## Online material

### 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, 3*σ*-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, d*I*/ d*λ*
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 1*σ* 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*