3 Membership determination

Accurate membership determination is of fundamental importance for further astrophysical studies of clusters. The fundamental mathematical model set up by Vasilevskis et al. (1958) and the technique based upon the maximum likelihood principle developed by Sanders (1971) have been refined continuously since then.

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 & Shao (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. (1998). Zhao et al. (1988, 1994) developed a statistical method on the same principle to determine the distribution parameters and membership of rich galaxy clusters. Shao & Zhao (1996) then extended the above method to the situation of multiple substructures and multiple criteria, and developed a strict, rigorous, and useful mathematical model. Tian et al. (1998) adapted this multi-substructure and multi-criterion maximum likelihood method in one-dimensional radial velocity to the case of two dimensional velocity space (relative proper motions), and determined successfully the distribution parameters and membership of a region with two open clusters.

As we pointed out in the introduction, there may be two open clusters, NGC 1817 and NGC 1807, in the region examined in the present paper. In order to confirm this point, we will follow the same method to determine the distribution parameters and membership of the two open clusters.

The frequency function for the i-th star of a cluster can be written as follows:

$$\begin{eqnarray} \Phi_{\rm c}^{\vec{v}} &=&\frac{1} {2\pi (\sigma_{\rm c}^2+\ep... ...y\rm c})^2} {\sigma_{\rm c}^2+\epsilon_{yi}^2}\right]\right\}, \end{eqnarray}$$ (3)

where ($\mu_{xi},\mu_{yi}$) are the proper motions of the i-th star, ($\mu_{x\rm c},\mu_{y\rm c}$) the cluster proper motion center, $\rm \sigma_c$ the intrinsic proper motion dispersions of member stars and ($\epsilon_{xi}$,$\epsilon_{yi}$) are the observed errors of the proper-motion components of the i-th star.

And for the field,

$$\begin{eqnarray} \Phi_{\rm f}^{\vec{v}} =&\frac{1} {2\pi (1-\gamma^2)^{1/2} (\s... ...\rm f})^2} {\sigma_{y\rm f}^2+\epsilon_{yi}^2}\right] \right \} ,\end{eqnarray}$$ (4)

where ($\mu_{xi},\mu_{yi}$) are the proper motions of the i-th star, ($\mu_{x\rm f},\mu_{y\rm f}$) the field proper motion center, ($\epsilon_{xi}$,$\epsilon_{yi}$) are the observed errors of the proper-motion components of the i-th star, ($\sigma_{x\rm f},\sigma_{y\rm f}$) the field intrinsic proper motion dispersions and $\gamma$ the correlation coefficient.

On the other hand, the surface-number-density of cluster members is a function of the position. A Gaussian profile is the chosen approximation,

$$\Phi_{\rm c}^{\vec{r}} = \frac{1}{2\pi r_{\rm c}^2}. \exp\le... ...left(\frac{y_i-y_{\rm c}}{r_{\rm c}}\right)^2\right] \right \},$$ (5)

where ($x_{\rm c}$,$y_{\rm c}$) is the center of the cluster, and $r_{\rm c}$ the characteristic radius.

And a uniform distribution of field stars is adopted,

$$\Phi_{\rm f}^{\vec{r}}=\frac{1} {\pi r_{\rm max}^2}.$$ (6)

The distribution of all stars in the region can, then, be described as follows:

$$\Phi = \Phi_{\rm c} + \Phi_{\rm f} = \sum_{c=1}^{2}n_{\rm c... ...m f} \cdot \Phi_{\rm f}^{\vec{v}} \cdot \Phi_{\rm f}^{\vec{r}}.$$ (7)

$$\sum_{c=1}^{2} n_{\rm c} + {n_{\rm f}} =1.$$ (8)

respectively, $\Phi_{\rm c}^{\vec{r}}$, $\Phi_{\rm f}^{\vec{r}}$,$\Phi_{\rm c}^ {\vec{v}}$, and $\Phi_{\rm f}^{\vec{v}}$, are the normalized distribution functions of cluster members and field stars in the position $(\vec{r})$ and relative proper motion $(\vec{v})$ spaces. With $n_{\rm c}$, the normalized number of cluster stars, and $n_{\rm f}$, the normalized number of field stars.