TIC 433545934: The first 2+2 type doubly eclipsing binary with extra, mutual eclipses

TIC 433545934 is identified as the first known 2+2-type doubly eclipsing quadruple stellar system that also exhibits mutual eclipses between the two constituent eclipsing binaries (EBs). The system consists of two EBs, designated A and B, which revolve around each other with an outer period of PAB = 224.84 days (Table 3), making it the fifth most compact known 2+2-type quadruple system. The inner periods are PA = 2.072891 days and PB = 1.412719 days (Table 3). The system was observed with the TESS space telescope during six sectors (Sectors 38, 64, 65, and 99–101), and a triple-dipped extra eclipsing event was discovered in the Sector 38 light curve, where the stars of binary B eclipsed the two components of binary A. Additional extra eclipses were identified in archival ground-based data from ASAS-SN and ATLAS, but these events occur only once per outer revolution, meaning binary A does not eclipse binary B. This is consistent with the high outer eccentricity of eAB = 0.6228 (Table 3).

The analysis combined TESS light curves, ground-based follow-up photometry (from the FRAM telescope at the Pierre Auger Observatory and the 1.54 m Danish telescope at La Silla), eclipse timing variation (ETV) curves, and spectral energy distribution (SED) fitting within a complex photodynamical model using the Lightcurvefactory software package. The photodynamical analysis simultaneously fitted the light curves, ETVs, and SED, using PARSEC isochrones as proxies for stellar parameters in the absence of radial velocity measurements. The system was found to be substantially flat, with mutual inclination angles below approximately 2 degrees (Table 3: (imut)A = 1.17 degrees, (imut)B = 0.98 degrees).

The stellar parameters derived are as follows (Table 3): Binary A consists of two quite similar, slightly evolved late A-type stars with masses MAa = 2.333 M⊙ and MAb = 2.150 M⊙ (mass ratio qA = 0.918), radii RAa = 2.932 R⊙ and RAb = 2.333 R⊙, and effective temperatures TAa = 8826 K and TAb = 8802 K. Binary B has a dominant primary star with mass MBa = 2.261 M⊙ (which lies between the masses of the two stars of binary A) and a less massive secondary with MBb = 1.348 M⊙ (mass ratio qB = 0.597). The primary of binary B is the hottest star in the system at TBa = 8864 K, while the secondary Bb is the coolest at TBb = 6693 K. The radii are RBa = 2.657 R⊙ and RBb = 1.343 R⊙. The system age is approximately 580 Myr (log(age) = 8.764 dex), with metallicity [M/H] = 0.079 dex, and a photometric distance of 1464 pc.

The outer orbit is highly eccentric (eout = 0.6228) with argument of periastron ωout = 273.5 degrees, and inclination iout = 88.92 degrees. The orientation of the outer periastron means the outer orbit is seen almost from the direction of pericenter, explaining why only near-pericenter outer eclipses are detected. The ETV analysis revealed that the light travel time effect (LTTE) accounts for only about half of the ETV amplitude, with the remainder due to dynamical delays from the mutual gravitational perturbations of the two binaries. The ratio of the dynamical to LTTE amplitudes was found to be Adyn/ALTTE = 1.11 from the preliminary analytic fit. The outer period was confirmed to be 224.52 days from BLS analysis of archival ATLAS and ASAS-SN data, ruling out the alternative half-period of 112 days.

The system is dynamically stable given the outer period is more than two orders of magnitude longer than the inner periods. The three orbital planes are mutually aligned to within 2 degrees, suggesting formation from a single flat disk with sequential disk collapse. The paper notes that the two more massive stars in binaries A and B will both overflow their Roche lobes in approximately 152 Myr, potentially at about the same time. The authors emphasize that the extra outer eclipses provide vital information about system parameters that could not be obtained even with radial velocity measurements, and that without any RVs, all system parameters were extracted with accuracies of 2-5% for stellar parameters, better than a degree for orbital angles, and better than 0.01 for the outer eccentricity.

Improvements to AI Systems:

1. Hierarchical Multi-Body Photodynamical Modeling

• Enhance AI models to simultaneously fit light curves, eclipse timing variations (ETVs), and spectral energy distributions (SEDs) for multi-tiered stellar systems (e.g., 2+2 quadruples).

• The improved AI can automatically disentangle mutual eclipses, dynamical delays (beyond light-travel-time), and eccentric orbital geometries without radial velocity data, recovering stellar masses, radii, and temperatures to 2–5% accuracy.

2. Rare Event Detection in Sparse Time-Series Data

• Train AI to identify single-occurrence, multi-dip eclipse signatures (like the triple-dipped event in Sector 38) across heterogeneous datasets (TESS, ASAS-SN, ATLAS).

• The improved system can flag rare outer-orbit eclipses in archival data, reject aliases (e.g., half-period alternatives), and predict optimal observation windows for future transits.

3. Orbital Architecture Inference from ETV Decomposition

• Implement AI that separates dynamical (gravitational) and light-travel-time (LTTE) contributions to ETV curves, using amplitude ratios (e.g., Adyn/ALTTE = 1.11) to constrain outer eccentricity and mutual inclination.

• The improved system can infer unseen orbital parameters (e.g., eout, ωout, iout) from ETV shapes alone, even when only partial phase coverage exists.

4. Formation Scenario Classification

• Develop AI classifiers that use mutual inclination (<2°), mass ratios, and orbital period ratios to distinguish between disk-fragmentation and dynamical capture origins for compact quadruples.

• The improved system can automatically categorize newly discovered systems as flat sequential collapse candidates, aiding population studies.

5. Stellar Evolution–Orbit Coupling Prediction

• Integrate AI with stellar isochrones (PARSEC) and Roche-lobe overflow models to forecast future mass-transfer or merger events (e.g., both primaries overflowing in 152 Myr).

• The improved system can simulate the coupled evolution of hierarchical multiples, predicting when and how inner binaries will interact, and whether outer stability is preserved.

6. Parameter Degeneracy Breaking via Multi-Observable Fusion

• Use AI to fuse sparse, non-simultaneous data (photometry, ETVs, SEDs) with physical priors (e.g., isochrone constraints) to break degeneracies that single-observable fits cannot resolve.

• The improved system can deliver precise orbital angles (better than 1°) and eccentricities (better than 0.01) even without radial velocities, as demonstrated for TIC 433545934.

7. Automated Detection of Mutual Eclipses in Future Surveys

• Train AI on synthetic light curves of 2+2 quadruples to recognize mutual-eclipse patterns in upcoming missions (e.g., PLATO, Roman).

• The improved system can pre-screen light curves for rare outer eclipses, prioritizing targets for follow-up and enabling statistical studies of compact quadruple frequency.

What the Improved AI System Can Do:

• Fully characterize hierarchical stellar systems (masses, radii, temperatures, orbits) from photometry alone, with accuracy comparable to RV-based methods.

• Automatically discover and model rare mutual-eclipse events in large survey archives, reducing manual search time by orders of magnitude.

• Predict dynamical evolution and merger timescales for compact multiples, aiding gravitational-wave and stellar-population forecasts.

• Provide robust formation-history classifications for new quadruple systems, enabling tests of star-formation theories.

In this work we identify and photodynamically analyze TIC 433545934, the very first doubly eclipsing 2+2-type quadruple stellar system, which shows outer eclipses, too. One such outer eclipse was discovered with TESS, which triggered a special interest in this system. Most of the data for this study come from TESS observations, but we also obtained supplemental ground-based photometric measurements for this quadruple system. The eclipse timing variation curves extracted from the TESS and ground-based follow-up data, the photometric light curves, and the spectral energy distribution are combined in a complex photodynamical analysis to yield the stellar and orbital parameters of this dynamically interesting system. The periods of the two inner, eclipsing binaries were found to be P A=2.07 d and P B=1.41 d, while the outer period is P AB=224.5 d. The outer period alone makes this system the fifth most compact known 2+2-type quadruple. Moreover, what really makes TIC 433545934 unique is that TESS observed a triple-dipped extra eclipsing event when the stars of binary B eclipsed the two components of binary A. We identified similar extra eclipsing events in the archival, ground-based data of ASAS-SN and ATLAS, as well. We found, however, that these extra events occur only once during an outer revolution, that is, binary A does not eclipse the stars of binary B. This is in accord with our finding that the outer eccentricity is quite high, being e AB=0.62. Our analysis reveals that binary A consists of two quite similar, but slightly evolved late A-type stars (q A=0.92), while in the case of binary B, the primary star, whose mass is between the masses of the two stars of binary A, is quite dominant, with q B=0.60. The system was found to be substantially flat, with mutual inclination angles below about2.