###### Abstract

We review the recent theoretical and observational developments concerning the
interaction of the Large Magellanic Cloud (LMC) with the Milky Way and its
neighbourhood. An emerging picture is that the LMC is a fairly massive
companion (10–20% of the Milky Way mass) and just passed the pericentre of its
orbit, likely for the first time. The gravitational perturbation caused by the
LMC is manifested at different levels. The most immediate effect is the
deflection of orbits of stars, stellar streams or satellite galaxies passing
in the vicinity of the LMC. Less well known but equally important is the
displacement (reflex motion) of central regions of the Milky Way about the
centre of mass of both galaxies. Since the Milky Way is not a rigid body, this
displacement varies with the distance from the LMC, and as a result, the
Galaxy is deformed and its outer regions (beyond a few tens kpc) acquire a net
velocity with respect to its centre. These phenomena need to be taken into
account at the level of precision warranted by current and future
observational data, and improvements on the modelling side are also necessary
for an adequate interpretation of these data.

###### keywords:

Galaxy: kinematics and dynamics; Galaxy: structure; Magellanic Clouds; Local
Group

11 1 59 Academic Editors: Vladimir Korchagin, Maxim Yu. Khlopov, Orchidea
Maria Lecian 4 March 2023 13 April 2023 14 April 2023 18 April 2023
https://doi.org/10.3390/galaxies11020059 The effect of the LMC on the Milky
Way system LMC and the Milky Way Eugene Vasiliev Eugene Vasiliev Vasiliev, E.

##  1 Introduction

The Magellanic Clouds are the best known and the largest galaxies in the Milky
Way neighbourhood, and the only ones prominently visible on the night sky.
They were always believed to be Galactic satellites, but the views on how much
they affect the Milky Way has changed over years. This review summarises the
recent observational and theoretical developments that together point to a
large impact of the LMC on the dynamical processes in the outer Galaxy and
even beyond.

The long-accepted view that the Clouds have been circling around our Galaxy
for many orbits (e.g., (Tremaine, 1976; Murai and Fujimoto, 1980; Lin and
Lynden-Bell, 1982; Gardiner et al., 1994)) gradually gave way to the now-
standard picture that their past orbits are highly eccentric with apocentres
well beyond 200 kpc and orbital periods in excess of 5 Gyr, or even unbound
(Besla et al., 2007). This conclusion is largely driven by the high measured
tangential velocity component of the LMC (Kallivayalil et al., 2006, 2013),
but would also naturally explain the continuous presence of the Small
Magellanic Cloud (SMC) and other more recently discovered satellites of the
LMC, which would have been stripped by the Milky Way tidal field if the Clouds
were on a much tighter orbit around the Galaxy. Nevertheless, as discussed in
Section 3, there remain large uncertainties in the past orbit of the LMC and a
number of caveats in reconstructing it.

Another recent revelation was the mounting evidence (reviewed in Section
2.2.1) that the LMC is fairly massive, only 5–10 times smaller than the Milky
Way itself. Interestingly, more than 50 years ago a similarly titled paper by
Avner and King (1967) adopted a 1:10 mass ratio, but back in the days both
galaxies were believed to be 10 times less massive than current estimates. The
high mass of the LMC naturally explains a number of independent dynamical
effects discovered recently, which are the major part of this review and are
discussed in Section 4.

##  2 Introducing the participants

Before proceeding to the discussion of various aspects of the Milky Way–LMC
interaction, we summarise the current knowledge about our Galaxy and the
Magellanic system, in as much as it is relevant for their interaction.

###  2.1 Milky Way

Naturally, the Galaxy that we live in happens to be the best studied one. The
quantity and precision of observational data have skyrocketed in the last
decade with the advent of large-scale spectroscopic surveys such as SEGUE
(Yanny et al., 2009), APOGEE (Majewski et al., 2017) and LAMOST (Zhao et al.,
2012), and since 2018, the Gaia satellite provides positions, parallaxes and
proper motion (PM) measurements for $\gtrsim 1.5\times 10^{9}$ stars across
the entire sky (Gaia Collaboration, 2018, 2021, 2022).

One of the major consequences of this wealth of data is that now it becomes
possible to explore not only the global structure of the Galaxy, but also
relatively small ($\sim 10-20\%$) deviations from equilibrium. The best-known
example is the so-called “Gaia snail” – a winding spiral pattern in the
$\\{z,v_{z}\\}$ phase space (Antoja et al., 2018), indicating ongoing phase
mixing in a perturbed disc. The leading (though not universally accepted)
theory is that it was created by a passage of a massive satellite (likely the
Sagittarius dSph) through the disc plane $\sim 1$ Gyr ago (e.g., Widrow et
al., 2012; Laporte et al., 2018). As will be discussed in this review, the LMC
is unlikely to significantly disturb the inner Galaxy, but creates a
comparable if not larger perturbation in the outer halo.

Our knowledge of Galactic structure and mass distribution is mostly derived
from measuring the kinematics of stars and other tracers, coupled with
suitable modelling procedures. The observational data on the kinematics of
tracers are abundant in the Solar neighbourhood, but increasingly scarce and
less precise at large distances. For instance, the line-of-sight velocity of
individual stars is usually measured to within a few km s-1, but the
uncertainty in the transverse velocity at 50 kpc is at the level of 100 km s-1
in the current Gaia DR3, comparable to the velocity dispersion in the halo.
Much better precision can be obtained for globular clusters and satellite
galaxies by averaging the PM of many stars, but these tracers themselves are
not very numerous, with only a few dozen galaxies and a handful of clusters
beyond 50 kpc. Last but not least, stellar streams formed by tidal stripping
of stars from clusters and satellites (including objects that by now are
completely disrupted) are very valuable probes of Galactic potential, since
stars in each stream approximately follow the orbit of its progenitor and span
a large range of Galactocentric distances. Refs. (Eyre and Binney, 2011;
Sanders and Binney, 2013; Bovy, 2014; Gibbons et al., 2014) and other studies
discuss the mechanics of stream formation and their use for constraining the
Galactic potential. Nearly a hundred streams in the Milky Way have been
catalogued so far (Mateu, 2023), although only a few reach large enough
Galactocentric distances to be affected by the LMC. Of these, the most
important are the Sagittarius stream (Majewski et al., 2003) produced by the
eponimous galaxy (one of the closest and most massive Milky Way satellites),
and the Orphan–Chenab stream without a known progenitor (Grillmair, 2006;
Belokurov et al., 2007; Shipp et al., 2018; Koposov et al., 2019). Both span
more than $180^{\circ}$ on the sky and tens of kpc in distance, and are
sensitive to the LMC perturbation, as discussed below.

Many of the “classical” modelling techniques such as Jeans equations or
distribution function-based models rely on the assumption of dynamical
equilibrium, and need a sufficient number of individual tracers to exploit
this assumption. Any single kinematic tracer provides very weak constraints
(essentially, only a lower limit on the escape velocity, assuming that the
object is bound to the Galaxy), and a joint distribution of many objects is
needed to exploit the assumption of phase-mixedness built into these modelling
approaches. On the other hand, even a single stream can provide interesting
constraints on the potential in the range of radii spanned by its track (e.g.,
Johnston et al., 1999; Bonaca and Hogg, 2018). Bland-Hawthorn and Gerhard
(2016) give a comprehensive review of Milky Way structure at all scales in the
pre-Gaia era, and Wang et al. (2020) summarize the more recent developments in
determining the total mass of our Galaxy and its spatial distribution. The
precision and scatter in these measurements greatly vary with Galactocentric
distance. Naturally, the best-studied region is around the Sun ($\sim 8$ kpc),
where the enclosed mass is known to within $\lesssim 10\%$ and is of order
$10^{11}\,M_{\odot}$. On the other hand, at larger distances the constraints
are generally weaker and display a large scatter between different studies.
Sometimes the well-measured mass distribution in the inner Galaxy is
extrapolated to large distances assuming a particular functional form (e.g., a
Navarro–Frenk–White profile), resulting in unrealistically small uncertaities
in absense of actual kinematic tracers. An emerging consensus is that the
stellar and total mass of the Milky Way are of order $5\times
10^{10}\,M_{\odot}$ and $10^{12}\,M_{\odot}$ respectively, with an optimistic
level of uncertainty being $\sim 20-30\%$. The mass enclosed within 50 kpc is
better constrained and is at the level $4\times 10^{11}\,M_{\odot}$,
corresponding to a circular velocity of 185 km s-1. On the other hand, the 3d
shape of the Galactic halo remains largely unknown, with different studies
arriving at conflicting results.

###  2.2 LMC

####  2.2.1 Mass

0123$\displaystyle M_{LMC}$$\displaystyle\phantom{>}2.50^{+0.90}_{-0.80}$
Peñarrubia et al. (2016)$\displaystyle\phantom{>}1.38^{+0.37}_{-0.24}$ Erkal
et al. (2019)$\displaystyle{>}1.24\\!$ Erkal and Belokurov
(2020)$\displaystyle\phantom{>}1.30^{+0.30}_{-0.30}$ Vasiliev et al.
(2021)$\displaystyle\phantom{>}1.88^{+0.35}_{-0.40}$ Shipp et al.
(2021)$\displaystyle\phantom{>}1.65^{+0.47}_{-0.49}$ Correa Magnus and
Vasiliev (2022)$\displaystyle\phantom{>}1.29^{+0.28}_{-0.23}$ Koposov et al.
(2023)$\displaystyle\times 10^{11}\,M_{\odot}$Local Groupstreamssat.
censussat. kinem. Figure 1: LMC mass estimates from different methods: blue –
perturbations inflicted on stellar streams (Erkal et al., 2019; Vasiliev et
al., 2021; Shipp et al., 2021; Koposov et al., 2023); cyan – momentum balance
in the Local Group (Peñarrubia et al., 2016); yellow – census of LMC
satellites (Erkal and Belokurov, 2020); magenta – kinematics of Milky Way
satellites (Correa Magnus and Vasiliev, 2022).

The LMC is undoubtedly the largest satellite of the Milky Way, but just how
large it is only became apparent relatively recently (see Figure 1 for a
compilation of estimates). The stellar mass of the LMC is $\sim 3\times
10^{9}\,M_{\odot}$ (van der Marel et al., 2002) and gas mass $\sim 5\times
10^{8}\,M_{\odot}$ (Fox et al., 2014), but its total mass ought to be
significantly larger. There are several lines of evidence suggesting that the
LMC mass is in the range $(1-2)\times 10^{11}\,M_{\odot}$.

  * •

Cosmological stellar mass–halo mass relation.

  * •

Mass modelling from internal kinematics.

  * •

Census of its satellites.

  * •

Interaction with the SMC.

  * •

Dynamical perturbation of stellar streams.

  * •

Kinematic and spatial distortions in the Milky Way halo.

  * •

Hubble flow in the Local Universe.

Let us consider these arguments in turn. In the standard cosmological
scenario, the baryonic mass of the LMC is expected to comprise about 1% of its
total mass (Moster et al., 2013; Read and Erkal, 2019), implying the latter to
be $\gtrsim 10^{11}\,M_{\odot}$. The caveat is that if the LMC has been
orbiting the Milky Way for some time, its dark halo would have been tidally
stripped, similarly to what has happened to the Sagittarius dSph, whose
present-day total mass is only a few times higher than its stellar mass
Vasiliev and Belokurov (2020). However, as discussed below, this scenario
appears to be unlikely.

The three-dimensional structure and kinematics of stars in the LMC have been
extensively studied in several large-scale photometric, spectroscopic and
astrometric surveys, and in combination with suitable modelling techniques
making it possible to determine its total mass distribution. Using the Hubble
space telescope (HST) astrometry, (van der Marel and Kallivayalil, 2014)
measured the asymptotic circular velocity to be $\sim 90$ km s-1 at the
distance of 9 kpc, implying an enclosed mass of $\sim 2\times
10^{10}\,M_{\odot}$. More recent analysis based on Gaia DR2 (Vasiliev, 2018;
Wan et al., 2020) and (e)DR3 (Luri et al., 2021; Jiménez-Arranz et al., 2023)
corroborates these findings. The mass profile at larger distances is difficult
to measure, not only due to the dearth of kinematic tracers, but also due to
apparent disturbances in the stellar motions likely caused by the interaction
with the SMC Choi et al. (2022); Cullinane et al. (2022). Nevertheless, the
measured peak circular velocity translates to the halo mass of order
$10^{11}\,M_{\odot}$ (Katz et al., 2019), in agreement with the stellar mass-
based estimate (a consequence of the baryonic Tully–Fisher relation).

The LMC possesses a number of satellites, as discussed in the next section,
but in order to hold them up, it needs to be massive enough. In particular,
the relative velocity of the LMC–SMC pair ($\sim 100$–120 km s-1) is quite
high, and would correspond to an unbound orbit if their total mass is
$\lesssim 10^{11}\,M_{\odot}$ (Kallivayalil et al., 2013). It is considered
unlikely that the two Clouds just happened to be both coming into the vicinity
of the Milky Way independently, without being a bound system.

Perturbations of stellar streams and other kinematic tracers in the outer
Milky Way halo are discussed at length in Section 4, but they all are best
explained if the LMC mass is in the range (1–2)${}\times 10^{11}\,M_{\odot}$.
Finally, Peñarrubia et al. (2016) estimated that an LMC mass of up to
$2.5\times 10^{11}\,M_{\odot}$ is needed to best match the local Hubble flow
(at a distances of a few Mpc); this argument is discussed at the end of
Section 4.3.

####  2.2.2 Satellites

Table 1: Possible satellites of the LMC, according to several studies: Jethwa et al. (2016), Sales et al. (2017), Kallivayalil et al. (2018), Erkal and Belokurov (2020), Patel et al. (2020), Battaglia et al. (2022), Correa Magnus and Vasiliev (2022). Highly likely, uncertain, unlikely and recently captured/interacted satellites are denoted by +, ?, – and $c$ respectively. Objects in brackets lacked line-of-sight velocity measurements at the time of writing of the corresponding paper, but are consistent with being satellites for some values of velocity (note that all objects in J16 and S17 also lacked PM measurements).  galaxy | J16 | S17 | K18 | E20 | P20 | B22 | C22
---|---|---|---|---|---|---|---
Carina |  |  |  | – | – | ? | –
Carina II |  |  | + | + | + | + | +
Carina III |  |  | + | + | + | + | +
Delve 2 |  |  |  |  |  |  | (?)
Eridanus III |  | (?) |  | (?) |  |  | (?)
Grus I | ? | ? | – | – |  | – | –
Grus II | (?) |  |  | (–) |  | $c$ | $c$
Horologium I | + | + | + | + | + | + | +
Horologium II | (+) | (?) |  | (–) |  | ? | ?
Hydrus I |  |  | + | + | + | + | +
Pictor I | (?) |  |  | (–) |  |  | (–)
Pictor II |  |  |  |  |  |  | (?)
Phoenix II | (+) | (?) | (?) | + | $c$ | + | +
Reticulum II | + | ? | – | + | $c$ | + | ?
Reticulum III | (?) | (?) |  | (–) |  | – | –
SMC |  |  |  | + |  |  | +
Tucana II | + | ? | – | – |  | – | –
Tucana III | (?) |  | – | – | – | – | –
Tucana IV | (+) | (?) |  | (–) |  | ? | $c$
Tucana V | (+) | (?) |  |  |  | – | –

The LMC comes not alone, but with an entourage of its own satellite galaxies.
One of them, the SMC, is well known, whereas other, much fainter ones, have
been discovered relatively recently, mostly in the Dark Energy Survey (Koposov
et al., 2015; Bechtol et al., 2015; Drlica-Wagner et al., 2015). Several
studies have estimated the probability of being a satellite of the LMC for a
number of classical and ultrafaint dwarf galaxies; Table 1 summarizes these
findings. SMC and 5–6 ultrafaint dwarfs have been universally accepted as
members of the LMC cohort, while a couple of other galaxies have experienced a
recent interaction with the LMC, but are unlikely to be its satellites in the
past. By comparing this number of satellites with cosmological simulations,
Erkal and Belokurov (2020) estimated the LMC mass to be at least $1.2\times
10^{11}\,M_{\odot}$, but noted that many more even fainter satellites might
await detection (see also (Jethwa et al., 2016)). Naturally, the association
of these galaxies with the LMC is conditioned on the LMC mass (see Figure 13
in (Correa Magnus and Vasiliev, 2022)), but for most of these candidate
satellites it stays fairly high in the entire range of reasonable masses.

###  2.3 SMC

The SMC, being in conspicuous proximity to the LMC and only a few times less
luminous, was long suspected to be a satellite of the latter. Reconstruction
of its 3d shape from photometry of red clump stars and other standard candles
indicates that it is fairly extended along the line of sight: with a mean
distance of $\sim 68$ kpc, its thickness is of order 10 kpc (see Table 1 in
(Tatton et al., 2021) for a summary of studies). This, together with kinematic
evidence for a strong perturbation, suggests a rather dramatic harassment by
its bigger sibling. The most recent close encounter between the two Clouds
occurred around 150 Myr ago at a distance of $\lesssim 10$ kpc (Besla et al.,
2012; Kallivayalil et al., 2013; Zivick et al., 2018), but in all likelihood,
they have orbited each other for several Gyr in the past. As mentioned
earlier, the requirement that the Clouds form a bound pair imposes a lower
limit on their total mass of order $10^{11}\,M_{\odot}$. It is quite likely
that the SMC had been much more massive in the past than it is now, having
been stripped of most of its dark matter halo and a considerable fraction of
its gas. Its present-day mass is estimated to be
$\mathcal{O}(10^{10})\,M_{\odot}$, thus it has negligible effect on the Milky
Way, but still an appreciable effect on the LMC. In particular, if both
galaxies should be considered as a binary system, then the reconstruction of
their past orbits should be performed for the centre-of-mass velocity rather
than the LMC velocity.

###  2.4 Magellanic stream, bridge and other structures

A long and prominent gas stream trailing behind the LMC was discovered by
Mathewson et al. (1974). In the earlier picture where the LMC was orbiting the
Milky Way for a long time, the stream was believed to be created either by the
ram pressure stripping or the tidal force of our Galaxy (Murai and Fujimoto,
1980). However, in the more recent scenario of the first approach, its origin
must be different, and the prevailing view is that it is formed primarily from
the gas stripped from the SMC while it orbits the LMC (Besla et al., 2010).
The leading arm of this gas stream is more difficult to explain in this
scenario, especially when including the hot gas corona of the LMC (Tepper-
García et al., 2019), though the debate about the role of hot corona is still
open (Lucchini et al., 2020). D’Onghia and Fox (2016) present a comprehensive
review of the Magellanic gas stream.

Unlike the Sagittarius galaxy, the LMC does not produce any noticeable stellar
stream, further strengthening the argument that it is likely on its first
passage and/or embedded in a massive protective dark halo envelope. However,
recently a tentative evidence for a stellar counterpart of the gas stream was
provided by Zaritsky et al. (2020) and Petersen et al. (2022) in the trailing
and leading arms, respectively. The measured microlensing optical depth
towards the LMC can be explained by the lensing of the faint stellar
counterpart of the Magellanic stream by stars of the LMC halo (Besla et al.,
2013). A young star cluster was discovered by Price-Whelan et al. (2019) near
the leading arm, but follow-up spectroscopic observations disagreed about its
possible association with the stream: Nidever et al. (2019) found it likely
while Bellazzini et al. (2019) placed it on a much tighter orbit around the
Galaxy with a pericentre $\lesssim 15$ kpc and a period of $\sim 0.5$ Gyr.

On the other hand, there are numerous stellar structures in the region between
the two Clouds (the Magellanic bridge) (Belokurov et al., 2017), likely formed
out of stars stripped from both galaxies in the recent collision (Belokurov
and Erkal, 2019; Zivick et al., 2019), although some perturbations in the
outer disc of the LMC are more likely to be caused by the Milky Way tidal
force (Mackey et al., 2016; Cullinane et al., 2022).

##  3 Orbit of the LMC

###  3.1 Present-day position and velocity

Table 2: Compilation of measurements of the LMC PM. $\alpha_{0}$ and $\delta_{0}$ are the ICRS coordinates of the measured centre point (or adopted from another study, shown in brackets); $\mu_{\alpha*}\equiv({\rm d}\alpha/{\rm d}t)\cos\delta$, $\mu_{\delta}\equiv{\rm d}\delta/{\rm d}t$ are the corresponding PM components in mas yr-1. See also Table 2 in D’Onghia and Fox (2016) for a summary of older measurements.  reference | $\alpha_{0}$ | $\delta_{0}$ | $\mu_{\alpha*}$ | $\mu_{\delta}$
---|---|---|---|---
Kallivayalil et al. (2006) | ($81.90^{\circ}$) | ($-69.87^{\circ}$) | $2.03\phantom{0}\pm 0.08$ | $0.44\phantom{0}\pm 0.05$
van der Marel and Kallivayalil (2014) | $78.76^{\circ}$ | $-69.19^{\circ}$ | $1.910\pm 0.02$ | $0.229\pm 0.05$
Helmi et al. (2018) | ($78.77^{\circ}$) | ($-69.01^{\circ}$) | $1.850\pm 0.03$ | $0.234\pm 0.03$
— (alt.centre) | ($81.28^{\circ}$) | ($-69.78^{\circ}$) | $1.890\pm 0.03$ | $0.314\pm 0.03$
Wan et al. (2020) | $80.90^{\circ}$ | $-68.74^{\circ}$ | $1.878\pm 0.03$ | $0.293\pm 0.03$
Luri et al. (2021) | ($81.28^{\circ}$) | ($-69.78^{\circ}$) | $1.858\pm 0.02$ | $0.385\pm 0.02$
— (alt.centre) | $81.07^{\circ}$ | $-69.41^{\circ}$ | $1.847\pm 0.02$ | $0.371\pm 0.02$

The LMC is currently located at a distance of $\sim 49.6\pm 0.5$ kpc
(Pietrzyński et al., 2019), its heliocentric line-of-sight velocity is $\sim
262.2\pm 3.4$ km s-1 (van der Marel et al., 2002), and its PM has been
measured with an increasing precision first by HST (Kallivayalil et al., 2006,
2013) and more recently by Gaia (Helmi et al., 2018; Luri et al., 2021). There
is some ambiguity regarding the position of the LMC centre (see section 4.1 in
van der Marel and Kallivayalil (2014) for a discussion), and different centre
locations correspond to different values of mean PM (Helmi et al., 2018; Wan
et al., 2020; Luri et al., 2021) due to both perspective effects and internal
motion in the LMC. The line-of-sight velocity also has a significant spatial
gradient of $\sim 10$ km s-1 per degree (mostly with declination $\delta$),
but unfortunately this factor is ignored in almost all studies, which adopt
the same value but attribute it to a different centre point. Table 2 lists a
number of recent measurements of the mean PM and the associated centre
locations (in some studies it was kept fixed while in others it was varied
during the fit). The LMC moves away from the Milky Way with a Galactocentric
radial velocity of $\sim 70$ km s-1, whereas its tangential velocity is around
310 km s-1 (well above the circular velocity at that distance). These values
imply that it just recently passed the pericentre of its orbit, but the
reconstruction of its orbit further into the past depends on a number of
factors, most importantly the masses of both galaxies.

Until the more precise space-based measurements of the LMC velocity became
available, it was commonly assumed that it had completed several orbits around
the Milky Way over the Hubble time (e.g., Murai and Fujimoto, 1980). However,
the high transverse velocity measured by Kallivayalil et al. (2006) implies a
very eccentric or even hyperbolic orbit, unless the Milky Way is very massive.
(Note however that the more recent PM measurements reduce the tangential
velocity by a few tens km s-1, weakening the evidence for a highly eccentric
or unbound orbit). Thus Besla et al. (2007) put forward the now favourite
scenario of a first infall (i.e., no previous pericentre passages within the
last 10 Gyr, although the precise definition varies between studies, e.g.,
Patel et al. (2020) denoted first-infall orbits as having a period above 6 Gyr
and apocentre exceeding the virial radius of the Milky Way). There are several
arguments supporting this scenario:

  * •

Vigorous ongoing star formation without signs of quenching indicates ample
reservoir of gas, while it is expected that a satellite galaxy would be
stripped of its gas shortly after infall into the main halo. The star
formation rate of the LMC has been unusually low until a recent burst starting
$\sim 3$–4 Gyr ago (Harris and Zaritsky, 2009; Meschin et al., 2014;
Hasselquist et al., 2021; Massana et al., 2022), which might have been
triggered by the compression of gas as it experiences a bow shock upon
entering the Milky Way gas corona, although the interaction with the SMC is
another possible explanation. A lack of evidence for an earlier episode of
elevated star formation rate (excluding the time shortly after the Big Bang)
may be seen as the argument against a previous pericentre passage.

  * •

Absence of a large-scale stellar tidal stream like that of the Sagittarius
galaxy, which has completed several orbits around the Milky Way. Signs of
tidal perturbation in the outer disc (Mackey et al., 2016; Belokurov and
Erkal, 2019; Cullinane et al., 2022) can be attributed to the interaction
between LMC and SMC. There is a prominent gas stream (Mathewson et al., 1974)
that roughly matches the past orbit of the Magellanic Clouds over
$>150^{\circ}$ on the sky; however, it is also better explained by the
interaction between the two Clouds (Besla et al., 2010; Diaz and Bekki, 2012).

  * •

LMC satellites, including the SMC, would have been tidally stripped from it if
it had similarly close pericentre passage around the Milky Way in the past.

###  3.2 Past orbit

The reconstruction of the past orbit of the LMC is a challenging task. If its
mass were negligible, one could simply integrate the equations of motion
backward in time for any assumed Milky Way potential. Even in this
oversimplified case, the orbit parameters (apocentre distance and period) are
surprisingly sensitive to minor variations in the assumed present-day phase-
space coordinates. Figure 2 shows the orbits computed in a fiducial Milky Way
potential for several choices of the LMC PM and associated centre locations
from Table 2. Although the difference in 3d velocity between these studies
(with the exception of the older HST measurement from Kallivayalil et al.
(2006)) is only $\sim 10$ km s-1, the orbital periods vary by a factor of two
because the 3d velocity is quite close to the local escape velocity. A
comparison between left and right panels shows that even a 1% variation in the
assumed distance changes the orbits very significantly; this change is driven
by a $\sim 5$ km s-1 increase in the tangential velocity (same PM multiplied
by a slightly larger distance) rather than a change in the position per se.
This complicates the comparison of studies that adopt slightly different
present-day positions and PM of the LMC without much discussion of the impact
of these choices.

The LMC orbit is also very sensitive to the assumed Galactic potential. Figure
3, left panel, shows the evolution of Galactocentric distance for several
choices of the potential. The adopted Milky Way density profile consists of
three components:
bulge: $\rho_{\rm
bulge}\propto(1+r/0.2\,\mbox{kpc})^{-1.8}\,\exp\big{[}-(r/1.8\,\mbox{kpc})^{2}\big{]},\quad
M_{\rm bulge}=1.2\times 10^{10}\,M_{\odot}$;
disc: $\rho_{\rm
disc}\propto\exp\big{[}-R/3\,\mbox{kpc}-|z|/0.3\,\mbox{kpc}\big{]},\quad
M_{\rm disc}=5\times 10^{10}\,M_{\odot}$;
halo: $\rho_{\rm halo}\propto r^{-1}\,(r+r_{\rm
halo})^{-2}\,\exp\big{[}-(r/300\,\mbox{kpc})^{4}\big{]}$.
The halo density normalization is chosen to produce a circular velocity of 233
km s-1 at $R=8$ kpc. The virial mass of this model (referred to as $M_{\rm
MW}$) is defined as the mass enclosed in a radius $r_{\rm vir}$ such that the
mean density in this volume is 100 times the critical density of the Universe;
$r_{\rm vir}=(M_{\rm MW}/10^{12}\,M_{\odot})^{1/3}\times 260$ kpc. For the
fiducial model, we set $r_{\rm halo}=14$ kpc, corresponding to $M_{\rm
MW}=0.95\times 10^{12}\,M_{\odot}$ and $M(<50\,\mbox{kpc})=0.41\times
10^{12}\,M_{\odot}$. In this case, a test-particle orbit has a period of 11
Gyr. A mere 10% variation in the Milky Way mass (well within the current
uncertainties) changes the period to 30 Gyr or 6 Gyr.

Figure 2: Past trajectory of the LMC for various choices of the present-day
position/velocity. Different lines correspond to sky positions and PM from
different studies listed in Table 2; shaded regions show the uncertainties
associated with the PM and line-of-sight velocity ($262.2\pm 3.4$ km s-1 in
all cases (van der Marel et al., 2002)). Left and right panels show the effect
of changing the current distance by 1%, which is comparable to the difference
in PM between studies and far exceeds the nominal PM uncertainty in each
study. The LMC follows a test-particle orbit in the fiducial potential
described in the text.  Figure 3: Past trajectory of the LMC for different
choices of the Milky Way potential (left panel) or LMC mass (right panel). The
black curve corresponds to the fiducial potential with $M_{\rm MW}=0.95\times
10^{12}\,M_{\odot}$, $M_{\rm LMC}=0$, present-day position and PM taken from
Luri et al. (2021), and heliocentric distance of 49.6 kpc (Pietrzyński et al.,
2019). Blue and green curves in the left panel are plotted for a slightly
lighter ($0.85\times 10^{12}\,M_{\odot}$) or heavier ($1.05\times
10^{12}\,M_{\odot}$) Milky Way, and the magenta curve represents the case when
the Milky Way mass grows linearly over the Hubble time. Shaded regions show
the 1$\sigma$ range of orbits arising from the uncertainty in the present-day
position/velocity (chiefly the distance with a 1% error bar). In the right
panel, dotted blue and dashed red curves show the trajectory of a massive LMC
($0.5\times 10^{11}\,M_{\odot}$ and $0.5\times 10^{11}\,M_{\odot}$) in live
$N$-body simulations correspond to the same Galactic potential and present-day
phase-space coordinates as the black (test-particle) orbit. The dot-dashed
yellow curve shows the orbit of a massless LMC, but with present-day
coordinates corresponding to the centre of mass of the LMC–SMC system with a
mass ratio of 10:1.  Figure 4: Past trajectory of the LMC for different
choices of the Milky Way halo shape parametrised by the axis ratios $p\equiv
Y/X$ and $q\equiv Z/X$, where the orientation of the coordinate system is such
that the LMC is currently located on the $X$ axis and its angular momentum
points in the $Z$ direction. The Galactic potential has the same spherically-
averaged mass profile in all cases, with the total mass $M_{\rm MW}=0.95\times
10^{12}\,M_{\odot}$, and the LMC follows a test-particle orbit; the
spherically symmetric case is identical to the one shown by the black curve in
Figure 3. The trajectory plane $X$–$Y$ plane is approximately perpendicular to
the Galactic disc, which is shown by a black dash in the top left panel. The
orbits for 9 different combinations of $p$ and $q$ are shown by different
colours and dashes. Centre and right columns show the equipotential contours
for different choices of $p$ and $q$, and the corresponding orbits. The strong
variation of orbital period and the apocentre radius is caused by the changes
in the value of the potential at the current location of the LMC. This is more
apparent for the centre column, where making the halo more flattened in the
$Y$ direction (from top to bottom) stretches the equipotential surfaces
horizontally, thereby lowering the value of the potential on the $X$ axis. On
the other hand, increasing the flattening in the $Z$ direction (right column,
from top to bottom) leaves the equipotential contours nearly round, but lowers
the potential in the entire $X$–$Y$ plane.

As already mentioned, the shape of the Galactic halo is poorly known, and it
also has a significant effect on the past orbit of the LMC. Figure 4 shows
test-particle orbits in several variants of a triaxial Milky Way halo, which
all have the same spherically averaged density profile. Naturally, the halo
shape and orientation affects the rate of precession of the orbit, but even
more importantly, it also significantly changes the orbital energy and hence
the period. To highlight this effect, we choose the orientation of the
triaxial halo such that the $X$ axis points towards the current location of
the LMC, and the $Z$ axis is parallel to its angular momentum vector. A halo
that is flattened in the $Y$ direction has equipotential surfaces elongated
along the $X$ axis, thereby lowering the value of the potential at the
location of the LMC and making it more tightly bound. Flattening in the $Z$
direction has a similar but weaker effect, because the potential has lower
values in the entire $X$–$Y$ plane compared to the value along the $Z$ axis at
the same radius. The surprising strength of this effect is again due to the
LMC being close to its pericentre, so that even small variations of the
potential ($\sim 10\%$ between the most extreme cases in this example)
significantly change the total energy (by a factor of two). Interestingly, the
studies of the Orphan–Chenab (Erkal et al., 2019; Koposov et al., 2023) and
Sagittarius (Vasiliev et al., 2021) streams, although disagreening in detail,
generally favour an oblate halo shape with a minor axis roughly in the
direction of the LMC angular momentum. A prolate halo with a major axis
pointing towards the LMC is also a viable solution for the Orphan–Chenab
stream. In both cases, the non-spherical shape of the halo effectively deepens
the potential at the current location of the LMC, making its energy more
negative and shortening its period.

The above analysis considered test-particle orbits of the LMC. However, a
sufficiently massive LMC necessarily affects the orbits of stars in its
vicinity. The most well-known manifestation of this influence is the
accumulation of stars behind the moving object and the associated dynamical
friction force created by this density wake. The classical Chandrasekhar
(1943) theory has been developed for the idealised case of motion on a
straight line in an infinite homogeneous background, which is obviously never
satisfied in real circumstances; nevertheless, it provides a reasonable order-
of-magnitude estimate of the friction force and the magnitude of the
overdensity in the wake. As is well known, the long-range nature of
gravitational interactions leads to a logarithmic divergence of integral over
the impact parameter of two-body interactions, which determines the overall
friction force. In reality, the upper limit of this integral is imposed by the
inhomogeneity of the background distribution, and the lower limit is set
either by the distance of strong scattering ($\gtrsim 90^{\circ}$ deflection
angle), or the characteristic size of the moving object, whichever is larger.
The logarithm of the ratio of these two scales, known as the Coulomb logarithm
$\ln\Lambda$, is customarily treated as a free parameter in the Chandrasekhar
formula, calibrated against $N$-body simulations on account of approximate
nature of the theory. In the simplest case, one would keep $\ln\Lambda$
constant; however, this approximation is rather crude, and a more refined
version would make $\ln\Lambda$ depend on the orbital phase, usually by
equating the maximum impact parameter with the instantaneous orbit radius
(Tremaine, 1976; Hashimoto et al., 2003; Just and Peñarrubia, 2005).

One may conclude that a more massive LMC needs to have a larger apocentre
distance and higher initial energy, since it would have lost more energy to
dynamical friction by present time. This argument led Kallivayalil et al.
(2013) to the conclusion that a high mass of the LMC provides a stronger
support for the first-infall scenario (see also Figure 1 in Besla (2015)).
However, this ignores another equally important factor, namely the reflex
motion of the Milky Way in response to the LMC’s gravitational pull. If both
galaxies were point masses moving on Keplerian orbits about the common centre
of mass, then increasing the mass of the LMC would shorten its period. This
counteracts the effect of dynamical friction, but it is not obvious which of
the two factors dominates. This question would be readily settled by $N$-body
simulations, but the major problem is that for a reliable comparison of
different runs, one needs to match the final (present-day) relative position
and velocity of the LMC and the Milky Way to high precision. As already
illustrated by Figure 2, even a 5 km s-1 difference in velocity can
dramatically alter the past orbit.

A popular technique for reconstruction of past orbits of both galaxies (Milky
Way and LMC) under mutual gravitational tug is based on the following
approximation (Gómez et al., 2015; Jethwa et al., 2016; Erkal et al., 2019;
Patel et al., 2020; Vasiliev et al., 2021; Koposov et al., 2023). Each galaxy
is represented by a static (non-evolving) analytic potential $\Phi_{\rm MW}$,
$\Phi_{\rm LMC}$ centered at $\boldsymbol{x}_{\rm MW}$, $\boldsymbol{x}_{\rm
LMC}$ respectively. The centres of both galaxies satisfy the equations of
motion

$\displaystyle\ddot{\boldsymbol{x}}_{\rm LMC}$ $\displaystyle=$
$\displaystyle\nabla\Phi_{\rm MW}(\boldsymbol{x}_{\rm LMC}-\boldsymbol{x}_{\rm
MW})+\boldsymbol{a}_{\rm DF}\big{(}\rho_{\rm MW}(\boldsymbol{x}_{\rm
LMC}-\boldsymbol{x}_{\rm MW}),\dot{\boldsymbol{x}}_{\rm
LMC}-\dot{\boldsymbol{x}}_{\rm MW}\big{)},$
$\displaystyle\ddot{\boldsymbol{x}}_{\rm MW}$ $\displaystyle=$
$\displaystyle\nabla\Phi_{\rm LMC}(\boldsymbol{x}_{\rm MW}-\boldsymbol{x}_{\rm
LMC}),$ (1)

where $\boldsymbol{a}_{\rm DF}$ is the dynamical friction acceleration acting
upon the LMC (usually incorporating a distance-dependent Coulomb logarithm).
This ODE system is integrated backward in time starting from the present-day
position and velocity of the LMC $\boldsymbol{x}_{\rm
LMC},\dot{\boldsymbol{x}}_{\rm LMC}$ and setting those of the Milky Way to
zero (in effect, using the inertial reference frame in which the Milky Way is
currently at rest). Note that the reference frame tied to the Milky Way centre
at all times is not inertial, which needs to be taken into account when
computing trajectories of other objects in the combined gravitational field of
the Milky Way and the moving LMC. These equations are manifestly asymmetric
(the mutual forces do not satisfy Newton’s third law) and ignore the tidal
distortions of both galaxies, yet they account for the reflex motion, at least
qualitatively.

Using this approach, Gómez et al. (2015) argued that accounting for the reflex
motion of the Milky Way significantly reduces the apocentre radius of the LMC
compared to the case with only the dynamical friction taken into account.
However, in their Figure 1, a higher-mass LMC still had a larger apocentre
than a lower-mass (in effect, the dynamical friction dominated over the reflex
motion). On the other hand, more recent applications of this method by Patel
et al. (2017, 2020), as well as full $N$-body simulations, seem to indicate
that the reflex motion is actually more important, i.e., higher-mass LMC has
smaller apocentre radius and shorter period (Figure 3, right panel). The
simulations shown on that figure were conducted in a carefully controlled way,
iteratively refining the initial conditions for the orbit, so that the
present-day position/velocity of the LMC matches the observations to within a
fraction of kpc and km s-1, ensuring a fair comparison with the test-particle
integrations. Note that this trend is opposite to the one shown in Figure 1 of
Besla (2015), which ignores the reflex motion correction. Therefore, an LMC of
mass (1–2)${}\times 10^{11}\,M_{\odot}$ can be on the first infall only if the
Milky Way mass is below $10^{12}\,M_{\odot}$, contrary to many previous
studies.

It should be noted that the reconstruction of past orbit of the LMC is usually
performed assuming an isolated (non-growing) Milky Way; however, there are
compelling reasons to believe that our Galaxy has substantially grown in mass
over the last 10 Gyr. Naturally, this implies a less bound (or more unbound)
orbit of the LMC in the past: Kallivayalil et al. (2013) find that the
inclusion of a cosmologically motivated Milky Way mass evolution
(unfortunately, without providing a detailed recipe) increases the LMC period
by $\sim 20$–30%, strengthening the case for the first infall. Magenta curve
in the left panel of Figure 3 shows the orbit in a very simple model of the
Milky Way mass growing linearly with time from zero at Big Bang to its
present-day value: this produces an effect comparable with a $10\%$ reduction
of its mass at present moment.

As mentioned in Section 2.3, even a comparatively puny SMC may cause a
noticeable offset in the LMC’s position and velocity relative to their common
centre of mass. Yellow curve in the right panel of Figure 3 demonstrates that
a mass ratio of 10:1 produces a reduction of the orbital period comparable to
the change of the LMC mass from 0 to $1.5\times 10^{11}\,M_{\odot}$ – in other
words, a non-negligible effect that should be taken into account in future
studies. Patel et al. (2020) found that including the SMC in their analysis
led to minor and occasionally moderate changes in the orbits of other Milky
Way satellites, but did not discuss how it affects the inferred orbit of the
LMC.

In summary, the past orbit of the LMC is very sensitive to various seemingly
minor changes in the model assumptions and the current coordinates. A more
massive LMC and a higher mass ratio between the SMC and the LMC both cause the
orbital period to decrease relative to the massless case, whereas a
cosmologically motivated growth of the Milky Way mass moderately increases the
period. A detailed study taking into account all these factors still has to be
performed, and the $N$-body simulations would be the most realistic approach,
but they are hampered by the need for extreme precision in matching the
present-day coordinates of all galaxies to ensure a fair comparison between
models. We conjecture that the first-approach scenario remains viable, but
only if the Milky Way mass is below $10^{12}\,M_{\odot}$. It is interesting to
note that a low mass of both the Milky Way and the LMC is advocated by Hammer
et al. (2015); Wang et al. (2019) based on their simulations of the Magellanic
stream, though this view is not shared by the majority of current studies. An
intriguing alternative would be that both galaxies are more massive than
usually assumed, therefore a previous pericentre passage would have occurred a
few Gyr ago, but at a much larger distance ($\sim 100$ kpc) than the most
recent one, especially when accounting for the time evolution of the Milky Way
mass. In this case, the tidal radius of the LMC at this point would remain
large enough ($30-40$ kpc) to retain most of its mass and satellites, negating
one of the arguments in favour of the first passage. Obviously, a reliable
simulation of this scenario is even more challenging (e.g., Guglielmo et al.,
2014).

##  4 Dynamical implications

Having established that the LMC is likely rather massive and just passed its
first pericentre around the Milky Way, we now consider the implications of
this scenario, which range from localised perturbations to far-reaching
consequences within the entire Local Group.

###  4.1 Local effects

The most obvious effect of a massive moving object is the perturbation of
stars in its immediate vicinity. As discussed above, it creates an overdensity
of stars behind the moving object (“wake”), which in turn is responsible for
dynamical friction. Belokurov et al. (2019) provided observational evidence
for such a wake in the form of the so-called Pisces plume, detected both as an
overdensity in the 3d distribution of RR Lyrae stars and as a kinematical
offset in the co-spatial population of BHB stars, qualitatively consistent
with the wake signatures predicted by Garavito-Camargo et al. (2019) from
$N$-body simulations.

A close passage of the LMC deflects the orbits of several Milky Way satellites
and other objects in the Galactic halo. Simon et al. (2020) determined that a
recently discovered ultrafaint dwarf galaxy Tucana IV approached to within a
few kpc from the LMC $\sim 120$ Myr ago, and possibly had a comparably close
encounter with the SMC, but in both cases with a high enough relative velocity
that a bound orbit about either Cloud is unlikely. Nevertheless, its orbit
around the Milky Way had been significantly altered as a result of this
interaction. A few other galaxies have also passed within 20–30 kpc from the
LMC in the last few hundred Myr, including Aquarius II, Grus II, Sagittarius
II, Tucana III, Tucana V, but again with a high enough velocity to exclude
long-term dynamical association (see Figure 9 in (Battaglia et al., 2022) and
Figure 12 in (Correa Magnus and Vasiliev, 2022)). This list does not include
the likely satellites of the LMC discussed in Section 2.2.2, nor the other
galaxies whose orbits have been indirectly affected by the LMC through the
induced reflex motion of the Milky Way, as discussed later in Section 4.3.

Last but not least, a close encounter with the LMC may have a significant
effect on thin stellar streams. In a static Milky Way potential, stars in a
tidal stream follow very similar (though not exactly the same) orbits (Eyre
and Binney, 2011; Sanders and Binney, 2013), and their velocity vectors are
expected to be aligned with the stream track (even in a non-spherical
potential). In practice, we do not measure the 3d velocities directly, but
only their sky-plane components from PM, and for a subset of stars with
spectroscopic follow-up observations, the line-of-sight component. Both have
contributions from the Solar velocity, which needs to be subtracted – this is
straightforward for the line-of-sight component, but compensating the
contribution of Solar reflex to the PM requires an accurate estimate of the
distance. Assuming that this can be done, the next step is to compare the
reflex-corrected values with the gradient of the distance to the stream (for
the line-of-sight component) and the stream track on the sky (for the PM); the
former is usually much less well known. Any misalignment could be taken as an
indication of some form of non-stationarity in the gravitational potential –
be it a close passage of the LMC (or another massive object such as the
Sagittarius dwarf galaxy (Dillamore et al., 2022)), acceleration and
deformation of the Milky Way induced by the LMC (Vasiliev et al., 2021;
Lilleengen et al., 2023), or a figure rotation of the potential (Valluri et
al., 2021).

The above mentioned Tucana III galaxy has a highly eccentric Galactocentric
orbit with a pericentre distance of only a few kpc, and thus undergoes tidal
disruption producing an observed stream. Erkal et al. (2018) conjectured that
the LMC may induce a PM component in the direction perpendicular to the
stream, and predicted that it should be detectable in the upcoming Gaia DR2
catalogue. However, a subsequent analysis of Gaia data by Shipp et al. (2019)
did not corroborate this prediction, which they attributed to a possible
systematic offset in the distance to the stream (which is crucial for the
Solar reflex correction) or due to oversimplifications of the model. On the
other hand, such a misalignment between the stream track and the reflex-
corrected PM vectors was clearly detected by Koposov et al. (2019) and Fardal
et al. (2019) for the very long Orphan–Chenab stream (which originally was
discovered separately in the Northern and the Southern hemispheres, hence the
dual name). Erkal et al. (2019) demonstrated that it can be well explained by
a perturbation from the LMC if its mass is $(1.3\pm 0.3)\times
10^{11}\,M_{\odot}$, simultaneously constraining the Milky Way mass within 50
kpc with a 10% uncertainty. In this model, the stream was deflected by the LMC
passing within 20–30 kpc from the southern portion of the stream. Shipp et al.
(2021) extended this analysis to a few other streams in the Southern
hemisphere, finding that four of them (including Orphan–Chenab) produce a
consistent estimate of the LMC mass (although they kept the Milky Way
potential fixed). The final value from all four streams combined was $M_{\rm
LMC}=(1.8\pm 0.2)\times 10^{11}\,M_{\odot}$, and given that the distance of
closest approach is different for each stream, such a combined analysis opens
up the possibility of constraining not just the total mass, but the mass
profile of the LMC. Koposov et al. (2023) repeated the fit of the
Orphan–Chenab stream with a more flexible model and using updated astrometric
and spectroscopic observational data. They obtained the total LMC mass of
(1–1.5)${}\times 10^{11}\,M_{\odot}$, constrained its mass profile (most
tightly around 30 kpc, with a 13% uncertainty) and provided even tighter
constraints on the Milky Way mass profile and shape in the radial range 20–50
kpc, with the smallest relative uncertainty of only 5% around 30 kpc. Thus the
emerging consensus is that a massive LMC with $M_{\rm LMC}=(1$–$2)\times
10^{11}\,M_{\odot}$ is needed to explain the deflection of several streams.

Another interesting “local effect” of the LMC is not dynamical, but rather has
implications for direct dark matter detection experiments. Namely, the dark
halo of the LMC itself likely extends beyond the 50 kpc distance to its
centre, and therefore we may expect to find dark matter particles from the LMC
in the Solar neighbourhood. Crucially, these particles have very high velocity
relative to the Sun ($\sim 500$–$800$ km s-1), and even if their density is
tiny compared to the Milky Way halo itself, the high velocities can produce a
prominent signal in direct detection experiments – or rather, the absence of
detection places stronger constraints on the properties of dark matter
particles (Besla et al., 2019; Donaldson et al., 2022; Smith-Orlik et al.,
2023).

###  4.2 Global effects on the Milky Way

####  4.2.1 Theory

Figure 5: Top panel: side view of the final snapshot in the $N$-body
simulation of a Milky Way–LMC interaction. The Galactic disc is oriented edge-
on (gray line), and the Sun’s position is directly in front of the image
plane, which is close to the orbital plane of the LMC. The past trajectories
of both galaxies are shown by red and green curves. Red and green contours
show the projected density of the Milky Way and LMC haloes, respectively,
while the dashed blue contours show the initial Milky Way halo (spherical).
Regions where the red contours are closer to or further away from the Galactic
centre than the blue ones appear as under- or overdensities, respectively.
Arrows display the mean velocity of Milky Way particles in the centre-of-mass
frame, which reaches 60 km s-1 in the central region. In the non-inertial
reference frame associated with the Galactic centre, the mean velocity in the
central 20-30 kpc is close to zero, but in the outer regions it is primarily
directed upward (one needs to subtract the central velocity vector from all
vectors shown in the plot).
Bottom panel: the same snapshot as viewed from the Solar location at rest
w.r.t. the Galactic centre (i.e., corrected for the Solar motion). Shown are
sky maps of over/under density (top row), mean velocity in the $l$ and $b$
directions (PM multiplied by distance, second and third rows), and mean line-
of-sight velocity (bottom panel). The snapshot is split into four bins in
heliocentric distance, shown in columns from left to right. The innermost one
has very little perturbation in either density or kinematics, but it becomes
progressively larger further out. Apart from localized overdensities along the
past LMC orbit, the dominant feature is the north–south dipole asymmetry in
density and line-of-sight kinematics, and the positive mean $v_{b}$ (upward
motion in the sky plane) in the outer Galaxy.

The deflection of nearby stars is only part of the story, however. A more
subtle but equally important effect is associated with the reflex motion of
the Milky Way induced by the LMC and the associated distortions in the
Galactic halo. Figure 5 illustrates this phenomenon, using the present-day
snapshot from the fiducial $N$-body simulation with $M_{\rm MW}=0.95\times
10^{12}\,M_{\odot}$ and $M_{\rm LMC}=1.5\times 10^{11}\,M_{\odot}$; the effect
remains qualitatively the same in different Milky Way potentials, but scales
in proportion to the LMC mass.

In the inertial frame, both galaxies move around the common centre of mass.
Top panel shows the edge-on view of the simulation. The LMC moves diagonally
towards the upper right corner on this plot, and the Milky Way,
correspondingly, in the opposite sense. However, the mean velocity of stars in
the Galactic halo is not uniform in space: whereas the inner regions (few tens
kpc) move with the same velocity as the Galactic centre, and therefore appear
stationary in the Galactocentric reference frame, the outer regions have
experienced different amounts of gravitational acceleration from the LMC. The
region on the left, for instance, was first pulled towards the LMC as it was
approaching from the left side of the plot, and then pulled in the opposite
direction (towards the Galactic centre) as the LMC moved right; the net
velocity change turns out to be close to zero. The region on the right, on the
other hand, was steadily accelerated towards the LMC (and hence the Galactic
centre) with a larger accumulated velocity change than the central region
itself. But these horizontal motions pale in comparison to the vertical
($z$-component) velocity: the inner Milky Way rapidly swings towards the LMC,
which passes below the Galactic disc (in the Southern hemisphere) in the last
few hundred Myr, but the outer halo did not have time to react to this sudden
change in velocity of the inner Galaxy, nor had it experienced a comparable
amount of downward acceleration from the LMC itself (since it was more
distant). As a result, the net velocity of the outer halo in the
Galactocentric reference frame is directed upwards in $z$ and slightly inwards
in $R$.

A comparison of red (present-day) and blue (initial) density contours shows
that this sudden downward displacement of the inner Galaxy w.r.t. the outer
halo resulted in an apparent overdensity in the Northern hemisphere and
underdensity in the South. The reason is again the same: stars at large
distances towards the North Galactic pole have a smaller mean $z$-component of
velocity than the inner Galaxy, so they have moved downward a smaller distance
than the central region, and therefore currently find themselves at a larger
Galactocentric radius than originally. Since the halo density drops with
radius, these “impostors” from smaller initial radii retain a higher-than-
expected density at their current location. The situation is opposite in the
Southern hemisphere.

Bottom panel of the same figure shows how these perturbations should appear to
an observer at the Solar location and at rest w.r.t. the Galactic centre. The
inner 30 kpc do not show significant perturbation, since all stars move in
sync with the origin of the reference frame itself. The asymmetries become
progressively larger further out. In the density plot (top row), one can
discern the local wake along the past trajectory of the LMC in each distance
bin, superimposed with the global north–south dipole asymmetry. The net upward
motion of the outer halo creates a similar dipole asymmetry in the line-of-
sight velocities (bottom row) and a positive bias in the $b$ component of PM
(third row), whereas the $l$ component (second row) largely shows the
localized signal from the wake.

To summarize, the large-scale perturbation from the LMC appears because the
Galaxy does not move as a rigid body in response to its gravitational pull.
Although the dynamical consequences of the reflex motion of the Milky Way
induced by the LMC were recognised long ago by Weinberg (1995) and emphasised
in Gómez et al. (2015), the importance of differential motion (i.e.,
distortion) was first pointed out by Erkal et al. (2019), who predicted a net
upward motion in the outer halo. Garavito-Camargo et al. (2019, 2021); Erkal
et al. (2020); Petersen and Peñarrubia (2020); Cunningham et al. (2020) and
Makarov et al. (2023) conducted simulations of the MW–LMC encounter and
presented maps of observable signatures similar to the ones described above
(note that the abscissa axis is flipped in some of these plots). Although the
details of these simulations differ somewhat, the predicted perturbation
features are qualitatively similar and not very sensitive to the particular
choice of the Milky Way potential (unlike the entire past trajectory of the
LMC), since they are mainly created in the last few hundred Myr as the LMC
plunges below 100 kpc. By and large, the inner part of the Milky Way remains
unperturbed; simulations conducted by Laporte et al. (2018) indicated that the
LMC passage induces a warp in the stellar and gas discs at distances $\sim 20$
kpc, but its amplitude ($\lesssim 1$ kpc) is significantly smaller than the
observed one even for the highest mass LMC they considered ($2.5\times
10^{11}\,M_{\odot}$). All these simulations assume a first approach
trajectory; the distortions in the Milky Way halo in the case of more than one
pericentre passage would likely be different, but remain unexplored.

####  4.2.2 Observations

Figure 6: Top panel: $z$-component of Galactocentric velocity for Milky Way
globular clusters (cyan crosses) and satellite galaxies (coloured circles; the
colour only serves to distinguish them more easily) plotted against
Galactocentric distance. The LMC is shown by a black square, and its
satellites are excluded from the plot; nevertheless, there is a conspicuous
concentration of a dozen unrelated galaxies with positive $v_{z}$ within 70
kpc. Further out, the bias towards positive $v_{z}$ is less prominent but
still noticeable. Red dashed line shows the mean $v_{z}$ in the present-day
snapshot of the fiducial $N$-body simulation shown in Figure 5. It does not
explain the coherent group of galaxies with $v_{z}\simeq 200$ km s-1 in the
inner region, but qualitatively matches the positive bias in the outer region.
Bottom left panel: orbits of selected satellites with current Galactocentric
distances under 70 kpc, computed for the last 100 Myr and shown in Mollweide
projection as viewed from the Galactic centre. Although many of them have
similar values of $v_{z}\simeq 200$ km s-1, they are not located in the same
region of space (except the three Tucana and three Boötes galaxies), nor share
common orbits.
Bottom right panel: directions of orbital poles for the same selection of
orbits (similar to the top panel of Figure 3 in (Fritz et al., 2018) or the
left panel of Figure 9 in (Correa Magnus and Vasiliev, 2022)). Again, there is
no significant non-uniformity in this distribution, suggesting that the
velocity values are purely coincidental.

These rather unambiguous and prominent predicted signatures have been
supported by observational evidence. As soon as accurate measurements of 3d
velocities of globular clusters and galactic satellites became available
(after Gaia DR2), it was noticed that there is a positive mean $v_{z}$ in the
population of satellites beyond 30 kpc (Erkal et al., 2020; Petersen and
Peñarrubia, 2021). Figure 6, top panel, demonstrates that a handful of
globular clusters and most of the 20-odd satellite galaxies beyond 80 kpc have
an average $v_{z}$ of a few tens km s-1, consistent with expected signature
from a massive LMC. On the other hand, the largest asymmetry in the
$z$-velocity of satellites is observed at smaller distances (20–60 kpc), where
many galaxies have $v_{z}\sim 200$ km s-1– including the LMC itself, but
excluding objects even tentatively marked as its satellites. However
conspicuous this concentration might be, it seems to be largely accidental,
since these objects are neither close in space nor share a common orbital
plane (bottom panel). Given this bizarre coincidence, which cannot be caused
by the LMC-induced net velocity, we might just as well disregard the weaker
positive bias further out in the halo, but it certainly doesn’t contradict the
model.

The bias towards positive $v_{z}$ should also be manifested in the line-of-
sight velocity as a dipole north–south asymmetry (bottom row in Figure 5). Its
detection has been independently announced by Erkal et al. (2021) and Petersen
and Peñarrubia (2021), using mostly the same spectroscopic sample of BHB stars
and K giants from the SEGUE survey (Xue et al., 2011, 2014) (for these types
of stars, the distance can be measured with a $\sim 10\%$ precision, though in
(Erkal et al., 2021) it was only used to restrict the sample to the outer
halo, and not for the analysis of transverse velocities due to relatively
large PM errors). Ref. (Petersen and Peñarrubia, 2021) also determined the
orientation of the velocity dipole (the apex vector, in their terminology) to
be significantly tilted (by $\gtrsim 55^{\circ}$) from the South pole towards
the direction where the LMC was a few hundred Myr ago, see their Figure 1. By
contrast, Erkal et al. (2021) only examine the north–south gradient in the
line-of-sight velocity. The models generally predict that the dipole points
towards a more recent location of the LMC, some 15–20∘ in the opposite
direction from the South Galactic pole than in (Petersen and Peñarrubia,
2021). At the moment, the reason for this tension is unclear and not much
discussed in the literature.

The signal in the $b$-component of the PM is more difficult to detect, given
relatively large measurement uncertainties ($\sim 0.2$ mas yr-1 for an RR
Lyrae at a distance of 50 kpc, corresponding to the velocity uncertainty of
100 km s-1) and the need to accurately measure the distance in order to
subtract the solar reflex velocity. Currently there is no conclusive evidence
for such a signal. It should be noted that a net rotation of the stellar halo
could also manifest itself in large-scale asymmetries in the line-of-sight
velocity and PM maps, but they have different decomposition into multipoles
than the LMC-induced linear motion. On the other hand, unmixed debris from
various accreted and tidally disrupted galaxies may confound the signal
encoded in the smooth halo (Cunningham et al., 2020); in fact, the removal of
the largest of these – the Sagittarius stream – is a perennial problem in the
studies of halo structure and kinematics.

The density asymmetries induced by the LMC (or any other features in the
density profile, for that matter) are generally more difficult to measure with
certainty. Unlike kinematic patterns, the inference about the density profile
is much more dependent on the precise knowledge of the selection function of
the survey (the probability of an actually existing star to enter the
catalogue), and the latter depends on many factors that are often difficult to
quantify and control. Even an all-sky survey such as Gaia does not have a
uniform sensitivity and coverage due to the way it scans the sky (Boubert and
Everall, 2020; Cantat-Gaudin et al., 2023), and signatures of this scanning
law are apparent in various star count maps at faint magnitudes. These
complications notwithstanding, Conroy et al. (2021) used Gaia (optical) and
WISE (infrared) photometry to select a sample of red giant stars beyond 60
kpc, which was found to display a prominent asymmetry pattern similar to the
one shown in the right two columns of top row of Figure 5. They interpreted it
as a signature of the LMC influence – both local (the wake along its past
trajectory in the south, also detected in (Belokurov et al., 2019) as
described earlier) and global (offset of the outer halo creating an
overdensity in the north, which they call “collective response”). However, the
amplitude of the observed asymmetries was $\sim 2\times$ higher than even the
heaviest-LMC model predicted. On the other hand, in Chandra et al. (2022) the
same team analysed their spectroscopic sample of distant halo stars in the
same regions as these overdensities, and conjectured that they can be
attributed to debris from an early massive merger in the Milky Way history,
which remain on highly radial orbits tilted at $\sim 45^{\circ}$ from the
Galactic disc.

It is worth emphasising that in this review we used the term “reflex motion”
in the purely mechanical sense, as the displacement and associated motion of
our Galaxy about the common centre of mass of the Milky Way–LMC system
(analogous to the wobbling of a star induced by its planets). However, this
definition is impractical because this centre of mass is not an
observationally accessible concept, and moreover the Galactic centre does not
coincide with the centre of mass of the entire Milky Way. Other studies
(Garavito-Camargo et al., 2021; Petersen and Peñarrubia, 2021) use a more
pragmatic approach and define the reflex motion as the velocity of the Milky
Way centre w.r.t the outer parts of its halo, implicitly assuming that the
latter stays fixed. As the red dashed line in Figure 6 shows, this velocity
difference reaches an asymptotic value beyond $\sim 100$ kpc, making that
region a good proxy for the inertial frame; on the other hand, the number of
suitable kinematic tracers drops rapidly beyond 50 kpc. Likewise, the
distortions of the Milky Way halo shape resulting from the accumulation of
kinematic differences with time have a variety of names in the literature.
(Weinberg, 1995; Garavito-Camargo et al., 2021; Conroy et al., 2021) use the
term “collective response”, but unlike collective modes and instabilities in
plasma and self-gravitating systems, self-gravity plays no role in this
phenomenon (Rozier et al., 2022) – stars respond similarly not because they
interact with each other, but because the potential in which they move changes
in the same way.

####  4.2.3 Implications for the Milky Way dynamics

The global upward motion of the outer halo w.r.t. the Milky Way centre affects
the dynamics of various components of our Galaxy, for instance, stellar
streams. The largest and most complex of these – the Sagittarius stream –
stretches across the entire great circle and spans a range of Galactocentric
distances from 15 to more than 100 kpc. Its 3d track, especially around the
apocentre of the leading arm, is difficult to explain: a widely used model
from Law and Majewski (2010) invoked a rather unusual shape of the Milky Way
potential, with its minor axis lying in the Galactic plane and roughly aligned
with the orbital pole of the LMC (which is also on a nearly polar orbit, but
orthogonal to that of the Sagittarius). They already conjectured that this
might be an artifact of neglecting the gravitational effect of a massive LMC –
a suspicion reinforced by Vera-Ciro and Helmi (2013) and Gómez et al. (2015).
However, neither of these studies considered the consequences of the Milky Way
deformation, nor they had access to full 6d phase-space coordinates of the
stream stars. After these became available in Gaia DR2, Vasiliev et al. (2021)
discovered that the reflex-corrected PM vectors are misaligned with the stream
track, like in the Orphan–Chenab stream (Koposov et al., 2019). A
comprehensive modelling effort performed in (Vasiliev et al., 2021)
demonstrated that this misalignment, along with the shape of the leading arm,
can only be explained by invoking a time-dependent perturbation of a massive
LMC with $M_{\rm LMC}=(1.3\pm 0.3)\times 10^{11}\,M_{\odot}$. Critically, the
main effect of the LMC is not a local deflection of the stream, as in the case
of Orphan–Chenab, but the downward reflex motion of the Milky Way. Figures 7
and 12 in that paper demonstrate that when the Milky Way potential is
artificially pinned down at origin, the shape of the stream is completely
wrong and the longitudinal variations of energy and angular momentum along the
stream even reverse sign.

Another example of a physical situation where the reflex motion is crucial to
take into account is the deflection of hypervelocity stars (Boubert et al.,
2020). Figure 1 in that paper shows that such a star ejected from the Galactic
centre on a nearly radial trajectory would be deflected downward by the LMC if
the Milky Way were not moving, but would appear to be deflected upward (like
the rest of the outer halo) when accounting for the reflex motion.

In the above examples, the displacement (reflex motion) of the Milky Way left
prominent signatures in the spatial distribution and kinematic properties of
tracer populations (halo stars, streams, satellites). However, as the dark
halo of the Galaxy would be subjected to the same deformations, the
gravitational potential of the Milky Way is also distorted, which would create
further perturbations in the tracer orbits. Lilleengen et al. (2023) examined
the importance of these next-order perturbations for the modelling of the
Orphan–Chenab stream, using a basis-set representation of potentials of both
galaxies. This technique allows one to selectively enable or disable different
terms in the spherical-harmonic expansion describing the deformation. First a
real $N$-body simulation of the Milky Way–LMC interaction is performed with
all deformations inherently present. Then a simulated stream is evolved in the
pre-recorded time-dependent potential of interacting galaxies under different
assumptions about their deformations. They find that the dominant contribution
comes from the Milky Way dipole (i.e., upward displacement of the outer halo),
followed by monopole and quadrupole perturbations in the LMC (tidal shock and
formation of two tails), and that neglecting these factors has a measurable
effect on the thin and cold stream. On the other hand, a similar analysis
performed in (Vasiliev et al., 2021) for the Sagittarius stream concluded that
these deformations are relatively unimportant, presumably due to this stream
being much broader and hotter. In any case, the distortions of the Milky Way
halo caused by the LMC undoubtedly complicate the analysis of its shape, which
so far has been inconclusive.

Using a different technique (matrix method from kinetic theory), Rozier et al.
(2022) found that accounting for the self-gravity of the Milky Way halo (i.e.
changes in the potential resulting from the distortions) does not
significantly affect the amplitude of its response to the LMC pull. The global
dipole perturbation (essentially, the differential reflex motion) is also
rather insensitive to the kinematic structure of the halo (velocity
anisotropy). On the other hand, the local wake is relatively stronger in
radially biased models than in tangentially biased ones, and its geometry also
varies with anisotropy. By contrast, Makarov et al. (2023) find that particles
on near-circular orbits at radii 50–100 kpc exhibit a larger kinematic offset
from the inner Milky Way than particles with low angular momentum in the same
radial range.

Given the substantial perturbations (on the order of few tens percent) in the
density and kinematics of the Milky Way halo, one may question the
applicability of classical dynamical modelling methods for determining the
Galactic gravitational potential, which are almost always based on the
equilibrium assumption. Erkal et al. (2020) found that neglecting these
perturbations biases up the inferred mass by 15–20% beyond 50 kpc (for an LMC
mass of $1.5\times 10^{11}\,M_{\odot}$). They used the tracer mass estimator
from (Watkins et al., 2010), which approximates the potential and the tracer
density as power laws, assumes a constant anisotropy, and invokes the
spherical Jeans equation to link these quantities. Despite its simplicity,
this method is widely used for measuring the mass profiles of the Milky Way
and other galaxies. They also found that the bias is slightly reduced when
using the velocity dispersion rather than the full second moment of velocity,
and could be minimised by choosing a particular region of the halo least
affected by velocity and density perturbations.

Deason et al. (2021) used a related class of power-law mass estimators, but
relying on the distribution function rather than the Jeans equation, to
measure the Milky Way mass profile out to 100 kpc from the kinematics of a few
hundred stars in the outer halo (the sample from (Erkal et al., 2021)). They
tested the method on simulated data including the LMC perturbation, and found
that it returns a value biased up by $\lesssim 10\%$; this bias is lower than
in the previous study in part due to a fortunate location of the observational
sample in the sky regions less affected by the LMC. In fact, biases from
unrelaxed structures in the halo remaining from other accretion events (in
particular, the Sagittarius stream) may be more important than the neglect of
the LMC.

Correa Magnus and Vasiliev (2022) implemented a more sophisticated
distribution function-based dynamical modelling method, which allows for more
flexible density and anisotropy profiles. They also introduced an additional
step in modelling to compensate the LMC perturbation. For any choice of the
Galactic potential and LMC mass, the orbits of the LMC and the Milky Way are
reconstructed in the approximation of analytic non-deforming extended bodies
(Equation 1). Then the orbits of tracer objects are rewound back in time,
starting from the present-day measured phase-space coordinates, in the
combined potential of the Milky Way and moving LMC (including the acceleration
from the reflex motion of the Galactocentric reference frame). Finally, the
positions and velocities of tracer objects at the time 2–3 Gyr ago are assumed
to be drawn from an equilibrium distribution function, whose parameters
(together with the parameters of the Milky Way potential) are optimised to
match the observations. They demonstrated that this method is able to produce
unbiased potential estimates when doing the extra compensatory step, while
neglecting the LMC perturbation biases the potential up by $\sim 15\%$, in
agreement with (Erkal et al., 2020). Finally, they applied the method to the
kinematic sample of globular clusters and satellite galaxies with 6d phase-
space coordinates, and determined a value of the Milky Way mass within 100 kpc
that is somewhat higher but compatible with the estimates from the smooth
stellar halo (Deason et al., 2021) and streams (Vasiliev et al., 2021; Koposov
et al., 2023) (only the studies accounting for the LMC perturbation are
listed). A possible reason for a higher estimated mass is that if the
population of tracers (satellite galaxies) is incomplete and misses objects
near apocentres of their orbits, this biases the derived mass up (see Appendix
A in (Correa Magnus and Vasiliev, 2022)). They also derived constraints on the
LMC mass that are comparable to those from stream modelling (Section 4.1,
Figure 1).

Interestingly, the 10–20% reduction in the estimated Milky Way mass when
accounting for the LMC is nearly cancelled when adding back the LMC mass
itself (which makes sense given that it is almost entirely within the Galactic
virial radius at present). It is natural to expect that the bias would be
proportional to the LMC mass, but the fact that it matches it even
quantitatively is probably coincidental and might depends on details of the
LMC orbit and its phase.

Figure 7: Illustration of tidal deformation and mass loss of the LMC. Left
panel shows the projected density contours of the LMC particles at present
time in the same orientation as in Figure 5, top panel. Inner blue-shaded
region contains particles still bound to the LMC (having negative energy in
the associated reference frame), while particles in the outer red-shaded
region are unbound. The Milky Way disc is shown by a gray line with the dot
marking its centre, and the past orbit of the LMC is shown by a green curve.
Top right panel shows the spherically-averaged density profile of all (red)
and bound (blue) particles, with the bound fraction dropping precipitously
beyond 30–40 kpc from the LMC centre. Bottom right panel shows the time
evolution of bound mass (blue) and Galactocentric distance (red) of the LMC.

Finally, we note that almost all studies quoted the LMC mass before its infall
into the Milky Way, but not all of this mass remains bound at present. The
concept of bound mass is ill-defined in the case of a highly dynamic
interaction, because the gravitational potential changes rapidly and it is
generally not possible to determine if a given particle will remain attached
to the satellite after it reaches the next apocentre of its orbit. A plausible
recipe is to declare a particle to be bound if it has negative total energy in
the reference frame of the moving LMC, $E_{i}=\Phi_{\rm
LMC}(\boldsymbol{x}_{i})+\frac{1}{2}|\boldsymbol{v}_{i}-\boldsymbol{v}_{\rm
LMC}|^{2}$, where $\Phi_{\rm LMC}$ is the potential created only by bound LMC
particles and does not include the Milky Way. As this definition is self-
referencing, the procedure needs to be repeated several times until the list
of bound particles stabilises. Figure 7 demonstrates that the bound mass
starts to rapidly decrease once the LMC comes within 100 kpc from the Milky
Way $\sim 0.5$ Gyr ago. Particles beyond a few tens kpc from the LMC centre
are no longer bound to it and form tidal tails, although they still contribute
to the gravitational perturbation of the Milky Way.

###  4.3 Effects on the Local Group

The reflex velocity that the inner Milky Way acquires with respect to the
outer halo has consequences for the dynamics of the entire Local Group (or
better say, affects our inference about these processes from the present-day
velocity measurements of various objects, which has been distorted by our
motion induced by the LMC). For instance, the past orbits of Milky Way
satellites computed from their current phase-space coordinates, assuming the
perfect knowledge of the Galactic potential but ignoring the LMC, may be
rather different from their true orbits in the presence of the LMC. On the
other hand, if we can reconstruct sufficiently reliably the past orbits of the
Milky Way and the LMC for any choice of their potential profiles, then the
computation of orbits of other satellites becomes a straightforward exercise –
marred, as usual, by the uncertainties in their current phase-space
coordinates.

The most obvious effect of including the LMC is on orbits of its own
satellites, which have been discussed in Section 2.2.2. After Gaia DR2 made
possible the measurement of PM for most dwarf galaxies in the Milky Way
neighbourhood, several studies provided the census of likely LMC satellites,
based either on the proximity of their angular momentum plane to that of the
LMC (Kallivayalil et al., 2018; Fritz et al., 2019) or on a negative total
energy in the LMC-centered reference frame for a significant interval of time
(Erkal and Belokurov, 2020; Patel et al., 2020). Because of uncertainties in
the present-day phase-space coordinates, orbits of even genuine LMC satellites
cannot be accurately reconstructed beyond a few Gyr (D’Souza and Bell, 2022),
but if a dwarf galaxy has been gravitationally bound to the LMC for at least a
Gyr in the majority of orbit samples, it can be considered a LMC satellite
with high confidence.

Battaglia et al. (2022) provided an updated catalogue of PM of almost all
known Milky Way satellites using Gaia eDR3 and then used these data to
reconstruct their past orbits for two choices of static Milky Way potentials
and a time-dependent Milky Way + LMC potential from (Vasiliev et al., 2021).
They confirmed the association with the LMC for a number of dwarfs, but also
demonstrated that orbits of many Milky Way satellites are significantly
changed in the presence of the LMC (their Appendix D). This comparison was
limited to a single choice of Milky Way potential and just one realization of
orbit for each object. Correa Magnus and Vasiliev (2022) performed a much more
detailed study of the effect of the LMC on satellite orbits, marginalizing
over the uncertainties in their present-day phase-space coordinates and
exploring a range of Milky Way potentials and LMC masses. Their Figure 11
illustrates the variety of outcomes: some galaxies are hardly affected by the
LMC, others increase or decrease their orbital energy and correspondingly have
longer or shorter periods when their orbits are corrected for the LMC
influence. For instance, Draco, Sculptor and Ursa Minor have been on less
bound orbits until the arrival of the LMC, whereas Leo I had a lower energy.
We stress that in most cases, these changes are driven not by a close
encounter with the LMC itself, but by modifying the Milky Way’s velocity
relative to that of the satellite. Leo I is a particularly noteworthy example,
as its current Galactocentric radial velocity likely exceeds the local escape
speed, unless the Milky Way is very massive. In the past, its high recession
speed was invoked as an argument in favour of a heavy Galaxy (significantly
more massive than preferred by most other techniques), but if it is reduced by
a few tens km s-1 on account of the LMC displacement of our Galaxy, this eases
the tension in the Milky Way mass estimates, as already noted by Erkal et al.
(2020). As discussed in the previous section, accounting for the LMC
perturbation also slightly reduces the mass profile of our Galaxy inferred
from the ensemble of satellites as kinematic tracers.

It is well known that a large fraction of dwarf galaxies around the Milky Way
(including the LMC) share a common plane, dubbed the “Vast Polar Structure”
(VPOS) (Kroupa et al., 2005; Pawlowski et al., 2012), and the PM provided by
Gaia confirm that orbits of most of these objects also lie in this relatively
thin plane (Fritz et al., 2018; Pawlowski and Kroupa, 2020), even after
excluding the satellites of the LMC. Garavito-Camargo et al. (2021) suggested
that the LMC-induced perturbation of satellite orbits may bring them closer to
the alignment due to several factors: the direct action of the massive LMC,
the kinematic offset of the outer halo with respect to the observer in the
inner Milky Way, and the associated deformations in the Galactic gravitational
potential (both the local wake and the global dipole perturbation). Other
studies (Pawlowski et al., 2022; Correa Magnus and Vasiliev, 2022) found the
effect to be rather weak and insufficient to explain the observed degree of
clustering of orbital planes, not least because many of these galaxies are
fairly distant to be significantly affected. However, a comprehensive analysis
of this phenomenon using suitably chosen Milky Way–LMC analogues from
cosmological simulations has not yet been performed.

Finally, the reflex motion of central parts of our Galaxy induced by the LMC
has implications for the so-called “timing argument” – the estimate of the
total mass of the Local Group (dominated by the Milky Way and Andromeda) from
their orbital period. The argument (Kahn and Woltjer, 1959) postulates that
since Andromeda is currently approaching the Milky Way, it must have completed
more than a half of its orbital period since the Big Bang. Assuming that the
(proto-)galaxies were at the pericentre of a Keplerian orbit 13.7 Gyr ago, one
can infer the sum of their masses by matching the orbital solution to the
present-day distance and relative velocity of Andromeda w.r.t. the Milky Way.
The radial component of this velocity (almost identical to the line-of-sight
velocity of Andromeda corrected for Solar motion around the Galactic centre)
is $-114\pm 1$ km s-1, and the tangential velocity has been measured with
increasing precision by HST and Gaia (van der Marel et al., 2019; Salomon et
al., 2021) in the range $30$–$100$ km s-1. However, both these velocity
components have contributions from the LMC-induced reflex motion of the Milky
Way. Peñarrubia et al. (2014) demonstrated that the velocities of galaxies
within a few Mpc closely follow the Hubble law with a constant offset
proportional to the mass of the Local Group, and determined the latter to be
$(2.3\pm 0.7)\times 10^{1}2\,M_{\odot}$. Peñarrubia et al. (2016) further
found that the scatter of measured velocities about this mean trend is
minimized when the radial velocity of Andromeda w.r.t. the Milky Way is
corrected for the reflex velocity induced by a fairly massive LMC ($\sim
2.5\times 10^{11}\,M_{\odot}$), which also decreases the inferred mass of
Milky Way + Andromeda and brings it into better agreement with the previous
estimate from the local Hubble flow. Undoing this perturbation, the radial
velocity of Andromeda should be reduced in magnitude, while the tangential
velocity increases (Figure 8, left panel); to first order, these changes are
proportional to the LMC mass. Benisty et al. (2022) revisited the classical
timing argument, taking into account several factors: (1) correction of
apparent radial and tangential velocity of Andromeda for the LMC perturbation
(using the approach of (Correa Magnus and Vasiliev, 2022)), (2) effect of dark
energy on the cosmological expansion, and (3) cosmic bias correction factor
for the total mass of the Local Group calibrated against cosmological
simulations (i.e., applying the timing argument to pairs of galaxies
resembling Milky Way and Andromeda, and determining the multiplicative factor
that brings the obtained result into agreement with the actual masses of these
galaxies). They find that the inclusion of the LMC reduces the inferred Local
Group mass by $\lesssim 10\%$, whereas the cosmic bias factor, which accounts
for the approximate nature of the timing argument itself, reduces the mass by
40%. Chamberlain et al. (2023) performed a similar analysis while only
accounting for the LMC perturbation (which also had a $\sim 10\%$ effect), and
obtained a narrower posterior distribution, but compatible with other
estimates except (Peñarrubia et al., 2016), who had a similarly narrow
distribution but shifted to even lower masses. These estimates are summarised
in the right panel of Figure 8 and are broadly consistent with the sum of
dynamically inferred masses of both galaxies.

Figure 8: Left panel: relative velocity of Andromeda and Milky Way. Red
contours show the measured values (using PM from (Salomon et al., 2021)),
green – corrected for the LMC-induced motion of the Milky Way centre. Right
panel: inferred mass of the Local Group (or more correctly, Milky Way and
Andromeda) from the timing argument (Benisty et al., 2022). Red dashed line is
the result using the measured relative velocity, green dot-dashed line – after
correcting for the LMC perturbation, blue solid – additionally multiplying by
a cosmic bias correction factor. Black dotted and red dash-double-dotted lines
show results of (Chamberlain et al., 2023) and (Peñarrubia et al., 2016), who
only account for the LMC but not the cosmic bias.

##  5 Conclusions

The Magellanic Clouds are not only spectacular and inspiring, but also quite
unusual – only a small fraction of Milky Way-like galaxies in the local
Universe have a comparably large pair of satellites (Liu et al., 2011;
Tollerud et al., 2011). Even more peculiar is that they are currently just
past the pericentre of their Galactic orbit, which amplifies their dynamical
effects on the Milky Way. Although still ranked as a minor merger and
remaining a perturbation, the dynamical consequences of this ongoing encounter
between our Galaxy and the LMC are significant and well detectable in the
currently available observational data. As the precision of our measurements
keeps improving, astronomers will be facing even more ambitious challenges in
exploring the structure and evolution of the outer regions of Milky Way, for
instance, detecting the traces of earlier accretion events or constraining the
3d shape of the Galactic halo. Obviously, the impact of the LMC cannot be
ignored at the level of detail warranted by these studies.

The following aspects of the LMC–Milky Way interaction are worth emphasising:

  * •

The total pre-infall mass of the LMC is likely to be (1–2)${}\times
10^{11}\,M_{\odot}$, i.e., only 5–10 times smaller than the Milky Way mass.
This estimate is supported by a number of empirical arguments reviewed in
Section 2.2.1, and the dynamical effects of the LMC on the Milky Way discussed
in Section 4 are best explained by a similar mass range. If the LMC is on its
first passage around the Galaxy, its dark halo becomes deformed but is still
almost entirely brought within 100 kpc from the Milky Way centre.

  * •

Despite the increasing precision of PM measurements, there remains a
considerable uncertainty in the orbital period and apocentre distance of the
LMC (Section 3.2). This is largely due to the orbit being only marginally (if
at all) bound to the Milky Way. Likewise, these parameters depend strongly on
the Milky Way mass profile. The evidence for the first-passage scenario is
less strong now than it was 15 years ago Besla et al. (2007), since the most
recent PM measurements reduce the tangential velocity by a few tens km s-1.
The orbital period most likely exceeds 5 Gyr, unless the Milky Way is
significantly more massive than $10^{12}\,M_{\odot}$ (which is disfavoured by
current models), but an earlier pericentre passage at a distance of $\sim 100$
kpc cannot be ruled out with certainty. The implications of this alternative
second-passage scenario for the LMC satellites, the Magellanic Stream, and the
Milky Way itself are poorly studied.

  * •

The most obvious dynamical consequences of the massive LMC are local
perturbations to objects that pass in its vicinity (Section 4.1), but equally
if not more important is its global effect on the Milky Way (Section 4.2). It
is often associated with the reflex motion of the Galaxy about the common
centre of mass of the Milky Way–LMC system, but this is only part of the
story. Stars and other objects in the outer halo of the Milky Way (roughly
beyond 30 kpc) are not displaced by the LMC in the same way as the inner
Galaxy; in other words, the differential perturbation causes a deformation of
the Milky Way both in space and in kinematics. This phenomenon only began to
be appreciated in the last few years, and is now clearly seen both in
simulations and in observations.

  * •

The LMC-induced reflex motion of the Milky Way leads to an overall reduction
of its inferred past orbital period in the case of a more massive LMC compared
to a test-particle orbit. This counteracts the more well-known effect of
dynamical friction, and has not been accounted for in many earlier studies.

Despite significant progress in recent years, several open questions and
associated challenges need to be addressed by future studies:

  * •

As already mentioned, the reconstruction of the past orbit of the LMC is still
uncertain and very sensitive to variations in its present-day phase-space
coordinates, parameters of the Milky Way potential and other factors.

  * •

Connected to the previous point, any modelling effort that aims at exploring
the dependence of the past LMC orbit on its mass or on the Milky Way potential
must match the present-day position and velocity of the LMC with very high
accuracy (better than 1 kpc and a few km s-1) to ensure a meaningful
comparison between different cases. This is nearly impossible to achieve in
large-volume cosmological simulations and is very difficult even in dedicated
simulations of the Milky Way–LMC[–SMC] system; this level of precision was
rarely attained or even mandated in previous studies. On the other hand, the
LMC trajectory in the last few hundred Myr (up to 100–150 kpc from the
Galactic centre) is only weakly sensitive to its current velocity or the
Galactic potential, so most of the dynamical effects on the Milky Way do not
strongly depend on these factors (but do scale with the LMC mass).

  * •

Given that full $N$-body simulations of the Milky Way–LMC interaction are
expensive and difficult to conduct with sufficient precision, alternative
computationally cheaper methods need to be more accurate and sophisticated.
The classical dynamical friction expression poorly describes the orbital
evolution of massive satellites (Vasiliev et al., 2022) even after a manual
tuning of the Coulomb logarithm, and the distortions in the gravitational
potential of both galaxies have non-negligible dynamical effects (Lilleengen
et al., 2023); these factors are ignored in the popular approximation of
orbital evolution of two extended but non-deforming bodies.

  * •

Although the observed spatial and kinematic signatures of the LMC-induced
perturbations in the outer halo qualitatively match the models, there appear
to be significant differences in the amplitude and direction of these
perturbations (Petersen and Peñarrubia, 2021; Conroy et al., 2021).

  * •

The analysis of the perturbations is hampered by the scarcity of available
kinematic tracers: existing major spectroscopic surveys with the exception of
SEGUE (Yanny et al., 2009) contain very few stars beyond 50 kpc. Fortunately,
the sample will expand significantly in the next years with the public data
releases from DESI (Cooper et al., 2023), WEAVE (Jin et al., 2023) and H3
(Conroy et al., 2019) surveys.

The LMC will continue orbiting the Milky Way and will inevitably merge with it
in a couple of Gyr. As discussed by Cautun et al. (2019), this will bringing
about further changes in the properties of our Galaxy, e.g., dramatically
increasing the mass and mean metallicity of its stellar halo. By the time this
merger is complete, however, future astronomers will face an even bigger
calamity – the impeding collision with Andromeda. Regardless of whether the
LMC is boon or bane for a current-generation dynamicist, we have to endure its
presence for the time being, and better learn to cope with its effects on our
Galaxy.

The snapshots, potentials and other files from $N$-body simulations of the
Milky Way–LMC interaction for several choices of the LMC mass and the Galactic
potential are provided at this zenodo repository:
https://doi.org/10.5281/zenodo.7832266. The main kinematic features induced by
the LMC for any other choice of these parameters (Figure 5) can be well
reproduced without running a full $N$-body simulation, using test-particle
integrations in a time-dependent potential of the moving but non-deforming
galaxies (Correa Magnus and Vasiliev, 2022). A script is provided at
https://github.com/GalacticDynamics-
Oxford/Agama/blob/master/py/example_lmc_mw_interaction.py

###### Acknowledgements.

I thank Gurtina Besla, Denis Erkal, Nico Garavito-Camargo, Jorge Peñarrubia,
Mike Petersen, Simon Rozier, the anonymous referees and other colleagues for
valuable discussions and suggestions. References

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