4 Discussion

4.1 Effectiveness of membership determination

The clustering of celestial bodies (such as star clusters or galaxy clusters) is an important research area in astronomy and astrophysics. As membership in clusters of celestial bodies is determined, contamination by background and foreground objects through the influence of the observational projection effect can not be avoided. Ever since the concept of membership probability was established to distinguish real cluster members from field objects on the basis of observational data (proper motions, radial velocities, photometry, polarization, etc.), the method suggested by Sanders (1971) has been a successful technique. The particular method of membership determination used in the present study is an improved one. Shao & Zhao (1996) set up the concept of the effectiveness of membership determination, which can be reasonably used to judge quantitatively how effective the results of membership determination of a cluster are. They suggested a widely applicable index E which can be used to measure the effectiveness of membership determination:

$$\begin{eqnarray} E & =&1-N \sum_{i=1}^{N} \left \{ P(i)\left[1-P(i) \right] \rig... ...=1}^{N} P(i) \sum_{i=1}^{N} \left[1-P(i)\right] \right\} \right.. \end{eqnarray}$$ (16)

The bigger E is, the more effective the membership determination is. If $\overline{P}$ is the average membership probability of all the bodies in a sample, i.e., $\overline{P}=\sum_{i=1}^N P(i)/N ,$then Eq. (16) can be written as follows:

$$E= \sum_{i=1}^N \left[P^2(i)-\overline{P}^2\right]\left/ \left(N\overline{P}-N\overline{P}^2\right) \right..$$ (17)

From Eq. (17) we can determine that the effectiveness of membership determination is 0.66 and 0.76 for NGC 1750 and NGC 1758 respectively, under the assumption of only one cluster, the effectiveness of membership determination is 0.60. It indicates that existence of two cluster is more reasonable than one cluster. It is shown in the 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 the their work, we can see that the effectiveness of membership determination for two open clusters present in this paper is now significantly higher in both cases.

4.2 Surface density distribution

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

$$\rho_{\rm c}=\frac{\sum{P_{\rm c}(i)}} {\Delta {S}} \pm \frac{\sqrt{\sum{P_{\rm c}(i)}}} {\Delta {S}}.$$ (18)

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:

$$\rho_{\rm f}=\frac {\sum{P_{\rm f}(i)}} {\Delta{S}}.$$ (19)

In Eqs. (18) and (19) the sums are performed for the stars in the area $\Delta S$ using the membership probabilities for each of the two clusters ($P_{\rm c}(i)$, i=1,2) and the field ($P_{\rm f}$) in turn. The surface densities $\rho_{\rm c}$ 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 8 gives the surface density distributions $\rho_{\rm c}$of the member stars and the corresponding uncertainty $\sigma$ in the two distributions.

  

Table 8: The surface densities of the member stars and the corresponding uncertainties in two open clusters

Figure 7 shows the surface density distributions of members of the two open clusters and of the common field stars respectively. It is seen that the surface densities of member stars in the two clusters decrease rapidly with distance from the cluster center, and the radial variation is more obvious for NGC 1758 than for NGC 1750. We can see from these figures that both NGC 1750 and NGC 1758 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, these figures indicate that the two star clusters defined in the present study actually exist independently, though they overlap each other on the sky.

  

Figure 7: The surface density distribution (dotted line is the field stars)

  

Figure 8: The fitting results obtained from King's empirical density law

4.3 The radii of NGC 1750 and NGC 1758

In order to study the fundamental dynamics, we can use the surface density distribution to fit the radius of a cluster on the basis of King's model. King (1962) gave an empirical formula for the surface density of a stellar system

$$\rho=\rho_0\left[\frac{1} {\left(1+r^2 / r_{\rm c}^2 \right)... ...1}{\left(1+r_{\rm t}^2 /r_{\rm c}^2 \right)^{1/2}} \right]^2 ,$$ (20)

where $\rho$ is the density, and $\rho_0$, $r_{\rm c}$ and $r_{\rm t}$ are the fitting parameters, which have clear physical meanings: $r_{\rm c}$ and $r_{\rm t}$ are the core radius and the tidal radius of a cluster, and $\rho_0$ is the central surface density; $c=r_{\rm t}/r_{\rm c}$ can be used to describe the central concentration of the cluster. The fitting parameters can be obtained from a $\chi^2$ test:

$$\chi^2=\sum_i \frac{1} {\sigma_i^2} \left[\rho_{\rm ob}(i)-\rho_{\rm exp}(i)\right]^2$$ (21)

where $\rho_{\rm ob}$ is the observed value of the surface density in an annulus and $\sigma_i$ is its uncertainty, which are defined in Eq. (18) and are listed in Table 7. $\rho_{\exp}$ is the theoretical value of the surface density from derived from Eq. (20). The fitting results are: $\rho_0=0.57/\hbox{\rm arcmin}^2$, $r_{\rm c}=17\hbox{$.\mkern-4mu^\prime$}2$ with a significance level of $89\%$ for NGC 1750; $\rho_0=5.26/\hbox{\rm arcmin}^2$, $r_{\rm c}=2\hbox{$.\mkern-4mu^\prime$}3$,$r_{\rm t}=10'.4$ with a significance level of $91\%$ for NGC 1758. In the previous section we obtained Gaussian characteristic radii for the two open clusters of 22${}^{\prime}$.70 $\pm$ 1${}^{\prime}$.35 for NGC 1750 and 2${}^{\prime}$.93 $\pm$ 0${}^{\prime}$.53 for NGC 1758 from the maximum likelihood solution. We can say that the results of two different methods are basically consistent. The solution for the dynamic radius $r_{\rm t}$ of NGC 1750 does not converge, and we believe that the main reason for this is that King's model is applicable to a star system with full relaxation, such as globular clusters or old open clusters with strong concentration, whereas the concentration of NGC 1750 is not very obvious. From the fitting results we see that the central concentration of NGC 1758 is 4.58, which means NGC 1758 is of higher concentration. This behavior can also be seen from the fitting curves shown in Fig. 8.

Acknowledgements

We would like to thank Dr. C.G. Su and Prof. J.J. Wang for their useful discussions. The present work is partially supported under the National Natural Science Fundation of China Grant Nos. 19673012 and 19733001 and in part by the astromical fundation of Astronomical Committee of CAS. This work is also supported under Joint Laboratory for Optional Astronomy of CAS. The National Research Council of Canada also supported the living expenses of J.L. Zhao and K.P. Tian while they visited the Dominion Astrophysical Observatory. We would specially like to express thanks to Dr. M. Geffert, for his very careful checking of and suggestions for this paper.