Until recently, such simulations had been published only for nearly spherical collapse. The spherical simulations of Shapiro and Teukolsky [220] produced collapse evolutions that were nearly homologous. In this case, the collapse time is roughly the freefall time at the horizon
The peak GW frequency is then , if the mass of the star is . This is in the middle of LISA’s frequency band of [243, 76].The amplitude of this burst signal can be roughly estimated in terms of the star’s quadrupole moment
Here is the distance to the star and is a measure of the star’s deviation from spherical symmetry. In this case, will be much less than one near the horizon, since the collapse is nearly spherical.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 [16] 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
when . The corresponding , again estimated in terms of the star’s quadrupole moment, is The LISA sensitivity curve is shown in Figure 18 (see [119] for details on the computation of this curve; a mission time of 3 years has been assumed). The GW signal from this dynamical barmode could be detected with LISA. Shibata and Shapiro [228] 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, nonaxisymmetric instabilities were unable to develop.The collapse of a uniformly rotating SMS has been investigated with postNewtonian hydrodynamics, in 3+1 dimensions, by Saijo, Baumgarte, Shapiro, and Shibata [208]. Their numerical scheme used a postNewtonian 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 [207]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 [16] 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).

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