CRITICAL EVALUATION OF MAGNETIC FIELD DETECTIONS REPORTED FOR PULSATING B-TYPE STARS IN LIGHT OF ESPaDOnS, NARVAL, AND REANALYZED FORS1/2 OBSERVATIONS^*

High-resolution (R  ∼  65, 000) circular polarization (Stokes V) spectra were obtained for six of the stars studied by H2011a and H2011b using the cross-dispersed échelle spectropolarimeters ESPaDOnS (at the Canada–France–Hawaii Telescope, CFHT) and its clone Narval (at the Telescope Bernardot Lyot (TBL), Pic du Midi Observatory) within the context of the MiMeS Large Program. Altogether, 18 independent observations were obtained between 2010 February and 2011 June. Each spectropolarimetric sequence consisted of four individual subexposures taken in different orientations of the half-wave retarders. From each set of four subexposures we derived Stokes I and Stokes V spectra following the double-ratio procedure (e.g., Donati et al. 1997). All frames were processed using the automated reduction package Libre-ESpRIT (Donati et al. 1997), with the Stokes I spectra normalized order-by-order using Legendre polynomials following extraction of Stokes V. The peak signal-to-noise ratios (S/Ns) per 1.8 km s⁻¹ spectral pixel in the reduced spectra range from about 450 to about 1300.

As discussed by Semel et al. (1993) and Schnerr et al. (2006), rapid temporal changes in line profile shapes and positions due to pulsation may also introduce spurious signal in the inferred polarization spectra. In the case of a non-magnetic star, such a phenomenon may result in a false magnetic detection. Spurious signatures may equally be introduced by pulsation when a magnetic field is present, combining with the real Zeeman signature and potentially modifying its shape and amplitude.

Since all targets are either confirmed or candidate β Cep or SPB stars, subexposures were timed, as far as possible, to correspond to no more than ∼1/80th of a pulsation cycle to ensure that no significant variability occurred during the observation. Of the five observed stars for which periods are known, three [ Lupi (pulsation periods of 0.155 days (primary) and 0.096 days (secondary); Uytterhoeven et al. 2005; Telting et al. 2006), 33 Eri (0.864 days; De Cat et al. 2005), and HY Vel (3.102 days; Lefévre et al. 2009)] were observed with subexposures of hundredths of a period. 15 CMa (0.093–0.181 days; Shobbrook et al. 2006) was observed with subexposures corresponding to less than 1/80th of the longer period, but about 1/60th of the shorter period. V1449 Aql is a multiperiodic star with a dominant mode period of 0.182 days (Briquet et al. 2009) and several lower-amplitude modes as rapid as 0.119 days. Longer subexposures were required due to its faint apparent magnitude, resulting in subexposures for this star that are somewhat longer than 1/80th of these modes. α Pyx is a candidate β Cep star for which the period is not yet known. However, its subexposure durations were 150 s, and therefore would satisfy the limit described above as long as its pulsation period is longer than 0.14 days.

Given the uncertainties described above, we have performed a number of tests to establish that pulsations did not significantly impact the magnetic diagnosis. We examined the Stokes I profiles associated with each of the subexposures of each observation. For Lupi, α Pyx, 33 Eri, 15 CMa, and HY Vel, we confirm that no variation of the I profile during an observation is detected above the noise. For V1449 Aql, on the other hand, the I profile is observed to vary during an observation by a maximum of about 10 km s⁻¹ in radial velocity, with a small (∼10%) variation in depth and negligible change in shape. To verify that instrumental instabilities or line profile variations such as those observed in V1449 Aql do not translate into a spurious Stokes V signal, we computed diagnostic null polarization spectra (labeled N) by combining the four subexposures in such a way that stellar polarization cancels out (see Donati et al. 1997 or Bagnulo et al. 2009 for more details on the definition of N). In no case is any signal detected in N above the level of the noise. Considering the numerical simulations of Schnerr et al. (2006) which demonstrate that spurious signatures due to pulsation appear in both V and N, this confirms in the case of V1449 Aql that the observed line variations introduce no spurious signal into Stokes V.

Notwithstanding this conclusion, variability of the profile of V1449 Aql during the observation may result in temporal smearing of the Stokes I and V profiles. To verify that this did not significantly reduce our sensitivity to any magnetic field, we have performed a test in which we generated fictitious non-zero Stokes V profiles by computing the numerical derivative of the Stokes I profile corresponding to each of the subexposures. While these profiles do not correspond to any real magnetic field (e.g., any prediction of the model of H2011b), they have the qualitative characteristics of a real Stokes V Zeeman signature, and in particular reflect the changing shape and position of the line profile of V1449 Aql during the course of a polarimetric observation. They therefore serve to understand the impact of the temporal smearing on Stokes V. The fictitious V profiles exhibit small changes from subexposure to subexposure, reflecting the radial velocity, depth, and shape variability of Stokes I. Nevertheless, we find that the characteristics of the mean V profile inferred by averaging the subexposures does not differ qualitatively from the V profiles of the individual subexposures. In particular,its overall shape and amplitude are not significantly modified. Therefore temporal smearing of the ESPaDOnS and Narval Stokes V and I profiles should not significantly affect either the detectability or the interpretability of any magnetic signatures present in the spectrum of V1449 Aql.

A related secondary effect involves the depth-dependence of the pulsational dynamics, and therefore a (weak) potential systematic difference in the shape of the line profiles as a function of line strength. To evaluate whether the depth of formation of spectral lines has any influence on the magnetic diagnosis of V1449 Aql, we performed a simple test using least-squares deconvolution (LSD; described in detail in Section 3.1) masks filtered for weak (unbroadened central depth <0.2 of the continuum) and strong (>0.2 of the continuum) spectral lines. The two masks contain similar numbers of spectral lines. We used these masks to extract Stokes V profiles of Lupi, the only star of the sample in which a magnetic field is detected. The LSD Stokes I profiles extracted using the two masks are reasonably similar in shape; the asymmetric structure of the line profile (see Figure 1) is marginally clearer in the "weak" line, while the wings of the "strong" line are more extended than those of the "weak" line. The two LSD profiles differ by about a factor of 1.5 in depth. Longitudinal fields measured from the "strong" and "weak" line LSD profiles both correspond to magnetic detections, and agree within the error bars.

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While the two tests described above do not represent a sophisticated evaluation of the temporal smearing and depth-dependence effects on magnetic diagnosis, they illustrate that no significant effect is present at the precision of the ESPaDOnS and Narval observations described in this paper.

In this paper, we present four observations of 33 Eri, 4 of 15 CMa, 3 of α Pyx, 2 of Lupi, 4 of V1449 Aql, and 1 of HY Vel. In the case of HY Vel, poor observing conditions limited the quality of the data and prevented us from coming to any firm conclusions.

The log of CFHT and TBL observations is reported in Table 1. For Lupi and V1449 Aql, co-added spectra were created from observations taken on the same night in order to increase the S/N. In the case of V1449 Aql, radial velocity differences (described above) between observations were significant enough that spectra were brought to a common velocity before being combined.

Table 1. Summary of Results from ESPaDOnS and Narval Data

Star	HD	Inst.	Exp.	Number	HJD	Peak	Phase	Range	〈Bz〉	Det. Prob.	〈Nz〉	Det. Prob.
	Number		Time (s)	Spectra	−2,400,000	S/N		(km s−1)	(G)	(V)	(G)	(N)
Lupi	136504	E	80	4	55634.1488	1088	...	−79–85	− 90 ± 17	1.000	−2 ± 17	0.001
		E	80	4	55727.8071	1241	...	−79–85	− 76 ± 16	1.000	−20 ± 16	0.011
α Pyx	74575	E	600	1	55549.0736	1202	0.643	−40–70	−5 ± 6	0.016	−4 ± 6	0.053
		E	600	1	55555.0931	1270	0.525	−40–70	−8 ± 6	0.246	−1 ± 6	0.236
		E	600	1	55561.0935	1320	0.402	−40–70	11 ± 5	0.693	4 ± 5	0.005
33 Eri	24587	N	1200	1	54373.9793	1105	0.316	−45–80	−46 ± 48	0.341	−4 ± 48	0.004
		E	600	1	55547.7521	809	0.074	−55–50	35 ± 25	0.001	−9 ± 25	0.055
		E	600	1	55549.9444	917	0.787	−55–50	−19 ± 22	0.001	9 ± 22	0.018
		E	600	1	55554.8367	745	0.611	−55–50	4 ± 34	0.290	−15 ± 34	0.015
15 CMa	50707	E	600	1	55250.9147	844	0.676	−50–100	−13 ± 13	0.353	4 ± 13	0.045
		E	600	1	55549.0566	869	0.261	−50–100	−38 ± 19	0.000	−15 ± 19	0.007
		E	600	1	55555.0197	785	0.733	−50–100	−4 ± 14	0.003	13 ± 14	0.189
		E	600	1	55557.9927	686	0.968	−50–100	−3 ± 16	0.022	11 ± 16	0.002
HY Vel	74560	E	240	1	55611.9666	460	...	−40–60	22 ± 217	0.750	4 ± 217	0.303
V1449 Aql	180642	E	2360	2	55721.9664	326	0.998	−90–90	28 ± 53	0.000	−91 ± 53	0.307
		E	960	3	55749.8108	441	0.002	−120–60	26 ± 53	0.009	−10 ± 53	0.103
		E	960	3	55756.0395	826	0.450	−100–80	2 ± 30	0.001	−11 ± 30	0.008
		N	960	6	55783.5336	847	0.429	−100–80	30 ± 50	0.000	−18 ± 49	0.094

Notes. Note that subexposure times correspond to 1/4 of the total exposure time.

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We have re-analyzed the FORS1/2 observations⁶ obtained by H2011a, both in published and re-reduced forms. H2011a presented 62 Stokes V observations of 5 of our program stars, obtained between 2003 November and 2010 March using grism 600B, delivering a spectral resolution R ∼ 2000. The observations sample this time period nonuniformly, and the number of observations varies significantly from star to star (7 observations each for α Pyx and Lup, 15 observations each for 15 CMa and HY Vel, and 18 observations for 33 Eri).

We also analyzed the 2 FORS1/2 and 13 SOFIN measurements of V1449 Aql reported by H2011b. The SOFIN spectra were obtained from 2009 August–September and in 2010 July and were made at intermediate spectral resolution (R ∼ 30,000).

The ESPaDOnS/Narval spectra were analyzed using the LSD multiline analysis method developed by Donati et al. (1997). LSD combines the information from essentially all metallic and He lines in the spectrum by means of the assumption that the spectrum can be reproduced by the convolution of a "mean" line profile (the "LSD profile") with an underlying spectrum of unbroadened atomic lines of given line depth, Landé factor, and wavelength (the "line mask," as described by Wade et al. 2000). This process allows the computation of averaged Stokes I, V, and N profiles with much higher S/Ns than those of the individual lines, dramatically improving the detectability of Zeeman signatures due to stellar magnetic fields. The application of LSD to spectra of pulsating B stars is discussed in detail by Silvester et al. (2009).

The line masks used in this analysis were modified from generic masks obtained using the Vienna Atomic Line Database (VALD; Piskunov et al. 1995; Ryabchikova et al. 1997; Kupka et al. 1999; Kupka et al. 2000) extract stellar requests. The VALD line lists are characterized by the stellar effective temperatures and gravities of the individual stars (as reported by H2011a), solar abundances, and a minimum line depth of 10% of the continuum. The masks were customized to the spectrum of each star by interactive comparison, removing weak lines, and those blended with Balmer or telluric lines. This "cleaning" of the mask ensures that lines included in the LSD procedure have shapes that are as similar and uniform as possible.

The lines remaining in the mask following the cleaning procedure—between 230 and 320 lines depending on the physical and spectral properties of the star—were then adjusted to better match the observed line depths in the associated observed spectrum. "Tweaking" the lines in this fashion improves the fit between the actual spectrum and the LSD spectrum model (that is, the convolution of the LSD profile and the line mask).

Two methods were used to evaluate the presence of a photospheric magnetic field: a statistical test performed on the Stokes V profile and direct inference based on the significant detection of a longitudinal magnetic field. The former method, described by Donati et al. (1992, 1997), employs the reduced χ² of the signal in Stokes V within the bounds of the line profile. It reports the detection of a magnetic signature as "definite" if the formal detection probability is greater than 99.999%, while the detection probabilities outside the Stokes V line profile, and inside the diagnostic null (N) line profile, are both negligible; a "marginal" detection corresponds to a detection probability between 99.9% and 99.999%.

The second, direct inference method involves computation of the mean longitudinal magnetic field from the first-order moment of the Stokes V profile within the line according to the expression:

$$\begin{equation} \langle B_z \rangle = -2.14\times 10^{11}\frac{\int \! vV(v)\mathrm{d}v}{\lambda g_{\rm {eff}}c\int \left[I_{\rm {c}}-I(v)\right] \mathrm{d}v}, \end{equation} \tag{ 1 }$$

where 〈B_z〉 is the mean longitudinal magnetic field in G, v is the velocity in km s⁻¹ within the profile measured relative to the center of gravity (Mathys 1989; Donati et al. 1997; Wade et al. 2000), and λ and g_eff are the reference values of the wavelength in nm and Landé factor used in computing the LSD profiles. This equation is also applied to the null profile, yielding a comparable measurement 〈N_z〉. The uncertainties associated with 〈B_z〉 and 〈N_z〉 were determined by propagating the formal (photon statistical) uncertainties of each pixel through Equation (1). This is described in more detail by Silvester et al. (2009).

The integration ranges employed in the evaluation of Equation (1) associated with each LSD profile were selected individually through visual inspection so as to include the entire span of the Stokes I profile. These ranges are indicated in Figure 1 and reported for each observation in Table 1. Integration ranges can differ significantly for different observations of the same star, especially when pulsations are a factor as they are here, and the ability to tailor the integration range to the shape of each line profile and thus optimize the magnetic diagnosis is an important strength of high-resolution spectropolarimetry. The pulsational character of the stars is clearly visible in the often rather asymmetric LSD Stokes I profiles. We tested the sensitivity of the inferred longitudinal field and its error bar to the integration range by adjusting the integration limits by ±10%. This has the effect of decreasing (for smaller range) or increasing (for larger range) the error bar by 15%–20%, but otherwise has no impact on the results. These results are qualitatively consistent with experiments carried out by Neiner et al. (2012; see their Figure 3).

Most stars show relatively broad wings, attributable to the inclusion of He lines in the line masks. Removing the He lines from the mask reduces this effect at the expense of sensitivity, yielding an increase in the error bars of 〈B_z〉 on order of 50%. Experiments using Lupi (the only star of the sample in which magnetic field is detected in the ESPaDOnS/Narval spectra) clearly show that masks including and excluding He lines yield congruent results in the presence of a magnetic field (apart from differences in error bars). The line masks from which the LSD profiles shown in Figure 1 were constructed to include He lines.

The mean longitudinal magnetic field 〈B_z〉 of FORS1/2 data was derived using the linear regression method developed by Bagnulo et al. (2002). The measurements were obtained in two ways: using only hydrogen Balmer lines and using the full spectrum (including H Balmer lines along with metal and He lines). No diagnostic null measurements were reported. We assume that uncertainties were derived from the formal uncertainties obtained from the regression. These measurements, which we re-analyze below, are reported in Tables 2 and 3 of H2011a.

〈B_z〉 was measured from SOFIN using the moment technique of Mathys (1994). Again no diagnostic nulls are reported. The measurements are provided in Table 1 of H2011b.

In order to test the robustness of the field measurements reported by H2011a, the archival data from which they were determined were obtained and re-reduced using a new FORS reduction pipeline, the details of which are described by Bagnulo et al. (2012). We have measured the longitudinal field using the method of Bagnulo et al. (2002) from H lines (Hβ to the Balmer jump), from the full spectrum (from 3800 Å to 4950 Å), as well as from "metal lines" (i.e., the full spectrum excluding H lines). The measurements were performed using a larger aperture for spectrum extraction, and omitting the region 3900–4000 Å, which has been found to be afflicted by internal reflections. For all three sets of measurements obtained from the Stokes V spectra, longitudinal fields from the corresponding diagnostic null spectra were also measured. Full spectrum and H line measurements are compared to the original published measurements in Figure 3.

At the time of this writing the SOFIN data remain proprietary and so a re-reduction cannot be undertaken.

Recall that of the seven pulsating B stars claimed to be magnetic by H2011a and H2011b, ephemerides are presented for five: ξ¹ CMa, 33 Eri, 15 CMa, α Pyx, and V1449 Aql. For each of these stars they derived a model of the magnetic field, including inclination of the rotation axis relative to the line of sight i, magnetic obliquity β, and dipolar field strength B_d.

ESPaDOnS and Narval longitudinal magnetic fields of 15 CMa, 33 Eri, α Pyx, and V1449 Aql are uniformly consistent with the absence of any magnetic fields, in contrast to the FORS1/2 results published by H2011a and H2011b. One possible explanation of this discrepancy is suggested by an examination of Figure 2. While the ESPaDOnS/Narval 〈B_z〉 measurements are all consistent with a null longitudinal field, many of these measurements were obtained at times when, according to the proposed ephemerides and field geometry models, the longitudinal field should be close to zero. Some data points, because of the relatively low amplitudes of the model longitudinal field variations and generally larger error bars of the FORS data, are in fact in reasonable agreement with the FORS measurements obtained at those same phases. So, while some of the ESPaDOnS/Narval 〈B_z〉 measurements are strongly inconsistent with the published variations—V1449 Aql in particular—one can reasonably accommodate many of the remaining differences within the measurement uncertainties and possibly the uncertainties of the model parameters.

However, the longitudinal field is but one (secondary) diagnostic of the magnetic field extracted from the ESPaDOnS and Narval data. This is because a large variety of magnetic configurations can produce a mean longitudinal field component that is formally null (i.e., for which the first-order moment of Stokes V is zero) at some phases. Nearly all of these configurations will generate a detectable Stokes V signature in the velocity-resolved line profile. The morphology of the Stokes V profile often allows these various magnetic configurations—all of which yield the same (zero) longitudinal field—to be distinguished.

None of the LSD Stokes V profiles for α Pyx, 15 CMa 33 Eri or V1449 Aql show any significant signal. To predict the Stokes V profiles we would expect to observe as a consequence of the proposed field geometry models, we calculate synthetic disc-integrated Stokes I and V profiles. The model is essentially identical to that described by Petit et al. (2008, 2011): the local Stokes I profiles are assumed to be Gaussian in shape, the local Stokes V profile is computed assuming the weak-field approximation, and an explicit disc integration is performed assuming a simple limb-darkening law. Various calculated Stokes I and V profiles computed by the simple model have been compared to calculations obtained using the Zeeman code (Landstreet 1988; Wade et al. 2001) and are found to be in acceptable agreement with the detailed polarized radiative transfer calculations.

We proceed to compute "synthetic observations" as follows, treating the LSD profiles as a single, mean spectral line. First, we match as closely as possible the observed and model Stokes I profile. We begin with the vsin i reported by H2011a and H2011b, scale the model profile to match the depth of the observed LSD profile, then add Gaussian turbulence as necessary to reproduce as closely as possible the line wings. Then, using the mean wavelength and Landé factor of the real LSD profile, we compute the model Stokes V profile corresponding to the appropriate phase and field geometry parameters (i, β, and B_d) of H2011a and H2011b. Finally, we resample the model profiles to the 1.8 km s⁻¹ pixels of the observed profiles and add synthetic Gaussian noise corresponding to the S/N of the real LSD profile. Clearly, these fits to the LSD profiles are not meant to represent a detailed line synthesis in the complex pulsating atmosphere. However, given the success of Donati et al. (2001) in applying a similar modeling procedure to self-consistently reproduce the Stokes V profiles and longitudinal field curve of β Cep itself (which we point out appears to exhibit much larger pulsational distortion of its line profiles than the stars discussed here), we are confident that this procedure provides a reasonable first-order calculation of the amplitude and shape of the Stokes V profiles predicted by the H2011a and H2011b models.

Analyzing the synthetic observations using the same procedures as used for the real observations, we find longitudinal fields that are (naturally) in good agreement with the model predictions (i.e., the solid curves in Figure 2). The Stokes V profiles of the synthetic observations, on the other hand, are strikingly different from those observed. In every observation of 33 Eri, 15 CMa, and α Pyx, the proposed field geometry models predict Zeeman signatures that would be easily observed and definitely detected (i.e., detection probability >99.999%) in our observations. These results are illustrated in Figure 4 for LSD profiles obtained from masks including He lines; if He lines are excluded, the results are essentially the same. The observations included in this figure were selected to be those for which the longitudinal field was most compatible with the models, as determined from Figure 2. The disagreement between the predicted and observed Stokes V profiles is quite striking. Given the S/Ns obtained in ESPaDOnS and Narval spectra, the models proposed by H2011a and H2011b predict definite detections of Zeeman signatures in all 15 observations of 33 Eri, 15 CMa, α Pyx, and V1449 Aql. Therefore, unlike the longitudinal field measurements, the incompatibility of the observed Stokes V profiles and those predicted by the models is totally unambiguous.

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One puzzling aspect of the results presented by H2011a and H2011b is the remarkably low values of the reduced χ² for some stars when the data are fit with a sinusoidal variation: in general a reduced χ² significantly below 1 for a sample of this size is indicative of fitting noise or overestimation of error bars. However, we recall that the re-reduction of the FORS1/2 spectra has yielded error bars systematically larger than those of the original reduction, suggesting that the H2011a error bars are not substantially overestimated (and are potentially underestimated.)

To investigate this matter further, we have constructed periodograms using both the full spectrum data and the H line data, both published and re-reduced, for all five stars for which detailed models were presented by H2011a and H2011b. We use a Lomb–Scargle-type approach, computing the reduced χ² as a function of adopted period by fitting a linear first-order sinusoid to the data. Examples of the derived periodograms are shown in Figure 5. The periodograms are quite complex, and in all cases numerous periods exist corresponding to fits to the data with reduced χ²s comparable to the periods reported by H2011a and H2011b. In other words, the H2011a and H2011b periods are not unique. None of these periods seems to be a priori preferable to another. In the cases of α Pyx, 15 CMa, and V1449 Aql, the periods identified by H2011a and H2011b are in fact those yielding the lowest reduced χ² (although in the case of α Pyx there are hundreds of comparable choices). In the case of 33 Eri, the period selected does not provide the best fit; a longer period of 10.437 days yields an improved value of χ²/ν. Ultimately, we see little objective justification for selecting one among the various possible period solutions for these data, as there are numerous solutions providing an acceptable fit to the time series.

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Because the temporal sampling of these data is diverse, the data window function is expected to be rather complex. Therefore to further explore possible periodicities in the data, we have performed an independent fast Fourier transform periodogram analysis using the clean procedure as implemented in clean-ng (Gutiérrez-Soto et al. 2009). The cleaned periodograms typically exhibit one dominant period, however in only one case (15 CMa) is that dominant period close to that selected by H2011a and H2011b. Moreover, the different reductions of the FORS data do not yield the same periods (see Figure 5). Application of our period analysis procedure to synthetic time series with the same window functions and error bars but consisting only of noise show that period solutions with similar amplitudes to those obtained from the published data can be easily obtained for all stars but 15 CMa. However, we find that when the most significant published measurements of 15 CMa are removed from the analysis (i.e., those with significance larger than 2σ), the resulting periodogram continues to show a strong peak at that same period, a persistence which can only suggest that the reported period arises from a fit to the noise.