Magnetic field, rotation, and emission of \xi^{1} CMa

The pulsating magnetosphere of the extremely slowly rotating magnetic Cep star CMa thanks: Based on observations obtained using the MuSiCoS spectropolarimeter at Pic du Midi Observatory; ESPaDOnS at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii; on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO program 292.D-5028(A) at the Paranal Observatory, ESO Chile, with the NIR interferometers AMBER and PIONIER; on observations obtained at the La Silla Observatory, ESO Chile, with the echelle spectrograph CORALIE at the 1.2m Euler Swiss telescope; and on NEWSIPS data from the IUE satellite.

M. Shultz, G.A. Wade, Th. Rivinius, C. Neiner, H. Henrichs, W. Marcolino, and the MiMeS Collaboration
Department of Physics, Engineering Physics & Astronomy, Queen’s University, Kingston, ON Canada, K7L 3N6
Department of Physics, Royal Military College of Canada, Kingston, Ontario K7K 7B4, Canada
ESO - European Organisation for Astronomical Research in the Southern Hemisphere, Casilla 19001, Santiago 19, Chile
Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala 75120
LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris Diderot,
Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
Department of Astrophysics/IMAPP Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
Universidade Federal do Rio de Janeiro, Observatório do Valongo, Ladeira Pedro AntUnio, 43, CEP 20.080-090, Rio de Janeiro, Brazil

CMa is a monoperiodically pulsating, magnetic Cep star with magnetospheric X-ray emission which, uniquely amongst magnetic stars, is clearly modulated with the star’s pulsation period. The rotational period has yet to be identified, with multiple competing claims in the literature. We present an analysis of a large ESPaDOnS dataset with a 9-year baseline. The longitudinal magnetic field  shows a significant annual variation, suggesting that is at least on the order of decades. The possibility that the star’s H emission originates around a classical Be companion star is explored and rejected based upon VLTI AMBER and PIONIER interferometry, indicating that the emission must instead originate in the star’s magnetosphere and should therefore also be modulated with . Period analysis of H EWs measured from ESPaDOnS and CORALIE spectra indicates  yr. All evidence thus supports that CMa is a very slowly rotating magnetic star hosting a dynamical magnetosphere. H also shows evidence for modulation with the pulsation period, a phenomenon which we show cannot be explained by variability of the underlying photospheric line profile, i.e. it may reflect changes in the quantity and distribution of magnetically confined plasma in the circumstellar environment. In comparison to other magnetic stars with similar stellar properties, CMa is by far the most slowly rotating magnetic B-type star, is the only slowly rotating B-type star with a magnetosphere detectable in H (and thus, the coolest star with an optically detectable dynamical magnetosphere), and is the only known early-type magnetic star with H emission modulated by both pulsation and rotation.

Stars : individual : CMa – Stars: magnetic field – Stars: early-type – Stars: oscillations – Stars: mass-loss.
pagerange: The pulsating magnetosphere of the extremely slowly rotating magnetic Cep star CMa thanks: Based on observations obtained using the MuSiCoS spectropolarimeter at Pic du Midi Observatory; ESPaDOnS at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii; on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO program 292.D-5028(A) at the Paranal Observatory, ESO Chile, with the NIR interferometers AMBER and PIONIER; on observations obtained at the La Silla Observatory, ESO Chile, with the echelle spectrograph CORALIE at the 1.2m Euler Swiss telescope; and on NEWSIPS data from the IUE satellite.LABEL:lastpagepubyear: 2002

1 Introduction

The bright ( mag), sharp-lined Cep pulsator CMa (HD 46328, B0.5 IV) was first reported to be magnetic by Hubrig et al. (2006), based upon FORS observations at the Very Large Telescope (VLT). Silvester et al. (2009) confirmed the detection using ESPaDOnS at the Canada-France-Hawaii Telescope (CFHT).

Despite intensive observation campaigns with different spectropolarimeters, there are conflicting claims regarding the star’s rotational period: Hubrig et al. (2011) derived a rotational period of 2.18 d based on FORS1/2 and SOFIN data, while Fourtune-Ravard et al. (2011) found a longer period of 4.27 d based on ESPaDOnS measurements. Both Hubrig et al. and Fourtune-Ravard et al. agreed that the star was likely being viewed with a rotational pole close to the line of sight. However, Fourtune-Ravard et al. noted that given the small variation in  they could not rule out intrinsically slow rotation.

Frost (1907) was the first to note CMa’s variable radial velocity. McNamara (1955) found a pulsation period of 0.2 d, identifying the star as a Cephei variable. While many Cep stars are multi-periodic, CMa appears to be a monoperiodic pulsator (Saesen et al., 2006) with an essentially constant period (Jerzykiewicz, 1999). In their analysis of CORALIE high-resolution spectroscopy, Saesen et al. (2006) found a pulsation period of 0.2095764(4) d (where the number in brackets refers to the uncertainty in the final digit). Saesen et al. also reported that CMa is one of the few Cep stars that achieves supersonic pulsation velocities. Several spectroscopic mode identification methods revealed that the oscillation frequency most likely corresponds to either a radial mode, or a dipolar mode, the latter viewed at small inclination. An mode was ruled out via modelling of the velocity moments. Photometric mode identification indicates that only the radial mode agrees with the frequency (Heynderickx et al., 1994).

Like many magnetic early-type stars, CMa displays the emission signatures of a magnetized stellar wind in optical, ultraviolet, and X-ray spectra. The star’s X-ray emission spectrum is very hard, and the brightest amongst all magnetic Cep stars, as demonstrated with Einstein (Cassinelli et al., 1994), ROSAT (Cassinelli et al., 1994), and XMM-Newton (Oskinova et al., 2011). The star’s X-ray emission has recently been shown to be modulated with the pulsation period (Oskinova et al., 2014), a so-far unique discovery amongst massive magnetic pulsators (Favata et al., 2009; Oskinova et al., 2015)111Short-term X-ray variability has also been reported for the magnetic Cep star Cen (Cohen et al., 1997; Alecian et al., 2011), and for the non-magnetic (Jason Grunhut, priv. comm.) Cep star Cru (Cohen et al., 2008). However, while subsequent analyses of the X-ray light curves of these stars have confirmed variability, they have not confirmed those variations to be coherent with the primary pulsation frequencies (Raassen et al., 2005; Oskinova et al., 2015), thus CMa is the only Cep star for which X-ray variation is unambiguously coherent with the pulsation period.. Various wind-sensitive ultraviolet lines show strong emission (Schnerr et al., 2008), which is typical for magnetic stars. CMa has also been reported to display H emission (Fourtune-Ravard et al., 2011), although the properties of this emission have not yet been investigated in detail.

Our goals in this paper are, first, to determine the rotational period based upon an expanded ESPaDOnS dataset, and second, to examine the star’s H emission properties.

The observations are presented in § 2. In § 3 we refine the pulsation period using radial velocity measurements. In § 4 we analyze the line broadening, determine the stellar parameters, and investigate the relationship between the star’s pulsations and its effective temperature. In § 5 we examine the possibility that the H emission may be a consequence of an undetected binary companion, constraining the brightness of such a companion using both interferometry and radial velocity measurements. The magnetic measurements and magnetic period analysis are presented in § 6. We examine the long- and short-term variability of the H emission in § 7, and investigate the influence of pulsation on wind-sensitive UV resonance lines. Magnetic and magnetospheric parameters are determined in § 8, § 9 presents a discussion of the paper’s results, and the conclusions are summarized in § 10.

2 Observations

2.1 ESPaDOnS spectropolarimetry

Under the auspices of the Magnetism in Massive Stars (MiMeS) CFHT Large Program (Wade et al., 2016), 29 ESPaDOnS Stokes spectra were acquired between 2008/01 and 2013/02. One additional observation was already published by Silvester et al. (2009). A further 4 ESPaDOnS observations were acquired by a PI program in 2014, and another 22 by a separate PI program in 2017222Program codes 14AC010 and 17AC16, PI M. Shultz.. ESPaDOnS is a fibre-fed echelle spectropolarimeter, with a spectral resolution , and a spectral range from 370 to 1050 nm over 40 spectral orders. Each observation consists of 4 polarimetric sub-exposures, between which the orientation of the instrument’s Fresnel rhombs are changed, yielding 4 intensity (Stokes ) spectra, 1 circularly polarized (Stokes ) spectrum, and 2 null polarization () spectra, the latter obtained in such a way as to cancel out the intrinsic polarization of the source. The majority of the data were acquired using a sub-exposure time of 60 s, with the exception of the first observation (75 s), and the data acquired in 2017, for which a 72 s sub-exposure time was used to compensate for degradation of the coating of CFHT’s mirror. The data were reduced using CFHT’s Upena reduction pipeline, which incorporates Libre-ESPRIT, a descendent of the ESPRIT code described by Donati et al. (1997). Wade et al. (2016) describe the reduction and analysis of MiMeS ESPaDOnS data in detail.

The log of ESPaDOnS observations is provided in an online Appendix in Table B4. The quality of the data is excellent, with a median peak signal-to-noise ratio (S/N) per spectral pixel of 828 in the combined Stokes spectrum. The 2 observations acquired on 2017/02/11 had a peak S/N below 500, and were discarded from the magnetic analysis. The log of sub-exposures, which were used for spectroscopic analysis, is provided in an online Appendix in Table C6. Note that while there are 56 full polarization sequences, due to one incomplete polarization sequence there are 227 rather than 224 sub-exposures listed in Table C6.

2.2 MuSiCoS spectropolarimetry

Three Stokes spectra were obtained in 2000/02 with the MuSiCoS spectropolarimeter on the Bernard Lyot Telescope (TBL) at the Pic du Midi Observatory. This instrument, one of several similar fibre-fed échelle multi-site continuous spectroscopy (hence MuSiCoS) instruments (Baudrand & Bohm, 1992) constructed at various observatories was uniquely coupled to a polarimeter (Baudrand & Bohm, 1992; Donati et al., 1999). It had a spectral resolution of 35,000 and covered the wavelength range 450–660 nm, across 40 spectral orders. As with ESPaDOnS, each polarimetric sequence consisted of 4 polarized subexposures, from which Stokes and spectra, as well as a diagnostic null spectrum, were extracted. The log of MuSiCoS data is provided in an online Appendix in Table B4. One of the MuSiCoS observations has a low S/N (below 100), and was discarded from the analysis. The data were reduced using ESPRIT (Donati et al., 1997).

Normal operation of the instrument was verified by observation of magnetic standard stars in the context of other observing programs (Shorlin et al., 2002; Kochukhov et al., 2004; Wade et al., 2006). In addition to this, we utilize the MuSiCoS spectra of 36 Lyn presented by Wade et al. (2006), one of which was obtained during the same observing run as those of CMa, in order to compare them to observations of the same star acquired recently with Narval, a clone of ESPaDOnS which replaced MuSiCoS at TBL, and which achieves essentially identical results (Wade et al., 2016). This comparison is provided in Appendix A.

2.3 CORALIE optical spectroscopy

A large dataset (401 spectra) was obtained between 2000/02 and 2004/10 with the CORALIE fibre-fed echelle spectrograph installed at the Nasmyth focus of the Swiss 1.2 m Leonard Euler telescope at the European Southern Observatory’s (ESO) La Silla facility (Queloz et al., 2000, 2001). The spectrograph has a spectral resolving power of 100,000, and covers the wavelength range 387–680 nm across 68 spectral orders. The data were reduced with TACOS (Baranne et al., 1996). The first analysis of these data was presented by Saesen et al. (2006).

The log of CORALIE observations is provided in an online Appendix in Table C10. The median peak S/N is 205. One observation, acquired on 10/12/2003, was discarded as it had a S/N of 16, too low for useful measurements.

2.4 IUE ultraviolet spectroscopy

CMa was observed numerous times with the International Ultraviolet Explorer (IUE). The IUE could operate in two modes, high-dispersion () or low-dispersion (), with two cameras, the Short Wavelength (SW) from 115 to 200 nm and the Long Wavelength (LW) from 185 to 330 nm. We retrieved the data from the MAST archive333Available at The data were reduced with the New Spectral Image Processing System (NEWSIPS). There are two simultaneous low-resolution spectra obtained with the Short Wavelength Prime (SWP) and Long Wavelength Redundant (LWR) cameras, along with 13 high-resolution spectra obtained with the SWP camera. The high-resolution data are of uniform quality as measured by the S/N, which is approximately 20 in all cases. Twelve of the spectra were obtained in close temporal proximity, covering a single pulsation cycle in approximately even phase intervals. The first observation was obtained 154 days previously. We used the absolute calibrated flux, discarded all pixels flagged as anomalous, and merged the various spectral orders.

2.5 VLTI near infrared interferometry

While the angular radii of CMa and its circumstellar material are certainly too small to be resolved interferometrically, the star is bright enough that the data can be used to search for a high-contrast binary companion.

We have acquired four Very Large Telescope Interferometer (VLTI) observations: low-resolution AMBER H and K photometry, a high-resolution AMBER NIR spectro-interferogram, and one low-resolution H band PIONIER observation. AMBER (Astronomical Multi-Beam Recombiner) offers three baselines and can operate in either low-resolution photometric mode or high-resolution () spectro-interferometric mode (Petrov et al., 2007). PIONIER (Precision Integrated-Optics Near-infrared Imaging ExpeRiment) combines light from 4 telescopes, offering visibilities across 6 baselines together with 4 closure phase measuremnts (Le Bouquin et al., 2011). All data were obtained using the 1.8 m Auxilliary Telescopes (ATs), which have longer available baselines, and hence better angular resolution than the 8 m Unit Telescopes (UTs). AMBER observations were obtained with baselines ranging from 80 to 129 m. PIONIER was configured in the large quadruplet, with a longest baseline of 140 m. The nearby star CMa (A0 III, ) was used as a standard star for calibration.

The observing log is given in Table D17. As the execution time for a full measurement (1 hr) is a significant fraction of the pulsation period, pulsation phases are not given.

Owing to the relatively faint magnitude, the wavelength edges of the AMBER H and K profiles were particularly noisy. These data points were edited out by hand before commencing the analysis. No such procedure was necessary for the PIONIER observation.

3 Pulsation Period

Figure 1: Frequency spectra for radial velocities (RVs) measured from CORALIE data (left), ESPaDOnS data (middle), and the combined (right) dataset, for the RVs (top row), and after pre-whitening with significant frequencies (subsequent rows). Dotted light blue lines indicate and its first two harmonics. In the bottom panels, the window function is overplotted in light blue. In the bottom right panel, the white frequency spectrum shows the results of prewhitening the combined data with a pulsation period increasing at a linear rate of . By contrast, using a constant period to prewhiten the combined dataset (black frequency spectrum) does not fully remove the peak at .
Figure 2: Top panel: Composite radial velocities (using N II 404.4 and 422.8 nm; N III 463.4 nm; O II 407.2, 407.9, 418.5, and 445.2 nm; Ne II 439.2 nm; Al III 451.3 nm; Si iii 455.3 nm; and Si IV 411.6 nm) as a function of pulsation phase, using the nonlinear ephemeris defined in Eqn. 1. Open black circles are CORALIE measurements, filled blue circles ESPaDOnS measurements. The solid red line indicates a -order sinusoidal fit. Bottom panel: residual radial velocities. Dashed red lines indicate 1 standard deviation, about 0.48 km s, and the dotted red lines show the median RV uncertainty.

3.1 Radial Velocities

We measured radial velocities (RVs) from the ESPaDOnS and CORALIE spectra using the centre-of-gravity method (Aerts et al., 2010). In addition to using the same line (Si iii 455.3 nm) used by Saesen et al. (2006), we have used: N ii 404.4 and 422.8 nm; N iii 463.4 nm; O ii 407.2, 407.9, 418.5, and 445.2 nm; Ne ii 439.2 nm; Al iii 451.3 nm; and Si iv 411.6 nm. For ESPaDOnS data we used individual sub-exposures rather than the Stokes profiles corresponding to the full polarization sequences: each polarization sequence encompasses about 2.3% of a pulsation cycle (460 s sub-exposures + 360 s chip readouts), as compared to about 0.33% for the sub-exposures (although the 2017 data had slightly longer subexposure times, this was compensated for by shorter readout times). The CORALIE data are not as uniform as the ESPaDOnS data in this regard: the median exposure time corresponds to about 2.8% of a pulsation cycle, and some are up to about 10% (all CORALIE measurements were retained, however HJDs were calculated at the middle rather than the beginning of each exposure).

Since centre-of-gravity measurements can be biased by inclusion of too much continuum, it is important to choose integration limits with care. Thus RVs were measured iteratively. A first set of measurements was conducted using a wide integration range ( km s about the systemic velocity), chosen to encompass the full range of variation. These initial RVs were then used as the central velocities for a second set of measurements, with an integration range of km s. These were in turn used a final time as the central velocities to refine the RVs with a third iteration using the same integration range. The error bar weighted mean RV across all lines was then taken as the final RV measurement for each observation, with the standard deviation across all lines as the uncertainty in the final RV. ESPaDOnS and CORALIE RV measurements are tabulated in an online Appendix in Tables C6 and C10, respectively. We determined the systemic velocity from the mean RV across all observations to be  km s.

Figure 3: Line profile fitting to determine  and , using Si iii 455.3 nm (top) and N II 404.4 nm (bottom). Left: landscapes (high values are dark, low values are light, where the reduced goes from approximately 1 to ), corresponding to the weighted mean across all observations. The best-fit solutions for individual observations are indicated by open red circles. Right: line profile fits at pulsation phases close to the RV minima and maxima () and km s(). Observations are shown by black circles. The solid red line shows the best overall fit for each observation. The dashed blue line shows the best fit model with  km s. The dash-dotted green line indicates the best fit model with km s. While the overall best-fit has a non-zero  of a few km s, excluding results in a noticeably worse fit, but excluding  does not substantially worsen the fit. The weaker N ii line yields a lower , but does not change the result that , or that equivalently good fits can be achieved with or without .

3.2 Frequency Analysis

We analyed the RVs using period04 (Lenz & Breger, 2005). Frequency spectra are shown in Fig. 1, where the top panels show the frequency spectra for the original dataset, and the panels below show frequency spectra after pre-whitening with the most significant frequencies from previous frequency spectra. A S/N threshold of 4 was adopted as the minimum S/N for significance (Breger et al., 1993; Kuschnig et al., 1997). The uncertainty in each frequency was determined using the formula from Bloomfield (1976), , where is the mean uncertainty in the RV measurements, is the number of measurements, is the amplitude of the RV curve, and is the timespan of observations.

Period analysis of the CORALIE dataset (left panels in Fig. 1) yielded the same results as those reported by Saesen et al. (2006), with significant frequencies at with an amplitude of 16.6 km s, and at the harmonics 2 and 3 with amplitudes of 0.6 km s and 0.3 km s, respectively. After pre-whitening with these frequencies, no significant frequencies remain (bottom left panel of Fig. 1), and all peaks are at the 1 aliases of the spectral window. The same analysis of the ESPaDOnS data (middle panels in Fig. 1) finds maximum power at , and at 2.

The combined dataset (right panels in Fig. 1) yields the strongest signal at , along with significant peaks at 2 and 3. In this case pre-whitening does not completely remove the peak corresponding to . This could indicate that the pulsation frequency is not the same between the datasets. The difference in frequencies between the CORALIE and ESPaDOnS datasets, , is about 30 times larger than the formal uncertainties in either frequency. This difference corresponds to an increase in the pulsation period of 0.1 s, approximately compatible with the increase of expected from the constant period change of reported by Jerzykiewicz (1999).

To explore this hypothesis, phases were calculated assuming a constant rate of period change , with taken as the frequency inferred from the CORALIE data acquired in 2000, and allowed to vary within the uncertainty in this frequency. The phases were calculated as


where is the initial period, and is the number of pulsation cycles elapsed between the observation and the reference epoch , defined as the time of the first RV maximum one cycle before the first observation in the dataset. The goodness-of-fit statistic was calculated for each combination of and using a 3-order least-squares sinusoidal fit in order to account for the pulsation frequency and its first two harmonics. The minimum solution was found for and , where the uncertainties were determined from the range over which does not change appreciably. The RVs are shown phased with Eqn. 1 using these parameters in Fig. 2. The bottom panel of Fig. 2 shows the residual RVs after subtraction of the 3-order sinusoidal fit. The standard deviation of the residuals is 0.48 km s, an improvement over the standard deviation of 0.94 km s obtained using the constant period determined from the full dataset. The bottom right panel of Fig. 1 shows the frequency spectrum obtained for the residual RVs in Fig. 2: in contrast to the results obtained via pre-whitening using a constant pulsation frequency, all power at and its harmonics is removed, with no significant frequencies remainining.

3.3 Comparison to previous results

Our analysis of the CORALIE dataset recovers the same pulsation frequency as that found by the analysis performed by Saesen et al. (2006) of the same data. However, we find to be almost 3 times larger than the rate found by Jerzykiewicz (1999). Whether this reflects an acceleration in the rate of period change will need to be explored in the future when larger datasets with a longer temporal baseline are available. Neilson & Ignace (2015) noted that CMa is one of the few Cep stars for which is low enough to be consistent with stellar evolutionary models. The higher suggested by our results brings CMa closer to the range observed for other Cep stars, which is to say, slightly above the predicted by evolutionary models for a star of CMa’s luminosity.

4 Stellar Parameters

4.1 Line Broadening

The high S/N, high spectral resolution, and wide wavelength coverage of the ESPaDOnS data, combined with the sharp spectral lines of this star, present numerous opportunities for constraining the line broadening mechanisms. As with measurement of RVs, in order to minimize the impact of RV variation, Stokes spectra from individual sub-exposures were used rather than the spectra computed from the combined exposures. We selected two lines for analysis: the Si iii 455.3 nm line, which is sensitive to the pulsational properties of Cep stars (Aerts & De Cat, 2003), and for comparison N ii 404.4 nm, a weaker line with lower sensitivity to pulsation.

We applied a model incorporating the projected rotation velocity , radial/tangential macroturbulence , and assumed radial pulsations, performing a goodness-of-fit (GOF) test on a grid spanning 0–20 km s in  and 0–30 km s in , in 1 km s increments. The disk integration model is essentially as described by Petit & Wade (2012), with the exception of radial pulsations. Pulsations were modeled as a uniform velocity component normal to the photosphere, with the local line profiles in each surface area element shifted by the projected line-of-sight component of the pulsation velocity. A limb darkening coefficient of was used, obtained from the tables calculated by Díaz-Cordovés et al. (1995) for a star with kK and . Local profiles were broadened with a thermal velocity component of 4 km s for Si iii, and 5.6 km s for N ii, and the disk integrated profile was convolved according to the resolving power of ESPaDOnS. Line strength was set by normalizing the EWs of the synthetic line profiles to match the EWs of the observed line profiles. In the course of this analysis we found a Baade-Wesselink projection factor of measured-to-disk-centre radial velocity of , which is consistent with determinations for other Cep variables (e.g., Nardetto et al. 2013).

Fig. 3 shows the resulting best-fit models. Taking the mean and standard deviation across all observations, both Si iii 455.3 nm and N ii 404.4 nm yield  km s. The two lines give different values for , however: 191 km s and 82 km s, respectively. The much lower value of determined from N ii 404.4 nm may indicate that the extended wings of Si iii 455.3 nm, which require a higher value of to fit, are at least partly a consequence of pulsations.

As demonstrated in Fig. 3, there is essentially no difference in quality of fit between models with nonzero  and , and zero ; conversely, a much worse fit is obtained by setting km s. For Si iii 455.3 nm, the of the model without is 13 times higher than the of the model without , while the of the model without  is only 1.4 times higher than the model with both turbulent and rotational broadening. The difference is not as great for N ii 404.4 nm, although the of the model with rotational broadening only is still higher than either the model with turbulent broadening only, or both rotational and turbulent broadening. Sundqvist et al. (2013) used magnetic O stars with rotational periods known to be extremely long (i.e. with   km s) and found that both the goodness-of-fit test and the Fourier transform methods severely over-estimate  when , as is the case for both of the lines we have examined. We conclude that the star’s line profiles are consistent with   km s.

Figure 4: Surface gravity determination using H. Top: model fits for , and 28 kK (cyan, blue, and pink, respectively) to the observed profile (black). The observation nearest (where the RV km s) was used. Bottom: residuals. The horizontal lines indicate the mean flux uncertainty. As explored in § 7.1, there is emission near the line core; this is more prominent in H. This region was not included in the fit.
Figure 5: Low-resolution IUE spectroscopy (solid lines) with the SWP (black) and LWR (blue) cameras, and photometry (filled symbols) compared to a synthetic photospheric spectrum (red dashed line). No dereddening correction has been applied to the data. Inset: Comparison between synthetic spectra (dashed lines) and observed spectra (solid lines) deredded using three values of : 0 (green), 0.02 (cyan), and 0.04 (pink). Both the observed spectrum and the best-fit synthetic spectra have been smoothed with a boxcar algorithm. The dot-dashed (purple) line is the reddening curve in arbitrary units. Even mild dereddening produces a bulge in the vicinity of the 220 nm silicate absorption feature.

4.2 Surface Gravity

The surface gravity was determined by fitting tlusty BSTAR2006 synthetic spectra (Lanz & Hubeny, 2007) to the H and H lines of the ESPaDOnS spectrum acquired at pulsation phases at which the RV was closest to the systemic velocity of 22.5 km s (i.e. at pulsation phases 0.25 or 0.75). As both of these lines are close to the edges of their respective orders, in order to avoid warping the line the orders were merged from un-normalized spectra, then normalized using a linear fit to nearby continuum regions. Five surface gravities were tested between and , and a low-order polynomial was fit to the resulting in order to identify the lowest solution. Uncertainties were determined by fitting models with  of 26, 27, and 28 kK, spanning the approximate range in the uncertainty in  (see below). Fig. 4 shows the best-fit models, , compared to the observed H line. Fitting to H yielded identical results.

4.3 Effective Temperature

The effective temperature  was determined using photometry, spectrophotometry, and spectroscopy. Using the Strömgren photometric indices obtained by Hauck & Mermilliod (1998), and the idl program uvbybeta.pro444Available at which implements the calibrations determined by Moon (1985), yields kK. Using the Johnson photometry collected by Ducati (2002) and the colour-temperature calibration presented by Worthey & Lee (2011) yields kK, where the uncertainty was determined from the range of  across the , and colours.

The  was measured using spectrophotometric data by fitting synthetic BSTAR2006 spectra to the IUE low-dispersion spectra, Johnson UBVRI photometry, and Strömgren photometry. The Johnson photometry was converted into absolute flux units using the calibration provided by Bessell et al. (1998), and the Strömgren photometry converted using the calibration determined by Gray (1998). The synthetic spectra were scaled by and , where the distance was fixed by the Hipparcos parallax and the stellar radius was left as a free parameter. The resulting fit is shown in Fig. 5. The best-fit model yields kK and . The uncertainty comes from the spread in  over the range of , and for two values of , 0 and 0.02. Fluxes were dereddened using the idl routine fm_unred555Available at, which utilizes the Fitzpatrick & Massa (1990) extinction curve parameterization. As demonstrated by the inset in Fig. 5, there is a clear absence of interstellar silicate absorption near 220 nm. The de-reddened spectra in the inset in Fig. 5 assume the strength of the absorption bump is , the minimum Galactic value (Clayton et al., 2003). Even with this low value of , using results in an increase of the flux near the silicate absorption bump which is not observed. Using the average Galactic value of instead would require that be even lower.

4.4 Comparison to previous results

Previous measurements of  have in general yielded somewhat higher, non-zero results than those obtained here ( km s, Saesen et al. 2006; 92 km s, Lefever et al. 2010; 145 km s, Aerts et al. 2014). Saesen et al. did not include in their profile fit, but did include pulsation and thermal broadening; Lefever et al. did not consider pulsations; and Aerts et al. included , but neither pulsation velocity nor thermal broadening. Our inclusion of thermal, macroturbulent, pulsational, and rotational velocity fields likely explains why our best-fit value of ,  km s, is somewhat lower than the values found previously, since all four sources of line broadening are of a similar magnitude in this star.

Stellar parameters for CMa have been determined numerous times using a variety of different spectral modelling methodologies, e.g., DETAIL/SURFACE (Morel et al., 2008), FASTWIND (Lefever et al., 2010), and PoWR (Oskinova et al., 2011). Niemczura & Daszyńska-Daszkiewicz (2005) employed an algorithmic method to simultaneously determine reddening, , metallicities, and surface gravities for Cep stars observed with IUE, and for CMa found   kK, , and , i.e. their analysis favoured a cooler, slightly less evolved star, but with the same very low level of reddening. Niemczura & Daszyńska-Daszkiewicz derived from photometric calibrations, rather than line profile fitting, which likely explains the different values of . Morel et al. (2008) found   kK and , Lefever et al. (2010) derived   kK and , and Oskinova et al. (2011) found   kK: all are in agreement with the results found here. The slightly higher  found in these works is likely a consequence of the higher value of reddening which, as is shown in § 4.3 and Fig. 5, is not consistent with the absence of silicate absorption. While reddening has an effect on the spectrophotometric method applied in § 4.3, we obtained essentially identical results using line strength ratios and ionization balances below in § 4.5, which should not be affected by reddening.

4.5 Effective Temperature Variation

Figure 6:  determined from EW ratios of various ion pairs as a function of pulsation phase. While the mean  differs by 2 kK between different ion pairs, a coherent and identical modulation with the pulsation period is seen in all cases. The semi-amplitude of the  variation, as determined from the least-squares sinusoidal fit to the data, is given at the bottom of each panel. The bottom right panel shows the weighted mean  across all ion pairs, with the dashed blue line indicating the  determined from SED fitting (Fig. 5).
Figure 7: Predicted variation in visual magnitude (red line) as determined from the variation in from RVs (Fig. 2) and in  from EW ratios (Fig. 6) as compared to the Hipparcos photometry (black circles). Pulsation phases were calculated using eqn. 1.

The photospheric temperature of a Cephei star is variable, a property which can be explored both photometrically and spectroscopically. The only public two-colour photometry for CMa are the Tycho observations. While pulsational modulation is visible in both passbands, the precision of the data is not sufficient to detect a coherent signal in the colours. This is not surprising, as colours are poorly suited to determining  for such hot stars. We therefore attempted to constrain the  change via the ionization balances of various spectral lines, under the assumption that all equivalent width (EW) changes are due to temperature variation.

We first measured EWs from the ESPaDOnS observations, and from appropriate synthetic spectra obtained from the tlusty BSTAR2006 library (Lanz & Hubeny, 2007). We used the ESPaDOnS Stokes spectra obtained from the combined sub-exposures, in order to maximize the S/N. The following spectral lines were used: He i 587.6 nm, He ii 468.6 nm, C ii 426.7 nm, C iii 406.8 nm, N ii 404.4 nm, N iii 463.4 nm, O i 777.4 nm, O ii 407.9 nm, Ne i 640.2 nm, Ne ii 439.2 nm, Si ii 413.1 nm, Si iii 455.3 nm, and Si iv 411.6 nm. Since the EW can depend on as well as , the model EW ratios were linearly interpolated to between the results for and . We then determined  by interpolating through the resulting grid to the EW ratios measured from the observed spectra. The results are illustrated in Fig. 6. While there is a considerable spread in values, from approximately 25 to 28 kK, the mean value of 26.8 kK agrees well with the spectrophotometric determination. A coherent variation with the pulsation period is seen for all ion strength ratios. Individual atomic species yield variations with semi-amplitudes up to K. Taking the weighted mean across the effective temperatures from all ion strength ratios at each phase yields a variation of K. If the zero-point of each  variation is first subtracted, the resulting mean  semi-amplitude is  K, slightly larger but overlapping within the uncertainty.

To check the validity of the temperature variation, we used it to model the Hipparcos light curve, as demonstrated in Fig. 7. With the uncertainty in the pulsation period and rate of period change determined in § 3, the accumulated uncertainty in pulsation phase for the Hipparcos photometry is between 0.009 and 0.012 cycles. We began by determining the mean radius from the mean  and the luminosity . was obtained from the Hipparcos parallax distance  pc, the apparent magnitude 4.06, and the bolometric correction  mag obtained by linear interpolation through the theoretical BSTAR2006 grid according to  and (Lanz & Hubeny, 2007). The absolute magnitude  mag, where  mag is the distance modulus and  mag is the extinction. The bolometric magnitude is then  mag, yielding where  mag. This yields , where kK. This radius agrees well with the value found via spectrophotometric modelling, , in which the radius was left as a free parameter (Fig. 5).

Integrating the radial velocity curve (Fig. 2) in order to obtain the absolute change in radius, and assuming radial pulsation, yields a relatively small change in radius of km, corresponding to between 1.0 and 1.5% of the stellar radius. The radius variation was then combined with the  variation from Fig. 6 to obtain and the at each phase. The apparent magnitude was then obtained by reversing the calculations in the previous paragraph. The solid red line in Fig. 7 shows the resulting model light curve compared to the Hipparcos photometry, where we made the assumption that and are approximately equivalent. The larger 500 K  semi-amplitude measured from individual pairs of ions is not consistent with the light curve, yielding a photometric variation larger than observed, with a semi-amplitude of 0.037 mag as compared to the observed semi-amplitude of 0.021 mag. The apparent phase offset of 0.05 cycles between the predicted and observed photometric extrema may suggest that may not in fact be constant; alternatively, there may simply be too few measurements to constrain the  variation well enough to obtain a close fit to the photometric data, as the latter is clearly not a perfect sinusoid.

Figure 8: CMa’s position on the HRD (top) and on the - diagram (bottom), relative to other magnetic stars with similar stellar parameters. The solid line indicates the Zero-Age Main Sequence (ZAMS), and the dot-dashed line the Terminal Age Main Sequence (TAMS). Dashed lines show evolutionary tracks from the (rotating, non-magnetic) models by Ekström et al. (2012), and dotted lines isochrones from the same models for (left) and 7.1 (right). The star’s fundamental parameters indicate and Myr. CMa is the most evolved magnetic B-type star above about 12 .

4.6 Mass and Age

Fig. 8 shows CMa’s position on the Hertzsprung-Russell diagram (HRD) (top) and the - diagram (bottom), where the mean  was used. Comparison to the evolutionary models calculated by Ekström et al. (2012) (which assume an initial rotational velocity of 40% of the critical velocity) indicates that the stellar mass , and that the absolute stellar age is  Myr. The Ekström et al. (2012) models do not include the effects of magnetic fields, however, grids of self-consistent evolutionary models including these effects in a realistic fashion are not yet available.

The positions of the star on the two diagrams are mutually consistent. The other known magnetic B-type stars with similar stellar parameters are also shown in Fig. 8. The stellar parameters of the majority of the other stars were obtained from the catalogue of magnetic hot stars published by Petit et al. (2013) and references therein; those of CMa and CMa were obtained from Fossati et al. (2015a). CMa is one of the most evolved stars in the ensemble, and is the most evolved star with a mass above about 12 . Its position on the - diagram indicates it may be a more evolved analogue of NU Ori, HD 63425, and HD 66665.

 (km s) 22.51 3
 (km s) 4.1
 (km s) 82 4.1
Sp. Type B0.5 IV Hubrig et al. (2006)
(mag) 4.3 Ducati (2002)
(mas) 2.360.2 van Leeuwen (2007)
(pc) 42435 van Leeuwen (2007)
(mag) -2.610.09 4.5
(mag) 4.5
(mag) -3.860.18 4.5
(mag) -6.470.27 4.5
4.490.11 4.5
 (kK) 271 4.3
3.780.07 4.2
7.90.6 4.3
14.20.4 4.6
Age (Myr) 11.10.7 4.6
0.770.04 4.6
1.80.1 4.6
Table 1: Observational and physical properties. The third column gives the reference, where numerical references indicate the section within this paper where the parameter is determined.

As an additional check on the stellar mass and radius, we utilized the pulsation period and the relation , where is the fundamental pulsation period, is the mean density, and is the pulsation constant. With , , and the theoretical value (Davey, 1973; Stothers, 1976), this yields d, very close to the true pulsation period  d. If the actual pulsation period is used to calculate , we obtain .

5 Binarity

5.1 Interferometry

Fourtune-Ravard et al. (2011) reported that CMa hosts weak H emission, which is atypical for a star of CMa’s stellar properties. There are two possibilities to explain this: first, that it originates in a stellar magnetosphere, and second, that it originates in the decretion disk of a heretofore undetected Be companion star. It seems reasonable to expect that CMa has a binary companion, as the binary fraction is 65% for B0 stars (Chini et al., 2012). Furthermore, early claims that the H emission of Cep, a similar magnetic early-B type pulsator, originated in its magnetosphere (Donati et al., 2001), proved unfounded following the detection of a Be companion star by Schnerr et al. (2006).

The peak H emission has a maximum strength of 28% of the continuum (see § 7.1). Be star emission lines can range up to of the continuum, although the emission is typically much less than this. We therefore expect that the putative companion must have a luminosity of of that of CMa, yielding or . Assuming the pair to be coeval, and therefore locating the star on the same isochrone as CMa, the companion’s mass should be near , compatible with a classical Be star.

CMa is listed in the Washington Double Star Catalogue as possessing a binary companion (Mason et al., 2001). However, the reported companion is both too dim to account for the H emission () and too far away, located approximately 28” from the primary, i.e. well outside the 1.6” ESPaDOnS aperture.

We acquired H and K low-resolution VLTI-AMBER data, together with VLTI-PIONIER data, in order to search for the presence of a binary star. The squared visibilities and closure phases of these data are shown as functions of spatial frequency in Fig. 9. Both the AMBER K band and PIONIER measurements are entirely flat, compatible with a single unresolved source. There is furthermore no signal in the PIONIER measurements, which are more precise than those available from AMBER.

We analyzed the data using the standard litpro package666litpro software is available at (Tallon-Bosc et al., 2008), fitting a two-point model, with one point fixed in the centre of the map and the (x,y) coordinates of the second free to move. The overall upper limit on the flux of a secondary component is 1.7% of the primary star’s flux. In order to constrain the maximum flux of a binary companion at different distances from the primary, we repeated the two-point model fit in successively wider boxes (as the current version of litpro does not support polar coordinates, a Cartesian approximation to annuli was used). As these flux ratios are in the and bands, but our estimated minimum flux ratio is in the band, we used synthetic tlusty SEDs (Lanz & Hubeny, 2007) to convert the and band flux ratios to band flux ratios. In this step we assumed that the secondary’s  is near 20 kK, appropriate to a 5-6 star near the main sequence. A companion of the required brightness is ruled out beyond AU.

Close binaries containing magnetic, hot stars are extremely rare, with % of close binary systems containing a magnetic companion earlier than F0 (Alecian et al., 2015). This does not mean that a close companion can be ruled out a priori. Such a companion may be detectable via radial velocities. As found in § 3, the RV curve of CMa is extremely stable, with a standard deviation of the residual RVs of 0.5 km s. To determine if a Be companion should have been detected, we computed radial velocities across a grid of models with secondary masses , semi-major axes , eccentricities , and inclinations of the orbital axis from the line of sight . We then phased the JDs of the observations with the orbital periods , using a single zero-point if was less than the time-span of the observations and multiple, evenly-spaced zero-points if was greater than this span. We then calculated the expected RV of the primary at each orbital phase, and compared the standard deviation of these RVs with the observed standard deviation (RV amplitudes were not used as, for orbits with periods longer than the timespan of observations, the full RV variation would not have been sampled). A given orbit was considered detectable if . Numerical experiments with synthetic RV curves including gaussian noise with a standard deviation of 0.5 km s indicate that this criterion is likely conservative: with 624 RV measurements, input periods can generally be recovered even when the semi-amplitude of the RV curve is similar to the noise level.

Figure 9: Low-resolution VLTI-AMBER H and K data and VLTI-PIONIER data, showing squared visibilities (top) and closure phases (bottom).

Each orbit was assigned a probability assuming a random distribution of over steradians (i.e., ), and a flat probability distribution for , , and . From this we obtained the 1, 2, and 3 upper limits on a companion’s mass as a function of . This mass upper limit was then transformed into a magnitude lower limit by interpolating along the 10 Myr isochrone in Fig. 8. Within the 40 AU inner boundary of the interferometric constraints, a binary companion of sufficient brightness to host the H emission can be ruled out entirely at 1 confidence, and almost entirely at 2.

We have proceeded under the very conservative assumption of a Be star with H emission of 10 its continuum level, however the majority of Be stars have emission of at most a few times the continuum, and many have much less than this. Thus, the upper limits on companion mass and brightness determined above rule out all but the most exceptional of classical Be stars as a possible origin for the H emission. We conclude that there is unlikely to be an undetected binary classical Be companion star that is sufficiently bright to be the source of the star’s H emission.

5.2 Spectro-interferometry

Figure 10: High-resolution VLTI/AMBER data showing (clockwise from top left), flux, closure phase , differential phases , and squared visibilities . Black shows the raw data. Red lines show data binned to 20% of the mean error bar in the raw data. Vertical dashed lines indicate the central wavelength of the Br line, solid vertical lines denote the extent of the line.

The formation of the line emission around the Cep star itself is supported by the absence of any spectrointerferometric signal across the Br line. Such observations were taken with AMBER, and no signature in phase was detected on the level of 2 (Fig. 10). This means that the photometric position as a function of wavelength remained stable on the level of 60 as (Lachaume, 2003; Štefl et al., 2009). An emission component of about 30% of the strength of the continuum (Br emission typically being comparable in strength to that of H in Be stars) must therefore have its photocenter within 60 as from the photocenter of the nearby continuum, as otherwise it would have produced a detectable offset of the phase signal. This not only excludes a general offset from the central star, i.e. formation around a companion, but also an extended orbiting structure, in which the blue emission would be formed at an offset opposite to the red emission.

6 Magnetic Field

6.1 Least Squares Deconvolution

Figure 11: Top: ESPaDOnS LSD profiles: Stokes (bottom), (middle), Stokes (top). Bottom: ESPaDOnS (black triangles) and MuSiCoS (red diamonds) LSD profiles, extracted with identical line masks, from observations acquired at similar pulsation phases (0.44). The broader and shallower MuSiCoS profile is a consequence of the much longer exposure times of the MuSiCoS observations, the instrument’s lower spectral resolution, and the intrinsic variability of the star. While the MuSiCoS LSD profile is obviously noisier, the Stokes signature is clearly of the opposite polarity of the ESPaDOnS observation.

While CMa’s sharp spectral lines and the high S/N of the ESPaDOnS observations mean that Zeeman signatures are visible in numerous individual spectral lines, in order to maximize the S/N we employed the usual Least Squares Deconvolution (LSD; Donati et al. 1997) multiline analysis procedure. In particular we utilized the ‘improved LSD’ (iLSD) package described by Kochukhov et al. (2010). iLSD enables LSD profiles to be extracted with two complementary line masks, improving the reproduction of Stokes and obtained from masks limited to a select number of lines.

We used a line mask obtained from the Vienna Atomic Line Database (VALD3; Piskunov et al. 1995; Ryabchikova et al. 1997; Kupka et al. 1999, 2000) for a solar metallicity star with   kK. While magnetic early-type stars are often chemically peculiar, CMa is of essentially solar composition (Niemczura & Daszyńska-Daszkiewicz, 2005), albeit with a mild N enhancement (Morel et al., 2008), thus a solar metallicity mask is appropriate. The mask was cleaned and tweaked as per the usual procedure, described in detail by Grunhut (2012), such that only metallic lines unblended with H, He, interstellar, or telluric lines remained, with the strengths of the remaining lines adjusted to match as closely as possible the observed line depths. Of the initial 578 lines in the mask, 338 remained following the cleaning/tweaking procedure. The resulting LSD profiles are shown in Fig. 11 (top panel). Note that, due to intrinsic line profile variability and the longer subexposure times used for the 2017 data, the line profiles of the most recent data are slightly broader than the data acquired previously.

In order to directly compare MuSiCoS and ESPaDOnS results, a second line mask was employed with all lines outside of the MuSiCoS spectral range removed, leaving 139 lines. The bottom panel of Fig. 11 shows a comparison between the highest S/N MuSiCoS LSD profile and the ESPaDOnS LSD profile with the closest pulsation phase (0.4377 vs. 0.4425). The Stokes profiles from the 2 instruments agree well, considering both the lower spectral resolution and the much longer exposure times of MuSiCoS (corresponding to 15% of a pulsation period, as compared to 2.3% of a pulsation period for ESPaDOnS). The key point of interest is in Stokes , which is clearly negative in the MuSiCoS LSD profile, but positive in all ESPaDOnS LSD profiles. The second MuSiCoS observation also yields a negative Stokes profile. The reliability of these MuSiCoS observations is evaluated in detail in Appendix A.

False Alarm Probabilities (FAPs) were calculated inside and outside of the line profile, and classified as Definite Detections (DDs), Marginal Detections (MDs), or Non-Detections (NDs) according the methodology and criteria described by Donati et al. (1992, 1997). Detection flags for Stokes and diagnostic null are given in Table B4. All MuSiCoS observations are formal non-detections. All ESPaDOnS observations are DDs (with the exception of the two discarded measurements from 2017/02/11, both of which yielded NDs due to their low S/N). However, in numerous ESPaDOnS observations, also yields a DD. This phenomenon is considered in greater detail in § 9.1.

6.2 Longitudinal Magnetic Field

The longitudinal magnetic field  was measured from the LSD profiles by taking the first-order moment of the Stokes profile normalized by the equivalent width of Stokes (Mathys, 1989). The same measurement using yields the null measurement , which should be consistent with 0 G. In order to ensure a homogeneous analysis, the LSD profiles were first shifted by their measured RVs to zero velocity, and an identical integration range of km s around line centre employed.  and  measurements are reported in Table B4. As expected from the LSD profiles, for both MuSiCoS measurements, while for all ESPaDOnS measurements. The ESPaDOnS data are of much higher quality, with a median  G, as compared to 57 G for the MuSiCoS data. In contrast to the FAPs, in which many ESPaDOnS profiles yield definite detections, is typically very low, with a maximum of 2.4 and a median of 0.6, i.e. statistically identical to zero.

Figure 12: Periodograms for ESPaDOnS  (top) and  (bottom). Many of the peaks in the  periodogram also appear in the  periodogram, with the exception of the maximum amplitude peak at 5100 d (vertical dashed line). The period determined from FORS1/2 data by Hubrig et al. (2011) is indicated by the black arrow; the period determined by Fourtune-Ravard et al. (2011) from an earlier, smaller ESPaDOnS dataset is indicated by the red arrow. There is very little amplitude at either period.
Figure 13: Top:  vs. time. The solid curve shows the least-squares sinusoidal fit to the annual mean measurements using a 30-year period, and the dashed curves indicate the uncertainty in the amplitude and mean value of the fit. The horizontal solid line shows  . Vertical lines indicate the times at which   (dotted), (dot-dashed), and (dot-dot-dashed). Bottom: H EWs. Curves show the least-squares -order sinusoidal fit to the annual means, which is performed independently of the fit to . Note that maximum emission corresponds to , and minimum emission corresponds to  . Also note that the fit predicts a second, weaker emission peak corresponding to .

Because  is expected to be modulated by stellar rotation, it can be used to determine the rotation period. The periodograms for  and  are shown in Fig. 12, where the latter shows the variation arising from noise. There are numerous peaks at periods of a few tens to a few hundreds of days which appear in both the  and the  periodograms. Phasing  with the periods corresponding to these peaks does not produce a coherent variation (e.g., phasing the data with the highest peak in this range, at 177 d, and fitting a first-order sinusoid, yields a reduced of 1162). The strongest peak in the  periodogram is at  d. This peak does not appear in the  periodogram. This is similar to the timespan of the ESPaDOnS dataset, and so the formal uncertainty is certainly under-estimated, with the 5100 d period representing a lower limit. However, the S/N of this peak is 26, above the threshold for statistical significance, indicating that the long-term variation is probably real.

Fig. 13 shows the  measurements, both individual (filled symbols) and in annual bins (open circles), as a function of time. There is an obvious long-term modulation, with  steadily declining from a peak of 330 G in 2010 (HJD 2455200) to value of 80 G in 2017 (HJD 2457800). The annual mean  measurements are provided in Table 2.

Month/Year HJD - H EW
2450000 (G) (nm)
02/2000 2451585 -12740
10/2000 2451822 0.14920.0002
11/2001 2452234 0.14620.0003
11/2002 2452580 0.14940.0007
12/2003 2452989 0.14400.0003
05/2004 2453156 0.14130.0002
01/2008 2454489 3146 0.1300.001
01/2010 2455203 3282 0.13140.0004
12/2010 2455556 3186 0.1360.001
02/2012 2455965 3053 0.13270.0007
01/2013 2456294 2813 0.13730.0006
01/2014 2456674 2535 0.13920.0008
02/2017 2457797 782 0.15070.0003
Table 2: Annual weighted mean  measurements and H EWs. Months/years and HJDs correspond to the mean of the cluster of observations within each bin.

The MuSiCoS observations, both of which are of negative polarity, indicate (in combination with the ESPaDOnS data) that the rotational period must be longer than 5100 d. The time difference between the MuSiCoS measurements and the maximum  ESPaDOnS observations is 3600 d. If these two epochs sample the  curve at its positive and negative extrema, then the rotational period must be at least 7200 d (20 years), or a half-integer multiple if there is more than one cycle between the observed extrema. The curvature of the ESPaDOnS  suggests that occurred at HJD 2455200, however as it cannot be ruled out that  G, it is possible that  d. Indeed, a longer period seems likely: phasing  with a 20-year period, and fitting a least-squares -order sinusoid (as expected for a dipolar magnetic field), produces a very poor fit as compared to longer periods. In Fig. 13, the illustrative sinusoudal fit was performed using a 30-year period, which achieves a reasonable fit to the data (the reduced is 2.6). Note that with this fit the MuSiCoS data do not define . Longer periods can also be accommodated, however at the expense of . We thus adopt  yr as the most conservative option allowed by the data.

6.3 Comparison to previous results

Figure 14: Top: ESPaDOnS, FORS1, FORS2, and SOFIN  measurements phased with the rotational period poposed by Hubrig et al. (2011). Bottom: ESPaDOnS  measurements phased with the rotational period proposed by Fourtune-Ravard et al. (2011). Neither of the previously proposed periods provides an acceptable phasing of the data.

There are two competing claims for rotational periods in the literature. The first, based on spectropolarimetry collected with FORS1, FORS2, and SOFIN, is approximately 2.18 d (Hubrig et al., 2011). The second period, based on a preliminary analysis of an earlier, smaller ESPaDOnS dataset, is  d (Fourtune-Ravard et al., 2011). Period analysis of the ESPaDOnS measurements using Lomb-Scargle statistics rules out both of these periods. The rotational periods provided by Hubrig et al. (2011) and Fourtune-Ravard et al. (2011) are respectively indicated with black and red arrows in the top panel of Fig. 12. There is no significant power in the periodogram at either period. The peak corresponding to the Fourtune-Ravard et al. (2011) period appears in both the  and  periodograms, suggesting it to be a consequence of noise. Phasing the data with the periods given by Hubrig et al. (2011) or Fourtune-Ravard et al. (2011) does not produce a coherent variation, as is demonstrated in Fig. 14. It impossible for a period on the order of days to account for the systematic decline of 200 G between the earliest data and the most recent data. We also note that the ESPaDOnS dataset presented here includes two epochs with superior time-sampling to that of the FORS1/2 datasets: 14 observations over 10 d in 2010, and 20 observations over 38 d in 2017, as compared to 13 FORS1 observations over 1075 d and 11 FORS2 observations over 60 d. The ESPaDOnS dataset thus enables a much better probe than the FORS1/2 dataset of short-term as well as long-term variability.

All FORS1/2 and SOFIN measurements are of positive polarity (Hubrig et al., 2006, 2011; Fossati et al., 2015b), in agreement with ESPaDOnS data. Furthermore, the magnitude, 300 G, is similar. The two SOFIN measurements also agree well with the ESPaDOnS data. The much shorter 2.18 d period determined by Hubrig et al. (2011) is due to an apparent  variation with a semi-amplitude that is significant at 1.8 in comparison with the median uncertainties in these measurements. Bagnulo et al. (2012) showed that systematic sources of uncertainty such as instrumental flextures must be taken into account in the evaluation of  uncertainties from FORS1 data, that the uncertainties in FORS1  measurements should thus be about 50% higher, and therefore that FORS1/2 detections are only reliable at a significance of 5, a much higher threshold than is satisfied by the variation the 2.18 d period is based upon. The catalogue of FORS1  measurements published by Bagnulo et al. (2015) additionally reveals differences of up to 150 G for observations of CMa between results from different pipelines. Using the published uncertainties, the FORS1/2 measurements presented by Hubrig et al. (2011) are generally within 2 of the sinusoidal fit to the ESPaDOnS and MuSiCoS data, and only 3 differ by greater than 5. Curiously, the FORS2 data published by Hubrig et al. (2011) are systematically 80 G lower than the FORS1 measurements. We conclude that the previously published low-resolution magnetic data are not in contradiction with the long-term modulation inferred from high-resolution  measurements.

7 Emission Lines

Having rejected the possibility that CMa’s H emission originates in the decretion disk of a classical Be companion star (§ 5), we proceed under the assumption that it is formed within the stellar magnetosphere. In this section we explore the star’s UV and H emission properties. We evaluate short-term variability with the stellar pulsations, as well as long-term variability consistent with the slow rotation inferred from the magnetic data.

7.1 H emission

7.1.1 Long-term modulation

Figure 15: H continuum-normalized flux (top), and residual flux (bottom) after subtraction of a photospheric model, at similar pulsation phases () in successive epochs. The thick red line shows a model line profile disk-integrated with the appropriate pulsation velocity. Vertical dashed lines indicate the integration range for EW measurement.

Fig. 15 shows H in 2000, 2004, 2010, 2014, and 2017, selected so as to share the same pulsation phase (). A synthetic line profile calculated using the physical parameters obtained in § 4 is overplotted as a thick line. Comparison of observed to synthetic line profiles demonstrates that emission is present at all epochs, although it is substantially weaker in the earlier CORALIE data. The agreement of H with the model is reasonable at all epochs, although there is also weak emission present in the line core (Fig. 4). Close analysis of the line core of H shows evidence of evolution, with the same trend as in H, although this evolution is of a much lower amplitude.

Figure 16: Periodogram for H EWs (top), and after pre-whitening with the highest-amplitude period (bottom). The highest-amplitude period is indicated by the dashed line. The red arrow indicates the pulsation period determined from the RVs.

H EWs were measured with an integration range of 0.4 nm of the rest wavelength in order to include only the region with emission (Fig. 15). The spectra were first shifted to a rest velocity of 0 km s by subtracting the RVs measured in § 3. The periodogram for these measurements is shown in the middle panel of Fig. 16. It shows a peak at 6700 d or 18 yr. The S/N of this peak is 29, indicating that it is statistically significant. This is consistent with the very long timescale of variation inferred from the ESPaDOnS  periodogram (Fig. 12), and with the minimum 7200 d period inferred from the positive and negative  extrema in the ESPaDOnS and MuSiCoS data.

The bottom panel of Fig. 13 shows the H EWs as a function of time. Maximum emission and maximum  occur at approximately the same phase, as do the  minimum and the minimum emission strength. The variation in EW is seen more clearly in the annual mean EWs (open circles), which are tabulated in Table 2. The relative phasing of  and EW is consistent with a co-rotating magnetosphere, and supports the accuracy of the adopted period. Since  is both positive and negative, two local emission maxima are expected at the positive and negative extrema of the  curve, as at these rotational phases the magnetospheric plasma is seen closest to face on. The solid curve in the bottom panel of Fig. 13 shows a -order sinusoidal fit to the annual mean EWs, which indeed yields two local maxima, the strongest corresponding to , and the second maximum predicted to occur at . Note that, due to the incomplete phase coverage (less than half of a rotational cycle), in the event that the H variation is indeed a double-wave a Lomb-Scargle periodogram should show maximum power at close to half of the rotational period. The periodogram peak at 18-yr would then indicate  yr, consistent with the rotational period inferred from .

7.1.2 Short-term modulation

Figure 17: Top: residual H EWs, after removal of the sinusoidal variation shown in the bottom panel of Fig. 13, phased with the pulsation period and binned with pulsation phase. The thick solid black line indicates the EW variation expected from a photospheric model calculated using the 300 K  variation determined in § 4.5, normalized by subtracting the mean EW of the model. The amplitude of the residual EW variation is 3 times larger than predicted by the photospheric model, and is about 0.5 cycles out of phase. Bottom: H V/R variation. Symbols and colors indicate the year of observation, the last two digits of which are given in the legend. The amplitude of the variation is much higher than predicted by the photospheric model, which is essentially flat on this scale and therefore is not shown. The amplitude furthermore changes between epochs, which is not expected for photospheric pulsations given the high stability of pulsation properties in other lines.
Figure 18: H dynamic spectra in the epochs of minimum (2000-2001) and maximum (2008-2010) emission. The synthetic spectrum shown in Fig. 15 was used as a reference spectrum. Observed spectra were shifted to 0 km s by subtracting their RVs, and then co-added in 20 phase bins. The white curves indicate the RV curve due to pulsations. In all epochs, the emission profile has three components: a central peak near 0 km s, which shifts slightly about line centre, but stays nearly constant in strength; a blue-shifted peak, which is at maximum strength at phase 0 and disappears by phase 0.5; and a red-shifted peak, which disappears at phase 0 and is at a maximum at phase 0.5. All three components increase in strength from 2000 to 2010. Note the greater amplitude of the variation in the secondary emission lobes in the 2008-2010 data.

As a first step to investigating whether H is affected by pulsation, residual EWs were obtained by subtracting the least-squares 2-order sinusoidal fit to the annual mean EWs. The bottom panel of Fig. 16 shows the period spectrum of the residual EWs. The strongest peak is at the stellar pulsation period, with a S/N of 12. After prewhitening the EWs with both the rotational and pulsational frequencies, the S/N of the highest peak in the periodogram is close to 4, suggesting that all significant variation is accounted for by these two frequencies.

While the semi-amplitude of the residual EW variation is similar to the median error bar, binning the residual EWs by pulsation phase does not change the semi-amplitude, but increases the significance of the variation to with respect to the mean error bar. The phase-binned residual EWs are shown phased with the pulsation period in the top panel of Fig. 17. There is a coherent variation of the phase-binned residual EWs with the pulsation period, with a semi-amplitude of approximately 0.003 nm.

One obvious candidate mechanism for producing this pulsational modulation is the change in the EW of the underlying photospheric profile due to the changing , as explored in § 4.5. To investigate this hypothesis we calculated synthetic spectra via linear interpolation between the grid of tlusty BSTAR2006 models (Lanz & Hubeny, 2007). We used the physical parameters from § 4, including the 300 K  variation found in § 4.5. Deformation of the line profile due to pulsation was accounted for using the RV curve from § 3, and modelling the (assumed radial) pulsations as described in § 4.1. Spectra were calculated at 20 pulsation phases, and the EW was measured at each phase using the same integration range as used for the observed data, and after moving the synthetic spectra to zero RV. The black line in the top panel of Fig. 17 shows the resulting variation, normalized by subtracting the mean EW across all models. The semi-amplitude of the EW variation expected due to changes in the photospheric profile due to pulsation is about 0.001 nm, much smaller than observed. Furthermore, it is out of phase with the observed variation by about 0.5 pulsation cycles. Note that the maximum and minimum EWs of the predicted variation correspond to the minimum and maximum of the  variation (Fig. 7), as expected for H lines, which grow weaker with increasing  in this  range.

Radial pulsation introduces asymmetry into the line profile, which can be quantified by , defined as the ratio of the EW in the blue half to the EW in the red half of the line. We measured from EWs calculated from 0.4 nm to line centre (defined at the laboratory rest wavelength shifted by the RV measured from metallic lines in § 3), and line centre to  nm. The bottom panel of Fig. 17 shows the variation. The semi-amplitude of the variation is about 0.25. This is much higher than the semi-amplitude of 0.0007 predicted by synthetic spectra calculated using a radially pulsating photospheric model; since this is essentially flat on the scale of Fig. 17, the photospheric variation is not shown. The low level of line asymmetry in the synthetic spectra is a consequence of the large Doppler broadening of the H line, approximately 30 km s or about twice the semi-amplitude of the RV curve. For narrower lines, e.g. the C ii 656.3 nm line for which the Doppler velocity is similar to the RV semi-amplitude, the predicted and observed variations are in reasonable agreement.

The semi-amplitude of the H variation increases steadily from 0.16 in 2000-2001, to 0.20 in 2002-2004, to 0.35 in 2008-2010, and then declines to 0.24 in 2012-2014 and 0.13 in 2017. This is the same pattern as the change in total emission strength, thus, the amplitude of the V/R variation correlates to the total emission strength and, therefore, to .

For a more detailed view of the pulsational modulation of H we calculated dynamic spectra phased with the pulsation period. These are shown in Fig. 18, using the synthetic line profile from Fig. 15 as a reference spectrum. Individual line profiles were moved to 0 km s by subtracting their RVs, and then binned by pulsation phase using phase bins of 0.05 cycles. The two panels of Fig. 18 show dynamic spectra in the epochs of minimum emission (2000-2001) and maximum emission (2008-2010). The emission line morphology and pattern of variability is essentially identical in other epochs. In both cases, the emission peaks near the centre of the line. In the CORALIE data, the H emission peak is 20% of the continuum. This rises to about 28% of the continuum in the ESPaDOnS data. The emission peak anticorrelates slightly with the RV, as shown by the overplotted white lines. The strong red and blue emission variability revealed by the variation in the bottom panel of Fig. 17 is due to secondary emission peaks which occur at phases 0.0 and 0.5, respectively blue- and red-shifted with respect to the line centre. As with the central emission peak, these are stronger in the ESPaDOnS data, reflecting the change in amplitude over time. Note that, in the data acquired at earlier epochs, these secondary emission bumps are apparently separated from the main emission peak, while in later epochs they are connected (although this depends on the choice of the reference spectrum).

Both the emission strength (as measured by the residual EWs after pre-whitening with the -order fit in Fig. 13) and the line asymmetry (as quantified by ) vary coherently with the pulsation phase. Synthetic photospheric spectra calculated using a radially pulsating model are unable to reproduce the residual EW variation, which is both 3 larger than predicted, and 0.5 cycles out of phase. The amplitude of the observed variation is about 300 larger than predicted by the model, which does not reproduce the prominent blue- and red-shifted secondary emission bumps. The amplitude of is furthermore variable with time, increasing and decreasing in strength in the same fashion as the total emission strength. As the star’s pulsation amplitude is extremely regular, a change in the amplitude of with epoch cannot be explained by photospheric pulsation. These discrepancies between model and observation suggest either that the origin of the pulsational modulation of H is either not photospheric, or that it cannot be explained due to  variation alone. Exotic processes, such as temperature inversions related to shockwaves produced by the star’s supersonic pulsations, may be one explanation. Alternatively, the origin of the pulsational modulation may reside within the magnetosphere.

7.2 UV emission lines

Figure 19: Comparison between the mean N v (top) and Al iii (bottom) emission lines of CMa (solid, black), and those of Cep at maximum emission (red, dashed) and maximum absorption (blue, dot-dashed). Vertical dashed lines indicate rest wavelengths. CMa’s emission lines are similar to those of Cep at maximum emission, albeit somewhat stronger. The lower panels show Temporal Variance Spectra (TVS, see text), calculated from all spectra, which characterize the variability of the spectra normalized to the continuum. The horizontal line indicates the continuum variation, in comparison to which there is no statistically significant variability inside the line profile.

Emission is present in four of the UV doublets often used to diagnose the wind properties of early-type stars: N v 1239, 1243 Å; Si iv 1394, 1403 Å; C iv 1548, 1551 Å; and Al iii 1854, 1863 Å. The emission profiles of these doublets are similar to those of the magnetic Cep pulsator Cep at maximum emission. Schnerr et al. (2008) performed this comparison for the C iv doublet. Fig. 19 compares the mean line profiles for CMa’s N v and Al iii lines to those of Cep, and demonstrates that these lines are also similar to those of Cep at maximum emission. Such emission is unique to magnetic stars, and as an indirect diagnostic of stellar magnetism has historically prompted the search for magnetic fields in such stars (e.g., Henrichs 1993; Henrichs et al. 2013; indeed, the detection of CMa’s UV emission motivated the collection of the MuSiCoS data).

The presence of emission in the N v line is particularly interesting. This line is not seen in normal early B-type stars (Walborn et al., 1995), nor is it present in most normal late O-type stars (e.g. Brandt et al. 1998). In their study of IUE data for 4 magnetic B-type stars, including Cep, Smith & Groote (2001) concluded that this doublet must be formed at 30 kK, somewhat higher than the photospheric . The presence of N v lines in the UV spectra of relatively cool stars is thought to be a consequence of Auger ionization due to the presence of X-rays (Cassinelli & Olson, 1979). Since CMa has a strong magnetic field, it is expected to be overluminous in X-rays due to magnetically confined wind shocks, and is indeed observed to be overluminous in X-rays to the degree predicted by models (Oskinova et al., 2011; Nazé et al., 2014).

Cep’s wind lines show clear variability synchronized with its rotation period (Henrichs et al., 2013). In contrast, CMa’s wind lines show only a very low level of variability, more similar to that seen in a normal (magnetically unconfined) stellar wind (Schnerr et al., 2008). The bottom panels of Fig. 19 show Temporal Variance Spectra (TVS), which compare the variance within spectral lines to the variance in the continuum (Fullerton et al., 1996). To minimize variation due to pulsation, spectra were moved to 0 km s by subtracting the RV computed on the basis of the RV curve and ephemeris determined in Section 3. With the uncertainty in and in (§ 3), the uncertainty in pulsation phase is about 0.028, corresponding to a maximum uncertainty in RV of 3 km s, less than the 7.8 km s velocity pixel of the IUE data. RV correction reduces the TVS pseudocontinuum level by a factor of about 2 to 3.

Comparison of EW measurements of these lines to EWs of synthetic spectra, using the same radially pulsating model described above in § 7.1, yielded ambiguous results, due to the small number of high-dispersion IUE observations (13 spectra), and formal uncertainties similar to the maximum level of variability, 0.02 nm. This low level of variability is consistent with the weak variability of the residual H EWs.

The lack of variability in CMa’s wind lines is consistent with a rotational pole aligned with the line of sight, an aligned dipole, a long rotation period, or some combination of these. The similarity to Cep’s emission lines at maximum emission suggests that the magnetic pole was close to being aligned with the line of sight when the UV data were acquired. The long-term modulations of both H and  favour an oblique dipole with a long rotational period, in which case the UV data should have been acquired at a rotational phase corresponding to one of the extrema of the  curve. Phasing the UV data with the same 30 yr rotational period as in Fig. 13 yields a phase close to 1.0, i.e. they would indeed have been obtained close to magnetic maximum. Assuming that the data must have been acquired near an extremum of the  curve (i.e. at a phase close to 0.5 or 1.0) would require a period of 10, 15, 20, 30, or 60 years, of which only 30 and 60 years are not excluded by the magnetic data. This assumes that the UV data were, in fact, acquired close to an extremum. Given that the UV emission is only slightly stronger than that of Cep, which has a weaker magnetic field and no H emission, it may be the case that the true maximum UV emission strength is significantly in excess of observations, in which case the UV data cannot be used to infer .

8 Magnetic and magnetospheric parameters

In this section, the  yr rotation period inferred from magnetic and spectroscopic data is used with the star’s physical parameters and  measurements to establish constaints on the properties of CMa’s surface magnetic field, circumstellar magnetosphere, and spindown timescales, using the self-consistent Monte Carlo method described by Shultz (2016). These results are summarized in Table 3.

Parameter Value Ref.
(d) 30 yr 6.1
-0.780.02 8.1
() 83 8.1
0.360.02 4.2
(kG) 8.1
 () 8.2
-8.00.1 8.2, Vink et al. (2001)
 (km s) 207060 8.2, Vink et al. (2001)
 () 8.2
41 8.3
4212 8.3
Table 3: Magnetic, magnetospheric, and rotational properties. The final column gives the section in which the parameter is derived.
Figure 20: Oblique Rotator Model parameters determined via a Monte Carlo approach. Red, blue, and pink show the 1, 2, and 3 contours of the grid. The bottom panels show probability density functions (PDFs) for , , and , where was drawn from a random distribution over a solid angle of 4.

The  variation of a rotating star with a dipolar magnetic field can be reproduced with a three-parameter model: the inclination angle between the rotational axis and the line of sight, the obliquity angle between the magnetic and rotational axes, and the polar strength of the magnetic dipole at the stellar surface . The two angular parameters are related via , where is the ratio defined by Preston (1967) as , with and the mean and semi-amplitude of the sinusoidal fit to  when phased with . We determined from the sinusoidal fit to the annual mean  measurements shown in the top panel of Fig. 13. In consequence these results rely on the 30-year period.

The degeneracy between and is usually broken by determining independently, e.g. from , , and . Using this method Hubrig et al. (2011) found , however, this was based on their rotation period of 2.18 d, which was shown in § 6.2 to be incorrect. Indeed, the equatorial rotational velocity implied by a 30-year period, km s, is much less than the upper limit on   km s found in § 4.1 (and indeed, much less than the 1.8 km s velocity resolution of ESPaDOnS data). Since cannot be constrained, we applied a probabilistic prior, requiring that be drawn from a random distribution over 4 steradians such that (e.g., Landstreet & Mathys 2000). The corresponding probability density function (PDF) is shown in the bottom right panel of Fig. 20. The bottom middle panel shows the resulting PDF for , which peaks at . Despite the inverse relationship of and obtained for (middle panel of Fig. 20), and the bias towards large in the prior, large are favoured overall. We note that we obtain a similar to that given by Hubrig et al. (2011), however this is simply a coincidence as their value was obtained with an incorrect rotation period and a very different value of the Preston parameter (0.56, as compared to -0.78).

For each () pair, was determined using Eqn. 1 from Preston (1967). This also requires knowledge of the limb darkening coefficient , which we obtained from the tables calculated by Díaz-Cordovés et al. (1995) as 4.1). The resulting PDF peaks sharply at the minimum value permitted by Preston’s equation, kG. As demonstrated in the middle and upper left panels of Fig. 20, kG over the range , with a low-probability tail extending out to several kG for very large and very small and . This is much lower than the value of 5.3 kG given by Hubrig et al. (2011), which arose from their very small .

If instead the ESPaDOnS and MuSiCoS measurements define the extrema of , we obtain . This does not affected , although a slightly smaller is favoured.

Magnetic wind confinement is governed by the balance of kinetic energy density in the radiative wind to the magnetic energy density. This is expressed by the dimensionless wind magnetic confinement parameter (ud-Doula & Owocki, 2002). If , the star possesses a magnetosphere.

Oskinova et al. (2011) analyzed IUE observations of CMa using the Potsdam Wolf-Rayet (PoWR) code in order to determine the wind parameters, obtaining and km s. However, magnetic confinement reduces the net mass-loss rate and, more seriously, strongly affects line diagnostics due to the departure from spherical symmetry in the circumstellar environment. Indeed, Oskinova et al. were unable to achieve a simultaneous fit to the Si iv and C iv doublets. Comparison of magnetohydrodynamic (MHD) simulations and spherically symmetric models to the magnetospheric emission of Of?p stars has demonstrated that MHD simulations yield a superior fit, and require mass-loss rates comparable to those obtained from the Vink et al. (2001) recipe, whereas spherically symmetric models in general require much lower mass-loss rates to achieve a relatively poor match to the observations (Grunhut et al., 2012b; Marcolino et al., 2012, 2013). In addition to this, magnetic wind confinement is not expected to modify the surface mass flux, which is the relevant quantity in calculating the strength of magnetic confinement (ud-Doula & Owocki, 2002). Thus, should be determined using the wind parameters as they would be in the absence of a magnetic field, rather than those measured via spectral modelling (e.g., Petit et al. 2013). Using the mass-loss recipe of Vink et al. (2001) with , , and from Table 1, assuming the metallicity , and determining the wind terminal velocity km s via scaling the star’s escape velocity by 2.6 as suggested by Vink et al., yields a mass-loss rate . From Eqn. 7 of ud-Doula & Owocki (2002), , so the wind is magnetically confined.

The physical extent of the magnetosphere is given by the Alfvén radius , defined as the maximum extent of closed magnetic field lines in the circumstellar environment.  can be calculated heuristically from (Eqn. 7 in ud-Doula et al. 2008): we find  . Unless is particularly large or small, in which case is significantly in excess of 2 kG, the magnetic and magnetospheric parameters will be close to the derived lower limits.  is almost certainly below 20 .

CMa has the highest X-ray luminosity and the hardest X-ray spectrum of any of the magnetic Cep stars (Cassinelli et al., 1994; Oskinova et al., 2011). The X-ray luminosity, corrected for interstellar absorption corresponding to , is (Nazé et al., 2014). Using 2D MHD simulations ud-Doula et al. (2014) calculated X-ray emission from magnetically confined wind shocks and developed an X-ray Analytic Dynamical Magnetosphere (XADM) scaling for  with , , , and . Comparison to available X-ray data for magnetic early-type stars has indicated that  predicted by XADM should be scaled by 5-20% to match the observed  (Nazé et al., 2014). This efficiency factor accounts for dynamical infall of the plasma. XADM successfully predicts the X-ray luminosity of magnetic, hot stars across 3 decades in , 2 decades in , and 5 decades in  (Nazé et al., 2014), i.e. the range of the model’s successful application is much larger than the uncertainty introduced by the efficiency factor. Assuming an efficiency of 10%, kG, and taking into account the uncertainties in the stellar parameters in Table 1, XADM predicts , in excellent agreement with the observed X-ray luminosity . Adopting the higher value of  kG suggested by Hubrig et al. (2011) yields  , slightly higher than observed (although this can be reconciled by lowering the efficiency factor to 5%). If the lower  and  determined by Oskinova et al. (2011) are used instead, XADM predicts