# The disturbed outer Milky Way disc

Paul J. McMillan1 , Jonathan Petersson1 , Thor Tepper-Garcia2,3 ,Joss Bland-
Hawthorn2,3 , Teresa Antoja4,5,6 , Laurent Chemin7 ,Francesca Figueras4,5,6 ,
Shourya Khanna8 , Georges Kordopatis9 , Pau Ramos4,5,6 , Merce Romero-
Gómez4,5,6 , George Seabroke10
1Lund Observatory, Department of Astronomy and Theoretical Physics, Lund
University, Box 43, SE-22100, Lund, Sweden
2Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW
2006, Australia
3Centre of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D),
Australia
4Departament de Física Quàntica i Astrofísica (FQA), Universitat de Barcelona
(UB), c. Martí i Franquès, 1, 08028 Barcelona, Spain
5Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), c.
Martí i Franquès, 1, 08028 Barcelona, Spain
6Institut d’Estudis Espacials de Catalunya (IEEC), c. Gran Capità, 2-4, 08034
Barcelona, Spain
7Centro de Astronomía, Universidad de Antofagasta, Avda. U. de Antofagasta
02800, Antofagasta, Chile
8INAF - Osservatorio Astrofisico di Torino, via Osservatorio 20, 10025 Pino
Torinese (TO), Italy
9Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice Cedex 4, France
10Mullard Space Science Laboratory, University College London, Holmbury St
Mary, Dorking, Surrey, RH5 6NT, United Kingdom.
E-mail<EMAIL_ADDRESS>
###### Abstract

The outer parts of the Milky Way’s disc are significantly out of equilibrium.
Using only distances and proper motions of stars from Gaia’s Early Data
Release 3, in the range $|b|<10^{\circ}$, $130^{\circ}<\ell<230^{\circ}$, we
show that for stars in the disc between around $10$ and $14\,\mathrm{kpc}$
from the Galactic centre, vertical velocity is strongly dependent on the
angular momentum, azimuth, and position above or below the Galactic plane. We
further show how this behaviour translates into a bimodality in the velocity
distribution of stars in the outer Milky Way disc. We use an $N$-body model of
an impulse-like interaction of the Milky Way disc with a perturber similar to
the Sagittarius dwarf to demonstrate that this mechanism can generate a
similar disturbance. It has already been shown that this interaction can
produce a phase spiral similar to that seen in the Solar neighbourhood. We
argue that the details of this substructure in the outer galaxy will be highly
sensitive to the timing of the perturbation or the gravitational potential of
the Galaxy, and therefore may be key to disentangling the history and
structure of the Milky Way.

###### keywords:

Galaxy: kinematics and dynamics – Galaxy: structure – Galaxy: evolution –
methods: numerical

††pubyear: 2022††pagerange: The disturbed outer Milky Way disc–B.1

##  1 Introduction

The European Space Agency’s Gaia mission (Gaia Collaboration et al., 2016) has
allowed us to see with new clarity the extent to which the Milky Way’s disc is
out of equilibrium. The data from Gaia includes photometry and spectra, but it
is the extraordinarily precise astrometry that sets Gaia furthest apart from
any other instrument.

With the releases of Gaia DR2 and EDR3 (Gaia Collaboration et al., 2018a,
2021a), new disturbances have been discovered, and ones that were already
known have been mapped in far greater detail. Substructure in velocities
parallel to the disc have been mapped locally and across the disc (e.g.,
Dehnen, 1998; Famaey et al., 2005; Antoja et al., 2008; Gaia Collaboration et
al., 2018b; Kawata et al., 2018; Trick et al., 2019; Friske & Schönrich, 2019;
Khanna et al., 2022). Vertical asymmetries have been found in both number
counts and velocities, and associated with both the Galactic warp and with
bending or breathing modes in the disc (e.g. Widrow et al., 2012; Williams et
al., 2013; Schönrich & Dehnen, 2018; Poggio et al., 2018; Bennett & Bovy,
2019; Carrillo et al., 2019; Khanna et al., 2019; Cheng et al., 2020).

The Gaia phase spiral111There is no consensus in the current literature on how
we should refer to this feature, which was originally dubbed the ‘snail
shell’. While we would prefer ‘kanelbulle’, reflecting the similarity with the
shape of the Swedish cinnamon roll, we choose ‘phase spiral’ in this study to
follow the most common choice. was discovered in Gaia DR2 data by Antoja et
al. (2018) and is an overdensity of stars along a spiral in the $Z-V_{Z}$
plane, where $Z$ is the direction perpendicular to the Galactic disc. It is
also found as a spiral shaped variation in the average $V_{R}$ or $V_{\phi}$
velocities in the $Z-V_{Z}$ plane. A number of studies have demonstrated that
this can be produced as a perturbation induced by the passage of the
Sagittarius (Sgr) dwarf galaxy phase-mixes over time (Binney & Schönrich,
2018; Darling & Widrow, 2019; Laporte et al., 2019; Bland-Hawthorn & Tepper-
García, 2021; Gandhi et al., 2022), and this is now the generally favoured
formation scenario. Other studies have investigated how this structure
persists and varies across the galaxy as a function of chemistry, position and
kinematics (e.g., Bland-Hawthorn et al., 2019; Xu et al., 2020; Li, 2021,
Alinder, McMillan & Bensby in prep.).

In connection with Gaia EDR3, Gaia Collaboration et al. (2021b, henceforth
AC21) looked at stars in the Galactic anticentre
($170^{\circ}<\ell<190^{\circ}$, $|b|<10^{\circ}$), and showed that in this
region the velocity distribution in the outer disc was significantly
disturbed. At Galactic radii between $10$ and $14\,\mathrm{kpc}$ stars
predominantly follow a bimodal velocity distribution. One of these modes is at
$V_{Z}\approx 10\,\mathrm{km\,s}^{-1}$ and is mostly populated by stars below
the Galactic plane, while the other is at
$V_{Z}\approx-5\,\mathrm{km\,s}^{-1}$, is mostly populated by stars above the
Galactic plane, and lags the rotation of the first group by around
$30\,\mathrm{km\,s}^{-1}$. We, perhaps inelegantly, refer to these two modes
as ‘clumps’ throughout this study. The strength of the two clumps varies
strongly with increasing radius, with the clump at positive $V_{Z}$ becoming
increasingly dominant at greater radii.

This behaviour is elegantly, if incompletely, summarised by plotting the
distribution in $V_{Z}$ velocity of stars at different angular momenta. AC21
showed that towards the Galactic anticentre this distribution has a clear
break from negative $V_{Z}$ to positive at an angular momentum around
$|L_{Z}|=2750\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$, corresponding to a guiding
centre radius of ${\sim}11.5\,\mathrm{kpc}$.

The AC21 study was limited to the anticentre region because in this region the
velocity in the plane of the sky closely corresponds to the Galactocentric
azimuthal and vertical velocities. This allowed the Gaia team to gain
significant insight into the velocity structure of the outer disc of the Milky
Way using a sample of stars without measured line-of-sight
velocities222Throughout this text we will refer to ‘line-of-sight’ velocities
rather than ‘radial’ velocities to avoid confusion with the velocity component
in the Galactocentric radial direction.. The productive use of proper motion
samples alone to study Galactic dynamics has a long history, including the
work by Eggen on moving groups (e.g., Eggen, 1958); the characterisation of
the local velocity distribution by Dehnen & Binney (1998), Dehnen (1998) and
recently for the white dwarf population by Mikkola et al. (2022); measurement
of velocity asymmetry in the Galactic disc (Antoja et al., 2017); and the
determination of the Milky Way’s escape velocity by Koppelman & Helmi (2021).

In this study we extend the approach of AC21 to include stars over a wide area
in the outer disc. Because we are not limited to stars with measured line-of-
sight velocities, we can use a far larger sample of stars than would otherwise
be available. This allows us to explore the region with a level of detail that
is not otherwise possible. We investigate whether the bimodality of the
velocity distribution is only found around the Galactic anticentre, or if it
can be found elsewhere in the galaxy. We ask whether we can see any change in
it as we move around the galaxy.

The paper is organised as follows: In Section 2 we describe the dataset we
study and the approximations we make to allow us to draw conclusions from
these data. We also demonstrate the validity of these approximations on mock
data. In Section 3 we look at the disturbances in this part of the outer Milky
Way disc. In Section 4 we present $N$-body simulations of an impulse-like
interaction with the Milky Way disc, and show that it can produce similar
behaviour in the outer disc to that which we see. Finally we discuss and
summarise our results in Secs. 5 & 6, respectively.

##  2 Data

In this work we investigate the dynamics of stars in the outer galaxy, with
Galactic longitudes in the range $130^{\circ}<\ell<230^{\circ}$ and latitudes
$|b|<10^{\circ}$. The range of Galactocentric azimuths covered by these data
are illustrated in Fig. 1. The data are taken from Gaia EDR3 and consist of
the proper motions $\mu_{\alpha}*$, $\mu_{\delta}$; and the photometric
magnitudes $G$, ${G_{\mathrm{BP}}}$, ${G_{\mathrm{RP}}}$. We take the distance
to the stars to be the photogeometric estimates from Bailer-Jones et al.
(2021). These estimates take into account the observed colours and magnitudes
of stars, along with their parallaxes (corrected following the recipe from
Lindegren et al., 2021) and a prior constructed from a three-dimensional model
of our Galaxy, to derive distance estimates that can be significantly more
precise than those derived without photometry.

Figure 1: Illustration of the range in Galactocentric $\phi$ angles covered by
ranges in $\ell$ at different points in the Galaxy (in the area we consider).
Our Gaia data come from $130^{\circ}<\ell<230^{\circ}$, while in Sec 4 we look
at particles in the simulation in the range $-20^{\circ}<\phi<20^{\circ}$.
Solid black line are lines of constant $\ell$, and converge at the position of
the Sun. Blue dashed lines are lines of constant $\phi$. Lines of constant
Galactocentric radius are shown as dotted lines at $R=11,12,13$ &
$14\,\mathrm{kpc}$ coloured, respectively, light-blue, orange, green and red.

Importantly, we do not restrict ourselves to only those stars with measured
line-of-sight velocities. Gaia EDR3 contains over 1.4 billion stars with
astrometric measurements, of which only 7 million (i.e., less than 0.5
percent) have published Gaia line-of-sight velocities. After Gaia DR3 in June
2022 this number will rise, but only to around 33 million (i.e., around 2.5
percent of the full catalogue). Our choice not to use them, therefore, enables
us to work with a very large sample of stars, but at the cost of losing one
component of the velocity. As we shall demonstrate, we are able to make
approximations that allow us to study the velocity distribution in the
vertical and azimuthal directions with just these data. However these
approximations become less reasonable away from $b=0^{\circ}$ and
$\ell=180^{\circ}$.

To ensure that we are using high quality data we require that the astrometric
‘Renormalised Unit Weight Error’ (RUWE) meets the criterion
$\mathtt{RUWE}<1.4$. We also require that the parallax divided by its error
$\varpi/\sigma_{\varpi}>3$, so that the photogeometric distance estimation is
not completely dominated by the photometric information. With these criteria,
we have $20\,041\,385$ stars in our full sample.

Bland-Hawthorn et al. (2019) argued that the kinematically colder populations
in the Milky Way disc are more strongly affected by the phase spiral than
kinematically warmer populations. Further, several authors (e.g., Amôres et
al., 2017; Poggio et al., 2018; Romero-Gómez et al., 2019) have shown that the
structure of the Galactic warp depends on the age of the tracer population
considered. We therefore construct a sub-sample dominated by young blue main-
sequence stars as indicated in Fig 2. Extinction is taken as the median
extinction at the quoted distance in the dust map of Green et al. (2019), and
the conversion from $E(B-V)$ to the Gaia bands is approximated as
$A_{G}=2.294\,E(B-V)$; $E(G_{BP}-G_{RP})=1.309\,E(B-V)$, figures we have taken
from Sanders & Das (2018).

We define our blue sample with a simple cut in extinction-corrected absolute
magnitude and colour

$\displaystyle M_{G}<3.0$ (1) $\displaystyle(G_{BP}-G_{RP})_{\,0}<0.5$ (2)

as illustrated in Fig.2. This leaves us with a sample of $1\,093\,192$ stars.

The selection of these stars is advantageous for two further reasons: Firstly,
because the stars are relatively bright, their Gaia parallax measurements will
have relatively small uncertainties; secondly, because their velocity
dispersion in the Galactocentric radial direction is typically very small, the
uncertainty due to the absence of a measured line-of-sight velocity is reduced
(see below).

Figure 2: Extinction-corrected colour-magnitude diagram for stars. The box at
the top left contains the stars in our young blue main-sequence sub-sample.
The quoted numbers are numbers of stars in a pixel of width $0.025$ mag in
colour and height $0.1$ mag in absolute magnitude. The arrow shown at the top
right corner of the diagram illustrates the assumed reddening vector. Stars
will be moved in this direction on the diagram depending on whether the
extinction to them has been over- or under-estimated.

###  2.1 Velocity estimates

If we have a measured distance and proper motion of a star, then we know two
components of the star’s velocity, which we can decompose into the direction
of increasing Galactic coordinates $\ell$ and $b$ (we refer to these as
$V_{\ell}$ and $V_{b}$ respectively). Without a measured line-of-sight
velocity for a star we cannot know its full 3D velocity.

Figure 3: Density in $V_{Z}^{*}$ at different $L_{Z}^{*}$ for mock data with
distance uncertainties and no line-of-sight velocity measurement. Each panel
shows the results for different $10^{\circ}$ ranges in $\ell$. Vertical dotted
lines are drawn at $L_{Z}^{*}={-}2500$ &
${-}3000\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ to guide the eye and to ease the
comparison between equivalent plots elsewhere in the text. Error bars in the
top right of each panel indicate the typical uncertainties in $L_{Z}^{*}$ and
$V_{Z}^{*}$. The underlying pattern is seen relatively clearly in panels near
$\ell=180^{\circ}$, but becomes more smeared out further from the anticentre.
Figure 4: True density in $V_{Z}$ at each $L_{Z}$ for our mock data. Dotted
lines are drawn at $L_{Z}^{*}={-}2500$ &
${-}3000\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ to guide the eye. The model is
based on one in equilibrium (which would have average $V_{Z}=0$ for all
$L_{z}$), perturbed as described in the text to mimic the structure seen in
the Milky Way.

To understand motions in a galaxy it is convenient to consider velocities in
terms of their Galactocentric components in cylindrical polar coordinates
$(V_{R},V_{Z},V_{\phi})$. The relationship between $(V_{\ell},V_{b})$ and
$(V_{R},V_{Z},V_{\phi})$ depends on a star’s position in the Galaxy, and
towards the Galactic anticentre $(\ell,b=180^{\circ},0^{\circ})$ and centre
$(\ell,b=0^{\circ},0^{\circ})$, we have $V_{\ell}=\pm V_{\phi}$ and
$V_{b}=V_{Z}$, while proper motions provide no information about $V_{R}$. This
convenience has led a number of authors to focus studies in these regions
(e.g., Kawata et al., 2018; Fernández-Alvar et al., 2021, AC21).

At other Galactic coordinates we do not have this convenient relationship.
However, we can make further progress by making a simple approximation: That
the typical $V_{R}\approx 0$ at any point in the disc. There is a dispersion
around this value, which will produce a corresponding uncertainty in the
derived values which increases as we move further from $\ell=180^{\circ}$. We
know that even the average $V_{R}$ is not exactly zero across the Galaxy, with
Gaia DR2 in particular demonstrating that it varies at the level of
${\sim}10\,\mathrm{km\,s}^{-1}$ in the extended Solar neighbourhood (Gaia
Collaboration et al., 2018b), while Eilers et al. (2020) showed that it varies
with an amplitude of ${\sim 7\,\mathrm{km\,s}^{-1}}$ in the outer disc.
However, this is much smaller than the typical $V_{\phi}$ velocity in the
disc, so makes a fairly small difference. We quantify the errors introduced by
this approximation in Section 2.2.

For a star in any position in the Galaxy with measured $(V_{\ell},V_{b})$,
there is a line-of-sight velocity, which we call ${V_{\parallel}}^{*}$, that
would give it $V_{R}=0$. We can show that this velocity

${V_{\parallel}}^{*}=\frac{Xv_{X}^{\bot}+Yv_{Y}^{\bot}}{(X\cos l+Y\sin l)\cos
b},$ (3)

where $(X,Y,Z)$ are Cartesian components of a star’s position in the
Galaxy333This coordinate system has the Galactic centre at $(0,0,0)$, and the
Sun at $(R_{\odot},0,Z_{\odot})$. $(R,\phi,Z)$ are the cylindrical coordinates
associated with this coordinate system., and
$(v_{X}^{\bot},v_{Y}^{\bot},v_{Z}^{\bot})$ are the components of the star’s
Galactocentric velocity in the $X,Y,Z$ coordinate system that come only from
its velocity in the plane of the sky. We can use this to find our estimates
for $V_{\phi}$ and $V_{Z}$, which are

$V_{\phi}^{*}=\frac{R(-v_{X}^{\bot}\sin\ell+v_{Y}^{\bot}\cos\ell)}{X\cos\ell+Y\sin\ell}$
(4) $V_{Z}^{*}=v_{Z}^{\bot}+{V_{\parallel}}^{*}\sin b$ (5)

A full derivation of these equations is in Appendix A.

One could consider taking more sophisticated approaches than the approximation
that $V_{R}=0$. The value of $V_{R}$ could be estimated as a function of
position using the stars for which there are measurements of the line-of-sight
velocity. This would have to be done carefully to avoid adding excess noise,
especially far from the Sun where there are few such stars. It would also be
taking its $V_{R}$ estimate from a different (brighter) population than the
population being considered. Recently, it has been proposed that a Bayesian
neural network can be used to estimate the missing line-of-sight velocities
for Gaia stars (Naik & Widmark, 2022), but it is not clear whether this
provides a better point estimate than our approach. In this study our simple
approach is sufficient, and has the benefit of being straightforwardly
reproducible and less susceptible to unexpected systematic errors. We defer
the study of improvements we could make for this approximation to future work.

We have to assume a position and velocity for the Sun in the Milky Way to
convert our measurements into the Galactocentric frame. We assume the Sun is
located a distance $R_{\odot}=8.178\,\mathrm{kpc}$ from the Galactic centre
(Gravity Collaboration et al., 2019) and a height above the Galactic plane of
$Z_{\odot}=0.02\,\mathrm{kpc}$ (Bennett & Bovy, 2019). We further assume
$(v_{X,\odot},v_{Y,\odot},v_{Z,\odot}=(-11.1,-248.5,7.25)\,\mathrm{km\,s}^{-1}$
for the solar motion (Schönrich, Binney & Dehnen, 2010; Reid & Brunthaler,
2020).

Throughout this paper we indicate that a quantity has been estimated without a
line-of-sight velocity measurement by giving it an asterisk. Our estimate of
the angular momentum of a star is $L_{Z}^{*}\equiv RV_{\phi}^{*}$, which for
most stars in the Milky Way disc is negative. In the interests of making our
results easier to interpret, we augment this throughout this article with a
simple estimate of the guiding centre radius
$R_{g}^{*}=|L_{Z}^{*}|/(236\,\mathrm{km\,s}^{-1})$.

###  2.2 Demonstration on mock data

To test our assumption that we can proceed using only proper motions, we apply
our method to mock data with known properties. We base this on a dynamical
model constructed using the agama software package (Vasiliev, 2019). This
model consists of two disc components and a stellar halo, and is designed to
be in equilibrium within a given axisymmetric gravitational potential (the
main model from McMillan, 2017). We sample $40\,000\,000$ particles from the
model which, since the model is axisymmetric, we can place at any position
angle in the Galaxy we are interested in. The samples we construct from this
model, which come from $10^{\circ}$ ranges in $\ell$ between $130^{\circ}$ and
$230^{\circ}$, contain between $2\,800\,000$ and $3\,200\,000$ particles when
we apply the constraint that they also feature only stars with
$|b|<10^{\circ}$.

Figure 5: The distribution in $V_{Z}^{*}$ of stars in the Gaia data as a
function of their estimated angular momentum $L_{Z}^{*}$ or, equivalently,
guiding centre radius $R_{g}^{*}$. These are shown for $10^{\circ}$ ranges in
$\ell$, ranging from $130^{\circ}$ to $180^{\circ}$ for the upper set of
panels and $180^{\circ}$ to $230^{\circ}$ for the lower set of panels. In each
case this is also divided into the distribution above and below the plane.
Vertical dotted lines are again drawn at $L_{Z}^{*}={-}2500$ &
${-}3000\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ to guide the eye and to ease the
comparison between equivalent plots elsewhere in the text. The green circle in
the panel for $Z>0$ at $180^{\circ}<l<190^{\circ}$ highlights a feature which
can be seen in a number of panels. The ‘break’ seen in the anticentre region
at $L_{Z}^{*}\approx 2750\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ is clearly
visible in many panels, with noticeable differences above and below the plane.

We perturb the vertical velocities of stars in this model to (very
approximately) mimic the structure seen by AC21. We change the vertical
velocity of each star by an amount $\Delta V_{Z}\,\mathrm{km\,s}^{-1}$ that is
a function of the $Z$ component of angular momentum, $L_{z}$. For convenience
we express this in terms of
$l_{Z}=|L_{Z}|/(\,\mathrm{km\,s}^{-1}\,\mathrm{kpc})$:

$\Delta V_{Z}=\begin{cases}0&\text{for }l_{Z}\leq 2000\\\
6\left(l_{Z}-2000\right)/250&\text{for }2000<l_{Z}\leq 2250\\\
6-20\left(l_{Z}-2250\right)/500&\text{for }2250<l_{Z}\leq 2750\\\ 5&\text{for
}2750<l_{Z}.\end{cases}$ (6)

To demonstrate that our method is robust against the variations in $V_{R}$ in
the Galaxy, we further perturb the model with a term that mimics the influence
of the spiral arms on the velocity field of the outer Milky Way, guided by the
results from Eilers et al. (2020). We assume that the influence of the spiral
arms is primarily on $V_{R}$, and perturb the velocities of all stars in our
model by an amount $\Delta V_{R}$ corresponding to an $m=2$ spiral
perturbation with

$\Delta V_{R}(R,\phi)=V_{\rm
max}\sin\left[m\left(\phi-\phi_{0}-\frac{\log(R/\,\mathrm{kpc})}{\tan(p)}\right)\right],$
(7)

where we take amplitude $V_{\rm max}=7\,\mathrm{km\,s}^{-1}$,
$\phi_{0}=0^{\circ}$ and pitch angle $p=12^{\circ}$. This produces a spiral
perturbation that is comparable to that found in the outer disc by Eilers et
al. (2020), or the $V_{R}$ variation found by Cheng et al. (2020).

Finally, we give each star a relative distance uncertainty of $15\%$, which is
slightly larger than the median relative uncertainty of the Bailer-Jones et
al. (2021) distance estimates. We then make our velocity estimates $V_{Z}^{*}$
and $V_{\phi}^{*}$ for these mock data.

The results are illustrated in Figure 3, which shows the angular momentum-
vertical velocity space $(L_{Z}^{*},V_{Z}^{*})$, coloured by density in
$V_{Z}^{*}$ at each $L_{Z}^{*}$, for bins between $\ell=130^{\circ}$ and
$230^{\circ}$. The choice to show the density in $V_{Z}^{*}$ rather than in
$(L_{Z}^{*},V_{Z}^{*})$ means that we can focus on the changes in $V_{Z}^{*}$
with $L_{Z}^{*}$, rather than having the structure of the plot being dominated
by the decreasing number of stars as a function of $L_{Z}^{*}$. These mock
observations can be compared to the model distribution, which is shown in Fig.
4. We also show in each panel the typical uncertainty in $L_{Z}$ and $V_{Z}$
(defined by the 16th and 84th percentile of the differences between the true
values and the derived values).

Figure 6: Like Fig 5, we show the distribution in $V_{Z}^{*}$ of stars in the
Gaia data as a function of $L_{Z}^{*}$ or $R_{g}^{*}$, divided into panels for
$130^{\circ}<\ell<180^{\circ}$ (upper) $180^{\circ}<230^{\circ}$ (lower) and
subdivided above and below the plane (as indicated). In this case we compute
this only for the young, bright, blue sample of stars indicated in Fig 2.
Again, dotted lines are at $L_{Z}^{*}={-}2500$ &
${-}3000\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ guide the eye. The green circle
in the panel for $Z>0$ at $180^{\circ}<l<190^{\circ}$ is identical to the one
in Fig. 5, and the feature it highlights is weaker for this kinematically cool
population. The ‘break’ at $L_{Z}^{*}\approx
2750\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ is even more clearly visible than for
all stars, and we now see that it extends to $\ell=130^{\circ}$, but still
shows no sign of being present for $\ell>210^{\circ}$.

It is obvious from first inspection that the structure in this figure is most
clear for bins near the anticentre. Similarly, the measurement uncertainties
grow as we move further from the anticentre. This is completely expected given
that the approximation described in Sec 2.1 is insensitive to the real line-
of-sight velocity at $\ell=180^{\circ}$ and becomes increasingly sensitive to
it as we move further away across the sky. Another subtler reason for this is
that the influence of distance errors becomes greater as we move further from
$\ell=180^{\circ}$, because a given Galactocentric radius $R$ corresponds to
an ever increasing distance from the Sun.

Despite the smearing out of the underlying structure, we are still able to see
some sign of it out to $50^{\circ}$ from the anticentre. This informs our
understanding when we look at the data – we should expect perturbations of the
galaxy to appear significantly smoothed further from the anticentre, not
introduced or accentuated by our approximations.

In these tests, the distance errors (and the velocity errors they introduce)
are the dominant source of blurring of the features of the distribution out to
around $\ell=150^{\circ}$ or $210^{\circ}$, at which point the uncertainties
due to having no line-of-sight velocity measurements begin to dominate. We
demonstrate this in Appendix B, where we isolate the influence of the two
effects separately. We also show that the influence of the spiral perturbation
is negligible. Reducing distance uncertainties is almost as important as
adding line-of-sight measurements for stars in our efforts to find this
structure, even far from the anticentre.

##  3 The outer Milky Way

First, in Figure 5 we show, for our full sample, angular momentum-vertical
velocity space $(L_{Z}^{*},V_{Z}^{*})$, coloured by density in $V_{Z}^{*}$ at
each $L_{Z}^{*}$. This is separated into panels each showing a $10^{\circ}$
range in Galactic longitude $\ell$, and also split above and below the
Galactic plane. The feature found by AC21 is again visible in these panels,
because in many of the panels there is a downturn in $V_{Z}^{*}$ at
$R_{g}^{*}\approx 10\,\mathrm{kpc}$ ($L_{Z}^{*}\approx
2400\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$) which goes down to a minimum at
$R_{g}^{*}\approx 11.5\,\mathrm{kpc}$ ($L_{Z}^{*}\approx
2750\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$) before what appears to be an abrupt
break, with $V_{Z}^{*}$ jumping to a positive value.

As we look in bins of increasing $\ell$, this pattern (a downturn followed by
an abrupt break) can first be seen below the plane in the $140-150^{\circ}$
bin (the high $R_{g}^{*}$ part of the $130-140^{\circ}$ bin is very poorly
populated below the plane). The break appears above the plane for the
$150-160^{\circ}$ bin, but remains stronger below the plane up to the
$160-170^{\circ}$ bin. The break is then stronger above the plane for
increasing $\ell$, and has faded away below the plane by about $190^{\circ}$
and above the plane by about $210^{\circ}$.

This extends our understanding beyond the results of AC21. In that study it
was found that the downturn towards lower $V_{Z}^{*}$ was dominated by stars
below the plane. We see the same for stars in the area of the sky probed by
AC21 ($170^{\circ}<\ell<190^{\circ}$), but can now see that this is not
consistent across the sky.

The panels of Figure 5 also have another feature worth noting, which is an
overdensity near $V_{Z}^{*}=-20\,\mathrm{km\,s}^{-1}$ at $R_{g}^{*}\approx
10\,\mathrm{kpc}$, particularly clear for the panels at $Z>0$ for
$180^{\circ}<\ell<200^{\circ}$, and visible elsewhere. We indicate this
feature with a circle in one panel of this figure. This feature is faintly
visible in the equivalent plot in AC21, but was not commented upon. This
feature is investigated further by Alinder, McMillan & Bensby (in prep.).

Figure 7: Velocity distributions in the $V_{\phi}^{*}{-}V_{Z}^{*}$ plane for
stars that are divided into bins that are $20^{\circ}$-wide in $\ell$,
$0.5\,\mathrm{kpc}$ in $R$, and contain stars on one side or another of the
Galactic plane (as indicated on the left in blue, at the top, and on each
panel, respectively). Dotted grid lines are added to ease comparisons between
panels. Bimodal structure is visible in many panels, especially below the
plane, with the gap being approximately at
$L_{Z}^{*}=2750\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$, which is marked on each
panel with a red cross at $V_{Z}^{*}=0$.

###  3.1 Young population

Previous studies have shown that the phase spiral is stronger for stars on
kinematically colder orbits, i.e., younger, bluer, stars. AC21 also showed
that the break at $R_{g}^{*}\approx 11.5$ stands out very clearly in their
young population (their Fig. E.8). We therefore show in Figure 6 the
$(L_{Z}^{*},V_{Z}^{*})$ plane for the bright young main-sequence sample
described in Sect.2, again coloured by density in $V_{Z}^{*}$. We note also
that these are all bright stars and therefore their Gaia parallax
uncertainties will be relatively small, allowing for more accurate distance
estimates even far from the Sun.

The vertical velocity dispersion of these sources is smaller than for the
whole population, which is clear in all of these plots. As seen by AC21, this
means that the break at $R_{g}^{*}\approx 11.5$ stands out clearly, and we
have extended this result to a far wider range in $\ell$. The feature near
$V_{Z}^{*}=-20\,\mathrm{km\,s}^{-1}$ at $R_{g}^{*}\approx 10\,\mathrm{kpc}$ is
harder to see for this limited population, presumably because it has a low
vertical velocity dispersion so does not sample the phase space around
$V_{Z}^{*}=-20\,\mathrm{km\,s}^{-1}$ well.

Using these stars as tracers we can see for the first time a hint that the
break extends to $\ell=130^{\circ}$, and therefore perhaps even beyond. The
panel for $130^{\circ}<\ell<140^{\circ}$ appears to have a break at around
$R_{g}^{*}=12\,\mathrm{kpc}$, but there is no sign of a downtrend before this.
There are still no real signs of any break for $\ell>210$, so we tempted to
say that it has disappeared. Certainly we can say with some confidence that it
is weaker at $\ell=210$ than at $\ell=150$ because, by symmetry, the
uncertainties at these two longitudes should be very similar (assuming
differences due to differing extinctions are negligible). However, we are very
much in the position of having limited data with which to make these
judgements, and a more detailed look, and a clearer perspective beyond this
range in $\ell$, will have to wait for a larger sample of stars with measured
line-of-sight velocities.

###  3.2 Velocity distributions

Finally, in Figure 7 we show the velocity distribution in the
$V_{\phi}^{*}$–$V_{Z}^{*}$ plane for $20^{\circ}$ slices in $\ell$, separated
into radial bins of width $0.5\,\mathrm{kpc}$ from $10.5\,\mathrm{kpc}$ to
$13.5\,\mathrm{kpc}$, and further divided into stars above and below the
plane.

These plots confirm the results seen by AC21, that the break seen in the
$L_{Z}^{*}-V_{Z}^{*}$ plane can equally be thought of as being made by a
bimodal velocity distribution, consisting of two clumps of stars in the
$V_{\phi}^{*}-V_{Z}^{*}$ plane, one tending to move downwards with a slower
rotational velocity, one moving upwards with a faster rotational velocity.

To guide the eye, we have put a cross in each panel of these figures at an
angular momentum of $2750\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ (for a star at a
radius in the middle of the range). The transition we see happens at around
this angular momentum, and the gap between the two clumps is at approximately
this point in many cases. This confirms that there is a connection between the
structure seen here and that seen in Figs 5 & 6. However it also demonstrates
that we lose some information by treating the disturbance as a function of
$L_{z}$, because in some panels the gap in the velocity distribution is at
different $L_{Z}^{*}$. For example, a number of the panels for
$(12.5<R<13)\,\mathrm{kpc}$ have a substantial clump of stars at the slower
rotational velocity and lower $V_{Z}^{*}$ that sits almost exactly at
$L_{Z}^{*}=2750\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$. Therefore it is clear
that the exact position of the break between the clumps is not simply a value
of $L_{z}$, and the behaviour of the outer Galaxy in this region can not be
completely simplified as in Figs. 5 & 6.

The break into two clumps can be seen below the plane in all $\ell$ ranges
from $130^{\circ}$ to $210^{\circ}$, with varying strength and clarity. Above
the plane it is only really clear in the range $150^{\circ}<\ell<190^{\circ}$.

Figure 8: Velocity distributions of a $20^{\circ}$ segment of the outer disc
of the simulated galaxy at three selected times (as indicated, note that the
disc crossing of the perturber occurs at $t=100\,\mathrm{Myr}$). The panels,
equivalent to those in Fig 7, show density in the $V_{\phi}$-$V_{Z}$ plane for
radial bins between $10.5$ and $13.5\,\mathrm{kpc}$, divided into particles
above and below the disc plane. Again, dotted grid lines are added to ease
comparisons between panels and with Fig 7. In all these selected times, the
velocity distribution is divided into two clumps, but only in the first case
are these two clumps displaced from one another in $V_{Z}$.

##  4 Simulations

In an effort to interpret what we are seeing in the outer Milky Way, we turn
to a numerical simulation of the interaction of an impulsive mass with a cold
stellar disc at a single transit point. This is based on the simulations
presented by Bland-Hawthorn & Tepper-García (2021), with the only differences
being that we have reduced the mass of the perturbing particle from $2\times
10^{10}\,{\rm M}_{\odot}$ to $10^{10}\,{\rm M}_{\odot}$ and reduced the number
of particles. The reduced number of particles allowed us to explore the
parameter space of impacts before settling on the simulation we present here,
while still allowing us to resolve the behaviour we sought to examine. We
found that a $2\times 10^{10}\,{\rm M}_{\odot}$ perturber produced an effect
in the outer disc that was significantly larger than that seen in our Milky
Way data, while the $10^{10}\,{\rm M}_{\odot}$ simulation provided a closer
match to the perturbation that we see, as the scale of this perturbation
scales approximately linearly with the mass of the perturber. We do not claim
to have made an exhaustive search of the possible impacts or timings, to have
pinned down the mass of the Sgr dwarf when it last had a major impact on the
Milky Way disc, or created an exact analogue of the behaviour seen in Sec 3.
That work is reserved for future studies.

Figure 9: The distribution in $V_{Z}$ as a function of angular momentum
$L_{z}$ in the outer disc of the simulated galaxy at the same three selected
times as Fig. 8. The panels, equivalent to those in Figs. 5 & 6, are again
divided into the distribution above and below the plane, and we have again
drawn dotted lines $L_{Z}^{*}={-}2500$ &
${-}3000\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}$ to ease the comparison between
these equivalent plots. We see corrugation of the distribution, which appears
over shorter and shorter scales as it winds up over the period shown here, but
we do not see a clean break of the kind seen in the Milky Way data.

The interested reader is directed to Bland-Hawthorn & Tepper-García (2021) for
a full description of this simulation but we will summarise the key elements
here. The galaxy model consists of a live $N$-body dark-matter halo, stellar
bulge and stellar disc with a total mass of ${\sim}1.45\times 10^{12}\,{\rm
M}_{\odot}$ and a nearly flat rotation curve out to $R=20\,\mathrm{kpc}$. It
is constructed from an action-based distribution function using agama such
that it is stable in its initial configuration without any relaxation, and its
disc is kinematically cold to enhance the signature of perturbations. We
populate this model with $10$ million particles in the halo, $5$ million
particles in the disc and $2.25$ million particles in the bulge, then evolve
it using the $N$-body code ramses (Teyssier, 2002). The perturbation is
modelled as a nearly impulsive interaction of a point mass which intersects
the Galactic plane at $R\approx 18\,\mathrm{kpc}$ travelling with a velocity
of ${\sim}330\,\mathrm{km\,s}^{-1}$, and a Galactic azimuth $\phi=0$.444Bland-
Hawthorn & Tepper-García (2021) place the impact and the Sun (at the present
day) at $\phi=180^{\circ}$, and their Galactic disc rotates in the positive
sense. We reverse these conventions for consistency with the rest of our
analyses. The disc crossing occurs at a simulation time $T\approx
100\,\mathrm{Myr}$, and after $T=150\,\mathrm{Myr}$ the mass of the perturber
begins to exponentially decrease, such that by the time of its second disc
crossing it has a mass of $\sim 10^{-6}$ times its original mass. This
simplifies the encounter and allows us (like Bland-Hawthorn & Tepper-García,
2021) to focus on the effect of a single impact.

Bland-Hawthorn & Tepper-García (2021), following the analytical models of
Binney & Schönrich (2018), showed that the impact of a perturber like this is
able to produce a phase spiral in the Solar neighbourhood that is
qualitatively similar to that found in Gaia data, and that it will have
produced similar phase spirals across the disc (mainly differing in their
phase). Here we focus instead on the influence on the outer disc and the
effects that we have shown and discussed in Sec. 3. Specifically, we focus on
a circular sector in the galactic disc of width $\Delta\phi=40^{\circ}$,
centred on $\phi=0$. This is roughly the same range in azimuth covered by our
Milky Way data (see Fig. 1 for a comparison).

Figure 8 shows the velocity distribution in this sector for 3 selected
snapshots at $t=0.35,0.68$ and $1.08\,\mathrm{Gyr}$ in our simulation. Like
Fig. 7 we show the velocity distribution in radial bins of width
$0.5\,\mathrm{kpc}$ from $10.5\,\mathrm{kpc}$ to $13.5\,\mathrm{kpc}$, and
show the distribution separately for stars above and below the plane. These
snapshots are selected because they are at times when the velocity
distribution in this part of the Galaxy separates into two clumps of the kind
seen in the Milky Way data in Fig. 7.

Of all of these velocity distributions, it is the one at $0.35\,\mathrm{Gyr}$
(i.e, $250\,\mathrm{Myr}$ after the disc crossing of the perturber) that bears
the strongest resemblance to the velocity distribution seen in the Milky Way.
The two clumps are at different $V_{Z}$, with the higher $|V_{\phi}|$ clump
being at positive $V_{Z}$ and the lower $|V_{\phi}|$ clump at negative
$V_{Z}$. There is a clear difference between the behaviour above and below the
Galactic plane. There is a clearer distinction between two clumps in the
velocity space for $Z<0$, and the lower velocity clump has essentially
disappeared by the $12.5$-$13.5\,\mathrm{kpc}$ bin for $Z<0$ while this is not
true above the plane.

The other two snapshots do show a velocity distribution divided into two
clumps, but they do not have a clear difference in their $V_{Z}$ values. We
have run this simulation to nearly $4\,\mathrm{Gyr}$, long after the perturber
has reduced to a negligible mass and have found that the velocity distribution
continues to divide into two clumps, then coalesce into a single clump, then
divide into two clumps again repeatedly throughout the simulation time.
However, the division does becomes weaker, concentrated in the outer bins, and
does not show a clear asymmetry either side of $z=0$ or difference in $v_{z}$
for the two clumps.

To understand this better, in Fig. 9 we look in the simulations at the
distribution in $V_{Z}$ as it changes with $L_{z}$, again in the sector of
width $\Delta\phi=20^{\circ}$ around the assumed Solar azimuth in the Galaxy,
and in the same times as the panels of Fig. 8. In each case there are
disturbances in the vertical velocities that are of similar scale to the
disturbances seen in the Milky Way. We also see, particularly in the two later
snapshots, breaks in the distribution which are comparable to those seen in
the Milky Way. The panel at $t=0.35\,\mathrm{Gyr}$ does not show a clear break
in the distribution, but, as we have seen in Fig. 8, is the snapshot with a
velocity distribution that best matches that seen in the Milky Way.

We would also note that the panels show that the wavelength (in angular
momentum) of the disturbance is relatively long shortly after the disc
crossing, and gets increasingly short as time passes. This occurs because the
disturbance winds up around the disc.

The choice of $\phi$ position to centre this analysis is, to an extent,
arbitrary. The initial disc is axisymmetric, and Bland-Hawthorn & Tepper-
García (2021) used their simulation to explore the behaviour of the phase
spiral at a range of positions in the Galaxy. We have explored other choices
of $\phi$ and found that the results are, qualitatively, rather similar. The
times at which the velocity distribution forms into two clumps varies somewhat
with $\phi$, as the disturbance wraps its way around the disc. The general
pattern remains the same as for the three snapshots illustrated in Figs 8 & 9.
For the interested reader we have made videos showing the time evolution of
the plot shown in Fig. 8 for choices of $\phi$ at $40^{\circ}$ intervals
available at https://www.astro.lu.se/~paul/MyResearch.

##  5 Discussion

###  5.1 Galactic seismology

For over a century, geologists have learned about the interior of the Earth
through seismology, studying the waves that pass through it. More recently,
helioseismology has revolutionised our understanding of the Sun, and
asteroseismology has brought us great insight into the stars of our Galaxy. We
have now entered the era of Galactic seismology (or Galactoseismology), with
much to learn about our Galaxy from its oscillations.555For a history of the
field and its terminology from Hunter (1963) through Iye (1985) and Widrow et
al. (2012) to the present day see Appendix A of Bland-Hawthorn & Tepper-García
(2021).

Galactic seismology differs from the other fields in many ways, of which the
two most important are perhaps timescale, and influence on the medium through
which the waves are passing. The timescale of the oscillations in the Galaxy
are measured in $\,\mathrm{Myr}$, as opposed to oscillations of the Earth or
stars, which can be measured in minutes or hours. We therefore can not wait to
follow the oscillations of the Galaxy as they go through their phases. The
oscillations of the Earth and the stars can be treated as non-invasive – not
changing the underlying structure of the medium through which they are
travelling. This is not true for the oscillations of the Galaxy. Darling &
Widrow (2019) note, following the argument of Hunter & Toomre (1969), that the
influence of the perturbation on the unperturbed disc is of the same order as
the influence of the unperturbed disc upon the perturbation. They further show
that the influence of self gravity slows the growth of the phase spiral. This
result that can also be seen by comparing the phase spiral found by Bland-
Hawthorn & Tepper-García (2021) with the one seen in the analytic models of
Binney & Schönrich (2018) which winds up quickly (see also Hunt et al., 2021).

The bimodality of the velocity distribution in the outer Galaxy discovered by
AC21 was, like the phase spiral before it, not predicted. Here we have shown
that it can be explained, at least qualitatively, by a model that also
produces a phase spiral. These different measurements, or phase space
projections in which we conduct our analysis, can be thought of as two
separate Galactic seismographs, but they must be intrinsically linked and
understanding of our Galaxy will come from models that can consistently
explain both.

For now we find ourselves like blindfolded people discovering an elephant,
each describing what is right in front of them (a trunk, a tail, a leg, etc.)
and not being able to agree on what they have found.666This is a parable that
dates back to at least 500 BCE, e.g., https://suttacentral.net/ud6.4 Similarly
we will probably need to simultaneously explain the phase spiral, the velocity
bimodality, the Galactic warp, the ridges in $R{-}V_{\phi}$ space (Kawata et
al., 2018; Ramos et al., 2018, AC21), the variations in $V_{R}$ as a function
of angular momentum (e.g. Friske & Schönrich, 2019) and Galactic $\phi$ (e.g.
Eilers et al., 2020), and even the structures of the outer Galaxy (e.g., Xu et
al., 2015; Laporte et al., 2022, AC21). This will be complicated by the fact
that some of these phenomena, especially those seen in the planar velocities,
are affected by the Galactic bar and spiral arms in a way that is likely to be
unrelated to the vertical velocity perturbations that we see. Future data from
Gaia, or spectroscopic surveys such as WEAVE (Dalton et al., 2016) and 4MOST
(de Jong et al., 2019), will no doubt allow us to identify a new tusk or two,
but our task will remain that of piecing all these elements into an over-all
picture.

This study has not described the entire elephant, but has extended our
understanding of one part of it. We have shown that the velocity bimodality
persists across a range of Galactic $\phi$ angle that goes far beyond those
probed by AC21, that it changes in character with varying $\phi$, and appears
to fade away by $\phi\approx 10^{\circ}$ ($\ell\approx 210^{\circ}$) although,
as we have discussed, this is at least partly due to the difficulty in
measuring it at these longitudes. The simulations we have performed show that,
like the phase spiral, bimodal velocity distributions can be induced by the
passage of the Sagittarius dwarf. However, we have not carefully matched the
timing of the velocity bimodality in our simulation to that of the formation
and evolution of the phase spiral, nor to the past history of Sagittarius.
That work is reserved for future study.

###  5.2 Density and bending waves

Bland-Hawthorn & Tepper-García (2021) argue that the impact of the Sgr dwarf
triggers two different $m=2$ modes in the inner disc: a density wave, and a
corrugated bending wave. They associate the phase spiral with the density wave
rolling up and down the bending wave. They note in addition that the outer
discs’ vertical displacement is better described instead as dominated by an
$m=1$ bending mode.

The density wave can be expected to create gaps (or simply low density
regions) in the angular momentum distribution, which translate into gaps in
$V_{\phi}$ for stars in a given radial bin. The bending wave can mean that the
two clumps either side of this gap have different $V_{Z}$. This is what we
appear to see in Fig. 8 for the snapshot at $t=0.35\,\mathrm{Gyr}$, but the
snapshots at $t=0.68$ and $1.08\,\mathrm{Gyr}$ have gaps, which we associate
with the density wave, which do not coincide with a significant vertical
velocity difference from the bending wave – the clumps in these two cases have
nearly identical vertical velocities. In this picture, the structure seen in
the Milky Way can only be produced by a coincidence of a minimum point in the
density wave and a point where the gradient of $V_{Z}$ in the bending wave is
very large.

It is certainly questionable whether this is exactly what we are seeing in the
outer Milky Way. The break in the $L_{Z}^{*}{-}V_{Z}^{*}$ plane seen in Figs.
5 & 6 does appear to be relatively clean at a given $L_{Z}^{*}$, but in
velocity space (Fig. 7) the divide does not appear to lie along a line of
constant $V_{\phi}^{*}$, as it does in the simulation (Fig. 8). Equally, we do
not see a clean break in the $L_{Z}{-}V_{Z}$ plane at $t=0.35\,\mathrm{Gyr}$,
as we do in the Milky Way. There is also a very clear difference in the Milky
Way between what we see above and below the plane. We do not see this here,
but this may be because we are averaging over a wider range in $\phi$.

The timing of the snapshot that seems to best mimic the behaviour in the Milky
Way ($t=0.35\,\mathrm{Gyr}$, i.e., $250\,\mathrm{Myr}$ after the disc
crossing) is also before Bland-Hawthorn & Tepper-García (2021) see a phase
spiral forming, though it is consistent with the timing of the phase spiral
found using a simpler, analytic, approach by Binney & Schönrich (2018). The
later snapshots do have breaks in the $L_{Z}{-}V_{Z}$ plane, but we can
clearly discern in each of them that there is an oscillation with an ever
decreasing scale length in $L_{Z}$, which we can associate with the winding up
of the bending wave across the Galactic disc. The equivalent distribution in
the Milky Way does not appear to have this oscillation on short scales in
$L_{Z}$, instead having a large scale variation with an abrupt break. We also
use a perturber that has a mass which half that of the Bland-Hawthorn &
Tepper-García (2021) simulation that reproduced the phase spiral, in order to
match the scale of perturbation seen in the outer Milky Way, so cannot claim
that we simultaneously model both.

Certainly we have only begun to interpret the velocity bimodality in the outer
Milky Way disc. One can hope that this will prove a fruitful area, both for
learning about the mass and timing of the most recent impacts on the Milky Way
disc, and for measuring the properties of the disc too. If the bimodality can
indeed only be explain by a coincidence of the density and bending wave then
this sets a strong constraint on the impact’s timing relative to orbital
frequencies in the disc. We might reasonably expect that in the outer disc, as
the disc itself becomes less dominant, the influence of self-gravity becomes
less important, allowing us to make a unambiguous estimate of these properties
and to use simpler, faster, modelling techniques of the kind employed by
Binney & Schönrich (2018).

##  6 Summary

In this paper we have shown that the velocity bimodality – or, equivalently,
break in the $L_{Z}{-}V_{Z}$ distribution – found in the Galactic anticentre
by AC21 persists across some significant fraction of the Galactic disc. It can
be found out to $\ell=130^{\circ}$, which corresponds to a Galactocentric
angle of $\phi\approx 20^{\circ}$, but appears to be fading away by
$\ell=210^{\circ}$ ($\phi\approx 10^{\circ}$). Its behaviour is, as seen by
AC21, different above and below the Galactic plane, with a stronger
break/bimodality usually visible below the plane. We note that selecting young
main-sequence stars allows us to see a stronger effect, which is consistent
with previous results that show younger, kinematically colder, stars are more
affected by disturbances in the vertical velocities of stars (e.g., Bland-
Hawthorn et al., 2019).

Using rather idealized $N$-body simulations we have shown that the passage of
a Sagittarius-like dwarf galaxy is capable of producing qualitatively similar
effects in a Galactic disc. These simulations are also capable of producing a
phase spiral akin to the one which has been discovered in the Solar
neighbourhood.

This study has been conducted entirely without measured line-of-sight
velocities for the stars considered. For the foreseeable future we will be in
the position that we have measured proper motions for far more stars than
line-of-sight velocities. This is due to both the two billion stars observed
by Gaia, and the still larger number which would be observed by a successor
mission conducting astrometry in the infrared (Hobbs et al., 2021). Exploiting
these incomplete 5D phase-space data are absolutely key to building up large
samples of stars for study, which allows us to either constrain our models far
more accurately or, as in the case of the velocity bimodality, find features
of our Galaxy that we have not predicted using our models. Our investigation
of the uncertainties inherent in our approach makes the further point that
distance uncertainties are as serious a barrier to characterising sharp
structure in velocity space as the absence of line-of-sight velocity
measurements.

## Software

This study utilised the Python packages numpy (Harris et al., 2020),
matplotlib (Hunter, 2007), pandas (Wes McKinney, 2010) and dustmaps (Green,
2018).

## Acknowledgements

PM and JP gratefully acknowledge support from project grants from the Swedish
Research Council (Vetenskaprådet, Reg: 2017-03721; 2021-04153). Some of the
computations in this project were completed on computing equipment bought with
a grant from The Royal Physiographic Society in Lund. TTG acknowledges partial
financial support from the Australian Research Council (ARC) through an
Australian Laureate Fellowship awarded to JBH. LC acknowledges financial
support from ANID/Fondecyt Regular Project 1210992. TA acknowledges the grant
RYC2018-025968-I funded by MCIN/AEI/10.13039/501100011033 and by “ESF
Investing in your future”. TA, MRG and FF acknowledge that this work was
partially funded by the Spanish MICIN/AEI/10.13039/501100011033 and by “ERDF A
way of making Europe” by the “European Union” through grant
RTI2018-095076-B-C21, and the Institute of Cosmos Sciences University of
Barcelona (ICCUB, Unidad de Excelencia ‘María de Maeztu’) through grant
CEX2019-000918-M. PR acknowledges funding from the University of Barcelona,
Margarita Salas grant (NextGenerationEU). This work has made use of data from
the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## Data availability

The observational data underlying this article were accessed from the Gaia
archive (https://gea.esac.esa.int/archive/). The derived data generated in
this research, mock data used in Sec. 2.2, and simulation data from Sec. 4
will be shared on reasonable request to the corresponding author.

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##  Appendix A Derivation of the approximations

Taking the usual convention for Heliocentric coordinates, that $U$ is the
component of velocity towards $(l,b)=(0,0)$, $V$ the component towards
$(l,b)=(90^{\circ},0)$ and $W$ towards $(l,b)=(0,90^{\circ})$, we can find the
components of Heliocentric velocity _excluding line-of-sight velocity_ as

$(U^{\bot},V^{\bot},W^{\bot})=(-V_{\ell}\sin l-V_{b}\cos l\sin b,\\\
V_{\ell}\cos l-V_{b}\sin l\sin b,\;V_{b}\cos b)$ (8)

where the missing line-of-sight velocity decomposes into
$(U^{\parallel},V^{\parallel},W^{\parallel})=v_{\parallel}(\cos l\cos b,\sin
l\cos b,\sin b)$, and $U=U^{\bot}+U^{\parallel}$ etc.

We can convert these into Galactocentric coordinates using the values for the
Sun’s position and velocity given in Section 2. We place the Sun at
$(X,Y,Z)=(R_{\odot},0,Z_{\odot})$, meaning that we have to flip the
orientation of the velocity axes in the plane from that of our Heliocentric
system. We have

$(v_{X}^{\bot},v_{Y}^{\bot},v_{Z}^{\bot})=(-(U^{\bot}+U_{\odot}),-(V^{\bot}+V_{\odot}),W^{\bot}+W_{\odot})$
(9)

To make further progress we have to introduce our approximation that
$V_{R}=0$. From the above, we can see that

$V_{R}=(v_{X}^{\bot}-{V_{\parallel}}\cos l\cos
b)\frac{X}{R}+(v_{Y}^{\bot}-{V_{\parallel}}\sin l\cos b)\frac{Y}{R}$ (10)

which, under the approximation that $V_{R}=0$, can be rearranged to give an
estimate of ${V_{\parallel}}$, which we refer to as ${V_{\parallel}}^{*}$
where

${V_{\parallel}}^{*}=\frac{Xv_{X}^{\bot}+Yv_{Y}^{\bot}}{(X\cos l+Y\sin l)\cos
b}$ (11)

This allows us to find our estimates

$\displaystyle V_{\phi}^{*}$
$\displaystyle=\frac{X(v_{Y}^{\bot}-{V_{\parallel}}^{*}\sin l\cos
b)-Y(v_{X}^{\bot}-{V_{\parallel}}^{*}\cos l\cos b)}{R}$
$\displaystyle=\frac{R(-v_{X}^{\bot}\sin l+v_{Y}^{\bot}\cos l)}{X\cos l+Y\sin
l},$ (12) $V_{Z}^{*}=v_{Z}^{\bot}+{V_{\parallel}}^{*}\sin b.$ (13)

We can find the error associated with a star having a true $V_{R}$ by placing
this in eq. 10 and following the equations through. We find errors
$\epsilon_{\phi}=V_{\phi}-V_{\phi}^{*}$ and $\epsilon_{Z}=V_{Z}-V_{Z}^{*}$ to
be

$\epsilon_{\phi}=V_{R}\;\frac{X\sin l-Y\cos l}{X\cos l+Y\sin l}$ (14)

and

$\epsilon_{Z}=-V_{R}\tan b\;\frac{R}{X\cos l+Y\sin l}.$ (15)

$\epsilon_{\phi}$ runs from ${\sim}-3V_{R}$ at $\ell=130^{\circ}$ to
${\sim}3V_{R}$ at $\ell=230^{\circ}$, with a mild dependence on distance (in
the sense that more distant stars are worse affected). $\epsilon_{Z}=V_{R}\tan
b$ at $\ell=180^{\circ}$, increasing symmetrically about $\ell=180^{\circ}$ to
${\sim}3V_{R}\tan b$ at $\ell=130^{\circ}$ and $\ell=230^{\circ}$, again with
somewhat greater errors at greater distances. We note that $|\tan b|<0.18$ for
all stars in our sample ($|b|<10^{\circ}$).

##  Appendix B Details of the mock data

Table 1: Parameters of the distribution functions used to sample the mock data used to validate our assumptions in Sec. 2.2. The parameters have the meanings described in App. B. Component | $\Sigma_{0}\,[\,{\rm M}_{\odot}\,\mathrm{kpc}^{-2}]$ |  $R_{\rm disc}\,$[$\,\mathrm{kpc}$] |  $h\,$[$\,\mathrm{kpc}$] |  $\sigma_{R,0}\,$[$\,\mathrm{km\,s}^{-1}$] |  $R_{\sigma_{R}}\,$[$\,\mathrm{kpc}$] | $\beta$ | $\tau_{\rm SFR}\,[\tau_{\rm max}]$ | $\xi$
---|---|---|---|---|---|---|---|---
Thin disc | $8.9\times 10^{8}$ | $2.5$ | $0.3$ | $90$ | $15$ | $0.33$ | $0.8$ | $0.225$
Thick disc | $1.8\times 10^{8}$ | $3.0$ | $0.7$ | $180$ | $6$ | - | - | -
- | $M\,[\,{\rm M}_{\odot}]$ | $J_{0}\,[\,\mathrm{km\,s}^{-1}\mathrm{kpc}]$ | $\Gamma$ | $\beta$ | $h_{r}$ | $h_{z}$ | $J_{\rm max}\,[\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}]$ | $\zeta$
Halo | $1.5\times 10^{10}$ | $500$ | $0$ | $3.5$ | $1.6$ | $0.7$ | $1\times 10^{5}$ | $2$
Figure 10: Velocity perturbation in $V_{R}$ applied to our mock data. The
perturbation is comparable to that seen in the Milky Way, but somewhat larger,
to provide a more stringent test.

The dynamical model on which we base the experiments in section 2.2 has two
discs and a halo, each of which is defined such that it is in equilibrium
within the axisymmetric Milky Way gravitational potential taken from McMillan
(2017). We use the Stäckel fudge approach of Binney (2012) to derive the
actions $\boldsymbol{J}=(J_{r},J_{z},J_{\phi}\equiv L_{Z})$ as a function of
position and velocity.

Figure 11: Density in $V_{Z}$ at different $L_{z}$ for mock data with
realistic distance uncertainties _but with precise line-of-sight velocities
and no spiral perturbation_. As with Fig. 3, each panel shows the results for
different $10^{\circ}$ ranges in $\ell$. Error bars indicate the typical
uncertainties in $L_{z}$ and $V_{Z}$. The error introduced is a significant
fraction of the total uncertainty seen in Fig. 3 in all panels, and is the
dominant source of uncertainty near the anticentre. Figure 12: Density in
$V_{Z}^{*}$ at different $L_{Z}^{*}$ for mock data with _no distance errors or
spiral perturbation_ , but with no line-of-sight velocity measurement. Each
panel shows the results for different $10^{\circ}$ ranges in $\ell$. The error
introduced by the lack of line-of-sight velocities is never much larger than
that introduced by the distance uncertainties (Fig 11), and only becomes the
dominant source of uncertainty ${\sim}30^{\circ}$ from the anticentre. Figure
13: Density in $V_{Z}^{*}$ at different $L_{Z}^{*}$ for mock data with _no
distance uncertainties_ , but with a spiral perturbation and no line-of-sight
velocity measurement (i.e., differing from Fig 12 only because of the re-
introduction of the spiral perturbation). The results are almost
indistinguishable from those in Fig 12, illustrating the very limited effect
of the spiral perturbation on these results.

The discs are based on the quasi-isothermal distribution function (Binney &
McMillan, 2011) as modified in agama,

$\displaystyle f(\boldsymbol{J})=$
$\displaystyle\frac{\Sigma_{0}\,\Omega(R_{\circ})}{2\pi^{2}\,\kappa(R_{\circ})^{2}}\exp\bigg{(}-\frac{R_{\circ}}{R_{\mathrm{disc}}}\bigg{)}\times
f_{\pm}(J_{\phi})$ (16)
$\displaystyle\times\frac{\kappa(R_{\circ})}{\tilde{\sigma}_{R}^{2}(R_{\circ})}\exp\bigg{(}-\frac{\kappa(R_{\circ})\,J_{r}}{\tilde{\sigma}_{R}^{2}(R_{\circ})}\bigg{)}$
$\displaystyle\times\frac{\nu(R_{\circ})}{\tilde{\sigma}_{z}^{2}(R_{\circ})}\exp\bigg{(}-\frac{\nu(R_{\circ})\,J_{z}}{\tilde{\sigma}_{z}^{2}(R_{\circ})}\bigg{)},$

with $(\kappa,\nu,\Omega)$ being the epicycle frequencies. $R_{\circ}$ is an
estimate of the typical orbital radius, which is given by radius of a circular
orbit with angular momentum $L_{\rm circ}=L_{z}+J_{r}+J_{z}/4$. The surface
density of this disc is $\Sigma\approx\Sigma_{0}\exp(-R/R_{\mathrm{disc}})$,
and the functions $\tilde{\sigma}_{R}$ and $\tilde{\sigma}_{z}$, which
approximately give the velocity dispersions in $R$ and $z$ (and therefore in
the latter case also determine the disc scale height), are of the form

$\displaystyle\tilde{\sigma}^{2}_{z}(R_{\circ})$
$\displaystyle=2\,h^{2}\,\nu^{2}(R_{\circ})+\sigma^{2}_{\rm min}$ (17)
$\displaystyle\tilde{\sigma}^{2}_{R}(R_{\circ})$
$\displaystyle=\sigma^{2}_{R,0}\,\exp(-2R_{\circ}/R_{\sigma_{R}})+\sigma^{2}_{\rm
min},$

where $\sigma_{\rm min}=5\,\mathrm{km\,s}^{-1}$ is introduced to avoid
numerical difficulties at large $R_{\circ}$. The term $f_{\pm}(J_{\phi})$ is a
function which is much larger for negative $J_{\phi}$ than the equivalent
positive $J_{\phi}$. This gives the model a negative sense of rotation, like
the Milky Way. Omitting it would give the distribution function of a disc that
has equal numbers of stars rotating in opposite directions.

The thick disc distribution function is taken to be a single quasi-isothermal
distribution, while the thin disc one is a superposition of quasi-isothermal
populations taken to be of ages $\tau$, weighted proportional to
$\exp({\tau/\tau_{\rm SFR}})$ in order to mimic the thin disc’s expected
decreasing star formation rate. The radial and vertical dispersions of the
quasi-isothermal populations both vary to reflect the age-velocity dispersion
relation in the form
$\sigma(\tau)=\sigma_{1}\big{[}\tau+(1-\tau)\xi^{1/\beta}\big{]}^{\beta}$,
with $\beta\simeq 0.33$ being the power law associated with the age-velocity
dispersion relation, and $\xi$ being the ratio of the velocity dispersion of
the oldest population and the youngest population.

The stellar halo is based on the double power-law form introduced by Posti et
al. (2015), as modified in agama

$\displaystyle f(\boldsymbol{J})$
$\displaystyle=\frac{M}{(2\pi\,J_{0})^{3}}\left[1+\left(\frac{J_{0}}{h(\boldsymbol{J})}\right)^{\eta}\right]^{\Gamma/\eta}\left[1+\left(\frac{g(\boldsymbol{J})}{J_{0}}\right)^{\eta}\right]^{\frac{\Gamma-\mathrm{B}}{\eta}}$
$\displaystyle\times\exp\bigg{[}-\left(\frac{g(\boldsymbol{J})}{J_{\mathrm{max}}}\right)^{\zeta}\bigg{]},$
(18)

where

$\displaystyle g(\boldsymbol{J})$ $\displaystyle\equiv
J_{r}+J_{z}+|J_{\phi}|,$ $\displaystyle h(\boldsymbol{J})$
$\displaystyle\equiv h_{r}J_{r}+h_{z}J_{z}+h_{\phi}|J_{\phi}|$

The interested reader is directed to Vasiliev (2018, 2019) for more details.
Parameters of the distribution functions of all three components are given in
table 1.

The samples drawn from this model are perturbed in the way described in 2.2,
which produces the disturbance in $V_{Z}$ illustrated in figure 4 and a
disturbance in $V_{R}$ which is illustrated in Fig 10, and is broadly similar
to that found by Eilers et al. (2020).

###  B.1 Different sources of error

Figures 11, 12 & 13 subdivide the effects which blur out the structure in the
$L_{Z}-V_{Z}$ plane. Figure 11 shows the effect of the 15 percent distance
uncertainty applied to the model on its own (i.e., assuming that line-of-sight
velocities are known). This causes blurring in all panels that increases
gradually as we move further from the anticentre, as the distance we have to
look to see stars at a given $R$ (which generally have $R_{g}{\approx}R$)
grows with distance from the anticentre. Note that we give quantities as, for
example, $L_{Z}$ rather than $L_{Z}^{*}$ because we are not having to estimate
these quantities without line-of-sight velocities.

Figure 12 shows the effect of assuming that there is no measured line-of-sight
velocity, even if the distance to the star is known precisely. The uncertainty
introduced is very small towards the anticentre – far smaller than the
uncertainty introduced by the distance uncertainty. However, the uncertainty
introduced by this approximation grows with increasing angle from the
anticentre faster than the uncertainty associated with the distance, and is
larger by $\ell\approx 150^{\circ}$. Figure 13 show that the effect of the
spiral in addition to the lack of line-of-sight velocity is so small as to be
completely negligible. We do not attempt to isolate the effect of the spiral
completely because, if the line-of-sight velocity is known, we no longer have
to to make the approximation that $V_{R}=0$, so the spiral perturbation has no
effect on our results.