Editing SSC/Dynamics/FreeFall (section)

====Falling from rest at a finite distance &hellip;====
In this case, we set <math>~v_i = 0</math> in the definition of <math>~k</math>, so,
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<math>~\frac{dr}{dt}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~- ~\biggl[\frac{2GM}{r} - \frac{2GM}{r_i}\biggr]^{1/2} =
\biggl(\frac{2GM}{r_i}\biggr)^{1/2} \biggl[\frac{r_i}{r}-1  \biggr]^{1/2}
\, , </math>
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and the relevant expression to be integrated is,
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<math>~dt </math>
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<math>~=</math>
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<math>~ - \biggl(\frac{2GM}{r_i} \biggr)^{-1/2} \biggl[ \biggl( \frac{r_i}{r} \biggr) - 1 \biggr]^{-1/2} dr  \, .</math>
  </td>
</tr>
</table>
</div>

Following [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L LMS65], we see that this equation can be straightforwardly integrated by first making the substitution,
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<math>~\cos^2\zeta \equiv \frac{r}{r_i}  \, ,</math>
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which also means,
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<math>~dr = - 2r_i \sin\zeta \cos\zeta d\zeta \, .</math>
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The relevant integral is, therefore,
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<math>~\int_0^t dt </math>
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<math>~=</math>
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<math>~+ \biggl(\frac{2r_i^3}{GM} \biggr)^{1/2} \int_0^\zeta \cos^2\zeta d\zeta \, ,</math>
  </td>
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</table>
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where the limits of integration have been set to ensure that <math>~r/r_i = 1</math> at time <math>~t=0</math>.  After integration, we have,
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<math>~ t </math>
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<math>~=</math>
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<math>~ \biggl(\frac{2r_i^3}{GM} \biggr)^{1/2} \biggl[ \frac{\zeta}{2} + \frac{1}{4}\sin(2\zeta) \biggr]  \, .</math>
  </td>
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</table>
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The physically relevant portion of this formally periodic solution is the interval in time from when <math>~r/r_i = 1 ~ (\zeta = 0)</math> to when <math>~r/r_i \rightarrow 0</math> for the first time <math>~(\zeta = \pi/2)</math>.  The particle's free-fall comes to an end at the time associated with <math>~\zeta = \pi/2</math>, that is, at the so-called "free-fall time,"
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<math>~\tau_\mathrm{ff} </math>
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<math>~\equiv</math>
  </td>
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<math>~ \biggl(\frac{2r_i^3}{GM} \biggr)^{1/2} \biggl[ \frac{\zeta}{2} + \frac{1}{4}\sin(2\zeta) \biggr]_{\zeta=\pi/2}  
= \biggl(\frac{\pi^2 r_i^3}{8GM} \biggr)^{1/2} \, .</math>
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<span id="Parametric">In summary,</span> then, the solution, <math>~r(t)</math>, to this simplified but dynamically relevant problem is provided by the following pair of analytically prescribable parametric relations (see also equations 2 &amp; 3 from LMS65, [[#Free-Fall_Collapse|reprinted above]]):

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<tr><th align="center">
Parametric &nbsp;<math>~r(t)</math>&nbsp; Solution
</th></tr>
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<math>~ \frac{r}{r_i} </math>
  </td>
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<math>~=</math>
  </td>
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<math>~ \cos^2\zeta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \frac{t}{\tau_\mathrm{ff}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{\pi} \biggl[ \zeta + \frac{1}{2} \sin(2\zeta) \biggr] </math>
  </td>
</tr>
</table>
</td></tr>
</table>
We note, as well, that the radially directed velocity is,
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<math>~v_r = \frac{dr}{dt} </math>
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<math>~=</math>
  </td>
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<math>~ - \biggl(\frac{2GM}{r_i} \biggr)^{1/2} \biggl[ \frac{1}{\cos^2\zeta}  - 1 \biggr]^{1/2} </math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~ - \biggl(\frac{2GM}{r_i} \biggr)^{1/2} \tan\zeta \, , </math>
  </td>
</tr>
</table>
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which formally becomes infinite in magnitude when <math>~\zeta \rightarrow \pi/2</math>, that is, when <math>~t \rightarrow \tau_\mathrm{ff}</math>.

It is worth demonstrating that the parametric solution derived here is identical to the solution published by [http://adsabs.harvard.edu/abs/1934QJMat...5...73M McCrea &amp; Milne (1934)] &#8212;  [[#McCreaMilne1934Solution|reprinted above]] &#8212; for the case, <math>~k > 0</math>, namely,
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<math>~t</math>
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<math>~=</math>
  </td>
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<math>~- \frac{\theta^{1/2} (\alpha - A' \theta)^{1/2}}{A'} + \frac{\alpha}{(A')^{3/2}} \sin^{-1} 
\biggl( \frac{A' \theta}{\alpha} \biggr)^{1/2} </math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~\frac{\alpha}{(A')^{3/2}} \biggl[ - \biggl(\frac{A' \theta}{\alpha} \biggr)^{1/2} \biggl(1 - \frac{A' \theta}{\alpha} \biggr)^{1/2} + \sin^{-1} 
\biggl( \frac{A' \theta}{\alpha} \biggr)^{1/2} \biggr] \, .</math>
  </td>
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</table>
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After reversing the substitutions detailed above, that is, after setting,
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<math>~\theta \rightarrow r \, ,</math> 
&nbsp; &nbsp; &nbsp; 
<math>~ \alpha \rightarrow 2GM \, ,</math> 
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
<math>~A' \rightarrow k   \, ,</math> 
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and remembering that, for this particular model example, we have set <math>~v_i = 0 ~\Rightarrow ~ k = 2GM/r_i</math>, the key dimensionless ratio in the McCrea &amp; Milne expression becomes,
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<math>~\biggl(\frac{A' \theta}{\alpha} \biggr)</math>
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<math>~=</math>
  </td>
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<math>~\frac{r}{r_i} = \cos^2\zeta \, ,</math>
  </td>
</tr>
</table>
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and the pre-factor on the righthand side becomes,
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<math>~\biggl[\frac{\alpha^2}{(A')^3} \biggr]^{1/2}</math>
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<math>~=</math>
  </td>
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<math>~\biggl[\frac{r_i^3}{2GM} \biggr]^{1/2} = \frac{2}{\pi} \tau_\mathrm{ff} \, .</math>
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</table>
</div>

Hence, the McCrea &amp; Milne solution becomes,
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<math>~\frac{t}{\tau_\mathrm{ff}}</math>
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<math>~=</math>
  </td>
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<math>~\frac{2}{\pi} \biggl[  -\cos\zeta \sin\zeta + \sin^{-1} (\cos\zeta)
\biggr]</math>
  </td>
</tr>

<tr>
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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~\frac{2}{\pi} \biggl[  -\frac{1}{2}\sin(2\zeta) + \biggl(\frac{\pi}{2} - \zeta \biggr)
\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 1 - \frac{t}{\tau_\mathrm{ff}}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~\frac{2}{\pi} \biggl[\zeta  +\frac{1}{2}\sin(2\zeta) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
We see that, by shifting the defined zero-point in time in this last expression such that <math>~t \rightarrow (\tau_\mathrm{ff} - t)</math>, which also reverses the ''sign'' on time, we have exact agreement between the solution that we have derived &#8212; designed to match the one published by {{ LMS65 }} &#8212; and the result for <math>k > 0</math> that was published by McCrea &amp; Milne in 1934.