Context: The Wilson depression is the difference in geometric height of unit
continuum optical depth between the sunspot umbra and the quiet Sun. Measuring
the Wilson depression is important for understanding the geometry of sunspots.
Current methods suffer from systematic effects or need to make assumptions on
the geometry of the magnetic field. This leads to large systematic
uncertainties of the derived Wilson depressions.
Aims: We aim at developing a robust method for deriving the Wilson depression
that only requires the information about the magnetic field that is accessible
from spectropolarimetry, and that does not rely on assumptions on the geometry
of sunspots or on their magnetic field.
Methods: Our method is based on minimizing the divergence of the magnetic
field vector derived from spectropolarimetric observations. We focus on large
spatial scales only in order to reduce the number of free parameters.
Results: We test the performance of our method using synthetic Hinode data
derived from two sunspot simulations. We find that the maximum and the umbral
averaged Wilson depression for both spots determined with our method typically
lies within 100 km of the true value obtained from the simulations. In
addition, we apply the method to Hinode observations of a sunspot. The derived
Wilson depression (about 600 km) is consistent with results typically obtained
from the Wilson effect. We also find that the Wilson depression obtained from
using horizontal force balance gives 110 - 180 km smaller Wilson depressions
than both, what we find and what we deduce directly from the simulations. This
suggests that the magnetic pressure and the magnetic curvature force contribute
to the Wilson depression by a similar amount.Comment: 12 pages, 8 figures. Accepted for publication in Astronomy &
Astrophysic
