THE ASTRONOMICAL JOURNAL, 117 :1792È1815, 1999 April1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

SPACE VELOCITIES OF GLOBULAR CLUSTERS. III. CLUSTER ORBITS AND HALO SUBSTRUCTURE

DANA I. DINESCU,1 TERRENCE M. GIRARD, AND WILLIAM F. VAN ALTENA

Department of Astronomy, Yale University, New Haven, CT 06520-8101 ; dana=astro.yale.eduReceived 1998 August 10 ; accepted 1999 January 6

ABSTRACTWe have compiled a catalog of absolute proper motions of globular clusters from various sources. The

sample consists of 38 clusters, from which most of the southern ones (15 clusters) were measured in ourprevious papers in this series. We have integrated orbits assuming two di†erent Galactic potentialmodels adopted from the literature and have calculated orbital parameters. The uncertainties associatedwith the orbital parameters were derived in a Monte Carlo approach, and we conclude that, overall, atthe present level of measurement errors, orbital di†erences due to Galactic potential models are notsigniÐcant.

Three metal-poor clusters are found to have orbits similar to prototypical metal-rich disk clusters.These clusters are NGC 6254 (M10), NGC 6626 (M28), and NGC 6752. We interpret this as a poten-tially signiÐcant constraint on the formation of the disk. It is thus possible that part of the inner metal-poor halo is the low-metallicity tail of the thick disk. In this case, the ages of these clusters indicate thatthe formation of the disk partially overlapped with that of the halo. The clusters classiÐed as ““ younghalo ÏÏ or ““ red horizontal-branch ÏÏ by Zinn show a radially anisotropic velocity distribution, their orbitsare of high total energy, with apocentric radii larger than 10 kpc and highly eccentric. In this sense theymay represent an accreted component of our Galaxy. We also discuss u CenÏs orbit characteristics in theview of an accreted origin.

We investigate the e†ect of the orbital motion on the internal dynamics of clusters. Adopting the for-malism from Gnedin & Ostriker and their destruction rates due to two-body relaxation, we Ðnd that, inmost cases, this internal process is more important than the destruction processes due to disk and bulgeshocking. Hubble Space Telescope (HST ) observations argue that NGC 6397Ïs luminosity function isdepleted at the faint end, and this is blamed on its high total destruction rate. We propose a list ofclusters with similar destruction rates that may also have depleted luminosity functions. We also notethe bias toward deriving higher destruction rates in studies that statistically assign tangential velocitiesbased on a kinematic model of the globular cluster system, in contrast to the rates derived from themeasured tangential velocities. Clusters prone to such biases are those that have circular orbits(kinematically thick-disk clusters) and some of those with orbits of high total energy.Key words : Galaxy : halo È Galaxy : structure È globular clusters : general

1. INTRODUCTION

Many physical properties of the globular clusters are, bynow, well determined, and signiÐcant progress in under-standing the globular cluster system and the Galaxy hasbeen made based on these properties (see, e.g., reviews byHesser 1995 and Majewski, Phelps, & Rich 1996). However,tangential velocities of globular clusters have long been amissing piece of information for these systems. The purposeof this paper, which is the third in our series, is to analyzethe present available data regarding tangential velocities ofglobular clusters and to investigate its limitations. In theprevious papers (Dinescu et al. 1997, 1999, hereafter PapersI and II, respectively) we have presented in more detail themotivation of such work and the variety of methods used tomeasure absolute proper motions. Therefore, here webrieÑy mention the issues we have investigated in this paper.These are the following : the kinematics of the globularcluster system as derived from the three components ofvelocities, types of orbits, and cluster subsystems as identi-Ðed from metallicity and horizontal-branch morphology,possible cluster streams, and the input of orbital motioninto the internal dynamics of clusters.

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ1 Current address : Department of Astronomy, University of Virginia,

P.O. Box 3818, Charlottesville, VA 22903.

Recent progress in the Ðeld of absolute proper motions ofglobular clusters has signiÐcantly enlarged the sample ofwell-measured clusters to 38, compared with the 26 clustersanalyzed by Dauphole et al. (1996). Our contribution some-what balances the initial northern-biased sample, with agroup of 15 southern clusters (Papers I and II). Althoughthe sample has increased, it is still rather small, especially ifone wishes to investigates cluster subsystems (see ° 7.1). Thissituation can be improved only by programs designed tomeasure absolute proper motions of clusters in a very effi-cient way, as opposed to the traditional ground-based,many-plate, long time baseline approach. Along with theproper motions, more accurate distances (good to a fewpercent, see ° 6) are also a concern if Galactic potentialmodels and cluster streams are to be investigated. Futurespace interferometric missions such as ESAÏs GlobalAstrometric Interferometer for Astrophysics (GAIA)(Tucholke & Brosche 1995) and NASAÏs Space Interferome-ter Mission (SIM) (NASA 1998) are very promising in thisregard.

We introduce our cluster sample and the collection ofdata in the following section. Section 3 analyzes the kine-matics of the sample. In ° 4, we present our Galactic poten-tial models, followed by orbital parameter determinationsand associated uncertainties (° 5). Di†erences in orbitalparameters due to di†erent potential models are presentedin ° 6. The orbital parameters are analyzed with respect to

1792

SPACE VELOCITIES OF GLOBULARS. III. 1793

halo subdivisions in ° 7, and in ° 8 we discuss the implica-tions for the GalaxyÏs formation. The orbital motionÏs e†ecton the internal dynamics is presented in ° 9. Our conclu-sions are summarized in ° 10.

2. THE CLUSTER SAMPLE

We have compiled a list of 38 clusters with well-measuredtangential velocities, and this is presented in Table 1. Themeasurements were chosen to be relatively modern (laterthan 1980, most of them made with high-precision measur-ing machines) and with formal errors less than D2 masyr~1. The correction to absolute proper motion is based ona variety of methods : tied directly to galaxies and quasi-stellar objects (Majewski & Cudworth 1993 ; Schweitzer,Cudworth, & Majewski 1993 ; Guo 1995 ; Scholz, Odenkir-chen, & Irwin 1993, 1994 ; Scholz et al. 1996, 1998 ; Paper I ;Paper II) ; using a kinematic model for the reference stars(Cudworth & Hanson 1993) ; using other calibrators such asstars with known absolute proper motion from the LickNorthern Proper Motion Program (Brosche et al. 1991), orSMC stars and a correction for the SMCÏs motion(Tucholke 1992a, 1992b) ; and Ðnally, Hipparcos stars(Odenkirchen et al. 1997 ; Paper II). Small changes betweenthe results presented here and the ones in Paper I for clus-ters NGC 288, 1851, 6362, 6584, and 6752 are due to theslightly di†erent transformation from plate coordinates toequatorial coordinates adopted in Paper II ; these changesare well below the formal error estimates. The resultsadopted here from Papers I and II are those that include thecorrections for Ðeld gradients in the mean motion of thereference systems (see Paper I, ° 6). For NGC 104 and NGC362 we have adopted, as one of the contributing measure-ments, the determinations made by Tucholke (1992a,1992b) with respect to SMC stars and adopted the correc-tion for the SMC motion from Kroupa & Bastian (1997,their Table 2, the combination of three determinations). Inour Table 1, some of the measurements of a single clustermay have used common plate material due to some overlapbetween studies such as Tucholcke (1992a, 1992b), Broscheet al. (1991), Ge†ert et al. (1993), and Odenkirchen et al.(1997). However, the correction to absolute proper motionis determined independently, using di†erent calibrators.Since, usually, it is this number that contributes most of themeasurement error, we have considered these values asindependent determinations.

The proper motions we use in the orbit integration areunweighted averages of the measurements listed in Table 1for each cluster. The formal error estimates adopted foreach cluster are the uncertainties in the average calculatedusing the error estimates quoted for each measurement [i.e.,if there are three measurements with error estimates v1, v2,and then the uncertainty of the average isv3, (v12] v22We therefore rely on the quoted uncertainties] v32)1@2/3].to derive a formal error estimate, but we do not rely onthem as weights in the averages, since potential systematicsin measurements of underestimated errors would signiÐ-cantly alter the mean result. Most of the clusters, though,have a single measurement, and thus it is difficult to judgethe actual uncertainty ; in these cases we simply rely on thesingle determination and its quoted error estimate. In anumber of cases in which several determinations arepresent, the formal uncertainty in the mean, which is basedon quoted individual error estimates, is smaller than thatgiven by the dispersion of all measurements, at least in one

coordinate. We did not use, however, the standard devi-ation as our estimated error of the mean, because of smallnumber statistics, and because, in cases with one measure-ment, we had to rely on the published uncertainty. Thus, theerror estimates for all clusters are somewhat on a similarsystem, which basically relies on the formal individual errorestimates provided by each study. In terms of disagreementbetween the formal error estimate based on quoted individ-ual uncertainties and that based on the standard deviationof the measurements, four cases are outstanding : NGC5272 (M3), NGC 5904 (M5), NGC 7078 (M15), and Pal 5. Inthese cases the di†erences between various measurementsare not accounted for by their formal error estimates, andpresumably systematic errors dominate. Therefore, careshould be taken when interpreting some of the results forthese clusters, since the proper-motion errors are underesti-mated and propagated into underestimated errors in theorbital parameters. For these clusters the orbit will bepoorly constrained, and reÐned work such as of therelationship between the orbit and the internal dynamics ofthe cluster cannot be done reliably for these cases. However,some derived information is still useful, such as the approx-imate total energy of the cluster and the crude shape of theorbit.

For Pal 5 we have adopted the average between the lasttwo measurements, where K. M. Cudworth (1998,unpublished) is an improvement over the same groupÏs Ðrstmeasurement (Schweitzer et al. 1993), quoted by Scholz etal. (1998). For NGC 6121 (M4) we have adopted only themeasurement of Paper II, although Cudworth & Hanson(1993) also have a measurement for this cluster. The mainsource of disagreement between the two studies is the cor-rection to absolute proper motion (see detailed discussionin Paper II). We believe that for the Ðeld of NGC 6121,which is heavily reddened, the correction based on the Hip-parcos system (Paper II) may be more reliable than a sta-tistically adopted correction for the mean motion of thereference stars, and thus we have adopted the former deter-mination.

Table 2 lists the adopted absolute proper motions, metal-licities, distances from the Sun, and Galactocentric radii (R.Zinn 1995, private communication). Heliocentric radialvelocities are from Pryor & Meylan (1993) and Harris(1996a). Local standard of rest (LSR) velocities were deter-mined adopting the solar motion ([11.0, 14.0, and 7.5 kms~1) from Ratnatunga, Bahcall, & Casertano 1989 wherethe U component is positive outward from the Galacticcenter. Velocities in a cylindrical coordinate system with itsorigin at the Galactic center were also determined by adopt-ing a solar circle radius of kpc and a rotationR0\ 8.0velocity of the LSR of km s~1. The % com-#0\ 220.0ponent is positive outward from the Galactic center, # ispositive in the direction of Galactic rotation, and W is posi-tive toward the north Galactic pole. Errors in velocitiesinclude errors in the proper motions, radial velocities, andan adopted 10% error in the distance.

Although recent papers such as Reid (1997) and Chaboy-er et al. (1998) claim that the error in the distance to globu-lar clusters has a systematic nature primarily because of thenew distance scale set by Hipparcos parallax measurementsof local subdwarfs, we have chosen not to explore the e†ectof a systematic error in the distance on the orbits. There aretwo main reasons for our choice. First, the exact amount ofsystematic shift is not precisely known, and it varies with

1794 DINESCU, GIRARD, & VAN ALTENA

TABLE 1

ABSOLUTE PROPER MOTIONS

ka cos d kdCluster (mas yr~1) (mas yr~1) Reference

NGC 104 (47 Tuc) . . . . . . 6.43 ^ 2.10 [2.99 ^ 2.11 Tucholke 1992a ] Kroupa & Bastian 19973.40 ^ 1.70 [1.90 ^ 1.50 Cudworth & Hanson 19937.00 ^ 1.00 [5.30 ^ 1.00 Odenkirchen et al. 1997

NGC 288 . . . . . . . . . . . . . . . . . 4.68 ^ 0.20 [5.25 ^ 0.19 Guo 19954.67 ^ 0.42 [5.95 ^ 0.41 Dinescu et al. 1997

NGC 362 . . . . . . . . . . . . . . . . . 4.43 ^ 1.02 [3.99 ^ 1.04 Tucholke 1992b ] Kroupa & Bastian 19975.70 ^ 1.00 [1.10 ^ 1.00 Odenkirchen et al. 1997

NGC 1851 . . . . . . . . . . . . . . . 1.28 ^ 0.68 2.39 ^ 0.65 Dinescu et al. 1997NGC 1904 . . . . . . . . . . . . . . . 2.12 ^ 0.64 [0.02 ^ 0.64 Dinescu et al. 1999NGC 2298 . . . . . . . . . . . . . . . 4.05 ^ 1.00 [1.72 ^ 0.98 Dinescu et al. 1999Pal 3 . . . . . . . . . . . . . . . . . . . . . . 0.33 ^ 0.23 0.30 ^ 0.31 Majewski & Cudworth 1993NGC 4147 . . . . . . . . . . . . . . . [2.70 ^ 1.30 0.90 ^ 1.30 Brosche et al. 1985, 1991

[1.00 ^ 1.00 [3.50 ^ 1.00 Odenkirchen et al. 1997NGC 4590 (M68) . . . . . . . . [3.76 ^ 0.66 1.79 ^ 0.62 Dinescu et al. 1999NGC 5024 (M53) . . . . . . . . 0.50 ^ 1.00 [0.10 ^ 1.00 Odenkirchen et al. 1997NGC 5139 (u Cen) . . . . . . [5.08 ^ 0.35 [3.57 ^ 0.34 Dinescu et al. 1999NGC 5272 (M3) . . . . . . . . . [3.10 ^ 0.20 [2.30 ^ 0.40 Scholz et al. 1993

0.90 ^ 1.00 [2.20 ^ 1.00 Odenkirchen et al. 1997NGC 5466 . . . . . . . . . . . . . . . [5.40 ^ 1.30 0.30 ^ 1.30 Brosche et al. 1983, 1991

[3.90 ^ 1.00 1.00 ^ 1.00 Odenkirchen et al. 1997Pal 5 . . . . . . . . . . . . . . . . . . . . . . [2.44 ^ 0.17 [0.87 ^ 0.22 Schweitzer et al. 1993

[2.55 ^ 0.17 [1.93 ^ 0.17 K. M. Cudworth 1998, unpublished[1.00 ^ 0.30 [2.70 ^ 0.40 Scholz et al. 1998

NGC 5897 . . . . . . . . . . . . . . . [4.93 ^ 0.86 [2.33 ^ 0.84 Dinescu et al. 1999NGC 5904 (M5) . . . . . . . . . 5.20 ^ 1.70 [14.20 ^ 1.30 Cudworth & Hanson 1993

6.70 ^ 0.50 [7.80 ^ 0.40 Scholz et al. 19963.30 ^ 1.00 [10.10 ^ 1.00 Odenkirchen et al. 1997

NGC 6093 (M80) . . . . . . . . [3.31 ^ 0.58 [7.20 ^ 0.67 Dinescu et al. 1999NGC 6121 (M4) . . . . . . . . . [12.50 ^ 0.36 [19.93 ^ 0.49 Dinescu et al. 1999NGC 6144 . . . . . . . . . . . . . . . [3.06 ^ 0.64 [5.11 ^ 0.72 Dinescu et al. 1999NGC 6171 (M107) . . . . . . [0.70 ^ 0.90 [3.10 ^ 1.00 Cudworth & Hanson 1993NGC 6205 (M13) . . . . . . . . [0.90 ^ 1.00 5.50 ^ 2.00 Cudworth & Hanson 1993

[0.90 ^ 1.00 5.50 ^ 1.00 Odenkirchen et al. 1997NGC 6218 (M12) . . . . . . . . 1.60 ^ 1.30 [8.00 ^ 1.30 Brosche et al. 1991

3.10 ^ 0.60 [7.50 ^ 0.90 Scholz et al. 1996[0.80 ^ 1.00 [8.00 ^ 1.00 Odenkirchen et al. 1997

NGC 6254 (M10) . . . . . . . . [6.00 ^ 1.00 [3.30 ^ 1.00 Odenkirchen et al. 1997NGC 6341 (M92) . . . . . . . . [4.60 ^ 1.10 [0.60 ^ 1.80 Cudworth & Hanson 1993

[4.40 ^ 0.70 1.10 ^ 0.40 Scholz et al. 1994[0.90 ^ 1.00 [1.50 ^ 1.00 Odenkirchen et. al. 1997

NGC 6362 . . . . . . . . . . . . . . . [3.09 ^ 0.46 [3.84 ^ 0.46 Dinescu et al. 1997NGC 6397 . . . . . . . . . . . . . . . 3.30 ^ 0.50 [15.20 ^ 0.60 Cudworth & Hanson 1993NGC 6584 . . . . . . . . . . . . . . . [0.22 ^ 0.62 [5.97 ^ 0.67 Dinescu et al. 1997NGC 6626 (M28) . . . . . . . . 0.30 ^ 0.50 [3.40 ^ 0.90 Cudworth & Hanson 1993NGC 6656 (M22) . . . . . . . . 8.60 ^ 1.30 [5.10 ^ 1.50 Cudworth & Hanson 1993NGC 6712 . . . . . . . . . . . . . . . 4.20 ^ 0.40 [2.00 ^ 0.40 Cudworth & Hanson 1993NGC 6752 . . . . . . . . . . . . . . . [0.69 ^ 0.42 [2.85 ^ 0.45 Dinescu et al. 1997NGC 6779 (M56) . . . . . . . . 0.30 ^ 1.00 1.40 ^ 1.00 Odenkirchen it et al. 1997NGC 6809 (M55) . . . . . . . . [1.42 ^ 0.62 [10.25 ^ 0.64 Dinescu et al. 1999NGC 6838 (M71) . . . . . . . . [2.30 ^ 0.80 [5.10 ^ 0.80 Cudworth & Hanson 1993NGC 7078 (M15) . . . . . . . . [0.30 ^ 1.00 [4.20 ^ 1.00 Cudworth & Hanson 1993

[1.00 ^ 1.40 [10.20 ^ 1.40 Ge†ert et al. 1993[0.10 ^ 0.40 0.20 ^ 0.30 Scholz et al. 1996[2.40 ^ 1.00 [8.30 ^ 1.00 Odenkirchen et al. 1997

NGC 7089 (M2) . . . . . . . . . 5.50 ^ 1.40 [4.20 ^ 1.40 Cudworth & Hanson 19936.30 ^ 1.00 [5.70 ^ 1.00 Odenkirchen et al. 1997

NGC 7099 (M30) . . . . . . . . 1.42 ^ 0.69 [7.71 ^ 0.65 Dinescu et al. 1999

cluster metallicity. Thus, Reid (1997) advocates that clusterswith [Fe/H]D [1.5 require a distance increase of 6%,while those with [Fe/H]D [2.0 require 15%; however, forNGC 6752 no distance adjustment is required. Also, thedistance modulus determined for NGC 6752 by Renzini et

al. (1996), using the white dwarf sequence, does not indicatethat previous distance determinations were underestimated.Second, uncertainties in the foreground extinction remainan issue (e.g., a a†ects distances by up to 7%,*E

B~V\ 0.02

Reid 1997), and they may vary from one cluster to another,

TA

BL

E2

KIN

EM

AT

ICD

AT

A

k acos

d[F

e/H

]D

_R

GCV rad

UV

W%

#N

GC

/Pal

(mas

yr~1

)k d

(mas

yr~1

)(k

pc)

(kpc

)(k

ms~

1)(k

ms~

1)(k

ms~

1)(k

ms~

1)(k

ms~

1)(k

ms~

1)

104

....

...

5.61

^0.

96[

3.40

^0.

93[

0.71

4.3

7.3

[18

.7^

0.2

75^

20[

77^

1657

^14

17^

1916

1^

1728

8..

....

.4.

40^

0.23

[5.

62^

0.23

[1.

407.

611

.1[

46.4

^0.

416

^09

[24

7^

1852

.9^

0.4

16^

09[

27^

1836

2..

....

.5.

07^

0.71

[2.

55^

0.72

[1.

287.

88.

922

3.5

^0.

521

^27

[27

8^

22[

80^

2055

^25

[29

^24

1851

....

..1.

28^

0.68

2.39

^0.

65[

1.16

11.3

16.0

320.

5^

0.6

229

^35

[21

7^

26[

106^

3118

6^

3213

3^

2919

04..

....

2.12

^0.

64[

0.02

^0.

64[

1.68

12.2

18.1

207.

5^

0.5

120

^30

[18

8^

297^

3493

^30

83^

2922

98..

....

4.05

^1.

00[

1.72

^0.

98[

1.82

9.3

14.4

149.

4^

1.3

[62

^41

[20

8^

2197

^44

[58

^36

[26

^29

3..

....

....

0.33

^0.

230.

30^

0.31

[1.

5781

.684

.983

.4^

8.4

[26

^10

166

^81

191^

77[

247

^88

148^

9441

47..

....

[1.

85^

0.82

[1.

30^

0.82

[1.

8016

.418

.718

3.0

^1.

077

^64

[17

9^

6514

7^

1557

^64

66^

6445

90..

....

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76^

0.66

1.79

^0.

62[

2.09

8.9

9.3

[95

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0.6

188

^30

34^

225^

22[

99^

2530

0^

2850

24..

....

0.50

^1.

00[

0.10

^1.

00[

2.04

17.9

18.4

[79

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4.1

[38

^84

38^

85[

75^

16[

106

^84

238^

8551

39..

....

[5.

08^

0.35

[3.

57^

0.34

[1.

594.

96.

323

2.5

^0.

7[

64^

11[

254

^09

4^

10[

31^

10[

65^

1052

72..

....

[1.

10^

0.51

[2.

30^

0.54

[1.

669.

511

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147.

1^

0.4

[17

^24

[11

6^

24[

126^

052

^24

105^

2454

66..

....

[4.

65^

0.82

0.80

^0.

82[

2.22

15.4

15.8

107.

7^

0.3

248

^65

[13

9^

6420

7^

1925

4^

65[

60^

645

....

....

..[

1.78

^0.

17[

2.32

^0.

23[

1.47

19.9

15.5

[55

.0^

16.0

10^

18[

262

^34

[45

^17

[11

^18

42^

3458

97..

....

[4.

93^

0.86

[2.

33^

0.84

[1.

9412

.47.

410

1.7

^1.

030

^31

[30

1^

5711

8^

4349

^50

71^

4159

04..

....

5.07

^0.

68[

10.7

0^

0.56

[1.

407.

26.

151

.8^

0.5

[32

3^

31[

140

^28

[20

4^

28[

313

^31

115^

2860

93..

....

[3.

31^

0.58

[7.

20^

0.67

[1.

648.

33.

07.

3^

4.1

[12

^10

[28

5^

37[

81^

2560

^36

[27

^13

6121

....

..[

12.5

0^

0.36

[19

.93

^0.

49[

1.28

1.8

6.3

70.4

^0.

4[

57^

03[

193

^22

[8^

05[

58^

0324

^22

6144

....

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3.06

^0.

64[

5.11

^0.

72[

1.86

9.0

2.7

189.

4^

1.1

170

^09

[26

0^

3812

^28

109

^35

[13

6^

1861

71..

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[0.

70^

0.90

[3.

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1.00

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995.

93.

5[

33.8

^0.

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^11

[67

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1115

1^

2862

05..

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[0.

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0.71

5.50

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1.65

6.8

8.2

[24

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^0.

925

0^

36[

85^

26[

117^

1827

9^

32[

54^

3062

18..

....

1.30

^0.

58[

7.83

^0.

62[

1.34

4.2

4.9

43.5

^0.

6[

50^

10[

98^

17[

106^

14[

21^

1013

0^

1762

54..

....

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00^

1.00

[3.

30^

1.00

[1.

604.

14.

875

.5^

1.1

[84

^09

[87

^22

97^

19[

53^

1014

9^

2163

41..

....

[3.

30^

0.55

[0.

33^

0.70

[2.

247.

49.

1[

120.

5^

1.7

26^

23[

147

^15

32^

1870

^19

33^

1963

62..

....

[3.

09^

0.46

[3.

83^

0.46

[1.

086.

85.

0[

13.3

^0.

681

^13

[11

2^

1837

^14

[40

^16

129^

1563

97..

....

3.30

^0.

50[

15.2

0^

0.60

[1.

912.

16.

118

.9^

0.1

34^

06[

91^

12[

99^

1118

^07

133^

1265

84..

....

[0.

22^

0.62

[5.

79^

0.67

[1.

5412

.96.

522

2.9

^15

.0[

78^

25[

354

^51

[18

1^

3915

0^

4039

^40

6626

....

..0.

30^

0.50

[3.

40^

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25

1796 DINESCU, GIRARD, & VAN ALTENA Vol. 117

thus adding more uncertainty to the amount of systematicshift in distance. Therefore, we have chosen our distanceerrors to have a normal distribution with a representativestandard deviation.

For the sample of 38 clusters, the distribution of the esti-mated relative error in the total proper motion, wherevk/k,

peaks at 12%. Including a 10%k \ [(ka cos d)2] kd2]1@2,error in the distance, the relative error in the tangentialvelocity is thus D16%, while the similar ratio for radialvelocities is D1%. Thus proper-motion errors and distanceerrors give roughly similar contributions to the tangentialvelocity errors for most of the clusters in our sample. Since,overall, the tangential velocity error is much larger than theradial velocity error, it is the former that will dominate theuncertainty in the orbit.

3. THE KINEMATICS OF THE SAMPLE

We show the relation between the velocity componentsand the Galactocentric radius in Figure 1 using the datafrom Table 2. Note that Pal 3 does not show in this Ðgure,because of its large Galactocentric radius ; its velocity,however, will be considered in our following discus-sion. Di†erent symbols indicate the type of the cluster asdeÐned by metallicity and horizontal-branch morphology(Zinn 1996 classiÐcation) : asterisk for disk clusters (D,[Fe/H][ [0.8), Ðlled circles for clusters with blue horizon-tal branch at a given metallicity (BHB), open circles formetal-poor (MP) clusters ([Fe/H]\ [1.8), triangles for

FIG. 1.ÈVelocity components as a function of Galactocentric radius.Asterisks are for disk clusters ([Fe/H] [ [0.8), Ðlled circles for BHB clus-ters (see text), open circles for MP clusters ([Fe/H]\ [1.8), triangles forRHB clusters, and the star for u Cen.

clusters with red horizontal branch at a given metallicity(RHB), and the star symbol for u Cen.

We wish to remind the reader that, based on propertiessuch as spatial distribution, line-of-sight velocity dispersion,mean rotation (from radial velocities alone), and age, thesegroups are hypothesized to have di†erent origins (Zinn1993, 1996). The RHB group is believed to represent clus-ters formed in satellite galaxies that were later accreted anddisrupted by our Galaxy, since it is a system with a slightlyretrograde motion, large velocity dispersion, sphericalspatial distribution, and whose members reside at Galacto-centric radii larger than 6 kpc. The BHB group may haveformed in a more ordered fashion, possibly in a dissi-pational collapse, since it has signiÐcant rotation, is spa-tially Ñattened, and shows evidence for a metallicitygradient with Galactocentric radius (Zinn 1993). The MPgroup is kinematically distinguishable from the BHB grouponly in the inner regions of the Galaxy. In the Galactocen-tric radius range 2.7È8.0 kpc, the MP group was found to bea hotter system km s~1,(Vrot \[3 ^ 65 plos\ 139 ^ 33km s~1) than the BHB group km s~1,(Vrot \ 59 ^ 23 plos \61 ^ 33 km s~1) according to Zinn (1996). It is thereforehypothesized that the MP clusters in the inner regions arethe oldest component of the Galaxy, since, in a dissipationalcollapse, with increasing metallicity and decreasing age thesystem becomes more Ñattened and acquires more rotation.

In order to investigate the kinematics of our sample wehave calculated, in each velocity component, the velocityaverages and dispersions for various groups of halo([Fe/H]\ [0.8) clusters, and these are summarized inTable 3. Since our total cluster sample is rather small, theresults for various subgroups may be prone to smallnumber statistics, and therefore they should be interpretedwith caution.

The Ðrst row of data in Table 3 lists values for the overallhalo sample. The following two lines show the results forclusters within and beyond the solar circle, respectively. Thenext lines show the results for BHB, RHB, and MP clusters,respectively. For the BHB and MP groups, we also showresults in the inner 8 kpc of the Galaxy.

Our velocity averages are listed in the top half of Table 3for several subgroupings of clusters. For the ““ all halo ÏÏsample we Ðnd that only the # component is signiÐcantlydi†erent from zero. This rotation is primarily due to clustersin the inner regions of the Galaxy, as seen in Figure 1 andTable 3. Table 3 also shows that this rotation is signiÐcantat a 3 p level only for BHB clusters, and its size increaseswhen the BHB sample is restricted to the inner 8 kpc of theGalaxy.

The velocity dispersions for the all halo sample, margin-ally indicate that the % component is somewhat larger thanthe other two components. The actual values (see Table 3)are in good agreement with dispersions obtained from thehalo stellar component by Carney & Latham (1986), (p%,

102 ^ 27, 107^ 15) km s~1, Norrisp#, pW) \ (154^ 18,

(1986), (131^ 6, 106^ 6, 85^ 4) km s~1, and Morrison,Flynn, & Freeman (1990), (133^ 8, 98 ^ 13, 94 ^ 6) kms~1. For each velocity component, dispersions for the innergroup are statistically similar to those in the outer group,given the size of the errors. However, all three componentsof the velocity dispersion for the inner group are smallerthan those for the outer group, thus suggesting that theinner halo is a cooler system than the outer halo. An inspec-tion of the BHB and MP groups in Table 3 indicates that

No. 4, 1999 SPACE VELOCITIES OF GLOBULARS. III. 1797

TABLE 3

VELOCITY AVERAGES AND DISPERSIONS

% # WGroup N (km s~1) (km s~1) (km s~1)

Averages :All halo ([Fe/H]\[0.8) . . . . . . 36 11 ^ 23 58^ 17 [29 ^ 18Inner (RGC ¹ 8 kpc) . . . . . . . . . . . 18 5 ^ 29 69^ 23 [42 ^ 22Outer (RGC [ 8 kpc) . . . . . . . . . . . 18 17 ^ 37 48^ 26 [15 ^ 30BHB (all) . . . . . . . . . . . . . . . . . . . . . . . 14 51 ^ 28 75^ 26 [53 ^ 28BHB (RGC¹ 8 kpc) . . . . . . . . . . . . 10 28 ^ 29 114^ 24 [36 ^ 23RHB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 [40 ^ 64 58^ 25 [55 ^ 46MP (all) . . . . . . . . . . . . . . . . . . . . . . . . . 12 14 ^ 36 50^ 39 16 ^ 27MP (RGC ¹ 8 kpc) . . . . . . . . . . . . . 5 2 ^ 49 4^ 52 [29 ^ 44

Dispersions :All halo ([Fe/H]\[0.8) . . . . . . 36 138 ^ 16 104^ 12 111 ^ 13Inner (RGC ¹ 8 kpc) . . . . . . . . . . . 18 122 ^ 20 99^ 17 92 ^ 15Outer (RGC [ 8 kpc) . . . . . . . . . . . 18 155 ^ 26 110^ 18 129 ^ 22BHB (all) . . . . . . . . . . . . . . . . . . . . . . . 14 103 ^ 20 96^ 18 105 ^ 20BHB (RGC¹ 8 kpc) . . . . . . . . . . . . 10 93 ^ 21 75^ 17 74 ^ 17RHB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 192 ^ 45 74^ 17 138 ^ 33MP (all) . . . . . . . . . . . . . . . . . . . . . . . . . 12 126 ^ 26 133^ 27 94 ^ 19MP (RGC ¹ 8 kpc) . . . . . . . . . . . . . 5 117 ^ 37 109^ 35 98 ^ 31

they have isotropic velocity dispersions, while the RHBgroup shows a somewhat higher velocity dispersion for the% component than for the other two components (i.e., the %component is larger than the W component although onlyat a 1 p level). Thus the RHB group displays some amountof radial anisotropy, suggesting that a di†erent formationscenario was at work for the RHB clusters than for the BHBand MP clusters.

Zinn (1996) shows that an important di†erence betweenthe BHB and MP groups is the run of the line-of-sightvelocity dispersion with Galactocentric radius : while(plos)the MP group displays a large line-of-sight velocity disper-sion km s~1) at any Galactocentric radius, the(plosD 150BHB group has a decreasing line-of-sight velocity disper-sion with decreasing Galactocentric radius (ZinnÏs Fig. 2).Our data show that the velocity dispersions of the BHBgroup are only suggestively smaller than those of the MPgroup, at the present level of the uncertainties (Table 3) ;however, in the inner 8 kpc of the Galaxy, this trend is moreprominent. The inner kpc) BHB clusters show a(RGC¹ 8larger velocity dispersion (Table 3) than the value derivedby Zinn (1996) km s~1), and a signiÐcantly(plos \ 61 ^ 10larger mean rotation velocity than that obtained by Zinn(1996) (our value is 114^ 24 km s~1, while his is 59^ 23km s~1). While the di†erences in these numbers may not besigniÐcant, since we had only 10 clusters compared withZinnÏs 18 clusters in the same Galactocentric radius range,it is more important to anticipate the results of ° 5 andremark that these clusters have very di†erent types of orbitsas is readily seen, for some clusters, from their velocity com-ponents (Table 2). For example the velocity componentsand present positions of NGC 6626 and NGC 6752 clearlyindicate circular orbits that are rather conÐned to theGalactic disk, while those of NGC 6121 indicate an eccen-tric orbit that is inclined with respect to the Galactic plane.Our data suggest that the group of BHB clusters in theinner regions of the Galaxy is, kinematically, rather inho-mogeneous. It may be that this group is a combination ofclusters that do indeed have thick-diskÈlike kinematics(circular motions with rotation velocities close to the rota-tion velocity of the thick disk) and clusters that have halo-

like kinematics with highly inclined, eccentric orbits, butwith most of them on relatively small-size orbits (e.g., NGC6121), such that the velocity dispersion of the inner BHBsystem is relatively low. Such a system may produce thekinematic characteristics derived by Zinn (1996) for theBHB group within the solar circle : mean rotation of59 ^ 23 km s~1 and a line-of-sight velocity dispersion of61 ^ 10 km s~1. It is therefore of great importance toobtain more data for clusters in the inner Galaxy, in orderto understand its components. Whether this population hasan intermediate mean rotation velocity of D60 km s~1 or isa mixture of two populations, one with zero rotation veloc-ity and one with the rotation velocity of the thick disk(D180 km s~1), is an open question with direct implicationsfor the GalaxyÏs formation.

Our conclusions based on the kinematic data obtainedfrom the three components of the velocities are as follows :the separation of halo globular clusters into ““ youngerhalo ÏÏ or RHB clusters and ““ older halo ÏÏ or BHB clusters isconÐrmed, in the sense that the RHB system is a pressure-supported system with a large velocity dispersion, radialanisotropy, and little rotation, while the BHB system dis-plays a signiÐcant amount of rotation, with possibly a lowervelocity dispersion than the RHB system. The exact natureof the BHB group is not understood ; it may be a mixture oftwo kinematically di†erent populations. The MP clustersshow an isotropic, relatively large velocity dispersion andno rotation.

The combination of the small size of the sample togetherwith the fact that velocities represent only a snapshot of aclusterÏs motion, make the kinematic analysis of halo sub-structure a rather limited approach. In what follows we willlook at orbitsÏ shapes and sizes in order to better under-stand halo subgroups.

4. GALACTIC GRAVITATIONAL POTENTIAL MODELS

In order to determine orbital parameters we have inte-grated the orbits for two realistic, yet relatively simple ana-lytical models of the GalaxyÏs potential. We have chosentwo models in order to investigate di†erences in the orbitalparameters due to di†erent potentials. The two models are

1798 DINESCU, GIRARD, & VAN ALTENA Vol. 117

from Johnston et al. (1995, hereafter JSH95) and from(1990, hereafter P90). Both models consist ofPaczyn� ski

axisymmetric potentials with three components : bulge,disk, and dark halo. The disks are Miyamoto & Nagai(1975) potentials with di†ering coefficients. Both dark halosare modeled as logarithmic potentials, which assure a Ñatrotation curve, but imply an inÐnite mass, and thereforeclusters will never escape these potentials. The bulges aremodeled as a Hernquist (1990) potential in JSH95 and as aPlummer potential in P90. The JSH95 bulge is moremassive and less centrally concentrated than that of P90.The disks are of comparable mass, but the JSH95 disk ismore Ñattened than the P90 one.

In the orbit calculations the conserved quantities are thez-component of the angular momentum, which is essentiallydetermined from the kinematic data, and the total energy,which is model dependent. As the logarithmic potentials ofthe dark halos are speciÐed only to within an additive con-stant, the total energy has an arbitrary zero point. We haveo†set the total energy in each potential by a negative con-stant that was determined such that the escape velocity at

kpc is 500 km s~1. This o†set attempts to set aR0\ 8.0similar scale for the total energies from the two models andhas no consequences on the orbits. For convenience, wesummarize in Table 4 the analytical form of the two modelpotentials. We also list the additive constant the period('0),of the LSR, and the circular velocity at R0.It is known that, given a certain potential, some orbitsmay exhibit a chaotic behavior. Carlberg & Innanen (1987)describe these as orbits with low-z angular momentum,which pass close to the center of the Galaxy. Such orbitschange their characteristics with time, since clusters getscattered by the nuclear region and motion in the Galacticplane may shift to vertical motion, conserving the totalenergy. For a detailed discussion see Schuster & Allen(1997) and references therein. Most studies agree that theonset of chaotic behavior is for orbits that penetrate to atleast 1 kpc from the center of the Galaxy. It is possibletherefore that some orbits integrated in the described poten-tials may have chaotic orbits ; however, this e†ect is difficultto quantify and isolate when uncertainties in the measuredquantities are considered. In addition, neither of thesepotentials is a realistic description of the mass distributionwithin D1 kpc of the Galactic center, as there is nowincreasing evidence that the Milky Way is a barred spiral

galaxy (see, e.g., chapter 2 of Blitz & Teuben 1996). Such anonaxisymmetric mass distribution is likely to alter theorbits that penetrate well within 1 kpc, as well as a†ect theamount of tide-induced shock (see ° 9), as the cluster passesthrough the bulge region. We have chosen not to considerthe contribution of a bar to the potential, as we believe thatuncertainties in orbits are still dominated by uncertaintiesin the measured velocities and positions. However, for clus-ters that have pericentric radii smaller than 1 kpc, oneshould keep in mind that uncertainties related to the inac-curate description of the potential in the inner regions maybe important.

5. CALCULATION OF THE ORBITAL PARAMETERS

5.1. Orbit Integration and Estimated Errors ofOrbital Parameters

The orbit integration is performed using the potentialsjust described and initial conditions from Table 2 (see also° 2). Initial cluster positions are calculated using the Galac-tic coordinates from the catalog of Djorgovski & Meylan(1993), distances from Table 2 (R. Zinn 1995, privatecommunication), and kpc. The integration routineR0\ 8.0is a Ðfth-order Runge-Kutta with adaptive step size (Presset al. 1992, p. 708). The relative change in the total energyover a 109 yr integration time is of the order of 10~11 for theJSH95 model and 10~12 for the P90 model. Examples oforbits for the clusters NGC 104, 1851, 6121, and 6752 arepresented in Figure 2 for the P90 model. The panels on theleft show the orbit in the disk plane of the Galaxy, while thepanels on the right show the orbit in the plane perpendicu-lar to the Galactic disk and containing the Sun. Note thedi†erent scale corresponding to the size of the orbit for acluster such as NGC 6121Èwhich spends most of its time inthe inner parts of the GalaxyÈand NGC 1851, which, as ayounger halo object, has large excursions into the outerparts of the Galaxy. Also note the similarity between theorbits of NGC 104, the standard thick-disk cluster, andNGC 6752, a relatively old, metal-poor cluster.

Orbital parameters are determined as averages over thenumber of cycles in a 10 Gyr integration time. This timeinterval was chosen such that a reasonable number of cyclesare performed, and therefore representative averages anddispersions of the orbital parameters are determined for themajority of the clusters. Pal 3, for example, would require a

TABLE 4

GALACTIC POTENTIAL MODELS

Parameter JSH95 P90

'b

(bulge) . . . . . . . . . [GM

br ] c

[GM

bJR2] (a

b] Jz2] b

b2)2

Mb\ 3.4] 1010 M

_, c\ 0.7 kpc M

b\ 1.12] 1010 M

_, a

b\ 0.0 kpc, b

b\ 0.277 kpc

'd

(disk) . . . . . . . . . . . [GM

dJR2] (a

d] Jz2] b

d2)2

[GM

dJR2] (a

d] Jz2] b

d2)2

Md\ 1011 M

_, a

d\ 6.5 kpc, b

d\ 0.26 kpc M

d\ 8.07] 1010 M

_, a

d\ 3.7 kpc, b

d\ 0.20 kpc

'h

(halo) . . . . . . . . . . . v02 lnA1 ]

r2d2B GM

hdC12

lnA1 ]

r2d2B

]dr

arctanrdD

v0\ 128 km s~1, d \ 12.0 kpc Mh\ 5 ] 1010 M

_, d \ 6.0 kpc

'0 (km s~1)2 . . . . . . [5.2] 104 [12.3] 104PLSR (108 yr) . . . . . . 2.18 2.23Vc(km s~1) . . . . . . . 222.5 220.1

No. 4, 1999 SPACE VELOCITIES OF GLOBULARS. III. 1799

FIG. 2.ÈExamples of orbits integrated in the potential model P90. Note the di†erent scales, particularly for NGC 1851.

larger integration time, since it barely performs one orbit in10 Gyr. We have, however, preferred a single time interval,since it accommodates most of the clusters in our sample,and since integration times longer than a Hubble time arenot physically justiÐed, when studying the conditions thatclusters have experienced.

The number of cycles is based on various periodicities inthe quantities r, R, and z, depending on the orbital param-eter in question. For example the orbital period, P, is the

period of revolution around the z-axis as determined by ther angle. Pericentric, and apocentric, radii are deter-R

p, R

a,

mined from the minimum and maximum of (R2] z2)1@2averaged over the number of cycles. Similarly, themaximum distance above (or below) the plane, iszmax (zmin),an average over the number of plane crossings. The numberof cycles for di†erent orbital parameters varies because ofthe precession of the pericenter, and the rate of precession ispotential model-dependent (see Odenkirchen et al. 1997).

1800 DINESCU, GIRARD, & VAN ALTENA

For instance, the number of revolutions is smaller than thenumber of pericenter (apocenter) passages. The inclinationangle with respect to the Galactic plane is deÐned as ( \

where is the largest absolutesin~1 (zmaxim/rzmaxim

), zmaximvalue of and and is (R2] z2)1@2 atzmin zmax, rzmaxim

zmaxim,and the angle is averaged over the number of plane cross-ings. Eccentricities are calculated as e\ (R

a[ R

p)/

where and are averages. The error esti-(Ra] R

p), R

aR

pmates in the averaged orbital parameters are the dispersionsover the number of cycles, while for the eccentricity theerror estimate is propagated from the dispersions of andR

aThe dispersions just mentioned are indicative of theRp.

intrinsic nature of the orbit, and this may be due to e†ectssuch as chaos as described by Carlberg & Innanen (1987) orSchuster & Allen (1997), and/or a complex distribution oforbit families. In principle a large intrinsic dispersion oforbital parameters shows that orbit characteristics varyconsiderably from one revolution to another. We willexplore this issue in the following subsection.

Besides the intrinsic variation within a 10 Gyr interval,we have also determined uncertainties in the orbital param-eters due to observational uncertainties. Repeated integra-tions were performed with slightly di†erent initialconditions for each cluster. The initial conditions were gen-erated in a Monte Carlo fashion by adding Gaussian devi-ates to the observable proper motions, radial velocities, anddistances. The standard deviations of the deviates are takento be uncertainties in Table 2 for the proper motions andradial velocities and 10% for the distances. The solarmotion, radius of the solar circle, and rotation of the LSRwere kept Ðxed, so that the variation of these quantities isnot explored at the present time as we believe that proper-motion and distance errors are the dominant source of errorin the orbital parameters. For each cluster, distributions ofthe orbital parameters have been constructed based on 300separate integrations. While the distributions of z-angularmomentum and total energy resemble normal distributions,since they are directly determined from quantities that wereassumed to have a Gaussian proÐle, the distributions of theorbital parameters, in many cases, are not well representedby Gaussians. Some display bimodality, high skewness,long tails, and wide wings. In some cases the initial condi-tions are so poorly constrained that di†erent families oforbits may occur for the same cluster. Figure 3 shows exam-ples of these distributions in model P90 for the clustersNGC 104 (a well-behaved case) and for NGC 6093 (anill-behaved case).

For the following analysis, we have adopted the resultingvalue as the representative value of the orbital parameter, asif no measuring uncertainties are present (i.e., from an inte-gration with initial conditions as exactly listed in Table 2).We have chosen to adopt one-half of the interquartile rangein the orbital parameterÏs Monte Carlo distribution as ameasure of the variation in the parameter. This is a betterrepresentation than the standard deviation, which would bean overestimate for most of the distributions.

We note that, in some cases, both the derived value andits error estimate are poor representations of a particulardistribution. This is because the input initial conditions,which are deÐned by both observable quantities and theiruncertainties, lead to orbital parameter distributions thatcannot be described simply by a ““mean ÏÏ and a““ dispersion.ÏÏ Nevertheless, for consistency, we haveadopted the same deÐnition for all clusters. Table 5 lists the

orbital parameters and their error estimatesÈas deÐnedaboveÈwith speciÐc notes for most non-Gaussian distribu-tions. There are two lines per cluster, the Ðrst correspondsto the JSH95 model, the second to the P90 model. Sinceboth models start with the same initial conditions, the zÈangular momentum is the same for both models. Quotederror estimates of zero are merely due to round o† to withinthe presented precision.

The notes following each orbital parameter in Table 5o†er a qualitative description of the shape of its distribu-tion. For the sake of comparison, for each parameter inTable 5 we have also calculated the median of the distribu-tion ; those with a footnote a refer to cases in which thedi†erence between the median and the adopted value inTable 5 is larger than the interquartile error estimate ; foot-note b refers to a bimodal distribution, and footnote c refersto unusual distributions that display shoulders, long tails,and multiple peaks, among other things. The plots of theactual distributions in each model can be found in Dinescu(1998). We have also included in Table 5 the type of thecluster as deÐned by metallicity and horizontal-branchmorphology (see ° 3).

From Table 5 it is clear that the adopted values for Pal 3are di†erent from the medians. This is because the distribu-tions are biased toward those orbits that cover a period in10 Gyr, as no orbital parameters were calculated for situ-ations in which there was not one complete cycle. Inspect-ing Table 5, there are a few clusters that display unusualdistributions in several orbital parameters. These are NGC362, 2298, 5024, 5139 (only in model JSH95), 5466, 5904,6093, 6121, 6144, 6341, 6397, 6584, 6656, 6712, 6779, and6934. Among these, the clusters with large total energy havepoorly constrained orbits (especially and becauseR

azmax),the kinetic energy is a large fraction of the total energy, and

the uncertainty in the kinetic energy is also large comparedwith the total energy. Such examples are NGC 5466, 5904,and 6934.

For clusters with low total energy (i.e., small-sized orbit),the unusual orbital distributions may exhibit a chaoticbehavior, superposed on uncertainties in the initial condi-tions. Examples of clusters that fall in this regime, based onthe orbital parameter distributions, are NGC 362, 6093,6121, 6144, and 5139.

5.2. E†ect of Improved Measurement Accuracy on theKnowledge of the Orbit

This subsection describes tests performed to measure thesensitivity of the orbit determination to improvement in themeasured accuracy in the input quantities. We will showthat, for some cases, even if the measurement accuracy inthe input quantities increases substantially, the orbit maystill be poorly deÐned. In other words, orbital parametersshow an intrinsic large scatter, and this intrinsic scatter isdependent on the model of gravitational potential. Thecause of this scatter is complex, and its study is beyond thescope of this paper. The cautionary note is that, regardlessof the accuracy with which we measure the motions of clus-ters and/or describe the Galactic potential, some clusterorbits will show signiÐcant variations in the 10 Gyr integra-tion time. This is important when studying tide-induceddestruction processes of globular clusters (° 9), as the bulge-shock destruction rate depends strongly on the pericentricradius, and an average pericentric radius may not be anappropriate representation.

FIG. 3.ÈDistributions of orbital parameters for NGC 104 (47 Tuc) and NGC 6093 (M80) in the potential model P90. and are apocentric andRa

Rppericentric radii, is the maximum distance above the plane, e is the eccentricity, and ( is the inclination. The distributions are accumulated from 300zmaxMonte Carlo realizations of the clustersÏ orbits.

TABLE 5

ORBITAL PARAMETERS

Lz

Etot P Ra

Rp

zmax (NGC/Pal (kpc km s~1) (102 km2 s~2) (106 yr) (kpc) (kpc) (kpc) e (deg) Type

104 . . . . . . . 1074 ^ 75 [872 ^ 10 190 ^ 4 7.3^ 0.1a 5.2 ^ 0.3 3.1 ^ 0.2 0.17 ^ 0.03 29 ^ 2 D[1094 ^ 9 193 ^ 4 7.3^ 0.1a 5.3 ^ 0.3 3.1 ^ 0.2 0.16 ^ 0.04 29 ^ 2

288 . . . . . . . [216 ^ 122 [787 ^ 18 224 ^ 7 11.2^ 0.3 1.7 ^ 0.5 5.8 ^ 0.3 0.74 ^ 0.06 44 ^ 2 BHB[1031 ^ 18 237 ^ 12c 11.1^ 0.4 1.8 ^ 0.5 6.1 ^ 0.3b 0.72 ^ 0.06 47 ^ 1b

362 . . . . . . . [203 ^ 87 [856 ^ 25 208 ^ 17 10.6^ 0.7 0.8 ^ 0.3a 2.1 ^ 1.2a,b 0.85 ^ 0.05 21 ^ 9b RHB[1089 ^ 22 209 ^ 16 10.5^ 0.7 0.8 ^ 0.3c 2.5 ^ 0.7a 0.86 ^ 0.07a,b 20 ^ 7a

1851 . . . . . . 1947 ^ 348 [340 ^ 59 584 ^ 72 30.4^ 3.9 5.7 ^ 1.1 7.6 ^ 1.4 0.69 ^ 0.03 22 ^ 2c RHB[601 ^ 62 685 ^ 114 34.7^ 5.9 5.7 ^ 1.2 7.9 ^ 1.8 0.72 ^ 0.02 21 ^ 2

1904 . . . . . . 1269 ^ 401 [526 ^ 35 388 ^ 25 19.9^ 1.0 4.2 ^ 1.3 6.2 ^ 2.2 0.65 ^ 0.08 28 ^ 8 BHB[793 ^ 36 422 ^ 32 20.4^ 1.3 4.4 ^ 1.7 6.6 ^ 2.9c 0.64 ^ 0.10c 29 ^ 10c

2298 . . . . . . [369 ^ 407 [655 ^ 45 304 ^ 21 15.3^ 1.0 1.9 ^ 1.4 6.7 ^ 3.5b 0.78 ^ 0.12 36 ^ 14c MP[902 ^ 37 336 ^ 23 15.7^ 0.9 2.3 ^ 1.8 7.9 ^ 3.1c 0.75 ^ 0.15c 44 ^ 15c

3 . . . . . . . . . . 9609 ^ 3833 656 ^ 106a 9740 ^ 1600a 419.8^ 74.8a 82.5 ^ 6.9 308.4 ^ 54.8a,b 0.67 ^ 0.11a 61 ^ 6 RHB307 ^ 111a 9840 ^ 1680a 412.1^ 75.0a 82.6 ^ 6.9a 324.5 ^ 63.6a,b 0.67 ^ 0.12a,c 60 ^ 6

4147 . . . . . . 645 ^ 381 [420 ^ 48 496 ^ 52c 25.3^ 2.6 4.1 ^ 2.2 13.1 ^ 1.7 0.72 ^ 0.10 45 ^ 4 RHB[699 ^ 47 551 ^ 74 26.8^ 3.4 4.0 ^ 1.9 13.3 ^ 2.1 0.74 ^ 0.08 43 ^ 4

4590 . . . . . . 2307 ^ 116 [396 ^ 48 504 ^ 56 24.4^ 3.1 8.6 ^ 0.3 9.1 ^ 1.3 0.48 ^ 0.03 30 ^ 1 MP[630 ^ 42 650 ^ 78 30.0^ 3.7 8.7 ^ 0.4 10.6 ^ 1.5c 0.55 ^ 0.03 30 ^ 1

5024 . . . . . . 1283 ^ 285 [203 ^ 141 779 ^ 310 36.0^ 16.8 15.5 ^ 1.9 24.2 ^ 8.1c 0.40 ^ 0.12 62 ^ 5b MP[482 ^ 152 964 ^ 468 42.7^ 24.8 15.9 ^ 1.8 27.9 ^ 12.2c 0.46 ^ 0.14c 61 ^ 5b

5139 . . . . . . [406 ^ 22 [1089 ^ 3 120 ^ 1 6.2^ 0.1a 1.2 ^ 0.2 1.0 ^ 0.4a,b 0.67 ^ 0.05c 16 ^ 2a,c *[1306 ^ 3 123 ^ 1 6.4^ 0.1 1.2 ^ 0.1 1.0 ^ 0.1 0.69 ^ 0.02 17 ^ 1

5272 . . . . . . 705 ^ 123 [649 ^ 22 297 ^ 15 13.4^ 0.8 5.5 ^ 0.8 8.7 ^ 0.5 0.42 ^ 0.07 55 ^ 2 RHB[897 ^ 22 321 ^ 18 14.0^ 0.8 5.4 ^ 0.8 8.8 ^ 0.5 0.44 ^ 0.06 54 ^ 2

5466 . . . . . . [339 ^ 273 [83 ^ 152 1060 ^ 434c 57.1^ 24.6 6.6 ^ 1.5 34.1 ^ 14.4b 0.79 ^ 0.03 42 ^ 5b MP[352 ^ 140 1340 ^ 595b 69.8^ 29.6c 6.7 ^ 1.4 40.7 ^ 14.0 0.83 ^ 0.03 48 ^ 6

5 . . . . . . . . . . 246 ^ 336 [628 ^ 88 318 ^ 53 15.9^ 2.5 2.3 ^ 2.3 9.0 ^ 1.9 0.74 ^ 0.18 47 ^ 5 RHB[896 ^ 87 335 ^ 61 15.9^ 2.9 2.3 ^ 2.3 9.0 ^ 2.3c 0.74 ^ 0.17 49 ^ 6b

5897 . . . . . . 271 ^ 164 [863 ^ 75 187 ^ 33 9.1^ 1.3 2.0 ^ 1.1 5.1 ^ 1.1 0.64 ^ 0.10 47 ^ 4b MP[1092 ^ 71 203 ^ 33 9.3^ 1.5 2.1 ^ 1.0 5.1 ^ 1.2 0.64 ^ 0.08c 50 ^ 4c

5904 . . . . . . 357 ^ 56 [289 ^ 103 722 ^ 204 35.4^ 9.4 2.5 ^ 0.2 18.3 ^ 5.5 0.87 ^ 0.02 33 ^ 2a,c RHB[510 ^ 102 995 ^ 286 46.1^ 12.8 2.5 ^ 0.2 24.1 ^ 7.3b 0.90 ^ 0.02 42 ^ 3

6093 . . . . . . [27 ^ 22 [1311 ^ 39 71 ^ 4a 3.5^ 0.2a 0.6 ^ 0.4c 1.5 ^ 0.5c 0.73 ^ 0.17b 39 ^ 13a,b BHB[1493 ^ 42 65 ^ 6a 3.2^ 0.2 1.0 ^ 0.6c 2.3 ^ 0.7b 0.54 ^ 0.21c 60 ^ 17b

6121 . . . . . . 154 ^ 99 [1121 ^ 13 116 ^ 3 5.9^ 0.3 0.6 ^ 0.1a 1.5 ^ 0.4 0.80 ^ 0.03a 23 ^ 6 BHB[1335 ^ 12 114 ^ 3a 5.8^ 0.3c 0.7 ^ 0.1 1.3 ^ 0.4b 0.79 ^ 0.03 24 ^ 6b

6144 . . . . . . [182 ^ 22 [1259 ^ 48 70 ^ 9 3.0^ 0.7 1.8 ^ 0.2 2.4 ^ 0.2b 0.25 ^ 0.15c 63 ^ 2b MP[1439 ^ 43 84 ^ 9 3.2^ 0.6 1.9 ^ 0.2c 2.6 ^ 0.3c 0.26 ^ 0.14c 61 ^ 2c

6171 . . . . . . 396 ^ 90 [1198 ^ 36 87 ^ 9 3.5^ 0.2 2.3 ^ 0.5 2.1 ^ 0.2 0.21 ^ 0.12 44 ^ 6 BHB[1398 ^ 25 99 ^ 7 3.3^ 0.2a 2.8 ^ 0.3 2.2 ^ 0.2 0.08 ^ 0.07 44 ^ 6

6205 . . . . . . [376 ^ 145 [476 ^ 88 429 ^ 86 21.5^ 4.7 5.0 ^ 0.5 13.2 ^ 2.1 0.62 ^ 0.06 54 ^ 5 BHB[705 ^ 89 526 ^ 132 25.3^ 6.9 5.7 ^ 0.5 15.4 ^ 3.2 0.63 ^ 0.07 54 ^ 4

6218 . . . . . . 587 ^ 58 [1063 ^ 16 125 ^ 5 5.3^ 0.1 2.6 ^ 0.3 2.3 ^ 0.2 0.34 ^ 0.05 33 ^ 3 BHB[1278 ^ 14 130 ^ 4 5.3^ 0.1 2.8 ^ 0.3 2.3 ^ 0.2 0.30 ^ 0.04 34 ^ 3

6254 . . . . . . 670 ^ 75 [1053 ^ 20 128 ^ 7 4.9^ 0.2 3.4 ^ 0.4 2.4 ^ 0.2 0.19 ^ 0.05 33 ^ 4 BHB[1269 ^ 23 132 ^ 7 5.0^ 0.2 3.4 ^ 0.4 2.3 ^ 0.2 0.18 ^ 0.05 32 ^ 4

6341 . . . . . . 269 ^ 89 [880 ^ 16 201 ^ 11 9.9^ 0.4 1.4 ^ 0.2 3.8 ^ 0.5b 0.76 ^ 0.03 23 ^ 1c MP[1110 ^ 15 208 ^ 12 9.9^ 0.4 1.3 ^ 0.1 3.9 ^ 0.3c 0.78 ^ 0.03 23 ^ 1c

6362 . . . . . . 591 ^ 23 [1090 ^ 7 120 ^ 2 5.5^ 0.2 2.4 ^ 0.2 1.4 ^ 0.0 0.39 ^ 0.04 22 ^ 0 BHB[1305 ^ 6 124 ^ 2 5.3^ 0.1 2.6 ^ 0.2 1.4 ^ 0.1b 0.35 ^ 0.04 24 ^ 0c

6397 . . . . . . 816 ^ 52 [1017 ^ 9 143 ^ 3 6.3^ 0.1 3.1 ^ 0.2 1.5 ^ 0.1b 0.34 ^ 0.02 18 ^ 2b MP[1232 ^ 10 143 ^ 2 6.3^ 0.1 3.1 ^ 0.2 1.5 ^ 0.1b 0.34 ^ 0.02 18 ^ 2c

6584 . . . . . . 209 ^ 254 [773 ^ 97 249 ^ 41 12.6^ 2.4 0.9 ^ 0.7c 3.1 ^ 2.3c 0.87 ^ 0.05a 20 ^ 7a RHB[994 ^ 90 266 ^ 61 12.7^ 2.9 1.3 ^ 0.7c 5.2 ^ 2.0 0.81 ^ 0.05 39 ^ 5a

6626 . . . . . . 502 ^ 67 [1313 ^ 46 75 ^ 9 3.0^ 0.3 2.1 ^ 0.4 0.6 ^ 0.0 0.19 ^ 0.06 13 ^ 3 BHB[1538 ^ 48 79 ^ 8 3.0^ 0.3 1.7 ^ 0.3a 0.5 ^ 0.0a 0.28 ^ 0.05a 10 ^ 2

6656 . . . . . . 895 ^ 81 [871 ^ 34 190 ^ 13 9.3^ 0.7 2.9 ^ 0.2 1.9 ^ 0.1a,c 0.53 ^ 0.01 18 ^ 3c BHB[1091 ^ 31 197 ^ 14 9.6^ 0.7 2.8 ^ 0.2 1.7 ^ 0.2c 0.55 ^ 0.01 16 ^ 3

6712 . . . . . . 130 ^ 110 [1092 ^ 23 131 ^ 9 6.2^ 0.3 0.9 ^ 0.1 0.9 ^ 0.2b 0.75 ^ 0.03 25 ^ 1c BHB[1319 ^ 22 126 ^ 11 5.9^ 0.3 0.9 ^ 0.1 1.1 ^ 0.3c 0.74 ^ 0.04 28 ^ 2c

6752 . . . . . . 1014 ^ 61 [977 ^ 16 156 ^ 6 5.6^ 0.2 4.8 ^ 0.3 1.6 ^ 0.1 0.08 ^ 0.02 18 ^ 2 BHB[1194 ^ 15 153 ^ 5 5.6^ 0.2 4.7 ^ 0.2 1.6 ^ 0.1 0.08 ^ 0.02 18 ^ 2

6779 . . . . . . [356 ^ 210 [791 ^ 57 230 ^ 18 12.4^ 1.5 0.9 ^ 0.3 1.1 ^ 0.7a,c 0.86 ^ 0.03 15 ^ 5 MP[1013 ^ 57 249 ^ 30 13.0^ 1.9 0.8 ^ 0.3 0.9 ^ 1.0a,c 0.88 ^ 0.03a 13 ^ 7c

6809 . . . . . . 196 ^ 59 [1038 ^ 11 122 ^ 5 5.8^ 0.3 1.9 ^ 0.2 3.7 ^ 0.3 0.51 ^ 0.04 56 ^ 6 MP

SPACE VELOCITIES OF GLOBULARS. III. 1803

TABLE 5ÈContinued

Lz

Etot P Ra

Rp

zmax (NGC/Pal (kpc km s~1) (102 km2 s~2) (106 yr) (kpc) (kpc) (kpc) e (deg) Type

[1248 ^ 8 136 ^ 5 6.0 ^ 0.3 1.7 ^ 0.2 3.7 ^ 0.2 0.56 ^ 0.04 51 ^ 66838 . . . . . . 1205 ^ 16 [957 ^ 5 165 ^ 1 6.7 ^ 0.0 4.8 ^ 0.1 0.3 ^ 0.0 0.17 ^ 0.01 3 ^ 0 D

[1170 ^ 5 157 ^ 2 6.7 ^ 0.1 4.5 ^ 0.1 0.3 ^ 0.0 0.19 ^ 0.01 3 ^ 06934 . . . . . . [651 ^ 488 [249 ^ 152 733 ^ 310c 37.5 ^ 15.2 6.0 ^ 1.6 21.2 ^ 9.5 0.72 ^ 0.07 55 ^ 5c RHB

[492 ^ 132 990 ^ 434 46.8 ^ 19.8 6.7 ^ 1.6 26.6 ^ 11.3b 0.75 ^ 0.06 43 ^ 5b7078 . . . . . . 1134 ^ 166 [752 ^ 51 242 ^ 25 10.3 ^ 0.7 5.4 ^ 1.1 4.9 ^ 0.8 0.32 ^ 0.05 36 ^ 4c MP

[986 ^ 48 253 ^ 29 10.4 ^ 0.8 5.6 ^ 1.2c 4.9 ^ 1.0 0.30 ^ 0.05 37 ^ 57089 . . . . . . [664 ^ 232 [290 ^ 145 654 ^ 239 33.6 ^ 12.5 6.4 ^ 1.1 20.0 ^ 6.5 0.68 ^ 0.06 53 ^ 6 BHB

[529 ^ 142 860 ^ 379c 42.2 ^ 17.9 6.3 ^ 1.2 24.1 ^ 8.4c 0.74 ^ 0.06 47 ^ 67099 . . . . . . [444 ^ 60 [937 ^ 24 159 ^ 10 6.9 ^ 0.3 3.0 ^ 0.4 4.4 ^ 0.3 0.39 ^ 0.06 52 ^ 2 MP

[1162 ^ 24 167 ^ 9 6.8 ^ 0.3 3.3 ^ 0.5 4.4 ^ 0.2 0.36 ^ 0.05 53 ^ 2

a The di†erence between the median and the adopted value in Table 5 is larger than the interquartile error estimate.b Bimodal distribution.c Unusual distributions that display, among other things, shoulders, long tails, and multiple peaks.

We have calculated the half-interquartile range of theorbital parameter distribution (here interpreted as thescatter in the orbital parameter) resulting from two assumedlevels of uncertainties in the input quantities. One run hadthe dispersions of the proper motions, distances, and radialvelocities artiÐcially set at 1%; the second one had the dis-persions of the proper motions and distances set at 10%,while that of the radial velocity was Ðxed at 1%. Thissecond run has errors whose properties are close to those ofthe actual sample (see ° 2). Naively, if the input errors areincreased by a factor of 10, a similar increase might beexpected in the output scatter. We have therefore calculatedthe ratios of the scatter in the orbital parameters betweenthe 1% and the 10% run. These ratios were calculated fororbital parameters e, and (. Then, as a repre-R

a, R

p, zmax,sentation of the total sensitivity of the output orbital

parameter scatter to input measuring errors, we have calcu-lated ““ distances ÏÏ in this Ðve-parameter space. The ““ origin ÏÏin this space was adopted as 0.1, the value that would be

FIG. 4.ÈSensitivity of the output and orbital parameter scatter to inputmeasuring errors. A large number in this plot represents little variation inthe output scatter as input errors are increased (see ° 5.2).

expected in the output scatter ratio if the process behaved ina simple, linear fashion.

Figure 4 shows these ““ distances ÏÏ for all the clusterswhere the horizontal axis is for the JSH95 model and thevertical one for the P90 model. In this plot, near the““ origin,ÏÏ there is a group of clusters for which the increasein the output scatter follows the increase in the input errorsin a predictable way.

The clusters with large values in this plot are those withintrinsic scatter in the orbit in the sense that in the givenpotential model and for the initial integration conditions,their orbital parameter scatter estimates do not change, aserrors in the input measured quantities are increased by afactor of 10.

In Figure 4, the NGC numbers of clusters with large““ distances ÏÏ are indicated. The distribution of these““ distances ÏÏ in model P90 is more concentrated than thedistribution in model JSH95. Clusters with large““ distances ÏÏ in model P90 are fairly isolated from the centerof the distribution, while, in model JSH95, such a clearseparation is not evident. From the standpoint of modelP90 and the error-sensitivity analysis, clusters with intrinsicscatter in the orbit are NGC 362, 6093, 6121, 6584, and6712. For model JSH95 such cluster candidates are NGC5139, 6656, and 6121.

6. MODEL DIFFERENCES IN THE ORBITAL PARAMETERS

We have calculated di†erences in the orbital parametersderived using the two di†erent potential models and com-pared these with the expected uncertainties in the di†er-ences. The uncertainties in the di†erences are v\ (vJSH952

where and are determined from the] vP902 )1@2, vJSH95 v

P90interquartile ranges as explained in ° 5. As before, severalruns were made with the input errors artiÐcially set to pre-determined values. Table 6 displays the ratio of the resultingmodel di†erence and its formal uncertainty. The uncer-tainties in the di†erences of the orbital parameters weredetermined for runs with relative errors of 1%, 5%, and10% in proper motions and distances, as well as for theactual error estimates as given in Table 2. The radial veloc-ity uncertainty was kept constant at the 1% level for theÐrst three runs. The 10% run is closest to the real inputmeasurement uncertainties (see ° 3). In Table 6, eachcolumn represents a di†erent value the relative error, whileeach line represents an orbital parameter. The values in

1804 DINESCU, GIRARD, & VAN ALTENA Vol. 117

TABLE 6

RATIOS OF MODEL DIFFERENCES OVER

FORMAL ERRORS

Parameter 1% 5% 10% Actual

P . . . . . . . . . . 12.0 2.5 1.3 1.0R

a. . . . . . . . . 1.8 0.8 0.7 0.4

Rp

. . . . . . . . 4.2 1.0 0.5 0.3zmax . . . . . . . 5.1 0.7 0.6 0.6e . . . . . . . . . . 11.7 4.2 1.3 0.8( . . . . . . . . . 5.0 1.2 0.5 1.3Vz

. . . . . . . . . 9.4 2.3 1.5 1.5Vp

. . . . . . . . . 10.9 2.8 1.5 0.6

Table 6 are the modes of the distributions of ratios for ourcluster sample.

The results indicate that, globally, model-induced di†er-ences in the orbital parameters are essentially comparableto or smaller than their formal error estimates for ourpresent level of input measurement uncertainty. The ratiosare signiÐcantly di†erent from 1 only if the input relativeerrors in proper motions and distances are lower thanD5%. We stress here that the di†erence between the twoGalactic potential models cannot be distinguished if oneconsiders the orbital parameters and their uncertainties.However, the di†erence between the models is signiÐcantwhen tide-induced destruction processes of globular clus-ters are studied (° 9).

7. ANALYSIS OF THE ORBITAL PARAMETERS

In the following section we will adopt model P90 as rep-resentative ; this is of minor importance in the analysis since,globally, for our sample, model di†erences in the orbitalparameters are not signiÐcant with respect to other sourcesof errors.

7.1. Orbital Parameters versus Metallicity and HB TypeWe have plotted the eccentricities as a function of metalli-

cities in Figure 5 (top). In the metallicity range from [1.7 to[1.0, one can clearly see two groups of clusters : one withlow eccentricities, resembling the standard metal-rich thick-disk clusters 47 Tuc (NGC 104) and NGC 6838, and onewith high eccentricities, more typical for halo clusters. Inthis plot we have also indicated the NGC numbers of theclusters with low-e and [Fe/H][ [1.7. At low metallicities([Fe/H]\ [1.8) there is a large range in eccentricities,which can get as low as 0.3, but, in the mean, the eccentricityis high. This is expected in a hot halo population, and it wasalso seen in the recent study of Chiba & Yoshii (1997),which makes use of Hipparcos high-accuracy propermotions for a sample of red giants and RR Lyrae variables.For metallicities lower than [1.6, their analysis of the dis-tribution of eccentricities as a function of z-distance, identi-Ðed a halo population including a fraction of 16%È20% ofthe stars with eccentricities lower than 0.4.

In order to select metal-poor clusters with thick-diskorbital characteristics we present the inclination as a func-tion of eccentricity in Figure 5 (bottom). The diagram isalmost evenly populated, with a very slight trend of increas-ing inclination with eccentricity. Since the halo populationis primarily a system with a large velocity dispersion, a widerange of orbit characteristics is expected, and this is indeedwhat the diagram shows. The clusters marked in the top

panel of Figure 5 are also indicated in the bottom panel. Asseen in this plot, if we adopt the metal-rich ([Fe/H][ [0.8)clusters 47 Tuc (NGC 104) and NGC 6838 as deÐning theorbital characteristics of thick-disk clusters, we can conser-vatively classify NGC 6254, 6626, and 6752 as thick-diskclusters in terms of their orbits. Here, the term conservativerelies on 1 pÈlevel errors as representative.

Also note the relatively high inclination of 47 TucÏs orbit ;this is consistent with its present distance below the Galac-tic plane (D3 kpc). Very similar to 47 TucÏs orbit is the orbitof NGC 6254 ; this would have not been readily seen in thevelocity data, since 47 Tuc has a low W -velocity as it islocated at the maximum distance from the Galactic plane(see Table 5), while NGC 6254 has a relatively largeW -velocity compared with that of 47 Tuc, as it is closer tothe Galactic plane. A Ðrst inspection of the velocity com-ponents might have discarded NGC 6254 from the popu-lation of metal-weak thick-disk clusters.

Thus, we can conclude that there are three metal-poorclusters that have orbital eccentricities and inclinations con-sistent with those of thick-disk clusters. Field star surveyshave also identiÐed a metal-poor thick-disk component (seereview paper by Norris 1996) ; however, they do not seem toagree upon the actual fraction of thick-disk stars in a givenmetallicity range. For example, Beers & Sommer-Larsen(1995) quote a fraction of 60% of kinematically thick-diskstars in the metallicity range from [1.6 to [1.0, while therecent study of Chiba & Yoshii (1997) advocates a fractionof 10% in the same metallicity range. Our fraction of metal-poor thick-disk clusters in the metallicity range from [1.6to [1.0 is D4%, and this can be regarded as a lower limit.Therefore, if the metal-weak thick-disk fraction is as low as10%, then we may expect to Ðnd D5 more metal-weakthick-disk clusters. This prediction is tentative, however,since it relies on numbers drawn from the Ðeld starcomponentÈwhich may not necessarily reÑect all theproperties of the globular cluster system. The actual frac-tion can be realistically determined only when orbit dataare available for most of the clusters. Also the metal-poorthick-disk clusters seem to form a slightly thicker disk(average of the three clusters in discussion is D1.5 kpc)zmaxthan the metal-poor, thick-disk Ðeld stars (scale height D1kpc ; Chiba & Yoshii 1997). What is meaningful thus far isthat within the cluster system a metal-weak thick-diskpopulation was also identiÐed, as in Ðeld star surveys.

In Figure 6 we represent the total energy as a func-Etottion of orbital angular momentum The types of clustersLz.

as deÐned by metallicity and HB-type are marked with dif-ferent symbols ; the convention for symbols is the same as inFigure 1 : asterisk for disk clusters (D, [Fe/H][ [0.8),Ðlled circles for BHB clusters, open circles for MP clusters([Fe/H]\ [1.8), triangles for clusters with RHB at a givenmetallicity, and the star for u Cen. While BHB and MPclusters Ðll the diagram at any and the RHB clustersEtot L

z,

have high total energy, all of them being located above acertain threshold. Among the RHB clusters in our sample,NGC 362 is the cluster with the lowest total energy. We canadopt its total energy (roughly [1.1] 105 km2 s~2) as thethreshold deÐning RHB clusters. The apocentric radii ofRHB clusters are all larger than 10 kpc (Table 5). Sinceclusters spend most of their time near apocenter, we conÐrmthat this population indeed resides in the outer regions ofthe Galaxy (Zinn 1993 ; see also van den Bergh 1993). Onaverage, the eccentricity of this group is high : 0.74^ 0.12,

No. 4, 1999 SPACE VELOCITIES OF GLOBULARS. III. 1805

FIG. 5.ÈEccentricity as a function of metallicity (top) and inclination as a function of eccentricity (bottom). Error bars are considered only in the orbitalparameters as derived in ° 5.

and seven of the nine clusters have eccentricities larger than0.7. Therefore, the RHB clusters are a hot population, withpredominantly high-energy, large-size, and highly eccentricorbits.

In order to investigate the Ñatness of these systems andthe amount of rotation, in Figure 7 we plot the inclination( as a function of the normalized orbital angular momen-tum represents the orbital angular momen-L

z/L

z,max. L z,maxtum of a circular orbit (in the Galactic plane) of the energyof the clusterÏs orbit and is considered as a scaling quantity,with zero uncertainty. The panels show the total sample, theBHB clusters, the MP clusters, and the RHB clusters. In theBHB panel we have also plotted and marked the two metal-rich thick-disk clusters 47 Tuc and NGC 6838 for purposesof comparison. It is important to note that, because of thedeÐnitions of inclination and orbital angular momentum,there are forbidden regions in this diagram, such as the one

at high inclinations and high The full sample dis-Lz/L

z,max.plays a large scatter in inclination at negative to moderatezÈangular momentum. At large orbital angular momentumthe inclinations become small according to the deÐnitions.If only the BHB clusters are considered, a clear correlationis seen between inclination and orbital angular momentum.This is partially due to the deÐnitions of inclination and

and partially due to the lack of highly retrogradeLz/L

z,max,orbits at low inclinations. The MP sample displays a largescatter in both coordinates and no trend. The RHB systemshows no trend, and an absence of clusters with L

z/L

z,max [0.5. The trend seen for the BHB system can be due to thedeÐnitions of ( and combined with small numberL

z/L

z,maxstatistics and/or due to a true correlation between Ñatteningand rotation. Given the present data, the BHB system, incomparison with the MP oneÈwhich can be thought of asrepresentative for a spherical system with isotropic velocity

1806 DINESCU, GIRARD, & VAN ALTENA Vol. 117

FIG. 6.ÈTotal energy as a function of orbital angular momentum The types of clusters as deÐned by metallicity and HB type are marked withEtot Lz.

symbols as in Fig. 1.

distribution (see ° 3)Èlooks like it lacks strongly retrogradeclusters at low inclinations, and it has more clusters with asigniÐcant amount of rotation. This can argue in favor of atrue Ñattening rotation correlation for the whole BHB

system. It is also possible that, within the BHB system,clusters with a signiÐcant amount of rotation are totallyunrelated to the ones at high inclinations, as the former maybe actually the metal-weak component of the thick disk.

FIG. 7.ÈInclination as a function of normalized orbital angular momentum

No. 4, 1999 SPACE VELOCITIES OF GLOBULARS. III. 1807

Thus the combination of two dynamically unrelated popu-lations may produce the trend seen in Figure 7 (top right).The trend seen for the BHB system is an intriguing feature ;its reality and nature can be further explored only if moredata are acquired.

7.2. T he Case of u CenThere are several lines of evidence that suggest that u

Cen is the ““ nucleus ÏÏ of a disrupted satellite galaxy. ItsÑattened shape, rotation, bimodal calcium abundance dis-tribution (Norris, Freeman, & Mighell 1996 ; see also Sun-tze† & Kraft 1996, who advocate a long-tailed rather thanbimodal distribution), and possible dependence of kine-matics on abundance (Norris et al. 1997) point to a di†erentformation scenario for u Cen than that for the GalaxyÏsother globular clusters. Here we investigate the orbit char-acteristics of u Cen in comparison with the other clusters.

The position of u Cen in the plot of the total energyversus orbital angular momentum (Fig. 6) is rather extremein the sense that at the largest amount of negative angularmomentum or retrograde motion it has the lowest amountof total energy. Also, in Figure 7 (top left), a group of threeclusters resides at low inclinations, with a fair amount ofretrograde motion. These are NGC 362, u Cen, and NGC6779, and they are indicated in the plot. Among these threeclusters, it is u Cen that has the smallest apocentric radius :6 kpc compared with 11 kpc for NGC 362 and 13 kpc forNGC 6779, the smallest orbit eccentricity (Table 5) and thestrongest retrograde motion. Therefore, a plausible scenariois that u Cen originated in a massive satellite that had arather strongly retrograde initial orbit, that decayed due tosigniÐcant dynamical friction. In this scenario, most of theorbital decay happened while the satellite was not com-pletely disrupted, such that dynamical friction played animportant role in modifying the initial orbit. N-body simu-lations of minor mergers of spiral galaxies with their satel-lite galaxies (Walker, Mihos, & Hernquist 1996, hereafterWMH96; see also Huang & Carlberg 1997) show thatbecause of dynamical friction with the disk, orbits of pro-grade satellites decay such that the orbit becomes moreconÐned to the Galactic plane as the satellite spirals to thecenter of the Galaxy. A similar behavior can be expected forretrograde orbit but perhaps not as prominently as forprograde ones. The decay time is longer for retrogradeorbits, and the e†ect of settling in the Galactic plane,although present, is not as strong as for a prograde orbit(I. R. Walker 1998, private communication). See, however,Huang & Carlberg (1997), who advocate a rather di†erentbehavior of prograde orbits from retrograde ones, which ismainly due to the response of the disk. Their simulationsshow that the disk tilts toward the orbital plane of a pro-grade satellite, while for a retrograde satellite, the disk tiltsaway from the orbital plane of the satellite. They predicttherefore that there should be a net excess of retrograderemnants with highly inclined orbits and a net excess ofprograde remnants with low orbital inclinations.

As the three clusters mentioned above are located ina rather conÐned and poorly populated region of the((, plane, one can ask whether they could haveL

z/L

z,max)originated in the same satellite. The common satellite originfor globular clusters has been a long-standing topic, and wewill discuss it in more detail in the following subsection.Had the three clusters in discussion originated in the same

satellite, then the smaller-sized orbit of u Cen might be dueto stronger dynamical friction on this massive cluster (afactor of D10 more massive than the other two clusters).The orbital decay time for u Cen for *R\R

a,NGC 362,6779kpc, due to dynamical friction, is D80 Gyr[ Ra,u CenD 6

based on formula 7.26 given in Binney & Tremaine (1987, p.428), with the adopted mass of u Cen taken to be 2.6] 106

(and adopted mass-to-light ratio of 3, see DjorgovskiM_1993). This argues that it is unlikely that the orbital di†er-

ences are due to dynamical friction and suggests that thethree clusters did not have a common origin. It is still pos-sible that all three clusters originated in the same satellite ifNGC 362 and NGC 6779 were peeled o†, due to tidal inter-action with the Galaxy, at a di†erent stage than u Cen, asthey could have been more loosely bound to the satellitethan u Cen. Therefore NGC 362 and NGC 6779 could havestarted their own independent orbital paths, with negligibledynamical friction a†ecting their orbits before the satellitewas disrupted, while u Cen still resided in the satellite thatcontinued to lose signiÐcant orbital energy due to dynami-cal friction.

Although a possible explanation for the orbital charac-teristics of u Cen is its origin in a massive satellite whoseorbit was signiÐcantly altered by dynamical friction withthe disk, more detailed modeling of such an event with thecharacteristics of a strongly retrograde orbit and a highmass and density of the satellite are particularly desirable inorder to clarify this issue. It may also be instructive to lookat the example of the Sagittarius dwarf galaxy. This satellitegalaxy is believed to have a rather large mass (D109 M

_)

and a high density (comparable to the mean density of theGalaxy material inside the orbit of Sagittarius) in order tohave survived tidal disruption given its currently knownorbit (Ibata et al. 1997). Also, associated with the Sagittariusgalaxy is the cluster M54 (Da Costa & Armandro† 1995),which is the second most massive globular cluster of ourGalaxy, after u Cen. It also has a rather large spread in theabundance distribution (Sarajedini & Layden 1995). Basedon these two facts and on its location at the center of theSagittarius galaxy, it is argued (see Larson 1996 and refer-ences therein) that M54 may represent the ““ nucleus ÏÏ of theSagittarius galaxy. If, by analogy, u Cen is the ““ nucleus ÏÏ ofa disrupted satellite, then this satellite presumably wouldhave been of comparable or larger mass than Sagittarius.

7.3. In Search of Cluster StreamsThe search for streams of globular clusters that orig-

inated in the same satellite and therefore share similarorbits has been a very popular subject, and several attemptsand predictions have been made, primarily based on radialvelocity data (see e.g., Rodgers & Paltoglou 1984 ; Lynden-Bell & Lynden-Bell 1995).

While the idea that such cluster streams exist is perfectlyvalid, to date there is no deÐnitive evidence of clusterstreams, and this is primarily due to the lack of accuratetangential velocities. Additional problems are the small sizeof the globular cluster system, which makes a phase-spaceanalysis based on overdensities in counts along orbitalpaths (Johnston, Hernquist, & Bolte 1996) impossible, andthe fact that the halo inside a radius of 40 kpc may be bettermixed (Zinn 1993) than the remote halo where such streamswere conÐrmed among the satellite population (e.g., theMagellanic stream; see Majewski, Phelps, & Rich 1996 and

1808 DINESCU, GIRARD, & VAN ALTENA Vol. 117

Schweitzer, Cudworth, & Majewski 1997 and referencestherein).

In this paper we will only qualitatively investigate pos-sible associations, based on the similarity of orbit character-istics. No clustering analysis of the orbital parameters isperformed since the sample is still too small and the errorsin the orbital parameters are still too large to warrant arobust quantitative approach. This latter statement is espe-cially valid for clusters on large-size orbits, which are thosemost likely to have been accreted (see ° 7.1).

We begin with the proposed association of clusters identi-Ðed by Rodgers & Paltoglou (1984). Among the seven clus-ters that they suggested belong to the same satellite, basedon a narrow range in metallicity and strong mean retro-grade motion, three now have tangential velocities andorbits : NGC 1851, 6584, and 6934. Among these, only NGC6934 has a retrograde motion. The three clustersÏ orbitalangular momenta span the entire range presented in Figure6, and their orbital parameters (Table 5) show completelydi†erent orbits. If similarity between orbits is taken as evi-dence for common origin, then these clusters show no signof having originated in the same satellite.

To further explore possible associations, we have lookedat plots of inclination versus eccentricity from which clus-ters on orbits with small mean radii were eliminated (Etot\[1.1] 105 km2 s~2, which is about the total energy ofNGC 362, the RHB cluster with the smallest total energyÈsee ° 7.1). We have also omitted clusters with errors ineccentricity larger than 0.1 and in inclination larger than 7¡.Possible groupings in this diagram were further searchedfor common values in and In this manner we haveEtot L

z.

found two groups of two clusters each : the pair NGC 362and NGC 6779, which was already mentioned in ° 7.3, andthe pair NGC 6934 and NGC 7089. Although NGC 6934and NGC 7089 have identical orbital parameters withintheir uncertainties (Table 5), the uncertainties in the totalenergy and the orbital angular momentum are rather large(see Fig. 6).

Therefore, we regard this grouping with caution andawait better data for them. The other pair of clusters seemsthe most signiÐcant association in the sense that theirorbital parameters are similar despite the modest size oftheir uncertainties. However, another determination of theabsolute proper motion of NGC 6779 based on a cali-bration other than the Hipparcos one is particularly desir-able and would help verify or invalidate this result ; a slightchange in the proper motions of distant clusters can com-pletely modify the orbit.

We end this section by remarking that the search forcluster streams in the halo is still tentative and it may not bea well-posed problem, unless remote clusters and satellitegalaxies are considered.

8. GLOBULAR CLUSTER ORBITS AND THE GALAXY

Armed with space velocities for about one-quarter of theglobular cluster system, an important question comes tomind regarding the global picture. What is the contributionof the globular cluster orbits to our knowledge and under-standing of the Galaxy?

The most signiÐcant result, in our opinion, is the exis-tence of metal-poor clusters with orbits consistent with thethick-disk motion (circular, low-inclination, with rotation

velocity close to the rotation velocity of the thick disk ; wemay also call these thick-diskÈlike orbits ; see ° 7.1). Theseclusters are NGC 6254, 6626, and 6752. A possible origin ofthese clusters is that they were produced in satellite galaxieswhose orbits were modiÐed and circularized due to dynami-cal friction before they were completely disrupted. In whatfollows we will compare the properties of these clusters withthe properties of the clusters associated with the Sagittariusdwarf galaxy, which is presently being accreted by ourGalaxy (Ibata et al. 1997). This is in order to see whether itis plausible that the clusters in discussion could haveformed in a satellite such as Sagittarius. While an age forNGC 6626 is not yet available, several comparative agedeterminations (Richer et al. 1996 ; Chaboyer, Demarque, &Sarajedini 1996 ; Buonanno et al. 1998) agree that bothNGC 6254 and NGC 6752 are prototypical old halo clus-ters. The cluster population of Sagittarius dwarf galaxyincludes Ter 7, Ter 8, Arp 2, and M54 (Da Costa & Arman-dro† 1995). While most studies now agree that Ter 7 is ayoung cluster (D7 Gyr younger than typical halo clusters,see Buonanno et al. 1998 and references therein), recentpapers still debate whether Arp 2 is signiÐcantly youngerthan Ter 8 and M54 (Buonanno et al. 1994 ; Montegri†o etal. 1998) or whether its age is comparable to the typical oldages of halo clusters (Layden & Sarajedini 1997). If we takeas a working hypothesis that the ages of the clusters Arp 2,Ter 8, and M54 are the same and also similar to those ofNGC 6254, 6626, and 6752 (this is not unreasonable : seeLayden & Sarajedini 1997 for Arp 2, Ter 8, and M54; Buon-anno et al. 1998 for Arp 2, NGC 6254, and NGC 6752, andthe discussion in Da Costa & Armandro† 1995 for NGC6626), then it seems that, at the same epoch in time, thesatellite systems in which NGC 6254, 6626, and 6752 pre-sumably originated were able to produce clusters that aremore metal-rich than those of the Sagittarius galaxy. Notethat the metallicities of Ter 8, Arp 2, and M54 are [1.99,[1.70, and [1.55, respectively (Da Costa & Armandro†1995), while those of NGC 6254, 6626, and 6752 are [1.60,[1.28, and [1.54, respectively (R. Zinn 1995, privatecommunication). Also note that cluster M54 is ratherspecial in terms of the abundance pattern and is suspectedto be the ““ nucleus ÏÏ of Sagittarius (see Sarajedini & Layden1995 and Da Costa & Armandro† 1995 for arguments proand con). Therefore, it may have had a di†erent enrichmenthistory than that of typical globular clusters. For the sake ofargument, had the three clusters of Sagittarius beenyounger than NGC 6254, 6626, and 6752, then the discrep-ancy would be even stronger, since, not even at a laterepoch, as chemical enrichment proceeds, was Sagittariusable to produce clusters as metal-rich as our three candi-dates. Therefore, it seems unlikely that clusters NGC 6254,6626, and 6752 were produced in satellite galaxies of thetype of Sagittarius. Intuitively, they should have formed insystems more massive than Sagittarius, if the mass of asystem correlates with its mean metallicity (Gallagher &Wyse 1994) and with the mean metallicity of its globularcluster system (Harris 1996b).

Another possibility is that these clusters were formed in arotationally supported system, such as a metal-poor,gaseous disk. Considering the age constraints as well, theimplications are that our Galaxy had material with rota-tional support, organized in a disk, fairly early in its history.

It also follows that the formation of the disk partiallyoverlapped with the halo formation and that the

No. 4, 1999 SPACE VELOCITIES OF GLOBULARS. III. 1809

conditionsÈgas surface densities and angular velocitiesÈinthis disk were such that it was able to form globular clusters(see Larson 1996), a process that has long since ceased.Since a metal-poor thick-disk population was also identi-Ðed in the Ðeld stars (see ° 7.1 and references therein), thecase for a primordial, metal-poor, gaseous disk may be aserious issue to be considered in Galaxy formation models.

Concerning the RHB group of clusters, Zinn (1993) pro-posed that they originated in satellite galaxies that escapedthe coalescence process of the gas-rich fragments fromwhich the outer halo was built.

Subsequent dynamical friction with the dark matter halocaused satellitesÏ orbits to decay, while tidal interactionswith the Galaxy and its disk caused the disruption of thesesatellites. The orbital data are consistent with this scenarioin the sense that indeed these clusters have large-size, high-eccentricity orbits (° 7.1), which points to their origin inmore energetic systems that probably escaped the process ofinitial collapse of the Galaxy and then evolved independent-ly (Zinn 1993). Their more energetic state may be due to lessloss of kinetic energy in inelastic collisions of the originalfragments in the outskirts of the proto-Galaxy, where thedensity of these fragments is lower than in the inner regions.However, some of the BHB and MP clusters display similarorbital characteristics to the RHB clusters. The problem indynamically separating these groups is that in the highorbital energy domain km2 s~2, see(Etot[ [1.1] 105° 7.1) no real distinction in the orbital parameters of thesegroups can be seen with the data available so far. Forinstance, the mean eccentricity for Ðve BHB clusters is0.66^ 0.03, for seven MP clusters it is 0.63^ 0.08, and fornine RHB clusters it is 0.74 ^ 0.04. The marginal indicationof a larger mean eccentricity for RHB clusters, as well as theradial velocity anisotropy remarked upon in ° 3, are theonly arguments in favor of a di†erent origin of RHB clustersfrom BHB and MP ones, at least in the high-energy domainor at large Galactocentric radii. Clearly, more data arenecessary in order to understand whether a signiÐcantdistinctionÈfrom the point of view of orbit typeÈcan bemade between these groups.

In the domain of small orbital energy, there is a distinc-tion between thick-diskÈlike and halo-like orbits. The thick-diskÈlike ones were found so far only among BHB clusters,while halo-like were found among BHB and MP clusters. Inthis regard our results are consistent with the MP clustersbeing a hot, pressure-supported system (see also the scatterin the diagram of inclination versus orbital angular momen-tum in Fig. 7, bottom left). Among the BHB clusters it is stillan open question as to whether the transition betweenthick-diskÈlike orbits and halo-like ones is continuous orrather abrupt and whether we see an independent systemwith rotational properties between those of the thick diskand those of the halo or if we see a mixed system thatcombines metal-poor thick-disk clusters with typical metal-poor halo ones.

To conclude, the results from space velocities andresulting orbits are very promising and provide a few out-standing cases that impose some revisions with respect tothe formation picture of the Galaxy. However, in order tolearn more about the globular cluster system and theGalaxy, a concentrated e†ort needs to be made to measureaccurate space velocities for a much larger sample of clus-ters. In this regard satellite missions such as ESAÏs GAIAand NASAÏs SIM will play a major role.

9. EFFECT OF ORBITAL MOTION ON THE INTERNAL

DYNAMICS

9.1. Formalism for the Destruction RatesComprehensive studies of globular cluster destruction

processes and the evolution of the globular cluster system asa whole include the pioneering work of Aguilar, Hut, &Ostriker (1988, hereafter AHO88), followed by two recentpapers that incorporate Fokker-Planck calculations. Theseare Gnedin & Ostriker (1997, hereafter GO97), whichfollows the approach of AHO88 including several new theo-retical results, and the study of Murali & Weinberg (1997),which includes only the e†ect of the bulge as tidal heating.Both recent papers agree that the system of globular clus-ters underwent signiÐcant evolution, mainly in the innerpart of the Galaxy. While these studies are interested in aglobal picture, where the orbital characteristics of the clus-ters are statistically determined or simply adopted, our dataallow a cluster-by-cluster approach to the problem. We cantherefore test the destruction rates in the context of ourGalactic potential models, as well as understand which pro-cesses are responsible for the depleted luminosity functionsseen recently in some clusters with deep color-magnitudediagrams derived from HST data (Piotto et al. 1997).

The main processes that induce the evaporation of starsare internally driven, such as two-body relaxation, andexternally driven, such as bulge and disk shocking (alsocalled tidal shocking) in which the energy input into thecluster leads to the escape of stars.

In addition to the energy input, the tidal shockinginduces relaxation. This speeds up the evolution of thecluster, further accelerating the mass loss. &Kundic�Ostriker (1995) Ðnd that at the half-mass radius the tidally-induced relaxation competes with the two-body relaxationin driving the cluster evolution.

Here we have adopted the formalism from GO97 todetermine destruction rates for the clusters in our sample.Besides the Fokker-Planck calculation, GO97 also providefunctional forms for the destruction rates due to bulge anddisk shocking ; these are deÐned in terms of the destructiontimescale at the half-mass radius. These formulations do notinclude adiabatic corrections, which are known to be sig-niÐcant in the inner parts of the clusters in which theimpulse approximation is violated. Also, for the bulge shockrate, no corrections are considered related to the bulge massdistribution, as described in GO97. These are signiÐcant forclusters that move well within 1 kpc (GO97). The destruc-tion rates include both the mass loss resulting directly fromthe shock and that resulting from the ““ tide-induced ÏÏ accel-eration of the relaxation. They represent the inverse of thetime to complete disruption of the cluster due to the energyinput. The expressions for the disk and bulge com-(l

D) (l

B)

ponents of the destruction rate are

lD

\ 1129

1G

rh3

Mcl

gm2

Vz2P

, (1)

lB\ 112

9G

rh3

Mcl

Mb2

Vp2R

p4P

. (2)

Here is the maximum gravitational acceleration in the zgmdirection due to the disk G is the gravitational([L'

d/Lz),

constant, the half-mass radius, the cluster mass,rh

Mcl Vzthe z-component of the velocity at plane crossing, theV

ptotal velocity at pericenter, the pericenter radius, P theRp

1810 DINESCU, GIRARD, & VAN ALTENA Vol. 117

orbital period, and the bulge mass. To calculate theMbrates, we have determined averages over the number of

cycles in 10 Gyr for the quantities and forgm2/V

z2 1/V

p2R

p4

each cluster. Uncertainties in these quantities were deter-mined similarly to uncertainties in the orbital parameters(° 5.1) by repeated integrations with slightly di†erent initialconditions, corresponding to uncertainties in the input,measured quantities. The period P and its uncertainty aretaken from Table 5, and model-dependent parameters arefrom Table 4. We have replaced the expression for theangular frequency of stars in the GO97 formulation with

Half-mass radii and absolute total magni-uh2\ GMcl/2r

h3.

tudes were taken from the catalog of Djorgovski (1993).Masses were derived assuming the mass-to-light ratio

and (Djorgovski 1993).Mcl/L V\ 3 M

_/L

V,_, MV,_ \ 4.79

The destruction rates due to two-body relaxation andevaporation were adopted from GO97 from their run withno tidal shock considered. It is important to note that whiletidal-shock rates are determined by both cluster character-istics and orbital parameters, the evaporation rate due totwo-body relaxation is determined only by cluster charac-teristics, such as total mass and concentration. These latterrates may be somewhat underestimated, since their Fokker-Planck calculations were of clusters with single-mass stars ;a mass spectrum will speed up the evolution as a result ofmass segregation.

The total destruction rate is the sum of the destructionrates of all three processes. The errors in the destructionrates are propagated from errors in the orbital parametersand, for the term the error considered is only due tor

h3/Mcl,the error in the distance to the cluster.

9.2. ResultsFigure 8 shows the log of the destruction rates as a func-

tion of the log of which can be thought of as therh3/Mcl,inverse of the mean cluster density. If we consider the uncer-

tainty in the term due only to the 10% error in therh3/Mcldistance to each cluster, then errors in the horizontal axis

are or about 2 times the size of the symbols. The rates~0.10`0.08,in the plot are calculated in the potential model P90 and arein units of (10 Gyr)~1. Thus, in this logarithmic plot a valueof zero indicates a rate of 1, which corresponds to a destruc-tion time of 10 Gyr, while a value of 1 indicates a destruc-tion time of 1 Gyr. The four panels correspond respectivelyto rates due to disk shock, bulge shock, disk]bulge shock,and disk]bulge]evaporation due to two-body relaxation.The bulge rate shows a larger scatter than the disk rate, andthis is mainly due to the strong dependence of the rate onthe pericentric radius. For most of the clusters, the two-body relaxation rate is larger than the tidal-shock rates ; thiscan be seen in the generally large increase in the total

FIG. 8.ÈDestruction rates as a function of the inverse ““ mean ÏÏ cluster density ; the line corresponding to a destruction time of 10 Gyr is indicated. The topleft panel shows the rate due to disk shocking, the top right panel, that due to bulge shocking ; the bottom left shows their e†ect combined, and the bottomright shows the total destruction rate, which includes, in addition to the tidal shocking, the two-body evaporation rate.

No. 4, 1999 SPACE VELOCITIES OF GLOBULARS. III. 1811

destruction rate in Figure 8 (bottom left to bottom right). Inour sample of 38 clusters there are only eight clusters forwhich tidal-shock rates are comparable or larger than two-body relaxation rates ; these clusters are : NGC 288, 5139(u Cen), 6121 (M4), 6144, 6362, 6712, 6779 (M56), and Pal 5.

The uncertainties in the destruction rates are proportion-al to the size of the rate (from eqs. [1] and [2]), with theconstant of proportionality for each cluster given by therelative errors in the orbital parameters (e.g., Thusp

P/P).

the bulge rates have larger estimated errors than do the diskrates. For the total rate, however, estimated errors are rela-tively smaller, as zero error was assumed for the two-bodyevaporation rate and this quantity dominates. The fewexceptions are clusters that have large tidal-shock rates,which are also larger that the two-body evaporation rates.One example is Pal 5, the cluster with the lowest density inour sample. This rate is so high (the destruction time isD0.1 Gyr) because the orbit has a very small pericentricradius for a cluster with such a small mean density. Clustersin the inner parts of the Galaxy, which su†er strong tidalshocks are much denser than Pal 5. It is also true that Pal 5has a large uncertainty in the total destruction rate, whichultimately is due to the uncertainty in the measured quan-tities. Notable is the radial velocity uncertainty for thiscluster km s~1 ; Table 2) as adopted from(Vrad\[55 ^ 16HarrisÏs (1996a) catalog.

Figure 9 shows a detail of Figure 8 (bottom right). Herewe have plotted the total destruction rates with Ðlledsymbols and the rates due to tidal shocks with opensymbols. Note that for those clusters where these symbolsare close or overlap, the destruction due to tidal shocks isdominant. Piotto et al. (1997) presented a comparativestudy of deep HST luminosity functions for the clustersNGC 6341, 6397, 7078, and 7099. They concluded thatNGC 6397 has a depleted luminosity function at the faintend, and they suspected that the orbit of NGC 6397 is suchthat tidal shocks contribute signiÐcantly to the mass loss, as

opposed to NGC 6341, 7078, and 7099. We have indicatedthese four clusters in Figure 9. Among the four clusters,NGC 6397 has the highest total destruction rate ; however,this is not due to signiÐcant tidal shocks, but rather due tothe two-body relaxation and evaporation process. This isnot unreasonable since its mass is the smallest among thefour clusters (Djorgovski 1993). Clusters with similar orlarger destruction rates than NGC 6397 (considering alsothe upper limits in destruction rates provided by theiruncertainties) are NGC 288, Pal 5, 6093, 6121, 6144, 6712,6779, and 6838. For all of these clusters except NGC 6838,the destruction rate due to tidal shocks is comparable to orlarger than the two-body relaxation rate. NGC 6144 isalready known to be a very sparse cluster, while the lumi-nosity function for NGC 6121 provided by Kanatas et al.(1995) from ground-based observations indicates depletionat the faint end.

We have also calculated the destruction rates in modelJSH95 ; these are on average larger than the ones in P90.This is primarily due to the more massive bulge (Table 4) inthe JSH95 model, which contributes directly to the destruc-tion rate. Therefore, we can regard model P90 as a lowerlimit model for the destruction rates.

9.3. Comparison with GO97 ResultsGO97 provide destruction rates for a set of 119 clusters in

four model combinations : two Galactic potential models(Ostriker & Caldwell 1983, hereafter OC, and Bahcall,Schmidt, & Soneira 1983, hereafter BSS) and two velocitydistributions of the globular cluster system: isotropic andanisotropic. We have compared their destruction rates withours in order to inspect the di†erences in initial conditionsor tangential velocities and their implications for the totaldestruction process.

It should be noted that di†erences in the rates shouldoccur due to two main reasons. First, there are the initialconditions : while their tangential velocities are derived in a

FIG. 9.ÈDetail of the bottom right panel of Fig. 8. Filled symbols represent the total destruction rates, while open symbols represent tidal shockdestruction rates. Clusters with deep HST luminosity functions are indicated.

1812 DINESCU, GIRARD, & VAN ALTENA Vol. 117

statistical way from an adopted kinematic model, we havethe actual measurements, and second, the Galactic potentialmodels are di†erent. It is also worth noting that ourdestruction rate is based on approximate formulae for tidal-shock rates, deÐned at the half-mass radius, while theirs isderived in a Fokker-Planck calculation, where tidal shocksare included in the di†usion coefficients, and is therefore amore rigorous approach.

For most of the clusters the destruction rate due to inter-nal, two-body relaxation is dominant ; therefore we shouldhave a population of clusters that show a one-to-one corre-

lation. Figure 10 shows di†erences between GO97 rates andours as a function of our rates. Error bars include only errorestimates from our rates. The left panels refer to the iso-tropic velocity distribution kinematic model, while the rightpanels refer to the anisotropic one. The top four panelsshow the GO97 rates from four combinations of modelsminus our rates from the P90 Galactic potential model. Inthe lower four panels the di†erences are with our rates cal-culated using the JSH95 model. All plots are on the samescale, and they show only a fraction of both the whole rangeof rates and of the di†erences in rates. We have plotted

FIG. 10.ÈDi†erences between GO97 destruction rates and ours as a function of our rates. The left panels refer to the isotropic velocity distributionkinematic model, while the right panels refer to the anisotropic one. The top four panels show the GO97 rates from four combinations of models minus ourrates from the P90 Galactic potential model. The bottom four panels are similar to the Ðrst four, except that our rates are calculated with the JSH95 model.OC refers to the Ostriker & Caldwell (1983) model, and BSS refers to the Bahcall, Soneira, & Schmidt (1983) model.

No. 4, 1999 SPACE VELOCITIES OF GLOBULARS. III. 1813

these ranges because we are interested in possible system-atic trends in the low-rate regime. At high rates, errors arelarger, as they increase with the size of the rate (see theabove comments). The range in the destruction rate corre-sponds to a destruction time between two Hubble times

yr) and practically inÐnite destruction time (l\(tH

\ 1010At destruction rates higher than the di†er-0.0t

H~1). 0.2t

H~1,

ences in the GO97 rates and ours bounce between positiveand negative values with slight dependences on the poten-tial models. In this range, not much global change is seen inthe di†erences as one changes from an isotropic to an aniso-tropic velocity distribution, at the same combination ofGalactic potential models. A few clusters may display somesigniÐcant change, but this is not a systematic trend.

At low destruction rates a systematic o†set(l\ 0.2tH~1),

can be seen in all model combinations, in the sense that theGO97 rates are overestimated with respect to ours. It isimportant to remark that, in this destruction rate range, noclusters were found with statistically signiÐcant negative dif-ferences, while a few clusters with positive di†erences largerthan the limit in the plot exist for some model(*l\ 1t

H~1)

combinations. In this low destruction-rate range we haveidentiÐed all clusters that have GO97 destruction ratesgreater than 150% our values in all eight model com-binations. Among these, we have selected those that showthese di†erences in three or more model combinations.These clusters and the corresponding number of modelcombinations are : NGC 104 (47 Tuc), four ; Pal 3, four ;NGC 5024 (M53), six ; NGC 5272, six ; NGC 5466, three ;NGC 5904 (M5), six ; NGC 6752, six ; and NGC 7078 (M15),six. It is unlikely that our Galactic potential models aresuch that they systematically underestimate destructionrates, since they are not substantially di†erent than thoseused by GO97. For instance, in terms of bulge destructionrate, a signiÐcant di†erence should be made by the mass ofthe bulge (see eq. [2]). Our P90 model has about the samebulge mass as BSS, while our JSH95 model has a massslightly smaller than the OC model (a factor of 0.74). Forthe disk destruction rates, it is the surface density that con-trols the process as it determines the z-component of thegravitational acceleration. A higher surface density of thedisk produces a larger destruction rate due to disk shocks.For comparison, the surface density of the disk at the SunÏslocation and within a column o z o\ 1.5 kpc is 34.5 pc~2M

_in the OC model, 90 pc~2 in the BSS model, 93.3M_

M_pc~2 in our JSH95 model, and 69.1 pc~2 in our P90M

_model. Therefore our disk models are more destructive thanthose used by GO97.

It is more likely that the initial conditions or tangentialvelocities are the cause of these di†erences. From our data,these are clusters in which the tidal shock destruction ratesare smaller or at most equal to the two-body relaxationdestruction rate. The GO97 data show total destructionrates of a few up to a hundred times larger (e.g., NGC 5466)than the two-body relaxation destruction rate. Since theselatter rates are roughly similar in both works (see ° 9.1), itfollows that the orbits considered by GO97 are prefer-entially more destructive than ours. This raises the questionof what kind of orbits these clusters are on, such that thestatistical approach of assigning tangential velocitiesappears to fail. It is perhaps not a coincidence that amongour eight selected clusters two have circular orbits (NGC104 and NGC 6752, see ° 7.1) and belong to a system with asigniÐcant amount of rotation : the thick disk. While NGC

104 and NGC 6752 have two-body relaxation destructionrates of and respectively, the other three0.07t

H~1 0.09t

H~1,

clusters with circular orbits (thick-disk kinematics, see ° 7.2)NGC 6254, 6626, and 6838 have much higher values :

and respectively (Table 3 in0.40tH~1, 0.52t

H~1, 3.40t

H~1,

GO97). For these last three clusters, the orbit-associateddestruction is less important than the internal destructionprocess, and the potentially wrong assigned tangentialvelocities are not relevant here. Thus, it is apparent that thekinematic model predicts orbits for thick-disk clusters thatsystematically lead to higher destruction rates than thosederived had the orbits been circular. This is not unreason-able, since the kinematic model treated the globular clustersystem as a whole, with kinematic features representative ofthe halo only (see GO97, their Table 2).

The rest of the six clusters that have systematically over-estimated rates are on orbits of high total energy and largesize (Table 5). The kinematic modelÈwhile possibly provid-ing a correct eccentricity most of the timesÈhas no know-ledge of the total orbital energy of a cluster and therefore ofthe frequency of pericentric and disk passages that producethe tidal shocks. GO97 have tested their initial conditionsfor consistency with observations using as a constraint thetheoretical relation between the tidal radius of the clusterand the orbital pericentric radius. However, the exact formof this relation is still under debate since the theory (King1962 ; Innanen et al. 1983 ; Oh & Lin 1992) does not agreewith the observations (Meziane & Colin 1996). Also, GO97did not apply constraints from tidal radius for core-collapsed clusters, and among our eight selected clusters,two (NGC 6752 and NGC 7078) are core-collapsed clusters.

If, indeed, some of the clusters in the low-destruction raterange have overestimated rates, then the shape of the dis-tributions of the destruction times in GO97ÈtheirFigures 25 and 26Èmay change in the range 5t

HÈ100t

H,

and thereafter some of the conclusions related to thedestruction of the globular cluster system, based on the Ðtsof these distributions, may be subject to revisions.

We would like to end this section cautioning that there isa potential for systematically overestimating destructionrates in cases in which tangential velocities are unknownand, instead, are treated statistically. The comparison pre-sented here warrants a more rigorous and consistentanalysis of this problem, which includes the best theory athand, as well as the accurately measured tangential veloci-ties.

10. CONCLUSIONS

We have used the most extensive sample of globular clus-ters with measured tangential velocities in order to investi-gate the contribution of this information to our generalpicture of the globular cluster system. A future signiÐcantincrease of this sample will be possible only if more efficient,nontraditional studies are designed for this purpose.

At the present level of errors in the measured quantitieswe Ðnd that orbital parameter di†erences due to di†erentGalactic potential models are not signiÐcant. If errors inproper motions and distances improve by a factor of 2,potential-model di†erences start to become signiÐcant. Thekinematics, based on all three components of the velocity,show that clusters outside the solar circle have a largervelocity dispersion, in each component, than those insidethe solar circle. Clusters classiÐed as RHB show radial

1814 DINESCU, GIRARD, & VAN ALTENA Vol. 117

velocity anisotropy, while BHB and MP clusters have iso-tropic velocity distributions.

Three metal-poor clusters (NGC 6254, 6626, and 6752)have orbits similar to those of metal-rich, thick-disk clus-ters. From age arguments it is suggested that materialbecame organized in a disk existed fairly early in the historyof the Galaxy, and pressure and density conditions in thisdisk were such that it was able to form globular clusters.

Among the three subsystems deÐned by Zinn (1996 ;BHBÈold halo, RHBÈyoung halo, and MP) it is only theBHB system that shows a correlation between Ñatness androtation. This correlation remains to be veriÐed as moreclusters of this type are measured. It is not clear, from ourdata, what the structure of the BHB system is. It is oftenconjectured to be a component of a more ordered collapse,with a mean rotation lower than that of the thick disk, butsigniÐcantly nonzero. Orbits of clusters in this system covera wide range in characteristics, from circular to highlyinclined and/or highly eccentric. More data for clusters ofthis type are required in order to make deÐnitive state-ments. Clusters classiÐed as RHB have highly energetic,large-size orbits and highly eccentric orbits. Their orbitcharacteristics are consistent with their hypothetical originin satellite systems. These satellites, originally located in theoutskirts of the Galaxy, su†ered less interfragment colli-sions and therefore less loss of kinetic energy, escaping themore uniform process of the GalaxyÏs collapse, and evolvingas independent systems. Also based on the characteristics oforbits, some evidence is presented in favor of an accretedorigin of u Cen.

We have made a tentative search for cluster streams inour sample and Ðnd only one hypothetical group of twoclusters : NGC 362 and NGC 6779. This problem is,

however, less tractable than the search for satellite streamsin the remote halo and among the Ðeld-star component.

Destruction processes of globular clusters are dominatedby internal relaxation and evaporation. Tidal shocks due tobulge and disk are more important or comparable to theinternal relaxation for only a handful of clusters : NGC 288,Pal 5, NGC 5139, 6121, 6144, 6362, 6712, and 6779. Amongfour clusters with deep HST luminosity functions (Piotto etal. 1997), NGC 6397 shows depletion at the faint end. Weshow that its orbit is not necessarily responsible for thisdepletion. Clusters with similar or larger destruction ratesthan that of NGC 6397 are NGC 288, Pal 5, NGC 6093,6121, 6144, 6712, 6779, and 6838. From the comparison ofour destruction rates with those of GO97, we show thatorbits derived based on tangential velocities statisticallyassigned from an adopted kinematic model are prefer-entially more destructive than the orbits derived from mea-sured tangential velocities. This is true both for clusters thathave thick-diskÈlike motion and for some clusters that havehigh orbital energy.

We would like to thank Robert Zinn and Richard Larsonfor helpful comments and discussions regarding the globu-lar cluster system, Oleg Gnedin for his data and discussions,and Ian Walker for his comments regarding retrogradeorbitsÏ decay. We are very grateful to Tad Pryor, whosesuggestions as referee of this paper have resulted in numer-ous improvements over its original version. In particular,his comments concerning the plots of inclination versusorbital angular momentum were most helpful. This researchwas supported in part by grants from the National ScienceFoundation to Yale University and the Yale SouthernObservatory.

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