The Curious Case of Palomar 13: The Influence of the Orbital Phase on the Appearance of Galactic ...

16 Pages·2013·1.79 MB·English

Mon.Not.R.Astron.Soc.000,1–16(2010) Printed11June2013 (MNLATEXstylefilev2.2)
The Curious Case of Palomar 13: The Influence of the
Orbital Phase on the Appearance of Galactic Satellites
3
1 Andreas H.W. Ku¨pper1,2⋆, Steffen Mieske2 and Pavel Kroupa1
0 1Argelander Institut fu¨r Astronomie (AIfA), Auf dem Hu¨gel71, 53121 Bonn, Germany
2 2European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile

n
u
J Accepted ....Received...;inoriginalform...

0
1
ABSTRACT

] We investigate the dynamical status of the low-mass globular cluster Palomar 13 by
A
means of N-body computations to test whether its unusually high mass-to-lightratio
G of about 40 and its peculiarly shallow surface density profile can be caused by tidal
. shocking. Alternatively, we test – by varying the assumed proper motion – if the
h
orbital phase of Palomar 13 within its orbit about the Milky Way can influence its
p
appearance and thus may be the origin of these peculiarities, as has been suggested
-
o by Ku¨pper et al. (2010b). We find that, of these two scenarios, only the latter can
r explaintheobservedmass-to-lightratioandsurfacedensityprofile.Wenote,however,
t
s thatthe particularorbitthatbestreproducesthoseobservedparametershasaproper
a
motioninconsistentwiththeavailableliteraturevalue.Wediscussthisdiscrepancyand
[
suggestthatitmaybecausedbyanunderestimationoftheobservationaluncertainties

3 in the proper motion determination. We demonstrate that Palomar 13 is most likely
v near apogalacticon, which makes the cluster appear supervirial and blown-up due to
3 orbital compression of its tidal debris. Since the satellites of the Milky Way are on
6 averagecloser to apo- than perigalacticon,their internal dynamics may be influenced
1
by the same effect, and we advocate that this needs to be taken into account when
3
interpreting their kinematical data. Moreover, we briefly discuss the influence of a
.
2 possible binary population on such measurements.
1
0 Key words: galaxies: kinematics and dynamics – galaxies: star clusters – globular
1 clusters: individual: Palomar 13 – methods: N-body simulations
:
v
i
X
r 1 INTRODUCTION Some of these uncertainties arise from peculiar sur-
a
face density profiles. That is, even though many objects
Therearemanyobjectsonthesky,especially inthehaloof
in the Milky Way halo are well limited and show a well
theMilkyWay(MW),whosenatureisnotcleartous.Some defined surface density profile with a slope of about R−4
ofthoseobjectsarehardtoaddressobservationally,andfor
in the region of the tidal radius, some objects obey shal-
others thereis just noconclusive theoretical explanation.
low surface density profiles in the outskirts, having slopes
In fact, simply by looking at a colour-selected sample
of about -1 to -2, like for example the MW globular clus-
of stars within a region of the sky it is sometimes not easy
ters Palomar 5 (Odenkirchenet al. 2003), NGC 5466, M
to determine the true extent of a stellar system, mostly
15, M 53, M 30, and NGC 5053 (Chunet al. 2010), AM
since it lacks a clear cut-off in its surface density profile.
4 (Carraro, Zinn & Moni Bidin 2007), Whiting 1 (Carraro
The same holds true for the determination of its velocity
2009), and NGC 1851 (Olszewski et al. 2009). The latter
dispersion through a sub-sample of stars with readily mea-
furthermore seems to be surrounded by a 500 pc halo of
sured radial velocities. These uncertainties typically result
stars whose origin is unknownup to now.
in discussions and speculations about a best-fitting density
profileaswellasasystem’struetidalradius(e.g.King1966; Other uncertainties arise from unusual mass-to-light
Elson, Fall & Freeman 1987; McLaughlin & van derMarel (M/L) ratios of some stellar systems. While most globu-
2005), and also about the true mass-to-light ratios of such larclustersshow mass-to-light ratios of1-2, Ultra-Compact
systems (e.g. Kroupa1997; Mieske et al. 2008). Dwarf galaxies (UCDs) have higher M/L by a factor of
about two, whereas dwarf spheroidal galaxies even show
valuesofupto103 (Dabringhausen, Hilker& Kroupa2008;
⋆ E-mail: [email protected] (AHWK); Geha et al. 2009). These differences are usually ascribed to
[email protected](SM);[email protected](PK) different dark matter contents, catastrophic tidal heating
2 A.H.W. Ku¨pper, S. Mieske and P. Kroupa
2 PALOMAR 13
Palomar 13 is an old and metal poor Galactic globular
cluster. From isochrone fits to its colour-magnitude dia-
gram, Cˆot´e et al. (2002) find Pal 13 to be about 13-14
Gyr old, and from spectroscopy they derive a metallicity
of [Fe/H] = −1.9±0.2. Moreover, it is among the faintest
objects listed in the Harris catalogue of Milky Way globu-
larclusters(Harris1996).Withanestimatedmassofabout
3000 M⊙(assumingamass-to-lightratioofunity)itisoneof
theleastmassiveglobularclustersoftheGalaxy(Cˆot´e et al.
2002). Furthermore, Siegel et al. (2001) argue with proper
motion measurements that Pal 13 is on an inclined, highly
eccentric orbit about the Milky Way. Thus, it most proba-
blywassubjecttostrongtidaldisruptionduringthelastfew
Gyr. Indeed, observations show further peculiarities about
this specific cluster. First, the cluster shows an unusually
high velocity dispersion and therefore a very high mass-to-
lightratio,andsecond,itssurfacedensityprofilediffersfrom
theones of most otherMilky Way globular clusters:
(i) Corresponding to data by Cˆot´e et al. (2002), which
Figure 1. Surface density profile of Pal 13 as obtained with we mainly use to compare with our computations, the
CFHTbyCoˆt´eetal.(2002).FittedtotheprofileareaKingtem- cluster is located at Galactic longitude of ℓ = 87.◦1 and
plate(King1962)andaKKBHtemplate (Ku¨pperetal.2010b). Galactic latitude of b = −42.◦7. Its distance from the
Giveninthefigurearethetemplatevaluesforthefittedtidalra-
diusforbothtemplates,RKingandRKKBH respectively,andfor Sun is RSun = 24.3+−11..21 kpc, placing it at a distance of
the extra-tidal slope, η (etta). The shatllow slopeat largeradii of RGC = 25.3+−11..21 kpc from the Galactic centre. Pal 13’s
η=1.9influencesthefitoftheKingtemplatesuchthatityields radial velocity was determined by Cˆot´e et al. to be
asignificantlylargervalueforthetidalradiusasdoestheKKBH Vr =(24.1±0.5) km/susingspectroscopicdataoftheHigh
template. 1 arcmin corresponds to about 7 pc at the assumed Resolution Echelle Spectrometer at the Keck telescope.
distanceofPal13. In the same investigation, its internal radial velocity
dispersion was found to be σ = (2.2 ± 0.4) km/s from
r
a sample of 21 stars located within the cluster’s inner
2 arcmin. Within their best estimate of Pal 13’s tidal
radius of 26 arcmin, Cˆot´e et al. furthermore measured
an absolute magnitude of M = −3.8 mag. Assuming
V
Pal 13 being in virial equilibrium, this would imply a
by gravitational shocks, a variation of the IMF, tidally re- mass-to-light ratio of M/L = 40+24. Cˆot´e et al. suggest
shaped stellar phase-space distribution functions, contam- −17
that this unusually high velocity dispersion could be the
inations from stellar streams in the MW halo, or alterna-
consequenceofeitheracatastrophicheatingduringarecent
tivegravitationaltheories(e.g.Kroupa1997,Gilmore et al.
perigalacticonpassageorthepresenceofadarkmatterhalo.
2007, Simon & Geha 2007, Mieske et al. 2008, Angus 2008,
Niederste-Ostholt et al. 2009).
(ii) Fig. 1 shows the surface density profile as obtained
Thelow-massGalactic globularclusterPalomar 13isa
with the 3.6 Canada-France-Hawaii telescope (CFHT) by
stellarsystemwhichshowsboth,anunclearextentduetoa
Cˆot´e et al. (2002). The profile shows a shallow slope η ≃
shallowsurfacedensityprofile,andahighvelocitydispersion
−2 out to large radii (10 arcmin correspond to about 70
resulting in a mass-to-light ratio of about 40 (Siegel et al.
pc at the distance of Pal 13), markedly different to the
2001; Cˆot´e et al. 2002). Further details on this cluster are
much steeper slope η ≃ −4 found in most other sur-
presentedinSec.2.Inthisinvestigation wedemonstrateby
face density profiles of globular clusters (compare, e.g.,
means of N-body calculations how these observational re-
McLaughlin & van derMarel 2005). This large extent of
sults can be explained without the need for dark matter,
Pal 13 can be interpreted in two ways. First, the stellar
tidal heating, binaries or changes in the law of gravity. To
population at large radii can be part of the cluster such
this end we compute models of Palomar 13 on various or-
that Pal 13 would be a very low concentrated cluster with
bits about the Galaxy that are consistent with its present-
alarge tidal radiusof about 26arcmin [180 pc](Cˆot´e et al.
day distance and radial velocity with respect to the Sun.
2002), or, second, Pal 13 can be interpreted as having a
We show how different such a stellar system can appear in
very pronounced tidal debris and the cluster itself having a
different phases of its orbit. Details on the models are de-
significantly smaller tidal radius of about 3 arcmin [20 pc]
scribedinSec.3.Theresultsofthesecomputationsandthe
(Siegel et al. 2001). To check these oppositional proposals
mock observations in which we show how this cluster may
for consistency with theoretical expectations, we can make
appearwhenobservedwithan8m-classtelescope,areshown
a first estimate of Pal 13’s truetidal radius, R , using
t
in Sec. 4. Sec. 5 is a short discussion on the plausibility of
ourfindings.Finally,inSec.6wegiveashortsummaryand GM 1/3
conclusions. Rt = 2Ω2 , (1)
(cid:16) (cid:17)
The Curious Case of Palomar 13 3
is shown in red solid lines for the case of combining the
proper motion of Siegel et al. with the radial velocity of
Cˆot´e et al. (2002). Note that we will refer to this orbit as
orbit 1 throughout the text. The plotted line corresponds
to the orbit integrated within an Allen & Santillan (1991)
Milky Way potential for the last 3 Gyr and 0.5 Gyr into
the future. Distance and orbital motion of the Sun where
taken from Dehnen& Binney (1998), that is, the Sun is lo-
cated at a Galactocentric distance of x=8 kpc,theorbital
velocity of the local standard of rest (LSR) is 220 km/s in
y-direction,andtheSunmoveswithrespecttotheLSRwith
V =10.0 km/s, V =5.3 km/s and V =7.2 km/s.
x y z
Aswecanseefrom Fig.2,correspondingtothesemea-
surements,Pal13hasaveryellipticalorbitwitheccentricity
R −R 82.8−11.1
ǫ= apo peri = =0.76, (3)
R +R 82.8+11.1
apo peri
where R is its apogalactic distance and R its peri-
apo peri
galactic distance, respectively. Therefore, with a Galacto-
centric distance of 25.3 kpc and theproper motion as mea-
sured by Siegel et al. (2001), Pal 13 is today quite close to
its perigalacticon. Because of this, Siegel et al. (2001) and
Cˆot´e et al. (2002) suggest that the high velocity dispersion
ofPal 13 andtheshallow slope ofits surface densityprofile
at large radii may well be due to the last pericentre pas-
Figure2.OrbitofPal13,asdeterminedusingthepropermotion sage which may have heated the cluster violently and may
measuredbySiegeletal.(2001)andtheradialvelocitymeasured have caused a rapid expansion and/or an overspilling over
byCoˆt´eetal.(2002),referredtoasorbit1,showninthemeriodal
thetidal boundaries.
planerepresentation,whererdepictstheradialdistancefromthe
A comprehensive N-body investigation of low-mass
GalacticcentrewithintheGalacticdiskandzistheheightabove
globularclustersoneccentricorbitshowevershowsthatperi-
the Galactic disk. Shown are the last 3 Gyr and 0.5 Gyr into
centrepassagesatsuchgreatgalactocentricdistancesbarely
the future. Also shown is the resulting orbit after setting the
transversevelocity,Vt,tozero,suchthattheclusterhasonlythe cause violent mass loss or rapid expansion (Ku¨pper et al.
(morepreciselydetermined) radialvelocity measuredbyCoˆt´eet 2010a). Furthermore, a follow-up investigation showed that
al. (orbit 2). The black dotted line depicts the orbit with the thesurface densityprofiles(a) cannot beusedtodrawcon-
transversevelocitychosensuchthatitminimizesPal13’sorbital clusions on the (theoretical) tidal radius of a cluster, and
velocity,Vorb,referredtoasorbit 3. (b)only showshallow slopes at large radiiwhen thecluster
isclosetoreachingitsapogalacticon (Ku¨pperet al.2010b).
The latter is due to the tidal tails of the cluster which get
whereGisthegravitationalconstant,M isPal13’spresent-
stretchedandcompressed alongtheorbit.Thatis,ifaclus-
day mass, and Ω its angular velocity on its orbit about the
ter and its tails move from apogalacticon to perigalacticon
Milky Way (Spitzer1987). This yields
theyget accelerated andstretched,whereas from perigalac-
R ≃43.7pc. (2) ticontoapogalacticontheygetdeceleratedandcompressed.
t
If the shallow slope in the surface density profile of
when we assume that Pal 13 has a mass of about 3000M⊙ Pal13isindeedduetotheeffectsdescribedinKu¨pper et al.
and is on a circular orbit with an orbital velocity of about
(2010a,b), then Pal 13 would need to be currently in a po-
V =220km/satR =25.3kpc.Infact,Pal13ismore
orb GC sition close to apogalacticon. This, however, disagrees with
likelytobeonaneccentricorbitandhencemayratherhave the position in its orbit (orbit 1) derived from the proper
a present-daytidalradius of about 50-100 pc when we take
motion measured bySiegelet al.andthat putsPal 13close
into account that its true angular velocity is likely to be
toperigalacticon (Fig.2).Inanattempttoresolvethiscon-
lower than that of a circular orbit. Thus, the theoretically
tradiction, we will in the following define two further test
expectedrangefor Pal13’stidalradiusdoesnotagreewith
orbits which differ in proper motion from the Siegel et al.
either of the two observational estimates of Cˆot´e et al. and
estimate. We will then fully integrate the dynamical evolu-
Siegel et al.
tionofPal13forallthreeorbitstocomparetheresultswith
the observed surface brightness profile, luminosity, and ra-
dialvelocitydispersion(Sec.4).Theunderlyingassumption
2.1 Orbit
for this procedure is that the proper motion measurements
The hypothesis that the above peculiarities are caused by are themost uncertain observed cluster properties.
tidal effects is supported by measurements of Siegel et al. We first force the cluster’s transverse velocity (proper
(2001), who, using CCD photometry and 40 years older motion) to zero, while keeping the radial velocity. In Fig. 2
photographic plates, find Pal 13’s proper motion to be wecanseethatthisresultsinanorbitwhichislesseccentric
µ cosδ = (2.30 ± 0.26) mas/yr and µ = (0.27 ± 0.25) (R =49.3 kpc, R =12.5 kpc, ǫ=0.60) and in which
α δ apo peri
mas/yr. In Fig. 2 theorbit of Pal 13 within theMilky Way theclustertodayisclosertoapogalacticon. Wewillreferto
4 A.H.W. Ku¨pper, S. Mieske and P. Kroupa
this orbit as orbit 2. In addition, we search for the proper component in the form of a power-law slope. In this way,
motion values which minimize Pal 13’s 3D orbital velocity, also more concentrated clusters can be represented for
V ,sincethiswouldyieldtheorbitinwhichPal13isclos- whichtheoriginal Kingtemplatefails, andfurthermore the
orb
est to apogalacticon. In fact, by doing this we get a more cluster profile can be fitted without being influenced by
eccentric orbit as the cluster falls deeper into the Galactic a dominant tidal debris. This effect can be seen in Fig. 1
centre(R =38.5kpc,R =3.5kpc,ǫ=0.83).Thisor- where we applied a regular King template fit to the CFHT
apo peri
bitwillbereferredtoasorbit3.ItisalsodepictedinFig.2, data by Cˆot´e et al. and also a fit of KKBH1. The King
with the corresponding values for the proper motion being templategetssignificantlyinfluencedbythestellarmaterial
µ cosδ = 0.72 mas/yr and µ = −1.2 mas/yr. Note that at large radii and yields a tidal radius of more than 15
α δ
thesevaluesdiffersignificantlyfromthevaluesmeasuredby arcmin. The KKBH template assigns a power-law slope of
Siegel et al.(2001)bymorethan1mas/yrineachdirection. η=1.91±0.15 to the tidal debris and yields a tidal radius
Forallthreeorbits,theclusteriscomingfromperigalac- of only 1.9 arcmin.
ticon and approaching apogalacticon. But all three predict
Pal13tobeinadifferentorbitalphase,p ,whichwehere By computingN-bodymodels forall threekindsofor-
orb
defineas bitsgivenabove,andfittingtheKKBHtemplateinthesame
R˙ R −R waytosimilarlyresolvedN-bodydata,wewilltrytorepro-
porb= |R˙GGCC|RaGpCo−Rppeerrii, (4) wduilclemtehaissuorbestehrveevdelsoucritfaycdeisdpeenrssiitoynsalonpdeaηb.soFluurttehmeramgonriteuwdee
and which is constructed to be zero in perigalacticon and ofthecomputedclustersinthesamewayasCˆot´eetal.have
unity in apogalacticon. This factor p gives the fraction done,andcomparethesevalueswiththeobservationalones.
orb
of the radial distance between R and R at which a
peri apo
clusteriscurrentlylocated.InthisdefinitionR˙ isthetime
GC
derivativeof thegalactocentric radius, which dividedby its
3 MODELS
magnitude adds a minus sign to the orbital phase in case
the cluster is moving from apogalacticon to perigalacticon. We computed 45 models of Pal 13 using the collisional N-
For circular orbits p is always zero. bodycodeNBODY6(Aarseth2003)ontheGPUcomputers
orb
We introduce the orbital phase, p , here in addition at AIfABonn. Weset up15 different clusterconfigurations
orb
totheorbital eccentricity,ǫ, since it appears crucial for the usingthepubliclyavailabletoolMcLuster2(Ku¨pperetal.,
appearanceoftheobservedeffects(Ku¨pperet al.2010b).A inprep.).WeusedatidallytruncatedPlummerprofilewhere
systematicstudyonthedependenceofthedescribedeffects we varied the cluster half-mass radius between 4, 6 and 8
onthosetwoparameterswillfollowinafutureinvestigation. pc,andtheinitialmassbetween3000,4000,5000,7500and
For orbit 1 we get porb =0.20, for orbit 2 porb =0.35, and 10000 M⊙,respectively.Themeanmassoftheclusterstars
for orbit 3 porb=0.62. was in all cases about 0.3M⊙, thus the number of objects
in the computations were a factor of three times higher.
Each of these 15 clusters was computed for the last 3 Gyr
2.2 Surface density profile on each of the three different orbits (orbit 1-3) mentioned
in Sec. 2 (see also Fig. 2). We focus on the last 3 Gyr of
Ku¨pper et al. (2010b) introducea template (KKBH)which
evolution since we are only interested in the nearby tidal
canbeusedtoreliablymeasuretheslopeofasurfacedensity
debris. Using equation 18 of Ku¨pper et al. (2010a) for the
profileat large radii. TheKKBH template reads as follows:
mean drift velocity of stars within thetidal tails,
−γ
f1(R) = k 1+R/RR/cR × vC =±(4GMΩ)1/3, (7)
c
(cid:20) (cid:21) whereGisthegravitationalconstant,M istheclustermass
2
1 1 andΩitsangularvelocityonitsorbitabouttheMilkyWay,
− (5)
" 1+(R/Rc)2 1+(Rt/Rc)2# we can estimate the length of the tidal tails after 3 Gyr of
evolution if the cluster was on a circular orbit. With the
for radii smallepr than µRt, and p samevalueswhichweusedineq.2wegetadriftvelocityof
about 0.77 pc/Myr and thusa minimum length of the tails
64 −η/64
R of 2.3 kpcin each direction from thecluster. Notethat this
f (R)=f (µR ) 1+ (6)
2 1 t µR estimate gets complicated through the fact that Pal 13 is
" (cid:18) t(cid:19) #
for R>µR , where k is a constant, R gives a core radius,
t c
γ the core slope inside R , and R a tidal radius (which 1 Ku¨pper et al. recommend using an additional constant back-
c t
Ku¨pperet al.nameedgeradiustoavoid confusion with the ground,b,fortheKKBHfittoallowformoreflexibilityatlarge
theoreticaltidalradius,sincethosetwocorrelateonlyunder radii,but as itturns out, this is only reasonable with highly re-
certain circumstances). For radii larger than afraction µ of solved (e.g., N-body) data. For less well resolved observational
datawithmuchfewerdatapoints,itismorereasonabletoreduce
R thetemplatechangesintoapower-lawwithslopeη.The
t thenumberoffitparameterstoaminimum.Wethereforesetthe
exponent64inf (R)causesthetemplatetochangeabruptly
2 background, b, mentioned in Ku¨pperetal. (2010b) to zero, and
into thepower-law slope at µR .
t inadditionfixthebreakradiusparameterµto0.5,aswasfound
KKBH is based on the template of King (1962) but is
byKu¨pperetal.tobethemostplausiblevalue.
modified in two steps: first it allows to have a power-law 2 www.astro.uni-bonn.de/~akuepper/mcluster/mcluster.html
cuspinthecore,andsecondithasanadditionalextra-tidal or www.astro.uni-bonn.de/~webaiub/german/downloads.php
The Curious Case of Palomar 13 5
mostlikelynotonacircularorbit.Throughtheacceleration
anddecelerationonaneccentricorbitthecluster-tailsystem
gets periodically stretched and compressed. This estimate
is therefore only a mean value of the length of the tails.
Anyway,forinvestigatingthevicinityofPal13thistimespan
seems to be sufficient.
Since we want to produce a realistic CMD of Pal 13
with the appropriate photometric observables, we use the
SSE code (Hurley,Pols & Tout 2000) in combination with
McLuster3 tosetupevolvedstellar populationsof10Gyr
age with a metallicity of [Fe/H] = −1.9. The populations
are evolved from a canonical Kroupa IMF (Kroupa 2001)
ranging from 0.08M⊙ to 100M⊙, where compact remnants
are only kept if their kick velocity which gets assigned to
them by SSE does not exceed Pal 13’s present-day escape
velocity,v , calculated using
esc
2GM
v = , (8)
esc R
r h
where M is again the cluster mass and R is the cluster’s
h
half-mass radius, respectively. This treatment is a bit arbi-
Figure 3. Mock colour-magnitude diagram of one of the com- trary since we do not dynamically model the first 10 Gyr
puted clusters at the end of 3 Gyr of N-body integration, i.e. of the cluster’s life and the true retention fraction could be
when the stellar population is 13 Gyr old. The y-axis gives the both,higherorlower.Anothersimpletreatmentwouldbeto
apparentVmagnitudeaswouldbeobservedfromthedistanceof keep all compact remnants which would havea slight effect
theSun.Randomerrorswhichgrowexponentiallywithincreasing ontheobservedmass-to-lightratioastheclusterwouldthen
magnitude were applied to both apparent magnitudes, mV and havemoremassthancanbeseeninstars.Butonlyabout80
mMB⊙(asneedtaexht)a.lfT-mhiasssparratdiciuuslaorfc8lusptce.rThhade adnasihneidtiablomxasshsoowfs50th00e compact remnantsget expelled from a 5000M⊙ cluster like
we model here in the way described above, hence we con-
regioninthecolour-magnitudediagraminwhichwedefine stars
siderthistobeofsecondaryimportanceandconcentrateon
tobeclustermembers.
thecase of low dark mass in theclusters.
Those evolved clusters we then feed to NBODY6 to
evolve them further, chemically and dynamically, up to a
total age of 13 Gyr. In this way we can concentrate on the
last few Gyr of dynamical evolution of the cluster, which
aremostimportantforitspresent-daystructureandnearby
tidal debris. Hence, we save computational time with this
technique. A similar approach has been successfully tried
byHurley et al. (2001) for theopen cluster M67.
ThestellarevolutionofsinglestarswithinNBODY6is
also calculated with SSE, a consistent treatment of stellar
evolution throughout the investigation is therefore guaran-
teed.FromNBODY6wefinallyextracttheluminositiesand
stellar radii of all stars within the calculations to compute
their effectivetemperatures and with this theircolours and
magnitudes in the Johnson-Cousins system (Bessell 1990).
We use the algorithm described in Flower (1996) to first
derivethebolometric correction,BC,andthecolourindex,
B−V,andwiththistheabsolutemagnitudeintheV-band,
M , and in the B-band, M . Together with the distance
V B
information of each star we can then derive the apparent
magnitudes, m and m , respectively, and can apply a re-
V B
alisticcut-offatamagnitudelimitof,e.g.,m =25magas
V
Figure 4. Colour-magnitude diagram of the Besan¸con model wouldbeachievedbyan8m-classtelescopeinafewminutes
(Robinetal. 2003) covering a 1 deg2 field around the position of integration.
of Pal 13. Within the region whichis occupied byPal 13inthis Sinceanyobservationobeysstatisticalandinstrumental
diagram(dashedbox)wecountabout1000starswhichwillpol- uncertainties, we furthermore apply a Gaussian-distributed
luteobservations whengoingdowntomV =25mag.
3 NotethatthisversionofMcLusterincludingSSEisalsoavail-
ablefromthegivenwebaddress.
6 A.H.W. Ku¨pper, S. Mieske and P. Kroupa
randomerror,dmtoeachapparentmagnitude,m andm ,
V B Table 1. Results for the N-body computations with the orbit
which increases with decreasing brightness as usingthetransversevelocitymeasuredbySiegeletal.(2001),i.e.
orbit 1.M0 givestheinitialmassoftheclusteratthebeginning
dm= 0.022+0.07×10.00.4×(m−25.0). (9) of the computations, and R0 its initial half-mass radius. MV is
themeasuredabsolute magnitudewithintheinner26arcminat
This gpives a minimum error of 0.02 mag, and an additional
an age of 13 Gyr, i.e. today. Coˆt´eetal. (2002) find a value of
uncertainty which is of the order of 0.07 mag at 25 mag
and decreases with increasing brightness. A final, synthetic MV = −3.8 mag for Pal 13. σr gives the velocity dispersion
within the inner 2 arcmin measured from a sample of 21 stars.
colour-magnitude diagram of one of the clusters is given in Thevaluegivesthemeanof106 independentmeasurements(see
Fig. 3. Sec.4.2fordetails),theuncertaintiesgivethelimitsinwhich67%
Objects with low surface densities such as tidal debris ofallmeasurementslie.Coˆt´eetal.(2002)findσr =2.2±0.4for
will be largely affected by background/foreground source Pal13. η istheslopeof thesurfacedensity profileatlargeradii
contamination. We therefore have to estimate the number measured with the KKBH template. The uncertainties give the
of stars which will pollute our mock observations. For this standard error from a least square fit. For Pal 13 we measure a
purposewegenerateanartificialstellarpopulationusingthe slopeofabout1.9basedontheobservationaldatabyCoteetal.
Besan¸conmodel(Robin et al.2003)fora1deg2fieldaround (2002).
the position of Pal 13 in the same filter set (Fig 4). Within
the CMD region occupied by the cluster model we count M0 R0 MV (R<26′) σr(R<2′) η
about 1000 stars which can be mistaken as cluster mem- [M⊙] [pc] [mag] [km/s]
bers, corresponding to 0.28 stars/arcmin2. That is, Pal 13 3000 4.0 -2.7 0.53+0.09 3.86±0.58
−0.09
anditstidaldebriswillonlybevisibleinthoseplaceswhere 3000 6.0 -2.6 0.54+0.09 3.55±0.34
−0.09
its surface density exceedsthis value.In addition to oures- 3000 8.0 -2.1 0.46+0.07 4.34±0.76
−0.07
timate and for the sake of being conservative, we will also 4000 4.0 -3.0 0.63+0.11 3.33±0.20
−0.11
discuss our results using the somewhat higher background 4000 6.0 -3.2 0.63+0.11 3.77±0.12
−0.10
surface density of 0.69 stars/arcmin2 found observationally 4000 8.0 -2.6 0.53+0.08 4.15±0.16
−0.08
by Cˆot´e et al. (2002). 5000 4.0 -4.0 0.69+0.12 3.58±0.14
−0.11
5000 6.0 -3.1 0.71+0.12 3.82±0.15
−0.11
5000 8.0 -4.0 0.58+0.09 3.86±0.14
−0.09
4 RESULTS 7500 4.0 -3.9 0.92+0.16 3.36±0.07
−0.16
7500 6.0 -4.2 0.84+0.14 4.00±0.16
From each computation we take a snapshot after 3 Gyr of 7500 8.0 -3.5 0.73+−00..1123 4.08±0.07
dynamicalevolutionasseenfromthelocationoftheSun.At 10000 4.0 -3.7 1.11−+00..2101 3.86±0.24
this point the stellar population is 13 Gyr old, and should 10000 6.0 -4.2 0.96−+00..1169 4.40±0.21
resemble the stellar population of Pal 13 within the given 10000 8.0 -3.6 0.83−+00..1135 4.24±0.15
uncertainties. Stellar maps of a 4 deg2 region around one −0.13
of the clusters (M = 5000,R = 8 pc) for each of the
0 0
threeorbitsareshownintheleftpanelsofFig.5-7.Inthese Moreover, comparing the same clusters but on the dif-
figures each dot represents a star above m = 25 mag. A ferent orbits, we find that the clusters evolve quite simi-
V
background of 0.28 stars/arcmin2 was added with random larlyinternallyandthattheirabsolutemagnitudesareonly
positions. By comparing the figures we see that the orbit marginallyinfluencedbytheorbitaltype.Infact,atthebe-
with the Siegel et al. proper motion (orbit 1, Fig. 5) and ginningofthecomputationstheclustersofagivenmassand
the one with zero proper motion (orbit 2, Fig. 6) produce sizeareexactlythesameclustersjustondifferentorbits.In
similar results, whereas the orbit with the minimal orbital this way we make sure that differences come from dynami-
velocity (orbit 3, Fig. 7) produces a cluster which appears calevolutionandnotfromstellarevolution.After3Gyrthe
largely extended. masses between the clusters of a given initial mass and size
differ by only 50−100M⊙. From this we can deduce that
the influence of the pericentre passages on all three orbits
4.1 Absolute magnitude areratherunimportant.Otherwise,themoreeccentricorbits
(orbit 3 and orbit 1) would have induced more dissolution
From these snapshots we measure the integrated absolute
onthoseclusters,andaltered thefinalabsolutemagnitudes
magnitude, M , of each cluster representation within a ra-
V
more significantly.
diusof26arcminarounditscentre,justasCˆot´e et al.(2002)
havedonefortheirobservationaldata.Theresultsarelisted
inTab.1-3forthethreeorbitaltypes.Weseethattheclus-
4.2 Velocity dispersion
ters starting off with smaller initial masses, independent
of the orbit, have lost too much mass within the 3 Gyr From the computations we also take radial velocity disper-
of evolution, such that today their absolute magnitude is sion measurements in the same way as Cˆot´e et al. (2002)
too low compared to theobservational valueof M =−3.8 have done. That is, we draw 21 stars from the sample of
V
mag. Clusterswith M0 >5000M⊙ lose just about theright stars within the inner 2 arcmin of the clusters, while mak-
amountofmasswithinthistime.Fromthetableswecanex- ingsurethatall21starsliewithin10km/sofeachother.A
pectclusterswithinitialmassesevenhigherthan10000M⊙ starwitharadialvelocitydifferingmorethan10km/sfrom
to exceed the observed absolute magnitude. This suggests theotherstars whichhavebeen drawnfrom thepopulation
that our range of initial parameters covers theright part of would therefore be regarded as a non-cluster member, even
theparameter space of initial conditions. though this does not necessarily hold true in our computa-
The Curious Case of Palomar 13 7
Figure 5. Left: stellar map of a 4 deg2 field around Pal 13, computed using the orbit with the transverse velocity measured by
Siegeletal. (2001), i.e. orbit 1. Each dot represents a star above mV =25 mag. A background of 0.28 stars/arcmin2 was added with
random positions. Right: underlying surface density map of Pal 13 for the same field. One bin corresponds to 9 arcmin2. The colour
coding shows log (N +1), where N is the number of Pal 13 stars in a bin. The expected background of 0.28 [0.69] stars/arcmin2
10
corresponds toavalueof0.5[0.9]inthisrepresentation. Atthe distance ofPal13, 1deg corresponds toroughly420 pc.Thecluster is
welllimitedandthedensityfallsoffsteeply,onlysmalltracesoftidaltailscanbeseen.
Figure 6. Left: stellar map of a 4 deg2 field around Pal 13, computed using the orbit with zero transverse velocity (orbit 2). Each
dot represents a star above mV = 25 mag. A background of 0.28 stars/arcmin2 was added with random positions. Right: underlying
surface density map of Pal 13 for the same field. One bin corresponds to 9 arcmin2. The colour coding shows log (N +1), where N
10
isthe number of Pal 13stars ina bin.The expected background of 0.28 [0.69] stars/arcmin2 corresponds to avalue of 0.5[0.9]inthis
representation.AtthedistanceofPal13,1degcorrespondstoroughly420pc.Theclusterisalsowelllimited,justasinFig.5.Onlythe
tidaltailsareabitmorepronouncedsincetheclustermovesatalowervelocityandhencethestellardensitywithinthetailsishigher.
tions.Thevelocitydispersion isthencomputedinthesame From Tab.1 we can see that the clusters on the Siegel
fashion as has been done in Ku¨pper & Kroupa (2010). We orbit(orbit 1)yield toolowvelocity dispersions incompar-
independently draw 106 sets of 21 stars from each cluster ison to theobservational valueof σ =2.2±0.4 km/s. The
r
andcomputeforeachsetthedispersion ofthestellarveloc- orbitwithzeropropermotionyieldssimilarresults(orbit 2,
ities.Fromthesevalueswetakethemean,whichisgivenin Tab.2).Inbothsetsofcomputationsweachievethehighest
Tab. 1-3. The uncertainties of these values give the bounds velocity dispersions of 1.1±0.2 km/s in the most massive
in which lie 67% (1σ) of all measurementsaboveandbelow and most compact cluster of 10000M⊙ and R0 = 4.0 pc.
themean. This is expected when we assume that the clusters are in
8 A.H.W. Ku¨pper, S. Mieske and P. Kroupa
Figure 7.Left: stellarmapofa4deg2 fieldaroundPal13, computed usingthe orbitwiththetransverse velocityminimizingPal13’s
orbitalvelocity(orbit 3).EachdotrepresentsastarabovemV =25mag.Abackground of0.28stars/arcmin2 wasaddedwithrandom
positions. Right: underlying surface density map of Pal 13 for the same field. One bin corresponds to 9 arcmin2. The colour coding
showslog (N+1),whereN isthenumberofPal13starsinabin.Theexpected backgroundof0.28[0.69]stars/arcmin2 corresponds
10
to a value of 0.5 [0.9] in this representation. At the distance of Pal 13, 1 deg corresponds to roughly 420 pc. The cluster is embedded
in a far-extending cloud of stars, which originates from the compressed tidal tails getting pushed back into the cluster vicinity as the
cluster-tailsystemisbeingdecelerated onitswaytoapogalacticon.
Table 2.The same as Tab. 1but forthe orbitwith zerotrans- Table3.ThesameasTab.1butfortheorbitwiththetransverse
versevelocity(orbit 2). velocity minimizing Pal 13’s orbital velocity (orbit 3). Velocity
dispersionvalueswhichagreewithin1σwiththeobservedveloc-
M0 R0 MV (R<26′) σr(R<2′) η itydispersionof2.2±0.4areboldfaced.
[M⊙] [pc] [mag] [km/s]
3000 4.0 -2.7 0.52+−00..0099 3.10±0.41 [MM⊙0] [Rpc0] MV [(mRa<g]26′) σr[(kRm</s]2′) η
3000 6.0 -2.5 0.52+0.09 4.04±0.13
1117775554443000555000000000000000000000000000000000000 8648648648648.............0000000000000 -------------3433434342332.............6275290116101 0010000000000.............89078956756642681326921344+−+−+−+−+−+−+−+−+−+−+−+−+−−000000000000000000000000000...........................111111011011011111101101100259136912811735913691281179 4434434334333.............40842505409347528035977857±±±±±±±±±±±±±0000000000000.............22222123041222294948370238 1177755544433300555000000000000000000000000000000000000000 46864864864864..............00000000000000 --------------34343434233222..............72529011620167 01100110100000..............8119762919998766040591593912+−++−−++−+−+−+−+−+−+−+−+−+−000000000001111100100000100...........................121131121210111099126742274806697907076404636383670458 12221211112111..............5150665956147996222610183810±±±±±±±±±±±±±±00000000000000..............2211131101001010150000909747
−0.17
10000 8.0 -3.6 0.87+0.56 1.96±0.31
−0.19
virial equilibrium. Note that the observed velocity disper-
sion is more than 5σ off. This fact led Cˆot´e et al. to the
assumption that Pal 13 may contain dark matter or got cmin of the cluster, which pollute the velocity dispersion
catastrophically heated bythe last pericentre passage. (Ku¨pperet al.2010b).Thiseffectismoresignificantforthe
But for the third kind of orbit, when the cluster is on clusterswhichareinitiallymoreextendedastheyshowmore
an orbit with a lower orbital velocity such that it is nowa- potential escapers. This is a consequence of them being en-
days closer to apogalacticon (orbit 3), the measured veloc- ergeticallymoreaffectedbythepericentrepassages(seee.g.
ity dispersion is significantly higher (Tab. 3). We get val- Gnedin,Lee & Ostriker1999).
ues of about 2.2 km/s within 1σ for many clusters in the The clusters on orbit 3 indeed show much more extra-
set. This is due to the number of unbound stars within tidalmaterial.LookingattherightpanelsofFig.5-7wesee
the cluster, so-called potential escapers, and stars outside that there is barely any stellar material outside the cluster
the tidal radius lying in projection within the inner 2 ar- fororbit 1 (Fig.5),andonlylittlemorefororbit 2 (Fig.6).
The Curious Case of Palomar 13 9
In contrast to this, orbit 3 shows a cluster with an unusual
extra-tidal extent of several hundred parsec (Fig. 7). This
extra-tidalmaterialresultsfromthecompressionofthetidal
tails as the cluster and its tails are being decelerated. This
decelerationissostrongthatthewholesystemconsistingof
cluster,leadingtail,andtrailingtail,iscompressedtoafew
hundred pc. In fact, the system extends even further than
can be seen in the figure and would extend much further
if the N-body computations would have been made for the
full 13 Gyr since the tidal tails need many Gyr to grow to
suchextent.Notealso,thattheshapeofthissystemisquite
irregular sincewelook at afolded stellar stream and notat
a bound structurein equilibrium.
Note that while our choice of the initial density profile
maywellaffectthemeasuredvelocitydispersion,e.g.incase
of a higher concentrated King model, it will not affect the
Figure12.Comparisonofpropermotionmeasurementfromour
appearance of thetidal debris.That is, thechoiceof profile
computationsandfromtheworkofSiegeletal.(2001).Thedata
may influence the internal structure of the cluster and also
points of orbit 1 - 3 show the proper motions ofall starswithin
its mass loss rate but not its tidal debris since the debris is
theinner6arcminoftheclusters.Asexpected,thedatapointsof
formedbyorbitalcompressionandnotbythemasslossrate orbit1 allmatchpreciselywiththeorbitdeterminedbySiegelet
(Ku¨pperet al. 2010b). al.,whilethedatapointsoforbit2 alllieconcentratedwithinthe
originat zero proper motion. The stars of the cluster on orbit 3
show asignificant intrinsicspread,though, whichisduetostars
4.3 Surface density profiles
belongingtothecluster’stidaldebrisandwhichthereforedonot
tightly follow the bulk cluster motion. The data from Siegel et
Thiseffectofcompressionofthetidaldebriscanalsobeseen
al. is shown with and without colour correction to illustrate the
as an increase of density within the surface density profiles
largespreadanduncertaintiesoftherawdata.Alsoshownisthe
of the modelled clusters, see Figs. 8-10. The figures show
resultingpropermotionfoundbythisgroup.
theprojectedstellarnumberdensityofthesnapshotsforall
clustersexceptforthemostmassiveones,measuredinrings
around the cluster centres, just as Cˆot´e et al. (2002) have
done it with their CFHT data (compare with Fig. 1). For
better comparison with the CFHT data, we subtract the
same background of 0.69 stars/arcmin2 from our data in-
stead of our lower estimate of 0.28 stars/arcmin2 since this
background estimate is important for the outermost data surface density profiles (Fig. 10). We get a shallow slope at
points and therefore may influence the fit of the KKBH large radii for all clusters with values as low as η = 1.5,
template. Error bars in the figures give the square-root of and for the steepest not more than η = 2.7. That is, the
theseresultingvaluesasstatisticaluncertainties.Differences clustersnearapogalacticondiffersignificantlyfromtheclus-
in the numbers of stars at small radii within this diagram ters which are closer to perigalacticon. Such a behaviour
mainly originate from the fact that Cote et al. go down to of the surface density profile has been found in observa-
m = 23.5 mag with their CFHT data whereas we cut at tions of other globular clusters as well. For instance, Palo-
V
m =25mag(assumingthatPal13 isobservedwithan8m mar 5, which is known to be close to its apogalacticon,
V
class telescope). We fit a KKBH template to these surface shows an η of 1.5 (Odenkirchenet al. 2003). Chun et al.
density profiles in order to measure the slope of the extra- (2010) find 5 Milky Way globular clusters to show shal-
tidal material. The results of these fits are displayed in the low values of η at large radii, that is 2.44 for NGC 5466,
figures (η, i.e., eta)as well as in Tab. 1-3. 1.59 for M 15, 1.58 for M 53, 1.41 for M 30, and 0.62 for
The Siegel orbit (orbit 1) yields well limited clusters NGC 5053. Moreover, AM 4 and Whiting 1 both show an
with steep slopes outside the tidal radius between η = 3.3 η of 1.8 (Carraro, Zinn & Moni Bidin 2007; Carraro 2009).
and η = 4.4 (Fig. 8), just as expected from clusters near Olszewski et al. (2009) furthermore find the Galactic glob-
perigalacticon (Ku¨pper et al. 2010b). Furthermore, there is ular cluster NGC 1851 to be surrounded by a 500 pc halo
no clear trend in the slopes with respect to the initial con- of stars. Its surface density profile shows a slope of η =
ditions. The differences are just the statistical fluctuations 1.24±0.66. Our investigation suggests that those clusters
which Ku¨pper et al. also found in their N-body data. The are all affected by this orbital effect.
sameholdstruefororbit 2 (Fig.9).Justthescatterisabit From Fig. 10 we furthermore find that fitting a King
larger from η=3.1toη=4.5 but,again, without anyclear (1962) template to these clusters with the shallow slopes
trend. If the slope at large radii was a consequence of the at large radii sometimes yields very different results for the
last pericentre passage, then we would expect a correlation tidal radius in comparison to the KKBH template (e.g. for
ofthisslopewiththeinitialhalf-massradius,R0,oftheclus- M0 = 3000M⊙ and R0 = 8 pc we get RtKKBH = 1.5 ar-
ter, since a more extended cluster should be more affected cmin and RKing = 15.3 arcmin). This we have observed as
t
bytidalshockingandthereforeproduceamorepronounced well in the original data (Fig. 1) and thus would be an ex-
tidal debris. planation for thelarge uncertainties in Pal 13’s tidal radius
In contrast to that, orbit 3 results in quite different (Siegel et al. 2001; Cˆot´e et al. 2002).
10 A.H.W. Ku¨pper, S. Mieske and P. Kroupa
Figure8.SurfacedensityprofilesoftheN-bodycomputationswiththetransversevelocitymeasuredbySiegeletal.(2001),i.e.orbit1,
for all clusters between 3000M⊙ and 7500M⊙. Uncertainties show the square-root of the number of stars in one bin after subtracting
abackgroundof0.69stars/arcmin2 likeCoˆt´eetal.(2002)havedone(comparewithFig.1).Theslopesatlargeradiiasmeasuredwith
theKKBHtemplatearegiveninthepanels(η,i.e.,eta).AlsogiveninthepanelsarethetidalradiiasfittedbytheKingandtheKKBH
template. Allslopesatlargeradiiarequitesteep, justaswouldbeexpected foraclusternearperigalacticon(Ku¨pperetal.2010b).