Editing SSC/Dynamics/IsothermalSimilaritySolution

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=Similarity Solution=
{| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
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! style="height: 150px; width: 150px; background-color:#ffff99;;" |[[H_BookTiledMenu#Nonlinear_Dynamical_Evolution|<b>Similarity<br />Solution</b>]]
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Much of the material in this chapter has been drawn from &sect;4.1 of a  [http://adsabs.harvard.edu/abs/1982FCPh....8....1T review article by Tohline (1982)] titled, ''Hydrodynamic Collapse''.


Several authors ([[#See_Especially|references given, below]]) have shown that when isothermal pressure gradients are important during a gas cloud's collapse, the equations governing the collapse admit a set of similarity solutions.  Certain properties of these solutions can be described analytically and are instructive models for comparison with more detailed, numerical collapse calculations.
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
==Establishing Set of Governing Equations==

Drawing from an [[SSC/Dynamics/IsothermalCollapse#IsothermalEulerianFrame|accompanying chapter's introductory discussion]], we begin with the set of governing equations that describe the collapse of isothermal spheres from an Eulerian frame of reference.

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">Eulerian Frame</th>
</tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial M_r}{\partial r} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r^2 \rho \, ,  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial M_r}{\partial t} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4\pi r^2 \rho v_r \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- c_s^2 \biggl( \frac{\partial \ln \rho}{\partial r}\biggr) - \frac{GM_r}{r^2}  \, .</math>
  </td>
</tr>
</table>

</td></tr></table>
</div>

Notice that, following [[SSC/Dynamics/IsothermalCollapse#Larson_.281969.29|Larson's (1969) lead]], we have replaced the standard continuity equation with the following equivalent statement of mass conservation:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dM_r}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math>
  </td>
</tr>
</table>
</div>

==Mathematical Solution==

===Summary===
A similarity solution becomes possible for these equations when the single independent variable,
<div align="center">
<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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r(r,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>
  </td>
  <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>
  </td>
  <td align="left">
<math>~- c_s U(\zeta) \, ,</math>
  </td>
</tr>
</table>
</div>

<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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dU}{d\zeta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m(\zeta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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, 
<div align="center">
<math>~x = \pm \frac{1}{\zeta} \, ,</math>
</div>
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.

<div align="center">
<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.

<div align="center">
<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>

===Proof===

Plugging the similarity solution expressions for <math>~M_r</math> and <math>~\rho</math> into the first of the three governing equations gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial r} \biggl[ \biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta)  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r^2 \biggl[ \biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \biggr]  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (c_s t )  \frac{\partial}{\partial r} \biggl[ m(\zeta)  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Rho (\zeta) \, .</math>
  </td>
</tr>
</table>
</div>

Plugging the similarity solution expressions for <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> into the second of the three governing equations gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial t} \biggl[ \biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta)  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4\pi r^2 \biggl[ \biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \biggr]  \biggl[ -c_s U(\zeta)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\partial}{\partial t} \biggl[ t m(\zeta)  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Rho (\zeta) U(\zeta) </math>
  </td>
</tr>

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

And, plugging the similarity solution expressions for <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> into the third of the three governing equations gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial }{\partial t} \biggl[ - c_s U(\zeta) \biggr] + \biggl[ - c_s U(\zeta) \biggr] \frac{\partial }{\partial r} \biggl[ - c_s U(\zeta) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- c_s^2 \biggl[\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta)   \biggr]^{-1} \frac{\partial }{\partial r}\biggl[  \biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \biggr] 
- \frac{G}{r^2}\biggl[ \biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\partial }{\partial t} \biggl[ U(\zeta) \biggr] - c_s U(\zeta) \frac{\partial }{\partial r} \biggl[ U(\zeta) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{c_s r^2}{\Rho (\zeta)}   \biggr]\frac{\partial }{\partial r}\biggl[  \biggl(\frac{\Rho (\zeta)}{r^2}\biggr)  \biggr] 
+ \biggl[ \frac{c_s^2 t}{r^2} \biggr] m(\zeta)  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\partial U}{\partial t} - (c_s U) \frac{\partial U}{\partial r} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{c_s}{\Rho}   \biggl[ \biggl( \frac{\partial \Rho}{\partial r}\biggr) -\frac{2\Rho}{r} \biggr]  
+ \biggl[ \frac{c_s^2 t}{r^2} \biggr] m(\zeta)  \, .</math>
  </td>
</tr>
</table>
</div>

Now, from the functional dependence of <math>~m(\zeta)</math> on <math>~\Rho(\zeta)</math> and <math>~U(\zeta)</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial m}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial r}
+ \Rho \biggl[ \frac{\partial U}{\partial r} - \frac{1}{\zeta^2} \frac{\partial \zeta}{\partial r}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial r}
+ \Rho \biggl[ \frac{\partial U}{\partial r} + \frac{1}{r \zeta} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial m}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t}
+ \Rho \biggl[ \frac{\partial U}{\partial t} - \frac{1}{\zeta^2} \frac{\partial \zeta}{\partial t}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t}
+ \Rho \biggl[ \frac{\partial U}{\partial t} - \frac{1}{t \zeta} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the first two governing equations become, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ r\zeta U + r \biggr] \frac{\partial\Rho}{\partial r}
+ (r\zeta \Rho ) \frac{\partial U}{\partial r} + \Rho 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \zeta U + 1\biggr] \frac{\partial\Rho}{\partial r}
+ (\zeta \Rho ) \frac{\partial U}{\partial r}  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rho (\zeta) U(\zeta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Rho\biggl[ U + \frac{1}{\zeta}\biggr] + t \biggl\{ 
\biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t}
+ \Rho \biggl[ \frac{\partial U}{\partial t} - \frac{1}{t \zeta} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
t \biggl\{ 
\biggl[ U + \frac{1}{\zeta} \biggr] \frac{\partial\Rho}{\partial t}
+ \Rho \biggl[ \frac{\partial U}{\partial t} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \zeta U + 1 \biggr] \frac{\partial\Rho}{\partial t}
+ (\zeta \Rho) \frac{\partial U}{\partial t}  \, .
</math>
  </td>
</tr>
</table>
</div>

Now, we can use these two relations to replace derivatives of <math>~\Rho</math> with derivatives of <math>~U</math> &#8212; or ''visa versa'' &#8212; in the third governing relation.  In the first case, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{\Rho}{c_s} \biggl[\frac{\partial U}{\partial t} - (c_s U) \frac{\partial U}{\partial r} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\partial \Rho}{\partial r}\biggr) -\frac{2\Rho}{r}   
+ \frac{\Rho^2}{r} \biggl[\zeta U + 1\biggr]   </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~   
\frac{\Rho^2(\zeta U + 1)}{r}  -\frac{2\Rho}{r}  - \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{\zeta \Rho}{(\zeta U + 1)} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~  \Rightarrow ~~~  \frac{1}{r} \biggl[
\Rho^2(\zeta U + 1)  - 2\Rho \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\Rho}{c_s} \biggl[\frac{\partial U}{\partial t}\biggr]  - (\Rho U) \frac{\partial U}{\partial r} 
+ \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{\zeta \Rho}{(\zeta U + 1)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~  \Rightarrow ~~~  \biggl[
\Rho(\zeta U + 1)  - 2 \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{r}{c_s} \biggl[\frac{\partial U}{\partial t}\biggr]  
+ \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{r \zeta }{(\zeta U + 1)} - (rU) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{t}{\zeta} \biggl[\frac{\partial U}{\partial t}\biggr]  
+ r \biggl( \frac{\partial U}{\partial r}\biggr) \biggl[ \frac{\zeta - U (\zeta U + 1)}{(\zeta U + 1)} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

And, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial U}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{dU}{d\zeta} \biggr)
\frac{\partial \zeta}{\partial t}
=
\biggl( \frac{dU}{d\zeta} \biggr)\frac{c_s}{r} = \biggl( \frac{dU}{d\zeta} \biggr)\frac{\zeta}{t} \, ;</math>
  </td>
</tr>
<tr><td colspan="3" align="center">and</td></tr>
<tr>
  <td align="right">
<math>~\frac{\partial U}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{dU}{d \zeta} \biggr) \frac{\partial \zeta}{\partial r}
=
- \frac{c_s t}{r^2} \biggl( \frac{dU}{d \zeta} \biggr) = -\frac{\zeta^2}{c_st} \biggl( \frac{dU}{d \zeta} \biggr) 
\, ,</math>
  </td>
</tr>
</table>
</div>

we can rewrite this as an ODE of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{d U}{d\zeta}\biggr)  \biggl[ \zeta^2U^2 + 2\zeta U + 1 - \zeta^2
\biggr]
</math>
  </td>
</tr>

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

In the second case, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{c_s}{\Rho} \biggl( \frac{\partial \Rho}{\partial r}\biggr)   
+ \frac{\zeta}{t}  \biggl[ \Rho (\zeta U + 1 ) - 2\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{\partial U}{\partial t} - (c_s U) \frac{\partial U}{\partial r} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\biggl[ \frac{\zeta U +1}{\zeta \Rho} \biggr] \frac{\partial \Rho}{\partial t}
+ (c_s U) \biggl[ \frac{\zeta U + 1}{\zeta\Rho} \biggr] \frac{\partial \Rho}{\partial r}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\frac{\zeta}{t}  \biggl[ 2- \Rho (\zeta U + 1 ) \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl\{ \frac{c_s}{\Rho}   
- (c_s U) \biggl[ \frac{\zeta U + 1}{\zeta\Rho} \biggr] \biggr\}\frac{\partial \Rho}{\partial r}
+ \biggl[ \frac{\zeta U +1}{\zeta \Rho} \biggr] \frac{\partial \Rho}{\partial t}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\frac{\zeta^2 \Rho}{c_s t}  \biggl[ 2- \Rho (\zeta U + 1 ) \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \zeta  
- U (\zeta U + 1 ) \biggr] \frac{\partial \Rho}{\partial r}
+ \frac{1}{c_s}\biggl[ \zeta U +1 \biggr] \frac{\partial \Rho}{\partial t} \, .
</math>
  </td>
</tr>
</table>
</div>

And, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \Rho}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{d\Rho}{d\zeta} \biggr)
\frac{\partial \zeta}{\partial t}
=
\biggl( \frac{d\Rho}{d\zeta} \biggr)\frac{c_s}{r} = \biggl( \frac{d\Rho}{d\zeta} \biggr)\frac{\zeta}{t} \, ;</math>
  </td>
</tr>
<tr><td colspan="3" align="center">and</td></tr>
<tr>
  <td align="right">
<math>~\frac{\partial \Rho}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{d\Rho}{d \zeta} \biggr) \frac{\partial \zeta}{\partial r}
=
- \frac{c_s t}{r^2} \biggl( \frac{d\Rho}{d \zeta} \biggr) = -\frac{\zeta^2}{c_st} \biggl( \frac{d\Rho}{d \zeta} \biggr) 
\, ,</math>
  </td>
</tr>
</table>
</div>

we can rewrite this as an ODE of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl\{( \zeta U +1 )  - \zeta [ \zeta  - U (\zeta U + 1 ) ] \biggr\} \frac{d\Rho}{d \zeta}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~ \Rightarrow~~~
\frac{d\Rho}{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>

Thus, we are able to understand the origin of the [[#CoupledODEs|pair of 1<sup>st</sup>-order ODEs, given above]], that describe the connected relationship between the two quantities, <math>~\Rho</math> and <math>~U</math>.

<!--
===Proof2===

First, note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \zeta}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{c_s}{r} = \frac{\zeta}{t} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \zeta}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{c_s t}{r^2} = -\frac{\zeta^2}{c_st} \, .</math>
  </td>
</tr>
</table>
</div>

Next, let's take partial derivatives, with respect to both <math>~r</math> and <math>~t</math>, of the three primary physical variables, <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math>.   
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial M_r}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{c_s^3 t}{G} \biggr)\frac{dm}{d\zeta} \biggl( \frac{\partial \zeta}{\partial r} \biggr)
=
\biggl( \frac{c_s^3 t}{G} \biggr)\frac{dm}{d\zeta} \biggl(- \frac{\zeta^2}{c_s t} \biggr)
=
-\biggl( \frac{c_s^2 \zeta^2}{G} \biggr)\frac{dm}{d\zeta}  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial M_r}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{c_s^3 t}{G} \biggr)\frac{dm}{d\zeta} \biggl( \frac{\partial \zeta}{\partial t} \biggr) + \biggl( \frac{c_s^3}{G}\biggr) m
=
\biggl( \frac{c_s^3 t}{G} \biggr)\frac{dm}{d\zeta} \biggl( \frac{\zeta}{t} \biggr) + \biggl( \frac{c_s^3}{G}\biggr) m
=
\biggl( \frac{c_s^3 }{G} \biggr) \biggl[ \zeta ~\frac{dm}{d\zeta}  + m \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial v_r}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-c_s \frac{dU}{d\zeta} \biggl( \frac{\partial \zeta}{\partial r} \biggr)
=
-c_s \frac{dU}{d\zeta} \biggl( - \frac{\zeta^2}{c_s t} \biggr)
=
\frac{\zeta^2}{t} \frac{dU}{d\zeta} \, ;
</math>
  </td>
</tr>
</table>
</div>
-->

==Limiting Behavior==

It can be shown by analytic manipulation of the pair of coupled ODEs that the dimensionless density, <math>~\Rho</math>, and the dimensionless radial velocity, <math>~U</math>, have the following behaviors in various limits:

* As, <math>\zeta \rightarrow - \infty</math>:
<p>&nbsp;</p>
{| class="Chap1A" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~U \approx \frac{2}{3} \biggl( - \frac{1}{\zeta}\biggr) + \frac{1}{45} \biggl[ \frac{2}{3} - e^{Q_0}\biggr] \biggl( - \frac{1}{\zeta}\biggr)^3 \, ,</math><p>&nbsp;</p>
|-
|<math>~Q \equiv \ln(\zeta^2 \Rho) \approx Q_0 + \frac{1}{6}\biggr[ \frac{2}{3} - e^{Q_0}\biggr] \biggl( - \frac{1}{\zeta}\biggr)^2 \, ,</math><p>&nbsp;</p>
|-
|where, <math>~Q_0</math> is a positive constant.
|}
<p>&nbsp;</p>

* [[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 13 July 2017:  In this expression for U, a "plus" sign has been inserted between the &zeta; term and the &zeta;-squared term, correcting a typographical error in equation 4.12a of Tohline (1982). And the expression for &Rho; has been expanded to include a &zeta;-cubed term.]]For, <math>\zeta \approx 0</math>:
<p>&nbsp;</p>
{| class="Chap1B" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~U \approx U_0 + \zeta(\Rho_0 - 2)+ \zeta^2 U_0 + \zeta^3\biggl[(\Rho_0-2)(1-\Rho_0/6) - \frac{2}{3} U_0^2 \biggr] \, ,</math><p>&nbsp;</p>
|-
|<math>~\Rho \approx \Rho_0 - \zeta^2\biggl[\frac{1}{2} \Rho_0(\Rho_0 -2)\biggr] + \frac{1}{3} \zeta^3 U_0 \Rho_0(\Rho_0-4) \, ,</math><p>&nbsp;</p>
|-
|where, <math>~U_0</math> and  <math>~\Rho_0</math> are positive constants.
|}
<p>&nbsp;</p>

* As, <math>\zeta \rightarrow + \infty</math>:
<p>&nbsp;</p>
{| class="Chap1C" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~U \approx (2m_0 \zeta)^{1 / 2} \, ,</math><p>&nbsp;</p>
|-
|<math>~\Rho \approx \biggl( \frac{m_0}{2\zeta} \biggr)^{1 / 2} \, ,</math><p>&nbsp;</p>
|-
|where, <math>~m_0</math> is a positive constant.
|}
<p>&nbsp;</p>
The values of the three constants, <math>~U_0</math>, <math>~\Rho_0</math>, and <math>~m_0</math> depend on the chosen value of <math>~Q_0</math>, as demonstrated by [http://adsabs.harvard.edu/abs/1977ApJ...218..834H Hunter (1977)].  In terms of the physical quantities, <math>~v_r(r,t)</math> and  <math>~\rho(r,t)</math>, these asymptotic behaviors translate into the following.

* For, <math>~t < 0</math> and <math>~r \ll c_s|t|</math>:
<p>&nbsp;</p>
{| class="Chap1D" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~v_r(r,t) \approx - \frac{2r}{3(-t)} \, ,</math><p>&nbsp;</p>
|-
|<math>~\rho(r,t) \approx \biggl[\frac{e^{Q_0}}{4\pi G}\biggr] \frac{1}{t^2} \, .</math><p>&nbsp;</p>
|}
<p>&nbsp;</p>

* For, <math>~r \gg c_s|t|</math> at any time:
<p>&nbsp;</p>
{| class="Chap1E" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~v_r(r,t) \approx - c_s U_0\, ,</math><p>&nbsp;</p>
|-
|<math>~\rho(r,t) \approx \biggl[\frac{c_s^2 \Rho_0}{4\pi G}\biggr] \frac{1}{r^2} \, .</math><p>&nbsp;</p>
|}
<p>&nbsp;</p>

* For, <math>~t > 0</math> and <math>~r \ll c_s|t|</math>:
<p>&nbsp;</p>
{| class="Chap1F" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
|- 
|<math>~v_r(r,t) \approx - \biggl( \frac{2m_0}{c_s} \biggr)^{1 / 2} \biggl( \frac{t}{r}\biggr)^{1 / 2} \, ,</math><p>&nbsp;</p>
|-
|<math>~\rho(r,t) \approx \frac{1}{4\pi G} \biggl[\frac{m_0 c_s^2}{2}\biggr]^{1 / 2} \biggl( \frac{1}{t r^3}\biggr)^{1 / 2} \, .</math><p>&nbsp;</p>
|}
<p>&nbsp;</p>

=Examine Connection With &hellip;=
Let's examine whether or not there is overlap between the properties of the above-discussed similarity solutions that give insight into the nonlinear dynamical behavior of collapse and the (a) known structure of the unperturbed, but marginally unstable Bonnor-Ebert sphere, and (b) eigenfunction that describes the radial profile of the marginally unstable radial pulsation mode.  Keep in mind that, as we have [[SSC/Stability/Isothermal#Overview|presented separately]], the truncation radius of the marginally unstable, Bonnor-Ebert sphere has, <math>~\xi_e \approx 6.4510534</math>.

==Pressure-Truncated Equilibrium Structure==
From our [[SSC/Stability/Isothermal#Groundwork|separate discussion of pressure-truncated isothermal spheres]], we can identify the following structural properties of the marginally unstable Bonnor-Ebert sphere.  The function, <math>~\psi(\xi)</math> satisfies the,
<div align="center">
<table border="0" cellpadding="8" align="center">
<tr><td align="center">
<font color="maroon">Isothermal Lane-Emden Equation</font> <p></p>
{{ Math/EQ_SSLaneEmden02 }}
</td></tr>
</table>
</div>

Given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from [[SSC/Structure/BonnorEbert#Pressure|our presentation]] that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^8}{4\pi G^3 M_{\xi_e}^2} \biggr) ~\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e} \, ;</math>
  </td>
</tr>
</table>
</div>

and, [[SSC/Structure/BonnorEbert#Radius|our expression]] for the truncated configuration's equilibrium radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_{\xi_e}}{c_s^2} \biggl[ \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-1} \, .</math>
  </td>
</tr>
</table>
</div>

Also, as has been summarized in our [[SSC/Structure/BonnorEbert#P-V_Diagram|accompanying discussion]], expressions that describe the general run of radius, pressure, and mass are, respectively,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_0 = c_s^2 \rho_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(c_s^2 \rho_c) e^{-\psi} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M_r  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  \, .</math>
  </td>
</tr>
</table>
</div>

Hence, for isothermal configurations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_0 \equiv \frac{GM_r}{r_0^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]
\biggl[ \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi\biggr]^{-2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_s^2  
\biggl( \frac{4\pi G \rho_c}{c_s^2} \biggr)^{1 / 2} 
\biggl( \frac{d\psi}{d\xi} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

From the [[#Summary|above summary of the Hunter (1977) similarity variables]], we also have,
A similarity solution becomes possible for these equations when the single independent variable,
<div align="center">
<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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r(r,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>
  </td>
  <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>
  </td>
  <td align="left">
<math>~- c_s U(\zeta) \, .</math>
  </td>
</tr>
</table>
</div>

Defining a new, dimensionless time as,
<div align="center">
<math>~\tau \equiv (4\pi G \rho_c t^2)^{1 / 2} \, ,</math>
</div>
then inserting the equilibrium structures into the expressions for the similarity variables gives, for example,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m(\zeta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{G}{c_s^3 t}\biggr) 
\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{\tau}\biggr) \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  \, ;
</math>
  </td>
</tr>
</table>
</div>

and,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\tau^2}{\zeta^2 }\biggr] e^{-\psi} \, ,</math>
  </td>
</tr>
</table>
</div>

while,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\zeta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{c_s t}\biggr) \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\xi }{\tau} \, .
</math>
  </td>
</tr>
</table>
</div>
Putting these last two expressions together also gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rho (\zeta) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi^2 e^{-\psi} \, .</math>
  </td>
</tr>
</table>
</div>


==Yabushita's Radial Pulsation Eigenvector==
As we have, separately, [[SSC/Perturbations#The_Eigenvalue_Problem|discussed in detail]], the eigenvalue problem is defined in terms of the following ''perturbed'' variables,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0(m) + P_1(m,t) = P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0(m) + \rho_1(m,t) = \rho_0(m) \biggl[1 + d(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0(m) + r_1(m,t) = r_0(m) \biggl[1 + x(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
And the following three linearized equations describe the physical relationship between the three dimensionless perturbation amplitudes <math>~p(r_0)</math>, <math>~d(r_0)</math> and <math>~x(r_0)</math>, for various characteristic eigenfrequencies, <math>~\omega</math>:

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
p = \gamma_\mathrm{g} d  \, .
</math>
</td></tr>
</table>
</div>

And, as was first [[SSC/Stability/InstabilityOnsetOverview#Yabushita.27s_Insight_Regarding_Stability|demonstrated by Yabushita (1975)]], 
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the Isothermal LAWE</b></font></td>
</tr>
<tr>
  <td align="right">
<math>~\sigma_c^2 = 0</math>
  </td>
  <td align="center">
&nbsp;and &nbsp;
  </td>
  <td align="left">
<math>~x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} \, .</math>
  </td>
</tr>
</table>
</div>

=Numerical Integration of Coupled ODEs=

Let's develop a finite-difference expression that allows us to straightforwardly integrate the [[#CoupledODEs|above pair of coupled ODEs]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dU}{d\zeta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>

Assume that the starting values of <math>~U</math> and <math>~\Rho</math> have been provided by, for example, one of the above detailed series expansions.  Let the subscript "1" denote these known values at coordinate-location, <math>~\zeta_1</math>, and let the subscript "2" denote the unknown values of <math>~U</math> and <math>~\Rho</math> at <math>~\zeta_2 = \zeta_1 + \Delta\zeta</math>.  We should be able to construct a 2<sup>nd</sup>-order accurate integration scheme by treating <math>~U</math> and <math>~\Rho</math> as ''average values'' everywhere they occur on the right-hand sides of the pair of ODEs.  That is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bar{U}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl(U_2 + U_1 \biggr) \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  <td align="right">
<math>~\bar{\Rho}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl(\Rho_2 + \Rho_1 \biggr) \, .</math>
  </td>
</tr>
</table>
</div>

==First ODE==
We'll begin by using the first ODE to provide one expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>.  We have,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\Rho_2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{(\zeta \bar{U} +1)} \biggl\{ 2+\biggl[ \frac{ (\zeta \bar{U} +1)^2 - \zeta^2  }{ (\zeta \bar{U} +1)  } \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - P_1 \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{\ell} \biggl\{ 2+\biggl[ \frac{ \ell^2 - \zeta^2  }{ \ell  } \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - P_1 \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\ell \equiv (\zeta \bar{U} + 1)</math> &nbsp;  &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; <math>~\bar{U} =\biggl[\frac{\ell - 1}{\zeta}\biggr] \, .</math>
</div>

==Second ODE==
The second ODE can presumably provide a second, independent expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>.  In this case we have, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{(P_2 - P_1)}{\Delta\zeta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\zeta \bar{\Rho} [2-\bar{\Rho} (\zeta \bar{U} +1)]}{[ (\zeta \bar{U} +1)^2 - \zeta^2]} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\zeta \tfrac{1}{2}(\Rho_1 + \Rho_2) [2-\tfrac{1}{2}(\Rho_1 + \Rho_2) (\zeta \bar{U} +1)]}{[ (\zeta \bar{U} +1)^2 - \zeta^2]} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{ 4[ (\zeta \bar{U} +1)^2 - \zeta^2]}{ \zeta \Delta\zeta}  \biggl(P_2 - P_1 \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\Rho_1 + \Rho_2) [4 - (\Rho_1 + \Rho_2) (\zeta \bar{U} +1)] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4(\Rho_1 + \Rho_2)
- (\Rho_1^2 + 2\Rho_1\Rho_2 +\Rho_2^2) (\zeta \bar{U} +1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Rho_2 [4
- 2\Rho_1 (\zeta \bar{U} +1) ]
- \Rho_2^2 (\zeta \bar{U} +1) 
+ [ 4\Rho_1 - \Rho_1^2  (\zeta \bar{U} +1) ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
4[ (\zeta \bar{U} +1)^2 - \zeta^2] P_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Rho_2 [4 - 2\Rho_1 (\zeta \bar{U} +1) ]\zeta \Delta\zeta
- \Rho_2^2 (\zeta \bar{U} +1)\zeta \Delta\zeta 
+ \biggl\{ [ 4\Rho_1 - \Rho_1^2  (\zeta \bar{U} +1) ]\zeta \Delta\zeta
+ 4[ (\zeta \bar{U} +1)^2 - \zeta^2] P_1 \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
This is a quadratic  equation of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~aP_2^2 + bP_2 + c </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ 
(\zeta \bar{U} +1)\zeta \Delta\zeta 
= \ell \zeta \Delta\zeta 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \{ [4 - 2\Rho_1 (\zeta \bar{U} +1) ]\zeta \Delta\zeta
- 4[ (\zeta \bar{U} +1)^2 - \zeta^2] \}
=
- [ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \{ [ 4\Rho_1 - \Rho_1^2  (\zeta \bar{U} +1) ]\zeta \Delta\zeta
+ 4[ (\zeta \bar{U} +1)^2 - \zeta^2] P_1 \} 
=
- [ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ] 
\, .
</math>
  </td>
</tr>
</table>
</div>

The pair of roots of this equation are, then,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2 \ell \zeta \Delta\zeta P_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ] 
\pm \biggl\{ 
[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2 
+ 4\ell \zeta \Delta\zeta[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ] 
\biggr\}^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
</div>
Or,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl\{ \frac{2 \ell \zeta \Delta\zeta}{ [ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ] } \biggr\}P_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
\pm \biggl\{ 1
+ \frac{4\ell \zeta \Delta\zeta[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]}{[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2 } 
\biggr\}^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
</div>

==Combined==
Now, before returning to the first ODE, let's write <math>~\Delta U</math> in terms of <math>~\ell</math> and, hereafter, use <math>~\ell</math> as the unknown instead of <math>~U_2</math>.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta U \equiv U_2 - U_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2\bar{U} - U_1) - U_1</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl[\bar{U} - U_1 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl[\frac{(\ell - 1)}{\zeta} - U_1 \biggr]</math>
  </td>
</tr>
</table>
</div>

Hence, the first ODE gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\ell^2 \Rho_2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\biggl\{ 2\ell +\biggl[ \ell^2 - \zeta^2 \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - \ell^2 P_1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\Delta\zeta \ell^2 \Rho_2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(4\ell - \ell^2 P_1)\Delta\zeta  + 2( \ell^2 - \zeta^2 ) \Delta U  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(4\ell - \ell^2 P_1)\Delta\zeta  + 4( \ell^2 - \zeta^2 ) \biggl[\frac{(\ell - 1)}{\zeta} - U_1 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
(2 \zeta \ell^2 \Delta\zeta )\Rho_2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2(4\ell - \ell^2 P_1) \zeta\Delta\zeta  + 8( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 ) \, . 
</math>
  </td>
</tr>
</table>
</div>

Finally, using this to replace <math>~\Rho_2</math> in the second ODE expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
2(4\ell - \ell^2 P_1) \zeta\Delta\zeta  + 8( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 ) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ell [ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\pm \ell \biggl\{ 
[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2 
+ 4\ell \zeta \Delta\zeta[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ] 
\biggr\}^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{8}{\ell} ( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 ) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-4 \zeta \Delta\zeta   - 4( \ell^2 - \zeta^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\pm \biggl\{ 
\biggl[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) \biggr]^2 
+ 4\ell \zeta \Delta\zeta \biggl[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 \biggr] 
\biggr\}^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{1}{\ell}  ( \ell^2 - \zeta^2 )\biggl[ 2  (\ell - 1 - \zeta U_1 ) + \ell \biggr] + \zeta \Delta\zeta
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pm \frac{1}{4} \biggl\{ 
\biggl[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) \biggr]^2 
+ 4\ell \zeta \Delta\zeta \biggl[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 \biggr] 
\biggr\}^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
( \ell^2 - \zeta^2 )\biggl[ 3\ell - 2 - 2\zeta U_1 \biggr] + \ell \zeta \Delta\zeta
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pm \frac{\ell}{4} \biggl\{ 
\biggl[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) \biggr]^2 
+ 4\ell \zeta \Delta\zeta \biggl[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 \biggr] 
\biggr\}^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
</div>
This doesn't look particularly useful because, after squaring both sides, it is a sixth-order polynomial in <math>~\ell</math>, which generally has no analytic solution.

=See Also=

* [http://adsabs.harvard.edu/abs/1969MNRAS.144..425P M. V. Penston (1969, MNRAS, 144, 425)]: &nbsp; ''Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases''
* [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)]: &nbsp; ''Numerical Calculations of the Dynamics of Collapsing Proto-Star''
* [http://adsabs.harvard.edu/abs/1977ApJ...214..488S F. H. Shu (1977, ApJ, 214, 488-497)]: &nbsp; ''Self-Similar Collapse of Isothermal Spheres and Star Formation''
* [http://adsabs.harvard.edu/abs/1977ApJ...218..834H C. Hunter (1977, ApJ, 218, 834-845)]: &nbsp; ''The Collapse of Unstable Isothermal Spheres''
* [http://adsabs.harvard.edu/abs/1985MNRAS.214….1W A. Whitworth &amp; D. Summers (1985, MNRAS, 214, 1 - 25)]: &nbsp; ''Self-Similar Condensation of Spherically Symmetric Self-Gravitating Isothermal Gas Clouds''


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