Sergey Mashchenko and Alison Sills- Globular Clusters with Dark Matter Halos. I . Initial Relaxation

8/3/2019 Sergey Mashchenko and Alison Sills- Globular Clusters with Dark Matter Halos. I . Initial Relaxation

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Navarro et al. (1997, hereafter NFW ) or Burkert (1995) prolesand have structural parameters taken from cosmological sim-ulations. We use the proto-GC model from Mashchenko & Sills(2004, hereafter MS04) to set up the initial nonequilibriumconguration of stellar clusters inside DM halos. In MS04 weshowed that the collapse of homogeneous isothermal stellar spheres leads to the formation of clusters with surface bright-ness proles very similar to those of dynamically young Ga-lactic GCs. In this model, all the observed correlations betweenstructural and dynamical parameters of Galactic GCs are ac-curately reproduced if the initial stellar density i,*and velocitydispersion i,*have the universal values i; 14 M pc3 and i; 1:91 km s 1.This paper is organized as follows. In x2 we describe our method of simulating the evolution of a hybrid GC and list the physical and numerical parameters of our models. In x3 weshow the results of the simulations. Finally, in x4 we discuss theresults and give our conclusions.

2. MODEL

2.1. Initial Considerations

In our model, GCs with a stellar mass m*are formed at thecenter of DM halos with a virial mass mDM at a redshift of z 7.We x the mass ratio m=mDM to 0.0088. The adoptedvalue of is between the universal baryonic-to-DM densityratio b= DM 0:20 (Spergel et al. 2003) and the fraction of baryons in GCs in the modern universe, 0.0025 (McLaughlin1999). This leaves enough room for such effects as a < 100%efciency of star formation in proto-GCs, mass losses due tostellar evolution (through supernovae and stellar winds), a de-cay of GC systems due to dynamical evolution of the clusters inthe presence of tidal elds, and any biased mechanism of GCformation (e.g., when GCs are formed only in DM halos with a low specic angular momentum, as in Cen 2001).

2.2. DM HalosWe consider a at CDM universe ( m 1), with thevalues of the cosmological parameters m 0:27 and H 71 km s 1 Mpc 1 (Spergel et al. 2003). In a at universe, the

critical density can be written as

c( z ) 3 H 2

8 G m(1 z )3 1 m : 1

The virial radius of a DM halo with a virial mass mDM is then

r vir 1

1 z 2mDM G z m c H 2 m

1=3

; 2where the spherical collapse overdensity c and the matter density of the universe in units of critical density at the red-shift of z , z m, are given by (Barkana & Loeb 2001)

c 18 2 82 x39 x2 3and

z m 1

1 mm(1 z )3 !

1: 4

Here x z m 1.We consider two types of DM density proles: NFW halosand Burkert halos. The NFW model describes reasonably well

DM halos from cosmological CDM simulations. It has a cuspyinner density prole with a slope of 1 and a steeper thanisothermal outer density prole,

N(r ) 0; N

r =r s (1 r =r s)2; 5

where

0; N mDM4 r 3 s

ln(1 c) c

1 c !1

: 6Here r s and c r vir =r s are the scale radius and concentration of the halo, respectively. Burkert halos, on the other hand, have a at core. Their outer density prole slope is identical to NFWhalos ( 3). This model ts well the rotational curves of disk galaxies (Burkert 1995; Salucci & Burkert 2000) and hasthe density prole

B(r ) 0; B

(1 r =r s)1 (r =r s)2; 7

where

0; B mDM2 r 3 s

ln (1 c) 12

ln (1 c 2) arctan c !1

: 8The concentration c of cosmological halos has a weak de-

pendence on virial mass (with less massive halos being moreconcentrated on average). Sternberget al. (2002)give the follow-ing expression for the concentration of low-mass DM halos in CDM cosmological simulations at z 0, which was obtainedfrom the analysis of halos with virial masses of 10 8 1011 M :c 27(mDM =109 M )0:08 . Combining this expression with theresult from Bullock et al. (2001) that the concentration scaleswith a redshift as (1

z )1 , we obtained the following formula

for the concentration of low-mass halos at different redshifts:

c 27

1 z mDM

109 M 0:08

: 9Zhao et al. (2003a, 2003b) showed that CDM halos do not haveconcentrations smaller than $3.5; hence equation (9) becomesinvalid for halos with c P 4. Sternberg et al. (2002) demon-strated that equation (9) describes very well concentrationsof four dwarf disk galaxies from Burkert (1995) tted by a Burkert prole, so it can be used for both NFW and Burkert halos.

2.3. Stellar Cores

The initial nonequilibrium conguration of proto-GCs wasmodeled after MS04 as a homogeneous isothermal stellar sphere with an isotropic Maxwellian distribution of stellar ve-locities. Stellar clusters of different mass m*initially had thesame universal values of stellar density and velocity dispersion

i;14 M pc3 and i;1:91 km s 1 (MS04). As in MS04,we use a mass parameter to describe different proto-GCmodels:

m10 3i;375

4 i;G 3

1=2

; 10where G is the gravitational constant. The connection between

the initial virial parameter and is 2 K =W 102 =3

,

MASHCHENKO & SILLS244 Vol. 619


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where K *and W *are initial kinetic and potential energies of the stellar cluster, respectively.We use models from MS04 to plot in Figure 1 thedependence

of the fraction of stars that form a gravitationally bound cluster after the initial relaxation phase of a proto-GC on the mass parameter (or the virial parameter ). Please note that this isthe case of no DM (bare stellar cores). As one can see, there arethree different regimes of the initial relaxation phase: (1) Hotcollapse ( 0:7 P P 0:35 or 3 k k 1:7) results in a sub-stantial loss of stars for which the initial velocity was above theescape speed for the system; only the slowest moving stars col-lapse to form a bound cluster. (2) Warm collapse ( 0:35 P P 0:85 or 1 :7 k k 0:27) is very mild and nonviolent andresults in virtually no escapers. (3) Cold collapse ( k 0:85or P 0:27) is violent and leads to an increasingly larger frac-tion of escapers for colder systems. In the last case, the natureof the unbinding of stars is very different from the hotcase. Dur-ing a cold collapse, the radial gravitational potential is wildlyuctuating. At the intermediate stages of the relaxation, thecentral potential of the cluster becomes much deeper than the

nal, relaxed value. As a result, a fraction of stars is acceleratedto speeds that will exceed the escape speed of the relaxed clus-ter. Analysis shows that most of the escapers are stars that ini-tially were predominantly in the outskirts of the cluster andmoving in the outward direction.

These three regimes of the initial relaxation of proto-GCs areof a very general nature and are not restricted to the GC for-mationscenario of MS04. To illustrate this, let us write down anexpression for the virial ratio of a homogeneous isothermalsphere, which is applicable to both the prestarburst gas phaseof a forming GC and the initial stellar conguration after thestarburst takes place:

5hV 2i R3GM

: 11Here M and R are the mass and the radius of the system, and

hV 2i is the mean-square speed of either gas molecules or stars.Assuming that newly born stars have the same velocity disper-sion as star-forming gas, equation (11) suggests the three fol-lowing physical scenarios resulting in our hot, warm, and coldstellar congurations (in all scenarios we start with an adiabat-ically contracting gas cloud of a Jeans mass, which correspondsto gas 1): (1) nonefcient star formation with a subsequent loss of the remaining gas leads to our hot case; (2) almost 100%efcient star formation results in the warm case; and (3) run-away cooling of the whole cloud on a timescale shorter than thefree-fall time, leading to high-efciency star formation, resultsin our cold case.

2.4. Physical Parameters of the Models

In this paper, we consider three initial stellar congurations,which can be considered to be typical hot, warm, and cold cases(see Fig. 1):

0:6 (model H, for hot),

0:4 (model W,

for warm), and 1:4 (model C, for cold). We study the re-laxation of stellar cores in either NFW (subscript N; e.g.,W N) or Burkert (subscript B) DM halos, which makes a to-tal of six different combinations of stellar cores and DM halos.The physical parameters of these six models are summarized inTable 1.

The stellar masses min our models cover the whole rangeof GC masses (see Table 1). The initial stellar radius r is muchsmaller than the scale radius of the DM halo r s in all the models.The masses of DM halos mDM range from 10 6 to 10 8 M . FromFigure 6 of Barkana & Loeb (2001), these DM halos correspondto 11.5 uctuations collapsing at the redshift z 7. As one

Fig. 1.Fraction of stars remaining gravitationally bound after the initialrelaxation phase in the MS04 proto-GC models. Ringed circles mark modelswith 0:6 (H), 0:4 (W ), and 1:4 (C).

TABLE 1Physical Parameters of the Models

Model m

( M )r

(pc)

(Myr)r ,min(pc)

m DM( M ) c

r vir (pc)

r s(pc)

r h, DM(pc)

DM(Myr) S

W N................... 0.4 0.54 8 :8 ; 104 11.2 0.49 2.98 10 7 4.88 885 181 395 52 4.7WB ................... 0.4 0.54 8 :8 ; 104 11.2 0.49 2.98 10 7 4.88 885 181 436 61 117C N .................... 1.4 0.12 8 :8 ; 105 24.2 0.078 1.57 10 8 4.06 1906 470 892 56 5.8CB .................... 1.4 0.12 8 :8 ; 105 24.2 0.078 1.57 10 8 4.06 1906 470 997 66 173H N.................... 0.6 2.5 8 :8 ; 103 5.21 17 4.14 10 6 5.87 411 70 174 48 3.7HB .................... 0.6 2.5 8 :8 ; 103 5.21 17 4.14 10 6 5.87 411 70 190 55 78

Note. Here and r are the initial virial ratio (assuming that there is no DM) and radius of the stellar core; is the crossing time at the half-mass radius for therelaxed, purely stellar models from MS04 rescaled to i;14 M pc3 and i;1:91 km s 1; r ,min is the minimum attained half-mass radius for relaxing stellar clusters from MS04 (rescaled to the above values of i, and i, ); r h, DM and DM are the half-mass radius and the crossing time at the half-mass radius for DM halos;and S m=mDM (r ) is the ratio of the stellar mass to the mass of DM within the initial stellar radius r .

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can see, halos in this mass range are populous enough at z $ 7to account for all observed GCs.In all our models, stars initially dominate DM in the core

[S m=mDM r > 1; see Table 1],whichcan be considered asa desirable property for star formation to take place. It is in-structive to check whether the above condition ( S > 1) holdsfor other plausible values of the GC formation redshift z andstellar mass fraction . In Figure 2 we show the redshift dependence of the minimum mass of a stellar core in a DM halo,satisfying the condition S ! 1, for a few xed values of :0.0025, 0.0088 (our ducial value), and 0.20. Both NFW ( thick lines ) and Burkert ( thin lines ) cases are shown. As one can see,only in the extreme case of an NFW halo at z 20 with a frac-tion of total mass in stars 0:0025 is the minimum mass of a dominant stellar core m; min 1:7 ; 103 M approaching themass range of GCs. The conclusion that we arrive at is that, for all plausible values of z and , stellar cores with the universalinitial density i;14 M pc3 and GC masses dominate DMat the center of DM halos.

2.5. Setting g Up N-Body ModelsDM halos in our models are assumed to have an isotropic

velocity dispersion tensor. To generate N -body realizationsof our DM halos, we sample the probability density function(PDF; Widrow 2000)

P ( E ; R) / R2( E )1=2 F ( E ): 12Here and E are the relative dimensionless potential and particle energy in units of 4 G 0; Nr 2 s (for NFW halos) and

2G 0; Br 2 s (for Burkert halos), F ( E ) is the phase-space distribution function, and R r =r s is the dimensionless radial co-ordinate of the particle. The dimensionless potential is equalto 1 at the center and 0 at innity. For NFW halos, R

1

ln(1 R). For Burkert halos, the corresponding expression

is given in Appendix A (eq. [A1]). We use the analytic ttingformulae from Widrow (2000) to calculate F ( E ) for the NFWmodel and derive our own tting formulae for the Burkert pro-le (see Appendix A). Our models are truncated at the virialradius r vir (eq. [2]): F ( E ) 0 for r > r vir . As we show in x3.2,this truncation results in an evolution in the outer DM den-sity prole (with the prole becoming steeper for r 3 r s) that should not affect our results.

We sample the PDF (eq. [12]) in two steps: (1) A uniformrandom number x 20; 1 is generated. The radial distance R isthen obtained by solving numerically one of the two followingnonlinear equations:

ln (1 R) R=(1 R) xln(1 c) c=(1 c) 13for the NFW prole and

ln(1 R) ln(1 R2) =2 arctan R xnln(1 c) ln(1 c 2) =2 arctan co 14

for the Burkert prole. (2) Given R, we can calculate

. In-stead of equation (12), we can now sample the PDF

P ( E ) / E

1=2 F ( E ) F ( )

: 15As F ( E ) is a monotonically increasing function and 0 E ,the inequality 0 1 holds. Two uniform random numbers, y20; 1 and E 20; , are generated. If y , we accept thevalue of the dimensionless energy E ; otherwise, we go back tothe previous step (generation of y and E ).

The velocity module can be obtained from the equation for the total energy of a particle, E 2=2 . Finally, two spher-ical coordinate angles and are generated for both radius andvelocity vectors using the expressions cos 2t 1 and 2 u (here t and u are random numbers distributed uniformly between 0 and 1).

The above method to generate N -body realizations of either NFW or Burkert halos does not use a local Maxwellian approx-imation to assign velocities to particles. Instead, it explicitlyuses phase-space distribution functions, which was shown to bea superior way of setting up N -body models (Kazantzidis et al.2004).

Stellar cores are set up as homogeneous spheres at the center of DM halos: R r x1=3 , where x 20; 1 is a uniformly dis-tributed random number. We use equal-mass stellar particles.The velocities of the particles have a Maxwellian distribution.The components of the radius and velocity vectors are gener-ated in the same fashion as for DM particles.

2.6. Numerical Parameters of the Models

In addition to the six models from Table 1, which consist of a live DM halo plus a stellar core, we also ran two more modelswith a static DM potential (marked with the letter S at the end).The numerical parameters for these eight models are listed inTable 2. In this table we also list parameters for the three prop-erly rescaled stars-only models from MS04 (H 0, W0, and C 0),which are used as reference cases.

To run our models, we use a parallel version of the multi-stepping tree code GADGET-1.1 (Springel et al. 2001). Thecode allows us to handle separately stellar and DM particles,

with each species having a different softening length: *and

Fig. 2.Redshift dependence of theminimum mass of a stellarcore,with theuniversal initial density i;14 M pc3, that dominates DM at the center of a DM halo. Thick lines correspond to stellar cores in NFW halos, and thin linescorrespond to clusters in Burkert halos. Long-dashed, solid, and short-dashedlines correspond to 0:0025, 0.0088, and 0.2, respectively. [ See the elec-tronic edition of the Journal for a color v ersion of this g ure .]



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DM , respectively. We slightlymodied the codeby introducingan optional static DM potential (either NFW or Burkert).

The values of the softening lengths were chosen to becomparable with the initial average interparticle distance. Weused the expression for the stellar particles 0:77r h N 1=3(Hayashi et al. 2003), where the half-mass radius r h was mea-sured at the initial moment of time (so r h r =21=3). The sameexpression was used to calculate DM for the H N,B models,whereas forthe models W N,B and C N,B we used halfof the valuegiven by the equation from Hayashi et al. (2003) to achieve a better spatial resolution within the stellar core.

Aarseth et al. (1988) showed that collapsing homogeneousstellar spheres witha nonnegligiblevelocity dispersion i,*do not experience a fragmentation instability (and, as a consequence, preserve the orbital angular momentumof stars)if the followingcondition is met: N k m=(r 2i;)

3 , where N *is the number of stars. For our models, this adiabaticity criterion can be rewrittenas

log N k 2 3 log5 : 16As we discussed in MS04, if real GCs indeed started off as iso-thermal homogeneous stellar spheres, they should have col-lapsed adiabatically. We also showed that if both the conditionin equation (16) and P 0:25r ; min are met, the results of sim-ulations of collapsing stellar spheres do not depend on the number of particles N *and the softening length *(here r ,min is theminimum half-mass radius of the cluster during the collapse).By comparing Tables 1 and 2, one can see that the values of N *and *that we use in our simulations meet both the aboveconditions.

The evolution time t 2 in our runs isat least 3 times longer thanthe crossing time for DM, DM , and hundreds or even thousandsof times longer than the crossing time forstellar particles *(seeTables 1 and 2). As we see in x3, this time is long enough for stellar and DM density proles to converge. On the other hand,it is short enough to avoid signicant dynamical evolution inthe stellar clusters caused by encounters between individual particles.

The individual time steps in the simulations are equal to(2 / a )1/2 , where a is the acceleration of a particle and the pa-

rameter controls the accuracy of integration. We used a very

conservative value of 0:0025 and set the maximum pos-sible individual time step t max to either 20 or 0.2 Myr (seeTable 2). As a result, the accuracy of integration was very high: E tot P 0:06%, where E tot is the maximum deviation of thetotal energy of the system from its initial value. One has to keepin mind, however, that numerical artifacts are the most pro-nounced in the central, densest area, where the frequency of strong gravitational interaction between particles is the highest.We estimate the severity of these effects by looking at the totalenergy conservation in our purely stellar models from MS04;in models H 0, W0, and C 0 the values of E tot are $0.05%,$0.85%, and $0.8%, respectively (for model W 0 we used a larger value of 0:02; hence the relatively large errors). Asone can see, although the numerical artifacts in the dense stellar cluster are more visible than in the DM + stellar core case, themagnitude of these effects is still reasonably low.

For completeness sake, we also give the values of other code parameters that control the accuracy of simulations: ErrTol-Theta=0.6 , TypeOfOpeningCriterion=1 , ErrTolForce-Acc=0.01 , MaxNodeMove=0.05 , TreeUpdateFrequency=0.1 , and DomainUpdateFrequency=0.2 (please see the codemanual 1 for explanation of these parameters). The code wascompiled with the option DBMAX enabled, which allowed it to use a very conservative node-opening criterion.

3. RESULTS OF SIMULATIONS

3.1. General Remarks

For each of our models, we generated 100200 individualtime frame snapshots. The last 20%60% of the snapshots (cor-responding to the time interval t 1 t 2; see Table 2) were used tocalculate the properties of relaxed models, including radial pro-les and the different stellar cluster parameters listed in Table 3.In allsnapshots, we removed in an iterative mannerthe escapers[particles whose velocity exceeds the escape velocity V esc 2

1=2 , where is the local value of the gravitational po-tential]. This procedure affected the models from MS04 (H 0 ,W0, and C 0) and DM particles only in the models presentedhere. One of the reasons for removing escapers was to make theresults of the simulations directly comparable with the results

TABLE 2Numerical Parameters of the Models

Model N

(pc) N DMDM

(pc)t 1

(Myr)t 2

(Myr) t max( Myr) Note

a

W0 .................. 10 5 0.15 . . . . . . 190 370 0.3 1W NS ............... 10 4 0.30 . . . . . . 95 190 0.2 2W N.................. 10 4 0.30 10 6 1.5 200 250 0.2 . . .WBS................ 10 4 0.30 . . . . . . 95 190 0.2 2WB .................. 10 4 0.30 10 6 1.5 150 250 0.2 . . .C0 ................... 10 5 0.32 . . . . . . 56 92 0.3 1C N................... 10 5 0.32 5 ; 105 4.7 130 190 0.2 . . .CB ................... 10 5 0.32 5 ; 105 5.2 95 190 0.2 . . .H0 ................... 10 5 0.069 . . . . . . 1300 1800 0.3 1H N................... 10 4 0.15 5 ; 105 1.7 1500 3700 20 . . .HB ................... 10 4 0.15 5 ; 105 1.8 1500 3700 20 . . .

Note. Here N and N DM are the number of stellar and DM particles, and DM are the softening lengths for starsand DM, and t 1, t 2, and t max are the moment of time when most model parameters converge to their quasisteady-state values, the total evolution time, and the maximum value for individual time steps, respectively.

a (1) Models from MS04 rescaled to i;14 M pc3 and i;1:91 km s 1 (stars only, no DM); (2) static DM potential.

1

Available at http://www.mpa-garching.mpg.de/gadget /.



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from Paper II, where this procedure is applied to discount par-ticles stripped by tidal forces.

The radial proles shown in this section were obtained byaveraging the corresponding proles from individual snapshotsin the time interval t 1 t 2. For each prole, we only show the part that is sufciently converged (the dispersion between dif-ferent snapshots is very small). For projected quantities (suchas surface brightness V and line-of-sight velocity dispersion

r ) we use the projection method described in Appendix B,which produces much less noisy radial proles than a straight-forward projection onto one plane or three orthogonal planes.This is crucial for the analysis of the properties of the outer,low-density parts of stellar clusters, which is the main purposeof this study.

The virial ratio for our models is consistent with unitywithin the measurement errors at the end of the simulations. Allthe global model parameters (listed in Table 3) converge to their nal values by t t 2. Similarly, radial density and velocity dispersion proles converge to their nal form over a wide range of radial distances. We explicitly checked that the evolution time t 2is more than three crossing times at the largest radius for thestellar density proles shown in Figures 3, 5, and 7. All of theabove considerations let us conclude that the results presentedhere are collisionless steady-state solutions.

3.2. Warm Collapse

In the absence of DM, an isothermal homogeneous stellar sphere with a mass parameter

0:4 relaxes to its equilibrium

TABLE 3Parameters for Relaxed Stellar Clusters

Modelr h,(pc)

c(km s1)

c

( M pc3) Rh,(pc)

Rhb(pc)

0(km s1)

V (mag arcsec 2)

r 0(pc) f 0

( M L1)r

(pc)r m

(pc)

W0 ....................... 4.64 4.07 310 3.64 2.77 3.84 18.71 3.01 0.226 1.43 . . . . . .W NS .................... 4.51 4.36 360 3.50 2.63 4.11 18.63 2.96 0.253 1.60 . . . . . .W N....................... 4.08 4.83 500 3.16 2.31 4.55 18.39 2.78 0.279 1.80 7.74 19.7WBS..................... 4.67 4.05 310 3.64 2.81 3.83 18.73 2.98 0.238 1.43 . . . . . .WB ....................... 4.65 4.12 350 3.61 2.54 3.89 18.65 2.86 0.229 1.52 17.8 54.7C0 ........................ 2.55 14.4 1 :3 ; 104 2.03 1.52 13.6 15.31 1.64 0.241 1.43 . . . . . .C N ........................ 4.50 14.3 1 :1 ; 104 3.54 1.55 13.4 15.45 1.75 0.166 1.55 10.0 44.0CB ........................ 4.50 14.4 1 :3 ; 104 3.52 1.49 13.5 15.30 1.62 0.168 1.42 27.6 125H0 ........................ 20.2 0.54 0.71 14.9 7.13 0.51 24.28 8.29 0.154 1.64 . . . . . .H N........................ 5.47 1.57 25 4.20 2.79 1.56 21.44 4.06 0.329 2.88 . . . . . .HB ........................ 11.5 0.97 4.8 8.84 4.43 0.97 22.70 5.71 0.212 2.23 10.4 18.2

Note. Here r h,, c , and c are the half-mass radius, central velocity dispersion, and central density of stellar clusters; Rh,, Rhb , 0 , andV are the projected

half-mass radius, half-brightness radius (where the surface brightness is equal to one half of the central surface brightness), projected central dispersion, and centralsurface brightness (see eq. [17]); r 0 and f 0 are the King core radius r 0 9 2c =(4 G c)

1=2 and the fraction of the total stellar mass inside r 0; is the apparent centralmass-to-light ratio (see eq. [18]); r is the radius at which the density of DM and stars becomes equal; and r m is the radius at which the enclosed mass of DM becomes equal to that of stars.

Fig. 3.Radial density proles for warm collapse models for ( a ) the case of an NFW halo and ( b) the case of a Burkert halo. Thick lines correspond to DMdensity; thin lines show stellar density. Solid lines correspond to the cases of a live DM halo + a stellar core (models W N and W B), long-dashed lines depict analyticDM proles and a relaxed stellar cluster prole in the absence of DM (model W 0), and dotted lines show stellar density proles for static DM models ( W NS andWBS). Vertical long-dashed, short-dashed, and dotted lines mark the values of DM , r m , and r s , respectively. [ See the electronic edition of the Journal for a color v ersion of this g ure .]



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state with virtually no stars lost (see Fig. 1). This makes it an in-terestingcase to test the result from Peebles (1984), that a stellar cluster inside a static constant-density DM halo can acquire a radial density cutoff similar to that expected from the action of tidal forces of the host galaxy, for the case ofliveDM halos withcosmologically relevant density proles and initially nonequi-librium stellar cores.

In Figure 3 we show the radial density proles for a relaxedstellar core and a DM halo both for an NFW model (Fig. 3 a)and a Burkert model (Fig. 3 b) for the case of a warm collapse(W models). Here long-dashed lines show the density prolesfor stars in the absence of DM ( thin lines ) and DM in the ab-sence of stars ( thick lines ). Solid lines, on the other hand, showthe density proles for stars and DM having been evolved to-gether. We also show as dotted lines the stellar radial density proles for the case of a static DM potential.

As can be seen from Figure 3, in the absence of DM (modelW0) the relaxed stellar cluster has a central at core and close toa power-law outer density prole. The radial density prole hasalso a small dent outside of the core. It is not a transient fea-ture (in the collisionless approximation), as the virial ratio for

the whole cluster is equal to unity within the measurement er-rors ( 1:001 0:005 for t t 1 t 2), and the evolution timeis >7 crossing times even at the very last radial density prole point. As we show in Paper II, collisional long-term dynami-cal evolution of a cluster will remove this feature, bringing theoverall appearance of the density prole closer to that of the ma- jority of observed GCs. The central stellar density in the modelW0 is $3 orders of magnitude larger than the central density of DM in the undisturbed Burkert halo and becomes comparableto the density of the undisturbed NFW halo only at very smallradii r P 0:1 pc (which is outside of the range of radial distancesin Fig. 3 a ).

In the case of a live DM halo coevolving with the stellar core,the DM density prole is signicantly modied within the

central area dominated by stars (see Fig. 3). In both NFW andBurkert halos, DM is adiabatically compressed by the potentialof the collapsing stellar cluster to form a steeper slope in theDM radial density prole. The slope of the innermost part of theDM density prole is 1:5 for the NFW halo and 1:0for the Burkert halo. The lack of resolution does not allow usto check whether the slope stays the same closer to the center of the DM halos.

In the outer parts of the DM halos (beyond the scale radius r s,which is marked by vertical dotted lines in Fig. 3), the DMdensity prole becomes steeper than the original slope of 3. This can be explained by the fact that our DM halo modelsare truncated at a nite radius r vir and hence are not in equi-librium. It should bear no effect on our results, as in all our

models stellar clusters do not extend beyond r s.As one can see in Figure 3, the impact of the presence of a liveDM halo on the stellar density prole is remarkably small (especially for the Burkert halo). In the presence of DM, the centralstellar density becomes somewhat larger ( by $60% and $10%for the NFW and Burkert halos, respectively; see Table 3).

The most interesting effect is observed in the outer parts of the stellar density proles, where the slope of the proles be-comes signicantly larger than that of a purely stellar cluster,which is similar to the result of Peebles (1984). The density prole starts deviating from the prole for the model W 0 some-where between the radius r and the radius r m (where the en-closed masses for stars and DM become equal; see Table 3).

There is a simple explanation for the steepening of the stel-

lar density prole seen in Figure 3 around the radius r m. In the

absence of DM, warm collapse of a homogeneous stellar sphere(our model W 0) is violent enough to eject a number of stars intoalmost-radial orbits, forming a halo of the relaxed cluster. Inthe case when the DM halo is present, the dynamics does not change much near the center of the cluster, where stars are thedominant mass component. Around the radius r m the enclosedDM mass becomes comparable to that of stars. As a result,gravitational deceleration, experienced by stars ejected beyondr m , more than doubles, resulting in a signicantly smaller number of stars populating the outer stellar halo at r > r m. Thisshould lead to a signicantly steeper radial density prole be-yond r m , as observed in our W N,B models.

As in the Peebles (1984) model, in our model the steepeningof the stellar density prole is caused by the fact that at largeradii the gravitational potential is dominated by DM. The prin-cipal difference between the two models is that in Peebles (1984)the stellar cluster is assumed to be isothermal and in equilib-rium; our clusters are not in equilibrium initially, and their nalequilibrium conguration is not isothermal (stars are dynami-cally colder in the outskirts of the cluster).

To allow a direct comparison of the model results with ob-

servedGCs, in Figure 4 we plot both the surface brightness (

V ) prole (Fig. 4 a) and the line-of-sight velocity dispersion ( r ) prole (Fig. 4 b) for the warm collapse models. The surface brightness V in units of mag arcsec 2 is calculated using theformula

V V 2:5log ( = GC ) 2 2 log (3600 ; 180 = ) :17

Here V 4:87 mag is the absolute V -band magnitude of theSun, is the projected surface mass density in units of M pc2,and GC is the assumed V -band mass-to-light ratio for baryonsin GCs. For GC we adopt the observationally derived averagedvalue of 1.45 from McLaughlin (1999), which is also between

the two stellar synthesis model values (1.36 for the Salpeter and 1.56 for the composite [with a zero slope for stellar masses< 0.3 M ] initial mass functions) of Mateo et al. (1998).

As can be seen in Figure 4, the presence of a DM halo in-troduces an outer cutoff in the surface brightness radial proleof the stellar cluster, which makes the prole look very similar to a King model prole. Interestingly, a stellar cluster evolvingin a live NFW DM halo acquires even steeper outer density and brightness proles than for the case of a static DM potential (seeFigs. 3 a and 4 a).

Analysis of the line-of-sight velocity dispersion proles for warm collapse models (Fig. 4 b) shows that the presence of a DM halo slightly inates the value of r in the core of the clus-ter (by P 20%; see Table 3). The outer dispersion prole either

stays unchanged (as in the case of the Burkert halo) or becomesslightly steeper (for the NFW model). The static DM models present proles of an intermediate type.

A slight increase of r in the central part of the cluster can bemisinterpreted observationally as a GC with no DM that has a slightly larger value of the baryonic mass-to-light ratio . Toquantify this effect, we apply a core-tting (or Kings) methodof nding the central value of in spherical stellar systems(Richstone & Tremaine 1986):

9 20

2 GI 0 Rhb: 18

Here I 0 is the central surface brightness in physical units. We

assume here that I 0 0= GC , where 0 is the projected surface



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mass density at the center of the model cluster. The core-ttingmethod assumes that the mass-to-light ratio is independent of radius, which is obviously not true for our stars + DM models.In our models the stellar particles have the same mass, so thereis no radial mass segregation between high- and low-mass starscausedby thelong-term dynamical evolutionof thecluster. As a result, equation (18) should not be used to compare our modelswith real GCs. Instead, we use it to check whether there is a

change in the apparent mass-to-light ratio

between our purelystellar models andthe modelscontaining DM. We list the valuesof for different models in Table 3. As one can see, in the caseof the NFW halo (model W N ) the presence of the DM halo leadsto a $25% increase in the value of the apparent mass-to-light ratio. Such a small increase is well within the observed disper-sion of values for Galactic GCs (Pryor & Meylan 1993). Inthe case of the Burkert halo, the apparent central mass-to-light ratio is only $6% larger than for the purely stellar model.

3.3. Cold Collapse

A principal difference between warm and cold collapsemodels from MS04 is that in the cold collapse case a signicant fraction of stars becomes unbound after the initial relaxation

phase (see Fig. 1). In our model C 0 with the mass parameter 1:4, the fraction of the stars lost is 30%. The radial density prole for the C 0 model is similar to that of the W 0 model (seeFig. 5). One can intuitively expect the escapers from the modelC0 to be trapped inside our models containing DM, forming a distinctive feature in the outer density and velocity dispersion proles.

As can be seen in Figures 5 and 6, our cold collapse modelsdo show such features. Both density and surface brightness pro-les become more shallow in the outer parts of the stellar clus-ters, where DM dominates stars. This is valid for both NFWandBurkert halos. For the NFW and Burkert cases, the slope of theouter stellar density prole is 2:6 and 2.2, respectively.For the purely stellar case (model C 0) the slope is 3:8, sotherelative changein the slope is 1:2 for the NFWand 1.6

for the Burkert case. This behavior is mimicked by the corre-sponding surface brightness proles (Fig. 6 a ). It is interestingto note that for the C N model the radial V prole exhibits a signicant steepening of the slope in the outermost parts of thecluster, creating the appearance of a tidal cutoff (similarly to thewarm collapse cases).

Even more pronounced are features in the line-of-sight velocity dispersion proles (Fig. 6 b). In both the NFW and

Burkert cases, there is a plateau in the radial r proles aroundthe radius r m, where DM becomes the dominant mass compo-nent. The apparent break in the surface brightness prole andaccompanying attening of the radial r prole seen in our models containing DM can be misinterpreted observationallyas the presence of extratidal stars heated by the tidal eld of thehost galaxy.

Similarly to warm collapse models, the apparent mass-to-light ratio for cold collapse models in the presence of a liveDM halo is very close to the purely stellar case C 0 (see Table 3).Interestingly, the half-mass radii for warm and cold collapsemodels with DM arealmost identical. This is consistent with the properties of the observed GCs, which have comparable half-mass radii for a wide range of cluster masses.

DM density proles for the cold collapse cases (Fig. 5) exhibit a very similar behavior to the warm collapse models in the star-dominated central region. For both NFW and Burkert halos, theinnermost slopes of the density proles become steeper ( 1:8 and 0.6, respectively) in the presence of a stellar core.The NFW DM density prole shows a break around the ra-dius r m .

As one can see in Table 3 and Figure 5, the nal stellar clus-ter half-mass radius is smaller than the DM softening length

DM $ 5 pc. We reran models C N and C B with a much smaller value of DM 1 pc to test the possibility that our results wereinuenced by the fact that DM is not resolved on the stellar cluster scale. For both C N and C B, the relaxed stellar proles arefound to be virtually identical to the cases with larger DM . In

particular, a kinkseen in Figures 5 a and 5 b around the radius

Fig. 4.Radial proles for warm collapse models of ( a) surface brightness V and ( b) line-of-sight velocity dispersion r . Thick lines correspond to NFW cases;thin lines correspond to Burkert cases. Solid lines show proles for models W N and W B, short-dashed lines correspond to models W NS and W BS, and long-dashedlines correspond to model W 0. Vertical short-dashed lines mark the values of r m. [See the electronic edition of the Journal for a color v ersion of this g ure .]



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DM is also present at the same location in the simulations withmuch smaller value of DM and is denitely not a numericalartifact. In the case of the NFW halo, the central stellar velocitydispersion becomes slightly larger, which results in somewhat larger value of 1:88. This could be in part because of ar-ticial mass segregation, which should be more pronounced inthe case of DM being signicantly lower than the optimal value(in our cold collapse models, DM particles are

$20 times more

massive than stellar particles). As a result, the actual value

could be even closer to the baryonic value. In the case of Burkert halo with DM 1 pc, the core mass-to-light ratio 1:45, which is identical to the case of no DM. We conclude that ourchoice of DM didnot affect the main results presented in thissection.

3.4. Hot Collapse

It is generally assumed that in an equilibrium star-formingcloud no bound stellar cluster will be formed after stellar winds

Fig. 5.Radial density proles for cold collapse models for ( a ) the case of an NFW halo and ( b) the case of a Burkert halo. Thick lines correspond to DMdensity; thin lines show stellar density. Solid lines correspond to the cases of a live DM halo + a stellar core (models C N and C B); long-dashed lines depict analyticDM proles and a relaxed stellar cluster prole in the absence of DM (model C 0). Vertical long-dashed, short-dashed, and dotted lines mark the values of DM , r m ,and r s , respectively. [ See the electronic edition of the Journal for a color v ersion of this g ure .]

Fig. 6.Radial proles for cold collapse models of ( a) surface brightness V and (b) line-of-sight velocity dispersion r . Thick lines correspond to NFW cases;thin lines correspond to Burkert cases. Solid lines show proles for models C N and C B; long-dashed lines correspond to model C 0. Vertical short-dashed lines mark

the values of r m. [See the electronic edition of the Journal for a color v ersion of this g ure .]



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and supernova explosions expel the remaining gas if the star formation efciency is less than 50% (which corresponds to theinitial virial parameter for the stellar cluster > 2). In MS04 weshowed that this is not true for initially homogeneous stellar clusters that have a Maxwellian distribution of stellar velocities.In such clusters with the initial virial parameter as large as 2.9

(corresponding to the mass parameter as low as 0.7), a bound cluster containing less than 100% of the total mass isformed by the slowest moving stars. (A similar conclusion wasreached by Boily & Kroupa [2003] for initially polytropic stel-lar spheres.)

In our model H 0 ( 0:6, 2:5), $57% of all starsstay gravitationally bound after the initial relaxation phase (seeFig. 1). Because in both cold and hot collapse models for purelystellar clusters a signicant fraction of stars become unbound,one might expect stellar density and velocity dispersion prolesto be similar for these two cases in the presence of DM. Anal-ysis of Figures 7 and 8 shows that this is denitely not the case;H N,B models show completely different behavior from C N,Bmodels. The explanation is simple: the C 0 model starts with a

stellar density that is already larger than the density of DMwithin the stellar core in models C N,B (in other words, S > 1;see Table 1) and after the relaxation phase reaches an evenhigher density that is $(r =r h;)3=2 430 times higher thantheinitial density. The hot collapse model H 0, on the other hand,having started with S > 1, acquires a density that is $120 timeslower than the initial density after the expansion and relaxation phase, resulting in S < 1 for the relaxed stellar cluster even for the Burkert halo. Under these circumstances, the radial prolesfor the H N and H B models should be signicantly different fromthe case with no DM (H 0), which is what is seen in Figures 7and 8. The only common feature between the models C N,B andH N,B is that in both cases all would-be escapers are trapped bythe potential of the DM halo.

In the case of the NFW halo, the central stellar density andcentral velocity dispersion become largerin the presence of DM byfactors of 35 and 3, respectively. In the Burkert halo case, theincrease is not as dramatic but still signicant: the centraldensity and the velocity dispersion become 7 and 2 times larger than those in the absence of DM, respectively. Interestingly, as

in the cold collapse cases, the presence of DM brings the half-mass radius of the stellar core in the H N,B models closer to thehalf-mass radius of the warm collapse models (see Table 3). Theapparent central mass-to-light ratio is noticeably inated bythe presence of DM (especially for the case of NFW halo) by76% and 36% for NFW and Burkert models, respectively.

As can be seen in Figure 7 a , in the model H N stars have a comparable density to DM in the core and become less densethan DM outside of the core radius r 0. The situation is different (and more in line with warm and cold collapse models) for theHB model (Fig. 7 b), where stars are the dominant (although not by a large margin) mass component in the core. There are noobvious features in the density and surface brightness prolescaused by the presence of DM: the proles look very similar to

those of the W 0 and C 0 models, which do not have DM.Similarly to the warm and cold collapse cases, the presenceof a stellar cluster makes DM denser in the central area. In thecase of the NFW halo, the innermost slope of the DM density prole is comparable to the slope 1 for the undisturbedhalo. For the Burkert halo, the slope becomes steeper, reachingthe value $ 0:4 at the innermost resolved point.The most interesting behavior is exhibited by the line-of-sight velocity dispersion proles (Fig. 8 b). In the presence of DM, the proles become remarkably at, changing by a mere0.20.3 dex over the whole range of radial distances (out to

$15 apparent half-mass radii). The combination of the lowmass, low central surface brightness, large apparent mass-to-light ratio, and almost at velocity dispersion proles seen in

Fig. 7.Radial density proles for hot collapse models for ( a ) the case of an NFW halo and ( b) the case of a Burkert halo. Thick lines correspond to DM density;thin lines show stellar density. Solid lines correspond to the cases of a live DM halo + a stellar core (models H N and H B); long-dashed lines depict analytic DM proles and a relaxed stellar cluster prole in the absence of DM (model H 0). Vertical long-dashed, short-dashed, and dotted lines mark the values of DM , r m (onlythe Burkert case is shown, as in the NFW case the enclosed DM mass is larger than that of stars at any radius), and r s, respectively. [ See the electronic edition of the Journal for a color v ersion of this g ure .]



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models H N,B can be mistakenly interpreted as a GC observed at the nal stage of its disruption by the tidal forces of the host galaxy.

4. DISCUSSION AND CONCLUSIONS

A signicant hindrance to a wider acceptance of the pri-mordial scenarios for GC origin is an apparent absence of DMin Galactic GCs. Many observational facts have been suggestedas evidence for GCs having no DM, including the presence of such features in the outer parts of the GC density prolesas apparent tidal cutoffs or breaks, relatively low values of theapparent central mass-to-light ratio that are consistent with purely baryonic clusters, the at radial distribution of the line-of-sight velocity dispersion in the outskirts of GCs believed to be a sign of tidal heating, and the nonspherical shape of theclusters in their outer parts.

Here we present the results of simulations of stellar clus-ters relaxing inside live DM minihalos in the early universe (at z 7). We study three distinctly different cases that can cor-respond to very different gasdynamic processes forming a GC:a mild warm collapse, a violent cold collapse resulting in a much

denser cluster with a signicant fraction of stars escaping theGC in the absence of DM, and a hot collapse resulting in a lower density cluster with many would-be escapers. We show that GCs forming in DM minihalos exhibit the same properties asone would expect from the action of the tidal eld of the host galaxy on a purely stellar cluster: King-like radial density cut-offs (for the case of a warm collapse) and breaks in the outer parts of the density prole accompanied by a plateau in thevelocity dispersion prole ( for a cold collapse). In addition, theapparent mass-to-light ratio for our clusters with DM is gen-erally close to the case of a purely stellar cluster. (The specialcase of a hot collapse inside a DM halo, which produces inatedvalues of , can be mistaken for a cluster being at its last stageof disruption by the tidal forces.)

We argue that the increasingly eccentric isodensity contoursobserved in the outskirts of some GCs could be created by a stellar cluster relaxing inside a triaxial DM minihalo and not by external tidal elds, as it is usually interpreted. Indeed, cos-mological DM halos are known to have noticeably nonsphericalshapes; a stellar cluster relaxing inside such a halo would haveclose to a spherical distribution in its denser part, where thestars dominate DM, and would exhibit isodensity contours of increasingly larger eccentricity in its outskirts, where DM be-comes the dominant mass component.

It is also important to remember that few Galactic GCs showclear signs of a tidal cutoff in the outer density proles (Trager et al. 1995). The tidal features of an opposite nature, breaks inthe outer parts of the radial surface brightness proles in someGCs, are often observed at or below the inferred level of con-tamination by foreground/background objects and could be,in many cases, an artifact of the background subtraction pro-cedure, which relies heavily on the assumption that the back-ground objects are smoothly distributed across the eld of view.A good example is that of the Draco dwarf spheroidal galaxy.Irwin & Hatzidimitriou (1995) used simple nonltered stellar

counts from photographicplates followedby a background sub-tractionprocedure to show that this galaxyappears to have a rel-atively small value of its radial density tidal cutoff r t 28A3 2A4 and a substantial population of extratidal stars. Odenkirchenet al. (2001) used a more advanced approach of multicolor l-tering of Draco stars from the Sloan Digital Sky Survey im-ages and achieved a much higher signal-to-noise ratio than inIrwin & Hatzidimitriou (1995). New, higher quality resultswere supposed to make the extratidal features of Draco muchmore visible. Instead, Odenkirchen et al. (2001) demonstratedthat Dracos radial surface brightness prole is very regular down to a very low level (0.003 of the central surface bright-ness) and suggested a larger value for the King tidal radius of r t 50 0.

Fig. 8.Radial proles for hot collapse models of ( a ) surface brightness V and ( b) line-of-sight velocity dispersion r . Thick lines correspond to NFW cases;thin lines correspond to Burkert cases. Solid lines show proles for models H N and H B; long-dashed lines correspond to model H 0. Vertical short-dashed lines mark the value of r m (only the Burkert case is shown, as in the NFW case the enclosed DM mass is larger than that of stars at any radius); vertical dotted lines correspondto r s. [See the electronic edition of the Journal for a color v ersion of this g ure .]



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We argue that the qualitative results presented in this paper are very general and do not depend much on the fact that weused the MS04 model to set up the initial nonequilibrium stellar core congurations or on the particular values of the model parameters (such as i,*, i,*, and ). As we discussed in x3, theappearance of tidal or extratidal features in our warm and coldcollapse models is caused by two reasons: (1) at radii r > r m the potential is dominated by DM, whereas in the stellar core the potential is dominated by stars from the beginning until the endof simulations; and (2) the collapse is violent enough to eject a fraction of stars beyond the initial stellar cluster radius.

As we showed in x2.4, for the initial stellar density i;14 M pc3 (from MS04) the whole physically plausible rangeof stars-to-DM mass ratios and GC formation redshifts z satisfy the above rst condition. For warm and cold collapse,this condition can be reexpressed as S m=mDM r > 1. Onecan easily estimate S for other values of i,*.The second condition is more difcult to quantify. In MS04we showed that collapsing homogeneous isothermal spheres produce extended halos for any values of the initial virial ratio (except for > 2:9 systems, which are too hot to form a bound

cluster in the absence of DM). Roy & Perez (2004) simulatedcold collapse for a wider spectrum of initial cluster congura-tions, including power-law density prole, clumpy, and rotatingclusters. In all their simulations an extended halo is formedafter the initial violent relaxation. It appears that in many (probablymost) stellar cluster congurations that are not in detailed equi-librium initially, our second condition can be met.

Of course, not every stellar cluster conguration will result in a GC-like object with a relatively large core and an extendedhalo after the initial violent relaxation phase. Spitzer & Thuan(1972) demonstrated that a warm ( 0:5) collapse of a homo-geneous isothermal sphere produces clusters with large cores(their models D and G). Adiabatic collapse of homogeneousisothermal spheres produces clusters in which the surface den-

sity proles are very close to those of dynamically young Ga-lactic GCs (MS04). Roy & Perez (2004) concluded that anycold stellar system that does not contain signicant inhomo-geneities relaxes to a large-core conguration. The results fromRoy & Perez (2004) can also be used to estimate the importanceof adiabaticity for core formation. Indeed, their initially ho-mogeneous models H and G span a large range of and include both adiabatic cases ( > 0:16 for their number of particles N 3 ; 104 , from eq. [16]) and nonadiabatic ones ( < 0:16).It appears that all their models (adiabatic and nonadiabatic)

form a relatively large core. The issue is still open, butit appearsthat the adiabaticity requirement (our eq. [16]) is not a veryimportant one forour problem (although this requirement is met automatically for real GCs for the values of i,*and i,*derivedin MS04).

An important point to make is that the simulations presentedin this paper describe the collisionless phase of GC formationand evolution and cannot be directly applied to GCs that haveexperienced signicant secular evolution due to encounters be-tween individual stars. In MS04 we showed that such collision-less simulations of purely stellar clusters describe very well thesurface brightness proles of dynamically young Galactic GCs(such as NGC 2419, NGC 5139, IC 4499, Arp 2, and Palomar 3;see Fig. 1 in MS04). In Paper II we will address (among other things) the issue of long-term dynamical evolution of hybridGCs. We will demonstrate that, at least for the warm-collapsecase, secular evolution does not change our qualitative results presented in this paper.

In the light of the results presented in this paper andthe abovearguments, we argue that additional observational evidence isrequired to determine with any degree of condence whether

GCs have any DM presently attached to them or whether theyare purely stellar systems truncated by the tidal eld of the host galaxy. Decisive evidence would be the presence of obvioustidal tails. A beautiful example is Palomar 5 (Odenkirchen et al.2003), where tidal tails were observed to extend over 10 in thesky.

Even if a GC is proven not to have a signicant amount of DM, it does not preclude it having been formed originally in-side a DM minihalo. In the semiconsistent simulations byBromm & Clarke (2002) of dwarf galaxy formation, proto-GCswere observed to form insideDM minihalos, with theDM beinglost during the violent relaxation accompanying the formationof the dwarf galaxy. Unfortunately, their simulations did not have enough resolution to clarify the fate of DM in proto-GCs.

In Paper II we will address the issue of the fate of DM in hybrid proto-GCs experiencing severe tidal stripping in the potentialof the host dwarf galaxy.

We would like to thank Volker Springel for his help withintroducing an external static potential in GADGET. S. M. is partially supported by SHARCNet. The simulations reportedin this paper were carried out on the McKenzie cluster at theCanadian Institute for Theoretical Astrophysics.

APPENDIX A

DISTRIBUTION FUNCTION FOR BURKERT HALOS

The dimensionless potential of Burkert halos with the density prole given by equations (7) and (8) is

1 2

(1 R1) arctan R ln(1 R) 12

(1 R1) ln (1 R2)& ': A1Here R r =r s is the radial distance in scale radius units, and is in 2G 0; Br 2 s units. The potential is equal to 1 at the center of the halo and zero at innity.

The phase-space distribution function for a Burkert halo with an isotropic dispersion tensor can be derived through an Abeltransform (Binney & Tremaine 1987, p. 237; we skip the second part of the integrand, which is equal to zero for Burkert halos):

F ( E ) 1

ffiffiffi8p 2 Z

E

0

d 2

d 2d

( E )1=2: A2

Here E is the relative energy in the same units as , and is the density in 0, B units, which results in ( r s)3 1=2

0; BG 3=2

units for F .



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The function d 2 / d 2 for a Burkert halo has the same asymptotic behavior for R 0 and R 1 as the model I from Widrow(2000), which has the density prole / (1 R)3:d 2

d 2( R 0) / R3 ;

d 2

d 2( R 1 ) /

(ln )3: A3

This allowed us to use the analytic tting formula from Widrow (2000) for the distribution function of Burkert halo,

F ( E ) F 0 E 3=2(1 E )1 ln E

1 E q

e P : A4Here P Pi pi E

i is a polynomial introduced to improve the t.We solve equation (A2) numerically for the interval of radial distances R 106 to 10 6. We obtained the following values for thetting coefcients in equation (A4):

F 0 5:93859 ; 104 ; q 2:58496 ; p1 0:875182 ; p2 24 :4945 ; p3 147 :871 ; p4 460 :351 ; p5 747 :347 ; p6 605 :212 ; p7 193 :621 : A5

We checked the accuracy of the tting formula (eq. [A4]) with the above values of the tting coefcients by comparing the analyticdensity prole of the Burkert halo (eq. [7]) with the density prole derived from the distribution function (Binney & Tremaine 1987, p. 236),

( R)

2

ffiffiffi8p 2

Z

0

F ( E )(

E )1=2 dE :

A6

The deviation of ( R) in equation (A6), with F ( E ) given by equations (A4) and (A5), from the analytic density prole was found to be less than 0.7% for the interval of the radial distances R 106 to 10 6.

APPENDIX B

PROJECTION METHOD

For N -body models of spherically symmetric stellar systems, such as the GC models presented in the paper, one can produce radial proles of projected observable quantities (such as surface mass density or velocity dispersion) by averaging these quantities over all possible rotations of the cluster relative to the observer. Radial proles created in this way preserve a maximum of information, whichis very important for studies of the outskirts of the clusters and for clusters simulated with a relatively small number of particles.

Let us consider the ith stellar particle P i in a spherically symmetric cluster, which is located at the distance r i from the center of

the cluster ( Fig. 9). We want to nd its averaged contribution to one projection bin, corresponding to the range of projected distances

Fig. 9.Scheme explaining the projection procedure. The stellar cluster is marked with its center O and a thin line as its circular boundary. The axis Z is arbitrarilychosen to point toward the observer. We show a sphere containing the ith particle P i , located at the distance r i from the center of the cluster, as a thick-lined circle. Therange of projected distances R1 R2 is shown as two horizontal shadowed bars. In three dimensions, it corresponds to space between two concentric cylinders of radii

R1 and R2 with the axis OZ as their central axis. These two cylinders cut segments out of the sphere containing the particle P i , which are marked as the four regions A, B, C , and D. In three dimensions, these four regions correspond to two segments of the sphere. For an arbitrary rotation of the cluster around its center O, the probability to nd the particle P i within the bin R1 R2 is equal to the ratio of the surface area of the segments to the total surface area of the sphere 4 r 2i .



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R1 R2. For an arbitrarily rotated cluster, the probability wi to detect our particle inside the bin (the shaded areas inFig. 9)is equal tothefraction of thesurface area of thesphere containing the particle P i cutout by twoconcentric cylinderswith the radii R1 and R2. I t iseasyto show that this probability is

wi 0; r i < R1;

1 ( R1=r i)2

1=2; R1 r i < R2;

1 ( R1=r i)2

1=2

1 ( R2=r i)2

1=2

; r i ! R2:8>>>:

B1

The projected surface mass density for the bin R1 R2 averaged over all possible rotations of the cluster is

( R1 ; R2) 1

( R22 R21)X N

i1miwi; B2

where N is the total number of stellar particles in the cluster and mi is the mass of the ith particle. The averaged value of a projectedscalar quantity Q is

Q( R1; R2) X N

i1wiQ i X

N

i1wi

1; B3

so the probability wi serves as a weight for scalar variables. For example, to calculate the square of the projected three-dimensionalvelocity dispersion 3D corresponding to the projected distance range R1 R2, one has to set Qi (V 2 x; i V 2 y; i V 2 z ; i) in equation (B3),where ( V x, i , V y, i , V z , i) are the velocity components of the ith particle. Observationally, one can determine 3D by measuring both line-of-sight velocities and two proper motion components for stars located within the interval R1 R2 of projected radial distances.

One can show that the above method is superior to a straightforward projection onto one plane or three orthogonal planes byconsidering a contribution made by one particle to the bin R1 R2. In the case of the straightforward projection, there is a high probability that the particle will make no contribution to the projection bin. In our method, every particle with r i > R1 will make a contribution to the bin, which will make the radial prole of the projected quantity signicantly less noisy. The central value of the projected quantity is obtained by setting R1 to zero in equations (B1)(B3). In this case, all particles make a contribution to theaveraged value of the quantity.

The situation with vector quantities is a bit more complicated. We designed a simplied procedure that is based on the assumptionthat the bin sizes are small (so the numberof bins is large). Each particlewith an associated vectorquantity(say, velocity vector) is rst rotated around the OZ axis to bring the particle into the plane XOZ . Next, the particle is rotated around the axis OY to make the

projected radial distance of the particle equal to the radial distance of the center of the bin R ( R1 R2)=2. (The observer is assumedto be at the end of the axis OZ .)The detailed procedure is as follows. For a particle with coordinates ( x, y, z ), after the two rotations the associated vector ( V x, V y, V z )

is transformed into the vector ( V 00 x ; V 00 y ; V 00 z ) with the components

V 00 x V 0 xz sin 0; V 00 y V xy sin 0; V 00 z V 0 xz cos 0: B4The parameters V 0 xz , V xy, 0, and 0can be obtained after a series of the following calculations:

r ( x2 y2 z 2)1=2 ; r xy ( x2 y2)1=2; V xy (V 2 x V 2 y)1=2 ; sin V y=V xy;cos V x=V xy; sin y=r xy; cos x=r xy; 0 ;

V 0 xz (V 2 xy cos 2 0V 2 z )1=2 ; sin V xy cos 0=V 0 xz ; cos V z =V 0 xz ;sin r xy=r ; cos z =r ; 0 arcsin : B5

Here is equal to R/ r if R=r < 1 and is equal to 1 otherwise; r , r xy, V xy, , , , and are intermediate variables.If the vector ( V x, V y, V z ) is the velocity vector, then substituting V 002 z in place of Q i in equation (B3) gives us the value of the square

of the line-of-sight velocity dispersion r , averaged over all possible rotations of the cluster, for the bin R1 R2. Similarly, replacing Q iwith V 002 x and V 002 y produces the averaged values of the square of the radialand tangential components of theproper motion dispersion,respectively.

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Sergey Mashchenko and Alison Sills- Globular Clusters with Dark Matter Halos. I . Initial Relaxation