Empirical chemical stratifications in magnetic Ap stars: questions of uniqueness

Over the last decades, modelling of the inhomogeneous vertical abundance distributions of various chemical elements in magnetic peculiar A type has largely relied on simple step-function approximations. In contrast, the recently introduced regularized vertical inversion procedure (VIP) is not based on parametrized stratification profiles and has been claimed to yield unique solutions without a priori assumptions as to the profile shapes. It is the question of uniqueness of empirical stratifications which is at the centre of this article. An error analysis establishes confidence intervals about the abundance profiles and it is shown that many different step functions of sometimes widely different amplitudes give fits to the observed spectra which equal the VIP fits in quality. Theoretical arguments are advanced in favour of abundance profiles that depend on magnetic latitude, even in moderately strong magnetic fields. Including cloud, cap and ring models in the discussion, it is shown that uniqueness of solutions cannot be achieved without phase-resolved high signal-to-noise ratio and high spectral resolution (R) spectropolarimetry in all four Stokes parameters.