4.6 Fragmentation

Fragmentation requires even more extreme spin rates than bar-mode instabilities. It is unlikely that any neutron-star–forming systems will have enough rotation to produce fragmentation. For most stellar models, the angular momentum increases with radius (or enclosed mass). When black holes form, material further out in the star (with higher angular momentum) becomes part of the compact remnant and it is possible to form proto black holes with higher spin rates than proto neutron stars. These proto black holes may be more likely to fragment. Even so, fragmentation requires extreme levels of rotation. As we shall see, some cases of GW emission from fragmentation is intimately related to black-hole ringing and some of the GW sources discussed here will also be discussed in Section 4.7.

Duez et al. [73Jump To The Next Citation Point] 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 GW amplitudes ≳ 10–21 at distances of 10 Mpc. Black hole ringing was also estimated by FHH [106Jump To The Next Citation Point], where they too assumed discrete accretion events. They found that, even with very optimistic accretion scenarios, such radiation will be of very low amplitude and beyond the upper frequency reach of LIGO-II (see [106Jump To The Next Citation Point] for details).

The general-relativistic hydrodynamics simulations of Zanotti, Rezzolla, and Font [344] 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 quasi-periodic 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 f ≈ 200 Hz GW. The values of h c and fGW quoted here are for a source located at 10 Mpc. This emission would be just outside the range of LIGO-II (see Figure 23View 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 [247].

Magnetized tori around rapidly-spinning black holes (formed either via core collapse or neutron-star–black-hole coalescence) have been examined in the theoretical study of van Putten and Levinson [321Jump 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 2 × 104 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 GRBs (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 [321].

By studying the results of current stellar-collapse models, FHH [106Jump To The Next Citation Point] predict that a fragmentation instability is unlikely to develop during core-collapse SNe because the cores have central density maxima (see also [108]). However, they do give estimates [calculated via Equations (7View Equation) and (6View Equation)] for the amplitude, power, and frequency of the emission from such an instability: −22 hpk ≈ 2 × 10, 54 −1 PGW = 10 erg s, 3 fGW ≈ 2 × 10 Hz. Again, this signal would fall just beyond the upper limit of LIGO-II’s frequency range.

Three-dimensional models of black-hole formation in a rotating stellar collapse are needed to truly study fragmentation. Zink et al. [348, 349] have found that with appropriate initial conditions, fragmentation can occur with even modest values of T ∕|W |: ∼ 0.2. But for current stellar models, fragmentation remains difficult to achieve. Rockefeller et al. [250Jump To The Next Citation Point] modeled in three dimensions the collapse of a 60M ⊙ star using a range of angular momenta to study the effects of the spin on the GW signal. For moderate spin rates, the instabilties in the disk grow and form pockets of denser material, but strong fragmentation does not occur (Figure 29View Image). The GW signal is far lower than the upper limits from FHH or those predicted by van Putten. Liu et al. [185] have performed axisymmetric calculations of the collapse of an SMS using a magneto-hydrodynamic code. As this group moves to 3-dimensional calculations, we will be able to test the development of these instabilities within strong magnetic fields.

View Image

Figure 29: The matter density in the equatorial plane of the rapidly-rotating collapsar simulation, 0.44 s after collapse. The spiral wave forms near the center of the collapsar 0.29 s after collapse and moves outward through the star. (Figure 4 of [250Jump To The Next Citation Point]; used with permission.)

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