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

===Setup===

====Continuous Form  of LAWE====
We begin by writing our generic version of the polytropic LAWE,
<div align="center">
{{ Math/EQ_RadialPulsation02 }}
</div>

then focus on the <math>n=1</math> case &#8212; setting <math>\gamma_g = 1 + 1/n = 2</math> and <math>\alpha = +1</math> &#8212; the relevant LAWE becomes,
<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} + \biggl\{ 4 - 2 Q \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^2}{\phi} 
- Q\biggr\}  \frac{x}{\eta^2} \, ,</math>
</td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <td align="center" bgcolor="pink">
{{ Chatterji51 }} &#8212; STEP 1
  </td>
</tr>
<tr><td align="left">
If we focus on the <math>n=1</math> case but leave <math>\gamma</math> (and, hence, <math>\alpha</math>) unspecified, the relevant LAWE becomes,
<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} + \biggl\{ 4 - 2 Q \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{6\gamma} \biggr)  \frac{\eta^2}{\phi} 
- \alpha Q\biggr\}  \frac{x}{\eta^2} \, .</math>
</td>
</tr>
</table>
If, in addition, we make the notation substitutions,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>Q</math></td>
  <td align="center"><math>~~~\rightarrow~~~</math></td>
  <td align="left">
<math>\frac{\mu}{n+1} = \frac{\mu}{2}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\frac{\sigma_c^2}{3\gamma}</math></td>
  <td align="center"><math>~~~\rightarrow~~~</math></td>
  <td align="left">
<math>\omega^2_\mathrm{Chatterji} \, ,</math>
  </td>
</tr>
</table>
the relevant LAWE becomes,
<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} + \biggl\{ 4 - \mu \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{3\gamma} \biggr)  \frac{\eta^2}{\phi} 
- \alpha \mu\biggr\}  \frac{x}{\eta^2} </math>
</td>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{d^2x}{d\eta^2} + \biggl[ \frac{4 - \mu}{\eta} \biggr] \frac{dx}{d\eta}  
+ \biggl[ \frac{\omega^2_\mathrm{Chatterji}}{\phi} 
- \frac{\alpha \mu}{\eta^2}\biggr]  x \, ,</math>
</td>
</tr>
</tr>
</table>
which, apart from notation, is identical to equation (1) of {{ Chatterji51 }}.
</td></tr>
</table>
Now, in the broadest context (see our [[SSC/Stability/n1PolytropeLAWE/Pt4#Beech88|related discussion]]),

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

<tr>
  <td align="right">
<math>\phi(\eta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
A \biggl[ \frac{\sin(\eta-B)}{\eta}\biggr] \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[[Appendix/References#Beech88|Beech88]], &sect;3, p. 221, Eq. (6)
  </td>
</tr>
</table>
Therefore, also, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\frac{d\phi}{d\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{A}{\eta^2} \biggl[\sin(\eta-B) - \eta\cos(\eta - B) \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ Q(\eta) \equiv - \frac{d\ln\phi}{d\ln\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{A}{\eta^2} \biggl[\sin(\eta-B) - \eta\cos(\eta - B) \biggr] \cdot \frac{\eta^2}{A\sin(\eta-B)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \eta\cot(\eta - B) \biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <td align="center" bgcolor="pink">
{{ Chatterji51 }} &#8212; STEP 2
  </td>
</tr>
<tr><td align="left">
In the context of an isolated, <math>n=1</math> polytrope, the appropriate parameter values are, <math>A=1</math> and <math>B=0</math>, in which case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\phi(\eta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sin\eta}{\eta} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\mu = 2Q</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(1 - \eta\cot\eta ) \, ,
</math>
  </td>
</tr>
</table>
and the relevant LAWE becomes,
<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} + \biggl[ \frac{4 + 2(\eta\cot\eta-1)}{\eta} \biggr] \frac{dx}{d\eta}  
+ \biggl[ \frac{\eta ~\omega^2_\mathrm{Chatterji}}{\sin\eta} 
+ \frac{2 \alpha (\eta\cot\eta-1)}{\eta^2}\biggr]  x </math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{d^2x}{d\eta^2} + \biggl[ \frac{2(1 + \eta\cot\eta)}{\eta} \biggr] \frac{dx}{d\eta}  
+ \biggl[ \frac{\eta ~\omega^2_\mathrm{Chatterji}}{\sin\eta} 
+ \frac{2 \alpha (\eta\cot\eta-1)}{\eta^2}\biggr]  x \, ,</math>
</td>
</tr>

</table>
which, again apart from notation, is identical to equation (2) of {{ Chatterji51 }}; see also the (unnumbered) equation in the middle of the left-hand-column of p. 223 in {{ MF85b }}.
</td></tr>
</table>

If we set <math>\gamma = (n+1)/n = 2</math> (and correspondingly set <math>\alpha = [3-4/\gamma] = +1)</math>, the <math>n=1</math> LAWE we becomes,
<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} + \biggl\{ 4 - 2 \biggl[1 - \eta\cot(\eta - B) \biggr] \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{A\sin(\eta-B)} 
- \biggl[1 - \eta\cot(\eta - B) \biggr]\biggr\}  \frac{x}{\eta^2}</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{d^2x}{d\eta^2} + 2\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{A\sin(\eta-B)} 
- 1 + \eta\cot(\eta - B) \biggr\}  \frac{x}{\eta^2} \, .</math>
</td>
</tr>
</table>
Multiplying through by <math>\phi</math>, we can write,
<table border=0 cellpadding=2 align="center">

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

====Discrete Form  of LAWE====
In order to integrate this 2<sup>nd</sup>-order ODE numerically, we will build from the [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center,_Outward|more general expression for polytropes used in our separate development of a finite-difference scheme]], namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta_i {x_i''}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \biggl[4\theta_i - (n+1)\xi_i (- \theta^')_i\biggr] \frac{x_i'}{\xi_i}  
- (n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g}  - 
\frac{\alpha}{\xi_i } (- \theta^')_i\biggr]  x_i \, .</math>
  </td>
</tr>
</table>
Making the notation substitutions, <math>(\xi, \theta) \rightarrow (\eta, \phi)</math>, we have instead,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\phi_i {x_i''}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \biggl[4\phi_i - (n+1)\eta_i (- \phi^')_i\biggr] \frac{x_i'}{\eta_i}  
- (n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g}  - 
\frac{\alpha}{\eta_i } (- \phi^')_i\biggr]  x_i \, .</math>
  </td>
</tr>
</table>
Now, adopting the finite-difference expressions,

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

<tr>
  <td align="right">
<math>x_i'</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - x_-}{2 \Delta_\eta} \, ,</math>&nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>
x_i'' 
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} \, ,</math>
  </td>
</tr>
</table>
</div>
<span id="DiscreteLAWE">the discrete form of the LAWE becomes,</span>

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

<tr>
  <td align="right">
<math>x_+ \overbrace{\biggl[2\phi_i +\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i\biggr]}^{\mathrm{TERM1}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_- \overbrace{\biggl[\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i - 2\phi_i\biggr]}^{\mathrm{TERM2}} 
+  x_i \overbrace{\biggl\{4\phi_i - \frac{\Delta_\eta^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 
2\alpha \biggl(- \frac{3\phi^'}{\eta}\biggr)_i\biggr]  \biggr\}}^{\mathrm{TERM3}} \, .</math>
  </td>
</tr>
</table>

When applied specifically to an <math>n=1</math>, polytropic configuration, we should insert the following specific expressions: 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\gamma_g</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \frac{1}{n} = 2
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\alpha</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3 - \frac{4}{\gamma_g} = +1
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\phi_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
A \biggl[ \frac{\sin(\eta_i-B)}{\eta_i}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(-\phi')_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{A}{\eta_i^2} \biggl[\sin(\eta_i-B) - \eta_i \cos(\eta_i - B) \biggr] \, .
</math>
  </td>
</tr>
</table>