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2.4 Numerical predictions of GW emission

2.4.1 Full collapse simulations

The accretion-induced collapse of white dwarfs has been simulated by a number of groups [1416626382Jump To The Next Citation Point]. The majority of these simulations have been Newtonian and have focused on mass ejection and neutrino and g-ray emission during the collapse and its aftermath (note that neglecting relativistic effects likely introduces an error of order (v/c)2 ~ 10% for the neutron star remnants of AIC; see below). The most sophisticated are those carried out by Fryer et al. [82Jump To The Next Citation Point], as they include realistic equations of state, neutrino transport, and rotating progenitors.

As a part of their general evaluation of upper limits to GW emission from gravitational collapse, Fryer, Holz, and Hughes (hereafter, FHH) [86Jump To The Next Citation Point] examined an AIC simulation (model 3) of Fryer et al. [82Jump To The Next Citation Point]. FHH used both numerical and analytical techniques to estimate the peak amplitude hpk, energy EGW, and frequency fGW of the gravitational radiation emitted during the collapse simulations they studied.

For direct numerical computation of the GWs emitted in these simulations, FHH used the quadrupole approximation, valid for nearly Newtonian sources [169]. This approximation is standardly used to compute the GW emission in Newtonian simulations. The reduced or traceless quadrupole moment of the source can be expressed as

integral ( ) 1 2 3 I-ij = r xixj - --dijr d r, (1) 3
where i,j = 1,2,3 are spatial indices and V~ ------------ r = x2 + y2 + z2 is the distance to the source. The two polarizations of the gravitational wave field, h+ and h×, can be computed in terms of ..-. Iij, where an overdot indicates a time derivative d/dt. The energy E GW is a function of ... I- ij. Equation (1View Equation) can be used to calculate I-ij from the results of a numerical simulation by direct summation over the computational grid. Numerical time derivatives of I-ij can then be taken to compute h+, h×, and EGW. However, successive application of numerical derivatives generally introduces artificial noise. Methods for computing .. I- ij and ... I- ij without taking numerical time derivatives have been developed [74Jump To The Next Citation Point25170Jump To The Next Citation Point]. These methods recast the time derivatives of I-ij as spatial derivatives of hydrodynamic quantities computed in the collapse simulation (including the density, velocities, and gravitational potential). Thus, instantaneous values for .. I-ij and ... I- ij can be computed on a single numerical time slice (see [170Jump To The Next Citation Point271Jump To The Next Citation Point] for details). Note that FHH define hpk as the maximum value of the rms strain V~ ---------- h = <h2+ + h2× >, where angular brackets indicate that averages have been taken over both wavelength and viewing angle on the sky.

Errors resulting from the neglect of general relativistic effects (in collapse evolutions as a whole and in GW emission estimations like the quadrupole approximation) are of order 2 (v/c). These errors are typically ~ 10% for neutron star remnants of AIC, < 30% for neutron star remnants of massive stellar collapse, and > 30% for black hole remnants. Neglect of general relativity in rotational collapse studies is of special concern because relativistic effects counteract the stabilizing effects of rotation (see Section 3.4).

Because the code used in the collapse simulations examined by FHH [82Jump To The Next Citation Point] was axisymmetric, their use of the numerical quadrupole approximation discussed above does not account for GW emission that may occur due to non-axisymmetric mass flow. The GWs computed directly from their simulations come only from polar oscillations (which are significant when the mass flow during collapse [or explosion] is largely aspherical).

In order to predict the GW emission produced by non-axisymmetric instabilities, FHH employed rough analytical estimates. The expressions they used to approximate the rms strain h and the power P = dE /dt GW of the GWs emitted by a star that has encountered the bar-mode instability are

V~ --- 2 2 h = 32-G-mr--w-- (2) bar 45 c4 d
32 G Pbar = ----5m2r4w6. (3) 45 c
Here m, 2r, and w are the mass, length, and angular frequency of the bar and d is the distance to the source. FHH vary the mass m assumed to be enclosed by the bar (which has a corresponding length 2r) and compute the characteristics of the GW emission as a function of this enclosed mass. For simplicity, FHH assumed that a fragmentation instability will cause a star to break into two clumps (although more clumps could certainly be produced). Their estimates for the rms strain and power radiated by the orbiting binary fragments are
---- V~ 128 G mr2w2 hbin = ------------- (4) 5 c4 d
128 G 2 4 6 Pbin = -5-c5 m r w . (5)
Here m is the mass of a single fragment, 2r is the separation of the fragments, and w is their orbital frequency.

For their computation of the GWs radiated via r-modes, FHH used the method of Ho and Lai [115] (which assumes amax = 1) and calculated only the emission from the dominant m = 2 mode. This approach is detailed in FHH. If the neutron star mass and initial radius are taken to be 1.4Mo. and 12.53 km, respectively, the resulting formula for the average GW strain is

( ) -24 ( ns ) 20 Mpc h(t) = 1.8 × 10 a ------ -------- , (6) 1 kHz d
where a is the mode amplitude and ns is the spin frequency.

FHH’s numerical quadrupole estimate of the GWs from polar oscillations in the AIC simulation of Fryer et al. [82Jump To The Next Citation Point] predicts a peak dimensionless amplitude -24 hpk = 5.9× 10 (for d = 100 Mpc). The energy 45 EGW = 3 × 10 erg is emitted at a frequency of fGW ~~ 50 Hz. This amplitude is about an order of magnitude too small to be observed by the advanced LIGO-II detector. The sensitivity curve for the broadband configuration of the LIGO-II detector is shown in Figure 2View Image (see Appendix A of [86Jump To The Next Citation Point] for details on the computation of this curve). Note that the characteristic strain h is plotted along the vertical axis in Figure 2View Image (and in LISA’s sensitivity curve, shown in Figure 18View Image). For burst sources, h = hpk. For sources that persist for N cycles, V~ --- h = N hpk.

View Image

Figure 2: A comparison between the GW amplitude h(f) for various sources and the LIGO-II sensitivity curve. See the text for details regarding the computations of h. The AIC sources are assumed to be located at a distance of 100 Mpc; 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).
The simulation of Fryer et al. [82Jump To The Next Citation Point] considered by FHH does produce an object with an off-center density maximum. However, because the maximum b reached in the AIC simulation of Fryer et al. was < 0.06 (and because the degree of differential rotation present in the remnant was low), the collapsing object is not likely to encounter dynamical rotational instabilities. The choice of initial angular momentum J for the progenitor in an AIC simulation can affect this outcome. Fryer et al.’s choice of 49 2 - 1 J = 10 g cm s is such that the period of the white dwarf progenitor is 10 s less than the shortest observed period 30 s of a cataclysmic variable white dwarf [139]. Higher angular momenta are certainly possible. If higher values of J exist in accreting white dwarfs, bar-mode instabilities may be more likely to occur (see the discussion of work by Liu and Lindblom below).

According to FHH, the remnant of this AIC simulation will be susceptible to r-mode growth. Assuming amax ~ 1 (which is likely not physical; see Section 2.3), they predict EGW > 1052 erg. FHH compute h(fGW) for coherent observation of the neutron star as it spins down over the course of a year. For a neutron star located at a distance of 100 Mpc, this track is always below the LIGO-II noise curve. The point on this track with the maximum h, which corresponds to the beginning of r-mode evolution, is shown in Figure 2View Image. The track moves down and to the left ( i.e., h and f decrease) in this figure as the r-mode evolution continues.

2.4.2 Analyses of equilibrium representations of collapse

In addition to full hydrodynamics collapse simulations, many studies of gravitational collapse have used hydrostatic equilibrium models to represent stars at various stages in the collapse process. Some investigators use sequences of equilibrium models to represent snapshots of the phases of collapse (e.g., [16Jump To The Next Citation Point185Jump To The Next Citation Point156Jump To The Next Citation Point]). Others use individual equilibrium models as initial conditions for hydrodynamical simulations (e.g., [231192Jump To The Next Citation Point184Jump To The Next Citation Point49Jump 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 [156Jump To The Next Citation Point155Jump 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 [156Jump 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 [98]. 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 rc = 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 rc = 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 [82]. The realistic equation of state used to construct the white dwarfs is a Coulomb corrected, zero temperature, degenerate gas equation of state [21052].

In the second step of their process, Liu and Lindblom [156Jump 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 [156Jump 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 b ranging from 0.23 -0.26. It is interesting to compare these results with those of Zwerger and Müller [271Jump To The Next Citation Point]. Zwerger and Müller performed axisymmetric hydrodynamics simulations of stars with polytropic equations of state (P oc rG). Their initial models were G = 4/3 polytropes, representative of massive white dwarfs. All of their models started with rc = 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 b < 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. Zwerger and Müller’s work will be discussed in much more detail in Section 3, as it was performed in the context of core collapse supernovae.

In a continuation of the work of Liu and Lindblom, Liu [155Jump To The Next Citation Point] used linearized hydrodynamics to perform a stability analysis of the cold neutron star AIC remnants of Liu and Lindblom [156]. 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 b = 0.26. Note that the m = 1 mode, observed by others to be the dominant mode in unstable models with values of b much lower than 0.27 [24726019249Jump To The Next Citation Point], did not grow in his simulation. Because Liu and Lindblom’s Models I and II had lower values of b, Liu identified the onset of instability for neutron stars formed via AIC as bd ~~ 0.25.

Liu estimated the peak amplitude of the GWs emitted by the Model III remnant to be hpk ~~ 1.4 × 10- 24 and the LIGO-II signal-to-noise ratio (for a persistent signal like that seen in the work of [184] and [34]) 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 tGW ~ 7 s (~ 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 2View Image). Details of the approximations on which these estimates are based can be found in [155Jump To The Next Citation Point].

Liu cautions that his results hold if the magnetic field of the proto-neutron star is B < 1012 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 8 > 10 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 ~ 107 - 3× 108 G [256].

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