MNRAS 000, 1–15 (2020) Preprint 8 September 2020 Compiled using MNRAS LATEX style file v3.0

Age dissection of the vertical breathing motions in GaiaDR2: evidence for spiral driving

Soumavo Ghosh1?, Victor P. Debattista2, Tigran Khachaturyants21 Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India2Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACTGaia DR2 has revealed breathing motions in the Milky Way, with stars on both sidesof the Galactic mid-plane moving coherently towards or away from it. The generatingmechanism of these breathing motions is thought to be spiral density waves. Herewe test this hypothesis. Using a self-consistent, high-resolution simulation with starformation, and which hosts prominent spirals, we first study the signatures of breathingmodes excited by spirals. In the model, the breathing modes excited by the spiralstructure have an increasing amplitude with distance from the mid-plane, pointing toan internal cause for them. The same behaviour is present also in the Gaia data. Wethen show that, at fixed height, the breathing motion amplitude decreases with age.We demonstrate that the Gaia data exhibit the same trend, which strengthens thecase that the observed breathing modes are driven by spiral density waves.

Key words: Galaxy: disc - Galaxy: kinematics and dynamics - Galaxy: structure -galaxies: interaction - galaxies: spiral - galaxies: kinematics and dynamics

1 INTRODUCTION

Spirals are common features of disc galaxies in the local Uni-verse (e.g Elmegreen et al. 2011; Yu et al. 2018; Savchenkoet al. 2020), as well as in high-redshift (z ∼ 1.8) disc galax-ies (e.g. Elmegreen & Elmegreen 2014; Willett et al. 2017;Hodge et al. 2019). The nature of spirals has been debatedextensively, and, while there is broad agreement that spi-rals are density waves, a full consensus is yet to emerge (fordetailed reviews, see e.g., Dobbs & Baba 2014; Sellwood &Carlberg 2019). A wide variety of physical mechanisms havebeen proposed for exciting spirals, including bars (e.g., Saloet al. 2010), tidal encounters, (e.g., Toomre & Toomre 1972;Dobbs et al. 2010), swing amplification of noise (Goldreich& Lynden-Bell 1965; Julian & Toomre 1966; Toomre 1981),giant molecular clouds (D’Onghia et al. 2013), other spirals(Masset & Tagger 1997), and recurrent groove modes (Sell-wood & Lin 1989; Sellwood & Kahn 1991; Sellwood 2012;Sellwood & Carlberg 2019). Star formation plays a pivotalrole in the persistence of spirals, by cooling the stellar discand facilitating the generation of fresh spirals (Sellwood &Carlberg 1984), while the interstellar gas helps spiral den-sity waves to survive for longer (Ghosh & Jog 2015, 2016).The vital dynamical role of spiral arms in transporting an-gular momentum (Lynden-Bell & Kalnajs 1972), as well as

? E-mail : soumavo@iucaa.in

in the radial mixing of stars without heating (Sellwood &Binney 2002; Roskar et al. 2008; Schonrich & Binney 2009)has kept the study of the formation and evolution of spiralsof continuing interest.

The Milky Way is a barred spiral galaxy (e.g. see Val-lee 2005, 2008). The presence of large-scale, non-zero meanvertical motions of Solar Neighbourhood stars has been re-ported in various surveys, for example, using main-sequencestars in the Sloan Extension for Galactic Understanding andExploration (SEGUE) survey (Widrow et al. 2012), F-typestars in the Large Sky Area Multi-Object Fibre Spectro-scopic Telescope (LAMOST) survey (Carlin et al. 2013), andred clump stars in the Radial Velocity Experiment (RAVE)data (Williams et al. 2013). These vertical motions displaytwo distinct modes: one in which stars on either side of themid-plane move coherently in the same direction along theperpendicular to the disc (bending modes) and the otherin which stars move coherently in the opposite direction,thereby compressing towards the mid-plane or expandingaway from it (breathing modes). In addition, Widrow et al.(2012) reported evidence for wave-like north-south asym-metries in the number counts of stars in the Solar Neigh-bourhood, which were further explored by Yanny & Gardner(2013).

The Gaia mission (Gaia Collaboration et al. 2016,2018a) has revolutionised the study of Galactic archaeol-ogy by measuring the stellar kinematics of stars in the So-

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lar Neighbourhood with an unprecedented precision. Thestellar kinematic study from the second Gaia Data Release(hereafter Gaia DR2) revealed rich kinematic substructurein the phase-space distribution of Solar Neighbourhood stars(Antoja et al. 2018), as well as the presence of large-scalenon-zero vertical motions (∼ 10 km s−1 in magnitude), andassociated bending and breathing modes (Gaia Collabora-tion et al. 2018b). Using the RAVE and the Tycho-Gaia as-trometric solution (TGAS) catalogue, Carrillo et al. (2018)showed the presence of both bending and breathing modes inSolar Neighbourhood. Bennett & Bovy (2019) revisited thenorth-south asymmetry in the Solar Neighbourhood starsusing the position-velocity measurements from Gaia DR2,and confirmed the presence of periodic over- and under-densities of stars in the vertical distribution, regardless oftheir colours. For an axisymmetric potential, the bulk radialand vertical motions should be zero (e.g. Binney & Tremaine2008). Therefore, the question arises what mechanism(s) caninduce these non-zero bulk vertical motions in our Galaxy.

Theoretical efforts to understand the cause of verticalmotions have explored both external and internal mecha-nisms. An interaction with a satellite or a dark matter sub-halo can excite large-scale coherent vertical bending motions(e.g., see Gomez et al. 2013; Widrow et al. 2014; D’Onghiaet al. 2016; Chequers et al. 2018). However bending motionscan be produced by purely internal mechanisms, such as thebar (Khoperskov et al. 2019) or the warp (Khachaturyantset al. in progress). On the other hand, internal causes suchas spiral density waves (Debattista 2014; Faure et al. 2014;Monari et al. 2016) or the Galactic bar (Monari et al. 2015)and external mechanism, e.g., fly-by encounter with a satel-lite or passing of a dark matter sub-halo (Widrow et al.2014) have been proposed for exciting the breathing modes.Strong spirals drive large-scale vertical breathing motions(|〈vz〉| ∼ 5 − 20 km s−1); the amplitude of these vertical mo-tions is small (but non-zero) near the mid-plane and it in-creases with distance from the mid-plane (Debattista 2014).The relative sense of the bulk motions, whether compressingor expanding changes across the corotation resonance (here-after CR) of the spiral. Inside the CR, the vertical motionsare compressive behind the peak of the spiral and expandingahead of the spiral’s peak. Outside the CR, the compressiveand expanding breathing modes reverse their sense of occur-rence (Debattista 2014). Similar changes in the sense of theoccurrence of compressive and expanding vertical motionsacross the CR has also been reported in the test-particlesimulations of Faure et al. (2014).

In this paper, we continue to explore the possibility thatthe breathing modes observed in Gaia DR2 are caused bythe Milky Way’s spiral structure. Using a high-resolution,self-consistent, star-forming simulation of a Milky Way-likegalaxy, we study the characteristics of Solar Neighbourhoodbreathing modes driven by spiral structure. Previous studiesof breathing modes either used perturbation theory (Monariet al. 2016) and test particle simulations (Faure et al. 2014)or used self-consistent simulations but lacked the high massresolution necessary to sub-sample the stellar populations(Debattista 2014). The novelty of our paper, aside from com-paring directly with the Gaia DR2 data, lies in the fact thatour high-resolution model allows us to explore how differentpopulations participate in the breathing modes.

The paper is organised as follows: Section 2 provides the

details of simulation, including quantifying the spiral struc-ture present at our chosen snapshot, which, for simplicity,we refer to as ‘the model’. Section 3 presents the breath-ing modes of the model. Section 4 compares the verticalkinematic signatures of Milky Way stars with the trendsobtained in the model. Section 5 discusses the unique fin-gerprint of spiral-driven vertical breathing motions, whileSection 6 summarises our main conclusions.

2 STAR-FORMING SIMULATION

We use an N-body+smooth particle hydrodynamics (SPH)simulation with gas and star formation to motivate some ofthe analysis of the Gaia data. This is a higher mass resolu-tion version of the models described in our previous papers(e.g. Roskar et al. 2008; Loebman et al. 2016). This modelstarts with a gas corona in pressure equilibrium with a co-spatial Navarro-Frenk-White (Navarro et al. 1996) dark mat-ter halo which constitutes 90% of the mass. The dark mat-ter halo has a virial radius r200 ' 200 kpc and a virial massM200 = 1012 M�. Gas velocities are given a spin of λ = 0.065(Bullock et al. 2001), with specific angular momentum pro-portional to radius. Both the gas corona and the dark matterhalo consist of 5 × 106 particles; gas particles have softeningε = 50 pc, while that of dark matter particles is ε = 100 pc.All stars form out of cooling gas, inheriting the softening ofthe parent gas particle. We evolve the simulation for 13 Gyrwith gasoline (Wadsley et al. 2004, 2017). As gas cools itsettles into a disc; once the gas density exceeds 0.1 cm−3

and the temperature drops below 15,000 K, star formationand supernova blast wave feedback commence (Stinson et al.2006). Supernovae feedback couples 40% of the 1051 erg persupernova to the interstellar medium as thermal energy. Gasmixing uses turbulent diffusion as described by Shen et al.(2010).

We use a base timestep of ∆t = 5 Myr with timestepsrefined such that δt = ∆t/2n < η

√ε/ag, where we set the

refinement parameter η = 0.175. We set the opening angleof the tree-code gravity calculation to θ = 0.7. Gas particletimesteps also satisfy the condition δtgas = ηcourant h/[(1 +α)c+ βµmax], where ηcourant = 0.4, h is the SPH smoothinglength set over the nearest 32 particles, α and β are the linearand quadratic viscosity coefficients and µmax is described inWadsley et al. (2004). Star particles represent single stellarpopulations with a Miller-Scalo initial mass function.

2.1 Spiral structure in the model

We use a snapshot from this model at t = 12 Gyr. We choosethis snapshot strictly because it allows a good division ofits stellar populations, not because of any special propertiesof the model at this time. While the simulation we use hasnot been published before, the general evolution of the spiralstructure in this class of models is explored in Roskar et al.(2012).

In order to facilitate comparison with the Milky Way,we quantify the spiral amplitude in the model to show thatthe spirals in the model are not unreasonably strong. We alsoquantify the strength of the spiral as a function of stellar age.We divide the stars into three broad age bins: a young stellar

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Spiral driven breathing modes in Milky Way 3

population (with ages between 0-4 Gyr), an intermediate-age stellar population (with ages between 4-8 Gyr), and anold stellar population (with ages between 8-12 Gyr). Weuse these age bins for the model throughout our analysis.The top panels of Fig. 1 shows the surface density of starsin the face-on, (x, y), view for the three age populations.The bottom panels show the residual surface density, Σ(R, φ),calculated as:

Σ(R, φ) =Σ(R, φ) − Σavg(R)Σavg(R)

, (1)

where Σavg(R) is the average density at radius R, and theaveraging is carried out over the whole azimuthal range.

The density plots of Fig. 1 show that the youngest pop-ulation exhibits a very prominent two-armed spiral struc-ture, which is also present in the intermediate-age popula-tion, but is indistinct in the old population. However theresidual density plots show that the m = 2 spiral is presenteven in the old population, albeit weaker. This therefore isa genuine density wave which propagates in all stellar pop-ulations. The estimated pitch angle, γ, of the spiral in thissnapshot is ∼ 33◦. In comparison, the Milky WayaAZs spi-rals are tighter, having a pitch angle γ ∼ 10◦ (Siebert et al.2012).

To quantify the strength of the spirals, we measure theradial profiles of the Fourier moments of the surface density,defined as:

Am/A0(R) =∑i mieimφi∑

i mi(2)

where Am is the coefficient of the mth Fourier moment of thedensity distribution, mi is the mass of the ith particle and φiis its cylindrical angle. The peak of the Fourier m = 2 ampli-tude, (A2/A0)max ' 0.19 for our model. In comparison, Rix &Zaritsky (1995) found that, for their sample of galaxies withm = 2 spirals, the peak value of A2/A0 varies in the range 0.15to 0.6, with ∼ 28% of their sample having (A2/A0)max ≤ 0.2.The spiral structure in the model therefore is sufficientlyrealistic to provide a basis for comparing the kinematic sig-nature of the spiral in the model with that in the Milky Way.The amplitudes for the m = 2 and m = 4 Fourier harmon-ics for all three stellar populations at t = 12 Gyr are shownin Fig. 2. The spiral has a significant m = 2 amplitude at4 ≤ R/ kpc ≤ 8. While the amplitude decreases with stellarage, all stellar populations take part in the spiral structure,as opposed to material spirals present in only the youngstellar populations (for detailed discussion, see e.g. Binney& Tremaine 2008; Dobbs & Baba 2014). The m = 4 ampli-tude is likewise strongest in the young population, and isalmost zero for the old population. Thus the spiral in theold population varies more gently in azimuth while that inthe young population is more sharply delineated, as can beseen in Fig. 1.

We also calculate the residual density distributionδΣ(φ, z) within a radial annulus of width ∆R as

δΣ(φ, z;∆R) =Σ(φ, z;∆R) − Σavg(z;∆R)

Σavg(z;∆R) , (3)

where Σavg(z;∆R) is the average density, and the averaging iscarried out over the whole azimuthal range within the radialannulus of radial width ∆R, at a fixed height z. Fig. 3 showsthe resulting residual maps in the (φ, z) plane at different an-nuli of ∆R = 1 kpc. At each annulus, the peak density in the

mid-plane (z = 0) is ahead (relative to the sense of rotation)of the peak density at larger heights from the mid-plane.Also, the density distribution at fixed R and z is azimuthallyasymmetric relative to the peak density, in agreement withthe findings of Debattista (2014). The spiral extends to thefull vertical extent, rather than being limited to close to themid-plane. This behaviour was also found in pure N-bodysimulations by Debattista (2014), and will be important forour subsequent analysis.

3 BREATHING MODES INDUCED BYSPIRALS

We start by studying the breathing modes induced by thespirals in the star-forming model in order to explore howthe breathing motions vary between different age popula-tions. Debattista (2014) showed that the amplitudes of thebreathing motion increases with height from the mid-plane;here we extend that earlier work but exploring how the am-plitudes change as a function of both height and stellar age.

We define the breathing velocity, Vbreath, as (Debattista2014; Widrow et al. 2014; Gaia Collaboration et al. 2018b):

Vbreath(x, y) =12[〈vz (x, y,∆z)〉 − 〈vz (x, y,−∆z)〉] , (4)

where 〈vz (x, y,∆z)〉 is the mean vertical velocity at position(x, y) in the galactocentric cartesian coordinate system, av-eraged over a horizontal layer of thickness ∆z (for details seeGaia Collaboration et al. 2018b). We choose ∆z = 400pc, andcalculate Vbreath in three such vertical layers: |z | = [0, 400] pc,|z | = [400, 800] pc and |z | = [800, 1200] pc. In this definitionof Vbreath, the average in a particular layer is carried out be-fore the difference between the two sides is taken. In theobservational data this helps reduce the effect of any selec-tion function differences between the two sides of the disc.Eqn. 4 implies that when Vbreath > 0 then stars are coher-ently moving away from the mid-plane (expanding breath-ing mode), while Vbreath < 0 when stars are moving towardsthe mid-plane (compressing breathing mode). For the sakeof comparison, we also consider the bending velocity:

Vbend(x, y) =12[〈vz (x, y,∆z)〉 + 〈vz (x, y,−∆z)〉] . (5)

The star-forming model, which is not externally perturbed,lacks substantial coherent bending waves, so the model’sbending velocities are not shown here. But the Milky Waydoes have significant bending motions, and the comparisonof these with the breathing motions provides an importantclue to their origin. The variation of Vbend(x, y) with stellarage and height for the Milky Way is discussed in Section 4.2.

3.1 Bulk vertical motions of the different agepopulations

We examine the star-forming model’s vertical motions asa function of stellar age. The distribution of the bulk ver-tical motions in the meridional, (R, z), plane, calculated atfour different azimuthal wedges, for the three stellar pop-ulations at t = 12 Gyr, are shown in Fig. 4. Large-scale,non-zero vertical motions are present in all three popula-tions. The absolute values of the mean vertical velocity,|〈vz〉|, are small, ∼ 5 km s−1, but non-zero, increasing with

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Figure 1. Star-forming model: distribution of stellar surface density (top panels), and the residual surface density (bottom panels),calculated using Eqn. 1 at t = 12 Gyr for (left to right) young (ages = 0− 4 Gyr), intermediate (ages = 4− 8 Gyr), and old (ages = 8− 12 Gyr)populations. Contours of constant surface density (black solid lines) are overlaid on the surface density maps. Unambiguous spiral

structure is present in the young and intermediate age populations, and the residuals reveal that the old population also supports thespiral. The sense of rotation is towards increasing φ.

height from the mid-plane. There are not enough stellarparticles in the young population at large heights to mea-sure a reliable mean vertical velocity for them there. Theintermediate-age population exhibits a large |〈vz〉| at largerheights; |〈vz〉| gets somewhat weaker for the old stellar pop-ulation. All three populations are coherently moving awayfrom, or towards, the disc mid-plane indicating breathingmodes predominate in all stellar populations. There are alsobending modes present, but these are weaker compared tothe breathing modes. (These bending modes will be studiedin detail elsewhere, Khachaturyants et al. in progress). Theintermediate-age population shows prominent bulk verticalbreathing mode motions at those locations which containthe densest parts of the spirals, e.g., at 90◦ ≤ φ ≤ 180◦ and−90◦ ≤ φ ≤ 0◦ for radial extent 4 ≤ R/ kpc ≤ 9. However, theazimuthal wedges used in Fig. 4 are large, therefore the coin-cidence of strong breathing motion with the spiral structureis not readily apparent in this figure.

We then select stars in the radial annulus 5 ≤ R/ kpc ≤7, calculate 〈vz〉 in the (φ, z) plane for the different stellarpopulations, and plot these in Fig. 5. Non-zero bulk vertical

motions are evident in all the stellar populations, althoughthe young population particle numbers are not large enoughat the uppermost slice to measure a meaningful 〈vz〉. Theazimuthal locations where the spirals are present exhibitlarger |〈vz〉| (compare with Fig. 1). These results indicatebreathing modes are present in this annulus. The verticalmotions switch from compressive to expanding at φ ' 0◦,which Fig. 1 shows is the location of the peak density in theyoungest population. The amplitude of the vertical motionsdecreases with age, with the old population having velocitiesroughly half those of the intermediate population.

3.2 Breathing modes of the different agepopulations

We therefore compute the breathing velocities for starsof different ages in three vertical slices: |z | = [0, 400] pc,|z | = [400, 800] pc, and |z | = [800, 1200] pc. We calculateVbreath using Eqn. 4 separately for each age population. Fig. 6shows Vbreath in the (x, y) plane whereas Fig. 7 shows Vbreathin the (R, φ) plane. Strong breathing modes are present in all

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Spiral driven breathing modes in Milky Way 5

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Figure 2. Star-forming model: radial profile of m = 2 (toppanel) and m = 4 (bottom panel) Fourier moments of the surface

density for the three stellar populations at t = 12 Gyr. The nearly

zero A4 in the old population indicates that the spiral in thesestars is more gently varying with azimuth than in younger pop-

ulations. Note the change of vertical axis scale between the two

panels. The black dashed lines denote the Fourier moments forall stars.

three age populations; Vbreath has small amplitude near themid-plane and increases with distance from it. This trend ispresent for all ages; at the largest heights the intermediate-age population has the strongest Vbreath amplitude, but theyoungest population does not reach these heights so can onlybe compared in the intermediate layer. The intermediatepopulation at the largest height shows that the compressivemotions are closely associated to the peak density of the spi-ral in the region corresponding to the Solar Neighbourhood.

In summary, we find large-scale, non-zero vertical mo-tions present both in the meridional plane as well as inthe (φ, z) plane at a fixed radial annulus. These non-zerobulk vertical motions appear preferentially at azimuthal lo-cations which contain the strongest spiral structure. The re-sulting breathing velocity, Vbreath, is small near the disc mid-plane, and increases with height, as found also by Debattista(2014). The breathing velocity is largest for the intermedi-ate age population at the largest heights, because the youngpopulation does not reach such heights; in the middle layerthe young population has the largest Vbreath, while the oldpopulation has the smallest.

4 COMPARISON WITH VERTICAL MOTIONSIN THE MILKY WAY

We now search for these signatures of spiral-driven breathingmodes in the Gaia DR2 data as a function of stellar ages,and compare with the results obtained from the star-formingmodel.

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Figure 3. Star-forming model: residual surface density,

δΣ(φ, z;∆R), calculated using Eqn. 3, shown in the (φ, z) planeat five different radial annuli of width ∆R = 1 kpc. The contours

indicate constant residual surface density, with the red contours

denoting δΣ(φ, z;∆R) = 0. The sense of rotation is towards in-creasing φ.

4.1 Sample selection of Milky Way stars

We use the publicly available sample of stellar ages ofSanders & Das (2018) (hereafter, SD18), which provides acatalogue of distance, mass, and age for approximately 3 mil-lion stars in the Gaia DR2. This sample allows us to studythe vertical motions for different age populations. The prob-ability distributions for distance, mass, and ages for each starwere calculated by SD18 via a Bayesian framework, based onthe photometric and spectroscopic parameters from a varietyof ground-based surveys (APOGEE, Gaia-ESO, GALAH,LAMOST, RAVE, and SEGUE), and the astrometric in-formation from Gaia DR2, simultaneously (for details seesection 2 of Sanders & Das 2018). This dataset includes thesix-dimensional Galactocentric cylindrical position-velocitycoordinates, with the Sun placed at a radius R0 = 8.2 kpcand a height above the plane z0 = 15 pc (Bland-Hawthorn &Gerhard 2016), and the values of the Solar peculiar motionsfrom Schonrich et al. (2010).

We choose stars in the radial range from 4 kpc to 13 kpc,azimuthal angle (as measured from the line joining the Sunand the Galactic Centre) between −30◦ to 30◦, and verticalheight above the mid-plane of |z | ≤ 3 kpc, which is similarto the region considered in the original Gaia DR2 kinemat-

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Figure 4. Star-forming model: distribution of bulk vertical velocity, 〈vz 〉, in the meridional, (R, z), plane, for the three different stellar

populations at t = 12 Gyr. Left to right: stars age = 0 − 4 Gyr (young), = 4 − 8 Gyr (intermediate), and 8 − 12 Gyr (old). Top to bottom:stars at different azimuthal wedges are shown. The age bins and the azimuthal wedges are listed above each sub-plot. Contours of thetotal density are overlaid in each panel.

ics presentation paper (see Gaia Collaboration et al. 2018b).Each stellar entry in the SD18 dataset is characterised bythree flags that, depending on the nature of the analysis,can aid in filtering out problematic data. The flag en-try characterises all issues related to the quality of the in-put data (spectroscopy, astrometry, photometry) from theabove listed surveys and of the output data from the SD18Bayesian frameworks (mass, age). If there are no issues inthe input or output data, then flag = 0. The duplicated

flag specifies if stellar entries are duplicates due to a cross-match within a single survey or between multiple surveys.Within a single survey, the duplicate with the smallest ver-tical velocity uncertainty is kept, while duplicates betweensurveys are kept based on the flag entry and a survey hi-erarchy set in SD18 (APOGEE, GALAH, GES, RAVE-ON,RAVE, LAMOST and SEGUE). For example, if APOGEEand RAVE contain a duplicate entry with proper input andoutput data (flag = 0) then, regardless of the relative er-

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Figure 5. Star-forming model: distribution of 〈vz 〉 in the

(φ, z) plane, for stars in the radial annulus 5 ≤ R/ kpc ≤ 7at t = 12 Gyr. Top: the young stellar population, middle: theintermediate-age stellar population, and bottom: the old stellar

population. The presence of large-scale, non-zero bulk vertical

velocities for stars in all three populations is visible, and are co-herent across age. The vertical motions are smallest in the old

population. The sense of rotation is towards increasing φ.

rors, the APOGEE entry is kept and labelled as duplicated= 0. Lastly, the best flag in SD18 encompasses the two priorflags and is 1 when flag = 0, duplicated = 0 and the starhas a valid cross-match in Gaia. Our selected sample forstars with best = 1 contains a total of 3,318,118 stars, andthough SD18 list the possibility of additional cuts based onwhich survey is being used, we consider the best = 1 cut tobe sufficient for our analysis.

Fig. 8 presents how the selected sample of stars is di-vided into three equal-sized populations by cutting the cu-mulative distribution function, F(age), of the stellar ages.The resulting sub-populations are: young (0 ≤ F(age) <1/3), intermediate (1/3 ≤ F(age) < 2/3), and old (2/3 ≤F(age) < 1) stellar populations, or, in terms of the stellarages, [0, 4.5] Gyr (young), [4.5, 6.9] Gyr (intermediate), and[6.9, 12.6] Gyr (old). Fig. 9 shows the distributions of these

stellar populations in the Galactocentric (x, y) plane, as wellas in the (R, z) plane.

4.2 Vertical motions for different ages

Fig. 10 shows the mean vertical velocity distribution in the(R, z) plane for the three stellar populations. Non-zero meanvertical motion is seen in all three stellar populations. Theamplitudes of the bulk vertical velocity is small near themid-plane, but at |z | ≥ 1 kpc, stars on either side of themid-plane are moving in opposite directions, i. e., with anon-zero Vbreath, while at moderate heights all populationsexhibit bending modes (i. e., stars on either side of the mid-plane are moving in the same direction).

We compute Vbreath in three vertical slices: |z | =[0, 400]pc, |z | = [400, 800]pc, and |z | = [800, 1200]pc. We com-pute the uncertainty on Vbreath, which is shown in Fig. 11,as:

ε(Vbreath) =12

√[ε2(〈vz (x, y,∆z)〉) + ε2(〈vz (x, y,−∆z)〉)

]. (6)

The uncertainties on 〈vz (x, y,∆z)〉 are given by

ε(〈vz (x, y,∆z)〉) = σvz (x, y,∆z)/√

N(x, y,∆z) , (7)

where N(x, y,∆z) denotes the number of stars in any (x, y,∆z)bin, and σvz is the velocity dispersion in vz . A similar expres-sion holds for ε(〈vz (x, y,−∆z)〉). We note that the uncertaintyon the bending velocity is identical to that in the breathingvelocity, ε(Vbend) = ε(Vbreath).

Fig. 12 shows these breathing motions. Prominentbreathing modes are present in all three age populations.Near the mid-plane, Vbreath is small. In the intermedi-ate layer, focusing on the broad region with ε(Vbreath) <1.5 km s−1, we see that Vbreath reaches the largest values,>∼ 3 km s−1, in the young population and the smallest in theold population, as predicted by the model. In the uppermostlayer, Vbreath is large over a broader swathe of the galaxy,with Vbreath > 1 km s−1 in a larger area in the young popula-tion and decreasing with age. However, due to the relativelysmall number of stars in our sample, it is not possible toachieve uniform Vbreath maps for all three stellar populationsat the largest |z |.

Fig. 13 shows the bending motions in the same threevertical slices and for the same stellar ages. Prominent bend-ing motions are seen in all three stellar populations. Unlikethe breathing motions, the bending motions do not varysignificantly either with height from the mid-plane, or withstellar ages.

5 SIGNATURES OF SPIRAL DRIVENBREATHING MODES

Using a star-forming simulation we have shown that a spiraldensity wave drives vertical breathing motions. The increas-ing amplitude of the breathing velocity with height abovethe mid-plane, previously reported by Debattista (2014), isconsistent with the trend seen in the Milky Way (Gaia Col-laboration et al. 2018a). This behaviour is an important clueto the source of breathing motions: it must be something lo-cal to affect stars near the mid-plane differently from thosefurther up. The reason why spirals have this effect is easy

MNRAS 000, 1–15 (2020)

8 Ghosh et al.

10

5

0

5

10

y [kp

c]

[0,4] Gyr [0,400] pc [4,8] Gyr [0,400] pc [8,12] Gyr [0,400] pc

10

5

0

5

10

y [kp

c]

[0,4] Gyr [400,800] pc [4,8] Gyr [400,800] pc [8,12] Gyr [400,800] pc

10 5 0 5 10x [kpc]

10

5

0

5

10

y [kp

c]

[0,4] Gyr [800,1200] pc

10 5 0 5 10x [kpc]

[4,8] Gyr [800,1200] pc

10 5 0 5 10x [kpc]

[8,12] Gyr [800,1200] pc

-5.0

-3.0

-1.0

1.0

3.0

5.0

Vbreath

[km

s−

1]

Figure 6. Star-forming model: breathing velocity, Vbreath, for stars of different ages at different vertical distances from the mid-plane,shown at t = 12 Gyr. From left to right: stars with ages 0 − 4 Gyr (young), 4 − 8 Gyr (intermediate), and 8 − 12 Gyr (old), respectively,

whereas from top to bottom: stars at |z | = [0, 400] pc, at |z | = [400, 800] pc, and at |z | = [800, 1200] pc, respectively. Contours of total

surface density are overlaid in each panel. The dashed circles (in maroon) indicate galactic radii ranging from 2 kpc to 10 kpc, in 2 kpcsteps.

to understand: on vertical scales small compared to the discscalelength, the vertical gravitational field can be approxi-mated, using Gauss’s law, as a uniform field from an infinitesheet of mass corresponding to the mass enclosed within ancylinder of the same height and small radius. Both the sim-ulation of Debattista (2014) and the one considered here

show that the spiral density wave extends to large heights(see Fig. 3). If the density of the disc is decomposed into anaxisymmetric part and a spiral perturbation part:

ρ(R, φ, z) = ρ0(R, z) + ρS(R, φ, z), (8)

then the vertical velocities are in equilibrium with the ax-isymmetric part of the potential and ρ0(R, z) gives rise

MNRAS 000, 1–15 (2020)

Spiral driven breathing modes in Milky Way 9

120

60

0

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120

φ [d

eg.]

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1.250

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0

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02.

750

3.00

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0

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500

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03.

000

[8,12] Gyr [0,400] pc

1.250

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0

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02.

750

3.00

0

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120

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eg.]

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1.250

1.250

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0

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750

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1.2501.2

50

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0

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eg.]

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1.250

1.250

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0

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02.

750

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0

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[4,8] Gyr [800,1200] pc

1.250

1.250

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750

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[8,12] Gyr [800,1200] pc

1.2501.2

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0

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2.50

02.

750

3.00

0 7

5

3

1

1

3

5

7

Vbreath

[km

s−

1]

Figure 7. Star-forming model: Same as Fig. 6, but shown in the (R, φ) plane. The age bins and the vertical layers are presented aboveeach panel. Contours of total surface density (in log scale) are overlaid in each panel. Large |Vbreath | is associated with density peaks and

troughs of the spiral.

to Vbreath = 0. Since the azimuthally varying part, ρS , isdue to spirals, it extends to large height. The enclosedmass therefore increases with height as the vertical inte-gral of ρS(R, φ, z). Therefore the gravitational field increaseswith distance from the mid-plane, and Vbreath increases withheight, as we see in the simulations, and in the Milky Way(see Figs. 6 and 12). It is noteworthy that this verticalincrease in Vbreath is absent in the model of Faure et al.

(2014), who used an analytic spiral potential with a veryshort scale height of 100 pc, much shorter than that of theunperturbed density distribution. As a result, the gravita-tional field reaches nearly constant value quite close to themid-plane in their model, and Vbreath does not increase withheight. The vertical variation of Vbreath in the Gaia datatherefore indicates that the spiral structure in the MilkyWay is vertically extended. The vertical increase of Vbreath is

MNRAS 000, 1–15 (2020)

10 Ghosh et al.

0.0 2.5 5.0 7.5 10.0 12.5Age [Gyr]

0

10000

20000

30000

40000

50000

60000

# st

ars

SD18

0.0 2.5 5.0 7.5 10.0 12.5Age [Gyr]

0.0

0.2

0.4

0.6

0.8

1.0

F(ag

e)

F(age) = 1/3

F(age) = 2/3

SD18

Figure 8. SD18 sample: histogram of stellar ages (left panel) and the corresponding cumulative distribution function (CDF), F(age)of stellar ages (right panel) of our selected sample from the SD18 dataset. The black dashed lines in the right panel indicate the cutsapplied on the CDF whereas the vertical blue dashed lines indicate the corresponding stellar ages.

different to the nearly constant Vbend as a function of height(compare Figs. 12 and 13). This indicates that the MilkyWay’s bending motions are probably not excited by spirals.

Moreover we examined the dependence of the breath-ing velocity on stellar age. The star-forming model exhibits adecreasing Vbreath with increasing stellar age indicating thatincreasing random motions decrease the vertical response ofthe stars. Our sample of stars with Gaia DR2 kinematics andages from the Sanders & Das (2018) catalogue have, at inter-mediate heights, Vbreath which is largest in the young popula-tion and smallest in the old. Young stars are not present ateven larger heights in enough numbers to reliably measureVbreath but the intermediate age population still has largerVbreath than the old one.

For these reasons we therefore argue that the breathingmotions seen in the Milky Way are very likely to be drivenby the spiral density wave structure. Widrow et al. (2014)proposed a scenario where a satellite galaxy, while plung-ing into the disc, can generate both bending and breathingmotions, depending on the value of the satellite’s velocityin the direction perpendicular to the disc. Such interactionsalso excite a strong spiral response within the disc which areprobably partly responsible for the breathing modes seen inthe simulations of Widrow et al. (2014). The vertical vari-ation of the amplitude of the breathing modes excited bysatellites is also unclear.

6 FUTURE PROSPECTS AND SUMMARY

Our requirement of stellar ages limits the number of starsavailable to us by almost a factor of two when comparedwith the Gaia main sample (which contains 6,376,803 stars;Gaia Collaboration et al. 2018b). In addition, the azimuthalcoverage in Gaia DR2 is not large (∆φ ∼ 30◦). This, in turn,prevents us from exploring the change from expanding tocompressing breathing motions as a function of azimuthalangle, which would help constrain the extent of the spi-

ral and its corotation radius (Debattista 2014; Faure et al.2014). Gaia DR3 holds the potential to permit us to inves-tigate the variation of breathing motions as a function ofazimuth, in order to constrain the Milky Way’s spiral struc-ture better.

Our main findings are:

• The spirals in the self-consistent star-forming modeldrive coherent vertical breathing motions, with amplitudewhich increases with the height from the disc mid-plane.We argue that this vertical variation is a consequence of thevertically extended spiral structure. Since Gaia DR2 alsoexhibits this increasing breathing amplitude with height, itimplies that the Milky Way’s spirals are also vertically ex-tended.

• Prominent breathing motions are present in stellar pop-ulations of all ages in the model, with the strongest motionsin the young stellar population, and weakest in the old one.

• Using Gaia DR2 with complementary age data fromSanders & Das (2018), we showed that breathing motionsare present for all ages in the Milky Way too. The amplitudeof the breathing motions decreases with stellar age, as in thesimulation model. This behaviour is consistent with spiralsbeing the driving mechanism of breathing modes.

• The bending motions in the Milky Way are very similarat all heights, which argues for a more distant source of thesemotions (e.g. satellite interactions, the bar, or the warp),rather than spirals. Moreover, the bending motions appearbarely affected by the stellar age.

The resemblance in the height and age variation of theamplitude of breathing velocity, in the star-forming modeland in the Gaia DR2 data, indicates that the observedbreathing modes in the Milky Way are likely to be excitedby spiral density waves.

MNRAS 000, 1–15 (2020)

Spiral driven breathing modes in Milky Way 11

4 6 8 10 12x [kpc]

5.0

2.5

0.0

2.5

5.0

y [kp

c]

young

4 6 8 10 12x [kpc]

5.0

2.5

0.0

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5.0intermediate

4 6 8 10 12x [kpc]

5.0

2.5

0.0

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5.0old

1

2

3

4

# st

ars (

log

scal

e)

4 6 8 10 12 14R [kpc]

3

2

1

0

1

2

3

z [kp

c]

4 6 8 10 12 14R [kpc]

4 6 8 10 12 14R [kpc]

0.00.51.01.52.02.53.03.54.0

# st

ars (

log

scal

e)

Figure 9. SD18 sample: distribution of our selected sample from SD18 is shown in the (x, y) plane (top panels) and in the (R, z) plane

(bottom panels) for three stellar populations: young (left panel), intermediate (middle panel), and old (right panel) stellar populations

(for details see section 4.1).

4 6 8 10 12 14R [kpc]

3

2

1

0

1

2

3

z [kp

c]

young

4 6 8 10 12 14R [kpc]

intermediate

4 6 8 10 12 14R [kpc]

old

9.9

6.6

3.3

0.0

3.3

6.6

9.9

〈vz〉 [

km s−

1]

Figure 10. SD18 sample: mean vertical velocity, 〈vz 〉, for different stellar populations in the (R, z) plane. From left to right: stars withages 0 − 4.5 Gyr (young), 4.5 − 6.9 Gyr (intermediate), and 6.9 − 12.6 Gyr (old), are shown (for details see section 4.1). A mixture of

bending and breathing modes are evident in the bulk vertical motions of each of these populations.

MNRAS 000, 1–15 (2020)

12 Ghosh et al.

4

2

0

2

4

y [kp

c]

young [0,400] pc intermediate [0,400] pc old [0,400] pc

4

2

0

2

4

y [kp

c]

young [400,800] pc intermediate [400,800] pc old [400,800] pc

4 6 8 10 12x [kpc]

4

2

0

2

4

y [kp

c]

young [800,1200] pc

4 6 8 10 12x [kpc]

intermediate [800,1200] pc

4 6 8 10 12x [kpc]

old [800,1200] pc

0.0

0.5

1.0

1.5

2.0

4.0

ε(Vbreath

) [km

s−

1]

Figure 11. SD18 sample: uncertainty in the breathing velocity, ε (Vbreath), for stars of different ages at different vertical distances fromthe mid-plane. From left to right: stars with ages 0 − 4.5 Gyr (young), 4.5 − 6.9 Gyr (intermediate), and 6.9 − 12.6 Gyr (old) (for details see

section 4.1), whereas from top to bottom: stars at |z | = [0, 400] pc, at |z | = [400, 800] pc, and at |z | = [800, 1200] pc.

ACKNOWLEDGEMENT

We thank the anonymous referee for useful commentswhich helped to improve this paper. S.G. acknowledges sup-port from an Indo-French CEFIPRA project (Project No.:5804-1). V.P.D. is supported by STFC Consolidated grantST/R000786/1. The simulation in this paper was run atthe DiRAC Shared Memory Processing system at the Uni-versity of Cambridge, operated by the COSMOS Projectat the Department of Applied Mathematics and Theoret-ical Physics on behalf of the STFC DiRAC HPC Facility(www.dirac.ac.uk). This equipment was funded by BIS Na-

tional E-infrastructure capital grant ST/J005673/1, STFCcapital grant ST/H008586/1, and STFC DiRAC Operationsgrant ST/K00333X/1. DiRAC is part of the National E-Infrastructure.

DATA AVAILABILITY

The simulation data underlying this article will be sharedon reasonable request to V.P.D (vpdebattista@gmail.com).The Sanders & Das dataset is publicly available from thisURL.

MNRAS 000, 1–15 (2020)

Spiral driven breathing modes in Milky Way 13

4

2

0

2

4

y [kp

c]

young [0,400] pc

0.50

0

0.50

0

1.000

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intermediate [0,400] pc

0.5001.000

1.000

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old [0,400] pc

0.50

0

0.500

1.000

1.500

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2

0

2

4

y [kp

c]

young [400,800] pc

0.500

0.500

0.500

1.000

1.000

1.00

0

1.000

1.00

0

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01.5

00

2.00

0

intermediate [400,800] pc0.500

0.500

0.500

1.000

1.000

1.000

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1.500

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old [400,800] pc

0.500

0.500

0.500

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0

1.5001.500

2.000

2.000

2.000

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4

2

0

2

4

y [kp

c]

young [800,1200] pc

0.500

0.500

0.500

1.0001.000

1.500

1.5001.5002.0

002.

000

4 6 8 10 12x [kpc]

intermediate [800,1200] pc

0.500

0.50

0

0.500

1.000

1.000

1.000

1.5001.500

1.500

1.500

1.500

2.000

2.00

0

2.000

4 6 8 10 12x [kpc]

old [800,1200] pc

0.500

0.500

0.500

1.000

1.000

1.000

1.500

1.500

2.000

2.000

-5.0

-4.0

-3.0

-2.0

-1.0

1.0

3.0

5.0

Vbreath

[km

s−

1]

Figure 12. SD18 sample: breathing velocity, Vbreath, for stars of different ages at different vertical distances from the mid-plane. Fromleft to right: stars with ages 0−4.5 Gyr (young), 4.5−6.9 Gyr (intermediate), and 6.9−12.6 Gyr (old), respectively (for details see section 4.1),

whereas from top to bottom: stars at |z | = [0, 400] pc, at |z | = [400, 800] pc, and at |z | = [800, 1200] pc, respectively. The black solid

contours indicate ε (Vbreath) ranging from 0.5 km s−1 to 2 km s−1, in 0.5 km s−1 steps (for details see Section 4.2).

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2.000

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Vbend [k

m s−

1]

Figure 13. SD18 sample: bending velocity, Vbend, for stars of different ages at different vertical distances from the mid-plane. From leftto right: stars with ages 0− 4.5 Gyr (young), 4.5− 6.9 Gyr (intermediate), and 6.9− 12.6 Gyr (old), respectively (for details see section 4.1),

whereas from top to bottom: stars at |z | = [0, 400] pc, at |z | = [400, 800] pc, and at |z | = [800, 1200] pc, respectively. The black solid

contours indicate ε (Vbend) ranging from 0.5 km s−1 to 2 km s−1, in 0.5 km s−1 steps (for details see section 4.2).

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