# Extra-Tidal Members and Dynamics of the Open Cluster NGC 6705
## 2 Data and Method
### Membership Determination  
To determine more reliable cluster members (including extra-tidal members) of NGC 6705, we employed the DBSCAN clustering algorithm, a widely used unsupervised clustering method [27]. The advantage of DBSCAN is that it does not require prior knowledge of the sample star distribution and can identify clusters of any shape while being resistant to noise and outliers [28; 29; 30; 31]. This makes it particularly well-suited for identifying reliable cluster members in star clusters contaminated by field stars, such as NGC 6705, which is located in a low galactic latitude region and suffers from significant field star contamination. Given that DBSCAN's clustering results are highly sensitive to the input parameters _Eps_ and _MinPts_, we carefully selected these parameters by constructing a parameter grid [19; 27; 32]. The _Eps_ parameter, which defines the neighborhood radius, was varied from 0.01 to 0.2 in steps of 0.01, while the _MinPts_ parameter, defining the minimum number of points required to form a dense region, ranged from 1 to 50 with a step of 1. This grid search approach ensured a thorough exploration of parameter space to identify the optimal settings for the clustering. To ensure that all dimensions were on a comparable scale, we applied min-max normalization to the 5D astrometric data, which included the parallax (\(\varpi\)), proper motion in right ascension (\(\mu_{\alpha}cos\delta\)), proper motion in declination (\(\mu_{\delta}\)), galactic longitude (\(l\)), and galactic latitude (\(b\)). Each dimension was rescaled to the range [0, 1] using the following formula:  
\[x_{ik}^{{}^{\prime}}=\frac{x_{k}-Min(X_{k})}{Max(X_{k})-Min(X_{k})}(k=1,2,3,4,5) \tag{1}\]  
where \(x_{ik}\) is the original value of the \(k\)-th dimension for the \(i\)-th star and \(x_{k}^{{}^{\prime}}\) is the normalized value. \(Min(X_{k})\) and \(Max(X_{k})\) represent the minimum and maximum values of the \(k\)-th dimension across all sample stars. We ran the DBSCAN clustering algorithm across the parameter grid to classify stars as cluster members or outliers. To further refine our member selection, we calculated the membership probability \(P_{i}\) for each star by running the DBSCAN clustering algorithm 1000 times across different parameter combinations. The membership probability of the \(i\)-th \begin{table}
\begin{tabular}{c c c c c c} \hline
**Age (Gyr)** & **Distance (pc)** & **[Fe/H]** & **AV (mag)** & **E (B-V)** & **Bibliography** \\ \hline
0.200 & 1900 & & & 0.42 & Solomon and McNamara 1980 [23] \\  & 2000 & \(-0.2\pm 0.4\) & & 0.42 & [PERSON] et al. 2015 [24] \\
0.288 & 1888 & 0.043 & 1.457 & & [PERSON] et al. 2021 [6] \\
0.310 & 2233 & & 1.3 & Hunt and Reffert 2023 [7] \\
0.309 & 2203 & 0.114 & & & [PERSON] et al. 2024 [25] \\
0.309 & 2203 & 0.114 & 1.2 & & [PERSON] et al. 2024 [26] \\ \hline \end{tabular}
\end{table} 
Table 1: Relevant parameters for NGC 6705.  
star is defined as follows: \(P_{i}=N_{ic}/N_{t}\), where \(N_{ic}\) is the total number of times that the \(i\)-th star is classified as a \"core point\" across all runs and \(N_{t}\) is the total number of runs. Stars with higher membership probabilities are retained as reliable cluster members, ensuring that only stars consistently identified as cluster members were included for further analysis. From Figure 1, we see that the stars can be divided into two groups, and the stars with \(R>18\) arcmin have lower membership probabilities (\(P_{5D}<0.1\)). Therefore, the upper limit for field star probability is \(P_{5D}\approx 0.1\). At this point, \(P_{5D}=0.15\) can be considered a safe threshold for selecting reliable cluster members (represented by the red dashed line), and these members extend up to a radial distance of \(\sim\)20 arcmin (\(\sim\)13 pc at a distance of 2200 pc). This radial distance can be defined as the detectable radius of the cluster. Based on the threshold, we identified 1969 likely cluster members.