Relativistic simulations of eccentric binary neutron star mergers: One-arm spiral instability and effects of neutron star spin

We perform general-relativistic hydrodynamical simulations of dynamical capture binary neutron star mergers, emphasizing the role played by the neutron star spin. Dynamical capture mergers may take place in globular clusters, as well as other dense stellar systems, where most neutron stars have large spins. We find significant variability in the merger outcome as a function of initial neutron star spin. For cases where the spin is aligned with the orbital angular momentum, the additional centrifugal support in the remnant hypermassive neutron star can prevent the prompt collapse to a black hole, while for antialigned cases the decreased total angular momentum can facilitate the collapse to a black hole. We show that even moderate spins can significantly increase the amount of ejected material, including the amount unbound with velocities greater than half the speed of light, leading to brighter electromagnetic signatures associated with kilonovae and interaction of the ejecta with the interstellar medium. Furthermore, we find that the initial neutron star spin can strongly affect the already rich phenomenology in the postmerger gravitational wave signatures that arise from the oscillation modes of the hypermassive neutron star. In several of our simulations, the resulting hypermassive neutron star develops the one-arm ($m=1$) spiral instability, the most pronounced cases being those with small but non-negligible neutron star spins. For long-lived hypermassive neutron stars, the presence of this instability leads to improved prospects for detecting these events through gravitational waves, and thus may give information about the neutron star equation of state.

Figure 1

Resolution study for the case with $r_p/M=8$ and $(a_{\mathrm{NS},1},a_{\mathrm{NS},2})=(0,0.4)$. The top panel shows the convergence of the $L2$ norm of the constraints $C_a≔\square x_a-H_a$, scaled assuming first-order convergence. Before the merger (when there are no shocks) and at later times, the convergence is closer to second order. The middle panels shows the magnitude and frequency (time derivative of the phase) of the $\ell =m=2$ component of ${\mathrm{\Psi }}_4$. During the turbulent-like postmerger phase, strict convergence of the amplitude of the waveform is lost, though there is still good agreement on the frequency. The bottom panel shows the amount of unbound rest mass, binned according to the asymptotic velocity, as a function of resolution.

Figure 2

Equatorial rest-mass density snapshots for representative cases. Top row ($r_p/M=5$, $a_{\mathrm{NS},1}=a_{\mathrm{NS},2}=0$) from left to right: ($t\approx 192M\approx 2.6\mathrm{ms}$) the NSs make contact; ($t\approx 216M\approx 2.9\mathrm{ms}$) a BH apparent horizon forms; ($t\approx 246M\approx 3.3\mathrm{ms}$) the BH accretes the small disk; ($t\approx 362M\approx 4.8\mathrm{ms}$) little matter is left outside the BH at the end of the simulation. Second row ($r_p/M=5$, $a_{\mathrm{NS},1}=a_{\mathrm{NS},2}=0.2$) from left to right: ($t\approx 192M\approx 2.6\mathrm{ms}$) the NSs make contact; ($t\approx 276M\approx 3.7\mathrm{ms}$) the two cores bounce, launching shocks and ejecting matter outwards; ($t\approx 621M\approx 8.3\mathrm{ms}$) the cores recoalesce and a HMNS forms with a bar-shaped core surrounded by an extended envelope and disk; ($t\approx 1140M\approx 15.2\mathrm{ms}$) the one-arm spiral instability is not evident. Third row ($r_p/M=6$, $a_{\mathrm{NS},1}=0.4$, $a_{\mathrm{NS},2}=0.0$) from left to right: ($t\approx 197M\approx 2.6\mathrm{ms}$) the NSs make contact (the spinning NS is the star on the left); ($t\approx 270M\approx 3.6\mathrm{ms}$) the two cores bounce while the spinning NS (now on the right) is being disrupted ejecting matter outward; ($t\approx 722M\approx 9.6\mathrm{ms}$) the cores recollapse forming a HMNS with an egg-shaped core, surrounded by an extended envelope and disk; ($t\approx 1165M\approx 15.5\mathrm{ms}$) the one-arm spiral instability is not evident. The radius of each NS prior to merger ($R_{\mathrm{NS}}\approx 12\mathrm{km}$) sets the scale.

Figure 3

Equatorial density snapshots for representative cases. Top row ($r_p/M=8$, $a_{\mathrm{NS},1}=a_{\mathrm{NS},2}=0.025$) from left to right: ($t\approx 212M\approx 2.8\mathrm{ms}$) the NSs make contact for the first time; ($t\approx 277M\approx 3.7\mathrm{ms}$) the two stars merge, shearing and ejecting matter outward from the outer edges of the two stars; ($t\approx 591M\approx 7.86\mathrm{ms}$) shortly after a HMNS forms with an ellipsoidal core surrounded by an extended envelope and disk; ($t\approx 1160M\approx 15.4\mathrm{ms}$) the one-arm spiral instability develops (see Sec. 3d) giving rise to an $m=1$ deformation. Bottom row ($r_p/M=8$, $a_{\mathrm{NS},1}=a_{\mathrm{NS},2}=-0.4$) from left to right: ($t\approx 207M\approx 2.8\mathrm{ms}$) the NSs make contact for the first time; ($t\approx 260M\approx 3.5\mathrm{ms}$) an apparent horizon is found for the first time; ($t\approx 288M\approx 3.8\mathrm{ms}$) the BH accretes matter while some matter is ejected; ($t\approx 500M\approx 6.6\mathrm{ms}$) little matter ($\sim 10^{-3}{M}_{\odot }$) is left outside the BH at the end of the simulation. The radius of each NS prior to merger ($R_{\mathrm{NS}}\approx 12\mathrm{km}$) sets the scale.

Figure 4

The fraction of internal energy that is thermal (as measured by an Eulerian observer) as a function of time for cases with $r_p/M=5$ and various spins. The truncated curves are the cases that promptly collapsed to BHs around merger. For the cases that do not promptly form BHs, most of the heating occurs later in the evolution, after the time when smaller spin cases have already collapsed to BHs.

Figure 5

The $l=m=2$ mode of the Newman-Penrose scalar ${\mathrm{\Psi }}_4$ representing the GWs. The top and bottom panels show the real part of ${\mathrm{\Psi }}_4$ from representative cases with $r_p/M=5$ and 8, respectively. The notation $a_{\mathrm{NS}1/2}=(A,B)$ implies that spin $a_{\mathrm{NS},1}=A$ and spin $a_{\mathrm{NS},2}=B$. Note the different vertical scale between the two panels.

Figure 6

The $l=m=2$ mode of the Newman-Penrose scalar ${\mathrm{\Psi }}_4$ representing the GWs. The top panel shows the real part of ${\mathrm{\Psi }}_4$ of the one $r_p/M=9$ case studied, and the bottom representative cases from $r_p/M=10$. The initial encounter in all these examples is a fly-by leading to a bound system, though only for the $r_p/M=9$, $a_{\mathrm{NS}1/2}=0.4$ case did we follow the subsequent evolution all the way through merger (here, as indicated by the GW signal, a second fly-by occurs roughly 10 ms after the first, and after that a couple of grazing close encounters before the merger at $\approx 19\mathrm{ms}$). The notation $a_{\mathrm{NS}1/2}=(A,B)$ implies that spin $a_{\mathrm{NS},1}=A$ and spin $a_{\mathrm{NS},2}=B$. Note the different vertical and horizontal scales between the two panels.

Figure 7

The characteristic strain, defined as $h_c=|\tilde{h}|f$ where $\tilde{h}$ is the Fourier transform of the strain and $f$ is frequency, as observed at a distance of 50 Mpc and on axis. In the top panel we show the $r_p/M=8$, $a_{\mathrm{NS},1}=0$, $a_{\mathrm{NS},2}=0.4$ case at the three different resolutions, and, for the medium resolution case, the resulting curve when the waveform is extended from 12.5 to 26.3 ms. In the bottom panel we show various cases with different spins. For comparison, we also show the proposed broadband aLIGO sensitivity curve [94] and proposed Einstein Telescope (ET-D) sensitivity curves [95].

Figure 8

Distribution of the asymptotic velocity of unbound rest-mass, binned in increments of $0.05c$, and computed $\approx 7.0\mathrm{ms}$ postmerger for $r_p/M=5$ (top) and $r_p/M=8$ (bottom) and various spins.

Figure 9

Equatorial density (left) and ${\mathrm{\Omega }}_{xy}$ (right) snapshots at select times for $r_p/M=8$, $a_{\mathrm{NS},1}=a_{\mathrm{NS},2}=0.025$. From top to bottom, at $t\approx 3.1\mathrm{ms}$ the collision of the NSs creates a vortex sheet at the shear interface that subsequently breaks apart into multiple small scale vortices. Most notably, two larger vortices form near the surface of the star due to shearing between the HMNS and the tidal tails, evident to the left and right of center on the snapshots in the second row ($t\approx 4.5\mathrm{ms}$). At $t\approx 5.5\mathrm{ms}$ these two vortices are in the process of migrating towards the center, while other smaller vortices are stretched away. By $t\approx 6.5\mathrm{ms}$ the two vortices have merged into one central vortex, giving rise to an underdense rotation axis. The one-arm instability then sets in, and by $t\approx 14.6\mathrm{ms}$ it is fully developed, with the vortex now offset from the center and corotating about it.

Figure 10

Equatorial density (left) and ${\mathrm{\Omega }}_{xy}$ (right) snapshots at select times for $r_p/M=8$, $a_{\mathrm{NS},1}=0.05$, $a_{\mathrm{NS},2}=0.075$. The description is similar to that in the caption of Fig. 9, and from top to bottom the corresponding coordinate times are the same. Some of the main differences with the symmetric spin case are: (a) while both of the larger vortices inspiral toward the center of the HMNS, only one of them reaches the center to create a central underdensity, whereas the other one is stretched out and eventually dissipates as are the other smaller scale vortices; (b) in this asymmetric case the vorticity is overall slightly larger throughout the star.

Figure 11

The phase of $C_1$ (thick lines) as a function of $\varpi$ in the equatorial plane and center of mass frame for four low-spin $r_p/M=8$ cases. In each plot the early time chosen corresponds to the growth phase of the instability, the intermediate time is near the time of saturation of the $m=1$ mode, and the late time is after the $m=1$ mode dominates over the $m=2$ mode. For $\varpi /M\lesssim 1.5M$ the pattern of the mode exhibits almost rigidlike rotation, with the characteristic spiral feature evident at larger radii. Overlayed on these plots are equatorial density contours (thin dashed lines) at the time of the intermediate-time phase line, normalized to the maximum density value at that time. The numbers inlined in the contours indicate the value of the level surface. All plots use data from the high-resolution runs.

Figure 12

Amplitude of $C_m$ normalized to $C_0$ for various cases. The two top rows correspond to low-spin $r_p/M=8$ cases and a high spin $r_p/M=6$ case where the $m=1$ dominates over all other $m\ne 0$ modes by the termination of the simulations. The bottom row corresponds to an $r_p/M=5$ (left) and $r_p/M=9$ case (right). The $r_p/M=9$ case is the one that we followed through several close encounters before merging. After merger the $m=1$ mode grows for $r_p/M=5$ and $r_p/M=9$, but unlike the other cases it never dominates over the $m=2$ mode by the end of the simulations. The merger time in the $r_p/M=5$, $r_p/M=6$ and $r_p/M=8$ cases is $\sim 3.0\mathrm{ms}$, whereas it is $\sim 18.0\mathrm{ms}$ in the $r_p/M=9$ case. Plots corresponding to $r_p/M=8$ use data from the high-resolution runs.

Figure 13

Amplitude of several spherical harmonic components of the GW signal for various cases following merger. These correspond to the same cases shown in Fig. 12. The $r_p/M=5$, $r_p/M=6$ and $r_p/M=8$ cases merge at $t-r\sim 3.0\mathrm{ms}$, while the $r_p/M=9$ case merges at $t-r\sim 18.0\mathrm{ms}$. Plots from the $r_p/M=8$ cases show data from the high-resolution runs.

Figure 14

Characteristic strain of a portion of the GW signal beginning at $t\approx 10\mathrm{ms}$, after the onset of the one-arm instability, and lasting for $\sim 15\mathrm{ms}$, as would be seen by an edge-on observer (in contrast to the on-axis signal shown in Fig. 7). Shown are various low-spin cases with $r_p/M=8$, as well as the aLIGO and proposed Einstein Telescope (ET-D) sensitivity curves [95] at a distance of 10 Mpc. The HMNS and the one-arm instability GW signal presumably last much longer than the 15 ms represented here (possibly on the order of $t_{\mathrm{HMNS}}\sim 0.1–1\mathrm{s}$) in which case the GW power would be multiplied by $t_{\mathrm{HMNS}}/(15\mathrm{ms})$, and the SNR for this part of the signal by roughly the square root of this factor.

Figure 15

Left: Azimuthally averaged angular velocity versus the cylindrical coordinate radius. The quantities are computed in the HMNS center of mass at select times. Lines without (with) markers correspond to times prior to (following) the development of the one-arm spiral instability. Also shown are a curve $\propto {\varpi }^{-0.29}$ which approximates the angular velocity profile for $\varpi \gtrsim 1.0M$, and the angular velocity ${\mathrm{\Omega }}_1$ of the $m=1$ spiral mode. Right: The arrows indicate the flow coordinate velocity and the black solid lines are contours of density normalized to its maximum value. The contour near the center of the image corresponds to a value of 0.98. The (red) “x” indicates the HMNS center of mass. The plot corresponds to the $t=6.1\mathrm{ms}$ curve shown on the left. The data for both plots are taken from the $r_p/M=8$, $a_{\mathrm{NS},1}=a_{\mathrm{NS},2}=0.025$ case.

Figure 16

Ratio of total kinetic to potential energy (top) and cumulative angular momentum emitted in GWs (bottom) for various low-spin $r_p/M=8$ cases following merger. Data from high-resolution runs are plotted here.

Figure 17

Evolution of the amplitude in the $m=1$ density mode for the high (HR), medium (MR), and low (LR) resolutions used in the $r_p/M=8$, $a_{\mathrm{NS},1}=a_{\mathrm{NS},2}=0.025$ case. We observe qualitative convergence of the growth time and early saturation amplitude with resolution.

Figure 18

Evolution of the amplitude in the $m=1$ density mode for the high, medium, and low resolutions used in the $r_p/M=8$, $(a_{\mathrm{NS},1},a_{\mathrm{NS},2})=(0.025,0.05)$ (top) and (0.05, 0.05) (bottom) cases. A time shift (indicated in the legend) is applied to each case so that curves are aligned at $t-\mathrm{\Delta }t=0$ with amplitude $1/4\times$ the maximum amplitude of the highest resolution (indicated by the lower horizontal line). The growth rate $\tau$ (also indicated in the legend) measures the subsequent time required for amplitude to grow by a factor of 2 (upper horizontal line).