A catalogue of 74 new open clusters found in Gaia Data-Release 2

The unsupervised machine learning method DBSCAN used in this work was derived from scikit-learn (Pedregosa et al. 2011). This algorithm was used to distinguish high-density groups from low-density regions in $n$-dimensional samples (Ester et al. 1996), including extracting the core members and border members located in different groups, and rejecting any outliers not located in any group. There are two parameters in the algorithm: the minimum number of data points $min\_samples$ and the radius $\epsilon$, of which the latter was measured by a distance function as follows:

where $D_{ij}$ is the distance between data points $x_{i}$ and $x_{j}$.

A core is defined as a sub-sample of the data set where $min\_samples$ neighbours exist within a distance equal to $\epsilon$; that is, the core members are located in a dense area of vector space. Border members themselves do not form cores, but they may be located close to core members (i.e. $D_{ij}$ < $\epsilon$). In addition, outliers cannot constitute a core sample, nor can a candidate be composed of just border members, and they must be located at least $\epsilon$ from any core member. A larger $min\_samples$ or a smaller $\epsilon$ means a higher density required to form a group.

The parameter $min\_samples$ mainly controls the tolerance to outliers in this algorithm. Schubert et al. (2017) found that $min\_samples$ has a weak effect on the clustering results. Sander et al. (1998) suggested that $min\_samples$ can be determined as twice the value of the dimension of the data set; that is, $min\_samples$ =2 $\cdot$ $dim$, where $dim$ = 5 in this work. We tried different values of $min\_samples$ around 10, and found that the detection efficiency was highest when using $min\_samples=8$, that is, we could obtain more cores without obvious outliers. At the same time, this value was also within the range of the best values of $min\_samples$ used by Castro-Ginard et al. (2018); Hao et al. (2020). We present the results of $min\_samples$ = 8 in this article.

The parameter $\epsilon$ controls the radius of the local area of the points in the data set, and cannot be set to the default value (Schubert et al. 2017; Castro-Ginard et al. 2018). When this parameter is too small, most of the data will not be clustered; when it is too large, any adjacent clusters or outliers will merge into one group. In previous studies, Ester et al. (1996); Sander et al. (1998) selected $\epsilon$ based on the $k_{th}$ nearest neighbour distance ($k_{th}$NND). On this basis, Schubert et al. (2017) obtained the corresponding $\epsilon$ value by observing the significant change of sorted $k_{th}$NND plot. In the algorithm of $k_{th}$NND, the reference point is not specified as a neighbor, while that is included in the DBSCAN for density estimation, so the $k$ value used in this work is $k$ = $min\_samples$ -1.

In this work, the main steps to get $\epsilon$ are:

• •

  Compute the $7_{th}$NND histogram via $ssklearn$ ¹¹1 https://github.com/scikit-learn/scikit-learn.

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  Fit the $7_{th}$NND using a bimodal Gaussian function. For gravitationally bound systems such as OCs, the intervals between the member stars in position and proper motion vectors are closer than those of field stars, so their $7_{th}$NNDs are significantly smaller than those of field stars (Castro-Ginard et al. 2018). In the $7_{th}$NND histogram, the $7_{th}$NND of the cluster is distributed on the leftmost end (i.e. the minimum end, Fig. 1) of the whole data set. Inspired by Castro-Ginard et al. (2018), in this work, we used a fitted $7_{th}$NND histogram to determine $\epsilon$ as:

  $$\mathbf{n}=a_{1}\cdot e^{\frac{-(\mathbf{d}_{k,nn}-\mu_{1})^{2}}{2\sigma_{1}^{2}}}+a_{2}\cdot e^{\frac{-(\mathbf{d}_{k,nn}-\mu_{2})^{2}}{2\sigma_{2}^{2}}}$$

  (3)

  where $\mathbf{n}$ is the statistical number of $7_{th}$NND, $\mathbf{d}_{k,nn}$ is the distance, the fitted range of $\mathbf{d}_{k,nn}$ is (0, $\mathbf{d}_{k,nn}$ [n=max (n)]), and $(a,\mu,\sigma)_{i}$ are coefficients of the Gaussian functions. As shown in Fig. 1, we used two functions (curves 1 and 2) obtained by Eq. 3 to fit the $7_{th}$NND histogram. For a data set that contains any contributions from OCs, we found that the two Gaussian functions have a real number solution (e.g. the vertical red line in Fig. 1), and it is set to be the $\epsilon$ value used in DBSCAN.

Our first cross-matched catalogue contained OCs derived from Gaia DR2. In the previous studies, Sim et al. (2019) found 207 new OCs within 1 kpc of solar system, while Liu & Pang (2019) found 2,443 star clusters and candidates, 76 of which were most probable new star clusters. Cantat-Gaudin et al. (2020) re-visited 2017 OCs, including the catalogues of Cantat-Gaudin et al. (2018, 2019); Cantat-Gaudin & Anders (2020); Castro-Ginard et al. (2018, 2019, 2020), and a portion of OCs published by Sim et al. (2019); Liu & Pang (2019). In addition, we considered the 28, 16 and four new OCs recently been found by Ferreira et al. (2019, 2020); Hao et al. (2020); Qin et al. (2020), respectively.

Most OCs are observed to have projected spatial extents up to nearly 15 pc (Gaia Collaboration et al. 2017) from the cluster centres, such as the apparent radii of OCs found in Gaia DR2 by Cantat-Gaudin & Anders (2020). Therefore, within the range of the maximum distance from the centre of an OC considered here (15 pc, 5$\sigma$ dispersion) within $(l,b)$, we extracted the reported clusters adjacent to the center of each identified group. Then, in the range of a 5$\sigma$ dispersion about $(\varpi,\mu_{\alpha^{*}},\mu_{\delta})$, the extracted clusters were matched, and the matched ones were removed. Finally, a total of 1,978 reported star clusters were removed.

We continue to match the remaining groups with the catalogues published by Dias et al. (2002, D02) and Kharchenko et al. (2013, K13). To do this we made use of the root mean square (RMS) differences of the ages and distance moduli between the K13 clusters and cross-matched clusters identified in Gaia DR2 by Cantat-Gaudin et al. (2020), which reflected the differences of these parameters between the re-identified clusters in Gaia DR2 and the previous known OCs. We selected clusters in a range of max(15 pc, 5$\sigma$ dispersion) about $(l,b)$, and then compared the ages and distance moduli of the remaining groups with those of the K13 and D02 clusters within a 2-RMS level. In all, 46 groups were removed.

Next, Bica et al. (2019, B19) constructed a sample 4,384 open/globular clusters, cluster candidates and cluster remnants, where most of them were previously catalogued by D02 and K13. We also matched the objects found by B19 with the remaining groups, where the radius of each matched cluster had to satisfy $d_{r}\leq radius\leq r_{15pc}$, where $d_{r}$ is the angular size between the matched group and the reported cluster. Although most star clusters could be physically distinguished, we also performed visual checks of the groups and reported clusters. In the end, we found that some groups were closely matched with the centre positions catalogued by B19, and we included two of them in the final table (Table LABEL:tab1). After all the checking and filtering, 986 groups remained in our collated sample.