2 Plate measurements and reduction of proper motions

2.1 Plate material and measurements

In the present study 24 plates of the NGC 1817 region are available. They were taken with the double astrograph at the Zo-Sè station of Shanghai Observatory. This telescope, built by Gaultier in Paris at the beginning of this century, has an aperture of 40 cm, a focal length of 6.9 m, and a plate scale of 30''/mm. The size of the plates is 24 by 30 cm, or 2${}^\circ$.0 $\times$ 2${}^\circ$.5. The oldest plate was taken in 1916, and the newest ones in 1997. The relevant information on these plates is given in Table 1. The hour angles are not provided in Table 1 because the starting time of the first-epoch plates was not recorded.

  

Table 1: Plate material

All the plates were measured on a Photometric Data Systems (PDS) model 1010 automatic measuring machine at the Purple Mountain Observatory in Nanjing (China). There is a total of 916 stars being measured with a photographic magnitude limit close to about 15.0. An aperture size of 20 by 20 microns, scanning step of 20 ${\mu}$m, scanning speed of 25 ${\mu}$m/s and R scanning type were adopted. The program for digital image centering is based on the algorithm developed by Lee & Van Altena (1983). In order to monitor and reduce any possible plate displacements and the resulting systematic errors, all the stellar images to be measured were divided into 18 groups, all the groups sharing four common images which were scanned twice, before and after all the other images for each group. A Gaussian fit to the density array marginal distribution F(x,y), was made following the software developed by Wang et al. (1995), from which the image center coordinates (x₀, y₀), the background density b, peak density D₀, and the image radius R can be obtained:

$$F(x,y) = D_0 \exp\left(\frac{-r^2}{2R^2}\right)+b ,$$ (1)

r²=(x-x₀)²+(y-y₀)² .

2.2 Proper motions

The reduction of the relative proper motions for 722 stars in the region of NGC 1817 and NGC 1807 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. 1993, 1980; Su et al. 1998). There are three steps in the whole process: the first is to transform the measured results for 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.

After two loops of the least-squares adjustment, 83 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 99 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. This defines a preliminary proper motion system in which the proper motions of the selected stars are collectively free of translation, rotation and expansion.

Owing to the limited number of reference stars and the accuracies of the proper motions of these stars, the plate pair 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 and the measuring accuracy of the stellar image.

  

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 1817/1807 region. Units are mas/yr

Table 2 gives the accuracies of final proper motions for stars in the NGC 1817 region with different numbers of measured pairs (greater than 2) and different distances from the field center. The units of the proper motions and their accuracies in this paper are mas/year. It is shown from the table 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 table that there is no obvious relation between the accuracies of the final proper motions and the distances of stars from the plate center, which shows that the imaging quality of the telescope has been very good and that the PDS machine was quite stable.

Figure 1 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.

  

Figure 1: The number of stars vs. the number of available plate pairs

The rms errors of the proper motions of all 722 stars are ${\epsilon}_x={\pm}$ 1.42 mas/yr, ${\epsilon}_y ={\pm}$ 1.37 mas/yr, and ${\epsilon}={\pm}$ 1.39 mas/yr, where $\epsilon=\sqrt{\frac{\epsilon ^{2}_{x}+\epsilon^{2}_{y}}{2}}$. In the most accurate case, the rms errors are found to be 0.81 mas/yr for stars with 10 to 12 plates and distances to the centre between 15' and 30'.

The rms proper-motions errors can be seen from Fig. 2, which shows the relations N versus $\epsilon _x$, N versus $\epsilon _y$ and N versus $\epsilon$. So what we can say is that the accuracies of the proper motions of stars in the region of NGC 1817 and NGC 1807 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.

  

Figure 2: The rms errors of proper motions vs. the number of stars