Test Problems for Potential Field Source Surface Extrapolations of Solar and Stellar Magnetic Fields

The potential field source surface (PFSS) equations are commonly used to model the coronal magnetic field of the Sun and other stars. As with any computational model, solving equations using a numerical scheme introduces errors due to discretisation. We present a set of tests for quantifying these errors by taking advantage of analytic solutions to the PFSS equations when the input field is proportional to a single spherical harmonic. From the spherical harmonic solutions we derive analytic equations for magnetic field lines traced through the three dimensional magnetic field solution. We propose these as a set of standard analytic solutions that all PFSS solvers should be tested against to quantify their inherent errors. We apply these tests to the pfsspy software package, showing that it reproduces spherical harmonic solutions well with a slight overestimation of the unsigned open magnetic flux. It is also successful at reproducing analytic field line equations, with errors in field line footpoints typically much less than one degree.