Multicolour photometry and Gaia EDR3 astrometry of two couples of binary clusters (NGC 5617 and Trumpler 22) and (NGC 3293 and NGC 3324) Free

ABSTRACT

This paper presents a comprehensive analysis of two pairs of binary clusters (NGC 5617 and Trumpler 22) and (NGC 3293 and NGC 3324) located in the fourth quadrant of our Galaxy. For this purpose, we use different data taken from VVV survey, WISE, VPHAS, APASS, and GLIMPSE along with Gaia EDR3 astrometric data. We identified 584, 429, 692, and 273 most probable cluster members with membership probability higher than |$80 {{\ \rm per\ cent}}$| towards the region of the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324. We estimated the value of |$R=\frac{A_{V}}{E(B-V)}$| as ∼3.1 for the clusters NGC 5617 and Trumpler 22, which indicates the normal extinction law. The values of |$R\, \sim 3.8$| and ∼1.9 represent the abnormal extinction law towards the clusters NGC 3293 and NGC 3324. Our kinematical analysis shows that all these clusters have circular orbits. Ages are found to be 90 ± 10 and 12 ± 3 Myr for the cluster pairs (NGC 5617 and Trumpler 22) and (NGC 3293 and NGC 3324), respectively. The distances of 2.43 ± 0.08, 2.64 ± 0.07, 2.59 ± 0.1, and 2.80 ± 0.2 kpc estimated using parallax are consistent with the values calculated by using the distance modulus. We have also identified 18 and 44 young stellar object candidates present in NGC 5617 and Trumpler 22, respectively. Mass function slopes are found to be in fair agreement with the Salpeter’s value. The dynamical study of these objects shows a lack of faint stars in their inner regions, which leads to the mass-segregation effect. Our study indicates that NGC 5617 and Trumpler 22 are dynamically relaxed but the other pair of clusters are not. Orbital along with the physical parameters show that the clusters in both pairs are physically connected.

1 INTRODUCTION

The open clusters (OCs) are considered excellent laboratories for studies of stellar evolution and the dynamics of stellar systems. The structure of the cluster is a result of its evolutionary processes such as initial physical conditions of the molecular clouds, external tidal perturbation, etc (Chen et al. 2004; Sharma et al. 2008). OCs become beneficial objects for the stellar evolution because they are formed by the collapse and fragmentation of a turbulent molecular cloud (Harris & Pudritz 1994; Bate et al. 2003). OCs are influenced by the contamination of field stars. In recent years, the detailed membership analysis of stars in the cluster field has become a subject of intense investigation, mainly in view to understand the cluster properties (Carraro et al. 2008; Yadav et al. 2008; Joshi et al. 2014; Cantat-Gaudin et al. 2018). Recently, many authors have estimated membership probability for the clusters using Gaia DR2 kinematical data (Cantat-Gaudin et al. 2018, 2019; Castro-Ginard et al. 2018, 2019; Bisht et al. 2019, 2020). The (early) Third Gaia Data Release (hereafter EDR3; Gaia Collaboration et al. 2020) was made public on 2020 December 3rd. EDR3 consists of the central coordinates, proper motions in right ascension and declination and parallax angles (α, δ, μ_αcosδ, μ_δ, π) for around 1.46 billion sources with a limiting magnitude of 3–21 mag in G band. The Gaia EDR3 data are much accurate than the second data release of theGaia mission.

Bhatia (1990) has suggested that the lifetime of the binary clusters depends on cluster separation, tidal force of the parental Galaxy, and encounters with giant molecular clouds. In the Large and Small Magellanic Clouds (LMC and SMC, respectively), |${\sim}10{{\ \rm per\ cent}}$| of the well-known OCs may be in pairs and around |$50{{\ \rm per\ cent}}$| of them are primordial binary clusters (Bhatia & Hatzidimitriou 1988; Dieball & Grebel 2000; Dieball, Muller & Grebel 2002). In our Milky Way Galaxy, around 10 |${{\ \rm per\ cent}}$| of total OCs have been proposed to be in binary or multiple systems (Subramaniam et al. 1995; de la Fuente Marcos & le la Fuente Marcos 2010). The main aim of this paper is to study the properties of the binary OCs NGC 5617, Trumpler 22 and NGC 3293, NGC 3324. The available information about these objects in the literature are as follows:

(a) NGC 5617 (Cl426-605) (α₂₀₀₀ = 14^h 29^m 48^s, δ₂₀₀₀ = −60°43′00″; l = 314|${_{.}^{\circ}}$|67, b = −0|${_{.}^{\circ}}$|11): Lindoff (1968) has estimated age of the cluster as ∼4.6 × 10⁷ yr using photographic data. Based on photographic-photoelectric photometry Haug (1978) obtained parameters for this cluster as; E(B−V) = 0.53, A_V = 1.69, and a distance of 1.8 kpc. Colour–colour diagram (CCD) UBV photometry has been reported by Kjeldsen & Frandsen (1991, hereafter KF91), who got a smaller reddening E(B−V) = 0.48 ± 0.02, a larger distance of 2.05 ± 0.2 kpc, and an age of 70 Myr. It is an intermediate age OC (8.2 × 10⁷ yr) containing red giants and blue straggler stars (Ahumada & Lapasset 2007) in its surroundings, which membership of the cluster is still in doubt.

(b) Trumpler 22 (α₂₀₀₀ = 14^h 31^m 02^s, δ₂₀₀₀ = −61°10′00″; l = 314|${_{.}^{\circ}}$|64, b = −0|${_{.}^{\circ}}$|58): Haug (1978) studied this object using photographic data. De Silva et al. (2015) have done photometric and spectroscopic analysis of both the clusters NGC 5617 and Trumpler 22. They have obtained common age, distance, and radial velocity for both the clusters as 70 ± 10 Myr, 2.1 ± 0.3 kpc, and 38.5 ± 2.0 km s^–1, respectively.

(c) NGC 3293 (α₂₀₀₀ = 10^h 35^m 51^s, δ₂₀₀₀ = −58°13′48″; l = 285|${_{.}^{\circ}}$|85, b = 0|${_{.}^{\circ}}$|07): This object is moderately younger and belongs to the rich Carina complex. Preibisch et al. (2017) studied this object using Chandra X-ray observations. They found the age of this object as 8–10 Myr. Delgado et al. (2011) have estimated parameters of pre-main-sequence stars in this cluster. They obtained flatter mass function (MF) slope than the Salpeter’s value. Slawson et al. (2007) studied the stellar mass spectrum of NGC 3293 using CCD UBVRI images. They found significantly fewer lower mass stars towards the region of NGC 3293. They confirmed the age of this cluster as 10 Myr on the basis of the presence of some intermediate-mass stars near the main sequence in the HR diagram. Tuvikene & Sterken (2006) checked the variability of stars in NGC 3293. Out of 15 candidates, they found 3 constant stars, 10 stars with significant variability, and 2 of them were considered as suspected variables. Photometric study has been done by Baume et al. (2003) using CCD photometric observations at UBVRI_CH α. They found distance as 2750 ± 250 pc and age as 8 ± 1 Myr. The initial MF slope was estimated as 1.2 ± 0.2, a bit flatter than the typical slope for field stars.

(d) NGC 3324 (α₂₀₀₀ = 10^h 37^m 20^s, δ₂₀₀₀ = −58°38′30″; l = 286|${_{.}^{\circ}}$|23, b = −0|${_{.}^{\circ}}$|18): This object is also situated in proximity with NGC 3293 in Carina complex. Carraro et al. (2001) reported the first CCD UBVRI photometry of NGC 3324, and found that this cluster is very young and contains several pre-main-sequence stars. Claria (1977) presented wide-band (UBV) and narrow-band (H α) photometry of this object. According to this study, NGC 3324 contains at least 20 O- and B-type members and it is located at 3.12 kpc in the Carina spiral feature. A mean colour excess and age are found as 0.47 mag and 2.2 × 10⁶ yr, respectively.

Apart from this available information for the clusters under study, membership is still a question of debate. Our main goal is to estimate the membership probability for these objects and determine the more precise fundamental parameters, Galactic orbits, luminosity function (LF), MF, and dynamical state of the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324 using multiwavelength photometric data along with high-precision astrometric data from the Gaia EDR3 catalogue. Gaia EDR3 involves photometric magnitudes in three bands (G, G_BP, G_RP), astrometric data at the sub milliarcsecond level along with parallax values (Gaia Collaboration et al. 2020).

Proper motion is a very important parameter of OCs. Another important implication of the cluster’s proper motion is the determination of membership probabilities for individual stars (Sanders 1971). The investigation of OCs also offers to understand the MF of stellar objects, which is an important tool to study the star formation history (Jose et al. 2017; Sharma et al. 2017; and references therein). In recent years, many authors have estimated the present day MF for plenty of OCs (Dib et al. 2017; Joshi et al. 2020). The spatial distribution of massive and faint stars within the clusters provides important information to understand the mass segregation in OCs (Bisht et al. 2017).

The outline of this paper is as follows. The brief description of the data used has been described in Section 2. Section 3 is devoted to the study of mean proper motion and estimation of membership probability of stars. In Section 4, orbits of the clusters are calculated. The cluster structure has been explained in Section 5. The main fundamental parameters of the clusters are discussed in Section 6. The dynamical properties of the clusters are described in Section 7. Binarity of the clusters have been discussed in Section 8. The conclusion of this paper has been given in Section 9.

2 DATA USED

We collected the astrometric and photometric data from Gaia EDR3 along with broad-band photometric data from VVV, WISE, APASS, GLIMPSE, and VPHAS data for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324. The finding charts for the clusters are taken from Digitized Sky Survey (DSS) and shown in Fig. 1. We cross-matched each catalogue for the clusters under study. The brief description has been given for each data sets as follows.

2.1 The multidimensional Gaia EDR3 data

We used Gaia EDR3 (Gaia Collaboration et al. 2020) data for the astrometric analysis of the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324. Data should be considered complete down to G = 18–19 mag. The G, G_BP, and G_RP bands cover the wavelength range from 330 to 1050, 330 to 680, and 630 to 1050 nm, respectively (Evans et al. 2018). We have plotted the photometric errors in G, G_BP, and G_RP versus G band as shown in the bottom panels of Fig. 3. The uncertainties in parallaxes have the range of ∼0.02–0.03 milliarcsecond (mas) for sources at G ≤ 15 mag and ∼0.07 mas for sources with G ∼ 17 mag. The uncertainties in the respective proper motion components are up to 0.01−0.02 mas yr⁻¹ (for G ≤ 15 mag), 0.05 mas yr⁻¹ (for G ∼ 17 mag), and 0.4 mas yr⁻¹ (for G ∼ 20 mag). The proper motion and their corresponding errors are plotted against G magnitude in Fig. 2.

2.2 WISE data

This data base is based on a NASA medium-class explorer mission that conducted a digital imaging survey of the entire sky in the mid-IR bands. The effective wavelength of mid-IR bands are 3.35 μm (W1), 4.60 μm (W2), 11.56 μm (W3), and 22.09 μm (W4) (Wright et al. 2010). We have taken data for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324 from the ALLWISE source catalogue. This catalogue has achieved 5σ point source sensitivities better than 0.08, 0.11, 1, and 6 mJy at 3.35, 4.60, 11.56, and 22.09 μm, which is expected to be more than |$99{{\ \rm per\ cent}}$| of the sky (Bisht et al. 2020). These sensitivities are 16.5, 15.5, 11.2, and 7.9 for W1, W2, W3, and W4 bands corresponding to Vega magnitudes.

2.3 VVV data

The VVV survey is an ESO infrared large public survey (Minniti et al. 2010; Saito et al. 2012b) that uses the 4-m VISTA telescope located at Cerro Paranal Observatory, Chile. The effective wavelength of near-infrared broad-band filters are 0.87 μm (Z), 1.02 μm (Y), 1.25 μm (J), 1.64 μm (H), and 2.14 μm (K). The telescope has a near-infrared camera, VIRCAM (Dalton et al. 2006), consisting of an array of 16 detectors with 2048 × 2048 pixels. The errors given in the VVV catalogue for the (J, H, K) bands and W1, W2 bands from the WISE catalogue are plotted against J magnitudes in the top panels of Fig. 3.

2.4 APASS data

The American Association of Variable Star Observers (AAVSO) Photometric All-Sky Survey (APASS) is based on five filters: B, V (Landolt), g′, r′, and i′, finding stars with theV-band magnitude ranging from 7 to 17 mag (Heden & Munari 2014). DR9 is the latest catalogue and covers about |$99{{\ \rm per\ cent}}$| sky (Heden et al. 2016). We have extracted these data from http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/336.

2.5 GLIMPSE data

The Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE; Benjamin et al. 2003; Churchwell et al. 2004) data have been used for the clusters NGC 5617 and Trumpler 22. The basic calibration of the GLIMPSE IRAC frames was performed by the Spitzer Science Center Pipeline (Spitzer Observers Manual 2004). This data base consists only of high-reliability sources with each source must be detected twice in any of the four IRAC bands (3.6, 4.5, 5.8, 8.0 μm).

2.6 VPHAS data

The VST/Omegacam Photometric H α Survey (VPHAS) is imaging the entire Southern Milky Way in visible light at ∼1 arcsec angular resolution down to ≥20 mag using the VLT Survey Telescope in Chile. We have extracted data from VPHAS catalogue (Drew et al. 2014) for the analysis of the clusters under study. This catalogue includes data in u, g, r, i, and H α passbands.

3 MEAN PROPER MOTION AND CLUSTER MEMBERSHIP

The proper motion of stars is very precious to differentiate member stars from field stars. We have used proper motion and parallax data from the Gaia EDR3 catalogue to remove field stars from the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324. We made a catalogue of common stars after matching the Gaia data with the above mentioned photometric data sets in this paper.

PMs, μ_αcosδ and μ_δ are plotted as vector point diagrams (VPDs) in the top panels of Figs 4 and 5 to see the distribution of cluster and field stars. The middle and bottom panels of Fig. 4 show the corresponding Z versus (Z−Y) and G versus (G_BP−G_RP) colour–magnitude diagrams (CMDs) for the clusters NGC 5617 and Trumpler 22. In Fig. 5, we used proper motion distributions of stars in the upper panels while their corresponding G, G_BP−G_RP CMDs are plotted in the lower panels for the clusters NGC 3293 and NGC 3324. The left-hand panel in the CMDs shows all stars present in the cluster’s area, while the middle and right-hand panels show the probable cluster members and non-member stars, respectively. By visual inspection, we define the centre and radius of the cluster members in VPD for a preliminary analysis. This selection was performed in a way to minimize the field star contamination and to keep the maximum possible number of lower mass stars. A circle of 0.6 mas yr⁻¹ for NGC 5617, NGC 3293, and NGC 3324 while 0.4 mas yr⁻¹ for Trumpler 22 around the centre of the member star distribution in the VPDs characterize our membership criteria. The picked radius is an agreement between losing cluster members with poor PMs and the involvement of non-member stars. We have also used parallax for the reliable estimation of cluster members. A star is considered as probable cluster member if it lies inside the circle in VPD and has a parallax value within 3σ from the mean cluster parallax. The CMDs of the probable members are shown in the middle and bottom row panels in each cluster CMDs as shown in Figs 4 and 5. The main sequence of the cluster is separated. These stars have a PM error of ≤0.4 mas yr⁻¹.

For the precise estimation of mean proper motion, we deal with only probable cluster members based on clusters VPDs and CMDs as shown in Fig. 6. By fitting the Gaussian function into the constructed histograms, we determined the mean proper motion in the directions of RA and Dec., as shown in Fig. 6. From the peak of the Gaussian distribution, we found mean-proper motion in RA and Dec. directions for all clusters and are listed in Table 7(given later). The estimated values of mean proper motions for each cluster are in fair agreement with the values given by Cantat-Gaudin et al. (2018). Cantat-Gaudin catalogue (2018) reports the membership probabilities of few stars towards the region of the clusters under study. We derived membership probabilities of each star in all the studied clusters and the adopted method has been described in the next section.

3.1 Membership probability

Stars with PM errors ≤0.5 mas yr⁻¹ have been used to determine |$\phi _{\rm c}^{\nu }$| and |$\phi _{\rm f}^{\nu }$|⁠. A group of stars is found at μ_xc = −5.66 mas yr⁻¹, μ_yc = −3.19 mas yr⁻¹ for NGC 5617, μ_xc = −5.13 mas yr⁻¹, μ_yc = −2.70 mas yr⁻¹ for Trumpler 22, μ_xc = −7.65 mas yr⁻¹, μ_yc = 3.36 mas yr⁻¹ for NGC 3293, and μ_xc = −7.06 mas yr⁻¹, μ_yc = 2.85 mas yr⁻¹ for NGC 3324. Assuming a distance of 2.52, 2.68, 2.65, and 2.85 kpc for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively, and radial velocity dispersion of 1 km s⁻¹ for the open star clusters (Girard et al. 1989), the expected dispersion (σ_c) in PMs would be ∼0.08 mas yr⁻¹ for the clusters NGC 5617 and Trumpler 22 while ∼0.10 mas yr⁻¹ for the other two clusters. For field region stars, we have estimated (μ_xf, μ_yf) = (−3.5, −5.2) mas yr⁻¹ for NGC 5617, (μ_xf, μ_yf) = (−3.2, −4.5) mas yr⁻¹ for Trumpler 22, (μ_xf, μ_yf) = (−5.5, 2.0) mas yr⁻¹ for NGC 3293, (μ_xf, μ_yf) = (−5.3, 1.3) mas yr⁻¹ for NGC 3324 and (σ_xf, σ_yf) = (4.5, 4.9), (3.8, 4.1), (5.5, 4.8), (4.9, 3.7) mas yr⁻¹ for NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively.

In this way, we identified 584, 429, 692, and 273 stars as cluster members for NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively, with membership probability higher than |$80{{\ \rm per\ cent}}$| and G ≤ 20 mag. In the top left- and right-hand panel of Fig. 7, we plotted membership probability versus G magnitude and parallax versus G magnitude, respectively, for the cluster NGC 5617. Same have been plotted from top to bottom panels in Fig. 7 for the clusters Trumpler 22, NGC 3293, and NGC 3324. For all the clusters, we have plotted G versus (G_BP−G_RP) CMD, the identification chart, and proper motion distribution using stars with membership probability higher than |$80{{\ \rm per\ cent}}$| in Fig. 8. The Cantat-Gaudin et al. (2018) catalogue reports membership probabilities for all the clusters under study. We matched our likely members with this catalogue having membership probability higher than 80 |${{\ \rm per\ cent}}$| and those have been denoted by blue dots in CMDs as shown in Figs 14 and 15.

3.2 Determination of the effectiveness of probabilities

4 ORBIT ANALYSIS OF THE CLUSTERS

4.1 Galactic potential model

4.2 Orbit calculation

To estimate the orbits and orbital parameters for the clusters under study, we have used the Galactic potential models. The input parameters, such as central coordinates (α and δ), mean proper motions (μ_αcosδ, μ_δ), parallax values, clusters age, and heliocentric distances (d_⊙) for the clusters used in this paper have been taken from Table 7 (given later). The radial velocity of clusters NGC 5617 and Trumpler 22 has been taken as −38.50 ± 2.15 km s^–1 calculated by De Silva et al. (2015). Radial velocity of the cluster NGC 3293 has been taken as −13.16 ± 0.55 km s^–1 from Soubiran et al. (2018) while for the cluster NGC 3324, we obtained it as −14.27 ± 0.70 km s^–1 taking weighted mean of probable members from Gaia EDR3.

We transformed position and velocity vectors into the Galactocentric cylindrical coordinate system using the transformation matrix given in Johnson & Soderblom (1987). In this system, (r, ϕ, z) indicates the position of an object in the Galaxy, where r is the distance from the Galactic Centre, ϕ is the angle relative to the Sun’s position in the Galactic plane, and z is the distance from the Galactic plane.

The right-hand coordinate system is adopted to transform equatorial velocity components into Galactic-space velocity components (U, V, W), where U, V, and W are radial, tangential, and vertical velocities, respectively. In this system, the x-axis is taken positive towards Galactic Centre, the y-axis is along the direction of Galactic rotation, and the z-axis is towards Galactic North Pole. The Galactic Centre is taken at (17^h 45^m 32|${_{.}^{\rm s}}$|224, −28°56′10″) and North Galactic Pole is at (12^h 51^m 26|${_{.}^{\rm s}}$|282, 27°7′42|${_{.}^{\prime\prime}}$|01) (Reid & Brunthaler, 2004). To apply a correction for Standard Solar Motion and Motion of the Local Standard of Rest (LSR), we used position coordinates of the Sun as (8.3, 0, 0.02) kpc and its space-velocity components as (11.1, 12.24, 7.25) km s^–1 (Schonrich et al. 2010). Transformed parameters in Galactocentric coordinate system are listed in Table 1.

In orbit determination, we estimated the radial and vertical components of gravitational force, by differentiating total gravitational potentials with respect to r and z. The second order derivatives of gravitational force describe the motion of the clusters. For orbit determination, the second order derivatives are integrated backwards in time, which is equal to the age of the clusters. Since potentials used are axis-symmetric, energy, and z component of angular momentum are conserved throughout the orbits.

Fig. 9 shows orbits of the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324. In left-hand panels, the motion of the clusters is shown in terms of distance from the Galactic Centre and Galactic plane, which manifests two dimensional side view of the orbits. In middle panels, the cluster motion is described in terms of x and y components of Galactocentric distance, which shows a top view of orbits. The clusters are also oscillating along Z-axis as shown in right-hand panels of these figures. (NGC 5617 and Trumpler 22) and (NGC 3293 and NGC 3324) are oscillating along Z-axis within the limit of −0.04 to 0.04 and −0.1 to 0.1 kpc, with a time period of 67 and 79 Myr, respectively. The time period of revolution around the Galactic Centre is 176, 172, 194, and 193 Myr for NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively. The time periods of these oscillations for both the binary clusters are very similar. We also calculated the orbital parameters for the clusters and listed it in Table 2. Here, e is eccentricity, R_a is the apogalactic distance, R_p is the perigalactic distance, Z_max is the maximum distance travelled by the cluster from Galactic disc, E is the average energy of orbits, J_z is z component of angular momentum, and T is the time period of the clusters in the orbits.

The orbits of the clusters under study follow a boxy pattern and eccentricities for all the clusters are nearly zero. Hence, they trace a circular path around the Galactic Centre. We have shown the birth and present day position of the clusters in the Galaxy which are represented by the filled circle and filled triangle, respectively, in Fig. 9. NGC 5617 and Trumpler 22 are intermediate age while NGC 3293 and NGC 3324 are young age OCs. The orbits are confined in a box of |${\sim}6.6 \lt R_{\rm gc} \le 6.7$| kpc for (NGC 5617 and Trumpler 22) and ∼7.9 < R_gc ≤ 8.6 kpc for the clusters (NGC 3293 and NGC 3324). Our analysis indicates that all the clusters under study are inside the solar circle in the thin disc and may interact with the molecular clouds present in the Galaxy. Carraro & Chiosi (1994) found that the clusters orbiting beyond the solar circle survive more as compared to the clusters which are in the inner solar circle. Webb et al. (2014) found that the clusters having circular orbits evolve slower as compared to the eccentric ones. Orbital parameters determined in the present analysis are similar to the parameters found by Wu et al. (2009), except their orbits, are more eccentric than what we found in the present analysis.

5 STRUCTURAL PARAMETERS OF THE CLUSTERS

5.1 Cluster centre

The central coordinates of OCs play an important role to describe cluster properties more accurately. In the previous studies, the centre has been determined just by the visual inspection (Becker & Fenkart 1971; Romanishim & Angel 1980). We applied the star-count method using probable cluster members based on proper motion and parallax data base. The histograms are built for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324 in both the RA and Dec. directions as shown in Fig. 10. The Gaussian curve-fitting is performed to the star counts profiles in RA and Dec. directions. All estimated centre coordinates are listed in Table 7. Our estimated values are in good agreement with the values given by Dias et al. (2002). Our derived centre coordinates for all objects are also coordinated with Cantat-Gaudin et al. (2018) catalogue within uncertainty.

5.2 Radial density profile

The density contrast parameter (⁠|$\delta _{\rm c} = 1 +\frac{f_{0}}{f_{\rm b}}$|⁠) is calculated for all the clusters under study using member stars selected from proper motion data. Current evaluation of δ_c (4.1, 4.8, 6.5, and 4.4 for NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively) are lower than the values (7 ≤ δ_c ≤ 23) given by Bonatto & Bica (2009). This estimation of δ_c indicates that all clusters are sparse. All the structural parameters are listed in Table 3 for the clusters under study.

6 THE FUNDAMENTAL PARAMETERS OF NGC 5617 AND TRUMPLER 22

6.1 Two-colour diagrams

The two-colour diagrams (TCDs) are very useful to determine the relation of various colour excesses and their variations towards the cluster region.

6.1.1 Optical to mid-infrared extinction law

In this section, we combined multiwavelength photometric data with Gaia astrometry for the clusters under study to check the extinction law from optical to mid-infrared region. The resultant (λ−G_RP)/(G_BP−G_RP) TCDs are shown in Fig. 12 for all the clusters. Here, λ denotes the filters other than G_RP. All stars showing in Fig. 12 are probable cluster members. A linear fit to the data points is performed and slopes are listed in Table 4. The estimated values of slopes are in good agreement with the value given by Wang & Chen (2019) only for the binary clusters NGC 5617 and Trumpler 22. We estimated |$R=\frac{A_{V}}{E(B-V)}$| as ∼3.1 for the clusters NGC 5617 and Trumpler 22. Our obtained values of R are ∼3.8 and 1.9 for clusters NGC 3293 and NGC 3324. Our analysis indicates that reddening law is normal towards the cluster region of NGC 5617 and Trumpler 22 while it is abnormal for the binary clusters NGC 3293 and NGC 3324.

6.1.2 Interstellar reddening from JHK colours

Using the above relationships, we obtained the interstellar reddenings, E(B−V) as, 0.55, 0.64, 0.23, and 0.45 for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively. Our derived value of E(B−V) is similar to Haug (1978) and slightly higher than KF91 for NGC 5617. Our E(B−V) value for NGC 3324 is in good agreement with the value obtained by Claria (1977).

6.2 Age and distance to the clusters

To trace the Galactic structure and chemical evolution of the Galaxy using OCs, the distance, and age of OCs play the most important role (Friel & Janes 1993). We have estimated the mean value of A_G for the studied clusters using the most probable members from Gaia DR2. Our values are 1.58, 1.65, 0.86, and 0.75 for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively. The main fundamental parameters (age, distance, and reddening) are estimated by fitting the solar metallicity (Z = 0.019) isochrones of Marigo et al. (2017) to all the CMDs |$(G, G_{\rm BP}-G_{\rm RP}), (G, G_{\rm BP}-G), (G, G-G_{\rm RP}) (Z, Z-Y), (J, J-H) \ {\rm and} \ (K, J-K)$| as shown in Figs 14 and 15. We have used only probable the cluster members in order to reduce the field star contamination in the cluster’s area.

The Galactocentric coordinates of the clusters X (directed towards the Galactic Centre in the Galactic disc), Y (directed towards the Galactic rotation), and distance from the Galactic plane Z (directed towards Galactic North Pole) can be estimated using clusters’ distances, longitude, and latitude. The Galactocentric distance has been calculated by considering 8.5 kpc as the distance of the Sun to the Galactic Centre. The estimated Galactocentric coordinates are listed in Table 7.

The estimation of the main fundamental parameters for the clusters are given below:

6.2.1 NGC 5617

We fitted the theoretical isochrones of different ages (log(age) = 7.90, 7.95, and 8.00) in all the CMDs for the cluster NGC 5617, shown in the upper panels of Fig. 14. The best global fit is favourable for the middle isochrone with log(age) = 7.95 to the high mass cluster members. A satsifactory fitting of isochrones provides an age of 90 ± 10 Myr. The apparent distance modulus ((m − M) = 13.70 ± 0.4 mag) provides a distance 2.5 ± 0.30 kpc from the Sun. The estimated distance is in good agreement with the value of 3.0 kpc as given by Cantat-Gaudin et al. (2018).

6.2.2 Trumpler 22

In the CMDs of Trumpler 22, we have fitted exactly the similar age isochrones as shown in Fig. 14. So, the age of this object is the same as that of NGC 5617. The inferred apparent distance modulus (m − M) = 14.20 ± 0.3 mag provides a heliocentric distance as 2.8 ± 0.2 kpc. This value of the distance is very close to the distance derived by Cantat-Gaudin et al. (2018).

6.2.3 NGC 3293

For the cluster NGC 3293, we have fitted the theoretical isochrones of different ages (log(age) = 7.00, 7.05, and 7.10) as shown in Fig. 15. Based on the best-fitted middle isochrone of log(age) = 7.05, we found the age of this object as 12 ± 2 Myr. The inferred apparent distance modulus (m − M) = 12.90 ± 0.2 mag provides a heliocentric distance as 2.6 ± 0.1 kpc. This value of the distance is very close to the distance derived by Cantat-Gaudin et al. (2018).

6.2.4 NGC 3324

For this cluster also, the isochrones of the same age values as NGC 3293 were fitted (see Fig. 15). The inferred apparent distance modulus ((m − M) = 13.00 ± 0.2) mag provides a heliocentric distance of 2.8 ± 0.2 kpc. This value of the distance is very close to the distance derived by Cantat-Gaudin et al. (2018).

We have used kinematical data from Gaia EDR3 to estimate the distances of the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324. We can estimate the cluster distance using the mean parallax of probable members (Lauri et al. 2018). The Gaia DR2 parallax has been corrected for these clusters after adopting zero-point offset (−0.05 mas) as given by Riess et al. (2018). The histograms of parallax using probable members in all clusters with 0.15 mas bins are shown in Fig. 16. The mean parallax is estimated as 0.41 ± 0.008, 0.38 ± 0.009, 0.39 ± 0.004, and 0.36 ± 0.01 mas for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively, and the corresponding distance values are 2.44 ± 0.05, 2.63 ± 0.06, 2.56 ± 0.02, and 2.77 ± 0.07 kpc. These obtained values of distance are reciprocal of cluster parallax. The mean parallax for all the clusters are very close to the parallax obtained by Cantat-Gaudin et al. (2018). We also determined the distance of the clusters according to the method discussed by Bailer-Jones et al. (2018). In this way, our estimated values are 2.43 ± 0.08, 2.64 ± 0.07, 2.59 ± 0.1, and 2.80 ± 0.2 kpc for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively. The estimated values of cluster distance are also in good agreement with the results obtained using the isochrone fitting method as described in the above paragraph.

6.3 Young stellar object candidates

7 DYNAMICAL STUDY OF THE CLUSTERS

7.1 Luminosity function and mass function

Luminosity function (LF) and mass function (MF) are primarily dependent on cluster membership and also connected with the well-known mass–luminosity relationship. To construct the LF for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, we used G versus (G_BP−G_RP) CMD. We converted the G magnitudes of main sequence stars into the absolute magnitudes using the distance modulus and reddening calculated in this paper for all the clusters. A histogram is constructed with 1.0 mag intervals as shown in Fig. 18. This figure exhibits that the LF continues to increase up to M_G ∼ 3.4, 2.0, 3.3, and 5.2 mag for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively.

Where dN is the probable cluster members in a mass bin dM with central mass M and x is MF slope. Since Gaia data (G mag) is not complete below G = 19 mag (Arenou et al. 2018), then we took stars brighter than this limit, which corresponds to stars more massive than 1 M_⊙. The estimated values of the MF slopes are x = 1.40 ± 0.16, 1.44 ± 0.24, 1.59 ± 0.22, and 1.51 ± 0.25 for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively. These obtained values are satisfactory with the Salpeter’s initial MF slope within error. The total mass has been estimated for clusters using the derived MF slope. All MF related parameters in this section, like mass range, MF slope, and the total mass estimated are listed in Table 5.

7.2 Mass segregation

In mass segregation, the higher mass stars gradually sink towards the cluster centre and transfer their kinetic energy to the more numerous lower mass stellar component. The mass-segregation effect in clusters can be due to the dynamical evolution or imprint of the star formation process or both. Considerable work has been done by many authors to check the mass-segregation effect in clusters (Lada & Lada 1991; Brandl et al. 1996; Hillenbrand & Hartmann 1998; Meylan 2000; Bisht et al. 2019, 2020). In this study, we used only probable cluster members based on membership probability as described in Section 3. To understand mass segregation, cluster stars are divided into three different mass ranges as shown in Table 6 for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324. The cumulative radial stellar distribution has been plotted for the main-sequence stars of all objects as shown in Fig. 20. This figure demonstrates the mass-segregation effect in these clusters based on the arrangement of massive and faint stars. We found the confidence level of mass segregation as 88, 75, 77, and 70 |${{\ \rm per\ cent}}$| for the clusters NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively, based on the Kolmogrov–Smirnov (KS) test.

In the above formula, N denotes the cluster members, R_h is the radius within which half of the cluster mass is accommodated, and m is the mean mass of the cluster stars (Spitzer & Hart 1971).

We estimated the value of T_R as 13.5, 24.5, 26, and 17 Myr for NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively. The dynamical evolution parameter (⁠|$\tau =\frac{\rm age}{T_{E}}$|⁠) are found to be greater than 1 for the clusters NGC 5617 and Trumpler 22, which concludes that these objects are dynamically relaxed. The value of τ is less than 1 for the clusters NGC 3293 and NGC 3324. Hence, our study demonstrates that the binary clusters NGC 3293 and NGC 3324 are not dynamically relaxed.

7.3 Dissociation time of clusters

7.4 Tidal radius of the clusters

8 BINARITY OF THE CLUSTERS

We analysed the orbits of both the cluster pairs. All the clusters are orbiting in a circular orbit. Their close values of orbital parameters indicate that they are moving together. Their distance and age also indicate that they have formed in a similar time-scale. Therefore, based on the motion, we can conclude that these clusters are a cluster pairs of our Galaxy.

9 CONCLUSIONS

One of the outcomes of this study is the estimation of membership probability of stars in the field of the two binary clusters (NGC 5617 and Trumpler 22) and (NGC 3293 and NGC 3324). We have estimated all the fundamental parameters of the clusters as shown in Table 7. The main conclusions of this study are as follows:

• The new centre coordinates are derived for all the clusters and are listed in Table 7.

• CCDs have been constructed after combining the Gaia  EDR3, VVV, VPHAS, APASS, and WISE data bases. The diagrams show that the interstellar extinction law is normal towards the cluster’s area of NGC 5617 and Trumpler 22. We found that the extinction law is abnormal for the clusters NGC 3293 and NGC 3324.

• The distance estimation from parallax are well supported by the values estimated using the isochrone fitting method to the clusters CMDs. We obtained a similar age of cluster pairs by comparing with the theoretical isochrones of solar metallicity taken from Marigo et al. (2017).

• We obtained 18 and 44 stars towards the cluster region of NGC 5617 and Trumpler 22 based on the YSO reddening parameter (Q) method.

• We determined LFs and MFs of both objects by considering the members as selected on the basis of membership probability of stars. The MF slopes are in fair agreement with the Salpeter (1955) value for the clusters under study.

• The presence of mass segregation is examined for these clusters using probable cluster members. We found that the massive stars are concentrated towards the inner regions of the clusters in comparison to faint stars. The confidence level of mass segregation is found as 88, 75, 77, and 70 |${{\ \rm per\ cent}}$| for NGC 5617, Trumpler 22, NGC 3293, and NGC 3324, respectively, on the basis of the KS test. Our study indicates that NGC 5617 and Trumpler 22 are dynamically relaxed, while NGC 3293 and NGC 3324 are not relaxed.

• Galactic orbits and orbital parameters are estimated using Galactic potential models. We found that these objects are orbiting in a boxy pattern in circular orbit. The different orbital parameters are listed in Tables 1 and 2 for the clusters under study. Present analysis shows that pairs of the clusters (NGC 5617 and Trumpler 22) and (NGC 3293 and NGC 3324) are physically connected and are cluster pairs of the Milky Way.

ACKNOWLEDGEMENTS

The authors are thankful to the anonymous referee for useful comments, which improved the content of the paper significantly. This work was financially supported by the Natural Science Foundation of China (NSFC-11590782, NSFC-11421303). Devesh P. Sariya and Ing-Guey Jiang are supported by the grant from the Ministry of Science and Technology (MOST), Taiwan. The grant numbers are MOST 105-2119-M-007 -029 -MY3 and MOST 106-2112-M-007 -006 -MY3. 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). Funding for the DPAC is provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. In addition to this, it is worth mentioning that this work was done by using WEBDA.

DATA AVAILABILITY

We have used the different data sets for the analysis of two pairs of binary clusters (NGC 5617 and Trumpler 22) and (NGC 3293 and NGC 3324), which are publicly available at the following links:

• GAIA EDR3 https://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=I/350/gaiaedr3

• WISEhttps://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=II/311/wise

• VVV http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/348

• APASS http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II

• GLIMPSE https://vizier.u-strasbg.fr/viz-bin/VizieR?-source=

• VPHAS http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II

REFERENCES