# Wide binaries from the H3 survey: the thick disk and halo have similar wide
binary fractions

Hsiang-Chih Hwang Institute for Advanced Study, Princeton, 1 Einstein Drive,
NJ 08540, USA  Department of Physics & Astronomy, Johns Hopkins University,
Baltimore, MD 21218, USA  Yuan-Sen Ting Research School of Astronomy &
Astrophysics, Australian National University, Cotter Rd., Weston, ACT 2611,
Australia  Research School of Computer Science, Australian National
University, Acton ACT 2601, Australia  Charlie Conroy Center for Astrophysics
$\mid$ Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Nadia L.
Zakamska Institute for Advanced Study, Princeton, 1 Einstein Drive, NJ 08540,
USA  Department of Physics & Astronomy, Johns Hopkins University, Baltimore,
MD 21218, USA  Kareem El-Badry Center for Astrophysics $\mid$ Harvard &
Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Phillip Cargile Center
for Astrophysics $\mid$ Harvard & Smithsonian, 60 Garden St, Cambridge, MA
02138, USA  Dennis Zaritsky Steward Observatory, University of Arizona, 933
North Cherry Avenue, Tucson, AZ 85721-0065, USA  Vedant Chandra Center for
Astrophysics $\mid$ Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138,
USA  Jiwon Jesse Han Center for Astrophysics $\mid$ Harvard & Smithsonian, 60
Garden St, Cambridge, MA 02138, USA  Joshua S. Speagle (沈佳士) Banting & Dunlap
Fellow  Department of Statistical Sciences, University of Toronto, Toronto, ON
M5S 3G3, Canada  David A. Dunlap Department of Astronomy & Astrophysics,
University of Toronto, Toronto, ON M5S 3H4, Canada  Dunlap Institute for
Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada
Ana Bonaca Observatories of the Carnegie Institution for Science, 813 Santa
Barbara Street, Pasadena, CA 91101, USA

###### Abstract

Due to the different environments in the Milky Way’s disk and halo, comparing
wide binaries in the disk and halo is key to understanding wide binary
formation and evolution. By using Gaia Early Data Release 3, we search for
resolved wide binary companions in the H3 survey, a spectroscopic survey that
has compiled $\sim$150,000 spectra for thick-disk and halo stars to date. We
identify 800 high-confidence (a contamination rate of 4%) wide binaries and
two resolved triples, with binary separations mostly between $10^{3}$-$10^{5}$
AU and a lowest [Fe/H] of $-2.7$. Based on their Galactic kinematics, 33 of
them are halo wide binaries, and most of those are associated with the
accreted Gaia-Sausage-Enceladus galaxy. The wide binary fraction in the disk
decreases toward the low metallicity end, consistent with Hwang et al. (2021).
Our key finding is that the halo wide binary fraction is consistent with the
thick-disk stars at a fixed [Fe/H]. There is no significant dependence of the
wide binary fraction on the $\alpha$-captured abundance. Therefore, the wide
binary fraction is mainly determined by the iron abundance, not their disk or
halo origin nor the $\alpha$-captured abundance. Our results suggest that the
formation environments play a major role for the wide binary fraction, instead
of other processes like radial migration that only apply to disk stars.

binaries: general — stars: kinematics and dynamics — Galaxy: halo — stars:
abundances

##  1 Introduction

Since wide binaries are weakly bound, they are sensitive to their formation
and evolution environments. Wide binary fractions reflect the density of star
formation environments because dense environments like star clusters can
easily disrupt wide binaries and thus reduce wide binary formation (Scally et
al., 1999; Parker et al., 2009; Geller et al., 2019; Deacon & Kraus, 2020). In
the Milky Way disk, gravitational interactions with passing stars, molecular
clouds, and the Galactic tide can disrupt wide binaries at separations
$\gtrsim 10^{4}$ AU, providing a unique probe for pc-scale gravitational
fields (Retterer & King, 1982; Bahcall et al., 1985; Weinberg et al., 1987;
Jiang & Tremaine, 2010). In the halo where the interactions with passing stars
and molecular clouds are less important, halo wide binaries provide a critical
constraint on the nature of dark matter (Chaname & Gould, 2004; Quinn et al.,
2009; Carr & Kühnel, 2020). Now with an increasing number of disrupted
accreted galaxies found in the Milky Way’s stellar halo (Belokurov et al.,
2018; Helmi et al., 2018; Conroy et al., 2019a; Li et al., 2019; Belokurov et
al., 2020; Naidu et al., 2020, see review by Helmi 2020), halo wide binaries
are critical in constraining wide binary formation and the formation
environments of these progenitor dwarf galaxies in the early Universe.

Our recent work shows that the wide binary fraction (defined at
$10^{3}$-$10^{4}$ AU) in the disk peaks at the solar metallicity and decreases
both toward the low- and high-metallicity ends (Hwang et al., 2021). We
propose several hypotheses that may explain the non-monotonic metallicity
dependence, including the different stellar density in the formation
environments at different epochs (Harris & Pudritz, 1994; Elmegreen & Efremov,
1997; Kravtsov & Gnedin, 2005; Kruijssen, 2014; Ma et al., 2020), the
dynamical unfolding of compact triples (Reipurth & Mikkola, 2012; Elliott &
Bayo, 2016), and the radial migration of Galactic orbits (Sellwood & Binney,
2002). In the radial migration scenario, stars with supersolar and subsolar
metallicities are formed at other Galactocentric radii and then migrated to
the solar neighborhood. Then the lower wide binary fraction at supersolar and
subsolar metallicities may be due to the different formation environments at
different Galactocentric radii, or due to that the radial migration process
(e.g. the enhanced interaction with molecular clouds) can efficiently disrupt
wide binaries.

To constrain these hypotheses, comparing properties of binaries in the disk
and the halo is particularly valuable because they probe a very different
metallicity range, formation environments, stellar ages, and evolution
environments. In particular, halo stars do not experience radial migration in
the Milky Way disk, and thus halo wide binaries do not get disrupted by the
radial migration process. Therefore, if radial migration plays an important
role in the wide binary fraction, we would expect a higher wide binary
fraction in the halo than in the disk.

Despite their importance, halo wide binaries have been challenging to identify
in the past. First, the density of halo stars is low compared to that of the
disk in the solar neighborhood, comprising less than 0.2 per cent of local
stars (Helmi, 2008). Therefore, assembling a large and clean halo star sample
is not straightforward. Second, identifying a resolved companion as the wide
binary companion was difficult before the Gaia era. Even though the Gaia
mission provides the proper motions and parallaxes for billions of stars (Gaia
Collaboration et al., 2016), most of the wide binary samples are still limited
within 1 kpc (El-Badry & Rix, 2018; Hartman & Lépine, 2020; El-Badry et al.,
2021). Furthermore, most of these wide binaries do not have radial velocities
available from Gaia, making differentiating disk and halo stars challenging.
Therefore, the number of robustly identified halo wide binaries remains
insufficient to place a useful constraint on the halo wide binary fraction
(Hwang et al., 2021).

In this paper, we overcome these difficulties by having a customized wide
binary search for stars in the H3 survey. The H3 survey is a large
spectroscopic survey targeting halo and thick-disk stars (Conroy et al.,
2019b). There are $\sim 150,000$ H3 stars in our current analysis, and it is
expected to have $\sim$300,000 stars at the end of the survey, with about 20%
of them being halo stars. The high-precision radial velocities measured from
H3 are critical to differentiate disk and halo stars by their Galactic
kinematics. Since most of the H3 stars are more than 1 kpc away, we optimize
the wide binary method for the H3 stars to maximize the resulting information.
With the H3 wide binaries, we investigate the difference in the wide binary
fractions of the halo and the disk to understand the formation and evolution
of wide binaries.

The paper is structured as follows. Sec. 2 describes the H3 and Gaia dataset
and the general criteria for the sample selection. Sec. 3 details the method
of wide binary search and presents the results. Sec. 4 explores the properties
of the wide binary candidates, and Sec. 5 investigates the metallicity
dependence of the wide binary fraction. We then discuss the implications in
Sec. 6 and conclude in Sec. 7. Throughout the paper, we use ‘wide binaries’
and ‘wide binary candidates’ interchangeably when referring to the selected
wide binaries. We use ‘metallicity’ and ‘iron abundance’ ([Fe/H])
interchangeably and always refer to [$\alpha$/Fe] as $\alpha$-captured
abundances. We use the notation ‘binary’ even if some of them can be
unresolved triples.

##  2 Sample selection

###  2.1 H3 survey

The H3 (‘Hectochelle in the Halo at High Resolution’) survey is a high-
resolution spectroscopic ($R\approx 32,000$) survey targeting thick-disk and
halo stars (Conroy et al., 2019b). The survey is conducted by the 6.5-m MMT
with a wavelength coverage of 5150-5300Å. The survey has been collecting data
since 2017. The main selection of H3 is composed of the following criteria:
(1) $15<r<18$ where $r$ is the $r-$band magnitude from Pan-STARRS (Chambers et
al., 2016; Flewelling et al., 2016); (2) $\pi-2\sigma_{\pi}<0.5$ mas (later
changed to $\pi<0.4$) where $\pi$ and $\sigma_{\pi}$ are parallaxes and
parallax uncertainties from Gaia Data Release 2 (Gaia Collaboration et al.,
2018); (3) $|b|>30^{\circ}$ to increase the halo star fraction; (4)
declination $>-20^{\circ}$ to be observable from MMT. The H3 survey has
secondary selections targeting stellar streams, K giants, blue horizontal
branch stars, and RR Lyrae. This simple selection of magnitudes and parallaxes
provides an unbiased view of the distant Milky Way.

The stellar parameters are measured from the H3 spectra using MINESweeper
(Cargile et al., 2020). MINESweeper is a Bayesian framework that incorporates
the information from H3 spectra, broadband photometry, and priors like Gaia
parallaxes to fit stellar models computed by MIST (v2.0) (Dotter, 2016; Choi
et al., 2016, Dotter et al. in preparation). Cargile et al. (2020) show that
the stellar parameters of benchmark stars and clusters measured by MINESweeper
are in good agreement with literature values.

In this paper, we use the radial velocities, distances, surface iron
abundances ([Fe/H]), surface $\alpha$-captured elemental abundances
([$\alpha$/Fe]), and extinction ($A_{V}$) measured from MINESweeper. The
surface elemental abundances may differ from the initial abundances due to
atomic diffusion and mixing. By including the model from Dotter et al. (2017),
MINESweeper derives the initial chemical abundances, denoted as [Fe/H]init and
[$\alpha$/Fe]init. The difference between [Fe/H] and [Fe/H]init depends on the
evolutionary stages, stellar ages, and initial metallicities, with the largest
difference of $0.5$ dex happening at the main-sequence turnoff of metal-poor
stars (Dotter et al., 2017). In our sample, the median difference between
[Fe/H] and [Fe/H]init is 0.1 dex, and we do not find significant changes in
our results between them. We present our results using surface abundances for
better comparison with the literature. The $\alpha$-captured elemental
abundances from MINESweeper scale the $\alpha$-elements O, Ne, Mg, Si, S, Ca,
and Ti, and the measurement is most sensitive to [Mg/Fe] due to H3’s
wavelength coverage of Mg I triplet (Cargile et al., 2020).

In this paper, we use the proprietary H3 data from
rcat_V4.0.3.d20201031_MSG.fits, with data collected between September 2017 and
October 2020. We require FLAG==0 and signal-to-noise ratios (SNR) $>3$ to
avoid unreliable stellar parameter measurements. Targets from the Sagittarius
stream (Sgr_FLAG==1 from Johnson et al. 2020) and tiles specifically for
streams (tileID starting with ‘tb’) are excluded because stream members moving
in similar directions may increase the chance-alignment probability. For stars
that were observed by H3 multiple times, we use the entries with the highest
SNR. These selections result in a parent sample of 100,967 unique H3 stars for
wide binary search. In this sample, the median uncertainty is 0.05 dex for
[Fe/H], 0.05 dex for [$\alpha$/Fe], 0.2 km s-1 for radial velocities, and 6%
for distances.

###  2.2 Gaia survey

Gaia is an all-sky survey that provides optical broad-band photometry and
high-precision astrometric measurements (Gaia Collaboration et al., 2016).
Gaia Early Data Release 3 (EDR3) (Gaia Collaboration et al., 2020) was
released in December 2020, with parallaxes and proper motions available for
1.5 billion sources down to Gaia G-band magnitudes of $\sim 21$ mag. In
addition, Gaia measures radial velocities for bright stars ($G\lesssim 15$ mag
for EDR3, and may reach $G\sim 16$ mag by the end of mission), but most halo
stars are fainter than this magnitude limit. This is one of the H3 survey’s
motivations to target for stars fainter than 15 mag. We use Gaia EDR3 to
search for the comoving companions around the H3 stars. Because H3 target
selection uses Gaia’s parallaxes which are derived together with proper
motions, all H3 targets have proper motions from Gaia.

The goal of this paper is to search for H3 stars’ resolved wide companions
that have nearly identical Gaia proper motions as the H3 targets. We query a
field star sample from Gaia EDR3 to conduct the wide binary search. For this
field star sample, we require that their proper motions be available in Gaia
EDR3 (astrometric_params_solved$==31$ or 95). We do not apply any criteria on
magnitudes and astrometric quality indicators (e.g. ruwe) to maximize the
chance of finding the companions. Our wide binary selection compares the
proper motions for each pair, which implicitly requires good astrometric
quality, and it is inevitable that we may miss some genuine wide binaries
because some of their stars have worse astrometric quality.

Gaia EDR3’s ability to resolve close pairs starts dropping significantly below
0.7″, and only about 50% of pairs can be spatially resolved at 0.5″ (Fabricius
et al., 2020). We do not use Gaia’s BP-RP colors for our wide binary
selection, but we use them when investigating the properties of the selected
wide binaries. When BP-RP colors are used in the analysis, we require their
bp_flux_over_error and rp_flux_over_error to be larger than 10. Unlike G-band
photometry, BP and RP fluxes do not have de-blending treatment, and therefore
their fluxes can be affected by nearby sources within about 2″ (Riello et al.,
2021). To ensure that BP-RP colors are not strongly affected by the nearby
sources, we apply an additional criterion of phot_bp_rp_excess_factor$<1.4$
when BP-RP colors are used in the analysis.

###  2.3 Calculations of Galactic kinematics

Radial velocities from the H3 survey are necessary for computing Galactic
kinematics parameters and to differentiating disk and halo stars. We measure
the parameters for Galactic kinematics, including the total energy ($E_{tot}$)
and the angular momentum along the Galactic $z$-direction ($L_{z}$), using
Gala v1.1 (Price-Whelan et al., 2020). We use the Galactocentric frame from
Astropy v4.0 (Astropy Collaboration et al., 2013, 2018): Sun’s Galactocentric
radius $R_{0}=8.122$ kpc (Gravity Collaboration et al., 2018), solar motion
with respect to the local standard of rest [$V_{R,\odot}$, $V_{\phi,\odot}$,
$V_{Z,\odot}$]=[$-12.9$, $245.6$, $7.78$] km s-1(Drimmel & Poggio, 2018), and
Sun’s current Galactic height $Z_{\odot}=20.8$ pc (Bennett & Bovy, 2019). The
Milky Way potential MilkyWayPotential (Bovy, 2015) is adopted. $E_{tot}$ and
$L_{z}$ are used to separate halo and disk stars, and using different Milky
Way potential models does not affect this selection significantly (Naidu et
al., 2020). A right-handed coordinate is used, and thus a star on a prograde
(retrograde) orbit has a negative (positive) $L_{z}$. The Galactic orbit is
integrated for 25 Gyr with a time step of 1 Myr using the Dormand & Prince
(1978) integration scheme. The eccentricity of Galactic orbits is then derived
from $e=(r_{apo}-r_{peri})/(r_{apo}+r_{peri})$ where $r_{apo}$ and $r_{peri}$
are the apocenter and pericenter of the Galactic orbit.

The observational inputs for kinematic parameter calculations are the radial
velocities and distances measured from H3 spectra using MINESweeper, and
celestial coordinates and astrometric solutions measured from Gaia EDR3.
Following Naidu et al. (2020), when $E_{tot}$ and $L_{z}$ are used in the
analysis, we require that (i) $|E_{tot}|/\sigma_{E_{tot}}>3$ $\land$
$|L_{z}|/\sigma_{L_{z}}>3$; or (ii) $\sigma_{E_{tot}}<0.1\times 10^{5}$ km2
s-2 $\land$ $\sigma_{L_{z}}<0.5\times 10^{3}$ kpc km s-1, where ‘$\land$’ is
the Boolean ‘and’ operator. The first condition is a normal 3-$\sigma$
uncertainty cut, and the second condition is to ensure that we keep halo stars
with small $L_{z}$ in the sample (otherwise they may be excluded by the
3-$\sigma$ uncertainty cut). Out of 100,967 H3 stars, 99,161 (98 per cent) of
them satisfy these kinematic criteria.

##  3 Wide binary search

Figure 1: The wide binary search in the proper motion difference-angular
separation space. Left: the distribution for all the pairs consisting of an H3
star and a Gaia field star. In addition to the chance-alignment pairs
dominating in the upper-right part, there is an enhanced wide binary
population with small proper motion differences and small angular separations.
Middle: the contamination test where the H3 stars’ Galactic latitudes are
offset by 1 degree. Therefore, all the pairs in this panel are chance-
alignment pairs. Right: the chance-alignment probability computed from the
left and middle panel. Based on the chance-alignment probability, we use an
empirical selection (the dashed lines in the three panels) to select high-
confidence wide binaries.

For a solar-mass wide binary with a semi-major axis of $>1000$ AU, its orbital
Keplerian velocity is $<0.7$ km s-1. Therefore, two resolved stars in a wide
binary would have nearly identical proper motions. For a given H3 star and a
nearby star from Gaia, we compute the proper motion difference by

$\delta{PM}=\sqrt{({\rm\texttt{pmra}_{1}}-{\rm\texttt{pmra}_{0}})^{2}+({\rm\texttt{pmdec}_{1}}-{\rm\texttt{pmdec}_{0}})^{2}},$
(1)

where ${\rm\texttt{pmra}_{i}}$ and ${\rm\texttt{pmdec}_{i}}$ are the Gaia EDR3
proper motions in right ascension direction ($\mu_{\alpha}\cos\delta$) and in
declination direction ($\mu_{\delta}$) for star $i$ ($=0$ or 1), respectively.
With Gaia’s proper motion precision at the relevant magnitudes, the typical
uncertainties of $\delta PM$ is $\sim 0.1$ mas yr-1. The relative velocity
difference projected in the plane of sky is $\delta V=4.74\times\delta
PM\times d$, where $\delta V$ is in units of km s-1, $\delta PM$ is in units
of mas yr-1, $d$ is the heliocentric distance of the pair in kpc, and the
factor of 4.74 is from the unit conversion. Therefore, for a wide binary at
distances of 1 to 10 kpc, Gaia can measure the difference in the projected
orbital velocity $\delta V$ with uncertainties of $0.5-5$ km s-1. This
precision is sufficient to distinguish genuine wide binaries where the orbital
velocities are $<1$ km s-1 from random chance-alignment pairs where the
typical velocity differences are $>50$ km s-1 for the thick-disk and halo
stars.

For every H3 target, we search for a possible resolved wide companion in Gaia
within 60 arcsec. In most cases, radial velocities are only available for H3
targets but not for Gaia field stars, so we cannot compute radial velocity
difference for the pair to help the binary search. Furthermore, limited by the
fiber allocation, the minimum angular separation between two H3 stars is 20
arcsec, making the wide binary search among H3 stars (when both stars were
observed by H3) difficult to result in a statistically useful sample. Most H3
targets are distant ($>1$ kpc) and therefore they do not have precise
distances from Gaia EDR3. The precision of the distance measurement from Gaia
and the MINESweeper fit is not sufficient to distinctly separate the wide
binary population from the random chance-alignment pairs in the relative
velocity-physical separation space (e.g. Hwang et al. 2020). Without
distances, proper motion differences and angular separations cannot be
converted to projected relative velocity differences and physical separations.
Therefore, we search the wide binaries in the proper motion difference-angular
separation space. The advantage is that their measurement uncertainties are
nearly negligible ($<0.1$ mas for angular separations and $\sim 0.1$ mas yr-1
for proper motion differences), with the caveat that the search has different
sensitivities for binary’s physical separations at different distances.

Fig. 1 (left panel) shows the proper motion differences and angular
separations of all the H3-Gaia pairs. To reduce the contamination, we require
that the parallax difference of the pair is consistent with zero within
$3\sigma_{\pi}$. An enhanced population with separations $<10$ arcsec and
proper motion differences $\lesssim 1$ mas yr-1can be seen, indicative of the
resolved wide binary populations. The rest of the pairs in the upper-right
part of the plot are chance alignments.

To better quantify the contamination from the chance-alignment pairs, we
conduct a test by offsetting the H3 targets’ Galactic latitudes by 1 degree to
the North (with the coordinates of field Gaia stars unshifted) and redo the
wide binary search (1 degree corresponds to 17 pc at a distance of 1 kpc).
Therefore, all the pairs from this test are chance-alignment pairs. The result
is shown in Fig. 1, middle panel. This contamination test well reproduces the
chance-alignment pairs in the left panel, except that the left panel has an
additional wide binary population at small angular separations and small
proper motion differences.

In the right panel of Fig. 1, we compute the chance alignment probability
(CAP) in each two-dimensional bin. Specifically, CAP in every bin is
calculated by the number of pairs in the middle panel divided by the number of
pairs in the left panel. The right panel of Fig. 1 shows that the wide
binaries can be selected by small proper motion differences with CAP less than
10% out to 30 arcsec. At separations $>30$ arcsec, CAP is still less than
$20$% at small proper motion differences ($\lesssim 0.3$ mas yr-1), indicative
of the existence of wide binaries beyond 30 arcsec ($\sim 0.1$ pc for stars at
1 kpc distances), consistent with the simulation and observation that a
significant fraction of comoving stars with separations up to $\sim 10$ pc are
conatal (Kamdar et al., 2019; Nelson et al., 2021). Some low CAP pixels at the
edge of large proper motion differences are due to small-number statistics.

In this paper, we focus on the wide binaries with angular separations $<30$
arcsec for their larger sample size and the lower CAP. We adopt an empirical
demarcation line (the dashed lines in Fig. 1) to select high-confidence wide
binaries with CAP$<10$%. The demarcation line starts from $\delta{PM}=1.6$ mas
yr-1 at 0″ separation to $\delta{PM}=0.1$ mas yr-1 at 30″, connected by a
straight line in the log-linear space. Pairs below this demarcation line and
at angular separations $<30$ arcsec are selected as wide binaries, resulting
in 800 wide binaries and 2 resolved triples. There are 33 pairs located below
this demarcation line in the contamination test in the middle panel of Fig. 1,
suggesting that the contamination rate from the chance-alignment pairs in our
wide binary sample is $4.1\pm 0.7$ per cent. A more stringent selection (e.g.
angular separations $<10$ arcsec instead of $<30$ arcsec) does not change our
main results and conclusions.

We provide an electronic catalog for the H3 wide binaries. Table 1 tabulates
the descriptions for the catalog. Field names starting with the prefix ‘0_’
are the information for the H3 stars, and those starting with the prefix ‘1_’
are for the companions. A resolved triple system would have two entries in the
table with the same H3 star and different companion stars.

Table 1: Descriptions for the wide binary catalog. Field | Description
---|---
0_source_id |  Gaia EDR3 source_id of the H3 star
0_ra | Right ascension of the H3 star from Gaia EDR3 (J2016.0; deg)
0_dec | Declination of the H3 star from Gaia EDR3 (J2016.0; deg)
0_parallax | Parallax of the H3 star from Gaia EDR3 (mas)
0_parallax_error | Uncertainty in 0_parallax from Gaia EDR3 (mas)
0_pmra | Proper motion in right ascension direction of the H3 star from Gaia EDR3 (mas yr-1)
0_pmra_error | Uncertainty in 0_pmra (mas yr-1) from Gaia EDR3
0_pmdec | Proper motion in declination direction of the H3 star from Gaia EDR3 (mas yr-1)
0_pmdec_error | Uncertainty in 0_pmdec (mas yr-1) from Gaia EDR3
0_G | Apparent G-band magnitude of the H3 star from Gaia EDR3 (mag)
0_H3_ID | H3 id for the H3 star
0_FeH | Iron abundance [Fe/H] of the H3 star measured by H3 (dex)
0_aFe |  $\alpha$-captured elemental abundance [$\alpha$/Fe] of the H3 star measured by H3 (dex)
0_Teff | Effective temperature of the H3 star measured by H3 (K)
0_logg | Surface gravity of the H3 star measured by H3 ($\log$ cgs)
0_Vrad | Radial velocity of the H3 star measured by H3 ( km s-1)
0_dist_adpt | Heliocentric distances of the H3 star measured by H3 (kpc)
0_Etot | Total orbital energy of the H3 star ($10^{5}$ km2 s-2)
0_Etot_err | Uncertainty in 0_Etot ($10^{5}$ km2 s-2)
0_Lz | Orbital momentum along the Galactic z-axis of the H3 star ($10^{3}$ kpc km s-1)
0_Lz_err | Uncertainty in 0_Lz ($10^{3}$ kpc km s-1)
1_source_id |  Gaia EDR3 source_id of the companion star
1_ra | Right ascension of the companion star from Gaia EDR3 (J2016.0; deg)
1_dec | Declination of the companion star from Gaia EDR3 (J2016.0; deg)
1_parallax | Parallax of the companion star from Gaia EDR3 (mas)
1_parallax_error | Uncertainty in 1_parallax from Gaia EDR3 (mas)
1_pmra | Proper motion in right ascension direction of the companion star from Gaia EDR3 (mas yr-1)
1_pmra_error | Uncertainty in 1_pmra (mas yr-1) from Gaia EDR3
1_pmdec | Proper motion in declination direction of the companion star from Gaia EDR3 (mas yr-1)
1_pmdec_error | Uncertainty in 1_pmdec (mas yr-1) from Gaia EDR3
1_G | Apparent G-band magnitude of the companion star from Gaia EDR3 (mag)
separation_arcsec | Angular separation of the wide binary (arcsec)
separation_AU | Projected physical separation of the wide binary (AU)
pm_diff | Proper motion difference of the wide binary ( mas yr-1)
triple | 1 if it is a resolved triple, otherwise 0
halo_disk | 1 if it is in the halo; 0 if it is in the disk; $-1$ if it does not pass the kinematic criteria

##  4 Properties of H3 wide binaries

###  4.1 Distances and binary separations

Figure 2: Binary separations and heliocentric distances of wide binaries,
color-coded by their iron abundances. The dashed line corresponds to an
angular separation of 1 arcsec, roughly Gaia’s spatial resolution. The dotted
line indicates 30 arcsec used in our selection criterion. Most of the wide
binaries are 1-5 kpc from the Sun with binary separations of $10^{3}$-$10^{5}$
AU.

Fig. 2 shows the binary separations and heliocentric distances of the wide
binaries, color-coded by their iron abundances. Most of the wide binaries have
angular separations between 1 (dashed line) and 30 arcsec (dotted line), where
the former is close to Gaia’s spatial resolution and the latter is due to our
selection criterion. Most of these wide binaries are located 1-5 kpc from the
Sun. Those closer to the observer tend to have higher iron abundances.

There are four wide binaries located at larger than 10 kpc from the Sun, with
the farthest one at 50 kpc. They all have very wide binary separations of
$\gtrsim 1$ pc. These distant wide binaries are valuable for probing the
gravitational interactions in the outer Milky Way. Even though these distant
wide binaries may suffer from a higher chance-alignment probability because of
their larger angular separations ($>15$ arcsec) and smaller total proper
motions ($<5$ mas yr-1), their companions’ photometry, as shown in the next
Section, are consistent with their H3 stars’ isochrones, suggesting that they
are not chance-alignment pairs. One possibility is that they are pairs with
other member stars in a yet unidentified stream, instead of being a wide
binary. This may explain why these four distant wide binaries all have large
angular separations. Further work is needed to investigate if they are wide
binaries or stream-member pairs.

###  4.2 Hertzsprung-Russell diagram

Fig. 3 shows the Hertzsprung-Russell (H-R) diagram of the wide binaries, where
the absolute G-band magnitudes are computed using the distances measured from
MINESweeper. The non-H3 stars adopt the same distances as their H3 companions.
We apply the criteria on BP and RP as described in Sec. 2.2 for more robust
photometry, resulting in 584 pairs. The gray background shows the distribution
of all stars in the H3 catalog, where most of them do not have wide
companions. Most of the wide binaries identified here are main-sequence stars,
and some are located on the giant branch. The companions scatter more around
the main sequence track because they are fainter and more likely to have
larger uncertainties in their photometry.

The solid green lines in Fig. 3 highlight the four wide binaries with
distances $>10$ kpc. They are rare double-giant wide binaries due to the
selection effect that we only see giant stars at large distances. One of them
has nearly the same G and BP-RP magnitudes for the two member stars, forming
an unusual twin giant binary where two stars have nearly identical
evolutionary stages. Fig. 3 shows that the non-H3 companions of these distant
wide binaries have Gaia photometry consistent with the H3 stars’ isochrones,
suggesting that they are not chance-alignment pairs.

Because there are limited regions in the H-R diagram where stars can exist,
the locations of the non-H3 companions in the H-R diagram are a useful
diagnostic of whether they are genuine binary companions. Since wide binaries
with separations $\lesssim$ a few pc are conatal, two stars of the same wide
binary have nearly identical stellar ages and chemical abundances (Andrews et
al., 2018; Kamdar et al., 2019; Hawkins et al., 2020; Nelson et al., 2021). We
use the distance, [Fe/H]init, and [$\alpha$/Fe]init measured from the H3 stars
to investigate whether the companion stars are located close to the
corresponding MIST isochrone in the H-R diagram. We define color deviation
$\Delta($BP-RP$)$ by:

$\Delta({\rm BP}-{\rm RP})_{i}=({\rm BP}-{\rm RP})_{i}-({\rm BP}-{\rm
RP})_{iso,i},$ (2)

where ${\rm(BP-RP)}_{i}$ is the observed Gaia BP-RP color for star $i$ (a
non-H3 companion), and ${\rm(BP-RP)}_{iso,i}$ is the expected BP-RP color from
the MIST isochrone for this star given its absolute G-band magnitude. If the
companion star $i$ is located close to the expected isochrone, then
$\Delta{\rm(BP-RP)}_{i}\approx 0$. We use
brutus111https://github.com/joshspeagle/brutus (Speagle et al. submitted) to
interpolate the MIST v2.0 grid (Choi et al., 2016) given the [Fe/H]init,
[$\alpha$/Fe]init, stellar age, and $A_{V}$ measured from the H3 star. Then
for a non-H3 companion, its expected color ${\rm(BP-RP)}_{iso,i}$ is computed
from the MIST isochrone given its absolute G-band magnitude, where we assume
that the non-H3 companion has the same distance as the H3 star. This step also
requires that an absolute G-band magnitude only maps to a single BP-RP value,
which is satisfied for the main-sequence phase and the early stage of the
giant phase. For the late giant phase, one absolute G-band magnitude can
intersect an isochrone at multiple BP-RP colors. Therefore, we only compute
$\Delta{\rm(BP-RP)}_{i}$ for stars where their ${\rm(BP-RP)}_{iso,i}$ can be
uniquely determined, resulting in 546 pairs. Because brutus includes the
extinction when computing ${\rm(BP-RP)}_{iso,i}$, the resulting
$\Delta{\rm(BP-RP)}_{i}$ is extinction corrected.

Fig. 4 shows the distribution of $\Delta{\rm(BP-RP)}$ for the non-H3 members
of the wide binaries, $\Delta{\rm(BP-RP)}_{companion}$, and chance-alignment
stars, $\Delta{\rm(BP-RP)}_{chance}$. The chance-alignment stars are selected
by large proper motion differences $>5$ mas yr-1 and within 10 arcsec around
the H3 stars. The $\Delta{\rm(BP-RP)}_{companion}$ (red histogram) is
clustered around 0, indicating that these companions are located close to the
expected isochrone. The width of $\Delta{\rm(BP-RP)}_{companion}$ is mainly
due to the uncertainties of distance estimates and photometry. For comparison,
chance-alignment companions (black histogram) have a broad distribution of
$\Delta{\rm(BP-RP)}_{chance}$. This is because these chance-alignment stars in
general have different distances as the H3 stars, resulting in non-zero
$\Delta{\rm(BP-RP)}_{chance}$ due to their erroneous absolute G-band
magnitudes when we assume they have the same distances as the H3 stars.
Therefore, Fig. 4 suggests that the majority of the wide binary candidates are
indeed genuine wide binaries instead of chance-alignment stars.

Figure 3: The H-R diagram of the wide binaries. The blue points show the H3
targets in the wide binaries, and the red open circles are their companions.
Most of the H3 wide binaries are main-sequence stars. The sold green lines
connect four wide binaries at distances $>10$ kpc, and all of them are double-
giant wide binaries, with one having nearly identical evolutionary stages for
two member stars (giant twin).  Figure 4: The distribution of color deviation
$\Delta$(BP-RP), defined in Eq. 2. The color deviation of the companions
around H3 stars (red) is strongly concentrated around 0. In contrast, the
companions from chance-alignment pairs have a wide range of color deviations
(black). This comparison suggests that most of the wide binaries are genuine
and are not chance-alignment pairs.

###  4.3 Galactic kinematics

Figure 5: The Galactic orbits of stars in the $E_{tot}$-$L_{z}$ space. The H3
giant stars (black points) represent the disk and the kinematic substructures
in the halo, with labels highlighting the prominent structures. H3 wide
binaries are shown as red points. The empirical demarcation line used to
distinguish the disk and halo stars is in blue. Most of the wide binaries are
in the disk. There are 33 halo wide binaries, and most of them are associated
with the accreted Gaia-Sausage-Enceladus remnant, which dominates the halo
stars in the inner Milky Way.

Galactic kinematics is key to revealing substructures in the Milky Way’s
stellar halo (Johnston et al., 1996; Helmi & White, 1999; Bullock & Johnston,
2005; Cooper et al., 2010). Fig. 5 shows the H3 giants (black points) and the
wide binaries (red points) in the $E_{tot}$-$L_{z}$ space. We select H3 giants
(black points) by $\log g<3.5$ and adopt the same criteria in Sec. 2.1 and
Sec. 2.3 to ensure reliable kinematic measurements, except that we include
giant stars from the Sagittarius (Sgr_FLAG==1). Applying our wide binary
search to stars with Sgr_FLAG==1 results in two additional wide binaries, but
both of them do not satisfy the kinematics criterion in Sec. 2.3, making no
change to Fig. 5. The distance distributions vary in the $E_{tot}$-$L_{z}$
space. In particular, the thick disk stars are on average closer to the Sun
than the halo stars. Using the H3 giants yields a relatively unbiased view
with respect to the distances of the substructures in the halo out to $\sim
50$ kpc (Naidu et al., 2020).

Two main structures in Fig. 5 are the thick disk and the Gaia-Sausage-
Enceladus (the overdensity centered at $L_{z}=0$) remnant. There are more
other substructures related to previous accretion events in the halo detailed
in Naidu et al. (2020). These substructures mark the rich accretion history of
the Milky Way. In particular, Gaia-Sausage-Enceladus is from a merger event
with a mass ratio of 1:4 about 10 Gyr ago (Belokurov et al., 2018; Helmi et
al., 2018; Bonaca et al., 2020; Koppelman et al., 2020; Naidu et al., 2021),
accounting for $\sim 50$% of the halo population within 20 kpc of the inner
Milky Way (Naidu et al., 2020). The merger of Gaia-Sausage-Enceladus likely
dynamically heated up the Milky Way’s disk and formed the thick disk (Gallart
et al., 2019). Then the Sagittarius dwarf galaxy started infalling around 8
Gyr ago with a mass ratio of $\sim 1$% (Ibata et al., 1994; Law et al., 2010;
Dierickx & Loeb, 2017; Fardal et al., 2019). Since then, the Milky Way has
undergone a relatively quiescent evolution and grown the low-$\alpha$ (thin)
disk (Wyse, 2001, 2009; Ting & Rix, 2019; Bonaca et al., 2020).

In Fig. 5, we show all 798 wide binaries that pass the kinematics quality
criteria. We use the $E_{tot}$-$L_{z}$ plane to separate the halo and the
thick-disk populations. Due to H3’s target selection, 90% of the H3 stars have
Galactic heights $|Z|>1$ kpc, and therefore we refer to the disk stars in the
H3 as thick-disk stars. The solid blue line in Fig. 5 is our empirical
demarcation line, with thick disk stars located in the lower-left part and the
halo stars for the rest. The disk-star selection is: (1) $E_{tot}<-0.7\times
10^{5}$ km2 s-2; $\land$ (2) $L_{z}<0$; $\land$ (3)
$E_{tot,5}<-1.45-0.22L_{z,3}$, where $E_{tot,5}=E_{tot}/$($10^{5}$ km2 s-2)
$L_{z,3}=L_{z}/$($10^{3}$ kpc km s-1), and $\land$ is the Boolean ‘and’
operator. With this selection, we have 763 (96%) thick-disk wide binaries and
33 (4%) halo wide binaries. If we apply this disk-halo selection to the entire
H3 catalog with the same quality criteria, 16% of the H3 stars are classified
as halo stars. One main reason for the lower efficiency of identifying wide
binaries in the halo than in the disk is that halo stars are more distant,
making Gaia more difficult to resolve or to detect the companion. The other
reason is their different wide binary fractions, which will be detailed in the
next Section.

Fig. 6 shows the normalized distribution of surface iron abundances for disk
and halo wide binaries. Halo wide binaries have [Fe/H] peaking around $-1$,
similar to the metallicity of Gaia-Sausage-Enceladus remnant (Naidu et al.,
2020). Wide binaries in the thick disk have [Fe/H] centering around $-0.5$.
Both resolved triples belong to the disk with a relatively high [Fe/H] of
$-0.38$ and $+0.01$. The most metal-poor wide binaries in our sample are
[Fe/H]$=-2.7$ in the disk and [Fe/H]$=-2.5$ in the halo (the distant giant
twin in Fig. 3), among the most metal-poor wide binaries known today (Reggiani
& Meléndez, 2018; Hwang et al., 2021; Lim et al., 2021).

It is particularly interesting that many halo wide binaries have $E_{tot}$ and
$L_{z}$ associated with the Gaia-Sausage-Enceladus remnant. The stars from the
Gaia-Sausage-Enceladus are characterized by their high Galactic eccentricities
$e>0.7$ (Belokurov et al., 2018; Naidu et al., 2020). Indeed, 26 out of the 33
halo wide binaries have $e>0.7$, suggesting that they are from the Gaia-
Sausage-Enceladus remnant. The rest of 7 low-eccentricity ($e<0.7$) halo wide
binaries may belong to the in-situ halo, and 3 of them have [Fe/H] and
[$\alpha$/Fe] satisfying the criteria of the in-situ halo in Naidu et al.
(2020). Together with a recent finding of a wide binary associated with the
Sequoia remnant (Lim et al., 2021), our results unambiguously show the
existence of wide binaries among the accreted stars.

The two halo wide binaries having the largest $E_{tot}$ in Fig. 5 are the ones
with distances $>10$ kpc discussed earlier in Sec. 4.1 and 4.2. The one with
the largest $E_{tot}$ has a highly eccentric Galactic orbit ($e=0.92$) and may
be considered as Gaia-Sausage-Enceladus’ member in Naidu et al. (2020),
despite of its unusually high $E_{tot}$. The other one has a modest Galactic
eccentricity of 0.47 does not belong to the Gaia-Sausage-Enceladus. Its
kinematics is similar to the Sagittarius in $E_{tot}$-$L_{z}$ space, but its
$L_{y}$ and $L_{z}$ do not fall in the selection for Sagittarius stars in
Johnson et al. (2020).

There are three metal-poor wide binaries with [Fe/H]$<-2$ in the disk. All of
them are main-sequence stars with ages $\sim 10$ Gyr according to the
MINESweeper spectroscopic-photometric estimates, high [$\alpha$/Fe] (0.56,
0.54, 0.37), and prograde orbits ($L_{z}<0$). Although the sample is limited,
they may be related to the ancient Milky Way disk (Carter et al., 2021).

Figure 6: The metallicity distribution of the H3 wide binaries. The most
metal-poor wide binary in our sample has [Fe/H]$=-2.7$. The [Fe/H] of the halo
wide binaries is consistent with that of Gaia-Sausage-Enceladus.

##  5 The abundance dependence of the wide binary fraction

###  5.1 Bayesian formulation

Our purposes are twofold: we aim to (1) compare the halo and disk wide binary
fraction at a fixed metallicity, and to (2) investigate the metallicity trend
of the disk wide binary fraction. Due to the small sample size and the limited
metallicity range, we are not able to investigate the metallicity dependence
of the wide binary fraction within the halo sample. These investigations need
to take distances and masses into account. At a fixed binary separation, it is
more difficult to spatially resolve more distant binaries, and companions of
more distant binaries are more likely to fall below the magnitude limit of
Gaia. Furthermore, because H3 selects targets based on apparent magnitudes,
different metallicity and mass ranges are probed at different distances.
Therefore, to compare wide binary fractions at different metallicities, we
need to include the distance and mass dependence in the model.

For an H3 star $i$, we model the probability that this star has a wide binary
companion detected by Gaia by

$\begin{split}&p_{i}(w_{i}=1|\bm{\theta})\\\
&=f_{wb}({\rm[Fe/H]}_{i},{\rm[\alpha/Fe]}_{i})\times m_{i}^{a}\times
d_{i}^{b},\end{split}$ (3)

where $w_{i}$ is a Boolean value indicating if the star $i$ is a wide binary
($w_{i}=1$) or not ($w_{i}=0$) in our sample, and $m_{i}$ and $d_{i}$ are the
mass and the distance of the star $i$ in units of solar mass and kpc. The
probability that a star $i$ does not have a resolved Gaia companion is
$p_{i}(w_{i}=0|\bm{\theta})=1-p_{i}(w_{i}=1|\bm{\theta})$. $\bm{\theta}$ is
the set of the model parameters including the exponents $a$ and $b$ and other
parameters involved in $f_{wb}$ which will be introduced shortly. We adopt a
power-law form for the mass and distance dependence, with exponents $a$ and
$b$. The distance dependence comes from the selection efficiency due to the
angular resolution and the magnitude limit, and their relation depends on the
underlying binary separation distribution. We run a simulation for wide
binaries with companions following an initial mass function and a power-law
binary separation distribution, finding that a power-law form is a robust
description for the distance dependence that includes the effect of the
angular resolution and the magnitude limit.

$f_{wb}({\rm[Fe/H]}_{i},{\rm[\alpha/Fe]}_{i})$ in Eq. 3 is the wide binary
fraction at different chemical abundances for star $i$. We assume that the
[Fe/H] and [$\alpha$/Fe] dependences are independent and therefore the
function is separable:

$f_{wb}({\rm[Fe/H]}_{i},{\rm[\alpha/Fe]}_{i})=g({\rm[Fe/H]}_{i})\times
h({\rm[\alpha/Fe]}_{i}).$ (4)

To investigate the different iron abundance dependence between halo and disk
stars, we define $g({\rm[Fe/H]}_{i})$ as

$g({\rm[Fe/H]}_{i})=\begin{cases}g_{halo},&\text{if halo star}.\\\
g_{disk}({\rm[Fe/H]}_{i}),&\text{if disk star}.\end{cases}$ (5)

We use a single value $g_{halo}$ for the halo sample because of its small
sample size and limited abundance range. For disk stars, we consider a multi-
step function such that $g_{disk}({\rm[Fe/H]}_{i})=g_{disk,j}$ if
$b_{j}<{\rm[Fe/H]}_{i}<b_{j+1}$. Based on the sample size, the metallicity bin
$b_{j}$ is chosen to be [$-3$, $-2$, $-1$, $-0.5$, 0, 0.5]. In this way, we
can measure the disk wide binary fraction in each metallicity bin.
$g({\rm[Fe/H]}_{i})$ can be understood as the observed wide binary fraction
with separations of about $10^{3}$-$10^{4}$ AU in the H3 data for 1 M⊙ stars
at a distance of 1 kpc from the Sun.

The dependence on the $\alpha$-captured abundance is encoded in
$h({\rm[\alpha/Fe]}_{i})$in Eq. 4 by

$h({\rm[\alpha/Fe]}_{i})=1+c\times({\rm[\alpha/Fe]}_{i}-0.25),$ (6)

where $c$ is the only free parameter for the slope of the dependence on
[$\alpha$/Fe], and $c=0$ means no dependence on [$\alpha$/Fe]. Because [Fe/H]
is the main focus here, we choose a simpler form for [$\alpha$/Fe] than [Fe/H]
to reduce the number of free parameters involved, and we use the same
[$\alpha$/Fe] relation for both disk and halo stars.

Therefore, $\bm{\theta}$ in Eq. 3 includes 9 free parameters: $a$ and $b$ for
the mass and distance dependence, $c$ for the [$\alpha$/Fe] dependence (Eq.
6), one from $g_{halo}$, and 5 from $g_{disk,j}$. We do not marginalize the
uncertainties of stellar parameters ([Fe/H]i, etc.) because their
uncertainties from H3 are small and do not significantly affect the results.

The Bayesian formulation is

$\begin{split}p(\bm{\theta}|\\{w_{i}\\})\ \propto
p(\bm{\theta})\prod_{i}p(w_{i}|\bm{\theta}),\end{split}$ (7)

where $p(\bm{\theta}|\\{w_{i}\\})$ is the posterior distribution of the
parameter set $\bm{\theta}$ given $\\{w_{i}\\}$, the observed wide binary
identification for the H3 sample. We use uninformative flat priors for
$p(\bm{\theta})$. Eq. 7 can be rewritten as

$\begin{split}p(\bm{\theta}|\\{w_{i}\\})\ \propto\
\prod_{w_{i}==1}p_{i}(w_{i}=1|\bm{\theta})\
\prod_{w_{i}==0}\left[1-p_{i}(w_{i}=1|\bm{\theta})\right],\end{split}$ (8)

where $p_{i}(w_{i}=1|\bm{\theta})$ is from Eq. 3. We use emcee (Foreman-Mackey
et al., 2013), an affine-invariant ensemble sampler for Markov chain Monte
Carlo, to sample the posteriors of Eq. 8. To distinguish halo and disk stars,
we require the sample to have good Galactic kinematics with criteria described
in Sec. 2.3. We limit the sample to heliocentric distances $<5$ kpc because
wide binary detectability beyond this distance is low and may not follow the
power-law distance dependence we adopt. Two resolved triples are excluded from
this analysis. Therefore, this analysis has 8610 halo stars (23 halo wide
binaries) and 80572 disk stars (757 disk wide binaries). The resulting corner
plot (the two-dimensional posterior distribution) made using corner (Foreman-
Mackey & Daniel, 2016) is shown in the Appendix, Fig. 1.

Our formulation includes the mass and distance dependence of binary
detectability in Eq. 3. This is important because halo and disk stars have
different distance and mass distributions due to their physical differences
and H3’s target selection. This formulation assumes that the wide binary
fractions of all stars follow the same mass, distance, and [$\alpha$/Fe]
dependence. This implicitly assumes that the mass-ratio distribution is not a
strong function of metallicities and binary separations. This mass-ratio
assumption is validated by Moe & Di Stefano (2017) that the mass-ratio
distribution for binaries at $>200$ AU is consistent with random pairings from
the initial mass function. In addition, we assume that the [Fe/H] and
[$\alpha$/Fe] are independent and therefore are separable in Eq. 4. Future
studies are needed to investigate these assumptions.

###  5.2 Wide binary fraction in the halo and the disk, and its metallicity
dependence

Fig. 7 shows the wide binary fraction as a function of [Fe/H]. The red circle
and blue triangles show the measurements for the halo and the thick disk,
respectively. Their [Fe/H] values are the mean metallicities in each sample,
and the horizontal error bar of the halo sample indicates the 16-84
percentiles of its metallicity distribution. Their wide binary fractions are
$g_{halo}$ and $g_{disk}$([Fe/H]) in Eq. 4, defined as the observed values at
1 kpc for 1 M⊙ stars. The wide binary fractions and their errors in Fig. 7 are
the 16-50-84 percentiles of their marginalized distributions.

For comparison, in Fig. 7 we overplot the results from Hwang et al. (2021) as
black crosses where the 500-pc sample is dominated by the thin-disk stars and
the wide binaries refer to separations between $10^{3}$ and $10^{4}$ AU, with
[Fe/H] values indicating the centers of the bins. The wide binary fractions of
the disk and halo stars from this work (red circle and blue triangles) can be
compared with each other in the absolute sense, meaning that the halo wide
binary fraction is consistent with that in the thick disk at a fixed [Fe/H],
taking into account the differences in the distance and mass distributions.
Although the thick-disk wide binary fractions from this work and those from
Hwang et al. (2021) happen to have similar values in Fig. 7, they should only
be compared in a relative sense (i.e. the overall metallicity trend) but not
in an absolute sense, because they use samples that have different detection
limits for the companions due to the different distance, binary separation,
and mass ranges.

Fig. 7 shows that the wide binary fraction in the thick disk decreases towards
the low-metallicity end. This is similar to the trend in the thin disk (Hwang
et al., 2021), but now this work extends the trend down to [Fe/H]$<-2$. At
[Fe/H]$>-0.5$, the H3 sample shows a flat relation between the wide binary
fraction and the metallicity, in contrast to the peak at [F/H]$=0$ for the
thin-disk stars. This is consistent with Hwang et al. (2021) where we found
that the peak at [F/H]$=0$ is more prominent for younger disk stars.

Hwang et al. (2021) differentiated the thick-disk stars and halo stars using
total 3-dimensional velocities and the maximum Galactic heights of the stellar
orbits. We found that at a fixed metallicity, the wide binary fraction of the
halo seems consistent with that of the thick disk. However, there are only two
halo wide binary candidates in Hwang et al. (2021), so the conclusion is still
limited by small number statistics.

Here with a better Galactic kinematics selection and a larger halo wide binary
sample from the H3 survey, we robustly show that the wide binary fraction of
the halo stars is consistent with that of the disk stars at a fixed
metallicity. This result suggests that iron abundance is the main driver of
the wide binary fraction, and that the disk and halo stars do not have
significantly different wide binary fractions once the metallicity, distance,
and mass are taken into account.

Figure 7: The wide binary fraction as a function of [Fe/H]. The blue and red
markers show the results from this study using the H3 survey, with [Fe/H]
values indicating the mean metallicity in each sample. The red point is for
the halo stars, and its horizontal error bar indicates the 16-84 percentiles
of its metallicity distribution. The blue triangles are for the thick-disk
stars. The black crosses show the results from a thin-disk-dominated sample
(Hwang et al., 2021). This plot shows that the wide binary fractions of the
halo and the thick disk are consistent at a fixed [Fe/H]. The different trend
between the thin and thick disk at [Fe/H]$>-0.5$ is mainly due to the younger
stellar population in the thin disk (Hwang et al., 2021).

The [$\alpha$/Fe] dependence is an important clue about binary formation.
Mazzola et al. (2020) find that the close binary ($a<$10 AU) fraction anti-
correlates with [$\alpha$/Fe]. This relationship may arise because the dust
formed from $\alpha$-captured elements affects the fragmentation of the
protostellar disks by changing its opacity. In contrast, Niu et al. (2021)
consider the total binary fraction and suggest a tentative positive
correlation between the binary fraction and [$\alpha$/Fe] at a fixed [Fe/H].

In contrast to the previous studies, here we investigate the [$\alpha$/Fe]
dependence of the wide binary fraction at separations $>10^{3}$ AU. The
parameter for the [$\alpha$/Fe] dependence in Eq. 6 is measured to be
$c=-0.05^{+0.06}_{-0.14}$, where the errors correspond to 16-84 percentiles.
Therefore, $c$ is consistent with zero and the wide binary fraction does not
appear to depend on the $\alpha$-captured abundances. The different
[$\alpha$/Fe] dependence of the close and wide binary fraction suggests that
different formation mechanisms are responsible for close and wide binaries,
similar to the conclusions from the [Fe/H] dependence (El-Badry & Rix, 2019;
Hwang et al., 2021).

##  6 Discussion

###  6.1 Comparison with the literature

In the pre-Gaia era when high-precision parallaxes were not available for most
of the stars, the reduced proper motion diagram was used to distinguish high-
velocity halo stars from lower-velocity disk stars and to identify halo wide
binaries. The reduced proper motion diagram uses proper motions as a proxy for
distances because closer stars have larger proper motions, and halo stars are
more separated from the disk stars because of halo stars’ lower metallicities
and higher spatial velocities (Salim & Gould, 2002). Using this method,
previous studies have found 100-200 halo(-like) wide binary candidates
(Chaname & Gould, 2004; Quinn & Smith, 2009; Allen & Monroy-Rodríguez, 2014;
Coronado et al., 2018). Wide binaries in the halo can be used to constrain the
mass of massive compact halo objects – for example, the contribution of free-
floating black holes and neutron stars to the mass budget of the Universe (Yoo
et al., 2004; Quinn et al., 2009; Monroy-Rodríguez & Allen, 2014).

In addition to proper motions, the reduced proper motion diagram uses
photometry to distinguish halo and disk stars. However, because lower-
metallicity stars have a higher close binary fraction (Moe et al., 2019), and
close binaries are more likely to have wide companions (Hwang et al., 2020),
metal-poor stars selected from the reduced proper motion diagram may have
binary-related selection effects because unresolved companions make the
systems deviate from the reduced proper motion diagram.

Gaia provides high-precision astrometry for billions of stars (Gaia
Collaboration et al., 2016), making the search for a large number of wide
binaries possible (Oh et al., 2017; El-Badry & Rix, 2018; Jiménez-Esteban et
al., 2019; Hartman & Lépine, 2020). Using Gaia EDR3, El-Badry et al. (2021)
have cataloged about a million wide binaries within 1 kpc. By cross-matching
this catalog the with LAMOST survey DR6 (Deng et al., 2012; Zhao et al., 2012)
and using LAMOST’s radial velocities to compute $E_{tot}$ and $L_{z}$, we find
about 50 halo wide binaries that satisfy the halo selection from Sec. 4.3 and
SNR $>10$ in g-band from LAMOST with a lower spectral resolution ($R\sim
1800$) than H3.

Our wide binaries feature the most distant ($>1$ kpc and four candidates at
$>10$ kpc), most metal-poor (down to [Fe/H]$=-2.7$) sample with detailed
abundance and kinematic measurements from a $R=32,000$ spectroscopy. The H3
survey is critical in providing a homogeneous sample of disk and halo stars to
compare the wide binary fractions among various subsamples. Since our wide
binary search depends only on astrometry and not on photometry, our selections
are less affected by unresolved companions. Our halo classifications and the
association with Gaia-Sausage-Enceladus remnant are robust thanks to the
precise $E_{tot}$ and $L_{z}$ measurements from H3.

###  6.2 Origin of the iron-abundance dependence

Hwang et al. (2021) identified 7671 wide binaries within 500 pc with projected
separations between $10^{3}$ and $10^{4}$ AU. These wide binaries are mostly
thin- and thick-disk stars. With the iron abundances measured from LAMOST (Wu
et al., 2011a, b; Ting et al., 2019; Xiang et al., 2019), we found that the
wide binary fraction at $10^{3}$-$10^{4}$ AU peaks at around [Fe/H]$=0$, and
decreases towards both low and high metallicity ends (black symbols in Fig.
7).

Hwang et al. (2021) proposed several possibilities to explain such metallicity
dependence. The positive correlation between the metallicity and the wide
binary fraction at [Fe/H]$<0$ may be due to the denser formation environments
in the earlier Universe that tend to disrupt the wide binaries. At [Fe/H]$>0$,
the anti-correlation between metallicity and the wide binary fraction may be
due to some wide binaries forming from the dynamical unfolding of compact
triples (Reipurth & Mikkola, 2012; Elliott & Bayo, 2016), and thus they
inherit the metallicity dependence from the close binary fraction (Raghavan et
al., 2010; Yuan et al., 2015; Badenes et al., 2018; Moe et al., 2019; El-Badry
& Rix, 2019). The radial migration of Galactic orbits (Sellwood & Binney,
2002) may cause such non-monotonic metallicity dependence, either due to
different formation environments at different galactocentric radii or due to
disruption of wide binaries during the enhanced interaction with high-density
gas clumps during the migration process. Wide binary disruptions by passing
stars are unlikely to play an important role because the disruption timescale
of $\sim 100$ Gyr for $\sim 1000$ AU binaries is too long (Weinberg et al.,
1987).

If the metallicity dependence of the wide binary fraction is due to the radial
migration, which is more important for disk stars, we would expect a higher
wide binary fraction in the halo than in the disk. Our H3 results suggest that
iron abundance is the main driver for the wide binary fraction (Fig. 7),
instead of their origin (disk versus halo) nor $\alpha$-captured abundance.
The similar wide binary fractions in the halo and the disk implies that radial
migration is not the main cause for the metallicity dependence of the wide
binary fraction at [Fe/H]$<0$.

Our leading explanation for the iron abundance dependence of the wide binary
fraction at [Fe/H]$<0$ is that the formation environments are denser at lower
[Fe/H], which are more disruptive for wide binaries. In particular, studies
have found that the star formation environments have higher pressures and
densities at higher redshifts, and higher-mass clusters tend to form in such
environments (Harris & Pudritz, 1994; Elmegreen & Efremov, 1997; Kravtsov &
Gnedin, 2005; Kruijssen, 2014; Ma et al., 2020). Therefore, the lower wide
binary fraction may directly reflect their denser formation environments in
the earlier epochs.

Although a lower-metallicity population is on average older than a higher-
metallicity sample (Casagrande et al., 2011; Bensby et al., 2014; Silva
Aguirre et al., 2018), our result is better explained by the metallicity
dependence instead of the age dependence. Following Bonaca et al. (2020), we
select main-sequence turn-off stars by $3.8<\log\ g<4.3$ and SNR$>10$ for
their precise age measurements. This sample shows that, for our H3 thick-disk
sample, the mean stellar age changes approximately linearly from $12$ Gyr at
[Fe/H]$=-2.0$ dex (Carter et al., 2021) to $8.5$ Gyr at [Fe/H]$=0$ dex. This
does not explain the metallicity trend in Fig. 7 where the wide binary
fraction becomes flat at [Fe/H]$>-1$ when the mean age changes by about 2 Gyr
from [Fe/H]$=-1$ to [Fe/H]$=0$. Therefore, our results suggest that stellar
ages play a minor role.

While halo wide binaries are currently not susceptible to disruptions via
radial migration and giant molecular cloud, they were once in presumably gas-
rich environments in the progenitor (dwarf) galaxies. It is intriguing that
halo and thick-disk wide binary fractions are similar at a fixed metallicity
despite their potentially different formation environments. The similar wide
binary fractions between the thick disk and the halo may hint that the
metallicity and the formation environment properties are tightly correlated,
and by controlling the metallicity, we mitigate the differences in their
formation environments. One important formation environment property is the
cluster mass function because wide binaries are sensitive to the density
(which correlates with mass) of clusters (e.g. Parker et al. 2009; Geller et
al. 2019). Therefore, one possibility is that the Milky Way and the Gaia-
Sausage-Enceladus dwarf may have similar cluster mass functions at a fixed
metallicity (Krumholz et al. 2019), resulting in the similar wide binary
fractions.

Since wide binaries are unlikely to survive in globular clusters (e.g. Ivanova
et al. 2005), our results can place a rough constraint on the contribution of
disrupted globular clusters to the stellar halo. Here we find that the wide
binary fraction of the halo is consistent with that of the disk (Fig. 7).
Assuming that the original halo wide binary fraction (halo stars that are not
from disrupted globular clusters) is the same as that in the disk at a fixed
metallicity and that the wide binary fraction from disrupted globular clusters
is zero, our result implies that the disrupted globular clusters contribute
less than $\sim 50$% of the halo. If the disrupted globular clusters
contribute more than $50$% of the halo, the halo wide binary fraction would be
significantly lower than that of the disk, inconsistent with our results. The
$<50$% constraint is consistent with other studies finding that $\sim 10$% of
halo stars originate in disrupted globular clusters (Martell & Grebel, 2010;
Martell et al., 2011, 2016; Koch et al., 2019), or likely an even smaller
fraction (Naidu et al., 2020). Our constraint serves as a useful independent
check but is likely tentative because of the various assumptions used, and it
may be possible that wide binaries can form during globular cluster disruption
(Peñarrubia, 2021).

##  7 Conclusions

What determines the wide binary fraction remains not well understood. Hwang et
al. (2021) found that the wide binary fraction of disk stars peaks around
solar metallicity ([Fe/H]$=0$) and decreases toward both low and high
metallicity end. We proposed three possible scenarios that may explain this
metallicity dependence, including the radial migration of Galactic orbits. In
this paper, armed with the thick-disk and halo wide binaries identified from
the H3 survey, we rule out the radial migration scenario at [Fe/H]$<0$. Our
findings include:

  1. 1.

With Gaia’s astrometry, we search for resolved wide companions around the H3
stars (Fig. 1). Our search results in a total of 800 wide binaries and two
resolved triples. The photometry of the non-H3 companions are consistent with
the isochrones of the H3 stars, suggesting that most of them are genuine wide
binaries instead of chance-alignment pairs (Fig. 4).

  2. 2.

Most of the wide binaries are main-sequence stars (Fig. 3), with binary
separations ranging from $10^{3}$ to $10^{5}$ AU and heliocentric distances
$\gtrsim 1$ kpc (Fig. 2). These wide binaries have [Fe/H] down to $-2.7$ (Fig.
6), among the most metal-poor wide binary currently known. There are four
distant wide binary candidates at $>10$ kpc. They are all rare double-giant
wide binaries, and one of them is an unusual metal-poor ([Fe/H]$=-2.5$) giant
twin where the member stars have very similar photometry (Fig. 3).

  3. 3.

Based on the Galactic kinematics, we classify the wide binaries into disk and
halo population (Fig. 5), resulting in a sample of 33 halo wide binaries. The
majority of these halo wide binaries have kinematics associated with the
accreted Gaia-Sausage-Enceladus remnant.

  4. 4.

The wide binary fraction in the disk decreases toward the low-[Fe/H] end (Fig.
7), consistent with the result from Hwang et al. (2021). We do not find a
significant dependence of the wide binary fraction on [$\alpha$/Fe]. Both
[Fe/H] and [$\alpha$/Fe] dependences of the wide binary fraction are different
from those of the close binary fraction, suggesting that different formation
mechanisms are responsible for close and wide binaries.

  5. 5.

We show that the wide binary fraction in the halo is consistent with that in
the thick disk at a fixed [Fe/H] (Fig. 7). Since halo stars are not subject to
radial migration, the similar wide binary fractions in the thick disk and the
halo imply that radial migration of Galactic orbits plays a minor role in the
metallicity dependence of the wide binary fraction at [Fe/H]$<0$. Our study
favors the explanation that lower-metallicity formation environments have
higher stellar densities that tend to disrupt wide binaries, thus lowering the
wide binary fraction.

The data underlying this article are available in the article and in its
online supplementary material.

HCH appreciates the inspiring discussion with Yao-Yuan Mao on the dark matter
subhalos, and with Amina Helmi on exploring other Galactic orbital parameters.
HCH acknowledges support from Space@Hopkins and the Infosys Membership at the
Institute for Advanced Study. YST acknowledges financial support from the
Australian Research Council through DECRA Fellowship DE220101520. The H3
Survey acknowledges funding from NSF grant AST-2107253.

Facilities: Gaia, MMT.

Software: IPython (Pérez & Granger, 2007), jupyter (Kluyver et al., 2016),
Astropy (Astropy Collaboration et al., 2013, 2018), numpy (Harris et al.,
2020), scipy (Virtanen et al., 2020), emcee (Foreman-Mackey et al., 2013;
Foreman-Mackey & Daniel, 2016), matplotlib (Hunter, 2007).

##  Appendix A The posterior distributions for the binary fraction model

Fig. 1 shows the corner plot of the model parameters described in Sec. 5.1.
Their measurements (marginalized 16-50-84 percentiles) are: $g_{halo}$
$=0.012^{+0.003}_{-0.002}$, $g_{disk,0}$ $=0.013^{+0.008}_{-0.005}$,
$g_{disk,1}$ $=0.012^{+0.003}_{-0.002}$, $g_{disk,2}$
$=0.026^{+0.002}_{-0.002}$, $g_{disk,3}$ $=0.027^{+0.002}_{-0.002}$,
$g_{disk,4}$ $=0.021^{+0.003}_{-0.003}$, a $=0.661^{+0.225}_{-0.252}$, b
$=-1.326^{+0.070}_{-0.055}$, c $=-0.052^{+0.063}_{-0.142}$.

Figure 1: Corner plot for the posterior distributions of the wide binary
fraction model in Sec. 5.1.

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