Editing ProjectsUnderway/CoreCollapseSupernovae (section)

===Nonrotating, Spherically Symmetric Collapse===

When describing the [[SSC/Dynamics/FreeFall#Free-Fall_Collapse|free-fall (pressure-free) collapse]] from rest of a uniform-density sphere of mass, <math>~M</math>, and initial radius, <math>~r_0</math> &#8212; hence, initial density <math>~\rho_0 = 3M/(4\pi r_0^3)</math> &#8212; it is convenient to use the Lagrangian radial coordinate, <math>~r(t)</math>, which tracks the radius of the sphere at any time, <math>~t</math>.  The relevant equation of motion is (see [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L Lin, Mestel &amp; Shu 1965] or our [[SSC/Dynamics/FreeFall#Uniform-Density_Sphere|accompanying discussion]]),
<div align="center">
<table border="0", cellpadding="5">

<tr>
  <td align="right">
<math>~\frac{d^2r}{dt^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM}{r^2} = - \frac{G}{r^2} \biggl[ \frac{4\pi}{3} ~r_0^3 \rho_0 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{1}{r_0} \frac{d^2r}{dt^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{4\pi G \rho_0}{3}  \biggl( \frac{r_0}{r} \biggr)^2 \, ,</math>
  </td>
</tr>

</table>
</div>
which integrates to give,
<div align="center">
<math>~r = r_0 \cos^2 \zeta \, ,</math>
</div>
where,
<div align="center">
<table border="0", cellpadding="5">

<tr>
  <td align="right">
<math>~\frac{2}{\pi} \biggl[ \zeta + \frac{1}{2} \sin(2\zeta) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{t}{\tau_\mathrm{ff}} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tau_\mathrm{ff}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[  \frac{3\pi}{32 G \rho_0} \biggr]^{1/2} \, .</math>
  </td>
</tr>

</table>
</div>
(Detailed steps through this derivation, as well as an in-depth discussion of the free-fall collapse problem, are provided [[SSC/Dynamics/FreeFall#Free-Fall_Collapse|in an accompanying chapter of this H_Book]].)  A set of dimensionless expressions drawn from this analytic solution that will prove useful to our discussion of gravitational-wave signals from core-collapse supernovae is provided in the following table.  All lengths have been normalized to <math>~r_0</math>, and times have been normalized to the free-fall time, <math>~\tau_\mathrm{ff}</math>.  In the lower half of the table, these analytic functions have been evaluated at six different times during the free-fall collapse.

<div align="center" id="FreeFallSpherical">
<table border="1" cellpadding="5" width="75%">
<tr>
  <th align="center" colspan="7">
Analytic Solution to Spherical Free-Fall Collapse
  </th>
</tr>
<tr><td align="center" colspan="7">

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

<tr>
  <td align="right">
<math>~x(\zeta) \equiv \frac{r}{r_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos^2 \zeta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x'(\zeta) \equiv \biggl( \frac{\tau_\mathrm{ff}}{r_0} \biggr) \frac{dr}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{\pi}{2} \tan \zeta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x''(\zeta) \equiv \biggl(\frac{\tau^2_\mathrm{ff}}{r_0} \biggr) \frac{d^2r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{\pi^2}{8} \cdot \frac{1}{\cos^4 \zeta} = ~-~\frac{\pi^2}{8} \cdot \frac{1}{x^2}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{2} \ddot{I} \equiv \frac{1}{2} \biggl(\frac{\tau_\mathrm{ff}}{r_0} \biggr)^2 \frac{d^2(r^2)}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+~\frac{\pi^2}{8} \cdot (\tan^2\zeta - 1) \, .</math>
  </td>
</tr>
</table>
</td></tr>
<tr> 
  <td align="center">
<math>~\zeta</math>
  </td>
  <td align="center">
<math>~\frac{t}{\tau_\mathrm{ff}}</math>
  </td>
  <td align="center">
<math>~x</math>
  </td>
  <td align="center">
<math>~x'</math>
  </td>
  <td align="center">
<math>~x''</math>
  </td>
  <td align="center">
<math>~\frac{1}{2}\ddot{I}</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~0</math>
  </td>
  <td align="center">
<math>~0.0</math>
  </td>
  <td align="center">
<math>~1.0</math>
  </td>
  <td align="center">
<math>~0.0</math>
  </td>
  <td align="center">
<math>~-1.2337</math>
  </td>
  <td align="center">
<math>~-1.2337</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~\frac{\pi}{8}</math>
  </td>
  <td align="center">
<math>~0.47508</math>
  </td>
  <td align="center">
<math>~0.85355</math>
  </td>
  <td align="center">
<math>~-0.65065</math>
  </td>
  <td align="center">
<math>~-1.69336</math>
  </td>
  <td align="center">
<math>~-1.02203</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~\frac{\pi}{6}</math>
  </td>
  <td align="center">
<math>~0.60900</math>
  </td>
  <td align="center">
<math>~0.75000</math>
  </td>
  <td align="center">
<math>~-0.90690</math>
  </td>
  <td align="center">
<math>~-2.19325</math>
  </td>
  <td align="center">
<math>~-0.82247</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~\frac{\pi}{4}</math>
  </td>
  <td align="center">
<math>~0.81831</math>
  </td>
  <td align="center">
<math>~0.50000</math>
  </td>
  <td align="center">
<math>~-1.57080</math>
  </td>
  <td align="center">
<math>~-4.93480</math>
  </td>
  <td align="center">
<math>~0.0000</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~\frac{\pi}{3}</math>
  </td>
  <td align="center">
<math>~0.94233</math>
  </td>
  <td align="center">
<math>~0.25000</math>
  </td>
  <td align="center">
<math>~-2.72070</math>
  </td>
  <td align="center">
<math>~-19.73921</math>
  </td>
  <td align="center">
<math>~+2.46740</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~\frac{\pi}{2} - 0.1</math>
  </td>
  <td align="center">
<math>~0.99958</math>
  </td>
  <td align="center">
<math>~0.00997</math>
  </td>
  <td align="center">
<math>~-15.6556</math>
  </td>
  <td align="center">
<math>~-1.24196\times 10^4</math>
  </td>
  <td align="center">
<math>~+121.315</math>
  </td>
</tr>
</table>
</div>

Note that the second time-derivative of the moment of inertia, <math>~\ddot{I}</math>, goes through zero when <math>~\zeta = \pi/4</math>, that is, at time,
<div align="center">
<math>~\frac{t}{\tau_\mathrm{ff}} = \frac{2}{\pi}\biggl[ \zeta + \frac{1}{2}\sin(2\zeta) \biggr]
= \biggl( \frac{1}{2} + \frac{1}{\pi} \biggr) \, .</math>
</div>

<!--       OPTIONAL INTRODUCTION OF POWER SPECTRUM CONCEPT
Because observers tend to think in terms of the power spectrum of a source's signal, it is worth pointing out that for the free-fall collapse the Fourier transform of the strain &#8212; that is, of the function <math>~f(t) = \ddot{I}</math> &#8212; is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\hat{f}(\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_{-\infty}^{\infty} f(t) e^{-2\pi i t \xi} dt \, .</math>
  </td>
</tr>
</table>
</div>
Then, because,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2} \sin(2\zeta) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~ dt</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ 1 + \cos(2\zeta)\biggr]d\zeta \, ,</math>
  </td>
</tr>
</table>
</div>
we may write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{f}(\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\pi^2}{4} \biggl( \frac{2\tau_\mathrm{ff}}{\pi} \biggr) \int_{-\infty}^{\infty} 
\biggl[ \tan^2\zeta - 1 \biggr] \biggl[1 + \cos(2\zeta)  \biggr] 
\exp\biggl\{  -2\pi i \xi \cdot \frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta) \biggr] \biggr\} d\zeta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\pi \tau_\mathrm{ff}}{2}  \int_{-\infty}^{\infty} 
\biggl[ \tan^2\zeta - 1 \biggr] \biggl[1 + \cos(2\zeta)  \biggr] 
\exp\biggl\{  -i \xi \cdot 4\tau_\mathrm{ff} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta) \biggr] \biggr\} d\zeta 
\, .</math>
  </td>
</tr>
</table>
</div>

END OF OPTIONAL DISCUSSION         -->