Rotating twisted magnetosphere of magnetars: approximate analytical solutions

TL;DR

This work tackles how a rotating magnetar magnetosphere with magnetic twist modifies open-field-line regions and associated emissions. It develops an approximate analytical solution by combining a self-similar twisted dipole poloidal field with a minimum-torque toroidal current, yielding explicit scalings for the Y-point radius $R_Y$, polar cap angle $\theta_{\rm pc}$, and particle outflow luminosity $L_{\rm twist}$, including the key relations $R_Y = n R_{\rm lc}$, $\sin\theta_{\rm pc}=(R/(n R_{\rm lc}))^{n/2}$, and $L_{\rm twist}/L_{\rm dipole}=(1/n)^{2n}(R_{\rm lc}/R)^{2-2n}$. The analytic results are validated against recent numerical simulations, showing consistent trends such as smaller $R_Y$, larger polar caps, and enhanced outflows with twist. The framework has implications for magnetar radio emission and fast radio bursts, suggesting that twisted magnetospheres create larger and evolving open-field regions that can power such emissions. Overall, the paper provides a tractable, physics-driven model for the magnetar magnetosphere during outbursts and its observational consequences.

Abstract

An approximate analytical solution for the rotating twisted magnetosphere of magnetars is presented. The poloidal flux is approximated by the self-similar twisted dipole field. The toroidal field is obtained by the minimum torque model. Under this approximation, it is found that: (1) The Y-point radius decreases with the increase of twist of the magnetic field. (2) The polar cap is larger for larger twist. (3) The particle outflow luminosity is larger for larger twist. (4) The maximum acceleration potential, pulse width of magnetar radio emission both increase with the twist. (5) For an untwisting magnetosphere, the physical properties evolve toward that of the normal pulsars. The above findings are consistent with previous analytical and numerical results. The larger polar cap may correspond to the hot spot during magnetar outburst. In general, a rotating twisted magnetosphere has larger open field line regions. The radio emission of magnetars and fast radio bursts may both originate in the larger and evolving open field line regions of magnetars.

Figures (5)

• Figure 1: Y-point radius vs. the twist. The Y-point radius is in units of the light cylinder radius. The blue dots, orange squares, and green diamonds are from Table 1 to 3 in Ntotsikas et al. (2024), respectively. The magenta triangle is from Table 1 in Ntotsikas & Gourgouliatos (2025). The black dashed line is the analytical approximation, eq.(\ref{['eqn_RY']}).
• Figure 2: Polar cap angular radius vs. the twist. The analytical modeling is from eq.(\ref{['eqn_thetapc']}). The labeling is similar to figure 1.
• Figure 3: Particle outflow luminosity vs. the twist. The particle outflow luminosity is normalized to that of the dipole case. The analytical modeling is from eq.(\ref{['eqn_L']}). The labeling is similar to figure 1.
• Figure 4: Maximum acceleration potential for an evolving twisted magnetosphere. The typical threshold potential for radio generation is about $10^{12} \ \rm V$.
• Figure 5: Pulse width as a function of emission height, for different values of $n$. The emission height is in units of the neutron star radius. The pulse width is multiplied by $\sin\alpha$, the unknown inclination angle. A typical pulse width of magnetar radio emission about $10 \ \%$ is shown.