space distribution and orbits of globular

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WANG Long et aL / Chinese stronomy and Astrophysics 27 2003) 42-65 43

1 . I N T R O D U C T I O N

T h e s t u d y o f t h e s p ace d i s t r i b u t i o n , o rb i t s an d d y n am ica l p ro p e r t i e s o f t h e g lo b u l a r c l u s t e r s

i n t h e Ga l ax y i s im p o r t an t fo r u n d e r s t an d in g t h e b as i c p h y s i ca l p ro ces s e s an d d y n am ic f ea -

tu res o f ga lax ies [1-31 . So fa r , ab ou t 147 g lobu lar c lus te rs hav e been d i scovere d in the G alaxy ,

an d fo r 1 20 o f t h e s e w i th g a l ac to cen t r i c d i s t an ces l e ss th an 4 0 k p c , t h e r ad i a l v e lo c i t ie s h av e

b een m eas u red . Us in g t h e s e d a t a , m an y p eo p l e h av e s t u d i ed th e i r s p a t i a l an d k in em a t i c

p ro p e r t i e s an d acq u i r ed s o m e im p o r t an t s t a t i s ti c a l r e s u l t s [4 -6 ].

In 1 9 96 , Ha r r i s [7] p re s en t ed h i s C a t a lo g u e o f Pa ram e te r s fo r G lo b u l a r C lu s t e r s i n t h e

M i lk y W ay , i n wh ich p rec i s e r ad i a l v e lo c i t y d a t a o f t h e 1 20 g lo b u l a r c l u s t e r s we re l i st ed .

In 1 99 6, D an p h o l e e t a l . Is] f i rs t i n t ro d u ced ab s o lu t e p ro p e r m o t io n s i n to t h e k in em a t i c

s tu d y , t h u s ex t en d in g t h e s t u d y f ro m o n e -d im en s io n a l r ad ia l v e lo c i t ie s t o 3 -d im en s io n a l

s p ace m o t io n s . T h en , Od en k i rch en e t a l . [9] c a l cu l a t ed an d d i s cu s s ed t h e o rb i t s o f g lo b u l a r

c l u s t e r s b a s e d o n t h e i r o w n m e a s u r e m e n t s a n d t h e H i p p a r c o s a b s o l u t e p r o p e r m o t i o n d a t a .

In 1997-1999 , D inescu e t a l . [ l -12] p ub l i shed a se r ies o f pap ers , p res en te d th e a bso lu te p rop er

m o t i o n d a t a m e a s u r e d b y t h e m s e l v e s f o r 1 5 s o u t h e r n s k y g l o b u l a r c l u s t e r s a n d c a l c u l a t e d

the i r o rb i t s . In the i r 1999 pap er , Dine scu e t a l . [12] p rese n ted a ca ta logu e o f the 38 g lobu lar

c lu s t e r s w i th m eas u red ab s o lu t e p ro p e r m o t io n s , an d m ad e s o m e m ean in g fu l co n c lu s io n s o n

the i r sp ace ve loc i t i es, o rb i ta l fea tu res , e tc . . A nd in Ju ne 1999 , H arr i s [13] ha d h i s par am ete r

ca t a lo g u e o f 1 9 9 6 r ev i s ed an d p u b l i s h ed o n t h e i n t e rn e t , i n wh ich t h e b as i c p a ram e te r s o f

147 known g lobu lar c lus te rs were p resen ted fo r epoch 2000 .0 , so p rov id ing a bas i s fo r deeper

s tud ies o f g lobu lar c lus te rs .

T h e p u rp o s e o f t h i s p a p e r is , fo r a s am p le o f 2 9 g lo b u l a r c lu s t e r s w i th i n t eg ra t ed s p ec t r a l

t y p e F , we ca lcu l a t e t h e ir s p a t i a l an d v e lo c i t y d i s t r i b u t i o n s u s in g t h e n ew p a ram e te r s g iv en

b y H a r r is a n d t h e a b s o l u t e p r o p e r m o t i o n d a t a o f D i n e s cu e t a l. W e t h e n t a k e t h e s e a s i n it ia l

co n d i t i o n s an d co n t i n u e t h e i r o rb i t s n u m er i ca l l y fo r t h ree d i f f e ren t g rav i t a t i o n a l p o t en t i a l

m o d e l s , an d d i s cu s s h o w th e o rb i t s an d r e l a t ed p a ram e te r s ev o lv e .

2 . T H E S A M P L E A N D D A T A P R O C E S S I N G

2 1 S o u r c e o f t h e S a m p l e

T h e c a t a l o g u e o f D i n e s c u e t a l. l is ts t h e m e a s u r e d a b s o l u t e p r o p e r m o t i o n s o f 3 8

g lo b u l a r c lu s t e r s. Am o n g t h e s e 2 9 a re o f i n t eg ra t ed s p ec t r a l t y p e F , w i th m e ta l l i c it i e s l es s

t h an -0.8 : t h ey a re h a lo c lu s t e rs . So , we t ak e t h e s e 2 9 g lo b u l a r c lu s t e r s a s o u r s am p le , an d

s tu d y t h e i r s p ace d i s t r i b u t i o n , v e lo c it i es an d o rb i t a l f ea tu re s .

T ab l e 1 l i st s t h e b as i c p a ra m e te r s o f o u r s am p le c lu s t e r s. In t h e t ab l e , t h e h e l i o cen tr i c

equ ato r ia l coor d ina te s , he l iocen t r ic d i s tances , rad ia l ve loc i t i es , e tc . com e f rom Ha rr i s [13],

and the abs o lu te p ro per m ot ion s co me f rom Dine scu e t a l . [12].

2 2 S o m e B a si c C o n v e n t i o n s i n t h e D a t a P r o ce s s i ng

I ) E p o c h : A l l t h e c o o r d i n a t e s a n d p a r a m e t e r s u s e d i n t h i s p a p e r c o r r e s p o n d t o e p o c h

2000.0[141;

(2) C o o r d in a t e s y s t em : W e u s e t h e l e f t -h an d , g a l ac to cen t r i c Ga l ac t i c r ec t an g u l a r co -

o rd in a t e s y s t em O -

XYZ, as

s h o wn in F ig . l , t h e X - Y p l an e is t h e G a l ac t i c p l an e , t h e

d i r ec ti o n f ro m th e Ga l ac t i c cen t e r t o t h e cen t e r o f t h e s u n i s t h e p o s i t i v e d i rec t i o n o f the


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X-axis , the posi t ive direct ion of the Y-axis is the direct ion of the Galac t ic rota t i on, and

according to the left -hand rule , the posi t ive Z-axis points to the north Galac t ic pole. U, V,

W are the three comp onen ts of the space veloci ty in the X, Y, Z direct ions, respect ively;

Ta b le 1 Ba s i c pa ra me te r s o f t he s a mpl e

l u s t e r s

O ~ ( ~ R ( : ) V r a d ~ c ~ C 0 8 ~ ~ /' 6

N G C S p T h m s o t k p c ( k m s - 1 ) m a s y r - I m a s y r - *

362 F 9 01 03 14.3 -70 50 54 8.54-0 .9 223.54-0.5 5.074-0.71 -2.554-0.72

1851 F 7 05 14 06.3 -40 02 50 12.14-1.2 320.5+0.6 1.284-0 .68 2.394-0.65

1904 F 5 05 24 10.6 -24 31 27 12.94-1.3 207.54-0.5 2.124-0.64 -0.024-0.64

2298 F 5 06 48 59.2 -36 00 19 10.74-1.1 149.44-1.3 4.054-1.00 -1.724-0.98

4147 F 2/3 12 10 06.2 18 32 31 19.34-1.9 183.04-1.0 -1.854-0.82 -1.304-0.82

,4590 F 2/3 12 39 28.0 -26 44 34 10.24-1.0 - 95.14-0.6 -3.764-0 .66 1.794-0.62

5024 F 6 13 12 55.3 18 10 09 18.34-1.8 -79.14-4.1 0.504-1.00 -0.10+1.00

5139 F 5 13 26 45.9 -47 28 37 5.34-0 .5 232.54-0.7 -5.084-0,35 -3.574-0.34

5272 F 6 13 42 11.2 28 22 32 10.44-1.0 -147.14-0.4 -1.104-0.51 -2.304-0.54

,5897 F 7 15 17 24.5 -21 00 37 12.84-1.3 101.74-1.0 -4.93+0.86 -2.334-0.84

,5904 F 7 15 18 33.8 02 04 58 7. 54 -0 .8 51.84-0.5 5.074-0.68 -10.74-0.56

6093 F 6 16 17 02.5 -22 58 30 10.0 4-1.0 7. 34-4 .1 -3.314-0.58 -7.204-0.67

6121 F 8 16 23 35.5 -26 31 31 2. 24 -0 .2 70.44-0.4 -12.54-0.36 -19.934-0.49

6144 F 5/6 16 27 14.1 -26 01 29 10.34-1.0 189.44-1.1 -3.06+0.64 -5.114-0.72

,6205 F 6 16 41 41.5 36 27 37 7.74-0.8 - 246.64-0.9 -0.904-0.71 5.504-1.12

6218 F 8 16 47 14.5 -01 56 52 4. 94 -0 .5 43 .54-0.6 1.304-0.58 -7.834-0.62

6254 F 3 16 57 08.9 -04 05 58 4. 44 -0 .4 75.54-1.1 -6.004-1.00 -3.304-1.00

,6341 F 2 17 17 07.3 43 08 11 8.24-0.8 - 120.54-1.7 -3.304-0.55 -0.334-0.70

6397 F 4 17 40 41.3 -53 40 25 2.3 4- 0.2 18.94-0.1 3.300.50 -15.204-0.60

6584 F 6 18 18 37.7 -52 12 54 13.44-1.3 222.94-0.5 -0.224-0.62 -5.794-0.67

6626 F 8 18 24 32.9 -24 52 12 5. 74 -0 .6 15.84-1.0 0.304-0.50 -3.404-0.90

6656 F 5 18 36 24.2 -23 54 12 3.24-0.3 - 149.14-0.6 8.604-1.30 -5.104-1.30

,6712 F 9 18 53 04.3 -08 42 22 6.94-0.7 - 107.74-0.6 4.204-0.40 -2.004-0.40

6752 F 4/5 19 i0 51.8 -59 58 55 4.04-0.4 - 32.14-1.5 -0.694-0.42 -2.854-0.45

6779 F 5 19 16 35.5 30 11 05 10.14-1.0 - 135.94-0.9 0.304-1.00 1.40+0.10

6809 F 4 19 39 59.4 -30 57 44 5.44-0 .5 174.94-0.4 -1.424-0.62 -10.254-0.64

7078 F 3/4 21 29 58.3 12 I0 01 10.34-1.0 - 106.64-0.6 -0.954-0.51 -5.634-0.50

7089 F 4 21 33 29.3 -00 49 23 11.54-1.2 - 3.14-0.9 5.904-0.86 -4.954-0.86

7099 F 3 21 40 22.0 -23 10 45 8.0+0.8 - 184.34-1.0 1.424-0.69 -7.714-0.65

(3) In the abov e co or din ate system , the solar mot ion is (-10.4, 14.8, 7 . '3 )k m/s [15], the

distance between the sun an d the Galact ic center is 8.5 kpc, and a t the posi t ion of the sun,

the veloci ty of the Galact i c r ota t ion is 220.0km/s[12];

(4) In the hel iocentric equatoria l rectangular coordinate system, the posi t ion angle of

the north Galact ic pole is :

O~NGP 12h51rn34'-15, ~NGP 265 1'23 .18, 80 = 1230'0 ,

in which, the thir d angle 80 is the included angle between the great c i rc le passing t hro ugh

the north celest ia l pole and the great c i rc le passing throug h b oth the no rth Galac t ic pole

and th e zero point of Galact i c longi tude;

(5) Notations:

7r + a~ repr esents the par all ax an d its uncertaint y, in units of arcsec;

Vrad av repr esents the }adial velo city and its uncertain ty, in units of km /s ;

#a a ,~ represents the absolute proper motion in r ight ascension and i ts uncerta inty, in

uni t s of mas / yr ;


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#6 ~ a ~ represents the absolute proper motion in declination and its uncertainty, in units

of mas/yr.

2.3 Derivat ion of the Co ordi nate s and

Velocities

To obtain the coordinates and ,velocities of

the sample clusters in the galactocentric Galac-

tic coordinate system requires some data pro-

cessing and reduction. In 1987, Johnson et

al.[15] proposed a method of calculating the

space velocity of Galact ic clusters and its uncer-

tainty. Our reduction used Johnson's method.

2.3.1 Derivation of the Coordinates

This involves the coordinate transformation

from the heliocentric equatorial rectangular co-

ordinates to the galactocentric Galactic rectan-

gular coordinates. Usually it can be realized

by the rotation matrix of the coordinate sys-

tem. If I and b are the Galactic longitude and

Galactic declination of a sample cluster, then its

N O R T H G A L A C T I C P O L E

Fig. 1 The variation of polar shift

galactocentric Galactic rectangular coordinates can be obtained by the following transfor-

mation:

i x ] [ 8 x o ] rR o c o s b c o s z ] r c o s o s

Y = Y o = I R o c o s b s i n l =T . /s in ac os 5 , (1)

Z Z0 L Ro sin b L sin 5

in which, T is the t ransformation matrix. From Table 1 and the parameters defined in

section 2.2, we have

-0.05272 -0.87243 -0.48589]

T = 0.49084 -0.44638 0.74821/ (2)

-0.86965 -0.19905 0.45176J

2.3.2 Derivation of the Velocities

In the same way, the space velocities of the sample clusters in the galactocentric Galactic

rectangular coordinate system can be derived:

[ -(V~ad + 10.4) ]

= B ~#a cos 5/7r + 14.8 + 220.0

,p~/~ + 7.3

(3)

in which,

B = T .

cos a cos 5 - sin a - cos a sin 5

sin a cos 5 cos a - sin a sin 5

sin 5 0 cos 5

(4)


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=4.74057) is the well- known conversion factor from AU/ yea r to km/ s.

2.3.3 Calculati on of the Uncert a int ies

Errors in the radial velocity, absolute proper motion, parallax and other observational

dat a will cause uncert a inti es in the calculated space coordinates an d velocities , and these

can be calculated from the varia tion of a multiv aria t e function.

If F x , y , z ) i s a mult i var ia te funct ion, then i ts uncer ta int y can be der ived f rom the

following varia tional equ ation:

2 O F 2 2 O F 2 2 O F 2 2

= + ) % + ) ( 5 )

By applying this equatio n to Formu lae 1) and 3), we can derive the unc erta int ies of the

space coordinates and the three components of velocity.

T a b l e 2 T h e c o o r d i n a t e s a n d v e l o c i ti e s o f t h e s a m p l e c l u s t e rs

NGC Fe/H X Y Z Rgc U V W

kpc) kpc) kpc) kpc) km s - 1 km s- 1 km s- 1

362 -1.16 5.44-0.6 -3.24-0.3 -7.2+0.8 9.64-0 .7 32.34-30 -71.84-29 -72.94-22

1851 -1.22 12.74-1.3 3.04-0.3 -10.9-4-1.1 17.14-1.2 238 .04-38 -3.84-26 -101.64-33

1904 -1.57 16.1+1.6 4.3 4-0 .4 -9.54-1.0 19.24-1.5 121.94-32 -29.04-31 12.74-36

2298 -1.85 12.74-1.3 1.24-0.1 -9 .84-1.0 16.1-4-1.2 -77 .44 -47 -0 .44-26 118.34-51

4147 -1.83 9.84-1 .0 -5.94-0.6 -18.34-1.8 21.64-1.6 91.04-75 -15.44-76 140.94-17

,4590 -2.06 4.44-0 .4 3.04-0.3 -8.94-0.9 10.44-0.8 211.84-34 248.14-23 12.04-25

5024 -1.99 5.74-0.6 16.24-1.6 -8.14-0.8 18.94-1.4 -38.24-86 258.94-87 -75.54-16

5139 -1.62 5.34-0.5 0.94-0.1 -4.14-0.4 6.74-0.5 -56.04-12 -41. 44-1 2 0.24-11

5272 -1.57 7.04-0 .7 7. 44-0 .7 7.24-0.7 12.54-0.7 -19.44-26 -93.44-29 -124.5-4-05

,5897 -1.80 -2.14-0.2 6.24-0 .6 -3.74-0.4 7.5 4-0 .6 35. 04-32 -88.84-58 121.64-45

,5904 -1.29 3.44-0 .4 5.54-0 .6 0.54-0 .1 6.54-0.5 -334.74-33 -72.74-27 -212.9-4-30

6093 -1.75 -0.84-0.1 3.3 4-0 .3 -1.34-0.1 3 .74-0.3 -10.24-11 -126.24-47 -97.1+30

6121 -1.20 6.44-0 .6 0.64-0 .1 -0.34-0.0 6.54-0.6 -51.54-03 -16.2-4-25 -14.04-06

6144 -1.73 -1.34-0A 2.84-0.3 -1.44-0.1 3.44-0.2 -165.6-4-10 -7 5. 64 -4 4 8.1-4-32

,6205 -1.54 5.54-0 .6 2.64-0.3 6.64-0.7 9.04-0.6 271.54-41 144.54-25 -113.44-20

6218 -1.48 4.34- 0.4 2.1 4-0 .2 1.34-0.1 5.04-0.4 -137.74-12 126.14-18 -81.44-16

6254 -1.52 4.64- 0.4 1.74-0 .2 1.24-0.1 5.04-0.4 -83.34-10 126.04-24 101.94-20

,6341 -2.29 6.0-4-0.6 1.74-0.2 7.64- 0.7 9.9+0.7 27.3-4-26 66.14-17 . 42.44-21

6397 -1.95 6.44-0.6 -0.44-0.0 -0.94-0.1 6.54-0.6 40.6+07 119.54-12 -107.7=h13

6584 -1.49 3.84-0 .4 -3.64-0.3 -4.14-0.4 6.64-0.4 -73.94-26 -146.74-51 -181.4-4-41

6626 -1.45 2.94-0.3 -0.54-0 .1 0.84-0 .1 3.04-0.3 -32.04-03 159.04-24 -43.14-17

6656 -1.64 5.44-0.5 -0.44-0.0 0.64-0 .1 5.44-0.5 152.44-05 194.24-19 -123.44-25

,6712 -1.01 2.34-0.2 -0.54-0.0 3.0 4-0 .3 3.8 4-0 .3 98.14-06 186.54-12 -136.44-20

6752 -1,56 5.24-0.5 -1.64-0.2 -1.64-0.2 5.7 4-0 .5 37. 14-05 194.94-09 24.04-07

6779 -1.94 3.94-0.4 0.74-0 .1 9.04-0 .9 9.84-0.8 111.44-18 143.54-29 4.64-43

6809 -1. 81 3.64-0.3 -2.14-0.2 0.84 -0.1 4.24-0.3 -190.14-07 -1.04-31 -106,14-15

7078 -2.25 4.64-0.4 -2.04-0.2 9.34-0.9 10.64-0.8 -162.84-30 -11.14-22 -66.14-25

7089 -1.62 2.94-0.3 -4.04-0.4 9.24-1.0 10.54-0.9 97 .2 4- 42 7.34-42 -328.84-51

7099 -2.12 3.64-0.4 -5.24-0.5 3.64-0 .4 7.34-0.5 64.14-21 -103.84-37 53.64-20

2 4 R e s u l t s o f D a t a R e d u c t i o n

The derived space coordinates, velocit ies and other re la ted par amet ers of the 29 sample

clusters are l is ted in Table 2. In this table , Col umn 1 is the N GC nu mb er of the c luster;

Column 2 is the metall ic ity [Fe/H]; Columns 3-5 are the galatocentric coordinates X, Y,

Z an d their uncerta int ies; Col um n 6 is the galactocentri c distan ce R g c and unce r ta in ty ;

Colu mns 7-9 are U, V, W, the comp one nts of the space velocity in the d irections of X,


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47

Y, Z, and uncertainties. Asterisks mark those clusters used in the discussion of orbital

morphologies below.

3. S PA CE D IS TR IBU T IO N A N D V ELO CITIES O F TH E S A MP LE

CLU S TERS

Based on a study of the metallicities and motions of the globular clusters in the Galaxy,

Zinn[4] and Armondroff 18] proposed that globular clusters consist of two subsystems belong-

ing respectively to the halo population and the disk population and that the halo subsystem

has a low metallicity ([Fe/H]< -0.8), a spherical space distribution, and properties similar

to the halo field stars while the disk subsystem has a high metallicity, and properties similar

to the thick-disk field stars. Their proposition has been confirmed by many observations

and theories. Further observations indicate that the number density of the globular clus-

ters varies with the galactocentric distance, and for galactocentric range 3--20 kpc, it varies

as Rg 35. Thus, most of the globular clusters, especially the halo clusters, are distributed

within galatocentric distance 10 kpc, and have a spherically symmetrical distribu tion around

the Galactic center.

Next, based on the results listed in Table 2, we will make some analyses and comparisons

on the space position and velocity distribution of the 29 halo clusters of our sample.

3.1 The Space Dis tri but ion

(1) The three-dimensional distribution of our sample clusters in

X Y Z

is shown in

Fig.2, and and their distribution in galactocentric distance is shown by the histogram of

Fig. 3 (the curve is the best Ganssian fit). These figures indicate tha t the clusters exhibit

a spherically symmetric distribution around the Galactic center, that they are somewhat

concentrated toward the Galactic plane, and that most of them are within 10 kpc, and the

number density has a peak between 5 and 10 kpc. These results are in accordance with the

conclusion of Zinn et al.

(2) The studies made by LIN Qing et al. [17 18] indicate that the metallicity distribution

of the globular clusters in the Galaxy exhibits a double-peak structure. The two peaks given

by Gauss fitting are: [Fe/H]= -1.58, a = 0.33 (metal-poor clusters), and [Fe/H]= -0.54,

a = 0.21 (metal-rich clusters). The metallicities of our sample clusters are [Fe/H] < -0 .8 , so

there is only one peak in our sample.

Fig.4 is the metallicity histogram and Gauss fitting of our sample clusters: it has a

peak at [Fe/H]-~ -1.6. This result indicates tha t for the halo globular clusters of the same

integrated spectral type, the number of the clusters as a function of metallicity is consistent

with the above conclusion.

3.2 Velocity Fea tur es

(1) The space velocities given in Table 2 are plot ted in the 3-dimensional U - V - W

space in Fig.5. I t shows tha t the space velocities of clusters with the same integrated spectral

type exhibit apparently an ellipsoidal distribution, with the dispersion obviously greater in

the Galactic plane than perpendicular to it.

(2) Based on his deep study on the distribution, motions and physical features of the

globular clusters in the Galaxy, Zinn[ 19,2] suggested th at subsystems of globular clusters


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4 8 W AN G Long e t a l. / Chinese As t ron om y and Astrophysics 27 2003) 42-6 5

20 :'- :

. . . . . . . . . . . . .

v - z ' ~ i ~ ~

15

. . . 5

0 ~Y,~YPlan. l ~ ' ; ~ . ~ L~--

- ' ~ . . ' , ~ ~

i . ~ , Y

-5 ~, ~

o

-20 L '~ X-Z P'ian: . . . . . . . i ~ 1 15

- L s . i ~ . . - : : ; ~ 5 '

o ' 6 5 x

Y ( k p ~ ) ~ (kpc)

Fig. 2 Distribution of the sample clusters in space

1 . 0 '

0 . 8

~ 0 . 6

o

0.4

0.2

0 . 0 -

. . . . . . . . . .

0 5 l0 15 20 25

RGC (KPC )

Fig. 3 Th e number of sample duste rs as a function

of the galatocentric distance

l.O

0 . 8

~ 0 . 6

0 . 4

0.2

0.C

-2.5 -2.0 -1.5 -1.0 -0.5 ~ 0

~em

F ig .4 T h e n u m b e r o f s a m p l e c l u s te r s a s a ~ n c t i o n

ofmetM l i c i~

3

: 1 I

1 1 I

:: 4

Fig. 5 Distribution of space velocities of the sam-

ple clusters

w i t h d i f fe r e n t p r o p e r t i e s ( s p a c e d i s t r i b u t i o n , r a d i a l v e l o c i ty , v e l o c i t y d i s p e r s i o n , a g e , e t c . )

m a y h a v e d i ff e r e n t m e c h a n i s m s o f o r ig i n . T h e h a l o c l u s te r s c a n b e f u r t h e r d i v i d e d i n t o t w o

s u b s y s te m s . O n e i s t h e h o r i z o n t a l - b r a n c h ( H B ) s u b s y s t e m w i t h m e t a l li c i ty [ F e / H ] < - 0 . 8 ,

i ts e l f f u r t h e r d i v i d e d i n to a r e d h o r i z o n t a l b r a n c h ( R H B ) a n d a b l u e h o r i z o n t a l b r a n c h

( B H B ) . T h e o t h e r is t h e m e t a l - p o o r ( M P ) s u b s y s t e m w i t h m e t a l li c i t y [ F e / H ] < - I . 8 .

I t is c o m m o n l y b e li e v e d t h a t t h e R H B c l u s te r s h a v e a sp h e r i c a l s p a c e d i s t r i b u t i o n ,

a l a rg e r v e lo c i t y d i s p e r s io n a n d s m a l le r r o t a t i o n a l m o t i o n s , a n d a r e d i s t r i b u t e d m o s t l y i n

t h e r e g io n m o r e t h a n 6 k p c f r o m t h e G a l a c t ic c e n te r , a n d t h a t t h e y m a y h a v e b e e n f o r m e d

l a t e r in s o m e s a t e ll it e g a l a x y f o ll o w in g a c c r e t i o n o r s p l i tt i n g o f t h e G a l a x y . I n c o n t r a s t , t h e

B H B c l u s t e r s h a v e m a r k e d r o t a t i o n s , a f l a t e l l i p s o i d a l s p a c e d i s t r i b u t i o n , a n d m e t a l l i c i t i e s

v a r y i n g o b v i o u s ly w i t h t h e g a l a c t o c e n t r ic d is ta n c e . T h e y m a y h a v e b e e n f o r m e d n o r m a l l y

i n t h e p r o c e s s e s o f g e n e r a l c o l la p s e . D i f f e re n t f r o m t h e R H B c l u s te r s , t h e M P c l u s t e r s a r e

d i s t r i b u t e d i n t h e i n n e r r e g i o n o f t h e G a l a x y , t h e i r g a l a c t o c e n t r i c d i s t a n c e s a r e g e n e r a l l y


8/24

W AN G Long e t a l. / Ch inese st r o n om y a n d A s t r o p h y s i c s 2 7 ( 2 0 0 3) 2 - 6 5

49

in the range 2.7--8kpc. As they were being formed during the collapse, their metallicities

gradually increased, their rotation speeded up, and their space distribution got flatter: they

belong to the oldest branch in the Galaxy.

Table 3 lists the type or subsystem RHB, BHB or MP) to which each of our sample

clusters belongs, and Figs.6 a--c) show, separately for the three subsystems, circles for

MP, triangles for BHB, filled circles for RHB), the variations of the three components of the

space velocity with galatocentr ic distance.

Table 3 The

b e l o n g i n g t y p e s o f t h e s a m p l e c l u s t e r s

362 RHB 5204 MP 6121 BHB 6397 MP 6779 MP

1851 RHB 5139 MP 6144 MP 6584 RHB 6809 MP

1904 BHB 5272 RHB 6205 BHB 6626 BHB 7078 MP

2298 MP 5897 MP 6218 BHB 6656 BHB 7089 BHB

4147 RHB 5904 RHB 6254 BHB 6712 BHB 7099 MP

4590 MP 6093 BHB 6341 MP 6752 BHB

From Figs. l-6 , showing the distributions and the correlations of the various parameters,

we find:

1) The clusters of the same spectral type but belonging to different subsystems are

essentially in accordance with their known classification properties.

2) The results of our data processing and reduction are correct. They provide a reliable

basis for further discussions on the orbital motions of the clusters.

4. THREt~ G A LA C TIC G RA V ITA TIO N A L P O TEN TIA L MO D ELS A N D

O R B I T C A L C U L A T I O N S

4 1 A B r i e f D e s c r i p t i o n o f T h r e e G a l a ct ic G r a v i ta t io n a l P o t e n t i a l M o d e l s

To calculate and analyse the orbits of the sample clusters, we must star t with an analyt ic

model of the Galaxy. We have selected three gravitational potential models in order to study

the effect of different models on the resulting orbital behaviour.

The first model is one proposed by Paczynski 21] P90 hereafter); the second, by John-

son et al.[22] JSH95 hereafter); the third , by Danphole et al. DC95 hereafter) in which the

clusters are restricted to galactocentr ic distances < 40 kpc. In fact, more and more observa-

tions have proved tha t globular clusters of the Galaxy are mostly dist ributed within 40 kpc,

and some recent studies take such clusters as having a common origin with the Galaxy. So

it seems that the DC95 model has a universal meaning for the study of globular clusters.

All the three models are axisymmetric and consist of three components: a bulge, a disk,

and a dark halo. For all three models, the disk potential follows the axisymmetric form

proposed by Miyamoto and Nagai in 1975. For the other two components, however, the

forms of the gravitational potentials are different: for the bulge, JSH95 uses the form given

by Hernquist [24], while P50 and DC95 use the tradit ional Plummer spherically symmetric

form[25]; for the dark halo, Pg0 and JSH95 use the logarithmic form derived directly from

fitting the rotation curve of the Galaxy, while DC95 uses again the Plummer form.

For convenience, the specific forms and parameters of the three gravi tational potential

models are listed in Table 4 in which r 2 = R 2 + z2).


9/24

5 0 W AN G Long e t a l. / Chinese As t ron om v and Ast rophys ics ~7 2003) ,~2-65

300 '

200 '

, . . . , 100'

o

- ' 1 0 0

-200

- 3 0 0

-40O

0

3 0 0

2 5 0

2 0 0

150

100

> 50-

0-

- 5 0

-100

-150

0

200

1 0 0

0

~: - lO0

- 2 0 0

-300

-400

0

O

~ A

O

u o

o~

C

5 lO 5 2 o

( a ) U v s . RG c ( K PC)

I B ~ I

~ ~ I P t t B I

0 ~

~ C

0

5

10 15 20 25

( b ) V v s . RG c (K PC)

I P H B I

O && O

o ..~

0

~r,

5 10 15 2 0 25

( c ) W v s RGC ( K PC)

F i g . 6 T h e t h r e e c o m p o n e n t s o f t h e s p a c e v el o c it ie s o f s a m p l e

c l u s t e r s p l o t t e d a g a i n s t g a l a c t o c e n t r i c d i s t a n c e


10/24

W A N G L o n g e t a l. / C h i n e s e A s t r o n o m y a n d A s t r o p h y s i c s Z 7 2 0 0 3 ) ,~ 2 - 65

5

T a b l e G a l a c t ic p o t e n t i a l m o d e l s

Model

P90

JSH95

DC95

potential model and parameters

P 9 0 :

x, y, z)

= ~ b u l g e + ~ d i s k + O h a l o -~ - 0 0 = - -

( I M b G M d . . .. . ..

R 2 - ~ ( a b +

z 2 + b 2 ) 2

R 2 + ( a d ~ -

z 2 + b 2 ) 2

M b = 1.12 x 1 0 1 M , a b = 0. k p , b b = 0 . 2 7 7 k p e , M d = . 07 x 1 0 1 M , a d = 3.7kpe

b d = 0 . 2 0 k p c , M h = 5.0 x 1 0 1 M , d = 6.0kpe, Oo = -1 2. 3 x 104 km s-l ) 2

O b u l g e , O d i s k , O h a l o , 0 0 )

G M b _ G M ~

J S H 9 5 : O ( x ' Y ' Z ) = O b u l g e + O d i s k }- O h a l + O O = - - r + c ~ / R 2 + ( a d + ~ ) 2

122 r 2

+ o ln 1 + d--~-) + O0

M b = 3.4 101 M, c = 0. 7kp c, Md = 10 1 0 1 M , a d = 6.5 kpc, bd = 0.26 kpc

uo : 128 km/ s,d = 12.0kpc,~o = -5. 2 104 km s- l) 2

I )bu l se , Cd i sk , I )ha l o , 00 )

= -- C~M , C4M d ,,

G M h

M b = 1.396 x 1 0 m M , b b = O . 3 5k p e, M d = 7.908 x 1 0 1 0 M O , a d ---- 3.55 kpc

b h = 2 4 . 0 k p e , b d = 0 . 2 5 k p c , M h = 6.978 x 1OHM

O b u l s e , O d i s k , O h a l o )

4.2 The Orbit Calcul at ions

1) Equation of motion

In the galatocentric coordinate system O -

X Y Z

shown in Fig.l, the equations of

motion of the cluster are:

d V x O ( x , y , z ) d x

d t Ox d t Vx ,

dV ~ _ O ~ ( x , y , z ) d Y = v y ,

d t Oy d t

d V z O ~ ( x , y , z ) d z

dt - Oz d- - t= V z

6 )

in which if x, y, z) is the Galact ic gravita tional potential, Vx = U, Vu = V, Vz = W are the

three components of the space velocity.

Based on the model parameters given in Table 6, we have:

For the P90 and JSH95 models, (x , y , z ) = butge + Cdisk + ha to + 40;

For the DC95 model,

~ (x , y , z ) = b~,lg~ +

~ d i s k - ~ ( ~ h a l o .

2) Numerical integration of the orbit.

Taking the coordinates

( X , Y , Z )

and velocity components

( U , V , W )

of the sample clus-

ters listed in Table 2 as initial values, and using the 4-order Runge-Kut ta method we numeri-

cally integrated the equations of motion and calculated the orbits and the related parameters

for all the sample clusters for the three different gravitational potential models. The step

length of integration was 105 yr, and the tota l integration time was 10 l yr 10 Gyr).


11/24

52 W A N G L ong e t a l / Ch i nese A st ron omy and A st roph ysi c s ~7 2003) 42-65

5 . R E S U L T S O F O R B I T C A L C U L A T I O N A N D D I S C U S S I O N

5 .1 R e s u l t s o f O r b i t C a l c u l a t i o n s

T h e r e s u l t s o f o u r o r b i t c a l c u l a t i o n s a re s u m m a r i z e d i n T a b l e 5 w h i c h l i s t s t h e a v e r a g e

v a l u e s o f t h e o r b i t a l p a r a m e t e r s a n d t h e i r u n c e r t a i n t ie s .

T a b l e 5 T h e o r b i t a l p a r a m e t e r s a n d t h e i r u n c e r t a i n t i e s f or 9 s a m p l e c l u s t e r s

N G C t p ~ a e P r P ~ [ J [ Etot

number kpc) kpc) kpc) 106yr) 108yr) kp cm s -I 102km2s -2

362 0.884-0.29 12.224-0.19 6.55 0.05 0.874-0.05 1391 2334-2 -10440.2

0.860.31 12.464-0.21 6.660 .06 0.870.08 1364-I 2271 43464 -8084-0.5

0.840.31 11.840.21 6.340.05 0.870.06 1281 2102 -14380.0

1851 3.090.04 41.310.02 22.200.01 0.860.00 5242 8274 -5500.1

3.040.07 36.664-0.04 19.850.02 0.854-0.00 4282 6944-3 148821 -2844-0.1

2.970.06 30.390.02 16.684-0.02 0.824-0.00 3144-0 5134-1 -8944-0.1

1904 3.270.09 23.070.05 13.170.02 0.754-0.01 2934-2 4473 -7640.1

3.190.16 22.870.09 13.034-0.04 0.760.01 2672 426 2 140336 -492 0.3

3.084-0.10 21.674-0.04 12.374-0.03 0.754-0.01 2334-1 3724-2 -10894-0.0

2298 1.580.24 21.294-0.15 11.444-0.05 0.860.04 2584-I 4254-2 -8084-0.5

1.544-0.28 20.984-0.19 11.264-0.06 0.860.07 2364-1 3921 76858 -5470 .8

1.500.25 19.654-0.16 10.574-0.05 0.864-0.05 2064-1 3434-2 -11600.6

4147 10.210.12 27.254-0.09 18.730.01 0.464-0.01 4164-3 5904-3 -6480.0

9.084-0.17 26.900.13 17.994-0.02 0.500.01 3594-2 5184-3 31744-22 -3644-0.0

8.884-0.10 24.50 0.07 16.694-0.02 0.470.01 2932 4554-3 -9524-0.0

*4590 7.874-0.05 40.894-0.03 24.38+0.01 0.680.00 5654-3 824-4 -5330.0

7.620.10 33.510.06 20.574-0.02 0.634-0.00 4222 6283 29914-19 -2924-0.0

7.820.08 26.544-0.05 17.180.02 0.554-0.00 3034-2 4734-2 -9244-0.0

5024 5.430.02 49.164-0.01 27.294-0.01 0.800.00 6424-3 9674-4 -4794-0.0

5.240.05 42.664-0.02 23.950.01 0.780.00 5172 8013 23784-11 -2100.1

5.164-0.03 34.450.02 19.814-0.01 0.740.00 3632 5764-2 -8060.1

5139 0.694-0.09 8.190.07 4.444-0.01 0.854-0.08 921 1524-0 -12281.0

0.680.03 8.350 .02 4.510.01 0.850.02 931 1530 3317 -10094-1.0

0.654-0.07 8.070.07 4.360.01 0.854-0.08 891 1440 -16321.1

5272 6.130.29 14.580.24 10.350.02 0.414-0.03 2184-2 3042 -8864-0.0

5.784-0.33 14.904-0.25 10.344-0.04 0.444-0.03 1942 2984-2 19254-43 -6400.0

5.870.26 13.920.19 9.904-0.04 0.414-0.02 1822 2762 -12640.0

-10992.6

-8781.6

-15002.0

-5290.0

-3184-0.0

-9310.1

-14294-0.6

-12180.6

-18454-0.7

-13322.0

-11220.9

-17341.5

-14084-0.1

-12020.1

-18260.1

-5904-0.0

-3574-0.0

-9870.0

6218 -12260.1

-10130.0

-16340.1

,5897 0.930.02 10.780.02 5.854-0.01 0.840.02 1250 2074-0

0.910.03 10.830.03 5.874-0.01 0.840.04 12 04 -0 1974-0 4434-6

0.890.03 10.454-0.03 5.670.01 0.840.04 1140 1860

*5904 5.900.03 42.464-0.02 24.180.01 0.760.00 5614-0 8314-0

5.910.07 32.464-0.04 19.180.01 0.690.00 3992 6084-3 246014

5.904-0.05 27.260.03 16.584-0.01 0.640.00 2994-2 4682

6093 1.470.16 5.004-0.10 3.230.03 0.550.08 63 0 1001

1.474-0.08 4.890 .07 3.180.01 0.544-0.06 654-1 990 50719

1.394-0.13 4.860.09 3.130.02 0.550.07 59 0 924-0

6121 0.220.00 6.70/:0.01 3.460 .00 0.944-0.01 72 0 1300

0.234-0.04 6.720.04 3.474-0.01 0.934-0.04 740 1394-0 1314

0.210.03 6.694-0.03 3.450.00 0.944-0.06 711 1310

6144 1.974-0.46 4.440.35 3.464-0.05 0.434-0.12 661 105+1

2.060.36 4.714-0.28 3.394-0.04 0.394-0.10 661 1011 62055

1.900.45 4.760.35 3.330.05 0.430.12 631 974-1

,6205 7.264-0.06 34.900.04 21.080.01 0.664-0.00 4862 6974-3

7.090.12 28.760.08 17.930.02 0.600.01 3630 5431 272823

7.250.10 23.880.06 15.570.02 0.530.01 2772 4292

2.810.41 7.040.34 4.920.04 0.430.08 921 1442

2.920.42 7.140.33 5.030.05 0.420.08 962 1501 91054

2.74-}-0.41 6.690.34 4.810.04 0.430.08 901 1371


12/24

W A N G L o n g e t a i . / Ch i n e se s tr o n o m y and A s t r o p h y s i cs 2 7 2 0 0 3 ) 4 2 - 6 5 5

T a b l e 5 T h e o r b i t a l p a r a m e t e r s a n d t h e ir u n c e r t a i n t i e s f or 2 9 s a m p l e

c l u s t e r s c o n t i n u e d )

N G C R p R a a e P r P C [ JI

Etot

n u m b e r ( k pc ) ( k p c ) ( k p c) ( 1 0 6 y r ) ( 1 0 S y r ) k p c m s - 1 1 O 2 k m 2 s - 2

6 2 5 4 3 . 0 7 4 - 0 . 4 7 6 . 1 1 4 - 0 . 3 8 4 . 5 9 4 - 0 . 0 4 0 . 3 3 4 - 0 .1 0 8 4 4 -1 1 3 2 4 - 2 - 1 2 6 8 4 - 0 . 0

3 .244-0 .42 6 .164-0 .32 4 .704-0 .05 0 .314-0 .08 884-1 1374-1 9154-6 0 -1054 4-0 .0

3 .004-0 .46 5 .984-0 .37 4 .494-0 .04 0 .334-0 .10 814-1 1264-1 -1676 4-0 .0

6341 1 .224-0 .29 11 .834-0 .20 6 .524-0 .05 0 .814-0 .05 1374-1 2264-2 -1054 4-0 .2

1 .914-0 .33 12 .164-0 .21 6 .674-0 .06 0 .824-0 .06 1364-1 2234-1 5594-65 -8174-0 .3

1 .174-0 .29 11 .534-0 .19 6 .354-0 .05 0 .824-0 .05 1274-1 2074-2 -14474 -0 .1

6 3 9 7 3 . 3 3 4 - 0 . 3 8 7 . 2 4 4 - 0 . 3 0 5 . 2 9 4 - 0 . 0 4 0 . 3 7 4 - 0 .0 7 9 9 4 -1 1 5 1 4 -2 - 1 1 9 5 4 - 0 . 1

3 .494-0 .36 7 .314-0 .27 5 .404-0 .05 0 .354-0 .07 1044-1 1594-1 10334-56 -9834-0 ,1

3 . 2 4 4 - 0 . 3 8 7 . 1 6 4 - 0 .3 1 5 . 2 1 + 0 . 0 4 0 . 3 8 4 - 0 . 0 8 9 7 4 -1 1 47 4- 1 - 1 5 9 7 4 - 0 . 1

6584 0 .954-0 .31 14 .824-0 .20 7 .894-0 .06 0 .884-0 .07 1734-1 2924-2 -9614-0 ,4

0 . 9 3 4 - 0 . 2 7 1 4 . 2 2 4 - 0 . 2 0 7 . 5 8 4 - 0 .0 5 0 . 8 8 4 - 0 .0 9 1 5 5 4 - I 2 5 9 4 -1 4 7 6 4 - 6 4 - 7 4 2 4 - 0 . 7

0 .914-0 .34 13 .644-0 .22 7 .284-0 .06 0 .884-0 .09 1464-1 2434-1 -1364 4-0 .5

6626 1 .454-0 .37 4 .014-0 .27 2 .734-0 .05 0 .474-0 .13 514-1 834-1 -15214 -0 .1

1 .444-0 .27 3 .844-0 .23 2 .644-0 .02 0 .464-0 .10 554-1 814-1 4634-35 -13064 -0 .1

1 .374-0 .33 3 .884-0 .26 2 .624-0 .04 0 .484-0 .12 504-1 774-1 -19434-0 .1

6656 4 .084-0 .25 10 .314-0 .20 7 .204-0 .02 0 .434-0 .04 1414-1 2104-2 -10444-0 .1

4 .174-0 .27 10 .234-0 .20 7 .204-0 .04 0 .424-0 .04 1364-1 2104-1 13364-44 -8304-0 .1

4 .054-0 .27 9 .814-0 .21 6 .934-0 .03 0 .424-0 .04 1324-1 1954-1 -14494-0 .1

6712 2 .534-0 .38 8 .674-0 .29 5 .604-0 .04 0 .554-0 .06 1114-1 1724-2 -1154 4-0 .0

2 .664-0 .39 8 .364-0 .27 5 .514-0 .06 0 .524-0 .07 1064-1 1664-1 9074-6 7 -9584-0 .1

2 .504-0 .40 8 .224-0 .31 5 .364-0 .05 0 .534-0 .07 1014-1 1564-2 -1570 4-0 .0

6752 3 .724-0 .45 7 .404-0 .40 5 .564-0 .02 0 .334-0 .08 1054-2 1604-1 -1172 4-0 .0

3 .874-0 .45 7 .524-0 .38 5 .704-0 .04 0 .324-0 .07 1074-2 1664-1 11194-44 -9584 -0 .0

3 .674-0 .45 7 .204-0 .39 5 .444-0 .03 0 .334-0 .08 1004-2 1514-1 -1577 4-0 .0

6779 5 .074-0 .33 13 .984-0 .27 9 .534-0 .03 0 .474-0 .03 1964-2 2844-2 -9184 -0 .0

4 .924-0 .37 14 .144-0 .29 9 .534-0 .04 0 .484-0 .04 1844-2 2804-2 16904-48 -6794 -0 .0

4 .984-0 .32 13 .014-0 .24 9 .004-0 .04 0 .454-0 .03 1684-2 2524-2 -1311 4-0 .0

6809 1 .284-0 .39 7 .024-0 .26 4 .154-0 .06 0 .694-0 .11 824-1 1334-1 -1285 4-0 .2

1 .274-0 .23 7 .144-0 .17 4 .204-0 .03 0 ,704-0 .08 854-1 1334 1 5 1 2 4 - 5 3 - 1 0 6 8 4 - 0 . 4

1 .224-0 .36 6 .844-0 .25 4 .034-0 .06 0 .704-0 .12 784-1 1234-1 -1696 4-0 .3

7 0 7 8 3 . 2 7 4 - 0 . 1 9 1 5 . 9 8 4 - 0 . 1 3 9 . 6 2 4 - 0 . 0 3 0 . 6 6 4 - 0 . 0 2 2 0 5 4 - 2 3 0 9 4 - 2 - 9 0 1 4 - 0 . 0

3 .254-0 .25 15 .884-0 .17 9 .564-0 .04 0 .664-0 .02 1894-1 2994-2 12934-47 -6584-0 .1

3 .194-0 .20 14 .754-0 .12 8 .974-0 .04 0 .644-0 .02 1734-1 2654-2 -1291 4-0 .0

7 0 8 9 5 . 1 0 4 - 0 . 0 4 5 0 . 1 8 4 - 0 . 0 2 2 7 . 6 4 4 - 0 .0 1 0 . 8 2 4 - 0 . 0 0 6 5 5 4 - 0 9 9 2 4 - 1 - 4 7 3 4 - 0 . 0

5 . 0 3 4 - 0 . 0 7 4 0 . 6 1 4 - 0 . 0 4 2 2 . 8 2 4 - 0 .0 1 0 . 7 8 4 - 0 .0 0 4 8 9 4 - 2 7 5 7 4 - 3 2 2 8 0 4 - 1 7 - 2 3 1 4 - 0 . 0

5 . 0 6 4 - 0 . 0 6 3 1 . 1 9 4 - 0 . 0 3 1 8 . 1 2 4 - 0 .0 2 0 . 7 2 4 - 0 .0 0 3 3 4 4 - 2 5 2 8 4 - 2 - 8 6 3 4 - 0 . 0

7 0 9 9 0 . 1 8 4 - 0 . 0 5 9 . 9 7 4 - 0 . 0 3 5 . 0 7 4 - 0 . 0 2 0 . 9 7 4 - 0 .0 4 1 1 2 4 -0 2 1 5 4 - 1 - 1 1 4 3 4 - 2 . 3

0 .194-0 .0 3 10 .074-0 .03 5 .13 ::h0 .01 0 .964-0 .02 1094-0 21 54-0 1144-11 -9224 -1 .6

0 .174-0 .03 9 .744-0 .02 4 .964-0 .01 0 .974-0 .03 1044-0 2024-1 -1544 4-2 .4

N o t e : T h e t a b l e l is t s f o r al l s a m p l e c l u s t e r s t h e i d e n t if i c a t io n n u m b e r ( C o l .1 ) , m e a n p e r i g a l a c t i c d i s t a n c e

R p a n d m e a n a p o g a l a c t i c d i s t a n c e R a ( C o l s .2 , 3 ), o r b i t a l s e m i - m a j o r a x is ( C o l A ) , o r b i t a l e c c e n t r i c it y (C o l . 5) ,

m e a n r a d i a l p e ri o d P r a n d m e a n a z i m u t h a l p e r i o d P ~ o f t h e o r b i t ( C o ls .6 ,7 ) , t h e a v e r a ge t o t a l a n g u l a r

m o m e n t u m c a l c u l a t e d w i t h t h e J S H 9 5 m o d e l ( C o l .8 ) , a n d a v e r a g e t o t a l e n e r g y ( C o l .9 ) . T h e u n c e r t a i n t i e s o f

t h e p a r a m e t e r s a r e g i v e n t o g e t h e r w i t h t h e c o r r e s p o n d i n g c a l c u l a t e d v a l u e s .

T h e d a t a i n t h e t h r e e l i n e s b e h i n d t h e i d e n t i f i c a t io n n a m e a r e t h e r e s u l t s c a l c u l a t e d b y t h e G a l a c t i c

p o t e n t i a l m o d e l s P 9 0 , J S H 9 5 , a n d D C 9 5 , r e s p e c t i v e l y .

T h e s a m p l e c l u s t e r s d e n o t e d b y t h e s y m b o l * a xe u se d f o r d i s c u ss i o n o f o r b i t a l m o r p h o l o g i e s .

5 . 2 A n a l y s i s a n d D i s c u s s i o n o f R e s u l t s

5 . 2 .1 T h e O r b i t a l M o r p h o l o g y

F o r d i s c u s s i n g t h e r e l a t i o n s h i p o f t h e o r b i t s , m e t a l l i c i t i e s a n d o r b i t a l m o r p h o l o g i e s ,

w e c h o s e f r o m o u r s a m p l e o f 2 9 g l o b u l a r c l u s t e r s , s i x r e p r e s e n t a t i v e s . T a b l e 6 l i s t s t h e


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54 W AN G Long et a l. / Chinese Astronomy and Astrophysics 27 2003) 42-65

identification number, type, galatocentric distance, and metallicity of these six representative

clusters. Other relevant da ta can be read off from Tables 2 and 5, where these six clusters are

marked with asterisks. Among the 6 representative clusters, the metallicities of NGC6712

and NGC6341 are respectively the largest and smallest in our sample, and the metallicity

of NGC5897 happens to be the critical value dividing the HB and MP subsystems. The

metallicities of the six representatives are in the range 0.23,~0.29.

Table 6 The b a s i c p a r a m e t e r s o f r e p r e s e n t a t i v e c l u s t e r s

NGC(No.) Type [Fe/H]

R s c k p c )

NGC(No.) Type [Fe/H]

n g c k p c )

6712 BHB -1.01

3.77

5897 MP -1.80 7.53

5904 RHB -1.29 6.46 4590 MP -2.06 10.35

6205 BHB -1.54 9.00 6341 MP -2.29 9.88

Figs.7 (a--f) illustrate the orbits of the 6 representative clusters in three different grav-

itational potentials, projections in the Galactic plane X - Y plane) are shown on the left,

those in the vertical plane passing through the sun (the

X - Z

plane), on the right. Analysing

and comparing Figs.7 (a- -f) , we find:

(1) The orbits of the sample clusters exhibit some periodicity, and most of them are

distr ibuted within galactocentric distance 40 kpc. Because of the continuous precession of

the perigalacticon, the radial period P r and azimuthal period PC are not equal: the orbits

are periodic, bound, but not closed.

(2) Fig.7 and the data in Table 5 indicate that using different potential models does not

make much difference in the resulting orbital morphology, but does affect the apogalactic

distance: the apogalactic distances calculated by the P90 model are greater than those

calculated by the other two models, and some of the apogalactic distances are greater tha n

40 kpc, not in agreement with the observational results. In comparison, the DC90 model

gives all the apogalactic distances < 40 kpc, and so is more reasonable.

(3) A large number of studies have revealed tha t the orbits of some clusters may exhibit

a type of abnormality, often called chaotic behavior. Schuster et al.[ 26] and Carlberg et

al.[27] have discussed in detail the orbital variation when the cluster approaches the Galactic

center. They found that because the clusters are affected by the radia tion from the Galactic

core, their motion in the Galactic plane will partially be transformed to vertical motion.

This leads to changes in the abnormal morphology. At present, most authors believe that

the chaotic behavior may take place when the cluster crosses the innermost ,-, 1 kpc region

around the Galactic center.

From Figs.7a, 7d, and 7f we can see that for three clusters with small perigalactic

distances, i.e. NGC6712 (Rp -~ 2.5kpc), NGC5897(Rp -,, 0.9kpc), and NGC6341(Rp ,,~

1.0kpc), the chaotic behavior does take place when they come within about 1 kpc of the

center. This is particular ly clear for NGC5897 with Rp ,,, 0.9 kpc.

Based on the results of our numerical calculations, we can consider that the chaotic

behavior may be caused by the following factors:

(i) In the region close to the Galactic core there exist the effect of radiation pointed

out by Schuster et al. [26] and Carlberg et al.[27] and the effect of other disturbances. (ii)

The selected Galact ic gravitational potential models may have some possible effects. In the

region close to the Galactic core these models may no longer be suitable. They may not

describe precisely the mass distribution in the inner region and the results given by the


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W AN G Long e t a l. / Chinese As t ro nom y and As t rophys ic s 27 2003) 42 -65 55

num er ica l in teg ra t ion wi l l no t re f lec t co r re c t ly the ac tu a l mot ion . ( ii i) Th e e f fec t re la te d to

t h e m e t h o d a n d s t e p o f t h e n u m e r ic a l i n t eg r a t io n is n o t v e r y i m p o r t a n t , b e c a u s e t h e s a m e

m et h o d was u s ed fo r a l l 6 c l u s t e r s , an d o n l y t h o s e w i t h s m a l l p e r i g a l ac t i c d i s t an ces s h o wed

t h e ab n o rm a l b eh av i o u r , wh i ch was t h e m o re m ark ed t h e s m a l l e r t h e p e r i g a l ac t i c d i s t an ce .

B es i d es t h e s e 6 c l u s te r s , w e h av e ca l cu l a t ed t h e o rb i t s o f t h e o t h e r c l u s t e r s i n o u r s am p l e ,

and the conc lus ion i s s imi la r .

In ad d i t i o n , fo r th o s e c l u s t e r s wh o s e g a l a t o cen t r i c d i s t an ces m ay b e l e ss t h an 1 k p c ,

t h e u n c e r t a i n t y o f t h e o r b i t d e t e r m i n a t i o n d e p e n d s g r e a t l y o n w h e t h e r o r n o t t h e G a l a c t i c

g rav i t a t i o n a l p o t en t i a l c an d es c r i b e p rec is e l y t h e i n n er r eg i o n o f t h e Ga l ax y . Ho w ev e r , wh e n

we ev a l u a t e q u a n t i t a t i v e l y t h e u n ce r t a i n t i e s i n th e o rb i t c a l cu l a ti o n s , t h e v a r i o u s ef f ec ts

a re d i ff icu lt t o d i s en tan g l e . Th e re fo re , w e s u p p o s e t h a t t h e u n c e r t a i n t i e s i n o u r ca l cu l a t ed

o rb i t a l p a ram e t e r s a r e m a i n l y cau s ed b y u n ce r t a i n t i e s i n t h e p o s i t i o n s , v e l o c i t i e s an d o t h e r

b as i c p a ra m e t e r s o f t h e c l u s t e rs .

(4) Fo r o u r s am p l e c l u s t e r s , t h e co r r e l a t io n b e t ween t h e m e t a l l ic i t y an d o rb i t a l m o r -

p h o l o g y is n o t ap p a re n t , b u t t h e o rb i t a l s i ze i s co r r e l a t ed w i t h t h e o rb i t a l m o rp h o l o g y . Fo r

e x a m p l e , N G C 6 2 0 5 a n d N G C 4 5 9 0 h a v e a b o u t t h e s a m e p e r ig a l ac t ic d i s t a n c e a n d a p o g a l a c t i c

d i s tance , hence a s imi la r o rb i ta l morpho logy , whi le the i r meta l l i c i t i es a re qu i te d i f fe ren t .

(5 ) A m o n g t h e 6 r e p r e s e n t a t i v e c l u st e rs , o n ly N G C 5 9 0 4 b el o n g s t o t h e R H B s u b s y s t e m

and i t s apogala c t ic d i s ta nce i s the l a rges t . As Dinescu e t a l.[ ll ] ind ica te d , the c lus te rs o f the

R H B s u b s y s t e m g en e ra l l y h av e l a rg e r ap o g a l ac t i c d i st an ces . No w , i n p laces fa r awa y f ro m

t h e G a l ac t i c cen t e r t h e weak en ed co n s t r a i n t o f t h e G a l ac t i c g rav i t a t i o n a l p o t en t i a l m ak es

t h e o rb i t a l p e r i o d s l o n g e r, an d t h e o rb i t a l cu rv es g e t s p a r s e r. Th e o rb i t a l m o rp h o l o g i e s

s h o wn in F i g . 7 (b ) h av e co n f i rm ed t h i s co n c l u s io n . In co m p a r i s o n , t h e o r b i t a l s i ze s o f t h e

c l u s te r s o f t h e B H B an d M P s u b s y s t em s a re m u ch s m a l l e r , an d t h i s p o i n t i s a l s o co n f i rm ed

by the o ther f ive f igures .

5 .2 .2 An a l y s i s o f t h e Orb i t a l Pa ra m e t e r s

(1 ) Th e r e l a t i o n s h i p b e t ween t h e o rb i t a l p a ram e t e r s an d m e t a l l i c i t y

U s i n g t h e c a l c u l a t e d r e s u l ts g i v e n in T a b l e 5 , w e h a ve d i s p la y e d s e p a r a t e l y i n F i g s . 8 -

1 2, a s fu n c t i o n s o f t h e m e t a l l i c it y [Fe / H] , t h e o rb i t a l p a ram e t e r s ( t h e o rb i t a l s em i -m a j o r ax i s

a , o rb i t a l e ccen t r i c i t y e , p e r i g a l ac t i c d i s t an ce R p , ap o g a l ac t i c d i s t an ce R ~ , an d az i m u t h a l

p e r i o d PC ) o f t h e 2 9 s am p l e c l u s t e r s . Ex am i n i n g t h e s e f ig u res , we can f i n d:

( i ) Th e o rb i t a l s em i -m a j o r ax i s a , ap o g a l ac t i c d i s t an ce R a , an d az i m u t h a l p e r i o d PC

vary wi th the m eta l l i c i ty in s imi la r way s (see F ig .8 , F ig .11 , and F ig .12) . For 22 c lus te rs

( ab o u t 76 o f t h e s am p l e ) w i t h d if f e ren t m e t a l l ic i ti e s , t h e s e o rb i t a l p a ra m e t e r s a r e re s p ec -

t iv e l y i n t h e f o ll ow i n g r a n ge s : 3 k p c < a < 1 5 k p c , 4 k p c < R a < 2 5 k p c , a n d 8 0 x 1 0 8 y r

< PC < 400 x 106 y r . Espec ia l ly , in the [Fe /H] in te rva l be tw een -1 .9 and -1 .4 a re conc en-

t r a t e d t h e 1 5 s am p l e c l u s t e r s wh o s e ap o g a l ac t i c d i s t an ces a r e le s s t h an 4 0 k p c , an d t h e re

is a p e a k a t [ F e / H ] = - 1 . 6 . T h e r e is a l so a n o ti c e a b le p e a k i n t h e r e l a t io n s h i p b e t w e e n

t h e p e r i g a l ac ti c d i s t an ce an d m e t a l l ic i t y , s h o wn i n F i g . 1 0. Th i s r e s u l t i s co n s i s t en t w i t h

t h e co n c l u si o n o f L IN Q i n g e t a l. an d t h e c u rv e s h o wn i n F i g .4 . Th i s i n d i ca t e s t h a t t h e

v a r i a t io n s o f t h e o rb i t a l p a ram e t e r s a r e r e l a t ed t o t h e i n it i a l s p ace p o s i t i o n s an d v e l o c i ti e s

o f t h e g i v en s am p l e c l u s te r s ;

( ii ) Bo th Dine scu (1999) and C hib a e t a l.[ 2s] have foun d th a t the o rb i ta l ecce n t r ic i ty

t en d s t o b e h ig h i n t h e r an g e [Fe/H]_< -1 . 8 . Th e r a t i o o f t h e c l u s t e r s w i t h l o w eccen t r ic i t ie s

wi ll increase in the rang e [F e /H ]> 1 .8 , and fo r ab ou t 16 ,,,20 o f the ha lo c lus te rs , the i r


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5 6

W AN G Long e t a l. / Chinese As t ron om y and Ast rophys ics 27 2003) 42-6 5

8 [NGC 6712 P90)[ [Fe/H]:- I .01 [NGC 671 2 P9 0) l [Fc /H ]:- l .01

6

-2

-4

6

8 6 4 2 0 2 4 6 8 8 6 4 2 0 2 4 6 8

x k p c ) x k p c )

6

4

-2

-4.

6

~qGC 6712 JSH 95~ [Fg/H]:-I .01

6

4

2

~o

N

-2

-4

-6

6 4 2 0 2 4 6

X kpc)

~IGC 6712 JSH9 5~ [Fe/I-I]:- l .01

-6 -4 -2 0 2 4 6

X kpc)

41 INGC 6712 DC95) ] [Fe /H ] : - l .01 6

4

0

-4.

6

-6 -4 -2 0 2 4 6

X kpc)

[NGC 6712 DC95)1 [Fe/H]:-l .01

6 4 2 0 2 4 6

x k p c )

a) The o rbi t of NGC6712


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W A N G L o n g e t a l. / C h i n e s e A s t r o n o m y a n d A s tr o p h y s i c s 2 7 2 0 03 ) 4 2 - 6 5 5 7

5 0

4 0

30

20

-10

-20

-30

-40

I NGC5904 P90)I [Fe/H ]:- l .29

- 4 0 - 3 0 - 2 0 - I 0 0 10 2 0 3 0 4 0

X kpc)

2 0

1 5

1 0

~ , 5

~ o

>.

-5

-10

-15

-20

[Fe /H] : - l .29

- 4 0 - 3 0 - 2 0 - 1 0 0 10 2 0 3 0 4 0

X kpe)

~NGC5 9 - ) 4 ~ [ F e /H l : -l .2 9 15 ~ N G C 5 9 0 4 J S H 9 5 ~ [ F e /H ] : -l .2 9

0 0 1 1 0 1

l O l

20 -10

-30 -15

-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30

X kpc) X kpc)

[ NGC5904 DC95)] [Fe/H ]:-I .29

3 0 [ - N G C 5 9 0 4 D C 9 5 ) [ F e /H ] : -l .2 9 1 5

-15

-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30

X kpc) X kpc)

b) The orbit o f NGC 5904


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58

W A N G L o ng e t a t. / C h i n e se A s t r o n o m y a n d A s t r o p h y s ic s 2 7 2 0 0 3 ) 4 2 - 6 5

1 5 ] [ F e / H ] : - I . 5 4 4 0

101 30

5 5 2 0

~ I 0

- i 0

-10[

- 2 0

-15 -30

-40

-40 -30 -20 -10 0 10 20 30 40

INGC 2 0 5 ( P 9 0 ) ~ [ F e / I - I ] : -. 5 4

-40 -30 -20 -10 0 10 20 30 40

X(kpc)

X k p c )

101 ~NGC6205(JSH95~

t

~ .5t

-10

-15

-30 -20 -10 0 10 20 30

[ F e/ H ]: - I. 5 4 ] I N G C 6 2 0 5 ( J S H 9 5 ~ [ F e / H] : - I . 54

3 0

l

20

-30

-30 -:20 -1'0 0 fo 2'0 3 0

X(kpc)

X(kpc)

[ N ~ : C ~ 0 ~ ~ J

[Fem]:-l.54

3 0 -

10 ~ 20

~ 5 ~ I 0 -

_ - l O -

.101 20J

~0 -20 .10 0 10 20 30

I N G C 6 2 0 5 ( D C 9 5 ) J

[Fe/H]:-I.54

3~30 -20 -10

0 10 20 30

x ~ c )

x ~ o

(c) The orbitof NGC6205


18/24

WANG Long et al. Chinese Astronomy and Astrophysics 27 2003) 42-65

9

-2 -1 0 1 2 3

X ( k p c )

8

6

4 -

2

O

- 2

-4

- 6

- 8

- l O

-3

~ N ~ } C 5 8 9 7 ~ q 0 ~ [ Fe / H ] : - 1 . 80

2 ]

l -

0 -

~ - 1

~ -2-

-3

-4-

5

-3

[NOC 5897(P90)1

[ F e / H ] : - 1 . 8 0

~- N G C5897( JSH 95~ [ Fe / H ] : - 1 . 80

-2 -1 0 i 2 3

x ~ , c )

10 .

8

6

4

~ 2

~ - o

- 2

-4

-6

8

. .10

-2 -1 0 1 2 3 4

X k p c )

t,q

4

3

2

1

0

-1

-2

-3

-4

-5

I N G C 5 8 9 7 ( J S H 9 5 ~ [ F e / H ] : - 1 .8 0

-2 -1 0 1 2 3 4

X ( k p c )

8

6

4

2

~- o I

~ -2

-4 .

-6

-8

-10.

[ ~ C 5 8 9 7 L - ~ D C ~ _ 5 [ F e / H ] : - 1 . 80

-3

-2 -1 0 i 2 3

X ( k p c )

3 - I N G C 5 8 9 7 ( D C 9 5 ~ [ F e / H ] :- 1 . 8 0

2

- 3 - 2 - 1 0 1

X ( k p c )

2 3

( d ) T h e o r b i t o f N G C 5 8 9 7


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6 0 W A N G L o n g e t a l. / C h i n e s e A s t r o n o m y a n d A s t r o p h y s ic s 2 7 2 0 03 ) 4 ~ - 6 5

i

N 4 5 9 o 9 0 : . 2 . o I 1 N 4 5 9 o o : . 2 . o 6

0 40

2O

2

1

-20

-20 -40

-3

40 40

40

:-2.06 :-2.06

30

20

21)

I 10

N -10

-10~

-20

-20 -30

4 0

-3 0 -2 0 - t O 0 10 20 30 40 -30 -20 - lO 0 lO 20 30 40

X kp0 X kpc)

[NGC4590(DC95)I [Fern]:-2.06 INGC4590(DC95)I [Fe/H]:-2.06

20

20

5

10 t

10t -t0]

-15 -20

-20

-30 -20 -I0 0 t0 20 30 3030 -20 -t0 0 t0 20 30

X kpc) X kpc)

(e) The o rbit of NGC 4590


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W A N G L o n g e t a l. / C h i n e s e A s t r o n o m y a n d A s t r o p h y s i cs 2 7 2 0 0 3 ) 4 2 - 6 5 6 1

10.

8

6.

- 4

o

; 0

8

-10

[NGC6341( Pg0 ~ [Fe/H]:-2.29

- 1 0 - 8 - 6 - . 4 - 2 0 2 4 6 8 10

X (kpc)

10-

8

6-

4

, - , 2

0

N -2.

-4.

- 6

. 8 .

- 1 0

[ ~ q ~ 6 3 4 I ( J S H 9 ~ [ F e/ H ]: -2.29

1 0 8 6 4 2 0 2 4 6 8 1 0

x kpc)

A v

8 .

6 .

4 .

2

N - 2

-4

-6

8

-10

INGC6341( P90 ] [Fe /H ]:-2.29

- 1 0 - 8 - 6 - 4

2 0 2 4 6 8

10

X ( k ~ )

10.

8

6

4

2

N -2

-4

-6

8

INGC6341(JSH95] [FePd]:-2.29

- 1 0 - 8 - 6 - 4 2 0 2 4 6 8 10

X ( k ~ )

10

8

6

,~ 4 2

..~ 2-

~ 0

. 2 -

[NGC6341(DC95)J [Fe /H]:-2.29

- 4 -

. 6 ~

-10-

- 1 0 - 8 6 4 2 0 2 4 6 8 10

X (kp)

10

8

6

4 -

N 0

.4-

[NGC6341(DC95)[ [Fe /H]:-2.29

6.

- 8

- 1 0 - 8 - 6 - 4 - 2 2 4 6 8 1.0

X kp)

(f) The orbit of NGC6341

F i g . 7 T h e o r b i t s o f 6 r e p r e s e n t a t i v e c l u s t e r s f o r 3 G a l a c t i c p o t e n t i a l m o d e l s ( t h e l e f t a n d

r i g h t p a n e l s sh o w t h e o r b i t s i n t h e X - Y a n d X - Z p l a n e s , r e s p e c ti v e l y )


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62 W AN G Long et al. / Chinese Astronom y and Astrophysics P7 2003) 42-65

orb i ta l eccen t r ic i t ies . a re l ess tha n 0 .4 . F ig .9 show s the re la t ionsh ip be tw een the o rb i ta l

eccen t r ic i t y an d m e t a l l ic i t y fo r t h e 2 9 s am p l e c l u s t e rs . Tak i n g t h e r e s u l t o f c a lcu l a t i o n w i t h

DC 9 5 m o d e l a s ex am p l e , am o n g t h e 1 2 s am p l e c l u s t e r s w i t h [Fe / H]_ < -1 . 8 t h e re i s o n l y

o n e c l u s te r wh o s e eccen t r i c i ty i s l o wer t h an 0 .4 , an d am o n g t h e 1 7 s am p l e c l u s t e r s w i t h

[Fe /H ]> - 1 .8 ther e a re 6 c lus te rs (o r 35 ) whose eccen t r ic i t ies a re _< 0 .4 . The refo re , fo r 7

of our 29 sam ple c lus te rs (24 ) , the e ccen t r ic i ty is _< 0 .4 . Ob v ious ly , ou r resu l t i s bas ica l ly

in ag reem ent w i th the f ind ing o f Dinesc u e t a l .

(2) E f fec t o f d i f fe r en t g rav i t a ti o n a l p o t en t i a l m o d e l s o n t h e o rb i t a l p a ram e t e r s .

Th e e f f ec t s o f d i f fe r en t g rav i ta t i o n a l p o t en t i a l m o d e l s o n t h e p e r i g a l ac t i c d i s t an ce an d

orb i ta l eccen t r ic i ty a re re la t ive ly smal l . Espec ia l ly , the e f fec t on the e ccen t r ic i ty is very

s m a ll , an d t h i s i s i n ag reem e n t w i t h t h e co n c lu s i on o f D i n es cu e t a l . B u t t h e e f f ec t s o n t h e

s em i -m a j o r ax i s a, r ad i a l p e r i o d

P r ,

az i m u t h a l p e r i o d PC , an d o t h e r s , an d , e s p ec i a l l y o n t h e

ap o g a l ac t i c d i s t an ce R a , are re la t ive ly c lear . From Fig . 11 we see th a t R a v a l u es g rea t e r t h an

4 0 k p c a re n o t fo u n d i n t h e DC 9 5 m o d e l ( t h ey a re fo u n d i n t h e o t h e r t wo ) : t h i s i l l u s t r a t e s

t h e s t ro n g co n s t r a i n t o f t h e D C 9 5 m o d e l o n t h e a p o g a l ac t i c d i s tan ce .

(3) T h e u n ce r t a i n t i e s o f t h e x ) rb i ta l p a ra m e t e r s

Th e u n ce r t a i n t i e s o f t h e o rb i t a l e l em en t s a r e a l so gi v en i n Tab l e 5 an d F i g . 8 - -F i g . 1 2 fo r

a l l t h e s am p l e c l u s t e r s . Ex cep t t h e u n ce r t a i n t y i n t h e eccen t r i c i t y , t h e u n ce r t a i n t i e s i n t h e

o t h e r o rb i t a l e l em en t s a r e a l l v e ry s m a ll , an d t h e e f fec t o f d i f fe r en t g rav i t a t i o n a l p o t en t i a l

m o d e l s is n o t o b v i o u s . I t p ro v es t h a t t h e u n c e r t a i n t ie s o f t h e o rb i t a l e l em en t s a r e m a i n l y

cau s ed b y t h o s e in t h e b a s i c p a r am e t e r s o f t h e c l u s te r s , su ch a s t h e ab s o l u t e p ro p e r m o t i o n ,

r ad i a l v e l o c i t y , i n i t i a l p o s i t i o n , an d o t h e r f ac t o r s , an d am o n g t h e v a r i o u s o rb i t a l e l em en t s

t h e o rb i t a l e ccen t r i c i t y is m o s t s en s i ti v e t o t h e s e f ac t o r s.

(4 ) Th e o rb i t a l t o t a l en e rg y an d an g u l a r m o m en t u m d i s t r i b u t i o n

I n F ig .1 3 , w e p r e s e n t t h e r e l a t io n s h ip b e t w e e n t h e t o t a l e n e r g y a n d a n g u l a r m o m e n t u m

s e p a r a t e l y f o r c l u st e rs b e l o n gi n g t o t h e t h r e e s u b s y s t e m s ( R H B , B H B , a n d M P ) a n d f o r t h e

t h ree g rav i t a t i o n a l p o t en t i a l m o d e l s . Th i s f i g ure s h o ws t h a t fo r a ll t h r ee p o t en t i a l m o d e l s ,

t h e R H B c l u s te r s t en d t o o ccu r i n t h e r eg i o n o f h i gh en e rg y an d an g u l a r m o m en t u m , wh i l e

t h e R H B a n d M P c l u st e r s a r e m o r e s p r e a d o u t i n t h e e n e r g y ra n g e .

C o m b i n in g t h e a b o v e a n a l y s e s o f t h e o r b i ta l m o r p h o l o g y a n d p a r a m e t e r s w e c a n f i nd

t h a t t h e R HB t y p e c l u s t e r s h av e h i g h e r m e t a l l i c i t i e s , e ccen t r i c i t i e s , o rb i t a l s i z e s an d t o t a l

en e rg i es , wh i l e t h e o t h e r t w o t y p es a r e n o t s o d i s ti n c t i n t h e s e p h y s i ca l q u an t i t i e s . Th i s

co n c lu s io n is ag a in s i m i l a r t o t h e r e s u l ts o f D i n es cu e t a l . Th e re fo re , o u r s t u d y o n c l u s t e r s

o f t h e s a m e i n t e g r a t e d s p e c t r a l t y p e a ls o s h o w s t h a t g l o b u la r c l u st e rs o f t h e R H B t y p e m a y

d if fe r f ro m t h e o t h e r t y p e s i n t h e i r o r i g i n an d ev o l u t i o n m ech an i s m .

6 . C O N C L U S I O N

W e h av e s e l ect ed a s am p l e o f 2 9 g l o b u l a r c l u s te r s w i t h i n t eg ra t ed s p ec t r a l t y p e F i n t h e

Galaxy . Ba sed on the i r b as ic pa ram ete rs suc h as d i s tance , rad ia l ve loc i ty , me ta l l i c i ty , e tc .

g i v en b y Har r i s i n 1 9 9 9 an d t h e i r ab s o l u t e p ro p e r m o t i o n d a t a g i v en b y Di n es cu e t a l . , we

d e r i v ed t h e i r s p a t i a l an d v e l o c i t y d i s t r ib u t i o n s . Tak i n g t h e s e a s in i ti a l co n d i t i o n s , t h e i r o rb i t s

were n u m er i ca l l y fo ll o wed fo r t h ree Ga l ac t i c g rav i t a t i o n a l p o t en t i a l m o d e l s . Th e r e s u l t s o f

the ca lcu la t ions ind ica te :


22/24

WANG Long et al. / Chinese Astronomy and Astrophysics 27

2 0 0 5 )

42 65 6

3 0

2 5

2 0

G ' I 5

1 0

I

5

6

P 9 0

J S H 9 5

* D C 9 5 1 . 0

0.9

0 . 8

* 0 . 7 -

.

0 . 6

U

0 . 5

t 0 . 4

0 . 3 .

, 0

0 . 2 -

O

0 . l

6

0 . 0

-2 . 4 -2 . 2 -2 . 0 -1 . 8 -1 . 6 -1 . 4 -1 . 2 -1 . 0

[Fc/I-I]

F i g . 8 O r b i t a l s e m i - m a j o r a x i s a s a f u n c t i o n o f

m e t a l l i c i t y f o r 2 9 s a m p l e c l u s t e r s

P 9 0

J S H 9 5

I

- 2.4 - 2 . 2 - 2 . 0 - 1 . 8 - 1 . 6 - 1 . 4 - 1 . 2 - 1.0

[Fe/H]

F i g . 9 O r b i t a l e c c e n t r i c i t y a s a f u n c t i o n o f m e t a l -

l i c i t y f o r 2 9 s a m p l e c l u s t e r s

P 9 0

1 2 -

J S H 9 5

1 0 - * * D C 9 5

8 0

I

~ . 6 l *

4 I I ~ | | * |

2 -

i t ~ i |

, I

O

-2 . 4 -2 . 2 -2 . 0 -1 . 8 -1 . 6 -1 . 4 -1 . 2 -1 . 0

[Fe/H]

F i g . 1 0 P e r i g a l a c t i c d i s t a n c e a s a f u n c t i o n o f


5 0

4 0

3 O

~ 2 0

l O

P 9 0

J S H 9 5

, * D C 9 5

S t *

| I

II

4* I i~ ii I I

I

-2 , 4 -2 . 2 -2 . 0 -1 . 8 -1 . 6 -1 . 4 -1 . 2 -1 , 0

[Feral

F i g . 1 1 A p o g a l a c t i c d i s t a n c e a s a f u n c t i o n o f


(1) T h e s am p le c lu s t e rs ex h ib i t a s p h e ri ca l ly s y m m et r i c d i s tr i b u t i o n a ro u n d t h e Ga l ac t i c

cen t e r. T h ey a re l o ca t ed w i th in g a l ac to cen t r i c d i s t an ce 4 0 k p c , an d a re co n ce n t r a t ed i n th e

5- -1 0 kpc range . The i r sp ace ve loc i t i es exh ib i t an e l lipso ida l d i s t r ibu t io n .

(2) O f t h e 2 9 s am p le c lu s t e r s, 1 7 b e lo n g to t h e h o r i zo n ta l b ran ch (HB ) s u b s y s t em

( [ F e / H ] < - 0 . 8 ) ( w i t h 6 b e l o n g i n g t o R H B , 1 1 b e l o n g i n g t o B H B ) , a n d 1 2 b e l o n g t o t h e

m e t a l - p o o r ( M P ) s u b s y s t e m ( [ F e / H ] < - 1 . 8 ) . T h e r e su l ts o f o u r c a l c u la t io n s d e m o n s t r a t e

th a t t h e i r p h y s i ca l ch a rac t e r i s t ic s , s u ch t h e i r d i s t r i b u t i o n i n s p ace an d v e lo c i t y d i s p e r s io n an d

s o o n a re acco rd an t w i th t h e k n o w n re s u l t s o f t h e g lo b u l a r c l u st e r s in g en e ra l . T h e n u m b er

d en s i t y o f o u r s am p le c lu s te r s v a r ie s w i th t h e m e ta l li c i ty an d p eak s a ro u n d [Fe /H ]= -1 . 6 .

(3) W e h av e an a ly zed t h e o rb i t a l m o rp h o lo g ie s o f 6 r ep re s en t a t i v e c lu s t e r s . I t i s fo u n d

th a t a lm o s t a l l si x m o v e i n o p en an d p e r i o d i c o rb i ts w i th in g a l ac to cen t r i c d i s t an ce 4 0 k p c .

Us in g d i f f e ren t g rav i t a t i o n a l p o t en t i a l m o d e l s d id n o t m u ch a f f ec t t h e s p ec i f i c o rb i t a l m o r -

p h o lo g i es , b u t d id a f f ec t t h e ap o g a l ac t i c d i s t an ce t o v a ry in g d eg rees . M o s t o f t h e c lu s t e r s

h av e m ax im u m g a l ac to cen t r i c d i s t an ces l e ss t h an 4 0 k p c . Fo r a g iv en g rav i t a t i o n a l p o t en t i a l

m o d e l , wh en t h e o rb i t p a s s e s w i th in ab o u t 1 k p c o f t h e Ga l ac t i c cen t e r , ch ao t i c b eh av io r


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6 W A N G Long e t a l / Ch i n e se A s t r o n o my and A st r o p h y si c s 2 7

2003) 42-65

I000

8OO

6OO

400

200 ~. .

%

0

- 2 ,4 -2 . 2 - 2 . 0 - 1 . 8 - 1 . 6 - 1 .4 - 1 . 2 - 1 . 0

[Fem]

F i g . 12 A z i m u t h p e r i o d a s a f u n c t i o n o f m e t a l l i c i t y

f or 2 9 s a m p l e c l u s t e r s

P 9 0

J S H 9 5

D C 9 5

A v I

i ~

0 .

t

* t

0.2

- 0 . 4

- 0 . 6 .

-0 .8 -

1.O

- 1 . 2 ,

- 1 . 4

1.6

l .g

- 2 . 0

0

F i g . 1 3

~ tJ

v ~

oD

o o

o | A

v ~

500 1000 1500 2000 250030003500

IJI (kpe* k in/s)

T o t a l e n e r g y a s a fu n c t i o n o f o r b i t a l a n -

g u l ar m o m e n t u m f or 2 9 sa m p l e c l u s t er s

m a y t a ke p la c e, e x a m p l e s ar e N G C 6 7 1 2 , N G C 5 8 9 7 , a n d N G C 6 3 4 1 . T h e c o rr e la ti on b e t w e e n

t h e m e t a l li c it y a n d t h e o r b i t a l m o r p h o l o g y is n o t o b v io u s . T h e c l u s te r s o f t h e R H B s u b -

s y s t e m h a v e g r e a t e r g a l a c t o c e n t r i c d i s t a n c e s , a n d m a j o r p a r t s o f t h e i r o r b i t s s tr e t c h i n t o

r e g io n s fa r f r o m t h e G a l a c t i c c o r e . I n co m p a r i s o n , c l u s te r s o f t h e B H B a n d M P s u b s y s t e m s

h a v e m u c h s m a l l e r o r b i t s .

( 4 ) F o r o u r 2 9 s a m p l e c l u s t e r s , t h e o r b i t a l s e m i - m a j o r a x i s , a p o g a l a c t i c d i s t a n c e , a n d

a z i m u t h a l p e r i o d v a r y w i t h t h e m e t a l l i c it y i n e s s e n t ia l l y s im i l a r w a y s . T h e o r b i t a l e c c e n t r i c -

i t y i s r e l a te d t o t h e m e t a l l ic i t y . A m o n g t h e s e le c t e d h a l o c l u s t e r s , a b o u t 2 4 h a v e o r b i t a l

e c c e n t r ic i ti e s l e s s t h a n 0 . 4 . T h e d i ff e r en t g r a v i t a t i o n a l m o d e l s h a v e l i t tl e e f f e c t s d n t h e p e r i -

g a l a c t i c d i s t a n c e a n d e c c e n t r i c i t y , b u t q u i t e n o t i c e a b l e e f f e c t s o n t h e a p o g a l a c t i c d i s t a n c e ,

o r b i t a l s e m i - m a j o r a x i s , r a d ia l p e r i o d , a z i m u t h a l p e r i o d . T h e i r e f fe c t o n t h e u n c e r t a i n t i e s

o f t h e o r b i ta l p a r a m e t e r s i s n o t a p p a r e n t . T h e u n c e r t a in t i e s o f t h e o r b i t a l p a r a m e t e r s a r e

m a i n l y c a u s e d b y u n c e r t a in t i e s i n t h e b a s i c p a r a m e t e r s s u c h a s t h e a b s o l u t e p r o p e r m o t i o n ,

r a d i a l v e l o c i t y , i n i t i a l p o s i t i o n , a n d s o o n , a n d t h e e f f e c t o n t h e o r b i t a l e c c e n t r i c i t y i s m o s t

o b v i o u s . C o m p a r e d t o t h e o t h e r t w o s u b s y s t e m s , t h e R H B c l u s te r s h a v e h ig h e r m e t a l l ic i t ie s ,

g r e a t e r e c c e n t r i c i t i e s , a s w e l l a s l a r g e r o r b i t s , g r e a t e r t o t a l e n e r g i e s a n d a n g u l a r m o m e n t a .

A l l t h i s fu r t h e r j u s t i f i e s t h e p o i n t t h a t g l o b u l a r c l u s t e r s o f t h e R H B t y p e d if fe r f r o m ' th e

o t h e r t y p e s i n t h e ir o r i g in a n d e v o l u t i o n m e c h a n i s m .


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R e f e r e n c e s

1 MA Er, ZHANG Mei, Progress in Astronomy, 1990, 8(4),317

2 MA Er, ZHANG Mei, Progress in Astronomy, 1991, 9(1), 22

3 WU Zhen-yu, et al., 'Progress in Astronimy, 2000, 18(1), 75

4 Zinn R., ApJ, 1985, 293, 424

5 Hesser J. E., Shawl S. J., Meyer J. E., PASP, 1986, 98, 403

6 Webbink R. F., In: Grindlay J. E., Davis Philips A. G., eds. IAU Symp. No.126, Kinematics of the

Galactic Globular Cluster System, Reidel, Dordrecht, 1988, 49

7 Harris W. E., A J, 1996, 112, 1487

8 Dauphole B., et al., A A, 1996, 313, 119

9 Odenkirchen M., Brosche P., Geffert M., et al., New Astron., 1997, 2, 477

10 Dinescu D. I., Girard T. M., Van Altena W. F., et al., AJ, 1997, 114, 1014

11 Dinescu D. I., Van Altena W. F., Girard T. M., et al., AJ, 1999, 117,

277

12 Dinescu D. I., Girard T. M., Van Altena W. F., AJ, 1999, 117, 1792

13 Harris W. E., http://physun .physics.mcmaster.ca/Globular.html,1999

14 Purple Mountain Observatory, Astronomical Almanac 2000, Beijing: Science Press, 1999

15 Johnson D. R. H., Soderblom D. R., A J, 1987, 93, 864

16 Almandroff, T. E., AJ, 1989, 97, 375

17 LIN Qing, ZHAO Jun-l iang, Progress in Astronomy, 2001, in press

18 LIN Qing, ZHAO Jun-liang, Progress in Astronomy, 2001, in press

19 Zinn R., in ASP Conf. Ser., 1993, 48, 38

20 Zinn R., in ASP Conf. Ser., 1996, 92, 211

21 Paczynski B., ApJ, 1990, 348,485 (Pg0)

22 Jonhston K. V., Spergel D. N., Hernquist L., ApJ, 1995, 451,598 (JSH95)

23 Dauphole B., Colin J. A., A A, 1995, 300, 117(DC95)

24 Hernquist L., ApJ, 1990, 356, 359

25 Binney J., Tremaine S., Galactic'Dynamics, Princeton: Princeton University Press, 1987

26 Schuster W. J., Allen C., A A, 1997, 319, 796

27 Carlberg R. G., Innanen K. A., AJ, 1987, 94, 666

28 Chiba M., Yoshii Y., ApJ, 1997, 490, L73

space distribution and orbits of globular

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