11institutetext:  IRAP, Université de Toulouse, CNRS, UPS, CNES, 57 Avenue
d’Azereix, 65000 Tarbes, France
11email<EMAIL_ADDRESS>22institutetext: IRAP, Université de
Toulouse, CNRS, UPS, CNES, 14 Avenue Edouard Belin, 31400 Toulouse, France
22email<EMAIL_ADDRESS>33institutetext: Institute of Astronomy and NAO,
Bulgarian Academy of Sciences, 72 Tsarigradsko shose, 1784 Sofia, Bulgaria
44institutetext: Geneva Observatory, University of Geneva, 51 Chemin des
Maillettes, 1290 Versoix, Switzerland 55institutetext: Department of Physics,
Royal Military College of Canada, PO Box 17000, Station ’Forces’, Kingston,
Ontario, Canada K7K 4B4

#  Pollux: A weak dynamo-driven dipolar magnetic field and implications for
its probable planet††thanks: Based on observations obtained at the the
Télescope Bernard Lyot (TBL) at Observatoire du Pic du Midi, CNRS/INSU and
Université de Toulouse, France, and the Canada-France-Hawaii Telescope (CFHT),
which is operated by the National Research Council of Canada, CNRS/INSU and
the University of Hawaii.

M. Aurière 11 P. Petit 22 P. Mathias 11 R. Konstantinova-Antova 33 C.
Charbonnel 4422 J.-F. Donati 22
O. Espagnet 11 C.P. Folsom 22 T. Roudier 22 G.A. Wade 55

(Received ??; accepted ??)

###### Abstract

Context. Pollux is considered as an archetype of a giant star hosting a planet
since its radial velocity (RV) presents very stable sinusoidal variations with
a period of about 590 d. We then discovered a weak magnetic field at its
surface using spectropolarimetry, questioning the planetary hypothesis.

Aims. We followed up our investigations on Pollux to characterize its magnetic
field and to infer the effects of magnetic activity on the RV variations.

Methods. We first used ESPaDOnS at CFHT and then Narval at TBL to obtain
Stokes $I$ and Stokes $V$ spectra of Pollux to study their variations for a
duration of 4.25 years, that is, for more than two periods of the RV
variations. We used the least-squares deconvolution (LSD) profiles to measure
the longitudinal magnetic field and to perform a Zeeman Doppler imaging (ZDI)
investigation.

Results.  The longitudinal magnetic field of Pollux is found to vary with a
sinusoidal behavior and a period similar to that of the RV variations. From
the ZDI investigation a rotation period of Pollux is determined to be equal to
660$\pm$15 days and possibly different than the period of variations of the
RV. As to the magnetic topology, the poloidal component is dominant and almost
purely dipolar with an inclination of 10.5∘ of the dipole with respect to the
rotation axis. The mean strength of the surface magnetic field is 0.44 G.
Pollux is found approximately as active as the Sun observed as a star and this
activity could induce moderate RV variations.

Conclusions. As to the origin of the magnetic field of Pollux, we favor the
hypothesis that it is maintained through contemporaneous dynamo action. Pollux
appears as the representative of a class of slowly rotating and weakly
magnetic G-K red giants. To explain the sinusoidal RV variations of Pollux,
two scenarios are proposed. If the RV period is different from the rotation
period, the observed periodic RV variations are due to the hosted planet and
the contribution of Pollux magnetic activity is not significantly detected. In
the peculiar case in which the two periods are equal, we cannot discard the
possibility that the activity of Pollux could explain the total RV variations
and that the planet hypothesis would appear unnecessary. In any case magnetic
activity could contribute significantly to RV variations in some intermediate
mass G-K red giants hosting planets, particularly those with small amplitude
RV variations.

###### Key Words.:

stars: individual: Pollux – stars: magnetic field – stars: late giant – stars:
rotation – planetary system

##  1 Introduction

Pollux ($\beta$ Geminorum, HD 62509) is a well-studied K0III giant neighbor of
the Sun (e.g., Gray, 2014). It is considered the archetype of a giant star
hosting a planet since it presents periodic sinusoidal radial velocity (RV)
variations of about 590 d period and 40 m s-1 semi-amplitude, which have been
stable for more than 25 years (Hatzes et al., 2006). More recently, Aurière et
al. (2009) discovered a weak magnetic field at the surface of Pollux, which
was the first detection of a class of weakly magnetic G-K giants (Aurière et
al. 2015, hereafter AKC; Konstantinova-Antova et al. 2014). This magnetic
field appeared significantly variable, and the magnetic variations could be
correlated with the RV (Aurière et al., 2009). Following this first result, it
is important to determine if the magnetic variations are periodic, possibly
resulting from rotation and if the RV variations could be impacted by, or even
reflect, this weak activity. Not only can stellar activity induce jitter,
which complicates planet detection, but the planetary hypothesis has been
discarded in some cases of red giant hosts (e.g., Hatzes et al., 2018; Delgado
Mena et al., 2018; Reichert et al., 2019). We therefore performed a Zeeman
long-term monitoring of Pollux to characterize its magnetic field and to
investigate its contribution to the RV measurements. We collected
spectropolarimetric data of Pollux using Narval at TBL and ESPaDOnS at CFHT
for a duration of 4.25 years, that is, for more than two periods of the RV
variations. The results of our investigation were briefly presented by Aurière
et al. (2014) and by AKC. In this work, we report the full data, updated
investigation, and conclusions, which differ in part from the preliminary
findings.

##  2 Spectropolarimetric survey of Pollux with Narval and ESPaDOnS

We observed Pollux on 41 epochs and acquired 265 Stokes V series, first using
ESPaDOnS (Donati et al., 2006a) at CFHT in a snapshot program and then its
twin Narval at the TBL in a long-term monitoring program, from 2007 September
to 2012 February, as described below. All data obtained as part of this
project are available through the PolarBase database (Petit et al. 2014). Each
instrument consists of a Cassegrain polarimetric module connected by optical
fibers to an echelle spectrometer. In polarimetric mode, the instrument
simultaneously acquires two orthogonally polarized spectra covering the
spectral range from 370 nm to 1000 nm in a single exposure, with a resolving
power of about 65000. The observational properties of the instruments, as well
as reduction procedures, are the same as described by Aurière et al. (2009)
and AKC.

A standard circular polarization observation consists of a series of four sub-
exposures between which the half-wave retarders (Fresnel rhombs) are rotated
in order to exchange the paths of the orthogonally polarized beams within the
whole instrument (and therefore the positions of the two spectra on the CCD),
thereby reducing spurious polarization signatures. The extraction of the
spectra, including wavelength calibration, correction to the heliocentric
frame, and continuum normalization, was performed using Libre-ESpRIT (Donati
et al. 1997), which is a dedicated and automatic reduction package installed
at CFHT and at TBL. The extracted spectra consist of the normalized Stokes $I$
($I/I_{\rm c}$) and Stokes $V$ ($V/I_{\rm c}$) parameters as a function of
wavelength, along with their associated Stokes $V$ uncertainty $\sigma_{V}$
(where $I_{\rm c}$ represents the continuum intensity). Also included in the
output are the ”null polarization” spectra $N$, which are in principle
featureless, and therefore serve to diagnose the presence of spurious
contributions to the Stokes $V$ spectrum. Each spectrum used in this work is
of good quality with a peak signal-to-noise ratio (S/N) in Stokes $I$ per 2.6
km s-1 spectral bin larger than 1000.

To obtain a high-precision diagnosis of the spectral line circular
polarization, least-squares deconvolution (LSD; Donati et al. 1997) was
applied to each reduced Stokes $I$ and $V$ spectra. We used a solar abundance
line mask calculated from ATLAS9 models (Kurucz 1993), for an effective
temperature of 5000 K, $\log g=3.0$, and a microturbulence of 2.0 km s-1,
which is consistent with the physical parameters of Pollux (Hekker and
Meléndez 2007, Takeda et al. 2008). The mean photospheric profile has an
effective Landé factor of 1.21 and is centered at 564 nm. To increase the
precision of our measurements, the data were time-averaged each night and, in
four cases, for successive nights; each group contained from 4 to 20 Stokes
$V$ (and thus also Stokes $I$) series. These data are presented in Table 1,
which provides the log of observations, with the dates, the number of averaged
Stokes $V$ (also Stokes $I$) series, the mean S/N for Stokes $V$ profiles, and
the mean Heliocentric Julian Date (HJD) of the binned measurement.

From these mean Stokes profiles we computed the surface-averaged longitudinal
magnetic field $B_{\ell}$ in gauss, using the first-order moment method (Rees
& Semel 1979), adapted to LSD profiles (Donati et al. 1997, Wade et al. 2000).
These measurements of $B_{\ell}$ are presented in Table 1 with their 1$\sigma$
error, in gauss, as well as the $B_{\ell,N}$ values computed from the
corresponding null polarization spectrum $N$.

We computed the $S$-index (defined from the Mount Wilson survey; Duncan et al.
1991) for the chromospheric Ca ii H&K cores to monitor the spectral line
activity indicators. These computations are described in detail by AKC. We
also computed the heliocentric RV of Pollux from the averaged LSD Stokes $I$
profiles using a Gaussian fit. The RV stability of ESPaDOnS and Narval is
about 20-30 m s-1 (Moutou et al. 2007; AKC). The averaged value for each date
of $S$-index and RV are presented in Table 1. The data up to 18 March 2009
were already presented by Aurière et al. (2009). All the new observations were
made with Narval. The $S$-index measurements were updated to use the procedure
described by AKC and those of $B_{\ell}$ and RV to use the method of Mathias
et al. (2018).

Table 1: Log of observations of Pollux

Date | Number | S/N | HJD | $B_{\ell}$ | $\sigma$ | $B_{\ell,N}$ |  $S$-index | RV
---|---|---|---|---|---|---|---|---
| Stokes $V$& $I$ | Stokes $V$ | (245 0000+) | (G) | (G) | (G) | Ca ii | (km s-1)
29 Sep-02 Oct07 | 4 | 45794 | 4375.65 | -0.72 | 0.26 | 0.43 | 0.123 | 3.646
12 Dec-18 Dec07 | 6 | 67831 | 4450.72 | -0.54 | 0.14 | -0.19 | 0.118 | 3.467*
31 Dec 07 | 5 | 54334 | 4466.99 | -0.57 | 0.20 | 0.05 | 0.120 | 3.536
05 Apr 08 | 4 | 76954 | 4562.43 | -0.15 | 0.15 | -0.31 | 0.119 | 3.529
15 Apr 08 | 8 | 36901 | 4572.42 | -0.11 | 0.23 | 0.09 | 0.120 | 3.506
16 Sep 08 | 8 | 49678 | 4726.72 | -0.45 | 0.13 | 0.11 | 0.118 | 3.548
20-21 Sep08 | 8 | 60062 | 4731.20 | -0.16 | 0.16 | 0.32 | 0.119 | 3.572
30 Sep 08 | 8 | 76737 | 4740.72 | -0.31 | 0.11 | 0.01 | 0.117 | 3.581
21 Dec 08 | 8 | 57649 | 4822.62 | -0.32 | 0.15 | -0.18 | 0.118 | 3.615
25 Feb 09 | 8 | 74324 | 4888.39 | -0.63 | 0.11 | 0.08 | 0.118 | 3.629
12 Mar 09 | 8 | 61004 | 4903.43 | -0.71 | 0.14 | -0.03 | 0.116 | 3.643
18 Mar 09 | 15 | 62958 | 4909.42 | -0.57 | 0.10 | -0.23 | 0.119 | 3.435*
23 Sep 09 | 8 | 61466 | 5098.62 | -0.71 | 0.14 | 0.06 | 0.118 | 3.536
25-26 Oct09 | 12 | 55612 | 5131.21 | -0.49 | 0.12 | -0.23 | 0.117 | 3.538
24 Nov 09 | 8 | 58795 | 5160.75 | -0.66 | 0.14 | 0.10 | 0.119 | 3.512
15 Dec 09 | 8 | 67564 | 5181.76 | -0.31 | 0.12 | -0.11 | 0.118 | 3.476
12 Mar 10 | 8 | 54953 | 5268.42 | -0.23 | 0.15 | 0.09 | 0.118 | 3.626
10 Apr 10 | 8 | 66035 | 5297.33 | -0.24 | 0.13 | -0.03 | 0.118 | 3.613
18 Oct 10 | 8 | 65555 | 5488.73 | -0.46 | 0.13 | 0.18 | 0.117 | 3.600
18 Nov 10 | 13 | 77848 | 5519.70 | -0.51 | 0.08 | -0.02 | 0.116 | 3.620
30 Nov 10 | 20 | 27310 | 5531.73 | -0.60 | 0.16 | 0.13 | 0.117 | 3.532*
18 Dec 10 | 8 | 52935 | 5549.64 | -0.48 | 0.16 | 0.16 | 0.117 | 3.645
03 Jan 11 | 10 | 62144 | 5565.58 | -0.58 | 0.12 | -0.02 | 0.119 | 3.605
19 Jan 11 | 8 | 45256 | 5581.47 | -0.67 | 0.19 | -0.11 | 0.117 | 3.434*
18 Mar 11 | 8 | 52109 | 5639.40 | -0.25 | 0.16 | -0.14 | 0.116 | 3.594
25 Sep 11 | 8 | 61396 | 5830.70 | -0.54 | 0.14 | 0.04 | 0.113 | 3.513
16 Oct 11 | 8 | 61620 | 5851.71 | -0.43 | 0.14 | -0.20 | 0.118 | 3.576
23 Nov 11 | 8 | 47510 | 5889.76 | -0.71 | 0.18 | -0.22 | 0.120 | 3.510
10 Dec 11 | 8 | 62981 | 5906.61 | -0.41 | 0.13 | -0.17 | 0.118 | 3.535
07 Jan 12 | 8 | 67918 | 5934.61 | -0.35 | 0.12 | -0.07 | 0.119 | 3.530
08 Feb 12 | 8 | 52775 | 5966.54 | -0.33 | 0.16 | -013 | 0.120 | 3.559

111Individual columns report date of observation, number of averaged Stokes
$V$ & $I$ series, mean S/N for Stokes $V$ profiles, mean HJD, $B_{\ell}$ and
its error in gauss, $B_{\ell,N}$ corresponding to the null polarization
spectrum $N$, the $S$-index for Ca ii H&K, and the heliocentric RV measured
from the LSD Stokes $I$ profile. For details, see Sect. 2. An asterisk
indicates that the measurement is not taken into account in this work.

Figure 1: Variations of the heliocentric RV (upper plot), the $B_{\ell}$
(middle plot), and the $S$-index (lower plot) of Pollux with HJD. Figure 2:
Radial velocity (upper plot) and unsigned $B_{\ell}$ (lower plot) of Pollux
phased with the 589.64 d and 660 d periods, respectively. The mean value of RV
and $|B_{\ell}|$ are shown. Two periods are plotted for clarity.

##  3 Variations of the RV, of the longitudinal magnetic field $B_{l}$, and
the $S$-index of Pollux

Extensive studies of the RV of Pollux have been performed following the
discovery of a RV variation with a total amplitude of about 100 m s-1 (Walker
et al. 1989) and a period of about 580 d (Larson et al. 1993, Hatzes & Cochran
1993). To interpret these variations, the planetary companion hypothesis is
now generally accepted (Hatzes et al. 2006, Reffert et al. 2006, Han et al.
2008). These three previous investigations give periods for the RV variations
near 590 d. Table 2 shows the period, its uncertainty, and the span time of
the observations for each reference. Baklanova et al. (2011) used all the RV
from these studies to derive a new period of 592.9$\pm$0.6 d and also provided
a few $B_{\ell}$ measurements. We used 589.64 d as the RV period of Pollux
(Hatzes et al. 2006) since this period comes from the longest span of
observations with homogeneously reduced data leading to the most accurate
period determination.

Figure 1 (upper plot) shows the variations of RV with HJD. The error bars are
fixed to 30 m s-1 (see Sect. 2). For four dates, the RV values were outliers
from the sine trend (being generally significantly smaller) and were
discarded. These values are denoted by an asterisk in Table 1 and they are not
presented in the plot. Figure 1 (middle plot) shows the variations of
$B_{\ell}$ with HJD (only our homogeneous $B_{\ell}$ measurements were used).
The error bars are those presented in Table 1. The lower plot of Fig. 1
presents the variations of the $S$-index. Error bars of 0.002, corresponding
to the largest deviation from the mean (apart for two cases), are shown.

Figure 1 clearly confirms the trend shown by Aurière et al. (2009), in which
$B_{\ell}$ appears to mimic the variations of RV. Compared to the observed
variations, the error bars on $B_{\ell}$ are however larger than for RV,
leading to a noisier plot. On the other hand, the $S$-index (lower plot) does
not show any clear trend, remaining at a very low level; this behavior is
typical of a very weakly active star as described by AKC.

Our Zeeman Doppler imaging (ZDI) study of Pollux using the Stokes $V$ data,
presented in Sect. 4, determines a rotation period $P_{\rm rot}$ of 660$\pm$15
d (statistical error), which is different from the 589.64 d period of the RV
variations at the 4$\sigma$ level. However we do not exclude that the two
periods could be equal. This possibility is discussed in Sect. 4 and Sect. 6.

We carried out a classical Fourier analysis of the $B_{\ell}$ and RV data and
found possible periods of 635 d and 661 d, but the errors are about 40 d and
27 d, respectively, because of our limited sample and poor relative precision
on $B_{\ell}$ and RV. For example, our relative error is about 30 m s-1 on RV
measurements, when it is a few m s-1 for the works referenced in Table 2.
These periods are therefore compatible with the two other periods (589.64 d
and 660 d). We consider in this work that the $P_{\rm rot}$ of 660 d is
responsible for the $B_{\ell}$ variations. The similarity of the curves of
variations in RV and $B_{\ell}$ is due to either chance if the two periods are
really different, or to a true physical relationship if the two periods are
equal.

Figure 2 then shows the RV (upper plot) and the $B_{\ell}$ (lower plot)
measurements of Pollux phased with the 589.64 d and 660 d periods,
respectively. Two periods are shown for clarity. Each sinusoid is fitted with
the least-squares method on each data set. The periods are fixed, while the
semi-amplitude, the mean value, and the phase origin (corresponding to the
mean value of RV and $B_{\ell}$) are the only free parameters. This gives
amplitudes of variations of 53.8 m s-1 and 0.19 G, respectively. For these
sine fits, the frequencies corresponding to the two periods account for 58$\%$
and 41$\%$ of the fraction of the variance for RV and $B_{\ell}$,
respectively. The difference between phase origins in RV and $B_{\ell}$ is
about 92 days; if this is due to the difference in periods this value should
vary with time, increasing by about this difference after each rotation of
Pollux or revolution of the planet. Because of the large errors on $B_{\ell}$
and RV, as described above, we were unable to confirm a possible variation of
this difference between phase origins during the span of our observations.

Table 2: Observed periods on Pollux and parameters for a hosted planet

Reference | Period | Error | Span | $m_{2}\sin i$ | $a$
---|---|---|---|---|---
| (Day) | (Day) | (Year) | ($M_{\rm Jup}$) | (AU)
Hatzes 2006 | 589.64 | 0.81 | 25.1 | 2.3 | 1.64
Reffert 2006 | 589.7 | 3.5 | 3.5 | 2.7 | 1.69
Han 2008 | 596.6 | 2.26 | 4.5 | 2.47 | 1.66
ZDI | 660 | 15 | 4.25 |  |
$B_{\ell}$ | 635 | 40 | 4.25 |  |
RV | 661 | 27 | 4.25 |  |
222Numbers from the 3 first references are used in Sect. 3 and Sect. 6. The
quantity $m_{2}\sin i$ is given for a mass of Pollux of 1.7 $M_{\odot}$ as
used in these 3 references. The ZDI work is described in Sect. 4. The
$B_{\ell}$ and RV work is presented in Sect. 3.

##  4 Zeeman Doppler imaging of Pollux

To fully exploit the Zeeman information included in our Stokes V data, we
employed the ZDI method, using the new implementation of Folsom et al. (2018).
This Python version implements the same physical model and inversion method as
the code detailed in Donati et al. (2006b). The two ZDI codes were extensively
tested to ensure they produced identical results for identical input
parameters. The ZDI method has already been successfully used to get magnetic
field distribution at the surface of five red giant stars, including both fast
rotators (V390 Aur; Konstantinova-Antova et al. 2012; 37 Com, Tsvetkova et al.
2017) and Ap-star descendant candidates (EK Eri; Aurière et al. 2011; $β$ Cet;
Tsvetkova et al. 2013; OU And; Borisova et al. 2016). The peculiarity of
Pollux with respect to these five red giants is that its rotation period is
expected to be longer and its magnetic field is much weaker giving very weak
Stokes $V$ signals. However the ZDI method gave sound results in the case of
EK Eri, which has $P_{\rm rot}$ = 309 d, and in the case of Vega, which has a
1 G magnetic field. Vega is even more difficult to detect since fewer spectral
lines are available (Petit et al. 2010).

###  4.1 Rotation period determination

Using ZDI we inferred the rotation period $P_{\rm rot}$ and the magnetic
topology of Pollux. We limited the number of spherical harmonics to $l<10$
since increasing the limit did not significantly change the results; almost
all the magnetic field is included in the spherical harmonics modes with
$l<3$. We followed the approach of Petit et al (2002) for determining the
$P_{\rm rot}$ of Pollux, and Fig. 3 shows the obtained periodogram. The
deepest minimum at 660 d is interpreted as corresponding to the $P_{\rm rot}$
of Pollux, with a statistical error of 15 d. Our investigation does not detect
either solar or antisolar differential rotation. One weak minimum at 365 d is
interpreted as an alias owing to the seasonal sampling. The 660 d period is
longer than the period of the RV variations (589.64 d, Hatzes et al. 2006) and
than the inferred $P_{\rm rot}=587.2$ d provided in Aurière et al. (2014) from
ZDI. The likely erroneous $P_{\rm rot}$ presented in the preliminary work by
Aurière et al. (2014) is due to an initially insufficient number of digits in
the coding of the rotational phase, in comparison to the very small changes of
phase between consecutive spectra of a nightly series.

In addition to statistical errors, the 660 d period found by ZDI can be
affected by systematic errors. In the case of Pollux we might be concerned by
changes in the magnetic configuration, since observations span over several
years and more than two rotations. Differential rotation might also cause
concern, although it is not detected by the ZDI process. Solar-like
differential rotation might provide the longer period obtained from ZDI with
Stokes $V$ data mainly sensitive to polar regions where the large-scale field
is strongest as presented by our magnetic map (Sect. 4.2 just below), whereas
RV data are mostly sensitive to equatorial regions. The difference between 660
d and 590 d would correspond to about 3% of the solar differential rotation or
to a small change in latitude. A total error of a few tens of days on the
$P_{\rm rot}$ would not change the conclusions of our magnetic study. On the
other hand, the possibility of equality between the rotation period and the RV
period (590 d) raises the hypothesis that the magnetic field could be the
origin of RV variations. In this case, the magnetic activity has to be stable
during the 25-year span of the Hatzes et al. (2006) RV observations. During
the two rotations, to check the stability of our magnetic model presented in
Sect. 4.2, we divided the observations in two subsets that cover about one
rotation each and we compared the two maps and models obtained with ZDI. The
uncertainties on each parameter determination are increased and we do not find
any significant change between the two models nor with the parameters obtained
for the full data. In particular we get the $P_{\rm rot}$ values of 649 d and
643 d, lower but consistent with the 660 d value and its statistical error of
15 d.

Finally, our ZDI work gives a value of about 660 d for the $P_{\rm rot}$ of
Pollux, but because of a statistical error of 15 d and possible systematic
errors including differential rotation effects, we cannot exclude that it
could be equal to the 590 d period of the RV variations.

###  4.2 Magnetic model

With the new version of the ZDI code, $l<10$ and $P_{\rm rot}$=660 d, our
preferred ZDI model is as follows. The magnetic field is found to be very
axisymmetric (95% of the total field). The inclination angle $i$ optimizing
the ZDI model is equal to 50∘. The poloidal and toroidal components contain
53% and 47%, respectively, of the reconstructed magnetic energy. About 95% of
the poloidal magnetic energy corresponds to the dipole component (2.3% and
1.7% corresponding to the quadrupole and octopole components, respectively).
Figure 4 presents the ZDI map corresponding to our model. Figure 5 compares
the observed Stokes $V$ and Stokes $I$ LSD profiles to the synthetic averaged
profiles of our magnetic model.

This fraction of toroidal field energy may appear high for a weakly magnetic
star (e.g., See et al. 2015 in the case of main-sequence stars). This toroidal
component could be due to a redshift of about 570 m s-1 between Stokes $V$ and
Stokes $I$ profiles, which the ZDI algorithm interprets as a ring of azimuthal
field. This redshift is illustrated in Fig. 6, which compares the derivative
of the averaged Stokes $I$ profile and the averaged Stokes $V$ profile. The
resulting sub-gauss toroidal component would be negligible for more active
stars, where surface large-scale magnetic fields can reach tens to hundreds of
gauss. In the case of Pollux, however, the consequence is an apparent balance
between the poloidal and toroidal field. This redshift of Stokes $V$ with
respect to Stokes $I$ could suggest a link of magnetic features with
convective downflows, as observed in Betelgeuse (Mathias et al. 2018). An
alternate ZDI reconstruction, where the redshift of Stokes V has been
corrected for, leads to a magnetic map shown in Fig. 7 that is vastly
dominated by the poloidal field component. The poloidal field contains 99.95 %
of the reconstructed magnetic energy and is dipolar for about 97%. The angle
$\beta$ between the rotation axis and the magnetic dipole axis is 10.5∘. The
mean surface magnetic field $B_{\rm mean}$ is 0.44 G, the radial dipole
strength is 0.8 G and its longitude corresponds to phase 0.4 in Fig. 7. The
maximum of $|B_{\ell}|$ of the fitted sinusoid shown in Fig. 2 occurs at
rotational phase 0.65 in Fig. 7.

Taking into account the $P_{\rm rot}$ of 660 d, $i$=50∘, a radius of 8.8
$R_{\odot}$ (Nordgren et al. 2001 and Hipparcos parallax) and within the solid
rotation hypothesis, the $v\sin i$ of Pollux is 0.52 km s-1. This value for
$v\sin i$ is significantly smaller than those presented in the literature
(e.g., Gray 2014, $v\sin i=1.70\pm 0.2$km s-1). This very small $v\sin i$ is
difficult to measure, as shown in the investigation by Aurière et al. (2009),
and it would require very high spectral resolution observations to be
detected. Using $v\sin i=1.7$ km s-1 (Gray 2014) would lead to a $P_{\rm rot}$
of about 200 d. As shown in Fig. 3, this period would give a ZDI solution with
a significantly worse reduced ${\chi}^{2}$.

Figure 3: Periodogram of Pollux obtained using the Stokes $V$ spectra and the
method of Petit et al. (2002) based on ZDI. Figure 4: Zeeman Doppler image of
Pollux. Each chart illustrates the field projection onto one axis of the
spherical coordinate frame. The magnetic field strength is expressed in gauss.
Ticks along the top chart indicate the observed rotational phases.

Figure 5: Comparison between the observed Stokes $V$ (upper plot) and Stokes I
(lower plot) LSD profiles and the synthetic averaged profiles of our magnetic
model. The X-axis is the RV and the ordinates are ($V/I_{\rm c}$) and
($I/I_{\rm c}$), respectively. The observations are illustrated in gray
(individual profiles) and in black (averaged profiles), while the synthetic
profiles are shown in red.  Figure 6: Derivative of the averaged Stokes $I$
profile (red) compared to the averaged Stokes $V$ profile (black). The Y-axis
is scaled as $V/I_{\rm c}$. Figure 7: Zeeman Doppler image of Pollux with the
Stokes $V$ profiles shifted by 570 m s-1 as explained in Sect. 4.2. Each chart
illustrates the field projection onto one axis of the spherical coordinate
frame. The magnetic field strength is expressed in gauss. Ticks along the top
chart indicate the observed rotational phases.

##  5 Origin of the magnetic field of Pollux

Pollux is the first red giant with a sub-gauss mean surface magnetic field for
which ZDI is performed and $P_{\rm rot}$ is determined. This $P_{\rm rot}$ is
also the longest determined up to now for a red giant (AKC). The high Rossby
number $\rm Ro$ of 2 inferred for Pollux using this $P_{\rm rot}$ and the
convective turnover time $\tau_{\rm conv}$ = 330 d derived using the
evolutionary models with rotation of Lagarde et al. (2012) and Charbonnel et
al. (2017) is the highest for the red giants with measured $P_{\rm rot}$ in
the sample of AKC. The magnetic field origin of the other giants than Pollux
was suggested to be an $\alpha$ $\Omega$ dynamo or to descend from magnetic Ap
stars (AKC).

###  5.1 Ap star descendant hypothesis

The high $\rm Ro$, the dominant dipolar component of the reconstructed
magnetic energy, and the stability of the magnetic field during two rotations
would suggest that Pollux is an Ap star descendant. However using 1 G for the
dipole strength and taking into account the magnetic flux conservation
hypothesis, we find that an Ap star progenitor of Pollux (taking a mass of 2.5
$M_{\odot}$, AKC) would have a magnetic strength of about 20 G, that is, far
below the 100-300 G lower limit found for an Ap star dipole strength by
Aurière et al. (2007). Unlike the present Ap star descendant candidates,
Pollux does not appear as a stronger magnetic star with respect to other red
giants of its class; the Ap star descendant candidates for which ZDI
investigations have been done are listed Sect. 4. Therefore we consider that
it is very unlikely that Pollux would be the descendant of a weak (and rare)
magnetic Ap star.

###  5.2 Dynamo operation hypothesis

While it falls below the overall trends, Pollux is not a strong outlier in the
plots presented by AKC showing the correlation of the magnetic field strength
with $P_{\rm rot}$, $\rm Ro,$ and $S$-index for red giants harboring an
$\alpha$ $\Omega$ dynamo-driven magnetic field.

The weak magnetic activity of Pollux compares well with that of the Sun
observed as a star. The solar mean magnetic field (SMMF) representing the
disk-averaged line-of-sight magnetic field of the Sun has a strength varying
between 0.15 and 1 G with respect to the solar cycle (Garcia et al. 1999),
when the $B_{\rm mean}$ of Pollux is about 0.4 G (Sect. 4) and $|B_{\ell}|$
varies in the range 0.1-0.7 G (Table 1). The $S$-index of the global Sun
varies during the solar cycle mainly in the range about 0.17-0.19 (Meunier
2018), that is, a little larger than that corresponding to the basal
chromospheric flux. The $S$-index of Pollux is stable at about 0.118 and very
close to the value corresponding to the basal chromospheric flux of red giants
(Schröder et al. 2012). The average level X-ray luminosity of the Sun (Judge
et al. 2003) is about that of Pollux (about $10^{27}$ erg s-1; Schröder et al.
1998). This means that the magnetic activity of Pollux is similar to that of
the Sun observed as a star, in which an $\alpha$ $\Omega$ dynamo is known to
be involved.

However the stellar parameters of Pollux are very different from those of the
Sun; the $P_{\rm rot}$ is 20 times longer, the mass 2 times larger, and the
radius is 9 times larger. An extreme case of a dynamo when rotation is not
involved is a local dynamo as inferred for the red supergiant Betelgeuse
(Aurière et al. 2010). In the case of Betelgeuse, the magnetic field is
observed to vary intrinsically on several timescales from weeks to months
(Mathias et al. 2018) when we observe a 660 d period, semi-sinusoidal
variation for Pollux. Pollux is in mass and radius, intermediate between the
Sun and Betelgeuse. Red giants and supergiants are expected to have giant
convective cells with respect to the Sun (Schwarzschild 1975). The convective
cells of Betelgeuse have been imaged by spectropolarimetry and both their
upflow and downflow speeds are derived (Mathias et al. 2018, Lòpez Ariste et
al. 2018). The upflows are traced by the bright center of convective cells and
linear polarization, while the downflows of the same convective cells
concentrate the observed magnetic field in circular polarization. Weak
magnetic fields are detected, which are concentrated in the downflow lanes in
between granules, similar to the quiet Sun magnetism. This suggests that this
process, occurring in the quiet Sun and in Betelgeuse, is at work in Pollux in
addition to a global component.

Three-dimensional nonlinear magnetohydrodynamic (MHD) simulations were
performed with the ASH code for a star representative of Pollux with
$\Omega_{*}=\Omega_{\odot}/20$ (Palacios & Brun 2014, Brun & Palacios 2015).
This study shows that this kind of slowly rotating red giant star is likely to
possess global magnetic field maintained through contemporaneous dynamo
action. The strength of the large-scale magnetic components is found to be on
the order of a few gauss. These simulations are consistent with the results of
this work and suggest that Pollux is not a unique case of a weakly magnetic
slowly rotating red giant.

A few other red giants have been detected with magnetic field strength at the
sub-gauss or gauss level (AKC, Konstantinova-Antova et al. 2014). These stars
are very similar in other respects to ordinary giants, with a small $S$-index,
near the value corresponding to the basal chromospheric flux.

Pollux is thus a representative of a class of slowly rotating and weakly
magnetic red giants. Among these objects Pollux presents a stability of its
magnetic configuration during two rotations. As to the origin of this magnetic
field, we favor the case of contemporaneous dynamo action in which rotation
plays a role. Since Pollux falls below the overall trends in the plots
presented in AKC for red giants harboring an $\alpha$ $\Omega$ dynamo-driven
magnetic field the dynamo regime could be weaker; for example, turbulent
dynamo as has been suggested to occur in asymptotic branch red giants (Soker &
Zoabi 2002), or a universal turbulence-related dynamo mecanism explaining
magnetic activity levels of both main sequence and evolved stars (Lehtinen et
al. 2020).

##  6 Origin of the radial velocity variations of Pollux

The variations of the RV of Pollux have been unanimously considered to be the
result of a hosted planet (e.g., Hatzes et al. 2006, Reffert et al. 2006, Han
et al. 2008). Table 2 shows that the planet hosted by Pollux inferred by these
authors has a mass higher than about 2.5 $M_{\rm Jup}$ (for a mass of Pollux
of 1.7 $M_{\odot}$ used in these three references) and an orbital semimajor
axis of about $a=1.64$ AU. We now have to consider how the weak magnetic field
and the associated stellar activity could contribute to the RV variations of
Pollux.

###  6.1 Surface magnetic field of Pollux: Incidence on RV measurements

The effect of magnetic activity on RV measurements of stars has been
investigated for a long time in the case of stellar spots or other magnetic
features (e.g., Hatzes 2002, Hébrard et al. 2014). However these studies
involved magnetic fields that were generally stronger than a few tens or
hundreds of G. In the context of the weak magnetic field of Pollux, since it
is comparable to that of the Sun observed as a star (see Sect. 5.2), it is
relevant to refer to the solar studies made in the scope of disentangling weak
activity effects and orbital effects on RV variations in the case of stars
hosting planets (e.g., Meunier et al. 2010a,b, Haywood et al. 2016, Collier
Cameron et al. 2019). These studies of the Sun observed as a star are based
both on simulations and observations of the Sun from space or the ground. From
these studies, several learnings can enlighten Pollux’s case.

1) Space observations using SDO/HMI and SDO/AIA (e.g., Bose and Nagaraju 2018)
or using MDI/SOHO (e.g., Meunier et al. 2010b) enabled us to disentangle
contributions from different magnetic features to the global solar properties.
Efficient magnetic activity contributions to the RV variations observed on the
global Sun are through attenuation of the convective blueshift (CB) and by
actual downflows (as linked with supergranules in the Sun or with the large
convective cells in Betelgeuse).

2) In the Sun as a star, variations of up to 10 m s-1 are observed along the
magnetic cycle (e.g., Meunier et al. 2010b). Variations of about 5 m s-1 are
linked to the transit of active regions (Haywood et al. 2016, Collier Cameron
et al. 2019).

The solar example presented above shows that a main-sequence star with an
activity level similar to that of Pollux can induce RV variations of a few m
s-1 to 10 m s-1, when including both rotation and magnetic cycle effects. The
process at work in Pollux should not be due to transit of strongly magnetic
structures such as solar spots or plages since no modulation near 600 days is
observed on magnetic indicators apart from RV and Stokes $V$ data (Hatzes et
al. 2006, Gray 2014, this work). Pollux is more massive and more evolved than
the Sun, but we noted in Sect. 5.2 that some similar processes exist
concerning weak magnetic fields between the Sun and the supergiant Betelgeuse,
which could also occur in Pollux. This includes actual downflows, which could
be linked to a redshift of Stokes $V$ profiles with respect to Stokes $I$,
revealed by our ZDI investigation in Sect. 4.2.

Investigating the effect of the weak surface magnetic field of a red giant as
Pollux on RV measurements is out of the scope of this work. The differences of
atmospheric conditions at the surface of Pollux with respect to the Sun, for
example lower gravity, are significant. The equipartition magnetic field is
going to be smaller in Pollux and a 1 G surface magnetic field is likely to
have a much larger dynamical impact on the convection than in the Sun. We then
consider that the surface magnetic field of Pollux might contribute
significantly to the sinusoidal RV variations of 40 m s-1 semi-amplitude.

###  6.2 Scenarios explaining the radial velocity variations of Pollux

####  6.2.1 Only one significant period detected for the RV variations

A mixing of contributions from one hosted planet and from magnetic activity is
expected to provide the periodic RV variations of Pollux. The observed period
of 589.64 d would be a combination of orbital and rotation periods. In this
context we have to check if the two periods appear in the periodogram of the
RV measurements. Hatzes et al. (2006) made a deep investigation in this
direction using their long timescale sample of Pollux spectra. On their Lomb-
Scargle periodogram of the residual RV measurements after subtracting the
contribution of the orbital motion due to the companion the highest peak
corresponds to a period of 121 days and is not considered as significant.
However since it is near the 135 d inferred for the photometric variations
observed by Hipparcos, Hatzes et al. (2006) suggested that the rotation period
of Pollux would be about 130 days. From our knowledge of the magnetic red
giants (AKC), this value is much too small for Pollux’s $P_{\rm rot}$ since it
would imply an active giant with a $|B_{\ell}|_{\max}$ that is five times
larger than that observed and much larger $S$-index and $L_{X}$ activity
indicators. Therefore we can conclude that only one period gives significant
contribution to the very stable periodic variations of RV, which is of about
590 d. We then consider two scenarios. In each scenario, one effect dominates
to produce the stable RV variations of Pollux.

####  6.2.2 Planet hypothesis

This corresponds to a general case in which the RV period is different from
the $P_{\rm rot}$ and the magnetic activity of the star is weak. The planet is
the main driver of the RV variations with a weak effect of the magnetic
activity of Pollux, which cannot be disentangled from the present
observations.

####  6.2.3 Magnetic activity hypothesis

This is the peculiar case in which the observed period of the RV variations
(about 590 d) is the $P_{\rm rot}$ of Pollux, which cannot be discarded
completely from our ZDI determination ($P_{\rm rot}$ about 660 d, see Sect.
4.1). Sect. 6.1 shows that RV variations of about 10 m s-1 are observed in the
solar case taken as an example of a star with a similar magnetic activity
level as that of Pollux. Then a planet could exist as well to fulfil the total
40 m s-1 semi-amplitude RV variations. The revolution of the planet would be
synchronized with the rotation of the star. Because of the large semimajor
axis and the long rotation period, the expected tidal effects appear very weak
for modifying the rotation period or the orbit (e.g., Pont 2009, Gallet et al.
2017), and the synchronous rotation would have to be explained. However the
atmospheric conditions in Pollux are different from that in the Sun as argued
in Sect. 6.1, and we cannot discard the possibility that the activity of
Pollux could explain the totality of the RV variations and that the planet
hypothesis would appear unnecessary.

##  7 Magnetic field and the RV variations in planet-hosting red giants other
than Pollux

About 112 credible substellar companion candidates orbiting 102 G and K giant
stars have been found so far (Reichert et al. 2019; a link to a catalog is
given in the introduction of the paper). From this catalog, a very well-marked
peak is seen at spectral type K0III, which is that of Pollux, and almost all
stars are between K0 and K2. The study of these systems is of importance since
it would enable us to explore the fate of planetary systems during stellar
evolution, as well as to access to exoplanets around stars with higher mass
than the Sun.

In addition to Pollux, three stars of the above list of credible candidates
for hosting planets, of spectral types between G9III and K1III, have been
observed with Narval or ESPaDOnS (AKC; Konstantinova-Antova et al. 2014) with
a sub-gauss accuracy. Two of these stars, $ι$ Dra and $γ$$_1$ Leo do not show
any detectable Zeeman-signal and have small $S$-indices. This means that they
do not host an observable magnetic field as Pollux does, and their magnetic
activity may not contribute significantly to the observed large RV variations
(amplitude six and four times larger than in Pollux). On the other hand, the
third, $ϵ$ Tau is Zeeman-detected and its magnetic properties are found to be
similar to those of Pollux; the data are still too scarce to enable any
conclusion with respect to eventual correlations to RV (AKC).

Recent studies show that for a number of intermediate-mass red giants, the
planet hypothesis inferred for explaining sinusoidal RV variations should be
abandoned because the RV signal does not remain perfectly stable when a long-
term study is performed (Delgado Mena et al. 2018, for several open cluster
stars; Hatzes et al. 2018, for $γ$ Dra; Reichert et al. 2019, for Aldebaran).
The alternative process (stellar oscillations?) is still unclear but it
appears clear that we should be cautious in interpreting RV variations as due
to companions in giant stars (Hatzes et al. 2015).

Aldebaran, as well as $\gamma$ Dra, are K5III and significantly cooler than
Pollux. Sinusoidal variations of RV were discovered for Aldebaran by Hatzes et
al. (1993) at the same time as for Pollux. Recently Hatzes et al. (2015)
argued that these variations could be explained by a planetary companion of
revolution period of 629 d, with stellar activity in addition. From
observations of activity indicators, these authors suggested that the rotation
period of Aldebaran is 520 d. Then, Reichert et al. (2019) presented a new
study that does not support the planet hypothesis; oscillatory convective
modes might be a plausible alternative explanation of the observed RV
variations.

The AKC work has Zeeman-detected Aldebaran. Aldebaran presents a variable weak
magnetic field with a reversing $B_{\ell}$ and a variable $S$-index
significantly larger than that of Pollux. A correlation between RV variations
and $B_{\ell}$ as well as activity indicators could not be concluded for
Aldebaran from their sparse sample measurements. The results cited above
suggest that RV variations coming from combined effects of a planet and
magnetic activity could occur in the case of Aldebaran.

A substantial fraction of the G-K red giants hosting planets are intermediate-
mass stars and a large percentage of these stars could be weakly magnetic
(AKC; Konstantinova-Antova et al. 2014). Our preliminary investigation
presented above concerns Pollux and three other stars of about the same
spectral type: Pollux and $\epsilon$ Tau present similar magnetic activity and
$\iota$ Dra and $\gamma$1 Leo did not present magnetic activity during our
observations. The magnetic activity would not significantly impact the planet-
driven RV variations in the case of $\iota$ Dra and $\gamma$1 Leo, and new
observations are required to clarify the situation in the case of Pollux and
$\epsilon$ Tau. In other cases, as possibly in Aldebaran, the observed RV
variations could be a combination of orbital and rotational effects. The case
of planet hosting red giant candidates with small amplitude RV variations
particularly need to be checked for a magnetic activity contribution.

##  8 Conclusions

We have monitored the magnetic field of Pollux for a duration of 4.25 years,
that is, more than two times the period of the RV variations. The longitudinal
magnetic field of Pollux is found to vary with a sinusoidal behavior and a
period that is similar to that of the RV variations. Our ZDI investigation
shows that the rotation period of Pollux is equal to 660$\pm$15 d, close to
the RV variation period of 590 d; the equality of the two periods is not
excluded, particularly if we take into account possible systematic errors. As
to the magnetic topology, the poloidal component is dominant and almost purely
dipolar, with an inclination of 10.5∘ of the dipole with respect to the
rotation axis. The mean strength of the surface magnetic field is about 0.4 G.
We suggest that a weak-regime dynamo, as supported by 3-D nonlinear MHD
simulations (Brun & Palacios 2015), is the origin of the magnetic field of
Pollux. In this way Pollux appears as the representative of a detected class
of slowly rotating and weakly magnetic G-K giants (AKC; Konstantinova-Antova
et al. 2014).

Pollux is about as active as the Sun observed as a star and this activity
induces RV variations of a few m s-1 to about 10 m s-1 for the global Sun when
both rotation and magnetic cycles are taken into account. We suggest in Sect.
6. that the magnetic activity of the more massive and more evolved star Pollux
could induce significant RV variations as well, which would be modulated by
the rotation period. Since from the long-term study of Hatzes et al. (2006),
only one period emerges to produce the stable sine shape of RV observations of
Pollux, we propose two scenarios to explain these RV observations. If the RV
period is different from the rotation period of Pollux, the observed periodic
RV variations are due to the hosted planet and the contribution of Pollux
magnetic activity is not significantly detected. In the peculiar case when the
two periods are equal, the weak magnetic field would contribute to the RV
variations and we cannot discard the possibility that the activity of Pollux
could explain the total RV variations and that the planet hypothesis would
appear unnecessary.

In any case, magnetic activity could contribute significantly to RV variations
in some intermediate-mass G-K red giants hosting planets, particularly those
with small amplitude RV variations. Therefore new Zeeman observations of a
number of red giant candidates for hosting planets (including Pollux) would be
desirable for disentangling the contributions of orbital and rotation periods.

###### Acknowledgements.

We thank the TBL and CFHT teams for providing service observing. We
acknowledge financial support from ’Programme National de Physique Stellaire’
(PNPS) of CNRS/INSU, France. JFD acknowledges funding from the European
Research Council under the H2020 research & innovation programme (grants
$\\#$740651 NewWorlds). GAW acknowledges support from the Natural Science and
Engineering Research Council of Canada (NSERC).

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