Vector tomography for the coronal magnetic field I. Longitudinal Zeeman effect measurements

Aims. We study the possibility of applying tomographic techniques in order to reconstruct the 3D magnetic field configuration in the solar corona. Our simulations are based on data which can be obtained from longitudinal Zeeman-effect (or from Faraday rotation) measurements obtained during a solar rotation. Methods. The Zeeman effect provides essentially the integrated line-of-sight component of the magnetic field. The reconstruction problem relates to a family of similar problems termed vector tomography. For inversion of this type problems it is known that the curl-free part of a vector field cannot be reconstructed from the integrated along line-of-sight data. To remove the resulting ambiguity of the reconstruction we include the additional constraint, ∇ · B = 0, into the inversion similar to smoothing constraints in Tikhonov regularization problems. Results. It is shown that Zeeman data obtained from vantage points in the ecliptic plane alone is sensitive only to certain magnetic field structures. For a full reconstruction it is necessary either to have also observations from viewing directions at higher heliographic latitudes, or to combine the longitudinal Zeeman-effect data with observations that provide the transverse component of the field, e.g. from the Hanle effect.