# Distant Relatives: The Chemical Homogeneity of Comoving Pairs Identified in
Gaia

Tyler Nelson Department of Astronomy, The University of Texas at Austin, 2515
Speedway Boulevard, Austin, TX 78712, USA  Yuan-Sen Ting Institute for
Advanced Study, Princeton, NJ 08540, USA  Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 08544, USA  Observatories of the
Carnegie Institution of Washington, 813 Santa Barbara Street, Pasadena, CA
91101, USA  Research School of Astronomy & Astrophysics, Australian National
University, Cotter Rd., Weston, ACT 2611, Australia  Keith Hawkins Department
of Astronomy, The University of Texas at Austin, 2515 Speedway Boulevard,
Austin, TX 78712, USA  Alexander Ji Observatories of the Carnegie Institution
of Washington, 813 Santa Barbara Street, Pasadena, CA 91101, USA  Harshil
Kamdar Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, 02138, USA
Kareem El-Badry  Department of Astronomy and Theoretical Astrophysics Center,
University of California Berkeley, Berkeley, CA 94720 Tyler Nelson
<EMAIL_ADDRESS>
(Accepted XX. Received YY; in original form ZZ)

###### Abstract

Comoving pairs, even at the separations of $\mathcal{O}(10^{6})\,$AU, are a
predicted reservoir of conatal stars. We present detailed chemical abundances
of 62 stars in 31 comoving pairs with separations of $10^{2}-10^{7}\,$AU and
3D velocity differences $<2\mathrm{\ km\ s^{-1}}$. This sample includes both
bound comoving pairs/wide binaries and unbound comoving pairs. Observations
were taken using the MIKE spectrograph on the Magellan/Clay Telescope at high
resolution ($\mathrm{R}\sim 45,000$) with a typical signal-to-noise ratio of
150 per pixel. With these spectra, we measure surface abundances for 24
elements, including Li, C, Na, Mg, Al, Si, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni,
Cu, Zn, Sr, Y, Zr, Ba, La, Nd, Eu. Taking iron as the representative element,
our sample of wide binaries is chemically homogeneous at the level of $0.05$
dex, which agrees with prior studies on wide binaries. Importantly, even
systems at separations $2\times 10^{5}-10^{7}\,$AU are homogeneous to $0.09$
dex, as opposed to the random pairs which have a dispersion of $0.23\,$dex.
Assuming a mixture model of the wide binaries and random pairs, we find that
$73\pm 22\%$ of the comoving pairs at separations $2\times 10^{5}-10^{7}\,$AU
are conatal. Our results imply that a much larger parameter space of phase
space may be used to find conatal stars, to study M-dwarfs, star cluster
evolution, exoplanets, chemical tagging, and beyond.

Stars: abundances, Stars: binaries, Stars: kinematics and dynamics, Stars:
late-type

††facilities: Magellan/Clay Telescope, Simbad††software: astropy (Astropy
Collaboration et al., 2013, 2018), NumPy (Harris et al., 2020), iPython (Perez
& Granger, 2007), Matplotlib (Hunter, 2007), Galpy (Bovy, 2015), SciPy
(Virtanen et al., 2020), Photutils (Bradley et al., 2020), BACCHUS (Masseron
et al., 2016), CarPy (Kelson, 2003), topcat (Taylor, 2005), iSpec (Blanco-
Cuaresma et al., 2014a)

##  1 Introduction

Stars which are born from the same gas, i.e., conatal, are critical in
astronomy. The conatal nature of open clusters and wide binaries have made
them indispensable laboratories for testing and improving our understanding of
various areas of Galactic and stellar astrophysics. Both open clusters and
wide binaries have been used to evaluate the current feasibility of chemical
tagging (De Silva et al., 2007; Ting et al., 2012; Ness et al., 2018; Andrews
et al., 2019; Hawkins et al., 2020b). Open clusters are used to calibrate
stellar parameter and chemical abundance pipelines for large surveys (e.g.,
García Pérez et al., 2016). Wide binaries can be used to calibrate the
metallicity of M-dwarfs (e.g., Lépine & Bongiorno, 2007; Rojas-Ayala et al.,
2010; Montes et al., 2018), constrain the age-magnetic activity relation
(e.g., Garcés et al., 2011; Booth et al., 2017), the age-metallicity relation
(e.g., Rebassa-Mansergas et al., 2016), and the initial-final mass relation
(e.g., Zhao et al., 2012; Andrews et al., 2015). Besides, we can also use wide
binaries to study exoplanet engulfment (e.g., Meléndez et al., 2017; Oh et
al., 2018).

Gravitationally bound binaries favor a conatal origin. For separations within
a few thousand AU, binaries are mostly formed via turbulent core fragmentation
(Offner et al., 2010; Lee et al., 2017). At separations up to $\sim
10^{5}\,$AU, wide binaries can be formed from dynamical unfolding of triple
systems (Reipurth & Mikkola, 2012), the evaporation of star clusters
(Kouwenhoven et al., 2010; Moeckel & Clarke, 2011), and chance gravitational
capture of prestellar cores (Tokovinin, 2017). The predicted conatal nature of
wide binaries motivated studies of their chemistry (Andrews et al., 2019;
Ramírez et al., 2019; Hawkins et al., 2020b). They have found wide binaries to
be chemically homogeneous to $\sim 0.02-0.1$ dex in
$\Delta\mathrm{[Fe/H]}$111Some systems have a larger metallicity difference
(e.g., $\Delta\mathrm{[Fe/H]}\sim 0.2$ dex, Oh et al., 2018). Nonetheless, the
studies of chemical homogeneity for binaries remain rather limited
($\mathcal{O}(10)$ pairs) as large surveys often do not observe both stars due
to the subsampling, and we have to resort to targeted studies.

Comoving pairs, on the other hand, are even less studied than wide binaries.
The investigation of detailed chemistry for comoving pairs remains largely
non-existent for separations beyond a few parsecs. Comoving pairs separated by
$\gtrsim 10^{5}\,$AU are no longer bound (Jiang & Tremaine, 2010). As outlined
in Oh et al. (2017), these unbound systems could arise from
disrupted/dissolved conatal systems (open clusters, stellar binaries) or
unrelated systems (chance alignments, gravitational resonance). Recently,
simulations from Kamdar et al. (2019a) argued comoving pairs have a high
probability of being conatal provided that their 3D velocity difference
($\Delta\mathrm{v_{3D}}$) is below 2 km s-1 at separations up to
$\mathcal{O}(10^{6})\,$AU. Their simulations predicted that pairs with
separations $\sim 10^{6}$ AU are $\sim 80\%$ conatal. If true, this could
dramatically expand the number of systems that could be conatal, which can
improve stellar atmospheric models, study exoplanet engulfment, and calibrate
surveys, among other applications. With the precise astrometry provided by the
recent data release of Gaia (Gaia Collaboration et al., 2018, 2020) sufficient
comoving systems can readily be located, and this study is set up to test this
proposition – are comoving stars with large separations conatal?

A key characteristic of conatal stars is their chemical composition’s
homogeneity; conatal stars are expected to have similar initial chemical
compositions as they are usually formed from well-mixed ISM gas (Feng &
Krumholz, 2014). Nonetheless, measured surface abundances can be changed by
internal processes (atomic diffusion, rotational mixing, dredge-up,
gravitational settling, radiative levitation, Schuler et al., 2011; Dotter et
al., 2017), model limitations (incomplete laboratory data, NLTE, 3D Ruchti et
al., 2013; Heiter et al., 2015; Nissen & Gustafsson, 2018; Jofré et al.,
2019), and methodological limitations (Jofré et al., 2014, 2019). All these
effects could complicate any interpretation of the chemical homogeneity of the
two stars.

To distinguish model systematic from astrophysical processes, in this study,
we will appeal to differential analysis of stellar twins – i.e., stellar pairs
with similar stellar parameters. Differential abundances can remove many of
these effects provided the stars are close in stellar parameters because the
systematic effects in the star of interest and reference star largely cancel
out (see Gray, 2008; Nissen & Gustafsson, 2018; Jofré et al., 2019).

In this study, we measure abundance differences in 24 elements for 33 comoving
pairs with $\Delta\mathrm{v_{3D}}<2\ \mathrm{km\ s^{-1}}$ and separations
between $\sim 10^{2}-10^{7}\,$AU. With access to homogeneous high resolution,
high signal-to-noise data, we uniquely positioned to explore the detailed
abundances from the wide binaries and the comoving stars and the chemical
homogeneity of the two components. In Section 2 we describe the observations.
Section 3 details the methods for estimating stellar parameters and
abundances. Our results are presented in Section 4, in which we explore the
conatal fraction of the comoving pairs. These results are discussed in Section
5 and summarized in Section 6.

##  2 Data Properties

Our target selection focused on comoving pairs of FG dwarfs within 300 pc of
the Solar neighborhood. FG dwarfs were selected because they are both
luminous, and their spectral models are best characterized. We create the
initial list of sources with the following ADQL query:

⬇

SELECT *

FROM gaiadr2.gaia_source

WHERE ra > 165

OR ra < 15

AND dec < 25

AND bp_rp BETWEEN 0.5 AND 1.5

AND 1000/parallax < 300

AND parallax_over_error > 5

AND phot_g_mean_mag IS NOT NULL

AND ((phot_g_mean_mag - 5*LOG10(1000/parallax) + 5) BETWEEN 2 AND 7)

AND radial_velocity IS NOT NULL

Our range of $\mathrm{bp\\_rp}$ was selected to exclude hot stars as some of
them could be rapid rotators with broad spectral features which complicate
spectral analysis. Next, to ensure that our systems are genuine (isolated)
comoving stars, we filter out members of open clusters and stellar
associations with a friend-of-friend (FoF) connectivity cut following Kamdar
et al. (2021), removing any pairs that belong to an aggregate with more than 4
FoF companions (with separations $<10\,$ pc, $\Delta{\rm v_{3D}}<5\,$ km s-1).
We verify the efficacy of this filtering by cross-matching our sample with the
open clusters and comoving groups catalogs from Kounkel & Covey (2019) and
Cantat-Gaudin & Anders (2020), finding no overlap. From this list of possible
sources, we calculate the 3D velocity separation $\Delta\mathrm{v_{3D}}$ and
spatial separation using Galpy (Bovy, 2015) using astrometry and radial
velocities measurements from Gaia DR2. The target selection prioritized stars
in similar locations on the color-magnitude diagram (CMD) to reduce the
influence of systematic errors on the derived elemental abundances. Together,
we consider the comoving pairs that satisfy the following criteria:

  * •

$|\Delta\mathrm{M}_{\rm G}|<0.5\mathrm{\ mag}$

  * •

$|\Delta(\mathrm{G_{BP}-G_{RP}})|<0.1\mathrm{\ mag}$

  * •

$\Delta\mathrm{v_{3D}}<2\mathrm{\ km\ s^{-1}}$

  * •

3D separation $<300\mathrm{\ pc}$

The remaining comoving pairs were binned in log separation and sorted by
apparent magnitude. The final sample was selected by taking the most luminous
$\sim 10$ pairs from each log separation bin.

Throughout this study, we will use the term comoving stars to signify both
bound and unbound pairs of stellar companions. In particular, for consistency,
we will use the words “close” comoving pairs and wide binaries interchangeably
to refer to (mostly bound) wide binaries and “far” comoving pairs to be the
primarily unbound systems. For simplicity, we define close comoving pairs as
those with 3D spatial separation of 1 pc (i.e., $2\times 10^{5}\,{\rm AU}$),
and the others far comoving pairs. This choice is partly motivated by the fact
that in simulations from Kamdar et al. (2019b), the minimum separation of
unbound comoving pairs have a separation of 1 pc. This is also consistent with
the previous study from Jiang & Tremaine (2010).

We observed 33 pairs of comoving stars using the MIKE spectrograph on the
_Magellan/Clay_ telescope (Bernstein et al., 2003) from June 13 to June 16,
2019. These 33 pairs constitute our “main sample”. We also observe two
additional pairs with $\Delta\mathrm{v_{3D}}>2\mathrm{\ km\ s^{-1}}$ as a
control sample. But unless otherwise stated, we will only refer to the main
sample throughout this study. Additionally, we also observed four Gaia
benchmark stars (Jofré et al., 2014; Blanco-Cuaresma et al., 2014b; Heiter et
al., 2015) dispersed over the sampled CMD to improve the data reduction. The
benchmark stars used were 18 Sco, bet Vir, HD 140283, and mu Ara. The
instrument employs a blue and red spectrograph to cover 3350–5000 Å and
4900–9500 Å respectively. We used the 0.5” slit with $2\times 1$ binning,
which gave the blue and red spectrographs typical resolving powers of 50,000
and 40,000, respectively. The median signal-to-noise ratio (SNR) per pixel for
the blue and red chips were 121 and 185. The observational details are given
in Table 1.

Our sample size of 33 pairs is comparable to previous detailed chemical
studies of wide binaries (Andrews et al., 2019; Ramírez et al., 2019; Hawkins
et al., 2020b), but expands the range of separations by two orders of
magnitude. Previous studies mostly restrict their sample to $\lesssim
10^{5}\,$AU, whereas in this study we have comoving pairs with separations up
to $10^{7}\,$AU. In the top panel of Figure 1 we show the CMD of our comoving
targets (white circles) with a random sample of 200,000 stars from Gaia DR2
plotted in the background for reference.

Among our sample, we found that, for two particular pairs, each of which had a
component that exhibit large deviations in the radial velocity measurements
from Gaia eDR3222Note that the radial velocity in eDR3 follows the one from
DR2 and our RV derived with the MIKE spectra. Both systems were flagged as
outliers based on the interquartile range (IQR) test333Consider a data
distribution, with $P_{i}$ denoting the ith percentile of said distribution. A
data point $D$ is labeled an outlier if $D$ is outside the interval $P_{50}\pm
1.5\times(P_{75}-P_{25})$, where $(P_{75}-P_{25})$ is the interquartile
range.. This deviation could indicate an unresolved companion star. We exclude
them from the primary sample because the unresolved binaries in hierarchical
triplets can bias metallicity estimates by 0.1 dex (El-Badry et al., 2018).
Excluding these pairs leaves us with 31 pairs of comoving stars.

Table 1: Abbreviated target list for the 33 comoving pairs investigated in this study. A complete machine-readable version is available online. For compactness, we only display a subset of the columns. Comoving pairs can be determined using the identifier column. The fourth and fifth columns are the parallax and parallax errors from Gaia eDR3. Gaia eDR3 ID | Identifier | RA | DEC | $\varpi$ | $\sigma_{\varpi}$ | G | BP - RP | RV | $\sigma_{\mathrm{RV}}$ | SNRBlue | …
---|---|---|---|---|---|---|---|---|---|---|---
|  | (o) | (o) | $\mathrm{(mas)}$ | $\mathrm{(mas)}$ | $\mathrm{(mag)}$ | $\mathrm{(mag)}$ | $\mathrm{(km\ s^{-1})}$ | $\mathrm{(km\ s^{-1})}$ |  | …
6160638833832941312 | CM00A | 191.8605 | -30.8986 | 9.931 | 0.017 | 4.75 | 0.84 | -12.50 | 0.05 | 86 | …
3471119180522314624 | CM00B | 189.8021 | -29.1721 | 9.613 | 0.014 | 4.77 | 0.83 | -11.23 | 0.06 | 75 | …
3613454637229633664 | CM01A | 206.5261 | -11.7535 | 4.092 | 0.015 | 3.45 | 0.79 | 22.96 | 0.03 | 93 | …
3613454637229634176 | CM01B | 206.5178 | -11.7507 | 4.085 | 0.016 | 3.05 | 0.81 | 22.17 | 0.03 | 120 | …
3639520621950395904 | CM02A | 215.6611 | -7.7688 | 23.742 | 0.025 | 4.16 | 0.74 | -32.62 | 0.03 | 209 | …
3639520621950395776 | CM02B | 215.6613 | -7.7704 | 23.802 | 0.024 | 4.22 | 0.74 | -33.26 | 0.03 | 167 | …
5897704951769034368 | CM03A | 218.3675 | -52.5846 | 5.880 | 0.017 | 3.47 | 0.84 | 43.98 | 0.03 | 114 | …
5897785667103983616 | CM03B | 217.2034 | -52.9914 | 5.645 | 0.018 | 3.71 | 0.80 | 44.24 | 0.03 | 94 | …
6224633983987510528 | CM04A | 225.8985 | -27.8432 | 19.433 | 0.027 | 4.70 | 0.79 | -15.77 | 0.03 | 131 | …
6224633983987511552 | CM04B | 225.9000 | -27.8416 | 19.413 | 0.029 | 4.68 | 0.78 | -14.99 | 0.03 | 145 | …
Figure 1: Top panel: The observed stars (white circles) are overplotted on
200,000 random stars from Gaia DR2 (background). Most of the samples are main-
sequence stars because giant-giant stellar twins are rare, and we focus on the
bright local sample. Bottom panel: Stellar parameter fits for the comoving
pairs from BACCHUS. Representative (median) error bars are given in the upper
right corner of the plot. For each pair, the two comoving components are
connected by a black line. Figure 2:  The upper panel shows continuum
normalized spectra of both components of two comoving pairs offset by a
constant with $\Delta$[Fe/H] of 0.15 dex and 0.03 dex for the upper and lower
set of spectra, respectively. Spectral lines which passed the initial quality
cut (see Section 3) are also shown in the upper panel. The lower panel shows
the difference between the components. Since we only target stellar twins with
$\Delta\mathrm{T_{eff}}\lesssim 100\ \mathrm{K}$ for both comoving pairs, the
differences in the spectra are primarily due to the element abundances of the
stars. Figure 3: The 2D projected pair separation plotted against the 3D pair
separation for both Gaia DR2 (blue) and eDR3 (orange). The square symbols
indicate pairs which are in the wide binary catalog from El-Badry et al.
(2021), and circles are used otherwise. All separations are calculated
directly from Gaia eDR3 astrometry. For a given pair, the grey arrow links the
DR2 data to the eDR3 calculation. The blue dashed line indicates our defined
boundary separating the close and far comoving pairs. For the far comoving
pairs, the parallax error is subdominant. As such, the 2D separations and 3D
separations line up. However, for close moving pairs, the parallax error
dominates over the separation calculation, and the calculated 3D separation is
much larger than the 2D projected separation. Since bound binaries are
expected to have smaller separations ($\lesssim 10^{5}\,{\rm AU}$), this
further validates that the parallax errors dominate for the close comoving
pairs. Due to this limitation, we impose a geometric prior (Appendix A) when
calculating the 3D separation for the close comoving pairs.

We reduce the raw MIKE data with CarPy (Kelson, 2003) which outputs
wavelength-calibrated multi-order echelle spectra for the science targets and
flats. After flat-fielding, the spectra were normalized by fitting a cubic
spline as a pseudo continuum for each order. We determine the pseudo continuum
by iterative sigma clipping. We discard 100 pixels on both ends of each order
because the blaze function is less well defined due to the lack of signal.
Orders are then combined using a flux error weighted average. The sigma
clipping process was tuned by comparing observations of four Gaia Benchmark
stars to the online library (Blanco-Cuaresma et al., 2014b). We employ iSpec
(Blanco-Cuaresma et al., 2014a) to determine the radial velocity (RV)
correction and remove cosmic rays.

The reduced spectra for two comoving pairs in the wavelength regime of
$5384\leq\lambda\leq 5400\mathrm{\ \AA}$ are shown in Figure 2. In the top
panel, we compare the spectra of both components for the two comoving pairs.
The bottom panel shows the difference in spectra between the comoving
components. Since we only target stellar twins, the difference predominantly
comes from the difference in elemental abundances. The two spectra exhibit
only minimal differences when they have similar chemistry. The careful
selection of stellar twins allows us to derive elemental abundances with high
fidelity for this homogeneous sample of comoving pairs.

###  2.1 Positions and Velocities

We calculate the positions and velocities for each star using Gaia eDR3
astrometry and spectroscopic RVs from our observations. Note that we made the
target selection with DR2 before eDR3 became available. However, since Gaia
eDR3 improves Gaia DR2 astrometry, we decide to use eDR3 astrometry for the
final determination and plots throughout this study. This update causes some
objects within our main sample to appear outside the $\Delta\mathrm{v_{3D}}<2\
\mathrm{km\ s^{-1}}$ cut, even though our our main sample are selected to be
within this boundary with the DR2 values.

We will consider two approaches to calculate the 3D separations between
members of comoving pairs. The first (“method 1”) uses the 6D phase space
information to calculate the 3D separation directly using a Monte Carlo
approach. We construct a multivariate normal distribution for each star with
mean of the reported values and a covariance matrix using the 5D measured
correlations in astrometry from Gaia eDR3 as well as the RV errors from the
iSpec RV fit. We sample from this distribution 10,000 times and apply a
transformation from equatorial to Galactic Cartesian coordinates to each
sample. Finally, we calculate the 3D separation and velocity difference of the
samples in Cartesian coordinates, taking the mean and standard deviation
across samples as the point estimate and uncertainty for these quantities.

This method does not invoke additional assumptions about the geometry or
separation distribution of pairs. However, for comoving pairs with 3D
separations, $\lesssim 10^{5}\,{\rm AU}$, the uncertainty in 3D separation is
often significantly inflated by parallax uncertainties, making it impossible
to accurately measure 3D separations using method 1. Consider two stars with
identical true parallax, $\varpi$, and parallax uncertainty,
$\sigma_{\varpi}$. In the limit of $\sigma_{\varpi}/\varpi\ll 1$, the typical
difference in apparent line-of-sight distance between the two stars is $\Delta
d_{{\rm apparent}}\sim\frac{1000\,{\rm pc}}{\left(\varpi/{\rm
mas}\right)}\times\sigma_{\varpi}/\varpi$. For typical pairs in our sample,
with $\varpi\sim 10\,\rm mas$ and $\sigma_{\varpi}\sim 0.02$ mas, this
translates to an apparent distance difference of $\sim 4\times 10^{4}$ AU,
which represents the amount by which the distance differences of typical pairs
will be inflated by parallax uncertainties. For the pairs in our sample with
the smallest $\varpi$ and largest $\sigma_{\varpi}$, this value is $\sim
3\times 10^{5}$ AU. This intuition also prompted us to divide the close
comoving pairs and the far comoving pairs with a 1pc (or $2\times 10^{5}\,$AU)
threshold.

It is expected that many of the closest pairs in the sample are
gravitationally bound wide binaries with true 3D separations smaller than this
distance “resolution” afforded by the Gaia parallaxes. Parallax uncertainties
effectively stretch these pairs out along the line of sight, in an effect that
is analogous to the “fingers of God” distortion in cosmological surveys, where
galaxy pairs and clusters are stretched out by redshift uncertainties. For
such pairs, the point estimate of the 3D separation from method 1 will
dramatically overestimate the true 3D separation. The projected separation
(which is not subject to inflation by parallax uncertainties) will be much
smaller than the point estimate of the 3D separation.

Figure 3 compares the 2D and apparent 3D pair separations for all 33 comoving
pairs. The 3D separations are calculated using method 1, as described above.
We also show the results separately using astrometry from DR2 and eDR3. At
large separations, where the parallax uncertainties are small compared to the
parallax differences, the 2D and 3D separations agree within a factor of a
few, and difference between the separations calculated using DR2 and eDR3
astrometry are minor. By contrast, pairs with 2D separations less than
$10^{5}$ AU fall far below the one-to-one line, with projected separations
that are much smaller than the calculated 3D separations. For a majority of
these pairs, the apparent 3D separation shrinks between DR2 and eDR3
astrometry. This reflects the fact that parallax uncertainties are smaller in
eDR3 data, causing the separation inflation to be less severe. Cross-matching
with El-Badry et al. (2021) (E21) shows that almost all of these pairs have a
high probability of being gravitationally bound wide binaries,444In
particular, 18 of the 21 pairs in our sample with projected 2D separations
below $10^{5}$ AU are in the E21 catalog; these all have a reported chance
alignment probability of less than 0.001. None of the pairs with 2D
separations above $10^{5}$ AU are in the catalog. The three pairs that have 2D
separations below $10^{5}$ AU (and a 3D separation of $\sim
10^{5}-10^{6}\,$AU) and are not in the catalog only narrowly fail the cuts on
parallax and/or proper motion consistency required for membership in the E21
catalog and thus may still be bound. suggesting that their 3D separations are
indeed over-estimated due to the parallax errors.

This suggests that the separations of the close comoving pairs are dominated
by parallax errors and that even Gaia eDR3 cannot “resolve” their line-of-
sight distance difference. Therefore, for close pairs, we instead derive
constraints on the 3D separation based on the projected separation, the
assumption of random viewing angles, and a prior on the 3D separation
distribution of wide binaries. This is describe in Appendix A. We refer to
this method of estimating the 3D separation, which only relies on the
projected 2D separation, as “method 2”. For method 2, the point estimate and
1$\sigma$ confidence interval on the 3D separation, $r_{\mathrm{3D}}$ are
$r_{\mathrm{3D}}=1.12_{-0.11}^{+0.49}\times r_{\mathrm{2D}}$, where
$r_{\mathrm{2D}}$ is the projected separation. Motivation for this expression
is given in Appendix A.

With different separation estimators for close pairs (thought to be binaries)
and far pairs (thought to be unbound), it is necessary to decide where to draw
the line between the two regimes. For simplicity, we use method 2 on all pairs
for which the 3D separations calculated using method 1 are less than $2\times
10^{5}$ AU (dashed blue line in Figure 3). It is possible that a few pairs
wider than this (i.e. with $10^{5}-10^{6}\,$AU), which also have projected
separations significantly smaller than their calculated 3D separations, are
also binaries. Improved parallax uncertainties in future Gaia data releases
will help determine whether they are bound. But we have checked that
separating the bound and unbound pairs with a $10^{6}\,$AU threshold leaves
most of our qualitative results unaltered.

As most studies usually operate at the projected space instead of the 3D space
adopted in this study, when comparing with the literature values (Ramírez et
al. (2019), and Hawkins et al. (2020b)), we calculate the 3D separations and
velocity differences of their targets with the same process as detailed above.
We adopt the latest Gaia eDR3 astrometry and the RV from the literature. RV
data was not provided in Ramírez et al. (2019). Hence we used Gaia eDR3 RVs
for the Ramírez et al. (2019) data set.

Table 2: The 3D separations ($\mathrm{S}$) and velocity differences ($\Delta\mathrm{v_{3D}}$) between comoving pairs. A full machine-readable version is available online. The subscript eDR3 indicates separations calculated solely from Gaia eDR3 astrometry and the spectroscopic RV. As discussed in Section 2.1, the separations calculated this way may overestimate the true separation between the close comoving components. For the subset of comoving pairs suspected to be wide binaries, we invoke a geometric prior based to infer the 3D separations based on the better measured 2D projected separations (Appendix A). We only perform this correction for the close comoving pairs. These separations are denoted with a subscript “geometric”. The uncertainty for the geometric separations is asymmetric, so the upper and lower error bars (16 and 84 percentiles) for this separation are listed separately. The results in this paper assumes $\mathrm{S_{geometric}}$ for close comoving pairs (i.e., $\mathrm{S_{eDR3}}<2\times 10^{5}$ AU) and $\mathrm{S_{eDR3}}$ for the far comoving pairs.  Gaia eDR3 ID for | Gaia eDR3 ID for | Identifier | $\mathrm{S_{eDR3}}$ | $\sigma_{\mathrm{S_{eDR3}}}$ | $\Delta\mathrm{v_{3D}}$ | $\sigma_{\Delta\mathrm{v_{3D}}}$ | $\mathrm{S_{geometric}}$ | $\sigma_{\mathrm{lower}_{\mathrm{S_{geometric}}}}$ | $\sigma_{\mathrm{upper}_{\mathrm{S_{geometric}}}}$
---|---|---|---|---|---|---|---|---|---
Component A | Component B |  | (AU) | (AU) | (km s-1) | (km s-1) | (AU) | (AU) | (AU)
6160638833832941312 | 3471119180522314624 | CM00 | 1.1 x 106 | 2.7 x 104 | 0.86 | 0.05 | – | – | –
3613454637229633664 | 3613454637229634176 | CM01 | 9.0 x 104 | 1.8 x 105 | 1.02 | 0.08 | 8.5 x 103 | 8.3 x 102 | 3.7 x 103
3639520621950395904 | 3639520621950395776 | CM02 | 2.2 x 104 | 9.3 x 103 | 1.29 | 0.04 | 2.8 x 102 | 27 | 1.2 x 102
5897704951769034368 | 5897785667103983616 | CM03 | 1.5 x 106 | 1.0 x 105 | 0.93 | 0.05 | – | – | –
6224633983987510528 | 6224633983987511552 | CM04 | 1.1 x 104 | 1.6 x 104 | 1.71 | 0.08 | 4.2 x 102 | 41 | 1.9 x 102
6004256909931771136 | 6004256802548180736 | CM05 | 1.2 x 103 | 8.6 x 103 | 0.82 | 0.06 | 8.2 x 102 | 80 | 3.6 x 102
5877155289930033024 | 5877155289930029184 | CM06 | 6.5 x 104 | 6.7 x 104 | 0.70 | 0.06 | 3.7 x 103 | 3.6 x 102 | 1.6 x 103
5798991008295109120 | 5798991008295120896 | CM07 | 3.2 x 105 | 7.1 x 104 | 1.43 | 0.07 | – | – | –
6214634887804201728 | 6214634887804233088 | CM08 | 6.4 x 103 | 4.8 x 104 | 0.58 | 0.04 | 5.0 x 102 | 49 | 2.2 x 102
6208919660724640896 | 6208919660723526272 | CM09 | 8.1 x 104 | 3.2 x 104 | 0.79 | 0.06 | 8.3 x 102 | 81 | 3.6 x 102

##  3 Stellar Parameters and Abundances

We use the Brussels Automatic Code for Characterizing High accUracy Spectra
(BACCHUS, Masseron et al., 2016) to determine stellar parameters (i.e., Teff,
log $g$, [Fe/H], vmicro) and abundances. An overview of the software is
provided below. For a more comprehensive description we refer the reader to
Section 2.2 of Hawkins et al. (2015), Masseron et al. (2016), and Appendix
A.12 of Smiljanic et al. (2014).

BACCHUS fits observations using spectral synthesis. These spectra were
constructed from MARCS model atmospheres (Gustafsson et al., 2008) using
TURBOSPECTRUM (Plez, 2012) for radiative transfer. MARCS models are calculated
in 1D LTE. If the surface gravity (log $g$) is $\geq 3.0$ dex, plane-parallel
models are used, and spherical models otherwise. The atmospheric composition
uses solar abundance from Grevesse et al. (2007) scaled by metallicity for
most elements. MARCs models use separate abundance estimates for C, N, and O
(see Gustafsson et al., 2008, Section 4 for more information)555Some MARCs
models take into account the Galactic trends in [$\alpha$/Fe] vs [Fe/H]. These
models are used for $\mathrm{[Fe/H]}\leq-1$ dex. However, over the metallicity
range of $0\pm 0.3$ dex spanned by this sample, such effect is minimal. We
thus assume the solar scaling for the $\alpha$-captured elements for this
study.. We use Gaia-ESO line list version 5 (Heiter et al., 2019) for atomic
transitions. The line list includes hyperfine structure splitting for Sc I, V
I, Mn I, Co I, Cu I, Ba II, Eu II, La II, Pr II, Nd II, Sm II. We also include
molecular data for CH (Masseron et al., 2014), C2, CN, OH, MgH (T. Masseron,
private communication), SiH (Kurucz, 1992), TiO, FeH, and ZrO (B. Pelz private
communication). Fe ionization-excitation equilibrium is used in BACCHUS to
derive effective temperature (Teff), surface gravity, iron abundance ([Fe/H]),
and microturbulent velocity (vmicro). Other broadening sources (e.g.,
rotation, instrument resolution, macroturbulence) are modeled by a Gaussian
convolution.

Stellar parameters and the convolution kernel size are derived in BACCHUS
using the param module. We first optimize for the convolution. The we
determine Teff by requiring a null trend,within 1$\sigma$ of the fit, in Fe I
abundance versus excitation potential. We determine log $g$ by enforcing the
abundances from Fe I and Fe II lines to agree within the line-by-line rms and
vmicro by enforcing that there is no trend in iron abundance to reduced
equivalent width (i.e., EW/$\lambda$). The metallicity is taken as the mean of
the Fe I lines relative to the Sun. The parameter fitting uses 94 Fe I lines
and 32 Fe II lines, and the details can be found in the full version of Table
3. The reported errors in Teff, log $g$, and vmicro are calculated through the
usual propagation of error from the determination of EW of individual lines.
We note that, currently, BACCHUS does not estimate the covariances between
stellar parameters.

After optimizing the stellar parameters, abundances for 24 elements are
measured in BACCHUS with the abund module. We measure abundances for the
following elements: Li, C, Na, Mg, Al, Si, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni,
Cu, Zn, Sr, Y, Zr, Ba, La, Nd, and Eu. The abund module synthesizes spectra at
different [X/H], fixing the atmospheric model parameters to the best fitting
stellar parameters. This model is generated through interpolation of nearby
MARCs atmospheres; we determine the abundance for each line by minimizing the
unweighted $\chi^{2}$ of the observed and synthesized spectrum, and we repeat
this process for all selected absorption lines for that species. The line list
selection is given in Table 3. We select lines based on visual inspection of
the synthesis compared with observed spectra for three stars, CM07A, CM11A,
and CM25A. We select these stars because they are representative for different
atmospheric parameters and rotational broadening. On top of that, for every
star in our sample, selected lines must pass an internal quality check to be
considered usable for that object. The internal quality check corresponds to a
decision tree regarding whether the abundance output is physically plausible
and constrained by the trial solutions (see Section 2.2 of Hawkins et al.,
2015, for more details on the internal quality check). As a consequence of the
internal quality check, the lines with measured abundances may vary between
pairs. But we emphasize that for each pair, we adopt the same set of lines for
the differential analysis.

Table 3: A portion of our line list selection. A full machine-readable version, including abundances for each star for each line, is available online. The lines used will vary between stars because of the quality checks. $\chi$ is the excitation potential, and log($A_{\chi}$) is the absolute abundance (before subtracting the Solar abundances) derived for this line. The solar abundances adopted are from Grevesse et al. (2007), except where described otherwise in Section 3. Reference keys can be matched with the full citation in Table 7 to determine the origin of the log $gf$ values used. Identifier | Element | $\lambda$ | $\log gf$ | Reference Key | $\chi$ | $\log(A_{X})$
---|---|---|---|---|---|---
|  | ($\mathrm{\AA}$) | (dex) |  | (eV) | (dex)
CM00A | Na I | 5682.630 | -0.706 | GESMCHF | 2.102 | 6.053
CM00A | Na I | 5688.200 | -0.404 | GESMCHF | 2.104 | 6.291
CM00A | Na I | 6154.220 | -1.547 | GESMCHF | 2.102 | 6.181
CM00A | Na I | 6160.740 | -1.246 | GESMCHF | 2.104 | 6.196
CM00B | Na I | 5682.630 | -0.706 | GESMCHF | 2.102 | 6.103
CM00B | Na I | 5688.200 | -0.404 | GESMCHF | 2.104 | 6.311
CM00B | Na I | 6154.220 | -1.547 | GESMCHF | 2.102 | 6.212
CM00B | Na I | 6160.740 | -1.246 | GESMCHF | 2.104 | 6.233
CM01A | Na I | 4751.820 | -2.078 | GESMCHF | 2.104 | 6.558
CM01A | Na I | 5682.630 | -0.706 | GESMCHF | 2.102 | 6.528

We study the abundance differences using both line-by-line (differential) and
non-line-by-line (nLBL) “global” methods. Both methods provide similar
conclusions; however, the differential method generally leads to better
precision because it mitigates systematic errors in the spectral models. We
will adopt the abundances derived with the differential method throughout this
study. But for completeness, we include results from both approaches in Table
4. For clarity, we denote abundances for individual lines with a subscript i.
For example, [Xi/H] indicates an abundance of species X for line i,
$\Delta$[Xi/H] is the difference of species X using the abundance of line i
between two stars. Species without subscripts are the overall estimate of the
abundance from all corresponding spectral features.

In the differential approach, we compute $\Delta$[Xi/H] for each transition
observed in both stars. The abundance difference is taken as the median of
these values (i.e., $\Delta$[X/H] = median($\Delta$[Xi/H])). The median is
used instead of the mean because it is more robust against outliers. As for
the certainty estimation, we assume the statistical errors for $\Delta$[X/H],
i.e., std($\Delta$[Xi/H])/$\sqrt{{\rm number\;of\;lines}}$, as the standard
error of the mean of the line abundance differences. However, if a single line
remains after quality cuts, we take the error as 0.1 dex. This is 1/3 the step
size used between trial values in [X/H] when calculating the abundance for
each line.

Abundance errors from the uncertainty in Teff, log $g$, and vmicro are
propagated according to Hawkins et al. (2020b). For each parameter, we perturb
the best fit model and derive abundances for this perturbed model atmosphere.
The difference between the abundances from the best fit and perturbed model is
the error introduced from that parameter. These abundance errors are added in
quadrature with the line-by-line statistical errors for $\Delta$[X/H] to
determine the total error for an abundance measurement. For simplicity, and
due to the BACCHUS code’s limitations, we choose to omit the covariances
following Hawkins et al. (2020b).

We perform the error analysis on a subset of 30 representative stars
distributed evenly over the stellar parameters of our data set. The offsets
for Teff, log $g$, and vmicro were $\pm 20$ K, $\pm 0.1$ dex, and $\pm 0.03$
km s-1, respectively. The values for Teff and vmicro are the median $1\sigma$
errors reported by BACCHUS for our sample. We assume a $\Delta\,$log $g$= 0.1
dex instead of median reported errors from BACCHUS (0.20 dex) because we find
that BACCHUS tend to overestimate the errors of log $g$ due to some outlying
Fe II lines. We have separately analyzed these spectra with Bayesian stellar
parameter code LoneStar (Nelson et al in prep), which agrees with BACCHUS
estimates on all quantities besides the log $g$ errors, which are found to be
$\sim 0.05$ dex. We chose log $g$ errors of 0.10 dex to be conservative since
BACCHUS favors larger log $g$ errors. We note however that our comparison
between the close comoving pairs, far comoving pairs, and random pairs, is
robust and independent of our error estimates.

When we estimate the errors for individual observations, we adopt
interpolation by inverse distance weighting (IDW) to estimate the abundance
errors resulting from the uncertainties in the stellar parameters for the
remainder of the data set. IDW interpolation approximates points outside the
reference set by constructing a weighted average of points from the reference
sets. These weights are the inverse rms distance. During the interpolation, we
transform the stellar parameters onto the interval (-1,1) to weigh each
equally. We tested the accuracy of the interpolation using a leave-one-out
cross-validation for each element. The average error introduced was typical
$<2\%$ of the true errors calculated for that particular star. Sr and Eu were
the only two elements with larger relative errors at $\lesssim 5\%$.
Nonetheless, these errors are at least one order of magnitude below the total
error budget for each element and are therefore negligible.

Table 4 summarizes the derived stellar parameters and abundance measurements
and uncertainties in this study. In addition to the total abundance errors, we
also report the errors from the individual errors contributed by the stellar
parameter error propagation and the line-to-line scatter of $\Delta$[X/H],
respectively. The stellar parameters are the main source of uncertainty. For
23 elements, the stellar parameters account for $>90\%$ of total error. For
Sr, the stellar parameters accounting for $\sim 70\%$ of the total errors. Of
the remaining species, Li and Eu typically only have a single line measured
and are therefore subjected to the 0.1 dex error floor on the abundance
scatter. Only in these rare cases, the “line-to-line” error is more dominant
than the stellar parameters’ uncertainty. Finally, for completeness, although
not shown, the nLBL abundances and errors are also provided in the full
machine-readable table.

##  4 Results

Table 4: Stellar parameters and elemental abundances derived from BACCHUS. A and B subscripts denote the two components from the comoving pairs. $\xi$ denotes the microturbulence. The electronic table contains measurements from nLBL and differential methods. All the differential measurements (e.g., $\Delta\mathrm{[Fe/H]},\sigma_{\Delta\mathrm{[Fe/H]}}$) are measured with differential method. But we also provide individual estimates of each component with the nLBL method, indicated with the subscripts A and B. By definition, the differential analysis only applies to the differences of the two components, and there will be no individual values with the differential method. The full table also separately lists the errors from differential stellar parameters (denoted with a subscript $\theta$) and the statistical line-to-line scatter (denoted with a subscript $\mathrm{line}$), but below we only show the total errors (adding in quadrature). $\Delta\mathrm{RV}$ refers to the difference in radial velocity measured from our observations and Gaia eDR3. The large $\Delta\mathrm{RV}$ column indicates whether the system was flagged for potentially being a hierarchical triplet with an unresolved binary. Component A ID | Component B ID | Identifier | Large $\Delta\mathrm{RV}$ | $\mathrm{T_{eff_{A}}}$ | $\sigma_{\mathrm{T_{eff_{A}}}}$ | $\mathrm{T_{eff_{B}}}$ | $\sigma_{\mathrm{T_{eff_{B}}}}$ | $\log g_{\mathrm{A}}$ | $\sigma_{\log g_{\mathrm{A}}}$ | $\log g_{\mathrm{B}}$ | $\sigma_{\log g_{\mathrm{B}}}$ | $\xi_{\mathrm{A}}$ |  | $\xi_{\mathrm{B}}$ | $\sigma_{\xi_{\mathrm{B}}}$ | $\Delta\mathrm{[Fe/H]}$ | $\sigma_{\Delta\mathrm{[Fe/H]_{Total}}}$ | $\Delta\mathrm{[Na/H]}$ | $\sigma_{\Delta\mathrm{[Na/H]_{Total}}}$ | $\Delta\mathrm{[Mg/H]}$ | $\sigma_{\Delta\mathrm{[Mg/H]_{Total}}}$ | $\Delta\mathrm{[Al/H]}$ | $\sigma_{\Delta\mathrm{[Al/H]_{Total}}}$ | $\Delta\mathrm{[Si/H]}$ | $\sigma_{\Delta\mathrm{[Si/H]_{Total}}}$ |  | $\sigma_{\Delta\mathrm{[Ca/H]_{Total}}}$ | $\Delta\mathrm{[Sc/H]}$ | $\sigma_{\Delta\mathrm{[Sc/H]_{Total}}}$ | $\Delta\mathrm{[Ti/H]}$ | $\sigma_{\Delta\mathrm{[Ti/H]_{Total}}}$ | $\Delta\mathrm{[V/H]}$ | $\sigma_{\Delta\mathrm{[V/H]_{Total}}}$ | $\Delta\mathrm{[Cr/H]}$ | $\sigma_{\Delta\mathrm{[Cr/H]_{Total}}}$ | …
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (K) | (K) | (K) | (K) | (dex) | (dex) | (dex) | (dex) | (km s-1) |  | (km s-1) | (km s-1) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) |  | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex) | …
6160638833832941312 | 3471119180522314624 | CM00 | False | 5809 | 9 | 5934 | 65 | 4.53 | 0.30 | 4.65 | 0.41 | 1.22 |  | 1.31 | 0.05 | 0.04 | 0.02 | 0.03 | 0.02 | 0.02 | 0.02 | -0.02 | 0.02 | 0.01 | 0.02 |  | 0.05 | 0.04 | 0.06 | 0.07 | 0.03 | 0.09 | 0.04 | 0.05 | 0.03 | …
3613454637229633664 | 3613454637229634176 | CM01 | False | 6058 | 25 | 6002 | 56 | 4.29 | 0.16 | 4.14 | 0.28 | 1.30 |  | 1.33 | 0.04 | 0.02 | 0.02 | 0.05 | 0.02 | 0.03 | 0.02 | 0.03 | 0.01 | 0.01 | 0.02 |  | 0.03 | 0.02 | 0.05 | 0.02 | 0.03 | 0.01 | 0.03 | 0.06 | 0.03 | …
3639520621950395904 | 3639520621950395776 | CM02 | False | 5962 | 11 | 5966 | 11 | 4.38 | 0.22 | 4.41 | 0.23 | 1.06 |  | 1.05 | 0.04 | 0.00 | 0.02 | 0.00 | 0.02 | 0.01 | 0.02 | -0.02 | 0.01 | 0.01 | 0.02 |  | 0.04 | 0.01 | 0.06 | -0.00 | 0.03 | -0.01 | 0.04 | 0.01 | 0.03 | …
5897704951769034368 | 5897785667103983616 | CM03 | False | 6036 | 37 | 6121 | 28 | 4.34 | 0.22 | 4.28 | 0.21 | 1.31 |  | 1.34 | 0.04 | -0.08 | 0.02 | -0.11 | 0.02 | -0.06 | 0.02 | -0.12 | 0.02 | -0.08 | 0.02 |  | 0.03 | -0.10 | 0.05 | -0.07 | 0.03 | -0.09 | 0.04 | -0.09 | 0.03 | …
6224633983987510528 | 6224633983987511552 | CM04 | False | 5763 | 26 | 5822 | 11 | 4.51 | 0.14 | 4.50 | 0.20 | 0.90 |  | 0.99 | 0.04 | -0.01 | 0.02 | -0.00 | 0.03 | 0.01 | 0.01 | 0.00 | 0.01 | -0.03 | 0.03 |  | 0.05 | -0.04 | 0.07 | 0.00 | 0.03 | 0.05 | 0.04 | 0.03 | 0.04 | …
Figure 4: A comparison of the comoving pairs (blue) with similar works on wide
binaries. Separation errors for pairs with separations $>10^{6}\mathrm{\ AU}$
are essentially negligible and smaller than the symbol size. Our close
comoving pairs (i.e., wide binaries) have a similar degree of chemical
homogeneity to the literature values. Far comoving pairs (pairs with
separations $>2\times 10^{5}\mathrm{\ AU}$) are also chemically homogeneous in
[Fe/H], with a scatter of $<0.1\,{\rm dex}$, albeit being slightly less
homogeneous than close comoving pairs.

As discussed in the introduction, this paper’s main goal is to assess whether
or not the comoving pairs in this study, especially for the far comoving
pairs, are conatal systems. Since conatalitly cannot be assessed directly
outside star-forming regions, we resort to studying the chemical homogeneity
(i.e., $\Delta\mathrm{[X/H]}$). Figure 4 displays the difference in
metallicity between comoving pairs (blue) as a function of pair separation.
Although not shown, similar trends are seen for the majority of the elements
studied.

For comparison, we also include the wide binaries results from Ramírez et al.
(2019), and Hawkins et al. (2020b). We note that Ramírez et al. (2019)
includes results for 12 systems, 11 of which are from other studies (i.e.,
Ramírez et al., 2010; Liu et al., 2014; Mack et al., 2014; Ramírez et al.,
2014; Tucci Maia et al., 2014; Teske et al., 2016; Saffe et al., 2016; Mack et
al., 2016; Saffe et al., 2017; Reggiani & Meléndez, 2018; Oh et al., 2018). In
the remainder of the paper, we will exclude the system from Mack et al. (2014)
from comparisons because it appears to be a hierarchical triplet with an
unresolved binary, and use Ramírez et al. (2019) as a shorthand for the
remaining 11 systems. We note that since the sample Ramírez et al. (2019) is a
heterogeneous sample, the comparison with Hawkins et al. (2020b) might be more
relevant. Furthermore, the samples in Hawkins et al. (2020b) have spectra with
spectra and were analyzed in the same way as this study. We also compare our
data to Andrews et al. (2019); however, they did not provide individual
measurements, so our comparison with this work is limited to summary
statistics.

As previously discussed, we split our comoving sample into the close (i.e.,
primarily wide binaries) and far comoving pairs to compare with these works.
Close comoving pairs are defined as those having separations below 1 pc, i.e.
$2\times 10^{5}$ AU, (with 17 pairs in this study), and far comoving pairs
have separations above 1 pc (14 pairs). For the close pairs, we use the
geometric prior when calculating the separations (see Section 2.1) because the
parallax error dominates. Our close comoving pairs sample (i.e., wide
binaries) show a similar degree of chemical homogeneity as other studies on
wide binaries with a dispersion of 0.05 dex in $\Delta$[Fe/H], which is
expected. Compared to the close comoving pairs, we find that the far comoving
pairs are less chemically homogeneous, with a $\Delta$[Fe/H] dispersion of
0.08 dex; however, the far comoving pairs are substantially more homogeneous
than the random pairs (0.23 dex), which we will further explore in the next
section. Table 5 summarizes the scatter in $\Delta$[Fe/H] comparing our data
with prior works.

| median $|\Delta[\mathrm{Fe/H}]|$ (dex) | std $\Delta[\mathrm{Fe/H}]$ (dex)
---|---|---
Wide Binaries |  |
Hawkins et al. (2020b) | 0.02 | 0.05
Ramírez et al. (2019) | 0.04 | 0.09
Andrews et al. (2019) | - | 0.04
close comoving pairs | 0.03 | 0.05
Unbound pairs |  |
far comoving pairs | 0.05 | 0.08
random pairs | 0.16 | 0.23
Table 5: A comparison of the $\Delta\mathrm{[Fe/H]}$ scatter for our study and
three other studies on wide binaries. Andrews et al. (2019) only provides
standard deviation without giving the individual measurements, which prohibits
us from calculating the median of the absolute difference. Andrews et al.
(2019) provides two values for the scatter in $\Delta[\mathrm{Fe/H}]$, one for
the entire sample (0.08 dex) and the other (0.04 dex) for wide binaries with
$\Delta$Teff$<100$ K. We only state the latter as it is more relevant to this
study. We caution that Ramírez et al. (2019) is a heterogeneous sample of
several studies. As such, it could incur a large scatter due to different
systematics from these studies, the comparison to the other studies might be
less direct.

###  4.1 Chemical Homogeneity of Comoving Pairs

To contextualize how homogeneous our comoving population is, we simulate pairs
of random field stars. In a similar fashion to Hawkins et al. (2020b), we
create a set of random pairs by assigning every star to another star, which is
not its comoving partner. Since the comoving stars are selected to be stellar
twins, for better comparison, we create random pairs which minimize the
difference in Teff and log $g$ (labeled Teff/log $g$ pairs) to mitigate
systematic effects. Since Teff and log $g$ have different units, when
minimizing the difference, we divide both quantities by the standard deviation
of the pairwise difference for all non-comoving pairs of stars in our sample.
The random pairs are shown in Figure 5. Across all elements, the average
dispersion in $\Delta$[X/H] for the random pairs is 0.24 dex, significantly
larger than the dispersion of both the close comoving pairs (0.05 dex) and the
far comoving pairs (0.10 dex).

Figure 5: The stellar parameters of the simulated random field pairs in this
study. To simulate unrelated pairs of field stars and compare that to the
comoving pairs, we assign a partner based on the nearest star in stellar
parameter space, which is not its comoving counterpart. The black lines
connect paired stars by minimizing the distance in Teff and log $g$. A single
star may be paired with multiple partners. This was allowed because we want to
prevent pathological assignments where two stars are paired over large
distances because all intermediates are unavailable, artificially inflating
systematic uncertainties. Figure 6: The distribution of difference in
elemental abundances normalized by their uncertainties. The comoving pairs are
divided into three groups (the entire comoving sample; violet thin dashed
line), close (violet thick solid line), and far (violet dashed line, medium
width). The thin black line indicates an expected distribution if the scatter
in abundance differences between comoving pairs were explained by the
measurement uncertainties. The solid teal line indicates the distribution for
the random pairs. The close comoving pairs are more chemically homogeneous
than the random groups and generally approaches a distribution consistent with
the observed spread being only a result of measurement errors. The far
comoving pairs are less chemically homogeneous than the close comoving pairs.
However, they are generally more chemically homogeneous than the random pairs,
indicating a mixed population of conatal pairs and chance alignments.

To better illustrate the difference in chemical homogeneity between these
groups, similar to Andrews et al. (2019), we visualize the homogeneity of our
sample using $|\Delta\mathrm{[X/H]}|/\sigma_{\Delta\mathrm{[X/H}]}$, which
will be abbreviated as $\Delta/\sigma$. The $\Delta$ here is the difference in
elemental abundances between the two components, and the $\sigma$ is the
quadrature sum of the uncertainties from the two components. Figure 6 shows
the cumulative distribution function (CDF) of $\Delta$[X/H] normalized by the
measurement error for the comoving pairs (violet/solid and dashed) and random
pairs (teal/solid). For reference, each panel includes the CDF of
$\Delta/\sigma$ drawn from the unit Gaussian – i.e., what a chemical
homogeneous would look like at the level of our current measurement
uncertainties (assuming that our uncertainty estimations are accurate).

There is a clear difference in the random pairs’ distribution and comoving
pairs, both close and far, largely attributed to the larger chemical
dispersion seen in the random pairs. The difference in the CDF between the
comoving and random pairs also varies between elements. This variation is
primarily caused by the difference in our measurement precision for those
elements. On the one hand, for better-measured elements, such as Fe, Mg, Al,
and Si, the comoving pairs exhibit a more distinct chemical homogeneity than
the random pairs. On the other hand, unsurprisingly, for less well-measured
elements, the difference in chemical homogeneity is less distinct, as shown in
the CDFs for La, Nd, and Eu. We do not include Li in this plot because its
surface abundance has a strong dependence on effective temperature and is
therefore not a useful indicator for conatality (see Gray, 2008; Ramírez et
al., 2012; Hawkins et al., 2020b).

For most elements, the close comoving pairs are consistent with a chemically
homogeneous distribution at our measurement precision, with the CDF for 16
elements exceeding 96% by $\Delta/\sigma=4$. In contrast, the random pairs are
less homogeneous, with the CDF for 13 elements $\lesssim 50\%$ at
$\Delta/\sigma=4$.

As already seen in Fig. 4, the far comoving pairs are more heterogeneous than
the close comoving pairs for most elements, albeit still being more
homogeneous than the random pairs. The slightly larger chemical inhomogeneity
is unsurprising because we expect some of the far comoving pairs might be
contaminated by chance alignment pairs (e.g., Kamdar et al., 2019a), which we
will quantify in the next section.

In this study, we focus on the abundance [X/H] instead of the abundance ratio
[X/Fe]. However, all elements correlate with iron in [X/H]. Therefore, the
difference between random and close comoving pairs in $\Delta/\sigma$ for
[X/H] may only reflect a metallicity difference. While [X/H] (or metallicity
alone) are tell-tale signs of conatality (or at least co-eval), studying the
abundance ratio [X/Fe] might, in principle, constitute a more stringent test
and might reveal subtle physics. Although not shown, we tried to examine
$\Delta/\sigma$ for [X/Fe]666For this test, we minimized the difference in
Teff and [Fe/H] when simulating the random field pairs to find pairs of stars
with the same metallicity.. We find little difference between the comoving and
random pairs of stars in the [X/Fe] space. This is not unexpected because the
residual variance in [X/Fe] is much more subtle than in [X/H]. For example,
Ting & Weinberg (2021) shows that when one subtracts the mean chemical track,
the residual correlation in [X/Fe] is small, with signal $\lesssim 0.01-0.02$
dex. Therefore, to see such a signal, we would need to measure [X/Fe] better
than $\sim 0.01\,$dex. With the current pipeline adopted in this study, we
found that achieving 0.01 dex remains challenging. Therefore, we only focus on
[X/H]. Future studies with even higher quality spectra and/or better analysis
will be needed to shed light on these subtle variations.

###  4.2 Conatal Fraction for Separations $>2\times 10^{5}\,$AU

The close comoving pairs (wide binaries) at separations below $2\times
10^{5}\,{\rm AU}$ are commonly believed to be conatal as chance capture is
unlikely. However, beyond such separation, the conatality for the far comoving
pairs remains unexplored. Armed with our high-resolution spectroscopic data
from these pairs, in this section, we will constrain how conatal the far
comoving pairs with separations $>2\times 10^{5}\,$AU are relative to the
close comoving pairs. Showing the conatality of far-comoving pairs can have
far-reaching consequences; if these comoving pairs are chemically homogeneous
and conatal, this would significantly expand the sample of calibrators beyond
open clusters and wide binaries.

As we have seen in the previous section, these far comoving pairs are more
heterogeneous than the close comoving pairs/wide binaries but more chemically
homogeneous than the random pairs. One explanation for the slightly larger
chemical inhomogeneity is that the far comoving pairs are a mixture of conatal
pairs and chance alignments. The interlopers create a tails of large
$\Delta[{\rm X/H}]$. If that is the case, we should be able to constrain the
conatality fraction through these far-comoving pairs’ chemical homogeneity,
which is what we will attempt next.

We only model $\Delta\,$[Fe/H] because iron has the most precise measurements
of any element in our sample, and, as we have argued, most elemental
abundances only trace [Fe/H] at the current precision. The basic idea is that,
if the $\Delta\,$[Fe/H] of the far comoving pairs are contributed by the two
populations – a conatal population that resembles the wide binaries population
and the random pairs interlopers, we would expect the distribution of
$\Delta\,$[Fe/H] would also be a mixture of the two. More specifically, we
model the PDF of $|\Delta\mathrm{[Fe/H]}|/\sigma_{\Delta\mathrm{[Fe/H}]}$ (or
in short, $\Delta/\sigma$, in the following) for the far comoving pairs as a
mixture model using the weighted average of the close comoving pairs’
$\Delta/\sigma$ PDF and the random pairs’ $\Delta/\sigma$ PDF. The best-
estimated weight signifies the fraction of pairs consistent with the close
comoving pairs among the far comoving populations, hence its conatal fraction.
Due to our small sample size ($\sim 10$ pairs), we expect considerable
uncertainty for estimating the conatal fraction because of the sampling noise.
Therefore, to properly model the sampling noise, we also estimate the
uncertainty of the estimated conatal fraction by bootstrapping the sample.

Operatively, first, we sample, with replacement, the data distribution of
$\Delta/\sigma$ for the close, far, and random pairs. The distributions of
points drawn from the close comoving pairs and random pairs are then
separately approximated as normal distributions through maximum likelihood
estimation (MLE). Subsequently, we fit for the weight of the far comoving
pairs distribution, also through MLE, assuming that the $\Delta/\sigma$
distribution of the far comoving pairs is a weighted mixture of the ones
determined for the close moving pairs and the random pairs. The bootstrapping
process is repeated 1000 times. The bootstrapping produces a distribution of
weights. The median and standard deviation are taken as the best estimate and
uncertainty of the conatal fraction, respectively.

In Figure 7 we demonstrate the nominal fit without the replacement. For this
estimate, we only consider comoving pairs with separations $2\times
10^{5}-10^{7}\,$AU, excluding the three pairs with separations $>10^{7}\,$AU,
leaving us with 11 pairs (out of 14 pairs). As we will see in the following,
these three pairs are clearly interlopers (based on predictions from
simulations). The $\Delta/\sigma$ CDF of the close moving pairs is shown in
violet, the far comoving pairs in violet/black, and the random pairs in teal.
A continuum of mixture model CDFs is displayed in the background as a
blue/yellow gradient ranging from close:random ratios of 1:4 to 9:1, or
equivalently, a conatal fraction 20%-90%. Besides, we highlight some fractions
with white dotted contours to guide the reader. We find the conatal fraction
for the 11 far comoving pairs with separations $2\times 10^{5}-10^{7}\,$AU to
be $73\pm 22\%$. In short, our result implies that $\sim 75\%$ of the far
comoving pairs are conatal based on their metallicity measurements.

Figure 7:  The normalized chemical homogeneity CDFs for different populations
in this study. The CDF from the close comoving pairs/wide binaries is plotted
in the bold violet line and the random field pairs in the bold teal line. The
colored region represents a mixture model of the close and random pairs using
different mixing ratios. The far comoving pairs (violet/black dashed) with
separations of $2\times 10^{5}-10^{7}\,$AU is best modeled with a mixture with
a 73%/27% ratio of the close comoving pairs versus the random pairs. If all
close comoving pairs are conatal, the chemistry of stars reveals that about
73% of the far comoving pairs are conatal. Figure 8: A comparison of our main
sample (red circles) with the theoretical predictions from simulation (Kamdar
et al., 2019a). The symbol sizes are scaled relative to the two stars’
metallicity differences. A smaller metallicity difference indicates a larger
likelihood of being a conatal pair. The background shows the expected conatal
fraction according to the simulation. The simulation does not include wide
binaries and therefore does not extend to separations below $10^{5}\,$AU. The
black dashed contours on the left indicate the maximum velocity difference a
bound system could have with a given total mass, assuming equal-mass
companions and no unseen tertiaries. Our data show excellent agreement with
the simulation, with smaller metallicity differences within the predicted
locus of conatal stars and more considerable metallicity differences outside
the locus. We include two systems (bluegreen) that have velocity difference
$>2\,$ km s-1 as references. The first is a system with a velocity difference
of $2.5$ km s-1 and a separation of $2\times 10^{2}\,$AU, this is well within
the parameter space for a binary system and is likely conatal. The second has
a larger spatial separation ($2\times 10^{6}\,$AU) and a velocity difference
greater than 4 km s-1. The simulation predicts that such a pair is likely a
chance alignment. The abundance differences of these two pairs concur with
these expectations.

##  5 Discussion

Conatal stars are the central hallmarks in modern-day Galactic Archaeology.
Absolute standards are hard to come by (Jofré et al., 2014; Blanco-Cuaresma et
al., 2014b; Heiter et al., 2015), but if we know the two stars are conatal,
the two components serve as each other references. This is why open clusters
and wide binaries have always been the golden calibrators to refine stellar
models, to study exoplanets, gas mixing, and to calibrate surveys. More
recently, the high precision astrometry measurements from Gaia now allow us to
identify wide binaries at high fidelity, which further propels the study of
wide binaries. However, most studies of wide binaries (e.g., El-Badry et al.,
2021) often assume a dichotomy between wide binaries and chance alignments,
and this picture is clearly too simplistic. Simulations (Kamdar et al., 2019a,
b) have suggested comoving stars from disrupted star clusters could have a
high probability of being conatal out to separations of
$\mathcal{O}(10^{6})\,$AU. If this is correct, unbound comoving stars could be
used in similar ways as wide binaries and open clusters. Based on modeling of
Gaia data by Ting et al (in prep), we expect to find roughly 3300 conatal
pairs with 3D separations $\lesssim 2\times 10^{6}\,$ AU (10 pc) and
$\Delta\mathrm{v_{3D}}<2$ km s-1.

In this study, we perform the first homogeneous study of comoving pairs that
span five orders of magnitude in separation. A homogeneous sample allows us to
study the chemical homogeneity of the comoving stars with different
separations consistently. We found that the close comoving pairs with
separations $<2\times 10^{5}\,{\rm AU}$ have a $\Delta$[Fe/H] scatter of
$0.05\,$dex. These values are comparable to the typical values seen in open
clusters and other wide binaries studies. In particular, we opt to perform the
same analysis as Hawkins et al. (2020b). With similar resolution and SNR,
Hawkins et al. (2020b) showed a dispersion of 0.05 dex for wide binaries,
largely consistent with this study. On top of that, we show that the abundance
differences for almost all elements are consistent with the measurement
uncertainties, further validating that close comoving pairs/wide binaries are
conatal. We note that some of the $\Delta\mathrm{[Fe/H]}$ scatter seen in the
systems from Ramírez et al. (2019) could be a consequence of a selection
effect. Many of those systems were published in the context of planet
engulfment signatures. Consequently, the systems which show the engulfment
signals (e.g., Kronos and Krios Oh et al., 2018), will also have larger metal
differences. Since exoplanet accretion changes the surface abundances, these
systems may not be representative of the true chemical homogeneity of conatal
systems.

More importantly, our study expanded on previous works by also examining
comoving pairs with separation $>10^{6}\,{\rm AU}$. The key result of our
study is that comoving pairs at these larger separations are still
significantly more chemical homogeneous than the random pairs, albeit with a
slightly larger dispersion in chemistry, with a scatter
$\Delta\mathrm{[Fe/H]}$ of $0.09$ dex. The random pairs have a scatter of 0.23
dex. If we assume that the far comoving pairs are comprised of both the
conatal stars and chance alignments, we estimate that about $73\pm 22\%$ of
pairs with separations $2\times 10^{5}-10^{7}\,{\rm AU}$ are conatal.

A caveat with this estimation is that, in this study, we select stellar twins
with a median $|\Delta\,$Teff$|$ of 105 K. The selection of stellar twins
enables the mitigation of any potential systematics in terms of the spectral
modeling by performing a line-by-line differential study. Conveniently, from
their construction, the simulated random pairs have a smaller difference in
stellar parameters (with a median $|\Delta\,$Teff$|$ of 47 K), so we would
expect these systematic effects to be similar or weaker in the random
population. However, for completeness, we investigated whether or not the more
significant dispersion in metallicity difference is real or simply due to
systematic errors when deriving abundances from stars that are more different
in stellar parameters.

To dissect that, we looked into any potential biases in
$\Delta\mathrm{[Fe/H]}$ as a function of the difference in stellar parameters.
We found weak trend between $\Delta\mathrm{[Fe/H]}$ and
$\Delta\mathrm{T_{eff}},\Delta\log g,$ and $\Delta\mathrm{v_{micro}}$. For
example, $\Delta\mathrm{[Fe/H]}$ shows a positive correlation with
$\Delta\mathrm{T_{eff}}$, with a gradient of $2.3\times 10^{-4}\mathrm{dex\
K^{-1}}$. We attempted to account for that by fitting a multivariate linear
regression between $\Delta\mathrm{[Fe/H]}$ as a function of
$\Delta\mathrm{T_{eff}},\Delta\log g,$ and $\Delta\mathrm{v_{micro}}$. We
found that this process typically leads to a correction of
$0.002\,\mathrm{dex}$ in $\Delta\mathrm{[Fe/H]}$, which is negligible for our
study. We conclude that it is unlikely the differences in chemistry between
the comoving and random pairs are attributable to this correlation. We opted
not to remove these correlations because it is hard to determine the exact
causal direction of this correlation.

Recall that, the observations in this study are largely motivated by the
simulation from Kamdar et al. (2019a), in which the authors argued that far
comoving pairs are also conatal. So, how does our study compare with the
simulation? In Figure 8, we compare our data with the Kamdar et al. (2019a)
simulation. Our comoving pairs are represented as red circles, and the symbol
size shows the metallicity difference of the two stars for individual pairs.
The black dashed lines indicate the escape velocity of bound systems, assuming
equal-mass companions, and with the total masses of $2{\rm M}_{\odot}$ and
$5{\rm M}_{\odot}$ respectively. The simulation does not extend below
$10^{5}\,$AU because the simulations did not attempt to model wide binaries
(only disrupted star clusters).

Our results exhibit excellent agreement with the simulations; our sample shows
a high degree of chemical homogeneity in the regime where the simulations
predict to have a high conatality rate (the region in yellow). In the phase
space regime where chance alignments should dominate (in blue background), our
observations also show the largest metallicity differences (largest symbol
sizes). Also more quantitatively, the simulation predicts that $\sim 80\%$ of
comoving pairs are conatal provided $\Delta\mathrm{v_{3D}}<2$ km s-1 and the
pair separation is between $2\times 10^{5}-10^{7}\,$AU . Our conatality
fraction estimate of $73\pm 22\%$ for separations is well aligned with the
simulation’s predictions.

On top of that, besides our (31 pairs of the main sample), as discussed in the
target selection, we also observed two pairs with large
$\Delta\mathrm{v_{3D}}$ ($>2\,$km/s) as the control sample. One of the two
pairs has a spatial separation of $2\times 10^{2}\,$AU. Even though they are
not within $\Delta\mathrm{v_{3D}}<2$ km s-1, this pair is clearly a bound
binary, and therefore is conatal. The other pair has a large separation
($2\times 10^{6}\,$AU) and a large velocity difference ($4\,$km s-1). The
simulation predicted this pair is firmly a chance alignment pair as disrupted
star cluster members are unlikely to exhibit such a large velocity difference.
As shown in Fig. 8, our abundance measurement indeed concurs with this
picture, the former has a metallicity difference that is typical of that of
wide binaries, and the latter has a dispersion of 0.24 dex, consistent with
what we expect from a random pair. With this in mind, we caution that all the
results presented in this study, including the conatal fraction, only strictly
applies to stars with $\Delta\mathrm{v_{3D}}<2\,$km s-1.

While the agreement is encouraging, due to the small sample size,
unfortunately, we could only settle on two separation bins – those with
separations $<2\times 10^{5}\,{\rm AU}$ and those with $2\times
10^{5}-10^{7}\,{\rm AU}$. It is clear from the simulation that there is no
sharp transition between the conatal to random pairs. But instead, the regime
of purely conatal pairs transitions gradually toward the regime dominated by
chance alignments. The transition depends critically on many Galactic
properties, including how stars form and disperse and the cluster mass
function. On top of that, as explored in Kamdar et al. (2019a), the number
density of comoving stars provide tell-tale signs on the number density of
Galactic perturbers, such as giant molecular clouds in the Milky Way disk, as
they disrupt comoving pairs. In short, understanding this transition with
critical and refining this boundary region with a larger sample will no doubt
shed insights into the formation and dispersion of star clusters and other
substructures in the Milky Way.

The exact value of the division between close pairs (i.e., wide binaries) and
far pairs (presumed unbound) is somewhat arbitrary. Bound pairs are not
expected to exist at separations beyond 1-2 pc, because at these separations
the Galactic tidal field is stronger than the internal acceleration within a
binary (e.g. Jiang & Tremaine, 2010). At separations of order 1 pc, there will
always be some ambiguity between still-bound and recently-dissolved pairs, and
the reliability with which bound pairs can be identified will depend on the
precision of the astrometry (e.g. El-Badry et al., 2021). A possible
alternative to adopting a strict threshold at ${\rm S_{eDR3}}=2\times
10^{5}\,\rm AU$ would be to only consider as binaries pairs that are members
of a predetermined wide binary catalog, such as the one produced by El-Badry
et al. (2021). The disadvantage of this approach is that such catalogs are
generally not 100% complete. Increasingly precise astrometry form future Gaia
data releases will make it possible to classify pairs with separations of
order 1 pc more reliably. For now, we note that because of our small sample
size, our estimates of the conatal fraction are considerable and are dominated
by small number statistics. Small changes in the adopted boundary between
close and far pairs thus lead to changes in our inferred conatal fraction that
are comfortably within our reported uncertainties. For completeness, we
provide a summary of the effects of choice of separations for the far pairs in
Table 6.

Threshold (AU) | Conatal Fraction
---|---
$10^{5}$ | $52\pm 26\%$
$2\times 10^{5}$ | $73\pm 22\%$
$10^{6}$ | $61\pm 31\%$
Table 6: This table details the effects of using different thresholds for the
close and far pairs. We always assume an upper separation threshold of
$10^{7}\,$AU for the far comoving pairs.

Finally, in this study, we assume that the difference in chemical homogeneity
for the far comoving pairs compared to the close conatal pairs stems from
random pairs interlopers. However, another potential source for these
differences in chemical homogeneity could be from the ISM being less mixed at
the largest scale. At the $\mathcal{O}(10^{7})\,$AU scale, ISM can have
abundance scatter of up to 0.3 dex (e.g. Sanders et al., 2012). If stars
formed in a poorly mixed cloud, they might inherit this scatter. We deem such
a scenario unlikely because hydrodynamic simulations have found that, in a
single star-forming region, even small amounts of turbulence can cause a
factor of five reductions in abundance to scatter in stars compared to the
progenitor ISM (Feng & Krumholz, 2014). And this chemical reduction scatter
applies to scales large as $\mathcal{O}(10^{6})-\mathcal{O}(10^{7})\,$AU, the
maximum scale probed in this study.

That said, if the stars formed in a filamentary structure at different star-
forming regions, the ISM might be less well mixed. For example, Hawkins et al.
(2020a) argue Pisces-Eridanus stellar stream could have been produced from a
stellar filament. They find a metallicity range of $\sim 0.2$ dex for the
stream, which is consistent with our metallicity difference in the far
comoving pairs. If the chemical inhomogeneity is due to the ISM physics
instead of random interlopers, our conatality fraction estimated would be a
conservative limit. The far comoving pairs would have an even higher conatal
fraction than what was is inferred here.

##  6 Summary

We obtained a homogeneous sample of high-resolution ($R\simeq 40000$), high
SNR ($\sim 150$ per pixel) observations of 62 FG-type stars residing in 31
comoving pairs with separations that span five orders of magnitude in
separation, between $10^{2}\,$AU to $10^{7}\,$AU. Previous studies mostly
restricted to wide binaries with separation $<10^{6}\,$AU, and we investigate
if the (unbound) comoving stars with separations $=2\times 10^{5}-10^{7}\,$AU
are conatal. We measured the stellar atmospheric parameters and chemical
abundances for 24 species, including covering the different nucleosynthetic
pathways. The derived elemental abundances: Li, C, Na, Mg, Al, Si, Ca, Sc, Ti,
V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Sr, Y, Zr, Ba, La, Nd, and Eu.

We separate our sample into the classical wide binaries regime (i.e., close
comoving pairs with separations $<2\times 10^{5}\,$AU), the far comoving pairs
(separations $>2\times 10^{5}\,$AU), as well as the random field pairs created
through random pairings of our observations. We find the wide binaries are
significantly more chemically homogeneous than field stars in [X/H] (see
Figure 6) with a typical scatter in $\Delta$[Fe/H] of 0.05 dex. Our results
agree with previous works on wide binaries and comoving pairs (see, e.g.,
Andrews et al., 2019; Hawkins et al., 2020b). For most elements in the close
comoving pairs, the dispersion in [X/H] is consistent with the measurement
errors, implying that the close comoving/wide binaries pairs are chemically
homogeneous at our measurement precision.

We demonstrate that the far comoving pairs with separations of $2\times
10^{5}-10^{7}\,$AU also exhibit substantial chemical homogeneity ($0.09\,$ dex
in the [Fe/H] scatter) compared to random pairs (0.23 dex). Nonetheless, the
far comoving pairs less homogeneous than the close comoving/wide binaries
population. If we assume that the far comoving pairs comprise of a mixture of
the conatal and random chance alignments population, modeling the distribution
of $\Delta$[Fe/H] as a mixture of these two populations implies that about
$73\pm 22\%$ of the unbound comoving stars with separations $2\times
10^{5}-10^{7}\,$AU and $\Delta\mathrm{v_{3D}}<2$ km s-1 are conatal. This
conatality fraction is in excellent agreement with the predictions from
simulations from Kamdar et al. (2019a).

Our study implies that most comoving stars are conatal, even though they are
well-separated. As well-separated pairs of stars are more common than wide
binaries, this new population of “clusters of two” enables many windows for
studies that we have thus far restricted to the wide binaries and open
clusters. Harnessing these vastly abundant comoving pairs will have broad
applications, ranging from calibrating surveys to understanding star
formation, planet engulfments, and beyond.

##  7 Acknowledgements

TN & KH have been partially supported by a TDA/Scialog (2018-2020) grant
funded by the Research Corporation and a TDA/Scialog grant (2019-2021) funded
by the Heising-Simons Foundation. TN & KH acknowledge support from the
National Science Foundation grant AST-1907417. YST is grateful to be supported
by the NASA Hubble Fellowship grant HST-HF2-51425.001 awarded by the Space
Telescope Science Institute. KH is partially supported through the Wootton
Center for Astrophysical Plasma Properties funded under the United States
Department of Energy collaborative agreement DE-NA0003843. APJ acknowledges
support from a Carnegie Fellowship and the Thacher Research Award in
Astronomy. HK acknowledges support from the DOE CSGF under grant number DE-
FG02-97ER25308. The authors thank Carnegie Observatory for granting us the
observing time to conduct this study.

This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France (Wenger et al., 2000) and NASA’s Astrophysics Data System
Bibliographic Services. This work has made use of data from the European Space
Agency mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

Table 7: Atomic Data References Reference Key | Reference
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1968PhFl…11.1002W | Wolnik et al. (1968)
1969AA…..2..274G | Garz & Kock (1969)
1970AA…..9…37R | Richter & Wulff (1970)
1970ApJ…162.1037W | Wolnik et al. (1970)
1980AA….84..361B | Biemont & Godefroid (1980)
1980ZPhyA.298..249K | Kerkhoff et al. (1980)
1982ApJ…260..395C | Cardon et al. (1982)
1982MNRAS.199…21B | Blackwell et al. (1982a)
1983MNRAS.204..883B | Blackwell et al. (1983)
1984MNRAS.207..533B | Blackwell et al. (1984)
1984MNRAS.208..147B | Booth et al. (1984)
1984PhST….8…84K | Kock et al. (1984)
1985AA…153..109W | Whaling et al. (1985)
1985JQSRT..33..307D | Doerr & Kock (1985)
1986JQSRT..35..281D | Duquette et al. (1986)
1986MNRAS.220..289B | Blackwell et al. (1986a)
1989AA…208..157G | Grevesse et al. (1989)
1989ZPhyD..11..287C | Carlsson et al. (1989)
1990JQSRT..43..207C | Chang & Tang (1990)
1991JPhB…24.3943H | Hibbert et al. (1991)
1992AA…255..457D | Davidson et al. (1992)
1993AAS…99..179H | Hibbert et al. (1993)
1993JPhB…26.4409B | Butler et al. (1993)
1993PhyS…48..297N | Nahar (1993)
1995JPhB…28.3485M | Mendoza et al. (1995)
1996PhRvL..76.2862V | Volz et al. (1996)
1998PhRvA..57.1652Y | Yan et al. (1998)
1999ApJS..122..557N | Nitz et al. (1999)
2000MNRAS.312..813S | Storey & Zeippen (2000)
2003ApJ…584L.107J | Johansson et al. (2003)
2006JPCRD..35.1669F | Fuhr & Wiese (2006)
2006JPhB…39.2861Z | Zatsarinny & Bartschat (2006)
2007AA…472L..43B | Blackwell-Whitehead & Bergemann (2007)
2007PhyS…76..577L | Li et al. (2007)
2009A2009AA…497..611M | Meléndez & Barbuy (2009)
2009JPhB…42r5002K | Kułaga-Egger & Migdałek (2009)
2013ApJS..205…11L | Lawler et al. (2013a)
2013ApJS..208…27W | Wood et al. (2013a)
2014ApJS..211…20W | Wood et al. (2014)
2014MNRAS.441.3127R | Ruffoni et al. (2014)
ABH | Arnesen et al. (1977)
AMS | Andersen et al. (1972)
AMb | Alonso-Medina (1997)
APH | Andersen et al. (1976)
APR | Andersen et al. (1975)
AS | Andersen & Soerensen (1973)
ASa | Andersen & Sorensen (1974)
ASb | Anisimova & Semenov (1974)
ATJL | Aldenius et al. (2007)
Astrophysical |  Kurucz (2007, 2009)777There are four lines with this designation. Two Fe lines and two V lines. The Fe lines come from Kurucz (2007) and the V lines from Kurucz (2009). The log$gf$ values were astrophysically calibrated.
BBC | Blanco et al. (1995)
BBEHL | Biémont et al. (2011)
BDMQ | Biémont et al. (1998)
BGF | Biemont et al. (1989)
BGHL | Biemont et al. (1981a)
BGHR | Baschek et al. (1970)
BGKZ | Biemont et al. (1984)
BHN | Bizzarri et al. (1993)
BIEMa | Biémont (1973)
BIEMb | Bièmont (1977)
BIPS | Blackwell et al. (1979a)
BK | Bard & Kock (1994)
BKK | Bard et al. (1991)
BKM | Biemont et al. (1982)
BKP,BKM | Blagoev et al. (1977); Biemont et al. (1982)
BKPb | Blagoev et al. (1978)
BKor | Bridges & Kornblith (1974)
BL | O’brian & Lawler (1991)
BLNP | Blackwell-Whitehead et al. (2006)
BLQS | Biémont et al. (2003)
BQR | Biémont et al. (2002)
BQZ | Biemont et al. (1993)
BRD | Biemont et al. (1981b)
BSB | Berzinsh et al. (1997)
BSScor | Blackwell et al. (1980)
BWL | O’Brian et al. (1991)
BWL,BK | O’Brian et al. (1991); Bard & Kock (1994)
BXPNL | Blackwell-Whitehead et al. (2005)
Bar,BBC | Barach (1970); Blanco et al. (1995)
CB | Corliss & Bozman (1962a)
CBcor | Corliss & Bozman (1962b)
CC | Cowley & Corliss (1983a)
CCout | Cowley & Corliss (1983b)
CM | Clawson & Miller (1973)
CRC | Cowley (1973)
CSE | Cocke et al. (1973)
DCWL | den Hartog et al. (1998)
DHL | Den Hartog et al. (2005)
DHWL | Den Hartog et al. (2002)
DIKH | Drozdowski et al. (1997)
DLSC | Den Hartog et al. (2006)
DLSSC | Den Hartog et al. (2011)
DLW | Dolk et al. (2002)
DLa | Duquette & Lawler (1982)
DLb | Duquette & Lawler (1985)
DSJ | Dworetsky et al. (1984)
DSLa | Duquette et al. (1981)
DSLb | Duquette et al. (1982a)
DSLc | Duquette et al. (1982b)
ESTM | Kurucz (1993a)
FDLP | Fedchak et al. (2000)
FMW | Fuhr et al. (1988)
GARZ | Garz (1973)
GC | García & Campos (1988)
GESB79b | Blackwell et al. (1979b)
GESB82c | Blackwell et al. (1982b)
GESB82d | Blackwell et al. (1982c)
GESB86 | Blackwell et al. (1986b)
GESG12 | Grevesse (2012)
GESHRL14 | Den Hartog et al. (2014)
GESMCHF | Froese Fischer & Tachiev (2012)
GESOP | Saraph & Storey (2012)
GHLa | Gough et al. (1982)
GHR | von der Goltz et al. (1984)
GHcor | Kurucz (1993b)
GKOPK | Gorshkov et al. (1980)
GKOa | Gorshkov et al. (1981)
GKOb | Gorshkov et al. (1983)
GNEL | Gurell et al. (2010)
GUES | Kurucz (1993c)
HLB | Hannaford et al. (1985)
HLGBW | Hannaford et al. (1982)
HLGN | Hannaford et al. (1992)
HLL | Hannaford et al. (1981)
HLSC | Den Hartog et al. (2003)
IAN | Ivarsson et al. (2003)
ILW | Ivarsson et al. (2001)
JMG | Migdalek (1978)
K03 | Kurucz (2003)
K04 | Kurucz (2004)
K06 | Kurucz (2006)
K07 | Kurucz (2007)
K08 | Kurucz (2008)
K09 | Kurucz (2009)
K10 | Kurucz (2010)
K11 | Kurucz (2011)
K12 | Kurucz (2012)
K13 | Kurucz (2013)
K14 | Kurucz (2014)
K75 | Kurucz (1975)
K99 | Kurucz (1999)
KG | Kling & Griesmann (2000)
KK | Kroll & Kock (1987)
KP | Kurucz & Peytremann (1975)
KR | Kock & Richter (1968)
KSG | Kling et al. (2001)
KZB | Kwiatkowski et al. (1984)
KZBa | Kwiatkowski et al. (1982)
LAW | Lawrence (1967)
LBS | Lawler et al. (2001a)
LCG | Lotrian et al. (1978)
LCV | Laughlin et al. (1978)
LD | Lawler & Dakin (1989)
LD-HS | Lawler et al. (2006)
LDLS | Lawler et al. (2007)
LGWSC | Lawler et al. (2013b)
LGb | Lotrian & Guern (1982)
LMW | Lambert et al. (1969)
LN | Lindgård & Nielson (1977)
LNAJ | Ljung et al. (2006)
LNWLX | Lundqvist et al. (2006)
LSC | Lawler et al. (2004)
LSCI | Lawler et al. (2009)
LSCW | Lawler et al. (2008)
LV | Laughlin & Victor (1974)
LWCS | Lawler et al. (2001b)
LWG | Lawler et al. (1990)
LWHS | Lawler et al. (2001c)
LWST | Lennard et al. (1975)
LWa | Lambert & Warner (1968)
MC | Meggers et al. (1975)
MFW | Martin et al. (1988)
MIGa | Migdalek (1976)
MIGb | Migdałek (1976)
MRB | Miller et al. (1971)
MRW | May et al. (1974)
MULT | Kurucz (1993d)
MW | Miles & Wiese (1969)
MWRB | Miller et al. (1974)
NG | Grevesse (1969)
NHEL | Nilsson et al. (2010)
NI | Nilsson & Ivarsson (2008)
NIJL | Nilsson et al. (2002a)
NIST10 | Ralchenko et al. (2010)
NWL | Nitz et al. (1998)
NZL | Nilsson et al. (2002b)
OK | Obbarius & Kock (1982)
PGBH | Pinnington et al. (1993)
PGHcor | Pauls et al. (1990)
PGK | Penkin et al. (1984)
PK | Penkin & Komarovskii (1976)
PN | Pitts & Newsom (1986)
PQB | Palmeri et al. (2001)
PQWB | Palmeri et al. (2000)
PRT | Parkinson et al. (1976)
PST | Pfennig et al. (1965)
PSa | Penkin & Shabanova (1963)
PTP | Pickering et al. (2001)
PV | Plekhotkin & Verolainen (1985)
QPB | Quinet et al. (1999a)
QPBM | Quinet et al. (1999b)
RHL | Ryabchikova et al. (1994)
RPU | Raassen et al. (1998)
RSa | Ryabchikova & Smirnov (1989)
RU | Raassen & Uylings (1998)
RW | Rosberg & Wyart (1997)
S | Smith (1988)
S-G,BBC | Schulz-Gulde (1969); Blanco et al. (1995)
SCRJ | Sansonetti & Reader (2001)
SDL | Salih et al. (1983)
SDcor | Schnehage et al. (1983)
SEN | Sengupta (1975)
SG | Smith & Gallagher (1966)
SK | Smith & Kuehne (1978)
SLS | Sobeck et al. (2007)
SLb | Salih & Lawler (1985)
SLd | Sigut & Landstreet (1990)
SN | Smith & O’Neill (1975)
SPN | Sikström et al. (2001)
SR | Smith & Raggett (1981)
Si2-av1 | Ryabchikova (2012)
Sm | Smith (1981)
T | Theodosiou (1989)
T83av | Ryabchikova et al. (1999)
TB | Seaton et al. (1994)
VGH | Vaeck et al. (1988)
WBW | Wolnik et al. (1971)
WBb | Whaling & Brault (1988)
WGTG | Werij et al. (1992)
WL | Wickliffe & Lawler (1997a)
WLN | Wickliffe et al. (2000)
WLSC | Wood et al. (2013b)
WLa | Wickliffe & Lawler (1997b)
WM | Wiese & Martin (1980)
WSG | Wiese et al. (1966)
WSL | Wickliffe et al. (1994)
WSM | Wiese et al. (1969)
WV | Ward et al. (1985)
Wa | Warner (1968a)
Wc | Warner (1968b)
XJZD | Xu et al. (2003a)
XSCL | Xu et al. (2003b)
XSQG | Xu et al. (2003c)
ZLLZ | Zhiguo et al. (2000)
ZZZ | Zhiguo et al. (1999)

##  Appendix A Estimating 3D separations for wide binaries

Consider a pair of stars with true 3D separation $r_{\rm 3D}$. If the pair is
viewed along a random viewing angle, the probability distribution of its
projected separation $r_{\rm 2D}$ can be calculated via a straightforward
geometric argument (e.g. Nottale & Chamaraux, 2018):

$p\left(r_{\rm 2D}|r_{\rm 3D}\right)=\frac{1}{r_{\rm 3D}}\frac{r_{\rm
2D}/r_{\rm 3D}}{\sqrt{1-\left(r_{\rm 2D}/r_{\rm 3D}\right)^{2}}}.$ (A1)

For the pairs in our sample suspected to be binaries, we have a precise
measurement of $r_{\rm 2D}$ and wish to constrain $r_{\rm 3D}$. By Bayes’
rule, the probability distribution of $r_{\rm 3D}$ given a measurement or
$r_{\rm 2D}$ depends both on the measured $r_{\rm 2D}$, and on the prior on
$r_{\rm 3D}$.

$p\left(r_{\rm 3D}|r_{\rm 2D}\right)\propto p\left(r_{\rm 2D}|r_{\rm
3D}\right)p\left(r_{\rm 3D}\right).$ (A2)

The separation distribution of wide binaries is well-constrained
observationally to be $p\left(r_{\rm 3D}\right)\sim r_{\rm 3D}^{-3/2}$ at
$r_{\rm 3D}>500$ AU (e.g. Andrews et al., 2017; El-Badry & Rix, 2018), so we
adopt this as the prior for the 3D separation distribution of pairs thought to
be binaries. In this case, Equation A2 becomes

$p\left(r_{\rm 3D}|r_{\rm
2D}\right)=\begin{cases}\frac{2}{\sqrt{\pi}}\frac{\Gamma\left(7/4\right)}{\Gamma\left(5/4\right)}\frac{1}{r_{\rm
3D}}\frac{\left(r_{\rm 2D}/r_{\rm 3D}\right)^{5/2}}{\sqrt{1-\left(r_{\rm
2D}/r_{\rm 3D}\right)^{2}}},&r_{\rm 3D}>r_{\rm 2D}\\\ 0&{\rm
otherwise}\end{cases},$ (A3)

where $\Gamma$ represents the Gamma function, and the coefficient is derived
from the normalization condition. The red line in Figure 9 shows this
distribution, with $r_{\rm 2D}=10^{4}$ AU adopted for concreteness.888For
better visual interpretability, we plot $p\left(\log r_{{\rm 3D}}|r_{{\rm
2D}}\right)=\ln 10\times r_{3D}\times p\left(r_{3{\rm D}}|r_{{\rm
2D}}\right)$. For this distribution, the median value of $r_{\rm 3D}/r_{\rm
2D}$ is 1.12, and the middle 68.2% range is (1.01, 1.61). Given a measurement
of $r_{\rm 2D}$, we thus report $r_{\rm 3D}=1.12_{-0.11}^{+0.49}\times r_{\rm
2D}$ if the pair is suspected to be a binary.

Figure 9: Conditional probability distribution of the logarithmic 3D
separation $r_{\rm 3D}$, given a measured projected separation of $r_{\rm
2D}=10^{4}$ AU. We compare distribution for three different choices of the
prior on the intrinsic 3D separation distribution, $p(r_{\rm 3D})$. Dashed red
line shows the prior we adopt for pairs suspected to be wide binaries. For
both this choice and a flat prior (black line), $p\left(\log r_{\rm 3D}|r_{\rm
2D}\right)$ is sharply peaked at $r_{\rm 2D}$; that is, the most likely 3D
separation is within a factor of a few of the projected separation. Dotted
cyan line shows $p(r_{\rm 3D})\propto r_{\rm 3D}^{2}$, as is expected for
random chance alignments. In this case, the probability mass is dominated by
pairs with $r_{\rm 3D}\gg r_{\rm 2D}$; that is, widely separated pairs viewed
almost end-on.

Of course, other choices are possible for the prior $p(r_{\rm 3D})$ when the
separation exceeds what we expect from the binary population. The black line
in Figure 9 corresponds to a “flat” prior, $p(r_{\rm 3D})=\rm const$. Such a
separation distribution might be expected for conatal pairs that are no longer
bound, if the birth and dissolution rates of such pairs are constant. In this
case, the median and 1$\sigma$ range in $r_{\rm 3D}$ would be $r_{\rm
3D}=1.41_{-0.38}^{+2.63}\times r_{\rm 2D}$, still implying that most pairs
have 3D separations within a factor of a few of their 2D projected separation.

Finally, the dotted cyan line in Figure 9 shows the results of assuming
$p(r_{\rm 3D})\propto r_{\rm 3D}^{2}$. This is the distribution expected for
random pairings of isotropically distributed stars. In this case, the
probability distribution is normalizable only if we assume that there are no
pairs beyond a 3D separation $r_{\rm max}$. The conditional probability
distribution in this case is

$p\left(r_{\rm 3D}|r_{\rm
2D}\right)=\begin{cases}\frac{1}{\sqrt{\left(1-r_{\rm 2D}^{2}/r_{\rm
3D}^{2}\right)\left(r_{{\rm max}}^{2}-r_{\rm 2D}^{2}\right)}},&r_{\rm
3D}>r_{\rm 2D}\\\ 0&{\rm otherwise}\end{cases},$ (A4)

which approaches a constant value at $r_{\rm 3D}\gg r_{\mathrm{2D}}$.
Intuitively, what happens in this case is that the larger number of possible
pairs at large $r_{\rm 3D}$ (which scales as $r_{\rm 3D}^{2}$) exactly
compensates for the smaller number of sightlines along which a pair can be
viewed to have a given projected separation (which scales as $r_{\rm
3D}^{-2}$). In Figure 9, we adopt $r_{\rm max}=10^{6}$ AU.

To summarize, we calculate constraints on the 3D separation of pairs thought
to be wide binaries based on their projected 2D separation using Equation A3.
This yields a median and 1$\sigma$ range of $r_{\rm
3D}=1.12_{-0.11}^{+0.49}\times r_{\mathrm{2D}}$, which at small
$r_{\mathrm{2D}}$ is much more constraining than the constraint on $r_{\rm
3D}$ that considers the distances to both stars independently (“method 1”).
The primary assumption of this constraint is that the separation distribution
of wide binaries falls off as $p(r_{\mathrm{3D}})\propto
r_{\mathrm{3D}}^{-3/2}$. It is not appropriate for pairs that are not binaries
(unbound comoving pairs or chance alignments), because their separation
distribution is expect to increase with separation due to the different
priors. We thus only apply this correction to pairs with separation $<2\times
10^{5}\,$AU.

##  Appendix B Line List references

A large part of this work is made possible by numerous heroic efforts by
various groups who perform the pain-stacking tasks of curating/calibrating the
atomic line list used in BACCHUS. Those efforts are unfortunately under
appreciated in the literature. As such, albeit long, we decide to include the
references to all original sources of the line list adopted in this study in
Table 7.

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