Dwarf Galaxy Planes in the Local Group

Dwarf Galaxy Planes: the discovery of symmetric structures in the Local Group

Marcel S. Pawlowski, Pavel Kroupa, Helmut Jerjen
Argelander Institute for Astronomy, University of Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany
Research School of Astronomy and Astrophysics, Australian National University, Mt Stromlo Observatory,
Cotter Rd., Weston ACT 2611, Australia
E-mail: mpawlow@astro.uni-bonn.de
Accepted 2013 July 23. Received 2013 July 23; in original form 2013 May 14

Both major galaxies in the Local Group (LG) are surrounded by thin planes of mostly co-orbiting satellite galaxies, the vast polar structure (VPOS) around the Milky Way (MW) and the Great Plane of Andromeda (GPoA) around M31. We summarize the current knowledge concerning these structures and compare their relative orientations by re-determining their properties in a common coordinate system. The existence of similar, coherent structures around both major LG galaxies motivates an investigation of the distribution of the more distant non-satellite galaxies in the LG. This results in the discovery of two planes (diameters of 1–2 Mpc) which contain almost all nearby non-satellite galaxies. The two LG planes are surprisingly symmetric. They are inclined by only 20 degrees relative to the galactic disc of M31, are similarly thin (heights of  kpc) and have near-to-identical offsets from the MW and from M31. They are inclined relative to each other by 35 degrees. Comparing the plane orientations with each other and with additional features reveals indications for an intimate connection between the VPOS and the GPoA. They are both polar with respect to the MW, have similar orbital directions and are inclined by about  degrees relative to each other. The Magellanic Stream approximately aligns with the VPOS and the GPoA, but also shares its projected position and line-of-sight velocity trend with a part of the dominating structure of non-satellite dwarf galaxies. In addition, the recent proper motion measurement of M31 indicates a prograde orbit of the MW-M31 system, the VPOS and the GPoA. The alignment with other features such as the Supergalactic Plane and the over-density in hypervelocity stars are discussed as well. We end with a short summary of the currently proposed scenarios trying to explain the LG galaxy structures as either originating from cosmological structures or from tidal debris of a past galaxy encounter. We emphasise that there currently exists no full detailed model which satisfactorily explains the existence of the thin symmetric LG planes.

Galaxy: halo – galaxies: dwarf – galaxies: individual: M31 – galaxies: kinematics and dynamics – Local Group – Magellanic Clouds
pagerange: Dwarf Galaxy Planes: the discovery of symmetric structures in the Local GroupReferencespubyear: 2012

1 Introduction

1.1 The Milky Way satellite system

Signs for the existence of what now is called the vast polar structure (VPOS) of satellite object around the MW were first reported by Lynden-Bell (1976) and Kunkel & Demers (1976). Lynden-Bell (1976) discovered that a number of MW satellite galaxies (Draco, LMC, SMC, Ursa Minor), globular clusters (Palomar 1 and 14) and streams align with the orbital plane of the Magellanic Clouds. Lynden-Bell (1982) identified a second possible group consisting of the dwarf spheroidal (dSph) galaxies Fornax, Leo I, Leo II and Sculptor, which all lie along a common great-circle as seen from the MW centre, and termed this second association the Fornax-Leo-Sculptor (FLS) stream.

These early discoveries of coherent structures, made before the advent of the currently prevailing cosmological model based on dark energy and cold dark matter (CDM), were thought to be related to planes of tidal debris from which new stellar systems formed. Thus, initially the planar structures were seen as an indication that many of the MW satellites are what today are termed tidal dwarf galaxies (TDGs, e.g. Barnes & Hernquist 1992, Elmegreen, Kaufman & Thomasson 1993, Duc & Mirabel 1998, Wetzstein, Naab & Burkert 2007, Bournaud 2010, Duc et al. 2011). Under this assumption, Kroupa (1997) modelled dark matter free MW satellite galaxies. One of these models, when compared to the Hercules satellite galaxy discovered later, turned out to be one of the few successful predictions concerning satellite galaxies (Kroupa et al., 2010).

However, later studies have focussed on comparing the MW satellite galaxies to the expectations for primordial dwarf galaxies residing in dark matter sub-haloes, revealing a number of unsuccessful predictions of the CDM model.These include the predicted central dark matter peak (core/cusp problem Dubinski & Carlberg, 1991), the large predicted total number of satellites (missing satellite problem, Klypin et al., 1999; Moore et al., 1999), the predicted existence of very concentrated, massive satellites (missing bright satellites or too big to fail problem, Bovill & Ricotti, 2011; Boylan-Kolchin et al., 2011), and the predicted internal dark matter distribution (Strigari et al., 2008).

Kroupa, Theis & Boily (2005) analysed the spatial distribution of the 11 brightest, ’classical’ MW satellites and compared it with the expected distribution derived from CDM models. They found that all MW satellites reside within a single thin plane, and that this distribution is inconsistent with a near-isotropic one expected for cosmological dark matter sub-haloes. From this they concluded that the MW satellite galaxies could be better understood as tidal dwarf galaxies with a common origin in a galaxy interaction instead of dark matter dominated primordial dwarf galaxies of cosmological origin, therefore returning the discussion back to the TDG scenario. More sophisticated studies of the spatial distribution of MW satellite galaxies, including the fainter MW satellites discovered in the meantime, confirmed the existence of the planar distribution (Metz, Kroupa & Jerjen 2007, Metz, Kroupa & Jerjen 2009, Kroupa et al. 2010, Pawlowski, Pflamm-Altenburg & Kroupa 2012a).

With increasing evidence that the satellite planarity is indeed significant (Metz et al., 2007) and that the satellites preferentially orbit within the plane (Metz, Kroupa & Libeskind 2008, Pawlowski & Kroupa 2013), the attention has shifted towards identifying possible mechanisms which could give rise to flattened and even coherently orbiting sub-halo populations within a CDM cosmology. One suggested mechanism is the accretion of primordial dwarf galaxies in groups (Li & Helmi, 2008; D’Onghia & Lake, 2008; Deason et al., 2011), such that the galaxies would then orbit their host in a common direction. Metz et al. (2009) have refuted these claims by demonstrating that observed dwarf galaxy associations are much too extended to be able to form thin VPOS-like planes. In addition, as almost all MW satellites lie close to the same plane they would have had to be accreted as a single group. However, Wang, Frenk & Cooper (2013) demonstrate that the majority of the 11 most-massive (in stellar mass) satellites in their high-resolution CDM simulations must have been accreted individually. Based on the dynamical friction time scale of the MW satellite galaxies, Angus, Diaferio & Kroupa (2011) also argue that a recent accretion of the satellites does not work out.

Another attempt to reconcile the planar MW satellite distribution with expectations from cosmological models was the suggestion that luminous sub-haloes are accreted along dark matter filaments (Libeskind et al., 2005; Libeskind et al., 2010; Lovell et al., 2011). This claim is disputed by Pawlowski et al. (2012b), who demonstrate that the distribution of orbital poles of sub-haloes that have been accreted onto a host via filaments does not reproduce quasi-planar distributions with a coherent rotation comparable to that of the VPOS. One of the underlying inconsistencies of the filamentary accretion scenario is that cosmological dark matter filaments are more extended than the virial radius of the host galaxy dark matter halo (Vera-Ciro et al., 2011), such that they are orders of magnitude too wide to be responsible for the formation of a structure only a few 10 kpc thin, a fact that has already been pointed out by Kroupa et al. (2005). Nevertheless, dark matter filaments have prevailed as a frequently mentioned mechanism related to the formation of thin planes of satellite galaxies (Keller, Mackey & Da Costa 2012, Tully 2013, Wang et al. 2013).

Recent studies populating simulated dark matter sub-haloes with luminous satellites via semi-analytic galaxy formation models demonstrate that the positional flattening of the MW satellite system is unlikely in a CDM context (Starkenburg et al., 2013; Wang et al., 2013). These studies over-estimate the agreement of the simulated results with the observed situation. One reason is that they only consider the flattening in the satellite positions, but not the comparable flattening in velocity space (Metz et al., 2008; Fouquet et al., 2012; Pawlowski & Kroupa, 2013), even though the sub-halo velocities can be extracted from the simulations. Furthermore, these studies only consider the distribution of the brightest satellites, but the less luminous ones follow the same polar structure (Metz et al., 2009; Kroupa et al., 2010).

In addition to the MW satellite galaxies, those globular clusters classified as young halo objects (Mackey & van den Bergh, 2005) follow a planar distribution which aligns with that of the satellite galaxies (Keller et al., 2012; Pawlowski et al., 2012a). The preferred alignment of stellar and gaseous streams within the MW halo with the VPOS provides additional evidence that many of the MW satellite objects orbit within the VPOS (Pawlowski et al., 2012a).

1.2 The M31 satellite system

The importance of VPOS-like satellite galaxy planes has been stressed by Wang et al. (2013), who write: ’[a] larger sample of satellites around other galaxies will test the tidal formation hypothesis of Pawlowski et al. (2012) in which highly flattened configurations are easily achieved and should therefore be the norm. If, on the other hand, the CDM model is a realistic description of nature, then the average satellite configurations should be only moderately flattened […]’. Such an additional sample of satellites around another galaxy is the nearby M31 system. Several early searches for a preferred planar distribution of M31 satellites have been carried out (McConnachie & Irwin, 2006; Koch & Grebel, 2006; Metz et al., 2007). These initial studies were hampered by the small number of known satellite galaxies but have already identified possible planar structures.

An analysis of the spatial distribution of satellite galaxies should be easier for our neighbouring galaxy than for the MW. In contrast to the searches for satellites around the MW not the entire sky must be surveyed for M31 satellite galaxies but only the region towards M31. The Pan-Andromeda Archaeological Survey (PAndAS, McConnachie et al., 2009) is such a survey covering an area of 150 kpc radius around M31 in projection. It has resulted in the discovery of numerous M31 satellite galaxies. In addition to the positions of galaxies on the sky, their distances have to be known in order to make the discovery of structures in their full spatial distribution possible. One of the first large catalogues of distances to Local Group dwarf galaxies, measured with the tip of the red giant branch method, was provided by McConnachie et al. (2005). Recently, accurate distances to the M31 satellites were determined in a homogeneous way by Conn et al. (2011); Conn et al. (2012). The resulting dataset of M31 satellite galaxies with coordinates and distances allowed a detailed analysis of their spatial distribution (Ibata et al., 2013; Conn et al., 2013). This resulted in the discovery of a vast thin plane of satellites around M31 at very high statistical significance (Conn et al., 2013). About half of M31’s satellite galaxies can be associated with this structure, which is seen edge-on from the MW. The line-of-sight velocities of the satellites in the structure indicate that most of them follow a common orbital sense (Ibata et al., 2013). Like the VPOS, the satellite plane around M31 is rotating around its host galaxy.

1.3 The satellite planes and the Local Group

In any scenario addressing the formation of thin planes of satellite galaxies, a causal connection of the planes to a larger-scale structure requires that structure to have a similarly narrow spatial extend as the planes. The distribution of the galaxies within the LG has been investigated in search for a preferred plane in several studies (Hartwick, 2000; Sawa & Fujimoto, 2005; Pasetto & Chiosi, 2007). They have in common that a preferred planar direction is generally found. However, they either focus on the overall distribution of all (isolated) LG galaxies, resulting in structures with a thickness of several 100 kpc (Hartwick, 2000; Pasetto & Chiosi, 2007), or they define a thin plane by visual inspection of the galaxy distribution only, resulting in many distant LG galaxies being outliers from the reported plane (Sawa & Fujimoto, 2005).

The aim of the current study is to compile our present knowledge of the satellite galaxy planes in the LG. We do this by re-determining the plane parameters in a common coordinate system in order to facilitate and encourage the comparison with other structures. We furthermore discuss the distribution of non-satellite galaxies in the LG in the context of the VPOS and GPoA and suggest the existence of two very symmetric planes of dwarf galaxies in the LG. At least one of these is essentially connecting the MW and M31 in position and velocity space. Almost all presently known nearby galaxies in the LG can be associated to one of several planes with heights of only a few 10 kpc each. We discuss the relative orientations of all planes and compare them with a number of prominent features, including the Magellanic Stream, the orbital pole of the MW-M31 system deduced from the recent proper motion measurement for M31 and the hypervelocity star over-density in the MW halo. These comparisons indicate that many of these features might be intimately related.

Sect. 2 presents the dataset used in the following analysis and describes the employed methods. In Sects. 3 and 4 we re-analyse the planes found in the satellite galaxy distributions around the MW and M31, respectively. In Sect. 5 we expand upon the previous works and investigate the distribution of known non-satellite dwarf galaxies in the LG, which reveals two symmetric planes of galaxies. Sect. 6 discusses the remaining dwarf galaxies not associated with any of the planes and briefly mentions a possible second preferred plane of M31 satellites. The results are discussed in Sect. 7, in particular analysing the (mutual) orientations of the found planar galaxy structures and their relation to other features. Possible avenues to be explored in order to find an explanation for the structured LG dwarf galaxy populations are then discussed in Sect. 8, and finally the conclusions are given in Sect. 9.

2 The dataset and methods

Table 1 compiles descriptions of frequently used symbols which will be introduces in the following section.

Symbol Description
Distance of a galaxy from the centre of the MW
Distance of a galaxy from the centre of M31
Distance from midpoint between MW and M31
Centroid of a galaxy sample
Normal direction to a plane-fit, expressed in Galactic
Longitude  and Latitude
Offset (minimum distance) of a plane from the MW
Offset (minimum distance) of a plane from M31
RMS height of galaxy sample from its best-fitting plane
Ratio of short to long axis of a galaxy sample
Ratio of intermediate to long axis of a galaxy sample

Symbols (first column) frequently used in this paper, their description (second column).

Table 1: Symbol definitions

2.1 The Local Group Galaxy Dataset

[kpc] [kpc] [kpc] [kpc] [kpc]

The Galaxy
193 -312 144 0 0
Canis Major 181 -318 143 1 1
Sagittarius dSph 210 -309 138 2 2
Segue (I) 173 -321 162 2 2
Ursa Major II 162 -300 163 4 5
Bootes II 199 -313 183 1 1
Segue II 161 -298 123 2 2
Willman 1 165 -304 176 6 8
Coma Berenices 182 -316 187 4 4
Bootes III 194 -305 189 2 2
LMC 192 -353 117 2 2
SMC 209 -350 99 3 4
Bootes (I) 207 -312 206 2 2
Draco 188 -249 187 6 6
Ursa Minor 170 -260 197 3 4
Sculptor 187 -321 59 5 6
Sextans (I) 156 -369 202 4 4
Ursa Major (I) 132 -292 223 4 5
Carina 168 -408 104 6 6
Hercules 277 -261 223 12 13
Fornax 151 -363 10 12 13
Leo IV 178 -396 273 6 7
Canes Venatici II 176 -293 303 4 4
Leo V 171 -404 296 10 10
Pisces II 208 -190 11 0 0
Canes Venatici (I) 195 -275 358 10 10
Leo II 115 -370 359 14 14
Leo I 69 -431 336 15 16
Andromeda -193 312 -144 25 26
M32 -202 328 -157 74 82
Andromeda IX -211 291 -114 24 25
NGC 205 -208 349 -153 26 27
Andromeda I -171 264 -169 24 24
Andromeda XVII -163 285 -87 26 38
Andromeda XXVII -215 370 -104 45 47
Andromeda III -145 273 -187 24 25
Andromeda XXV -197 371 -78 44 46
Andromeda XXVI -163 338 -49 41 43
Andromeda V -256 290 -58 28 29
Andromeda XI -153 234 -213 17 17
Andromeda XIX -130 345 -233 147 31
Andromeda XXIII -278 220 -163 45 47
Andromeda XX -73 298 -191 53 42
Andromeda XIII -212 299 -274 19 20
Andromeda X -188 205 -63 39 25
Andromeda XXI -108 414 -128 26 23
Andromeda XXXII -202 344 -7 48 52
NGC 147 -142 257 -22 27 29
Andromeda XXX -153 261 -11 77 32
Andromeda XIV -177 246 -291 181 22
Andromeda XII -248 381 -299 135 39
Andromeda XV -165 137 -116 34 80
Andromeda II -173 131 -173 18 18
NGC 185 -121 201 -10 25 26
Andromeda XXIX -28 279 -230 70 78
Triangulum -292 189 -277 22 23
Andromeda XXIV -169 143 -24 32 34
Andromeda VII -66 396 12 34 36
IC 10 -200 382 98 43 45
Andromeda XXXI 44 401 -74 41 43
LGS 3 -164 154 -359 24 25
Andromeda VI 10 295 -320 25 26
Andromeda XXII -331 249 -372 141 30
Andromeda XVI -51 25 -98 30 44
Andromeda XXVIII 173 297 -113 58 171
IC 1613 -53 -26 -514 41 43
Phoenix 190 -461 -243 19 20
NGC 6822 578 -125 -1 17 17
Cetus 140 -94 -578 24 25
Pegasus dIrr 129 353 -490 29 30
Leo T -63 -484 432 19 20
WLM 249 -57 -751 34 35
Andromeda XVIII -286 750 -209 44 40
Leo A -281 -453 776 43 45
Aquarius 943 201 -413 39 40
Tucana 664 -674 -509 48 50
Sagittarius dIrr 1140 56 -155 85 92
UGC 4879 -778 -48 1071 25 25

Positions of the LG galaxies in -, - and -coordinates of the Cartesian coordinate system defined in Sect. 2.1. The position uncertainties are along the line connecting the Sun (situated at  kpc) and the respective galaxy. They are given as , denoting the 1 radial distance uncertainty towards the Sun, and , denoting the 1 radial distance uncertainty away from the Sun.

Table 2: Positions of galaxies in the Local Group

The analysis presented in the following is based on the catalogue of nearby galaxies as compiled by McConnachie (2012) (see also Mateo 1998). It includes information on all known galaxies within 3 Mpc from the Sun, which have distance estimates based on resolved stellar populations. We use the galaxy positions, radial distances and line-of-sight velocities of the LG galaxies as provided by the most-recent online version of the tables by McConnachie (2012)111We make use of the data tables updated on June 17, 2013, as provided online at https://www.astrosci.ca/users/alan/Nearby_Dwarfs_Database.html.. To this we add the recently-published line-of-sight velocity for Andromeda XXIX (Tollerud et al., 2013) for which no velocities are provided in the catalogue yet. For consistency with the catalogue provided by McConnachie (2012) we treat the Canis Major over-density as a MW satellite (e.g. Martin et al., 2004), but note that it might be a substructure of the MW caused by the warp of the Galactic disc (e.g. Momany et al., 2006). Due to its close proximity to the centre of the MW, excluding this object from the analysis of the VPOS would not significantly affect our results.

The catalogue is most likely still incomplete because it was not constructed based on a deep, homogeneous all-sky survey (which does not exist yet). In the following analysis, this prevents us from determining the significances of the known and suggested galaxy planes in a meaningful way. We therefore will not attempt a detailed statistical analysis like that carried out for the M31 satellites by Conn et al. (2013). However, as the catalogue compiles all known, nearby dwarf galaxies, it provides us with the most-complete picture of our neighbourhood to date and is therefore the best available dataset to search for possible associations of dwarf galaxies in planar structures. Future discoveries of additional galaxies will then provide observational tests of these suggested structures.

Throughout this paper, we adopt a Cartesian coordinate system (, , ), with the -axis pointing toward the Galactic north pole, the -axis pointing in the direction from the Sun to the Galactic centre, and the -axis pointing in the direction of the Galactic rotation. We chose the origin of the coordinate system to be the midpoint between the MW and M31, which we denote the centre of the LG. We assume a distance of 8.5 kpc between the Sun and the centre of the MW, such that M31 is at a distance of 788 kpc from the MW centre. We decided to chose the midpoint and not the LG barycentre for simplicity. The exact mass distribution in the LG is still uncertain, in particular the reported mass-ratio  between M31 and the MW varies between about 0.8 and 2.0 (van der Marel & Guhathakurta, 2008). As the two galaxies have approximately similar total (halo) masses in any case, the midpoint between the two galaxies can be assumed to approximate the barycentre. All positions given relative to this origin can be converted to a MW- or M31-origin by subtracting the respective galaxy’s position in  coordinates, i.e.  kpc or  kpc. The positions of the LG galaxies in this Cartesian coordinate system are compiled in Tab. 2.

We chose this MW-based coordinate system to ease the comparison of the different galaxy structures. In particular, directions (for example of normal vectors) can then also be expressed in Galactic longitude and latitude. This allows the comparison of different datasets and shall encourage the reader to compare the found dwarf galaxy planes with additional data not included in this study. As an example, we express the orientations of the galactic discs of the MW and M31 via their spin directions. The MW spin points into the direction of the negative -axis (), which in Galactic coordinates corresponds to . We determine M31’s spin vector direction by adopting the same parameters as Conn et al. (2013): a position angle  and an inclination of  (de Vaucouleurs, 1958). This results in a M31 spin direction pointing to  (e.g. Gott & Thuan, 1978; Raychaudhury & Lynden-Bell, 1989), i.e. .

In the following sections, all galaxies of the McConnachie (2012) catalogue which are within 1.5 Mpc of the origin will be considered LG members. This radius was chosen because it is about half way to the next nearby galaxy groups such as Sculptor (fig. 9 of Jerjen et al., 1998). Thus, beyond  Mpc the LG’s gravitational influence can be expected to become less important (assuming a similar mass for the LG and these galaxy groups). In addition, this distance corresponds to approximately two times the current distance between the MW and M31, the most prominent distance scale in the LG. It is also similar to the radius of the zero-velocity surface around the LG. The observationally inferred radius is  Mpc (Karachentsev et al., 2009), while the turnaround radius of  Mpc predicted from the LG mass estimated via the timing argument (van der Marel & Guhathakurta, 2008) coincides well with our adopted radial cut. Finally, there is a gap in the distribution of galaxies in our sample at about this radius. The galaxy furthest from the origin but within our adopted radial cut is UGC 4879, which lies at a distance of 1.3 Mpc from our adopted origin, while NGC 3109, the next galaxy further away, is already at a distance of 1.6 Mpc.

With this distance cut, our sample consists of 78 galaxies. We split these up into three categories: hosts, satellites and non-satellites. The two most-massive galaxies in the LG, the MW and M31, are considered to be hosts. They each harbour a large number of satellite galaxies. To determine which galaxies we consider to be satellites, we introduce another distance criterion. For each galaxy (except the MW and M31), the distance to the MW () and to M31 () is determined. The minimum of these two values is the distance to the nearest host (). In Fig. 1 we plot the cumulative distribution of galaxies against . The vertical dashed line indicates our distance criterion for satellite galaxies: all galaxies closer than 300 kpc to either host galaxy are considered to be satellites, all other galaxies are considered to be relatively isolated non-satellite galaxies. This distance of 300 kpc was chosen for three reasons: (1) The cumulative distance distribution shows that there is a small gap in distances close to this value (no known galaxy lies at a distance between 270 and 320 kpc). (2) At this distance the slope of the cumulative distribution changes, becoming shallower for larger radii222This change might, in part, be due to the fact that the surveys searching for dwarf galaxies in the LG focus on the volumes around the two host galaxies. (3) The radius corresponds to the viral radii of the dark matter haloes assumed to surround the MW (308 kpc) and M31 (300 kpc) (van der Marel et al., 2012).

To conclude, our sample consists of two host galaxies, 15 non-satellite galaxies, 27 MW satellites and 34 M31 satellites.

Figure 1: Cumulative distribution of dwarf galaxies within a certain distance from their host, which is the closest major LG galaxy (MW or M31). Galaxies left of the dashed vertical line (within 300 kpc of either the MW or M31) are categorized as satellite galaxies, galaxies at larger distances are categorized as non-satellite LG galaxies.

2.2 Plane-fitting technique

The best-fitting plane for a set of  galaxy positions  is determined following the method described in Metz et al. (2007). The positions of the galaxies are not weighted, i.e. each galaxy has the same weight in the plane fit.

At first, the centroid  of all positions is calculated as

Then the moments of inertia tensor around this centroid is constructed as

where  is the unit matrix and  is the transposed version of the vector . The eigenvalues and eigenvectors of  are determined. The eigenvector corresponding to the largest eigenvalue is the normal to the plane containing the centroid333Metz et al. (2007) wrongly state that the eigenvector corresponding to the smallest eigenvalue gives the normal of the best-fitting plane., while the eigenvectors corresponding to the intermediate and smallest eigenvalue correspond to the intermediate and longest axis of the distribution. This method has been tested against the one used in Kroupa et al. (2010) and Pawlowski et al. (2012a) and both were found to give the same results, with the method used in this analysis being much more efficient.

In addition to the centroid  and the normal vector to the best-fitting plane , we determine , the root-mean-square (RMS) thickness perpendicular to the best-fitting plane. We furthermore measure the RMS extend of the distribution along the intermediate and the long axis. With this, we determine the RMS axis ratios between shortest and longest () and intermediate and longest axis () of the distribution. A small value of  indicates either an oblate distribution (a thin plane) if  is large, or a narrow prolate distribution (filament-like) if is similarly small (). Finally, we also determine the offset (distance perpendicular to the plane) of each galaxy in our LG sample from the plane.

We want to caution the reader that the assumption of a perfect planar alignment of the satellites might be too simplistic. A coherent distribution of satellite galaxies might be affected by precession, which is more significant for the satellites close to a host galaxy than for those far away, or a satellite plane might be ’bend’ if it is initially offset from the centre of mass, to name only two possibilities.

2.3 Effects of galaxy distance uncertainties

All dwarf galaxy positions are determined from their heliocentric distance modulus, as reported by McConnachie (2012), and the distance uncertainties are determined from the reported distance modulus uncertainties. We assume that the MW centre is at a distance of 8.5 kpc from the Sun, but the exact distance is not important for our analysis because a change in this value simply translates all galaxy positions by the same distance, preserving their mutual orientations. It only changes the position of the MW and results in a minor change in the position of the origin of our coordinate system, i.e. the reported distances between the planes and the MW would change. However, these changes are on the order of less than 1 kpc, which is negligible for the typical distances of many 100 kpc in the LG.

The uncertainties in the position of the galaxies are dominated by their uncertain radial distance from the Sun. We assume that the angular position on the sky is accurate. To determine the effect of the distance uncertainties, in the following all parameters are determined by sampling the galaxy distances 1000 times. For each realisation, the galaxy distances are generated by starting with their most-likely distance as reported in the catalogue by McConnachie (2012). This distance is then increased (or decreased) by a value sampled from a Gaussian distribution with a width set to the positive (negative)  distance uncertainty. We therefore assume that the distance uncertainties follow a Gaussian distribution, which can be asymmetric around its peak to account for differing positive and negative uncertainties.

The parameters determined in the analysis are stored for each of the 1000 realisations. Unless mentioned otherwise, the parameters reported in the following (e.g. in Tables 3, 4 and 5) are the mean values determined by averaging over the values of all realisations. We report the standard deviation of the parameters around this average as the uncertainties.

For the normal vector directions, we determine the spherical standard distance  (Metz et al., 2007; Pawlowski et al., 2012b) of the normal directions for the realisations relative to the normal direction determined for the most-likely galaxy positions. The spherical standard distance is a measure for the clustering of vectors. It is defined as

where  are the normal direction unit vectors,  is the normal vector determined by fitting a plane to the most-likely galaxy positions and ’’ denotes the scalar product of the vectors. Note that the formula deals with axial data and therefore includes the absolute value of the scalar product, in contrast to the case in which vectorial data such as an orbital pole (direction of angular momentum) is used.

2.4 4-galaxy-normal density plots

The plane-fitting method described in Sect. 2.2 determines the parameters of a plane fitted to a pre-chosen group of galaxies. This method is not suitable to determine whether the chosen group defines the most-prominent planar arrangement in a sample of galaxies. We therefore also investigate whether, in a given galaxy sample, there are signs for a dominant plane defined by a sub-sample only. This allows a consistency check by comparing whether this sub-sample constituting the dominant plane is similar to the chosen group of galaxies.

This task is approached with a method based on constructing planes for many small sub-samples of galaxies. We draw all possible combinations of four galaxies from a given galaxy sample. For each combination, we fit a plane as described in Sect. 2.2 and record the normal-axis direction  to the plane (which we call the 4-galaxy-normal) and the axis ratios. For a total sample of  galaxies, there are possible combinations of 4 galaxies each. If several galaxies in a sample lie within a common plane, different combinations of four galaxies from this plane will result in very similar 4-galaxy-normal directions. Thus, when plotting the density of 4-galaxy-normal directions on a sphere we can identify the dominant plane orientation by looking for an over-density of normal directions. The density distribution for all possible normal directions  is plotted in a Galactic coordinate system, weighting each 4-galaxy-normal with the logarithm of the ratio between the shortest and the sum of the intermediate and long axis

This weighting emphasizes those normal directions which are associated to plane-like distributions, i.e. short dimensions along the short axes  and large dimensions along the two remaining axes  and  of the distribution.

By determining which galaxies contribute to an over-density in the 4-galaxy-normal distribution we can identify galaxies as likely members of the dominant plane. Galaxies which do not contribute at all to an over-density can be excluded, because they can not lie within the respective plane: there is no combination with any other galaxies which would give a 4-galaxy-normal direction close to the plane normal.

To determine which galaxies contribute to a 4-galaxy-normal density peak, we proceed as follows. For a 4-galaxy-normal which lies within a defined angle (typically ) of a density peak, we determine the four contributing galaxies. Each of these galaxies is counted as contributing the 4-galaxy-normal’s weight to the peak. This is repeated for all 4-galaxy-normals which are close to the density peak. In the end, for each galaxy the sum of all its associated plane weights is determined. We then express this as a relative weight for each galaxy by normalising the weight contribution of the most-dominant galaxy to one. The resulting relative weights are plotted on a common axis for the satellite galaxies’ contribution to different peaks, revealing which galaxies contribute most to which peak direction.

To account for the uncertainties in the galaxy positions, 100 realisations sampling from the galaxy distance uncertainties as described in Sect. 2.3 are generated for each galaxy sample. Each of these 100 realizations contributes equally to the 4-galaxy-normal density plots and the determination of the dwarf galaxy contributions to the peak directions.

We have also tested this method (without weighting) using all possible combinations of only three galaxies, which always define a perfect plane. The resulting normal distribution plots look similar and also reveal the same preferred normal directions (4-galaxy-normal density peaks). However, as three points always define a plane, this method can not weight the normal directions. We have therefore chosen to use combinations of four points. It is possible to extend the analysis to combinations of more points, but this then de-emphasizes planes consisting of only few satellites, in particular when we deal with small-number samples of dwarf galaxies like the only 15 non-satellite galaxies.

The method we use is similar to the one employed recently by Conn et al. (2013), but there are important differences. As we investigate the distribution of dwarf galaxies in the whole LG, we do not assume a fixed point which the planes have to contain. In contrast to this, Conn et al. (2013), interested only in the satellite galaxies of M31, forced all planes to run through the centre of M31. Another difference is that we only consider combinations of four galaxies, while Conn et al. (2013) have used combinations of 3 to 7 galaxies in the plane construction. They found that the smaller combination sizes (3-4 satellites per plane fit) are particularly useful for identifying the thinnest planes, which thus supports our approach. Finally, Conn et al. (2013) have used a different weight, the inverse of the thickness of the fitted planes. We, however, investigate the distribution of dwarf galaxies not only around a host, but also within the much larger LG. We therefore decided to weight by the axis ratios of the fitted planes because this is a scale-free representation of the thinness of a planar distribution. As discussed in Sect. 4, our approach reproduces the findings of Conn et al. (2013), giving further confidence in the agreement of both methods.

3 The vast polar structure (VPOS) around the MW

Figure 2: Density distribution of the 4-galaxy-normal directions for our sample of 27 MW satellite galaxies, determined and weighted as explained in Sect. 2.4. They show one distinct peak (Peak 1) at , close to the MW equator, indicating a preferred polar orientation of the planes. A second, smaller peak (Peak 2) is found at , also close to the equator. Note that normal directions are axial, so each 4-galaxy-combination has two normals in opposite directions. For clarity and easier comparison, we only plot the 4-galaxy-normals in the centre of the plot between  and . Several other directions are also plotted: The direction of the normal to the plane fitted to all 27 MW satellites (VPOSall, dark-blue hexagon) lies in-between the two peaks, while the normal to the plane fitted to all satellites but Leo I, Hercules and Ursa Major (I) (VPOS-3, smaller, light-blue hexagon) coincides with the more-pronounced peak. The Magellanic stream normal (blue diamond, Pawlowski et al., 2012a) is close to this peak, too. This is also the case for the average direction of the orbital poles of the MW satellites for which proper motions have been measured (dark-blue star, Pawlowski & Kroupa, 2013). The blue circle around it indicates the concentration of the orbital poles around their average direction, as measured by the spherical standard distance (see Sect. 2.3) of the orbital pole distribution, which is . The normal to the plane fitted to all 30 young-halo globular clusters of the MW (cyan hexagon, Pawlowski et al., 2012a) coincides well with the second peak, but restricting the sample to the 20 young-halo GCs within 20 kpc of the MW results in a plane normal which is close to the major peak (small light cyan hexagon, Pawlowski et al., 2012a). Finally, the average of the normal directions fitted to streams in the MW halo (cyan diamond, Pawlowski et al., 2012a) also points into the general direction of the two peaks, but the spherical standard distance of these stream normals, illustrated by the cyan circle, is large ().
Figure 3: Relative contributions of the different MW satellites to the regions within  of the 4-galaxy-normal peaks in Fig. 3.

Within our sample of galaxies, 27 objects are satellites of the MW with a maximum radius from the MW of  kpc. A fit to all of them results in a best-fitting plane of RMS height  kpc, which is offset from the MW centre by  kpc. The distribution has axis ratios of  and . The normal  to the best-fitting plane points to , the uncertainty in this direction is given by the spherical standard distance of the normal directions for the different galaxy distance samples. It is . We adopt this plane fit as the ’VPOSall’. This plane fit reproduces the earlier results of Kroupa et al. (2010) ( kpc,  kpc,  pointing to ), even though the current sample includes updated radial distances for some of the satellites, four additional objects (Canis Major, Segue I and II and Bootes III) and lacks Pisces I.

Does the 4-galaxy-normal technique confirm this plane orientation as the preferred one? In Fig. 2 we plot the direction of the normal vector of the plane fitted to all 27 satellites. In addition, we include the normal vector to the plane of all 30 young halo globular clusters (YH GC), that of only those YH GCs closer than 20 kpc from the MW centre, and the average stream normal direction, as reported in Pawlowski et al. (2012a). We also plot the average orbital pole of the eight classical MW satellite galaxies which co-orbit in the VPOS (Pawlowski & Kroupa, 2013), which represents the normal direction to the average orbital plane of the MW satellites in the VPOS. The contours indicate the density distribution of the 17550 different possible 4-galaxy-normal directions. There is a pronounced density peak (Peak 1) at . This is close, but inclined by about , to the normal of the plane fitted to all satellite positions. As second, much shallower peak (Peak 2) at  coincides with the position of the YH GC normal.

Interestingly, these two peaks approximately agree with the two ’stream’ axis directions already discussed by Lynden-Bell (1982): his Magellanic stream axis, pointing to , and his FLS stream axis, pointing to . That the outer YH GCs coincide with the second peak / FLS stream axis direction has already been noticed by Majewski (1994). However, the first peak in Fig. 2 is much more pronounced than the second.

As discussed in Sect. 2.4, we can check which satellite galaxies contribute to the peaks. Fig. 3 plots the contributions of the different satellite galaxies to the 4-galaxy-normals in the regions  around the two peaks. Overall, the weight contributions to Peak 1 are larger than those to Peak 2, because the second peak is less pronounced than the first.

Almost all satellite galaxies contribute to Peak 1, with three marked exceptions: Leo I, Hercules and Ursa Major (I). For these three satellites there is almost no combination with any three of the other satellites which describes a plane with a normal pointing close to Peak 1. They are also the three satellite galaxies which have the largest vertical distance (,  and  kpc, respectively) to the best-fitting plane to all 27 MW satellites. All other satellites have distances of less than 40 kpc from the best-fitting plane. We have checked that the 4-galaxy-normal distribution for the 24 MW satellites without the three outlying ones does not show Peak 2 any more, while Peak 1 is still present.

It might therefore be worthwhile to exclude Leo I, Hercules and Ursa Major (I) from the plane fit. We have done so and fitted a plane to the remaining 24 satellites only, referring to this as the ’VPOS-3’. This plane is much thinner than the VPOSall,  kpc, and slightly more offset from the MW centre,  kpc. The distribution has axis ratios of  and . The normal  to this plane now points close to the 4-galaxy-normal density peak in Fig. 2, into the direction of  and the standard deviation of this direction for the different galaxy distance samples is . Thus, by excluding the three outlying galaxies, the orientation of the satellite galaxy plane fit changes by , but the best-fitting plane is polar in both cases. The VPOS-3 normal direction is much closer to the direction of the average orbital pole of the MW satellites (Metz et al., 2009; Pawlowski & Kroupa, 2013), to the normal of the best-fitting plane to the inner YH GCs and to the Magellanic Stream normal (Pawlowski et al., 2012a). These alignments might be seen as indications that the VPOS-3 is a better representation of the satellite structure surrounding the MW.

A similar analysis of the contributions to Peak 2 remains inconclusive. All MW satellite galaxies contribute to this peak to some degree. Excluding the five satellites contributing the least (Ursa Major (I), Ursa Minor, Pisces II, Leo V and Leo IV) results in a plane fit which has a normal pointing to  with an uncertainty of . It is offset from the MW centre by  kpc, has a RMS height of  kpc and axis ratios of , .

Among the seven satellites contributing the most to Peak 2 (Leo I, Leo II, Canes Venatici, Fornax, Canes Venatici II, Sextans and Sculptor) are all galaxies which Lynden-Bell (1982) identified to be in the FLS stream (Fornax, Leo I, Leo II and Sculptor), and also Sextans, which was reported to be in the FLS stream by Majewski (1994). Fitting a plane to these gives a normal pointing to  with an uncertainty of . It is offset from the MW centre by  kpc, has a RMS height of  kpc and axis ratios of , . Most of these seven satellites, however, substantially contribute to Peak 1, too.

In a similar manner we can select the seven satellites contributing most to Peak 1 (Pisces II, Carina, Leo V, Leo IV, Canes Venatici, Draco and Canes Venatici II). Fitting a plane to these gives a normal pointing to  with an uncertainty of . It is offset from the MW centre by  kpc, has a RMS height of  kpc and axis ratios of , .

It will require a more-complete census of the satellite galaxy population in the southern hemisphere, such as the Stromlo Missing Satellites Survey (Jerjen, 2010, 2012), to reveal whether the two-peak structure in the 4-galaxy-normal distribution becomes more pronounced. More tightly constrained proper motions for the dSphs will then allow to test whether the VPOS consists of two separate polar streams (the satellite orbital poles would cluster around the two peaks), whether the VPOS is better interpreted as one structure with a few unrelated objects (most orbital poles would point into one preferred direction), or whether the VPOS is one dynamical structure with an opening angle defined by the two peaks (the orbital poles would be distributed in between the two peaks). For many satellites, the current uncertainties in proper motion determinations result in orbital pole directions which are uncertain to  or more (Pawlowski & Kroupa, 2013), and are therefore still inconclusive.

For the following discussion, we adopt the parameters for the VPOSall (fitted to all 27 MW satellites) and the VPOS-3 (fitted to all MW satellites except Leo I, Hercules and Ursa Major). As the normal to the VPOSall lies in-between the two peaks in Fig. 2, we focus on this fit. If the two peaks indeed suggest the existence of two separate planar distributions around the MW, these planes would be inclined by  with respect to each other, and by  with respect to the VPOSall. Therefore, the error in the plane orientation we make by adopting the VPOSall is only  if there are indeed two planes. We also consider the VPOS-3, as its normal direction coincides with the dominant peak of the 4-galaxy-normal distribution, and it also agrees better with a number of additional features. In particular, it is aligned with the Magellanic Stream and the average orbital pole of the MW satellites, indicating that at least a number of satellites orbit preferentially within this plane. The resulting parameters of the VPOSall and VPOS-3 plane fits are compiled in Table 3, the distances of individual galaxies to the best-fitting planes are given in Table 4.

Do any of the non-satellite galaxies in the LG lie close to the satellite galaxy plane around the MW? The galaxy closest to the VPOSall is Phoenix, which has a distance of only  kpc. WLM, the next-nearest galaxy, already has a distance of more then twice this value ( kpc). Interestingly, both Phoenix and WLM are closer to the VPOS-3. Phoenix then has a distance of only  and WLM of  kpc, which is quite remarkable given the VPOS-3’s RMS height of only 20 kpc. Thus, Phoenix and WLM, which have distances from the MW of 415 and 930 kpc, respectively, are within  of the VPOS-3. All remaining dwarf galaxies are offset by more than 100 kpc from the VPOS-3, but due to their large distances from the MW some have relatively small angular distances from the VPOS-3 ( for Cetus,  for Andromeda XXVIII and Pegasus dIrr).

4 The Great Plane of Andromeda (GPoA)

Figure 4: Density distribution of the 4-galaxy-normal directions for our sample of 34 M31 satellite galaxies, determined and weighted as explained in Sect. 2.4. They show one very pronounced peak at . As in Fig. 2 only the 4-galaxy-normals in the centre of the plot between  and  are shown for clarity, not their mirrored counterparts. The spin direction of the galactic disc of M31 is indicated by the black diamond, and the M31 equator lies along the great circle  offset from this direction, plotted as a black line. Also shown are the directions of the normals for planes fitted to the whole sample of M31 satellites (light-red hexagon) and to the M31 satellite subsample defining the GPoA (dark-red hexagon). The light-red circle indicates the spherical standard distance of the normal direction distribution for all M31 satellites, resulting from varying the satellite distances. Its large extend () indicates that the best-fit plane for the full sample is only poorly defined, the full M31 satellite distribution is only mildly anisotropic.
Figure 5: Contributions of M31 satellites to the region within  of the dominant 4-galaxy-normal peak at  in Fig. 4. Compare with the similar plot shown in figure 13 of Conn et al. (2013). Our and their dwarf contributions are comparable, despite the slightly different method and our more numerous but less homogeneous M31 satellite sample.

In analogy to the MW satellite galaxies, we start by fitting a plane to all 34 dwarf galaxies that are considered to be M31 satellites. The parameters of the fit reveal that this distribution is only mildly anisotropic. The axis ratios are very similar to each other ( and ) and the RMS height of  kpc is comparable to the RMS radius of the M31 satellite distribution  kpc. The direction of the normal vector to the plane fit is very uncertain, the best-fitting direction for the different galaxy distance realizations varies by . Interestingly, the average plane normal  of the fits to the M31 satellites points to , roughly in the direction of the VPOSall normal and the normal of the plane defined by the YH GCs around the MW.

Following a detailed analysis of the M31 satellite galaxy positions, Ibata et al. (2013) and Conn et al. (2013) have identified a sub-sample of 15 out of their 27 M31 satellites which lie within a thin plane. Their analysis gives a very high significance for this discovery. The probability that a similar alignment occurs at random is only 0.13 per cent (Ibata et al., 2013). The structure’s significance rises to 99.998 per cent when also taking into account the line-of-sight velocities which reveal that 13 of the 15 plane members co-orbit. Most M31 satellites in the northern part of the plane recede from the MW while most in the southern part approach the MW relative to M31. We therefore tentatively adopt their sample of galaxies: Andromeda I, Andromeda III, Andromeda IX, Andromeda XI, Andromeda XII, Andromeda XIV, Andromeda XVII, Andromeda XXV, Andromeda XXVI, NGC 147, NGC 185, Andromeda XIII, Andromeda XXVII and Andromeda XXX.

Andromeda XVI, which is in the plane sample by Ibata et al. (2013) and Conn et al. (2013), has a distance of 323 kpc from M31 and is therefore considered a non-satellite according to our criteria (Sect.2.1). For reasons of consistency of our distance criterion we exclude Andromeda XVI from the galaxy sample, but note that the plane fitting results do not change significantly if Andromeda XVI is included.

The plane fitted to the resulting 14 dwarf satellite galaxies has a normal vector  pointing to , with a standard deviation of this direction of only  (see Fig. 4). The RMS height of the plane members around the best-fitting plane is  kpc and the plane is offset by  kpc from the centre of M31. The axis ratios of the dwarf galaxy distribution are  and . This plane is inclined by  from the galactic disc of M31. Despite the differences in our dataset and disc fitting analysis, our plane fit is very similar to that of Conn et al. (2013), who report a slightly smaller RMS plane height of  kpc and an inclination from M31’s galactic disc of .

The analysis of Ibata et al. (2013) and Conn et al. (2013) concentrates on those M31 satellites which are found within the PAndAS (McConnachie et al., 2009) survey region. Which additional M31 satellites (i.e. dwarf galaxies within 300 kpc from M31) are close to this plane? The five closest which are not in the sample of Ibata et al. (2013) are NGC 205 ( kpc from the best-fit plane), M32 ( kpc), IC 10 ( kpc) and LGS 3 ( kpc), whose likely alignment is also mentioned by Conn et al. (2013), and in addition the recently discovered satellite galaxy Andromeda XXXII ( kpc) (Martin et al., 2011). All remaining satellites have a distance of more than  kpc from this plane, about four times the plane’s RMS height . We therefore add these four objects to the sample. The parameters of a plane fitted to this extended sample of 19 galaxies444Andromeda I, Andromeda III, Andromeda IX, Andromeda XI, Andromeda XII, Andromeda XIII, Andromeda XIV, Andromeda XVII, Andromeda XXV, Andromeda XXVI, Andromeda XXVII, Andromeda XXX, Andromeda XXXII, IC 10, LGS 3, M32, NGC 147, NGC 185, NGC 205. We keep Andromeda XVI excluded as we consider it a non-satellite, but including it does not change the results significantly. are only minimally different from the fit to the 14 objects. The orientation of the best-fitting plane to the larger sample differs by only , its normal  points to , with a standard deviation of this direction of only . The RMS height of the plane members around the best-fitting plane is slightly smaller for the larger sample ( kpc) and the plane passes closer to the centre of M31,  kpc. The axis ratios are  and . For the following discussion, we will adopt this sample and the resulting plane parameters as the GPoA.

In addition to Andromeda XVI, there are two other non-satellite dwarf galaxies which lie close to the GPoA: IC 1613 ( kpc,  from the GPoA) and Phoenix ( kpc, ). They have a distance of more than 500 and 800 kpc from M31, respectively. All remaining non-satellite dwarfs in our sample have offsets from the GPoA of more than 100 kpc, but Cetus and Andromeda XVIII are at angular distances of only . Interestingly, the non-satellites Phoenix and Cetus are also close to the VPOS-3 plane.

The normal direction to the GPoA is also prominent as a strong peak in the density-contours of the 4-galaxy-normal distribution of the 34 M31 satellites (Fig. 4). This is consistent with the similar analysis by Conn et al. (2013), even though we consider a slightly different sample consisting of all currently known M31 satellite galaxies, but without objects at distances larger than 300 kpc from M31. The normal direction of the plane fitted to all M31 satellites does not coincide with a feature in the 4-galaxy-normal plot, which is another indication that the full M31 satellite population does not follow a single preferred plane.

Fig. 5 shows how much the different satellites of M31 contribute to the 4-galaxy-normals within of the GPoA peak. Sorted according to their relative weighted contribution, the 21 satellites contributing the most are (those written in italics are in the GPoA satellite galaxy sample): Andromeda XII, IC 10, LGS 3, NGC 185, Andromeda XIV, M32, Andromeda XXVII, Andromeda XIII, Andromeda XXXII, NGC 147, Andromeda XXX, NGC 205, Andromeda I, Andromeda XI, Andromeda XXV, Andromeda XVII, Andromeda XXVI, Andromeda II, Andromeda IX, Andromeda XV, Andromeda III. Thus, among the 21 galaxies contributing most to the peak are all 19 M31 satellites that make up our GPoA sample.

The plane fit parameters for the GPoA, which will be used for the later discussion, are compiled in Table 3. Before we investigate the possibility that some of the remaining M31 satellite galaxies constitute a second common plane we first turn our attention to the distribution of the non-satellite galaxies in the LG.

5 Local Group Planes

Figure 6: The distribution of LG galaxies as seen from the midpoint between the MW and M31. Note that in contrast to the previous plots, this is not plotted in Galactic coordinates  and . Instead, the orientation of the coordinate system was chosen such that the MW and M31 lie on the equator and the normal to the plane fitted to all 15 non-satellite galaxies points to the north pole. The positions and orientations of the MW and M31 discs are indicated by black ellipses. The Galactic disc of the MW is seen from the south, the Galactic south pole points to the upper right of the plot. Satellite galaxies are plotted as crosses (+ for MW,  for M31), non-satellites are plotted as filled circles. The one-sigma distance uncertainties for the galaxies result in position uncertainties in this projection, which are indicated by the grey lines. For most galaxies they are smaller than the symbols. Galaxies within a common plane are marked with the same color. All MW satellites are assumed to lie in the VPOSall are plotted in blue, while the M31 satellites assigned to the GPoA are plotted in red. Most of the non-satellite galaxies in the LG lie along one of two ’bands’, one above and one below the plot’s central axis. The only LG galaxy not along one of the bands is the Pegasus dwarf irregular (dIrr). It is, however, very close to the plane of M31 itself. We have indicated this by marking the satellites close to the M31 disc plane, but not in the GPoA, in magenta.

Our galaxy sample contains 15 objects which we consider non-satellite galaxies as they are more distant than 300 kpc from both the MW and M31. Fitting a single plane to their distribution results in a best-fit normal vector pointing to , with an uncertainty of . The best-fit plane runs through both the MW ( kpc) and M31 ( kpc). However, the fit results in an RMS height of  and axis ratios of  and . Therefore, the distribution is not planar, but rather a triaxial ellipsoidal.

We can compare this ’plane’ with the LG galaxy planes discussed in earlier works. Hartwick (2000) determine the spatial distribution of 13 galaxies which they consider to be ’relatively isolated’ LG galaxies. In contrast to our sample, their galaxies have LG distances of up of to 2.5 Mpc. They describe the galaxy distribution with a flat ellipsoid which has a short axis pointing to , which is relatively close to the normal vector we determined for our LG non-satellite galaxy sample.

Sawa & Fujimoto (2005) determine a planar distribution of LG galaxies by first investigating their positions on the sky, plotted in Galactic coordinates as seen from the Sun. They identify a ring-like distribution traced by most LG galaxies. To avoid parallax effects due to the projected view, they then look at the three-dimensional positions of the LG galaxies and identify a thin plane (they report a thickness of 50-100 kpc without stating how it was measured) of galaxies which they claim to be responsible for the ring-like distribution. This plane’s normal points to . A look at their figure 3 reveals that those galaxies agreeing best with their LG plane are mostly members of the GPoA, the non-satellites IC 1613 and Phoenix which lie very close to the GPoA, and the MW satellites, which also lie within the GPoA because it is seen edge-on from the MW (see Sect. 7). Consequently, the normal direction of the Sawa & Fujimoto (2005) LG plane is close to the GPoA normal direction.

Pasetto & Chiosi (2007) have also determined a best-fitting plane to the same sample of LG galaxies used by Sawa & Fujimoto (2005) by applying a principal component analysis technique. They report a plane normal direction of , corresponding to  in our notation of non-negative Galactic longitude, and a plane thickness estimate of 200 kpc without specifying how this thickness was measured. Using a second method which assumes that the line connecting the MW and M31 lies within the LG plane, they repeat their plane fit, resulting in a plane normal pointing to . As this normal direction points close to the position of M31 (), it can not describe a plane including both the MW and M31. We therefore have to assume that the -component of their second normal direction lacks a minus sign, which would agree with the statement by Pasetto & Chiosi (2007) that the difference between their two planes is small. If this is the case, their second plane fit would have a normal pointing to  in our notation. Thus, their results agree well with our plane fitted to all non-satellite galaxies in the LG.

With a RMS height of almost 300 kpc, the single plane fitted to all non-satellite galaxies is much wider than the satellite galaxy planes around the MW and M31. Motivated by the GPoA, which consists of only a sub-sample of M31 satellites, we look for the possibility that there are sub-samples of non-satellite galaxies in the LG which lie in a thinner plane. Fig. 6 shows an Aitoff projection of the distribution of all LG galaxies as seen from the midpoint between the MW and M31 (the origin of our Cartesian coordinate system). The angular coordinate system for this plot is chosen such that the normal-vector of the plane fitted to all 15 non-satellite galaxies defines the north pole, and such that the MW and M31 lie along the equator at longitudes of  and , respectively. All non-satellite galaxies are plotted as filled points in Fig. 6, the MW satellite positions are indicated with plus signs and those of the M31 satellites with crosses.

Galaxies which lie within a common plane that contains or passes close to the midpoint between the MW and M31 will lie along a common great-circle in Fig. 6. This is, for example, the case for the M31 satellites in the GPoA (red symbols), because the GPoA is oriented such that it is seen edge-on from the MW and therefore also from the midpoint between the MW and M31. Two groupings are obvious for the non-satellites. Mostly contained in the upper half of the plot, the LG galaxies UGC 4879, Leo A, Leo T, Phoenix, Tucana, Cetus, WLM, IC 1613 and Andromeda XVI (plotted in yellow) lie along a common ’band’ (below, this group will be referred to as LGP1). A second, smaller grouping can be identified in the lower half of the plot, consisting of NGC 6822, Sagittarius dIrr, Aquarius, Andromeda XXVIII and Andromeda XVIII (plotted in green, will be referred to as LGP2). Only the Pegasus dwarf irregular (dIrr) seems to be unrelated to these two bands, as it lies in-between them. It is, however, very close to a number of M31 galaxies (plotted in magenta, see Sect. 6) which lie close to the plane defined by M31’s galactic disc (black ellipse in the plot).

Fitting planes to the two groups of non-satellite galaxies demonstrates that the galaxies indeed lie within two thin planes.

5.1 Local Group Galaxy Plane 1 (LGP1)

Figure 7: Face-on view of LG Plane 1 (LGP1). Galaxies assigned to this plane are plotted as yellow dots. The horizontal axis is parallel to the major axis of the distribution of galaxies in the plane, while the vertical axis is parallel to the intermediate axis. The black ellipses indicate the positions and orientations of the MW (upper) and M31 (lower). The black arrows show the direction and amount of the Galactocentric velocity  of the plane members and M31, a length of 100 kpc represents a velocity of 100  (note that the tangential velocities are unknown). All other LG galaxies are plotted as crosses, with their colour representing plane membership as in Fig. 6. The plot is centred on the centroid position  of LGP1, which in this projection is close to the midpoint between the MW and M31.

For the first group, which we denote the Local Group Plane 1 (LGP1), we determine the following plane parameters. The normal vector  points to , with an uncertainty of . The plane is offset by  kpc from the MW and by  kpc from M31. It has a RMS height of  kpc, less than one fifth of the RMS height of the total non-satellite sample. The axis ratios of  and indicate a very thin, planar distribution. A face-on view of this plane is plotted in Fig. 7.

The only galaxy associated with LGP1 which has a distance of more than two times the RMS plane height is Andromeda XVI, which is offset by  kpc from the best-fit plane. Andromeda XVI lies within the GPoA around M31 (see Sect. 4), follows the GPoA velocity trend (i.e. is co-orbiting with the majority of the GPoA satellites, see Ibata et al., 2013), and is only barely classified as a non-satellite by our radial cut at 300 kpc distance (it has a distance of about 320 kpc from M31). Therefore, it might indeed be unrelated to the LGP1, but rather belong to the GPoA.

None of the MW satellite galaxies is closer than  from LGP1, but the best-fit plane runs right through the M31 satellite galaxy LGS 3 ( kpc) and passes close to Triangulum/M33 ( kpc). Fitting a plane to the LGP1 sample without Andromeda XVI results in a better alignment of M33 ( kpc) and adds the possible M33 satellite galaxy Andromeda XXII (Chapman et al., 2013) to the well-aligning M31 satellites ( kpc), but LGS 3 is then more offset ( kpc). Two of these galaxies are no GPoA members (Triangulum/M33 and Andromeda XXII), but LGS 3 is the southernmost known M31 satellite which lies within the GPoA. Its line-of-sight velocity, which is similar to the line-of-sight velocity of M31, does not follow the strong co-rotating trend of the majority of GPoA satellites. Therefore, we consider these three galaxies possible members of LGP1. A plane-fit to the larger sample, now consisting of 11 objects (Andromeda XVI excluded; Andromeda XXII, Triangulum/M33, and LGS 3 included) results in a normal vector  pointing to , with an uncertainty of . It is offset from the MW by  and from M31 by  kpc. The RMS height of the dwarf galaxies around the best-fit plane is remarkably small given that the plane diameter is 1–2 Mpc:  kpc and the axis ratios are  and .

For the following discussion, we will use the LGP1 satellite galaxy sample and plane fit parameters determined for the 9 non-satellite galaxies only, but keep the possibly associated M31 satellites in mind. The resulting fit parameters are compiled in Table 3.

5.2 Local Group Galaxy Plane 2 (LGP2)

Figure 8: Same as Fig. 7, but for LG plane 2. The centroid of this plane is offset considerably from the line connecting the MW (upper) and M31 (left).

Similarly, the second group of non-satellite galaxies seen in Fig. 6, which consists of NGC 6822, Sagittarius dIrr, Aquarius, Andromeda XXVIII and Andromeda XVIII, is denoted the Local Group Plane 2 (LGP2). The best-fit plane parameters are: normal vector  pointing to , with an uncertainty of ,  kpc,  kpc,  kpc and axis ratios of  and . A face-on view of this plane is plotted in Fig. 8.

The LGP1 member Leo T lies close to LGP2 ( kpc), indicating that the two LG planes intersect close to that galaxy. Nevertheless, we keep Leo T assigned to LGP1 because it is very close to the best-fit LGP1 ( kpc) and together with the nearby LGP1 members Leo A and UGC 4879 traces a common trend in radial distance and line-of-sight velocity (Figs. 9 and 10).

All three most-recently discovered M31 satellites are close to LGP2: Andromeda XXX ( kpc), Andromeda XXXI ( kpc) and Andromeda XXXII ( kpc), but Andromeda XXX and XXXII are also members of the GPoA. Of the other outer M31 satellites, IC 10, Andromeda VII and Andromeda XXIV are within  (,  and  kpc, respectively), but also several of the inner M31 satellites are close. Similarly, many MW satellites are close to the plane. Most satellite galaxies are, however, closer to one of the satellite galaxy planes than to LGP1 or LGP2, as can be studied in detail by comparing the dwarf galaxy distances from the plane fits compiled in Table 4.

The most-distant LGP2 member from the plane fit is Sagittarius dIrr, which is offset by  kpc. Removing it results in the following plane parameters: normal vector  pointing to , with an uncertainty of ,  kpc,  kpc,  kpc and axis ratios of  and . Thus, the remaining four dwarf galaxies lie within a very thin plane, but given the small number of galaxies this might not be unexpected. The distant M31 satellite Andromeda XXXI is somewhat closer to this plane fit ( kpc).

5.3 Comparing LGP1 and LGP2

Figure 9: Edge-on view of both LG planes. The orientation of the MW and M31 are indicted as black ellipses in the centre. Members of the LGP1 are plotted as yellow points, those of LGP2 as green points. MW galaxies are plotted as plus signs (+), all other galaxies as crosses (), the colours code their plane membership as in Fig. 6. The best-fit planes are plotted as solid yellow and green lines (for LGP1 and LGP2, respectively), and the dashed lines of the same colours indicate the planes’ RMS heights . The view direction was determined from the cross-product of the two plane normal vectors, resulting in a projection in which both planes are seen edge-on. The view direction is along , and the Galactic north points up. The view, thus, is along the line connecting the MW and M31, such that the two major galaxies and their surrounding satellites overlap in the centre. Therefore both LG planes are parallel to the MW-M31 line. The two planes cross at about the position of Leo T (Leo A if removing Sagittarius dIrr from LGP2), which therefore might be a member of either plane. As Leo T (Leo A) falls onto the line extending from (connecting) the two nearby LGP1 members UGC 4879 and Leo A (Leo T), we nevertheless consider it to be a member of LGP1. The black arrows indicate the line-of-sight velocities of the non-satellite galaxies in this projection as in Figs. 7 and 8. The grey arrow in the lower right indicates the motion of the LG with respect to the CMB rest-frame (Sect. 7). It points approximately into the direction where LGP1 and LGP2 intersect in this projection, but the major component of this velocity is directed along the MW-M31 line (perpendicular to the figure).

The two bands of galaxies seen in Fig. 6 are indeed indicative of two thin, planar structures within which almost all known non-satellite galaxies in the LG are found (14 out of 15). As expected, the Pegasus dIrr lies in-between the two planes, having a distance of  kpc from LGP1 and  kpc from LGP2.

The most striking property of these planes is their symmetry. Much of this symmetry is visible in Fig. 9, which shows a view of the LG such that both LG planes are seen edge-on:

  1. Both planes have a similar RMS height (LGP1:  kpc, LGP2:  kpc), but are very wide (diameters of 1-2 Mpc). This is indicated by the dashed lines in Fig. 9.

  2. Both planes have a similar offset from the MW (LGP1:  kpc, LGP2:  kpc).

  3. Both planes have similar offsets from M31 (LGP1:  kpc, LGP2:  kpc), which at the same time are similar to their offsets from the MW.

  4. Thus, both planes are parallel to the line connecting the MW and M31. In Fig. 9, the MW and M31 (whose positions and orientations are shown as black ellipses in the centre of the plot) as well as most of their satellite galaxies are within a ’wedge’ formed by LGP1 and LGP2.

  5. Both planes have a similar inclination from the galactic disc of M31 (LGP1: , LGP2: ), but different inclinations from the Galactic disc of the MW (LGP1: , LGP2: ). The black ellipse representing M31 in Fig. 9 is also seen almost edge-on and its orientation is similar to the two planes (major axis running from lower left to upper right).

  6. Both planes cross the outer parts of the satellite galaxy distribution of the MW and M31. In particular the northernmost (IC 10) and southernmost (LGS 3) M31 satellites in the GPoA (respectively the uppermost and lowermost red cross in Fig. 9) are each close to one of the planes (LGP2 and LGP1, respectively). LGS 3, the southernmost GPoA member lies close to LGP1 and does not follow the strong line-of-sight velocity trend of the GPoA. Similarly, IC 10 is the northernmost GPoA member, lies relatively close to the LGP2 and does not follow the strong line-of-sight velocity trend of the GPoA either.

Figure 10: Other properties of the LG galaxies shown in Fig. 6. The symbols and colours are the same, and the positions on the horizontal axis are given by the longitude  also used in Fig. 6. The upper panel plots the radial distance  from the midpoint between the MW and M31 (i.e. the distance of each galaxy from the point of view adopted in Fig. 6). Uncertainties are again shown as grey lines. The LG galaxies follow a pronounced trend. From left to right, they are first found at large distances ( Mpc), then approach the MW distance. To the right of the MW, the distances of non-satellites increase again to more than 1 Mpc, but then approach the M31 distance. Thus, the non-satellite galaxies are not only ordered in their projected position, but also follow a common radial distance behaviour. It is also interesting that no galaxies are known in the region spanning about  to the right of the M31 satellite system. The lower panel plots , the line-of-sight velocity with respect to the Galactic standard of rest. This is the only velocity component available for the non-satellite galaxies. It is measured from the position of the Sun, so the angle between this velocity direction and the line-of-sight adopted for Fig. 6 varies for the different objects. There appears to be a trend of decreasing  with increasing  for the non-satellites, with a break at the MW position (see lower panel of Fig. 17, too).

Looking for further indications of coherence in the structure of the LG planes, in the upper panel of Fig. 10 we have plotted the radial distance from the midpoint between the MW and M31 against the longitude  in the coordinate system of Fig. 6, approximately along the two bands. This reveals a seemingly ordered behaviour of the radial distances of the non-satellite galaxies: Starting with UGC 4879, the most-distant LG galaxy in our sample, on the left near , the LG galaxy distance decreases systematically as as we move closer to the MW in . The MW satellites then follow the same trend of decreasing distance with increasing . To the right of the MW the LGP1 and LGP2 members seem to follow a similar radial behaviour. The galaxy distances now increase with increasing , towards a maximum of about 1.2 Mpc (Sagitarrius dIrr) between the MW and M31 position. At larger , the galaxy distances decrease almost monotonously towards M31 and its satellite galaxies. Only Andromeda XVI (rightmost yellow point) is much closer to the origin/point of view than M31, and Andromeda XVIII (rightmost green point) lies further away than M31. The face-on view of LGP1 in Fig. 7 reveals this arc-like distribution, too, which starts with UGC 4879 at the tip of the long axis, and then passes through the position of the MW from where it bends down to end at the position of M31.

The lower panel of Fig. 10 plots the Galactocentric line-of-sight velocities of the LG galaxies against . Qualitatively, the trend of the line-of-sight velocities is similar to that of the distances in the upper panel: from left-to-right, the non-satellite galaxy velocities first become more negative (approaching the MW) with increasing , then rise and again drop almost monotonically between the MW and M31. Again, the LGP1 and LGP2 members follow a similar trend. The lower panel of Fig. 10 also shows the velocity trend of the GPoA satellites (red crosses). On the left of M31, most velocities are more-negative than those of M31, while the opposite is the case on the right side. The two outermost GPoA satellites, LGS 3 (leftmost red cross) and IC 10 (rightmost red cross) do not follow this trend, but they lie close to LGP1 and LGP2, respectively, and have velocities similar to the LGP1 and LGP2 members in the vicinity of M31. This might be another indication that they are better understood as LG plane members or as a connection between the LG planes and the GPoA.

5.4 Consistency check

Figure 11: Density distribution of the 4-galaxy-normal directions for our sample of 15 non-satellite galaxies in the LG, determined and weighted as explained in Sect. 2.4. They show two very pronounced peaks at  (Peak 1) and  (Peak 2), and a smaller one at (Peak 3), which are are close to each other. The contributions of the different non-satellite galaxies to these three peaks are shown in Fig. 12. As in Figs. 2 and 4 only the 4-galaxy-normals in the centre of the plot between  and  are shown, not their mirrored counterparts. The black point indicates this direction of the line connecting the MW and M31, with the black line being the great circle around this direction. It illustrates all possible directions of normal vectors for planes parallel to the line connecting the MW and M31. Also plotted are the normal to the plane fitted to all 15 non-satellite galaxies (grey hexagon) and the normals of the LG planes 1 and 2 (yellow and green hexagons, respectively, i.e. the same colours in which their members are marked in Figs. 6 and 10). The normal to LG plane 1 coincides with Peak 1, and the 4-galaxy-normal contribution plot for this peak in Fig. 12 demonstrates that only LG plane 1 members contribute to this peak.
Figure 12: Relative weight contributions of the different non-satellite LG galaxies to the three peaks identified in Fig. 11. For each peak, the contributions of the 4-galaxy-normals pointing to within  (instead of  as used for Figs. 3 and 5) around the peak positions are shown. This smaller angle was chosen to avoid an overlap in the peak areas due to the closeness of the three peaks.

The following discussion (Sect 7) is restricted to the LGP1 and LGP2, but before proceeding we have to discuss whether there might be a different dominant plane in the LG non-satellite galaxy distribution. As a consistency test, Fig. 11 shows the density distribution of all 1365 possible 4-galaxy-normal directions for the 15 non-satellite galaxies. As before, 1000 realisations of the galaxy positions varying their distances within the uncertainties have been combined. The plot reveals that the LGP1 normal direction coincides with a pronounced peak (Peak 1 at ) in the density distribution, lending further support to our discovery. However, there is a second strong peak at  (Peak 2), and another nearby but smaller peak at  (Peak 3).

We have determined which dwarf galaxies contribute to each of the three peaks (Fig. 12). As expected, the nine galaxies contributing significantly to Peak 1 are identical to those we assign to LGP1. The contributions to the other two peaks are dominated by only seven galaxies each. These are (sorted by their relative contribution): Tucana, UGC 4879, Pegasus dIrr, Andromeda XVIII, Leo A, Leo T and Andromeda XVI for Peak 2 and Andromeda XVIII, WLM, Andromeda XVI, Leo T, Leo A, Phoenix, and Cetus for Peak 3. Five out of the seven galaxies contributing to Peak 2 are members of LGP1, and even six of seven galaxies contributing to Peak 3 are LGP1 members. The galaxy samples contributing to Peaks 2 and 3 are therefore very similar to the LGP1 sample and trace a similar structure, as indicated by their proximity to the LGP1 peak. Fitting planes to the Peak 2 and Peak 3 galaxy samples confirms the similar orientation. For Peak 2, the fit results in a normal vector  pointing to , with an uncertainty of ,  kpc,  kpc,  kpc and axis ratios of  and . For Peak 3, the fit gives a normal vector  pointing to , with an uncertainty of ,  kpc,  kpc,  kpc and axis ratios of  and .

With the presently-known galaxies, the previously-determined LGP1 and LGP2 are therefore not the only possible planar structures in the LG. However, they contain all but one of the 15 non-satellite LG dwarf galaxies (which in turn might be related to the M31 disc plane, see the following Section) and at the same time exhibit a striking symmetry in their parameters. This is not the case for the two dwarf galaxy planes which give rise to Peak 2 and Peak 3, which each contain only 7 out of 15 galaxies.

In the future it will be important to study the statistical significance of the here discovered symmetries of the LG, but currently no conclusive meaningful test is available since a model is required as a null hypothesis. Furthermore, such a test has to take observational biases like the sky coverage of surveys searching for LG dwarf galaxies into account. Due to the very inhomogeneous nature of the galaxy data, this is currently not feasible. It would appear rather clear though that a distribution process which is inherently stochastic will not be able to deliver present-day positions of galaxies within the LG which end up being as symmetrically distributed as is observed.

6 The remaining dwarf galaxies

Out of the 76 non-host galaxies in our LG sample, 16 galaxies are currently not associated with any plane. Except for Pegasus dIrr, all of them are M31 satellites. Some of these non-associated galaxies have already been discussed as possible LG plane members due to their closeness to the best-fit planes. For LGP1, the closest non-associated M31 satellites are Triangulum/M33 ( kpc), Andromeda XXIII ( kpc), Andromeda II ( kpc) and Andromeda XXII ( kpc). For LGP2, these are Andromeda XXI ( kpc), Andromeda XXXI ( kpc) and Andromeda VII ( kpc).

Is there any other planar structure to be found? Of the 16 non-associated galaxies, 5 are within less than  of the galactic disc plane of M31 (Andromeda V, Andromeda VI, Andromeda XIX, Andromeda XXIX and Pegasus dIrr). In particular the non-satellite Pegasus dIrr lies perfectly within the plane defined by M31’s disc orientation. Another three (Andromeda X, Andromeda XX and Andromeda XXIV) are within seven to 16 degree of M31’s disc plane. These eight objects are marked with magenta symbols in the Figures. Except for Andromeda V, which has a distance of  kpc from LGP2, none of these galaxies are closer than 140 kpc from the two LG planes555This is expected because the two LG planes are inclined by only  to M31’s galactic disc plane and offset from M31’s centre by more than 100 kpc. Therefore, the planes do not come close to M31’s galactic disc plane.. All non-GPoA members are further than  kpc () from the GPoA.

For an object with a randomly chosen position, the chance to be within 4 degree of an independently given plane is 7 per cent. The probability to find at least five out of 16 objects within 4 degree of such a plane is about 0.4 per cent, assuming that the objects are randomly distributed. This is not the case here, because all galaxies within the GPoA have already been excluded from the sample, such that the probability will be considerably higher. More importantly, the existence of the co-orbiting GPoA also disproves the assumption that the satellites are randomly distributed.

If we nevertheless group the eight galaxies close to M31’s galactic disc in a tentative ’M31 disc plane’, a plane fitted to them has a normal vector  pointing to , with an uncertainty of . The best-fit plane is therefore inclined by  to M31’s galactic disc. It has only a small offset of  kpc from the centre of M31. The RMS height of the galaxies around the best-fit plane is  kpc and the axis ratios are  and .

While these eight galaxies thus lie within a common, thin plane, their line-of-sight velocities do not indicate a preferentially co-orbiting association. This can be seen in Fig. 10. In contrast to the GPoA members (red crosses), which have preferentially faster line-of-sight velocities than M31 in one side of M31 and slower line-of-sight velocities on the other, the few galaxies associated to the M31 disc plane for which line-of-sight velocities are known (magenta crosses) do not show a pronounced trend, but a mixture of co- and counter-orbiting objects can not be ruled out. We therefore caution against overrating the possible existence of this second plane of M31 satellites.

A similar plane of galaxies aligned with the Galactic disc of the MW would probably remain undetected due to the difficulty in discovering satellite galaxies obscured by the Galactic disc. The lowest-latitude MW satellite is the Sagittarius dSph at . The nearby (7 kpc) Canis Major over-density is situated at even lower Galactic latitude (), but it might be a substructure in the Galactic disc of the MW and not a MW satellite galaxy (Momany et al., 2006).

Name VPOSall VPOS-3 GPoA LGP1 LGP2 M31 disc plane
Introduced in Sect. 3 Sect. 3 Sect. 4 Sect. 5.1 Sect. 5.2 Sect. 6
Type MW satellites MW satellites M31 satellites non-satellites non-satellites M31 satellites
and one non-satellite
 [] 1.12 0.43 0.79 0.41 1.72 2.87
27 24 19 9 5 8

Parameters of the plane fits discussed in Sects. 3 to 6. These are: : -, - and -position of the centroid of the plane in the coordinate system introduced in Sect. 2.1. : The direction of the normal vector (minor axis) of the best-fit plane in Galactic longitude  and latitude . : Uncertainty in the normal direction. This and all other uncertainties were determined by varying the galaxy positions within their uncertainties and then determining the standard deviation in the resulting plane parameters.  and : offset of the planes from the MW and M31 position. : RMS height of the galaxies from the best-fit plane.  and : ratios of the short and intermediate axis to the long axis, determined from the RMS heights in the directions of the three axes. : Number of galaxies associated with the planes used for the fitting. In particular LGP1 and LGP2 might have additional satellite galaxies as members, but these were not included in the plane fits compiled here.

Table 3: Satellite and Local Group dwarf galaxy planes
Name category MW VPOSall MW VPOS-3 GPoA LGP1 LGP2 M31 disc plane
The Galaxy 0.0 787.6 host
Canis Major 13.6 786.3 MW sat.
Sagittarius dSph 18.2 791.7 MW sat.
Segue (I) 28.0 792.4 MW sat.
Ursa Major II 38.1 771.1 MW sat.
Bootes II 39.4 807.0 MW sat.
Segue II 40.9 753.4 MW sat.
Willman 1 43.0 780.7 MW sat.
Coma Berenices 44.9 802.7 MW sat.
Bootes III 45.8 800.5 MW sat.
LMC 50.0 811.4 MW sat.
SMC 61.2 811.7 MW sat.
Bootes (I) 63.9 820.0 MW sat.
Draco 75.9 754.9 MW sat.
Ursa Minor 77.9 758.3 MW sat.
Sculptor 86.0 765.8 MW sat.
Sextans (I) 89.1 839.0 MW sat.
Ursa Major (I) 101.7 777.2 MW sat.
Carina 106.9 842.0 MW sat.
Hercules 126.0 826.6 MW sat.
Fornax 149.4 772.6 MW sat.
Leo IV 154.8 901.2 MW sat.
Canes Venatici II 160.6 837.5 MW sat.
Leo V 178.7 915.1 MW sat.
Pisces II 181.1 660.2 MW sat.
Canes Venatici (I) 217.5 863.9 MW sat.
Leo II 236.0 901.5 MW sat.
Leo I 257.5 922.1 MW sat.
Andromeda 787.6 0.0 host
M32 809.5 22.7 M31 sat.
Andromeda IX 770.0 40.5 M31 sat.
NGC 205 828.2 41.6 M31 sat.
Andromeda I 748.8 58.4 M31 sat.
Andromeda XVII 731.9 70.0 M31 sat.
Andromeda XXVII 832.1 74.2 M31 sat.
Andromeda III 751.9 75.2 M31 sat.
Andromeda XXV 816.8 88.8 M31 sat.
Andromeda XXVI 766.0 102.7 M31 sat.
Andromeda V 777.6 109.5 M31 sat.
Andromeda XI 738.5 110.6 M31 sat.
Andromeda XIX 823.6 114.0 M31 sat.
Andromeda XXIII 774.3 126.4 M31 sat.
Andromeda XX 744.3 129.8 M31 sat.
Andromeda XIII 843.5 132.1 M31 sat.
Andromeda X 674.6 133.8 M31 sat.
Andromeda XXI 831.0 134.0 M31 sat.
Andromeda XXXII 780.5 140.9 M31 sat.
NGC 147 680.2 142.9 M31 sat.
Andromeda XXX 686.6 147.7 M31 sat.
Andromeda XIV 798.2 161.3 M31 sat.
Andromeda XII 933.0 178.6 M31 sat.
Andromeda XV 630.0 178.9 M31 sat.
Andromeda II 656.3 184.1 M31 sat.
NGC 185 620.9 187.7 M31 sat.
Andromeda XXIX 733.7 188.3 M31 sat.
Triangulum 814.1 206.5 M31 sat.
Andromeda XXIV 604.8 208.2 M31 sat.
Andromeda VII 764.9 218.3 M31 sat.
IC 10 798.5 252.1 M31 sat.
Andromeda XXXI 760.2 263.0 M31 sat.
LGS 3 773.0 268.5 M31 sat.
Andromeda VI 785.4 269.0 M31 sat.
Andromeda XXII 925.2 274.0 M31 sat.
Andromeda XVI 480.7 323.2 non-sat.
Andromeda XXVIII 660.9 367.8 non-sat.
IC 1613 757.8 520.1 non-sat.
Phoenix 414.9 867.6 non-sat.
NGC 6822 451.9 897.5 non-sat.
Cetus 755.6 680.4 non-sat.
Pegasus dIrr 921.0 474.3 non-sat.
Leo T 422.0 990.7 non-sat.
WLM 932.7 836.2 non-sat.
Andromeda XVIII 1216.7 452.5 non-sat.
Leo A 803.0 1200.0 non-sat.
Aquarius 1065.5 1172.1 non-sat.