Equation of the field lines of an axisymmetric multipole with a source surface

TL;DR

This paper derives the equations for the magnetic field lines of axisymmetric multipole fields around stars, providing analytical solutions assuming potential fields outside the star.

Contribution

It presents a general analytical derivation of the field line equations for arbitrary degree axisymmetric multipoles in stellar magnetospheres.

Findings

Derived the equations of field lines for axisymmetric multipoles.

Provided analytical expressions for magnetic field components.

Applicable to modeling stellar magnetic fields with multipolar components.

Abstract

Optical spectropolarimeters can be used to produce maps of the surface magnetic fields of stars and hence to determine how stellar magnetic fields vary with stellar mass, rotation rate, and evolutionary stage. In particular, we now can map the surface magnetic fields of forming solar-like stars, which are still contracting under gravity and are surrounded by a disk of gas and dust. Their large scale magnetic fields are almost dipolar on some stars, and there is evidence for many higher order multipole field components on other stars. The availability of new data has renewed interest in incorporating multipolar magnetic fields into models of stellar magnetospheres. I describe the basic properties of axial multipoles of arbitrary degree and derive the equation of the field lines in spherical coordinates. The spherical magnetic field components that describe the global stellar field topology are obtained analytically assuming that currents can be neglected in the region exterior to the star, and interior to some fixed spherical equipotential surface. The field components follow from the solution of Laplace's equation for the magnetostatic potential.