11institutetext:  Institut de Recherche en Astrophysique et Planétologie,
Université de Toulouse, CNRS, CNES, 14 avenue Edouard Belin, 31400 Toulouse,
France 22institutetext: Department of Physics & Space Science, Royal Military
College of Canada, PO Box 17000 Station Forces, Kingston, ON, Canada K7K 0C6
33institutetext: Departamento de Física Teórica e Experimental, Universidade
Federal do Rio Grande do Norte, Natal, RN 59072-970, Brazil 44institutetext:
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA
02138, USA 55institutetext: Universität Göttingen, Institut für Astrophysik,
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 66institutetext: Max-Planck-
Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen,
Germany 77institutetext: University of Southern Queensland, Centre for
Astrophysics, Toowoomba, QLD 4350, Australia 88institutetext: Laboratoire
Univers et Particules de Montpellier, Université de Montpellier, CNRS,
F-34095, France 99institutetext: School of Physics, Trinity College Dublin,
University of Dublin, Dublin-2, Ireland

# Multi-instrumental view of magnetic fields and activity of $\epsilon$
Eridani with SPIRou, NARVAL, and TESS

P. Petit 11 C.P. Folsom 1122 J.-F. Donati 11 L. Yu 11 J.-D. do Nascimento Jr
3344 S. Jeffers 5566 S.C. Marsden 77
J. Morin 88 A.A. Vidotto 99

###### Abstract

Aims. We report on observations of the active K2 dwarf $\epsilon$ Eridani
based on contemporaneous SPIRou, NARVAL and TESS data obtained over two months
in late 2018, when the activity of the star was reported to be in a non-cyclic
phase.

Methods. Near-infrared spectropolarimetry was obtained using SPIRou over 4
nights in late September, while visible spectropolarimetry was collected with
NARVAL over 20 nights, from 18 Sept to 07 Nov. We first recover the
fundamental parameters of the target from both visible and nIR spectral
fitting. The large-scale magnetic field is investigated from polarimetric
data. From unpolarized spectra, we estimate the total magnetic flux through
Zeeman broadening of magnetically sensitive nIR lines and the chromospheric
emission using the CaII H&K lines. The photometric monitoring, secured with
TESS between 19 Oct and 15 Nov, is modelled with pseudo-periodic Gaussian
Process Regression.

Results. Fundamental parameters of $\epsilon$ Eridani derived from visible and
near-infrared wavelengths provide us with consistent results, also in
agreement with published values. We report a progressive increase of
macroturbulence towards larger nIR wavelengths. Zeeman broadening of
individual lines highlights an unsigned surface magnetic field $B_{\rm
mono}=1.90\pm 0.13$ kG, with a filling factor $f=12.5\pm 1.7$% (unsigned
magnetic flux $Bf=237\pm 36$ G). The large-scale magnetic field geometry,
chromospheric emission and broadband photometry display clear signs of non-
rotational evolution over the course of data collection. Characteristic decay
times deduced from the light curve and longitudinal field measurements fall in
the range 30-40 d, while the characteristic timescale of surface differential
rotation, as derived through the evolution of the magnetic geometry, is equal
to $57\pm 5$ d. The large-scale magnetic field exhibits a combination of
properties not observed previously for $\epsilon$ Eridani, with a surface
field among the weakest previously reported, but also mostly axisymmetric, and
dominated by a toroidal component.

###### Key Words.:

stars:activity, stars:magnetic field, stars:rotation, stars: solar-type

##  1 Introduction

Most Sun-like stars experience a strong magnetic activity during the first Gyr
of their evolution, as a consequence of an efficient global dynamo triggered
by their high spin rate. The variable phenomena induced at the stellar surface
and in upper atmospheric layers span a wide range of spatial scales, temporal
scales, and spread their signatures over most of the electromagnetic spectrum,
shaping the extended environment of the star where young planets may orbit (do
Nascimento et al., 2016). Widely studied manifestations of stellar activity
include core emission in chromospheric lines (Noyes et al., 1984; Boro Saikia
et al., 2018), spectroscopic and broad band photometric signatures of dark
spots and plages (see Berdyugina 2005, for a review), polarized Zeeeman
signatures (e.g. Donati et al. 1990; Marsden et al. 2014) and Zeeman
broadening of magnetically-sensitive spectral lines (Saar, 1988; Lavail et
al., 2019). Owing to the diverse instrumentation and modelling tools needed to
study these fragmented diagnoses, most studies concentrate on a specific
tracer of activity, getting in return a very incomplete view of a complex
phenomenon. Here, we propose to grasp a more diverse view of stellar activity
for the specific case of $\epsilon$ Eridani, taking advantage of
contemporaneous observations gathered with three different instruments.

With a distance of $3.2028\pm 0.0047$ pc (Gaia Collaboration et al., 2018),
$\epsilon$ Eridani is one of the closest solar-type stars. Being in the solar
neighborhood, it benefited from interferometric radius measurements
($R=0.74\pm 0.01R_{\odot}$, di Folco et al. 2007; Baines & Armstrong 2012),
confirming its status as a main sequence dwarf. From the combined knowledge of
the distance and radius, it is possible to infer its surface black body
temperature ($T_{\rm eff}$$=5076\pm 30$ K, Heiter et al. 2015). This estimate
is consistent with its K2V spectral type (Keenan & McNeil, 1989) and mostly
agrees with its spectroscopic effective temperature ($T_{\rm eff}=5146\pm 30$
K, Valenti & Fischer 2005). Spectroscopic and interferometric studies also
agree with each other on a mass estimate, with $0.856\pm 0.08M_{\odot}$ given
by Valenti & Fischer (2005) or $0.80\pm 0.06M_{\odot}$ from Heiter et al.
(2015).

Radio observations have unveiled a debris disc (Gillett, 1986) taking the form
of a narrow ring with arc-like azimuthal structures (Booth et al., 2017). This
circumstellar residual from the stellar formation is indicative of a young age
($439\pm 52$ Myr was proposed by Barnes 2007), and the geometrical ring
properties may reveal a resonant interaction with massive planetary
companions. A first planet was actually detected at 3.4 AU by Hatzes et al.
(2000) with a mass $M.\sin i=0.86\leavevmode\nobreak\ M_{\rm J}$, and later
confirmed by Anglada-Escudé & Butler (2012). This eccentric planetary
companion ($e=0.6$) may share the neighborhood of $\epsilon$ Eridani with a
second, unconfirmed planet of $\approx 0.1\leavevmode\nobreak\ M_{\rm J}$ at
40 AU from the star, according to numerical simulations of the debris disc
(Quillen & Thorndike, 2002).

Due to its young age, $\epsilon$ Eridani is rotating relatively fast, with
sustained CaII H&K emission exhibiting a 11.68 d period (Donahue et al.,
1996). The long term variability of the chromospheric emission was first
reported to be chaotic in nature (Baliunas et al., 1995), before new
observations revealed that the magnetic activity has become more regular
during the last two decades, following a 2.95 yr chromospheric cycle (Metcalfe
et al., 2013). This cyclic pattern lasted until about 2017, with less regular
variations reported after this date (Coffaro et al., 2020). The surface
magnetic field of $\epsilon$ Eridani was first detected through Zeeman
broadening at infrared wavelengths (Valenti et al., 1995) and then in the
visible domain (Rueedi et al., 1997). Zeeman-Doppler Imaging (ZDI hereafter)
was applied to spectropolarimetric observations of $\epsilon$ Eridani, with
monitoring of the evolution of the large-scale magnetic geometry over
timescales as long as several years (Jeffers et al., 2014) and as short as a
few months (Jeffers et al., 2017).

We present here quasi-simultaneous observations of this prototypical young,
active solar-type star using spectropolarimetric data collected from the
ground with SPIRou and NARVAL, and space-borne photometric data delivered by
the TESS spacecraft. We first describe the data sets (Sec. 2), then present
our determination of fundamental stellar parameters (Sec. 3), as well as the
measurements and modeling of the longitudinal magnetic field (Sec. 4), Zeeman
broadening (Sec. 5), chromospheric emission (Sec. 6), radial velocities (Sec.
7), brightness fluctuations (Sec. 8), large-scale magnetic geometry (Sec. 9)
and surface differential rotation (Sec. 10). We finally summarize, compare and
discuss these different measurements (Sec. 11).

##  2 Observations

Table 1: Journal of SPIRou and NARVAL observations. We list here the Julian date, rotational phase, exposure time, stellar effective radial velocity, longitudinal magnetic field, and the S-index. RV and $B_{\rm eff}$ measurements with SPIRou are obtained from a night average of LSD profiles. Typical uncertainties for NARVAL radial velocities and S-index measures are 30 m s-1 and 0.002, respectively. The RV precision refers to the relative RV, not the absolute RV. There is likely to be a systematic RV shift (larger than the SPIRou RV error bar) between the two instruments. Instrument | HJD | phase | exp. time (sec) | RV (km s-1) |  $B_{\rm eff}$ (G) | S-index
---|---|---|---|---|---|---
SPIRou | 2458384.1653 | 0.6991 | $5\times 4\times 11.1$ | $16.441\pm 0.002$ | $-2.6\pm 0.8$ | –
| 2458385.1645 | 0.7846 | $4\times 4\times 33.4$ | $16.432\pm 0.001$ | $-3.1\pm 1.0$ | –
| 2458387.1576 | 0.9553 | $5\times 4\times 33.4$ | $16.451\pm 0.001$ | $+0.6\pm 0.8$ | –
| 2458388.1564 | 0.0408 | $1\times 4\times 33.4$ | $16.456\pm 0.002$ | $+2.5\pm 1.0$ | –
NARVAL | 2458380.6620 | 0.3992 | $4\times 200$ | 16.53 | $-2.7\pm 0.2$ | 0.495
| 2458381.6358 | 0.4825 | $4\times 200$ | 16.54 | $-2.8\pm 0.3$ | 0.479
| 2458384.6252 | 0.7385 | $4\times 200$ | 16.54 | $-4.2\pm 0.2$ | 0.428
| 2458386.6208 | 0.9093 | $4\times 200$ | 16.51 | $-0.6\pm 0.3$ | 0.444
| 2458387.6261 | 0.9954 | $4\times 200$ | 16.52 | $+0.4\pm 0.2$ | 0.452
| 2458388.6513 | 0.0832 | $4\times 200$ | 16.50 | $+1.1\pm 0.2$ | 0.459
| 2458389.5919 | 0.1637 | $4\times 200$ | 16.52 | $+0.9\pm 0.2$ | 0.479
| 2458395.6595 | 0.6832 | $4\times 200$ | 16.46 | $-4.8\pm 0.3$ | 0.446
| 2458396.6073 | 0.7643 | $4\times 200$ | 16.46 | $-3.3\pm 0.2$ | 0.443
| 2458397.6046 | 0.8497 | $4\times 200$ | 16.45 | $-2.0\pm 0.2$ | 0.439
| 2458400.5678 | 0.1034 | $4\times 200$ | 16.51 | $-0.3\pm 0.5$ | 0.447
| 2458404.5693 | 0.4460 | $4\times 200$ | 16.52 | $-2.5\pm 0.4$ | 0.465
| 2458414.5792 | 0.3030 | $4\times 200$ | 16.48 | $+2.3\pm 0.2$ | 0.473
| 2458415.5853 | 0.3892 | $4\times 200$ | 16.50 | $+0.9\pm 0.3$ | 0.475
| 2458416.5284 | 0.4699 | $4\times 200$ | 16.50 | $-3.0\pm 0.2$ | 0.451
| 2458417.5236 | 0.5551 | $4\times 200$ | 16.49 | $-4.9\pm 0.2$ | 0.452
| 2458430.4804 | 0.6644 | $4\times 200$ | 16.47 | $-2.8\pm 0.3$ | 0.439
| 2458436.5098 | 0.1806 | $4\times 200$ | 16.53 | $+3.0\pm 0.2$ | 0.444
| 2458437.4856 | 0.2642 | $4\times 200$ | 16.52 | $+2.3\pm 0.2$ | 0.436
| 2458439.4952 | 0.4362 | $4\times 200$ | 16.51 | $-2.1\pm 0.2$ | 0.433

We report on three time-series of observations of $\epsilon$ Eridani taken
with three different instruments in 2018, over a period of about two months
spread from late September to late November. Spectropolarimetric data were
obtained in the visible and near-infrared (nIR) domain, using NARVAL and
SPIRou, respectively. Photometric monitoring was performed onboard the TESS
spacecraft.

###  2.1 SPIRou near-infrared spectra

SPIRou is a cryogenic, near-infrared, high-resolution échelle
spectropolarimeter and velocimeter recently installed at CFHT (Donati et al.,
2020). Each spectrum covers the Y, J, H, and K bands (nominal spectral range
from 0.98 to 2.35 $\mu$m), at a spectral resolution of 70,000, with a radial
velocity precision of between 1 and $2$ m s-1 RMS. Similar to its optical
predecessors ESPaDOnS and NARVAL, Stokes $V$ sequences produced by SPIRou
consist of four sub-exposures collected with different rotation angles of the
half-wave Fresnel rhombs in the Cassegrain mounted polarimeter. This procedure
ensures that the two polarimetric states exchange their position on the H4RG
detector, so that spurious polarimetric signatures can be removed at first
order (Semel et al., 1993; Donati et al., 1997).

The data reduction was performed with an adapted version of the LibreEsprit
pipeline (Donati et al., 1997). The spectral domain of SPIRou being affected
by a large number of telluric lines, their subtraction was performed with a
Principal Component Analysis approach inspired by Artigau et al. (2014), using
a large number of observations of hot stars (with very few photospheric lines
in the near infrared) as a learning data set featuring a variety of
configurations of the telluric spectrum.

Observations of $\epsilon$ Eridani were obtained as part of the science
verification of the instrument. One to five Stokes $V$ spectra were obtained
daily between 21 Sep and 25 Sep, with one missing night over this period (23
Sep). The integration time, first set to 11.1 sec per sub-exposure on Sep 21,
was then increased to 33.4 sec per sub-exposure. The peak signal-to-noise
ratio (S/N hereafter) in Stokes $I$ spectra ranges from about 800 on 21 Sep to
1,300 during the following nights (this peak value is reached at a wavelength
of around 2.1 $\mu$m). The polarimetric sequence was obtained only once on 25
Sep, and was repeated four times on 22 Sep and five times on 21 Sep and 24 Sep
(for a total of 15 spectra over the four nights).

###  2.2 NARVAL visible spectra

NARVAL is a high-resolution échelle spectropolarimeter installed at Télescope
Bernard Lyot (Pic du Midi Observatory, France, Aurière 2003). Its spectral
resolution in polarimetric mode is close to 65,000, and it is able to cover,
in a single exposure, a spectral domain ranging from 370 nm to 1,000 nm
(except for small wavelength gaps between the redmost spectral orders). We
have collected here circularly polarized sequences only (Stokes parameters I
and V), because the amplitude of Zeeman signatures is maximal in this
polarization state. All spectra were reduced by the automated LibreEsprit
pipeline (Donati et al., 1997).

The NARVAL data set is constituted of 20 visits to the target, obtained as
part of the Bcool Large Program (Marsden et al., 2014). This observing
sequence of observations started on Sep. 18, and we accumulated new data until
Nov. 16. Four NARVAL spectra were obtained during the SPIRou observing window,
and 6 spectra are collected during the TESS monitoring (see Sec. 2.3).

The choice to adopt integration times of $4\times 200$ sec in polarimetric
sequences results in peak S/N (reached in order #31, at about 730 nm) mostly
comprised between 1,200 and 2,000. Only two spectra, taken on Oct. 08 and 12,
have their peak S/N slightly below 1,000. The collected temporal sequence
suffers from a few gaps, the most extended one spanning 13 nights between Oct
25 and Nov 07 (immediately followed by a new 5 night gap).

All NARVAL data used here are publicly available on the PolarBase archive
(Petit et al., 2014).

###  2.3 TESS photometry

The Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015) performed
observations of $\epsilon$ Eridani as part of the Sector 4 sequence, lasting
from Oct 20 to Nov 19 (HJD 2458410.9 to 2458436.8). The public, reduced light
curve was retrieved from the MAST archive.

Each data point in the light curve is delivered with a 12-bit quality flag,
with each bit indicating an abnormal data condition if positive. All the
flagged data points were discarded from our study. We also rejected data
obtained between Julian dates 2458420 and 2458424 because (a) this fraction of
the light curve exhibited a periodicity and amplitude inconsistent with the
rest of the time series, and (b) other light curves of stars on the same
detector (e.g. $\delta$ Eri, 7 Eri, HD 19349) also show a similarly abnormal
behaviour at the same dates, which lead us to conclude that TESS was
undergoing sub-nominal conditions while taking these measurements. The
resulting time series is a constituted of 13989 points (Fig 1).

Figure 1: Normalized TESS light curve of $\epsilon$ Eridani, as a function of
the barycentric Julian date. Red points show the observations affected by a
non-zero quality flag. Grey points were impacted by measurement instabilities
that were also recorded on other targets observed by the same detector. Blue
points are the ones retained for our study.

##  3 Fundamental parameters

Figure 2: Zeeman adjustment (red) of spectral lines in the SPIRou wavelength
domain after removal of telluric lines (black line). The dashed line shows the
SPIRou spectrum prior to the subtraction of tellurics. Vertical blue ticks
show the wavelength position of spectral lines included in the Zeeman
analysis.

The physical parameters of $\epsilon$ Eri are generally well known. There have
been numerous spectroscopic studies, some reaching very good precision (e.g.
Valenti & Fischer, 2005). Furthermore, there are two interferometric diameter
estimates (di Folco et al. 2007, and Baines & Armstrong 2012), providing
independent information. The star was included in the meta-analysis of Heiter
et al. (2015) of Gaia FGK benchmark stars. In this context, an additional
spectroscopic analysis is not urgent, however with both visible and IR spectra
we can perform a detailed comparison between parameters derived in widely
separated spectral regions using contemporaneous observations.

Our analysis proceeded by directly fitting synthetic spectra to the
observations by $\chi^{2}$ minimization. The stellar parameters fit were
effective temperature ($T_{\rm eff}$), surface gravity ($\log g$),
macroturbulence ($\xi_{\rm mac}$), microturbulence ($\xi_{\rm mic}$),
metallicity and radial velocity. Line broadening from macroturbulence and
$v\sin i$ is ambiguous for such a slow rotator, thus we used the
interferometric radius of di Folco et al. (2007, $0.74\pm 0.01R_{\odot}$) and
the equatorial rotation period and inclination of Jeffers et al. (2014, Peq
$=10.33$ d, $i=46^{\circ}$) to infer a $v\sin i$ of 2.59 km s-1. We calculated
synthetic spectra using the Zeeman spectrum synthesis code (Landstreet, 1988;
Wade et al., 2001). This calculates atomic line spectra in LTE, using plane-
parallel model atmospheres. Input model atmospheres from the MARCS grid of
Gustafsson et al. (2008) were used. Input atomic data from VALD (Piskunov et
al., 1995; Ryabchikova et al., 1997; Kupka et al., 1999, 2000; Ryabchikova et
al., 2015) were used. The analysis proceeded by fitting several spectroscopic
windows independently, then taking the average as the best value and the
standard deviation as the uncertainty on that value. This may overestimate the
uncertainty, since the standard deviation may be driven by the worst
spectroscopic window. However, we prefer it to simply using the covariance
matrix, since this more completely accounts for possible systematic errors in
the model and atomic data.

To achieve a high degree of precision in spectrum modelling across many lines,
empirical corrections to atomic line oscillator strengths are commonly needed.
We find this is particularly true in the IR, possibly due to this less
commonly used data being less accurate, or possibly due to the lower density
of lines per nm causing errors to average out less effectively. We derive
empirical oscillator strength corrections by modelling the solar spectrum. We
used observations of sunlight reflected from the moon with NARVAL and SPIRou.
A synthetic spectrum was calculated for solar parameters, then discrepant
lines were identified by hand, and oscillator strengths for those lines were
iteratively fit, further discussed in Sec. 3.2.

###  3.1 NARVAL visible analysis

For the analysis in the visible we used the NARVAL spectrum from the night of
Sep. 25, 2018. The Stokes $I$ spectrum did not vary noticeably and the S/N was
high enough that coadding spectra was unnecessary. The fitting procedure
closely followed Folsom et al. (2016) and Folsom et al. (2018a). We used six
spectral windows each roughly 100 Å long: 6025–6100, 6100–6200, 6200–6275,
6314–6402, 6402–6500, and 6590–6700 Å. Regions contaminated with telluric
lines, as well as a few other features not present in our line list, were
avoided. The Li line at 6707.8 Å was not unambiguously detectable in the
observation, so a reliable Li abundance could not be derived. We used a fixed
$v\sin i$ of 2.59 km s-1 and fit macroturbulence. However, if we were to
assume no macroturbulence and fit for $v\sin i$ we would get $2.87\pm 0.26$ km
s-1, which is consistent with the value based on the rotation period and
radius plus a small amount of turbulent broadening. The resulting best fit
stellar parameters averaged over the windows are presented in Tab. 2. Our
results are generally in good agreement with the results of Valenti & Fischer
(2005), and Heiter et al. (2015).

Table 2: Physical parameters for $\epsilon$ Eri derived from visible and IR spectra. References for literature values: 1 Heiter et al. (2015), 2 Valenti & Fischer (2005), 3 Luck & Heiter (2005), 4 Jofré et al. (2014).  | visible | IR | literature
---|---|---|---
$T_{\rm eff}$ (K) | $5010\pm 64$ | $4991\pm 59$ | $5076\pm 30^{1}$
$\log g$ (cgs) | $4.53\pm 0.08$ | $4.48\pm 0.14$ | $4.61\pm 0.03^{1}$
$v\sin i$ (km s-1) | $2.59$ | $2.59$ | $2.4\pm 0.5^{2}$
$\xi_{\rm mic}$ (km s-1) | $0.99\pm 0.13$ | $1.18\pm 0.22$ |  $0.7^{3}$, $1.14\pm 0.05^{4}$
$\xi_{\rm mac}$ (km s-1) | $0.93\pm 0.38$ | $3.03\pm 0.85$ |
metallicity | $-0.08\pm 0.04$ | $-0.09\pm 0.04$ | $-0.09\pm 0.06^{4,1}$

###  3.2 SPIRou infrared analysis

For the analysis in the IR we used the SPIRou spectra from 21 Sep. and co-
added them to produce one high S/N spectrum for the night, using the telluric
correction provided by our upgraded LibreEsprit pipeline. We found no clear
variability between nights in the absorption lines, and results of this
analysis from different nights were consistent within the noise. Typically
there are fewer spectral lines per Å in the IR than in the visible for a K
star, so we used larger spectral windows. Since regions of the spectrum that
are dominated by telluric lines are very difficult to fully correct, such
regions were avoided entirely. This lead to using large windows of:
10500–10920, 11760–12600, 15110–15697, 15815–16390, 16439–17140, and
21017–22850 Å. This spans most of the usable spectral range of SPIRou, in 6
independent windows. Within these windows there were several gaps, avoiding
stronger telluric features (most notably from 15985 to 16145 Å), a few broad
features not present in our model spectra, and stronger molecular lines.
Comparing the telluric corrected and uncorrected spectra was very helpful for
identifying regions where the telluric correction may not have been
sufficiently accurate, and such regions were avoided.

A major limitation of Zeeman is that it does not currently compute molecular
lines in stellar spectra. In the visible for K stars molecular lines can
easily be identified and avoided, particularly since VALD version 3 contains
extensive molecular line lists. However in some regions of the SPIRou domain
this becomes more difficult, even for a star of $\sim$5000 K. Care was taken
to identify and avoid molecular features, however a few very weak lines of CN
and OH could not be avoided in the 11760–12600, 15815–16390, and 16439–17140 Å
regions, and a larger number of very weak CN lines were unavoidable in the
15110–15697 Å region. (A number of CN lines are also present in the
21017–22850 Å region but they can be more practically avoided due to the large
spacing between lines). Since these very weak lines are at or near the noise
level, we do not expect them to have a large impact on the results, but they
may contribute to our uncertainties by increasing the standard deviation of
results from different windows. The 15110–15697 Å region may be viewed as the
least reliable due to the larger number of contaminating molecular lines.

The atomic line data in the SPIRou domain appears to be less reliable than in
the visible, or at least obtaining accurate results is more reliant on
corrections to the input atomic data. There is a good agreement on the
presence of lines in VALD and in the observation, with a few exceptions, and
the wavelengths generally appear to be correct. However line strengths are in
many cases wrong, and in some stronger lines the width of the Lorentzian
component of the line also appears to be incorrect. To correct for this, we
used a spectrum of sunlight reflected off the Moon obtained with SPIRou. We
computed synthetic spectra with solar parameters ($T_{\rm eff}$ $=5777$ K,
$\log g$ $=4.4$ (with $g$ in cm s-1), $v\sin i$ $=2$ km s-1, $\xi_{\rm mic}$
$=0.9$ km s-1, and macroturbulence of 3 km s-1), and identified discrepant
lines visually. We then fit the oscillator strength ($\log gf$) of the lines
by $\chi^{2}$ minimization. For some lines it was also necessary to fit the
van der Waals broadening coefficient (van der Waals broadening is typically
the dominant Lorentzian broadening source in K stars), and it was included in
a simultaneous fit with $\log gf$. These empirical corrections were essential
for deriving accurate stellar parameters and achieving reasonable fits to the
infrared observations. Corrections for 480 lines were used in our final
results, as detailed in Appendix C. To verify the reliability of these
empirical corrections, we fit the solar spectrum using the corrections to
derive $T_{\rm eff}$, $\log g$, $\xi_{\rm mic}$, macroturbulence and
metallicity, and ensured we obtained results consistent with the solar values.

The best fit parameters for individual windows in the SPIRou spectrum are
presented in Tab. 6, and the final average and standard deviation are in Tab.
2. We find a good agreement in $T_{\rm eff}$, $\log g$, and metallicity across
all six spectral windows in the infrared, and a very good agreement between
the average results for the visible and infrared spectra. For comparison we
include well established literature values in Tab. 2, and find good agreement
within our uncertainty. For microturbulence, there are two windows where we
find anomalously low values, this may be due to the influence of weak
unidentified blends or incompletely removed telluric lines. Rejecting the most
extreme microturbulence value as likely wrong, we obtain an average that is in
good agreement with the optical and literature values, albeit with a larger
uncertainty. The larger macroturbulence value found in the infrared is
discussed in Appendix B.

##  4 Longitudinal field measurements

The Least-Square Deconvolution method (LSD hereafter, Donati et al. 1997;
Kochukhov et al. 2010), was applied to every NARVAL and SPIRou reduced
spectrum to get the S/N boost necessary to the detection of polarized Zeeman
signatures and to precise radial velocity measurements. We used a visible and
infrared list of photospheric spectral lines computed for effective
temperature and surface gravity closely matching the stellar parameters
derived in Sec. 3 (and ignoring portions of the spectra blended with
chromospheric lines or heavily affected by telluric absorption). The adopted
lists feature close to 8,000 lines in the SPIRou domain, and 11,000 lines in
the visible domain. The outcome of this procedure is a single, pseudo line
profile with a typical S/N in Stokes $V$ as large as 18,000 for SPIRou
observations, versus about 50,000 for NARVAL data.

The projection on the line of sight of the disc-integrated magnetic field
($B_{\rm eff}$) can be estimated through the first moment of Stokes $V$ LSD
profiles, following Rees & Semel (1979); Donati et al. (1997):

$B_{\rm eff}=-2.14\times 10^{-11}\frac{\int vV(v)dv}{\lambda_{0}g_{\rm
eff}c\int(1-I(v))dv}\leavevmode\nobreak\ G.$ (1)

where $v$ is the radial velocity, $\lambda_{0}$ is the central wavelength of
LSD pseudo-profiles, and $g_{\rm eff}$ its effective Landé factor.

Figure 3: Longitudinal magnetic field measurements, as a function of the
rotational phase. Red symbols show SPIRou estimates. Blue/green symbols
correspond to NARVAL measurements, with bluer (resp. greener) symbols showing
older (resp. more recent) observations.

The series of $B_{\rm eff}$ measurements is plotted in Fig. 3, as a function
of the rotational phase. The phases were computed using a rotation period
equal to 11.68 d (Donahue et al., 1996), and the null phase is set at
HJD=2458399.36, which corresponds to the mean date of the NARVAL observations.
We note that the four SPIRou measurements are consistent with NARVAL estimates
obtained at close-by phases, although the shorter exposure times, smaller
number of lines used to compute the SPIRou LSD pseudo-profiles, and lower line
depth in the nIR lead to larger error bars.

The rotational modulation of $B_{\rm eff}$ is obvious, with a smooth
transition from values close to -5 G around phase 0.6 to +3 G at phase 0.2.
Some scatter is also observed, showing that the temporal variability is not
entirely controlled by stellar rotation. We note that a fair part of the
scatter is due to the most recent NARVAL observations, with differences as
large as 2 G between the oldest and most recent observations, with a global
trend to get larger field estimates (negative fields becoming weaker and
positive fields becoming stronger) in more recent data.

To model both the rotational modulation and its temporal evolution, we fit the
series of $B_{\rm eff}$ measurements by means of Gaussian Process Regression
(GPR, Haywood et al. 2014; Yu et al. 2017). We use here a pseudo-periodic co-
variance function, defined for times $t$ and $t^{\prime}$ and for the four
hyperparameters $\theta_{1}$ to $\theta_{4}$ by the following equation:

$K(t,t^{\prime})=\theta_{1}^{2}\exp\left[-\frac{(t-t^{\prime})^{2}}{\theta_{3}^{2}}-\frac{\sin^{2}\left(\frac{\pi(t-t^{\prime})}{\theta_{2}}\right)}{\theta^{2}_{4}}\right]$
(2)

where $\theta_{1}$ is the amplitude (in Gauss) of the GP, $\theta_{2}$ the
cycle length ($i.e.$ the rotation period, in days), $\theta_{3}$ the decay
timescale (in days, describing the lifetime of magnetic spots contributing to
the large-scale magnetic geometry), and $\theta_{4}$ a smoothing parameter in
the [0,1] interval.

Figure 4: Outcome of the MCMC run for the four GP hyperparameters of the $B_{\rm eff}$ model. The decay time and cycle length are expressed in days, the GP amplitude in Gauss, and the smoothing parameter is dimensionless. Dashed lines show the best parameters, while dotted lines indicate the error bars. Table 3: Priors used for the MCMC exploration of the hyperparameter space. Hyperparameter | Prior type | adopted value
---|---|---
$\theta_{1}$ (amplitude) | Modified Jeffreys | MJ$(0.01,\,700)$
$\theta_{2}$ (rotation period) | Gaussian | $\mathcal{N}(11.5,\,0.1)$
$\theta_{3}$ (decay time) | Jeffreys | J$(12,\,700)$
$\theta_{4}$ (smoothing) | Uniform | $\mathcal{U}[0.001,\,1]$

The hyperparameter domain is explored through the Markov Chain Monte Carlo
(MCMC) method. The result of the MCMC simulation is illustrated in Fig. 4. The
resulting cycle length is equal to $11.4\pm 0.3$ d, with an amplitude of
$2.6^{+0.7}_{-0.6}$ G, a decay time equal to $31^{+9}_{-7}$ d, and a smoothing
parameter of $0.7\pm 0.2$.

##  5 Zeeman broadening

Figure 5: Infrared spectral lines of $\epsilon$ Eridani observed with SPIRou
(black points) and modelled with Zeeman. The wavelength of lines is shown as
green vertical ticks, and the effective Landé $g$ factors are indicated to the
right of each modelled line name. The best two-component magnetic model (blue)
provides an acceptable fit to all lines, while a model with no magnetic field
but otherwise the same parameters (red) only fits the magnetically insensitive
line (upper right panel). A multi-component model (orange) provides a
comparable fit to the two-component model.

The observation before telluric correction (dashed line) is shown for
reference.

The strength of the stellar magnetic field can be estimated from the impact of
Zeeman splitting in Stokes $I$. Stokes $V$ observations are sensitive to the
sign of the line of sight component of the magnetic field, thus they are
relatively sensitive to the geometry of the magnetic field, but nearby regions
of opposite sign can cancel out, effectively becoming undetectable. Stokes $I$
is relatively insensitive to geometry, being sensitive to the strength but not
the orientation of the magnetic field111Zeeman broadening in Stokes $I$ also
typically lacks Doppler resolution across the surface of the star, since it is
only reliably detectable when it is comparable to, or larger than, the
rotational broadening. This is another important difference with Stokes $V$
measurements, which can still reliably detect Zeeman splitting when it is much
less than rotational or other line broadening processes.. Thus for a magnetic
field distribution similar to that of the Sun, Stokes $I$ provides a measure
of small intense magnetic field regions that cover a small fraction of the
stellar surface and largely cancel out in Stokes $V$. Since the Zeeman effect
scales as $\lambda^{2}$ while most other broadening processes scale with
$\lambda$, lines further into the infrared provide a more sensitive
diagnostic. The approach we take here is to directly fit a small number of
carefully selected lines with synthetic spectra computed by Zeeman.

This method relies on a selection of good lines with a range of Landé factors,
and the wide spectral domain of SPIRou offers many possibilities. We looked
for lines with particularly large or small effective Landé factors ($g_{\rm
eff}$), minimal blending, preferred wavelengths longer than 15000 Å, and
avoided anything blended with strong telluric lines in case of imperfections
in the telluric correction. We focused on lines that are strong enough to
limit the impact of noise, but not so strong that they have large Lorentzian
wings. The S/N of the observation begins to decrease beyond 2 $\mu$m, so lines
in the K band are less optimal.

We adopt four Fe i lines for this analysis, at 15343.79 ($g_{\rm eff}=2.63$),
15381.96 ($g_{\rm eff}=0.01$), 15611.14 ($g_{\rm eff}=1.83$), and 15648.51
($g_{\rm eff}=2.98$) Å. The Fe i 15381.96 Å line has an effective Landé factor
near zero, and provides a magnetically insensitive diagnostic for turbulent
and rotational broadening. The Fe i 15648.51 and 15343.79 Å lines have
exceptionally large effective Landé factors, while the 15611.14 line is also
quite sensitive. The Fe i 15648.51 Å line has been used by several other
authors (e.g. Valenti et al., 1995; Lavail et al., 2017).

Some alternate lines that were considered but not used include the Fe i
15534.3 Å line (Valenti et al., 1995), which has a large effective Landé
(1.95) factor but may have weak CN blends in the red wing, so we excluded it.
Similarly, the very low Landé factor Fe i 15560.8 Å line (Valenti et al.,
1995) is weak and appears to have a weak OH or CN blend in the blue wing
making it unsuitable. The Fe i 15621.65 and 15662.0 Å lines (Lavail et al.,
2017) are strong with large Loretzian wings in $\epsilon$ Eridani, which makes
distinguishing pressure and Zeeman broadening more difficult, thus they were
not used. The Ti i 22211.2, 22232.9, 22274.0, 22310.6 Å lines (Johns-Krull et
al., 2004) have large effective Landé factors (up to 2.5 for 22310.6 Å), but
are relatively weak and in a lower SNR portion of the $\epsilon$ Eridani
spectrum, and in some observations can be blended with strong telluric lines.
Thus they would be good choices for cooler stars, but are not optimal for
$\epsilon$ Eridani.

Table 4: Atomic line data for the Fe i lines used in the Zeeman broadening analysis, including the effective Landé $g$ factor and van der Waals damping parameter.  $\lambda$ (Å) | $\log gf$ | $g_{\rm eff}$ | van der Waals
---|---|---|---
15343.788 | $-0.67\pm 0.04$ | 2.63 | -7.52
15381.960 | $-0.69\pm 0.04$ | 0.01 | -7.43
15611.145 | $-3.30\pm 0.07$ | 1.83 | -7.79
15647.413 | $-1.08\pm 0.11$ | 1.00 | -7.29
15648.510 | -0.63 | 2.98 | -7.49
Table 5: Best fit Zeeman broadening results for $\epsilon$ Eridani. Results are presented for the two-component model, and for a multi-component model that provides limits on the presence of stronger magnetic fields. Parameters without uncertainties were fixed based on a fit to the visible spectrum. The final column contains averages over the four nights, with the standard deviation as the uncertainty. | two-component model
---|---
parameter | 21 Sept. | 22 Sept. | 24 Sept. | 25 Sept. | average
$T_{\rm eff}$ (K) | 5010
$\log g$ (cgs) | 4.53
$v\sin i$ (km s-1) | 2.59
$\xi_{\rm mic}$ (km s-1) | 0.99
$\xi_{\rm mac}$ (km s-1) | $4.93\pm 0.24$ | $3.03\pm 0.24$ | $3.34\pm 0.18$ | $3.19\pm 0.22$ | $3.26\pm 0.17$
metallicity | $0.038\pm 0.007$ | $0.024\pm 0.010$ | $0.015\pm 0.008$ | $0.012\pm 0.009$ | $0.022\pm 0.010$
$B_{\rm mono}$ (G) | $1898\pm 129$ | $1812\pm 136$ | $1846\pm 133$ | $1783\pm 135$ | $1835\pm 43$
filling factor | $0.125\pm 0.017$ | $0.159\pm 0.023$ | $0.126\pm 0.018$ | $0.141\pm 0.021$ | $0.138\pm 0.014$
| multi-component model
parameter | 21 Sept. | 22 Sept. | 24 Sept. | 25 Sept. | average
$T_{\rm eff}$ (K) | 5010
$\log g$ (cgs) | 4.53
$v\sin i$ (km s-1) | 2.59
$\xi_{\rm mic}$ (km s-1) | 0.99
$\xi_{\rm mac}$ (km s-1) | $5.02\pm 0.21$ | $3.22\pm 0.19$ | $3.44\pm 0.19$ | $3.40\pm 0.18$ | $3.40\pm 0.12$
metallicity | $0.048\pm 0.012$ | $0.029\pm 0.016$ | $-0.008\pm 0.015$ | $0.005\pm 0.014$ | $0.019\pm 0.022$
$f_{\rm 0kG}$ | $0.831\pm 0.043$ | $0.840\pm 0.057$ | $0.850\pm 0.054$ | $0.859\pm 0.051$ | $0.845\pm 0.011$
$f_{\rm 2kG}$ | $0.118\pm 0.017$ | $0.131\pm 0.023$ | $0.140\pm 0.022$ | $0.129\pm 0.020$ | $0.130\pm 0.008$
$f_{\rm 4kG}$ | $0.000\pm 0.022$ | $0.028\pm 0.028$ | $0.009\pm 0.026$ | $0.010\pm 0.025$ | $0.012\pm 0.010$
$f_{\rm 6kG}$ | $0.050\pm 0.033$ | $0.000\pm 0.044$ | $0.000\pm 0.042$ | $0.000\pm 0.040$ | $0.013\pm 0.021$

The oscillator strengths of the four lines we focused on for Zeeman
broadening, as well as a weak blending Fe i line at 15647.41 Å, were corrected
by fitting synthetic spectra to a solar spectrum, as was done in Sec. 3.2. For
the Fe i 15648.51 line, we adopt the oscillator strength of Valenti et al.
(1995), since it provided an adequate fit for $\epsilon$ Eridani and was
consistent with our best fit solar value. The adopted oscillator strengths, as
well as the effective Landé factors and van der Waals damping coefficients are
provided in Tab. 4.

The model spectra assume a radial magnetic field uniformly distributed over
some fraction of the stellar surface, thus the spectra are effectively the
combination of a uniform radial magnetic model and a non-magnetic model. The
fraction of the surface containing magnetic field is parameterized as a
filling factor. This is clearly much simpler than the real magnetic field of a
star, but Zeeman broadening in Stokes $I$ is relatively insensitive to the
magnetic geometry, so a more realistic geometry would add more free parameters
but not improve the fit to the observations. We use the $T_{\rm eff}$, $\log
g$, $v\sin i$, and microturbulence derived from the visible range in Sec. 3,
however, we allow metallicity and macroturbulence to be free parameters. This
allows the model to fit line strength through metallicity, and line width
through macroturbulence, so that assumptions in these parameters do not unduly
influence the derived magnetic field strength.

We consider two approaches to fitting Zeeman broadening parameters. First we
consider a model with free parameters for a characteristic magnetic field
strength ($B_{\rm mono}$) covering a filling factor ($f$), and the remaining
surface is non-magnetic (the ‘two-component’ model). This is consistent with
the approach of, e.g., Marcy (1984), Gray (1984), and Valenti et al. (1995),
and is reasonable for $\epsilon$ Eridani since Zeeman broadening does not
dominate over other broadening processes. Later, we consider a second approach
using a model with a grid of fixed magnetic field strengths, and filling
factors for those strengths as free parameters (the ‘multi-component’ model,
e.g. Johns-Krull et al. 1999; Shulyak et al. 2019). Since the magnetic field
of $\epsilon$ Eridani is not extremely strong, we cannot derive an extended
distribution of filling factors, but we can place limits on the presence of
multi-kilogauss magnetic fields.

For the coadded spectrum from 21 Sept., the best fit two-component model we
find has magnetic field $B_{\rm mono}=1898\pm 129$ G with a filling factor of
$0.125\pm 0.017$. The full set of parameters are presented in Tab. 5, and the
fits are presented in Fig. 5, contrasted with a non-magnetic model that fits
the Landé factor 0.01 line. The uncertainties are based on the diagonal of the
covariance matrix, scaled by the square root of the reduced $\chi^{2}$, thus
they account for noise but not systematic uncertainties in input parameters.
We repeat this analysis for the nightly coadded spectra for all four nights,
and find an average $B_{\rm mono}$ of 1835 G and a standard deviation of 43 G,
the average filling factor is 0.138 with a standard deviation of 0.014. The
small standard deviation in $B_{\rm mono}$ suggests we may have overestimated
the random uncertainties, although it may simply be due to small number
statistics. We find no statistically significant variation in $B_{\rm mono}$
or $f$ over the four coadded spectra.

To estimate the impact of uncertainties in other parameters, we try varying
the input parameters by a reasonable uncertainty, rerun the fit, and check the
impact on the magnetic field and filling factor. Varying $T_{\rm eff}$ by $\pm
80$ K changes $B_{\rm mono}$ by $\pm 30$ G and the filling factor by $\pm
0.005$, and $\log g$ has a smaller impact. Varying $v\sin i$ by $\pm 0.3$ km
s-1 changes $B_{\rm mono}$ by $\pm 8$ G and the filling factor by $\pm 0.001$,
since the free macroturbulence parameter offsets this change. Changing the
microturbulence by $\pm 0.15$ km s-1 changes $B_{\rm mono}$ by $\pm 10$ G and
the filling factor by $\pm 0.007$, but has a stronger impact on the inferred
metallicity. If we assume an error in the line oscillator strengths of $\pm
0.1$ dex in $\log gf$, the impact is between 50 and 150 G in $B_{\rm mono}$,
and 0.02 to 0.03 in filling factor. The formal uncertainties from fitting the
oscillator strengths are less than 0.1 (except for the very weak 15647.413 Å
line), but this may still be the dominant source of systematic error.

We then considered a multi-component model, largely to constrain the
possibility of very strong magnetic fields covering small areas. For the
multi-component model, we chose a grid of magnetic field strengths of 0, 2, 4
and 6 kG222For the practical purpose of implementing this within a $\chi^{2}$
minimization routine, the filling factors for the non-zero field regions are
treated as the free parameters and the filling factor for the 0 G region is
calculated as $1-\sum_{i}f_{i}$. Experiments using a grid spaced only 1 kG
apart suggests such small bins are not fully resolved, leading to strong
covariances between filling factors and potentially a poorly constrained
model. Fitting the coadded 21 Sept. spectrum (see Tab. 5) produces a
significant filling factor in the 2 kG bin ($f_{\rm 2kG}=0.118\pm 0.017$),
while the filling factors for the 4 kG bin ($f_{\rm 4kG}=0.001\pm 0.022$) and
6 kG bin ($f_{\rm 6kG}=0.050\pm 0.033$) are consistent with zero, and most of
the surface is in the 0 kG bin ($f_{\rm 0kG}=0.831\pm 0.043$). The coadded 22
Sept. spectrum produces results consistent within $1\sigma$ with $f_{\rm
0kG}=0.840\pm 0.057$, $f_{\rm 2kG}=0.131\pm 0.023$, $f_{\rm 4kG}=0.028\pm
0.028$, and $f_{\rm 6kG}=0.000\pm 0.044$, and there is no statistically
significant variability on the Sept. 24 or Sept. 25. From this we find no
evidence for a magnetic field stronger than 2 kG on the star.

The magnetic field we derive here broadly agrees with previous measurements of
the magnetic field through Zeeman broadening. Valenti et al. (1995) determine
a magnetic field and filling factor from infrared lines, finding $B=1445\pm
58$ G and a filling factor $f=0.088\pm 0.008$. This is similar to, but not
formally consistent with our values, however they adopted a somewhat larger
$\log g$ (4.70) and $T_{\rm eff}$(5133 K) than we did. In exploring the
systematic impact of $\log g$ and $T_{\rm eff}$ they tested model with $\log
g$$=4.55$ and $T_{\rm eff}$$=4960$ K, finding $B=1553$ G and $f=0.096$, which
agrees with our results within $3\sigma$ in $B$ and $2\sigma$ in $f$ (these
differences being possibly linked to intrinsic magnetic variability). A number
of earlier magnetic estimates based on Zeeman broadening in the visible exist
(Marcy, 1984; Gray, 1984; Saar, 1988; Mathys & Solanki, 1989; Marcy & Basri,
1989, e.g.), but produce significantly larger values of $B$ or $f$ and Valenti
et al. (1995) question their reliability. Rueedi et al. (1997) provide a more
consistent analysis of Zeeman broadening in the visible, although they prefer
to report the product $Bf$ since they consider it less vulnerable to
systematic errors (supported by the analysis of Valenti et al. (1995) and we
also find a possible negative covariance between the parameters). Rueedi et
al. (1997) find $Bf=165\pm 30$ G, consistent with Valenti et al. (1995)
$Bf=127\pm 13$ G, and somewhat smaller than the product of the values from our
two-component model $Bf=237\pm 36$ G, but within $2\sigma$ of our joint
uncertainties. From our multi-component model we calculate
$\sum_{i}B_{i}f_{i}=542\pm 219$, although this quantity is dominated by the
larger field bins with small, uncertain filling factors, thus we consider it
effectively an upper limit. More recently Lehmann et al. (2015) investigated
Zeeman broadening in $\epsilon$ Eridani spectra spanning 5 years, using a
principal component analysis based approach. They report $B\sqrt{f}$ values
varying between $124\pm 25$ and $230\pm 21$ G, possibly varying with a 3 year
cycle (e.g. Metcalfe et al. 2013). They interpret this as a surface average
magnetic field strength for a filling factor of 1, although for Zeeman
broadening studies the filling factor is likely smaller, thus a direct
comparison with our results is less obvious. This quantity is consistent with
our $Bf$, as well as Valenti et al. (1995) and Rueedi et al. (1997), but is
much smaller than our $B\sqrt{f}$ and that of Valenti et al. (1995). While
some discrepancies between studies are likely due to systematic errors or
underestimated uncertainties, the results of Lehmann et al. (2015) suggest
that there is also important temporal variability to the Zeeman broadening of
$\epsilon$ Eridani.

##  6 CaII H and K emission

Figure 6: NARVAL S-index measurements, as a function of the rotational phase.
Bluer (resp. greener) symbols show older (resp. more recent) observations.

NARVAL spectra cover the CaII H & K lines at 396.847 and 393.366 nm. Their
core is seen in emission in all our set of $\epsilon$ Eridani spectra, as an
effect of the sustained chromospheric activity. As a standard way to estimate
the chromospheric emission and compare its value with archival measurements,
we extract S-index values from our data following the guidelines of Noyes et
al. (1984). This chromospheric indicator integrates the light flux in the
cores of CaII H & K using triangular bandpasses, and normalizes the core flux
over two broader continuum bands (using a rectangular bandpass) taken on both
sides of the doublet. Our measurements are calibrated according to Marsden et
al. (2014) to produce NARVAL S-index estimates that can be directly compared
to Mount Wilson values. The resulting S-index time-series is reported in Tab.
1. The average value is close to 0.45, which is in the lower half of typical
chromospheric emission measures for $\epsilon$ Eridani for which S-index
values over the last five decades generally range from 0.4 to 0.6 (Metcalfe et
al., 2013). Our values are in overall agreement with the contemporaneous
monitoring of Coffaro et al. (2020), who report an unexpected drop in
chromospheric activity with respect to the previously regular S-index
fluctuations (so that our set of observations, that was expected to be close
to an S-index maximum, are in fact typical of a lower activity state).

Statistical error bars can be computed for every individual estimate but these
formal uncertainties are, in practice, dominated by residual inaccuracies in
continuum normalisation around the calcium doublet (even if, by definition,
the S-index corrects a fair part of normalization issues). To obtain an
empirical estimate of typical error bars, we consider the dispersion of
S-index measurements for Pollux, a quiet red giant with chromospheric emission
close to the basal level and a K0 spectral type relatively close to the
spectral type of $\epsilon$ Eridani. From a series of 266 observations
performed with NARVAL and ESPaDOnS over several years, at a S/N close to the
one achieved here, we measure a standard deviation of the S-index of $\sim
0.002$ (Aurière et al., 2021). This value is taken as a proxy of the S-index
uncertainty for $\epsilon$ Eridani.

The S-index variations are plotted, against rotational phases, in Fig. 6. The
phase dependence is not as prominent as the one recorded for the longitudinal
field, to the point that the rotation period cannot be unambiguously
identified using a GPR model similar to the one described in Sec. 4, or using
a simple Lomb-Scargle approach. Fast evolution of chromospheric structures may
be responsible for the relatively large scatter, with non-rotational evolution
especially obvious between phases 0.2 and 0.4, where S-index levels have
sharply dropped during the time span of observations. In the phase range
$[0.16,0.18]$, we record a decrease from $S\approx 0.48$ at $HJD=2458389.6$ to
$S\approx 0.44$ at $HJD=2458436.5$, which is well above uncertainties. During
a comparable time interval, a more limited drop is observed at phases
0.67-0.68, with S-index values decreasing from 0.45 to 0.44. We therefore
observe a sudden transition from a marked rotational modulation of
chromospheric emission to a flatter one, with most of the recorded evolution
taking place during the last three days of the monitoring. Prior to this
specific event, the rotational modulation is more visible, with the strongest
emission around phase 0.4 and the weakest chromospheric flux around phase 0.7.
Even during this first part of the time series, a significant scatter is
present and may be a hint that frequent transitory events contribute to the
S-index.

We also estimated an $H_{\alpha}$ index, following the methodology of Gizis et
al. (2002), and obtain a similar trend, although the contrast between high and
low emission is not as large using this specific chromospheric tracer (see
Fig. 10). We finally checked the chromospheric emission in the CaII IRT
triplet, using the index defined by Petit et al. (2013). In this last case
(not shown here), neither the early rotational modulation or the late drop are
visible in the time-series.

##  7 Radial velocities

Radial velocities have been estimated for both sets of SPIRou and NARVAL LSD
profiles, with values reported in Tab. 1. The accuracy of NARVAL radial
velocity estimates is mainly limited by the wavelength calibration using
telluric lines, as part of the standard reduction process. From observations
of $\tau$ Boo, Moutou et al. (2007) reported a precision of the order of 30 m
s-1. Uncertainties for SPIRou are taken equal to the standard deviation of the
values obtained for a given night, and range between 1 and 2 m s-1. SPIRou
values tend to be smaller than NARVAL values. Although the difference stays
mostly within uncertainties, a different absolute calibration between the two
instrument is likely causing this offset.

All available values are plotted against the rotation phase in Fig. 10. The
rotational signal is not detected in the NARVAL time-series, which is in
agreement with a peak-to-peak activity jitter of about 30 m s-1 reported by
Giguere et al. (2016), below our detection threshold. SPIRou measurements are
too sparse to reveal any rotational dependence. Variations observed over the
four nights reach 24 m s-1peak-to-peak, which remains at a level that we
should not expect to detect with NARVAL.

##  8 Brightness fluctuations

The temporal variability of the optical lightcurve across the TESS observing
window is dominated by quasi-periodic, smooth variations that, to the naked
eye, look roughly consistent with the known rotation period of 11.68 d. The
$\sim 2$ mmag amplitude is smaller than the $\sim 6$ mmag reported by Giguere
et al. (2016), suggesting that our observing epoch was characterized either by
a smaller number of spots and plages, or by a more axisymmetric distribution
of surface brightness inhomogeneities. We model the photometric variations
through a GPR model, based on a pseudo-periodic type of GP and an MCMC
exploration of the hyperparameter space similar to the one described in Sec.
4.

With this aim, we first reduce the number of data points by rebinning the
light curve, as the temporal sampling offered by TESS exceeds by far the
number of observations required to sample the rotation period of $\epsilon$
Eridani. In each bin, the average is weighted with the inverse square of the
error bars, for both time and flux, and the error is the weighted standard
deviation, as described by the following equations.

$<{\rm BJD}>_{\rm bin}=\dfrac{\sum_{{\rm BJD}_{i}\in{\rm bin}}\dfrac{{\rm
BJD}_{i}}{\sigma_{i}^{2}}}{\sum_{{\rm BJD}_{i}\in{\rm
bin}}\dfrac{1}{\sigma_{i}^{2}}}\\\ $ (3)

$<F>_{\rm bin}=\dfrac{\sum_{{\rm BJD}_{i}\in{\rm
bin}}\dfrac{F_{i}}{\sigma_{i}^{2}}}{\sum_{{\rm BJD}_{i}\in{\rm
bin}}\dfrac{1}{\sigma_{i}^{2}}}\\\ $ (4)

$\sigma_{<F>,{\rm bin}}=\sqrt{\dfrac{\sum_{{\rm BJD}_{i}\in{\rm
bin}}{\dfrac{(F_{i}-<F>_{\rm bin})^{2}}{\sigma_{i}^{2}}}}{\sum_{{\rm
BJD}_{i}\in{\rm bin}}\dfrac{1}{\sigma_{i}^{2}}}}$ (5)

where BJD is the barycentric Julian date, $F$ is the flux, and $\sigma_{i}$ is
the error bar on each photometric measurement. We chose to represent the local
error $\sigma_{<F>,{\rm bin}}$ by the standard deviation because the observed
dispersion is around 5 times higher than the photon noise level. We generated
one light curve with 0.2 d bins (114 points in total), and a second one with
0.05 d bins (456 points).

Using 0.2 d bins, the most likely value of the GP hyperparameters include a
rotation period of 11.62 $\pm$ 0.15 d (best: 11.63 d) and a decay time of
43${}^{+18}_{-13}$ d (best: 42 d). Using 0.05 d bins, the same procedure leads
to a rotation period of 11.56$\leavevmode\nobreak\ \pm$ 0.19 d (best: 11.42 d)
and a decay time equal to 33${}^{+13}_{-9}$ d (best: 35 d), in agreement
(within uncertainties) with the values obtained using 0.2 d bins.

While intrinsic evolution of the spot pattern is likely to dominate the
observed non-rotational variability of the light curve, a few transitory
events visible in Fig. 1 (e.g. at Julian date $\sim 2458417$) may also
contribute to reduce the decay time. This may be reflected in the slightly
smaller decay time derived with 0.05 d bins, since short-term fluctuations are
better filtered out using 0.2 d bins.

##  9 Tomographic mapping of the large-scale surface magnetic field

Figure 7: Time-series of NARVAL Stokes $V$ LSD profiles (black dots), and
synthetic profiles produced by the ZDI model (red lines). Successive profiles
are translated vertically for display purposes, with vertical shifts
proportional to phase gaps. The rotation cycle is indicated on the right part
of the plot. Dashed blue horizontal lines depict the zero level of each
profiles.

The series of NARVAL Stokes $V$ pseudo-profiles was used to model the large-
scale magnetic field geometry by means of ZDI. This tomographic approach was
first proposed by Semel (1989) and is based on a maximum entropy
regularisation of the ill-posed problem of inverting a set of circularly
polarised Zeeman signatures. More specifically, we use here a magnetic model
where the surface magnetic field is decomposed over a spherical harmonics
frame (Donati et al., 2006), through the Python implementation of Folsom et
al. (2018a). We also assume that the local Stokes $I$ line profiles
(associated to each surface pixel of our model) take the shape of a Voigt
profile, following Morin et al. (2008) and Folsom et al. (2018b).

The input projected rotational velocity and the inclination angle are taken
equal to the values selected by Jeffers et al. (2014, 2017), with $v\sin
i$$=2.59$ km s-1 (see Sec. 3.2) and $i=46$∘. The differential rotation
parameters were refined compared to the earlier estimate of Jeffers et al.
(2014), but this specific point will be discussed in Sec. 10. The set of
observed Stokes $V$ LSD profiles, as well as the set of synthetic profiles
produced by the ZDI model, are plotted in Fig. 7.

Figure 8: ZDI reconstruction of the large-scale surface magnetic geometry of
$\epsilon$ Eridani, using NARVAL data. Every chart displays a different
component of the field in spherical coordinates, color coded according to the
field strength (expressed in gauss). The vertical ticks on top of the radial
field map show the rotational phases of observations.

The magnetic map of Fig. 8 (including differential rotation) highlights a
complex distribution of magnetic regions, although the small $v\sin i$ value
limits the spatial resolution of the ZDI inversion. The average field strength
is equal to 9.2 G, while the peak field modulus reaches 20 G. The toroidal
magnetic component hosts most of the surface magnetic energy (68%). The field
geometry is also predominantly axi-symmetric, with as much as $73\%$ of the
magnetic energy in spherical harmonics modes with $m=0$. A majority of the
magnetic energy is obtained in the lowest-order components of the spherical
harmonics expansion, with $92\%$ in modes with $1\leq\ell\leq 3$. If we
consider the poloidal field component alone, we find $34\%$ of its magnetic
energy in the dipole, $28\%$ in the quadrupole and $17\%$ in the octupole.

Comparing these general field properties to values previously published by
Jeffers et al. (2014) and Jeffers et al. (2017) is hampered by the different
local profile shape used in our study (Voigt versus gaussian), as well as a
slightly different $v\sin i$ value (2.2 km s-1 in these previous studies). To
allow for a more direct comparison, we reconstructed a magnetic model based on
these alternate ZDI settings. The fit to Stokes $I$ profiles is significantly
better with a Voigt profile that allows for a good adjustment of both the core
and wings of LSD pseudo-profiles, while a gaussian profile provides only a
good fit to the line core. The consequence on the Stokes $V$ adjustment is, in
practice, less dramatic with general reconstructed field characteristics
reasonably close to the ones found using a Voigt profile. The alternate model
leads to a mean field strength of 8 G, 64% of the magnetic energy in the
toroidal component, or 68% of the energy in axisymmetric modes. The average
field value obtained here is on the lower end of previously published values
(obtained close to activity minimum). On the other hand, the fraction of
energy in the toroidal component is the highest reported to date. As noticed
by See et al. (2015), higher toroidal fields generally correspond to higher
field axisymmetry, which is also observed here with a magnetic geometry among
the most axisymmetric recorded so far for $\epsilon$ Eridani. This atypical
combination of magnetic properties may suggest that the dynamo activity of
$\epsilon$ Eridani, which was already reported in the past to change from
chaotic to cyclic (Metcalfe et al., 2013) may have entered another new phase,
as suggested by the absence of the expected activity maximum in early 2019
(Coffaro et al., 2020) and low flaring activity in radio observations obtained
in 2019 by Suresh et al. (2020).

##  10 Surface differential rotation

Reconstructing the magnetic field geometry of $\epsilon$ Eridani under the
simple assumption of solid-body rotation leads to a disappointing reduced
$\chi^{2}$ ($\chi^{2}_{\rm r}$ hereafter) equal to 3.2. We try here to improve
the model by assuming that the surface experiences a latitudinal shear.

The ZDI model includes the possibility for the surface field geometry to
change with time, under the progressive shear imposed by a solar-like
differential rotation law described by the following equation:

$\Omega(l)=\Omega_{\rm eq}-d\Omega.\sin^{2}(l)$ (6)

ı

Here $\Omega(l)$ is the rotation rate at stellar latitude $l$, $\Omega_{\rm
eq}$ the rotation rate at equatorial latitude, and $d\Omega$ stands for the
difference in rotation rate between the equator and pole. Following Donati et
al. (2000) and Petit et al. (2002), the two free parameters of this simple law
are estimated by running a large number of ZDI models over a grid of values of
the two parameters ($\Omega_{\rm eq};d\Omega$), searching for values that
optimize the ZDI model (i.e. that minimize the $\chi^{2}_{\rm r}$ of the
model, at fixed entropy value).

Figure 9: Reduced $\chi^{2}$ landscape obtained for a grid of ZDI models
implementing Eq. 6. The three concentric, white contours depict the $1\sigma$,
$2\sigma$ and $3\sigma$ limits around the $\chi^{2}_{\rm r}$ minimum.

A clear minimum is found in the $\chi^{2}$ landscape (Fig. 9), for an
equatorial rotation period $P_{\rm eq}=2\pi/\Omega_{\rm eq}=10.77\pm 0.06$ d
and $d\Omega=0.11\pm 0.01$ rad d-1. The resulting $\chi^{2}_{\rm r}$ value is
close to 1.5, unveiling a much better fit to the data when the surface is
assumed to be sheared by differential rotation. The fact that our best
$\chi^{2}_{\rm r}$ is still larger than one is indicative that other phenomena
(e.g. continuous emergence and decay of magnetic spots) contribute as well to
the magnetic evolution.

The periods derived from longitudinal field measurements and photometry (Sec.
4 and 8) are longer than the equatorial period obtained through the ZDI model.
They are also shorter than the rotation period we can extrapolate at polar
latitude from Eq. 6 (about 13.3 d). This is expected if the main surface
features contributing to the longitudinal magnetic field or to the photometric
variability are located at intermediate latitudes. The shear level obtained
here is about twice the solar value, and leads to a laptime (the time it takes
for the equator to be one rotation cycle ahead of the pole) of
$2\pi/d\Omega=57\pm 5$ d, providing us with a third typical timescale of
surface intrinsic evolution.

##  11 Discussion

###  11.1 Phase dependence of activity tracers

The different tracers investigated here of the magnetic activity at
photospheric and chromospheric levels produce a diversity of temporal
signatures, primarily highlighted by their different rotational dependence. In
absolute value, the longitudinal magnetic field is maximal around phase $0.6$,
which on the magnetic map is translated as a peak in the radial field strength
(with negative polarity). In the light curve, this specific rotation phase is
off any extrema, with a brightness maximum at phase 0.75, while the minimum in
the light flux is recorded around phase 0.45.

The longitudinal field is switching from positive to negative at phase 0.4,
close to the minimum light flux recorded by TESS. On the magnetic map, this
phase is mostly characterized by a minimum in the azimuthal field, as well as
a transition from a positive to a negative radial field polarity at
intermediate and low latitudes (which may suggest that the magnetic equator
preferentially hosts cooler spots than the magnetic pole). Another zero-
crossing of the longitudinal field is recorded around phases 0.95-1, where the
radial field at intermediate latitudes on the ZDI map goes from a negative to
a positive polarity. This second field minimum has no remarkable counterpart
on the light curve, which stays close to its average value in the same phase
interval.

Prior to its weakening during the last days of the NARVAL time-series, the
S-index of $\epsilon$ Eridani was maximal around phase 0.4, where in November
the light curve was near its minimum, and where the radial component of the
large-scale magnetic field was switching polarity. During the last rotation
period covered in the time-series, the phase dependence of the S-index seems
to become flatter, although late observations of phases above 0.7 are missing
to track this fast evolution throughout the whole rotation cycle. We note that
this sharp evolution, taking place within a few days, is not reflected in the
longitudinal field values, for which the phase modulation is mostly stable
over the whole time-series. The TESS data, which are representative of the
second half of the time-span covered by NARVAL, display a same peak-to-peak
amplitude (about 5 mmag) in the first and second observed rotation period
(blue and orange symbols in the upper panel of Fig. 10). The incomplete phase
coverage of the first period (with no available data for phases above 0.65,
missing the phase of maximal flux) leads to this apparent stability, while the
photometric amplitude seems to have mostly decreased between the two periods,
at least for the brightest phases (phases 0.0 to 0.35, and phases above 0.6),
while the dimmest phases seem to be unaffected by this evolution.

Among the possible explanations of the apparently discrepant phase dependence
of the different magnetic tracers, one obviously at play is their different
spatial resolution. Magnetic fields on $\epsilon$ Eridani are likely
distributed in a complex pattern featuring both field polarities, and nearly
identical radial velocities (due to the relatively small $v\sin i$). The
visible magnetic spots have most of their Stokes $V$ signatures mutually
canceled, and the remaining signal is limited to the largest spatial scales of
the surface field. Zeeman broadening, on the opposite, does not depend on
field polarity, explaining the much weaker field strength reconstructed in the
ZDI map, compared to the Zeeman broadening of nIR lines (a detailed discussion
of this aspect can be found in See et al. 2019). The absence of any detectable
phase dependence of the Zeeman broadening tends to support the picture of
complex distribution of magnetic regions. The broad band stellar photometry is
a degenerate observation, with a disc-integrated brightness reflecting the
surface balance between dark and bright surface features. Finally, the
chromospheric emission is a cumulative effect, unaffected by the local
orientation of the chromospheric magnetic field.

Another effect contributing to a different phase dependence between the
different quantities investigated here is their different limb visibility.
While polarized Zeeman signatures produced by spots with radial magnetic
fields are best seen close to disc center, azimuthal magnetic fields will have
larger Stokes $V$ amplitudes at intermediate limb angles (Donati & Brown,
1997). Similar considerations impact the interpretation of the TESS light
curve, with the contribution of dark spots being maximal close to disc center
while, by analogy with the Sun, faculae may be brighter close to the limb
(Hirayama & Moriyama, 1979), which may also affect the phase dependence of the
S-index.

###  11.2 Characteristic timescales for short-term surface evolution

Longitudinal field measurements, photometric variations and the differential
rotation model provide us with three independent estimates of evolution
timescales. Two decay times are obtained from GPR applied to $B_{\rm eff}$ and
TESS data, while $2\pi/d\Omega$ gives a characteristic timescale of the
surface shear. All three estimates are consistent within uncertainties. The
longest one is the differential rotation laptime ($57\pm 5$ d). The decay time
deduced from photometry comes second and is equal to 43${}^{+18}_{-13}$ d,
while the same quantity estimated from $B_{\rm eff}$ measurements is the
shortest and is equal to $31^{+9}_{-7}$ d. The laptime is linked to the
specific component of the surface evolution driven globally by differential
rotation, while the two other estimates include as well the contribution of
the limited lifetime of surface structures (note that the laptime estimated
for differential rotation can also be biased whenever the shear tracers come
and go, see Petit et al. 2002). We interpret this difference as the cause of
the shorter timescales obtained out of the light curve and longitudinal field
data.

##  12 Conclusions and prospects

This multi-instrumental view of $\epsilon$ Eridani reveals how different
tracers of magnetism and activity carry different and complementary
information about the surface activity linked to the vivid dynamo action of
young solar-like stars. Each available measurement brings its own set of clues
about the underlying emergence and decay of active regions, each specific
tracer being limited by its own degeneracy, spatial resolution, temporal
resolution, or limb dependence. The conclusions that we can draw from the
diverse data presented here are also limited by the non-simultaneity of the
observations. This would advocate the future development of high-resolution
spectropolarimeters covering both the optical and near-infrared domain, as
would be offered by a combination of SPIRou and ESPaDOnS. In addition to help
reach a better understanding of photospheric and chromospheric stellar
activity, such instruments would help progress in the filtering of stellar
activity, as part of RV exoplanet search and characterization around cool
active stars.

###### Acknowledgements.

JFD acknowledges funding from from the European Research Council (ERC) under
the H2020 research & innovation programme (grant agreement # 740651
NewWorlds). AAV acknowledges funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 817540, ASTROFLOW). This research made use of the SIMBAD
database operated at CDS, Strasbourg, France, and the NASA’s Astrophysics Data
System Abstract Service.

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##  Appendix A Synthetic view of all activity tracers

Figure 10: Synthetic view of all activity tracers, as a function of the
rotational phase. From top to bottom: normalized TESS light curve, with the
first period in blue and the second one in orange. The longitudinal magnetic
field, the S index, the H$\alpha$ index, radial velocities.

##  Appendix B Extra source of line broadening in the infrared domain

Table 6: Parameters derived from individual windows in the IR spectrum, for fits using macroturbulence controlling broadening (upper portion) with $v\sin i$ inferred from rotation period and radius, and no magnetic field. Fits using the magnetic field from the Zeeman broadening analysis (and $v\sin i$ from the visible spectrum analysis) are also presented (bottom portion).  | 10500–10920 | 11760–12600 | 15110–15697 | 15815–16390 | 16439–17140 | 21017–22850
---|---|---|---|---|---|---
$T_{\rm eff}$ (K) | 5082 | 5051 | 4909 | 4942 | 4983 | 4981
$\log g$ | 4.65 | 4.45 | 4.49 | 4.46 | 4.61 | 4.21
$v\sin i$ (km s-1) | 2.59 | 2.59 | 2.59 | 2.59 | 2.59 | 2.59
$\xi_{\rm mic}$ (km s-1) | 1.09 | 1.25 | 0.89 | 1.48 | 0.62 | 2.28
$\xi_{\rm mac}$ (km s-1) | 2.08 | 2.30 | 3.91 | 4.34 | 3.84 | 4.81
metallicity | -0.10 | -0.03 | -0.05 | -0.12 | -0.07 | -0.14
B=1700 G, f=0.141
$T_{\rm eff}$ (K) | 5066 | 5081 | 4947 | 5048 | 5067 | 5044
$\log g$ | 4.60 | 4.51 | 4.50 | 4.65 | 4.62 | 4.55
$v\sin i$ (km s-1) | 2.8 | 2.8 | 2.8 | 2.8 | 2.8 | 2.8
$\xi_{\rm mic}$ (km s-1) | 0.72 | 0.85 | 0.51 | 0.89 | 0.0 | 0.92
$\xi_{\rm mac}$ (km s-1) | 1.90 | 1.28 | 3.19 | 3.46 | 3.55 | 4.01
metallicity | -0.06 | -0.02 | -0.04 | -0.06 | -0.02 | -0.03

We find an unexpected trend in macroturbulence towards larger values for
longer wavelengths. Alternatively modelling the line broadening as $v\sin i$
produces the same trend. The other best fit parameters do not vary with
wavelength, thus this appears to be a feature of the observations, not an
error in our methodology. To investigate if this is real, we fit the solar
spectrum obtained with SPIRou for the same spectral windows, assuming $v\sin
i$ $=2$ km s-1 and using macroturbulence as a free parameter. We find a trend
that is similar but weaker and less consistent, from 2.5 km s-1 in the blue to
3.8 km s-1 in the red. This is unlikely to be an artefact of the instrument or
data reduction process, since calibration images show a consistent line width
across the spectrum. To further check for instrumental effects we fit Gaussian
profiles to 77 telluric lines in the observation of $\epsilon$ Eridani,
distributed across the SPIRou domain. While there is some scatter in Gaussian
widths, we find consistent widths as a function of wavelength, and widths of
the narrower lines are consistent with an R of 70000. Thus this does not
appear to be an instrumental or data reduction effect.

Rotational broadening ($v\sin i$) should be consistent across a spectrum,
apart from a wavelength dependence in limb darkening, which our models account
for. Turbulent broadening may be depth dependent, and widely separated
wavelengths have different opacities, thus the physical depth of formation of
spectra lines varies with wavelength. However it is not clear if changes in
turbulence with depth could explain a 3.5 km s-1 difference between 6000 Å and
22000 Å. The contribution of cool spots to the spectrum increases with
wavelength, however it is not clear that cool spots would have much large
turbulent broadening than the rest of the photosphere (convection is generally
considered to be suppressed in these regions), thus this does not offer an
obvious explanation.

The relative impact of Zeeman broadening increases with wavelength (the Zeeman
effect scales as $\lambda^{2}$ while most other broadening processes scale as
$\lambda$), offering a possible explanation. In Sect 5 using Zeeman broadening
we find a 1800 G magnetic field covering 14% of the stellar surface. We
repeated the above spectroscopic analysis assuming this magnetic model, rather
than no magnetic field. This produced a lower reduced $\chi^{2}$ (by 1 or 2)
than for the model with no magnetic field in all windows, except one where
reduced $\chi^{2}$ was largely unchanged. We find $T_{\rm eff}$, $\log g$, and
metallicity that are virtually unchanged, a microturbuence that is reduced to
$0.78\pm 0.15$ km s-1, however there still is a systematic trend in
macroturbulence with wavelength, from 1.90 km s-1 in the blue-most window up
to 4.01 km s-1 in the red-most. Thus including this magnetic field only
slightly reduced the wavelength dependence of macroturbulence and is not
sufficient to explain it.

To further investigate whether this wavelength dependent broadening could be
Zeeman broadening, we fit the 10500–10920 and 21017–22850 Å windows
simultaneously, adding a magnetic field strength and filling factor to the
free parameters of $T_{\rm eff}$, $\log g$, microturbulence, and metallicity.
Macrotrublence was fixed to the value of the bluer window from above (1.90 km
s-1), since if left free it tends to a larger intermediate value, too broad
for the bluer window. This produced a best fit magnetic field of 1587 G and
filling factor of 0.291, while $T_{\rm eff}$, $\log g$, and metallicity were
consistent with above (microturbulence was smaller at 0.36 km s-1). As a
second attempt the 11760–12600 and 16439–17140 Å windows were fit
simultaneously using the same approach. This produced a magnetic field of 2107
G and a filling factor of 0.284, and otherwise consistent stellar parameters.
These two models did a good job fitting the line widths in the redder and
bluer windows, but both produced filling factors that are inconsistently large
for the Zeeman broadening analysis in Sec. 5. Specifically, the Fe i 15343.79,
15611.14, and 15648.51 Å lines analyzed below all show wings that are too deep
with these two models. Thus including Zeeman broadening is important for
accurately determining line broadening in the infrared, but apparently not
sufficient to explain the wavelength dependence in line broadening we find
here.

We tried allowing the temperature of the magnetic region to differ from the
non-magnetic area. This involves effectively calculating spectra for the two
regions using different model atmospheres, interpolated from the same grid and
assuming the same $\log g$, but using different $T_{\rm eff}$. The flux ratio
between cool to warm regions increases further to the infrared. If the
magnetic regions were cooler, perhaps this could help produce extra Zeeman
broadening further into the infrared without increasing the filling actor.
Fitting the 10500–10920 and 21017–22850 Å windows with this model we found a
magnetic field of 1652 G, a filling factor of 0.262, a temperature of 4731 K
in the magnetic region, and a temperature of 5182 K in the non-magnetic
region. Fitting the 11760–12600 and 16439–17140 Å windows we found a magnetic
field of 2066 G, filling factor of 0.307, temperature in the magnetic region
of 4615 K, and temperature in the non-magnetic region of 5321 K. However, on
closer inspection, the temperature in the cooler region appears to be driven
largely by the strength of a few lines with very low excitation potentials.
This suggests a possible spot temperature of 4600-4700 K, but does a poor job
of providing a model that could explain the wavelength dependent line
broadening.

From these tests, it appears that the wavelength dependent line broadening is
real, and cannot be fully explained by Zeeman broadening. This suggests that a
depth dependent turbulent velocity should be investigated. However, that goes
beyond the simple micro- macro-turbulence approximation, and may require 3D
hydrodynamic model atmospheres to properly investigate.

##  Appendix C Corrections to atomic line data

The corrections to atomic line data that we adopted in the SPIRou wavelength
range are presented here. The data were initially extracted from VALD version
3, on 14 Feb. 2019, using an ‘extract stellar’ request for the parameters of
$\epsilon$ Eridani, and the default ‘line list configuration’ (i.e. selection
of input line lists). An extensive list of empirical corrections to the
oscillator strengths were derived by fitting a solar spectrum, as discussed in
Sect. 3.2. The modified line data are presented in Tab. 7. Lines that did not
require modification are omitted for brevity. A number of theoretical
transitions were predicted to be detectable but were not present in our
observations. These are indicated in Tab. 7 by ‘*’. A few lines in VALD were
apparent duplicates from different sources, specifically components of Mg i
blends. These are also listed with a ‘*’. For some Si lines we adopted $\log
gf$ from the NIST Atomic Spectra Database, with the original data from
Zatsarinny & Bartschat (2006). Since these provided adequate fits to the
observation, these adopted values are included in the Table. Also included are
empirical $\log gf$ corrections for a few Fe i lines from Valenti et al.
(1995), and Ti i lines from Johns-Krull et al. (2004), since these provided an
adequate match to the observations.

Additionally, for lines with visible disagreement between the model and
observation in the widths of the wings, empirical corrections to the van der
Waals damping parameter ($\gamma_{6}$) were derived. In some cases
$\gamma_{6}$ values were unavailable, and calculations in the Unsöld
approximation appeared to be insufficient, so empirical values were derived.
The empirical corrections to the transition data likely depend on the
parameters of star being investigated and the limitations of the model being
employed, thus they should be treated with caution.

While there are a significant number of apparent errors in the oscillator
strengths currently available from VALD in the infrared, the list of atomic
lines is largely complete. Very few atomic lines in our observations were
missing a theoretical counterpart in VALD. While we did not investigate them
in detail, the molecular line list also appears to be largely complete.
However, outside of G and K spectral types, especially towards cooler M
dwarfs, the completeness of the line list may become an issue.

Table 7:  Empirical modifications to the line data adopted. Lines with a $\log gf$ showing up as ‘*’ were predicted but not detected in the observations (or apparent duplicates). When a correction to the van der Waals damping parameter ($\gamma_{6}$) was used, it is indicated. $\Delta\log gf$ and $\Delta\gamma_{6}$ are differences with respect to the values from VALD. Lines without a $\Delta\gamma_{6}$ value did not have a $\gamma_{6}$ value available in VALD.  $\lambda$ (Å) | ion | $\log gf$ | $\Delta\log gf$ | $\gamma_{6}$ | $\Delta\gamma_{6}$
---|---|---|---|---|---
10511.588 | P i | 0.10 | 0.23 | – | –
10532.234 | Fe i | -1.67 | -0.19 | – | –
10535.709 | Fe i | -0.14 | -0.03 | – | –
10541.227 | C i | -1.09 | 0.30 | – | –
10555.649 | Fe i | -1.30 | -0.19 | – | –
10577.139 | Fe i | -3.06 | 0.08 | – | –
10582.160 | Si i | -1.04 | 0.13 | – | –
10596.903 | P i | 0.11 | 0.32 | – | –
10602.816 | Si i | -2.17 | -1.23 | – | –
10617.877 | Fe i | * | * | – | –
10622.592 | Fe i | * | * | – | –
10627.648 | Si i | -0.30 | 0.56 | – | –
10633.080 | S i | * | * | – | –
10661.623 | Ti i | -1.88 | 0.04 | – | –
10667.380 | Fe i | -1.90 | -0.44 | – | –
10667.520 | Cr i | -1.80 | -0.32 | – | –
10677.047 | Ti i | -3.00 | -0.48 | – | –
10694.251 | Si i | 0.39 | 0.34 | -7.23 | -0.25
10709.942 | Fe i | * | * | – | –
10717.806 | Fe i | -1.50 | -1.06 | – | –
10726.391 | Ti i | -2.00 | 0.06 | – | –
10727.406 | Si i | 0.50 | 0.28 | -7.07 | -0.11
10731.947 | Fe i | -2.30 | -0.46 | – | –
10741.728 | Si i | -0.81 | 0.08 | – | –
10753.004 | Fe i | -2.05 | -0.20 | – | –
10754.281 | Fe i | -1.80 | -0.61 | – | –
10761.445 | Fe i | * | * | – | –
10768.365 | Al i | -2.10 | -0.60 | – | –
10780.694 | Fe i | -3.60 | -0.31 | – | –
10782.045 | Al i | -1.90 | -0.65 | – | –
10783.050 | Fe i | -2.74 | -0.17 | – | –
10784.562 | Si i | -0.68 | 0.16 | – | –
10786.849 | Si i | -0.10 | 0.20 | -7.56 | -0.29
10796.106 | Si i | -1.34 | -0.08 | – | –
10811.053 | Mg i | 0.02 | 0.00 | -6.75 | -0.07
10811.084 | Mg i | -0.14 | 0.00 | -6.75 | -0.07
10811.097 | Mg i | -1.04 | 0.00 | -6.75 | -0.07
10811.122 | Mg i | -1.04 | 0.00 | -6.75 | -0.07
10811.158 | Mg i | -0.30 | 0.00 | -6.75 | -0.07
10811.198 | Mg i | * | * | – | –
10811.219 | Mg i | * | * | – | –
10818.274 | Fe i | -2.03 | -0.08 | – | –
10825.079 | Ca i | * | * | – | –
10827.088 | Si i | 0.75 | 0.45 | -8.12 | -0.86
10831.938 | Ca i | * | * | – | –
10832.595 | Ca i | * | * | – | –
10838.970 | Ca i | 0.07 | -0.17 | – | –
10846.792 | Ca i | -0.31 | 1.01 | – | –
10849.465 | Fe i | -0.66 | 0.78 | – | –
10853.001 | Fe i | -1.22 | 0.23 | – | –
10869.536 | Si i | 0.55 | 0.18 | -7.39 | -0.16
10881.758 | Fe i | -3.41 | 0.20 | – | –
10882.809 | Si i | -0.64 | 0.18 | – | –
10884.262 | Fe i | -2.07 | -0.14 | – | –
10885.333 | Si i | 0.15 | -0.08 | -8.13 | -0.80
10891.736 | Al i | -0.98 | 0.12 | – | –
10905.710 | Cr i | -0.78 | -0.22 | – | –
10914.244 | Mg ii | 0.05 | 0.03 | – | –
10914.887 | Sr ii | -0.25 | 0.39 | – | –
10915.284 | Mg ii | -1.67 | -0.74 | – | –
11783.265 | Fe i | -2.00 | -0.43 | -7.50 | 0.32
11811.558 | Mg i | * | * | – | –
11820.982 | Mg i | * | * | – | –
11828.171 | Mg i | -0.18 | 0.15 | -7.27 | -0.08
11838.997 | Ca ii | 0.57 | 0.26 | – | –
11848.710 | C i | -0.60 | 0.10 | – | –
11879.580 | C i | -0.48 | 0.13 | – | –
11882.844 | Fe i | -2.44 | -0.78 | -6.99 | 0.83
11884.083 | Fe i | -2.79 | -0.71 | -6.78 | 1.04
11949.547 | Ti i | -2.28 | -0.71 | – | –
11949.744 | Ca ii | -0.08 | -0.09 | -6.83 | 0.73
11973.046 | Fe i | -3.26 | -1.78 | -6.13 | 1.69
11973.847 | Ti i | -1.46 | -0.07 | – | –
11984.198 | Si i | 0.30 | 0.06 | -7.40 | -0.10
11991.568 | Si i | -0.09 | 0.02 | -7.37 | -0.07
12005.397 | Fe i | -1.14 | -0.60 | – | –
12005.547 | Fe i | -0.91 | 0.01 | – | –
12010.578 | Fe i | -1.80 | -0.63 | – | –
12031.504 | Si i | 0.72 | 0.25 | -7.81 | -0.52
12053.082 | Fe i | -1.68 | -0.13 | – | –
12081.972 | Si i | -0.61 | -0.09 | – | –
12083.278 | Mg i | * | * | – | –
12083.346 | Mg i | -1.56 | -0.77 | – | –
12083.649 | Mg i | 0.41 | 0.00 | -7.08 | -0.10
12084.976 | C i | -0.87 | -0.41 | – | –
12100.181 | Si i | -1.37 | -0.24 | -6.10 | 0.88
12105.841 | Ca i | -0.42 | -0.11 | -6.27 | 0.82
12110.659 | Si i | -0.59 | -0.46 | – | –
12119.494 | Fe i | -1.86 | -0.23 | – | –
12133.995 | Si i | -1.67 | 0.43 | – | –
12189.241 | Si i | -0.88 | 0.13 | – | –
12190.098 | Fe i | -2.80 | -0.47 | – | –
12227.112 | Fe i | -1.59 | -0.23 | – | –
12342.916 | Fe i | -1.67 | -0.21 | – | –
12390.154 | Si i | -1.85 | -0.08 | – | –
12395.832 | Si i | -1.81 | -0.16 | – | –
12417.936 | Mg i | -1.80 | -0.14 | – | –
12423.029 | Mg i | -1.35 | -0.17 | – | –
12432.273 | K i | -0.31 | 0.13 | – | –
12433.452 | Mg i | -1.06 | -0.09 | – | –
12433.748 | Ca i | -0.39 | -0.32 | – | –
12457.132 | Mg i | -1.59 | 0.67 | – | –
12510.519 | Fe i | -2.01 | -0.40 | – | –
12521.810 | Cr i | -1.57 | 0.01 | – | –
12522.134 | K i | 0.02 | 0.16 | – | –
12556.996 | Fe i | -4.08 | -0.46 | – | –
12569.634 | Co i | -1.29 | -0.30 | – | –
15112.331 | Fe i | -0.85 | -0.23 | – | –
15120.504 | Fe i | -1.52 | -0.72 | – | –
15122.380 | Fe i | -0.49 | -0.28 | -6.92 | 0.59
15122.549 | Si i | -1.47 | -0.20 | – | –
15135.306 | Mg i | -1.81 | -0.15 | – | –
15136.124 | Fe i | -0.51 | -0.39 | -6.90 | 0.55
15143.089 | Fe i | -1.16 | -0.27 | – | –
15144.051 | Fe i | -0.50 | -0.10 | – | –
15155.208 | Fe i | -1.40 | -0.87 | – | –
15163.067 | K i | 0.51 | -0.18 | – | –
15168.376 | K i | 0.36 | -0.11 | – | –
15176.713 | Fe i | -0.76 | -0.26 | -6.80 | 0.67
15182.924 | Fe i | -0.78 | -0.25 | – | –
15183.435 | Fe i | -1.25 | -0.18 | – | –
15201.561 | Fe i | -0.56 | -0.40 | – | –
15207.526 | Fe i | -0.20 | -0.53 | -6.72 | 0.77
15213.020 | Fe i | -0.69 | -0.22 | – | –
15219.618 | Fe i | -0.06 | 0.77 | -7.13 | 0.32
15231.593 | Mg i | -1.90 | -0.74 | – | –
15231.681 | Mg i | -2.30 | -0.90 | – | –
15231.776 | Mg i | -2.00 | -0.50 | – | –
15239.712 | Fe i | -0.07 | -0.04 | – | –
15243.588 | Si i | -1.14 | -0.27 | – | –
15244.973 | Fe i | -0.08 | -0.01 | -6.96 | 0.49
15246.394 | Fe i | -3.06 | 0.55 | – | –
15259.363 | Fe i | -1.60 | -0.35 | – | –
15260.642 | Fe i | -0.70 | -0.23 | – | –
15271.550 | Fe i | -2.04 | -1.29 | – | –
15293.135 | Fe i | 0.01 | -0.13 | – | –
15294.560 | Fe i | 0.38 | -0.34 | -7.02 | 0.47
15323.555 | Fe i | -0.71 | -0.12 | – | –
15334.847 | Ti i | -1.10 | -0.14 | – | –
15343.788 | Fe i | -0.67 | -0.09 | – | –
15348.367 | Fe i | -1.66 | -0.56 | – | –
15348.966 | Fe i | -0.86 | 0.40 | – | –
15375.346 | Fe i | -1.39 | -0.40 | – | –
15375.428 | Si i | -1.50 | 0.20 | – | –
15376.831 | Si i | -0.74 | -0.05 | -6.70 | 0.69
15381.960 | Fe i | -0.69 | -0.23 | – | –
15394.673 | Fe i | -0.35 | -0.36 | -6.47 | 0.98
15398.485 | Fe i | -2.04 | -1.96 | – | –
15400.077 | S i | 0.38 | -0.17 | – | –
15402.331 | S i | * | * | – | –
15403.724 | S i | -0.35 | -0.67 | – | –
15403.791 | S i | 0.55 | 0.38 | – | –
15405.978 | S i | -1.09 | -0.71 | – | –
15422.261 | S i | -0.35 | -0.71 | – | –
15422.276 | S i | 0.71 | 0.34 | – | –
15427.619 | Fe i | -0.86 | -0.20 | – | –
15444.354 | Fe i | * | * | – | –
15444.376 | Fe i | * | * | – | –
15451.298 | Fe i | -0.40 | -0.13 | – | –
15469.816 | S i | -0.20 | -0.15 | – | –
15475.182 | Fe i | -0.70 | 0.71 | – | –
15475.204 | Fe i | -2.10 | -0.09 | -6.50 | 1.04
15475.616 | S i | -0.68 | -0.16 | – | –
15475.897 | Fe i | -1.87 | -0.76 | – | –
15476.500 | Fe i | -1.05 | -0.24 | – | –
15478.482 | S i | 0.02 | -0.16 | – | –
15479.603 | Fe i | -0.90 | -0.56 | – | –
15484.334 | Fe i | * | * | – | –
15485.454 | Fe i | -0.81 | 0.29 | – | –
15486.078 | Fe i | * | * | – | –
15490.337 | Fe i | -4.90 | -0.32 | – | –
15490.526 | Fe i | * | * | – | –
15490.881 | Fe i | -0.62 | -0.05 | – | –
15497.000 | Si i | -2.27 | -0.25 | – | –
15497.041 | Fe i | -1.52 | -0.74 | – | –
15500.799 | Fe i | -0.14 | -0.09 | – | –
15501.320 | Fe i | 0.05 | -0.28 | – | –
15506.978 | Si i | -1.63 | -0.34 | – | –
15510.642 | Fe i | * | * | – | –
15524.308 | Fe i | -1.28 | -0.40 | – | –
15531.751 | Fe i | -0.63 | -0.39 | -6.68 | 0.77
15531.802 | Fe i | -0.93 | -0.09 | – | –
15532.449 | Si i | -1.78 | -0.38 | – | –
15534.245 | Fe i | -0.46 | -0.08 | -6.67 | 0.84
15537.453 | Fe i | -1.67 | -0.59 | – | –
15537.695 | Fe i | -0.28 | -0.25 | – | –
15542.079 | Fe i | -0.65 | -0.31 | -6.54 | 0.91
15543.761 | Ti i | -1.29 | -0.21 | – | –
15547.711 | Fe i | -1.28 | -0.63 | – | –
15550.435 | Fe i | -0.30 | -0.20 | – | –
15551.433 | Fe i | -0.21 | 0.16 | – | –
15557.778 | Si i | -0.78 | 0.03 | -6.97 | 0.42
15560.784 | Fe i | -0.36 | 0.11 | – | –
15565.222 | Fe i | -0.77 | -0.21 | – | –
15566.725 | Fe i | -0.37 | 0.31 | – | –
15586.927 | Fe i | * | * | – | –
15588.259 | Fe i | 0.48 | 0.06 | – | –
15590.046 | Fe i | -0.42 | 0.41 | – | –
15591.490 | Fe i | 0.69 | -0.18 | -6.97 | 0.36
15593.749 | Fe i | -1.76 | 0.17 | – | –
15594.396 | Fe i | * | * | – | –
15598.769 | Fe i | -1.00 | -0.72 | – | –
15598.869 | Fe i | -1.07 | -0.83 | – | –
15602.842 | Ti i | -1.67 | -0.23 | – | –
15604.220 | Fe i | 0.45 | -0.09 | – | –
15605.684 | Ni i | -0.26 | -0.28 | – | –
15611.145 | Fe i | -3.30 | 0.47 | – | –
15613.625 | Fe i | -0.15 | 0.52 | – | –
15617.701 | Fe i | * | * | – | –
15621.654 | Fe i | 0.01 | -0.58 | -6.61 | 0.84
15629.364 | Fe i | -1.82 | -0.81 | – | –
15631.947 | Fe i | 0.10 | -0.02 | – | –
15639.477 | Fe i | -0.87 | -0.80 | – | –
15647.413 | Fe i | -1.08 | 1.21 | – | –
15648.510 | Fe i | -0.63 | -0.03 | – | –
15649.674 | Fe i | * | * | – | –
15650.563 | Si i | * | * | – | –
15652.871 | Fe i | -0.04 | 0.12 | – | –
15662.013 | Fe i | 0.12 | -0.25 | -6.92 | 0.53
15665.240 | Fe i | -0.41 | -0.08 | – | –
15670.124 | Fe i | -0.80 | 0.18 | – | –
15671.004 | Fe i | -0.45 | -0.23 | – | –
15671.866 | Fe i | -1.19 | -0.13 | – | –
15673.151 | Fe i | -0.58 | 0.15 | – | –
15674.652 | Si i | -0.98 | 0.35 | – | –
15677.012 | Fe i | -0.66 | -0.52 | – | –
15677.519 | Fe i | 0.29 | 0.84 | – | –
15680.060 | Cr i | 0.04 | -0.11 | – | –
15682.016 | Fe i | * | * | – | –
15682.513 | Fe i | -0.25 | 0.02 | – | –
15683.387 | Fe i | -1.61 | 0.57 | – | –
15686.020 | Fe i | -0.07 | -0.05 | – | –
15686.441 | Fe i | 0.20 | -0.37 | – | –
15687.140 | Fe i | -0.80 | -0.31 | – | –
15691.853 | Fe i | 0.47 | -0.18 | -6.85 | 0.48
15693.311 | Mg i | -1.70 | -0.68 | – | –
15693.454 | Mg i | -1.90 | -0.72 | – | –
15693.555 | Mg i | -1.30 | 0.05 | – | –
15816.631 | Fe i | -0.67 | -0.34 | – | –
15822.816 | Fe i | -0.02 | -0.20 | – | –
15827.213 | Si i | -0.75 | -0.10 | – | –
15833.602 | Si i | -0.33 | -0.14 | -7.52 | -0.14
15834.164 | Fe i | -0.70 | -0.25 | – | –
15840.190 | Fe i | -0.38 | 0.78 | – | –
15852.580 | C i | -0.44 | -0.18 | – | –
15852.807 | Fe i | -0.84 | -0.29 | – | –
15853.315 | Fe i | -0.76 | -0.20 | – | –
15854.029 | Fe i | -2.69 | -0.68 | – | –
15858.656 | Fe i | -1.33 | -0.85 | – | –
15868.524 | Fe i | -0.03 | -0.12 | – | –
15868.572 | Fe i | -0.09 | -0.52 | – | –
15873.843 | Ti ii | -2.01 | -0.20 | – | –
15878.444 | Fe i | -0.31 | 0.71 | – | –
15884.454 | Si i | -0.91 | -0.08 | -6.83 | 0.56
15886.188 | Mg i | -1.70 | -0.18 | – | –
15888.409 | Si i | 0.06 | 0.00 | -7.40 | 0.18
15891.160 | Fe i | -0.35 | -0.33 | – | –
15892.395 | Fe i | 0.16 | 0.15 | – | –
15892.769 | Fe i | 0.16 | 0.04 | – | –
15896.555 | Fe i | -0.77 | -1.02 | – | –
15898.016 | Fe i | 0.35 | 0.08 | -6.46 | 0.86
15898.890 | Fe i | -1.83 | -1.34 | – | –
15899.252 | Fe i | -0.34 | -0.06 | – | –
15899.586 | Fe i | -1.00 | -0.88 | – | –
15899.710 | Si i | -1.27 | -0.33 | – | –
15901.518 | Fe i | -0.46 | 0.42 | – | –
15906.044 | Fe i | -0.09 | -0.42 | – | –
15909.084 | Fe i | -0.70 | -0.77 | – | –
15909.241 | Fe i | -0.70 | -0.24 | – | –
15912.591 | Fe i | -0.61 | -0.67 | – | –
15912.594 | Mg i | -1.84 | -0.58 | – | –
15913.627 | Fe i | -1.33 | -1.04 | – | –
15914.116 | Si i | -1.73 | -0.46 | – | –
15920.642 | Fe i | 0.38 | 0.01 | – | –
15921.096 | Fe i | -1.36 | -0.44 | – | –
15922.442 | Fe i | -1.10 | -0.49 | – | –
15922.600 | Fe i | -1.00 | -0.48 | – | –
15928.670 | Fe i | * | * | – | –
15929.472 | Fe i | -0.54 | -0.15 | – | –
15929.843 | Fe i | * | * | – | –
15932.171 | Fe i | * | * | – | –
15938.918 | Fe i | -0.10 | -0.17 | – | –
15941.848 | Fe i | 0.03 | -0.24 | – | –
15954.085 | Fe i | -0.58 | -0.08 | – | –
15954.477 | Mg i | -0.95 | -0.20 | – | –
15960.063 | Si i | 0.01 | -0.08 | -6.83 | 0.56
15962.558 | Fe i | 0.09 | 0.17 | -6.67 | 0.65
15964.865 | Fe i | -0.02 | -0.30 | – | –
15980.725 | Fe i | 0.70 | -0.26 | -6.77 | 0.55
15982.072 | Fe i | -0.39 | 0.37 | – | –
16150.762 | Ca i | -0.24 | -0.21 | – | –
16152.714 | Ni i | -1.74 | -0.28 | – | –
16155.236 | Ca i | -0.70 | -0.21 | – | –
16156.557 | Fe i | -0.40 | -0.10 | – | –
16157.364 | Ca i | -0.18 | -0.43 | – | –
16163.691 | Si i | -0.93 | -0.07 | -7.09 | 0.35
16165.029 | Fe i | 0.64 | -0.35 | -6.73 | 0.59
16174.975 | Fe i | 0.16 | -0.03 | -6.86 | 0.46
16186.475 | Si i | -1.28 | 0.34 | – | –
16195.060 | Fe i | 0.14 | -0.33 | – | –
16197.075 | Ca i | 0.10 | -0.15 | -6.55 | 0.70
16201.513 | Fe i | -0.51 | -0.18 | – | –
16202.330 | Fe i | * | * | – | –
16203.328 | Fe i | -0.68 | 0.33 | – | –
16204.252 | Fe i | 0.06 | -0.15 | – | –
16213.537 | Fe i | 0.24 | -0.08 | – | –
16215.670 | Si i | -0.81 | -0.18 | -6.89 | 0.55
16225.618 | Fe i | 0.09 | -0.20 | – | –
16227.151 | Fe i | -0.84 | 1.08 | – | –
16232.518 | Fe i | * | * | – | –
16235.966 | Fe i | -0.23 | -0.21 | – | –
16238.952 | Fe i | * | * | – | –
16240.870 | Fe i | -0.74 | -0.52 | – | –
16241.833 | Si i | -0.74 | 0.03 | -7.33 | 0.11
16245.763 | Fe i | -0.73 | 0.94 | – | –
16246.460 | Fe i | -0.12 | -0.18 | – | –
16252.550 | Fe i | -0.42 | -0.11 | – | –
16258.912 | Fe i | -0.81 | 0.22 | – | –
16272.468 | Fe i | -0.75 | 0.51 | – | –
16284.769 | Fe i | 0.15 | 0.32 | – | –
16292.840 | Fe i | -0.48 | -0.32 | – | –
16310.501 | Ni i | 0.06 | -0.16 | – | –
16316.320 | Fe i | 0.74 | -0.33 | -6.72 | 0.60
16324.451 | Fe i | -0.56 | 0.00 | -7.11 | 0.38
16331.524 | Fe i | -0.58 | -0.18 | – | –
16333.141 | Fe i | -1.44 | -0.80 | – | –
16333.928 | C i | -1.40 | -0.27 | – | –
16346.857 | Si i | -0.70 | 0.39 | – | –
16363.103 | Ni i | 0.62 | 0.03 | -8.13 | -0.65
16364.748 | Mg i | -1.30 | -0.46 | – | –
16364.850 | Mg i | -1.30 | -0.30 | – | –
16364.960 | Mg i | -0.68 | 0.49 | -5.42 | none
16366.337 | Fe i | -0.36 | 1.00 | – | –
16377.388 | Fe i | -0.23 | 0.24 | – | –
16380.176 | Si i | -0.70 | -0.23 | – | –
16381.204 | Fe i | -0.14 | 0.11 | – | –
16381.534 | Si i | -0.50 | -0.04 | – | –
16381.814 | Fe i | -0.90 | -0.88 | – | –
16382.251 | Fe i | 0.35 | -0.04 | -7.00 | 0.33
16384.141 | Fe i | -0.19 | 0.55 | – | –
16412.982 | Si i | -1.60 | -0.87 | – | –
16444.816 | Fe i | 0.18 | -0.48 | -6.74 | 0.71
16466.921 | Fe i | 0.08 | 0.08 | – | –
16468.533 | C i | -1.08 | 0.07 | – | –
16471.753 | Fe i | -0.68 | -0.71 | – | –
16474.077 | Fe i | -0.42 | 0.54 | – | –
16476.933 | Fe i | -0.47 | 0.12 | – | –
16481.228 | Fe i | -0.30 | -0.14 | – | –
16486.666 | Fe i | 0.24 | -0.54 | -6.62 | 0.83
16489.788 | Fe i | * | * | – | –
16489.987 | Mn i | * | * | – | –
16494.427 | Fe i | -0.80 | -0.28 | – | –
16494.500 | Fe i | -0.84 | -0.38 | – | –
16494.702 | Fe i | -1.10 | -0.47 | – | –
16504.140 | Fe i | * | * | – | –
16506.293 | Fe i | -0.37 | 0.09 | – | –
16517.223 | Fe i | 0.48 | -0.20 | -7.03 | 0.29
16518.940 | Fe i | * | * | – | –
16519.147 | Fe i | * | * | – | –
16522.074 | Fe i | -0.05 | -0.38 | – | –
16524.466 | Fe i | 0.45 | -0.24 | -6.70 | 0.62
16531.983 | Fe i | -0.05 | 0.77 | – | –
16537.994 | Fe i | -0.42 | 0.45 | – | –
16539.193 | Fe i | -0.10 | 0.02 | -7.00 | 0.33
16540.870 | Fe i | -0.62 | 0.05 | – | –
16541.423 | Fe i | -0.39 | 1.31 | – | –
16541.962 | Fe i | -0.36 | -0.51 | – | –
16542.660 | S i | -0.39 | -0.32 | – | –
16544.667 | Fe i | -0.32 | -0.29 | – | –
16550.383 | Ni i | 0.33 | 0.06 | – | –
16551.994 | Fe i | 0.15 | -0.19 | – | –
16556.347 | Si i | -1.00 | -0.05 | – | –
16556.484 | Fe i | * | * | – | –
16556.674 | Fe i | * | * | – | –
16557.148 | Fe i | -0.37 | 0.71 | – | –
16559.677 | Fe i | -0.26 | -0.47 | – | –
16570.510 | Fe i | * | * | – | –
16575.271 | Fe i | -2.35 | -0.86 | – | –
16578.064 | Fe i | -2.77 | -0.76 | – | –
16581.383 | Fe i | * | * | – | –
16584.480 | Ni i | -0.38 | 0.33 | – | –
16586.051 | Fe i | -1.31 | -0.56 | – | –
16587.493 | Fe i | -1.03 | -0.29 | – | –
16589.439 | Ni i | -0.57 | -0.23 | – | –
16607.634 | Fe i | -0.57 | 1.03 | – | –
16612.761 | Fe i | 0.09 | -0.20 | – | –
16619.737 | Fe i | -1.43 | -0.57 | – | –
16624.881 | Mg i | -1.78 | -0.41 | – | –
16629.836 | Fe i | * | * | – | –
16632.019 | Mg i | * | * | – | –
16632.230 | Mg i | -1.19 | -0.09 | – | –
16632.503 | Fe i | -1.32 | -1.08 | – | –
16640.640 | Fe i | * | * | – | –
16647.246 | Fe i | * | * | – | –
16648.203 | Fe i | -0.29 | 0.20 | – | –
16649.877 | Ca ii | 0.72 | 0.08 | – | –
16652.387 | Fe i | -0.80 | -0.37 | – | –
16652.798 | Fe i | -0.60 | -0.12 | – | –
16661.379 | Fe i | 0.22 | 0.97 | – | –
16666.773 | Fe i | -0.87 | -0.22 | – | –
16673.706 | Ni i | 0.14 | -0.25 | – | –
16680.770 | Si i | -0.10 | 0.04 | -7.20 | 0.24
16693.072 | Fe i | -0.28 | -0.14 | – | –
16711.282 | Fe i | * | * | – | –
16718.957 | Al i | 0.05 | -0.10 | -7.09 | none
16721.462 | Fe i | -0.39 | 0.19 | – | –
16724.685 | Fe i | -0.73 | -0.32 | – | –
16725.440 | Fe i | -0.90 | -0.34 | – | –
16728.309 | Fe i | -1.40 | -0.65 | – | –
16729.672 | Si i | * | * | – | –
16737.240 | Fe i | * | * | – | –
16739.311 | Fe i | -1.17 | -0.37 | – | –
16750.564 | Al i | 0.38 | -0.03 | -7.16 | none
16753.065 | Fe i | 0.31 | -0.09 | – | –
16757.642 | Co i | -1.50 | -0.58 | – | –
16760.218 | Mg ii | 0.47 | -0.01 | – | –
16763.360 | Al i | -0.43 | 0.12 | – | –
16828.159 | Si i | -1.05 | -0.02 | -7.21 | 0.23
16833.052 | Fe i | -1.02 | -0.13 | – | –
16843.228 | Fe i | -1.22 | 0.10 | – | –
16843.877 | Fe i | -1.90 | -1.27 | – | –
16853.089 | Ni i | * | * | – | –
16853.467 | Fe i | * | * | – | –
16854.936 | C i | -0.93 | -0.12 | – | –
16856.442 | Fe i | -0.93 | -0.84 | – | –
16857.135 | Fe i | -1.75 | -1.50 | – | –
16858.523 | Fe i | -1.54 | -1.68 | – | –
16864.079 | Fe i | -0.86 | 0.68 | – | –
16865.513 | Fe i | -0.87 | -0.12 | – | –
16869.950 | Fe i | -0.74 | -0.32 | – | –
16874.116 | Fe i | -0.80 | -0.64 | – | –
16878.540 | Fe i | -1.33 | -0.55 | – | –
16883.606 | Fe i | -1.41 | -0.47 | – | –
16884.809 | Fe i | -1.07 | 0.65 | – | –
16889.473 | Fe i | -1.31 | -0.47 | – | –
16890.380 | C i | 0.32 | -0.25 | -6.90 | 0.61
16892.384 | Fe i | -0.61 | 0.19 | – | –
16893.954 | Fe i | -1.01 | 0.47 | – | –
16898.883 | Fe i | -0.72 | 0.19 | – | –
16900.231 | Fe i | -0.94 | -0.13 | – | –
16910.683 | Fe i | * | * | – | –
16927.611 | Fe i | -0.31 | -0.15 | – | –
16928.623 | Fe i | -0.73 | 0.33 | – | –
16930.962 | Fe i | -1.12 | -0.19 | – | –
16947.417 | Ca i | * | * | – | –
16954.102 | C i | -0.71 | 0.25 | – | –
16957.794 | Si i | -1.20 | -0.13 | – | –
16996.265 | Ni i | 0.44 | -0.02 | -7.56 | -0.09
17001.025 | Ni i | 0.34 | -0.04 | -6.93 | 0.57
17004.631 | Fe i | * | * | – | –
17007.489 | Fe i | -0.94 | -0.37 | – | –
17012.728 | Fe i | * | * | – | –
17018.033 | Ti i | 0.50 | -0.26 | – | –
17018.624 | Fe i | -1.30 | -0.33 | – | –
17025.116 | Fe i | -1.31 | -0.94 | – | –
17032.896 | Fe i | -1.36 | -0.43 | – | –
17033.659 | Fe i | -0.72 | -0.47 | – | –
17037.787 | Fe i | -0.28 | 0.58 | – | –
17040.099 | Fe i | * | * | – | –
17047.651 | Fe i | * | * | – | –
17052.181 | Fe i | -0.53 | 0.21 | – | –
17052.876 | Fe i | * | * | – | –
17061.249 | Fe i | -0.93 | 0.36 | – | –
17064.887 | Fe i | -0.39 | -0.16 | – | –
17065.265 | Fe i | -0.69 | -0.19 | – | –
17067.529 | Fe i | -0.03 | 1.36 | – | –
17070.548 | Fe i | -0.83 | -0.16 | – | –
17072.825 | Fe i | -2.19 | -0.28 | – | –
17075.120 | Fe i | -0.92 | 1.20 | – | –
17085.630 | Mg i | -1.69 | 0.22 | – | –
17086.250 | C i | -1.54 | 0.34 | – | –
17091.304 | Fe i | * | * | – | –
17094.434 | Fe i | -0.60 | -0.23 | – | –
17108.631 | Mg i | -0.17 | -0.24 | -6.91 | none
17112.447 | P i | 1.03 | 0.53 | – | –
17115.719 | Fe i | -1.07 | 0.46 | – | –
17120.503 | Ni i | -0.42 | -0.22 | – | –
17130.952 | Fe i | -0.45 | 0.45 | – | –
17132.928 | Fe i | -0.73 | 0.36 | – | –
17134.200 | Fe i | -1.01 | 0.28 | – | –
17137.105 | Fe i | -0.83 | -0.67 | – | –
17138.897 | Fe i | -0.88 | 0.66 | – | –
17151.667 | Fe i | -1.07 | -0.70 | – | –
17232.229 | Fe i | -0.88 | 0.54 | – | –
17233.171 | Fe i | -0.79 | 0.84 | – | –
21047.143 | Si i | -0.33 | 0.13 | – | –
21056.253 | Si i | -0.47 | -0.11 | – | –
21060.711 | Mg i | -0.45 | 0.04 | -5.97 | none
21060.891 | Mg i | -0.37 | 0.00 | -6.00 | none
21061.091 | Mg i | -0.29 | -0.04 | -5.76 | none
21093.029 | Al i | -0.48 | -0.17 | -6.96 | none
21139.759 | Si i | -0.47 | -0.03 | – | –
21144.154 | Si i | -0.68 | -0.11 | – | –
21162.035 | Fe i | -0.31 | 0.03 | – | –
21163.755 | Al i | 0.11 | 0.12 | -7.05 | none
21204.829 | Si i | -0.15 | 0.18 | – | –
21208.141 | Mg i | -1.00 | 1.04 | – | –
21211.510 | C i | 0.13 | 0.20 | – | –
21233.658 | Mg i | * | * | – | –
21238.466 | Fe i | -1.50 | -0.08 | – | –
21246.817 | Ca i | * | * | – | –
21259.897 | C i | 0.59 | 0.09 | -6.48 | 0.85
21354.198 | Si i | 0.13 | -0.06 | -7.34 | 0.04
21425.785 | Si i | 0.08 | 0.30 | – | –
21441.924 | Si i | 0.15 | 0.45 | – | –
21489.572 | Fe i | * | * | – | –
21523.114 | Si i | * | * | – | –
21756.947 | Fe i | -0.70 | 0.15 | – | –
21761.020 | Mg i | * | * | – | –
21779.660 | Si i | 0.02 | -0.39 | -6.86 | 0.47
21782.944 | Ti i | -1.18 | -0.01 | – | –
21816.566 | Fe i | -0.20 | 0.25 | – | –
21820.661 | Fe i | -0.92 | -1.05 | – | –
21851.381 | Fe i | -3.76 | -0.15 | – | –
21858.065 | Fe i | 0.10 | 0.40 | – | –
21879.324 | Si i | 0.18 | -0.23 | -6.68 | 0.65
21880.871 | Fe i | * | * | – | –
21882.987 | Fe i | -0.04 | -0.18 | – | –
21894.983 | Fe i | -0.04 | 0.31 | – | –
21897.391 | Ti i | -1.30 | 0.17 | – | –
21951.294 | Fe i | * | * | – | –
22056.400 | Na i | 0.27 | -0.02 | -7.13 | none
22072.550 | Si i | -1.15 | -0.21 | – | –
22083.662 | Na i | -0.02 | -0.01 | -7.09 | none
22139.693 | Fe i | * | * | – | –
22178.155 | Fe i | * | * | – | –
22211.238 | Ti i | -1.64 | 0.14 | – | –
22232.858 | Ti i | -1.56 | 0.13 | – | –
22260.179 | Fe i | -0.99 | -0.05 | – | –
22274.022 | Ti i | -1.63 | 0.17 | – | –
22310.617 | Ti i | -1.94 | 0.13 | – | –
22419.976 | Fe i | -0.22 | -0.07 | – | –
22493.671 | Fe i | -1.14 | -0.25 | – | –
22537.534 | Si i | -0.31 | -0.08 | -6.66 | 0.60
22563.828 | S i | -0.00 | 0.26 | – | –
22609.238 | Fe i | -1.33 | -0.64 | – | –
22619.838 | Fe i | -0.65 | -0.28 | -6.51 | 1.03
22626.723 | Ca i | -0.38 | -0.16 | – | –
22651.177 | Ca i | 0.55 | -0.30 | -6.73 | 0.60
22665.757 | Si i | -0.26 | 0.42 | – | –
22707.738 | S i | 0.21 | -0.23 | – | –
22807.745 | Mg i | -0.32 | -0.34 | -6.04 | none
22807.775 | Mg i | -1.25 | -0.21 | -6.30 | none
22807.775 | Mg i | -1.25 | -0.20 | -6.30 | none
22808.025 | Mg i | -0.29 | -0.14 | -6.48 | none
22808.033 | Mg i | * | * | – | –
22808.265 | Mg i | -0.50 | -0.19 | -6.30 | none
22812.586 | Fe i | -1.50 | -0.60 | – | –
22832.364 | Fe i | -1.34 | -0.07 | – | –