the evolution of globular clusters

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Advances in Space Research 46 (2010) 500–508

The evolution of globular clusters

Chenggang Shu a,*, Zhijian Luo a, Mao-an Han b, Abraham C.-L. Chian c,Wen-Ping Chen d, Zhen-yu Wu e

a Shanghai Key Lab for Astrophysics, Shanghai Normal University, Shanghai 200234, Chinab Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

c National Institute for Space Research (INPE), S~ao Jos�e dos Campos SP 12227-010, Brazild Graduate Institute of Astronomy, National Central University, Taiwan 32054, China

e National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China

Received 3 March 2008; received in revised form 10 June 2009; accepted 16 October 2009

Abstract

Taking a wide range of the initial conditions, including spatial density distribution and mass function etc, the dynamical evolution ofglobular clusters in the Milky Way is investigated in detail by means of Monte Carlo simulations. Four dynamic mechanisms are con-sidered: stellar evaporation, stellar evolution, tidal shocks due to both the disk and bulge, and dynamical friction. It is found that stellarevaporation dominates the evolution of low-mass clusters and all four are important for massive ones. For both the power-law and log-normal initial clusters mass functions, we can find the best-fit models which can match the present-day observations with their main fea-tures of the mass function almost unchanged after evolution of several Gyr. This implies that it is not possible to determine the initialmass function only based on the observed ones today. Moreover, the dispersion of the modelled mass functions mainly depends on thepotential wells of host galaxies with the almost constant peaks, which is consistent with current observations.� 2010 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Globular clusters; Evolution; Dynamics

1. Introduction

Globular clusters (GCs) in our Galaxy are the oldeststellar population which have survived for a long term evo-lution ( J 12 GyrÞ. It is of great importance to investigatethe evolution of GCs which provides clues to understandstar formation processes and dynamical evolution of stellarsystems. Although luminosity functions (LFs) of youngGCs in starburst galaxies display as a power-law (Zhangand Fall, 1999; Whitmore et al., 1999; Meurer, 1995) sim-ilar to that of open clusters in the Milky Way (Battinelliet al., 1994), the LFs of GCs of the Galaxy and many othergalaxies show to be lognormal with the peaks taken as thestandard candles to measure the extragalactic distances,

0273-1177/$36.00 � 2010 COSPAR. Published by Elsevier Ltd. All rights rese

doi:10.1016/j.asr.2010.03.009

* Corresponding author.E-mail address: [email protected] (C. Shu).

which provide strong constraints on the galaxy formationand evolution in the universe.

While the formation and evolution of globular clustersare still poorly known, many theoretical approaches andobservations have clearly shown that the initial propertiessuch as the mass function of GCs have been altered signif-icantly today (Gnedin and Ostriker, 1997; Heggie, 2001;Lin et al., 2001). Given a constant mass-to-light ratio whichhas been confirmed by observations (McLaughlin, 2000),Vesperini (1998) and his collaborators (Vesperini et al.,2003; Vesperini and Zepf, 2003) investigated the evolutionof the mass function, i.e., LF, of the globular cluster system(GCMF) taking into account the stellar evolution, two-body relaxation, tidal shocks due to the disk, and dynam-ical friction. Adopting an initial lognormal GCMF, hefound within a wide range of initial values for the parame-ters that a particular initial GCMF can keep its initialshape and parameters unaltered during the entire evolution

rved.

http://dx.doi.org/10.1016/j.asr.2010.03.009

mailto:[email protected]

0010111.0-8

-6

-4

-2

0

2

R (kpc)

Fig. 1. The number density distribution of GCs in the Milky Way withdifferent types of lines denoting results of different bin sizes selected.

C. Shu et al. / Advances in Space Research 46 (2010) 500–508 501

through a subtle balance between disruption of clusters andevolution of the masses of those which survive; Baumgardt(1998) investigated the evolution of GCs in our Galaxyassuming a power-law initial mass function. Consideringthe dynamical friction and evaporation of cluster stars,he found that the resulted mass function after enough longdynamical evolution ð� 16 GyrÞ can match the present-dayGCMF in the Galaxy for the initial power index between1.8 and 2.0, similar to those observed in starburst galaxies;Fall and Zhang (2001) presented a series of simple analyt-ical models to compute the effects of disruption on the massfunction of star clusters including evaporation by two-bodyrelaxation and gravitational shocks, and mass loss bystellar evolution. They found that, for a variety of initialconditions, after 12 Gyr, the mass function developed aturnover or peak, very close to the observed GCMF inthe Galaxy.

In this paper, for various initial conditions, we will inves-tigate the mass function evolution of the GCs in our Galaxyin more detail than the previous work by considering fourmajor mechanisms including mass evaporation by two-bodyrelaxation, tidal shocks due to the disk and bulge, dynamicalfriction and stellar evolution. Throughout the present study,a constant mass-to-light ratio M=L ¼ 3 is assumed for indi-vidual GCs whenever it needs to be stated. The paper isstructured as follows. In Section 2, the main observationalfeatures of GCs in the Galaxy, i.e., their radial distributionand mass function, are briefly summarized which will beadopted for the model predictions to compare. In Section3, the prescriptions of our model are discussed in detail.The model predictions for a single GC and GC populationin the Galaxy are presented in Section 4. Finally, the conclu-sions are in Section 5.

2 3 4 5 6 7 80

1

2

3

Log(M)

M/L=2.0

Fig. 2. The mass function of GCs with the histogram denoting theobservations and the dashed line denoting the best-fit to an exponentialdistribution, respectively.

2. Observational features

In the present paper, the observational features of theGCs in the Galaxy adopted for the model comparisonfocus on the density distribution and mass function ofGCs due to the limitation of the paper. It must be pointedout that the other observational features, such as the distri-bution in velocity space, are also taken into account in thepresent study.

The observational features of GCs in the Milky Way arefrom the GC catalog compiled by Harris (1996), whichcontains 147 GCs. The spatial distribution of GCs isalmost isotropic in the Galaxy at present day and theirnumber density can be well described by

qðRÞ / 1þ RRc

� ��a

; ð1Þ

with R being the Galactocentric distance, Rc the scale ra-dius and a the slope of the density.

By minimizing v2 to Eq. (1) based on the GC catalog,the best-fit results are shown in Fig. 1. Note that thebest-fit results of Rc and a depend on the bin sizes as usual

without significant changes. We conclude that the slopea � 3:5–4 and Rc � 0:5–2 kpc which are consistent withthe previous study (Djorgovski and Meylan, 1994).

After investigating the fundamental plane of GCs indetail, McLaughlin (2000) concluded that there is almostno “dark” matter in GCs and the mass-to-light ratioM=L of GCs is � 2. The luminosity function, hence themass function, can be obtained from Harris (1996) and isdisplayed as histogram in Fig. 2, within which the mass-to-light ratio is assumed to be 3. The mass function canbe well fitted by

/ðLogðMÞÞ / exp �Log2ðM=M�Þ2r2

� �; ð2Þ

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

R (kpc)

bulge+disk+halobulgediskhalo

Fig. 3. The predicted rotation curve of the Galaxy and the correspondingcontributions of individual components of P90.

502 C. Shu et al. / Advances in Space Research 46 (2010) 500–508

with M� ¼ 1:4� 105 M� and r ¼ 0:8 and is shown asdashed line in Fig. 2.

3. Model

3.1. The Galactic model

To investigate the evolution of GCs in our Galaxy, themotion of GCs must be considered which needs the Galaxymodel to establish the background gravitational potential.In the present paper, we adopt the Galaxy model suggestedby Paczynski (1990, hereafter P90). Note that his modelcan well describe the three main components of the Galaxy,i.e., the bulge, disk and halo, respectively. Especially, themodel can well reproduce the observed rotation curve.

In cylindric coordinates the density distributions of theGalactic bulge, disk and halo and their correspondingpotentials in the P90 model are

qbðbulgeÞ ¼ 3b2bMb

4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ z2 þ b2

b5

q ; ð3Þ

UbðbulgeÞ ¼ � GMbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ z2 þ b2

b

q ; ð4Þ

with Mb ¼ 1:12� 1010 M� and bb ¼ 0:277 kpc for thebulge, and

qdðdiskÞ ¼ b2dMd

4p

� � ad R2þ ad þ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2þ b2

d

q� �ad þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2þ b2

d

q� �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2þ ad þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2þ b2

d

q� �25

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2þ b2

d3

q

266664

377775ð5Þ

UdðdiskÞ ¼ � GMdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ ðad þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ b2

d

qÞ2

r ð6Þ

with Md ¼ 8:07� 1010 M�; ad ¼ 3:7 kpc and bd ¼ 0:20 kpcfor the disk, and

qhðhaloÞ ¼ qc

1þ r=rcð Þ2; Mg ¼ 4pqcr

3c ð7Þ

UhðhaloÞ ¼ GMg

rc

1

2ln 1þ r2

r2c

� �þ rc

rarctan

rrc

� �ð8Þ

with Mg ¼ 5:0� 1010 M�; rc ¼ 6:0 kpc, and r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ z2

pfor the halo, respectively. Here R is the distance to the Galac-tic center in the Galactic plane and z is the distance perpen-dicular to the Galactic plane. Moreover, the Galacticdistance of the Sun is taken as 8 kpc. The predicted rotationcurve of P90 is shown in Fig. 3. As can be seen, P90 can wellreproduce the main observational features of the Galaxy.

3.2. Two-body relaxation

Since the number density of member stars within a GCis very high, stellar encounters happen frequently, which

leads to the energy exchange among cluster stars. Starswithin a GC with energy larger than the cluster bindingenergy will escape from the cluster, i.e., the mass of a GCwill decrease with time. This mechanism is named as two-body relaxation. The mass loss dMc;ev due to this mecha-nism for a GC of mass Mc within the external tidal fieldis given by (Baumgardt, 1998; Aarseth and Heggie, 1993)

dMc;ev ¼ �nevMcdttrh; nev ¼ 0:016

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ aev

rh

rt

� �3s

; ð9Þ

with the half-mass relaxation time scale trh as (Spitzer andHart, 1971)

trh ¼ 0:138

ffiffiffiffiffiffiffiMcp

r3=2h

< m� >ffiffiffiffiGp

lnð0:4NÞ: ð10Þ

Here G is the gravitational constant, rh is the half-mass ra-dius, < m� >¼ 0:41 M� is the mean mass of the clusterstars (Baumgardt, 1998), N is the number of cluster stars,

aev ¼ 14:9 (Baumgardt, 1998) and rt ¼ Mc3MG

� �1=3

RG is the

tidal radius of the cluster (Binney and Tremaine, 1987) withMG being the mass of the Galaxy within the Galactocentricdistance of the cluster RG.

3.3. Tidal shock

When a cluster encounters the bulge or disk with time-scale shorter than its internal dynamical time, the clusterstars will gain energy and speed up the evaporation of clus-ter stars. Such an interaction is referred to be the tidalshock (Spitzer, 1987). The first- and second-order energiesenhanced for individual cluster stars due to the bulge shockare (Gnedin and Ostriker, 1997; Gnedin et al., 1999a)


hDEib ¼2GMb

vpR2p

!2r2

h

3AðxÞvðRpÞkðe;RpÞ ð11Þ

and

hDE2ib ¼2GMb

vpR2p

!20:8GMcrh

9AðxÞvðRpÞkðe;RpÞ ð12Þ

respectively, where Mb is the bulge mass, vp is the clustervelocity at the perigalactic distance Rp, r and v are therms (root mean square) of positions and velocities of clus-ters stars with a given energy E, and e is the eccentricityof cluster orbit. Here, A(x) is the adiabatic correction fac-tor for cluster stars in different positions, x ¼ xðrÞs is theadiabatic parameter, xðrÞ is the star orbital frequency, sis the timescale of the shock. Usually,

AðxÞ ¼ ð1þ xÞ�c ð13Þ

with c ¼ 2:5 for x 6 1:0 and c ¼ 1:5 for x > 1:0 (Gnedinand Ostriker, 1999); vðRpÞ is the correction factor for tak-ing the bulge as a point source which can be found in detailin the reference (Gnedin et al., 1999b); k is the correctionfactor for the tidal force along an elliptic orbit of the clus-ter. Following Aguilar et al. (1988),

kðe;RpÞ ¼MðRaÞMðRpÞ

1� e1þ e

� �3

� 1

" #2

ð14Þ

with MðRaÞ and MðRpÞ the Galactic mass inside perigalacticand apogalactic distances of cluster respectively. Definethat the characteristic timescale of the bulge shock by tak-ing into account the first- and second-order energy changestsh is

1

tsh¼ 1

tsh;1þ 1

tsh;2; ð15Þ

tsh;1 ¼jEhj

dEh=dt¼ P

jEhjhDEib

; ð16Þ

tsh;2 ¼E2

h

dE2h=dt¼ P

E2h

hDE2ib; ð17Þ

where jEhj � 0:2GMc=rh is the total energy of a cluster gi-ven by virial theorem (Spitzer, 1987), P is the cluster orbitperiod. Then the mass loss of a cluster dMc;sh due to tidalshock of the bulge can be written as (Fall and Zhang, 2001)

dMc;sh ¼ �Mcdttsh: ð18Þ

For the tidal shock caused by the disk, the first- and sec-ond-order energies enhanced for individual cluster starsare (Gnedin et al., 1999a)

hDEid ¼gm

vd

� �22r2

h

3AðxÞ ð19Þ

and

hDE2id ¼gm

vd

� �21:6GMcrh

9AðxÞ ð20Þ

respectively. Here gm is the maximum vertical gravitationalacceleration produced by the disk, vd is the vertical compo-nent of the cluster velocity in relation to the disk. Similar tothe bulge, we can also estimate the tidal shock caused bythe disk based on Eqs. (15)–(17), except that the period P

now must be modified to be the time for a cluster passingthrough the Galactic disk.

3.4. Dynamical friction

When a GC moves in the Galaxy, it loses its orbitalenergy and angular momentum due to dynamical frictioncaused by the background and will spiral towards theGalactic center. Assuming that the velocity distributionof background particles is isotropic with the velocity dis-persion rv, the deceleration due to the dynamical frictionwhich acts on a cluster can be expressed as (Binney andTremaine, 1987)

adf ¼ �4pG2Mcq lnðKÞjvcj3

erfðX Þ � 2Xffiffiffipp e�X 2

� �vc

( )ð21Þ

with erfðX Þ the error function and X ¼ vc=ffiffiffi2p

rv

, where

vc is the velocity of the cluster relative to the backgroundparticles. K ¼ bmax=bmin with bmax and bmin being the maxi-mum and minimum of impact parameters, respectively. q isthe spatial density of background particles. The dynamicalfrictions caused by the bulge, the disk and the halo of Gal-axy are all taken into account in the present paper.

3.5. Stellar evolution

The fourth mechanism we consider is that the cluster masswill decrease with increasing time due to the stellar evolu-tion, which will be very significant during the early evolu-tionary epoch. Because the star formation timescale for aGC is much shorter than its evolved timescale, we can rea-sonably assume that all stars within a cluster formed instan-taneously at the same time. We adopt the Starburst99(Leitherer et al., 1999) model given the Salpeter stellar initialmass function with the upper limit of the stellar mass to be100 M� assuming that the metallicity of a GC is Z =0.001. Note that the mass loss of a GC is not significant after100 Myr because massive stars with M P 8 M� haveevolved already.

4. Results

4.1. A single globular cluster

To clarify, the predicted evolution of a single GC is dis-cussed firstly in this subsection. The initial physical inputsfor a cluster include its mass, position and velocity. For


simplicity, we only discuss the variety of masses, positionsas illustrations with a given initial velocity as follows.

Taking the ratio of cluster mass to its initial mass M=M0

as an indicator of its evolution, we show in Fig. 4 theevolution of a cluster with time under the consideration ofdifferent dynamical mechanisms given its mass to be104 M�; 105 M� and 106 M�, the position of ðR; zÞ ¼ð3:0 kpc;0Þ, and the circular velocity of 150 km s�1, respec-tively, as initial conditions.

It can be found from the figure that evaporation of clus-ter stars due to two-body relaxation is more crucial for lessmassive clusters. Tidal shock is more important for themore massive clusters. Under the consideration of stellarevolution, a significant fraction of mass ð� 30%Þ will loosefor a cluster in the early evolution stage. As expected, thedynamical friction plays an important role for the evolu-tion of massive clusters. A cluster with mass of 106 M� will,for instance, rapidly spiral into the Galactic center after� 5 Gyr with the consideration of only “eva” and dynam-ical friction which is shown in the left figure of the lowerpanel in Fig. 4.

0

0.2

0.4

0.6

0.8

1

eva eva

0 5 100

0.2

0.4

0.6

0.8

1

t (Gyr)0 5

t (G

Fig. 4. The evolution of a cluster in the Galaxy with the consideration of dmechanisms of two-body relaxation, tidal shock and stellar evolution, respeconsidering the dynamical friction respectively.

Furthermore, given a cluster with the initial mass of106 M� and initial velocity of 150 km s�1, its evolutionwith time at different initial position is shown in Fig. 5. Itcan be found from the figure that except the stellar evolu-tion, all the other three dynamical mechanisms are now lessimportant when clusters locate more distant from theGalactic center.

4.2. GC population

For any given set of initial distributions of the positions,velocities and masses of the GCs, we can randomly gener-ate a sufficiently large GC sample to investigate their evo-lution with time in the Galaxy under the consideration ofthe four mechanisms discussed in the previous section byMonte Carlo simulations over 12 Gyr which is consistentwith the typical age of GCs in the Galaxy (Carrettaet al., 2000). In order to minimize the statistical fluctua-tions, we have run more than 2000 times for individual setsof initial conditions. The best-fit model of GC population

+sh eva+sh+evo

10yr)

0 5 10t (Gyr)

ifferent dynamical mechanisms with“eva”, “sh” and “evo” denoting thectively. The upper and lower panels are the situation with and without

0

0.2

0.4

0.6

0.8

1

eva eva+sh eva+sh+evo

0 5 100

0.2

0.4

0.6

0.8

1

t (Gyr)0 5 10

t (Gyr)0 5 10

t (Gyr)

Fig. 5. The evolution of a cluster with the initial mass 106 M� and velocity of 150 kms�1 at different initial positions, where the notations of individuallines are as in Fig. 4.


in the Milky Way are obtained by comparing the predictedglobal features, i.e., LF and spatial distribution of mod-elled GC population, with the observational results shownin Section 2.

The brief prescriptions of the initial conditions we adoptare summarized as follows.

� The initial radial distribution of GCs in the presentpaper is parameterized as Eq. (1) with a and Rc beingfree parameters, which is similar to Djorgovski andMeylan (1994);� For the initial velocity distribution, two typical situa-

tions are considered:(1) isotropic with a constant dispersion � 120 km s�1

which is consistent with current observations, andwith a systematic rotation velocity 50 km s�1 inde-pendent of position which is consistent with Frenkand White (1980);

(2) anisotropic with the velocity distribution from Edd-ington’s anisotropic velocity distribution (Binneyand Tremaine, 1987);

� Two kinds of initial mass function of GCs areconsidered:

(1) a power-law as

/ðMÞ / M�b; ðMl < M < MuÞ; ð22Þ

where Ml; Mu and b are free parameters;(2) a lognormal function as Eq. (2) with median M� and

dispersion rM being parameters.

The following three criteria are adopted in the simula-tion to define that a cluster will no longer be a GC duringits evolution (destruction).

� whenever the mass of a cluster is less than 103M�;� whenever the ratio of the tidal to half-mass radius of a

cluster is less than 2;� when a cluster locates at R < 0:1 kpc and it will not

move out of this region.

For a power-law initial mass function, the correspondingparameters for the best-fit model are a ¼ 4:5; Rc ¼1:5 kpc; b ¼ 1:5; Ml ¼ 104 M� and Mu ¼ 107 M�, respec-tively. The yielded radial distribution and mass function ofGCs at present day is shown in Fig. 6. It can be found thata power-law initial mass function of GCs can evolve into a

1 10 100

-8

-6

-4

-2

0

2 R (kpc)

2 3 4 5 6 7 8

0

1

2

3

4Log(M)

2 3 4 5 6 7 80

1

2

3

4

Log(M)2 3 4 5 6 7 8

0

1

2

3

4

Log(M)

Fig. 6. The predicted radial distribution and mass function of GCs (solid) at present day for the best-fit model with a power-law initial mass function. Thedotted and solid lines are the initial and current distributions with the dashed lines denoting the best-fitted exponential form.


lognormal mass function successfully after 12 Gyr. The pre-dicted spatial distribution of GCs in the Galaxy is initiallysteeper. With the decreasing masses due to the four mecha-nisms implemented, the total number of GCs also decreaseswith the increasing time as well. Only � 1=6 of the initialGCs have survived after 12 Gyr. For further comparison,we plot in the figure the results in the regions of R > 5 kpcand R < 5 kpc respectively since they initially have almostthe same number of GCs. It also can be concluded that thenumber of GCs decreases more seriously in the inner regionthan that in the outer region, since the destruction of clustersduring their evolution in the inner region is more significant.

For a Gaussian initial mass function, the correspondingparameters for the best-fit model are a ¼ 4:5; Rc ¼1:5; M� ¼ 1:5� 105 M� and r ¼ 0:8, respectively. The pre-dicted radial distribution and the mass function of GCs atpresent day is also shown in Fig. 7. As expected, the log-normal initial mass function can certainly result a lognor-mal mass function for GCs at present day. Same as thatof the power-law initial mass function, the number ofdestroyed clusters in the inner region is larger than thatof the outer region.

5. Conclusions

In the present paper, four major dynamical mechanismshave been considered for the evolution of GCs in the Gal-axy. It can be found that the stellar evaporation induced bytwo-body relaxation is the dominant process for low-mass

clusters and the other three are more important for massiveones, especially when clusters locate in the inner region.

Taking a large variety range of initial conditions, i.e.,model parameters, we run many simulations to investigatethe global evolution of globular clusters in the Galaxy. Themain observed features of GC population, such as massfunction and spatial distribution, are adopted to comparethe model predictions with observations. It is found thatthe best-fit simulations of both the power-law and lognor-mal initial cluster mass functions can well match theobserved present-day space distribution and the mass func-tion. This implies that we cannot make any crucial conclu-sion on the profile of the initial GCMF only based on theirobserved ones today. Moreover, the spatial distribution ofGCs will change with time. For instance, the characteristicradius Rc and a for the GC distribution are larger initiallythan that of present-day ones.

Because of the limitation of the paper, it is worthy listingfurther conclusions of the simulations which are notincluded above. We find that after evolution of several Gyrthe main features of the mass function, i.e., the profile, willchange to be lognormal-like and keep almost unchanged.The dispersion of the modelled GCMF mainly depends onthe model of the host galaxy but the peak is very insensitive.It implies that the peaks of LFs for GCs should be the samefor different galaxies but the dispersions could be different,which is consistent with current observations.

For the power-law initial GCMFs, the peak values ofthe resulted GCMFs are sensitive to the lower limit Ml ofcluster’s mass adopted initially. For the best-fit model, it

1 10 100

-8

-6

-4

-2

0

2 R (kpc)

2 3 4 5 6 7 8

0

1

2

3

4Log(M)

2 3 4 5 6 7 80

1

2

3

4

Log(M)

2 3 4 5 6 7 80

1

2

3

4

Log(M)

Fig. 7. Same as Fig. 6 except that the initial mass function of GCs is lognormal.


is � 104 M�, which is consistent with the lower limitobtained from the dynamical analysis for the giant molec-ular clouds to form observed GCs (Elmegreen, 1993).Meanwhile, the power index b for an exponential initialGCMF will influence the peak value after evolution. Itshould be between 1.5 and 2 for the best-fit model whichis consistent with the recent observations in starburst gal-axies (Zhang and Fall, 1999; Whitmore et al., 1999).

Finally, for the best-fit models of both power-law andlognormal initial GCMFs, there are about 107 M� withinthe Galactocentric distance R < 0:1 kpc today contributedfrom the evolution of GCs. This could relate to the growthof the central black hole. On the other hand, about3� 108 M� of field halo stars were original from globularclusters.

Acknowledgements

We thank the referees for helpful discussions. This pro-ject is partly supported by the Chinese National ScienceFoundation Nos. 10878003 and 10778725, 973 ProgramNo. 2007CB 815402, Shanghai Science Foundations andLeading Academic Discipline Project of Shanghai NormalUniversity (DZL805).

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