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Appendix A: Measuring the mean longitudinal magnetic field from low-resolution FORS 2 spectropolarimetry

For each star, four consecutive observations were obtained with pairs of position angles separated by 90°, namely [− 45°, + 45°, + 45°,−45°,−45°, + 45°, + 45°,−45°]. The exposure time for each position angle was 20 s for HD 23478 and 600 s for HD 345439. A peak signal-to-noise ratio (S/N) of 1900 in the final Stokes I spectrum was achieved for HD 23478, while for HD 345439 we obtained a peak S/N of ~900.

The V/I spectrum is calculated using (A.1)where + 45° and − 45° indicate the position angle of the retarder waveplate and f^o and f^e are the ordinary and extraordinary beams, respectively. Null profiles, N, are calculated as pairwise differences from all available V profiles. From these, 3σ-outliers are identified and used to clip the V profiles. This removes spurious signals, which mostly come from cosmic rays, and also reduces the noise. The mean longitudinal magnetic field, ⟨B_z⟩, is measured on the rectified and clipped spectra based on the relation (A.2)where V is the Stokes parameter that measures the circular polarization, I is the intensity in the unpolarised spectrum, g_eff is the effective Landé factor, e is the electron charge, λ is the wavelength, m_e is the electron mass, c is the speed of light, dI/ dλ is

the wavelength derivative of Stokes I, and ⟨B_z⟩ is the mean longitudinal (line-of-sight) magnetic field. The longitudinal magnetic field is usually measured in two ways: using only the hydrogen Balmer lines or using the entire spectrum including all available lines.

To identify any systematic differences that could exist in treating the FORS 2 data by different research groups, the mean longitudinal magnetic field, ⟨B_z⟩, was derived in both stars using independent software packages (one developed in Bonn and the other one in Potsdam). For the first reduction, we used a suite of IRAF (Tody 1993)¹ and IDL routines that follow the technique, recipes, and recommendations by Bagnulo et al. (2002; 2012; 2013)². The determination of the mean longitudinal magnetic field using low-resolution FORS spectropolarimetry with the second software package developed in Potsdam is described by Hubrig et al. (2014; 2015) and by Schöller et al. (in prep.). In general, the measurement method is the same for both pipelines. A few minor differences refer to the clipping procedure, rectification, and the choice of the wavelength regions including the hydrogen lines.

Furthermore, Monte Carlo bootstrapping tests are carried out in the second software package (e.g. Rivinius et al. 2010). In these tests, we generate 250 000 statistical variations of the original dataset by the bootstrapping technique and analyse the resulting distribution P(⟨B_z⟩) of the regression results. Mean and standard deviation of this distribution are identified with the most likely mean longitudinal magnetic field and its 1σ error, respectively. The main advantage of this method is that it provides an independent error estimate. The measurement uncertainties for both stars obtained before and after Monte Carlo bootstrapping tests were found to agree closely, indicating the robustness of the measurement method.

© ESO, 2015