4.3 Bar modes

Rotational instabilities in proto neutron stars, if they exist, could be very powerful GW sources. Global rotational instabilities in a collapsed core lead to rapid variations in the quadrupole moment, and hence, strong GW emission. Global rotational instabilities in fluids arise from non-axisymmetric modes e±imϕ, where m = 2 is known as the “bar mode” [302, 5Jump To The Next Citation Point]. It is convenient to parameterize a system’s susceptibility to these modes by the stability parameter β = Trot∕|W |. Here, Trot is the rotational kinetic energy and W 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 βd ≈ 0.27. These instabilities have been confirmed by a number of studies [276, 192, 74, 313, 331, 314Jump To The Next Citation Point, 147, 289Jump To The Next Citation Point, 148, 146, 237Jump To The Next Citation Point, 315, 154, 221Jump To The Next Citation Point, 30Jump To The Next Citation Point, 184Jump To The Next Citation Point, 232, 233], and we will discuss some of these results here. At lower rotation rates, a secular instability may develop that produces bar modes if viscous or gravitational radiation reaction forces can redistribute the angular momentum [52, 173Jump To The Next Citation Point, 171Jump To The Next Citation Point, 236Jump To The Next Citation Point].
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Figure 23: A comparison between the GW amplitude h(f) for various sources and the LIGO-II sensitivity curve based on the analytic estimates from FHH [106Jump To The Next Citation Point]. See the text for details regarding the computations of h. The SNe sources at 10 Mpc; and the Population III sources at a luminosity distance of ∼ 50 Gpc. Secular bar-mode sources are identified with an (s), dynamical bar modes with a (d). This assumes strong bar modes exist, which has not been demonstrated robustly.

FHH [106Jump To The Next Citation Point] (and [108Jump To The Next Citation Point]) compute an upper limit (via Equation (9View Equation)) to the emitted amplitude from their dynamically unstable model of h ∼ 3 × 10−22 (if coherent emission from a bar located at 10 Mpc persists for 100 cycles). The corresponding frequency and maximum power are fGW ≈ 103 Hz and PGW = 1053 erg s−1. LIGO-II should be able to detect such a signal (see Figure 23View Image, where FHH’s upper limit to h 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 h ∼ 10− 21, if the bar persists for 102 – 104 cycles. The maximum f GW 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 [157], due to the complicated phase evolution of the emission). Ou et al. [236Jump To The Next Citation Point] 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 24Watch/download Movie.

Many stellar models models (e.g., [135, 138]) do not produce stars with sufficiently high spin rates to produce fast-rotating cores [245Jump To The Next Citation Point, 103Jump To The Next Citation Point, 113, 325, 231, 69]. Fryer & Heger [103] 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, 104Jump To The Next Citation Point, 319, 42], and the collaspe of merging white dwarfs [341, 339, 342, 118, 63, 61]. For example, Fryer & Heger [104] merged helium stars in an effort to increase this angular momentum (Figure 9View Image). For Population III 300M ⊙ stars, the situation may be better. Figure 25View Image shows the β value and growth time for such a 300M ⊙ 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. [115Jump To The Next Citation Point] argued that bar modes could well develop in these systems.

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Figure 24: mov-Movie (9566 KB) Evolution of a secular bar instability, see Ou et al. [236] for details.
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Figure 25: β and instability growth time versus mass in the proto black hole formed in the collapse of a 300M ⊙ star at bounce (dotted line) and just prior to black-hole formation (solid line). Figure 10 of [115].

First, scientists have learned that dynamical instabilities can excite m = 1 modes as well as the well-studied, m = 2 bar modes [46Jump To The Next Citation Point, 258Jump To The Next Citation Point, 234Jump To The Next Citation Point, 235Jump To The Next Citation Point, 260Jump To The Next Citation Point]. Second, scientists have discovered that non-axisymmetric instabilities can occur at much lower values of β when the differential rotation is very high [278Jump To The Next Citation Point, 279Jump To The Next Citation Point, 282, 326Jump To The Next Citation Point, 257, 260Jump To The Next Citation Point]. For some models, cores with high differential rotation have exhibited non-axisymmetric instabilities for values of β ≈ 0.01. These results may drastically change the importance of these modes in astrophysical observations.

4.3.1 Equilibrium models to study instabilities

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, 237Jump To The Next Citation Point, 221Jump To The Next Citation Point, 46Jump To The Next Citation Point]). 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 [184Jump To The Next Citation Point, 183Jump To The Next Citation Point] have applied this equilibrium approach to AIC. Their investigation began with a study of equilibrium models built to represent neutron stars formed from AIC [184Jump To The Next Citation Point]. These neutron star models were created via a two-step process, using a Newtonian version of Hachisu’s self-consistent field method [128]. 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 10 ρc = 10 and 9 −3 6 × 10 g cm, 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 ρc = 4 × 109 g cm −3 (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 [102]. 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 [184Jump To The Next Citation Point] 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 [184Jump To The Next Citation Point] 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 β ranging from 0.23 – 0.26. It is interesting to compare these results with those of Villain et al. [323Jump To The Next Citation Point] or of Zwerger and Müller [350Jump To The Next Citation Point]. Villain et al. [323] found maximum β 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 (P ∝ ρΓ). Their initial models were Γ = 4∕3 polytropes, representative of massive white dwarfs. All of their models started with ρc = 1010 g cm −3. 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 β < 0.07. 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 [183Jump To The Next Citation Point] used linearized hydrodynamics to perform a stability analysis of the cold neutron-star AIC remnants of Liu and Lindblom [184]. He found that only the remnant of the O-Ne-Mg white dwarf (Liu and Lindblom’s Model III) developed the dynamical bar mode (m = 2) instability. This model had an initial β = 0.26. Note that the m = 1 mode, observed by others to be the dominant mode in unstable models with values of β much lower than 0.27 [314, 332, 237, 46Jump To The Next Citation Point], did not grow in this simulation. Because Liu and Lindblom’s Models I and II had lower values of β, Liu identified the onset of instability for neutron stars formed via AIC as βd ≈ 0.25.

Liu estimated the peak amplitude of the GWs emitted by the Model III remnant to be − 24 hpk ≈ 1.4 × 10 and the LIGO-II signal-to-noise ratio (for a persistent signal like that seen in the work of [221] and [30]) to be S ∕N ≤ 3 (for fGW ≈ 450 Hz). 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 τ ≈ 7 s GW (∼ 3 × 103 cycles). Thus, h ≈ 8 × 10−23. (Note that this value for h 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 23View Image). Details of the approximations on which these estimates are based can be found in [183].

Liu cautions that his results hold if the magnetic field of the proto neutron star is 12 B ≤ 10 G. 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 B field was ≥ 108 G. 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 7 8 ∼ 10 –3 × 10 G [330]. Strong toroidal magnetic fields of 14 B ≥ 10 G will also suppress the bar instabilities [119].

4.3.2 Hydrodynamic models

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 hpk 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 [245Jump To The Next Citation Point]. These authors started their Newtonian simulations with the only model (A4B5G5) of Zwerger and Müller [350Jump To The Next Citation Point] that had a post-bounce value for the stability parameter β = 0.35 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 (Γ r = 1.28), highest βi = 0.04, 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 β 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 m = 4. The growth of this particular mode was instigated by the cubical nature of the computational grid. When an m = 3 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 m = 2 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. [46Jump To The Next Citation Point] indicate it is possible that some of the post-bounce configurations of Zwerger and Müller, which have lower values of β than the model studied by Rampp, Müller, and Ruffert [245Jump To The Next Citation Point], may also be susceptible to non-axisymmetric instabilities. Centrella et al. have performed 3D hydrodynamics simulations of Γ = 1.3 polytropes to test the stability of configurations with off-center density maxima (as are present in many of the models of Zwerger and Müller [350]). 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 β ≳ 0.14. The observed instability had a dominant m = 1 mode. Centrella et al. estimate that if a stellar core of mass M ∼ 1.4M ⊙ and radius R ∼ 200 km encountered this instability, the values of hpk from their models would be ∼ 2 × 10 −24–2 × 10 −23, for d = 10 Mpc. The frequency at which hpk 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 [31] 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 [245Jump To The Next Citation Point]. He performed 3D hydrodynamical simulations of the post-bounce configurations resulting from 2D simulations of core collapse. His pre-collapse initial models are Γ = 4∕3 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 [137Jump To The Next Citation Point]. 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 [137]. The model evolved by Rampp, Müller, and Ruffert [245Jump To The Next Citation Point] had much stronger differential rotation than any of Brown’s models. To induce collapse, Brown reduced the adiabatic index of his models to Γ = 1.28, the same value used by [245Jump To The Next Citation Point].

Brown found that β 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 [245Jump To The Next Citation Point]. 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 Ω24 and Ω20, both of which had β > 0.27 after bounce) and of the model of Rampp et al. (which, although it starts out with β = 0.35, has a sustained β < 0.2). 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 m = 1 mode observed by Centrella et al. [46Jump To The Next Citation Point]. 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 [245Jump To The Next Citation Point]).

Brown’s simulations determined that both his most rapidly-rotating model Ω24 (with post-bounce β > 0.35) and model RMR are unstable to growth of the m = 2 bar mode. However, his model Ω20 (with post-bounce β > 0.3) was stable. Brown observed no dominant m = 3 or m = 4 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 m = 2 mode begins to grow in model RMR at about the same time as Rampp et al. stopped their evolutions. No substantial m = 1 growth was observed.

The results of Brown’s study indicate that the overall β of the post-bounce core may not be a good diagnostic for the onset of instability. He found, as did Rampp, Müller, and Ruffert [245], that only the innermost portion of the core (with ρ > 1010 g cm −3) is susceptible to the bar mode. This is evident in the stability of his model Ω20. This model had an overall β > 0.3, but an inner core with βic = 0.15. Brown also observed that the β of the inner core does not have to exceed 0.27 for the model to encounter the bar mode. Models Ω24 and RMR had βic ≈ 0.19. 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 βd < 0.27 for such configurations.

The GW emission from non-radial quasinormal mode oscillations in proto neutron stars has been examined by Ferrari, Miniutti, and Pons [85]. They found that the frequencies of emission fGW 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, EGW must be −3 −2 2 ∼ 10 –10 M ⊙c. It is unlikely that this much energy is stored in these modes (the collapse itself may only emit −7 2 ∼ 10 M ⊙c in gravitational waves [67]).

Shibata et al. [278, 279] found that, with extremely differentially-rotating cores, a bar mode instability can occur at β 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 (m = 1) instabilities. The work of Centrella et al. [46Jump To The Next Citation Point] has been followed by a large set of results, varying the density and angular velocity profiles [46, 258, 234, 235Jump To The Next Citation Point, 260]. Ou & Tohline [235] argued that these instabilities are akin to the Rossby wave instabilities studied in black-hole accretion disks [176]. These one-armed spirals could also produce a considerable GW signal.

Watts et al. [326] argue that these low-β 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.

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