The outcome of SMS collapse can be determined only with numerical, relativistic 3D hydrodynamics
simulations. Until recently, such simulations had been published only for nearly spherical collapse. The
spherical simulations of Shapiro and Teukolsky [275] produced collapse evolutions that were nearly
homologous. In this case, the collapse time is roughly the free-fall time at the horizon
The amplitude of this burst signal can be roughly estimated in terms of the star’s quadrupole
moment
There are two possible aspherical collapse outcomes that have been studied. The first outcome is direct
collapse to a SMBH. In this case, will be on the order of one near the horizon. Thus, according to
Equation (31
), the peak amplitude of the GW burst signal will be
Alternatively, the star may encounter the dynamical bar mode instability prior to complete collapse.
Baumgarte and Shapiro [12] have estimated that a uniformly-rotating SMS will reach
when
. The frequency of the quasiperiodic gravitational radiation emitted by the bar can be
estimated in terms of its rotation frequency to be
Shibata and Shapiro [284] have published a fully general-relativistic, axisymmetric simulation of the collapse of a rapidly, rigidly-rotating SMS. They found that the collapse remained homologous during the early part of the evolution. An apparent horizon does appear in their simulation, indicating the formation of a black hole. Because of the symmetry condition used in their run, non-axisymmetric instabilities did not develop.
The collapse of a uniformly-rotating SMS has been investigated with post-Newtonian hydrodynamics, in
3+1 dimensions, by Saijo, Baumgarte, Shapiro, and Shibata [259]. Their numerical scheme used a
post-Newtonian approximation to the Einstein equations, but solved the fully relativistic hydrodynamics
equations. Their initial model was an
polytrope.
The results of Saijo et al. (confirmed in conformally-flat simulations [256]) indicate that the collapse of
a uniformly-rotating SMS is coherent (i.e., no fragmentation instability develops). The collapse evolution of
density contours from their model is shown in Figure 31. Although the work of Baumgarte and Shapiro [12]
suggests that a bar instability should develop prior to black-hole formation, no bar development was
observed by Saijo et al. They use the quadrupole approximation to estimate a mean GW amplitude from
the collapse itself:
, for a
star located at a distance of 50 Gpc. Their estimate for
at the time of black-hole formation is
. This signal would be detectable with LISA (see
Figure 30
).
Saijo et al. also consider the GW emission from the ringdown of the black-hole remnant. For the
quasi-normal mode of a Kerr black hole with
, they estimate the characteristic
amplitude of emission to be
at
for an
source located at a luminosity distance of 50 Gpc (see [174, 307, 285] for details). Here,
is the radiated energy efficiency and may be
[291]. This GW signal is within
LISA’s range of sensitivity (see Figure 30
).
http://www.livingreviews.org/lrr-2011-1 |
Living Rev. Relativity 14, (2011), 1
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