Twisted magnetar magnetospheres: a class of semi-analytical force-free non-rotating solutions

Abstract

Magnetospheric twists, that is magnetospheres with a toroidal component, are under scrutiny due to the key role the twist is believed to play in the behaviour of neutron stars. Notably, its dissipation is believed to power magnetar activity, and is an important element of the evolution of these stars. We exhibit a new class of twisted axi-symmetric force-free magnetospheric solutions. We solve the Grad-Shafranov equation by introducing an ansatz akin to a multipolar expansion. We obtain a hierarchical system of ordinary differential equations where lower-order multipoles source the higher-order ones. We show that analytical approximations can be obtained, and that in general solutions can be numerically computed using standard ODE solvers. We obtain a class of solutions with a great flexibility in initial conditions, and show that a subset of these asymptotically tend to vacuum. The twist is not confined to a subset of field lines. The solutions are symmetric about the equator, with a toroidal component that can be reversed. This symmetry is supported by an equatorial current sheet. We provide a first-order approximation of a particular solution that consists in the superposition of a vacuum dipole and a toroidal magnetic field sourced by the dipole, where the toroidal component decays as $1/r^4$. As an example of strongly multipolar solution, we also exhibit cases with an additional octupole component.

Figures (7)

• Figure 1: Solution for $\dot F_1= B_1 =1, c_2=6$ between 1 and 5 stellar radii, where $B_1$ is the dipole field strength at the pole. All quantities are multiplied by $r^3$ for better visualisation. Left, left hemisphere: poloidal cross-section of the current amplitude $\alpha|\vec{B}| r^3$, normalised by its maximum. Left, right hemisphere: poloidal cross-section of the toroidal field $B_\varphi r^3$, normalised by its maximum. Poloidal magnetic field lines are shown as solid black lines, and vacuum-dipole field lines are shown for comparison as grey dashed lines. Top right: Magnetic-field components at the stellar surface, $B_r$ dashed line, $B_\theta$ dot-dashed line, and $B_\varphi$ dotted line in units of $B_1$. For comparison, vacuum-dipole components are plotted as grey lines with corresponding styles. Bottom right: current-sheet current components $\sigma_r$, dashed line, and $\sigma_\varphi$, dotted line, rescaled by a factor $r^3$.
• Figure 2: Components (left) and convergence (right) of the solution to Eq. \ref{['eq:hierarchy']} for the conditions of Fig. \ref{['fig:B1c2']}. Left: the components $F_i$ up to the truncation at order 30. For clarity only the lowest 9 orders are shown in colour and labelled, and the remaining ones are shown in grey. The thick black line represents their sum up to order 30 such that it represents the potential ${\cal P}$, Eq. \ref{['eq:ppot']}, on the stellar surface at $r=1$. Right: convergence of the series defining ${\cal P}$, Eq. \ref{['eq:ppot']}, for the conditions of Fig. \ref{['fig:B1c2']}. The maximum of each $|F_i(\mu)|$ for $\mu$ between 0 and 1 is plotted against its order $i$, where $F_i$ is solution of Eq. \ref{['eq:hierarchy']}. Only odd orders are shown, as even orders are all equal to 0.
• Figure 3: Same as Fig. \ref{['fig:B1c2']} for $\dot F_1 =1, \dot F_3 =4, c_2=3.2$ in the anti-twisted configuration.
• Figure 4: Same as Fig. \ref{['fig:B1c2comp']} for the parameters of Fig. \ref{['fig:b1b3c2']}. The thick black (left panel) line shows the sum of the first 50 orders.
• Figure 5: Same as Fig. \ref{['fig:B1c2']} for $\dot F_1 =1, \dot F_3 =2, c_1=15.9$ and regular twist.