Editing SSC/Dynamics/IsothermalSimilaritySolution (section)

===Summary===
A similarity solution becomes possible for these equations when the single independent variable,
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<math>~\zeta = \frac{c_s t}{r} \, ,</math>
</div>
is used to replace both <math>~r</math> and <math>~t</math>.  Then, if <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> assume the following forms,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r(r,t)</math>
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  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~\biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho(r,t)</math>
  </td>
  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_r(r,t)</math>
  </td>
  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~- c_s U(\zeta) \, ,</math>
  </td>
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</table>
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<span id="CoupledODEs">the three coupled partial differential equations reduce to two coupled ordinary differential equations for the functions,</span> <math>~\Rho (\zeta)</math> and <math>~U(\zeta)</math>, namely,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\frac{dU}{d\zeta}</math>
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  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~
\frac{(\zeta U +1) [\Rho (\zeta U +1) -2)]}{[ (\zeta U +1)^2 - \zeta^2]} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{dP}{d\zeta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\zeta \Rho [2-\Rho (\zeta U +1)]}{[ (\zeta U +1)^2 - \zeta^2]} \, ,</math>
  </td>
</tr>
</table>
</div>

and a single equation defining <math>~m(\zeta)</math>,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m(\zeta)</math>
  </td>
  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~\Rho \biggl[ U + \frac{1}{\zeta} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

The parameters <math>~\zeta, m, \Rho</math>, and <math>~U</math>, and this summary set of equations are exactly those used by [http://adsabs.harvard.edu/abs/1977ApJ...218..834H Hunter (1977)] in his analysis of this problem.  But they differ in form from the relations used by [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Larson (1969)], [http://adsabs.harvard.edu/abs/1969MNRAS.144..425P Penston (1969)], and [http://adsabs.harvard.edu/abs/1977ApJ...214..488S Shu (1977)] primarily because these authors chose to use a similarity variable, 
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<math>~x = \pm \frac{1}{\zeta} \, ,</math>
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instead of <math>~\zeta</math>.  Hunter's analysis is the most complete and his relations will be used here, but a transformation between his presentation and those of the other authors can be easily obtained from Table 1 of [http://adsabs.harvard.edu/abs/1977ApJ...218..834H Hunter (1977)] which, for convenience, is reproduced here.

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<table border="1" cellpadding="5" align="center" width="60%">
<tr>
  <th align="center" colspan="5">
Analogous to Table 1 from [http://adsabs.harvard.edu/abs/1977ApJ...218..834H Hunter (1977)]<br />
''Relations Between the Variables Used by Different Authors''
  </th>
</tr>
<tr>
  <td align="center" width="20%">Physical<br />Quantity</td>
  <td align="center" width="20%">Herein</td>
  <td align="center" width="20%">[http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Larson (1969)]</td>
  <td align="center" width="20%">[http://adsabs.harvard.edu/abs/1969MNRAS.144..425P Penston (1969)]</td>
  <td align="center">[http://adsabs.harvard.edu/abs/1977ApJ...214..488S Shu (1977)]</td>
</tr>
<tr>
  <td align="center"><math>~\frac{c_s t}{r}</math></td>
  <td align="center"><math>~\zeta</math></td>
  <td align="center"><math>~- \frac{1}{x}</math></td>
  <td align="center"><math>~- \frac{1}{x}</math></td>
  <td align="center"><math>~+ \frac{1}{x}</math></td>
</tr>
<tr>
  <td align="center"><math>~- \frac{v_r}{c_s}</math></td>
  <td align="center"><math>~U</math></td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~- V</math></td>
  <td align="center"><math>~-v</math></td>
</tr>
<tr>
  <td align="center"><math>~\frac{4\pi G\rho r^2}{c_s^2}</math></td>
  <td align="center"><sup>&dagger;</sup><math>~\Rho</math></td>
  <td align="center"><math>~x^2\eta</math></td>
  <td align="center"><math>~x^2 e^Q</math></td>
  <td align="center"><math>~x^2\alpha</math></td>
</tr>
<tr>
  <td align="center"><math>~\frac{GM_r}{c_s^3 t}</math></td>
  <td align="center"><math>~m</math></td>
  <td align="center">&hellip;</td>
  <td align="center"><math>~-N</math></td>
  <td align="center"><math>~m</math></td>
</tr>
<tr>
  <td align="center"><math>~\ln(4\pi G\rho t^2)</math></td>
  <td align="center"><math>~Q</math></td>
  <td align="center"><math>~\ln\eta</math></td>
  <td align="center"><math>~Q</math></td>
  <td align="center"><math>~\ln\alpha</math></td>
</tr>
<tr>
  <td align="center"><math>~\frac{r}{(- c_s t)}</math></td>
  <td align="center"><math>~y</math></td>
  <td align="center"><math>~x</math></td>
  <td align="center"><math>~x</math></td>
  <td align="center"><math>~-x</math></td>
</tr>
<tr>
  <td align="left" colspan="5">
<sup>&dagger;</sup>Adopting Hunter's notation, this dimensionless variable name, <math>~\Rho</math> (the capital Greek letter, <math>~\rho</math>), should not be confused with the variable name, <math>~P</math>, that represents herein the ideal gas pressure.
  </td>
</tr>
</table>
</div>

The following pair of images are reproductions of (left) Figure 1 and (right) Figure 3 from [http://adsabs.harvard.edu/abs/1977ApJ...218..834H Hunter (1977)].  The solid curves show how (left) the dimensionless velocity, <math>~U</math>, and (right) the dimensionless density, <math>~\Rho</math>, behave as a function of the similarity variable, <math>~\zeta</math>, for models having several different prescribed values of Hunter's parameter, <math>~Q_0</math>.  For each value of <math>~Q_0</math>, the table of numbers immediately below the pair of images provides corresponding values of several other numerical constants.

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<table border="1" cellpadding="5">
<tr><td align="center" colspan="2">
Figures extracted from [http://adsabs.harvard.edu/abs/1977ApJ...218..834H Hunter (1977)]<p></p>
"''The Collapse of Unstable Isothermal Spheres''"<p></p>
ApJ, vol. 218, pp. 834 - 845 &copy; American Astronomical Society
</td></tr>
<tr>
<td>
[[File:Hunter77Fig1.png|400px|center|Figure 1 from Hunter (1977, ApJ, 218, 836]]
</td>
<td>
[[File:Hunter77Fig3.png|400px|center|Figure 3 from Hunter (1977, ApJ, 218, 836]]
</td>
</tr>
<tr>
  <td align="center" colspan="2">
{| class="wikitable" style="text-align:center;"
|- 
| style="width:60px; text-align:center; border-bottom:2px solid black; "| Model 
| style="width:5px; text-align:center; "| &nbsp; 
| style="width:60px; text-align:center; border-bottom:2px solid black; "| <math>~Q_0</math> 
| style="width:5px; text-align:center; "| &nbsp; 
| style="width:60px; text-align:center; border-bottom:2px solid black; "| <math>~U_0</math> 
| style="width:5px; text-align:center; "| &nbsp; 
| style="width:60px; text-align:center; border-bottom:2px solid black; "| <math>~\Rho_0</math> 
| style="width:5px; text-align:center; "| &nbsp; 
| style="width:60px; text-align:center; border-bottom:2px solid black; ;"| <math>~m_0</math>
|- 
| LP || &nbsp;
| 0.5139  || &nbsp;
| 3.278  || &nbsp;
| 8.854  || &nbsp;
| 46.915
|- 
| H(b) || &nbsp;
| 11.236  || &nbsp;
| 0.295  || &nbsp;
| 2.378  || &nbsp;
| 2.577
|-  
| H(d) || &nbsp;
| 20.975  || &nbsp;
| 0.026 || &nbsp;
| 2.023  || &nbsp;
| 1.138
|-  
| EW || &nbsp;
| <math>~+ \infty</math>  || &nbsp;
| 0.000  || &nbsp;
| 2.000  || &nbsp;
| 0.975
|}
  </td>
</tr>
</table>
</div>