# Extra-Tidal Members and Dynamics of the Open Cluster NGC 6705
## 3 Structural Parameters and Dynamical State
Figure 7: Positions of the 5D cluster members (blue dots) and field stars (gray dots), where the green circle is the tidal radius of the cluster (20.02 arcmin). halo by analyzing both AIC and BIC values. This suggests that NGC 6705 can be naturally divided into core and halo using a two-component GMM (see Figure 10). The diagonal elements of the \(2\times 2\) full covariance matrix of the two-component GMM provide valuable insights into the cluster structure; it is evident that the core (\(\sigma_{ll}=2.2^{\prime},\sigma_{bb}=2.4^{\prime}\)) is more concentrated than the halo (\(\sigma_{ll}=7.0^{\prime},\sigma_{bb}=5.6^{\prime}\)). This indicates that NGC 6705 has developed a distinct core-halo structure through dynamical evolution. We have identified 1279 stars in the core and 690 stars in the halo. In addition, we computed the proper motions for members of the core and halo regions of the NGC 6705 cluster. The results show that the mean proper motion directions of the core and halo regions are very close to each other, with a mean RA of \(-1.543\) mas/yr and a DEC of \(-4.167\) mas/yr in the core region and a mean RA of \(-1.539\) mas/yr and a DEC of \(-4.163\) mas/yr in the halo region. This suggests that, although these two regions are different in terms of their spatial distributions, their directions of motion are almost the same. The arrows represent the direction of the stars' total proper motions (see Figure 11). The key parameters for both the core and halo components are provided in Table 2.  
Next, we used principal component analysis (PCA) to estimate the lengths of the semi-major axis (\(a\)) and semi-minor axis (\(b\)) for the core and halo. PCA is a commonly used dimensionality reduction technique that projects high-dimensional data onto a few principal components through linear transformation [35; 36]. These principal components represent the directions of maximum variance in the data. The first principal component (PC1) represents the direction with the greatest variance in the data distribution, while the second principal component (PC2) represents the direction with the second greatest variance; these major and minor axes reflect the primary distribution patterns of the data (see Figure 12). At the same time, they also allow for an accurate estimation of the ellipticites (\(e=b/a\)) of the core and halo. For the core, we obtained standard deviations of \(\sigma_{1}=2.4^{\prime}\) and \(\sigma_{2}=2.2^{\prime}\) for PC1 and PC2, respectively. The core has a semi-major axis of \(a=3\sigma_{1}\sim 7^{\prime}\) (\(\sim\)4.5 pc) and a semi-minor axis of \(b=3\sigma_{2}\sim 6^{\prime}\) (\(\sim\)3.9 pc). The ellipticity of the core was determined to be \(e\)\(\sim\)0.89, suggesting that NGC 6705 has a nearly circular core that is less affected by the tidal field. For the halo, we calculated the standard deviations of PC1 and PC2 as \(\sigma_{1}=7.0^{\prime}\) and \(\sigma_{2}=5.6^{\prime}\), respectively. These values are appropriate for estimating the spatial size of the halo; it has a semi-major axis of \(a=3\sigma_{1}\sim 21^{\prime}\) (\(\sim\)13.6 pc) and a semi-minor axis of \(b=3\sigma_{2}\sim 17^{\prime}\) (\(\sim\)11.0 pc). The ellipticity of the halo (\(e=b/a\)\(\sim\)0.80) indicates that its shape is significantly influenced by the tidal field of the galaxy. The fundamental parameters (\(\sigma_{1}\), \(\sigma_{2}\), \(e\)) for both the core and halo, derived from PCA, are also listed in Table 2. As shown in Figure 12, the gap between the halo and core regions represents areas where member stars have been lost because of the influence of the galactic tidal force or the internal dynamical evolution of the cluster. This phenomenon is related to the effect of mass segregation in the cluster, where higher-mass member stars tend to gather in the core region, while lower-mass member stars are more likely to be distributed in the outer regions.