Orbits of galactic globular clusters

Astron. Nachr. 313 (1992) 2, 69-81

Orbits of galactic globular clusters

M. ODENKIRCHEN and P. BROSCHE, Bonn, Germany

Sternwarte der Universitat Bonn

Received 1992 January 28; accepted 1992 February 17

New absolute proper motions of the two globular clusters NGC 4147 and NGC 6218 and a specific model of galactic mass distribution are used to integrate their orbits numerically. The resulting values of orbital parameters and their variation due to the uncertainty of the initial values are discussed. Furthermore, the deviations from the time average of the virial theorem are determined. Unter Verwendung von absoluten Eigenbewegungen und einem bestimmten Modell der galaktischen Massenverteilung werden Bahnen der Kugelsternhaufen NGC 4147 und NGC 6218 und deren charakteristische Parameter numerisch berechnet und diskutiert. Dabei werden auch mogliche Variationen der Bahnen und Bahnparameter beriicksichtigt, welche sich aus der vorgegebenen Unsicherheit der Anfangswerte ergeben konnen. SchlieBlich wird festgestellt, welche Abweichungen vom zeitlichen Mittelwert des Virial-Theorems auftreten

Key words: galaxy: kinematics and dynamics - clusters: globular

A A A subject classification: 151; 154

1. Introduction

During the last decade, proper motions of a small number of globular clusters have been measured with respect to an extragalactic reference frame (Brosche et.al. 1983, 1985, 1992; Tucholke 1992; Scholz 1990). This method seems to be free of drastic systematic errors and yields tangential motion in an inertial frame with r.m.s. errors of about 0.13 arcsec/century. These proper motions, combined with measurements of the cluster’s distance, position and radial velocity lead to a complete set of space- and velocity-components with respect to a heliocentric inertial frame. For further reduction to a galactocentric inertial frame the sun’s location and motion in the very same frame has to be estimated and added. The resulting galactocentric space- and velocity-components of the cluster and a given force field or potential determine uniquely the orbit. If both are sufficiently accurate, the corresponding orbit provides the geometrical history of the cluster as a whole for times in which our Galaxy itself has not changed its gravitational field.

2. The mass model and its implication

We aimed to calculate the orbit of a globular cluster due to the net gravitational interaction between the cluster and the rest of the Milky Way. The galactic mass distribution was represented by a model from Allen and Martos (1986) with the components bulge (mass-point in the center), disk (system of spheroids according to Schmidt(l956)) and halo (spherical model of dark matter). This model produces a gravitational potential 0 , which 1. is static (time-independent), 2. is nonspherical, but has 3. an axis of rotational symmetry to be identified with the north- south-axis of the galaxy, 4. is symmetric with respect to the galactic plane. In more detail, using cylindrical space-coordinates p,z ,(p adapted to the above symmetries, the corresponding nonvanishing components of specific force F p , F, fulfill the relations:

Fp < 0

70 Astron. Nachr. 313 (1992) 2

These properties are rather common among galactic mass models and allow some general conclusions to be drawn for any orbit in this potential: (For simplicity the quantities energy, potential, force, angular momentum and virial in this paper are meant t o be specific, i. e. divided by the unknown mass of the cluster!) a) There are only two classical integrals of motion, namely the energy E (due to time-independence of 0) and the z-component J, of the angular momentum vector (due to rotational symmetry of 0). No other independently conserved quantities are known so far. b) Fixing the absolute value of J z , the equations of motion can be reduced to motion in the meridional plane with coordinates p,z and an effective potential := 0 + J,2/2p2 with mimimum Emin. Fixing also the value of E to Emin < E < 0, motion in the meridional plane is confined to the inner part of the closed curve ‘Pel, = E (dashed line in Fig. 1,2,7,13 and 15). c) The sense of rotation of the orbit around the z-axis is constant and determined by the sign of J , , but the period of revolution will generally vary with time. The vector of angular momentum J := r x r and thus the plane perpendicular t o it known to be the momentaneous plane of motion are also subject to variations with time, since J, and J, are not conserved. d) Concerning the precession of J around the z-axis we find

j . (e, x J ) = - ~ + ( Z ’ F ~ - z ~ F , ) (2)

where as a result of eq. (1) the expression in the bracket on the right is always positive. Thus J precedes always opposite to the sense of revolution of the orbit (sign of +) around the z-axis of the galaxy. e) The quantity J := IJI is expected to oscillate, this can be analysed by

d - J 2 = 2 5 . 5 d t

= 2(r x vm)(r x F)

= o - (rllF or rllvm), (3)

where v, := vpep + v,e, denotes the velocity of the orbit in the meridional plane. Using eq. (1) we conclude, that J takes a local maximum if and only if the orbit intersects the galactic plane ( z = 0 ) and a local minimum if and only if r is tangential to the orbit in the meridional plane, i.e. r 1 1 v,, which happens only a t z # 0. ( Here 11 stands for paral le l or a n t i p a r a l l e l . ) f) There exists a group of operations, which lead from one solution of the equations of motion, i . e. one orbit of the model, to another one. It consists of spatial rotations about z-axis (due t o rotational symmetry of a), spatial reflection with respect to the galactic plane (due to north-south symmetry of a), reversal and translation of time (due to time-independence of 0). g) As for any particle bound to an external field of force F ( i . e. 1x1 5 R,,,) the virial theorem is valid here in the form

1

- - r2 + r . F = 0

where overlining of quantities means averaging along the orbit over a time-interval of infinite length or over an integer number of periods in case of a periodic orbit. The first term in the sum (4) is known as 2 times the kinetic energy (denoted by 2Tk,,), the second is called the virial (denoted by Vir) . The last two conclusions have to be commented: Concerning f ) we emphasize that despite the symmetry of the potential an orbit is not forced to show symmetry with respect to the galactic plane, neither in a finite time-interval nor in the limit t --+ cm. In fact, many types of orbits with envelopes having north-south-asymmetry have been found when investigating systematically the phase-space structure of the dynamical system defined by the potential described above (Odenkirchen 1991). As will be shown in the following chapters, north-south-asymmetry may also occur among the orbits of globular clusters.

Concerning g) i t depends on the particular orbit and is not known a priori, how much the terms 2Tkin and Vir vary from their mean values and which length of the time-interval is necessary to find the theorem eq. (4) approximately fulfilled. In practice, observations provide us with a snap-shot of the present state. Thus we ask how large the deviation of the virial sum from its mean value is a t such an arbitrary instant.

(4)

3. Orbits of two galactic globular clusters

As examples of an outer and an inner halo object we discuss the orbits of the globular clusters NGC 4147 and NGC 6218 (M 12) according to the galactic model mentioned before. They have been obtained by numerical

M. Odenkirchen and P. Brosche: Orbits of galactic globular clusters 71

NGC

4147 6218

integration with the extrapolation method of Stoer and Bulirsch (1973) (cf. also Hussels 1973) and are based on the mean initial values shown in Tab. 3. The latter were drawn from observational data given by Brosche et al. (1983,1985,1991), Harris (1976), Peterson et al. (1986), Peterson and Latham (1986), Harris et al. (1983) and exhibited in Tab. 1, the solar parameters of Tab. 2, and conventional equatorial positions of galactic north pole and galactic center ( cf. Johnson and Soderblom 1987 ). All space coordinates I , y, z and velocity components U , V, W, as well as U L S R refer to a galactocentric righthanded Cartesian coordinate system with its tangent vector e, directed towards the galactic north pole and e, directed towards the anticenter. Cylindrical coordinates p, 'p, z also correspond to this system.

The orbits were generally calculated backwards in time over a standard interval of 10 Gyr taking into account the possible age of globular clusters as well as the time for which the galaxy may be assumed to be stationary. The orbital parameters given here always refer to this interval of time whereas plots may also refer to longer or shorter periods.

a( 1950) 6( 1950) D V r pa cos 6 p6 [kpcl [kmlsl [arcsecll OOyr] 0 , h m

12 07.6 18 49 17.3 k 3 . 1 181.4f2.0 -0.27k0.13 0.09f0.13 16 44.6 -1 52 5.75f1.0 -41.3f2.0 0.16f0.13 -0.80f0.13

Table 1 Globular clusters: data from observations

Zsun Ysun Zsun

[ kPC 1 -8 .0f0 .5 0.0 0.0

u s u n K u n Wsun VLSR

[ kmls I [km/sI

10.Ofl.O 15 .0f3 .0 7 . 5 f 0 . 5 225f20

Table 2

NGC I z Y Z

Sun: estimated location and motion relative to the galactocentric inertial frame

U V W [ kPC 1

For sources of the data and definition of x , y, z, U, V, W see text

[ k m l s 1

Table 3 Globular clusters: derived space- and velocity-components

6218 I -3.0f1.0 1.4f0.2 2 .5f0 .4 I 8 3 f 26 9 2 f 47 -148f40 For definition of x, y, z and U, V, W see text

3.1.

With the values of the constants of motions defined by the initial values of Tab. 3 the resulting limiting curve for the orbit of NGC 4147 in the meridional plane ( dashed line in Fig. 1 ) extends far into the outer halo of the galaxy. According to Fig. 1 the same holds for the orbit itself, which in the standard interval of time appears to be nearly periodical in the meridional plane, thus filling the inner part of the limiting curve only scarcely and in an asymmetric manner. The corresponding frequency diagrams of Fig. 4 reveal the following: The cluster spends more than 65% of the time interval at distances R larger than 40 kpc from the galactic center and at absolute velocities u smaller than 200 km/s. The velocity uv of rotation around the galactic z-axis is smaller than 100 km/s in absolute value over 75% of the time. Consequently, in 10 Gyrs the orbit revolves only 7 times in eastern direction around the galactic center, with periods of revolution T alternating between 1.0 and 1.8 Gyr. This can be recognized in Fig. 3, where the projection of the orbit into the galactic plane is plotted. Again the orbit appears to be nearly periodic, deviating from periodicity by a slow secular rotation of opposite sense, yet keeping up also an east-west-asymmetry during the standard interval.

The orbit of NGC 4147

72 Astron. Nachr. 313 (1992) 2

Fig. 5 shows the behaviour of the angular momentum vector during the standard interval: Its absolute value J varies around the mean ’J by less than 10%’ adopting maxima and minima according to the rules described above (Fig. 5a). I ts precession within 10 Gyr amounts to only of a full turn (Fig. 5c) and the inclination angle i towards the z-axis remains roughly constant (Fig. 5b). This is due to the orbit’s extension into the outer halo, where the potential of the model tends towards spherical symmetry.

Table 4 NGC 4147: Orbital parameters in the standard time-interval

a) Geometry of the orbit -

Rmzn R m a z R R(t = 0) e Zmcn Zmoz

12.5 72. 51. 19.5 0.704 -62. 42, [ kPC I [ kPC I

b) Absolute value of velocity -

Vmin Vmaz V v ( t = 0) v e s c ( l = 0) [ km/s 1

63. 381. 179. 324. 461.

c) Rotation around z-axis -

Vv,mtn Vp,mas vv v9(t = 0) T nu -

I kmls 1 l G d . . - 1

-357. -35. -82. -237. 1.43 7.0

d) Angular momentum and inclination of the orbit - - J A J I J i Ai n P

[kpc.km/s] [degree]

0.24 +2.2 4849. % 118.8 0 0

e) Kinetic energy and virial * - - 2Tktn 2Tktn,rnos 2Tktn.msn Vir Virmaz Virmtn

[ km2/s2 ]

40284. 145702. 3985. -40216. -35341. -42721.

- Q(t = 0) Q “ Q Qmaz Qmin

[ kmz/sZ 1 1.71 1.4. 1.00 2.60 -0.92

* averages taken over 3 orbital (quasi-)periods

Table 5 NGC 4147: Variation of orbital parameters with initial values

- initial T R m t n R m o z R e Zmrn Zmoz

values [Gyr] [ kPC 1 [ kPC I mean 1.43 7.0 12.5 72. 51. 0.704 - 62. 42.

l a 0.91 11.0 8.3 44. 31. 0.680 - 39. 39. lb 2.80 3.6 14.6 157. 108. 0.830 -122. 133. 2a 2.70 3.7 13.4 144. 101. 0.830 -113. 128. 2b 1.02 9.8 12.7 48. 35. 0.580 - 40. 40.

In Fig. 6a the terms 2 T k , , and Vir of the virial sum eq. (4) are seen to oscillate quasiperiodically with the same period of 2.86 Gyr (corresponding to the approximate orbital period of 2 revolutions around the galactic center), but with surprisingly different amplitudes. Compared to the deviations of 2Tk,,, from its mean, which result from

M.

7

0 a 3 N

0.2[, r r I , , , I , , , , , , ,

-

-

-I

Odenkirchen and P. Brosche: Orbits of

I " " I " " I " " I

1 b) 0.15 -

0.1 -

0.06 -

0-

0 20 40 60 80 P [kpcl

Fig. 1.

- 0 a

h 3

-

galactic globular

Fig. 3.

01'1'1'"'''''''

clusters

50

7 0 a 0 3 N

-50

0 20 40 60 80 P [kpcl

Fig. 2.

Fig. 1 NGC 4147: Meridional orbit in the standard time-interval of (-10 Gyr; 0 Gyr). Fig. 2 NGC 4147: Meridional orbit in the time-interval (-40 Gyr; 0 Gyr). Fig. 3 NGC 4147: Orbit in the standard interval, projected onto the galactic plane. Fig. 4 NGC 4147: Distribution of values of selected orbital parameters along the orbit with time in the standard interval. a) distance from the galactic center, b) absolute value of velocity, c) pcomponent of velocity.

J-

100 200 300 400 v b / s I

0.3 -

0.2 -

-

0.1 -

-300 -200 -100 YPM r w s i

i

0

73

Fig. 4

74 Astron. Nachr. 313 (1992) 2

5400

p 5200 E

5000

$4800 - 4800

4400

7

0

122

3 120 0)

g 118 ;I

.- 118

-2 0 2 Jx/l Jz I

Fig. 5. a) absolute value of the angular momentum vector, b) inclination of the momentaneous plane of motion, c) projection of J/IJzI onto the galactic plane for the interval (-40 Gyr; 0 Gyr).

NGC 4147: Behaviour of angular momentum quantities in the standard interval.

1.5 105

F;f 2 105

E 5 lo4

s

- X Y

C 0

cp

0 .4.

0.3

0.2

0.1

0

Fig. 6. a) variation of 2Tk,, (upper curve) and Vir (lower curve) with time, b) distribution of values of the scaled virial sum Q along the orbit.

NGC 4147: Behaviour of Virial quantities in the standard time-interval.

the orbit's large spatial extension] the virial remains nearly constant with time. Consequently, the virial sum is found to vary by nearly the same amount as 2Tkin. Wishing to quantify these variations in a relative measure, we - defined the quotient Q := (2Tki, + Vir)/2Tki,. ( Since the mean value 2Tk,, + Vir x 0 cannot be used for scaling, 2Tkin is an appropriate choice here. In order to minimize deviations - from the theoretical long-time-average, we make use of the quasiperiodicity of the orbit by taking the average 2Tkin over the maximum of 3 (quasi-)periods ( 8 . 5 8 Gyr) within 10 Gyr. ) The distribution of Q along the orbit (during 10 Gyr) ranges from -0.9 to 2.6 ( cf. Tab. 4e ) and is shown in Fig. 6b. Its strongly asymmetric shape can be explained by the close relationship to the distribution of velocity in Fig. 4b ( Vir x const, 2Tk,,, = v2 ). The rms-deviation UQ of Q, which may serve as an estimate of IQI for arbitrary observation, is equal to 1. In fact, with the clusters distance R(t = 0) being relatively near to Rmin the present value of Q is considerably higher. Due to periodicity, the virial theorem should be valid for time-averages over multiples of the period of 2.86 Gyr. This is confirmed by calculation of 2Tki, and Vir for averaging intervals of increasing length r. For comparison, Tab. 4e gives the values of 2Tkin, V i r and

- - --


for an interval of 3 (quasi-)periods ( T = 8.58 Gyr ). But apart from such discrete values of T we find that, while Vir converges rapidly with increasing T, more than the whole standard interval is needed for averaging in order to confine 2Tki,, permanently to its limit within 3%.

Extrema, mean values (overlined symbols) and present values (argument t=O) of the orbital parameters mentioned so far are given in Tab. 4. One sees that the present values are atypical for the orbit compared with the distributions of Fig. 4 and 6b.

In order to find the envelope of the orbit, the equations of motion were integrated over a longer time-interval. Fig. 2 shows the meridional orbit calculated backwards so far ( namely to -40 Gyr ) that we can recognize that the orbit is box-like, i. e. it keeps on filling a (strongly bent) box in the meridional plane, which is symmetrical about the galactic plane and touches the limiting curve only with its four corners. In other words, the meridional orbit is topologically equivalent to a Lissajous-figure. Therefore the asymmetries in the distribution of the orbit in space vanish in the limit t -+ 00, yet they are relevant for times comparable with the age of the cluster.

- -

3.2.

According to Tab. 3 NGC 6218 is located much nearer to the galactic center with lower values of the velocity- components than NGC 4147. Therefore it has lower values of E and IJ, I and the limiting curve for the meridional orbit ( dashed line in Fig. 7, note the change in the scale with respect to Fig. 1 ) encloses only a small area well within the cylinder which contains the solar orbit. Because of the small distance from the galactic center the number of revolutions around the center in the standard interval must be high, in fact it is 10 times as high as for NGC 4147. Thus, because of overcrowding the orbit within 10 Gyr cannot be plotted in one diagram. Instead we restricted its meridional projection (Fig. 7) and its projection onto the galactic plane (Fig. 8) to the interval (-1 Gyr; 0 Gyr). In doing so, there is no loss of information to the reader, because it turned out that all important features of the orbit appear within this interval already: Fig. 7 shows the meridional orbit to be box-like again, filling the box almost symmetrically already in & of the standard time-interval. This box takes only about half of the area allowed to the orbit by the limiting curve. So the cluster keeps at a minimum distance of about 2.4 kpc from the assumed orbit of the sun and remains within 4 kpc distance from the galactic plane. Fig. 8 suggests an interpretation of the orbit by rapidly preceding ellipses with medium eccentricity ( cf. the value of e in Tab. 6 ). Obviously, the orbit will take values of 'p and z in an absolutely symmetrical way during the standard interval. The distributions of distance and velocity along the orbit with time are given in Fig. 9 ( these again refer to the full interval of 10 Gyrs length ). They show, that NGC 6218, while being much nearer to the center moves only a bit faster on average than NGC 4147. The same can be seen in Tab. 6, where extrema, mean values etc. of selected orbital parameters are presented. Remarkably, the present values of R, v and vv are quite near to the mean values and maxima of the corresponding frequency distributions in this case.

An analysis of the behaviour of the angular momentum vector of the orbit of NGC 6218 in Fig. 10 reveals the same rapid precession as seen in Fig. 8. During the standard interval J makes 22 turns around the z-axis. At the same time its absolute value J varies by up to 55% around 7, the inclination angle i oscillates with an amplitude Ai of 10 to 14 degrees around ( cf. Fig. 10a,b and Tab. 6d ). The considerable amount of change in the angular momentum vector can be explained by the fact that the cluster is continously located in the area where the galactic potential deviates strongly from spherical symmetry; the rapidity of this change is due to the rapid movement of the cluster from one passage through the galactic plane to the next.

For the same reason, 2Tki, and Vir vary fast (Fig. l la) . The amplitude of 2Tkin is smaller for NGC 6218 than for NGC 4147 because of the comparatively small spatial extension of the orbit, still it is much larger than the amplitude of Vir and again dominates the variation of the virial sum. The distribution of Q ( here: virial sum scaled by 2Tki, taken over 10 Gyr ) in Fig. l l b and the corresponding Tab. 6e show that, while the present value is relatively close to the average ( the latter complies well with the theoretical zero for T -+ 00 ), values of Q found in 10 Gyr are scattered asymmetrically from -0.66 to 2.10 with an rms-deviation of 0.55 from zero. Concerning the averages we find, that on interval lengths 2 0.5 Gyr both remain permanently within 5% of their limiting value. Thus we call the virial theorem eq. (4) approximately fulfilled when averages are taken over such times.

The orbit of NGC 6218

-

and

3.3.

Having in mind the completely different orbital parameters of the two clusters, i t may seem surprisingly at first sight, that the long-time averages of 2Tk,, (resp. of Vir) for both objects agree within 1% to a value of ~ 4 0 0 0 0 km2/s2. In fact, this only reveals an effective resemblance of the chosen galactic model to a much simpler one. For if we consider a spherical distribution of mass in the galaxy with M ( R ) ( mass inside sphere of radius R ) increasing proportional to R and scale it by the velocity v,,, of the solar vicinity, we get for the radial force component

Comment on the average kinetic energy

76 Astron. Nachr. 313 (1992) 2

FR = -V~,,,,/R, (4)

which means that the rotation curve of this simple model is completely flat ( i. e. all circular orbits in the galactic plane have the same velocity v,,, ). In this case the virial becomes

( 5 ) 2 Vir = R . FR = -vs,,

and it follows from eq. (4), that for any orbit in this field (as well as -E) has the very same value v:,,,. Adopting v,,, = 200 km/s in accordance with the flat outer part of the rotation curve of the more complicated model we consistently arrive at the value vz,, = 40000 km2/s2 found above for m.

Table 6 NGC 6218: Orbital parameters in the standard time-interval

a) Geometry of the orbit

Table 7

- R m t n R m a s R R(t = 0) e Zmin zmaz

2.1 5.6 4.4 4.1 0.455 -4.0 4.0 [ k P C 1 [ k P C I

b) Absolute value of velocity -

Vmtn Vmaz V v ( t = 0) uesc( t = 0) [ km/s 1

100. 358. 193. 193. 575.

c) Rotation around z-axis -

Vq,min Vy.mas VV U V ( t = 0) T nu -

[ km/s 1 P y r I -217. -71. -122. -118. 0.146 68.3

d) Angular momentum and inclination of the orbit

A J / T i Ai n p [kpc.krn/s] [degree]

22 $10 732. -23 124 - 1 4

+55 %

e) Kinetic energy and virial - - 2Tk,n 2Tkin,mas 2Tkin,mtn Vir V i r m a z V i r m i n

[ km2 /sz ]

40576. 128423. 9965. -40584. -37018. -50357.

- Q(t = 0) Q OQ Qmas Qmin

[ km2/s2 ] -0.05 - 2 . 1 0 - ~ 0.55 2.10 -0.66

NGC 6218: Variation of orbital parameters with initial values

- - initial T nu R m i n R m a s R e Zm:n Zmas

values [Gyr] [ kPC 1 kPC 1 mean 0.146 68.3 2.1 5.6 4.4 0.455 -4.0 4.0 la 0.145 69.1 3.0 5.1 4.3 0.259 -3.2 3.2 l b 0.164 61.1 1.6 6.7 4.9 0.614 -5.3 5.3 2a 0.129 77.5 1.3 6.1 4.0 0.649 -2.1 3.4 2b 0.179 55.9 2.8 7.1 5.4 0.434 -5.2 5.2


4. The influence of initial data uncertainty on the orbits

In Tab. 1 and 2 we provide error estimates for the quantities needed to determine the cluster's initial values and constants of motion. These errors have to be propagated somehow into errors of the latter. Tab. 3 gives error-bars for the initial values obtained by the usual linear formula ( cf. Brandt 1981) assuming the errors in Tab. 1 and 2 to be uncorrelated. Here, we encounter two complications: 1. The derived errors may well be correlated, thus for example the errors in Tab. 3 are not useful for describing the true uncertainty of the vector (2, y, z , U , V, W). 2. Since not all of the errors in Tab. 1 and 2 can be regarded as being small, the validity of a linear propagation is questioned and must be checked.

7 0

3

N

a A

4 -

2 -

0 -

-2 -

-4 -

7 0 a

x 3

0 2 4 0 P [kpcl

Fig. 7. (left) NGC 6218: Meridional orbit in the time-interval (-1 Gyr; 0 Gyr). Fig. 8. (right) NGC 6218: Orbit in the interval (-1 Gyr; 0 Gyr), projected into the galactic plane.

0.4

0.3

0.2

0.1

0

Fig. 9. NGC 6218: Distribution of values of selected orbital parameters along the orbit with time in the standard interval. a) distance from the galactic center, b) absolute value of velocity, c) pcomponent of velocity.

78 Astron. Nachr. 313 (1992) 2

Fig. 10. NGC 6218: Behaviour of angular momentum quantities during the interval (-1 Gyr; 0 Gyr). a) absolute value of the angular momentum vector, b) inclination of the momentaneous plane of motion, c) projection of J/IJzI onto the galactic plane.

3. c N

-5. lo4

-1 -0.5 t [Gyrl

0

m 0 L

0.2

0.1

0 -1 0 1 2

9

Fig. 11. NGC 6218: Behaviour of virial quantities in the standard time-interval. a) variation of 2Tktn (upper curve) and Vir (lower curve) with time, b) distribution of values of the scaled virial sum Q along the orbit.

Concerning the second point, a multidimensional random Gaussian distribution of values with means and standard deviations according to Tab. 1 and 2 was produced and transformed into the initial values and constants of motion. It turns out, that for determining the uncertainty a linear formula is adequate in case of the initial values, but not in case of the constants of motion. Therefore Fig. 12 and 14 show the uncertainty of the pair (J,, E ) according to the point distribution method.

The ellipsoidal shape of the 'error distribution' in (2, y, z , U , V, W ) being confirmed, the first point can be treated by finding the semi-major axis vectors of the error ellipsoid given by the covariance matrix of the initial values. Due to the predominant error in proper motion two of these six vectors carry the major variability of the initial values. Consequently, they have been added and subtracted from the mean initial values to produce four alternative sets of initial values ( enumerated l a to 2b in Tab. 5 and 7 ) and the corresponding orbits for each cluster. Fig. 13a,b and 15a,b show some of these orbits in meridional projection, Tab. 5 and 7 exhibit the dependence of some important orbital parameters on the input quantities (within the standard time interval).

In the case of NGC 4147 the set of likely values of J , and E covers a large and elongated area in the (Jz, E)- plane. So there remains some doubt about whether the cluster is really bound to the model galaxy ( E < 0),


whereas we can be rather sure about the cluster's sense of rotation around the z-axis to comply with galactic rotation (J, < 0). Shape and size of the meridional orbit are subject to large variations with the maximum distance of the cluster from the galactic centre ranging from 40 to 160 kpc in the four selected examples given in Tab. 5 . Special orbital features like asymmetries appear at random in the large variety of possible orbits. Therefore, at present they cannot be regarded as being typical for this cluster.

The influence of distance D ( object to observer ) on the determination of initid values is twofold: A small D produces a small error of D itself, but due to (on average) large values of proper motion it also produces high relative accuracy of proper motion. This explains, why in the case of NGC 6218 errors of the initial values are smaller, thus the values of the constants of motion are defined more accurate ( cf. Fig. 14 ). For example, there is no doubt about J, and E both being negative in this case. On any of the four orbits la to 2b the cluster stays within the solar circle and within 5.5 kpc from the galactic plane, the number of revolutions lying roughly between 50 and 80 in the standard interval. Still it remains uncertain if the type of the orbit is box- (Fig. 15b) or tube-like (Fig. 15a), the latter being asymmetric with respect to the galactic plane.

5

-5.

-1.

3 -6000 -4000 -2000 0

Jz [kpc km/s]

Fig. 12 by the point, the lines refer to SO%, 66% and 90% likelihood.

NGC 4147: Uncertainty in the determination of the constants of motion. The mean value of J , and E is marked

9 0 a 3 N

0 20 40 P b P C 1

- 0 a X -i

N

0 50 100 150 P ~ P C I

Fig. estimated error. a) initial values la , b) initial values Ib.

13. NGC 4147: Meridional orbits in the standard interval as produced by initial values varied according to the

80 Astron. Nachr. 313 (1992) 2

-1.2 lo6 L 7

cf ~ 1 . 4 lo5 - 3

\ N

w

-1.6 lo5 -

-1000 -800 -600 -400 -200 Jz [kpc km/s]

Fig. 14

\ \ \ \

I \ 7 I \ V a I

3 O - \ N

I \ I

I /

/ /

/ /

._-r-

-5 0 L 2 P [kpcl 4 6

0

Fig. 14. NGC 6218: Uncertainty in the determination of the constants of motion. The mean value of J , and E is marked by the point, the lines refer to SO%, 66% and 90% likelihood. Fig. 15. NGC 6218: Meridional orbits in the interval ( -1 Gyr; 0 Gyr) as produced by initial values varied according to the estimated error. a) initial values 2a, b) initial values 2b.

V' a

N

Fig. 15.

5 . Summary

We started from proper motions with the very same absolute accuracy and used them to trace the dynamical history of two globular clusters. In the case of NGC 6218 we arrived at rather definite results while in the case of NGC 4147 we are left with only more general insights. This is due to differences in distance and proper motion values between the two clusters. For NGC 4147 and other distant clusters, a further increase in the accuracy of proper motions would be desirable. Nevertheless, with the available accuracy clear distinctions between the possible orbits of NGC 4147 and NGC 6218 can already be achieved. Being capable of such qualitative characterization, the calculation of theoretical orbits could for example lead to a decision on the existence of thick-disk clusters ( cf. Zinn 1985 ). However, the determination of absolute proper motions for such clusters has to be postponed until an extragalactically calibrated positional reference system is available at low galactic latitudes. The forthcoming Hipparcos system will be the one of choice.


References

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Address of the authors:

Michael Odenkirchen Sternwarte der Universitat Bonn Auf dem Hiigel 71 W-5300 Bonn Germany

Peter Brosche Observatorium Hoher List W-5568 Daun Germany

6 Astron. Nachr 313 (1992) 2

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Orbits of galactic globular clusters