Topological methods in astrophysics

Most objects in astrophysics are filled with highly conducting plasma and hence easily carry magnetic fields. The topological properties of these fields have important physical consequences. The atmospheres of the Sun, many types of stars, and accretion disks have magnetic fields rooted at the surface. The topological structure of the magnetic lines of force determines the possible equilibrium configurations of the field. Solar and stellar atmospheres are much hotter than expected given the surface temperature. A proposed model of heating involves tangled magnetic field lines, which release their energy in small flares. The degree of topological complexity of a magnetic field helps to determine how much energy it stores. Flares simplify the topology of the field and thereby release the stored energy. Topology is also important in understanding large–scale properties of the solar dynamo that generates the solar magnetic field. The magnetic helicity integral, which measures linking properties of the field, can be decomposed into contributions from different regions of the Sun and space. Transport of helicity from one region to another underlies many important processes in solar activity.