Global Magnetohydrodynamic Simulations of Monster Shocks in Neutron Star Magnetospheres

Abstract

Waves launched from the neutron star surface or inner magnetosphere propagate through the magnetosphere as small perturbations, but can grow relative to the background magnetic field and steepen into ``monster shocks'' -- ultra-relativistic magnetized shocks which can power high-energy emission from magnetars, neutron star mergers and collapse. They occur in magnetically dominated plasma and are described by relativistic magnetohydrodynamics (MHD). We present global relativistic MHD simulations of monster shocks in unperturbed and perturbed (``wrinkled'') backgrounds with a global dipolar geometry. Our simulations confirm analytical predictions for equatorial shocks and provide new insight into the behavior of oblique shocks off the equator. Simulations where the shock is formed through Alfvén mode to fast mode conversion are also presented, demonstrating the generic nature of the monster shock mechanism. We explore how the presence of additional modes in the magnetosphere modifies the shock behavior. Modes of comparable amplitude can fragment the shock front, substantially reduce the magnetization, produce localized enhancements in the Lorentz factor relative to an unperturbed dipole background, and intermittently generate additional shocks along a line of sight.

Figures (18)

• Figure 1: The three setups presented in this paper. Left: a spherical FMS wave, $E^\phi = R_{\rm{NS}} E_{\rm{w}} \sin(\omega t)/r$, is launched from the neutron star surface into a dipolar background magnetic field. Middle: toroidal A waves, $B^\phi \propto (R_{\rm{NS}}/r)^{3/2}$, of opposite polarity on either side of the equator are launched through a localized twist on the stellar surface (Equation \ref{['eq:twist']}). The waves collide at the equator producing a fast mode which propagates radially outwards and shocks. Right: the initial background dipolar magnetic field is perturbed with a harmonic standing wave (Equation \ref{['eq:wiggles']}), resulting in a wrinkled background. A fast mode is then launched from the stellar surface and interacts with this wave perturbation as it begins to shock. The ratio of wrinkle and background field, $\delta B/B$, is shown for two values of the wrinkle amplitude $\tilde{a}$ and wave number $m$ (Equation \ref{['eq:wiggles']}). White lines show linearly spaced magnetic flux contours.
• Figure 2: Comparison of the equatorial monster shock to analytic predictions 2023ApJ...959...34B. Left: The upstream Lorentz factor of the monster shock on the equator as a function of cylindrical radius, plotted for multiple values of $\sigma_\times$ and $\lambda=2\pi c/\omega$. The black dashed line is the expected analytical scaling (Equation \ref{['eq:lfac_equator_vs_r']}). Right: The maximum Lorentz factor of the equatorial monster shock as a function $c \sigma_\times/\omega R_\times$. The wave amplitude $E_{\rm{w}}$ (green circles), wave frequency $\omega$ (blue diamonds), and magnetization $\sigma_\times$ (red squares) are varied. The black dashed line is a linear function with slope set by the analytical expectation (Equation \ref{['eq:lfac_equator']}) and a fitted non-zero $y$-intercept. Details of the spherical FMS wave simulations can be found in Table \ref{['tab:sims']}. All values are taken on the equator, $z/R_{\times}=0$. Individual linear fits where a single parameter is varied are presented in Appendix \ref{['app:fitting']}, showing excellent agreement with the analytical expectation.
• Figure 3: Visualizations of spherical FMS wave simulation sigma100 at $t = 2 R_{\rm{NS}}/c$. Top three rows are two-dimensional (2D) visualizations of: the toroidal ideal electric field $E^\phi$ (top left); the radial ${\bm E}\times{\bm B}$ velocity $v^r$ (middle left); the angular velocity $v^\theta$ (bottom left); the Lorentz factor $\Gamma$ (top right); the fluid temperature $T$ (middle right); and the magnetization accounting for the fluid temperature $\sigma_{\rm{hot}}$ (bottom right). Contours of poloidal magnetic flux are plotted as white solid lines; the contours are linearly spaced. A white (black) vertical dashed line is placed at $x=1.15R_\times$. Bottom row shows 1D equatorial slices ($z/R_{\times}=0$) of: the FMS electromagnetic wave components $B^\theta$ and $E^\phi$ (left); and the radial four velocity (right).
• Figure 4: The radial ${\bm E}\times{\bm B}$ velocity at $t=4 R_{\rm{NS}}/c$ in spherical FMS wave simulation +sigma100+. White arrows show the direction of flow in regions with Lorentz factor over a fixed threshold. Solid white lines are placed at $\sin\theta = \pm 2/\sqrt{5}$, the angle at which the radial velocity is expected to change sign for the $\bm{B}_{\rm w} \cdot \bm{B}_{\rm bg} < 0$ portion of the wave.
• Figure 5: The maximum ${\bm E}\times {\bm B}$ Lorentz factor, as a function of time and polar angle, for the spherical FMS wave simulations listed in Table \ref{['tab:sims']}; these vary the number of FMS wavelengths launched from one to four. The Lorentz factor in the secondary shock fronts is seen to peak off the equator. The analytical expectation for the non-linearity radius, $r_\times(\theta)$ (Equation \ref{['eq:rx']}), is plotted as a dashed white line for each wavelength, assuming propagation at the speed of light.