A Mathematical Model for an Hourglass Magnetic Field

TL;DR

This paper presents an analytic model for hourglass-shaped magnetic fields generated by specific current distributions, providing a tool for fitting observed magnetic patterns and comparing with numerical simulations.

Contribution

It introduces a new analytic solution for hourglass magnetic fields that avoids cusps and fits simulation data better than previous models.

Findings

Analytic solution matches simulation hourglass patterns.

Model can be used for source-fitting observed magnetic fields.

Provides a cusp-free, general form for hourglass magnetic fields.

Abstract

Starting with a mathematical boundary value problem for the magnetic vector potential in an axisymmetric cylindrical coordinate system, we derive a general solution for any arbitrary current distribution using the method of Green's functions. We use this to derive an analytic form for an hourglass magnetic field pattern created by electrical currents that are concentrated near (but not confined within) the equatorial plane of a cylindrical coordinate system. Our solution is not characterized by a cusp at the equatorial plane, as in previous solutions based on a current sheet. The pattern we derive provides a very good fit to hourglass magnetic field patterns emerging from three-dimensional numerical simulations of core formation, and can in principle be used for source-fitting of observed magnetic hourglass patterns.