Editing SSC/Dynamics/FreeFall (section)

====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.