Structure of stable binary neutron star merger remnants: A case study

In this work, we study the merger of two neutron stars with a gravitational mass of $1.4M_{\odot }$ each, employing the Shen-Horowitz-Teige equation of state. This equation of state is a corner case, allowing the formation of a stable neutron star with the given total baryonic mass of $3.03M_{\odot }$. We investigate in unprecedented detail the structure of the remnant, in particular the mass distribution, the thermal structure, and the rotation profile. We also compute fluid trajectories both inside the remnant and those relevant for the formation of the disk. We find a peanut-shaped fluid flow inside the remnant following a strong $m=2$ perturbation. Moreover, the flow is locally compressive, causing the appearance of dynamic hot spots. Further, we introduce new diagnostic measures that are easy to implement in numeric simulations and that allow one to quantify mass and compactness of merger remnants in a well-defined way. As in previous studies of supra- and hypermassive stars, we find a remnant with a slowly rotating core and an outer envelope rotating at nearly Keplerian velocity. We compute a Tolman-Oppenheimer-Volkoff star model which agrees well with that of the remnant in the core, while the latter possesses extensive outer layers rotating close to Kepler velocity. Finally, we extract the gravitational wave signal and discuss the detectability with modern observatories. This study has implications for the interpretation of gravitational wave detections from the postmerger phase and is relevant for short gamma-ray burst models.

Figure 1

Spacetime diagram showing the world tubes of different isocontours in the orbital plane. The time coordinate runs from right to left. Top panel: isodensity contour containing (at each time) 25% of the total baryon mass, showing the evolution of the core(s). Bottom panel: the red surface is the contour of constant entropy density chosen to highlight the evolution of the hot spots. For comparison, the surface in the top panel is shown again, but rendered transparent. The size of the tiles in the background is $2\mathrm{ms}\times 20\mathrm{km}$, and the lower panel is zoomed to show the postmerger phase.

Figure 2

Snapshots of entropy distribution at times (from top-left to bottom-right) 0.1, 1.8, 3.5, 8.5, and 18.3 ms after merger. The red surface is the isosurface of entropy density which contains 85% of the entropy. The transparent surface depicts the isosurface of mass density which contains a mass of $2.7M_{\odot }$. The size of the tiles in the background is 10 km.

Figure 3

Evolution of baryonic mass contained in the bulk and separate cores. The latter is defined as the mass of all matter with density higher than the one at the origin.

Figure 4

Evolution of bulk compactness $M_b/R_V$ (solid line), where $R_V=((3/(4\pi ))V{)}^{1/3}$ is the volumetric radius of the bulk (see main text), $V$ its proper volume, and $M_b$ its baryon mass. For comparison, we show the same measure for a configuration with the same bulk mass, but consisting either of a single TOV star (dotted line) or of 2 TOV stars (dashed line). The EOS for the latter is the same as for the initial data.

Figure 5

Evolution of average specific entropy for matter inside and outside the bulk mass, labeled tentatively as remnant and disk. The average specific entropy is defined as the ratio of total entropy and total baryon number. The cross marks the activation of the thermal evolution.

Figure 6

Average specific entropy of matter versus rest mass density. The time in the labels is the time after merger. The vertical lines mark the density defining the bulk.

Figure 7

Mass distribution of the remnant 18.7 ms after merger. Plotted is the baryonic mass versus the proper volume contained in surfaces of constant mass density. For comparison, we also show the same profile for one of the initial stars in isolation, and for the TOV core equivalents (see main text) with zero temperature as well as constant specific entropy of $1k_B$. The symbols mark, from left to right, the bulk (see main text) of initial star, cold and hot core equivalent, and remnant. Further, we show the bulk mass versus the bulk volume for the two sequences of TOV stars with $T=0$ and $s=1k_B$.

Figure 8

Evolution of $\varphi$-averaged angular velocity in the equatorial plane. $r_c$ is the circumferential radius and $t$ the coordinate time relative to the merger time. The location of the maximum after the merger phase is marked by the solid curve. The dashed horizontal lines mark the boundaries of the time-average shown in Fig. 9.

Figure 9

Rotation profile of the remnant 14 ms after merger. Shown is the angular velocity $\mathrm{\Omega }$ in the equatorial plane averaged in the $\varphi$ direction as well as in time over a 2 ms window (13–15 ms). For comparison, the green dashed line is the orbital angular velocity ${\mathrm{\Omega }}_K$ of test particles in corotating circular orbits. The dotted grey line is the frame-dragging contribution. The horizontal line is the pattern angular velocity ${\dot{\mathrm{\Phi }}}_{22}=\dot{\varphi }/2$ of the $l=m=2$ component of the gravitational signal at the same (retarded) time. The vertical line marks the radius where the density falls below 5% of the maximum one.

Figure 10

Evolution of mass density moments in the equatorial plane.

Figure 11

Spacetime diagram showing the fluid evolution in the orbital plane. The time coordinate runs from right to left. Each time slice has been rotated to the phase of the $m=2$ component of the density perturbation in the orbital plane. The size of the tiles in the background is $2\mathrm{ms}\times 20\mathrm{km}$. Top panel: the red solid surface is the contour of constant entropy density chosen to highlight the evolution of the hot spots, and the green transparent surface is the world tube of the isodensity contour containing (at each time) 25% of the total baryon mass. Bottom panel: traced trajectories of fluid elements. Only elements not ejected from the remnant are shown. The colors highlight groups of trajectories discussed in the main text.

Figure 12

Fluid motion in the frame corotating with the $m=2$ component of the density perturbation. The color plot shows the temperature in the orbital plane 8.5 ms after merger. The curves show fluid trajectories during the time interval $\pm 2\mathrm{ms}$ around the snapshot. The solid blue line marks the entropy density isocontour whose world tube is shown also in Fig. 11, and the thick white line marks the bulk isodensity surface.

Figure 13

Like Fig. 12, but showing the specific entropy.

Figure 14

Gravitational wave strain at 100 Mpc (top panel) and instantaneous frequency (bottom panel). The latter is computed from the phase velocity of the complex strain, smoothed over 0.1 ms to avoid amplification of high-frequency noise.

Figure 15

Top panel: gravitational wave spectrum at 100 Mpc, both for the full simulation (solid green line) and for the strain starting 4 ms after the merger (dotted blue line). For comparison, we show the sensitivity curves of GW detectors. Bottom panel: strain amplitude versus instantaneous frequency. The part starting 4 ms after the merger is plotted in blue, the earlier evolution in red.

Figure 16

Disk structure at the end of the simulation, averaged in time over 4 ms. The colors correspond to the specific entropy, while the lines are the isocontours of mass density $1.43\times 10^{14}$, $8.43\times 10^{13}$, $5.43\times 10^{12}$, $1.20\times 10^{12}$, and $1.25\times 10^{11}{\mathrm{g}/\mathrm{cm}}^3$, from innermost to outermost, respectively. The total mass of matter with higher density than the plotted density levels is equal to the bulk mass of the TOV core equivalent, the bulk mass of the remnant itself, 90%, 92%, and 95% of the total baryon mass, respectively.

Figure 17

Fluid trajectories in the orbital plane with respect to the coordinate system corotating with the $m=2$ density perturbation. The trajectories are shown for the time interval 2 to 12 ms after merger. The trajectories have been tracked backwards in time starting 14 ms after merger from seeds arranged in a regular grid with an extent of 70 km. The trajectories are colored green if they stay in the remnant until the end of the simulation, or else they are black. Note the trajectories move predominantly clockwise.