Magnetic field topology of τ Scorpii The uniqueness problem of Stokes V ZDI inversions

Context. The early B-type star tau Sco exhibits an unusually complex, relatively weak surface magnetic field. Its topology was previously studied with the Zeeman Doppler imaging (ZDI) modelling of high-resolution circular polarisation (Stokes V) observations. Aims. Here we assess the robustness of the Stokes V ZDI reconstruction of the magnetic field geometry of tau Sco and explore the consequences of using different parameterisations of the surface magnetic maps. Methods. This analysis is based on the archival ESPaDOnS high-resolution Stokes V observations and employs an independent ZDI magnetic inversion code. Results. We succeeded in reproducing previously published magnetic field maps of tau Sco using both general harmonic expansion and a direct, pixel-based representation of the magnetic field. These maps suggest that the field topology of tau Sco is comprised of comparable contributions of the poloidal and toroidal magnetic components. At the same time, we also found that available Stokes V observations can be successfully fitted with restricted harmonic expansions, by either neglecting the toroidal field altogether, or linking the radial and horizontal components of the poloidal field as required by the widely used potential field extrapolation technique. These alternative modelling approaches lead to a stronger and topologically more complex surface field structure. The field distributions, which were recovered with different ZDI options, differ significantly and yield indistinguishable Stokes V profiles but different linear polarisation (Stokes Q and U) signatures. Conclusions. Our investigation underscores the well-known problem of non-uniqueness of the Stokes V ZDI inversions. For the magnetic stars with properties similar to tau Sco (relatively complex field, slow rotation) the outcome of magnetic reconstruction strongly depends on the adopted field parameterisation, rendering photospheric magnetic mapping and determination of the extended magnetospheric field topology ambiguous. Stokes Q and U spectropolarimetric observations represent the only way of breaking the degeneracy of surface magnetic field models.