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

===Profile===

Once the values of the key set of parameters have been determined as illustrated in Table 1, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in [[#Steps_2_.26_3|step &#35;4]] and [[#Step_7|step &#35;8]], above.  Table 2 summarizes the mathematical expressions that define the profile throughout the core (column 2) and throughout the envelope (column 3) of the normalized mass density, <math>\rho^*(r^*)</math>, the normalized gas pressure, <math>P^*(r^*)</math>, and the normalized mass interior to <math>r^*</math>, <math>M_r^*(r^*)</math>.  For all profiles, the relevant normalized radial coordinate is <math>r^*</math>, as defined in the <math>2^\mathrm{nd}</math> row of Table 2.  Graphical illustrations of these resulting profiles can be viewed by clicking on the thumbnail images posted in the last few columns of Table 2.  

<div align="center">
<b>Table 2:  Radial Profile of Various Physical Variables</b>
<table border="1" cellpadding="6">
<tr>
  <td align="center" rowspan="2">
Variable
  </td>
  <td align="center" rowspan="2">
Throughout the Core<br>
<math>0 \le \xi \le \xi_i</math>
  </td>
  <td align="center" rowspan="2">
Throughout the Envelope<sup>&dagger;</sup><br>
<math>\eta_i \le \eta \le \eta_s</math>
  </td>

  <td align="center" colspan="3">
Plotted Profiles
  </td>
</tr>

<tr>
  <td align="center">
<math>\xi_i = 0.5</math>
  </td>
  <td align="center">
<math>\xi_i = 1.0</math>
  </td>
  <td align="center">
<math>\xi_i = 3.0</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>r^*</math>
  </td>
  <td align="center">
<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta</math>
  </td>

  <td align="center" colspan="3">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi(\eta)</math>
  </td>

  <td align="center">
<!--  [[File:PlotDensity_xi_0.5.jpg|thumb|75px]] -->
[[Image:DenXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>P^*</math>
  </td>
  <td align="center">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}</math>
  </td>
  <td align="center">
<math>\theta^{6}_i [\phi(\eta)]^{2}</math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:PresXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>M_r^*</math>
  </td>
  <td align="center">
<math>\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:MassXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="left" colspan="6">
<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>\phi(\eta)</math> and its first derivative using the information presented in Step 6, above.
  </td>
</tr>
</table>
</div>

[As of 28 April 2013] For the interface locations <math>\xi_i = 0.5, 1.0,\mathrm{and}~3.0</math>, Table 2 provides profiles for three values of the molecular weight ratio:  <math>\mu_e/\mu_c = 1.0, 1/2,\mathrm{and}~1/4</math>.  In all nine graphs, blue diamonds trace the structure of the <math>n_c=5</math> core; the core extends to a radius, <math>r^*_\mathrm{core}</math>, that is independent of molecular weight ratio but varies in direct proportion to the choice of <math>\xi_i</math>.  Specifically, as tabulated in the fourth row of Table 1, <math>r^*_\mathrm{core} = 0.34549, ~0.69099, \mathrm{and} ~2.07297</math> for, respectively, <math>\xi_i = 0.5,~1,~\mathrm{and}~3</math>.  Notice that, while the pressure profile and mass profile are continuous at the interface for all choices of the molecular weight ratio, the density profile exhibits a discontinuous jump that is in direct proportion to the chosen value of <math>\mu_e/\mu_c</math>.  

Throughout the <math>n_e = 1</math> envelope, the profile of all physical variables varies with the choice of the molecular weight ratio.  In the Table 2 graphs, red squares trace the envelope profile for <math>\mu_e/\mu_c = 1.0</math>; green triangles trace the envelope profile for <math>\mu_e/\mu_c = 1/2</math>; and purple crosses trace the envelope profile for <math>\mu_e/\mu_c = 1/4</math>.  The surface of the bipolytropic configuration is defined by the (normalized) radius, <math>R^*</math>, at which the envelope density and pressure drop to zero; the values tabulated in row 16 of Table 1 &#8212; <math>1.35550, ~1.62766, ~\mathrm{and} ~3.35697</math> for, respectively, <math>\xi_i = 0.5,~1,~\mathrm{and}~3</math> &#8212; correspond to a molecular weight ratio of unity and, hence also, to the envelope profiles traced by red squares in the Table 2 graphs.  As the molecular weight ratio is decreased from unity to <math>1/2</math> and, then, <math>1/4</math> for a given choice of <math>\xi_i</math>, the (normalized) radius of the bipolytrope increases roughly in inverse proportion to <math>\mu_e/\mu_c</math> as suggested by the formula for <math>R^*</math> shown in Table 1.  This proportional relation is not exact, however, because the parameter <math>\eta_s</math>, which also appears in the formula for <math>R^*</math>, contains an implicit dependence on the chosen value of the molecular weight ratio through the parameter <math>\eta_i</math>.

For a given choice of the interface parameter, <math>\xi_i</math>, the (normalized) mass that is contained in the core is independent of the choice of the molecular weight ratio.  However, the (normalized) total mass, <math>M_\mathrm{tot}^*</math>, varies significantly with the choice of <math>\mu_e/\mu_c</math>; as suggested by the expression provided in row 17 of Table 1, the variation is in rough proportion to <math>(\mu_e/\mu_c)^{-2}</math> but, as with <math>R^*</math>, this proportional relation is not exact because the parameters <math>\eta_s</math> and <math>A</math> which also appear in the formula for <math>M_\mathrm{tot}^*</math> harbor an implicit dependence on the molecular weight ratio.