Editing Apps/RotatingPolytropes (section)

====Focus on Compressible Configurations====
* [https://ui.adsabs.harvard.edu/abs/1978PASJ...30..507E/abstract Y. Eriguchi (1978)], PASJapan, 30, 507:  ''Hydrostatic Equilibria of Rotating Polytropes''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">This paper is based on the author's dissertation, submitted to the Univerrsity of Tokyo, in partial fulfillment of the requirements for the doctorate.</font>

Results 4a:  n = 1.5, 4.0, and 4.9, all with uniform rotation; compared to published results of James and of 

Results 4b:  n = 1.5 only, with a <math>~\dot\varphi(\varpi)</math> rotation law &#8212; obtained from combining eqs. (30) and (7) &#8212; that ''resembles'' the so-called j-constant [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profile]], namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\dot\varphi}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{A}{[e^{2t}\sin^2\theta + \alpha^2]^{3/2}} 
= 
\frac{A}{[(r/R_0)^2\sin^2\theta + \alpha^2]^{3/2}}
= 
\frac{A}{[(\varpi/R_0)^2 + \alpha^2]^{3/2}} \, .
</math>
  </td>
</tr>
</table>
After employing this "equation (30)" rotation law, <font color="green">&hellip; Rapid rotation near the central region results in density inversion and a "ring"-like structure appears in figure 7.  No other author has used the rotation law (30), and therefore a comparison cannot be made.  The structure in figure 6 resembles the results of [https://ui.adsabs.harvard.edu/abs/1968ApJ...154..627M/abstract Mark (1968)], and density inversion appears also in [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract Stoeckly's (1965)] results.</font>

</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1982PThPh..68..191H/abstract I. Hachisu, Y. Eriguchi &amp; D. Sugimoto (1982)], Progress of Theoretical Physics, 68, 191:  ''Rapidly Rotating Polytropes and Concave Hamburger Equilibrium''
* [https://ui.adsabs.harvard.edu/abs/1982PThPh..68..206H/abstract I. Hachisu &amp; Y. Eriguchi (1982)], Progress of Theoretical Physics, 68, 206:  ''Bifurcation and Fission of Three Dimensional, Rigidly Rotating and Self-Gravitating Polytropes''
* [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Y. Eriguchi &amp; I. Hachisu (1983)], Progress of Theoretical Physics, 69, 1131: ''Two Kinds of Axially Symmetric Equilibrium Sequences of Self-Gravitating and Rotating Incompressible Fluid:  Two-Ring Sequence and Core-Ring Sequence''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">The computational scheme is much the same as that used in the computation of one-ring equilibrium sequence</font> &hellip; see [https://ui.adsabs.harvard.edu/abs/1981PThPh..65.1870E/abstract Eriguchi &amp; Sugimoto (1981)], above.
</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1983MNRAS.204..583H/abstract I. Hachisu &amp; Y. Eriguchi (1983)], MNRAS, 204, 583:  ''Bifurcations and phase transitions of self-gravitating and uniformly rotating fluid''
* [https://ui.adsabs.harvard.edu/abs/1983PThPh..70.1534E/abstract Y. Eriguchi &amp; I. Hachisu (1983)], Progress of Theoretical Physics, 70, 1534:  ''Gravitational Equilibrium of a Multi-Body Fluid System''
* [https://ui.adsabs.harvard.edu/abs/1984Ap%26SS..99...71H/abstract I. Hachisu &amp; Y. Eriguchi (1984)], Astrophysics &amp; Space Sciences, 99, 71:  ''Fission Sequence and Equilibrium Models of Rigidity [sic] Rotating Polytropes''
* [https://ui.adsabs.harvard.edu/abs/1988ApJS...66..315H/abstract I. Hachisu, J. E. Tohline &amp; Y. Eriguchi (1988)], ApJS, 66, 315:  ''Fragmentation of Rapidly Rotating Gas Clouds.  II. Polytropes &#8212; Clues to the Outcome of Adiabatic Collapse''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">We find a fission sequence from an ellipsoidal configuration to a binary by way of dumb-bell equilibrium.</font>
</td></tr></table>