Extrapolation of the solar magnetic field within the potential-field approximation from full-disk magnetograms

This paper is concerned with the Laplace boundary-value problem with the directional derivative, corresponding to the specific nature of measurements of the longitudinal component of the photospheric magnetic field. The boundary conditions are specified by a distribution on the sphere of the projection of the magnetic field vector into a given direction, i.e., they exactly correspond to the data of daily magnetograms distributed across the full solar disk. It is shown that the solution of this problem exists in the form of a spherical harmonic expansion, and uniqueness of this solution is proved. A conceptual sketch of numerical determination of the harmonic series coefficients is given. The field of application of the method is analyzed with regard to the peculiarities of actual data. Results derived from calculating magnetic fields from real magnetograms are presented. Finally, we present differences in results derived from extrapolating the magnetic field from a synoptic map and a full-disk magnetogram.