© The Authors 2024

1 Introduction

During the last few years, studies have shown that the ‘doubly eclipsing nature’ (i.e. two periodic eclipsing signals coming from one point source on the sky) is probably much more common in stellar populations than originally thought. The first such example, named V994 Her, was found by Lee et al. (2008); however, since then quite an extensive collection of similar candidates have been detected. This is true namely due to the large photometric surveys, automated telescopes and satellites (see e.g. Kostov et al. 2024 or Zasche et al. 2022b). Nowadays, we know more than 900 such doubly eclipsing candidate systems.

However, the reason why we should still consider these systems as only candidates is the fact that the pure detection of two sets of eclipses coming from one point source on the sky cannot be taken as proof that these really constitute the bound quadruple system. For solid proof, one needs some additional information. Owing to their typical lower brightness, there exists no spec-troscopy for most of them; moreover, the pairs were not resolved into the double. Therefore, an easy and straightforward method would be the detection of eclipse-timing variations (hereafter ETVs) for both pairs on their mutual orbit. If we are able to collect an adequately large compilation of photometry and derive times of eclipses for both pairs, these would behave in opposite manners, and we would definitely be dealing with a real 2+2 quadruple. We used this approach in our previous studies (Zasche et al. 2022a,b, 2023), proving the quadruple nature of several such systems. However, it should be said that the number of such confirmed 2+2 quadruples is still very limited, and only a small fraction of the candidates were analysed in detail. Hence, any new contribution to the topic would be welcome.

A great benefit of such systems of four stars in one gravita-tionally bound system is that they put rather strict limitations on the light-curve (hereafter LC) solution and modelling of such a system. Both inner pairs must share the same age, same initial chemical composition, same distance, and so on. All of these are strict conditions that should be taken into account when constructing a proper model of both eclipsing pairs.

Moreover, the need for deeper analyses of such triple and quadruple systems with one, two, or even three eclipsing binaries (Powell et al. 2021) shows that their origin or formation mechanism has not yet been explained satisfactorily. There still exist competing theories involving the close encounters, disc fragmentation, and small-N-body dynamics (see e.g. Whitworth 2001; Kostov et al. 2021; Tokovinin 2021). Finally, such complex multiple systems are ideal astrophysical laboratories for studying the dynamical effects like the nodal precession, eccentricity variations, apsidal motion due to the Kozai-Lidov cycles, or the three-body dynamics in general (Pejcha et al. 2013; Borkovits et al. 2022).

2 Selected stars

Our system-selection method is quite straightforward and is part of our long-term effort. We are still collecting the data for several (dozens of) interesting systems both in the northern and southern hemispheres. When an adequate number of data points are collected, we proceed to the more detailed analysis of the particular star system. All of the targets were our new discoveries, and their basic information is summarised in Table 1. There, one can see the name of the particular star, its position on the sky, and some magnitude and temperature estimates. The latter should be only taken as a rough estimate with large uncertainty.

What should be noted here is the fact that most of the systems presented contain at lest one binary of a contact-like (or near contact) LC shape. This is still quite rare, as one can easily see from existing papers, since most of the detected, doubly eclipsing quadruples (via ETVs, or spectroscopy) still belong to the group of detached systems. This is quite a natural consequence of the topic; that is, the detached systems are generally easier to analyse, their LCs are not too blended for LC disentangling, and their spectral lines are typically not overly blended with each other.

What is clearly seen in Table 1 is that we are dealing with two different types of stars. Four stars from the Small Magellanic Cloud (hereafter SMC) galaxy are part of the early-type stars with high temperatures, while the four others are the northern-hemisphere stars detected in the TESS data and are of a later spectral type that is cooler and smaller than our Sun.

3 Data used for the analysis

Owing to the relatively low brightness of the stars, we only used the photometric data for our whole analysis. The best quality photometry is provided by the TESS satellite (Ricker et al. 2015). It provides us with the uninterrupted data sequence of 27 days in one TESS sector. For different stars, several sectors of data are sometimes available covering a few years.

Apart from the TESS data, the older photometric archives also were used for extracting the reliable data for each of the stars. This was not easy in cases where the amplitude of the photometric variation is small. The other older photometry is almost unavailable (for the southern-sky stars), especially for the fainter targets (see Table 1), while for the northern-sky stars pair B sometimes suffers from larger scatter in the ETV diagrams (due to the poor quality LC for pair B). A colour-coding scheme is also given below in our Fig. 1 for distinguishing between the different data sources. These were on the northern sky mainly the following surveys: ASAS-SN (Shappee et al. 2014; Kochanek et al. 2017), Atlas (Heinze et al. 2018), WASP (Pollacco et al. 2006), ZTF (Masci et al. 2019), NSVS (Woźniak et al. 2004), CRTS (Drake et al. 2017), and KWS (Maehara 2014).

For the stars in the SMC galaxy, we used the OGLE data obtained in filter I (Pawlak et al. 2013). These data were used in its III and IV phases, sometimes accompanied from phase II when available. The MACHO photometry (Alcock et al. 1997) was also used when available. Only for one system in the SMC (OGLE SMC-ECL-4756) was it possible to also extract useful TESS data for the analysis and use these data for both A and B pairs. For the rest of the SMC targets, their magnitudes are too low for TESS to provide reliable photometry.

Aside from these freely available data, we also used our own data obtained especially for this study. The new data obtained in recent years at different observatories are the following (plotted in red in Fig. 1):

• The 1.54 m Danish telescope on La Silla observatory in Chile, equipped with a CCD camera, data obtained usually in R, and I filters.

• The 65 cm telescope at Ondrejov observatory, Czech Republic, using a G2-MII CCD camera equipped with an R photometric filter.

• The 30 cm telescope at a private observatory in Veltěže u Loun, Czech Republic, equipped with an MII G2-8300 CCD camera.

4 Analysis

For the LC model, we used the best available dataset, which was the TESS one (the best sector with the shortest cadence data) for the northern targets, and the OGLE (I) one for the southern SMC targets. We carried out the LC analysis on these data using the PHOEBE program (Prša & Zwitter 2005).

The whole analysis followed this procedure. A preliminary LC fit of the more dominant pair was done, producing the residuals for the analysis of the other pair. The derived LC template was then used for our AFP method (Zasche et al. 2014) to derive the times of eclipses. With these eclipse times, better ephemerides of the binary were derived. With such an ephemerides, the LC was modelled again. With a better LC fit, the residuals were re-computed. Such an approach was used several times itera-tively until the individual fitting steps provided reasonable stable results.

We usually started with the assumption of equal masses (i.e. mass ratio 1.0) and the equal luminosities of both pairs (i.e. third light fraction 50%). Then, the third light was freed from constraints, as was also the mass ratio for some of the stars with better data and larger out-of-eclipse variations.

5 Results

In this section, we focus on the individual systems presented in our analysis. However, stars and their analyses are only briefly described since for most of them our study still represents their first publication and the method used was nearly the same for all of them.

Fig. 1 Fits for the LC and ETV data for both pairs for all of our analysed systems. Full dots denote the primary eclipses, open circles the secondary ones. Different sources of data points are distinguished by different colours.

5.1 OGLE SMC-ECL-2339

The first star studied was OGLE SMC-ECL-2339, which is part of the Small Magellanic Cloud galaxy. This star had never been studied in detail, probably due to its low brightness. It is the faintest star in our sample. The star was not recognised as a doubly eclipsing system before, and its only period is listed as 0.7288 days (Pawlak et al. 2016).

Due to our limited knowledge about the star, many assumptions have to be made. However, our analysis of both the LCs of pairs A (0.728836 d) and B (3.39576 d), together with the complete ETV analysis of both pairs, can be seen in Fig. 1. We plot the disentangled LCs from OGLE data (I filter) for both pairs A and B and the period variations of both pairs. It is obvious that these ETVs behave in opposite manners, confirming a quadruple 2+2 nature. Both inner pairs are circular. Parameters of our fits are given in Tables 2 and 3.

5.2 OGLE SMC-ECL-3075

The next system under our analysis was OGLE SMC-ECL-3075, which is somewhat similar to the previous one considering its faintness and depth of eclipses. The only detected period by Pawlak et al. (2016) is 1.35887 days.

Our detailed study revealed that the system also shows the second eclipsing period of about 2.41586 days for pair B. This pair is also slightly eccentric (eccentricity about 0.105; long term apsidal motion about 113 years). Besides the OGLE III and IV data, the older OGLE II and MACHO (Faccioli et al. 2007) data are also available. Thanks to these older data, it was much easier to detect the long-term period variation of both A and B pairs, as plotted in Fig. 1.

5.3 OGLE SMC-ECL-4756

The system named OGLE SMC-ECL-4756 is the brightest among the SMC stars in our sample. However, it had not been studied before, and its second eclipsing period is presented here for the first time.

Our analysis (see Fig. 1) shows that the second eclipsing pair B has a lower amplitude and longer period than pair A (for these reasons, our attempts to observe pair B were not successful at all). Both pairs show a contact-like shape of the LC, but the pair B is only marginally eclipsing. There was a surprisingly high difference between the amplitudes from the ETV analyses of A and B, indicating rather different masses of both pairs. However, our LC analysis led to rather comparable third-light fractions. The reason for this discrepancy remains a mystery. This is the only SMC system for which we also used the TESS data for the ETV analysis.

5.4 OGLE SMC-ECL-6093

The last SMC system was OGLE SMC-ECL-6093, which is the only SMC system observed spectroscopically. Its spectral type was published as B1.5V (Ramachandran et al. 2019). No other detailed information about the star was found in the published literature.

The analysis in Fig. 1 led to the parameters given in Tables 2 and 3. Here, the amplitude of the ETV for both pairs is in good agreement with the detected third-light values for A and B. Pair B was found to be eccentric (eccentricity 0.284; apsidal period of about 70 years). This system shows the longest mutual A-B period in our sample; it is over 30 years long, but it is still affected by large uncertainties.

5.5 GSC 01949-01700

The first presented system from our Galaxy is GSC 0194901700. This is probably the system with the latest spectral type classification according to the Gaίa DR3 information (Gaia Collaboration 2023).

We found both pairs of A and B LCs to be asymmetric. Therefore, we introduced a hypothesis of spots for both pairs for a proper modelling of the LC shape. Both pairs show circular orbits and are short-period pairs. There was quite a large difference between the resulting third-light fractions, while the amplitudes from the ETV analysis led to more similar pairs. Here, the predicted angular separation of the double to be resolved (according to Gaίa DR3 parallax) was computed to be of about 47 mas, which is probably feasible with the current high-angular-resolution technique. However, its low brightness would be problematic.

5.6 ZTF J171602.61+273606.5

The star ZTF J171602.61+273606.5 was discovered in the ZTF survey as an eclipsing binary by Chen et al. (2020). No detailed analysis was published.

Our analysis led to the findings in Fig. 1 and results given in Tables 2 and 3. A slightly asymmetric pair A was found; however, we did not use any spot hypothesis here. Pair B shows only a very shallow secondary eclipse, indicating a low luminosity of such a component. The predicted angular separation resulted in about 24 mas here.

5.7 WISE J210935.8+390501

The system named WISE J210935.8+390501 was detected as an eclipsing binary thanks to the WISE data (see Chen et al. 2018). Only the shorter orbital period of pair A with a period of about 0.33228 days was discovered, despite the fact that pair B shows even deeper eclipses.

Our analysis as presented below revealed that both A and B pairs are circular, and their masses are also similar to each other (both regarding the similar luminosities of both pairs and the similar amplitudes in the ETV diagrams). Here, we deal with the shortest mutual period of the quadruple, which is only approximately two years long. Thanks to this fact, we have several outer periods covered with data today, and the bound quadruple 2+2 hypothesis is definitely proven.

5.8 V597 And

The last system in our sample is named V597 And. It is also probably the most interesting one. This star was incorrectly listed as an RR Lyr star in Simbad according to Dimitrov & Popov (2007), who gave a period of about 0.2338 days. Its correct period is double this, and it is definitely an eclipsing type. This is the brightest star in our sample.

We followed more or less the same procedure as for the previous systems, using the TESS data as the best for the LC fitting. There appears that the LC shape of pair A shows asymmetry, probably due to surface spots. However, the different sectors of TESS data show a slightly different shape of the LC; hence, some spot evolution can be traced here. This complication makes the analysis more difficult; however, its long-term evolution of both the orbital periods is easily detectable here. This is especially true thanks to available data from the older photometric databases such as ASAS-SN, SuperWASP, Atlas, and KWS. The star has deep enough eclipses and a bright enough magnitude that both pairs were detectable in the older data. Despite the mutual A-B period of about 20 years, it is well covered with regards to data today, and its 2+2 quadruple nature has been proven.

There appears to be one problem with the most distant data point in our Fig. 1 near the year 2000, which comes from the published observation of the star in the GCVS catalogue (Samus’ et al. 2017). We are not able to identify this one observation with either pair A or pair B. Therefore, we mark this data point with a question mark in Fig. 1 and did not take it into account in our analysis. Assuming that it was the observation of pair A, it should indicate a longer mutual orbital period. On the contrary, if it belongs to pair B, then the ETV curve needs to be shorter or have a higher amplitude. We leave this as an open question since the star was classified as RR Lyr originally, so there is still a possibility that this observation indicates a maximum and not a minimum brightness.

The system is also closest to the Sun, according to Gaia DR3 (Gaia Collaboration 2023) only about 400 pc distant. Thanks to this value, the predicted angular separation of the double should be of about 57 mas. Such a separation is within the capabilities of the current technique, and the high-resolution methods should be used to resolve both components. Finally, the star was also found to contain probably another close faint component at about 3.6″. In the Gaia DR3 catalogue both the components share the same parallax and proper motion. The result is that this system is a rare quintuple system of (2+2)+l architecture.

6 Conclusions

We performed the very first analysis of eight new quadruple systems and proved their gravitational coupling. All of them are now proven to constitute rare 2+2 quadruples thanks to the long-term analysis of their ETV curves for both pairs. This was especially possible due to older photometric data from various databases, as well as thanks to our new dedicated observations of these systems.

All of the systems presented here are new detections; that is, they were not listed as candidate doubly eclipsing systems before. Most of the stars have their mutual A-B orbital periods of the order of years to decades. Such periods are nowadays short enough to be well covered with reliable observations. On the other hand, they are long enough that the dynamical interactions of the two doubles are small enough to be almost negligible. For the most compact system (i.e. this one, where P_AB/P_A, or P_AB/P_B is the smallest), WISE J210935.8+390501, the period of these long-term perturbations ($P_{AB}^2/P_A,\text{ or }{P}_{AB}^2/P_B$, see e.g. Borkovits et al. 2015), is several centuries long. Hence, it can definitely be ignored in our dataset spanning a few decades at maximum.

However, one can ask whether it is still important to discover such systems of 2+2 architecture and derive their physical and orbital properties. Ten years ago, this was new, and only a few such 2+2 stars were known at that time. Nowadays, we have several dozen of them, with new ones being discovered every year. Besides filling the statistics these systems can help us to understand their formation mechanisms, indicating that there are probably different formation routes for the 2+1+1 and for 2+2 quadruples (Tokovinin 2021). Moreover, such complicated tight multiples can sometimes offer us a possibility of detecting effects that are otherwise undetectable (Borkovits & Mitnyan 2023).

Acknowledgements

We do thank the SuperWASP, ZTF, ASAS-SN, NSVS, KWS, CRTS, Atlas, OGLE, MACHO, and TESS teams for making all of the observations easily public available. We are also grateful to the ESO team at the La Silla Observatory for their help in maintaining and operating the Danish 1.54m telescope. I. Šándorová is acknowledged for some preliminary reduction of the data. 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 research of P.Z., J.K., and J.M. was also supported by the project COOPERATIO - PHYSICS of Charles University in Prague. 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. 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 Fits for the LC and ETV data for both pairs for all of our analysed systems. Full dots denote the primary eclipses, open circles the secondary ones. Different sources of data points are distinguished by different colours.
In the text