Editing SSC/Structure/PolytropesEmbedded (section)

===Horedt's Presentation===
It appears as though {{ Horedt70 }} &#8212; hereafter, {{ Horedt70hereafter }} &#8212; was the first to draw an analogy between the mass limit that is associated with bounded isothermal spheres &#8212; the so-called [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]] &#8212; and the limiting mass that can be found in association with equilibrium sequences of embedded polytropes that have polytropic indexes <math>~n > 3</math>.  Using a tilde to denote values of parameters at the (truncated) edge of a pressure-bounded polytropic sphere, {{ Horedt70hereafter }} (see the bottom of his p. 83) derives the following set of parametric equations relating the configuration's dimensionless radius, <math>~r_a</math>, to a specified dimensionless bounding pressure, <math>~p_a</math>:
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~r_a \equiv \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~p_a \equiv \frac{P_\mathrm{e}}{P_\mathrm{Horedt}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, ,
</math>
  </td>
</tr>
</table>
</div>
where it is understood that, [[SSC/Structure/Polytropes|as discussed elsewhere]], <math>~\theta_n(\xi)</math> is the solution to the Lane-Emden equation for a polytrope of index {{Math/MP_PolytropicIndex}},
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\tilde\theta'
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\frac{d\theta_n}{d\xi} ~~~\mathrm{evaluated}~\mathrm{at}~\tilde\xi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~R_\mathrm{Horedt}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\alpha_r \biggl( \frac{\alpha_M}{M} \biggr)^{(1-n)/(n-3)} = 
\biggl[ \frac{4\pi}{(n+1)^n}\biggl( \frac{G}{K_n} \biggr)^n M^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~P_\mathrm{Horedt}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
K_n \biggl( \frac{\alpha_M}{M} \biggr)^{2(n+1)/(n-3)} = 
K_n^{4n/(n-3)}\biggl[ \frac{(n+1)^3}{4\pi G^3 M^2} \biggr]^{(n+1)/(n-3)}  \, .
</math>
  </td>
</tr>
</table>
</div>
Notice that, via these normalizations, Horedt chose to express <math>~R_\mathrm{eq}</math> and <math>~P_\mathrm{e}</math> in terms of {{Math/MP_PolytropicConstant}} and the system's total mass, <math>~M</math>.