## Main

The original discovery of the Milky Way’s ‘plane of satellites’ (then just five galaxies)1 preceded the advent of the Λ cold dark matter (ΛCDM) model, where Λ is the cosmological constant, as the paradigm for galaxy and structure formation2. A key ΛCDM prediction is that galaxies such as the Milky Way (MW) are surrounded by a dispersion-supported dark matter halo and by satellite galaxies formed within its substructures. However, while several ΛCDM predictions, including the discovery of dozens of additional MW satellites, have since been borne out3, the ‘plane of satellites problem’4,5 has emerged as its most persistent challenge6,7,8,9,10,11.

That the ‘plane of satellites’ problem has so far eluded resolution is not for lack of trying. Planes of satellites form with the same (low) frequency in collisionless and hydrodynamic cosmological simulations12,13,14 and in MW analogues in isolation or in pairs15,16, with no significant correlation with other properties of the host halo16. There is evidence that filamentary accretion17,18, a compact satellite system19 or the presence of massive satellites20 can generate some anisotropy, but systems as thin as the Milky Way’s are still very rare12. Moreover, any planes that do form in ΛCDM are transient, chance alignments of substructures14,20,21,22 rather than long-lived, rotationally supported disks. We examine here the contention that the MW contains an exceptional plane of satellites, explain the origin of the observed anisotropy and study its time evolution in light of proper motion measurements by the Gaia space telescope23.

## The present MW plane of satellites

The Milky Way’s ‘plane of satellites’ canonically consists of the 11 ‘classical’ satellites, the brightest within a radius of r = 300 kpc of the Galactic Centre, believed to constitute a complete sample. To characterize the spatial anisotropy of a system of N satellites, it is customary to consider the inertia tensor, defined as

$${I}_{ij}=\mathop{\sum }\limits_{n=1}^{N}{x}_{n,i}{x}_{n,j},$$
(1)

where xn are the coordinates of the nth satellite relative to the centre of positions, and i and j index the three spatial dimensions. We label the square roots of its eigenvalues as a, b and c, corresponding to the dispersions in position along the unit eigenvectors xa, xb and xc. A related metric is the ‘reduced’ inertia tensor24, defined after projection of the positions onto a unit sphere. We label the square roots of its eigenvalues as ared, bred and cred. Both c/a and (c/a)red ≡ cred/ared parametrize the spatial anisotropy but differ in the weight attached to each galaxy. c/a measures the full spatial anisotropy, whereas (c/a)red considers only its angular component, independently of the radial distribution. For both c/a and (c/a)red, smaller values imply greater anisotropy; for example, a sphere has c/a = 1 while a perfectly thin disk has c/a = 0. Note that for small N, the expectation values of c/a and (c/a)red decrease, regardless of the underlying anisotropy25.

Figure 1 shows the present positions and estimated orbits of the 11 brightest MW satellites projected along the principal axes. We measure c/a = 0.183 ± 0.004 and (c/a)red = 0.3676 ± 0.0004, mean ± s.d.

## The effect of the radial distribution

Earlier work26 has found that only 0.7% of ΛCDM simulations produce systems as anisotropic as the MW. However, we find this likely to be a severe underestimate caused by the artificial disruption of satellites in numerical simulations, which can cause artificially extended radial profiles27,28,29,30. To identify analogues of the classical MW satellites in simulations, it is common to select at z = 0 the 11 satellites with the highest value of vpeak, the peak value of the maximum circular velocity, vmax, across a halo’s history17. In our own analysis of 202 MW analogues resimulated in the SIBELIUS constrained simulations intended to reproduce the structures observed in the Local Universe31, including only surviving subhaloes, we find only 2 (1%) systems with c/a as low as the MW, and none that reproduce the radial distribution. However, this is an ‘incomplete’ sample. Accounting for artificial disruption32 and selecting the 11 satellites among a ‘complete’ sample containing both the surviving and the recovered, artificially disrupted satellites, we find radial distributions resembling the MW’s, as shown in the left panel of Fig. 2. This reproduces the results of very-high-resolution simulations30,33. As each satellite contributes to the inertia (equation (1)) in proportion to $${r}_{i}^{2}$$, c/a is sensitive to the radial profile. To quantify this relationship, we introduce the Gini coefficient formalism. The central panel of Fig. 2 shows the summed weights of the closest i satellites from the centre, $$\mathop{\sum }\limits_{j=1}^{i}{r}_{j}^{2}$$, normalized by the total weight of all 11 satellites, $$\mathop{\sum }\limits_{j=1}^{11}{r}_{j}^{2}$$. The area between each curve and the diagonal measures the inequality of the satellites’ contributions to the inertia, or the sample Gini coefficient of inertia,

$$G=\frac{1}{N-1}\mathop{\sum }\limits_{i=1}^{N}(2i-N-1){r}_{i}^{2}/\mathop{\sum }\limits_{i=1}^{N}{r}_{i}^{2}.$$
(2)

Compared to a more equal distribution, the Milky Way’s centrally concentrated radial profile is equivalent to sampling a system with fewer points, lowering the expectation value of c/a.

The right panel of Fig. 2 shows the relationship between G and c/a. Systems with higher central concentration (higher G) tend to be more anisotropic (lower c/a). Accounting for artificial disruption (filled circles), 58% of ΛCDM systems have G above the MW and 11 (~5%) have c/a < 0.183. Neglecting this effect (faint crosses) produces no systems with G as high the MW and only two (1%) with as low c/a.

The Milky Way’s anisotropy additionally results from the fact that its two outermost satellites, Leo I and Leo II, which contribute two thirds of the total inertia, are currently in close proximity. However, as is already apparent from Fig. 1, this constellation is short-lived.

## The clustering of orbital poles

Supporting the notion that the satellite plane constitutes a spinning disk34, the orbital poles of 7 of the 11 classical satellites (the Large Magellanic Cloud (LMC), the Small Magellanic Cloud (SMC), Fornax, Leo II, Carina, Draco and Ursa Minor) are reported to cluster with a standard deviation in direction of only 16.0° (ref. 26), found in only 0.04% of ΛCDM systems. Using the more precise proper motions from Gaia Early Data Release 3 (EDR3) for the same satellites, we find that this angle increases to $$23.{2}^{\circ }{\,}_{-2.8}^{+3.5}$$. We also repeated the analysis, and find that a different subset (that includes Leo I instead of Leo II) yields a smaller dispersion of $$18.{9}^{\circ }{\,}_{-1.4}^{+1.9}$$. Among our sample of 202 simulated systems, adopting either 18.9° or 23.2° and accounting for a minimum look-elsewhere effect, we find three or five systems with subsets of satellites with a smaller probability to arise from isotropic distributions. That is, we find that ~2% of ΛCDM systems contain satellites whose orbital poles are even more anisotropic than the most clustered subset of the Milky Way, an ~50-fold increase over previous results. The orbital clustering of a subset of the Milky Way satellites is unusual in ΛCDM, but not astronomically improbable.

Notably, one of our systems has both a c/a value below the MW value (0.166 compared to 0.183 for MW) and an orbital pole clustering more unlikely to arise out of isotropy than the MW’s (probability of 0.005 compared to 0.009 for the MW). With no strong correlation between orbital pole clustering and anisotropy, and estimated probabilities below 10−3 and 10−2, respectively26, the combination of the two had previously been considered extremely unlikely.

Importantly, while the ‘plane of satellites’ includes all 11 classical satellites, the orbital anisotropy only concerns a subset, which is in fact more spatially isotropic than the system as a whole. The orbital pole clustering does not drive the spatial anisotropy.

## The plane of satellites is transient

Another defining feature of a rotationally supported disk would be a significantly higher velocity dispersion parallel to the plane (σv) than perpendicular to it (σv). However, for the MW’s classical satellites, we measure σv = 165.1 ± 1.2 km s−1 and σv = 121.6 ± 0.4 km s−1. The ratio, σv/σv = 1.36, is indistinguishable from the purely geometrical factor of $$\sqrt{2}$$. By this basic measure, the plane is not rotation supported.

The longevity of the plane can also be tested directly via orbital integration. This method was first, but inconclusively, applied using pre-Gaia data35. In this work, we benefit from significantly more precise observations, including Gaia EDR3 proper motions and more accurate distances.

In Fig. 1, we saw that several satellites are presently crossing the plane, while Leo I and II, which dominate the inertia, are moving apart. To elucidate the impact of such ‘fortuitous alignments’ on the anisotropy, we show in Fig. 3 the effect on c/a and (c/a)red of randomizing the position of each galaxy on the sky at the observed radius, keeping all other galaxies at their observed positions. For Sagittarius, the satellite with the smallest distance, the c/a distribution is extremely narrow: Sagittarius contributes less than 1% to the inertia tensor. However, randomizing the angular position of either of the two outer satellites, Leo I or Leo II, raises the median value of c/a to 0.28 and 0.31, respectively, with maxima of 0.53 and 0.63, far above the ΛCDM median.

While placing satellites at random sky coordinates highlights the sensitivity of c/a to individual galaxies, it is not a physical process. However, we also show at the top of Fig. 4 the anisotropy distributions when each satellite simply moves along its orbit. The time-averaged anisotropy of the system is then calculated over one full orbital period centred on the present time. Depending on the orbital phase of Leo II alone, c/a could be as high as 0.39, more isotropic than most ΛCDM systems. In other words, almost the entire anisotropy of the classical MW satellites is due to the orbital phases of Leo I and Leo II. Simply omitting both Leo I and Leo II from the analysis would yield c/a = 0.279, also more isotropic than 37% of ΛCDM systems of 9 satellites.

The bottom panels of Fig. 4 show time-averaged probability densities of c/a and (c/a)red when all satellites evolve simultaneously. The current value of c/a is significantly lower than in the recent past: over the past 0.5 and 1 Gyr, the time-averaged medians of c/a are 0.23 and 0.27 respectively, greater than 13% and 23% of ΛCDM systems. (c/a)red has varied widely. Neither metric is an invariant of the satellite system, both are sensitive to the orbital phases.

The four panels of Fig. 5 show the evolution of c/a, (c/a)red and of the orientations of the planes defined by the full and reduced inertia tensors, which we parametrize by the angles between the vectors normal to the planes, xc and xc,red, and their present day equivalents, xc,0 and xc,0,red. The value of c/a approaches the ΛCDM median within a lookback time of 0.5 Gyr. (c/a)red evolves more rapidly, twice exceeding the ΛCDM median. That c/a and (c/a)red vary on such different timescales is a further consequence of the radial distribution and of the fact that c/a and (c/a)red are determined by satellites at different radii and on orbits with different periods: while Leo I and Leo II, which largely determine c/a, have orbital periods of 3.4 ± 0.2 and 7.2 ± 0.3 Gyr, respectively, the eight inner satellites, which largely determine (c/a)red, all have orbital periods under 2 Gyr.

We further see that the orientation of the Milky Way’s plane of satellites is not stable, but has tilted by ~17° over the past 0.5 Gyr (and ~40° for the reduced definition). This is readily understood considering that the plane is largely defined by only two outlying satellites. This is demonstrated clearly in Fig. 6, where we show the orbits and positions of the 11 classical satellites projected edge-on, in the frame defined by the minor and major axes of the inertia tensor at z = 0. The central panel, which shows the positions of the satellites at z = 0, shows the plane aligned with the major axis, analogous to the bottom left panel of Fig. 1. The other four panels show the evolution up to 1 Gyr in the past and the future. The orientation of the plane of satellites is changing, so that at each moment, it points toward the current position of the outermost satellites. Rather than satellites orbiting inside a stable plane, the plane tilts as it tracks the positions of its most distant members.

## Discussion

The high reported anisotropy of the MW satellite system can largely be attributed to its high central concentration, not previously reproduced in simulations, combined with the close but fleeting contiguity of its two most distant members. Accounting for the radial distribution reveals the MW satellites to be consistent with ΛCDM expectations. Compared with previous work, we also find a much higher likelihood of subsets whose orbital poles are as clustered as the MW. Although the Milky Way contains such a subset, the plane of satellites does not constitute a rotationally supported disk. Instead, it evolves on timescales similar to the ‘transient’ alignments previously found in ΛCDM simulations.

Our orbital integration assumes a static MW potential with satellites as massless test particles. We Monte Carlo sample all sources of observational error and also vary the components of the potential within the uncertainties (Methods). We find our results to be robust, and while the real potential is more complex (for example, due to the presence of the LMC), these simplifications are valid within a dynamical time (~2 Gyr) of the halo, particularly for the important outer satellites36,37. Furthermore, a more complex potential would only accelerate the dissolution of a planar configuration38. Our simulations do not include the potential disruption of satellites by the disk of the MW. This might slightly extend the radial distributions, but not enough to change our conclusions. Importantly, among systems that have radial distributions similar to the MW, flattened ‘planes of satellites’ are quite common.

This work only directly addresses the archetypal ‘plane of satellites’ around the MW. Anisotropic satellite distributions have also been reported around several other galaxies14,39 with different criteria, which can exacerbate the look-elsewhere effect. Assessing their significance requires a careful statistical analysis12,40. While not all criteria are equally sensitive to the radial distribution, we also expect that the much higher anisotropy we report here for simulated MW analogues will apply to ΛCDM analogues of other, similarly defined systems.

After centuries of observations, the Milky Way and its satellites are the best studied system of galaxies in the Universe. Viewed in sufficient detail, every system inevitably reveals some remarkable features. However, based on the best currently available data, there is no evidence for a plane of satellites incompatible with, or even particularly remarkable in, ΛCDM. On the contrary, as judged by the spatial anisotropy of the brightest satellite galaxies, we appear to live in a fairly regular ΛCDM system.

## Methods

### Observations

We adopt the sky positions and radial velocities from the McConnachie41 catalogue, and combine these, where available, with McConnachie and Venn23 proper motion measurements based on Gaia EDR3 (ref. 42). The systemic proper motions were measured within a Bayesian framework that combines information from stars with full astrometric data with information from stars with only photometric and proper motion data. The method is a mixture model that associates a probability for each candidate star to be associated with the target galaxy taking into account foreground and background contaminants. It constitutes the best technique currently available in the literature to determine proper motions and provides the most precise measurements so far37. For the three innermost satellites (Sagittarius dSph, the LMC and the SMC), where the McConnachie & Venn catalogue does not include proper motions, we use the Gaia Data Release 2 (DR2) proper motions of ref. 43. We further compiled the most recent estimates of the distance moduli of each satellite. The sky coordinates41, proper motions42,44 and distance moduli45,46,47,48,49,50 used in this study, including their sources, are listed in Supplementary Table 2.

We also repeated our analysis using the Gaia EDR3 proper motions of Battaglia et al.37 and the Gaia DR2 proper motions described in Riley et al.43. A comparison of the evolution of c/a and (c/a)red, analogous to Fig. 5, is shown in Extended Data Fig. 1. Unsurprisingly, the evolution based on the two EDR3 datasets are in excellent agreement. The main difference when using the DR2 data is the larger uncertainty (the astrometry errors of EDR3 are reduced by approximately a factor of two compared with DR2), but the evolution of both c/a and (c/a)red is essentially the same in all three datasets, and the results are consistent within the respective errors.

#### Monte Carlo samples

We account for measurement errors by generating Monte Carlo samples of the satellites in the space of observed quantities: distance modulus, radial velocity and proper motions, as well as the position of the Sun relative to the Galactic Centre. We model each observable as a Gaussian distribution with the mean value and standard deviation given by the measurements and their quoted errors. For the Sun’s distance from the Galactic Centre we assume R = (8.178 ± 0.022) kpc (ref. 51), for the circular velocity at the Sun’s position Vcirc = (234.7 ± 1.7) km s−1 (ref. 52) and for Sun’s motion with respect to the local standard of rest (U, V, W) = (11.10 ± 0.72, 12.24 ± 0.47, 7.25 ± 0.37) km s−1 (ref. 53), where U is defined positive toward the Galactic center, V is positive in the direction of Galactic rotation and W is positive toward the North Galactic Pole.

#### The clustering of orbital poles

To characterize the clustering of orbital poles, we adopt the orbital pole dispersion for a subset of Ns satellites, Δstd, defined by Pawlowski and Kroupa26 as:

$${{{\varDelta }}}_{{{{\mathrm{std}}}}}({N}_{{\mathrm{s}}})=\sqrt{\frac{1}{{N}_{{\mathrm{s}}}}\mathop{\sum }\limits_{i=1}^{{N}_{{\mathrm{s}}}}{\theta }_{i}^{2}},$$
(3)

where θi is the angle between the orbital pole of the ith satellite and the mean orbital pole of the satellites in the subset. To compute the clustering of an observed system relative to expectation, the same analysis is performed on the observations and simulations.

Based on earlier Gaia DR2 and Hubble Space Telescope proper motions, Pawlowski and Kroupa26 calculated orbital pole dispersions for all possible satellite subsets in the MW and in simulations with Ns = 3...11, and discovered that Ns = 7 yielded the most unusual configuration. However, there is no a priori reason to specifically consider Ns = 7. When considering only a proper subset of satellites, the interpretation of Δstd(Ns) as evidence for unusual clustering is subject to the ‘look-elsewhere effect’. To account for this, we follow here the method of ref. 12, which involves performing the same analysis for the simulated systems. As in the observations, we consider all subsets of size Ns = 3...11 in each simulated system, and identify the most unlikely to arise by chance from an isotropic distribution, which we calculate based on 105 isotropic distributions of N = 11 points, and the probability distributions of Δstd(Ns) for all Ns = 3...11 possible subsets.

In Extended Data Fig. 2, we show a Hammer equal area projection of the most likely orbital poles and their Monte Carlo sampled uncertainty. The 3 black circles identify the clustering of 7 of the 11 satellites. The dotted line corresponds to Δstd(7) = 16.0°, the dispersion calculated by Pawlowski and Kroupa using pre-Gaia EDR3 data. The solid line shows our results, Δstd(7) = 23.2°, for the same satellites using Gaia EDR3 data. The dashed line shows our result for the most clustered subset Δstd(7) = 18.9°, exchanging Leo II for Leo I.

### Time evolution and time integration

To infer the time evolution of the Milky Way satellite system, the orbits of the satellites are integrated numerically as massless test particles in a static potential using the Gala package54. The potential consists of a disk, stellar nucleus and bulge, and a dark matter halo. The disk is modelled as an axisymmetric Miyamoto–Nagai disk55, which, for our default model, has disk mass 5.17 × 1010M, a = 3 kpc and b = 0.028 kpc (ref. 56). The nucleus and stellar bulge are both modelled as spherically symmetric Hernquist profiles57. For the nucleus we assume a mass of 1.71 × 109M and a scale radius a = 0.07 kpc, and for the bulge we assume a mass of 5.0 × 109M and a = 1.0 kpc. For the dark matter halo, we assume a spherically symmetric Navarro, Frenk and White (NFW)58 potential.

Until recently, the Milky Way halo mass may have been a prohibitive source of uncertainty for calculating the orbital evolution of the satellites, as its value was known only to within a factor of two59. However, the Galactic halo mass has now been estimated with an uncertainty of only about 20% using Gaia data. Multiple dynamical probes, such as the stellar rotation curve, the escape velocity, the motions of halo stars, globular clusters and satellite galaxies60,61,62,63,64, consistently imply a dark matter halo mass for the MW of M200 = (1.0 ± 0.2) × 1012M and NFW concentration, c200 = 11 ± 2.

Based on these results, we adopt a reference MW halo of mass 1.0 × 1012M and a concentration parameter c200 = 11, corresponding to an NFW scale radius of rs = 19.2 kpc. The positions and velocities relative to the plane of satellites, and the orbital periods and apocentre distances for the default potential, are listed in Supplementary Table 1, where the quoted uncertainties reflect 68% confidence intervals for all quantities based on Monte Carlo sampling. Varying the MW potential within the observational uncertainties does not significantly affect the conclusions of our study, as we show in Extended Data Fig. 3 where we vary the halo mass between 0.8 and 1.2 × 1012M. The inferred anisotropies, as measured by c/a and (c/a)red, begin to diverge for lookback times beyond ~500 Myr, but the behaviour is qualitatively similar across the halo mass range. As shown in Extended Data Fig. 4, we also tested the impact of varying the other three mass components and the concentration parameter within the observational uncertainties, and found no significant effects. The true Milky Way potential evolves with time, but the dynamical time of the halo (~2 Gyr at z = 0) is much longer than the timescale for the reported dissolution of the plane of satellites (several hundred Myr). A further possible source of uncertainty may be the mass of the LMC and its perturbation of the potential. However, at a distance of 49 kpc from the centre, even a massive LMC does not significantly perturb the acceleration field at distances above 150 kpc (ref. 36) and would not significantly affect the orbits of the two most distant satellites, Leo I and Leo II.

### Numerical simulations

#### Initial conditions

The simulations used in this work are cosmological zoom-in-constrained simulations, based on initial conditions created for the SIBELIUS project31 and designed to reproduce Local Group (LG) analogues within the observed large-scale structure. The simulations assume a ΛCDM cosmology with density parameters Ω0 = 0.307 and ΩΛ = 0.693 for matter and dark energy, respectively, an r.m.s. density variation on a scale of 8h1 Mpc of σ8 = 0.8288 and a Hubble parameter of h = 0.6777. We use physical units throughout this work. Building on an octree representation of the phase information65, we used the methods described in ref. 66 to supplement the observationally constrained scales in the initial density field67,68 with independent random information below 3.2 comoving Mpc (cMpc). In total, we generated 60,000 simulations, resulting in several thousand loosely defined Local Group analogues. From these, we selected 112 for the high-resolution resimulations used in this work. All initial conditions refine a Lagrangian region extending to at least r = 3 Mpc around the centre of the LG at z = 0, with the remainder of a 1,0003 cMpc3 volume populated with progressively more massive particles. The simulations start at z = 127. More details about the initial conditions may be found in ref. 31.

#### Simulations

All simulations used in this work were performed with the public version of the Gadget-4 code69, on the Cosmology Machine at the University of Durham and at the Finnish IT Center for Science. Extended Data Fig. 5 shows a comparison of one LG analogue at two different resolutions. The left panel corresponds to the resolution of the set of 60,000 simulations of the SIBELIUS project (particle mass 2.0 × 109M). At this resolution, a MW analogue halo contains approximately 500 particles, and only the largest substructures are identifiable. The right panel shows a simulation with a particle mass of 1.0 × 106M. At this resolution, a MW analogue halo contains approximately 106 particles, and an average of ~200 subhaloes down to 2 × 107M can be identified within 300 kpc from the centre. All results presented in this paper are based on simulations at this resolution.

#### Structure finding

Structures and self-bound substructures were identified using the Friends-of-Friends and Subfind algorithms implemented in Gadget-4 at 60 snapshots equally spaced in time, from z = 4 until a lookback time of 1 Gyr, and a further 40 snapshots equally spaced over the final 1 Gyr up to z = 0. Given their mass and separation, the two most massive self-bound substructures of the LG analogues can either belong to the same or to separate Friends-of-Friends structures. Throughout this work, we refer to the two principal self-bound substructures of each LG analogue at z = 0 simply as ‘haloes’ and to the lower mass substructures within 300 kpc of the centre of potential of each halo as ‘satellites’. We select Local Group analogues as pairs of haloes with individual masses in the range 0.6–2.2 × 1012M, separated by 500–1,000 kpc, with radial velocity −150 < vr <−50 km s−1 and transverse velocity vt < 70 km s−1. In total, our set of high-resolution simulations contains 101 LG analogues, and, for the purposes of this work, we consider both haloes as a MW analogue.

We use Gadget’s on-the-fly merger tree construction to find the progenitors of these subhaloes at previous snapshots. We cut the chain of links when a subhalo’s progenitor is no longer found, or when a clear discontinuity in mass and position indicates that a satellite’s progenitor has been erroneously identified as the main halo. At each snapshot, we record the maximum circular velocity of each subhalo, xc,red and define vpeak as the highest value of vmax of a subhalo and its progenitors over time. Following ref. 17, we use the standard procedure to rank satellites by vpeak, and identify the top 11 within 300 kpc of each MW analogue at z = 0 as analogues to the classical MW satellites.

#### Obtaining a complete satellite sample

As noted above, the radial distribution of satellites is important for the anisotropy. Numerical simulations suffer from the artificial disruption of substructures, that can affect subhaloes far beyond the particle number limit at which they can theoretically be identified28,70. According to van den Bosch and Ogiya71, the main cause of this artificial tidal disruption is inaccurate force softening, which can cause force errors that, for particles within substructures, do not cancel out: once a particle is lost to the main halo, it cannot be recovered. Additionally, the amplification of discrete noise in the presence of a strong tidal field near the centre of a halo can lead to a runaway instability that can lead to the complete, but purely numerical, disruption of a subhalo.

These effects can, however, be mitigated using semi-analytical models (which populate merger trees constructed from simulated dark matter subhaloes with galaxies). These models include so-called ‘orphan’ galaxies, that is, galaxies whose dark matter subhalo has been numerically disrupted. After the subhalo is disrupted numerically, its subsequent evolution is followed by tracing the positions of its most bound particle32. Our ‘complete’ sample includes these ‘orphan’ subhaloes.

One important result of this work is that the ‘incomplete’ and ‘complete’ samples of satellite haloes have different radial distributions. Even though our high-resolution simulations resolve, on average, 200 surviving satellite haloes inside 300 kpc of each MW analogue at z = 0 and we rank the satellites by vpeak (vmax being more strongly affected by tidal stripping), we find that the radial distribution of the top 11 surviving satellites in the ‘incomplete’ samples are systematically and significantly less centrally concentrated than the MW’s brightest satellites.

Figure 2 in the main text showed consistency between the radial distributions of the complete sample and the Milky Way satellites. It is reproduced in the top row of Extended Data Fig. 6. By contrast, the bottom row of Extended Data Fig. 6 shows how our comparison between simulations and observations would have looked had we considered only subhaloes from the incomplete samples. In the bottom left, it can be seen that their radial distribution is systematically offset from the MW data (shown as thick black line). For example, the innermost nine MW satellites are found within a distance of 140 kpc from the Galactic Centre, but none of the 202 incomplete samples has 9 satellites within this radius. The bottom-centre panel shows that the more uniform radial distributions of the incomplete samples lead to more equal contributions to the moment of inertia than in the case of the MW satellites. In fact, as shown on the bottom-right panel, none of the incomplete satellite systems have Gini coefficients as high as the MW’s and only two have c/a as low or lower. This is in line with previous studies (for example, refs. 22,26), which, presumably using incomplete subhalo populations, have found c/a values as low as the MW’s to be very rare.

To demonstrate the dependence of the anisotropy on the Gini coefficient and the radial concentration independently of our simulations, we also repeat the analysis on synthetic random data. The distributions shown in the top row of Extended Data Fig. 7 use a parent radial distribution that is uniform in r1/2, while those in the bottom row use one that is uniform in r; both parent angular distributions are isotropic. The relation between c/a and G is independent of the radial distribution, but the more centrally concentrated r1/2 distribution attains larger values of G and higher anisotropies than the more extended one.

#### Baryonic effects

In hydrodynamic simulations, Milky Way mass haloes tend to be slightly more spherical than their counterparts in collisionless simulations72, which might reduce the anisotropy of subhaloes. Additionally, the adiabatic contraction induced by baryons can increase the concentration of the dark matter halo73, which can lead to a more central concentration of subhaloes but, conversely, could also enhance the disruption of subhaloes near the centre.

The presence of a stellar disk could also lead to additional disruption of subhaloes close to the centre of the galaxy74,75,76,77. To estimate its possible impact, we have calculated the fraction of subhaloes and orphans that pass through a cylinder whose radius, r = 4.3 kpc, is twice the scale-length and whose height, h = 0.5 kpc, is twice the scale-height of the MW disk during the past 1.3 Gyr. We find that in approximately half of the systems, none of the 11 satellites that we consider analogues of the classical satellites have passed through this disk, and at most one satellite has in ~83% of systems. Since satellites that passed through the disk are more likely to be found near the centre, this disruption could make the radial profiles slightly more extended. We find that this does not have a significant effect on our results: removing all satellites identified as having passed through the disk and replacing them with the next highest by vpeak only reduces the median Gini coefficient from 0.63 to 0.61, while it increases the median value of c/a from 0.35 to 0.36.

Some authors have argued that, in aggregate, the various baryonic effects that could individually lead to higher or lower anisotropy can increase the anisotropy of satellite systems in ΛCDM13,35,78. On the other hand, refs. 26 and 20 concluded that baryons only have a negligible effect. A fully realistic simulation of the Milky Way satellite system would clearly include baryons, but we believe that the inclusion of baryons would, at the very least, not decrease the anisotropy significantly. In other words, including baryons is unlikely to make the ‘plane of satellite problem’ worse.