1.1. NGC 1193

In 1786 William Herschel discovered the open cluster NGC 1193 (α = 03^h 05^m 56^s.64, δ = +44° 22' 58’’.80, l = 146°.8143, b = -12°.1624), located in the constellation of Perseus (^{Dreyer 1888}). Together with an angular size of 2^′, NGC 1193 has a dense central stellar concentration and is classified as II3m (^{Ruprecht 1966}). ^{King (1962)} reported that NGC 1193 is likely to be old. In the study of ^{Janes & Adler (1982)}, NGC 1193 was identified as a dense, poorly studied open cluster with an angular diameter of 2’. ^{Kaluzny (1988)} presented the first CCD BV photometric study of NGC 1193, identifying five possible blue straggler stars in the cluster. By BV isochrone fitting to the color magnitude diagram (CMD), they determined the color excess and distance to be 0.12 ≤ E(B - V) ≤ 0.23 mag and 4.2 ≤ d ≤ 4.9 kpc, respectively. Additionally the metallicity, distance module, and age of the cluster were adopted as Z = 0.01, (m - M)_V = 13.8 mag, and t = 8 × 10⁹ years. Through investigation of the cluster’s color-magnitude diagram, ^{Kaluzny (1988)} indicated that subgiant branch stars are more populous than red giant branch stars. Utilizing spectroscopic observations, ^{Friel, Liu, & Janes (1989)} calculated the first radial velocity estimate for the cluster as 〈V_r〉 = -82 km s⁻¹. ^{Friel & Janes (1993)} performed medium resolution spectroscopic analyses and estimated the cluster metallicity as [Fe/H] = -0.50 ± 0.18 dex from four giant members. They also calculated the radial velocities of stars whose values lie within -64 ≤ V_r ≤ 103 km s⁻¹. ^{Tadross (2005)} used photometric data of ^{Kaluzny (1988)} and astrometric data from the USNO-B1.0 catalog of ^{Monet et al. (2003)} to determine the cluster’s color excess E(B - V) = 0.10 ± 0.06, distance modulus µ = 13.90 ± 0.10 mag, distance d = 5.25 ± 0.24 kpc, age t = 8 Gyr, and metallicity as Z = 0.008. Moreover, ^{Tadross (2005)} analysed the cluster with regard to the radial profile of ^{van den Bergh & Sher (1960)} and estimated the core radius as r_c = 1'.4 and the limiting radius as r_lim = 6'.5. ^{Kyeong et al. (2008)} applied a fitting procedure of the theoretical isochrones of ^{Bertelli et al. (1994)} to the color-magnitude diagrams based on CCD UBVI photometric data of NGC 1193. They calculated the color excess as E(B − V) = 0.19 ± 0.04 mag, the metallicity [Fe/H] = −0.45 ± 0.12 dex, the true distance module (m − M_V)₀ = 13.30 ± 0.15 mag, and the cluster age as log t(yr) = 9.7 ± 0.1.

The Gaia mission (^{Gaia collaboration et al. 2016}) has led to substantial improvements in the quality and precision of astrometric, photometric, and spectroscopic data. Gaia has provided precise astrometric, photometric, and spectroscopic data of nearly 1.8 billion stars. ^{Cantat-Gaudin et al. (2018)} identified 215 most likely cluster members, using astrometric and photometric data of stars across the locality of NGC 1193. In the study, they determined the mean proper motion of the cluster as (µ_α cos δ, µ_δ) = (-0.125 ± 0.023, -0.329 ± 0.019) mas yr⁻¹ and the trigonometric parallax as ϖ = 0.159 ± 0.009 mas. ^{Soubiran et al. (2018)} used the second Gaia data release (^{Gaia DR2; Gaia collaboration et al. 2018}) spectroscopy to identify a radial velocity measurement for one member star of NGC 1193, calculating its radial velocity as 〈V_r〉 = -83.24 ± 0.51 km s⁻¹. In addition, ^{Carrera et al. (2019)} determined the mean radial velocity of the cluster as 〈V_r〉 = -85.16 km s⁻¹, based on APOGEE spectroscopic data for two member stars of the cluster. ^{Donor et al. (2020)} analysed three cluster member stars using APOGEE DR16 spectroscopic data and calculated the radial velocity and metallicity of the NGC 1193 as 〈V_r〉 = -84.7 ± 0.2 km s⁻¹ and [Fe/H] = -0.34 ± 0.01 dex, respectively. Using Gaia DR2 data, they determined the mean proper motion components of the cluster as (µ_α cosδ, µ_δ) = (−0.22 ± 0.10, −0.36 ± 0.07) mas yr⁻¹.

1.2. NGC 1798

The open cluster NGC 1798 (α = 05^h11^m39^s. 36, δ = +47° 41' 27’’. 60, l = 60°. 7043, b = +04°.8500) was discovered in 1885 by Edward Barnard, located in the Auriga constellation (^{Dreyer 1888}). With an angular size of about 5 arcmin, this cluster is classified as II2m with a central dense stellar concentration (^{Ruprecht 1966}). Examination of the cluster’s CMD reveals that the regions of the main sequence and red clump (RC) stars are more distinct than the red giant branch (RGB). Based on this morphological feature, ^{Janes & Phelps (1994)} gave the age of NGC 1798 as 1.5 Gyr and the distance as 3.44 kpc.

The first CCD UBVI photometric observations of the NGC 1798 were made by ^{Park & Lee (1999)}. The angular diameter of the cluster was given as 8.3 arcmin (10.2 pc), the color excess E(B - V) = 0.51 ± 0.04 magnitude, the distance d = 4.2 ± 0.3 kpc, the metallicity [Fe/H] = 0.47 ± 0.15 dex, and the age t = 1.4 ± 0.2 Gyr. ^{Lata et al. (2002)} used the data of ^{Park & Lee (1999)} to determine the absolute magnitude and color indices for the I band as I(M_V) = −4.86, I(U − V)₀ = 0.97, I(B - V )₀ = 0.82, and I(VI)₀ = 1.14 mag. ^{Maciejewski & Niedzielski (2007)} obtained the structural and astrophysical parameters of 42 open clusters with CCD BV photometry. They determined the cluster’s limiting radius r_lim = 9 arcmin, the core radius r_c = 1.3 ± 0.1 arcmin, the central stellar density f₀ = 9.5 ± 0.28 star arcmin⁻², and the background stellar density f_bg = 3.14 ± 0.05 stars per arcmin². These researchers used the isochrones of ^{Bertelli et al. (1994)}, obtaining the color excess of the cluster as E(B − V) = 0.37 ^+0.10_-.09, the distance modulus as (m − M) = 13.90^+0.26_-0.63 mag, the distance as d = 3.55^+0.64_−1.22 kpc, and the age being log t (yr) = 9.2. ^{Ahumada & Lapasset (2007)} examined 1,887 blue straggler star (BSS) candidates in 427 open clusters and identified 24 BSS in the direction of NGC 1798. They indicated that six of these BSS are massive and 18 are low mass stars. ^{Carrera (2012)} calculated the radial velocities of four open clusters including NGC 1798 by analyzing spectroscopic data of their member stars. By measuring Ca II lines, ^{Carrera (2012)} determined the mean radial velocity of NGC 1798 as 〈V_r〉 = 2 ± 10 km s⁻¹. They used six member stars in total, consisting of five RGB stars and one main sequence turn-off star. Oralhan et al. (2015) analyzed CCD UBVRI photometric observations of 20 open clusters and obtained their astrophysical parameters. They determined the reddening, photometric metallicity, distance modulus, distance, and age of the NGC 1798 as E(B − V ) = 0.47 ± 0.07 mag, [Fe/H] = −0.50 ± 0.28 dex, (m − M)₀ = 12.70 ± 0.04 mag, d = 3.47 ± 0.06 kpc, and t = 1.78 ± 0.22 Gyr.

^{Cantat-Gaudin et al. (2020)} used photometric and astrometric data from the Gaia DR2 (Gaia collaboration et al. 2018) to determine astrometric and astrophysical parameters of 2,017 open clusters. They identified 218 member stars in NGC 1798. Considering these members they calculated mean proper-motion components and trigonometric parallaxes of the cluster as (µ_α cos δ, µ_δ) = (0.913 ± 0.011, -0.318 ± 0.010) mas yr⁻¹ and ϖ = 0.178 ± 0.005 mas. ^{Liu & Pang (2019)} used astrometric and photometric data of 78 member stars of NGC 1798 to calculate the mean proper-motion components, trigonometric parallaxes, and age of the cluster as (µ_α cosδ, µ_δ) = (0.903 ± 0.026, −0.400 ± 0.295) mas yr⁻¹, ϖ = 0.241 ± 0.026 mas, and t = 1.7 ± 0.1 Gyr.

A number of studies explored open clusters using ground-based telescopes within the scope of spectroscopic survey programs (^{Gilmore et al. 2012}; ^{Conrad et al. 2014}; ^{Maciejewski & Niedzielski 2007}; ^{Kos et al. 2018}). Within the context of the APOGEE survey, ^{Donor et al. (2018)} utilized spectral observations of 259 cluster member stars in 19 open clusters including NGC 1798 and obtained the radial velocity and different metal abundance values of the stars. Analysing nine member stars in NGC 1798 ^{Donor et al. (2018)} determined the mean radial velocity as 〈V_r〉 = 2 ± 1.7 km s⁻¹ and the iron abundance [Fe/H] = -0.18 ± 0.02 dex. ^{Soubiran et al. (2018)} analysed Gaia DR2 spectroscopic data of four member stars in the cluster and obtained the mean radial velocity as ⟨V_r⟩ = 2.60 ± 0.41 km s⁻¹. ^{Donor et al. (2020)} analysed eight cluster member stars using APOGEE DR16 spectroscopic data and calculated the radial velocity and metallicity of the NGC 1798 as 〈V_r〉 = 2.7 ± 0.8 km s⁻¹ and [Fe/H] = -0.27 ± 0.03 dex, respectively. Using Gaia DR2 data, they determined the mean proper motion components of the cluster as (µ_α cosδ, µ_δ) = (0.83 ± 0.04, −0.31 ± 0.04) mas yr⁻¹.

2.1. CCD UBV Photometric Data

The observations of these two clusters, along with many others, were carried out at the San Pedro Martir Observatory,⁶ as part of an ongoing UBVRI photometric survey of Galactic stellar clusters. The 84-cm (f/15) Ritchey-Chretien telescope was employed in combination with the Mexman filter wheel.

NGC 1193 was observed on 2013-09-19 with the ESOPO CCD detector (a 2048 × 2048 13.5-µm square-pixels E2V CCD42-40 with a gain of 1.65 e⁻/ADU and a readout noise of 3.8 e^−| at the 2 × 2 binning employed, providing an unvignetted field of view of 7.4 9.3 arcmin²). Short and long exposures were taken to properly measure both the bright and faint stars of the fields. Exposure times were 2, 12, 120s for both I and R; 6, 30, 200 for V; 30, 100, 700s for B; and 60 and 1800s for U.

NGC 1798 was observed on 2009-11-01 with the SITE3 detector (a Photometrics 1024 1024 24-µm square-pixels with a gain of 1.3 e⁻/ADU and a readout noise of 6.8 e⁻, giving an unvignetted field of view of 6.8 × 6.8 arcmin²). Exposure times for I and R were 2, 12 and 120s in duration; 6, 30 and 200s for V; 30, 100 and 700s for B; and 60 and 1800s for U.

Landolt’s standard stars (^{Landolt 2009}) were also observed in good sky conditions, at the meridian and at about two air masses, to properly determine the atmospheric extinction coefficients. Flat fields were taken at the beginning and the end of each night and bias images were obtained between cluster observations. Data reduction with point spread function (PSF) photometry was carried out by one of the authors (RM) with the IRAF/DAOPHOT packages (^{Stetson 1987}) and employing the transformation equations recommended, in their Appendix B, by ^{Stetson et al. (2019)}.

3.1. Gaia Astrometric and Photometric Data

The (early) third data release of Gaia (hereafter Gaia EDR3, ^{Gaia collaboration et al. 2021}) provides high quality astrometric and photometric data of nearly 1.5 billion celestial objects. Together with ground-based CCD UBV photometry we took into account Gaia EDR3 astrometric and photometric data to perform astrometric, photometric, and kinematic analyses of NGC 1193 and NGC 1798. We extracted such EDR3 data for all stars within regions of 20 arcmins about the centres of each cluster, using the coordinates given by ^{Cantat-Gaudin et al. (2020)} (α = 03^h05^m56^s. 64, δ = +44° 22' 58’’.80 for NGC 1193 and α = 05^h11^m39^s. 36, δ = +47° 41' 27’’.60 for NGC 1798). Thus we reached 9,141 stars within the magnitude range 7 < G < 23 mag for NGC 1193 and 14,834 stars within 8 < G < 21 mag for NGC 1798, respectively. 20 arcmin field of view optical images for the two clusters are presented in Figure 1. To construct photometric and astrometric catalogues for each cluster, we matched the UBV data to that from the Gaia EDR3 catalogue using stellar equatorial coordinates considering distances less than 5 arcsec. The mean difference in distances between the coordinates of stars in the matched catalogues was ≈ 0.08 arc seconds for both clusters. Both resulting catalogues contain positions (α, δ), UBV observational data (apparent V magnitudes, color indices U − B, B − V), Gaia EDR3 astrometric (µ_α cosδ, µ_δ, ϖ) and photometric data (G, G_BP - G_RP), and membership probabilities (P) as calculated in this study (Table 1). Catalogues of CCD UBV photometric as well as Gaia photometric and astrometric data for all the detected stars in the cluster regions are available electronically for NGC 1193 and NGC 1798⁷. Errors of the UBV and Gaia EDR3 photometric data were adopted as internal errors, being the uncertainties in the determination of the instrumental magnitudes of the stars. We calculated the mean photometric errors separately as functions of V and G intervals, listing the results in Table 2 (on page 338). It can be seen from the table that the mean internal UBV errors reach 0.08 mag for stars brighter than V = 20 mag for both clusters. The mean internal errors of Gaia EDR3 photometry for stars brighter than G = 21 mag reach 0.011 mag for NGC 1193 and 0.007 mag for NGC 1798.

Fig. 1 Identification charts for NGC 1193 (left panel) and NGC 1798 (right panel), taken from the Leicester database and archive service (LEDAS). 

To obtain precise astrophysical parameters, we identified photometric completeness limits for each cluster. Stars fainter that these limits were not included in further analyses. G and V magnitude histograms were constructed to determine the photometric completeness limits for each clusters (see Figure 2). Stellar counts decrease for magnitudes fainter than G = 20 for both NGC 1193 (Figure 2a) and NGC 1798 (Figure 2c). Stellar counts decrease for magnitudes fainter than V = 19 for NGC 1193 (Figure 2b) and NGC 1798 (Figure 2d), indicating that incompleteness (of stellar recovery) has set in. Thus, for both clusters, we adopted these values as the cluster photometric completeness limits.

Fig. 2 Interval G and V -band magnitude histograms of NGC 1193 (a, b) and NGC 1798 (c, d): The red arrows show the faint limiting apparent magnitudes in G and V -bands. The color figure can be viewed online. 

3.2. Structural Parameters of the Clusters

We utilized Radial Density Profile (RDP) analysis to determine the structural parameters of the clusters. First, we specified many concentric rings outwards from the cluster center, using the central coordinates given by ^{Cantat-Gaudin et al. (2020)}. Stellar densities (ρ) were estimated for each ring by dividing the number of stars within the photometric completeness limit (G ≤ 20 mag) in it by the ring area. The resulting RDPs were fitted with ^{King (1962)} models via χ² minimisation, giving estimates for the core, limiting, and effective radii of each cluster. The King (1962) model is described as ρ(r) = f_bg + [f₀/(1 + (r/r_c)²)] where r is the radius from the cluster centre, f_bg the background density, f₀ the central density, and r_c the core radius. See Figure 3 for each cluster’s RDP together with the best fitting ^{King (1962)} model to it. As a result of the fitting procedure, we inferred central stellar density, core radius and background stellar density as f₀ = 166.865 ± 1.573 stars arcmin⁻², r_c = 0.526 ± 0.009 arcmin and f_bg = 5.225 ± 0.124 stars arcmin⁻² for NGC 1193 and f₀ = 53.597 ± 3.789 stars arcmin⁻², r_c = 1.190 ± 0.056 arcmin and f_bg = 11.318 ± 0.321 stars arcmin⁻² for NGC 1798, respectively. At the r = 8 arcmin limiting radius, the stellar density becomes similar to the background density (a grey horizontal line) as seen in Figure 3a (NGC 1193) and Figure 3b (NGC 1798). Therefore, we concluded that the limiting radii for both clusters are r_lim = 8 arcmin. We considered only the stars inside these limiting radii in further analyses.

Fig. 3 Radial density profiles for NGC 1193 (a) and NGC 1798 (b). Errors were derived using the equation of 1 / √N, where N represents the number of stars used in the density estimation. The solid lines represents the optimal ^{King (1962)} profiles. The background density level and its errors are the horizontal grey bands. The King fit uncertainty (1σ) is shown by the red shaded region. The color figure can be viewed online. 

3.2. CMDs and Membership Probabilities of Stars

The membership probabilities (P) of stars located in each of two cluster regions were calculated applying the Unsupervised Photometric Membership Assignment in Stellar Cluster program (^{UPMASK; Krone-Martins & Moitinho 2014}). UPMASK uses k-means clustering, where k is the number of clusters, to detect spatially concentrated groups and identify the most likely cluster members. An integer k-means is not adjusted directly by the user and the best result from the upmask methodology is achieved when the k-means value is within 6 to 25 (^{Krone-Martins & Moitinho 2014}; ^{Cantat-Gaudin et al. 2020}). We applied UPMASK to calculate stellar membership probabilities by considering each star’s five-dimensional astrometric parameters from Gaia EDR3 (^{Gaia collaboration et al. 2021}), which contains equatorial coordinates (α, δ), proper motion components (µ_α cos δ, µ_δ), trigonometric parallaxes (ϖ), and their uncertainties. During application we scaled these five parameters to unit variance and ran 100 iterations for each clusters to assess cluster membership. The membership probability of a star is defined by the frequency of the group in which it is clustered. We reached the best results when k was set to 12 for NGC 1193 and 15 for NGC 1798. We identified as possible cluster members those stars brighter than G = 20 mag with membership probabilities P ≥ 0.5 that we identified as possible members of clusters. This led to 735 possible members for NGC 1193 and 1,536 for NGC 1798. ^{Cantat-Gaudin et al. (2020)} give the number of stars brighter than G = 18 mag with the membership probabilities P > 0.5 as 215 for NGC 1193 and 218 for NGC 1798. The dissimilarity can be explained by lower precision in the astrometric Gaia DR2 data compared to Gaia EDR3, as well as the G magnitude limit of stars used in the analyses. With the release of Gaia EDR3 data, the precision of astrometric and photometric measurements increased with respect to Gaia EDR2 data. For the Gaia EDR3 release the accuracy of trigonometric parallaxes increased by 30 percent and the uncertainties decreased by nearly 40%, the proper motion accuracy increased by a factor of 2 and the associated uncertainties improved by a factor 2.5. Moreover, the precision of photometric data and celestial positions are better in terms of homogeneity (^{Gaia collaboration et al. 2021}).

To take into consideration the impact of binary stars in the main-sequences of the studied clusters, we plotted the V × (B − V) CMDs of the stars within the cluster limiting radii (r_lim) which we had obtained for the clusters and then fitted the Zero Age Main-Sequence (ZAMS) of ^{Sung et al. (2013)} to these diagrams. The ZAMS fitting was by eye according to the stars with the membership probability P ≥ 0.5 and shifted 0.75 mag towards brighter magnitudes in order to account for the most likely cluster binary stars (4a and c). During the ZAMS fitting we made sure for each cluster that the main-sequence, turn-off, and giant stars with membership probabilities P ≥ 0.5 were selected. The process resulted in 181 likely member stars for NGC 1193 and 161 for NGC 1798 which lie between the fitted ZAMS curves and are located inside the r_lim radii. We used these stars to determine astrophysical parameters of the two clusters. Figure 4 shows the V × (B − V) CMDs with the best fitted ZAMS (Figures 4a and c) and G(G_BP - G_RP) CMDs (Figures 4b and d) with the background and most likely member stars. Figure 5 presents histograms of the number of stars located through the two cluster fields versus their membership probabilities. Vector-Point Diagrams (VPDs) were plotted for the stars within the limiting radii and are shown as Figure 6. It can be seen from the figure that NGC 1193 (Figure 6a) and NGC 1798 (Figure 6b) are affected by field stars but with the membership selection criteria, the ‘most likely’ cluster stars (shown as the colorscaled points in Figure 6) can be separated from field stars (grey dots in Figure 6). The mean proper motion components of the most likely cluster members are (µ_α cosδ, µ_δ) =(−0.207 ± 0.009,−0.431 ± 0.008) for NGC 1193 and (µ_α cosδ, µ_δ) = (0.793 ± 0.006, −0.373 ± 0.005) mas yr⁻¹ for NGC 1798. Moreover, using these members we obtained mean trigonometric parallaxes of NGC 1193 and NGC 1798 as ϖ_Gaia = 0.191±0.157 mas and ϖ_Gaia = 0.203 ± 0.099 mas, respectively.

Fig. 4 V × (B − V) and G × (GBP − GRP) CMDs of NGC 1193 (a, b) and NGC 1798 (c, d). The blue dot-dashed lines represent the ZAMS (Sung et al. 2013) including the binary star effect. The membership probabilities of stars that lie within the fitted ZAMS are shown with different colors according to the color scales shown to the right of the figure. These member stars are located within r_lim = 8 arcmin of the cluster centres calculated for NGC 1193 and NGC 1798. Grey dots indicate low probability members (P < 0.5), or field stars (P = 0). The color figure can be viewed online. 

Fig. 5 Histograms of the membership probabilities for NGC 1193 (a) and NGC 1798 (b). The red colored shading denotes the stars that lie within the main-sequence band and effective cluster radii (r_lim ≤ 8’), while the white colored bars indicate the membership probabilities of all stars in each cluster’s direction. The color figure can be viewed online. 

Fig. 6 VPDs of NGC 1193 (a) and NGC 1798 (b). Colored dots identify the membership probabilities of the most likely cluster members according to the color scale shown on the right. The zoomed box in the panels represents the region of condensation for both clusters in the VPD. Dashed lines are the intersection of the mean proper motion values. The color figure can be viewed online. 

4.1. Reddening

The E(U - B) and E(B - V ) color excesses for NGC 1193 and NGC 1798 were derived using (U - B) (B - V) TCDs. We selected the mainsequence stars for which simultaneous U, B, and V magnitudes were available, as well as with membership probabilities P ≥ 0.5. As shown in Figure 7, we constructed TCDs for these stars and compared their positions by fitting the solar metallicity dereddened ZAMS of Sung et al. (2013). The ZAMS was fitted according to the equation E(U - B) = 0.72 × E(B - V) + 0.05 × E(B - V)² (^{Garcia et al. 1988}) by applying χ² optimisation with steps of 0.001 mag. The best solutions for E(B - V) and E(U - B) values are those corresponding to the minimum χ², being E(B - V) = 0.150 ± 0.037 mag for NGC 1193 and E(B - V) = 0.505 ± 0.100 mag for NGC 1798. The errors of color excesses are determined as ±1σ deviations, and are presented as the green lines in Figure 7. When we compared the reddening estimated for NGC 1193, we concluded that it is in a good agreement within the errors with the values (0.10 ≤ E(B - V) ≤ 0.19 mag) given by different authors (^{Kaluzny 1988}; ^{Tadross 2005}; ^{Kyeong et al. 2008}). For NGC 1798, our finding result is compatible with the values given by ^{Park & Lee (1999}, E(B − V) = 0.51 ± 0.04 mag) and ^{Oralhan et al. (2015}, E(B − V) = 0.47 ± 0.07 mag).

Fig. 7 Two-color diagrams of the most probable member main-sequence stars in the regions of NGC 1193 (a) and NGC 1798 (b). Red dashed and green solid curves represent the reddened ZAMS given by Sung et al. (2013) and ±1σ standard deviations, respectively. The color figure can be viewed online. 

4.2. Metallicities

The determination of photometric metallicities of the two clusters employed the method given by ^{Karaali et al. (2003a}, ^b, ²⁰¹¹). This method is based on F and G type main-sequence stars and their UV-excesses as well as on stars whose color index range correspond to 0.3 ≤ (B - V)₀ ≤ 0.6 mag (^{Eker et al. 2018}, ²⁰²⁰). We selected F-G type main-sequence stars within the range 0.3 ≤ (B − V)₀ ≤ 0.6 mag after calculating the intrinsic (B − V )₀ and (U − B)₀ colors of the most likely cluster member (P ≥ 0.5) stars. To determine the difference between the (U B)₀ color indices of cluster stars and the Hyades main sequence which corresponds to the same (B − V)₀ color indices, we constructed (U - B)₀ (B - V)₀ TCDs. This difference between cluster and Hyades stars is defined as the UV-excess which is expressed by the equation of δ = (U - B)_0,H - (U - B)_0,S, where H and S denote the Hyades and cluster stars respectively, which implies the same (B − V)₀ color indices. By calibrating (B − V)₀ of stars to (B − V)₀ = 0.6 mag (i.e., δ_0.6) we normalised the UV excess and plotted the histogram of normalised δ_0.6 values. To calculate the mean δ_0.6, we fitted a Gaussian to the distribution. Taking into account the Gaussian peak, the photometric metallicities of the studied clusters are obtained from the equation given by ^{Karaali et al. (2011)}:

We identified 12 and 7 F-G type main-sequence stars to calculate the photometric metallicity of NGC 1193 and NGC 1798, respectively. TCDs and the distributions of normalised δ_0.6 UV excesses for two clusters are shown in Figure 8. The calculated mean δ_0.6 values of NGC 1193 and NGC 1798 are 0.085±0.010 mag and 0.068±0.011 mag, respectively. The photometric metallicity [Fe/H] value for NGC 1193 is [Fe/H] = −0.30 ± 0.06 dex and for NGC 1798 it is [Fe/H] = −0.20 ± 0.07 dex, which correspond to their peak values in the δ_0.6 distribution.

The [Fe/H] metallicities were transformed to the mass fraction Z to derive ages of the clusters. For this, the analytic equations of Bovy^{8, 9} for parsec (^{Bressan et al. 2012}) models were used, namely:

and

z and z_x are the elements heavier than helium and the intermediate operation function, respectively. z_⊙ is the solar metallicity which was adopted as 0.0152 (^{Bressan et al. 2012}). We calculated z = 0.008 for NGC 1193 and z = 0.010 for NGC 1798.

Fig. 8 Two-color diagrams (upper panels) and the distributions of normalised δ0.6 (lower panels) for NGC 1193 (a) and NGC 1798 (b). The solid blue lines in the upper and lower panels represent the main-sequence of Hyades and Gaussian models which were fitted to the histograms, respectively. The color figure can be viewed online. 

Many authors obtained spectroscopic metallicities of NGC 1193 and NGC 1798 based on groundbased observations, as listed in Table 3. Photometric metallicities calculated in this study are well supported by the spectroscopic studies presented in the literature. We conclude that our metallicity findings are reliable. Thus, we adopted our results for the determination of distance moduli and age.

4.3. Distance Moduli and Age Estimation

We used parsec isochrones (^{Bressan et al. 2012}), which contain UBV filters as well as Gaia passbands, to obtain the distance moduli and ages of the studied clusters simultaneously. To do this, we selected the parsec models considering the mass fractions (z) estimated for each cluster and compared them to the V × (U −B), V × (B−V), and G × (G_BP− G_RP) CMDs according to member stars (P ≥ 0.5). Selected isochrones were fitted to CMDs visually by attaching importance to ‘most likely’ member stars which make up the main-sequence, turn-off and giant regions of each cluster. During the fitting process of parsec models to the UBV data, we used the E(B − V) values derived above by this study, while for the Gaia EDR3 data we considered the equation of E(G_BP - G_RP) = 1.41 × E(B - V) (^{Sun et al. 2021}). We obtained the error of the distance moduli and distances using the relation given by ^{Carraro et al. (2017)}. We fitted two more isochrones to estimate age uncertainties considering the spread of the most likely member stars in the turn-off and sub-giant regions of the cluster. The ages of such selected isochrones give the higher and lower acceptable values for the estimated cluster ages. The best fit with z = 0.008 gave the distance moduli and age of NGC 1193 as µ = 14.191 ± 0.149 mag and t = 4.6 ± 1.0 Gyr. For NGC 1798, the best fit of z = 0.010 gave these values as µ = 14.808 ± 0.332 mag and t = 1.3 ± 0.2 Gyr, respectively. The distances of the clusters corresponding to the estimated distance moduli are also d_iso = 5562 ± 381 pc for NGC 1193 and d_iso = 4451 ± 728 pc for NGC 1798. The V × (U - B), V × (B - V), and G ×(G_BP - G_RP) CMDs

with the best fit isochrones and associated errors are shown in Figure 9.

Fig. 9 CMDs for the NGC 1193 (Panels a, b, and c) and NGC 1798 (Panels d, e, and f). The differently colored dots represent the membership probabilities according to the color scales shown on the right side of the diagrams. Grey dots indicate low probability members (P < 0.5), or field stars (P = 0). The blue lines show the PARSEC isochrones, while the shaded areas surrounding these lines are their associated errors. The color figure can be viewed online. 

The isochrone-based distance for NGC 1193 as estimated by this study is compatible with the result given by ^{Tadross (2005}, d = 5.25 ± 0.24 kpc). As well, the estimated age of the cluster is in a good agreement with the value of ^{Kyeong et al. (2008}, t = 5.0 ± 1.3 Gyr). For NGC 1798, the derived distance matches well within the errors with the result of ^{Park & Lee (1999}, d = 4.2 ± 0.3 kpc). The age of the cluster is coherent with the findings given by ^{Park & Lee (1999}, t = 1.4 ± 0.2 Gyr) and ^{Maciejewski & Niedzielski (2007}, t = 1.6 Gyr).

Applying the linear equation of α (mas) = 1000/d (pc), we converted isochrone distances to trigonometric parallaxes for the two clusters. This indicated that the parallax distances of NGC 1193 and NGC 1798 are ϖ_iso = 0.180 ± 0.012 mas and ϖ_iso = 0.225 ± 0.037 mas, respectively. It is concluded that these values are in good agreement with the Gaia EDR3 trigonometric parallax distances for both clusters.