# Magnetic field evolution of the K2 dwarf V471 Tau

B. Zaire1, J.-F. Donati1, and B. Klein2
1IRAP, Université de Toulouse, CNRS / UMR 5277, CNES, UPS, 14 avenue E. Belin,
Toulouse, F-31400 France
2 Sub-department of Astrophysics, Department of Physics, University of Oxford,
Oxford, OX1 3RH, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Observations of the eclipsing binary system V471 Tau show that the time of the
primary eclipses varies in an apparent periodic way. With growing evidence
that the magnetically active K2 dwarf component might be responsible for
driving the eclipse timing variations (ETVs), it is necessary to monitor the
star throughout the predicted $\sim 35$ yr activity cycle that putatively
fuels the observed ETVs. We contribute to this goal with this paper by
analysing spectropolarimetric data obtained with ESPaDOnS at the Canada-
France-Hawaii Telescope in December 2014 and January 2015. Using Zeeman-
Doppler Imaging, we reconstruct the distribution of brightness inhomogeneities
and large-scale magnetic field at the surface of the K2 dwarf. Compared to
previous tomographic reconstructions of the star carried out with the same
code, we probe a new phase of the ETVs cycle, offering new constraints for
future works exploring whether a magnetic mechanism operating in the K2 dwarf
star is indeed able to induce the observed ETVs of V471 Tau.

###### keywords:

Magnetic fields – stars: magnetic field – stars: imaging – stars: individual:
V471 Tau – binaries: eclipsing – techniques: polarimetric

††pubyear: 2022††pagerange: Magnetic field evolution of the K2 dwarf V471
Tau–B

##  1 Introduction

Several eclipsing binary systems display periodic eclipse timing variations
(ETVs) when considering a linear ephemeris to predict the time of mid-eclipse
(Lanza et al., 1998; Lanza & Rodonò, 1999; Zorotovic & Schreiber, 2013; Bours
et al., 2016). It is estimated that around 90 per cent of the post-common-
envelope binary (PCEB) systems display ETVs (Zorotovic & Schreiber, 2013). The
main explanations that have been proposed to account for the existence of ETVs
are associated with the presence of circumbinary bodies perturbing the orbit
of the system (Irwin, 1952) or magnetically-induced gravitational modulations
caused by an active star in the system (Applegate & Patterson, 1987;
Applegate, 1992; Lanza et al., 1998; Lanza, 2005, 2006, 2020; Völschow et al.,
2016, 2018). In most cases, ETVs are attributed to circumbinary planet/sub-
stellar components that, given their mass and orbital distance, can explain
the periodicity and amplitude of ETVs (Parsons et al., 2010; Rappaport et al.,
2013; Conroy et al., 2014; Marsh et al., 2014; Hajdu et al., 2019; Marcadon et
al., 2020; Papageorgiou et al., 2021). However, recent investigations showed
that caution must be taken when interpreting ETVs as caused by circumbinary
objects (e.g., Marsh, 2018). In particular, some of the circumbinary objects
inferred from the ETVs have been refuted afterwards using dynamical stability
analysis (Horner et al., 2011, 2012, 2014; Wittenmyer et al., 2012; Marsh,
2018; Mai & Mutel, 2021) or high-resolution direct imaging of the systems
(e.g. V471 Tau, Hardy et al., 2015).

V471 Tau is a close binary system consisting of a K2 dwarf main-sequence star
and a hot white dwarf (Nelson & Young, 1970). The system has a short orbital
period of $P_{\mathrm{orb}}=0.5211833875$ day (Vaccaro et al., 2015) and due
to tides the K2 dwarf is forced to rotate nearly synchronously with the
orbital period ($P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$). As in most PCEBs,
cyclic ETVs are observed in V471 Tau with typical modulations of semi-
amplitude $\Delta P/P_{\mathrm{orb}}~{}\approx~{}8.5\times 10^{-7}$ (where
$\Delta P$ is the difference between the observed orbital period minus the
mean orbital period $P_{\mathrm{orb}}$) and periodicity of 30–35 yr (Kundra &
Hric, 2011; Vaccaro et al., 2015; Marchioni et al., 2018; Lanza, 2020). Guinan
& Ribas (2001) analysed whether the gravity influence of a hypothetical third
body could lead to the ETVs of the system. The authors found that V471 Tau
would need a brown dwarf component with a mass of $\approx 0.0393\pm 0.0038$
M⊙ and a semi-major axis of $11.2\pm 0.4$ AU to reconcile the amplitude and
periodicity of the ETV cycle. However, an image of V471 Tau obtained with
SPHERE at the Very Large Telescope (VLT) refuted the existence of the brown
dwarf (Hardy et al., 2015). This view is supported by Vanderbosch et al.
(2017), who dismissed the brown dwarf component using different arguments
based on the lack of temporal variations of the rotational period of the white
dwarf (that otherwise should vary with the same periodicity of the ETVs due to
the barycenter wobbling).

Alternative effects of magnetic origin have thus been put forward as the most
probable cause of ETVs in V471 Tau (e.g., Applegate, 1992; Völschow et al.,
2016; Navarrete et al., 2018, 2020; Lanza, 2020). Despite differences between
the proposed models, a common feature that they all share relies on the
magnetism of the active component in V471 Tau – i.e., the K2 dwarf star. The
Applegate effect (Applegate, 1992) explains ETVs as an indirect outcome of the
redistribution of angular momentum within the convective zone of the K2 dwarf
throughout a magnetic cycle. The main idea behind the model is that the
redistribution of angular momentum causes temporal modulation of the
gravitational quadrupole moment of the K2 dwarf. This increases (resp.
decreases) the gravitational field at the orbital plane forcing the white
dwarf component to orbit closer to (resp. further from) the K2 dwarf and with
shorter (resp. longer) periods to conserve the total angular momentum of the
system (thus creating ETVs). Besides, activity studies suggest a cyclic nature
for the magnetism of the K2 dwarf yielding a putative period of about 13 yr
(İbanoǧlu et al., 2005; Kamiński et al., 2007; Pandey & Singh, 2008; Kővári et
al., 2021). However, the feasibility of the Applegate mechanism in V471 Tau
has been debated ever since Applegate (1992) as it requires significant
variations of the differential rotation that are yet to be detected at the
surface of the K2 dwarf (see discussions of Lanza, 2005, 2006; Völschow et
al., 2016, 2018; Zaire et al., 2021).

Lanza (2020) proposed a new mechanism (hereinafter the Lanza mechanism) that
requires lower variations of the differential rotation at the surface of the
K2 dwarf to explain the ETVs. This new model is based on the existence of a
non-axisymmetric gravitational quadrupole moment induced by a non-axisymmetric
stationary field throughout the convective zone of the K2 dwarf. Similar to
the Applegate effect, the idea behind the Lanza effect is that the modulation
of the gravitational field along the line joining both stars generates ETVs.
However, the Lanza effect provides a novel approach to the source of variation
of the gravitational field, which results from a non-axisymmetric stationary
magnetic field that is forced to librate around the Lagrange L1 point of the
system or to circulate monotonically in the orbital plane. The Lanza mechanism
has been shown to reduce by at least an order of magnitude the required
fluctuation amplitude of the differential rotation with respect to the
Applegate effect. Nevertheless, in order for the Lanza effect to explain the
ETVs of V471 Tau the non-axisymmetric field needs to librate/circulate with a
period of $70$ yr, which disagrees with the 13 yr activity cycle proposed from
recent observations of the K2 dwarf (Kővári et al., 2021). Therefore, the
origin of ETVs on V471 Tau is still unclear and demonstrating whether an
Applegate effect, a Lanza effect, or another effect of magnetic origin
operates in the system requires dedicated studies of the K2 dwarf magnetism.

Recently, Zaire et al. (2021, hereafter Paper I) reported first large scale
surface magnetic maps and offered new differential rotation measurements of
the K2 dwarf for two different epochs (November/December 2004 and December
2005). They found that the K2 dwarf exhibits significant fluctuations in its
differential rotation amplitude (ranging from the solar value to about twice
the solar differential rotation in a year) and it is not always rotating as a
solid body as it was reported to in an early study (Hussain et al., 2006).
Despite providing useful information to disentangle the magnetic effects
proposed to explain ETVs in V471 Tau, this initial study only probed a maximum
of the ETVs cycle at which differential rotation is not expected to peak in
the Applegate scenario. Additional surface maps and shear measurements probing
different phases of the ETV cycle are thus still needed to determine the
fluctuation amplitude of the surface shear and to search for a possible long
term evolution, perhaps following the prediction of Applegate or Lanza, of the
surface magnetic field of the K2 dwarf.

In this study, we reconstruct new large-scale magnetic field maps and perform
new differential rotation measurements of the K2 dwarf of V471 Tau in December
2014/January 2015, probing a new phase of the ETV modulation cycle in which
the observed orbital period is close to the mean orbital period
$P_{\mathrm{orb}}=0.5211833875$ day (Vaccaro et al., 2015). Section 2
describes the spectropolarimetric observations and, Section 3, presents the
tomographic reconstructions and the differential rotation measurements.
Finally, we discuss our results and conclude in Section 4.

##  2 Observations

We use spectropolarimetric observations of V471 Tau collected with `ESPaDOnS`
at the Canada-France-Hawaii Telescope. The optical spectropolarimeter
`ESPaDOnS` covers wavelengths from $370$ to $1,000$ nm at a resolving power of
$65,000$ (Donati, 2003; Donati et al., 2006a). Our data set consists of 236
unpolarised (Stokes $I$), and 59 circularly polarised (Stokes $V$) profiles
acquired in 11 nights spread between 20 December 2014 and 12 January 2015.
Circularly-polarised spectra are computed combining 4 sub-exposures of 200 s
each taken at different orientations of the polarimeter retarders combined in
an optimal way to minimize potential spurious signatures and to remove
systematics in the circularly polarised spectra (Donati et al., 1997). The
data reduction was carried out with the pipeline `Libre-ESpRIT` optimized for
`ESPaDOnS` observations (Donati et al., 1997). The observational logbook is
given in Table 2. Circularly polarised spectra show peak signal-to-noise
ratios (SNRs) ranging from 122 to 212 (per 1.8 km s-1 spectral pixel), with a
median of 184. Orbital cycles $E$ are computed according to the ephemeris of
Vaccaro et al. (2015):

$\mathrm{HJED}\mbox{\quad=\quad}2445821.898291+0.5211833875\times E,$ (1)

where phase 0.5 corresponds to the K2 dwarf mid-eclipse (i.e., when the white
dwarf is in front of the K2 star). Moreover, because the K2 dwarf rotates
nearly synchronously, its rotational cycle is equal to the orbital cycle $E$.

Figure 1: Observed Stokes $I$ LSD profiles at the rotational phase 0.6 (dotted
lines) and absorption line profile computed for an unspotted star with
$v\sin{i}=89.3\pm 0.1$ km s-1 (black continuous line). The two profiles
correspond to observations at the same rotational phase in 2014.9 (red) and
2015.1 (blue). Rotational cycles (starting from cycle 21470) are indicated.
Figure 2: Tomographic reconstruction of the K2 dwarf surface using both 2014.9
and 2015.1 data sets. The star is shown in a flattened polar view with
concentric circles representing $30\degr$ steps in latitude. Ticks outside the
star indicate the rotational phase of our observations. The first plot shows
the brightness distribution, where cool spots are shown as brown shades and
warm plages as blue shades. The following plots show respectively the radial,
azimuthal, and meridional components of the large-scale magnetic field in
spherical coordinates. Magnetic fields are expressed in Gauss, with positive
values represented in red shades and negative in blue.

In order to generate averaged photospheric lines of the K2 dwarf with enhanced
SNRs, we applied Least-Squares Deconvolution (LSD Donati et al., 1997) on all
absorption lines with a relative depth of at least $40\%$ with respect to the
continuum. Figure 1 shows two example Stokes $I$ LSD profiles obtained at
rotational phase 0.6 using the same K2 dwarf absorption line mask detailed in
Paper I. Observed Stokes $I$ LSD profiles show clear distortions with respect
to the absorption line shape of an unspotted stellar surface assuming a line-
of-sight projected equatorial velocity of $v\sin{i}=89.3\pm 0.11$ km s-1
(Vaccaro et al. 2015, Paper I). These Stokes $I$ signatures provide evidence
for brightness inhomogeneities at the surface of the K2 dwarf star (i.e.,
signatures within $\pm v\sin{i}$), similar to what was found in previous
Doppler images of this star (Ramseyer et al. 1995; Hussain et al. 2006; Paper
I; Kővári et al. 2021). Moreover, the shape difference of line profiles
collected at the same rotational phase but different rotation cycles suggests
that the brightness distribution evolves on a timescale of a few weeks.

##  3 Results

We apply the Zeeman-Doppler imaging technique (`ZDI`) to the time series of
Stokes $I$ or Stokes $V$ LSD profiles to simultaneously reconstruct the
surface brightness distribution and the large-scale magnetic field topology.
To do so, `ZDI` models the stellar surface as a grid of a few thousand cells,
whose individual contributions to the total synthetic Stokes profiles are
computed using the analytical solution of Unno-Rachkovsky to the polarised
radiative transfer equations in a Milne-Eddington atmosphere (see Landi
Degl’Innocenti & Landolfi, 2004). The `ZDI` code inverts the observed LSD
profiles into surface images using a conjugate gradient algorithm that
searches for the maximum-entropy image that reproduces the data down to a
reduced $\chi^{2}$ of about unity (Donati et al., 1989; Brown et al., 1991;
Donati & Brown, 1997; Donati et al., 2006b). The entropy of each image is
computed considering individual cells for the brightness maps, while it is a
function of spherical harmonics coefficients for the magnetic maps. As in
Paper I, the magnetic field expansion is limited to spherical harmonics with
order $\ell\leq 15$.

Our tomographic reconstruction follows closely the procedures described in
Paper I, where we reconstructed brightness and large-scale magnetic surface
maps of the K2 dwarf V471 Tau at two early epochs (2004.9 and 2005.9). In a
first step, we use `ZDI` to optimise the orbital motion correction by
reconstructing surface spots from our set of Stokes $I$ profiles. Using a
fixed semi-amplitude of $K=149.3$ km s-1 (Paper I), we reconstruct several
brightness surface maps by varying the systemic velocity ($v_{\gamma}$) and
phase offset ($\phi_{0}$) assuming the ephemeris of Equation 1. We find that,
at constant information at the surface of the star, the best parameters
reproducing the observations are $v_{\gamma}=35.0\pm 0.10$ km s-1 and
$\phi_{0}=0.0025\pm 0.0005$.

In all the image reconstructions that follow, we use the orbital parameters
derived above to correct the spectra from Doppler shifts before applying
`ZDI`. Akin to Paper I, we fix the line-of-sight projected equatorial velocity
and the stellar inclination angle to $v\sin{i}=89.3\pm 0.11$ km s-1 and
$i=78.755\pm 0.030\,\degr$ (Vaccaro et al., 2015), respectively. We also
recall that, due to the difficulty of `ZDI` to distinguish features from
Northern and Southern hemispheres in nearly equator-on stars (Vogt et al.,
1987; Rice et al., 1989; Unruh & Collier Cameron, 1995; Şenavcı et al., 2018;
Lehmann et al., 2019; Hahlin et al., 2021), the reconstructed images may be
subject to some mirroring effect with respect to the equator, to the same
extent as those presented in Paper I. In practice, this effect is expected to
be mitigated by the excellent phase coverage of our data set (e.g. see Vincent
et al., 1993).

###  3.1 Brightness and magnetic imaging

We first attempt at reconstructing the surface maps of the K2 dwarf star using
the LSD profiles collected in 2014.9 and 2015.1. Applying `ZDI` to the Stokes
$I$ LSD profiles (Stokes $V$ LSD) shows that the data can only be fitted down
to a reduced $\chi^{2}$ of 1.47 (1.15) when assuming that the star rotates as
a solid body. When assuming differential rotation (see Section 3.2), the
Stokes $I$ data can now be fitted down to a reduced $\chi^{2}$ of $1.10$ and
Stokes $V$ data to $1.07$.

Figure 2 shows the maps obtained after including differential rotation in our
image reconstruction process. The brightness map recovered for the combined
2014.9 and 2015.1 data set shows inhomogenieties with respect to the
unperturbed photosphere (with an effective temperature of about 5066 K). It
features a cool polar cap with low-latitude appendages that extend down to
$30\degr$ latitude. As in Paper I, we find that warm low-contrast plages
forming a partial ring structure are also present at low latitudes. From the
brightness map we obtain that $10\%$ of the stellar surface is covered with
cool spots and $8\%$ with warm plages.

The reconstructed large-scale magnetic field is also shown in Figure 2. We
find an average magnetic field strength of 360 G. It shows up from the surface
maps that strong negative radial fields (reaching strengths up to 500 G)
overlap with the high-contrast cool spots forming the polar cap. The overall
magnetic topology that we obtain is dominated by the poloidal component,
whereas the toroidal magnetic energy accounts for $25\%$ of the total energy.
The poloidal field features a strong dipole mode (containing $60\%$ of the
poloidal energy), while other spherical harmonics modes with order $\ell\geq
4$ contribute altogether to $30\%$ of the poloidal energy. We also find that
$75\%$ of the poloidal energy is stored in axisymmetric modes with $m<\ell/2$.
The dipolar component has a polar strength of 335 G and is tilted by $7\degr$
towards phase 0.87.

####  3.1.1 Short-term variability

We find that even after including differential rotation in our image
reconstruction process, the total data set can only be fitted down to a
reduced $\chi^{2}$ of 1.1 when using Stokes $I$ profiles and 1.07 when using
Stokes $V$. This suggests that the surface brightness and magnetic maps evolve
on a timescale of a few weeks. To explore whether a short-term evolution
indeed occurs, we split the total data set in two. The first subset gathers
spectra from 2014.9 (totalling 132 Stokes $I$ profiles and 33 Stokes $V$
spread over 6 non-consecutive nights) and the other combines spectra from
2015.1 (104 Stokes $I$ profiles and 26 Stokes $V$ collected over 5 nights).

Figure 3 shows the spot coverage at different iterations of the `ZDI`
reconstruction process when we aim at fitting the time series of Stokes $I$
profiles at a progressively lower reduced $\chi^{2}$. The data illustrates how
`ZDI` adds spots at the stellar surface to better fit the observations. We can
see in this figure that the spot coverage sharply increases below a given
reduced $\chi^{2}$ (whose value depends on the data set). This behavior
suggests that below this reduced $\chi^{2}$ threshold, the tomographic imaging
process starts to fit noise features present in the data. Using the slope of
the curves in Figure 3 as a criteria to define the reduced $\chi^{2}$ at which
the `ZDI` reconstruction process aims (e.g., see Alvarado-Gómez et al., 2015,
for a detailed explanation about using derivatives as a stop-criteria), we
find that the Stokes $I$ subsets of 2014.9 and 2015.1 can be fitted down to a
reduced $\chi^{2}$ of 1.0 and 0.9, respectively.

Figure 3: Spot coverage ($f_{\text{spot}}$) as a function of the reduced
$\chi^{2}$ for a tomographic reconstruction process aiming at a low value of
reduced $\chi^{2}$ and including differential rotation (see Section 3.2).
Curves with different colors show reconstructions using different Stokes $I$
data sets (see legend). Black circles highlight the values when the spot
coverage sharply rises for a decreasing reduced $\chi^{2}$. Figure 4: ZDI
reconstruction of the individual subsets of 2014.9 (top panels) and 2015.1
(bottom panels). Surface maps are illustrated in a similar fashion to Figure
2.

Figure 4 shows the brightness and magnetic maps obtained for the individual
subsets including differential rotation. Whereas the maps derived in 2014.9
and 2015.1 look similar at first order, we observe small differences
reflecting an intrinsic evolution of the brightness and magnetic field of the
K2 dwarf. Starting from the brightness maps, we note a visible decrease in the
contrast of the cool spot at the polar cap and tiny azimuthal rearrangements
in the distribution of warm plages. As a result, the K2 dwarf surface appears
slightly less spotted in 2015.1. We find that warm plages cover $7\%$ of the
surface in both maps, while dark spots covered $10\%$ of the stellar surface
in 2014.9 and $9\%$ in 2015.1. Regarding magnetic maps, we find that the
negative radial field covers a larger portion of the North pole in 2014.9 than
2015.1 (see Figure 4). As expected, the brightness and magnetic maps derived
from the original data set (Figure 2) resemble an average of the individual
maps derived from the subsets. Table 1 summarises the magnetic properties
derived from the split and original data sets. We assess the uncertainties in
the image reconstruction process using the bootstrap technique detailed in
Paper I.

Table 1: Magnetic field properties of the K2 dwarf star. Brms is the root-mean-square field, Bdip is the dipolar strength, and Epol is the fractional energy in the poloidal field. Eℓ=1, Eℓ=2, Eℓ=3 and Eℓ≥4 are, respectively, the fractional energies of the dipolar, quadrupolar, octupolar, and multipolar (defined as $\ell\geq 4$) components. | Data set
---|---
| 2014.9/2015.1 | 2014.9 | 2015.1
Brms (G) | $360\pm 7$ | $415\pm 5$ | $335\pm 6$
Bdip (G) | $-335\pm 20$ | $-440\pm 35$ | $-250\pm 26$
$\theta_{\mathrm{dip}}\left(\degr\right)$ | $7\pm 5$ | $9\pm 2$ | $9\pm 3$
E${}_{\mathrm{pol}}(\%)$ | $75\pm 5$ | $80\pm 5$ | $70\pm 3$
E${}_{\ell=1}(\%)$ | $55\pm 5$ | $60\pm 9$ | $45\pm 7$
E${}_{\ell=2}(\%)$ | $5\pm 5$ | $5\pm 2$ | $5\pm 2$
E${}_{\ell=3}(\%)$ | $5\pm 2$ | $5\pm 2$ | $5\pm 2$
E${}_{\ell\geq 4}(\%)$ | $35\pm 4$ | $30\pm 9$ | $45\pm 7$

###  3.2 Differential rotation

As mentioned in Subsection 3.1, the global data set 2014.9/2015.1 cannot be
fitted down to a reduced $\chi^{2}$ of 1 as a result of temporal evolution of
surface maps. One of the potential sources for this evolution is the presence
of differential rotation at the surface of the K2 dwarf star.

The `ZDI` code allows one to explore whether stars rotate differentially by
searching for recurrent distortions in the line profiles of our
spectropolarimetric time series (Donati et al., 2000). To do so, `ZDI`
incorporates in the image reconstruction process a predefined latitudinal
differential rotation law given by

$\Omega(\theta)=\Omega_{\mathrm{eq}}-\mathrm{d}\Omega\sin^{2}(\theta),$ (2)

where $\Omega$ is the latitudinal angular velocity profile, $\theta$ is the
latitude, $\Omega_{\mathrm{eq}}$ is the angular velocity at the equator, and
$\mathrm{d}\Omega$ is the difference between $\Omega_{\mathrm{eq}}$ and the
angular velocity at the pole. Brightness and magnetic maps are thus
individually reconstructed for each pair of
($\Omega_{\mathrm{eq}},\mathrm{d}\Omega$) values, with a $\chi^{2}$ value
attributed to each tomographic reconstruction (carried out at constant
information content for all pairs of differential rotation parameters).

Figure 5: Differential rotation measurements obtained with the sheared-imaging
method using Stokes $I$ or Stokes $V$ profiles. Contour levels represent
confidence levels up to $5\,\sigma$. Black circles represent the best-fit
obtained after assuming a paraboloid distribution.

Figure 5 shows the resulting confidence levels for the differential rotation
parameters, when reconstructing the brightness surface distribution (i.e.,
using Stokes $I$ alone) and the magnetic topology (i.e., using Stokes $V$
alone). Using the minimum of the paraboloid and its local curvature to
retrieve the optimal shear parameters and corresponding error bars (Donati et
al., 2003), we obtain $\mathrm{d}\Omega=60\pm 1$ mrad d-1 and
$\Omega_{\mathrm{eq}}=12.084\pm 0.001$ rad d-1 from the $\chi^{2}$
distribution using Stokes $I$, and $\mathrm{d}\Omega=42\pm 3$ mrad d-1 and
$\Omega_{\mathrm{eq}}=12.077\pm 0.001$ rad d-1 from the $\chi^{2}$
distribution using Stokes $V$. As in Paper I, these results again suggest that
brightness inhomogeneities and magnetic structures are sheared by different
amounts.

Moreover, we use ten bootstrapped data sets to estimate the differential
rotation parameters uncertainty associated with the image reconstruction
process (see details in Paper I). To do that, we repeat the steps above to
determine the differential rotation parameters using bootstrapped Stokes $I$
or Stokes $V$ profiles. From the $\chi^{2}$ maps obtained using bootstrapped
data sets, we find that the mean values of the ten error bars obtained using
Stokes $I$ and Stokes $V$ profiles are similar to the error bars obtained from
the original data.

###  3.3 H$\alpha$ variability

Figure 6: Dynamical spectra of H$\alpha$ shown in the rest frame of the K2
dwarf. Red shades indicate that H$\alpha$ is in emission and green shades in
absorption. The vertical dashed lines correspond to the stellar rotational
broadening of $\pm v\sin(i)$. Sine waves of semi-amplitudes $150$ km s-1
(center of mass), $205$ km s-1 (prominence position), and $320$ km s-1 (white
dwarf position) are over-plotted as dashed-dotted lines.

The variability of H$\alpha$ in V471 Tau has been reported by several authors
(Young et al. 1991; Rottler et al. 2002; Kővári et al. 2021, Paper I). The
dynamical spectra of the H$\alpha$ line in 2014.9/2015.1 is plotted in Figure
6. Starting with the radial velocity range within $\pm v\sin(i)$, we identify
that H$\alpha$ exhibits the typical rotational modulation at the the K2 dwarf
surface. H$\alpha$ is in emission on the stellar hemisphere that faces the
white dwarf and in absorption on the opposite hemisphere. The H$\alpha$
equivalent width reveals a peak-to-peak amplitude of about 1.2 Å with a
maximum emission of $-0.5$ Å at phase 0.5.

Further, we observe a modulated emission with an amplitude of $205\pm 40$ km
s-1 in the rest frame of the K2 dwarf (see Figure 6). We speculate that this
emission is due to a stable prominence trapped at $2.30\pm 0.45$ R⋆ from the
K2 dwarf (or, equivalently, at $1.29\pm 0.45$ R⋆ from the white dwarf
component). We find a full width at half maximum (FWHM) of $1.89$ Å and an
equivalent width of about $-0.17$ Å when fitting a Gaussian to the prominence
emission at phase 0.75. Assuming that the prominence is spherical, we estimate
a prominence radius of about $0.50$ R⋆ from its FWHM emission in H$\alpha$.

##  4 Discussions and conclusions

In this paper, we analysed new spectropolarimetric data of the binary system
V471 Tau collected from 20 December 2014 to 12 January 2015 with `ESPaDOnS`.
Using Zeeman-Doppler imaging, we modelled time series of LSD Stokes $I$ and
$V$ profiles to recover new brightness and magnetic maps of the K2 dwarf
component of V471 Tau.

###  4.1 Brightness map, magnetic field topology, and differential rotation

Our brightness image reveals a strong cool polar cap in 2014.9/2015.1. This
result is confirmed by an independent Doppler imaging reconstruction using a
different inversion code (Kővári et al., 2021). Along with previous brightness
maps (Ramseyer et al. 1995; Hussain et al. 2006; Paper I), we find that the
cool polar cap seen in the star surface is stable in a timescale of years. The
spot coverage of $\approx 18\%$ derived in 2014.9/2015.1 is in good agreement
with what is expected from photometry (in the range 15–25$\%$, see Paper I, )
suggesting that most of the brightness spots generating photometric
fluctuations in V471 Tau are large enough to be detected and resolved by
Doppler imaging.

The reconstructed large scale magnetic field shows a dominant poloidal
component that accounts for about $75\%$ of the magnetic energy in
2014.9/2015.1. This value is slightly larger than those observed in 2004.9 and
2005.9, whose fractional poloidal energy corresponded to $70\%$ and $60\%$,
respectively. Moreover, we find that the dipole strength in 2014.9/2015.1 is
about 3.6 times stronger than that in 2004.9 and 2005.9.

We also confirmed that the surface of the K2 dwarf is differentially rotating.
We measured an equatorial to pole angular velocity difference of 60 and 42
mrad d-1 from spot and magnetic structures, respectively. This finding
confirms the solar-like differential rotation profile obtained for the star in
2004.9 and 2005.9 (Paper I). Interestingly, the shear level inferred from our
2014.9/2015.1 data set resembles closely those obtained nine years before (73
and 48 mrad d-1 in 2005.9; Paper I, ).

###  4.2 Magnetic activity

Studies of chromospheric/coronal activity indicators (Rottler et al., 2002;
Kamiński et al., 2007; Pandey & Singh, 2008; Kővári et al., 2021) and long-
term photometry (Skillman & Patterson, 1988; İbanoǧlu et al., 2005) of the K2
dwarf suggest an activity cycle of about 13 yr. This possible activity cycle
indicates that the two data sets analysed in Paper I (2004.9 and 2005.9)
occurred at activity minimum (spanning from late-2004 to late-2007), whereas
the data set analysed in this paper (2014.9/2015.1) took place close to
activity maximum (spanning from late-2011 to late-2014). Such scenario is
indeed corroborated by our large-scale magnetic field maps. We find that the
averaged unsigned magnetic field strength increased by about 2.2 times from
the two first epochs (at activity minimum) to the last epoch (at activity
maximum). No such modulation is visible in the brightness maps, which display
a spot coverage of $14\%$, $17\%$, and $18\%$ in 2004.9, 2005.9, and
2014.9/2015.1, respectively. This result emphasizes that spot coverage may not
always be an appropriate observable to study activity cycles in very active
rapidly rotating stars.

The analysis of the H$\alpha$ emission in 2014.9/2015.1 shows a prominence
located farther than the Lagrange point L1 towards the white dwarf component,
and that remained stable during the entire observation window (44 rotation
cycles). The prominence size and location we infer are consistent within error
bars with the prominence properties derived in 2004.9 (Paper I). Using the
prominence flux in H$\alpha$ of $1.1\times 10^{-13}$ erg s-1 cm-2, we derive a
prominence mass of $4\times 10^{18}$ g in 2014.9/2015.1 slightly smaller than
that identified in 2004.9 of $6\times 10^{18}$ g (see Equation 3 in Steeghs et
al., 1996). The prominence mass-range of 4–6$\times 10^{18}$ g is broadly
consistent with those derived for other K dwarf stars hosting prominences –
e.g. K0 dwarf AB Dor (2-10$\times 10^{17}$ g, e.g., Collier Cameron &
Robinson, 1989; Collier Cameron et al., 1990) and the K3 dwarf Speedy Mic
(0.5–2.3$\times 10^{17}$ g, Dunstone et al., 2006).

Figure 7: Potential field extrapolation of the large-scale radial magnetic
field reconstruction of the K2 dwarf obtained with ZDI in 2014.9/2015.1. Field
lines are seen at rotational phase 0.75 and are shown in yellow/black when the
lines are open/closed. The prominence is illustrated as a red circle, and the
field lines crossing the prominence are coloured in magenta. The local surface
field strength (G) of the star is shown in colours and follows the colour
scale on the right. A black circle indicates the white dwarf (WD) position;
however, its magnetic field is not considered in the potential field
extrapolation.

As illustrated in Figure 7, the potential field extrapolation of the radial
magnetic field map that we derived for the K2 dwarf shows closed loops of
magnetic field that extend out from the surface and reach the prominence
location. This result is consistent with that of Paper I and offers further
qualitative proof that a slingshot mechanism is likely responsible for
confining the prominence further away from the center of mass of the system
and from the Lagrange point L1 (located at $1.679\pm
0.004\,\mathrm{R_{\star}}$ and $1.84\pm 0.02\,\mathrm{R_{\star}}$ from the
center of the K2 dwarf star, respectively). Slingshot mechanisms have been
also suggested to operate in single fast-rotating stars hosting prominences at
a few stellar radii above the stellar surface (see discussion in Jardine &
Collier Cameron, 2019), such as AB Dor (Collier Cameron & Robinson, 1989;
Waugh & Jardine, 2019), HK Aqu (Byrne et al., 1996), LQ Lup (Donati et al.,
2000), Speedy Mic (Dunstone et al., 2006; Waugh & Jardine, 2019), V374 Peg
(Vida et al., 2016), and V530 Per (Cang et al., 2020; Cang et al., 2021).

Figure 8: Observed minus computed eclipse timing variations of V471 Tau
assuming the linear ephemeris given by Equation 1. Crosses represent O-C
measurements from eclipse timings and circles the O-C values estimated from
the phase offset $\phi_{0}$ multiplied by the orbital period (shown with
1-$\sigma$ errorbars). The black square gives the single eclipse timing
measure using K2 data (Muirhead et al., 2021), corresponding to HJD
2457097.1816484 in Terrestrial Time (TT) scale (see Eastman et al., 2010). The
vertical dashed line marks the recent observation campaign of V471 Tau with
ESPaDOnS (2021B).

Altogether, the potential field extrapolations available for the K2 dwarf V471
Tau show that when a prominence is seen in the system (2004.9 and
2014.9/2015.1) close loops of magnetic lines reach the prominence location,
whereas when no prominence is detected (2005.9) only open field lines are
found at the expected prominence location (see details about previous
reconstructions in Paper I). This finding indicates that the evolution of the
large-scale magnetic field controls the rate at which stable prominences are
generated in V471 Tau.

###  4.3 ETVs in V471 Tau

The observed minus computed (O-C) eclipse timings available in the literature
for V471 Tau (cross symbols) are illustrated in Figure 8. The data evidence
the periodic behaviour of the ETVs in V471 Tau with current observations
yielding a modulation period of 30-40 yr (Guinan & Ribas, 2001; İbanoǧlu et
al., 2005; Kundra & Hric, 2011; Marchioni et al., 2018) and an O-C amplitude
ranging from 130 to 200 s (Kundra & Hric, 2011; Marchioni et al., 2018)
depending on the ephemeris employed. To compare this trend with independent
measures, we use the phase offsets $\phi_{0}$ available in the literature for
V471 Tau to infer the O-C amplitude. These two quantities are expected to
scale as

$\text{O-C}=\phi_{0}\times P_{\mathrm{orb}}.$ (3)

We find an O-C amplitude of $158\pm 90$ s in 2002.9 (green circle, Hussain et
al., 2006), $180\pm 9$ s in 2004.9 (red, Paper I, ), $158\pm 9$ s in 2005.9
(blue, Paper I, ), and $113\pm 23$ s in 2014.9/2015.1 (yellow, this paper).

The O-C values that we infer from Equation 3 agree with the trend found using
long-term photometry. They offer an independent validation of the ETVs in V471
Tau as all the four phase offset measurements considered take into account the
presence of spots at the surface of the K2 dwarf (see Section 3) that may
otherwise affect the eclipse timing measurements from photometry (Kalimeris et
al., 2002). Furthermore, the intermediate value of O-C that we infer in
2014.9/2015.1 suggests that although the orbital period of the system was
decreasing from 2002.9 to 2015.1, it did not reach the minimum orbital period
recorded for V471 Tau (which took place around 1980, i.e. when O-C crosses 0
going to negative values). As a result, the observation reported in this paper
did not probe the ETV cycle at the phase of largest surface differential
rotation as predicted if an Applegate mechanism is indeed operating on V471
Tau. This result is corroborated by the eclipse timing measure using
photometric data from the $K2$ mission acquired around 2015.2, which yields
O-C = 169 s.

Similar to the findings of Paper I, we detect a relative differential rotation
in 2014.9/2015.1 of $\mathrm{d}\Omega/\Omega_{\mathrm{eq}}=0.5\%$ and $0.4\%$
using Stokes $I$ and Stokes $V$, respectively. These values are weaker than
what is needed for the feasibility of an Applegate mechanism in V471 Tau.
Considering the range of shears currently measured at the surface of the K2
dwarf (ranging from 0.4 to 1.1$\%$), the Applegate mechanism would drive ETVs
with a semi-amplitude of $\Delta P/P_{\mathrm{orb}}\lesssim 10^{-7}$ (Völschow
et al., 2018) whereas V471 Tau displays $\Delta
P/P_{\mathrm{orb}}~{}\approx~{}8.5\times 10^{-7}$. Nevertheless, our O-C
estimation in 2014.9/2015.1 indicates that the system was not orbiting at the
minimum orbital period expected for V471 Tau (i.e., when the largest surface
shear is expected in the framework of the Applegate mechanism). It may be
possible that higher values of $\mathrm{d}\Omega/\Omega_{\mathrm{eq}}$ occur
at the surface of the K2 dwarf and thus that the Applegate mechanism may
indeed be at work.

We suggest that spectropolarimetric observations in the upcoming years will
help understand whether the ETVs in V471 Tau are magnetically-driven
especially if they can probe the ETV cycle at the expected phase of largest
differential rotation. Along with the tomographic maps already reconstructed
for the K2 dwarf, it will be possible to further investigate whether the ETVs
of V471 Tau are caused by the mechanism proposed by Applegate (1992) or by
Lanza (2020). For this purpose, observations of V471 Tau were recently
collected with ESPaDOnS in 2021B.

## Acknowledgements

We thank the anonymous referee for helping improve and clarify this
manuscript. This project received funding from the European Research Council
(ERC) under the H2020 research $\&$ innovation programme (grant agreements
$\\#740651$ New-Worlds and $\\#865624$ GPRV). This paper is based on
observations obtained 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 Scientique of
France, and the University of Hawaii. The observations at the CFHT were
performed with care and respect from the summit of Maunakea which is a
significant cultural and historic site.

## Data Availability

This paper includes data collected by the ESPaDOnS spectropolarimeter, which
is publicly available from the Canadian Astronomy Data Center (program IDs:
15AP15 $\&$ 14BP15).

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##  Appendix A Journal of observations

The logbook of the spectropolarimetric observations of V471 Tau used in this
study is shown in Table 2.

Table 2: Summary of ESPaDOnS/CFHT observations for V471 Tau from December 2014 to January 2015. Columns 1 to 4 respectively record (i) the date of observation, (ii) the UT time at mid sub-exposure, (iii) the time in Heliocentric Julian Date (HJD), and (iv) the rotation cycle of each observation. Column 5 illustrates peak SNR values for the Stokes $V$ spectrum (per $1.8$ km/s spectral pixel). Column 6 shows the RMS noise level of the Stokes $V$ LSD profile. Date | UT | HJD | $E$ | SNR | $\sigma_{LSD}$
---|---|---|---|---|---
| (h:m:s) | $(2,453,337+)$ | $(21,470+)$ |  | $(10^{-4})$
20 Dec 2014 | 06:17:40 | 3674.76735 | 0.118441 | 202 | 1.5
20 Dec 2014 | 06:35:24 | 3674.77967 | 0.142079 | 207 | 1.5
20 Dec 2014 | 06:53:09 | 3674.79199 | 0.165718 | 200 | 1.5
20 Dec 2014 | 12:05:19 | 3675.00877 | 0.581656 | 137 | 2.5
20 Dec 2014 | 12:23:04 | 3675.02109 | 0.605294 | 134 | 2.5
20 Dec 2014 | 12:40:48 | 3675.03340 | 0.628914 | 122 | 2.9
21 Dec 2014 | 05:41:49 | 3675.74239 | 1.989260 | 194 | 1.6
21 Dec 2014 | 05:59:33 | 3675.75471 | 2.012899 | 195 | 1.6
21 Dec 2014 | 06:17:18 | 3675.76704 | 2.036556 | 193 | 1.6
21 Dec 2014 | 11:29:48 | 3675.98404 | 2.452917 | 182 | 1.8
21 Dec 2014 | 11:47:33 | 3675.99636 | 2.476555 | 179 | 1.8
21 Dec 2014 | 12:05:17 | 3676.00868 | 2.500194 | 165 | 2.0
22 Dec 2014 | 06:40:09 | 3676.78284 | 3.985582 | 187 | 1.7
22 Dec 2014 | 06:57:53 | 3676.79515 | 4.009202 | 196 | 1.6
22 Dec 2014 | 07:15:38 | 3676.80748 | 4.032859 | 198 | 1.6
22 Dec 2014 | 12:26:57 | 3677.02366 | 4.447646 | 176 | 1.9
22 Dec 2014 | 12:44:42 | 3677.03598 | 4.471285 | 178 | 1.9
22 Dec 2014 | 13:02:26 | 3677.04830 | 4.494923 | 157 | 2.1
28 Dec 2014 | 08:56:52 | 3682.87737 | 15.679221 | 205 | 1.5
28 Dec 2014 | 09:14:36 | 3682.88968 | 15.702840 | 209 | 1.5
28 Dec 2014 | 09:32:20 | 3682.90200 | 15.726479 | 184 | 1.7
29 Dec 2014 | 06:02:07 | 3683.75595 | 17.364961 | 196 | 1.6
29 Dec 2014 | 06:19:52 | 3683.76827 | 17.388600 | 198 | 1.6
29 Dec 2014 | 06:37:36 | 3683.78059 | 17.412238 | 189 | 1.6
29 Dec 2014 | 11:46:02 | 3683.99477 | 17.823188 | 194 | 1.6
29 Dec 2014 | 12:03:47 | 3684.00709 | 17.846826 | 191 | 1.7
29 Dec 2014 | 12:21:32 | 3684.01941 | 17.870465 | 182 | 1.8
30 Dec 2014 | 04:34:15 | 3684.69486 | 19.166458 | 193 | 1.6
30 Dec 2014 | 04:52:01 | 3684.70719 | 19.190115 | 196 | 1.6
30 Dec 2014 | 05:09:45 | 3684.71951 | 19.213754 | 203 | 1.6
30 Dec 2014 | 10:24:43 | 3684.93823 | 19.633414 | 210 | 1.5
30 Dec 2014 | 10:42:28 | 3684.95055 | 19.657053 | 210 | 1.5
30 Dec 2014 | 11:00:12 | 3684.96287 | 19.680691 | 212 | 1.5
07 Jan 2015 | 05:43:50 | 3692.74255 | 34.607644 | 133 | 2.5
07 Jan 2015 | 06:01:37 | 3692.75490 | 34.631341 | 138 | 2.4
07 Jan 2015 | 06:19:24 | 3692.76725 | 34.655037 | 168 | 1.9
07 Jan 2015 | 11:29:05 | 3692.98230 | 35.067655 | 162 | 2.0
07 Jan 2015 | 11:46:50 | 3692.99461 | 35.091275 | 152 | 2.2
07 Jan 2015 | 12:04:35 | 3693.00694 | 35.114932 | 144 | 2.3
08 Jan 2015 | 04:44:36 | 3693.70134 | 36.447285 | 202 | 1.6
08 Jan 2015 | 05:02:30 | 3693.71377 | 36.471134 | 202 | 1.6
08 Jan 2015 | 05:20:14 | 3693.72609 | 36.494773 | 202 | 1.5
08 Jan 2015 | 10:30:48 | 3693.94174 | 36.908543 | 191 | 1.7
08 Jan 2015 | 10:48:33 | 3693.95406 | 36.932181 | 188 | 1.7
08 Jan 2015 | 11:06:18 | 3693.96639 | 36.955839 | 191 | 1.7
09 Jan 2015 | 04:39:52 | 3694.69797 | 38.359529 | 189 | 1.7
09 Jan 2015 | 04:57:37 | 3694.71030 | 38.383187 | 189 | 1.7
09 Jan 2015 | 05:15:22 | 3694.72262 | 38.406825 | 193 | 1.6
09 Jan 2015 | 10:24:48 | 3694.93749 | 38.819099 | 191 | 1.7
09 Jan 2015 | 10:42:34 | 3694.94982 | 38.842756 | 190 | 1.7
09 Jan 2015 | 11:00:19 | 3694.96215 | 38.866414 | 178 | 1.8
10 Jan 2015 | 04:40:42 | 3695.69846 | 40.279180 | 182 | 1.7
10 Jan 2015 | 04:58:27 | 3695.71079 | 40.302838 | 177 | 1.8
10 Jan 2015 | 05:16:11 | 3695.72311 | 40.326476 | 183 | 1.8
10 Jan 2015 | 10:26:58 | 3695.93890 | 40.740515 | 190 | 1.7
10 Jan 2015 | 10:44:43 | 3695.95123 | 40.764172 | 190 | 1.7
10 Jan 2015 | 11:02:29 | 3695.96356 | 40.787830 | 180 | 1.8
12 Jan 2015 | 10:25:28 | 3697.93769 | 44.575614 | 198 | 1.6
12 Jan 2015 | 10:43:13 | 3697.95002 | 44.599271 | 195 | 1.7

##  Appendix B Stokes signatures

Stokes $I$ and Stokes $V$ profiles are given in Figures 9 and 10,
respectively. Observed Stokes LSD profiles are shown in red, and modelled
Stokes profiles are given in black. Modelled Stokes signatures are associated
with the surface maps obtained through independent `ZDI` reconstructions using
either the spectropolarimetric data set of December 2014 (top panels in Figure
4) or January 2015 (bottom panels in Figure 4).

Figure 9: Stokes $I$ profiles in 2014.9 (columns 1 and 2) and 2015.1 (columns
3 and 4) data sets. Observed Stokes $I$ LSD profiles are shown in red, whereas
modelled observations are shown in black (see Section 3.1.1 for further
details). All profiles are equally shifted for illustration purposes. The
rotation cycle of each observation is shown in the right. Figure 10: Stokes
$V$ profiles in 2014.9 (left) and 2015.1 (right) data sets. Observed Stokes
$V$ LSD profiles are shown in red, whereas modelled observations are shown in
black (see Section 3.1.1 for further details). All profiles are equally
shifted for illustration purposes. The rotation cycle of each observation is
shown in the right and 1 $\sigma$ error bars in the left.