Relativistic Expansion of Magnetic Loops at the Self-similar Stage

TL;DR

This paper derives self-similar solutions for relativistically expanding magnetic loops with azimuthal magnetic fields, revealing how high-speed expansion influences magnetic and plasma structures, and estimating energies comparable to magnetar giant flares.

Contribution

It introduces new self-similar solutions for relativistic magnetic loop expansion considering azimuthal fields and displacement current effects.

Findings

Displacement current reduces toroidal current at high speeds.

Magnetic energy estimates match magnetar giant flare energies.

Solutions describe free, self-similar expansion of magnetic loops.

Abstract

We obtained self-similar solutions of relativistically expanding magnetic loops taking into account the azimuthal magnetic fields. We neglect stellar rotation and assume axisymmetry and a purely radial flow. As the magnetic loops expand, the initial dipole magnetic field is stretched into the radial direction. When the expansion speed approaches the light speed, the displacement current reduces the toroidal current and modifies the distribution of the plasma lifted up from the central star. Since these self-similar solutions describe the free expansion of the magnetic loops, i.e., $Dv/Dt=0$, the equations of motion are similar to those of the static relativistic magnetohydrodynamics. This allows us to estimate the total energy stored in the magnetic loops by applying the virial theorem. This energy is comparable to that of the giant flares observed in magnetars.