Evolution of the magnetized, neutrino-cooled accretion disk in the aftermath of a black hole-neutron star binary merger

Black hole–torus systems from compact binary mergers are possible engines for gamma-ray bursts (GRBs). During the early evolution of the postmerger remnant, the state of the torus is determined by a combination of neutrino cooling and magnetically driven heating processes, so realistic models must include both effects. In this paper, we study the postmerger evolution of a magnetized black hole–neutron star binary system using the Spectral Einstein Code (SpEC) from an initial postmerger state provided by previous numerical relativity simulations. We use a finite-temperature nuclear equation of state and incorporate neutrino effects in a leakage approximation. To achieve the needed accuracy, we introduce improvements to SpEC’s implementation of general-relativistic magnetohydrodynamics (MHD), including the use of cubed-sphere multipatch grids and an improved method for dealing with supersonic accretion flows where primitive variable recovery is difficult. We find that a seed magnetic field triggers a sustained source of heating, but its thermal effects are largely cancelled by the accretion and spreading of the torus from MHD-related angular momentum transport. The neutrino luminosity peaks at the start of the simulation, and then drops significantly over the first 20 ms but in roughly the same way for magnetized and nonmagnetized disks. The heating rate and disk’s luminosity decrease much more slowly thereafter. These features of the evolution are insensitive to grid structure and resolution, formulation of the MHD equations, and seed field strength, although turbulent effects are not fully converged.

Figure 1

Comparison of the total density-averaged entropy for different numerical methods for nonmagnetized and stronger magnetized cases. The $\tau$ method for energy evolution shows extra heating happening at the late time evolution for the nonmagnetized case using both puncture and multipath methods.

Figure 2

Convergence test on the specific entropy in the first 10 ms of evolution for the magnetized disk with puncture-tau and multipatch-entropy methods. A smoothing of scalar primitive variables after the interpolation onto multipatch grids causes slightly higher initial average entropy in these runs, but the difference quickly decreases and the subsequent evolutions for both methods are in good agreement at different resolutions.

Figure 3

Comparison of the magnetic energy to the thermal energy ratio for different magnetized cases with different methods. The $\beta 36\text{−}\mathrm{P}\text{−}\tau \text{−}\mathrm{L}1$ case reaches the same saturation level as the stronger field cases, but the magnetic energy starts to dissipate in the $\beta 36\text{−}\mathrm{M}\text{−}\mathrm{Ent}\text{−}\mathrm{L}2$ case, leading the ratio to decrease over time and finally saturate at a lower level (see table 1 for simulation labels).

Figure 4

The evolution of the accretion rate $\dot{M}$ (first panel from the top), the electron fraction $Y_e$ (second panel), the specific entropy $s$ (third panel), and temperature $T$ (last panel) for magnetized $\beta 13\text{−}\mathrm{P}\text{−}\tau \text{−}\mathrm{L}1$ (dashed dot line), $\beta 36\text{−}\mathrm{P}\text{−}\tau \text{−}\mathrm{L}1$ (dashed line), and nonmagnetized B0-M-Ent-L1 cases (solid line). The accretion rate is higher by about one order of magnitude for the magnetized cases due to magnetic winding and magnetorotational instablity. The entropy grows higher as a result of effective viscous heating, while the temperature decreases over time because of adiabatic cooling for the magnetized cases.

Figure 5

Snapshot of the rest-mass density in the meridional x-z plane at $t=45\mathrm{ms}$ for $\beta 13$ case. The solid line shows magnetic field magnitude contours correspond to $\approx [10^{12},10^{13},10^{14},10^{15},10^{16}]\mathrm{G}$.

Figure 6

Snapshot of the rest-mass density in the meridional x-z plane at $t=45\mathrm{ms}$ for the nonmagnetized case.

Figure 7

Vertically and azimuthally averaged electron fraction profiles at the initial time, $t=25\mathrm{ms}$, and $t=55\mathrm{ms}$. $Y_e$ decreases at densest regions and increases at low-density outer radii regions in nomnagnetized case. In the magnetized case, $Y_e$ starts decreasing in the high-density regions only at late times.

Figure 8

Vertically and azimuthally averaged radial profiles of gas pressure, total and poloidal component magnetic pressures at $t=25\mathrm{ms}$, and $t=55\mathrm{ms}$ for $\beta 13$ (top panel) and $\beta 36$ (bottom panel) magnetized cases. The toroidal component becomes dominant after the magnetic field saturates in all the magnetized cases.

Figure 9

Neutrino luminosity evolution for electron-flavor neutrinos and antineutrinos in the $\beta =13$ and nonmagnetized simulations. The magnetized run has systematically higher electron neutrino and antineutrino luminosities.

Figure 10

The energy-averaged optical depth of electron neutrinos radial profiles at the initial time, $t=25\mathrm{ms}$, and $t=55\mathrm{ms}$. The nonmagnetized case becomes more opaque in the high-density region. This makes the neutrino cooling less efficient than in the magnetized case, which becomes more transparent during the evolution.

Figure 11

The evolution of the total heating rate ${\dot{S}}^{+}$, neutrino cooling rate ${{\dot{S}}^{-}}_{\nu }$ and advection cooling rate ${{\dot{S}}^{-}}_{\mathrm{Adv}}$ ratios to the total entropy for the nonmagnetized case. Heating and neutrino cooling rates drop significantly around $t=30\mathrm{ms}$. The advection cooling is almost zero at the end of the simulation.

Figure 12

The evolution of the total heating rate ${\dot{S}}^{+}$, neutrino cooling rate ${{\dot{S}}^{-}}_{\nu }$ and advection cooling rate ${{\dot{S}}^{-}}_{\mathrm{Adv}}$ ratios to the total entropy for the case with weaker seed field ($\beta =36$). Like in the nonmagnetized case, the total heating and neutrino cooling rates decrease significantly comparing with the early time. The advection cooling rate is considerably higher due to the MHD effects.

Figure 13

The evolution of the total heating rate ${\dot{S}}^{+}$, neutrino cooling rate ${{\dot{S}}^{-}}_{\nu }$ and advection cooling rate ${{\dot{S}}^{-}}_{\mathrm{Adv}}$ ratios to the total entropy for the standard, stronger seed field. The disk shows the same qualitative thermal behavior as in the weaker seed field case.

Figure 14

Convergence study of the total neutrino luminosity and heating rate for nonmagnetized and strong seed field multipatch runs. The results are qualitatively convergent in both cases.

Figure 15

An illustration of the synchronization of ghost zone regions at internal patch boundaries. As is standard practice with uniform grids, one grid is extended the full ghost zone width (three points, in our case) beyond the interface, while the other grid extends two points. For a cubed spheres setup, ghost zone extensions must be chosen to guarantee sufficient overlaps on 3-patch edges. Above, open circles are ghost zone points; filled circles are live points. The points B and H overlap and mark the interface. First H is set to B (which does not require interpolation). Then H can be used in the interpolation to get C.

Figure 16

Magnetized Riemann problem evolved on both a cubed-sphere multipatch grid (with about 240 grid points across the diameter of the spherical computational domain), together with the results for the same problem evolved on a Cartesian one-dimensional grid which is able to utilize the planar symmetry. The interface between the inner cube and outer cubed spheres is at $\pm 0.27$.

Figure 17

Convergence test for Bondi accretion with a radial magnetic field. Shown here is the error in the $\tau$ conservative variable at $t=5M$, by which time it has settled. The error plotted is the absolute change in $\tau$. The relative change of $\tau$ is about $2\times {10}^{-4}$ at the lower resolution. Second-order convergence breaks down at the sonic radius at $r=8$, as expected. The grid consists of 48 domains, with each of the six patches split in two on each of its axes.

Figure 18

Meridional snapshot at $t=1600M$ of a turbulent accretion torus. Shown on the left is the density (on a logarithmic scale covering the four decades up to the maximum). On the right is the grid, with resolution reduced by about a factor of 4 for clarity. The effects of the radial and angular maps are visible, as is the regularity of the poles. The unusually close radial lines are nonmatching ghost zones.