Period ratios and observation of noticeable resonance at 3:2 for 2+2 quadruple systems

ABSTRACT

Quadruple stellar systems with two mutually orbiting eclipsing pairs (2+2 quadruples) are currently of great scientific interest because they offer a unique opportunity to determine the physical parameters of their constituent stars with high precision. In this study, we enlarge their numbers and present an analysis of the orbital periods and their ratios for a sample of 781 quadruple candidates with a 2+2 structure. Moreover, we compare the observed distribution of period ratios to a simulated prediction based on a uniform distribution of inner periods. We find a preference for a 3:2 resonance between the inner orbital periods, while the 1:1, 4:3, and 5:3 resonances do not deviate significantly from the predicted distribution model. Higher resonant values are on a slight decline, probably due to a lack of data. These results, derived from the largest statistical ensemble of 2+2 quadruples to date, provide evidence for the hypothesis that gravitational interactions between the star pairs can preferentially lead to a 3:2 resonance during their evolution.

1 INTRODUCTION

Stars are categorized into single stars, binaries, or higher order multiple systems. Binary and high-order fraction increases for more massive objects (Offner et al. 2023). Usually, we observe multiple stellar systems in stable configurations. In the case of four component systems, two possible hierarchical structures exist. The first possibility is a 3+1 configuration, a triple star with an additional distant component. The 2+2 system consists of two binary pairs, A and B (Fig. 1), which orbit together around their common centre of mass. More known quadruples exist in the 2+2 configuration than in the 3+1 (Tokovinin 2014, 2023), because their geometry sometimes allows both pairs to be eclipsing binaries. Therefore, this object class is often referred to as a doubly eclipsing system. In fact, 2+2 quadruples, especially those containing Algol-like systems, enable us to obtain the physical parameters of individual stars and a deeper understanding of the evolution of these higher order multiple systems.

Zasche et al. (2019) first introduced the general statistics for 2+2 doubly eclipsing and non-eclipsing systems and increased their numbers. Photometric surveys, satellites, and ground-based observations represent a powerful tool for detecting and calculating the pair's mutual motion around their common centre of mass. Using the $O-C$ diagrams (also known as eclipse time variations ETV), we can study the doubly eclipsing star on a long time-scale and detect the light time travel effect (LiTE) for both pairs. The LiTE periods of the binaries have to be the same. However, the $O-C$ variations are in antiphase. The LiTE amplitudes $A_{\mathrm{A}}$ and $A_{\mathrm{B}}$ can be different but in most cases they are similar. The amplitude ratio gives us the mass ratio $q_{\mathrm{AB}}$ between both binary pairs

where $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are the corresponding binary masses. Deriving the mass ratio is one of the main parameters of the quadruple stars research. Moreover, we can obtain the following parameters from the light-curve analysis, such as relative radii or surface temperatures, and thus, it is fully desirable to obtain as large a database of these 2+2 quadruples as possible.

Nowadays, the number of doubly eclipsing candidates has started growing rapidly (Kostov et al. 2022; Zasche, Henzl & Mašek 2022b), especially with TESS (Ricker et al. 2015) and other photometric surveys. However, only a few candidates (60 systems in our used sample) are confirmed 2+2 quadruple stars with at least estimated orbital parameters for their common outer orbit. The first statistics of 146 2+2 systems and their candidates were done by Zasche et al. (2019). They defined the period ratio $R=\frac{P_{\mathrm{A}}}{P_{\mathrm{B}}}$⁠, where $P_{\mathrm{A}}$ and $P_{\mathrm{B}}$ are inner periods for binary pairs A and B, respectively. The period ratio distribution indicated some interesting possibilities for resonant capture: The increase around 1:1 resonance surroundings was noticeable together with 3:2 resonant ratio overabundance. On the other hand, another first-order orbital resonance of 2:1 was in decline.

Breiter & Vokrouhlický (2018) studied the dynamics of the 1:1 resonance case. Their calculations stated that the 1:1 resonant capture was improbable and not highly expected. They also suggested to expand their work and to add the cases of 2:1 and 3:2 resonances. Tremaine (2020) described the conditions for resonant capture of 2:1 and 3:2 and discussed which period ratios are near or directly at the resonance. He also analysed and compared the results with the period ratio distribution from Zasche et al. (2019) and pointed out issues of systems near 1:1 resonance but not generally at the precise resonant value. Interestingly, decreased density in the 2:1 ratio was observed instead of the expected enhancement. Only two systems were at the 3:2 resonance, and the other six were in its surroundings. However, these data sets still need to be larger to provide specific results. Tremaine (2020) presented essential questions at the end of the paper, i.e. what is the exact shape of the period ratio distribution, how far from the given ratio can we still consider a resonance, and what the statistics look like for each spectral type. We open up and add one more fundamental question to which there is currently no known answer: What this distribution would look like for systems in different evolutionary states? More 2+2 candidates with well-described inner orbital periods are needed for proper analysis and statistics. Enhancing the confirmed candidate numbers is also essential to improve the current research state.

In this paper, we comprehensively analyse the periods and resonances of currently available systems of 2+2 quadruples and their candidates. Moreover, we present newly detected candidates and additional used data. For the acquired data, we describe the general period statistics through observations and show the period ratio distribution and resonance statistics. Our results are evidence to specify the systems' closeness to resonant values at 3:2 and, therefore, resolve the issues suggested by Tremaine (2020). This work can be used to compare theoretical models and the observations in the planned paper mentioned by Breiter & Vokrouhlický (2018).

2 DATA COLLECTING

2.1 Small Magellanic Cloud

We searched for new doubly eclipsing quadruple candidates. We selected the OGLE photometric survey (Pawlak et al. 2013, 2016). Zasche et al. (2019) already inspected the light curves of the Large Magellanic Cloud (LMC). However, the Small Magellanic Cloud (SMC) was still not carefully and adequately searched for new candidates. We decided to manually inspect the phase-folded light curves of known eclipsing binaries (pair A in our notation) in the OGLE data base, looking for the detection of additional periodic variations corresponding to another eclipsing pair (B). Our criteria were system brightness constraints (up to 20 mag in I filter), a reasonable orbital period value of the known pair (our period range was set to 0.5–20 d), and a sufficient variations' amplitude 0.08 mag for pair A primary minimum. The 0.5 d minimum limit allows us to detect the additional brightness changes due to eclipses and distinguish between other possible variability causes. Our maximum period limit was set so that the brightness changes could be well measured against further surveys and ground-based observations. The amplitudes of both pairs had to be clearly visible through the scatter, which depended on the brightness. Thus, our typical brightness values were between 13 and 17 mag.

On the phase-folded light curve (according to pair A period) we could easily see other brightness decreases that are not phase-folded with a known pair A period. Usually, these eclipses go through the whole phase curve as random brightness dimming (Fig. 2 upper panel). We could see some kind of regularity if the binary pairs are in resonance.

It was necessary to check every phase curve visually to prevent instrumental error influences and false positive signals. A common issue was that the pair A orbital period in OGLE was half or twice the real value. This factor could lead to a strange phase curve shape, but it is usually well recognizable (the secondary minimum misses or the eclipses are repeating during one falsely determined cycle). The orbital period was also determined inaccurately in some cases, or there were other effects such as period change or apsidal motion. We could not clearly state the situation for eclipsing systems where the other possible pair B changes were lost in the noise (below $\sim 0.01$ mag), and sometimes, we detected outliers that were probably caused by instrumental faults. For some candidates, we were not able to detect the additional period. The cause could be a lack of data points or also not obvious outliers that do not have a physical origin. Another case of false positives was when the other signal had a different nature than the eclipses, typically pulsations, rotation effects, or unclear causes. We did not include these systems in our analysis.

The suspicious systems were further analysed using light-curve disentangling in silicups software (Cagaš 2017). Phenomenological models from Mikulášek (2015) were used for modelling the known pair A. After pair A subtraction, unphased pair B remains. We searched for its orbital period, and in some cases, the period was easily found by our estimations. After excluding false positive cases, we have found eight new doubly eclipsing candidates: OGLE SMC-ECL-1086 (with OGLE SMC-ECL-1087 as the possible pair B), 2417, 2541, 4569, 4595, 5925, 8061, and 8098. Two other systems studied by Zasche et al. (2024) were also independently found in our searching. We consider that $P_{\mathrm{B}}$ of OGLE SMC-ECL-2339 is half the value mentioned in paper above (1.69788 days). From the $O-C$ variations (Kolář et al., in preparation), we could state that OGLE SMC-ECL-6093 is probably a new confirmed quadruple system (also confirmed in Zasche et al. 2024). Detailed information about objects, especially their periods, are listed in Table 1.

2.2 TESS

Another main source of our search was the Transiting Exoplanet Survey Satellite (TESS) (Ricker et al. 2015). Due to its continuous measurements for a period of 27 d for each sector and very high precision, it is an excellent tool for detecting new periodic variations that could not possibly be seen or detected by other surveys. For the reasons stated above, we could detect binary pairs with brightness changes to 0.01 mag (depending on the total system brightness and current sector) and orbital periods with sometimes even more than 10 or 15 d with the careful combination of more TESS sectors. The light curves were disentangled using silicups software with the same process as in the previous subsection.

The disadvantage of TESS images is the low angular resolution. Thus, we had to deal with possible blends and false positive candidates. We checked the candidates pixel-by-pixel and used ground-based photometric surveys to identify the blends (follow-up observations would be the other helpful method, but it was not feasible due to the large total candidates number). Some of these targets were already published in Zasche et al. (2022b). Here, we introduce new doubly eclipsing systems without duplicity with other available mentioned sources.

2.3 Additional data

We took additional quadruple candidates available in the literature. We used data from Kostov et al. (2022, 2024), Zasche et al. (2019, 2022a, b, 2023, 2024), Vaessen & van Roestel (2024), Pawlak et al. (2013), Southworth (2022), Kounkel et al. (2021), Fedurco & Parimucha (2018), Fezenko et al. (2022), and Tokovinin (2023) together with our new candidates' parameters for the analysis.

For our purposes of period analysis, we selected the following system criteria: both inner periods have to be known for every object, the period lengths are a maximum of hundreds of days (usually several days), and also already known blends were discarded. Based on these parameters, our sample consists of 781 systems in total, mostly in our Galaxy and a small amount in the LMC and the SMC (Fig. 3).

3 ANALYSIS

3.1 Period ratio distribution

We investigated the known inner periods' relations in every given system. We calculated the period ratio R:

Period values were ordered to be $R\ge 1$ (longer period divided by shorter one). We made this sorting to get a uniform order. The pair naming, A or B, has no physical significance; pair A is usually the first detected binary. The detection depends also on brightness amplitudes, observation sampling, etc.

The period ratio distribution was plotted in the histogram in Fig. 4. The data show a decline with increasing R value with small deviations. We calculated the model prediction of the R distribution and compared it with the used data set. We assumed two uniform distributions for the inner periods $P_{\mathrm{A}}$ and $P_{\mathrm{B}}$ in the range between 0.2 and 20 d. The lower limit of the range can be set to 0.2 d because these are simulated data and we do not have observational constraints. We took two period values and calculated their ratio R. We repeated the procedure for 781 systems to get the same amount as the data sample. By this method we obtained one simulated period ratio distribution. We generated $500000$ simulations to derive the average model distribution. We plotted this model as our prediction together with the data histogram. Using the Kolmogorov–Smirnov (K–S) test, there is a high probability of $93\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ that data and model statistically correlate (the comparison was made to the maximum value of $R=5$ as in the histogram in Fig. 4, because higher R values had too few known systems). We could still see some minor differences, probably caused by a lack of data.

3.2 Resonant values

The relatively large sample of the candidates allows us to investigate resonant ratios in more detail. First, we had to choose which resonant values could be reliable and have sufficient data for analysis. We decided to study the smallest integers and half-integers and added the nearest third values. Thus, the studied resonances were 1:1, 4:3, 3:2, 5:3, 2:1, 5:2, 3:1, 7:2, and 4:1. Higher values had no sufficient coverage in the sample. We aimed primarily for the most discussed resonances, 1:1, 3:2, and 2:1, which could have, according to previous literature, some kinds of deviation compared with the predicted expectations.

Another crucial parameter is the condition under which R goes into some of the mentioned resonant ratio. We defined the closeness C to the resonance, represented by the following equation, and two closeness vicinity ranges

The critical role of the C vicinity range was to have sufficient system numbers to detect some possible phenomena (overabundance/decrease). On the other hand, too wide a range is not suitable because the possible deviations disappear in the scatter of the wide vicinity. We defined the shorter range of resonance vicinity as $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ to fulfill the condition of having large enough systems. We took double the range with $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ for comparison. We considered systems that fit within these intervals for the selected resonance values to be close to the given resonance. The vicinity limits for both ranges are shown in Table 2.

We tested the properties of the resonant values. For each C range (⁠$1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ and $2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠), we plotted the resonance diagrams (Figs 5 and 6) using our data together with the simulated model. The data and model comparison is also shown in Table 3. The model predicts how many systems we should detect for every resonance. The model uncertainty is $\pm 1$ for every resonant ratio. Resonance comparison diagrams indicate some interesting behaviour.

• The 1:1 resonance corresponds very well with the assumed prediction, with no significant differences. The number of systems is not large compared to the following resonances, because the C ranges are the smallest for 1:1 resonance (depending on the C definition as the fraction between the R ratio and the given resonance).

• Resonant values 4:3 and 5:3 are in agreement with the model. There is only a negligible deviation for 4:3 in the case of $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠.

• We can observe an overabundance for the 3:2 resonant ratio. This excess is the most significant within our sample for both C ranges.

• 2:1 and 5:2 indicate a slight decline compared with the assumed model.

• The 3:1 and 7:2 ratios show slight changes between the C ranges. There are only two systems close to the 3:1 resonance with $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠, the 7:2 deviations are less significant.

• There is a small decrease in the 4:1 resonant ratio. However, 4:1 is the least represented element in the analysis and could be crucially influenced by selection effects.

3.3 Confirmed quadruple systems

Long-term monitoring of the selected candidates led to the confirmation of mutual motion around the common centre of mass. The outer period $P_{\mathrm{AB}}$ and other orbit parameters are usually challenging to derive. There are only a few well-described 2+2 system outer orbits (i.e. Zasche et al. 2019, 2022a, 2024)) with short orbital outer periods (in order of years). We can consider the system as confirmed when we can see non-linear changes for both binaries in the $O-C$ diagrams that correspond to each other. The $O-C$ variations can lead to confirmation even though the LiTE is covered partially. We included these cases in our analysis as sufficient confirmation of the quadruple nature.

Only 60 (⁠$7.5\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠) indisputably confirmed 2+2 quadruples are known in our sample. This small percentage is caused mainly by the orbital outer periods, which are usually very long (compared to the inner periods and the available observing time). There are 13 quadruples close to the resonances for $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ and 6 for $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠. Statistical calculations of the period ratio R and the resonances were not possible from such a small number.

3.4 Galaxies

We investigated the sky positions of the studied candidates (Fig. 3). The vast majority of objects are located in our Galaxy, and only $12\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ are in LMC and SMC. Systems' distribution in the Galaxy follows primarily the position of the Galactic plane. We found 107 systems close to the resonance for $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ and 52 for $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠. The LMC and SMC contain 22 systems for $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ and 10 for $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠. We observe mostly more luminous stars in the Magellanic Clouds. Thus, statistics could be biased (towards more luminous stars).

4 DISCUSSION

Our sample contains mostly Algol-type binary stars. Usually, at least one of the inner periods is long, around 1 d or more. $27\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ of the inner periods were shorter than 1 d. Selection effects strongly influence the candidates' detection. Quadruples with longer periods (several days) are significantly more prevalent in photometric surveys, and light curve disentangling is usually easier. Algol-type binaries are thus preferred in our sample. Therefore, the distribution between Algol, $\beta$ Lyr, and W UMa type binary stars could not be covered. We suggest searching for new candidates with inner orbital periods of up to 1 d to provide these statistics.

The used data set consists of various literature sources and our new objects. Thus, periods precision is not uniform. The most accurately derived periods' precision is in the order of 0.01 s. Furthermore, the sample includes data of lower precision, about several seconds and even several minutes or hours for the extreme cases. The period ratio R accuracy also varied, depending on the quality of the measured inner periods. In most cases, especially for the statistically sufficiently covered part between ratios 1 and 5, the periods belonged to the better-defined group. Larger R could sometimes have poorer accuracy, because longer inner periods may not always be determined precisely.

The period ratio distribution (Fig. 4) shows that the used data mainly correspond to the model prediction. Zasche et al. (2019) described this distribution for 146 candidates and discussed the deviations against their model. There were primarily three of the most significant phenomena: an increase of the 3:2 resonance, a decline between the ratio 1.2 and 1.5, and a minor dip around the 2:1 resonant ratio. Most of the deviations disappeared using the current numbers of 781 candidates compared with the simulation. A 3:2 increase and a potential 2:1 decline are visible during resonance analysis, where we used shorter C ranges of the vicinity than the histogram bins. We do not observe the dip between the 1.2 and 1.5 values. There are two slight decreases between 1.7 and 1.9 and 2.0 and 2.2. The exact explanation of this phenomenon is not clear yet. The reasons could be data fluctuations and bin width influences in the histogram.

The data and model agreement for the 1:1 ratio (Figs 5 and 6) shows that there is no clear preference for long-term stay in this resonance, it occurs here within the expected state. The same is observed for the 4:3 and 5:3 resonant ratios. The 3:2 overabundance was first detected by Zasche et al. (2019) and was discussed as a potential long-term stable solution. This ratio has the largest increase in our data sample. However, this difference is prevalent in only several systems. The given deviations indicate a possible preference for the 3:2 state as the long-term stable option. A precise analysis with certain conclusions would require more data than is currently available.

The 2:1 ratio has a slight decrease that almost disappears within the C range up to $1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠. Interestingly, a similar decline is also observed for the 5:2 resonance. The 3:1 and 7:2 ratios are not sufficiently covered to state specific properties and they vary between the two C ranges. The 3:1 overabundance for $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ changed for $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ to have only two known systems. On the other hand, the 7:2 resonance has a decline for $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ and is in good agreement for $C<1\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠. As we stated in Section 3.2, the selection effects could be extremely strong for the 4:1 ratio. The R value is relatively large, and the longer inner period could be hard to detect even for the TESS satellite and other surveys. Overall, the larger R values comprise too small an amount of data to draw clear conclusions.

Fabrycky et al. (2014) studied features of first-order resonant period ratios for multitransiting exoplanet candidates in KEPLER data. They analysed the first-order resonance offset from all the planetary pair candidates. There are practically no systems within the ratio between 1.00 and 1.25. Nevertheless, our sample contains 162 2+2 candidates within this R range. Thus, we can observe first-order resonant values like 6:5, 7:6, 8:7, etc., that could contain misleading results. The configuration of multitransiting exoplanet systems is significantly different than in the 2+2 quadruple. For these reasons, this method is not suitable for multiple stellar objects.

There is a relatively small representation close to the 3:2 resonance for the systems with one or both inner periods up to 1 d. We can see only three binaries (⁠$11\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠) with a longer inner period and five binaries (⁠$19\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠) with a shorter one close to the 3:2 ratio within $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠. It is the least represented case out of the mentioned resonances. The 1:1 value has five binaries (⁠$33\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠) for both cases, with longer and shorter inner periods. Interestingly, the 4:3 ratio has the largest sample of the resonances within the inner period values up to 1 d, eight binaries (⁠$35\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠) for the longer periods and nine (⁠$39\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠) for the shorter ones. The 5:3 ratio has three binaries (⁠$18\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠) for both categories. We see here the largest sample of the periods up to 1 d for the 4:3 resonance and the smallest one for the 3:2 resonance, the ratio with the largest overabundance compared to the used model. Thus, most 3:2 candidates (more than expected) have their inner periods longer than 1 d.

13 of 60 confirmed 2+2 systems are close to the studied resonant values within $C<2\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$⁠. The relative coverage is similar to the whole sample of candidates, but in absolute numbers there are only a few systems to state any properties. Most of the confirmed candidates have relatively short outer periods $P_{\mathrm{AB}}$⁠. Thus, the binaries should be close to each other. We suggest a question of whether the dynamical interactions between the binary pairs could lead to breaking their resonant state. We can observe (i.e. Kostov et al. 2021, 2023) that dynamical effects can be stronger than LiTE for very close quadruple stars with short outer periods. On the contrary, very wide binaries could evolve independently and possibly not create and stay in the resonant ratio.

The possible resonance could be broken by another scenario. Each binary evolves within its subsystem. Mass transfer plays a key role during the evolution process of close binaries, and the inner orbital period changes. Thus, the resonant ratio can be potentially disturbed by the inner evolution of one or both pairs. The eccentric orbits could also contribute to the resonance breaking. However, there are no observational evidences for these suggestions within the used data set.

5 SUMMARY

In summary, we assembled the largest sample of available 2+2 quadruple candidates up to date, and we complemented it with our newly discovered doubly eclipsing systems into a comprehensive analysis of inner orbital periods and period ratios, comprising a sample of 781 systems. Most analysed objects are located in our Galaxy following the galactic plane (Fig. 3). The remaining candidates are in the LMC and SMC. We determined the inner period ratio R distribution (Fig. 4) and compared it with the simulated prediction, assuming a uniform period distribution. Using two vicinity ranges for the chosen resonant values, we created resonance comparison diagrams (Figs 5 and 6) for our data and model. The majority of the observing ratios corresponded well with the prediction model. However, we observed the preference resonance at 3:2, which is also supported by previous studies. On the other hand, the ratios at 1:1, 4:3, and 5:3 show a remarkably close match with the prediction, and therefore there are no signs of preference for these resonant values. A slight decline compared to the model can be seen for the 2:1 ratio, similar to the 5:2 resonance. The higher examined resonances have too few systems to state their properties. Interestingly, the 3:2 ratio has most of the inner periods longer than 1 d, more than we could expect from the studied sample. Further searching and analysis of a larger sample of 2+2 candidates (in the order of thousands of systems) will show us more period and resonance features. For future work, we plan to observe the sets of candidates to confirm their quadruple nature and exclude the blends.

ACKNOWLEDGEMENTS

The authors thank T. Plšek for helpful discussions, SNdeV for the text suggestions and comments, and the anonymous referee for the review and analysis of this paper. JK was partly supported by the project MUNI/A/1419/2023.

DATA AVAILABILITY

The data underlying this article are available in the article.

REFERENCES