# Stellar kinematics of dwarf galaxies from multi-epoch spectroscopy:
application to Triangulum II ††thanks: New observations reported here were
obtained at the MMT Observatory, a joint facility of the Smithsonian
Institution and the University of Arizona. Some of the data presented herein
were obtained at the W. M. Keck Observatory, which is operated as a scientific
partnership among the California Institute of Technology, the University of
California and the National Aeronautics and Space Administration. The
Observatory was made possible by the generous financial support of the W. M.
Keck Foundation.

Rachel Buttry1, Andrew B. Pace1, Sergey E. Koposov2,3,1, Matthew G. Walker1,
Nelson Caldwell4, Evan N. Kirby5, Nicolas F. Martin6,7, Mario Mateo8, Edward
W. Olszewski9, Else Starkenburg10, Carles Badenes11, and Christine Mazzola
Daher11
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 Rd, Cambridge, CB3
0HA, UK
4Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
5California Institute of Technology, 1200 E. California Blvd., MC 249-17,
Pasadena, CA 91125, USA
6Observatoire astronomique de Strasbourg, Université de Strasbourg, CNRS, UMR
7550, 11 rue de l’Université, F-67000 Strasbourg, France
7Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany
8Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
9Steward Observatory, The University of Arizona, 933 N. Cherry Avenue, Tucson,
AZ 85721, USA
10Kapteyn Astronomical Institute, University of Groningen,Postbus 800, 9700
AV, Groningen, the Netherlands
11Department of Physics and Astronomy and Pittsburgh Particle Physics,
Astrophysics and Cosmology Center (PITT PACC),
University of Pittsburgh, 3941 O‘Hara Street, Pittsburgh, PA 15260, USA
E-mail<EMAIL_ADDRESS>(RB)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present new MMT/Hectochelle spectroscopic measurements for 257 stars
observed along the line of sight to the ultra-faint dwarf galaxy Triangulum
II. Combining with results from previous Keck/DEIMOS spectroscopy, we obtain a
sample that includes 16 likely members of Triangulum II, with up to 10
independent redshift measurements per star. To this multi-epoch kinematic data
set we apply methodology that we develop in order to infer binary orbital
parameters from sparsely sampled radial velocity curves with as few as two
epochs. For a previously-identified (spatially unresolved) binary system in
Tri II, we infer an orbital solution with period
$296.0_{-3.3}^{+3.8}\rm~{}days$ , semi-major axis
$1.12^{+0.41}_{-0.24}\rm~{}AU$, and a systemic velocity $-380.0\pm
1.7\rm~{}km~{}s^{-1}$ that we then use in the analysis of Tri II’s internal
kinematics. Despite this improvement in the modeling of binary star systems,
the current data remain insufficient to resolve the velocity dispersion of
Triangulum II. We instead find a 95% confidence upper limit of
$\sigma_{v}\lesssim 3.4\rm~{}km~{}s^{-1}$.

###### keywords:

(stars:) binaries: spectroscopic – galaxies: kinematics and dynamics

††pubyear: 2021††pagerange: Stellar kinematics of dwarf galaxies from multi-
epoch spectroscopy: application to Triangulum II ††thanks: New observations
reported here were obtained at the MMT Observatory, a joint facility of the
Smithsonian Institution and the University of Arizona. Some of the data
presented herein were obtained at the W. M. Keck Observatory, which is
operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation.–B

##  1 Introduction

Dwarf galaxies are of great importance for astrophysics. From a galaxy
formation perspective, dwarf galaxies are among the oldest and least
chemically evolved objects (Mateo, 1998; Tolstoy et al., 2009; McConnachie,
2012). From a dark matter perspective, they include the most dark matter
dominated systems known, with published dynamical mass-to-light ratios
reaching as high as $10^{4}$ in solar units (Simon, 2019, and references
therein). In this vein, dwarf galaxies are believed to be key components in
unpacking the mystery of dark matter, as they probe the small-scale structure
(< 1 Mpc) regime of $\Lambda$CDM cosmology (Bullock & Boylan-Kolchin, 2017).

In order to place dwarf galaxies into their proper cosmological context, we
must obtain accurate estimates of their dark matter content. The simplest
dynamical mass estimators, based on the assumption of dynamic equilibrium, are
functions of the effective radius and line-of-sight velocity dispersion
measured for the stellar component (e.g., Illingworth, 1976; Walker et al.,
2009; Wolf et al., 2010; Errani et al., 2018). Yet measurements of the stellar
velocity dispersion can be challenging. One reason is the small number and low
luminosities of stellar tracers in especially the ‘ultrafaint’ dwarf galaxies.
Another challenge is the existence of unresolved binary stars, whose orbital
motions add a time-dependent component to the velocities measured for
individual stars, and—if unaccounted for—can thereby inflate measurements of
dwarf galaxy velocity dispersions. Binary orbital motions alone can generate
apparent velocity dispersions of a few km s-1 (McConnachie & Côté, 2010).
While this effect is negligible for the more luminous dwarf spheroidals, which
have intrinsic velocity dispersions of $\sim 10$ km s-1 (Olszewski et al.,
1996), it can potentially contribute significantly to the $\la 3$ km s-1
dispersions observed for the least luminous galaxies (McConnachie & Côté,
2010; Minor et al., 2010).

Various strategies have been used to account for the effect of binary stars on
velocity dispersions and the dynamical masses derived therefrom. When multi-
epoch spectroscopy is available, one can identify probable binary systems via
their observed accelerations (e.g., Olszewski et al., 1996; Martinez et al.,
2011; Koposov et al., 2011; Minor et al., 2019); indeed modeling of multi-
epoch spectroscopic data sets for luminous dwarf spheroidals suggests typical
binary fractions near $\sim 50\%$ (Minor 2013, Spencer et al. 2017, Spencer et
al. 2018), consistent with studies of Galactic binaries that indicate
relatively high multiplicity fractions at low metallicity (Badenes et al.,
2018). In some cases the removal of suspected binary stars has significant
impact on the measured velocity dispersion (e.g., Kirby et al., 2017; Venn et
al., 2017).

The Triangulum II (Tri II) ultra-faint dwarf galaxy provides an interesting
case study. The original kinematic study of Tri II, based on single-epoch
spectroscopy of six member stars, measured a velocity dispersion of
$5.1^{+4.0}_{-1.4}\rm~{}km~{}s^{-1}$ , suggesting a dynamical mass-to-light
ratio of $3600_{-2100}^{+3500}$ in solar units, and an extremely high dark
matter density of $4.8_{-3.5}^{+8.1}\rm~{}M_{\odot}/\mathrm{pc}^{3}$ (Kirby et
al., 2015). An independent study by Martin et al. (2016) obtained a
spectroscopic sample of 13 member stars, finding complicated kinematics in
which a central velocity dispersion of
$\sigma_{v}=4.4^{+2.8}_{-2.0}\rm~{}km~{}s^{-1}$ gives way to a larger value of
$14.1_{-4.2}^{+5.8}$ km s-1 at large radius. Both studies found evidence for
nonzero metallicity dispersion, supporting the conclusion that Tri II is a
dwarf galaxy embedded in a massive dark matter halo, and not a self-
gravitating star cluster.

However, follow-up spectroscopy soon provided a time domain and revealed the
presence of at least one star with significant velocity variability. From
high-resolution spectra obtained primarily to analyze chemical abundances,
Venn et al. (2017) measured a change in velocity for one star (Star46) of
$\sim 25$ km s-1 with respect to the initial epoch measured by Martin et al.
(2016). Kirby et al. (2017) added additional epochs for this star,
independently confirming its velocity variability and finding that, when they
excluded the likely binary from their analysis, the velocity dispersion was
unresolved. These circumstances leave the case for a dominant dark matter halo
in Tri II resting on the indirect argument provided by its metallicity spread
(Venn et al., 2017).

Here we add to the saga of Tri II in two ways. First, we present new
spectroscopic data acquired with the Hectochelle spectrograph at the 6.5-m
MMT. Second, we combine with the previously-published spectroscopic data in
order to obtain a multi-epoch data set that then lets us model the orbital
parameters of the likely binary star. Our orbital solution includes an
inference for the binary system’s center-of-mass motion, allowing us properly
to include this star in our analysis of Tri II’s stellar kinematics.

To date, only one star system within a dwarf spheroidal galaxy has a full
orbital solution, based on 34 independent velocity measurements taken over a
two-year baseline (Koch et al., 2014). Here we develop methodology for
inferring orbital solutions with as few as two velocity epochs. The problem of
finding orbital parameters for a binary system given a small number of radial
velocity (RV) measurements has been undertaken previously by Price-Whelan et
al. (2017) to create the JOKER. Like the JOKER, the binary model we present in
this paper takes the approach of performing rejection sampling with likelihood
function marginalized over some orbital parameters. However, our method has
the added modifications such as the marginalization over inclination rather
than semi-amplitude (allowing for the calculation of semi-major axis), the
ability to take non-trivial priors over binary parameters, and using parameter
samples for hierarchical models of binary populations.

In the next section we discuss the MMT and Keck catalogues used in this
analysis as well as the calculation of a zero-point correction between the two
instruments. In section 3 we present our methodology for modeling of binary
and non-binary star systems as well as the galaxy kinematics. We then detail
the resulting orbital parameter for the Tri II binary system and the overall
Tri II kinematics in section 4. Lastly, we discuss the findings from our work
in section 5 and we suggest a hierarchical model building off the methods
used.

##  2 Data

###  2.1 MMT Hectochelle

During Dec. 2015 and Oct. - Nov. 2016, we acquired new spectra of stars in Tri
II using the Hectochelle spectrograph (Szentgyorgyi et al., 2011) at the 6.5-m
MMT Observatory on Mt. Hopkins, Arizona. Hectochelle deploys up to 240 optical
fibers, each with aperture $1.5$ arcsec, over a field of diameter $1^{\circ}$.
We observed using the ‘RV31’ filter, isolating the wavelength range of
$5150-5300$ at resolution $\mathcal{R}\approx 34,000$.

We observed five different Hectochelle fiber configurations, each centered on
the published center of Tri II, allowing us to observe up to $\sim 500$ unique
targets, with many stars included in multiple targeting configurations.

Targets were selected using metallicity-sensitive CaHK narrow-band photometry
to prioritize low metallicity stars that are more likely to be members of the
very metal-poor Tri II. We obtained a series of deep broadband $g$ and $i$
observations with the $\sim 1$-deg2 imager MegaCam on the Canada France Hawaii
Telescope for total integration times of 4,050 s and 3,150 s, respectively,
along with CaHK images totaling 4,800 s. All were observed in service mode by
the CFHT observing staff between 2015-07-18 and 2016-02-13. Images were then
processed with a version of the CASU pipeline (Irwin & Lewis, 2001) tailored
to the processing of MegaCam images (e.g., McConnachie et al. 2018).

To isolate metal-poor point sources extracted from the resulting stacks, we
follow the same strategy put in place for the Pristine survey (Starkenburg et
al., 2017). Most stars in the field are metal-rich and therefore follow a
well-defined stellar locus whereas metal-poor stars deviate from this locus.
In the case of Tri II, members previously confirmed via spectroscopy are
clearly offset by 0.2–0.5 magnitudes from the locus of contaminating Milky Way
stars. Given the large number of available Hectochelle fibers we were able to
prioritize metal-poor stars and remove obviously metal-rich stars, but we
remained generous in our selection. This strategy worked well for previous
observing campaigns (Longeard et al., 2020) to ensure that we did not miss
member stars with potentially higher metallicities. In addition to known Tri
II members, this selection includes potential members out to $\sim 0.5\deg$
from the center of the dwarf galaxy.

We processed all raw Hectochelle spectra using the CfA pipeline (HSRED
v2.1111https://bitbucket.org/saotdc/hsred/ ). Following the procedure
described in detail by Walker et al. (2015), we then analyzed each individual
spectrum by fitting a model based on a library of synthetic template spectra
that span a regular grid in effective temperature, surface gravity and [Fe/H]
metallicity. In addition to the stellar-atmospheric parameters, we fit for
line-of-sight velocity as well as several free parameters that specify the
continuum shape and correct for wavelength-dependent velocity shifts.

With respect to the procedure documented by Walker et al. (2015), for present
purposes we update our estimation of systematic errors associated with line-
of-sight velocity and metallicity. For this task we use our entire catalog of
MMT/Hectochelle observations of dwarf galaxies and globular clusters,
including observations spanning the years 2005 to 2020. This sample includes
12517 independent observations of 7906 unique stars, including 2501 stars with
up to 13 individual measurements. We model the pair-wise velocity and
metallicity differences as a mixture of a Gaussian with an outlier model (see
Section 4.1 of Li et al. 2019 and Section 2.2 of Pace et al. 2021).The final
uncertainty ($\sigma_{v,\rm calib}$) is treated as a systematic error
($\sigma_{v,\,{\rm systematic}}$) plus a scaling parameter ($k_{v}$) and
$\sigma_{v,\rm calib}^{2}=\sigma_{v,\,{\rm
systematic}}^{2}+(k_{v}\sigma_{v,{\rm mcmc}})^{2}$. We find $k_{v}=1.03\pm
0.02$ and $\sigma_{v,\,{\rm systematic}}=0.35\pm 0.02$ km s-1for the velocity
systematic errors, and $k_{\rm[Fe/H]}=1.33\pm 0.01$ and
$\sigma_{{\rm[Fe/H]},\,{\rm systematic}}=0.0\pm 0.01$ for metallicity
systematic errors.

###  2.2 Keck DEIMOS

Our sample also includes spectroscopy obtained with the Deep Extragalactic
Imaging Multi-Object Spectrograph (DEIMOS; Faber et al., 2003). First, Kirby
et al. (2015); Kirby et al. (2017) observed six slitmasks in 2015 and 2016.
They used the 1200G grating, which achieves a spectral resolution of $R\sim
7000$ at 8500 Å, in the spectral vicinity of the Ca ii infrared triplet.
Second, Martin et al. (2016) observed two DEIMOS slitmasks with a similar
spectral configuration as Kirby et al.

Kirby et al. (2015) selected stars using Keck/LRIS (Oke et al., 1995)
photometry. They chose a generous selection region in the CMD around the red
giant branch as defined by the ridgeline of the globular cluster M92. Martin
et al. (2016) used a similar selection technique with photometry (Laevens et
al., 2015) from the Large Binocular Camera. In general, the field of Tri II is
sparse enough that most candidate member stars in the field of the slitmask
could be observed. As a result, the samples have little selection bias due to
color (or stellar age or metallicity). Kirby et al. (2017) were mainly
interested in quantifying radial velocity variability, not in finding new
members. As a result, they designed their slitmasks to target stars already
identified as members by Kirby et al. (2015) and Martin et al. (2016).

The velocities were measured in slightly different ways. Kirby et al. (2015);
Kirby et al. (2017) reduced the spectra with custom modifications to the
spec2d data reduction pipeline (Cooper et al., 2012). They matched empirical
spectral templates observed with DEIMOS to the Tri II spectra and varied the
velocity until $\chi^{2}$ was minimized. Martin et al. (2016) used their own
custom pipeline (Ibata et al., 2011) to reduce the spectra. They determined
radial velocities from the mean wavelengths of Gaussian fits to the Ca ii
triplet.

Slit imaging spectrographs can experience radial velocity zeropoint shifts if
the star is not perfectly centered in the slit. This effect can be mitigated
by observing the wavelengths of telluric absorption lines (e.g., Sohn et al.,
2007). The slit centering correction is taken to be the deviation of the
wavelengths of these lines from the geocentric rest frame. All of the studies
used in this work performed such a correction.

###  2.3 Zero-point offset

In this section, we describe the calculation of the zero-point velocity offset
between Keck and MMT. Though our complete set of Tri II Keck observations is a
concatenation of the Kirby et al. (2017) and Martin et al. (2016) datasets,
the zero-point offset between the two appear consistent with zero (see Section
3.2 of Kirby et al., 2017). For this analysis, we have defined the offset as
the MMT zero-point minus the Keck zero-point. There are 8 objects in our
dataset around the Tri II galactic center with radial velocity measurements in
both MMT and Keck catalogues. The measurements for a given object and
instrument are combined into a single weighted mean value. We then calculate
the offset between the sets of combined instrument velocities while assuming
Gaussian errors.

The methods described here and in section 3 are based on a Gaussian likelihood
function, which is described in terms of model prediction $\mu$, model error
$\sigma$, given data in the form of the observed velocity, $v$:

$\mathcal{G}(v|\mu,\sigma)=\frac{1}{\sqrt{2\pi(v_{\rm
error}^{2}+\sigma^{2})}}\exp\left(-\frac{(v-\mu)^{2}}{2(v_{\rm
error}^{2}+\sigma^{2})}\right)$ (1)

This is the general likelihood for a given velocity prediction and the total
likelihood function is the product of the velocity likelihoods.

In the context of finding a zero-point offset, we apply this likelihood to a
set of velocity differences $\\{v_{i}\\}$, where $v_{i}$ represents the
difference between the $i$th star’s MMT weighted mean velocity and its Keck
weighted mean velocity, $v_{i}=\bar{v}_{i,\rm MMT}-\bar{v}_{i,\rm Keck}$. We
assume that these velocity differences are consistent with the zero-point
offset between instruments $\delta_{v}$ and that there is no dispersion in
$\delta_{v}$, only observational errors. Thus the total likelihood is the
product of individual object likelihoods,
$\prod_{i=1}^{N}\mathcal{G}(v_{i},\delta_{v},0)$.

However, some objects have a very large velocity difference between the
instruments, possibly due to low signal-to-noise or unconfirmed binarity. A
convenient way to account for these objects is to construct a mixture model to
treat them as outliers, a similar model is used in Li et al. (2019). The new
likelihood for a given object is the sum original Gaussian likelihood and
outlier model likelihood weighted by the probability the object is an outlier.
The new likelihood is written as…

$\begin{split}L(\\{v_{i}\\}|\delta_{v},\gamma,p)=\prod_{i=1}^{N}[\gamma\mathcal{G}(v_{i}|0,p)+(1-\gamma)\mathcal{G}(v_{i}|\delta_{v},0)]\end{split}$
(2)

where $\gamma$ represents the outlier fraction and $\delta_{v}$ represents the
offset correction. The outlier model is applied to the difference in the
combined weighted mean velocities for each instrument and is taken to be a
Gaussian with a large standard deviation. The priors are as follows:

  * •

offset $\delta_{v}$: uniform($-10\rm~{}km~{}s^{-1}$, $10\rm~{}km~{}s^{-1}$)

  * •

outlier fraction $\gamma$: uniform(0, 0.5)

  * •

outlier model standard deviation $p$: uniform($4\rm~{}km~{}s^{-1}$,
$20\rm~{}km~{}s^{-1}$)

We define the posterior as the Gaussian likelihood multiplied by our set of
priors and look for a $\delta_{v}$ that maximizes this posterior. The error on
the offset is taken as the width of this offset posterior distribution. We
derive the offset value from the resulting posterior, sampled using Markov
Chain Monte Carlo (MCMC) by the Metropolis–Hastings algorithm via the emcee
python package (Foreman-Mackey et al., 2013). Applying this procedure to the 8
stars in the overlapping MMT-Keck datasets observed around the Tri II galactic
center, we find an offset value of $\delta_{v}=-0.11\pm
1.02\rm~{}km~{}s^{-1}$. While this value could be determined using only our
Tri II dataset, the large error means that the offset is not resolved within
$1\rm~{}km~{}s^{-1}$ and the introduction of a large offset error can make it
more difficult to resolve the Tri II velocity dispersion.

To improve the error on our zero-point offset between the instruments, we also
include Keck/DEIMOS (Pace et al., 2020) and MMT/Hectochelle (Spencer et al.,
2018) observations from an additional 288 Ursa Minor objects. The combined Tri
II and Ursa Minor dataset results in an offset correction of
$\delta_{v}=-1.33\pm 0.33\rm km~{}s^{-1}$.

We opt to use the offset that is calculated while including Ursa Minor
observations, as the larger amount of data gives a more precise value for the
offset. Thus, we bring all observations onto a common zero-point by
subtracting a fixed amount of $1.33\pm 0.33\rm~{}km~{}s^{-1}$ from each
velocity measurement obtained with Keck/DEIMOS. We do not propagate the error
on the offset as the total error is dominated by the combination of
instrument-specific systematics and random errors.

###  2.4 Membership

To determine Tri II membership, we sigma clip our dataset at $3.5\sigma$ from
both the galaxy mean line-of-sight velocity and proper motion. This step
removes objects whose observations are differ from the Tri II measurements
$>3.5$ times the root sum of the squared of the galaxy and measurement errors,
$\sigma=\sqrt{\sigma_{\rm Tri~{}II}^{2}+\sigma_{\rm obs}^{2}}$. We use proper
motions from Gaia eDR3 (Gaia Collaboration et al., 2021, Figure 1). All
members are consistent with zero parallax (i.e, $\varpi-3\sigma_{\varpi}<0$).
The Tri II mean velocity is taken as $-381.7\rm~{}km~{}s^{-1}$ (Kirby et al.,
2017) and the galaxy proper motion as $\mu_{\alpha\star}=0.58\pm
0.06\rm~{}mas~{}yr^{-1},\mu_{\delta}=0.11\pm 0.07\rm~{}mas~{}yr^{-1}$ (Pace et
al. in prep ). While the velocity dispersion is unresolved, we use
$\sigma_{\rm Tri~{}II}=4~{}\rm~{}km~{}s^{-1}$ (Figure 2). Our resulting sample
consists of 16 member stars, one of which is the previously confirmed binary
star (which we refer to as Star46 or MIC2016-46 in this work). There is one
star that is observed by MMT only, two stars in the overlap of MMT and Keck,
and the remaining 13 are observed by Keck only.

Figure 1: Proper motion distribution of stars within 10 ′($\sim 4$ half-light
radii) of Tri II center from Gaia eDR3 after a parallax cut (blue). The green
point marks the Tri II proper motion value that is used to determine
membership. The black triangle and star are the two stars with uncertain
membership in Kirby et al. (2017). The red dots are the remaining members that
have Gaia proper motion measurements.

Figure 3 shows the location of our final sample of member stars on a color-
magnitude diagram as compared to all the stars within the Tri II half light
radius after a star-galaxy separation cut observed by the Subaru/Hyper
Suprime-Cam (HSC; Miyazaki et al., 2018) and Pan-STARRS1 (Flewelling et al.,
2020), the isochrone parameters are taken from Carlin et al. (2017). In
Figures 1, 2, and 3, we indicate the locations of the binary star (Star46) and
another member, referred to as Star31 (or MIC2016-31) in this analysis, with
black markers of different shapes. Star31 is specifically marked because of
its uncertainty as a member in Kirby et al. (2017) (also called Star31 in
their work). Its membership is brought into question due to it being (1) far
from the the galactic center, (2) the most metal rich star in the sample (to
the point of driving Tri II’s [Fe/H] dispersion), and (3) the only star whose
velocity is $>1\sigma$ from the galaxy mean velocity. However, the now
available Gaia eDR3 indicate its proper motion is consistent with Tri II,
bolstering its case as a member.

Figure 2: Mean metallicities and velocities of all stars observed in MMT
around the Tri II field (black). The green area represents the The Tri II mean
velocity from Kirby et al. (2017) used in membership cuts. Our final sample of
Tri II members is plotted with two colors/markers to distinguish between Keck
(blue circles) and MMT (red squares) measurements. The binary star, alone, is
plotted with the systemic velocity found in this work rather than the weighted
mean of the velocity observations. Figure 3: Left: Color-magnitude diagram of
Tri II stars within $\sim 4$ half-light radii ($r_{h}=$2.5′) taken by Hyper
Suprime-Cam (HSC) and Panstarrs with a MIST isochrone overlaid. The location
of the Tri II binary star and Star31 is indicated by a star and triangle,
respectively. The isochrone is defined by the following values: $m-M=17.44$,
[Fe/H]=$-2.2$, and Age=13 Gyr. Right: Sky positions of members and stars
observed with MMT. The dotted lines represents the 1 and 2 half-light radii of
Triangulum II. In both plots, the members are plotted with either a red square
or blue circle for MMT and Keck observations respectively.

##  3 Methods

###  3.1 Binary Stars

When a star is a part of a binary system, the orbital motion can give rise to
periodic variability in the line-of-sight velocity. If one star is much
brighter than its companion (e.g., a red giant with a main sequence companion)
then a single-epoch spectrum may not show any evidence of variability.
Conventional methods of combining observations, such as a weighted mean, can
misrepresent the true systemic velocity if the system is sparsely observed
and/or result in inflated error in the combined velocity. Thus, binary stars
can directly inflate the calculated velocity dispersion if not accounted for
properly. Though the true line-of-sight velocity will vary with time, it will
oscillate around the center-of-mass (systemic) velocity.

In terms of true anomaly $\nu$ (angular position in orbital plane from
periastron direction), argument of periapsis $\omega$ (angle in orbital plane
between ascending node and periastron) and systemic velocity $v_{0}$, a member
of a binary stars system has line-of-sight velocity that varies with time
according to…

$V=K(\cos(\omega+\nu)+e\cos\omega)+v_{0}$ (3)

(Murray & Correia, 2010). We can further expand the semi-amplitude $K$ in
terms of the binary parameters period $P$, semi-major axis $a$, inclination
$\sin i$, and eccentricity $e$ as

$K=\frac{2\pi}{P}\frac{a\sin i}{\sqrt{1-e^{2}}}.$ (4)

This is the model that we use to fit to the observed radial velocity curves.
It is worth mentioning that instead of using $\nu$ as a free parameter, we
sample in $\Phi_{0}$, representing the phase, which is related to a constant
called the time of periastron passage, $T$, by a factor of $2\pi/P$. We can
relate $\nu$ and $\Phi_{0}$ using the eccentricity anomaly, $\zeta$.

$\tan\left(\frac{1}{2}\nu\right)=\sqrt{\frac{1+e}{1-e}}\tan\left(\frac{1}{2}\zeta\right)$
(5)

$\frac{2\pi}{P}(t-T)=\frac{2\pi}{P}t+\Phi_{0}=\zeta-e\sin\zeta$ (6)

We furthermore include a jitter parameter that acts as secondary error to
account for velocity variability not due to periodic orbital motion. We note
that Hekker et al. (2008) found a correlation between velocity jitter and
surface gravity for red-giant branch stars, where stars with $\log g\lesssim
1$ can have excess jitter $\sim 1\rm~{}km~{}s^{-1}$. However, the majority of
our members, including Star46, exist on the lower red giant branch where $\log
g\gtrsim 1$. This is commonly the case for stars in ultra-faint dwarfs, so
this is little concern that internal mechanisms will inflate the galaxy
velocity dispersion.

We continue to assume a Gaussian error on velocity measurements, thus allowing
us to use the previously defined Gaussian likelihood function, equation 1. We
generate samples of the posterior on the orbital binary parameters using
rejection sampling. This sampling is done over the course of $10^{5}$
iterations with each iteration starting with $10^{6}$ points before rejection.
After rejection, the surviving samples across all iterations are combined. The
choice to use rejection sampling instead of Monte-Carlo-Markov-Chain (MCMC) is
due to the potential multi-modality posterior. A sparsely observed binary can
have multiple possible orbital period solutions, which have corresponding
total mass and systemic velocity values, represented as peaks in the
posterior. This specific scenario can lead to untouched areas of the posterior
when using MCMC.

The priors are quite uninformative, ensuring that we can fully sample all the
possible binary parameters. Note that there is a pre-defined mass range, such
that the prior in total mass is loguniform from $\log(0.1\rm~{}M_{\odot})$ to
$\log(10\rm~{}M_{\odot})$, which is used to bound the semi-major axis prior.
The priors on all the angles are uniform. We adopt the uniform prior on
$\cos(i)$ corresponding to random orientation of binary orbit and close to
uniform between 0 and 1 beta distribution of eccentricities, $\rm
Beta(1.5,1.5)$. For compactness, we explicitly state the priors on each of the
orbital parameters alongside the resulting Star46 binary parameter posteriors
in Table 1.

Binary Star Orbital Parameters
---
Parameter |  Prior |  Short Per. |  Long Per.
Period $P$ |  linearly decreasing in log($P$) from 1 day to $10^{10}$ days |  $148.1_{-1.4}^{+2.8}$ days |  $296.1_{-3.3}^{+3.8}$ days
Eccentricity $e$ |  Beta(1.5,1.5) |  $0.29_{-0.07}^{+0.10}$ |  $0.50_{-0.04}^{+0.06}$
Systemic velocity $v_{0}$ |  Normal(0,$474\rm~{}km~{}s^{-1}$) |  $-387.6_{-1.2}^{+0.8}$ $\rm~{}km~{}s^{-1}$ |  $-380.0_{-1.7}^{+1.8}$ $\rm~{}km~{}s^{-1}$
Phase $\Phi_{0}$ |  uniform(0,2$\pi$) |  $3.12_{-2.10}^{+2.21}$ |  $3.25_{-2.23}^{+2.04}$
Argument of periapsis $\omega$ |  uniform(0,2$\pi$) |  $4.04_{-0.65}^{+0.35}$ |  $3.43_{-0.24}^{+0.28}$
Jitter |  Normal($0\rm~{}k~{}ms^{-1},2\rm~{}k~{}ms^{-1})$ |  $1.3_{-0.9}^{+1.4}$ $\rm~{}km~{}s^{-1}$ |  $0.9_{-0.6}^{+1.1}$ $\rm~{}km~{}s^{-1}$
Semi-major axis $a$ |  loguniform, bounds determined from mass |  $0.71_{-0.15}^{+0.26}$ AU |  $1.12_{-0.24}^{+0.41}$ AU
Inclination $\sin i$ |  uniform(0,2$\pi$) in $\cos i$ |  $0.31_{-0.09}^{+0.08}$ |  $0.39_{-0.11}^{+0.11}$
Table 1: Priors used in binary modeling and resulting orbital parameters for
the Tri II binary star. The posterior is separated into short and long period
solutions at the 0.6 years (219.2 days) boundary.

We are able to make further assumptions about the system by modifying the
surviving posterior samples. We remove nonphysical solutions corresponding to
close binaries where the pericenter is less than the stellar radius or where
the mass of the companion star is negative. These stellar radii and masses are
calculated using the Isochrones python package (Morton, 2015) to determine
stellar parameters for the binary star of age 13 Gyr, [Fe/H] of -2.2, and
distance modulus of 17.24 using the MESA Isochrones and Stellar Tracks (MIST)
grid (Dotter 2016; Choi et al. 2016). Figure 3 shows the relevant isochrone
fit on a color-magnitude diagram with Tri II members plotted. For the binary
star Star46, we determine a mass of $M_{\rm bin}=0.777\rm~{}M_{\odot}$ and
stellar radius of $r_{\rm bin}=0.027\rm~{}AU$ from an isochrone fit to Tri II
stars. We equivalently derive a mass and radius for the other Tri II members
by finding the point on the isochrone closest to each member star using the
same steps.

Similarly, Raghavan et al. (2010) looked at 259 confirmed solar-type binaries
in the Hipparcos catalog and found the periods fit are distributed according
to a log-normal distribution with $\mu_{\log P}=5.03$ and standard deviation
of $\sigma_{\log P}=2.28$, where the $P$ is in days. This is a more
informative prior on the period than our default period prior. We refer to
this lognormal distribution as the "Raghavan prior" for the remainder of this
paper. We present this distribution with the minor caveat that most of the
detectable member stars of dwarf galaxies are not solar-type (main sequence)
stars, but rather red giant branch stars.

We are able to infer the posterior under this prior by re-sampling from the
current surviving samples with non-uniform weights. These weights are
proportional to the ratio of the Raghavan period distribution to the default
prior. All of these assumptions shift the posterior away from shorter orbital
periods, but as we’ll see in section 4.2, the application of this prior on
Star46 significantly alters the parameter posterior. As such, we present the
results both before and after the application of the Raghavan prior.

###  3.2 Non-Binary Stars

In the case of non-binary stars, we expect line-of-sight velocity observations
to be consistent with a constant velocity model with some scatter in each
measurement proportional to the observational error. Our goal is to determine
a star’s true velocity by combining multiple epochs of radial velocity
measurements into one stellar velocity. A common approach to this is
calculating the weighted mean—i.e., summing the measurements weighted by the
inverse of the squared errors. This approach has the benefits of being
analytic and thus not computationally intensive. However, it has the downside
of underestimating the error when dealing with potentially variable systems.
In this section we offer another method to calculate the true stellar velocity
by finding its posterior distribution.

Once again, the assumption of Gaussian error allows us to use the previously
defined Gaussian likelihood function, equation 1, and a set of priors in true
velocity and velocity error to sample from the true velocity posterior. The
sampling is done using MCMC sampling and is performed with 24 walkers, 1000
steps each, and 100 step burn-in. For a set of line-of-sight velocity
observations $\\{v_{i}\\}$ with errors $\\{\epsilon_{i}\\}$, we define a
uniform prior in true velocity that is non-zero from
$\min(\\{v_{i}\\})-4\max(\\{\epsilon_{i}\\})$ to
$\max(\\{v_{i}\\})+4\max(\\{\epsilon_{i}\\})$. We also use Jeffrey’s prior for
a the standard deviation of a Gaussian distribution ($1/\sigma$ or uniform in
$\log\sigma$) for the true velocity error. The error on the combined velocity
measurement is taken as one standard deviation from the posterior on the mean.
We use the mixture sub package found in scikit-learn (Pedregosa et al., 2011)
to fit a Gaussian mixture model (GMM) of a singular Gaussian to the resulting
posterior samples. Figure 4 shows the radial velocity curve and resulting
posteriors using this method, referred to as the Gaussian fit method, and
binary model method for a star with multiple velocity observations.

Figure 4: Left: RV curve of an example member star with more than one
observation. Right: Corresponding posterior on the mean velocity resulting
from both the Gaussian fit and binary model. The binary model posterior
samples have had both modifications that remove short period solutions to
reduce the posterior scatter.

This Gaussian Fit method has the advantage in that it does not assume
binarity, but does retain increased error with larger variability in the RV
curve, more than the weighted mean. For comparison, we also calculate the
combined velocities under the weighted mean and binary model methods (Figure
5). Applying the binary model to the, assumed non-binary, member stars takes
up significantly more computation time and results in a larger uncertainty in
the combined velocity. Over short time-scales, the observations can be
consistent with either a constant velocity model or the RV curve of a long
period binary.

Figure 5: Radial velocity curves of non-binary member stars with multiple
epochs. The different values are results from different methods of combining
observations. $\bar{v}$: weighted mean; $B$: fitting a Gaussian distribution
to the binary model $v_{0}$ posterior; $\mathcal{G}$: assuming the
observations are taken from a Gaussian distribution centered on a true
systemic velocity. For comparison, the standard deviation of the observations
is also listed ($\sigma$). The stars are identified by their Gaia ID when
applicable. All values listed and on the y-axis are in $\rm km~{}s^{-1}$. The
dates are in the format Year-Month-Day.

###  3.3 Mean Velocity and Velocity Dispersion

The Tri II velocity dispersion can be inferred from the posterior using the
same Gaussian likelihood function as in previous sections (Equation 1). We
define another Gaussian model in terms of the mean velocity $\bar{v}$ and
velocity dispersion $\sigma_{v}$ of the galaxy. The total likelihood is the
product of the likelihood of the model on the stellar velocity of each member.
For stars with multiple velocity measurements, the measured velocities are
combined using one of the methods presented in the previous sections. In the
case of a multi-modal posterior, such as for Star46, we must fit a one-
dimensional GMM to the systemic velocity posterior and the likelihood for that
object becomes…

$\centering
L=\sum_{j}^{n}w_{j}\mathcal{G}(\mu_{j}|\bar{v},\sigma_{v})\@add@centering$ (7)

where $\mu_{j}$ and $w_{j}$ represent the mean and weight of the $j$th
Gaussian in the mixture model with $n$ Gaussians. The error on $\mu_{j}$ is
taken as as the standard deviation of the Gaussian $\sigma_{j}$ and the number
of Gaussians in the GMM is determined by manually inspecting the $v_{0}$
posterior. Figure 6 shows the GMM fits to the Star46 systemic velocity
posterior before and after the application of the Raghavan prior.

Figure 6: The posterior on the systemic velocity of the binary star and the
Gaussian mixture model fit under different conditions. These conditions are a
re-weight of the samples such that the period prior is the Raghavan
distribution and removing non-physical samples.

The velocity dispersion prior is log-uniform in from $\log_{10}(0.05\rm
km~{}s^{-1})$ to $\log_{10}(100\rm km~{}s^{-1})$, which is equivalent to
Jeffrey’s prior. The minimum value corresponds to the case where the galactic
dynamics are fully determined by visible matter (See Appendix A). We have
chosen to find the posterior of $\log_{10}\sigma_{v}$ rather than $\sigma_{v}$
to better sample smaller velocity dispersion values. Once again, we use a
uniform prior from $\min(\\{v_{i}\\})-4\max(\\{\epsilon_{i}\\})$ to
$\max(\\{v_{i}\\})+4\max(\\{\epsilon_{i}\\})$ as our prior on mean velocity.

##  4 Results

###  4.1 Binary Orbital Parameters

Within our data set, we are able to spectroscopically identify one star,
Star46, as a binary. Attempting to fit a constant velocity model to the
observations of this object, we find a reduced chi-squared of $\chi_{\rm
red}=157.2$. Previous analyses by Venn et al. (2017) and Kirby et al. (2017)
also confirmed this star to be part of a binary system. For comparison, the
member with the second highest reduced chi-squared is MIC2016-27 with
$\chi_{\rm red}=4.4$. While this value could suggest possible binarity for
MIC2016-27, its velocity variations are more consistent with a constant
velocity model than Star46.

In this analysis, we use the 4 velocity measurements of this star from MMT
(Table B) and 5 from Keck (Table 2). There are 2 additional velocity
measurements made in Geminin/GRACES data (Venn et al., 2017; Ji et al., 2020)
that we have opted to not use for two reasons. First, the inclusion of this
data would add an additional unknown zero-point correction that is more
difficult to quantify due to less overlap in number of stars. Second, we have
explored including them and found that they do not impact the resulting
orbital parameter posterior.

After an initial pass with our model on Star46, we find that there are two
noticeable peaks in the posterior corresponding to periods of $\sim$ 0.4 and
$\sim$ 0.8 years. We are able to eliminate the sub-harmonics (e.g., 0.2, 0.1
years) from our restrictions on pericenter and Kepler’s third law both
limiting the allowed semi-major axis values. Because the period modes are
distinct, we re-run the sampling with a more restricted period prior bound at
0.3 and 0.95 years to ensure a larger number of surviving posterior samples.
The resulting binary orbital parameters presented in this section are derived
from the posterior after removing samples that correspond to a negative
companion mass, but before the application of the Raghavan prior. Figure 7 is
a corner plot of the posterior of select parameters and shows two distinct
peaks corresponding to a long and short period solution. We separate the
posterior samples at the 0.6 year boundary and present the orbital parameter
values taken from the truncated posteriors in Table 1. Figure 8 shows the RV
curve for this star as well as the orbital solutions that correspond to
surviving samples.

Figure 7: Orbital parameters’ posterior for the binary star derived from our
binary model. There are two distinct modes corresponding to  Figure 8:
Possible fits to the radial velocity curve of the binary star, Star46,
calculated using our binary model. The orange and purple lines correspond to
solutions with orbital periods of approximately 0.4 years and 0.8 years
respectively. The dates are in the format Year-Month-Day.

The semi-major axis posteriors for both solutions are consistent with values
less than $a=2\rm AU$, indicating that this is a close binary. The resulting
overall binary parameter space is that of a common binary star system. The
companion mass posterior is not distinct enough from the prior to be
informative, but all of the samples correspond to stellar masses potentially
on the main sequence.

For comparison, the only other binary star in an ultra-faint dwarf spheroidal,
Her-3 in the Hercules dwarf, has a smaller eccentricity ($e_{\rm Her-3}=0.18$)
and period ($P_{\rm Her-3}=135.28\pm 0.33\rm days$) values (Koch et al.,
2014). Plugging into equation 4 we find the corresponding semi-amplitudes,
$K_{\rm short}=16.24_{-1.17}^{+1.59}~{}\rm km~{}s^{-1}$ and $K_{\rm
long}=18.72_{-1.59}^{+2.67}~{}\rm km~{}s^{-1}$ which are only slightly larger
than to Herc-3’s solution, $K_{\rm Her-3}=14.48\pm 0.82~{}\rm km~{}s^{-1}$.

The truncation in companion mass slightly shifts the posterior towards longer
periods, but no more than $0.2\rm~{}km~{}s^{-1}$, much smaller than the width
of the posterior peaks ($\sim 2\rm~{}km~{}s^{-1}$). Re-weighting the samples
such that the prior is the Raghavan period distribution weights the posterior
in favor of the longer period solutions and the corresponding larger systemic
velocities, removing the multi-modality of the posterior (Figure 6). Even
before applying the Raghavan prior, there is a strong preference towards the
long period solutions as the ratio of the long period posterior to short
period posterior is approximately 5:2. After applying the Raghavan prior, the
posterior becomes almost entirely in favor of the long period solution as the
ratio becomes 80:1, effectively removing the short period mode. As such, we
present the parameters corresponding to the long period mode as the orbital
solution, but we also explore the effect using the Raghavan prior has on the
velocity dispersion.

This binary star is only 1 of the 16 stars in our sample. Moe et al. (2019)
and Mazzola et al. (2020) found that intrinsic close binary fraction is anti-
correlated with metallicity such that Tri II’s metallicity value of
[Fe/H]=-2.1 implies a close binary fraction $f_{\rm bin,close}\approx 0.5$.
This suggests that there are more close binary members, but the current data
is not enough to determine binarity for any other binaries in the galaxy.
Though, with growing observation power, they may one day be observed
sufficiently.

###  4.2 Triangulum II Dynamics

While we have briefly explored applying the binary model to the non-binary
members and applying the Gaussian fit model to Star46, we unsurprisingly find
that the best variation is to treat confirmed the binary star under the binary
model and the remaining members under the Gaussian fit model. This option
assumes variability for only Star46. These additional variations made it more
difficult to resolve the dispersion and instead result in a posterior with a
maximum at the $\sigma_{v}$ prior’s minimum value. Thus, we focus our final
results on the best variation.

The mean velocity posterior is consistent with the previously found mean
velocity of Tri II, $\langle v\rangle=-381.7\pm 1.1\rm~{}km~{}s^{-1}$ (Kirby
et al., 2017) within $2\sigma$, where $\sigma$ is the error on the mean
velocity from the posterior. We present a mean velocity of $\langle
v\rangle=-382.3\pm 0.7\rm~{}km~{}s^{-1}$ from this analysis. However, even in
this best variation, were we unable to completely resolve the velocity
dispersion.

The posterior has a small peak at $\log_{10}\sigma_{v}=0.2$ (which corresponds
to $\sigma_{v}=1.6\rm km~{}s^{-1}$), but a large probability tail that
stretches back to the minimum value. We see that applying the Raghavan prior
only slightly reduces this tail (Figure 9). Figure 10 shows the complete
posterior with the binary under the Raghavan prior. This remaining tail of
probability means we were unable to resolve the velocity dispersion of Tri II,
and we instead present upper limits on $\sigma_{v}$. Before applying the
Raghavan prior, we find a 90% confidence limit of $2.7\rm~{}km~{}s^{-1}$ and
95% confidence limit of $3.5\rm~{}km~{}s^{-1}$. After applying the Raghavan
prior, we find a 90% confidence limit of $2.6\rm~{}km~{}s^{-1}$ and 95%
confidence limit of $3.4\rm~{}km~{}s^{-1}$. Due to the differences between
these values being so small, we again choose to present the values determined
using the more informative period prior.

Figure 9: Velocity dispersion posterior with and without the use of the
Raghavan prior on Star46. The 16th, 50th, and 84th percentiles are marked by
the blue, orange and green lines respectively. Figure 10: Complete posterior
Tri II mean velocity and velocity dispersion when treating the binary star
under the binary model and other non-binary members under the Gaussian fit
with the Raghavan prior.

##  5 Conclusions

We have found orbital parameters for the Tri II binary star member, making it
the second binary star within a ultra-faint dwarf with an orbital solution
after Her-3 in Hercules Dwarf (Koch et al., 2014). In doing so, we have
demonstrated a method of accounting for binary star systems in velocity
dispersion calculations that does not demand the removal of such systems. We
also see that using our binary model when unnecessary, such as with non-binary
stars, not only adds lengthy computation time, but can lead to an
overestimation of velocity error.

The classification of Tri II as either an ultra-faint dwarf or globular
cluster remains an open question. Though the use of this model has offered an
improved analysis of the dwarf galaxy, we were unable to resolve a velocity
dispersion that would confirm the existence of a dark matter halo in Tri II.
However, Kirby et al. (2017) found a metallicity dispersion of $\sigma([\rm
Fe/H])=0.53^{+0.38}_{-0.12}\rm dex$ when including all available metallicity
measurements for the dataset. They make clear that this value relies on the
membership two stars (labelled Star31 and Star46 in this analysis), one of
which was the now confirmed binary star (Star46). The Gaia proper motions of
both stars confirm that they are Tri II members (Figure 1). We briefly tested
performing the same $\sigma([\rm Fe/H])$ calculation while including MMT
metallicities. In doing so, we assumed the existence of another zero-point
offset which was determined using the non-outlier offset model described in
Section 2.3 with the two stars that are in the overlap of the MMT and Keck
catalogues. We found that the inclusion of MMT measurements did not
significantly impact the resulting dispersion where the difference after
inclusion was $\lesssim 0.03\rm~{}dex$ from the previous value. This
metallicity dispersion greatly strengthens the case for Tri II’s
classification as a dwarf galaxy, indicating the system is embedded in a dark
matter halo (Willman & Strader, 2012).

The choice to apply our binary model to only one star means that we are
operating with a hard binary fraction of $f_{\rm bin}\approx 0.06$. A more
informed model would be to assume that there is some non-zero fraction of the
“non-binary” member stars that are actually binaries. Each star would have an
associated binary and non-binary likelihoods that are calculated from
marginalizing over the binary parameter posterior. These likelihoods would
allow for the sampling of binary fraction as a parameter alongside with mean
velocity and velocity dispersion, forming a hierarchical model that builds
from the set of binary posteriors. This likelihood would be functionally
similar to the mixture model used to determine the offset between the
instrumental zero-point velocities. However, this approach introduces
complications as marginalization becomes less straightforward when modifying
the posterior. For instance, removing binary samples based on a parameter,
such as pericenter, makes the posterior no longer normalized to the calculated
binary/non-binary likelihoods. The likelihoods must be re-normalized after
truncation for every additional condition that is imposed. Though it is
outside the scope of this paper, we find the hierarchical model to be an
interesting problem and important step to explored in future works.

## Acknowledgements

This work is supported in part by NSF grants AST-1813881 and AST-1909584.
E.N.K. is supported by NSF grant AST-1847909, and he gratefully acknowledges
support from a Cottrell Scholar award administered by the Research Corporation
for Science Advancement. EO is partially supported by NSF grant AST-1815767.
NC is supported by NSF grant AST-1812461. MM is supported by U.S. National
Science Foundation (NSF) grants AST-1312997, AST-1726457 and AST-1815403. ES
acknowledges funding through VIDI grant "Pushing Galactic Archaeology to its
limits" (with project number VI.Vidi.193.093) which is funded by the Dutch
Research Council (NWO). CB and CMD are supported by NSF grant AST-1909022.

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.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been
made possible through contributions by the Institute for Astronomy, the
University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society
and its participating institutes, the Max Planck Institute for Astronomy,
Heidelberg and the Max Planck Institute for Extraterrestrial Physics,
Garching, The Johns Hopkins University, Durham University, the University of
Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for
Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation Grant No.
AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the
Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

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).

Observations reported here were obtained at the MMT Observatory, a joint
facility of the University of Arizona and the Smithsonian Institution. The
authors wish to recognize and acknowledge the very significant cultural role
and reverence that the summit of Maunakea has always had within the indigenous
Hawaiian community. We are most fortunate to have the opportunity to conduct
observations from this mountain.

## Data Availability

The processed MMT/Hectochelle spectra for Triangulum II targets are publicly
available at the Zenodo database, http://.

## References

  * Badenes et al. (2018) Badenes C., et al., 2018, ApJ, 854, 147
  * Bullock & Boylan-Kolchin (2017) Bullock J. S., Boylan-Kolchin M., 2017, ARA&A, 55, 343
  * Carlin et al. (2017) Carlin J. L., et al., 2017, AJ, 154, 267
  * Choi et al. (2016) Choi J., Dotter A., Conroy C., Cantiello M., Paxton B., Johnson B. D., 2016, ApJ, 823, 102
  * Cooper et al. (2012) Cooper M. C., Newman J. A., Davis M., Finkbeiner D. P., Gerke B. F., 2012, spec2d: DEEP2 DEIMOS Spectral Pipeline (ascl:1203.003)
  * Dotter (2016) Dotter A., 2016, ApJS, 222, 8
  * Errani et al. (2018) Errani R., Peñarrubia J., Walker M. G., 2018, MNRAS, 481, 5073
  * Faber et al. (2003) Faber S. M., et al., 2003, in Iye M., Moorwood A. F. M., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4841, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes. pp 1657–1669, doi:10.1117/12.460346
  * Flewelling et al. (2020) Flewelling H. A., et al., 2020, ApJS, 251, 7
  * Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
  * Gaia Collaboration et al. (2021) Gaia Collaboration et al., 2021, A&A, 649, A1
  * Hekker et al. (2008) Hekker S., Snellen I. A. G., Aerts C., Quirrenbach A., Reffert S., Mitchell D. S., 2008, in Journal of Physics Conference Series. p. 012058 (arXiv:0710.4134), doi:10.1088/1742-6596/118/1/012058
  * Ibata et al. (2011) Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., Dalessandro E., 2011, ApJ, 738, 186
  * Illingworth (1976) Illingworth G., 1976, ApJ, 204, 73
  * Irwin & Lewis (2001) Irwin M., Lewis J., 2001, New Astron. Rev., 45, 105
  * Ji et al. (2020) Ji A. P., Simon J. D., Frebel A., Venn K. A., Hansen T. T., 2020, VizieR Online Data Catalog, p. J/ApJ/870/83
  * Kirby et al. (2015) Kirby E. N., Cohen J. G., Simon J. D., Guhathakurta P., 2015, The Astrophysical Journal, 814, L7
  * Kirby et al. (2017) Kirby E. N., Cohen J. G., Simon J. D., Guhathakurta P., Thygesen A. O., Duggan G. E., 2017, ApJ, 838, 83
  * Koch et al. (2014) Koch A., Hansen T., Feltzing S., Wilkinson M. I., 2014, ApJ, 780, 91
  * Koposov et al. (2011) Koposov S. E., et al., 2011, ApJ, 736, 146
  * Laevens et al. (2015) Laevens B. P. M., et al., 2015, The Astrophysical Journal, 802, L18
  * Li et al. (2019) Li T. S., et al., 2019, Monthly Notices of the Royal Astronomical Society, 490, 3508
  * Longeard et al. (2020) Longeard N., et al., 2020, MNRAS, 491, 356
  * Martin et al. (2016) Martin N. F., et al., 2016, The Astrophysical Journal, 818, 40
  * Martinez et al. (2011) Martinez G. D., Minor Q. E., Bullock J., Kaplinghat M., Simon J. D., Geha M., 2011, ApJ, 738, 55
  * Mateo (1998) Mateo M. L., 1998, ARA&A, 36, 435
  * Mazzola et al. (2020) Mazzola C. N., et al., 2020, MNRAS, 499, 1607
  * McConnachie (2012) McConnachie A. W., 2012, AJ, 144, 4
  * McConnachie & Côté (2010) McConnachie A. W., Côté P., 2010, ApJ, 722, L209
  * McConnachie et al. (2018) McConnachie A. W., et al., 2018, ApJ, 868, 55
  * Minor (2013) Minor Q. E., 2013, ApJ, 779, 116
  * Minor et al. (2010) Minor Q. E., Martinez G., Bullock J., Kaplinghat M., Trainor R., 2010, ApJ, 721, 1142
  * Minor et al. (2019) Minor Q. E., Pace A. B., Marshall J. L., Strigari L. E., 2019, MNRAS, 487, 2961
  * Miyazaki et al. (2018) Miyazaki S., et al., 2018, PASJ, 70, S1
  * Moe et al. (2019) Moe M., Kratter K. M., Badenes C., 2019, ApJ, 875, 61
  * Morton (2015) Morton T. D., 2015, isochrones: Stellar model grid package (ascl:1503.010)
  * Murray & Correia (2010) Murray C. D., Correia A. C. M., 2010, Keplerian Orbits and Dynamics of Exoplanets. pp 15–23
  * Oke et al. (1995) Oke J. B., et al., 1995, PASP, 107, 375
  * Olszewski et al. (1996) Olszewski E. W., Pryor C., Armandroff T. E., 1996, AJ, 111, 750
  * 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, arXiv e-prints, p. arXiv:2105.00064
  * Pedregosa et al. (2011) Pedregosa F., et al., 2011, Journal of Machine Learning Research, 12, 2825
  * Price-Whelan et al. (2017) Price-Whelan A. M., Hogg D. W., Foreman-Mackey D., Rix H.-W., 2017, ApJ, 837, 20
  * Raghavan et al. (2010) Raghavan D., et al., 2010, ApJS, 190, 1
  * Simon (2019) Simon J. D., 2019, ARA&A, 57, 375
  * Sohn et al. (2007) Sohn S. T., et al., 2007, ApJ, 663, 960
  * Spencer et al. (2017) Spencer M. E., Mateo M., Walker M. G., Olszewski E. W., McConnachie A. W., Kirby E. N., Koch A., 2017, AJ, 153, 254
  * 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
  * Starkenburg et al. (2017) Starkenburg E., et al., 2017, MNRAS, 471, 2587
  * Szentgyorgyi et al. (2011) Szentgyorgyi A., et al., 2011, PASP, 123, 1188
  * Tolstoy et al. (2009) Tolstoy E., Hill V., Tosi M., 2009, ARA&A, 47, 371
  * Venn et al. (2017) Venn K. A., Starkenburg E., Malo L., Martin N., Laevens B. P. M., 2017, MNRAS, 466, 3741
  * Walker et al. (2009) Walker M. G., Mateo M., Olszewski E. W., Peñarrubia J., Evans N. W., Gilmore G., 2009, ApJ, 704, 1274
  * Walker et al. (2015) Walker M. G., Olszewski E. W., Mateo M., 2015, MNRAS, 448, 2717
  * Willman & Strader (2012) Willman B., Strader J., 2012, AJ, 144, 76
  * 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

##  Appendix A Lower limit on $\log_{10}\sigma_{v}$ prior

From the virial theorem, we know that the mass contained within the half-light
radius of a dwarf galaxy in virial equilibrium can be described by…

$M(<R_{1/2})=C\frac{R_{1/2}\sigma_{v}^{2}}{G}$ (8)

Where $\sigma_{v}$ is the galaxy velocity dispersion and $G$ is the
gravitational constant. $C$ is a proportionality constant. Assuming that all
of the mass is accounted for by the total stellar mass, the following relation
becomes true.

$M(<R_{1/2})=\frac{\Upsilon L}{2}$ (9)

Where $\Upsilon$ is the stellar mass-to-light ratio and L is the total
luminosity. Combining these, we can solve for $\sigma_{v}$.

$\sigma_{v}=\sqrt{\frac{G}{2C}\frac{\Upsilon L}{R_{1/2}}}$ (10)

It is estimated that Tri II has a total luminosity $L\sim 400\rm~{}L_{\odot}$
and a half light radius $R_{1/2}\sim 34\rm~{}pc$ (Laevens et al., 2015).
Stellar mass to light ratios usually exist in the rage $\Upsilon\sim 0.5$ to
$3.0\rm~{}M_{\odot}~{}L_{\odot}^{-1}$ and the proportionality constant is in
the range $C\sim 2$ to $4$. Taking the lower limit for $\Upsilon$ and the
upper limit for $C$, we find that $\sigma_{v}$ can go as small as
$0.05\rm~{}km~{}s^{-1}$. This value will serves as the lower limit of our
velocity dispersion prior and corresponds to a system where the dynamics are
fully determined by visible matter.

##  Appendix B Data Tables

Gaia ID | KCS2015 | MIC2016 | Member | RA (ICRS) | Dec (ICRS) | HJD (days) | S/N | $v_{\rm los}(\rm km~{}s^{-1})$ | [Fe/H] | $\rm T_{eff}$ | $\log g$
---|---|---|---|---|---|---|---|---|---|---|---
331085117451894528 |  |  | True | 33.4272917 | 36.0950281 | 2457683.72 | 8.67 | -386.9$\pm$0.7 | -3.0$\pm$0.3 | 4760$\pm$226 | 0.7$\pm$0.5
331085117451894528 |  |  | True | 33.4272917 | 36.0950281 | 2457688.77 | 10.68 | -385.5$\pm$0.9 | -3.2$\pm$0.4 | 4684$\pm$255 | 1.4$\pm$0.5
331085117451894528 |  |  | True | 33.4272917 | 36.0950281 | 2457711.69 | 6.07 | -384.4$\pm$1.7 | -3.2$\pm$0.6 | 4772$\pm$453 | 0.9$\pm$0.7
331085117451894528 |  |  | True | 33.4272917 | 36.0950281 | 2457685.74 | 4.74 | -383.4$\pm$1.7 | -3.0$\pm$0.7 | 4783$\pm$495 | 1.7$\pm$0.9
331086526201161088 | 65 | 46 | True | 33.33975 | 36.1659431 | 2457374.8 | 8.84 | -400.4$\pm$0.7 | -2.5$\pm$0.3 | 4823$\pm$159 | 2.0$\pm$0.4
331086526201161088 | 65 | 46 | True | 33.33975 | 36.1659431 | 2457683.72 | 6.78 | -405.9$\pm$0.8 | -2.9$\pm$0.3 | 4535$\pm$215 | 0.9$\pm$0.4
331086526201161088 | 65 | 46 | True | 33.33975 | 36.1659431 | 2457688.77 | 5.08 | -405.1$\pm$1.0 | -2.0$\pm$0.7 | 5416$\pm$588 | 2.0$\pm$0.7
331086526201161088 | 65 | 46 | True | 33.33975 | 36.1659431 | 2457711.69 | 2.88 | -386.4$\pm$2.1 | -0.8$\pm$0.9 | 6811$\pm$855 | 2.2$\pm$1.0
331089446778920576 | 106 | 40 | True | 33.3189583 | 36.17939 | 2457374.8 | 27.68 | -382.8$\pm$0.6 | -3.5$\pm$0.1 | 4367$\pm$87 | 0.9$\pm$0.2
331089446778920576 | 106 | 40 | True | 33.3189583 | 36.17939 | 2457683.72 | 21.62 | -381.8$\pm$0.5 | -3.3$\pm$0.2 | 4507$\pm$95 | 1.0$\pm$0.3
331089446778920576 | 106 | 40 | True | 33.3189583 | 36.17939 | 2457688.77 | 23.86 | -381.9$\pm$0.5 | -3.3$\pm$0.1 | 4505$\pm$82 | 0.8$\pm$0.2
331089446778920576 | 106 | 40 | True | 33.3189583 | 36.17939 | 2457711.69 | 13.11 | -381.9$\pm$0.7 | -3.6$\pm$0.2 | 4253$\pm$135 | 0.7$\pm$0.3
331089446778920576 | 106 | 40 | True | 33.3189583 | 36.17939 | 2457685.74 | 12.85 | -382.2$\pm$0.8 | -3.2$\pm$0.3 | 4597$\pm$195 | 1.1$\pm$0.5

Individual observations of Tri II members taken with MMT Hectochelle. This
table is a subset of the full MMT catalogue.

KCS2015 |  MIC2016 |  RA (ICRS) |  Dec (ICRS) |  HJD (days) |  $v_{\rm los}(\rm km~{}s^{-1})$
---|---|---|---|---|---
N/A |  25 |  33.3214149 |  36.1205826 |  2457284.07 |  $-364.1\pm 5.6$
N/A |  8 |  33.2591667 |  36.2090836 |  2457284.07 |  $-388.4\pm 7.7$
128 |  N/A |  33.3093333 |  36.1641944 |  2457303.10 |  $-386.3\pm 3.2$
- |  - |  - |  - |  2457416.70 |  $-385.4\pm 2.1$
- |  - |  - |  - |  2457416.80 |  $-384.2\pm 2.0$
- |  - |  - |  - |  2457639.10 |  $-383.8\pm 2.1$
N/A |  31 |  33.4694167 |  36.2233611 |  2457370.70 |  $-377.6\pm 1.8$
- |  - |  - |  - |  2457416.70 |  $-381.5\pm 2.6$
- |  - |  - |  - |  2457639.10 |  $-377.0\pm 2.8$
- |  - |  - |  - |  2457284.07 |  $-377.1\pm 3.1$
N/A |  29 |  33.3789583 |  36.1988889 |  2457370.70 |  $-387.5\pm 4.7$
- |  - |  - |  - |  2457284.07 |  $-398.4\pm 7.8$
N/A |  27 |  33.3389583 |  36.1414167 |  2457370.70 |  $-373.5\pm 5.2$
- |  - |  - |  - |  2457416.70 |  $-389.9\pm 8.3$
- |  - |  - |  - |  2457284.07 |  $-402.7\pm 6.6$
N/A |  26 |  33.3534583 |  36.1727222 |  2457370.70 |  $-376.9\pm 11.2$
N/A |  24 |  33.3416667 |  36.1738611 |  2457416.70 |  $-371.8\pm 17.1$
- |  - |  - |  - |  2457284.07 |  $-384.4\pm 4.9$
76 |  23 |  33.3358750 |  36.1629167 |  2457303.10 |  $-391.0\pm 3.0$
- |  - |  - |  - |  2457370.70 |  $-384.6\pm 3.1$
- |  - |  - |  - |  2457639.10 |  $-384.0\pm 3.0$
- |  - |  - |  - |  2457284.07 |  $-389.2\pm 3.6$
N/A |  22 |  33.3028750 |  36.1470556 |  2457370.70 |  $-381.5\pm 3.4$
- |  - |  - |  - |  2457416.70 |  $-381.4\pm 3.3$
- |  - |  - |  - |  2457416.80 |  $-384.9\pm 3.7$
- |  - |  - |  - |  2457569.10 |  $-381.1\pm 23.5$
- |  - |  - |  - |  2457639.10 |  $-376.6\pm 6.6$
- |  - |  - |  - |  2457284.07 |  $-388.3\pm 3.8$
116 |  21 |  33.3165000 |  36.1710556 |  2457303.10 |  $-378.9\pm 3.7$
- |  - |  - |  - |  2457370.70 |  $-383.3\pm 2.3$
- |  - |  - |  - |  2457416.70 |  $-387.0\pm 3.2$
- |  - |  - |  - |  2457416.80 |  $-382.5\pm 3.6$
- |  - |  - |  - |  2457639.10 |  $-380.6\pm 3.1$
- |  - |  - |  - |  2457284.07 |  $-384.1\pm 3.1$
91 |  20 |  33.3305000 |  36.1925833 |  2457303.10 |  $-387.3\pm 3.1$
- |  - |  - |  - |  2457370.70 |  $-379.1\pm 1.9$
- |  - |  - |  - |  2457569.10 |  $-388.2\pm 3.9$
- |  - |  - |  - |  2457639.10 |  $-379.2\pm 2.4$
- |  - |  - |  - |  2457284.07 |  $-380.0\pm 2.9$
N/A |  9 |  33.3639183 |  36.2251930 |  2457416.70 |  $-388.9\pm 7.7$
- |  - |  - |  - |  2457284.07 |  $-406.0\pm 5.1$
65 |  46 |  33.33975 |  36.1659431 |  2457282.50 |  $-373.8\pm 2.4$
- |  - |  - |  - |  2457303.10 |  $-375.8\pm 1.7$
- |  - |  - |  - |  2457416.70 |  $-386.7\pm 1.6$
- |  - |  - |  - |  2457569.10 |  $-376.3\pm 4.0$
- |  - |  - |  - |  2457639.10 |  $-384.6\pm 1.6$
Table 2: Individual observations of Tri II members from Keck DEIMOS used in
this analysis after adding a $\delta_{v}=-1.33\rm~{}km~{}s^{-1}$ zero-point
offset correction. The first epochs for each star are listed with the star’s
ID numberings and sky position while any additional epochs for the same star
are in subsequent rows with "-" in the ID and sky position columns.