P}{P_c} \biggr] 
+
\ln \biggl[ \frac{\rho}{\rho_c} \biggr]^{-\gamma} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{a}{(\gamma-1)}\biggl\{ 
\ln \biggl[ \frac{1}{(\gamma-1)}\biggr] 
+
\ln \biggl[ \frac{\sin^2\xi}{\xi^2} \biggr] 
+
\ln \biggl[ \frac{\sin\xi}{\xi} \biggr]^{-\gamma} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{a}{(\gamma-1)}\biggl\{ 
\ln \biggl[ \frac{1}{(\gamma-1)}\biggr] 
+
(2-\gamma)\ln \biggl[ \frac{\sin\xi}{\xi} \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>\frac{d\Upsilon}{d\xi} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a}{(\gamma-1)}\biggr] \frac{d}{d\xi}\biggl\{ 
(2-\gamma)\ln \biggl[ \frac{\sin\xi}{\xi} \biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]  
\frac{\xi}{\sin\xi} \cdot  \frac{d}{d\xi}\biggl[ \frac{\sin\xi}{\xi} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]  
\frac{\xi}{\sin\xi} \cdot  \biggl[ \frac{\cos\xi}{\xi} - \frac{\sin\xi}{\xi^2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]  
\biggl[ \frac{\cos\xi}{\sin\xi} - \frac{1}{\xi}\biggr] 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d^2\Upsilon}{d\xi^2} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]  
\frac{d}{d\xi}\biggl[ \frac{\cos\xi}{\sin\xi} - \frac{1}{\xi}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]  
\biggl[ -1 - \frac{\cos^2\xi}{\sin^2\xi} + \frac{1}{\xi^2}\biggr] 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl(\frac{d\Upsilon}{d\xi} \biggr)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]^2  
\biggl[ \frac{\cos\xi}{\sin\xi} - \frac{1}{\xi}\biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]^2  
\biggl[ \frac{\cos^2\xi}{\sin^2\xi} - \frac{2\cos\xi}{\xi \sin\xi} + \frac{1}{\xi^2}\biggr] 
\, .
</math>
  </td>
</tr>
</table>
So, the RHS of the Lane-Emden expression becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl\{~~~\biggr\}_\mathrm{RHS}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{2}{\xi} \cdot \frac{d\Upsilon}{d\xi} 
+
\frac{d^2\Upsilon}{d\xi^2}  
+
\biggl(\frac{d\Upsilon}{d\xi} \biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr] \biggl\{  
\biggl[ \frac{\cos\xi}{\sin\xi} - \frac{1}{\xi}\biggr] 
\frac{2}{\xi} 
+
\biggl[ -1 - \frac{\cos^2\xi}{\sin^2\xi} + \frac{1}{\xi^2}\biggr]  
+
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]  
\biggl[ \frac{\cos^2\xi}{\sin^2\xi} - \frac{2\cos\xi}{\xi \sin\xi} + \frac{1}{\xi^2}\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr] \biggl\{  
\biggl[- \frac{\cos^2\xi}{\sin^2\xi} + \frac{2\cos\xi}{\xi\sin\xi} - \frac{1}{\xi^2} 
-1  \biggr]  
+
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]  
\biggl[ \frac{\cos^2\xi}{\sin^2\xi} - \frac{2\cos\xi}{\xi \sin\xi} + \frac{1}{\xi^2}\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ \frac{a(2-\gamma)}{(\gamma-1)}\biggr]^2 \biggl\{  
\biggl[ \frac{(n + 1 -n\gamma)}{(2-\gamma)}\biggr]  
\biggl[- \frac{\cos^2\xi}{\sin^2\xi} + \frac{2\cos\xi}{\xi\sin\xi} - \frac{1}{\xi^2} 
-1  \biggr]  
+
\biggl[ \frac{\cos^2\xi}{\sin^2\xi} - \frac{2\cos\xi}{\xi \sin\xi} + \frac{1}{\xi^2}\biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Now, since,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>Ae^\Upsilon</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\Theta = \frac{\sin\xi}{\xi} \, ,
</math>
  </td>
</tr>
</table>
the LHS of the Lane-Emden expression is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-(Ae^\Upsilon)^{(n-1)}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \biggl(\frac{\sin\xi}{\xi}\biggr)^{(n-1)} \, .
</math>
  </td>
</tr>
</table>
As a result, the entire expression reads,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> 
- \biggl(\frac{\sin\xi}{\xi}\biggr)^{(n-1)} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ \frac{(2-\gamma)}{(n+1-n\gamma)}\biggr]^2 \biggl\{  
\biggl[ \frac{(n + 1 -n\gamma)}{(2-\gamma)}\biggr]  
\biggl[- \frac{\cos^2\xi}{\sin^2\xi} + \frac{2\cos\xi}{\xi\sin\xi} - \frac{1}{\xi^2} 
-1  \biggr]  
+
\biggl[ \frac{\cos^2\xi}{\sin^2\xi} - \frac{2\cos\xi}{\xi \sin\xi} + \frac{1}{\xi^2}\biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
If we leave <math>\gamma</math> unspecified but set n = 1, both sides of the expression become "-1", so the Lane-Emden expression is satisfied for all values of <math>\xi</math> and for any choice of <math>\gamma</math>.

==Example Fundamental Modes for Isolated Configurations==
For an isolated polytrope whose surface does not extend to infinity &#8212; that is, for <math>0 < n < 5</math> &#8212; the eigenvector for the fundamental mode of radial oscillation depends on the specification of a single parameter: <math>\gamma</math>.  Then, for virtually any choice of the square of the radial oscillation frequency, <math>\sigma_c^2</math>, the governing polytropic LAWE can be integrated (usually, numerically) to obtain the radial-displacement, <math>x(\xi)</math>, that is consistent with that choice of <math>\sigma_c^2</math>.  While this function, <math>x(\xi)</math>, satisfies the LAWE, its slope at the surface of the polytrope usually will not satisfy the physically relevant boundary condition.  Other "guesses" for <math>\sigma_c^2</math> must be made until the <math>x(\xi)</math> function satisfies the proper boundary condition; the result provides the eigenfrequency and eigenfunction (together, the eigenvector) that are associated with the specified value of <math>\gamma</math>.

As an example, consider specifying <math>\gamma = \tfrac{5}{3}</math> for an isolated, <math>n = 1</math> polytrope. The following table records the value of the square of the eigenfrequency that has been independently determined by three different research groups:  1.155 by {{ Chatterji51 }}; 1.1499 by {{ HRW66 }}; and 1.1492896    [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|herein]].  Also  for comparison, the corresponding ''eigenfunction''  obtained from two of these investigations has been displayed graphically [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|herein]].

Not unexpectedly, when a different value of <math>\gamma</math> is specified, the result is a different radial oscillation eigenfrequency along with a different eigenfunction.  However, as was first demonstrated by {{ Sterne37 }}, for an <math>n = 0</math> (uniform-density) polytrope, even though the eigenfrequency varies with the choice of <math>\gamma</math>, the radial displacement ''eigenfunction'' is identically the same for all chosen <math>\gamma</math>.

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="6">Published Fundamental-Mode Oscillation Frequencies</td>
</tr>

<tr>
  <td align="center"><math>n</math></td>
  <td align="center"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>\gamma</math></td>
  <td align="center"><math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math>
  <td align="center">Publication</td>
  <td align="center">Relevant<br />JETohlineWiki<br />Chapter</td>
</tr>

<tr>
  <td align="center"><math>0</math></td>
  <td align="center">1</td>
  <td align="center">Any <math>\gamma</math></td>
  <td align="center"><math>6(\gamma - 4/3)</math></td>
  <td align="left"><sup>c</sup>{{ Sterne37 }}</td>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>2</math></td>
  <td align="left"><sup>b</sup>{{ HRW66 }}</td>
  <td align="center">[[SSC/Stability/Polytropes/Pt3#Tables|here]]</td>
</tr>

<tr>
  <td align="center" bgcolor="lightgrey" colspan="6">&nbsp; </td>
</tr>

<tr>
  <td align="center"><math>1</math></td>
  <td align="center"><math>\frac{\pi^2}{3}</math></td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>1.155</math></td>
  <td align="left"><sup>d</sup>{{ Chatterji51 }}</td>
  <td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>1.1499</math></td>
  <td align="left"><sup>b</sup>{{ HRW66 }}</td>
  <td align="center">[[SSC/Stability/Polytropes/Pt3#Tables|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>1.1492896</math></td>
  <td align="center" colspan="2">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|Our imposed surface B.C.]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{20}{13}</math></td>
  <td align="center"><math>0.715</math></td>
  <td align="left"><sup>d</sup>{{ Chatterji51 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{10}{7}</math></td>
  <td align="center"><math>0.334</math></td>
  <td align="left"><sup>d</sup>{{ Chatterji51 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center" bgcolor="lightgrey" colspan="6">&nbsp; </td>
</tr>

<tr>
  <td align="center"><math>3</math></td>
  <td align="right">54.18248</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>0.34175</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">[[SSC/Stability/n3PolytropeLAWE#Schwarzschild_(1941)|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>0.34161</math></td>
  <td align="left"><sup>b</sup>{{ HRW66 }}</td>
  <td align="center">[[SSC/Stability/Polytropes/Pt3#Tables|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{20}{13}</math></td>
  <td align="center"><math>0.23979</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{10}{7}</math></td>
  <td align="center"><math>0.12604</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{4}{3}</math></td>
  <td align="center"><math>0.0</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="left" colspan="6">
NOTES:<br />
<ol type="a">
<li>&nbsp;<math>\sigma_c^2 = \tfrac{3}{2}\gamma \omega^2_\mathrm{Sch}</math></li>
<li>&nbsp;<math>\omega^2 = s^2_\mathrm{HRW66}</math></li>
<li>&nbsp;<math>\omega^2 = n^2_\mathrm{Sterne37}</math></li>
<li>&nbsp;<math>\sigma_c^2 = 3\gamma \omega^2_\mathrm{Chatterji}</math></li>
</ol>
  </td>
</tr>
</table>

=How Does Stability Change with P<sub>e</sub>?=

=In Bipolytropes, How Does Stability Change with &xi;<sub>i</sub>=

Taken from [[SSC/Stability/BiPolytropes/Pt3#MuRatio_0.310|an accompanying discussion]].


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="2">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = Fun031FirstOvertone]]
Variation of Oscillation Frequency with <math>\xi_i</math> for <math>(5, 1)</math> Bipolytropes Having <math>\mu_e/\mu_c = 0.310</math>
  </td>
</tr>
<tr>
  <td align="center">
[[File:VariationOf2Modes.png|450px|Variation of 2 Modes]]
  </td>
  <td align="center">

  <table border="1" align="left" cellpadding="8">
  <tr>
    <td align="center" rowspan="2"><math>\xi_i</math></td>
    <td align="center">Fundamental<br /><font color="red"><b>(red)</b></font></td></td>  
    <td align="center" colspan="3">1<sup>st</sup> Overtone<br /><font color="darkblue"><b>(blue)</b></font></td>  
  </tr>
  <tr>
    <td align="center"><math>\Omega^2</math></td>
    <td align="center"><math>\sigma_c^2</math></td>
    <td align="center"><math>\frac{\rho_c}{\bar\rho}</math></td>
    <td align="center"><math>\Omega^2 \equiv \frac{\sigma_c^2}{2}\biggl( \frac{\rho_c}{\bar\rho} \biggr)</math></td>
  </tr>
  <tr>
<td align="right">1.60</td>
<td align="right">3.8944</td>
<td align="right">0.498473</td>
<td align="right">58.398587</td>
<td align="right">14.555059</td>
  </tr>
  <tr>
<td align="right">2.00</td>
<td align="right">3.81053</td>
<td align="right">0.236047</td>
<td align="right">108.69129</td>
<td align="right">12.828126</td>
  </tr>
  <tr>
<td align="right">2.40</td>
<td align="right">2.79491</td>
<td align="right">0.0870005</td>
<td align="right">199.16363</td>
<td align="right">8.6636677</td>
  </tr>
  <tr>
<td align="right">2.609509754</td>
<td align="right">0.00000</td>
<td align="right">0.048214</td>
<td align="right">270.5922</td>
<td align="right">6.5231608</td>
  </tr>
  <tr>
<td align="right">3.00</td>
<td align="right">- 13.287</td>
<td align="right">0.0232907</td>
<td align="right">468.15</td>
<td align="right">5.4517612</td>
  </tr>
  <tr>
<td align="right">3.50</td>
<td align="right">- 44.63801</td>
<td align="right">0.0117478</td>
<td align="right">902.64028</td>
<td align="right">5.3020065</td>
  </tr>
  <tr>
<td align="right">4.00</td>
<td align="right">- 98.215</td>
<td align="right">0.0064276</td>
<td align="right">1656.926</td>
<td align="right">5.3250395</td>
  </tr>
  <tr>
<td align="right">5.00</td>
<td align="center">---</td>
<td align="right">0.0022154</td>
<td align="right">4900.105</td>
<td align="right">5.4278831</td>
  </tr>
  <tr>
<td align="right">6.00</td>
<td align="center">---</td>
<td align="right">0.0008785</td>
<td align="right">12544.67</td>
<td align="right">5.5100707</td>
  </tr>
  <tr>
<td align="right">9.014959766</td>
<td align="center">---</td>
<td align="right">9.61 &times; 10<sup>-5</sup></td>
<td align="right">116641.6</td>
<td align="right">5.6036778</td>
  </tr>
  <tr>
<td align="right">12.0</td>
<td align="center">---</td>
<td align="right">1.86 &times; 10<sup>-5</sup></td>
<td align="right">6.01 &times; 10<sup>+5</sup></td>
<td align="right">5.5796084</td>
  </tr>
  </table>

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