SPECTROPOLARIMETRY OF MAGNETIC STARS .5. THE MEAN QUADRATIC MAGNETIC-FIELD

Systematic determinations of the mean quadratic magnetic field of Ap stars have been performed for the first time. The mean quadratic magnetic field (or, in short, the quadratic field) is the square root of the sum of the mean square magnetic field modulus and of the mean square longitudinal magnetic field. The latter are averages over the visible stellar disk of the square of, respectively, the modulus of the magnetic field and its component along the line of sight. These averages are weighted by the local emergent line intensities. The quadratic field is diagnosed from the study of the magnetic broadening of the stellar spectral lines as observed in unpolarized light, through the characterization of the widths of the lines by the second-order moments of their profiles (in the Stokes parameter I) about their centre. The theoretical basis of the interpretation of these moments in terms of magnetic field and the strategy followed in the analysis are presented. It is shown that this analysis yields, as a by-product, the projected equatorial velocity v(e) sin i of the studied stars. Observations of a sample of 29 stars are analyzed. For 22 of them, meaningful values or upper limits of the quadratic field can be determined. The lower limit of detection of the quadratic fields, set by the spectral resolution of the observations, is of the order of 5 kG. The observed quadratic fields range from this value up to 37 kG, in the star HD 137509. The magnetic field of this star is likely the second strongest known field in Ap stars. Quadratic field values derived for stars where resolved magnetically split lines are observed in higher-dispersion spectra are consistent with the values of the mean field modulus measured in those stars from the line splitting. For the stars of the sample repeatedly observed through their rotation cycle, the variations of the quadratic field are well represented by a cosine with the rotation frequency of the star, or by the superposition of such a cosine and of a cosine with twice that frequency. However, it appears that it is essential to have a large number of observations distributed sufficiently uniformly and sufficiently densely over the rotation phases to determine unambiguously the shape of the variations. The extrema of the quadratic field tend to occur at phases close to those of the extrema of the longitudinal field, but in some stars, the two quantities definitely vary out of phase. The ratio between the maximum and the minimum of the quadratic field is always smaller than 1.7.