Merger of black hole and neutron star in general relativity: Tidal disruption, torus mass, and gravitational waves

We study the properties of the merger of black hole–neutron star (BH-NS) binaries in fully general relativistic simulation, focusing on the case that the NS is tidally disrupted. We prepare BH-NS binaries in a quasicircular orbit as the initial condition in which the BH is modeled by a nonspinning moving puncture. For modeling the NS, we adopt the $\Gamma$-law equation of state with $\Gamma =2$ and the irrotational velocity field. We change the BH mass in the range $M_{\mathrm{BH}}\approx 3.3M_{\odot }–4.6M_{\odot }$, while the rest mass of the NS is fixed to be $M_{*}=1.4M_{\odot }$ (i.e., the NS mass $M_{\mathrm{NS}}\approx 1.3M_{\odot }$). The radius of the corresponding spherical NS is set in the range $R_{\mathrm{NS}}\approx 12–15\mathrm{km}$ (i.e., the compactness $G{M}_{\mathrm{NS}}/R_{\mathrm{NS}}{c}^2\approx 0.13–0.16$). We find for all the chosen initial conditions that the NS is tidally disrupted near the innermost stable circular orbit. The numerical results indicate that, for the model of $R_{\mathrm{NS}}=12\mathrm{km}$, more than $\sim 95\%$ of the rest mass is quickly swallowed by the BH and the resultant torus mass surrounding the BH is less than $\sim 0.05M_{\odot }$. For the model of $R_{\mathrm{NS}}\approx 14.7\mathrm{km}$, by contrast, the torus mass is $\sim 0.15M_{\odot }$ for the BH mass $\approx 4M_{\odot }$. The thermal energy of the material in the torus increases due to the shock heating that occurs in the collision between the spiral arms, resulting in the temperature ${10}^{10}–{10}^{11}\mathrm{K}$. Our results indicate that the merger between a low-mass BH and its companion NS may form a central engine of short gamma-ray bursts of the total energy of order ${10}^{49}\mathrm{ergs}$ if the compactness of the NS is fairly small, $\lesssim 0.145$. However, for the canonical values $M_{\mathrm{NS}}=1.35M_{\odot }$ and $R_{\mathrm{NS}}=12\mathrm{km}$, the merger results in small torus mass, and hence it can be a candidate only for the low-energy, short gamma-ray bursts of total energy of order ${10}^{48}\mathrm{ergs}$. We also present gravitational waveforms during the inspiral, tidal disruption of the NS, and subsequent evolution of the disrupted material. We find that the amplitude of gravitational waves quickly decreases after the onset of tidal disruption. Although the quasinormal mode is excited, its gravitational wave amplitude is much smaller than that of the late inspiral phase. This is reflected in the fact that the spectrum amplitude sharply falls above a cutoff frequency which is determined by the tidal disruption process. We also find that the cutoff frequency is 1.25–1.4 times larger than the frequency predicted by the study for the sequence of the quasicircular orbits, and this factor of the deviation depends on the compactness of the NS.

Figure 1

Angular momentum ($J/M_0^2$) vs angular velocity ($M_0\Omega$) for model A (filled circle). [For models B and C, which have the same mass ratio as that of model A ($q=M_{\mathrm{NS}}/M_{\mathrm{BH}}=1/3.06$), the location of the plots is approximately the same.] For comparison, the relation for the binary of $q=1/3.06$ in the third PN theory (solid curve) and numerical results of Ref. 3 for $q=1/3$ are shown together (note that the mass ratio $q$ is slightly different from that of models A–C, and hence the relation for $q=1/3$ in the third PN theory is plotted together by the dotted curve). Because $J/M_0^2$ is approximately proportional to the ratio of reduced mass to total mass, the angular momentum for the hypothetical sequence of $q=1/3.06$ computed by the formulation of Ref. 3 would be by $\approx 1\%$ systematically smaller than those presented in this figure, and would approximately coincide with our results within 1% disagreement.

Figure 2

Snapshots of the density contour curves for $\rho$ and the three-velocity for $v^i(=d{x}^i/dt)$ in the equatorial plane for model A. The solid contour curves are drawn for $\rho =2i\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ ($i=1–4$) and for ${10}^{14-i}\mathrm{g}/{\mathrm{cm}}^3$ ($i=1\sim 5$). The maximum density at $t=0$ is $\approx 8.86\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$. (The blue, cyan, magenta, and green curves denote ${10}^{14}$, ${10}^{13}$, ${10}^{12}$, and ${10}^{11}\mathrm{g}/{\mathrm{cm}}^3$, respectively.) The vectors shows the velocity field $(v^x,v^y)$, and the scale is shown in the upper right-hand corner. The thick (red) circles are the apparent horizons. The second and third panels denote the states after one orbit and one-and-half orbits, respectively. The last panel plots the density contours and velocity vectors in the $x=0$ plane at the same time as that for the eighth panel.

Figure 3

Profiles of ${\epsilon }_{\mathrm{th}}$, ${\epsilon }_{\mathrm{th}}/\epsilon$, and $\rho$ along the $x$ axis (solid curves) and $y$ axis (dashed curves) at $t=7.117\mathrm{ms}$ (left), 9.070 ms (middle), and 11.02 (right).

Figure 4

Profiles of ${\epsilon }_{\mathrm{th}}$, ${\epsilon }_{\mathrm{th}}/\epsilon$, and $\rho$ along the $x$ axis (solid curves) and $y$ axis (dashed curves) at $t=11.02$ for models B (left panel) and C (right panel).

Figure 5

The same as Fig. 1 but for models B and C at $t=11.02\mathrm{ms}$. The upper two panels plot the contour plots and velocity vectors for the $x\mathrm{\text{−}}y$ and $y\mathrm{\text{−}}z$ planes for model B. The lower two panels show the plots for model C.

Figure 6

The evolution of the rest mass of the material located outside the apparent horizon (upper panel of each figure) and the evolution of the area of the apparent horizon in units of $16\pi M^2$ (lower panel of each figure) (a) for models A, D, and F (for $R_{\mathrm{NS}}=13.2\mathrm{km}$), (b) for models B and E (for $R_{\mathrm{NS}}=12\mathrm{km}$), and (c) for models A, B, and C (for $q\approx 0.33$). $t_{\mathrm{disr}}$ in the horizontal axis denotes the approximate time at which the tidal disruption sets in, and we determine it to be 4.5 ms for model A, 4.2 ms for model B, 4.3 ms for model C, 4.2 ms for model D, 3.9 ms for model E, and 5.6 ms for model F, respectively. (d) $M_{r>r_{\mathrm{AH}}}$ at the end of the simulations as a function of $R_{\mathrm{NS}}$.

Figure 7

(a) Upper panel: $+$ and $\times$ modes of gravitational waveforms observed along the $z$ axis for model A (solid and dashed curves, respectively). $t_{\mathrm{ret}}$ denotes the retarded time [see Eq. (21)] and $M_0$ is the total mass defined in the caption of Table . The amplitude at a distance of observer can be found from Eq. (22). (a) Lower panel: The real and imaginary parts of ${\psi }_4$ (solid and dashed curves). (b) The real (upper panel) and imaginary (lower panel) parts of ${\psi }_4$ for model A together with the hypothetical curves of gravitational waves associated with the QNM (dotted curves) for $5\mathrm{ms}\lesssim t_{\mathrm{ret}}\lesssim 6.5\mathrm{ms}$. (c) $h_{+}$ (upper panel) and $\mathrm{Re}({\psi }_4)$ (lower panel) for models A (dashed curve) and C (solid curve). (d) The real (solid curve) and imaginary (dashed curve) parts of ${\psi }_4$ for model B (upper panel). The waveforms for $4.8\mathrm{ms}\lesssim t_{\mathrm{ret}}\lesssim 6\mathrm{ms}$ are magnified in the lower panel together with the hypothetical curves of gravitational waves associated with the QNM (dotted curves). For all the plots, only the $l=|m|=2$ modes are taken into account.

Figure 8

(a) Spectrum of gravitational waves $h(f)f$ for model A (solid curve). The dashed and dotted curves are the spectra for the purely numerical (short-term) data and for the analytic third PN waveform. (b) The same as (a) but for models A, D, and F. The vertical dotted lines denote the expected frequencies at which the tidal disruption sets in for models D and F; $f_{\mathrm{tidal}}\approx 0.84\mathrm{kH}z$ for model F and 0.88 kHz for model D. (c) The same as (b) but for models D and E. The vertical dotted line denotes the expected frequency at which the tidal disruption sets in, $f_{\mathrm{tidal}}\approx 1.0\mathrm{kH}z$. (d) The same as (b) but for models A–C. For (b)–(d), the solid circles denote the location of $f_{\mathrm{cut}}$.

Figure 9

The same as Fig. 6c but for models A, C, A0, A1, A2, and C2. We choose the values of $t_{\mathrm{disr}}=4.30\mathrm{ms}$, 4.17 ms, 4.75 ms, and 4.25 ms for A2, A1, A0, and C2, respectively. For models A and C, see the caption of Fig. 6.

Figure 10

The same as Fig. 7c but (a) for models A (solid curves) and A2 (dashed curves) and (b) for models C (solid curves) and C2 (dashed curves), respectively.

Figure 11

The same as Fig. 9 but for models A (solid curves), Ab (long-dashed curves), and Ac (dashed curves). For models Ab and Ac, $t_{\mathrm{disr}}$ is chosen to be 3.4 and 2.85 ms, respectively.

Figure 12

The same as Fig. 10 but for models A (solid curves) and Ab (dashed curves). For model Ab, we plot $-h_{+}$ and $-h_{\times }$ as a function of $t_{\mathrm{ret}}+1\mathrm{ms}$.