3 Membership determination

The determination of reasonable membership criteria for open clusters is an essential prerequisite for further astrophysical research. The analysis of photometric and/or kinematic data is usually used for this purpose. Because there are a lot of binaries in open clusters, the uncertainty for photometric membership determination can be quite large (Mathieu 1984). The most popular way to distinguish cluster members from field stars is therefore based on kinematic data, especially on radial velocities and on relative proper motions obtained with a number of plates with large epoch differences. The latter technique can be more powerful than the former because it exploits the motion in two dimensions rather than in only one, and because it is less sensitive to orbital motion in unrecognized binary systems. The fundamental mathematical model set up by Vasilevskis et al. (1957) and the technique based upon the maximum likelihood principle developed by Sanders (1971) have been devised to obtain the distribution of stars in the region of a cluster and the membership probabilities of individual stars. Since then many astronomers -- including those in our group -- have refined this method continuously. An improved method for membership determination of stellar clusters based on proper motions with different observed accuracies was developed by Stetson (1980) and Zhao & He (1990). Then Zhao & Zhao (1994) added the correlation coefficient of the field star distribution to the set of parameters describing their distribution on the sky. The spatial distribution of cluster stars and the dependence of the distribution parameters on the magnitudes of stars were considered by Su et al. (1997). In the meantime, the fundamental principle of Sanders' method was successfully used for membership determination of clusters of galaxies. Zhao et al. (1988) and Zhao & Zhao (1994) established and developed a statistical method that can be used to determine the distribution parameters and membership of rich galaxy clusters by using radial velocities and positions of galaxies as the observational criteria. In view of possible multiple substructures in galaxy clusters, in his doctoral thesis Shao (1996) extended the above method to the situation of multiple substructures and multiple criteria. He developed a strict, rigorous, and useful mathematical model, and successfully determined the distribution parameters and membership of a galaxy cluster with a complex structure.

As we pointed out in the introduction, there may be two open clusters, NGC 1750 and NGC 1758, in the region examined in the present paper. In order to confirm this point, we will extend the maximum-likelihood method available for the multi-substructure and multi-criterion case in one-dimensional velocity space (radial velocity) to the case of two-dimensional velocity space (relative proper motions), to determine the distribution parameters and membership of the two open clusters.

3.1 Basic hypotheses of the model

Assume that the observational data consist of K kinds of components, including $K_{\rm c}$ subclusters and $K_{\rm f}$ field populations (foreground or background), where K=$K_{\rm c}+K_{\rm f}$. Then, the star distribution ${\Phi}$ in the observational data space being used as criteria, such as positions and proper motions, can be expressed as a mixture of K sub-distributions ${\Phi}_{\rm c}$ and ${\Phi}_{\rm f}$:

$$\Phi =\sum_{i=1}^{K} \Phi_{\rm k}(i) = \sum_{i=1}^{K_{\rm c}} \Phi_{\rm c}(i) + \sum_{i=1}^{K_{\rm f}} \Phi_{\rm f}(i).$$ (1)

Furthermore, if we use positions (two dimensions) and proper motions (two dimensions) as criteria, ${\Phi}_{\rm c}$ and ${\Phi}_{\rm f}$ can be expressed as follows

$$\Phi_{\rm c} = \sum_{i=1}^{K_{\rm c}}n_{\rm c}(i) \cdot \Phi_{\rm c}^{\bf v}(i) \cdot \Phi_{\rm c}^{\bf r}(i),$$ (2)

$$\Phi_{\rm f} = \sum_{i=1}^{K_{\rm f}} n_{\rm f}(i) \cdot \Phi_{\rm f}^{\bf v}(i) \cdot \Phi_{\rm f}^{\bf r}(i).$$ (3)

For the case of star clusters, which is different from that of galaxy clusters, only one field population should be considered, which means $K_{\rm f}=1$. Therefore Eq. (3) can be simplified as

$$\Phi_{\rm f} = n_{\rm f} \cdot \Phi_{\rm f}^{\bf v} \cdot \Phi_{\rm f}^{\bf r}.$$ (4)

In the above equations $n_{\rm c}$ and $n_{\rm f}$ are the normalized numbers of subcluster members and field stars. They should satisfy the following condition:

$$\sum_{i=1}^{K_{\rm c}} n_{\rm c}(i) + {n_{\rm f}} =1.$$ (5)

Respectively, $\Phi_{\rm c}^{\bf r}$, $\Phi_{\rm f}^{\bf r}$, $\Phi_{\rm c}^ {\bf v}$, and $\Phi_{\rm f}^{\bf v}$, are the normalized distribution functions of subcluster members and field stars in the position $({\bf r})$ and relative proper motion $({\bf v})$ spaces. Obviously, n refers to the relative number of members of each of the different components, and ${\Phi}$ is refers to the shape of each distribution. Usually, the distribution of subcluster members in proper motion space can be assumed to be a (2-dimensional, isotropic) Gaussian function, and that of field stars is also Gaussian (also 2-dimensional), but with an elliptical shape. Projected onto the surface of the celestial sphere, we have no reason to reject a uniform distribution of field stars. On the other hand, the projected number-density of subcluster members should be a function of position. Some approximate formulae can be used to describe the function: for example, the King model profile or -- more simply -- a Gaussian (this paper) with characteristic radius $r_{\rm c}$ is often used. Thus,

$$\Phi_{\rm c}^{\bf r} = \frac{1}{2\pi r_{\rm c}^2}. \exp\left... ...left(\frac{y_i-y_{\rm c}}{r_{\rm c}}\right)^2\right] \right \},$$ (6)

$$\Phi_{\rm f}^{\bf r}=\frac{1} {\pi r_{\rm max}^2}$$ (7)

and

$$\begin{eqnarray} \Phi_{\rm c}^{\bf v}&=&\frac{1} {2\pi (\sigma_{\rm c}^2+\epsilo... ...y{\rm c}})^2} {\sigma_{\rm c}^2+\epsilon_{yi}^2}\right]\right\},\end{eqnarray}$$ (8)

$$\begin{eqnarray} \Phi_{\rm f}^{\bf v}&=&\frac{1} {2\pi (1-\gamma^2)^{1/2} (\sig... ... f}})^2} {\sigma_{y{\rm f}}^2+\epsilon_{yi}^2}\!\right]\!\right\},\end{eqnarray}$$ (9)

where $\epsilon_{xi}$ and $\epsilon_{yi}$ are the observed errors of the proper-motion components of the i-th star; and $x_{\rm c}$,$y_{\rm c}$ (center of subcluster), $r_{\rm c}$ (characteristic radius), $\mu_{x{\rm c}}, \mu_{y{\rm c}},\mu_{x{\rm f}},\mu_{y{\rm f}}$ (mean values of proper motions of member and field stars), $\sigma_{\rm c},\sigma_{x{\rm f}},\sigma_{y{\rm f}}$ (intrinsic proper motion dispersions of member and field stars) and $\gamma$ (correlation coefficient) are the spatial and kinematic distribution parameters (Shao & Zhao 1996).

3.2 Solution and results

There are nineteen unknown parameters q_j(j=1,2,.....19) in Eqs. (6)-(9): $(n_{\rm c}(i)$, $x_{\rm c}(i)$, $y_{\rm c}(i)$, $r_{\rm c}(i)$,  $\mu_{x{\rm c}}(i)$,$\mu_{y{\rm c}}(i)$, $\sigma_{\rm c}(i))_{i=1,2}$, $(\mu_{x{\rm f}}$, $\mu_{y{\rm f}}$,$\sigma_{x{\rm f}}$, $\sigma_{y{\rm f}})$, and $\gamma$. The standard maximum likelihood method can be used to obtain the values of these parameters. The likelihood function of the sample can be written as:

$$L=\prod_{i=1}^N\Phi(i) .$$ (10)

Now according to the maximum likelihood principle we have

$$\frac{\partial{\ln{L}}}{\partial{q_{i}}} = \frac{\partial}{\... ...{q_{i}}} (\sum\ln{\Phi_{j}})=0 \quad\quad\quad (j=1,2......19).$$ (11)

From the above equation the nineteen unknown distribution parameters can be found. Then we can determine the probability that the i-th star belongs to either of the two different open clusters by the following equations:

$$P_{\rm c}(i)=\frac{\Phi_{\rm c}(i)} {\Phi(i)}\quad\quad\quad ({c}=1,2).$$ (12)

The uncertainties of the distribution parameters can be found from a square matrix A composed of $m\times m$ second-order derivatives $\frac{\partial^2\ln{L}} {\partial{q_{\rm l}}\partial{q_{\rm t}}}$,$(l,t=1,2,\ldots,m)$, q referring in turn to each of the parameters and m=19 being the order number of the matrix:

$$\hbox{\bf A}=(\frac{\partial^2{\ln{L}}} {\partial{q_{\rm l}} \partial{q_{\rm t}}}).$$ (13)

Let the inverse matrix of A be

$$\hbox{\bf B} = \hbox{\bf A}^{-1}=(-b_{\rm lt}),$$ (14)

then the uncertainty of the parameter $q_{\rm l}$ is

$$\Delta{q_{\rm l}}=(-b_{\rm ll})^{1/2}.$$ (15)

The distribution parameters of the two open clusters and their corresponding uncertainties can be obtained and are shown in Table 4, where the units of the proper motions and proper motion intrinsic dispersions are  mas/yr. The two proper motion dispersions of the cluster members in Table 4 reflect mainly the internal velocity dispersions of the two clusters. This would also explain the two different values for the proper motion dispersion, which have also different distances from the Sun. We will present the further research about photometry, H-R diagram, distance and another astrophysical parameters of the two open clusters in the next paper. Table 5 (only available in electronic form) lists the results for all 540 stars in the region of the two open clusters: Col. 1 is the ordinal star number; Cols. 2 and 3 are $\alpha_{\rm J2000.0}$ and $\delta_ {\rm J2000.0}$, based on 27 stars in the PPM Catalogue (the cross-identifications of the 27 stars are given in Table 6); 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 1750 (P₁), NGC 1758 (P₂), and the field $(P_{\rm f})$ respectively; and Col. 11 is the number of plate pairs used in the present study. Table 7 gives the cross-identifications of 32 stars between Table 5 and Straizys(Straizys et al. 1992). Figures 4 and 5 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 1750 with $P_1\ge0.7$, "$\circ$" a member of NGC 1758 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 4. We can also see from the diagrams that the central concentration of NGC 1758 in positional space is more obvious than its central concentration in velocity space, which indicates that the spatial distribution of NGC 1758 plays a dominant role in its definition. The membership probability histogram (Fig. 6) 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 1750 and NGC 1758 are 332 and 23 respectively, and their average membership probabilities are 0.93 and 0.88 respectively, i.e., contamination by field stars is expected to be only $7\%$ and $12\%$ for the two clusters. All of our work indicates that the determination of two open clusters is successful: there exist two real open clusters NGC 1750 and NGC 1758.

  

Table 4: Distribution parameters and their uncertainties for NGC 1750 and NGC 1758 (the units of  $\mu$ and $\sigma$ in mas/yr)

  

Table 5: Proper motions and membership probabilities of stars in the region of NGC 1750 and NGC 1758 (the units of $\mu$ and $\sigma$ in mas/yr)

  

Table 6: The cross-identification of stars between the PPM catalogue and Table 5

  

Table 7: The cross-identification of 32 stars between Table 5 and Straizys (Straizys et al. 1992)

  

Figure 4: The proper motion vector-point diagram of NGC 1750 and NGC 1758 ("$\bullet$" denotes a member of NGC 1750 with $P_1\ge0.7$, "$\circ$" a member of NGC 1758 with $P_2\ge0.7$,"$\times$" a field star)

  

Figure 5: The position distribution of stars in NGC 1750 and NGC 1758 ("$\bullet$" denotes a member of NGC 1750 with $P_1\ge0.7$, "$\circ$" a member of NGC 1758 with $P_2\ge0.7$, "$\times$" a field star)

  

Figure 6: The histogram of membership probability of NGC 1750 and NGC 1758 (solid line is the stars of NGC 1750, dotted line NGC 1758)