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 ˆ and the offset of the plane away ∗ [email protected] from the origin (center of M31) alongn ˆ, 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 ˆ 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 | · | 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 | | LGS 3 01 03 55.0 48 20 15 769−24 −203.8−1.1 +43 +1.2 nˆ = [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 ˆ only varies by 1o (68% while the lower section consists of satellites whose member- confidence interval) under the same sampling. ship is debatable. Givenn ˆ 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. √15 ± 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. Because orbits in the potential of an NFW halo are not closed, after the C. Determining Satellite Orbits next periapsis the major axis of the orbit changes significantly and the orbit is no longer close to the plane. These orbits represent temporary alignments between the orbit Each choice of proper motion (µα∗, µδ) is equivalent to a different orbit. As we will see in section III, we are and a plane unrelated to the orbit. interested in three properties of each orbit: the maximum Since we are interested in finding orbits that are not distance of the satellite from the plane in the next 10 temporarily aligned with the plane of satellites, we would periapses like to avoid accepting these orbits. In order to remove them, we choose to integrate over a fixed number of pe- dmax = max ~r(t) nˆ + d , (5) riapses: we chose this fixed number to be 10. t | · | For comparison with other work we also compute the and the minimum and maximum distances of the satellite angle between the normal to the plane from Eq. (3) and from the center of M31 (also over the next 10 periapses) the angular momentum of the satellite L~ = ~r ~v around the center of M31: × rmin = min ~r(t) (6) t | | ~ ~ rmax = max ~r(t) . (7) cos θ = L n/ˆ L . (8) t | | · | | Here we should briefly mention our choice of integra- In order to compute these quantities we use GALPOT tion time. A possible choice would be to set a fixed inte- [16] to integrate the orbits in the gravitational field of gration time of order the age of M31, and we believe that M31. We model M31 as re-scaled DM halo of the Milky this is an acceptable choice. However, there are orbits, Way. We use an NFW dark matter halo with density 3 profile from M31 is less than the virial radius of M31 (500kpc). Otherwise, the satellite would not be identified as M31’s ρ0 ρ(r) = 2 , (9) satellite and would eventually cross into the Milky Way’s (r/r0)(1 + r/r0) gravitational potential. This criterion is then 7 −3 where ρ0 = 1.7 10 M kpc (corresponds to MM31 × 12 ∼ r < 500kpc, (12) 2MMW 2.6 10 M ) and the scale radius rs = 19kpc max [17]. We∼ have× also produced results for a Milky Way-like which replaces the previous criterion from Eq. (11). We potential with MM31 MMW. We have omitted the∼ M31 baryonic disk and bulge as would like to note that if a satellite is orbiting in the well as the fact that its DM halo is elliptical. While the M31 plane of satellites, it is only a matter of time before effects of the baryonic disk can be neglected, the effects its orbit’s major axis is pointing at the MW, because of the non-spherical DM halo may be significant. Devia- orbits in the NFW profile precess fairly rapidly and the tion from spherical symmetry of the background poten- M31 plane is nearly edge-on to the MW. As a result, tial will contribute to changes of the angular momentum excursions past the virial radius are very likely to result of each satellite. However, we leave this discussion to fu- in an eventual loss of the satellite. ture work, where we would like to explore this effect in If a satellite galaxy comes too close to M31 the tidal order to place a bound on the ellipticity of the M31 DM interactions strip its content and disrupt the satellite. halo. Therefore, we impose a bound on the satellite’s closest approach to M31. In this work we choose to implement this bound as: D. Uncertainties rmin > 15kpc. (13) We have found that varying the line-of-sight velocities within the uncertainty band in Table I does not signifi- We can think of this scale as an approximate Roche limit 7 cantly alter our results, so we do not include the impact for a satellite of mass of order 10 M of size 0.5kpc or- 11 of these variations in our work. biting an enclosed mass of order 10 M . This is a con- On the other hand, the uncertainties in the heliocentric servative bound, because in reality such close encounters distance measurements to the satellites are much more are still somewhat dangerous. significant. To include this effect, we evaluate rmax, rmin The third criterion can be implemented in two differ- and dmax for five values of distance: ent ways. We could require that in order to be a permanent member of the plane the angular momentum of the ∆ ∆0 ++, ∆0 ++/2, ∆0, ∆0 −/2, ∆0 − , (10) satellite has to be aligned with the normal to the plane. ∈ { − − } This is equivalent to demanding that the θ from Eq. (8) where ∆0 is the central value of the distance and + and is smaller than some reference angle θ0. The angle θ0 is − are the upper and lower edges of the one sigma unrelated to the scatter of these angles for satellites we con- certainty band from Table I. sider in-plane. The authors of [9, 10] have done just that We have also propagated the uncertainty due to the ◦ and have chosen θ0 = 37 for the MW plane (the some- proper motion of M31. This offset can be treated as con- what large size of this angle is driven by the fact that stant across the whole range of proper motion we show in the MW plane is thicker than the M31 plane). We could our plots. In order to make this systematic uncertainty determine θ0 from the scatter of the ~n on the unit sphere apparent, we show the M31 PM uncertainty in every plot and this method will be discussed in the upcoming work in Figs. 1 and 2 as a green ellipse. [18]. However, even if θ < θ0, it is still possible for a satellite to stray far from the plane by a large radial excursion, III. CONSTRAINTS which would make it appear as a non-member. For this reason we require that each satellite does not make ex- If the plane of satellites is a permanent feature, we cursions far away from the plane by requiring that: expect it to be rotationally stabilised. This implies that the orbits of the in-plane satellites are constrained to be dmax < 1, 2, 3 σd 13, 25, 37 kpc, (14) bound, the satellites must be able to survive on their { } × ≈ { } orbits and they do not make large excursions away from where σ is defined in Eq. (4). Another advantage of the plane. d expressing the third criterion this way is that σd is a The first two criteria are not hard to implement: the directly observable scale and can be determined from the orbit is bound if the energy of the satellite is negative data. E < 0. (11) A choice of (µα∗, µδ) that satisfies the conditions in Eqs. (12), (13) and (14) leads to an orbit that is consis- A slightly more stringent condition would be to re- tent with a stable plane composed of the first 15 satellites quire that the maximum excursion of the satellite away in Table I. 4

IV. RESULTS AND DISCUSSION We have also tested every other satellite of M31 (those not considered in-plane by Ibata et al. [1]). None of the A. Proper Motion Predictions other satellites have PMs corresponding to orbits within the plane. The plots in Figs. 1 and 2 can be grouped into three We present our results as a phase plot in µα∗ and µδ in Figs. 1 and 2. The trajectory for each PM is tested broad categories with similar features: against the constraints for five heliocentric distances from 1. M32, NGC205 and Andromeda IX are M31’s three the set in Eq. (10), as discussed in section II D. closest satellites, with distances 23kpc, 42kpc and The coloured regions are defined as follows: 40kpc from the centre of M31 respectively. As long 1. The blue regions contain values of PMs that pass as their velocity is small enough, there are orbits the constraints, and are consistent with the satel- consistent with the plane for all angles between the satellite’s angular momentum and the plane nor- lite staying within σd, 2σd and 3σd of the plane in at least one distance realisation (from darkest to mal. It is likely these satellites are not truly a part lightest). of the planar structure, with their position only consistent with the plane due to their close prox- 2. The orange region contains trajectories which stray imity to the origin of M31. Indeed, Ibata et al. [1] further than 500kpc from M31, i.e. which fail con- considered M32 and NGC205 as less likely planar dition (12) for all five distance realisations. The members for this reason. The discontinuous na- boundaries of these regions lie on velocity contours, ture of the contours in plots for M32 and NGC205 so the constraint essentially imposes a maximum is caused by the fact that the distance uncertain- velocity. ties are significant compared to their actual distance from M31 and hence our five samples from 3. The red region indicates trajectories which come the uncertainty band are not fine enough to pro- closer than 15kpc to the centre of M31, i.e. that duce smooth results. However, given most likely fail condition (13) in all five distance realisations. accidental membership in the plane, we feel it is un- As expected these regions lie around the singularity necessary to determine the precise nature of these of the angle contours, corresponding to orbits with satellites’ orbits. small angular momentum. The uncertainty in the PM of the M31 galaxy itself is 2. IC10, LGS3, IC1613 and Andromeda XVI are indicated by a green circle in each plot, as discussed in very far from M31, with distances 252kpc, 269kpc, sec. II D. 520kpc and 279kpc respectively. As a result, only We include grey dashed lines which indicate the re- a thin strip of PM’s, which correspond to angular gions for which the angle, θ, between the satellite’s an- momenta very closely aligned to the plane normal, gular momentum and the normal to the plane of satellites produce trajectories which remain closer than 3σD is less than 30 degrees (for the central value of satellite from the plane. IC1613 is so far from the origin of distance). In Pawlowski and Kroupa [9, 10], Pawlowski M31 that no PM’s correspond to planar orbits, so et al. [11], MW satellites with θ < 37o were considered to we don’t include it in Figs. 1 and 2. Andromeda be co-orbiting within the MW’s plane of satellites. This XVI on the other hand currently lies almost per- approach produces predictions which are independent of fectly on the plane, so there is a relatively wide the mass of M31, as a larger M31 mass allows higher band of PM’s consistent with it remaining less than velocity orbits along the same angular direction. How- σD from the plane. IC10, LGS 3 and IC1613 are ever higher velocity orbits can deviate from the plane more likely accidentally aligned with the plane. o significantly for even a small value of θ, and the θ < 37 3. Andromeda III, XXV and XXVI are the furthest region on its own fails to include a restriction on such offset from the plane of those considered likely to trajectories. For this reason, we believe Eq. (14), which be members by Ibata et al. [1]. As a result the defines our blue regions, gives a better indication of or- regions consistent with the plane are much smaller bits which should truly be considered as stable members (in the Andromeda XXV plot there are no orbits of the plane. within 1σd of the plane), with too large a velocity or The blue regions for most of the satellites are centred o o too large an angular momentum deviation from the on θ = 0 or 180 , indicating a direction of rotation. pane normal resulting in orbits which stray further The directions of rotation are consistent with those sug- than 3σD from the plane. gested by Ibata et al. [1], where Andromeda XIII and Andromeda XXVII counter-rotate and the other 13 satel- 4. The remaining satellites were all considered likely lites co-rotate. Our results indicate it may also be possi- members of the plane by Ibata et al. [1], and have ble for Andromeda I and XVII to counter-rotate within similar spade-like blue regions in Figs. 1 and 2. This the plane, though the counter-rotating regions pass closer is due to a relationship between the variable dmax than 15kpc to the centre of M31 for at least one distance in constraint (14) and cos θ as defined in Eq. (8). realisation. From the geometry of the setup it is obvious that 5

dmax sin(θ)rmax. However, since the orbits are and therefore it is safe to conclude the IC10 is unlikely not closed,≤ over time this inequality is saturated. to be a member of the plane. This explains the shape of the contours: instead of wedges of constant angle, we get shapes that narrow towards larger velocities because larger ve- C. Experimental Feasibility locities imply larger rmax and hence to keep dmax constant, the acceptable range of θ must become Fig. 1 illustrates that in order to distinguish the ran- smaller. dom alignment hypothesis from the coherent structure Finally, we would like to discuss the effect of mass of hypothesis it is necessary to measure one of the linear the host galaxy: the plots in Figs. 1 and 2 represent the combinations of the PM to accuracy of order 10 µas/yr. Naturally we need to ask if this is feasible. choices MM31 2MMW and MM31 MMW. The results are almost identical,∼ up to a rescaling∼ by a factor of √2. There are two planned measurements by HST and This is expected because doubling the mass increases∼ the JWST. The HST has already observed the proper mo- escape velocities by a factor of √2. If the line-of-sight ve- tions of NGC147 and NGC 185, with planned accuracy locity is relatively small, then the escape velocity is sat- of order 25km/s, which corresponds to PMs of order urated by the proper-motion which results in the size of 10 µas/yr [20]. The JWST has planned guaranteed time the orange regions to scale as M 1/2 as we see. However, if observation for the PMs of Andromeda I, III, XIV, and the line-of-sight velocity component is close to saturating XVII [21] with similar planned accuracy. As a result in the escape velocity alone, then the proper motion com- a couple of years we should have information about six ponents are much more influenced by the change of mass out of the fifteen members of the M31 plane. of M31. As an example the blue regions for Andromeda Unfortunately, GAIA is unlikely to deliver in this di- XII and XIV are more than a factor of √2 larger in Fig. 1 rection. The horizontal branch (HB) in Andromeda I has compared to Fig. 2. This is because Andromeda XII and magnitude around 25 [22] in the V band. This is similar XIV have a large velocity relative to M31, at -272 and for most of the M31 satellites as the distance modulus -208 km/s (for zero relative proper motion). A larger does not vary too much (it is between 23.9 and 24.9 for M31 mass therefore prevents these satellites from mak- all the M31 satellites). In the G band this corresponds to ing large excursions from M31 and the plane. For the magnitude 25.5 for the center of Horizontal Branch stars. same reason, the red regions in Fig. 1 encroach further The GAIA satellite has not been directly tested to on the blue regions, proportionally, than in Fig. 2. This measure PMs of such dim stars, but naively extrapolat- is most noticeable for Andromeda IX and XI. ing the PM uncertainty relation from Gaia Collaboration This brings about an interesting opportunity. When et al. [23] we arrive at an expected measurement uncer- 5 the proper motions of M31’s satellites are measured, we tainty of order 10 µas/yr. We would need to observe 8 can infer a bound on mass of M31 under the assump- close to 10 HB stars in order to determine the PM of tion that each satellite is a member of a stable plane of Andromeda I to the desired accuracy. Since Andromeda satellites. I does not have this many HB stars (or this many stars Similarly, once the proper motions are measured, it for that matter), the situation seems hopeless even with- is possible to put a constraint on the proper motion of out dealing with additional caveats associated with our M31, by finding the M31 proper motion that minimises naive extrapolation. However, the proposed THEIA mission [24] has bet- the dmax for all the measured satellites. ter prospects. With 40 hours of observation per satellite over four years, THEIA should be able to achieve PM B. Proper Motion of IC10 measurement with uncertainty of order 200 µas/yr for stars of the 25th magnitude. As a result only a hand- The existence of a maser in IC10 allowed the authors ful of stars (roughly 100) per satellite galaxy would be of Brunthaler et al. [19] to measure the PM of this dwarf needed to start resolving the question of stability of the galaxy: M31 plane. Given that even JWST is likely to fly before THEIA is approved, THEIA would only need to observe

µα∗,IC10 = 20 5µas/yr the remaining 9 in-plane satellites. As a result we would − ± advocate that the THEIA mission would be able to re- µδ,IC10,exp = + 31 8µas/yr. ± solve whether or not the M31 plane is a kinematically stable feature with a dedicated total of (400) hours of We have shown the value of this measurement in both O Figs. 1 and 2. The 1σ, 2σ and 3σ error-ellipsoids are observation. the result of combining the error bars from the PM of M31 and PM of IC10 in quadrature. For heavier M31, Fig. 1, the bound orbit is within 2σ and technically still V. CONCLUSION not ruled out. However, for light M31, all bound orbits with apsis less than 500kpc are ruled out by 3σ. In both We have collected the available data on the three- scenarios, all in-plane orbits are ruled out by at least 3σ dimensional position and line-of-sight velocity for 20 of 6

Andromeda I Andromeda III Andromeda IX Andromeda XI Andromeda XII 100 50 25 50 50 50 0 0 0 0 0 25 −

yr] 50 50 / 50 50 50 − − as − − − µ [ 75 δ − µ 100 100 100 100 − − − − 100 − 150 150 150 150 125 − − − − − 150 200 200 200 − − − − 200 − 175 − 100 0 100 200 0 100 200 0 100 200 50 0 50 100 150 200 0 50 100 150 − − Andromeda XIII Andromeda XIV Andromeda XVI Andromeda XVII Andromeda XXV

50 100 0 0 50 50 0 50 50 − 0 0 − yr]

/ 50 − 100 as − 50 µ 50 [ − − δ 100

µ − 100 150 100 − − − 100 − 150 150 150 − 200 − − − 150 200 − − 200 250 50 0 50 100 150 − 0 50 100 150 200 − 0 100 200 100 0 100 200 50 0 50 100 150 200 − − − Andromeda XXVI Andromeda XXVII And XXX NGC 147 NGC 185

50 50 50 50 50

0 0 0 0 0 yr] / 50 as 50 50 50 50 − µ − [ − − − δ µ 100 100 100 100 100 − − − − −

150 150 150 − 150 150 − − − − 200 200 − 0 100 200 − 0 100 200 50 0 50 100 150 200 50 0 50 100 150 200 50 0 50 100 150 200 − − − M32 NGC 205 IC 10 LGS 3 150 75 100 100 0 50 50 25 50 25 − 0 50 0 0 − yr]

/ 75 as 50 50 25 − µ [ − − − δ 100 µ 100 100 50 − − − − 125 150 150 75 − − − − 150 200 100 − − 200 − − 175 250 125 − − − 100 0 100 200 100 0 100 200 0 50 100 150 0 50 100 150 − − µα [µas/yr] µα [µas/yr] µα [µas/yr] µα [µas/yr] ∗ ∗ ∗ ∗

FIG. 1. Predicted PMs of satellites of M31. The blue region indicates PMs that are consistent with the satellite staying within {1, 2, 3}σD of the plane from darkest to lightest. The red region indicates PMs for which the satellite comes closer than 15kpc to the center of the M31 galaxy and is very likely to get disrupted. The orange region indicates PMs for which the satellite wonders off more than 500kpc from the M31 galaxy. All of the previous regions are marginalised over one sigma band of distance measurements. The grey lines indicate the regions for which the angle between the satellite’s angular momentum and the normal to the plane of satellites is less than 30 degrees (for the central value of satellite distance). The dotted lines indicate L~ andn ˆ are almost aligned, while the dashed lines enclose a region in which they are anti-aligned. Finally, the green circle indicates the uncertainty on the PM of the M31 galaxy itself. The top three rows (first 15 satellites) have been identified by Ibata et al. [1] as very likely members of the plane. The bottom row (last five satellites) are possible members. The plot for IC1613 is missing because there are no proper motions that keep IC1613 within 500 kpc of M31. The plot of IC10 also shows the measurement of its proper motion by Brunthaler et al. [19] and the 1σ, 2σ and 3σ combined error-ellipsoids are shown as solid black, dash-dotted black and dotted black.

the satellites of M31, 15 of which are believed to form a satellites and provide a benchmark for future astronomi- narrow planar structure. We have calculated the range cal measurements. Some of these proper motions will be of possible PMs for each satellite consistent with this published soon (NGC147 and NGC185) and others have plane being a stable structure and the satellite being a been given guaranteed observation time by JWST (An- member of the structure, by requiring the corresponding dromeda I, III, XIV, and XVII). On top, the proposed orbits to lie close to the plane. The PMs are consistent THEIA mission [24] would be capable of delivering the with the direction of rotation reported in Ibata et al. resolution required to determine whether the true PMs [1]. Our results consist of predictions of PMs of these are consistent with the sets presented here. This would 7 help resolve the question of whether the plane is rota- Pawlowski for many insights that made this paper much tionally stabilised or a temporary structure. We have better. We would also like to thank Roeland van der compared the previous measurement of the proper mo- Marel for his help. We would like to thank Steven Abel tion of IC10 and we conclude it is highly unlikely to be for support for this project as BH’s summer advisor. JS a proper member of the M31 plane of satellites. would like to thank the COFUND scheme and the IPPP Furthermore, if the measured PMs indicate the plane for his funding. BH would also like to thank the IPPP is at least partially stabilised, we could use the PMs to summer fellowship program for partial support. We have derive a bound on the mass of M31 and form an inde- also beneﬁted from the availability of the Astropy soft- pendent estimate of the PM of M31. ware package [25, 26].

ACKNOWLEDGEMENTS

We would like to thank Jan Scholtz and Carlos Frenk for fruitful discussions. We would like to thank Marcel

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Andromeda I Andromeda III Andromeda IX Andromeda XI Andromeda XII 50 50 50 20 0

0 0 0 25 40

] 50 r

y 60

/ 50 50 50 s

a 75 [ 80 100 100 100 100

125 100 150 150 150 150 120 50 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 20 40 60 80 100

Andromeda XIII Andromeda XVI Andromeda XVII Andromeda XXV 25 25 50 20 0 50 0 25 0 75 20 ] r

y 50 / 100 40 s

a 50

[ 75 125 60 80 100 150 100 100 125 175 150 120 150 200 0 50 100 150 50 100 150 200 50 0 50 100 150 0 50 100 150

Andromeda XXVI Andromeda XXVII And XXX NGC 147 NGC 185

50 25 25 20 25 25 0 0 0 0 0 20 25

] 25 25 r 25 y / 40 50 s a 50 50 50 [ 60 75 75 75 75 80 100 100 100 100 100 125 125 125 120 125 150 150 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150

M32 NGC 205 IC 10 LGS 3 20 50 40 50 40 20 0 0 0 60 ] r y / s 50 20 80

a 50 [ 40 100 100 100 60 120 150 150 80 140 100 200 50 0 50 100 150 200 50 0 50 100 150 200 50 0 50 100 25 50 75 100 125 * [ as/yr] * [ as/yr] * [ as/yr] * [ as/yr]

FIG. 2. Predicted proper motions of satellites of M31 with the ansatz MM31 = MMW. The color scheme is identical to the one used in Fig. 1. The plots for IC 1613 and Andromeda XIV are missing because there are no bound solutions for light MM31.

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