Editing DarkMatter/CK2015 (section)

==Looking Back Through My Research Files==
===Discovery===
Yesterday, while cleaning out some of the file draws from my office at LSU, I stumbled upon a file folder dated 1987 with the label, "Analytic Model for Rotating Polytropes."  This folder contained the following items:
* A hand-written derivation from Izumi Hachisu dated 26 May 1987 and titled, "Semi-analytic form of Rapidly Rotating Polytropes."
* Two (typed) rough drafts of a paper authored by J. E. Tohline, I. Hachisu, &amp; D. M. Christodoulou and titled, "A General Analytic Model for Rotating Polytropes" &#8212; one draft dated 2 July 1987 and the other dated 15 July 1987.
* A manuscript authored by F. Schmitz (Institut f&uuml;r Astronomie und Astrophysik, der Universit&auml;t W&uuml;rzburg, Federal Republic of Germany) titled, "Equilibrium Structures of Differentially Rotating Self-gravitating Gases," and dated 17 July 1987.
* A two-page referee's report, written by Izumi Hachisu and typed on LSU letterhead, that evaluates the paper by F. Schmitz.

While I don't remember the entire story that underpins the contents of this file folder, I do remember discussing this research topic with Izumi and Dimitris.  At that time we were quite familiar with the published work by [[Apps/HayashiNaritaMiyama82#Rotationally_Flattened_Isothermal_Structures|Hayashi, Narita and Miyama (HNM82)]] which presents an analytic model for rotating ''isothermal'' gas clouds that have a constant-velocity rotation profile.  We were wondering whether analytic models also could be determined for rotating gas clouds with a more general polytropic equation of state if a ''particular'' rotation profile was selected for each polytropic index.  Evidently we dropped our pursuit of this problem when Izumi received the F. Schmitz manuscript to referee.  Presumably this was because:  (1) While we had formulated the problem mathematically, we had not discovered any closed-form solutions; and (2) by addressing a very similar problem in his manuscript, Schmitz had scooped us!

When I came across this file folder yesterday, I immediately thought about the CK15 manuscript and wondered to what extent there might be overlap between the CK15 work and this earlier work by F. Schmitz.  I wondered, as well, what overlap there was between the CK15 work and the mathematical formulation of the problem as developed by Tohline, Hachisu, &amp; Christodoulou over 25 years ago.

===Examination of Overlap===
We note, first, references to four separate papers by Schmitz can be found in the first paragraph of &sect;4 in CK15.

CK15 present the following 2<sup>nd</sup>-order ODE to describe the equilibrium of rotating, polytropic configurations:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~nc_o^2 \cdot \frac{1}{x}\frac{d}{dx} \biggl[ x\frac{d\tau^{1/n}}{dx} \biggr] + \tau</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x}\frac{dv^2}{dx} \, .</math>
  </td>
</tr>
</table>
</div>

If we write the velocity profile as,
<div align="center">
<math>v^2 = c_o^2 f(x) \, ,</math>
</div>
and switch <math>~\tau</math> to the more familiar, <math>~\theta^n</math>, the CK15 governing equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~nc_o^2 \cdot \frac{1}{x}\frac{d}{dx} \biggl[ x\frac{d\theta}{dx} \biggr] + \theta^n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{c_o^2}{x}\frac{d}{dx}\biggl[  f(x) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ c_o^2 \cdot \frac{1}{x}\frac{d}{dx} \biggl[ nx\frac{d\theta}{dx} -f(x)\biggr] + \theta^n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
which is the same as equation (10) of Tohline, Hachisu, & Christodoulou (1987), if you ignore variations in the vertical direction.