Triply eclipsing triple stars in the northern TESS fields: TICs 193993801, 388459317, and 52041148 Free

ABSTRACT

1 INTRODUCTION

Thus, as a general rule, if the orbital eccentricities are small to modest in size, the outer orbital period of the triple system must be greater than ∼5 times the period of the inner binary, and more typically a factor of 10 times the period of the inner binary for a common value of the outer eccentricity of e_out ≃ 0.3. The shortest known ratio of P_out/P_bin in a triple system is about 5.4 for LHS 1070 (Xia, Fu & Wang 2019), which consists of three very low mass red dwarfs with inner and outer periods of 18.2 and 99 yr, respectively. Due to their long periods, however, even the short period perturbations (i.e. those that are comparable with the periods of the inner and outer binary periods) are not easily measurable on a human time-scale. Moreover, the outer period is known only with a large uncertainty and therefore the period ratio is uncertain as well (see Tokovinin 2021, for further discussion). Amongst compact stellar triples, where both periods are known with much higher accuracy, the record-holder is KIC 7668648 with P_out/P_bin ∼ 7, which ultimately turned out to be a triply eclipsing triple system (Borkovits et al. 2015; Orosz 2015).¹

Outer periods in triple systems can range all the way from millions of years in some visual systems (Tokovinin 2018), down to 33.03 d in the case of λ Tau (Ebbighausen & Struve 1956). Interestingly, the second and third shortest known outer period systems have outer periods that are only a few percent longer, namely, 33.92 d in the case of KOI 126 (Carter et al. 2011) and 33.95 d for HD 144548 (Alonso et al. 2015). However, in principle, there is no dynamical reason for triple stars not to have stable outer orbital periods as short as a couple of days with, e.g. P_bin ≃ 0.25 d and P_out ≃ 2 d.

Compact triple star systems are quite fascinating objects to study because all their interesting timescales for dynamical interaction can happen over just a few observing seasons, or even on times as short as months and weeks. Another feature of compact triples is that the a priori probability for third-body (or ‘outer’) eclipses² to appear grows with decreasing outer orbital period roughly as |$\propto P_{\rm out}^{-2/3}$|⁠. There are two advantages of finding third-body eclipses in triple systems. First, it immediately reveals the triple nature of the system without the need for radial velocity or eclipse timing variation studies. Secondly, it carries a considerable amount of important information for the analysis of these systems, in particular for determining the properties of the outer orbit (see e.g. Carter & et 2011; Borkovits et al. 2013; Masuda, Uehara & Kawahara 2015; Orosz 2015; Alonso et al. 2015).

We show in Fig. 1 an updated summary of the known compact triples first presented in Borkovits et al. (2020b). The information is plotted in the plane of P_out versus P_bin. All but the triply eclipsing triples shown in this diagram were found by eclipse timing variations (ETVs), and the ≈200 found in this way with Kepler are shown as grey dots (Borkovits et al. 2016). Superposed on the plot are lines of constant Roemer delays as well as dynamical delays where the period of inner binary is lengthened by the presence of the third body (see e.g. Rappaport et al. 2013; Borkovits et al. 2015; Borkovits et al. 2016). The filled black circles are the known triple systems which exhibit third-body eclipses. The three red squares are the newly discovered systems reported in this work.

Figure 1.

Summary of compact triple systems updated from the work of Borkovits et al. (2020b). The results are plotted in the plane of the outer third body orbital period versus the inner binary period. We also overlay illustrative lines of constant Roemer delays (coloured horizontal lines) and dynamical delays (coloured diagonal lines) that would show up in eclipse timing variations of binary stars with third-body companions (see e.g. Borkovits et al. 2015). Horizontal dashed lines represent rough estimates of the eclipse probability for a third body orbiting the binary. The 17 filled black circles are known triply eclipsing triple systems, while the three red squares represent the new triply eclipsing triples reported in this work. The fainter dots are some 200 other triple systems found with Kepler that have measured or estimated outer orbital periods (see Borkovits et al. 2016).

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Here we present the discovery and detailed analyses of three new compact triply eclipsing triple star systems. In Section 2 we present the TESS observations (Section 2.1) that led to the discovery of these objects; the use of archival photometric data to help determine the outer orbital period via third-body eclipses (Section 2.2); follow-up ground-based observations to gather more timing data for the binary and third-body eclipses (Section 2.3); and radial velocity measurements for one of the objects (Section 2.4). The detailed photodynamical model by which we analyse jointly the photometric light curves, eclipse timing variations, spectral energy distributions, and radial velocities (where available) is reviewed in Section 3. The net results are in the form of three comprehensive tables, one for each triple system, of detailed extracted parameters for the masses, radii, and effective temperatures, as well as the orbital parameters for both the inner and outer orbits. We summarize our results and discuss a few of the salient findings from our study in Section 4.

2 OBSERVATIONS

2.1 TESS light curves

Our ‘Visual Surveyors Group’ (VSG) continues to search for multistellar systems in TESS light curves. We estimate that, thus far, we have visually inspected some 3 million light curves from TESS. Such visual searches are a complement to more automated ones using machine learning algorithms (see e.g. Powell et al. 2021). The light curves are displayed with Allan Schmitt’s lctools and lcviewer software (Schmitt, Hartman & Kipping 2019), which allows for an inspection of a typical light curve in just a few seconds. For TESS’s first excursion into the Northern hemisphere, we largely made use of the light curve pipeline QLP (Huang et al. 2020).

The first signatures that are looked for in terms of identifying triples, quadruples, and higher order stellar systems are: (i) an eclipsing binary (EB) light curve with an additional strangely shaped extra eclipse or rapid succession of isolated eclipses; and (ii) two (or possibly more) sets of EB light curves with different periods. Many of the latter detections (⁠|$\gtrsim 80{{\ \rm per \, cent}}$|⁠) turn out to be false positives due to the large TESS pixel size where two completely independent EBs may accidentally land near each other on the sky. Most false positives are identified using the lightkurve software (Lightkurve Collaboration 2018). We normally generate a 15 × 15 pixel mask and use the software’s interactive aperture feature in order to either (i) rapidly locate the individual EBs when their positions are sufficiently separated or (ii) compare how the eclipse depths scale with aperture location. For those cases where the eclipses do not scale equally with aperture position, we reject the candidates as possible bound systems that are likely to exhibit interesting dynamical interactions within a human lifetime (i.e. ∼100 yr). Sources that pass this check still require further vetting (e.g. with studies of eclipse timing variations, radial velocities, or high-resolution imaging) to ascertain more robustly whether the EBs are physically associated, and if there are, or will be, detectable dynamical interactions on interesting time-scales.

One gratifying aspect of finding triply eclipsing triples is that they are in a sense ‘self-vetting’. In particular, there is no way for a single binary, or sets of independent stars or binaries to produce such ‘extra’ eclipsing events. Therefore, additional vetting becomes largely unnecessary.

While searching through the light curves obtained from the first northerly TESS observations we have found more than ∼30 of these triply eclipsing triples. We report on three of them in this work: TIC 193993801, TIC 388459317, and TIC 52041148. (See Table 1 for the main catalogue data of the targets.)

All three targets were measured in TESS’s 30-min cadence mode. TIC 193993801 was observed in Sectors 16, 23, and 24. A set of extra eclipses was detected in each sector (see Fig. 2), which made the identification of the target as a potential triply eclipsing triple system candidate quite trivial. TIC 388459317 was observed in Sectors 16 and 17. We have identified one complex, |$\sim \, 1.5$|-d-long extra eclipsing event in Sector 17 (see Fig. 3). Finally, TIC 52041148 was observed by TESS in Sectors 18, 24, and 25. This target exhibited an extra eclipsing event in both Sectors 18 and 24 (see Fig. 4).

Figure 2.

Three third-body, or outer eclipses of TIC 193993801 observed with TESS in sectors 16, 23, and 24, respectively. Blue dots represent the observed TESS full frame image (FFI) fluxes, while red lines represent the best-fitting spectro-photodynamical model. In the case of the first two third-body eclipses (upper and middle panels) the distant, third star eclipsed the members of the inner, close pair, while in the bottom panel the third star was eclipsed separately by the two inner stars.

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Figure 3.

The sole third-body eclipse of TIC 388459317 observed with TESS in Sector 17. As above, blue dots represent the observed FFI fluxes, while the red curve represents the best-fitting spectro-photodynamical model. During this event the third star was eclipsed by the two inner binary stars.

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Figure 4.

Two third-body, or outer eclipses of TIC 52041148 observed with TESS in Sectors 18 and 24, respectively. Blue dots represent the observed FFI fluxes, while red and grey curves stand for the best-fitting photodynamical models with and without additional fits using SED and parsec isochrones constraints, respectively. In the case of both outer eclipses, the distant third star eclipsed the members of the inner, close binary pair.

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In the following steps of the analysis, more sophisticated photometric data processing was carried out by employing convolution-based differential image analysis implemented using the tasks of the fitsh package (Pál 2012). The processing steps are based on the ones used in the extraction of fluxes from moving objects in the Solar System (Pál et al. 2020) as well as other types of pulsating variables (Plachy et al. 2021).

Our next step in analyzing systems with detected third-body eclipsing events is to search for ETVs. Such triply eclipsing triple systems typically have outer periods in the range of 40–200 d or else the probability for an outer eclipse diminishes rapidly with increasing size of the outer orbit. Therefore, we anticipate the likelihood of also finding detectable ETVs. The expected dynamical delays for a coplanar system are proportional to |$(3e_{\rm out}/2 \pi) \zeta P_{\rm bin}^2/P_{\rm out}$|⁠, where the ‘out’ subscript refers to the outer orbit, and ζ is the mass ratio of the third component to the total triple-star system. Thus for typical parameters of P_bin ∼ 3 d, P_out ≃ 100 d, and e_out ∼ 0.3 dynamical delays of ∼0.004 d (5.5 min) are expected. For combinations of shorter-period inner binaries and longer outer periods we expect the classical light traveltime delays (LTTE) to dominate the ETVs. To measure the ETVs we use our standard procedure as described in Borkovits et al. (2016). We list the times of minima of the three targets in Tables 2–4 and plot the ETV curves in Fig. 5. These ETV data carry a number of valuable pieces of information for the further analyses of these systems.

Figure 5.

Eclipse timing variations of TICs 193993801 (upper), 388459317 (middle), and 52041148 (bottom). Red and blue dots represent the primary and secondary binary eclipses, respectively, while the correspondingly coloured curves connect the best-fitting photodynamical model ETV points. Solid and dashed vertical black lines indicate the times of the third-body (i.e. outer) eclipses detected by TESS and during ground-based follow-up observations, respectively. The lower panel shows the residuals.

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For example, in the case of TIC 193993801 the ETV curve itself, even without the third-body eclipses, reveals a ∼49-d-period third companion. From the nearly perfectly sinusoidal shape of the ETV curve, one can assume that the outer orbit may be nearly circular. Furthermore, the first and second TESS-observed third-body eclipses are located close to the upper extrema of the ETV curve (i.e. close to the maximum delay), while the third one is near the lower extremum. Thus, in so far as the ETV is dominated by the geometrical light-traveltime effect, the first and second extra events occurred during the inferior conjunction of the distant third star, while the third one was a superior conjunction event.

In the case of TIC 52041148 the similar curvature of the two sections of the ETV curves around the times of the two third-body eclipses suggested that the two TESS-observed events belong to the same type of the outer eclipses (i.e. primary versus secondary). Hence, we readily got an upper limit for the outer period of this triple, as it cannot be longer than the duration between the two events.

In contrast to the previous two systems, in the case of TIC 388459317, the TESS ETV curve by itself did not carry any readily usable information about the outer orbit.

2.2 Probing the outer orbit with Archival Data

After discovering a triply eclipsing triple system with TESS data, the question immediately arises as to the nature of its outer orbit, e.g. period and eccentricity. We therefore routinely examine the ASAS-SN (Shappee et al. 2014; Kochanek et al. 2017) and ATLAS (Tonry et al. 2018; Smith et al. 2020) archival data sets. The former typically have ∼1500–3000 photometric measurements of a given source, while ATLAS provides approximately 1500 measurements. However, the latter set has somewhat better photometric precision, and therefore the two data sets are of comparable statistical precision. Of course, we also make use of KELT (Pepper et al. 2007), WASP (Pollacco et al. 2006), and/or HAT (Bakos et al. 2002) data when they are available.

For TIC 193993801, TIC 388459317, and TIC 52041148, there are both ASAS-SN and ATLAS photometry available for all three. Additionally, there are KELT data available for TIC 193993801. For this latter source, the ATLAS data are somewhat affected by saturation effects and therefore we do not include them in the analysis of that source. Our procedure for combining the ATLAS and ASAS-SN data sets is as follows. We first median normalize the different colours within the ATLAS data set to each other, and the same for the ASAS-SN data set. We then median normalize the ATLAS to the ASAS-SN data sets. Outliers to the very bright and very faint side of the median value are eliminated, being careful not to exclude any low points that might conceivably be part of an eclipse.

We next apply a Box Least Squares transform (Kovács, Zucker & Mazeh 2002) to the combined archival data set to determine the long-term (over several years) average sidereal period of the eclipsing binary. Once we know that period accurately we remove that signal from the data train as follows. We fit simultaneously for between 30 and 100 harmonics of the EB, depending on how sharp its eclipsing features are. These best-fitting sines and cosines are evaluated at each time of the archival data set, and then that value is subtracted from the flux at that point.

We illustrate the results of the above procedure with a set of three figures for each of our sources (Figs. 6–8). In the top panel we show the fold of the archival data for the eclipsing binary in the triple. In the middle panel we show the BLS transform of the data after the EB light curve has been removed via its Fourier components. Finally, the bottom panel displays the fold of the data about the outer orbital period as deduced from the BLS transform. In all cases, zero phase for the outer orbit is taken to be the time of one of the third-body eclipses in the TESS data.

Figure 6.

The outer orbit of TIC 193993801 as diagnosed by the ASAS-SN and KELT archival photometric data. Top panel: Fold of the data about the period of the eclipsing binary. Middle panel: BLS transform of the data after subtracting out the light curve of the EB. The highest six peaks in the BLS are all due to the 49.285-d outer orbit and its harmonics. Bottom panel: Fold of the data about the outer orbital period of 49.285 d as determined from the BLS transform of the cleaned data set. One can see that the primary and secondary outer eclipses are separated by close to half the orbital cycle making ecos ω_out ≲ 0.02.

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Figure 7.

The outer orbit of TIC 388459317 as diagnosed by the ASAS-SN and ATLAS archival photometric data. Top panel: Fold of the data about the period of the eclipsing binary. Middle panel: BLS transform of the data after subtracting out the light curve of the EB. The highest three peaks in the BLS are all due to the 88.6-d outer orbit and its harmonics. Bottom panel : Fold of the data about the outer orbital period of 88.85 d as determined from the BLS transform of the cleaned data set. The eccentricity of the outer orbit is self-evident from the folded light curve.

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Figure 8.

The outer orbit of TIC 52041148 as diagnosed by the ASAS-SN and ATLAS archival photometric data. Top panel: Fold of the data about the period of the eclipsing binary. Middle panel: BLS transform of the data after subtracting out the light curve of the EB. The highest three peaks in the BLS are all due to the 176.99-d outer orbit and its harmonics. Bottom panel: Fold of the data about the outer orbital period of 176.99 d as determined from the BLS transform of the cleaned data set. One can see from the phasing of the primary outer eclipse that it occurs ∼0.21 outer orbital phases prior to the secondary outer eclipse which is the one observed by TESS.

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2.3 Photometric follow up

Following the discovery and the preliminary characterization of these (and some other) triply eclipsing triple systems, we organized photometric follow-up observational campaigns with the participation of amateur and professional astronomers at different sites in Hungary and the US, as follows:

Gothard Astrophysical Observatory (Szombathely, Hungary): Observations were carried out with a newly installed 0.8-m alt-azimuth Ritchey-Chrétien-type ASA AZ800 f/7 telescope, equipped with an FLI MicroLine ML 16803 CCD camera with 4096 × 4096 pixels and 9 × 9 μm pixel size. The field of view of the instrument is 22.6 × 22.6 arcmin, and the pixel scale is 0.33 arcsec per pixel. The CCD uses Sloan g′, r′, i′, z′, and Clear filters. TIC 193993801 was observed in the Sloan z′ filter with an exposure time of 120 s. The remaining two fainter objects, TIC 52041148 and TIC 388459317, were observed in white light using the Clear filter and had an exposure time of 60 s. The photometric observations were reduced using the software package iraf. The reduction included dark and flat corrections, astrometric calibrations, and differential photometry. During the differential photometry several close comparison stars of similar brightness were used.

Baja Astronomical Observatory of Szeged University (Baja, Hungary): An identical ASA AZ800 f/7 telescope to that at Gothard Astrophysical Observatory is operated here as well. It has an FLI ProLine PL23042 back-illuminated CCD camera with 2048 × 2048, 15 μm square pixels. This setup provides an 18.9 × 18.9 arcmin field of view, with a plate scale of 0.55 arcsec per pixel. The Sloan r′ and z′ filters with exposure times between 10 and 120 s were used for the triple-star observations (there was no ’Clear’ filter available). The usual photometric calibration procedures were performed using site-specific packages written in iraf and python. The reduction included bias, dark and flat corrections, followed by astrometric calibration (using astrometry.net) and aperture photometry. The closest few comparison stars were used for obtaining differential magnitudes.

Piszkéstető Observatory (Piszkés-tető, Hungary): We used the 1-m Ritchey-Chrétien-Coudé telescope located at the observatory and equipped with an SI 1100S CCD. The camera has a 4 k × 4 k pixel frame with 15 micron pixel-size resulting in an effective field-of-view of 16 × 16 arcmin and effective resolution of 0.23 arcsec per pixel. We obtained photometric images of TIC 193993801 in Johnson-Cousins R and TIC 388459317 in a Clear filter both with 60 s exposure time. The reduction of the images was performed with the fitsh (Pál 2012) software package applying bias, dark, and flat-field corrections, astrometric calibrations, and differential photometry using the nearby Gaia DR2 1594781593922650624 and Gaia DR2 2003450996324196480 as comparison stars, respectively.

Junk Bond Observatory (Arizona, US): JBO is in southeast Arizona at 1210-m elevation. It houses a domed 0.8-m Ritchey-Chrétien at F4.6, robotically controlled with astronomers control panel (acp) software. The camera is an SBIG STL6303E CCD and the images are unfiltered. The aperture photometry was pipelined through astroimagej software with standard darks, flats, and bias frames.

Patterson Observatory (Arizona, US): Patterson Observatory is in southeast Arizona at an elevation of 1412 meters. It houses a domed, 0.5 meter Ritchey-Chrétien at F8, robotically controlled with Astronomers Control Panel (ACP) software. The camera is an SBIG STL-1001e unfiltered. The aperture photometry was pipelined through AstroimageJ software with standard darks, flats and bias frames.

Hereford Arizona Observatory (Arizona, US): The Hereford Arizona Observatory consists of an AstroTech 0.41-m Ritchey-Chretien telescope located at a Southern Arizona site with an altitude of 1420 m. An SBIG ST-10XME CCD camera was used without filter. Image sets with exposure times of 10 seconds were obtained of TIC 193993801 on three dates in 2020 September and October. Standard bias, dark and flat-field calibrations were performed before photometry measurements were made with a single reference star with an r′ = 12.338 magnitude from the APASS catalog.

During the whole multisite photometric campaign, TIC 193993801 was observed 42 nights between 2020 September 18 and 2021 October 11. These measurements cover portions of seven third-body eclipses (Fig. 9). Moreover, 29 regular eclipses of the inner binary were also observed. The mid-eclipse times for these eclipses are also listed in Table 2.

Figure 9.

Seven third-body eclipses of TIC 193993801 partially observed during our ground-based follow up photometric campaign. Red, claret and dark brown dots refer to observations made with R_C, r′ and z′ filters, respectively. The lighter dots stand for the original observations, while the darker ones show their 900-s averages that have been used for the photodynamical fits. The black line represents the best-fitting spectro-photodynamical model solution. (Note that the first contact of the last event around BJD 2 459 484.3 was observed after finalizing the spectro-photodynamical modelling and therefore naturally, it was not taken into account for the parameter search.) Residuals against this photodynamical model is also shown below.

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The photometric campaign observations of TIC 388459317 covered 16 nights between 2020 July 9 and 2021 October 13. Sections of three additional third-body eclipses (Fig. 10) were obtained. Additionally, we observed 11 regular, inner binary eclipses, of which the mid-eclipse times can be found in the last rows of Table 3.

Figure 10.

Unfiltered ground-based follow up photometric observations of three third-body (i.e. outer) eclipses of TIC 388459317. Note that during the photodynamical analysis the data set was modelled in the R_C-band. The lighter dots stand for the original observations, while the darker ones show their 900-s averages that have been used for the photodynamical fits. The black line represents the best-fitting spectro-photodynamical model solution. Residuals against this photodynamical model are also shown in the bottom panel.

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Finally, TIC 52041148 was observed on 28 nights during the interval between 2020 June 11 and 2021 October 24. In addition to the sections of three third-body eclipses (Fig. 11), we also obtained light curves of 16 regular eclipses of the inner binary (see Table 4 for the mid-eclipse times determined from these observations).

Figure 11.

Three third-body eclipses of TIC 52041148 together with the ground-based follow up photometric observations that partially cover the events. Note that during the photodynamical analysis the data set was modelled in R_C-band, though different sections of the observations were carried out with different filters, or even without any filter. The lighter red dots stand for the original observations, while the darker ones show their 900-s averages which have been used for the photodynamical fits. Moreover, in the case of the first event, we plot also the ASAS-SN points (light blue dots), though these latter data were not used for our analysis. The black line represents the best-fitting spectro-photodynamical model solution. The substantial discrepancies between the observations and the model during the last two nights of the third event (right-hand panel) are discussed in the text. Residuals against this photodynamical model are also shown in the bottom panel.

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2.4 Radial-velocity data

18 high-resolution spectra of TIC 193993801 were obtained from 2021 March 26 to June 26. The observations were obtained with a 1.3 m, f/8.36 Nasmyth-Cassegrain telescope equipped with a fibre-fed Échelle spectrograph at the Skalnaté Pleso Observatory at an altitude of 1783-m above sea level. Its layout follows the MUSICOS design (Baudrand & Bohm 1992). The spectra were recorded by an Andor iKon-L DZ936N-BV CCD camera with a 2048 × 2048 array, 13.5 μm square pixels, 2.9 e⁻ read-out noise, and gain close to unity. The spectral range of the instrument is 4250–7375 Å  (56 échelle orders) with the maximum resolution of R = 38 000. The wavelength stability is 100–200 m s⁻¹. To increase the SNR three consecutive 900-s exposures were combined. This resulted in the SNR ranging from 10 to 18 in the yellow part of the spectrum.

The raw spectroscopic data were reduced using iraf package tasks, linux shell scripts, and fortran programs as described in Pribulla, Garai & Hambálek et (2015).In the first step, master dark frames were produced. In the second step, the photometric calibration of the frames was done using dark and flat-field frames. Bad pixels were cleaned using a bad-pixel mask, and cosmic hits were removed using the program of Pych (2004). Order positions were defined by fitting sixth order Chebyshev polynomials to tungsten-lamp and blue LED spectra. In the following step, scattered light was modeled and subtracted. Aperture spectra were then extracted for both object and ThAr lamp and then the resulting 2D spectra were dispersion solved. Finally, 2D spectra were combined to 1D spectra.

1D cross-correlation functions (CCFs) were calculated for each observed spectrum with the ispec software (Blanco-Cuaresma et al. 2014; Blanco-Cuaresma 2019) using the built-in NARVAL Sun linelist as a template. As was expected from the preliminary model, the system was found to be a triple-lined (SB3) spectroscopic triple star based on the CCFs, with two rotationally broadened components (the inner pair), and a sharp-lined tertiary. The CCFs were then shifted into the barycentric reference frame and fitted with the sum of three Gaussians in order to derive the RVs of the constituent stars. The RV data identified as the centers of the fitted Gaussians are tabulated in Table 5, while a section of them is plotted in Fig. 12 together with the best-fitting spectro-photodynamical model (see below, in Section 3).

Figure 12.

Radial velocity measurements of TIC 193993801 Aa, Ab, and B (red, blue and black dots, respectively), together with the best-fitting photodynamical solutions (the corresponding smooth curves) between BJDs 2 459 344 and 2 459 394 (i.e. one complete orbital cycle of the outer orbit). The lower panel shows the residuals.

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High dispersion spectra from the Skalnaté Pleso observatory were independently analyzed by the broadening-function (BF) technique (Rucinski 1992). According to the J − K = 0.274(28) colour index, the system seems to be of the F5V spectral type. Several slowly rotating templates of the F spectral type were used in an attempt to deconvolve the spectra. The best match of the object and template spectrum was found for HD 102870 (F8V). For this star the BF integral was close to unity hence the strength of the metallic lines of the object and template is also similar (see Rucinski, Pribulla & Budaj 2013). The BFs were extracted from the 4950 Å to 6046 Å spectral region devoid of strong hydrogen Balmer lines. In addition to the wide profiles of the close-binary components, all BFs clearly show a third slowly-rotating component which indicates a variable radial velocity (RV). While the third RV component was very well defined, the RV components of the eclipsing pair were rather noisy.

The BFs were modeled by a multicomponent Gaussian fit. In the next step, the Gaussian fit to the third component was subtracted from the BFs. The relative light contribution of the third component, estimated from the multiprofile fit, is about L₃/(L₁ + L₂) = 0.599 ± 0.034. In the following step, the resulting cleaned BFs were modeled by two rotational profiles which are more appropriate for stars where the rotation determines the line profiles. This resulted in RVs and rotational velocities of the components, vsin i. While the third component rotates slowly and its |$v \sin i \lt 8\, \mathrm{kms}^{-1}$| (limited by the spectral resolution), the primary and secondary components rotate at |$v \sin i = 59.5\pm 1.6\, \mathrm{kms}^{-1}$| and |$v \sin i = 47.4\pm 2.6\, \mathrm{kms}^{-1}$|⁠, respectively. Assuming that the stellar equators are aligned with the orbital planes, and taking the stellar radii (Table 6) from the spectro-photodynamical solution discussed below in Section 3, one can calculate the spin periods of the three stars to be |$P_\mathrm{rot,Aa}=1\rm{.\!\!^{\rm {\mathrm{d}}}}44\pm 0\rm{.\!\!^{\rm {\mathrm{d}}}}05$|⁠, |$P_\mathrm{rot,Ab}=1\rm{.\!\!^{\rm {\mathrm{d}}}}40\pm 0\rm{.\!\!^{\rm {\mathrm{d}}}}10$|⁠, |$P_\mathrm{rot,B}\gt 8\rm{.\!\!^{\rm {\mathrm{d}}}}5$|⁠, respectively. These findings indicate that the rotation of the inner pair is synchronized to the orbital period (⁠|$P_\mathrm{in}=1\rm{.\!\!^{\rm {\mathrm{d}}}}43$|⁠). (Since the RVs determined via the BF fitting method agree with the CCF results (Table 5) to well within the tabulated uncertainties for each RV point, we do not list them separately.)

3 SPECTRO-PHOTODYNAMICAL MODELING

We carried out combined (spectro-)photodynamical analyses with the software package lightcurvefactory (see e.g. Borkovits et al. 2019a, 2020a, and further references therein). This code, amongst other features, contains (i) a built-in numerical integrator to calculate the three-body perturbed coordinates and velocities of the three bodies; (ii) multiband light curve, ETV, and RV curve emulators, and (iii) for the inverse problem, a Markov chain Monte Carlo (MCMC)-based parameter search routine, which employs an implementation of the generic Metropolis–Hastings algorithm (see e.g. Ford 2005). The usage of this software package and the philosophy and consecutive steps of the whole process were previously explained in detail for a variety of systems (Borkovits et al. 2018, 2019a,b, 2020a,b, 2021; Mitnyan et al. 2020). These included tight and less tight triple systems (either exhibiting extra eclipses or not) as well as quadruple systems (having hierarchies of either the 2+2 or 2+1+1 types) and therefore here we discuss only the points specific to the current studies.

For all three of the triple systems considered in this work, we made two different kinds of analyses. The first of these we refer to as the ‘astrophysical model-independent solution’. Here we fit simultaneously the (multiple band) light curves, the ETV curves deduced from them, and, in the case of TIC 193993801, the RV data as well. This means that we use almost exclusively geometrical and dynamical information coded into the observational data without any a priori astrophysical model assumption.³ As a consequence, accurate results from this type of analysis (generally, precisions of |$1-2{{\ \rm per\ cent}}$| or better in the fundamental stellar parameters, e.g. in masses and radii) can be used as strong constraints for stellar evolutionary models (see e.g Torres, Andersen & Giménez 2010; Maxted et al. 2020; Southworth 2020; and further references therein).

For such an accurate analysis of an EB, however, one needs RV data for both components as these provide the information about the masses of the components. In the absence of this knowledge, one cannot convert the dimensionless quantities – such as the fractional radii of the stars, even if precisely determined from the light curve solution – into absolute, physical ones. Moreover, a further fundamental question in such an accurate analysis is the precision of the effective temperatures of the binary members. Recently Miller, Maxted & Smalley (2020) have developed a new method to determine the temperatures of eclipsing binary stars with a precision better than |$1{{\ \rm per\ cent}}$|⁠, via a careful analysis of the system’s net SED while taking into account the components’ flux ratio, as ‘quasi observables’, that can be measured from high quality multiband occultation light curves of some totally eclipsing binaries. Unfortunately, however, due to the significant light contributions of the third stellar components in our systems, as well as the lack of flat-bottomed binary eclipses and the strongly varying profiles of the third-body eclipses, this latter rigorous method for precise SED analysis cannot be applied directly to our triples. Moreover, amongst these three triply eclipsing triple systems RV data are available only for TIC 193993801. Thus, in the cases of the current triples, and especially for TIC 388459317 and 52041148, we require some other means to deduce these important parameters and therefore be able to characterize the system’s true physical nature and evolutionary status.

In the case of a dynamically interacting, not too distant third stellar component, however, its gravitational perturbations to the Keplerian motion of the inner EB carries information about the inner and outer mass ratios of the system. And, in special cases, even the masses of the individual stellar components (see e.g. Borkovits et al. 2015) can be deduced in this way. Unfortunately, however, in most cases the necessary precision cannot be reached via perturbations induced by the third star, and RV measurements are required. Therefore, we follow a different strategy. We introduce into the analysis some a priori knowledge about stellar astrophysics and evolution with the use of parsec isochrones and evolutionary tracks (Bressan et al. 2012). We use tabulated 3D grids of parsec isochrones that contain stellar temperatures, radii, surface gravities, luminosities, and magnitudes in different passbands of several photometric systems for [age, metallicity, initial stellar mass] triplets. Then, allowing these latter three parameters to be freely adjustable variables, the stellar temperature, radius, and actual passband magnitude are calculated through trilinear interpolations from the grid points and these values are used to generate synthetic light curves, and SED that can be compared to their observational counterparts. This process is described in detail in Borkovits et al. (2020a).

Regarding the technical details of the two different kind of analyses, in the case of the stellar evolution model independent runs, the freely adjusted (i.e. trial) parameters were as follows:

• Five plus 0–4 light curve related parameters: the temperature ratios of T_Ab/T_Aa and T_B/T_Aa; the durations of the primary eclipses of the inner pair (Δt_pri); the ratios of the radii of the inner stars (R_Ab/R_Aa); the fractional radius of the third component (r_B = R_B/a_out) and, the contaminated extra light in different passbands (ℓ₄). Here we tabulate also the effective temperature of the primary of the inner binary (T_Aa) which was also adjusted in the MCMC process with a uniform prior. The initial value of this parameter in the initial runs was set according to the catalogue values (given in Table 1), but later, after getting the results of the preliminary model-dependent runs, it was set according to these results.

• Three of six orbital-element related parameters of the inner, and six parameters of the outer orbits, i.e. the components of the eccentricity vectors of the two orbits (esin ω)_{in, out}, (ecos ω)_{in, out}, the inclinations relative to the plane of the sky (i_{in, out}), and moreover, three other parameters for the outer orbit, including the period (P_out), time of the first (inferior or superior) conjunction of the third component observed in the TESS data (⁠|$\mathcal {T}_\mathrm{out}^\mathrm{inf,sup})$| and finally, the longitude of the node relative to the inner binary’s node (Ω_out).

• Three mass-related parameters: the mass of the primary of the inner binary (m_Aa), and the mass ratios of the two (inner and outer) binaries (q_{in, out}).

In the case of the model dependent runs, the light curve-related parameters (i above), apart from the extra light (ℓ₄), were no longer adjusted as all three radii and temperatures are now calculated from the parsec tables. On the other hand, the following new adjustable parameters are introduced: the metallicity of the system ([M/H]), the (logarithmic) age of the three stars (log τ), the interstellar extinction E(B − V) toward the given triple, and its distance. Here additional notes about the ‘age’ and the ’distance’ are in order. First, regarding the ’age’ parameter, our former experience has shown that it is better to allow the ages of the three stars to be adjusted separately instead of demanding strictly equal ages. This issue was partly discussed in Rowden et al. (2020) and Borkovits et al. (2021), and we will return to this question in the discussion below. Turning now to the distance of the system, one can argue that the accurate trigonometric distances calculated from the Gaia parallax measurements (Bailer-Jones et al. 2021) should be used as Gaussian priors to penalize the model solutions. However, unfortunately, neither DR2 nor the recently released EDR3 Gaia parallaxes have been corrected for the binary or multiple natures of the targets and therefore the published parallaxes and determined distances are not necessarily accurate or, even reliable for our systems. For example, in the case of TIC 193993801 the EDR3 and DR2 parallaxes differ by about 5 of their mutual σ’s. Therefore, we decided not to utilize the Gaia distances. Instead, we constrained the distance by minimizing the |$\chi ^2_\mathrm{SED}$| value a posteriori, at the end of each trial step.

Some other parameters were also constrained instead of being adjusted or fixed during our analyses. Thus, the orbital period of the inner binary (P_in) and its orbital phase (through the time of an arbitrary primary eclipse or, more strictly, the time of the inferior conjunction of the secondary star – |$\mathcal {T}^\mathrm{inf}_\mathrm{in}$|⁠) were constrained internally through the ETV curves in both kinds of analyses. Moreover, in the case of TIC 193993801 the systemic radial velocity (γ) was also constrained internally via minimization of the |$\chi ^2_\mathrm{RV}$| contribution a posteriori in each trial step.

Regarding the atmospheric parameters of the stars under analysis, we handled them in a similar manner as in our previous works. Therefore, we used the logarithmic limb-darkening law (Klinglesmith & Sobieski 1970) for which the passband-dependent linear and non-linear coefficients were interpolated in each trial step with the use of the tables from the original version of the phoebe software (Prša & Zwitter 2005). We set the gravity darkening coefficients for all late type stars to β = 0.32 in accordance with the classic model of Lucy (1967) valid for convective stars and keep them as constant. In the case of TIC 388459317, however, the analysis of the net SED has revealed that the system consists of hotter stars, having radiative envelopes. Therefore, we set β = 1.0 for the radiative components of TIC 388459317. This parameter, however, has only minor significance, as the stars under the present investigations are close to spheroids.

While preparing the observational data for analysis, in order to save computational time, we dropped out the out-of-eclipse sections of the 30-min cadence TESS light curves, retaining only the ±0.15º phase-domain regions around the binary eclipses themselves. However, during sections of the data containing the third-body (i.e. ‘outer’) eclipses, we kept the data for an entire binary period both before and after the first and last contacts of the given third-body eclipse.

Regarding the data obtained during the follow-up campaigns, these data mostly cover times around predicted third-body eclipses and several regular eclipses as well. These data are quite inhomogeneous. They are, however, extremely important for constraining the dynamical properties of the systems very precisely. Independent of the original exposure times, we binned these data into 900 s and normalized the fluxes from one data set to another. For the two fainter targets, most of the measurements were carried out without a passband filter. For the analysis, however, for practical reasons, we consider them as Cousins-R (R_C) data. For TIC 193993801, the bulk of the observations were carried out with two similar 80-cm telescope with r′ and z′ filters. These data were analysed accordingly. The remaining, inhomogeneous data set were also considered to be taken in R_C. We emphasize again, however, that these data have only a minor influence on the stellar properties, e.g. the flux (or temperature) ratios, and constrains almost exclusively the dynamical parameters through the shapes and timings of the third-body eclipses.

In addition to the photometric light curves, ETV data (Tables 2–4), and RV data in the case of TIC 193993801 (Table 5), were also simultaneously included into both kinds of analyses. Moreover, in the case of the second type of analysis, using SED fitting and parsec tables, the observed passband magnitudes (Table 1) were also fitted against the code-generated synthetic SEDs. Note that, however, during the SED fitting, in order to avoid the over-dominance of the very precise Gaia magnitudes the uncertainties for each passband magnitude were not allowed to be less than 0.03 mag.

The median values of the orbital and physical parameters of the three triple systems, derived from the MCMC posteriors and their 1σ uncertainties are tabulated in Tables 6–9. Furthermore, the observed versus model light curves are plotted in Figs. 2–4, 9–11. The observed versus model ETV, RV curves, and summed SEDs are shown in Figs. 5, 12, and 13, respectively.

Figure 13.

The summed SED of the stars of TICs 193993801 (upper panel), 388459317 (middle panel), and 52041148 (lower panel) in the flux domain. The dereddened observed magnitudes are converted into the flux domain (red filled circles), and overplotted with the quasi-continuous summed SEDs for the multiple star systems (thick black line). These SEDs are computed from the Castelli & Kurucz (2004) ATLAS9 stellar atmospheres models (http://wwwuser.oats.inaf.it/castelli/grids/gridp00k2odfnew/fp00k2tab.html). The separate SEDs of the component stars are also shown with thin red, cyan, and purple lines, respectively.

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4 DISCUSSION

4.1 TIC 193993801

4.1.1 Use as a ‘calibration system’ with and without RVs

We use this triple system as a benchmark to test how well the use of theoretical parsec isochrones and SED fitting can serve as a partial substitute for RV data, and at what level we can trust the results of the former model-dependent analysis. This is instructive since we do have RV information for TIC 193993801 while we do not for TIC 388459317 or TIC 52041148. Therefore, we carried out four different kinds of analyses for this system, namely both model-dependent and model-independent runs with and without the use of the RV data set. The results of the model-independent and model-dependent runs with the use of the RV data (hereafter ‘MIR’ and ‘MDR’, respectively) are tabulated in columns 2–4 and 5–7 of Table 6, while their non-RV data counterparts (hereafter ‘MIN’ and ‘MDN’, respectively) can be found in Table 7. First we compare and discuss the results of these models, and then will discuss the astrophysical and orbital results for TIC 193993801 itself.

The most important question is, how the dynamical and astrophysical parameters that can be directly obtained and/or inferred from the MIR and MDN solutions compare to each other. One can see from a comparison of columns 2–4 in Table 6 and columns 5–7 in Table 7 that the majority of the parameters, including practically all the directly measurable orbital, atmospheric (temperature ratios), and geometric (i.e. fractional radii of the stars) parameters agree well within their 1σ uncertainties. The masses and the other mass-related parameters, however, show larger discrepancies. The masses in the MDN solution were found to be systematically larger. Namely, |$\Delta m_\mathrm{Aa}\approx 0.12\, \mathrm{M}_\odot$| (that is about |$\sim 9{{\ \rm per\ cent}}$| of the MIR mass or, 7σ_MIR and, 2.5σ_MDN), |$\Delta m_\mathrm{Ab}\approx 0.06\, \mathrm{M}_\odot$| (⁠|$\sim 5{{\ \rm per\ cent}}$|⁠, 3.5σ_MIR, 1.5σ_MDN), and |$\Delta m_\mathrm{B}\approx 0.05\, \mathrm{M}_\odot$| (⁠|$\sim 4{{\ \rm per\ cent}}$|⁠, 1.5σ_MIR, 1.5σ_MDN). These comparisons may be easier to visualize graphically, as we show in Fig. 14. Here we plot the radius and mass for all three stars in TIC 19399380 for each of the three different types of solutions, i.e. MIN, MDN, MIR, and MDR, in roughly increasing order of accuracy.

Figure 14.

Comparison of the masses and radii for the three stars in TIC 193993801 as determined by four different model fits. (The radii for star B have been shifted upward by 0.2 R_⊙ to avoid overlap with the points for star Aa.) The red points are for the model-independent analysis of the ETV and light curves, but without the use of the RVs (‘MIN’); the orange points result when we add the SED analysis plus the use of stellar isochrone models (‘MDN’); blue points are from the ‘MIR’ analysis which are model-independent with the use of RVs; and green points are from the ‘full’ model using all the information available (‘MDR’).

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Regarding the mass ratios, in the case of the inner binary (q_in), the MDN value was found to be lower with Δq_in = −0.031, i.e. almost 3σ_MIR from the MIR solution, while for the outer mass ratio (q_out), the MDN value is lower with Δq_out = 0.015 (1.5σ_MIR). In this context one naturally should consider the stellar radii, as well. As the fractional radii agree in both solutions within 1σ, and the orbital semimajor axes (a_{in, out}) scales with |$m_\mathrm{A,AB}^{1/3}$| the discrepancy in the physical stellar sizes remains below |$2.1{{\ \rm per\ cent}}$| for all the three stars (see Fig. 14 for a graphical comparison).

In order to investigate the origin of the tendency for higher masses to be deduced from the MDN solution, we include the MDR (i.e. full) solution in our discussion. In this model, the RV data set with the implied lower masses forces the parsec+SED fitting section of the whole process to find some evolutionary tracks that belong to less massive stars and consistent with the overall solution. However, we were then unable to find a satisfactory coeval solution. Therefore, we allowed the ages of all three stars to vary independently. In such a manner we were able to find an MDR (i.e. model-dependent ‘full’) solution with masses consistent with the base MIR solution results within |$1-3{{\ \rm per\ cent}}$| (⁠|$m_\mathrm{Aa}^\mathrm{MIR}=1.31\pm 0.02\, \mathrm{M}_\odot$| versus |$m_\mathrm{Aa}^\mathrm{MDR}=1.33\pm 0.01\, \mathrm{M}_\odot$|⁠; |$m_\mathrm{Ab}^\mathrm{MIR}=1.19\pm 0.02\, \mathrm{M}_\odot$| versus |$m_\mathrm{Ab}^\mathrm{MDR}=1.16\pm 0.01\, \mathrm{M}_\odot$|⁠; |$m_\mathrm{B}^\mathrm{MIR}=1.34\pm 0.04\, \mathrm{M}_\odot$| versus |$m_\mathrm{B}^\mathrm{MDR}=1.33\pm 0.03\, \mathrm{M}_\odot$|⁠). Note that the total mass of the inner binary, however, was found to be almost equal in the two cases (⁠|$m_\mathrm{A}^\mathrm{MIR}=1.495,\mathrm{M}_\odot$| versus |$m_\mathrm{A}^\mathrm{MDR}=1.489\, \mathrm{M}_\odot$|⁠). The mass of the third stellar component in the two solutions differs by less than 1σ or, about |$1.3{{\ \rm per\ cent}}$|⁠. The better agreement in the masses of the wide binary (and, accordingly the outer mass ratio Δq_out = −0.006) can be explained by the fact that the strongly rotationally broadened spectral lines of the inner binary stars result in several kms⁻¹ uncertainties in the determination of their RV values, which are 10–15 times larger uncertainties than those of the RVs of the third stellar component. Therefore, the RV solution is mostly dominated by the orbital solution of the outer orbit. The cost for this less massive, MIR-consistent MDR solution was, however, that while the two similar mass stars Aa and B were found to be essentially coeval (well within 1σ; τ_{Aa, B} = 2.7 ± 0.2 Gyr), for the less massive inner secondary star Ab we obtained a significantly older age (τ_Ab = 3.7 ± 0.4 Gyr), i.e. the discrepancy is ∼3σ.

Comparing the two different kinds of model-dependent solutions (i.e. the coeval MDN and non-coeval MDR models), we find that the latter led to an older (τ^MDR = 2.7 − 3.7 Gyr versus τ^MDN = 2.0 ± 0.2 Gyr) and less metal-rich solution ([M/H]^MDR = 0.0 ± 0.2 [dex] versus [M/H]^MDN = 0.2 ± 0.2 [dex]). This is a natural consequence of the fact that, for a given stellar mass, a star with lower metallicity is hotter than a more metal-rich one.

Finally, we turn to the MIN model, i.e. where only the light curves and the ETV curves were utilized in the analysis. In general, the light curve of a detached eclipsing binary is not sensitive to either the masses or to the mass ratio of the two components. In our case, however, the presence of the third star on a tight orbit and, especially the third-body eclipses, should carry information about both the inner and outer mass ratios, and even on the individual masses as well. Note that the source of this extra information is not restricted to only the timing variations of the inner and third-body eclipses. In fact, the shapes of the third body eclipses also carry substantial information on the global system geometry and therefore on both mass ratios, as was discussed in the appendix A of Borkovits et al. (2013). The TESS observations of TIC 193993801 cover continuously one full and another half outer orbit, three completely observed third-body eclipses and, moreover, during our ground-based follow up photometric campaign, we were able to measure at least sections of seven other third-body eclipses. It is therefore a relevant question to ask whether these data make it possible to obtain reliable fundamental stellar parameters in a model-independent way, even in the absence of RV data and, if so, what precision can be achieved. This latter, MIN solution is tabulated in columns 2–4 of Table 7. Comparing the masses and their statistical 1σ uncertainties to the base solution MIR, one can see that the σ_MIN uncertainties are about one order of magnitude larger than those of the MIR solutions (and also of the other two, MDR and MDN) solutions, and they represent |$\sim \, 10-12{{\ \rm per\ cent}}$| of the masses themselves. Moreover, the deviation of the median masses (and other parameters) of the MIN solution from the ‘true’ MIR results in most cases are well within the corresponding 1σ_MIN uncertainties.

These results, in our interpretation, suggest that in the absence of reliable RV measurements, the MDN model can be used as a tolerably good substitute. However, the MDN solutions lead to lower accuracies of |$5-10{{\ \rm per\ cent}}$| instead of the |$1-2{{\ \rm per\ cent}}$| or better precision that can be reached in the case of high quality RV data. Moreover, careful application of the parsec isochrones and the correspondingly generated theoretical SEDs, i.e. the inclusion of some a priori stellar astrophysical knowledge in addition to the RV data (the MDR model), neither alters the model independent (MIR) results nor does it reduce or increase their uncertainties. Rather, the MDR results can help to improve the information content of the entire solution, making it possible to find stellar temperatures, evolutionary states, and some other parameters (see below) of the stars under investigation. In addition, the MDR results may help to eliminate some possibly false solutions from the MIR results, which might arise from combinations of the fitted parameters that lead to astrophysically unrealistic stars.

4.1.2 Results for TIC 193993801

Turning now to the discussion of our findings for TIC 193993801, both models (MIR and MDR) reveal that this triple system consists of two very similar, mid F-type stars (components Aa and B), and a slightly less massive late F-type component (Ab). The stars in both solutions are slightly evolved (i.e. a bit oversized for their masses). The discrepancies between the MDR and the MDN solutions, and also the non-existence of satisfactory coeval solutions in the MDR fit may imply some small systematic deviations from the single-star evolutionary tracks that were used. We may speculate that these discrepancies might have arisen from the rapid rotation of the members of the inner binary or, from some prior mass transfer phase between the close binary pair, though the latter seems to be unlikely.

The MDR model has yielded a metallicity [M/H] and interstellar extinction (E(B − V)) that are in good agreement with the catalogue values (Table 1). The photometric distance of TIC 193993801 was found to be d_phot = 669 ± 9 pc which is in perfect agreement with the Gaia EDR3 derived geometric distance of |$d_\mathrm{geo}^\mathrm{EDR3}=676\pm 26$| pc (Bailer-Jones et al. 2021).

Regarding the dynamical properties of TIC 193993801, it has the fourth shortest outer period known for a triply eclipsing triple star (see Fig. 1), with its outer anomalistic period of |$P_\mathrm{out}=49\rm{.\!\!^{\rm {\mathrm{d}}}}402\pm 0\rm{.\!\!^{\rm {\mathrm{d}}}}001$|⁠.⁴ On the other hand, given the short period of the inner binary (⁠|$P_\mathrm{in}=1\rm{.\!\!^{\rm {\mathrm{d}}}}43081\pm 0\rm{.\!\!^{\rm {\mathrm{d}}}}00004$|⁠) the outer-to-inner period ratio is P_out/P_in ≈ 34.5, which implies only moderate third-body perturbations to the Keplerian motion. Moreover, similar to the majority of previously investigated triply eclipsing systems, this one is also found to be very flat, with |$i_\mathrm{mut}=0{_{.}^{\circ}} 47_{-0{_{.}^{\circ}} 26}^{+0{_{.}^{\circ}} 55}$|⁠. Furthermore, both the inner and outer orbits are almost circular (e_in = 0.0008 ± 0.0001; e_out = 0.003 ± 0.002), where the latter is certainly atypical, even for such tight triple systems. For example, the even more compact triply eclipsing triples KOI-126 (⁠|$P_\mathrm{out}=33\rm{.\!\!^{\rm {\mathrm{d}}}}92$|⁠) and HD 144548 (⁠|$P_\mathrm{out}=33\rm{.\!\!^{\rm {\mathrm{d}}}}95$|⁠) have outer eccentricities e_out = 0.31 (Carter et al. 2011; Yenawine et al. 2021) and e_out = 0.27 (Alonso et al. 2015), respectively. On the other hand, however, as a rare counterexample, we note that the much longer outer period (⁠|$P_\mathrm{out}=235\rm{.\!\!^{\rm {\mathrm{d}}}}55$|⁠) triply eclipsing triple star TIC 278825952 was also found to be doubly circular and flat (Mitnyan et al. 2020).

Finally, given the small eccentricities and the flatness of the system, we conclude that the present configuration of this triple system is dynamically stable and, moreover, we do not expect significant, measurable non-Keplerian variations in the orbital elements and therefore in the system observables. In accordance with the analytical findings of Borkovits et al. (2003, 2015), the ETV curve is fully dominated by the light-traveltime effect (since for a doubly circular and flat configuration the most prominent quadrupole perturbation terms in the ETV curve disappear), and the same holds for the RV measurements. Moreover, on longer time-scales, in the present system neither apsidal motion nor orbital plane precession is expected.

4.2 TIC 388459317

The TESS observations of TIC 388549317 have shown only a single third-body eclipse and, moreover, the ETV curve generated from the satellite data was found to be featureless. Therefore, the determination of even the outer orbital period was a great challenge. However, having a strong clue about the outer period from the analysis of the archival ATLAS and ASAS-SN data (see Section 2.2), and observing segments of three additional third-body eclipses during our photometric campaign, made it possible to carry out a full photodynamical analysis of this target. In the absence of RV data for this target we utilized only the MIN and MDN modeling in our analysis (the results of the latter tabulated in Table 8). The orbital and the other dimensionless light curve parameters (fractional radii, temperature ratios) obtained from the two solutions agree fairly well, mostly within their mutual 1σ uncertainties. The solutions reveal a slightly less tight (⁠|$P_\mathrm{in}=2\rm{.\!\!^{\rm {\mathrm{d}}}}18487\pm 0\rm{.\!\!^{\rm {\mathrm{d}}}}00001$|⁠; |$P_\mathrm{out}=89\rm{.\!\!^{\rm {\mathrm{d}}}}031\pm 0\rm{.\!\!^{\rm {\mathrm{d}}}}007$|⁠; P_out/P_in ≈ 40.7) triple star system with almost circular inner and moderately eccentric outer orbits (e_in = 0.004 ± 0.001 and e_out = 0.103 ± 0.002, respectively. This system was also found to be almost coplanar with |$i_\mathrm{mut}=1{_{.}^{\circ}} 3_{-0{_{.}^{\circ}} 8}^{+1{_{.}^{\circ}} 5}$|⁠. In order to check the consistency of the orbital elements of the outer orbit that we obtained, we calculated synthetic light curves for the earlier times of the ASAS-SN observations. Some illustrative sections of this back-integration can be seen in the panels of Fig. 15.

Figure 15.

Sections of the spectro-photodynamical light curve solution of TIC 388459317 (red lines) integrated back to the times of the ASAS-SN observations (blue dots). As one can see, despite the large scatter, the lower-flux data points are in agreement with the back-projected times of third-body eclipses in the past.

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