Three-Dimensional Numerical Simulations of Magnetar Crust Quakes

TL;DR

This work investigates the three-dimensional dynamics of magneto-elastic waves generated by magnetar crust quakes, using a spectral 3D approach to propagate high-frequency waves through the crust and couple them to the magnetosphere and core. The study finds that crustal disturbances predominantly drain energy into the liquid core on a timescale of about $\tau_{\rm core} \sim 10\ \,\mathrm{ms}$, while only a small fraction enters the magnetosphere as Alfvén and fast magnetosonic waves, with the emission pattern strongly dependent on the quake location relative to the magnetic axis. The characteristic crustal bounce frequency is $\sim 1\ \mathrm{kHz}$, and the magnetospheric emission is localized near the quake epicenter due to limited lateral spreading before damping. These results have implications for interpreting magnetar bursts and the trigger mechanisms of fast radio bursts, offering a framework to connect crustal dynamics with magnetospheric and radiative phenomena.

Abstract

Crust quakes are frequently invoked as a mechanism to trigger sudden transients in the magnetospheres of magnetars. In this picture, a mechanical failure of the crust excites seismic motions of the magnetar surface that launch force-free waves into the magnetosphere. We first investigate this problem analytically and then perform three-dimensional numerical simulations. Our simulations follow the propagation of high-frequency magneto-elastic waves in the entire crust, and include magnetic coupling to the dipolar magnetosphere and liquid core through simplified radiation boundary conditions. We observe seismic waves bouncing between the crust-core interface and the surface with a characteristic frequency $\sim 1$~kHz, which could appear as a modulation of the magnetospheric radiation. Both the star quake and its associated magnetospheric wave emission are strongly damped on a timescale $\sim 10 \ \rm ms$ by magnetic coupling to the liquid core. Since the seismic waves are significantly damped before they can spread laterally around the crust, magnetospheric wave emission occurs primarily near the initial epicenter of the quake. Our simulations suggest that non-axisymmetric quakes will launch a mixture of Alfvén and fast magnetosonic waves into the magnetosphere. The results will be important for interpreting magnetar bursts and understanding the possible trigger mechanisms of fast radio bursts.

Figures (6)

• Figure 1: The values of effective wave speed ($\tilde{v}_s$, black line) and Alfvén wave speed ($v_A$, blue dashed line) as a function of mass density ($\rho$) in the magnetar magnetosphere, ocean, crust and core. Characteristic mass densities are presented. The following parameters are adopted: magnetic field strength $B_\star=4\times10^{14} \ \rm G$, a magnetar of mass $M_\star = 1.4$$M_\odot$, magnetar radius $11.69 \ \rm km$. The crust-core boundary is located at $r_{\rm core} = 10.8 \ \rm km$, and the mass density at the core $\rho_{\rm core}=1.27\times10^{14} \ {\rm g \ cm^{-3}}$.
• Figure 2: Toy model of a crust quake. The incompressible displacement of field lines $\xi_x$ (red wiggler) is generated by the sudden yielding of a strain layer in the deep crust, and propagates vertically along $z$-axis with wave vector $k_z$ (green arrow) towards the magnetar surface. The blue shaded region shows the liquid ocean and the gray shaded region shows the fault zone of thickness $\Delta \ell$.
• Figure 3: Relative directions of the background magnetic field $\pmb{B}$ (red arrow), wave vector $\pmb{k}$ (green arrow), and displacement perturbation of elastic waves $\pmb{\xi}$ (blue arrow) in the crust of the magnetar. The wave vector $\pmb{k}$ is aligned along the $z$-axis. $\pmb{B}$ is assumed to be in the $x-z$ plane (indicated by the yellow plane) with the angle $\theta_0$ with respect to the $z$-axis. The displacement along the $y$-axis generates pure Alfvén waves (left panel), whereas displacement along the $x$-axis generates pure fast magnetosonic waves (right panel).
• Figure 4: Evolution of the star quake triggered in the deep crust at $\theta_Q = 3\pi/10$. The left column shows a snapshot during the first elastic crossing time at $t=0.2$ ms, the middle column at $t=6$ ms, and the right column at $t=9$ ms. The first and second row show the velocity fields $v_\theta$ and $v_\phi$ on the neutron star surface. The white arrow indicates the location of the magnetic pole, and the dashed white curve indicates the location of the cross-sectional slice that is displayed in the third row. Red indicates positive velocities, blue indicates negative velocities, and gray indicates zero velocity. All panels are normalized to the same color scale. The axis labels display the $x$ and $z$ coordinate in units of $10 \ \rm km$.
• Figure 5: Time evolution of energy for different locations of the initial magnetar quake. The yellow curve represents the energy in the crust, the green curve represents the energy emitted into the core as Alfvén waves, and the blue solid and red dashed curves represent the energy emitted into the magnetosphere as Alfvén waves and fast magnetosonic waves respectively. The dotted black line represents the the analytic model for the decay of quake energy (Equation (\ref{['decay']})), and the solid black line demonstrates the conservation of the total quake energy.