1983ApJ. . .274. . .53W The AstrophysicalJournal,274:53-61,1983November1 © 1983.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. to bemorethananoccasionalannoyancetheirallies,theycansufferconsiderabledamageasaresult of this unequalassociation.Theirperipheralregionsmaybestrippedbythegiant’sforcefield,andtheirorbits in thedisintegrationandtotalabsorptionoflatter.Thisorbitaldecayisatrootamanifestationgeneral gradually decay,causinganevermoredisruptiverelationshipbetweengiantandsatellitewhichmayculminate tendency ofclosedsystemstoeliminatediversity.Thedetailedmechanisminvolvedhasbeendescribedintwo known asdynamicalfrictionwhichprovidesapurelylocaldescriptionoftheinteraction(Chandrasekhar1942). the satellitetoindividualstarsinhaloofgiant;transferratecanbemodeledbyaneffectiveforce complementary ways.Ona“microscopic”levelencounterslead,onaverage,totransferofkineticenergyfrom An apparentlymorerigorous“macroscopic”descriptionconsiderstheeffectonitsorbitofglobalresponse be inphasewiththesatelliteandsosuggestnegligibleorbitaldecay.Theresolutionoftheseconflictingpredictions satellite stimulatesinitscompanion(e.g.,Kalnajs1970).Unfortunatelywhileastraightforwardapplication of Chandrasekhar’s formulapredictsasignificantdecayrate,calculationsoftheglobalresponseusuallyfindit to numerical simulationoffersausefulcomplementtopurethought.AnearlyfullA-bodyofmineseemed to may lieinapropertreatmentofresonantstars,butfullunderstandingstilleludesus(Tremaine1981). with asmallnumberofparticlescangreatlystrengthenprejudices,itneverbeconclusive.LinandTremaine agree withasimplemodelbasedondynamicalfriction(White1978).Nevertheless,althoughsingleexperiment Two massiveparticlesrepresentthesatelliteandcore oftheparentgalaxy.Intheircombinedfieldmovea simulations andtosuppresscertainspuriouseffects.Their methodisanextensionoftherestricted3-bodyapproach. but notoneachother;inthiswaythehaloresponsecan influencethesatellite,buttimeneededtocompute (1983) recentlytackledtheprobleminafarmoresystematic wayusinganoveltechniquetospeeduptheir particle accelerationsismuchlessthaninafulliV-body calculation.LinandTremainetermthismethoda large numberofparticlesrepresentingtheouterhalo theparent.Theseparticlesexertforcesonsatellite “ semirestrictedA-body”approach.Thedecayratesintheir experimentsagreedwiththepredictionsofChandrasekhar’s history. local formulainsize,dependenceonsatellitemassand diameter,andintheirinsensitivitytopreviousorbital forces betweenthehaloparticles,andwroteaprogram to followmotionsintheresultingforcefield.Anobvious visit totheInstituteforAdvancedStudy,Idulysetout equationsforacore-halo-satellitesystem,eliminatedall first testwasarepeatofone ofLinandTremaine’sexperiments,soIconstructeda galaxy-satellite system,loaded it intothecomputer,andwaited eagerlyfortheresults.Firstrunsrarelycomeout well,andIwasdisappointed to findthatmysatellite’sorbit didnotdecayasfastexpected;indeed,itappeared tohavelittlewish reach thecenterofitsparent atall.Somechecksshowedthatmycode,although unabletoproducefriction, could conserveenergyand angular momentumverywell.Iretiredbaffledtoread LinandTremaine’spreprint Satellite galaxiesorbitdeepwithinthesphereofinfluencetheirgiantneighbors.Althoughusuallytoopuny © American Astronomical Society • Provided by the NASA Astrophysics Data System It iscertainlyofinteresttoknowhowfastsatellitesaredraggedintotheirparentgalaxies,anddirect Much impressedbytheseresults,Idecidedtowritea semirestrictedA-bodycodeofmyown.Whileona follow thedynamicalprocessinavarietyofdifferentapproximations.Acomparisonresults enhanced byfixingthecenterofparent,anditissuppressedneglectingself-gravity shows thatthedecayratedependsonglobalresponseofparentgalaxy.Itisartificially description oftheunderlyingphysics,althoughitpredictsdecayrateswhichareapproximately of theresponse.Chandrasekhar’slocaldynamicalfrictionformulacannot,therefore,beacomplete correct. Asimulationmethodwhichcalculatesthegalacticgravitationalfieldtosecondorder based onageneralizationoftherestrictedthree-bodyproblem. in amultipoleexpansiongivesresultsagreementwithfulliV-bodyexperiments,butitruns several hundredtimesfasterforN=5000;itisonlyafactorof2or3slowerthanmethods Subject headings:galaxiesclustersof—numericalmethodsstarsstellardynamics The orbitaldecayofsatellitegalaxiescanbeinvestigatedusingsimulationmethodswhich Department ofAstronomyandSpaceSciencesLaboratory,UniversityCalifornia,Berkeley SIMULATIONS OFSINKINGSATELLITES Received 1983February11;acceptedApril6 Simon D.M.White I. BACKGROUND ABSTRACT 53 1983ApJ. . .274. . .53W 2 2 2 produced amuchhigherdecayrateconsistentwiththeirresults.Apparentlythedependedstronglyon the coreoftheirgalaxy.Theoffendingequationmotionwaseasilyremovedfrommycode,andanewrun declared thereasonformydiscrepantresultstobeself-evident:afterall,ifcenterofEarthwerenailed chance Tremainehimselfwasvisiting,andsoIshowedhimmyresults.Afterafewmomentsofintrospectionhe did myfullAT-bodysimulationagreewithdynamicalfrictiontheoryandLinTremaine’sexperiments, down, whatwouldhappentothetides?Thisinsightdid,indeed,offeraplausibleexplanation—butwhythen while itdisagreedwiththenewexperimentinwhichcenterwasfree?Clearlysomethingremainedtobe understood. Letuslookatthesituationinmoredetail. whether thecoreofgiantgalaxycouldpursueitsownlittleorbitaboutbarycenterpair.By ingredient oftheirmethod;inordertoeliminateallparticle-particlerelaxation,theyhadelectednaildown more carefully.AnexaminationofthetextbetweenAbstractandConclusionsshowedthatIhadmissedakey 54 WHITEVol.274 respectively, the“softened”lengthsaredefinedbyp=\x—x\+eandanalogousformulae,Newton’s In theseequationsthesubscriptsc,s,andi(=1,N)denotepropertiesofcore,satellite,haloparticles constant hasbeensetequaltoone.Theyarederivedfromthetotalpotentialenergy, p ccr~truncatedatr=20.Ithengaveeachparticlearandomlydirectedvelocityequaltothecircular linear andangularmomentum. to theforceonaparticleisbalancedbyanequalandoppositeanotherparticle,systemconservesboth and asaresultsystemwhichobeysthemconservesenergyitevolves.Inadditionbecauseeachcontribution initial systemwhichwasinvirialequilibriumbutnotproperlyphasemixed.Ithereforeletitevolvefor200 at itsdistancefromthecore(takingintoaccountimposedsofteninge=10).Thisprocedureproduced an N =5000haloparticles.Theirtotalmasswasm10,andtheyhadasphericaldistributionwithdensityprofile time units(about10crossingtimes)withthesatellitemasssettozero.Atthisithadreachedasteady cut-off. ThissystemhasahighercentralconcentrationthanLinandTremaine’sstandard“galaxy”is a state, hadretaineditsinitialdensityprofilewithinr=14,andspreadoutattheedgestoblursharp “galaxy” withm=2,r30;itthushadone-tenththetotalmassofitscompanion,andorbitenclosed95% closer approximationtowhatoneexpectsofarealsystem.Isetthesatelliteoncircularorbitaroundthis roughly withthatexpectedforagalaxyinorbitaboutcompanionof10timesitsmass. simple fourthorderRunge-Kuttaroutine.Inadditionitimpliedaneffectivesizeforthesatellitewhichagreed of thehalomass.ThelargesofteningIchoseallowedmetointegrateequationsmotionrapidlywith a scsc h predicts thatoneortwoorbitsshouldbesufficientto reach thecenterfromr=11.Nailingdowncoreto satellite hadspiraledinonlyasfarr=11,andittook 22orbitstogetr=7;thedynamicalfrictionformula do notconservelinearmomentumbecausetheforcesexerted bythecorearenolongerreflectedinitsacceleration. The remainingequationsofmotionstillconserveenergy sincetheyarederivedfromequation(4).However, mimic LinandTremaine’sexperimentswasquiteeasy. I justremovedequation(1)frommycodeandsetx=0. previous experiment(usinganidenticalinitialcondition), eightorbitswereenoughtobringthesatelliteinto Angular momentumaboutthecoreisconservedbecause thecoreforceispurelyradial.WhenIrepeatedmy is quadrupolarandhassmalleramplitude.(Rememberwhy theEarthhastwosmalltidesadayratherthanone difference intidalresponsepointedoutbyTremaine:when thecenterisfixed,satellitepullshaloparticles core ofitsparent.Thesubstantialdifferencebetween the twosimulationsseemedplausiblytoresultfrom h big one.)Aftersomediscussion Itriedatestofthisinterpretation;reranthefixed centersimulationwitha quadrupolar inthissituation and,asexpected,thedecayratewasagainveryslow. Thecore-satelliteseparation away fromitinapredominantlydipolarresponse,whereas whenthecenterisfree,responseinitsneighborhood second satelliteofidentical mass orbitingattheantipodalpointtofirst.(The symmetry wasmaintainedby s in thesethreeexperimentsis plotted asafunctionoftimeinFigure1;thedifferences areobvious. appropriate averagingofthe twoaccelerationsateachtimestep.)Theresponse ofthehaloisforcedtobe c The equationsofmotionwhichIoriginallychosetorepresentacore-halo-satellitesystemwerethefollowing: © American Astronomical Society • Provided by the NASA Astrophysics Data System I setupanequilibriumgalaxyformyexperimentasfollows.Aroundacoreparticlewithm=10,put The abovespecificationsresultedinthesloworbitaldecay whichsosurprisedme.Aftereightfullorbitsthe c 3 Xi =m(x-Xi)p+{xjpr;i1,N.(3) ccish mx3 = Äs-c)Psc~+m'£(x)p;(1) hjicic xX 1 Mc s)Pcs"I”^(is)Pis?(2) W =mp-+Y,(rnP~),(4) scschcisi II. THEPROBLEM i i i No. 1, 1983 SIMULATIONS OF SINKING SATELLITES 55

Fig. 1.—The distance of the satellite from the center of its parent galaxy is plotted as a function of time for three 5000-body simulations which used versions of the semirestricted iV-body code. Triangles show results from my first model in which the center of the galaxy was allowed to move freely; squares show evolution from the same initial conditions when the center was fixed; circles give the results of an experiment in which two identical satellites were forced to remain at diametrically opposed positions. The initial orbital period of the satellite was about 220 units.

At this point at least one thing had become clear. The relatively rapid orbital decay in the experiment with a fixed center depended crucially on the global response of the halo particles; a local dynamical friction description of the interaction could not be adequate. The agreement between Lin and Tremaine’s experiments and the predictions of Chandrasekhar’s formula must, then, have been partly fortuitous, at least for such soft and massive satellites. As a result, when he went home Tremaine added several disclaimers to the Conclusions of his manuscript. A disquieting problem remained, however; my old iV-body results seemed to disagree with the most realistic of the new simulations and instead gave a decay rate similar to that in the models with a fixed core. Was some further artificiality of our approach suppressing the physical response we were investigating? III. THE PLOT THICKENS The most obvious artificiality of the semirestricted models lay in the structure of the large galaxy. The satellite could become more bound only by weakening the binding of halo particles to the core. The self-binding of the halo and of the core did not appear in the models, even though they may be important in real galaxies. In addition the kinematics of the halo particles were not consistent with the mass distribution felt by the satellite. Thus the satellite in my simulations initially responded to twice as much mass as a halo particle in its vicinity, and, as a result, its orbital frequency did not match those of the particles it encountered; this mismatch seemed a plausible source for the discrepancy between the new experiments and my old full AT-body simulation. A test of this explanation clearly required some generalization of the semirestricted method which would make it possible to construct a more realistic galaxy. The evolution of a spherical system can be simulated efficiently using Newton’s discovery that the force at each point is the same as if all the interior mass were concentrated at the center; it seemed to me that this might provide the generalization I needed. Consider replacing equation (3) by, 3 3 Xi = (mc+ N¡mh)(xc - xjp^ + ms(xs - x;)ps¡“ , i = 1, Nh; (5) where ATf is the number of particles with |jc — jcc| < 1^ — jcc|. This equation differs from equation (3) in that the halo particles now exert forces on each other. These forces are calculated as though the mass of each particle were distributed evenly over a spherical shell centered on the core, and (apart from the softening) they clearly give a close approximation to the exact force field in a near spherical model. Provided a fast sorting routine is available, the integration of equations (1), (2) and (5) is not much slower than that of equations (1-3), even for as many as 5000 halo particles. If we fix the core, the remaining equations can be derived from the total potential energy, 1 1 l W = mcmspc~ + mh £ [(mc + N¡mh)pc¡- + mspsr ], (6)

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1983ApJ. . .274. . .53W 56 WHITEVol.274 and so,asbefore,theyconserveenergyangularmomentumbutnotlinearmomentum.However,equation(1), in usingequations(1),(2),and(5)appearedtobetheneglectofself-gravitynonsphericaldistortions;this variations oftotalenergyandangularmomentumturnedouttobesmallthemainapproximationinvolved as aresultthefullsetofequationsconservesneitherenergynorangularmomentum.Insatelliteproblem, although apparentlythemostreasonableequationofmotionforcore,cannotbeobtainedfrom(6); seemed unlikelytobeimportant. from understandingwhatwashappening.Asexpected,experimentswiththecenterfixedhadastrongerresponse massive Itookm=20/(N-hi).Whentheresultscameoutofcomputertheyprovedthatwasstillfar already described.Theonlydifferencelayinthefactthatsincemethodnolongerrequiredcoretobe when thesemirestrictedequationswereused.Figure2showscore-satelliteseparationasafunctionoftimeintwo and evenmorerapidorbitaldecaythanbefore.Unfortunatelyevolutionwiththecenterfreewasslower between differentrealizationsofthesameinitialconditions.Theseresultsforcedmetoconcludethatunrealistic simulations ofeachtype;withN=5000,statisticalfluctuationswereclearlytoosmalltocausesignificantdifferences the decayrateactuallyseemedtodecreaseatlatetimes;whenoneofexperimentsinFigure2wasrunfrom galactic structurewasnotresponsibleforthesloworbitaldecayinmyearlierexperiment.Innewsimulations in-phase responsepredictedbysometheoreticalcalculations?Ifso,whywastherestilladiscrepancywiththefull integrator andruna1000-bodyversionofmystandardsatellitedecayexperiment.Luckilycomputertimeatthe with sofewparticlesmightnotbeveryreliable.TocheckthisIdecidedtodustoffmycopyofAarseth’sV-body iV-body results? t =1000to2000,thecore-satelliteseparationdecreasedbyonlyfourunits.Couldthisbeamanifestationof was fixed.Trianglesshowevolution from thesesameinitialconditionswhenthecenterwasallowedtomove. Circlesaretheresultsofa1000-body ch spherical shellscodewhichwouldhavetaken25minutestodothesame1000-bodyexperiment.Thedecayof simulation. ThisdemonstratednotonlytheadvantagesofowningaVAX11/780,butalsoefficiency Institute wasnothardtoget,andinonlyafewdaysIhadthe41hoursofCPUtimeneededcomplete satellite orbitinthisnewsimulationisshownFigure2;whileitwasslowerthansphericalshellsmodelswith iV-body methodtosimulatesatellitedecaywasclearlyamoresubtlematterthanpoorstatistics. a fixedcenter,itwasmuchfasterthaninthemodelswherecenterfree.Anyproblemwithusingdirect simulation usingAarseth’sfulliV-body integrator. h simulation, evenaheavilysoftened1000-bodyexperiment,mightdestroytheabilityoforbitstolineupwith are toostronglyperturbedbystochasticeffects.Itseemedpossiblethattwo-bodyrelaxationinafullV-body I setupinitialconditionsforsomenewexperimentsinasimilarwayandwithidenticalparameterstothose By thistimeIwasbeginningtowonderwhethermy250-bodyfullV-bodysimulationatfault;oneexperiment © American Astronomical Society • Provided by the NASA Astrophysics Data System Fig. 2.—Squaresinthisdiagramshow theevolutionofsatelliteorbitintwo5000-bodysphericalshells simulationsinwhichthecenter A simulatedgalaxymaybeunabletosetupacoherentresponseperiodicperturbationifitsparticleorbits IV. SOMEDEADENDS Time 1983ApJ. . .274. . .53W 23 No. 1,1983SIMULATIONSOFSINKINGSATELLITES57 some ofmysphericalshellsmodels.InanefforttoinvestigatethisImodifiedthecodeincludea response; itmight,therefore,preventtheformationofanunevolvingbinarysystemkindapproachedby random deflectionofthevelocityeachhaloparticleattimestep.Ichosermsvalue introduced velocitychangessimilartothosefoundinarealiV-bodysystembutdidnotaffectthetotalenergyof at eachradiustomimictheeffectofencountersina1000-bodysystemwithsimilarstructure.Thisprocedure this newprogram,Iwasyetagaindisappointedbytheresults.Theorbitofsatellitecompletelyunaffected simulation. WhenItooktheinitialconditionsofmyprevioussphericalshellsexperimentsandevolvedthemwith energy andangularmomentummustgiverisetoaneffectiveviscosityiniV-bodyexperiment,somight by thechange.Theabilitytosuppressrandomperturbationoforbitswasclearlynotcrucialslowdecay them hesuggestedthatthefullAT-bodyresultsmightbesuspect,butnotbecauseofuncorrelatedfluctuationsin rate oftheearliersimulations. particle velocity,butbecauseencounterscausecorrelatedchangesinthevelocitiesoftwoparticles.Thisexchange cause thetidalbulgeraisedbyasatellitetolagbehinditinsamewaythatdissipationcauses of theEarthtobedisplacedfromsublunarpoint.Tremaineprovidedaroughestimatethiseffectwhich showed itsamplitudetobeinverselyproportionalNandsuggestedthatitmightbecomedominantbelowavalue of Nintherange10-10.ThisseemedworthtestingsoIraneightmorefullA-bodyrealizationsmystandard particularly intheearlystageswhensatellitewasonouteredgeofmassdistribution.Second,these seem significant,andtheAAbodyresultsretainedanuncompromised,ifnotunquestioned,claimtovalidity. galaxy (r<20)itsorbitdecayedatasimilarrateinallthefullAT-bodymodels.Thusspuriousviscositydidnot small Nexperimentsdidnotevolvefasterthanthe1000-bodysimulation;infactonceasatellitewaswellwithin its Unfortunately, thereasonfortheirdisagreementwithsphericalshellssimulationswasstillnotexplained. experiment usingjust200particles;ifspuriousviscouseffectswereimportant,thesesimulationswouldevolvefaster lines aretheresultsofeight200-body simulationsofthesamephysicalsituation. than the1000-bodyexperimentwhichIhadrunearlier.TheresultsareshowninFigure3andallowonetodraw generalize mycodeagainsothatitcouldaccount,atleasttolowestorder,fortheself-gravityofnonspherical of thegalaxyisstillonlyapproximate.Inparticularasymmetricpartgalaxy’sresponsetosatellite, distortions. Thiswasaconsiderableincreaseincomplexity,buttechniquefirstexploitedbyVillumsen(1982) although abletoactdirectlyonthelatter,isunableinfluencestructureofgalaxyitself.Itseemedunlikely two conclusions.First,N=200wassufficientlysmallthatthevariationsbetweenrealizationsbecamesignificant, offered afeasiblewaytoattacktheproblem. that thislackofself-consistencycouldbeimportant,buthowonesure?Theonlyreliabletestseemedto to Conversations withTremaineperiodicallyrekindledmyattemptstounderstandthisproblem,andduringoneof © American Astronomical Society • Provided by the NASA Astrophysics Data System Fig. 3.—Thedecayofthesatelliteorbit inthe1000-bodyfullN-bodysimulationisindicatedbyfilled circlesinthisdiagram.Theother What couldbethematterwithsphericalshellscode?AsInotedabove,itstreatmentofgravitationalfield V. ARESOLUTION? 58 WHITE Vol. 274 Consider the following integral expression for the softened potential due to a finite continuous distribution of matter:

(j)(r) = j d3r'p(r')(\r'- r\2 + e2) 1/2 . (7)

In the region defined by r' > r the factor p' = (r'2 + e2)1/2 can be removed from the denominator of the Green’s function, and the remainder may be expanded in powers of r/p'. Similarly in the interior region, r' < r, the factor p = (r2 + £2)1/2 can be removed and the remainder expanded in powers of r'/p. If these expansions are carried out to second order the result is: 4>{r) = [ d3r'p(r')[p'~1 + (r • - |r2p'“3 + l(f ' »")V~5] *r >r

+ J dh'pir'lp-1 + (r • ^)p-3 - V2P"3 + !(»• • r'fp-5]. (8)

For a discrete system the potential at the position of the ith particle is obtained by replacing the continuous p(r) by a sum of delta-functions. If the particles are assumed to be numbered in order of increasing r, we obtain: M 1 3 3 5

r, = X; — xr r,- = \r: The potential felt by each particle is thus a sum of monople, dipole, and quadrupole terms which represent separately the effect of particles inside and outside its position. Note that this expansion is not a second order version of the expansion used by Villumsen (1982). If equation (9) is multiplied by mh and summed over all particles, the total potential energy is found to be: - IF mcmsPcS + Wlh ^ [(mc “I ^in)Pci F ^sPsi F (fy* ^in l^lnjPci F 2^i ^in ^iPci ] • (1^) i If this is adopted as the self-energy of the galaxy, and if its interaction with the satellite and the motion of its center are treated as before, the resulting equations of motion are (1), (2), and 5 - 3 _j_ It Xi mcripci -i ms(xs x¡ )psi -I- Pex "b ~ Min''¡Pci + Pin Pci - ¿(Pin ' ‘•¡ViPic + 2 -Mn npu r 5 J 7 + 3(^i„ ' i)Pic~ — r(*'¡ ‘ ^in »•.>¡P¡c“ i =lNh. (ii) As in the case of the spherical shells model, these equations conserve energy and angular momentum if the core is fixed, but not if it is allowed to obey equation (1). Notice that this equation now specifies the acceleration of the core to be identical to that of a halo particle in its neighborhood; this requirement seems reasonable but is not obeyed if either equation (3) or equation (5) is used. The new equations move each particle as if it were a point mass, but calculate its gravitational influence as if it were a certain continuous density distribution localized within a distance e of a spherical shell of radius r. Once my spherical shells code was modified to integrate these new equations of motion, an obvious test was to run an isolated equilibrium galaxy with the center fixed. When I did this, I was gratified to find that energy and angular momentum were indeed conserved, but I was puzzled because the output showed the system to be less quiescent than expected. Pictures of the particle distribution showed what was wrong; for some reason the center of the halo density distribution did not stay on top of the center of the harmonic expansion, but instead jumped to a position about 1.5 softening lengths away. Although initially perplexed, I soon found an explanation for this strange behavior. The problem lay in the truncation of the multipole expansion. It is easy to show that the total force on any close pair of particles due to their mutual interaction is Ft = ?mhr2P~5(9 — 15r2p_2)»‘, (12) where the pair separation is assumed to be much less than their distance, r, from the center of expansion. While Ft is inwardly directed for large r, it becomes outwardly directed when r < e. This outwardly directed mean force drives the density center away from the center of expansion when the latter is fixed and leads to quite unstable behavior even when the center is free. While isolating the source of instability was relatively easy, finding a cure

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1983ApJ. . .274. . .53W 2 No. 1,1983SIMULATIONSOFSINKINGSATELLITES59 for itwasalotmoretricky.Ieventuallyhitontheideaofsofteningmonopoletermlessthandipoleand After aninitialmixingperioditsstructurewasquitestableandIcouldseenosignofevolution.Indeedtherms of £inthehigherordertermsissufficientforFtobeinwardlydirectedatallradii.WithconsiderablereliefI quadropole termsin(11)andthusenhancingtheattractiontocenter.Itturnsoutthatdoublingvalue ready totacklethesatelliteproblem. predicts foraheavilysoftenedsystemof5000particles.Usingthisinitialgalaxytorunsatelliteexperimentwith rms deviationgrewasthesquarerootoftime,itsgrowthwasabout50timesslowerthanstandardrelaxationtheory crossing times,showingaquiteremarkablesuppressionoftwo-bodyrelaxationeffects.Although,asexpected,this change inthebindingenergyofanindividualhaloparticlewasonly5%averagevalueoverlast40 found thatwhenthisextrasofteningwasincluded,thequadrupolecodedidindeedbecomestableandsoseemed As afinaltestIranthissysteminisolationwiththecenterfreefor1000timeunits(orabout50crossingtimes). when Irananexperimentwiththecenterfreeitagreedquitecloselyresultsofdirect1000-body the centerfixedled,asbefore,toastrongcollectiveresponseofgalaxyandrapiddecay,butnow,finally, simulation. Tomakesuremyluckwouldhold,Iranasecondrealizationofthesameinitialconditionsandgot results inFigure4.TheagreementbetweenthequadrupolecodeandAf-bodyisexcellent.Atlastthings almost identicalresults.Plotsoftheevolutionsatelliteorbitinthesetwoexperimentsarecomparedwithearlier seemed tomakesense;thediscrepantresultsofsphericalshellscodeapparentlystemmedfromitsneglect too small,orithadanincorrectphaselagwithrespecttothesatellite.Thesimplestwaydistinguishbetween effect onthesatellitedecayrate.Twopossibilitiespresentedthemselves;eitheramplitudeoftidalbulgewas self-gravity oftheresponse.Whatstillneededtobedeterminedwashowthisneglectcouldhavesuchalarge them wasjusttoplottheresponseandseewhatitlookedlike. full iV-bodyexperiment.Inthelastcasehighnoiselevelforcedmetosuperposethreesnapshotsofsame onto theorbitalplane,smoothingresultingparticledistribution,subtractingitsaxiallysymmetriccomponent,and t experiment separatedbyabouthalfasatelliteorbit.Inallfourcasesthewas15unitsawayfrom then contouringtheresult.Plotsaregivenforthreeof“shortcut”simulationmethods,and1000-body well thecrosseswhichdenotethese experiments agreewiththecircleswhichdenoteresultsobtained thefulliV-bodycode. include highermomentsoftheforcefield,butitsphaselag,andthereforetorqueitexertedonsatellite, clear inthefirstpictureandcontrastsdramaticallywithessentiallyquadrupolarresponseofothermodels. core ofthegalaxyattimeshown.Thestrongdipoleresponseinmodelswherecenterwasfixedisvery In thesphericalshellsexperimentsamplitudeofresponsewassimilartothatinsimulationswhich that producedbythequadrupolecode.Thisgavemeatleastsomeconfidenceshortcutcodewasnow was small.Thusthephaselagofresponse,ratherthanitsamplitude,seemstodependonapropertreatment of self-gravity. Althoughobscuredbythehighernoiselevel,responseinfullAT-bodysystemwasquitesimilar to I setupaninitialgalaxyinequilibriumthestandardmannerandagaintookN=5000,m20/5001. © American Astronomical Society • Provided by the NASA Astrophysics Data System Figure 5showsplotsobtainedfrompairsofsimilarexperimentsbysuperposingsnapshotstheirprojection hc Fig. 4.—Theorbitaldecayintwo5000-body simulationswiththequadrupolecodeiscompared resultsshowninFig.2.Notehow 1983ApJ. . .274. . .53W by filledcircles,andthesatelliteisorbitinginacounterclockwise direction intheplaneofplot.Thesmoothingisapproximately gaussian withadispersionof3units.Theresponseobtainedthe sphericalshellscodewhenthecenterisfixedshowninFig.5a; In eachdiagramtheboxhasside50unitsandiscenteredongalaxy. Thepositionsofthegalaxycoreandsatelliteareshown zero. Themainmaximaandminimaarelabeledwiththelocalsurface density;theunitsaresuchthatmeansurfacedensitywithin projected halfmassradiusoftheunperturbedgalaxyis600. Fig. 5d,thatfoundwiththefullJV-bodycode.Contoursarelinearlyspaced andaredashedatnegativevalues,solidpositivevalues Fig. 5bshowstheresponsefoundwithsamecodewhencenter isfree;Fig.5cgivestheresponsefoundwithquadrupolecode,and © American Astronomical Society Fig. 5.—Contourplotsoftheasymmetricpartgalaxyresponsein simulationsofasatellite-galaxysystemusingfourdifferenttechniques. Provided bythe NASA Astrophysics Data System 1983ApJ. . .274. . .53W fourth orderandcheckedthatthismadenodifference.However,theuncertaintyhasnotyetconvincedme sufficiently generaltotreattheproblemcorrectly.Tobecertain,Ishould,perhaps,havegeneralizedit(say) such alargeincreaseincomplexityisjustified. discoveries? Thedifferencesinbehaviorbetweenarealgalaxyandsystemwhosegravitationalfieldisdescribedby increased byanotherfactorof2replacingtheRunge-Kuttaintegrationschemeapredictor-correctormethod, and Aarseth’sfullN-bodyintegratorwouldtakeabout500timesaslong.Theextratimeneededtorunthequadrupole spherical shellscodetakes1.4timesaslongthesemirestrictedcode;quadrupole2.2long, Chandrasekhar, S.1942,PrinciplesofStellarDynamics(NewYork: code seemswellworthspendinginviewofitsmorerealistictreatmenttheforcefield.Itsspeedhasrecentlybeen and ithasalsobeenmodifiedtoallowtheuseofshorttimestepsnearcenterexpansion;thisallows associated withsuchshortcutsisverylarge.Forthestandard5000-bodyproblemIhavebeendiscussing, that “cautionshouldbeexecutedwhenundertakingsuchsimulations.”Nevertheless,thegainincomputertime a truncatedsetofequationscanbequitesubtle,andIwouldadviseagainstnotedauthor'srecentsuggestion Kalnajs, A.1970,inGravitationalN-bodyProblems,ed.M.Lecar inferred fromtheexperimentsofLinandTremaine.Theagreementbetweentheirpredictions problems ingalacticdynamicsforwhichsuchasimulationmethodcouldgiveusefulresults. the codetooperateefficientlywithamuchsmallersofteningparameterthanbefore.Thereiswiderangeof Lin, D.N.C,andTremaine,S.1983,Ap.J.,264,364. by myexperiments(relativelymassiveandfuzzysatellites);thisis,however,theregimewhichisofmostinterestfor for theseexperimentalresults.AlthoughthedependenceofdecayrateonsatellitesizeinLinandTremaine’s effect onthesatelliteistobemodeledcorrectly;neglectingself-gravityofresponseleadsanincorrect application torealgalaxysystems.LinandTremaine’sdecisionnaildownthecenteroftheirentirely of Chandrasekhar’slocalformulafordynamicalfrictionnowseemstohavebeenfortuitousintheregimeexplored experiments demonstratestheimportanceoflocaleffects,apurelydescriptionsatellitedecayclearlymisses phase lagandtoamuchreducedtorqueontheorbit.Itwouldbeofgreatinterestfindtheoreticalexplanation altered theglobalpatternofitsresponse,andinmyexperimentsthisalterationgreatlychangedorbitaldecayrate. predictions ofChandrasekhar’sformulaandtheresultsnumericalexperimentmayturnouttobelittlemore some essentialaspectsoftheunderlyingphysics.Forsoftandmassivesatellites,agreementbetween More surprisingly,mysimulationsshowthattheresponsemustbecalculatedinaself-consistentmannerif its than acombinationofdimensionalanalysisandcoincidence. when startingacomplexproject,andthatgrantfromtheNSF(AST-8114715)canbegreathelpinfinishing it. Simon D.M.White:DepartmentofAstronomy,601Campbell Hall,UniversityofCalifornia,Berkeley,CA94720 Dover). (Dordrecht: Reidel),p.13. What canbelearnedaboutgalaxysimulationsandtheorbitaldecayofsatellitesfromthissagadeadends The orbitaldecayofsatellitegalaxieshasturnedouttobeamorecomplexproblemthanmighthavebeen © American Astronomical Society • Provided by the NASA Astrophysics Data System A finalconclusionfromthisstudyisthattheatmosphereofInstituteforAdvancedStudygreatbenefit SIMULATIONS OFSINKINGSATELLITES61 VI. EPILOGUE REFERENCES Tremaine, S.D.1981,inStructureandEvolutionofNormalGalaxies, White, S.D.M.1978,M.N.R.A.S.,184,185. Villumsen, J.V.1982,M.N.R.A.S.,199,493. ed. S.M.FallandO.Lynden-Bell,(Cambridge:Cambridge University Press)p.67.