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
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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
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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.
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Young e ical phase 3069
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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).
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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)
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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 ).
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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.
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Young e ical phase 3073
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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.
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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 ),
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