Eccentric black hole-neutron star mergers: Effects of black hole spin and equation of state

There is a high level of interest in black hole-neutron star binaries, not only because their mergers may be detected by gravitational wave observatories in the coming years, but also because of the possibility that they could explain a class of short duration gamma-ray bursts. We study black hole-neutron star mergers that occur with high eccentricity as may arise from dynamical capture in dense stellar regions such as nuclear or globular clusters. We perform general relativistic simulations of binaries with a range of impact parameters, three different initial black hole spins (zero, aligned and antialigned with the orbital angular momentum), and neutron stars with three different equations of state. We find a rich diversity across these parameters in the resulting gravitational wave signals and matter dynamics, which should also be reflected in the consequent electromagnetic emission. Before tidal disruption, the gravitational wave emission is significantly larger than perturbative predictions suggest for periapsis distances close to effective innermost stable separations, exhibiting features reflecting the zoom-whirl dynamics of the orbit there. Guided by the simulations, we develop a simple model for the change in orbital parameters of the binary during close-encounters. Depending upon the initial parameters of the system, we find that mass transfer during nonmerging close-encounters can range from essentially zero to a sizable fraction of the initial neutron star mass. The same holds for the amount of material outside the black hole post-merger, and in some cases roughly half of this material is estimated to be unbound. We also see that nonmerging close-encounters generically excite large oscillations in the neutron star that are qualitatively consistent with $f$-modes.

Figure 1

$L^2$-norm of the constraint violation, $C_a\equiv H_a-\square x_a$, in units of $1/M$ from the $r_p=10M$ case in the $100M\times 100M$-region around the center of mass in the equatorial plane (i.e. $\sqrt{\int \Vert C_a{\Vert }^2{d}^{2}x/\int d^{2}x}$). This is shown for low, medium, and high resolutions for the standard initial separation between the BH and NS of $d=50M$; the relative magnitudes at a given time are consistent to good approximation with second-order convergence. Also shown is a medium resolution run with an initial separation of $d=100M$. The time is shifted so that the point of closest approach occurs at approximately $t=0$ for all cases.

Figure 2

Trajectories of the NS center-of-mass for various simulations. The $r_p=7.0$ with 2H EOS, $r_p=7.0$ with B EOS, and $r_p=5.5$ $a=0.5$ (HB EOS) both undergo a close-encounter followed by a short elliptic orbit before merging. Note that while in both the $r_p=7.0$-cases the NS approaches the BH on essentially the same orbit, the dynamics around the close-encounter and subsequent orbits are very different due to EOS effects (see Sec. 4c). The $r_p=10$ (HB EOS) undergoes a long-period elliptical orbit following the initial periapsis passage. The remainder of the cases shown merge on the first encounter while displaying various degrees of whirling behavior.

Figure 3

Rest-mass density in the equatorial plane from various BH-NS simulations, left to right, top to bottom: (1) the BH and NS undergoing a close-encounter ($t=242M$, $r_p=7.0$, B EOS), (2) the NS undergoing a whirling-phase before merging ($t=269M$, $r_p=8.13$, $a=-0.5$), (3) the NS stretched into a long tidal stream during merger ($t=506M$, $r_p=7.5$, $e=0.75$), (4) a mass-transfer episode ($t=292M$, $r_p=7.0$, 2H EOS), (5) towards the end of a mass-transfer episode during the NS’s first periapsis passage ($t=272M$, $r_p=5.5$, $a=0.5$), and (6) a nascent accretion disk ($t=388M$, $r_p=5$, $a=0.5$). Recall that $r_p$ is reported in units of total mass $M$. The color scale is logarithmic from ${10}^{-6}$ to 1 times the initial maximum density (${\rho }_{\max }=8.3(3.7,9.8)\times {10}^{14}\mathrm{g}{\mathrm{cm}}^{-3}$ for the HB (2H, B) EOS). The BH is roughly the same coordinate size (with diameter $\approx 3M$) in all panels, which can be used to infer the relative scale of each snapshot.

Figure 4

Convergence of the NS trajectory for the $r_p=10$, $a=0$-case (see Sec. 4a). What is plotted is the magnitude of the difference in the center of rest-mass position of the NS between the low and medium-resolution runs, and between the medium and high-resolution runs scaled assuming second-order convergence. The close-encounter occurs near $t=230M$, where the largest “perturbation” of the orbit due to truncation error happens; the relatively large growth of the difference in trajectories following this is mostly just a reflection of this change in the orbital parameters that occurred around periapsis.

Figure 5

Effect of resolution on the disruption and subsequent accretion of NS material, as measured by the total rest-mass exterior to the BH horizon, for the $r_p=5$, $a=0.5$-case (see Sec. 4b). The amount of rest-mass remaining at late times at the different resolutions is consistent with approximately second-order convergence.

Figure 6

Top: The amount of rest-mass (normalized to the total rest-mass of the NS) accreted by the BH as a function of time for the $r_p=7.0$ 2H-case and for the $r_p=5.5$, $a=0.5$ HB-case. In both cases there are two significant episodes of mass transfer. Bottom: The angular momentum of the BH horizon in units of the total ADM-mass squared, $M^2$, as a function of time for the same two cases.

Figure 7

The fraction of rest-mass outside a given radius $(1/M_0){\int }_r^R\rho dV$ for an isolated star with the HB or 2H EOS and $M=1.35M_{\odot }$. Here radius $r$ is normalized to the radius of the HB star ($R_{\mathrm{HB}}$). The horizontal, dotted-lines indicate the amount of material accreted into the BH during the initial close-encounter for the $r_p=5.5$-case with HB EOS (bottom) and the $r_p=7.0$-case with 2H EOS (top) as shown in Fig. 6.

Figure 8

NSs exhibiting large-amplitude $f$-mode oscillations following the first interaction with the (nonspinning) BH. The cases shown are $r_p=6.95$ with the HB EOS at $t=341M$ (left) and $r_p=7.0$ with the 2H EOS at $t=356M$ (for the latter case, see the bottom left-panel of Fig. 3 for a snapshot of the density near the time of closest approach to the BH when this large oscillation is excited). The color map shows the rest-mass density in the equatorial plane on a logarithmic scale, normalized to the instantaneous maximum density. The arrows show the velocity in the NS center-of-mass frame. The longest velocity arrows correspond to velocity magnitudes 0.084 (left) and 0.17 (right). Though the two cases have similar periapsis radii, the lower compaction in the 2H-case leads to a much stronger tidal interaction and an $f$-mode, which shows nonlinear characteristics.

Figure 9

The real and imaginary components (black diamonds and red squares) of the $l=2$, $m=2$ spherical-harmonic of $r{\Psi }_4$ for $e=0.75$ and $r_p=8.125$ (top) and $r_p=10$ (bottom). For comparison the NQA analytical results are shown multiplied by an overall factor so that the magnitude and phase match at peak ($t=0$).

Figure 10

The real and imaginary components (solid, black-lines and red, dotted-lines) of ${\Psi }_4$ on the $z$-axis (perpendicular to the orbital plane) during merger for the following cases (top to bottom): $r_p=5$, HB, $r_p=6.81$, HB; $r_p=7.5$, $e=0.75$, HB;$r_p=8.13$, $a=-0.5$, HB; and $r_p=7.0$, 2H (see Fig. 11 for the full signal). The waveforms are aligned so that the peak occurs at $t=0$ with zero phase.

Figure 11

The real and imaginary components (solid, black-lines and red, dotted-lines) of ${\Psi }_4$ on the $z$-axis (perpendicular to the orbital plane) at $r=70M$ for the $r_p=7.0$ 2H simulation. The large burst at $t\approx 300M$ comes from the initial fly-by, where the NS becomes extremely distorted (Fig. 3) and loses a significant portion of its mass (Fig. 6). The smaller pulse at $t=600M$ comes from the merger. In between, there is a smaller component of the signal coming from the tidally induced oscillation of the NS.

Figure 12

Top: Energy lost to GWs during the initial close-encounter (i.e. excluding merger) as a function of $r_p$ for initial BH-spin $a=0$,$-0.5$, and 0.5 and for initial eccentricity $e=1$ and 0.75. The functional form $E_0(1-(\delta r_p/\Delta {)}^{\gamma })$ (lines) motivated by zoom-whirl dynamics is a fit to the simulation results (points). $\delta r_p=r_p-r_c$ where$r_c$ is the threshold value for merger during the encounter. $E_0$ is the difference in energy between a quasicircular orbit and an $e\approx 1$ (0.75) orbit, both having $r_p=r_c$. $\Delta$ is the range over which zoom-whirl-like behavior dominates the GW-emission energetics. $\gamma$ is a parameter that in the geodesic analogue is related to the instability exponent of the corresponding unstable circular orbit, here, we use it as our fitting parameter. We obtain $\gamma =0.19$, 0.13, 0.25, and 0.16 for $(e,a)=(1,0)$, ($1,-0.5$), (1,0.5), and (0.75,0), respectively. Bottom: This shows the same simulation data points as the top figure, though here we use the $e=1$, $a=0$case (solid, black points) to determine the free parameters for the method described in the text to extrapolate the values of $r_c$ and $\gamma$ to the other three cases.

Figure 13

The evolution of the eccentricity and periapsis separation of various $4:1$ mass ratio BH-NS binaries that begin marginally unbound and undergo a series of close-encounters (large red points) before merging. For comparison, we also plot the results using the NQA expressions from 14, 65 (magenta $x$’s). We also plot the critical eccentricity for a given $r_p$ for a close-encounter to result in merger (blue, dotted-line). Hence, the points below this curve correspond to merger events. From left to right the plots correspond to an initial BH-spin of $a=-0.5$, 0, and 0.5, respectively.