Editing SSC/Structure/BiPolytropes/Analytic1.53/Pt2 (section)

===Power-Law Density Distribution===

In an [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|accompanying discussion]], we have demonstrated that power-law density distributions can provide analytic solutions of the Lane-Emden equation, although the associated boundary conditions do not naturally conform to the boundary conditions that are suitable to astrophysical configurations.  We have just shown that the point-source envelope configuration appears to admit a power-law temperature (alternatively, enthalpy) solution. Via the polytropic relation, <math>H = K\rho^{1 / n}</math>, we can convert to the density-radius relation,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>d\biggl[\xi^3\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl[ \frac{K\rho_c^{1/n}}{H_\mathrm{norm}}  \biggr]^{(2n - 13 ) / 2} d\biggl[ \frac{\rho}{\rho_c} \biggr]^{(2n - 13 ) / 2n}
</math>
  </td>
</tr>
</table>
which, upon integration gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">constant</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl[ \frac{\rho}{\rho_c} \biggr]^{(2n - 13 ) / 2n} - \xi^3 \, ,
</math>
  </td>
</tr>
</table>
if we adopt the definition,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\rho_c</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>
\biggl(\frac{H_\mathrm{norm}}{K}\biggr)^n \, .
  </math>
  </td>
</tr>
</table>

Setting the integration constant to zero, our result gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{\rho}{\rho_c}</math></td>
  <td align="center"><math>\propto</math></td>
  <td align="left"><math>
\xi^{6n/(2n-13)} \, .
</math>
  </td>
</tr>
</table>
In astrophysically relevant configurations, the exponent on <math>\xi</math> must be negative, which means that we are confined to models for which <math>n < \tfrac{7}{2}</math>.

Now, from our [[SSC/Structure/PowerLawDensity#Derivation|associated discussion of power-law density distributions]] in polytropes, we discovered that hydrostatic balance can be established at all radial positions within a spherically symmetric configuration for power-law density distributions of the form,
<div align="center">
<math>
\frac{\rho}{\rho_c} \propto \xi^{- 2n/(n-1)}
</math>
</div>
This matches our just-derived point-source model if,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>6n/(2n-13)</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- 2n/(n-1)</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ 6(n-1)</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>2(13 - 2n)</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ n</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{16}{5} \, ,</math>
  </td>
</tr>
</table>
which ''is'' less than <math>\tfrac{7}{2}</math>, so it is an astrophysically viable result.