Membership and fundamental parameters of the intermediate-age open clusters NGC 2281 and NGC 2539 based on Gaia-DR2

List of the selection criteria of sample stars for the two clusters.*

Name	Cluster center	D _D02	PM _D02	μ_αcos δ	μ_δ	R _s	R _d	N _s
	(°)	(pc)	(mas yr⁻¹)	(mas yr⁻¹)	(mas yr⁻¹)	(′)	(pc)
NGC 2281	102.0708, +41.0783	558	(−2.48, −6.76)	[−10 10]	[−10 10]	90	[0 1500]	23508
NGC 2539	122.6542, −12.8183	1363	(−3.20, −1.24)	[−10 10]	[−10 10]	40	[500 3000]	17694

Name	Cluster center	D _D02	PM _D02	μ_αcos δ	μ_δ	R _s	R _d	N _s
	(°)	(pc)	(mas yr⁻¹)	(mas yr⁻¹)	(mas yr⁻¹)	(′)	(pc)
NGC 2281	102.0708, +41.0783	558	(−2.48, −6.76)	[−10 10]	[−10 10]	90	[0 1500]	23508
NGC 2539	122.6542, −12.8183	1363	(−3.20, −1.24)	[−10 10]	[−10 10]	40	[500 3000]	17694

*D_D02: The cluster distance in Dias et al. (2002). PM_D02: The cluster proper motion in Dias et al. (2002). μ_αcos δ, μ_δ: The proper motion range used for selecting sample stars. R_s: The angular radius used for selecting sample stars. R_d: The distance range used for selecting sample stars. N_s: The number of sample stars.

Table 1.

List of the selection criteria of sample stars for the two clusters.*

Name	Cluster center	D _D02	PM _D02	μ_αcos δ	μ_δ	R _s	R _d	N _s
	(°)	(pc)	(mas yr⁻¹)	(mas yr⁻¹)	(mas yr⁻¹)	(′)	(pc)
NGC 2281	102.0708, +41.0783	558	(−2.48, −6.76)	[−10 10]	[−10 10]	90	[0 1500]	23508
NGC 2539	122.6542, −12.8183	1363	(−3.20, −1.24)	[−10 10]	[−10 10]	40	[500 3000]	17694

Name	Cluster center	D _D02	PM _D02	μ_αcos δ	μ_δ	R _s	R _d	N _s
	(°)	(pc)	(mas yr⁻¹)	(mas yr⁻¹)	(mas yr⁻¹)	(′)	(pc)
NGC 2281	102.0708, +41.0783	558	(−2.48, −6.76)	[−10 10]	[−10 10]	90	[0 1500]	23508
NGC 2539	122.6542, −12.8183	1363	(−3.20, −1.24)	[−10 10]	[−10 10]	40	[500 3000]	17694

*D_D02: The cluster distance in Dias et al. (2002). PM_D02: The cluster proper motion in Dias et al. (2002). μ_αcos δ, μ_δ: The proper motion range used for selecting sample stars. R_s: The angular radius used for selecting sample stars. R_d: The distance range used for selecting sample stars. N_s: The number of sample stars.

2.2 Method

Reliable identification of cluster members is the critical first step for further investigation of the two clusters. We have developed a ML method for identifying reliable cluster members in high-dimensional parameter space (Gao 2018a, 2018b). This method has been used to investigate several open clusters based on the high-precision astrometric and photometric data of Gaia-DR2 (Gao 2018a, 2018b, 2019). This supervised ML method includes two key steps: (1) constructing a reliable training set, and (2) estimating membership probabilities for sample stars—readers are advised to refer to Gao (2018a). A training set must include a sufficient number of reliable cluster members and field stars, which are necessary to build a random forest (RF; Breiman 2001). In this work, we use a two-component Gaussian mixture model (GMM) to construct a reliable training set for each cluster. Given n data points (x₁, x₂, x₃, ..., x_n) in an m-dimensional parameter space, the two-component GMM f(x_i) can be written as

\begin{eqnarray} f(x_{i};\theta ) = \sum _{j=1}^{2}w_{j}G_{j}^{m}(x_{i};u_{j}, \Sigma _{j}) \end{eqnarray}

(1)

\begin{eqnarray} = \sum _{j=1}^{2}w_{j}\frac{1}{(2\pi )^{m/2}}{\exp }\left[-\frac{1}{2}(x_{i}-u_{j})^{\rm T}\Sigma _{j}^{-1}(x_{i}-u_{j})\right], \end{eqnarray}

(2)

where θ indicates the distribution parameters (w_j, u_j, Σ_j) of the jth (j = 1, 2) Gaussian component (⁠|$G_{j}^{m}$|⁠), u_j and Σ_j are the mean value and full covariance matrix (m × m) of |$G_{j}^{m}$| respectively, and w_j is the weight of |$G_{j}^{m}$|⁠. The weights of the two Gaussian components are subject to the following relationship:

\begin{eqnarray} \sum _{j=1}^{2}w_{j}&\equiv &1. \end{eqnarray}

(3)

The distribution parameters (w_j, u_j, Σ_j) of each Gaussian component can be determined based on the expectation maximization (EM) algorithm (Dempster et al. 1977), and the membership probability (P_GMM) of each sample star can also be easily determined. It should be noted that the two-component GMM is a poor description for the distribution of our sample stars, especially for the parallax distribution. For each cluster, the GMM is a valid first-order approximation for the sample stars within a limited radius from the cluster center (see table 2), if the parallaxes are excluded. We therefore segregate cluster and field stars for each cluster using the two-component GMM, taking into account only four astrometric parameters (positions and proper motions). The fundamental parameters of the GMMs for the two clusters are listed in table 2, and the full covariance matrices (4 × 4) of the two clusters are listed in equations (4)–(7). The first two are for NGC 2281 and the other two for NGC 2539. For each cluster, the elements on the main diagonal in the cluster component [Σ_c; see equations (4) and (6)] are significantly smaller than those in the field component [Σ_f; see equations (5) and (7)], suggesting the existence of a star cluster in the four-dimensional parameter space (positions and proper motions).

Table 2.

Fundamental parameters of the GMMs for the two clusters.*

Name	R _GMM	N _s	μ_c	μ_f	C _c	C _f	w _c	w _f
	(′)		(mas yr⁻¹)	(mas yr⁻¹)	(°)	(°)
NGC 2281	40	5191	(−2.997, −8.373)	(0.431, −3.328)	(102.0627, 41.0609)	(102.0642, 41.0669)	0.1013	0.8987
NGC 2539	19	4523	(−2.321, −0.595)	(−2.178, 0.547)	(122.6579, −12.8328)	(122.6484, −12.8176)	0.1199	0.8801

Name	R _GMM	N _s	μ_c	μ_f	C _c	C _f	w _c	w _f
	(′)		(mas yr⁻¹)	(mas yr⁻¹)	(°)	(°)
NGC 2281	40	5191	(−2.997, −8.373)	(0.431, −3.328)	(102.0627, 41.0609)	(102.0642, 41.0669)	0.1013	0.8987
NGC 2539	19	4523	(−2.321, −0.595)	(−2.178, 0.547)	(122.6579, −12.8328)	(122.6484, −12.8176)	0.1199	0.8801

*R_GMM: The limited radius within which sample stars have GMM membership probabilities. N_s: The number of sample stars within the limited radius. μ_c: The mean proper motion of the cluster component. μ_f: The mean proper motion of the field component. C_c: The central position of the cluster component. C_f: The central position of the field component. w_c: The weight of the cluster component. w_f: The weight of the field component.

Table 2.

Fundamental parameters of the GMMs for the two clusters.*

Name	R _GMM	N _s	μ_c	μ_f	C _c	C _f	w _c	w _f
	(′)		(mas yr⁻¹)	(mas yr⁻¹)	(°)	(°)
NGC 2281	40	5191	(−2.997, −8.373)	(0.431, −3.328)	(102.0627, 41.0609)	(102.0642, 41.0669)	0.1013	0.8987
NGC 2539	19	4523	(−2.321, −0.595)	(−2.178, 0.547)	(122.6579, −12.8328)	(122.6484, −12.8176)	0.1199	0.8801

Name	R _GMM	N _s	μ_c	μ_f	C _c	C _f	w _c	w _f
	(′)		(mas yr⁻¹)	(mas yr⁻¹)	(°)	(°)
NGC 2281	40	5191	(−2.997, −8.373)	(0.431, −3.328)	(102.0627, 41.0609)	(102.0642, 41.0669)	0.1013	0.8987
NGC 2539	19	4523	(−2.321, −0.595)	(−2.178, 0.547)	(122.6579, −12.8328)	(122.6484, −12.8176)	0.1199	0.8801

*R_GMM: The limited radius within which sample stars have GMM membership probabilities. N_s: The number of sample stars within the limited radius. μ_c: The mean proper motion of the cluster component. μ_f: The mean proper motion of the field component. C_c: The central position of the cluster component. C_f: The central position of the field component. w_c: The weight of the cluster component. w_f: The weight of the field component.

\begin{eqnarray} \Sigma_{\rm c}=\begin{array}{c} \begin{array}{c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c} &\mu _{\alpha} \cos \delta & \mu _{\delta} & {\rm RA} & {\rm Dec} \end{array}\\ \begin{array}{c}\mu _{\alpha} \cos \delta \\ \mu _{\delta} \\ {\rm RA} \\ {\rm Dec} \end{array} \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0.2154 & -0.0033 & -0.0413 & -0.0022 \\ -0.0033 & 0.1749 & -0.0007 & 0.0058 \\ -0.0413 & -0.0007 & 0.1156 & -0.0019 \\ -0.0022 & 0.0058 & -0.0019 & 0.0693 \end{array}\right)\end{array}, \end{eqnarray}

(4)

\begin{eqnarray} \Sigma_{\rm f}=\begin{array}{c} \begin{array}{c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c} &\mu _{\alpha} \cos \delta & \mu _{\delta} & {\rm RA} & {\rm Dec} \end{array}\\ \begin{array}{c}\mu _{\alpha} \cos \delta \\ \mu _{\delta} \\ {\rm RA} \\ {\rm Dec} \end{array} \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 12.4029 & 0.6166 & 0.0231 & 0.0275 \\ 0.6166 & 10.9181 & -0.0007 & -0.0149 \\ 0.0231 & -0.0007 & 0.1962 & 0.0079 \\ 0.0275 & -0.0149 & 0.0079 & 0.1086 \end{array}\right)\end{array}, \end{eqnarray}

(5)

\begin{eqnarray} \Sigma_{\rm c}=\begin{array}{c} \begin{array}{c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c} &\mu _{\alpha} \cos \delta & \mu _{\delta} & {\rm RA} & {\rm Dec} \end{array}\\ \begin{array}{c}\mu _{\alpha} \cos \delta \\ \mu _{\delta} \\ {\rm RA} \\ {Dec} \end{array} \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0.0399 & -0.0074 & -0.0028 & -0.0007 \\ -0.0074 & 0.0333 & 0.0001 & -0.0002 \\ -0.0028 & 0.0001 & 0.0165 & -0.0014 \\ -0.0007 & -0.0002 & -0.0014 & 0.0165 \\ \end{array}\right)\end{array}, \end{eqnarray}

(6)

\begin{eqnarray} \Sigma_{\rm f}=\begin{array}{c} \begin{array}{c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c@{\quad\qquad}c} &\mu _{\alpha} \cos \delta & \mu _{\delta} & {\rm RA} & {\rm Dec} \end{array}\\ \begin{array}{c}\mu _{\alpha} \cos \delta \\ \mu _{\delta} \\ {\rm RA} \\ {\rm Dec} \end{array} \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 7.2747 & -2.5612 & -0.0155 & -0.0068 \\ -2.5612 & 9.4136 & 0.0065 & -0.0044 \\ -0.0155 & 0.0065 & 0.0261 & 0.0002 \\ -0.0068 & -0.0044 & 0.0002 & 0.0244 \\ \end{array}\right)\end{array}, \end{eqnarray}

(7)

Figure 1 shows the histograms of the GMM membership probabilities for the two clusters. The GMM exhibits good performance in segregating cluster members and field stars within the limited radii of the two clusters (see table 2). For each cluster, only the high-probability members (P_GMM ≥ 0.9) and low-probability field stars (P_GMM < 0.2) are used to constructed a training set. The training sets of NGC 2281 and NGC 2539 contain 4997 and 4321 sample stars, respectively. Based on the training sets, we can build two RFs to identify likely cluster members for all the sample stars (see subsection 2.1). In this work, we build two RFs using the Python task RandomForestClassifier in the scikit-learn package² (Pedregosa et al. 2011). For each cluster, the RF is built using 11 parameters, since each sample star in the training set has five astrometric (positions, parallax, proper motions) and six photometric parameters (G, BP, RP, |$\mathit {BP-RP}$|⁠, |$\mathit {BP-G}$|⁠, and |$\mathit {G-RP}$|⁠). The RF for NGC 2281 contains 5000 decision trees, while the one for NGC 2539 contains 4500 decision trees. These numbers are chosen to roughly match the numbers of stars used in the training sets. Each decision tree randomly selects four parameters from the 11 parameters, and makes a definite decision on whether or not a sample star is a cluster member. Note that it is common practice to use |$m\sim \sqrt{N}$| when applying random forests, where m indicates the number of selected parameters and N indicates the total number of parameters. The fundamental parameters of the two RFs are listed in table 3. We use the out-of-bag (OOB) score to evaluate the prediction accuracies of the two RFs, and two high scores are obtained (see table 3), indicating that the RFs have high prediction accuracies. The RFs can also calculate membership probabilities (P_RF) for all the sample stars by majority votes, indicating that the RFs can identify more spatially complete cluster members beyond the limited radii (see table 2).

Fig. 1.

Histograms of the GMM membership probabilities for the sample stars within the limited radii of the two clusters (see table 2). (Color online)

Table 3.

Fundamental parameters of the RFs.*

Name	N _ts	N _tr	N _s	OOB score
NGC 2281	4997	5000	23508	0.999
NGC 2539	4321	4500	17694	0.998

*N_ts: The number of stars in the training set. N_tr: The number of decision trees in the RF. N_s: The total number of sample stars for the cluster. OOB score: The out-of-bag (OOB) score of the RF.

Table 3.

Fundamental parameters of the RFs.*

Name	N _ts	N _tr	N _s	OOB score
NGC 2281	4997	5000	23508	0.999
NGC 2539	4321	4500	17694	0.998

*N_ts: The number of stars in the training set. N_tr: The number of decision trees in the RF. N_s: The total number of sample stars for the cluster. OOB score: The out-of-bag (OOB) score of the RF.

2.3 Likely cluster members

Based on the combined ML method (GMM+RF), we determine 696 and 664 likely cluster members with P_RF ≥ 0.6 for NGC 2281 and NGC 2539, respectively. Figures 2 and 3 show color–magnitude diagrams (CMDs) and proper motion vector point diagrams (PM-VPDs) of the likely cluster members and field stars in the fields of the two clusters. The CMDs of likely cluster members in the two clusters show well-defined main sequences and turn-off points. Figures 4 and 5 plot the parallax of our likely cluster members versus the distance to the cluster center. The likely cluster members located outside the limited radii can also be identified, indicating that the ML method has a strong prediction ability. It should be noted that Cantat-Gaudin et al. (2018) determined 414 and 518 likely cluster members for NGC 2281 and NGC 2539, respectively. However, they did not take into account faint stars with G > 18 mag. The likely cluster members of NGC 2281 and NGC 2539 (P_RF ≥ 0.6) are listed in the Appendix (see tables 7 and 8).

Fig. 2.

CMDs (left panel) and PM-VPDs (right panel) of the 696 likely cluster members (P_RF ≥ 0.6) of NGC 2281 and field stars (P_RF < 0.6). In each panel, the blue circles and black dots represent the likely cluster members and field stars, respectively. (Color online)

Fig. 3.

CMDs (left panel) and PM-VPDs (right panel) of the 664 likely cluster members (P_RF ≥ 0.6) of NGC 2539 and field stars (P_RF < 0.6). The symbols are the same as those in figure 2. (Color online)

Fig. 4.

Parallax vs. distance to the cluster center for the 696 likely cluster members of NGC 2281 (P_RF ≥ 0.6; blue circles) and field stars (P_RF < 0.6; black dots). The red dots indicate the GMM-based likely cluster members (P_GMM ≥ 0.6; see subsection 2.2). For illustration purposes, only stars with 0.7 ≤ π ≤ 4 mas (250–1429 pc) are shown in this figure. (Color online)

Fig. 5.

Parallax vs. distance to the cluster center for the 664 likely cluster members of NGC 2539 and field stars. The symbols are the same as those in figure 4. For illustration purposes, only the stars with 0.333 ≤ π ≤ 2 mas (500–3000 pc) are shown in this figure. (Color online)

3 Spatial structure

We have obtained likely cluster members for the two clusters using the combined ML method (see subsection 2.3). These cluster members are highly suitable for investigating the spatial structure of the two clusters, since they are selected over large sky areas (see subsection 2.1). Figures 6 and 7 show the the radial surface density profiles of the two clusters; only stars with P_RF ≥ 0.1 are taken into account. We estimate the core radii of the two clusters by fitting the observational data to the King model (King 1962):

\begin{eqnarray} \rho (r)=\rho _{\rm b}+\frac{\rho _{\rm c}}{1+\left(\frac{r}{R_{\rm c}}\right)^{2}}, \end{eqnarray}

(8)

where r indicates the angular distance to the cluster center, ρ_b is the background density, ρ_c is the central peak density, and R_c the core radius of the cluster. The limiting radii of the two clusters are also estimated using the following formula (Bukowiecki et al. 2011):

\begin{eqnarray} R_{\rm lim}=R_{\rm c}\sqrt{\frac{\rho _{\rm c}}{3\sigma _{\rm b}}-1}, \end{eqnarray}

(9)

where R_lim indicates the limiting radius of the cluster, R_c is the core radius of the cluster, ρ_c is the central peak density, and σ_b is the uncertainty of the background density ρ_b. The concentration parameters C = log(R_lim/R_c) of the two clusters are also estimated (Peterson & King 1975). The spatial structure parameters of the two clusters are listed in table 4.

Fig. 6.

Radial surface density profile of NGC 2811; only stars with P_RF ≥ 0.1 are taken into account. The solid curve is a King model fitted to the data, and the error bars represent the 1 σ Poisson errors. (Color online)

Fig. 7.

Radial surface density profile of NGC 2539; only stars with P_RF ≥ 0.1 are taken into account. The symbols and lines are the same as those in figure 6. (Color online)

Table 4.

Spatial structure parameters of the two clusters.

Name	R _c	R _lim	C*	ρ_c	ρ_b
	(′)	(′)		(stars arcmin⁻²)	(stars arcmin⁻²)
NGC 2281	8.89\|$\, \pm \,$\|0.27	79.26\|$\, \pm \,$\|2.86	0.95	0.73\|$\, \pm \,$\|0.01	0.015\|$\, \pm \,$\|0.003
NGC 2539	5.62\|$\, \pm \,$\|0.28	33.65\|$\, \pm \,$\|1.96	0.78	2.37\|$\, \pm \,$\|0.07	0.044\|$\, \pm \,$\|0.021

Name	R _c	R _lim	C*	ρ_c	ρ_b
	(′)	(′)		(stars arcmin⁻²)	(stars arcmin⁻²)
NGC 2281	8.89\|$\, \pm \,$\|0.27	79.26\|$\, \pm \,$\|2.86	0.95	0.73\|$\, \pm \,$\|0.01	0.015\|$\, \pm \,$\|0.003
NGC 2539	5.62\|$\, \pm \,$\|0.28	33.65\|$\, \pm \,$\|1.96	0.78	2.37\|$\, \pm \,$\|0.07	0.044\|$\, \pm \,$\|0.021

*C: The concentration parameter of the cluster.

Table 4.

Spatial structure parameters of the two clusters.

Name	R _c	R _lim	C*	ρ_c	ρ_b
	(′)	(′)		(stars arcmin⁻²)	(stars arcmin⁻²)
NGC 2281	8.89\|$\, \pm \,$\|0.27	79.26\|$\, \pm \,$\|2.86	0.95	0.73\|$\, \pm \,$\|0.01	0.015\|$\, \pm \,$\|0.003
NGC 2539	5.62\|$\, \pm \,$\|0.28	33.65\|$\, \pm \,$\|1.96	0.78	2.37\|$\, \pm \,$\|0.07	0.044\|$\, \pm \,$\|0.021

Name	R _c	R _lim	C*	ρ_c	ρ_b
	(′)	(′)		(stars arcmin⁻²)	(stars arcmin⁻²)
NGC 2281	8.89\|$\, \pm \,$\|0.27	79.26\|$\, \pm \,$\|2.86	0.95	0.73\|$\, \pm \,$\|0.01	0.015\|$\, \pm \,$\|0.003
NGC 2539	5.62\|$\, \pm \,$\|0.28	33.65\|$\, \pm \,$\|1.96	0.78	2.37\|$\, \pm \,$\|0.07	0.044\|$\, \pm \,$\|0.021

*C: The concentration parameter of the cluster.

According to the precise distances of the two clusters (see section 4), the linear sizes of the core and limiting radii of the two clusters can also be easily determined. Our obtained linear sizes of the core radii are consistent with the values obtained by Kharchenko et al. (2013). However, Kharchenko et al. (2013) determined a tidal radius of 5.13 pc for NGC 2281, which is significantly smaller than our result (12.0 pc). This may be due to the fact that Kharchenko et al. (2013) identified only 193 likely cluster members within a radius of ∼24′ from the cluster center. However, our investigation shows that NGC 2281 may actually contain more than 600 cluster members within a radius of ∼79′ from the cluster center. Our obtained limiting radius (12.4 pc) for NGC 2539 is significantly larger than the result (7.4 pc) obtained by Lapasset, Clariá, and Mermilliod (2000).

The dynamical evolution of an open cluster drives the massive stars to concentrate toward the cluster center, while low-mass stars remain in the outer region of the cluster (Bonnell & Davies 1998; Raboud & Mermilliod 1998). This process is also known as dynamical mass segregation. NGC 2811 and NGC 2539 are intermediate-age (500–600 Myr) open clusters containing more than 600 member stars, so they have had enough time to reach internal dynamical relaxation. Therefore, mass segregation is expected to be detected in the two clusters. Yoshizawa (1978) found that the binary stars in NGC 2281 are spatially more concentrated than the single stars, indicating the presence of mass segregation in this cluster. To our knowledge, mass segregation in NGC 2539 has not yet been investigated. In this work, we use the radial cumulative distribution function (RCDF) of the likely cluster members (P_RF ≥ 0.6) to investigate mass segregation in the two clusters. For each cluster, the likely cluster members are divided into three groups according to their G-band magnitudes. The spatial concentration of massive (bright) member stars relative to less massive (faint) ones is apparent in figures 8 and 9, indicating the presence of clear mass segregation in the two clusters. We also determine the core radii (R_c) for the two clusters based on the same magnitude ranges as those in the RCDFs (see figures 8 and 9); clear evidence for the presence of mass segregation in the two clusters can also be confirmed (see table 5). Our investigations demonstrate that the two clusters have reached or approached internal dynamical relaxation.

Fig. 8.

RCDF of the 696 likely cluster members in NGC 2281.

Fig. 9.

RCDF of the 664 likely cluster members in NGC 2539.

Table 5.

Core radii in different magnitude ranges for the two clusters.*

Name	R _c1 (′)	R _c2 (′)	R _c3 (′)
NGC 2281	5.26\|$\, \pm \,$\|0.10	12.18\|$\, \pm \,$\|0.59	16.95\|$\, \pm \,$\|1.60
NGC 2539	3.75\|$\, \pm \,$\|0.14	6.26\|$\, \pm \,$\|0.60	7.94\|$\, \pm \,$\|0.81

Name	R _c1 (′)	R _c2 (′)	R _c3 (′)
NGC 2281	5.26\|$\, \pm \,$\|0.10	12.18\|$\, \pm \,$\|0.59	16.95\|$\, \pm \,$\|1.60
NGC 2539	3.75\|$\, \pm \,$\|0.14	6.26\|$\, \pm \,$\|0.60	7.94\|$\, \pm \,$\|0.81

*R_c1, R_c2, and R_c3 indicate the core radii for the high-mass, medium-mass, and low-mass stars, respectively (see text for details).

Table 5.

Core radii in different magnitude ranges for the two clusters.*

Name	R _c1 (′)	R _c2 (′)	R _c3 (′)
NGC 2281	5.26\|$\, \pm \,$\|0.10	12.18\|$\, \pm \,$\|0.59	16.95\|$\, \pm \,$\|1.60
NGC 2539	3.75\|$\, \pm \,$\|0.14	6.26\|$\, \pm \,$\|0.60	7.94\|$\, \pm \,$\|0.81

Name	R _c1 (′)	R _c2 (′)	R _c3 (′)
NGC 2281	5.26\|$\, \pm \,$\|0.10	12.18\|$\, \pm \,$\|0.59	16.95\|$\, \pm \,$\|1.60
NGC 2539	3.75\|$\, \pm \,$\|0.14	6.26\|$\, \pm \,$\|0.60	7.94\|$\, \pm \,$\|0.81

*R_c1, R_c2, and R_c3 indicate the core radii for the high-mass, medium-mass, and low-mass stars, respectively (see text for details).

4 Distance, proper motion and radial velocity

We have determined 696 and 664 likely cluster members for NGC 2281 and NGC 2539, respectively (see subsection 2.3). Based on these cluster members, we can accurately estimate the distances and proper motions of the two clusters. However, Chen, Chen, and Shu (2004) found that most open clusters are spatially elongated, which may have a negative impact on accurate distance estimates for the two clusters. In this work, a Monte Carlo simulation technique (Gao 2019) is used to estimate the most likely distances and proper motions for the two clusters. For each cluster, only the high-probability cluster members (P_RF ≥ 0.9) located in the inner region of the cluster (⁠|$\lt 3\, R_{\rm c}$|⁠; see section 3) are taken into account.

We randomly simulate the median parallax and proper motion of the high-probability cluster members based on their observational values and uncertainties. We repeat the simulation 100000 times, and obtain 100000 simulated median parallaxes and proper motions for each cluster (see figures 10 and 11). According to the simulated median values, the mean parallaxes of NGC 2281 and NGC 2539 are determined to be 〈π〉 = 1.8855|$\, \pm \,$|0.0066 (sd) and 0.7577|$\, \pm \,$|0.0042 mas, respectively. Note that a parallax zero point of ∼− 0.029 mas has been confirmed (Lindegren et al. 2018; Vasiliev 2018). Taking into account the parallax zero point (⁠|$-0.029\, \pm \,$|0.043 mas; see Lindegren et al. 2018), the distances of NGC 2281 and NGC 2539 are 〈D〉 = 522|$\, \pm \,$|12 and 1271|$\, \pm \,$|70 pc, respectively. The mean proper motions of NGC 2281 and NGC 2539 are found to be (〈μ_αcos δ〉, 〈μ_δ〉) = (⁠|$-2.947\, \pm \,$|0.014, |$-8.344\, \pm \,$|0.014) and (⁠|$-2.335\, \pm \,$|0.007, |$-0.583\, \pm \,$|0.006) mas yr⁻¹, respectively. In addition, we also estimate the radial velocities of the two clusters using only the high-probability cluster members (P_RF ≥ 0.9) with radial velocities published in Gaia-DR2. We find that NGC 2811 and NGC 2539 have 42 and 11 high-probability cluster members with radial velocities, respectively. We randomly simulate (100000 times) the median radial velocities of the two clusters (see figure 12) and obtain 〈RV〉 = |$+20.0\, \pm \,$|0.7 and |$+29.6\, \pm \,$|0.4 km s⁻¹ for NGC 2281 and NGC 2539, respectively.

Fig. 10.

Monte Carlo simulations for the 294 high-probability cluster members in NGC 2281. Panels (a), (c), and (e) show the histograms of the observational parallaxes and proper motions (μ_αcos δ, μ_α). Panels (b), (d), and (f) show the histograms of the 100000 simulated median values. (Color online)

Fig. 11.

Monte Carlo simulations for the 360 high-probability cluster members in NGC 2539. Panels (a), (c), and (e) show the histograms of the observational parallaxes and proper motions (μ_αcos δ, μ_α). Panels (b), (d), and (f) show the histograms of the 100000 simulated median values. (Color online) .

Fig. 12.

Panels (a) and (b) show, respectively, the histograms of the 42 observational radial velocities and 100000 simulated median values for NGC 2281. Panels (c) and (d) show, respectively, the histograms of the 11 observational radial velocities and 100000 simulated median values for NGC 2539. (Color online)

Generally, our distances are consistent with the values in Dias et al. (2002) and Cantat-Gaudin et al. (2018). Our proper motions are also consistent with the values in Cantat-Gaudin et al. (2018), but somewhat different to those in Dias et al. (2002). However, Cantat-Gaudin et al. (2018) did not determine the radial velocities of the two clusters. Our radial velocities are also consistent with the results in Dias et al. (2002) and Soubiran et al. (2018). The study of Soubiran et al. (2018) was based on the same data as the present study (Gaia-DR2 radial velocities). Our radial velocity for NGC 2539 also agrees well with the result derived by Mermilliod and Mayor (1989). Our simulated results demonstrate that the Monte Carlo simulation technique is effective. The simulated results of the two clusters are listed in table 6.

Table 6.

Simulated results for the two clusters.*

Name	〈π〉 (mas)	〈D〉 (pc)	〈μ_αcos δ〉 (mas yr⁻¹)	〈μ_δ〉 (mas yr⁻¹)	〈RV〉 (km s⁻¹)	N
NGC 2281	1.9145\|$\, \pm \,$\|0.0435	522\|$\, \pm \,$\|12	\|$-2.947\, \pm \,$\|0.014	\|$-8.344\, \pm \,$\|0.014	\|$+20.0\, \pm \,$\|0.7	100000
NGC 2539	0.7867\|$\, \pm \,$\|0.0432	1271\|$\, \pm \,$\|70	\|$-2.335\, \pm \,$\|0.007	\|$-0.583\, \pm \,$\|0.006	\|$+29.6\, \pm \,$\|0.4	100000

Name	〈π〉 (mas)	〈D〉 (pc)	〈μ_αcos δ〉 (mas yr⁻¹)	〈μ_δ〉 (mas yr⁻¹)	〈RV〉 (km s⁻¹)	N
NGC 2281	1.9145\|$\, \pm \,$\|0.0435	522\|$\, \pm \,$\|12	\|$-2.947\, \pm \,$\|0.014	\|$-8.344\, \pm \,$\|0.014	\|$+20.0\, \pm \,$\|0.7	100000
NGC 2539	0.7867\|$\, \pm \,$\|0.0432	1271\|$\, \pm \,$\|70	\|$-2.335\, \pm \,$\|0.007	\|$-0.583\, \pm \,$\|0.006	\|$+29.6\, \pm \,$\|0.4	100000

*〈π〉: The zero-point-corrected parallax for the cluster. 〈D〉: The zero-point-corrected distance for the cluster. N: The number of copies of the simulation.

Table 6.

Simulated results for the two clusters.*

Name	〈π〉 (mas)	〈D〉 (pc)	〈μ_αcos δ〉 (mas yr⁻¹)	〈μ_δ〉 (mas yr⁻¹)	〈RV〉 (km s⁻¹)	N
NGC 2281	1.9145\|$\, \pm \,$\|0.0435	522\|$\, \pm \,$\|12	\|$-2.947\, \pm \,$\|0.014	\|$-8.344\, \pm \,$\|0.014	\|$+20.0\, \pm \,$\|0.7	100000
NGC 2539	0.7867\|$\, \pm \,$\|0.0432	1271\|$\, \pm \,$\|70	\|$-2.335\, \pm \,$\|0.007	\|$-0.583\, \pm \,$\|0.006	\|$+29.6\, \pm \,$\|0.4	100000

Name	〈π〉 (mas)	〈D〉 (pc)	〈μ_αcos δ〉 (mas yr⁻¹)	〈μ_δ〉 (mas yr⁻¹)	〈RV〉 (km s⁻¹)	N
NGC 2281	1.9145\|$\, \pm \,$\|0.0435	522\|$\, \pm \,$\|12	\|$-2.947\, \pm \,$\|0.014	\|$-8.344\, \pm \,$\|0.014	\|$+20.0\, \pm \,$\|0.7	100000
NGC 2539	0.7867\|$\, \pm \,$\|0.0432	1271\|$\, \pm \,$\|70	\|$-2.335\, \pm \,$\|0.007	\|$-0.583\, \pm \,$\|0.006	\|$+29.6\, \pm \,$\|0.4	100000

*〈π〉: The zero-point-corrected parallax for the cluster. 〈D〉: The zero-point-corrected distance for the cluster. N: The number of copies of the simulation.

5 Summary and conclusions

In this work, we aim to improve the details of the cluster members and fundamental astrophysical parameters of NGC 2281 and NGC 2539 based on the high-precision astrometric and photometric data of Gaia-DR2. The ML method (GMM+RF) allows us to identify reliable cluster members in an 11-dimensional parameter space, starting from a limited training set. The obtained reliable cluster members allow us to determine more accurate fundamental astrophysical parameters for the two clusters. Our main results and conclusions can be summarized as follows:

We obtain more complete cluster members for the two clusters. We find that the two clusters each contain more than 600 likely cluster members.
The spatial structure of the two clusters are investigated based on the obtained reliable cluster members. The core and limiting radii of the clusters are estimated (see section 3). We find that the limiting radius of NGC 2281 is significantly larger than the value derived by Kharchenko et al. (2013). In addition, we find strong evidence for the existence of significant mass segregation in the two clusters. This indicates that the two intermediate-age clusters have reached or approached dynamical relaxation.
We estimate the distances, proper motions, and radial velocities of the two cluster based on a Monte Carlo simulation technique (see section 4). Our simulated results are robust and less sensitive to outliers, since our results are derived based on the simulated median values for the high-probability cluster members located in the inner regions of the clusters. The obtained geometric distances have significantly smaller relative uncertainties (<6%). The obtained radial velocities of the two clusters are robust and reliable, even though the observational data may contain several significant outliers (see figure 12).

Acknowledgements

We thank the anonymous referee for substantial and detailed comments which greatly improved this work. This research was supported by the National Natural Science Foundation of China (NSFC, Grant No. 11403004). This work has made use of data from the European Space Agency (ESA) mission Gaia 〈https://www.cosmos.esa.int/gaia〉, processed by the Gaia Data Processing and Analysis Consortium (DPAC) 〈https://www.cosmos.esa.int/web/gaia/dpac/consortium〉. This research has made use of the VizieR catalog access tool, CDS, Strasbourg, France. This research has made use of the WEBDA database, operated at the Department of Theoretical Physics and Astrophysics of the Masaryk University.

Appendix. Fundamental parameters of likely cluster members in the two clusters

Fundamental information for the likely members of clusters NGC 2281 and NGC 2539 are shown in tables 7 and 8, respecively.

Table 7.

Fundamental information for the 696 likely cluster members of NGC 2281.*

ID	RA (°)	Dec (°)	P _RF
951483213394919808	102.33803612767	+41.13296136694	1.000
951481083091177600	102.08352925624	+41.07253947933	1.000
951483625711795456	102.22748370769	+41.10605339959	1.000
951462597551931520	102.31908890231	+40.86738969180	1.000
951459711334054528	102.38270767415	+40.83971766595	1.000
951670400952043776	102.01583118264	+41.20894889123	1.000
951466308403648896	102.47285389840	+40.93339625154	1.000
951483797510477184	102.27000131614	+41.14929611941	1.000
945468575553054592	102.11591972486	+40.81670395501	1.000
951478536173055744	102.09572225891	+41.01431216737	1.000
951464521697267968	102.29754379537	+40.98983345407	1.000
951698786392577664	101.82486275930	+41.57233285404	0.999
945469468906259328	101.99930676771	+40.84967025574	0.999
951463353466163840	102.35090300698	+40.92929419682	0.999
951478948489863552	101.98607828217	+40.97922096312	0.999
951669576318384384	102.03325554893	+41.19147394575	0.999
945464933420822912	101.70700962744	+40.73059130598	0.999
951472729378195968	102.44768444542	+41.12721879672	0.999
951418307849289088	102.58310300857	+40.89781604617	0.999
951478811050907136	102.08050654776	+41.04113186602	0.999

ID	RA (°)	Dec (°)	P _RF
951483213394919808	102.33803612767	+41.13296136694	1.000
951481083091177600	102.08352925624	+41.07253947933	1.000
951483625711795456	102.22748370769	+41.10605339959	1.000
951462597551931520	102.31908890231	+40.86738969180	1.000
951459711334054528	102.38270767415	+40.83971766595	1.000
951670400952043776	102.01583118264	+41.20894889123	1.000
951466308403648896	102.47285389840	+40.93339625154	1.000
951483797510477184	102.27000131614	+41.14929611941	1.000
945468575553054592	102.11591972486	+40.81670395501	1.000
951478536173055744	102.09572225891	+41.01431216737	1.000
951464521697267968	102.29754379537	+40.98983345407	1.000
951698786392577664	101.82486275930	+41.57233285404	0.999
945469468906259328	101.99930676771	+40.84967025574	0.999
951463353466163840	102.35090300698	+40.92929419682	0.999
951478948489863552	101.98607828217	+40.97922096312	0.999
951669576318384384	102.03325554893	+41.19147394575	0.999
945464933420822912	101.70700962744	+40.73059130598	0.999
951472729378195968	102.44768444542	+41.12721879672	0.999
951418307849289088	102.58310300857	+40.89781604617	0.999
951478811050907136	102.08050654776	+41.04113186602	0.999

*This table contains the Gaia-DR2 identifier, positions, and RF membership probabilities for the 696 likely cluster members (≥0.6). Only a portion of this table is shown here to demonstrate its form and content; a machine-readable version of the full table is available. ID: Gaia-DR2 identifier. RA, Dec: Right ascension and declination. P_RF: RF membership probability.

Table 7.

Fundamental information for the 696 likely cluster members of NGC 2281.*

ID	RA (°)	Dec (°)	P _RF
951483213394919808	102.33803612767	+41.13296136694	1.000
951481083091177600	102.08352925624	+41.07253947933	1.000
951483625711795456	102.22748370769	+41.10605339959	1.000
951462597551931520	102.31908890231	+40.86738969180	1.000
951459711334054528	102.38270767415	+40.83971766595	1.000
951670400952043776	102.01583118264	+41.20894889123	1.000
951466308403648896	102.47285389840	+40.93339625154	1.000
951483797510477184	102.27000131614	+41.14929611941	1.000
945468575553054592	102.11591972486	+40.81670395501	1.000
951478536173055744	102.09572225891	+41.01431216737	1.000
951464521697267968	102.29754379537	+40.98983345407	1.000
951698786392577664	101.82486275930	+41.57233285404	0.999
945469468906259328	101.99930676771	+40.84967025574	0.999
951463353466163840	102.35090300698	+40.92929419682	0.999
951478948489863552	101.98607828217	+40.97922096312	0.999
951669576318384384	102.03325554893	+41.19147394575	0.999
945464933420822912	101.70700962744	+40.73059130598	0.999
951472729378195968	102.44768444542	+41.12721879672	0.999
951418307849289088	102.58310300857	+40.89781604617	0.999
951478811050907136	102.08050654776	+41.04113186602	0.999

ID	RA (°)	Dec (°)	P _RF
951483213394919808	102.33803612767	+41.13296136694	1.000
951481083091177600	102.08352925624	+41.07253947933	1.000
951483625711795456	102.22748370769	+41.10605339959	1.000
951462597551931520	102.31908890231	+40.86738969180	1.000
951459711334054528	102.38270767415	+40.83971766595	1.000
951670400952043776	102.01583118264	+41.20894889123	1.000
951466308403648896	102.47285389840	+40.93339625154	1.000
951483797510477184	102.27000131614	+41.14929611941	1.000
945468575553054592	102.11591972486	+40.81670395501	1.000
951478536173055744	102.09572225891	+41.01431216737	1.000
951464521697267968	102.29754379537	+40.98983345407	1.000
951698786392577664	101.82486275930	+41.57233285404	0.999
945469468906259328	101.99930676771	+40.84967025574	0.999
951463353466163840	102.35090300698	+40.92929419682	0.999
951478948489863552	101.98607828217	+40.97922096312	0.999
951669576318384384	102.03325554893	+41.19147394575	0.999
945464933420822912	101.70700962744	+40.73059130598	0.999
951472729378195968	102.44768444542	+41.12721879672	0.999
951418307849289088	102.58310300857	+40.89781604617	0.999
951478811050907136	102.08050654776	+41.04113186602	0.999

*This table contains the Gaia-DR2 identifier, positions, and RF membership probabilities for the 696 likely cluster members (≥0.6). Only a portion of this table is shown here to demonstrate its form and content; a machine-readable version of the full table is available. ID: Gaia-DR2 identifier. RA, Dec: Right ascension and declination. P_RF: RF membership probability.

Table 8.

Fundamental information for the 664 likely cluster members of NGC 2539.

ID	RA (°)	Dec (°)	P _RF
5727491996210260992	122.79607163656	−12.75839368064	1.000
5727495333406159104	122.77221166332	−12.71195201600	1.000
5727489973287041664	122.67577074191	−12.82749684154	1.000
5727501273339613952	122.76746486370	−12.68954361638	1.000
5727495569623956864	122.73989523202	−12.71144166422	1.000
5727492962580743680	122.64966279496	−12.81828355219	1.000
5727490179445451904	122.69797733365	−12.79654138703	1.000
5727490282524648832	122.77028000402	−12.84132727733	1.000
5727491622554445952	122.73328370387	−12.78633711340	0.999
5727497115814392192	122.85356351808	−12.74990816938	0.999
5727491588194718336	122.71963117535	−12.80548036166	0.999
5727506431601749760	122.54512333755	−12.72157279765	0.999
5727489354811757696	122.67585345547	−12.84843318760	0.999
5727506191083568896	122.57435929319	−12.72993172810	0.999
5726731963095795072	122.87230553925	−13.03894086852	0.999
5727469185645395072	122.55763846023	−12.85810553540	0.999
5727493718498559872	122.57800086172	−12.79352528310	0.999
5727491996210261888	122.80209971813	−12.76453288585	0.999
5727493374901139712	122.65105011178	−12.78757693909	0.999
5727492172310238464	122.76505846347	−12.74919263331	0.999

ID	RA (°)	Dec (°)	P _RF
5727491996210260992	122.79607163656	−12.75839368064	1.000
5727495333406159104	122.77221166332	−12.71195201600	1.000
5727489973287041664	122.67577074191	−12.82749684154	1.000
5727501273339613952	122.76746486370	−12.68954361638	1.000
5727495569623956864	122.73989523202	−12.71144166422	1.000
5727492962580743680	122.64966279496	−12.81828355219	1.000
5727490179445451904	122.69797733365	−12.79654138703	1.000
5727490282524648832	122.77028000402	−12.84132727733	1.000
5727491622554445952	122.73328370387	−12.78633711340	0.999
5727497115814392192	122.85356351808	−12.74990816938	0.999
5727491588194718336	122.71963117535	−12.80548036166	0.999
5727506431601749760	122.54512333755	−12.72157279765	0.999
5727489354811757696	122.67585345547	−12.84843318760	0.999
5727506191083568896	122.57435929319	−12.72993172810	0.999
5726731963095795072	122.87230553925	−13.03894086852	0.999
5727469185645395072	122.55763846023	−12.85810553540	0.999
5727493718498559872	122.57800086172	−12.79352528310	0.999
5727491996210261888	122.80209971813	−12.76453288585	0.999
5727493374901139712	122.65105011178	−12.78757693909	0.999
5727492172310238464	122.76505846347	−12.74919263331	0.999

*This table contains the Gaia-DR2 identifier, positions, and RF membership probabilities for the 664 likely cluster members (≥0.6). Only a portion of this table is shown here to demonstrate its form and content; a machine-readable version of the full table is available. ID: Gaia-DR2 identifier. RA, Dec: Right ascension and declination. P_RF: RF membership probability.

Table 8.

〈http://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html〉.

Fundamental information for the 664 likely cluster members of NGC 2539.

ID	RA (°)	Dec (°)	P _RF
5727491996210260992	122.79607163656	−12.75839368064	1.000
5727495333406159104	122.77221166332	−12.71195201600	1.000
5727489973287041664	122.67577074191	−12.82749684154	1.000
5727501273339613952	122.76746486370	−12.68954361638	1.000
5727495569623956864	122.73989523202	−12.71144166422	1.000
5727492962580743680	122.64966279496	−12.81828355219	1.000
5727490179445451904	122.69797733365	−12.79654138703	1.000
5727490282524648832	122.77028000402	−12.84132727733	1.000
5727491622554445952	122.73328370387	−12.78633711340	0.999
5727497115814392192	122.85356351808	−12.74990816938	0.999
5727491588194718336	122.71963117535	−12.80548036166	0.999
5727506431601749760	122.54512333755	−12.72157279765	0.999
5727489354811757696	122.67585345547	−12.84843318760	0.999
5727506191083568896	122.57435929319	−12.72993172810	0.999
5726731963095795072	122.87230553925	−13.03894086852	0.999
5727469185645395072	122.55763846023	−12.85810553540	0.999
5727493718498559872	122.57800086172	−12.79352528310	0.999
5727491996210261888	122.80209971813	−12.76453288585	0.999
5727493374901139712	122.65105011178	−12.78757693909	0.999
5727492172310238464	122.76505846347	−12.74919263331	0.999

ID	RA (°)	Dec (°)	P _RF
5727491996210260992	122.79607163656	−12.75839368064	1.000
5727495333406159104	122.77221166332	−12.71195201600	1.000
5727489973287041664	122.67577074191	−12.82749684154	1.000
5727501273339613952	122.76746486370	−12.68954361638	1.000
5727495569623956864	122.73989523202	−12.71144166422	1.000
5727492962580743680	122.64966279496	−12.81828355219	1.000
5727490179445451904	122.69797733365	−12.79654138703	1.000
5727490282524648832	122.77028000402	−12.84132727733	1.000
5727491622554445952	122.73328370387	−12.78633711340	0.999
5727497115814392192	122.85356351808	−12.74990816938	0.999
5727491588194718336	122.71963117535	−12.80548036166	0.999
5727506431601749760	122.54512333755	−12.72157279765	0.999
5727489354811757696	122.67585345547	−12.84843318760	0.999
5727506191083568896	122.57435929319	−12.72993172810	0.999
5726731963095795072	122.87230553925	−13.03894086852	0.999
5727469185645395072	122.55763846023	−12.85810553540	0.999
5727493718498559872	122.57800086172	−12.79352528310	0.999
5727491996210261888	122.80209971813	−12.76453288585	0.999
5727493374901139712	122.65105011178	−12.78757693909	0.999
5727492172310238464	122.76505846347	−12.74919263331	0.999

*This table contains the Gaia-DR2 identifier, positions, and RF membership probabilities for the 664 likely cluster members (≥0.6). Only a portion of this table is shown here to demonstrate its form and content; a machine-readable version of the full table is available. ID: Gaia-DR2 identifier. RA, Dec: Right ascension and declination. P_RF: RF membership probability.

Footnotes

1

〈https://www.univie.ac.at/webda/〉.

2

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