198 9MNRAS.238. .407H The interactionofalargegalaxywithitssystemsatellitescanhavenumberobservable used asemi-restrictedN-body techniquetoexaminethedependenceofdecay ratesonthe changes inresponsetotheaccretionofevenlow-masscompanions.Sucheventsmaybe consequences. Thisisespeciallytruefordiscs,whichcanundergolarge-scalemorphological periods ofenhancedstarformation,granddesignspiralstructure,andverticalthickening responsible forthegrowthofnormaland/orboxybulges,triggeringnuclearactivityand haloes, satelliteorbitsmaynotbestableoveraHubble time.Usinganorder-of-magnitude cores. discs. Theoutcomesofencountersbetweenellipticals andsmallergalaxies,thoughless dynamical friction.Dependingontheorbitalradiiof thesatellitesandstructureofgalaxian evolution. Ingeneral,theaccretionofsatellitesby a largegalaxyisdirectconsequenceof anomalous amountsofgasordust,shells,multiple nuclei, X-structures,andcounter-rotating spectacular, arealmostasdiverse.Cannibalism may accountforellipticalscontaining investigate thedecayofsatellite orbitsaroundvariousmassdistributionsinsome detail. the MilkyWaycouldhave accretedupto10percentofitsmassinthe formofsmall estimate appropriateforanisothermalhalo,Tremaine (1981)computedthatagalaxysuchas companions sincebeingformed. Giventhepotentialimportanceofcannibalism, itisnaturalto 1 Introduction Most previousstudieshave reliedontheuseofparticlesimulations.Lin&Tremaine (1983) In ordertostudysuchscenarios,itisnecessary tounderstandthesatellite’sorbital © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Mon. Not.R.astr.Soc.(1989)238,407-416 Accepted 1988November29.Received8;inoriginalform Lars HernquistandMartinD.WeinbergInstituteforAdvanced Simulations ofsatelliteorbitaldecay Study, Princeton,NJ08540,USA The inferredsinkingratesareingoodagreementwithanalyticpredictionsfrom numerically usingbothself-consistentandsemi-restrictedN-bodymethods. Summary. Theorbitaldecayofsatellitesaroundsphericalgalaxiesisstudied misuse oftheterm‘non-self-gravitating’. discussions ofthisproblem.Wesuggestthatthediscrepancycanbetracedto a linearperturbationanalysis. satellite isnotalocalprocess. self-gravity oftheresponseisincluded,indisagreementwithotherrecent 1988 June27 In particular,weverifythattheorbitaldecayisstronglysuppressedif Finally, wedemonstratethatthelossoforbitalangularmomentumby 198 9MNRAS.238. .407H p{r)32 (1) 408 L.HemquistandM.D.Weinberg orbital geometryandparametersassociatedwiththesatelliteprimarygalaxy.White(1983) unimportant. White’s claim,andsuggested,rather,thattheself-gravityofresponseprimaryisnot consistent studiesbyBontekoe&vanAlbada(1987;hereafterBvA)failedtosubstantiate errors offactors~2-3,becausetheapproximatenaturetheirmethod.However,self- of self-consistentalgorithms.Theyconcludedthattheself-gravityresponseis White (1988,hereafterZW)recentlyre-examinedthesinking-satelliteproblemusingavariety subsequently notedthatthesinkingtimesinferredbyLin&Tremainecouldbesubjectto by BvAandZW.Weemphasize,however,thatthisdisagreementismainlyovertheinterpreta- relevant findingsare: an importantfactorindeterminingthedecayrates.PromptedbyBvA’ssimulations,Zaritsky& In ordertofacilitatecomparisonswithothernumericalstudies,weusetheprimarymodel tion oftheterm‘non-self-gravitating’andnotreliabilitysimulations. important ingredientinthedynamicalfrictionprocess,contrarytoconclusionsexpressed Chandrasekhar (1943)may,insomesituations,providethecorrectscalings, and sinkingtimesanalytically,usinglinearperturbationtheory.Forpresentpurposes,hismost chosen byBvAandZW.Thegalaxyisan«=3polytrope,withmassM=2radiusR=l, presented inSection3.Finally,conclusionsandadiscussionfollow4. semi-restricted methods.Ourresultsindicatethattheself-gravityofresponseisan a sphericalgalaxyusingthemodelconsideredbyBvAandZW,withfullyself-consistent and theunitoftimeisfixedbysettingG=1.ThesatellitemodelledasarigidPlummer Following BvAandZWunlessotherwisestated, wetakethesatellitemass,ra,tobe 4*r[l+Wr)F' sphere withdensityprofile 2.1 PARAMETERSANDUNITS 2 Method in theprimary.Typically,r(0)=1,1.2,or1.5.For runswithr(0)>1,wemeasurethedecay The self-consistentsimulations performedbyBvAandZWwereobtained using expansion techniques. In ordertotestthedependence oftheresultsonmethod itself,ourself-gravitat- time fromtheinstantatwhichsatellitereaches edgesofthegalaxy,r=1. satellite orbit,a*(0),inordertoexaminethesensitivity ofthedecayratetotransientresponses M: ra=10:1andtheratiooftotalsizeprimary tothesatellitescale-lengthisalways Æ:r=10:l. m =0.2,andthesatellitescale-length,>,tobe r =0.1.Thus,themassratioisusually 2.2 SELF-CONSISTENTMETHOD s c s s s s s c sc c A differentapproachhasbeentakenbyWeinberg(1986,1989)whocalculatestheresponse (ii) theself-gravityofresponsewillincreasesinkingtimebyfactors~2-3. In anattempttoclarifythepresentsituation,weexaminedecayofsatelliteorbitsaround In thefollowingsectionwedescribeourmethodinsomedetail.Numericalresultsare (i) theresponseisalwaysglobal,althoughasimpleexpressionsuchasthatderivedby In allcases,theorbitofsatelliteisinitiallycircular. Wevarytheinitialradiusof © Royal Astronomical Society • Provided by theNASA Astrophysics Data System , _3ml1 s 198 9MNRAS.238. .407H 2 ing runswereperformedusingahierarchicaltreealgorithm(Barnes1986;Barnes&Hut is computedbydirectlysummingtherelevanttwo-bodyinteractions.Theinfluenceofremote hierarchy, subjecttoaprescribedaccuracycriterion.Asrule,theforcefromnearbyparticles tional accelerationisobtainedbyallowingeachparticletointeractwithvariouselementsofthe cells andthenmultiplemomentsofeachcell,uptoafixedorder,arecomputed.Thegravita- particles isincludedbyevaluatingthemultipoleexpansionsofcellssatisfyingaccuracy ^0, (2) requirements atthelocationofparticleonwhichtocomputeacceleration. where sisthesizeofacell,ddistancefromcentremasscelltoparticlein Tremaine 1983). decay tominimizerelaxationeffectsandanyassociatednumerical‘viscosity’(e.g.Lin& question, and6isaninputparameterwhichtypicallyoforderunity,thentheinternal in theexpansionsisusuallysmallcomparedwithnumberofparticlescorresponding tree methodallowsfortheuseofalargeA,whichisdesirableincasesatelliteorbital manipulate thehierarchyreducesoperationscountperstepfrom~0(N),whichwould cell andasignificantimprovementinefficiencyisrealized.Theuseoftreedatastructureto structure ofthecellisignoredandmultipoleapproximationused.Thenumberterms 1986). result fromadirectsumoverallparticles,to~0(N\ogN).Inprinciple,then,thehierarchical d the particlenumber,A,timestep,A/,andaccuracyofforcecomputation,whichis length equaltothescale-lengthofsatellite,r.Since,typically,>e,individualparticles evolution ofthesatelliteorbitstoeachthesequantitieswasstudiedinorderestablishthat determined by6andthenumberoftermsinmultipoleexpansions.Thesensitivity motion are a convergedsolutionhadbeenobtained. effectively interactwiththesatelliteasiftheywerepoint-like.Fordefiniteness,equationsof particle /,andthe^areparticle-cellparticle-particle interactionpotentials. where isthemassofeachparticle,^number ofcellsandparticlesthatinteractwith ?=- z M ^i’ijr/„_-\2j.„213/2 The non-self-gravitatingsimulations wereperformedusingavariationofthe semi-restricted determined from theLane-Emdenequation, withpolytropicindex n=3.Thepotentialfield is N-body techniqueintroduced byLin&Tremaine(1983).Thepotentialfor eachparticleis c 2.3 SEMI-RESTRICTEDMETHOD In thehierarchicaltreemethod,particlesarefirstorganizedintoanestedhierarchyof Specifically, if The freeparametersintheself-consistentsimulationsaretwo-bodysofteninglength,e, The satelliteandparticlescomprisingthegalaxyalwaysinteractdirectlywithasoftening © Royal Astronomical Society • Provided by theNASA Astrophysics Data System r mr /-i IA¡r)+rj j-i [(fj-rs)+r sc c v js(<~r) s 2-13/2 ? Satellite orbitaldecay409 (4) (3) 198 9MNRAS.238. .407H mr 5 410 L.HernquistandM.D.Weinberg fixed intimeandisnotallowedtomovespatially,whichtheappropriatenumericalanalogue particles interactdirectlyandtheequationsofmotionare of thenon-self-gravitatinganalysisperformedbyWeinberg(1986,1989). _Vi(s~i) VAX 11/780orSUN3/50.Thesemi-restrictedcodeisalsofullyvectorizedandprovedtobe if^ -Ví>(r¡,/)—: where istheexactexpressionforpotentialofan«=3polytrope.ValuesVOare A fullyvectorizedversion,identicalinallotherrespectstotheoriginalcode,wasusedforsome Pittsburgh SupercomputingCenter.Themajorityoftheself-consistentmodelswereevolved All simulationsdiscussedinthispaperwereperformedontheCRAYXMP-48at obtained fromalook-uptable. using thetreecodedescribedbyHernquist(1987,1988),whichisnotcompletelyvectorized. 2.4 DETAILS highly efficient,allowingustoroutinelyuseV~10. Hernquist (1989),leadstoanimprovementintheoverallefficiencyofmethodbyafactor of thesimulations.Theprocedureusedtovectorizetreetraversals,whichisdescribedby of ~2-3,implyingatotalspeed-upfactor400-500onanXMPcomparedto An exampleofthesinkingcurvesfoundbymethodsoutlinedinSection2isshownFig.1 (solid curves),wheretheyarecomparedwiththecorrespondingresultsfromWeinberg(1989) we plotthedistanceofsatellitefromorigincoordinatesystemwhichcoincides (dashed curves).Fortheself-gravitatingmodel,weplotdistancefromsatelliteto 3 Results f 3.1 SINKINGRATES with thecentreoffixedpotentialforalltime. centre ofmasstheparticledistributionasafunctiontime.Innon-self-consistentcase, expansions, implyingtypicalerrorsintheacceleration oneachparticle,relativetoafulldirect and £=0.01.Termsupto,including,quadrupole orderwereincludedinthemultipole do notinteractwithoneanotheronanindividualbasis, thetimestepinthiscaseisnotlimited by theparticledensity. number ofparticleswas7V=80000andthetimestep wasagainAi=0.01.Sincetheparticles energy wasapproximately0.55percent,while the changeinangularmomentumwas sum, oforderafewtenths1percent.Overthecourse ofthesimulation,changeintotal both BvAandZW;forthe samevalueofN,decaytimesagreeto«2percent. However,the approximately 0.47percent.Forthesemi-restricted simulationshowninFig.1,thetotal numerically. Thetimesfor thesatellitetoreachr=0.2differbynearly50per cent.Speculat- sinking timeimpliedby the analyticcalculationissomewhatlargerthan thatobtained s The onlynumericalparametersinthiscaseareNandAt.Again,thesatellitegalaxy The parametersfortheself-consistentmodelinFig. 1were20000;0=0.95;A0.01, © Royal Astronomical Society • Provided by theNASA Astrophysics Data System The sinkingrateinferredfrom oursimulationisinexcellentagreementwiththat obtainedby r—23/ y=i LVjr)+Tc]’ s N /\ TtlAIi is) 2]3/2 (6) (5) Satellite orbital decay 411

ing that this might be attributable to non-linear and/or time-dependent effects which are not included in the analytic calculations, we performed a number of non-self-gravitating simulations with satellites of increasingly smaller masses. In the linear regime the decay time

scales exactly as rdecay

varying mass, where we choose to define rdecay as the time for the satellite orbit to decay from

rs = 1.0 to t\ = 0.2. As is clear from this figure, non-linear effects are significant for ms ä 0.05. For smaller masses the inferred decay times appear to be converging to a scaled value within the error of the analytic prediction of « 12.7.

Ideally, we would like to repeat the self-gravitating simulations with small ms to verify that the discrepancy in this case is also due to non-linear and/or time-dependent effects. Unfortunately, the computational expense is prohibitive for the particle number required to minimize discreteness effects for the time-scales of interest. These issues might be more efficiently studied using a technique optimized for spherical distributions such as an expansion method. Nonetheless, we emphasize that the ratios of self-gravitating to non-self-gravitating decay times are nearly identical for both the linearized calculation and the simulation. The robustness of the sinking curves in Fig. 1 to numerical effects was examined by varying other simulation parameters. For smaller A the decay times are slightly smaller than the models shown in Fig. 1, for both simulations with and without self-gravity. For example, the sinking time in a self-consistent model with N= 5000, as typically adopted by BvA and ZW, is approximately 5 per cent shorter than that of the model in Fig. 1. Reducing the number of particles from 80 000 to 20 000 in the semi-restricted code also reduces the sinking time by about 5 per cent, compared with the corresponding curve in Fig. 1. Improving the accuracy of the force computation in the hierarchical tree calculation by reducing 6 does not result in a noticeable change in the sinking curve.

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 198 9MNRAS.238. .407H 412 L.HernquistandM.D.Weinberg negligible. Forexample,innon-self-gravitatingrunswithN=80000,thetimesfororbitto galaxy. the satellitecompletes~4revolutionsand20revolutions,respectively,beforereaching tively. Inthelattertwocases,transientspresumablyhavelesseffectonorbitaldecay,since decay fromr=1.0to0.2are8.1,8.0andfor(0)=l,1.2,1.5,respec- satellite orbit.Thedifferencesintheinferredsinkingtimesbetweenvariousrunsare Linear perturbationtheorydemonstratesthatthewakeinprimarygalaxydueto orbiting atr=1/3inaself-gravitatingn3polytrope isshowninFig.3.Thedipole(/=1) analytic prediction(solidhorizontalline)anditsestimatederror(shaded)areshownforcomparison. Figure 2.Scaleddecaytimeasfunctionofsatellitemassfornon-self-gravitatingsimulations(opencircles).The maximum densitypeak,locatednearx=-0.1,y =-0.1,whilethesecondarymaximum, higher-order (/=2,3,4,...)termsinamplitude.The dipoletermisresponsibleforthe component isreducedbythebarycentricmotionand isroughlycomparabletothesumof 3.2 GLOBALRESPONSE immediately behindthesatellite,resultsfromhigher-order components. satellite hasglobalextent(Weinberg1989).Forexample, thewakeproducedbyasatellite observed inthesimulation, p_,andthedensityinabsenceofsatellite, p.Locating discussed above,weexpecttoseesuchwakesinthe particledistributions.Forthepurposeof wakes iscomplicatedby both discretenesseffectsandambiguityindetermining theproper comparison withanalyticresults,wedefinethewake asthedifferencebetweendensity diametrically opposedsatellites ofequalmass,ra=0.03.Theirorbitsareforced toremain origin forp,owingtobarycentric motion.Thelatterdifficultyisespeciallyirksome. sdecay s Nbody 0 s 0 Finally, weconsideredthepossibleeffectsoftransientsbyvaryinginitialradius To circumventthedifficulties arisingfromthebarycentricmotion,weopt toorbittwo Given theagreementbetweenperturbation theory andtheN-bodyexperiments © Royal Astronomical Society • Provided by theNASA Astrophysics Data System m s 198 9MNRAS.238. .407H T decay curveshowninFig. 1andthosecomputedearlierbyBvAZWusing otherA-body welcome checkonbothtechniques. Inaddition,theagreementbetween self-gravitating The agreementbetweenthe numericalsimulationsandtheanalyticdecaycurves providesa techniques demonstrates that thehierarchicaltreemethodisaviabletool forevolving multipoles. established byapplyingtheharmonicdecomposition routinetoA-particlerealizationsofpure reasonably well.Thedifferencesareconsistentwith fluctuationnoise,thelevelofwhichwas decomposition insphericalharmonicsYand Besselfunctions;uptoorders1=4, primary. Wecomputethewakefromparticle distributionbyperformingaharmonic prediction usingthesemi-restrictedmethod.ThedecaycurveissimilarinshapetoFig.1,but will vanish.Accordingtoanalytictheory,thedecaytimeshouldthenincreasebyafactorof~ wake mustbeeveninazimuth,andallharmoniccontributionswithoddorder(/=1,3,5,...) 4 Discussionandconclusions identical totheone-satellitecase,assumingra

-1 -.5 0 .5 1

-1 -.5 0 .5 1 X Figure 4. Wakes derived from (a) analytic calculation and (b) non-self-gravitating simulation involving two diametrically opposed satellites each of mass ras = 0.03. Satellite positions are indicated by solid circles. The units and contour levels are as in Fig. 3.

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 198 9MNRAS.238. .407H =+ =+ the hierarchicaltreemethodrelativetoexpansiontechniquesHernquist&Barnes1989). times, dependinguponwhetherornottheresponseistreatedself-consistently.Accordingto collisionless systems,atleastinsomecircumstances.(Wedeferadiscussionoftheefficiency gravitating case,theeffectofself-consistentresponseonevolutionsystemisigno- Fig. 1,theself-gravitytypicallyincreasesdecaytimebyafactorof«2-2.5.Innon-self- potential perturbationfa-.Inthenon-self-gravitatingcase,contributionoffato satellite. HerewhichdeterminesfaaccordingtoPoisson’sequation,istheresponse where ^isthepotentialofprimary,pandareunperturbedterms, y°A) Pi-(7) red, bydefinition.Thatis,supposeweexpandthedensityandpotentialinperturbationseries the lowest-ordercorrectionsand^,whichisofsameorderaspotential potential inthecollisionlessBoltzmannequationistobeignored.Thecorrectnumerical and Acknowledgments from misuseoftheterm‘non-self-gravitating’. between ourconclusionsandthoseofotherrecentnumericaltreatmentsapparentlyresults large galaxiesissensitivetothefullself-gravitatingresponseofprimary.Thediscrepancy mass are,essentially,attemptstoincludetheinfluenceofhigher-ordertermsinpotential procedure istosettheprimarypotentialequalvaluethatitwouldhaveinabsenceof We thankSimonWhiteforstimulatingdiscussionand JoshBarnes,PietHut,ScottTremaine the effectofnon-lineartermsistakenintoaccount.Theorbitaldecaytimessatellitesaround Thus, thevalidityoftheseapproximationsmustbejustifiedonacase-by-casebasis. will yieldmeaningfulresultssincethehigher-ordertermsdistortpotentialaswellshift expansion oftheprimaryresponse,equation(8).Itisnotclear,apriori,thatsuchprocedures the centreofmass;thisshiftdoesnotcorrespondtoanysingleterminperturbationseries. aspect oftheself-gravityresponse. as iftherewerenoperturbationpresent.Strictlyspeaking,then,thecentreofmassshiftisan in partbyNewJerseyHighTechnologygrant88-240090-2. ^extpri0 -“(8) supplied throughagrantprovidedbythePittsburgh SupercomputingCenter.MDWwas and ananonymousrefereeforusefulcommentson themanuscript.Supercomputertimewas Bontekoe, Tj.R.&vanAlbada, T.S.,1987.Mon.Not.R.astr.Soc.,224,349. Barnes, J.,1986.In:TheUseofSupercomputersinStellarDynamics, p.175,ed.Hut,P.&McMillan,S.,Springer- supported inpartbyNationalScienceFoundationgrant AST-8802533andLHwassupported Hernquist, L.,1987.Astrophys. J.Suppl,64,715. Chandrasekhar, S.,1943.Astrophys. J.,97,255. Barnes, J.&Hut,P.,1986.Nature, 324,446. References the satellite.Thatis,^=fa)andcentreofprimarypotentialistoremainfixedinspace, x pri{)x ext 1? pri More importantly,boththeanlayticandnumericalcalculationsgiveratherdifferentsinking In summary,theanalyticandnumericalresultspresentedhereareingoodagreement,when Modifications tothesemi-restrictedapproachwhichincludemotionofcentre © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Verlag, Berlin. Satellite orbitaldecay415 198 9MNRAS.238. .407H 416 Tremaine, S.,1981.In:TheStructureandEvolutionofNormalGalaxies,p.67,edsFall,S.M.&Lynden-Bell, Hernquist, L.,1988.Comp.Phys.Comm.,48,107. Hernquist, L.,1989.J.Comp.Phys.,inpress. Weinberg, M.D.1989.Mon.Not.R.astr.Soc.,inpress. Hernquist, L.&Barnes,J.,1989.Astrophys.submitted. White, S.D.M.,1983.Astrophys.J.,274,53. Weinberg, M.D.,1986.Astrophys.J.,300,93. Lin, D.N.C.&Tremaine,S.,1983.Asirop/rys./.,264,364. Zaritsky, D.&White,S.M.,1988.Mon.Not.R.astr.Soc.,235,289. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System D., CambridgeUniversityPress,Cambridge. L. HernquistandM.D.Weinberg