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Rotationally driven nonaxisymmetric instabilities may be important during the collapse of the cores of massive rotating stars or during the accretion induced collapses of accreting white dwarfs in mass transfer binary systems. The former occur during Type II SN and are known as fizzlers. Fizzlers were initially proposed in the 1970s by Tommy Gold. Interest in them has recently been revived because of the possibilty they may be sources of gravitational radiation accessible to LIGO (see Figure to the left). The collpase of accreting white dwarfs may lead to a Type Ia SN or to the formation of a rapidly rotating neutron star or black hole. The theoretical properties of Type Ia SN have also received renewed attention recently because of their importance to our understanding of the ultimate fate of the Universe. |
| Sensitivity of LIGO. The red dumbbell is the strain range for typical fizzlers |
Understanding the onset and development of the secular and dynamic global nonaxisymmetric instabilities in rapidly rotating dense objects has thus recently become more than of academic interest. (Here, by secular we mean instabilities which grow on a dissipative time scale of the system and by dynamic we mean instabilities which grow on the dynamic time scale of the system.) We consider both secular and dynamic instabilities of dense objects.
SECULAR INSTABILITIES
Using a Lagrangian Variational Principle, we determined the secular stability limits for the low order nonaxisymmetric Kelvin-like modes for fizzlers. Interestingly, as Friedman & Schutz first pointed out, all rotating polytropic objects are subject to nonaxisymmetric secular instabilities. This result may also be extended to fizzlers. However, for fizzlers, we showed that secular instabilities driven by gravitational radiation reaction grow only very slowly and are, in fact, damped by viscoity.
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Secular instabilities apparently do not play a role in fizzler evolution
DYNAMIC INSTABILITIES
We study the dynamic instabilities of fizzlers using linear, quasi-linaer, and nonlinear techniques. We are able to map out the relevant regions of parameter space and to study the onset and ultimate endpoints of unstable fizzlers. We find remarkable consistency between our linear, quasi-linear, and nonlinear results.
The initial fizzlers which form are typically stable. They are forced to contract because the hot dense fizzler matter slowly deleptonizes and cools. The slow contraction causes the fizzler to spin-up until its T/|W| reaches 0.27, the canonical bar mode instability point.The fizzler then becomes unstable and the bar mode grows dynamically. The fizzler forms a central bar and ejects spiral arms. The strong Newtonian gravitational torque between the central bar and spiral arms causes the bar to shed angular momentum allowing it to settle into an apparently stable dynamic state. A typical final state bar will be similar to structure shown in the upper panel. The structure is based on our linear/quasi-linear theory. A typical nonlinear fizzler evolution is summarized in the middle panel to the right. We show the evolution of the low order (m=1-8) Fourier powers of the nonaxisymmetric disturbance. The bar mode component, the fastest growing mode, has m=2. The bottom panel shows the strain evolution for the fizzler evolution shown in the middle panel. The strain is the distortion of space-time produced by the passage of a gravity wave. The presented strains are normalized. The unnormalized strain is roughly several parts of 10-22, strains well below the LIGO I threshold for a fizzler at VIRGO distances. If the apparently stable bar manages to persist for 500-1,000 cycles, then it is conceivable that fizzlers could be accessbile by LIGO I out to Virgo distances.
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