Editing Appendix/Ramblings/51BiPolytropeStability/RethinkEnvelope (section)

==Earlier Example==

In our [[SSC/Stability/BiPolytropes/HeadScratching#Through_the_Envelope|earlier analysis]], we determined that the following relations hold in an equilibrium bipolytrope.

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Keep in mind that, once <math>\mu_e/\mu_c</math> and <math>\xi_i</math> have been specified, other [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values at the interface]] are:
<table border="0" cellpadding="5" align="center">

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

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

<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{d\phi}{d\eta}\biggr)_i 
= 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{1}{\sqrt{3} \xi_i \theta_i^2} - \frac{\xi_i}{\sqrt{3}} \, ,
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
B + \pi \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>
As a test case, let's draw from the [[SSC/Stability/BiPolytropes/HeadScratching#Selected_Models|accompanying <b>B2</b> model]] for which, <math>\mu_e/\mu_c = 0.25</math> and <math>\xi_i = 2.4782510</math> and &hellip;
<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center"><math>\theta_i</math></td>
  <td align="center"><math>\eta_i</math></td>
  <td align="center"><math>\Lambda_i</math></td>
  <td align="center"><math>A</math></td>
  <td align="center"><math>B</math></td>
  <td align="center"><math>\eta_s</math></td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center"><math>Q_\rho</math></td>
  <td align="center"><math>Q_m</math></td>
  <td align="center"><math>Q_r</math></td>
</tr>

<tr>
  <td align="center">0.572857</td>
  <td align="center">0.352159</td>
  <td align="center">1.408807</td>
  <td align="center">0.608404</td>
  <td align="center">-0.265127</td>
  <td align="center"><math>2.876465</math></td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center">0.00938349</td>
  <td align="center">13.558308</td>
  <td align="center">7.0373055</td>
</tr>
</table>

The following pair of plots show how the normalized density, <math>\rho^*</math>, and normalized integrated mass, <math>M_r^*</math>, varies over the radial-coordinate range, <math>0 \le \eta \le 3</math>, for both the core description and the envelope description for Model B2.  Both plots present the same four curves except, in the "first plot", the density has been magnified by a factor of 35 to aid in visualizing the shapes of the curves.  In the "first plot" the maximum ordinate value is 40, which comfortably accommodates the maximum value of both mass curves. In the "second plot" the maximum ordinate value is 0.09, which permits us to zoom in on the behavior of the (unmagnified) density curves in the vicinity of the core-envelope interface.


More specifically, here are the expressions that were used to generate each of the four curves (in both plots).

<b>Grey dotted curve:</b>  After setting <math>\xi = Q_r\eta</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>\rho^*\biggr|_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl[1 + \frac{\xi^2}{3}  \biggr]^{-5/2} \, .</math></td>
</tr>
</table>

<b>Orange curve:</b>  After setting <math>\xi = Q_r\eta</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>M_r^*\biggr|_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl( \frac{6}{\pi} \biggr)^{1/2} \xi^3\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} \, .</math></td>
</tr>
</table>

<b>Dark-blue dotted curve:</b>  Acknowledging that <math>B = -0.265127</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>\rho^*\biggr|_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>Q_\rho \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr] \, .</math></td>
</tr>
</table>

<b>Red curve:</b>  Acknowledging that <math>B = -0.265127</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>M_r^*\biggr|_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>Q_m \biggl[ \sin(\eta-B) - \eta\cos(\eta-B)\biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">
  <tr>
<td align="center"><b>Model B2</b> &#8212; first plot<br />[[File:ModelB2firstAnnotated.png|400px|First Plot]]</td>
<td align="center"><b>Model B2</b> &#8212; second plot<br />[[File:ModelB2secondAnnotated.png|400px|Second Plot]]</td>
  </tr>
  <tr>
<td align="left" colspan="2">
<b>Things to notice:</b>
<ol>
  <li>Because <math>B \ne 0</math> and <math>\rho^*|_\mathrm{env}</math> is proportional to <math>\eta^{-1}</math>, the envelope-density (dark-blue dotted) curve shoots up to infinity as <math>\eta \rightarrow 0</math>.  Nevertheless, as the red curve in the "first plot" shows, the integrated envelope mass, <math>M_r^*|_\mathrm{env}</math>, is well behaved; it goes to <math>Q_m\sin(-B)=3.5527</math> as <math>\eta \rightarrow 0</math>.</li>
  <li>As seen in the "second plot," the envelope-density (dark-blue dotted curve) first goes to zero when <math>\eta \rightarrow \eta_s = \pi + B = 2.876465</math>.  As the red curve in the "first plot" shows, this is also where <math>M_r^*|_\mathrm{env}</math> reaches its maximum value, <math>Q_m \eta_s = 39.00000</math>.</li>
  <li>The gray-dotted curve in the "first plot" shows how the "core density" varies over the entire examined range.  At the center &#8212; where <math>\eta \rightarrow 0</math> and, hence, <math>\xi \rightarrow 0</math> &#8212; the core density is unity; as <math>\eta</math> climbs, the core density drops smoothly toward zero, but always remains positive.</li>
  <li>As the orange curve in the "first plot" shows, the integrated core mass is zero at <math>\eta =0</math>; as <math>\eta</math> increases, the integrated core mass smoothly increases, heading toward a limiting value of <math>M_r^*|_\mathrm{core} \rightarrow 3^{3/2}(6/\pi)^{1 / 2}= 7.18096</math> as <math>\eta \rightarrow \infty</math> and, hence, <math>\xi \rightarrow \infty</math>.</li>
  <li>As the "first plot" shows, the (red) curve representing the envelope mass intersects the (orange) curve representing the envelope mass ''twice''.  Moving from the center, outward, the first intersection occurs at the <b>Model B2</b> core-envelope interface, where <math>\eta = \eta_i = 0.352159</math> and <math>\xi = \xi_i = 2.47825</math>.  As can be seen in the "second plot," the two "density" curves do not intersect at the interface.  However, by design and construction, at the core-envelope interface the value of <math>\rho^*|_\mathrm{env}</math> is precisely a factor of <math>\mu_e/\mu_c = 0.25</math> smaller than <math>\rho^*|_\mathrm{core}</math>; in the "second plot," the vertical red line-segment highlights this discontinuous drop in the density at the interface.</li>
</ol>
</td>
  </tr>
</table>


<table border="1" align="center" cellpadding="8">
  <tr>
<td align="center"><b>Model B2</b> &#8212; third plot<br />[[File:ModelB2thirdAnnotated.png|400px|Third Plot]]</td>
<td align="center"><b>Model B2</b> &#8212; fourth plot<br />[[File:ModelB2fourthAnnotated.png|400px|Fourth Plot]]</td>
  </tr>
  <tr>
<td align="left" colspan="2">
<b>Things to notice:</b>
</td>
  </tr>
</table>