Investigating pulsating variables and eclipsing binaries in NGC 2126 using ground- and space-based photometry, astrometry, spectroscopy, and modelling

ABSTRACT

Pulsating variables are prevalent in the classical |$\delta$| Scuti instability strip of intermediate-age open star clusters. The cluster membership of these stars facilitates a comparative analysis of their evolution in analogous environments. In this study, we integrate ground-based observations, Transiting Exoplanet Survey Satellite (TESS) Full Frame Images (FFIs), and Gaia Data Release 3 (DR3) data to investigate variable stars in the intermediate-age open star cluster NGC 2126. We performed ground-based time-series observations of NGC 2126 to identify variable stars within its vicinity. Next, we determined the membership of these stars using parallax and the proper motions from Gaia DR3 archive. Then, we searched the TESS FFIs for counterparts to the variables identified above and performed their frequency analysis and classification. Finally, we modelled the light curves (LCs) of detected eclipsing binaries (EBs), including V551 Aur, which has a pulsating component. We found 25 members and 85 field variable stars. In TESS FFIs, we found LCs for 11 known variables and a new rotational variable. We determined that the pulsating EB V551 Aur is a member of the cluster. The low- and medium-resolution spectra revealed the line profile variation and the basic parameters for the star, respectively. Simultaneous modelling of the eclipses and the embedded pulsations resulted in improved orbital parameters for the binary system. We also report the determination of orbital parameters for the previously uncharacterized EB system UCAC4 700−043174.

1 INTRODUCTION

Open star clusters comprise stars that have emerged from a common molecular cloud and are gravitationally bound to one another. Consequently, they possess analogous ages, distances, and initial chemical compositions. The membership of stars within an open cluster imposes constraints, especially regarding their age in comparison to the ambiguously determined ages of field stars. Thus, examining stars that are constituents of an open cluster is more advantageous than analysing field stars. Diverse independent methodologies for stellar characterization, such as asteroseismology (Aerts 2021), can gain from open clusters by imposing supplementary constraints on parameters such as age and mass (Bedding et al. 2023; Pamos Ortega et al. 2023; Li et al. 2024). The intermediate-age open star clusters possess their turn-off point adjacent to the classical instability strip, thereby hosting a variety of pulsating variable stars of spectral type A and F, including |$\delta$| Scuti stars and |$\gamma$| Doradus stars (Chehlaeh et al. 2018).

Ground-based studies of open clusters yield ensemble photometry for a collection of stars within a defined area. The Aryabhatta Research Institute of Observational Sciences (ARIES) in Nainital, India, possesses photometric observational facilities in the Northern hemisphere, which are very suitable for studying open clusters. Details regarding the telescopes at the site will be presented in Section 3 later in this paper. To capitalize on the advantages of studying open clusters, ARIES, Nainital is conducting long-term photometric monitoring using 1-m class telescopes to examine pulsating variable stars in both young open clusters (Joshi et al. 2020a; Lata et al. 2023) and intermediate-age open clusters (Joshi et al. 2020b; Maurya et al. 2023). Additionally, we also perform long-term surveys for pulsating A- and F-type field stars (Ashoka et al. 2000; Martinez et al. 2001; Joshi et al. 2003, 2006, 2009, 2010, 2012, 2016b, 2017). We have recently augmented this exploration with space-based observations from extended Kepler mission (K2) and Transiting Exoplanet Survey Satellite (TESS) concerning field stars (Trust et al. 2021, 2023; Joshi et al. 2022b; Dileep, Joshi & Kurtz 2024; Sarkar et al. 2024). In this study, we integrate the space-based observations for open star clusters with the ground-based observations.

Similar to open star clusters, eclipsing binaries (EBs) provide an independent assessment of fundamental stellar parameters. They provide accurate measurements of stellar masses and radii through orbital dynamics and eclipse depths (Panchal et al. 2023). If one of the binary components is pulsating, the target becomes increasingly unusual. The examination of pulsations in EB components provides insights into the internal structure of the pulsating star, system evolution, and the influence of tidal forces inducing oscillations (Pigulski 2006; Murphy 2019; Lampens 2021). The existence of these unusual variable systems in open clusters serves as valuable test beds for validating various theoretical models for pulsation and stellar evolution.

We conducted a comprehensive ground- and space-based investigation of the intermediate-age open star cluster NGC 2126 (Gáspár et al. 2003). We utilized ground-based photometric observations from telescopes of varying diameters at ARIES, Nainital, low- to medium-resolution spectra, TESS Full Frame Images (FFIs), and Gaia Data Release 3 (DR3) data.

The manuscript is structured as follows. Section 2 delineates the sample selection, while Section 3 examines ground-based observations for the detection of variable stars. In Section 4, we ascertain the membership of the detected variables within the cluster, and in Section 5, we evaluate the cluster parameters. Subsequently, in Section 6, we examine the TESS light curves (LCs) for all stars and categorize the identified variables in Section 7. Ultimately, we present the modelling of the binary LCs in Section 8 and provide a summary of our findings in Section 9. We present our conclusions in Section 10.

2 SAMPLE SELECTION

We selected NGC 2126 (RA = |$06^{\mathrm{ h}} 02^{\mathrm{ m}} 37 {_{.}^{\rm s}} 9$|⁠, Dec. = |$+49^{\circ} 52^{\prime } 59^{\prime \prime }$|⁠), which is an intermediate-age open star cluster hosting various pulsating stars located in the |$\delta$| Scuti instability strip. The first CCD photometric study of the cluster was performed by Gáspár et al. (2003). Later, it was studied by Liu et al. (2012) and more recently Chehlaeh et al. (2018) presented new ground-based photometric observations for NGC 2126 and reported 11 variables, including 2 new |$\delta$| Scuti variables. This makes this cluster as a potential target for asteroseismic investigation of pulsating variables of A and F type. Another reason that makes this cluster interesting is the presence of the EB V551 Aur, which shows pulsational variability on top of the eclipses. The membership of this star to the cluster will place strong constraints on its evolution and pulsation mechanism. All of the previous ground-based studies had limitations due to their short and discrete observations. Thus, aliasing and frequency resolution limit the accuracy of detected frequencies. Furthermore, NGC 2126 has one EB, UCAC4 700–043174, but there were insufficient data to model the system or report an orbital period (Chehlaeh et al. 2018). Thanks to the nearly continuous TESS observations spanning approximately a month in each sector, we can address the limitations due to lack of continuous data in ground-based observations. The long and short cadence of TESS enables us to study both faint and bright targets of the cluster.

3 GROUND-BASED PHOTOMETRIC OBSERVATIONS

Time-series photometric observations for NGC 2126 were acquired in the V and R bands with the 1.3-m Devasthal Fast Optical Telescope (DFOT), Nainital, India (Joshi et al. 2022a). This telescope is equipped with a 2k |$\times$| 2k CCD, and gives a field of view of about 18 × 18 arcmin|$^2$| with a plate scale of 0.54 arcsec pixel⁻¹. For standardization of our targets, the standard field SA 98 of Landolt (1992) was also observed in UBVRI filters. For the pre-processing of the images, several bias frames and twilight flats were taken during the observing nights. We furthermore observed the cluster in the V and R bands during one more night using the 1.04-m Sampurnanand Telescope (ST), Nainital, India. This telescope is equipped with a 4k |$\times$| 4k CCD, which gives a field of view of about 15 × 15 arcmin|$^2$| (Yadav et al. 2022). The CCD was used in 4 |$\times$| 4 binning mode to increase the signal-to-noise ratio (SNR); this gives a plate scale of 0.92 arcsec pixel⁻¹ after binning. Additionally, we also observed the pulsating EB V551 Aur in NGC 2126 for a couple of nights with the 3.6-m Devasthal Optical Telescope (DOT; Kumar et al. 2018). This telescope is equipped with a 4k |$\times$| 4k CCD Imager that covers a field of view of 6.5 × 6.5 arcmin|$^2$| (Pandey et al. 2018) and has a plate scale of 0.2 arcsec pixel⁻¹ after 2 |$\times$| 2 binning. An overview with the details of the collected ground-based observations is given in Table 1.

The images were pre-processed with Image Reduction and Analysis Facility (iraf) tasks: zerocombine, flatcombine, and CCDPROC. We applied the method of point-spread function (PSF) photometry to obtain the instrumental magnitudes for the stars in the target field; for this, we used the code daophotii by Stetson (1992).

The standard field SA 98 was used to convert the instrumental magnitudes to standard magnitudes. The following equations define the resulting transformations (Stetson 1987):

where v, b, i, r, and u are the instrumental magnitudes; |$V, B, I, R, \mathrm{ and} \ U$| refer to the standard magnitudes; and Q is the airmass.

The search for variables in the cluster field was done based on the LCs obtained with 1.3-m DFOT. The V-band photometric LCs from each night were combined together and then phase-folded with the period that matches the first main peak in the Lomb–Scargle periodogram. The same period was used to phase-fold the combined R-band LCs to see any corresponding variations in the R band. The stars that showed brightness variation greater than three standard deviations with respect to the mean brightness, at least in the V band, were considered to be variable stars and were chosen for further analysis. The LCs for newly detected member variables from the ground are given in Fig. 1 and the field stars in Figs. A1–A5. We also list the detected dominant frequencies and the corresponding amplitudes of the field stars in Table A1.

4 CLUSTER MEMBERSHIP

For the determination of membership, we used astrometric data from Gaia DR3. The Gaia mission (Gaia Collaboration 2016) is an all-sky survey that collects multiband photometric, astrometric, and spectroscopic data. Gaia’s astrometric data have proven to be very successful in identifying and determining the membership of open clusters and globular clusters. The proper motion and parallax of all stars within a radius of 18 arcmin around the cluster centre were obtained by querying the Gaia DR3 catalogue (Gaia Collaboration 2023). Stars with large error bars were excluded from our study. We constructed the vector point diagram (VPD) of the cluster region to identify stars with similar proper motion (Fig. 2). Furthermore, we used the parallaxes to select samples with similar distances. For the membership probability, we used the method described in Wu et al. (2002) by identifying a cluster population and field population from the VPD. For this purpose, we plotted the radial density profile (RDP) of the VPD by calculating the number density in concentric annular regions from the centre of the VPD. For the centre of VPD, we smoothed the proper motion space with a 2D Gaussian kernel and chose the point with the maximum kernel density. The resultant RDP is shown in Fig. 3 along with a fitted King’s profile (King 1962). The number density was seen to approach the background density at a radial distance |$\sim$|0.8 mas yr⁻¹. We chose this as the limiting radius for probable cluster members and the stars outside this radius as field stars.

The Gaia colour–magnitude diagram (CMD) for stars having membership probability greater than 50 per cent is shown in Fig. 4. The members that are highly probable are situated along the main sequence, the evolved red giant branch, and the turn-off point. These members will introduce substantial constraints on the system’s age by means of isochrone fitting (see Section 5).

5 CLUSTER PARAMETERS

The CMD can be used to fit theoretical isochrones and determine cluster parameters such as age, metallicity, and reddening. The age and metallicity of the member stars should be comparable as a result of their shared origin. Thus, by comparing theoretical isochrones of varying ages and metallicities, we can estimate the cluster parameters. The Gaia CMD consists of the color index between gaia blue and red photometer bands (BP-RP colour) and the G-band mean magnitude. We used the isochrones downloaded from the CMD service CMD 3.7 in the Gaia early data release 3 (Gaia EDR3) photometric system, and the evolutionary tracks from the parsec release v1.2S (Bressan et al. 2012; Chen et al. 2014, 2015; Tang et al. 2014). We utilized the Gaia DR3 parallaxes for the cluster members to determine an initial distance modulus and used prior literature values for an initial value for reddening. Utilizing these corrections, we drew a series of isochrones next to the observed CMD. Subsequently, the fit was optimized using the asteca code (Perren, Vázquez & Piatti 2015) by selecting the above guessed range of values (metallicity = [0.010, 0.030], log age = [9.0, 9.3], distance modulus = [9, 12], and extinction = [0.2, 1]) as uniform priors. asteca computes synthetic isochrones between the given grid of isochrones and uses a genetic algorithm (GA; Charbonneau 1995) to minimize the negative log likelihood, −log[|$L_{i}$|(z, a, d, e)], where z is the metallicity, a is the age, d is the distance, and e is the extinction. The GA is an optimization technique derived from natural selection, wherein a population of candidate solutions evolves over generations through selection (optimal fit), crossover (various combinations of parent sets), and mutation (random alterations introduced at each stage) to identify the most effective solution for a specific problem. It is particularly effective in exploring complex, multidimensional spaces and avoiding local minima. For more thorough details regarding the algorithm, please refer to Charbonneau (1995). Additionally, to quantify parameter uncertainties we employed the Markov chain Monte Carlo method within asteca; the resultant posterior distribution is shown in Fig. 5. By fitting a Gaussian to the parallax histogram (Fig. 6), we calculated the mean parallax, which allowed us to calculate the distance to the cluster, which came out to be 1.3 |$\pm$| 0.1 kpc. Also, we calculated distance to the cluster from the distance modulus of the isochrone fit using standard relations. We used the best-fitting coefficients published by Gaia team based on the empirical relation used by Danielski et al. (2018) to convert extinction in Gaia G band (⁠|$A_{G}$|⁠) to extinction in V band (⁠|$A_{V}$|⁠), and used |$R_{v}$| = 3.1 to get the reddening |$E(B-V)$| for the cluster. We also verified this estimate using colour–colour diagrams from ground-based observations (Fig. B1). Apart from that, a medium-resolution spectrum (see Section 8.2) of a member variable, V551 Aur, was used to estimate the reddening towards the cluster. We used the calibration published by Poznanski, Prochaska & Bloom (2012), between the equivalent width of Na doublets and the reddening (Fig. 7). The final values for the Hertzsprung-Russell (HR) diagram were found by taking the average of the distance estimates from the isochrone fit and the parallax fit, as well as the reddening estimates from the isochrone fit and the Na doublets. The best-fitting parameters for the cluster are tabulated in Table 2 and the best-fitting isochrone is plotted in Fig. 8.

6 TESS PHOTOMETRY

The TESS is an all-sky survey telescope whose prime objective is to detect exoplanets orbiting nearby bright stars using the transit method (Ricker 2014). Apart from its primary mission, it also provides time-series photometry for asteroseismic studies of stars with magnitudes up to 12. TESS target pixel files (TPFs) and LC files are available with short cadences (20 and 120 s), while FFIs are taken with a longer cadence (30 min). Recently, the FFIs were taken at a shorter cadence of 200 s, which we used in our study. These data products can be downloaded from Mikulski Archive for Space Telescopes (MAST).

Given the large pixel size of the TESS CCD, which is 21 arcsec pixel⁻¹, resolving crowded fields like we see in open star clusters will be difficult, so our search for variability was primarily based on ground-based observations. The higher spatial resolution of ground-based observations enables the resolution of individual stars in the crowded field. Consequently, we concentrated our investigation of variability in the TESS FFIs on sources that had confirmed variable signals from the ground.

TESS observed the field of NGC 2126 during Sector 19 with an 1800-s cadence and during Sectors 59, 60, and 73 with a 200-s cadence. The observations from Sector 60 were excluded from this study because the targets were too close to the edge of the TESS CCD, preventing useful photometry. The log of observations taken from the TESS archive is given in Table 3. We performed the FFI photometry of our targets in the field using the python packages eleanor (Feinstein et al. 2019) and lightkurve (Lightkurve Collaboration 2018). To perform the photometry, we relied on the cut-outs taken from the FFI using the package tesscut (Brasseur et al.2019).

eleanor is a python package that automates the extraction of LCs from FFIs. For Sector 19, the eleanor package was used and the resulting sky-subtracted LCs were free of telescope systematic and suitable for analysis. To identify our targets in the FFI, we overplotted the Gaia sources in the field and the apertures were carefully selected to reduce the contamination from nearby targets. eleanor did not perform well in Sectors 59 and 73, necessitating a manual approach utilizing the lightkurve package. Here, we also conducted an investigation into the FFI by overplotting the Gaia sources to minimize contamination from nearby sources and used custom apertures to extract the sky-subtracted LCs. As a final step, we used the RegressionCorrector in lightkurve package. The task creates a design matrix, which is a matrix of flux time series of the background pixels. For which a principal component analysis (Deb & Singh 2009) is performed to reduce its dimensionality and select the first few principal components, which contain most of the instrumental and systematic effects in that particular part of the sector. Then, a linear combination of these components is used to remove the long-term trends in the LC.

Furthermore, we attempted a global photometric search for variables in TESS FFIs independent of the ground-based observations with a PSF-based technique utilizing the flux distribution from Gaia observations, implemented in the python package tglc (Han & Brandt 2023). We used tglc for only long-cadence data (Sector 19), since the computational expense for short cadence was high and not possible for us at the moment. This resulted in the discovery of one new variable in addition to the known variables. Due to contamination-related uncertainty, we had to discard a lot of other targets with variable signals. We discuss how we dealt with the effect of contamination in our data in Section 6.1.

The frequency analysis for the variable stars in TESS data was done using the program PERIOD04 (Lenz & Breger 2004). We adopted the criterion SNR |$> =\!\! 5.6$| given by Zong et al. (2016) for space-based observations. The SNR was calculated after pre-whitening all of the dominant frequencies to obtain the residual spectra, followed by calculating the noise using a box size of one cycle per day along both sides of the target frequency. The errors in frequency and amplitude were calculated using the analytical relations provided by Montgomery & O’Donoghue (1999). Tables C1 and C2 list the frequencies for the known variables detected using TESS and from the ground and the LCs in the respective sectors are plotted in Figs C1, C2, and C3.

6.1 Contamination

The large pixel size of TESS causes the blending of signals from nearby sources. The resulting LCs for a target can only be trusted when (i) the source is much brighter than the nearby contaminating sources so that the background source only adds to the noise and (ii) the contaminating sources are non-variables and, hence, only increase the noise level rather than introduce signals in the periodogram.

In order to investigate the contamination in TESS LCs, we first overplotted the Gaia sources on the TPF in the vicinity of each target to see whether there are any bright nearby objects that might potentially contribute to contamination. After that, we selected a variety of custom apertures to determine the origin of a specific set of frequencies within the frequency spectrum. An example of this investigation is shown in Fig. 9, where V551 Aur (V6) is a pulsating EB that is contaminated with a |$\delta$| Scuti star (N1). Next, we cross-checked whether any of these frequencies are pre-defined in the ground-based data obtained from Chehlaeh et al. (2018). For example, in case of V6 in Fig. 9, the two strong blue peaks towards the right were detected by Chehlaeh et al. (2018) for N1 and few of the dominant red peaks to the left were reported by Chehlaeh et al. (2018) for V6.

We found that two of the pulsating stars ZV2 (hybrid star) and N1 (⁠|$\delta$| Scuti star) were contaminated significantly by the nearby binary stars V4 and V6, respectively, and found that the contamination can be addressed with the above technique. Furthermore, to quantify the extent of contamination endured by the variable stars detected in our study, we calculated a contamination factor for each of these stars using Gaia DR3 positions and magnitudes, where we took the ratio of the target flux to the total flux in a 21 arcsec radius.

7 CLASSIFICATION OF VARIABLE STARS

Variable stars are classified according to the shape of their LCs, amplitude of variability, frequency of variability, and location on the HR diagram. The pulsating stars can be identified from the HR diagram by plotting the instability strips for |$\delta$| Scuti and |$\gamma$| Doradus pulsator. In our study, we used the theoretical |$\delta$| Scuti and |$\gamma$| Doradus instability strips, computed by Dupret et al. (2005).

The standard B- and V-band magnitudes from our ground-based observations with the 1.3-m DFOT were used to generate the HR diagram for the cluster members. After extinction correction, the colour index, which is the difference between B-band magnitude and V-band magnitude (B-V), was used to find the effective temperature (⁠|$T_{\rm eff}$|⁠), which was based on an empirical relationship from Torres (2010). Next, the empirical relation provided by Torres (2010) was employed to calculate the bolometric correction (⁠|$\mathrm{ BC}_{v}$|⁠) for the stars in the V band. Then, the luminosity can be estimated using the standard relations, where we took bolometric magnitude of sun as |$M_{\mathrm{ bol}_{\odot }}$| = 4.73 (Torres 2010). The HR diagram in Fig. 10 shows the known variables, the newly found variables, and the instability strips for the |$\delta$| Scuti and |$\gamma$| Doradus pulsations.

|$\gamma$| Doradus variables are mainly non-radial g-mode pulsators showing periodic variations in their brightness with pulsation frequencies ranging from 0.5 to 3 d|$^{-1}$| (Aerts, Christensen-Dalsgaard & Kurtz 2010) and amplitude of variation up to 0.1 mag in the V band. The convective blocking drives the g-mode pulsations, with buoyancy as the restoring force (Dupret et al. 2005). In the HR diagram, the |$\gamma$| Doradus instability strip is seen overlapped with the cooler edge of the |$\delta$| Scuti instability strip in the main sequence with an early F spectral type.

|$\delta$| Scuti stars are primarily p-mode pulsators with brightness variations ranging from 0.001 to 0.8 mag in the V band and pulsation frequencies ranging from 5 to 50 d|$^{-1}$| (Aerts et al. 2010). The |$\kappa$|-mechanism drives the pulsations in these stars, which occur at the partial ionization zones in the outer radiative region (Pamyatnykh 2000). These stars are found at the lower part of the classical instability strip, which intersects the main sequence, called the |$\delta$| Scuti instability strip. They are A- to early F-type stars with masses ranging from 1.5 to 2.5 |${\rm M}_{\odot }$|⁠. They pulsate in both radial and non-radial p modes. Some of the stars pulsate both in g modes and p modes, with characteristics similar to |$\gamma$| Doradus and |$\delta$| Scuti stars. These stars are known as hybrid pulsators (Handler et al. 2009).

The G- and K- spectral type stars are located below the |$\delta$| Scuti and |$\gamma$| Doradus instability strips. These stars, with a convective envelope, exhibit rotational variability due to spots on their surface (Mamajek & Hillenbrand 2008). The LCs of these rotational variables produce a frequency spectrum with the rotational frequency below 5 d|$^{-1}$| and at least one harmonic of the rotational frequency (Balona 2013). Thus, we classified the main-sequence variables with a frequency less than 5 d|$^{-1}$| falling in the G to K spectral type as rotational variables. Also, in this region one can find the solar-type oscillations. The stars whose LCs show (partial or full) eclipses can be classified as EBs.

All of the member stars, as well as those with LCs in both TESS and ground-based observations, were classified according to the criteria listed above. Table 4 shows the results for newly discovered member stars, while Table 5 shows the results for stars with variability in TESS and ground-based observations. We observe that ground-based classification is incomplete due to insufficient data.

7.1 Intrinsic variables

7.1.1 |$\gamma$| Doradus stars

NGC 2126 contains two known |$\gamma$| Doradus variables (V1 and V2). Our membership analysis revealed that V2 is not likely to be a cluster member. The two stars were detected by TESS. Both stars have a frequency lower than 3 c/d. Fig. 11 shows a comparison of our results with previous frequency estimates from the literature. There were no harmonics of the fundamental frequency in the frequency spectrum, showing that they are not rotational variables. The position of V1 in the HR diagram is within the |$\gamma$| Doradus instability strip. Since V2 is not a member star, we used the Gaia temperature |$T_{\rm eff}$| = 7134 K, indicating that it is an early F-type star. Thus, we classify V1 and V2 as |$\gamma$| Doradus variables. From our ground observations, we classified the star 875, as |$\gamma$| Doradus variable, based on the frequency and position in the HR diagram. The TESS data for this star produced null results.

7.1.2 |$\delta$| Scuti stars and hybrid pulsators

Chehlaeh et al. (2018) reported five |$\delta$| Scuti stars (V3, V5, ZV1, N1, N2) and a hybrid pulsator (ZV2). All of these stars were detected in TESS data. ZV1, N1, and N2 were determined to be probable cluster members. Figs 11 and 12 show a comparison between the frequencies detected in literature and those we detected from TESS’s three sectors. Pulsation frequencies in ZV1, N1, and N2 were similar to those of a |$\delta$| Scuti star, ranging from 5 to 30 c/d. They belong to the |$\delta$| Scuti instability strip on the HR diagram. Thus, we classify these stars as |$\delta$| Scuti stars. However, N1 is contaminated by a nearby binary star V6. To address this, we used the method described in the previous section. Since V3 and V5 are not cluster members, we used the Gaia temperatures; V3 has |$T_{\rm eff}$| = 7454 K, while V5 has |$T_{\rm eff}$| = 6166 K, which corresponds to a spectral type F. The spectral type along with the pulsation frequencies from TESS indicates that these are |$\delta$| Scuti stars. According to Zhang et al. (2012), ZV2 is a hybrid pulsator because it has a low frequency of 1.812 d|$^{-1}$| in addition to the |$\delta$| Scuti-type frequencies. Our analysis of TESS data recovered the dominant frequency 14.851 d|$^{-1}$|⁠, as well as the low frequency reported by Zhang et al. (2012). As ZV2 is not a member, we used the Gaia temperature, |$T_{\rm eff}$| = 7081 K, indicating an F-type star. This means that the star is in the |$\gamma$| Doradus instability strips where |$\delta$| Scuti instability strip also overlaps. Thus, we classify it as a hybrid pulsator. This star was also contaminated by a nearby EB, V4. We used the method described in previous section in this case also to identify the frequencies. Using ground-based data, we identified three new |$\delta$| Scuti variables, 143, 568, and 614. The classification was made based on their positions in the HR diagram and oscillation frequencies. TESS detected no signals from these stars due to high contamination.

7.2 Extrinsic variables

7.2.1 Rotational variable

We discovered a new variable, 839, in the field of NGC 2126. The star shows variable signals on the ground, as well as TESS data. Fig. 13 compares the TESS LC and frequency spectra of this star. This star demonstrates TESS’s improved frequency resolution. The dominant frequency is the same in both spectra; however, the ground-based spectrum has many aliases. The frequency is less than 5 c/d and appears as harmonics in the TESS frequency spectra. The Gaia temperature, |$T_{\rm eff}$| = 5725, indicates a G-type star. Thus, we classify this star as a rotational variable. From the ground-based observations, we also identified three additional rotational variables, 377, 881, and 1203. We were unable to detect any of these stars using TESS due to contamination.

7.2.2 Eclipsing binaries

Chehlaeh et al. (2018) identified three EBs in NGC 2126, V4, V6, and ZV3. They were able to cover a large enough phase range for two stars, V4 and V6, allowing them to model the LCs and derive basic parameters such as orbital periods, inclination, mass ratio, and temperature ratio. For ZV3, however, modelling and period determination were not possible due to a lack of data covering the full eclipse. We used continuous time-series TESS observations to fully cover the three EBs’ phase range in our study. ZV3 and V6 are members of the cluster, while V4 is not a member. Given the cluster membership, the case of V6 is more interesting because it is a pulsating EB.

According to available data, Chehlaeh et al. (2018) estimated an orbital period of 1–1.5 d for ZV3. However, with the help of TESS data that included multiple eclipses, we were able to calculate the orbital period for this star. We used all of the primary minima found in Sectors 19, 59, and 73 by fitting a parabola to their dips. To determine the system’s linear ephemeris, the orbital cycle versus time of primary minima (TOM) curve was fitted with a linear relation.

The resulting linear ephemeris for ZV3 is

where |$\mathrm{ BJD_{min}}$| is the time of primary minima at epoch E. Fig. 14 shows the ephemeris fit.

For V4, Chehlaeh et al. (2018) discovered an orbital period of 3.300 281 d. We collected LCs for this star from TESS Sectors 19, 59, and 73. In addition to the primary minima detected from TESS sectors, two minima reported in Chehlaeh et al. (2018) were included in the ephemeris fit.

The subsequent linear ephemeris for V4 is

Gáspár et al. (2003) described V6 as an EB with a pulsating component. Chehlaeh et al. (2018) revisited the nature of this variable star, as well as the parameters of the open cluster that hosts it. They concluded that the pulsation could be caused by tidally excited g modes, which exhibit tidal resonance with the orbital period. Although previous research concluded that V551 Aur is unlikely to belong to the open cluster, our analysis of Gaia DR3 data revealed that it most likely does. As in the previous case, we determined the linear ephemeris from all the primary minima detected in TESS sectors, as well as the ground-based minima listed in table 9 of Chehlaeh et al. (2018).

The final linear ephemeris for V6 is

Fig. 15 depicts the phased and binned TESS LCs for three EBs. Figs D1 and D2 show the ephemeris fit for V6 and V4, respectively.

8 ECLIPSING BINARY MODELLING

EBs provide an independent method for calculating stellar masses and radii. By modelling the eclipsing LCs and the radial velocity curve, we can determine the best-fitting parameters for masses, radii, inclination, and so on. However, we only have TESS LCs for our EBs. Due to the faintness of these stars, no high-resolution spectra could be obtained using the facilities we have direct access to. Thus, we used the eclipsing LCs alone to model our EBs. While modelling the EB LC, one should take care of the intrinsic variability that it may possess due to either some intrinsic process in the system such as pulsations, mass transfer, star-spots, etc. (Pigulski 2006) or external reasons such as telescope systematics or contamination in case of TESS. Therefore, it was imperative to incorporate the intrinsic variability or long-term systematics into the LC modelling process.

The primary and secondary minima of all three of our EBs (V4, V6, and ZV3) are clearly distinct, with the out-of-eclipse region flat and the secondary minima precisely located at the 0.5 phase of the orbital period. Consequently, it is reasonable to assume that they are detached binaries in a circular orbit. We modelled the intrinsic variability in EB LCs using the celerite package (Foreman-Mackey et al. 2017; Foreman-Mackey 2018). This package is part of the exoplanet¹ package (Foreman-Mackey et al. 2021) and features a scalable Gaussian process model with a SHOTerm (a term representing a stochastically driven, damped harmonic oscillator). This package has the capability to simultaneously model the intrinsic variability with a scalable Gaussian process and the eclipses for a detached binary system. The exoplanet code employs pymc (Fonnesbeck et al. 2015) for its probabilistic modelling. It utilizes the starry package (Luger et al. 2019) to generate limb-darkened LCs and employs a fast and efficient solver for Kepler’s equations.

The q-search method (Wilson 1994; Joshi, Jagirdar & Joshi 2016a) can be employed to estimate the mass ratio (q-value) of the system in the absence of spectroscopic data. Previous studies by Liu et al. (2012) and Chehlaeh et al. (2018) calculated the mass ratio for V6 and V4, respectively, using the photometric q-search method using the phoebe 1.0 legacy version based on Wilson–Devinney code (Prša & Zwitter 2005). We used the q-values calculated for V4 and V6 from previous studies. For ZV3, we used a similar approach and calculated the q-value with phoebe 1.0. The best-fitting q-value was q = 0.48 based on the q-value versus chi-square plot, as shown in Fig. 16.

These photometrically derived q-values served as priors for modelling the EB LCs. The primary star’s mass was determined by comparing evolutionary tracks on the HR diagram, and the radius was calculated using the Stefan–Boltzmann’s relation, with the primary star’s temperature being the B-V colour temperature in the case of V4 and ZV3. For V6, we used the spectroscopic temperature obtained through medium-resolution spectroscopy.

Table E1 contains the prior distributions that were chosen for modelling in the exoplanet code. Table 6 provides the best-fitting parameters for binary modelling for all stars, which were compared to previous estimates by Chehlaeh et al. (2018). Figs 17, 18, and 19 depict the best-fitting LCs. Figs E1, E2, and E3 and Figs 20, E4, and E5 show the Gaussian process fits and corner plots, respectively.

8.1 V551 Aur: a pulsating eclipsing binary

V551 Aur is an EB that exhibits pulsational variability in its LC. Previous authors proposed that the variability could be due to pulsations in the binary system’s primary component. However, the nature and origin of this star’s pulsation remain poorly understood. The associated frequency, 7.7248(5), is very close to the 9th harmonic of the orbital frequency, so the pulsation may be a higher order resonance with the orbital frequency. This could indicate a tidally excited g mode on the primary star. The companions’ nature is unknown because no spectroscopic observations have been made.

As described in the previous section, we modelled both eclipses and intrinsic variability simultaneously. The pulsational frequency, first harmonics, second harmonics, and sub-harmonics were identified in the residuals of the eclipse model subtracted LC (refer to Fig. 21). The presence of a sub-harmonic frequency in the pulsation indicates that the pulsations are non-linear, which is not expected from a pure stable oscillator. This lends credence to the theory that the pulsation in one of the companions is caused by tidal forces. The ground-based phased LC (Fig. 22) shows that during the secondary eclipse, pulsations dominate, indicating that the pulsating member is the primary companion.

8.2 Spectroscopy

Low-resolution spectroscopic observations for V551 Aur were carried out with Himalayan Chandra telescope (HCT), Hanle Faint Object Spectrograph Camera (HFOSC; Cowsik, Srinivasan & Prabhu 2002), during 2021 and 2022. A total of 16 spectra with a resolution of R|$\sim$| 1200 were obtained during different phases of the orbital period. The radial velocities proved to be unreliable for our study because of the low resolution and due to the fact that the components’ profiles are always blended and furthermore affected by the pulsation. However, close examination of the strong Balmer lines shows significant line profile variations across different phases (see Fig. 23 for the H |$\alpha$| lines) indicating a spectroscopic binary. A series of high-resolution spectra is required to construct the star’s radial velocity curves, which will provide accurate system parameters. Because we do not have direct access to the spectroscopic observational facilities required to observe a 14th magnitude star, we propose this as a future project. We used the low-resolution spectra to determine the basic parameters of the star by comparison with synthetic spectra (Table F1), using the chi-square minimization technique employed in gssp code (Tkachenko 2015).

A medium-resolution spectrum was taken with the 2.4-m telescope, Thailand National Observatory, Thailand, Medium-Resolution Spectrograph with a resolution of R  |$\sim$| 18 000 over a range of 380–900 nm. The H |$\beta$| line (Fig. 24) was used to calculate the |$T_{\rm eff}$| and |$\log g$| by comparison with synthetic spectra using the gssp code. Due to binarity and poor resolution, the metallicity estimation is unreliable and cannot be compared to the cluster metallicity. To improve the SNR of the line profile, we applied the least-squares deconvolution (LSD) method (Donati et al. 1997). The LSD profile (Fig. 25) was used to calculate the |$v\sin i$| and radial velocity of the star. The best-fitting results of our analysis are presented in Table 7.

9 SUMMARY

Our ground-based observations revealed 25 member variables and 85 field variables within a 9 arcmin radius field of NGC 2126. Our research was focused on member variables and stars with promising LCs in the TESS data. We discovered 18 new variable cluster members from the ground.

In TESS, we identified counterparts for 11 known variables, including 6 members and 5 field stars. Furthermore, we identified a new field star as a rotational variable based on TESS LC. TESS frequency spectra show distinctive pulsation peaks, especially for |$\delta$| Scuti stars. These frequencies can be used to conduct asteroseismic studies on member pulsating stars. Aside from that, the cluster has a higher metallicity than the solar metallicity, implying the presence of chemically peculiar stars in the cluster. Thus, a spectroscopic investigation can benefit our understanding of the member variables.

We attempted to solve the TESS contamination problem by using custom apertures for our sources and cross-matching detected frequencies to ground-based frequencies. This method was useful for stars that produced strong signals in TESS.

We used TESS data to model the EBs V4, V6, and ZV3 by combining their TESS LCs. However, this model still lacks the support of spectroscopic radial velocity measurements. Due to a lack of data, ZV3 had not previously been modelled or its orbital period determined. We utilized the continuous LCs from TESS data to model this star, thereby establishing the linear ephemeris and determining the orbital period.

The intrinsic variability and eclipses of the pulsating EB V551 Aur were modelled simultaneously to determine its parameters. This resulted in slightly different parameter estimates than those reported in previous literature. The low-resolution spectroscopy confirms the spectral line profile variation for this system, indicating that it is a spectroscopic binary. The medium-resolution spectra provided better estimates of spectroscopic parameters for V551 Aur.

10 CONCLUSIONS

This project was initiated to search and study pulsating variables in the open star cluster NGC 2126, utilizing ground-based observations, Gaia data, and TESS FFIs for improved analysis. This study led us to draw the following conclusions:

• Our study demonstrates that TESS LCs alone are unreliable in the case of open clusters due to contamination. To address this issue and fully utilize the TESS data, one must integrate Gaia data and ground-based observations. This allows us to identify contaminated sources and make better use of the frequency resolution provided by TESS.

• V551 Aur, which was known to exhibit pulsations in one of its components, was found to be a spectroscopic binary with membership to the cluster. To understand this system and model it accurately, high-resolution spectroscopic follow-up observations covering this star’s orbital phase are required. Due to the star’s faintness, we intend to use observations with a larger aperture telescope.

From isochrone fitting, we determined log age |$\sim$| 9.2, a metallicity |$\sim$|0.016, and a distance |$\sim$|1.2 kpc for the cluster. Our results are in agreement with previous determinations from Chehlaeh et al. (2018), but with slight differences (log age |$\sim$| 9.1, Z|$\sim$| 0.019, and d|$\sim$| 1.4 kpc in Chehlaeh et al. 2018), possibly due to the improved magnitude, colour, proper motion, and parallax from Gaia.

Our observations of NGC 2126 are limited to a few nights. More observations are required for a detailed frequency analysis of the detected targets. Therefore, we plan to conduct further observations for the cluster using the observing facilities at ARIES, Nainital.

ACKNOWLEDGEMENTS

We are grateful to the Indian and Belgian funding agencies Department of Science and Technology (DST) (DST/INT/Belg/P-09/2017) and Belgian Science Policy Office (BELSPO) (BL/33/IN12) for providing financial support to carry out this work. AD acknowledges the financial support received from Department of Science and Technology's Innovation in Science Pursuit for Inspired Research (DST-INSPIRE) Fellowship Programme (No. DST/INSPIRE Fellowship/2020/IF200245).

DATA AVAILABILITY

The data underlying this article will be shared upon request to the corresponding author. The observations reported in this paper were obtained using the 1.04 m ST, 1.3 m DFOT and 3.6 m DOT telescopes at ARIES, Nainital, India , the 2 m HCT at IAO, Hanle, the High Altitude Station of Indian Institute of Astrophysics, Bangalore, India and 2.4 m telescope at Thai National Observatory (TNO), Chiang Mai, Thailand. This research has made use of the 'Set of Identifications, Measurements and Bibliography for Astronomical Data' (SIMBAD) database operated at CDS, Strasbourg, France. 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 has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This paper includes data collected by the TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI).

Footnotes

REFERENCES

APPENDIX A: FIELD STARS

APPENDIX B: COLOUR–COLOUR DIAGRAM

APPENDIX C: TESS PHOTOMETRY

APPENDIX D: BINARY EPHEMERIS

APPENDIX E: EB MODELLING

APPENDIX F: LOW-RESOLUTION SPECTROSCOPY