Axisymmetric general relativistic simulations of the accretion-induced collapse of white dwarfs

The accretion-induced collapse (AIC) of a white dwarf may lead to the formation of a protoneutron star and a collapse-driven supernova explosion. This process represents a path alternative to thermonuclear disruption of accreting white dwarfs in type Ia supernovae. In the AIC scenario, the supernova explosion energy is expected to be small and the resulting transient short-lived, making it hard to detect by electromagnetic means alone. Neutrino and gravitational-wave (GW) observations may provide crucial information necessary to reveal a potential AIC. Motivated by the need for systematic predictions of the GW signature of AIC, we present results from an extensive set of general-relativistic AIC simulations using a microphysical finite-temperature equation of state and an approximate treatment of deleptonization during collapse. Investigating a set of 114 progenitor models in axisymmetric rotational equilibrium, with a wide range of rotational configurations, temperatures and central densities, and resulting white dwarf masses, we extend previous Newtonian studies and find that the GW signal has a generic shape akin to what is known as a “type III” signal in the literature. Despite this reduction to a single type of waveform, we show that the emitted GWs carry information that can be used to constrain the progenitor and the postbounce rotation. We discuss the detectability of the emitted GWs, showing that the signal-to-noise ratio for current or next-generation interferometer detectors could be high enough to detect such events in our Galaxy. Furthermore, we contrast the GW signals of AIC and rotating massive star iron core collapse and find that they can be distinguished, but only if the distance to the source is known and a detailed reconstruction of the GW time series from detector data is possible. Some of our AIC models form massive quasi-Keplerian accretion disks after bounce. The disk mass is very sensitive to progenitor mass and angular momentum distribution. In rapidly differentially rotating models whose precollapse masses are significantly larger than the Chandrasekhar mass, the resulting disk mass can be as large as $\sim 0.8M_{\odot }$. Slowly and/or uniformly rotating models that are limited to masses near the Chandrasekhar mass produce much smaller disks or no disk at all. Finally, we find that the postbounce cores of rapidly spinning white dwarfs can reach sufficiently rapid rotation to develop a gravitorotational bar-mode instability. Moreover, many of our models exhibit sufficiently rapid and differential rotation to become subject to recently discovered low-$E_{\mathrm{rot}}/|W|$-type dynamical instabilities.

Figure 1

Average electron fraction $Y_e$ in the innermost 2 km in the collapsing WD as a function of density obtained from 2D MGFLD simulations with the VULCAN/2D code for models $1.46M_{\odot }$ and $1.92M_{\odot }$ of Dessart et al. [28]. Both models were set up with the same initial dependence of temperature on density and a temperature $T_0=1.0\times {10}^{10}\mathrm{K}$ (see Sec. 2b for details).

Figure 2

The radial profile of the rest-mass density for nonrotating white dwarf models AU0, BU0, CU0, and DU0.

Figure 3

Upper panels: angular velocity as a function of equatorial radius (left panel) and enclosed mass coordinate (right panel) for three representative precollapse WD models AD3, AD5, and AD10. Lower panels: angular velocity normalized to the local Keplerian value as a function of equatorial radius and enclosed mass for the same models.

Figure 4

Color map of the rest-mass density for the precollapse white dwarf models AD1 (left panel), AD5 (center panel), and AD10 (right panel). The apparent ruggedness of the WD surface layers is a result of the finite resolution of our computational grid and the mapping procedure in the visualization tool. The ruggedness has no influence on the collapse and postbounce dynamics of the inner core.

Figure 5

Angular velocity as a function of equatorial radius for model AD10 and varying values of the dimensionless shear parameter $f_{\mathrm{sh}}$ that controls the rate at which the angular velocity increases with $\varpi$ in the region $\varpi <{\varpi }_{\mathrm{p}}$.

Figure 6

Parameter ${\beta }_{\mathrm{i}}$ versus central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$ for our AIC progenitor WD model set.

Figure 7

Evolution of the maximum (central) density for the nonrotating low-$T$ models AU0, BU0, CU0, and DU0. The inset plot displays a zoomed-in view of the maximum density around the time of core bounce on a linear scale. As is clearly discernible from this figure, the collapse dynamics in the plunge and bounce phase are essentially independent of the initial WD central density. Time is normalized to the time of bounce $t_{\mathrm{b}}$.

Figure 8

Times to core bounce from the onset of collapse as a function of initial central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. Shown are the results of the high-$T$ model sequence (the low-$T$ models exhibit identical systematics). Models denoted by an unfilled (filled) circle undergo a pressure-dominated bounce with (without) significant prompt postbounce convection. Models marked by a cross undergo centrifugal bounce at subnuclear densities, and models marked with a small (large) symbol are of set A (D). The colors correspond to various precollapse rotational configurations (see the legend in Fig. 9). Note that due to their much higher initial compactness, the high-density D models (shown in the inset plot) have much shorter collapse times than their lower-density A counterparts.

Figure 9

The maximum density ${\rho }_{\max ,\mathrm{b}}$ at bounce as a function of the inner-core parameter ${\beta }_{\mathrm{ic},\mathrm{b}}$ at bounce for the entire set of high-$T$ AIC models. Because of the increasing role of centrifugal support, ${\rho }_{\max ,\mathrm{b}}$ decreases monotonically with increasing rotation (see the main text for details). The symbol convention for the various sets is explained in the caption of Fig. 8.

Figure 10

The inner-core parameter ${\beta }_{\mathrm{ic},\mathrm{b}}$ at the time of bounce for all low-$T$ AIC models plotted as a function of the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. For models with slow to moderately rapid rotation, ${\beta }_{\mathrm{ic},\mathrm{b}}$ increases roughly linearly with ${\Omega }_{\mathrm{c},\mathrm{i}}$ (at fixed ${\rho }_{\mathrm{c},\mathrm{i}}$ and rotation law). In more rapidly rotating models, the growth of ${\beta }_{\mathrm{ic},\mathrm{b}}$ saturates at $\sim 24.5\%$, and further increase of the progenitor rotation results in a decrease of ${\beta }_{\mathrm{ic},\mathrm{b}}$. Since D models experience less spin-up during collapse than A models, an increase of ${\rho }_{\mathrm{c},\mathrm{i}}$ at fixed ${\Omega }_{\mathrm{c},\mathrm{i}}$ and rotation law results in a decrease of ${\beta }_{\mathrm{ic},\mathrm{b}}$. A uniformly rotating model with a given ${\Omega }_{\mathrm{c},\mathrm{i}}$ and ${\rho }_{\mathrm{c},\mathrm{i}}$ reaches smaller ${\beta }_{\mathrm{ic},\mathrm{b}}$ than the differentially rotating model with the same ${\Omega }_{\mathrm{c},\mathrm{i}}$ and ${\rho }_{\mathrm{c},\mathrm{i}}$. The symbol convention for the various sets is explained in the caption of Fig. 8.

Figure 11

Mass $M_0(r_{\mathrm{e}})$ in units of solar masses of the WD inner region plotted as a function of the equatorial radial coordinate $r_{\mathrm{e}}$ for a number of AIC models with slow and rapid rotation as well as high and small central densities. The inset plot shows the same on a linear radial scale. The mass distribution of the inner $M_0(r_{\mathrm{e}})\lesssim 0.5M_{\odot }$ region is largely independent of the rotational configuration, while an increase (decrease) of the central density leads to homologous contraction (expansion) of the inner regions. See Sec. 2c for details of the model setups.

Figure 12

Angular velocity profiles in the equatorial plane at the time of core bounce and time evolution of the central density ${\rho }_{\mathrm{c}}$ (in the inset) for models AU2 and DU7. The initial angular velocity of model DU7 is larger by a factor of $\sim 5.3$ than that of model AU2, but the latter experiences a $\sim 5.3$ greater spin-up during collapse. As a result, these models produce inner cores with almost identical rotational configurations and similar masses in the bounce phase. This is reflected in an identical evolution of the central densities at bounce.

Figure 13

Mass $M_{\mathrm{ic},\mathrm{b}}$ of the inner core at bounce for all high-$T$ models versus the parameter ${\beta }_{\mathrm{ic},\mathrm{b}}$ of the inner core at bounce. The systematics of $M_{\mathrm{ic},\mathrm{b}}$ with ${\beta }_{\mathrm{ic},\mathrm{b}}$ are identical for the set of low-$T$ models, but their $M_{\mathrm{ic},\mathrm{b}}$ are generally $\sim 10\%$ larger. The symbol convention for the various sets is explained in the caption of Fig. 8.

Figure 14

Profiles of radial velocity (top panels) and specific entropy per baryon (bottom panels) at different postbounce times for AIC model DD7.

Figure 15

Evolution of the dimensionless GW strain $h$ (in units of ${10}^{-21}$ at a source distance of 10 kpc) as a function of postbounce time for representative models with different precollapse rotation profiles, central densities, and temperatures (low-$T$ models with solid black lines and high-$T$ with dashed red lines). Models with slow and (almost) uniform precollapse rotation (e.g., AD1f4 or DU3) develop considerable prompt postbounce convection visible as a dominating lower-frequency contribution in the waveform. Centrifugal effects damp this prompt convection and the waveforms of models with moderately rapid rotation (e.g., AD1, AU4, DD7, and AD8) and of rapidly rotating models (e.g., AD9 or AD10) exhibit no such contribution to the signal.

Figure 16

Peak value $|h{|}_{\max }$ of the GW amplitude at a source distance of 10 kpc distance for the burst signal of all models versus the parameter ${\beta }_{\mathrm{ic},\mathrm{b}}$ of the inner core at the time of bounce. At slow to moderately rapid rotation, $|h{|}_{\max }$ scales almost linearly with ${\beta }_{\mathrm{ic},\mathrm{b}}$ (marked by the dotted straight line), while for ${\beta }_{\mathrm{b}}\gtrsim 16\%$ centrifugal effects reduce $|h{|}_{\max }$. The symbol convention for the various sets is explained in the caption of Fig. 8.

Figure 17

Spectral energy density of the GW signal for representative AIC models AD1f3 (top panel), AD4 (center panel), and AD10 (bottom panel). $f_{\max }$ is the peak frequency of the GWs emitted at core bounce.

Figure 18

Frequency $f_{\max }$ at the maximum of the GW spectral energy density in pressure-dominated bounce and a subset of centrifugal bounce models versus the parameter ${\beta }_{\mathrm{ic},\mathrm{b}}$ of the inner core at bounce. The symbol convention for the various sets is explained in the caption of Fig. 8.

Figure 19

Time-frequency color maps of the GW signals of models AD1f3 (type IIIa), AD4 (type IIIb), and AD10 (type IIIc, see also Fig. 17). Plotted is the “instantaneous” spectral GW energy density $d{E}_{\mathrm{GW}}/df$ in a 2 ms Gaussian window as a function of postbounce time. Note the prebounce low-frequency contribution in the moderately rapidly rotating models (model AD4, center panel) and rapidly rotating models (model AD10, right panel). The range of the colormap of the left panel (model AD1f3) is smaller by one dex than those of the other panels.

Figure 20

Evolution of GW signals in the high-$T$ AIC models DU2, AU5, AD10 (black solid lines) and three massive star iron-core-collapse models s20A1O05, s20A3O07, and s20A3O15 (red dashed lines) from the model set of Dimmelmeier et al. [50]. The AIC model DU2 and the iron-core-collapse model s20A1O05 undergo pressure-dominated bounce with significant prompt convection. Models AU5 and s20A3O7 experience pressure-dominated bounce without significant convection, and models AD10 and s20A3O15 undergo centrifugal bounce. The inner cores of models DU2, AU5, and AD10 (s20A1O05, s20A3O07, and s20A3O15) reach values of ${\beta }_{\mathrm{ic},\mathrm{b}}$ of about 0.4%, 10.2%, and 21.3% (0.7%, 10.1%, and 21.6%). Times are given relative to the time of core bounce which we mark with a vertical line.

Figure 21

Detector-dependent characteristic amplitudes of the GW signals of all models at an assumed distance of 10 kpc. The symbol convention for the various sets is described in the caption of Fig. 8. See the text for a discussion of the numbers and arrows.

Figure 22

Location of the GW signals from core bounce in the $h_{\mathrm{c}}-f_{\mathrm{c}}$ plane relative to the sensitivity curves of various interferometer detectors (as color-coded) for an extended set of models AD. The sources are at a distance of 10 kpc for LIGO, 0.8 Mpc for advanced LIGO, and 5 Mpc for the Einstein telescope.

Figure 23

The inner-core parameter ${\beta }_{\mathrm{ic},\mathrm{b}}$ at bounce is plotted as a function of the precollapse parameter ${\beta }_{\mathrm{i}}$ for high-$T$ models. Because of different central densities and rotation profiles of the precollapse WD models, there is no one-to-one correspondence between ${\beta }_{\mathrm{ic},\mathrm{b}}$ and ${\beta }_{\mathrm{i}}$. Hence, although one can extract ${\beta }_{\mathrm{ic},\mathrm{b}}$ accurately from the bounce AIC GW signal, it is impossible to put strong constraints on ${\beta }_{\mathrm{i}}$ using the GW signal.