The effect of the surface distribution of elements on measuring the magnetic field of chemically peculiar stars

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A non-homogeneous distribution over the stellar surface of the chemical elements and the presence of a large-scale magnetic field are invoked in order to explain the periodic line strength, photometric and magnetic variability of magnetic chemically peculiar stars. In such a framework, the variability period is identical to the rotational period. In magnetic chemically peculiar stars, the so-called effective magnetic field Heff, the average over the visible stellar disk of the longitudinal field component weighted by the local line strength, is routinely measured from Stokes I and V profiles of selected spectral lines. In spite of evidence that the distribution of the various chemical elements over the stellar surface can be inhomogeneous and different from element to element, Heff values obtained from lines of different elements are often statistically combined to improve the accuracy of effective field measurements. Similarly, mean high S/N profiles are obtained from the profiles of lines of different elements. We have established, by means of circular spectropolarimetry of the magnetic chemically peculiar star HD 24712 in the 4700-7000 Å range, the dependence of the measurements of Heff on the atomic weight of 24 elements (from carbon to erbium, if possible at different ionisation states). At all 3 rotational phases considered, Heff values derived from different elements can differ by up to 800 G. We find an overall increase in Heff with atomic number and a maximum near . The behaviour of sodium is quite singular inasmuch as it always exhibits a negative value of the field, peaking at -0.39 kG when the rare earths give kG. Under the assumption of a dipolar field, we conclude that the elements giving the largest values of Heff are concentrated near the positive polar region and that the other elements are more homogeneously distributed over the stellar surface or concentrated in belts around the magnetic equator. Sodium seems to be localised in the negative magnetic hemisphere only. This picture is corroborated by the equivalent width variability of the lines: up to iron, equivalent widths changes out-of-phase with respect to the Heff variability; elements heavier than iron present equivalent widths that are variable in-phase. We do not find any relation between the respective amplitudes of the equivalent width variations and the atomic numbers of the elements. For iron, the observed equivalent width variability does not seem to be simply related to any non-homogeneous distribution over the stellar surface. We conclude that measurements originating from different elements cannot in general be combined to improve the precision of Heff measurements. Indeed, any modelling attempt based on the periodic variations in Heff is subject to the risk that the sampling of the magnetic field over the stellar surface by the lines of a given chemical element is uneven or incomplete.