2.1. Data and Analysis

In the present work, we use both the GAIA DR2 and 2MASS databases. Astrometric, photometric and spectroscopic measurements of the entire sky have been obtained by the ESA mission Gaia (^{Gaia Collaboration et al. 2016}). The latest version of the Gaia data (Gaia DR2) contains over 1.3 billion sources in three photometric bands (G,GBP,GRP), in addition to their proper motion and parallax (^{Gaia Collaboration et al. 2018}, ^{Lindegren et al. 2018}). The limiting G-band magnitude of Gaia DR2 data is 21 mag. At the bright end (G < 14mag), the uncertainties reach 0.02 mas in parallax and 0.05 mas/yr in proper motions, while for sources near G ∼ 21 mag the uncertainties reach 2 mas and 5 mas/yr respectively. The 2MASS survey is the catalog of choice for astronomical studies near the Galactic plane (^{Skrutskie et al. 2006)}. The limiting magnitudes of 2MASS data J(1.25 µm), H(1.65 µm) and K(2.17µm) are 15.8, 15.1 and 14.3 mag, respectively.

We extracted the data around NGC 2158 inserting the coordinates of the cluster NGC 2158 as RA=06h 07m 25s.00, DEC=+24 ∘ 05 ' 48 .''00, l=186 .∘634 and b=+1 .∘781 (^{Dias et al. 2002}) in the Virtual Observatory tool TOPCAT (^{Taylor 2005}), within a radius of 300. Cross-matching the Gaia DR2 and 2MASS data we obtain a matched number of stars of 17977.

To obtain the best astrometric precision of the proper motions, we used the sources brighter than G = 18 mag, which correspond to typical astrometric uncertainties smaller than 0.3 mas/yr in proper motion (^{Cantat-Gaudin et al. 2018}, ^{Lindegren et al. 2018)}. Using the Vector Point Diagram (VPD) of the proper motions in RA and DEC, we select the overdensity in the cluster region, obtaining 2103 stars (red region on the left panel, Figure 2). To obtain the candidate cluster members, we removed stars from the selected cluster region (red region) if their proper motions in RA and DEC were inconsistent with a median proper motion by more than three median absolute deviations (MAD). The color magnitude diagram (CMD) in the right panel of Figure 2 shows that the candidate members (red dots) fall on a well-defined main sequence. The selected cluster region of the candidate members has a nearly ellipsoidal shape. The equivalent radius Re=a2+b2 (with a and b the semi-major and semi-minor axes of the ellipse, respectively) of the selected cluster region is 0.7 mas/yr. This value is well within the range of the best-precision Gaia proper motions of 2 mas/yr (^{Ferreira et al. 2019}).

Fig. 2 The left panel shows the Vector Point Diagram (VPD), and the right panel shows the Color Magnitude Diagram (CMD) of the open cluster NGC 2158. The selected red region from the VPD marks the candidate member stars. The blue triangle and the black dots represent the eclipsing binary [NBN2015]78 and the field stars respectively. The red dots on the CMD represent candidate members. The color figure can be viewed online. 

Our membership criterion depends on the chosen overdensity region to define the members of the cluster from the VPD, which obviously is a compromise between losing cluster members and including field stars (see ^{Bisht et al. 2019}). To minimize the field contamination, we used the Blob subset from TOPCAT for selecting the cluster region in the VPD.

Stars are considered members if they lie inside the overdensity cluster region in VPD and have proper motions in RA and DEC within ±3 MAD from the mean proper motion; they must also have a clear main sequence in the CMD (Figure 2). These stars must be inside the limiting radius and they should have the same direction of proper motion vectors (Figure 4). The cluster members must have the same angular speed in the space.

To determine the mean proper motion of the cluster, we constructed the histograms and made a Gaussian fitting of the proper motions in both directions RA and DEC, see Figure 3. We found that the mean proper motion of NGC 2158 is pmRA =−0.159 ± 0.165mas/yr and pmDEC = −2.000 ± 0.175mas/yr. These values are in good agreement with the values obtained by ^{Cantat-Gaudin et al. (2018)}, see Table 1.

Fig. 3 The histograms of the proper motion in RA (mas/yr) and DEC (mas/yr), and the Gaussian fits are shown as blue dashed lines. The color figure can be viewed online. 

The measured proper motions of the eclipsing binary [NBN2015]78 in Gaia DR2 are pmRA=−3.252 ± 0.673mas/yr and pmDEC =−2.63 ± 0.596mas/yr with relative uncertainties smaller than 20%. The position of the eclipsing binary [NBN2015]78 in VPD is found to be very far away from the cluster region, close to the main sequence of the CMD (Figure 2). It is clearly seen that the direction of the proper motion vector of [NBN2015]78 in Figure 4 is different from the direction of the cluster members. Therefore, in spite of its position in the color magnitude diagrams in the optical and IR (2MASS, Figure 7) this eclipsing binary does not seem to be a member of the cluster.

Fig. 4 Projection on the sky of the proper motion vectors. The red vectors (red triangles) represent the cluster members, the blue vector (blue triangle) represents the eclipsing binary [NBN2015]78 and the gray vectors (gray arrows) represent the field stars. The color figure can be viewed online. 

2.2. Cluster Center and Radius

To determine the cluster center we count the stars in right ascension (RA) and declination (DEC). The center of the open cluster NGC 2158 is calculated at the point where the maximum stellar density of the cluster’s area is reached, using a Gaussian fitting. The center of the cluster NGC 2158 is found to be at RA = 91°.866±0°.077 and DEC = 24°.109±0°.070, and the corresponduing Galactic coordinates are l=186.∘629 and b=+1∘.79 (Figure 5). We find that that the difference between our estimation of the centre of the cluster in RA and DEC and that of ^{Cantat-Gaudin et al. (2018)} is 14''4 and 0'.60, respectively. The difference with the values from ^{Anderson et al. (2013)}; ^{Wu et al. (2009)}; ^{Dias et al. (2002)} is 43''.2 and 0'.72, respectively (Table 2).

Fig. 5 Inferring the center of NGC 2158 in RA (left panel) and DEC (right panel). The blue lines are Gaussian fits. The color figure can be viewed online. 

To calculate the core and limiting radii, we measure the radial density profile (RDP) of the open cluster NGC 2158, extracting the brightest stars G ≤ 18 mag within a square region of side 2 mas/yr around the center of the overdensity in the VPD, see Figure 2. We then divide the square area around the cluster into concentric rings. The stars are counted within each ring and their number divided by the area of each ring. A ^{King (1966)} model is then fitted to the radial density profile of NGC 2158 as shown in Figure 6.

where f_bg, f₀ and Rcore are the background density, the central star density and the core radius of the cluster, respectively. The bestfit values are f₀ = 67.06 ± 2.47 stars/arcmin₂ and Rcore = 1'.70 ± 00.08 see Table 3. The limiting radius of NGC 2158 is estimated as R_lim = 17'.00, beyond which stars begin to merge with the background population (see Figure 6, ^{Tadross and Hendy 2016}; ^{Hendy 2018}). Following ^{Peterson and King (1975)}, we find that the concentration parameter C = logs(Rlim/Rcor) of NGC 2158 is C = 1.00.

Fig. 6 The radial density profile of NGC 2158. The red curved line is our best-fit King profile). The value of the limiting radius is R_lim = 17'.00. The solid line represents the background density. The color figure can be viewed online. 

Our estimations of the core and limiting radii are in very good agreement with the ones obtained by ^{Anderson et al. (2013)} and ^{Kharchenko et al. (2013)}, see Table 3, but the limiting radius obtained by ^{Glushkova et al. (2010)} is significantly smaller than the one we find.

2.3. Color Magnitude Diagrams

To obtain photometric data for multi-color magnitude diagrams (MCMDs), we cross-matched our cluster members in Figure 4 with Gaia, 2MASS and BV data from ^{Nardiello et al. (2015)}. We obtained 1223 common stars which are members of the cluster. The MCMDs of NGC 2158 are built using optical data from Gaia DR2 [G,(GBP −GRP)], ^{Nardiello et al. (2015)} [V,(B − V )] and infrared 2MASS data [J,(J − H) and J−,J − K)] are shown in Figure 7.

Fig. 7 CMDs of NGC 2158: [G − (GBP − GRP)] from Gaia DR2, [V − (B − V )] from ^{Nardiello et al. (2015)} and [J − (J − H) & J − (J − K)] from 2MASS data respectively. The blue circles in the CMDs refer to the position of the eclipsing binary [NBN2015]78. The dots represent 1223 member stars in the optical (Gaia & [V − (B − V )]), and the 2MASS data using our criteria. The open circles represent 1031 membesr stars obtained by ^{Cantat-Gaudin et al. (2018)} using their UPMASK code to determine membership probabilities. The color figure can be viewed online. 

To obtain the best fit for each CMD, we use the Padova isochrones of ^{Marigo et al. (2017)} with a solar metallicity of 0.0152 (^{Bressan et al. 2013)}. Using the turn-off point, we find an age of logage = 9.350 ± 0.050, that is, and age of 2.240 ± 0.260 Gyr. The extinction ratios used for correcting the magnitudes come from the following transformation equations, based on the work by ^{Cardelli et al. (1989)} and ^{O’Donnell (1994)}:

and hence

Table 4 lists the comparison between the fundamental parameters of NGC 2158 obtained here and those obtained in other studies. Using the inferred values of the photometric color excess of NGC 2158, we found that E(G_BP − G_RP) = 0.590, E(J − H) = 0.160 and E(J − K) = 0.220. The mean color excess E(B − V ) and the intrinsic distance modulus (m − M)₀ are 0.420 ± 0.050mag and 12.540 ± 0.130mag respectively. E(B − V ) and (m − M)₀ are in a good agreement with those estimated by ^{Glushkova et al. (2010)}, ^{Tadross (2001)}; ^{Loktin et al. (2001)}; ^{Dias et al. (2002)} and ^{Bedin et al. (2010)}, see Table 4. The distance of NGC 2158 from the Sun is d⊙=3224±200 200pc, in good agreement with that of ^{Glushkova et al. (2010)}.

^{Cantat-Gaudin et al. (2018)} calculated the distance of 1229 open clusters by using the UPMASK code of probable cluster members with probabilities larger than 50%. ^{Lindegren et al. (2018)} found that the Gaia DR2 parallaxes are affected by a zero-point offset of −0.029 mas. ^{Cantat-Gaudin et al. (2018)} reported in their Table 1 (Vizier database reference J/A+A/618/A93) the distance and its uncertainties d, d+ and d− as 4535.1, 3119.8 and 8301 pc respectively⁴. We estimate a distance of d⊙=3224±200 pc using the best fit of the isochrones in the multi-color magnitude diagrams of optical and 2MASS data (Figure 7), in a good agreement with the smaller value of d+ = 3119.8 pc of ^{Cantat-Gaudin et al. (2018)}.

The published parallax of the eclipsing binary [NBN2015]78 in Gaia DR2 is ϖ=2.202±0.409 mas, a relative uncertainty smaller than 20%. We determine the distance of this binary as dEB(Gaia) = 454.2 pc using the inverse of the Gaia parallax. This distance to [NBN2015]78 is much smaller than the mean distance to NGC 2158 d⊙=3224±200200 pc, and therefore [NBN2015]78 is a foreground field star and not a cluster member.

Following ^{Tadross (2011)}we assume that the Sun lies at distance 8.2 kpc from the Galactic center, and find, for the following parameters: the distance of NGC 2158 from the Galactic center (Rgc) , the projected distances on the Galactic plane from the Sun ( X⊙ and Y⊙ ) and the distance from the Galactic plane (Z⊙) the values Rgc=11407±394 pc, X⊙=−3201±278 pc, Y⊙=−372±32 pc and Z⊙=101±009 pc, respectively. These values for Rgc , X⊙ , Y⊙ and Z⊙ are fully consistent with those of ^{Cantat-Gaudin et al. (2018)} using d_+=3119.8 d+=3119.8  pc, see  Table4.

3.1. Data Analysis

The photometric analysis of [NBN2015]78 was carried out using the data obtained by ^{Nardiello et al. (2015)}, using the SBIG STL-11000M camera attached to the Asiago 67/92 cm Schmidt Telescope, available at the Astronomical Observatory of Padova (OAPD-Osservatorio Astronomico di Padova). The V , R data were collected from a number of images gathered over the three seasons. Table 5 presents the total number of observations in each filter as obtained from ^{Nardiello et al. (2015)} and Figure 8 shows the histogram of the observations in each filter.

Fig. 8 Histogram of the number of images collected per night during the three campaigns. The histograms refer to observations in the V and R filters. The color figure can be viewed online. 

The period and the times of primary (I) and secondary (II) minima of [NBN2015]78 were obtained for the V and R filters (see Table 6). Using the method of ^{Kwee and Woerden (1956)}, we infer that the new ephemeris for the system is given by

Fig. 9 The observed phase diagram of [NBN2015]78 in the V and R filters. The color figure can be viewed online. 

3.2. Light Curve Analysis

The light curve analysis of the system [NBN2015]78 is performed using the light curves in both V and R bands and is carried out using the program of Wilson-Devinney (W-D, ^{Nelson 2009}; ^{Wilson and Devinney 1971}). We assumed gravity darkening and bolometric albedo exponents appropriate for the convective envelopes of late-type stars (T_eff < 7500K). We adopted g₁ = g₂ = 0.32 (^{Lucy 1967)} and A₁ = A₂ = 0.5 (^{Rucinski 1969)}. The limb-darkening coefficients were interpolated from the tables of van ^{Hamme (1993)} using the logarithmic law and the values X1 = X2 and Y1 = Y2 for the over-contact mode. The effective temperature of the primary star (T1) is adopted as 4925 K (^{Flower 1996)}, corresponding to a B−V color index of 0.959 from the combined light of both components (^{Nardiello et al. 2015)}. The adjusted parameters are the orbital inclination (i), the mean temperature of the secondary star (T2), the potential of the two components Ω = Ω₁ = Ω₂, the mass ratio (q), and the luminosity of the primary star (L1). We used Mode 3 in the Wilson-Devinney (W-D) program (over-contact mode not in thermal contact). The orbital solutions using both V and R light curves are performed and the accepted parameters are listed in Table 7. Figure 10 shows the best match between the model and the observed light curves. The solution shows that the primary component is the more massive and the hotter one, with a difference in effective temperature equal to 145 K. According to this solution, the two components of [NBN2015]78 are of spectral types K1 and K2 (^{Covey 2007)}.

Fig. 10 V , R filters for the system [NBN2015]78 as fitted with the synthetic model (sold line). The color figure can be viewed online. 

Figure 11 shows the geometric configuration of [NBN2015]78 at different phases and the corresponding Roche lobe geometry. The absolute physical parameters of the system are calculated using the empirical relations of ^{Harmanec (1988)}. The mass of the primary component M1=0.817M⊙, while the mass of the secondary component is directly calculated from the estimated mass ratio of the system (q=M2/M1) as M2=0.214M⊙. The radii of the two components R1(R⊙); R2(R⊙) and the bolometric magnitudes M1bol and M2bol also calculated and listed in Table 8.

Fig. 11 Geometric configuration at different phases of [NBN2015]78. The color figure can be viewed online. 

3.3. Evolutionary Status of [NBN2015]78

Using the calculated physical parameters listed in Table 8 we investigated the current evolutionary status of the system. Figure 12 shows the components of the system in the mass-luminosity (M-L, left panel), mass-radius (M-R, middle panel) and masstemperature (M-T, right panel) relations, along with evolutionary tracks computed by ^{Mowlavi et al. (2012)} for both zero age main sequence (ZAMS) and terminal age main sequence (TAMS) with solar metallicity ( Z⊙=0.014)

Fig. 12 The position for the components of the system on the mass-luminosity (left), mass-radius (middle) and empirical M − Teff relation (right) for intermediate mass stars. The color figure can be viewed online. 

The right panel also shows the location of the components on the empirical M − Teff relation for intermediate mass stars from ^{Malkov (2007)}. As it is clear from Figure 12 (left and middle panels) the primary component is lying on the ZAMS track while the secondary component is an evolved star.