Editing SSC/Virial/PolytropesEmbeddedOutline (section)

====Virial Equilibrium====
Plugging these coefficient assignments into the [[SSC/Virial/PolytropesEmbeddedOutline#Equilibrium_Configurations|above mathematical prescription of the virial theorem]] gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressure-truncated polytropic configurations,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\mathcal{A} + \frac{4\pi}{3} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{norm}} \biggr) 
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \, ,
</math>
  </td>
</tr>

</table>
</div>
which matches the [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|statement of virial equilibrium presented in our accompanying, more detailed analysis]].  A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~
\frac{P_\mathrm{e}}{P_\mathrm{norm}} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{4\pi} 
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{-4}
\biggl[ \mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} -\mathcal{A} \biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
or (see the [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.29|accompanying derivation]] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_e </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}} 
- \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr)\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>
  </td>
</tr>
</table>
</div>

Notice that this equilibrium relation exactly matches the one derived by Whitworth &#8212; and [[SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium|rederived above]] &#8212; when all three structural form factors are set to unity.  This is as it should be because all of Whitworth's results were derived assuming uniform-density configurations.  Also notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math> (see the left-hand column of our [[SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]]).