6 Proper motion, space velocity and orbital parameters of the cluster

We have obtained three values for absolute proper motion of the cluster, from (1) the maximum likelihood fitting $(-1.65 \pm 0.39,-2.69 \pm 0.38)$ mas/yr; (2) the mean value of proper motions of all stars, weighted by membership probabilities $(-2.08 \pm 0.41,-3.07 \pm 0.39)$ mas/yr; (3) the mean value of proper motions of all members, equally weighted $(-1.94 \pm 0.47,-2.86 \pm 0.39)$ mas/yr. In comparison, Geffert et al. (1997) gave the value (-1.0,-3.5) mas/yr (this value is also used by Odenkirchen et al. 1997). All these results can be regarded as consistent within the uncertainties. We take the second set as our final estimation of the absolute proper motion of the cluster. Taking into account a systematic error of the Hipparcos proper motions of $\pm 0.25$ mas/yr, we have the absolute proper motion of the cluster as $(-2.08 \pm 0.48,-3.07 \pm 0.46)$ mas/yr. We calculated the space velocity of NGC 4147 in a system of galactic standard of rest using a distance of the Sun from the galactic center of 8 kpc, a rotation of 225 km s^-1 at the place of the Sun, a peculiar velocity of the Sun relative to LSR of U = 10 km s^-1, V= 15 km s^-1 and W= 8 km s^-1 (note that U points from the Sun to the galactic center, V is parallel to the galactic plane in the direction of the galactic rotation and W perpendicular to the galactic plane), a heliocentric distance of NGC 4147 of 18.32 kpc and a heliocentric radial velocity of NGC 4147 (from Harris 1996) of 183.2 km s^-1. The resulting space velocity of NGC 4147 is $164 \pm 24$ km s^-1. Taking the proper motion value from just the maximum likelihood fitting, we obtained the space velocity as $142 \pm 12$ km s^-1, which is, within the uncertainty range, in a good agreement with the above.

We didn't take into account the uncertainty of the heliocentric distance determination of the cluster in the above reductions. Let us assume the adopted distance of the cluster has a relative error of 10%. With this error assumption, we used two extreme distance values (that is, by adding $\pm1\sigma$variations to the adopted one) as input data for a similar calculation. The derived space velocity values are $183 \pm 30$ km s^-1 and $149 \pm 18$ km s^-1, respectively, which are consistent with our adopted value of $164 \pm 24$ km s^-1 within the uncertainty range.

Therefore we take the space velocity result derived without considering the distance uncertainty as input value in the following kinematic discussions.

   

Table 10: Space motion and orbital parameters of NGC 4147

PM value	$\Pi$	$\Theta$	Z	$R_{\rm a}$	$R_{\rm p}$	e	i
	(km s-1)	(km s-1)	(km s-1)	(kpc)	(kpc)
(2)	+70	-91	116	22.5	4.2	0.69	$89\hbox{$.\!\!^\circ$ }2$
(2)$+1\sigma$	+104	-58	123	22.0	2.3	0.81	84.2
(2)$-1\sigma$	+26	-124	109	23.5	6.9	0.55	91.7

The velocity components of NGC 4147 in the galactic standard of rest as well as the orbital parameters calculated using the potential model from Dauphole & Colin (1995) are listed in Table 10. In addition, we varied the cluster proper motion by $\pm$ 1 $\sigma$ and re-calculated the corresponding velocity and orbital parameters, which are also listed in Table 10. The velocity components $\Pi, \Theta, Z$ are in a system of galactic standard of rest. $\Pi$ points radially outwards from the galactic center to the cluster, $\Theta$ is in the direction of the galactic rotation, Z is perpendicular to the galactic plane and towards the galactic north pole; $R_{\rm a}$ is the apogalactic distance, $R_{\rm p}$ is the perigalactic distance, e is the eccentricity, and i is the mean inclination angle of the orbit plane.