11institutetext: INAF-Osservatorio di Astrofisica e Scienza dello Spazio, via
P. Gobetti 93/3, 40129 Bologna (Italy)
11email<EMAIL_ADDRESS>22institutetext: Department of Physics and
Astronomy, Bologna University, via P. Gobetti 93/2, 40129 Bologna (Italy)
33institutetext: Fundación Galileo Galilei - INAF, Breña Baja, La Palma
(Spain) 44institutetext: INAF-Osservatorio Astronomico di Roma - Sede di
Monteporzio Catone, via di Frascati 33, 00040, Monte Porzio Catone (Italy)

# The chemical composition of the oldest nearby open cluster Ruprecht 147.

Angela Bragaglia 11 Xiaoting Fu 2211 Alessio Mucciarelli 2211 Gloria Andreuzzi
3344 Paolo Donati 11

(accepted by A&A)

###### Abstract

Context. Ruprecht~147 (NGC 6774) is the closest old open cluster, with a
distance of less than 300 pc and an age of about 2.5 Gyr. It is therefore well
suited for testing stellar evolution models and for obtaining precise and
detailed chemical abundance information.

Aims. We combined photometric and astrometric information coming from
literature and the Gaia mission with very high-resolution optical spectra of
stars in different evolutionary stages to derive the cluster distance, age,
and detailed chemical composition.

Methods. We obtained spectra of six red giants using HARPS-N at the Telescopio
Nazionale Galileo (TNG). We also used European Southern Observatory (ESO)
archive spectra of 22 main sequence (MS) stars, observed with HARPS at the
3.6m telescope. The very high resolution (115000) and the large wavelength
coverage (about 380-680nm) of the twin instruments permitted us to derive
atmospheric parameters, metallicity, and detailed chemical abundances of 23
species from all nucleosynthetic channels. We employed both equivalent widths
and spectrum synthesis. We also re-derived the cluster distance and age using
Gaia parallaxes, proper motions, and photometry in conjunction with the PARSEC
stellar evolutionary models.

Results. We fully analysed those stars with radial velocity and proper
motion/parallax in agreement with the cluster mean values. We also discarded
one binary not previously recognised, and six stars near the MS turn-off
because of their high rotation velocity. Our final sample consists of 21 stars
(six giants and 15 MS stars). We measured metallicity (the cluster average
[Fe/H] is +0.08, rms=0.07) and abundances of light, $\alpha$, Fe-peak, and
neutron-capture elements. The Li abundance follows the expectations, showing a
tight relation between temperature and abundance on the MS, at variance with
M67, and we did not detect any Li-rich giant.

Conclusions. We confirm that Rup 147 is the oldest nearby open cluster. This
makes it very valuable to test detailed features of stellar evolutionary
models.

###### Key Words.:

Stars: abundances - stars: evolution - open clusters and association: general)
- open clusters and associations: individual (Ruprecht 147)

##  1 Introduction

Figure 1:  Left-hand panel: $T_{\rm eff}$, $\log(g)$ diagram for the 21 stars
analysed spectroscopically. Right-hand panel: CMD for Rup 147 based on Gaia
DR2 photometry, with the stars fully analysed indicated by light blue filled
squares (MS stars) and orange filled diamonds (giants). The stars with high
rotation are indicated by pink filled triangles and the binary by a dark green
plus symbol; stars considered members by Curtis et al. (2013) and candidate
members based on Gaia astrometry and without HARPS/HARPS-N spectra are shown
as empty black circles. In both panels we plot PARSEC isochrones for
metallicity Z=0.017 ([M/H]=0.07), distance 308 pc (i.e. intrinsic distance
modulus 7.44), and age 2.5 and 3.0 Gyr. A reddening $E(G_{BP}-G_{RP})=0.15$
and an extinction on the Gaia G band A(G)=0.30 are applied to the isochrones
in the CMD.

The Gaia mission (Gaia Collaboration et al., 2016a) with its legacy of
astrometric and photometric data for more than 1.3 billion objects in the
Milky Way and beyond is bringing us what is often referred to as a revolution
in Galactic astrophysics. However, even if the Gaia Radial Velocity
Spectrometer (RVS) will provide radial velocity for a few million stars (e.g.
Katz et al., 2018; Marchetti et al., 2018) and chemical abundances for the
brightest among them, the spectroscopic capabilities of Gaia are limited. This
leaves space for complementary projects from the ground, such as for example
the large spectroscopic surveys Gaia-ESO (Gilmore et al., 2012; Randich et
al., 2013), GALAH (Martell et al., 2017) and others both on-going and future,
and to smaller programmes concentrating on high precision velocities and
detailed chemical composition. The synergy with Gaia will enhance results for
all Galactic populations, in particular for the stellar clusters within the
few kiloparsecs where the Gaia astrometry reaches the highest precision and
high-resolution spectra of good quality are obtainable. This is the case for
many open clusters (OCs) which can then be used to test the stellar
evolutionary models on which, ultimately, age determinations are based; see
for example a first application combining Gaia and Gaia-ESO in Randich et al.
(2018). Furthermore, detailed abundances of elements of all nucleosynthetic
chains in different evolutionary phases are important to test the “subtleties”
of stellar models, such as diffusion and mixing (e.g. Önehag et al. 2014;
Smiljanic et al. 2016; Bertelli Motta et al. 2018). Ruprecht~147, an old and
very close OC, represents an ideal case for these studies.

Ruprecht 147 received very little attention until recently, despite being
recognised as an old (age about 2.5 - 3 Gyr) and very close (175-300 pc)
cluster in the Dias et al. (2002) and Kharchenko et al. (2005, 2013)
catalogues. High-resolution spectra of three giant stars were obtained by
Pakhomov et al. (2009), who derived atmospheric parameters, a metallicity
slightly above solar ([Fe/H]=$0.11\pm 0.07$ averaging the three stars), a mean
radial velocity (RV) of $42.5\pm 3$ km s-1, and abundances of many elements
(light, $\alpha$, Fe-peak, and n-capture). Spectra of eight giant stars were
obtained by Carlberg (2014) to measure radial and rotation velocities; she
derived a mean RV=$42.5\pm 1.0$ km s-1 and found that five of the targets are
good candidate members, based on their RV. Two of the stars have also been
studied by Pakhomov et al. (2009) and four are in common with our sample;
comparison of results will be presented below. Two stars in Rup 147 were
studied by Brewer et al. (2016) among about 1600 F, G, and K stars observed in
a search for planets. Spectra were obtained with HIRES@Keck and analysed using
spectral synthesis, determining atmospheric parameters, projected rotational
velocity, and abundances for 15 elements. The stars are CWW 21 = SPOCS 3038
and CWW 22 = SPOCS 3049, where SPOCS is the identification in Brewer et al.
(2016), and they are not among our targets. They are solar-type stars, with
[Fe/H]=+0.23.

The most relevant paper on this cluster is by Curtis et al. (2013). The
authors, identifying its possible role as a “benchmark” cluster, given its
proximity and old age, presented a comprehensive study of Rup 147, combining
photometry, high-resolution spectroscopy, and literature astrometry. Curtis et
al. (2013) selected possible astrometric members and conducted an RV survey
with three different high-resolution spectrographs, finding about 100
candidate members and about 10 binaries or suspected binaries. The average RV
for Rup 147 is 41.1 km s-1. They also collected higher-signal-to-noise(S/N)
spectra of six stars, mostly in the main sequence (MS) evolutionary phase,
which could be used for chemical analysis. On the basis of three of these
stars, they derived an average [Fe/H]$=0.07\pm 0.03$, in good agreement with
the Pakhomov et al. (2009) result. Using deep MegaCam@CFHT photometry and
different sets of stellar models they also studied the cluster colour-
magnitude diagram (CMD) and deduced an age of about 2.5-3.0 Gyr and a distance
of about 300 pc. We used information on membership based on Curtis et al.
(2013) to select our targets (see following section).

The first Gaia data release (Gaia DR1, Gaia Collaboration et al. 2016b)
contained also the Tycho-Gaia astrometric solution (TGAS), that is, a subset
of bright stars for which proper motions (PM) and parallaxes ($\varpi$) are
derived using Hipparcos and Tycho-2 positions as first epoch. While Rup 147 is
not among the 19 validation OCs studied by Gaia Collaboration et al. (2017),
data for many stars towards its position were available in TGAS. Cantat-Gaudin
et al. (2018a) tried to characterise the open clusters in the solar
neighbourhood (within 2 kpc) using TGAS parallaxes and PMs complemented by
UCAC4 PMs and 2MASS photometry (Zacharias et al., 2012; Skrutskie et al.,
2006). They found 63 astrometric members within a radius of 3 deg and
determined the following average values: $\varpi=3.26\pm 0.09$ mas,
$\mu_{\alpha}=-1.04\pm 0.18$, and $\mu_{\delta}=-26.85\pm 0.20$ mas yr-1 (PMs
come from UCAC4). They also determined age from isochrones for about one fifth
of their 129 clusters, but unfortunately Rup 147 is not among them. Yen et al.
(2018), combining information from all-sky ground-based photometry, TGAS, and
HSOY (Altmann et al., 2017), derived fundamental parameters for 24 nearby OCs.
For Rup 147 they found: $\varpi=3.53\pm 0.23$ mas (i.e. distance 265 pc),
$\mu_{\alpha}=-1.48\pm 0.26$, $\mu_{\delta}=-26.92\pm 0.18$ mas yr-1,
E($B-V$)=0.059, and an age of 725 Myr. They adopted PARSEC isochrones with
solar metallicity (Bressan et al., 2012) for their analysis; their values for
reddening and especially age are not consistent with literature values or with
the findings in the present paper. They acknowledge the discrepancy, but do
not give a fully convincing explanation. In fact, their procedure initially
found a young age for the cluster (the one published), due to the inclusion of
blue stragglers in the fit, which they try to manually exclude. However, they
might have excluded too many stars close to the turn-off, resulting in an old
age (about 6 Gyr). Ruprecht 147 is present in the second Gaia data release
(Gaia DR2) (Gaia Collaboration et al., 2018a; Cantat-Gaudin et al., 2018b) and
we used those $\varpi$ and PM values in the present paper. Parameters based on
Gaia DR2 astrometry and photometry were derived by one of the validation
papers (Gaia Collaboration et al., 2018b, see their Table 2, where the cluster
is indicated by the alternate name of NGC 6774) using PARSEC isochrones and
literature metallicity. They found distance modulus=7.455, log(age)=9.3, and
$E(B-V)$=0.08, based on 154 candidate members. These values compare well with
our findings (see Sect. 3); the age and reddening are slightly smaller, while
the adopted metallicity, [Fe/H]=0.16, is higher.

The paper is organised as follows: Section 2 describes the data, both
proprietary and archival; Section 3 concerns cluster parameters derived using
photometry and astrometry from Gaia; Section 4 deals with atmospheric
parameters and chemical abundances; Section 5 presents a comparison with
literature results; Section 6 discusses some elements in more details; and
finally Sect. 7 summarises and puts our results in the context of our current
understanding.

Figure 2: Proper motion distribution for stars in a region of radius 80 arcmin
around the nominal centre of Rup 147, coloured according to parallax (Gaia
Collaboration et al., 2018a). Candidate member stars in Rup 147 show a well
clustered distribution roughly centred on
$PM_{\alpha},PM_{\delta}\approx-1,-26$ and $\varpi\approx 3$.  Figure 3:  A
small spectral region close to the Li i 607.8nm line for our 15 MS sample
stars (first and second panels), and the spectral region close to the Na i
568.8nm line for the 6 giant stars observed with HARPS-N (third panel). The
vertical lines indicate all the Y/U lines we used in the analysis. We also
show examples of synthetic spectrum fits for the 21 stars analysed (see text),
while the 6 stars close to the MSTO excluded from analysis are shown
separately (fourth panel).

##  2 The data

We gathered spectra of six evolved stars of Rup 147, selected among the most
probable single members according to Curtis et al. (2013). A log of the
observations and of basic parameters taken from Gaia DR2 is given in Table 4.

We used the very-high-resolution fibre High-Accuracy Radial velocity Planet
Searcher in North hemisphere (HARPS-N) spectrograph, mounted at the Telescopio
Nazionale Galileo (TNG) in La Palma, Canary Islands (programme A33DDT0).
HARPS-N covers the spectral range 383-693 nm, with resolution R=115000. The
spectra were reduced automatically using the Data Reduction Software (DRS)
which supplies science-quality data. The basic processing steps comprise bias
subtraction, spectrum extraction, flat fielding, and wavelength calibration.
The spectra were corrected for barycentric motion.

Furthermore, we downloaded the ESO archive spectra of 22 MS stars obtained
with HARPS at the ESO 3.6m telescope to search for Neptune-size planets
(original programmes 091.C-0471, 093.C-0540, and 095.C-0947). Also in this
case, stars were selected among good candidate members from Curtis et al.
(2013). These spectra are generally of low S/N (average value about 20) but
there are several/many exposures for each star (from 2 to 58 individual
spectra, with a mean of 15). Information on the archive stars based on Gaia
DR2 is found in Table 5. We downloaded the Advanced Data Products (ADP)
spectra. The HARPS echelle data were reduced automatically using the DRS
pipeline developed by the HARPS consortium, corrected for barycentric motion,
and sky subtracted. The spectral coverage is essentially the same as HARPS-N,
that is, 378-691 nm, and the resolution is also R=115000. The HARPS spectra
were combined to enhance the S/N (see Table 6) and the chemical analysis was
done on the combined spectra.

We measured the RV on the individual spectra using iSpec (Blanco-Cuaresma et
al., 2014b) and a line list; results and errors are given in Table 6, where we
show the average values for the MS stars. We also obtained v$\sin i$, again
using iSpec; most of the stars are slow rotators (see Table 6). However, as
shown in the right panel of Fig. 1, we eliminated the six stars closer to the
MS turn-off (MSTO), because their rotation velocity makes their lines wider
and more subject to blends.

The MS stars show generally a constant RV, however, we found two interesting
cases among them: a) CWW 58 is clearly a binary, with an RV variation $>5$ km
s-1 in the 9 spectra obtained over a two-year interval; and b) CWW 71 shows a
linear trend in its RV, which changes from 42.17 to 41.23 km s-1 for the 11
spectra, again obtained over approximately 2 years. Neither one was indicated
as problematic in Curtis et al. (2013). We excluded star CWW 58 from further
analysis but retained star 71. We have only one spectrum for the giants, so we
cannot state that they are single stars; comparison with literature values
cannot be conclusive because they are based on spectra of lower resolution and
precision. Furthermore, systematics between the different analyses could hide
small differences such as the ones we found for the two stars discussed above.

##  3 Cluster parameters

The stars observed are shown in Fig. 1 (right panel) in the CMD based on Gaia
G band, $BP$ and $RP$ data. The targets were selected among high-probability
members, based on RV and ground based proper motions (see Curtis et al., 2013,
for details) and confirmed as members by Gaia DR2 PM, $\varpi$ values a
posteriori. The stars observed define the cluster sequence very well.

For this further assessment of their membership we downloaded $\varpi$ and PM
values for stars in a region of 80 arcmin in radius around the cluster centre.
In Fig. 2 we show the proper motions for stars with G ¡15 mag, colour-coded
using the parallax; Rup 147 is well isolated from field stars. All stars in
our spectroscopic sample are also included in Gaia DR2 (see Tables 4, 5) and
their mean $\varpi$ (3.25$\pm 0.09$ mas) and PMs ($PM_{\alpha}=-0.95\pm 0.78$
mas/yr, $PM_{\delta}=-26.53\pm 0.65$ mas/yr) are in very good agreement with
the cluster averages based on TGAS or DR2 (Cantat-Gaudin et al., 2018a, b) .
All other stars that satisfy the Rup 147 parallax and PM are considered as
member candidates (black circles in Fig. 1).

The averages were computed as simple mean values, without taking into account
the correlations between astrometric parameters, which do not have a relevant
impact. Furthermore, we used the derived distance mainly to find good starting
points for $T_{\rm eff}$ and gravity for the spectroscopic analysis, in
combination with the PARSEC isochrones (version 1.2S from CMD 3.0 web input
form 111http://stev.oapd.inaf.it/cgi-bin/cmd_3.0, Bressan et al., 2012; Chen
et al., 2014).

From the isochrone fit in Fig. 1 we estimate that Rup 147 is 2.5 to 3.0 Gyr
old, adopting distance=308 pc from Gaia DR2, which translates to
$(m-M)_{0}$=7.44, metallicity Z=0.017 ([M/H]=0.07, see Curtis et al. 2013),
which is in good agreement with what we find (see Sect. 4), extinction
$E(G_{BP}-G_{RP})=0.15$, and absorption in the Gaia G band A(G)=0.30. By
assuming a standard extinction law (RV=3.1, Cardelli et al.,, 1989) and taking
$A(G)=0.85926\,A(V),\leavevmode\nobreak\
A(G_{BP})=1.06794\,A(V),\leavevmode\nobreak\ A(G_{RP})=0.65199\,A(V)$ from the
PARSEC website, the reddening in (B-V) colour is
$E(B-V)=0.775\,E(G_{BP}-G_{RP})=0.113$, and the extinction in V band is
$A_{V}=0.35$, which is in between the values of 0.46 and 0.25 from Pakhomov et
al. (2009) and Curtis et al. (2013), respectively. Further refinements are not
required for the main goal of this paper, which focuses on the detailed
chemical properties.

Adopting the Gaia DR2 parallax of the member candidates, we calculate the
heliocentric Galactic coordinates of Rup 147: $X=280.21\pm 8.14$ pc (towards
the Galactic centre), $Y=106.90\pm 4.23$ pc (towards the local direction of
rotation in the Galactic plane), and $Z=-70.00\pm 2.38$ pc (towards the north
Galactic pole). The Galactic radius of this cluster is $R_{GC}=8.28$ kpc. Its
iron abundance ([Fe/H]=0.08) is in good agreement with the expectations at its
Galactocentric radius (see e.g. the homogeneous samples in Donati et al.,
2015; Netopil et al., 2016; Reddy et al., 2016).

We then confirm once more that Rup 147 is the only old and nearby OC; next OC
close-by and older than 1 Gyr is NGC 752 (age and distance about 1.6 Gyr and
450 pc, respectively) and we need to reach approximately 900 pc to find an OC
older than Rup 147, that is M67. Rup 147 is therefore very important as a
benchmark cluster, as remarked by Curtis et al. (2013), and efforts to
determine its detailed properties through photometry, astrometry, high-
resolution spectroscopy, and modelling are welcome.

##  4 Atmospheric parameters and chemical abundances

To derive the atmospheric parameters we used the equivalent widths (EWs) of
iron lines, both neutral and ionised, employing MOOG (Sneden, 1973) via iSpec.
Our analysis was done assuming local thermodynamic equilibrium (LTE) and using
the MARCS model atmospheres (Gustafsson et al., 2008). We used the public
Gaia-ESO line list (Heiter et al., 2015, 2018), which is based on VALD3 data
(Ryabchikova et al., 2011), selecting only the Y/Y lines, that is, the most
isolated ones with the most robust atomic data. We followed the classical
spectroscopic method to derive temperature $T_{\rm eff}$, gravity $\log g$,
microturbulent velocity $\xi$, and the iron abundance [Fe/H]. $T_{\rm eff}$ is
obtained eliminating trends between the line abundances and the excitation
potentials (excitation equilibrium), $\log(g)$ requiring that Fe i and Fe ii
give the same abundance (ionisation equilibrium), and $\xi$ was obtained by
minimising the slope of the relation between line abundances and EWs. The
stellar parameters are given in Table 6, together with the uncertainties,
based on the uncertainties in the slopes of the three relations. With the
$T_{\rm eff}$ and $\log(g)$ values, we derive stellar mass from isochrones for
our sample stars, the results are also listed in Table 6.

We obtained an average [Fe/H]=0.08 (rms 0.07) dex for Rup 147. If we divide
the giants from the dwarfs to take into account possible (small) effects of
diffusion (see e.g. Önehag et al., 2014; Bertelli Motta et al., 2018, both on
M67), we have [Fe/H]=0.10 (rms 0.06) and 0.07 (rms 0.08) dex for the six
giants and the 15 MS stars, respectively.

We derived the abundances of 23 species, including Li, light, $\alpha$, Fe-
peak, and neutron capture elements. We employed iSpec, again using the MOOG
choice and the GES public line list, choosing Y/U lines. Given the (much)
smaller number of lines available, we relaxed the criterium adopted for iron
and also used lines for which blending had not been checked by the GES
consortium; however, our spectra have a larger resolution and we inspected
dubious cases. We employed spectrum synthesis for all lines, including hyper-
fine structure (HFS). In Fig. 3 we show examples of the region near the Li i
line for the 15 MS stars and near Na i for the 6 giants and the 6 MS stars
excluded from further analysis because of their larger rotation velocity. We
checked that the line list and the synthesis reproduced the solar abundances
using the spectrum “HARPS.Archive_Sun-4” from the library of the Gaia FGK
benchmark stars 222https://www.blancocuaresma.com/s/benchmarkstars (Blanco-
Cuaresma et al., 2014a). We found a difference for only three elements (Cu,
Ba, and Eu), so we corrected the cluster abundances by these offsets based on
our derived solar abundance. Finally, we visually checked a few lines in case
of large dispersion in the line-by-line abundances.

All abundances were obtained using LTE and are reported in Tables 4, 7, and 8.
Oxygen was measured from the forbidden [O i] 630.3nm line in the six giants,
after making sure it was free from telluric contamination. The O triplet near
777nm is not present in the HARPS wavelength range, so we did not measure O in
the MS stars. For Li and Na we also corrected the LTE abundances with the
prescription in Lind et al. (2009, 2011); we used the INSPECT web
page333http://inspect.coolstars19.com/ deriving the non-LTE (NLTE) corrections
line by line. In Table 4 we give both LTE and NLTE abundances.

We derived the sensitivity to changes in stellar parameters by repeating the
analysis for one typical giant and MS star, changing one parameter while
holding the other fixed. Results are presented in Table 9.

We show in Fig. 4 the run of [X/H] values with $\log g$ for all elements, with
the exception of Li, which will be discussed in Sect. 6. We see that dwarfs
and giants have slightly different levels in some cases. This is expected for
Na (see e.g. Smiljanic et al., 2016, 2018, and Sect. 6) and can be explained
for the other elements by the larger uncertainties associated to the analysis
of the dwarfs (less and weaker lines) and by the systematic differences
expected from their different atmospheres and sensitivity to details in the
analysis (see e.g. Dutra-Ferreira et al. 2016 on the Hyades cluster). In
principle, evolutionary differences may also be expected as a result of
diffusion processes; they have been found for the older cluster M67 (see
Bertelli Motta et al., 2018; Souto et al., 2018; Gao et al., 2018, for Gaia-
ESO, APOGEE, and GALAH results, respectively). However, the efficiency of the
diffusion depends on the cluster age and we checked that only very small
variations are expected for an age of 2.5-3 Gyr (less than $\sim 0.1$ dex in
most of the cases) using both PARSEC and MIST (Choi et al., 2016) stellar
models.

We computed the average abundance ratios [X/Fe] for Rup 147, given in Table
10, together with the root mean square (rms), both all together and separating
dwarfs and giants. They were obtained adopting the reference solar values from
Grevesse et al. (2007).

Figure 4: We show here the run of the derived elements ([X/H]) with gravity.
Different colours indicate light, $\alpha$, Fe-peak, n-capture (slow and
rapid). Starred symbols indicate the mean values for dwarfs and giants (error
bars are the standard deviation). Figure 5:  Li evolution as a function of
log(g) (left panel), $T_{\rm eff}$(middle panel), and stellar mass (right
panel). Three clusters with similar metallicity but different ages are
displayed. The red-filled dots are our Rup 147 (2.5-3.0 Gyr) sample stars with
NLTE correction, the blue-filled squares are NLTE-corrected NGC 752 ($\sim$
1.6 Gyr) single stars originally from Castro et al. (2016), and the grey-
filled diamonds are Li-corrected M67 ($\sim$ 3.7 Gyr) data from Pace et al.
(2012). The Li abundance derived from chondrite meteorites (A(Li)=3.25 dex,
Grevesse et al., 2007) is indicated by a dashed line.

##  5 Literature comparison

We have only one star in common with Pakhomov et al. (2009), CWW10/HD180112,
for which we have $4745\pm 33$/$2.38\pm 0.08$/$0.07\pm 0.11$ for $T_{\rm
eff}$, $\log g$, and [Fe/H], compared to 4733/2.53/$0.14\pm 0.06$ (see Table
11). Our average [Fe/H], based only on the giants for consistency with their
work, is in perfect agreement: $0.10\pm 0.06$ (six giants) compared to
$0.11\pm 0.07$ (three giants).

We have four stars in common with Carlberg (2014); the difference in RV is
within 0.5 km s-1 and also the $v\sin i$ values are in agreement within the
errors; see Table 11.

Curtis et al. (2013) have the most complete analysis of Rup 147 to date. We
agree with them on age (2.5 to 3 Gyr), distance (about 300 pc), and
metallicity. For the last, their average is $0.07\pm 0.03$, based on five MS
stars observed with Keck/HIRES, while we have $0.07\pm 0.08$ from 15 MS stars.
We have two stars in common (CWW 78, 91, see Table 11), their $T_{\rm eff}$
and $\log g$ are larger in our study than in theirs, but the metallicities are
in better agreement.

Curtis et al. (2018) studied star CWW 93 (hosting a sub-Neptune planet) in
detail by means of photometry and spectroscopy and also obtained spectra of a
further six solar-type stars in the cluster. All spectra were obtained with
MIKE@Magellan and were analysed using SME (Spectroscopy Made Easy, Valenti &
Piskunov 1996). For the seven solar-type stars they derived [Fe/H]$=0.10\pm
0.04$, while the spectroscopically derived parameters for CWW 93 are $T_{\rm
eff}$=5697 K, $\log\leavevmode\nobreak\ g$=4.453, [Fe/H]=0.141, and $v\sin
i$=1.95 km s-1.The mass and radius of the star were obtained combining
spectroscopic results with photometry and the distance modulus in Curtis et
al. (2013) and adopting three different isochrone sets and methods. The
procedures gave consistent values and they adopted the mean values as final
choice: mass=$1.009\pm 0.027$ M⊙ and radius=$0.945\pm 0.027$ R⊙. For
comparison, for CWW 93 we obtain $T_{\rm eff}$=5841$\pm 57$ K, $\log
g$=4.53$\pm 0.09$, [Fe/H]=0.18$\pm 0.13$, and $v\sin i=1.21\pm 1.29$ km s-1.
The implied stellar mass for this star is 1.02 $\pm 0.01$ M⊙.

Finally, Gaia DR2 contains the RV for 18 of the 21 stars in our final list,
obtained by the Gaia RVS instrument (Cropper et al., 2018). The RVs are
generally in agreement, especially when the error on the RVS measurements is
small (see Table 11). For all cases with a large difference, the RVS value has
an error (much) larger than 1 km s-1, while all our errors are one order of
magnitude smaller. The RV of the binary star CWW 58 is also similar between
our measurements and Gaia’s (36.35, rms=2.21 and $35.38\pm 2.06$ km s-1,
respectively); the Gaia pipelines did not detect this star as a possible
binary. The validation of RVs for DR2 discards stars with very high errors (20
km s-1) or suspect SB2 systems (Katz et al., 2018) and they do not apply to
star CWW 58. However, the Gaia RV is based only on two transits, so we believe
that ours is a more robust indication.

Figure 6: Run of [Na/Fe] with mass at MSTO, where each point represents a
cluster average, based only on evolved stars. Our determination for Rup 147
(average of the 6 giants) is shown as a filled light blue circle with error
bars highlighted by a larger circle. We show literature values coming from
three homogeneous groups: the Gaia-ESO survey clusters in Smiljanic et al.
(2016), Overbeek et al. (2017), and Tang et al. (2017) are shown in red, the
young clusters in Smiljanic et al. (2018) in blue, both with error bars, and
clusters in Reddy et al. (2012, 2013, 2015, 2016) in grey. Figure 7:
Comparison of some of our abundance ratios (black crosses) with those of Gaia-
ESO open clusters from Magrini et al. (2017), shown as open squares of
different colours (see legend). For both samples we show the individual stars
value, not cluster averages. The agreement is good, especially once
differences in solar reference is taken into account (their solar O and Mg is
higher by 0.12 dex and V lower by 0.11 dex).

##  6 Discussion

It has been found that Li abundance decreases as a function of age in solar
twin stars and in open clusters (see e.g. Carlos et al., 2016; Castro et al.,
2016, and references therein). However, age itself does not play a key role in
this correlation, the genuine driver of the abundance dispersion is Li burning
at the bottom of the surface convection zone, which could be indicated by
$T_{\rm eff}$. For instance, Xiong & Deng (2009) point out that for stars with
the same temperature ($T_{\rm eff}$$\lessapprox$6000 K), the Li abundance
decreases as age increases. In Fig. 5 we compare the Li abundance of Rup 147
to that of two other open clusters with similar metallicity but different
ages. Data for NGC 752 is from Castro et al. (2016) who provide [Fe/H]=0.0 dex
and an age of $\sim$1.6 Gyr. Lithium abundances in Castro et al. (2016) are
originally given in LTE, but in Fig. 5 we applied NLTE corrections (Lind et
al., 2009) to them considering a uniform microturbulence velocity $\xi$=2 km
s-1. This does not introduce spurious effects, since the NLTE correction for
Li is not sensitive to microturbulence; for instance, using $\xi$=1 km s-1
changes the final results by $<0.01$ dex. In the figure, only stars marked as
single are plotted. Lithium data on M67 ([Fe/H]=0.01, age=3.7 Gyr) from Pace
et al. (2012) are already NLTE-corrected. Since several works in the
literature conclude that the Li abundance scatter in M67 may be an exception
for Li evolution in open clusters (see e.g. Sestito et al., 2004; Xiong &
Deng, 2009), we put the M67 data in the figure only for reference. The left
panel of Fig. 5 illustrates the Li evolution from the MS to the giant branch
for the three clusters. Though the dwarf stars show a minor difference on Li,
the giants stars of Rup 147 and NGC 752 have a very similar Li abundance
level. Here we also notice that there is an un-reported Li-rich giant star
(H77) in NGC 752 with A(Li)NLTE=1.60 dex. In the middle panel, Rup 147 shows a
tight A(Li)-$T_{\rm eff}$ relation for the MS stars ($T_{\rm eff}$ from
$\sim$5300 K to 6300 K); there is no large Li scatter as seen in M67. This
tight relation supports the conclusion of Sestito et al. (2004) that M67 is
the only cluster showing a large Li spread for solar-type stars, and the Li
scatter is not typical of an old open cluster. Furthermore, all of the Rup 147
dwarf stars in our sample have $T_{\rm eff}$ ¡ 6300 K, which is around the
$T_{\rm eff}$ border of the Li-dip at solar metallicity (see NGC 752 in the
middle panel of Fig. 5). Considering that the turn-off $T_{\rm eff}$ of Rup
147 is $\lessapprox$ 6400 K (see the HR diagram in Fig. 1), one can very
hardly expect a dip-like pattern in the Li-$T_{\rm eff}$ figure of Rup 147
even if more turn-off stars are observed in the future.

Compared to the MS stars of NGC 752, which is 1 Gyr younger, Rup 147 dwarfs
present a lower Li abundance at the same temperature. The age difference on Li
abundance is also seen in the right panel of Fig. 5; with the same stellar
mass, Rup 147 MS stars (M ¡ 1.3 M⊙) have lower Li abundance compared to NGC
752 dwarfs, a difference mostly caused by microscopic diffusion. However, the
stellar mass was derived using different stellar models for each cluster, and
this may introduce some systematic uncertainty.

The average [Na/Fe] value for MS stars is 0.08 dex, while for giants this is
0.24 dex. This enhancement for giants is not uncommon among OCs (see e.g.
MacLean et al., 2015, for references) but is not universal (e.g. Sestito et
al., 2008; Bragaglia et al., 2012, all for old OCs). After excluding cases due
to neglecting NLTE effects, the enhancement may be attributed to mixing of Na
to the stellar photosphere after the first dredge-up (Iben, 1967). The amount
of mixing is then dependent on the stellar mass (and metallicity), with low-
mass, low-metallicity stars showing no changes (see the observations by
Gratton et al., 2000) and higher-mass stars showing increasing indications
(see e.g. the models by Charbonnel & Lagarde, 2010). The presence and extent
of Na enhancement among OC giants has recently been studied systematically by
Smiljanic et al. (2016, 2018), with comparison to various stellar models. For
Rup 147, given the age, we should in principle not expect a large effect, but
it seems on the contrary to show a higher Na enhancement than two OCs of
similar age in the Gaia-ESO survey (see Fig. 6). However, our solar Na is 6.17
(Grevesse et al., 2007), while Smiljanic et al. (2016) use 6.30; had we used
the latter, [Na/Fe] would be at the same level as the Gaia-ESO clusters (the
solar reference iron is 7.45 for both samples).

Apart from Li and Na, which are known to vary with evolutionary phase, how do
other elements behave in comparison to other open clusters? We compare
$\alpha$ and Fe-peak elements with the results of 11 OCs homogeneously
analysed by the Gaia-ESO survey covering almost the whole interval in
metallicity of OCs and with ages from 120 Myr to 4 Gyr, published in Magrini
et al. (2017). Figure 7 shows that Rup 147 behaves well, with abundance ratios
in line with clusters of similar metallicity. Only three species look slightly
discrepant (O, Mg, and V), but the differences between Rup 147 and the Gaia-
ESO results can be explained by the different solar reference values adopted
(see figure caption).

##  7 Summary and conclusions

We observed six evolved stars in the old, nearby cluster Rup 147 using HARPS-N
at the TNG and retrieved spectra of 22 MS stars from the ESO HARPS archive.
Our final sample comprises the six giants and 15 MS stars (one of the MS stars
was excluded because we found it to be a binary, and six more because they
rotate and display wide lines). We measured RVs, determined atmospheric
parameters, and derived abundances for iron, light, $\alpha$, iron-peak, and
neutron-capture elements. Comparisons to extant measurements reveal general
agreement. We did not find evidence of significant differences between dwarfs
and giants, with two exceptions. Sodium is enhanced in giants with respect to
MS stars, as expected from mixing mechanisms. Lithium shows a normal depletion
pattern as a function of evolutionary phase; in particular Li and $T_{\rm
eff}$ follow a tight relation for MS stars, at variance with M67, a cluster of
similar age and metallicity, which shows an unusual (and maybe unique)
dispersion in A(Li) at each $T_{\rm eff}$.

We reassessed the membership status of all our targets using Gaia parallaxes
and proper motions. Combining Gaia photometry with distance, metallicity,
$T_{\rm eff}$ and $\log g$ values from our spectroscopic analysis, and PARSEC
stellar models, for Rup 147 we derived distance, Galactic coordinates,
reddening, and age. Ruprecht 147 has metallicity Z=0.017 ([Fe/H]$\approx$0.08
dex), reddening $E(G_{BP}-G_{RP})=0.15$ (i.e. $E(B-V)=0.113$), an age of 2.5
to 3 Gyr, a distance about 308 pc from the Sun and 8.28 kpc from the Galactic
centre, and lies 0.07 kpc below the Galactic plane.

With the present paper we add another object to the short list of clusters for
which both giants and dwarfs are analysed, an important test of consistency
between the analysis of the different kinds of stars and of evolutionary
processes affecting the surface abundances. However, existing spectroscopy is
limited to the brighter part of the MS. Using Gaia data, Cantat-Gaudin et al.
(2018b) found many more high-probability candidate members of Rup 147 on the
single-star MS and on the well-separated binary sequence, down to the present
limit for precise Gaia astrometry (G=18). There is still room to improve the
understanding of this important cluster and solidify its standing as a
benchmark for stellar evolution studies.

Table 1: Information for the six stars observed with HARPS-N.

CWW | Gaia source ID | RA | DEC | expt | Date-obs | UT | PLX | ePLX | PMra | ePMra | PMde | ePMde | G | $G_{BP}$ | $G_{RP}$ | $RV_{Gaia}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (J2000) | (J2000) | (s) | (y:m:d) | (hh:mm:ss) | (mas) | (mas) | (mas/yr) | (mas/yr) | (mas/yr) | (mas/yr) | (mag) | (mag) | (mag) | (km/s)
1 | 4184143881214538112 | 19:15:26.120 | -16:05:57.10 | 900 | 2016-05-24 | 01-37-46.061 | 3.163 | 0.049 | 1.709 | 0.085 | -25.438 | 0.071 | 7.04 | 7.71 | 6.30 | 40.56 $\pm$1.89
2 | 4183949198935967232 | 19:17:23.840 | -16:04:24.30 | 900 | 2016-05-24 | 01:56:40.485 | 3.237 | 0.049 | -0.537 | 0.077 | -26.453 | 0.065 | 7.18 | 7.93 | 6.38 | 38.51 $\pm$ 0.17
4 | 4184137077986034048 | 19:17:11.300 | -16:03:08.20 | 1200 | 2016-05-24 | 02:15:32.621 | 3.250 | 0.056 | -0.797 | 0.089 | -26.997 | 0.077 | 8.02 | 8.66 | 7.31 | 41.33 $\pm$ 0.18
6 | 4087762027643173248 | 19:17:03.430 | -17:03:13.80 | 1200 | 2016-05-24 | 02:38:39.041 | 3.291 | 0.068 | -1.365 | 0.108 | -26.523 | 0.093 | 8.03 | 8.62 | 7.33 | 41.77 $\pm$ 0.19
10 | 4184125807991900928 | 19:15:51.290 | -16:17:59.10 | 1200 | 2016-05-24 | 03:03:21.454 | 3.197 | 0.044 | -0.629 | 0.076 | -26.895 | 0.071 | 8.12 | 8.73 | 7.42 | 40.80 $\pm$ 0.22
11 | 4183930438518525184 | 19:18:09.780 | -16:16:22.20 | 1200 | 2016-05-24 | 04:44:48.073 | 3.368 | 0.042 | -1.243 | 0.084 | -27.019 | 0.077 | 8.25 | 8.88 | 7.53 | 41.62 $\pm$ 0.14

444Observing Programme A33 DDT0. CWW is the identification in Curtis et al.
(2013). Parallax, proper motions, G, GBP, GRP, and RVGaia come from Gaia DR2
(Gaia Collaboration et al., 2018a).

Table 2: Information for the stars in the ESO/HARPS archive.

CWW | Gaia source ID | RA | DEC | N. exp | PLX | ePLX | PMra | ePMra | PMde | ePMde | G | $G_{BP}$ | $G_{RP}$ | $RV_{Gaia}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (J2000) | (J2000) |  | (mas) | (mas) | (mas/yr) | (mas/yr) | (mas/yr) | (mas/yr) | (mag) | (mag) | (mag) | (km/s)
23 | 4087853875535923200 | 19:15:42.69 | -16:33:05.0 | 15 | 3.388 | 0.070 | -0.707 | 0.157 | -26.717 | 0.104 | 11.38 | 11.71 | 10.91 | 42.46 $\pm$1.34
54 | 4184136906187904128 | 19:16:55.73 | -16:03:22.0 | 5 | 3.226 | 0.034 | -0.180 | 0.065 | -27.718 | 0.061 | 11.11 | 11.43 | 10.64 | 38.64 $\pm$0.73
57 | 4183934355528796672 | 19:17:04.33 | -16:23:18.5 | 8 | 3.143 | 0.056 | -1.246 | 0.083 | -26.339 | 0.068 | 10.93 | 11.26 | 10.47 | 42.84 $\pm$1.88
58 | 4184188961192147840 | 19:17:21.72 | -15:35:59.2 | 5 | 3.781 | 0.067 | 4.143 | 0.171 | -29.425 | 0.122 | 10.98 | 11.31 | 10.50 | 35.58 $\pm$2.06
59 | 4087799621507741312 | 19:15:12.60 | -17:05:12.1 | 8 | 3.394 | 0.058 | -1.125 | 0.096 | -26.862 | 0.077 | 10.91 | 11.21 | 10.46 | 43.73 $\pm$0.84
63 | 4184157659469838592 | 19:15:29.81 | -15:51:04.7 | 13 | 3.174 | 0.062 | -0.553 | 0.086 | -26.958 | 0.068 | 11.20 | 11.55 | 10.71 | 40.50 $\pm$1.40
70 | 4183932534462716928 | 9:16:38.27 | -16:25:03.9 | 2 | 3.169 | 0.052 | 1.136 | 0.081 | -26.436 | 0.070 | 11.19 | 11.51 | 10.74 | 48.14 $\pm$1.06
71 | 4087860506965490560 | 19:15:45.11 | -16:23:15.7 | 19 | 3.196 | 0.044 | -1.350 | 0.085 | -26.558 | 0.062 | 11.64 | 11.99 | 11.15 |
74 | 4184198788077655936 | 19:15:09.25 | -15:52:24.1 | 12 | 3.299 | 0.044 | -0.868 | 0.070 | -27.248 | 0.063 | 11.26 | 11.59 | 10.77 | 41.02 $\pm$1.57
75 | 4183936795070404352 | 19:16:11.21 | -16:21:48.5 | 11 | 3.321 | 0.042 | -0.701 | 0.068 | -26.871 | 0.063 | 12.73 | 13.17 | 12.14 | 39.71 $\pm$4.79
76 | 4088004611707768320 | 19:13:43.34 | -16:49:10.9 | 11 | 3.194 | 0.095 | -0.696 | 0.108 | -25.737 | 0.078 | 12.27 | 12.66 | 11.73 | 42.86 $\pm$1.42
78 | 4184245586042699008 | 19:16:08.79 | -15:24:27.9 | 58 | 3.257 | 0.086 | -1.166 | 0.240 | -27.336 | 0.150 | 11.41 | 11.74 | 10.93 |
79 | 4088057521393630848 | 19:14:28.16 | -16:20:02.3 | 57 | 3.175 | 0.096 | -0.902 | 0.257 | -26.688 | 0.162 | 12.17 | 12.56 | 11.64 | 43.63 $\pm$1.34
81 | 4087847067995609728 | 19:15:18.97 | -16:39:24.4 | 21 | 3.216 | 0.050 | -2.166 | 0.090 | -25.484 | 0.073 | 11.43 | 11.75 | 10.96 | 41.59 $\pm$1.33
83 | 4088110332311492224 | 19:13:41.26 | -16:10:20.1 | 5 | 3.267 | 0.085 | -2.413 | 0.241 | -26.055 | 0.158 | 11.81 | 12.19 | 11.27 |
85 | 4087838959097352064 | 19:16:59.40 | -16:35:27.1 | 34 | 3.279 | 0.040 | -1.238 | 0.079 | -25.733 | 0.066 | 12.41 | 12.82 | 11.86 | 42.80 $\pm$2.97
90 | 4087736159069458304 | 19:16:36.72 | -17:13:10.1 | 8 | 3.252 | 0.038 | -0.618 | 0.069 | -25.935 | 0.062 | 12.09 | 12.45 | 11.57 | 42.91 $\pm$0.74
91 | 4184135394358918656 | 19:16:47.25 | -16:04:09.3 | 7 | 3.231 | 0.033 | -0.727 | 0.059 | -26.952 | 0.055 | 12.35 | 12.76 | 11.79 | 42.20 $\pm$0.68
93 | 4184182737768311296 | 19:16:22.03 | -15:46:15.9 | 6 | 3.085 | 0.040 | -0.937 | 0.065 | -26.009 | 0.057 | 12.55 | 12.98 | 11.98 | 42.18 $\pm$2.03
94 | 4184146900561610880 | 19:15:21.41 | -16:00:10.7 | 11 | 3.107 | 0.062 | -0.676 | 0.082 | -26.241 | 0.071 | 12.76 | 13.21 | 12.17 | 42.52 $\pm$0.31
97 | 4184136558285344000 | 19:17:02.85 | -16:05:16.6 | 13 | 3.250 | 0.038 | -0.805 | 0.062 | -27.640 | 0.057 | 12.59 | 13.01 | 12.02 | 39.82 $\pm$1.17
98 | 4183940127965076224 | 19:16:26.56 | -16:14:54.5 | 7 | 3.489 | 0.154 | -1.927 | 0.203 | -27.787 | 0.178 | 12.96 | 13.47 | 12.25 | 44.97 $\pm$2.20

555Observing Programmes 091.C-0471, 093.C-0540, and 095.C-0947. CWW is the
identification in Curtis et al. (2013). Parallax, proper motions, G, GBP, GRP,
and RVGaia come from Gaia DR2 (Gaia Collaboration et al., 2018a).

Table 3: Radial velocity, v$\sin\leavevmode\nobreak\ i$, atmospheric
parameters, and isochrone-derived stellar mass.

CWW | S/N | RV | rms | nr | $T_{\rm eff}$ | err | log(g) | err | [M/H] | err | $\xi$ | err | N FeI | N FeII | v$\sin\leavevmode\nobreak\ i$ | err | Instr | mass | err
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (km s-1) | (km s-1) |  | (K) | (K) | (dex) | (dex) | (dex) | (dex) | (km s-1) | (km s-1) |  |  | (km s-1) | (km s-1) |  | (M⊙) | (M⊙)
1 | 40 | 37.29 | 0.01 | 1 | 4586 | 34 | 2.47 | 0.08 | 0.20 | 0.14 | 1.55 | 0.01 | 145 | 11 | 3.28 | 0.56 | H-N | 1.56 | 0.04
2 | 32 | 38.68 | 0.01 | 1 | 4373 | 30 | 1.96 | 0.08 | 0.04 | 0.12 | 1.66 | 0.01 | 129 | 9 | 2.94 | 0.62 | H-N | 1.55 | 0.05
4 | 42 | 41.20 | 0.01 | 1 | 4725 | 33 | 2.60 | 0.08 | 0.07 | 0.12 | 1.61 | 0.01 | 148 | 12 | 2.40 | 0.68 | H-N | 1.52 | 0.04
6 | 47 | 41.38 | 0.01 | 1 | 4736 | 33 | 2.51 | 0.08 | 0.07 | 0.11 | 1.51 | 0.01 | 164 | 11 | 2.39 | 0.68 | H-N | 1.53 | 0.05
10 | 50 | 40.77 | 0.01 | 1 | 4744 | 33 | 2.38 | 0.08 | 0.07 | 0.11 | 1.50 | 0.01 | 161 | 12 | 2.22 | 0.71 | H-N | 1.53 | 0.05
11 | 47 | 41.44 | 0.01 | 1 | 4718 | 32 | 2.60 | 0.08 | 0.15 | 0.10 | 1.48 | 0.01 | 138 | 10 | 2.44 | 0.65 | H-N | 1.52 | 0.05
23 | 115 | 41.41 | 0.01 | 15 | 6273 | 80 | 4.27 | 0.09 | -0.01 | 0.13 | 1.79 | 0.03 | 214 | 14 | 2.39 | 2.20 | H | 1.18 | 0.03
71 | 114 | 41.70 | 0.34 | 19 | 6178 | 74 | 4.32 | 0.10 | 0.05 | 0.12 | 1.57 | 0.02 | 218 | 10 | 2.78 | 1.69 | H | 1.14 | 0.02
75 | 50 | 42.19 | 0.01 | 11 | 5602 | 50 | 4.45 | 0.08 | 0.11 | 0.12 | 1.14 | 0.03 | 246 | 13 | 0.47 | 1.61 | H | 0.96 | 0.01
76 | 57 | 42.67 | 0.01 | 11 | 5825 | 61 | 4.26 | 0.08 | 0.14 | 0.11 | 1.21 | 0.02 | 235 | 12 | 0.00 | 1.60 | H | 1.05 | 0.01
78 | 182 | 41.00 | 0.30 | 59 | 6279 | 80 | 4.29 | 0.09 | 0.06 | 0.10 | 1.57 | 0.03 | 212 | 11 | 5.19 | 1.55 | H | 1.18 | 0.02
79 | 150 | 41.75 | 0.01 | 58 | 5932 | 62 | 4.47 | 0.09 | 0.20 | 0.11 | 1.14 | 0.03 | 246 | 14 | 0.00 | 1.60 | H | 1.05 | 0.01
81 | 148 | 41.37 | 0.05 | 21 | 6163 | 79 | 4.15 | 0.09 | 0.03 | 0.09 | 1.43 | 0.02 | 240 | 14 | 0.00 | 1.60 | H | 1.26 | 0.07
83 | 49 | 41.71 | 0.00 | 5 | 5987 | 77 | 4.43 | 0.10 | -0.04 | 0.13 | 1.42 | 0.03 | 202 | 8 | 2.00 | 1.48 | H | 1.06 | 0.01
85 | 133 | 42.66 | 0.02 | 34 | 5767 | 62 | 4.42 | 0.09 | 0.07 | 0.11 | 1.27 | 0.02 | 237 | 16 | 2.08 | 1.09 | H | 1.0 | 0.01
90 | 59 | 42.75 | 0.04 | 8 | 5994 | 68 | 4.48 | 0.10 | 0.09 | 0.12 | 1.21 | 0.02 | 232 | 13 | 0.00 | 1.60 | H | 1.07 | 0.01
91 | 55 | 41.67 | 0.01 | 7 | 5825 | 66 | 4.51 | 0.09 | 0.11 | 0.11 | 1.16 | 0.02 | 220 | 11 | 0.00 | 1.60 | H | 1.02 | 0.01
93 | 43 | 41.61 | 0.01 | 6 | 5841 | 57 | 4.53 | 0.09 | 0.18 | 0.13 | 1.19 | 0.03 | 221 | 9 | 1.21 | 1.29 | H | 1.02 | 0.01
94 | 70 | 42.06 | 0.01 | 20 | 5612 | 49 | 4.49 | 0.09 | 0.15 | 0.14 | 1.15 | 0.03 | 236 | 14 | 1.67 | 0.99 | H | 0.96 | 0.01
97 | 73 | 40.36 | 0.02 | 13 | 5649 | 53 | 4.29 | 0.09 | -0.04 | 0.11 | 1.27 | 0.03 | 237 | 11 | 1.74 | 1.10 | H | 0.98 | 0.01
98 | 39 | 40.68 | 0.04 | 7 | 5380 | 43 | 4.60 | 0.07 | -0.04 | 0.16 | 0.85 | 0.05 | 219 | 7 | 1.78 | 0.89 | H | 0.9 | 0.01
58 | 67 | 36.34 | 2.21 | 9 |  |  |  |  |  |  |  |  |  |  |  |  | H (bin.) |  |
54 | 48 | 38.84 | 0.02 | 5 |  |  |  |  |  |  |  |  |  |  | 9.25 | 2.98 | H |  |
57 | 87 | 41.32 | 0.02 | 8 |  |  |  |  |  |  |  |  |  |  | 6.78 | 3.60 | H |  |
59 | 71 | 42.18 | 0.10 | 8 |  |  |  |  |  |  |  |  |  |  | 12.31 | 2.73 | H |  |
63 | 92 | 41.02 | 0.06 | 13 |  |  |  |  |  |  |  |  |  |  | 10.06 | 2.74 | H |  |
70 | 28 | 44.78 | 0.05 | 2 |  |  |  |  |  |  |  |  |  |  | 14.16 | 3.23 | H |  |
74 | 96 | 41.54 | 0.07 | 12 |  |  |  |  |  |  |  |  |  |  | 8.12 | 3.12 | H |  |

666For HARPS stars the S/N is given for the co-added spectra. For HARPS-N we
give the RV and uncertainty on the single RV, for HARPS we give the average of
the multiple spectra and the rms.

Table 4: Light and $\alpha$-element abundances; the Sun is from Grevesse et al. (2007). CWW | Li iLTE | upper limit | Li iNLTE | O i | Na iLTE | err | Na iNLTE | err | Mg i | err | Al i | err | Si i | err | Ca i | err | Ti i | err | Ti ii | err
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 0.12 | ¡ | 0.41 | 8.80 | 6.66 | 0.09 | 6.63 | 0.10 | 7.91 | 0.09 | 6.58 | 0.11 | 7.79 | 0.15 | 6.42 | 0.22 | 4.91 | 0.13 | 5.08 | 0.13
2 | 0.29 | ¡ | 0.62 | 8.63 | 6.51 | 0.11 | 6.48 | 0.15 | 7.85 | 0.09 | 6.55 | 0.12 | 7.64 | 0.15 | 6.34 | 0.23 | 4.86 | 0.16 | 4.89 | 0.15
4 | 0.30 | ¡ | 0.54 | 8.77 | 6.50 | 0.08 | 6.47 | 0.12 | 7.82 | 0.10 | 6.50 | 0.15 | 7.68 | 0.14 | 6.37 | 0.20 | 4.88 | 0.15 | 4.97 | 0.13
6 | 0.08 | ¡ | 0.32 | 8.66 | 6.49 | 0.07 | 6.46 | 0.09 | 7.81 | 0.09 | 6.49 | 0.14 | 7.63 | 0.13 | 6.38 | 0.21 | 4.86 | 0.13 | 4.92 | 0.13
10 | 0.32 | ¡ | 0.57 | 8.61 | 6.47 | 0.08 | 6.45 | 0.10 | 7.82 | 0.07 | 6.49 | 0.13 | 7.60 | 0.13 | 6.39 | 0.20 | 4.86 | 0.14 | 4.87 | 0.15
11 | 0.46 | ¡ | 0.71 | 8.74 | 6.55 | 0.07 | 6.56 | 0.13 | 7.90 | 0.10 | 6.59 | 0.12 | 7.72 | 0.13 | 6.46 | 0.20 | 4.94 | 0.14 | 5.03 | 0.14
23 | 2.63 |  | 2.60 |  | 6.33 | 0.10 | 6.26 | 0.13 | 7.66 | 0.16 | 6.31 | 0.09 | 7.48 | 0.12 | 6.36 | 0.12 | 4.88 | 0.19 | 4.92 | 0.08
71 | 2.61 |  | 2.59 |  | 6.33 | 0.12 | 6.29 | 0.13 | 7.65 | 0.17 | 6.40 | 0.05 | 7.52 | 0.13 | 6.41 | 0.14 | 4.88 | 0.13 | 4.96 | 0.08
75 | 0.50 | ¡ |  |  | 6.38 | 0.08 | 6.34 | 0.10 | 7.78 | 0.11 | 6.48 | 0.18 | 7.56 | 0.15 | 6.50 | 0.11 | 5.00 | 0.11 | 4.99 | 0.11
76 | 1.52 | ¡ | 1.57 |  | 6.43 | 0.07 | 6.37 | 0.06 | 7.78 | 0.10 | 6.51 | 0.16 | 7.62 | 0.12 | 6.53 | 0.12 | 4.99 | 0.11 | 5.00 | 0.10
78 | 2.67 |  | 2.65 |  | 6.32 | 0.08 | 6.27 | 0.08 | 7.67 | 0.13 | 6.35 | 0.08 | 7.51 | 0.13 | 6.39 | 0.11 | 4.87 | 0.16 | 4.98 | 0.13
79 | 2.18 |  | 2.21 |  | 6.42 | 0.07 | 6.37 | 0.08 | 7.79 | 0.12 | 6.50 | 0.15 | 7.60 | 0.14 | 6.53 | 0.10 | 5.02 | 0.10 | 5.07 | 0.10
81 | 2.65 |  | 2.63 |  | 6.33 | 0.09 | 6.27 | 0.09 | 7.68 | 0.12 | 6.38 | 0.08 | 7.50 | 0.14 | 6.41 | 0.12 | 4.84 | 0.13 | 4.96 | 0.10
83 | 2.48 |  | 2.47 |  | 6.35 | 0.11 | 6.31 | 0.11 | 7.68 | 0.15 | 6.51 | 0.12 | 7.52 | 0.12 | 6.47 | 0.16 | 4.99 | 0.21 | 4.98 | 0.09
85 | 0.72 | ¡ |  |  | 6.36 | 0.06 | 6.31 | 0.09 | 7.72 | 0.12 | 6.44 | 0.12 | 7.52 | 0.15 | 6.46 | 0.11 | 4.92 | 0.11 | 4.93 | 0.10
90 | 2.32 |  | 2.33 |  | 6.33 | 0.09 | 6.29 | 0.12 | 7.68 | 0.10 | 6.45 | 0.16 | 7.51 | 0.25 | 6.46 | 0.12 | 4.95 | 0.12 | 5.02 | 0.10
91 | 1.78 | ¡ | 1.82 |  | 6.37 | 0.04 | 6.32 | 0.06 | 7.77 | 0.11 | 6.52 | 0.16 | 7.57 | 0.16 | 6.50 | 0.10 | 4.98 | 0.15 | 5.05 | 0.13
93 | 1.59 | ¡ | 1.64 |  | 6.50 | 0.19 | 6.46 | 0.19 | 7.84 | 0.15 | 6.58 | 0.12 | 7.63 | 0.15 | 6.60 | 0.09 | 5.13 | 0.12 | 5.11 | 0.15
94 | 1.06 | ¡ | 1.13 |  | 6.41 | 0.06 | 6.40 | 0.08 | 7.81 | 0.14 | 6.52 | 0.16 | 7.60 | 0.15 | 6.53 | 0.12 | 5.02 | 0.09 | 5.04 | 0.12
97 | 1.26 | ¡ | 1.31 |  | 6.35 | 0.05 | 6.30 | 0.06 | 7.69 | 0.11 | 6.34 | 0.12 | 7.45 | 0.12 | 6.39 | 0.12 | 4.82 | 0.11 | 4.84 | 0.08
98 | 1.01 | ¡ | 1.10 |  | 6.34 | 0.08 | 6.30 | 0.09 | 7.62 | 0.12 | 6.39 | 0.14 | 7.52 | 0.14 | 6.45 | 0.13 | 4.93 | 0.12 | 4.94 | 0.23
Sun | 1.05 |  |  | 8.66 | 6.17 |  |  |  | 7.53 |  | 6.37 |  | 7.51 |  | 6.31 |  | 4.90 |  | 4.90 |

Table 5: Iron peak element abundances; the Sun is from Grevesse et al. (2007).

CWW | Sc1 | err | V1 | err | Co1 | err | Ni1 | err | Cu1 | err | Zn1
---|---|---|---|---|---|---|---|---|---|---|---
1 | 3.08 | 0.06 | 3.95 | 0.14 | 4.98 | 0.14 | 6.35 | 0.22 | 4.25 | 0.12 | 4.31
2 | 3.01 | 0.06 | 3.88 | 0.17 | 4.87 | 0.17 | 6.20 | 0.25 | 4.06 | 0.08 | 4.07
4 | 3.10 | 0.07 | 3.94 | 0.13 | 4.92 | 0.14 | 6.24 | 0.20 | 4.20 | 0.07 | 4.21
6 | 3.05 | 0.07 | 3.89 | 0.12 | 4.87 | 0.14 | 6.22 | 0.20 | 4.15 | 0.06 | 4.24
10 | 3.03 | 0.09 | 3.89 | 0.12 | 4.85 | 0.14 | 6.21 | 0.19 | 4.12 | 0.06 | 4.20
11 | 3.13 | 0.06 | 3.98 | 0.13 | 4.96 | 0.15 | 6.32 | 0.20 | 4.26 | 0.11 | 4.30
23 | 3.36 | 0.20 | 3.92 | 0.21 | 4.89 | 0.25 | 6.18 | 0.16 | 4.22 | 0.07 | 4.30
71 | 3.30 | 0.24 | 3.93 | 0.18 | 4.91 | 0.18 | 6.22 | 0.16 | 4.21 | 0.07 | 4.36
75 | 3.28 | 0.25 | 4.05 | 0.10 | 4.97 | 0.11 | 6.30 | 0.16 | 4.29 | 0.01 | 4.47
76 | 3.28 | 0.23 | 4.02 | 0.14 | 5.00 | 0.12 | 6.32 | 0.15 | 4.29 | 0.04 | 4.51
78 | 3.28 | 0.17 | 3.91 | 0.20 | 4.93 | 0.21 | 6.19 | 0.20 | 4.22 | 0.08 | 4.32
79 | 3.34 | 0.21 | 4.06 | 0.10 | 5.03 | 0.13 | 6.35 | 0.16 | 4.34 | 0.03 | 4.49
81 | 3.32 | 0.24 | 3.87 | 0.19 | 4.84 | 0.18 | 6.20 | 0.14 | 4.18 | 0.06 | 4.40
83 | 3.55 | 0.18 | 4.10 | 0.20 | 4.90 | 0.34 | 6.23 | 0.17 | 4.20 | 0.08 | 4.29
85 | 3.29 | 0.21 | 3.97 | 0.08 | 4.91 | 0.15 | 6.23 | 0.15 | 4.24 | 0.01 | 4.45
90 | 3.27 | 0.20 | 3.97 | 0.15 | 4.98 | 0.16 | 6.29 | 0.16 | 4.29 | 0.09 | 4.48
91 | 3.37 | 0.27 | 4.06 | 0.15 | 5.03 | 0.15 | 6.32 | 0.16 | 4.28 | 0.07 | 4.47
93 | 3.46 | 0.25 | 4.21 | 0.10 | 5.15 | 0.17 | 6.40 | 0.18 | 4.40 | 0.10 | 4.55
94 | 3.31 | 0.17 | 4.09 | 0.09 | 5.01 | 0.12 | 6.32 | 0.16 | 4.24 | 0.08 | 4.49
97 | 3.20 | 0.20 | 3.88 | 0.13 | 4.84 | 0.12 | 6.15 | 0.15 | 4.13 | 0.04 | 4.39
98 | 3.20 | 0.13 | 3.98 | 0.13 | 4.90 | 0.15 | 6.25 | 0.18 | 4.26 | 0.10 | 4.28
Sun | 3.17 |  | 4.00 |  | 4.92 |  | 6.23 |  | 4.21 |  | 4.60
777Cu i abundances corrected, see text.

Table 6: Neutron-capture element abundances; the Sun is from Grevesse et al.
(2007).

CWW | Y2 | err | Zr1 | Zr2 | err | Ba2 | err | La2 | err | Ce2 | err | Eu2
---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 2.25 | 0.27 | 2.62 | 2.68 | 0.08 | 2.49 | 0.24 | 1.26 | 0.10 | 1.92 | 0.23 | 0.79
2 | 2.08 | 0.28 | 2.55 | 2.41 | 0.08 | 2.19 | 0.37 | 1.04 | 0.18 | 1.69 | 0.13 | 0.66
4 | 2.13 | 0.27 | 2.64 | 2.68 | 0.05 | 2.22 | 0.24 | 1.17 | 0.07 | 1.83 | 0.23 | 0.67
6 | 2.09 | 0.27 | 2.60 | 2.60 | 0.04 | 2.24 | 0.24 | 1.10 | 0.07 | 1.75 | 0.23 | 0.65
10 | 2.01 | 0.20 | 2.57 | 2.54 | 0.01 | 2.21 | 0.24 | 1.02 | 0.05 | 1.66 | 0.23 | 0.58
11 | 2.22 | 0.32 | 2.67 | 2.67 | 0.01 | 2.31 | 0.24 | 1.14 | 0.09 | 1.85 | 0.25 | 0.75
23 | 2.08 | 0.20 | 3.22 | 2.70 | 0.13 | 2.01 | 0.13 | 1.21 | 0.06 | 1.79 | 0.37 | 0.71
71 | 2.16 | 0.22 | 3.13 | 2.78 | 0.21 | 2.04 | 0.21 | 1.14 | 0.04 | 1.71 | 0.42 | 0.71
75 | 2.22 | 0.19 | 3.04 | 2.85 | 0.16 | 2.22 | 0.11 | 1.26 | 0.06 | 2.08 | 0.26 | 0.76
76 | 2.22 | 0.16 | 3.13 | 2.78 | 0.18 | 2.22 | 0.13 | 1.18 | 0.01 | 1.91 | 0.33 | 0.69
78 | 2.13 | 0.19 | 2.97 | 2.67 | 0.14 | 2.10 | 0.18 | 1.21 | 0.10 | 1.77 | 0.42 | 0.63
79 | 2.27 | 0.24 | 2.97 | 2.82 | 0.10 | 2.31 | 0.13 | 1.31 | 0.06 | 2.04 | 0.25 | 0.84
81 | 2.11 | 0.19 | 2.82 | 2.67 | 0.13 | 2.18 | 0.13 | 1.08 | 0.01 | 1.92 | 0.13 | 0.63
83 | 2.20 | 0.27 | 3.45 | 2.79 | 0.08 | 2.16 | 0.14 | 1.09 | 0.08 | 2.12 | 0.06 | 1.01
85 | 2.28 | 0.14 | 3.10 | 2.83 | 0.09 | 2.30 | 0.15 | 1.05 | 0.13 | 2.02 | 0.28 | 0.69
90 | 2.21 | 0.26 | 3.02 | 2.72 | 0.15 | 2.17 | 0.23 | 1.24 | 0.08 | 1.96 | 0.31 | 0.56
91 | 2.30 | 0.33 | 2.74 | 2.74 | 0.16 | 2.21 | 0.06 | 1.39 | 0.11 | 2.02 | 0.37 | 0.84
93 | 2.36 | 0.29 | 3.44 | 2.88 | 0.17 | 2.34 | 0.15 | 1.53 | 0.15 | 2.17 | 0.35 | 0.96
94 | 2.25 | 0.27 | 2.78 | 2.72 | 0.21 | 2.21 | 0.15 | 1.22 | 0.02 | 2.03 | 0.45 | 0.63
97 | 2.27 | 0.15 | 2.89 | 2.88 | 0.11 | 2.28 | 0.13 | 1.22 | 0.05 | 1.92 | 0.33 | 0.64
98 | 2.20 | 0.25 | 2.67 | 2.39 | 0.34 | 2.16 | 0.19 | 0.86 | 0.24 | 2.28 | 0.18 | 0.75
Sun | 2.21 |  | 2.58 | 2.58 |  | 2.17 |  | 1.13 |  | 1.70 |  | 0.52
888Ba ii and Eu ii abundances corrected, see text.

Table 7: Sensitivity to errors in atmospheric parameters.

element | lines |  $\Delta$ A(X) (CWW 10) |  $\Delta$ A(X) (CWW 81)
---|---|---|---
$\Delta$$T_{\rm eff}$=+33 K |  $\Delta$log(g)=+0.08 dex |  $\Delta$$T_{\rm eff}$=+79 K |  $\Delta$log(g)=+0.09 dex |  $\Delta$[M/H]=+0.06 dex
Li i | 1 |  |  | 0.070 | 0.000 | 0.010
O i | 1 | 0.010 | 0.040 |  |  |
Na i | 8 | 0.029 | -0.005 | 0.035 | -0.005 | -0.006
Mg i | 5 | 0.018 | -0.010 | 0.028 | -0.012 | -0.004
Al i | 3 | 0.033 | 0.003 | 0.030 | 0.000 | 0.000
Si i | 29 | -0.012 | 0.009 | 0.019 | 0.002 | -0.003
Ca i | 25 | 0.041 | -0.008 | 0.046 | -0.012 | -0.007
Sc i | 3 | 0.050 | 0.007 | 0.070 | -0.003 | -0.003
Sc ii | 17 | 0.001 | 0.027 | 0.003 | 0.039 | 0.012
Ti i | 75 | 0.052 | 0.004 | 0.066 | -0.004 | -0.006
Ti ii | 17 | -0.002 | 0.027 | -0.001 | 0.036 | 0.014
V i | 28 | 0.057 | 0.006 | 0.074 | 0.002 | -0.013
Co i | 28 | 0.020 | 0.014 | 0.061 | -0.001 | -0.004
Ni i | 84 | 0.013 | 0.011 | 0.047 | -0.001 | -0.004
Cu i | 3 | 0.023 | 0.020 | 0.053 | 0.003 | -0.010
Zn i | 1 | -0.010 | 0.010 | 0.030 | 0.010 | -0.010
Y ii | 13 | 0.012 | 0.025 | 0.013 | 0.028 | 0.015
Zr i | 1 | 0.060 | 0.000 | 0.080 | 0.000 | -0.010
Zr ii | 2 | 0.010 | 0.040 | 0.010 | 0.040 | 0.005
Ba ii | 3 | 0.013 | 0.010 | 0.033 | 0.020 | 0.017
La ii | 3 | 0.008 | 0.033 | 0.015 | 0.035 | 0.015
Ce ii | 3 | 0.003 | 0.033 | 0.005 | 0.035 | 0.015
Eu ii | 1 | 0.000 | 0.030 | 0.010 | 0.040 | 0.020

999Sensitivity computed for star CWW 10, a giant star with $T_{\rm eff}$=
4744$\pm$33 K, log(g)= 2.38$\pm$0.08 dex, [M/H]=-0.02$\pm$0.06 dex; and a
dwarf star CWW 81, with $T_{\rm eff}$=6163$\pm$79 K, log(g)=4.15$\pm$0.09 dex,
[M/H]=+0.02$\pm$0.06 dex.

Table 8: Average abundance ratios for giants, dwarfs, and entire sample.

[X/Fe] | mean | rms | num | mean | rms | num | mean | rms | num
---|---|---|---|---|---|---|---|---|---
| giants | dwarfs | all
Fe | 0.10 | 0.06 | 6 | 0.07 | 0.08 | 15 | 0.08 | 0.07 | 21
O1 | -0.06 | 0.05 | 6 |  |  |  | -0.06 | 0.04 | 6
Na1 | 0.24 | 0.02 | 6 | 0.08 | 0.05 | 15 | 0.13 | 0.08 | 21
Mg1 | 0.22 | 0.03 | 6 | 0.12 | 0.04 | 15 | 0.15 | 0.06 | 21
Al1 | 0.06 | 0.05 | 6 | 0.01 | 0.04 | 15 | 0.02 | 0.06 | 21
Si1 | 0.07 | 0.03 | 6 | -0.04 | 0.04 | 15 | -0.01 | 0.06 | 21
Ca1 | -0.02 | 0.03 | 6 | 0.09 | 0.05 | 15 | 0.06 | 0.07 | 21
Sc1 | -0.20 | 0.05 | 6 | 0.08 | 0.11 | 15 | 0.00 | 0.16 | 21
Sc2 | -0.09 | 0.04 | 6 | -0.05 | 0.05 | 15 | -0.06 | 0.05 | 21
Ti1 | -0.11 | 0.03 | 6 | -0.02 | 0.06 | 15 | -0.05 | 0.07 | 21
Ti2 | -0.04 | 0.03 | 6 | 0.02 | 0.04 | 15 | 0.00 | 0.05 | 21
V1 | -0.18 | 0.04 | 6 | -0.07 | 0.08 | 15 | -0.10 | 0.08 | 21
Co1 | -0.11 | 0.02 | 6 | -0.04 | 0.04 | 15 | -0.06 | 0.05 | 21
Ni1 | -0.07 | 0.01 | 6 | -0.04 | 0.04 | 15 | -0.05 | 0.04 | 21
Cu1 | -0.14 | 0.04 | 6 | -0.03 | 0.05 | 15 | -0.06 | 0.07 | 21
Zn1 | -0.48 | 0.04 | 6 | -0.25 | 0.04 | 15 | -0.32 | 0.11 | 21
Y2 | -0.18 | 0.04 | 6 | -0.06 | 0.07 | 15 | -0.10 | 0.08 | 21
Zr1 | -0.07 | 0.05 | 6 | 0.37 | 0.23 | 15 | 0.25 | 0.28 | 21
Zr2 | -0.08 | 0.07 | 6 | 0.10 | 0.11 | 15 | 0.05 | 0.13 | 21
Ba2 | 0.01 | 0.05 | 6 | -0.05 | 0.08 | 15 | -0.03 | 0.08 | 21
La2 | -0.11 | 0.05 | 6 | 0.00 | 0.11 | 15 | 0.04 | 0.11 | 21
Ce2 | -0.02 | 0.05 | 6 | 0.21 | 0.16 | 15 | 0.15 | 0.17 | 21
Eu2 | 0.06 | 0.03 | 6 | 0.15 | 0.14 | 15 | 0.12 | 0.12 | 21
101010All abundances are [X/Fe], with the exception of [Fe/H]. Li is not
reported here. The value for Na is in NLTE.

Table 9: Comparison with literature for RV and atmospheric parameters.

CWW | RV | $T_{\rm eff}$ | $\log g$ | [Fe/H] | $\xi$ | RVC14 | RVC | RVG | $T_{\rm eff}$ | $\log g$ | [Fe/H] | $\xi$ | Ref
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Present paper |  | Literature |
1 | 37.29 |  |  |  |  | 38.0 | 38.5 | $40.56\pm 1.89$ |  |  |  |  | 2,3,5
2 | 38.68 |  |  |  |  | 39.0 | 43.4 | $38.51\pm 0.17$ |  |  |  |  | 2,3,5
4 | 41.20 |  |  |  |  |  | 42.7 | $41.33\pm 0.18$ |  |  |  |  | 2,5
6 | 41.38 |  |  |  |  | 42.1 | 46.2 | $41.77\pm 0.19$ |  |  |  |  | 2,3,5
10 | 40.77 | 4744 | 2.38 | 0.07 | 1.50 |  | 40.1 | $40.80\pm 0.22$ | 4633 | 2.53 | 0.14 | 1.28 | 1,5
11 | 41.44 |  |  |  |  | 41.8 | 44.2 | $41.62\pm 0.14$ |  |  |  |  | 2,3,5
23 | 41.41 |  |  |  |  |  |  | $42.46\pm 1.34$ |  |  |  |  | 5
75 | 42.19 |  |  |  |  |  |  | $39.71\pm 4.79$ |  |  |  |  | 5
76 | 42.67 |  |  |  |  |  |  | $42.86\pm 1.42$ |  |  |  |  | 5
78 | 41.00 | 6279 | 4.29 | 0.06 | 1.57 |  | 41.02 |  | 6129 | 3.60 | -0.01 |  | 2
79 | 41.75 |  |  |  |  |  |  | $43.63\pm 1.34$ |  |  |  |  | 5
81 | 41.37 |  |  |  |  |  |  | $41.59\pm 1.33$ |  |  |  |  | 5
85 | 42.66 |  |  |  |  |  |  | $42.80\pm 2.97$ |  |  |  |  | 5
90 | 42.75 |  |  |  |  |  |  | $42.91\pm 0.74$ |  |  |  |  | 5
91 | 41.67 | 5825 | 4.51 | 0.09 | 1.16 |  | 40.35 | $42.20\pm 0.68$ | 5747 | 4.35 | 0.06 |  | 2,5
93 | 41.61 | 5841 | 4.53 | 0.09 | 1.19 |  | 41.58 | $42.18\pm 2.03$ | 5697 | 4.453 | 0.141 |  | 4,5
94 | 42.06 |  |  |  |  |  |  | $42.52\pm 0.31$ |  |  |  |  | 5
97 | 40.36 |  |  |  |  |  |  | $39.82\pm 1.17$ |  |  |  |  | 5
98 | 40.68 |  |  |  |  |  |  | $44.97\pm 2.20$ |  |  |  |  | 5

111111 (1) Pakhomov et al. (2009) ; (2) Curtis et al. (2013), Tables 3, 5 ;
(3) Carlberg (2014) ; (4) Curtis et al. (2018); (5) Gaia DR2

###### Acknowledgements.

This paper is based on observations made with the Italian Telescopio Nazionale
Galileo (TNG) operated on the island of La Palma by the Fundaciíon Galileo
Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de
Canarias. This paper is based on data obtained from the ESO Science Archive
Facility under request number 299084. XF acknowledges funding by Premiale 2015
MITiC (PI B. Garilli) and the EU COST Action CA16117 (ChETEC). This work
presents results from the European Space Agency (ESA) space mission Gaia. Gaia
data are being processed by the Gaia Data Processing and Analysis Consortium
(DPAC). Funding for the DPAC is provided by national institutions, in
particular the institutions participating in the Gaia MultiLateral Agreement
(MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia
archive website is https://archives.esac.esa.int/gaia. This research has made
use of Vizier and SIMBAD, operated at CDS, Strasbourg, France, NASA’s
Astrophysical Data System, and TOPCAT (http://www.starlink.ac.uk/topcat/,
Taylor 2005). This research made use of the cross-match service provided by
CDS, Strasbourg.

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# New Nonrelativistic Quantum Theory of Cold Dark Matter

Z. E. Musielak  Department of Physics, The University of Texas at Arlington,
Arlington, TX 76019, USA<EMAIL_ADDRESS>
###### Abstract

Cold dark matter is conceived as a gas of massive particles that undergo
collisions, interact gravitationaly, and exchange quanta of energy. A new
nonrelativistic quantum theory is presented for this model of dark matter,
based on recently discovered equation for a spinless, no charge, and free
particle. This theory describes the quantum processes undergoing by the
particles, specifies the required characteristic wavelength of the quanta of
energy, gives constraints on mass of dark matter particles, and predicts a
detectable gravitational wave background associated with dark matter halos.

##  1 Introduction

The existence of dark matter (DM) is inferred from gravitational effects.
While the Universe contains 4.9% of ordinary matter (OM), data from the Planck
2018 mission [1] suggests that 26.8% of DM constitutes the total mass-energy
density.

Although different theories of DM have been proposed, the physical nature of
the particles that make up DM remains unknown [2-8]. The most commonly
accepted theory proposes a weakly interacting massive particle (WIMP). In the
past several years, the search for WIMPs has intesified but thus far
instruments have failed to detect them [8-10].

In this paper, a new nonrelativistic quantum theory of DM is presented, based
on an equation describing a spinless, without charge, and free particle [11]
that may be considered a candidate for DM. The existence of this equation is
supported by the irreducible representations (irreps) of the extended Galilean
group of the metric [12,13]. The irreps of the group also allow deriving the
Schrödinger equation of quantum mechanics (QM) [14]. It was demonstrated that
there is a special type of symmetry between these two equations and, since the
Schrödinger equation describes the quantum structure of OM, the new equation
may be used to represent DM [11].

The preliminary quantum model of DM reported in [11] is significantly extended
here by taking into account collisions between massive DM particles and their
gravitational interaction, as well as exchange of quanta of energy. The
developed theory describes the quantum emission and absorption of the quanta
of energy and the resulting equilibrium, and it also specifies the required
characteristic wavelength of the quanta of energy, gives constraints on the
mass of dark matter particles, and predicts a gravitational wave background
for DM halos.

This paper is organized as follows: the basic equations are given in Section
2; a model of dark matter is described in Section 3; a quantum theory of dark
matter is presented in Section 4; physical implications of the theory are
discussed in Section 5; and Conclusions are given in Section 6.

##  2 Basic equations of non-relativistic quantum physics

In Galilean relativity, space and time are represented by different metrics
that remain invariant with respect to all transformations that form the
Galilean group of the metric $\mathcal{G}=[T(1)\otimes
O(3)]\otimes_{s}[T(3)\otimes B(3)]$, where $T(1)$, $O(3)$, $T(3)$ and $B(3)$
are subgroups of translations in time, rotations, translations in space, and
boosts, respectively [12]. However, the Schrödinger equation is invariant with
respect to the extended Galilean group [15-17], whose structure is
$\mathcal{G}_{e}=[O(3)\otimes_{s}B(3)]\otimes_{s}[T(3+1)\otimes U(1)]$, where
$T(3+1)$ is an invariant Abelian subgroup of combined translations in space
and time, and $U(1)$ is a one-parameter unitary subgroup [13].

Classification of the irreps of $\mathcal{G}_{e}$ by Bargmann [18]
demonstrated that only the scalar and spinor irreps are physical, but vectors
and tensors are not because they do not allow for elementary particle
localizations. According to Wigner [19], a wave function must transform like
one of the irreps of the group because, only in this case, all inertial
observers identify the same physical object and agree on the description of
its physical properties [20].

It was shown [14] that the Wigner condition in the Galilean space and time can
be expressed mathematically by two eigenvalue equations [14,17], which were
used [11] to derive the following equations

$\left[i{{\partial}\over{\partial
t}}+C_{s}\nabla^{2}\right]\phi(t,\mathbf{x})=0\ ,$ (1)

and

$\left[{{\partial^{2}}\over{\partial
t^{2}}}-iC_{w}{\mathbf{k}}\cdot\nabla\right]\phi(t,\mathbf{x})=0\ .$ (2)

where $\mathbf{x}=(x,y,z)$, with $x$, $y$ and $z$ being the Cartesian
coordinates, $C_{s}=\omega/k^{2}$, and $C_{w}=\omega^{2}/k^{2}$, with
$k^{2n}=(\mathbf{k}\cdot\mathbf{k})^{n}$. In addition, $\omega$ and
${\mathbf{k}}$ are labels of the irreps of $\mathcal{G}_{e}$, which means that
they can be any real numbers. As a result, there is an infinite number of
equations given by Eqs (1) and (2).

Since the metrics for space and time in Galilean relativity are separated, it
is required that the derived equations are asymmetric with respect to the
space and time derivatives, which is the necessary condition to make the
equations Galilean invariant. Because of its form, Eq. (1) is called a
Schrödinger-like equation, while Eq. (2) is referred to as a new asymmetric
equation [11].

Among an infinite number of Schrödinger-like equations, the Schrödinger
equation of QM can be obtained by specifying the constant $C_{s}$, which can
be done when the de Broglie relationship is used [11]. The result is

$\left[i{{\partial}\over{\partial
t}}+\frac{\hbar}{2m}\nabla^{2}\right]\phi(t,\mathbf{x})=0\ ,$ (3)

which is the Schrödinger equation for a free particle. By additing different
potentials, the equation can be used to describe quantum states of OM in
different physical settings [15].

There are also infinitely many new asymmetric equations. It is easy to verify
that the de Broglie relationship cannot be used to evaluate the constant
$C_{w}$. This means that $C_{w}$ does not depend on $\hbar$ but instead it
requires a new constant of Nature denoted as $\varepsilon_{o}$, which
represents a quanta of energy. The resulting new asymmetric equation can be
written as

$\left[{{\partial^{2}}\over{\partial
t^{2}}}-i\frac{\varepsilon_{o}}{2m}{\mathbf{k}}\cdot\nabla\right]\phi(t,\mathbf{x})=0\
,$ (4)

and it describes a free, spinless, and without charge particle. There are
differences between this equation and the Schrödinger equation, namely, Eq.
(4) has the second-order derivative in time instead of the first-order, and
the first-order derivative in space instead of the second-order; thus, there
is a special kind of space and time symmetry between these two equations. In
addition, the Schrödinger equation depends on the Planck constant that makes
any quantized energy levels to depend on frequency, but the presence of quanta
of energy $\varepsilon_{o}$ in the new asymmetric equation makes any quantized
energy levels to be independent from frequency.

Since the new asymmetric equation (Eq. 4) does not depend on the Planck
constant, it cannot describe properly the quantum structure of OM. Therefore,
it was suggested that the equation may represent the quantum structure of DM
[11]. In this paper, the equation is used to develop a new nonrelativistic
quantum theory of DM.

##  3 Dark matter model and governing equations

Let $\mathcal{S}$ be a sphere of DM particles, or DM halo, and $m$ denotes the
mass of each particle, which is charge and spin free. For the particles
uniformly distributed inside the halo, the gravitational potential $V_{h}(R)$
[21] is given by

$V_{h}(R)=\frac{GM_{h}}{2R_{h}^{3}}\left(R^{2}-3R_{h}^{2}\right)\ ,$ (5)

where $M_{h}$ and $R_{h}$ are the mass and radius of the halo, and $R$ is the
spherical coordinate. The acceleration of DM particles inside the halo
resulting from this potential is

$g_{h}(R)=\frac{dV_{h}(R)}{dR}=\frac{GM_{h}}{R_{h}^{3}}R\ ,$ (6)

which shows that $g_{h}(R)$ is a linear function of $R$ [21]. As a result, the
derivative of $g_{h}(R)$ with respect to $R$ gives

$\Omega^{2}_{h}=\frac{dg_{h}(R)}{dR}=\frac{GM_{h}}{R_{h}^{3}}={\rm const}\ ,$
(7)

which shows that $\Omega^{2}_{h}$ remains the same at every point inside the
halo.

A stable DM halo requires hydrostatic equailibrium, which means that

$\frac{dp_{h}(R)}{dR}=-\rho_{h}(R)g_{h}(R)\ ,$ (8)

where $p_{h}(R)$ and $\rho_{h}(R)$ the pressure and density of DM inside the
halo. By writing the above equation in the form

$g_{h}(R)=-\frac{1}{\rho_{h}}\frac{dp_{h}(R)}{dR}\ ,$ (9)

the accelaration $g(R)$ becomes a pressure-gradient force per unit mass [22].

The particles of the spherical DM halo are allowed to collide, interact
gravitationally, and also exchange energy by quantum processes. To account for
gravitational interaction between a pair of DM particles, a local spherical
coordinate system $(r,\theta,\phi)$ centered at one of these particles is
considered, with $r$ representing the distance between the particles. In this
model, the only spatial variable considered is $r$, which means that there are
neither changes with respect to $\theta$ nor $\phi$.

The gravitational potential $V(r)$ is the work that is needed to bring the DM
particle from infinity to its location, and is given as $V(r)=-Gm/r$. By
taking the negative gradient of $V(r)$, it yields the acceleration per unit
mass of the particle, $a(r)=Gm/r^{2}$ (see Eq. 6). Taking the derivative with
respect to $r$ one more time, the result is

$\Omega^{2}_{g}(r)=\frac{da(r)}{dr}=-\frac{2Gm}{r^{3}}\ .$ (10)

which can be included into Eq. (4) to account for gravitational interaction
between a pair of DM particles. Then, Eq. (4) becomes

$\left[{{\partial^{2}}\over{\partial
t^{2}}}-i\frac{\varepsilon_{o}}{2m}(\mathbf{k}\cdot\mathbf{\hat{r}})\frac{\partial}{\partial
r}-\Omega^{2}_{g}(r)\right]\phi(t,r)=0\ ,$ (11)

with $\mathbf{r}=r\mathbf{\hat{r}}$. Since $\Omega_{h}$ = const anywhere
inside the DM halo (see Eq. 7), the above new asymmetric equation with the
$\Omega^{2}_{g}(r)$ term remains valid for any pair located at any point
inside the DM halo. This is an important result as it shows that the developed
model of gravitationally interacting pair of DM particles is valid for the
entire DM halo. The model also allows for the particles of the pair to change
their kinetic energy by colliding with other particles in the DM halo.

According to [1-8,23-26], DM is not a source of any form of electromagnetic,
weak or strong interactions, but it is known to interact gravitationally with
OM. However, in the model of DM considered in this paper, OM is not included,
but only gravitationally interacting DM particles are taken into account.

##  4 Quantum theory of dark matter

The quantum theory of DM is formulated by considering a pair of DM particles,
whose evolution in time and space is described by Eq. (11). By separation of
variables, $\phi(t,r)=\chi(t)\eta(r)$, the following equation is obtained

$\frac{1}{\chi}\frac{d^{2}\chi}{dt^{2}}=i\frac{\varepsilon_{o}}{2m}(\mathbf{k}\cdot\mathbf{\hat{r}})\frac{1}{\eta}\frac{d\eta}{dr}+\frac{2Gm}{r^{3}}=-\mu\
,$ (12)

where $\mu$ is a separation constant to be determined. For a time-independent
model of DM, the equation to be solved is

$\frac{d\eta}{\eta}=-4i\frac{Gm^{2}}{\varepsilon_{o}}\frac{dr}{(\mathbf{k}\cdot\mathbf{\hat{r}})r^{3}}-2i\frac{m\mu}{\varepsilon_{o}}\frac{dr}{(\mathbf{k}\cdot\mathbf{\hat{r}})}\
,$ (13)

and the resulting solution is

$\eta(r)=\eta_{o}\exp{\frac{2i(\varepsilon_{g}-2\varepsilon_{k})}{\varepsilon_{o}(\mathbf{k}\cdot\mathbf{r})}}\
,$ (14)

where $\eta_{o}$ is an integration constant, $\varepsilon_{g}=Gm^{2}/r$ is the
gravitational potential energy of DM particles with mass $m$, and
$\varepsilon_{k}=mv^{2}/2$ is the kinetic energy of the particles with their
thermal velocity $v(r)$. The particle’s thermal velocity is determined by the
temperature of DM and rate of collisions between the particles in the DM halo.

Let $\Delta\varepsilon=\varepsilon_{g}-2\varepsilon_{k}$, and $\eta_{r}(r)$ be
the real part of the solution of $\eta(r)$ given by

$\eta_{r}(r)=\eta_{o}\cos\left(\frac{2\Delta\varepsilon}{\varepsilon_{o}(\mathbf{k}\cdot\mathbf{r})}\right)\
.$ (15)

The following quantum processes are allowed:

(i) $\Delta\varepsilon=0$ requires $\varepsilon_{g}=2\varepsilon_{k}$, which
means that

$\cos\left(\frac{2\Delta\varepsilon}{\varepsilon_{o}(\mathbf{k}\cdot\mathbf{r})}\right)=1\
,$ (16)

and that a pair of DM particles reaches its dynamical equilibrium, which is
established between the gravitational potential and kinetic energies of the
particles.

In the equlibrium, the velocities of both particles of the pair are the same
and given by $v_{e}=\sqrt{Gm/r_{e}}$, where the subscript ’e’ stands for the
equilibrium. Thus, the time $t_{e}$ required by a particle with its thermal
velocity $v_{e}$ to travel the distance $r_{e}$ is $t_{e}=r_{e}/v_{e}$. This
allows finding the separation constant to be $\mu=1/t_{e}^{2}$ or
$\mu=Gm/r^{3}_{e}$, whose value is fixed. Particles whose $v(r)\neq v_{e}$
travel in time $t_{e}$ the distance $r=v(r)t_{e}$.

(ii) $\Delta\varepsilon>0$ corresponds to emission of the quanta
$\varepsilon_{o}$ by the pair ($2\Delta\varepsilon$). However, the condition
for the emission process for one DM particle of the pair is

$\frac{\Delta\varepsilon}{\varepsilon_{o}(\mathbf{k}\cdot\mathbf{r})}=n\pi\ ,$
(17)

with $n$ = 1, 2, 3, …, as it guarantees that after this process takes place
the pair reaches its dynamical equlilibrium.

(iii) $\Delta\varepsilon<0$ represents absorption of the quanta
$\varepsilon_{o}$ by the pair, and the condition for this process to occur for
one DM particle of the pair is

$\frac{\Delta\varepsilon}{\varepsilon_{o}(\mathbf{k}\cdot\mathbf{r})}=-n\pi\
,$ (18)

because after this process the pair returns to its dynamical equlilibrium.

In general, the quantization rules for one DM particle of the pair can be
written as

$\Delta\varepsilon=\pm n\pi\varepsilon_{o}(\mathbf{k}\cdot\mathbf{r})\ ,$ (19)

where $n$ = 0, 1, 2, 3, …, with $n=0$ representing the dynamical equilibrium
of the pair.

A new result, when compared to QM, is the presence of the term
$(\mathbf{k}\cdot\mathbf{r})$, which shows that the emission and absorption
processes depend on the direction between the vectors $\mathbf{k}$ and
$\mathbf{r}$. The latter connects the particles of the pair but the former is
the label of the irreps of $\mathcal{G}_{e}$ and it represents the inverse of
the characteristic wavelength $\lambda_{o}$ associated with the quanta
$\varepsilon_{o}$. The wavelength is given by

$\lambda_{o}=\frac{Gm^{2}}{\varepsilon_{o}}\ ,$ (20)

with $\mathbf{k}=(1/\lambda_{o})\mathbf{\hat{k}}$, and plays the same role as
the Compton wavelength $\lambda_{c}=h/mc$ in QM. However, while $\lambda_{c}$
depends on the Planck constant $h$ and the speed of light $c$, the
characteristic wavelength for DM particles depends on the gravitational
constant $G$ and on the new constant $\varepsilon_{o}$, implying that
$\varepsilon_{o}$ may become important in quantization of gravity, instead of
the Planck constant.

##  5 Physical implications

In the developed quantum theory of cold DM, massive particles are confined to
a sphere (DM halo), and they form pairs bounded gravitationally. The particles
within the pairs may collide with other DM particles, changing the kinetic
energy of the colliding particles. A pair reaches its equlibrium when the
gravitational potential energy equals the particle’s kinetic energy. If the
gravitational energy exceeds the kinetic energy, a pair emits the quanta of
energy $\varepsilon_{o}$; in the opposite case, a pair absorbs the same
quanta. This shows that $\varepsilon_{o}$ plays the same role for DM as the
Planck constant does for OM.

However, the main difference between $\hbar$ and $\varepsilon_{o}$ is that
while the former is present in all quantum theories of OM that deal with
electromagnetic, weak and strong interactions, the latter appears in the
quantum theory of DM that deals with gravitational interaction, which is
treated here classically. From a physical point of view, the situation
resembles that known in QM, where the electron described by the Schrödinger
equation with the Coulomb potential requires photons to change orbits in the
atom; however, nonrelativistic QM does not deal with photons, which are
introduced by relativistic QM or quantum electrodynamics.

Similarly, in the presented quantum theory of DM, the emission and absorption
processes take place when the quanta $\varepsilon_{o}$ are exchanged. Since
the gravitational interaction is the only one considered in the theory, it is
suggested that the quanta $\varepsilon_{o}$ are callled here dark gravitons,
to emphasize that they are associated with DM, as gravitons are associated
with OM.

Dark gravitons may be abundant in DM halos. Such a large number of
$\varepsilon_{o}$ in the sea of dark gravitos may be responsible for
generating a gravitational wave background that would be specific for DM halos
and, therefore, observable by the gravitational wave detectors as clearly
distinct from the other proposed forms of stochastic gravitational wave
background [26].

Finally, it is interesting to consider the Compton wavelength $\lambda_{c}$ to
be of the same order as the characteristic wavelength $\lambda_{o}$ of the
quanta of $\varepsilon_{o}$ given by Eq. (20). More specifically, let
$\lambda_{c}=\lambda_{o}$, which gives

$m=\sqrt[3]{\frac{h\varepsilon_{o}}{Gc}}\ .$ (21)

This shows that the mass of DM particles is fully determined by all four
constants of Nature, with $\varepsilon_{o}$ being currently unknown. However,
if Eq. (21) is valid, then $\varepsilon_{o}$ would be known when the mass of
DM particles could be experimentally established. The four constants give also
the following characteristic length
$l_{c}=\sqrt[3]{Gh^{2}/(\varepsilon_{o}c^{2})}$.

##  6 Conclusions

A spherical DM halo filled with massive DM particles is considered. The
particles are allowed to form gravitationally bounded pairs, whose dynamical
stability can be impaired by particle collisions. By using a recently
discovered asymmetric equation for spinless, no charge, and free particles
[11], a quantum theory of cold DM is developed, showing that the dynamically
unstable pairs can regain their stability by emitting or absorbing the quanta
of energy $\varepsilon_{o}$. The quantum rules for these processes are
presented and the characteristic wavelength corresponding to $\varepsilon_{o}$
is obtained. Since the rules are independent from frequency, $\varepsilon_{o}$
is both a new constant of Nature as well as the quanta called dark gravitons
associated exclusively with DM. The abundance of $\varepsilon_{o}$ in DM halos
forms the dark gravitons sea, which generate a gravitational wave background
(GWB) that is specific for these halos and, therefore, potentially observable
by the gravitaional wave detectors, as clearly distinct from other proposed
forms of stochastic GWB [26].

Acknowledgment: The author thanks Dora Musielak for valuable comments on the
earlier version of this manuscript. This work was partially supported by
Alexander von Humboldt Foundation.

##  7 References

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