© ESO 2022

1. Introduction

The role of classical eclipsing binaries in current astrophysical research has been discussed elsewhere (see for example Southworth 2012). However, in the era of huge surveys and extremely precise satellite data, the objects with double eclipsing features can also be discovered and studied. The use of such objects brings us new insight into stellar formation mechanisms (see for instance Tokovinin 2021). However, many more constraints should be taken into account when dealing with two binaries in one system – independent analysis should lead to the same distance, same age, same metallicity, and so on.

The doubly eclipsing binaries as an independent topic is still rather new. The first of its kind (V994 Her) was discovered by Lee et al. (2008). It was about ten years later when the mutual orbit of the two pairs was definitively confirmed; hence, it comprised a still rather rare 2+2 quadruple system. This is quite a typical situation due to the long time taken to collect the data needed for the confirmation of the mutual orbit via eclipse-timing-variation (ETV). The only comprehensive study of such fascinating objects is that published by our group in Zasche et al. (2019). Since then, several new discoveries of doubly eclipsing objects have been published in the last years, mostly based on new and extremely precise data from the TESS satellite (Ricker et al. 2015).

The complex nature of these quadruple systems make them ideal astrophysical laboratories, especially for the celestial mechanics and the dynamical evolution of their orbits, the role of their orbit-orbit interaction, the role of Kozai cycles, the long-term evolution of the orbits such as the precession of the two eclipsing pairs, or the detection of small changes of other their orbital parameters. All of these are ongoing problems and only a thorough dynamical modelling would we be able to answer all these questions. However, for a proper modelling, one needs precise input data concerning individual physical parameters of the components, as well as of their orbits. Hence, our present study should be considered as a starting point in such an analysis.

2. System selection

Most of the systems showing two sets of eclipses published in the last few years were detected in the TESS data (and sometimes also proved to orbit each other after carefully analysing the photometric TESS data). These data have a huge advantage not only due to their precision, but also thanks to their cadency and time coverage. For these reasons, the detected periods of their mutual orbits are usually rather short (sometimes only hundreds of days, see Kostov et al. 2021), but the depths of eclipses are sometimes rather shallow. What is obviously not a problem for TESS photometry can sometimes be quite tricky for a normal ground-based observatories.

Knowing all these reasons and limitations of the TESS data (only a short time span is currently covered), we decided to follow a different method. Our new discovered systems on the northern sky are those found by chance as a by-product in some fields already monitored with our small-aperture amateur telescopes located in the Czech Republic and controlled by one of the authors (Z.H.). The one complementary system from the southern sky is located in the SMC galaxy and was included in this study due to the fact that our photometric monitoring of the target during the last years finally led to definite proof that both the ETVs show evidence of their orbit, hence constituting a real quadruple.

Some of the candidates for new doubly eclipsing systems were omitted due to a possibility that the two periodic signals come from different sources on the sky. When two very close companions are located on the sky, and we are not able to define the common origin with our technique, we skip such a star for the moment. For all of the presented systems, no close companion is evident in various sky surveys, and such systems are very likely doubly eclipsing quadruples.

3. Data availability

Dealing only with the photometry, we mainly used the TESS data, together with our own ground-based ones. The TESS photometry was extracted using the raw TESS data with the lightkurve tool (Lightkurve Collaboration 2018). The usual problem with too large TESS pixels (hence a possible influence of other close-by targets in the vicinity of the main star), other than the necessity of additional flattening of the TESS fluxes, which is discribed elsewhere (Feinstein et al. 2019), one has to be cautious with the interpretation of the results and its comparison with other photometric data.

The ground-based observations were obtained by one of the authors, Z.H., at his private observatory in Veltěže u Loun, Czech Republic. A quite untypical observational setup was used due to the filter used for that data, namely the astrophotographic Baader UV/IR Sperrfilter. It very effectively cuts almost all signal below 400 nm and above 700 nm. Maybe the most similar among other standard photometric filters is the transmission curve of the former SuperWASP filter (Pollacco et al. 2006). Therefore, for the proper modelling of the limb-darkening in the PHOEBE light-curve analysis program (Prša & Zwitter 2005), we adopted coefficients for the SWASP band in our modelling, which we do not expect to affect our analysis significantly.

For the one southern object, the follow-up data were secured using the 1.54-m Danish telescope located in La Silla, Chile. A standard I band Cousins filter was used (Bessell 1990). For the reduction of the data, the dark frames, flat fields, and standard aperture photometry were used to extract the data.

4. Analysis

Due to the fact that our whole analysis is solely based on photometry (unfortunately, the systems are rather faint for obtaining good spectra), we used the program PHOEBE, which is based on the Wilson-Devinney method (Wilson & Devinney 1971). It uses classical Roche geometry, and this approach was adequate for our study. However, several simplifying assumptions have to be made due to missing spectroscopy. The problem of deriving the mass ratio for detached binaries solely from the photometry was discussed elsewhere (see for instance Terrell & Wilson 2005). However, for the precise TESS photometry and more compact binaries with large out-of-eclipse ellipsoidal variations, we can also try to fit the mass ratio as a free parameter.

Another way to simplify assumptions is to keep the synchronous rotation (that is F = 1.0), albedo, and gravity brightening fixed at their suggested values according to a particular temperature. The second most important issue is the temperature. Due to not having known the spectral types of either of the components, we used an effective temperature estimation based on Gaia data (Gaia Collaboration 2016) (see our Table 1). We are aware of the fact that these numbers could be shifted from reality due to several effects such as interstellar extinction or a combination of signals from four components into one flux, which we then analysed with a simple single-component approach. However, this rough estimation is still better than using a reddened (B − V) index, and the resulting temperature ratio should at least be viable in this aspect.

Our entire approach comes from the fact that both the periods of these doubly eclipsing systems are already known (from our rough photometric data), and we can plot the phased light curves with both periods. Our method was the following. The dominant eclipsing binary (pair A) was first modelled in PHOEBE, but still using complete photometry of both pairs together. Then, this model was subtracted from the original photometry. On the residuals, pair B was analysed and modelled. With this model, we made a preliminary period analysis using our automatic-fitting-procedure (AFP) method (Zasche et al. 2014). This approach uses a light-curve template on shorter time intervals to derive both primary and secondary times of eclipses simply via shifting the template over the data in the x and y axes. With the resulting times of eclipses, one can easily see whether curvature of the ETV curve is evident or not. One can obviously also correct the orbital period of pair B. With this corrected period, a new analysis of pair B was carried out. Such a fit of pair B was then subtracted from the complete photometry of both pairs. Hence, we obtain only the residuals containing solely pair A data. The analysis of pair A was done in a similar way. This approach was repeated several times to obtain really symmetric and flat residuals when subtracting both pairs from the complete photometry. Obviously, when the curvature of the ETV is too large and adequately short, we have to use different ephemerides and orbital periods for different epochs of data, and the whole process is repeated independently several times on shorter time intervals.

The fitting process of light-curve analysis starting from the easiest assumption that both pairs are similar, that is third light is 50% for both. Then, this assumption was left and the third light was also set free and computed as an independent parameter. However, its significance is rather low in poor data and these ones with only shallow eclipses. Naturally, the resulting third light ratio should be in agreement with the resulting mass ratio of both pairs from our analysis. However, this is sometimes rather problematic due to additional light in the aperture in large TESS pixels.

5. Results

Here, we focus on the individual systems of our analysis in more detail.

5.1. CzeV2647

The very first system in our analysis is named CzeV2647, or UCAC4 642-089788, discovered as a variable about a year ago by the author Z.H. at his private observatory. The dominant variation with a period of about 0.57 days is also clearly visible on the data from surveys ASAS-SN (Shappee et al. 2014, and Kochanek et al. 2017), ZTF (Masci et al. 2019), and of course by TESS. The object was discovered by chance in some already monitored fields and was only recognised as a variable from the ATLAS survey (Heinze et al. 2018), which gave an incorrect period about half of the true one. Due to its low brightness, unfortunately only very little is known about the star. Even the temperature estimations range between 4900 and 8530 K. Its distance seems to be about 1 kpc from the Sun (Bailer-Jones et al. 2021).

We collected all available data for the star, finding that the best for the analysis are our own ground-based data together with the TESS satellite data (especially Sector 41 with 2-min cadence). The latter is for the precise light-curve analysis to derive the physical parameters, while the former was used to derive many useful times of eclipses of both pairs for ETV analysis, proving that the system is a real quadruple one. As one can see from the final light-curve fits (see Fig. 1), both the binaries show very different eclipse depths. Pair A shows about five times deeper eclipses, which is the reason why the star was being missed as a doubly eclipsing one.

Fig. 1. Light-curve fits of CzeV2647 of both eclipsing pairs, as resulting from PHOEBE (see the text for details). TESS data are compared with our ground-based data obtained by one of the authors (Z.H.).

The analysis in PHOEBE yielded the parameters given in Table 2, showing that we are dealing with two detached binaries. Pair A is the dominating one in luminosity (from photometry) as well as in mass (from ETV, see below). Due to good coverage and precision of the TESS data for pair A and its deep eclipses, we also tried to fit the mass ratio here. As can be seen from Table 2, such a task also for the shallow pair B is much more tricky, and the result is affected by large errors. Owing to the missing spectroscopy, only relative values of radii can be derived properly. Moreover, the star was found to be located close to the resonant ratio of its orbital periods at only 1% from 3:5.

What is quite surprising is the fact that both solutions provided with TESS and our ground-based data (see Fig. 1 for comparison) resulted in quite different parameters, and the picture of the system is still questionable. We are aware of the fact that the PHOEBE code itself usually provides rather unreliably small errors of individual parameters, but a real uncertainty of the particular one can better be estimated as a difference between these two solutions. As one can see, values such as mass ratios or inclinations resulted in very different values, and hence the solution should be taken with caution. However, its overall shape as detached binaries with the pair A being the dominant one is in agreement within both these solutions. We can only speculate about the origin of such a discrepancy. The ground-based data are of much worse quality; moreover, they also show slight asymmetry of both light curves outside of minima, which can only be fitted with a spot on the surface. However, due to the non-detection of anything similar in the TESS data, we did not include any such hypothesis in our analysis. Quite surprising is the fact that we detected much higher contributing third light for both binaries from our ground-based photometry, despite the fact that it uses smaller pixels and higher angular resolution than the TESS data. High values of both third lights are certainly responsible for both inclination angles being higher than from TESS photometry due to their strong correlation. However, we were not able to trace why is it so. The explanation may be that the ground-based photometry itself is not precise enough for independent derivation of its mass ratio, and this value should also be kept fixed from the TESS solution. Fixing it to its proper value would also shift the whole solution and maybe diminish the third-light inclination problem.

The long-term evolution of both A and B orbits can be studied using the eclipse timing data (so-called ETV analysis, or the O − C diagram analysis). We used the same approach in our former analysis of doubly eclipsing systems CzeV1731 (Zasche et al. 2020), TIC 168789840 (Powell et al. 2021), or all systems in Zasche et al. (2019). Such a method is very powerful and effective when good data are available, adequately distributed in time, and well covered for both A and B binaries. This is partly true here for the dominant pair A, but quite problematic for pair B. As one can see from Fig. 2, the variation of eclipses with the orbital period of about 4.55-yr is definitely present in pair A. For pair B, this is still barely visible. The very first eclipses around the year 2000 show us that no mass transfer is present in the system and its orbital period is stable over longer time intervals. The only definitely detectable shapes of the ETV for pair B are both periastron passages on the long orbit, but the overall shape of the variation is derived mainly from the precise pair-A data. The amplitudes of both ETVs of about 0.01 days are about the minimum of what can be detected with our method with such a quality of data. The lower amplitude of ETV for pair A is to be expected due to its higher luminosity (and hence also expected higher mass), but the amplitude of pair B is still affected by large uncertainty now. More precise data secured in the upcoming years would be very welcome to constrain the orbit with higher fidelity.

Fig. 2. ETV diagram of CzeV2647 showing two sets of eclipses of A and B pairs as resulting from our analysis. Theoretical curve representing the fit given in Table 3 is plotted as a red solid curve, while the black dots stand for primary eclipses and the open circles for secondary ones: the larger the symbol, the higher the weight.

5.2. CzeV1645

The next stellar system discovered as suspected for doubly eclipsing nature is named CzeV1645, or UCAC4 755-079509. In the ASAS-SN database, it is also detected as variable and named ASASSN-V J232332.11+605312.9. Hence, its dominant 1.09-day variation was already being recognized. We call this pair A, while another periodicity with period of about 1.66 days was also detected. This is quite a typical situation – the pair already detected is usually the one with larger photometric amplitude, or the one with shorter orbital period. Only very limited information was published for this star, and it even lacked a proper parallax value from the Gaia DR2. However, the TESS Input Catalog (Stassun et al. 2019) gave its distance as 1413 pc.

Firstly, we used the TESS data for our light-curve modelling of both pairs, using a similar approach as for the previous system. The results with the theoretical fit are given in Fig. 3, while the parameters are written in Table 2. As one can see, both pairs have well-covered light curves, and their eclipse depths are also comparable and relatively large. It is therefore quite surprising that the secondary periodicity was not detected earlier. However, when downloading and processing the TESS data, we found a problematic issue with that photometry. There were different trends in the data, which we needed to subtract with a polynomial fitting, but what we were not able to remove is the different amplitude of pair B in different time epochs. Such an effect is even more visible when comparing sector 1 and sector 3 of the TESS data (very incomparable eclipse depths), but an artifact of this is also visible in Fig. 3 regarding pair B. What is quite surprising is the fact that the secondary eclipse of pair B is not affected, but the primary one is. We have no clear explanation for this strange behavior. Rapid inclination change due to dynamical effects of the other pair would cause both eclipses to change their depths. An instrumental effect of the TESS data reduction would be a possible source of this discrepancy. Perhaps a more sophisticated approach for the TESS data or the emergence of new TESS data is needed here.

Fig. 3. Light-curve fits of CzeV1645 of both eclipsing pairs, as resulting from PHOEBE; see the text for details. TESS data are being compared with our ground-based data obtained by one of the authors (Z.H.).

For this reason, we also used another source of the photometry. Namely, our new ground-based data obtained by the author (Z.H.). The comparison of these data with the former TESS ones is being plotted in Fig. 3. As one can see, both shapes are similar, but not equal (due to different eclipse depths due to different pixels and apertures used), but their relative precision is well-comparable. What we found on our ground-based data is some slight asymmetry on the light curve of pair A (in the out-of-eclipse region close to phases 0.35 and 0.7), possibly explainable with the presence of a spot on the surface of the star. However, the strange change of the depth of primary eclipse of pair B is definitely not visible in our data, despite these ranging over a longer time interval than the TESS ones.

Our light-curve modelling resulted in two well-detached systems of both A and B pairs, but whose two orbital periods are surprisingly close to a 2:3 mean motion resonance. Therefore, it should be studied in more detail. Not so surprising is the fact that in using the TESS data, the overall sum of the two third-light values of both pairs was over 100%. This is something naturally explainable by the fact of large TESS pixels, and some additional light from some close-by source, not connected with the quadruple itself. Nothing like that is visible in our data with a much better angular resolution. There is some discrepancy between these two solutions (see Table 2), mainly for pair A. This can also be caused by larger luminosity contamination of the TESS data, and perhaps also by the fact that with such a shape of the light curve one cannot reliably derive the mass ratio from photometry alone due to the small out-of-eclipse variations (Terrell & Wilson 2005).

An analysis of all available photometry to derive the eclipse times of both pairs was also carried out. Besides the TESS data, we also used our new ground-based photometry of the target, together with the ASAS-SN data and the ZTF as well. On all of these datasets both eclipses are clearly detectable due to their relatively large depths. This is also the reason why the derived times of eclipses are of good precision for the ETV analysis (see Fig. 4, showing that practically no visible variation is evident here). Pair B would show some mild ETV signal, only very barely visible for A. However, its nature and period is still very hard to estimate. It is probably rather long (over 6 years), and its almost non-detection for pair A would be caused by its lower amplitude (probably due to the higher mass of pair A). Therefore, we call for another observation of both pairs in the upcoming years for its final confirmation.

Fig. 4. Long-term study of eclipse times of CzeV1645; see the text for details.

5.3. CzeV3436

The only one system in our sample obviously showing two short, periodic, contact-binary-shaped light curves is CzeV3436 (or also UCAC4 631-070565), which should be visually classified as EW+EW or EB+EW. However, due to its low brightness we have only very limited information for a proper analysis. Not only is the effective temperature unsure, but for the distance different authors also give very different values: 8.2 kpc (Gaia EDR3, Gaia Collaboration 2021), 5.7 kpc (Anders et al. 2019), 5 kpc (Bailer-Jones et al. 2021), and 11.6 kpc (Stassun et al. 2019).

There is no variation visible in the ASAS-SN data or on ZTF. This is caused by its faintness and low amplitude of photometric variations. Hence, the only source of photometric information here is the TESS satellite and our new ground-based photometry. The analysis was carried out iteratively in several steps subtracting the particular pair for analysis of the other one. However, after a few iterations, we finally reached a solution, which we feel is a reasonable one (see Table 2 for the final parameters and a plot of both curves in Fig. 5). The light curve solution based on the TESS data provide quite a reasonable picture of the whole system. In agreement with a similar depth of eclipses of both pairs, their luminosity contributions are also close to 50%. However, similarly to the previous case, we found an asymmetric light curve from our ground-based data here. For this reason, its fit is not as good as the TESS one, and it is affected by larger errors. We can only speculate about the reason for such deviation.

Fig. 5. Light-curve fits of CzeV3436 of both eclipsing pairs, as resulted from PHOEBE; see the text for details. TESS data are compared with our ground-based data obtained by one of the authors (Z.H.).

However, despite all these problems, we also performed the long-term analysis of its period changes. Both pairs were studied separately, but only based on TESS and our own ground-based data. For this reason (see Fig. 6), we were not able to trace any visible variation in either of the pairs. Their mutual orbit should be longer (or face-on oriented orbit), and only future observations would help to answer the question about their mutual gravitational interaction.

Fig. 6. Long-term constancy of period of both pairs of CzeV3436.

5.4. OGLE SMC-ECL-1758

The only system of our Galaxy, located in the SMC field of the OGLE survey, is named OGLE SMC-ECL-1758. It was also added into our study due to the fact that it was observed several times during our monitoring of other targets in the SMC fields. Therefore, we have adequately large observational material besides the OGLE data to perform a thorough analysis of both detected pairs and to reveal its structure. The star is already known as a doubly eclipsing one since Pawlak et al. (2013) detected both eclipsing periods. It is quite a typical star in the SMC galaxy, but very little is known about it at present, and no spectral type estimation was published.

We analysed all OGLE data (OGLE II+III+IV phases of the survey), together with the older MACHO data. In all of these observations, both eclipses are clearly visible because its amplitude is quite large, of about 0.2 mag (for pair A, slightly less for pair B). The parameters of the light curve fits are given again in Table 2, while the fits of the OGLE IV data are plotted in Fig. 7. We only used the more precise OGLE IV data for the light-curve modelling due to their best quality among all available datasets. The observation from MACHO provides more data points, which are spread over longer time interval, but they are of worse quality. As for the curiosity, we also plotted the TESS data of the target in Fig. 7 to verify that these data definitely cannot be used for any light curve modelling. However, we used them at least to derive the eclipse times of both pairs for construction of the ETV diagrams. The bad quality and uselessness of the TESS data is due to their low angular resolution and very large pixels. This always causes problems of contamination of other close-by sources in such dense fields (as this one is in central parts of SMC), making all variations much smaller. Here the amplitude of photometric variability is about seven times lower than for the OGLE IV data.

Fig. 7. Light-curve fits of OGLE SMC-ECL-1758 of both eclipsing pairs, as resulting from PHOEBE; see the text for details. The TESS data are only shown here to compare their quality, and they are not used for the light-curve solution.

Because of much worse quality of the data comparing with the previous three systems, we used a slightly different approach here. To compute the mass ratio of the eclipsing pairs, the photometry itself is only slightly sensitive and usually derived with large uncertainty. This especially applies for the detached systems (see for example the study by Terrell & Wilson 2005). For this reason, we used an alternative method of mass ratio computation, introduced by Graczyk (2003). Instead of directly computing the mass ratio as a free parameter in a light-curve fitting, the mass ratio was computed from the luminosity ratio of both components (each step of the light-curve fitting in PHOEBE provides bolometric magnitudes of both components). Naturally, it works with the assumption that both components are the main sequence stars following a standard mass-luminosity relation. Both of the inner eclipsing systems were analysed in this way. Quite surprisingly, both of these inner mass ratios were about the same (of about 0.62–0.63). Moreover, as found from the ETV analysis of both pairs (see Fig. 8), both pairs are of about the same mass (q_A/B = 0.95). This is quite a surprising finding, possibly telling us something about the formation mechanism of such a system. As stated, for example, in the recent study by Tokovinin (2021), the 2+2 quadruples may be formed via several mechanisms acting together: core fragmentation, disc instability, dynamical interactions, and accretion-driven orbital migration. In this aspect, our presented system can shed new light on these interesting systems and their statistics. Such a system having roughly the same mass of both binaries (that is q_A/B ≈ 1), but with quite different components within these binaries (q ≈ 0.6), is not very typical (see for example Fig. 5 in Tokovinin 2021).

Fig. 8. ETV diagram of OGLE SMC-ECL-1758 showing that both inner binaries A and B show significant period variations. This is a consequence of their mutual motion around a common barycenter.

Moreover, it was also found that pair B has slightly eccentric orbit (barely detectable at a level of e ≃ 0.01, with possible apsidal motion about 100 yr long). However, such a small eccentricity is barely detectable with the quality of the available data. Finally, it is quite interesting to note that both inner periods, P_A, and P_B, are less than 0.5% away from the 1:4 resonance. Whether it says something about the resonances themselves, or whether it is pure coincidence, we leave as an open question.

6. Discussion and conclusions

We carried out an analysis of four doubly eclipsing systems that had never been studied before, among which three were discovered and published for the first time here. These new systems are being added to a still rather limited group of doubly eclipsing systems (today with 159 members). However, it is interesting that a large majority of these systems are detached and have longer periods, rather than being short-period contact stars. Two such detached light curves are more easily being disentangled from each other; hence, their analysis is more straightforward. Moreover, to prove that these are not pulsations but two real contact binaries can also sometimes be rather tricky. For this reason, we feel that our system CzeV3436 is quite rare in this aspect. The same also applies for the star CzeV2647, which could easily be missed as a doubly eclipsing one because of its very low photometric amplitude of pair B, and only a careful visual inspection of very precise data would be able to detect such a system. All of these should be taken into account when considering the statistics of these rare systems.

Two out of four analysed systems were detected to orbit around a barycenter, hence constituting a real 2+2 quadruple. The other two systems are waiting for more data to become available in the next few years to prove such orbit. Due to our limited data coverage in time for all these systems, we believe that most of these doubly eclipsing systems orbit each other, and only a very limited portion of them will later be found as chance alignments. For example, for CzeV1645 we believe that new observations in the upcoming years will confirm our hypothesis of mutual motion and detect the period (maybe several decades long).

One may also ask whether we can also detect the mutual movement and the orbit using another means. But spectroscopy needed for such a confirmation would be rather time-consuming due to the detection of changes in systemic velocities over a longer time interval (because the orbits are rather long), and the targets are quite faint for good spectroscopy. Another possibility is to detect the two pairs as a visual double. However, both detected quadruples are rather distant (that is their angular separation is too small: a few mas for CzeV2647, and below 1 mas for OGLE SMC-ECL-1758), but a more serious limitation is that both the stars are too faint for good interferometry. Hence, we cannot hope for any interferometric detection of the two close components to prove our hypothesis independently with another method. Therefore, the photometry remains the last remaining and most suitable method for proving such binaries orbit and constitute real quadruples.

Acknowledgments

An anonymous referee is being acknowledged for useful comments and suggestions, greatly improving the manuscript. We do thank the NSVS, ZTF, ASAS-SN, TESS, and OGLE teams for making all of the observations easily public available. The research of P.Z. was supported by the project Progress Q47 Physics of the Charles University in Prague. We are also grateful to the ESO team at the La Silla Observatory for their help in maintaining and operating the Danish telescope. 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. The observations by Z.H. in Veltěže were obtained with a CCD camera kindly borrowed by the Variable Star and Exoplanet Section of the Czech Astronomical Society. We would like to also thank M.Mašek and H.Kučáková for obtaining the photometric data for CzeV2647, and for observations of OGLE SMC-ECL-1758 J.Merc, J.Janík, E.Paunzen, and M.Zejda as well. This research made use of Lightkurve, a Python package for TESS data analysis (Lightkurve Collaboration 2018). This research has made use of the SIMBAD and VIZIER databases, operated at CDS, Strasbourg, France and of NASA Astrophysics Data System Bibliographic Services.

References

All Tables

All Figures

	Fig. 1. Light-curve fits of CzeV2647 of both eclipsing pairs, as resulting from PHOEBE (see the text for details). TESS data are compared with our ground-based data obtained by one of the authors (Z.H.).
In the text

	Fig. 2. ETV diagram of CzeV2647 showing two sets of eclipses of A and B pairs as resulting from our analysis. Theoretical curve representing the fit given in Table 3 is plotted as a red solid curve, while the black dots stand for primary eclipses and the open circles for secondary ones: the larger the symbol, the higher the weight.
In the text

	Fig. 3. Light-curve fits of CzeV1645 of both eclipsing pairs, as resulting from PHOEBE; see the text for details. TESS data are being compared with our ground-based data obtained by one of the authors (Z.H.).
In the text

	Fig. 4. Long-term study of eclipse times of CzeV1645; see the text for details.
In the text

	Fig. 5. Light-curve fits of CzeV3436 of both eclipsing pairs, as resulted from PHOEBE; see the text for details. TESS data are compared with our ground-based data obtained by one of the authors (Z.H.).
In the text

	Fig. 6. Long-term constancy of period of both pairs of CzeV3436.
In the text

	Fig. 7. Light-curve fits of OGLE SMC-ECL-1758 of both eclipsing pairs, as resulting from PHOEBE; see the text for details. The TESS data are only shown here to compare their quality, and they are not used for the light-curve solution.
In the text

	Fig. 8. ETV diagram of OGLE SMC-ECL-1758 showing that both inner binaries A and B show significant period variations. This is a consequence of their mutual motion around a common barycenter.
In the text