Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1 (section)

==Detailed Force-Balanced Solution==

As has been [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|discussed in detail in another chapter]], {{ Horedt70full }}, {{ Whitworth81full }} and {{ Stahler83full }} have separately derived what the equilibrium radius, <math>R_\mathrm{eq}</math>, is of a polytropic sphere that is embedded in an external medium of pressure, <math>P_e</math>.  Their solution of the detailed force-balanced equations provides a pair of analytic expressions for <math>R_\mathrm{eq}</math> and <math>P_e</math> that are parametrically related to one another through [[SSC/Structure/Polytropes#Lane-Emden_Equation|the Lane-Emden function]], <math>\theta</math>, and its radial derivative.  For example &#8212; see our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|related discussion for more details]] &#8212; from {{ Horedt70 }} we obtain the following pair of equations:
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = r_a \cdot \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggl[ \frac{4\pi}{(n+1)^n} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{P_\mathrm{e}}{P_\mathrm{norm}} = p_a \cdot \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} 
\biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)}  \, ,
</math>
  </td>
</tr>
</table>
</div>
where we have introduced the normalizations,

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

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  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  \, .</math>
  </td>
</tr>

</table>
</div>

In the expressions for <math>r_a</math> and <math>p_a</math>, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, <math>\xi_1</math>, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the ''isolated polytrope,'' but at the radial coordinate, <math>~\tilde\xi</math>, where the internal pressure of the isolated polytrope equals <math>P_e</math> and at which the ''embedded'' polytrope is to be truncated.  The coordinate, <math>\tilde\xi</math>, therefore identifies the surface of the embedded &#8212; or, pressure-truncated &#8212; polytrope.  We also have taken the liberty of attaching the subscript "limit" to <math>M</math> in both defining relations because it is clear that {{ Horedt70 }} intended for the normalization mass to be the mass of the pressure-truncated object, not the mass of the associated ''isolated'' (and untruncated) polytrope.  

From these previously published works, it is not obvious how &#8212; or even ''whether'' &#8212; this pair of parametric equations can be combined to directly show how the equilibrium radius depends on the value of the external pressure.  Our examination of the free-energy of these configurations and, especially, an application of the viral theorem shows this direct relationship.  Foreshadowing these results, we note that,

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

<tr>
  <td align="right">
<math>~\biggl[ \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr) \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4\biggr]_\mathrm{Horedt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}} \biggr] 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, ;
</math>
  </td>
</tr>
</table>
</div>
or, given that <math>P_\mathrm{norm}R_\mathrm{norm}^4 = GM_\mathrm{tot}^2</math>, this can be rewritten as,

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}}   \, .
</math>
  </td>
</tr>
</table>
</div>