Until recently, such simulations had been published only for nearly spherical collapse. The spherical simulations of Shapiro and Teukolsky  produced collapse evolutions that were nearly homologous. In this case, the collapse time is roughly the free-fall time at the horizon[243, 76].
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 can be discussed. 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 8, 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  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 for details on the computation of this curve; a mission time of 3 years has been assumed). The GW signal from this dynamical bar-mode could be detected with LISA. Shibata and Shapiro  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 were unable to 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 . 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 Update) 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 19. Although the work of Baumgarte and Shapiro  suggests that a bar instability should develop prior to BH 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 . Their estimate for at the time of BH formation is . This signal would be detectable with LISA (see Figure 18).
© Max Planck Society and the author(s)