# DELVE-ing into the Jet: a thin stellar stream on a retrograde orbit at 30
kpc

P. S. Ferguson George P. and Cynthia Woods Mitchell Institute for Fundamental
Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
Department of Physics and Astronomy, Texas A&M University, College Station, TX
77843, USA  N. Shipp Department of Astronomy and Astrophysics, University of
Chicago, Chicago IL 60637, USA  Kavli Institute for Cosmological Physics,
University of Chicago, Chicago, IL 60637, USA  A. Drlica-Wagner Fermi National
Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA  Kavli Institute
for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
Department of Astronomy and Astrophysics, University of Chicago, Chicago IL
60637, USA  T. S. Li Observatories of the Carnegie Institution for Science,
813 Santa Barbara St., Pasadena, CA 91101, USA  Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 08544, USA  NHFP Einstein Fellow
W. Cerny Kavli Institute for Cosmological Physics, University of Chicago,
Chicago, IL 60637, USA  Department of Astronomy and Astrophysics, University
of Chicago, Chicago IL 60637, USA  K. Tavangar Kavli Institute for
Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
Department of Astronomy and Astrophysics, University of Chicago, Chicago IL
60637, USA  A. B. Pace McWilliams Center for Cosmology, Carnegie Mellon
University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA  J. L. Marshall George
P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,
Texas A&M University, College Station, TX 77843, USA  Department of Physics
and Astronomy, Texas A&M University, College Station, TX 77843, USA  A. H.
Riley George P. and Cynthia Woods Mitchell Institute for Fundamental Physics
and Astronomy, Texas A&M University, College Station, TX 77843, USA
Department of Physics and Astronomy, Texas A&M University, College Station, TX
77843, USA  M. Adamów National Center for Supercomputing Applications,
University of Illinois, 1205 West Clark St., Urbana, IL 61801, USA  Center for
Astrophysical Surveys, National Center for Supercomputing Applications,
Urbana, IL, 61801, USA  J. L. Carlin Rubin Observatory/AURA, 950 North Cherry
Avenue, Tucson, AZ, 85719, USA  Y. Choi Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA  D. Erkal Department of
Physics, University of Surrey, Guildford GU2 7XH, UK  D. J. James ASTRAVEO,
LLC, PO Box 1668, Gloucester, MA 01931  Sergey E. Koposov Institute for
Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill,
Edinburgh EH9 3HJ, UK  Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, UK  Kavli Institute for Cosmology,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK  N. Kuropatkin
Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA
C. E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, NSF’s National
Optical-Infrared Astronomy Research Laboratory,
Casilla 603, La Serena, Chile  S. Mau Department of Physics, Stanford
University, 382 Via Pueblo Mall, Stanford, CA 94305, USA  Kavli Institute for
Particle Astrophysics & Cosmology, P.O. Box 2450, Stanford University,
Stanford, CA 94305, USA  B. Mutlu-Pakdil Kavli Institute for Cosmological
Physics, University of Chicago, Chicago, IL 60637, USA  Department of
Astronomy and Astrophysics, University of Chicago, Chicago IL 60637, USA  K.
A. G. Olsen NSF’s National Optical-Infrared Astronomy Research Laboratory, 950
N. Cherry Ave., Tucson, AZ 85719, USA  J. D. Sakowska Department of Physics,
University of Surrey, Guildford GU2 7XH, UK  G. S. Stringfellow Center for
Astrophysics and Space Astronomy, University of Colorado, 389 UCB, Boulder, CO
80309-0389, USA  B. Yanny Fermi National Accelerator Laboratory, P.O. Box 500,
Batavia, IL 60510, USA Peter Ferguson<EMAIL_ADDRESS>
###### Abstract

We perform a detailed photometric and astrometric analysis of stars in the Jet
stream using data from the first data release of the DECam Local Volume
Exploration Survey (DELVE) DR1 and Gaia EDR3. We discover that the stream
extends over ${\sim}\,29{}^{\circ}$ on the sky (increasing the known length by
$18{}^{\circ}$), which is comparable to the kinematically cold Phoenix, ATLAS,
and GD-1 streams. Using blue horizontal branch stars, we resolve a distance
gradient along the Jet stream of 0.2 kpc/deg, with distances ranging from
$D_{\odot}{\sim}\,27-34$ kpc. We use natural splines to simultaneously fit the
stream track, width, and intensity to quantitatively characterize density
variations in the Jet stream, including a large gap, and identify substructure
off the main track of the stream. Furthermore, we report the first measurement
of the proper motion of the Jet stream and find that it is well-aligned with
the stream track suggesting the stream has likely not been significantly
perturbed perpendicular to the line of sight. Finally, we fit the stream with
a dynamical model and find that the stream is on a retrograde orbit, and is
well fit by a gravitational potential including the Milky Way and Large
Magellanic Cloud. These results indicate the Jet stream is an excellent
candidate for future studies with deeper photometry, astrometry, and
spectroscopy to study the potential of the Milky Way and probe perturbations
from baryonic and dark matter substructure.

††software: astropy (Astropy Collaboration et al., 2013, 2018), emcee
(Foreman-Mackey et al., 2013), fitsio,555https://github.com/esheldon/fitsio
HEALPix (Górski et al., 2005),666http://healpix.sourceforge.net
healpy,777https://github.com/healpy/healpy Matplotlib (Hunter, 2007), numpy
(Van Der Walt et al., 2011), pandas (pandas development team, 2020; Wes
McKinney, 2010), scipy (Jones et al., 2001), STAN (Carpenter et al., 2017)
ugali (Bechtol et al., 2015).888https://github.com/DarkEnergySurvey/ugali

FERMILAB-PUB-21-205-AE-LDRD

##  1 Introduction

Stellar streams form through the tidal disruption of dwarf galaxies and
globular clusters as they accrete onto a larger host galaxy (e.g., Newberg &
Carlin, 2016). The formation of stellar streams is an expected feature of
hierarchical models of galaxy formation where large galaxies grow through
mergers of smaller systems (Lynden-Bell & Lynden-Bell, 1995; Johnston et al.,
2001). Due to their formation mechanism, transient nature, and dynamical
fragility, stellar streams provide a direct and powerful probe of the
gravitational field in galactic halos at both large and small scales (e.g.,
Johnston et al., 1999, 2002; Ibata et al., 2002). Within our Milky Way in
particular, stellar streams have been proposed as sensitive probes of the
large- and small-scale distributions of baryonic and dark matter within the
Galactic halo (Johnston et al., 2001; Carlberg, 2013; Erkal et al., 2016;
Bonaca & Hogg, 2018; Banik et al., 2019).

Milky Way stellar streams form when stars are unbound from the progenitor at
the Lagrange points between the progenitor stellar system and the Milky Way.
Stars that are unbound from the inner Lagrange point have lower energy and
thus shorter orbital periods than the progenitor whereas those at the outer
Lagrange point have higher energy and shorter orbital periods. Thus, as the
progenitor is disrupted, leading and trailing streams of stars will form
roughly tracing the orbit of the progenitor within the Milky Way potential
(Sanders & Binney, 2013). The width of a stellar stream is proportional to the
velocity dispersion of its progenitor (Johnston et al., 2001; Erkal et al.,
2019), implying that stellar streams formed through the disruption of globular
clusters are narrow (${\sim}\,100\mathrm{\,pc}$) and dynamically cold, while
streams originating from dwarf galaxies are broader ($>500\mathrm{\,pc}$) and
dynamically hot. The population of cold stellar streams with small internal
velocity dispersions provides a sensitive probe of the gravitational field far
from the Milky Way disk.

In a smooth gravitational potential, stellar streams form as coherent
structures spanning tens of degrees on the sky (e.g., Newberg & Carlin, 2016).
Long stellar streams can be used to trace the local gravitational field over
tens of kpc (Bovy, 2014). In conjunction with orbit modeling and simulations,
streams can constrain the total mass enclosed inside their orbits (e.g.,
Gibbons et al., 2014; Bowden et al., 2015; Bovy et al., 2016; Bonaca & Hogg,
2018; Malhan & Ibata, 2019), and the shapes and radial profiles of the
gravitational field (e.g., Law & Majewski, 2010; Koposov et al., 2010; Bowden
et al., 2015; Malhan & Ibata, 2019). Bonaca & Hogg (2018) find that a dozen
cold stellar streams with full 6D kinematic measurements should contain enough
information to constrain the mass and shape of a simple Milky Way potential
with $\sim 1\%$ precision. Additionally, perturbations from large structures
can induce a misalignment between the orbit and track of a stream, which can
be used to constrain the mass of the perturbing object (Erkal et al., 2018;
Shipp et al., 2019). For example, Erkal et al. (2019) and Vasiliev et al.
(2021) used the Orphan and Sagittarius streams, respectively, to
simultaneously measure the mass of the Milky Way and Large Magellanic Cloud
(LMC).

Stellar streams can also probe the clustering and distribution of dark matter
at small scales. The dark energy plus cold dark matter ($\rm\Lambda CDM$)
model predicts that dark matter should clump into gravitationally bound halos
on scales that are much smaller than the smallest galaxies (Green et al.,
2004; Diemand et al., 2005; Wang et al., 2020). Dark matter subhalos that pass
close to stellar streams may gravitationally perturb the stream by altering
the stream track and inducing small-scale density fluctuations. Discrete gaps
in stellar streams, such as those found in the Pal 5 and GD-1 streams
discovered from data collected by the Sloan Digital Sky Survey (Odenkirchen et
al., 2001; Grillmair & Dionatos, 2006), can probe the population of compact
subhalos with $10^{6}\mathrm{\,M_{\odot}}<M<10^{8}\mathrm{\,M_{\odot}}$ that
contain no luminous matter (e.g., Erkal et al., 2016; Bonaca et al., 2019).
Additionally, the power spectrum of density fluctuations along a cold stream
can place limits on the number of dark subhalos (e.g., Banik et al., 2019) and
the mass of warm dark matter candidates (e.g., sterile neutrinos; Dodelson &
Widrow, 1994; Shi & Fuller, 1999). However, baryonic structures such as giant
molecular clouds (Amorisco et al., 2016) or Milky Way substructure such as the
disk and bar (Erkal et al., 2017; Pearson et al., 2017; Banik et al., 2019)
can induce perturbations that mimic the observational signature of dark matter
subhalos. It is thus crucial to characterize cold streams at large
Galactocentric radii where they are less likely to be affected by baryonic
structures (e.g., Li et al., 2020).

Despite the importance of Milky Way stellar streams as probes of galaxy
formation in a cosmological context, they remain difficult to detect due to
their low surface brightness (fainter than $28.5\;\mathrm{mag/arcsec}^{2}$)
and large spatial extent across the sky ($\gtrsim 10{}^{\circ}$). The phase
space signature of streams at large Galactocentric distances is often
difficult to detect from space-based observatories (e.g., Gaia). Stars in
these streams are either too faint to have well-measured proper motions or
their proper motions overlap with the locus of faint foreground stars at small
distances (Ibata et al., 2020). Distant streams have only recently been
detected thanks to deep, wide-area imaging by ground-based digital sky surveys
(e.g. SDSS, Pan-STARRS1, and DES, Belokurov et al., 2006; Bernard et al.,
2016; Shipp et al., 2018).

The Jet stream is one such dynamically cold stellar stream that was discovered
by Jethwa et al. (2018) (hereafter referred to as J18) in the Search for the
Leading Arm of Magellanic Satellites (SLAMS) survey. This stream was found to
have a width of 0.18∘ and a length of 11∘ (truncated on one end by the survey
footprint). They found the stellar population of Jet to be well-described by
an old (12.1 Gyr), metal-poor ($\text{[Fe/H]}=$$-1.57$) isochrone. Fits to the
main sequence turn-off (MSTO) and the distribution of blue horizontal branch
(BHB) stars in the central portion of the stream place its heliocentric
distance at $\sim$29 kpc. At this distance the physical width of the stream
corresponds to $\sim$ 90 pc, placing the stream firmly in the dynamically cold
category. This narrow width also suggests the progenitor of Jet was likely a
globular cluster, although no progenitor was found by J18.

We further investigate the Jet stream using data from the DECam Local Volume
Exploration Survey (DELVE) Data Release 1 (DR1) (Drlica-Wagner et al., 2021).
This catalog covers over $\sim$ 4,000 $\text{deg}^{2}$ in four photometric
bands ($g,r,i,z$) and over $\sim$ 5,000 $\text{deg}^{2}$ in each band
independently. The sensitivity of DELVE has been demonstrated by the discovery
of a Milky Way satellite galaxy candidate with $M_{V}=-5.5$ at a distance of
${\sim}\,116\mathrm{\,kpc}$ (Centaurus I; Mau et al., 2020) and two faint star
cluster candidates (DELVE 1, DELVE 2; Mau et al., 2020; Cerny et al., 2021).
DELVE DR1 contiguously and homogeneously covers a large region including and
extending the SLAMS survey footprint. Thus, the DELVE data are ideal to
further characterize the Jet stream.

To dynamically model the stream and extract local properties of the
gravitational field, additional phase space information is needed. With full
3D kinematic information, the Jet stream can become an even better tool for
measuring the properties of the Milky Way, thereby allowing us to probe its
interaction history. A combination of proper motion and radial velocity
measurements are required to obtain the full 6D phase space information of the
Jet stream. The high-precision astrometric survey Gaia has revolutionized this
field and allowed for measurements of the proper motion of faint stream stars
for the first time. The early third data release from Gaia (Gaia EDR3; Gaia
Collaboration et al., 2016, 2020) provides proper motion measurements for more
than 1.4 billion stars down to a magnitude of $G\sim 21$. Gaia has previously
been used to characterize the proper motions of many stellar streams (e.g.
Shipp et al., 2019; Price-Whelan & Bonaca, 2018; Koposov et al., 2019) and
discover tens of candidate stellar streams (e.g. Malhan & Ibata, 2018; Malhan
et al., 2018; Ibata et al., 2020). In this paper we use astrometric
measurements from Gaia and photometry from DELVE to measure the proper motion
of the Jet stream for the first time and quantitatively characterize its
shape. These measurements, along with future spectroscopic observations, will
allow for a full characterization and dynamical modeling of this stream.

This paper is organized as follows. In Section 2 we briefly present the DELVE
DR1 dataset. We then describe our analysis of the Jet stream in Section 3,
initially using the DELVE DR1 dataset to characterize the stream track over an
extended region of the sky. Next, we measure a distance gradient along the
stream using blue horizontal branch stars, and use this distance gradient to
optimize our matched-filter. Then, we model the observations to quantitatively
characterize the structure of the stream. To further characterize the stream,
we use the DELVE DR1 photometry along with proper motion measurements from
Gaia EDR3 to measure the proper motion of the Jet stream for the first time.
In Section 4 we fit the stream with a dynamical model to determine the best-
fit orbital parameters and to determine whether the stream is likely to have
been significantly perturbed by large substructure such as the Milky Way bar
or presence of the LMC. We discuss our results in Section 5 and conclude in
Section 6.

##  2 DELVE DR1 Data

DELVE seeks to provide contiguous and homogeneous coverage of the high-
Galactic-latitude ($|b|>10{}^{\circ}$) southern sky ($\delta<0{}^{\circ}$) in
the $g,r,i,z$ bands (Drlica-Wagner et al., 2021). This is done by assembling
all existing archival DECam data and specifically observing regions of the sky
that have not been previously observed by other community programs. These data
are consistently processed with the same data management pipeline to create a
uniform dataset (Morganson et al., 2018). The DELVE DR1 footprint consists of
the region bounded by $\delta<0{}^{\circ}$ and $b>10{}^{\circ}$ with an
additional extension to $b=0{}^{\circ}$ in the region of
$120{}^{\circ}<\alpha<140{}^{\circ}$ to search for extensions of the Jet
stream. This footprint is shown in Figure 1 as a light blue shading.
Additionally, we searched the Jet Bridge region below the Galactic plane
(light orange shading). However, no evidence of a continuation of the Jet
stream was found in this region.

Figure 1: The DELVE DR1 region where our search was performed is shown in
light blue. The additional region (the Jet Bridge) that was searched is shown
as a light orange patch. The solid black line indicates the plane of the Milky
Way ($b=0{}^{\circ}$) and the two dashed lines indicate $b=\pm 10{}^{\circ}$.

The DELVE DR1 dataset consists of $\sim$30,000 DECam exposures, with exposure
times between $30\mathrm{\,s}<{\rm t_{exp}}<350\mathrm{\,s}$. Additionally,
the following quality cuts were applied to individual exposures: a minimum cut
on the effective exposure time scale factor $\mathrm{t_{eff}}>0.3$ (Neilsen et
al., 2015) and a good astrometric solution relative to Gaia DR2 (For each
exposure $>250$ astrometric matches, $\chi^{2}_{\rm astrom}<500$, where a
match has $\Delta(\alpha)<150\mathrm{\,mas}$, and
$\Delta(\delta)<150\mathrm{\,mas}$). All exposures were processed with the DES
Data Management (DESDM) pipeline (Morganson et al., 2018), enabling sub-
percent-level photometric accuracy by calibrating based on seasonally averaged
bias and flat images and performing full-exposure sky background subtraction
(Bernstein et al., 2018). Automatic source detection and photometric
measurement is performed on each exposure using SExtractor and PSFex (Bertin &
Arnouts, 1996; Bertin, 2011). Astrometric calibration was performed against
Gaia DR2 using SCAMP (Bertin, 2006). Photometric zeropoints for each CCD were
derived by performing a $1\mathrm{\,arcsec}$ match between the DELVE
SExtractor catalogs and the ATLAS Refcat2 catalog (Tonry et al., 2018), and
using transformation equations derived by comparing stars in ATLAS Refcat2 to
calibrated stars from DES DR1 to convert the ATLAS Refcat2 measurements into
the DECam _griz_ -bandpass (Drlica-Wagner et al., 2021). The zeropoints
derived from this processing were found to agree with the DES DR1 zeropoints
with a scatter of $\lesssim 0.01\mathrm{\,mag}$. Dust extinction corrections
were applied using extinction maps from Schlegel et al. (1998) assuming
$R_{V}=3.1$ and a set of $R_{\lambda}$ coefficients derived by DES (DES
Collaboration et al., 2018) including a normalization adjustment from Schlafly
& Finkbeiner (2011). Hereafter, all quoted magnitudes have been corrected for
interstellar extinction. For more details on the catalog creation and
validation see Drlica-Wagner et al. (2021).

A high-quality stellar sample is selected based on the SExtractor quantity
SPREAD_MODEL (Desai et al., 2012) measured in the DELVE _g_ -band.
Specifically, we select objects with $|\texttt{{SPREAD\\_MODEL\\_G}}|<0.003$.
The performance of this classifier was evaluated by matching sources in the
DELVE DR1 catalog with the W04 HSC-SSP PDR2 catalog (Aihara et al., 2019). For
our analysis we choose a limiting magnitude of $g=23.1$ mag where the stellar
completeness drops to $\sim 60\%$ and contamination rapidly rises to $\sim
40\%$ as estimated from the HSC catalog. For more information on morphological
classification in this catalog see Drlica-Wagner et al. (2021). bright-end
limit of 16th magnitude in the $g$-band is chosen to avoid saturation effects
from bright stars. Additionally, since we are primarily interested in Main
Sequence (MS) and Red Giant Branch (RGB) stars associated with old, metal-poor
populations, we restrict the color range of our dataset to be
$0.0<(g-r)_{0}<1.0$. Only objects passing the above morphological, magnitude,
and color cuts are used in the following matched-filter analysis.

To account for missing survey coverage over our footprint, we quantify the sky
area covered by DELVE DR1 in the form of HEALPix maps. These maps account for
missing survey coverage, gaps associated with saturated stars and other
instrumental signatures. They are created using the healsparse
111https://github.com/LSSTDESC/healsparse tool developed for the Legacy Survey
of Space and Time (LSST) at the Vera C. Rubin Observatory, and its DECam
implementation decasu 222https://github.com/erykoff/decasu to pixelize the
geometry of each DECam CCD exposure. The coverage is calculated at a high
resolution (nside =16384; ${\sim}\,0.01\mathrm{\,arcmin}^{2}$) and degraded to
give a fraction of the lower resolution pixel area that is covered by the
survey.

###  2.1 Gaia cross-match with DELVE DR1

To enable a characterization of the proper motion of the Jet stream, we use
the Gaia EDR3 dataset (Gaia Collaboration et al., 2020). We begin by
performing an angular cross match between the Gaia EDR3 dataset and DELVE DR1
with a matching radius of 0$\farcs$5\. This results in a catalog containing
${\sim}\,143$ million sources. Subsequently, a number of quality cuts are
applied. Nearby sources are removed by applying a parallax cut similar to Pace
& Li (2019) of $\varpi-3\sigma_{\varpi}<0.05$. To remove sources with bad
astrometric solutions we place a cut on the renormalized unit weight error
(ruwe) of $\texttt{ruwe}<1.4$. Then, a cut on BP and RP excess is applied
following equation 6 of Riello et al. (2020)
($|C^{\star}|<3\sigma_{C^{\star}})$. Additionally, only sources with
$\texttt{astrom\\_chi2\\_al}<2$ are kept to avoid sources with bad astrometric
fits in Gaia. We check that no known Active Galactic Nuclei (AGN) are in our
sample by removing all sources that appear in the Gaia table
gaiaedr3.agn_cross_id. Finally, we remove faint sources with $G>20$ mag to
avoid contamination from stars with low signal-to-noise proper motion
measurements. The resulting catalog is used for the analyses described in
Sections 3.2 and 3.5.

##  3 Methods and Analysis

In this section we describe our procedure to fit the track, distance gradient,
proper motion, and morphology of the Jet stream. We begin by performing an
initial matched-filter selection for the Jet stream assuming the best-fit
isochrone parameters and distance modulus from J18 (Section 3.1). This allows
us to determine an initial estimate for the Jet stream track. We then select
candidate BHB stars that lie along the track and are clustered in proper
motion space to determine a distance gradient as a function of angular
distance along the stream (Section 3.2). Then we create a new optimized
matched-filter map of the Jet stream using the distance gradient of the
candidate BHB stars, and refitting an isochrone to a Hess difference diagram
(Section 3.3). This map is fit with a spline-based generative model to
quantitatively characterize the track, intensity and width of the stream as a
function of angular distance along the stream (Section 3.4). Finally, we
select RGB and BHB stars consistent with being members of the Jet stream and
fit a two component Gaussian mixture model to the selected stars determining
the proper motion for the Jet stream including a linear gradient term (Section
3.5).

Figure 2: A stellar density map from the DELVE photometry showing the Jet
stream at a distance modulus of $m-M=17.28$
($\text{D}_{\odot}=28.6\mathrm{\,kpc}$). The orange points show the extent of
the stream identified by Jethwa et al. (2018), and the blue points denote the
new extent of the stream as detected in MSTO and BHB stars as determined in
this study. The solid black line in the bottom right indicates the plane of
the Milky Way ($b=0{}^{\circ}$), and the dashed line shows $b=10{}^{\circ}$.

###  3.1 Initial matched-filter Search

To investigate the Jet stream in DELVE DR1, we began by applying a matched-
filter algorithm in color–magnitude space similar to Shipp et al. (2018,
2020). The matched-filter is derived from a Dotter et al. (2008) synthetic
isochrone as implemented in ugali (Drlica-Wagner et al.,
2020).333https://github.com/DarkEnergySurvey/ugali Candidate MS stars are
selected within a range of colors around the isochrone (Equation 4 in Shipp et
al. 2018) taking into account photometric uncertainty. To select stars
consistent with the Jet stream, we create a matched-filter based on the best-
fit parameters (including distance modulus) taken from J18: an age of 12.1
Gyr, a metallicity of $\text{[Fe/H]}=$$-1.57$, and a distance modulus of
$m-M=17.28\mathrm{\,mag}$.

Our selection is conducted using the DELVE DR1 catalog described in Section 2.
Stars are selected using the matched-filter and then objects are pixelized
into HEALPix pixels with nside = 512 (pixel area of ${\sim}\,0.01\deg^{2}$).
The pixelized filtered map is corrected by the geometric survey coverage
fraction for each pixel to account for survey incompleteness. This is done by
dividing the number of counts in each pixel by the observed fraction of that
pixel (Shipp et al., 2018). The coverage maps are created using the method
described in Section 4.4 of DES Collaboration et al. (2021), and pixels with a
coverage fraction less than 0.5 are removed from the analysis.

Figure 2 shows the results of this matched-filter selection. The Jet stream
can be clearly seen to extend beyond the initial discovery bounds (marked by
orange circles) in both directions. At high declination the stream becomes
fainter and more diffuse and appears to fan out, and at lower declination an
additional prominent component can be seen with obvious density variations.

Figure 3: _Top:_ Jet stream density after applying the same matched-filter as
Figure 2. Additionally, deviations from great circle path
$\phi_{2}=0{}^{\circ}$ are clearly seen. The orange points show the extent of
the stream identified by Jethwa et al. (2018), and the blue points denote the
new extent of the stream as detected in MSTO and BHB stars. For the range
$\phi_{1}<-12.7{}^{\circ}$ or $>10{}^{\circ}$ the stream is only detected
using BHB stars. _Middle:_ SFD dust map of the same region. _Bottom:_ Survey
coverage map where the color shows the fraction of each pixel that is observed
in the DELVE survey. White corresponds to observed fractions lower that 0.5
which are excluded from our analysis, and the greyscale ranges from light to
dark with observed fractions of 0.7 to 1. The red dashed lines show the track
of the Jet stream.

In absence of an obvious stream progenitor, we choose the stream-centered
coordinate frame to be the same as J18 defined by pole
$(\alpha_{pole},\delta_{pole})=64.983{}^{\circ},34.747{}^{\circ}$ and a
$\phi_{1}$ center of $\phi_{1}=63{}^{\circ}$
($\phi_{1},\phi_{2}=0{}^{\circ},0{}^{\circ}$ at
$\alpha,\delta=138.62{}^{\circ},22.10{}^{\circ}$). We define the rotation
matrix to convert $\alpha,\delta$ to $\phi_{1},\phi_{2}$ to be:

$\boldsymbol{R}=\begin{bmatrix}-0.69798645&0.61127501&-0.37303856\\\
-0.62615889&-0.26819784&0.73211677\\\
0.34747655&0.74458900&0.56995374\end{bmatrix}.$ (1)

The top panel of Figure 3 shows the transformed matched-filter stellar density
map. This map has been smoothed by a Gaussian kernel with a size of
$\sigma=0.06{}^{\circ}$, and each column has been normalized to have the same
median to correct for variable background stellar density along the field. The
track of the stream clearly deviates from the great circle path defined by
$\phi_{2}=0{}^{\circ}$. We fit a fourth-order polynomial to the peak intensity
of the stream at each $\phi_{1}$ for the range
$-14{}^{\circ}<\phi_{1}<14{}^{\circ}$ giving the following relation for
$\phi_{2}$ as a function of $\phi_{1}$:

$\displaystyle\phi_{2}(\phi_{1})=$ $\displaystyle
0.07247+0.01475\times(\phi_{1})-0.00138\times(\phi_{1})^{2}$ (2)
$\displaystyle+0.00006\times(\phi_{1})^{3}-0.00002\times(\phi_{1})^{4}.$

The middle panel of Figure 3 shows the Schlegel et al. (1998, SFD) dust map in
the transformed frame of Jet, and the bottom panel shows the survey coverage
map in this region. These two maps demonstrate that the detection of the Jet
stream does not correlate with any linear extinction features or variations in
survey coverage.

###  3.2 Distance Gradient

Using the DELVE DR1 catalog cross-matched with Gaia EDR3 (Section 2.1), we
identify candidate BHB stars and use them to measure a distance gradient along
the Jet stream. BHB stars are useful for determining a distance gradient
because of the tight color-luminosity relation that allows for distance
estimates with ${\sim}\,10\%$ uncertainty to an individual BHB star (Deason et
al., 2011). To determine a distance gradient we use a similar method to Li et
al. (2020) who use BHB stars to measure the distance gradient of the ATLAS-
Aliqa Uma stream. For the Jet stream, the gradient derived from BHB stars can
be used to refine the matched-filter selection from Section 3.1, make a
reflex-corrected proper motion measurement in Section 3.5, and improve
dynamical modeling of the stream (Section 4). Hereafter, all proper motions
($\mu_{\phi_{1}}^{\star},\mu_{\phi_{2}}$) are assumed to be reflex corrected
unless explicitly stated otherwise. We select probable BHB stars along the
track of the stream using a few criteria. Initially we select all sources with
$-12{}^{\circ}<\phi_{1}<10{}^{\circ}$ and separation from the stream track
$\Delta\phi_{2}<0.5{}^{\circ}$ ( Equation 2). Then, a color cut is applied to
select blue stars keeping only sources with $(g-r)_{0}$ between $-$0.3 and 0.0
mag. We then cut all sources with $g$-band magnitudes less than 17.0 mag or
greater than 18.5 mag to reduce contamination from the Milky Way foreground.
For each candidate we derive an estimate of its distance modulus, $m-M$, by
assuming it is a BHB star and using the relation for $M_{g}$ vs. $(g-r)_{0}$
from Belokurov & Koposov (2016).

To further remove contaminant stars from the Jet stream BHB stellar sample, we
use the Gaia EDR3 proper motions of candidate BHB stars along the Jet stream
track. We use the distance estimated for each BHB star to correct their proper
motions for the solar reflex motion assuming a relative velocity of the Sun to
the Galactic standard of rest to be
$(U_{\odot},V_{\odot},W_{\odot})=(11.1,240.0,7.3)\text{ km}\text{/s}$ (Bovy et
al., 2012). Figure 4 (left panel) shows the resulting measured proper motion
of our BHB candidate sample. The proper motion signal of the Jet stream is
seen at $(\mu_{\phi_{1}}^{\star},\mu_{\phi_{2}}){\sim}\,(-1,0)$ mas/yr, where
we define $\mu^{\star}_{\phi_{1}}=\mu_{\phi_{1}}\cos(\phi_{2})$. Additionally,
contributions due to the Milky Way foreground and stellar halo are seen at
$(\mu_{\phi_{1}}^{\star},\mu_{\phi_{2}}){\sim}\,(0.5,2)$ and $(0.5,0)$ mas/yr
respectively The BHB candidates within the green box are selected as likely
members of the Jet stream to be used to estimate the distance gradient. Figure
4 (right panels) shows on-sky positions and distances of the likely member
candidate BHB stars.

Using these derived distances and assuming an uncertainty of 0.1 mag on the
distance modulus for each BHB star (Deason et al., 2011), we fit for the
distance modulus as a function of position along the stream using a simple
linear fit. We find the following relation for the distance modulus as a
function of $\phi_{1}$ along the Jet stream,

$(m-M)=17.45-0.014\times(\phi_{1}).$ (3)

The heliocentric distance of the Jet stream is found to vary from
$26\mathrm{\,kpc}$ to $34.5\mathrm{\,kpc}$ over its observed length with a
gradient of $-0.2$ kpc/deg.

Figure 4:  _Left:_ proper motions of candidate BHB stars along the Jet stream.
The green box shows our proper motion selection for likely members.
Contributions from the Milky Way foreground and stellar halo are seen at
$(\mu_{\phi_{1}}^{\star},\mu_{\phi_{2}})\sim(0.5,2)$ and $(0.5,0)$ mas/yr
respectively _Right:_ on-sky distribution (top) and distances modulus (bottom)
of candidate BHB stars that are likely associated with the Jet stream. The
matched-filter stream track (Equation 2 is shown as green dashed line on the
top plot, and the linear fit on distance modulus, $(m-M)$, of the candidate
BHB stars (Equation 3) is shown as a green dashed line on the bottom plot.

###  3.3 Creation of optimized matched-filter map

We next use our measured distance gradient to find a best-fit isochrone, and
create an optimized matched-filter map to study the morphology of the stream.
We begin by creating a Hess difference diagram shown in Figure 5. This is done
by selecting stars along the stream track with
$-11{}^{\circ}<\phi_{1}<-8{}^{\circ}$ as well as stars with
$-5{}^{\circ}<\phi_{1}<6{}^{\circ}$, excluding the area around the observed
under-density in the stream at $\phi_{1}\sim-6.5{}^{\circ}$ to increase
signal-to-noise. In $\phi_{2}$ we select stars within 2 times the observed
width of the stream ($w$) as a function of $\phi_{1}$ (the width is derived in
Section 3.4). At $\phi_{1}=0$ we find a width of $w=0.16{}^{\circ}$ and this
value varies from $w=0.13{}^{\circ}$ at $\phi_{1}=-11{}^{\circ}$ to
$w=0.19{}^{\circ}$ at $\phi_{1}=6{}^{\circ}$. Additionally, a background
region is selected to be along the same $\phi_{1}$ range but above and below
the stream, with $1{}^{\circ}<|\phi_{2}|<2{}^{\circ}$. For the stars in each
of these regions we compute the absolute $g$-band magnitude ($M_{g}$) assuming
a distance modulus derived from the observed BHB stars’ gradient (Equation 3).
Then we select only stars with $M_{g}<5.47$; this corresponds to the faint
limit of our catalog ($g_{0}=23.1$) at $\phi_{1}=-13{}^{\circ}$, the most
distant portion of the detected stream. This absolute magnitude selection
ensures that the observed density variations in the matched-filter map are not
affected by the survey completeness. We then create binned color–magnitude
diagrams (CMDs) for the on-stream and background selections and subtract the
background from the on-stream region correcting for relative areas. The result
of this process is shown in Figure 5.

Next we fit an isochrone to this Hess difference diagram using a similar
methodology to Shipp et al. (2018). Briefly, we model the observed binned CMD
of the stream region as a linear combination of the background region and a
stellar population following a Dotter et al. (2008) isochrone. Then we compute
the likelihood that the binned CMD is a Poisson sample of the model. This
likelihood is then sampled using emcee (Foreman-Mackey et al., 2013). We find
a best-fit isochrone consistent with the J18 result and distance modulus in
strong agreement with the distance derived from candidate BHB stars
($(m-M)_{MS}-(m-M)_{BHB}=0.01^{+0.05}_{-0.05}$).

Since we find consistent results, we use the J18 isochrone to create an
optimized matched-filter that is used in Section 3.4 to quantitatively
characterize distance variation. This map covers the region defined by
$-16{}^{\circ}\leq\phi_{1}\leq 17{}^{\circ}$, and
$-3{}^{\circ}\leq\phi_{1}\leq 1{}^{\circ}$. We choose a pixel size of 0.2 deg
in $\phi_{1}$ and 0.05 deg in $\phi_{2}$. The distance modulus of the matched-
filter follows Equation 3. The result of this more optimal matched-filter is
shown in the top panel of Figure 6. We note that while the image in Figure 6
has been smoothed with a 0.08 deg Gaussian kernel, we do not apply any
smoothing when fitting our model to the stream data.

Figure 5: Hess difference diagram created by subtracting a background region
from the on-stream region. The main sequence of the Jet stream is clearly
seen. The Dotter et al. (2008) isochrone we use for our analysis with a
metallicity of [Fe/H]=$-1.57$ and age of $\tau=12.1\mathrm{\,Gyr}$ is shown as
a solid black line. The red line shows the matched-filter used to create the
optimized map shown in the top row of Figure 6. The left y-axis shows the
absolute magnitude of sources and the right y-axis shows the corresponding
apparent magnitude at $\phi_{1}=0{}^{\circ}$.

###  3.4 Fitting Stream Morphology

To quantitatively characterize the observed features of the stream morphology
we use a generative stream model developed by Koposov et al. (2019) and Li et
al. (2020), which is similar to that of Erkal et al. (2017). This model uses
natural cubic splines with different numbers of nodes to describe stream
properties. This model is implemented in STAN (Carpenter et al., 2017) and is
fit to the data using the Hamiltonian Monte Carlo No-U-Turn Sampler (NUTS) to
efficiently sample the high dimensional parameter space.

The stream is modeled by a single Gaussian in $\phi_{2}$ with central
intensity, width, and track that are allowed to vary as functions of
$\phi_{1}$. The parameters of the model are $\mathcal{I}(\phi_{1})$,
$w(\phi_{1}),\,\Phi_{2}(\phi_{1}),\,\mathcal{B}_{1}(\phi_{1}),\,\mathcal{B}_{2}(\phi_{1}),$
and $\mathcal{B}_{3}(\phi_{1})$, which describe the logarithm of the stream
central intensity, the logarithm of the stream width, the stream track, the
log-background density, the slope of the log-background density, and the
quadratic term of the log-background density, respectively. For more details
see Koposov et al. (2019) (Section 3.1 and equation 2) The model is fit to the
binned matched-filter data described above using Equation 3 to describe the
distance modulus as a function of $\phi_{1}$. We assume the number of stars in
an individual pixel of the matched-filter map is a Poisson sample of the model
density at that location.

Following Li et al. (2020), we use Bayesian optimization to determine model
complexity in a data-driven way. In particular, the number of nodes for all
parameters except the stream width are hyper-parameters of this model, and
they are determined through model selection. Where Bayesian optimization
(Gonzalez et al., 2016; The GPyOpt authors, 2016) of the cross-validated
($k=3$) log-likelihood function is used. For the parameters
[$\mathcal{I}(\phi_{1})$, $w(\phi_{1})$, $\Phi_{2}(\phi_{1})$,
$\mathcal{B}_{0}(\phi_{1})$, $\mathcal{B}_{1}(\phi_{1})$,
$\mathcal{B}_{2}(\phi_{1})$] we find the optimal number of nodes to be [11, 3,
8, 28, 25, 5], respectively. For each parameter, the range of allowable nodes
is 3 to 30 except for the width, $w(\phi_{1})$, and quadratic term of the log-
background density, $\mathcal{B}_{3}(\phi_{1})$, which have their maximum
number of nodes constrained to 15 and 10, respectively. The model is run for
1500 iterations with the first 700 discarded as burn in.

Figure 6: Modeling the track, width, and intensity of the Jet stream from
DELVE DR1 photometric data. _Top:_ The density of stars that pass the
optimized matched-filter selection that takes into account the observed
distance gradient in the region of the Jet stream. _Middle:_ The maximum a
posteriori (MAP) model of the data shown in the top panel containing both
stream and background components. _Bottom:_ The residual density map showing
the observed density minus the model.

The results of the model fit are shown in Figures 6 and 7. For Figure 6 the
top row shows the observed matched-filter map, the best-fit model is shown on
the second in the same color scale, and the residual of the model subtracted
from the data is shown on the bottom row. The key features captured by this
model are the variations in the density of stars. A large gap can be seen at
$\phi_{1}=-6{}^{\circ}$, and peaks in the intensity are found at at
$\phi_{1}=-9{}^{\circ},-2.5{}^{\circ}$ and $4{}^{\circ}$. The model does not
capture all of the observed small-scale substructure. In particular, the off-
track structure seen crossing the stream at $\phi_{1}=-12$ or the overdensity
above the stream at $\phi_{1}=5\,{}^{\circ}$ are discussed in more detail in
Section 5.2.

The nature of the on-stream structure can be better evaluated by looking at
the extracted stream parameters in Figure 7. These plots show the stream
surface brightness, the on-sky track, stream width and linear density. In each
panel, the best-fit value calculated as the maximum _a posteriori_ (MAP) of
the posterior for each parameter as a function of $\phi_{1}$ is shown as a
black line, and the 68% containment peak interval is shown as the blue shaded
region. The apparent width of the stream increases with $\phi_{1}$, consistent
with expected projection effects due to a constant width and the observed
distance gradient. This is supported by the relatively constant linear density
over large scales.

Figure 7: Measurement of Jet stream parameters as a function of position along
the stream ($\phi_{1}$) as derived from modeling the DELVE DR1 stellar density
maps. From top to bottom are stream surface brightness, stream track, stream
width, and linear density. The shaded area shows the 68% containment peak
interval, and the black line shows the best-fit estimate of each parameter.

###  3.5 Proper Motion of the Jet Stream

In this section, we use the cross-matched DELVE DR1 and Gaia EDR3 catalog to
measure the proper motion of the Jet stream. In Section 3.2 we demonstrated
that the proper motion signature of BHB stars could be clearly separated from
the Milky Way foreground. We seek to extend our analysis to the full stellar
population of Jet, applying several additional physically motivated cuts to
reduce Milky Way foreground contamination. Then we perform a Gaussian mixture
model fit with stream and Milky Way components to measure the proper motion of
Jet.

####  3.5.1 Data Preparation

Starting with the stellar catalog from Section 2.1, we apply several cuts to
reduce Milky Way contamination in our sample and to highlight the Jet stream
population. These cuts are depicted visually in the left two panels of Figure
8. The following selection process largely follows the methodology set out by
Shipp et al. (2019) and Li et al. (2019).

We begin by calculating the absolute magnitude in the $g$-band ($M_{g}$) for
each star assuming a distance given by the fit to the BHB stars (Equation 3;
$m-M(\phi_{1})$). A magnitude cut is made keeping only sources with $M_{g}<2$
to remove faint sources with large proper motion uncertainties. Then, a
color–magnitude selection is applied selecting stars in $(g-r)_{0}$ vs.
$M_{g}$ color–magnitude space (Fig. 8 left panel). RGB stars are selected
based on the Dotter et al. (2008) isochrone used in Section 3.1. We select
stars that meet either of the following conditions:

$\displaystyle-0.08$ $\displaystyle\leq$ $\displaystyle(g-r)_{0}-$
$\displaystyle(g-r)_{\rm Dotter}$ $\displaystyle\leq 0.02$ (4)
$\displaystyle-0.5$ $\displaystyle\leq$ $\displaystyle g_{0}-$
$\displaystyle(g_{0})_{\rm Dotter}$ $\displaystyle\leq 0.5.$

where $(g-r)_{\rm Dotter}$ is the isochrone color at a given observed
magnitude and $(g_{0})_{\rm Dotter}$ is the isochrone magnitude at a given
observed color. Next, we applied a $(g-r)_{0}$ vs. $(r-i)_{0}$ color–color cut
to select metal poor stars based on an empirical stellar locus that is derived
from dereddened DES data (second panel; Pace & Li, 2019; Li et al., 2018).
This locus gives the median $(r-i)_{0}$ colors for each $(g-r)_{0}$ bin, For
each star we compute $\Delta_{ri}=(r-i)_{0}-(r-i)_{\rm med}$ where $(r-i)_{0}$
is the observed color of a star and $(r-i)_{\rm med}$ is the median
$(r-i)_{0}$ color of stars with the same $(g-r)_{0}$ color taken from the
empirical stellar locus. Only stars within $-0.02<\Delta_{ri}<0.1$ are kept.
Finally, a spatial cut is applied only keeping stars within $3\times
w(\phi_{1})$ of the stream track where $w(\phi_{1})$ is the stream width and
track taken from the modeling in Section 3.4.

To select candidate horizontal branch members for the proper motion fit, we
use an empirical horizontal branch of M92 initially derived in Belokurov et
al. (2007) and transformed to the DES photometric system (Li et al., 2019;
Pace & Li, 2019). We select stars with $(g-r)_{0}$ colors within $\pm 0.1$ mag
and $M_{g}$ within $\pm 0.5$ mag of the empirical horizontal branch.

After applying these cuts, we perform a reflex correction on the proper motion
measurements of the remaining sources (assuming the distance fit from Equation
3). The proper motion signal of Jet is easily identified in the third panel of
Figure 8 as the overdensity at
$(\mu_{\phi_{1}}^{\star},\mu_{\phi_{2}})\sim(-1,0)$ mas/yr. To quantitatively
measure the proper motion and any gradient in the proper motion, we use a
Gaussian mixture model based analysis described in the following section.

Figure 8: Selection of stars used to measure the proper motion of the Jet
stream. _Left:_ A color–magnitude diagram demonstrating the selection applied
to the data. The orange lines show the empirical M92 horizontal branch and
Dotter isochrone ($[\text{Fe/H}]=$$-1.57$) used to select data. The blue
dashed lines show our selection range in distance, and the red dashed lines
our selection range in color. _Center:_ A color-color plot showing our
selection range in blue around an empirical stellar locus of dereddened DES
photometry in orange. _Right:_ Proper motion in stream coordinates of the
sample of stars passing all selection cuts. The signal from the Jet stream is
easily seen at $(\mu_{\phi_{1}}^{\star},\mu_{\phi_{2}})\sim(-1,0)$ mas/yr
(circled in red).

####  3.5.2 Mixture Model

To determine the proper motion of the Jet stream, we use a simple mixture
model consisting of Gaussian distributions for the stream and Milky Way
foreground. The fit is performed on the candidate RGB and BHB stellar sample
defined in the previous section and shown in the right panel of Figure 8. The
likelihood and fitting methodology follows that of Pace & Li (2019) and Shipp
et al. (2019). The complete likelihood is given by:

$\mathcal{L}=(1-\lambda)\mathcal{L}_{\rm Jet}+\lambda\mathcal{L}_{\rm MW}$ (5)

where $\mathcal{L}_{\rm Jet}$ is the likelihood that a star belongs to the Jet
stream component, and $\mathcal{L}_{\rm MW}$ refers to the likelihood that a
star belongs to the Milky Way foreground component. The fraction of stars that
belong to the Milky Way component is denoted by $\lambda$. Each likelihood
term is made up of the product of both spatial and proper motion likelihoods,

$\mathcal{L}_{\rm Jet/MW}=\mathcal{L}_{\rm spatial}\;\mathcal{L}_{\rm PM}.$
(6)

For the stream spatial component we use the results from Section 3.4 for the
stream track, $\Phi_{2}(\phi_{1})$, and width, $w(\phi_{1})$, assuming the
stream follows a Gaussian distribution in $\phi_{2}$ around this track (i.e.
$\phi_{2,\text{obs}}-\Phi_{2}(\phi_{1})$ with standard deviation
$w(\phi_{1})$. The Milky Way spatial component is assumed to be uniform.

$\mathcal{L}_{\rm
spatial}=\begin{cases}\mathcal{N}(\phi_{2}\,|\,\Phi_{2}(\phi_{1}),w_{\mathrm{Jet}})&\text{Jet
component}\\\ \mathrm{Uniform}&\text{MW component}\\\ \end{cases}$ (7)

The proper motion term of the likelihood is modeled by the combination of
several multivariate Gaussians.

Each Gaussian is defined to have a mean proper motion vector
$\boldsymbol{\chi}$ given by
$\boldsymbol{\chi}=(\mu_{\phi_{1},Jet}^{\star}(\phi_{1}),\mu_{\phi_{2},Jet}(\phi_{1}))$.
We model both components of $\boldsymbol{\chi}$ as a linear functions of
$\phi_{1}$. The covariance, $\boldsymbol{C}$, is defined by an observational
error component and an intrinsic component:

$\displaystyle\boldsymbol{C}$
$\displaystyle=\boldsymbol{C}_{obs}+\boldsymbol{C}_{intrinsic}$
$\displaystyle=\begin{bmatrix}\epsilon_{\mu_{\phi_{1}}\,\cos\phi_{2}}^{2}&\epsilon_{\mu_{\phi_{1}}\,\cos\phi_{2}\times\mu_{\phi_{2}}}\\\
\epsilon_{\mu_{\phi_{1}}\,\cos\phi_{2}\times\mu_{\phi_{2}}}&\epsilon_{\mu_{\phi_{2}}}^{2}\
\end{bmatrix}+$ (8)
$\displaystyle\quad\;\begin{bmatrix}\sigma_{\mu_{\phi_{1}}\,\cos\phi_{2}}^{2}&\sigma_{\mu_{\phi_{1}}\,\cos\phi_{2}}\times\sigma_{\mu_{\phi_{2}}}\times\rho\\\
\sigma_{\mu_{\phi_{1}}\,\cos\phi_{2}}\times\sigma_{\mu_{\phi_{2}}}\times\rho&\sigma_{\mu_{\phi_{2}}}^{2}\\\
\end{bmatrix}$

where $\epsilon$ represents the proper motion errors, $\sigma$ is the
intrinsic proper motion dispersions, and $\rho$ is a correlation term. For the
stream component, the intrinsic dispersion is assumed to be
$5\mathrm{\,km}/\mathrm{\,s}$ which at the distance of the stream corresponds
to $\sim 0.04\mathrm{\,mas}/\mathrm{\,yr}$ and the correlation terms are
assumed to be zero.

The model then has 9 free parameters: The systemic proper motions of the
stream measured at $\phi_{1}=0{}^{\circ}$
($\bar{\mu}_{\phi_{1}}\cos(\phi_{2}),\bar{\mu}_{\phi_{2}}$), the proper motion
gradients in each coordinate direction for the stream
($d\mu_{\phi_{1}}/d\phi_{1},d\mu_{\phi_{2}}/d\phi_{1}$) in units of
$\mathrm{\,mas}/10{}^{\circ}$, the mean proper motion of the Milky Way
foreground Gaussian ($\mu_{\phi_{1}\,{\rm MW}},\mu_{\phi_{2}\,{\rm MW}}$), the
dispersion of the Milky Way foreground Gaussian ($\sigma_{\phi_{1}\,{\rm
MW}},\sigma_{\phi_{2}\,{\rm MW}}$), and the fraction of stars that belong to
the stream component ($\lambda$). The proper motion of the stream component as
a function of $\phi_{1}$ is given by

$\displaystyle\mu_{\phi_{1},\mathrm{Jet}}(\phi_{1})$
$\displaystyle=\bar{\mu}_{\phi_{1}}+\,\frac{d\mu_{\phi_{1}}}{d\phi_{1}}\times(\phi_{1}/10\deg)$
(9) $\displaystyle\mu_{\phi_{2},\mathrm{Jet}}(\phi_{1})$
$\displaystyle=\bar{\mu}_{\phi_{2}}+\,\frac{d\mu_{\phi_{2}}}{d\phi_{1}}\times(\phi_{1}/10\deg).$

The total proper motion likelihood is then given by

$\mathcal{L}_{\rm
PM}=\sum_{N=1}^{k}\mathcal{N}((\boldsymbol{\mu}_{\phi_{1},obs}^{\star},\boldsymbol{\mu}_{\phi_{2},obs})_{N}\,|\,\boldsymbol{\chi}_{true,N}\;,\boldsymbol{C_{N}}).$
(10)

Parameter inference is conducted using a Hamiltonian Monte Carlo No-U-Turn
Sampler (NUTS) implemented in STAN (Carpenter et al., 2017). We use 10
parallel chains with 2000 iterations each (1000 of the iterations are
discarded as a burn in). Convergence is verified using the Gelman-Rubin
$\hat{R}<1.1$ diagnostic (Gelman & Rubin, 1992).

The results of our fit are listed in Table 1. We find the results from the
mixture model
($\bar{\mu}_{\phi_{1},\mathrm{Jet}}^{\star},\bar{\mu}_{\phi_{2},\mathrm{Jet}})=(-0.933\pm
0.018,-0.023\pm 0.016)$ mas/yr, which agrees with our rough estimate from the
observed overdensity of candidate BHB stars in Section 3.2. The
$\bar{\mu}_{\phi_{2}}$ value is near zero as expected for a stable stream that
has not been heavily perturbed. We detect gradients in both proper motion
coordinates that are similar in magnitude. Based on these results, the
tangential velocity of the Jet stream at $\phi_{1}=0{}^{\circ}$ is
$v_{tan}=195\pm 3$ km/s.

A membership probability is calculated for each star by taking the ratio of
the stream likelihood to the total likelihood:
$p_{mem}=\lambda\mathcal{L}_{\mathrm{Jet}}/(\lambda\mathcal{L}_{\rm
Jet}+(1-\lambda)\mathcal{L}_{\rm MW})$. To determine the value of $p_{mem,i}$
for each star we calculate $p_{mem}$ for each point in the posterior of our
fit and take $p_{mem,i}$ to be the median of the calculated values for each
star. A star is then considered a high (medium) probability member if
$p_{mem,i}>0.8\;(0.5)$. For our sample, $89\;(97)$ candidate RGB stars and
$28\;(28)$ candidate BHB stars pass this criterion.

Figure 9: Proper motion of high probability member RGB (red) and BHB (blue)
stars ($p_{mem}>0.8$). Additionally the line of best-fit proper motion is
shown as a black dashed line. All the BHB stars used to measure the distance
gradient are also high probability members.

The proper motion of the high probability stars are shown in Figure 9 along
with lines showing the best-fit stream proper motion
($\bar{\mu}_{\phi_{1}}^{\star}(\phi_{1}),\bar{\mu}_{\phi_{2}}(\phi_{1}))$ from
our analysis. The BHB stellar sample (blue points) is very similar to the
sample selected by a rough cut in Figure 4, and it can be seen that these
stars closely follow the proper motion gradient ($d\mu/d\phi_{1}$) found in
our analysis. In Appendix A, we include Table A.1 that contains the properties
of all candidate BHB and RGB stars with membership probability ($p_{mem}$)
higher than 10%. With the current dataset we are only able to fit for linear
evolution of the proper motion with $\phi_{1}$, but with future spectroscopic
datasets can test for a more complex evolution of the proper motion as a
function of $\phi_{1}$ (e.g., quadratic).

Previous studies such as Shipp et al. (2019) have looked for signs of large
scale perturbation of stellar streams from the influence of the LMC or Milky
Way bar (Li et al., 2018; Erkal et al., 2018; Koposov et al., 2019; Vasiliev
et al., 2021). Evidence of these interactions sometimes appears as a mismatch
between the proper motion of the stream ($\mu_{\phi_{2}}/\mu_{\phi_{1}}$) and
the derivative of the stream track ($d\phi_{2}/d\phi_{1}$) (Erkal et al.,
2019). In the case of Jet, we find that the ratio of the proper motions to the
stream track ($\mu_{\phi_{2}}/\mu_{\phi_{1}})/(d\phi_{2}/d\phi_{1}$) has an
average value of $2.00\pm 1.18$ over the extent of the stream. This is
consistent with a value of one which indicates the proper motions are largely
aligned with the track of the stream.

Table 1:  Results of the proper motion fit in reflex corrected proper motion coordinates Parameter | Value | Prior | Range | Units
---|---|---|---|---
$\bar{\mu}_{\phi_{1}}^{\star}$ | $-0.933\pm 0.018$ | Uniform | -10,10 | mas/yr
$\bar{\mu}_{\phi_{2}}$ | $-0.023\pm 0.016$ | Uniform | -10,10 | mas/yr
$d\mu_{\phi_{1}}^{\star}/d\phi_{1}$ | $-0.109\pm 0.030$ | Uniform | -3,3 | mas/yr/10 deg
$d\mu_{\phi_{2}}/d\phi_{1}$ | $0.120\pm 0.021$ | Uniform | 3,3 | mas/yr/10 deg
$\lambda$ | $0.086\pm 0.011$ | Uniform | 0,1 |

##  4 Dynamical Modeling

Using our measurements of the stream track, distance, and proper motion, we
can fit a dynamical model to the data. The Jet stream is modeled in the same
method as Erkal et al. (2019) and Shipp et al. (2021). We make use of the
modified Lagrange Cloud Stripping (mLCS) technique developed in Gibbons et al.
(2014) adapted to include the total gravitational potential of the stream
progenitor, the Milky Way, and the LMC. Following Erkal et al. (2019), the
Milky Way and LMC are modeled as independent particles with their respective
gravitational potentials which allows us to capture the response of the Milky
Way to the LMC. The Milky Way potential is modeled with 6 axisymmetric
components, namely bulge, dark matter halo, thin and thick stellar disk, and
HI and molecular gas disk components following McMillan (2017). Following
Shipp et al. (2021), we normalize the Milky Way potential to the realization
of the McMillan (2017) potential that yields the best-fit from the ATLAS data
($M_{\rm MW}=8.3\times 10^{11}M_{\rm\odot}$; Li et al., 2020). We evaluate the
acceleration from the potential using galpot (Dehnen & Binney, 1998). We take
the Sun’s position ($R_{0}=8.23$ kpc) and 3D velocity,
($\text{U}_{\odot},\text{V}_{\odot},\text{W}_{\odot})=(8.6,232.8,7.1)$ km/s,
from McMillan (2017).

We model the mass distribution of the LMC as a stellar disk and a dark matter
halo. The stellar disk is modelled as a Miyamoto-Nagai disk (Miyamoto & Nagai,
1975) with a mass of $3\times 10^{9}M_{\odot}$, a scale radius of $1.5$ kpc,
and a scale height of $0.3$ kpc. The orientation of the LMC disk matches the
measurement of van der Marel & Kallivayalil (2014). The LMC’s dark matter halo
is modelled as a Hernquist profile (Hernquist, 1990). We fix the total infall
mass of the LMC to $1.5\times 10^{11}\mathrm{\,M_{\odot}}$, consistent with
the value derived in Erkal et al. (2019) and Shipp et al. (2021). We fix the
scale radius to match the circular velocity measurement of $91.7$ km/s at
$8.7$ kpc from van der Marel & Kallivayalil (2014). Note that this is in
agreement with more recent measurements of the LMC’s circular velocity (e.g.,
Cullinane et al., 2020). We account for the dynamical friction of the Milky
Way on the LMC using the results of Jethwa et al. (2016). We also fix the
LMC’s present-day proper motion, distance, and radial velocity to measured
values (Kallivayalil et al., 2013; Pietrzyński et al., 2013; van der Marel et
al., 2002). The LMC mass remains fixed throughout each simulation.

We model the potential of the Jet stream’s progenitor as a Plummer sphere
(Plummer, 1911) with a mass and scale radius chosen to match the observed
stream width. During the course of tidal disruption, the progenitor’s mass
decreases linearly in time to account for tidal stripping. Since Jet does not
have a known progenitor, we assume that the progenitor has completely
disrupted, i.e., that its present day mass is zero. Furthermore, we assume
that the remnant of the progenitor is located at $\phi_{1}=0{}^{\circ}$.

We calculate the likelihood for the stream model by producing a mock
observation of a simulated stream and comparing it with the data described in
the previous sections. For each stream model, we calculate the track on the
sky, the radial velocity, the proper motions in $\phi_{1}$ and $\phi_{2}$, and
the distance as functions of $\phi_{1}$, the observed angle along the stream.

We assign the mass of the progenitor in order to reproduce the observed width
of the stream. Our best-fit model uses a progenitor mass of $M_{\rm
prog}=2\times 10^{4}\mathrm{\,M_{\odot}}$, and a Plummer scale radius of
$r_{\rm plum}=10\mathrm{\,pc}$. We note that these values are highly dependent
on the location of the progenitor.

We perform a Markov Chain Monte Carlo (MCMC) fit using emcee (Foreman-Mackey
et al., 2013). Our model includes 5 free parameters. We fit the present-day
progenitor $\phi_{2}$ position, distance, radial velocity, and proper motion.
The prior distributions on each parameter are listed in Table 2. The position
of the progenitor along the stream is fixed to $\phi_{1}=0{}^{\circ}$ (i.e.,
the middle of the stream’s observed extent). We show the Jet data and the
best-fit stream models in Figure 10. In each panel we show the observations in
red, and simulated stream in blue. The radial velocity panel contains no
observations, but can be used to predict the radial velocity of the stream. We
find the best-fit model is a good fit to the observations of the distance
modulus, stream track and proper motions.

We have tried fits that include/exclude the effect of the LMC and Milky Way
bar. For the Milky Way bar we assume an analytic model with the same
parameters as used in Li et al. (2020) (described in their Section 5.2.1) and
Shipp et al. (2021). For both cases, the LMC and Milky Way bar, we find that
it is unlikely that the Jet stream has been significantly affected by these
substructures.

This model emphasizes some of the observed features of the Jet stream
discussed in Sections 3.4 and 3.5. In particular, we note the observed
curvature of the stream track away from $\phi_{2}=0{}^{\circ}$ is due both to
Galactic parallax and the non-spherical potential of the MW. None of the
intensity features gaps/peaks are seen in the model, and we also fail to
replicate the off-track features seen in the photometry. Additionally, this
model predicts a stream component in the Jet Bridge region (figure 1) that is
not detected in our search. This non-detection is likely due to the increased
Milky Way foreground contamination and reddening in this region

Table 2:  Priors on the dynamical model  Parameter | Prior | Range | Units | Description
---|---|---|---|---
$\phi_{\rm 2,prog}$ | Uniform | (-1, 1) | deg | Location of the progenitor perpendicular to the stream track.
$\mu_{\rm\alpha,prog},\mu_{\rm\delta,prog}$ | Uniform | (-10, 10) | mas/yr | Reflex-corrected proper motion of the progenitor.
$v_{\rm r,prog}$ | Uniform | (-500, 500) | km/s | Radial velocity of the progenitor.
$(m-M)_{\rm prog}$ | Normal | $(m-M)_{0}\pm 0.2$ | mag | Distance modulus of the progenitor.
$\phi_{\rm 1,prog}$ | Fixed | 0 | deg | Location of the progenitor along the stream track.
$M_{\rm LMC}$ | Fixed | $1.5\times 10^{11}$ | $\mathrm{\,M_{\odot}}$ | Total mass of the LMC.
$\mu_{\rm\alpha,LMC}$ | Normal | $1.91$ | mas/yr | Proper motion of the LMC in RA.
$\mu_{\rm\delta,LMC}$ | Normal | $0.229$ | mas/yr | Proper motion of the LMC in Dec.
$v_{\rm r,LMC}$ | Normal | $262.2$ | km/s | Radial velocity of the LMC.
$d_{\rm LMC}$ | Normal | $49970.0$ | pc | Distance of the LMC.
Figure 10:  Best-fit stream model to the Jet stream. In each row the dark blue
points show the best-fit stream model and the red points show the observations
to which the model was fit. _First row:_ the on-sky distribution of the
stream. _Second row:_ the distance modulus of the stream. _Third and Fourth
rows:_ The reflex-corrected proper motions of the stream in $\phi_{1}$ and
$\phi_{2}$ respectively. _Fifth row:_ predicted heliocentric velocities of the
stream.

##  5 Discussion

###  5.1 Properties of the Jet stream

The Jet stream is now detected from $-15.2{}^{\circ}\leq\phi_{1}\leq
13.7{}^{\circ}$ increasing its known length from $\sim 11{}^{\circ}$ to nearly
$29{}^{\circ}$. For $\sim 23{}^{\circ}$ ($-13{}^{\circ}\leq\phi_{1}\leq
10{}^{\circ}$) the main sequence turn off of the stream is strongly detected
in the DELVE photometry; at $\phi_{1}<-13{}^{\circ}$ and $>10{}^{\circ}$ the
intensity of the stream decreases greatly, and so is only significantly
detected using BHB and RGB stars with measured proper motions. At observed
distances ranging from $\sim 26-34.5\mathrm{\,kpc}$, the stream has a physical
length of 16 kpc, with a strong photometric detection covering 13.4 kpc. This
makes the extent of the Jet stream comparable to the kinematically cold
Phoenix, ATLAS, and GD-1 streams (Balbinot et al., 2016; Koposov et al., 2014;
Grillmair & Dionatos, 2006), which span ${\sim}\,5\mathrm{\,kpc}$
($13.6{}^{\circ}$; Shipp et al., 2018), ${\sim}\,12\mathrm{\,kpc}$
($34{}^{\circ}$ including Aliqa Uma; Li et al., 2020) and
${\sim}\,15.4\mathrm{\,kpc}$ (${\sim}\,100{}^{\circ}$; de Boer et al., 2020;
Webb & Bovy, 2019; Price-Whelan & Bonaca, 2018; Malhan et al., 2018),
respectively. Our dynamical models place Jet on a retrograde orbit with a
pericenter of $12.7\mathrm{\,kpc}$ which is comparable to the pericenters of
Phoenix (${\sim}\,13\mathrm{\,kpc}$; Wan et al., 2020) ATLAS
($13.3\mathrm{\,kpc}$; Li et al., 2020), Kshir (${\sim}\,14\mathrm{\,kpc}$;
Malhan et al., 2019), and GD-1 (${\sim}\,14\mathrm{\,kpc}$; Koposov et al.,
2010; Malhan & Ibata, 2019).

As discussed in Section 3.5, the proper motion of the Jet stream members and
the observed track of the stream are fairly well aligned, suggesting that the
Jet stream has not been strongly perturbed perpendicular to the line of sight
from interactions with large Milky Way substructures. However, perturbations
in the radial or track direction are difficult to measure from proper motion
alone.

###  5.2 Small-Scale Features

This detailed view of the Jet stream has started to reveal its complexity,
adding it to the group of streams that show small-scale features (e.g., Erkal
et al., 2017; Price-Whelan & Bonaca, 2018; Li et al., 2020; Caldwell et al.,
2020). Most noticeably, a ${\sim}\,4{}^{\circ}$ gap in the stream is seen
centered on $\phi_{1}=-6{}^{\circ}$ extending from -. This structure may be
due to interactions between the Jet stream and its environment (e.g., a dark
matter subhalo passing by and perturbing the stream), or the environment of
the progenitor parent satellite (e.g., Malhan et al., 2021). Alternatively,
this structure could also be the result of a complete dissolution of the
progenitor as suggested by Webb & Bovy (2019) in relation to the GD-1 stream.
Understanding the nature of this gap will be important for future studies with
deeper photometry and radial velocities of member stars near this region. We
also note that in the top row of Figure 6 the stream looks extremely clumpy on
smaller scales than our model probes. Deeper photometric observations, such as
those possible with the Vera C. Rubin Observatory’s Legacy Survey of Space and
Time (LSST), will increase the signal-to-noise of these features, allowing
better modeling and therefore a more complete understanding of the system.
Finally, there are density features seen off the main track of the stream. At
$\phi_{1}\sim 5{}^{\circ}$ a signal is seen in the matched-filter map just
above the stream (Figure 6). This could be a substructure similar to the
“spur” of GD-1 (Price-Whelan & Bonaca, 2018) or evidence of some other type of
interaction.

At $\phi_{1}\sim-12.5{}^{\circ}$, a feature is seen both above and below the
stream; it is possible that the increased reddening in this region is causing
this feature, or it could be even more evidence of past interactions between
the Jet stream and other substructure.

To fully understand these features, followup spectra will be crucial, as these
observations will enable the use of radial velocities and metallicities to
robustly identify members, as well as allow for the use of the 6D information
of members in these off track features to determine their origin.

The observations of small-scale structure in the Jet stream are particularly
interesting given its orbital properties. The Jet stream’s current
Galactocentric radius ($r_{G}=30-37$ kpc), orbital pericenter ($r_{peri}=12.7$
kpc), lack of perturbation from the Milky Way bar in our simulations, and
retrograde orbit suggest the Jet stream is less likely to have been perturbed
on small scales due to interaction with baryonic matter (Pearson et al., 2017;
Banik et al., 2021). This indicates that the Jet stream is likely to be one of
the best known streams for constraining dark matter substructure in the Milky
Way.

###  5.3 Stream Mass and Progenitor Properties

Based on the DELVE photometry, Jet appears to be another stellar stream with
no obvious detected progenitor (e.g., Phoenix, ATLAS, GD-1, and many others).
Although we do not detect a progenitor for Jet, we can use our observations
and modeling to further constrain the properties of the progenitor of the Jet
stream. J18 determined the current stellar mass of the Jet stream by fitting
the observed stream-weighted CMD and found that the total stellar mass is
$2.5\pm 0.2\times 10^{4}\mathrm{\,M_{\odot}}$. We can set a lower limit on the
stellar mass of the Jet stream using the number of high confidence BHB
candidates we detect. Based on the color–magnitude and proper motion
selections applied in Sections 3.2 and 3.5, we detect 28 high confidence BHB
candidates along the stream. Assuming a Chabrier (2001) initial mass function,
an age of 12.1$\mathrm{\,Gyr}$ and a metallicity of [Fe/H]$=$$-1.57$, we find
that a stellar mass of $2.6_{-0.4}^{+0.6}\times 10^{4}\mathrm{\,M_{\odot}}$ is
required to produce the observed number of BHB stars. This estimate is in good
agreement with the previous results of J18.

Erkal et al. (2016) suggested that the total dynamical mass of a stream can be
estimated from its width. This was used by Shipp et al. (2018) to estimate
dynamical masses of the DES streams, and J18 applied the same procedure to
estimate that the total dynamical mass of the Jet progenitor is expected to be
${\sim}\,6.4\times 10^{4}\mathrm{\,M_{\odot}}$. In this analysis, we find the
observed angular width of Jet varies by a factor of ${\sim}\,2$ over its
extent (Section 3.4). However, if we account for the measured distance
gradient (Section 3.2), the observations are consistent with a constant
physical width of $\sim 90\mathrm{\,pc}$ over the range
$\phi_{1}=-12{}^{\circ}-10{}^{\circ}$. The stream appears to fan out even more
in the region $\phi_{1}>10{}^{\circ}$ where the intensity of the stream drops
greatly. The observed complex physical structure makes it difficult to
motivate the simple scaling between stream width and dynamical mass from Erkal
et al. (2016). Thus, we instead estimate the dynamical mass from the best-fit
orbital model of the Jet stream described in Section 4. We find that these
simulations prefer a total dynamical mass of ${\sim}\,2\times
10^{4}\mathrm{\,M_{\odot}}$ which is a factor of ${\sim}\,3$ less massive than
the estimate of J18, but our mass estimate is highly dependent on the location
of the progenitor which we assume is $\phi_{1}=0{}^{\circ}$. For a progenitor
at $\phi_{1}=20{}^{\circ}$ we find a worse fit to the overall stream
properties but recover a dynamical mass consistent with J18. The ratio of
stellar and dynamical mass (${\sim}\,1$) supports the hypothesis that the
progenitor of Jet was a globular cluster ($\mathrm{M/L}\,{\sim}\,1-2$;
Kruijssen, 2008) rather than a dwarf galaxy
($\mathrm{M/L}\,{\sim}\,10^{2}-10^{4}$ for $\mathrm{L}\,{\sim}\,2\times
10^{4}$; Simon, 2019).

The results of our MCMC modeling can be used to estimate the heliocentric
velocity ($\mathrm{v}_{\odot}$) of the stream and other orbital parameters. We
find a predicted heliocentric velocity at $\phi_{1}=0{}^{\circ}$ of
$\mathrm{v}_{\odot}=286\pm 10\mathrm{\,km}/\mathrm{\,s}$.444This agrees with
unpublished spectroscopic data from AAT/2dF (T. S. Li, private communication).
For the orbit of Jet we find a pericenter of $r_{\rm peri}=12.7\pm
0.9\mathrm{\,kpc}$, an apocenter of $r_{\rm apo}=38\pm 1\mathrm{\,kpc}$, and
an eccentricity of 0.59 and an orbital period of 0.5 Gyr.

These orbital properties can be used to explore whether Jet could be
associated with other known globular clusters or streams. The predicted
$R_{v}$ estimate, along with the measured proper motion of the Jet stream,
give an expected angular momentum perpendicular to the Galactic disk
($\hat{z}$ direction) $L_{z}$ and total energy $E_{\rm tot}$. These two
quantities, $L_{z}$ and $E_{\rm tot}$, are both conserved assuming a static
axis-symmetric Milky Way potential. To compute these parameters we use the
same Milky Way plus LMC potential from Section 4. We randomly draw from the
posteriors of our fit values for the proper motions, $\phi_{2}$ position and
radial velocity of the Jet stream at $\phi_{1}=0{}^{\circ}$, and repeat this
1000 times. Then for each draw we compute the $L_{z}$ and $E_{tot}$ of the Jet
stream at $\phi_{1}=0{}^{\circ}$. Doing this we find the predicted $L_{z}$ and
$E_{\rm tot}$ of the Jet stream to be $L_{z}=1.9\pm
0.1\mathrm{\,kpc}^{2}/\text{Myr}$ and predicted $E_{\rm tot}=-0.103\pm
0.001\mathrm{\,kpc}^{2}/\text{Myr}^{2}$. Using these results we look for
globular clusters with similar $E_{\rm tot}$ and $L_{z}$ properties that could
have been the progenitor of the Jet stream. From the Vasiliev (2019) catalog
of globular cluster orbital properties we find no close matches suggesting
that the progenitor of the Jet stream is either fully disrupted or
undiscovered.

Comparing these results to Figures 1 and 2 in Bonaca et al. (2020), we find
that the Jet stream is on a retrograde orbit with orbital parameters closest
to Phelgethon ($L_{z}{\sim}\,1.93\mathrm{\,kpc}^{2}/\mathrm{\,Myr}$, $E_{\rm
tot}{\sim}\,-0.10\mathrm{\,kpc}^{2}/\mathrm{\,Myr}^{2}$) and nearby Wambelong
and Ylgr as well. It seems likely that the progenitors of Phelgethon and Jet
were accreted onto the Milky Way in the same accretion event. The work of
Naidu et al. (2020) with the H3 survey identified a number of Milky Way
accretion events, and localized them in the $E_{tot}-L_{z}$ paramter space.
The $E_{tot}-L_{z}$ properties of the Jet stream places it in the region of
parameter space likely associated with the Sequoia, I’itoi, and Arjuna
progenitors (their Figure 2), suggesting that the progenitor of the Jet stream
was a globular cluster associated with one of these accretion events (Naidu et
al., 2020; Bonaca et al., 2020; Myeong et al., 2019).

Similarly to J18, we note that the stellar stream PS1-B (Bernard et al., 2016)
is well aligned to the on-sky track of the Jet stream, but at a much different
distance ($D_{\odot}=14.5\mathrm{\,kpc}$). In fact, with our extended
detection to large $\phi_{1}$ with BHB stars the two on-sky stream tracks come
within $0.3{}^{\circ}$ of each other. Our matched filter analysis did not
detect the PS1-B stream; however, our filter was not optimized for the
detection of PS1-B, and future studies could further investigate this
potential association.

##  6 Conclusions

We have presented deep photometric and astrometric measurement of the Jet
stream. We utilized the deep, wide-field, homogeneous DELVE DR1 data, which
allowed us to discover substantial extensions of the Jet stream. We used both
DELVE photometry and proper motions from Gaia EDR3 to select a sample of
candidate BHB member stars. These stars allow us to resolve a distance
gradient along the stream. The DELVE photometry is then used to model the
stream intensity, track, and width, quantitatively characterizing the observed
density variations. Additionally, we are able to use BHB and RGB stars to
measure the systemic proper motion and proper motion gradient of Jet for the
first time. Finally, we fit the stream with a dynamical model to constrain the
orbit of the Jet stream.

The results of these analyses are summarized as follows:

  * •

We extend the known extent of the Jet stream from $11{}^{\circ}$ to
${\sim}\,29{}^{\circ}$ corresponding to a physical length of ${\sim}\,16$ kpc.

  * •

We measure a distance gradient of $-0.2$ kpc/deg along the stream ranging from
$D_{\odot}\sim 34.2\mathrm{\,kpc}$ at $\phi_{1}=-15{}^{\circ}$ to
$D_{\odot}\sim 27.4\mathrm{\,kpc}$ at $\phi_{1}=13.7{}^{\circ}$.

  * •

We model the stream morphology to quantitatively characterize the stream
track, width and linear density. We identify a gap in the stream and two
features off the main track of the stream.

  * •

We measure the proper motion of the Jet stream for the first time, and
identify likely member RGB/BHB stars from their proper motions.

  * •

Our modeling suggests Jet is on a retrograde orbit, unlikely to have been
significantly affected by the LMC or Milky Way bar, and has an orbital
pericenter of $r_{peri}=12.7$ kpc.

Our analysis of the Jet stream has already been used to target spectroscopic
measurements with AAT/2dF as part of the Southern Spectroscopic Stellar Stream
Survey ($S^{5}$; Li et al., 2018). Medium-resolution spectroscopic
measurements with $S^{5}$ will confirm stream membership, provide radial
velocities for stream members, and measure metallicities from the equivalent
widths of the calcium triplet lines (Li et al., 2018). Such measurements have
already yielded interesting dynamical information for the ATLAS stream (Li et
al., 2020) and measured an extremely low metallicity for the Phoenix stream
(Wan et al., 2020). These measurements will further allow the targeting of
high-resolution spectroscopy, which can provide detailed elemental abundances
for Jet member stars (Ji et al., 2020), and help to determine the nature of
the Jet stream progenitor.

The future of resolved stellar studies is bright with ongoing and future deep
and wide-area photometric surveys. In particular, detailed studies of stellar
streams will provide important information for modeling both the large and
small-scale structure of the Milky Way halo, ultimately helping to constrain
the fundamental nature of dark matter (Drlica-Wagner et al., 2019). In the
near future, DELVE will significantly improve the extent and homogeneity of
the southern sky coverage, setting the stage for the LSST-era. Our work on the
Jet stream provides an important precursor legacy to similar measurements that
will be possible with LSST.

##  7 Acknowledgments

PSF acknowledges support from from the Visiting Scholars Award Program of the
Universities Research Association. ABP acknowledges support from NSF grant
AST-1813881. The DELVE project is partially supported by Fermilab LDRD project
L2019-011 and the NASA Fermi Guest Investigator Program Cycle 9 No. 91201.

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the DOE and NSF (USA), MISE (Spain), STFC (UK),
HEFCE (UK), NCSA (UIUC), KICP (U. Chicago), CCAPP (Ohio State), MIFPA (Texas
A&M University), CNPQ, FAPERJ, FINEP (Brazil), MINECO (Spain), DFG (Germany),
and the collaborating institutions in the Dark Energy Survey, which are
Argonne Lab, UC Santa Cruz, University of Cambridge, CIEMAT-Madrid, University
of Chicago, University College London, DES-Brazil Consortium, University of
Edinburgh, ETH Zürich, Fermilab, University of Illinois, ICE (IEEC-CSIC), IFAE
Barcelona, Lawrence Berkeley Lab, LMU München, and the associated Excellence
Cluster Universe, University of Michigan, NSF’s National Optical-Infrared
Astronomy Research Laboratory, University of Nottingham, Ohio State
University, OzDES Membership Consortium University of Pennsylvania, University
of Portsmouth, SLAC National Lab, Stanford University, University of Sussex,
and Texas A&M University.

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 is based on observations at Cerro Tololo Inter-American Observatory,
NSF’s National Optical-Infrared Astronomy Research Laboratory (2019A-0305; PI:
Drlica-Wagner), which is operated by the Association of Universities for
Research in Astronomy (AURA) under a cooperative agreement with the National
Science Foundation.

This manuscript has been authored by Fermi Research Alliance, LLC, under
contract No. DE-AC02-07CH11359 with the US Department of Energy, Office of
Science, Office of High Energy Physics. The United States Government retains
and the publisher, by accepting the article for publication, acknowledges that
the United States Government retains a non-exclusive, paid-up, irrevocable,
worldwide license to publish or reproduce the published form of this
manuscript, or allow others to do so, for United States Government purposes.

Blanco, Gaia.

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##  Appendix A Membership Probability

Table A.1 includes probable stream member stars with membership probability
greater than 0.1 from the likelihood analysis described in Section 3.5.

ID (DELVE)aafootnotemark:  |  ID (Gaia)aafootnotemark:  |  R.A.bbfootnotemark:  |  Dec.bbfootnotemark:  |  $g_{0}$ccfootnotemark:  |  $r_{0}$ccfootnotemark:  | $\mu_{\alpha}\cos{\delta}$ | $\mu_{\delta}$ |  $D_{\phi_{1}}$ddfootnotemark:  |  $p_{i}$eefootnotemark:
---|---|---|---|---|---|---|---|---|---
|  | (deg) | (deg) | (mag) | (mag) | (mas/yr) | (mas/yr) | (kpc) |
10728400085912 | 5692169948945471488 | 147.32085 | -10.90436 | 18.06 | 17.51 | $-0.81\pm 0.11$ | $-1.85\pm 0.08$ | 27.28 | $0.83\pm 0.23$
10728500064937 | 3769827073557279232 | 148.25261 | -10.92857 | 18.84 | 18.34 | $-0.93\pm 0.21$ | $-1.29\pm 0.28$ | 27.17 | $0.48\pm 0.17$
10728500080043 | 3769884424255747584 | 147.84902 | -10.82876 | 18.25 | 17.71 | $-1.25\pm 0.12$ | $-1.56\pm 0.11$ | 27.20 | $0.38\pm 0.22$
10728500026630 | 3769900397239081472 | 148.34539 | -10.75644 | 19.10 | 18.65 | $-0.92\pm 0.21$ | $-2.20\pm 0.24$ | 27.13 | $0.99\pm 0.01$
10728500315132 | 3769918715274616320 | 148.13991 | -10.57946 | 19.01 | 18.47 | $-1.43\pm 0.25$ | $-1.43\pm 0.30$ | 27.12 | $0.87\pm 0.07$
10728500024967 | 3770009145810545664 | 148.73686 | -10.33853 | 18.97 | 18.47 | $-1.13\pm 0.23$ | $-2.04\pm 0.23$ | 27.01 | $0.99\pm 0.00$
10754000190703 | 5690650591379140608 | 145.22603 | -13.49643 | 18.82 | 18.35 | $-1.38\pm 0.17$ | $-1.93\pm 0.15$ | 27.99 | $0.74\pm 0.17$
10754000071117 | 5690723434024204288 | 146.34768 | -13.10071 | 19.10 | 18.66 | $-0.96\pm 0.20$ | $-1.55\pm 0.17$ | 27.78 | $0.94\pm 0.04$
10754000135902 | 5690754357789028352 | 146.29135 | -12.69526 | 18.68 | 18.15 | $-0.61\pm 0.16$ | $-1.65\pm 0.13$ | 27.72 | $0.90\pm 0.13$
10754000148096 | 5690771365860339712 | 146.78020 | -12.57939 | 19.16 | 18.70 | $-1.63\pm 0.23$ | $-1.56\pm 0.21$ | 27.64 | $0.32\pm 0.14$
… | … | … | … | … | … | … | … | … | …

Table A.1: this table includes all stars with $p_{i}>0.1$ for our analysis of
the proper motion of the Jet stream.
(This table is available in its entirety in machine-readable form.)
(a) DELVE ID’s are from the QUICK_OBJECT_ID column in DELVE-DR1, and Gaia IDs
are from SOURCE_ID column in _Gaia_ EDR3.
(b) R.A. and Dec. are from Gaia EDR3 catalog (J2015.5 Epoch).
(c) $g$, $r$ band magnitudes are reddening corrected PSF photometry
(MAG_PSF_DERED) from DELVE DR1 catalog.
(d) The $D_{\phi_{1}}$ column gives the distance in kpc derived from equation
3, except for candidate bhb stars whose distances are estimated from their
predicted absolute magnitude $M_{g}$ as discussed in section 3.2

(e) The probability that a star is a member of the Jet stream.