Editing SSC/Stability/BiPolytropes/RedGiantToPN

__FORCETOC__
=Red Giant to Planetary Nebula=

==Following the Lead of Yabushita75==

Here in the context of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we want to construct mass-versus-central density plots like the one displayed for truncated isothermal spheres in [[SSC/Stability/InstabilityOnsetOverview#Fig1|Figure 1 of an accompanying discussion]], and as displayed for a <math>(n_c, n_e) = (\infty, 3/2)</math> bipolytrope in Figure 1 (p. 445) of {{ Yabushita75full }}.

In our [[SSC/Structure/BiPolytropes/Analytic51/Pt2#BiPolytrope_with_nc_=_5_and_ne_=_1_(Pt_2)|accompanying chapter]] that presents example models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we have adopted the following [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Normalization|normalizations]]:
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{\rho}{\rho_0}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>M_r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>H^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{H}{K_c\rho_0^{1/5}}</math>
  </td>

  <td align="center">. &nbsp;&nbsp;&nbsp;</td>

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

</table>
</div>

Also, from the relevant [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface conditions]], we find,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \, .</math>
  </td>
</tr>
</table>

And, inverting this last expression gives,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\rho_0^{4/5}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \rho_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i \biggl( \frac{K_e}{K_c} \biggr)^{-1} \biggr]^{5/4} \, . </math>
  </td>
</tr>
</table>


for our normalizations we can replace <math>\rho_0</math> with <math>K_e/K_c</math> everywhere, via the relati

=Related Discussions=
* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes emdeded in an external medium]]

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