Global rotational instabilities in fluids arise from non-axisymmetric modes , where is known as the “bar-mode” [239, 6]. It is convenient to parameterize a system’s susceptibility to these modes by the stability parameter . Here, is the rotational kinetic energy and is the gravitational potential energy. Dynamical rotational instabilities, driven by Newtonian hydrodynamics and gravity, develop on the order of the rotation period of the object. For the uniform-density, incompressible, uniformly rotating MacLaurin spheroids, the dynamical bar-mode instability sets in at . For differentially rotating fluids with a polytropic equation of state, numerical simulations have determined that the stability limit is valid for initial angular momentum distributions that are similar to those of MacLaurin spheroids [221, 63, 162, 125, 192, 118, 248]. If the object has an off-center density maximum, could be as low as [247, 260, 192, 49]. General relativity may enhance the dynamical bar-mode instability by slightly reducing [223, 209]. Secular rotational instabilities are driven by dissipative processes such as gravitational radiation reaction and viscosity. When this type of instability arises, it develops on the timescale of the relevant dissipative mechanism, which can be much longer than the rotation period (e.g., . The secular bar-mode instability limit for MacLaurin spheroids is .
In an attempt to reduce these high rotation requirements, there has been increasing work studying bar-mode instabilities driven by dynamical sheer in differentially rotating neutron stars. Sheer instabilities excite the co-rotating -mode. If viscous forces don’t damp this instability altogether, it is possible that this instability can occur for -values as low as for stars with a large degree of differential rotation.
In rotating stars, gravitational radiation reaction drives the -modes toward unstable growth [5, 77]. In hot, rapidly rotating neutron stars, this instability may not be suppressed by internal dissipative mechanisms (such as viscosity and magnetic fields) . If not limited, the dimensionless amplitude of the dominant () -mode will grow to order unity within ten minutes of the formation of a neutron star rotating with a millisecond period. The emitted GWs carry away angular momentum, and will cause the newly formed neutron star to spin down over time. The spindown timescale and the strength of the GWs themselves are directly dependent on the maximum value to which the amplitude is allowed to grow [153, 154]. Originally, it was thought that . Later work indicated that may be [153, 236, 213, 154]. Some research suggests that magnetic fields, hyperon cooling, and hyperon bulk viscosity may limit the growth of the -mode instability, even in nascent neutron stars [136, 135, 201, 202, 154, 151, 102, 6] (significant uncertainties remain regarding the efficacy of these dissipative mechanisms). Furthermore, a study of a simple barotropic neutron star model by Arras et al.  suggests that multimode couplings could limit to values . If is indeed (see also ),Update GW emission from -modes in collapsed remnants is likely undetectable. For the sake of completeness, an analysis of GW emission from -modes (which assumes ) is presented in the remainder of this paper. However, because it is quite doubtful that is sizeable, -mode sources are omitted from figures comparing source strengths and detector sensitivities and from discussions of likely detectable sources in the concluding section.
There is some numerical evidence that a collapsing star may fragment into two or more orbiting clumps . If this does indeed occur, the orbiting fragments would be a strong GW source.
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