Modeling the disruption of the globular cluster Pal5 by Galactic tides

1.8 MB·English

by  W. Dehnen

MODELINGTHE DISRUPTION OFTHE GLOBULARCLUSTER PAL5 BY GALACTIC TIDES
WALTERDEHNEN123, MICHAEL ODENKIRCHEN3,EVA K. GREBEL43 AND HANS-WALTERRIX3
[email protected];[email protected];[email protected];[email protected]
ABSTRACT
The globular cluster Pal5 is remarkable not only because of its extendedmassive tidal tails, but also for its very low
massandvelocitydispersionanditssize, whichis muchlargerthanitstheoreticaltidalradius. Inorderto understand
theseextremeproperties,weperformedmorethan1000N-bodysimulationsofclusterstraversingtheMilkyWayonthe
orbitofPal5. Tidalshocksatdiskcrossingsnearperigalacticondominatetheevolutionofextendedlow-concentration
clusters, resultingin massive tidaltails and oftenin a quickdestructionof the cluster. The overlargesize of Pal5 can
4
beexplainedastheresultofanexpansionfollowingtheheatinginducedbythelaststrongdiskshock 150Myrago.
0 ∼
Someofthemodelscanreproducethelowobservedvelocitydispersionandtherelativefractionsofstarsinthetailsand
0
betweentheinnerandouterpartsofthetails. Oursimulationsillustratetowhichextenttheobservabletidaltailstraceout
2
theorbitoftheparentobject.ThetidaltailsofPal5showsubstantialstructurenotseeninoursimulations.Wearguethat

n
thisstructureisprobablycausedbyGalacticsubstructure,suchasgiantmolecularclouds,spiralarms,anddark-matter
a
clumps,whichwasignoredinourmodeling.
J
Clusters initially larger than their theoreticaltidal limit remain so, because, after being shocked, they settle into a
1
newequilibriumnearapogalacticon,wheretheyareunaffectedbytheperigalactictidalfield. Thisimpliesthat,contrary
2
to previous wisdom, globular clusters on eccentric orbits may well remain super-tidally limited and hence vulnerable

to strong disk shocks, which dominate their evolution until destruction. Our simulations unambiguously predict the
1
v destructionofPal5atitsnextdiskcrossingin 110Myr.Thiscorrespondstoonly1%oftheclusterlifetime,suggesting
∼
2 thatmanymoresimilarsystemscouldoncehavepopulatedtheinnerpartsoftheMilkyWay,buthavebeentransformed
2 intodebrisstreamsbytheGalactictidalfield.
4
1 Keywords: stellardynamics—Galaxy: halo—globularclusters: general—globularclusters: individual: (Palomar5)—methods:
0 N-bodysimulations
4
0 1. INTRODUCTION stretching creates drains through which stars from the outer
/ parts of the cluster are carried away. This limits the bound
h AmongsttheglobularclustersoftheMilkyWaygalaxy,Pal5is
partofaclusterwhichisinequilibriumwiththetidalfieldtoa
p exceptionalinmanyrespects. First,itisoneofthefaintestand
- leastmassiveoftheseobjects;itsvelocitydispersionissosmall tidalradius,rtid,whosesizemaybeestimatedbyequatingthe
o outwardly directed stretching tidal force to the cluster’s inter-
thatitcouldonlyrecentlybedetermined,usinghigh-resolution
tr spectroscopy, to be below 1kms−1 (Odenkirchenetal. 2002, nalgravitationalattraction. ForaclusterofmassM orbitingat
s galactocentricradiusRinagalaxywithconstantcircularspeed
a hereafterpaperI).Secondandmostremarkable,ithasbeende-
v ,weget5
: tected, using data from the Sloan Digital Sky Survey (SDSS; c
v GM
Xi Yteonrdkaettlaela.s2t040k0p),ction poopspsoessisteadipraeicrtioofnstifdraolmtatihlse,cwluhsitcehr aenxd- rt3id ≃ vc2 R2. (1)
r contain more stars then the cluster itself (Odenkirchenetal. Foreccentricorbits, rtid variesalongthe orbit,andissmallest
a 2001; Rockosietal. 2002; Odenkirchenetal. 2003, hereafter at perigalacticon. The strong tidal force at perigalacticonacts
paper II). Pal5 is actually the first and only globular cluster only for a short fraction of the orbital period, and passages of
so far for which such tails have been detected at a compa- theperigalacticonareoftenreferredtoastidalshocksor‘bulge
rable level of significance. Thus, this is the first direct ev- shocks’.
idence that mass-loss induced by the Galactic tidal field can Thetidalfieldgeneratedbythestellardiskwithitssteepver-
substantiallyaffecttheevolutionofglobularclusters,aprocess tical gradient is different. It is much stronger than that of the
whichisspeculatedtohavedestroyedmanylow-massclusters smoothhaloandbulge,anditiscompressive. Thelatterisbe-
of an initially much richer Galactic system of clusters (e.g., cause, whilst the cluster is crossing the disk, its stars feel the
Vesperini&Heggie1997). additionalattraction of the disk stars within the cluster. How-
The tidal field generated by a smooth Galactic mass distri- ever, the strong compression only acts for a short time. This
bution stretches the cluster in a direction towards the Galactic allows to use the impulse approximation, which gives for the
center and weakly compresses it in tangential directions. The velocitychangeofastar atverticaldistancez fromthecluster
center(Ostriker,Spitzer&Chevalier1972;Spitzer1987)
1Department ofPhysics & Astronomy, University ofLeicester, Leicester ∆v = 2gmz/V,
LE17RH,UnitedKingdom −
2Astrophysikalisches Institut Potdsam, An der Sternwarte 16, D-14482 whereV istheverticalvelocityoftheclustercrossingthedisk
Potdsam,Germany andg 2πGΣ(R)themaximalverticalaccelerationexerted
m
3Max-PlanckInstitutfu¨rAstronomie,Ko¨nigstuhl17,D-69117Heidelberg, ≈
Germany 5Intheliteraturethisequationoftencomeswithanextrafactor1/2,which
4Astronomisches Institut der Universita¨t Basel, Venusstrasse 7, CH-4102 occurseitheriftheorbitisassumedtobecircularortheGalacticpotentialtobe
Basel,Switzerland thatofapointmass.Ifbothassumptionsaremade,thefactoris1/3.
2 Dehnen,Odenkirchen,Grebel,&Rix
by a (thin) disk with surface density Σ(R). These changes in donotcoverthe globular-clusterparametersrelevantfor Pal5.
velocityresultinrelativechangesofthespecificenergiesofthe Gnedin et al., for example, assumed in their study that disk
starsoftheorderof shocksareslowandrare,i.e.thecluster-internaldynamicaltime
isshorterthanthedurationoftheshockandmuchshorterthan
∆E σg r 2g r
h| |i m h m h, (2) tdisk, whereas for Pal5 shocks are fast and frequent (see sec-
E ≈ V E ≈ σV tion2.4). Also,theorbitofPal5differssignificantlyfromthat
h| |i h| |i
ofNGC6254,asadoptedbytheseauthors.
wherewehaveused z rh/2withrh denotingthecluster Moreoverasfarasweareaware,allstudiesofglobularclus-
h| |i ≈
half-massradius.Low-concentrationsystemstendtohavelarge ter evolution start with clusters limited by their (perigalactic)
half-massradiiandlowvelocitydispersions,whichmakesthem tidalradius,assumingthatthislimitationisquicklyachievedby
very susceptible to disk shocks, in particular at small galacto- thetidalforcefield.Aswewillseebelow,thisassumptionisnot
centricradiiwhereΣ(R)andhencegm arelarge. Notsurpris- justified, neither theoretically as our simulations will demon-
ingly therefore,low-concentrationclusters are hardlyfoundin strate,norobservationally,sincePal5isanexcellentcounterex-
theinnerGalaxy,wherenotonlytheshocksaremuchstronger ample,seesection2.3.
but also occur much more frequently than for orbits in the Theprimarygoalofthepresentpaperistostudy,viadetailed
Galactichalo. N-body simulations, the effect of Galactic tides on a globu-
The velocity changes also result in an average heating per lar cluster moving on the orbit of Pal5 and to quantitatively
star of h∆Ei ≈ 2gm2rh2/3V2 (Ostriker,Spitzer&Chevalier comparetheobservablepropertiesofthemodelswiththoseof
1972; Spitzer 1987). One may use this result to estimate the the cluster in an attempt to understand its current dynamical
“shockheatingtime”tsh,thetimescaleoverwhichdiskshocks state and to constrain the formation history of this object. As
significantlyaffecttheclusterdynamics: a byproduct, our simulations will also give valuable insights
into the evolution of low-concentration globular clusters with
E 3σ2V2 β 1 experiencingstrong disk shocks in conjunctionwith a
t =t −h i =t (3)
sh disk ∆E disk4 g2r2 tim≫evaryingtidalfield.
h i m h
In section 2 we summarize the observed structural and dy-
(Gnedin,Lee&Ostriker1999). Here,tdisk isthetimebetween namical propertiesof Pal5. The N-body simulations are pre-
strong disk shocks, which for eccentric orbits, such as that of sented in section 3. The dynamics of the simulated tidal tails
Pal5,equalstheorbitalperiodP. are discussed in detail in section 4. In section 5, we compare
Apart from tidal forcing, the evolution of globular clusters oursimulationsdirectlytothedataforPal5. Theresultsofthe
is also driven by internal processes, such as stellar mass loss, N-bodysimulationsandtheimplicationforPal5arediscussed,
two-bodyrelaxationandbinaryinteractions.Theseinternalpro- respectively,insections6and7andsummarizedinsection8.
cessescausetheclusterto(eventually)undergoacorecollapse
andresultinmass-segregationandevaporationofpreferentially 2. THEOBSERVEDPROPERTIESOFPAL5
low-mass stars. While the evolution of isolated globular clus-
Weshallnowsummarizethestructuralandkinematicproperties
tershasbeenthesubjectofmanystudies,thecombinedeffects
oftheglobularclusterPal5, whicharerelevantforthe present
oftheseinternalandtheexternalprocesseshavebeenrarelyin-
study. Most of these properties have been presented and are
vestigated. Gnedin&Ostriker(1997)usedFokker-Plancksim-
discussedinpapersI&II.
ulationsincludingtwo-bodyrelaxation,tidallimitationanddisk
and bulge shocks to investigate the dissolution of the individ- 2.1. TheOrbit
ual Galactic globular clusters (though their adopted mass for As shown in paper II, the orbit of Pal5 is rather tightly con-
Pal5isfivetimeslargerthanourbestcurrentvalue). Withthe strained byits position andradialvelocityin conjunctionwith
samemethod,Gnedin,Lee&Ostriker(1999)studiedtheevolu- the observedorientationandcurvatureofthe tidaltail. Thisis
tionofglobularclusterswith variousconcentrationsc, defined thecase,becausethetailstarsdeviatefromtheclusterorbitonly
as c log(rlim/rcore) fromthe cluster’s limitingand corera- verylittle,asisevident,forinstance,forthesimulatedtidaltail
≡
dius, and ratios β trh/tsh between the two-bodyrelaxation in Figure 6 below. Figure 1 shows the meridional projection
≡
timetrh andtheshockheatingtime(3). Theyconsideredclus- of that orbit obtained with our standard model for the Galac-
terswithc [0.6, 2.6]andβ [10−5, 102]orbitingtheMilky tic gravitational potential (see section 3.1 for details). Differ-
∈ ∈
WayontheorbitoftheclusterNGC6254andconcludedthat(i) entassumptionsfortheGalacticpotential,e.g.differentcircular
forβ &0.1diskshocksdominatetheclusterevolutionandmay velocities,resultinverysimilarorbits,exceptfortheirperiods
leadtoquickdestruction,(ii)forsmallervaluesofβ tidalforc- (paper II). In particular, the perigalactic and apogalactic radii
ing accelerates the two-body-relaxation driven evolution, but arealwaysatabout5.5and19kpc.
also that (iii), in response to disk shocking, the cluster com-
pacts,substantiallyreducingβ anddiminishingthefurtherim- 2.2. MassandVelocityDispersion
portanceofdiskshocks.Pal5hasc 0.6(paperII)andβ 10 WithatotalluminosityofonlyM = 4.77 0.20(paperI)
V
∼ ∼ − ±
(section2.4),whichmeansthat,accordingtoGnedinetal.,the Pal5isoneofthefaintestGalacticglobularclusters. Italsohas
evolutionofPal5isentirelydominatedbydiskshocks. anunusuallyflatstellarluminosityfunction(Grillmair&Smith
CollisionalN-body-simulationstudieswhichinvestigatein- 2001) implying that its mass-to-light ratio is atypically low.
ternal processes along with a time varying tidal field either Together, this let us estimate in paper I the total mass to be
ignore disk shocks (Baumgardt&Makino 2003) or account 5200 700M ,substantiallylessthanpreviousestimates.
⊙
±
for disk shocks but assume an otherwise constant tidal field InpaperI,wemeasuredtheline-of-sightvelocitydispersion
(Vesperini&Heggie1997). σ fromhigh-resolutionspectra of18giantstars within6′ of
los
Unfortunately,neitherofthesestudiesisdirectlyapplicable theclustercentertobeatmost1.1 km s−1. However,theline-
toPal5,sincetheyeitherexcluderelevantexternalprocessesor of-sightvelocitydistributionissignificantlynon-normalbuthas
ModelingthedisruptionoftheglobularclusterPal5Galactictides 3
Fig. 1.—OrbitofPal5,projectedintothemeridionalplane,fortheassumed
Galactic potential. Thecurrent position ofPal5isindicated bythe star; the
trajectoryisplottedforthelast2.95Gyr(solid),correspondingto10radialpe-
riods,andthefuture0.5Myr(dotted).
extendedhigh-velocitytails,whichdominatethecalculationof
σlos. WhenmodelingthisdistributionasGaussianfortheclus- Fig. 2.—Radialprofileoftheazimuthallyaveragedsurfacenumberdensityof
terdynamicsplusacontributionfrombinariestoaccountforthe SDSSstarsinPal5anditstailsasmeasuredinpaperII(top),andcumulative
tails, weobtainedforthedynamicallyrelevantvelocitydisper- numberdistributionaveragedoverbothtails(bottom).ThetypicalmassofPal5
sionσ =0.22+0.19kms−1. starsinSDSSisabout0.8M⊙.
los −0.10
AswediscussedinpaperI,theseestimatesforthemassand respecttotheclustercenterisgivenby
velocitydispersionare consistentwith the best-fit King model
and the assumption of dynamical equilibrium. However, this Ftid(r,t)=(r ∇)∇Φ(R(t),t) (4)
·
does not imply that the cluster is in equilibrium and we shall
where Φ is the gravitational potential of the Milky Way and
nowseethatitcannotpossiblybe.
R(t)thegalactocentricpositionoftheclustercenterattime t.
Equating F to the cluster internalattractionGM/r2 gives
2.3. ExtentandTidalRadius tid
| |
relation(1)fortheradiusrifΦistakentobethepotentialofa
Figure2showsthesurfacedensity,averagedoverannuli,ofthe
singularisothermalsphere.
clusterandthetidaltail. Theprofileoftheclusteritself(which
InFigure3,weplotthestrengthofthetidalforcefield,quan-
caneasilybeobtainedfromdataindirectionsperpendicularto
tifiedbytheeigenvaluesof∂2Φ/∂x ∂x asfunctionoftimeon
the tail) appears to be truncated at a limiting radius of about i j
Pal5’sorbitinourmodelfortheGalacticpotential(seesection
16′.1 107pc,seeFig.5ofpaperII.Thisradiusmustbecom-
≈ 3.1). On top of a smooth underlying tidal field, whose dom-
paredtothetheoreticaltidalradius(equation1)forthecluster,
inant effect is a stretching (the absolute largest eigenvalue is
whichforv = 220kms−1 andthemassestimategivenabove
c negative, i.e. dotted in Fig. 3) and whose strength varies by a
is 54pcatthecurrentlocationofPal5andonly 24pcatthe
∼ ∼ factorof 10betweenapogalacticonandperigalacticon,there
perigalacticonofitsorbit.Thislattervalueismuchsmallerthan ∼
are shortspikes coincidingwith disk crossings, when the tidal
r andactuallyequalsthecluster’spresent-daycoreradius.
lim field is compressive and & 10 times stronger than otherwise.
Thus, the cluster is much larger than its tidal radius and
The duration of these shocks is given by the disk scale height
cannotpossiblybe in equilibriumwith the Galactic tidal force
dividedbythecluster’sverticalvelocityandamountsto10Myr
field6. Thisissurprisingandshowsthatthecurrentdynamical
orless. Thestrongestshocks,whichoccuratdiskcrossingsnear
stateofPal5mustbeapeculiarone. Inparticular,anyestima-
perigalacticon,happenroughlyonceperorbit.Actually,thelast
tionwhichisbasedontheassumptionofequilibriummaywell
suchshockoccurred146Myrago,whilethenextoneisduein
beinerror.
110Myr.
Withnumberscollectedintheprevioussubsections,wefind
2.4. TheImportanceofDiskShocks
from equation (2) that a typical relative change of kinetic en-
AfterhavingassessedthevariouspropertiesofPal5anditsor-
ergy amounts up to 30% for a disk shock near perigalacti-
bit,wemaynowconsidertheimportanceofdiskshocksatdisk ∼
crossings. The tidal force acting on a star at position r with con,whileequation(3)givestsh ∼ 2Gyrforthecharacteristic
shock-evolutiontimescale.
Another quantity of interest is the internal dynamicaltime,
6Inpreviousstudies,thisdiscrepancywasnotdetectedbecause(i)themass
whichwemayestimatefromthevelocitydispersiontocurrently
oftheclusterwasoverestimatedbyafactorof4-6and(ii)theperigalacticradius
wasunknown. be 80Myratthehalf-massradiusrh =30pc. Afterastrong
≈
4 Dehnen,Odenkirchen,Grebel,&Rix
shocksandeffectsdrivenbytwo-bodyrelaxation(partlycaused
bycloseencounters)aremuchlessimportant(stellarmassloss
affectsclustersonlyinthefirstGyroftheirlife).
3.1. TheGalacticGravitationalField
Inoursimulations,weconsideredonlyonechoicefortheGalac-
tic gravitational potential; any effects due to variations of the
Galactic potential are beyond the scope of this paper, but see
thediscussioninsection7.3. Becausetheevolutionofthesim-
ulated cluster is dominated by disk shocks, it is important to
model the Galactic disk as realistically as possible. We used
model 2 of Dehnen&Binney (1998), which contains a com-
pound of three exponential disks for the thick and thin stellar
diskandfortheISM,respectively,andtwo spheroidalcompo-
nents for the bulge and halo. The potential parameters have
been determined to fit all observational constraints known in
Fig. 3.—Strength ofthetidal fieldalongtheorbitofPal5(Fig.1)asmea- 1998andhasadiskscalelengthof2.4kpc,consistentwithre-
suredbytheeigenvalues of∂2Φ/∂xi∂xj. Positive(solid)andnegative(dot- centstudiesoftheinfraredbackgroundlightdistributioninthe
ted)eigenvaluescorrespondtocompressiveandstretchingtidalforces,respec-
tively. Thelittle arrows onthetopandbottomoftheplot indicate moments MilkyWay(Drimmel&Spergel2001).
ofdiskcrossingandpericentric passages,respectively. t = 0correspondsto
today. 3.2. InitialConditions
To create suitable initial conditions, we used King models
TABLE1
TIMESCALESFORPAL5 (Michie 1963; Michie&Bodenheimer 1963; King 1966) with
barycenteratthepositionandvelocityoftheorbitofPal5(see
dynamicalprocess timescale section2.1andFig.1)tenradialorbitalperiods(corresponding
durationofshocks . 10Myr to2.95Gyr)ago. Kingmodelshavethreefreeparameters,two
internaldynamics tdyn ∼ 80Myr scales(sizeandmass)andashapeparameterW ,whichisadi-
timebetweenshocks tdisk ≈300Myr 0
shock-drivenevolution tsh ∼ 2Gyr mensionlessmeasureforthedepthofthegravitationalpotential
two-bodyrelaxation trh ∼ 20Gyr andequivalenttotheconcentrationc. Asindependentparame-
NOTE.—Cluster internal time scales are esti- ters,weusedW0,thetotalinitialmassM0,andthegalactocen-
matedatthehalf-massradiusrh≈30pc. tricradius
r3
R lim v (6)
t c
diskshock,theclusterneedsafewdynamicaltimestosettleinto ≡sGM0
anewequilibrium.Sincethedynamicaltimeisnotsubstantially
shorterthanthetimebetweensubsequentshocksweanticipate with vc = 220kms−1, which means that the limiting radius7
thattheclusterhardlyeverisindynamicalequilibrium. rlim ofthemodelequalsitstidalradiusrtid (equation1)when
Yetanotherimportanttimescale isthe two-bodyrelaxation atadistanceRtfromtheGalacticcenter. WiththechoiceofRt
timetrh, whichwecanestimateusingSpitzer&Hart’s(1971) asparameter(ratherthan,say,rlim)theimportanceofthetidal
formula(Binney&Tremaine1987, eq.8.71).Atthehalf-mass field is largelyindependentofthe cluster mass whenthe other
radius,wefind twoparametersarekeptfixed.
Weconsidered1056modelsfromagridofpointsinthe3D
σ 3 ρ(rh) −1 M∗ −1 parameter space. For Rt, we took twelve values between 7.5
t 20Gyr ,
rh and 13kpc at steps of 0.5kpc. These are all larger than the
∼ (cid:20)0.22kms−1(cid:21) (cid:20)0.01M⊙pc−3(cid:21) (cid:20)0.3M⊙(cid:21) (5) perigalacticradiusofPal5, guaranteeingthattidalinteractions
which is too long for relaxation related processes to currently are important. For W0, we used eleven values between 1.75
play an importantrole and implies β 10 for Gnedin et al.’s and4.25in stepsof0.25,correspondingto lowconcentrations
shock-importanceparameter. ∼ c between0.46and0.88. Finally,fortheinitialmasstheeight
We have summarized these various time scales in Table 1. values8,10,12,16,20,24,28,and32thousandSolarmasses
Note that the estimates for internal time scales (t , t , t ) havebeenemployed.
dyn sh rh
mustberegardedascrudeones,becausetheyarebasedonthe Clearly, these initial conditions are, as often with N-body
assumptionofdynamicalequilibrium,whichwejustsawisnot simulations,somewhatartificial.Wedonotmeanthat3Gyrago
reallysatisfied. Weconclude,nonetheless,thatdiskshocksfor Pal5waswellfitbyaKingmodel,ratherweexpectthatitmight
Pal5 are fast (shock duration < t ) and frequent (t alreadyhaveundergoneseveretidaldisturbances.However,the
dyn dyn
t )andentirelydominatetheevolutionofthisobject(t 6≪< tidalfieldislikelytoquicklyeraseanyspecialitiesofourinitial
disk sh
t ). models,except,ofcourse,thelackofpossibletidaltailsorigi-
rh
natingfromearlierepochs. Moreover,evenifonewerewilling
3. THESIMULATIONS toimproveontheinitialconditionsbyintegratingforthewhole
lifetime of a cluster, one had to know the detailed history of
Inordertomodelthetidaldisruptionoflow-concentrationglob-
ular clusters in general and Pal5 in particular, we performed
collision-less N-body simulations, i.e. with softened gravity, 7TheoutermostradiusofaKingmodelisoftencalledits‘tidalradius’with
theideathataglobularclusteroughttobetidallylimited. However, asPal5
suppressing short-rangeinteractions. This is entirely justified,
clearlydemonstrates,thelimitingradiusmaybedifferent,actuallylarger,than
because the evolution of these systems is dominated by disk thetheoreticaltidalradius,seesection2.3.
ModelingthedisruptionoftheglobularclusterPal5Galactictides 5
Fig.4.—Timeevolutionofthemass(left),velocitydispersion(middle),andvirialradius(right)ofthesimulatedglobularclusterasfunctionoftimeforfiverepresen-
tativemodelswithparameterRtandW0asindicatedandM0=12000M⊙.Thedottedverticallinesindicatediskcrossingswherethethetidalforceisexceptionally
large,quantifiedbytheabsolutelargesteigenvalueof∂2Φ/∂xi∂xj exceeding7000(kms−1kpc−1)2,seeFig.3.Thetwosimulationswith(Rt,W0)=(11,2.5)
and(12,2)havebeenstoppedbeforet=0,becausethenumberofbodiesintheclusterdroppedbelow1000.
theGalactictidalfieldandwouldrequiremuchmoreCPUtime schemethatallowedupto8timessmallersteps. Theindividual
withoutgainingmuchscientificsignificance. step sizes τ where adjusted in an almost time-symmetric way
suchthatonaverageτ = a−1kpcGyr−1, whereadenotesthe
3.3. TechnicalDetails
modulusof the acceleration. This ensuredthat disk crossings,
WeusedthepubliclyavailableN-bodycodegyrfalcON,which which have a duration of only a few Myr, are accurately inte-
isbasedonDehnen’s(2000,2002)forcesolverfalcON,a tree grated. With these settings, the energy of an isolated cluster
code with mutual cell-cell interactions and complexity (N). was conserved to 0.8% over the period of 2.95Gyr. One full
O
falcONnotonlyconservesmomentumexactly,butalsoisabout simulationcorrespondstoabout6000blockstepsandrequired
10 times faster than an optimally coded Barnes & Hut (1986) about 75min of CPU time on a linux PC (AMD, 1800MHz).
treecode. 1190 hoursof CPU time were spent on the whole set of 1056
Each simulation used N=32000 bodies and a softening simulations, 361 of which were halted before t = 0, because
length of ǫ = 3pc with what we call the P2 softening kernel, theclusterwasfoundtobedissolved.
i.e.theNewtonianGreensfunctionΦ= G/rwasreplacedby
− 3.4. ClusterEvolutionintheTidalField
G ǫ2 3ǫ4 Forfiverepresentativesimulations,rangingfrommediumcon-
Φ(r)= 1+ + .
−√r2+ǫ2 2(r2+ǫ2) 4(r2+ǫ2)2 centration and small extent to low concentrationand large ex-
(cid:20) (cid:21)
tent,i.e.fromleasttomostvulnerabletoGalactictides,Figure4
Thedensityofthiskernelfallsoffmoresteeplyatlarger than
showsthetimeevolutionoftheclustermassM,velocitydisper-
thatofthestandardPlummerkernelandbecomesactuallyneg-
sionσ, andtheratioGM/σ2. Theyhavebeencomputedfrom
ative(thoughwithabsolutevaluesmallerthanforthePlummer
thosestarsthatarewithintheoriginallimitingradiusfromthe
kernel) such that force bias is substantially reduced, see also
cluster center, which was determined iteratively as barycenter
Dehnen(2001).Withthesesofteningparameters,themaximum
ofthesamestarsstartingtheiterationwiththesetofstarsfrom
possible force between two bodies is equal to that for Plum-
theprevioustimestep.
mersofteningwithǫ 1.4pc,whichhencemightbecalledthe
‘equivalentPlummer≈softeninglength’. 3.4.1. TheMechanicsofTidalDiskShocks
Thetime integrationwas performedeitherfor2.95Gyr, i.e. As is evident from this Figure, the cluster evolution is driven
until today, or until cluster destruction. In practice, a cluster by disk shocks (the instants of the strongest of which are in-
was considered destroyed if the number of bodies within one dicated by dotted vertical lines), which may actually quickly
initial limiting radius from the cluster center dropped below destroy the cluster, depending on its initial state. Each strong
1000 3% of the initial number. We used the leap-frog in- diskshockcausesanalmostinstantaneousincreaseoftheclus-
tegrato≃rwithstepsizeof2−11Gyr 0.5Myrandablock-step ter’s velocity dispersion (middle panels of Fig. 4) by 8–25%.
≈
6 Dehnen,Odenkirchen,Grebel,&Rix
Thelinearmasslossisaccompaniedbyanaveragedecrease
invelocitydispersion. IntherightpanelsofFig.4,weplotthe
virialradius
r GM/σ2 (7)
vir
≡
ofthestarswithintheinitiallimitingradiusofthecluster9. For
most simulated clusters that surviveuntil the presenttime, we
findamodestdecreaseofthevirialradius.Inthetoprightpanel
ofFigure5,weplottherelativeshrinkingrate, r˙ /r (t =
vir vir
−
0), versus W for various values of R but fixed M (the de-
0 t 0
pendenceonM isveryweakatfixedR andW ). Mostmodel
0 t 0
clustersshrinkslightlyinresponsetothediskshockswitharate
thatisprimarilyafunctionofW ,or,equivalently,oftheinitial
0
clusterconcentration. Onlyverylow-concentrationclustersdo
notshrinkbutmayevenexpand,seealsothelowertwosimula-
tionsinFig.4. ThetopleftpanelofFigure5plotstherelative
ratesversuseachother.Whilethereisacleartendencyformass
losstoanti-correlatewithshrinking,thespreadinthisdiagram
isconsiderable.
We like to pointoutthatthe shrinkingratesare verysmall,
of the order of a few percentper Gyr, too small to protectthe
clusterfromcontinuedtidalstripping,i.e.self-limitationofdisk
Fig. 5.—Relativemass-lossandshrinkingratesplottedversuseachother(top shocks does not apply here, as is also evident from the un-
left)andversusW0formodelclustersthathavenotbeendissolvedafter2Gyr dampedmass-lossrates.
(formodelsdissolvedearliertheuncertaintiesintheshrinkingratecanbecon-
siderable). M˙ andr˙virhavebeenobtainedbystraight-linefitsovertheentire 3.5. TidalTailMorphology
timeinterval,exceptingthe0.5Gyrbeforeclusterdissolution,ifapplicable.For
clarity,onlymodelswithM0 = 24000M⊙ areshown(othervaluesforM0 Thestarslostfromtheclusterformtwotidaltails, oneleading
yieldverysimilarresults).ThesymbolsrefertodifferentvaluesforRtasindi- theclusterandonetrailingit. Wewilldiscussthedynamicsof
cated. thetidaltailindetailinthenextsection. Here,wewanttogive
animpressionofthemorphologyofthesetails.
Thiscorrespondstoanincreaseoftheclusterkineticenergyby
ForthemodelwhoseremnantclusterbestfitsPal5(seesec-
16–50%andpushestheclusteroutofvirialequilibrium. Stars
tion5.1),hereafter‘modelA’,Figure6showssnapshotsof(one
which have been accelerated beyond their escape velocity are
outoften)starsintailandcluster: theparticledistributionpro-
lostfromthecluster,resultinginadropoftheclustermass(left
jectedontothe(instantaneous)orbitalplaneattimeintervalsof
panelsofFig.4),whichisdelayedfromtheinstantoftheshock
125Myr. At the first time shown, the cluster has not yet been
bythetimeneededforthestarstodriftoutofthecluster.
tidally shocked and all the particles are still inside the cluster,
Inresponsetotheheatingbythediskshock,theclusteralso
afterthat,thetwotailsdevelop.
expands. This together with the loss of the fastest stars re-
There are several notable observations possible from this
sultsinasubstantialdropofthevelocitydispersionbyapprox-
Figure.First,nearthecluster,thetailshowsthetypicalS-shape,
imatelytwicetheamountoftheinitialincrease. Thisdropoc-
alsovisibleintheobservationsofPal5, whichoriginatesfrom
cursroughlyonadynamicaltimescale, i.e.slowerfortheless
the trailing tail being at larger and the leading tail at smaller
concentratedand/ormoreextendedclusters. Subsequently,the
distancefromtheGalacticcenter.
velocitydispersionshowsdampedoscillationsuntilthecluster
Second, the tail morphology varies between thin and long
settles into a new equilibrium with velocity dispersion lower
near perigalacticon (e.g. t = 0.45Gyr), and short and thick
than before the shock. The damping is due to the fact that −
near apogalacticon (e.g. t = 2.075, 0.075Gyr), which is
stars oscillate at different frequencies and is weakest for low- − −
duetovariationsoftheorbitalvelocityandhencetheseparation
concentrationclusters, since for those the range in orbital fre-
between tail stars and cluster. This is analogous to cars on a
quencies is smallest. For clusters of low concentration and/or
motorway:slowdrivingwithshortseparationsasinatrafficjam
large extent, the settling into a new equilibrium is forestalled
andfastdrivingwithlargedistancescorrespondtothesituations
bythenextdiskshock,i.e.theseclustersareneverinastateof
nearapogalacticonandperigalacticon,respectively.
dynamicalequilibrium.
Third,thetailneatlyhuddlesagainsttheclusterorbitatmost
3.4.2. MassLossandTimeEvolution of the times; only rarely near apogalacticon do some stars of
AremarkableobservationfromFig.4isthefactthattheorbit- thetrailing(leading)tailappearcloser(further). Fourth,occa-
averagedclustermassdecreasesalmostlinearlywithtimeover sionallynearapogalacticon,thetidaltailshowsastreakystruc-
the durations simulated8. This implies that, at least for low- ture. Eachofthesestreakscorrespondstoaswarmofstarsthat
concentrationclusters,disk-shockinginducedmasslossquickly has beenset loose by the same disk shock andfills apparently
destroysthecluster. Theorbit-averagedmass-lossratedepends a streak-like phase-space volume, which occasionally projects
stronglyon the size, but also on the concentrationof the clus- alsoontoastreakinconfigurationspaceshownintheFigure.
ter, spanning one order of magnitude in M˙ /M amongst our
0
simulations,seethebottomleftpanelofFig.5. 9Onemightinsteadwanttouseonlystarsthatareboundtotheclusterin
thepresenceofthetidalfield.However,duetostrongchangesofthetidalfield
duringshocks,astarmayswitchfromboundtounboundandback,resultingin
8We found the second time derivate M¨ to be small in the sense that spuriousvariationsoftheestimatedquantities.Moreover,thecontaminationby
|M˙/M¨|>7Gyrwithnopreferenceforasign. unboundstarsissmallanddoesnotsignificantlyaffectourestimates.
ModelingthedisruptionoftheglobularclusterPal5Galactictides 7
Fig. 6.—MorphologyofthesimulatedtidaltailinthesimulationwithparametersRt = 10kpc,W0 = 2.75,andM0 = 20000M⊙(alsocalled‘modelA’)at
varioustimes,giveninGyrineachpanel. Foreverytenthbody,weplottheposition(inkpc)projectedontotheinstantaneous orbitalplaneofthecluster, whose
motionistowardstherightineachpanel.Thethinlineindicatestheclusterorbit.
4. THEDYNAMICSOFTIDALTAILS orbitalperiod. Overtime,thelonger(shorter)orbitalperiodre-
sultsinalag(lead)ofthestarw.r.t.thecluster,i.e.positiveand
The gravitational influence of the cluster ceases not abruptly
negativeδE resultinatrailingandleadingtidaltail,respec-
beyondits tidalradius, butit iscompletelynegligibleforstars orb
tively.Sincethetrailingtailmovesonahigherorbitalenergy,it
thathavedriftedawayfromtheclusterbymorethanafewtidal
hasalargerradius.
radii. Similarly,theself-gravitywithinthetidaltailisunimpor-
If we assume thata tail star leavesthe cluster at a largeror
tant,simplybecausethetidal-tailstarsmovetoofastw.r.t.each
other10.Thus,toverygoodapproximation,themotionofatidal smallergalactocentricradius,butthesameGalacticazimuthand
latitude,andthesamevelocityasthecluster,thenweexpectit
tailstarisgovernedbythegravitationalpotentialΦoftheMilky
tohavethesameorbitaleccentricityastheclusterandtomove
Wayonly.
inthesameorbitalplane(paperII).Inreality,weexpectdevia-
Foratime-invariantandsmoothGalacticgravitationalfield,
tionsfromthisideal,but,ofcourse,theorbitaleccentricityand
suchasadoptedinthesimulations(butseesection7.3foradis-
inclination should only slightly differ from that of the cluster
cussionofdeviationsfromthisideal),atidal-tailstarmoveson
orbit.
aGalacticorbitwhichisverysimilartothatoftheclusteritself.
The offset in orbital energy may be estimated to be (Johnston 4.1. CoordinatesforTidal-TailStars
1998)
Sincetheorbitsofatidal-tailstarandoftheclusteraresosim-
δEorb rtid ∇Φ. (8) ilar, the position of a tail star may be referred to the cluster’s
≃± | |
trajectory. In paper II, we gave a simple model for the kine-
Thisfollowsfromthefactthatwhenthestarleavesthecluster
matics of stars in the tail, which was based on the assumption
its cluster-internal energy approximately vanishes and its ve-
thatitsorbithasthesameeccentricityasthatoftheclusterand
locity is very similar to that of the cluster, which leaves only
differsonlyinenergy.IfwefurtherassumethattheMilkyWay
the differencein potentialenergyas significantcontributionto
hasaflatrotationcurve,wehave
δE .
orb
In reality, the star may leave the cluster with a higher than δx δP
| | | | (9)
jtuosltaergscearpvealeuneesr,gbyu,taninda|δnEyocrabs|efoδlElowsadistEributi.onThwuisth,aasttaairl |xorb| ≈ P
| orb| ≪ | orb| whereδP isthedifferencebetweentheorbitalperiodofthetail
inthetidaltailmovesonanorbitwithslightlyhigher(lower)or-
star and the period P of the cluster, while δx x x is
bitalenergythanthecluster,andhenceslightlylonger(shorter) ≡ − orb
the positionaloffset of a tail star from the cluster trajectory at
the same orbital phase, not the same time, see Fig. 7 for an
10This may be different in tidal tails emerging from galaxy interactions, illustration.
whereself-gravityinthetails,possiblyinconjunctionwithgas-dynamics,may Sinceatthesamephaseδx x ,equation(9)givesasim-
leadtotheformationofboundclumpsthatthenbecomeso-called‘tidaldwarf orb
k
galaxies’(e.g.Ducetal.2000). ple relation between the perpendicular positionaloffsetof the
8 Dehnen,Odenkirchen,Grebel,&Rix
and ζ 0. That is, not only should these coordinates be
⊥
≈
conserved,i.e.allthekinematicsiscontainedintheirconstruc-
tion, but they are related in a very simple way. However, the
modelonwhichequation(9)isbasedignoresdifferencesinor-
bital eccentricity and inclination between tail-star and cluster.
Since the orbitalperiod is mainly a functionof orbitalenergy,
whereasthe orbitalposition atfixed phase also dependssensi-
tivelyontheeccentricityandinclination,weexpectdeviations
from our ideal to mainly affect (13a) and ζ 0, but not so
⊥
≈
much(13b).
4.2. KinematicsoftheTidalTail
We are now going to investigate in some detail the distribu-
Fig. 7.—Illustrationforthedefinitionoftailcoordinates, seesection4.1for tionoftidal-tailstarsovertherelativecoordinates(η, ζ , ζ ),
details. k ⊥
δE , and t, thereby also testing the simple relations (13)
orb l
tail star from the cluster trajectory and the period difference, above. Forthispurpose,weconcentrateontheonesimulation
which is directly related to the positional offset along the tra- already used in Fig. 6 above, which is one that gives a good
jectory. InpaperII,weusedthissimplerelationtoestimatethe description of Pal5 (see Section 5). In this model the cluster
mean drift rate, and consequentlythe mass-loss rate, from the originallyhadrlim =53pc.
meanradialorbitaloffsetofthetidaltailofPal5fromitsorbit. In the top left panelof Figure 8, we plotthe dimensionless
Foraflat-rotation-curveGalaxy,wefurtherhave(forδP P) driftrateη,computedattheendofthesimulation,vs.thetime
≪ t of loss from the cluster (evaluated as the last instant when
l
δP =exp δEorb 1 δEorb. (10) the star crossed rlim outwards). Clearly, each disk shock trig-
P v2 − ≈ v2 gers the loss of a swarm of stars with a spread in η which is
(cid:18) c (cid:19) c almost symmetric for trailing and leading tail. The morphol-
Motivated by this model, we now introduce simple coordi- ogyin(η,t)ofeachoftheseshocksisthesame. Firstthestars
natesforstarsin thetidal-tail. Fora tailstar atpositionx, we withlargestl η escape,becausetheyhavehighestvelocity,and
first find the reference point xorb on the local cluster orbit by somewhatla|te|rthosewith smaller η . Thesmallest η ofany
minimizing x xorb(τ) w.r.t.thetimeparameterτ alongthe tailstardecreasestowardslatertime|s|,becausetheclu|st|erloses
| × |
cluster orbit, see Figure 7 for an illustration. With τ defined massandescapingfromitbecomeseasier.
such thatxorb(0) = x0, we can use the time τ directlyas the Thedriftratesη followabroaddistributionwithafactorof
tail star’s (relative) coordinate along the tail. For the trailing fourbetweenminimumandmaximum. Therefore,a swarmof
(sketched in Fig. 7) and leading stars, τ < 0 and τ > 0, re- tailstarssetloosebyonediskshockquicklydispersesalongthe
spectively. Note that, in contrast to the true positional offset tail. Inparticular,thefastestdriftingtailstarswillsoonruninto
betweenclusterandtailstars,τ isnotsubjectto‘seasonal’vari- theslowestdriftingoneslostatthepreviousshock.Thisimplies
ationsbetweenapogalacticonandperigalacticonseeninFig.6, thatatanydistancefromtheclusterthetailisasuperpositionof
sincethesearecausedbyvariationsinorbitalvelocity. starslostatvariousepochs,illustratedalsoinFig.10below.We
Asfurthercoordinatesweusethecomponents mayestimatethenumberofshockscontributingtothetidaltail
at some ‘distance’ τ by dividing the difference between max-
δx δx x /x , (11a)
k ≡ · orb | orb| imum and minimum t tl by the average time tdisk between
δx δx L /L , (11b) shocks −
⊥ orb orb
≡ · | |
ofδxparalleltoxorb (intheorbitalplane)andparalleltothe n (η−1 η−1 ) τ 3 τ . (14)
angularmomentumvectorL x x˙ (perpendicular ≈ min− max t ∼ 4η t
orb orb orb disk min disk
≡ ×
totheorbitalplaneandthepaperinFigs.6and7).
Wemaygoastepfurtheranddefinethedimensionlesscoor- With ηmin 0.005 and tdisk 300Myr, this gives n
∼ ≈ ∼
dinates 0.5(τ/Myr). For the current situation of Pal5, this translates
into n 2(l/deg) with angular separation l from the cluster
ζη ≡ τδx/(t/−xtl),, ((1122ba)) (assTuhmei∼mngaxviomrbu∼m1d0ri0ftkrmatse−i1sfsotirllthraethcleursltoewr’s:oarbboiutatl2vepleorccietyn)t,.
k ≡ k | orb| i.e. it would take 50 orbits 15Gyrfor the fastest first es-
ζ δx /x , (12c) ∼ ∼
⊥ ⊥ orb capeestocoverafullorbitalperiod,i.e.tofillagreatcircleon
≡ | |
thesky.
where t is the time when the star was lost from the cluster.
l The top-right panel of Fig. 8 shows that there is indeed a
η essentially is a dimensionlessdriftrate, while ζ andζ are
k ⊥ tightrelationbetweenδE andη,asexpectedfromourequa-
orb
dimensionlessoffsetsinandoutoftheorbitalplane.
tion(13b).However,thereisaconsiderablespread,inparticular
The time τ corresponds to a phase difference δθ = τω,
at small η , which persists if we restrict the analysis to larger
whichoriginatesfromthe frequencydifferenceδω actingover | |
t (excluding stars still in the neighbourhoodof the cluster).
l
thetimet t,i.e.δθ =(t t)δω. Wethushaveη =δω/ω = | |
− l − l There is also a systematic deviation in the sense that the lin-
δP/P andequations(9)and(10)yield
− earrelationsforthetrailingandleadingtailhavenon-vanishing
zeropoints.
ζ η (13a)
k ≈ − The bottom panels of Fig. 8 show histograms of tl, i.e. the
η δE /v2 (13b) mass-lossrate,andδE . AcomparisonwithFig.3showsthat
− ≈ orb c orb
ModelingthedisruptionoftheglobularclusterPal5Galactictides 9
Fig.8.—Distributionoftidal-tailstarsoverthedimensionlessdriftrateη,timetloflossfromthecluster,andorbitalenergyoffsetδEorbfromthecluster.Thevalues
areobtainedatt = 0frommodelA(Rt = 10kpc,W0 = 2.75,andM0 = 20000M⊙). Tidal-tailstarswithtl > −0.2Gyrhavebeenomitted. Thethinlower
histograminthebottomrightpanelisforstarswithtl∈[−2.5,−2.4]Gyr.Intheupperrightpanelonlyoneoutoffourbodiesisplotted.
theamountofmasslostinashockcorrelateswiththestrength
of the tidal field, as expected. The distribution in δE is, of
orb
course, a superposition of the distributions created by each of
the disk shocks and the thin histogram shows the distribution
from the single shock at 2.45Gyr. The fact that the inner
−
edge of the distribution (towards small δE ) is not abrupt
orb
| |
ismainlyduetothefactthattheclusterweakenedwithtime,al-
lowing smaller δE at later epochs. The typical scale for
orb
| |
δE is 300(kms 1)2. This value indeed follows from
orb
∼ −
equation (8) for a cluster with r 60pc at R 10kpc in
tid
alogarithmicpotentialwithv 22∼0kms−1. The∼distribution
c
∼
inδE isnotquite,butstillremarkablysymmetric.
orb
InFigure9,weplotthedimensionlessorbitaloffsetsζ and
k
ζ vs.ηcomputedattwodifferentepochs,oncenearapogalac-
⊥
ticonandoncenearperigalacticon.Quiteobviously,therelation
η ζ (13a)doesnotgenerallyhold,onlyforthetrailingtail
k
≈−
at t = 0 we find reasonable agreement. We also find that ζ
⊥
deviates from zero substantially in the sense that typical val-
ues are significant compared to η. A comparison between the
two epochs clearly shows that ζ and ζ are in fact not con-
k ⊥
served.This,togetherwiththeviolationofζ ηandζ 0
k ⊥
≈− ≈
implies,inaccordancewiththediscussionfollowingequations
(13), thattheorbitaleccentricitiesandinclinationsof tailstars
differ significantly and systematically from that of the cluster.
On the other hand, δE shows only marginal changes with
orb
time(notshowninaFigure),whileηchangessignificantlywith
time,butstillmuchlessthan100%,incontrasttoζ andζ .
k ⊥
Wemaythusconcludethatinpracticethedimensionlessdrift
rate η is reasonably well conserved and correlates well with Fig. 9.—Plotsofζ⊥ (top)andζk (bottom)vs.η formodelA(thesameas
δE , while ζ and ζ are subject to substantial time evolu- in Fig. 8) at t = 0 (left, near apocenter) and t = −0.45Gyr (right, near
tiono.rbThese coknclusion⊥s are supported by all simulations, not apocenter). The thin lines indicate the relations (13a) and ζ⊥ = 0, which
areexpectedfromnaivemodeling. Forclarity, onlyoneoutoffourbodiesis
justtheonepresentedinFigures8and9. plotted.
4.3. StellarDensityalongtheTail
The linear stellar density ̺ = dM/ds along the tail, or, simi- AstudybyJohnston,Sigurdsson&Hernquist(1999)ofthefor-
larly,thecircularlyaveragedsurfacedensityΣ(R)ofstars(tail mationoftidaltailsduetoGalactictidesactingonglobularclus-
&cluster)areeasilyobservablefromPal5(seeFig.2,paperII). tersorsatellitegalaxiessuggestthatΣ R−1,orequivalently
∝
10 Dehnen,Odenkirchen,Grebel,&Rix
Fig.10.—Distributionoftidal-tailstarsinτandtimetloflossfromthecluster
formodelA(thesameasinFig.8).Thetwoverticallinesbrackettherangein
τinaccessibletostarswithtl <−2.95Gyr.Thatis,starslostpriortothestart
ofthesimulation,whichare,ofcourse,notcontainedinourmodel,contribute
onlyto|τ|>19Myr.
Fig. 11.—Forall 695simulations withacluster surviving until t = 0, we
̺ const11 plot against each other: the average offset in orbital energy δEorb between
≈Wemayidentifythreefactorsthatdetermine̺(s): themass- tailstarsandcluster; theaverageabsolutedimensionless driftrateη; andthe
loss rate M˙ ; the distributionf(η) of dimensionlessdriftrates, bweeeinghctaeldcumlaeteadnaotftth=ei0n.itTiahleatnhdinficnuarlvemsaisnstohfetthweoclleufsttpear.neδlEsfoorlbloawndapηohwaevre-
whichmaybeafunctionofthetimet oflossfromthecluster; lawwithexponent1/3.
l
and the orbital kinematics that map the phase-distance τ into
sincetailstarswitht < 2.95Gyrarelackingfromourmod-
physicaldistancesalongtheorbit. Thefirsttwofactors,which l −
els, see also the discussion at the end of section 3.2). For the
dependmainlyonthegeneralpropertiesoftheorbit(periodand
lineardensity̺,thelatterimpliesaslightdecreasewithsanda
perigalacticradius)andthecluster,determinethedistributionof
steeperthanR−1decreaseforΣ. Thus,weexpectnon-constant
tailstarsoverτ
̺(s) even for a (orbit-averaged)constant mass-loss rate and a
dM t ∞ tailspanningasmallrangeof vorb ,if themasslosswassub-
= dtl M˙ (tl) dη f(η;tl)δ τ η(t tl) . (15) stantial. | |
dτ − −
Z−∞ Z−∞ Forclustersorsatelliteswhichundergoonlynon-substantial
(cid:0) (cid:1)
massloss,thedistributionoverηshouldindeedbeindependent
Thelastfactordependsonthecurrentpositionontheorbitand
oftimeandconsequentlydM/dτ const,consistentwiththe
yields
≈
resultsoftheaforementionedstudybyJohnstonetal..
1 dM
̺(s;t)= (16) The bumpiness of dM/dτ, which of course translates in a
v (τ) dτ
| orb | bumpinessof̺(s), iscausedbyindividualshocksandthefact
wheres= τ v (τ′) dτ′ calculatedatthecurrenttimet. thatforsmallτ onlyfewofthemcontribute. Thesebumpsare
A constan0t|̺(osr)b, suc|h as foundin the simulationsby John- expectedtobesymmetric,i.e.occurintrailingandleadingtail
stonetal.,eRmergesfromaconstant(orbit-averaged)M˙ incon- at about the same distance from the cluster. For larger τ or
distancessfromtheclustertheseinhomogeneitiesareaveraged
junctionwithatime-invariantdistributionf(η)ofdriftratesand
v constovertheextentofthetail. Thislattercondition outbythesuperpositionofmanyswarmsofstarslostfromthe
orb
| | ≈ clusteratdifferentshockevents.
isnotsatisfiedwheneverthetidaltailspansa significantrange
inphasesonaneccentricorbit. 4.4. DriftRatesandTailKinematicsoftheModels
AswesawinFig.8,thedistributionoverdriftratesη isac-
In orderto investigatethe dependenceof the tailpropertieson
tuallynottimeinvariantforourmodels:thedriftratesdecrease,
themodelparameters,wecomputedforeachmodelofacluster
because the cluster is significantlyweakenedby the continued
thatsurviveduntiltodaythemeananddispersionofη, ζ , ζ ,
loss of stars. With a constant (orbit averaged) mass-loss rate, k ⊥
and δE evaluatedat t = 0 for boththe leadingand trailing
orb
this results in dM/dτ slightly decreasing with τ . This is il-
tail.
| |
lustratedin Figure 10, which plotsthe distributionoftail stars
The estimate (8) for δE suggests that δE depends
orb orb
over τ and t. In the top panel, it is evident that the typical h| |i
l only on the mass M of the cluster at the instant of the star
τ increasesslightlyfasterthanlinearwitht t (theinneren-
l leavingit. Because ourmodelsare subjectto substantialmass
| | −
velopesarenotstraightlines).TheplotofdM/dτ inthebottom
loss, their mass is not conserved. However, as the bottom-
panelshows(i)some‘bumpiness’atthe.20%leveland(ii)a
left panel of Figure 11 demonstrates, the average δE
orb
slightdecreasewith τ (datafor τ > 19Myrareincomplete h| |i
over all tail stars (lost over the whole integration interval) is
| | | |
strongly correlated with some time-averaged cluster mass. In
11In another study Johnston,Choi&Guhathakurta (2002) reported much fact δE M1/3,exactlyasequations(8)and(1)predict.
steeperprofiles(Σ ∝ R−3). However,theseoccurredinmuchmoremassive h| orb|i∝
The constantof proportionalityin this relation dependson the
satellitesandoverlargerpartsoftheorbitandarenotrelevanttothesituation
studiedhere. clusterorbitandtheGalacticpotentialonly,butnotontheclus-