# The magnetic variability of the $\beta$ Cep star $\xi^{1}$ CMa

S. P. Järvinen<EMAIL_ADDRESS>S. Hubrig  M. Schöller  I. Ilyin  Leibniz-
Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam,
Germany  European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748
Garching, Germany

###### Abstract

$\xi^{1}$ CMa is a known magnetic star showing rotationally modulated magnetic
variability with a period of 2.17937 d. However, recent work based on high-
resolution spectropolarimetry suggests that the rotation period is longer than
30 years. We compare our new spectropolarimetric measurements with FORS 2 at
the VLT acquired on three consecutive nights in 2017 to previous FORS 1/2
measurements of the longitudinal magnetic field strength. The new longitudinal
magnetic field values are in the range from 115 to 240 G and do not support
the presence of a long period.

###### keywords:

stars: magnetic fields , stars: oscillations , stars: individual ($\xi^{1}$
CMa)

††journal: New Astronomy

##  1 Introduction

$\xi^{1}$ CMa is known to be a bright (m${}_{\rm V}=4.3$) sharp-lined $\beta$
Cephei variable with spectral type B0.5–B1 IV (e.g. Morel et al., 2008).
$\beta$ Cephei variables are massive (7 to 20 solar masses), B0–B3 main-
sequence stars pulsating in several pressure and gravity modes with periods
between 3 and 8 hours. Most uncertainties in the stellar structure of these
massive stars are related to the amount of core convective overshooting and
rotational mixing. The lack of knowledge on the internal structure of these
progenitors of core-collapse supernovae has large consequences for the whole
duration and the end of the stellar life, and, consequently, for any
astrophysical results relying on it, such as the chemical enrichment of
galaxies. The presence of large-scale organised magnetic fields in $\beta$
Cephei stars is well established (e.g. Hubrig et al., 2006, 2009; Briquet et
al., 2012). It is also generally accepted that these magnetic fields are
frozen-in fields that were acquired during stellar formation on an early pre-
main sequence stage. Pulsating stars with magnetic fields are considered as
promising targets for asteroseismic analysis (e.g. Shibahashi & Aerts, 2000),
which requires as input the observed parameters of the magnetic field
geometry.

To determine the magnetic field geometry and its strength, the rotation period
has to be known. However, only for very few $\beta$ Cephei stars has the
behaviour of the magnetic field been studied over the rotation cycle. Using
low-resolution spectropolarimetric observations with the FOcal Reducer low
dispersion Spectrograph (FORS 1/2; Appenzeller et al., 1998) in
spectropolarimetric mode at the Very Large Telescope (VLT) and the Soviet-
Finnish echelle spectrograph (SOFIN; Tuominen et al., 1999) installed at the
2.56 m Nordic Optical Telescope on La Palma, Hubrig et al. (2006, 2009) were
able to detect magnetic fields in a few $\beta$ Cephei stars. The first
detection of a magnetic field in $\xi^{1}$ CMa was achieved by Hubrig et al.
(2006).

Among the targets with detected magnetic fields, a few $\beta$ Cephei
variables were selected for follow-up VLT multi-epoch magnetic field
measurements with the aim to detect their most probable periods and to put
constraints on their magnetic field geometry (Hubrig et al., 2011a, b). The
simplest models of the magnetic field geometry are usually based on the
assumption that the studied stars are oblique dipole rotators, i.e. their
magnetic field can be approximated by a dipole with the magnetic axis inclined
to the rotation axis. Using the dipole approximation, Hubrig et al. (2011a)
showed that $\xi^{1}$ CMa is a rather fast rotating star with a rotation
period of 2.17937$\pm$0.00012 d seen close to pole-on and having a rather
strong dipole magnetic field of the order of 5.3 kG.

A few years later, Fossati et al. (2015) presented three additional
measurements of $\xi^{1}$ CMa obtained with FORS 2 in 2013 and suggested a
rotation period of 2.17950 d. On the other hand, Shultz et al. (2017) used
high-resolution ($R\approx 67\,000$) spectropolarimetric observations made
with the Multi-Site Continuous Spectroscopy (MuSiCoS; Baudrand & Bohm, 1992)
between 2000 and 2002 and with the Echelle SpectroPolarimetric Device for the
Observation of Stars (ESPaDOnS; Donati, 2003; Donati et al., 2006) between
2008 and February 2017 and proposed that the longitudinal magnetic field is
gradually decreasing since 2010. Correspondingly, the rotation period of
$\xi^{1}$ CMa should be longer than 30 years. Notably, the determination of
this long rotation period was partly based on the MuSiCoS observations that
were carried out at an airmass never below 2.45. Only these observations show
the presence of a longitudinal magnetic field with negative polarity, whereas
all other measurements with both FORS 1/2 and ESPaDOnS exhibit positive
polarity. To verify the proper operations of MuSiCoS during that period,
Shultz et al. (2017) refer to observations of 36 Lyn. However, since 36 Lyn is
an object that crosses Pic-du-Midi close to zenith, it is rather questionable
if this star can actually support the validity of the results obtained on
$\xi^{1}$ CMa. For example, Petit et al. (2011), who used NARVAL and ESPaDOnS
spectropolarimetric observations for a sensitive magnetic field search in
Sirius, dismissed all observations obtained at high airmass from their study.
Therefore, it is not clear whether the detection of a magnetic field of
negative polarity using MuSiCoS at high airmass can be considered genuine.

Since previous low-resolution FORS 2 observations did not show any hint at a
long-term periodicity, we obtained three additional low-resolution FORS 2
observations for this target on three consecutive nights in October 2017, to
investigate the possible presence of a much longer rotation period. In the
following, we discuss the results of our analysis of these three new
observations and compare all currently available FORS 1/2 measurements of
$\xi^{1}$ CMa with those obtained using high-resolution spectropolarimetry.

##  2 Observations and magnetic field analysis

The FORS 2 multi-mode instrument is equipped with polarisation analysing
optics comprising super-achromatic half-wave and quarter-wave phase retarder
plates, and a Wollaston prism with a beam divergence of 22′′ in standard
resolution mode. We used the GRISM 600B and the narrowest available slit width
of 0$.\\!\\!^{\prime\prime}$4 to obtain a spectral resolving power of
$R\approx 2000$. The observed spectral range from 3250 to 6215 Å includes all
Balmer lines, apart from H$\alpha$, and numerous helium lines. For the
observations, we used a non-standard readout mode with low gain
(200kHz,1$\times$1,low), which provides a broader dynamic range, hence
allowing us to reach a higher signal-to-noise ratio (S/N) in the individual
spectra.

A description of the assessment of the presence of a longitudinal magnetic
field using FORS 1/2 spectropolarimetric observations was presented in our
previous work (e.g. Hubrig et al., 2004a, b, and references therein).
Rectification of the $V/I$ spectra was performed in the way described by
Hubrig et al. (2014). Null profiles, $N$, are calculated as pairwise
differences from all available $V$ profiles so that the real polarisation
signal should cancel out. From these, 3$\sigma$-outliers are identified and
used to clip the $V$ profiles. This removes spurious signals, which mostly
come from cosmic rays, and also reduces the noise. A full description of the
updated data reduction and analysis will be presented in a separate paper
(Schöller et al., in preparation, see also Hubrig et al., 2014). The mean
longitudinal magnetic field, $\left<B_{\rm z}\right>$, is measured on the
rectified and clipped spectra based on the relation following the method
suggested by Angel & Landstreet (1970):

$\displaystyle\frac{V}{I}=-\frac{g_{\rm eff}\,e\,\lambda^{2}}{4\pi\,m_{\rm
e}\,c^{2}}\,\frac{1}{I}\,\frac{{\rm d}I}{{\rm d}\lambda}\left<B_{\rm
z}\right>\,,$ (1)

where $V$ is the Stokes parameter that measures the circular polarization, $I$
is the intensity in the unpolarized spectrum, $g_{\rm eff}$ is the effective
Landé factor, $e$ is the electron charge, $\lambda$ is the wavelength, $m_{\rm
e}$ is the electron mass, $c$ is the speed of light, ${{\rm d}I/{\rm
d}\lambda}$ is the wavelength derivative of Stokes $I$, and $\left<B_{\rm
z}\right>$ is the mean longitudinal (line-of-sight) magnetic field.

Figure 1:  F-statistics periodogram for the longitudinal magnetic field
measurements of $\xi^{1}$ CMa using the entire spectrum, published in the work
by Hubrig et al. (2011a). The window function is presented in red.

The longitudinal magnetic field was measured in two ways: using the entire
spectrum including all available lines or using exclusively hydrogen lines.
Furthermore, we have carried out Monte Carlo bootstrapping tests. These are
most often applied with the purpose of deriving robust estimates of standard
errors (e.g. Steffen et al., 2014). The measurement uncertainties obtained
before and after the Monte Carlo bootstrapping tests were found to be in close
agreement, indicating the absence of reduction flaws. In Fig. 1 we show the
periodogram calculated from the measurements presented by Hubrig et al.
(2011a) using the entire spectrum and utilizing the Levenberg–Marquardt method
(Press et al., 1992), and which was not presented in this previous work.

Table 1:  New longitudinal magnetic field measurements of $\xi^{1}$ CMa obtained using FORS 2 observations.  HJD | Date | S/N | $\left<B_{\rm z}\right>_{\rm all}$ | $\left<B_{\rm z}\right>_{\rm hyd}$ | $\left<B_{\rm z}\right>_{\rm N}$ | Phase
---|---|---|---|---|---|---
2 450 000+ |  | at 4700Å | [G] | [G] | [G] |
8029.810749 | 2017–10–03 | 1875 | 116$\pm$ | 69 | 115$\pm$ | 102 |  $-$31$\pm$ | 59 | 0.42
8030.790235 | 2017–10–04 | 2610 | 181$\pm$ | 57 | 240$\pm$ | 77 |  $-$56$\pm$ | 61 | 0.87
8031.740386 | 2017–10–05 | 2720 | 157$\pm$ | 76 | 207$\pm$ | 95 | 45$\pm$ | 65 | 0.30

For the new observations, we present in Table 1 the heliocentric Julian date
together with the observing date, followed by the achieved signal-to-noise
ratio, and the results of the longitudinal magnetic field measurements
obtained again in two ways: using the entire spectrum or exclusively the
hydrogen lines. The measurements using the diagnostic null spectra $N$
calculated using the entire spectrum and the rotational phases corresponding
to $P_{\rm rot}=2.17937$ d are presented in the last two columns.

We note that low-resolution spectropolarimetric observations are most
appropriate for the detection of weak magnetic fields in $\beta$ Cep pulsating
stars, for which the impact of short-time variability on the magnetic field
measurements can be quite significant. The work of Järvinen et al. (2017)
indicated that ignoring the velocity shifts leads to an underestimation of the
magnetic field strength roughly by a factor of two. Furthermore, the
pulsational radial velocity amplitude of the order of 70 km s-1 for $\xi^{1}$
CMa is one of the largest known among $\beta$ Cep stars (e.g. Saesen et al.,
2006). While a full HARPS sequence of subexposures for $\xi^{1}$ CMa requires
about 30 min, one FORS 2 observation of the same star lasts less than 10 min.
Owing to the strong changes in the positions and shapes of line profiles in
the spectra of pulsating stars, a method using high-resolution spectra
averaged over all subexposures usually leads to erroneous wavelength shifts
and thus to wrong values for the longitudinal magnetic field. Interestingly,
Shultz et al. (2017) report on the detection of apparent scatter in the
measurements of the mean longitudinal magnetic field with a standard deviation
of up to 15 G. To explain this scatter, the authors suggest the presence of a
modulation of the field with pulsation phase or some unidentified instrumental
effect, but also consider the possibility of the impact of line profile
variability.

Figure 2:  Phase folded measurements of the longitudinal magnetic field for
$\xi^{1}$ CMa. The filled circles represent previously published values from
Hubrig et al. (2006, 2011a) and Fossati et al. (2015). The different colors
correspond to the different time bins presented in Fig. 3. The red filled
squares are the new measurement values. The sinusoidal fit parameters and the
ephemeris MJD$55140.73332\pm 0.03794+2.17937\pm 0.00012E$ used for the folding
of the phase are from Hubrig et al. (2011a).

In Fig. 2 we present the phase diagram for the magnetic field measurements of
$\xi^{1}$ CMa, using the period of 2.17937 d, including the three recent FORS
2 measurements together with the results from Hubrig et al. (2006, 2011a) and
Fossati et al. (2015). Our new measurements acquired on three consecutive
nights in 2017 indicate a longitudinal magnetic field strength in the range
from 115 to 240 G. Two measurements have rather large uncertainties and still
fit the magnetic phase curve well. Only the measurement obtained on 2017
October 4 ($\left<B_{\rm z}\right>_{\rm all}=181\pm 57$ G, ($\left<B_{\rm
z}\right>_{\rm hyd}=240\pm 77$ G) has a significance level of about 3$\sigma$.
Therefore, the obtained new observations do not provide much evidence for the
presence of the period of more than 30 years reported by Shultz et al. (2017).

To investigate the temporal evolution of the magnetic field strength in our
data, we used time binned averages of the measured field strengths between
2005 and 2017. The distribution of the binned average field strengths and the
number of measurements used in each bin is shown in Fig. 3 and the comparison
of our field strength distribution with that of Shultz et al. (2017) is shown
in Fig. 4. In contrast to the measurements presented by Shultz et al., we
observe in both figures a slight decrease of the magnetic field strength in
2009. Comparing the identical colors between Figs. 2 and 3, it is obvious that
the observed magnetic field decrease in the binned data is due to the
rotational modulation of the magnetic field.

Figure 3:  Temporal evolution of the longitudinal magnetic field. The time is
given both in HJDs and years. Measurements belonging to different time bins
are plotted with different colors. Identical colors are used in Fig. 2 for the
phase folded points. The number of individual measurements in each bin is
given in the plot. The blue line connects the time binned averages.  Figure 4:
The same as Fig. 3. The black diamonds represent the annual weighted mean
$\left<B_{\rm z}\right>$ measurements by Shultz et al. (2017), connected by
the dashed line.

##  3 Discussion

Insufficient knowledge of the strength, geometry, and time variability of
magnetic fields in hot pulsating stars prevented until now important
theoretical studies on the impact of magnetic fields on stellar rotation,
pulsation, and element diffusion. Morel et al. (2006, 2008) performed a NLTE
abundance analysis of a sample of early-type B dwarfs with detected magnetic
fields and low $v\sin i$–values, among them $\xi^{1}$ CMa. Their analysis
strongly supported the existence of a population of nitrogen-rich and boron-
depleted early B-type stars. It is of interest that the obtained abundance
patterns for this group of stars are well-reproduced by the theoretical models
of Heger & Langer (2000), but only assuming initial velocities exceeding
$\sim$200 km s-1 which would imply strong rotational mixing and surface
nitrogen enrichment. The existence of a rotation period of the length of more
than 30 years in $\xi^{1}$ CMa would suggest an extreme loss of angular
momentum over the fractional main-sequence lifetime of 0.76 reported in the
study of Hubrig et al. (2006). Very long rotation periods of the order of
decades are currently only known for a few chemically peculiar A-type (Ap)
stars with kG magnetic fields. As evolutionary changes of the rotation periods
of Ap stars during their main-sequence lifetime are at most of the order of a
factor 2 (Hubrig et al., 2007), it is believed that the period differentiation
took place already before Ap stars arrive at the main sequence. The typical
rotation periods of more massive early B-type stars with much shorter main-
sequence lifetimes are usually of the order of a few days and less and no
early B-type star is known to have long rotation periods of the order of years
or more.

While rotational modulation with a short period of 2.18 d was previously
clearly detected in FORS 1/2 data, the rotation period longer than 30 years
reported by Shultz et al. (2017) based on high-resolution spectropolarimetric
observations requires additional observational constraints. First of all, the
impact of the strong pulsational variability on the magnetic field
measurements in high-resolution spectropolarimetric observations has to be
taken into account. Apart from spectropolarimetric monitoring of $\xi^{1}$ CMa
in the coming years using circularly polarised spectra, a useful approach to
proof the presence of an extremely long rotation could be to measure broad-
band linear polarisation, which is caused by different saturation of the $\pi$
and $\sigma$ components of a spectral line in the presence of a magnetic
field. This differential effect is qualitatively similar for all lines, so
that in broad-band observations the contributions of all lines add up. A model
for the interpretation of such observations developed by Landolfi et al.
(1993) does not allow to derive the field strength, but provides very useful
constraints on the geometry of the magnetic field.

## Acknowledgements

We would like to thank the anonymous referee for his useful comments. Based on
observations made with ESO Telescopes at the La Silla Paranal Observatory
under programme ID 0100.D-0110.

## References

## References

  * Angel & Landstreet (1970) Angel, J. R. P. & Landstreet, J. D. 1970, ApJL, 160, L147
  * Appenzeller et al. (1998) Appenzeller, I., Fricke, K., Fürtig, W., et al. 1998, The Messenger, 94, 1
  * Baudrand & Bohm (1992) Baudrand, J. & Bohm, T. 1992, A&A, 259, 711
  * Briquet et al. (2012) Briquet, M., Neiner, C., Aerts, C., et al. 2012, MNRAS, 427, 483
  * Donati (2003) Donati, J.-F. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 307, Solar Polarization, ed. J. Trujillo-Bueno & J. Sanchez Almeida, 41
  * Donati et al. (2006) Donati, J.-F., Catala, C., Landstreet, J. D., & Petit, P. 2006, in Astronomical Society of the Pacific Conference Series, Vol. 358, Astronomical Society of the Pacific Conference Series, ed. R. Casini & B. W. Lites, 362
  * Fossati et al. (2015) Fossati, L., Castro, N., Schöller, M., et al. 2015, A&A, 582, A45
  * Heger & Langer (2000) Heger, A. & Langer, N. 2000, ApJ, 544, 1016
  * Hubrig et al. (2009) Hubrig, S., Briquet, M., De Cat, P., et al. 2009, Astronomische Nachrichten, 330, 317
  * Hubrig et al. (2006) Hubrig, S., Briquet, M., Schöller, M., et al. 2006, MNRAS, 369, L61
  * Hubrig et al. (2011a) Hubrig, S., Ilyin, I., Schöller, M., et al. 2011a, ApJL, 726, L5
  * Hubrig et al. (2004a) Hubrig, S., Kurtz, D. W., Bagnulo, S., et al. 2004a, A&A, 415, 661
  * Hubrig et al. (2007) Hubrig, S., North, P., & Schöller, M. 2007, Astronomische Nachrichten, 328, 475
  * Hubrig et al. (2011b) Hubrig, S., Schöller, M., Kharchenko, N. V., et al. 2011b, A&A, 528, A151
  * Hubrig et al. (2014) Hubrig, S., Schöller, M., & Kholtygin, A. F. 2014, MNRAS, 440, 1779
  * Hubrig et al. (2004b) Hubrig, S., Szeifert, T., Schöller, M., Mathys, G., & Kurtz, D. W. 2004b, A&A, 415, 685
  * Järvinen et al. (2017) Järvinen, S. P., Hubrig, S., Ilyin, I., Schöller, M., & Briquet, M. 2017, MNRAS, 464, L85
  * Landolfi et al. (1993) Landolfi, M., Landi Degl’Innocenti, E., Landi Degl’Innocenti, M., & Leroy, J. L. 1993, A&A, 272, 285
  * Morel et al. (2006) Morel, T., Butler, K., Aerts, C., Neiner, C., & Briquet, M. 2006, A&A, 457, 651
  * Morel et al. (2008) Morel, T., Hubrig, S., & Briquet, M. 2008, A&A, 481, 453
  * Petit et al. (2011) Petit, P., Lignières, F., Aurière, M., et al. 2011, A&A, 532, L13
  * Press et al. (1992) Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C. The art of scientific computing
  * Saesen et al. (2006) Saesen, S., Briquet, M., & Aerts, C. 2006, Communications in Asteroseismology, 147, 109
  * Shibahashi & Aerts (2000) Shibahashi, H. & Aerts, C. 2000, ApJL, 531, L143
  * Shultz et al. (2017) Shultz, M., Wade, G. A., Rivinius, T., et al. 2017, MNRAS, 471, 2286
  * Steffen et al. (2014) Steffen, M., Hubrig, S., Todt, H., et al. 2014, A&A, 570, A88
  * Tuominen et al. (1999) Tuominen, I., Ilyin, I., & Petrov, P. 1999, in Astrophysics with the NOT, ed. H. Karttunen & V. Piirola, 47