Long-term evolution of a merger-remnant neutron star in general relativistic magnetohydrodynamics: Effect of magnetic winding

Long-term ideal and resistive magnetohydrodynamics (MHD) simulations in full general relativity are performed for a massive neutron star formed as a remnant of binary neutron star mergers. Neutrino radiation transport effects are taken into account as in our previous papers. The simulation is performed in axial symmetry and without considering dynamo effects as a first step. In the ideal MHD, the differential rotation of the remnant neutron star amplifies the magnetic-field strength by the winding in the presence of a seed poloidal field until the electromagnetic energy reaches $\sim 10\%$ of the rotational kinetic energy, $E_{\mathrm{kin}}$, of the neutron star. The timescale until the maximum electromagnetic energy is reached depends on the initial magnetic-field strength and it is $\sim 1\mathrm{s}$ for the case that the initial maximum magnetic-field strength is $\sim {10}^{15}\mathrm{G}$. After a significant amplification of the magnetic-field strength by the winding, the magnetic braking enforces the initially differentially rotating state approximately to a rigidly rotating state. In the presence of the resistivity, the amplification is continued only for the resistive timescale, and if the maximum electromagnetic energy reached is smaller than $\sim 3\%$ of $E_{\mathrm{kin}}$, the initial differential rotation state is approximately preserved. In the present context, the post-merger mass ejection is induced primarily by the neutrino irradiation/heating and the magnetic winding effect plays only a minor role for the mass ejection.

Figure 1

Evolution of the electromagnetic energy for all the ideal MHD models, Bv-R0-Y, Bh-R0-Y, Bh-R0-N, Bm-R0-Y, Bm-R0-N, and for a resistive model with a high conductivity, Bh-Rw-Y (left panel) and for all the resistive models employed in this paper (right panel).

Figure 2

Evolution of the angular velocity profile for models Bh-R0-Y (top left), Bm-R0-Y (top right), Bm-Rl-Y (middle left), Bm-Rm-Y (middle right), Bm-Rh-Y (bottom left), and Bh-Rw-Y (bottom right). We note that due to the neutrino cooling, the remnant neutron star contracts and its central region spins up. For model Bm-Rh-Y, the increase of the angular velocity is predominantly due to this effect.

Figure 3

Poloidal (left) and toroidal (right) magnetic-field strength along the $x$-direction with $z=1\mathrm{km}$ at selected time slices for models Bm-R0-Y (top panel), Bm-R0-N (2nd top panel), and Bm-Rm-Y (3rd panel). The bottom panel shows the poloidal and toroidal magnetic-field strengths along the $z$-direction with $x=1\mathrm{km}$ for model Rm-B0-Y. The field strengths of the poloidal and toroidal components are defined by $B^P=\sqrt{{\mathcal{B}}^x{\mathcal{B}}_x+{\mathcal{B}}^z{\mathcal{B}}_z}{\gamma }^{-1/2}$ and $B^T=\mathrm{sign}({\mathcal{B}}^y)\sqrt{{\mathcal{B}}^y{\mathcal{B}}_y}{\gamma }^{-1/2}(\approx {\mathcal{B}}^y{\gamma }^{-1/3})$, respectively.

Figure 4

Profiles for the density, temperature, specific entropy, electron fraction (left for these four plots), poloidal and toroidal magnetic-field strength, the ratio of the rest-mass to magnetic energy density, and the ratio of the gas pressure to the magnetic pressure (right for these four plots) after the saturation of the growth of the electromagnetic field for models Bm-R0-Y (top panels) and Bm-Rm-Y (bottom panels).

Figure 5

Ejecta mass as a function of time for the ideal MHD cases. Model B0-Y refers to the result in the absence of the magnetic-field effect but with the neutrino effects.

Figure 6

Average velocity (left panel) and electron fraction (right panel) of the ejecta as functions of time for the ideal MHD simulations and for one simulation without the MHD effect (B0-Y).

Figure 7

Numerical solution of $B^y$ at $t=1$ (dotted curve) and 10 (solid curve) with the analytic solution of Eq. (a1) at $t=10$ (dashed curve).

Figure 8

Numerical solutions of $\rho$ (upper panel) and $B^y$ (lower panel) for the shock tube test with ${\sigma }_{\mathrm{c}}={10}^{10}$, ${10}^2$, 10, 1, and ${10}^{-3}$. The lower panel plots $B^y$ (not ${\widehat{B}}^y$).

Figure 9

The results of the resistive rotor problem for ${\sigma }_{\mathrm{c}}={10}^{10}$ (left) and 1 (right). The upper and lower panels show $p$ and $E^z$, respectively.

Figure 10

Left: propagation of electromagnetic waves ($B^z$) in the flat spacetime for ${\sigma }_{\mathrm{c}}=0$ at $t=0$, 5, 10, and 15. At $t=15$, the amplitude is much smaller than unity entirely. The solid and dashed curves show the numerical and exact solutions, respectively. Right: dispersion of electromagnetic waves in the flat spacetime for $4\pi {\sigma }_{\mathrm{c}}=50$ at $t=0$, 50, and 100. The solid and dashed curves show the numerical and exact solutions, respectively (both curves approximately agree). For $x<0$, the solution has the reflection symmetry for both panels (and we do not plot it).

Figure 11

The maximum value of $B^y$ for the steady dynamo test for ${\sigma }_{\mathrm{c}}=10$ and ${\alpha }_{\mathrm{d}}=0.2$ with ${\lambda }_x=2$, 1, $1/2$, $1/3$, $2/7$, $1/4$, $2/9$, $a=0.1$, and $b=0$ (no perturbation in the $z$ direction) and with ${\lambda }_x=1/4$, ${\lambda }_z=1/2$, $a=0.1$, and $b=0.01$. Note that the fastest growing mode has the wavelength of $1/2$ while for the marginally stable mode it is $1/4$. $\lambda$ in the inset of the plot implies ${\lambda }_x$.

Figure 12

Evolution of the electromagnetic energy for the remnant massive neutron star initially with a purely toroidal magnetic field for ${\sigma }_{\mathrm{c}}={10}^7$, ${10}^8$, ${10}^9$, and ${10}^{11}{\mathrm{s}}^{-1}$ (solid curves). The dashed lines denote $\propto \exp (-t/{\tau }_{\mathrm{diss}})$ with ${\tau }_{\mathrm{diss}}=0.01$, 0.1, and 1.0 s for ${\sigma }_{\mathrm{c}}={10}^7$, ${10}^8$, and ${10}^9{\mathrm{s}}^{-1}$, which are consistent with Eq. (59). The result by the ideal MHD implementation is also shown by the dotted curve, which agrees approximately with the curve for ${\sigma }_{\mathrm{c}}={10}^{11}{\mathrm{s}}^{-1}$.

Figure 13

Evolution of the electromagnetic energy for two ideal MHD models Bh-R0-Y, Bh-R0-N and for two resistive MHD models Bh-Rw-Y and Bh-Rl-Y with two different grid resolutions. The solid and dashed curves show the results for $\mathrm{\Delta }x=200\mathrm{m}$ and $\mathrm{\Delta }x=250\mathrm{m}$, respectively.