4 Results and discussion

The unknown parameters for this distribution are ($n_{\rm c}$,$x_{\rm c}$, $y_{\rm c},r_{\rm c},\mu_{x\rm c},\mu_{y\rm c},\sigma_{\rm c})_{c=1,2}$ and ($n_{\rm f}$,$\mu_{x{\rm f}}$, $\mu_{y{\rm f}}$,$\sigma_{x{\rm f}}$,$\sigma_{y{\rm f}}$,$\gamma)$. Membership probabilities of the i-th star belonging to the c-th cluster can be calculated from the following,

$$P_{\rm c}(i)=\frac{\Phi_{\rm c}(i)} {\Phi(i)} {\hspace{3cm}} (c=1,2).$$ (9)

According to the standard maximum likelihood method we obtained the distribution parameters and their corresponding uncertainties, given in Table 3, where the units of the proper motions and proper motion dispersions are mas/yr.

  

Table 3: Distribution parameters and their uncertainties for NGC 1817 and NGC 1807. The coordinates are given in J2000, epoch 1991.25. The units of ${\mu}$ and $\sigma$ are in mas/yr

Equatorial coordinates were computed using the Tycho Catalog (ESA 1997) as reference stars. Forty stars from this catalog are in the region under study. Following Galadí-Enríquez et al. (1998), the best fitting of these reference stars was a second order pair of equations. The cross-identifications of these 40 stars are given in Table 4. Only 15 stars were found in this region from the PPM Catalog, and the respective identification is also included in the same table.

  

Table 4: The cross-identification of stars in Table 5 with the TYCHO and PPM catalogues

Table 5 lists the results for all 722 stars in the region of the two open clusters: Col. 1 is the ordinal star number; Cols. 2 and 3 are $\alpha_{\rm J2000}$ and $\delta_ {\rm J2000}$ with an epoch of 1991.25; Cols. 4 and 5 are the proper motions; Cols. 6 and 7 are the standard errors of the proper motions; Cols. 8, 9, and 10 are probabilities of stars belonging to NGC 1817 (P₁), NGC 1807 (P₂), and the field $(P_{\rm f})$ respectively; and Col. 11 is the number of plate pairs used in the present study.

Figures 3 and 4 show the proper motion vector-point diagram and the position distribution on the sky for all the measured stars respectively, where $''\bullet''$ denotes a member of NGC 1817 with $P_1\ge0.7$, $''\circ''$ a member of NGC 1807 with P₂ $\ge0.7$, and all another stars are considered field stars indicated by "$\times$". It can be noted from the two diagrams that the centers in positional space and the centers in velocity (proper motion) space for the two open clusters are very clearly separated, which can be confirmed from the distribution parameters listed in Table 3. The dispersion of the cluster proper motions in this table is due to two reasons: the intrinsic velocity dispersion of the cluster members on one hand, and the precision of the proper motions (that in any case is poorer than usual, maybe because the epoch differences of plates are not long enough) on the other hand. The dispersion of proper motions compared to the mean internal errors seems to indicate that the velocity dispersion of members is larger than in other clusters, if the distance of 1800 pc (HH 1997) is assumed.

The angular diameter of NGC  1817 presented here is significantly larger than the values found by previous authors. Cuffey (1938) gave an estimation of the angular diameter based on his photometric study of the central region. We found a bigger diameter based on a complete astrometric study of all the region. NGC 1817 has an halo more extended that can be seen in a visual inspeccion of the area. Further studies using photometric data and radial velocities will be very helpful on this subject.

  

Figure 3: The proper motion vector-point diagram of NGC 1817 and NGC 1807. ("$\bullet$" denotes a member of NGC 1817 with $P_1\ge0.7$, "$\circ$" a member of NGC 1807 with $P_2\ge0.7$,"$\times$" a field star)

  

Figure 4: The position distribution of stars in NGC 1817 and NGC 1807 area. ("$\bullet$" denotes a member of NGC 1817 with $P_1\ge0.7$, "$\circ$" a member of NGC 1807 with $P_2\ge0.7$, "$\times$" a field star)

The membership probability histogram (Fig. 5) shows a very clear separation between cluster members and field stars. We find that the numbers of stars with membership probabilities higher than 0.7 for NGC 1817 and NGC 1807 are 416 and 14 respectively, and their average membership probabilities are 0.93 and 0.83 respectively, i.e., contamination by field stars is expected to be only $7\%$ and $17\%$ for the two clusters. If these stars were considered as belonging to the field, the apparent deficiency on field stars in the central region should vanish.

Till this point, all of our work indicates that the determination of two open clusters is successful: there exist two real open clusters NGC 1817 and NGC 1807. But, unfortunately, the fact of the NGC 1807 cluster being near to the edge of the plates makes the number of stars measured very small for what might be desirable. Conclusive results for this cluster should come from photometry or radial velocities studies. In this sense our group is working on a complete photometric study of the region that could be highly helpful in this respect.

  

Figure 5: The histogram of membership probability of NGC 1817 (solid line for the cluster stars, dashed line for the field stars)

4.1 Effectiveness of membership determination

Contamination by background and foreground objects through the influence of the observational projection effect can not be avoided. Following Shao & Zhao (1996) we can judge quantitatively how effective the results of our membership determination was. The effectiveness of membership determination is set as:

$$E=1- \frac{N \sum_{i=1}^{N} \left\{ P(i)\left[1-P(i) \right]... ...t\}} {\sum_{i=1}^{N} P(i) \sum_{i=1}^{N} \left[1-P(i)\right] }$$ (10)

where the bigger E is, the more effective the membership determination is.

So we can determine that the effectiveness of membership determination is 0.68 and 0.63 for NGC 1817 and NGC 1807 respectively. It is shown in Fig. 3 of Shao's paper (Shao $\&$ Zhao 1996) that the effectiveness of membership determination of 43 open clusters are from 0.20 to 0.90 and the peak value is 0.55. Compared with previous works (Shao $\&$ Zhao 1996; Tian et al. 1998), we can see that the effectiveness of membership determination for the two open clusters is significantly high in both cases.

4.2 Surface density distribution

The surface density distribution for the cluster members can be defined by the following equations:

$$\rho_{\rm c}=\frac{\sum_{i=1}^{N}{P_{\rm c}(i)}} {\Delta {S}} \pm \frac {\sqrt{\sum_{i=1}^{N}{P_{\rm c}(i)}}} {\Delta {S}}.$$ (11)

The second term of the right side of the above equation is the uncertainty, $\sigma_i$, which follows the Poisson distribution; at the same time the surface density distribution of the field stars is:

$$\rho_{\rm f}=\frac {\sum_{i=1}^{N}{P_{\rm f}(i)}} {\Delta{S}}.$$ (12)

In Eqs. (11) and (12) the sums are performed for the stars in the area $\Delta S$ using the membership probabilities for each of the two clusters ($P_{\rm c}(i)$, c=1,2) and the field ($P_{\rm f}$) in turn. $\rho_{\rm c}$ and $\rho_{\rm f}$ are calculated for each different $\Delta S$, which is defined as an annulus with varying radial distance from the cluster center, and $\rho_{\rm c}$ is calculated separately for each of the two clusters. Table 6 gives the surface density distributions $\rho_{\rm c}$ of the member stars and the corresponding uncertainty $\sigma$ in the two distributions.

  

Table 6: The surface densities of the member stars and the corresponding uncertainties in two open clusters

Figure 6 shows the radial density profiles of members of the two open clusters and of the field stars. We can see from this figure that both NGC 1817 and NGC 1807 have good central concentration, while on the other hand the surface density of field stars is quite uniform in the whole region. At the same time, this figure indicates that the two star clusters defined in the present study actually exist independently.

  

Figure 6: The radial density profiles

Acknowledgements

We would like to thank Prof. J.J. Wang, Dr. Z.Y. Shao and Dr. C.G. Su for their useful discussions. We also would like to thank Dr. C.Jordi and Dr. D.Galadí-Enríquez for their kind help and useful comments on the final steps of this paper.

This work was partially supported under the National Natural Science Foundation of China Grant No. 19673012 and No. 19733001, and by the Centro de Estudios de Asia Oriental (FGUAM), Universidad Autónoma de Madrid. This work was also partially supported under Joint Laboratory for Optional Astronomy of CAS.