6 Results

For 140 clusters no members could be found in the Hipparcos Catalogue. In the remaining 205 clusters, we found altogether 630 certain and 130 possible members. They are listed in Table 1 together with some non-members. The majority of non-members were never supposed to be cluster stars since they have colours, magnitudes and angular distances that clearly rule out their membership. In order to keep Table 1 short, we therefore list only non-members which could be found in the "Database for Galactic Open Clusters'' (BDA) by Mermilliod (1995). These stars are presumably much closer to the cluster centre than the majority of non-members and many of them were previously thought to be members.

Column 8 of Table 1 gives a membership probability derived from the proper motions. It was calculated as follows: If a star was not used to calculate the mean cluster motion, we took the difference x between the proper motion of the cluster (taken from Table 2) and the star and calculated the product of this difference with the sum $\Sigma$ of the covariance matrices of the star and the cluster according to:

$$c = x' \; \Sigma^{-1} x .$$ (1)

The dimensionless number c was then transformed into a membership probability under the assumption that it is distributed like a chi-square distribution with two degrees of freedom. If a star was used for the derivation of the mean cluster motion, we first calculated a solution for the cluster without using the star in question and compared the proper motion of the star with this new solution.

Most non-members were classified as such due to their large proper motion differences relative to the mean cluster motion. The remaining stars that were classified as non-members were either too far away from the cluster centre to be bound or had a photometry which was incompatible with the assumption of cluster membership. Possible members with a high probability for membership are often not well-studied, outlying stars. They may be members according to their proper motion and photometry.

Table 2 presents our final solution for the astrometric cluster parameters, determined as outlined in Sect. 5. The unit weight standard deviations show some scatter around the theoretical value of one. This is caused by the small number of stars available for most clusters. The mean over all clusters that contain at least two stars is 1.01, which is very close to the theoretical value. We note that the unit weight standard deviations are generally smaller than one for clusters with more than 5 stars in the astrometric parameter determination. Since the correlations of the abscissae residuals become more important for clusters containing many stars, this may indicate that these correlations were slightly overestimated when the mean values were calculated. If the small-scale correlations are not taken into account, we obtain a mean unit weight standard deviation of 0.97 for all clusters, which is also very close to the theoretical value of one.

Figure 1: Proper motion errors derived by taking small-scale correlations into account compared to the uncorrelated solution as a function of the number of cluster stars N. The crosses show individual clusters, the dots mark the mean values for a given N. The solid line shows a dependence proportional to N-0.15 (exponent derived from Lindegren 1988), the dashed line shows one proportional to N-0.14

Figure 1 compares the errors obtained by taking small-scale correlations into account with the errors of the uncorrelated solution. Shown is the ratio of the sums $\sigma_\mu$ of the proper motion errors $\sigma_\mu = \sqrt{\sigma^2_{\mu \alpha*} + \sigma^2_{\mu \delta}}$ as a function of the number of cluster stars N. If a cluster contains only one star, this ratio must be unity, since both solutions are taken directly from Hipparcos. For clusters with more than one star, the mean error should be proportional to N^-0.5 in the uncorrelated case. For the correlated case, Lindegren (1988) estimated an N^-0.35 dependence, so the ratio of the errors should be proportional to N^-0.15. Clusters containing less than 6 Hipparcos stars are best fitted by a N^-0.14 law, very close to the expected value. An N^-0.36 decrease can therefore be taken as a rule of thumb to estimate the errors of the astrometric parameters for correlated measurements in Hipparcos.

Robichon et al. (1999) determined absolute proper motions and parallaxes of all clusters that are closer than 300 pc or have more than 4 members in the Hipparcos Catalogue. Figure 2 compares our proper motions with theirs for the clusters in common. Although they also use the Hipparcos Catalogue and a similar method to derive the proper motions, the final results differ by typically 0.5 mas/yr, which is of the same order as the quoted errors. The differences are due to differences in the stars selected as cluster members and slight differences in the abscissae formal errors and correlations that are used in the methods of Robichon et al. (1999) and van Leeuwen & Evans (1998).

Figure 2: Plot of the proper motions derived in this work minus the proper motions from Robichon et al. (1999) for the clusters in common

6.1 Comparison of the Hipparcos parallaxes with photometric distances

Loktin et al. (1994, 1997) and Dambis (1999) have determined distances and ages of open clusters on the basis of published photometry. Their data represent large and homogeneous parameter sets. With the help of the Hipparcos parallaxes, we can check for global errors f in their distance scales:

$$R_{\rm Phot} = f \cdot R_{\rm true}.$$ (2)

Such errors can be detected with a test similar to that used by Feast & Catchpole (1997): the (true) distance of a cluster is connected with the cluster parallax $\pi$ through the equation

$$\pi = 1/R_{\rm true} ,$$ (3)

so that an estimate for f can be obtained by inserting Eq. (3) into (2):

$$f = 0.001 \; \pi_{\rm Hip} \; R_{\rm Phot} .$$ (4)

Here the Hipparcos parallax $\pi_{\rm Hip}$ is measured in mas and the photometric distance is in parsecs. The mean over all clusters was taken to derive f:

$$<\!\!f\!\!>\ = \frac{\sum \frac{f_i}{\sigma^2_i}}{\sum \frac{1}{\sigma^2_i}}\cdot$$ (5)

The errors $\sigma_i$ on the right-hand side are a combination of the errors in the Hipparcos parallaxes and the (random) errors in the photometric distances to individual clusters, taken to be 10% for the Loktin sample. Since the error estimate of the photometric distance is done with the observed distance and not with the true one, the above test leads to a slightly biased estimate for f. Monte-Carlo simulations show that this bias remains small as long as the errors in the photometric distances are not of the order of 40% or larger.

From 186 clusters in common between Loktin et al. (1997) and this work, we derive a correction of $f = 1.12 \pm 0.05$ to their distance scale, i.e. their distances should be decreased by 12%. Part of this decrease may be explained by the fact that Loktin et al. assumed an Hyades distance modulus of (M-m)₀ = 3.42, which is too large since the Hipparcos data indicate (M-m)₀ = 3.33 (Perryman et al. 1998). The errors in the photometric distances of Dambis (1999) are taken from their work. From 117 clusters in common, we obtain a correction factor of $f = 0.99 \pm 0.06$. Their overall distance scale is therefore in very good agreement with Hipparcos.

Using the test of Arenou & Luri (1999), we can also check the parallax errors in the Hipparcos Catalogue: the difference $\Delta \pi$ between the Hipparcos parallax and the photometric parallax $\pi_{\rm Phot} = 1/R_{\rm Phot}$

$$\Delta \pi = \pi_{\rm Hip} - \pi_{\rm Phot}$$ (6)

is mainly caused by the error in the Hipparcos parallax, since the errors in the photometric parallaxes are only of the order of 0.1 mas. If the errors in the Hipparcos Catalogue are normally distributed and show no correlations other than those that were already accounted for in Sect. 5, the ratio of $\Delta \pi/\sigma_{\rm Hip}$ should also be normally distributed.

Figure 3 shows the distribution of $\Delta \pi/\sigma_{\rm Hip}$ for the Loktin et al. distances. A normal distribution provides a very good fit to the data. Narayanan & Gould (1999) proposed correlated errors extending over angular scales of 2 to 3 degrees and with amplitudes of up to 2 mas in the Hipparcos parallaxes as the reason for the discrepancy between recent photometric distances and the Hipparcos distance to the Pleiades. Our clusters have small angular sizes and would be effected by such errors as a whole. If they exist in the entire Hipparcos Catalogue, such errors would significantly broaden the cluster distribution in Fig. 3 (note that typical parallax errors of our clusters are only 0.5 mas). Since such a broadening is not observed we can rule out correlated errors with amplitudes of more than a few tenths of a mas for the vast majority of our clusters. A similar conclusion was drawn by Arenou & Luri (1999). We confirm their results with a larger database. We finally note that a similar result is obtained if the distances of Dambis (1999) are used.

Figure 3: Histogram of the normalised differences $(\pi _{\rm Hip}-$ $\pi _{\rm Loktin})/\sigma _{\rm Hip}$. A Gaussian provides a very good fit to the data. Correlated errors of the order of 0.5 mas or larger would significantly broaden the observed distribution and can therefore be ruled out

6.2 Cepheids in open clusters

The Hipparcos proper motions (and to a lesser degree also the parallaxes) confirm or reject the cluster membership of stars which are of astrophysical interest, like e.g. Wolf-Rayet stars, red giants and various types of variable stars. This may help to better define their physical parameters. As an example, we discuss the membership of Cepheids in our clusters.

11 Cepheids are included in Table 1. Many of them are the only Hipparcos star in their cluster, so our classification is entirely based on results found in the literature. Eight Cepheids are in clusters with two or more members (see Table 3 and Fig. 4).

The membership of U Sgr (HIP 90836) in IC 4725 and of S Nor (HIP 79932) in NGC 6087 was already discussed by Lyngå & Lindegren (1998) on the basis of the Hipparcos data. We confirm the cluster membership of both Cepheids. DL Cas (HIP 2347) is a highly probable member of NGC 129 on the basis of its photometry (Turner et al. 1992), radial velocity (Mermilliod et al. 1987) and proper motion (Lenham & Franz 1961). The Hipparcos data confirm these results: from two cluster members in the Hipparcos Catalogue we derive a cluster motion which gives a membership probability of 34% for DL Cas. The Cepheid is therefore very likely a cluster member.

V Cen (HIP 71116) is a member of NGC 5662 according to Claria et al. (1991). Figure 4a shows the proper motions of the Cepheid and the other cluster stars. The agreement is excellent and the Cepheid has a high membership probability of 90%. We conclude that it is a member of NGC 5662.

The membership of SZ Tau (HIP 21517) in NGC 1647 was proposed by Efremov (1964a, 1964b) and confirmed by Turner (1992) on the basis of UBV photometry and spectroscopy. Geffert et al. (1996), based on proper motions from photographic plates, denied the cluster membership of the Cepheid. We could find five members of NGC 1647 (three certain, two possible) in the Hipparcos Catalogue. Their mean proper motion differs significantly from the motion of the Cepheid and rules out its membership. Despite a rough agreement in the radial velocities of the cluster and the Cepheid (see the discussion in Turner 1992), we conclude that SZ Tau is not a member of NGC 1647.

Figure 4: Proper motions of Cepheids in clusters with at least two other Hipparcos members, see Table 4. Not shown are U Sgr in IC 4725 and S Nor in NGC 6087, which were already discussed by Lyngå & Lindegren (1998), and EV Sct and Y Sct in NGC 6664. Cluster stars are shown by filled circles, Cepheids without symbols

EV Sct (HIP 91239) and Y Sct (HIP 91366) are possible members of NGC 6664. The membership of EV Sct is well established by its radial velocity (Mermilliod et al. 1987). The membership of Y Sct is also very likely since its mean radial velocity of $\gamma = 17.8$ km s^-1(Moffett & Barnes 1987) is in good agreement to the mean cluster velocity of $r_{\rm v} = 17.8 \pm 0.2$ km s^-1 given by Mermilliod et al. (1987). The Hipparcos data is in agreement with the radial velocities, since both Cepheids have high membership probabilities of 81.8%. The cluster motion is however not very well established since the two Cepheids are the only cluster members in the Hipparcos Catalogue.

The situation is less clear for the Cepheids BB Sgr (HIP 92491) and GH Car (HIP 54621). BB Sgr was proposed to be a coronal member of Col 394 by Tsarevsky et al. (1966). Its membership was later confirmed by Turner & Pedreros (1985) on the basis of UBVRI photometry and by Gieren et al. (1998) on the basis of new calibrations of the surface brightness (Barnes-Evans) method. The Hipparcos data are inconclusive. Two Hipparcos members give a low membership probability of 2.7% for BB Sgr, which is not completely inconsistent with the assumption of a cluster membership. If additional members from the TRC Catalogue are taken into account, the membership probability of BB Sgr drops to below 0.1%, raising serious doubts on its cluster membership. We note here that the relative proper motion of the Cepheid is pointing towards the cluster, which advocates against a common origin of both. Given also the large angular distance from the cluster centre, we conclude that BB Sgr is unlikely to be a member of Col 394.

GH Car is a member of Tru 18 according to Vazquez & Feinstein (1990). From three stars in the Hipparcos and TRC catalogues (1 from Hipparcos, 2 from the TRC Catalogue), we derive a mean proper motion of ( $\mu_{\alpha*}$/ $\mu_\delta$) = ( $-6.79 \pm 0.83$/ $1.79 \pm 0.75$) mas/yr for the cluster. This gives a relatively low membership probability of 10% for GH Car. However, there is a discrepancy in the photometric distances: from their UBVRI-photometry, Vazquez & Feinstein (1990) found a cluster distance of D = 1550 pc. The Cepheid seems to be located further away, since the PL-relation of Feast & Catchpole (1997) gives an absolute magnitude of M_V = -3.55 and a distance of D = 2313 pc assuming that the period, mean V magnitude and reddening of GH Car are given by P = 5.72557 d, $<\!V\!>\ = 9.17$ mag and E(B-V) = 0.29 mag (Vazquez & Feinstein 1990). GH Car is probably not physically related to Tru 18. Radial velocities would help to confirm our conclusions concerning the last two Cepheids.

 

 

Table 3: Proper motions of Cepheids and open clusters. The proper motions of the clusters are calculated without taking the motions of the Cepheids into account. The constants c were calculated according to Eq. (1) and the membership probabilities in Col. 12 were derived under the assumption that c is distributed like a chi-square distribution with two degrees of freedom

		Proper motion Cepheid				Proper motion Cluster
Cepheid	Cluster	$\mu_{\alpha*}$	$\sigma_\mu$	$\mu_\delta$	$\sigma_\mu$	$\mu_{\alpha*}$	$\sigma_\mu$	$\mu_\delta$	$\sigma_\mu$	c	Mem.	Hipparcos stars used to calculate
		[mas /yr]		[mas /yr]		[mas /yr]		[mas /yr]			Prob.	the proper motion of the cluster
U Sgr	IC 4725	-4.15	0.94	-6.05	0.70	-1.68	0.87	-6.39	0.62	3.83	14.7	90801, 90900
S Nor	NGC 6087	-1.10	0.77	-1.20	0.70	-1.67	0.71	-1.60	0.66	0.51	77.3	79891, 79907, 79973
DL Cas	NGC 129	-0.75	1.05	-1.38	0.73	-2.67	0.94	-1.85	0.65	2.16	33.9	2354, 2382
V Cen	NGC 5662	-5.97	0.78	-7.18	0.67	-5.60	0.50	-7.33	0.51	0.21	90.0	71163, 71326, 71334, 71378, 71397,
												71398
SZ Tau	NGC 1647	-3.76	0.72	-6.77	0.52	-1.37	0.97	-1.02	0.77	41.6	0.0	21875, 22112, 22161, 22185, 22211
EV Sct	NGC 6664	1.16	2.31	-2.84	1.73	-0.63	1.65	-2.20	1.29	0.49	81.8	91366
Y Sct	NGC 6664	-0.63	1.65	-2.20	1.29	1.16	2.31	-2.84	1.73	0.49	81.8	91239
BB Sgr	Col 394	-0.58	1.10	-4.93	0.68	-4.80	1.12	-4.78	0.69	7.23	2.7	92505, 92650
BB Sgr	Col 394	-0.58	1.10	-4.93	0.68	-4.79	0.44	-6.22	0.37	15.43	0.0	92505, 92650, 1,6,12,22,26,27,28,33,
												52,53,55,57,58,59,60,62,63,66,67,76
GH Car	Tru 18	-8.56	1.08	3.74	0.84	-6.79	0.83	1.79	0.75	4.68	9.6	54668, 11, 16

Notes: Col 394: The numbers of the TRC stars in Col. 13 are taken from Claria et~al. (1991). Tru 18: The numbers of the two TRC stars

are from Vazquez & Feinstein (1990).