Editing SSC/Dynamics/FreeFall (section)

====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,
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<math>~f = \sin\zeta_i \, .</math>
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Similarly, an inspection of the two, separately derived, expressions for time suggests that,
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<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>
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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,
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<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>
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we can rewrite the ODE as,
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<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,
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<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>
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Hence, the left-hand side of the rewritten ODE becomes,
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<table border="0" cellpadding="5" align="center">

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  <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:

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<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>
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