General relativistic simulations of magnetized binary neutron star mergers

Binary neutron stars (NSNS) are expected to be among the leading sources of gravitational waves observable by ground-based laser interferometers and may be the progenitors of short-hard gamma-ray bursts. We present a series of general relativistic NSNS coalescence simulations both for unmagnetized and magnetized stars. We adopt quasiequilibrium initial data for circular, irrotational binaries constructed in the conformal thin-sandwich (CTS) framework. We adopt the BSSN formulation for evolving the metric and a high-resolution shock-capturing scheme to handle the magnetohydrodynamics. Our simulations of unmagnetized binaries agree with the results of Shibata, Taniguchi and Uryū [M. Shibata, K. Taniguchi, and K. Uryū, Phys. Rev. D 68, 084020 (2003).]. In cases in which the mergers result in a prompt collapse to a black hole, we are able to use puncture gauge conditions to extend the evolution and determine the mass of the material that forms a disk. We find that the disk mass is less than 2% of the total mass in all cases studied. We then add a small poloidal magnetic field to the initial configurations and study the subsequent evolution. For cases in which the remnant is a hypermassive neutron star, we see measurable differences in both the amplitude and phase of the gravitational waveforms following the merger. For cases in which the remnant is a black hole surrounded by a disk, the disk mass and the gravitational waveforms are about the same as the unmagnetized cases. Magnetic fields substantially affect the long-term, secular evolution of a hypermassive neutron star (driving “delayed collapse”) and an accretion disk around a nascent black hole.

Figure 1

Magnetic field configurations in a widely separated, spherical NS companion constructed from the vector potential in Eq. (18) for $n_p=0$ and $P_{\mathrm{cut}}=0.04P_{\max }$ (upper panel), and for $n_p=3$ and $P_{\mathrm{cut}}=0.001P_{\max }$ (lower panel). Here $P_{\max }$ is the maximum pressure. The compaction of this NS is $M_{*}/R=0.14$. Dotted (black) concentric circles are rest-mass density contours drawn for ${\rho }_0/{\rho }_0^{\max }=0.9$, 0.7, 0.5, 0.3, 0.1, and 0.001. Solid (green) lines are contours of $A_{\phi }$, which coincide with the magnetic field lines in axisymmetry. Contour levels of $A_{\phi }$ are drawn for $A_{\phi }=(A_{\phi }^{\max }-A_{\phi }^{\min })(i/10{)}^2+A_{\phi }^{\min }$ with $i=1,2,\dots 9$, where $A_{\phi }^{\max }$ and $A_{\phi }^{\min }$ are the maximum and minimum value of $A_{\phi }$, respectively.

Figure 2

Evolution of maximum density ${\rho }_0^{\max }$ and minimum lapse ${\alpha }_{\min }$ for unmagnetized (black solid line) and magnetized (blue dash line) runs of model M1414. The time $t$ is normalized by the initial orbital period $P_0=193M=1.3\times {10}^{-5}s(M_0/2.8M_{\odot })$. The initial maximum rest-mass density is ${\rho }_0^{\max }(0)=0.118/\kappa =7.9\times {10}^{14}{\mathrm{gcm}}^{-3}(2.8M_{\odot }/M_0{)}^2$. The merger occurs at $t\approx 1P_0$.

Figure 3

Nondimensional rest-mass density ${\overline{\rho }}_0=\kappa {\rho }_0$ along the $x$-axis (black solid lines) and $y$-axis (blue dash lines) in the equatorial plane at three different times for the unmagnetized (left) and magnetized (right) cases.

Figure 4

Snapshots of density and velocity field in the equatorial plane from run M1414B0 (unmagnetized run). Density contours are drawn for ${\rho }_0/{\rho }_0(0)=0.9$, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.01, 0.001, and 0.0001.

Figure 5

Same as Fig. 4 but for run M1414B1 (magnetized run).

Figure 6

Constant geodesic proper distance from the center $D$ in the equatorial plane at $t=4P_0$ for run M1414B0. Contours are drawn for $D/M=1.86i$ ($i=1$, 2, 3, 4, 5).

Figure 7

Gravitation waveforms for model M1414. (a) $\mathrm{Re}(rM{\psi }_4^{22})$ extracted at $r=43M$ (black solid line), $r=38M$ (blue dotted line), and $r=32M$ (red dash line). (b) Same as (a) but for $\mathrm{Im}(rM{\psi }_4^{22})$. (c) $\mathrm{Re}(rM{\psi }_4^{22})$ extracted at $r=43M$ for the unmagnetized (black solid line) and magnetized (cyan dash line) cases. (d) Same as (c) but for $\mathrm{Im}(rM{\psi }_4^{22})$. Note that in (a) and (b), the lines are hardly distinguishable, showing good agreement of waveforms at various extraction radii.

Figure 8

Constraint violations for the unmagnetized run of model M1414. Upper panel: Normalized L2 norm of the Hamiltonian constraint. Lower panel: Normalized L2 norm of the $x$ (black solid line), $y$ (blue dotted line) and $z$ (red dash line) components of the momentum constraint.

Figure 9

Same as Fig. 8 but for the magnetized run of model M1414.

Figure 10

Evolution of maximum density ${\rho }_0^{\max }$ and minimum lapse ${\alpha }_{\min }$ for runs M1616B0 (black solid line), M1616B1 (blue dotted line) and M1616B2 (red dash line). The merger occurs at $t\approx 150M$, and an apparent horizon appears at $t=192M$ for both magnetized and unmagnetized cases.

Figure 11

Same as Fig. 4 but for run M1616B0 (unmagnetized run). The black region near the center in the last three panels denotes the apparent horizon.

Figure 12

Same as Fig. 11 but for run M1616B1 (magnetized run).

Figure 13

Same as Fig. 11 but for run M1616B2 (magnetized run).

Figure 14

Rest mass of the material outside the apparent horizon $M_{\mathrm{out}}$ for runs M1616B0 (black solid line), M1616B1 (blue dotted line) and M1616B2 (red dash line).

Figure 15

Gravitation waveform ${\psi }_4^{22}(t-r)$ for runs M1616B0 (black solid line), M1616B1 (blue dotted line) and M1616B2 (red dash line). Gravitation waves are extracted at radius $r=43M$.

Figure 16

Gravitation waveform $h_{+}$ and $h_{\times }$ for run M1616B0 observed in the direction 45° to the $z$-axis.

Figure 17

Constraint violations for the unmagnetized run M1414B0. Upper panel: Normalized L2 norm of the Hamiltonian constraint. Lower panel: Normalized L2 norm of the $x$ (black solid line), $y$ (blue dotted line) and $z$ (red dash line) components of the momentum constraint. Note that the large violations corresponding to the peaks at $t=192M$ are trapped inside the event horizon. When the apparent horizon is detected ($t>192M$), the constraints are computed only in the region outside the horizon.

Figure 18

Same as Fig. 17 but for the magnetized run M1414B1.

Figure 19

Evolution of maximum density ${\rho }_0^{\max }$ and minimum lapse ${\alpha }_{\min }$ for unmagnetized (black solid line) and magnetized (blue dash line) runs of model M1418. The merger occurs at $t\approx 180M$, and an apparent horizon appears at $t=232M$ for both magnetized and unmagnetized cases.

Figure 20

Same as Fig. 11 but for run M1418B0 (unmagnetized run).

Figure 21

Same as Fig. 11 but for run M1418B1 (magnetized run).

Figure 22

Rest mass of the material outside the apparent horizon $M_{\mathrm{out}}$ for the unmagnetized (black solid line) and magnetized (blue dash line) runs of model M1418.

Figure 23

Gravitation waveform ${\psi }_4^{22}(t-r)$ for unmagnetized (black solid line) and magnetized (blue solid line) runs of model M1418. Gravitation waves are extracted at radius $r=43M$.