Editing ProjectsUnderway/CoreCollapseSupernovae (section)

==Free-Fall Collapse==
===Nod to Lynden-Bell's Early Contributions===
[[Image:LyndenBell1964.png|300px|right|Lynden-Bell (1964, ApJ, 139, 1195)]]I want to begin this section by paying tribute to [http://en.wikipedia.org/wiki/Donald_Lynden-Bell Donald Lynden-Bell] who, in 1962 ([http://dx.doi.org.libezp.lib.lsu.edu/10.1017/S0305004100040767 Mathematical Proceedings of the Cambridge Philosophical Society, vol. 58, pp. 709-711]), was the first to appreciate the relatively simple behavior that should be exhibited by the free-fall collapse of a uniformly rotating, uniform-density spheroid.  In an article less than two pages in length, Lynden-Bell first noted that the governing dynamical equations (written in cylindrical coordinates) take the form,
<div align="center">
<table border="0", cellpadding="5">

<tr>
  <td align="right">
<math>~\ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ 2A_L(t) R + \frac{h_{LB}^2}{R^3} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\ddot{Z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ 2C_L(t) Z \, ,</math>
  </td>
</tr>

</table>
</div>
where the dots denote differentiation with respect to time, <math>~h_{LB}</math> is a constant and the two time-dependent coefficients, <math>~A(t)</math> and <math>~C(t)</math>, come from the gravitational potential that, according to Lyttleton (1953), has the form,
<div align="center">
<table border="0", cellpadding="5">

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ 2A_L(t) R^2 ~-~ C_L(t) Z^2 \, .</math>
  </td>
</tr>

</table>
</div>
Then Lynden-Bell deduced that, (a) "the result of the motion is merely a change of scales"; (b) the collapsing system "remains uniform [in density], and the boundary remains spheroidal"; and (c) "the collapse &hellip; will be through a series of [uniformly rotating] spheroids."  Two years later in a separate article, [http://adsabs.harvard.edu/abs/1964ApJ...139.1195L Lynden-Bell (1964], ApJ, 139, 1195) presented results from the numerical integration of this governing set of dynamical equations (see his Figure 1, reprinted to the right, here).  The various publications by other authors who also have modeled the free-fall collapse of rotating or nonrotating spheroids in various contexts (see our discussion that follows) have not always acknowledged Lynden-Bell's pioneering analysis of this problem.

===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         -->

===Nonrotating, Oblate-Spheroidal Collapse===

The analogous collapse from rest of a nonrotating, uniform-density, oblate spheroid with equatorial radius, <math>~\varpi_\mathrm{eq}(t)</math>, and polar radius, <math>~Z_p(t)</math>, is governed by the equations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{\partial\Phi}{\partial \varpi} \biggr|_{\varpi_{eq}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d^2 Z_\mathrm{p}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{\partial\Phi}{\partial Z} \biggr|_{Z_{p}} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, to within an additive constant,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\varpi,Z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G \rho [ A_1(e) \varpi^2 + A_3(e) Z^2] \, .
</math>
  </td>
</tr>
</table>
</div>

We should clarify and emphasize that this expression for the time-dependent gravitational potential has been written in terms of the (time-varying) eccentricity of the spheroid, <math>~e</math>, as measured in the meridional plane.  Specifically, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl( 1 - \frac{Z_p^2}{\varpi_\mathrm{eq}^2} \biggr)^{1/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~~ Z_p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\varpi_\mathrm{eq} \biggl( 1 - e^2 \biggr)^{1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
and, as is derived in our accompanying discussion of the [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Oblate_Spheroids_.28a1_.3D_a2_.3E_a3.29|properties of homogeneous ellipsoids]],

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_1(e)
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_3(e)
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1/2} \, .
</math>
  </td>
</tr>

</table>
Hence,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~\nabla\Phi
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~2\pi G \rho \biggl[ \hat{e}_\varpi A_1(e) + \hat{e}_Z A_3(e) Z \biggr] \, ,
</math>
  </td>
</tr>

</table>
and the pair of governing dynamical equations become,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\pi G \rho A_1(e) \varpi_\mathrm{eq} = - \frac{3}{2} \biggl[ \frac{GM}{\varpi_\mathrm{eq} Z_\mathrm{p}} \biggr] A_1(e) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d^2 Z_\mathrm{p}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\pi G \rho A_3(e) Z_\mathrm{p} = - \frac{3}{2} \biggl[ \frac{GM}{\varpi_\mathrm{eq}^2} \biggr] A_3(e) \, ,
</math>
  </td>
</tr>
</table>
</div>
where we have used the relation that is valid for uniform-density, oblate spheoids,
<div align="center">
<math>
~\rho = \frac{3M}{4\pi \varpi_\mathrm{eq}^2 Z_\mathrm{p}} \, .
</math>
</div>