Gravitational waves and mass ejecta from binary neutron star mergers: Effect of the spin orientation

We continue our study of the binary neutron star parameter space by investigating the effect of the spin orientation on the dynamics, gravitational wave emission, and mass ejection during the binary neutron star coalescence. We simulate seven different configurations using multiple resolutions to allow a reasonable error assessment. Due to the particular choice of the setups, five configurations show precession effects, from which two show a precession (“wobbling”) of the orbital plane, while three show a “bobbing” motion; i.e., the orbital angular momentum does not precess, while the orbital plane moves along the orbital angular momentum axis. Considering the ejection of mass, we find that precessing systems can have an anisotropic mass ejection, which could lead to a final remnant kick of $\sim 40\mathrm{km}/\mathrm{s}$ for the studied systems. Furthermore, for the chosen configurations, antialigned spins lead to larger mass ejecta than aligned spins, so that brighter electromagnetic counterparts could be expected for these configurations. Finally, we compare our simulations with the precessing, tidal waveform approximant imrphenompv2_nrtidalv2 and find good agreement between the approximant and our numerical relativity waveforms with phase differences below 1.2 rad accumulated over the last $\sim 16$ gravitational wave cycles.

Figure 1

Orbital dynamics and GW emission for all simulations. Column 1 shows the coordinate tracks of each NS in the binary. Column 2 shows the corresponding precession cones. The spin evolution of the individual stars (blue and green), and the orbital angular momentum of the system (red) are shown here. Column 3 shows the (2, 2)- and (2, 1)-modes of the GW strain $rh$.

Figure 2

A schematic of the Lense-Thirring effect in a binary system. Left column: A system with symmetrically misaligned spins of the NSs A and B with respect to the orbital angular momentum direction. Right column: A system with asymmetrically misaligned spins of the NSs A and B.

Figure 3

Precession cone for the ${\mathrm{SLy}}^{(\nearrow \nearrow )}$ configuration for the orbital angular momentum (red). Additionally, as a green dashed line we show also the precession cones for $({\widehat{L}}_x,{\widehat{L}}_y,{\widehat{L}}_z)$ for the ${\mathrm{SLy}}^{(\searrow \searrow )}$ configuration. The opening angles for both the configurations are almost identical, due to the symmetry of the systems.

Figure 4

Radiated angular momentum (${\overrightarrow{\mathbf{J}}}_{\mathrm{rad}}$) in GWs computed using the relations given in Appendix pp1. For symmetrically misaligned configurations the angular momentum is only radiated in the $z$-component and the $x$, $y$-components remain identically zero during the inspiral. Whereas for the asymmetrically misaligned systems there is radiation in all the components.

Figure 5

Reduced binding energy $E_b$ as a function of the specific orbital angular momentum $\ell$ for all configurations considered in the article. Additionally, we also include the curve for an irrotational case ${\mathrm{SLy}}^{(00)}$ taken from the CoRe Database (ID:-BAM:0095:R02) for comparison. As expected, the irrotational curve matches the ${\mathrm{SLy}}^{(\leftarrow \to )}$ case nicely.

Figure 6

Top panel: Estimate of the spin orientation effects on the conservative dynamics by taking the difference between all the configurations and the ${\mathrm{SLy}}^{(00)}$ (irrotational case taken from CoRe Database) configuration. The shaded region marks the difference in results obtained with a lower resolution and takes into account the uncertainty of the initial data. Bottom panel: Spin and orbital contributions to the binding energy estimated following the discussion in the text.

Figure 7

Maximum of $\rho$ vs coordinate time $t$. The cases that form a black hole (BH) after the merger show a sharp change in the density where the density drops to zero for such cases, because matter is removed inside the BH. Note that we report the merger remnant properties for the R3 resolution setups as the simulations could be evolved for longer times owing to the reduced computational costs.

Figure 8

2D plots for the ${\mathrm{SLy}}^{(\swarrow \searrow )}$ configuration showing the density and velocity field at different times close to the merger, with the unbound material shown in the brown to green color scale, while the bound material is shown in a blue to red color scale. Top row: Plots show the $xy$-plane covering a distance of $\sim 88\mathrm{km}$ in each direction. Bottom row: Plots show the $xz$-plane, where each direction is covering a distance of $\sim 293\mathrm{km}$. Columns one to three: Time snapshots when the surfaces of the stars touch until the cores of the NSs finally merged. Fourth column: Postmerger phase when a hypermassive NS has been formed. Interestingly, we see that in the final phase of the merger unbound matter is ejected asymmetrically due to the “bobbing” motion that this system undergoes. Such an asymmetrical matter ejection is capable of imparting a kick velocity to the merger remnant.

Figure 9

Top panel: Kick estimates for the ${\mathrm{SLy}}^{(\uparrow \uparrow )}$ case. The aligned/antialigned cases do not show the “bobbing” or the “wobbling” motion of the orbital plane. Bottom panel: Kick estimates for the ${\mathrm{SLy}}^{(\searrow \searrow )}$ case that shows the “wobbling” motion of the orbital plane. The kicks are estimated using the recoil from the ejecta and the GWs and are shown for the R4 setup. The merger time, corresponding to the peak in the (2, 2)-mode of GW strain is shown as a “gray” line.

Figure 10

Gravitational wave strains $h_{+}$ (first and third panels) and $h_{\times }$ (second and fourth panels) for the inclinations $𝚤=0$ (face on, top panels) and $𝚤=\pi /2$ (edge on, bottom panels).

Figure 11

Top panel: Phase differences for all spinning configurations with respect to the irrotational case for the (2, 2)-mode of the GW strain. The errors represented by the shaded regions are estimated by computing the phase differences for different resolutions. Note again that the irrotational case data used, namely R3 ($0.118M_{\odot }$) and R4 ($0.078M_{\odot }$) resolutions, are not of exactly the same resolution as the other configurations simulated and therefore the error estimates should be taken as conservative estimates. Bottom panel: Estimate of spin-orbit and spin-spin contribution to the phase from the aligned/antialigned configurations as described in the text.

Figure 12

$\varphi (\widehat{\omega })$ accumulated in $\widehat{\omega }\in$ [0.038, 0.18] for all the configurations considered for the R4 resolution.

Figure 13

Rescaled GW strains for the precessing systems ${\mathrm{SLy}}^{(\nearrow \nearrow )}$ (left panel) and ${\mathrm{SLy}}^{(\searrow \searrow )}$ (right panel) (blue, solid curves) for the R4 resolution compared with the imrphenompv2_nrtidalv2 model (black, dashed curve). Results for $h_{+}$ are shown in the first and third panels and for $h_{\times }$ are shown in the second and fourth panels for the inclinations $𝚤=0$ (face on) in top panels and $𝚤=\pi /2$ (edge on) in bottom panels. Note that we find small differences in the amplitudes (as visible in the $h_{\times }$ panels for the $𝚤=\pi /2$ case) indicating the importance of future GW waveform model development.

Figure 14

Spectrograms and the corresponding contours for all the configurations computed using the GW strain $h$. An inclination angle of $𝚤=\pi /2$ is assumed for all the plots and a logarithmic color scale is used. All individual chunks of the spectrogram have a length of $\sim 2\mathrm{ms}$ and a tapering with a tanh function is applied before Fourier transforming to minimize oscillations.

Figure 15

Hamiltonian constraint (first and third panels) and rest mass conservation (second and fourth panels) for the ${\mathrm{SLy}}^{(\swarrow \searrow )}$ case (top) and the ${\mathrm{SLy}}^{(\nearrow \nearrow )}$ case (bottom). The merger time, corresponding to the peak in the (2, 2)-mode of GW strain is shown as vertical dashed line for each resolution.

Figure 16

Real part of the (2, 2)-mode (top panel) and (2, 1)-mode (bottom panel) for resolution R4 as well as the phase difference between different resolutions for the ${\mathrm{SLy}}^{(\nearrow \nearrow )}$ configuration shown versus retarded time. We multiply the amplitude of the (2, 1)-mode by a factor of 20 for better visibility.