Editing SSC/Structure/Polytropes/Numerical (section)

==Numerical Solutions==
Here we explain how an Excel workbook can be used to numerically solve the Lane-Emden equation, evaluating the Lane-Emden function across a one-dimensional, discrete grid.  

===Techniques===
====HSCF Technique====
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On the first spreadsheet within the workbook, we establish the following columns of number:
<ul>
  <li>'''Column A:''' &nbsp; Labeled <math>~r_i/R</math> (for ''i'' between 1 and N), that represents a discrete radial grid of spacing, <math>~~\Delta = (N-1)^{-1}</math>; each row gives the radial coordinate location of the ''i''<sup>th</sup> zone, starting from <math>~r_1/R = 0</math> and ending at <math>~r_N/R = 1</math>.</li>
  <li>'''Column B:''' &nbsp; Labeled <math>~\mathrm{rhf}_i</math> (for ''i'' between 1 and N-1); each row gives the radial coordinate of the mid-point of a grid zone.</li>
<div align="center">
<math>~\mathrm{rhf}_i \equiv \frac{1}{2}\biggr[\frac{r_i}{R} + \frac{r_{i+1}}{R}  \biggr] \, .</math>
</div>
  <li>'''Column C:''' &nbsp; Labeled <math>~\rho_i</math> (for ''i'' between 1 and N-1); each row provides an initial ''guess'' for the mass-density of the grid zone.  Usually it is sufficient to guess, <math>~\rho_i = 1</math> throughout.  For an <math>~n=0</math> polytrope, this proves also to be the correct ''final'' density profile.</li>
  <li>'''Column D:''' &nbsp; Labeled <math>~M_i</math> (for ''i'' between 1 and N); the ''i''<sup>th</sup> row gives the integrated mass enclosed interior to the radial grid coordinate, <math>~r_i/R</math>.  Specifically, <math>~M_1 = 0</math>, and thereafter, beginning with zone, <math>~i = 2</math>,</li>
<div align="center">
<math>~M_i = M_{i-1} + \frac{4\pi \rho_{i-1}}{3}\biggl[ \biggl(\frac{r_{i}}{R} \biggr)^3 - \biggl(\frac{r_{i-1}}{R} \biggr)^3  \biggr] \, .</math>
</div>
  <li>Note that, <math>~M_\mathrm{tot} = M_N \, .</math></li>
  <li>'''Column E:''' &nbsp; Labeled <math>~g_i</math> (for ''i'' between 2 and N); each row tabulates the inwardly directed gravitational acceleration that is felt at the outer edge of each grid zone.  Specifically,</li>
<div align="center">
<math>g_i = \frac{GM_i}{ (r_i/R)^2} \, .</math>
</div>
  <li>'''Column F:''' &nbsp; Labeled <math>~\Phi_i</math> (for ''i'' between 1 and N); each row gives the value of the gravitational potential at the mid-point of a grid zone.  Here, we start by specifying the ''value'' of the potential just (specifically, half a radial grid-zone) outside the surface of the configuration, where it should be, <math>~\Phi_N = -GM_\mathrm{tot}/(1+\Delta/2)</math>.  Then, working from the surface, inward and, given that, <math>~g = d\Phi/dr</math>, we use the corresponding finite-difference representation of the radial derivative and set,</li>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Phi_i - \Phi_{i-1}}{\Delta} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~g_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \Phi_{i-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \Phi_i - g_i \Delta \, .</math>
  </td>
</tr>
</table>
</div>
<li>Note that the value of the gravitational potential ''at'' the surface is not <math>~\Phi_N</math> but, rather, must be <math>~\Phi_\mathrm{surf} = -GM_\mathrm{tot}/R</math>.</li>
<li>Furthermore, note that a lop-sided Taylor-series expansion about the center of the configuration provides the following good approximation to the gravitational potential ''at'' the center: &nbsp; <math>~\Phi_c \approx (9\Phi_1 - \Phi_2)/8</math>.</li>
<li>Note as well that all of these numerically determined values of the gravitational potential can be checked against the [[SSC/Structure/UniformDensity#UniformSpherePotential|known analytic expression]] for the radial profile of the potential in a uniform-density sphere.</li>
<li>'''Column G:''' &nbsp; Labeled <math>~H_i</math> (for ''i'' between 1 and N-1); each row provides the value of the fluid enthalpy at the center of a grid cell.  Adopting the convention that the enthalpy is zero at the surface of the configuration, and given that [[SSCpt2/SolutionStrategies#Technique_3|the enthalpy and the gravitational potential must sum to zero]] throughout the configuration, we have,</li>
<div align="center">
<math>H_i = \Phi_\mathrm{surf} - \Phi_i \, .</math>
</div>
<li>At the center of the configuration, we have, <math>~H_c = \Phi_\mathrm{surf} - \Phi_c</math>. </li>
<li>'''Column H:''' &nbsp; Labeled <math>~H_\mathrm{norm}</math> (for ''i'' between 1 and N-1); each row provides the value of the fluid enthalpy, renormalized to the central value, specifically,</li>
<div align="center">
<math>~[H_\mathrm{norm}]_i = \frac{H_i}{H_c} \, .</math>
</div>
</ul>


The second spreadsheet within the workbook should be initially created by generating a copy of the first spreadsheet.  Then:
<ul>
  <li>'''Column C:''' &nbsp; Labeled <math>~\rho_i</math> (for ''i'' between 1 and N-1); generate a new, improved ''guess'' for the normalized mass-density at each grid zone based on the corresponding value of the normalized enthalpy from the previous spreadsheet/iteration. Specifically, given that the [[SR#Barotropic_Structure|relationship between the density and enthalpy in a polytrope]] of index, <math>~n</math>, is, <math>~\rho \propto H^n</math>, we should set,</li>
<div align="center">
<math>~\biggl\{ \frac{\rho_i}{\rho_c} \biggr\}_\mathrm{sheet2}= \biggr\{[H_\mathrm{norm}]_i^n \biggr\}_\mathrm{sheet1} \, .</math>
</div>
</ul>

====Straight Numerical Integration====
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<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>via<br />Direct<br />Numerical<br />Integration</b>]]</font>
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The [[#LaneEmdenEquation|above governing relation]] may be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
\xi \theta^{''} + 2 \theta^' 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \xi \theta^n \, .</math>
  </td>
</tr>
</table>
</div>

We'll adopt the following finite-difference approximations for the first and second derivatives on a grid of radial spacing, <math>~\Delta_\xi</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta_i'</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{\theta_+ - \theta_-}{2\Delta_\xi}</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta_i''</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{\theta_+ - 2\theta_i +\theta_-}{\Delta_\xi^2} \, .</math>
  </td>
</tr>
</table>
</div>
Our finite-difference approximation of the governing equation is, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_i \biggl[ \frac{\theta_+ - 2\theta_i +\theta_-}{\Delta_\xi^2} \biggr] + 2\biggl[  \frac{\theta_+ - \theta_-}{2\Delta_\xi}\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \xi_i \theta_i^n </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi_i [ \theta_+ - 2\theta_i +\theta_-] + \Delta_\xi [  \theta_+ - \theta_- ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_\xi^2\xi_i \theta_i^n </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\theta_+
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\xi_i \theta_i  + \theta_-(\Delta_\xi - \xi_i)  - \Delta_\xi^2\xi_i \theta_i^n }{\Delta_\xi + \xi_i} \, .</math>
  </td>
</tr>
</table>
</div>

Now, for the first two steps away from the center &#8212; where, <math>~\theta_i = \theta_0 = 1</math> and <math>~\xi_i = \xi_0 = 0</math> &#8212;  we will use the following [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicLaneEmden|power-series expansion]] (see, for example, eq. 62 from &sect;5 in Chapter IV of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>]) to determine the value of <math>~\theta_i</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>1 - \frac{\Delta_\xi^2}{6} + \frac{n \Delta_\xi^4}{120}  - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \Delta_\xi^6 \, ,</math> 
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>1 - \frac{(2\Delta_\xi)^2}{6} + \frac{n (2\Delta_\xi)^4}{120}  - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) (2\Delta_\xi)^6 \, .</math> 
  </td>
</tr>
</table>
</div>

===Results===
====Tabulated Global Properties====

Here, drawing from tables that have been previously published by other authors, we record numerically determined properties of polytropic models having a fairly wide range of polytropic indexes.  First, we draw from Table 4 (p.96) of [[Appendix/References#C67|[<b><font color="red">C67</font></b>] ]].  To convert from his tabulated variables to our desired [[SSCpt1/Virial/FormFactors#PTtable|3 structural form-factors]], our normalized equilibrium radius (see [[SSC/Structure/Polytropes#MassRadiusRelation|earlier ASIDE]]), and the "virial" (drawn from a [[SSC/FreeEnergy/PowerPoint#Pressure-Truncated_Polytropes|more general overview]]), note that for ''isolated'' polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_\mathrm{M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho_c}{\bar\rho}\biggr)^{-1}  = \biggl[ - \frac{3\theta^'}{\xi}\biggr]_{\xi_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(5-n)\mathfrak{f}_\mathrm{W}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5 \mathfrak{f}_\mathrm{M}^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(5-n)\mathfrak{f}_\mathrm{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\biggl( \frac{4\pi}{3}\biggr) W_n\biggr]^{-1}  = 3(n+1)(-\theta^')^2_{\xi_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \xi_1 (- \xi^2 \theta^')^{(1-n)/(n-3)}_{\xi_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
Virial
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5-n)\biggl[ \frac{b}{n} \cdot x_\mathrm{eq}^{(n-3)/n} - \frac{a}{3} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{3}{4\pi} \biggr)^{1/n} \frac{(5-n)\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}  \cdot x_\mathrm{eq}^{(n-3)/n} - 1 \, .</math>
  </td>
</tr>
</table>
</div>


<div align="center" id="Chandrasekhar">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 4 of [[Appendix/References#C67|[<font color="red">C67</font>] ]]</th></tr>
<tr>
  <th align="center">Copied Directly from Table (1<sup>st</sup> 5 columns) &nbsp; &nbsp; &hellip; &nbsp; &nbsp; Implied Values of 3 Structural Form Factors, <math>x_\mathrm{eq}</math>, and Virial (last 5 columns)</th>
</tr>
<tr>
  <td align="left" colspan="1">
<pre>
n           xi_1            "mass"       rho_c/rho_avg       W_n             f_M             (5-n)f_W          (5-n)f_A        x_eq        Virial

0          2.4494           4.8988         1              0.119366       1                 5                 2.00000         0.620335        ---
0.5        2.7528           3.7871         1.8361         0.26227        0.544632645       1.483123592       0.910254        0.831089     -0.19009
1          3.14159          3.14159        3.28987        0.392699       0.303963378       0.461968677       0.607927        1.253313      3.0E-06
1.5        3.65375          2.71406        5.99071        0.77014        0.166925122       0.139319982       0.309857        2.357285      2.4E-06
2          4.35287          2.41105       11.40254        1.63818        0.087699758       0.038456238       0.145730        7.517481      1.4E-06
2.5        5.35528          2.18720       23.40646        3.90906        0.042723248       0.00912638        0.061072      186.3666        1.6E-08
3          6.89685          2.01824       54.1825        11.05066        0.018456144       0.001703146       0.0216035          ---          ---
3.25       8.01894          1.94980       88.153         20.365          0.011343913       0.000643422        0.0117227      3.3265E-06    2.9E-06
3.5        9.53581          1.89056      152.884         40.9098         0.006540907       0.000213917       0.00583558      0.00166854    2.2E-06
4         14.97155          1.79723      622.408        247.558          0.001606663       1.29068E-05       0.00096435      0.051854      6.1E-06
4.5       31.83646          1.73780     6189.47        4922.125          0.000161565       1.30516E-07       4.8502E-05      0.284868     -5.9E-05
4.9      169.47             1.73205        9.348E+05      3.693E+06      1.06975E-06       5.7218E-12        6.4645D-08      2.129056      1.5E-04
</pre>
  </td>
</tr>
<tr>
  <td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
  </td>
</tr>
</table>
</div>




<div align="center" id="Horedt2004">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 2.5.2 (p. 77) of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]
 &nbsp; &mdash; &nbsp; "Polytropic Spheres (N = 3)"</th></tr>
<tr>
  <th align="center">Copied Directly from Table (1<sup>st</sup> 3 columns) &nbsp; &nbsp; &hellip; &nbsp; &nbsp; Implied Values (last 7 columns)</th>
</tr>
<tr>
  <td align="left" colspan="1">
<pre>
n	    xi_1	    theta'	 "mass"	       rho_c/rho_avg	    W_n	           f_M	          (5-n)f_A	   x_eq	        Virial
0	2.44948974	-8.164966E-01	4.898980	1.000000	1.193662E-01	1.000000E+00	2.000000E+00	6.2035049E-01	
0.5	2.75269805	-4.999971E-01	3.788651	1.835143	2.122091E-01	5.449168E-01	1.124987E+00	8.3099030E-01	0.0E+00
1	3.14159265	-3.183099E-01	3.141593	3.289868	3.926990E-01	3.039636E-01	6.079272E-01	1.2533141E+00	0.0E+00
1.5	3.65375374	-2.033013E-01	2.714055	5.990704	7.701402E-01	1.669253E-01	3.099856E-01	2.3572860E+00	0.0E+00
2	4.35287460	-1.272487E-01	2.411047	1.140254E+01	1.638182E+00	8.769977E-02	1.457301E-01	7.5164793E+00	0.0E+00
2.5	5.35527546	-7.626491E-02	2.187199	2.340646E+01	3.909062E+00	4.272324E-02	6.107153E-02	1.8636634E+02	0.0E+00
3	6.89684862	-4.242976E-02	2.018236	5.418248E+01	1.105068E+01	1.845615E-02	2.160341E-02		
3.5	9.53580534	-2.079098E-02	1.890557	1.528837E+02	4.090983E+01	6.540920E-03	5.835575E-03	1.6685566E-03	0.0E+00
4	1.49715463E+01	-8.018079E-03	1.797230	6.224079E+02	2.475594E+02	1.606664E-03	9.643439E-04	5.1854394E-02	0.0E+00
4.5	3.18364632E+01	-1.714549E-03	1.737799	6.189473E+03	4.921842E+03	1.615646E-04	4.850469E-05	2.8486849E-01	0.0E+00
4.99	1.75818915E+03	-5.598955E-07	1.730765	1.046736E+09	4.237887E+10	9.553503E-10	5.633289E-12	2.3460204E+01	2.2E-15</pre>
  </td>
</tr>
<tr>
  <td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
  </td>
</tr>
</table>
</div>

====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>

====Emden's (1907) Tabulated Data====

<div align="center" id="Horedt2004">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 13 (p. 84) of [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)]
 &nbsp; &mdash; &nbsp; "Global Properties"</th></tr>
<tr>
  <td align="left" colspan="1">
<pre>
n	xi_1		- theta'	2nd deriv.	"mass"		rho_c/rho_avg
0	2.4494		0.81647		-0.33333	4.8988		1
0.5	2.7528		0.49975		0.36309		3.7871		1.8361
1	3.14159		0.31831		0.20264		3.14159		3.2899
1.5	3.6571		0.20316		0.11355		2.7176		6.0003
2	4.3518		0.12729		0.06262		2.4107		11.396
2.5	5.4172		0.075		0.02795		2.201		24.076
3	6.9011		0.04231		0.01282		2.015		54.36
4	14.999		0.00803		0.00107		1.8064		623.4
4.5	32.14		0.00168		0.000104	1.7354		6377.7
4.9	169.47		6.04E-05	4.208E-07	1.73554		9.485E+05
5	infinity	0		0		sqrt(3)		infinity
</pre>
  </td>
</tr>
<tr>
  <td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
  </td>
</tr>
</table>
</div>


====Horedt's (1986) Tabulated Data====

[https://ui.adsabs.harvard.edu/abs/1986Ap%26SS.126..357H/abstract G. P. Horedt (1986)], Astrophysics and Space Science, Vol. 126, Issue 2, pp. 357 - 408:  ''Seven-digit tables of Lane-Emden functions''

<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">
In Table I we present seven digit numerical solutions of the Lane-Emden equation for the plane-parallel (N = 1), cylindrical (N = 2), and spherical (N = 3) case for polytropic indices of <math>~n = -10, -5, -4, -3, -2, -1.5, -1.01, -0.9, -0.5, 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 10, 20, \pm \infty</math>, supplemented by <math>~n = 2.5, 3.5, 4.5,</math> and <math>~4.99</math> for the spherical case.

In Table II some finite boundary values of polytropic slabs, cylinders, and spheres are summarized.  For polytropic spheres (N = 3) we have also quoted boundary values near the minimum of the dimensionless mass <math>~-\xi_1^2\theta_1</math>, occurring at n &asymp; 4.823 [https://ui.adsabs.harvard.edu/abs/1978SvA....22..711S/abstract (Z. F. Seidov and R. Kh. Kuzakhmedov, 1978)].
</font>
</td></tr></table>

Focusing specifically on the spherically symmetric (N = 3) configurations, we list here the page number(s) on which the table associated with each individual polytropic index can be found in [https://ui.adsabs.harvard.edu/abs/1986Ap%26SS.126..357H/abstract Horedt (1986)].
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="3">Spherical (N = 3)<br />Configurations</th>
</tr>
<tr>
  <td align="center">n</td>
  <td align="center">page(s)</td>
  <td align="center" colspan="1"><math>~\xi_1</math></td>
</tr>
<tr>
  <td align="center">- 10</td>
  <td align="center">386 &rarr; 387</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">- 5</td>
  <td align="center">387 &rarr; 388</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">- 4</td>
  <td align="center">388 &rarr; 389</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">- 3</td>
  <td align="center">389</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">- 2</td>
  <td align="center">390</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">- 1.5</td>
  <td align="center">390 &rarr; 391</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">- 1.01</td>
  <td align="center">391 &rarr; 392</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">- 0.9</td>
  <td align="center">392 &rarr; 393</td>
  <td align="center">2.05040073E+00</td>
</tr>
<tr>
  <td align="center">- 0.5</td>
  <td align="center">393</td>
  <td align="center">2.20858842E+00</td>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">393 &rarr; 394</td>
  <td align="center">&radic;6 = 2.44948974E+00</td>
</tr>
<tr>
  <td align="center">0.5</td>
  <td align="center">394</td>
  <td align="center">2.75269805E+00</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">394 &rarr; 395</td>
  <td align="center">&pi; = 3.14159265E+00</td>
</tr>
<tr>
  <td align="center">1.5</td>
  <td align="center">395</td>
  <td align="center">3.65375374E+00</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">395 &rarr; 396</td>
  <td align="center">4.35287460E+00</td>
</tr>
<tr>
  <td align="center">2.5</td>
  <td align="center">396 &rarr; 397</td>
  <td align="center">5.35527546E+00</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">397 &rarr; 398</td>
  <td align="center">6.89684862E+00</td>
</tr>
<tr>
  <td align="center">3.5</td>
  <td align="center">398 &rarr; 399</td>
  <td align="center">9.53580534E+00</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">399</td>
  <td align="center">1.49715463E+01</td>
</tr>
<tr>
  <td align="center">4.5</td>
  <td align="center">399 &rarr; 400</td>
  <td align="center">3.18364632E+01</td>
</tr>
<tr>
  <td align="center">4.99</td>
  <td align="center">400 &rarr; 401</td>
  <td align="center">1.75818915+03</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">401 &rarr; 402</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">402 &rarr; 403</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">10</td>
  <td align="center">403 &rarr; 404</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center">20</td>
  <td align="center">404 &rarr; 405</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
<tr>
  <td align="center"><math>~\pm \infty</math></td>
  <td align="center">405 &rarr; 406</td>
  <td align="center"><font size="+2">&infin;</font></td>
</tr>
</table>