Editing SSC/Structure/Polytropes/Analytic (section)

===Other (All) Solutions===
In a very clearly written article titled, ''All Solutions of the n = 5 Lane-Emden Equation'', [http://adsabs.harvard.edu/abs/2012JMP....53f2503M Patryk Mach (2012, J. Math. Phys., 53, 062503)] has pointed out that there are other families of solutions to the Lane-Emden equation of index, <math>~n=5</math>, in addition to the two solutions that have just been detailed, which he includes as his equations (3) and (5):  

<!--
<div align="center">
<table border="2" cellpadding="10">
<tr>
  <th align="center">
Extracted (with minor editing) from [http://adsabs.harvard.edu/abs/2012JMP....53f2503M Mach (2012)]
  </th>
<tr>
  <td>
[[File:Mach2012Eqs.png|300px|center|Mach (2012)]]
[[Image:AAAwaiting01.png|300px|center|Mach (2012)]]
  </td>
</tr>
</table>
</div>
-->
<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equations extracted<sup>&dagger;</sup> from pp. 062503-1 &amp; -2 of [http://adsabs.harvard.edu/abs/2012JMP....53f2503M Mach (2012)]<p></p>
"''All Solutions of the n = 5 Lane-Emden Equation''"<p></p>
Journal of Mathematical Physics, vol. 53, pp. 062503-062503-6 &copy; American Institute of Physics
</td></tr>
<tr>
  <td align="center">
<!-- [[File:Mach2012Eqs.png|400px|center|Mach (2012)]] -->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta(\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \frac{1}{\sqrt{1 + \xi^2/3}}</math>
  </td>
  <td align="right">
&nbsp; &nbsp; &nbsp; <math>~(3)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\theta(\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \frac{\sin(\ln \sqrt{\xi})}{\sqrt{3\xi - 2\xi\sin^2(\ln\sqrt{\xi}) }}</math>
  </td>
  <td align="right">
&nbsp; &nbsp; &nbsp; <math>~(5)</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Equations displayed here, with layout modified from the original publication.</td></tr>
</table>
</div>


For completeness, Mach mentions a well-known solution that works for all indexes, <math>~n > 3</math>, which we have discussed separately in the context of  [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|power-law density distributions]], namely,
<div align="center">
<math>\theta^n(\xi) = 
\frac{\rho}{\rho_c} = \biggl[ \frac{2(n-3)}{(n-1)^2} \biggr]^{n/(n-1)} \xi^{- 2n/(n-1)} \, .
</math>
</div>

In addition, Mach identifies the rarely referenced work of [http://adsabs.harvard.edu/abs/2000JMP....41.7029G H. Goenner &amp; P. Havas (2000, J. Math. Phys., 41, 7029)], which presents a family of solutions that is expressed in terms of the Weierstrass elliptic function; and he derives a new family of solutions &#8212; see equation (10) in ''his'' &sect;2.1 &#8212; that can be expressed entirely in terms of Jacobi elliptic functions.  Mach's new solutions, in particular, are oscillatory (like Srivastava's solution) but have no zeros, so in isolation they are not likely to be useful for astrophysical models.  But, as Mach suggests, they "can be used in composite stellar models on the same footing as Srivastava's solution" &#8212; see our [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution|accompanying description of a composite model using Srivastava's solution]].