Editing SSC/Structure/Polytropes/Numerical (section)

====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>