Proper Motions of the Satellites of M31 Ben Hodkinson and Jakub Scholtz∗ IPPP, Durham University, DH1 3LE, UK (Dated: April 9, 2019) We predict the range of proper motions of 19 satellite galaxies of M31 that would rotationally stabilise the M31 plane of satellites consisting of 15-20 members as identified by Ibata et al. (2013). Our prediction is based purely on the current positions and line-of-sight velocities of these satellites and the assumption that the plane is not a transient feature. These predictions are therefore independent of the current debate about the formation history of this plane. We further comment on the feasibility of measuring these proper motions with future observations by the THEIA satellite mission as well as the currently planned observations by HST and JWST. I. INTRODUCTION pared to determine whether the plane is indeed a stable structure or a temporary alignment of satellites of M31. Ibata et al. [1] reported the existence of a planar sub- We do feel the need to re-iterate two points: First, the group of 15 satellites of the M31 galaxy. Of the 15 in- results of this work are independent of the debate about plane satellites, 13 are co-rotating, which suggests, but the nature of this plane: it could be a rare structure of does not show, this plane is a kinematically stable struc- the ΛCDM model or a manifestation of a deviation from ture. Moreover, the small width of the plane, roughly the ΛCDM. As long as the M31 plane is a stable feature, 13kpc, is a challenge to explain. As a result, it is a mat- our results remain valid. Second, whether the plane is ter of active debate if this structure may be an indica- kinematically stabilised or an accidental alignment only tion of phenomena not predicted within standard ΛCDM observable in the current era is not a dichotomy: the cosmology [2{5]. In particular, Gillet et al. [6], Buck answer may be that a subset of satellites form a kine- et al. [7] have argued that kinematically stable planes matically stable plane, while the rest of the satellites are are less common than accidental alignments in the stan- aligned accidentally. dard ΛCDM scenario. Therefore, the question of whether This paper is organised as follows: in sec. II we discuss this plane is a truly stable structure has important con- our methods for determining properties of the satellites' sequences for our understanding of the ΛCDM model. orbits. In sec. III we define the constraints an orbit needs A similar plane of satellites has previously been dis- to satisfy in order to belong to the plane. In sec. IV we covered for the Milky Way [8], and the proper motions discuss our results and the experimental program needed (PMs) of the satellites for which data was not already to probe these results. We conclude in sec. V. available were predicted under the assumption that the MW plane is rotationally stabilised [9{12]. In this paper we carry out a similar analysis for the II. METHOD satellite members of the plane of M31 based on our current knowledge of the satellites' orbital parameters, A. Positions and Velocities of the M31 System which includes their full three-dimensional position and line-of-sight velocities [13, 14]. At first glance, a perma- We use the data available in McConnachie [13] and nent membership of an individual satellite in the plane in Conn et al. [14] as presented in Table I. We take the can be reduced to the condition that the satellite's angu- velocity of M31 with respect to the Sun, according to lar momentum is aligned with the normal to the plane. McConnachie [13] and van der Marel et al. [15], to be This fixes one of the components of proper motion. The remaining component determines the size of the angular vlos;M31 = 300 4 km=s momentum of the satellite and is limited by the condi- − ± µα∗;M31 = +65 18 µas=yr (1) arXiv:1904.03192v1 [astro-ph.GA] 5 Apr 2019 tion that the satellite is on a bound orbit. As a result the ± µδ;M31 = 57 15 µas=yr: constraints implied by the stability of the plane result in − ± a finite range of allowed values of their proper motions. In this paper, we identify the range of possible PMs for In this paper, we will adopt a Cartesian coordinate sys- the 13 co-rotating satellites, two counter-rotating satel- tem, aligned with the standard Galactic coordinate sys- lites and five additional satellites that were also reported tem, but centered on M31. in Ibata et al. [1] as being consistent with a stable plane. Our results are a prediction and provide a benchmark against which measurements of the PMs can be com- B. Properties of the Plane of M31 Satellites For our purposes, it is best to describe the plane by a unit normal vectorn ^ and the offset of the plane away ∗ [email protected] from the origin (center of M31) alongn ^, d. Although each 2 individual satellite orbit is contained in a plane that also Name R.A. Dec. D[kpc] vlos[km/s] +18 +2:2 contains the origin (the satellites orbit around M31), it And I 00 45 39:8 38 02 28 727−17 −376:3−2:2 +18 +1:7 does not mean that the plane of satellites itself has to And III 00 35 33:8 36 29 52 723−24 −344:3−1:7 +91 +2:5 And IX 00 52 53:0 43 11 45 600−23 −209:4−2:5 contain the origin unless all the orbits are exactly co- +29 +4:4 And XI 00 46 20:0 33 48 05 763−106 −419:6−4:4 planar. +40 +3:2 And XII 00 47 27:0 34 22 29 928−136 −558:4−3:2 We choosen ^ and d such that the sum of the squares +126 +2:4 And XIII 00 51 51:0 33 00 16 760−154 −185:4−2:4 of distances of the satellites from the plane is minimised. +23 +1:2 And XIV 00 51 35:0 29 41 49 793−179 −480:6−1:2 Note, that we exclude M32, NGC 205, LGS 3, IC10 and +44 +2:8 And XVI 00 59 29:8 32 22 36 476−29 −367:3−2:8 IC1613 in this calculation, as their planar membership +39 +1:8 And XVII 00 37 07:0 44 19 20 727−25 −251:6−2:0 is debatable, and consider just the first 15 satellites in +23 +1:0 And XXV 00 30 08:9 46 51 07 736−69 −107:8−1:0 Table I. Denoting the position vectors of satellites by ~ri, +218 +3:0 And XXVI 00 23 45:6 47 54 58 754−164 −261:6−2:8 then the sum of the squares of the distances is given by +42 +4:7 And XXVII 00 37 27:1 45 23 13 1255−474 −539:6−4:5 +32 +6:0 And XXX 00 36 34:9 49 38 48 681−78 −139:8−6:6 15 15 +21 +0:8 2 X 2 X 2 NGC 147 00 33 12:1 48 30 32 712−19 −193:1−0:8 D = Di = ~ri ~n + d : (2) +19 +1:1 j · j NGC 185 00 38 58:0 48 20 15 620−18 −203:8−1:1 i=1 i=1 +82 +6:0 M32 00 42 41:8 40 51 55 805−74 −199:0−6:0 +27 +0:3 We minimise D, subject to the normalisation constraint NGC 205 00 40 22:1 41 41 07 824−26 −286:5−0:3 +45 +1:0 IC 10 00 20 17:3 59 18 14 794−43 −348:0−1:0 ~n = 1. This yields the normal vector and plane offset: +25 +1:1 j j LGS 3 01 03 55:0 48 20 15 769−24 −203:8−1:1 +43 +1:2 n^ = [0:887; 0:443; 0:132] (3) IC 1613 01 04 47:8 02 07 04 755−41 −231:6−1:2 − d = 3:5 1:5kpc; − ± TABLE I. Data from [13, 14] for the satellites reported in [1] to plausibly lie in the planar structure. The heliocentric where the uncertainty bands come from sampling the he- distances are given in kpc, with upper and lower errors. The liocentric distances of M31 and all the satellites from heliocentric velocities are given as line-of-sight velocities rela- Gaussians with widths given by the uncertainty bands tive to the sun with upper and lower errors. The top section in Table I. We omit the uncertainty bands for the pole, of this table contains presumed members of the M31 plane, because the pole of the planen ^ only varies by 1o (68% while the lower section consists of satellites whose member- confidence interval) under the same sampling. ship is debatable. Givenn ^ and d, we can also compute the thickness of the plane: which are only accidentally aligned with the plane and D σd = = 12:4 0:8kpc; (4) stay aligned for up to a couple billion years. p15 ± An example of such an orbit would be a highly eccen- where the uncertainty band comes from the same dis- tric orbit such that the plane of this orbit is not aligned tance sampling as described above. with the plane of M31, but the major axis of this orbit lies along the intersect of these two planes.