Editing SSC/Structure/Polytropes/Numerical (section)

====Plotted Structural Profiles====
Using the just-described numerical techniques, we have solved the polytropic Lane-Emden equation on a 200-zone, uniform grid for a variety of values of the polytropic index.  In each case we have recorded how the dimensionless enthalpy, <math>~\theta_n(\xi)</math>, and its first radial derivative, <math>~\theta_n^'(\xi) \equiv d\theta_n/d\xi</math>, vary with <math>~\xi</math>, from the center of the polytropic configuration to its surface.  For the record, these tabulated results reside in the following DropBox files:

* <b>n = 2.5:</b> (10 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n25.xlsx
* <b>n = 3:</b> (19 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n300.xlsx
* <b>n = 3.005:</b> (15 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n3005.xlsx
* <b>n = 3.05:</b> (15 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n305.xlsx
* <b>n = 3.5:</b> (18 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n25.xlsx
* <b>n = 6:</b> (direct integration) &nbsp; WorkFolder/Wiki edits/EmbeddedPolytropes/N6.xlsx

For each of these models, as indicated (n = 2.5, 3, 3.005, 3.05, 3.5, 6), [[#Fig4|Figure 4]] illustrates how the normalized mass, <math>~M/M_\mathrm{SWS}</math>, varies with the normalized radius, <math>~R/R_\mathrm{SWS}</math>, where the definition of these two functions,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{M}{M_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \theta_n^{(n-3)/2} \xi^2 
\biggl| \frac{d\theta_n}{d\xi} \biggr|  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{R}{R_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \xi \theta_n^{(n-1)/2}  \, ,
</math>
  </td>
</tr>
</table>
</div>

has been drawn from an [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion of pressure-truncated polytropic configurations]].  In four of the Figure 4 panels, we have compared the profile of our numerically determined polytropic function (curve defined by <math>~\sim 200</math> small, black circular markers) to results (7 - 9 larger, blue circular markers) taken from Table 2.5.1 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] &#8212; see, specifically the segment of his table on pp. 74 - 75 that applies to polytropic spheres &#8212; in an effort to demonstrate that our numerically determined solutions are accurate.   

<div align="center" id="Fig4">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="3">
Figure 4: &nbsp; Numerically Determined Solutions to the Polytropic Lane-Emden Equation
  </th>
</tr>
<tr>
  <td align="center">[[File:n25SequenceA.png|250px|n = 2.5 equilibrium sequence]]</td>
  <td align="center">[[File:n30SequenceA.png|250px|n = 3 equilibrium sequence]]</td>
  <td align="center">[[File:n3005SequenceA.png|250px|n = 3.005 equilibrium sequence]]</td>
</tr>
<tr>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n25.xlsx --- worksheet = Horedt_n25]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n300.xlsx --- worksheet = Horedt_n300]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n3005.xlsx --- worksheet = Horedt_n3005]] <font size="+2">&#x21b2;</font></td>
</tr>
<tr>
  <td align="center">[[File:n305SequenceA.png|250px|n = 3.05 equilibrium sequence]]</td>
  <td align="center">[[File:n35SequenceA.png|250px|n = 3.5 equilibrium sequence]]</td>
  <td align="center">[[File:n600SequenceA.png|250px|n = 6 equilibrium sequence]]</td>
</tr>
<tr>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n305.xlsx --- worksheet = Horedt_n305]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n35.xlsx --- worksheet = Horedt_n35]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/N6.xlsx --- worksheet = PolytropeN6 (2)]] <font size="+2">&#x21b2;</font></td>
</tr>
<tr>
  <td align="left" colspan="3">
Data examples from Table 2.5.1 (pp. 74 - 75) of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]:
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\theta_n</math></td>
  <td align="center"><math>~\frac{d\theta_n}{d\xi}</math></td>
  <td align="center"><math>~\frac{R}{R_\mathrm{SWS}}</math></td>
  <td align="center"><math>~\frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr>
  <td align="center">2.5</td>
  <td align="right">4.00000</td>
  <td align="right">0.1376807</td>
  <td align="left">- 0.1340534</td>
  <td align="right">0.4032551</td>
  <td align="right">3.926310</td>
</tr>
<tr>
  <td align="center">3.0</td>
  <td align="right">5.00000</td>
  <td align="right">0.1108198</td>
  <td align="left">- 0.08012604</td>
  <td align="right">0.2707342</td>
  <td align="right">2.936234</td>
</tr>
<tr>
  <td align="center">3.5</td>
  <td align="right">5.00000</td>
  <td align="right">0.1786843</td>
  <td align="left">- 0.07362030</td>
  <td align="right">0.3065541</td>
  <td align="right">2.210326</td>
</tr>
<tr>
  <td align="center">6.0</td>
  <td align="right">5.00000</td>
  <td align="right">0.3973243</td>
  <td align="left">- 0.05113662</td>
  <td align="right">0.3437981</td>
  <td align="right">1.327430</td>
</tr>
</table>
  </td>
</tr>
</table>
</div>