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

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.

💡 Research Summary

The paper presents a new semi‑analytical class of solutions for non‑rotating, axisymmetric, force‑free magnetospheres, a problem relevant to magnetars and other highly magnetised neutron stars. Starting from the force‑free condition ∇×B = α B, the authors introduce an Euler potential P that describes the poloidal field and derive the Grad‑Shafranov equation for P. Unlike previous works that rely on self‑similar or single‑multipole ansätze, the authors propose a multipolar expansion of the form

 P(r, μ) = Σ_{i≥1} F_i(μ) r^{‑i},

where μ = cos θ and the functions F_i(μ) are dimensionless angular profiles. They also expand the current‑function α(P) and its integral A(P) as power series in P, with dimensionless coupling constants c_i. By matching the radial powers on both sides of the Grad‑Shafranov equation, they obtain a hierarchy of ordinary differential equations (ODEs) for each multipole order i:

 ‑i(i+1) F_i − (1‑μ²) F_i’’ =