A NUMERICAL STUDY OF STELLAR DISCS IN GALACTIC NUCLEI

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AdvanceAccesspublication2022November25

A numerical study of stellar discs in galactic nuclei

Taras Panamarev

¹^,²^,³^‹

and Bence Kocsis

¹^‹

1RudolfPeierlsCentreforTheoreticalPhysics,ParksRoad,OxfordOX13PU,UK

2FesenkovAstrophysicalInstitute,Observatory23,Almaty050020,Kazakhstan

3EnergeticCosmosLaboratory,NazarbayevUniversity,53KabanbayBatyrave.,Astana010000,Kazakhstan

Accepted2022October19.Received2022September29;inoriginalform2022July13

ABSTRACT

Weexplorethedynamicsofstellardiscsintheclosevicinityofasupermassiveblackhole(SMBH)bymeansofdirectN-body simulations.Weshow that anisolated nuclear stellardisc exhibitsanisotropic mass segregationmeaning that massivestars settletolowerorbitalinclinationsandmorecircularorbitsthanthelightstars.However,insystems inwhichthe stellardisc isembedded inamuch moremassiveisotropicstellarcluster, anisotropicmass segregationtendstobe suppressed. Inboth cases,aninitiallythinstellardiscbecomesthicker,especiallyintheinnerpartsduetothefluctuatinganisotropyinthespherical component.Wefindthatvectorresonantrelaxationisquenchedinthediscbynodalprecession,butitisstillthemostefficient relaxationprocessaround SMBHsofmass10⁶M andabove. Two-bodyrelaxationmaydominatefor lessmassiveSMBHs foundindwarfgalaxies.Stellardiscsembeddedinmassiveisotropicstellarclustersultimatelytendtobecomeisotropiconthe localtwo-bodyrelaxationtime-scale.OursimulationsshowthatthedynamicsofyoungstarsatthecentreoftheMilkyWayis mostlydrivenbyvectorresonantrelaxationleadingtoananticorrelationbetweenthescatteroforbitalinclinationsanddistance fromtheSMBH.IftheS-starsformedinadisclessthan10Myrago,theymaycoexistwithacuspofstellarmassblackholes oranintermediatemassblackholewithmassupto1000Mtoreproducetheobservedscatterofangularmomenta.

Keywords: methods:numerical– stars:kinematicsanddynamics– Galaxy:centre– galaxies:nuclei.

1 I N T R O D U C T I O N

Morethantwodecadesofrepeatedmonitoringofstellarorbitsin the Galactic centre revealed the presence of a compact massive objectthatcoincideswiththeradiosourceSgrA∗(Ghezetal.2000; Gillessenetal.2009;Genzel,Eisenhauer&Gillessen2010;Gillessen etal.2017).Thehighmass(M4×10⁶M)and compactsize (R<10⁻⁶pc)suggestthattheobjectisasupermassiveblackhole (SMBH; see Eckart etal. 2017 fora discussion). TheSMBH is surroundedbyadenseclusterofstars,mostofwhichareold(>5Gyr old),butsomestarsareveryyoung(<10Myrold).Themajority ofyoungandmassivestarsaredistributedin adisc-likestructure asseenfromtheirangularmomentum vectordirections(Levin&

Beloborodov2003;Paumardetal.2006;Bartkoetal.2009;Yelda etal.2014;vonFellenberg etal.2022).This kinematicstructure iscalledtheclockwisestellardiscandislocatedbetween0.04and 0.5 pc(Levin & Beloborodov 2003). Another distinct kinematic structureistheS-starcluster:aclusterofyoungmassivestarslocated within the inner arcsecond (0.04 pc) from the SMBH. Detailed spectroscopicstudiesoftheS-starsindicatetheiragesarecomparable withthoseoftheclockwisestellardiscsuggestingthesameorigin forbothsystems(Habibietal.2017).Recentobservationssuggest thattheS-starclusterislikelytobearrangedintwoorthogonaldiscs (Alietal.2020;Peißkeretal.2020)whichmaybeidentifiedfrom

E-mail:[email protected](TP);[email protected](BK)

thedistributionsofthepositionanglesofthesemimajoraxesofthe sky-projectedorbits(Alietal.2020).¹

TheMilkyWaygalaxyisnottheonlygalaxythatfeaturesastellar disc.AtthecentreoftheAndromedagalaxy,twodistinctbrightness peaksareobserved(Laueretal.1993)whichmaybeexplainedby theso-calledeccentricnucleardisc(Tremaine1995)whereorbitsof starshavealignedargumentsofperiapsides.Observationsofnuclear starclustersinnearbyedge-ongalaxiessuggestthatsomeofthem hoststellardiscsassociatedwithmultiplestellarpopulations(Seth et al.2006, 2008).Therefore, the coexistenceof the nuclearstar clusterswithSMBHsandstellardiscsappearstobecommoninthe Universemotivatingstudiesofthesesystems.Themainfocusofthis paperisthenuclearstellardiscoftheMilkyWay,butwealsodiscuss stellardiscsinnucleiofdwarfgalaxies.

Theinteractionbetweenayoungstellardiscandtheoldspherical cluster maybe describedbysecular processesthat takeplace on time-scalessignificantlyshorterthantwo-bodyrelaxation.Duetothe finitenumberofstarsevenasphericalclusterexhibitsafluctuating stochasticanisotropythatgeneratesastrongnetgravitationaltorque on stellarorbits, giving rise to rapiddiffusion of orbital angular momentainaprocesscalledresonantrelaxation(Rauch&Tremaine 1996;Hopman&Alexander2006;Eilon,Kupi&Alexander2009; Kocsis&Tremaine2011,2015;GiralMart´ınez,Fouvry&Pichon 2020).Innear-Keplerpotentialsinwhichtheorbitaltimeismuch shorterthantheapsidalprecessiontime,thedynamicsofstarscan

1NotethattheexistenceoftwoorthogonaldiscsinS-starsisdebated(von Fellenbergetal.2022).

PublishedbyOxfordUniversityPressonbehalfofRoyalAstronomicalSociety.ThisisanOpenAccessarticledistributedunderthetermsoftheCreative CommonsAttributionLicense(http://creativecommons.org/licenses/by/4.0/),whichpermitsunrestrictedreuse,distribution,andreproductioninanymedium,

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berepresentedastheinteractionofquasi-stationaryellipticalwires exerting mutual gravitational torques.In this case, the individual orbitalenergiesareapproximatelyconserved,butthetorqueschange boththemagnitudesandthe directionsoftheangularmomentum vectorsduetoscalarresonantrelaxation(SRR).Innon-Keplerian sphericalmean-fieldpotential,whicharisesintheGalacticcentredue totheextendedstellarmassdistributionand/orgeneralrelativistic precession,theellipticalorbitsarenotclosed,butexperiencerapid apsidalprecession.Forthesesystems,thedynamicalrelaxationof orbital parameters is further accelerated by the coherent torques betweenNringsorannulicoveredbytheindividualstellarorbits.

Thisreorientstheangularmomentumvectordirectionsevenmore rapidlyinaprocesscalledvectorresonantrelaxation(VRR)while bothorbitalenergy andangular momentummagnitudeare nearly conserved(Rauch&Tremaine1996).

TheoreticalstudiesofVRRbenefitfromtheHamiltonianformal- ismwheretheHamiltonianrepresentsthegravitationalenergyfrom thestellarpotentialexcludingtheKeplerianorbitalenergyaround theSMBH (Kocsis& Tremaine2015).Thismay beachievedby orbit-averagingovertheprecessiontime-scale.Thefinalequilibrium statemay be found by means of mean field theory, the Markov chainMonteCarlo(MCMC)method,kinetictheory,orbyintegrating Hamilton’sequationsofmotionintimeusingorbit-averagedN-ring ordirectN-bodysimulations.First,usingthemeanfieldapproach, thedistributionfunctionoftheangularmomentumvectordirections can be found by maximizing the entropy of the system using calculusofvariations(Roupas,Kocsis&Tremaine2017;Tak´acs&

Kocsis2018;Magnanetal.2021).Theequationshavebeensolved analytically in the idealized case where all stars have identical masses,semimajoraxes,andeccentricities.Roupasetal.(2017)and Tak´acs&Kocsis(2018)foundthatthestellardiscsmayrepresent statisticalequilibriumstructures.Moreover,dependingonthetotal energyandangularmomentumthesystemexhibitsaphasetransition betweendiscandsphericalphasesshowingananalogywithliquid crystals.Recently,thesemodelsweregeneralizedbyMagnanetal.

(2021)toincludethemassspectrumofstarsshowingthatmassive starstendtoarrangeinthinnerdiscsthanlightstarsinaprocesscalled verticalmasssegregation.Thisconfirmstheoriginalexpectationof Rauch&Tremaine(1996).

A similar conclusion was reached earlier using the MCMC method. Szölgyén & Kocsis (2018) showedthat fora particular anisotropicinitial conditionthe massive starsin the cluster form a disc. The study was recently extended by Máthé,Szölgyén&

Kocsis(2022)wheretheauthorsexploredtheVRRequilibriumfora rangeofinitialconfigurationsinenergy– angularmomentumspace.

Bothofthesestudies includedorbit-averagedinteractionsbut did notconsiderthediffusionarisingfromtwo-bodyencounters.They foundthatmassiveobjectsformdiscsevenincaseswheretheinitial levelofanisotropyisonlyafewpercent.

Masssegregationmayalsooccurintheeccentricitydistribution, butin this case drivenby SRR. SRR isthe dominant processto randomizethe eccentricities of the S-stars in the Galactic centre (Peretsetal.2009).Fouvry,Pichon&Chavanis(2018)showedthat massivestarstendtobecomemorecircularthanlightstarsindis- cretequasi-Keplerianaxisymmetricdiscs.Insphericallysymmetric systems,mass segregationin eccentricitymaytake place in both directions:theorbitsofmassivestarsbecomemorecircularandlight starsbecomemoreeccentricorviceversa dependingonthe total energyofthesystem(Gruzinov,Levin&Zhu2020).

Thetime-evolutionofthesystemtowardsVRRequilibriummay bedescribedbykinetictheorysolvingtheBoltzmannequation.This approachhasbeenusedtoelucidateSRR(Bar-Or&Fouvry2018) andVRRprocesses(Fouvry,Bar-Or&Chavanis2019b).

The time-evolution leading to mass-dependent anisotropy was demonstrated in a set of direct N-body and N-ring simulations featuringa stellardisc, anintermediate massblack hole(IMBH) and a spherically symmetriccluster of stars (implementedas an externalpotential)withanSMBH.Szölgyén,Máthé&Kocsis(2021) showedthattheorbitoftheIMBHalignsrapidlywiththediscof starswithin3–10Myr(dependingontheIMBHmassandtheinitial inclinationangle)andtheIMBHeccentricitydecreasesrapidlydueto VRRandSRRbytheeffectcalledresonantdynamicalfriction.This workfeatureddirectintegrationoftwo-bodyencountersbetweenthe SMBH,IMBH,andthestarsinthedisc,butneglectedthetwo-body interactionsbetweenstarsinthediscandinthesphericalclusterand deviationsfromsphericalsymmetry.

Masssegregationeffectsinthevicinityofamassiveblackhole wereoriginallydescribedinthecontextoftwo-bodyrelaxationin isotropic spherically symmetric stellar systems (Bahcall & Wolf 1977) which were later confirmed by direct N-body simulations (Preto & Amaro-Seoane 2010; Panamarev et al. 2019). For an isotropic system two-body relaxation is much slower than VRR by the ratio of the central mass to the individual stellar mass times N^1/2, i.e. MSMBH/(N^1/2m), where N is the number of stars.

It drivesmasssegregationslowlybothinsemimajoraxesand,as shownbyMikhaloff&Perets(2017),itleadstomasssegregation in orbital inclinations and eccentricities in isolated stellar discs.

Recently,N-bodymodellingofFoote,Generozov&Madigan(2020) demonstrated verticaland eccentricmasssegregationin eccentric nucleardiscs.Itwasnotclearfromthisstudywhethertheseeffects werecausedbytwo-bodyorresonantrelaxationorboth.Anisotropic masssegregationwasalsoobservedindirectN-bodysimulationsof rotatingglobularclusters(Sz¨olgy´en,Meiron&Kocsis2019),where VRRdominatesovertwo-bodyrelaxationforN10⁴(Meiron&

Kocsis2019).

Peretsetal.(2018)showedthatthecollectiveeffectofstarsin asphericaldistribution(intheircaseacuspofstellarblackholes) mayleadtotheformationofclumps,warps,andspiralarmsinthe stellardisc.TheycomparedresultsfromdirectN-bodysimulations ofisolatedstellardiscs,stellardiscsembeddedinasmoothpotential, ahybridself-consistentfieldmodellingofdisc– sphereinteractions (Meironetal.2014)anddirectN-bodyintegrationofthewholesys- tem.Whileisolateddiscsanddiscsembeddedinasmoothpotential showed steady increase in discthickness, both hybrid and direct N-body modelsledto the formationofclumps,warps, andspiral arms.Thequalitativeagreementbetweenhybridanddirectmodels suggeststhattheseeffectsmaybecausedbyresonantrelaxation.

Mastrobuono-Battistietal.(2019)useddirectN-bodysimulations tostudythe co-evolutionofmultiplestellardiscsembeddedin an analyticalstellarcuspandadiscretepopulationofstellarblackholes.

Byintroducinganewdiscevery100Myr,theyfoundthatthediscs evolvetowardsauniformdistributioninorbitalinclinations,butat the end oftheirsimulations (500Myr)eachof thediscs showed differentmorphologiesandkinematics.

Kocsis & Tremaine (2015) and Giral Mart´ınez et al. (2020) showed that the fluctuatinganisotropy ofa sphericaldistribution leadstodiffusioninangularmomentumdirectionspaceinanearly spherical system due to VRR. Thus, as longas the gravitational interaction between disc particles may be neglected, a spherical distribution drives the disruption of a stellar disc. Furthermore, two-bodyrelaxationmayfurtheracceleraterapiddiffusion,rapidly increasing the thickness of an initially very thin disc (Cuadra, Armitage& Alexander2008).In the oppositelimitof astrongly self-interactingthinstellardiscwithnotwo-bodyrelaxation,thedisc actsasacoupledsystemofharmonicoscillators,counteractingthe externaltorquessuchthatthediscremainsintactandexhibitsnormal

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modeoscillations(Kocsis&Tremaine2011).Inthispaper,weaimto studytheinteractionofanuclearstellardiscwithasphericalnuclear starclusteraroundacentralmassiveblackholeself-consistentlyby meansofdirectN-bodysimulations.Weimprovethephysicalrealism andparticlenumberresolutionoverpreviousdirectN-bodymodels tounderstandifstellardiscsorblackholediscsmaybelonglivedin nuclearstarclusters.

Thepaperisorganizedasfollows.In Section2,wereviewthe Galactic centre time-scales.In Section 3, wedescribe the initial setupforournumericalmodels.Section4isdevotedtotheanalysis ofisolatedstellardiscswithoutasphericalstellarpopulation,and Section5to the effectscausedby thedynamicalinteractionwith thesphere.InSection6,weapplyourfindingstocomparewiththe observedpopulationofS-starsand,finally,wesummarizethepaper inSection7.

2 T H E T I M E - S C A L E S

Inthissection,wereviewtherelaxationprocessesingalacticnuclei andtheassociatedtime-scalessimilartoKocsis&Tremaine(2011) andRauch&Tremaine(1996).

2.1 Two-bodyrelaxation

Two-bodyrelaxationarisesfrom thefluctuatingforceactingon a subjectstarovertheorbitalperiod.Asthetotalimpulsesreceived byastarovertheorbitalperiodareuncorrelated,therelaxationrate occursinarandom-walkfashion andisoftencallednon-coherent relaxation.The two-bodyrelaxationchanges boththe energyand thecorrespondingangularmomentumattherate(seee.g.Rauch&

Tremaine1996orBinney&Tremaine2008):

E

E =αm2N¹^/² Mbh

t torb

₁_/₂

, L

L =βm2N¹^/² Mbh

t torb

₁_/₂ , (1)

whereNisthetotalnumberofstars,Mbhisthemassofthecentral massiveblackhole,E∼2GM_bh/Risthe Keplerianenergy,m₂= m²/mistheeffectivemassandα∼β∼(ln)^1/2withinfactors oforderunitywherelnln(M_bh/m)istheCoulomblogarithm,m isthestellarmass,andt_orbistheorbitalperiod.

Thetwo-bodyrelaxationtime-scaleforasphericalstellarsystem withacentralmassiveblackholecanbecomputedby(Binney&

Tremaine2008):

t_relax=0.34 σ³(r)

G²ρ(r)m₂ln= M_bh²

β²m²₂Nt_orb, (2) whereσ isthe1Dvelocitydispersion,ρisthestellardensity.

2.2 Scalarresonantrelaxation

Contrarytotwo-bodyrelaxation,SRRoccursinacoherentwayover theapsidalprecessiontime-scale.Innear-Keplerpotentials,theorbit- averagedinteractionmaybeapproximatedasellipticwiresexerting mutualtorques.Inthiscase,theKeplerianenergyisconserved,but boththemagnitudeandthedirectionofangularmomentumvectors Larechangedatthefollowingrate:

L L_c=ηs

m₂N¹^/² M_bh

t_prect t_orb²

1/ 2

, (3)

whereL_c=L/√

1−e²,η_sisadimensionlesscoefficientoforder unityandtprecistheapsidalprecessiontime.Thetotalrelaxationrate occursinarandomwalkfashionwiththe apsidalprecessiontime beingthestepsize(durationofthecoherentphase).Thelongduration

ofthestepsizecomparedtotheorbitalperiodmakesthisprocess more efficientthan two-body relaxationin near-Keplerpotentials wheretprectorb.

TheSRRtimeinasphericalstellarsystemcanbefoundby:

trr,s= 4π|ω| β_s²²

M_bh²

M(r)m₂, (4)

whereω=2π/t_precistheapsidalprecessionrate(sumofNewtonian and relativistic), = 2π/torb is the orbital frequency, and βs is a dimensionlesscoefficientestimatedbyEilon etal.(2009)to be 1.05±0.02.

2.3 Vectorresonantrelaxation

Insphericalpotentialswheretheprecessiontimeisshort,thestellar orbits may be approximatedas annuli that exert mutual torques.

In this case, the torques change the direction of orbital angular momentumvectorsattherate:

L/Lc=ην

m₂N¹^/² M_bh

t t_orb

1/2

+βν

m₂N¹^/² M_bh

t

t_orb, (5) where ην is a dimensionless coefficient that corresponds to the contribution oftwo-body relaxation and SRR,and the term with βν=1.83±0.03representsthecontributionfromthecoherentphase ofVRR(linearwitht/t_orb)(Eilonetal.2009).Kocsis&Tremaine (2015)foundthatVRRisslowerbyafactor3duetorapidapsidal precessionconsistentwithearlierwork(Rauch&Tremaine1996).

ItisexpectedthatVRRmaybethemostefficientwaytorandomize thestellarorbitalinclinationsasthestepsizeofthecoherentphase isthelargestamongallrelaxationprocesses.

Forasphericalstellarsystem,theVRRtimeis(Eilonetal.2009):

t_rr,_v= Mbh

√M(r)m₂ torb

β_ν². (6)

Kocsis& Tremaine(2015)foundthatm2isreplacedbytheRMS massforVRR.

2.4 Two-bodyrelaxationinastellardisc

Two-bodyrelaxationtime-scaleforastellardisccanbecomputed by(Stewart&Ida2000):

t_rx_,_disc= e²2

4.5 M_bh²

m₂r²ln, (7)

whereisthesurfacedensityofthedisc,<e²>^3/2M_bh/m.The formulaassumes<i²>^1/20.5<e²>^1/2.

2.5 Vectorresonantrelaxationinastellardisc

VRRmayalsooccurinstellardiscs.Sincestarsexerttorquesfrom thediscplaneleadingtoprecessioninthelineofnodesattherate (Kocsis&Tremaine2011):

ν

i²1/2

M_disc

M_bh, (8)

thenodalprecessionwilllimitthestepsizeforthecoherentphase ofVRR.TocomputeVRRinastellardisc,wereplacetheapsidal precessionrateinequation(4)bythenodalprecessionrateandM_disc byM(r):

t_vrr,_disc 4π

i²₁_/₂Mbh

m2

. (9)

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Figure1. Thetime-scalesofdynamicalprocessesasafunctionofdistancefromtheSMBHinoursimulationsoftheGalacticcentre.Computedfromanalytical expressionspresentedinSection2usingthedatafromtheinitialconditionsofthemodelsdescribedinSection3.Allthicklinesshowthetime-scalesrelatedto theinteractionofthestellardiscwithasphericalcomponentwitha3Ddensitydistributionρ∝r^−1.75.Allthinlinesshowtherelaxationtime-scaleswithinthe discneglectingcontributionfromasphericalcomponent.Blacklinesillustratetwo-bodyrelaxationwithinthediscs,purplelinesshowVRRwithinthediscs.

Differentlinestylescorrespondtodensitydistributionsofthestellardiscswithcorrespondingpower-lawdensityslopeaccordingtothelegend.

Thisexpressionshowsrelaxationoftheangularmomentumvectors whichin this case isdominatedby relaxation in azimuthalcom- ponentsdrivenbythe nodalprecession(asshownin section2of Kocsis&Tremaine2011).NotethatVRRintheverticaldirection may bemuchslower due to kineticblocking (Fouvry,Bar-Or&

Chavanis2019a).Furthermore,tvrr,disc estimatesthe time-scalefor therelaxationofadiscbyneglectingthefluctuatingtorquesfromthe sphericalcomponentofthestellardistribution.

Werefer to Tremaine(1998) and Fouvryet al. (2018) for the discussionandanalysisofSRRindiscs.

Fortherelaxationprocessesthatoccurmuchfasterthantwo-body relaxation,itisoftenusefultocomparethetime-scaleswithrespect totheseculartime,definedas:

tsec = M_bh

M_totPinner, (10)

whereP_inneristheorbitalperiodoftheinnermoststar(inourmodels determinedbytheinneredgeofthestellardisc)andM_totisthetotal stellarmassofthesystem.Thistime-scalesetstheshortestapsidal precessiontime.

Fig. 1 shows the time-scales described above applied to the Galacticcentreusingdatafromoursimulations(seeSection3).The sphericalcomponentcorrespondsto theBahcall–Wolfcusp(Bah- call&Wolf1976)whilestellardiscsfeature variousdistributions of3Ddensitiesandorbitalparametersadoptedinoursimulations asdescribedinthefollowingsection.Thefigurecomparesthetime- scalesof dynamicalprocesseswithinthe sphere (thicklines)and withinthediscs(thinlines).Aswesee,VRRwithinthesphere(thick redline)isthefastestprocessfollowedbyVRRindiscs(althoughfor somediscmodelstwo-bodyrelaxationwithinthedisciscomparable

insomeregions;seepurpleandblacklines).Ontheotherhand,if thetotalmassofthewholestellarsystemisincreasedbyafactorof 30(labelledas30Xinthelegend),whilekeepingthesamenumber ofparticles,two-bodyrelaxationwithinthediscbecomesthefastest process(seethesectionbelowforamotivationonthe30Xmodels).

Notethatthetime-scalespresentedinFig.1(andtheequivalent analyticalexpressions)arederivedeitherneglectingthecontribution from the disc(time-scales within the sphere)or from the sphere (time-scaleswithinthediscs),butinrealitythedynamicsofastellar discembeddedinaspheremaybeshapedbythecontributionfrom boththediscandthesphere.Thetorqueactingonatestparticlein thepresenceofanisotropicclusterduetothefluctuatingstochastic anisotropyisoftheorderof(Kocsis&Tremaine2015)

L˙_sphere∼βν

N_sphere¹^/²m_rms_,_sphere Mbh

Lc

torb

, (11)

whileastellardiscdrivesnodalprecessionattherateoftheorderof L˙disc∼ N_discm_av_,_disc

M_bh L_c

t_orb. (12)

Here, mrms,sphere = m²^1/2and m_av,disc= m for objects in the sphericalclusterandthedisc,respectively.Thus,theeffectofthedisc dominatesoverthesphereifN_discm_av_,_discN_sphere¹^/²m_rms_,_sphereandthe discexhibitsnormalmodeoscillations(Kocsis&Tremaine2011), andintheoppositelimitthediscdissolvesonthet_rr,vVRRtime-scale dueto thesphere(Kocsis&Tremaine2015;GiralMart´ınezetal.

2020).Toexplorethedynamicsandthedominantrelaxationprocess fordifferentsystems,weperformdirectN-bodysimulationsofstellar discs embedded in a spherical cusp of stars in the intermediate

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regime whereNdiscmav,disc and N_sphere¹^/² mrms,sphere are comparableas wedescribeinthefollowingsection.

3 S I M U L AT I O N S

Weadoptthefollowingsystemofunitsforallthemodels:

G=M_bh=R_out=1, (13)

whereGisthegravitationalconstant,M_bhistheinitialSMBHmass, andR_outistheinitialouterradiusofthestellarsystemwhichisdefined asthe orbitalsemimajoraxisof theoutermoststarin thesystem.

WhenconvertingtophysicalunitswetypicallyassumeR_out=0.5pc, Mbh=4×10⁶Munlessindicatedotherwise,andinsomecaseswe adoptR_out=1pc,M_bh=1.3×10⁵M.

3.1 Thecode

WeuseamodifieddirectN-bodycodePHI-GRAPE(Harfstetal.2007) thatuses fourth-orderHermiteintegrationmethod (Makino1991; Makino& Aarseth1992; Aarseth2003) to solvethe equation of motion.ThecodewasoriginallydesignedfortheGRAPEcardsand nowutilizesanemulationlibrarytorunonmodernGPUs(Nitadori&

Makino2008).Themodifiedversionoftheoriginalcodeincludesthe gravitationalinteractionwiththemassivecentralobjectimplemented asafixedexternalpoint-masspotentialandtheaccretionofstarson tothecentralobject(Justetal.2012;Lietal.2012;Zhong,Berczik&

Spurzem2014).Theequationofmotionis

¨

r_i=−

i= j

Gmjrij

(r_ij²+_ss²)³^/²− GMbhri

r_i³ , (14)

where r_ij=r_i−r_jwith r_i, r_jthe positions of stars i and j, respectively,_ss= 1.0 ×10⁻⁴is the stellarsoftening parameter.

The value forthe softening between stars is chosen to be small enough to resolve relevant close encounters but large enough to preventformationofthecompactbinarysystems.Lowervaluefor thesofteningmayresultinalargernumberofverycloseencounters betweenstars,buttheyarerareandarenotrelevantontheresonant relaxationtime-scaleswhicharethemainfocusofthiswork.

Thecentralmassiveblackholecangrowinmassbyconsumption ofstars.Thecriterionfortheaccretionistheinstantaneousdistance tothestarislessthantheaccretionradiuswhichwassettobeequalto thetidaldisruptionradiusofa2Rstarbya4×10⁶Mblackhole.

Aftertheaccretioneventthetotalmassofthestarisinstantaneously addedtothemassoftheSMBHandthestarisremovedfromthe simulation(Justetal.2012;Lietal.2012;Zhongetal.2014).The accretionradiussetstheinnermostresolutionofthesimulationsand, thus,allowsnottosoftentheinteractionbetweenstarsandtheSMBH (Khanetal.2018).

The accuracyof the simulationsis controlledby the time-step factorη(Aarseth1985;Makino&Aarseth1992).Wechooseη= 0.01asacompromisebetweentheaccuracyandthecomputingtime.

Toensure that η = 0.01isthe optimal choice, one canmeasure the total energy exchangebetween particlescausedby two-body relaxationover theapsidal precessiontimeand compareit to the totalabsoluteenergyerrorof thesystem overthe sameperiod of time.Forall ofourmodels,theratiooftheabsoluteenergyerror over thetotalenergyexchangebetweenparticlesdoesnotexceed 10⁻⁵overtheapsidalprecessiontimeforagivenparticleensuring thatη=0.01istheoptimalchoice.Thetotalrelativeenergyerrorat theendofthesimulationsisoftheorderofE= ^E⁻_E^E₀⁰ ≈10⁻⁴,the totalangularmomentumerrorisoftheorderofL= ^|^L_|⁻_L₀^L_|⁰^|≈10⁻³.

Reducingthevalueforηimprovestheerrortolerance,butslowsdown thecomputationsandqualitativelyshowsthesameresults.

3.2 Initialconditions

Westudythegravitationalinteractionofagalacticnucleuswiththree components:acentralmassiveblackhole,asphericalclusterofold stars,andapopulationofstarsresemblingadisc.

Werunone-to-onesimulationsmeaningthatoneparticleinthe simulation represents one realistic star. This canbe achieved by modellingasystemof10⁵particleswithanaverageparticlemassof 10⁻⁶Mbh.Usingatop-heavyinitialmassfunction(IMF,equation15 below) and applying the parameters to the Milky Way Galaxy centregivesthetotalstellarmassMtot2×10⁵Mfortheinner 0.5pc.Thisvalueiscomparableto the totalstellarmassinferred from observations: Sch¨odel etal. (2018) find M1.3×10⁴M within0.1pcandM1.0×10⁶Mwithin1pc.²Themostrecent estimatesbasedoninterferometricastrometryindicatethatthetotal extendedmasswithin0.1pcdoesnotexceedM10⁵M(Gravity Collaboration2022).

Wegeneratetheinitialpositionsandvelocitiesforthespherical stellarsystemtofollowKeplerianorbitswithspatialdensitydistri- butionresemblingaBahcall–Wolfcuspwithρ∝r⁻^7/4whereristhe distancefromtheSMBH(Bahcall&Wolf1976).Thedistribution oforbitalparametersforthesphericalclusteristhesameinallour models whilewe varythe spatial densitydistributionand orbital parametersforthediscstarsasdescribedinSection3.2.Inallthe models,wekeepthestellardiscembeddedinasphericalcomponent.

Tomodelthemassspectrumofstars,weadopttheKroupa(2001) top-heavyIMFforthesphere:

dN

dm∝m^−α, αsphere=

⎧⎨

⎩

1.3,if0.08M≤m<0.5M 2.3,if0.5M≤m<1.0M 1.5,ifm≥1.0M

. (15)

Thetop-heavyIMFismotivatedbytheexpectedmasssegregation ingalacticnuclei(seee.g.Panamarevetal.2019),andtheobserved stellarmassfunctionintheGalacticcentrefollowingm^−1.7^±^0.2(Lu etal.2013).AftertheIMFisgeneratedweusethestellarevolution code(SSE;Hurley,Pols&Tout2000)toevolvethewholesystem up to 1 Gyrand use stellarmasses at 1Gyr as the initial mass distributionforbothdiscandsphericalcomponents.Thisallowsus toignorethemass-lossduetothestellarevolutioninthecodeduring thedynamicalevolution.

We use a slightly shallower slope for the heavier masses but keepthesamebreakpointstogeneratetheIMFforthestellardisc motivatedbyobservations(Bartkoetal.2010)³:

αdisc=

⎧⎨

⎩

1.3,if0.08M≤m<0.5M 2.3,if0.5M≤m<1.0M 1.3,ifm≥1.0M

. (16)

We explore several models for the distribution of orbital parameters in the discas summarizedin Table 1. We considertwo main scenarios forthe origin of the stellar disc.The first one is the formationofthe discdueto thestar – discinteractionsin an activegalacticnucleus(AGN).Panamarevetal.(2018)showedthat thegaseousaccretiondiscmaycapturestarsfromthesurrounding

2Notethattheseestimatesdonotincludestellarremnantsmeaningthatthe actualenclosedmasswithintheregionsmaybehigher.

3Galactic centre observations suggest an even more top-heavy profile dN/dm∝m⁻^0.45^±^0.3(Bartkoetal.2010).

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Table1. Listofsimulationswithanuclearstellardiscandsphere.

Fiducialmodels:

Massfactor Initialorbital Disc3Ddensity parameters

1;10;30 stardisc 1.75

2.4 3.3

1;10;30 stardisc-random 2.4

1;10;30 thermal 2.4

Additional models:

Nd Ns Md/Ms

10⁴ 10⁵ 0.14

10³ 10⁵ 0.04(massivedisc)

5×10⁴ 5×10⁴ 1.0

9×10⁴ 10⁴ 8.8

Note.Listofmodelswithdifferentinitialconditionsforstellardiscs.The defaultnumberofstarsinthediscandthesphereareNd=10³andNs=10⁵, respectively;theradialnumberdensityprofileexponentofthesphereand thediscare−1.75andγ=−2.4.Forthestardiscinitialconditionswealso adoptedtwoadditionalγvaluesasshown.Foreachofthesemainmodels, weadoptedthreedifferentmassfactorstoscalethestellarmassdistribution asshowntoacceleratethecode(seetext).Intotal,forthemainmodelswe have9stardiscmodels,3stardisc-random,and3thermalmodels.Forthe additionalmodels,themassfactoris30,thediscradialdensityprofileslope isγ=2.4andinitialorbitalparametersarethermal.

star cluster with the captured stars following the disc-like shape resemblingtheshapeoftheunderlyinggaseousaccretiondisc(see alsoBartosetal.2017).Theformedstellardiscisinsteadystate balancedby theaccretionofstarsontothe SMBHandcapturing newstarsbytheaccretiondisc.Togeneratetheinitialpositionsand velocities,wetakedatafromPanamarevetal.(2018)at1relaxation time (enough to form the steady state disc) and make statistical bootstrappingtoincreasethenumberofstars(inPanamarevetal.

2018 the authors had to use the superparticle approach where 1 particle representeda group ofstars).First, weconvertpositions and velocities to 6 Keplerian orbital parameters (this is a good approximation for orbits deep inside the influence radius of the SMBH),then generate alarger number of objects corresponding

to the distribution functionof orbital parameters, and finally, we converttheorbitalparametersbacktopositionsandvelocities.This way wegenerate1000 particlesforourmodelsfrom theoriginal

≈100particlestakenfromPanamarevetal.(2018).Werefertothe initial orbitalparametersof thediscstarsderivedthisway asthe stardiscinitialconditions.Fig.2(bluelinesinbothpanels)shows notable features: nearly circular orbits formost of the stars and low orbitalinclinations. Thereis alsoalineardependence of the orbitalinclination,eccentricityandsemimajoraxisthatresembles theouterwarpofthestellardisc(seetheleft-handpanelofFig.3 that shows the correlation between the inclination angles and eccentricities).

As this type of initial conditions may seem specific to the underlyingaccretiondiscmodelusedinPanamarevetal.(2018),we exploredanotherfamilyofthestardiscinitialconditionswherewe keptthesamedistributionsoftheorbitalparametersasinFig.2,but randomizedtheinclination– eccentricity– semimajoraxisrelation as shownin the middle panelof Fig. 3. Werefer to theseinitial conditionsasthestardisc-randominitialconditions.Inthestardisc initialconditionmodels,wevarythe3Ddensitypower-lawslopefor semimajoraxesasdescribedinSection3.2.

Inadditiontothestardiscandstardisc-randominitialconditions, wealsoexplorethe casewherethe stellardiscfollowsathermal eccentricity distribution,uniformly distributed orbitalinclinations betweencos10^◦andcos0^◦,anda3Dpower-lawdensityslopeforthe semimajoraxesρ∝r^−2.4implyingthatdN/da=a^−0.4.Orangelines inbothpanelsofFig.2andtheright-handpanelofFig.3highlightthe differencesbetweenthemodels.Werefertotheseinitialconditionsas thermalinitialconditions.TheremainingKeplerianorbitalelements, namelylongitudesofthe ascendingnodes,argumentsofperiapsis andmeananomaliesaredrawnfromauniformdistributionwithin thewholerangeoftheirallowedvalues.

WeperformasetofsimulationswithN_s=10⁵totalnumberof starsinthesphere,N_d=10³totalnumberofstarsinthediscand averagemassratioofm_∗/Mbh=5×10⁻⁷.Giventheslightlydifferent massfunctionsforthediscandforthespherethetotalmassfraction ofthediscisMd/Ms 0.015.Toexplorethe effectsoftheinitial orbitalparametersdistributionweusethreesetsofmodels:stardisc, stardisc-random,andthermal,asdescribedabove.Forthestardisc model,wevarythepower-lawslopeforthe3Ddensitydistribution ρ ∝ r⁻^γ with γ = 1.75 to represent the standard Bahcall–Wolf cusp(Bahcall&Wolf1976),γ=2.4tomatchtheobserveddensity

Figure2. Left-handpanel:Initialdistributionofeccentricitiesforthestellardisc.Bluehistogramshowstheinitialconditionsoriginatingfromthestardisc simulations(Panamarevetal.2018)ofAGNswhiletheorangehistogramrepresentsthermaleccentricitydistribution.Right-handpanel:Distributionofcosines ofinclinationanglesforthestellardisc.Blueshowsthestardiscinitialconditionsandorangelinecorrespondstothethermalmodel:uniformdistributionin cos(i)correspondingtoanglesbetween0and10^◦.

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Figure3. Scatterplotofstellardisceccentricitiesandinclinationanglesforthethreetypesofinitialconditionsadopted.Left-handpanel,labelledstardisc, showstherelationbetweeneccentricityandinclinationanglearisingfrompreviousstardiscsimulationsinAGNs(Panamarevetal.2018).Middlepanel,labelled stardisc-random,showsthemodelinwhichthecorrelationisremovedbyindependentlyassigninginclinationsandeccentricitiesfromthestardiscmodel,and theright-handpanelshowsthethermalmodel(seetext).

distributionoftheclockwisestellardiscintheGalacticcentre(Yelda etal.2014)andγ =3.3– thesteepestdensityprofileinourmodels whichoriginatesfromthestar– discsimulationsofPanamarevetal.

(2018),forothermodelswefixγ=2.4.Thisgivesusfivedifferent modelswhicharereferredas1Xmodels.Duetothehighnumerical cost,thesetypesofsimulationscanbeadvancedup to5–10Myr whenappliedtotheGalacticcentrecorrespondingtotheobserved ageofthenuclearstellardiscandS-stars(Habibietal.2017).

Tostudylong-termevolutionofthesystem,weincreasethetotal stellarmassofthesystembyfactorsof10and30,respectively,while keepingthesamenumberofparticles.Thisgives10moremodels.We refertothesemodelsas10Xand30Xmodels.AswesawinSection2, thedynamicaltime-scalesarereducedforalargertotalstellarmass.

Duetothefactthatthescalingwithmassisdifferentfortheresonant relaxationandforthetwo-bodyrelaxation(seeequations2and6), wecanstudythe contributionfrom theserelaxationprocessesby comparingthe1X,10X,and30Xmodels.Table1listsallthemodels andtheirparameters.

ThebottompartofTable1listsseveraladditionalmodelsthatwe simulatedwiththethermalinitialconditionsandγ=2.4discdensity exponent.Firstweincludeadditionalvariantsofthe 30Xmodels, whicharenumericallytheleastexpensiveandallowustoexplorethe parameterspaceofthesystem.Inparticular,werunadditionalmodels with(i)alargernumberofstarsinthediscN_d=10⁴;and(ii)withthe samenumberofstarsinthediscbutincreasedtotalmassofthedisc.

Furthermore,weexaminetwoadditionalmodelswherethenumber ofstarsinthediscwasequaltothenumberofstarsinthesphereand wherethenumberofstarsinthediscwas90percentofthetotalnum- berofparticleswiththetotalnumberofparticlesN=10⁵inbothruns.

Inaddition,inordertostudytheeffectofthesphereonthedynamics ofstarswithinthedisc,werunthefiducial1X,10X,and30Xmodels withoutthesphere,withonlyastellardiscofN_d=10³starsaround theSMBH.Werefertothesemodelsastheisolateddiscmodels.

4 DY N A M I C S O F T H E I S O L AT E D S T E L L A R D I S C S

Inthis section, wedescribethe evolutionof isolatedstellardiscs rotating around anSMBH with 100 percent of stars initially on progradeorbitsand no sphericalstellar component.As reference models,wechoosethethermalmodelswiththepower-lawdensity

slope ofthediscγ =2.4andthemassfactors1,10,and 30.We examinehowthetotalstellarmass(withfixednumberofparticles) affectsthedynamicsoftherelaxationprocesses.

Thedynamicalrelaxationprocessesare expectedto changethe distributionoforbitalinclinationanglesbywarping, twisting,and affectingthethicknessofthedisc.Theleft-handpanelofFig.4shows the10percent,50percent,and90percentcumulativedistribution levels of orbital inclination angles as a function of semimajor axis. Theinnermoststarstendtohave higherorbitalinclinations, whichisexplainedbytheshorterrelaxationtime-scalesatsmaller distancesfromtheSMBH(seeSection2).Theright-handpanelof thefigureshowstheaverageinclinationangleasafunctionofmass indicatingthatthehigh-massstars(blackholes)havesystematically lower inclinationsformingathindisc. Thiseffectdevelopsin all modelswithanisolatedstellardisc.Thetimeinstancescorresponding tothe1X,10X,and30XmodelsinFig.4arechosentohavethesame averageinclinationangleforthelightstars(m≤10M),⁴implying thatthecurvesintheright-handpanelofFig.4overlapforlightstars byconstruction.Theinclinationversussemimajoraxisshowsvery similartrendsintheleft-handpanelofthefigureimplyingthatall modelsareatthesamelevelofrelaxation.

In the top panels of Fig. 5, we compare the distribution of cosinesoftheorbitalinclinationsformassive(m≥10M)andlight (m≤10M)stellarobjects.Eachpanelcorrespondstothemodel with differentmassfactors(1X, 10X,and 30X)atthe sametime as in Fig.4. Whilethe distributionoflow-mass starsisidentical byconstruction,the1Xmodelclearlyshowsthestrongesteffectin verticalmasssegregationcomparedto10Xand30Xmodels.Bottom panelsofthesamefiguredemonstratethattheisolatedstellardiscs alsofeaturemasssegregationintheeccentricitydistributionasseen fromthenormalizeddistributionoforbitaleccentricitiesforlightand massivestars.Butinthiscasethehigher-massmodelsshowstronger masssegregationthanthe1Xmodel.

Toexaminethetimedependenceofmasssegregationininclination andeccentricityanditsdependenceonthe1X,10X,30Xmodels,we trackthetimeevolutionoftheroot-mean-square(rms)inclination

4Thischoiceissomewhatarbitrary,butasweseefromtheright-handpanel ofFig.4thevaluem=10Misinthemass-gapproducedbythestellar evolutionandallobjectswithhighermassesinthesimulationarestellarmass blackholes.

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Figure4. Left-handpanel:Movingaverageofinclinationanglesinsemimajoraxesforthethermalisolateddiscmodels(seeTable1)withmassfactors accordingtothelegend.Right-handpanel:Movingaverageofinclinationanglesinmassforthesamemodels.Bothpanelscorrespondtothesametimesnapshot.

Blue,red,andgreencoloursindicate1X,10X,and30Xthermalisolateddiscmodels,respectively.Fadedlines(points)ofthesamecolourshow90and 10percentquantiles.Thetimesnapshotsforeachmodelarechosensuchthataverageorbitalinclinationsmatchforthelow-massstars(thiscorrespondsto56, 5,and0.9Myrfor1X,10X,and30Xmodels,respectively;seedottedlinesinFig.6).Forthe10Xand30Xmodels(redandgreenlinesintheright-handpanel), weshowm/10andm/30,respectively,sothatthemassrangesoverlap.Thewindowforthemovingaverageswaschosentobe100datapoints.

Figure5. Normalizedhistogramsoforbitalinclinationsandeccentricitiesforlightandmassiveparticles.Toppanelsshowthecosinesoforbitalinclinationsfor 1X,10X,and30Xmodels.Bottompanelsshoweccentricitiesforthesamemodels.Solidanddashedlinesindicatemassive(m≥10M)andlight(m<10M) particles,respectively.ThehistogramscorrespondtothesametimesnapshotsasinFig.4.Theshadedhistograminthebottomrightpanelshowstheinitial eccentricitydistributionforallofthesemodels.

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Figure6. EvolutionofrmsinclinationanglesandeccentricitiesforthesamethermalisolateddiscmodelsasinFigs4and5afunctionofseculartime(defined inequation10).Solidanddashedlinesindicatemassiveandlightparticles,respectively.Dottedverticallinesshowthetimeatcorrespondingtothesnapshots ofFigs4and5.

anglesandeccentricitiesforallthemodelsasafunctionofsecular time.Fig.6confirmstheexpectationthatverticalmasssegregation isstrongestforthe1Xmodelsandtheweakestforthe30Xmodels whilemassdependenceineccentricitiesistheopposite.

Toexplore thelong-termevolution ofisolatedstellardiscs,we focusonthe30Xmodelwhichisnumericallytheleastexpensive.

Fig. 7showsthat massiveandlightstarsdevelopa differentrms inclinationandeccentricityduringthe firststagesoftheevolution andcontinuewiththesamepaceafterafewthousandseculartimes.

Asaresult,masssegregationeffectsareexpectedtobepresentin suchsystems(seeFig.8).

VerticalmasssegregationingalacticnucleimaybecausedbyVRR as shownfirstby Szölgyén & Kocsis(2018)and later confirmed by otherstudies(Fouvry etal.2020;Magnanetal. 2021;Máthé etal.2022).On the otherhand,angular momentum conservation duringpairwiseinteractionsimpliesthattwo-bodyrelaxationmay alsocauseverticalmass segregationin the long-run especiallyin highlyanisotropicsystems(Ernstetal.2007;Tiongco,Collier &

Varri2021).Themasssegregationineccentricitiesmaybecaused bybothSRR(Fouvryetal.2018;Gruzinovetal.2020)and two- bodyrelaxation. Asshownby Alexander,Begelman& Armitage (2007),thermseccentricityofastellardiscisrelatedtoitsvelocity dispersionas:

e_rms=√ 2σ

v_K, (17)

where vK is the Keplerian orbital speed. Following this logic, Mikhaloff&Perets(2017)showedthattheevolutionofrmseccen- tricitiesisdifferentforlightandheavystarsasaresultoftwo-body interactions.

To explore which relaxation process drives anisotropic mass segregationpredominantlyinourmodelsofisolatedstellardiscswith nosphericalcomponent,weperformthecorrelationcurveanalysis (Rauch& Tremaine1996; Eilonetal. 2009;Kocsis& Tremaine

Figure7. Long-termevolutionofthermsinclinationanglesandeccentrici- tiesforthe30Xmodel.Toplinesineachpanelshowlightparticles.

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Figure8. Normalizedhistogramsofthecosinesoforbitalinclinationsand eccentricitiesforlightand massiveparticles. For the30Xmodelsatthe momentof28000tsec.

2015).Wemeasurechangesinenergiesandangularmomentafor eachparticleto computetherateofdiffusionin energy– angular momentum for the whole system (see Appendix A for details).

Fig.9showsthermschangeinKeplerianenergy(totracktwo-body relaxation),angularmomentummagnitude(totrackSRR),angular momentumvectordirection(totrackVRR),andtheZ-componentof theangularmomentumvector(VRRinverticaldirection)relativeto theinitialstate.Clearly,VRRstronglydominatesinthe1Xmodels:

therelative changein angularmomentum vector directionoccurs fasterthan the change in other quantities (thered line is always above).However, due to the strongnodalprecession, the change is predominantly along the azimuthal component of the angular momentumvector,whiletheorbitalinclinationisnearlyconstant.

Themixingoforbitalinclinationanglesisrepresentedbythechange in theZ-componentof the angularmomentum vectors(shown as a black line in Fig. 9). Thisis suppressed initially compared to the changein the energy,but becomesmoreprominent after10² orbitalperiods.For10Xand30Xmodelstwo-bodyrelaxationisthe mostefficientrelaxationprocess,atleastduringthefirst10³periods.

Fig.9alsoshowsthecomparisonoftheefficiencyoftherelaxation processesformassive(dashedlines ofthesamecolour)andlight (dottedlinesofthesamecolour)stars.Aslightstarsrepresentthe majorityofthesystem,theyarealmostindistinguishablefromthe overallclusterproperties.Ontheotherhand,thedifferencebetween the change in energy and angular momentum for massive stars indicatesthatthediffusioninenergyandangularmomentumforthe massivestarsislessefficient.SincerelaxationisdrivenbyVRRin the1Xmodel,thisexplainsthestrongestverticalmasssegregation comparedto 10X and 30X models discussedabove(seeFig. 6).

Onthe otherhand,energyand angularmomentumchangesinthe 10X and 30X models are mostlydriven by two-body relaxation.

This explainsthe more prominent masssegregation effect in the eccentricitiesin10Xand30Xmodelscomparedtothe1Xmodel.

SincethecontributionfromSRRistheleastsignificantforthestudied models (especially the 10X and 30X models, see green lines in Fig. 9),weconcludethatthe anisotropicmasssegregationeffects in thermal isolateddiscsare causedby both VRRand two-body relaxation.

5 I N T E R AC T I O N O F A N U C L E A R S T E L L A R D I S C W I T H A S P H E R I C A L C U S P O F S TA R S We analyse the shapeand thickness of the stellar discusing the quadrupolemomentmatrix(seee.g.Roupasetal.2017,Sz¨olgy´en etal.2021)definedasfollows:

Qαβ= _N

i= 1LiαLiβ

_N

i= 1 |Li|² , (18)

whereLiistheangularmomentumvectorofthei-thstar,αandβ arethecorrespondingCartesiancomponents.

Thelargesteigenvalueofthematrixcorrespondstotheshapeof thediscwhilethecorrespondingprincipaleigenvectordescribesthe orientationofthesystem

Qαβν=λν. (19)

In this normalization, the trace of the matrix satisfies TrQ = 1 meaningthatequaleigenvaluesλ₁=λ₂=λ₃= ¹₃representasphere withzeroangularmomentum,andarazor-thindischas(λ1,λ2,λ3)= (1,0,0).Thus,thelargesteigenvaluewhichtakesthevalues1/3≤λ

≤1quantifiesthethicknessofthestellardisc.

Inthissection,theinclinationanglesofthediscstarsrefertothe mean inclinationswithrespectto theprincipaleigenvectorof the disc.Thisway,theorbitalinclinationsarealwayscomputedrelative to the instantaneous mid-planeofthe discin angular momentum spaceevenifthediscasawholeistiltedwithrespecttoitsinitial position.

5.1 Secularevolutionoftheembeddednuclearstellardiscs Wefollow thesame stepsas inSection 4to study the dynamics of discsembedded in a spherical cuspof starson secular time- scales.But inthissubsectionweusethestardiscmodelswiththe power-lawslope γ =3.3and comparethe isolateddisc,the disc embeddedinasphere,andthe30Xmodelofthesamediscembedded inasphere.Thesemimajoraxes– inclinationdependence(left-hand panelofFig.10)isqualitativelysimilarforallthreemodels,butthe modelswithasphericalcomponentextendtohigherinclinationsin theinnermostpart.

Theright-hand panelof Fig. 10shows a strikingdifference in theaverageinclinationsasafunctionofmassforhigh-massstars:

whiletheisolateddiscmodelshowslowerinclinationangleswith increasing mass, there is almost no correlation between stellar massandorbitalinclinationsformodelswithanisotropicspherical component.SimilartoFig.4,thetimeinstancesshowninFig.10 have the same average inclination angle for the low-mass stars (m≤10M)byconstruction.Whiletheverticalmasssegregation effectvanishes,thedependenceoftheinclinationonthesemimajor axisismoreprominent.Thelattereffectdevelopsfasterandextends tohigherinclinations,andsomestarsevenfliptocounter-rotating orbits(i>90^◦).