# Extra-Tidal Members and Dynamics of the Open Cluster NGC 6705
## 3 Structural Parameters and Dynamical State  
We obtained 1969 cluster members (\(P_{5D}\geq 0.15\)) for NGC 6705 based on the membership probabilities (see Figure 1). Their CMDs, PM-VPDs, and parallax data indicate that the majority, if not all, of these stars are reliable members of NGC 6705 (see Figures 3-5). As these cluster members extend up to 19 arcmin from the cluster center (equivalent to 12 pc at a distance of 2300 pc), they are well-suited for investigating the spatial structure of the cluster (see Figure 5). Figure 4: PM-VPDs of the 5D cluster members (red dots) and field stars (gray dots). Figure 5: Parallax (\(\alpha\)) of 5D cluster members (blue dots) and field stars (gray dots) as a function of radial distance (R). We estimated the core (\(R_{c}\)) and tidal radii (\(R_{t}\)) for NGC 6705 by fitting the radial density profile (RDP) of the cluster members (see Figure 6) to the King model [33]:  
\[\Sigma(r)=k[\frac{1}{\sqrt{1+(r/R_{c})^{2}}}-\frac{1}{\sqrt{1+(R_{t}/R_{c})^{2}}} ]^{2} \tag{2}\]  
where \(r\) is the radial distance to the cluster center and \(k\) is a constant. We take logarithms to see the density of the core and halo structure in more detail. The core and tidal radii were determined to be \(R_{c}\)= 3.11 \(\pm\) 0.21 and \(R_{t}\) = 20.02 \(\pm\) 0.71 arcmin, respectively. A centrally concentration parameter of \(C=log(R_{c}/R_{t})\approx\) 0.81 was determined for the cluster, indicating that NGC 6705 has a high density in its central region, meaning that the stars in the core of the cluster are very closely packed, with strong mutual gravitational interactions between the member stars, possibly forming a relatively tight core. [PERSON] and [PERSON] [7] obtained core and tidal radii of 5.9 arcmin and 10.73 arcmin, respectively, while our results indicate a smaller core radius and a larger tidal radius. This discrepancy may be due to differences in member stars, as our method identifies a larger number of reliable member stars. In addition, our tidal radius is smaller than those reported by [PERSON] and [PERSON] et al. [34], who estimated a tidal radius of 25.3 arcmin. This may be due to the different data usage, leading to differences in the results. Moreover, [PERSON] and [PERSON] et al. did not specify the errors. We used stricter member selection, excluding stars further from the cluster, resulting in a smaller tidal radius. Figure 7 shows the spatial distribution of the cluster; cluster members can be seen to be spatially concentrated. We used the standard deviation ellipse (SDE) to analyze the shape of NGC 6705 (see Figure 8). NGC 6705 exhibits an elongated form, which can be roughly characterized by an ellipse with a semi-major axis of \(a\) = 13.7 (\(\sim\)9.2 pc) and a semi-minor axis of \(b\) = 11:6 (\(\sim\)7.8 pc). \(a\) and \(b\) were estimated based on three times the standard deviations of the spatial positions along the major and minor axes, respectively. We speculate that the elongated shape of the cluster is influenced by the tidal field. Figure 6: RDP of the cluster members, where the black dots represent the observed data of cluster members, and the red curve shows the fit to the King model. In this study, to effectively divide the core and halo of the NGC 6705 cluster, we employed the GMM to perform clustering analysis on the spatial distribution of the cluster members. We extracted the 2D spatial positions \((l,b)\) of the 5D cluster members and calculated their relative positions with respect to the cluster center. Based on different numbers of Gaussian components, we calculated the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to evaluate the performance of the models (see Figure 9). AIC and BIC are commonly used model selection criteria. We explored GMM with component numbers ranging from 1 to 10 and successfully distinguished the core and  
Figure 8: SDE analysis (green line) for the spatial positions of the 5D cluster members (gray dots).