Editing SSC/Dynamics/FreeFall (section)

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

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  <td align="right">
<math>~A_{0,i}  </math>
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  <td align="center">
<math>~=</math>
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  <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,
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<math>~\biggl(\frac{\pi}{2}\biggr)^2  \biggl[ \frac{\tanh \chi^3}{\chi^3} \biggr]</math>
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<math>~=</math>
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  <td align="left">
<math>~\biggl[ \biggl(\frac{2}{\pi}\biggr) \frac{t}{\tau_{ffc}} \biggl]^{-2}</math>
  </td>
</tr>

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  <td align="right">
<math>~\Rightarrow ~~~ 0</math>
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  <td align="center">
<math>~=</math>
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  <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.