Editing SSC/Dynamics/FreeFall (section)

===Example Density Profiles Examined by Tohline (1982)===

By way of illustration, [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)] adopted several density profiles parameterized by the integer index, <math>~m</math>, of the form,

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<math>~
\frac{\rho_{0,i}}{\rho_c}
</math>
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<math>~=</math>
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<math>~1 - \biggl(\frac{r_{0,i}}{R}\biggr)^m \, .</math>
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In this case, we have, 
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<math>~M_{0,i}</math>
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<math>~=</math>
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<math>~4\pi R^3 \rho_c \int_0^{r_{0,i}} \biggl( \frac{r}{R}\biggr)^2 \biggl[1 - \biggl(\frac{r}{R}\biggr)^m \biggr] \frac{dr}{R} </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{4\pi R^3 \rho_c}{3}  \biggl( \frac{r_{0,i}}{R}\biggr)^3 \biggl[1 - \frac{3}{m+3} \biggl(\frac{r_{0,i}}{R}\biggr)^{m} \biggr] </math>
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<math>~\Rightarrow~~~
A_{0,i}^2
</math>
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<math>~=</math>
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<math>~
\biggl[ \frac{3\pi }{16 \rho_c r_{0,i}^3 } \biggr]
\frac{4\pi R^3 \rho_c}{3}  \biggl( \frac{r_{0,i}}{R}\biggr)^3 \biggl[1 - \frac{3}{m+3} \biggl(\frac{r_{0,i}}{R}\biggr)^{m} \biggr] 
</math>
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&nbsp;
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<math>~=</math>
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<math>~\biggl(\frac{\pi}{2}\biggr)^2
\biggl[1 - \frac{3}{m+3} \biggl(\frac{r_{0,i}}{R}\biggr)^{m} \biggr] 
</math>
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<math>~\Rightarrow~~~
\frac{d\ln A_{0,i}}{d\ln r_{0,i}} = \frac{r_{0,i}}{A_{0,i}} \biggl[ \frac{d A_{0,i}}{d r_{0,i}} \biggr]
</math>
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<math>~=</math>
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<math>~
r_{0,i}\biggl[1 - \frac{3}{m+3} \biggl(\frac{r_{0,i}}{R}\biggr)^{m} \biggr]^{-1 / 2} \frac{d }{d r_{0,i}} \biggl[1 - \frac{3}{m+3} \biggl(\frac{r_{0,i}}{R}\biggr)^{m} \biggr]^{1 / 2}
</math>
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&nbsp;
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<math>~=</math>
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<math>~
\frac{r_{0,i}}{2} \biggl[1 - \frac{3}{m+3} \biggl(\frac{r_{0,i}}{R}\biggr)^{m} \biggr]^{-1 }\biggl[- \frac{3m}{R(m+3)} \biggl(\frac{r_{0,i}}{R}\biggr)^{m-1} \biggr]
</math>
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&nbsp;
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<math>~=</math>
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<math>~
-\frac{3m}{2(m+3)} \biggl[1 - \frac{3}{m+3} \biggl(\frac{r_{0,i}}{R}\biggr)^{m} \biggr]^{-1 }\biggl(\frac{r_{0,i}}{R}\biggr)^{m} \, .
</math>
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Using these functional expressions &#8212; and guided by [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline's (1982)] presentation &#8212; we have modeled the early free-fall collapse of configurations whose initial density profiles are defined by setting the index, m = 2, 5, and 25.  In three of the four panels of the following composite Figure 1 we have used semi-log plots to show how the density profile varies with time for each of these three models at four times:  <math>~t/\tau_{ffc} = 0.0, 0.5, 0.75, 0.99</math>; they replicate the three panels that appear in Figure 3 of Tohline (1982).  [The fourth figure panel similarly displays the evolutionary behavior of the model labeled "Case A" that has subsequently been studied by [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)]; see further discussion, below.]  As the images illustrate, there is inevitably a run-away collapse of the central core, even for configurations that are initially only mildly centrally condensed,  

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  <th align="center" colspan="2">Composite Figure 1
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[[File:FreeFallRhoM2.png|350px|center|Figure 3a from Tohline (1982]]
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[[File:FreeFallRhoM5.png|350px|center|Figure 3b from Tohline (1982]]
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[[File:FreeFallRhoM25.png|350px|center|Figure 3b from Tohline (1982]]
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[[File:FreeFallRhoCaseA2.png|350px|center|Figure 3c from Coughlin (2017]]
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In large measure, the three panels labeled, respectively, m = 2, m = 5, and m = 25, are intended to replicate the three panels that appear in Figure 3 of [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)]. For comparison, the fourth panel displays in a similar fashion the early evolutionary behavior of the model labeled ''Case A'' that has been studied by [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)].
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