# Four New Compact Triply Eclipsing Triples found with Gaia and TESS

Donát R. Czavalinga,1,2 Tamás Borkovits,1,2,3,4,5 Tibor Mitnyan,1,2 Saul A.
Rappaport,6 András Pál 3
1 Baja Astronomical Observatory of University of Szeged, H-6500 Baja, Szegedi
út, Kt. 766, Hungary
2 HUN-REN-SZTE Stellar Astrophysics Research Group, H-6500 Baja, Szegedi út,
Kt. 766, Hungary
3 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences,
H-1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary
4 ELTE Gothard Astrophysical Observatory, H-9700 Szombathely, Szent Imre h. u.
112, Hungary
5 HUN-REN-ELTE Exoplanet Systems Research Group, H-9700 Szombathely, Szent
Imre h. u. 112, Hungary
6Department of Physics, Kavli Institute for Astrophysics and Space Research,
M.I.T., Cambridge, MA 02139, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

This paper presents a comprehensive analysis of four triply eclipsing triple
star systems, namely TIC 88206187, TIC 14839347, TIC 298714297, and TIC
66893949. The four systems with third-body eclipses were found in the TESS
lightcurves from among a sample of $\sim$400 matches between known eclipsing
binaries and the Gaia DR3 Non-Single Star (NSS; Gaia Collaboration, 2022;
Pourbaix et al., 2022) solution database. We combined photometric lightcurves,
eclipse timing variations, archival spectral energy distributions, and
theoretical evolution tracks in a robust photodynamical analysis to determine
the orbital and system parameters. The triples have outer periods of 52.9,
85.5, 117, and 471 days, respectively. All dozen stars have masses $\lesssim
2.6$ M⊙. The systems are quite flat with mutual inclination angles between the
inner and outer orbital planes that are all $\lesssim 4^{\circ}$. The outer
mass ratios $(q\equiv M_{3}/M_{\rm bin})$ range from 0.39–0.76, consistent
with our earlier collection of compact triply eclipsing triples. TIC 88206187
exhibits a fractional radius of the outer tertiary component $(r_{B}\equiv
R_{B}/a_{\rm out})$ exceeding 0.1 (only the third such system known), and we
consider its future evolution. Finally, we compare our photodynamical analysis
results and the orbital parameters given in the Gaia DR3 NSS solutions,
indicating decent agreement, but with the photodynamical results being more
accurate.

###### keywords:

binaries:eclipsing – binaries:close – stars:individual: TIC 14839347, TIC
66893949, TIC 88206187, TIC 298714297

††pubyear: 2023††pagerange: Four New Compact Triply Eclipsing Triples found
with Gaia and TESS–A

##  1 Introduction

Triply eclipsing triple star systems are outstanding objects in which to study
the properties and dynamics of multistellar systems. These are triple systems
in which the inner binary eclipses the tertiary star, or vice versa, during
the course of their outer orbit. The outer orbital periods range from a month
to a year or two, and therefore dynamical interactions can usually be observed
within the duration of a PhD study. Furthermore, even in the absence of radial
velocity (RV) observations, readily available spaced-based photometric
observations, such as with Kepler (Borucki et al., 2010) or TESS (Ricker et
al., 2015) can yield precise eclipse timing measurements, as well as the
photometry of the third-body eclipses. These can be combined with the archival
spectral energy distribution, and processed in a comprehensive spectro-
photodynamical analysis to yield a full set of accurately determined stellar
and orbital parameters for the system. In terms of the spatial configuration
of the system, this analysis can provide such quantities as the ratio of
orbital periods, the orbital eccentricities, and the mutual inclination of the
two orbits. In turn, these quantities play a crucial role in the long-term
evolution of the whole system. These are therefore excellent laboratories for
testing theories regarding the formation, evolution and final evolutionary
states of multi-star systems.

The first such system was discovered in Kepler data by Carter et al. (2011),
and the second one followed very shortly thereafter (Derekas et al., 2011).
Within a few years, more than a dozen publications had reported the discovery
of 15 more similar objects, among them the first detections using other
observing facilities than Kepler, including CoRoT (Hajdu et al., 2017) and
OGLE (Hajdu et al., 2022). Since then, TESS observations have accelerated the
search for these objects and have led the way in such studies. Borkovits et
al. (2020b) reported the first identification and analysis of a triply
eclipsing triple based on TESS data, and in less than three years since then,
20 additional similar objects have been identified and analyzed in detail
(Mitnyan et al., 2020; Borkovits et al., 2022; Powell et al., 2022; Rappaport
et al., 2022, 2023). That means, thanks to Kepler and then nowadays TESS, the
discovery of triply eclipsing triples has become more or less routine, and the
numbers of known triply eclipsing triple systems have been steadily growing.

In our previous work (Czavalinga et al., 2023), we utilized the Gaia Data
Release 3 (DR3; Gaia Collaboration et al., 2023; Babusiaux et al., 2023) Non-
Single Star (NSS; Gaia Collaboration, 2022; Pourbaix et al., 2022) solutions
to search for tertiary stars in eclipsing binary (EB) systems that had
previously been reported in the literature. Gaia DR3 provides a unique
opportunity to search for close hierarchical triple stars across the entire
sky based on long-term astrometric and spectroscopic observations. If we
compare the eclipsing period with the period obtained from the Gaia NSS
solutions, and the period ratio is higher than 5, we can infer that the object
is likely a triple-star system candidate with the inner period being that of
the EB and the outer period of the tertiary star the one determined by Gaia’s
NSS solutions (see Czavalinga et al., 2023, for details). Using this method,
we identified 403 potential compact hierarchical triple-star system candidates
among $\sim$1 million known EBs, including four newly identified triply
eclipsing triples. Here we provide a detailed analysis of the latter four
systems. In Section 2, we describe the observational data and the methods
utilized for the discovery and validation of the corresponding objects. In
Section 3, we construct a photodynamical model for each system, and then in
Section 4, we give a detailed discussion of the resulting models. In Section
5, we compare the orbital parameters from our photodynamical models with those
from the Gaia NSS solutions, and finally we summarize our findings in Section
6.

##  2 Discovery and Observations

###  2.1 TESS observations

To validate the triple nature of our candidate systems we started by
constructing a TESS photometric lightcurve (LC) for each one. Our aim was to
search for eclipse timing variations (ETVs) and possible third-body eclipses
in the LC. After downloading all the available Full-Frame Images (FFIs) from
the MAST portal111https://mast.stsci.edu/, we applied the same convolution-
aided image subtraction photometry pipeline based on FITSH (Pál, 2012) along
with the same detrending process based on the Wōtan (Hippke et al., 2019) and
lightkurve (Lightkurve Collaboration et al., 2018) Python packages that we
used in our previous paper (see details in Czavalinga et al., 2023).

We identified six LCs with potential extra third-body eclipses. Two of them
had already been discovered by other teams, with one system being HD 181068,
the second triply eclipsing triple found in Kepler observations (Derekas et
al., 2011), and the other one being TIC 229785001 recently analyzed in detail
by Rappaport et al. (2023). The remaining four systems were completely unknown
in the literature. These are as follows:

TIC 14839347 was observed by TESS in Sectors 14, 15, 41, and 55. We found a
well-defined third-body eclipse in Sector 41, and then, after making use of
the outer period from the Gaia NSS solution, we additionally found two very
shallow secondary eclipses in Sectors 14 and 55 (see upper left panel of Fig.
1).

TIC 66893949 was observed in Sectors 15, 41, and 55. In Sector 15, we
identified an extra dip in the LC, which we interpreted as third-body eclipse.
No eclipses occurred in Sector 41, but we observed two extra eclipsing events
with different depths in Sector 55 (upper right panel of Fig. 1). We note that
recently Rowan et al. (2023) also independently discovered this object as a
triply eclipsing triple, but they have not analyzed the system in any detail.

TIC 88206187 was observed by TESS in Sectors 19 and 59. We discovered a very
deep and long, flat bottomed eclipse in the middle of Sector 19. The source
also produced another, similarly long in duration but much shallower, third-
body eclipse in Sector 59 (lower left panel of Fig. 1).

Finally, TIC 298714297 was observed in Sectors 15, 55, and 56 by TESS. We
found evidence for one third-body eclipse in Sector 55, accompanied by what
are likely prominent, stellar-activity induced variations in the LC (lower
right panel of .Fig. 1).

For the ETV calculation, we applied the same process as in our previous paper
(Czavalinga et al., 2023), and that is described in detail in Borkovits et al.
(2015). In short, we fitted template polynomials to both the primary and
secondary eclipses in the phase-folded, binned LC of each system. The mid-
eclipse times were thereby determined for each orbital cycle. We list the
determined individual mid-eclipse times in Tables 5–8 separately for all the
four systems.

Figure 1: Sections of TESS LCs with third-body eclipse events for each of the
four triple systems The blue points are the TESS observations, while the
overplotted smooth red curves are the models from the photodynamical fit (see
Sect. 3). In the left and right upper panels (the two very shallow secondary
outer eclipses of TIC 14839347), the additional black lines represent just the
flux occulations of the giant tertiary component as the EB passes in front of
it. In the second row, the lighter blue points in the out-of-eclipse regions
of TIC 66893949 were omitted from the photodynamical fits to save computation
time. In the case of TIC 298714297 (bottom row) the gray curve displays the
pure EB model (without the Fourier-modeled rotational distortions).

###  2.2 ASAS-SN, ATLAS observations

We looked up all four triply eclipsing triples in the ASAS-SN (Shappee et al.,
2014; Kochanek et al., 2017) and ATLAS (Tonry et al., 2018; Smith et al.,
2020) archives. All four systems have good data in the ASAS-SN archives, and
two have useful data in the ATLAS archives (TIC 298714297 and TIC 66893949 are
too bright for ATLAS). We readily see the EB LC in the archival data for all
four systems.

After we determine an accurate EB period for data that span of order a decade,
we remove the EB LC by fitting for, and then subtracting out, between 50 and
100 orbital harmonics. This allows for a more sensitive BLS search (Box Least
Squares; Kovács et al., 2002) for the eclipses of the outer orbit.

For TIC 66893949 and TIC 298714297 (with outer periods of 470.6 and 118.6
days, respectively) there are insufficient archival data, and too few outer
eclipses observed, to make a statistically significant detection of the outer
eclipses. However, for TIC 14839347 and TIC 88206187, the outer periods of
85.47 days and 52.84 days, respectively, were robustly detected in a
combination of the ASAS-SN and ATLAS data. The results are shown in Figs. 2
and 3, respectively. In the case of TIC 88206187, Fig. 3 shows a shallow but
clear secondary outer eclipse. The phasing of the two outer eclipses yields
$e_{\rm out}\cos\omega_{\rm out}=0.003\pm 0.005$, with an implication that the
outer orbit is likely fairly circular.

Figure 2: Folded outer orbit of TIC 14839347. Figure 3: Folded outer orbit of
TIC 88206187. Table 1: Main parameters of the four systems from the
literature.

Parameter | TIC 14839347 | TIC 66893949 | TIC 88206187 | TIC 298714297
---|---|---|---|---
Gaia Source $\mathrm{ID^{a}}$ | 2058085351143518464 | 1870771652581668352 | 207943285475761280 | 1849392443551822848
$\mathrm{RA^{a}}$ [deg] | 306.408 | 311.711 | 82.739 | 324.463
$\mathrm{DEC^{a}}$ [deg] | 37.895 | 36.352 | 44.803 | 29.474
$\mathrm{T^{b}[mag]}$ | 12.2117 $\pm$ 0.0448 | 11.6134 $\pm$ 0.0101 | 11.8518 $\pm$ 0.0078 | 10.8136 $\pm$ 0.0069
$\mathrm{G^{a}[mag]}$ | 13.070626 $\pm$ 0.002975 | 11.9993 ± 0.0002 | 12.478771 $\pm$ 0.002864 | 11.542984 $\pm$ 0.003035
$\mathrm{G_{bp}^{a}[mag]}$ | 13.866420 $\pm$ 0.004369 | 12.2891 $\pm$ 0.0008 | 13.006023 $\pm$ 0.003710 | 12.203759 $\pm$ 0.004678
$\mathrm{G_{rp}^{a}[mag]}$ | 12.175914 $\pm$ 0.005246 | 11.5470 $\pm$ 0.0005 | 11.783351 $\pm$ 0.004393 | 10.743742 $\pm$ 0.004989
$\mathrm{B^{c}[mag]}$ | 14.837 $\pm$ 0.048 | 12.47 $\pm$ 0.255 | 13.403 $\pm$ 0.275 | 13.309 $\pm$ 0.399
$\mathrm{V^{c}[mag]}$ | 13.807 $\pm$ 0.149 | 12.082 $\pm$ 0.057 | 12.794 $\pm$ 0.069 | 11.584 $\pm$ 0.138
$\mathrm{g^{\prime c}[mag]}$ | 14.211$\pm$ 0.016 | 12.408$\pm$0.000 | 13.007 $\pm$0.220 | 12.493$\pm$0.040
$\mathrm{r^{\prime c}[mag]}$ | 13.176$\pm$ 0.033 | 12.058$\pm$0.000 | 12.487$\pm$0.072 | 11.529$\pm$0.045
$\mathrm{i^{\prime c}[mag]}$ | 12.562$\pm$ 0.019 | 11.869$\pm$0.000 | 12.135$\pm$0.084 | 11.055$\pm$0.051
$\mathrm{J^{d}[mag]}$ | 10.81 $\pm$ 0.021 | 11.004 $\pm$ 0.021 | 10.828 $\pm$ 0.022 | 9.717 $\pm$ 0.028
$\mathrm{H^{d}[mag]}$ | 10.231 $\pm$ 0.018 | 10.737 $\pm$ 0.017 | 10.351 $\pm$ 0.021 | 9.092 $\pm$ 0.034
$\mathrm{K^{d}[mag]}$ | 9.999 $\pm$ 0.016 | 10.697 $\pm$ 0.016 | 10.206 $\pm$ 0.016 | 8.965 $\pm$ 0.02
$\mathrm{W1^{e}[mag]}$ | 9.827 $\pm$ 0.023 | 10.685 $\pm$ 0.023 | 10.099 $\pm$ 0.023 | 8.877 $\pm$ 0.023
$\mathrm{W2^{e}[mag]}$ | 9.857 $\pm$ 0.02 | 10.72 $\pm$ 0.021 | 10.129 $\pm$ 0.02 | 8.878 $\pm$ 0.02
$\mathrm{W3^{e}[mag]}$ | 9.673 $\pm$ 0.088 | 10.666 $\pm$ 0.099 | 10.011 $\pm$ 0.062 | 8.752 $\pm$ 0.024
$\mathrm{W4^{e}[mag]}$ | 7.836 $\pm$ 0.191 | 9.015 $\pm$ NaN | 8.67 $\pm$ 0.36 | 8.82 $\pm$ 0.378
$\mathrm{T_{eff}^{a}[K]}$ | - | 5883 $\pm$ 27 | - | 5498 $\pm$ 121
$\mathrm{Distance^{f}[pc]}$ | $1970_{-41}^{+48}$ | $587_{-7}^{+9}$ | $2490_{-78}^{+96}$ | $111.9_{-0.7}^{+0.7}$
$\mathrm{E[B-V]^{c}[mag]}$ | 0.772 $\pm$ 0.065 | 0.03929 $\pm$ 0.0124 | 0.2585 $\pm$ NaN | 0.009 $\pm$ 0.00595
$\mathrm{\mu_{\alpha}^{a}}$ | -2.657 $\pm$ 0.011 | 14.4171 $\pm$ 0.0202 | 0.720 $\pm$ 0.015 | -28.768 $\pm$ 0.044
$\mathrm{\mu_{\delta}^{a}}$ | -7.034 $\pm$ 0.012 | -15.2782 $\pm$ 0.0241 | -2.416 $\pm$ 0.011 | -58.138 $\pm$ 0.036
$\mathrm{RUWE^{a}}$ | 1.221 | 1.9273869 | 0.905 | 3.162
NSS $\mathrm{model^{a}}$ | SB1 | Orbital | SB1 | Orbital
$\mathrm{P_{binary}^{g}[days]}$ | $1.154060_{-0.000052}^{+0.000050}$ | $4.805309_{-0.000005}^{+0.000004}$ | $1.184592_{-0.000063}^{+0.000055}$ | $1.072891_{-0.000019}^{+0.000010}$
$\mathrm{P_{triple}^{g}[days]}$ | $85.530_{-0.017}^{+0.017}$ | $471.03_{-0.07}^{+0.10}$ | $52.922_{-0.039}^{+0.041}$ | $117.24_{-0.31}^{+0.36}$

Notes. ’-’ means that the value is not available (a) - Gaia DR3 (Gaia
Collaboration et al., 2023), coordinates: ICRS J2016.0; (b) - TESS Input
Catalog V8.2 (Paegert et al., 2021); (c) - AAVSO Photometric All Sky Survey
(APASS) DR9 (Henden et al., 2015); (d) - 2MASS All-Sky Catalog (Skrutskie et
al., 2006); (e) - ALLWISE Data Release (Cutri et al., 2021); (f) - (Bailer-
Jones et al., 2021); (g) - parameters from this paper

##  3 Photodynamical analysis

Similar to other TESS-discovered triply eclipsing triple stars, we carried out
a joint, simultaneous LC, ETV curve and spectral energy distribution (SED)
analysis with the software package Lightcurvefactory. The software package
itself, and the consecutive steps of the complex analysis, were described in
several papers, e.g., in Borkovits et al. (2020a, 2022); Rappaport et al.
(2023) and, hence, we will not repeat the details here, but rather restrict
ourselves to some particular notes about the current, specific systems.

Our analyses are mainly based on the TESS LCs, which were processed in the
manner described in Sect 2.1. In contrast to the vast majority of our former
analyses of triple and quadruple systems, in the case of 3 of the 4 systems
reported here, we did not restrict our analyses only to the narrow regions of
the inner and outer eclipses, dropping out the majority of the out-of-eclipse
portions of the LCs, but rather we kept the complete TESS time series. The
reason is that, apart from the somewhat wider 4.8-day inner EB in TIC
66893949, the other three, much more compact inner EBs, display significant
ellipsoidal light variations in the out-of-eclipse LC sections, which we found
to be worthwhile to retain for the analysis. Moreover, we note the special
case of TIC 298714297, in which the LC exhibits further remarkable rotational
variations with periods similar to the eclipsing period of the inner EB. We
modelled these variations simultaneously with the full, eclipsing LC solution
(but separately for Sectors 15 and 55-56), with the use of additional fits of
harmonic functions in the manner described, e.g., in Borkovits et al. (2018,
2021).

As usual, we fit the ETV curves of the four EBs simultaneously with their LCs.
As our literature search did not turn up any additional previously recorded
eclipse times, our ETV curves were restricted to only those times that we
determined from the TESS observations themselves.

Due to the lack of publicly available RV timeseries for any of the four
currently investigated systems, with the exception of TIC 14839347, we used a
combination of (i) observed composite SED values (tabulated from the available
catalog passband magnitudes, and listed in Table 1) and (ii) pre-computed,
tabulated PARSEC isochrones (Bressan et al., 2012) to find stellar masses and
corresponding effective temperatures, as well. The use of PARSEC isochrone-
based SED analysis for this purpose was explained in Borkovits et al. 2020a,
while a comparison of the accuracy and efficiency of such an astrophysical
model-dependent analysis with the classical, astrophysical model-independent
analysis (the latter of which is based on RV data), was carried out and
discussed in Borkovits et al. 2022.

In the case of one of the four systems, TIC 14839347, our preliminary analysis
run, however, revealed a situation where the secondary is fairly low in mass,
but large enough to fill or nearly fill its Roche lobe. We therefore suspect
that the secondary star in TIC 14839347 may have already transferred a portion
of its envelope to the (current) primary star. As a consequence, for this
triple we cannot use this proxy method because the precomputed PARSEC
evolution grids are valid only for single stars, i.e., for such binary
components which have not previously exchanged mass. Therefore, in the case of
this particular triple, we followed an iterative method as follows. First, we
carried out Lightcurvefactory fitting, excluding the SED analysis section. In
such a way, we obtained strong constraints for the relative (or, fractional)
radii of the three stars, as well as their temperature ratios. Then, using
these relative quantities, as well as the Gaia EDR3 derived distance (Bailer-
Jones et al., 2021), we made an independent SED analysis without the use of
any astrophysical preassumptions, with a slightly modified version of the
method described in Rappaport et al. (2022), Sect. 3. In this way, we obtained
likely mass and temperature ranges for the three constituent stars. Then we
used the SED-determined values for the mass of the primary of the inner binary
($m_{\mathrm{Aa}}$) and the effective temperature of the tertiary
($T_{\mathrm{B}}$) with their statistical uncertainties as Gaussian priors,
and reiterated the whole simulatneous LC and ETV curve fitting procedure with
Lightcurvefactory. With this iterative procedure, we were able to find the
system geometry and dynamics robustly, while also inferring the physical
properties of the constituent stars at a reasonable level of accuracy.

In regard to TIC 14839347, we also had to introduce some further modification
to the light-curve fitting part of the complex analysis. We found that, in
this case, the reflection/irradiation effect gives a significant contribution
to the LC of the inner EB. Hence, we “switched on” this effect for the
modeling. First we set the bolometric albedos according to the theoretically
expected values of $A=1.0$ and $0.5$ for the primary (Aa) and secondary stars
(Ab), (i.e., radiative and convective envelopes) respectively. We found,
however, that the fit becomes distinctly better when one sets
$A_{\mathrm{Ab}}\approx 0.8$. For this reason, finally we decided to adjust,
as an exception to our usual approach, the bolometric albedos and gravity
darkening exponents ($\beta$) for these two stars.

Finally we note the fact that none of the four investigated triples is a
‘tight’ system (the outer-to-inner period ratios,
$P_{\mathrm{out}}/P_{\mathrm{in}}$ are somewhat large, ranging from 45 for TIC
88206187, to 109 TIC 298714297) and, hence, one cannot expect significant
third-body perturbations which would produce significant departures from
simple Keplerian motions. On the other hand, considering the fact that the
third-body eclipses are extremely sensitive to the current system geometry,
similar to our former analyses (where much tighter triples were analysed) we
integrated the motion of the three stellar components numerically, instead of
approximating the motion with the superposition of two Keplerian orbits. The
numerical integrator built into Lightcurvefactory is a seventh order Runge-
Kutta-Nystrom integrator. This integrator provides the gravitationally (pure
Newtonian two- and three-body terms), tidally (within the framework of the
equilibrium tide model) and, optionally, the relativistically perturbed
Cartesian Jacobian coordinates (and velocities) for each observational
instant. A detailed description of the integrator itself, and its
implementation in the photodynamical software package can be found in
Borkovits et al. (2004); Borkovits et al. (2019a, b).

We initialized and ran several MCMC chains for each of the four triple
systems. The best fit parameters (median values of the posteriors), their
$1-\sigma$ uncertainties, as well as several derived parameters, are tabulated
in Tables 2 and 3. Here we note, that while most of the given derived
parameters (e.g., the semi-major axes, radial velocity amplitudes, bolometric
lumonisities, etc.) do not require any further explanations, we discuss
briefly the calculations and significances of the different apsidal and nodal
motion parameters, tabulated in between the orbital elements and the stellar
parameters.

We give the theoretical apsidal motion periods both in the observational and
the dynamical frames of reference ($P_{\mathrm{apse}}$ and
$P_{\mathrm{apse}}^{\mathrm{dyn}}$). While the formulae for their
calculations, from the initial osculating orbital elements at each accepted
MCMC trial step, are discussed in detail in Sect. 6.3 of Kostov et al. (2021),
here we emphasize only the fact that the observational and dynamical apsidal
motions and, hence, their timescales are substantially different. From an
observational point of view, what is significant is the revolution of the
argument of periastron in the observational frame of reference.222The basic
plane of the observational reference frame is the tangential plane of the sky,
which is perpendicular to the line of sight to the target as seen by the
observer. The argument of pericenter in this frame is the angle measured from
one of the intersections of the orbit and the nodal line of this tangential
plane and the orbital plane, to the pericenter point, along the orbit. This
revolution manifests itself, e.g., in the periodic, quasi-sinusoidal and
anticorrelated shifts of the primary and secondary eclipses, in the case of
eccentric EBs. In contrast to this, the revolution of the argument of
pericenter in the dynamical frame333The basic plane of the dynamical reference
frame is the invariable plane of the triple system, i.e., the plane whose
normal is parallel to the constant total angular momentum of the triple. The
argument of pericenter in this frame is measured from one of the intersections
of the orbit and the nodal line of this invariable plane and the orbital
plane, to the pericenter point, along the orbit. cannot be observed directly,
however, this dynamical argument of periastron and, hence, the dynamical-frame
apsidal motion are significant for dynamical studies, as this parameter is one
that occurs in the perturbation equations. The observable-frame apsidal
motion, in general, is a non-linear combination of the dynamical-frame apsidal
motion and the (dynamical) nodal regression, and hence, their periods (i.e.,
$P_{\mathrm{apse}}$ and $P_{\mathrm{apse}}^{\mathrm{dyn}}$) may be
substantially different (see, e.g., Borkovits et al. 2015 for further
details). Note also, that besides the apsidal motion and nodal regression
($P_{\mathrm{node}}^{\mathrm{dyn}}$) periods, we give separately the three
components of the apsidal advance rates (classical tidal, $\Delta\omega_{\rm
tide}$; general relativistic, $\Delta\omega_{\rm GR}$; and dynamical third-
body, $\Delta\omega_{\rm 3b}$, respectively) for one orbital revolution of the
inner and outer orbits, respectively. These contributions are calculated in
the dynamical frame of reference.

Finally, note that the third-body eclipse sections of the model LCs from the
best-fit complex photodynamical solutions are plotted in Fig. 1, while the
model ETV curves of the best-fit solutions are shown in Figs. 4–7. We briefly
discuss our results for each system separately in Sect. 4.

Table 2: Orbital and astrophysical parameters of TICs 14839347 and 66893949
from the joint photodynamical TESS, ETV, SED and PARSEC isochrone solution.
Note that the orbital parameters are instantaneous, osculating orbital
elements and are given for epoch $t_{0}$ (first row).

| TIC 14839347 | TIC 66893949
---|---|---
orbital elements
| subsystem | subsystem
| Aa–Ab | A–B | Aa–Ab | A–B
$t_{0}$ [BJD - 2400000] | 58 683.0 | 58 711.0
$P$ [days] | $1.154060_{-0.000052}^{+0.000050}$ | $85.530_{-0.017}^{+0.017}$ | $4.805309_{-0.000005}^{+0.000004}$ | $471.03_{-0.07}^{+0.10}$
$a$ [R⊙] | $6.769_{-0.034}^{+0.043}$ | $144.2_{-0.9}^{+1.7}$ | $16.29_{-0.13}^{+0.16}$ | $386.4_{-3.4}^{+4.2}$
$e$ | $0.0005_{-0.0003}^{+0.0005}$ | $0.042_{-0.013}^{+0.013}$ | $0.0050_{-0.0003}^{+0.0013}$ | $0.4016_{-0.0038}^{+0.0039}$
$\omega$ [deg] | $147_{-97}^{+138}$ | $269.6_{-3.1}^{+2.8}$ | $317_{-35}^{+32}$ | $25.8_{-1.5}^{+1.3}$
$i$ [deg] | $88.75_{-1.13}^{+1.13}$ | $86.51_{-0.13}^{+0.12}$ | $90.18_{-0.29}^{+0.34}$ | $90.222_{-0.009}^{+0.010}$
$\mathcal{T}_{0}^{\mathrm{inf}}$ [BJD - 2400000]a | $58\,684.0427_{-0.0001}^{+0.0001}$ | ${59\,430.6949_{-0.0210}^{+0.0213}}$ | $58\,714.2412_{-0.0002}^{+0.0002}$ | ${58\,729.4535_{-0.0245}^{+0.0225}}$
$\tau$ [BJD - 2400000]b | $58\,683.56_{-0.46}^{+0.20}$ | $59\,429.6_{-0.7}^{+0.8}$ | $58\,710.46_{-0.42}^{+0.38}$ | $58\,349.1_{-1.6}^{+1.4}$
$\Omega$ [deg] | $0.0$ | $-2.9_{-3.1}^{+1.7}$ | $0.0$ | $0.60_{-0.48}^{+0.53}$
$i_{\mathrm{mut}}$ [deg] | $3.5_{-1.0}^{+3.3}$ | $0.69_{-0.35}^{+0.49}$
mass ratio $[q=m_{\mathrm{sec}}/m_{\mathrm{pri}}]$ | $0.250_{-0.015}^{+0.016}$ | $0.757_{-0.019}^{+0.050}$ | $0.670_{-0.005}^{+0.005}$ | $0.388_{-0.006}^{+0.007}$
$K_{\mathrm{pri}}$ [km s-1] | $59.4_{-3.1}^{+3.3}$ | $36.7_{-0.7}^{+1.7}$ | $68.84_{-0.40}^{+0.44}$ | $12.67_{-0.23}^{+0.28}$
$K_{\mathrm{sec}}$ [km s-1] | $237.5_{-2.3}^{+2.1}$ | $48.4_{-0.8}^{+0.5}$ | $102.80_{-1.06}^{+1.30}$ | $32.67_{-0.24}^{+0.27}$
Apsidal and nodal motion related parametersc
$P_{\mathrm{apse}}$ [year] | $0.80_{-0.02}^{+0.02}$ | $889_{-33}^{+33}$ | $417_{-6}^{+6}$ | $2821_{-29}^{+30}$
$P_{\mathrm{apse}}^{\mathrm{dyn}}$ [year] | $0.79_{-0.02}^{+0.02}$ | $48.0_{-1.6}^{+0.6}$ | $204_{-3}^{+2}$ | $350_{-3}^{+3}$
$P_{\mathrm{node}}^{\mathrm{dyn}}$ [year] | $50.7_{-0.7}^{+1.8}$ | $399_{-4}^{+5}$
$\Delta\omega_{\mathrm{3b}}$ [arcsec/cycle] | $156_{-3}^{+7}$ | $6321_{-79}^{+224}$ | $79.4_{-0.9}^{+1.0}$ | $4779_{-45}^{+47}$
$\Delta\omega_{\mathrm{GR}}$ [arcsec/cycle] | $3.81_{-0.04}^{+0.05}$ | $0.315_{-0.004}^{+0.008}$ | $1.26_{-0.05}^{+0.04}$ | $0.088_{-0.004}^{+0.003}$
$\Delta\omega_{\mathrm{tide}}$ [arcsec/cycle] | $5014_{-131}^{+123}$ | $1.7_{-0.1}^{+0.1}$ | $2.93_{-0.12}^{+0.15}$ | $0.00102_{-0.00004}^{+0.0005}$
stellar parameters
| Aa | Ab | B | Aa | Ab | B
Relative quantities and atmospheric properties
fractional radius [$R/a$] | $0.4239_{-0.0041}^{+0.0037}$ | $0.2676_{-0.0045}^{+0.0046}$ | $0.0573_{-0.0016}^{+0.0018}$ | $0.0949_{-0.0006}^{+0.0007}$ | $0.0549_{-0.0007}^{+0.0009}$ | $0.00223_{-0.00006}^{+0.00008}$
temperature relative to $(T_{\mathrm{eff}})_{\mathrm{Aa}}$ | $1$ | $0.548_{-0.022}^{+0.016}$ | $0.559_{-0.020}^{+0.024}$ | $1$ | $0.803_{-0.006}^{+0.006}$ | $0.786_{-0.009}^{+0.008}$
fractional flux [in TESS-band] | $0.3703_{-0.0125}^{+0.0122}$ | $0.0372_{-0.0015}^{+0.0012}$ | $0.5352_{-0.0265}^{+0.0478}$ | $0.7644_{-0.0140}^{+0.0107}$ | $0.1230_{-0.0016}^{+0.0016}$ | $0.1039_{-0.0082}^{+0.0011}$
grav. dark. exponent [$\beta$]d | $0.41_{-0.25}^{+0.39}$ | $0.17_{-0.13}^{+0.20}$ | … | … | … | …
albedo [$A$]d | $0.97_{-0.06}^{+0.05}$ | $0.76_{-0.05}^{+0.05}$ | … | … | … | …
Physical Quantities
$m$ [M⊙] | ${2.497_{-0.025}^{+0.030}}^{d}$ | $0.625_{-0.038}^{+0.041}$ | $2.359_{-0.061}^{+0.166}$ | $1.503_{-0.038}^{+0.050}$ | $1.007_{-0.021}^{+0.026}$ | $0.974_{-0.034}^{+0.043}$
$R$ [R⊙] | $2.871_{-0.032}^{+0.031}$ | $1.812_{-0.038}^{+0.039}$ | $8.242_{-0.191}^{+0.313}$ | $1.547_{-0.023}^{+0.023}$ | $0.894_{-0.018}^{+0.023}$ | $0.861_{-0.029}^{+0.039}$
$T_{\mathrm{eff}}$ [K] | $8977_{-371}^{+380}$ | $4906_{-108}^{+95}$ | ${5019_{-44}^{+40}}^{c}$ | $6756_{-157}^{+147}$ | $5421_{-90}^{+81}$ | $5298_{-119}^{+150}$
$L_{\mathrm{bol}}$ [L⊙] | $48.0_{-6.9}^{+8.2}$ | $1.71_{-0.14}^{+0.13}$ | $38.7_{-2.6}^{+3.7}$ | $4.47_{-0.49}^{+0.55}$ | $0.618_{-0.059}^{+0.074}$ | $0.523_{-0.077}^{+0.117}$
$M_{\mathrm{bol}}$ | $0.54_{-0.17}^{+0.17}$ | $4.16_{-0.08}^{+0.09}$ | $0.77_{-0.10}^{+0.14}$ | $3.14_{-0.13}^{+0.13}$ | $5.29_{-0.12}^{+0.11}$ | $5.47_{-0.22}^{+0.17}$
$M_{V}$ | $0.60_{-0.11}^{+0.12}$ | $4.50_{-0.12}^{+0.15}$ | $1.06_{-0.10}^{+0.08}$ | $3.09_{-0.13}^{+0.13}$ | $5.42_{-0.15}^{+0.14}$ | $5.65_{-0.27}^{+0.22}$
$\log g$ [dex] | $3.920_{-0.008}^{+0.009}$ | $3.718_{-0.009}^{+0.009}$ | $2.978_{-0.025}^{+0.034}$ | $4.235_{-0.004}^{+0.005}$ | $4.537_{-0.011}^{+0.008}$ | $4.555_{-0.020}^{+0.015}$
Global system parameters
$\log$(age) [dex] | $-$ | $8.77_{-0.21}^{+0.12}$
$[M/H]$ [dex] | $-$ | $0.36_{-0.06}^{+0.04}$
$E(B-V)$ [mag] | $-$ | $0.115_{-0.034}^{+0.030}$
extra light $\ell_{4}$ [in TESS-band] | $0.054_{-0.036}^{+0.025}$ | $0.007_{-0.005}^{+0.009}$
$(M_{V})_{\mathrm{tot}}$ | $0.03_{-0.09}^{+0.10}$ | $2.88_{-0.14}^{+0.13}$
distance [pc] | $...$ | $623_{-15}^{+19}$

Notes.  a: $\mathcal{T}_{0}^{\mathrm{inf}}$ denotes the moment of an inferior
conjunction of the secondary (Ab) and the tertiary (B) along their inner and
outer orbits, respectively; b: time of periastron passage; c: for the meaning,
significance and discussion of these parameters, see Sect. 3; d: gravitational
darkening exponents ($\beta$) and bolometric albedos ($A$) were adjusted only
for the inner pair of TIC 14839347 and, hence, these are given only in this
case; d: taken from independent SED analysis with Gaussian priors.

##  4 Discussion

###  4.1 TIC 14839347

As was mentioned above, the analysis of this triple system had to be carried
out iteratively, and with extra care. We adopted $m_{\mathrm{Aa}}=2.50\pm
0.13\,\mathrm{M}_{\odot}$ and $T_{\mathrm{B}}=5000\pm 200$ K for the mass of
the primary and effective temperature of the tertiary, from the separate SED
analysis. The inner mass ratio was found to be as low as
$q_{\mathrm{in}}=0.25\pm 0.02$, which, together with a near Roche-lobe-filling
secondary, make this inner binary appear to be a typical Algol-type system
with a reversed mass ratio. The distant third star was found to be a little
less massive than the current primary of the inner binary
($m_{\mathrm{B}}=2.36_{-0.09}^{+0.17}\,\mathrm{M}_{\odot}$).

This finding again emphasizes that, most likely, intensive mass exchange has
occurred between the two inner binary stars in the near past of this triple
system, as the tertiary component is evidently an evolved, red giant star
($R_{\mathrm{B}}=8.2\pm 0.3\,\mathrm{R}_{\odot}$), in contrast to the
similarly massive, hot ($T_{\mathrm{Aa}}=8975\pm 380$ K) and much less evolved
($R_{\mathrm{Aa}}=2.87\pm 0.03\,\mathrm{R}_{\odot}$) primary component.

Here we emphasize again that, in the case of this particular system, we took
into account the reflection/irradiation effect and, moreover, both the two
gravity-darkening exponents and the bolometric albedos of the inner pair of
stars were freely adjusted MCMC parameters. In the case of the hot and, hence,
radiative primary star (component Aa) we obtained an albedo of
$A_{\mathrm{Aa}}=0.97\pm 0.05$ which fits nicely with the theoretically
expected value of $A_{\mathrm{Aa}}^{\mathrm{theo}}=1$.444In this regard, note
that in their study Southworth et al. (2011) have obtained a clearly
unphysical value of $A>2$ during the analysis of another semi-detached system
with a hot primary. The authors explained this by the incompleteness of the
physical model that they used – which happens to be very close to ours. Hence,
this agreement we find with the theoretical value is far from trivial. On the
other hand, the bolometric albedo obtained for the convective secondary
($A_{\mathrm{Ab}}=0.76\pm 0.05$) significantly differs from the theoretically
expected value of $A_{\mathrm{Ab}}^{\mathrm{theo}}=0.5$. According to our
knowledge, such a high albedo for a convective star is quite unusual, however,
a deeper investigation of this question is beyond the scope of this paper.

Turning to the orbital properties of this triple system, we find that the
inner orbit is circular, as is expected in the case of a nearly semi-detached
system.555Note that though our statistical results give an inner eccentricity
of $e_{\mathrm{in}}=0.0005\pm 0.0004$, this is only for the instantaneous
osculating orbital elements, which, in general, cannot result in exactly zero
eccentricity in a perturbed Keplerian problem. This question was discussed in
detail e.g. in Kiseleva et al. (1998); Borkovits et al. (2002). By contrast,
the outer orbit displays some small, but significant, eccentricity
($e_{\mathrm{out}}=0.04\pm 0.01$). What makes this result, however, a bit less
robust is that the secondary outer eclipses (in Sectors 14 and 55) are located
exactly at phase 0.5. This is reflected by the fact that the outer argument of
pericenter is $\omega_{\mathrm{out}}=270\degr\pm 3\degr$ and, hence,
$e_{\mathrm{out}}\cos\omega_{\mathrm{out}}\approx 0$. In turn, this would
suggest that the outer orbit is seen exactly from the direction of its major
axis. However, one should keep in mind that, in this particular case, the
outer eccentricity is determined only via the weakly determined parameter
$e_{\mathrm{out}}\sin\omega_{\mathrm{out}}$, which primarily manifests itself
in the durations of the outer eclipses and has strong correlations with the
outer inclination, the fractional radius of the tertiary, the node of the
outer orbit relative to the inner one, and even some other parameters.666As
one can readily see in Fig. 4, the ETV curve is currently so poorly covered
that it does not carry any information about the eccentricity of the outer
orbit.

As both the inner and outer orbits are fairly circular and almost coplanar
($i_{\mathrm{m}}=3\aas@@fstack{\circ}5_{-1}^{+3}$), one cannot expect large
perturbations and, hence, there are only small departures from Keplerian
motions for the inner and outer orbits. This is true despite the fact that the
nominal apsidal motion period of the inner orbit is shorter than a year
($P_{\mathrm{apse}}=0.80\pm 0.02$ yr). As one can see from the different kinds
of apsidal advance rates $\Delta\omega$, in the case of the inner orbit the
classic tidal ($\Delta\omega_{\mathrm{tide}}$) contribution is the dominant
one—larger by one order of magnitude than that of the third-body driven
apsidal motion. We emphasize again, however, that as the inner orbit is
practically circular, this very short apsidal motion is only virtual, stemming
from the above mentioned slight imperfections in the use of osculating orbital
elements in this particular case. On the other hand, for the small but clearly
non-zero mutual inclination, one can expect a very low (few degrees) amplitude
inclination variation with a period of $P_{\mathrm{node}}^{\mathrm{dyn}}=51\pm
1$ yr, which might manifest itself in very small eclipse depth variations in
the future.

Here we note also that the SED portion of the iterative solution for this
system allowed for an age determination of the system of $520\pm 115$
Myr777Here we assumed that even though the inner binary stars may have
previously exchanged mass, the tertiary star has not been affected and is
evolving normally as a single star., and an independent fit for the
interstellar extinction of $A_{V}=2.2\pm 0.16$.

Figure 4: Photodynamical fit to the primary TESS ETV curve for TIC 14839347.
The larger and darker red circles represent the observed primary times of EB
eclipses, while the smaller, lighter symbols, connected with straight lines
are taken from the photodynamical model ETV curve. The three thin vertical
lines denote the locations of the three third-body outer eclipses. Residuals
are also shown in the lower panel, where the uncertainty on each point is also
noted.

###  4.2 TIC 66893949

This system has the longest inner ($P_{\mathrm{in}}=4.805$ d) and outer
($P_{\mathrm{out}}=471.0$ d) periods. Because the TESS observations cover only
16-17% of the full outer orbit (see Fig. 5) it is quite surprising (and, of
course, fortuitous) that both primary and secondary third-body eclipses were
observed.

According to our photodynamical results, the system consists of three sun-
like, main sequence stars ($m_{\mathrm{Aa}}=1.51\pm 0.04\,\mathrm{M}_{\odot}$,
$m_{\mathrm{Ab}}=1.01\pm 0.02\,\mathrm{M}_{\odot}$, $m_{\mathrm{B}}=0.97\pm
0.04\,\mathrm{M}_{\odot}$). Note, amongst our four triple systems, this is the
only one where the tertiary star was found to be the least massive.

In connection with the somewhat larger inner orbital separation, the inner
binary pair has a small, but significant, eccentricity
($e_{\mathrm{in}}=0.005\pm 0.001$). The outer orbit also has the largest
eccentricity in our sample with $e_{\mathrm{out}}=0.402\pm 0.004$.
Interestingly, despite the somewhat wide configuration, this triple was found
to be quite flat ($i_{\mathrm{mut}}=0\aas@@fstack{\circ}7\pm
0\aas@@fstack{\circ}5$).

The dynamical timescales in this systems exceed several centuries for both the
inner and outer subsystems. Moreover, as one can see from the separate apsidal
advance rates, both the classic tidal and the relativistic contributions to
the apsidal motions are negligible relative to the dynamical (forced by third-
body perturbations) apsidal motions of the inner and outer orbits.

Finally, we note that our analysis resulted in a slightly larger distance than
that of Bailer-Jones et al. (2021), i.e., $d_{\mathrm{phot}}=625\pm 17$ pc,
vs. $d_{\mathrm{EDR3}}=587\pm 9$ pc, a $2-\sigma$ discrepancy.

Figure 5: Photodynamical fit to the TESS ETV curves for TIC 6893949. The
larger and darker red circles and blue squares represent the observed primary
and secondary times of EB eclipses, while the smaller, lighter symbols,
connected with straight lines are taken from the photodynamical model ETV
curve. The two thin vertical lines denote the locations of the two third-body
outer eclipses. Residuals are also shown in the lower panel, where the
uncertainty on each point is also noted.

###  4.3 TIC 88206187

Despite the fact that this triple was observed only in two TESS sectors, we
were able to obtain the most robust solution for this system among those
studied in this work. The reason is that, fortunately, TESS observed both
primary and secondary third-body eclipses and, moreover, both the inner and
outer eclipses are total (i.e., having flat-bottom mid-eclipse sections) which
strongly constrain the surface brightness ratios of the constituent stars.
Moreover, the combination of the TESS observations with the archive ground-
base data (see Sect. 2.2) we were able to determine the outer period with
considerable accuracy ($P_{\mathrm{out}}=52.84\pm 0.01$ days) which was found
to be in reasonable agreement with that of the somewhat less accurate Gaia SB1
solution ($P_{\mathrm{out}}=53.03\pm 0.06$ days; Table 4). This makes this
triple the most compact in our sample.

Already, a very first inspection of the 2-day-long primary third-body eclipse
around BJD 2 458 829 (at the middle of Sector 19, see lower left panel of Fig.
1) reveals that the tertiary is likely the largest component and, most
probably, is a red giant star.888Note that amongst other TESS-discovered
triply eclipsing triples, analyzed in former papers, it is TIC 54060695 whose
LC (more strictly speaking, the third-body eclipses) most closely resembles
that of the current triple system (see the middle left panel of Fig. 1 of
Rappaport et al., 2022). The unequal primary and secondary third-body eclipse
depths, with the flat bottom of the much deeper (primary) third-body eclipse,
indicate that the members of the inner EB are much smaller, but substantially
hotter, than the tertiary star. From this, one can directly infer that the
tertiary component should be the most massive amongst the three stars, being
the most evolved (assuming, of course, coeval evolution for the three
objects). Finally, the fact that, in the case of the primary third-body
eclipse (at Sector 19) the tertiary component passes its inferior conjunction
point, reveals the outer orbital phase at that (or any other) moment uniquely,
despite the absence of any informative ETV data (see Fig. 6).

The photodynamical solution then confirmed all these preliminary assessments,
as it was found that the tertiary star was actually a moderately massive
($m_{\mathrm{B}}=2.6\pm 0.1\,\mathrm{M}_{\odot}$) red giant star with the
basic parameters of $R_{\mathrm{B}}=11.7\pm 0.3\,\mathrm{R}_{\odot}$ and
$T_{\mathrm{B}}=4950\pm 70$ K. The members of the inner binary pair, however,
were found to be A and F-type, but still MS stars, with parameters of
$m_{\mathrm{Aa,Ab}}=2.05\pm 0.05\,\mathrm{M}_{\odot}$, $1.35\pm
0.03\,\mathrm{M}_{\odot}$; $R_{\mathrm{Aa,Ab}}=2.25\pm
0.05\,\mathrm{R}_{\odot}$, $1.35\pm 0.03\,\mathrm{R}_{\odot}$;
$T_{\mathrm{Aa,Ab}}=8\,360^{+270}_{-160}$ K, $6\,590^{+170}_{-70}$ K for the
primary and secondary binary components, respectively.

The future evolution of this triple seems to be quite interesting. Due to the
compactness of both the inner and outer subsystems, we may expect that both
the currently more massive red giant tertiary and the TAMS-aged primary of the
inner binary will fill out their respective Roche-lobes. The current size of
the tertiary and its Roche lobe radius are 11.7 R⊙ and 38.3 R⊙, respectively.
The same quantities for the primary EB star, Aa, are 2.25 R⊙ and 2.94 R⊙,
respectively. Thus, star Aa needs to increase its radius only by $\simeq 30\%$
in order to overflow its Roche lobe, whereas the tertiary would need to triple
its current radius to overflow its Roche lobe. Nonetheless, according to MIST
stellar evolution tracks (Dotter, 2016; Choi et al., 2016) it will take star
Aa some 260 Myr to reach a state where it overflows its Roche lobe with
respect to star Ab. By contrast, the more massive and currently more evolved
tertiary star will evolve much more rapidly and will reach the tip of the red
giant branch (RGB) in a few Myr. At that time, its radius will not quite be
sufficiently large to fill its Roche lobe. However, in less than 180 Myr after
that, it will ascend the asymptotic giant branch (AGB) and overfill its Roche
lobe for certain.

The detailed evolution of this system, after mass transfer from the giant
tertiary to the inner binary commences, is beyond the scope of this paper.
However, we can speculate that since the tertiary will be a convective giant
at that time, and the outer mass ratio ($m_{B}/(m_{\rm Aa}+m_{\rm Ab}$)) is
0.76, the mass transfer could be dynamically stable. But, formally, for a
completely convective donor star, this ratio should be $\lesssim 2/3$.

Apart from its future evolution with mass transfer, this triple may harbour
other interesting effects in its present configuration as well. Recently Gao
et al. (2023) called attention to the fact that such compact triple stars,
where the tertiaries are red giants, may produce remarkable tidal effects
whereby tidal dissipation could lead to observable orbital shrinkage within
some decades-long timescales. In this regard, we note that amongst all the
known analysed compact triple star systems, TIC 88206187 is only the third
case in which the fractional radius of the distant tertiary exceeds 0.1
($r_{\mathrm{B}}=R_{\mathrm{B}}/a_{\mathrm{out}}=0.109\pm 0.002$)999The other
two systems are TIC 242132789 (Rappaport et al., 2022) and HD 181068
(Borkovits et al., 2013; Fuller et al., 2013) with $r_{\mathrm{B}}=0.151$ and
$0.138$, respectively., and thus, tertiary tides might currently be effective,
at least marginally. In this regard, also note that, despite the relatively
more significant apsidal advance rate of the outer orbit due to tidal forcing
($\Delta\omega_{\mathrm{tide}}=9^{\prime\prime}\pm 1^{\prime\prime}$/cycle),
it is clearly negligible relative to the third-body effects
($\Delta\omega_{\mathrm{tide}}=11400^{\prime\prime}\pm
100^{\prime\prime}$/cycle) and does not alter the timescale of the apsidal
precession ($P_{\mathrm{apse}}=185\pm 1$ yr).

Finally, we note that the photometric distance inferred from our analysis is
$d_{\mathrm{phot}}=2580\pm 60$ pc, which is in quite a good agreement with the
Gaia-parallax inferred distance of $d_{\mathrm{EDR3}}=2490\pm 90$ pc (Bailer-
Jones et al., 2021).

Figure 6: Photodynamical fit to the TESS ETV curves for TIC 88206187.

###  4.4 TIC 298714297

In contrast to the other systems in this study, TIC 298714297 is found to be a
triplet of three low-mass cool red (K and M-type) dwarfs. Besides the three
dips of a complex third-body eclipse, the other remarkable feature of the LC
is the likely rotational distortions, caused by stellar spots, which are most
pronounced in the Sector 55, 56 LCs. Moreover, the system shows three sudden,
short $\sim$$3-4\%$ brightenings which we attribute to stellar flares.

All these features suggest strong chromospheric/photospheric stellar activity
on at least one of the stellar components. In the case of the rotational
modulation, since the observed period is equal, or very close to, the period
of the inner EB, it seems quite likely that its origin is component Aa (which
is the most massive and brightest amongst the three stars). In the case of the
three stellar flare events, their origin, strictly speaking, is less certain.
But, considering the other signals of strong magnetic activity are likely from
star Aa, we may tentatively assume that the flare events are also hosted by
this star. It is noteworthy, however, that the two larger flares were observed
during Sector 15, when the rotational modulation (i.e., due to the
spottedness) was much less pronounced.

Regarding the astrophysical parameters, this triple was found to be not only
the least massive ($m_{\mathrm{Aa}}=0.83\pm 0.03\,\mathrm{M}_{\odot}$,
$m_{\mathrm{Ab}}=0.50\pm 0.02\,\mathrm{M}_{\odot}$, $m_{\mathrm{B}}=0.68\pm
0.02\,\mathrm{M}_{\odot}$) of our set, but also the most aged ($\tau\sim 10$
Gyr). Note, however, that some caution is necessary, as our model solution
resulted in a reddening of $E(B-V)=0.3\pm 0.08$ which looks quite unrealistic
for such a close object ($d_{\mathrm{phot}}=138\pm 4$ pc vs.
$d_{\mathrm{EDR3}}=111.9\pm 0.7$ pc Bailer-Jones et al. 2021).

The inner orbit was found to be practically circular
($e_{\mathrm{in}}=0.002_{-0.001}^{+0.007}$) which is naturally expected for
such an old and close binary, while the outer orbit was found to be moderately
eccentric ($e_{\mathrm{out}}=0.24\pm 0.03$). In this regard, we note that the
secondary’s ETV curve in the inner EB is offset a little from the primary’s
one (Fig. 7). However, in our interpretation, in this case such an offset is
not due a slight orbital eccentricity, but rather is caused by the LC
variations due to the starspots on at least one of the stars. Such shifts in
the ETVs were found by Tran et al. (2013).

Finally, note also that besides TIC 14839347, this is the other system in our
sample where the apsidal motion of the inner orbit is dominated by the tidal
forces instead of third-body perturbations
($\Delta\omega_{\mathrm{tide}}=105^{\prime\prime}\pm 10^{\prime\prime}$/cycle
vs. $\Delta\omega_{\mathrm{3b}}=63^{\prime\prime}\pm 2^{\prime\prime}$/cycle).
But, again, due to the very low inner eccentricity one cannot expect strong,
non-Keplerian variations during the apsidal motion cycle of
$P_{\mathrm{apse}}=28\pm 2$ yr.

Figure 7: Photodynamical fit to the TESS ETV curves for TIC 298714297. Table
3: Orbital and astrophysical parameters of TICs 88206187 and 298714297 from
the joint photodynamical TESS, ETV, SED and PARSEC isochrone solution. Note
that the orbital parameters are instantaneous, osculating orbital elements and
are given for epoch $t_{0}$ (first row).

| TIC 88206187 | TIC 298714297
---|---|---
orbital elements
| subsystem | subsystem
| Aa–Ab | A–B | Aa–Ab | A–B
$t_{0}$ [BJD - 2400000] | 58 816.0 | 58 711.0
$P$ [days] | $1.184592_{-0.000063}^{+0.000055}$ | $52.922_{-0.039}^{+0.041}$ | $1.072891_{-0.000019}^{+0.000010}$ | $117.24_{-0.31}^{+0.36}$
$a$ [R⊙] | $7.085_{-0.061}^{+0.053}$ | $107.7_{-1.0}^{+1.1}$ | $4.863_{-0.077}^{+0.050}$ | $127.5_{-1.8}^{+1.3}$
$e$ | $0.0012_{-0.0005}^{+0.0005}$ | $0.026_{-0.013}^{+0.017}$ | $0.0019_{-0.0009}^{+0.0072}$ | $0.241_{-0.028}^{+0.029}$
$\omega$ [deg] | $75_{-27}^{+21}$ | $147_{-99}^{+100}$ | $84_{-10}^{+71}$ | $28_{-6}^{+8}$
$i$ [deg] | $89.86_{-1.98}^{+2.36}$ | $89.54_{-0.75}^{+0.73}$ | $89.00_{-0.79}^{+0.93}$ | $89.86_{-0.05}^{+0.10}$
$\mathcal{T}_{0}^{\mathrm{inf/sup}}$ [BJD - 2400000] | $58\,816.9391_{-0.0002}^{+0.0002}$ | $58\,828.8819_{-0.0081}^{+0.0081}$ | $58\,711.4972_{-0.0001}^{+0.0002}$ | ${59\,804.6649_{-0.0385}^{+0.0396}}^{*}$
$\tau$ [BJD - 2400000] | $58\,816.292_{-0.094}^{+0.067}$ | $58\,800.2_{-13.8}^{+15.1}$ | $58\,710.942_{-0.029}^{+0.213}$ | $59\,791.8_{-1.6}^{+2.0}$
$\Omega$ [deg] | $0.0$ | $-1.17_{-2.18}^{+3.01}$ | $0.0$ | $4.23_{-4.13}^{+4.33}$
$i_{\mathrm{mut}}$ [deg] | $2.90_{-1.37}^{+1.63}$ | $4.41_{-2.70}^{+4.18}$
mass ratio $[q=m_{\mathrm{sec}}/m_{\mathrm{pri}}]$ | $0.661_{-0.009}^{+0.009}$ | $0.765_{-0.017}^{+0.010}$ | $0.617_{-0.016}^{+0.013}$ | $0.512_{-0.009}^{+0.007}$
$K_{\mathrm{pri}}$ [km s-1] | $120.36_{-0.92}^{+1.02}$ | $44.48_{-0.73}^{+0.97}$ | $87.26_{-2.02}^{+2.14}$ | $19.12_{-0.27}^{+0.34}$
$K_{\mathrm{sec}}$ [km s-1] | $182.26_{-2.40}^{+1.83}$ | $58.47_{-0.37}^{+0.41}$ | $141.79_{-1.94}^{+1.14}$ | $37.48_{-0.60}^{+0.57}$
Apsidal and nodal motion related parameters
$P_{\mathrm{apse}}$ [year] | $7.3_{-0.3}^{+0.2}$ | $185_{-1}^{+1}$ | $27.8_{-2.2}^{+1.9}$ | $1124_{-34}^{+30}$
$P_{\mathrm{apse}}^{\mathrm{dyn}}$ [year] | $5.2_{-0.2}^{+0.1}$ | $16.41_{-0.09}^{+0.16}$ | $22.4_{-1.5}^{+1.2}$ | $104_{-3}^{+2}$
$P_{\mathrm{node}}^{\mathrm{dyn}}$ [year] | $18.0_{-0.2}^{+0.1}$ | $114_{-2}^{+3}$
$\Delta\omega_{\mathrm{3b}}$ [arcsec/cycle] | $443_{-5}^{+4}$ | $11433_{-113}^{+62}$ | $63.4_{-1.5}^{+1.6}$ | $4013_{-85}^{+105}$
$\Delta\omega_{\mathrm{GR}}$ [arcsec/cycle] | $3.96_{-0.07}^{+0.06}$ | $0.459_{-0.008}^{+0.10}$ | $2.28_{-0.07}^{+0.05}$ | $0.014_{-0.004}^{+0.004}$
$\Delta\omega_{\mathrm{tide}}$ [arcsec/cycle] | $362_{-17}^{+24}$ | $9.0_{-0.8}^{+0.9}$ | $104_{-9}^{+12}$ | $0.025_{-0.002}^{+0.002}$
stellar parameters
| Aa | Ab | B | Aa | Ab | B
Relative quantities
fractional radius [$R/a$] | $0.3185_{-0.0039}^{+0.0047}$ | $0.1905_{-0.0028}^{+0.0030}$ | $0.1086_{-0.0020}^{+0.0021}$ | $0.1728_{-0.0025}^{+0.0026}$ | $0.1042_{-0.0043}^{+0.0050}$ | $0.0053_{-0.0001}^{+0.0001}$
temperature relative to $(T_{\mathrm{eff}})_{\mathrm{Aa}}$ | $1$ | $0.789_{-0.010}^{+0.010}$ | $0.592_{-0.012}^{+0.010}$ | $1$ | $0.678_{-0.007}^{+0.006}$ | $0.840_{-0.013}^{+0.015}$
fractional flux [in TESS-band] | $0.1600_{-0.0026}^{+0.0027}$ | $0.0294_{-0.0011}^{+0.0011}$ | $0.800_{-0.015}^{+0.008}$ | $0.6794_{-0.0335}^{+0.0386}$ | $0.0532_{-0.0026}^{+0.0029}$ | $0.2062_{-0.0217}^{+0.0191}$
Physical Quantities
$m$ [M⊙] | $2.047_{-0.061}^{+0.051}$ | $1.349_{-0.026}^{+0.031}$ | $2.589_{-0.090}^{+0.107}$ | $0.827_{-0.036}^{+0.024}$ | $0.508_{-0.025}^{+0.024}$ | $0.678_{-0.022}^{+0.026}$
$R$ [R⊙] | $2.253_{-0.034}^{+0.048}$ | $1.349_{-0.028}^{+0.030}$ | $11.71_{-0.30}^{+0.28}$ | $0.839_{-0.023}^{+0.021}$ | $0.506_{-0.028}^{+0.030}$ | $0.669_{-0.018}^{+0.022}$
$T_{\mathrm{eff}}$ [K] | $8358_{-154}^{+279}$ | $6587_{-72}^{+169}$ | $4947_{-59}^{+70}$ | $5332_{-198}^{+106}$ | $3620_{-121}^{+55}$ | $4463_{-137}^{+133}$
$L_{\mathrm{bol}}$ [L⊙] | $22.6_{-2.0}^{+2.5}$ | $3.13_{-0.23}^{+0.28}$ | $74.0_{-4.3}^{+4.6}$ | $0.504_{-0.040}^{+0.037}$ | $0.039_{-0.004}^{+0.004}$ | $0.159_{-0.016}^{+0.019}$
$M_{\mathrm{bol}}$ | $1.39_{-0.12}^{+0.10}$ | $3.53_{-0.09}^{+0.08}$ | $0.10_{-0.07}^{+0.07}$ | $5.51_{-0.08}^{+0.09}$ | $8.30_{-0.11}^{+0.11}$ | $6.77_{-0.12}^{+0.11}$
$M_{V}$ | $1.35_{-0.08}^{+0.09}$ | $3.50_{-0.09}^{+0.09}$ | $0.38_{-0.09}^{+0.07}$ | $5.67_{-0.10}^{+0.15}$ | $9.82_{-0.19}^{+0.33}$ | $7.41_{-0.23}^{+0.23}$
$\log g$ [dex] | $4.040_{-0.011}^{+0.010}$ | $4.307_{-0.010}^{+0.010}$ | $2.713_{-0.016}^{+0.016}$ | $4.505_{-0.012}^{+0.010}$ | $4.733_{-0.030}^{+0.027}$ | $4.617_{-0.012}^{+0.009}$
Global system parameters
$\log$(age) [dex] | $8.81_{-0.05}^{+0.04}$ | $10.02_{-0.04}^{+0.04}$
$[M/H]$ [dex] | $0.17_{-0.16}^{+0.07}$ | $-0.01_{-0.18}^{+0.28}$
$E(B-V)$ [mag] | $0.28_{-0.02}^{+0.04}$ | $0.30_{-0.09}^{+0.05}$
extra light $\ell_{4}$ [in TESS-band] | $0.010_{-0.008}^{+0.015}$ | $0.05_{-0.04}^{+0.05}$
$(M_{V})_{\mathrm{tot}}$ | $-0.03_{-0.08}^{+0.07}$ | $5.45_{-0.12}^{+0.17}$
distance [pc] | $2584_{-68}^{+53}$ | $138_{-4}^{+4}$

Notes. $\mathcal{T}_{0}^{\mathrm{inf/sup}}$ denotes the moment of an inferior
or superior conjunction of the secondary (Ab) and the tertiary (B) along their
inner and outer orbits, respectively. Superior conjunctions are noted with ∗.

##  5 Comparing orbital parameters with Gaia

In general, we find substantial agreement between the Gaia and photodynamical
results for the four systems studied in this work. As far as we can judge,
however, our results are generally the more accurate ones. Having said this,
however, there are a number of caveats to discuss. First, we note that a
comparison of the orbital elements from the Gaia NSS solution and those from
the Lightcurvefactory photodynamical models is limited due to the differences
in their methodologies. The Gaia NSS solutions adopt a Keplerian two-body
orbit model, whereas the calculated orbital elements from Lightcurvefactory
represent instantaneous osculating elements. Consequently, the orbital
elements obtained from these two approaches are not exactly comparable.
However, for the four sources studied in this work, the perturbations to the
motions along the outer orbits are negligible on the timescale of the Gaia
observation interval, and therefore they do not result in significant
discrepancies on this account. For example, the apsidal motion timescales of
the outer orbits in all our four triples range from 185 yr to 2820 yr. Hence,
the orientations of the astronomical orbits measured by Gaia will vary by only
a negligible amount during the full operation of the space telescope.
Moreover, due to the flatness of the four systems, no significant variations
in the outer inclinations will occur. One can therefore expect that the orbits
will remain (almost) pure Keplerian over the entire life cycle of Gaia.

Figure 8: Astrometric orbits for each stellar component in all four
investigated triple systems, together with our calculated photocenter orbits,
the latter of which are only measurable by Gaia. The sky positions and
therefore, the astrometric orbits of all the components were taken directly
from the numerical integrator of Lightcurvefactory. These were then converted
into milliarcseconds from their original physical dimensions according to the
photometric distances obtained from the photodynamical solutions. The
positions of the photocenters were calculated from the flux-weighted averages
of the individual stellar positions, according to Eqs. (C1–C2) of Rappaport et
al. (2022). For each plot there are also two smaller, insert figures. In the
left inserts we show again the astrometric orbits, but with unequally scaled
$x$ and $y$ axes for a better view of the nearly edge-on and, consequently,
near straight line-like sky-projected orbits. In the right inserts we display
zoomed short sections of the photocenter orbits, where the rapid flux
variations caused by the inner eclipses are also taken into account. As one
can see, these eclipses produce sudden, quasi discontinuous shifts in the
photocenter, which may result in higher scatter and, hence, increased
uncertainties in the astrometric orbital elements.

Another issue is that in the Gaia DR3 catalogue, the NSS solutions are limited
to spatially unresolved stellar systems, primarily due to the specific data
processing constraints employed by the Gaia team. In the case of astrometric
solutions (i.e., for TIC 298714297 and TIC 66893949) Gaia measures only the
orbit of the photocenter of the three stars, and not of any of the individual
constituent members within the triple system (Marcussen & Albrecht 2023; Eq.
C2 of Rappaport et al. 2022). This fact, however, may produce some departures
from pure Keplerian astrometric orbits, as during the relatively deep primary
eclipses of the four inner pairs, one may expect some quasi-discontinuous
shifts in the positions of the photocenter.101010Naturally, the same holds for
the astrometric measurements during third-body eclipses, where these
discontinuities might be even much larger. However, due the rarity of the
extra eclipse events, observations during these events are less likely. In
order to illustrate these effects, we plot the theoretically expected
photocenter orbits (together with the above mentioned potential
discontinuities in the small, inserted plots) for all four investigated
systems in Fig. 8. Keeping in mind all of the above discussion, we conclude
that the only direct comparisons that can be made for these systems with the
photodynamical results are the period, the eccentricity, and the inclination
of the outer orbit. In the cases involving the Gaia spectroscopic orbits
(i.e., TIC 88206187 and TIC 14839347), Gaia primarily observes the spectral
lines of the more luminous tertiary component. Thus we can also compare the
Gaia RV semi-amplitude ($K1$) of the outer tetriary with the value from the
photodynamical fit.

A third issue is that generally the argument of periastron of the outer orbit
from the Gaia measurements and those of the photodyamical results typically
differ by 180∘. This is due to the fact that Gaia is generally tracking the
brighter tertiary, while the photodynamical analysis utilizes the ETV points
from the inner binary.

Finally, with these caveats in mind, we compare the Gaia and photodynamical
analysis in Table 4. We find generally good agreement for the outer orbital
period and its inclination angle. However, in the case of the outer
eccentricity, the differences are mostly larger, but in the case of TIC
66893949, for example, the two values are in good agreement.

Table 4: Comparison of orbital parameters of the outer orbit between LightcurveFactory and Gaia DR3 NSS solution. For each object, the upper values belong to the LightcurveFactory, while the lower values correspond to the Gaia DR3 NSS solution. Name | NSS model | P[days] | e | $\mathrm{\omega[deg]}$ | $\mathrm{a[R_{\odot}]}$ | $\mathrm{i[deg]}$ | $\mathrm{K_{sec}[km/s]}$
---|---|---|---|---|---|---|---
TIC 14839347 | SB1 |  | $85.524_{-0.0015}^{+0.0017}$
---
$85.315\pm 0.2458$
| $0.047_{-0.013}^{+0.011}$
---
$0.022\pm 0.057$
| $269.9_{-3.3}^{+2.8}$
---
$44.0\pm 156.3$
- | - |  | $46.4_{-0.7}^{+0.5}$
---
$50.3\pm 2.6$
TIC 66893949 | Orbital |  | $471.03_{-0.07}^{+0.10}$
---
$470.72\pm 3.37$
| $0.4016_{-0.0038}^{+0.0039}$
---
$0.404\pm 0.065$
| $25.8_{-1.5}^{+1.3}$
---
$237.011\pm 8.744$
| $386.4_{-3.4}^{+4.2}$
---
$72.09\pm 2.48$
| $90.222_{-0.009}^{+0.010}$
---
$91.39\pm 1.37$
-
TIC 88206187 | SB1 |  | $52.922_{-0.039}^{+0.041}$
---
$53.035\pm 0.058$
| $0.026_{-0.013}^{+0.017}$
---
$0.100\pm 0.044$
| $147_{-99}^{+100}$
---
$212\pm 9$
- | - |  | $58.47_{-0.37}^{+0.41}$
---
$65.23\pm 3.54$
TIC 298714297 | Orbital |  | $117.25_{-0.31}^{+0.36}$
---
$118.60\pm 0.28$
| $0.242_{-0.029}^{+0.029}$
---
$0.185\pm 0.065$
| $28_{-7}^{+8}$
---
$161\pm 20$
| $127.3_{-1.8}^{+1.3}$
---
$16.37\pm 0.50$
| $89.86_{-0.05}^{+0.09}$
---
$89.53\pm 2.21$
-

##  6 Summary and Conclusions

The most common method for discovering triply eclipsing triple star systems
involves searching for evidence of a third-body eclipse in the extended time
series of EB systems. These utilize the extensive time series datasets
obtained from the TESS or Kepler photometric sky survey telescopes. The
searches are usually done by eye or with the assistance of machine learning
tools (see, e.g., Rappaport et al., 2022, 2023). It goes without saying that
the longer the duration of a data set the greater the likelihood of
identifying eclipses involving third bodies within a given system.

In our previous paper (Czavalinga et al., 2023), we reported the discovery of
four triply eclipsing triple systems using an alternative method that utilizes
the Gaia DR3 data set. In that study, we crossmatched a catalog of
approximately 1 million binaries with Gaia stars that had longer-period NSS
orbital solutions. Of the $\sim$400 triple systems found in that study, four
of them turned out to also be triply eclipsing, as discovered through the use
of TESS data. Though the fractional yield was small (1%), this still
demonstrated the effectiveness of alternative methods in the search for triply
eclipsing triples.

In this work, we studied these four triply eclipsing systems in detail using
precision ETV curves and photometry from TESS data, archival photometry from
ASAS-SN and ATLAS, and archival SED curves. These input data sets undergo a
comprehensive analysis via a spectro-photodynamics code (LightcurveFactory) to
yield both the stellar and orbital parameters of the systems. In general, the
four new compact triples are not dissimilar to ones our group has been
studying over the past number of years.

TIC 66893949 consists of stars with masses similar to our Sun. This system has
the longest outer period among the triples investigated in our sample, and it
has the most eccentric outer orbit $e_{\rm out}\simeq 0.4$. Even though it has
a wide configuration, the mutual inclination angle is fairly low at
$i_{\mathrm{mut}}=0.6\pm 0.5$ degrees.

TIC 88206187 has the shortest outer period in our sample
$P_{\mathrm{out}}=52.92\pm 0.04$ days, and contains a tertiary red giant star
with radius of $11.7\pm 0.3\mathrm{R}_{\odot}$. This represents only the third
system discovered thus far in which the fractional radius (i.e., $R/a$) of the
distant tertiary component exceeds 0.1, suggesting the presence of ongoing
tidal effects. Our photodynamical model benefitted from the ATLAS and ASAS-SN
LCs which show both the outer primary and secondary eclipses, and therefore
provide a precise period and a value for $e_{\rm out}\cos\omega_{\rm out}$.
From the MIST stellar evolution tracks, we find that it is likely that the
outer tetriary will be the first star in the system to fill its Roche lobe
when it becomes an AGB star and transfer mass to the inner binary pair.

TIC 298714297 is one of the closest triply eclipsing triple systems ever found
with a distance of $138\pm 4$ pc. It contains low-mass stars that exhibit
pronounced spot activity in one of the components of the inner binary, with
several flares identified. The most probable host of these flares is star Aa
with the largest luminosity in the system.

We encountered some difficulties with modelling TIC 14839347. The low mass
secondary component of the inner pair was found to be strongly non-spherical,
filling or nearly filling its Roche lobe, which hints at a previous episode of
mass transfer. Consequently, the use of the PARSEC single-star evolutionary
tracks within the Lightcurvefactory analysis could not be used. However, we
were able to use a separate SED fitting code (Rappaport et al., 2022) that
does not assume any physical relation between the radius and $T_{\rm eff}$ of
either binary star, and which can estimate the stellar properties of all three
stars. That code, in turn, used estimates of a number of dimensionless ratios
such as of the radii and $T_{\rm eff}$ which we could get from
Lightcurvefactory. Since neither code could proceed without results from the
other, we ran them iteratively until a satisfactory solution was achieved.
From this analysis we find that the inner binary has a mass ratio of $0.25\pm
0.04$ and, as mentioned above, the secondary is nearly filling its Roche lobe.
Therefore the inner binary appears as an Algol system with an inverted mass
ratio. The outer stellar component is an evolved red giant star, while the
primary star of the inner binary appears to be less evolved than the tertiary
though these two stars are similar in mass. This feature may also be explained
by the fact that a mass exchange event has occurred between the inner binary
stars in the near past.

The use of Gaia DR3 NSS solutions have demonstrated the value of exploring
triple systems using alternative methodologies. The upcoming release of Gaia
DR4 will provide even more extensive data, making it extremely worthwhile to
search for signs of triples within these datasets and subsequently perform
photodynamical analyses on them. Such a greatly enhanced statistical sample of
compact triples should yield a more comprehensive understanding of these
important, intriguing, and fun systems.

## Acknowledgements

A. P. acknowledges the financial support of the Hungarian National Research,
Development and Innovation Office – NKFIH Grant K-138962.

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Science Mission directorate. Some of the data
presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST). STScI is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST
for non-HST data is provided by the NASA Office of Space Science via grant
NNX09AF08G and by other grants and contracts.

This work has made use of data from the European Space Agency (ESA) mission
Gaia111111https://www.cosmos.esa.int/gaia, processed by the Gaia Data
Processing and Analysis Consortium
(DPAC)121212https://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.

We have made use of the All-Sky Automated Survey for Supernovae archival
photometric data. See (Shappee et al., 2014) and (Kochanek et al., 2017) for
details of the ASAS-SN survey. We also acknowledge use of the photometric
archival data from the Asteroid Terrestrial-impact Last Alert System (ATLAS)
project. See (Tonry et al., 2018) and (Heinze et al., 2018) for specifics of
the ATLAS survey.

This publication makes use of data products from the Wide-field Infrared
Survey Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration.

This publication makes use of data products from the Two Micron All Sky
Survey, which is a joint project of the University of Massachusetts and the
Infrared Processing and Analysis Center/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation.

We used the Simbad service operated by the Centre des Données Stellaires
(Strasbourg, France) and the ESO Science Archive Facility services (data
obtained under request number 396301).

This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI : 10.26093/cds/vizier). The original description of
the VizieR service was published in (Ochsenbein et al., 2000).

## Data Availability

The TESS data underlying this article were accessed from MAST (Barbara A.
Mikulski Archive for Space Telescopes) Portal
(https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html), including the
data products found in the bulk download website
(http://archive.stsci.edu/tess/bulk_downloads/bulk_downloads_ffi-tp-lc-
dv.html). Part of the data were derived from sources in public domain as given
in the respective footnotes. The derived data generated in this research and
the code used for the photodynamical analysis will be shared on reasonable
request to the corresponding author.

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##  Appendix A Mid-eclipse times of the four eclipsing binaries

Table 5: Times of minima of TIC 014839347. BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev.
---|---|---|---|---|---|---|---|---|---|---|---
$-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$
58683.46208 | -0.5 | 0.00036 | 58712.27472 | 24.5 | 0.00042 | 59422.20711 | 640.5 | 0.00022 | 59799.07319 | 967.5 | 0.00018
58684.04020 | 0.0 | 0.00009 | 58712.85171 | 25.0 | 0.00013 | 59422.78730 | 641.0 | 0.00008 | 59799.64946 | 968.0 | 0.00011
58684.61498 | 0.5 | 0.00123 | 58713.42656 | 25.5 | 0.00023 | 59423.36554 | 641.5 | 0.00029 | 59800.22534 | 968.5 | 0.00025
58685.19280 | 1.0 | 0.00020 | 58714.00445 | 26.0 | 0.00015 | 59423.94014 | 642.0 | 0.00010 | 59800.80247 | 969.0 | 0.00010
58685.76391 | 1.5 | 0.00036 | 58714.58008 | 26.5 | 0.00055 | 59424.51522 | 642.5 | 0.00027 | 59801.37868 | 969.5 | 0.00025
58686.34518 | 2.0 | 0.00019 | 58715.15689 | 27.0 | 0.00015 | 59425.09262 | 643.0 | 0.00013 | 59801.95434 | 970.0 | 0.00008
58686.92020 | 2.5 | 0.00063 | 58715.73324 | 27.5 | 0.00083 | 59425.66792 | 643.5 | 0.00026 | 59802.52871 | 970.5 | 0.00025
58687.49786 | 3.0 | 0.00028 | 58716.30855 | 28.0 | 0.00022 | 59426.24497 | 644.0 | 0.00015 | 59803.10674 | 971.0 | 0.00012
58688.07271 | 3.5 | 0.00060 | 58716.88273 | 28.5 | 0.00017 | 59426.82292 | 644.5 | 0.00034 | 59803.68408 | 971.5 | 0.00028
58688.64952 | 4.0 | 0.00012 | 58717.46216 | 29.0 | 0.00017 | 59427.39750 | 645.0 | 0.00013 | 59804.25859 | 972.0 | 0.00010
58689.22368 | 4.5 | 0.00056 | 58718.03791 | 29.5 | 0.00072 | 59427.97102 | 645.5 | 0.00047 | 59804.83306 | 972.5 | 0.00026
58689.80177 | 5.0 | 0.00023 | 58718.61384 | 30.0 | 0.00021 | 59428.55059 | 646.0 | 0.00011 | 59805.41148 | 973.0 | 0.00008
58690.37606 | 5.5 | 0.00056 | 58719.19201 | 30.5 | 0.00026 | 59429.12418 | 646.5 | 0.00041 | 59805.98763 | 973.5 | 0.00032
58690.95388 | 6.0 | 0.00014 | 58719.76702 | 31.0 | 0.00034 | 59431.43166 | 648.5 | 0.00018 | 59806.56397 | 974.0 | 0.00008
58691.52642 | 6.5 | 0.00044 | 58720.34276 | 31.5 | 0.00021 | 59432.00803 | 649.0 | 0.00008 | 59807.13972 | 974.5 | 0.00034
58692.10763 | 7.0 | 0.00021 | 58721.49532 | 32.5 | 0.00057 | 59432.58124 | 649.5 | 0.00161 | 59807.71629 | 975.0 | 0.00013
58692.67565 | 7.5 | 0.00170 | 58722.07231 | 33.0 | 0.00011 | 59433.73796 | 650.5 | 0.00120 | 59808.29119 | 975.5 | 0.00023
58693.83076 | 8.5 | 0.00055 | 58722.64775 | 33.5 | 0.00078 | 59434.31238 | 651.0 | 0.00008 | 59808.86828 | 976.0 | 0.00010
58694.41220 | 9.0 | 0.00016 | 58723.22456 | 34.0 | 0.00026 | 59434.88859 | 651.5 | 0.00034 | 59809.44182 | 976.5 | 0.00039
58694.98602 | 9.5 | 0.00039 | 58723.80602 | 34.5 | 0.00164 | 59435.46581 | 652.0 | 0.00009 | 59811.17408 | 978.0 | 0.00009
58695.56427 | 10.0 | 0.00012 | 58725.52948 | 36.0 | 0.00020 | 59436.04049 | 652.5 | 0.00027 | 59811.74866 | 978.5 | 0.00028
58696.13822 | 10.5 | 0.00063 | 58726.10307 | 36.5 | 0.00038 | 59436.61758 | 653.0 | 0.00009 | 59812.32618 | 979.0 | 0.00009
58697.86901 | 12.0 | 0.00018 | 58726.68231 | 37.0 | 0.00010 | 59437.19277 | 653.5 | 0.00041 | 59812.90057 | 979.5 | 0.00029
58698.44024 | 12.5 | 0.00036 | 58727.25884 | 37.5 | 0.00021 | 59437.76986 | 654.0 | 0.00010 | 59813.47942 | 980.0 | 0.00010
58699.02193 | 13.0 | 0.00017 | 58727.83522 | 38.0 | 0.00020 | 59438.34494 | 654.5 | 0.00028 | 59814.05354 | 980.5 | 0.00015
58699.59538 | 13.5 | 0.00079 | 58728.41084 | 38.5 | 0.00022 | 59438.92271 | 655.0 | 0.00013 | 59814.63170 | 981.0 | 0.00012
58700.17426 | 14.0 | 0.00015 | 58728.98760 | 39.0 | 0.00012 | 59439.49674 | 655.5 | 0.00031 | 59815.20384 | 981.5 | 0.00035
58700.75026 | 14.5 | 0.00052 | 58729.56338 | 39.5 | 0.00037 | 59440.07554 | 656.0 | 0.00021 | 59815.78414 | 982.0 | 0.00010
58701.32582 | 15.0 | 0.00015 | 58730.14071 | 40.0 | 0.00022 | 59440.65165 | 656.5 | 0.00023 | 59816.35846 | 982.5 | 0.00018
58701.90006 | 15.5 | 0.00019 | 58730.71633 | 40.5 | 0.00038 | 59441.22686 | 657.0 | 0.00011 | 59816.93536 | 983.0 | 0.00011
58702.47884 | 16.0 | 0.00018 | 58731.29283 | 41.0 | 0.00014 | 59441.80305 | 657.5 | 0.00041 | 59817.51193 | 983.5 | 0.00026
58703.05672 | 16.5 | 0.00099 | 58731.86939 | 41.5 | 0.00035 | 59442.37988 | 658.0 | 0.00011 | 59818.08789 | 984.0 | 0.00010
58704.20398 | 17.5 | 0.00257 | 58732.44572 | 42.0 | 0.00014 | 59442.95537 | 658.5 | 0.00028 | 59818.66672 | 984.5 | 0.00022
58704.78336 | 18.0 | 0.00015 | 58733.02129 | 42.5 | 0.00023 | 59443.53204 | 659.0 | 0.00013 | 59819.24122 | 985.0 | 0.00011
58705.35753 | 18.5 | 0.00045 | 58733.59824 | 43.0 | 0.00026 | 59444.10607 | 659.5 | 0.00036 | 59819.81796 | 985.5 | 0.00026
58705.93572 | 19.0 | 0.00020 | 58734.17401 | 43.5 | 0.00028 | 59444.68495 | 660.0 | 0.00009 | 59820.39417 | 986.0 | 0.00011
58706.50396 | 19.5 | 0.00117 | 58735.32918 | 44.5 | 0.00115 | 59445.25979 | 660.5 | 0.00027 | 59820.96853 | 986.5 | 0.00023
58707.67101 | 20.5 | 0.00074 | 58735.90361 | 45.0 | 0.00020 | 59445.83675 | 661.0 | 0.00011 | 59821.54589 | 987.0 | 0.00009
58708.24049 | 21.0 | 0.00018 | 58736.47388 | 45.5 | 0.00151 | 59446.41231 | 661.5 | 0.00025 | 59822.12237 | 987.5 | 0.00036
58708.81818 | 21.5 | 0.00024 | 59420.48162 | 639.0 | 0.00012 | 59797.34480 | 966.0 | 0.00009 | 59822.69894 | 988.0 | 0.00010
58709.39326 | 22.0 | 0.00023 | 59421.05827 | 639.5 | 0.00022 | 59797.91970 | 966.5 | 0.00024 | 59823.27240 | 988.5 | 0.00021
58709.96928 | 22.5 | 0.00051 | 59421.63472 | 640.0 | 0.00013 | 59798.49733 | 967.0 | 0.00012 | 59823.85117 | 989.0 | 0.00010
58711.69848 | 24.0 | 0.00020 |  |  |  |  |  |  |  |  |
Table 6: Times of minima of TIC 066893949. BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev.
---|---|---|---|---|---|---|---|---|---|---|---
$-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$
58711.83201 | -0.5 | 0.00072 | 58733.46997 | 4.0 | 0.00018 | 59435.09934 | 150.0 | 0.00007 | 59802.71738 | 226.5 | 0.00018
58714.24737 | 0.0 | 0.00019 | 58735.85916 | 4.5 | 0.00032 | 59437.48807 | 150.5 | 0.00018 | 59805.13517 | 227.0 | 0.00009
58716.63577 | 0.5 | 0.00183 | 59420.68260 | 147.0 | 0.00009 | 59439.90506 | 151.0 | 0.00008 | 59812.33065 | 228.5 | 0.00020
58719.05333 | 1.0 | 0.00014 | 59423.06986 | 147.5 | 0.00019 | 59442.29382 | 151.5 | 0.00021 | 59814.74714 | 229.0 | 0.00007
58721.44275 | 1.5 | 0.00038 | 59425.48859 | 148.0 | 0.00008 | 59444.71088 | 152.0 | 0.00009 | 59817.13492 | 229.5 | 0.00022
58726.24731 | 2.5 | 0.00061 | 59427.87673 | 148.5 | 0.00015 | 59797.91148 | 225.5 | 0.00025 | 59819.55200 | 230.0 | 0.00010
58728.66411 | 3.0 | 0.00016 | 59430.29490 | 149.0 | 0.00009 | 59800.32911 | 226.0 | 0.00007 | 59821.93961 | 230.5 | 0.00027
58731.05216 | 3.5 | 0.00087 |  |  |  |  |  |  |  |  |
Table 7: Times of minima of TIC 088206187. BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev.
---|---|---|---|---|---|---|---|---|---|---|---
$-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$
58816.34459 | -0.5 | 0.00033 | 58830.55929 | 11.5 | 0.00067 | 59914.20263 | 926.5 | 0.00017 | 59925.45387 | 936.0 | 0.00013
58816.93757 | 0.0 | 0.00079 | 58831.15198 | 12.0 | 0.00024 | 59914.79520 | 927.0 | 0.00011 | 59926.04698 | 936.5 | 0.00017
58817.53133 | 0.5 | 0.00045 | 58831.74569 | 12.5 | 0.00050 | 59915.38830 | 927.5 | 0.00015 | 59926.63933 | 937.0 | 0.00013
58818.12388 | 1.0 | 0.00049 | 58832.33502 | 13.0 | 0.00017 | 59915.97845 | 928.0 | 0.00012 | 59927.23320 | 937.5 | 0.00019
58818.71385 | 1.5 | 0.00093 | 58832.92766 | 13.5 | 0.00169 | 59916.57213 | 928.5 | 0.00020 | 59927.82292 | 938.0 | 0.00015
58819.30876 | 2.0 | 0.00018 | 58833.51975 | 14.0 | 0.00029 | 59917.16356 | 929.0 | 0.00012 | 59928.41569 | 938.5 | 0.00020
58819.90018 | 2.5 | 0.00084 | 58834.11457 | 14.5 | 0.00110 | 59917.75583 | 929.5 | 0.00020 | 59929.00834 | 939.0 | 0.00010
58820.49284 | 3.0 | 0.00025 | 58834.70346 | 15.0 | 0.00013 | 59918.34719 | 930.0 | 0.00010 | 59929.60009 | 939.5 | 0.00027
58821.08425 | 3.5 | 0.00207 | 58835.29747 | 15.5 | 0.00130 | 59918.93905 | 930.5 | 0.00018 | 59930.19175 | 940.0 | 0.00013
58821.67783 | 4.0 | 0.00035 | 58835.88897 | 16.0 | 0.00017 | 59919.53184 | 931.0 | 0.00010 | 59930.78609 | 940.5 | 0.00013
58822.26863 | 4.5 | 0.00045 | 58836.48139 | 16.5 | 0.00042 | 59920.12605 | 931.5 | 0.00022 | 59931.37704 | 941.0 | 0.00012
58822.86037 | 5.0 | 0.00021 | 58837.07319 | 17.0 | 0.00038 | 59920.71698 | 932.0 | 0.00012 | 59931.97030 | 941.5 | 0.00014
58823.45424 | 5.5 | 0.00038 | 58837.66618 | 17.5 | 0.00029 | 59921.30883 | 932.5 | 0.00022 | 59932.56131 | 942.0 | 0.00008
58824.04631 | 6.0 | 0.00062 | 58838.25782 | 18.0 | 0.00086 | 59921.90097 | 933.0 | 0.00009 | 59933.15368 | 942.5 | 0.00017
58824.63694 | 6.5 | 0.00056 | 58838.84976 | 18.5 | 0.00033 | 59922.49323 | 933.5 | 0.00016 | 59933.74530 | 943.0 | 0.00012
58825.22950 | 7.0 | 0.00024 | 58839.44051 | 19.0 | 0.00072 | 59923.08616 | 934.0 | 0.00015 | 59934.33831 | 943.5 | 0.00043
58825.82227 | 7.5 | 0.00021 | 58840.03120 | 19.5 | 0.00044 | 59924.27054 | 935.0 | 0.00015 | 59934.92946 | 944.0 | 0.00014
58826.41376 | 8.0 | 0.00018 | 58840.62481 | 20.0 | 0.00047 | 59924.86028 | 935.5 | 0.00025 | 59935.52280 | 944.5 | 0.00025
58827.00838 | 8.5 | 0.00043 | 59913.61059 | 926.0 | 0.00013 |  |  |  |  |  |
Table 8: Times of minima of TIC 298714297. BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev. | BJD | Cycle | std. dev.
---|---|---|---|---|---|---|---|---|---|---|---
$-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$ | $-2\,400\,000$ | no. | $(d)$
58711.49994 | 0.0 | 0.00037 | 58737.24706 | 24.0 | 0.00024 | 59816.56348 | 1030.0 | 0.00007 | 59835.33955 | 1047.5 | 0.00009
58712.03597 | 0.5 | 0.00063 | 59797.25118 | 1012.0 | 0.00005 | 59817.09956 | 1030.5 | 0.00028 | 59835.87540 | 1048.0 | 0.00005
58712.57261 | 1.0 | 0.00040 | 59797.78710 | 1012.5 | 0.00022 | 59817.63617 | 1031.0 | 0.00018 | 59836.41296 | 1048.5 | 0.00011
58713.64553 | 2.0 | 0.00032 | 59798.32436 | 1013.0 | 0.00010 | 59818.17406 | 1031.5 | 0.00031 | 59836.94845 | 1049.0 | 0.00005
58714.18127 | 2.5 | 0.00043 | 59798.86084 | 1013.5 | 0.00022 | 59818.70930 | 1032.0 | 0.00007 | 59837.48540 | 1049.5 | 0.00014
58714.71826 | 3.0 | 0.00054 | 59799.93378 | 1014.5 | 0.00038 | 59819.24606 | 1032.5 | 0.00021 | 59838.02128 | 1050.0 | 0.00005
58715.79108 | 4.0 | 0.00017 | 59800.46966 | 1015.0 | 0.00015 | 59819.78208 | 1033.0 | 0.00019 | 59839.09430 | 1051.0 | 0.00004
58716.32573 | 4.5 | 0.00092 | 59801.00665 | 1015.5 | 0.00028 | 59820.31865 | 1033.5 | 0.00033 | 59839.63180 | 1051.5 | 0.00010
58716.86396 | 5.0 | 0.00038 | 59801.54269 | 1016.0 | 0.00007 | 59820.85502 | 1034.0 | 0.00008 | 59840.16712 | 1052.0 | 0.00004
58717.93698 | 6.0 | 0.00020 | 59802.07904 | 1016.5 | 0.00047 | 59821.39111 | 1034.5 | 0.00019 | 59840.70440 | 1052.5 | 0.00011
58718.47299 | 6.5 | 0.00059 | 59802.61562 | 1017.0 | 0.00015 | 59821.92778 | 1035.0 | 0.00019 | 59841.24016 | 1053.0 | 0.00005
58719.00964 | 7.0 | 0.00038 | 59803.15228 | 1017.5 | 0.00020 | 59822.46474 | 1035.5 | 0.00025 | 59841.77669 | 1053.5 | 0.00010
58720.08276 | 8.0 | 0.00044 | 59803.68825 | 1018.0 | 0.00009 | 59823.00079 | 1036.0 | 0.00008 | 59842.31323 | 1054.0 | 0.00004
58720.61764 | 8.5 | 0.00054 | 59804.76143 | 1019.0 | 0.00010 | 59823.53781 | 1036.5 | 0.00035 | 59842.84990 | 1054.5 | 0.00009
58721.15530 | 9.0 | 0.00040 | 59805.29855 | 1019.5 | 0.00022 | 59824.07355 | 1037.0 | 0.00036 | 59843.38607 | 1055.0 | 0.00005
58722.22824 | 10.0 | 0.00009 | 59805.83441 | 1020.0 | 0.00013 | 59825.68383 | 1038.5 | 0.00014 | 59843.92310 | 1055.5 | 0.00009
58722.76369 | 10.5 | 0.00080 | 59806.37126 | 1020.5 | 0.00031 | 59826.21946 | 1039.0 | 0.00005 | 59844.45912 | 1056.0 | 0.00006
58723.30097 | 11.0 | 0.00042 | 59806.90729 | 1021.0 | 0.00014 | 59826.75659 | 1039.5 | 0.00011 | 59844.99635 | 1056.5 | 0.00011
58725.44656 | 13.0 | 0.00025 | 59807.44376 | 1021.5 | 0.00027 | 59827.29228 | 1040.0 | 0.00004 | 59845.53190 | 1057.0 | 0.00004
58726.51958 | 14.0 | 0.00009 | 59807.98022 | 1022.0 | 0.00005 | 59827.82956 | 1040.5 | 0.00014 | 59846.60491 | 1058.0 | 0.00005
58727.05501 | 14.5 | 0.00107 | 59808.51656 | 1022.5 | 0.00020 | 59828.36514 | 1041.0 | 0.00005 | 59847.14169 | 1058.5 | 0.00016
58727.59211 | 15.0 | 0.00019 | 59809.05307 | 1023.0 | 0.00018 | 59828.90253 | 1041.5 | 0.00012 | 59847.67775 | 1059.0 | 0.00005
58728.66521 | 16.0 | 0.00011 | 59809.58973 | 1023.5 | 0.00030 | 59829.43803 | 1042.0 | 0.00005 | 59848.21495 | 1059.5 | 0.00010
58729.20005 | 16.5 | 0.00225 | 59811.19906 | 1025.0 | 0.00014 | 59829.97538 | 1042.5 | 0.00012 | 59848.75079 | 1060.0 | 0.00004
58729.73778 | 17.0 | 0.00018 | 59811.73565 | 1025.5 | 0.00027 | 59830.51100 | 1043.0 | 0.00005 | 59849.28744 | 1060.5 | 0.00014
58730.81024 | 18.0 | 0.00046 | 59812.27176 | 1026.0 | 0.00008 | 59831.04834 | 1043.5 | 0.00010 | 59849.82361 | 1061.0 | 0.00005
58731.34656 | 18.5 | 0.00033 | 59812.80619 | 1026.5 | 0.00019 | 59831.58384 | 1044.0 | 0.00006 | 59850.36021 | 1061.5 | 0.00009
58731.88324 | 19.0 | 0.00004 | 59813.34481 | 1027.0 | 0.00016 | 59832.12125 | 1044.5 | 0.00013 | 59850.89676 | 1062.0 | 0.00005
58732.95598 | 20.0 | 0.00039 | 59813.88129 | 1027.5 | 0.00019 | 59832.65671 | 1045.0 | 0.00005 | 59851.43322 | 1062.5 | 0.00016
58734.02900 | 21.0 | 0.00009 | 59814.41770 | 1028.0 | 0.00007 | 59833.19406 | 1045.5 | 0.00010 | 59851.96958 | 1063.0 | 0.00004
58735.10149 | 22.0 | 0.00046 | 59814.95407 | 1028.5 | 0.00022 | 59833.72953 | 1046.0 | 0.00004 | 59852.50629 | 1063.5 | 0.00010
58735.63702 | 22.5 | 0.00033 | 59815.49060 | 1029.0 | 0.00020 | 59834.26648 | 1046.5 | 0.00013 | 59853.04255 | 1064.0 | 0.00007
58736.17434 | 23.0 | 0.00002 | 59816.02711 | 1029.5 | 0.00026 | 59834.80254 | 1047.0 | 0.00005 |  |  |