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

We present new $(3+1)\mathrm{D}$ numerical relativity simulations of the binary neutron star (BNS) merger and postmerger phase. We focus on a previously inaccessible region of the binary parameter space spanning the binary’s mass ratio $q\sim 1.00–1.75$ for different total masses and equations of state, and up to $q\sim 2$ for a stiff BNS system. We study the mass ratio effect on the gravitational waves (GWs) and on the possible electromagnetic (EM) emission associated with dynamical mass ejecta. We compute waveforms, spectra, and spectrograms of the GW strain including all the multipoles up to $l=4$. The mass ratio has a specific imprint on the GW multipoles in the late-inspiral-merger signal, and it affects qualitatively the spectra of the merger remnant. The multipole effect is also studied by considering the dependency of the GW spectrograms on the source’s sky location. Unequal mass BNSs produce more ejecta than equal mass systems with ejecta masses and kinetic energies depending almost linearly on $q$. We estimate luminosity peaks and light curves of macronova events associated with the mergers using a simple approach. For $q\sim 2$ the luminosity peak is delayed for several days and can be up to 4 times larger than for the $q=1$ cases. The macronova emission associated with the $q\sim 2$ BNS is more persistent in time and could be observed for weeks instead of a few days ($q=1$) in the near infrared. Finally, we estimate the flux of possible radio flares produced by the interaction of relativistic outflows with the surrounding medium. Also in this case a large $q$ can significantly enhance the emission and delay the peak luminosity. Overall, our results indicate that the BNS merger with a large mass ratio has EM signatures distinct from the equal mass case and more similar to black hole–neutron star binaries.

Figure 1

The four different EOSs employed in this work. The EOSs are modeled by piecewise-polytropic fits to the tabulated EOSs. The fits are taken from [52], see also Table 1. Markers correspond to spherical individual stars in our configurations, and dashed black lines the total masses $M$ of our configurations for which we investigate all four EOSs. In the figure, we do not include the total mass of the setup MS1b-094194, since for those masses only the EOS MS1b was employed.

Figure 2

BNS parameter space in terms of $(q,{\kappa }_2^T)$. Different colors represent different EOSs, while different markers stand for different total masses.

Figure 3

Mass transfer between the two neutron stars. Top: The panels show the time evolution of the baryonic mass and the “mass defect” $|\mathrm{\Delta }M|=|M(t)-M(t=5000M_{\odot })|$ inside coordinate spheres of radius $10.5M_{\odot }$ around the two neutron stars as a function of the simulation time $t$. While the mass is increasing for the primary star (solid lines) the secondary star “loses” mass (dashed lines). For comparison we rescale everything to the mass at $t=5000M_{\odot }$, with this approach only influences at the late stage of the inspiral are taken into account. Note that the dashed line can be seen as an upper bound for the mass transfer, since mass is also lost due to ejecta. Bottom: Mass transfer as a function of the shifted time $\tilde{t}$, such that the moment of merger happens at the same time ${\tilde{t}}^{\mathrm{mrg}}$ for all models. The gray vertical dashed lines refer to the times shown in Fig. 4.

Figure 4

Density and entropy on the orbital plane for MS1b-094194 in the final merger phase. Left plots: Density profile inside the orbital plane. The presented times are (2562.3,2648.9,2735.5,2822.0) $M$ for the upper left, upper right, lower left, lower right panel respectively. Those times correspond to the times marked in Fig. 3 as vertical dashed lines. We color the density from blue to red and the unbound density from brown to green. During the inspiral mass is ejected from the tidal tail of the less massive star due to torque. The upper right panel corresponds to the moment of merger. Right plots: Entropy indicator $\widehat{S}$ inside the orbital plane for MS1b-094194. The times are the same as in the right panels. We also include white contour lines for densities $\rho ={10}^{-6}$, ${10}^{-5}$, ${10}^{-4}$ and black lines for the ejecta material with ${\rho }_u={10}^{-12}$, ${10}^{-11}$, ${10}^{-10}$, ${10}^{-9}$, ${10}^{-8}$, ${10}^{-7}$, ${10}^{-6}$, ${10}^{-5}$. During the simulation we observe that the entropy between the two neutron stars is small, i.e. no shock heating produced ejecta can be observed for this setup at the chosen times.

Figure 5

Entropy indicator $\widehat{S}$ for SLy-137137 in the early postmerger phase. The panels represent snapshots at $t=2708M$, $t=2714$, $2721M$ from top to bottom in the $x\text{−}z$ plane. The last panel shows the entropy inside the orbital plane at $t=2721M$. We also include white contour lines for densities $\rho ={10}^{-6}$, ${10}^{-5}$, ${10}^{-4}$ and black lines for the ejecta material with ${\rho }_u={10}^{-12}$, ${10}^{-11}$, ${10}^{-10}$, ${10}^{-9}$, ${10}^{-8}$, ${10}^{-7}$, ${10}^{-6}$, ${10}^{-5}$.

Figure 6

Ejected properties as a function of the mass ratio for masses $M=2.75$ and $M=2.888$. We show the difference between two adjacent resolutions as an error bar in our plot. Top: Ejecta mass, where we see that for an increasing mass ratio stiffer EOSs produce significantly more ejecta, while a similar effect cannot be observed for softer EOSs. Upper middle: Kinetic energy of the ejecta, where for an increasing mass ratio ejecta have significantly more kinetic energy, in particular for stiff EOSs. Lower middle: $⟨|v_{\rho }|⟩$, Eq. (10), as a function of q. Bottom: $⟨|v_z|⟩$, Eq. (11), as a function of q. While the velocity inside the orbital plane seems to be almost constant independent of the mass ratio, the velocity orthogonal to the orbital plane decreases for higher mass ratios.

Figure 7

Ejected mass as a function of the total mass of the binary system for a mass ratio of $q=1.5$. We show the difference between two adjacent resolutions as an error bar. The influence of the total mass is smaller than the effect of the mass ratio, however, less ejecta are produced for higher total masses in most cases.

Figure 8

Rest mass of the disk surrounding the final BH for the ALF2 configurations with a mass of $M=2.75$ and mass ratios $q=1.0$; 1.25, 1.5. Clearly visible is that for an increasing mass ratio the disk is more massive. However, ALF2-100175 and ALF-110165 give almost the same disk masses. We have not included ALF2-100175 in this figure since the simulation after the collapse to the BH is much shorter than for the other configurations.

Figure 9

Real part and amplitude of the (2,2)-multipole $h_{lm}(u)$ of the GWs for different configurations and the highest resolution. Different colors correspond to different EOS: ALF2 is orange, H4 green, MS1b red, and SLy blue. The lower resolution runs are shown as a black dotted line to understand the influence of the resolution in our configurations.

Figure 10

Spectra $\tilde{h}(\theta =\pi /4,\varphi =0)$ for all configurations. Clearly visible is the chirp signal of the inspiral-merger phase followed by the postmerger signal (if no prompt collapse). In most of the cases, the spectrogram highlights that several frequencies, other than the $f_2$, are excited in the postmerger. Note also that configurations like ALF2-137137, H4-137137, and SLy-122153, in which a HMNS forms and collapses within dynamical times, clearly shows that the postmerger spectrum is not discrete, but rather continuous, and similar to the inspiral-merger chirp.

Figure 11

Effect of mass ratio on the GW spectra. Top: ${\tilde{h}}_{22}$ for MS1b setups for mass ratios $q=1.00$; 1.25; 1.50; 1.75. Bottom: ${\tilde{h}}_{22}$ for H4 setups for mass ratios $q=1.00$; 1.25; 1.50; 1.75. We mark the $f_2$ frequencies as diamonds, the $f_{\mathrm{mrg}}$ as triangles and the secondary peak frequencies $f_s$ as circles.

Figure 12

Spectra of individual GW modes for the H4-137137-R2 setup. We present ${\tilde{h}}_{lm}$ for $(lm)=(22)$, (21), (33), (44). The amplitudes are rescaled for better visibility, i.e. ${\tilde{h}}_{21}$ and ${\tilde{h}}_{33}$ are rescaled by a factor of $10^{2.5}$ and ${\tilde{h}}_{44}$ by $10^{3.5}$.

Figure 13

Angular dependence of the GW spectra for H4-110165-R2. We present $\tilde{h}$ for $(\theta =0,\varphi =0)$ (top panel), $(\theta =\pi /4,\varphi =0)$ (middle panel), and $(\theta =\pi /2,\varphi =0)$ (bottom panel).

Figure 14

Total GW energy emitted for different EOSs and mass ratios. For the total energy we also take into account contributions before the beginning of the simulation [$M-M_{\mathrm{ADM}}(t=0)$]. Shown as dashed lines are estimates for the released energy during BBH mergers (see text for details). The left panel shows the energies at the moment of merger, i.e. peak of the GW amplitude, right panels represent data 20 ms after merger. We have removed those setups for which the initial linear momentum, Table 7, was large and an artificial drift of the center of mass was present.

Figure 15

GW energy emitted by different modes (up to $l=m=4$) and rescaled by the amount of energy emitted by the (2,2) mode. Black lines correspond to $l=2$ contributions, red lines the $l=3$ contributions, and $l=4$ is represented by blue lines. Different dashing corresponds to different $m$. We present the configurations MS1b-137137 (upper left panel), MS1b-122153 (upper right panel), MS1b-110165 (lower left panel), and MS1b-100175 (lower right panel).

Figure 16

GW energy emitted by different modes $E_m=\sum _{l=m}^8{E}_{lm}$. We show $E_m$ for $m=1$; 3; 4 rescaled by $E_2$. Left panels show the energies at the moment of merger, i.e. peak of the GW amplitude, right panels represent data 20 ms after merger. We have removed those setups for which the initial linear momentum, Table 7, was large and an artificial drift of the center of mass was present.

Figure 17

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 ratio. As before we consider configurations with $M=2.75M_{\odot }$ and $M=2.888M_{\odot }$.

Figure 18

Top panel: Predicted light curve (absolute bolometric luminosity) for the predicted macronova produced by the merger of the configurations employing the MS1b EOS. The light curve is produced with the publicly available code of [78]. Middle and bottom panels: Absolute magnitudes for the ugrizJHK-band assuming that the macronova is produced 200 Mpc away for MS1b-094194 (middle panel) and MS1b-137137 (bottom panel).

Figure 19

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 ratio. We consider BNS configurations with $M=2.75M_{\odot }$ and $M=2.888M_{\odot }$.

Figure 20

Mass conservation of MS1b-094194. The top panel shows the mass evolution of the baryonic mass rescaled to its initial value. The bottom panel shows $\mathrm{\Delta }M=M(t)-M(t=0)$. We mark the moment of merger as gray shaded regions, where the light gray region marks the interval $[t_{R1s}^{\mathrm{mrg}},t_{R2s}^{\mathrm{mrg}}]$, and where the dark shaded region is $[t_{R2s}^{\mathrm{mrg}},t_{R3s}^{\mathrm{mrg}}]$. The error in the total mass stays below 0.2% even for the lowest resolution until the time where material leaves the computational domain as ejecta.

Figure 21

ADM constraints for MS1b-094194. The top panel shows the $L_2$-volume norm of the Hamiltonian constraint $|\mathcal{H}{|}_2$. The lower panel shows the square magnitude of the momentum constraint $\parallel {\mathcal{M}}^i{\parallel }_2=\sqrt{|{\mathcal{M}}^x{|}_2^2+|{\mathcal{M}}^y{|}_2^2+|{\mathcal{M}}^z{|}_2^2}$. The constraints are evaluated on level $l=1$. The constraints are decreasing for increasing resolution during the inspiral of the neutron stars. For the whole simulation the constraints stay below or at the constraint violation of the initial slice due to the constraint damping of the Z4c scheme. We have filtered our data with an average filter to allow a better visualization and reduce high frequency noise.

Figure 22

ADM quantities for MS1b-094194. The top panel shows the ADM energy, which we correct with the amount of GW energy emitted by the system. The quantity is conserved at or below the 0.5% level for R2s and R3s until material is leaving the computational domain. The bottom panel shows the ADM-angular momentum corrected by the amount of angular momentum extracted via GW emission. For R2s and R3s the ADM-angular momentum is conserved better than 0.05%. In both panels we show the numerical data as semitransparent lines and the results after applying an average filter as solid lines. This filtering reduces oscillations caused by the orbital motion and allows an easier interpretation of the data.

Figure 23

Top panel: Real part of the dominant (2,2) GW mode of MS1b-094194. Bottom panel: Phase difference between different resolutions. The dotted red line corresponds to a convergence order of 1.3. As for the previous discussions the different moments of merger are represented as a shaded region.