Editing SSC/Structure/BiPolytropes/AnalyzeStepFunction (section)

====Step 5====
<font color="red"><b>STEP 5:</b></font> &nbsp; Guess the eigenvector, <math>{\delta r}_i</math>, remembering that a reasonably good trial eigenfunction for the core is one that has a "parabolic dependence on the radius," namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{x}{\alpha_\mathrm{scale}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \frac{\xi^2}{15} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp; 
  <td align="right">
<math>\xi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2\pi}{3}\biggr)^{1 / 2} r_0 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>x(r_0)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale}\biggl[1 - \biggl(\frac{2\pi}{45}\biggr)r_0^2  \biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{dx}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\alpha_\mathrm{scale}\biggl(\frac{4\pi}{45}\biggr)r_0 
\, .
</math>
  </td>
</tr>
</table>

Note as well that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln r_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\alpha_\mathrm{scale}\biggl(\frac{4\pi}{45}\biggr) \frac{r_0^2}{x}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{4\pi}{45}\biggr) \frac{3\xi^2}{2\pi} \biggl[\frac{\xi^2}{15}-1\biggr]^{-1}
\, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><b>Finite-Difference Representation of <math>\delta r</math></b></div>
Once a "guess" for the fractional displacement vector, <math>(\delta r)_j = [x \cdot r_0]_j\, ,</math> has been specified, we recognize that the perturbed location of each radial shell is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r_j</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(r_0)_j + (\delta r)_j \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>


----


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="13">Parabolic Displacement Function w/ <math>\alpha_\mathrm{scale} = -0.001</math></td>
</tr>

<tr>
  <td align="center" rowspan="2">Shell</td>
  <td align="center" rowspan="2"><math>m </math></td>
  <td align="center" rowspan="2"><math>r_0</math></td>
  <td align="center" rowspan="2"><math>\frac{x}{\alpha_\mathrm{scale}}</math></td>
  <td align="center" rowspan="2"><math>r_j</math></td>
  <td align="center" rowspan="1" colspan="2">Analytic</td>
  <td align="center" rowspan="1" colspan="3">FD</td>
  <td align="center" rowspan="2"><math>P^*</math></td>
  <td align="center" rowspan="2"><math>-~\frac{dP^*}{dm}</math></td>
  <td align="center" rowspan="2"><math>-4\pi (r^*)^2\frac{dP^*}{dm} = g_0</math></td>
</tr>

<tr>
  <td align="center" rowspan="1"><math>\rho_0</math></td>
  <td align="center" rowspan="1"><math>d_\mathrm{analytic}</math></td>
  <td align="center" rowspan="1"><math>[\rho_{j - 1 / 2}]_0</math></td>
  <td align="center" rowspan="1"><math>d_{j-1 / 2}</math></td>
  <td align="center" rowspan="1"><math>\rho_{j - 1 / 2}</math></td>
</tr>

<tr>
  <td align="center">0</td>
  <td align="right">0.000000</td>
  <td align="right">0.000000</td>
  <td align="right">1.000000</td>
  <td align="right">0.000000</td>
  <td align="right">1.000000</td>
  <td align="right">+ 0.003000</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.995868</td>
  <td align="right">0.003004</td>
  <td align="right">0.998860</td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">1</td>
  <td align="right">0.001039</td>
  <td align="right">0.062922</td>
  <td align="right">0.999447</td>
  <td align="right">0.062859</td>
  <td align="right">0.993123</td>
  <td align="right">0.002997</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.991754</td>
  <td align="right">5.275715</td>
  <td align="right">0.262476</td>
</tr>

<tr>
  <td align="center">1&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.981896</td>
  <td align="right">0.002999</td>
  <td align="right">0.984840</td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">2</td>
  <td align="right">0.008211</td>
  <td align="right">0.125843</td>
  <td align="right">0.997789</td>
  <td align="right">0.125718</td>
  <td align="right">0.972886</td>
  <td align="right">0.002989</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.967552</td>
  <td align="right">2.605474</td>
  <td align="right">0.518508</td>
</tr>

<tr>
  <td align="center">2&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.955465</td>
  <td align="right">0.002988</td>
  <td align="right">0.958319</td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">3</td>
  <td align="right">0.027155</td>
  <td align="right">0.188765</td>
  <td align="right">0.995025</td>
  <td align="right">0.188577</td>
  <td align="right">0.940420</td>
  <td align="right">0.002975</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.928937</td>
  <td align="right">1.701968</td>
  <td align="right">0.762083</td>
</tr>

<tr>
  <td align="center">3&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.917695</td>
  <td align="right">0.002971</td>
  <td align="right">0.920421</td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">4</td>
  <td align="right">0.062586</td>
  <td align="right">0.251686</td>
  <td align="right">0.991155</td>
  <td align="right">0.251437</td>
  <td align="right">0.897462</td>
  <td align="right">0.002956</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.878252</td>
  <td align="right">1.241164</td>
  <td align="right">0.988001</td>
</tr>

<tr>
  <td align="center" colspan="13"><font size="+1"><b>&vellip;</b></td>
</tr>

<tr>
  <td align="center">95</td>
  <td align="right">6.769823</td>
  <td align="right">5.977546</td>
  <td align="right">- 3.988996</td>
  <td align="right">6.001390</td>
  <td align="right">0.0002917</td>
  <td align="right">- 0.021945</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.000057</td>
  <td align="right">0.000422</td>
  <td align="right">0.189466</td>
</tr>

<tr>
  <td align="center">95&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.0002844</td>
  <td align="right">- 0.021852</td>
  <td align="right">0.0002782</td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">96</td>
  <td align="right">6.777942</td>
  <td align="right">6.040467</td>
  <td align="right">- 4.094580</td>
  <td align="right">6.065200</td>
  <td align="right">0.0002773</td>
  <td align="right">- 0.022473</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.000054</td>
  <td align="right">0.000405</td>
  <td align="right">0.185762</td>
</tr>

<tr>
  <td align="center">96&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.002705</td>
  <td align="right">- 0.022366</td>
  <td align="right">0.0002644</td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">97</td>
  <td align="right">6.785828</td>
  <td align="right">6.103389</td>
  <td align="right">- 4.201270</td>
  <td align="right">6.129031</td>
  <td align="right">0.0002638</td>
  <td align="right">- 0.023006</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.000051</td>
  <td align="right">0.000389</td>
  <td align="right">0.182163</td>
</tr>

<tr>
  <td align="center">97&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.0002574</td>
  <td align="right">- 0.022884</td>
  <td align="right"> 0.0002515 </td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">98</td>
  <td align="right">6.793488</td>
  <td align="right">6.166310</td>
  <td align="right">- 4.309066</td>
  <td align="right">6.192881</td>
  <td align="right">0.0002511</td>
  <td align="right">- 0.023545</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.000048</td>
  <td align="right">0.000374</td>
  <td align="right">0.178666</td>
</tr>

<tr>
  <td align="center">98&frac12;</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.0002450</td>
  <td align="right">- 0.023408</td>
  <td align="right"> 0.0002393 </td>
  <td align="right">1.000000</td>
  <td align="center"><math>\infty</math></td>
  <td align="right">0.000000</td>
</tr>

<tr>
  <td align="center">99</td>
  <td align="right">6.800930</td>
  <td align="right">6.229232</td>
  <td align="right">- 4.417967</td>
  <td align="right">6.256752</td>
  <td align="right">0.0002391</td>
  <td align="right">- 0.024090</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="right">0.000045</td>
  <td align="right">0.000359</td>
  <td align="right">0.175267</td>
</tr>
</table>


----

[[File:DensityPert51Core02.png|325px|right|Density perturbation]]Hence, from the [[SSC/Perturbations#Continuity_Equation|linearized continuity equation]],

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

<tr>
  <td align="right">
<math>d</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- r_0 \frac{dx}{dr_0} - 3 x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale} \biggl\{
\biggl(\frac{4\pi}{45}\biggr)r_0^2  
- 3 \biggl[1 - \biggl(\frac{2\pi}{45}\biggr)r_0^2  \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale} \biggl\{
2\biggl(\frac{2\pi}{45}\biggr)r_0^2  
-3  + 3 \biggl(\frac{2\pi}{45}\biggr)r_0^2  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale} \biggl\{
\biggl(\frac{2\pi}{9}\biggr)r_0^2  
-3    
\biggr\} \, .
</math>
  </td>
</tr>
</table>
The solid black curve in the figure shown here, on the right, displays this analytically specified density perturbation, <math>d(m)</math>, for the case <math>\alpha_\mathrm{scale} = - 0.001</math>.

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center"><b>Finite-Difference Representation of <math>d</math></b></div>
Our finite-difference representation of the mass-density at each radial shell in the equilibrium configuration (subscript "0") is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[{\bar\rho}_{j-1/2} \biggr]_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[m_j - m_{j-1}\biggr]_0\biggl[\frac{4\pi}{3} (r_j^3 - r_{j-1}^3)  \biggr]_0^{-1} \, .
</math>
  </td>
</tr>
</table>
After perturbing the radial location of each shell &#8212; that is, after setting <math>r_j = (r_0)_j + [x \cdot r_0]_j</math> &#8212; the resulting finite-difference representation of the perturbed mass density of each shell is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>{\bar\rho}_{j-1/2} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[m_j - m_{j-1}\biggr]_0\biggl[\frac{4\pi}{3} (r_j^3 - r_{j-1}^3)  \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
(Note that we retain a subscript "0" on the mass, <math>m_j</math>, because it serves as our Lagrangian identifier for each shell.)  The fractional density perturbation at each discrete shell is, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
d_{j-1/2} \equiv \biggl[ \frac{\delta \rho}{\rho_0} \biggr]_{j-1/2}
=
\biggl\{ \frac{ {\bar\rho}_{j-1 / 2} }{ [{\bar\rho}_{j-1 / 2}]_0 } - 1 \biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{(r_j^3 - r_{j-1}^3)_0}{(r_j^3 - r_{j-1}^3)}  -1 
\, .
</math>
  </td>
</tr>
</table>
The red dots in the above "density perturbation" figure display how <math>d_j</math> varies with mass shell when <math>x</math> is specified by the parabolic dependence on radius.  The  dots lie virtually on top of the solid black curve in this figure, indicating that our finite-difference representation of the perturbed mass density matches well the analytically specified perturbed mass density.
</td></tr></table>

And, from [[SSC/Perturbations#Entropy_Conservation|linearized entropy conservation]],

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

<tr>
  <td align="right">
<math>p</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale} \biggl\{
\biggl(\frac{2\pi}{9}\biggr)r_0^2  
-3    
\biggr\} \gamma_c 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale} \biggl\{\biggl(\frac{2\pi}{9}\biggr)r_0^2  -3    \biggr\} \frac{6}{5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{dp}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale} \biggl(\frac{8\pi}{15}\biggr)r_0  
\, .
</math>
  </td>
</tr>
</table>

Now, [[#Model_Amodel2|from below]] we find that,
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right"><math>\frac{P_0}{\rho_0} = \frac{P^*}{\rho^*}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5 / 2}
</math>
  </td>
</tr>

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

<tr>
  <td align="right"><math>g_0 = \frac{m}{(r^*)^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] 
\biggl[ \biggl(\frac{3}{2\pi}\biggr)^{1 / 2}\xi \biggr]^{-2}
</math>
  </td>
</tr>

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

Hence, the [[SSC/Perturbations#Euler_+_Poisson_Equations|linearized Euler + Poisson Equations]] expression gives,

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

<tr>
  <td align="right">
<math>\omega^2 r_0 x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0} 
- (4x + p)g_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\alpha_\mathrm{scale} \biggl(1 + \frac{\xi^2}{3}\biggr)^{-1 / 2}  
\biggl(\frac{8\pi}{15}\biggr)r_0 
- \biggl\{
4\alpha_\mathrm{scale}\biggl[1 - \biggl(\frac{2\pi}{45}\biggr)r_0^2  \biggr] 
+ 
\alpha_\mathrm{scale} \biggl[\biggl(\frac{2\pi}{9}\biggr)r_0^2  -3    \biggr] \frac{6}{5}
\biggr\}
\biggl( \frac{8\pi}{3 } \biggr)^{1/2} \biggl[ \xi \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl(\frac{\omega^2}{\alpha_\mathrm{scale}}\biggr)\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{3/2}  r_0 x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(1 + \frac{\xi^2}{3}\biggr)  
\biggl(\frac{8\pi}{15}\biggr)r_0 
- \biggl\{
4 - \biggl(\frac{8\pi}{45}\biggr)r_0^2  
+ \biggl(\frac{12\pi}{45}\biggr)r_0^2  - \frac{18}{5} 
\biggr\}
\biggl( \frac{8\pi}{3 } \biggr)^{1/2} \xi  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(1 + \frac{\xi^2}{3}\biggr)  
\biggl(\frac{8\pi}{15}\biggr)r_0 
- \biggl[
\frac{2}{5}   
+ \biggl(\frac{4\pi}{45}\biggr)r_0^2  
\biggr]
\biggl( \frac{8\pi}{3 } \biggr)^{1/2} \xi  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(1 + \frac{\xi^2}{3}\biggr)  
\biggl(\frac{8\pi}{15}\biggr)\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \xi 
- \biggl[
\frac{2}{5}   
+ \biggl(\frac{4\pi}{45}\biggr)\biggl(\frac{3}{2\pi}\biggr) \xi^2  
\biggr]
\biggl( \frac{8\pi}{3 } \biggr)^{1/2} \xi  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2^5\pi}{3 \cdot 5^2}\biggr)^{1 / 2} \xi 
- \frac{2}{5} \biggl( \frac{8\pi}{3 } \biggr)^{1/2} \xi  
+
\biggl(\frac{2^5\pi}{3^3\cdot 5^2}\biggr)^{1 / 2} \xi^3 
- \biggl( \frac{2^5\pi}{3^3\cdot 5^2 } \biggr)^{1/2} \xi^3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>&nbsp; &nbsp; &nbsp; &nbsp; <font color="red">Exactly!</font>
  </td>
</tr>
</table>