A Template for Gravitational Wave Signals from Core-Collapse Supernovae

29 August 2014: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.

Our principal objective is to help the gravitational-wave community better understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. This discussion is intended to supplement and complement reviews that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the 2009 review by Christian Ott and the 2012 review by Logue et al..

Free-Fall Collapse

Nod to Lynden-Bell's Early Contributions

I want to begin this section by paying tribute to Donald Lynden-Bell who, in 1962 (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,

<math>~\ddot{R}</math>	<math>~=</math>	<math>~-~ 2A_L(t) R + \frac{h_{LB}^2}{R^3} \, ,</math>
<math>~\ddot{Z}</math>	<math>~=</math>	<math>~-~ 2C_L(t) Z \, ,</math>

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,

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 … will be through a series of [uniformly rotating] spheroids." Two years later in a separate article, Lynden-Bell (1964, ApJ, 139, 1195) presented results from the numerical integration of this governing set of dynamical equations (see §X and Figure 1 of his article). 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 free-fall (pressure-free) collapse from rest of a uniform-density sphere of mass, <math>~M</math>, and initial radius, <math>~r_0</math> — hence, initial density <math>~\rho_0 = 3M/(4\pi r_0^3)</math> — 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 Lin, Mestel & Shu 1965),

<math>~\frac{d^2r}{dt^2} </math>	<math>~=</math>	<math>~- \frac{GM}{r^2} = - \frac{G}{r^2} \biggl[ \frac{4\pi}{3} ~r_0^3 \rho_0 \biggr]</math>
<math>~\Rightarrow~~~~\frac{1}{r_0} \frac{d^2r}{dt^2} </math>	<math>~=</math>	<math>~- \frac{4\pi G \rho_0}{3} \biggl( \frac{r_0}{r} \biggr)^2 \, ,</math>

which integrates to give,

where,

<math>~\frac{2}{\pi} \biggl[ \zeta + \frac{1}{2} \sin(2\zeta) \biggr]</math>	<math>~=</math>	<math>~\frac{t}{\tau_\mathrm{ff}} \, ,</math>
<math>~\tau_\mathrm{ff}</math>	<math>~\equiv</math>	<math>~\biggl[ \frac{3\pi}{32 G \rho_0} \biggr]^{1/2} \, .</math>

(Detailed steps through this derivation will be provided elsewhere.) 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.

Analytic Solution to Spherical Free-Fall Collapse

<math>~x(\zeta) \equiv \frac{r}{r_0}</math>	<math>~=</math>	<math>~\cos^2 \zeta \, ,</math>
<math>~x'(\zeta) \equiv \biggl( \frac{\tau_\mathrm{ff}}{r_0} \biggr) \frac{dr}{dt}</math>	<math>~=</math>	<math>~-~\frac{\pi}{2} \tan \zeta \, ,</math>
<math>~x(\zeta) \equiv \biggl(\frac{\tau^2_\mathrm{ff}}{r_0} \biggr) \frac{d^2r}{dt^2}</math>	<math>~=</math>	<math>~-~\frac{\pi^2}{8} \cdot \frac{1}{\cos^4 \zeta} = ~-~\frac{\pi^2}{8} \cdot \frac{1}{x^2} \, ,</math>
<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>	<math>~=</math>	<math>~+~\frac{\pi^2}{8} \cdot (\tan^2\zeta - 1) \, .</math>

<math>~\frac{t}{\tau_\mathrm{ff}}</math>

<math>~-1.24196\times 10^4</math>

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,

<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>

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,

<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>	<math>~=</math>	<math>~ -~\frac{\partial\Phi}{\partial \varpi} \biggr\|_{\varpi_{eq}} \, , </math>
<math>~\frac{d^2 Z_\mathrm{p}}{dt^2}</math>	<math>~=</math>	<math>~ -~\frac{\partial\Phi}{\partial Z} \biggr\|_{Z_{p}} \, , </math>

where, to within an additive constant,

<math>~\Phi(\varpi,Z)</math>

<math>~ \pi G \rho [ A_1(e) \varpi^2 + A_3(e) Z^2] \, . </math>

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,

<math>~e </math>	<math>~\equiv</math>	<math>~ \biggl( 1 - \frac{Z_p^2}{\varpi_\mathrm{eq}^2} \biggr)^{1/2} </math>
<math>~\Rightarrow~~~~~ Z_p</math>	<math>~=</math>	<math>~ \varpi_\mathrm{eq} \biggl( 1 - e^2 \biggr)^{1/2} \, , </math>

and, as is derived in our accompanying discussion of the properties of homogeneous ellipsoids,

<math> ~A_1(e) </math>	<math> ~= </math>	<math> ~\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, , </math>
<math> ~A_3(e) </math>	<math> ~= </math>	<math> ~\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1/2} \, . </math>

Hence,

<math> ~\nabla\Phi </math>

<math> ~2\pi G \rho \biggl[ \hat{e}_\varpi A_1(e) + \hat{e}_Z A_3(e) Z \biggr] \, , </math>

and the pair of governing dynamical equations become,

<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>	<math>~=</math>	<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>
<math>~\frac{d^2 Z_\mathrm{p}}{dt^2}</math>	<math>~=</math>	<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>

where we have used the relation that is valid for uniform-density, oblate spheoids,

<math> ~\rho = \frac{3M}{4\pi \varpi_\mathrm{eq}^2 Z_\mathrm{p}} \, . </math>

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