2 Plate measurements and reduction of proper motions

2.1 Plate material and measurements

Twenty plates of the region of NGC 1750 and NGC 1758 which were taken with the double astrograph at the Zo-Sè station of Shanghai Observatory are available for this study. The telescope, built by Gaultier in Paris at the beginning of this century, has an aperture of 400 mm, a focal length of 6.9 m, and a plate scale of 30$.\mkern-4mu^\prime$/mm. The size of the plates is 240 by 300 mm, or 2$.\!\!^\circ$0 $\times$ 2$.\!\!^\circ$5. The first-epoch plates were taken from 1918 to 1961, some of which were used by Li (1954) in his study of this area. The second-epoch plates were all taken from 1983 to 1986. Table 1 lists the details for the 20 plates used in this paper. The quality in Table 1 gives the sharpness of star images on the plate. The hour angles of all second-epoch plates are within the range $\pm$1 hour. The hour angles are not provided in Table 1 because the starting times of the first-epoch plates were not recorded. In the column "G" and "SL" indicate respectively the image "good" and "slightly length".

  

Table 1: Plate material

All the plates were measured on a Photometric Data Systems (PDS) model 1010 automatic measuring machine at the Dominion Astrophysical Observatory in Victoria, Canada. This microdensitometer combines an accurate, high-speed photometer with a precise x - y coordinate scanning system to allow the acquisition of density and position information from photographic images. Approximate positions of all the stars were obtained from measurements of one plate using a two-screw Mann measuring machine, and stored in a disk file. Then a small square area around each stellar position was scanned, using a 17 $\mu$m (0$.\!\!^{\prime\prime}$51) square aperture stepped by 17 $\mu$m in x and y. A 30 $\times$ 30, 40 $\times$ 40, 50 $\times$ 50, or 60 $\times$ 60 box was scanned at each stellar position depending on the brightness of the star, which determines the density and apparent diameter of its image. In order to monitor the scanning stability, a "reference loop" consisting of 15 stars spread uniformly over the plate was rescanned at the beginning, middle, and end of the measurement of each plate.

2.2 Proper motions

The reduction of the relative proper motions for 540 stars to a limiting magnitude $V\simeq 15.0$ in the region of NGC 1750 and NGC 1758 was made on the basis of the PDS measurements by means of an approach we have adopted many times before (Tian et al. 1982, 1983; Zhao et al. 1981, 1993; Su et al. 1997). There are three steps in the whole process: the first is to transform the measured results of all the plates to a common system, in order to eliminate the errors due to small differences in the orientation of different plates in scanning; the second step is to establish a reference frame, i.e. to decide upon the reference stars; the last step is to calculate proper motions of all the stars with respect to this reference frame, and their corresponding uncertainties.

Generally, any stars can be chosen as reference stars for determining relative proper motions. However, in order to obtain a good plate solution and to make the absolute proper motions of the reference frame as small as possible, our principle is to choose as many stars common to all the plate pairs as possible, except for any stars with extraordinarily large proper motions and stars located in the crowded central region. On the other hand, the distribution of star images on the plate and the magnitude distribution of the reference stars should be homogeneous. For these reasons, after two loops of the least-squares adjustment, 300 stars with residuals in both x and y coordinates less than 2${\sigma}_x$ and 2${\sigma}_y$ respectively were chosen to be reference stars from the 370 stars common to all the plate pairs, where ${\sigma}_x$ and ${\sigma}_y$ are the rms residuals in the x and y coordinates obtained from the least-squares adjustment.

There are two ways that can be used to determine proper motions. One is known as the plate-pairs method, and another is called the central overlap technique. Owing to the limited number of reference stars and the accuracies of the proper motions of these stars, the plate-pairs technique is used in the present study. All the linear and quadratic coordinate-dependent terms and the coma term are included in the plate solutions. The weighted mean of the proper motion of a star obtained from all of the available plate pairs should be taken as the final value of the proper motion of the star. The proper motion weight for a star in a plate pair is determined from the epoch difference of the pair. As we know, accuracies of proper motions for individual stars are different, since the time baselines, number of available pairs, weather conditions of observation, exposure times, and plate washing can be different for different plate pairs. The corresponding internal standard errors can be estimated from a comparison of the proper motions obtained from different plate pairs for individual stars; the standard errors of the proper motions are very important for membership determination and dynamic studies of an open cluster.

Tables 2 and 3 give the accuracies of the final proper motions for stars in the region of NGC 1750 and NGC 1758 with different numbers of measured pairs (greater than 2) and different distances from the cluster center, and in different magnitude ranges, respectively. The units of the proper motions and their accuracies in this paper are mas/year. It is shown from the tables that the accuracies depend strongly on the number of plate pairs, and the greater the number of pairs, the higher the accuracies of the final proper motions of the stars. This shows that increasing the number of available plate pairs is very important for improving the accuracy of proper motions. It can also be seen from the tables that there is no obvious relation between the accuracies of the final proper motions and the distances of stars from the plate center or between the accuracies and the magnitudes of stars, which shows that the imaging quality of the telescope has been very good and that the PDS machine was quite stable. Figure 2 gives the number of stars for which different numbers of plate-pairs are available. More than $70\%$ of proper motions are obtained from more than 6 plate pairs. The rms errors of the proper motions of all 540 stars are ${\epsilon}_x$=$\pm$ 0.63 mas/yr, ${\epsilon}_y$ =$\pm$ 0.70 mas/yr, and ${\epsilon}$=$\pm$ 0.67 mas/yr, where $\epsilon=\sqrt{\frac{\epsilon ^{2}_{x}+\epsilon^{2}_{y}}{2}}$ the rms proper-motions errors of the $70\%$ of all stars that were observed in more than 6 plate pairs (see above) are better than $\pm$0.50 mas/yr. This can be seen from Fig. 3, which shows the relations ${\epsilon}_x$ versus N, ${\epsilon}_y$ versus N and ${\epsilon}$ versus N. So what we can say is that the errors of the proper motions of stars in the region of NGC 1750 and NGC 1758 obtained by us are relatively high, because of the good stellar images taken with the 40 cm double astrograph and the excellent positioning behavior of the PDS scanning machine.

  

Table 2: Accuracies of proper motions for stars in different numbers of plate pairs and at different distances from the plate center in the NGC 1750/1758 region (units in mas/yr)

  

Table 3: Accuracies of proper motions for stars in different magnitude ranges in the region of NGC 1750 and NGC 1758 (units in mas/yr)

  

Figure 2: The number of stars vs. the number of plate pairs available

  

Figure 3: Mean error of proper motions vs. proper motions