Editing SSC/Dynamics/FreeFall (section)

===Example Mass Profiles Examined by Coughlin (2017)===

====Approach====

Via an independent analysis, [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] has also examined the free-fall collapse of initially (mildly) centrally condensed, spherically symmetric configurations.  When viewed from a Lagrangian frame of reference &#8212; see the paragraph accompanying his equation (16) &#8212; he concludes that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{r_i(t)}{r_{0,i}}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - f^2(\xi_i) \, ,</math>
  </td>
</tr>
</table>
</div>
where, in his derivation, time is referenced parametrically through <math>~\xi_i</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~t \biggl[ \frac{2GM_{0,i}}{r^3_i(t)}\biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
</div>

While Coughlin does not provide an explicit expression for the function, <math>~f(\xi_i)</math>, he shows &#8212; see his equation (11) &#8212; that it must satisfy the following nonlinear differential equation:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{df}{d\xi_i}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1 - f^2}{2 + 3\xi_i f} \, .
</math>
  </td>
</tr>
</table>
</div>

====Relationship to Tohline's Approach====
A comparison between the expression for <math>~r_i(t)/r_{0,i}</math> that appears in [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] derivation and the one published by [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)] suggests that,
<div align="center">
<math>~f = \sin\zeta_i \, .</math>
</div>
Similarly, an inspection of the two, separately derived, expressions for time suggests that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_i = t \biggl[ \frac{2GM_{0,i} }{r^3_i(t)} \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~t \biggl[ \frac{8\pi G \rho_c}{3}\biggr]^{1 / 2} \biggl[ \frac{3M_{0,i}}{4\pi \rho_c r^3_{0,i}}\biggr]^{1 / 2} \biggl[ \frac{r_{0,i}}{r_i(t)}\biggr]^{3 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~t \biggl[ \frac{A_{0,i} }{\tau_{ffc}}\biggr] \biggl[ \frac{1}{\cos^3\zeta_i}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \zeta_i + \frac{1}{2} \sin(2\zeta_i) \biggr] \biggl[ \frac{1}{\cos^3\zeta_i}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Let's see whether these explicit expressions for <math>~f(\zeta_i)</math> and <math>~\xi_i(\zeta_i)</math> satisfy Coughlin's nonlinear ODE.  


Given that,
<div align="center">
<math>~\frac{df}{d\zeta_i} = \frac{d\xi_i}{d\zeta_i} \cdot \frac{df}{d\xi_i} ~~~\Rightarrow ~~~ \biggl[ \frac{df}{d\xi_i} \biggr]^{-1} =  \biggl[ \frac{df}{d\zeta_i} \biggr]^{-1} \frac{d\xi_i}{d\zeta_i}  \, ,</math>
</div>
we can rewrite the ODE as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
(1-f^2)\biggl[ \frac{df}{d\zeta_i} \biggr]^{-1} \frac{d\xi_i}{d\zeta_i}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 + 3\xi_i f \, .
</math>
  </td>
</tr>
</table>
</div>

After adopting our proposed parameter mapping, we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\xi_i}{d\zeta_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\sin\zeta_i }{\cos^4\zeta_i}\biggl[ \zeta_i + \frac{1}{2} \sin(2\zeta_i) \biggr] 
+ \frac{1}{\cos^3\zeta_i} \biggl[1 + \cos(2\zeta_i) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\sin\zeta_i }{\cos^4\zeta_i}\biggl[ \zeta_i + \sin\zeta_i \cos\zeta_i \biggr] 
+ \frac{2}{\cos\zeta_i} \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the left-hand side of the rewritten ODE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos^2 \zeta_i \biggl[\cos\zeta_i \biggr]^{-1} \biggl\{
\frac{2}{\cos\zeta_i}  +\frac{3\sin\zeta_i }{\cos^4\zeta_i}\biggl[ \zeta_i + \sin\zeta_i \cos\zeta_i \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2  +\frac{3\sin\zeta_i }{\cos^3\zeta_i}\biggl[ \zeta_i + \sin\zeta_i \cos\zeta_i \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

It is easy to see that, after adopting the prescribed parameter mapping, the right-hand side of this ODE presents exactly the same expression.  We conclude, therefore, that the pair of functions, <math>~f</math> and <math>~\xi_i</math>, introduced by [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] may be straightforwardly expressed in terms of [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline's (1982)] parameter, <math>~\zeta_i</math>, as follows:

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

<tr>
  <td align="right">
<math>~f</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin\zeta_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \zeta_i + \frac{1}{2} \sin(2\zeta_i) \biggr] \biggl[ \frac{1}{\cos^3\zeta_i}\biggr] </math>
  </td>
</tr>
</table>

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

====Models A, B, &amp; D, with Focus on Case A====
By way of illustration, [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] examined the pressure-free collapse of configurations having the following three initial mass profiles (see his equations 28, 29, &amp; 30):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_{0,A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_\mathrm{tot} \tanh \biggl(\frac{r_{0,i}}{R_C}\biggr)^3 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M_{0,B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_\mathrm{tot} \biggl[ 1 - e^{-(r_{0,i}/R_C )^3} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M_{0,D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_\mathrm{tot} \biggl(\frac{2}{\pi}\biggr) \tan^{-1} \biggl[ \frac{\pi}{2} \biggl(\frac{r_{0,i}}{R_C}\biggr)^3\biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where, in all three cases, "<math>~R_C</math> should be interpreted as a scale length over which the density decreases appreciably."  Adopting the shorthand notation, 
<div align="center">
<math>~\chi \equiv \frac{r_{0,i}}{R_C} \, ,</math>&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~a \equiv \frac{3M_\mathrm{tot}}{4\pi\rho_c R^3_C} \, ,</math>
</div>

we have deduced that the mathematical expressions that define the corresponding initial density profiles, <math>~\rho_{0,i}</math>, are as recorded in the left-hand portion of composite Figure 2.  The right-hand portion of this composite figure presents a plot of <math>~\rho_{0,i}/\rho_c</math> versus <math>~0 \le \chi \le 2</math> for all three of these initial models, in addition to a piecewise constant description of a uniform-density sphere of radius, <math>~R_C</math>; this diagram has been constructed to replicate Figure 2 from [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)].

<!-- CASE "A" Density
<div align="center">
<table border="1" align="center" cellpadding="8">
<tr><th>Case "A" from [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] </th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\frac{\rho_{0,i}}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{\cosh^2(\chi^3)} 
</math>
  </td>
</tr>
</table>
</div>

</td></tr></table>
</div>
END CASE "A" Density -->

<!-- DERIVE MASS PROFILE FOR CASE "A"
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_{0,i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \int_0^{r_{0,i}} r^2 \rho_{0,i}(r) dr</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R_C^3 \rho_c \int_0^{\chi} \chi^2 \biggl[ \frac{\rho_{0,i}(\chi)}{\rho_c}\biggr] d\chi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R_C^3 a\rho_c \int_0^{\chi} \frac{\chi^2}{\cosh^2(\chi^3)} d\chi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi R_C^3 a\rho_c}{3} \biggl[ \tanh(\chi^3)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
a
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3M_\mathrm{tot}}{4\pi \rho_c R_C^3}</math>
  </td>
</tr>
</table>
</div>

END CASE "A" MASS DERIVATION -->
<!-- CASE "B" DENSITY
<div align="center">
<table border="1" align="center" cellpadding="8">
<tr><th>Case "B" from [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] </th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\frac{\rho_{0,i}}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a e^{-\chi^3}
</math>
  </td>
</tr>
</table>
</div>

</td></tr></table>
</div>
END CASE "B" Density -->
<!-- DERIVE MASS PROFILE FOR CASE "B"
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_{0,i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \int_0^{r_{0,i}} r^2 \rho_{0,i}(r) dr</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R_C^3 \rho_c \int_0^{\chi} \chi^2 \biggl[ \frac{\rho_{0,i}(\chi)}{\rho_c} \biggr] d\chi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R_C^3 a\rho_c \int_0^{\chi}  \chi^2 e^{-\chi^3} d\chi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R_C^3 a\rho_c \biggl[ C_0 - \frac{1}{3} e^{-\chi^3} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi R_C^3 a\rho_c}{3} \biggl[ 1 -  e^{-\chi^3} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
a
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3M_\mathrm{tot}}{4\pi \rho_c R_C^3}</math>
  </td>
</tr>
</table>
</div>

END CASE "B" MASS DERIVATION -->
<!-- CASE "C" Density
<div align="center">
<table border="1" align="center" cellpadding="8">
<tr><th>Case "D" from [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] </th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\frac{\rho_{0,i}}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a \biggl[ 1+ \biggl(\frac{\pi}{2} \chi^3\biggr)^2\biggr]^{-1} 
</math>
  </td>
</tr>
</table>
</div>

</td></tr></table>
</div>
END CASE "C" Density -->
<!-- DERIVE MASS PROFILE FOR CASE "D"
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_{0,i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \int_0^{r_{0,i}} r^2 \rho_{0,i}(r) dr</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R_C^3 \rho_c \int_0^{\chi} \chi^2 \biggl[ \frac{ \rho_{0,i}(\chi) }{\rho_c}\biggr] d\chi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R_C^3 a\rho_c \int_0^{\chi} \chi^2 \biggl[ 1+ \biggl(\frac{\pi}{2} \chi^3\biggr)^2\biggr]^{-1}d\chi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi R_C^3 a\rho_c}{3} \biggl[ \biggl(\frac{2}{\pi}\biggr) \tan^{-1}\biggl( \frac{\pi \chi^3}{2} \biggr)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
a
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3M_\mathrm{tot}}{4\pi \rho_c R_C^3}</math>
  </td>
</tr>
</table>
</div>

END CASE "D" MASS DERIVATION -->

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="3">Composite Figure 2</th>
</tr>
<tr>
  <th align="center" colspan="2">
[http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] Example Density Profiles<sup>&dagger;</sup>
  </th>
  <th align="center" colspan="1">
Plot Constructed to Replicate Figure 2 of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)]
  </th>
</tr>
<tr>
  <td align="center" rowspan="1">
&nbsp; &nbsp; '''Case A''' &nbsp; &nbsp; 
  </td>
  <td align="center" rowspan="1">
<math>~\frac{\rho_{0,i}}{\rho_c} = \frac{a}{\cosh^2(\chi^3)} </math>
  </td>
  <td align="center" rowspan="3">
[[File:CoughlinFig2.png|450px|center|Figure 2 from Coughlin (2017]]
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
&nbsp; &nbsp; '''Case B''' &nbsp; &nbsp; 
  </td>
  <td align="center" rowspan="1">
<math>~\frac{\rho_{0,i}}{\rho_c} = a e^{-\chi^3} </math>
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
&nbsp; &nbsp; '''Case D''' &nbsp; &nbsp; 
  </td>
  <td align="center" rowspan="1">
<math>~\frac{\rho_{0,i}}{\rho_c} = a \biggl[ 1+ \biggl(\frac{\pi}{2} \chi^3\biggr)^2\biggr]^{-1} </math>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
<sup>&dagger;</sup>As stated in the paragraph that follows his equation (27), Coughlin set the leading coefficient, <math>~a = 1</math>, in all three cases.
  </td>
</tr>
</table>
</div>

We have deduced, as well, that the corresponding functional forms of the coefficient, <math>~A_{0,i}</math>, are:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A^2_{0,i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\pi M_{0,i}}{16 \rho_c r_{0,i}^3 } = a \biggl(\frac{\pi}{2}\biggr)^2 \chi^{-3} \biggl[ \frac{M_{0,i}}{M_\mathrm{tot}} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
that is, for the three example cases,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
&nbsp; &nbsp; '''Case A:''' &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~A^2_{0,A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a \biggl(\frac{\pi}{2}\biggr)^2  \biggl[ \frac{\tanh \chi^3}{\chi^3} \biggr] 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~\frac{d\ln A_{0,A}}{d\ln \chi} = 
- \frac{3}{2}\biggl[ 1 - \frac{\chi^3}{ \sinh (\chi^3) \cdot \cosh (\chi^3)} \biggr]  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; &nbsp; '''Case B:''' &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~A^2_{0,B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a \biggl(\frac{\pi}{2}\biggr)^2 \chi^{-3} \biggl[ 1 - e^{-\chi^3} \biggr] 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~\frac{d\ln A_{0,B}}{d\ln \chi} = 
-\frac{3}{2} \biggl[ 1 - \chi^3 e^{-\chi^3}(1 - e^{-\chi^3})^{-1} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; &nbsp; '''Case D:''' &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~A^2_{0,D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a \biggl(\frac{\pi}{2}\biggr) \chi^{-3}  \tan^{-1} \biggl[ \frac{\pi \chi^3}{2} \biggr] 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~\frac{d\ln A_{0,D}}{d\ln \chi} = 
-\frac{3}{2}\biggl\{ 1 - \biggr(\frac{\pi \chi^3}{2}\biggr) \biggl[1 + \biggl(\frac{\pi \chi^3}{2}\biggr)^2  \biggr]^{-1} \biggl[\tan^{-1}\biggl( \frac{\pi \chi^3}{2}\biggr) \biggr]^{-1} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Plugging these functional expressions into the [[#KeyExpressions|above-defined set of Lagrangian evolution equations]] and following the [[#ModelingSteps|above-defined set of modeling steps]], we have modeled the early free-fall collapse of these three models.  Guided  by [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] presentation, we will focus our discussion on the evolution of the model whose initial density profile has been labeled, "Case A."  

The right-hand panel of our composite Figure 3 shows how the radial density profile changes over time during the initial free-fall collapse of Coughlin's "Case A" model.  This panel also has been included in [[#Figure1|our composite Figure 1, above]] to facilitate comparison with similar segments of the model evolutions presented by [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)].  The left-hand panel of our composite Figure 3 presents a plot that is largely intended to replicate the left-most panel in [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] Figure 3.   It presents a plot of the model's mass profile &#8212; normalized to the total mass &#8212; at time <math>~t = 0</math> (dashed black curve) and at the following five evolutionary times: 
<div align="center">
<math>~\biggl(\frac{\pi}{2} \biggr) \frac{t}{\tau_{ffc}} = 0.3\pi, 0.4\pi, 0.5\pi, 0.6\pi, 0.7\pi \, .</math>
</div>


<table align="center" cellpadding="8" border="1">
<tr>
  <th align="center" colspan="2">
Composite Figure 3: &nbsp; Time-Evolution of "Case A" from [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)]
  </th>
</tr>
<tr>
<td align="center" colspan="1">[[File:CoughlinFig3a.png|510px|center|Figure 3a from Coughlin (2017]]</td>
<td align="center" colspan="1">[[File:FreeFallRhoCaseA3.png|290px|center|Figure 3c from Coughlin (2017]]</td>
</tr>
<tr>
  <td align="left" colspan="2" width="805px">
''Left:'' &nbsp; Radial mass profile initially (dashed black curve) and at five additional evolution times, as annotated; this plot has been constructed to replicate the left-most panel in Figure 3 of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)]. &nbsp; ''Right:''&nbsp; Semi-log plot of the radial density profile initially (dashed black curve) and at three additional evolution times, as annotated.  This same plot is displayed in [[#Figure1|composite Figure 1, above]], to aid in comparison with models presented by [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)]; it also can appropriately be compared with the middle panel from Figure 3 of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)], which presents three of the same density profiles, but on a log-log plot.
  </td>
</tr>
</table>

These time-evolving mass profiles &#8212; interspersed among others from our analysis of Coughlin's "Case A" evolution &#8212; have also been displayed as an animation sequence in our composite Figure 4.  Actually, they appear as the top segment of two separate animations &#8212; one on the left, the other on the right of the figure &#8212; along with the time-evolving density profile (middle segment) and the time-evolving velocity profile (bottom segment).  In these animated diagrams, the density is scaled to the model's ''intial'' central density, <math>~\rho_c</math>, and, following [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] &#8212; see one of the paragraphs following shortly after his equation (27) &#8212; the velocity is everywhere, and at all times, normalized to,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_C</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2GM_C}{R_C} \biggr]^{1 / 2} = a\biggl(\frac{\pi}{2}\biggr) \frac{R_C}{\tau_{ffc}} \, .
</math>
  </td>
</tr>
</table>
</div>

These additional "density profile" and "velocity profile" panels present evolutionary information that, for the most part, may be extracted, respectively, from the middle panel and the right-most panel in [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] Figure 3; in order to accommodate large dynamic ranges, the density profiles and velocity profiles are presented as log-log plots.

<table align="center" cellpadding="8" border="1">
<tr>
  <th align="center" colspan="2">Composite Figure 4</th>
</tr>
<tr>
  <th align="center" colspan="2">
Animation of ''Case A'' Evolution: &nbsp; Radial profiles of (top) mass, (middle) density &amp; (bottom) velocity
  </th>
</tr>
<tr>
<td align="center">[[File:Coughlin2017FreeFallMovie.gif|350px|center|Animation of Figure 3 from Coughlin (2017]]</td>
<td align="center">[[File:Coughlin2017NoLoopMovie.gif|350px|center|Animation of Figure 3 from Coughlin (2017]]</td>
</tr>
<tr>
  <td align="left" width="710px" colspan="2">
In both animation sequences, the red circular marker identifies the radial location of the Lagrangian mass shell that was initially located at <math>~\chi = 1</math>; in the animation on the right the red marker disappears after this mass shell has fallen into, and become part of the central core.  After one initial ''central'' free-fall (blue curves in animation on the right), a black square marker appears on the vertical axis in the top panel to identify the mass of the central, point-mass core.
  </td>
</tr>
</table>

The animation that appears in the left-hand column of our composite Figure 4 loops back and forth between the initial state, <math>~t = 0 \tau_{ffc}</math>, and (nearly) one ''central'' free-fall time, <math>~t = 0.9999\tau_{ffc}</math>.  It provides a focus on the behavior of the free-fall collapse and, in particular, illustrates the run-away development of a centrally condensed structure, as [[#Example_Density_Profiles_Examined_by_Tohline_.281982.29|highlighted earlier in the model evolutions presented by Tohline]].  Each loop of the animation that appears in the right-hand column of composite Figure 4 also proceeds from the initial state to one initial free-fall time (drawn as solid black curve segments), but then it continues to display (as solid blue curve segments) evolution ''through'' the first free-fall time and formation of a point-mass core, and &#8212; see the discussion that follows &#8212; into a phase where continued accretion of matter onto the core results in a rapid increase of the core's mass.

====Accretion Onto a Point-Mass Core====

The mathematical model that has been used, above, to describe the development of nonlinear structure in configurations that undergo free-fall collapse must break down as <math>~t \rightarrow \tau_{ffc}</math> because the volume enclosing the central-most mass shell shrinks to zero &#8212; concomitantly, the central density formally climbs to infinity &#8212; while the inward-directed radial velocity remains nonzero.  At the very least, in the central region of the collapsing configuration it is physically unreasonable to ignore the effects of pressure (and/or general relativity) as <math>~t \rightarrow \tau_{ffc}</math>.  

[[File:CommentButton02.png|right|100px|Note from J. E. Tohline:  Throughout his article, Coughlin (2017) actually asserts that the time of singularity formation is set by the initial ''average'' density of the collapsing cloud, rather than by the cloud's initial ''central'' density. We disagree with this assertion, as the mathematical model establishes the ''central'' free-fall time as the point in time at which the singularity forms.]]As [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] has suggested, however, it may not be unreasonable to consider that <math>~\tau_{ffc}</math> marks the point in time when a central point mass core forms and to continue to use this mathematical model to represent the behavior of the "envelope" material as it continues to free-fall toward the core.  We acknowledge that this point of view can provide additional insight especially if the extended examination is confined to regions well outside the core.  It is with this in mind that the animation sequence on the right-hand side of our composite Figure 4 has been extended to times, <math>~(\pi/2)t/\tau_{ffc} > 0.5\pi</math>; at these later times, the accretion envelope profiles are depicted by solid blue curve segments.  

As the evolution proceeds past the initial free-fall time, the radius of a larger and larger number of mass shells will shrink to zero and, under the scenario just outlined, must be considered part of the central point-mass core.  Presumably, as Coughlin has illustrated, the above-developed free-fall model can continue to be used to estimate how rapidly the core mass grows.  Suppose that, at times <math>~t > \tau_{ffc}</math>, we want to determine the mass of the central core.  This reduces to determining at what time <math>~\zeta_i \rightarrow \tfrac{\pi}{2}</math> for each mass shell; or, for a given time, <math>~t/\tau_{ffc}</math>, determining ''for which mass shell'', 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A_{0,i}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl(\frac{2}{\pi}\biggr) \frac{t}{\tau_{ffc}} \biggl]^{-1} \, .</math>
  </td>
</tr>
</table>
</div>

For "Case A", for example, after setting the coefficient, <math>~a=1</math>, this is equivalent to determining the root, <math>~\chi</math>, of the equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{\pi}{2}\biggr)^2  \biggl[ \frac{\tanh \chi^3}{\chi^3} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl(\frac{2}{\pi}\biggr) \frac{t}{\tau_{ffc}} \biggl]^{-2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{t}{\tau_{ffc}} \biggl]^{2} \tanh \chi^3 - \chi^3 \, .</math>
  </td>
</tr>
</table>
</div>

The core masses that have been determined in this fashion at times, <math>~t/\tau_{ffc} > 1</math>, have been marked by the small, black squares along the vertical axis in the the left panel of our composite Figure 3 and in the top panel of the animation sequence that appears on the right-hand side of our composite Figure 4.