Gravitational instability in a planet-forming disk

58 | Nature | Vol 633 | 5 September 2024
Article
Gravitationalinstabilityinaplanet-forming
disk
Jessica Speedie1✉, Ruobing Dong1,2✉, Cassandra Hall3,4
, Cristiano Longarini5,6
,
Benedetta Veronesi7
, Teresa Paneque-Carreño8,9
, Giuseppe Lodato5
, Ya-Wen Tang10
,
Richard Teague11
& Jun Hashimoto12,13,14
Thecanonicaltheoryforplanetformationincircumstellardisksproposesthat
planetsaregrownfrominitiallymuchsmallerseeds1–5
.Thelong-considered
alternativetheoryproposesthatgiantprotoplanetscanbeformeddirectlyfrom
collapsingfragmentsofvastspiralarms6–11
inducedbygravitationalinstability12–14
—
ifthediskisgravitationallyunstable.Forthistobepossible,thediskmustbemassive
comparedwiththecentralstar:adisk-to-starmassratioof1:10iswidelyheldasthe
roughthresholdfortriggeringgravitationalinstability,incitingsubstantialnon-
Kepleriandynamicsandgeneratingprominentspiralarms15–18
.Althoughestimating
diskmasseshashistoricallybeenchallenging19–21
,themotionofthegascanrevealthe
presenceofgravitationalinstabilitythroughitseffectonthedisk-velocitystructure22–24
.
HerewepresentkinematicevidenceofgravitationalinstabilityinthediskaroundAB
Aurigae,usingdeepobservationsof13
COandC18
OlineemissionwiththeAtacama
LargeMillimeter/submillimeterArray(ALMA).Theobservedkinematicsignals
stronglyresemblepredictionsfromsimulationsandanalyticmodelling.From
quantitativecomparisons,weinferadiskmassofuptoathirdofthestellarmass
enclosedwithin1″to5″onthesky.
We targeted the disk around AB Aurigae (hereafter AB Aur), a
2.5–4.4-Myr-old25–28
Herbig Ae (ref. 29) star of intermediate mass
(M⋆ = 2.4 M⊙)26,27,30
at a distance of 155.9 ± 0.9 pc (ref. 31). AB Aur is at
a relatively late stage of protostellar evolution, classified as a Class II
young stellar object32,33
. To investigate the velocity structure of the
disk,weobtaineddeepALMABand6observationsofmolecularemis-
sion lines 13
CO ( J = 2–1) and C18
O ( J = 2–1) with high velocity resolu-
tion (channel widths of vchan = 42 m s−1
and 84 m s−1
, respectively). The
observations were taken in two array configurations with baselines
rangingfrom14to2,216 m,reachingatotalon-sourceintegrationtime
of 5.75 h. Imaging with a Briggs robust value of 0.5 provided image
cubes with a spatial resolution or beam size of 0.237″ × 0.175″ (beam
positionangle(PA) = 1.2°),equivalentto37 × 27 au.Wecollapsethe3D
image cubes into 2D moment maps to expose the velocity-integrated
intensity (moment 0), intensity-weighted line-of-sight velocity (vlos,
moment1)andemissionlinewidth(moment2).Thiscollectionisshown
in Extended Data Fig. 1.
Torevealthespiralarmsinthedisk,weapplyahigh-passfilter34
(see
Methods) to the ALMA 13
CO moment maps (Fig. 1b–d). In the filtered
line-of-sight velocity (moment 1) map, we observe spiral-shaped dis-
turbances in the gas velocity field throughout the disk (Fig. 1b). With
the filtered velocity-integrated intensity (moment 0) and linewidth
(moment 2) maps, we visually highlight regions of peak density and
temperature(Fig.1c,d).Compressionandshockheatingareexpected
toleadtotemperatureenhancements(andthuslocalizedlinebroad-
ening) within gravitational instability (GI)-induced density spirals
in self-regulating disks13,23
. The VLT/SPHERE H-band scattered-light
image of AB Aur originally presented in ref. 35 is shown for compari-
son (Fig. 1a). Scattered light comes from the disk surface, tracing the
distribution of (sub-)micrometre-sized dust usually well coupled
with the gas. Previous simulations have shown that GI-induced den-
sity spirals are prominent in scattered light16,36
. At least seven spiral
structures (S1–S7) have been previously identified in the H-band
image35,37
, although not all occupy the same radial region and some
may be branches of adjacent arms38
. The disk rotates anticlockwise
(the spiral arms are trailing) and the south side is the near side, tilted
towards us38–40
.
To provide a qualitative comparison with the ALMA observations,
we run 3D smoothed-particle hydrodynamic (SPH) simulations of a
gravitationally unstable disk (see Methods). The simulations were
post-processed with radiative transfer and then further processed to
havethesameviewingangle,sensitivity,spectralandangularresolution
as the AB Aur data. To place the disk comfortably within the gravita-
tionally unstable regime (Mdisk/M⋆ ≳ 0.1), we set the total gas mass to
https://doi.org/10.1038/s41586-024-07877-0
Received: 11 December 2023
Accepted: 25 July 2024
Published online: 4 September 2024
Check for updates
1
Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia, Canada. 2
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing, People’s Republic of
China. 3
Department of Physics and Astronomy, The University of Georgia, Athens, GA, USA. 4
Center for Simulational Physics, The University of Georgia, Athens, GA, USA. 5
Università degli Studi
di Milano, Milan, Italy. 6
Institute of Astronomy, University of Cambridge, Cambridge, UK. 7
Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574,
Saint-Genis-Laval, France. 8
Leiden Observatory, Leiden University, Leiden, The Netherlands. 9
European Southern Observatory, Garching, Germany. 10
Institute of Astronomy and Astrophysics,
Academia Sinica, Taipei, Taiwan. 11
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA. 12
Astrobiology Center, National
Institutes of Natural Sciences, Mitaka, Japan. 13
Subaru Telescope, National Astronomical Observatory of Japan, Mitaka, Japan. 14
Department of Astronomy, School of Science, Graduate
University for Advanced Studies (SOKENDAI), Mitaka, Japan. ✉e-mail: jspeedie@uvic.ca; rbdong@uvic.ca

Nature | Vol 633 | 5 September 2024 | 59
0.3 times the mass of the star. For sustained spiral arms, we set the
cooling timescale to ten times the local dynamical timescale (β = 10).
ThesimulatedGIdiskshowsspiralstructuresinallthreemomentmaps,
resemblingthoseintheABAurdisk(ExtendedDataFigs.1and2).Over-
all,theABAurdiskhostsaglobalarchitectureofspiralarmsat100-au
to1,000-auscalesacrossallazimuthsinmultiwavelengthobservations
tracingdifferentdiskcomponentsandquantities,stronglyindicating
ongoing GI.
OnecharacteristickinematicfeatureintheABAurdiskcanbefound
in the isovelocity curve at the systemic velocity vsys in the moment 1
map.Figure2ashowsasinusoidalpatternatvlos = vsys (alongtheminor
axis;whitecolour),moreprominenttowardsthesouth.Thissignature,
knownasa‘minor-axisGIwiggle’22
,hasbeenpredictedinhydrodynamic
simulations22,24
andanalytictheory23
asaclearkinematicsignatureofGI
(Fig.2b,c).ItisoneofaglobalsetofGIwigglesinisovelocitycurvesthat
we observe throughout the AB Aur disk (Extended Data Fig. 3). These
wigglesaregeneratedbyself-gravitatingspiralarms,whichconstitute
localminimainthegravitationalpotentialfieldandinducecorrespond-
ing oscillations in the gas velocity field. The synthetic moment 1 map
oftheSPHGIdisksimulationshowsaminor-axisGIwigglewithsimilar
morphology as the observed one (Fig. 2c), completely distinct from
the linear pattern found in a disk undergoing Keplerian rotation with
no radial motions (Fig. 2b,c insets).
AmongallGIwiggles,theminor-axisGIwigglehasbeenknownand
targetedinpaststudiesforitsconvenienceinquantitativeanalysis23,24
.
Owingtoprojectioneffects,onlytheradialandverticalcomponentsof
the disk-velocity field (vr or vz) contribute to vlos at the systemic veloc-
ity traced by this wiggle. In the case of GI-induced velocity perturba-
tions, the vr contribution is expected to dominate22
. As we show with
2D analytic calculations of gravitationally unstable disks (see Meth-
ods),aself-gravitatingspiralarminducesradialmotionconvergenton
itself,appearingasawiggleinthemoment1mapatvsys wherethespiral
crossestheminoraxis(seeExtendedDataFig.5).Thefilteredmoment
1 map in Fig. 1b shows redshift and blueshift patterns corresponding
to convergent flows towards spiral S5 (visible in both scattered-light
and13
COmoments0and2;Fig.1a,c,d),supportingtheinterpretation
thattheGIwigglealongthesouthernminoraxisinFig.2aisgenerated
by a self-gravitating spiral arm.
HavingidentifiedevidenceofGIindiskkinematicsandinthedetec-
tions of spirals across several tracers and moment maps, we now
−6
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(arcsec)
VLT/SPHERE
H-band image
a
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i
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a
x
i
s
Major axis
ALMA 13CO
Filtered moment 2
ALMA 13CO
Filtered moment 0
c d
RA offset (arcsec)
dec.
offset
(arcsec)
RA offset (arcsec)
0 100 200 300
–0.05 0 0.05
(log10 mJy per beam km s–1)
–50 0 50
(m s–1)
0
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Filtered moment 1
b
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(m s–1)
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S5
S5 S5
S5
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RA offset (arcsec) RA offset (arcsec)
dec.
offset
(arcsec)
dec.
offset
(arcsec)
Fig.1|GlobalspiralsintheABAurdisk.a,VLT/SPHEREH-bandscattered-
lightimageoftheABAurdisk35
tracingaspiralstructurein(sub-)micrometre-
sizeddustgrains.ThelabelledspiralsS1–S7aretakenfrompreviousworks37,38
.
b,FilteredALMA13
COintensity-weightedmeanvelocity(moment1)map,
revealingresidualgasmotionwithinthebulkflow.Thesynthesizedbeamis
showninthebottom-leftcornerasanellipse.Theinsetzoomsintotheregion
aroundwhereS5crossestheminoraxis,highlightingconvergingflowsonthe
twosidesofS5indicatedbyarrows.c,FilteredALMA13
COintegratedintensity
(moment0)map,highlightingpeaksinthegasdensityand/ortemperature.
d,FilteredALMA13
COemissionlinewidth(moment2)map,showinglocalized
linebroadeningwithinthespiralarms.Insetsincanddzoomintothesame
regionastheinsetinb,showingenhancedgasdensityand/ortemperature
causedbytheradiallyconvergingflowsaroundS5.

Article
quantitatively analyse the GI wiggle along the southern minor axis to
constrainthediskmass.Weextractthe13
COandC18
Oemissionspectra
along the southern disk minor axis (Fig. 3a,b) and detect the wiggle
in position–velocity space (hereafter referred to as the ‘PV wiggle’),
whichisadifferentviewoftheposition–positionwiggleinFig.2a.Slic-
ing the 3D image cubes this way more comprehensively exposes the
gas velocity structure and enables us to quantify the perturbation in
unitsofvelocity.Wemeasuretheemissionlinecentresbyperforming
aquadraticfittothespectrumineachspatialpixeloftheimagecube41
.
This method achieves sub-spectral-resolution precision on the line
centre and yields statistically meaningful and robust uncertainties42
.
We find remarkably similar sinusoidal morphology between the PV
wiggles in 13
CO and C18
O emission (Fig. 4a).
Theoretical studies have shown that the dynamical response of a
disk to its own self-gravity is sensitive to the disk-to-star mass ratio
and the cooling rate23,24
. Specifically, the amplitude of the induced
radial-velocity perturbations is proportional to (Mdisk/M⋆)2
and β−1/2
(equations (11) and (18) in the Methods). This allows us to use the
observed minor-axis PV wiggle to infer the disk mass once we make
assumptions on the disk cooling rates. Following ref. 23, we use a sta-
tisticalmetrictoquantifythe‘magnitude’oftheminor-axisPVwiggle,
defined as the standard deviation of the line-centre velocities over a
radial range. Bounded by the inner central cavity and outer edge of
recoveredC18
Oemission,ourradialrangespans1″to5″(155to780 au).
Wefindamagnitudeof37.4 ± 2.9 m s−1
forthesouthernminor-axisPV
wiggle in 13
CO and 44.2 ± 1.3 m s−1
in C18
O (Fig. 4b). For comparison,
the gravitationally unstable disk in the SPH simulation has a south-
ern minor-axis PV wiggle in 13
CO emission with quantitatively similar
amplitude and sinusoidal morphology (Fig. 3c) and a magnitude of
39.1 ± 1.8 m s−1
(Extended Data Fig. 7a).
AB Aur 13CO
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RA offset (arcsec)
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dec.
offset
(arcsec)
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M
i
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o
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i
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Analytic models
b
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6
SPH simulations
c
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0
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0
0
0
6
–1.0
–0.5
0
0.5
1.0
v
los
−
v
sys
(km
s
–1
)
dec.
offset
(arcsec)
dec.
offset
(arcsec)
Fig.2|DetectionoftheGIwiggleintheABAurdisk.a,ALMA13
COintensity-
weightedmeanvelocity(moment1)mapshowingvlos ofgasintheABAurdisk.
Theobservationsshowthe‘GIwiggle’alongtheminoraxis(arrow)predicted
inref. 22 asaclearkinematicsignatureofGI.b,vlos mapofagravitationally
unstablediskattheinclinationandPAoftheABAurdisk,computedwith2D
analyticmodelling23
.Self-gravitatingspiralarmscrossingtheminoraxis
induceradialmotionthatappearsasawiggle(arrow).c,SyntheticALMA13
CO
moment1mapofthe3DSPHGIdisksimulation,revealingthesameGIwiggle
signature(arrow).TheinsetsinbandcshowcorrespondingimagesforKeplerian
diskswithnoradialgasmotion,inwhichtheisovelocitycurveatthesystematic
velocityappearsasastraightlinealongtheminoraxis.
AB Aur
ALMA 13CO
−1.0
−0.5
0
0.5
1.0
0 2 4 6
0 2 4 6
Position along the minor axis (arcsec)
0
50
100
Intensity
(mJy
per
beam)
v
los
−
v
sys
(km
s
–1
)
SPH simulations
GI disk
Synthetic ALMA 13CO
−1.0
−0.5
0.5
1.0
0
50
100
Intensity
(mJy
per
beam)
v
los
−
v
sys
(km
s
–1
)
Keplerian disk
Synthetic ALMA 13CO
SPH simulations
0 2 4 6
−1.0
−0.5
0.5
1.0
0
50
100
Intensity
(mJy
per
beam)
v
los
−
v
sys
(km
s
–1
)
AB Aur
ALMA C18O
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Intensity
(mJy
per
beam)
0
25
50
75
v
los
−
v
sys
(km
s
–1
)
a
c d
b
0 0
8 10
Fig.3|ThePVwiggle.Emissionspectra(intensityasafunctionofvelocity)
extractedalongthesouthernminoraxisofthedisk,plottedwithdistance
fromthestar.Thelinecentresareshownasyellowpoints.Theinsetsshowthe
correspondingline-centremap,withthecirclesdelineating1″radialincrements.
Theyellowlinealongthesouthernminoraxisisthenarrow(0.5°-wide)wedge-
shapedmaskwithinwhichthespectraandlinecentresareextracted.InallPV
diagrampanels,thegreyboxinthebottom-leftcornerhashorizontalwidth
equaltothebeammajoraxisandverticalheightequaltothechannelwidth.
a,b,ALMAobservationsoftheABAurdiskin13
CO(a)andC18
O(b).c,Synthetic
ALMA13
COobservationsgeneratedfrom3DSPHsimulationsofagravitationally
unstablediskwithadisk-to-starmassratioof0.3andacoolingratedescribed
byβ = 10.d,Thesimulateddiskhasitsvelocitystructureartificiallypost-
processedtobeKeplerian.

Nature | Vol 633 | 5 September 2024 | 61
Quantifying the minor-axis PV wiggle magnitude as above, we per-
form comparisons against analytic models to identify the combina-
tions of disk mass (Mdisk/M⋆) and cooling timescale (β) that satisfy the
AB Aur observations. A proof of concept of this technique with the
SPH simulation is shown in Extended Data Fig. 7b. Using the analytic
modellingcodegiggle(https://doi.org/10.5281/zenodo.10205110)from
ref. 23 (Methods), we calculate the minor-axis PV wiggle magnitude
in gravitationally unstable disk models for 60 × 60 combinations of
Mdisk/M⋆ and β, letting each vary within the ranges 0 ≤ Mdisk/M⋆ ≤ 0.4
and 10−2
≤ β ≤ 102
. A demonstrative analytic curve for the minor-axis
PVwigglefromthesamemodelshowninFig.2bisunderlaidinFig.4a
for qualitative comparison. Figure 4c shows the resulting map of
60 × 60 analytic minor-axis PV wiggle magnitudes. Overlaying con-
tours in this map at the magnitude values measured for the AB Aur
13
CO and C18
O southern minor-axis PV wiggles, we find a disk mass
in the gravitationally unstable regime with 0.1 ≲ Mdisk/M⋆ ≲ 0.3 for
a cooling timescale of 0.1 < β < 10. This result is robust to plausible
variations in the analytic model parameter choices (Extended Data
Fig. 8). This disk mass range is broadly consistent with the observed
spiral morphology—a lower disk mass may result in a large number
of more tightly wound spirals than we observe, and vice versa12,43
.
To demonstrate that the implied cooling timescales are compatible
with the constrained disk-mass values, Fig. 4c also shows the ranges
of β derived from independent radiative cooling prescriptions
(see Methods).
ThedetectionofGIinthediskaroundABAur,aClassIIyoungstellar
object32,33
, demonstrates that GI can take place during later evolu-
tionary stages. This result, together with previous reports of several
protoplanetcandidatesinandamongspiralarmsinthesystem35,44–46
(Extended Data Fig. 9), provides a direct observational connection
betweenGIandplanetformation.Lookingforward,theABAursystem
can be an ideal test bed for understanding how planet formation is
facilitatedbyGI-inducedspiralarms—whetherbyfragmentationinto
gas clumps enabled by rapid cooling7–10
(β ≲ 3) or by dust collapse of
solidsconcentratedwithinspiralarmssustainedbyslowcooling47–50
(β ≳ 5).
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/s41586-024-07877-0.
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a b
c
0 1 2 3 4 5
−0.2
−0.1
0
0.1
0.2
0 1 2 3 4 5
−0.2
−0.1
0
0.1
0.2
10−2 10−1 100 101 102
Cooling timescale β
Disk-to-star
mass
ratio
M
disk
/M
*
2σ
3σ
0 10 20 30 40 50 60 70 80
PV wiggle magnitude (m s–1)
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
v
los
−
v
sys
(km
s
–1
)
v
los
−
v
sys
(km
s
–1
)
ALMA 13CO
Analytic GI model
ALMA C18O
Analytic GI model
1σ from Keplerian in
13CO
C18O
13CO
C
18 O
13 CO
(13σ)
Plausible β range for r = 1−5″
Fig.4|PVwigglemorphology,magnitudeandconstraintsontheABAur
diskmass.a,TheALMA13
COandC18
Olinecentresalongthesouthernminor
axisfromFig.3a,b,afterquadraticdetrending(seeMethods).Uncertaintieson
thelinecentresareshownbyyellow-shadedregions.Forqualitativecomparison,
the minor-axis PV wiggle of the analytic GI model disk from Fig. 2b is shown
inthebackgroundinlightgrey.b,Themagnitudeofthesouthernminor-axisPV
wiggleinABAurismeasuredtobe37.4 ± 2.9 m s−1
in13
COand44.2 ± 1.3 m s−1
in
C18
O.c,Mapoftheminor-axisPVwigglemagnitudeof3,600analyticGImodel
disks,calculatedfora60 × 60gridofdisk-to-starmassratiosandcooling
timescales.Eachcellinthemaprepresentstheminor-axisPVwigglemagnitude
fromadifferentmodel.AyellowcontourisdrawnateachoftheABAur13
CO
andC18
Omeasuredmagnitudevaluesanddashedlinesrepresentthequoted
uncertainties.White-shadedregionsdenote1σ,2σand3σdeparturesfrom
aKepleriansignalin 13
CO(seeFig.3d).Horizontalbarsindicateindependently
derived βrangesataselectionofMdisk/M⋆ values(seeMethods).

Article
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Methods
Furtherinformationonthesource
AB Aur is accreting from the disk at a rate of M M
˙ ≈ 10 yr
−7 −1
⊙ (ref. 51),
within the range expected for modest GI-driven accretion (10−7
–
10−6
M⊙ yr−1
)52
.Thisaccretionrate,takentogetherwiththecurrentage
t0 = 2.5–4.4 Myr (refs. 25–28), implies a high ‘latent disk mass’:
M M t t M
= ( ) × = 0.25 − 0.44
disk
latent
0 0
̇
⊙, or M M
/ ≈ 0.1 − 0.2
disk
latent
⋆ . Mdisk
latent
provides an accretion-rate-based assessment of disk mass, assuming
a constant stellar accretion rate Ṁ and we are observing the system
midwaythroughthelifetimeofthedisk53,54
.Thisisaconservativeesti-
mate as the accretion rate at earlier epochs is probably higher55
. In
millimetre continuum observations, the disk shows a dust ring at
about 1″ and a cavity inside56
, probably caused by the trapping of
millimetre-sizeddustatapressurebump.Thedustringislocatedinside
themainspiralsinboththescatteredlightandgasemission.Lateinfall
fromaboveorbelowthemaindiskplane56–59
isprobablyencouraging
GI by providing a source of mass to maintain a high Mdisk/M⋆ value18,38
.
ALMAobservations
We observed AB Aur with ALMA in April, May and September 2022
under ALMA programme ID 2021.1.00690.S (PI: R. Dong). Measure-
ments were taken with the Band 6 receivers60
in array configurations
C-3 (two execution blocks) and C-6 (six execution blocks). In total,
the eight execution blocks reached an on-source integration time of
5.75 h, making this the longest fine-kinematics (vchan < 100 m s−1
) pro-
gramme towards a single protoplanetary disk so far. Extended Data
Table 1 provides details of the observations. We centred one spectral
window(SPW)atthe13
COJ = 2–1molecularemissionlinetransitionrest
frequency(220.3986 GHz),coveringabandwidthof58.594 MHzwith
1,920channels,resultinginthehighestachievablespectralresolution
of 41.510 m s−1
after default spectral averaging with N = 2 by Hanning
smoothing within the correlator data processor. A second SPW was
centred at the C18
O J = 2–1 rest frequency (219.5603 GHz), covering
thesamebandwidthwithhalfasmanychannels(960channels;owing
to sharing a baseband with another SPW), achieving a 83.336 m s−1
spectral resolution. To enable self-calibration, our correlator setup
sampled the continuum in another SPW centred at 233.012 GHz with
128 channels each 15.625 MHz in width, obtaining the full available
2.0-GHz bandwidth. Using the continuum data, all execution blocks
werealignedtoacommonphasecentreintheu–vplane.Weperformed
aseriesofphase-onlyself-calibrationiterationsandavoidedcombin-
ing by SPW in the first two rounds to remove any potential per-SPW
phase offsets. We also carried out one round of amplitude and phase
self-calibration.Finally,weappliedthephase-centrerealignmentsand
calibration-gaintables(thatwegeneratedwiththecontinuumdata)to
the line data. We performed continuum subtraction in the u–v plane
using the uvcontsub task.
All imaging was performed with the CASA tclean task. We used the
multiscaledeconvolutionalgorithm61
with(Gaussian)deconvolution
scales [0.02″, 0.1″, 0.3″, 0.6″, 1.0″]. We did not image with a Keplerian
mask so as not to restrict our ability to observe non-Keplerian emis-
sion. After experimentation with CASA’s auto-multithresh masking
algorithm62
, we used an imaging strategy similar to PHANGS-ALMA63
,
in which we clean conservatively, with a broad mask (usemask=‘pb’
and pbmask=0.2), forcing frequent major cycles. (The 13
CO robust
0.5 cube underwent 198 major cycles and the C18
O cube underwent
76.)Toachievefrequentmajorcycles,wesetthemaximumnumberof
minor-cycle iterations per channel to cycleniter=80, the minor cycle
thresholdtomax_psf_sidelobe_level=3.0andminpsffraction=0.5,and
the maximum assigned clean component to gain=0.2 times the peak
residual.WeusedaBriggsrobustweightingschemeandgeneratedtwo
sets of image cubes; one with a robust value of 0.5 and a second with
robust 1.5. The corresponding beam sizes for 13
CO are 237 × 175 mas,
1.2° for robust 0.5 and 390 × 274 mas, −1.4° for robust 1.5. We imaged
withafieldofviewouttotheprimarybeamfullwidthathalfmaximum
(FWHM; 38″) with 0.02″ pixels (9 or 12 pixels per synthesized beam
minor or major axis, respectively). We imaged in LSRK velocity chan-
nelsat42 m s−1
for13
COand84 m s−1
forC18
O,respectively(nearlynative
channelspacing).TheCLEANthresholdwassettofivetimesther.m.s.
noise measured in 20 line-free channels of the dirty image cube. We
applied JvM correction64,65
and primary-beam correction. The r.m.s.
noise in the resulting 13
CO cubes imaged with robust 0.5 and robust
1.5is2.0 mJyperbeamand1.2 mJyperbeam,respectively,and0.6 mJy
per beam in the C18
O cube imaged with robust 1.5.
Weusedtherobust0.5imagecubesforourposition–positionanaly-
sis (moment maps; Figs. 1 and 2) and the robust 1.5 cubes for our PV
analysis (PV diagrams and line centres; Figs. 3 and 4). We made the
moment 0, 1 and 2 maps using the bettermoments41,66
methods col-
lapse_zeroth, collapse_first and collapse_percentiles, respectively.
We note that we calculate our ‘moment 2’ maps as the average of the
redshifted and blueshifted linewidths about the intensity-weighted
median line centre (that is, as the average of the wpdVr and wpdVb
maps).Mathematically,thisisadifferentapproachtofindthelinewidth
thantheclassicmoment2approach,althoughinourcase,wefindthat
the two yield nearly identical outcomes. We applied sigma clipping
at five times the r.m.s. noise and performed no spectral smoothing.
Geometricproperties
WeusedthePythonpackageeddy67
toinfergeometricpropertiesofthe
disk,namelytoconstrainthediskcentrex0,y0,thediskinclinationi,the
PA, the vsys and the dynamical stellar mass M⋆. We performed Markov
chainMonteCarlotofittheC18
Omoment1map(ExtendedDataFig.4)
with a geometrically thin Keplerian disk rotation profile:
v
GM
r
i ϕ v
= sin cos + , (1)
0 sys
⋆
in which r is the disk radius, ϕ is the azimuthal angle around the disk
and G is the gravitational constant. Following convention, we fix the
inclination to the value found from fitting the continuum, i = 23.2°
(refs. 45,56),andthedistanceto155.9 pc(refs. 31,68).Weassumedflat
priorsforallvaluesandspatiallydownsampledtherotationmaptothe
beam FWHM before the likelihood calculation so that only spatially
independentpixelswereconsidered.Thecalculationoftheposterior
distributions was run with 128 walkers and an initial burn-in period
of 10,000 steps before the posterior distributions were sampled for
a further 10,000 steps. The resulting posterior distributions were
x0 = −5 ± 7 mas, y0 = −17 ± 7 mas, PA = 236.7 ± 0.3°, M⋆ = 2.23 ± 0.02 M⊙
and vsys = 5,858 ± 5 m s−1
, for which we report the uncertainties rep-
resented by the 16th and 84th percentiles about the median value.
The last three values are consistent with constraints from previous
observations26,27,45,56,69
.
HydrodynamicsimulationsandsyntheticALMAobservations
We performed 3D global SPH simulations with the PHANTOM code70
using1millionSPHparticles.Weassumedacentralstarmassof2.4 M⊙
(refs. 26,27), represented by a sink particle71
with accretion radius set
to60 au.Theinitialinnerandouterdiskradiiweresettorin,SPH = 80 au
androut,SPH = 500 au,respectively.Wesettheinitialgasmassto0.7 M⊙,
corresponding to Mdisk/M⋆ = 0.29. The surface density profile follows
Σ ∝ r−p
(inwhichthepower-lawindexp = 1.0)andthesoundspeedpro-
file follows cs ∝ r−q
(in which q = 0.25). The initial disk aspect ratio was
set to H/r = 0.05 at 80 au. We set αSPH such that αmin ≤ αSPH ≤ αmax, with
αmin = 0.001 and αmax = 1.0, with the value of αSPH set by the Cullen and
Dehnen72
switchthatincreasesviscosityonlyinthecaseofconverging
flows.ThisresultsinaShakura–SunyaevviscosityofαSS ≈ 0.01through-
out the disk.
Weassumedanadiabaticequationofstate,withheatingfromcom-
pressional P dV work and shock heating. The disk cools by Gammie

Article
cooling7
(also known as β cooling), for which the cooling timescale
is proportional to the local dynamical time by the factor β, such that
tcool(r) = βΩ−1
(r), in which Ω(r) = (GM⋆/r3
)1/2
is the Keplerian frequency.
We set β = 10, a typical value used or found in simulations19,22–24
. We
let the simulation evolve for five orbital periods of the outermost
particle,atwhichpointthedisksettlesintoastateinwhichtheToomre
Q parameter is between 1 and 2 from rin,SPH to 1.1rout,SPH.
Wecomputedthediskthermalstructureand13
CO( J = 2–1)modelline
cubes using the Monte Carlo radiative transfer code MCFOST73,74
. We
assumedthatthe13
COmoleculeisinlocalthermodynamicequilibrium
with its surroundings and that the dust is in thermal equilibrium with
thegas(Tgas = Tdust).Wesetthe13
CO/H2 abundanceto7 × 10−7
(refs. 22,75)
and we used approximately 107
photon packets to calculate Tdust. We
performed Voronoi tessellation on 990,972 SPH particles, which cor-
responded to 99% of the mass in the simulation. We set the total dust
massto1%ofthetotalSPHgasmassandusedadust-grainpopulation
with 50 logarithmic bins ranging in size from 0.1 μm to 3.0 mm. The
dust optical properties are computed using Mie theory. The central
starwasrepresentedasasphereofradius2.5 R⊙ radiatingisotropically
at an effective temperature Teff = 9,770 K, set to match AB Aur46,76–78
.
The disk was given an inclination of 23.2°, a PA of 236.7° (for which PA
ismeasuredeastofnorthtotheredshiftedmajoraxis)andplacedata
distance of 155.9 pc, all consistent with the AB Aur system.
We used the same PHANTOM simulation to create both the GI and
Keplerian model line cubes shown in Figs. 2c and 3. We created the
KepleriancounterpartwithMCFOST,usingtheflags-no_vrand-no_vzto
forcetheradialandverticalvelocitiestobezero,and-vphi_Keptoforce
the azimuthal velocities to be Keplerian. Both 13
CO model line cubes
weregeneratedwithMCFOST,binnedattheobservedspectralresolu-
tion of 42 m s−1
and gridded in the image plane to have 2,048 × 2,048
pixels of size 0.02″. We assumed a turbulent velocity of 0.05 km s−1
.
WegeneratedsyntheticALMAimagecubesfromthe13
COmodelline
cubesusingsyndisk(https://github.com/richteague/syndisk)tomatch
thepropertiesoftheobservedABAur13
COimagecubes(robust0.5and
1.5).Inthelattercase,themodellinecubewasconvolvedwithabeam
ofsize0.390″ × 0.274″andPA−1.4°.Correlatednoisewasaddedwithan
r.m.s.of1.2 mJyperbeam.Themodeldatawerethensmoothedwitha
Hanningspectralresponsefunctionwitharesolutionof42 m s−1
.Effects
associated with interferometric or spatial filtering are not captured
by this process and our synthetic ALMA image cubes are effectively
fully sampled in the u–v plane. The synthetic cubes were collapsed
into moment maps following the same procedure as the AB Aur data
(Extended Data Fig. 1).
Analyticmodelling
Weanalyticallycomputethevelocityfieldsofgravitationallyunstable
disksusingthegiggle(https://doi.org/10.5281/zenodo.10205110)pack-
age developed by Longarini et al.23
. Working in 2D polar coordinates
(r, ϕ), giggle considers a geometrically thin disk with surface density
profile Σ0 ∝ r−p
and inclination i, centred on a star of mass M⋆. It com-
putes the projected line-of-sight velocity field as:
v v ϕ v ϕ i v
= ( sin + cos ) sin + , (2)
r ϕ
los sys
in which vr and vϕ are the radial and azimuthal components of the
disk-velocity field, respectively. The basic state of the disk (that is,
considering only the gravitational potential contribution from the
centralstar)isassumedtobeKeplerian:vr = 0andvϕ = vKep.Thescheme
of the model is to determine the perturbations in vr and vϕ generated
byGIbytakingintoaccounttheextragravitationalcontributionfrom
thedisk,whichisinitializedasmarginallyunstableandimprintedwith
globalspiraldensityperturbations.Themodelcomputesthevelocity
fieldundertheassumptionthatthediskisself-regulated.Thisstateis
imposedbyassumingabalancebetweenheating(bycompressionand
shocks within the spiral arms) and cooling (by radiative processes).
As such, the amplitude of the spiral density perturbations A Σ
/
Σ 0
spir
saturated to a finite value proportional to the cooling timescale β
(refs. 13,79) is:
A
Σ
χβ
= , (3)
Σ
0
−1/2
spir
inwhichtheproportionalityfactorχisoforderunity13,23
.Theimprinted
spiraldensityperturbationisassumedtobesmallrelativetotheback-
ground surface density, so that all the relevant quantities (density Σ,
gravitational potential Φ, velocities vr and vϕ and enthalpy h) can be
written as a linear sum of the basic state and the perturbation:
X r ϕ X r X r ϕ
( , ) = ( ) + ( , ). (4)
0 spir
The spiral perturbation in density is given the form:
R
Σ r ϕ A
( , ) = [ e ], (5)
Σ
mϕ ψ r
spir
j( + ( ))
spir
in which j = −1 (as we are using i to represent the disk inclination)
and m is the azimuthal wavenumber. The ‘shape function’ ψ(r) is
described by m and the spiral pitch angle αpitch as:
ψ r
m
α
r
( ) =
tan
log , (6)
pitch
which is related to the radial wavenumber k by dψ/dr = k. The spiral
densityperturbationnecessarilyintroducesacorrespondingpertur-
bation to the gravitational potential:
Φ r ϕ
G
k
Σ r ϕ
( , ) = −
2π
( , ). (7)
spir spir
The negative proportionality Φspir ∝ −Σspir is the definition of
self-gravitatingspiralarms.Asaresult,correspondingperturbations
in the azimuthal and radial velocities are driven:
R
v r ϕ A r
( , ) = [ ( ) e ], (8)
r v
mϕ ψ r
j( + ( ))
r
v r ϕ A r rΩ
( , ) = [ ( ) e ] + , (9)
ϕ v
mϕ ψ r
j( + ( ))
ϕ
R
in which we note rΩ ≠ vKep because the angular frequency Ω includes
super-Keplerian rotation from the disk mass contribution:
Ω
GM
r r
Φ
r
= +
1 ∂
∂
. (10)
2
3
disk
⋆
By assuming that the disk is marginally unstable, and by maintaining
theself-regulatedstatecondition,theamplitudeoftheradialandazi-
muthal velocity perturbations A r
( )
vr
and A r
( )
vϕ
are determined23
:
A r mχβ
M r
M
v r
( ) = 2j
( )
( ), (11)
v
−1/2 disk
2
Kep
r






⋆
⋆
A r χβ
M r
M
v r
( ) = −
1
2
j
( )
( ), (12)
v
−1/2 disk
Kep
ϕ






inwhichMdisk(r)isthediskmassenclosedwithinradiusr.Withasurface
density profile Σ0(r) ∝ r−p
, then Mdisk(r) ∝ r−p+2
and the amplitude of
the radial perturbation is described by A r r
( ) ∝
v
p
−2 +7/2
r
. For p < 7/4,
A r
( )
vr
isanincreasingfunctionofradius.Thefactorofimaginarynum-
ber j in equation (11) has important physical consequences: when the
real component of A r
( )
vr
is taken (equation (8)), the radial-velocity

perturbation is π/2 out of phase with the spiral density perturbation
(equation (5)) and convergent at the locations at which Σspir takes a
maximum. Explicitly,
   








v r m ψ r
,
π
2
∝ − sin
π
2
+ ( ) , (13)
r
   








Σ r m ψ r
,
π
2
∝ cos
π
2
+ ( ) . (14)
spir
For qualitative visual comparison with the AB Aur moment 1 map
in Fig. 2a, we compute the projected line-of-sight velocity field of a
gravitationally unstable disk with β = 10 and Mdisk/M⋆ = 0.3 in Fig. 2b.
We set m = 3 and αpitch = 15° to approximately match the 13
CO spirals
in the AB Aur disk (Fig. 1) and assume p = 1.0 and χ = 1.0 (ref. 13). The
dominant azimuthal wavenumber is expected to be inversely related
tothedisk-to-starmassratioq,roughlyobeyingm ≈ 1/q(refs. 12,13,16),
so our choice of m = 3 is consistent with Mdisk/M⋆ ≈ 0.3.
Revealingglobalspiralstructure
WeobtaintheresidualmomentmapsshowninFig.1usingavariationon
theconventionalhigh-pass-filtering(alsoknownasunsharpmasking)
technique. The conventional method is to convolve the image with a
Gaussiankernelandsubtracttheblurredimagefromtheoriginal.Itis
acommontechniquetoincreasethevisualcontrastofvariationsinan
image and has been used successfully to reveal spiral structure disks
(for example, refs. 28,34,35,80–83). Here we perform the convolu-
tion with a radially expanding kernel (https://github.com/jjspeedie/
expanding_kernel)—that is, with a Gaussian kernel whose FWHM, w,
increases with radial distance from the image centre with a simple
power-law dependence:
w r w r r
( ) = × ( / ) , (15)
γ
0 0
in which w0 is the kernel width at r0 = 1″. A radially expanding kernel
providesawaytohighlightvariationsmoreevenlythroughoutthedisk,
given the spatial scales of the variations (which are expected to track
with the local scale height and increase with radius) and the dynami-
cal range of the variations, which fall with radius. After experimenta-
tion, we use w0 = 0.3″ and γ = 0.25, although we emphasize that this is
aqualitativechoiceandthekeyspiralfeatures,suchastheirlocations,
are robust against a variety of choices in kernel parameters. The high-
pass-filtertechniqueisalsoflexibletothediskemissionsurfacemorphol-
ogy and can capture global-scale deviations from Keplerian rotation
in the background disk. Extended Data Fig. 4 compares the residual
moment1mapsin13
COandC18
Oobtainedaftersubtractingtheaxisym-
metric geometrically thin Keplerian model (equation (1)) versus after
subtracting a blurred version of the moment 1 map made with the
expanding kernel filter. The Keplerian residuals (panels c and h) show
signs of global-scale deviation from Keplerian: the east (west) side is
generallyblueshifted(redshifted),hintingatsuper-Keplerianrotation,
signatures of disk mass contributing to the total mass of the system.
AlthoughspiralstructureisindeedalsovisibleintheKeplerianresidu-
als,theexpandingkernelresiduals(panelseandj)revealtheunderlying
spiralstructureinaspatiallyevenmanner,indicatingthattheexpanding
kernel background model (panels d and i) more successfully captures
thequasi-localbackgrounddiskvelocity.Wenotethatthisbackground
modelisnon-axisymmetric;itshowsexcessblueshiftedvelocityinthe
southeastquadrantofthedisksuchthatthecontourofvlos = vsys diverges
westwardfromtheminoraxissouthofthestar,possiblyindicativeofa
globaldiskwarp.Thisiswhatnecessitatesadetrendingofthelinecentres
toisolatethesinusoidalcomponentofthesouthernminor-axisPVwiggle
inFig.4a(seethesection‘Measuringthemagnitudeoftheminor-axisPV
wiggle’).FilteredmomentmapsforthesyntheticALMAobservationsof
thesimulatedSPHGIdiskareshowninExtendedDataFig.2.
Globalkinematicsofself-gravitatingspiralarms
Radiallyconvergentmotion(asinFig.1b–dinsets)servesasakinematic
signature for the location of self-gravitating spiral arms at disk azi-
muthsatwhichtheradial-velocityperturbationcontributessufficiently
stronglytotheobservedvelocityfieldandthuscannotbeafullyunam-
biguous locator at disk azimuths away from the minor axis. Extended
DataFig.5c,gprovidesmapsofvelocityresidualsfromKeplerianforthe
2DanalyticGIdiskmodelandtheSPHGIdisksimulation.Theconver-
gentmotiontowardsthespiralspinesisvisibleforarangeofazimuths
around the minor axis but becomes progressively less clear moving
towardsthemajoraxisastheazimuthalvelocity(super-Keplerianrota-
tion) contributes progressively more to the line of sight. However,
high-pass filtering (panel h) captures and removes the background
super-Keplerian rotation, leaving a residual map that resembles the
isolated radial component (panel d). Extended Data Fig. 5i–l overlays
the locations of 13
CO spirals in the AB Aur disk (from filtered moment
0/2; Fig. 1c,d) onto the filtered moment 1 maps, to illustrate where
convergent motion does or does not serve as a locator throughout
the disk. Ambiguity occurs around the major axis, which is a location
of transition in the sign of vrsinisinϕ (first term of equation (2)), and
when two spirals are not well separated and their motions superim-
pose. Three of the seven spiral structures in VLT/SPHERE scattered
light seem to be spatially associable with those in 13
CO (S1, S5 and S7;
panel l inset). Offsets in the southeast quadrant of the disk (S2, S3
and S4) may be further indication of a disk warp (Extended Data
Fig.4d,i)orothernon-trivialphenomena(forexample,verticaldensity
and temperature gradients, or projection effects84
).
The kinematic signatures observed in the present ALMA dataset—
probingdiskscalesfromabout100to1,000 au—arerecognizablydif-
ferentfromwhatisexpectedforplanet-drivenperturbations.Planetary
wakes are dampened and become nearly circular as they propagate
away from the planet85–87
, whereas GI-driven spirals maintain their
modestpitchangleswithradiusandtheamplitudeoftheinducedveloc-
ity perturbations depends on the enclosed disk mass (equations (11)
and(12)).Intheplanetarycase,thedensityandradial-velocitypertur-
bations are in phase (their peaks spatially coincide) and the pattern
of motion within an arm along a radial cross-section is divergent88,89
.
Overall, the essential characteristic of GI-induced spirals is that they
occurglobally22,23
(seeFig.1andExtendedDataFigs.2,3and5).Inpre-
vious datasets investigating smaller spatial scales (within the central
cavityoftheABAurdisk)planetarycandidatesP1/f1(refs. 35,45),P2/b
(refs. 45,46,90–92)andf2(ref. 35)areknowntobeassociatedwith—or
driving—spiralarms,asobservedinVLT/SPHEREscatteredlightand/or
ALMA12
COemission.AsshowninExtendedDataFig.9,owingtotheir
smallseparations(≲0.7″),kinematicsignaturesfromthesecandidates
are inaccessible to our ALMA observations. Clump-like signals ‘c’ and
‘d’seenbyHST/STIS(ref. 46)atwideseparations(approximately2.75″
andapproximately3.72″,respectively)areinlocationstentativelysug-
gestiveofconstitutingspiralarmfragmentsandmaywarrantfurther
investigation.
PVanalysis
Weusetherobust1.5imagecubesforourPVanalysistomaximizethe
recoveryofemissionatlargediskradii.Owingtotheclearassociation
withaself-gravitatingspiralarm(Fig.1b–dinsets),wetargetthewiggle
onthesouthernminoraxis.Aclearspiralarminmoment0/2crossing
the northern minor axis is also observed, but at the outer edge of the
recovered13
COandC18
Oemission(about3″;seeExtendedDataFig.5k,l).
WeobtainthePVdiagramsshowninFig.3usingeddy67
toextractspectra
frompixelswithina0.5°-widewedge-shapedmaskoriented90°clock-
wiseoftheredshiftedmajoraxis(showninFig.3insets).Ourquantita-
tiveanalysisoftheminor-axisPVwigglesisperformedwithmapsofthe
line centres made using the quadratic method of bettermoments41,66
,
which fits a quadratic curve to the spectrum in each pixel of the cube:

Article
I v a a v v a v v
( ) = + ( − ) + ( − ) , (16)
0 1 peak 2 peak
2
in which vpeak is the channel of peak intensity in the spectrum. We
select this approach over the traditional intensity-weighted mean
velocity (moment 1) method specifically for its ability to provide
well-characterized, statistically meaningful uncertainties on the line
centre, σvlos (ref. 41). The statistical uncertainty on each line centre is
computed as:








σ
σ
a
a
a
=
8
3
+ , (17)
v
I
los
2
2
2
1
2
2
4
in which σI is the r.m.s. noise of the intensities (see ref. 41 for a deriva-
tion). The quadratic method also has the advantage of being unaf-
fectedbysigmaclippingandofautomaticallydistinguishingthefront
side of the disk from the back side41
. Before the quadratic fitting, we
spectrallysmooththedatawithaSavitzky–Golayfilterofpolynomial
order1andfilterwindowlengthoftenchannels(420 m s−1
)inthecase
of 13
CO and three channels (252 m s−1
) in the case of C18
O. The former
was also applied to the two synthetic ALMA 13
CO image cubes gener-
atedfromtheSPHsimulations.Weextractthevaluesfromtheresulting
line-centre and line-uncertainty maps within the same wedge mask
described above. The extracted line-centre values are shown as yel-
low points in Fig. 3 and the uncertainties are shown as yellow-shaded
regions in Fig. 4a.
Measuringthemagnitudeoftheminor-axisPVwiggle
Followingref. 23,wemeasurethe‘magnitude’ofaminor-axisPVwiggle
asthestandarddeviationoftheline-centrevaluesoveraradialrange.
Bounded by the inner central cavity and the outer edge of C18
O emis-
sion, we use a radial range of 1″ to 5″. We estimate the uncertainty on
the magnitude measurement using a resampling procedure: we take
10,000drawsfromGaussiandistributionscentredontheobservedline
centres with standard deviation σvlos (equation (17)) to create 10,000
instancesoftheminor-axisPVwiggle,computetheirmagnitudesand
thenreporttheuncertaintyasthestandarddeviationofthose10,000
magnitude estimates.
As well as the wiggle, the 13
CO and C18
O emissions on the southern
diskminoraxisalsoexhibitanunderlyingmonotonicbluewardtrend
withdiskradius,seeninFig.3a,basasubtledownwardbendwithradius
ofthelinecentresorequivalentlyinFig.2aasawestwardorclockwise
shiftinthecontourofvlos = vsys.Weearmarkthisfeatureasapossibledisk
warp(ExtendedDataFig.4d,i)andusealeast-squares-fittingapproach
to isolate the sinusoidal component of the PV wiggle. This approach
yieldsthebackgroundtrendlinethatminimizesthestandarddeviation
oftheresiduals,thusprovidingthemostconservativeestimateforthe
magnitude of the detrended PV wiggle. We fit a quadratic trend line
(ExtendedDataFig.6a)asitmorecloselyresemblesthehigh-pass-filter
backgroundcurvethanalinearone(ExtendedDataFig.6b,c).Weshow
thequadraticallydetrendedPVwigglesinFig.4aandreporttheirmag-
nitudes in Fig. 4b. We find very similar magnitudes for both the 13
CO
and C18
O wiggles, despite C18
O probably tracing lower optical depths
in the AB Aur disk. This empirically substantiates comparisons with
the 2D analytic model (next section).
Performing the same procedure outlined above on the synthetic
13
COminor-axisPVwiggleoftheGIdiskintheSPHsimulation,wefind
a wiggle magnitude of 39.1 ± 1.9 m s−1
Constrainingdiskmasswithquantitativecomparisonswith
analyticmodels
Weperformquantitativecomparisonsbetweentheobserved13
COand
C18
O minor-axis PV wiggles and the projected radial-velocity compo-
nentinouranalyticmodel,vrsini(ref. 23).Fromequations(8)and(11),
the projected radial velocity on the minor axis (ϕ = π/2) is:
(18)
v r i mχβ
M r
M
v r m ψ r i
,
π
2
sin = − 2
( )
( ) sin
π
2
+ ( ) sin .
r
−1/2 disk
2
Kep
⋆
   














This curve reflects the disk mass enclosed within the inner and outer
radiiofthemodel,whichwesettospanthesameprojectedradialrange
astheobservedPVwiggles(1″to5″).Wecompute3,600ofthesecurves
for a 60 × 60 grid of models with (total enclosed) Mdisk/M⋆ linearly
spaced ∈ [0, 0.4] and β logarithmically spaced ∈ [10−2
, 102
]. Again, we
set m = 3 and αpitch = 15° to match the AB Aur disk and assume p = 1.0
and χ = 1.0 (ref. 13). For qualitative comparison, we plot an example
analyticminor-axisPVwigglebehindthedatainFig.4a;themodelhas
β = 10 and Mdisk/M⋆ = 0.3. We show in Extended Data Fig. 8 that m = 3
reproduces the observed wiggles better than other choices and that
p = 1.5couldalsoprovideasatisfyingmatch,whereasp = 2.0istoosteep.
Because the wiggle amplitude is independent of αpitch (equation (11)),
themagnitudeisconstantwithαpitch whensampledoverthesamerange
in phase (not shown).
Wemeasuretheminor-axisPVwigglemagnitudeofthe3,600mod-
els and present the resulting magnitude map in Fig. 4c. By drawing
contours in the Fig. 4c map at the magnitude values measured for AB
Aur (37.4 ± 2.9 m s−1
in 13
CO and 44.2 ± 1.3 m s−1
in C18
O), we find every
combinationofMdisk/M⋆ andβthatsatisfiestheobservations.Repeat-
ing this procedure with our synthetic ALMA observations of the SPH
GI disk simulation shown in Fig. 3c, we find that this technique suc-
cessfully recovers the disk mass set in the underlying SPH simulation
For independent physical estimates of plausible β values between
1″and5″(155to780 au),werelyonradiativecoolingprescriptions93,94
.
From equation (39) in ref. 94, β is a function of r and depends on Mdisk
through the surface density Σ. We assumeT =
ϕL
r σ
8π
1/4
2
SB
⋆
 




, in which σSB
is the Stefan–Boltzmann constant, L⋆ = 59 L⊙ is the stellar luminosity
of AB Aur46
and ϕ = 0.02 represents the flaring angle95
. We use the
DSHARP Rosseland mean opacity96
κR = κR(T, amax) for a power-law
grain-sizedistributiontruncatedatamax.Wesetamax to0.1 mmandthe
dust-to-gasmassratiotof = 0.1%,basedonradialdriftargumentsand
lack of (sub-)millimetre emission at these large radii. We compute a
β(r) profile for each Mdisk/M⋆ ∈ [0, 0.4] and extract the values at 1″ and
5″.Weoverlaytheresultingβ(Mdisk/M⋆)rangesaswhite-shadedregions
inExtendedDataFig.8(inwhichthedependenceonparisesfromthe
dependence on Σ) and in Fig. 4 as white horizontal bars at a selection
of Mdisk/M⋆ values. For example, for Mdisk/M⋆ = 0.2 and p = 1.0, we find
β(1″) = 5.3andβ(5″) = 3.6 × 10−2
.Althoughknowledgeofcoolingindisks
is very limited, these estimates help to emphasize that not all values
of β are equally likely.
Dataavailability
All observational data products presented in this work are avail-
able through the CANFAR Data Publication Service at https://doi.
org/10.11570/24.0087. All simulated data products are available at
https://doi.org/10.5281/zenodo.11668694.TherawALMAdataarepub-
liclyavailableattheALMAarchive(https://almascience.nrao.edu/aq/)
underprojectID2021.1.00690.S.TherawVLT/SPHEREdataarepublicly
availablefromtheESOScienceArchiveFacility(https://archive.eso.org/
eso/eso_archive_main.html)underprogramme0104.C-0157(B).Source
data are provided with this paper.
Codeavailability
ALMA data-reduction and imaging scripts are available at https://
jjspeedie.github.io/guide.2021.1.00690.S.ThefollowingPythonpack-
ageswereusedinthiswork:bettermoments(https://github.com/rich-
teague/bettermoments),eddy(https://github.com/richteague/eddy),

giggle v0 (https://doi.org/10.5281/zenodo.10205110), PHANTOM
(https://github.com/danieljprice/phantom) and MCFOST (https://
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Acknowledgements We thank our referees for their careful and insightful comments that
improved the manuscript. We thank K. Kratter for enlightening discussions and valuable
suggestions. J.S. thanks R. Loomis, S. Wood and T. Ashton at the North American ALMA
Science Center (NAASC) for providing science support and technical guidance on the ALMA
data as part of a data reduction visit to the NAASC, which was financed by the NAASC. The
reduction and imaging of the ALMA data were performed on NAASC computing facilities.
J.S. thanks C. Pinte, D. Price and J. Calcino for support with MCFOST, L. Keyte and F. Zagaria
for discussions on self-calibrating ALMA data and C. White for sharing perceptually uniform
colour maps. J.S. acknowledges financial support from the Natural Sciences and Engineering
Research Council of Canada (NSERC) through the Canada Graduate Scholarships Doctoral
(CGSD) programme. R.D. acknowledges financial support provided by the NSERC through
a Discovery Grant, as well as the Alfred P. Sloan Foundation through a Sloan Research
Fellowship. C.L. and G.L. acknowledge funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant agreement no.
823823 (RISE DUSTBUSTERS project). C.L. acknowledges funding from the UK Science and
Technology Facilities Council (STFC) through the consolidated grant ST/W000997/1. B.V.
acknowledges funding from the ERC CoG project PODCAST no. 864965. Y.-W.T. acknowledges
support through National Science and Technology Council grant nos. 111-2112-M-001-064- and
112-2112-M-001-066-. J.H. was supported by Japan Society for the Promotion of Science (JSPS)
KAKENHI grant nos. 21H00059, 22H01274 and 23K03463. This paper makes use of the
following ALMA data: ADS/JAO.ALMA#2021.1.00690.S. ALMA is a partnership of ESO
(representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada),
MOST and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of
Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National
Radio Astronomy Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. This work has made use of data from
the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed
by the Gaia Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/
web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions,
in particular the institutions participating in the Gaia Multilateral Agreement. Based on data
products created from observations collected at the European Organisation for Astronomical
Research in the Southern Hemisphere under ESO programme 0104.C-0157(B). This work has
made use of the SPHERE Data Centre, jointly operated by OSUG/IPAG (Grenoble), PYTHEAS/
LAM/CESAM (Marseille), OCA/Lagrange (Nice), Observatoire de Paris/LESIA (Paris) and
Observatoire de Lyon. This research used the Canadian Advanced Network for Astronomical
Research (CANFAR) operated in partnership by the Canadian Astronomy Data Centre and the
Digital Research Alliance of Canada, with support from the National Research Council of
Canada, the Canadian Space Agency, CANARIE and the Canada Foundation for Innovation.
Author contributions R.D. led the ALMA proposal. J.S. processed the ALMA data. J.H.
processed the VLT/SPHERE data. C.H. performed the SPH simulations. J.S. performed the
radiative-transfer calculations. C.L. and G.L. developed the analytic model. J.S. performed all
presented analyses. J.S. and R.D. wrote the manuscript. All co-authors provided input to the
ALMA proposal and/or the manuscript.
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/s41586-024-07877-0.
Correspondence and requests for materials should be addressed to Jessica Speedie or
Ruobing Dong.
Peer review information Nature thanks Jonathan Williams and the other, anonymous,
reviewer(s) 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.

Article
ExtendedDataFig.1|Momentmaps:ABAurobservationsandGIdisk
simulations.a–c, Integratedintensity(moment0),intensity-weightedmean
velocity(moment1) andintensity-weightedlinewidth(moment2)mapsfor
theALMA13
COobservationstowardsABAur.Panelbappearsinthemaintextas
Fig.2a.d–f,Moment0,1and2mapsforthesyntheticALMA13
COobservations
oftheSPHGIdisksimulation.LiketheABAurobservations,thesimulated
GIdiskshowsaprominentGIwigglealongthesouthernminoraxis(indicated
bywhitearrows).

ExtendedDataFig.2|Filteredmomentmaps:ABAurobservationsand
GIdisksimulations. Expandingkernelfilterresidualsofthemapsshownin
ExtendedDataFig.1,highlightingglobalspiralsandvelocitydisturbances
generatedbyGI.Panelsa–cappearinthemaintextasFig.1b–d.Theminor-
axisGIwiggleindicatedbyarrowsinExtendedDataFig.1b,eisshownhereasan
isovelocitycontouratvlos = vsys ± vchan inallpanels.

Article
ExtendedDataFig.3|GlobalGIwigglesinanalyticmodels,SPHsimulations
andtheABAurdisk.Isovelocitycontoursinline-of-sightvelocitymapsatthe
velocityvaluesindicatedbythecolourbar.a,vlos mapofthe2DanalyticGIdisk
model(showninFig.2b).b, vlos mapofthe2DanalyticKepleriandiskmodel
(showninFig.2b inset).c,SyntheticALMA13
COmoment1mapforthe3DSPHGI
disksimulation(showninFig.2c).d,SyntheticALMA13
COmoment1mapfor
the3DSPHKepleriandisksimulation(showninFig.2c inset).e,ObservedALMA
13
COmoment1mapfortheABAurdisk,imagedwithrobust0.5(showninFig.2a).
f,Likeebutimagedwithrobust1.5.

Extended Data Fig. 4 | Obtaining velocity residuals in the AB Aur disk.
a, ALMA 13
CO moment 1 map, imaged with robust 0.5, as shown in Fig. 2a.
b, Background model made with a Keplerian rotation profile, assuming a
geometricallythinaxisymmetricdisk(equation(1)). c,Velocityresidualsafter
subtractingthemodelinpanelb.Globalspiralsubstructureisvisiblebut
unevenlyso.Themodeldoesnotcapturethenon-axisymmetricemission
surfacemorphologyand/orsuper-Keplerianrotation.d,Background model
madewiththeexpandingkernelfilter(equation(15)). e,Velocityresidualsafter
subtractingthemodelinpaneld,asshowninFig.1b.f–j,Likea–ebutwiththe
ALMAC18
Omoment1map,imagedwithrobust1.5.

Article
ExtendedDataFig.5|KinematicsofGI-drivenspiralarms.a–d,2Danalytic
modelling23
.e–h,SyntheticALMA13
COobservationsofthe3DSPHGIdisk
simulation.i–l,ALMAobservationsoftheABAurdisk.a,Disksurfacedensity
(equations(4)and(5)). b,Line-of-sightvelocity(equation(2)),asinFig.2b.
c,VelocityresidualsfromKeplerian(thatis,subtractingFig.2binset).d,Line-of-
sightcomponentoftheradialvelocity(firsttermofequation(2)). e,Filtered
moment0.f,Moment1.g,Moment1residualsfromKeplerian.h,Filtered
moment1.i,ALMA13
COfilteredmoment0.j,ALMA13
COfilteredmoment2.
k,ALMA13
COfilteredmoment1.l,ALMAC18
Ofilteredmoment1(robust1.5).
PanellinsetoverlaystheVLT/SPHEREH-bandscattered-lightspiralsS1–S7
(refs. 37,38)inredand13
COspiralsS1–S9weidentifyinblack.

ExtendedDataFig.6|Methodsforisolatingthesinusoidalcomponent
ofthesouthernminor-axisPVwiggleintheABAurdisk.a,Detrendingthe
ALMA13
COlinecentresfromFig.3awithlinearandquadratictrendlinesfound
byaleast-squaresfit.b,Detrendingwiththeexpandingkernelhigh-passfilter,
varyingthekernelwidthparameterw0 andkeepingthekernelradialpower-law
indexfixedtoγ = 0.25(equation(15)).Wefindthebackgroundtrendlinesby
extractingthevelocityvaluesfromthehigh-pass-filterbackgroundmap(for
example,ExtendedDataFig.4d)withinthesame0.5°-widewedge-shaped
maskaswedoforthelinecentres,positionedalongthesoutherndiskminor
axis.c,Likebbutvaryingγandkeepingw0 fixedtow0 = 0.30″.Thehigh-pass-
filterdetrendingapproachconvergestothesamemeasuredPVwiggle
magnitudeasthequadraticfitapproach.

Article
ExtendedDataFig.7|PVwigglemorphology,magnitudeanddiskmass
recoveryintheSPHGIdisksimulation.LikeFig.4butforthesyntheticALMA
observationsoftheSPHGIdisksimulation.a,ThesyntheticALMA13
COline
centresalongthesouthernminoraxisfromFig.3cafterquadraticdetrending.
Uncertaintiesonthelinecentresareshownbyyellow-shadedregions.The
magnitudeofthisPVwiggleismeasuredtobe39.1 ± 1.9 m s−1
.Theanalyticmodel
showninthebackgroundforqualitativecomparisonhasthesameparameters
astheunderlyingSPHsimulation(Mdisk/M⋆ = 0.29andβ = 10)anditsPVwiggle
magnitude is 39.0 m s−1
. b, As in Fig. 4c, a map of the minor-axis PV wiggle
magnitudeof60 × 60analyticmodelsonagridofdisk-to-starmassratiosand
cooling timescales. A contour is drawn at the measured magnitude of the
synthetic 13
CO PV wiggle in panel a and dashed lines represent the quoted
uncertainties.ThetechniquesuccessfullyrecoversthediskmasssetintheSPH
simulation.

ExtendedDataFig.8|Comparisonswithfurthersetsofanalyticmodels.
LikeFig.4butvaryingtheazimuthalwavenumbermandsurfacedensity
power-lawindexpinthecomparisongridofanalyticGImodeldisks.Each
uppersubpanelshowsthequadraticallydetrended13
COandC18
Olinecentres
(yellow)behindademonstrativeanalyticPVwiggle(black)computedwiththe
combinationofmandpindicatedbytherowandcolumnlabels(keeping
Mdisk/M⋆ = 0.3andβ = 10fixed).Eachlowersubpanelshowsthecorresponding
mapofPVwigglemagnitudecomputedfora60 × 60gridofanalyticmodelsin
Mdisk/M⋆ andβ,againwiththecombinationofmandpindicatedbytherowand
columnlabels.Thetwoyellowcontoursaredrawnatthemagnitudevalues
measuredfortheobservedABAur13
COandC18
Osouthernminor-axisPVwiggles.
Thewhite-shadedregionbetweentwowhitecurvesrepresentsplausibleβranges
from r = 1–5″.ThecombinationshowninFig.4cism = 3,p = 1.0.

Article
ExtendedDataFig.9|Candidatesitesofplanetformation.Colouredcrosses
markthelocationsofcandidateprotoplanetsreportedintheliterature35,45,46
.
A table providing the coordinates of the candidates on the sky, estimated
masses and the reporting references is available as source data. a, Filtered
ALMA13
COmoment0map,asinFig.1c.b,VLT/SPHEREH-bandscattered-light
image(ref. 35),asinFig.1a.Theinsetzoomsintothecentral2″ × 2″regionto
showthespiralstructuresindifferenttracersatspatialscalesunresolvedby
thepresentALMAobservations.TheH-bandscattered-lightimageisshown
afterhigh-passfilteringandorangecontoursshowthetwospiralsidentifiedin
ALMA12
COJ = 2–1moment0(ref. 45)atlevelsfrom25to50 mJyperbeamkm s−1
inincrementsof5 mJyperbeamkm s−1
.

Extended Data Table 1 | Details of the ALMA Band 6 observations

Gravitational instability in a planet-forming disk

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