11institutetext:  Institut für Astrophysik, Georg-August-Universität,
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
11email<EMAIL_ADDRESS>22institutetext: University of Applied
Sciences Technikum Wien, Höchstädtplatz 6, 1200 Wien, Austria 33institutetext:
Fakultät fur Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090
Wien, Austria 44institutetext: Departamento de Física de la Tierra y
Astrofísica and IPARCOS-UCM (Instituto de Física de Partículas y del Cosmos de
la UCM), Facultad de Ciencias Físicas, Universidad Complutense de Madrid,
28040, Madrid, Spain 55institutetext: Centro de Astrobiología (CSIC-INTA),
ESAC, Camino bajo del castillo s/n, 28692 Villanueva de la Cañada, Madrid,
Spain 66institutetext: Institut de Ciències de l’Espai (ICE, CSIC), Campus
UAB, Can Magrans s/n, 08193 Bellaterra, Spain 77institutetext: Institut
d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
88institutetext: Landessternwarte, Zentrum für Astronomie der Universität
Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany 99institutetext:
Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía s/n,
18008 Granada, Spain 1010institutetext: Centro Astronómico Hispano-Alemán,
Observatorio de Calar Alto, Sierra de los Filabres, 04550 Gérgal, Spain
1111institutetext: Instituto de Astrofísica de Canarias, 38205 La Laguna,
Tenerife, Spain 1212institutetext: Departamento de Astrofísica, Universidad de
La Laguna, 38206 La Laguna, Tenerife, Spain 1313institutetext: Thüringer
Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany
1414institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany 1515institutetext: Max-Planck-Institut für
Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
1616institutetext: Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, United Kingdom 1717institutetext: Hamburger
Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany
1818institutetext: Homer L. Dodge Department of Physics and Astronomy,
University of Oklahoma, 440 West Brooks Street, Norman, OK 73019, United
States of America

# The CARMENES search for exoplanets around M dwarfs

Not-so-fine hyperfine-split vanadium lines in cool star spectra

Y. Shan The CARMENES search for exoplanets around M dwarfsThe CARMENES search
for exoplanets around M dwarfs A. Reiners The CARMENES search for exoplanets
around M dwarfsThe CARMENES search for exoplanets around M dwarfs D. Fabbian
The CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfsThe CARMENES search for exoplanets around M dwarfsThe CARMENES search
for exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfs E. Marfil The CARMENES search for exoplanets around M dwarfsThe
CARMENES search for exoplanets around M dwarfs D. Montes The CARMENES search
for exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfs H. M. Tabernero The CARMENES search for exoplanets around M dwarfsThe
CARMENES search for exoplanets around M dwarfs I. Ribas The CARMENES search
for exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfsThe CARMENES search for exoplanets around M dwarfsThe CARMENES search
for exoplanets around M dwarfs J. A. Caballero The CARMENES search for
exoplanets around M dwarfsThe CARMENES search for exoplanets around M dwarfs
A. Quirrenbach The CARMENES search for exoplanets around M dwarfsThe CARMENES
search for exoplanets around M dwarfs P. J. Amado The CARMENES search for
exoplanets around M dwarfsThe CARMENES search for exoplanets around M dwarfs
J. Aceituno The CARMENES search for exoplanets around M dwarfsThe CARMENES
search for exoplanets around M dwarfsThe CARMENES search for exoplanets around
M dwarfsThe CARMENES search for exoplanets around M dwarfs V. J. S. Béjar The
CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfsThe CARMENES search for exoplanets around M dwarfs M. Cortés-Contreras
The CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfs S. Dreizler The CARMENES search for exoplanets
around M dwarfsThe CARMENES search for exoplanets around M dwarfs A. P. Hatzes
The CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfs Th. Henning The CARMENES search for exoplanets
around M dwarfsThe CARMENES search for exoplanets around M dwarfs S. V.
Jeffers The CARMENES search for exoplanets around M dwarfsThe CARMENES search
for exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfsThe CARMENES search for exoplanets around M dwarfs A. Kaminski The
CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfs M. Kürster The CARMENES search for exoplanets
around M dwarfsThe CARMENES search for exoplanets around M dwarfs M. Lafarga
The CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfsThe CARMENES search for exoplanets around M dwarfsThe CARMENES search
for exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfs J. C. Morales The CARMENES search for exoplanets around M dwarfsThe
CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfsThe CARMENES search for exoplanets around M dwarfs
E. Nagel The CARMENES search for exoplanets around M dwarfsThe CARMENES search
for exoplanets around M dwarfsThe CARMENES search for exoplanets around M
dwarfsThe CARMENES search for exoplanets around M dwarfs E. Pallé The CARMENES
search for exoplanets around M dwarfsThe CARMENES search for exoplanets around
M dwarfs V. M. Passegger The CARMENES search for exoplanets around M dwarfsThe
CARMENES search for exoplanets around M dwarfsThe CARMENES search for
exoplanets around M dwarfsThe CARMENES search for exoplanets around M dwarfs
C. Rodriguez-López The CARMENES search for exoplanets around M dwarfsThe
CARMENES search for exoplanets around M dwarfs A. Schweitzer The CARMENES
search for exoplanets around M dwarfsThe CARMENES search for exoplanets around
M dwarfs M. Zechmeister The CARMENES search for exoplanets around M dwarfsThe
CARMENES search for exoplanets around M dwarfs

(Received 11 June 2021 / Accepted 6 August 2021)

###### Abstract

Context. M-dwarf spectra are complex and notoriously difficult to model,
posing challenges to understanding their photospheric properties and
compositions in depth. Vanadium (V) is an iron-group element whose abundance
supposedly closely tracks that of iron, but has origins that are not
completely understood.

Aims. Our aim is to characterize a series of neutral vanadium atomic
absorption lines in the 800–910 nm wavelength region of high signal-to-noise,
high-resolution, telluric-corrected M-dwarf spectra from the CARMENES survey.
Many of these lines are prominent and exhibit a distinctive broad and flat-
bottom shape, which is a result of hyperfine structure (HFS). We investigate
the potential and implications of these HFS split lines for abundance analysis
of cool stars.

Methods. With standard spectral synthesis routines, as provided by the
spectroscopy software iSpec and the latest atomic data (including HFS)
available from the VALD3 database, we modeled these striking line profiles. We
used them to measure V abundances of cool dwarfs.

Results. We determined V abundances for 135 early M dwarfs (M0.0 V to M3.5 V)
in the CARMENES guaranteed time observations sample. They exhibit a
[V/Fe]-[Fe/H] trend consistent with that derived from nearby FG dwarfs. The
tight ($\pm 0.1$ dex) correlation between [V/H] and [Fe/H] suggests the
potential application of V as an alternative metallicity indicator in M
dwarfs. We also show hints that neglecting to model HFS could partially
explain the temperature correlation in V abundance measurements observed in
previous studies of samples involving dwarf stars with $T_{\rm eff}\lesssim
5300$ K.

Conclusions. Our work suggests that HFS can impact certain absorption lines in
cool photospheres more severely than in Sun-like ones. Therefore, we advocate
that HFS should be carefully treated in abundance studies in stars cooler than
$\sim 5000$ K. On the other hand, strong HFS split lines in high-resolution
spectra present an opportunity for precision chemical analyses of large
samples of cool stars. The V-to-Fe trends exhibited by the local M dwarfs
continue to challenge theoretical models of V production in the Galaxy.

###### Key Words.:

atomic data – line: profiles – techniques: spectroscopic – stars: abundances –
stars: low-mass

##  1 Introduction

The analysis of detailed chemical abundances in stellar photospheres can help
constrain the mechanisms of cosmic nucleosynthesis, reveal the evolution of
galaxies and their stellar populations, and shed light on the formation of
exoplanets (e.g., Tinsley, 1980; Timmes et al., 1995; Kobayashi et al., 2006;
Fabbian et al., 2009; Nomoto et al., 2013; Kobayashi et al., 2020; Fischer &
Valenti, 2005; Meléndez et al., 2009; Adibekyan et al., 2012; Dorn et al.,
2015; Adibekyan et al., 2021). Accurate measurements require access to spectra
with high resolution and high signal-to-noise (S/N), as well as robust stellar
atmosphere models and comprehensive atomic data (see e.g., reviews by Allende
Prieto, 2016; Jofré et al., 2019).

Vanadium (23V) is an odd-$Z$ iron-peak element whose origin is not completely
understood. It is thought to be chiefly produced in explosive silicon and
oxygen burning in supernovae (SNe) of type II (Woosley & Weaver, 1995), with
type Ia SNe supposedly also contributing to a lesser extent (e.g., Clayton,
2003; Bravo & Martínez-Pinedo, 2012). However, when the theories are
propagated through nucleosynthesis and Galactic chemical evolution (GCE)
models, the predicted vanadium-to-iron ratios, [V/Fe] $\sim-0.2$ to $-0.5$
dex, fall significantly below those observed in representative stellar
populations, where [V/Fe] remains close to solar or slightly super-solar over
a large range of metallicities (e.g., Kobayashi et al., 2011, 2020; Nomoto et
al., 2013; Sneden et al., 2016).

To date, most measurements of V abundance have been performed in populations
of Sun-like stars (e.g., Gratton & Sneden, 1991; Feltzing & Gustafsson, 1998;
Prochaska et al., 2000; Reddy et al., 2003; Battistini & Bensby, 2015, to name
a few) or giants and subgiants (e.g., Liu et al., 2012; Roederer et al., 2014;
Hawkins et al., 2016; Lomaeva et al., 2019), almost exclusively with $T_{\rm
eff}\gtrsim 4000$ K. The majority of studies use observations in the visual
wavelength range: 400–700 nm. A number of neutral and ionized V lines lie in
this region, from which abundances can be constrained (and, in some cases,
from as few as one single line) with either equivalent width (EW) analysis or
direct spectral synthesis fits (SSF). While these combinations of data and
methodology seem to generate robust results for FG-type stars, several studies
whose samples included K dwarfs have found a large scatter in [V/Fe] and a
systematic upward trend with decreasing effective temperature ($T_{\rm eff}$)
below $\sim 5300$ K (e.g., Bodaghee et al., 2003; Gilli et al., 2006; Neves et
al., 2009; Adibekyan et al., 2012; Tabernero et al., 2012; Montes et al.,
2018). This correlation has often been ascribed to modeling deficiencies, such
as effects that are not accounted from non-local thermodynamic equilibrium
(NLTE).

Incidentally, ${}^{51}_{23}$V, the most abundant isotope accounting for
$>99\,\%$ of naturally occurring vanadium, has a rather high nuclear spin
(+7/2) and shows pronounced hyperfine structure (HFS), which are shifts in
atomic energy levels. These shifts split atomic transitions into multiple,
usually closely spaced components. The manifestation of HFS in a number of
absorption lines, including those of V, was already noticed in solar spectra
by Abt (1952). However, due to limitations in atomic data, and perhaps also in
awareness, not all stellar population studies to date have completely
accounted for HFS in their spectral modeling. Interestingly, all the
aforementioned studies of K dwarfs that have observed a correlation between V
abundance and $T_{\rm eff}$ neglected to include HFS components in their line
lists.

A number of authors have cautioned against inappropriate modeling of HFS in
stellar abundance analysis, either in terms of the use of inaccurate HFS
atomic data or a complete absence of consideration for HFS. For example,
Prochaska & McWilliam (2000) and Jofré et al. (2015b) found notable abundance
systematics for certain elements (including V) arising from the suboptimal
treatment of HFS. When presenting the Hypatia Catalog (a large compilation of
detailed elemental abundance measurements for stars in the solar
neighborhood), Hinkel et al. (2014) also commented on the increased scatter
seen in literature abundances in elements such as Mn and V, speculating that a
“casual” treatment of HFS could be to blame. On the other hand, several
investigations have reached the broad conclusion that abundance measurements
tend to be relatively insensitive to the inclusion of HFS (e.g., del Peloso et
al., 2005; Takeda, 2007; Hinkel et al., 2016; Jofré et al., 2017), which may
appear to contradict the former set of studies. However, these latter works
were mostly based on the analysis of a handful of elements in FG-type dwarfs
over limited wavelength ranges, and their conclusions chiefly apply to the
final results, which are averages over particular ensembles of chosen lines.
For specific individual lines and, especially, for cooler stars, the latter
studies have also noted significant HFS-related deviations.

Until very recently, elemental abundance studies have largely shunned M-type
stars, as their cool temperatures promote the presence of molecules, giving
rise to forests of poorly modeled lines that veil the continuum and distort
atomic features. Elemental abundances other than iron have been characterized
for only a handful of M dwarfs (Veyette et al., 2017; Souto et al., 2017,
2018; Abia et al., 2020; Ishikawa et al., 2020), mostly from high-resolution
spectra in the $YJH$ bands in the near-infrared (NIR) region ($\lambda\gtrsim$
960 – 1700 nm). Unfortunately, in M dwarfs, the $YJH$ bands host very few
useful V lines. As a result, V abundance has been quantified for only one M
dwarf, namely Kepler-138 (M1 V, $T_{\rm eff}\sim 3800$ K), using one line in
the APOGEE $H$ band ($\lambda\sim 1592.4$ nm; Souto et al. 2017). More
recently, Maldonado et al. (2020) determined abundances for a number of
elements, including V, for $\sim 200$ M dwarfs. Their approach was not based
on the study of individual line profiles, but rather on broad features in
severely degraded HARPS spectra ($\sim$ 400–700 nm, $\mathcal{R}\sim$ 115 000
to $\mathcal{R}\sim$ 1000–2000). The features that best describe the degraded
spectra were identified using principal component analysis, then correlated to
abundances using a calibration sample of 19 FGK+M binaries. While their novel
technique achieved impressive success for many of the elements considered, the
V results presented in Maldonado et al. (2020) were dubious and significantly
deviated from patterns measured for ensembles of FGK stars. Therefore, it
appears that no robust analysis of vanadium on a large sample of M dwarfs has
been performed yet.

Figure 1: Representative vanadium features in normalized, telluric absorption
corrected, and co-added CARMENES spectra for M0.0 V (black, J14257+236W) and
M2.0 V (gray, J03213+799) stars. Overplotted are PHOENIX-ACES spectra (1D LTE)
convolved to the CARMENES instrumental resolution ($\mathcal{R}$ = 94 600) for
4000 K (blue) and 3500 K (red) dwarfs for comparison. Wavelengths are given in
terms of the rest frame as measured in air.

Meanwhile, the CARMENES survey has provided high-resolution, high-quality
spectra for hundreds of nearby M dwarfs. Some of these data were recently used
to study rubidium, strontium, and zirconium in 57 local M dwarfs (Abia et al.,
2020). In this present work, we focus on vanadium. In the wavelength region of
800 nm to 910 nm, which hosts a relatively low density of molecular lines,
there exist about a dozen features of V i, many of them being extremely strong
and prominently split by HFS. In some cases, the line shape deviations in
these high-resolution spectra are so large that they are very far from being
degenerate with any other conventional broadening parameters. We demonstrate
that the striking vanadium line profiles seen in the CARMENES spectra can be
excellently reproduced using the latest available atomic data and standard
stellar models, assuming reasonable stellar parameters. From these features,
we extracted robust vanadium abundances for 135 early M dwarfs and show that
they track iron with little scatter, matching the patterns seen for local FG-
type dwarfs.

This paper is organized as follows. Section 2.1 describes the CARMENES survey
and the spectroscopic data products used for this work. Section 2.2 offers a
heuristic description of the HFS-split vanadium features. We give an overview
of HFS and its role in spectral modeling and abundance studies in Section 3.
In Section 4, we document our effort to determine vanadium abundances for a
subsample of early M dwarfs from the CARMENES survey. A comparison with the
literature and discussion can be found in Section 5. In particular, the
apparent overabundance of vanadium measured in some studies of K dwarfs is
explored in Section 5.2, and Section 5.5 briefly examines the potential of
vanadium abundance as a metallicity indicator for late-type stars. We give our
conclusions in Section 6.

##  2 Vanadium lines in CARMENES spectra

###  2.1 The CARMENES survey

CARMENES is a spectroscopic survey of nearby M dwarfs to search for planets
using the radial velocity method. The observations are conducted with the 3.5
m telescope at the Calar Alto Observatory in Almería, Spain, using the
homonymous state-of-the-art fiber-fed high-resolution spectrograph with
optical (VIS: $\Delta\lambda\approx$ 520–960 nm, $\mathcal{R}\sim 94\,600$)
and near-infrared (NIR: $\Delta\lambda\approx$ 960–1710 nm, $\mathcal{R}\sim
80\,400$) channels (Quirrenbach et al., 2014, 2018). Since January 2016, the
CARMENES project has collected high-quality spectra over multiple epochs of
$\sim 380$ stars across the full M-dwarf range as a part of its guaranteed
time observations (GTO) program (Reiners et al., 2018; Schweitzer et al.,
2019, hereafter Schw19). The utility of this data set has been demonstrated in
numerous planet discoveries and atmospheric characterizations, as well as
studies of stellar properties, such as activity and fundamental parameters
(see Quirrenbach et al., 2020, for an overview)

As multiple spectra are taken over time for each target, they can be combined
into a single “template spectrum” optimized for detailed stellar
characterization, as follows: the raw spectra have been processed with the
CARMENES pipeline caracal, which is based on flat-relative optimal extraction
(Zechmeister et al., 2014) and wavelength calibration, which utilizes spectra
from hollow cathode lamps and Fabry-Pérot etalons (Bauer et al., 2015). The
spectra and, thus, the templates are corrected for telluric absorption
following the template division telluric modeling approach (Nagel, 2019),
which employs
molecfit111http://www.eso.org/sci/software/pipelines/skytools/molecfit (Smette
et al., 2015). In a further iteration, the
serval222https://github.com/mzechmeister/serval code (Zechmeister et al.,
2018) coadds all spectra of each star via a B-spline regression and creates a
telluric-free high S/N template spectrum (Caballero et al., 2016b). Each
template spectrum, covering from 520 nm to 1710 nm with a few inter-order gaps
mostly in the $H$ band, comprises on average $\sim 50$ individual spectra, and
reaches S/N of $>100$ virtually throughout the entire wavelength range. The
high resolution, high S/N, and virtual absence of telluric lines make these
template spectra suitable for identifying and modeling fine features intrinsic
to a given stellar spectrum.

###  2.2 Heuristic description of “square-bucket” features associated with V i
lines

Compared to shorter wavelengths, the 800–910 nm region of M dwarf spectra is
noticeably less impacted by molecular absorption in the stellar photosphere.
After correction for telluric absorption, a series of features stand out in
this wavelength range in the template spectra of many CARMENES GTO stars (see
Fig. 1 for examples, and Figs. 14 and 15 for the whole 800–910 nm region).
These features are striking in appearance owing to their strength and breadth,
and in particular their well-defined sharp corners, which cannot be reconciled
with conventional broadening mechanisms of a single absorption line. Rather,
each “bucket” resembles the smearing out of a forest of narrow, adjacent
components. Some of these components are even marginally resolved into
“wiggles”. These absorption profiles are ostensibly mismatched with synthetic
spectra found in standard libraries (e.g., PHOENIX-ACES, Husser et al. 2013).
Many of these features persist throughout the early M-dwarf range. They are
associated with transitions of neutral vanadium atom.

The fact that these particular V i transitions are affected by such
substantial splitting in cool dwarfs may have received little attention
because no previous program has conducted a dedicated M-dwarf survey at high
resolution, with high S/N, and at multiple epochs so as to enable robust
telluric corrections in the red optical wavelength range, as CARMENES has. The
specific lines belonging to the V i transitions that we consider here diminish
in strength toward higher temperatures, nearly vanishing at about $5000$ K, as
most neutral vanadium becomes ionized (we note that V ii transitions also
exist, but are extremely weak in M dwarfs in the optical wavelength range).
These lines also lose their “corners” in lower resolution ($\mathcal{R}\sim
20\,000$) spectra (see Fig. 2). In cool dwarfs, many V i transitions are also
visible at shorter wavelengths (e.g., 500–700 nm), but generally appear less
broadened and significantly more blended with forests of molecular absorption.

Figure 2: Synthetic spectra around the dramatically split V i $\lambda 811.7$
nm synthesized by iSpec using the latest VALD3-HFS line list for solar
vanadium abundance. This feature comprises 21 components, indicated by gray
vertical lines whose heights scale with their expected relative strengths at
$T=4000$ K, with EW used as a proxy for strength, $\log({\rm
EW})=\log(gf\lambda)-\theta\chi$, where $\lambda$ is the transition wavelength
in Å, $\theta=5040/T$, temperature $T$ is in K, and $\chi$ is the lower
excitation potential of the transition in eV. The strongest components toward
the left side of the feature manifest themselves as marginally resolved
“wiggles” in spectra with CARMENES-like resolution.

##  3 Modeling the broad vanadium features with hyperfine structure

###  3.1 Hyperfine splitting in stellar abundance studies

The broad V i features are a consequence of HFS splitting, a well-understood
quantum effect arising from the interaction between electron angular momentum
and nuclear magnetic moment, resulting in (typically) hyper-finely split
energy levels within an atom (e.g., Condon & Shortley, 1963; Sobelman, 2006).
HFS manifests itself in a number of atomic transitions, most notably in
odd-$Z$ elements where an unpaired proton gives rise to a nonzero nuclear spin
that, together with a large nuclear $g$ factor, can result in a large nuclear
magnetic moment. Besides V, other examples of prominent elements of
astrophysical interest whose most common isotopes exhibit a high nuclear
magnetic dipole moment include Mn, Co, Cu, Sc, Eu, and Rb. For these elements,
failure to account for HFS could result in inadequately modeled line profiles
and systematically incorrect abundances.

The importance of accurate HFS modeling in spectral analysis has gained
recognition through works such as that of Prochaska & McWilliam (2000), who
pointed out that incorrect HFS data for Mn and Sc had led to claims of
spurious trends between abundances and metallicity and inappropriate
conclusions about the mechanisms of Galactic nucleosynthesis (see also, e.g.,
Booth & Blackwell, 1983; Vitas & Vince, 2003; Bergemann et al., 2010; North et
al., 2012; Jofré et al., 2019). HFS is now routinely incorporated in stellar
abundance determinations of many elements. However, it has also been neglected
in other notable studies (e.g., Feltzing & Gustafsson, 1998; Neves et al.,
2009; Adibekyan et al., 2012, but see Adibekyan et al. 2016 which does account
for HFS). One reason for the lack of universal consideration is that, for many
lines, the abundance biases introduced by neglecting HFS are minuscule because
the shape distortions due to HFS are subtle and, for weak lines in the
unsaturated regime, HFS preserves the equivalent widths. Another important
obstacle in the way of accounting for HFS effects in spectral modeling might
simply be the incompleteness or lack of accessibility to atomic data. The
basic physics measurements required to characterize HFS patterns are not
available for many split transitions and, even for those energy levels that
have been measured, it is nontrivial to convert raw laboratory data into line
information directly readable by standard software for atmosphere
synthesis333For a transition affected by HFS, laboratory data, if existent,
are often given in terms of the magnetic dipole, $A$, and electric quadrupole,
$B$, constants for the unsplit energy levels involved. Together with knowledge
of the angular momentum quantum number, $J$, and nuclear spin, $I$, the HFS
line component parameters can be computed via the Casimir formula and the
Hönl-Kronig intensity rule (prescriptions described in, e.g., Woodgate 1980;
McWilliam et al. 2013; Pakhomov et al. 2019).. On the other hand, databases
that do include HFS information tend to supply the HFS components in files
separate from their default line lists (e.g., Kurucz,
2011)444http://kurucz.harvard.edu/linelists.html. This could hinder widespread
usage, since it gives HFS an air of exoticism and requires extra initiative
from the user to incorporate them into their studies.

The precise impact of unmodeled HFS on abundance measurements is correlated
with line strength and depends strongly on the spectral type, the element, and
the specific set of lines used in the analysis. However, the exact nature of
these dependences is difficult to characterize, as the split patterns
appearing for each transition are not simple functions of wavelength,
temperature, or composite strength, and must be determined on a case-by-case
basis. Several authors have investigated the magnitude of systematic errors
introduced by neglecting HFS in FG- (and occasionally K-) type stars in the
visual wavelength range for various elements, using small sets of lines (e.g.,
del Peloso et al., 2005; Takeda, 2007; Jofré et al., 2015b, 2017; Hinkel et
al., 2016). At least for the FG-type stars, the biases are often within
typical uncertainties, especially when an abundance measurement is based on
the mean over several lines. However, significant deviations were detected in
specific individual lines, especially in strong lines toward the cool ends of
the considered stellar samples. Empirically, the deviations were particularly
evident in the works by Jofré et al. (2015b) and Jofré et al. (2017), where
stars cooler than $T_{\rm eff}\lesssim 5000$ K like Arcturus were also
studied. Therefore, to obtain accurate elemental abundance measurements for
arbitrary types of stars, it is important to properly model all the HFS-
affected lines whenever possible.

###  3.2 Using HFS components for V i from the latest update in VALD3

Figure 3: Illustration of synthetic spectra generated with iSpec from the new
VALD3 HFS atomic data for V compared with the CARMENES telluric-corrected
composite spectrum of J14257+236W (gray) in several hyperfine-split V lines.
Red solid, blue dashed, and magenta dotted lines are generic models with
$T_{\rm eff}=4000$ K, $\log{g}$ = 4.5, [M/H] = 0.0, and V abundances of solar
(i.e., $A_{\odot}({\rm V})=\log(N_{\rm V}/N_{\rm H})_{\odot}=-8.11$, Asplund
et al. 2009), +0.3 dex, and –0.3 dex. The best-matching model in green dash-
dotted lines takes the basic stellar parameters from Schw19: $T_{\rm
eff}=4024$ K, $\log{g}$ = 4.64, [M/H] = +0.17, and $A({\rm V})$ = –7.87 (i.e.,
[V/H] = +0.24), which simultaneously reproduces the overall shape of all the
features. We note that the feature at 891.98 nm is blended with an Fe i line
at 892.0 nm.

HFS data have recently been added to the Vienna Atomic Line Database (VALD3),
as detailed by Pakhomov et al. (2017, 2019). This supplementary database
appears to be the result of an extensive effort to gather the latest atomic
data from the literature, compute the associated HFS patterns, and output them
into user-friendly forms. As of May 2020, these data have been readily
accessible in the familiar format via the standard VALD3 web
interface555http://vald.astro.uu.se by selecting the “HFS splitting” option.
The atomic data collection for vanadium therein includes, among others, recent
results from the Wisconsin atomic group (Lawler et al., 2014; Den Hartog et
al., 2014; Wood et al., 2014).

We downloaded and converted these line lists into a format recognized by the
stellar spectroscopy software package iSpec (Blanco-Cuaresma et al., 2014;
Blanco-Cuaresma, 2019). Using iSpec, we simulated a number of broad vanadium
features between 800 nm and 910 nm, seen in the CARMENES VIS spectra. With the
settings described in Sect. 4.3 and 4.6, we generated model spectra that
satisfactorily reproduce the observed prominent V i line profiles in early M
dwarfs, an example of which is shown in Fig. 3. The depths of the features
appear to exhibit a high sensitivity to the exact V abundance value assumed.
By tweaking the V abundance in the synthetic spectra, excellent matches can be
made simultaneously by the model to all the features that we considered,
suggesting the potential effectiveness of these features for abundance
determination using the spectral synthesis technique.

##  4 Measuring vanadium abundances of cool stars

The presence of strong and modelable vanadium features in a relatively clean
region of cool star spectra points to an opportunity to characterize V
abundances of all stars of similar type. We used spectral synthesis fits to
derive vanadium abundances for a large number of early M dwarfs in the
CARMENES GTO sample.

###  4.1 Atomic data

We used the “extract stellar” option of the VALD3 web interface to download a
list of all line data, including HFS-split-line components, in the vicinity of
the regions defined in Table 1 with theoretical depth exceeding $10^{-3}$ for
4000 K and 3500 K dwarf stars. The 4000 K line list was used for stars with
$T_{\rm eff}\gtrsim 3700$ K and the 3500 K line list was for those with
$T_{\rm eff}\lesssim 3700$ K. These line lists include all atomic species, as
well as several molecules including TiO, MgH, CH, CO, C2, OH, and CN.

###  4.2 Line region selection

Through systematic visual inspection of V i lines in the M-dwarf spectra at
800–910 nm, we identified 13 regions that are well modeled using the new VALD3
atomic line lists for stars around 4000 K. For stars with $T_{\rm eff}$ closer
to 3500 K, where molecular contamination becomes more significant, there are
nine such regions for which the profile match quality is tolerable. Therefore,
in choosing the set of lines to fit, we made a distinction between earlier-
($T_{\rm eff}\gtrsim 3700$ K) and later- ($T_{\rm eff}\lesssim 3700$ K) type M
dwarfs. Overall, our selected line regions have a variety of strengths and
breadths attributed to HFS, and are potentially suitable for V abundance
determination. These regions and the number of split components that compose
them are listed in Table 1 and illustrated in Figs. 16 and 17. We note that
the line regions with fewer split components tend to look less “square”.

Table 1: Wavelength regions used for the V abundance fits.

$\lambda_{\rm central}$ | $\lambda_{\rm low}$ | $\lambda_{\rm upp}$ | # comps. | $T_{\rm eff}\gtrsim$ | $T_{\rm eff}\lesssim$
---|---|---|---|---|---
(nm) | (nm) | (nm) | modeleda | 3700 K | 3700 K
802.7366 | 802.70 | 802.90 | 18 | yes | yes
809.3468 | 809.32 | 809.37 | 12 | yes | yes
811.6789 | 811.64 | 811.72 | 21 | yes | yes
814.4560 | 814.44 | 814.47 | 6 | yes | yes
816.1062 | 816.06 | 816.15 | 16 | yes | yes
818.6728 | 818.63 | 818.70 | 10 | yes | no
824.1599 | 824.13 | 824.17 | 6 | yes | yes
825.3506 | 825.30 | 825.39 | 18 | yes | yes
825.5896 | 825.56 | 825.61 | 12 | yes | yes
891.9847 | 891.95 | 892.02 | 18 | yes | yes
893.2947 | 893.26 | 893.34 | 12 | yes | no
903.7613 | 903.74 | 903.79 | 4 | yes | no
904.6693 | 904.64 | 904.70 | 10 | yes | no

Notes: a: Number of HFS components included in the line model.

###  4.3 Synthetic spectra and fitting algorithm

As described in Section 3.2, we used the iSpec stellar spectroscopy software
to synthesize and fit model spectra generated from the new line lists. iSpec
is a highly versatile program developed for stellar parameter and abundance
analysis. It allows users to mix and match suites of atmosphere structure
grids and radiative transfer codes (Blanco-Cuaresma, 2019), and comes
preloaded with several popular options of both. The user provides the program
with a solar elemental abundance pattern and line lists in the wavelength
regions under consideration. Then, given a set of stellar properties ($T_{\rm
eff}$, $\log g$, [M/H], $\alpha$), broadening parameters ($v_{\rm mic}$,
$v_{\rm mac}$, $v\sin i$, instrumental resolution) and individual abundances,
iSpec can synthesize the stellar spectra by interpolating the atmosphere
structure grid. It can also perform the inverse problem, deriving fundamental
parameters and abundances from an input spectrum and line region definitions,
using either SSF (based on an iterative $\chi^{2}$ minimization algorithm) or
EW. The continuum model can be found by fitting splines to a version of the
spectra smoothed with maximum and median filters, as described by Blanco-
Cuaresma et al. (2014) and Blanco-Cuaresma (2019). The spline and window
parameters are specified by the user. For broadening, iSpec uses a radial-
tangential kernel for macroturbulence and the standard Gray kernel for
rotation (Gray, 1992). For comparison with observed spectra, the synthesized
spectra are convolved with a Gaussian line profile corresponding to the user-
specified resolution of instrument. For all of our analyses, we chose the
MARCS 1D, plane-parallel, LTE atmosphere models (Gustafsson et al., 2008)
paired with the radiative transfer code Turbospectrum version 15.1 (Plez,
2012), both as provided in iSpec. Our solar abundance pattern followed Asplund
et al. (2009). We used the iSpec SSF scripts to perform line-by-line fits to
the applicable V i line regions, varying only V abundance and fixing all other
fit parameters to those externally determined or assumed. Where only [Fe/H] is
available (e.g., in the literature), we take it as a proxy for [M/H], iSpec’s
scaling factor for the overall elemental abundance pattern ([M/H] is a measure
of the average overall metal content, which is expected to deviate from
[Fe/H], the iron abundance, at low metallicities, mainly due to an enhancement
of $\alpha$-elements. However, the distinction between [Fe/H] and [M/H] is
sometimes casually treated in the M dwarf literature, meaning that some
published [Fe/H] values may be closer to [M/H] – see Marfil et al. 2021 for a
relevant discussion).

###  4.4 Test case: Arcturus

Arcturus is a K giant with publicly-accessible, high-resolution
($\mathcal{R}\sim 150\,000$), high signal-to-noise (S/N $\sim 1000$) échelle
spectra across the entire optical wavelength range (373–930 nm, Hinkle et al.,
2000). It is also the only late-type star with V abundance exquisitely
measured from 55 lines at optical wavelengths (550–910 nm) with a complete
accounting of the most up-to-date knowledge of the atomic parameters and HFS
effects (Wood et al., 2018). This makes Arcturus a suitable benchmark target
on which to test our method. Our independent selection of line regions turns
out to be very similar to those used by Wood et al. (2018) in the 800–910 nm
range. We excluded the V i $\lambda$897.1673 nm and $\lambda$908.5231 nm lines
because they appear to be affected by relatively heavy molecular blending in
early M-dwarf spectra.

Where possible, we adopted identical stellar parameters as used in the
analysis of Wood et al. 2018: $T_{\rm eff}=4275$ K, $\log{g}=1.3$, $v_{\rm
mic}=1.6$ km s-1, and [Fe/H] = $-0.55$ dex, taken from Peterson et al. (2017).
In addition, we fixed $v\sin i=1.5$ km s-1 and $v_{\rm mac}=5.2$ km s-1, as
measured by Gray & Brown (2006) for Arcturus. The fitted synthetic spectra
were convolved to $\mathcal{R}=150\,000$ using a Gaussian profile to match the
resolution of the Arcturus spectrum. We used iSpec’s default continuum
settings whereby quadratic B-splines were fitted to 5 nm segments filtered
using the “median+max” scheme, where the median filter was 0.05 nm and the
maximum filter was 1.0 nm wide. We performed line-by-line fits to the 13 lines
listed in Table 1, arriving at $\langle\log\epsilon({\rm V})\rangle=3.55\pm
0.01$ (sample standard deviation $\sigma=0.04$)666$\log\epsilon({\rm
X})\equiv\log_{10}(N_{\rm X}/N_{\rm H})+12$.. Our value compares very well
with the determination by Wood et al. (2018) of $\langle\log\epsilon({\rm
V})\rangle=3.54\pm 0.01$ from 55 lines ($\sigma=0.04$).

###  4.5 CARMENES sample selection

We visually inspected the telluric-corrected template spectra of 331 CARMENES
GTO targets to identify stars that prominently exhibit the V HFS features of
interest and in which these features can be adequately reproduced by the
models. We observed that spectra of M dwarfs with $T_{\rm eff}\lesssim 3400$ K
tend to be very contaminated in a majority of the lines and surrounding
continuum regions that we considered (from Section 4.2), whereas those with
$T_{\rm eff}\gtrsim 3500$ K and $v\sin i<4$ km s-1 are well-fit in most of the
line regions under consideration. Based on the $T_{\rm eff}$ determined for
this sample from Schw19 whenever possible and, otherwise, from Passegger et
al. (2019) and Passegger et al. (2018), we removed 117 stars with $T_{\rm
eff}<3400$ K. A further 34 stars had no measured $T_{\rm eff}$ in any of the
references given above because they are fast rotators, are very late-type
stars, or have an S/N that is too low. Therefore, we discarded those as well.
Of the 117 stars with $T_{\rm eff}>3500$ K, we retained all except J16102–193,
which has a $v\sin i=7.3$ km s-1 (Reiners et al., 2018), J11201–104 and
J18174+483, which show large broadening in the V features in excess to that
accountable by the measured $v\sin i$, J00162+198W and J05532+242, which are
double-lined spectroscopic binaries (Baroch et al., 2018), J15474–108, which
is a spectroscopic triple (Baroch et al., 2021), and J04219+213, which
exhibits large spurious features. This left 110 stars. In the remaining sample
of 63 stars with 3400 K $\leq T_{\rm eff}\leq$ 3500 K we first eliminated four
stars that have a measured $v\sin i>4$ km s-1, leaving 59 stars. We assessed
these by eye and picked out a further 25 stars where a majority of the
features appear to be fit decently with the model spectra. The resulting
sample consisted of $110+25=135$ early M dwarfs, all of which have $T_{\rm
eff}\gtrsim 3400$ K and $v\sin i<4$ km s-1.

###  4.6 Vanadium abundances for the CARMENES GTO subsample

We fit to the V regions as listed in Table 1 for this subsample of early M
dwarfs from the CARMENES GTO program777The following V i lines were excluded
from the fit for the following objects: J12248–182: $\lambda$825.3506 nm (too
weak); J14082+805: $\lambda$816.1062 nm (contaminated); J16167+672S:
$\lambda$818.6728 nm and $\lambda$825.3506 nm (contaminated); J17355+616:
$\lambda$818.6728 nm (contaminated).. In these fits, the stellar parameters
$T_{\rm eff}$, $\log g$, and [Fe/H] were fixed to those derived by Schw19
based on PHOENIX-ACES model fits to metal lines in the same spectra. iSpec
picks an atmosphere structure model consistent with the expected $\alpha$
enhancement as a function of [M/H] through the usual relation ([$\alpha$/Fe] =
0.0 for [M/H] = 0.0, increasing to +0.4 for [M/H] = –1.0; Gustafsson et al.
2008). Since macroturbulence velocities, $v_{\rm mac}$, are not well-
characterized for cool dwarfs, but are likely to be small (e.g., extrapolation
from Fig. 17.10 in Gray 1992 extends below 1 km s-1 for stars later than K3 V
– see also Lindgren et al. 2016), we set the $v_{\rm mac}$ parameter to 1.0 km
s-1 in our fits. The microturbulence velocity $v_{\rm mic}$ is determined by
iSpec’s built-in function and depends on $T_{\rm eff}$, $\log g$, and [M/H].
For M dwarfs, $v_{\rm mic}$ as assigned by iSpec is generally between 0.8 km
s-1 and 0.9 km s-1. The values recommended by Husser et al. (2013) based on
convective velocities from the PHOENIX-ACES models are approximately $v_{\rm
mic}\sim 0.5$ km s-1. We checked that the final fit results are insensitive to
such small $v_{\rm mic}$ variations. Where possible, we used the $v\sin i$
values measured by Reiners et al. (2018) for this sample. When only an upper
limit of 2 km s-1 is available, we set $v\sin i$ = 2 km s-1. We convolved the
synthesized spectra with a Gaussian instrumental line profile corresponding to
the resolving power of the CARMENES VIS spectrograph. As in the Arcturus fit,
the local continuum normalization was done using the “median+max” scheme with
window length of 0.05 nm and 1.0 nm respectively.

The high-S/N input template spectra are already shifted into the rest frame
with wavelength given in air, with the velocity shift determined from the
fitted peak of a cross-correlation function with the PHOENIX-ACES spectra for
the corresponding stellar parameters. Therefore, in all our abundance fits,
the radial velocity ($v_{\rm r}$) was set to 0 km s-1.

We present the mean V abundances derived from our individual line-by-line fits
in Table LABEL:t:v-abunds, given in terms of [V/H]888We used the standard
definition $[{\rm X/H}]\equiv\log\epsilon({\rm X})-\log\epsilon_{\odot}({\rm
X})$, where $\log\epsilon({\rm X})\equiv\log_{10}(N_{\rm X}/N_{\rm H})+12$ and
$\log\epsilon_{\odot}({\rm V})=3.93$ (Asplund et al., 2009).. Figure 4 shows
the V-to-Fe abundance ratio of this sample as a function of metallicity (i.e.,
[Fe/H] from Schw19). Since the [V/Fe] value is not directly constrained in our
fits, we calculated it as [V/H] – [Fe/H]. Our [V/Fe] values lie close to 0.00
dex (sample standard deviation = 0.09 dex), consistent with that of FG-type
stars in the solar neighborhood (e.g., Battistini & Bensby, 2015, hereafter
BB15). The later-type stars in the sample (i.e., $T_{\rm eff}<3700$ K) present
a larger scatter than the earlier-type stars (i.e., $T_{\rm eff}>3700$ K).
This could be a consequence of cooler star V abundances being more sensitive
to metallicity errors (see Section 4.7) and systematically poorer line fits
(see Section 4.8).

Figure 4: [V/Fe] versus [Fe/H] diagram of the 135 CARMENES GTO stars derived
from our line-by-line analysis (red diamonds for $T_{\rm eff}>3700$ K and pink
diamonds for $T_{\rm eff}<3700$ K). Each vertical error bar represents the
sample standard deviation of the abundances derived from the set of V lines
used for each star. The typical error in the literature metallicity used in
our analysis is 0.16 dex (Schw19), also shown in the upper left of the figure.
Overplotted are selected representative V measurements in populations of
nearby FGK stars from the literature. Gray filled circles are measurements
from Battistini & Bensby (2015) for local FG dwarfs. The triangles show
results from Montes et al. (2018) for local FGK dwarfs, with blue denoting the
K-dwarf subsample therein ($T_{\rm eff}<5300\,$K). The V abundances measured
for these K dwarfs are generally significantly higher than for the rest of the
stars, possibly a consequence of not having accounted for HFS in the modeling.
The purple line shows the predicted trend for the solar neighborhood from the
Galactic chemical evolution models of Kobayashi et al. (2011). The dashed
magenta line shows a model from Sneden et al. (2016), which is built upon the
Kobayashi et al. (2011) model but incorporates a hypernova jet effect.

###  4.7 Error quantification

One standard way to estimate the error in abundance measurements is through
the line-to-line scatter. Column $\delta$[V/H] in Table LABEL:t:v-abunds lists
the sample standard deviation of the line abundances over all the lines
considered for each star. This is typically 0.03 dex to 0.07 dex (typically
$\sim$0.04 dex for $T_{\rm eff}>3700$ K and $\sim$0.06 dex for $T_{\rm
eff}<3700$ K, see also Fig. 5). The standard error of the mean (SEM) line
abundances (sample standard deviation over square root of the number of lines)
is another commonly used measure of abundance error. For all but two stars in
our sample, the SEM lies between 0.01 dex and 0.03 dex.

Errors in [V/H] arising from uncertainties in the input stellar parameters
were calculated for two representative stars and listed in Table 2. To derive
these values, we varied the $T_{\rm eff}$, $\log g$, and [Fe/H] one-by-one
within their formal $1\sigma$ errors as given by Schw19 (51 K, 0.07 dex, and
0.16 dex, respectively). The total expected errors contributed by the
incertitude of stellar parameters is calculated from summing each source of
error in quadrature (i.e., they are treated independently from one another).
Adding the line-to-line standard deviations, 0.06 dex and 0.08 dex are
representative total uncertainties in our V measurements for typical M0.0 V
and M2.0 V stars, respectively.

Table 2: Abundance sensitivity to uncertainties in stellar parameters (from Schw19) for two representative stars in the sample. Stellar | J14257+236W | J03213+799
---|---|---
parameters | ($T_{\rm eff}\sim 4000$ K) | ($T_{\rm eff}\sim 3500$ K)
$T_{\rm eff}\pm 51$ K |  $\pm$0.01 dex |  $\pm$0.02 dex
$\log g\pm 0.07$ dex |  $\pm$0.03 dex |  $\pm$0.02 dex
$\rm{[Fe/H]}\pm 0.16$ dex |  $\pm$0.03 dex |  $\pm$0.05 dex
Total | 0.04 dex | 0.06 dex

###  4.8 Line-to-line abundance systematics

We now examine the quality of the line selections and how our final abundance
results depend on them. In Fig. 5 we plot the deviations of individual line
abundances from the mean over all lines for the whole sample. On the whole,
the abundances derived from our choice of lines show decent agreement. The
scatter is typically within $\pm 0.05$ dex, which is comparable to or better
than that of similar studies. However, in every line there exists various
degrees of systematic deviation, in general worsening for cooler stars. The
largest differences are found in the V i $\lambda$818.6728 nm,
$\lambda$824.1599 nm, and $\lambda$891.9847 nm lines. Abundances derived from
these lines are on average 0.05 dex above or below the mean across all lines,
and can differ more severely for individual stars. As shown in Figure 17,
these line regions are more contaminated by molecules and Fe i lines, which
could affect the accuracy of the modeling by introducing line blending or
complicating the determination of the pseudo-continuum level, as well as make
the fits more sensitive to uncertainties in metallicity. The V i
$\lambda$814.4560 nm, $\lambda$816.1062 nm, $\lambda$825.3506 nm, and
$\lambda$825.5896 nm lines show good agreement with the average at
temperatures higher than $\sim 3700$K, but fare more poorly for cooler stars.
This is again indicative of inadequate modeling of molecules. The presence of
systematic abundance discrepancies from line to line calls for caution in
interpreting abundance constraints from single or small sets of lines unless
they are very well-understood.

Examples of single lines that could potentially be used effectively on their
own are the V i $\lambda$809.3468 nm and $\lambda$811.6789 nm lines. Both
appear to trace the mean abundance very well and exhibit very little scatter.
The fact that they are useful throughout the $T_{\rm eff}$ range of 3400 K to
4200 K makes them the most dependable single V abundance indicator lines in
this wavelength and temperature range. By extension, they could be promising
metallicity proxies as well (see Section 5.5).

Figure 5: Individual line abundance deviations from the mean abundance across
the lines for the star sample. At each given line wavelength (in air), the
black circles denote the line [V/H] $-$ mean [V/H] for each star in which that
line was studied. The thick orange squares show the median abundance deviation
and sample standard deviation for each line (over all the CARMENES stars for
which that line was used). The blue and red squares mark the same metric for
the warmer ($T_{\rm eff}>3700$ K) and cooler ($T_{\rm eff}<3700$ K)
subsamples, respectively. The horizontal dotted lines bracket the median
abundance error (in terms of the line-to-line standard deviation) for the
whole sample ($\pm$0.047 dex, blue squares), the warmer subsample ($\pm$0.036
dex, red diamonds), and the cooler subsample ($\pm$0.056 dex, pink diamonds).
The V i $\lambda$818.6728 nm, $\lambda$893.2947 nm, $\lambda$903.7613 nm, and
$\lambda$904.6693 nm lines were not fitted for stars with $T_{\rm eff}<3700$
K.

###  4.9 Fits using alternative stellar parameters

Accurate determination of M-dwarf stellar parameters is an active area of
research and method-dependent disagreements are commonplace. There is some
evidence that the existing published metallicity values for the CARMENES GTO
stars (Passegger et al. 2018, 2019, Schw19) derived from PHOENIX model grid
fits predominantly to strong Fe, Ti, Mg, and K lines in the optical and NIR,
may be somewhat systematically overestimated. For Passegger et al. (2019), who
used the latest PHOENIX-SESAM model grids, the systematic offset is especially
evident for the earliest M dwarfs and most metal-poor stars (also see a
literature comparison in Figs. 5 and 6 of Passegger et al. 2019). The problem
is less severe, but possibly still persists, for the parameters given by
Schw19, which were based on PHOENIX-ACES grids. For example, the apparent
dearth of relatively metal-poor stars ([Fe/H] $<$ –0.3) in the sample could be
suggestive of metallicity measurements being biased to higher values. A
detailed investigation of such deviations is beyond the scope of this paper.
It may be a combination of saturation effects in the lines, spectral model
inadequacies, the neglect of $\alpha$ enhancement, NLTE effects, or something
else (see also Abia et al., 2020; Ishikawa et al., 2020; Olander et al., 2021,
for discussions). Therefore, it is worthwhile to check the robustness of our V
abundances to alternative sets of stellar parameters.

One further effort to re-determine the fundamental stellar parameters
uniformly for the entire CARMENES GTO sample will be presented in a
forthcoming publication by Marfil et al. (2021), using the spectral analysis
software SteParSyn (Tabernero et al., 2018, 2021). SteParSyn is an automatic
code designed to infer the stellar atmospheric parameters $T_{\rm eff}$, $\log
g$, [Fe/H]999Denoted as [Fe/H]corr in Marfil et al. (2021)., and $v\sin i$ of
late-type stars (FGKM) following the spectral synthesis method using a Markov
Chain Monte Carlo algorithm. This code has already been used in spectroscopic
studies that involve late FGKM stars in a variety of astrophysical scenarios
(e.g., Tabernero et al., 2018; Palle et al., 2021; Borsa et al., 2021). The
work of Marfil et al. (2021) is based on a synthetic spectral grid generated
from BT-Settl model atmospheres (Allard et al., 2012) coupled to Turbospectrum
(Plez, 2012) in a large number of Fe i and Ti i line regions, as well as TiO
bandheads. Currently, some of the $\log g$ values determined with SteParSyn
appear to be possibly too large. Therefore, for our work, we assigned new
$\log g$ values based on $T_{\rm eff}$ and [Fe/H] via the PARSEC isochrones
(Bressan et al., 2012; Chen et al., 2014), similar to the procedure used by
Passegger et al. (2019). The isochrone model inputs were the fitted $T_{\rm
eff}$ and [Fe/H] from Marfil et al. (2021), and age estimates as given by
Passegger et al. (2019). We present the SteParSyn-derived $T_{\rm eff}$ and
[Fe/H], as well as the $\log g$ constrained from the isochrones, for our
CARMENES GTO subsample in Table LABEL:t:v-abunds. For simplicity, we refer to
this collection of alternative stellar parameters, consisting of $T_{\rm eff}$
and [Fe/H] from Marfil et al. (2021) and the modified $\log(g)$’s, as the
“Marfil+” values.

As shown in Fig. 6, the metallicities given in Marfil+ and in Schw19
systematically differ. The discrepancy is largest for relatively metal-poor
stars, which also tend to be cooler: those deemed by SteParSyn to have [Fe/H]
$<-0.2$ have on average 0.19 dex higher value according to Schw19. Another
puzzling pattern is that the majority of stars with super-solar metallicities
according to Schw19 are part of the warmer (¿ 3700 K) subset, while the metal-
poor stars according to Marfil+ are almost exclusively cooler (¡ 3700 K).
These discrepancies are discussed in more detail by Marfil et al. (2021).

Figure 6: Comparison between [Fe/H] from Schw19 and Marfil+. The pink and red
diamonds denote stars with $T_{\rm eff}<3700$ K and $T_{\rm eff}>3700$ K,
respectively. The thin gray dashed line is the one-to-one line.

Despite the large systematic differences, the final derived [V/H] abundances
assuming each set of stellar parameters are very similar. Figure 7 compares
the [V/H] values determined from line-by-line fits where $T_{\rm eff}$, $\log
g$, and [Fe/H] were fixed to the Marfil+ values with those based on the Schw19
values. Although there is a small systematic offset of 0.034 dex overall
(0.056 dex for the metal-poor stars with [Fe/H] $<-0.2$), the agreement is
within 0.10 dex in the vast majority of cases, even when the [Fe/H]s differ by
up to $\sim$ 0.4 dex. The relative insensitivity of V abundance derived from
the lines used in this work to the exact stellar parameters is not surprising
given the error analysis presented in Section 4.7. Therefore, in spite of
systematic uncertainties in the exact stellar parameters, our [V/H] values
(relative to the solar V abundance of Asplund et al. 2009) could be taken as
robust.

Figure 7: [V/H] measured with stellar parameters fixed to those from Marfil+
versus Schw19. Black and blue points denote stars with [Fe/H] ¿ -0.2 and
[Fe/H] ¡ -0.2, respectively. The thick and thin gray dashed lines mark the
one-to-one line and that offset by 0.1 dex.

Finally, we show [V/Fe] versus [Fe/H] for the Marfil+ stellar parameters in
Figure 8, which exhibits overall good agreement with the distribution of stars
from BB15. Since the fitted [V/H] values for the two sets of stellar
parameters are very similar, the qualitative differences between Fig. 8 and
Fig. 4 chiefly reflect the differing metallicity measurements between Schw19
and Marfil+. One curious dissimilarity concerns the relative scatter in [V/Fe]
between the cooler and warmer subsets: the notably smaller [V/Fe] scatter in
the warmer stars in Fig. 4 is not reproduced in Fig. 8. These patterns could
arise from a complex interplay between the errors from the line fits,
systematic uncertainties in the stellar parameters, and perhaps also
astrophysical causes. Disentangling these factors is beyond the scope of this
work.

Figure 8: Same as Fig. 4, but for fits based on the Marfil+ stellar parameters
for 135 stars. The vertical error bars represent the sample standard deviation
of the line-to-line abundances.

##  5 Discussion

###  5.1 Comparison with local FG dwarfs and K giants

The [V/Fe] versus [Fe/H] trend for the local CARMENES M-dwarf sample is
consistent with the behavior observed in populations of local Galactic disk
FG-dwarfs stars (e.g., BB15) for the overlapping metallicity range (–0.6 dex
to +0.3 dex), as shown in Figs. 4 and 8. BB15 performed a comprehensive study
of odd-$Z$ iron peak element abundance in the solar neighborhood. In
particular, they carried out a large spectroscopic study of 714 FG-dwarfs
(Bensby et al., 2014), 466 of which have a measured $\log\epsilon({\rm V})$.
The analysis was done using a line-by-line approach and differentially with
respect to the Sun. The authors modeled HFS effects in five V i line regions
between 550 nm and 630 nm, taking the line component data from Prochaska et
al. (2000). An abundance is considered by the authors to be determined if at
least one of the five lines appears to be well fit. The BB15 results are also
largely compatible with other large studies of nearby stars that have
accounted for HFS, including the NIR spectroscopic study of K giants in the
SDSS-APOGEE survey of Hawkins et al. (2016) and the local giant sample of
Lomaeva et al. (2019).

###  5.2 A potential partial solution to the issue of abundance-temperature
relation in local K dwarfs

A number of well-known abundance studies whose samples also included K dwarfs,
many of them dealing with distinguishing the chemical compositions of planet
hosts from barren stars, noticed a puzzling systematic trend with regard to V
abundance. For example, in a sample of about 1000 FGK stars, Adibekyan et al.
2012 (hereafter AD12) measured [V/Fe] of around 0.0 dex throughout the FG
spectral type range, but rising steadily from 0.0 dex to +0.6 dex as $T_{\rm
eff}$ decreases from 5300 K to 4500 K (as shown in Fig. 10). Incidentally, a
similar trend is observed therein for Sc, another element known to exhibit HFS
(see Section 3). Recognizing it to be unphysical, the authors attributed this
behavior to a variety of possible causes, including increased line blending
for cool stars, NLTE, inaccurate model atmosphere structures, and imprecise
$\log(gf)$ values.

Larger scatter and a systematic correlation with $T_{\rm eff}$ below 5300 K
was also observed in, for example, Bodaghee et al. 2003; Gilli et al. 2006;
Neves et al. 2009; Tabernero et al. 2012; Jofré et al. 2015a, and Montes et
al. 2018, prompting some authors to either empirically correct the
measurements in this temperature range, or to disregard them altogether. This
trend is weaker but still visible in the sample of GK-type giants of Adibekyan
et al. (2015). In Fig. 4, we overplot the results of Montes et al. (2018), who
measured elemental abundances for FGK-type dwarfs in multiple systems with
M-dwarf common proper motion companions. The distribution of [V/Fe] for stars
with $T_{\rm eff}>5300$ K and $T_{\rm eff}<5300$ K are far removed from each
other, with those below 5300 K enhanced by more than $\sim+0.5$ dex on average
compared to solar.

Figure 9: Comparison of simulated profiles of vanadium lines used in Adibekyan
et al. (2012), without and with HFS components, for a K dwarf ($T_{\rm eff}$ =
4800 K) and a G dwarf ($T_{\rm eff}$ = 5700 K) with a solar abundance pattern
at CARMENES VIS resolution ($\mathcal{R}\sim 94\,600$). The red and orange
solid lines are generated from the VALD3 atomic line list including the latest
HFS components, for the K and G dwarf respectively. The dotted blue and green
lines are simulated from a line list without HFS components. The lines are
much stronger in later-type stars. While most lines show very small
deviations, a few lines (notably $\lambda$567.085 nm, $\lambda$573.707 nm, and
$\lambda$625.183 nm) in the K dwarf exhibit significant systematic differences
when HFS is considered.

The oscillator strengths for many V i transitions were improved by Lawler et
al. (2014), which could impact works that used older line lists. However, many
of the aforementioned analyses were conducted differential with respect to the
Sun and, therefore, uncertainties in $\log(gf)$ are expected to cancel. More
interesting is the fact that none of the works that found an abundance-
temperature trend appear to have used HFS components in their spectral
analysis. Inspired by our present work with V lines in cool dwarfs, we
hypothesized that a failure to account for HFS could be a key factor
responsible for the apparent V overabundance in K dwarfs. Indeed, HFS was
already proposed as a possible reason for the large scatter in [V/Fe] of
nearby stars compiled from the literature in the Hypatia Catalog (Hinkel et
al., 2014). A hint could also be found in the work of Maldonado et al. (2015),
who also determined elemental abundances in FGK stars. The line regions used
in that work for the V analysis were identical to those of Adibekyan et al.
(2012), but Maldonado et al. (2015) included HFS components for the subset of
V i lines redward of 600 nm (Ramírez et al., 2014). The temperature dependence
below 5300 K was still somewhat visible, but far less severe.

To investigate our hypothesis within our existing analysis framework, we
performed a simple experiment. We simulated G- and K-dwarf spectra in the V i
line regions used in the abundance work of AD12 with iSpec. We did this once
with the new HFS components, to mimic reality, and once with an older line
list likely resembling what would have been used in the previous analysis, to
mimic past methodologies. The line profiles are compared in Fig. 9 for a K
dwarf $T_{\rm eff}=4800$ K, $\log g=4.4$, [M/H] = 0.0 dex, $v\sin i=1.0$ km
s-1, and $v_{\rm mic}=1.0$ km s-1, and for a G dwarf with $T_{\rm eff}=5700$
K, $\log g=4.4$, [M/H] = 0.0 dex, $v\sin i=1.05$ km s-1, and $v_{\rm
mic}=3.92$ km s-1. The first thing to note is that, unlike the bucket-shaped
lines presented in this work, these line shapes are visually not markedly
different from Gaussian and, therefore, they may not raise a red flag. Another
thing to note is that all the lines in the K dwarf indeed appear broader (and
in some cases also deeper) when HFS is taken into account. Though the effect
is subtler in some lines than others, large deviations are apparent for V i
$\lambda$567.0848 nm, $\lambda$573.7062 nm, and $\lambda$625.1823 nm. On the
other hand, deviations are almost invisible across all the lines in the G
dwarf. A third point is that these lines at K dwarf temperatures, $T_{\rm
eff}\approx 4800$ K, are much stronger than in G dwarfs and, therefore, more
likely to be affected by saturation, which would mean HFS would no longer
conserve their EWs.

To quantify the impact of the difference between expected and actual line
profiles on abundance measurements, we performed line-by-line synthetic
spectral fits to the mock spectra generated using HFS components on a
temperature grid from 4500 K to 5500 K in increments of 100 K. All mock
spectra were synthesized with solar metallicity, $\log g=4.6$ for $T_{\rm
eff}\leq 5000$ K and $\log g=4.5$ for $T_{\rm eff}>5000$ K, and $v_{\rm mic}$
= 0.88 km s-1 to 0.98 km s-1, with $v_{\rm mac}$ and $v\sin i$ set to 0. The
models used for the abundance fits were generated with a line list ignoring
HFS. We ran a trial retrieval using the most current $\log(gf)$ values from
Lawler et al. (2014). In addition, we repeated the trials with the $\log(gf)$
data used by Neves et al. (2009) (Table 2 therein), which are systematically
lower than the updated values by about 10 %. In all the fits, only the V
abundance was allowed to vary. We took the mean of the abundances retrieved
from the seven lines and used the standard error of the mean as an indication
of measurement error. Figure 10 overplots the simulated [V/Fe] versus $T_{\rm
eff}$ on the results from AD12.

The upward trend with decreasing temperature starting from 5300 K is
reproduced by our simulations with a very similar slope and inflection point.
Moreover, the worsening line-to-line scatter seen in AD12 also quantitatively
resembles that of our simulation. A few “bad” lines have a big impact.
Exchanging the outdated oscillator strengths with the new values results in a
systematic offset of 0.05 dex to 0.10 dex, although this difference should
only manifest when considering abundance retrievals on an absolute scale. The
remaining offset suggests that other sources of systematics still need to be
identified, for example NLTE effects or stellar model differences. While no
dedicated NLTE study of neutral vanadium has been conducted to date,
systematic differences between abundances derived from V ii versus V i lines
in metal-poor stars do hint that NLTE might be important (Ou et al., 2020). It
has also been suggested that, at least in Sun-like photospheres, NLTE effects
may be significant in V i due to the majority of V atoms being in the singly-
ionized state (Scott et al., 2015). The shape and magnitude of the deviation
pattern may also differ when using an EW method, as was done by AD12 and
similar works. Nevertheless, the results from our simulated SSF abundance
retrievals indicate that failure to take HFS into account could play a key
role in introducing artificial trends and enhancing the measured V abundances
of K dwarfs.

Figure 10: Vanadium abundances based on line-by-line synthetic spectral
fitting analysis without HFS on mock spectra generated with an HFS line list
as a function of temperature, compared to the abundance trend in AD12. The
gray filled circles are measurements and error (computed as line-to-line
RMS/$\sqrt{7}$ for seven lines) by AD12 and the blue squares show the median
of these measurements and error binned by temperature. The dark green diamonds
show the mock retrieval results for model spectra with $\log(gf)$ values from
Neves et al. (2009) for the V lines, whereas the neon green diamonds use the
latest $\log gf$ values. Below 5300 K, the “measured” [V/Fe] increases with
decreasing effective temperature in a very similar manner as found by
Adibekyan et al. (2012) and others. The observed line-to-line scatter as a
function of $T_{\rm eff}$ also resembles the simulated behavior.

The effect of unmodeled HFS on vanadium abundance measurements in Sun-like
stars was already identified in the appendix of Takeda (2007). They quantified
the expected systematic abundance bias from the EW method for nine V i lines
between 560 nm and 660 nm, two of which are in common with the lines we
examined (see Fig. 15 therein). They also found significant corrections to be
required for individual lines such as the one at 567.0848 nm, especially at
large strengths. Takeda (2007)’s final conclusion that this correction is
“insignificantly small” (i.e., $\lesssim 0.05$ dex when averaged over all nine
lines) seemingly contradicts our findings. However, considering that their
study was limited to cases where $T_{\rm eff}\gtrsim 5000$ K, the results of
Takeda (2007) are actually quite compatible with ours. An extrapolation of the
steepening trend of the correction factor with $T_{\rm eff}$ below $T_{\rm
eff}=5000$ K (in their Fig. 16) would imply much larger offsets at cooler
temperatures. Further discrepancies could also arise from differences in the
exact choice of lines. Relatedly, for red clump giants (4400 K $<T_{\rm eff}<$
5100 K), Liu et al. (2007) demonstrated that HFS introduced up to +0.5 dex of
bias in V abundances derived from the three lines that they considered
($\lambda$567.0854 nm, $\lambda$572.7057 nm, $\lambda$621.6358 nm). The
deviation worsens with increasing metallicity, presumably because the
magnitude of the correction tends to grow with line strength.

###  5.3 Trends within the thin and thick disks

Figure 11: [V/Fe] versus [Fe/H] of the 135 CARMENES-GTO stars derived from our
line-by-line analysis, color- and symbol-coded for their membership
attribution to Galactic kinematic groups (CC21). Top: analysis based on
stellar parameters from Schw19. Bottom: analysis based on stellar parameters
from Marfil+. The thick-disk members appear to be better distinguished in
metallicity space with the parameters estimated by Marfil+.

The literature is divided on the existence and nature of fundamental
differences in V abundance trends as a function of Galactic population. In a
study of ten thick-disk stars, Prochaska et al. (2000) measured an enhancement
of about +0.2 dex in [V/Fe] over solar, which is notably higher than the
typical [V/Fe] measurements for thin disk stars available at the time.
However, using the same atomic data, Brewer & Carney (2006) found no
significant distinction between V abundance patterns of 23 G dwarfs in the
thin and thick disk. Combining data from Reddy et al. (2003), Reddy et al.
(2006) leveraged $\sim 360$ nearby thin- and thick-disk FG stars to find a
$\sim 0.1$ dex offset between the distribution of [V/Fe] for the thick disk
over the thin disk. With 466 FG dwarfs, BB15 showed that [V/Fe] of disk stars
in their sample is mostly flat around solar with a slight uptick at the lower
end of its metallicity range, whereas a more pronounced sloped relationship of
decreasing metallicity and increasing [V/Fe] emerges for the thick disk. While
the two segments are visually dissimilar, the authors argued that the thick
disk is merely a continuous extension of the thin disk trend into lower
metallicities. Recently, Lomaeva et al. (2019) resolved a clear, $\sim 0.1$
dex parallel separation between trends in [V/Fe] versus Fe abundance for $\sim
300$ giant stars belonging to the thin and thick disk (see Fig. 3 therein),
even for overlapping metallicity ranges between $-0.4$ dex and $-0.1$ dex. We
note that the criteria for disk membership assignment are not uniform across
the studies. For example, in BB15, the classification was made using
isochronal age (with the thin-thick disk boundary drawn at 7–9 Gyr), while
Lomaeva et al. (2019) assigned stars to the thin or thick disk based on its
[Fe/H], [Ti/Fe], and Galactic kinematics.

Using kinematic designations from the Carmencita catalog (Caballero et al.,
2016a; Cortés-Contreras, 2017), we investigate whether there are thin/thick
disk distinctions in V abundance trends in this sample, as shown in Fig. 11.
We mark three different kinematic categories: the thin disk, the thick disk,
and the thin-thick disk (i.e., stars with intermediate characteristics, as
defined by Montes et al. 2001). While the Marfil+ [Fe/H] values appear to be
better correlated with the kinematic designations than the Schw19
metallicities, we are unable to identify a meaningful trend using either sets
of stellar parameters. This could partially be due to small number statistics.
All but 14 out of the 135 stars studied here belong to the thin disk.

###  5.4 Implication for Galactic nucleosynthesis

Our measurements reinforce the disagreement between observational and
theoretical expectations from GCE models, which predict a significantly lower
vanadium-to-iron abundance ratio across this metallicity range. Overplotted on
Fig. 4 are the relations between [V/Fe] and [Fe/H] in the solar neighborhood
from the GCE model of Kobayashi et al. (2011) and its variant presented in
Sneden et al. (2016). The Kobayashi et al. (2011) model is calculated from the
then-updated theoretical nucleosynthetic yields for SNe types Ia and II
(including hypernovae), as well as for asymptotic giant branch (AGB) stars, in
conjunction with observationally constrained initial mass functions (IMFs) and
star formation history (SFH) models. The predicted vanadium abundance is
around $-0.5$ dex below that observed for a large range of stellar types and
metallicities. Similar deficiencies are also present for Sc and Ti (Kobayashi
et al., 2011; Nomoto et al., 2013). In addition to the need to better
understand SNe explosion parameters and yields, proposed physics solutions
include possible production enhancement facilitated by neutrino interactions
(Kobayashi et al., 2011), the use of multidimensional nucleosynthetic models
(Nomoto et al., 2013), incorporation of hypernovae jet effects (Sneden et al.,
2016), and revised parameters for the IMF and assumptions for SFH and Galaxy
formation scenarios (Sneden et al., 2016). The Sneden et al. (2016) model
shown in Fig. 4 is based on the Kobayashi et al. (2011) model and includes
bipolar jets from hypernovae in addition, which reduces the tension with
observation to $<0.3$ dex. Despite hints of progress, no model to date has
satisfactorily reproduced the observed trend. Our work further highlights
persistent problems with the current understanding of nucleosynthesis
mechanisms and supernova yields for vanadium in the context of the Milky Way’s
formation history.

###  5.5 Potential as a metallicity indicator in cool stars

Figure 12: [V/H] versus [Fe/H] measured for stars in the solar neighborhood.
The thick gray dashed line is the one-to-one line, while the thin gray dashed
lines bracket the $\pm 0.2$ dex regions. In this metallicity range, [V/H]
appears to correlate tightly with [Fe/H], reflecting its potential as an
alternative metallicity indicator.

Traditionally, M-dwarf spectra have the notoriety of being riddled with
multifarious mysteries (e.g., molecules, magnetic fields, mismatched
features). As models grow more sophisticated and data become more exquisite,
the question of spectroscopic metallicities for M dwarfs has become the
subject of many investigations over the past decade. While many avenues (i.e.,
spectral ranges and stellar models) have been explored, and progress has been
made, modern high-resolution studies still struggle to agree on metallicities
for individual cool stars (e.g., Veyette et al., 2017; Passegger et al., 2018,
2019; Souto et al., 2020; Sarmento et al., 2021) and are also in tension with
low-resolution counterparts (e.g., Rojas-Ayala et al., 2012; Mann et al.,
2013; Newton et al., 2014). Among other implications, the question of
metallicity has a significant impact for exoplanet demographics studies (e.g.,
Fischer & Valenti, 2005; Johnson et al., 2010; Buchhave et al., 2014; Petigura
et al., 2018).

Above [Fe/H] $\sim-0.5$ dex, V appears to trace Fe abundance in lockstep in a
large variety of settings (e.g., Feltzing & Gustafsson, 1998; Battistini &
Bensby, 2015; Ernandes et al., 2018), with dispersion on the order of 0.1 dex.
Figure 12 shows that [V/H] versus [Fe/H] for ours and the BB15 sample of stars
in the solar neighborhood fall largely within 0.2 dex to the one-to-one line.
Below [Fe/H] $\sim-0.5$ dex, [V/H] deviates from an exact correspondence, but
maintains a linear correlation with [Fe/H]. In both regimes, the scatter is
comparable to or better than precisions currently achievable for metallicity
measurements for M-dwarfs as well as study-to-study variations for individual
stars. Furthermore, we have demonstrated in Section 4 that [V/H] can be easily
and robustly determined nearly independently of other fundamental stellar
parameters. Even for cases where a large discrepancy exists between the
[Fe/H]s inferred by different procedures (up to $\sim 0.4$ dex), the measured
[V/H]s agree within 0.1 dex. Therefore, methods described in this work could
represent an alternative, indirect way to estimate the metallicity for M
dwarfs. A growing number of high-resolution spectrographs (e.g., 2.2 m La
Silla-FEROS, VLT-UVES, Keck-HIRES, Lick-Hamilton, Gemini-MAROON-X, NEID, HET-
HPF, LBT-PEPSI) cover the wavelength range considered in this work. Analyses
using observations from these instruments should be capable of delivering
reliable vanadium measurements and potentially metallicity proxies for M
dwarfs.

###  5.6 Implications for planets

####  5.6.1 Planet hosts

To date, 19 of the stars in our sample are known planet hosts. As shown in
Fig. 13, the [V/H] distribution of known planet hosts and non-planet hosts do
not appear to be significantly different. This is not surprising, as most of
the planets hosted by our stars are Neptune-sized or smaller, for which a
notable correlation between occurrence rate and the detailed chemical
composition of the host star is not expected (e.g., Adibekyan et al., 2012).

Figure 13: Normalized distributions of [V/H] for known planet hosts (red)
versus the rest of the stars (black) in the sample based on stellar parameters
from Schw19 (top) and Marfil+ (bottom). The medians of each distribution are
marked by the vertical dotted lines and given in the legends.

####  5.6.2 Planetary atmospheres

As better and more precise spectrographs are being developed at large aperture
telescopes, V and other atomic species can be searched for in planetary
atmospheres. In fact, V has already been detected in at least one planetary
atmosphere (WASP-121b Borsa et al., 2021; Ben-Yami et al., 2020). HFS does not
seem to pose a problem for the detection of individual lines in transmission
spectroscopy, as the analyses are centered on the stronger lines, and the
lines are typically broadened in the planetary atmospheres. However, the use
of an HFS line list in the calculation of atomic cross-sections could affect
the S/N of the signals. Future studies dealing with retrieving atmospheric
abundances of exoplanets might have to take this effect into account.

It is also interesting to consider how the effect of HFS might manifest in
molecules such as VO. While VO, TiO, and metallic species such as Fe ii are
believed to be responsible for the thermal inversion of hot exoplanet
atmospheres (Yan et al., 2020), there is still no clear detection of VO in a
planetary atmosphere. Given that the detection of VO relies on the cross-
correlation technique (Snellen, 2004), where hundreds of VO spectral lines are
cross-correlated to a template, perhaps such methods are more sensitive to HFS
effects. However, since the effect of HFS in molecules is more difficult to
characterize, and does not appear to be well-studied for VO, a detailed
exploration of this topic is beyond the scope of this paper.

##  6 Conclusion

Detailed chemical analysis for cool stars stands at the cusp of exploration.
In this work, we have performed one of the first elemental abundance studies
for a large sample of M dwarfs. In the high-resolution, high S/N, telluric-
corrected CARMENES spectra, we located about a dozen hyperfine-split atomic V
i lines in the 800–910 nm region, which have clear presentations and are
relatively free of molecular contamination. Using the latest hyperfine
components for V i from VALD, we have shown that these lines can be modeled
well and that they are sensitive to abundances and relatively insensitive to
uncertainties in other fundamental stellar parameters. Using spectral
synthesis fits to these line regions, we obtained robust [V/H] measurements
for 135 nearby, early M dwarfs ($T_{\rm eff}>3400$ K) in the CARMENES GTO
sample, with typical errors of $\sim 0.07$ dex. The [V/Fe] versus [Fe/H] trend
closely resembles that observed for local FG-type stars, lying significantly
above predictions from GCE models based on the current knowledge of Galactic
star formation history and nucleosynthesis channels for this element.
Moreover, the tight correlation between Fe- and V- abundances in nearby stars
and the relative ease to measure photospheric [V/H] in high-resolution cool
star spectra suggests that V has the potential to be an effective metallicity
indicator for local M dwarfs.

We have discussed at length the risks of not properly accounting for HFS,
which could bias abundance measurements in various ways. A number of large
stellar abundance surveys have reported a systematic trend of increasing
[V/Fe] with decreasing $T_{\rm eff}$ for K dwarfs ($T_{\rm eff}\lesssim 5300$
K). Through simple toy simulations, we have shown that the neglect of HFS is
likely responsible for this phenomenon to a large extent. However, the exact
impact of HFS on abundance measurements cannot be generalized, as the degree
of splitting depends on the exact atomic transition parameters of the lines
considered and the properties of the atmosphere. The amount of correction
required must be evaluated on a case-by-case basis by considering the stellar
type and exact set of lines involved. The safest course of action in any
abundance analysis is to use only lines that are well understood (i.e., have
complete atomic data available), whenever possible, and to fully model this
effect.

Our investigation underlines the fact that HFS can have a dramatic impact on
line profiles and on abundance measurements. We suggest that this quantum
effect could play an important role in cool star spectroscopy where a number
of elements appear to exhibit stronger and wider splitting patterns than in
Sun-like stars. We have exploited this property to provide a reasonable
assessment of V abundance in the cool photospheres of the most abundant type
of stars in the Galaxy. This type of study is made possible with the
accessibility of constantly improving laboratory measurements, which are
essential and powerful tools for interpreting increasingly complex stellar
spectra in new temperature and wavelength regimes. Together with Abia et al.
(2020) for Rb, Sr, and Zr, and other forthcoming papers, our findings in the
CARMENES spectra open new windows for elemental analysis in cool stars.

###### Acknowledgements.

We thank Nikola Vitas and Andrew McWilliam for pedagogical discussions on the
nature of HFS splitting. We are grateful to our referee Vardan Adibekyan for
insightful feedback that helped to improve the quality of this manuscript.
CARMENES is an instrument at the Centro Astronómico Hispano-Alemán (CAHA) at
Calar Alto (Almería, Spain), operated jointly by the Junta de Andalucía and
the Instituto de Astrofísica de Andalucía (CSIC). CARMENES was funded by the
Max-Planck-Gesellschaft (MPG), the Consejo Superior de Investigaciones
Científicas (CSIC), the Ministerio de Economía y Competitividad (MINECO) and
the European Regional Development Fund (ERDF) through projects FICTS-2011-02,
ICTS-2017-07-CAHA-4, and CAHA16-CE-3978, and the members of the CARMENES
Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de
Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai,
Institut für Astrophysik Göttingen, Universidad Complutense de Madrid,
Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias,
Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-
Alemán), with additional contributions by the MINECO, the Deutsche
Forschungsgemeinschaft (DFG) through the Major Research Instrumentation
Programme and Research Unit FOR2544 “Blue Planets around Red Stars”, the Klaus
Tschira Stiftung, the states of Baden-Württemberg and Niedersachsen, and by
the Junta de Andalucía. We acknowledge financial support from the Agencia
Estatal de Investigación of the Ministerio de Ciencia, Innovación y
Universidades and the ERDF through projects
PID2019-109522GB-C5[1:4]/AEI/10.13039/501100011033 and the Centre of
Excellence “Severo Ochoa” and “María de Maeztu” awards to the Instituto de
Astrofísica de Canarias (CEX2019-000920-S), Instituto de Astrofísica de
Andalucía (SEV-2017-0709), and Centro de Astrobiología (MDM-2017-0737), the
Generalitat de Catalunya/CERCA programme, the Austrian Science Fund (P
33140-N), the Ministerio de Universidades (FPU15/01476), the DFG (SPP 1992 JE
701/5-1), and NASA (NNX17AG24G). This work has made use of the VALD database,
operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and
the University of Vienna.

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##  Appendix A Long Table

Table LABEL:t:v-abunds is available in its entirety in electronic form at the
CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.

Table 3: Measured vanadium abundances for the sample Karmn | SpT |  $v\sin i$a | Schw19 Stellar Parameters | Marfil+ Stellar Parameters | Kinematic
---|---|---|---|---|---
|  | (km/s) |  $T_{\rm eff}$ (K) | $\log g$ | [Fe/H] | [V/H] |  $\delta$[V/H]b |  $T_{\rm eff}$ (K) | $\log g$ | [Fe/H] | [V/H] |  $\delta$[V/H]b | Membership
J00051+457 | M1.0V | 2.0 | 3675 | 4.84 | -0.16 | -0.01 | 0.03 | 3759 | 4.72 | -0.02 | 0.02 | 0.04 | YD
J00183+440 | M1.0V | 2.0 | 3615 | 4.91 | -0.25 | -0.25 | 0.07 | 3613 | 4.86 | -0.47 | -0.31 | 0.06 | D
J00389+306 | M2.5V | 2.0 | 3537 | 4.89 | -0.05 | -0.04 | 0.06 | 3559 | 4.81 | -0.18 | -0.10 | 0.06 | D
J01013+613 | M2.0V | 2.0 | 3529 | 4.94 | -0.16 | -0.15 | 0.06 | 3562 | 4.81 | -0.24 | -0.20 | 0.06 | D
J01025+716 | M3.0V | 2.0 | 3488 | 4.91 | 0.02 | 0.06 | 0.07 | 3523 | 4.76 | -0.06 | 0.00 | 0.07 | D
J01026+623 | M1.5V | 2.0 | 3805 | 4.69 | 0.13 | 0.13 | 0.04 | 3806 | 4.71 | 0.02 | 0.12 | 0.04 | YD
J01433+043 | M2.0V | 2.0 | 3532 | 4.91 | -0.09 | -0.09 | 0.06 | 3550 | 4.82 | -0.20 | -0.14 | 0.06 | D
J01518+644 | M2.5V | 2.0 | 3555 | 4.90 | -0.08 | 0.06 | 0.03 | 3648 | 4.55 | -0.02 | -0.05 | 0.04 | YD
J02015+637 | M3.0V | 2.0 | 3489 | 4.93 | -0.05 | 0.09 | 0.05 | 3563 | 4.81 | -0.13 | 0.05 | 0.04 | D
J02123+035 | M1.5V | 2.0 | 3663 | 4.81 | -0.05 | -0.24 | 0.07 | 3595 | 4.87 | -0.49 | -0.35 | 0.04 | TD
J02222+478 | M0.5V | 2.0 | 3935 | 4.67 | 0.07 | 0.10 | 0.03 | 3899 | 4.67 | 0.00 | 0.09 | 0.03 | D
J02358+202 | M2.0V | 2.0 | 3585 | 4.89 | -0.13 | 0.08 | 0.04 | 3715 | 4.55 | 0.00 | 0.02 | 0.05 | YD
J02442+255 | M3.0V | 2.0 | 3472 | 4.95 | -0.06 | -0.06 | 0.07 | 3495 | 4.88 | -0.14 | -0.09 | 0.07 | YD
J02565+554W | M1.0V | 2.0 | 3896 | 4.66 | 0.18 | 0.16 | 0.03 | 3828 | 4.68 | -0.03 | 0.14 | 0.02 | D
J03181+382 | M1.5V | 2.0 | 3861 | 4.66 | 0.20 | 0.27 | 0.03 | 3845 | 4.68 | 0.10 | 0.25 | 0.03 | D
J03213+799 | M2.0V | 2.0 | 3549 | 4.93 | -0.18 | -0.14 | 0.06 | 3592 | 4.80 | -0.23 | -0.18 | 0.06 | D
J03217-066 | M2.0V | 2.0 | 3555 | 4.92 | -0.15 | 0.00 | 0.06 | 3653 | 4.75 | -0.04 | -0.01 | 0.05 | YD
J03463+262 | M0.0V | 3.3 | 3993 | 4.65 | 0.12 | 0.13 | 0.03 | 3965 | 4.69 | -0.01 | 0.14 | 0.02 | YD
J03531+625 | M3.0V | 2.0 | 3488 | 4.93 | -0.04 | -0.13 | 0.07 | 3496 | 4.84 | -0.18 | -0.20 | 0.07 | TD
J04225+105 | M3.5V | 2.0 | 3445 | 4.95 | 0.00 | 0.16 | 0.05 | 3526 | 4.79 | -0.04 | 0.13 | 0.05 | YD
J04290+219 | M0.5V | 3.9 | 4199 | 4.59 | 0.23 | 0.31 | 0.02 | 4143 | 4.63 | 0.12 | 0.30 | 0.02 | D
J04376+528 | M0.0V | 3.4 | 4034 | 4.68 | -0.09 | -0.07 | 0.03 | 4031 | 4.66 | -0.06 | -0.07 | 0.03 | D
J04376-110 | M1.5V | 2.0 | 3621 | 4.85 | -0.07 | -0.02 | 0.04 | 3651 | 4.81 | -0.11 | -0.04 | 0.04 | D
J04429+189 | M2.0V | 2.0 | 3574 | 4.89 | -0.10 | 0.07 | 0.04 | 3674 | 4.71 | 0.00 | 0.04 | 0.04 | D
J04429+214 | M3.5V | 2.0 | 3437 | 4.97 | -0.03 | 0.05 | 0.06 | 3511 | 4.77 | -0.01 | 0.02 | 0.06 | D
J04538-177 | M2.0V | 2.0 | 3572 | 4.90 | -0.12 | -0.14 | 0.06 | 3573 | 4.88 | -0.31 | -0.20 | 0.06 | D
J04588+498 | M0.0V | 2.0 | 4022 | 4.65 | 0.09 | 0.10 | 0.03 | 3994 | 4.67 | 0.02 | 0.09 | 0.03 | D
J05127+196 | M2.0V | 2.0 | 3582 | 4.90 | -0.13 | -0.07 | 0.05 | 3614 | 4.79 | -0.26 | -0.14 | 0.04 | D
J05314-036 | M1.5V | 2.0 | 3891 | 4.64 | 0.23 | 0.29 | 0.04 | 3871 | 4.65 | 0.13 | 0.29 | 0.03 | D
J05365+113 | M0.0V | 3.8 | 4074 | 4.65 | 0.05 | 0.03 | 0.03 | 4064 | 4.68 | 0.00 | 0.02 | 0.02 | YD
J05415+534 | M1.0V | 2.0 | 3875 | 4.69 | 0.08 | 0.09 | 0.04 | 3859 | 4.54 | 0.03 | 0.00 | 0.04 | YD
J06103+821 | M2.0V | 2.0 | 3521 | 4.92 | -0.11 | -0.06 | 0.06 | 3546 | 4.82 | -0.17 | -0.10 | 0.06 | D
J06105-218 | M0.5V | 2.0 | 3837 | 4.70 | 0.07 | 0.09 | 0.04 | 3829 | 4.71 | 0.00 | 0.09 | 0.04 | YD
J06421+035 | M3.5V | 2.0 | 3454 | 4.95 | 0.00 | 0.00 | 0.06 | 3498 | 4.77 | -0.08 | -0.06 | 0.06 | D
J06548+332 | M3.0V | 2.0 | 3451 | 4.96 | -0.03 | -0.01 | 0.07 | 3528 | 4.86 | -0.17 | -0.05 | 0.06 | YD
J07044+682 | M3.0V | 2.0 | 3461 | 4.95 | -0.04 | 0.04 | 0.06 | 3543 | 4.82 | -0.10 | 0.01 | 0.06 | D
J07287-032 | M3.0V | 2.0 | 3466 | 4.94 | -0.03 | 0.00 | 0.06 | 3504 | 4.84 | -0.18 | -0.06 | 0.06 | D
J07353+548 | M2.0V | 2.0 | 3520 | 4.94 | -0.16 | -0.15 | 0.06 | 3567 | 4.84 | -0.25 | -0.18 | 0.06 | YD
J07361-031 | M1.0V | 3.1 | 3894 | 4.69 | 0.05 | -0.04 | 0.05 | 3834 | 4.70 | 0.00 | -0.05 | 0.04 | D
J07393+021 | M0.0V | 2.0 | 4014 | 4.66 | 0.07 | 0.10 | 0.03 | 3991 | 4.53 | -0.01 | 0.02 | 0.03 | YD
J08161+013 | M2.0V | 2.0 | 3568 | 4.89 | -0.10 | -0.06 | 0.05 | 3586 | 4.80 | -0.25 | -0.12 | 0.05 | D
J08293+039 | M2.5V | 2.0 | 3585 | 4.88 | -0.08 | 0.07 | 0.04 | 3675 | 4.55 | -0.01 | -0.03 | 0.04 | YD
J08358+680 | M2.5V | 2.0 | 3461 | 4.98 | -0.11 | -0.10 | 0.07 | 3498 | 4.77 | -0.06 | -0.13 | 0.08 | D
J09133+688 | M2.5V | 2.0 | 3555 | 4.92 | -0.16 | 0.02 | 0.04 | 3699 | 4.73 | 0.09 | 0.05 | 0.05 | YD
J09140+196 | M3.0V | 2.0 | 3493 | 4.95 | -0.10 | 0.08 | 0.05 | 3565 | 4.56 | -0.04 | -0.04 | 0.05 | YD
J09143+526 | M0.0V | 2.0 | 4024 | 4.68 | -0.05 | -0.05 | 0.03 | 3992 | 4.71 | -0.14 | -0.04 | 0.02 | YD
J09144+526 | M0.0V | 2.3 | 4005 | 4.68 | -0.03 | -0.03 | 0.03 | 3992 | 4.69 | -0.06 | -0.03 | 0.03 | YD
J09163-186 | M1.5V | 2.0 | 3585 | 4.91 | -0.18 | -0.05 | 0.06 | 3707 | 4.73 | -0.01 | -0.02 | 0.06 | YD
J09360-216 | M2.5V | 2.0 | 3505 | 4.94 | -0.12 | -0.16 | 0.09 | 3494 | 4.84 | -0.27 | -0.24 | 0.08 | D
J09411+132 | M1.5V | 2.0 | 3613 | 4.87 | -0.11 | -0.02 | 0.05 | 3695 | 4.73 | -0.07 | -0.05 | 0.04 | YD
J09425+700 | M2.0V | 2.0 | 3515 | 4.94 | -0.12 | 0.10 | 0.05 | 3574 | 4.77 | -0.07 | 0.08 | 0.04 | YD
J09468+760 | M1.5V | 2.0 | 3680 | 4.78 | -0.01 | -0.10 | 0.05 | 3628 | 4.78 | -0.27 | -0.18 | 0.05 | TD
J09511-123 | M0.5V | 2.0 | 3777 | 4.76 | -0.07 | -0.10 | 0.04 | 3737 | 4.74 | -0.29 | -0.16 | 0.04 | TD
J09561+627 | M0.0V | 2.0 | 3980 | 4.66 | 0.07 | 0.09 | 0.03 | 3957 | 4.69 | 0.00 | 0.11 | 0.02 | YD
J10023+480 | M1.0V | 2.0 | 3785 | 4.72 | 0.05 | 0.01 | 0.03 | 3758 | 4.73 | -0.17 | -0.04 | 0.02 | TD
J10122-037 | M1.5V | 2.0 | 3610 | 4.87 | -0.13 | 0.07 | 0.04 | 3722 | 4.73 | -0.04 | 0.08 | 0.04 | YD
J10167-119 | M3.0V | 2.0 | 3532 | 4.88 | 0.01 | 0.08 | 0.05 | 3561 | 4.81 | -0.10 | 0.04 | 0.05 | D
J10251-102 | M1.0V | 2.0 | 3780 | 4.72 | 0.06 | 0.08 | 0.05 | 3746 | 4.72 | -0.06 | 0.05 | 0.04 | YD
J10289+008 | M2.0V | 2.0 | 3579 | 4.88 | -0.09 | -0.10 | 0.05 | 3586 | 4.59 | -0.18 | -0.24 | 0.06 | YD
J10350-094 | M3.0V | 2.0 | 3469 | 4.94 | -0.02 | 0.03 | 0.06 | 3546 | 4.82 | -0.16 | -0.02 | 0.05 | D
J10396-069 | M2.5V | 2.0 | 3554 | 4.88 | -0.02 | 0.11 | 0.05 | 3596 | 4.76 | -0.08 | 0.05 | 0.04 | YD
J11000+228 | M2.5V | 2.0 | 3499 | 4.94 | -0.11 | -0.08 | 0.06 | 3518 | 4.60 | -0.11 | -0.21 | 0.07 | YD
J11026+219 | M1.0V | 2.6 | 3905 | 4.69 | 0.04 | -0.01 | 0.04 | 3872 | 4.70 | 0.00 | -0.02 | 0.04 | D
J11054+435 | M1.0V | 2.0 | 3639 | 4.92 | -0.30 | -0.34 | 0.07 | 3639 | 4.92 | -0.53 | -0.38 | 0.06 | TD-D
J11110+304 | M2.0V | 2.0 | 3753 | 4.71 | 0.13 | 0.18 | 0.04 | 3739 | 4.69 | -0.04 | 0.13 | 0.04 | D
J11126+189 | M1.5V | 2.0 | 3645 | 4.86 | -0.14 | 0.05 | 0.03 | 3735 | 4.70 | -0.08 | 0.05 | 0.04 | D
J11302+076 | M2.5V | 2.0 | 3513 | 4.91 | -0.03 | 0.07 | 0.05 | 3563 | 4.59 | -0.10 | -0.07 | 0.05 | D
J11306-080 | M3.5V | 2.0 | 3422 | 4.98 | -0.01 | 0.02 | 0.07 | 3491 | 4.78 | -0.04 | -0.02 | 0.07 | YD
J11467-140 | M3.0V | 2.0 | 3570 | 4.82 | 0.11 | 0.24 | 0.05 | 3629 | 4.72 | 0.00 | 0.17 | 0.04 | D
J11511+352 | M1.5V | 2.0 | 3645 | 4.87 | -0.17 | -0.14 | 0.05 | 3704 | 4.70 | -0.06 | -0.14 | 0.06 | D
J12123+544S | M0.0V | 2.0 | 3933 | 4.70 | -0.03 | -0.04 | 0.03 | 3910 | 4.73 | -0.10 | -0.05 | 0.03 | D
J12230+640 | M3.0V | 2.0 | 3540 | 4.86 | 0.05 | 0.08 | 0.06 | 3578 | 4.80 | -0.15 | 0.01 | 0.05 | D
J12248-182 | M2.0V | 2.0 | 3485 | 4.98 | -0.17 | -0.34 | 0.11 | 3476 | 4.93 | -0.48 | -0.45 | 0.08 | TD
J12312+086 | M0.5V | 2.0 | 3914 | 4.71 | -0.07 | -0.04 | 0.03 | 3919 | 4.67 | -0.06 | -0.05 | 0.03 | D
J12350+098 | M2.5V | 2.0 | 3572 | 4.85 | 0.02 | -0.03 | 0.07 | 3563 | 4.81 | -0.21 | -0.11 | 0.07 | D
J12388+116 | M3.0V | 2.0 | 3439 | 4.96 | 0.00 | 0.14 | 0.05 | 3511 | 4.77 | 0.00 | 0.11 | 0.05 | D
J13196+333 | M1.5V | 2.0 | 3812 | 4.67 | 0.18 | 0.24 | 0.04 | 3792 | 4.54 | -0.02 | 0.15 | 0.03 | YD
J13209+342 | M1.0V | 2.0 | 3763 | 4.73 | 0.03 | -0.01 | 0.04 | 3760 | 4.73 | -0.21 | -0.06 | 0.04 | D
J13283-023W | M3.0V | 2.0 | 3482 | 4.92 | 0.01 | 0.11 | 0.07 | 3554 | 4.75 | -0.07 | 0.07 | 0.06 | D
J13299+102 | M0.5V | 2.0 | 3714 | 4.81 | -0.15 | -0.06 | 0.05 | 3779 | 4.73 | -0.11 | -0.06 | 0.05 | D
J13450+176 | M0.0V | 2.3 | 3806 | 4.85 | -0.43 | -0.44 | 0.05 | 3816 | 4.83 | -0.60 | -0.46 | 0.04 | TD
J13457+148 | M1.5V | 2.0 | 3678 | 4.79 | -0.04 | -0.15 | 0.06 | 3619 | 4.86 | -0.40 | -0.23 | 0.04 | D
J14010-026 | M1.0V | 2.0 | 3724 | 4.77 | -0.04 | -0.05 | 0.04 | 3710 | 4.75 | -0.22 | -0.10 | 0.04 | D
J14082+805 | M1.0V | 2.0 | 3841 | 4.67 | 0.16 | 0.11 | 0.04 | 3803 | 4.68 | -0.03 | 0.09 | 0.03 | D
J14251+518 | M2.5V | 2.0 | 3521 | 4.91 | -0.07 | -0.03 | 0.06 | 3558 | 4.81 | -0.20 | -0.09 | 0.06 | D
J14257+236E | M0.5V | 2.0 | 3947 | 4.65 | 0.17 | 0.20 | 0.03 | 3907 | 4.67 | 0.01 | 0.19 | 0.02 | TD-D
J14257+236W | M0.0V | 2.0 | 4024 | 4.64 | 0.17 | 0.22 | 0.03 | 3983 | 4.67 | 0.07 | 0.22 | 0.02 | D
J14294+155 | M2.0V | 2.0 | 3671 | 4.76 | 0.06 | 0.07 | 0.04 | 3653 | 4.72 | -0.08 | 0.01 | 0.04 | TD
J14307-086 | M0.5V | 2.4 | 4089 | 4.63 | 0.14 | 0.18 | 0.02 | 4026 | 4.66 | 0.02 | 0.17 | 0.01 | TD-D
J14524+123 | M2.0V | 2.0 | 3548 | 4.89 | -0.08 | 0.19 | 0.04 | 3714 | 4.70 | 0.07 | 0.21 | 0.05 | D
J15095+031 | M3.0V | 2.0 | 3492 | 4.92 | -0.01 | 0.05 | 0.06 | 3545 | 4.82 | -0.10 | 0.01 | 0.05 | D
J15194-077 | M3.0V | 2.0 | 3441 | 4.98 | -0.08 | -0.10 | 0.09 | 3488 | 4.88 | -0.18 | -0.13 | 0.07 | YD
J15598-082 | M1.0V | 2.0 | 3625 | 4.89 | -0.19 | -0.06 | 0.04 | 3728 | 4.70 | -0.01 | -0.03 | 0.05 | D
J16167+672N | M3.0V | 2.0 | 3488 | 4.93 | -0.03 | 0.10 | 0.08 | 3575 | 4.74 | -0.04 | 0.06 | 0.07 | YD
J16167+672S | M0.0V | 2.7 | 4090 | 4.62 | 0.15 | 0.19 | 0.04 | 4054 | 4.66 | 0.07 | 0.20 | 0.03 | D
J16254+543 | M1.5V | 2.0 | 3525 | 4.97 | -0.26 | -0.30 | 0.08 | 3588 | 4.90 | -0.34 | -0.31 | 0.06 | YD
J16462+164 | M2.5V | 2.0 | 3506 | 4.92 | -0.06 | 0.03 | 0.05 | 3558 | 4.81 | -0.11 | 0.00 | 0.05 | D
J16581+257 | M1.0V | 2.0 | 3734 | 4.78 | -0.08 | 0.01 | 0.04 | 3780 | 4.73 | -0.11 | 0.00 | 0.04 | D
J17052-050 | M1.5V | 2.0 | 3642 | 4.81 | -0.01 | -0.14 | 0.07 | 3597 | 4.87 | -0.37 | -0.21 | 0.06 | TD-D
J17166+080 | M2.0V | 2.0 | 3543 | 4.91 | -0.10 | -0.07 | 0.06 | 3567 | 4.81 | -0.23 | -0.13 | 0.06 | YD
J17303+055 | M0.0V | 2.0 | 3812 | 4.77 | -0.15 | -0.06 | 0.03 | 3825 | 4.57 | -0.19 | -0.15 | 0.03 | YD
J17355+616 | M0.5V | 2.0 | 3893 | 4.69 | 0.06 | 0.07 | 0.05 | 3878 | 4.67 | 0.00 | 0.04 | 0.05 | D
J17378+185 | M1.0V | 2.0 | 3656 | 4.87 | -0.23 | -0.22 | 0.05 | 3675 | 4.83 | -0.37 | -0.25 | 0.05 | D
J18051-030 | M1.0V | 2.0 | 3663 | 4.86 | -0.20 | -0.16 | 0.06 | 3725 | 4.74 | -0.19 | -0.15 | 0.06 | D
J18198-019 | K7.0V | 2.0 | 4143 | 4.66 | -0.11 | -0.06 | 0.02 | 4162 | 4.69 | -0.12 | -0.04 | 0.02 | YD
J18353+457 | M0.5V | 2.0 | 3932 | 4.68 | 0.06 | 0.02 | 0.03 | 3896 | 4.67 | -0.05 | 0.00 | 0.03 | D
J18409-133 | M1.0V | 2.0 | 3812 | 4.70 | 0.08 | 0.11 | 0.04 | 3782 | 4.69 | -0.05 | 0.09 | 0.03 | D
J18480-145 | M2.5V | 2.0 | 3496 | 4.94 | -0.11 | -0.08 | 0.07 | 3540 | 4.82 | -0.15 | -0.12 | 0.07 | D
J18580+059 | M0.5V | 2.0 | 3923 | 4.68 | 0.08 | 0.08 | 0.03 | 3898 | 4.67 | -0.01 | 0.07 | 0.03 | D
J19070+208 | M2.0V | 2.0 | 3536 | 4.94 | -0.20 | -0.29 | 0.11 | 3541 | 4.90 | -0.49 | -0.37 | 0.08 | D
J19072+208 | M2.0V | 2.0 | 3526 | 4.96 | -0.23 | -0.30 | 0.10 | 3576 | 4.88 | -0.43 | -0.34 | 0.08 | D
J19169+051N | M2.5V | 2.0 | 3534 | 4.90 | -0.04 | 0.09 | 0.05 | 3587 | 4.80 | -0.14 | 0.03 | 0.04 | D
J19346+045 | M0.0V | 3.9 | 4062 | 4.69 | -0.17 | -0.15 | 0.04 | 4047 | 4.68 | -0.27 | -0.17 | 0.03 | D
J20450+444 | M1.5V | 2.0 | 3586 | 4.89 | -0.15 | -0.10 | 0.07 | 3615 | 4.82 | -0.25 | -0.15 | 0.06 | YD
J20533+621 | M1.0V | 2.0 | 3843 | 4.71 | 0.04 | 0.03 | 0.04 | 3815 | 4.68 | -0.07 | 0.01 | 0.03 | D
J20567-104 | M2.5V | 2.0 | 3514 | 4.91 | -0.03 | 0.09 | 0.05 | 3571 | 4.74 | -0.08 | 0.03 | 0.05 | TD
J21019-063 | M2.5V | 2.0 | 3507 | 4.92 | -0.08 | 0.05 | 0.04 | 3560 | 4.75 | -0.06 | 0.02 | 0.04 | D
J21152+257 | M3.0V | 2.0 | 3675 | 4.68 | 0.29 | 0.28 | 0.05 | 3700 | 4.67 | 0.13 | 0.27 | 0.04 | D
J21164+025 | M3.0V | 2.0 | 3485 | 4.94 | -0.08 | 0.05 | 0.06 | 3552 | 4.60 | -0.11 | -0.07 | 0.06 | YD
J21221+229 | M1.0V | 2.0 | 3705 | 4.83 | -0.20 | -0.11 | 0.05 | 3755 | 4.73 | -0.21 | -0.13 | 0.05 | D
J21348+515 | M3.0V | 2.0 | 3497 | 4.91 | 0.00 | 0.00 | 0.06 | 3523 | 4.86 | -0.13 | -0.04 | 0.06 | YD
J22021+014 | M0.5V | 2.0 | 3915 | 4.69 | 0.03 | 0.05 | 0.03 | 3904 | 4.67 | -0.02 | 0.03 | 0.03 | D
J22057+656 | M1.5V | 2.0 | 3653 | 4.86 | -0.17 | -0.08 | 0.04 | 3727 | 4.74 | -0.11 | -0.07 | 0.05 | D
J22096-046 | M3.5V | 2.0 | 3467 | 4.94 | -0.02 | 0.16 | 0.05 | 3579 | 4.77 | -0.03 | 0.13 | 0.05 | YD
J22115+184 | M2.0V | 2.0 | 3550 | 4.93 | -0.13 | 0.18 | 0.03 | 3723 | 4.70 | 0.00 | 0.19 | 0.04 | D
J22125+085 | M3.0V | 2.0 | 3501 | 4.92 | -0.04 | -0.05 | 0.07 | 3513 | 4.83 | -0.16 | -0.12 | 0.07 | D
J22330+093 | M1.0V | 2.0 | 3669 | 4.86 | -0.20 | -0.15 | 0.04 | 3732 | 4.78 | -0.26 | -0.14 | 0.04 | YD
J22503-070 | M0.5V | 2.0 | 3904 | 4.72 | -0.10 | -0.06 | 0.03 | 3911 | 4.70 | -0.10 | -0.07 | 0.03 | D
J22559+178 | M1.0V | 2.0 | 3834 | 4.70 | 0.06 | 0.06 | 0.04 | 3808 | 4.68 | -0.03 | 0.05 | 0.04 | D
J22565+165 | M1.5V | 2.0 | 3805 | 4.69 | 0.12 | 0.15 | 0.04 | 3787 | 4.69 | 0.00 | 0.14 | 0.03 | D
J23245+578 | M1.0V | 2.0 | 3842 | 4.68 | 0.14 | 0.18 | 0.03 | 3803 | 4.68 | -0.02 | 0.15 | 0.02 | D
J23340+001 | M2.5V | 2.0 | 3559 | 4.86 | 0.00 | -0.09 | 0.06 | 3531 | 4.83 | -0.20 | -0.18 | 0.05 | D
J23381-162 | M2.0V | 2.0 | 3551 | 4.91 | -0.13 | -0.16 | 0.08 | 3583 | 4.91 | -0.35 | -0.20 | 0.07 | D
J23492+024 | M1.0V | 2.0 | 3653 | 4.84 | -0.13 | -0.35 | 0.09 | 3572 | 4.96 | -0.61 | -0.45 | 0.06 | TD
J23556-061 | M2.5V | 2.0 | 3694 | 4.71 | 0.18 | 0.19 | 0.04 | 3698 | 4.70 | 0.03 | 0.16 | 0.03 | D
101010
$a$$a$footnotetext: a: the $v\sin i$’s given here are the rotational
broadening parameters used for each star in the spectral fits. It is set to
2.0 km/s when the measurement (from Reiners et al. 2018) only gives an upper
limit of 2.0 km/s, i.e., the true $v\sin i\leq 2$ km/s.
$b$$b$footnotetext: b: [V/H] error is given as the sample standard deviation
of the fitted abundances over all line regions used for that particular star.

##  Appendix B Representative CARMENES spectra in the wavelength region
800–910 nm

Figure 14: View of a representative, normalized, telluric-corrected and co-
added CARMENES spectrum in the wavelength ranges considered in this work for a
typical M0.0 V star (J14257+236W). The V i lines used in this study are marked
in red.  Figure 15: Same as Fig. 14, but for a typical M2.0 V star
(J03213+799).

##  Appendix C All fitted vanadium line regions

Figure 16: All V i line regions fitted to the observed CARMENES spectrum
(thick gray) of a typical M0.0 V star (J14257+236W). The model corresponding
to the best-fit V abundance, using the Schw19 stellar parameters, is shown in
green dot-dashed lines. Individual V i HFS components and their relative
strengths, as given by $\log(gf\lambda)-\theta\chi$, are shown in vertical
gray lines above each feature. Notable contaminating lines (i.e., expected
depth $>0.05$) inside the V i features from other species are marked in blue
(for TiO) and red.  Figure 17: Same as Fig. 16, but for a typical M2.0 V star
(J03213+799).