FHH  (and ) compute an upper limit (via Equation (9)) to the emitted amplitude from their dynamically unstable model of (if coherent emission from a bar located at 10 Mpc persists for 100 cycles). The corresponding frequency and maximum power are and . LIGO-II should be able to detect such a signal (see Figure 23, where FHH’s upper limit to for this dynamical bar mode is identified).
The GW emission from proto neutron stars that are secularly unstable to the bar mode has been examined by Lai and Shapiro [173, 171]. They predict that such a bar located at 10 Mpc would emit GWs with a peak characteristic amplitude , if the bar persists for 102 – 104 cycles. The maximum of the emitted radiation is in the range of 102 – 103 Hz. This type of signal should be easily detected by LIGO-I (although detection may require a technique like the fast chirp transform method of Jenet and Prince , due to the complicated phase evolution of the emission). Ou et al.  found that a bar instability was maintained for several orbits before sheer flows disintegrate the instability, producing GW emissions that would have a signal-to-noise ratio greater than 8 for LIGO-II out to 32 Mpc. A movie of this simulation is shown in Figure 24.
Many stellar models models (e.g., [135, 138]) do not produce stars with sufficiently high spin rates to produce fast-rotating cores [245, 103, 113, 325, 231, 69]. Fryer & Heger  argued that the explosion phase could eject a good deal of low angular-momentum material along the poles in their evolutions. Roughly 1 s after the collapse, the angular momentum in the faster cores will exceed the secular bar instability limit. It is also possible that standard stellar evolution models underestimate the rotation rate. Many scenarios for GRBs require much higher rotation rates than these models predict. This requirement has led to a host of new models predicting much faster rotation rates: mixing in single stars [338, 340], binary scenarios [114, 240, 104, 319, 42], and the collaspe of merging white dwarfs [341, 339, 342, 118, 63, 61]. For example, Fryer & Heger  merged helium stars in an effort to increase this angular momentum (Figure 9). For Population III stars, the situation may be better. Figure 25 shows the value and growth time for such a star at bounce and just prior to the collapse to a black hole. This bounce is caused by a combination of thermal and rotational support and hence the bounce is very soft. Fryer et al.  argued that bar modes could well develop in these systems.
First, scientists have learned that dynamical instabilities can excite modes as well as the well-studied, bar modes [46, 258, 234, 235, 260]. Second, scientists have discovered that non-axisymmetric instabilities can occur at much lower values of when the differential rotation is very high [278, 279, 282, 326, 257, 260]. For some models, cores with high differential rotation have exhibited non-axisymmetric instabilities for values of . These results may drastically change the importance of these modes in astrophysical observations.
One way to determine whether a specific star collapses to develop bar modes is through equilibrium models as initial conditions for hydrodynamical simulations (e.g., [289, 237, 221, 46]). Such simulations represent the approximate evolution of a model beginning at some intermediate phase during collapse or the evolution of a collapsed remnant. These studies do not typically follow the intricate details of the collapse itself. Instead, their goals include determining the stability of models against the development of non-axisymmetric modes and estimation of the characteristics of any resulting GW emission.
Liu and Lindblom [184, 183] have applied this equilibrium approach to AIC. Their investigation began with a study of equilibrium models built to represent neutron stars formed from AIC . These neutron star models were created via a two-step process, using a Newtonian version of Hachisu’s self-consistent field method . Hachisu’s method ensures that the forces due to the centrifugal and gravitational potentials and the pressure are in balance in the equilibrium configuration.
Liu and Lindblom’s process of building the nascent neutron stars began with the construction of rapidly-rotating, pre-collapse white dwarf models. Their Models I and II are C-O white dwarfs with central densities and , respectively (recall this is the range of densities for which AIC is likely for C-O white dwarfs). Their Model III is an O-Ne-Mg white dwarf that has (recall this is the density at which collapse is induced by electron capture). All three models are uniformly rotating, with the maximum allowed angular velocities. The models’ values of total angular momentum are roughly 3 – 4 times that of Fryer et al.’s AIC progenitor Model 3 . The realistic equation of state used to construct the white dwarfs is a Coulomb-corrected, zero temperature, degenerate gas equation of state [261, 51].
In the second step of their process, Liu and Lindblom  built equilibrium models of the collapsed neutron stars themselves. The mass, total angular momentum, and specific angular momentum distribution of each neutron-star remnant is identical to that of its white dwarf progenitor (see Section 3 of  for justification of the specific angular momentum conservation assumption). These models were built with two different realistic neutron-star equations of state.
Liu and Lindblom’s cold neutron-star remnants had values of the stability parameter ranging from 0.23 – 0.26. It is interesting to compare these results with those of Villain et al.  or of Zwerger and Müller . Villain et al.  found maximum values of 0.2 for differentially-rotating models and 0.11 for rigidly-rotating models. Zwerger and Müller performed axisymmetric hydrodynamics simulations of stars with polytropic equations of state (). Their initial models were polytropes, representative of massive white dwarfs. All of their models started with . Their model that was closest to being in uniform rotation (A1B3) had 22% less total angular momentum than Liu and Lindblom’s Model I. The collapse simulations of Zwerger and Müller that started with model A1B3 all resulted in remnants with values of . Comparison of the results of these two studies could indicate that the equation of state may play a significant role in determining the structure of collapsed remnants. Or it could suggest that the assumptions employed in the simplified investigation of Liu and Lindblom are not fully appropriate.
In a continuation of the work of Liu and Lindblom, Liu  used linearized hydrodynamics to perform a stability analysis of the cold neutron-star AIC remnants of Liu and Lindblom . He found that only the remnant of the O-Ne-Mg white dwarf (Liu and Lindblom’s Model III) developed the dynamical bar mode () instability. This model had an initial . Note that the mode, observed by others to be the dominant mode in unstable models with values of much lower than 0.27 [314, 332, 237, 46], did not grow in this simulation. Because Liu and Lindblom’s Models I and II had lower values of , Liu identified the onset of instability for neutron stars formed via AIC as .
Liu estimated the peak amplitude of the GWs emitted by the Model III remnant to be and the LIGO-II signal-to-noise ratio (for a persistent signal like that seen in the work of  and ) to be (for ). These values are for a source located at 100 Mpc. He also predicted that the timescale for gravitational radiation to carry away enough angular momentum to eliminate the bar mode is ( cycles). Thus, . (Note that this value for is merely an upper limit as it assumes that the amplitude and frequency of the GWs do not change over the 7 s during which they are emitted. Of course, they will change as angular momentum is carried away from the object via GW emission.) Such a signal may be marginally detectable with LIGO-II (see Figure 23). Details of the approximations on which these estimates are based can be found in .
Liu cautions that his results hold if the magnetic field of the proto neutron star is . If the magnetic field is larger, then it may have time to suppress some of the neutron star’s differential rotation before it cools. This would make bar formation less likely. Such a large field could only result if the white dwarf progenitor’s field was . Observation-based estimates suggest that about 25% of white dwarfs in interacting close binaries (cataclysmic variables) are magnetic and that the field strengths for these stars are . Strong toroidal magnetic fields of will also suppress the bar instabilities .
The GW emission from non-axisymmetric hydrodynamics simulations of stellar collapse was first studied by Bonazzola and Marck [194, 23]. They used a Newtonian, pseudo-spectral hydrodynamics code to follow the collapse of polytropic models. Their simulations covered only the pre-bounce phase of the collapse. They found that the magnitudes of in their 3D simulations were within a factor of two of those from equivalent 2D simulations and that the gravitational radiation efficiency did not depend on the equation of state.
The first use of 3D hydrodynamics collapse simulations to study the GW emission well beyond the core bounce phase was performed by Rampp, Müller, and Ruffert . These authors started their Newtonian simulations with the only model (A4B5G5) of Zwerger and Müller  that had a post-bounce value for the stability parameter that significantly exceeded 0.27 (recall this is the value at which the dynamical bar instability sets in for MacLaurin spheroid-like models). This model had the softest equation of state (), highest , and largest degree of differential rotation of all of Zwerger and Müller’s models. The model’s initial density distribution had an off-center density maximum (and therefore a torus-like structure). Rampp, Müller, and Ruffert evolved this model with a 2D hydrodynamics code until its reached 0.1. At that point, 2.5 ms prior to bounce, the configuration was mapped onto a 3D nested cubical grid structure and evolved with a 3D hydrodynamics code.
Before the 3D simulations started, non-axisymmetric density perturbations were imposed to seed the growth of any non-axisymmetric modes to which the configuration was unstable. When the imposed perturbation was random (5% in magnitude), the dominant mode that arose was . The growth of this particular mode was instigated by the cubical nature of the computational grid. When an perturbation was imposed (10% in magnitude), three clumps developed during the post-bounce evolution and produced three spiral arms. These arms carried mass and angular momentum away from the center of the core. The arms eventually merged into a bar-like structure (evidence of the presence of the mode). Significant non-axisymmetric structure was visible only within the inner 40 km of the core. Their simulations were carried out to 14 ms after bounce.
The amplitudes of the emitted gravitational radiation (computed in the quadrupole approximation) were only 2% different from those observed in the 2D simulation of Zwerger and Müller. Because of low angular resolution in the 3D runs, the energy emitted was only 65% of that emitted in the corresponding 2D simulation.
The findings of Centrella et al.  indicate it is possible that some of the post-bounce configurations of Zwerger and Müller, which have lower values of than the model studied by Rampp, Müller, and Ruffert , may also be susceptible to non-axisymmetric instabilities. Centrella et al. have performed 3D hydrodynamics simulations of polytropes to test the stability of configurations with off-center density maxima (as are present in many of the models of Zwerger and Müller ). The simulations carried out by Centrella and collaborators were not full collapse simulations, but rather began with differentially-rotating equilibrium models. These simulations tracked the growth of any unstable non-axisymmetric modes that arose from the initial 1% random density perturbations that were imposed. Their results indicate that such models can become dynamically unstable at values of . The observed instability had a dominant mode. Centrella et al. estimate that if a stellar core of mass and radius encountered this instability, the values of from their models would be , for . The frequency at which occurred in their simulations was 200 Hz. This instability would have to persist for at least 15 cycles to be detected with LIGO-II.
Brown  carried out an investigation of the growth of non-axisymmetric modes in post-bounce cores that was similar in many respects to that of Rampp, Müller, and Ruffert . He performed 3D hydrodynamical simulations of the post-bounce configurations resulting from 2D simulations of core collapse. His pre-collapse initial models are polytropes in rotational equilibrium. The differential rotation laws used to construct Brown’s initial models were motivated by the stellar evolution study of Heger, Langer, and Woosley . The angular velocity profiles of their pre-collapse progenitors were broad and Gaussian-like. Brown’s initial models had peak angular velocities ranging from 0.8 – 2.4 times those of . The model evolved by Rampp, Müller, and Ruffert  had much stronger differential rotation than any of Brown’s models. To induce collapse, Brown reduced the adiabatic index of his models to , the same value used by .
Brown found that increased by a factor 2 during his 2D collapse simulations. This is much less than the factor of 9 observed in the model studied by Rampp, Müller, and Ruffert . This is likely a result of the larger degree of differential rotation in the model of Rampp et al.
Brown performed 3D simulations of the two most rapidly-rotating of his post-bounce models (models and , both of which had after bounce) and of the model of Rampp et al. (which, although it starts out with , has a sustained ). Brown refers to the Rampp et al. model as model RMR. Because Brown’s models do not have off-center density maxima, they are not expected to be unstable to the mode observed by Centrella et al. . He imposed random 1% density perturbations at the start of all three of these 3D simulations (note that this perturbation was of a much smaller amplitude than those imposed by ).
Brown’s simulations determined that both his most rapidly-rotating model (with post-bounce ) and model RMR are unstable to growth of the bar mode. However, his model (with post-bounce ) was stable. Brown observed no dominant or modes growing in model RMR at the times at which they were seen in the simulations of Rampp et al. This suggests that the mode growth in their simulations was a result of the large perturbations they imposed. The mode begins to grow in model RMR at about the same time as Rampp et al. stopped their evolutions. No substantial growth was observed.
The results of Brown’s study indicate that the overall of the post-bounce core may not be a good diagnostic for the onset of instability. He found, as did Rampp, Müller, and Ruffert , that only the innermost portion of the core (with ) is susceptible to the bar mode. This is evident in the stability of his model . This model had an overall , but an inner core with . Brown also observed that the of the inner core does not have to exceed 0.27 for the model to encounter the bar mode. Models and RMR had . He speculates that the inner cores of these later two models may be bar-unstable because interaction with their outer envelopes feeds the instability or because for such configurations.
The GW emission from non-radial quasinormal mode oscillations in proto neutron stars has been examined by Ferrari, Miniutti, and Pons . They found that the frequencies of emission during the first second after formation (600 – 1100 Hz for the first fundamental and gravity modes) are significantly lower than the corresponding frequencies for cold neutron stars and thus reside in the bandwidths of terrestrial interferometers. However, for first generation interferometers to detect the GW emission from an oscillating proto neutron star located at 10 Mpc, with a signal-to-noise ratio of 5, must be . It is unlikely that this much energy is stored in these modes (the collapse itself may only emit in gravitational waves ).
Shibata et al. [278, 279] found that, with extremely differentially-rotating cores, a bar mode instability can occur at values of 0.01. They found that such an instability was weakly dependent on the polytropic index describing the equation of state and on the velocity profile (as long as the differential rotation is high). They predict an effective amplitude of roughly 10–22 at a distance of 100 Mpc.
Studies of systems with extreme differential rotation have also discovered the development of one-arm () instabilities. The work of Centrella et al.  has been followed by a large set of results, varying the density and angular velocity profiles [46, 258, 234, 235, 260]. Ou & Tohline  argued that these instabilities are akin to the Rossby wave instabilities studied in black-hole accretion disks . These one-armed spirals could also produce a considerable GW signal.
Watts et al.  argue that these low- instabilities can be explained if the corotating f-mode develops a dynamical shear instability. This occurs when the corotating f-mode enters the corotation band and when the degree of differential rotation exceeds a threshold value (that is within those produced in some collapse progenitors). These new instabilities are drastically changing our view of non-axisymmetric modes.
Living Rev. Relativity 14, (2011), 1
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