Gravitational waves and mass ejecta from binary neutron star mergers: Effect of the stars’ rotation

We present new ($3+1$)-dimensional numerical relativity simulations of the binary neutron star (BNS) mergers that take into account the NS spins. We consider different spin configurations, aligned or antialigned to the orbital angular momentum, for equal- and unequal-mass BNSs and for two equations of state. All the simulations employ quasiequilibrium circular initial data in the constant rotational velocity approach, i.e. they are consistent with the Einstein equations and in hydrodynamical equilibrium. We study the NS rotation effect on the energetics, the gravitational waves (GWs) and on the possible electromagnetic (EM) emission associated to dynamical mass ejecta. For dimensionless spin magnitudes of $\chi \sim 0.1$ we find that both spin-orbit interactions and spin-induced quadrupole deformations affect the late-inspiral merger dynamics. The latter is, however, dominated by finite-size effects. Spin (tidal) effects contribute to GW phase differences up to $\sim 5$ (20) radians accumulated during the last eight orbits to merger. Similarly, after merger the collapse time of the remnant and the GW spectrogram are affected by the NSs rotation. Spin effects in dynamical ejecta are clearly observed in unequal-mass systems in which mass ejection originates from the tidal tail of the companion. Consequently kilonovae and other EM counterparts are affected by spins. We find that spin aligned to the orbital angular momentum leads to brighter EM counterparts than antialigned spin with luminosities up to a factor of 2 higher.

Figure 1

The $(q,{\chi }_{\mathrm{mw}},{\kappa }_2^T)$ parameter space coverage. Different colors refer to the EOSs ALF2 (orange) and H4 (green). Different markers correspond to different spin configurations: circles (00), triangles pointing down ($\uparrow \downarrow$), triangles pointing right $(\uparrow 0)$, and triangles pointing up $(\uparrow \uparrow )$.

Figure 2

Rest-mass density profile inside the orbital plane for simulations employing the H4 EOS and using the R2 grid setup. The snapshots represent the moments of merger. The panels refer to (from top to bottom) mass ratios $q=1.00$, $q=1.25$, $q=1.50$ and (from left to right) spin configurations $(00),(\uparrow \downarrow ),(\uparrow 0),(\uparrow \uparrow )$. The rest-mass density $\rho$ is shown on a logarithmic scale from blue to red. The rest-mass density of unbound material is colored from brown to green. Most material gets ejected from the tidal tails of the NS inside the orbital plane.

Figure 3

Quasilocal measurement of the individual spin of the two NSs for $\mathrm{H}4\text{−}{137137}^{(\uparrow \downarrow )}$. The gray dashed dotted lines in the diagram represent the value computed from the initial data solver and given in Table 1. Different colors refer to different radii of the coordinate spheres. The increasing quasilocal spin is most likely caused by the choice of the center of the integration surface and by using a coordinate sphere that does not take tidal deformations into account.

Figure 4

Binding energy vs specific angular momentum curves $E(\ell )$ for the equal-mass configurations (upper panels). The circles mark the moment of merger for all configurations. We also include a nonspinning BBH configuration from the public SXS catalog; see text for more details. The bottom panels show the individual contributions to the binding energy, Eq. (11); we present the $SO$ (green), $S^2$ (orange), $S\leftrightarrow S$ (blue) configurations obtained from the NR data as solid lines. For those contributions we also include 3.5 PN (Appendix pp2) estimates as dashed lines. The tidal contributions are shown as cyan lines. We also include the SO contribution from the EOB model of Ref. [78]. We mark the difference between resolutions R2 and R1 as a colored shaded region. The vertical gray areas correspond to the merger of the $\mathrm{ALF}2\text{−}{137137}^{(\uparrow \downarrow )}$ (left) and the $\mathrm{H}4\text{−}{137137}^{(\uparrow \downarrow )}$ configurations (right).

Figure 5

Quantity ${\mathcal{E}}_{\text{Spin}}$ for different EOSs and the same mass ratio ($q=1.0$) in the top panel and for the same EOS (ALF2), but mass ratios $q=1.0$ and $q=1.5$ in the bottom panel. The colored shaded regions mark the difference between the results for resolutions R2 and R1. Notice that for an unequal-mass merger the $(\uparrow \downarrow )$ configuration also contains SO contributions and therefore could be included for $q=1.5$ although the late-time behavior is dominated by the SS interactions. The vertical gray areas correspond to the moment of merger for $\mathrm{H}4\text{−}{137137}^{(\uparrow 0)}$ (upper panel) and $\mathrm{ALF}2\text{−}{110165}^{(\uparrow \downarrow )}$ (lower panel). We also include the 3.5 PN SO contribution as a dashed line and the EOB ${\mathcal{E}}_{\text{Spin}}$ as presented in the EOB model of Ref. [78] as a dashed dotted line.

Figure 6

Binding energy (top panels) and specific angular momentum (bottom panels) as a function of the PN parameter $x$. The circles mark the moment of merger for all configurations. We also include a nonspinning BBH configuration from the public SXS catalog as in Fig. 4. We have applied a Savitzky-Golay filter on $E(x)$ and $\ell (x)$ to reduce numerical noise and eccentricity oscillations.

Figure 7

Individual contributions to the binding energy (top panels) and specific angular momentum (bottom panels) as a function of the PN parameter $x=(M\mathrm{\Omega }{)}^{2/3}$ for the equal-mass systems employing the H4 EOS. The colored shaded regions mark the difference between resolutions R2 and R1. The initial oscillations are caused mostly by remaining eccentricity effects not fully removed after applying the Savitzky-Golay filter to $E(x)$ and $\ell (x)$. We also include 3.5 PN estimates for the binding energy (Appendix pp2) as dashed dotted lines. The vertical gray line corresponds to the merger point of the $\mathrm{H}4\text{−}{137137}^{(\uparrow \downarrow )}$.

Figure 8

Binding energy vs specific angular momentum curves after the merger of the two neutron stars for the H4 setups with mass ratios $q=1.0$ (left) and $q=1.50$ (right). The merger is marked as a circle for all setups. The bottom panels present the frequency $M\mathrm{\Omega }$ estimated from the binding energy curves.

Figure 9

Mass of the ejecta (first row), kinetic energy of the ejecta (second row), ejecta velocities inside the plane (third row) and orthogonal to it (fourth row) as a function of the spin of the configurations. Green points represent data for the H4-EOS, while orange data points correspond to ALF2. Left panels refer to the mass ratio $q=1.00$ and right panels to the mass ratio $q=1.50$. The markers characterize the spin of the configurations as in Fig. 1, where circles correspond to (00) setups, downward-pointing triangles to ($\uparrow \downarrow$), right-pointing triangles to ($\uparrow 0$), and upward-pointing triangles to ($\uparrow \uparrow$). The spin of the secondary star influences the amount of ejected material and the kinetic energy, where aligned spin leads to larger ejecta. The velocity inside the orbital plane does not depend notably on $q$ or ${\chi }_{\mathrm{eff}}$, and the velocity perpendicular to the orbital plane decreases for increasing $q$.

Figure 10

Gravitational-wave signal for all considered configurations employing the R2 resolution. Top panels (yellow lines) refer to the ALF2 EOS, while bottom panels (green lines) refer to the H4 EOS. The mass ratios from top to bottom are $q=1.00$, $q=1.25$, and $q=1.50$. The columns refer to the setups (00),$(\uparrow \downarrow )$, $(\uparrow 0)$, and $(\uparrow \uparrow )$.

Figure 11

Top panel: $\varphi (\widehat{\omega })$ accumulated in $\widehat{\omega }\in [0.04,0.11]$ for $\mathrm{ALF}2\text{−}{137137}^{(00)}$ (blue), $\mathrm{ALF}2\text{−}{137137}^{(\uparrow \uparrow )}$ (red), and a nonspinning, equal-mass BBH setup (black). Bottom panel: Individual contributions $\mathrm{\Delta }\varphi (\widehat{\omega })$. We include tidal effects for ALF2 (orange) and H4 (green) EOSs, as well as spin effects for ALF2 (blue) and H4 (cyan). We also include estimates from the EOB models of Ref. [78] for the spinning contribution and Ref. [80] for the tidal contribution as dashed lines.

Figure 12

Spectrogram of the GW signal for simulations with the H4 EOS and resolution R2. The spectrogram only considers the dominant (2, 2) mode. Horizontal blue dashed lines refer to the $f_2$ frequency extracted from the entire post-merger GW signal. Black dashed lines refer to the frequency of the (2, 2) mode and white lines refer to the frequency extracted from the binding energy ${\partial }_{\ell }E$ (Fig. 8). The spectrograms show the last part of the inspiral signal (left bottom corners of the spectrograms) and evolution of the HMNS.

Figure 13

PSD for the H4-137137 (upper panel) and H4-110165 (lower panel) setups. We mark the merger frequency with triangles and the post-merger peak frequency $f_2$ as diamonds.

Figure 14

Peak time $t_{\mathrm{peak}}$ (top panel), peak luminosity $L_{\mathrm{peak}}$ (middle panel), and peak temperature $T_{\mathrm{peak}}$ (bottom panel) of macronovae produced by the BNS mergers considered in this article as a function of the mass-weighted spin ${\chi }_{\mathrm{mw}}$. We mark different EOSs with different colors: green (H4) and orange (ALF2). Different markers refer to different spin configurations; see Fig. 1.

Figure 15

Peak time in the radio band $t_{\mathrm{peak}}^{\mathrm{rad}}$ (top panel) and corresponding radio fluence $F_{\mathrm{peak}}^{\nu \mathrm{rad}}$ (bottom panel), as a function of the mass weightes spin ${\chi }_{\mathrm{mw}}$. We mark different EOSs with different colors: green (H4) and orange (ALF2). Different markers refer to different spin configurations; see Fig. 1.

Figure 16

Bolometric luminosity for the H4-110165 setups with different spin orientations. The luminosities are computed following the approach of Ref. [88].

Figure 17

Quasilocal spin measure [Eq. (1)] for a single rotating NS in equilibrium. The NS has similar mass and spin as the components of $\mathrm{H}4\text{−}137137^{(\uparrow \uparrow )}$. Different colors refer to different coordinate radii, where solid lines represent results for resolution R2 and dashed lines for resolution R1. The ADM angular momentum measured with sgrid is given as a black dashed line.

Figure 18

Example of frequency fit with template (c1). The fitting interval is $I_{\omega }=[0.039,0.129]$, covering about 18 cycles of a total of 23. The GW frequency at $u_{\mathrm{mrg}}$ is $\sim 0.144$ (vertical line). Data refer to ALF2 $q=1$ $M=2.75$ and no spin.

Figure 19

$Q_{\widehat{\omega }}(\omega )$ plot for the irrotational (blue) and $(\uparrow \uparrow )$ configurations (red) employing the ALF2 EOS. $Q_{\widehat{\omega }}$ is computed as described in Appendix pp3. We show as shaded regions the difference between different resolutions $|Q_{\widehat{\omega }}^{R2}-Q_{\widehat{\omega }}^{R1}|$. As a black line we include $Q_{\widehat{\omega }}$ for a nonspinning, equal-mass BBH setup obtained from the EOB code [78].