Zeeman Doppler mapping revisited: Force-free fields, regularisation functions, and abundance maps

Abstract

Aims. For the observational modelling of horizontal abundance distributions and of magnetic geometries in chemically peculiar (CP) stars, Zeeman Doppler mapping (ZDM) has become the method of choice. Comparisons between abundance maps obtained for CP stars and predictions from numerical simulations of atomic diffusion have always proved unsatisfactory. This study is intended to explore the reasons for the discrepancies. Methods. We cast a cold eye (evoking the epitaph on Nobel laureate W.B. Yeats’ gravestone: Cast a cold Eye / On Life, on Death. / Horseman, pass by) on essential assumptions underlying ZDM, in particular, the formulae governing the magnetic field geometry, but also the regularisation functionals. Results. Recognising that the observed strong magnetic fields in most well-mapped stars require the field geometry to be force free, we show that the formulae used so far to describe the magnetic geometry do not meet this condition. It follows that the published magnetic maps and the abundance maps of these stars are all spurious. Conclusions. To obtain observational constraints for the modelling of atomic diffusion, the use in ZDM of the correct formulae for force-free or potential magnetic fields is paramount. Extensive simulations are required to quantify the effects of chemical stratifications and of regularisation functions on the recovered magnetic and abundance maps.