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Centrifugal Hang-up of Core Collapse Supernovae

Massive stars which have burnt their nuclear fuel will collapse under the pull of their gravitational self attraction, sometimes triggering a supernova event. This collapse can lead to situations in which significant gravitational radiation is produced. In particular, there are various scenarios in which the collapsing core is thought to ``hang-up'' in a non-axisymmetric configuration and radiate this assymmetry away through gravitational waves.

A promising scenario of this type was explored by Lai and Shapiro [42] and proceeds as follows:a stellar core with some initial angular momentum collapses. As the collapse proceeds, the ratio $\beta=T/W$ of the rotational energy ($T$) to the gravitational potential energy ($W$) will vary inversely with the core radius. If $\beta$ becomes large enough, the evolution of non-axisymmetric modes of the core become unstable. Such instabilities are well known, being first described by Chandrasekhar [43]. There are two possibilities. If $\beta$ lies in a critical band of values between $\sim 0.17$ and $\sim 0.27$, the non-axisymmetric bar mode ($\ell=m=2$) of the core will be secularly unstable and grow due to either radiation reaction or viscosity. This long lasting instability is expected to produce significant gravitational radiation. If $\beta$ exceeds the upper limit of this band ($\beta > 0.27$) then non-axisymmetric modes become dynamically unstable. The transition through dynamical instability is rapid and results in a nearly axisymmetric configuration with $\beta < 0.27$ but still much larger than $0.17$. The core therefore again enters the regime of secular instability. In either scenario the onset of the secular instability occurs at approximately neutron star densities. If $\beta < 0.17$ for the entire collapse, then no significant gravitational radiation is expected from this mechanism.

Lai and Shapiro have calculated the waveform of the gravitational radiation emitted by a secularly unstable core based on crude Newtonian fluid ellipsoids models without viscosity. They find [44]:

\begin{displaymath}
\left. \begin{array}{l}
h_+(t) = A(f_{max},r,M,d)~u(t)^{2....
...{2.1}\sqrt{1-u(t)}\,\cos i~\sin \Phi(t).
\end{array} \right\}
\end{displaymath} (14.1.316)

where:
$f_{max}$ is the initial frequency of the gravitational wave (typically $\stackrel{<}{\sim}$ 800 Hz).
$r$ is the radius of the core (typically $\sim$ 10 km).
$M$ is the mass of the core (typically $\sim$ 1.4 $M_\odot$).
$d$ is the distance from the core (source).
$i$ inclination angle (see section [*]).
$A(f_{max},r,M,d)$ is the amplitude function
\begin{displaymath}
A=\frac{1}{2} \frac{M^2}{rd}\left\{\begin{array}{lr}
\left...
....0},&
\overline{f_{max}} > 415 {\rm Hz}.
\end{array} \right.
\end{displaymath} (14.1.317)

$\overline{f_{max}}$ = $f_{max} \sqrt{r_{10}^3 / M_{1.4}}$ where $r_{10} =
r/10$km and $M_{1.4}=M/(1.4M_\odot)$.
$u(t)$=$f(t)/f_{max}$ is given by the frequency evolution equation
\begin{displaymath}
\frac{du}{dt} = \left(\frac{M}{r}\right)^{5/2} \left(\frac{A}{0.16}\right)^2
\sqrt{\frac{M_{1.4}}{r_{10}}} ~~u^{5.2} (u-1).
\end{displaymath} (14.1.318)

$\Phi(t)$ is the phase, given implicitly by
\begin{displaymath}
f(t)=\frac{1}{2\pi}\frac{d\Phi}{dt}(t).
\end{displaymath} (14.1.319)

This model is implemented below.


next up previous contents
Next: Structure: LS_physical_constants Up: GRASP Routines: Supernovae and Previous: GRASP Routines: Supernovae and   Contents
Bruce Allen 2000-11-19