2. Plates and reduction of the data

2.1. Plate material and PDS measurement

There are 20 plates of the open cluster M 11 region used in the present study. All the plates were taken with the double astrograph at the Zo-Se station of Shanghai Observatory. This old telescope and its site have a detailed description (Chevalier 1905). Zo-Se is a small hill, which is one of the highest ones in Shanghai. The distance from Zo-Se to the center of Shanghai is nearly 30 km. Now it is the modern observation base of Shanghai Observatory.

The telescope, built by Gaultier in Paris, has an aperture of 40 cm and a focal length of 6.9 m, with a plate scale of . Each plate is 24 by 30 cm. The plates can be divided roughly into three epochs. The first one is between 1916 and 1923, the second is in 1964, and the third is between 1980 and 1986. Details of the plate pairs are listed in Table 1 (click here).

 

Pair No. Plates Epoch Exp. time Hour angle Quality Baselines Star No.

(1900+) (min) (a.m) (year)

1 435 16.72 90 -13 SL 70.00 832

86020 86.72 30 4 G

2 459 17.72 132 8 SL 69.02 864

86030 86.74 25 19 G

3 457 17.71 44 -10 G 69.02 787

86028 86.73 20 21 G

4 456 17.70 130 14 G 69.02 723

86025 86.72 20 18 G

5 518 23.71 100 12 G 63.01 703

86021 86.72 20 14 G

6 519 23.73 70 4 G 63.01 793

86023 86.74 20 10 G

7 64033 64.74 30 14 G 21.99 804

86024 86.73 20 -17 G

8 64029 64.70 20 5 G 17.00 828

81005 81.70 20 -14 G

9 64025 64.70 20 -7 G 15.97 831

80017 80.67 25 18 G

10 64028 64.70 35 -15 G 15.97 824

80018 80.67 25 11 G

 

These plates were measured with PDS machines in the Purple Mountain Observatory in Nanjing and the Institute of Technology and Communication in Luoyang, China. A pixel size of 20 by 20 microns, a step length of 20 micron, a speed of s and a scan type of R were used throughout. All the scanned programs are provided by Wang et al. (1990) and Wang & Chen (1992), Wang (1993, 1994) which are based on Lee & Van Altena (1983).

2.2. Proper motions

According to the method adopted by Zhao et al. (1993, 1980); Tian et al. (1982, 1983) and the references therein, we can derive the relative proper motions of stars based on the results of the PDS measurements. The whole process can be divided into three steps: the first one is to determine the reference stars, i.e., to establish a reference frame; the second to calculate proper motions; the last to estimate uncertainties of the data. Many authors, including ourselves, have done efforts in this field, so we give only a brief description here.

Theoretically, one can choose any stars freely to be reference stars to reduce relative proper motions. Reference stars are, in fact, normally chosen such that their proper motions are small. If this is not the case, program star proper motions might be distorted if they are located near a large proper motion star on the plate. To obtain a good result and make the absolute proper motion of the frame as small as possible, we should choose as many stars common to all plate pairs as possible, and stars with extraordinarily large proper motions and those in the crowded central region should be discarded. At the same time, the distribution of stars on the plates and the distribution of their magnitudes should also be chosen to be homogeneous. There are 618 stars in all the plate pairs in our study, 503 of which are chosen to be reference stars based on the above principles. One can see the reference frame adopted in the present study in Fig. 4 (click here) of the vector point diagram in the next section.

There are two ways used for reduction of relative proper motions. One is the central overlap technique, another the plate constant technique. In the present study, the central overlap technique is used. This approach has been adopted in our group since 1982. First, the error equations are limited only to first-order coordinate and magnitude terms. Second, the solutions must consist of quadratic coordinate and magnitude terms. Because accuracies are functions of time baselines of different pairs, the final results of relative proper motions must be weighted by the time baselines, as mentioned by Zhao et al. (1993) and Zhao & He (1988).

It is important to estimate the accuracies of proper motions for individual stars. In the early examples of such work, only the total accuracy of relative proper motions was given. As we now know, it is not enough to discuss only on the measurements, because the accuracies for individual stars depend on time baselines, number of pairs, exposure time, zenith distance, weather conditions and plate washing. The detailed description can be found in Zhao & He (1990, 1988) and Zhao et al. (1993).

  
Figure 1: The histograms of accuracies (in unit of as/100 yrs) of proper motions in X and Y directions for stars in the M 11 region respectively

Figure 1 (click here) shows the distribution of accuracies of the relative proper motions of stars in the M 11 region. Most of the stars (85) have proper motion accuracies better than 1 mas/yr. We also list total accuracies for stars appearing in different pairs in Table 2 (click here). It clearly shows that stars appearing in fewer pairs have poorer accuracies.

 

pair number star number

 2   2 14.01 12.39 13.22

 3   2 13.11 11.68 12.42

 4   2 11.59 9.72 10.69

 5   4 8.70 7.39 8.08

 6  13 4.18 4.23 4.21

 7  28 3.67 3.34 3.51

 8  50 2.79 2.58 2.69

 9 117 1.49 1.25 1.37

10 618 0.76 0.81 0.79

total 836 1.11 1.16 1.13

Table 2: Accuracies of relative proper motions(in unit of mas/yr) for the stars with different pair numbers in the M 11 region

 

The proper motion errors for 618 stars available in all ten plate pairs in different magnitude ranges are shown in Table 3 (click here).

 

< 12.5  64 0.80 0.89 0.84

12.5-13.0  86 0.44 0.48 0.46

13.0-13.5 164 0.56 0.54 0.55

13.5-14.0 151 0.65 0.67 0.66

14.0-14.5 114 0.88 1.01 0.95

 39 1.39 1.60 1.50

Table 3: Errors of relative proper motions (in unit of mas/yr) for the stars with all 10 plate pairs available in different magnitude ranges

 

The method for obtaining magnitudes will be mentioned in next subsection. From Table 3 (click here), the accuracies of proper motions for bright stars are generally better than those for faint stars.

  
Figure 2: The histograms of proper motions (in unit of as/100 yrs) in X and Y directions for stars in the M 11 region respectively

The histograms of the relative proper motions in X and Y directions are plotted in Fig. 2 (click here). It can be seen that the average values in both X and Y directions are all close to zero. This means that the reference frame used here is quite good.

2.3. Magnitudes

For the V band, we can obtain values from McNamara et al. (1977) after careful cross-identification between our catalogue and theirs. There are 435 common stars in the two catalogues, i.e., 435 stars in our sample have V magnitude values.

Because our plates in the 1980s (Table 1) were taken in the B band, we can use them to reduce B magnitudes of individual stars according to the density and size for each star obtained in the PDS measurements. Since no standard stars were observed at that time, we use the reduction method suggested by Stetson (1979) to calculate B magnitudes, in which the number of stars on the plates with known magnitudes is required to be large enough for statistical analysis. In the present study, we use 72 stars common to the study of Mathieu (1984) to do this, in which all stars have highly accurate magnitude values through CCD photometry.

The equation for the reduction is as follows:


where for the ith, B_i is its B magnitude, R_i radius of its image, d_i its density and b₀ a constant. Using our 72 stars, we obtain . Thus we have a B magnitude for each star based on Eq. (1).

The final results of relative proper motions and magnitudes for stars in the M 11 region are listed in Table 4 (available in electric form). The first column is the star number, among which with asterisk``^*" are reference stars. The second is the star number in the catalogue of McNamara et al. (1977). The total number of the stars common with theirs is 435. The third and fourth denote X and Y coordinates (cm); the 5th and 6th are B and V magnitudes; the 7th and 8th are relative proper motions in X and Y respectivel Y, and their corresponding uncertainties are in the 9th and 10th (all in unit of as/100 yr); the 11th is the number of plate pairs on which individual stars are available.

There are still 36 stars that are available on only one plate pair. So, we can only reduce their proper motion values but not their accuracies. These stars are all fainter than . In the present paper, the average accuracy for stars fainter than available in at least two plate pairs is estimated as the accuracy of these stars.