Editing SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2 (section)

===Pressure-Truncated n = 1 Polytrope===

In the case of an ''isolated'' <math>n=1</math> polytrope, we must set <math>B = 0</math>; in addition, it is customary to set <math>A = 1</math>.  The relevant LAWE is, then,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>0</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{d^2x}{d\eta^2} + 2\biggl\{ 1 + \eta\cot(\eta) \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{\sin(\eta)} 
- 1 + \eta\cot(\eta) \biggr\}  \frac{x}{\eta^2} \, .</math>
</td>
</tr>
</table>

====Review of Trial Analytic Eigenfunction====
This is the same 2<sup>nd</sup>-order ODE that we derived in a [[SSC/Stability/n1PolytropeLAWE#WorkInProgress|separate discussion]]; there it was accompanied by the [[SSC/Stability/Polytropes#Boundary_Conditions|surface boundary condition]],
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>- \frac{d\ln x}{d\ln \xi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\biggl( \frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)}
\biggl[ \frac{\xi}{\theta'}\biggr]_\mathrm{surf}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ - \frac{d\ln x}{d\ln \eta} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
1 + \frac{\sigma_c^2}{12}
\biggl[ \frac{\eta^3}{(\eta \cos\eta - \sin\eta)}\biggr]_{\eta=\pi}
=
1 - \frac{\pi^2 \sigma_c^2}{12} \, .
</math>
  </td>
</tr>
</table>

From, for example, a separate [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration|succinct demonstration]], we appreciate that if the displacement function is assumed to be,

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

<tr>
  <td align="right">
<math>x_P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{\eta^2} \biggl[ 1 - \eta \cot\eta \biggr] </math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
&hellip; that is,

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

<tr>
  <td align="right">
<math>x_P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{\eta^2}  - \frac{3 \cos\eta}{\eta \sin\eta} \, , </math>
  </td>
</tr>
</table>

in which case,

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

<tr>
  <td align="right">
<math>\frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{6}{\eta^3} + 3\biggl[\frac{\cos\eta}{\eta^2\sin\eta} + \frac{1}{\eta} + \frac{\cos^2\eta}{\eta\sin^2\eta}\biggr] 
\, ,
</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right">
<math>\frac{d^2x_P}{d\eta^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
+ 
3\frac{d}{d\eta}\biggl[ \frac{\cos\eta}{\eta^2\sin\eta} \biggr] 
+ 3\frac{d}{d\eta}\biggl[ \frac{1}{\eta} \biggr] 
+ 3\frac{d}{d\eta}\biggl[ \frac{\cos^2\eta}{\eta\sin^2\eta} \biggr]

</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
- 
3\biggl[ \frac{ 2\cos\eta}{\eta^3\sin\eta} + \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{\sin\eta}{\eta^2\sin\eta} \biggr] 
- \biggl[ \frac{3}{\eta^2} \biggr] 
- 3\biggl[ \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{2\cos^3\eta}{\eta\sin^3\eta} + \frac{2\cos\eta}{\eta\sin\eta} \biggr]
\, .
</math>
  </td>
</tr>
</table>
Hence, 
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">LAWE</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \frac{2}{\eta} + \frac{2\cos\eta}{\sin\eta} \biggr\}  \biggl\{
-\frac{6}{\eta^3} + \biggl[\frac{3\cos\eta}{\eta^2\sin\eta} + \frac{3}{\eta} + \frac{3\cos^2\eta}{\eta\sin^2\eta}\biggr]
\biggr\}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{\sin(\eta)} 
- 1 + \frac{\eta\cos\eta}{\sin\eta} \biggr\} \biggl[ \frac{3}{\eta^4}  - \frac{3 \cos\eta}{\eta^3 \sin\eta} \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
- 
3\biggl[ \frac{ 2\cos\eta}{\eta^3\sin\eta} + \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{\sin\eta}{\eta^2\sin\eta} \biggr] 
- \biggl[ \frac{3}{\eta^2} \biggr] 
- 3\biggl[ \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{2\cos^3\eta}{\eta\sin^3\eta} + \frac{2\cos\eta}{\eta\sin\eta} \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{2}{\eta} + 2\cot\eta \biggr]  \biggl[
-\frac{6}{\eta^3} + \frac{3\cot\eta}{\eta^2} + \frac{3}{\eta} + \frac{3\cot^2\eta}{\eta}
\biggr]  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{6} \biggr)  \frac{\eta^3}{\sin\eta} 
\biggr\} \biggl[ \frac{3}{\eta^4}  - \frac{3 \cot\eta}{\eta^3 } \biggr]
+ \biggl[- 2 + 2\eta\cot\eta \biggr] \biggl[ \frac{3}{\eta^4}  - \frac{3 \cot\eta}{\eta^3 } \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
- 
\biggl[ \frac{ 6\cot\eta}{\eta^3} + \frac{3\cot^2\eta}{\eta^2} + \frac{3}{\eta^2} \biggr] 
- \biggl[ \frac{3}{\eta^2} \biggr] 
- \biggl[ \frac{3\cot^2\eta}{\eta^2} + \frac{6\cot^3\eta}{\eta} + \frac{6\cot\eta}{\eta} \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{\sigma_c^2}{2} \biggr)  \frac{1}{\eta \sin\eta} \biggr\} 
\biggl[ 1  - \eta \cot\eta \biggr]
+
\biggl[ -\frac{12}{\eta^4} + \frac{6}{\eta^2} \biggr]  
+
\biggl[ 2\cot\eta \biggr]\biggl[ \frac{3}{\eta^3}  + \frac{3\cot\eta}{\eta^2} \biggr]  
+
\biggl[ 2\cot\eta \biggr]  \biggl[
-\frac{6}{\eta^3} + \frac{3\cot\eta}{\eta^2} + \frac{3}{\eta} + \frac{3\cot^2\eta}{\eta}
\biggr]  
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl[ -\frac{6}{\eta^4}  + \frac{6 \cot\eta}{\eta^3 } \biggr]
+ \biggl[ 2\cot\eta \biggr] \biggl[ \frac{3}{\eta^3}  - \frac{3 \cot\eta}{\eta^2 } \biggr]
+\frac{18}{\eta^4}
- \frac{6}{\eta^2} 
+ \biggl[ 2\cot\eta \biggr]\biggl[
- \frac{3\cot^2\eta}{\eta} 
- \frac{3\cot\eta}{\eta^2} 
- \frac{3}{\eta} 
-\frac{ 3}{\eta^3} 
\biggr]
</math>
</td>
</tr>

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

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

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

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

the <math>n=1</math> LAWE reduces to &hellip;

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">LAWE</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{\sigma_c^2}{2} \biggr)  \frac{1}{\eta \sin\eta} \biggr\} 
\biggl[ 1  - \eta \cot\eta \biggr] \, .
</math>
</td>
</tr>
</table>

ASSESSMENT:&nbsp; 
<ol>
  <li>
If we set <math>\sigma_c^2 = 0</math>, the right-hand-side of this expression goes to zero &#8212; and, hence, the <math>n=1</math> LAWE is satisfied &#8212; for any chosen truncation radius in the range, <math>0 < \eta_i < \pi</math>.  (We have not included the ''isolated'' <math>n=1</math> polytrope because <math>x_P</math> blows up at its surface, <math>\eta_i = \pi</math>.)
  </li>
  <li>
At the surface, <math>\eta_i</math>, the slope of this trial eigenfunction is,

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

<tr>
  <td align="right">
<math>\frac{1}{3} \cdot \frac{dx_P}{d\eta} \biggr|_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{\cot\eta_i}{\eta_i^2} + \frac{1}{\eta_i} + \frac{\cot^2\eta_i}{\eta_i}\biggr] -\frac{2}{\eta_i^3} 
\, .
</math>
  </td>
</tr>
</table>
By contrast, as stated above, the eigenvalue problem will be properly solved only if the surface slope is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>- \frac{d\ln x}{d\ln \eta} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
1 + \frac{\cancelto{0}{\sigma_c^2}}{12}
\biggl[ \frac{\eta^3}{(\eta \cos\eta - \sin\eta)}\biggr]_{\eta=\pi}
= 1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{1}{3} \cdot \frac{dx_P}{d\eta} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
- \frac{x_P}{3\eta}\biggr|_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
- \frac{1}{\eta_i^3} \biggl[ 1 - \eta_i \cot\eta_i \biggr] 
=
\frac{\cot\eta_i}{\eta_i^2} - \frac{1}{\eta_i^3} \, .
</math>
  </td>
</tr>
</table>
These two slopes do not appear to be the same, for any allowed choice of <math>\eta_i</math>.  We conclude, therefore, that no model along the sequence of pressure-truncated <math>n=1</math> polytropes is marginally unstable.
  </li>
</ol>

====Determining Discrete Representation of Eigenfunction====

Let's numerically integrate the [[#DiscreteLAWE|discrete form of the <math>n=1</math> LAWE]] over the radial coordinate range, <math>0 \le \eta_i \le \eta_s</math>.  Following our [[SSC/Stability/Polytropes#KickStart|discussion of the more general polytropic case]], we will kickstart integration from the center, outward, via the expression,

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

<tr>
  <td align="right">
<math>
x_2 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F}  \Delta_\eta^2}{60} \biggr] \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>
\mathfrak{F}
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 2\alpha\biggr] \, .</math>
  </td>
</tr>
</table>
Here, we will restrict our investigation to the case where <math>\gamma_g = (n+1)/n = 2</math>, in which case, <math>\alpha = (3-4/\gamma_g) = +1</math>, <math>\mathfrak{F} = (\sigma_c^2 -4)/2</math>, and 

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

<tr>
  <td align="right">
<math>
x_2 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_1 \biggl[ 1 - \frac{(\sigma_c^2 -4) \Delta_\eta^2}{60} \biggr] \, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">EXAMPLE:</font> &nbsp; <math>A=1</math>, <math>B=0</math>, <math>\Delta_\eta = \pi/99 = 0.031733259</math>; evaluated over range, <math>0 \le \eta_i \le \pi</math>.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center"><math>\sigma_c^2</math></td>
  <td align="center"><math>\mathfrak{F}</math></td>
  <td align="center"><math>x_2</math></td>
  <td align="center"><math>\phi_2</math></td>
  <td align="center"><math>(-\phi')_2</math></td>
  <td align="center">TERM1</td>
  <td align="center">TERM2</td>
  <td align="center">TERM3</td>
  <td align="center"><math>x_3</math></td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">+0.5</td>
  <td align="center">0.999983217</td>
  <td align="center">0.999832175</td>
  <td align="center">0.010576686</td>
  <td align="center">5.998321784</td>
  <td align="center">1.998993085</td>
  <td align="center">3.998992898</td>
  <td align="center">0.999932829</td>
</tr>
</table>
</td></tr></table>

=====Isolated n = 1 Polytrope=====

If we integrate all the way out to the natural, zero-pressure surface of our <math>n = 1</math> polytrope, then <math>\eta_s = \pi</math> and &#8212; as derived in our [[SSC/Structure/Polytropes#Summary|discussion of the equilibrium structure of n = 1 polytropes]] &#8212; <math>(\rho_c/\bar\rho) = \pi^2/3</math>.  In line with our discussion of [[SSC/Stability/n3PolytropeLAWE#Schwarzschild_(1941)|Schwarzschild's model of oscillations in <math>n=3</math> polytropes]], we therefore expect the boundary condition at the surface of our <math>n=1</math> configurations to be given by the expression,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>-\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\frac{1}{2}\biggl\{ \biggl[ \mathfrak{F} + 2\alpha\biggr]\biggl(\frac{\rho_c}{\bar\rho}\biggr) - 2\alpha \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
1 - \biggl(\frac{\sigma_c^2 \pi^2}{12}\biggr)  \, ,
</math>
  </td>
</tr>
</table>
as [[#Review_of_Trial_Analytic_Eigenfunction|reviewed immediately above]].  


<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <td align="center" bgcolor="pink">
{{ Chatterji51 }} &#8212; STEP 3
  </td>
</tr>
<tr><td align="left">
Here we examine what the boundary condition should be at the surface of an isolated <math>n=1</math> polytrope.  Given that, quite generally in the context of isolated polytropes,

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

<tr>
  <td align="right">
<math>\alpha</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3 - \frac{4}{\gamma} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\mathfrak{F}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sigma_c^2}{\gamma} - 2\alpha \, ,
</math>
  </td>
</tr>
</table>
the surface boundary condition is,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>-\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\frac{1}{2}\biggl\{ \biggl[ \mathfrak{F} + 2\alpha\biggr]\biggl(\frac{\rho_c}{\bar\rho}\biggr) - 2\alpha \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\frac{\sigma_c^2}{2\gamma}\biggl(\frac{\rho_c}{\bar\rho}\biggr)  + 3 - \frac{4}{\gamma}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - \frac{1}{\gamma}\biggl[ 4 + \frac{\sigma_c^2}{2}\biggl(\frac{\rho_c}{\bar\rho}\biggr) \biggr] \, .
</math>
  </td>
</tr>

</table>
<font color="maroon"><b>Considerations:</b></font>
<ol type="A">
<li>For an isolated <math>n=1</math> polytrope, the central-to-mean density is, <math>\rho_c/\bar\rho = \pi^2/3</math>.  Hence,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - \frac{1}{\gamma}\biggl[ 4 + \frac{\pi^2 \sigma_c^2}{6}\biggr] \, .
</math>
  </td>
</tr>

</table>

</li>
<li>If, in addition, we set <math>\gamma = (n+1)/n = 2 ~\Rightarrow ~\alpha=+1</math>, then,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1, ~\alpha=+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - \frac{1}{2}\biggl[ 4 + \frac{\pi^2 \sigma_c^2}{6}\biggr] = 1 - \frac{\pi^2 \sigma_c^2}{12} \, .
</math>
  </td>
</tr>

</table>

</li>
<li>If we set <math>\gamma = [4/(3-\alpha)]</math> for all other values of <math>\alpha</math>, we can write,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1, ~\alpha}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - (3-\alpha)\biggl[ 1 + \frac{\pi^2 \sigma_c^2}{24}\biggr]
=
\alpha - (3-\alpha)\biggl[ \frac{\pi^2 \sigma_c^2}{24}\biggr]\, .
</math>
  </td>
</tr>

</table>

</li>
</ol>


----


At the bottom of p. 469 of his article, {{ Chatterji51 }} states that, <font color="darkgreen">"&hellip; the condition for the Node to fall at the surface of the star</font> is,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>3f + z~\frac{df}{dz}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
</tr>
</table>
which we interpret to mean,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln f}{d\ln z}\biggr|_\mathrm{surf} \biggr\}_{n=1,~\alpha}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 \, .
</math>
  </td>
</tr>
</table>
He goes on to say, <font color="darkgreen">"As the adiabatic approximation breaks down near the boundary we have not strictly followed this condition."</font>

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


This should be compared with the finite-difference representation of the logarithmic derivative, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>+\frac{\Delta\ln x}{\Delta\ln \xi} \biggr|_\mathrm{surface}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{\xi_\mathrm{max}}{x_N} \biggl[ \frac{x_{N+1}-x_{N-1}}{2\Delta_\xi} \biggr] \, .</math>
  </td>
</tr>
</table>
<font color="red">CAUTION!</font> &nbsp; &nbsp; Because, for each ''guess'' of <math>\sigma_c^2</math>, the eigenfunction climbs (or plummets) rapidly as we approach the surface, in practice we evaluated the finite-difference representation of the logarithmic derivative at a zone location that is a bit inside of the actual surface; for example, when we divided the equilibrium configuration into <math>N = 100</math> grid zones, we evaluated the "surface" derivative at zone number 97.

Here we have adopted an analysis that closely resembles our [[SSC/Stability/n3PolytropeLAWE#Numerical_Integration|discussion of the analysis of <math>n=3</math> polytropes]] that was published by {{ Schwarzschild41 }}. Here we have divided our model into <math>N = 100</math> radial zones and, using this algorithm, integrated the LAWE from the center of the configuration to the surface, for <math>\alpha = +1</math>, and approximately 40 different chosen values of the frequency parameter across the range, <math>-2 \le \mathfrak{F} \le + 18</math>.  The radial displacement functions resulting from these integrations are presented in the following figure as an animation sequence.  The specified value of <math>\mathfrak{F}</math> is displayed at the top of each animation frame, and the resulting displacement function, <math>x(\eta)</math>, is traced by the small, red circular markers in each frame.
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="4">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = n1Oscillations]]Four Modes of Oscillation<br />of an Isolated, <math>n=1</math> Polytrope</td>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\sigma_c^2</math></td>
  <td align="center">Neg. Slope<br><math>1 - (\sigma_c^2\pi^2/12)</math></td>
  <td align="center"><math>\mathfrak{F} = \frac{\sigma_c^2}{\gamma_g} - 2\alpha</math></td>
</tr>
<tr>
  <td align="center">Fundamental</td>
  <td align="center">2.2405295</td>
  <td align="center">3.1287618</td>
  <td align="center">-0.879735</td>
</tr>
<tr>
  <td align="center">1<sup>st</sup> Overtone</td>
  <td align="center">6.340767</td>
  <td align="center">-32.06757</td>
  <td align="center">1.1703835</td>
</tr>
<tr>
  <td align="center">2<sup>nd</sup> Overtone</td>
  <td align="center">13.694927</td>
  <td align="center">-153.2545</td>
  <td align="center">4.8474635</td>
</tr>
<tr>
  <td align="center">3<sup>rd</sup> Overtone</td>
  <td align="center">28.462829</td>
  <td align="center">-665.3074</td>
  <td align="center">12.231415</td>
</tr>

<tr><td align="center" colspan="4">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n1Eigenvectors/MovieFrames/KeyAnimation/n1osc06.gif]]
[[File:N1osc06.gif|600px|Animated gif showing oscillation modes for n = 1 polytrope]]
</td></tr></table>



<table border="1" align="center" cellpadding="8" width="90%">
<tr>
  <td align="center" colspan="3" bgcolor="pink">{{ Chatterji51figure }}</td>
</tr>
<tr>
  <td align="center" colspan="3">''Amplitudes and frequencies of the displacement functions for three modes, assuming'' <math>\alpha = 0.6</math> and <math>\gamma = 5/3</math></td>
</tr>
<tr>
  <td align="center" colspan="1">Amplitudes extracted from Chatterji's Table II (p. 468)</td>
  <td align="center" colspan="1">Plot for comparison with Chatterji's Fig. 1 (p. 469)</td>
  <td align="center" colspan="1">Eigenfrequencies extracted<br />from Chatterji's Table I (p. 468)</td>
</tr>
<tr><td align="left" width="50%">
<div id="Chatterji51Table2Data" style="width: 100%; height: 30em; overflow: auto;">
<pre>
                                  1st            2nd
Abscissa	Fundamental	Overtone	Overtone
0.0	         1.000000	1.000000	1.000000
0.1	         1.000169	0.998882	0.996820
0.2	         1.000677	0.995518	0.987281
0.3	         1.001525	0.989874	0.971386
0.4	         1.002716	0.981890	0.949136
0.5	         1.004251	0.971495	0.920555
0.6	         1.006143	0.958596	0.885671
0.7	         1.008389	0.943052	0.844515
0.8	         1.011009	0.924713	0.797168
0.9	         1.014006	0.903389	0.743728
1.0	         1.017390	0.878859	0.684350
1.1	         1.021175	0.850865	0.619257
1.2	         1.025374	0.819105	0.548757
1.3	         1.030004	0.783221	0.473277
1.4	         1.035102	0.742868	0.393396
1.5	         1.040853	0.697654	0.309941
1.6	         1.047084	0.646920	0.223844
1.7	         1.053810	0.590042	0.136419
1.8	         1.061066	0.526318	0.049379
1.9	         1.068886	0.454939       -0.035065
2.0	         1.077309	0.374974       -0.114079
2.1	         1.086374	0.285357       -0.184050
2.2	         1.096138	0.184876       -0.240444
2.3	         1.106766	0.072271       -0.277903
2.4	         1.118204      -0.054152       -0.288854
2.5	         1.130496      -0.196154       -0.264401
2.6	         1.143708      -0.355598       -0.193931
2.7	         1.157893      -0.534282       -0.065351
2.8	         1.173099      -0.733397	0.133422
2.9	         1.189288      -0.951614	0.408698
3.0	         1.206063      -1.175893	0.734701
3.1	         1.218532      -1.251806	0.780444
</pre>
</div>
  </td>
  <td align="center">
[[File:Chatterji51Summary2.png|400px|Chatterji's Figure 1]]
  </td>
  <td align="center">
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\omega^2_\mathrm{Chatterji}</math></td>
  <td align="center"><math>\sigma_c^2</math></td>
</tr>
<tr>
  <td align="left">Fundamental</td>
  <td align="center" bgcolor="pink">0.231</td>
  <td align="center">1.155</td>
</tr>
<tr>
  <td align="left">1<sup>st</sup> Overtone</td>
  <td align="center" bgcolor="pink">1.517</td>
  <td align="center">7.585</td>
</tr>
<tr>
  <td align="left">2<sup>nd</sup> Overtone</td>
  <td align="center" bgcolor="pink">3.580</td>
  <td align="center">17.900</td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
The solid circular markers in the plot (center panel) show how the amplitude of the displacement function varies with radius <math>0 \le \eta < \pi</math> for three separate radial modes, according to the data provided in Table II of {{ Chatterji51 }}, which has been reproduced here in the ''scrollable'' left-hand panel.  In the plot, ''blue'' is the fundamental mode, ''red'' is the 1<sup>st</sup> overtone, ''green'' is the 2<sup>nd</sup> overtone.  The (square of the) eigenfrequency corresponding to each mode, according to Table I of {{ Chatterji51 }}, is provided in the column of the right-hand panel that is (highlighted in pink and) labeled <math>\omega^2_\mathrm{Chatterji}</math>; also listed are the corresponding values of <math>\sigma_c^2 = 3\gamma \omega^2_\mathrm{Chatterji}</math>.
<br />&nbsp;<br />
The smooth, solid curves in the middle-panel plot are ''not'' fits to Chatterji's data.  Rather, they result from our own, independent numerical integration of the relevant LAWE, assuming that Chatterji's published values of the (square of the) eigenfrequency are correct for all three modes.  In all three cases for the specified eigenfrequency, there is excellent agreement between our determination of the radial eigenfunction and the determination obtained by {{ Chatterji51 }}.
  </td>
</tr>
</table>

We are exceptionally pleased to find that, for each of the three modes of oscillation, the displacement function obtained via our integration of the LAWE (solid curves in the figure) runs through the discrete points recorded by {{ Chatterji51 }} (solid circular markers in the figure).  But in doing so, we find from our higher resolution model that there is an inflection point just inside the surface of the model; this is not the smooth behavior that is expected as the surface is approached.  In an effort to correct this behavior, we have changed the constraint that is applied while integrating the LAWE from the center, outward:  Instead of forcing <math>\omega^2_\mathrm{Chatterji}</math> to match the value published by {{ Chatterji51 }}, we have let the value of this oscillation frequency vary while enforcing the surface boundary condition describe above as <font color="maroon">CONSIDERATION "C"</font>.  The resulting "improved" solution is shown in the figure that follows.


<table border="1" align="center" cellpadding="8" width="90%">
<tr>
  <td align="center" colspan="3" bgcolor="yellow">Our Imposed Surface Boundary Condition:  <br />
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1, ~\alpha}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\alpha - (3-\alpha)\biggl[ \frac{\pi^2 \sigma_c^2}{24}\biggr]\, .
</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">''Amplitudes and frequencies of the displacement functions for three modes, assuming'' <math>\alpha = 0.6</math> and <math>\gamma = 5/3</math></td>
</tr>
<tr>
  <td align="center" colspan="1">Amplitudes determined from our numerical integration of the LAWE</td>
  <td align="center" colspan="1">Our version of Chatterji's Fig. 1</td>
  <td align="center" colspan="1">Eigenfrequencies determined from our integration</td>
</tr>
<tr><td align="left" width="50%">
<div id="Chatterji51Table2Data" style="width: 100%; height: 30em; overflow: auto;">
<pre>
Abscissa	Fundamental	1st Overtone	2nd Overtone
0.000000000	1.000000000	 1.000000000	 1.000000000
0.031733259	1.000047844	 0.999943498	 0.999786141
0.063466518	1.000089016	 0.999601959	 0.998867632
0.095199777	1.000174709	 0.999061021	 0.997382877
0.126933037	1.000294726	 0.998302963	 0.995304066
0.158666296	1.000449107	 0.997327028	 0.992631032
0.190399555	1.000637906	 0.996132244	 0.989363571
0.222132814	1.000861191	 0.994717421	 0.985501447
0.253866073	1.001119039	 0.993081149	 0.981044389
0.285599332	1.001411544	 0.991221792	 0.975992105
0.317332591	1.001738809	 0.989137492	 0.970344285
0.34906585	1.002100951	 0.986826157	 0.964100608
0.38079911	1.002498098	 0.984285466	 0.957260757
0.412532369	1.002930394	 0.981512860     0.949824425
0.444265628	1.003397993	 0.978505538	 0.94179133
0.475998887	1.003901064	 0.975260454	 0.933161228
0.507732146	1.004439786	 0.971774312	 0.923933924
0.539465405	1.005014356	 0.968043557	 0.914109296
0.571198664	1.005624981	 0.964064374	 0.903687306
0.602931923	1.006271884	 0.959832677	 0.892668024
0.634665183	1.006955299	 0.955344103	 0.881051648
0.666398442	1.007675478	 0.950594005	 0.868838527
0.698131701	1.008432684	 0.945577446	 0.856029188
0.72986496	1.009227198	 0.940289184	 0.842624363
0.761598219	1.010059312	 0.934723671	 0.828625017
0.793331478	1.010929336	 0.928875035	 0.814032388
0.825064737	1.011837596	 0.922737076	 0.798848013
0.856797996	1.01278443	 0.916303251	 0.783073774
0.888531256	1.013770197	 0.909566661	 0.766711937
0.920264515	1.01479527	 0.902520043	 0.749765195
0.951997774	1.015860038	 0.895155752	 0.732236722
0.983731033	1.016964908	 0.887465751	 0.714130222
1.015464292	1.018110307	 0.879441592	 0.695449985
1.047197551	1.019296676	 0.8710744	 0.676200956
1.07893081	1.020524478	 0.862354861	 0.656388792
1.110664069	1.021794193	 0.853273198	 0.636019943
1.142397329	1.023106322	 0.843819155	 0.615101726
1.174130588	1.024461383	 0.833981977	 0.593642407
1.205863847	1.025859919	 0.82375039	 0.5716513
1.237597106	1.027302491	 0.813112573	 0.549138858
1.269330365	1.028789682	 0.802056142	 0.526116785
1.301063624	1.030322099	 0.790568118	 0.502598146
1.332796883	1.03190037	 0.778634906	 0.478597499
1.364530142	1.033525148	 0.766242261	 0.454131023
1.396263402	1.035197111	 0.753375263	 0.429216666
1.427996661	1.03691696	 0.740018284	 0.403874305
1.45972992	1.038685425	 0.726154954	 0.378125913
1.491463179	1.04050326	 0.711768126	 0.351995745
1.523196438	1.042371249	 0.696839839	 0.325510537
1.554929697	1.044290203	 0.681351278	 0.298699722
1.586662956	1.046260964	 0.665282732	 0.271595661
1.618396215	1.048284403	 0.64861355	 0.244233898
1.650129475	1.050361424	 0.631322093	 0.216653434
1.681862734	1.052492964	 0.613385687	 0.188897019
1.713595993	1.054679993	 0.594780567	 0.161011472
1.745329252	1.056923518	 0.575481822	 0.133048033
1.777062511	1.059224581	 0.555463336	 0.105062731
1.80879577	1.061584262	 0.534697725	 0.077116794
1.840529029	1.064003681	 0.513156272	 0.049277088
1.872262289	1.066484	 0.490808851	 0.021616595
1.903995548	1.069026422	 0.467623859	-0.005785076
1.935728807	1.071632194	 0.443568129	-0.032841123
1.967462066	1.074302612	 0.418606853	-0.059456951
1.999195325	1.077039017	 0.392703483	-0.085529513
2.030928584	1.079842802	 0.365819646	-0.110946599
2.062661843	1.082715411	 0.337915031	-0.135586067
2.094395102	1.085658345	 0.308947292	-0.159315006
2.126128362	1.08867316	 0.278871926	-0.181988834
2.157861621	1.091761474	 0.247642155	-0.203450318
2.18959488	1.094924965	 0.215208794	-0.223528513
2.221328139	1.098165381	 0.181520118	-0.242037616
2.253061398	1.101484536	 0.146521711	-0.258775724
2.284794657	1.104884319	 0.110156314	-0.273523491
2.316527916	1.108366697	 0.072363659	-0.286042681
2.348261175	1.111933717	 0.033080298	-0.296074601
2.379994435	1.115587515	-0.007760587	-0.303338425
2.411727694	1.11933032	-0.050229374	-0.307529379
2.443460953	1.12316446	-0.094400211	-0.308316806
2.475194212	1.127092372	-0.140351229	-0.305342096
2.506927471	1.131116609	-0.188164774	-0.298216493
2.53866073	1.135239852	-0.237927637	-0.286518776
2.570393989	1.139464924	-0.289731302	-0.269792846
2.602127248	1.143794808	-0.343672193	-0.247545241
2.633860508	1.148232666	-0.39985192	-0.219242641
2.665593767	1.152781872	-0.458377516	-0.184309461
2.697327026	1.157446048	-0.519361653	-0.142125667
2.729060285	1.162229122	-0.582922804	-0.092025093
2.760793544	1.167135405	-0.649185328	-0.033294642
2.792526803	1.172169712	-0.718279394	 0.034824895
2.824260062	1.17733755	-0.790340637	 0.113135186
2.855993321	1.18264542	-0.865509304	 0.202471616
2.887726581	1.188101346	-0.943928474	 0.303685139
2.91945984	1.193715839	-1.025740391	 0.417605887
2.951193099	1.199503802	-1.111078809	 0.54496891
2.982926358	1.205488688	-1.200051936	 0.686253524
3.014659617	1.2117128	-1.292700149	 0.841298052
3.046392876	1.21826791	-1.388871861	 1.008209416
3.078126135	1.225416475	-1.487741753	 1.179292479
3.109859394	1.234427405	-1.584566325	 1.314705175
</pre>
</div>
  </td>
  <td align="center">
[[File:OurImprovedSummary2.png|400px|Chatterji's Figure 1]]
  </td>
  <td align="center">
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\omega^2_\mathrm{Chatterji}</math></td>
  <td align="center"><math>\sigma_c^2</math></td>
</tr>
<tr>
  <td align="left">Fundamental</td>
  <td align="center" bgcolor="yellow">0.2298579</td>
  <td align="center">1.1492896</td>
</tr>
<tr>
  <td align="left">1<sup>st</sup> Overtone</td>
  <td align="center" bgcolor="yellow">1.4733124</td>
  <td align="center">7.366562</td>
</tr>
<tr>
  <td align="left">2<sup>nd</sup> Overtone</td>
  <td align="center" bgcolor="yellow">3.3484654</td>
  <td align="center">16.742327</td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
The solid circular markers in the plot (center panel) show how the amplitude of the displacement function varies with radius <math>0 \le \eta < \pi</math> for three separate radial modes, according to the data provided in Table II of {{ Chatterji51 }}, which has been reproduced here in the ''scrollable'' left-hand panel.  In the plot, ''blue'' is the fundamental mode, ''red'' is the 1<sup>st</sup> overtone, ''green'' is the 2<sup>nd</sup> overtone.  The (square of the) eigenfrequency corresponding to each mode, according to Table I of {{ Chatterji51 }}, is provided in the column of the right-hand panel that is (highlighted in pink and) labeled <math>\omega^2_\mathrm{Chatterji}</math>; also listed are the corresponding values of <math>\sigma_c^2 = 3\gamma \omega^2_\mathrm{Chatterji}</math>.
<br />&nbsp;<br />
The smooth, solid curves in the middle-panel plot are ''not'' fits to Chatterji's data.  Rather, they result from our own, independent numerical integration of the relevant LAWE, assuming that Chatterji's published values of the (square of the) eigenfrequency are correct for all three modes.  In all three cases for the specified eigenfrequency, there is excellent agreement between our determination of the radial eigenfunction and the determination obtained by {{ Chatterji51 }}.
  </td>
</tr>
</table>

=====Pressure-Truncated n = 1 Polytrope=====
Drawing from an [[SSC/Stability/NeutralMode#Part_2|accompanying discussion]], if the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,

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

<tr>
  <td align="right">
<math>
-\frac{d\ln x}{d\ln\eta} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3</math> &nbsp; &nbsp; at &nbsp; &nbsp; <math>\eta = \tilde\eta</math>.
  </td>
</tr>
</table>