Our Sun lies within 300 parsecs of the 2.7-kiloparsecs-long sinusoidal chain of dense gas clouds known as the Radclife Wave1 . The structure’s wave-like shape was discovered using three-dimensional dust mapping, but initial kinematic searches for oscillatory motion were inconclusive2–7 . Here we present evidence that the Radclife Wave is oscillating through the Galactic plane while also drifting radially away from the Galactic Centre. We use measurements of line-of-sight velocity8 for 12CO and threedimensional velocities of young stellar clusters to show that the most massive star-forming regions spatially associated with the Radclife Wave (including Orion, Cepheus, North America and Cygnus X) move as though they are part of an oscillating wave driven by the gravitational acceleration of the Galactic potential. By treating the Radclife Wave as a coherently oscillating structure, we can derive its motion independently of the local Galactic mass distribution, and directly measure local properties of the Galactic potential as well as the Sun’s vertical oscillation period. In addition, the measured drift of the Radclife Wave radially outwards from the Galactic Centre suggests that the cluster whose supernovae ultimately created today’s expanding Local Bubble9 may have been born in the Radclife Wave.

1. Nature | www.nature.com | 1
Article
TheRadcliffeWaveisoscillating
Ralf Konietzka1,2,3✉, Alyssa A. Goodman1
, Catherine Zucker1,4
, Andreas Burkert3,5
, João Alves6
,
Michael Foley1
, Cameren Swiggum6
, Maria Koller6
& Núria Miret-Roig6
OurSunlieswithin300 parsecsofthe2.7-kiloparsecs-longsinusoidalchainofdense
gascloudsknownastheRadcliffeWave1
.Thestructure’swave-likeshapewas
discoveredusingthree-dimensionaldustmapping,butinitialkinematicsearchesfor
oscillatorymotionwereinconclusive2–7
.HerewepresentevidencethattheRadcliffe
WaveisoscillatingthroughtheGalacticplanewhilealsodriftingradiallyawayfrom
theGalacticCentre.Weusemeasurementsofline-of-sightvelocity8
for12
COandthree-
dimensionalvelocitiesofyoungstellarclusterstoshowthatthemostmassive
star-formingregionsspatiallyassociatedwiththeRadcliffeWave(includingOrion,
Cepheus,NorthAmericaandCygnusX)moveasthoughtheyarepartofanoscillating
wavedrivenbythegravitationalaccelerationoftheGalacticpotential.Bytreating
theRadcliffeWaveasacoherentlyoscillatingstructure,wecanderiveitsmotion
independentlyofthelocalGalacticmassdistribution,anddirectlymeasurelocal
propertiesoftheGalacticpotentialaswellastheSun’sverticaloscillationperiod.
Inaddition,themeasureddriftoftheRadcliffeWaveradiallyoutwardsfromthe
GalacticCentresuggeststhattheclusterwhosesupernovaeultimatelycreatedtoday’s
expandingLocalBubble9
mayhavebeenbornintheRadcliffeWave.
InFig.1a(SupplementaryFig.1),wepresentathree-dimensional(3D)
mapshowingthemostmassivestar-formingregions(includingOrion,
Cepheus, North America and Cygnus X) and embedded young stellar
clusters associated with the Radcliffe Wave1
. As expected, the young
clusters and star-forming molecular clouds are co-located in three
dimensions. The clusters were selected from a top-down view of the
Galaxy,withoutanaprioriselectioncriterionforheightofftheGalactic
plane (Methods), suggesting that the clusters’ stars were born in the
molecularcloudsalongtheRadcliffeWave.Therefore,theseclusters’
motionscanserveasatraceroftheWave’skinematics.Usingthemotion
ofclustersratherthanofindividualstarsallowsformoreprecisecon-
clusions about kinematics, as the motion of individual stars within a
host cluster are averaged out, reducing uncertainty.
The kinematic counterpart of the Radcliffe Wave’s spatial undula-
tion is shown in Fig. 1b (Supplementary Fig. 1), where the z axis shows
the vertical velocity of the stellar clusters, after accounting for the
Sun’smotion10
.Onthebasisofthesinusoid-likeshapeoftheRadcliffe
Wave, one could imagine it as either a travelling wave or a standing
wave. Extended Data Fig. 1 (and the animated version in Supplemen-
taryFig.4)showshowtravellingandstandingversionsoftheRadcliffe
Wavewouldappear.Foratravellingwave,thewave’sspatialshapefully
determinesverticalvelocities,assumingagravitationalpotential11–13
,as
explainedinMethods.Forastandingwave,regionslocatedatthezero
crossingsofthewaveareatrest,andthevelocitiesofregionslocatedat
spatialextremaarenotconstrainedbythewave’sspatialstructure.The
fit shown as a solid black line in Fig. 1 (compare Supplementary Fig. 1
andMethods)indicatesthattheRadcliffeWaveoscillatesthroughthe
Galacticplanelikeatravellingwave,suchthatregionscurrentlyatthe
zeropointsofthewave(neartheGalacticplane)movethroughattheir
greatest vertical velocity, while molecular clouds located at spatial
extrema(farthestfromtheplane)areattheirturningpoints,withzero
verticalvelocity.Whileatravellingwaveprovidesanexcellentfittothe
observations,witha90°phaseoffsetbetweenposition(z)andvelocity
(vz),asevidentinFig.1(andintheinteractiveversioninSupplementary
Fig.1),italsoprovidesanextremumforthewavemotion.Whenfitting
amixturemodelthatallowsforsuperpositionsofatravellingwaveand
a standing wave, we find that a travelling wave is strongly preferred
over a standing wave (Methods). A standing-wave model would have
azerophaseoffsetbetweenpositional(z)andvelocity(vz)variations,
which is inconsistent with the data (Methods).
ForstarsnotfarofftheGalaxy’sdiskcomparedwithitsscaleheight,
aclassicalharmonicoscillatorisoftenusedtodescribestars’responses
to the vertical Milky Way potential14
. By analogy to a pendulum, how-
ever,whentheamplitudeofoscillationislarge,oneneedstoconsider
nonlineareffectsthatarenotaccountedforintheformalismdescribing
apurelyharmonicoscillator.InthecaseoftheGalaxy,thesenonlineari-
tiesarisefromtheverticaldecreaseinthemidplanedensity,becoming
significant at vertical positions on the order of the Galaxy’s vertical
scaleheight.Therefore,indescribingthemotionofapendulumoscil-
latingfarfromvertical,wemodelthewave’smotionwithintheGalactic
potential11–13
asananharmonicoscillator.Inthemodelling,weaccount
for the Galaxy’s midplane density declining with height and also with
galactocentricradius(Methods).TreatingtheRadcliffeWaveasasingle
coherent structure in space and velocity, responding to the Galactic
potential,thestructureiswellmodelledasadampedsinusoidalwave
withamaximumamplitudeofabout220 pcandameanwavelengthof
about 2 kpc. The corresponding maximum vertical velocity is about
14 km s−1
.
https://doi.org/10.1038/s41586-024-07127-3
Received: 23 June 2023
Accepted: 26 January 2024
Published online: xx xx xxxx
Check for updates
1
Harvard University Department of Astronomy and Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA, USA. 2
Ludwig-Maximilians-Universität München, Munich, Germany.
3
Max Planck Institute for Extraterrestrial Physics, Garching, Germany. 4
Space Telescope Science Institute, Baltimore, MD, USA. 5
University Observatory Munich, Munich, Germany. 6
Department
of Astrophysics, University of Vienna, Vienna, Austria. ✉e-mail: ralf.konietzka@cfa.harvard.edu

2. 2 | Nature | www.nature.com
Article
As described in Methods, the transparency of the blue points in
Fig. 1 are coded by a fitting procedure in which statistical inliers
(opaque)andoutliers(transparent)wereexplicitlytakenintoaccount.
WefindthatthemostoftheoutliersbelongtothePerseusandTaurus
molecularclouds,whichlieonthesurfaceoffeedbackbubbles9,15
.The
expansionmotionsofthosebubblesprobablyoverwhelmthekinematic
imprint of the wave in the present day.
A schematic view of the vertical oscillation of the stellar clusters in
theRadcliffeWaveisshowninFig.2(andintheinteractiveviewinSup-
plementaryFig.6).InthestaticversionofFig.2,weshowtheexcursion
of the wave at the present day along with the best fit at phases cor-
respondingtotheminimum(60°)andmaximum(240°)deflectionof
thewaveaboveandbelowtheGalacticDisk.Giventhetravellingwave
nature of the Radcliffe Wave, the interactive version of Fig. 2 shows
that the extrema appear to move from right to left as seen from the
Sun. Using the modelled amplitude and wavelength at each end of
thewave,wecancalculatetheso-calledphasevelocitydescribingthis
apparent motion of the wave’s extrema (Methods). We find that the
phase velocity changes from about 40 km s−1
(near Canis Major OB1
(CMa OB1)) to about 5 km s−1
(near Cygnus X).
Inadditiontoitsverticaloscillation,wefindevidencethattheRad-
cliffeWaveisdriftingradiallyoutwardsfromtheGalacticCentrewith
avelocityofabout5 km s−1
.Thisdriftoccursinaco-rotatingreference
framedictatedbyGalacticrotation,modelledbythesameMilkyWay
potentials11–13
used to model the wave’s oscillation. In Extended Data
Fig. 2b (and in the interactive version in Supplementary Fig. 2), we
showthevectorsrepresentingthewave’ssolidbodydriftintheGalac-
tic plane. The direction of the drift bolsters the previously proposed
idea9
that the Radcliffe Wave served as the birthplace for the Upper
Centaurus Lupus and Lower Centaurus Crux star clusters, home to
the supernovae that generated the Local Bubble about 15 Myr ago9
.
An exact traceback of the wave’s position over the approximately
15 Myr since the birth of the Local Bubble would require model-
ling the deceleration of the wave as its dense clouds move through
a lower-density interstellar medium, which is beyond the scope of
this work.
Weconfirmthemotionfoundinthe3Dvelocitiesofthestellarclus-
ters using 12
CO observations of dense clouds along the wave, from
which we can measure line-of-sight velocities8
. As the Radcliffe Wave
(length2.7 kpc)issoclosetous(0.25 kpcattheclosestpoint),thelines
ofsightfromtheSuntovariouscloudsinthewaveareorientedatawide
variety of angles relative to an x, y, z heliocentric coordinate system,
as shown in Extended Data Fig. 3b (and in the interactive version in
Supplementary Fig. 3). Given the Radcliffe Wave’s vertical excursions
a Spatial view
b Kinematic view
–1,500
–1,000
–500
0
500
1,000 –1,000 –500
0 500 1,000 1,500 2,000
200
100
0
–100
–200
–300
15
10
5
0
–5
–10
–15
–1,500
x (pc) y (pc)
–1,500
–1,000
–500
0
500
1,000 –1,000 –500
0 500 1,000
1,500 2,000
–1,500
x (pc) y (pc)
z
(pc)
v
z
(km
s
–1
)
Fig.1|Aspatialandkinematicviewofthesolarneighbourhood.Forthebest
experience,pleaseviewtheinteractiveversioninSupplementaryFig.1(only
oscillation)andSupplementaryFig.5(totalmotion,combiningoscillationand
in-planedrift).Thevariationofthewavewithphase/timeintheinteractive
versionsisobtainedbyevolvingthebestfitwithphase/time(Methods),while
keepingthedistancesofthedatatothefitconstant.Ontimescalesofseveral
tensofmillionyears,theRadcliffeWaveislikelytobeaffectedbyinternal
(molecularcloud destruction28
)andglobal(kinematicphasemixing)effects.
Astheseeffectsmaybesuperimposedonthewave’smotioninthefuture,the
molecularcloudandstarclusterdataintheinteractiveSupplementaryFig.5
arefadingoutovertime.a,A3Dspatialviewofthesolarneighbourhoodin
Galactic Cartesiancoordinates (position–position–positionspace:x, y, z).We
showthemostmassivelocalstar-formingregionsspatiallyassociatedwiththe
RadcliffeWaveinred.AsweidentifythePerseusandTaurusmolecularclouds
asoutliers(Methods),wedonotshowthesecloudshere.Theyoungstellar
clustersareshowninblue(inliersareopaqueandoutliersaretransparent),
theSunisshowninyellowandthebest-fit-modelisshowninblack.Aswehave
nostarclustermeasurementsattherightendofthewaveandtheamplitude
oftheripplesinthisregionapproachestheerrorofthedustdistance
measurements29,30
,thissectionisshowningrey,implyingthatripplesandno
ripplesareindistinguishableatthisendofthewave.b,A3Dkinematicview
ofthesolar neighbourhood(position–position–velocityspace:x, y,vertical
velocityvz).Thecoloursandsymbolsarethesameasina.Forinteractive
versionsofthisfigurethatofferviewsfromanydirection,seeSupplementary
Information,https://faun.rc.fas.harvard.edu/czucker/rkonietzka/radwave/
interactive-figure1.htmlandhttps://faun.rc.fas.harvard.edu/czucker/
rkonietzka/radwave/interactive-figure5.html.

3. Nature | www.nature.com | 3
aboveandbelowtheSun’sposition,evenpurelyverticalcloudveloci-
ties are probed by a line-of-sight velocity measurement. Therefore, if
we treat the wave as a kinematically coherent structure, we can study
3D motion of its gas just by investigating observed ‘one-dimensional’
radial(localstandardofrest(LSR))velocities.Formoredetailsonthe
velocity modelling, see Methods.
Extant observations combined with our modelling can constrain
possible formation mechanisms for the Radcliffe Wave. A gravita-
tional interaction with a perturber seems a natural possible origin16
,
but the wave’s stellar velocities are not fully consistent with models
of a perturber-based scenario3
. In particular, recent studies3,17
have
suggested that the dominant wavelength resulting from such a per-
turbationisanorderofmagnitudelargerthanthepatternweobserve,
challenging this scenario for the wave’s origin. Gas streamers falling
ontothedisk18
potentiallyleadtoshorterwavelengths,butmodelling
hasnotyetbeendonetoseewhetherthisinflowinggascouldoscillate
onscalescommensuratewiththeRadcliffeWave.Internaltothedisk,
a hydrodynamic instability19
may be able to generate waves on the
right scale, but additional work is needed to determine whether such
an instability could push gas about 220 pc above the disk and/or pro-
duceatravellingwave.Asuperpositionoffeedback-drivenstructures
couldreproducetheobservedwavelengthandamplitudeofthewave,
but might require (too much) fine-tuning to also explain the wave’s
travelling nature and order-of-magnitude change in phase velocity.
Insupportoffeedback-drivenscenarios,wenotethatrelevantgalaxy
modelsdoshownearlystraightfilamentsreminiscentofthewave(as
viewedfromthetopdown)driftingradially20
inafashionsimilartothe
observedmotionoftheRadcliffeWave.Analysesoffutureastrometric
andspectroscopicmeasurementsofyoungstarsandhigh-resolution
imagingofexternalgalaxiesinconcertwithimprovedhydrodynamic
simulations of galactic-scale features should be able to discriminate
among potential formation scenarios.
The spatially and kinematically coherent Radcliffe Wave serves as
a unique environment for obtaining insights into local Galactic
dynamics. As explained above, the kinematic and spatial signature
of the Radcliffe Wave can be described in a self-consistent manner,
using its response to the local Galactic potential. Near the Sun, this
potential is dominated by the gravitational attraction of baryonic
matter in the form of stars and gas21
. In addition, dark matter, which
isknowntoenvelopegalaxiesinaspherical‘darkhalo’isalsoexpected
toeffectthelocaldiskkinematics22
.InMethods,weusethecoherent
motion of the Radcliffe Wave, assuming that it is controlled by the
combinedgravityofbaryonicanddarkmatter,toestimatetheprop-
ertiesofthelocalGalacticpotential.Wefindatotalmidplanedensity
of M
0.085 pc
−0.017
+0.021
☉
−3
, consistent with conventional approaches to
deriving properties of the local Galactic mass distribution21,23–26
. By
incorporating direct observations of the gaseous and stellar-mass
components21
,andtherebyconstrainingtheamountofthebaryonic
mass component in the solar neighbourhood, we also infer the local
amountofdarkmatter(Methods).Ouranalysisyieldsalocaldensity
of M
0.001 pc
−0.021
+0.024
☉
−3
for a spherical dark halo. If dissipative dark
matter exists, it is possible that dark matter may also accumulate in
theformofaverythindiskrelativetothebaryonicmasscomponent27
.
Wefindanupperboundforahypotheticaldarkdiskof M
0.1 pc
−3.3
+3.3
☉
−2
.
Simultaneously, we also utilize the Radcliffe Wave to derive the fre-
quencyoftheSun’soscillationthroughtheGalacticplane(Methods).
We find that the Sun oscillates through the disk of the Milky Way
with a period of 95 Myr
−10
+12
, implying that our Solar System crosses
the Galactic Disk every 48 Myr
−5
+6
, consistent with standard values14
(Methods).
The measurements of the oscillation and drift of the Radcliffe
Wave presented here offer constraints on the formation of dense
star-forming structures within the Milky Way, and the possible ori-
ginsofkiloparsec-scalewave-likefeaturesingalaxies.Upcomingdeep
and wide surveys of stars, dust and gas will probably uncover more
wave-like structures, and measurements of their motions should
provide insights into the star formation histories and gravitational
potentials of galaxies.
Fig.2|AviewoftheRadcliffeWaveanditsoscillatorypattern.Thelight
bluecurveshowsthetravellingwavemodelpresentedhereandthebluefuzzy
dotsshowthecurrentpositionsofthestellarclusters.Themagentaandgreen
tracesshowthewave’sminimumandmaximumexcursionsaboveandbelow
theplaneoftheMilkyWay,separatedby180°inphase.Formorephase
snapshots,seeExtendedDataFig.1,whichalsocomparesthemotionofthe
travellingwavewithpredictionsforastandingwave.TheSunisshowninyellow.
Thebackgroundimageisanartist’sconceptionofthesolarneighbourhood,
asseeninWorldWideTelescope.Aninteractivefigurecorrespondingtothis
staticviewisavailableonlineasSupplementaryFig.6:https://faun.rc.fas.
harvard.edu/czucker/rkonietzka/radwave/interactive-figure6/.

4. 4 | Nature | www.nature.com
Article
Onlinecontent
Anymethods,additionalreferences,NaturePortfolioreportingsumma-
ries,sourcedata,extendeddata,supplementaryinformation,acknowl-
edgements, peer review information; details of author contributions
andcompetinginterests;andstatementsofdataandcodeavailability
are available at https://doi.org/10.1038/s41586-024-07127-3.
1. Alves, J. et al. A Galactic-scale gas wave in the solar neighbourhood. Nature 578, 237–239
(2020).
2. Tu, A. J. et al. Characterizing the 3D kinematics of young stars in the Radcliffe Wave.
Astrophys. J. 936, 57 (2022).
3. Thulasidharan, L. et al. Evidence of a vertical kinematic oscillation beyond the Radcliffe
wave. Astron. Astrophys. Suppl. Ser. 660, L12 (2022).
4. Bobylev, V. V., Bajkova, A. T. & Mishurov, Y. N. Parameters of the Radcliffe Wave from
masers, radio stars, and T Tauri stars. Astron. Lett. 48, 434–441 (2022).
5. Li, G.-X. & Chen, B.-Q. Discovery of a coherent, wave-like velocity pattern for the Radcliffe
wave. Mon. Not. R. Astron. Soc. Lett. 517, L102–L107 (2022).
6. Alfaro, E. J. et al. Topography of the young Galactic Disk: spatial and kinematic patterns
of clustered star formation in the solar neighborhood. Astrophys. J. 937, 114 (2022).
7. Donada, J. & Figueras, F. Phase space analysis of the young stellar component of the
Radcliffe Wave. Preprint at https://arxiv.org/abs/2111.04685 (2021).
8. Dame, T. M., Hartmann, D. & Thaddeus, P. The Milky Way in molecular clouds: a new
complete CO survey. Astrophys. J. 547, 792 (2001).
9. Zucker, C. et al. Star formation near the Sun is driven by expansion of the Local Bubble.
Nature 601, 334–337 (2022).
10. Kerr, F. J. & Lynden-Bell, D. Review of galactic constants. Mon. Not. R. Astron. Soc. 221,
1023–1038 (1986).
11. Irrgang, A., Wilcox, B., Tucker, E. & Schiefelbein, L. Milky Way mass models for orbit
calculations. Astron. Astrophys. Suppl. Ser. 549, A137 (2013).
12. McMillan, P. J. The mass distribution and gravitational potential of the Milky Way. Mon.
Not. R. Astron. Soc. 465, 76–94 (2017).
13. Bovy, J. galpy: a Python library for galactic dynamics. Astrophys. J. Suppl. Ser. 216, 29 (2015).
14. Binney, J. & Tremaine, S. Galactic Dynamics 2nd edn (Princeton Univ. Press, 2008).
15. Bialy, S. et al. The Per-Tau shell: a giant star-forming spherical shell revealed by 3D dust
observations. Astrophys. J. Lett. 919, L5 (2021).
16. Tepper-Garcia, T., Bland-Hawthorn, J. & Freeman, K. Galactic seismology: joint evolution
of impact-triggered stellar and gaseous disc corrugations. Mon. Not. R. Astron. Soc. 515,
5951–5968 (2022).
17. Laporte, C. F. P., Johnston, K. V., Gómez, F. A., Garavito-Camargo, N. & Besla, G. The
influence of Sagittarius and the Large Magellanic Cloud on the stellar disc of the Milky
Way Galaxy. Mon. Not. R. Astron. Soc. 481, 286–306 (2018).
18. Trapp, C. et al. Gas infall and radial transport in cosmological simulations of milky
way-mass discs. Mon. Not. R. Astron. Soc. 509, 4149–4170 (2022).
19. Fleck, R. The ‘Radcliffe Wave’ as a Kelvin–Helmholtz instability. Nature 583, E24–E25
(2020).
20. Jeffreson, S. M. R. et al. Momentum feedback from marginally resolved Hii regions in
isolated disc galaxies. Mon. Not. R. Astron. Soc. 505, 3470–3491 (2021).
21. McKee, C. F., Parravano, A. & Hollenbach, D. J. Stars, gas, and dark matter in the solar
neighborhood. Astrophys. J. 814, 13 (2015).
22. Mo, H., van den Bosch, F. & White, S. Galaxy Formation and Evolution (Cambridge Univ.
Press, 2010).
23. Oort, J. H. The force exerted by the stellar system in the direction perpendicular
to the Galactic plane and some related problems. Bull. Astron. Inst. Neth. 6, 249
(1932).
24. Kuijken, K. & Gilmore, G. The Galactic Disk surface mass density and the Galactic force Kz
at z=1.1 kiloparsecs. Astrophys. J. 367, L9–L13 (1991).
25. Zhang, L., Rix, H. W., van de Ven, G. & Bovy, J. The gravitational potential near the sun from
SEGUE K-dwarf kinematics. Astrophys. J. 772, 108 (2013).
26. Widmark, A., Laporte, C. F. P., de Salas, P. F. & Monari, G. Weighing the Galactic disk using
phase-space spirals—II. Most stringent constraints on a thin dark disk using Gaia EDR3.
Astron. Astrophys. Suppl. Ser. 653, A86 (2021).
27. Kramer, E. D. & Randall, L. Updated kinematic constraints on a dark disk. Astrophys. J.
824, 116 (2016).
28. Jeffreson, S. M. R. & Kruijssen, J. M. D. A general theory for the lifetimes of giant molecular
clouds under the influence of galactic dynamics. Mon. Not. R. Astron. Soc. 476, 3688–3715
(2018).
29. Zucker, C. et al. A large catalog of accurate distances to local molecular clouds: the Gaia
DR2 edition. Astrophys. J. 879, 125 (2019).
30. Zucker, C. et al. A compendium of distances to molecular clouds in the Star Formation
Handbook. Astron. Astrophys. Suppl. Ser. 633, A51 (2020).
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© The Author(s), under exclusive licence to Springer Nature Limited 2024

5. Methods
Stellarclustercatalogue
The basis of the stellar cluster analysis comprises a large sample size
catalogue31
that uses Gaia32
data, omitting lower-quality parallaxes,
propermotionsandradialvelocities.Inaddition,weincorporateclus-
ters from other works33–37
that rely on Gaia data, ensuring that only
non-duplicate clusters were included in the final catalogue. We also
create a stellar member catalogue that encompasses all known stars
withintheidentifiedclusters.Weperformedcross-matchingbetween
the stellar members catalogue and the latest Gaia data release (Gaia
Data Release 338
). To enhance the number of radial velocity measure-
ments in our final sample, we also make use of the APOGEE survey39
.
Finally, for each cluster, we compute the mean position (x, y, z) and
velocity (u, v, w) as well as the standard error of the mean, removing
clusterswithkinematicanomalies(radialLSRvelocitiesabove50 km s−1
,
typicallyassociatedwithclusterswithalownumberofradialvelocity
measurements). We adopt a peculiar solar motion10
of (u☉, v☉, w☉) =
(10.0, 15.4, 7.8) km s−1
and correct the (u, v, w) values of each stellar
cluster for this solar motion to obtain its current 3D space velocity
with respect to the LSR frame (vx, vy, vz).
For all clusters, we use the ages as provided in the literature31,33–37
.
We restrict our analysis to only the youngest clusters (ages smaller
than 30 Myr), guaranteeing that these associations are still tracing
thevelocitiesofthemolecularcloudsthatservedastheirbirthplaces.
We make a conservative selection on the x–y plane by enclosing all
stellar clusters within ±5 times the radius of the wave (47 pc) from the
best fit of the dust component (see ‘Modelling the spatial molecular
clouddistribution’).WefindthattheLacerta OB1associationistheonly
complexthatisvertically(intheheliocentricGalacticzdirection)not
associatedwiththecoldmolecularcomponentoftheRadcliffeWave.In
addition,theLacerta OB1complexiskinematicallydriftingtowardsthe
SolarSystem,potentiallyconnectedtotheCepheusspur40
,whichlies
behindthewavewithrespecttotheSun.Thismotionisdistinctfromthe
motionoftheRadcliffeWave,whichshowsevidenceofdriftingcoher-
ently away from the Sun. As this implies that Lacerta OB1 is the only
stellarclustercomplexbeingspatiallyandkinematicallydistinctfrom
the Radcliffe Wave, we do not include its clusters in the final analysis.
We compare our cluster sample to a recent open cluster catalogue41
,
restrictingtheanalysisagaintoonlytheyoungestclusters,findingno
effect on our results.
Fullspatialandkinematicmodel
In our analysis of the Radcliffe Wave’s kinematics, we distinguish
betweentwovelocitycomponentswithrespecttotheGalacticplane:
(1) ‘in-plane’ motion and (2) vertical ‘z’ motion. The decoupling of the
in-planeandzmotionoftheRadcliffeWaveissatisfiedbythefactthat
themaximalverticaldeflectionofregionsalongthewaveisatmostof
the order of the scale height of the Galactic Disk14
. We also examined
the effect of a misalignment between the orientation of our adopted
Galactic ‘disk’ (parallel to z = 0 pc) and the ‘true’ Galactic Disk, which
couldbeslightlytiltedwithrespecttotheGalacticCartesiancoordinate
frame, finding it has no effect on our results.
For the first component, we take into account Galactic and differ-
ential rotation11–13
and allow the wave to have an additional overall
two-dimensional velocity component, meaning that we permit the
structure to move as a solid body through the Galactic plane with a
fixed x and y velocity (vx,plane, vy,plane).
Inthecaseofthezcomponent,wefindthattheRadcliffeWaveoscil-
lateslikeatravellingwavedrivenbytheMilkyWay’sgravitationalaccel-
eration. Adopting a typical density ratio of 104
between the wave and
thesurroundingmoretenuousmedium(basedon3Ddustmapping42
),
wecanassumethatthegravitationalaccelerationinducedbytheGalac-
tic Disk dominates over the deceleration caused by the surrounding
medium. We therefore model the forces acting on the wave solely by
gravity.AstheRadcliffeWave’samplitudeisontheorderoftheGalaxy’s
vertical scale height, the vertical gravitational potential Φ resulting
inthisaccelerationcanberepresentedbyananharmonicoscillator43
.
To capture the anharmonic nature of the oscillation requires that we
describe the height variation in the potential up to a fourth order in z
(as in equation (1), below). To ensure that the potential is symmetric
withrespecttotheGalacticDisk,wesettheprefactorsofthefirstorder
andthirdorderequaltozero.WeassumetheGalacticdensitydistribu-
tion to be axisymmetric and model the dependence of the density on
thegalactocentricradiusasadecayingexponentialfunction22
.Asthe
influenceoftheradialforceterminPoisson’sequationcanbeneglected
in our region of interest14,22
(7 kpc to 10 kpc in galactocentric radius
andupto300 pcinverticaldisplacementfromtheGalacticplane)the
potential can be expressed as
Φ z r ω r r r z μ z
( , ) =
1
2
exp(( − ) )( − ) (1)
0
2
☉ 0
−1 2
0
2 4
where r is the galactocentric radius and z is the Galactic Cartesian z
coordinate. We set the distance of the Sun to the Galactic Centre, r☉
equalto8.4 kpc,accordingtoouradoptedGalacticpotentialmodel11
.
It is noted that the distance from the Sun to the Galactic Centre can
alwaysbeabsorbedinω0 andr0,thezeroth-orderoscillationfrequency
andtheradialscalelengthofthepotential,respectively.Inourregion
of interest, μ0 is proportional to the inverse scale height of the poten-
tial.Asthescaleheightcanbeconsideredindependentofthegalacto-
centricradiusforsmallvariationsinradius22,44
,wecanassumeμ0 tobe
constant in our region of interest. Our model of the potential has,
therefore,intotalthreefreeparameters:ω0,μ0 andr0.Theseparameters
are adopted by comparing equation (1) with an existing model of the
Galacticpotential45
includingitsupdatedparameters11
usingastandard
chi-squaredminimizationintherangeofgalactocentricradiusvarying
from7 kpcto10 kpcwithastepsizeof120 pcandverticalheightabove
andbelowtheGalacticplanevaryingfrom0to300 pcwithastepsize
of12 pc.Thisleadstoω0 = 74 km s−1
kpc−1
,r0 = 3.7 kpcandμ0 = 1.3 kpc−1
.
The Oort constants AOort and BOort for the adopted Galactic potential11
are 15.1 km s−1
kpc−1
and −13.7 km s−1
kpc−1
. The circular velocity at the
positionoftheSunis242 km s−1
.Fromouradoptedvaluesofω0 andμ0,
we infer a local matter density ρ0 = 0.1 M☉ pc−3
and a local scale height
z0 = ( 6μ0)−1
= 314 pc. This is consistent with literature values21
of
ρ M
= 0.097 pc
0 −0.013
+0.013
☉
−3
and z = 280 pc
0 −50
+70
. We rerun our analysis
using two additional models12,13
of the Galactic potential including
their respective adopted r☉ values (so in total, the same three Milky
Way potentials applied to model the wave’s in-plane motion), finding
no effect on our results.
By assuming vertical energy conservation, the period T with which
each part of the Radcliffe Wave oscillates through the Galactic Disk
can be computed by:
∫
T z r Φ z r Φ z r z
( , ) = 2 2 ( ( , ) − ( , )) d (2)
z
max
0
max
−0.5
max
where zmax is the maximum deflection a subregion of the wave can
reach above the Galactic plane. Inserting equation (1) into equa-
tion (2) using the definition of the frequency of vertical oscillation
ω(zmax, r) = 2πT−1
(zmax, r)andintroducingtheellipticintegraloffirstkind
K,wecanderiveafullyanalyticexpressionoftheoscillationfrequency:
ω z r ω r r r μ z
K μ z μ z
( , ) =
π
2
exp(( − )(2 ) ) (1 − )
( (1 − ) ).
(3)
max 0 ☉ 0
−1
0
2
max
2 0.5
−1
0 max 0
2
max
2 −0.5
Introducing the distance s along the wave (projected onto the x–y
plane)suchthats = 0 pcatthebeginningofthewave(nearCMa OB1),
we note that based on the position of the wave in the Galactic plane

6. Article
(pitchangle ≠ 0°)thegalactocentricradiusrcanbeparameterizedasa
functionofthisdistancesalongthewave,implyingω(zmax(s), r(s)) = ω(s).
Itisnotedthat,consistentwiththeliterature14
,wetreattheunderly-
ing gravitational potential as stationary. This choice is furthermore
motivated by the distribution of the old stellar component around
the Radcliffe Wave, which is expected to dominate the local Galactic
potential14
. In the old stellar component, there is no evidence2,3
for
perturbationswithwavelength<10 kpc.InthevicinityoftheRadcliffe
Wave, a perturbation of the potential with wavelength >10 kpc would
correspond to a local tilt in the potential. The Galactic warp also con-
stitutes a large-scale change in the potential. However, we already
examined the inclusion of a tilt and found that including two extra
angle parameters has no effect on our results. In addition, perturba-
tions with wavelength >10 kpc, which can also be seen as a warping of
theGalacticDisk,areexpectedtochangeontimescaleslargerthanthe
period of the Radcliffe Wave46
.
Thereisalsorecentevidenceforradiallyevolvingverticalwavesnear
the Sun47,48
with wavelengths <10 kpc. As these waves appear radially
(pitchanglecloseto90°)whereastheRadcliffeWaveevolvespredomi-
nantly tangentially (pitch angle close to 0°) in the Milky Way, these
waves should not significantly affect the Radcliffe Wave.
TheoriginalmodelfortheRadcliffeWave1
waspurelyspatial.Now,as
wehavebothspatialandkinematicinformationofthewave,weapply
amodeltodescribetheoverallspatialandkinematicbehaviourofthe
RadcliffeWave.Althoughourmodelhasfewerfreeparametersthanthe
original one1
, it provides a better fit to the 3D spatial data (see details
in ‘Modelling the spatial molecular cloud distribution’). In the new
framework,theRadcliffeWave’sundulationismodelledbyadamped
sinusoidal wave perpendicular to the plane of the Milky Way. We use
a quadratic function in x and y (heliocentric Galactic Cartesian coor-
dinates) described by three anchor points (x0, y0), (x1, y1) and (x2, y2)
to model the slightly curved baseline of the structure in the Galactic
plane, as a linear model is insufficient to account for the observed
Radcliffe Wave’s curvature in the x–y plane. We examine the effect of
adding in additional parameters used to account for an inclination of
the structure with respect to the Galactic Disk in the original (spatial)
modelfortheRadcliffeWave1
.However,wefindtheresultstobefully
consistentwiththestructureoscillatingthroughthex–yplane,sothese
additional parameters describing the inclination are not included in
the final analysis.
Wemodelthewaveastwocounterpropagatingwaveswithfrequen-
cies of ±ω but different amplitudes, to allow motion in the form of a
travellingwave,astandingwaveorintermediatecases.Inthiscontext,
we introduce a parameter B that determines the mixture between a
standingandtravellingwavemodel,leadingtothefollowingequation:
(4)
z s t ζ s B ω t Λ s φ B ω t Λ s φ
( , ) = ( ) ( sin( + 2π ( ) + ) + (1 − )sin(− + 2π ( ) + ))
0 0
(5)
v s t ζ s ω s B ω t Λ s φ
B ω t Λ s φ
( , ) = ( ) ( )( cos( + 2π ( ) + )
−(1 − )cos(− + 2π ( ) + ))
z 0
0
where t denotes the time, s denotes the distance along the wave, ω is
definedbyequation(3),ω0 isthezeroth-orderoscillationfrequency,φis
thephase,andΛ(s)andζ(s)describedampingfunctionsalongthewave.
In principle, Λ(s) is a function of the time t. The dependence of t
resultsfromthesecond-andhigher-ordertermsinequation(3),which
we denote as |f(s)ω0|. Using our adopted gravitational potential, we
computeffromequation(3)toshowthatfbecomesatmostabout0.25
foranydistancesalongthewave.Ultimately,wefind|ω0t| ≈ 0.4,imply-
ingthat|f(s)ω0t|isoftheorderofabout0.03π.Thisallowsustoneglect
thedependenceofthetimetandapproximateΛ(s)asapurefunctionof
s.Itisnotedthatthetime-independentapproximationbreaksdownas
soonastheoscillationproceedsintothefuture,implyingthatwecanno
longersetanupperlimiton|ω0t|.Deviationsfromthetime-independent
approximation would be comparable to the noise level after roughly
40 Myr.Therefore,weletthedatainSupplementaryFig.5fadeoutwith
time.Weevolvethemodelwiththehorizontaldampingtimeindepend-
ent,todemonstratethekeykinematicfeatureofthewave,thatis,that
its kinematic behaviour is consistent with a travelling wave.
To characterize the horizontal damping of the wave, we define the
phaseϕk = arctan(zkωkvz k
,
−1
)ofaclusterkalongthewave.Ultimately,we
findthatBiscloseto1.Nevertheless,letusassumethatB = 0.5,mean-
ingthatthewavecorrespondstoastandingwave.Fromequations(4)
and (5), we conclude that the phases ϕk should be roughly constant
(varying at most on the order of |f(s)ω0t| ≈ 0.03π) as a function of dis-
tances sk along the wave (where sk is the distance along the wave s for
a cluster k). In contrast, we find that the phases ϕk change over the
whole range of −π/2 to π/2, such that B = 0.5 is inconsistent with the
data. We therefore concentrate on B ≈ 1 from this point onwards.
ForthecaseB ≈ 1(correspondingtoatravellingwave),weconclude
from equations (4) and (5) that the phase ϕ(s) should change accord-
ing to Λ(s) + ω0t + φ. By examining the overall distribution of stellar
clusters in position and velocity space (Fig. 1), we can conclude that
the horizontal damping function as a function of s, that is, Λ(s), is a
monotonicallyincreasingandconvexfunctionofthedistancealongthe
waves.Asthearctanfunctionprojectstophasesbetweenonly−π/2to
π/2,weshifteachϕk bynkπ(withnk beinganintegerforagivenclusterk)
such that the change of phases ϕk with distance sk is consistent with
thefactthatchangeinphaseisamonotonicallyincreasingandconvex
functionofthedistancealongthewave.WefindthatΛ(s)describedby
Λ s
s
p γs
( ) =
−
(6)
where s denotes the distance along the wave, p denotes the period
of the wave and γ sets the rate of the period decay, is able to explain
the observed change in phases ϕk, while keeping the number of free
parameters low. Given the change of Λ(s) with the distance along the
wave s, we use the following equation to compute the change in the
wave’s wavelength λ(s):
λ s Λ s p p γs
( ) = (∂ ( ′)| = ( − ) . (7)
s s s
−1
′ ′=
−2
Wecouldinprincipletaketheparabolicmodelintroducedintheorig-
inalwork1
tofitthedatausingoptimizedvaluesforthefreeparameters
given our kinematic constraints. However, in this case the first-order
periodandthesecond-orderscalingfactorbothgotoinfinity,meaning
thatthefirst-orderterminthemodelvanishesandthewavelengthλ(s)
goestoinfinityforsgoingtozero.AsforΛ(s)describedbyequation(6),
the wavelength λ(s) (see equation (7)) is well defined for s = 0 pc, we
adopt equation (6) for the horizontal damping instead. However, the
distribution of the phases allows for a variation of the model. With
the advent of even more precise data, future work should be able to
better constrain how the wavelength changes with distance along
the wave.
Next, we motivate our choice of the vertical damping function ζ(s).
Using the vertical gravitational potential introduced in this work, we
calculatetheverticalenergyofthewaveandfromthatthemaximalver-
ticaldeflectionzmax ofasinglestarclusterkundertheverticalpullofthe
gravitationalpotentialgivenitsverticalpositionandverticalvelocity.
Followingequations(1),(4)and(5),weapproximatezmax(s)asfollows:
z s ζ s Λ s φ B B
( ) ≈ | ( )| (1 + 4 cos (2π ( ) + ) ( − )) . (8)
max
2 2 0.5
Ultimately, we find that B is close to 1 and in particular B > 0.5. This
implies that the second part in equation (8) will always be larger than
zeroandcanbeseenasascattertermperturbing|ζ(s)|.Inotherwords,
theoverallbehaviourofzmax(s)iscapturedin|ζ(s)|.Inparticular,inthe
caseofB = 1(correspondingtoatravellingwave)zmax(s)isequalto|ζ(s)|.
WefindthataLorentziandescribesthebehaviourofzmax(s)margin-
ally better than a Gaussian, which is why we adopt the former for ζ(s)

7. compared with the original study1
. However, especially around the
peakregion,thedataareinfavouroftheGaussianmodel.Weconclude
that a mixture model with an additional free parameter (for example,
realizedbyaVoigtprofile)isprobablythetrueunderlyingdamping.As
ourdataaretoolimitedtoconstrainanextrafreeparameterproperly,
we keep the Lorentzian damping profile. However, the evidence in
favour of a mixture model might imply that the Radcliffe Wave origi-
nallyoriginatedfromtwodistinctstructures,whoseoriginaldamping
profilesmightstillbeimprintedonthedata.Thefinalformofζ(s)can
be described by the following equation:
ζ s A
s s
δ
( ) = − 1 +
−
(9)
0
2
−1
 












whereAdenotestheamplitude,s0 denotesthepositionofthemaximal
deflection and δ sets the rate of the amplitude decay.
As mentioned earlier, the vertical motion is decoupled from the
Radcliffe Wave’s in-plane motion. Therefore, the entire model is
described by 16 parameters, which can be separated into two uncor-
related subgroups θ1 = (x0, y0, x1, y1, x2, y2, A, p, φ, s0, δ, γ, ω0t, B) and
θ2 = (vx,plane, vy,plane).
In the following, we apply our overall model for the Radcliffe Wave
oscillation to just the molecular clouds as traced by 3D dust mapping
(as in the original study1
), just the star clusters and then the clouds
togetherwiththestarclusters.Wefindthatallthreefitsusingdifferent
observationalcomponentsandtheircombinationsleadtoconsistent
results,confirmingthattheRadcliffeWaveisaspatiallyandkinemati-
cally coherent structure composed of molecular clouds and young
stellar clusters. Ultimately, we investigate the wave’s in-plane motion
using the kinematic stellar information.
Modellingthespatialmolecularclouddistribution
Asintheoriginalspatial-onlyfitting1
,inthissectionweapplyouroverall
wavemodeltotheRadcliffeWave’sdustcomponent,describedbythe
corresponding 3D cloud positions from the Major Cloud Catalog29,30
andtheTenuousConnectionsCatalog1
.Weincludeinouranalysisonly
thosemolecularcloudsforwhichwecanobtainaradialvelocitymeas-
urement (see ‘Line-of-sight gas velocities’ section, below).
Itisnotedthatforthecloudcomponent,wehaveonlyspatialinfor-
mation. In the absence of any kinematic constraint, we can always
set B and ω0t to fixed values, as equation (4) can be formulated as
ζ(s) η sin(2πΛ(s) + φ + κ) where η and κ are functions of B and ω0t. We
choose B = 1 and ω0t = 0, leading to η = 1 and κ = 0.
Tomodelthedata,wedefinethespatialdistancevectordcloud,i ofthe
ithmolecularcloudoftheRadcliffeWaverelativetothewavemodelas
d s x y z x s y s z s
( ) = ( , , ) − ( ( ), ( ), ( )) (10)
i i i i
cloud,
wheresisthedistancealongthewaveintheGalacticplane.Theoriginal
sampling of the Radcliffe Wave1
was not chosen to provide equal cov-
erage over the entire structure, so some regions (for example, Orion)
have much higher sampling than other regions (for example, North
American Nebula). To consider each part of the Radcliffe Wave with
equalimportance,weintroduceweightswcloud,i derivedusingakernel
densityestimationapproachwithaGaussiankernelwhosebandwidth
ischosentobeafewpercentofthetotallengthofthewave.Givenour
definitionofthelikelihood,L,theweightsaresettosumtothenumber
ofmolecularcloudsinoursample.Assumingthatthepositionsofthe
molecularcloudshavebeenderivedindependently,thelog-likelihood
for a given realization of our model is
∑
L w
s
σ
σ
ln( ( )) = −
( )
2
+ ln(2π ) (11)
i
i
i
1 cloud,
cloud,
2
min
2
2








θ
d
where smin is defined as the distance along the wave that minimizes
d i
cloud,
2
foragivencloudiandthecorrespondingdistancevectorfrom
equation(10).Inaddition,weintroducedascatterparameterσ,which
indicates the spread of the clouds around the model. Assuming that
the relative cloud model distance vectors are normally distributed,
s
= ∑ ( )
i i
min
2
cloud,
2
min
d d is chi-squared distributed with two degrees of
freedom such that σ2
is calculated as 0.5 times the standard deviation
of min
2
d . Inferring the values of θ1 in a Bayesian framework, we sample
forourfreeparametersusingthenestedsamplingcodedynesty49
.On
the basis of initial fits, we adopt pairwise independent priors on each
parameter, which are described in Extended Data Table 1, and run a
combination of random-walk sampling with multi-ellipsoid decom-
positionsand1,000livepoints.Theresultsofoursamplingprocedure
aresummarizedinExtendedDataTable2.Wefindaradiusofthewave
of 47 pc defined by the wave’s scatter parameter σ. By calculating the
weighted and unweighted sum of the squared cloud model distances
for the new and the original1
wave model using equation (10), we find
in both cases (weighted and unweighted) that the ratio of the sums
between our model and the original model is less than one, implying
thatourmodel,despitehavingfewerfreeparameters,actuallyfitsthe
3D spatial molecular cloud data better.
Modellingtheoscillationofthewave’sstellarcomponent
Inthissection,weapplyouroverallwavemodeltotheRadcliffeWave’s
stellar component, as traced by star clusters. To fit our model to the
wave’sstarclusters,wedefinethespatialandkinematicdistancevector
dcluster,k of the kth cluster relative to the wave model as
s x y z v ω x s y s z s v s ω
( ) = ( , , , ) − ( ( ), ( ), ( ), ( ) ). (12)
k k k k z k z
cluster, , 0
−1
0
−1
d
Toconsiderpositionsandvelocitieswithequalimportance,wemul-
tiplied the fourth entry of dcluster,k by ω0
−1
. Assuming that the positions
and velocities of the clusters have been derived independently, the
log-likelihood in this case is given by
θ
d














∑
L w C
s
σ
σ
ln( ( )) = −
1
2
( )
+ 3 ln(2π ) (13)
k
k
k
1 cluster,
cluster, min
2
2
where smin is defined as the distance along the wave that minimizes
d k
cluster,
2
for a given cluster k and the corresponding distance vector
from equation (12). The scatter parameter σ remains the same as in
the molecular cloud discussion, and, to again consider each part of
theRadcliffeWavewithequalimportance,weightswcluster,k wereimple-
mented, derived using a kernel density estimation approach with a
Gaussiankernelwhosebandwidthischosentobeafewpercentofthe
total length of the wave. Given our definition of the log-likelihood,
the weights are set to sum to the number of clusters in our sample. In
addition, to search for outliers, we introduced an outlier model rep-
resented by a truncation function C. It is noted that in the case of the
Radcliffe Wave’s molecular clouds, no outlier model was applied as
the selection used for the dust already included outlier modelling in
the original study1
. We choose C(xk) = (∑nck,n)0.5
with ≔
c x
tanh( )
k n k n
, ,
2
,
where xk,n describes the nth component of xk (n going from 1 to 4),
satisfying that C(xk) =‖xk‖holds for small‖xk‖, and that clusters with
large distances between observation and model have little effect on
the log-likelihood.
Toallowforatravellingwave,astandingwaveandintermediatecases,
we apply a uniform prior from 0 to 1 for the parameter B (Extended
Data Table 1). We infer the values of θ1 in a Bayesian framework, sam-
plingforourfreeparametersusingthenestedsamplingcodedynesty.
We adopt pairwise independent priors on each parameter (Extended
Data Table 1), and run a combination of random-walk sampling with
multi-ellipsoid decompositions and 1,000 live points. The results of
our sampling procedure are summarized in Extended Data Table 2.

8. Article
Wefindabest-fit-valueforB = 0.84.ThevalueofBcloserto1indicates
thatthedatafavouratravellingwave.Inaddition,weredothesamefit
withthesamefreeparametersθ1 whilefixingB = 1andω0t = 0.Bycom-
paring the logarithm of the Bayes factor of the two realizations (B = 1
andBvaryingbetween0and1)giventheirevidencesZ1 andZ2,wefind
Δln(Z) = ln(Z2) − ln(Z1) ≈ 1.Thus,bothmodelsareanequivalentlygood
fittothedata,whiletherealizationwithfixedB = 1isslightlypreferred.
Wethereforeconcludethatthewave’smotionisoffirst-orderconsist-
ent with a travelling wave, implying that we can set B = 1 in all further
performed calculations, such that θ1 is reduced by two parameters,
implying θ1 = (x0, y0, x1, y1, x2, y2, A, p, φ, s0, δ, γ).
To investigate the effect of the Sun’s current vertical position with
respecttotheGalacticplane,wererunouranalysisincludingtheverti-
calpositionz0 oftheSunasafreeparameter.Wesetauniformpriorwith
alowerlimitofz0 = −10 pcandupperlimitz0 = 25 pc.Aftermarginalizing
over z0, we find that the amplitude A, the phase φ and the damping
parameterδchangebylessthanathirdofthe1σuncertaintyreported
in the case where z0 is fixed. All remaining parameters in θ1 change
by less than 10% of the 1σ uncertainty reported in the case where z0 is
fixed. Therefore, we perform all calculations with the Sun located in
the plane, consistent with recent work in this regard50
.
Modellingtheoscillationofthewave’sstellarandmolecular
cloudcomponent
Finally,wemodelthespatialpositionsofcloudstogetherwiththestellar
clusterspositionsandvelocities.Thecombinedlog-likelihoodsforboth
casesgivethebest-fit-modelshowninFig.1aswellasitscorresponding
parameterssummarizedinExtendedDataTable2.InFig.1,weencode
the transparency of the young stellar clusters as opaque for statisti-
cal inliers and transparent for outliers. On the basis of the previous
section, we define a cluster k to be an outlier if for any value of n the
valueofck,n islargerthantanh(3),meaningthatweincludeonlyclusters
thatlieinsidea3σradius,whereσisthewave’sscatterparameter.The
majorityoftheclustersidentifiedasoutliersbelongtothePerseusand
Taurus molecular clouds. In particular, as all clusters associated with
Perseus and Taurus are identified as outliers, we argue that Perseus
andTaurusseemtobekinematicallydistinctfromtheRadcliffeWave.
Theirkinematicsisprobablydominatedbytheexpansionmotionsof
theLocalandPerseus–Taurusfeedbackbubbles9,15
onwhosesurfaces
thecloudsarelocated.Mostoftheremainingoutlierscanbeassigned
totheOrionstar-formingregion.Itisimportanttonote,however,that
thefractionofoutlierscomparedwithinliersinOrionisnomorethan
a quarter, implying that it follows the general Radcliffe Wave motion,
soonlyafractionofOrion’sclusters’motionsisdominatedbyinternal
feedback51,52
.
In Fig. 1, the typical uncertainty in the positions and in the vertical
velocity of the stellar clusters and the typical uncertainty in the posi-
tions of the molecular clouds are of the same order of magnitude as
thesizeofthecorrespondingsymbols.Themethodusedtoderivethe
cloud distances introduces an additional systematic cloud distance
uncertainty of about 5% (refs. 29,30). It is noted that the small ripples
at the right end of the wave naturally result from the linear damping
model of the wave’s period with distance. As we have no star cluster
measurementsattherightendofthewave(aroundCygnus X)andthe
amplitudeoftheripplesinthisregiongetsclosetotheerrorofthedust
distance measurements29,30
, we can not distinguish between ripples
andnoripplesbasedonthedata,indicatedbythegreycolourofthefit
inFig.1.Thisdoesnotaffecttheoverallwave’sbestfitortheobtained
properties of the wave, as our results are dominated by the data with
larger deflections above and below the Galactic plane.
Using the best-fit-parameters and equation (7), we find a mean
wavelengthofaround2kpc,withanunderlyingrangeofwavelengths
varyingfromaround400 pc(nearCygnus X)to4kpc(nearCMa OB1).
Giventhechangeinwavelengthandtheverticaldampingofthewave,
wecalculateitsphasevelocityusingtheratiobetweenthewavelength
λ(s) (equation (7)) and the period T(s) (equation (2)) at any distance s
along the wave. We find that the phase velocity changes from about
40 km s−1
(near CMa OB1) to about 5 km s−1
(near Cygnus X). We
explore whether a variation in the local matter density may explain
thisorder-of-magnitudechangeinthephasevelocity.Wefindthatthe
localmatterdensityρ0 wouldneedtochangebyatleasttwoordersof
magnitude, which is inconsistent with observations or models of the
MilkyWay14,21
.However,thischangeinphasevelocitymayprovideclues
to the wave’s origin. As, in addition to the large change in the phase
velocity, a two-component damping function provides the best fit to
the vertical damping of the wave, we hypothesize that the Radcliffe
Wave may have originated from the interplay of two different physi-
calprocesses.Futuresimulationsshouldshedlightonwhethersucha
combinationcouldproducestructuresontherightscalethatarealso
kinematically consistent with a travelling wave.
Modellingthewave’sin-planemotion
Inthissection,weinvestigatethein-planemotionoftheRadcliffeWave
describedbythevaluesofθ2 = (vx,plane, vy,plane).Forthispurpose,wesub-
tract Galactic and differential rotation using an existing model of the
Galacticpotential45
includingitsupdatedparameters11
fromtheobser-
vational values of the x and y velocity (vx,k, vy,k) of the kth cluster. We
testedtwoalternativemodelsfortheGalacticpotential12,13
,findingno
effect on our results. We apply a standard chi-squared minimization
with independent priors on each vx,plane and vy,plane (Extended Data
Table1)tosearchforthebestvaluestodescribetheobservedin-plane
motion.Forthechi-squaredminimization,weadoptatypicalerrorof
2 km s−1
,derivedbytakingthemedianvx andvy erroracrossthesample.
By computing the 16th, 50th and 84th percentiles of the samples we
find v = −3.9 km s
x,plane −2.7
+2.6 −1
and v = −3.1 km s
y,plane −2.6
+2.7 −1
. This corre-
sponds to a motion of 4.9 km s−1
radially away from the Solar System
aswellasof1.0 km s−1
paralleltotheRadcliffeWave,implyingthatthe
Radcliffe Wave is radially drifting away from the Galactic Centre with
avelocityofaround5 km s−1
.InExtendedDataFig.2,weshowthewave’s
radial and tangential velocity vectors as well as the underlying stellar
clusterdatausedtoderivethekinematicproperties.InExtendedData
Fig.2a,c,theGalacticcoordinatex–yframeisrotatedanticlockwiseby
62°.ThexaxisinthisrotatedframeisparalleltoanaxisalongtheRad-
cliffe Wave and is referred to as ‘position along the wave’ throughout
this work. The direction of this motion suggests that in the past, the
RadcliffeWavemayhavebeenatthesamelocationwherethestarclus-
ters Upper Centaurus Lupus and Lower Centaurus Crux were born
15–16 Myr ago9
, potentially providing the reservoir of molecular gas
needed for their formation.
Line-of-sightgasvelocities
Inthissection,weconsiderthedynamicsofthewaveusingline-of-sight
velocity measurements8
of 12
CO, a tracer of gas motion independent
from that used in our 3D dust and clusters best-fit-analysis described
above. One may wonder how modelling 3D motion is possible when
spectral-linemeasurementsprobeonlyaone-dimensionalline-of-sight
velocity,vLSR.Asthewaveissoclose(250 pcatclosest),solong(about
3 kpc)andextendssofarabovetheGalacticDisk(about200 pc),various
lines of sight to its clouds actually sample a range of combinations of
vx,vy andvz.ExtendedDataFig.3bandtheinteractiveSupplementary
Fig. 3 illustrate this lucky circumstance with yellow lines of sight to
thewavethataviewerwillnoteareinclinedatawidevarietyofangles
relative to an x–y–z frame.
Wemeasureline-of-sightcloudvelocities(vLSR)usingaGalaxy-wide
12
CO survey8
with an angular resolution of 0.125° and an LSR velocity
resolution of 1.3 km s−1
. We exclude regions far beyond the Radcliffe
Wave by applying an LSR velocity cut-off of 30 km s−1
. For each set of
spatial cloud coordinates, we compute a 12
CO spectrum over a region
witharadiusof0.3°atthecorrespondingpositiononthesky.Aradial
velocity is then assigned to each spatial position using nonlinear

9. least-squaresfittingofGaussiansbasedontheLevenberg–Marquardt
algorithm53
.ForeachGaussian,weassignthemeanastheradialveloc-
ity and the 1σ deviation as the corresponding error. Nine per cent of
the clouds fall outside the limits of the 12
CO survey or show no emis-
sion above the noise threshold, so we exclude these clouds from our
phase-space analysis. In Extended Data Fig. 3a, we show the results
of the spectral fitting alongside our best model for the motion of the
Radcliffe Wave. The computation of the errors shown in Extended
Data Fig. 3a involved combining multiple measurements of a single
molecular cloud into a single data point. To this end, we average over
eachsinglecloudandpresenttheobtained16thand84thpercentiles.
Thecomparisonbetweenthedataandthemodelinthisfigureisbased
ontheprojectionofoursix-dimensional(x, y, z, vx, vy, vz)best-fit-model
onto the four-dimensional phase space (x, y, z, vLSR) of the gas data.
The theoretical description of the radial velocity vrad is derived using
x v
x v
v l b v l b v b v
( , ) = cos( )cos( ) + sin( )cos( ) + sin( )
= ⋅
(14)
i i i i i x i i i y i i z i
i i
LSR, , , ,
whereli andbi aretheGalacticlongitudeandlatitudeoftheithmolec-
ular cloud, xi denotes its normalized position in Galactic Cartesian
coordinatesandvi isthecorrespondingvelocitywithcomponentsvx,i,
vy,i and vz,i. We show in Extended Data Fig. 3a that our model explains
the 12
CO observations, meaning that the velocities of the molecular
clouds,forwhichwecanstudyonlyanincompletephasespace,match
the motion of the young stellar clusters. As a consequence, we can
transfer our results from the full phase-space study of the stars to all
componentsoftheRadcliffeWave,includingthegas,andcharacterize
its full structure and kinematics.
TheSun’sverticaloscillationperiodandlocalpropertiesofthe
Galacticpotential
In this section, we present a first application of the Radcliffe Wave’s
oscillation,usingittomeasuretheSun’sverticaloscillationfrequency
and local properties of the Galactic potential.
The Radcliffe Wave shows a systematic pattern in its gas and star
clustervelocities.Above,weshowhowthatpatternisconsistentwith
oscillation in the Galaxy’s gravitational potential. But we can also use
thatpattern—withoutpriorknowledgeabouttheverticalgravitational
potential—toinferamaximumamplitudeaswellasamaximumveloc-
ityforeachstar-formingregionalongthewave.Asaconsequence,we
can investigate the vertical oscillation frequency without needing to
adopt a standard model of the Milky Way gravitational potential11–13
.
We can essentially reverse the set-up of our study and directly meas-
ure the oscillation frequency described by equation (3), using only
the observed Radcliffe Wave oscillation. For this purpose, we fit our
model for the coherent oscillation of the Radcliffe Wave, described
by equations (4) and (5), to the observations by treating the variables
ω0,μ0 andr0,whichdefinetheoscillationfrequencycalculatedinequa-
tion(3),asfreeparameters.Earlier,theseparametersweredetermined
by comparing the Galactic potential described by equation (1) with a
full model of the Galactic potential to show that the Radcliffe Wave’s
oscillationisconsistentwiththecurrentunderstandingoftheGalactic
potentialoftheMilkyWay11–13
.Theentiremodelisnowdescribedby15
parameters θ3 = (x0, y0, x1, y1, x2, y2, A, p, φ, s0, δ, γ, ω0, μ0, r0).
Inferringthevaluesofθ3 inaBayesianframework,wesampleforour
freeparametersusingthenestedsamplingcodedynesty49
and,based
oninitialfits,weadoptpairwiseindependentpriorsoneachparameter,
which are described in Extended Data Table 1. We run a combination
of random-walk sampling with multi-ellipsoid decompositions and
1,000 live points. The results of our sampling procedure are summa-
rized in Extended Data Table 2. We find a median value and 1σ errors
(computed using the 16th, 50th and 84th percentiles of the samples)
of ω = 68 km s kpc
0 −7
+8 −1 −1
, μ = 1.4 kpc
0 −0.2
+0.2 −1
and r = 3.6 kpc
0 −0.5
+0.5
. These
valuesareconsistentwiththepreviouslyadoptedvaluesforω0,μ0 and
r0 withinthe1σerror.BycomparingthelogarithmoftheBayesfactor54
ofthetworealizations(fixedandvaryingoscillationfrequency)using
their evidences Z1 and Z2, we find Δln(Z) = ln(Z2) − ln(Z1) ≈ 1, meaning
that both models are an equivalently good fit to the data54
.
Following the vertical density model of an isothermal sheet based
onaMaxwellianverticalvelocitydistribution55
,wecalculatetheparam-
eters defining the Galactic mass distribution: the midplane density
ρ0 =ω0
2
(4πG)−1
andtheverticalscaleheightz0 = ( 6μ0)−1
.Thisleadsto
ρ M
= 0.085 pc
0 −0.017
+0.021
☉
−3
and z = 294 pc
0 −43
+63
. To compute errors on ρ0
and z0, we sample from the probability distribution of ω0 and μ0 to
obtain 2,000 realizations of the overall probability distributions.
Fromthat,wecomputethemedianvalueand1σerrorsusingthe16th,
50th and 84th percentiles of the samples. Our results are consistent
with the conventional approach of deriving properties of the local
Galactic mass distribution21,24–26
. However, we caution that our con-
straint on the scale height of z0 ≈ 294 pc is tempered by the fact that
a scale height of about 294 pc leads to a 7% change of the total oscil-
lation frequency. Given that we determine ω0 with an approximately
10%accuracy,weemphasizethatforz0 > 300 pc,thesensitivityofour
model to z0 would be indistinguishable from the noise level of
this study.
The derived midplane density contains both the local amount of
dark matter and the baryonic matter component. By considering
observations of the baryonic mass distribution near the Sun21
of ρ M
= 0.084 pc
0,baryons −0.012
+0.012
☉
−3
, we infer the amount of local dark
matter.Assumingasphericaldarkhalo,weobtainalocaldark-matter
densityof ρ M
= 0.001 pc
0,darkhalo −0.021
+0.024
☉
−3
,consistentwiththeconven-
tionalapproachtodeterminetheGalacticpotential’spropertiesatthe
1σ level21,24–26
. Moreover, we obtain a total surface density Σ0 = 2z0ρ0
of 50.3 M☉ pc−2
.
On the basis of equations (2) and (3), we additionally find the Sun’s
oscillationperiodtobe95 Myr
−10
+12
,whereweassumedaverticalveloc-
ityoftheSun10
of7.8 km s−1
.Tocomputetheperiod’serrorbars,weuse
the probability distribution of each free parameter reflecting their
measurementuncertaintiesandthenrandomlysamplefromtheover-
all multivariate probability distribution 2,000 times to create 2,000
mock realizations. From that, we compute the median value and 1σ
errorsusingthe16th,50thand84thpercentilesofthesamples.Again,
we allow the current position of the Sun to vary from −10 pc to 25 pc
below or above the Galactic plane. We find that the effect resulting
fromtheSun’sverticalpositionissmallerthan0.5 Myr,whichismore
than an order of magnitude smaller than the observed 1σ deviation.
Therefore,allcomputationsweredonewiththeSunlyingintheplane.
The maximal deflection zmax of the Sun above the plane used in equa-
tion (3) is computed by assuming vertical energy conservation. The
galactocentric radius of the Solar System is chosen to be 8.4 kpc. We
variedthesolarradiusfrom7.9 kpcto8.7 kpc,findingnoeffectonour
results.Thisisexpectedasouranalysismainlydependsonthedistance
betweentheSunandtheRadcliffeWave,whichiswellconstrained1,29,30
and not on our exact position in the Milky Way.
We also infer the period of the Sun following up-to-date measure-
ments of the baryonic matter component21
. Using equation (2) with a
potentialcorrespondingtotheverticaldensitymodelofanisothermal
sheet, based on a Maxwellian vertical velocity distribution55
Φ z G ρ z z z
( ) = 4π ln(cosh( )) (15)
0 0
2
0
−1
with midplane density21
ρ0 = 0.084 M☉ pc−3
, gravitational constant
G, and the vertical scale height21
z0 defined via the surface density21
Σ0 = 2z0ρ0 = 47.1 M☉ pc−2
,wecomputetheoscillationperiodthatwould
follow purely from the visible matter as 95.3 Myr. To compare our
method with the conventional approach of analysing stellar dynam-
ics by measuring the vertical gravitational acceleration23–26
, we add
to equation (15) the potential of a spherical dark halo, which can be
expressed in our region of interest as a harmonic oscillator24

10. Article
Φ z G ρ z
( ) = 2 π (16)
0
2
with a density21
of ρ0 = 0.013 M☉ pc−3
. This leads to a total period of
88.2 Myr, consistent with our result within the 1σ error.
Toaccountforapossibledarkdisk,weobservethatthesmallerthe
scale height of the dark disk, the smaller is the resulting oscillation
period, while keeping the surface density of the dark disk constant.
Therefore,weassumeacomparatively27,56
largeverticaldiskscaleheight
of25 pcandusethistoderiveanupperlimitonthesurfacedensitythat
a hypothetical dark disk may have to be consistent with our measure-
ment of the solar oscillation period of 95 Myr. Inserting the sum of
equation(15)withvaluesforthebaryonicmasscomponent21
andequa-
tion(15)withz0 = 25 pcandΣ0,darkdisk asafreeparameterinequation(2),
we find an upper bound for the dark disk of Σ0,darkdisk = M
0.1 pc
−3.3
+3.3
☉
−2
.
Itisnotedthatthisupperboundisconstrainedwithoutanyadditional
dark halo, which would only lower the upper bound further.
Dataavailability
The datasets generated and/or analysed during the current study are
publicly available at the Harvard Dataverse. The 12
CO radial veloci-
ties for the Radcliffe Wave are available at https://doi.org/10.7910/
DVN/Q7F4PC. The stellar cluster catalogue is available at https://doi.
org/10.7910/DVN/XSCB9N. The best-fit-model displayed in Fig. 1 is
available at https://doi.org/10.7910/DVN/F98QHY.
Codeavailability
The code used to derive the results is available from the correspond-
ingauthoruponreasonablerequest.Inthiscontext,publiclyavailable
software packages, including dynesty49
and astropy57
, were used. The
visualization,explorationandinterpretationofdatapresentedinthis
work were made possible using the glue58
visualization software. The
interactive figures were made possible by the plot.ly python library.
31. Cantat-Gaudin, T. et al. Painting a portrait of the Galactic Disc with its stellar clusters.
Astron. Astrophys. Suppl. Ser. 640, A1 (2020).
32. Prusti, T. et al. The Gaia mission. Astron. Astrophys. Suppl. Ser. 595, A1 (2016).
33. Sim, G., Lee, S. H., Ann, H. B. & Kim, S. 207 new open star clusters within 1 kpc from Gaia
Data Release 2. J. Korean Astron. Soc. 52, 145–158 (2019).
34. Hao, C. J. et al. Newly detected open clusters in the Galactic disk using Gaia EDR3. Astron.
Astrophys. 660, A4 (2022).
35. He, Z., Wang, K., Luo, Y., Li, J. & Liu, X. A blind all-sky search for star clusters in Gaia EDR3:
886 clusters within 1.2 kpc of the Sun. Astrophys. J. 262, 7 (2022).
36. Szilágyi, M., Kun, M. & Ábrahám, P. The Gaia view of the Cepheus flare. Mon. Not. R.
Astron. Soc. 505, 5164–5182 (2021).
37. Liu, L. & Pang, X. A Catalog of newly identified star clusters in Gaia DR2. ApJS 245, 32
(2019).
38. Gaia Collaboration et al. Gaia Data Release 3. Summary of the content and survey
properties. Astron. Astrophys. 674, A1 (2023).
39. Jönsson, H. et al. APOGEE data and spectral analysis from SDSS data release 16: seven
years of observations including first results from APOGEE-South. Astron. J 160, 120 (2020).
40. Pantaleoni González, M., Maíz Apellániz, J., Barbá, R. H. & Reed, B. C. The ALMA catalogue
of OB stars—II. A cross-match with Gaia DR2 and an updated map of the solar
neighbourhood. Mon. Not. R. Astron. Soc. 504, 2968–2982 (2021).
41. Hunt, E. L. & Reffert, S. Improving the open cluster census. II. An all-sky cluster catalogue
with Gaia DR3. Astron. Astrophys. 673, A114 (2023).
42. Leike, R. H., Glatzle, M. & Enßlin, T. A. Resolving nearby dust clouds. Astron. Astrophys.
639, A138 (2020).
43. Antoja, T. et al. A dynamically young and perturbed Milky Way disk. Nature 561, 360–362
(2018).
44. van der Kruit, P. C. & Searle, L. Surface photometry of edge-on spiral galaxies. I - A model
for the three-dimensional distribution of light in galactic disks. Astron. Astrophys. 95,
105–115 (1981).
45. Allen, C. & Santillan, A. An improved model of the galactic mass distribution for orbit
computations. Rev. Mex. Astron. Astrofis. 22, 255–263 (1991).
46. Poggio, E. et al. Evidence of a dynamically evolving Galactic warp. Nat. Astron. 4, 590–596
(2020).
47. Schönrich, R. & Dehnen, W. Warp, waves, and wrinkles in the Milky Way. Mon. Not. R.
Astron. Soc. 478, 3809–3824 (2018).
48. Xu, Y. et al. Rings and radial waves in the disk of the Milky Way. Astrophys. J. 801, 105
(2015).
49. Speagle, J. S. DYNESTY: a dynamic nested sampling package for estimating Bayesian
posteriors and evidences. Mon. Not. R. Astron. Soc. 493, 3132–3158 (2020).
50. Reid, M. J. et al. Trigonometric parallaxes of high-mass star-forming regions: our view of
the Milky Way. Astrophys. J. 885, 131 (2019).
51. Foley, M. M. et al. A 3D view of Orion: I. Barnard’s Loop. Astrophys. J. 947, 66 (2023).
52. Großschedl, J. E., Alves, J., Meingast, S. & Herbst-Kiss, G. 3D dynamics of the Orion cloud
complex. Discovery of coherent radial gas motions at the 100-pc scale. Astron.
Astrophys. Suppl. Ser. 647, A91 (2021).
53. Moré, J. J. The Levenberg-Marquardt algorithm: Implementation and theory. In Numerical
Analysis. Lecture Notes in Mathematics (ed. Watson, G. A.) 105–116 (Springer, 1978).
54. Jeffreys, H. The Theory of Probability (OUP, 1998).
55. Spitzer, L. Jr The dynamics of the interstellar medium. III. Galactic distribution. Astrophys. J.
95, 329 (1942).
56. Randall, L. & Reece, M. Dark matter as a trigger for periodic comet impacts. Phys. Rev.
Lett. 112, 161301 (2014).
57. Astropy Collaboration. The Astropy Project: Building an open-science project and status
of the v2.0 core package. Astron. J. 156, 123 (2018).
58. Beaumont, C., Goodman, A. & Greenfield, P. Hackable user interfaces in astronomy
with glue. In Astronomical Data Analysis Software an Systems XXIV (ADASS XXIV)
(eds Taylor, A. R. & Rosolowsky, E.) 101 (2015).
Acknowledgements We thank J. Carifio, S. Jeffreson, E. Koch, V. Semenov, G. Beane, M. Rugel,
S. Meingast, A. Saydjari, J. Speagle, C. Laporte, S. Bialy, J. Großschedl, T. O’Neill, L. Randall
and B. Benjamin for useful discussions. The visualization, exploration and interpretation
of data presented in this work were made possible using the glue visualization software,
supported under NSF grant numbers OAC-1739657 and CDS&E:AAG-1908419. A.A.G. and C.Z.
acknowledge support by NASA ADAP grant 80NSSC21K0634 ‘Knitting Together the Milky
Way: An Integrated Model of the Galaxy’s Stars, Gas, and Dust’. C.Z. acknowledges that
support for this work was provided by NASA through the NASA Hubble Fellowship grant
HST-HF2-51498.001 awarded by the Space Telescope Science Institute (STScI), which is
operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under
contract NAS5-26555. J.A. was co-funded by the European Union (ERC, ISM-FLOW, 101055318).
Views and opinions expressed are, however, those of the author(s) only and do not necessarily
reflect those of the European Union or the European Research Council. Neither the European
Union nor the granting authority can be held responsible for them. A.B. was supported by the
Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG;
German Research Foundation) under Germany’s Excellence Strategy – EXC-2094-390783311.
Author contributions R.K. led the work and wrote the majority of the text. All authors
contributed to the text. R.K., A.A.G., C.Z. and J.A. led the data analysis. R.K., A.A.G., C.Z., A.B.
and J.A. led the interpretation of the results, aided by M.F. and C.S. J.A., M.K. and N.M.-R. led
the compilation of the stellar cluster catalogue. R.K. and C.Z. led the statistical modelling.
R.K. and A.B. led the theoretical analysis. R.K., A.A.G. and C.Z. led the visualization efforts.
Competing interests The authors declare no competing interests.
Additional information
Supplementary information The online version contains supplementary material available at
https://doi.org/10.1038/s41586-024-07127-3.
Correspondence and requests for materials should be addressed to Ralf Konietzka.
Peer review information Nature thanks Chervin Laporte, Erik Rosolowsky and Lawrence
Widrow for their contribution to the peer review of this work. Peer reviewer reports are
available.
Reprints and permissions information is available at http://www.nature.com/reprints.

11. ExtendedDataFig.1|Comparisonbetweenatravelingandastandingwave.
Selectedphasesnapshotsareshownhere.InallPanels,thexaxiscorresponds
tothexaxisofacoordinateframeinwhichtheGalacticcoordinatex–yframe
hasbeenrotatedanticlockwiseby62°.LeftPanels: Theevolutionofthe
RadcliffeWave(stellarclusterinblue,molecularcloudsinred,thebest-fit-
modelinblack)withphase.TheSunisshowninyellow.Theverticalmotionthe
RadcliffeWaveisshowingcorrespondstoatravelingwave.Weshowinmagenta
(60°)andingreen(240°)thesamesnapshotsasinFig.2.RightPanels: The
evolutionoftheRadcliffeWavestartingfromthebest-fit-modelinblackif
theWave’smotioncorrespondedtoastandingwave.Forthebestexperience,
pleaseviewtheanimatedversioninSupplementaryFig.4andhttps://faun.rc.
fas.harvard.edu/czucker/rkonietzka/radwave/interactive-figure4.html.

12. Article
ExtendedDataFig.2|In-planeclustervelocities.Panela:Theradialin-plane
clustervelocitiesareshowninblue(inliersopaque/outlierstransparent).
ThebestfitincludingGalacticrotationaswellastheWave’ssolidbodymotion
ofabout5kms−1
intheradialdirection(radiallyawayfromtheSolarSystem,
seePanelb)isshowningreen.Thexaxiscorrespondstothex-axisofa
coordinateframeinwhichtheGalacticcoordinatex–yframehasbeenrotated
anticlockwiseby62°.Panelb(Interactive):ThebestfitoftheRadcliffeWave
isshowninblack.ThedirectionoftheWave’sradialvelocity(Panela)isshown
bythegreenvector.ThedirectionoftheWave’stangentialvelocity(Panelc)
isshownbythemagentavector.TheSunisshowninyellow.Thegrayline
correspondstothexaxisofPanelaandc.Forthebestexperience,pleaseview
theinteractiveversioninSupplementaryFig.2andhttps://faun.rc.fas.harvard.
edu/czucker/rkonietzka/radwave/interactive-figure2.html.Panelc: The
tangentialin-planeclustervelocitiesareshowninblue(inliersopaque/outliers
transparent).ThebestfitincludingGalacticrotationaswellastheWave’s
solidbodymotionofabout1kms−1
inthetangentialdirection(paralleltothe
RadcliffeWave,seePanelb)isshowninmagenta.TheeffectofGalacticrotation
isagainshowningray.ThexaxisisthesameasinPanela.Thetypicaluncertainty
(1σerrors)ofthein-planemotionisshowninblackintherightlowercorner.

13. ExtendedDataFig.3|Radial12
COVelocities.Panela:12
COradialvelocitiesof
entiremolecularcloudsalongtheRadcliffeWavederivedfromspectral-line
fittingareshowninred,includingtheir16thand84thpercentiles(1σerrors).
SinceweidentifiedthePerseusandTaurusmolecularcloudsasoutliers
(seeMethods),wedonotshowthesecloudsinthisfigure.Theprojectionof
our6Dbest-fit-modelontothe4Dphasespaceofthegasdataisshowninblack.
Wefindthatourmodelexplainstheobserved12
COspectrum,whichmeansthe
velocitiesofthemolecularclouds,forwhichwecanonlystudyanincomplete
phase space, match the motion of the young stellar clusters. The xaxis
correspondstothexaxisofacoordinateframeinwhichtheGalacticcoordinate
x–yframehasbeenrotatedanticlockwiseby62°.Panelb(Interactive):The
RadcliffeWaveisshownspatiallyin3Dincludingthemolecularcloudsinredas
wellasthebestfitinblack.TheSunisshownwithayellowdot.Theyellowlines
representasampleoflinesofsighttoillustratethewidevarietyofanglesalong
whichthe12
COradialvelocitieswerederived.Forthebestexperience,please
viewtheinteractiveversioninSupplementaryFig.3andhttps://faun.rc.fas.
harvard.edu/czucker/rkonietzka/radwave/interactive-figure3.html.

14. Article
Extended Data Table 1 | Priors on model parameters
We set our priors on our parameters to be independent for each parameter, based on initial fits. N (μ,σ) denotes a normal distribution with mean μ and standard deviation σ and U (b1,b2) denotes
a uniform distribution with lower bound b1 and upper bound b2.

15. Extended Data Table 2 | Best-fit-parameters
Best-fit-parameters for different underlying datasets and realizations are shown. See Methods section for more details. The errors are computed using the 16th, 50th and 84th percentiles of
their samples. (1) Model parameter. (2) Corresponding unit. (3) Best-fit-parameters obtained by applying our model only to the spatial molecular cloud observations. (4) Best-fit-parameters
obtained by applying our model only to the spatial and kinematic stellar cluster observations. (5) Same approach as in (4) but including a mixture model to allow for both, a standing and a
traveling wave. (6) Best-fit-parameters obtained by applying our model to the spatial and kinematic molecular cloud and stellar cluster observations. (7) Same approach as in (6) but leaving the
parameters describing the vertical oscillation frequency as free parameters.