arXiv subjects

R. Canbay

Publications and source records attributed to R. Canbay.

16 recordsLinked to original sources

arXiv2026

Precise physical and kinematical parameters of massive eclipsing binaries in the Small Magellanic Cloud

The formation of massive stars serves an important field of study, requiring research into whether these stars need to form together within clusters of low-mass stars or if they are able to form independently. Currently, there has been an increase in the number of massive stars found far from clusters in the Milky Way (MW) and Small Magellanic Cloud (SMC). The existence of isolation and composition in binary systems is an uncommon phenomenon among massive stars, and their rare occurrence presents valuable insights that are regarded as evidence in this field. This research looks for ways to increase the catalogue of identified massive stars in eclipsing binaries, particularly those that show significant indications of isolation. For the definition of the first step, we are analysing the eight double-lined spectroscopic binaries in the SMC, as detailed in the OGLE variable star catalogues, with the goal of identifying their astrophysical properties. Later, we examine the expected locations and characteristics of binary stars thought to be part of a cluster, using absolute parameters and a cross-validation approach to determine the connection between kinematic analysis and these parameters. If the kinematic connection with the cluster and the physical parameters match precisely with the cluster's isochrone, such stars could act as reliable standard candles for determining the cluster's distance and age.

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arXiv2026

Tracing the Galactic Disk with Gaia DR3: A Deep Study of Berkeley 17, 18, and 39 Open Star Clusters

We report a detailed investigation of three intermediate-to-old age open clusters, Berkeley 17, Berkeley 18, and Berkeley 39, utilizing precise astrometric and photometric data from Gaia DR3. Cluster membership was robustly determined through a probabilistic proper-motion analysis, yielding statistically significant samples of 600, 1042, and 907 stars, respectively. From the mean parallaxes of these members, we determine astrometric distances ranging from approximately 3.40 kpc for Berkeley 17 to 5.80 kpc for Berkeley 18. Isochrone fitting applied to the decontaminated color-magnitude diagrams constrains the cluster ages to 9.12 +/- 1.00 Gyr, 3.36 +/- 0.50 Gyr, and 5.10 +/- 0.50 Gyr, respectively. Interstellar reddening spans a wide range, from E(B-V) = 0.17 mag in Berkeley 39 to 0.58 mag in Berkeley 17. Structural parameters derived from King model fits to the radial density profiles, combined with mass function analyses, indicate that the clusters are dynamically relaxed systems with mass distributions broadly consistent with the canonical Salpeter slope. Our kinematic analysis reveals that Berkeley 17, Berkeley 18, and Berkeley 39 are part of the outer disk population.

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arXiv2025

Exploring the Structure and Evolution of Four Young Open Clusters Near the Galactic Mid-plane via Gaia DR3

We present a comprehensive analysis of four young open clusters, NGC 663, NGC 2301, NGC 2384, and NGC 7510, utilizing high-precision astrometric and photometric data from Gaia DR3. Cluster membership was determined using the UPMASK algorithm, resulting in probable member counts ranging from 337 to 1498 across the clusters. Bayesian MCMC isochrone fitting yielded cluster ages in the range $\log t \sim 7.0$ -$8.15$, with uncertainties of $\sim 0.11$-$0.18$. Reddening values ranged from $E(B-V)=0.093$ mag in NGC 2301 to $1.24$ mag in NGC 7510, consistent with their positions near the Galactic plane. The stellar mass function slopes ($\alpha \approx 2.00$-$2.26$) closely match the Salpeter IMF, with total stellar masses spanning nearly an order of magnitude, from $\sim 486\,M_\odot$ in NGC 2301 to $\sim 3584\,M_\odot$ in NGC 663. Dynamical relaxation times indicate that only NGC 2301 ($\tau \approx 10.00$) and NGC 2384b ($\tau \approx 2.03$) are dynamically relaxed; the others remain dynamically evolving. Orbital integrations in the MWPotential2014 model reveal nearly circular Galactic orbits with very low eccentricities ($e \approx 0.003$-$0.014$) and small vertical distances ($Z_{\rm max} < 0.142$ kpc), confirming their confinement to the thin disk. SED and kinematic analysis show that NGC 2384a and NGC 2384b are separated by $\sim 0.55$ kpc, indicating an optical pair.

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arXiv2025

A Comprehensive Study of Czernik 41 and NGC 1342 Using CCD UBV and Gaia DR3 Data

In this study, the structural, astrophysical, kinematic, and Galactic orbital parameters of the open clusters Czernik 41 and NGC 1342, as well as their dynamical evolution, are investigated using CCD UBV photometry and Gaia data. By applying the UPMASK algorithm to Gaia astrometric data for the estimation of cluster membership probabilities, we have determined that 382 stars in Czernik 41 and 111 stars in NGC 1342 exhibit the highest statistical likelihood of being cluster members. Fundamental parameters (including reddening, metallicity, distance, and age) were derived using both classical methods, where parameters are determined separately, and Markov Chain Monte Carlo (MCMC) methods, where parameters are estimated simultaneously. The results obtained from both approaches are in agreement, confirming the reliability of the derived parameters and demonstrating their robustness against potential degeneracies. The distances to Czernik 41 and NGC 1342 were determined as 2485$\pm$151 pc and 645$\pm$42 pc, respectively, while their ages were estimated to be 69$\pm$10 Myr and 1000$\pm$50 Myr. The metallicity values ([Fe/H]) were found to be 0.07$\pm$0.09 dex for Czernik 41 and -0.14$\pm$0.07 dex for NGC 1342. The stellar mass functions for both clusters were derived, yielding slopes of $\Gamma$=1.67$\pm$0.23 for Czernik 41 and $\Gamma$ =1.56$\pm$0.41 for NGC 1342. Kinematic orbit analysis indicates that Czernik 41 originated within the Solar circle, whereas NGC 1342 formed outside it.

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arXiv2025

A Multi-Data Approach to Open Clusters: Roslund 3 and Ruprecht 174 in CCD UBV and Gaia DR3 Context

A detailed analysis of the structural, astrophysical, kinematic, and dynamical properties of the open clusters Roslund 3 and Ruprecht1 74 is carried out using CCD UBV photometry in conjunction with astrometric and photometric data from Gaia DR3. Membership probabilities were computed via the UPMASK algorithm applied to Gaia proper motions and trigonometric parallaxes, leading to the identification of 198 likely members for Roslund 3 and 397 for Ruprecht 174. Astrophysical parameters were derived using both the classical approach, where parameters are independently determined, and a MCMC technique, which estimates them simultaneously. The agreement between the results from both methods confirms their reliability and highlights the robustness of the classical method. The reddening values were determined as $E(B-V)=0.410\pm 0.046$ mag for Roslund 3 and $E(B-V)=0.615\pm 0.042$ mag for Ruprecht 174. The estimated distances are $d=1687 \pm 121$ pc for Roslund 3 and $d=2385 \pm 163$ pc for Ruprecht 174. Both clusters exhibit metallicities close to the solar value, with [Fe/H] = $0.030 \pm 0.065$ dex for Roslund 3 and [Fe/H] = $0.041 \pm 0.064$ dex for Ruprecht 174. The corresponding ages were found to be $\tau=60\pm 6$ and $\tau=520\pm 50$ Myr, respectively. The present day mass function slopes were found to be $1.18 \pm 0.13$ for Roslund 3 and $1.53 \pm 0.30$ for Ruprecht 174, consistent with the canonical Salpeter value within uncertainties. Galactic orbital analyses indicate that both clusters are thin-disk members confined within the Solar circle. Additionally, relaxation times and spatial distributions of stars suggest that both clusters have reached dynamical relaxation and exhibit clear signs of mass segregation.

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arXiv2025

Distribution of Cataclysmic Variables in our Galaxy and Their Position in the HR Diagram in the Gaia Era

In this study, the distances of stellar systems classified as cataclysmic variables in the literature were determined by using the distance compiled from Bailer-Jones et al. (2021). The spatial distributions of cataclysmic variables in the heliocentric Galactic coordinate system are obtained and their positions in the Hertzsprung-Russell (HR) diagram constructed from Gaia colors are discussed.

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arXiv2024

Kinematics of Cataclysmic Variables in the Solar Neighborhood in the Gaia Era

Using high-precision astrometric data from Gaia DR3 and updated systemic velocities from the literature, kinematical properties of cataclysmic variables (CVs) were investigated. By constraining the data according to the total space velocity error and Galactic population class, a reliable sample of data was obtained. Non-magnetic CVs located in the thin disk have been found to have a total space velocity dispersion of $\sigma_{\nu} = 46.33\pm4.23$ km s$^{-1}$, indicating that the thin disk CVs with a mean kinematical age of $\tau = 3.95\pm0.75$ Gyr are much younger than the local thin disk of the Galaxy with $\tau\sim$6-9 Gyr. Total space velocity dispersions of non-magnetic CVs belonging to the thin disk component of the Galaxy were found to be $\sigma_{\nu}=47.67\pm3.94$ and $\sigma_{\nu}=44.43\pm4.33$ km s$^{-1}$ for the systems below and above the orbital period gap, respectively, corresponding to kinematical ages of $\tau=4.19\pm0.71$ and $\tau=3.61\pm0.74$ Gyr. $\gamma$ velocity dispersions of the thin disk CVs below and above the gap were obtained $\sigma_{\gamma} = 27.52\pm2.28$ and $\sigma_{\gamma} = 25.65\pm2.44$ km s$^{-1}$, respectively. This study also shows that the orbital period is decreasing with increasing age, as expected from the standard theory. The age-orbital period relation for non-magnetic thin disk CVs was obtained as $dP/dt=-2.09\pm0.22\times10^{-5}$ sec yr$^{-1}$. However, a significant difference could not be found between the $\gamma$ velocity dispersions of the systems below and above the gap, which were calculated to be $\sigma_{\gamma} = 27.52\pm2.28$ and $\sigma_{\gamma} = 25.65\pm2.44$ km s$^{-1}$, respectively.

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arXiv2024

Unveiling the Origins and Dynamics of the Hierarchical Triple Star System CN Lyn

In this study we present a detailed analysis of CN Lyn, an overlooked triple star system, by combining spectroscopic data from the literature, photometric \textit{TESS} data, and kinematic techniques. We updated the fundamental parameters of the known eclipsing components in the system with high precision. The chemical composition of both eclipsing components (Aab) and the third component (B) in the system were calculated with great accuracy. According to our analysis the mass, radius, and metallicity of the eclipsing components are $1.166_{-0.012}^{+0.013}\,M_\odot$, $1.786_{-0.014}^{+0.013}\,R_\odot$, and $-0.78_{-0.02}^{+0.02}$ dex for Aa and $1.143_{-0.012}^{+0.013}\,M_\odot$, $1.651_{-0.013}^{+0.014}\,R_\odot$, and $-0.55_{-0.02}^{+0.03}$ dex for Ab. The pair's age is $3.89_{-0.10}^{+0.10}$ Gyr. The mass, radius, metallicity, and age for B are $0.85_{-0.23}^{+0.23}\,M_\odot$, $1.436_{-0.023}^{+0.026}\,R_\odot$, $-1.83_{-0.11}^{+0.09}$ dex, and $12.5_{-2.5}^{+2.5}$ Gyr, respectively. It is also found that the triple system (AabB) satisfies the stability criteria for the hierarchical triple system. Kinematic and Galactic orbital parameters of CN Lyn were obtained from the astrometric and spectroscopic data of the system. Dynamical orbital analyses, taking into account the ages of the component stars in the central binary system (A) show that the CN Lyn originated at the metal-poor edge of the Galactic disk. The third component of the system was found to be a member of the halo population in terms of age, $\alpha$ elements and metal abundance. Given the different chemical abundances and age of B compared to A, this suggests that the third component was captured by the central system in a region with weak gravitational interactions far beyond the Galactic disc.

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arXiv2024

Transformation relations for UBV photometric system of 1m telescope at the T\"{U}B\.{I}TAK National Observatory

UBV CCD observations of standard stars selected from Landolt (2009, 2013) were performed using the 1-meter telescope (T100) of the T\"{U}B\.{I}TAK National Observatory equipped with a back-illuminated and UV enhanced CCD camera and Bessell UBV filters. Observations span a long time from the years 2012 to 2024, 50 photometric nights in total. Photometric measurements were used to find the standard transformation relations of the T100 photometric system. The atmospheric extinction coefficients, zero points and transformation coefficients of each night were determined. It could not be found time dependence of the secondary extinction coefficients. However, it was determined that the primary extinction coefficients decreased until the year 2019 and increased after that year. It could not be found a strong seasonal variation of the extinction coefficients. Small differences in seasonal median values of them were used to attempt to find the atmospheric extinction sources. We found calculated minus catalogue values for each standard star, $\Delta(U-B)$, $\Delta(B-V)$ and $\Delta V$. Means and standard deviations of $\Delta(U-B)$, $\Delta(B-V)$ and $\Delta V$ were estimated to be 1.4$\pm$76, 1.9$\pm$18 and 0.0$\pm$36 mmag, respectively. We found that our data well matched Landolt's standards for $V$ and $B-V$, i.e. there are no systematic differences. However, there are systematic differences for $U-B$ between the two photometric systems, which is probably originated from the quantum efficiency differences of the detectors used in the photometric systems, although the median differences are relatively small ($|\Delta(U-B)|$< 50 mmag) for stars with $-0.5<U-B~{(\rm mag)} <1.6$ and $0.2<B-V~{(\rm mag)} <1.8$. As an overall result, we conclude that the transformation relations found in this study can be used for standardized photometry with the T100 photometric system.

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arXiv2024

Updated Absolute Parameters and Kinematics of IS CMa

Eclipsing binary systems are significant objects for astrophysics in that direct observations can determine the fundamental parameters of stars. In this study, we determined precisely the fundamental parameters of the binary component stars obtained by simultaneous analysis of radial velocities and the {\it TESS} light curve using the Wilson and Devinney code. Following the analysis, the masses and radii of the primary and secondary components were determined as $M_{1}= 1.58\pm 0.01M_\odot$, $M_{2}= 0.48\pm0.02M_\odot$, and $R_{1}=1.93\pm 0.01R_\odot$, $R_{2}= 1.14\pm 0.01 R_\odot$, respectively. Furthermore, the distance of IS CMa is calculated as $92.7\pm6.5$ pc. On the basis of the analysis of the mid-eclipse times, it was found that the variation in the orbital period is represented by an upward parabola. It has an increasing rate of $dP/dt$ = 1.09 $\times$ 10$^{-7}$ day yr$^{-1}$. Using PARSEC stellar evolutionary tracks and isochrones with solar metallicity were estimated the age of IS CMa as $1.3\pm0.1$ Gyr. Kinematic and Galactic orbital parameters of IS CMa were obtained from the astrometric and spectroscopic data of the system. The Galactic orbit analysis reveals that IS CMa formed inside the solar circle and it is a member of the young thin-disc population.

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arXiv2023

Comprehensive Analysis of the Open Cluster Collinder 74

In this study, we have used the {\it Gaia} Third Data Release (Gaia DR3) to investigate an intermediate-age open cluster Collinder 74. Taking into account the stars with membership probabilities over 0.5 and inside the limiting radius of the cluster, we identified 102 most likely cluster members. The mean proper-motion components of Collinder 74 are estimated as ($μ_α\cos δ, μ_δ)=(0.960 \pm 0.005, -1.526 \pm 0.004$) mas yr$^{-1}$. We detected previously confirmed four blue straggler stars which show flat radial distribution. Colour excess, distance, and age of the cluster were estimated simultaneously by fitting {\sc PARSEC} isochrones to the observational data on {\it Gaia} based colour magnitude diagram. These values were derived as $E(G_{\rm BP}-G_{\rm RP})=0.425\pm 0.046$ mag, $d=2831 \pm118$ pc and $t=1800 \pm 200$ Myr, respectively. The mass function slope was estimated as $Γ= 1.34 \pm 0.21$ within the mass range $0.65\leq M/ M_{\odot}\leq 1.58$ which is well matched with that of Salpeter. Stellar mass distribution indicated that the massive and most likely stars are concentrated around the cluster center. The total mass of the cluster was found to be 365 $M/M_{\odot}$ for the stars with probabilities $P>0$. Galactic orbit integration shows that the Collinder 74 follows a boxy pattern outside the solar circle and is a member of the thin-disc component of the Galaxy.

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arXiv2023

CCD UBV and Gaia DR3 based analysis of NGC 189, NGC 1758 and NGC 7762 open clusters

This paper presents photometric, astrometric, and kinematic analyses of the open clusters NGC 189, NGC 1758 and NGC 7762 based on CCD UBV photometric and Gaia Data Release 3 (DR3) data. According to membership analyses, we identified 32, 57 and 106 most probable member stars with membership probabilities $P\geq 0.5$ in NGC 189, NGC 1758 and NGC 7762, respectively. The color excesses and photometric metallicities of each cluster were determined separately using UBV two-color diagrams. The color excess $E(B-V)$ is $0.590 \pm 0.023$ mag for NGC 189, $0.310 \pm 0.022$ mag for NGC 1758 and $0.640 \pm 0.017$ mag for NGC 7762. The photometric metallicity [Fe/H] is $-0.08 \pm 0.03$ dex for both NGC 189 and NGC 1758, and $-0.12 \pm 0.02$ dex for NGC 7762. Distance moduli and ages of the clusters were obtained by comparing PARSEC isochrones with the color-magnitude diagrams constructed from UBV and Gaia photometric data. During this process, we kept as constant color excess and metallicity for each cluster. The estimated isochrone distance is $1201 \pm 53$ pc for NGC 189, $902 \pm 33$ pc for NGC 1758 and $911 \pm 31$ pc for NGC 7762. These are compatible with the values obtained from trigonometric parallax. Ages of the clusters are $500\pm 50$ Myr, $650\pm 50$ Myr and $2000\pm 200$ Myr for NGC 189, NGC 1758 and NGC 7762, respectively. Galactic orbit integration of the clusters showed that NGC 1758 completely orbits outside the solar circle, while NGC 189 and NGC 7762 enter the solar circle during their orbits.

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arXiv2023

Galactic Model Parameters and Space Density of Cataclysmic Variables in Gaia Era: New Constraints to Population Models

The spatial distribution, Galactic model parameters and luminosity function of cataclysmic variables (CVs) are established using re-estimated trigonometric parallaxes of {\it Gaia} DR3. The data sample of 1,587 CVs in this study is claimed to be suitable for Galactic model parameter estimation as the distances are based on trigonometric parallaxes and the {\it Gaia} DR3 photometric completeness limits were taken into account when the sample was created. According to the analysis, the scale height of All CVs increases from 248$\pm$2 to 430$\pm$4 pc towards shorter periods near the lower limit of the period gap and suddenly drops to 300$\pm$2 pc for the shortest orbital period CVs. The exponential scale heights of All CVs and magnetic systems are found to be 375$\pm$2 and 281$\pm$3 pc, respectively, considerably larger than those suggested in previous observational studies. The local space density of All CVs and magnetic systems in the sample are $6.8^{+1.3}_{-1.1}\times$10$^{-6}$ and $2.1^{+0.5}_{-0.4}\times10^{-6}$ pc$^{-3}$, respectively. Our measurements strengthen the 1-2 order of magnitude discrepancy between CV space densities predicted by population synthesis models and observations. It is likely that this discrepancy is due to objects undetected by CV surveys, such as the systems with very low $\dot{M}$ and the ones in the period gap. The comparisons of the luminosity function of white dwarfs with the luminosity function of All CVs in this study show that 500 times the luminosity function of CVs fits very well to the luminosity function of white dwarfs. We conclude that the estimations and data sample in this study can be confidently used in further analysis of CVs.

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arXiv2022

A study of the NGC 1193 and NGC 1798 open clusters using CCD UBV photometric and Gaia EDR3 data

We present photometric, astrometric, and kinematic studies of the old open star clusters NGC 1193 and NGC 1798. Both of the clusters are investigated by combining data sets from Gaia EDR3 and CCD UBV observational data. Analysis of the radial distribution of stars through the cluster regions indicates that the cluster limit radii are $r_{\rm lim}=8'$ for both of the clusters. We determine the membership probabilities of stars considering Gaia EDR3 proper motion and trigonometric parallax data, resulting in 361 stars in NGC 1193 and 428 in NGC 1798 being identified as most likely cluster members, having membership probabilities greater than P>0.5. Mean proper motion components are estimated as ($μ_α\cos δ$, $μ_δ) = (-0.207(0.009), -0.431(0.008)$) for NGC 1193 and ($μ_α\cos δ$, $μ_δ)=(0.793(0.006), -0.373(0.005)$) mas/yr for NGC 1798. E(B-V) color excesses were derived for NGC 1193 as $0.150(0.037)$ and for NGC 1798 as 0.505(0.100) mag through the use of two-color diagrams. Photometric metallicities are also determined from two-color diagrams with the results of [Fe/H] = -0.30(0.06) dex for NGC 1193 and [Fe/H]=-0.20(0.07) dex for NGC 1798. The isochrone fitting distance and age of NGC 1193 are 5562(381) pc and 4.6(1) Gyr, respectively. For NGC 1798, these parameters are 4451(728) pc and 1.3(0.2) Gyr. These ages indicate that NGC 1193 and NGC 1798 are old open clusters. The overall present-day mass function slopes for main-sequence stars are found as 1.38(2.16) for NGC 1193 and 1.30(0.21) for NGC 1798, which are in fair agreement with the value of Salpeter (1955). Kinematic and dynamic orbital calculations indicate that NGC 1193 and NGC 1798 belong to the thick-disk and thin-disk populations, respectively. In addition, both of the clusters were born outside the solar circle, and both orbit in the metal-poor region of the Galactic disk.

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arXiv2022

A photometric and astrometric study of the open clusters NGC 1664 and NGC 6939

This study calculated astrophysical parameters, as well as kinematic and galactic orbital parameters, of the open clusters NGC 1664 and NGC 6939. The work is based on CCD UBV and Gaia photometric and astrometric data from ground and space-based observations. Considering Gaia Early Data Release 3 (EDR3) astrometric data, we determined membership probabilities of stars located in both of the clusters. We used two-color diagrams to determine $E(B-V)$ color excesses for NGC 1664 and NGC 6939 as $0.190 \pm 0.018$ and $0.380 \pm 0.025$ mag, respectively. Photometric metallicities for the two clusters were estimated as [Fe/H] = $-0.10 \pm 0.02$ dex for NGC 1664 and as [Fe/H] = $-0.06 \pm 0.01$ dex for NGC 6939. Using the reddening and metallicity calculated in the study, we obtained distance moduli and ages of the clusters by fitting PARSEC isochrones to the color-magnitude diagrams based on the most likely member stars. Isochrone fitting distances are $1289 \pm 47$ pc and $1716 \pm 87$ pc, which coincide with ages of $675 \pm 50$ Myr and $1.5 \pm 0.2$ Gyr for NGC 1664 and NGC 6939, respectively. We also derived the distances to the clusters using Gaia trigonometric parallaxes and compared these estimates with the literature. We concluded that the results are in good agreement with those given by the current study. Present day mass function slopes were calculated as $Γ=-1.22\pm0.33$ and $Γ=-1.18\pm0.21$ for NGC 1664 and NGC 6939, respectively, which are compatible with the Salpeter (1955) slope. Analyses showed that both of clusters are dynamically relaxed. The kinematic and dynamic orbital parameters of the clusters were calculated, indicating that the birthplaces of the clusters are outside the solar circle.

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arXiv2020

A Study of Open Clusters Frolov 1 and NGC 7510 using CCD UBV Photometry and Gaia DR2 Astrometry

We present reddening, photometric metallicity, age and distance estimates for the Frolov 1 and NGC 7510 open clusters based on CCD UBV photometric and Gaia data. Photometric observations were collected using the 1-m telescope of the TÜBİTAK National Observatory. Gaia DR2 proper motion data in the direction of two groupings were used to identify cluster membership. We determined mean proper motion values ($ μ_α\cosδ, μ_δ$) = ($-3.02\pm 0.10$, $-1.75 \pm 0.08$) and ($-3.66 \pm 0.07$, $-2.17 \pm 0.06$) mas yr$^{-1}$ for Frolov 1 and NGC 7510, respectively. We used two-colour diagrams to obtain $E(B-V)$ colour excesses for Frolov 1 and NGC 7510 as $0.65\pm0.06$ and $1.05\pm0.05$ mag, respectively. We derived the photometric metallicity of Frolov 1 as [Fe/H] = 0.03$\pm$0.03 dex and adopted a solar metallicity for NGC 7510. Based on these reddening and metallicities we determined the distance moduli and ages of the clusters via fitting PARSEC isochrones to the cluster colour-magnitude diagrams. Isochrone fitting distances for Frolov 1 and NGC 7510 are $2864 \pm 254$ and $2818 \pm 247$ pc, respectively, which correspond to the ages $35\pm 10$ Myr and $18\pm 6$ Myr. We also calculated mean Gaia distances and compared them with those given in the literature and in this study, concluding that our results are in good agreement with previous work. Finally, we calculated the mass function slopes as being $X=-1.21\pm0.18$ for Frolov 1 and $X=-1.42\pm0.27$ for NGC 7510.

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