Astrophysics. JN also acknowledges support from the Harvard Radcliffe
Fellowship Program of the Radcliffe Institute for Advanced Study at Harvard
University. JN, GM, and JJ-Z acknowledge support from the Harvard University’s
Radcliffe Research Partners Program of the Radcliffe Institute for Advanced
Study. EFB is grateful for support from the National Science Foundation
through grant NSF-AST 2007065. LBS acknowledges NASA-ATP award 80NSSC20K0509
and Science Foundation AAG grant AST-2009122. This research has made use of
the SIMBAD database, operated at CDS, Strasbourg, France. This research is
supported by the Director, Office of Science, Office of High Energy Physics of
the U.S. Department of Energy under Contract No. DE–AC02–05CH11231, and by the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility under the same contract; additional support for DESI is provided
by the U.S. National Science Foundation, Division of Astronomical Sciences
under Contract No. AST-0950945 to the NSF’s National Optical-Infrared
Astronomy Research Laboratory; the Science and Technologies Facilities Council
of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-
Simons Foundation; the French Alternative Energies and Atomic Energy
Commission (CEA); the National Council of Science and Technology of Mexico
(CONACYT); the Ministry of Science and Innovation of Spain (MICINN), and by
the DESI Member Institutions: https://www.desi.lbl.gov/collaborating-
institutions. The authors are honored to be permitted to conduct scientific
research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance
to the Tohono O’odham Nation. 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 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). This research also used the
PAndAS data hosted by the facilities of the Canadian Astronomy Data Centre,
which is operated by the National Research Council of Canada with the support
of the Canadian Space Agency. For the purpose of open access, the author has
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising from this submission. This work was performed in
part at Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1607611. This work was partially supported by a grant
from the Simons Foundation. KPNO:Mayall (DESI), WISE, Gaia, CFHT:Megacam

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##  Appendix A Quasars and Galaxies Behind M31

Figure 20: The selection of the DESI QSO targets. The left panel shows the
criteria used (dashed line) to select QSO candidates. The known QSOs from
Massey et al. (2019) and Huo et al. (2010, 2013, 2015) are shown as red
squares and the QSOs confirmed by DESI spectroscopy are shown as filled blue
dots. The right panel shows the Gaia $G$ vs $B_{p}-R_{p}$ diagram as a
greyscale for all Gaia stars within 5∘ of M31. The solid line shows the
criterion used to exclude M31 sources.

QSO candidates were selected using a combination of the Gaia DR2 (Gaia
Collaboration et al., 2016b, 2018) and the deep combined imaging data from the
unWISE catalogs (Lang, 2014; Meisner et al., 2017a, b, 2019; Schlafly et al.,
2019) using the following criteria:

  * •

$\pi-\sigma(\pi)\leq 0.1$

  * •

$|\mu_{\alpha}-2\sigma(\mu_{\alpha})|\leq 0.1$ and
$|\mu_{\delta}-2\sigma(\mu_{\delta})|\leq 0.1$

  * •

($G<19$ and AEN$<10^{0.5}$) or ($G\geq 19$ and AEN$<10^{0.5+0.2(G-19)})$

  * •

$(W1-W2)>0.5$

  * •

$(W1-W2)>(1.0-0.125(G-W1))$

  * •

$G\leq 26.46-5.991(BP-RP)+1.313(BP-RP)^{2}-0.07856(BP-RP)^{3}$

where $\pi$, $\mu_{\alpha}$, $\mu_{\delta}$ $\sigma(\pi)$,
$\sigma(\mu_{\alpha})$, $\sigma(\mu_{\delta})$ are the parallax, proper
motion, and associated uncertainties from the Gaia DR2 catalog; $G$, $BP$,
$RP$ are the Gaia DR2 mean photometric magnitudes; $W1$, $W2$ are the WISE
channel 1 and 2 magnitudes from the unWISE catalogs; and AEN is the
Astrometric Excess Noise parameter from the Gaia DR2 catalog.

The first three criteria are used to distinguish QSOs from Milky Way stars on
the basis of parallaxes and proper motions consistent with zero in the Gaia
DR2 catalog; these are generalized versions of the criteria used by van der
Marel et al. (2019) to select stars in M31 from the Gaia catalog. The AEN
criterion is the same used by the DESI program to separate point sources from
extended sources (i.e., galaxies) for Gaia DR2666See
https://github.com/desihub/desitarget/blob/2.5.0/py/desitarget/gaiamatch.py#L207-L210.
The $(W1-W2)>0.5$ criterion is a more relaxed version of the WISE AGN
selection discussed in Stern et al. (2012). The Gaia \- WISE criteria were
determined based on identifying the known spectroscopically confirmed QSOs
(from Massey et al., 2019; Huo et al., 2010, 2013, 2015) in $G-W1-W2$ color-
color space. Finally, the Gaia $G-BP-RP$ color criterion is an attempt to
avoid stars from the M31 RGB in the QSO selection (see Figure 20).

Figure 21: Left: Redshift distribution of the spectroscopically confirmed QSOs
(blue hashed histogram) and galaxies (red hashed histogram). Right: The sky
distribution of the spectroscopically-confirmed QSOs.

183 QSO candidates were targeted successfully (i.e., without fiber positioning
errors) on the three DESI tiles discussed in this paper, 172 of which were
spectroscopically confirmed as QSOs. The remaining 11 includes 8 stars and 3
galaxies. This represents a $\approx$94% success rate in the QSO selection
criteria. In addition, 12 of our M31 stellar candidates turned out to be QSOs,
and 683 were background galaxies. The spectroscopically confirmed QSOs and
galaxies are presented in Tables 5 and 6 respectively. Figure 21 shows the
redshift distribution of the extragalactic sources and the sky distribution of
the QSOs.

These targets are useful probes of the interstellar and circumgalactic media
of M31, and in particular provide a way of investigating any gas that may be
associated with the various kinematic structures traced by the stellar debris
(e.g., Koch et al., 2015).

Table 5: QSOs Behind M31aaSee Online Version for complete Table. The columns
are: (1) a running index; (2,3) J2000 RA and Dec in decimal degrees; (4)
Redshift; (5) Gaia $G$-band magnitude from Gaia DR3 (NaN if not available);
(6,7) the PAndAS $g$ and $i$ magnitude (NaN if not available); (8) Gaia DR3
identifier.

ID | RA (∘) | Dec (∘) | Redshift | $G_{\rm Gaia}$ | $g_{\rm PAndAS}$ | $i_{\rm PAndAS}$ | Alternate Name
---|---|---|---|---|---|---|---
1 | 10.0373918 | 40.1050291 | 2.196 | 19.88 | 19.68 | 19.46 | Gaia DR3 369102106371697792
2 | 10.3592920 | 40.8907780 | 1.159 | 19.96 | 19.87 | 19.27 | Gaia DR3 381161584267072896
3 | 10.0770974 | 39.8989696 | 0.284 | 19.76 | 19.90 | 19.54 | Gaia DR3 368710061756016896
4 | 10.4851362 | 39.9701449 | 1.834 | 19.50 | 19.84 | 19.42 | Gaia DR3 369044622529595520
5 | 10.0417886 | 39.7983967 | 0.675 | 20.06 | NaN | NaN | Gaia DR3 368707720998989184
6 | 11.2855672 | 37.7439449 | 1.934 | 19.90 | 20.38 | 20.00 | Gaia DR3 367495681227400192
7 | 10.8090715 | 37.6107886 | 2.489 | 19.61 | 19.98 | 19.86 | Gaia DR3 367443454424848768
8 | 11.0832708 | 37.6063001 | 2.533 | 19.04 | 19.48 | 19.19 | Gaia DR3 367490218028868352
9 | 11.2844517 | 37.5931928 | 1.287 | 20.28 | 20.74 | 20.23 | Gaia DR3 367492142174203904
10 | 10.8512314 | 37.7872685 | 2.199 | 18.98 | NaN | NaN | Gaia DR3 367543651717272064

Note. — Table 5 is published in its entirety in the machine-readable format. A
portion is shown here for guidance regarding its form and content.

Table 6: Galaxies Behind M31aaSee Online Version for complete Table. The
columns are: (1) a running index; (2,3) J2000 RA and Dec in decimal degrees;
(4) Redshift; (5,6) the PAndAS $g$ and $i$ magnitude (NaN if not available);
(7) Alternate name, where available.

ID | RA (∘) | Dec (∘) | Redshift | $g_{\rm PAndAS}$ | $i_{\rm PAndAS}$ | Alternate Name
---|---|---|---|---|---|---
1 | 10.2211266 | 40.0121625 | 0.283 | NaN | NaN | GC7461,SK090C ,SK090C
2 | 9.7017682 | 40.0505458 | 0.135 | NaN | NaN | GC7429,SK058C ,SK058C
3 | 10.3558474 | 40.5148375 | 0.236 | NaN | NaN | GC7191,SK078B ,SK078B
4 | 9.8651259 | 39.8292622 | 0.749 | 24.41 | 21.92 | PANDAS 95102
5 | 10.6379474 | 40.0817180 | 0.212 | NaN | NaN | GC7481,SK110C ,SK110C
6 | 10.8841125 | 37.6855167 | 0.560 | 23.81 | 21.52 | PSUPP 20989
7 | 11.0291593 | 37.6711566 | 0.676 | 24.29 | 21.83 | PANDAS 90320
8 | 11.0145968 | 37.7888733 | 0.765 | 24.67 | 22.08 | PANDAS 90419
9 | 11.0623134 | 37.7701455 | 0.674 | 24.99 | 21.89 | PANDAS 90457
10 | 11.1578093 | 37.8240122 | 0.801 | 24.54 | 22.03 | PANDAS 90472

Note. — Table 6 is published in its entirety in the machine-readable format. A
portion is shown here for guidance regarding its form and content.

##  Appendix B N body simulation details

For the simulations described in Section 4.2, we use a coordinate system that
is aligned with the disk of M31, such that $x_{\rm M31}$ and $y_{\rm M31}$ are
along the projected major and minor axes of the galaxy. The $x_{\rm M31}$ axis
is oriented approximately towards the northeast in the plane of sky; the
$y_{\rm M31}$ points out of the plane of the sky towards us but in a southeast
direction; and the $z_{\rm M31}$ axis is perpendicular to the M31 disk plane,
pointing in a northwesterly direction and tilted out of the plane of the sky
slightly towards us. The transformation between the M31 aligned coordinate
system and a sky-oriented coordinate system in which $x$ is pointing east, $y$
is pointing north, and $z$ is pointing away from us along the line of sight
can be done with the matrix $M$ (constructed assuming the position angle of
the M31 line of nodes of 37∘ and and inclination of 77∘):

$M=\begin{pmatrix}0.60181502&-0.1796539&-0.77816653\\\
0.79863551&0.13537892&0.58639054\\\ 0.&-0.97437006&0.22495105\end{pmatrix}$

We start the simulation with the progenitor at $X_{\rm
M31}=(-5.44,22.5,35.25)$ kpc with velocity $V_{\rm M31}=(19.66,-28.79,-64.68)$
${\rm km\,s^{-1}}$ (in the coordinate system aligned with the disk), where the
initial coordinates and velocities are taken from (Kirihara et al., 2017) and
rotated using the matrix $M$.

We run the model for 977 Myr using the gyrfalcON integrator (Dehnen, 2000,
2014) from the NEMO software package (Teuben, 1995) using the following
command:

    mkplum - 300000 r_s=1 seed=1 mass=9000 | snapshift rshift=-5.44,22.5,35.25
    vshift=19.66,-28.79,-64.68  in=- out=- | gyrfalcON - out.snp
    accname=nfw,miyamoto,miyamoto,miyamoto,hernquist
    accpars=’0,7.63,215;10.68,.72,3.07e5;22.99,.72,-2e5;3.49,0.72,.329e5;0,1.39e5,.6’
    tstop=1 kmax=18 eps=0.1 step=0.001

Some 1000 snapshots of the simulation made are provided on zenodo at
https://doi.org/10.5281/zenodo.6977494. After running the simulations, we
convert the outputs back into the space of observables, i.e., the coordinate
system aligned with the sky by applying the inverse rotation matrix and
assuming that M31 is at a distance of 750 kpc. We also compute the energies of
each particle using the gala package (Price-Whelan, 2017).

Figure 22 shows the resulting energy as a function of projected distance for
the particles in the simulation associated with the structure which matches
M31’s Giant Stellar Stream. In the range between 1∘ and 4∘, where we fit for
the GSS in § 5.2, the total energy is approximately linear with radius.

Figure 22: Total energy vs. projected galactocentric radius for the simulation discussed in Appendix B and § 4. The solid red line represents a slope of $\approx-380\leavevmode\nobreak\ ({\rm km\,s^{-1})^{2}\,kpc^{-1}}$. This shows that a linear relationship between energy and projected radius is reasonable approximation for projected radii $>1.5^{\circ}$. Table 7: GSS radial velocity measurements R | $V_{GSR}$ \- $V_{GSR,M31}$ | $\sigma_{V}$
---|---|---
deg | km/s | km/s
1.00 | -269.23 | 7.01
1.35 | -225.67 | 4.20
1.70 | -197.83 | 2.86
2.05 | -169.69 | 2.78
2.40 | -148.52 | 2.57
2.75 | -132.03 | 2.49
3.10 | -109.78 | 2.17
3.45 | -92.01 | 2.29
3.80 | -71.10 | 3.67

##  Appendix C DESI M31 Measurement Catalog

The measurements resulting from our analysis of the DESI spectroscopic
observations are available as a FITS data table
(DESI_M31_MEASUREMENT_CATALOG.fits). Table 8 contains a description of the
columns in the FITS data table. The data table and other information are also
available online at https://doi.org/10.5281/zenodo.6977494.

Table 8: Data Model for the M31 Data FITS File Column Name | Data Type | Units | Column Description
---|---|---|---
TARGETID | LONG64 | - | DESI target identification
RA | DOUBLE | deg | Right Ascention (J2000) in decimal degrees
DEC | DOUBLE | deg | Declination (J2000) in decimal degrees
DUPLICATE | LONG64 | - | DESI TARGETID of the duplicate entry for this source if one exists; 0 otherwise
PRIMARY | INT | - | 1 if this is the primary entry; 0 otherwise; only for duplicate entries
VRAD_BEST | DOUBLE | ${\rm km\,s^{-1}}$ | ”Best” heliocentric radial velocity in ${\rm km\,s^{-1}}$
VRAD_BEST_ERR | DOUBLE | ${\rm km\,s^{-1}}$ | Uncertainty on ”best” heliocentric radial velocity in ${\rm km\,s^{-1}}$
VRAD | DOUBLE | ${\rm km\,s^{-1}}$ | Heliocentric radial velocity from DESI’s RV Pipeline in ${\rm km\,s^{-1}}$
VRAD_ERR | DOUBLE | ${\rm km\,s^{-1}}$ | Uncertainty on heliocentric radial velocity from DESI’s RV Pipeline in ${\rm km\,s^{-1}}$
VRAD_SKEW | DOUBLE | ${\rm km\,s^{-1}}$ | Skew on the heliocentric radial velocity from DESI’s RV Pipeline in ${\rm km\,s^{-1}}$
VRAD_KURT | DOUBLE | ${\rm km\,s^{-1}}$ | Kurtosis on the heliocentric radial velocity from DESI’s RV Pipeline in ${\rm km\,s^{-1}}$
LOGG | DOUBLE | ${\rm log(cm\,s^{-2})}$ | Logarithm of the acceleration due to gravity from DESI’s RV Pipeline in ${\rm log(cm\,s^{-2})}$
TEFF | DOUBLE | K | Effective temperature from DESI’s RV Pipeline in degrees Kelvin
FEH | DOUBLE | dex | [Fe/H] in units of solar metallicity
LOGG_ERR | DOUBLE | ${\rm log(cm\,s^{-2})}$ | Uncertainty on log(g)
TEFF_ERR | DOUBLE | K | Uncertainty on Teff (K)
FEH_ERR | DOUBLE | dex | Uncertainty on [Fe/H] (dex)
CLASS | STRING | - | Selection class(es) for the target
DESIGNATION | STRING | - | Alternate designation of the target
SPECTYPE | STRING | - | Spectroscopic type: STAR, GALAXY or QSO; blank if unknown
GDR3_ID | LONG64 | - | Gaia DR3 unique identification number
TILEID | STRING | - | DESI Tiles contributing to the spectrum
SN_B | FLOAT | - | Signal-to-noise ratio in the blue arm of the DESI spectrograph
SN_R | FLOAT | - | Signal-to-noise ratio in the red arm of the DESI spectrograph
SN_Z | FLOAT | - | Signal-to-noise ratio in the near-infrared arm of the DESI spectrograph
VRAD_REDROCK | DOUBLE | ${\rm km\,s^{-1}}$ | Heliocentric radial velocity from the DESI Redrock pipeline in ${\rm km\,s^{-1}}$
VRAD_REDROCK_ERR | DOUBLE | ${\rm km\,s^{-1}}$ | Uncertainty on the heliocentric radial velocity from the DESI Redrock pipeline in ${\rm km\,s^{-1}}$
REDROCK_ZWARN | LONG64 | - | Warning flags for Redrock pipeline reductions (0=good)
RVS_WARN | INT | - | Warning flags for the RVS pipeline reductions (0=good)
VRAD_VI | DOUBLE | ${\rm km\,s^{-1}}$ | Heliocentric radial velocity from the Visual Inspection in ${\rm km\,s^{-1}}$
VI_QUALITY | INT | - | Quality flag from the Visual Inspection ($\geq$3 is good; $\leq$2 is bad; $<$0 is not inspected)
VI_SPECTYPE | STRING | - | Spectral type from the Visual Inspection
M31DIST | DOUBLE | deg | Projected radial distance from the center of M31aaThe center of M31 is assumed to lie at RA=10.68470833∘ and DEC=41.26875∘. in degrees
GAIA_G | FLOAT | mag | Gaia $G$ magnitude from the Gaia DR3 catalog
PANDAS_G | FLOAT | mag | PAndAS $g$-band magnitude from the PAndAS catalog
PANDAS_I | FLOAT | mag | PAndAS $i$-band magnitude from the PAndAS catalog
PRED_Z | FLOAT | mag | Predicted Legacy Surveys $z$-band magnitude
XI | DOUBLE | deg | Standard $\xi$ coordinate in frame centered on M31aaThe center of M31 is assumed to lie at RA=10.68470833∘ and DEC=41.26875∘. in degrees
ETA | DOUBLE | deg | Standard $\eta$ coordinate in frame centered on M31aaThe center of M31 is assumed to lie at RA=10.68470833∘ and DEC=41.26875∘. in degrees
THETA | DOUBLE | deg | Polar angle centered on M31 and measured clockwise from the west (as in Figures 6,7,11, and 12)