Revised and New Proper Motions for Confirmed and Candidate Milky Way Dwarf Galaxies

We use the updated and curated list of nearby galaxies from McConnachie (2012) to define our target sample.³ This list contains both confirmed dwarf galaxies and possible candidates. The latter category includes systems that may actually be globular clusters (such as Eridanus 3; see Conn et al. 2018a), or objects that may not actually be stellar systems (e.g., see the discussion of Tucana 5 and Cetus 2 in Simon et al. 2020 and Conn et al. 2018b, respectively). We stress that it is not the purpose of this paper to discuss the reality or otherwise of all these systems, but rather to determine if a systemic proper motion can be derived based on the assumption that they are real stellar systems.

We consider all galaxies within 450 kpc of the Sun (i.e., out to and including Leo T). We do not consider the Magellanic Clouds; excellent and comprehensive studies of the proper motions of these galaxies can be found in Gaia Collaboration et al. (2018c) as well as Kallivayalil et al. (2013) and related work. Similarly, we do not consider the Sagitarrius dwarf galaxy, whose stars extend across the entire sky and which has recently been explored using Gaia DR2 by Ibata et al. (2020). Further, we do not consider the Canis Major (Martin et al. 2004), Hydra 1 (Grillmair 2011), and Bootes 3 (Grillmair 2009) stellar overdensities. The nature of these structures, especially for the first two, remains uncertain (but see Hargis et al. 2016). For the latter, Bootes 3 is likely the projenitor of the Styx stellar stream (Carlin & Sand 2018). Reliable structural parameters are not available for any of these three systems; these parameters are a prerequisite for our method.

Table 1 gives a list of all Milky Way satellites considered in this paper, along with positional information, structural parameters, mean metallicities, and radial velocity information, where these data exist. Figure 1 shows an Aitoff projection of the spatial distribution of all the satellites in equatorial coordinates.

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Table 1.  All Milky Way Dwarf Galaxies and Candidates that Are Considered in This Paper, along with Relevant Positional, Structural, Radial Velocity, and Metallicity Parameters

Galaxy	R.A. (deg)	Decl. (deg)	${(m-M)}_{0}$	rh (arcmin)	$e=1-b/a$	θ°	vh (km s−1)	σv (km s−1)	$\langle [\mathrm{Fe}/{\rm{H}}]\rangle$
Antlia2	143.8867	−36.7672	${20.6}_{-0.11}^{+0.11}$	${76.2}_{-7.2}^{+7.2}$	${0.38}_{-0.08}^{+0.08}$	${156.0}_{-6.0}^{+6.0}$	${290.7}_{-0.5}^{+0.5}$	${5.71}_{-1.08}^{+1.08}$	$-{1.36}_{-0.04}^{+0.04}$
Aquarius2	338.4812	−9.3275	${20.16}_{-0.07}^{+0.07}$	${5.1}_{-0.8}^{+0.8}$	${0.39}_{-0.09}^{+0.09}$	${121.0}_{-9.0}^{+9.0}$	$-{71.1}_{-2.5}^{+2.5}$	${5.4}_{-0.9}^{+3.4}$	$-{2.3}_{-0.5}^{+0.5}$
Bootes1	210.025	14.5	${19.11}_{-0.08}^{+0.08}$	${11.26}_{-0.27}^{+0.27}$	${0.25}_{-0.02}^{+0.02}$	${7.0}_{-3.0}^{+3.0}$	${99.0}_{-2.1}^{+2.1}$	${2.4}_{-0.5}^{+0.9}$	$-{2.55}_{-0.11}^{+0.11}$
Bootes2	209.5	12.85	${18.1}_{-0.06}^{+0.06}$	${3.05}_{-0.45}^{+0.45}$	${0.24}_{-0.12}^{+0.12}$	$-{70.0}_{-27.0}^{+27.0}$	$-{117.0}_{-5.2}^{+5.2}$	${10.5}_{-7.4}^{+7.4}$	$-{1.79}_{-0.05}^{+0.05}$
Bootes4	233.6892	43.7261	${21.6}_{-0.2}^{+0.2}$	${7.6}_{-0.8}^{+0.8}$	${0.64}_{-0.05}^{+0.05}$	${3.0}_{-4.0}^{+4.0}$	⋯	⋯	⋯
CanesVenatici1	202.0146	33.5558	${21.69}_{-0.1}^{+0.1}$	${8.9}_{-0.4}^{+0.4}$	${0.39}_{-0.03}^{+0.03}$	${70.0}_{-4.0}^{+4.0}$	${30.9}_{-0.6}^{+0.6}$	${7.6}_{-0.4}^{+0.4}$	$-{1.98}_{-0.01}^{+0.01}$
CanesVenatici2	194.2917	34.3208	${21.02}_{-0.06}^{+0.06}$	${1.51}_{-0.23}^{+0.23}$	${0.46}_{-0.11}^{+0.11}$	${10.0}_{-11.0}^{+11.0}$	$-{128.9}_{-1.2}^{+1.2}$	${4.6}_{-1.0}^{+1.0}$	$-{2.21}_{-0.05}^{+0.05}$
Carina	100.4029	−50.9661	${20.11}_{-0.13}^{+0.13}$	${11.43}_{-0.12}^{+0.12}$	${0.37}_{-0.01}^{+0.01}$	${60.0}_{-1.0}^{+1.0}$	${222.9}_{-0.1}^{+0.1}$	${6.6}_{-1.2}^{+1.2}$	$-{1.72}_{-0.01}^{+0.01}$
Carina2	114.1067	−57.9992	${17.79}_{-0.05}^{+0.05}$	${8.69}_{-0.75}^{+0.75}$	${0.34}_{-0.07}^{+0.07}$	${170.0}_{-9.0}^{+9.0}$	${477.2}_{-1.2}^{+1.2}$	${3.4}_{-0.8}^{+1.2}$	$-{2.44}_{-0.09}^{+0.09}$
Carina3	114.63	−57.8997	${17.22}_{-0.1}^{+0.1}$	${3.75}_{-1.0}^{+1.0}$	${0.55}_{-0.18}^{+0.18}$	${150.0}_{-14.0}^{+14.0}$	${284.6}_{-3.1}^{+3.4}$	${5.6}_{-2.1}^{+4.3}$	$-{1.8}_{-0.2}^{+0.2}$
Centaurus1	189.585	−40.902	${20.33}_{-0.1}^{+0.1}$	${2.88}_{-0.4}^{+0.5}$	${0.4}_{-0.1}^{+0.1}$	${20.0}_{-11.0}^{+11.0}$	⋯	⋯	$-{1.8}_{-999.0}^{+999.0}$
Cetus2	19.47	−17.42	${17.1}_{-0.1}^{+0.1}$	${1.9}_{-0.5}^{+1.0}$	⋯	⋯	⋯	⋯	$-{1.28}_{-0.07}^{+0.07}$
Cetus3	31.3308	−4.27	${22.0}_{-0.1}^{+0.2}$	${1.23}_{-0.19}^{+0.42}$	${0.76}_{-0.08}^{+0.06}$	${101.0}_{-6.0}^{+5.0}$	⋯	⋯	⋯
Columba1	82.86	−28.03	${21.3}_{-0.22}^{+0.22}$	${1.9}_{-0.4}^{+0.5}$	⋯	⋯	${153.7}_{-4.8}^{+5.0}$	⋯	⋯
ComaBerenices	186.7458	23.9042	${18.2}_{-0.2}^{+0.2}$	${5.63}_{-0.3}^{+0.3}$	${0.37}_{-0.05}^{+0.05}$	$-{58.0}_{-4.0}^{+4.0}$	${98.1}_{-0.9}^{+0.9}$	${4.6}_{-0.8}^{+0.8}$	$-{2.6}_{-0.05}^{+0.05}$
Crater2	177.31	−18.4131	${20.35}_{-0.02}^{+0.02}$	${31.2}_{-2.5}^{+2.5}$	⋯	⋯	${87.5}_{-0.4}^{+0.4}$	${2.7}_{-0.3}^{+0.3}$	$-{1.98}_{-0.1}^{+0.1}$
DES J0225+0304	36.4267	3.0695	${16.88}_{-0.05}^{+0.06}$	${2.68}_{-0.7}^{+1.33}$	${0.61}_{-0.23}^{+0.14}$	${31.25}_{-13.39}^{+11.48}$	⋯	⋯	$-{1.26}_{-0.03}^{+0.03}$
Draco	260.0517	57.9153	${19.4}_{-0.17}^{+0.17}$	${9.93}_{-0.09}^{+0.09}$	${0.3}_{-0.01}^{+0.01}$	${87.0}_{-1.0}^{+1.0}$	$-{291.0}_{-0.1}^{+0.1}$	${9.1}_{-1.2}^{+1.2}$	$-{1.93}_{-0.01}^{+0.01}$
Draco2	238.1983	64.5653	${16.67}_{-0.05}^{+0.05}$	${3.0}_{-0.5}^{+0.7}$	${0.23}_{-0.15}^{+0.15}$	${76.0}_{-32.0}^{+22.0}$	$-{342.5}_{-1.2}^{+1.1}$	⋯	$-{2.7}_{-0.1}^{+0.1}$
Eridanus2	56.0879	−43.5333	${22.9}_{-0.2}^{+0.2}$	${1.81}_{-0.17}^{+0.17}$	${0.37}_{-0.06}^{+0.06}$	${82.0}_{-7.0}^{+7.0}$	${75.6}_{-3.3}^{+3.3}$	${6.9}_{-0.9}^{+1.2}$	$-{2.38}_{-0.13}^{+0.13}$
Eridanus3	35.6896	−52.2836	${19.7}_{-0.2}^{+0.2}$	${0.29}_{-0.23}^{+0.23}$	${0.32}_{-0.13}^{+0.13}$	${62.0}_{-11.0}^{+11.0}$	⋯	⋯	⋯
Fornax	39.9971	−34.4492	${20.84}_{-0.18}^{+0.18}$	${18.4}_{-0.2}^{+0.2}$	${0.28}_{-0.01}^{+0.01}$	${46.0}_{-1.0}^{+1.0}$	${55.3}_{-0.1}^{+0.1}$	${11.7}_{-0.9}^{+0.9}$	$-{0.99}_{-0.01}^{+0.01}$
Grus1	344.1767	−50.1633	${20.4}_{-0.2}^{+0.2}$	${2.08}_{-0.87}^{+0.87}$	${0.54}_{-0.26}^{+0.26}$	${11.0}_{-32.0}^{+32.0}$	$-{140.5}_{-1.6}^{+2.4}$	9.8	$-{1.42}_{-0.42}^{+0.55}$
Grus2	331.02	−46.44	${18.62}_{-0.21}^{+0.21}$	${6.0}_{-0.5}^{+0.9}$	⋯	⋯	$-{110.0}_{-0.5}^{+0.5}$	⋯	$-{2.51}_{-0.11}^{+0.11}$
Hercules	247.7583	12.7917	${20.6}_{-0.2}^{+0.2}$	${5.99}_{-0.58}^{+0.58}$	${0.69}_{-0.04}^{+0.04}$	$-{73.0}_{-2.0}^{+2.0}$	${45.2}_{-1.1}^{+1.1}$	${3.7}_{-0.9}^{+0.9}$	$-{2.41}_{-0.04}^{+0.04}$
Horologium1	43.8821	−54.1189	${19.5}_{-0.2}^{+0.2}$	${1.54}_{-0.34}^{+0.34}$	${0.31}_{-0.16}^{+0.16}$	${50.0}_{-26.0}^{+26.0}$	${112.8}_{-2.6}^{+2.5}$	${4.9}_{-0.9}^{+2.8}$	$-{2.76}_{-0.1}^{+0.1}$
Horologium2	49.1338	−50.0181	${19.46}_{-0.2}^{+0.2}$	${2.83}_{-1.31}^{+1.31}$	${0.86}_{-0.19}^{+0.19}$	${130.0}_{-16.0}^{+16.0}$	${168.7}_{-12.6}^{+12.9}$	⋯	−2.1
Hydra2	185.4254	−31.9853	${20.64}_{-0.16}^{+0.16}$	${1.5}_{-0.32}^{+0.32}$	${0.17}_{-0.13}^{+0.13}$	${29.0}_{-25.0}^{+25.0}$	${303.1}_{-1.4}^{+1.4}$	⋯	$-{2.02}_{-0.08}^{+0.08}$
Hydrus1	37.3892	−79.3089	${17.2}_{-0.04}^{+0.04}$	${7.42}_{-0.54}^{+0.62}$	${0.21}_{-0.07}^{+0.15}$	${97.0}_{-14.0}^{+14.0}$	${80.4}_{-0.6}^{+0.6}$	${2.69}_{-0.43}^{+0.51}$	$-{2.52}_{-0.09}^{+0.09}$
Indus1	317.2046	−51.1656	${20.0}_{-0.2}^{+0.2}$	${0.87}_{-0.45}^{+0.45}$	${0.72}_{-0.29}^{+0.29}$	${5.0}_{-20.0}^{+20.0}$	⋯	⋯	⋯
Indus2	309.72	−46.16	${21.65}_{-0.16}^{+0.16}$	${2.9}_{-1.0}^{+1.1}$	⋯	⋯	⋯	⋯	⋯
Leo1	152.1171	12.3064	${22.02}_{-0.13}^{+0.13}$	${3.3}_{-0.03}^{+0.03}$	${0.3}_{-0.01}^{+0.01}$	${78.0}_{-1.0}^{+1.0}$	${282.5}_{-0.1}^{+0.1}$	${9.2}_{-1.4}^{+1.4}$	$-{1.43}_{-0.01}^{+0.01}$
Leo2	168.37	22.1517	${21.84}_{-0.13}^{+0.13}$	${2.48}_{-0.03}^{+0.03}$	${0.07}_{-0.02}^{+0.02}$	${43.0}_{-8.0}^{+8.0}$	${78.0}_{-0.1}^{+0.1}$	${6.6}_{-0.7}^{+0.7}$	$-{1.62}_{-0.01}^{+0.01}$
Leo4	173.2375	−0.5333	${20.94}_{-0.09}^{+0.09}$	${2.61}_{-0.31}^{+0.31}$	${0.19}_{-0.09}^{+0.09}$	$-{29.0}_{-27.0}^{+27.0}$	${132.3}_{-1.4}^{+1.4}$	${3.3}_{-1.7}^{+1.7}$	$-{2.54}_{-0.07}^{+0.07}$
Leo5	172.79	2.22	${21.46}_{-0.16}^{+0.16}$	${1.0}_{-0.22}^{+0.22}$	${0.35}_{-0.07}^{+0.07}$	$-{65.0}_{-21.0}^{+21.0}$	${173.3}_{-3.1}^{+3.1}$	${3.7}_{-1.4}^{+2.3}$	$-{2.0}_{-0.2}^{+0.2}$
LeoT	143.7225	17.0514	${23.1}_{-0.1}^{+0.1}$	${1.25}_{-0.14}^{+0.14}$	${0.23}_{-0.09}^{+0.09}$	$-{107.0}_{-16.0}^{+16.0}$	${38.1}_{-2.0}^{+2.0}$	${7.5}_{-1.6}^{+1.6}$	$-{2.02}_{-0.05}^{+0.05}$
Pegasus3	336.0942	5.42	${21.56}_{-0.2}^{+0.2}$	${1.3}_{-0.4}^{+0.5}$	${0.46}_{-0.26}^{+0.18}$	${133.0}_{-17.0}^{+17.0}$	$-{222.9}_{-2.6}^{+2.6}$	${5.4}_{-2.5}^{+3.0}$	−2.1
Phoenix	27.7762	−44.4447	${23.06}_{-0.12}^{+0.12}$	${2.3}_{-0.07}^{+0.07}$	${0.3}_{-0.03}^{+0.03}$	${8.0}_{-4.0}^{+4.0}$	$-{21.2}_{-1.0}^{+1.0}$	${9.3}_{-0.7}^{+0.7}$	$-{1.49}_{-0.04}^{+0.04}$
Phoenix2	354.9975	−54.4061	${19.6}_{-0.2}^{+0.2}$	${1.61}_{-0.27}^{+0.27}$	${0.61}_{-0.15}^{+0.15}$	$-{19.0}_{-14.0}^{+14.0}$	${32.4}_{-3.8}^{+3.7}$	${11.0}_{-5.3}^{+9.4}$	⋯
Pictor1	70.9475	−50.2831	${20.3}_{-0.2}^{+0.2}$	${0.66}_{-0.32}^{+0.32}$	${0.24}_{-0.19}^{+0.19}$	${72.0}_{-10.0}^{+10.0}$	⋯	⋯	⋯
Pictor2	101.18	−59.8969	${18.3}_{-0.15}^{+0.12}$	${3.8}_{-1.0}^{+1.5}$	${0.13}_{-0.13}^{+0.22}$	${14.0}_{-66.0}^{+60.0}$	⋯	⋯	$-{1.8}_{-0.3}^{+0.3}$
Pisces2	344.6292	5.9525	${21.31}_{-0.17}^{+0.17}$	${1.22}_{-0.2}^{+0.2}$	${0.4}_{-0.1}^{+0.1}$	${99.0}_{-13.0}^{+13.0}$	$-{226.5}_{-2.7}^{+2.7}$	${5.4}_{-2.4}^{+3.6}$	$-{2.45}_{-0.07}^{+0.07}$
Reticulum2	53.9254	−54.0492	${17.4}_{-0.2}^{+0.2}$	${5.59}_{-0.21}^{+0.21}$	${0.56}_{-0.03}^{+0.03}$	${69.0}_{-2.0}^{+2.0}$	${64.7}_{-0.8}^{+1.3}$	${3.22}_{-0.49}^{+1.64}$	$-{2.46}_{-0.1}^{+0.09}$
Reticulum3	56.36	−60.45	${19.81}_{-0.31}^{+0.31}$	${2.4}_{-0.8}^{+0.9}$	⋯	⋯	${274.2}_{-7.4}^{+7.5}$	⋯	⋯
Sagittarius2	298.1688	−22.0681	${19.32}_{-0.02}^{+0.03}$	${1.7}_{-0.05}^{+0.05}$	${0.06}_{-0.06}^{+0.06}$	${103.0}_{-17.0}^{+28.0}$	$-{177.3}_{-1.2}^{+1.2}$	${2.7}_{-1.0}^{+1.3}$	$-{2.28}_{-0.03}^{+0.03}$
Sculptor	15.0392	−33.7092	${19.67}_{-0.14}^{+0.14}$	${12.33}_{-0.05}^{+0.05}$	${0.37}_{-0.01}^{+0.01}$	${94.0}_{-1.0}^{+1.0}$	${111.4}_{-0.1}^{+0.1}$	${9.2}_{-1.4}^{+1.4}$	$-{1.68}_{-0.01}^{+0.01}$
Segue1	151.7667	16.0819	${16.8}_{-0.2}^{+0.2}$	${3.95}_{-0.48}^{+0.48}$	${0.34}_{-0.11}^{+0.11}$	${75.0}_{-16.0}^{+16.0}$	${208.5}_{-0.9}^{+0.9}$	${3.9}_{-0.8}^{+0.8}$	$-{2.72}_{-0.4}^{+0.4}$
Segue2	34.8167	20.1753	${17.7}_{-0.1}^{+0.1}$	${3.64}_{-0.29}^{+0.29}$	${0.21}_{-0.07}^{+0.07}$	${166.0}_{-15.0}^{+15.0}$	$-{39.2}_{-2.5}^{+2.5}$	${3.4}_{-1.2}^{+2.5}$	$-{2.22}_{-0.13}^{+0.13}$
Sextans1	153.2625	−1.6147	${19.67}_{-0.1}^{+0.1}$	${27.8}_{-1.2}^{+1.2}$	${0.35}_{-0.05}^{+0.05}$	${56.0}_{-5.0}^{+5.0}$	${224.2}_{-0.1}^{+0.1}$	${7.9}_{-1.3}^{+1.3}$	$-{1.93}_{-0.01}^{+0.01}$
Triangulum2	33.3225	36.1783	${17.4}_{-0.1}^{+0.1}$	${3.9}_{-0.9}^{+1.1}$	${0.21}_{-0.21}^{+0.17}$	${56.0}_{-24.0}^{+16.0}$	$-{381.7}_{-1.1}^{+1.1}$	⋯	$-{2.24}_{-0.05}^{+0.05}$
Tucana2	342.9796	−58.5689	${18.8}_{-0.2}^{+0.2}$	${9.83}_{-1.11}^{+1.66}$	${0.39}_{-0.2}^{+0.1}$	${107.0}_{-18.0}^{+18.0}$	$-{129.1}_{-3.5}^{+3.5}$	${8.6}_{-2.7}^{+4.4}$	$-{2.23}_{-0.12}^{+0.18}$
Tucana3	359.15	−59.6	${17.01}_{-0.16}^{+0.16}$	${6.0}_{-0.6}^{+0.8}$	⋯	⋯	$-{102.3}_{-2.4}^{+2.4}$	${0.1}_{-0.1}^{+0.7}$	$-{2.42}_{-0.08}^{+0.07}$
Tucana4	0.73	−60.85	${18.41}_{-0.19}^{+0.19}$	${9.3}_{-0.9}^{+1.4}$	${0.39}_{-0.1}^{+0.07}$	${27.0}_{-8.0}^{+9.0}$	${15.9}_{-1.7}^{+1.8}$	${4.3}_{-1.0}^{+1.7}$	$-{2.49}_{-0.16}^{+0.15}$
Tucana5	354.35	−63.27	${18.71}_{-0.34}^{+0.34}$	${2.1}_{-0.4}^{+0.6}$	${0.51}_{-0.18}^{+0.09}$	${29.0}_{-11.0}^{+11.0}$	$-{36.3}_{-2.2}^{+2.5}$	⋯	$-{2.17}_{-0.23}^{+0.23}$
UrsaMajor1	158.72	51.92	${19.93}_{-0.1}^{+0.1}$	${8.34}_{-0.34}^{+0.34}$	${0.57}_{-0.03}^{+0.03}$	${67.0}_{-2.0}^{+2.0}$	$-{55.3}_{-1.4}^{+1.4}$	${7.6}_{-1.0}^{+1.0}$	$-{2.18}_{-0.04}^{+0.04}$
UrsaMajor2	132.875	63.13	${17.5}_{-0.3}^{+0.3}$	${13.95}_{-0.46}^{+0.46}$	${0.56}_{-0.03}^{+0.03}$	$-{77.0}_{-2.0}^{+2.0}$	$-{116.5}_{-1.9}^{+1.9}$	${6.7}_{-1.4}^{+1.4}$	$-{2.47}_{-0.06}^{+0.06}$
UrsaMinor	227.2854	67.2225	${19.4}_{-0.1}^{+0.1}$	${17.32}_{-0.11}^{+0.11}$	${0.55}_{-0.01}^{+0.01}$	${50.0}_{-1.0}^{+1.0}$	$-{246.9}_{-0.1}^{+0.1}$	${9.5}_{-1.2}^{+1.2}$	$-{2.13}_{-0.01}^{+0.01}$
Virgo1	180.0379	0.681	${19.8}_{-0.1}^{+0.2}$	${1.76}_{-0.4}^{+0.49}$	${0.59}_{-0.14}^{+0.12}$	${62.0}_{-13.0}^{+8.0}$	⋯	⋯	⋯
Willman1	162.3375	51.05	${17.9}_{-0.4}^{+0.4}$	${2.53}_{-0.22}^{+0.22}$	${0.47}_{-0.06}^{+0.06}$	${74.0}_{-4.0}^{+4.0}$	$-{12.3}_{-2.5}^{+2.5}$	${4.3}_{-1.3}^{+2.3}$	−2.1

Note. Objects in italics are those satellites for which we are unable to derive proper motions.

Download table as:  ASCIITypeset images: 1 2

In this era of Gaia (Gaia Collaboration et al. 2016), any star that is bright enough to be detected by the spacecraft has information available regarding its spatial position (ξ, η)_i, color and magnitude ${(G,{BP}-{RP})}_{i}$, and proper motion ${({\mu }_{\alpha }\cos \delta ,{\mu }_{\delta })}_{i}$. Note that (ξ, η)_i are the coordinates of star i projected on a tangent plane to the celestial sphere, and these are a function of the R.A. and decl. (α, δ)_i as measured by Gaia.

For every satellite listed in Table 1, we consider all stellar sources in Gaia DR2 (Gaia Collaboration et al. 2018b) that are within 2° of the center of the galaxy. In 53 out of the 59 cases, this area corresponds to a much larger area than subtended by the dwarf itself (i.e., hundreds or more times its half-light radius). However, for six of the larger (in terms of subtended area) satellites, we consider sources within 4–8 degrees from their centers (Antlia 2, Carina, Crater 2, Fornax, Sextans 1, and Ursa Minor).

As the color–magnitude distribution of sources is critical to our method, we require the $G,{BP},{RP}$ photometry to be reliable, and therefore follow Lindegren et al. (2018; their Equation (C.2)) to consider only stars that meet the following criterion:

$$\begin{eqnarray}&&1.0+0.015{\left({BP}-{RP}\right)}^{2}\lt E\lt 1.3+0.06{\left({BP}-{RP}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

where E is the flux excess factor, as defined in Lindegren et al. (2018). We use dereddened Gaia DR2 stellar magnitudes in our analysis, using the extinction maps of Schlegel et al. (1998) and following the definition of the relevant extinction coefficients described in Equation (1) and Table 1 of Gaia Collaboration et al. (2018a).

We only consider stars with full five-parameter astrometric solutions and high-quality astrometry as defined via the renormalized unit-weighted error (ruwe; see Lindegren et al. 2018 and discussion in the Gaia DR2 Documentation Release 1.2). Inspection of the distribution of ruwe for sources that otherwise meet our selection criteria motivated us to adopt ruwe < 1.3.

Finally, only stars with parallaxes that are consistent with the distance to the satellite are included. Specifically, the 3σ parallax range measured by Gaia DR2 must overlap the 3σ parallax range implied by the distance modulus of the dwarf, as given in Table 1. We correct for the global zero-point of the parallax in Gaia DR2 of −0.029 mas (Lindegren et al. 2018).

Member stars in Milky Way dwarf galaxies are generally too faint to have been observed with the Gaia/RVS. Therefore, unlike photometry and proper motions, radial velocities are only available for the stars in dwarf galaxies that were specifically targeted for followed-up spectroscopy from ground-based observatories. There is a wealth of such studies in the literature; a comprehensive compilation of the known radial velocity publications for the satellites in this paper is given in Table 2 (see also Table 1 of Fritz et al. 2018). For all except the brightest objects, these references are intended to be a complete list of published radial velocity measurements for these satellites, and any omissions are unintentional.⁴ For the brightest ("classical") objects—Fornax, Ursa Minor, Carina, Sextans 1, Draco, Sculptor, Leo I, and Leo 2—the references are not comprehensive. These bright dwarfs typically have hundreds to thousands of stars with radial velocity information, compared to typically tens to hundreds of stars in the fainter systems.

For each paper in Table 2, we have cross-matched the stars with measured radial velocities with Gaia DR2. For those papers listed in Table 2 that give multiple measurements per star, we have taken the weighted mean velocity for each star. For those satellites that have been observed by multiple groups, we have combined data sets, and taken the weighted mean velocities of any stars in common.

In general, the membership of a star in a dwarf galaxy can be judged by the position of the star (is the star near the dwarf?), by the color or metallicity of the star (does the star appear to have color or metallicity properties that are consistent with the stellar populations of the dwarf?), and by the dynamics of the star (is the motion of the star consistent with the motion of the rest of the dwarf?). The latter can be broken up into both a proper motion component and a radial velocity component. All approaches to the derivation of the systemic proper motion of satellites consider some weighted combination of these criteria.

One of the ultimate science goals of our analysis is to develop comprehensive membership lists for the faint Milky Way satellites (these will be discussed in forthcoming contributions). As such, we favor approaches in the derivation of the systemic proper motions that simultaneously allow us to calculate stellar membership probabilities (as opposed to cuts, which imply either that a star is definitely a member or definitely not a member). Furthermore, we want to incorporate as much information as is available in the data, while being aware that some types of data (i.e., radial velocities) are only available for a subset of stars. We are also aware that the uncertainties on some important parameters (e.g., the projected size of the satellite) are quite uncertain in some cases.

Our adopted approach is inspired by Pace & Li (2019). As described in that paper, for any star in Gaia, it is either a member of the Milky Way satellite being studied, or it is a member of the Milky Way foreground or background. We can therefore define the total likelyhood ${ \mathcal L }$ for a star as

$$\begin{eqnarray}&&{ \mathcal L }={f}_{\mathrm{sat}}{{ \mathcal L }}_{\mathrm{sat}}+(1-{f}_{\mathrm{sat}}){{ \mathcal L }}_{\mathrm{MW}},\end{eqnarray} \tag{ 2 }$$

where ${{ \mathcal L }}_{\mathrm{sat}}$ and ${{ \mathcal L }}_{\mathrm{MW}}$ are the likelihoods for the satellite and Milky Way foreground or background, respectively, and f_sat is the fraction of stars in the satellite. ${{ \mathcal L }}_{\mathrm{sat}}$ can be broken down as

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{sat}}={{ \mathcal L }}_{s}{{ \mathcal L }}_{\mathrm{CM}}{{ \mathcal L }}_{\mathrm{PM}},\end{eqnarray} \tag{ 3 }$$

where ${{ \mathcal L }}_{s},{{ \mathcal L }}_{\mathrm{CM}}$ and ${{ \mathcal L }}_{\mathrm{PM}}$ are the likelihoods from the spatial information, color–magnitude information, and proper motion information, respectively. ${{ \mathcal L }}_{\mathrm{MW}}$ can similarly be broken down into the product of its three constituent likelihoods.

Within a Bayesian framework, the probability of the data, D, given a set of model parameters, θ, is given by

$$\begin{eqnarray}&&P\left(D| \theta \right)\propto { \mathcal L }\times P(\theta ).\end{eqnarray} \tag{ 4 }$$

P(θ) is our prior on the model parameters. We aim to determine the set of model parameters that maximizes $P\left(D| \theta \right)$.

If it were the case that all stars in our sample also had radial velocity information, then the radial velocity information could be incorporated in Equation (4) via the incorporation of a fourth term, ${{ \mathcal L }}_{\mathrm{RV}}$, in Equation (3). However, only a tiny fraction of our data has radial velocity information. As such, only the spatial, color–magnitude, and proper motion information are incorporated into Equation (4) via the likelihood, ${ \mathcal L }$. The radial velocity data, where it exists, will be incorporated into Equation (4) via the prior, P(θ).

We note that once all of the relevant likelihoods have been calculated, the probability that a star is a member of the satellite is given by

$$\begin{eqnarray}&&{P}_{\mathrm{sat}}=\displaystyle \frac{{f}_{\mathrm{sat}}{{ \mathcal L }}_{\mathrm{sat}}}{{f}_{\mathrm{sat}}{{ \mathcal L }}_{\mathrm{sat}}+(1-{f}_{\mathrm{sat}}){{ \mathcal L }}_{\mathrm{MW}}}.\end{eqnarray} \tag{ 5 }$$

Within this framework, the problem of obtaining systemic proper motions for satellites (and the related problem of identifying member stars within satellites) becomes one of defining the appropriate likelihood functions for the satellite and the background. Here, we are conscious of two driving considerations. The first is that many of the satellites are intrinsically faint, and as such, there are significant uncertainties on their basic parameters (especially distance). Thus, any model of their structural and color–magnitude properties should seek to incorporate these uncertainties. The second consideration is that the structure of the Milky Way foreground and background is, to put it mildly, immensely complex. While variations in the global structure as a function of position can potentially be parameterized (e.g., see Pace & Li 2019), variations on a smaller scale that are due to known or unknown structures or substructures are especially problematic for proper motion analyses. Thus, we make absolutely no attempt to construct a parameterized model of the foreground or background. Instead, an empirical model based entirely on the data in hand is constructed with no unknown parameters.

Our model parameters, θ, therefore consist only of the unknown systematic proper motion components, ${({\mu }_{\alpha }\cos \delta ,{\mu }_{\delta })}_{\mathrm{sat}}$, in addition to f_sat introduced in Equation (2). This parameter space is explored using emcee (Foreman-Mackey et al. 2013, 2019). We now discuss our likelihood models for each term in Equation (3), and the form of our adopted prior, P(θ).

3.2.1. Satellite

The spatial likelihood function for a star to be a member of a dwarf galaxy is determined from the structural parameterization of the host galaxy, as estimated from its resolved stellar distribution. These are generally adequately described by an exponential function, adopted here for simplicity. The necessary parameters to (completely) describe the relevant 2D exponential distributions are given in Table 1.

To account for (large) uncertainties, we construct a 2D lookup map for the satellite spatial likelihood. This is made by coadding a thousand realizations of the dwarf galaxy stellar density distribution, where each realization uses parameters drawn from Gaussian distributions, centered on the reported values for r_h, e, and θ in Table 1. The standard deviation is set by the reported uncertainties on each parameter. Systems that do not have reported values of ellipticity or position angle are assumed to be circular. We note that the centers of these satellites are assumed to be fixed, although a few of the faintest satellites have relatively large uncertainties in their positions. However, these satellites also have large uncertainties in their other structural parameters, such that inclusion of the uncertainties in their positions does not change their spatial likelihood functions significantly, and does not affect the systemic proper motions that are calculated.

An example of the resulting likelihood function is shown in the left panel of Figure 2 for DES J0225+0304. For comparison, the right panel shows what the spatial likelihood function would look like if the uncertainties had not been considered. DES J0225+0304 has relatively poorly defined structural parameters; for systems with well-measured structural parameters, the distinction between the two panels becomes negligible.

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3.2.2. Foreground or Background

3.3.1. Satellite

The majority of the satellites under consideration are dominated by relatively old, metal-poor stellar populations. Mean metallicities (estimated using a variety of techniques) exist for most of them, and so it is relatively easy to construct a simple model of the expected color–magnitude likelihood function for the satellite.

In the first instance, we adopt a 12 Gyr Padova isochrone (Girardi et al. 2002) in the Gaia photometric bands using the filter definitions from Weiler (2018). For each dwarf satellite, the mean metallicity in Table 1 is adopted, and the isochrone shifted to the appropriate distance. Post-helium flash stars are not considered. At every magnitude, the stellar population is modeled as a Gaussian function with an intrinsic full width at half maximum of 0.1 mag in color, centered on the color of the isochrone. This is combined with the mean uncertainty in color at each magnitude for the stars under consideration. For a few of the dwarfs (Carina, Fornax, Leo 1, Leo 2, and Phoenix), a younger age is adopted for the isochrone to better match the color of the red giant branch stars.

In a similar way to the spatial distribution, a 2D lookup map is constructed for the color–magnitude likelihood function using the above methodology (with stars brighter than the tip of the red giant branch having zero likelihood of being a member). However, we coadd a thousand realizations, where the isochrone is moved to a distance selected from a Gaussian distribution centered on the recorded distance modulus of the dwarf, with a standard deviation equal to the uncertainty on the distance modulus. In this way, the (sometimes significant) distance uncertainties on the ultrafaint dwarf galaxies are taken into account, which otherwise makes a single realization of color–magnitude space unsatisfactory.

The left panel of Figure 3 shows an example of the resulting satellite likelihood model for color–magnitude space for the case of DES J0225+0304. We only consider stars defined within our likelihood grid; $-0.5\lt ({BP}-{RP})\lt 2.5$ and $22\lt G\lt {G}_{\mathrm{TRGB}}+5{\sigma }_{{(m-M)}_{o}}$, where ${\sigma }_{{(m-M)}_{o}}$ is the uncertainty in the distance modulus, given in Table 1.

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Padova isochrones for metallicities below [Fe/H] = −2.19 dex were not available. As such, we adopted this, the most metal-poor isochrone, for any galaxy that was more metal-poor than this limit. This is acceptable as the change in color at these low metallicities is very small. For any satellites that lack a metallicity estimate, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-2$ was adopted. We note that it would also be possible to draw the metallicity from a Gaussian function (also for age); however, this does not have a significant impact on our the results. Only the red giant branch and upper main sequence are considered when constructing these models, and we ignore stars on the horizontal branch and asymptotic giant branch. Thus, by construction, a star lying on an isochrone at any magnitude is considered a likely member of the dwarf satellite. It is clearly possible to develop a more sophisticated model, e.g., taking into account additional stellar populations and the relative number of sources as a function of magnitude or stellar evolutionary phase; however, we did not find this to be necessary for the task in hand.

3.3.2. Foreground or Background

3.4.1. Satellite

3.4.2. Foreground or Background

A 2D lookup map for the Milky Way proper motion likelihood function is constructed using the same approach as for the Milky Way CMD likelihood function (above). Specifically, we excise the same area as before and construct a proper motion map, where each remaining star is a bivariate Gaussian, using the reported stellar proper motion uncertainties and their correlation to define the relevant covariance matrices. Figure 4 shows an example of the Milky Way proper motion likelihood function for the case of DES J0225+0304. Note that we only consider stars in the likelihood grid with an absolute proper motion lower than or equal to 10 mas yr⁻¹ in each direction. For the closest satellites in our sample, this corresponds to tangential velocities of around 1000 km s⁻¹, ensuring that the possible proper motion space for the satellite is fully explored.

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There are three unknown parameters in our model: f_sat (the fraction of stars belonging to the satellite), and the two components of the systemic proper motion of the satellite, ${\left({\mu }_{\alpha }\cos \delta ,{\mu }_{\delta }\right)}_{\mathrm{sat}}$. For the former, we adopt a uniform prior between 0 and 1.

Two priors are selected on the systemic proper motions:

• 1.  
  Prior A: The first prior assumes that the dispersion in the set of tangential velocities of the satellites is somewhat similar to the dispersion in the radial velocities of known halo-tracer populations (including the satellites). The radial velocity dispersion profile of the outer Galaxy has been quantified by numerous authors (e.g., Deason et al. 2012b; Bhattacharjee et al. 2014) and is reasonably characterized as a roughly constant velocity dispersion at σ_r  ≃  100 km s⁻¹. Therefore, we adopt a Gaussian prior on the 2D proper motion, such that it is centered on the equivalent of a Galactocentric velocity of zero (after the reflex motion of the Sun has been taken into account) and has a dispersion equivalent to 100 km s⁻¹ at the distance of the satellite. This prior is not particularly restrictive, as its primary purpose is to inhibit unphysically high tangential velocities; for example, velocities greatly in excess of the escape velocity of the Galaxy. However, it does have the effect of creating an "envelope" of ∼100 km s⁻¹ in the maximum uncertainties on the systemic proper motions that we derive, where this value is driven by the prior, not the data. In addition, we note that for the three most distant galaxies in our sample—Eridanus 2, Phoenix, and Leo T—it is not clear if these objects should trace the tangential velocity dispersion of the halo of the Milky Way. As such, for these three galaxies, we provide estimates with and without this prior.
• 2.  
  Prior B: For satellites with radial velocity follow-up of individual stars, we have additional information to assess whether the stars are members of the satellite. For those stars that appear likely members, we modify the above prior to favor values of the systemic proper motion that will increase the probability of those stars being considered members.We search for stars with radial velocities that are within 2σ of the mean radial velocity of the satellite. For the satellites in Table 1, where the mean radial velocities are known but the radial velocity dispersions are unknown, we assume σ_v = 5 km s⁻¹. Simultaneously, a 2σ cut around the isochrone in the color–magnitude likelihood is adopted, as well as a cut inside two half-light radii in the spatial likelihood map.For stars that satisfy these three criteria (radial velocity likelihood, CMD likelihood, and spatial likelihood), the weighted mean of their proper motions is calculated. One iteration of sigma-clipping is performed to remove any obviously deviant proper motions. This is in line with the approach taken by many authors to calculate the proper motions of satellites from radial velocity data (e.g., Fritz et al. 2018; Simon et al. 2020). The resulting mean and uncertainities are interpreted as a bivariate Gaussian that we dub "the velocity prior." This velocity prior is multiplied into Prior A to create Prior B. This will heavily favor results that are consistent with stars that appear to be members in all of their other characteristics, including radial velocities, while still maintaining consistency with our original prior. Importantly, stars without radial velocities still contribute to the selection of the preferred model through the likelihood function, ensuring that the full Gaia data set is still used.

4.1.1. Comparison of Results with Prior A and B

The internal consistency of our approach is examined for satellites with radial velocity information. In Figure 5, a comparison of the systemic proper motions for satellites derived with (Prior B; blue circles) and without (Prior A; red squares) radial velocity information is shown. μ_α cos δ is shown on the x-axis, and μ_δ is shown on the y-axis. Error bars show 1σ uncertainties, excluding systematic errors. The scales are the same on each axis in each panel, but are different between panels, and were chosen to encompass all of the relevant points and their uncertainties. This figure is intended to highlight the relative agreement, or otherwise, between estimates.

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Without exception, Figure 5 shows extremely good agreement between the two sets of estimates. In all cases, the sets agree to within their combined 1σ uncertainties. This is extremely encouraging and suggests that the adopted methodology is robust in the absence of radial velocity data, in line with the findings from Pace & Li (2019). This will be of increasing importance in the coming years, with the expected flood of discoveries from the Legacy Survey of Space and Time (LSST, e.g., Hargis et al. 2014). Inspection of Table 2 shows that even with the current sample, radial velocity follow-up is not immediate or complete.

In general, it appears that radial velocity information is not required to obtain a reasonable estimate of the proper motions of dwarf satellites, but it can play an important role in some cases. For example, the large error bars on the systemic proper motion for Carina 3 using Prior A is the result of a bimodal probability distribution function (PDF). Carina 3 is at low latitude (b  ≃ 18°), where there is considerable foreground contamination in the field, including the satellite Carina 2, which is responsible for the second peak in the PDF. Incorporation of a velocity prior in this case helps to break the degeneracy in the solutions, and leads to a well-defined single-peaked solution. This is demonstrated in Figure 6, which shows the corner plots for Carina 3 in the case of Prior A (left panels) and Prior B (right panels). The same is also true, to a less dramatic extent, for Segue 1, Triangulum 2, and Tucana 4. For these systems, the PDFs without velocity information show some bimodality (although the two peaks are much closer to each other than for Carina 3), but the incorporation of radial velocity information via Prior B collapses these to single-peaked solutions.

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Despite the obvious benefits, we find that radial velocity information should be used with caution for systems with only a few radial velocities. It is entirely possible for a star to have a consistent radial velocity (and color–magnitude position, and spatial location), but not actually be a member of the satellite. Indeed, it is because of this consideration that we created Prior B by multiplication of the radial velocity prior with Prior A, rather than a straight substitution of the radial velocity prior for Prior A. For a galaxy with radial velocity follow-up of a large number of stars, this is not generally an issue, but it can be a concern for systems with only a few radial velocity members. For example, the velocity prior for Horologium 2 is based on a single star with $({\mu }_{\alpha }\cos \delta ,{\mu }_{\delta })=(5.81\pm 0.97,-2.05\pm 1.58)$ mas, which passes all of our quality control criteria, including parallax, CMD location within 2σ of the isochrone, the mean radial velocity, and on-sky location within two half-light radii. As a member of Horologium 2, this star would imply a tangential velocity greatly in excess of the escape velocity of the Milky Way. However, by weighting the velocity prior with the original Prior A, our algorithm is able to converge to a more acceptable value. Horologium 2 is the most extreme example of this effect in our analysis, but the principle remains true for each of the systems under study.

4.1.2. Sanity Checks Using Confirmed Radial Velocity Members

We now turn our attention to examining the contamination and completeness fractions of this new technique. This provides an indirect test on the robustness of our proper motion estimates. For example, if this technique assigned many stars as members that cannot be members (or many stars as nonmembers that are clearly members), then this technique would not be trustworthy.

4.2.1. Contamination: Radial Velocity Nonmembers

Most stars that are identified as a member of a dwarf satellite, and which also have a radial velocity measurement, should have a radial velocity that is consistent with membership. The number of stars with deviant radial velocities helps us to estimate our contamination fraction, F_cont.

Some radial velocity information is available for 47 satellites for which we derive systemic proper motions (see Table 2) for a total of n_DR2 = 14,675 stars that are also present in Gaia DR2. Of these, n_QC = 8912 stars pass our quality cuts and lie within our likelihood grids (defined in Section 3). However, these stars are not evenly distributed across the satellites: 6525 of them are found in only 6 galaxies (Fornax, Sculptor, Carina, Draco, Ursa Minor, and Sextans). Table 5 describes the distribution of radial velocity measurements between galaxies.

Table 5.  Summary of Membership Contamination Rates Estimated from Radial Velocity Data

Galaxy	nDR2	nQC	${n}_{\mathrm{QC},\mathrm{cont}}$	nm	${n}_{m,\mathrm{cont}}$	Fcont
Overall	14675	8912	3280	5173	173	0.05
Excluding galaxies with ${n}_{m}\gt 100$:
	5595	2387	1592	462	25	0.02
Fornax	2482	1951	37	1922	28	0.76
Sculptor	1457	1243	70	1185	30	0.43
Carina	1854	1246	676	544	56	0.08
Draco	1419	950	467	414	6	0.01
UrsaMinor	958	580	172	377	4	0.02
Sextans1	910	555	266	269	24	0.09
Leo2	219	59	3	56	1	0.33
Crater2	404	189	122	55	2	0.02
CanesVenatici1	144	55	10	45	2	0.2
Bootes1	275	122	75	37	7	0.09
Hydrus1	139	95	60	30	3	0.05
Reticulum2	59	38	12	24	0	0.0
Hercules	81	39	15	18	0	0.0
Tucana3	675	294	265	18	5	0.02
Leo1	381	22	2	15	0	0.0
Antlia2	221	202	57	11	0	0.0
UrsaMajor2	128	39	24	11	2	0.08
ComaBerenices	47	21	6	10	0	0.0
Tucana2	95	53	34	10	0	0.0
UrsaMajor1	107	32	13	10	0	0.0
CanesVenatici2	41	15	6	9	0	0.0
Eridanus2	42	17	7	9	0	0.0
Phoenix	116	13	3	9	0	0.0
Draco2	44	18	11	7	0	0.0
Grus2	254	105	83	7	0	0.0
Carina2	283	217	200	6	1	0.0
Horologium1	19	12	6	6	0	0.0
Segue1	310	113	104	6	2	0.02
Sagittarius2	120	31	24	5	0	0.0
Grus1	70	21	14	5	0	0.0
Hydra2	18	10	4	5	0	0.0
LeoT	39	10	4	5	0	0.0
Bootes2	8	4	0	4	0
Phoenix2	75	18	4	4	0	0.0
Segue2	214	55	33	4	0	0.0
Columba1	49	29	21	3	0	0.0
Tucana4	209	82	58	3	0	0.0
Aquarius2	10	3	1	2	0	0.0
Carina3	283	219	213	2	0	0.0
Leo5	124	33	29	2	0	0.0
Reticulum3	45	26	22	2	0	0.0
Tucana5	29	17	12	2	0	0.0
Willman1	66	11	3	2	0	0.0
Leo4	25	5	4	1	0	0.0
Pisces2	7	4	3	1	0	0.0
Triangulum2	28	19	14	1	0	0.0
Horologium2	92	20	11	0	0	0.0

Note. n_DR2 is the number of Gaia stars with ground-based radial velocity data for each satellite; n_QC is the number of these stars that pass our quality control cuts; ${n}_{\mathrm{QC},\mathrm{cont}}$ is the number of these stars that have radial velocities more than 3σ from the mean velocity of the satellite; n_m is the number of stars that are considered high-probability members (${P}_{\mathrm{sat}}\geqslant 0.5$) by our algorithm; ${n}_{m,\mathrm{cont}}$ is the number of these stars with radial velocities that are more than $3\sigma$ different from the mean radial velocity of the satellite. F_cont is the percentage contamination per galaxy, estimated as the ratio of ${n}_{m,\mathrm{cont}}$ to ${n}_{\mathrm{QC},\mathrm{cont}}$.

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When we define member stars as those stars with P_sat ≥ 0.5, there are a total of n_m = 5173 member stars with radial velocities across all satellites. Of these, only ${n}_{m,\mathrm{cont}}=173$ of them have velocities that are more than 3σ from the mean systemic radial velocity (as given in Table 1). When we only consider the dwarf satellite galaxies with fewer than 100 members with radial velocity measurements (i.e., excluding the 6 galaxies mentioned previously), the sample of 462 stars includes a total of 25 contaminants. To understand what this means for the average contamination rate, it is important to recognize that each galaxy had different levels of contamination prior to the application of the algorithm. In particular, there were a total of ${n}_{\mathrm{QC},\mathrm{cont}}=3280$ stars that passed our initial cuts and had velocities more than 3σ from the mean systemic radial velocity of the satellites. Because only 173 of these remain after application of the algorithm, we find ${F}_{\mathrm{cont}}={n}_{m,\mathrm{cont}}/{n}_{\mathrm{QC},\mathrm{cont}}=0.05$. When the brightest satellites are ignored, F_cont = 0.016.⁵

Of course, there can be significant variations in these results between the dwarf satellite galaxies. Fornax and Sculptor have high values of F_cont, but the radial velocity sample we used is already extremely clean, and so the algorithm cannot significantly improve on the high purity. In contrast, Draco initially has 467 stars out of 950 with discrepant velocities, and our algorithm identifies only 6 stars out of 414 members with discrepant velocities. Table 5 breaks these numbers down on a galaxy-by-galaxy basis. Based on these numbers, we have high confidence in the ability of our algorithm to select member stars with only modest (≤1 in 20) contamination. This also implies that the systemic proper motion estimates are robust to contamination effects.

4.2.2. Completeness: Confirmed Members via High-resolution Spectroscopy

To the extent that it is possible to "know" that a star is a member of a satellite, stars that have had their stellar parameters determined and chemical abundances derived from high-resolution spectroscopy are the gold standard. Prior to Gaia DR2, such stars were usually targeted for follow-up spectroscopy based on spatial coincidence, color–magnitude consistency, and radial velocity matching (post Gaia DR2, proper motion consistency is also usually required). For stars with published high-resolution spectroscopy, their subsequent analyses would confirm their memberships, and provide metallicity and abundance characteristics. We examine the completeness of our algorithm by determining our ability to retrieve these "known" members.

A search of the literature was carried out for all stars in the ultrafaint satellites (here defined as M_V  ≲ −8) for which high-resolution spectroscopic analyses exist and that are present in Gaia DR2. This list is intended to be complete up to 2020 May, and contains a total of 91 stars. These were cross-matched with our stellar catalogs to calculate the implied values of P_sat for each star based on our algorithm. Of these 91 stars, 16 stars are assigned P_sat < 0.5 and are listed in Table 6; the other 75 stars are listed in a table in the Appendix, Table A1, in the same format.

Table 6.  Membership Probabilities for Stars in Ultra-faint Dwarf Galaxies Studied in the Literature. This Table Includes All Stars with ${P}_{{sat}}\lt 0.5$. See Table A1 for Stars with ${P}_{{sat}}\geqslant 0.5$

Galaxy	Reference	Star ID	RA	Dec	G	$r/{r}_{h}$	Gaia DR2 source ID	Psat	Comments
Bootes1	Frebel et al. (2016)	Boo-980	13:59:12.68	+13:42:55.7	17.88	4.4	1230649834160653440	0.01	At several rh
Carina2	Ji et al. (2020)	CarII-V3	07:35:09.12	−57:57:14.8	18.47	1.7	5293940924860019584	$\lt {10}^{-10}$	RR Lyrae variable
Carina2	Li et al. (2018b)	J073646.47-575910.2	07:36:46.47	−57:59:10.1	19.41	0.5	5293948273547045248	0.36	Binary star
Carina2	Li et al. (2018b)	J073729.30-580447.8	07:37:29.28	−58:04:47.7	19.14	1.5	5293900513510499584	0.08
Carina2	Li et al. (2018b)	J073737.04-574925.5	07:37:37.02	−57:49:25.4	19.79	2.2	5293959680980519424	$\lt {10}^{-14}$	BHB
Carina2	Li et al. (2018b)	J073745.86-580406.7	07:37:45.85	−58:04:06.7	18.85	1.8	5293900857107911808	$\lt {10}^{-31}$	BHB
Hercules	Koch et al. (2013)	12175	16:31:15.82	+12:34:56.5	18.34	6.0	4460295812884988672	0.01	At several rh
Hercules	Koch et al. (2013)	12729	16:31:07.49	+12:31:33.7	19.6	8.1	4460293953161686656	0.00033	At several rh
Horologium2	Fritz et al. (2019)	horo2_2_48	03:17:21.08	−50:03:40.6	18.74	8.2	4749201525396641408	0.00094	At several rh
Segue1	Frebel et al. (2014)	J100702+155055	10:07:02.46	+15:50:55.2	18.0	5.2	621924492960734464	0.00142	At several rh
Segue1	Frebel et al. (2014)	J100742+160106	10:07:42.71	+16:01:06.9	18.16	3.1	621933460851966976	0.3
Segue1	Frebel et al. (2014)	J100639+160008	10:06:39.33	+16:00:08.5	19.04	2.1	621941535390988928	0.37
Segue1	Norris et al. (2010a)	SegueI-7	10:08:14.44	+16:05:01.1	17.45	4.5	621934805177217920	0.00024	At several rh
Triangulum2	Venn et al. (2017), Ji et al. (2019)	Star46	02:13:21.54	+36:09:57.4	18.84	0.4	331086526201161088	0.04	Binary star
Tucana3	Marshall et al. (2019)	J000549-593406	00:05:48.72	−59:34:06.2	16.2	11.7	4917991682841282176	$\lt {10}^{-7}$	At many rh (in tails)
Tucana3	Marshall et al. (2019)	J234351-593926	23:43:50.85	−59:39:25.8	17.22	16.1	6488511925629603200	$\lt {10}^{-8}$	At many rh (in tails)

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Analysis of the stars in Table 6 reveals the following relevant details for membership considerations:

• 1.  
  Eight of these stars are at radii corresponding to more than four half-light radii (accounting for the elliptical shape of the satellites). When we assume that the spatial distribution of stars in these satellites are very well described by the parameters listed in Table 1, it is increasingly likely that the most distant stars are not members. However, the structure of faint dwarf galaxies, especially at large radius, can be very complex, and it is known that some of these systems have extended tidal features that are not well captured by the simple parameterizations in Table 1 (especially Tucana 3; see Li et al. 2018a). As currently constructed, this algorithm will preferentially assign low-probability measurements to stars at large radius, and this may include stars that are actually members. If these stars are subsequently shown to be members of the satellite, then this will imply that the satellite's structure is more complex than currently parameterized in the spatial likelihood functions.
• 2.  
  Three stars are horizontal branch stars, either blue horizontal branch or RR Lyrae. As our algorithm considers membership of the satellite in color–magnitude space in relation to the proximity of the star to the main isochrone locus (main-sequence, subgiant, or red giant branch stars), then it is not surprising that these horizontal branch stars are not identified as members.
• 3.  
  Two stars appear to be in binary systems, one in Carina 2 (Li et al. 2018b) and another in Triangulum 2 (Venn et al. 2017). It is unclear if (or how) these affected the ability of the algorithm to determine their memberships.

If we consider stars with spatial and color–magnitude characteristics that should be well captured by the algorithm (specifically, stars within four half-light radii that are not on the horizontal branch), there are 80 relevant stars in Tables 6 and A1. Of these, 75 stars (94% of the sample) have P_sat ≥ 0.5. Interestingly, two of the five stars missed by the algorithm in this case are known binaries. With the necessary caveat relating to the structure of galaxies at large radii, we conclude that the algorithm is reasonably complete in correctly identifying main-sequence, subgiant, and red giant branch stars as members of the dwarf satellites under consideration.

Figure 7 shows a comparison between the adopted systemic proper motion from Table 4 and all literature estimates for those satellites based on Gaia DR2 for which previous estimates are published. ${\mu }_{\alpha }\cos \delta$ is shown on the x-axis, and μ_δ is shown on the y-axis. The adopted systemic proper motions derived in this paper are shown as blue points with blue error bars. Error bars or the semiaxes of the ellipses correspond to 1σ uncertainties, excluding systematic errors. The scales are the same on each axis in each panel, but different between panels, and were chosen to encompass all of the relevant points and their uncertainties. This figure is intended only to highlight the relative agreement, or otherwise, between estimates. Literature estimates are shown either as error bars or ellipses:

• 1.  
  Solid black ellipses correspond to estimates from Gaia Collaboration et al. (2018c);
• 2.  
  Dotted red ellipses correspond to estimates from Simon (2018);
• 3.  
  Dashed green ellipses correspond to estimates from Fritz et al. (2018);
• 4.  
  Dot–dashed cyan ellipses correspond to estimates from Kallivayalil et al. (2018);
• 5.  
  Dashed gray ellipses correspond to estimates from Massari & Helmi (2018);
• 6.  
  Green error bars correspond to estimates from Pace & Li (2019);
• 7.  
  Dot–dashed magenta ellipses correspond to estimates from Fritz et al. (2019; Horologium 2, Reticulum 3, Columba 1, Phoenix 2), Simon et al. (2020; Grus 2, Tucana 4, Tucana 5), Longeard et al. (2018; Draco 2), Longeard et al. (2020b; Sagittarius 2), Torrealba et al. (2019; Antlia 2), Mau et al. (2020; Centaurus 1), Fu et al. (2019; Crater 2, Hercules), and Pace et al. (2020; Ursa Minor);
• 8.  
  Dotted blue ellipses correspond to estimates from Gregory et al. (2020; Hercules) and Chakrabarti et al. (2019; Antlia 2).

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Inspection of Figure 7 shows good consistency between literature estimates for the satellites using Gaia DR2 and the new estimates derived here. Indeed, while there can be considerable spread in the measurements for each satellite between different studies, none of the values derived here are farther than ∼1σ from at least one of the literature estimates. It is also interesting to note that two of the brightest satellites—Fornax and Sextans—appear to have some of the most discrepant measurements in this plot. However, this is largely a result of the absence of a scale in the panels of Figure 7. Fornax and Sextans have mean random uncertainities in Table 4 of ∼2 μas yr⁻¹ and 20 μas yr⁻¹, respectively. Systematic uncertainties in these proper motions are on the order of several tens of μas yr⁻¹ (Lindegren et al. 2018), i.e., an order of magnitude larger than the random errors for Fornax, and the same order as the random errors for Sextans.

Almost all of the measurements that are highlighted by ellipses in Figure 7 are derived from the weighted mean proper motion of a subset of stars that were previously identified as members. In contrast, our approach is inspired by Pace & Li (2019; green error bars), in which determination of the most likely systemic proper motion is made by the analysis of the full data set in the vicinity of the satellite, members and nonmembers alike. As such, our uncertainties are in general smaller. Additionally, the maximum uncertainties that we measure on the systemic proper motions are of order 100 km s⁻¹ in each direction, and this is a result of our prior, as discussed in Section 3.5. While this is reasonable for most of the objects under discussion, it is important to realize that Eridanus 2, Phoenix, and Leo T are sufficiently distant that the assumption that they trace the velocity dispersion of the Milky Way halo may be incorrect. As such, we also list in Table 4 the systemic proper motions for these three galaxies in the absence of this prior. It is notable that the uncertainties increase significantly, especially for Leo T.

While our uncertainties for most satellites are generally smaller than previous literature estimates, this is not the case for Antlia 2 (Torrealba et al. 2019; Chakrabarti et al. 2019). We think that this is due to the extreme foreground contamination for Antlia 2; i.e., the value of f_sat that we calculate corresponds to only a few in every million stars that we would consider to be actual members. Thus, for individual stars, the probability that it is a member of Antlia 2 is much lower than for most other galaxies, and this propagates through to our estimate of the systematic proper motion. For those stars that have estimates by Pace & Li (2019), the difference in the size of the uncertainties between this study and Pace & Li (2019) is likely a result of our Milky Way contamination model, in which we do not have additional parameters that need to be marginalized over.

Only estimates using Gaia DR2 are shown in Figure 7. Many of the brighter satellites also have previous estimates of their proper motions derived using Hubble Space Telescope observations. These are compared to estimates from Gaia DR2 in Figure 15 of Gaia Collaboration et al. (2018c). These authors note agreement between their estimates and previous estimates at the 2σ level, and also note the much smaller error bars that are made possible by the Gaia DR2 data. Encouraging, they also find that the Gaia-based estimates agree best with those galaxies for which the space-based data have the longest baselines. The results from Gaia Collaboration et al. (2018c) are shown in Figure 7 as black ellipses, and it is clear that we agree closely with their estimates, especially when systematic uncertainties are taken into account.

Finally, we compare the 1σ (random) uncertainties in our estimates of the systemic proper motions to the uncertainties on the previous literature estimates using Gaia DR2. For each satellite in Figure 7, we selected the literature estimate that had the smallest quoted random uncertainties (averaged in both directions), σ_lit. This value is compared to our average uncertainty for each galaxy, σ_McVenn. In the median, σ_McVenn ≃ 0.74σ_lit (${\sigma }_{{McVenn}}\simeq 0.77{\sigma }_{{lit}}$ excluding those galaxies for which the prior is imposing a maximum uncertainty of around 100 km s⁻¹). Overall, this suggests that our algorithm provides robust proper motion values with significantly improved random uncertainties.

In addition to the 49 satellites shown in Figure 7 that have previous estimates from the literature, we derived estimates of the systemic proper motions for the first time for 5 other satellites: Indus 1, DES J0225+0304, Leo T, Cetus 2, and Pictor 2. The latter three are less robust than for the other systems. In addition, for Bootes 4, Cetus 3, Indus 2, Pegasus 3, and Virgo 1, we cannot resolve any signal for the systemic proper motions of these satellites in Gaia DR2. We discuss each of these systems in more detail.

In Figure 8 the 1D PDFs for f_sat for the 10 satellites with new or no systemic proper motions are shown as the blue (filled) histograms. The top row shows the systems with new proper motions that we argue are robust, the second row shows those systems with new proper motions that are less robust (for reasons we discuss below), and the remaining rows show the systems for which no proper motion is derived. As f_sat is the fraction of stars that belong to the satellite, then when the most likely value for f_sat is zero, this implies that no stars belong to that satellite, and we cannot derive a robust systemic proper motion.

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For the satellites in the top row of Figure 8, it is clear that the most likely value for f_sat is greater than zero (the peak of the PDF is resolved). This is especially clear for Indus 1. For DES J0255+0304, the peak is more marginally resolved, but it is at a value greater than zero. For all of the remaining satellites, the most likely value of f_sat is zero.

As f_sat was not resolved for several satellites, we decided to rerun the algorithm for all of the 10 satellites in Figure 8, but with looser selections on the data under consideration. In particular, the cut on the astrometric quality of the data was loosened (very slightly), such that ruwe  <  1.4, and the cut on the photometric quality of the data was removed. This latter change is the most significant: by no longer applying the cut described in Equation (1), many more stars were included for the analysis, including potential member stars of satellites. However, we did this knowing that there are concerns with the consistency of the G, BP, and RP photometry, which may affect the robustness of the CMD likelihood analysis.

The orange (empty) histogram in each panel of Figure 8 shows the corresponding PDFs for f_sat using this revised data set. For the satellites in the top row, the position of the peak of the PDF is relatively unchanged. In each of these two cases, the corresponding systemic proper motion is statistically the same as previously, giving us confidence in the robustness of the result, especially for DES J0255+0304. This system is relatively close at only ∼20 kpc, suggesting that there are just not very many stars in this galaxy in the Gaia magnitude range.

The situation for DES J0255+0304 should be compared with the satellites in the second row of Figure 8, where there is a clear difference in the PDF for f_sat between the two versions of the data set. For Leo T and Cetus 2, the peak in f_sat is clearly resolved with the looser cuts on the data. For Pictor 2, the situation is more ambiguous, but even here it appears that the favored model has f_sat > 0.

Leo T has radial velocity data available, and the use of the looser quality cuts enables a solution to be determined using both Prior A and Prior B. Indeed, the weighted mean proper motion of those members (P_sat > 0.5) with radial velocities is entirely consistent with the derived value (as discussed for all the satellites in Section 4.1.2). However, in all cases, the individual uncertainties in the proper motions of each member star are very large (greater than 1 mas yr⁻¹) and the relatively small uncertainty in the systemic proper motion derived under the assumption that Leo T traces the tangential velocity dispersion of the halo is due to the prior (as comparison to the final entries in Table 4 makes clear). Nevertheless, no stars are assigned membership of Leo T that have inconsistent radial velocities. Most encouraging of all, all five stars with radial velocities that we determine to be members are also, completely independently, determined to be members by Simon & Geha (2007), where they use a combination of SDSS photometry and a selection of spectral characteristics to assign membership. These findings give us confidence that despite the poorer quality photometry, our estimates of the proper motion of Leo T are reasonable given the data and the priors.

We conclude from this analysis that the measurements for Indus 1 and DES J0255+030 are robust. We present the measurements for Leo T, Cetus 2, and Pictor 2 using the looser cuts on the data alongside the other measurements in Tables 3 and 4. We stress that the proper motions for Leo T, Cetus 2, and Pictor 2 are based in part on photometry that may not be reliable. Any analyses that use these three measurements should proceed with caution, and we expect that future data releases from Gaia—potentially including the imminent EDR3—may contain sufficiently improved photometry to improve these three measurements. Table 7 lists all those stars in these five objects for which P_sat ≥ 0.5.