#  The Kinematics, Metallicities, and Orbits of Six Recently Discovered
Galactic Star Clusters with Magellan/M2FS Spectroscopy††thanks: This paper
presents data gathered with the Magellan Telescopes at Las Campanas
Observatory, Chile.

Andrew B. Pace,1 Sergey E. Koposov,2,3,4,1 Matthew G. Walker,1 Nelson
Caldwell,5 Mario Mateo, 6 Edward W. Olszewski, 7 Ian U. Roederer, 6,8 John I.
Bailey, III, 9 Vasily Belokurov,3 Kyler Kuehn, 10 Ting S. Li, 11 Daniel B.
Zucker 12,13
1McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Ave,
Pittsburgh, PA 15213, USA
2Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
4Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
5Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS-15,
Cambridge, MA 02138, USA
6Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
7Steward Observatory, The University of Arizona, 933 N. Cherry Avenue, Tucson,
AZ 85721, USA
8Joint Institute for Nuclear Astrophysics – Center for the Evolution of the
Elements (JINA-CEE), USA
9Department of Physics, UCSB, Santa Barbara, CA 93016, USA
10 Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff, AZ 86001, USA
11Department of Astronomy and Astrophysics, University of Toronto, 50 St.
George Street, Toronto ON, M5S 3H4, Canada
12 School of Mathematical and Physical Sciences, Macquarie University, Sydney,
NSW 2109, Australia
13 Macquarie University Research Centre for Astronomy, Astrophysics &
Astrophotonics, Sydney, NSW 2109, Australia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present Magellan/M2FS spectroscopy of four recently discovered Milky Way
star clusters (Gran 3/Patchick 125, Gran 4, Garro 01, LP 866) and two newly
discovered open clusters (Gaia 9, Gaia 10) at low Galactic latitudes. We
measure line-of-sight velocities and stellar parameters ([Fe/H], $\log{g}$,
$T_{\rm eff}$, [Mg/Fe]) from high resolution spectroscopy centered on the Mg
triplet and identify 20-80 members per star cluster. We determine the
kinematics and chemical properties of each cluster and measure the systemic
proper motion and orbital properties by utilizing Gaia astrometry. We find
Gran 3 to be an old, metal-poor (mean metallicity of ${\rm[Fe/H]}=-1.83$)
globular cluster located in the Galactic bulge on a retrograde orbit. Gran 4
is an old, metal-poor (${\rm[Fe/H]}=-1.84$) globular cluster with a halo-like
orbit that happens to be passing through the Galactic plane. The orbital
properties of Gran 4 are consistent with the proposed LMS-1/Wukong and/or
Helmi streams merger events. Garro 01 is metal-rich (${\rm[Fe/H]}=-0.30$) and
on a near circular orbit in the outer disk but its classification as an open
cluster or globular cluster is ambiguous. Gaia 9 and Gaia 10 are among the
most distant known open clusters at $R_{GC}\sim 18,~{}21.2~{}\mathrm{\,kpc}$
and most metal-poor with [Fe/H] $\sim-0.50,-0.34$ for Gaia 9 and Gaia 10,
respectively. LP 866 is a nearby, metal-rich open cluster ([Fe/H]$=+0.10$).
The discovery and confirmation of multiple star clusters in the Galactic plane
shows the power of Gaia astrometry and the star cluster census remains
incomplete.

###### keywords:

Stars: kinematics and dynamics – globular clusters: general – open clusters
and associations: general

††pubyear: 2023††pagerange:  The Kinematics, Metallicities, and Orbits of Six
Recently Discovered Galactic Star Clusters with Magellan/M2FS
Spectroscopy††thanks: This paper presents data gathered with the Magellan
Telescopes at Las Campanas Observatory, Chile.– The Kinematics, Metallicities,
and Orbits of Six Recently Discovered Galactic Star Clusters with
Magellan/M2FS Spectroscopy††thanks: This paper presents data gathered with the
Magellan Telescopes at Las Campanas Observatory, Chile.

##  1 Introduction

Star clusters are among the smallest stellar structures in the universe and
are a key component of hierarchical structure assembly. They are valuable for
studying stellar populations and their evolution at a variety of ages,
metallicities, and environs (e.g., Krumholz et al., 2019; Adamo et al., 2020).
Star clusters in the Milky Way (MW) are typically divided into two categories:
the older, denser, and more luminous globular clusters (Gratton et al., 2019),
and the younger clusters in the MW disk, referred to as open clusters (e.g.,
Cantat-Gaudin, 2022).

While the census of bright halo clusters is mostly complete (Webb & Carlberg,
2021), there has been a number of faint star clusters discovered in optical
wide-field imaging surveys, pushing the luminosity and surface brightness
boundary (e.g, Koposov et al., 2007; Belokurov et al., 2014; Torrealba et al.,
2019; Mau et al., 2019; Cerny et al., 2023). The census of star clusters in
the MW mid-plane is incomplete due to the high extinction and large stellar
foreground. With recent near-infrared surveys such as the VISTA Variables in
the Via Láctea Survey (VVV) (e.g., Minniti et al., 2011; Garro et al., 2020)
and astrometric data from the Gaia mission the number of star cluster
candidates has significantly increased (e.g., Koposov et al., 2017; Torrealba
et al., 2019; Garro et al., 2020; Gran et al., 2022). However, a number of the
star cluster candidates found pre-Gaia DR2 have been shown to be false
positives once proper motions and kinematics are considered (Gran et al.,
2019; Cantat-Gaudin & Anders, 2020).

Ultimately, stellar spectroscopy is required to validate candidate star
clusters and confirm they are not a mirage of MW stars (e.g., Gran et al.,
2022). Furthermore, spectroscopic radial velocities and metallicities will
identify star cluster members. With the systemic radial velocity, the orbit
and origin of a star cluster can be determined (e.g., Massari et al., 2019;
Kruijssen et al., 2019)and the internal dynamics analyzed with large radial
velocity samples (e.g., Baumgardt & Hilker, 2018; Garro et al., 2023).
Spectroscopic metallicities can assist in determining the classification and
origin of a star cluster (e.g., Gran et al., 2022).

We present spectroscopic confirmation of three recently discovered globular
cluster candidates (Gran 3, Gran 4, Garro 01) and three newly discovered open
clusters (Gaia 9, Gaia 10, LP 866). In Section 2, we discuss our search
algorithm and independent discovery of the star cluster candidates with Gaia
DR2. In Section 3, we discuss our spectroscopic observations, velocity and
metallicity measurements, and the auxiliary data analyzed. In Section 4, we
identify members of each star cluster, measure the general kinematic and
metallicity properties, measure the spatial distribution, and determine the
orbital properties. In Section 5, we analyze the globular cluster internal
kinematics, compare the globular clusters to other MW globular clusters,
discuss the origin and potential association to accretion events, analyze the
open clusters in the context of the Galactic metallicity gradient, and compare
our results to the literature. We summarize our conclusions in Section 6.

##  2 Discovery and Candidate identification

The search for the stellar overdensities was carried out in 2018 after the
release of the Gaia DR2 using the satellite detection pipeline broadly based
on the methods presented in Koposov et al. (2008); Koposov et al. (2015), but
extended into space of proper motions.

We describe here briefly the basics behind the detection algorithm, leaving a
more detailed description to a separate contribution (Koposov et al. in
prep.).

The algorithm consists of several steps.

  * •

Looping over all proper motions. In the search used here we ran overdensity
search for subsets of stars with proper motions $|\mu_{\alpha}-X_{i}|<1$,
$|\mu_{\delta}-X_{j}|<1$ where $X_{i}$,$X_{j}$ span the range of proper
motions from -15 to 15 mas/year with 1 mas/year steps.

  * •

Looping over all possible distance moduli to overdensities. We perform the
search for stars selected based on an isochrone filter placed at distances
from $\sim$ 6 kpc to 160 kpc. We used the extinction corrected BP, RP and G
magnitudes and an old metal-poor PARSEC isochrone (Bressan et al., 2012) with
an age of 12 Gyr and $[{\rm Fe/H}]=-2$. We also run a single search without
any isochrone colour magnitude selection.

  * •

Looping over overdensity sizes from 3 arcmin to 48 arcmin.

  * •

Segmentation of the sky into HEALPIX (Górski et al., 2011) tiles. To avoid
having to work with the dataset for the entire sky the overdensity search
algorithm works with approximately rectangular-shaped HEALPIX tiles, that we
also increase in size by 20% with respect to the standard HEALPIX scheme to
ensure overlap between tiles and avoid dealing with edge effects. The exact
Nside resolution parameter and pixel scale of tiling were different for
different runs depending on memory limitations and size of the overdensity
being searched for.

  * •

Creation of a pixelated stellar density map inside each tile. We use
tangential projection to map the stars into a rectangular x,y pixel grid, and
then make a 2-D histogram of stellar counts of stars selected by proper
motions, colours and magnitudes.

  * •

Creation of a stellar overdensity significance map based on a stellar count
map. This step is described below in more detail.

  * •

Identification of overdensities based on the significance map and merging of
candidate lists from various search configurations.

  * •

Cross-matching the candidate lists with external catalogues and construction
of validation plots.

Below we provide a brief description of the algorithm that provides an
overdensity significance map given the binned 2D stellar density map. The
algorithm requires three main parameters – the kernel size, which corresponds
to the size of the overdensity $k$ we are looking for, and two background
apertures, $b_{1}$ and $b_{2}$. Here we assume that we have a rectangular grid
of stellar number counts $H(x,y)$ and we estimate the significance of the
overdensity at pixel x=0, y=0. The key difference of our approach compared to
previous approaches (i.e., Koposov et al., 2008) is that we do not rely on the
assumption of Gaussianity or even a Poisson distribution of number counts in
the map.

We first compute the number of stars $N$ in a circular aperture with radius k
around the pixel 0,0. We then need to characterize what is the probability
distribution of $P_{null}(N)$ under a null hypothesis of no overdensity to
compute the tail probability/significance.

Our model for $P_{null}(N)$ is the negative-binomial distribution, which is a
discrete Poisson-like distribution (that is, an infinite mixture of Poisson
distributions with means having a Gamma distribution). The negative binomial
distribution can be parameterized with mean $\mu$ and $\sigma^{2}$, where
$\sigma^{2}\geq\mu$ (note that the variance of the negative-binomial
distribution is a parameter, as opposed to a Poisson distribution, where it is
equal to the mean). The $\mu$ and $\sigma$ are estimated based on the number
count distribution between two background apertures $b_{1}$ and $b_{2}$.

The significance (or the Z-score) in each pixel is then assigned as
$Z=F^{-1}(P_{null}(\geq N))$, where $F^{-1}()$ is the inverse of the CDF of a
normal distribution. The significant overdensities are selected as those where
Z is larger than a certain threshold. For this work we used $Z>6$ selected
candidates.

The application of the algorithm summarized above to Gaia DR2 in 2018 yielded
a few hundred significant distinct overdensities. The absolute majority of
them were known, but around 30 objects were deemed to be likely real dwarf
galaxies or globular clusters and were selected for further inspection. The
spectroscopic follow-up of six of these objects is the subject of this paper.
Some objects from the list, such as the Eridanus IV object (Cerny et al.,
2021) have been discovered and independently followed up since.

We began our spectroscopic follow-up in 2018 and note that 4 star clusters in
our sample have since been independently discovered. We refer to these
clusters by their name in the first discovery analysis. Garro 01 was
independently discovered by Garro et al. (2020) in the near-IR VISTA Variables
in the Via Láctea Extended Survey (VVVX). Gran et al. (2022) independently
discovered Gran 3 (also known as Patchick 125) and Gran 4 with Gaia DR2
astrometry, and they were confirmed with the VVV survey. Gran 3 was
independently discovered by the amateur astronomer Dana Patchick and named
Patchick 125. We used the literature open cluster compilation from Hunt &
Reffert (2023) to search for literature cross matches for our open clusters
which includes most post-Gaia open cluster discoveries (e.g., Bica et al.,
2019; Liu & Pang, 2019; Cantat-Gaudin & Anders, 2020; Kounkel et al., 2020;
Castro-Ginard et al., 2022). One cluster, internally KGO 8, was independently
discovered by Liu & Pang (2019) and referred to as LP 866 (although in some
catalogs it is referred to as FoF 866) and Kounkel et al. (2020) and referred
to as Theia 4124. We refer to this open cluster as LP 866 here. The two
remaining open clusters in our sample are new discoveries, and we name them
Gaia 9 and Gaia 10.

##  3 Spectroscopic Follow-Up

Table 1: Spectroscopic Observations of Star Clusters. object | R.A. (deg) | Dec. (deg) | Telescope/Instrument | UT Date | Exp. Time | $N_{\mathrm{obs}}$ | $N_{\mathrm{good}}$
---|---|---|---|---|---|---|---
Gran 3 | $256.135833$ | $-35.471278$ | Magellan/M2FS | 2018-08-11 | 6900 | 54 | 41
Gran 4 | $278.092083$ | $-23.211194$ | Magellan/M2FS | 2018-08-13 | 6100 | 118 | 115
Gran 4 | $278.112121$ | $-23.103756$ | AAT/AAOmega | 2018-06-24 | 1800 | 67 | 67
Garro 01 | $212.246250$ | $-65.738333$ | Magellan/M2FS | 2018-08-11 | 5400 | 205 | 193
Gaia 9 | $119.607083$ | $-38.984639$ | Magellan/M2FS | 2018-12-06 | 5400 | 96 | 50
Gaia 10 | $121.172500$ | $-38.984444$ | Magellan/M2FS | 2018-08-15 | 5800 | 56 | 43
LP 866 | $261.651250$ | $-39.280889$ | Magellan/M2FS | 2018-12-05 | 7200 | 164 | 160

###  3.1 Spectroscopic targeting

The possible member stars from candidate stellar overdensities discovered were
selected for spectroscopic observations by using the information about the
objects that was available from their detection, such as approximate object
angular size and proper motion. We did not have a uniform target selection
strategy from object to object, so we provide a broad overview of the
selection. We typically targeted stars using Gaia DR2 astrometry and
photometry, selecting stars with proper motions within 1-3 mas/yr of the
center of the detection. We applied the astrometric_excess_noise <1 cut and
selected stars with small parallaxes $\varpi<{\rm
Max}(0.1,3\,\sigma_{\varpi})$. Since the majority of followed-up overdensities
had small angular sizes we tried to maximise the number of fibers on each
object by assigning higher priority to central targets. We also did not apply
any colour-magnitude or isochrone selection masks to the targets, other than a
magnitude limit to ensure sufficient signal to noise, and prioritising
brighter stars, such as $G<16-18$.

###  3.2 M2FS Spectroscopy

We present spectroscopic observations of six star cluster candidates that we
obtained using the Michigan/Magellan Fiber System (M2FS; Mateo et al., 2012)
at the 6.5-m Magellan/Clay Telescope at Las Campanas Observatory, Chile. M2FS
deploys 256 fibers over a field of diameter $0.5^{\circ}$, feeding two
independent spectrographs that offer various modes of configuration. We used
both spectrographs in identical configurations that provide resolving power
$\mathcal{R}\sim 24,000$ over the spectral range $5130-5190$ Å. For all six
clusters, Table 1 lists coordinates of the M2FS field center, UT date and
exposure time of the observation, the number of science targets and the number
that yielded ‘good’ observations that pass our quality-control criteria.

We process and model all M2FS spectra using the procedures described in detail
by Walker et al. (2023). Briefly, we use custom Python-based software to
execute standard processing steps (e.g., overscan, bias and dark corrections),
to identify and trace spectral apertures, to extract 1D spectra, to calibrate
wavelengths, to correct for variations in pixel sensitivity and fiber
throughput, and finally to subtract the mean sky level measured from $\sim 20$
fibers per field that are pointed toward regions of blank sky. To each
individually-processed spectrum, we fit a model based on a library of
synthetic template spectra computed on a regular grid of stellar-atmospheric
parameters: effective temperature ($T_{\rm eff}$), surface gravity ($\log g$),
metallicity ([Fe/H]) and magnesium abundance ([Mg/Fe]). Including parameters
that adjust the resolution and continuum level of the template spectra, our
spectral model has 16 free parameters. We use the software package MultiNest
to draw random samples from the 16-dimensional posterior probability
distribution function (PDF) (Feroz & Hobson, 2008; Feroz et al., 2009). We
summarize 1D posterior PDFs for each of the physical parameters according to
the mean and standard deviation of the sample returned by MultiNest.

We consider stars with ${\rm S/N}>0$ and $\sigma_{v_{\rm
los}}>5\mathrm{\,km}\mathrm{\,s}^{-1}$ as good quality measurements (Walker et
al., 2023). We consider stars with ${\rm S/N}>2$ to be good quality [Fe/H]
measurements. Our selection of cuts for good quality [Fe/H] is based on repeat
measurements and dwarf galaxy data in the M2FS catalog (Walker et al., 2023).

###  3.3 AAT Observations

One of the objects detected in the Gaia search was submitted for follow-up
observations by the Two-degree Field (2dF) spectrograph (Lewis et al., 2002)
at the Anglo-Australian Telescope (AAT). The observations were conducted
during the observing run of the Southern Stellar Stream Spectroscopic Survey
($S^{5}$) (Li et al., 2019). In particular, an internal data release (iDR3.1)
was used for this work. We refer to Li et al. (2022) for a detailed
description of the data and processing and provide a brief summary here. The
stars were observed with two arms of the spectrograph: the red arm with the
1700D grating that covers a wavelength range from 8400 to 8800 Å (including Ca
ii near-infrared triplet with $\lambda\lambda$8498, 8542, and 8662) with a
spectral resolution of $R\sim 10000$, and the blue arm with the 580V grating
that provides low resolution $R\sim 1300$ spectra covering a broad wavelength
range from 3800 to 5800 Å. The blue and red spectra for each star were then
forward modeled by the rvspecfit code (Koposov, 2019) to provide estimates of
stellar parameters, radial velocities and their uncertainties. In addition to
the velocities, we also acquired the calcium triplet (CaT) metallicities from
equivalent widths and the Carrera et al. (2013) calibration for all the member
stars as detailed in Li et al. (2022).

###  3.4 Additional Data

We use photometric and astrometric data from the Gaia EDR3 catalog (Gaia
Collaboration et al., 2021). We only utilize astrometric data that passed the
following cuts: ruwe $<1.5$ (Lindegren et al., 2021), and
astrometric_excess_noise_sig$<3$. We note that some stars that were targeted
spectroscopically do not pass these quality cuts (partly due to the Gaia DR2
target selection) and we exclude those stars from any astrometry based
analysis. For parallax measurements, we apply the parallax offset from
Lindegren et al. (2021) and include an additional offset of
$\Delta\varpi=0.007~{}{\rm mas}$ based on the globular cluster analysis of
Vasiliev & Baumgardt (2021).

We use Gaia DR3 RR Lyrae (RRL) catalog to search for candidate RRL star
cluster members (Clementini et al., 2022). We use the DECam Plane Survey
(DECaPS) DR1 griz photometric data for Gran 3, Garro 01 and LP 866 (Schlafly
et al., 2018). We search for additional spectroscopic members of our star
cluster sample in large spectroscopic surveys including SDSS APOGEE DR17
(Abdurro’uf et al., 2022), GALAH (Buder et al., 2021), and Gaia RVS DR3 (Katz
et al., 2022).

##  4 Results

Table 2: Properties of the Star Clusters. Literature $M_{V}$ measurements of the Gran 3, Gran 4, and Garro 01 are from Garro et al. (2022a); Gran et al. (2022); Garro et al. (2020).  | Gran 3/Patchick 125 | Gran 4 | Garro 01 | Gaia 9 | Gaia 10 | LP 866
---|---|---|---|---|---|---
R.A. (J2000, deg) | 256.24 | 278.113 | 212.25 | 119.707 | 121.168 | 261.766
Dec (J2000, deg) | -35.49 | -23.105 | -65.62 | -39.011 | -38.928 | -39.439
l (deg) | 349.75 | 10.20 | 310.83 | 254.65 | 255.17 | 349.09
b (deg) | 3.44 | -6.38 | -3.94 | -4.97 | -3.96 | -2.42
$r_{h}$ (arcmin) | $1.7\pm 0.2$ | $2.2_{-0.4}^{+0.5}$ | $2.4_{-0.4}^{+0.6}$ | $1.4\pm 0.2$ | $1.6_{-0.2}^{+0.3}$ | $4.6_{-0.6}^{+0.7}$
$r_{h}$ (parsec) | $5.3_{-0.6}^{+0.7}$ | $14.2_{-2.5}^{+3.3}$ | $10.9_{-2.0}^{+2.6}$ | $5.5_{-0.7}^{+0.9}$ | $7.8_{-1.2}^{+1.6}$ | $3.1_{-0.4}^{+0.5}$
$r_{c}$ (arcmin) | $1.1_{-0.2}^{+0.3}$ | $1.4_{-0.4}^{+0.5}$ | $1.8_{-0.5}^{+0.7}$ | $0.8\pm 0.2$ | $1.0_{-0.2}^{+0.3}$ | $3.3_{-0.7}^{+0.8}$
$r_{t}$ (arcmin) | $>5.3$ | $>5.8$ | $>5.3$ | $>5.4$ | $>5.0$ | $>11.4$
$D$ (kpc) | $10.5$ | $21.9$ | $15.5$ | $13.8$ | $17.4$ | $2.3$
$(m-M)_{0}$ | $15.11$ | $16.70$ | $15.95$ | $15.70$ | $16.20$ | $11.80$
$R_{GC}$ (kpc) | $2.7$ | $13.7$ | $11.9$ | $18.0$ | $21.2$ | $6.3$
$M_{V}$ | -3.8 $\pm$ 0.8 | -6.45 | -5.62 $\pm$ 1 |  |  |
E(B-V) | $1.09$ | $0.45$ | $0.61$ | $0.86$ | $1.17$ | $1.36$
age (Gyr) | $>10$ | $<10$ | 11 $\pm$1 |  $\sim$1.5 |  $\sim$1 |  $\sim$3
$\overline{v_{\rm los}}~{}(\mathrm{\,km}\mathrm{\,s}^{-1})$ | $90.9\pm 0.4$ | $-266.4\pm 0.2$ | $31.0\pm 0.1$ | $159.0\pm 0.3$ | $135.9\pm 0.4$ | $-9.8\pm 0.1$
$\sigma_{v}~{}(\mathrm{\,km}\mathrm{\,s}^{-1})$ | $1.9\pm 0.3$ | $1.4\pm 0.2$ | $0.4\pm 0.3$ | $1.0\pm 0.3$ | $1.4_{-0.3}^{+0.4}$ | $0.6\pm 0.1$
$\overline{{\rm[Fe/H]}}$ | $-1.83_{-0.04}^{+0.03}$ | $-1.84\pm 0.02$ | $-0.30\pm 0.03$ | $-0.50\pm 0.05$ | $-0.34\pm 0.06$ | $0.10\pm 0.03$
$\sigma_{\rm[Fe/H]}$ | $<0.16$ | $<0.10$ | $<0.14$ | $<0.16$ | $<0.14$ | $<0.22$
$\overline{\mu_{\alpha\star}}~{}(\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}})$ | $-3.74\pm 0.03$ | $0.51\pm 0.01$ | $-4.35\pm 0.02$ | $-1.08\pm 0.03$ | $-0.73\pm 0.03$ | $2.93_{-0.02}^{+0.01}$
$\overline{\mu_{\delta}}~{}(\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}})$ | $0.71_{-0.02}^{+0.01}$ | $-3.51\pm 0.01$ | $-1.09\pm 0.02$ | $1.50\pm 0.03$ | $1.60_{-0.03}^{+0.04}$ | $0.44\pm 0.02$
$\varpi~{}(\mathrm{\,mas})$ | $0.12\pm 0.01$ | $0.07\pm 0.01$ | $0.09\pm 0.01$ | $0.08\pm 0.01$ | $0.10\pm 0.02$ | $0.437\pm 0.005$
$N_{\rm v_{\rm los}},~{}N_{\rm[Fe/H]}~{}N_{\rm\mu}$ | 35, 29, 33 | 62, 52, 65 | 43, 34, 42 | 19, 19, 19 | 23, 16, 21 | 80, 79, 86
$r_{\rm peri}$ (kpc) | $2.9\pm 1.0$ | $7.6_{-1.5}^{+1.6}$ | $9.8_{-1.8}^{+1.7}$ | $13.9_{-2.4}^{+2.7}$ | $19.9_{-2.5}^{+1.9}$ | $5.85\pm 0.07$
$r_{\rm apo}$ (kpc) | $3.3_{-0.4}^{+1.0}$ | $33.9_{-6.7}^{+8.8}$ | $13.3_{-1.4}^{+2.1}$ | $17.7\pm 1.1$ | $23.9_{-3.4}^{+4.8}$ | $8.09\pm 0.05$
ecc | $0.07_{-0.02}^{+0.15}$ | $0.63_{-0.00}^{+0.01}$ | $0.16_{-0.02}^{+0.04}$ | $0.12\pm 0.06$ | $0.10_{-0.01}^{+0.04}$ | $0.161\pm 0.003$
$P$ (Myr) | $-46_{-6}^{+7}$ | $-462_{-130}^{+97}$ | $-235_{-42}^{+35}$ | $-329_{-44}^{+39}$ | $-470_{-81}^{+68}$ | $-133\pm 1$
$z_{\rm max}$ (kpc) | $2.1_{-0.5}^{+0.9}$ | $20.5_{-3.7}^{+4.9}$ | $1.3\pm 0.2$ | $1.4_{-0.2}^{+0.3}$ | $1.7_{-0.3}^{+0.4}$ | $0.18\pm 0.01$
${\rm E~{}(10^{5}~{}km^{2}~{}s^{-2}})$ | $-1.77_{-0.13}^{+0.15}$ | $-0.84\pm 0.09$ | $-1.12\pm 0.07$ | $-0.99\pm 0.05$ | $-0.85\pm 0.06$ | $-1.371\pm 0.004$
${\rm L_{Z}~{}(10^{3}~{}kpc~{}km~{}s^{-1}})$ | $0.47_{-0.13}^{+0.14}$ | $-0.49_{-0.39}^{+0.33}$ | $-2.51_{-0.38}^{+0.35}$ | $-3.39_{-0.41}^{+0.39}$ | $-4.60_{-0.59}^{+0.55}$ | $-1.59\pm 0.01$
Figure 1: Summary of the spectroscopic observations of the six star clusters
and with members coloured for each star cluster. MW stars are blue points. The
star clusters are clearly identified based on the narrow $v_{los}$ peaks. Top
left panel: line-of-sight velocity ($v_{los}$) versus metallicity ([Fe/H]).
Only stars with good quality metallicity are included. Top middle panel:
Projected radial distance from the center for the cluster versus velocity in
the Galactic center of rest ($v_{GSR}$). Top right panel: vector point diagram
($\mu_{\alpha\star}$ vs $\mu_{\delta}$). Bottom left panel: $v_{los}$ versus
surface gravity ($\log{g}$). Bottom middle panel: $v_{los}$ versus effective
temperature ($T_{\rm eff}$). Bottom right panel: [Fe/H] versus [Mg/Fe]. Only
stars with good quality ${\rm[Fe/H]}$ and $[{\rm Mg}/{\rm Fe}]$ are included.

In Figure 1, we summarize the kinematics, chemistry, and stellar parameters of
cluster members and MW foreground stars in our follow-up spectroscopic
observations. The top panels from left to right compare the line-of-sight
velocities ($v_{los}$) to the stellar metallicity ([Fe/H]), the radial
distance from the center of the cluster versus the $v_{los}$, and the proper
motion ($\mu_{\alpha\star}$, $\mu_{\delta}$). The member stars in each cluster
are highlighted in different colours. Note in the proper motion panel, only
stars with good quality astrometry are included. In the bottom panels we
compare $v_{los}$ to the surface gravity (left, $\log{g}$), $v_{los}$ to
effective temperature ($T_{\rm eff}$, center), and compare [Mg/Fe] vs [Fe/H].
Excluding LP 866, all stars are red giant branch/red clump stars (with several
horizontal branch stars in Gran 3 and Gran 4). The derived properties of the
star clusters are summarized in Table 2.

###  4.1 Cluster Properties and Spectroscopic Membership

We are able to identify the members of four clusters, Gran 3, Gran 4, Gaia 9,
and Gaia 10, based purely on the line-of-sight velocity as the cluster mean
velocity is distinct from the MW foreground. The stellar parameters ([Fe/H],
$T_{\rm eff}$, and $\log{g}$) and photometry ($G$, $G_{BP}$, and $G_{RP}$) of
the members identified from the velocities further reinforce their membership
and they are consistent with single stellar populations. While there is a
clear overdensity in the $v_{\rm los}$ distribution in the LP 866 and Garro 01
fields, there is overlap with the MW foreground and we construct mixture
models to quantitatively identify members in these objects.

####  4.1.1 Gran 3/Patchick 125

Figure 2: Summary of the M2FS observations of Gran 3/Patchick 125. Left: line-
of-sight velocity ($v_{\rm los}$) versus metallicity ([Fe/H]) for Gran 3 M2FS
members (orange circles) and MW foreground stars (blue x’s). Center-left:
projected radius versus $v_{\rm los}$. We denote the M2FS observation of a RRL
with a green square. The red triangles and purple circles are APOGEE and Gaia
RVS members, respectively. Center: Vector point diagram for the same stars. We
include two additional RRL candidate members without spectroscopy. Center-
right: Gaia colour magnitude diagram ($G_{BP}-G_{RP}$ versus $G$). We include
all stars within 6′ that are consistent with the proper motion and parallax of
Gran 3 (small black points). We include an isochrone with age = 13 Gyr and
[Fe/H] = $-1.9$. For the photometry to match the isochrone, we increased the
standard MW reddening law to $R_{V}=3.3$. Right:  DECam g-r vs g photometry
from DECaPS.

In Figure 2, we summarize the Gran 3 members and the MW foreground stars from
our M2FS observations of the Gran 3 field. In addition to the primary M2FS
sample, we include 3 RRL members from the Gaia DR3 catalog (Clementini et al.,
2022), 2 APOGEE members111First identified in Fernández-Trincado et al.
(2022)., and 7 Gaia  DR3 RVS members2226 of the 7 of the members were first
identified in Garro et al. (2023).. The 36 M2FS members of Gran 3 are
identified by selecting stars in the $82<v_{\rm
los}<97\mathrm{\,km}\mathrm{\,s}^{-1}$ velocity range. The M2FS stars all are
consistent with a single metallicity and the stellar parameters ($\log{\rm g}$
and $T_{\rm eff}$) are consistent with red giant stars or horizontal branch
stars. There are two stars outside this range within $\sim
20~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ of the mean velocity of Gran 3. The
first, source_id333Here and throughout the paper source_id refers to Gaia DR3
source_id.=5977223144516980608, is a RRL star and the distance of this star
agrees with the star cluster. We consider it a cluster member but we exclude
this star from all kinematic analysis due to the velocity variability of RRL
stars. The second, source_id=5977223144516066944, is a $7\sigma$ outlier in
velocity but the stellar parameters agree with the cluster mean metallicity
and the proper motion agrees with the cluster. We exclude it from our analysis
and suggest that if it is a member, it is likely a binary star (e.g., Spencer
et al., 2018). Additional multi-epoch data is required to confirm this.

To determine the kinematics and chemistry we use a two-parameter Gaussian
likelihood function (Walker et al., 2006) and to use emcee to sample from the
posterior (Foreman-Mackey et al., 2013). We use a uniform prior for the
average and a Jeffreys prior for the dispersion. From the 35 stars in the M2FS
sample that are non-variable, we measure $\overline{v_{\rm los}}=+90.9\pm
0.4~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=1.9\pm
0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. With the 29 members with good quality
[Fe/H] measurements, we measure $\overline{{\rm[Fe/H]}}=-1.83_{-0.04}^{+0.04}$
and $\sigma_{\rm[Fe/H]}=0.09\pm 0.4$ ($\sigma_{\rm[Fe/H]}<0.16$). For limits
throughout this work, we list values at 95% confidence intervals. We note that
the non-zero metallicity dispersion is due to one star
(source_id=5977224587625168768; $[{\rm Fe/H}]=-1.58\pm 0.07$) that is
$3.5\sigma$ larger than the mean metallicity of Gran 3. This star has stellar
parameters and mean velocity that are otherwise consistent with Gran 3. If
this star is removed, the kinematics and mean metallicity are unchanged but
the metallicity dispersion is constrained to less than
$\sigma_{\rm[Fe/H]}<0.10$. We have opted to include this star in our sample.
From the 33 spectroscopically (M2FS, APOGEE, and Gaia RVS) identified members
with good quality astrometry, we measure:
$\overline{\mu_{\alpha\star}}=-3.74\pm
0.03\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=0.71_{-0.02}^{+0.01}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.10_{-0.02}^{+0.03}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.03_{-0.01}^{+0.02}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.12\pm 0.01\mathrm{\,mas}$. The parallax measurement corresponds
to $d=8.6_{-0.8}^{+0.9}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=14.7\pm 0.2$, which
is closer than the isochrone or RRL distance (see below). Assuming a distance
of 10.5 kpc (in agreement with these latter measurements), the proper motion
dispersions correspond to
$\sigma_{\mu_{\alpha\star}}=5.0_{-1.1}^{+1.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=1.7_{-0.7}^{+0.9}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.
While $\sigma_{\mu_{\alpha\star}}$ is larger than expected,
$\sigma_{\mu_{\delta}}$ agrees with $\sigma_{v}$. With the 7 Gaia RVS members
we measure: $\overline{v_{\rm
los}}=+93.5_{-1.5}^{+1.7}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and
$\sigma_{v}<6.6~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ (95% c.i.). The Gaia RVS
sample is consistent with the M2FS sample.

There are 3 stars in the secondary samples (1 APOGEE, 2 Gaia RVS) that are
$\sim 9~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ offset ($1-5\sigma$ outliers in
$v_{\rm los}$) from the bulk of Gran 3. 2 of these stars have repeat
measurements with other samples and those measurements are in good agreement
with the bulk velocity of the system and suggests that those stars could be
binary stars. The APOGEE star (source_id=5977223316333009024; $v_{\rm
los,~{}APOGEE}=100.2\pm 0.2\mathrm{\,km}\mathrm{\,s}^{-1}$) overlaps with Gaia
RVS ($v_{\rm los,~{}{\it Gaia}~{}RVS}=92.2\pm
2.2\mathrm{\,km}\mathrm{\,s}^{-1}$) and one of the Gaia RVS members
source_id=5977224587625168768; ($v_{\rm los,~{}{\it Gaia}~{}RVS}=99.2\pm
3.5\mathrm{\,km}\mathrm{\,s}^{-1}$) overlaps with M2FS ($v_{\rm
los,~{}M2FS}=92.6\pm 0.7\mathrm{\,km}\mathrm{\,s}^{-1}$). Both these stars may
be binary stars which would explain their offset.

We identify three RRL in the Gaia DR3 RRL catalog
(source_id=5977223144516980608, 5977224553266268928444We note that this star
is a 3-$\sigma$ outlier in $\mu_{\delta}$ compared to the systemic proper
motion, however, the large value of astrometric_excess_noise_sig suggests that
the astrometric solution may not be reliable., 5977224557581335424) that are
consistent with the proper motion and spatial position (all three have
$R<2\arcmin$) of Gran 3. One star (source_id=5977223144516980608) was observed
with M2FS. It is offset from the mean velocity of Gran 3 by $\sim
20\mathrm{\,km}\mathrm{\,s}^{-1}$. As RRL stars are variable in velocity and
vary more than $50\mathrm{\,km}\mathrm{\,s}^{-1}$ over the period (Layden,
1994), we consider this star a member. With more spectroscopic epochs the
systemic velocity of the star could be measured (e.g., Vivas et al., 2005). We
apply the metallicity correction to the absolute magnitude of a RRL in Gaia
bands to determine the absolute magnitude: $M_{G}=0.32{\rm[Fe/H]}+1.11$
(Muraveva et al., 2018). From the three RRL, we find a mean distance modulus
of $(m-M)_{0}=15.1$ corresponding to a distance of $d=10.5\mathrm{\,kpc}$.
This is slightly smaller than other distance measurements for this cluster:
$d=12.02~{}\mathrm{\,kpc}$ (Gran et al., 2022), $d=11\pm 0.5~{}\mathrm{\,kpc}$
(Fernández-Trincado et al., 2022), and $d=10.9\pm 0.5,~{}11.2\pm
0.5~{}\mathrm{\,kpc}$ (Garro et al., 2022a).

In Figure 2, we compare an old (age $=13$ Gyr) and metal-poor isochrone
([Fe/H] = $-1.9$) to Gran 3 with both Gaia and DECaPS photometry. We are able
to match the horizontal branch and the colour of the RGB if we use
$(m-M)_{0}=15.2$ and a $R_{V}=3.3$ dust law (compared to a standard of
$R_{V}=3.1$) and find it difficult to match the horizontal branch using the
RRL distance. As noted by Garro et al. (2022a), some studies suggest a lower
dust law is favored in the bulge regions (e.g., Souza et al., 2021; Saha et
al., 2019) which would disagree with the RGB of Gran 3. This larger distance
modulus is in better agreement with the literature distance measurements of
Gran 3 (Gran et al., 2022; Fernández-Trincado et al., 2022; Garro et al.,
2022a). We note that the brighter stars are bluer than the isochrone but the
bulk of the RGB matches the isochrone.

####  4.1.2 Gran 4

Figure 3: Similar to Figure 2 but for Gran 4. We include AAT spectroscopic
observations with brown and blue open triangles corresponding to Gran 4
members and MW stars, respectively. An isochrone with age = 13 Gyr and [Fe/H]
= $-1.9$ is included as a black curve.

The summary of our spectroscopic observations of Gran 4 is shown in Figure 3.
We identify 64, 22, and 3 Gran 4 members in the M2FS, AAT and Gaia RVS sample,
respectively, with the $-280<v_{\rm los}<-260\mathrm{\,km}\mathrm{\,s}^{-1}$
selection. 12 stars in the AAT sample overlap with the M2FS sample.

Within the M2FS sample, further examination of the velocity distribution
reveals two outlier stars. The first (source_id=4077796986282497664) is a RRL
star and is offset from the mean velocity by $\sim
8~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and by $\sim 3\sigma$ once the velocity
dispersion is considered. It is a cluster member but due to its variable
nature, we exclude it from any kinematic analysis. The second
(source_id=4077796810168905344) is a $\sim 8\sigma$ outlier in velocity and if
it is considered a member the velocity dispersion increases from
$1.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ to
$2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. It is consistent with the mean
metallicity and proper motion of Gran 4, and the stellar parameters ($T_{\rm
eff}$ and $\log{g}$) are consistent with a red giant branch star. It seems
unlikely for this star to be MW star based on the MW velocity and metallicity
distribution, however, as it is a $\sim 8\sigma$ outlier it is either a binary
star or a MW interloper and we exclude it from the analysis.

From the 62 non-variable M2FS members we find, $\overline{v_{\rm
los}}=-266.4\pm 0.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=1.4\pm
0.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\overline{{\rm[Fe/H]}}=-1.84\pm 0.02$
and $\sigma_{\rm[Fe/H]}<0.10$. Due to tail to zero dispersion and a lack of a
clear peak, we do not consider the metallicity dispersion resolved and list an
upper limit. The stellar parameters of these stars ($T_{\rm eff}$ and
$\log{g}$) are consistent with red giant branch stars or horizontal branch
stars which further confirms our membership identification. With the 3 Gaia
RVS members we measure: $\overline{v_{\rm
los}}=-262.7_{-3.7}^{+3.6}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\sigma_{v}<6.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ (95% c.i.).

From the 22 AAT members, we find $\overline{v_{\rm los}}=-265.9\pm
0.4~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and
$\sigma_{v}=1.5_{-0.3}^{+0.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. 12 of these
stars overlap with the M2FS sample. There is one velocity outlier
(source_id=4077796397852026240) with $\sim 3\sigma$, however, this star is in
the M2FS sample with almost the exact same velocity. Removing this star
decreases the velocity dispersion to $\sim 1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
from $\sim 1.5~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ in the AAT sample but its
inclusion or exclusion does not affect the M2FS kinematics and we opt to
include it. From the calcium triplet metallicities, we compute
$\overline{{\rm[Fe/H]}}=-1.82\pm 0.06$ and
$\sigma_{\rm[Fe/H]}=0.21_{-0.05}^{+0.06}$. The metallicity dispersion is
clearly resolved in contrast to our expectation of a single stellar population
in a star cluster and the lack of a metallicity dispersion in the M2FS sample.
The source of the metallicity dispersion in the AAT data is unclear. As the
M2FS sample is larger, we adopt the velocity and metallicity results from the
M2FS sample as our primary results.

From the combined M2FS, AAT, and Gaia RVS sample there are 65 stars with good
quality astrometric measurements. From these stars we measure:
$\overline{\mu_{\alpha\star}}=+0.51\pm
0.01\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$, and
$\overline{\mu_{\delta}}=-3.51\pm
0.01\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.02_{-0.01}^{+0.03}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.05_{-0.03}^{+0.02}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.07\pm 0.01\mathrm{\,mas}$. The parallax measurement corresponds
to $d=14.6_{-1.9}^{+2.5}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.8\pm 0.3$ which
is closer than the other distance measurements (see below). Assuming a
distance of 21.9 kpc (from the best-fit isochrone distance), the proper motion
dispersion correspond to:
$\sigma_{\mu_{\alpha\star}}=2.6_{-1.2}^{+2.7}\mathrm{\,km}\mathrm{\,s}^{-1}$
and $\sigma_{\mu_{\delta}}=5.4_{-2.8}^{+2.5}\mathrm{\,km}\mathrm{\,s}^{-1}$.

We identify two RRL (source_id=4077796986282497664, 4077796573965756928) in
the Gaia DR3 RRL catalog (Clementini et al., 2022) as members of Gran 4 based
on their proper motion and distance. The first RRL is also in the VVV RRL
(Molnar et al., 2022) and OGLE RRL catalogs (Soszyński et al., 2019). There is
a third candidate RRL (source_id=4077796608325479168) from the PanSTARRS1 RRL
catalog (Sesar et al., 2017), however, it is not in the Gaia DR3 RRL catalog
so we do not include it in the analysis. It is considered a variable star in
Gaia DR3 but only has a best_class_score=0.4 for being an RRL. Additional time
series data are required to confirm the status of this star. From the two RRL
members, $\mu=16.5,16.6$ corresponding to $d=19.9,21~{}\mathrm{\,kpc}$. This
is slightly closer than our best-fit $\mu\sim 16.7$ from matching the
horizontal branch to a metal-poor isochrone. It is also closer than
$\mu=16.84$, $d=22.49~{}\mathrm{\,kpc}$ from Gran et al. (2022).

####  4.1.3 Garro 01

Figure 4:  Same as Figure 2 but for Garro 01. In contrast to the other
globular clusters (Gran 3 and Gran 4) we use a mixture model to identify Garro
01 members and stars with membership $>0.01$ are coloured according to their
membership probability in the colour bars in the center-right and right
panels. We include an isochrone from the photometric analysis of Garro et al.
(2020) with an age = 11 Gyr and [Fe/H] = $-0.7$ (black) and an isochrone with
our best estimate using the spectroscopic metallicity ([Fe/H]= -0.3) and an
age = 4 Gyr (red). There are no candidate RRL stars in Garro 01.

There is overlap in the MW foreground and star cluster velocity distributions
for Garro 01 and LP 866 and we construct mixture models to account for this
overlap. The total likelihood for our mixture models is:

$\mathcal{L}=f_{\rm cluster}\mathcal{L}_{\rm cluster}+(1-f_{\rm
cluster})\mathcal{L}_{\rm MW}\,,$ (1)

where $\mathcal{L}_{\rm cluster}$, $\mathcal{L}_{\rm MW}$, and $f_{\rm
cluster}$ correspond to the cluster population, the MW population, and the
fraction of stars in the star cluster, respectively (e.g., Pace et al., 2020,
2021). We assume the probability distributions of each data component are
separable:

$\mathcal{L}_{\rm cluster/MW}=\mathcal{L}_{\rm spatial}\mathcal{L}_{\rm
PM}\mathcal{L}_{\rm v_{los}}\mathcal{L}_{\rm[Fe/H]}\,.$ (2)

Where $\mathcal{L}_{\rm spatial}$, $\mathcal{L}_{\rm PM}$, $\mathcal{L}_{\rm
v_{los}}$, and $\mathcal{L}_{\rm[Fe/H]}$ are the spatial likelihood, the
proper motion likelihood, the line-of-sight velocity likelihood, and the
metallicity likelihood, respectively. To compute the membership of each star,
we compare the ratio of the cluster likelihood to total likelihood for each
star: $p_{\rm member}=\mathcal{L}_{\rm cluster}/\mathcal{L}$ (e.g., Martinez
et al., 2011).

For Garro 01, we primarily analyze a mixture model with $v_{\rm los}$ and
[Fe/H] but also consider a second model with spatial information. We assume
the $v_{\rm los}$ and [Fe/H] likelihood distributions are Gaussian for both
the star cluster and MW components. We apply the following cuts to the Garro
01 spectroscopic sample to remove MW foreground stars: $T_{\rm
eff}-2\times\sigma_{T_{\rm eff}}<6000K$,
$\log_{10}{g}-2\times\sigma_{\log_{10}{g}}<4$, a parallax cut
($\varpi-3\sigma_{\varpi}<0.064$), and a loose $G_{BP}-G_{RP}$ colour cut of
0.25 around an age = 11 Gyr and [Fe/H] = $-0.6$ MIST isochrone (Dotter, 2016).
The isochrone selection is applied to remove blue MW main sequence stars from
the sample which have no overlap in colour-magnitude space with the cluster
members. From our primary mixture model, we find: $\overline{v_{\rm
los}}=+31.0\pm 0.1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=0.4\pm
0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
($\sigma_{v}<0.8~{}\mathrm{\,km}\mathrm{\,s}^{-1}$),
$\overline{{\rm[Fe/H]}}=-0.30\pm 0.03$ and $\sigma_{\rm[Fe/H]}<0.14$. In
Figure 4, we summarize the properties of the spectroscopic members identified
in the mixture model. Stars are coloured by their membership probability.

For the second spatial model we use a conditional likelihood and we assume
that the fraction of stars is spatially dependent (e.g., Martinez et al.,
2011; Pace et al., 2021):

$f_{\rm cluster}(R)=\Sigma_{\rm cluster}(R)/(\Sigma_{\rm
cluster}(R)+N\Sigma_{\rm MW}(R)).\\\ $ (3)

Where $\Sigma$ is the projected stellar distribution and $N$ is the relative
normalization between the cluster and MW spatial distributions We assume a
King (1962) distribution for the Garro 01 distribution with parameters from
Garro et al. (2020) and assume that the MW distribution is constant over the
small region examined. We utilize a conditional likelihood as there is an
unknown spatial selection function.

While the chemodynamic properties of Garro 01 from the two models are nearly
identical, there are some minor differences in the membership of individual
stars between the two models. The conditional likelihood model has larger
membership for stars near the center and lower membership for the two most
distant stars. The difference in membership between the models for individual
stars is small ($\sum|p_{\rm standard}-p_{\rm conditional}|=1.7$) and the
overall membership is similar for the two models; both models have $\sum
p=46.8\pm 1.9$. We consider stars with $p>0.9$ as high confidence members and
there are 39 and 42 members identified in the standard model and conditional
likelihood model, respectively and note there are 43 high confidence members
with $p>0.9$ in either model.

From the 42 members with good astrometry (42 M2FS and 1 Gaia RVS), we measure:
$\overline{\mu_{\alpha\star}}=-4.35\pm
0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=-1.09\pm
0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.09_{-0.02}^{+0.02}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and
$\sigma_{\mu_{\delta}}=0.08_{-0.02}^{+0.03}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$.
We measure $\varpi=0.08\pm 0.01\mathrm{\,mas}$ corresponding to
$d=11.9_{-1.5}^{+2.1}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.4\pm 0.3$. The
distance from the parallax is closer than the measurement derived from
isochrone fits. Assuming a distance of 15.3 kpc from Garro et al. (2020), the
proper motion dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=6.6_{-1.4}^{+1.5}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=5.8_{-1.8}^{+1.8}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.
Our proper motion measurement agrees with the Gaia DR2 proper motion
measurement, $\overline{\mu_{\alpha\star}}=-4.68\pm
0.47~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$, and
$\overline{\mu_{\delta}}=-1.35\pm
0.45~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$ from Garro et al. (2020).

We identify 2 members in the Gaia RVS sample with velocities and proper motion
consistent with Garro 01. Both stars are more evolved than the M2FS sample but
roughly match the isochrone. Both stars are included in Figure 4 as purple
circles.

We do not identify any RRL that have the same proper motion or a consistent
distance with Garro 01. With the distance modulus of Garro et al. (2020), we
can match the red clump of our spectroscopic sample with an isochrone and we
adopt this distance for our analysis.

In Figure 4, we show optical colour-magnitude diagrams using Gaia and DECaPS
photometry. The prominent feature in the CMDs is the red clump and complements
the near-infrared discovery photometry from Garro et al. (2020). Our
spectroscopic metallicity measurement is more metal-rich than the isochrone
analysis of Garro et al. (2020) which found [Fe/H] = $-0.7$ with their CMDs
fits (black isochrone in Figure 4). However, we cannot match the colour of the
system with this age and spectroscopic metallicity. If we assume the
spectroscopic metallicity, the isochrone is redder than the photometry. We
discuss this in more detail in Section 5.1.

####  4.1.4 Gaia 9

Figure 5: Similar to Figure 2 but for Gaia 9. The best fit isochrone is age =
1.5 Gyr and [Fe/H] = $-0.5$.

We identify 19 M2FS members of Gaia 9 with the velocity selection: $150<v_{\rm
los}<170~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. There is one star inside this
velocity range (source_id=5537860050401680000) that is a non-member based on
its proper motion. The properties of the members and proper motion selected
stars are displayed in Figure 5. From the 19 members, we measure:
$\overline{v_{\rm los}}=+159.0\pm 0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\sigma_{v}=1.0\pm 0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\overline{{\rm[Fe/H]}}=-0.50\pm 0.06$ and $\sigma_{\rm[Fe/H]}<0.16$. There is
one additional member Gaia RVS that we identify
(source_id=5537859848550503168) based on the radial velocity and proper motion
and we include it in Figure 5. From the 19 members with good astrometry, we
measure: $\overline{\mu_{\alpha\star}}=-1.08\pm
0.03~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=+1.50\pm
0.03~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.07\pm
0.04~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.08_{-0.03}^{+0.03}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.08\pm 0.01\mathrm{\,mas}$. The parallax measurement corresponds
to $d=12.9_{-2.0}^{+3.0}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.6_{-0.4}^{+0.5}$
which agrees with our isochrone derived distance (see below). Assuming a
distance of 13.8 kpc (from our isochrone derived distance), the proper motion
dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=4.8_{-2.4}^{+2.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=5.0_{-2.1}^{+2.3}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.

In the Gaia colour-magnitude diagram (Figure 5), the majority of the
spectroscopic members are red clump stars. We estimate the distance of the
cluster to be $(m-M)_{0}=15.7$ or $d=13.8~{}\mathrm{\,kpc}$ based on the red
clump using an MIST isochrone with age = 1.5 Gyr and [Fe/H] = $-0.5$. The blue
stars at $G_{0}\sim 17$ are likely the top of the main sequence and we use
this feature to assist in estimating the age of the system. Gaia 9 is younger
than the other clusters examined thus far and it likely an open cluster.

####  4.1.5 Gaia 10

Figure 6: Similar to Figure 2 but for Gaia 10. The best fit isochrone is age =
1 Gyr and [Fe/H] = $-0.5$. Similar to Gran 3, we increase the extinction to
find an adequate fit ($R_{V}=3.3$).

Gaia 10 has similar properties to Gaia 9 but is more distant. We summarize our
spectroscopic sample in Figure 6. We identify 23 members of Gaia 10 with a
velocity selection: $v_{\rm los}>120\mathrm{\,km}\mathrm{\,s}^{-1}$. Two stars
(source_id=5534905976200827776 and 5540909996882971520) have kinematics (line-
of-sight velocity and proper motion) that agree with the cluster but both
stars are significant outerliers in metallicity. 5534905976200827776 is
$3.2\sigma$ more metal-rich while 5540909996882971520 is $4.9\sigma$ more
metal-poor and the inclusion of either star results in an offset mean
metallicity and non-zero metallicity dispersion. We consider both stars non-
members. The inclusion of the star as a members would infer to a non-zero
metallicity dispersion. We note that this same star is in the Gaia RVS catalog
with a similar velocity. From the spectroscopic members, we measure:
$\overline{v_{\rm los}}=+135.9\pm 0.4~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\sigma_{v}=1.4_{-0.3}^{+0.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\overline{{\rm[Fe/H]}}=-0.34\pm 0.06$ and $\sigma_{\rm[Fe/H]}<0.14$.

From the 21 members with good astrometry, we measure:
$\overline{\mu_{\alpha\star}}=-0.74\pm
0.03~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=+1.60_{-0.03}^{+0.04}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.05_{-0.03}^{+0.05}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.06_{-0.03}^{+0.04}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.09\pm 0.02\mathrm{\,mas}$. The parallax measurement corresponds
to $d=10.8_{-1.9}^{+3.1}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.2_{-0.4}^{+0.6}$
which is much closer than our isochrone derived distance (see below). Assuming
a distance of 17.4 kpc (from our isochrone derived distance), the proper
motion dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=4.2_{-2.6}^{+3.8}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=5.0_{-2.8}^{+3.5}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.

We compare theoretical isochrones based on the spectroscopic metallicity to
the Gaia colour-magnitude diagram Figure 6. With an age of $1~{}{\rm Gyr}$ and
distance modulus of $(m-M)_{0}=16.2$ ($d=17.4~{}\mathrm{\,kpc}$) we can fit
the red clump and the possible main sequence turnoff based on the proper
motion selected sample. Similar to Gran 3, to match the colour of the
isochrone a larger extinction coefficient of $R_{V}=3.3$ is required. There
remains considerable spread in the colour of the members. This may be due to
differential reddening. We consider Gaia 10 an open cluster.

####  4.1.6 LP 866

Figure 7:  Same as Figure 4 but for LP 866. Similar to Garro 01, we include a
colourbar for the membership from the mixture model. The best fit isochrone is
age = 3 Gyr and [Fe/H] = $+0.1$ but with a reduced extinction ($\sim 53\%$ of
the E(B-V) value). We include DECam g-r vs g photometry from DECaPS.

LP 866 is the only star cluster where the entirety of the M2FS spectroscopic
sample is located on the main-sequence. Similar to Garro 01, we run a mixture
model to account for the MW foreground distribution with $v_{\rm los}$, proper
motion, and [Fe/H] components. We do not include a conditional likelihood run
as the spatial distribution was not known beforehand. We assume the proper
motion distribution is a truncated multi-variate Gaussian for both components
with limits: $1.5<\mu_{\alpha\star}<3.5~{}{\rm mas~{}yr^{-1}}$,
$-0.4<\mu_{\delta}<1.2~{}{\rm mas~{}yr^{-1}}$. While the proper motion was
included in the spectroscopic target selection (based on Gaia DR2 astrometry),
the proper motion dispersion is resolved in contrast to the other star
clusters. We excluded stars outside of this proper motion limit and two bright
stars with discrepant parallax measurements.

With the mixture model, we identify 80 high confidence members ($p>0.9$) and
measure the following properties for LP 866: $\overline{v_{\rm los}}=-9.8\pm
0.1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=0.6\pm
0.1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\overline{{\rm[Fe/H]}}=0.10\pm 0.03$,
$\sigma_{\rm[Fe/H]}=0.15\pm 0.04$ ($\sigma_{\rm[Fe/H]}<0.22$),
$\overline{\mu_{\alpha\star}}=2.93_{-0.02}^{+0.01}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=0.44_{\pm}0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.08_{-0.02}^{+0.02}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and
$\sigma_{\mu_{\delta}}=0.10_{-0.01}^{+0.02}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$.
The overall membership is $\sum p=86.4\pm 4.8$. As there is a non-zero
metallicity dispersion, it is possible that our model has incorrectly
identified some MW stars as cluster members. From the 86 members with good
astrometry, we measure $\varpi=0.437\pm 0.005\mathrm{\,mas}$ corresponding to
$d=2.29\pm 0.03~{}\mathrm{\,kpc}$ and $(m-M)_{0}=11.80\pm 0.02$. Assuming a
distance of 2.29 kpc, the proper motion dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=0.9\pm 0.02~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and
$\sigma_{\mu_{\delta}}=1.1\pm 0.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.

In the Gaia RVS sample there is a clear overdensity and velocity peak distinct
from the MW population within the central 8′of LP 866. We find 14 stars in the
Gaia RVS catalog that are consistent with the velocity, proper motion, and
parallax of LP 866. All 14 stars are more evolved than the M2FS sample and aid
in matching a stellar isochrone. With the 14 Gaia RVS members we find:
$\overline{v_{\rm los}}=-10.6\pm 0.8~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, and
$\sigma_{v}=0.2_{-0.1}^{+0.9}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
($\sigma_{v}<2.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$); and from the 12 stars
with good quality astrometry we find: $\overline{\mu_{\alpha\star}}=2.88\pm
0.01~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$, and
$\overline{\mu_{\delta}}=0.44\pm
0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$. These results are
consistent with the M2FS sample.

We include Gaia and DECaPs colour magnitude diagrams in Figure 7. We match
theoretical isochrones to estimate the age of the system. Unlike the other two
open clusters, we have a confident distance and metallicity measurement. We
find that reducing the E(B-V) by $\sim 53\%$ and setting the to age =
$10^{9.5}$ = $\sim$ 3 Gyr provides an adequate match. We note that the we
varied the extinction to match the RGB of the Gaia RVS stars and varied the
age to match the main sequence turn-off of the M2FS sample with the Gaia
photometry. The same isochrone provided an similarly adequate match to the g-r
vs g DECaPs photometry. More in depth modeling is required to improve
constraints on the age and extinction of the cluster. Regardless, we consider
LP 866 an open cluster.

###  4.2 Spatial Distribution

Figure 8:  Projected radial stellar density profile of our star cluster
sample. Top: the globular clusters, from left to right: Gran 3, Gran 4, Garro
01. Bottom: the open clusters, from left to right: Gaia 9, Gaia 10, LP 866.

To measure the spatial distribution of each star cluster, we construct a
larger proper motion selected sample from the Gaia DR3 catalog based on the
systemic proper motion found from our spectroscopic sample and apply a spatial
mixture model (Equations 1, 2). We do not apply this methodology to the
spectroscopic sample as it is not spatially complete and has an unknown
spatial target selection. For the spatial likelihood we model the star cluster
with two density profiles. The first profile is a Plummer distribution
(Plummer, 1911):

$\Sigma(R)=\frac{1}{\pi
r_{p}^{2}}\frac{1}{\left(1+\left(R/r_{p}\right)^{2}\right)^{2}}$ (4)

where $r_{p}$ is the Plummer scale radius (for a Plummer profile $r_{p}$ is
equivalent to the 2D deprojected half-light radius). The second is the King
profile (King, 1962):

$\Sigma_{\star}(R)\propto\left[\left(1+\frac{R^{2}}{r_{c}^{2}}\right)^{-1/2}-\left(1+\frac{r_{t}^{2}}{r_{c}^{2}}\right)^{-1/2}\>\right]^{2}$
(5)

where $r_{c}$ is the core radius and $r_{t}$ is the tidal radius. We model a
small region near each cluster and assume that the MW background is constant
within that small area after a proper motion selection is applied.

For the Gaia selected sample, we apply the following cuts: a 3-$\sigma$
selection in proper motion, a parallax selection ($\varpi-\varpi_{\rm
cluster}-3\sigma_{\varpi}<0$), $G<20$, $R<R_{\rm max}$, and stars with good
astrometry (i.e., satisfy our astrometric cuts in Section 3.4). We will refer
to this Gaia selected sample and utilize the same sample for examining the
colour-magnitude diagrams of the clusters. We use $R_{\rm max}=12\arcmin$ for
LP 866 and $R_{\rm max}=6\arcmin$ for all other clusters. For Garro 01 we
additionally apply a loose $G_{BP}-G_{RP}$ colour cut of 0.25 around an age =
11 Gyr and [Fe/H] = $-0.6$ MIST isochrone (Dotter, 2016) following the
spectroscopic selection. For the other clusters, the above selection primarily
identifies stars with a stellar population that agrees with the spectroscopic
sample. Any photometric outliers (i.e., MW stars) will be roughly distributed
uniformly within the small area examined and not bias the spatial distribution
calculations.

The Plummer and King fits along with the binned stellar profile of all six
clusters are shown in Figure 8. In general, the results from the Plummer and
King profile fits agree and provide adequate fits. Due to the low number of
stars, there is no preference for one profile over the other. We are unable to
constrain $r_{t}$ and generally only provide lower limits. For the globular
clusters, we find $r_{h}=1.7\pm 0.2\arcmin$, $r_{h}=2.2_{-0.4}^{+0.5}\arcmin$,
and $r_{h}=2.4_{-0.4}^{+0.6}~{}\arcmin$ corresponding to
$r_{h}=5.3_{-0.6}^{+0.7}~{}\mathrm{\,pc}$,
$r_{h}=14.2_{-2.5}^{+3.3}~{}\mathrm{\,pc}$, and
$r_{h}=10.9_{-2.0}^{+2.6}~{}\mathrm{\,pc}$ from the Plummer profile fits for
Gran 3, Gran 4, and Garro 01, respectively. With the King profile, we find
$r_{c}=1.1_{-0.2}^{+0.3}\arcmin$, $r_{c}=1.4_{-0.4}^{+0.5}\arcmin$, and
$r_{c}=1.8_{-0.5}^{+0.7}~{}\arcmin$ for Gran 3, Gran 4, and Garro 01,
respectively. For comparison, Gran et al. (2022) find $r_{h}=1.05\pm
0.04\arcmin$ and $r_{h}=1.14\pm 0.02\arcmin$, for Gran 3 and Gran 4,
respectively. These are smaller than the sizes we infer. Garro et al. (2020)
measure $r_{c}=2.5\pm 1.5\arcmin$ for Garro 01 and a poorly constrained
$r_{t}$ which agrees with our measurement.

For the open clusters, we find $r_{h}=1.4\pm 0.2\arcmin$,
$r_{h}=1.6_{-0.2}^{+0.3}\arcmin$, and $r_{h}=4.6_{-0.6}^{+0.7}\arcmin$ with
the Plummer profile fits for Gaia 9, Gaia 10, and LP 866, respectively. With
the King profile, we find $r_{c}=0.8\pm 0.2\arcmin$,
$r_{c}=1.0_{-0.2}^{+0.3}\arcmin$, and $r_{c}=3.3_{-0.7}^{+0.8}~{}\arcmin$ for
Gaia 9, Gaia 10, and LP 866, respectively. The results for the six clusters
are included in Table 2.

###  4.3 Orbital Properties

Figure 9: Five example orbits of Gran 3 (top), Gran 4 (middle), and Garro 01
(bottom) drawn from the observational uncertainties and integrated for 1 Gyr
(Gran 3) or 2 Gyr (Gran 4 and Garro 01).

Figure 10: Five example orbits of Gaia 9 (top), Gaia 10 (middle), LP 866
(bottom) drawn from the observational uncertainties integrated for 2 Gyr (Gaia
9 and Gaia 10) or 1 Gyr (LP 866).

We use the gala package to compute the orbits of the six star clusters and
compare them to other MW globular clusters. We use the default
MilkyWayPotential Galactic potential from Price-Whelan (2017). This potential
consists of two Hernquist (1990) spheroids to model the stellar bulge and
nucleus, a Miyamoto & Nagai (1975) axisymmetric stellar disk, and a NFW dark
matter halo (Navarro et al., 1996). For each cluster we compute the integrals
of motion, $E$ and $L_{z}$, the approximately conserved quantity $L_{\perp}$
(Massari et al., 2019) and the orbital pericenter ($r_{\rm peri}$), apocenter
($r_{\rm apo}$), and eccentricity. We list the results in Table 2. We apply
the same analysis to the MW globular clusters using the phase space results
from Vasiliev & Baumgardt (2021). We compute 1000 orbits drawn from each
satellite’s 6D phase distribution in Table 2 and compute statistics from these
runs. We use the astropy v4.0 frame (Astropy Collaboration et al., 2013, 2018)
for the Sun’s position and velocity: distance to Galactic Center,
$D_{\odot}=8.122\mathrm{\,kpc}$,
$v_{\odot}=(12.9,245.6,7.78)~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ (Drimmel &
Poggio, 2018; Gravity Collaboration et al., 2018; Reid & Brunthaler, 2004).

In Figures 9 and 10, we show 5 example orbits drawn from the observational
errors of each cluster. Gran 3 is on a near circular orbit ($e\sim 0.12$) in
the inner bulge ($R_{GC}\sim 2.7\mathrm{\,kpc}$) that is confined to the plane
of the disk ($z_{max}\sim 1.8\mathrm{\,kpc}$). Gran 3 is on a retrograde orbit
($L_{Z}\sim+0.45~{}{\rm kpc^{2}~{}Myr^{-2}}$). Gran 4 has an eccentric orbit
($e\sim 0.63$) with a small pericenter ($r_{\rm peri}\sim
8~{}\mathrm{\,kpc}$), large apocenter ($r_{\rm apo}\sim 35~{}\mathrm{\,kpc}$),
and is not confined to the MW midplane ($z_{\rm max}\sim
21~{}\mathrm{\,kpc}$). Gran 4 is an halo globular cluster that is currently
near passing the MW midplane. We find that Garro 01 is on a circular orbit
($e\sim 0.16$) that is confined to the Galactic midplane ($z_{\rm max}\sim
1.3~{}\mathrm{\,kpc}$) at a relatively large Galactocentric radius ($\sim
10-13~{}\mathrm{\,kpc}$). The orbits of the three younger open clusters are
all circular (${\rm ecc}\sim 0.1-0.16$), disk-like orbits. Relative to other
open clusters Gaia 9 and Gaia 10 are at large Galactic distances ($R_{\rm
GC}\sim 18~{}{\rm and}\sim 21~{}\mathrm{\,kpc}$) and have higher angular
momentum $L_{Z}\sim-3.5--4.7~{}{\rm kpc^{2}~{}Myr^{-2}}$. The largest source
of uncertainty in our orbit modeling comes from the distance measurement.

##  5 Discussion

We have presented accurate kinematics and metallicity measurements from
Magellan/M2FS spectroscopy for three recently discovered globular clusters,
Gran 3, Gran 4 and Garro 01, and the discovery and spectroscopic confirmation
of three young open clusters, Gaia 9, Gaia 10, and LP 866. Here we consider
our results in the context of the MW star cluster population.First, we comment
on the nature of Garro 01 and whether it is an old globular cluster or open
cluster (Section 5.1). In particular, how do Gran 3, Gran 4 and Garro 01
relate to the MW globular cluster population and other recently discovered
clusters (Section 5.3)? Are these new globular clusters connected to accretion
events or were they formed in-situ (Section 5.4)? How do the open clusters
compare to the Galactic radial metallicity gradient (Section 5.5)? We conclude
by comparing our results to the literature.

###  5.1 The Nature of Garro 01

Garro et al. (2020) classify Garro 01 as a globular cluster based on its close
similarity to the globular cluster 47 Tuc but several magnitudes fainter. Our
spectroscopic metallicity is more metal-rich ([Fe/H]$=-0.3$) than the
photometric analysis ([Fe/H]$=-0.7$). The orbit of Garro 01 is a disk-like
orbit and Garro 01 is confined to the Galactic plane ($z_{\rm max}\sim
1.3~{}\mathrm{\,kpc}$, ${\rm ecc}\sim 0.16$). Both properties are consistent
with the open cluster population. The age of a star cluster can be key for
determining its origin as a globular cluster or open cluster (e.g., Garro et
al., 2022a).

As previously noted, we had difficultly matching the spectroscopic metallicity
and literature age ($11\pm 1$ Gyr) with Gaia and DECaPS photometry as the
isochrone was redder than the photometry. To estimate the age, we vary the age
at a fixed metallicity ([Fe/H]$=-0.3$) and check whether the color of the red-
giant branch is matched. For this exercise, we examine both Gaia,
$G_{BP}-G{RP}$, and DECaPS, $g-r$, and $r-i$ color. Our best estimate for the
age between $2-13$ Gyr is 4 Gyr (red isochrone in Figure 4). For younger ages
$<3.5~{}{\rm Gyr}$, the main sequence turn-off would be apparent in our sample
which we do not observe. The isochrones with older ages ($>6$ Gyr) are redder
than the observed data. In addition, we find that the Gaia RVS candidate
members are better fit with the younger age.

An age of $\sim 4~{}{\rm Gyr}$ suggests that Garro 01 is an open cluster. This
agrees with the metallicity and disk-like orbit of Garro 01. An accurate age
measurement from deeper photometry would confidently classify this star
cluster as either an open or globular cluster. For the reminder of the
discussion, we will analyze Garro 01 with both the globular clusters and open
clusters in our sample.

###  5.2 Globular Cluster Kinematics

Figure 11:  Rotation curves with sinusoidal by-hand fits for Gran 3, Gran 4,
and Garro 01.  Figure 12:  Projected radius versus velocity dispersion
($\sigma_{\rm los}$) profile for Gran 3 (left), Gran 4 (center), and Garro 01
(right). The spatial errorbar corresponds to the size of the bin and the bins
have an equal number of stars (excluding the last bin). The clusters have 17,
16, and 15 stars per bin. We include velocity dispersion model fits with black
lines and the shaded bands correspond to the error. For Gran 4 (center) we
include a combined M2FS and AAT binned profile (orange bins).

Globular clusters have more complex kinematics than the simple constant
velocity dispersion model we have explored including rotation (e.g., Sollima
et al., 2019) and velocity dispersion profiles (Baumgardt & Hilker, 2018). We
search for rotation by comparing the difference between the mean line-of-sight
velocity across different position angles. We show the results of this
exercise in Figure 11. There is potential rotation on the order of $\sim
1-2\mathrm{\,km}\mathrm{\,s}^{-1}$ in the three clusters, however, when
considering the mean velocity errors it is not significant.

In general, globular clusters have velocity dispersion profiles that decrease
with radius (e.g., Baumgardt & Hilker, 2018) and we search for a radial
dependence in the velocity dispersion by binning the data. We show the binned
velocity dispersion profiles of the three globular clusters in Figure 12. Each
bin contains 18 (Gran 3), 16 (Gran 4), and 15 stars (Garro 01). For Gran 4 we
show the results with the M2FS data (blue) and combined M2FS + AAT
data555There is a $-1.36\mathrm{\,km}\mathrm{\,s}^{-1}$ offset applied to the
AAT data based on the repeat measurements. (orange). Combining the AAT and
M2FS data only increases the sample by 12 stars but there are 10 stars with
improved velocity precision due to multiple measurements. The M2FS velocity
dispersion profiles of Gran 3 and Gran 4 clearly decrease with radius. The
combined M2FS + AAT profile of Gran 4 is more consistent with the constant
dispersion model but the central bin has a larger velocity dispersion. With
Garro 01 the binned profile only measures an upper limit, similar to the
global fit in our mixture model.

To better constrain and quantify the velocity dispersion profile and/or
rotation we explore detailed models. We model the velocity dispersion with a
Plummer profile (Plummer, 1911) and the radially dependent velocity dispersion
profile is: $\sigma^{2}(R)=\sigma_{0}^{2}/\sqrt{1+(r/r_{0})^{2}}$, where
$\sigma_{0}$, and $r_{0}$ are free parameters. We use the following rotation
profile (e.g., Mackey et al., 2013; Cordero et al., 2017; Alfaro-Cuello et
al., 2020): $V_{\rm rot}=\frac{2V_{\rm max}}{r_{\rm
peak}}\frac{X_{PA}}{1+\left(X_{PA}/r_{\rm peak}\right)^{2}}$, where $V_{\rm
max}$, $r_{\rm peak}$, and $\theta_{PA}$ (which determines $X_{PA}$) are free
parameters. We use the dynesty package to sample the posterior and compute
Bayesian evidence for model comparison (Speagle, 2020; Koposov et al., 2022).

We apply the velocity dispersion and rotation models both separately and
together for all three clusters. The inferred velocity dispersion profile fits
are included in Figure 12 with their corresponding errors. For Garro 01 we
examine the stars with membership $>0.9$. For all three clusters the
functional velocity dispersion profile is favored over the constant, non-
rotating model but it is not favored at a statistically significant level. The
dispersion models for Gran 3 and Gran 4 both favor a larger dispersion in the
center of the clusters that decreases with radius. In terms of Bayesian
evidence666We use the scale of Trotta (2008) to quantify significance.
$\ln{Z}$ ranges of 0-1, 1-2.5, 2.5-5.5, $>5.5$ correspond to inconclusive,
weak, moderate, and substantial evidence in favor of the new model., we find
$\ln{Z}=1.7,~{}0.7,~{}0.3$ in favor of the $\sigma(R)$ model for Gran 3, Gran
4, and Garro 01, respectively. Gran 3 is favored with weak evidence whereas
the others are inconclusive. No rotation models are favored and the rotation
models produce an upper limit to the rotation. The rotation models do have a
non-zero peak but large portions of the posterior remain consistent with no
rotation. While there are coherent rotation signals in Figure 11, the non-zero
velocity dispersion and errors in the rotation curve are consistent with no
rotation. To improve constraints on the velocity dispersion profile or probe
potential rotation, larger samples of stars are required.

Last, we estimate the dynamical mass and corresponding mass-to-light ratio of
the three globular clusters. Specifically, we compute the dynamical mass using
the dispersion supported mass estimator from Errani et al. (2018):
$M(r<1.8R_{h})\approx 3.5\times 1.8R_{h}\langle\sigma^{2}_{\rm los}\rangle
G^{-1}$. This approximator is insensitive to the unknown underlying velocity
anisotropy (Walker et al., 2009; Wolf et al., 2010; Errani et al., 2018) but
assumes that the velocity dispersion is approximately constant with radius
which may not be true for globular clusters or the globular clusters in our
sample. With our line-of-sight velocity dispersion and half-light radii
measurements, we measure $M(r<1.8R_{h})=2.7\times 10^{4}\mathrm{\,M_{\odot}}$,
$4.0\times 10^{4}\mathrm{\,M_{\odot}}$, and $2.3\times
10^{3}\mathrm{\,M_{\odot}}$ ($<1.1\times 10^{4}\mathrm{\,M_{\odot}}$) for Gran
3, Gran 4, and Garro 01, respectively. The corresponding mass-to-light ratios
are777For reference, a Plummer profile at $r=1.8r_{\rm p}$ encloses 66.8% of
the total mass. are 1.8 and $0.2$ ($<1.1$), for Gran 4 and Garro 01,
respectively. There are two literature $M_{V}$ values for Gran 3: $M_{V}=-3.8$
(Garro et al., 2022a) and $M_{V}=-6.02$ (Gran et al., 2022). These correspond
to mass-to-light ratios of 1.8 (for $M_{V}=-6.02$) and 14.2 (for
$M_{V}=-3.8$). The mass-to-light ratios for Gran 3 and Gran 4 agree with old
stellar populations. More detailed dynamical modeling, focused on star
clusters would improve this analysis (e.g., Gieles & Zocchi, 2015; Song et
al., 2021).

###  5.3 Comparison to the Globular Cluster Population

In Figure 13, we compare the sizes (based on the 2D projected half-light
radii, $R_{h}$), the metallicity ([Fe/H]), and absolute magnitudes ($M_{V}$)
of our globular cluster sample (Gran 3, Gran 4, Garro 01) to the MW globular
cluster population (Harris, 1996) and to other recently discovered globular
clusters (or candidates) at low Galactic latitude. The (incomplete) list of
recently discovered globular clusters primarily in the Galactic disk and bulge
includes: FSR 1758 (Barbá et al., 2019; Vasiliev & Baumgardt, 2021; Myeong et
al., 2019; Romero-Colmenares et al., 2021), FSR 19 (Obasi et al., 2021), FSR
25 (Obasi et al., 2021), Garro 2 (Garro et al., 2022b), ESO456-29/Gran 1 (Gran
et al., 2019), Gran 2, Gran 5 (Gran et al., 2022), Patchick 99 (Garro et al.,
2021), Pfleiderer 2 (Ortolani et al., 2009), Ryu 059, Ryu 879 (Ryu & Lee,
2018), VVV CL001 (Minniti et al., 2011; Olivares Carvajal et al., 2022), VVV
CL002 (Moni Bidin et al., 2011), VVV CL160 (Minniti et al., 2021), Sagittarius
II (Mutlu-Pakdil et al., 2018; Longeard et al., 2021), and Crater 1 (Weisz et
al., 2016; Kirby et al., 2015). We note that the classification of some
objects is uncertain and spectroscopy is needed.

Our globular cluster sample is generally fainter than the MW globular cluster
population which explains their recent discovery with Gaia astrometry. Gran 4
and Garro 01 are both larger than most clusters ($r_{h}>10~{}\mathrm{\,pc}$).
The large size of Garro 01 is particularly unusual as almost all metal-rich
([Fe/H] $>-1.5$) MW globular clusters have smaller sizes
($r_{h}<10~{}\mathrm{\,pc}$). The exceptions to this are Palomar 12, which is
associated with Sagittarius (e.g., Law & Majewski, 2010; Massari et al.,
2019), and the Fornax 6 globular cluster associated with the Fornax dwarf
spheroidal galaxy (Wang et al., 2019; Pace et al., 2021). Unlike the other
large clusters, Garro 01 is on a circular disk-like orbit making it less
likely to have an ex-situ origin. Gran 3 and Gran 4 are in the metal-poor tail
of the MW globular cluster population as they are more metal-poor than $\sim
83\%$ of the globular clusters in the Harris catalog. In contrast, Garro 01 is
one of the more metal-rich globular clusters. If Garro 01 is confirmed as a
younger open cluster that could explain its large size compared to other
metal-rich globular clusters. In summary, Gran 3, Gran 4, and Garro 01 have
properties consistent with the MW globular cluster population.

Figure 13: Comparison of Gran 3, Gran 4, and the ambiguous open/globular
cluster Garro 01 with MW globular cluster population. Blue are clusters from
the Harris (1996) catalog, and orange are other recently discovered globular
clusters in the MW disk and bulge (see text for name and citations). Left:
Absolute magnitude ($M_{V}$) vs metallicity ([Fe/H]). Center: 2D half-light
radius ($R_{h}$) vs absolute magnitude ($M_{V}$). Right: 2D half-light radius
($R_{h}$) vs metallicity ([Fe/H]).

###  5.4 Origin and Connection to Accretion Events

Figure 14:  Comparison of Gran 3, Gran 4, and the ambiguous open/globular
cluster Garro 01 with the MW globular cluster population. Left: energy ($E$)
versus angular momentum in the z-direction ($L_{z}$). Center: angular momentum
in the z-direction ($L_{z}$) versus angular momentum in the perpendicular-
direction ($L_{\perp}$). Right: orbital pericenter ($r_{\rm peri}$) versus
apocenter ($r_{\rm apo}$). The MW globular clusters are coloured according to
their MW infall merger event (see text for details). Gran 4 is a candidate for
the LMS-1/Wukong merger or Helmi stream merger and Garro 01 is a candidate for
the Aleph merger. Gran 3 is a candidate member of the Galactic bulge
component.

The MW has a population of in-situ and accreted/ex-situ globular clusters
(e.g., Myeong et al., 2018; Massari et al., 2019; Kruijssen et al., 2019). To
determine whether Gran 3, Gran 4, and Garro 01 are associated with any
accretion events we compare the orbital properties of our sample to globular
clusters associated with known events. In Figure 14, we compare the orbital
properties of our globular cluster sample with the MW globular cluster
population. Specifically we examine the energy, angular momentum in the
z-direction, and the angular momentum in the perpendicular direction, and the
pericenter and apocenter. We group the MW clusters based on accretion/merger
events888We have not included the Sequoia+Arjuna+Iiloi structures (Myeong et
al., 2019; Naidu et al., 2020), or Pontus structure (Malhan, 2022) mergers as
there is little overlap with the three globular clusters in our sample.
Furthermore, Thamnos is not included as there are no known globular clusters.
, including: Sagittarius (Sgr), Gaia-Sausage/Enceladus (GSE) (Belokurov et
al., 2018; Helmi et al., 2018), LMS-1/Wukong (Yuan et al., 2020; Naidu et al.,
2020), Aleph (Naidu et al., 2020), Cetus (Newberg et al., 2009), the Helmi
stream (Helmi et al., 1999; Koppelman et al., 2019a), low-
energy/Koala/Kraken/Heracles (Massari et al., 2019; Kruijssen et al., 2019,
2020; Forbes, 2020; Horta et al., 2021) and the in-situ bulge population. We
use Malhan et al. (2022) to associate globular clusters and merger events for
GSE (we additionally include globular clusters from Massari et al. 2019 for
the GSE sample), Sgr, Cetus, LMS-1/Wukong, and the in-situ bulge population.
We follow Naidu et al. (2020) to associate Palomar 1 and the Aleph structure.
The association of the Helmi stream globular clusters is taken from Callingham
et al. (2022). We note between different analyses there is overlap between the
globular clusters assigned to the Helmi stream, GSE, and LMS-1/Wukong
accretion events. The Malhan (2022) analysis does not associate any globular
clusters with the Helmi streams and the Callingham et al. (2022) analysis does
not associate any globular clusters with the LMS-1/Wukong merger. For the
Kraken merger, we use the identification from Callingham et al. (2022). The
Kraken globular clusters overlap with the low-energy globular clusters from
Massari et al. (2019), the Heracles accretion event (Horta et al., 2021) and
the Koala merger from Forbes (2020) and all may be the same merger event. For
simplicity we only include the Kraken merger. We note the
Kraken/Koala/Heracles mergers (and associated globular clusters) in terms of
their chemistry are consistent with being born in-situ, in a pre-disk
population known as Aurora (Belokurov & Kravtsov, 2022). The identification of
each globular cluster with a particular merger event depends on the
methodology and sample (e.g., globular cluster, stellar stream, halo star),
and different analyses have assigned the same globular cluster to different
events or the in-situ population.

In the $E$-$L_{Z}$ plane Gran 3 is located near globular clusters associated
with the Galactic bulge and the low-energy/Koala/Kraken merger (Massari et
al., 2019; Kruijssen et al., 2019, 2020; Forbes, 2020). Gran 3 may be an
extension the Galactic bulge component to higher energy. The Kraken globular
clusters generally have smaller $L_{Z}$ than Gran 3 and have prograde orbits
with larger eccentricity (Massari et al., 2019; Kruijssen et al., 2019, 2020;
Forbes, 2020). As this was one of the first MW mergers the globular clusters
have low metallicities, similar to Gran 3. While there is overlap, the
retrograde orbit of Gran 3 disfavors an association with the Kraken merger.
There are several retrograde accretion events in the stellar halo including
the Sequoia+Arjuna+Iiloi event (Malhan et al., 2022) and Thamnos structure
(Koppelman et al., 2019b). The Thamnos structure has the lowest energy of the
retrograde structures and has a similar metallicity to Gran 3 (Naidu et al.,
2020; Horta et al., 2023), however, the energy is larger than Gran 3 and it is
unlikely for Gran 3 to be associated with the Thamnos merger. We consider Gran
3 to be a member of the Galactic bulge globular cluster group.

We find that Gran 4 is closet to the LMS-1/Wukong merger event in integral of
motion space ($E$, $L_{Z}$, $L_{\perp}$) and in orbital parameters ($r_{\rm
peri}$, $r_{\rm apo}$). While the apocenter and energy is higher than other
the clusters in LMS-1/Wukong merger, the agreement becomes better if the
distance of Gran 4 decreases. We note there is not agreement in the number or
assignment of globular clusters to merger events. In particular, Callingham et
al. (2022) assigns the same LMS-1/Wukong globular clusters here to the Helmi
streams. While Gran 4 is close in the $E$-$L_{Z}$ space to globular clusters
associated with the GSE merger, the GSE globular clusters have smaller
$L_{\perp}$ and $r_{\rm peri}$. We consider Gran 4 to be a candidate member of
the LMS-1/Wukong merger or Helmi streams.

Garro 01 has broad agreement with the energy, angular momentum in the
z-direction, eccentricity, and metallicity of the Aleph structure (Naidu et
al., 2020). There is only one candidate globular cluster in this structure,
Palomar 1 (Naidu et al., 2020). However, Garro 01 is confined to the disk
plane ($z_{\rm max}<1.5$) and the Aleph structure has a strong vertical action
and orbits with larger $z_{\rm max}$. We consider Garro 01 a candidate member
of the Aleph structure but it is more likely to be an in-situ outer disk
cluster.

Garro et al. (2020) suggested that Garro 01 could be associated with the
Monoceros ring (MRi) structure (Newberg et al., 2002). The MRi is proposed to
be either the remnants of tidally disrupted dwarf galaxy (e.g., Peñarrubia et
al., 2005) or a Galactic warp and flare (e.g., Sheffield et al., 2018). While
Garro 01 is not near the previously detected component of MRi, orbital
analysis of MRi suggests there is overlap at location of Garro 01 (Conn et
al., 2008; Grillmair et al., 2008). However, the radial velocity of the model
predictions and Garro 01 are offset. We measure $v_{\rm
gsr}\sim-143\mathrm{\,km}\mathrm{\,s}^{-1}$ for Garro 01 and the radial
velocity of different models varies between $v_{\rm gsr}\sim 0--100$ (see
Figure 17 in Li et al., 2012). The MRi models also predict larger distances
$D\gtrsim 20~{}{\rm kpc}$ than has been inferred for the cluster and the
observed metallicity distribution of MRi is more metal-poor than Garro 01 (see
Figure 13 of Zhang et al., 2022). The modeling and analysis of the MRi and its
connection to Garro 01 would benefit from a dedicated search in this region of
sky but the radial velocity disagreement suggests they are not associated.

###  5.5 Tracing the Galactic Metallicity Gradient with Open Clusters

Figure 15: Comparison of Gaia 9, Gaia 10, LP 866, and the ambiguous
open/globular cluster Garro 01 with MW open cluster population. We include
literature measurements from Spina et al. (2022) as blue points. Top:
Galactocentric radius ($R_{GC}$) versus metallicity ([Fe/H]). Bottom: angular
momentum in the Z-direction ($L_{Z}$) versus metallicity ([Fe/H]). In both
panels, we include the best fit relations derived from the literature open
cluster sample in Spina et al. (2022).

Open clusters are an important tracer of the Galactic radial metallicity
gradient as each cluster can be age-dated with individual chemical elements
studied (e.g., Jacobson et al., 2016; Donor et al., 2020; Spina et al., 2021;
Gaia Collaboration et al., 2023). The Galactic metallicity gradient traces
Galactic formation and evolution scenarios and the complex interplay between
star formation, stellar evolution, stellar migration, gas flows, and cluster
disruption in the Galactic disk (e.g., Chiappini et al., 1997; Schönrich &
Binney, 2009; Kubryk et al., 2015; Spitoni et al., 2019).

We compare the three new open clusters (Gaia 9, Gaia 10, LP 866) to the
literature open clusters (from Spina et al., 2022) and the Galactic radial
metallicity distribution in Figure 15. The Spina et al. (2022) sample includes
open cluster data from several different literature surveys including: APOGEE,
Gaia-ESO, GALAH, OCCASO, and SPA. We include the best fit relation relation
between $R_{GC}-{\rm[Fe/H]}$ and $L_{Z}-{\rm[Fe/H]}$ for literature open
clusters from Spina et al. (2022). We include comparisons to the Galactic
radius and the the angular momentum in the z-direction ($L_{Z}$). $L_{Z}$ is
conserved and the current Galactic radius may not be not representative of
their birth radius as open clusters may have undergone radial migration (e.g.,
Chen & Zhao, 2020). In both cases, the best fit relation becomes shallower at
large radius. Previous measurements have suggested that the relation flattens
out at large radii (e.g. Frinchaboy et al., 2013; Donor et al., 2020).

The open clusters analyzed here are in general agreement with the open cluster
population trends with metallicity, Galactocentric radius, and $L_{Z}$. While
our spectroscopic follow-up has only measured [Fe/H] for three more open
clusters, Gaia 9 and Gaia 10 are among the most metal-poor open clusters in
the MW open cluster population. Gaia 10 has bluethe largest $L_{Z}$ of any MW
open cluster. The properties of Garro 01 are consistent with the Galactic
metallicity gradient as traced by open clusters. Future analyses of the
Galactic radial metallicity gradient will be improved by including Garro 01,
Gaia 9, and Gaia 10 and the metallicity measured from our Magellan/M2FS
spectroscopy.

###  5.6 Comparison to Previous Studies

Of the six star clusters studied only Gran 3 has previous spectroscopic
follow-up. Gran et al. (2022) presented VLT/MUSE spectroscopy of Gran 3 and
found $v_{\rm los}=74.32\pm 2.70\mathrm{\,km}\mathrm{\,s}^{-1}$ and
${\rm[Fe/H]}=-2.37\pm 0.18$. Both measurements are discrepant with our results
and other Gran 3 spectroscopic studies (Fernández-Trincado et al., 2022; Garro
et al., 2023). There is a $\sim 20~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ offset
between the mean radial velocities measured in Gran et al. (2022) compared to
our results and literature. While velocity zero-point offsets of a few
$\mathrm{\,km}\mathrm{\,s}^{-1}$ are common between different
instruments/methods, this value is too large to be caused by a zero point
offset. It is unclear what the origin of this offset is. We note that all our
members are consistent with the same radial velocity, stellar parameters from
a single stellar population, a single metallicity, and consistent proper
motions. In Figure 10 of Gran et al. (2022), the proper motions are consistent
with the majority of their members having similar proper motions but there are
only a few radial velocity members. Some members may be missing due to the
field-of-view of Gran 3 but it is possible that the radial velocity peak was
misidentified.

Fernández-Trincado et al. (2022) analyzed high resolution APGOEE spectroscopy
of two stars in Gran 3. Due to their sample of two stars, the mean velocity of
Gran 3 they measure is offset from our by $\sim
4\mathrm{\,km}\mathrm{\,s}^{-1}$. The metallicity from Fernández-Trincado et
al. (2022) is [Fe/H]$=-1.7\pm 0.09$ is larger than our measurement but it is
consistent within uncertainties. Garro et al. (2023) identified 6 members in
the Gaia RVS sample and their velocity measurement ($v_{\rm los}=93.1\pm
3.6\mathrm{\,km}\mathrm{\,s}^{-1}$) agrees with our measurement within
uncertainties. In our analysis of the Gaia RVS data we identified one
additional Gran 3 member (Section 4.1.1).

We find larger angular sizes for Gran 3 and Gran 4 compared to Gran et al.
(2022). For reference, Gran et al. (2022) find $R_{h}=1.05\pm 0.04~{}{\rm
arcmin}$ and $R_{h}=1.14\pm 0.02~{}{\rm arcmin}$ for Gran 3 and Gran 4,
respectively, compared to our values of $R_{h}=1.7\pm 0.2~{}{\rm arcmin}$ and
$R_{h}=2.2_{-0.4}^{+0.5}~{}{\rm arcmin}$ for Gran 3 and Gran 4, respectively.
The source of this discrepancy could be due to different photometry (Gaia
versus near-IR) or fitting methodology. For Garro 01 there is excellent
agreement between our King profile fits ($r_{c}=1.8_{-0.5}^{+0.7}\arcmin$) and
the results ($r_{c}=2.1\pm 1.5\arcmin$) in Garro et al. (2020). We note that
the absolute magnitude of Gran 3 is discrepant between Garro et al. (2022a)
($M_{V}\sim-3.8$) and Gran et al. (2022) ($M_{V}\sim-6.02$).

Our orbital analysis of Gran 3 is similar to literature results (Gran et al.,
2022; Fernández-Trincado et al., 2022; Garro et al., 2023). Both Fernández-
Trincado et al. (2022) and Garro et al. (2023) include a rotating bar in their
modeling which is not included in our modeling. Compared to the other studies
the value of $z_{\rm max}$ is smaller and the energy lower. We attribute this
to the lower distance we assumed in this work. Compared to Fernández-Trincado
et al. (2022) and Garro et al. (2023) we have a more circular orbit (lower
eccentricity) which agrees with Gran et al. (2022). We note that compared to
other studies we have more precise velocity and proper motion measurements.

##  6 Conclusion

We have presented the spectroscopic follow-up of three recently discovered
globular clusters and three recently discovered open clusters. Our main
findings are as follows:

  * •

We have independently discovered three globular clusters (Gran 3/Patchick 125,
Gran 4, Garro 01) and three open clusters (Gaia 9, Gaia 10, LP 866) with Gaia
astrometry. Gaia 9 and Gaia 10 are new discoveries presented here.

  * •

We have presented spectroscopic follow-up with Magellan/M2FS and measured
stellar parameters of 601 stars and identified 273 members across 6 star
clusters and confirmed the legitimacy of all six clusters. In addition, we
have presented AAT/AAOmega spectroscopy of Gran 4 which confirms our M2FS
results.

  * •

We find Gran 3 (Patchick 125) is an old, metal-poor globular cluster on a
retrograde orbit trapped within the Galactic bulge. From our M2FS
spectroscopy, we identified 37 members and measured a heliocentric velocity of
$v_{\rm los}=90.9\pm 0.4\mathrm{\,km}\mathrm{\,s}^{-1}$ and metallicity of
${\rm[Fe/H]}=-1.83_{-0.03}^{+0.04}$. In addition, there are 2 APOGEE and 7
Gaia RVS members. From our orbital analysis, Gran 3 has a near circular orbit
(${\rm ecc}\sim 0.07$) and orbital pericenter and apocenter of $2.9~{}{\rm
kpc}$ and $3.3~{}{\rm kpc}$, respectively. Gran 3 is likely an in-situ bulge
globular cluster.

  * •

Gran 4 is an old, metal-poor globular cluster with a halo-like orbit that is
passing though the Galactic mid-plane. We identified 63 members from our M2FS
spectroscopy and 22 members (12 unique) from our AAT/AAOmega spectroscopy. We
measured a heliocentric velocity of $v_{\rm los}=-266.4\pm
0.2\mathrm{\,km}\mathrm{\,s}^{-1}$ and metallicity of ${\rm[Fe/H]}=-1.84\pm
0.02$. In addition, there are 3 Gaia RVS members. From our orbital analysis,
Gran 4 has an eccentric orbit circular orbit (${\rm ecc}\sim 0.63$) and
orbital pericenter and apocenter of $7.6~{}{\rm kpc}$ and $33.9~{}{\rm kpc}$,
respectively. Gran 4 is a candidate member of the LMS-1/Wukong and/or Helmi
stream merger events.

  * •

Garro 01 is a metal-rich star cluster on an outer disk-like orbit. We
identified 43 members with our M2FS spectroscopy and measured a heliocentric
velocity of $v_{\rm los}=31.0\pm 0.1\mathrm{\,km}\mathrm{\,s}^{-1}$ and
metallicity of ${\rm[Fe/H]}=-0.30\pm 0.03$. There is more overlap in velocity
with the MW foreground and we constructed a mixture model to quantitatively
account for the MW foreground. In addition, there are 2 candidate Gaia RVS
members. We found that Garro 01 has a relatively large size ($R_{h}\sim
11~{}{\rm pc}$) compared to other metal-rich globular clusters
($R_{h}<5~{}{\rm pc}$). From our orbital analysis, Garro 01 has a circular
orbit (${\rm ecc}\sim 0.16$) and orbital pericenter and apocenter of
$9.8~{}{\rm kpc}$ and $13.3~{}{\rm kpc}$, respectively. We estimated an age of
4 Gyr, which is younger than previous analysis (11$\pm 0.5$ Gyr Garro et al.,
2020). Combined with the metallicity and orbit, this suggests that Garro 01 is
an open cluster but a confident classification requires a more detailed age
measurement and we consider the classification ambiguous.

  * •

Both Gran 3 and Gran 4 have evidence for radially declining velocity
dispersion profiles (Figure 12). There is inconclusive evidence for rotation
in the globular clusters (Figure 11).

  * •

We have confirmed Gaia 9, Gaia 10, and LP 866 as open clusters and identified
19-83 spectroscopic members from our M2FS spectroscopy. We measured
metallicities of $-0.50$, $-0.34$, and $+0.10$ and estimated ages of 1.5, 1,
and 3 Gyr from isochrone fits for Gaia 9, Gaia 10, and LP 866, respectively.
All three open clusters are on circular, disk-like orbits. Gaia 9 and Gaia 10
are among the most distant ($R_{GC}\sim 18,~{}21.2~{}\mathrm{\,kpc}$) and most
metal-poor open clusters known and have some of the largest angular momentum
in the z-direction. These clusters will assist in tracing the Galactic
metallicity gradient to larger radii (Figure 15).

The Milky Way star cluster population remains incomplete and Gaia astrometry
has revolutionised our understanding of star clusters. We have
spectroscopically confirmed six star clusters and there remain many more
candidate star clusters that require spectroscopic follow-up.

## Acknowledgements

We thank the referee for their helpful comments. ABP is supported by NSF grant
AST-1813881. M.G.W. acknowledges support from NSF grants AST-1813881 and
AST-1909584. SK was partially supported by NSF grants AST-1813881 and
AST-1909584. EO was partially supported by NSF grant AST-1815767. NC is
supported by NSF grant AST-1812461. MM was supported by U.S. National Science
Foundation (NSF) grants AST-1312997, AST-1726457 and AST-1815403. IUR
acknowledges support from NSF grants AST-1613536, AST-1815403, AST-2205847,
and PHY-1430152 (Physics Frontier Center/JINA-CEE). DBZ acknowledges support
from Australian Research Council grant DP220102254. We thank Lorenzo Spina for
sharing their open cluster catalog. EO wants to remember Jill Bechtold here.

For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.

This work has made use of data from the European Space Agency (ESA) 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.

This work has used data acquired at the Anglo-Australian Telescope. We
acknowledge the traditional custodians of the land on which the AAT stands,
the Gamilaraay people, and pay our respects to elders past and present.

This research has made use of NASA’s Astrophysics Data System Bibliographic
Services. This paper made use of the Whole Sky Database (wsdb) created by
Sergey Koposov and maintained at the Institute of Astronomy, Cambridge by
Sergey Koposov, Vasily Belokurov and Wyn Evans with financial support from the
Science & Technology Facilities Council (STFC) and the European Research
Council (ERC).

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.

SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss4.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

Software: dynesty (Speagle, 2020; Koposov et al., 2022), astropy (Astropy
Collaboration et al., 2013, 2018), matplotlib (Hunter, 2007), NumPy (Walt et
al., 2011), iPython (Pérez & Granger, 2007), SciPy (Virtanen et al., 2020),
corner.py (Foreman-Mackey, 2016), emcee (Foreman-Mackey et al., 2013) , Q3C
(Koposov & Bartunov, 2006), gala (Price-Whelan, 2017), galpy (Bovy, 2015),
MultiNest (Feroz & Hobson, 2008; Feroz et al., 2009).

## Data Availability

We provide our Magellan/M2FS and AAT/AAOmega catalogs and a machine readable
version of Table 2 at Zenodo under a Creative Commons Attribution license:
doi:10.5281/zenodo.7809128. The other catalogs used in our analysis (Gaia DR3,
DECaPS, APOGEE) are publicly available.

## References

  * Abdurro’uf et al. (2022) Abdurro’uf et al., 2022, ApJS, 259, 35
  * Adamo et al. (2020) Adamo A., et al., 2020, Space Sci. Rev., 216, 69
  * Alfaro-Cuello et al. (2020) Alfaro-Cuello M., et al., 2020, ApJ, 892, 20
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Astropy Collaboration et al. (2018) Astropy Collaboration et al., 2018, AJ, 156, 123
  * Barbá et al. (2019) Barbá R. H., Minniti D., Geisler D., Alonso-García J., Hempel M., Monachesi A., Arias J. I., Gómez F. A., 2019, ApJ, 870, L24
  * Baumgardt & Hilker (2018) Baumgardt H., Hilker M., 2018, MNRAS, 478, 1520
  * Belokurov & Kravtsov (2022) Belokurov V., Kravtsov A., 2022, MNRAS, 514, 689
  * Belokurov et al. (2014) Belokurov V., Irwin M. J., Koposov S. E., Evans N. W., Gonzalez-Solares E., Metcalfe N., Shanks T., 2014, MNRAS, 441, 2124
  * Belokurov et al. (2018) Belokurov V., Erkal D., Evans N. W., Koposov S. E., Deason A. J., 2018, MNRAS, 478, 611
  * Bica et al. (2019) Bica E., Pavani D. B., Bonatto C. J., Lima E. F., 2019, AJ, 157, 12
  * Bovy (2015) Bovy J., 2015, ApJS, 216, 29
  * Bressan et al. (2012) Bressan A., Marigo P., Girardi L., Salasnich B., Dal Cero C., Rubele S., Nanni A., 2012, MNRAS, 427, 127
  * Buder et al. (2021) Buder S., et al., 2021, MNRAS, 506, 150
  * Callingham et al. (2022) Callingham T. M., Cautun M., Deason A. J., Frenk C. S., Grand R. J. J., Marinacci F., 2022, MNRAS, 513, 4107
  * Cantat-Gaudin (2022) Cantat-Gaudin T., 2022, Universe, 8, 111
  * Cantat-Gaudin & Anders (2020) Cantat-Gaudin T., Anders F., 2020, A&A, 633, A99
  * Carrera et al. (2013) Carrera R., Pancino E., Gallart C., del Pino A., 2013, MNRAS, 434, 1681
  * Castro-Ginard et al. (2022) Castro-Ginard A., et al., 2022, A&A, 661, A118
  * Cerny et al. (2021) Cerny W., et al., 2021, ApJ, 920, L44
  * Cerny et al. (2023) Cerny W., et al., 2023, ApJ, 953, 1
  * Chen & Zhao (2020) Chen Y. Q., Zhao G., 2020, MNRAS, 495, 2673
  * Chiappini et al. (1997) Chiappini C., Matteucci F., Gratton R., 1997, ApJ, 477, 765
  * Clementini et al. (2022) Clementini G., et al., 2022, arXiv e-prints, p. arXiv:2206.06278
  * Conn et al. (2008) Conn B. C., Lane R. R., Lewis G. F., Irwin M. J., Ibata R. A., Martin N. F., Bellazzini M., Tuntsov A. V., 2008, MNRAS, 390, 1388
  * Cordero et al. (2017) Cordero M. J., Hénault-Brunet V., Pilachowski C. A., Balbinot E., Johnson C. I., Varri A. L., 2017, MNRAS, 465, 3515
  * Donor et al. (2020) Donor J., et al., 2020, AJ, 159, 199
  * Dotter (2016) Dotter A., 2016, ApJS, 222, 8
  * Drimmel & Poggio (2018) Drimmel R., Poggio E., 2018, Research Notes of the American Astronomical Society, 2, 210
  * Errani et al. (2018) Errani R., Peñarrubia J., Walker M. G., 2018, MNRAS, 481, 5073
  * Fernández-Trincado et al. (2022) Fernández-Trincado J. G., Minniti D., Garro E. R., Villanova S., 2022, A&A, 657, A84
  * Feroz & Hobson (2008) Feroz F., Hobson M. P., 2008, MNRAS, 384, 449
  * Feroz et al. (2009) Feroz F., Hobson M. P., Bridges M., 2009, MNRAS, 398, 1601
  * Forbes (2020) Forbes D. A., 2020, MNRAS, 493, 847
  * Foreman-Mackey (2016) Foreman-Mackey D., 2016, The Journal of Open Source Software, 1, 24
  * Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
  * Frinchaboy et al. (2013) Frinchaboy P. M., et al., 2013, ApJ, 777, L1
  * Gaia Collaboration et al. (2021) Gaia Collaboration et al., 2021, A&A, 649, A1
  * Gaia Collaboration et al. (2023) Gaia Collaboration et al., 2023, A&A, 674, A38
  * Garro et al. (2020) Garro E. R., et al., 2020, A&A, 642, L19
  * Garro et al. (2021) Garro E. R., Minniti D., Gómez M., Alonso-García J., Palma T., Smith L. C., Ripepi V., 2021, A&A, 649, A86
  * Garro et al. (2022a) Garro E. R., et al., 2022a, A&A, 659, A155
  * Garro et al. (2022b) Garro E. R., Minniti D., Gómez M., Fernández-Trincado J. G., Alonso-García J., Hempel M., Zelada Bacigalupo R., 2022b, A&A, 662, A95
  * Garro et al. (2023) Garro E. R., et al., 2023, A&A, 669, A136
  * Gieles & Zocchi (2015) Gieles M., Zocchi A., 2015, MNRAS, 454, 576
  * Górski et al. (2011) Górski M., Pietrzyński G., Gieren W., 2011, AJ, 141, 194
  * Gran et al. (2019) Gran F., et al., 2019, A&A, 628, A45
  * Gran et al. (2022) Gran F., et al., 2022, MNRAS, 509, 4962
  * Gratton et al. (2019) Gratton R., Bragaglia A., Carretta E., D’Orazi V., Lucatello S., Sollima A., 2019, A&ARv, 27, 8
  * Gravity Collaboration et al. (2018) Gravity Collaboration et al., 2018, A&A, 615, L15
  * Grillmair et al. (2008) Grillmair C. J., Carlin J. L., Majewski S. R., 2008, ApJ, 689, L117
  * Harris (1996) Harris W. E., 1996, AJ, 112, 1487
  * Helmi et al. (1999) Helmi A., White S. D. M., de Zeeuw P. T., Zhao H., 1999, Nature, 402, 53
  * Helmi et al. (2018) Helmi A., Babusiaux C., Koppelman H. H., Massari D., Veljanoski J., Brown A. G. A., 2018, Nature, 563, 85
  * Hernquist (1990) Hernquist L., 1990, ApJ, 356, 359
  * Horta et al. (2021) Horta D., et al., 2021, MNRAS, 500, 1385
  * Horta et al. (2023) Horta D., et al., 2023, MNRAS, 520, 5671
  * Hunt & Reffert (2023) Hunt E. L., Reffert S., 2023, arXiv e-prints, p. arXiv:2303.13424
  * Hunter (2007) Hunter J. D., 2007, Computing In Science & Engineering, 9, 90
  * Jacobson et al. (2016) Jacobson H. R., et al., 2016, A&A, 591, A37
  * Katz et al. (2022) Katz D., et al., 2022, arXiv e-prints, p. arXiv:2206.05902
  * King (1962) King I., 1962, AJ, 67, 471
  * Kirby et al. (2015) Kirby E. N., Simon J. D., Cohen J. G., 2015, ApJ, 810, 56
  * Koposov (2019) Koposov S. E., 2019, RVSpecFit: Radial velocity and stellar atmospheric parameter fitting, Astrophysics Source Code Library, record ascl:1907.013 (ascl:1907.013)
  * Koposov & Bartunov (2006) Koposov S., Bartunov O., 2006, in Gabriel C., Arviset C., Ponz D., Enrique S., eds, Astronomical Society of the Pacific Conference Series Vol. 351, Astronomical Data Analysis Software and Systems XV. p. 735
  * Koposov et al. (2007) Koposov S., et al., 2007, ApJ, 669, 337
  * Koposov et al. (2008) Koposov S., et al., 2008, ApJ, 686, 279
  * Koposov et al. (2015) Koposov S. E., Belokurov V., Torrealba G., Evans N. W., 2015, ApJ, 805, 130
  * Koposov et al. (2017) Koposov S. E., Belokurov V., Torrealba G., 2017, MNRAS, 470, 2702
  * Koposov et al. (2022) Koposov S., et al., 2022, joshspeagle/dynesty: v2.0.1, doi:10.5281/zenodo.7215695, https://doi.org/10.5281/zenodo.7215695
  * Koppelman et al. (2019a) Koppelman H. H., Helmi A., Massari D., Roelenga S., Bastian U., 2019a, A&A, 625, A5
  * Koppelman et al. (2019b) Koppelman H. H., Helmi A., Massari D., Price-Whelan A. M., Starkenburg T. K., 2019b, A&A, 631, L9
  * Kounkel et al. (2020) Kounkel M., Covey K., Stassun K. G., 2020, AJ, 160, 279
  * Kruijssen et al. (2019) Kruijssen J. M. D., Pfeffer J. L., Reina-Campos M., Crain R. A., Bastian N., 2019, MNRAS, 486, 3180
  * Kruijssen et al. (2020) Kruijssen J. M. D., et al., 2020, MNRAS, 498, 2472
  * Krumholz et al. (2019) Krumholz M. R., McKee C. F., Bland-Hawthorn J., 2019, ARA&A, 57, 227
  * Kubryk et al. (2015) Kubryk M., Prantzos N., Athanassoula E., 2015, A&A, 580, A126
  * Law & Majewski (2010) Law D. R., Majewski S. R., 2010, ApJ, 718, 1128
  * Layden (1994) Layden A. C., 1994, AJ, 108, 1016
  * Lewis et al. (2002) Lewis I. J., et al., 2002, MNRAS, 333, 279
  * Li et al. (2012) Li J., Newberg H. J., Carlin J. L., Deng L., Newby M., Willett B. A., Xu Y., Luo Z., 2012, ApJ, 757, 151
  * Li et al. (2019) Li T. S., et al., 2019, MNRAS, 490, 3508
  * Li et al. (2022) Li T. S., et al., 2022, ApJ, 928, 30
  * Lindegren et al. (2021) Lindegren L., et al., 2021, A&A, 649, A2
  * Liu & Pang (2019) Liu L., Pang X., 2019, ApJS, 245, 32
  * Longeard et al. (2021) Longeard N., et al., 2021, MNRAS, 503, 2754
  * Mackey et al. (2013) Mackey A. D., Da Costa G. S., Ferguson A. M. N., Yong D., 2013, ApJ, 762, 65
  * Malhan (2022) Malhan K., 2022, ApJ, 930, L9
  * Malhan et al. (2022) Malhan K., et al., 2022, ApJ, 926, 107
  * Martinez et al. (2011) Martinez G. D., Minor Q. E., Bullock J., Kaplinghat M., Simon J. D., Geha M., 2011, ApJ, 738, 55
  * Massari et al. (2019) Massari D., Koppelman H. H., Helmi A., 2019, A&A, 630, L4
  * Mateo et al. (2012) Mateo M., Bailey J. I., Crane J., Shectman S., Thompson I., Roederer I., Bigelow B., Gunnels S., 2012, M2FS: the Michigan/Magellan Fiber System. p. 84464Y, doi:10.1117/12.926448
  * Mau et al. (2019) Mau S., et al., 2019, ApJ, 875, 154
  * Minniti et al. (2011) Minniti D., et al., 2011, A&A, 527, A81
  * Minniti et al. (2021) Minniti D., Fernández-Trincado J. G., Gómez M., Smith L. C., Lucas P. W., Contreras Ramos R., 2021, A&A, 650, L11
  * Miyamoto & Nagai (1975) Miyamoto M., Nagai R., 1975, PASJ, 27, 533
  * Molnar et al. (2022) Molnar T. A., Sanders J. L., Smith L. C., Belokurov V., Lucas P., Minniti D., 2022, MNRAS, 509, 2566
  * Moni Bidin et al. (2011) Moni Bidin C., et al., 2011, A&A, 535, A33
  * Muraveva et al. (2018) Muraveva T., Delgado H. E., Clementini G., Sarro L. M., Garofalo A., 2018, MNRAS, 481, 1195
  * Mutlu-Pakdil et al. (2018) Mutlu-Pakdil B., et al., 2018, ApJ, 863, 25
  * Myeong et al. (2018) Myeong G. C., Evans N. W., Belokurov V., Sanders J. L., Koposov S. E., 2018, ApJ, 863, L28
  * Myeong et al. (2019) Myeong G. C., Vasiliev E., Iorio G., Evans N. W., Belokurov V., 2019, MNRAS, 488, 1235
  * Naidu et al. (2020) Naidu R. P., Conroy C., Bonaca A., Johnson B. D., Ting Y.-S., Caldwell N., Zaritsky D., Cargile P. A., 2020, ApJ, 901, 48
  * Navarro et al. (1996) Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563
  * Newberg et al. (2002) Newberg H. J., et al., 2002, ApJ, 569, 245
  * Newberg et al. (2009) Newberg H. J., Yanny B., Willett B. A., 2009, ApJ, 700, L61
  * Obasi et al. (2021) Obasi C., Gómez M., Minniti D., Alonso-García J., 2021, A&A, 654, A39
  * Olivares Carvajal et al. (2022) Olivares Carvajal J., Zoccali M., Rojas-Arriagada A., Contreras Ramos R., Gran F., Valenti E., Minniti J. H., 2022, MNRAS, 513, 3993
  * Ortolani et al. (2009) Ortolani S., Bonatto C., Bica E., Barbuy B., 2009, AJ, 138, 889
  * Pace et al. (2020) Pace A. B., et al., 2020, MNRAS, 495, 3022
  * Pace et al. (2021) Pace A. B., Walker M. G., Koposov S. E., Caldwell N., Mateo M., Olszewski E. W., Bailey John I. I., Wang M.-Y., 2021, ApJ, 923, 77
  * Peñarrubia et al. (2005) Peñarrubia J., et al., 2005, ApJ, 626, 128
  * Pérez & Granger (2007) Pérez F., Granger B. E., 2007, Computing in Science and Engineering, 9, 21
  * Plummer (1911) Plummer H. C., 1911, MNRAS, 71, 460
  * Price-Whelan (2017) Price-Whelan A. M., 2017, The Journal of Open Source Software, 2
  * Reid & Brunthaler (2004) Reid M. J., Brunthaler A., 2004, ApJ, 616, 872
  * Romero-Colmenares et al. (2021) Romero-Colmenares M., et al., 2021, A&A, 652, A158
  * Ryu & Lee (2018) Ryu J., Lee M. G., 2018, ApJ, 863, L38
  * Saha et al. (2019) Saha A., et al., 2019, ApJ, 874, 30
  * Schlafly et al. (2018) Schlafly E. F., et al., 2018, ApJS, 234, 39
  * Schönrich & Binney (2009) Schönrich R., Binney J., 2009, MNRAS, 396, 203
  * Sesar et al. (2017) Sesar B., et al., 2017, AJ, 153, 204
  * Sheffield et al. (2018) Sheffield A. A., Price-Whelan A. M., Tzanidakis A., Johnston K. V., Laporte C. F. P., Sesar B., 2018, ApJ, 854, 47
  * Sollima et al. (2019) Sollima A., Baumgardt H., Hilker M., 2019, MNRAS, 485, 1460
  * Song et al. (2021) Song Y.-Y., Mateo M., Bailey John I. I., Walker M. G., Roederer I. U., Olszewski E. W., Reiter M., Kremin A., 2021, MNRAS, 504, 4160
  * Soszyński et al. (2019) Soszyński I., et al., 2019, Acta Astron., 69, 321
  * Souza et al. (2021) Souza S. O., et al., 2021, A&A, 656, A78
  * Speagle (2020) Speagle J. S., 2020, MNRAS, 493, 3132
  * Spencer et al. (2018) Spencer M. E., Mateo M., Olszewski E. W., Walker M. G., McConnachie A. W., Kirby E. N., 2018, AJ, 156, 257
  * Spina et al. (2021) Spina L., et al., 2021, MNRAS, 503, 3279
  * Spina et al. (2022) Spina L., Magrini L., Cunha K., 2022, Universe, 8, 87
  * Spitoni et al. (2019) Spitoni E., Silva Aguirre V., Matteucci F., Calura F., Grisoni V., 2019, A&A, 623, A60
  * Torrealba et al. (2019) Torrealba G., Belokurov V., Koposov S. E., 2019, MNRAS, 484, 2181
  * Trotta (2008) Trotta R., 2008, Contemporary Physics, 49, 71
  * Vasiliev & Baumgardt (2021) Vasiliev E., Baumgardt H., 2021, MNRAS, 505, 5978
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Vivas et al. (2005) Vivas A. K., Zinn R., Gallart C., 2005, AJ, 129, 189
  * Walker et al. (2006) Walker M. G., Mateo M., Olszewski E. W., Bernstein R., Wang X., Woodroofe M., 2006, AJ, 131, 2114
  * Walker et al. (2009) Walker M. G., Mateo M., Olszewski E. W., Peñarrubia J., Wyn Evans N., Gilmore G., 2009, ApJ, 704, 1274
  * Walker et al. (2023) Walker M. G., Caldwell N., Mateo M., Olszewski E. W., Pace A. B., Bailey J. I., Koposov S. E., Roederer I. U., 2023, ApJS, 268, 19
  * Walt et al. (2011) Walt S. v. d., Colbert S. C., Varoquaux G., 2011, Computing in Science & Engineering, 13, 22
  * Wang et al. (2019) Wang M. Y., et al., 2019, ApJ, 875, L13
  * Webb & Carlberg (2021) Webb J. J., Carlberg R. G., 2021, MNRAS, 502, 4547
  * Weisz et al. (2016) Weisz D. R., et al., 2016, ApJ, 822, 32
  * Wolf et al. (2010) Wolf J., Martinez G. D., Bullock J. S., Kaplinghat M., Geha M., Muñoz R. R., Simon J. D., Avedo F. F., 2010, MNRAS, 406, 1220
  * Yuan et al. (2020) Yuan Z., Chang J., Beers T. C., Huang Y., 2020, ApJ, 898, L37
  * Zhang et al. (2022) Zhang Z., Shi W. B., Chen Y. Q., Zhao G., Carrell K., Zhang H. P., 2022, ApJ, 933, 151