Editing SSC/Structure/BiPolytropes/Analytic51/Pt2 (section)

===Parameter Values===

The <math>2^\mathrm{nd}</math> column of Table 1 catalogues the analytic expressions that define various parameters and physical properties (as identified, respectively, in column 1) of the <math>(n_c, n_e) = (5, 1)</math> bipolytrope.  We have evaluated these expressions for various choices of the dimensionless interface radius, <math>\xi_i</math>, and have tabulated the results in the last few columns of the table.  The tabulated values have been derived assuming <math>\mu_e/\mu_c = 1</math>, that is, assuming that the core and the envelope have the same mean molecular weights.  

<!--  BEGIN TABLE OF PARAMETERS --->
<div align="center">
<b>Table 1:  Properties of <math>(n_c, n_e) = (5, 1)</math> BiPolytrope Having Various Interface Locations, <math>\xi_i</math></b><br>
[[File:BiPolytropeParametersV01.xml|Accompanying spreadsheet with parameter values]]

<table border="1" cellpadding="5" width="80%">
<tr>
  <td align="center">
Parameter
  </td>
  <td align="center">
<math>\xi_i</math>
  </td>
  <td align="center">
0.5
  </td>
  <td align="center">
1.0
  </td>
  <td align="center">
1.66864602<sup>&dagger;</sup>
  </td>
  <td align="center">
3.0
  </td>
  <td rowspan="21">
[[File:Bipolytrope51Boundaries02.png|500px|Examples]]<br />
<table border="0" align="left" cellpadding="10"><tr><td align="left">
For bipolytropic models having <math>\mu_e/\mu_c = 1.0</math>, this figure shows how the interface location, <math>\eta_i</math> (solid purple curve), the surface radius, <math>\eta_s</math> (green circular markers), and the parameter, <math>\tan^{-1}\Lambda_i</math> (orange circular markers), vary with <math>\xi_i</math> (ordinate) over the range, <math>0 \le \xi_i \le 12</math>.  The three horizontal, red-dashed line segments identify the values of <math>\xi_i</math> for which numerical values of these (and other) parameters have been listed in the table shown here on the left.</td></tr></table>
  </td>
</tr>

<tr>
  <td align="center">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2}</math>
  </td>
  <td align="center">
0.96077
  </td>

  <td align="center">
0.86603
  </td>
  <td align="center">
0.72016538
  </td>
  <td align="center">
0.50000
  </td>
</tr>

<tr>
  <td align="center">
<math>-\biggl(\frac{d\theta_i}{d\xi}\biggr)_i</math>
  </td>
  <td align="center">
<math>\frac{1}{3} \xi_i \biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-3/2}</math>
  </td>
  <td align="center">
0.14781
  </td>

  <td align="center">
0.21651
  </td>
  <td align="center">
0.20774935
  </td>
  <td align="center">
0.12500
  </td>
</tr>

<tr>
  <td align="center">
<math>r^*_\mathrm{core} \equiv r^*_i</math>
  </td>
  <td align="center">
<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi_i</math>
  </td>
  <td align="center">
0.34549
  </td>

  <td align="center">
0.69099
  </td>
  <td align="center">
1.15301487
  </td>
  <td align="center">
2.07297
  </td>
</tr>

<tr>
  <td align="center">
<math>\rho^*_i \biggr|_c = \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \rho^*_i \biggr|_e</math>
  </td>
  <td align="center">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-5/2}</math>
  </td>
  <td align="center">
0.81864
  </td>

  <td align="center">
0.48714
  </td>
  <td align="center">
0.19371408
  </td>
  <td align="center">
0.03125
  </td>
</tr>

<tr>
  <td align="center">
<math>P^*_i</math>
  </td>
  <td align="center">
<math>\biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-3}</math>
  </td>
  <td align="center">
0.78653
  </td>

  <td align="center">
0.42188
  </td>
  <td align="center">
0.13950617
  </td>
  <td align="center">
0.01563
  </td>
</tr>

<tr>
  <td align="center">
<math>H^*_i \biggr|_c = \frac{n_c+1}{n_e+1} \biggl( \frac{\mu_e}{\mu_c} \biggr) H^*_i \biggr|_e</math>
  </td>
  <td align="center">
<math>6 \biggl( 1+\frac{1}{3}\xi_i^2 \biggr)^{-1/2}</math>
  </td>
  <td align="center">
5.76461
  </td>

  <td align="center">
5.19615
  </td>
  <td align="center">
4.32099225
  </td>
  <td align="center">
3.00000
  </td>
</tr>

<tr>
  <td align="center">
<math>M^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>\biggl( \frac{6}{\pi}\biggr)^{1/2} (\xi_i \theta_i)^3</math>
  </td>
  <td align="center">
0.15320
  </td>

  <td align="center">
0.89762
  </td>
  <td align="center">
2.39822567
  </td>
  <td align="center">
4.66417
  </td>
</tr>

<tr>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}\eta_i</math>
  </td>
  <td align="center">
<math>\sqrt{3} ~\theta_i^2 \xi_i</math>
  </td>
  <td align="center">
0.79941
  </td>

  <td align="center">
1.29904
  </td>
  <td align="center">
1.49895749
  </td>
  <td align="center">
1.29904
  </td>
</tr>

<tr>
  <td align="center">
<math>-\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
  </td>
  <td align="center">
<math>\sqrt{3} ~\theta_i^{-3} \biggl( - \frac{d\theta}{d\xi} \biggr)_i = \frac{\xi_i}{\sqrt{3}}</math>
  </td>
  <td align="center">
0.28868
  </td>

  <td align="center">
0.57735
  </td>
  <td align="center">
0.96339323
  </td>
  <td align="center">
1.73205
  </td>
</tr>

<tr>
  <td align="center">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>\frac{1}{\eta_i} + \biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
  </td>
  <td align="center">
0.96225
  </td>

  <td align="center">
0.19245
  </td>
  <td align="center">
-&nbsp; 0.2962629
  </td>
  <td align="center">
-0.96225
  </td>
</tr>

<tr>
  <td align="center">
<math>A</math>
  </td>
  <td align="center">
<math>\eta_i (1 + \Lambda_i^2)^{1/2}</math>
  </td>
  <td align="center">
1.10940
  </td>

  <td align="center">
1.32288
  </td>
  <td align="center">
1.56335712
  </td>
  <td align="center">
1.80278
  </td>
</tr>

<tr>
  <td align="center">
<math>B</math>
  </td>
  <td align="center">
<math>\eta_i - \frac{\pi}{2} + \tan^{-1}( \Lambda_i)</math>
  </td>
  <td align="center">
- 0.00523
  </td>

  <td align="center">
-0.08163
  </td>
  <td align="center">
-0.359863583
  </td>
  <td align="center">
-1.03792
  </td>
</tr>

<tr>
  <td align="center">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>\pi + B</math>
  </td>
  <td align="center">
3.13637
  </td>

  <td align="center">
3.05996
  </td>
  <td align="center">
2.781729071
  </td>
  <td align="center">
2.10367
  </td>
</tr>

<tr>
  <td align="center">
<math>- \biggl( \frac{d\phi}{d\eta} \biggr)_s</math>
  </td>
  <td align="center">
<math>\frac{A}{\eta_s}</math>
  </td>
  <td align="center">
0.35372
  </td>

  <td align="center">
0.43232
  </td>
  <td align="center">
0.562009126
  </td>
  <td align="center">
0.85697
  </td>
</tr>

<tr>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \cdot \biggl[ R^* \equiv r^*_s \biggr]</math>
  </td>
  <td align="center">
<math>\frac{\eta_s}{\sqrt{2\pi} ~\theta_i^2}</math>
  </td>
  <td align="center">
1.35550
  </td>

  <td align="center">
1.62766
  </td>
  <td align="center">
2.139737125
  </td>
  <td align="center">
3.35697
  </td>
</tr>

<tr>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^2 M^*_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>\biggl(\frac{2}{\pi}\biggr)^{1/2} \theta_i^{-1} \biggl( -\eta^2 \frac{d\phi}{d\eta} \biggr)_s = \biggl(\frac{2}{\pi}\biggr)^{1/2} \frac{A\eta_s}{\theta_i}</math>
  </td>
  <td align="center">
2.88959
  </td>

  <td align="center">
3.72945
  </td>
  <td align="center">
4.818155932
  </td>
  <td align="center">
6.05187
  </td>
</tr>

<tr>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>~\frac{\eta_s^2}{3A\theta_i^5}</math>
  </td>
  <td align="center">
3.61035
  </td>

  <td align="center">
4.84326
  </td>
  <td align="center">
8.517046605
  </td>
  <td align="center">
26.1844
  </td>
</tr>

<tr>
  <td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \cdot \biggl[ \nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} \biggr]</math>
  </td>
  <td align="center">
<math>\sqrt{3} ~\biggl( \frac{\xi_i^3 \theta_i^4}{A\eta_s} \biggr)</math>
  </td>
  <td align="center">
0.05302
  </td>

  <td align="center">
0.24068
  </td>
  <td align="center">
0.497747627
  </td>
  <td align="center">
0.77070
  </td>
</tr>

<tr>
  <td align="center">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \cdot \biggl[ q \equiv \frac{r_\mathrm{core}}{R} \biggr]</math>
  </td>
  <td align="center">
<math>\sqrt{3}~\biggl[\frac{\xi_i \theta_i^2}{\eta_s}\biggr]</math>
  </td>
  <td align="center">
0.25488
  </td>

  <td align="center">
0.42453
  </td>
  <td align="center">
0.538858190
  </td>
  <td align="center">
0.61751
  </td>
</tr>

<tr>
  <td align="left" colspan="6">
<sup>&dagger;</sup>This choice of the value of <math>\xi_i = 1.66864602</math> is motivated by our [[SSC/Stability/BiPolytropes/Pt3#Fundamental_Modes|discussion of the fundamental mode of oscillation]] of the marginally unstable model that has <math>\mu_e/\mu_c = 1.0</math>; see especially row 1 of Table 2 in that associated chapter.
</tr>

</table>
</div>
<!--  END TABLE OF PARAMETERS --->


Alternatively, if given <math>\mu_e/\mu_c</math> and the value of the parameter, <math>\eta_i</math>, then we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1}\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3^{3/2}\xi_i}{3 + \xi_i^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\xi_i^2 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3^{3/2}}{\eta_i }\biggr] \xi_i + 3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\xi_i </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sqrt{3}
\biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3}{2\eta_i } \biggl\{ 1 \pm  
\sqrt{1 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{2 \eta_i }{3}\biggr]^2  }
\biggr\} \, .
</math>
  </td>
</tr>
</table>

It must be understood, therefore, that the interface location is restricted to the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~\le \eta_i \le</math>
  </td>
  <td align="left">
<math>~\frac{3}{2}\biggl(\frac{\mu_e}{\mu_c} \biggr)\, ,</math>
  </td>
</tr>
</table>

and that this upper limit on <math>\eta_i</math> is associated with a model whose core radius is, <math>\xi_i = \sqrt{3}</math>.  Also,

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

<tr>
  <td align="right">
<math>\Lambda_i </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta_i} -
\biggl(\frac{\mu_e}{\mu_c} \biggr) \frac{3}{2\eta_i } \biggl\{ 1 \pm  
\sqrt{1 - \biggl[ \biggl(\frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{2 \eta_i }{3}\biggr]^2  }
\biggr\} \, .
</math>
  </td>
</tr>
</table>