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4.4 Numerical Predictions of GW emission Mechanisms

The first fully general relativistic investigations of stellar core collapse were Nakamura’s 2D simulations of rotating collapse [179180]. However, because of the limits of his numerical formalism and computational resources, he was unable to compute the emitted gravitational radiation (the energy of this emission is quite small compared to the rest mass energy and thus was difficult to extract numerically). The results of this work indicate that collapse does not lead to black hole formation if the parameter q = J/M 2 exceeds unity (here J and M are the angular momentum and gravitational mass of the remnant).

Fully general relativistic collapse simulations (i.e., without the conformally flat approximation) have also been performed by Shibata [222Jump To The Next Citation Point]. He used an axisymmetric code that solves the Einstein equations in Cartesian coordinates and the hydrodynamics equations in cylindrical coordinates. The use of the Cartesian grid eliminates the presence of singularities and allows for stable, long-duration axisymmetric simulations [4]. The focus of this work was the effect of rotation on the criteria for prompt black hole formation. Shibata found that if the parameter 2 q = J/M is less than 0.5, black hole formation occurred for rest masses slightly greater than the maximum mass of spherical stars. However, for 0.5 < q < 1, the maximum stable rest mass is increased by ~ 70 - 80%. The results are only weakly dependent on the initial rotation profile. More recent results suggest that this limit can be eased for differentially rotating massive stellar cores, and systems with spin parameter q as high as 2.5 may collapse to form black holes [217]. Shibata did not compute the GW emission in his collapse simulations, but in a recent study using axisymmetric calculations, GW signals have been calculataed focusing on this collapse phase [267].

Duez et al. [62] found that if a black hole does form, but the disk is spinning rapidly, that the disk will fragment and its subsequent accretion will be in spurts, causing a “splash” onto the black hole, producing ringing and GW emission. Their result implies very strong gravitational wave amplitudes >~~ 10-21 at distances of 10 Mpc. Black hole ringing was also estimated by FHH, where they too assumed discrete accretion events. They found that even with very optimistic accretion scenarios, that such radiation will be of very low amplitude and beyond the upper frequency reach of LIGO-II (see [86Jump To The Next Citation Point] for details).

The new general relativistic hydrodynamics simulations of Zanotti, Rezzolla, and Font [269] suggest that a torus of neutron star matter surrounding a black hole remnant may be a stronger source of GWs than the collapse itself. They used a high resolution shock-capturing hydrodynamics method in conjunction with a static (Schwarzschild) spacetime to follow the evolution of “toroidal neutron stars”. Their results indicate that if a toroidal neutron star (with constant specific angular momentum) is perturbed, it could undergo regular oscillations. They estimate that the resulting GW emission would have a characteristic amplitude hc ranging from 6 × 10- 24 -5 × 10- 23, for ratios of torus mass to black hole mass in the range 0.1 - 0.5. (These amplitude values are likely underestimated because the simulations of Zanotti et al. are axisymmetric.) The corresponding frequency of emission is fGW ~ 200 Hz. The values of hc and f GW quoted here are for a source located at 10 Mpc. This emission would be just outside the range of LIGO-II (see Figure 2View Image). Further numerical investigations, which study tori with non-constant angular momenta and include the effects of self-gravity and black hole rotation, are needed to confirm these predictions. Movies from the simulations of Zanotti et al. can be viewed at [200].

Magnetized tori around rapidly spinning black holes (formed via either core collapse or neutron star-black hole coalescence) have recently been examined in the theoretical study of van Putten and Levinson [251Jump To The Next Citation Point]. They find that such a torus-black hole system can exist in a suspended state of accretion if the ratio of poloidal magnetic field energy to kinetic energy EB/Ek is less than 0.1. They estimate that ~ 10% of the spin energy of the black hole will be converted to gravitational radiation energy through multipole mass moment instabilities that develop in the torus. If a magnetized torus-black hole system located at 10 Mpc is observed for 4 2 × 10 rotation periods, the characteristic amplitude of the GW emission is ~ 6× 10-20. It is possible that this emission could take place at several frequencies. Observations of x-ray lines from gamma-ray bursts (which are possibly produced by these types of systems) could constrain these frequencies by providing information regarding the angular velocities of the tori: preliminary estimates from observations suggest fGW ~ 500 Hz, placing the radiation into a range detectable by LIGO-I [251Jump To The Next Citation Point].

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