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The Radcliffe Wave Of Milk Way is oscillating
The document discusses the Radcliffe Wave, a sinusoidal chain of gas clouds within 300 parsecs of the Milky Way, demonstrating evidence of its oscillatory motion and outward drift from the galactic center. Using three-dimensional mapping, the study shows star clusters associated with the wave move coherently as part of this oscillation, providing insights into the galactic potential and the formation of local star structures. The findings also measure the sun's oscillation period and suggest a significant interaction with both baryonic and dark matter in the local galactic dynamics.
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The Radcliffe Wave Of Milk Way is oscillating
- 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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- 5. Methods Stellarclustercatalogue The basis ofthe 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 theoriginal 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 withthesamefreeparametersθ1whilefixingB = 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 the1σ 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 Gρ 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.
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- 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, matchthe 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 Table1 | 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 Table2 | 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.