Vertical kinematics of the young Galactic clusters

MNRAS 537, 3066–3077 (2025) h ps://doi.o g/10.1093/mn as/s a 155 Ad ance Access publica ion 2025 Janua y 27 Ve ical kinema ics o he young Galac ic clus e s Emilio J. Al a o, 1 M. Ca men S ´ anchez-Gil 2 ‹and B uce Elmeg een 3 1 Ins i u o de As o ´ ısica de Andaluc ´ ıa, (CSIC), Glo ie a de la As onom ´ ıa, S/N, E-18008 G anada, Spain 2 Uni e sidad de C ´ adiz, Depa amen o de Es adis ica e I.O. A enida de la Uni e sidad N4, Je ez de la F on e a, E-11406 C ´ adiz, Spain 3 IBM Resea ch, IBM T.J. Wa son Resea ch Cen e , Yo k own Heigh s, NY, USA Accep ed 2024 Decembe 20. Recei ed 2024 Decembe 17; in o iginal o m 2024 Oc obe 14 A B S T R A C T The young disc e ical phase is pa amoun in ou unde s anding o Galaxy e olu ion. Analysing he e ical kinema ics a di e en Galac ic egions p o ides impo an in o ma ion abou he space– ime a ia ions o he Galac ic po en ial. The e ical phase snail shell s uc u e iden i ied in Gaia Da a Release 2 includes a wide age ange. Ho we e , he s uc u e o he V Z e sus Z diag am appea s linea when he analysis is limi ed o s udying objec s younge han 30 Ma. Based on he e ical eloci y and heigh -o e -disc maps ob ained o a sample o young open clus e s, his me hod also allows he ma e densi y in he sola neighbou hood o be es ima ed using a comple ely di e en app oach han p e iously ound in he li e a u e. We use wo di e en ca alogues o s a clus e s o con i m he p e ious esul and s udy new age anges. The linea pa e n be ween V Z and Z shows di e en slopes, ∂ V Z / ∂ Z, o a ious age g oups. The esul s i a simple model (ha monic oscilla o ) o in-plane decoupled e ical dynamics up o a ce ain age limi , co esponding o ∼30 Ma. This wo k also analyses he ela ionship be ween he local olume ic densi y o ma e ( ρ0 ) and he disc e ical kinema ics o di e en age anges, all below 50 Ma. The bes es ima es o he e ec i e olume ic mass densi y in he sola neighbou hood, 0.09–0.15 M pc −3 , ag ee wi h hose gi en by o he au ho s, assessing he eliabili y o he p oposed dynamical model. These alues a e a mino an o he ac ual ma e densi y in he egion. Key wo ds: me hods: s a is ical – ca alogues – Galaxy: kinema ics and dynamics – open clus e s and associa ions: gene al – sola neighbou hood – Galaxy: s uc u e. 1 INTRODUCTION The Gaia da a (Gaia Collabo a ion 2016 , 2018 , 2021 , 2023 ) ha e p o ided su p ising esul s in many b anches o as ophysics. One o he mos conspicuous was de ec ing a spi al pa e n in he e ical phase space ( V Z , Z) o disc s a s in he sola neighbou hood (An oja e al. 2018 ). Subsequen s udies ha e con i med his esul by ex ending i o a ious age, Galac ocen ic adius, and me allici y anges (Bland-Haw ho n e al. 2019 ; Li & Shen 2020 ; Li 2021 ; An oja e al. 2023 ). In all he ci ed cases, he younges age o he analysed sample exceeded 500 Ma, and he empo al in e al was o e 1 Ga. Se e al explana ions ha e been gi en o he gene a ion o his conspicuous pa e n, which can be di ided in o wo main scena ios: (a) he combina ion o a ansien and highly ene ge ic e en wi h an asymme ic Galac ic po en ial (An oja e al. 2018 ; Bland-Haw ho n & Teppe -Ga c ´ ıa 2021a ; Teppe -Ga c ´ ıa, Bland- Haw ho n & F eeman 2022 ), and (b) dynamical esonances associ- a ed, in his case, wi h he egion close o he co o a ion (Mich chenko e al. 2018 , 2019 ; L ´ epine 2022 ). In bo h cases, i is he dynamical e ec o an asymme ic Galac ic po en ial, wi h di e en bounda y condi ions, ac ing o e a wide ime span. Recen ly, Al a o e al. ( 2022 ) de e mined he ou -dimensional ( X , Y , Z , V Z ) phase subspace s uc u e o he young Galac ic disc de ined by a sample o Galac ic  E-mail: [email p o ec ed] clus e s wi h ages less han 30 Ma. The sample o young open clus e s (G-YOC) was ob ained om hese objec s’ posi ion and eloci y ca alogues compiled om Gaia da a (Soubi an e al. 2018 ; Can a - Gaudin e al. 2020 ; Ta icq e al. 2021 ). The e ical phase shown by he young clus e s (see ig. 12 o Al a o e al. 2022 ) p esen s a pa e n qui e di e en om ha o he old s a s in he Galac ic disc disco e ed by An oja e al. ( 2018 ). The e ical phase de ined by G-YOC p esen s a e y high linea co ela ion be ween V Z and Z . The samples used in bo h s udies a e also e y di e en in wo undamen al aspec s: (a) he disc s a s a e, on a e age, old objec s wi h ages exceeding he Ga, and (b) he i s de ec ion o he phase space spi al s uc u e (An oja e al. 2018 ) was es ic ed o a cylinde wi h a ci cum e ence o 200 pc adius cen ed on he Sun, while he young clus e s a e dis ibu ed in a squa e o 7 kpc side cen ed on he Sun. Fu he s udies ha e iden i ied he p esence o hese spi als o e a wide ange o Galac ocen ic adii, al hough he o ien a ion o he main axis seems o a y wi h he Galac ic adius, a kind o apsidal mo ion (see An oja e al. 2023 , and e e ences he ein). In his wo k, we wan o con i m he esul s ob ained in Al a o e al. ( 2022 ) wi h o he samples o young clus e s and p opose a scena io o gene a ing he obse ed linea pa e n. This simple model allows us o ob ain a mino an o he olume ic mass densi y in he sola neighbou hood ( ρ0 ), ep esen ing a new app oach o his p oblem. The pape is di ided in o ou sec ions, he i s o which is his in oduc ion. Sec ion 2 p esen s he new as ome ic da a ha o m he ini ial sample o ou analysis, as well as he me hodology used © 2025 The Au ho (s). Published by Ox o d Uni e si y P ess on behal o Royal As onomical Socie y. This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License ( h ps:// c ea i ecommons.o g/ licenses/ by/ 4.0/ ), which pe mi s un es ic ed euse, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 Young e ical phase 3067 MNRAS 537, 3066–3077 (2025) o ob ain he K iging maps. In Sec ion 3 , we desc ibe he dynamical model and compa e he p ope ies o he obse ed e ical phase wi h hose de i ed om he e ical accele a ion model. I also p esen s he p oposed dynamical model and es ima es he alue o he e ec i e olume ic mass densi y, while, inally, Sec ion 4 discusses he esul s and summa izes he conclusions o he wo k. 2 YOUNG OPEN CLUSTER DATA AND THE OBSERVATIONAL VERTICAL PHASE DIAGRAMS In addi ion o he G-YOC sample de ined in Al a o e al. ( 2022 ), we include he e a new sample o young open clus e s ob ained om he ca alogue o open clus e s (OCs) o Dias e al. ( 2021 ), which we call D-YOC. Dias e al. ( 2021 ) lis a se o undamen al pa ame e s o 1743 OCs in ou Galaxy. In o ma ion abou dis ance, p ope mo ions, and age has been de i ed om he Gaia Da a Release 2 da a in a homogeneous way. Ho we e , he selec ion o clus e membe s has been compiled om he li e a u e, which may include some inhomogenei ies ha may esul in a highe in e nal imp ecision o he de i ed pa ame e s. They also include he mean adial eloci y o 831 clus e s, 198 o which we e new addi ions a he ime o publica ion. The adial eloci ies, oge he wi h he as ome ic da a om he ca alogue, yield a eliable se o space eloci ies o de e mine he V Z ( X , Y ) map o he Galac ic disc in he sola neighbou hood. This ac has led us o selec he ca alogue o Dias e al. ( 2021 ) o ex end he s udy o he phase diag am o he young e ical disc. We ake he Galac ic plane as he main e e ence plane, and bo h he Ca esian space and eloci y coo dina es a e e e enced o he Sun. Tha is, ≡( X, Y , Z) = (0 , 0 , 0) pc and V ≡ ( V X , V Y , V Z ) = (0 , 0 , 0) km s −1 . The axes a e posi i e in he usual di ec ions: X owa ds he Galac ic cen e, Y in he di ec ion o Galac ic o a ion, and Z owa ds he No h Galac ic pole – he same o eloci ies. The selec ion c i e ia a e also simila o hose ha de ined he G- YOC sample: Clus e s wi hin a adius ≤3 . 5 kpc a ound he Sun, and age ≤10 7 . 7 y . He e, we ex end he age uppe limi by 0.2 dex wi h espec o G-YOC. Wi h hese selec ion c i e ia, we ob ained a sample o 485 clus e s wi h Galac ic Ca esian coo dina es and 135 wi h he h ee spa ial eloci ies. The in e es in inco po a ing new clus e ca alogs, c ea ed using a ious obus me hodologies, a ises om he necessi y o con i m whe he he linea pa e n obse ed in he ( V Z , Z ) diag am is an in insic cha ac e is ic o he young Galac ic clus e popula ion a he han an a i ac o sample bias. On he o he hand, as we will see below, he maps ob ained o di e en age subse s will allow us o es ablish a mo e con iden lowe bound o ρ0 . 2.1 The obse a ional e ical phase diag am The young Galac ic disc’s spa ial and kinema ical e ical maps, Z( X, Y ) and V Z ( X , Y ), a e es ima ed by he K iging me hod, an e x ensi e amily o spa ial es ima ion me hods o mul idimensional s ochas ic p ocesses. I op imizes exis ing knowledge by explo ing how a a ge a iable a ies in space h ough a a iog am model, conside ing spa ial co ela ion, and p o iding he bes linea unbi- ased es ima o (BLUE; Oli e & Webs e 2014 ). Al hough a mo e e x ended e xplana ion o his ma hema ical ool can be ound in he p e ious wo k (Al a o e al. 2022 , and e e ences wi hin), le us b ie ly e ie w his echnique. Fo he OC samples, we ha e { ( i , Z( i ) , V Z ( i )) } i= 1 , ... ,n measu e- men s o he a ge a iables, i.e. e ical dis ance Z( i ) and e ical eloci y V Z ( i ), a a ce ain numbe o poin s o loca ions i ( x i , y i ), o i = 1 , . . . , n , wi hin a adius o 3.5 kpc app oxima ely a ound he Sun. No e ha we will only e e o Z( i ) in he ollowing o mula ion o simplici y and cla i y. Suppose we wan o ob ain a spa ial map o such magni udes, es ima ed om a disc e e se o measu emen s a di e en posi ions in he plane i . In ha case, we need o es ima e he alue o his a iable ˆ Z ( ) a any spa ial coo dina e . This is based on he sample da a and some assump ions abou he o mula end o Z( ), i s a iance, and some spa ial co ela ion unde lying he sample da a. Rega ding Z( ) as a andom p ocess, i can be decomposed as Z( ) = m + e( ) , (1) whe e e( ) is a ze o mean andom p ocess wi h known co a iance C( h ) = E[ e ( ) e ( + h )], h ( i , j ) is he ec o sepa a ion be ween pai s o sample poin s, and m = E[ Z( )] is he mean o he p ocess, which a ies spa ially, does no depend on unde he assump ion o s a iona i y, and i can be modelled as linea unc ion (so-called Uni e sal K iging , UK) as m = p  j= 0 j ( ) βj = F β, (2) whe e j ( ) = ( j ( 1 ) , . . . , j ( n )) a e p known p edic o s o spa ial eg esso s e alua ed a each obse a ion i , and F = ( 1 ( ) , . . . , p ( )). β= ( β1 , . . . , βp ) T a e p unknown eg ession coe icien s. Gi en his model, he K iging p edic ion o Z( ) is he bes linea unbiased p edic o , gi en as ˆ Z ( 0 ) = ( 0 ) ˆ β+ T V −1 ( z( ) −F ˆ β) . (3) The i s e m is he es ima ed mean alue o loca ion 0 , whe e ( 0 ) = ( 1 ( 0 ) , . . . , p ( 0 )), and ˆ β= ( F T V −1 F ) −1 F T V −1 z( ) is he gene alized leas -squa es es ima e o he end coe icien s β. The second e m is a weigh ed mean o he esiduals om he mean unc ion, whe e T V −1 a e known as he K iging weigh s, = Co ( Z( 0 ) , z( )), V = Co ( e( )) is he known co a iance ma ix o z ( ), and z ( ) = ( Z ( 1 ) , . . . , Z ( n )) T . The p edic ion a iance, o K iging a iance, is compu ed as σ2 Z ( 0 ) = Va ( Z( 0 ) −ˆ Z ( 0 )) = Va ( Z( 0 )) − T V −1 +  ( 0 ) − T V −1 F ( F T V −1 F ) −1  ( 0 ) − T V −1 F T . (4) Besides he s a iona i y assump ion, which implies ha m = E[ Z( )] does no depend on and C( h ) only depends on he dis ance (iso opy), i is also assumed ha E[ Z( + h ) −Z( )] = 0. This is known as he in insic model, a weak hypo hesis equi ed in many p ac ical si ua ions o compu ing he a iance o he p ocess ha yields Va [ Z( + h ) −Z( )] = 2 γ( h ) , (5) whe e γ( h ) is he a iog am o he andom p ocess, and i is assumed o measu e he spa ial co ela ion in he ac ual ealiza ion o Z( ). The e o e, i s compu a ion is a c ucial s ep o a co ec o eliable es ima ion since i plays an essen ial ool p o iding he spa ial dependence o spa ially co ela ed a iables, and i le s us Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 3068 E. J. Al a o, M. C.Sanchez-Gil and B. Elmeg een MNRAS 537, 3066–3077 (2025) es ima e he alue o he spa ial a iable a an unsampled loca ion, ˆ γ( h ) = 1 2 p( h ) p( h )  j= 1  Z( j ) −Z( j + h )  2 , (6) whe e p( h ) is he numbe o poin pai s sepa a ed by a dis ance h ∈ R (in he pa icula case unde iso opy). Appendix A1 explains in mo e de ail i s es ima ion and model selec ion om he c oss- alida ion be ween se e al heo e ical a iog am models. We mainly chose he sphe ical models o bo h a iables, Z and V Z . Only in a ew cases is he exponen ial model sligh ly be e . Once he a iog am es ima o assessmen ˆ γ( h ) is done, p o iding he spa ial dependence o au oco ela ion in ou da a, we can inally pe o m he las s age o he K iging app oach o use he da a o make p edic ions wi h equa ion ( 3 ), le ing us c ea e a con inuous su ace o he phenomenon, as we can see in he le panels o Fig. 1 . These es ima ions we e pe o med on a spa ial g id encompassing he sampled a ea, speci ically a egion spanning 7 kpc a ound he Sun, di ided in o s eps o 10 pc along each axis. The calcula ions we e based on he i ed semi a iog am model, equa ions ( A2 )–( A5 ), and he spa ial a angemen o he nea es inpu da a. This app oach esul s in a modelled Galac ic plane su ace g id comp ising 700 × 700 poin s, wi h he co esponding es ima ed alues o Z and V Z , espec i ely. Fig. 1 p esen s he e ical phase diag ams o he ou age g oups o he D-YOC sample (in blue-g een colou poin densi y maps) and G-YOC da a (in o ange) p e iously compu ed in Al a o e al. ( 2022 ). The age in e als ha e been chosen so ha he cen al age alue sho ws a lo w dispe sion and can be assimila ed in o a coe al sample. The D-YOC sample was di ided in o ou sub-samples by age o analyse he p ope ies o he linea ela ionship be ween hese wo a iables wi h age and, in pa icula , o es ima e he sample’s age, a e which he linea pa e n disappea ed. We can see ha he sample aged be ween 30 and 50 Ma shows no linea s uc u e and seems o depic an incipien spi al pa e n, as obse ed in olde s a s (An oja e al. 2018 , 2023 ). Table 1 summa izes he sub-samples used in ou s udy, cha ac- e ized by hei age. I shows he main s a is ics o he o iginal sub- samples, he K iged es ima ed Z( X, Y ) and V Z ( X, Y ) maps, and he linea i o he ( V Z , Z) dis ibu ions. As we inc ease he cen al alue o he age o he ime in e al o he sample, he dispe sion o he ( V Z , Z ) linea dis ibu ion inc eases. The linea ela ionship disappea s o he oldes age g oup [log( ) be ween 7.5 and 7.7], and a quasi-ellip ical, o spi al, shape is d awn by he denses zones. The diag ams show he linea i o he dis ibu ion as a solid line. This i has been ob ained wi h a weigh p opo ional o he su ace densi y in he e ical phase space. Compa ing he e ical phase o age g oup log ( ) ≤7 . 5 o bo h samples (G-YOC and D-YOC), i is clea ha he dispe sion o he dis ibu ion is highe o he D-YOC sample. This esul should no be su p ising as he e a e ze o-poin and scale di e ences be ween he wo samples in bo h age and dis ance a iables (Dias e al. 2021 ). The mos conspicuous age di e ences a e ound o he younges clus e s. Ho we e , he slope o he e ical phase space dis ibu ion is e y simila o bo h diag ams. Two obse a ional conclusions ollow: he e ical phase space shows a linea shape o objec s younge han 30 Ma, and he dis ibu ion slope dec eases wi h age un il ∼40 Ma, whe e he linea s uc u e e ol es in o mo e complex shapes. Ve y young objec s (age ≤30 Ma) show a linea e ical phase pa e n, while he disc s a s, wi h an a e age age olde han 1 Ga, display a snail-shell s uc u e (An oja e al. 2023 ). Mo eo e , supe imposed on he linea dis ibu ion obse ed in he Z−Y and V Z –Y maps, cha ac e ized by a highe poin concen a ion (middle panels o Figs 1 and 2 a), he e is a w a e-lik e s uc u e o g ea e ampli ude wi h an appa en wa eleng h o app oxima ely 1 kpc. Included in his w a e-lik e s uc u e a e he Gould Bel young clus e s wi hin 500 pc o he Sun shown as cyan- illed iangles in Fig. 2 (a) ( om Dzib e al. 2018 ). This beha iou is unsu p ising, as e ical co uga ions along he Y -axis a e well documen ed. These co uga ions o igina e om he undula ions obse ed in se e al spi al a ms, whose p edominan o ien a ion uns pa allel o he Y - axis in his egion o he Galac ic plane. The exis ence o such co uga ed a ms has been ecognized o o e 50 y (see Dixon 1967 ; Va sa sky & Qui oga 1970 ; Qui oga 1974 ; Lockman 1977 ; Spicke & Fei zinge 1986 ; Al a o, Cab e a-Cano & Delgado 1991 , 1992 , among o he s) and has been in e p e ed h ough a ious heo e ical amewo ks (Nelson 1976 ; Weinbe g 1991 ; Al a o & E emo 1996 ; Bland-Ha w ho n & T eppe -Ga c ´ ıa ; T eppe -Ga c ´ ıa e al. 2022 , and e e ences he ein). Ne e heless, his s udy concen- a es on he ze o-o de e ms, ep esen ing he linea co ela ions be ween Z –Y and V Z –Y , esul ing in a linea dis ibu ion in he V Z –Z plane. Assessing how well he ini ial da a align wi h he su aces p o ided by Z ( X , Y ) and V Z ( X, Y ) K iging es ima ion is c ucial o alida ing he obus ness o ou me hodology. We ake he ma ix da a ( X , Y , ˆ Z , σZ , ˆ V Z , σV Z ) as he bes ep esen a ion o ou disc. Fig. 2 , simila o ig. 12 o Al a o e al. ( 2022 ), shows he V Z –Y , Z –Y , and V Z – Z diag ams, whe e he colou maps ep esen he densi y o poin s on he diag am. We plo ed he sample G-YOC (black poin s) wi h eloci y and posi ion da a in his image. We ha e d awn he su ace densi y con ou s o hese objec s (da k lines) supe imposed in he las diag am o enhance he linea i y gi en by he dense egions. In ac , despi e he appa en ou sp ead, he ligh e colou co esponds o densi y alues below he 10 h pe cen ile o he densi y poin s’ dis ibu ion. Mo eo e , hese diag am maps clea ly show ha he sample poin densi y maxima co esponds o he es ima ed poin densi y maxima. In o he wo ds, he V Z –Z ela ion de i ed om he models no only ep esen s he sample poin dis ib u ion b u also highligh s he global beha iou o he Galac ic disc. I we d aw on his same plo he s a - o ming egions wi hin a adius o 500 pc used by Dzib e al. ( 2018 ) o analyse he Gould Bel , we see ha he poin s (g een iangles) align pe ec ly wi h ou map’s egions o maximum densi y. To p o ide a clea e unde s anding o he unce ain ies associa ed wi h de i ing he slopes m ( V z /Z) om he K iging maps, we include Fig. 2 (b), which shows he V Z −Z ela ionship ob ained o he G- YOC da a. The le panel displays he G-YOC sample poin s (black ci cles) o e laid on he dis ibu ion de i ed om he K iging maps. The pixel colou ep esen s he su ace densi y o poin s, wi h a bin size o 2.5 pc in Z and 0.25 km s −1 in V Z . Despi e he dispe sion o he sample poin s, he a eas o he highes clus e concen a ion coincide wi h he densi y maxima o he K iging maps. Addi ionally, we include (g een iangles) he s a - o ming egions associa ed wi h he Gould Bel (Dzib e al. 2018 ), which a e cons ained o wi hin a 500 pc adius a ound he Sun. In he igh panel, we inco po a e a new sample o clus e s selec ed om he Hao e al. ( 2021 ) ca alogue using he same c i e ia as o G-YOC. The dis ibu ion o hese clus e s ollows a pa e n consis en wi h he G-YOC sample, wi h he highes concen a ion loca ed a he same posi ion as he densi y maxima in he K iging maps. These plo s indica e ha he V Z −Z dis ibu ion de i ed om he K iging maps p o ides a eliable and obus ep esen a ion o he Galac ic plane as de ined by he sample clus e s. Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 Young e ical phase 3069 MNRAS 537, 3066–3077 (2025) Figu e 1. Le panels: Z( X, Y ) and V Z ( X, Y ) K iged es ima ed maps o he D-YOC sample (Dias e al. 2021 ), di ided by age in o ou sub-samples, and G-YOC sample (Al a o e al. 2022 ). On op o hese maps, he illed poin s co espond wi h he loca ions and alues o he o iginal sample da a. Colou maps a e se o a common scale o make s aigh o wa d compa isons. V Z ( X, Y ) we e binned in o a disc e e scale due o hei small ange o alues, enhancing he eloci y g adien s ound. The h ee black lines co espond o, om le o igh , Pe seus, Local, and Sagi a ius spi al a ms (Cas o-Gina d e al. 2021 ), wi h ixed a ms’ wid h based on Reid e al. ( 2014 ). The Sun is a (0, 0). Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 3070 E. J. Al a o, M. C.Sanchez-Gil and B. Elmeg een MNRAS 537, 3066–3077 (2025) Figu e 1. ( Con inued ). Middle and igh panels: The K iged es ima ed V Z and Z e sus Y , and V Z e sus Z diag ams, espec i ely, o he (D-YOC) and (G-YOC) samples, ep esen ed as hea maps o 2D bin (o 10 pc ×1 km s −1 ) coun s. The highe densi y con ou lines a e also plo ed a he igh panels. Despi e he appa en wide ou sp ead, he ligh e colou s co espond o alues al w ays below he 10 h pe cen ile pe cen ile o he poin s’ densi y dis ibu ion. The a iable anges a e he same o all he plo s, so we can easily compa e he di e en age sub-samples. 3 THE VERTICAL ACCELERATION IN A YOUNG, THIN DISC: COMPARISON WITH OBSERVATIONAL VERTICAL PHASE DIAGRAMS The sample o clus e s used in his wo k is aged less han log ( ) ≤7 . 7 and de ines a young, hin disc wi h | Z| alues less han 200 pc. Fo his ange o Z, and assuming a cons an o a ional eloci y in he egion and a s eady and axisymme ic po en ial, i is sa is ied ha (a) he e ical accele a ion K Z depends only on Z since he in-plane mo ions a e decoupled om he e ical mo ion, and (b) K Z ≈−4 πG ρ0 Z, which ollows om he in eg a ion o Poisson’s equa ion unde he bounda y condi ions de ined by ou clus e sample. ρ0 is he e ec i e g a i a ional mass densi y ep esen a i e o he analysed disc olume (i.e. Bahcall 1984 ; Binney & T emaine 1987 ). This linea ela ionship be ween he e ical accele a ion and he e ical dis ance o he Galac ic plane is co obo a ed by p e ious dynamic s udies o he hin Galac ic disc o | Z| ≤200 pc (i.e. Siebe , Bienaym ´ e & Soubi an 2003 ; Salomon e al. 2020 ). Unde hese assump ions, we can w i e K Z = −2 Z, (7) whe e 2 = 4 πGρ0 , wi h G = 4 . 3009 ×10 −3 pc M −1 km 2 s −2 and ρ0 gi en in M pc −3 . Sol ing equa ion ( 7 ), we ge he mo ion equa ions as Z( ) = Z 0 cos (  ) + V 0 sin (  ) = A cos (  + φ) , (8) V Z ( ) = V 0 cos (  ) −Z 0 sin (  ) = −A sin (  + φ) , (9) whe e he ime is measu ed in Ma, Z ( ) in pc, and V Z ( ) in km s −1 . Besides, a any ime , i holds ha Z 2 ( ) + V 2 Z ( ) 2 = A 2 , (10) Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 Young e ical phase 3071 MNRAS 537, 3066–3077 (2025) Table 1. Summa y o he sample da a o di e en age anges, hei es ima ed K iging maps Z ( X , Y ) and V Z ( X, Y ), and he e ical phase slope linea i s. Summa y o obse a ions Summa y o K iging es ima ions Fi ed slope Subse Age (Ma) Z (pc) V Z (km s −1 ) m ( pc −1 km s −1 ) R 2 (pe cen ) # Da a log ( )   ±σ P 10 −P 90  Z ±σZ IQR  V Z  ±σV Z IQR 1 D-YOC < 7 7.40 ±1 . 52 (5.48, 9.62) −25 . 25 ±34.26 ( −43 . 80 , −5 . 57) −8 . 11 ±1 . 80 ( −9 . 71 , −6 . 62) 0.104 ±2 . 04 e −2 98.67 2 D-YOC 7 −7 . 5 17 . 15 ±6 . 01 (11 . 56 , 26 . 79) −20 . 299 ±40 . 51 ( −49 . 22 , 7 . 27) −7 . 28 ±1 . 55 ( −8 . 41 , −6 . 16) 0 . 045 ±1 . 03 e −2 78.26 3 D-YOC < 7 . 5 13.95 ±8 . 28 6 . 338 −24 . 95 −20 . 83 ±39.70 ( −46 . 99 , 5 . 49) −7 . 78 ±1 . 73 ( −9 . 15 , −6 . 38) 0 . 046 ±6 . 75 e −3 47.71 4 G-YOC < 7 . 5 17.66 ±6 . 78 6 . 31 −31 . 62 −17 . 54 ±45.98 ( −49 . 21 , 14 . 64) −8 . 52 ±1 . 62 ( −9 . 73 , −7 . 20) 0 . 035 ±1 . 16 e −3 99.3 5 D-YOC 7 . 5 −7 . 7 40.78 ±5 . 22 33 . 20 −46 . 92 −3 . 96 ±23.05 ( −21 . 15 , 12 . 16) −7 . 28 ±1 . 56 ( −8 . 41 , −6 . 16) −9 . 78 e −4 ±2 . 65 e −3 0.05 V Z ( ) Z( ) = − an (  + φ) . (11) The ampli ude and phase ( A , φ) a e ela ed wi h he cons an s o he mo ion equa ions ( 8 ) and ( 9 ) as ollows: Z 2 0 + V 2 0 2 = A 2 , V 0 Z 0 = − an ( φ) . (12) One way o look a equa ions ( 8 ) and ( 9 ) is o conside a dis ibu ion o coe al clus e s synch onized in he Y di ec ion, i.e. which is mos ly along he local spi al a m, ha d i e ically om he same ini ial posi ion, Z(0), s a ing a he same ime, bu wi h sligh ly di e en eloci ies, V Z (0). O e ime, hese clus e s will sp ead ou in Z , wi h he ini ially as e ones mo ing o g ea e Z . Fo example, i hey all s a in he mid-plane wi h ne ga i e V Z , hen φ= π/ 2. The ini ial dispe sion in e ical eloci y is ep esen ed by a dispe sion in A , V Z = A, and Z = 0 ini ially. A e a ime and hey sp ead ou , he il o hei dis ibu ion on a ( V Z , Z) plane is V Z Z = −co (  ) . (13) Fo  << 1, his a io is −1 / , and he ex en o hei heigh is Z( ) = A . The slopes o he clus e comple x es on he ( V Z , Z) plane, m , a e gi en in Table 1 . In his example, he p oduc o he a e age clus e age and he slope should equal  co (  ) wi h Z 0 = 0. Fo ep esen a i e m = 0.03 pc −1 km s −1 and   ∼10 Ma, m  ∼ 0.3, and hen  is sol ed o be ∼1.3. Di iding his by he age gi es ∼ 0.13 Ma −1 , and a co esponding a e age mid-plane densi y ρ0 ∼0 . 3 M pc −3 . These olume ic densi y alues, compu ed as abo e, a e highe han mos o he calcula ed by o he au ho s (e.g. Hughes 1974 ; Veede 1974 ; Binney & T emaine 1987 ). A de ailed i o he dis ibu ion o clus e eloci ies and posi ions equi es bo h ini ial posi ions and eloci ies. Fig. 3 sho ws se e al solu ions o equa ions ( 8 ) and ( 9 ) wi h sligh ly a ying pa ame e s, including a easonable i wi h  = 0 . 8, = 7 . 4 Ma, Z 0 = 37 pc, and V 0 in he ange om −4 o −12 km s −1 , which goes h ough he ci cles ep esen ing he main peaks in Fig. 1 . The di e en clus e s o cu es a e o di e en ages (lowe ages o he lowe igh , highe ages o he uppe le ), and di e en s a ing posi ions (highe posi i e Z 0 o he lowe le and lowe posi i e Z 0 o he uppe igh ). Fo each clus e o cu es, he dashed, solid, and do ed lines a e o  = 1 , 0.9, and 0.8. The bes i wi h  = 0 . 9 and = 7 . 4 Ma implies a mid-plane densi y o 0.22 M pc −3 . F om equa ion ( 10 ), V Z can be ew i en as a unc ion o Z ( ), and we can de i e he unc ion slope a any ime as m ( ) = ∂ V Z ( ) ∂ Z( ) =  an (  + φ) ( pc −1 km s −1 ) (14) = 1 − an (  ) an ( φ) an (  ) + an ( φ) , (15) = − an (  ) V 0 Z 0 an (  ) + V 0 Z 0  . (16) We ha e ha he slope o he e ical phase diag am o a se o coe al objec s bo n nea he Galac ic plane sa is ies he condi ions o hei e ical mo ion o obey he equa ion ( 7 ). In o he wo ds, s a s o clus e s o a ce ain age , bo n a a simila oscilla o y phase ( φ), wi h | Z 0 | << 200 pc, will o m a s aigh line in he ( V Z , Z ) diag am wi h a slope gi en by equa ion ( 14 ). Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 3072 E. J. Al a o, M. C.Sanchez-Gil and B. Elmeg een MNRAS 537, 3066–3077 (2025) Figu e 2. K iging maps o he OC sub-sample wi h log ( age ) < 7 . 5 (Al a o e al. 2022 ). Va ious OC samples om he li e a u e, including he o iginal one, a e plo ed on hese maps o e alua e hei consis ency wi h he K iging model’s da a dis ibu ion. 3.1 Model obse a ions’ compa ison The slope o he ( V Z , Z ) diag am is a unc ion o ime and phase φ, which implici ly depends on he ini ial condi ions and he e ec i e g a i a ional mass densi y in he egion. Only o a se o coe al objec s a he same oscilla o y phase φcan a linea dis ibu ion be obse ed in he e ical phase diag am, and ye his is wha we see in he ( V Z , Z ) planes de ined by he younge clus e s. In he le panel o Fig. 4 , we show m ( ) o i e di e en alues o ρ0 wi h φ= 0. The es ima ed alues o m ( ) o he di e en age g oups a e supe imposed in he plo . The D-YOC V Z −Z diag am o log ( ) be ween 7.5 and 7.7 de ia es om linea i y, and he co ela ion coe icien is almos ze o. The cen al alue o he age is 40 Ma, which seems o de ine he ime limi o which he linea dis ibu ion deg ades and a mo e s uc u ed pa e n appea s. Thus, gi en he obse ed dis ibu ion, he slope ob ained does no ep esen a ho izon al line bu a low co ela ion coe icien . In he le panel o Fig. 4 , we ep esen he model slopes (equa ion 14 ) o h ee di e en heo e ical s ella densi ies ρ0 , based on he dynamical mass densi y in he sola neighbou hood es ima ed in he ange 0.09–0.12 M pc −3 by Binney & T emaine ( 1987 ), 0 . 14 M pc −3 by Hughes ( 1974 ), o up o 0.18 by Veede ( 1974 ). We added he smalles alues, ρ0 = 0 . 03 and ρ0 = 0 . 055, o illus a i e pu poses o check he equi ed alue o achie e he oldes sub-sample o OCs. The la e is indeed excluded om i ing his pa ame e , as shown in he igh panel o Fig. 4 since hei V Z −Z ela ionships depa om linea i y as al eady discussed abo e. The black poin s ep esen he di e en slopes measu ed o he ou sub-samples om D-Y OC and G-Y OC samples, and he e o ba s conside bo h he K iging and model i ing unce ain ies. Table 2 shows hese i s. Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 Young e ical phase 3073 MNRAS 537, 3066–3077 (2025) Figu e 3. Se e al solu ions o equa ions ( 8 ) and ( 9 ) wi h sligh ly a ying pa ame e s,  , in Ma, Z 0 in pc, and V 0 a ying along he cu es in he ange om −4 o −12 km s −1 . The ρ0 es ima es, al hough wi h a la ge unce ain y gi en he limi ed numbe o poin s o a non-linea i , ep oduce he alues published by o he au ho s. This ag eemen seems o suppo he in e nal consis ency o ou dynamic model. 4 DISCUSSION Ou s udy begins wi h a sample o young clus e s, log(age) < 7 . 7, wi hin a squa e ( X, Y ) o 7 kpc on each side om a ca alogue wi h well-de e mined dis ances. This ca e ul selec ion ensu es he eliabili y o ou da a om he ou se . We conside hese objec s o Table 2. Summa y o he pa ame e es ima ion om model (equa ion 14 ). No. o samples Weigh s ˆ ρ0 RMSE 4 None 0.15 ±1.75 0.0377 σ−1 0.13 ±0.101 0.0377 3 None 0.094 ±2.4 0.0313 σ−1 0.092 ±0.108 0.0313 be samples o he 3D s uc u e o he young disc, which is a disc e e collec ion o 3D objec s. Applying he K iging me hodology, we ob ain he BLUE su aces Z( X , Y ) and V Z ( X , Y ), which ep esen ou Galac ic disc model. The ac ha he dis ibu ion o sample poin s is no homogeneous only means ha he es ima e’s unce ain y will be g ea e in he a eas o lowe sample densi y. This me hodology also es ima es he unce ain ies σZ and σV Z a each loca ion ( X, Y ) o he g id ma ix, whe e Z( X, Y ) and V Z ( X, Y ) alues ha e been compu ed. This unce ain y encompasses di e en aspec s, such as he Z heigh scale o he clus e s’ dis ibu ion, he spa ial dis ibu ion o he sample, and he sampling unce ain y o he a iables Z and V Z (see Al a o e al. 1991 , o mo e de ails). We ecall ha his modelling is ou bes ep esen a ion o he Galac ic disc, and he ma ices Z( X, Y ) and V Z ( X, Y ) se up he da a base on which we analyse he V Z −Z ela ionship. The spa ial dis ibu ion o he ini ial da a can be in e p e ed in di e en ways, ei he as h ee con inuous spi al a ms o as a disc e e collec ion o s a - o ming egions, whe e each g ouping con ains o he smalle ones in a cascade ep esen a i e o hie a chical s a o ma ion, as we al eady discussed in he p e ious wo k Al a o e al. ( 2022 ) whe e we analysed he e ical s uc u e along di e en lines Figu e 4. Le : Models e sus da a compa ison o he slope be ween V Z and Z, m ( ) (equa ion 14 ). The di e en a e aged s ella densi y ρ0 alues o he mid-plane a e examined om he li e a u e. The black poin s ep esen he es ima ed ˆ m ( ) o he YOC samples, oge he wi h hei mean ages (see Table 1 ). Righ : The model es ima ion o i de i ed om ou da a poin s, emo ing he olde s ella popula ion, i.e. he sub-sample wi hin he age ange 7 . 7 ≤log(age) ≤7 . 7 . In blue a e he co esponding i s o he ou poin s: non-weigh ed (con inuous line) and weigh ed (dashed line). In he same way, he ed lines ep esen he co esponding i s o he h ee poin s. The shaded colou s ep esen he oo mean squa ed e o (RMSE) om he di e en models. Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025 3074 E. J. Al a o, M. C.Sanchez-Gil and B. Elmeg een MNRAS 537, 3066–3077 (2025) on he plane and i s compa ison wi h obse a ional da a om o he au ho s. We now conside a su ace, no a disc e e collec ion o lines o egions on he plane. Fo ins ance, a linea ela ionship also de i es om Z−Y and V Z −Y , whe e Z( Y ) and V Z ( Y ) a e ob ained p ojec ing Z( X, Y ) and V Z ( X, Y ) on he Y axis collapsing he X axis (see Figs 1 and 2 ). The Galac ic quad an s I and II show mo e posi i e alues han quad an s III and IV o bo h a iables. This obse a ional ac de i ed he e o he young plane (age < 30 Ma) has been p e iously de ec ed in o he s udies (e.g. Rome o-G ´ omez e al. 2019 ). I appea s o be a global phenomenon o he disc, on which small local a ia ions a e obse ed, which could ep esen bubbles, co uga ions, o o he phenomenologies associa ed wi h ac i e s a - o ming egions. I is wo h highligh ing ha he s a - o ming egions associa ed wi h he Gould Bel (Dzib e al. 2018 ) align closely wi h he dense egions o he V Z −Z diag am (cyan iangles in Fig. 2 a). Ho we e , his alignmen does no ex end o he la ge scale V Z −Y and Z –Y diag ams, whe e he Gould Bel da a exhibi a signi ican ly s eepe dis ibu ion han he K iging maps’ main da a se . Li e al. ( 2023 ) ha e ecen ly analysed he V Z −Z diag ams o h ee di e en s ella popula ions, inding oughly linea ela ionships be ween he wo a iables. These au ho s in e p e hese diag ams in e ms o Galac ic wa p. In e es ingly, he dis ibu ion o ho , massi e s a s o spec al ypes O o ea ly- ype B, OB s a s, wi h an a e age age simila o he D-YOC sample, p esen s a V Z −Z g adien simila o he one ound in his wo k o | Z| alues less han 200 pc. We p opose ano he in e p e a ion o ou young OCs’ da a based on he simples dynamical model, he ha monic oscilla ion o each objec . Fo a g oup o coe al s ella objec s subjec ed o his po en ial and bo n in phase, we expec a linea ela ionship be ween V Z and Z, he slope o which depends on he age o he objec s, he phase, and he e ec i e mass densi y in ha olume. As ime p og esses and he sample’s age ange inc eases, he slope o he linea ela ionship changes and becomes noisie un il i eaches a ce ain age [ log ( ) > 7 . 5], a which poin he linea s uc u e b eaks do wn, gi ing ise o a quasi-ellip ical dis ibu ion. We can conside ha an indi ec p oo ha his simple model explains he obse a ions is ha he olume ic mass densi ies ob ained om his model a e compa ible wi h hose ob ained by o he au ho s. Ho we e , we de e mine hem wi h g ea unce ain y. Ano he ac ha cons ains he p oblem is he Galac ic po en ial, which bes desc ibes he physics o he sys em. Many p e ious s udies conside a decoupling be ween in-plane mo ions and e ical mo ions unde ce ain condi ions. Fo | Z| < 200 pc, he e ical accele a ion K Z is p opo ional o Z (Salomon e al. 2020 ), as long as we a e in a egion o he Galaxy whe e he o a ional eloci y can be conside ed cons an . In o he wo ds, we can add wha e e we wan o he e ical Galac ic po en ial, such as spi al a ms, ansien injec ions o momen um and ene gy, e c., bu i is e iden ha he e ical ha monic po en ial is he basic ing edien . Unde his simple dynamical model, we can explain he linea co ela ion be ween V Z and Z o coe al clus e s and he slope a ia ion wi h age. The alues o ρ0 d i ing he oscilla ion equency a e compa ible wi h p e ious es ima es ob ained wi h di e en me hods. We also ind ha V Z −Z, o clus e s olde han 40 Ma, de ia es om linea i y, sho wing a densi y dis ibu ion ha mo e closely esembles an eme ging spi al o ellipse on he V Z −Z plane. Fo ages olde han 40 Ma, he dynamic p oblem needs ing edien s o he han he simple ha monic oscilla o . In pa icula , he linea dis ibu ions be ween V Z −Y and Z−Y (Fig. 1 , middle e ical panel, and Fig. 2 a, op igh and cen al panels) canno be simply explained by a Galac ic po en ial as K Z ≈−2 Z o clus e s olde han 40 Ma. Finally, we conclude ha o ages less han 40 Ma, he mos signi ican e ical d i ing o ce de i es solely om he ha monic oscilla o po en ial, gene a ed by he in-plane ma e dis ibu ion and cha ac e ized by an e ec i e mass densi y ρ0 . The obse ed linea beha iou suppo s he hypo hesis ha clus e s o simila age a e o med wi h a compa able phase in his oscilla ion. The de ia ion om linea i y obse ed in olde clus e s may be a ibu ed o g ea e phase dispe sion o e ime in hese objec s caused by ansien phenomena. Such phenomena would in oduce highe o de and non- s a iona y e ms in o he Galac ic po en ial, dis up ing he decoupling be ween he e ical phase and in-plane mo ion. ACKNOWLEDGEMENTS Help ul commen s om he e e ee a e g a e ully acknowledged. This wo k has been pa ially unded by he Spanish MCIN/AEI g an PID2022-136640NB-C21. EJA acknowledges inancial suppo om he Se e o Ochoa g an CEX2021-001131-S unded by MCIN/AEI. EJA has made con inuous use o TOPCAT (Taylo 2005 ). This wo k p esen s esul s om he Eu opean Space Agency (ESA) space mission’s Gaia da a. Gaia da a a e being p ocessed by he Gaia Da a P ocessing and Analysis Conso ium (DPAC). Funding o he DPAC is p o ided by na ional ins i u ions, in pa icula , he ins i u ions pa icipa ing in he Gaia Mul iLa e al Ag eemen (MLA). The Gaia mission websi e is h ps://www.cosmos.esa.in /gaia . The Gaia a chi e websi e is h ps:// a chi es.esac.esa.in / gaia . DATA AVAILABILITY The da a unde lying his a icle we e accessed om Dias e al. ( 2021 ), a h ps:// doi.o g/ 10.1093/ mn as/ s ab770 , and Al a o e al. ( 2022 ) a DOI: 10.3847/ 1538-4357/ ac8b0c . The de i ed da a gene a ed in his esea ch will be sha ed on easonable eques o he co esponding au ho . REFERENCES Al a o E. J. , E emo Y. N., 1996, in Falco E., Fe nandez J. A., Fe e o R. F., eds, Re is a Mexicana de As onomia y As o isica Con e ence Se ies, Vol. 4. p. 1 Al a o E. J. , Cab e a-Cano J., Delgado A. J., 1991, ApJ , 378, 106 Al a o E. J. , Cab e a-Cano J., Delgado A. J., 1992, ApJ , 399, 576 Al a o E. J. , Jim ´ enez M., S ´ anchez-Gil M. C., S ´ anchez N., Gonz ´ alez M., Ma ´ ız Apell ´ aniz J., 2022, ApJ , 937, 114 An oja T. e al., 2018, Na u e , 561, 360 An oja T. , Ramos P., Ga c ´ ıa-Conde B., Be ne M., Lapo e C. F. P., Ka z D., 2023, A&A , 673, A115 Bahcall J. N. , 1984, ApJ , 276, 169 Binney J. , T emaine S., 1987, Galac ic Dynamics. P ince on Uni . P ess, P ince on, NJ Bland-Haw ho n J. , Teppe -Ga c ´ ıa T., 2021a, MNRAS , 504, 3168 Bland-Haw ho n J. e al., 2019, MNRAS , 486, 1167 Can a -Gaudin T. e al., 2020, A&A , 640, A1 Cas o-Gina d A. e al., 2021, A&A , 652, A162 Dias W. S. , Mon ei o H., Moi inho A., L ´ epine J. R. D., Ca a o G., Paunzen E., Alessi B., Villela L., 2021, MNRAS , 504, 356 Dixon M. E. , 1967, AJ , 72, 429 Dzib S. A. , Loina d L., O iz-Le ´ on G. N., Rod ´ ıguez L. F., Galli P. A. B., 2018, ApJ , 867, 151 Gaia Collabo a ion , 2016, A&A , 595, A1 Gaia Collabo a ion , 2018, A&A , 616, A10 Gaia Collabo a ion , 2021, A&A , 649, A1 Gaia Collabo a ion , 2023, A&A , 674, A1 Hao C. J. e al., 2021, A&A , 652, A102 Downloaded om h ps://academic.oup.com/mn as/a icle/537/4/3066/7984502 by Consejo Supe io de In es igaciones Cien i icas (CSIC) use on 04 Ma ch 2025