L I M I T S O N T H E O R B I T A L E C C E N T R I C I T I E S OF

G L O B U L A R C L U S T E R S

F. HOUSE and R. WIEGANDT Astronomisches lnstitut der Ruhr-Universitiit, D-4630 Bochum, F.R.G.

(Received 27 September, 1976)

Abstract. By use of an inverse-square mass-model for the Galaxy, the range in eccentricities for the orbits of 57 globular clusters is computed. On the assumption that all clusters have the same apo- galacticon distance, various values of this distance are considered. It is found that low eccentricities are possible for small apogalacticon distances.

1. Introduction

Globular clusters are high-velocity objects in the galactic halo. Because of their large distances their proper motions cannot be determined (with the exception of co

Centauri, a nearby cluster; cf. Murray et aL, 1965). Previous studies have usually concentrated on the four available kinematic parameters: the three positional co- ordinates and the velocity along the line-of-sight. Von Hoerner (1955) showed statistically that the orbits of clusters have a high probability of having a large eccentricity, and thus have narrow orbits with small perigalacticon distances. In estimating the mass of the Galaxy Lohmann (1956) assumed that clusters have extremely elongated orbits. Similar analyses by Kinman (1959) and Matsunami (1964) seem to confirm that the orbits of globular clusters have a high eccentricity. More recently, Peterson (1974) determined the minimum value for the orbital eccentricities of 41 clusters. As a fifth parameter he used the perigalacticon distance determined from tidal radii. His results indicate that low values for the eccentricities cannot be excluded.

In the theory of galaxy formation first suggested by Eggen et al. (1962), the Galaxy collapses from a sphere with a radius a few times the radius of the present galactic disk. The time-scale of the collapse is rapid (~10 s yr) and the eccentricity of the orbits of clusters will, on the average, be higher than they were originally. For a slower rate of collapse the eccentricities would not change appreciably. Unknown factors remain the original dimensions of the proto-galaxy and the way the density and the potential changed during the collapse.

We consider a number of possible values for the original dimensions of the proto- galaxy. This corresponds to the maximum distance from the galactic centre that each cluster reaches in its orbit. As a rough approximation we take the potential to be a point source.

Astrophysics and Space Science 48 (1977) 191-197. All Rights Reserved Copyright �9 1977 by D. Reidel Publishing Company, Dordrecht-Holland

192 F. HOUSE AND R. WIEGANDT

2. Equations

The clusters are in keplerian orbits about the galactic centre. We can thus use the theory developed in celestial mechanics for the motion of planets and comets in

the solar system. We consider an inertial system such that the galactic centre is at the origin, the galactic plane is the x, y-plane and the x-axis is in the direction of the Sun. The mass of the Galaxy is assumed much greater than that of the cluster. The equations of motion require the determination of 6 independent constants for their complete solution. In our case we have only 4 constants: the three spatial components at the present time and one velocity component - the radial velocity. One further parameter can be given, the maximum distance of the clusters from the galactic centre, which we assume to be the same for all clusters.

We have the equations (cf. Brouwer and Clemence, 1961)

f = -GMr/ l r [ s (1)

where

r = (x, y, Zo) r 2 _ x 2 + y2 + Z2, 1

V 2 = 2 2 + p 2 + ~ 2 ~ (2)

rf" = x2 + y~ + z~;J

R, = a(1 + e), (3)

where a and e are the semi-major axis and eccentricity of the orbit, respectively. If E is the eccentric anomaly, then

r = a(1 - e cos E), r~ = (aGM)~/2e sin E, (4)

From Equations (4) we can eliminate E and using Equation (3) we obtain a quadratic equation in e with the solution

e = Ao + (A~ - Bo) 1/2, (5)

where

1 (rr)2\ / 2 ] [ = , , . , - r=

\ I I

Bo ( ( R - r ) ' + R . ~ ) / ( R I - r 2 ) . J (6)

We can now use the vectorial orbital constants which are used to determine the orientational elements co, 12, i in celestial mechanics. The vectorial constants are the direction cosines which define the orientation of the orbital coordinate system to the inertial system. They can be written in terms of the eccentric anomaly: i.e.,

x ( a ) cos E - 2a 3/2 sin E/~/-G--M, (7) aP= =

x ( a l s i n E + : caa /2 (cos E - e)/~/-G---M, (8) bQ~ = \ r /

LIMIT S O N T H E O R B I T A L E C C E N T R I C I T I E S OF G L O B U L A R C L U S T E R S 193

with similar expressions for Py, P~, Qy and Q,, and where b=a~/1-e z. These

constants are also subject to the boundary conditions

2 2 2 __ Px + Py + P~ - 1 , + + =

p2 + Q2 = 1. J (9)

We consider the coordinate system S(x', y', z') with origin at the Sun, the x', y'-plane in the galactic plane and the x'-axis directed away from the galactic centre. The trans- formation from O(x, y, z) to S(x', y', z') is then

X = x' + Ro, y = y', z = z', (10)

where Ro is the distance from the Sun to the galactic centre. The transformation of the velocity is, similarly,

2 = 2' + 20, ? = Y + ? o , ~ = x' + Xo, (11)

where (2o, 3~o, ~o) are the components of the solar motion in the O(x, y, z) system. Rotational terms are not included since we consider the motion only at the time to.

The radial velocity of an object in the S(x', y', z') system is given by

v" = (x':, ' + y' .V + z'~')/r' , (12)

where r'2=x'2+y'2+z '2. Transforming this to the O(x, y, z) system, after some reduction we obtain

# = Co + Ro2, (13)

where

Co = ~;[(x - Ro) ~ + y2 + z2 ] .2 + (x + Ro)2o + y~o + Z~o.

Equation (13) can now be substituted in Equations (5) and (6) and the eccentricity can be directly determined; 2 remaining the only unknown parameter.

If we assume that the orbits are ellipses the velocity of the cluster must remain within the limits of the parabolic velocity. Thus we have

- < 2 < ~ �9 ( 1 4 )

and 2 can take any value within these limits.

With each value of x the eccentricity can be found from Equations (5) and (6). P~ and Q~ can be determined from Equations (7) and (8). We know that [Pxl ~< 1, [Qx[ ~< 1 and 2 2 IP~ + Qxl ~< t. These provide a further restriction of the possible values for the eccentricity.

Figure 1 shows a typical result for the eccentricity as a function of 2. The relatively narrow range is due to the constraints of the vectorial constants.

194

1.0

F. H O U S E A N D R . W I E G A N D T

e

0.9

0.8

0.7

0.6

0.5

0.4

0.3

NGC 362

I I I I

-200 -100 0 100 200 Mkm/sec}

Fig. 1. Plot of eccentricity vs 2 for NGC 362 for R = 20 kpc.

3. Results and Discussion

The coordinates and radial velocities were taken from the catalogue of Alter et al.

(1970). The distances were taken from Peterson and King (1975). The limits on the eccentricities were computed for the following values of the

apogalacticon distances: R~=20, 25, 30, 40, 50, 60, 70, 80, 90 and 100 kpc. They should bracket the real value. The larger values encompass the Magellanic Clouds.

Table I lists the maxima and minima of the eccentricities for two values of R,. Figures 2 and 3 illustrate the way the limits vary with R~. The dashed line is the mini- mum found by Peterson (1974). That his values are consistently below the minima in Table I is due to his using Schmidt's (1965) galactic mass model for the potential.

A cluster spends most of its orbit near apogalacticon, so the clusters should be near the maximum distance at any time. That most of the clusters are within 20 kpc suggests that the maximum distance is relatively smali. For eight of the clusters in Table I the eccentricity is greater than 1.0 for R, = 20 kpc. These clusters are marked with an asterisk. In Table II the minimum value of R, for which the eccentricity is less than 1.0 is given for these clusters. In the case of three of them no solution can be found even at R, = 100 kpc. These could be intergalactic ' t ramps' , not gravitationally

bound to our Galaxy. An extended and time-independent mass-model would change the orientation of the

orbit and the elliptic elements in the sense that the eccentricity would decrease. A time-dependent model would have a drastic effect on the elements, but if the collapse

LIMITS ON THE ORBITAL ECCENTRICITIES OF GLOBULAR CLUSTERS

TABLE I

Limits of eccentricities for two values of Ra

195

R . = 2 0 kpc R . = 100 kpc R . = 2 0 kpc Ra= 100 kpc

NGC emln emax emin emax N G C emln emax emin emax

104 0.42 0.62 0.85 0.96 6284 0.70 0.76 0.93 0.94 288 0.15 0.68 0.74 0.86 6293 0.72 0.80 0.94 0.95 362 0.32 0.92 0.81 0.97 6304 0.80 0.87 0.96 0.96

1261" - - 0.70 0.95 6333 0.77 0.96 0.95 0.99 1851 0.15 0.70 0.74 0.92 6341 0.33 0.92 0.82 0.99 1904 0.04 0.73 0.68 0.75 6356 0.58 0.68 0.90 0.91 2419" - - 0.12 0.14 6362 0.57 0.93 0.90 0.96 2808 0.32 0.87 0.81 0.99 6397 0.48 0.75 0.87 0.92 3201 0.36 0.53 0.83 0.94 6402 0.68 0.97 0.93 0.99 4147" - - 0.72 0.98 6541 0.71 0.91 0.93 0.98 4590* . . . . 6626 0.60 0.65 0.90 0.91 4833 0.45 0.98 0.86 0.99 6637 0.76 0.89 0.95 0.97 5024* - - 0.69 0.95 6638 0.74 0.82 0.94 0.96 5139 0.47 0.95 0.87 0.99 6656 0.53 0.66 0.88 0.91 5272 0.22 0.97 0.77 0.95 6712 0.65 0.95 0.92 0.99 5634 0.26 0.84 0.79 0.95 6715 0.57 0.81 0.89 0.95 5694* . . . . 6723 0.71 0.89 0.93 0.98 5824 0.53 0.93 0.80 0.96 6752 0.56 0.94 0.89 0.99 5904 0.49 0.93 0.87 1.00 6779 0.32 0.93 0.81 0.97 5986 0.64 0.95 0.92 0.99 6809 0.64 0.94 0.92 1.00 6093 0.72 0.90 0.94 0.98 6838 0.45 0.91 0.86 0.98 6121 0.48 0.62 0.87 0.90 6864* - - 0.73 0.87 6171 0.69 0.96 0.93 0.99 6934* - - 0.77 0.98 6205 0.39 0.94 0.84 0.99 6981 0.50 0.83 0.78 0.97 6218 0.61 0.93 0.91 0.99 7006* . . . . 6229 0.04 0.96 0.69 0.83 7078 0.31 0.91 0.81 0.97 6254 0.54 0.96 0.90 0.98 7089 0.29 0.88 0.80 1.00 6266 0.83 0.93 0.96 0.99 7099 0.44 0.96 0.85 0.99 6273 0.54 0.67 0.89 0.91

* Eccentricity greater than 1.0 for Ra = 20 kpc. Minimum values for Ra in Table II.

TABLE g

Mim mum v a l u e s o f R , for w h i c h e ~ l . 0

NGC R. NGC R.

1261 40 5694 > 100 2419 80 6864 25 4147 30 6934 25 4590 > 100 7006 > 100

196 r . HOUSE AND R. VVIEGANDT

e

1.0

0.9

0.8

0,7

0.6

0.5

0.4.

0.3

0.2

0.1

0

Fig. 2.

I I I I ~ I l I I I

J

f

NGCI04 147 Tuc)

f I , , I , I I I I I I I

10 20 30 4.0 50 60 70 80 90 Ra(kpc)

Limits of eccentricity for NGC 104. Dashed line is minimum eccentricity found by Peterson (1974).

e

1.0 ~

0.9

0.8

0.7

0.6

0.5

0.4.

0.3

0.2!

0.1

0

Fig. 3.

i i i i i i i i i i

f _ I

j J

J

NGC 6205 (M13}

I I I I I I I I I I

10 20 30 40 50 60 70 80 90 Ra {kpc)

Limits of eccentricity for NGC 6205. Dashed line is minimum eccentricity found by Peterson (1974).

LIMITS ON THE ORBITAL ECCENTRICITIES OF GLOBULAR CLUSTERS 197

of the proto-galaxy was short, as envisaged by Eggen et aL (1962), then only the initial parameters would have changed appreciably.

Our results indicate that the orbits of the globular clusters have not necessarily a large eccentricity, and if the apogalactieon of the clusters is small then the eccentricities are indeed quite low.

References

Alter, G., Ruprecht, J., and Van#sek, V. : 1970, Catalogue of Star Clusters and Associations, Budapest. Brouwer, D. and Clemence, G. : 1961, Methods of Celestial Mechanics, Academic Press, New York. Eggen, O., Lynden-Bell, D., and Sandage, A. : 1962, Astrophys. J. 136, 748. Hoerner, S. von: 1955, Z. Astrophys. 35, 255. Innanen, K. : 1973, Astrophys. Space Sei. 22, 393. Kinman, T.: 1959, Monthly Notices Roy. Astron. Soc. 119, 559. Lohmann, W.: 1956, Z. Phys. 144, 66. Matsunami, N. : 1964, Pub. Astron. Soc. Japan 16, 141. Murray, C. H., Jones, D. H. P., and Candy, M. P." 1965, Roy. Obs. Bull. 100. Ostriker, J. and Peebles, P. : 1973, Astrophys. d. 186, 467. Peterson, C. : 1974, Astrophys. J. 190, L17. Peterson, C. and King, I. : 1975, Astron. d. 80, 427. Schmidt, M. : 1965, in A. Blaauw and M. Schmidt (eds.), Stars andStellar Systems, Vol. V, Chicago.