Characterizing non-transiting exoplanets through transit timing variations using deep learning

Nowadays, transit timing variations (TTVs) are proving to be a very valuable tool in exoplanetary science to detect exoplanets by observing variations in transit times. To study the TTV of the hot Jupiter TrES-2b, we have combined 64 high-quality transit light curves from all seven sectors of NASA's Transiting Exoplanet Survey Satellite along with 60 best-quality light curves from the ground-based facility Exoplanet Transit Database and 106 midtransit times from the previous works. From the precise transit timing analysis, we have observed a significant improvement in the orbital ephemerides, but we did not detect any short-period TTVs that might result from an additional body. The inability to detect short-term TTVs further motivates us to investigate long-term TTVs, which might be caused by orbital decay, apsidal precession, the Applegate mechanism, and the Rømer effect, and the orbital decay appeared to be a better explanation for the observed TTV with ΔBIC = 4.32. The orbital period of the hot Jupiter TrES-2b appears to be shrinking at a rate of ∼–5.58 ± 1.81 ms yr−1. Assuming this decay is primarily caused by tidal dissipation within the host star, we have subsequently calculated the stellar tidal quality factor value to be ∼9.9 × 103, which is 2–3 orders of magnitude smaller than the theoretically predicted values for other hot-Jupiter systems, and its low value indicates more efficient tidal dissipation within the host star. Additional precise photometric and radial velocity observations are required to pinpoint the cause of the change in the orbital period.

We visually analyzed the transit timing variation (TTV) data of 5930 Kepler Objects of Interest (KOIs) homogeneously. Using data from Rowe et al. and Holczer et al., we investigated TTVs for nearly all KOIs in Kepler’s Data Release 24 catalog. Using TTV plots, periodograms, and phase-folded quadratic plus sinusoid fits, we visually rated each KOI’s TTV data in five categories. Our ratings emphasize the hundreds of planets with TTVs that are weaker than the ∼200 that have been studied in detail. Our findings are consistent with statistical methods for identifying strong TTVs, though we found some additional systems worth investigation. Between about 3–50 days and 1.3–6 Earth radii, the frequency of strong TTVs increases with period and radius. As expected, strong TTVs are very common when period ratios are near a resonance, but there is not a one-to-one correspondence. The observed planet-by-planet frequency of strong TTVs is only somewhat lower in systems with one or two known planets (7% ± 1%) than in systems with three or more known planets (11% ± 2%). We attribute TTVs to known planets in multitransiting systems but find ∼30 cases where the perturbing planet is unknown. Our conclusions are valuable as an ensemble for learning about planetary system architectures and individually as stepping stones toward more-detailed mass–radius constraints. We also discuss Data Release 25 TTVs, investigate ∼100 KOIs with transit duration and/or depth variations, and estimate that the Transiting Exoplanet Survey Satellite will likely find only ∼10 planets with strong TTVs.

Transit Timing Variations (TTVs) are deviations from the time an observer would expect to see an exoplanet transit its host star. In multiplanetary systems, significant TTVs may indicate the presence of another body in the system gravitationally interacting with the transiting exoplanet. T. Holczer et al. catalog 2599 Kepler Objects of Interest (KOIs) and provide a statistical analysis of their TTVs for candidates that have transited at least seven times. However, this conservative limit on the number of transits neglects long-period KOIs. Therefore, we extend the statistical analysis performed by T. Holczer et al. to the population of KOIs that have between three and six transits. We identify six KOIs, three of which have Kepler names (Kepler–103 c, Kepler–90 g, and Kepler–1662 c), with significant TTV signals that were originally overlooked by T. Holczer et al. Additionally, we search for trends regarding the planetary radius and orbital period of KOIs with significant TTVs. Through a survival analysis, we determine that planets with shorter orbital periods require larger TTV signals to be detected compared to longer period planets, regardless of planetary size. Uncovering trends in TTV statistics such as this one will provide targets for future forward modeling of planetary architectures.

We conduct a uniform analysis of the transit timing variations (TTVs) of 145 planets from 55 Kepler multiplanet systems to infer planet masses and eccentricities. Eighty of these planets do not have previously reported mass and eccentricity measurements. We employ two complementary methods to fit TTVs: Markov chain Monte Carlo simulations based on N-body integration, and an analytic fitting approach. Mass measurements of 49 planets, including 12 without previously reported masses, meet our criterion for classification as robust. Using mass and radius measurements, we infer the masses of planets’ gaseous envelopes for both our TTV sample and transiting planets with radial velocity observations. Insight from analytic TTV formulae allows us to partially circumvent degeneracies inherent to inferring eccentricities from TTV observations. We find that planet eccentricities are generally small, typically a few percent, but in many instances are nonzero.

It has been more than 25 years since the first light curve of a planetary transit has been published (Charbonneau et al. 2000). Since then, thousands of light curves have been published for many of the discovered planets. Although photometric observation of a planetary transit is conceptually straightforward, variations in data reduction, modeling approaches, and analysis tools have introduced significant heteroscedasticity into the resulting light curves — and consequently into the derived parameters. Among these, the timing of mid-transit is particularly sensitive, as its measurement and associated uncertainty are highly model-dependent. Yet mid-transit times form the backbone of Transit Timing Variation (TTV) analyses, through which one can probe the presence of additional planetary companions, tidal interactions with the host star, apsidal precession, and magnetic activity.Properly addressing the sources of heteroscedasticity in TTV datasets is essential for disentangling genuine physical signals from observational systematics. In practice, this manifests in TTV diagrams as short-term scatter exceeding the quoted timing uncertainties — particularly between datasets acquired with different instruments, reduction pipelines, or modeling techniques.We have been monitoring a sample of hot Jupiter systems (Basturk et al. 2020, 2022, 2023; Yalcinkaya et al. 2024), some of which are candidates for orbital decay due to strong tidal interactions. To date, definitive evidence for orbital decay has been reported in only two systems: WASP-12b (Yee et al. 2020) and WASP-4b (Basturk et al. 2025, and references therein). To systematically investigate the heteroscedasticity issue in TTV data, we focused on two well-observed systems, HAT-P-7 and TrES-2. Both were continuously monitored by the Kepler spacecraft during its primary mission, supplemented by numerous ground-based observations and, more recently, multiple sectors of TESS photometry.We compiled all available transit light curves for these systems, modeled them uniformly using the EXOFAST suite (Eastman et al. 2013), and derived mid-transit times to construct homogeneous TTV datasets for each planet. Our analysis revealed significant variance discrepancies between different data subsets — often exceeding the reported timing uncertainties. In this study, we present our methodology for addressing this issue and discuss the implications for interpreting the observed TTV signals in the context of tidal interaction theory.

Objective To investigate the clinical value of multi-slice CT angiography (MSCTA) of lower limbs in patients with peripheral arterial occlusive disease (PAOD) using the test-bolus technique.Methods Forty-four patients with PAOD were enrolled consecutively in the study.In group I, 18 subjects underwent CTA by bolus triggering method and in group 2, 26 subjects underwent CTA by test-bolus technique.During scanning procedure in group 2 subjects, the bolus transit time to aorta ( TAO ), popliteal arteries (Tpop ) and aorto-popliteal bolus transit time (T,) were calculated through dynamic acquisition at their respective level and the delay time were immediately set as TAO and scan time as double Tt.Two independent senior attending physicians with training experience in interpreting CTA determined the quality of each arterial segment visualization based on 5 parameters (1.visible farthest branch, 2.clarity of vessels border, 3.presence of venous contamination, 4.grading of stenosis, 5.CT value at 4 arterial segments).Inter-observer agreement on imaging quality between readers was evaluated using Cohen's k statistic by calculating K values.X2 test and t test were used to compare the quality of images in both groups.Results In group 2 patients,a larger individual variation in transit time of the contrast to reach aorta was obserued [ TAO = ( 17.1 ± 2.6) s with a range of 12.0—22.0 s ] and aorto-popliteal transit time [ Tt = ( 14.8 ± 5.5 ) s with a range of 8.0—24.0 s ].CTA of group 2 patients demonstrated bettor quality over group I patients' CTA, especifieally in the infra-pop|iteal and foot area arteries. There was an excellent inter-observer agreement for group 2 patients ( K ＞ 0.80 ) whereas in group 1 agreement in infra-popliteal segments for venous contamination ( K value 0.60 ) and stenosis degree ( K value 0.50 ) were not satisfactory enough.Group 1 patients were reported to have more severe stonosis in infra-popliteal and foot arteries( X2 = 30.55 and22.41,P＜0.01).Conclusion There was a wide interindividual variation in transit time for contrast medium to reach aorta and pollteal artery. Adaptive method by using two low-dose test bolus injection determined interindividual variation in delay time and scan time and thus above parameters was able to produce better quality images than using bolus triggering technique in below knee and foot region arteries. Key words: Tomography,spiral computed; Angiography; Arterial occlusive diseases

Context. The Kepler Object of Interest Network (KOINet) is a multi-site network of telescopes around the globe organised for follow-up observations of transiting planet candidate Kepler objects of interest with large transit timing variations (TTVs). The main goal of KOINet is the completion of their TTV curves as the Kepler telescope stopped observing the original Kepler field in 2013. Aims. We ensure a comprehensive characterisation of the investigated systems by analysing Kepler data combined with new ground-based transit data using a photodynamical model. This method is applied to the Kepler-82 system leading to its first dynamic analysis. Methods. In order to provide a coherent description of all observations simultaneously, we combine the numerical integration of the gravitational dynamics of a system over the time span of observations with a transit light curve model. To explore the model parameter space, this photodynamical model is coupled with a Markov chain Monte Carlo algorithm. Results. The Kepler-82b/c system shows sinusoidal TTVs due to their near 2:1 resonance dynamical interaction. An additional chopping effect in the TTVs of Kepler-82c hints to a further planet near the 3:2 or 3:1 resonance. We photodynamically analysed Kepler long- and short-cadence data and three new transit observations obtained by KOINet between 2014 and 2018. Our result reveals a non-transiting outer planet with a mass of mf = 20.9 ± 1.0 M⊕ near the 3:2 resonance to the outermost known planet, Kepler-82c. Furthermore, we determined the densities of planets b and c to the significantly more precise values ρb = 0.98−0.14+0.10 g cm−3 and ρc = 0.494−0.077+0.066 g cm−3.

KOI-227, KOI-319 and KOI-884 are identified here as (at least) two planet systems. For KOI-319 and KOI-884, the observed Transit Timing Variations (TTVs) of the inner transiting planet are used to detect an outer non-transiting planet. The outer planet in KOI-884 is 2.6 Jupiter masses and has the orbital period just narrow of the 3:1 resonance with the inner planet (orbital period ratio 2.93). The distribution of parameters inferred from KOI-319.01's TTVs is bimodal with either a 1.6 Neptune-mass (M_N) planet wide of the 5:3 resonance (period 80.1 d) or a 1 Saturn-mass planet wide of the 7:3 resonance (period 109.2 d). The radial velocity measurements can be used in this case to determine which of these parameter modes is correct. KOI-227.01's TTVs with large 10 hour amplitude can be obtained for planetary-mass companions in various major resonances. Based on the Bayesian evidence, the current TTV data favor the outer 2:1 resonance with a companion mass 1.5 M_N, but this solution implies a very large density of KOI-227.01. The inner and outer 3:2 resonance solutions with sub-Neptune-mass companions are physically more plausible, but will need to be verified.

Following on from Paper I in our series (Xie 2013), we report the confirmation by Transit Timing Variations (TTVs) of a further 30 planets in 15 multiple planet systems, using the publicly available Kepler light curves (Q0-Q16). All of these fifteen pairs are near first-order Mean Motion Resonances (MMR), showing sinusoidal TTVs consistent with theoretically predicted periods, which demonstrate they are orbiting and interacting in the same systems. Although individual masses cannot be accurately extracted based only on TTVs (because of the well known degeneracy between mass and eccentricity), the measured TTV phases and amplitudes can still place relatively tight constraints on their mass ratios and upper limits on their masses, which confirm their planetary nature. Some of these systems (KOI-274, KOI-285, KOI-370 and KOI-2672) are relatively bright and thus suitable for further follow-up observations.

While various indirect methods are used to detect exoplanets, one of the most effective and accurate methods is the transit method, which measures the brightness of a given star for periodic dips when an exoplanet is passing in front of the parent star. For systems with multiple transiting planets, the gravitational perturbations between planets affect their transit times. The difference in transit times allows a measurement of the planet masses and orbital eccentricities. These parameters help speculating on the formation, evolution, and stability of the system. Using transit timing variations (TTVs), we measure the masses and eccentricities of two planets orbiting K2-21, a relatively bright K7 dwarf star. These two planets exhibit measurable TTVs, have orbital periods of about 9.32 and 15.50 d, respectively, and a period ratio of about 1.66, which is relatively near to the 5:3 mean motion resonance. We report that the inner and outer planets in the K2-21 system have properties consistent with the presence of a hydrogen- and helium-dominated atmosphere, as we estimate their masses to be $1.59^{+0.52}_{-0.44}$ and $3.88^{+1.22}_{-1.07}\, \mathrm{ M}_\oplus$ and densities of $0.22^{+0.05}_{-0.04}$ and $0.34^{+0.08}_{-0.06}\, \rho _\oplus$, respectively (M⊕ and ρ⊕ are the mass and density of the Earth, respectively). Our results show that the inner planet is less dense than the outer planet; one more counterintuitive exoplanetary system such as Kepler-105, LTT 1445, TOI-175, and Kepler-279 systems.

We present 12 new transit light curves of the hot-Jupiter TrES-3b observed during 2012−2018 to probe the transit timing variation (TTV). By combining the midtransit times determined from these 12 transit data with those reestimated through uniform procedure from 71 transit data available in the literature, we derive new linear ephemeris and obtain the timing residuals that suggest the possibility of TTV in the TrES-3 system. However, the frequency analysis shows that the possible TTV is unlikely to be periodic, indicating the absence of an additional body in this system. To explore the other possible origins of TTV, the orbital decay and apsidal precession ephemeris models are fitted to the transit time data. We find the decay rate of TrES-3b to be ms yr−1, and the corresponding estimated modified stellar tidal quality factor of is consistent with the theoretically predicted values for the stars hosting the hot-Jupiters. The shift in the transit arrival time of TrES-3b after 11 years is expected to be T shift ∼ 69.55 s, which is consistent with the rms of the timing residuals. Besides, we find that the apsidal precession ephemeris model is statistically less probable than the other considered ephemeris models. It is also discussed that despite the fact that the linear ephemeris model appears to be the most plausible model to represent the transit time data, the possibility of the orbital decay cannot be completely ruled out in the TrES-3 system. To confirm this, further high-precision and high-cadence follow-up observation of transits of TrES-3b would be important.

Ground-based simultaneous multiband transit observations allow an accurate system parameters determination and may lead to the detection and characterization of additional bodies via the transit timing variations (TTVs) method. We aimed to (i) characterize the heavily bloated WASP-4b hot Jupiter and its star by measuring system parameters and the dependence of the planetary radius as a function of four (Sloan g', r', i', z') wavelengths and (ii) search for TTVs. We recorded 987 images during three complete transits with the GROND instrument, mounted on the MPG/ESO-2.2m telescope at La Silla Observatory. Assuming a quadratic law for the stellar limb darkening we derive system parameters by fitting a composite transit light curve over all bandpasses simultaneously. To compute uncertainties of the fitted parameters, we employ the Bootstrap Monte Carlo Method. The three central transit times are measured with precision down to 6 s. We find a planetary radius Rp = 1.413+/-0.020 RJup, an orbital inclination i = 88deg57'+/-0.45deg and calculate a new ephemeris, a period P = 1.33823144+/-0.00000032days and a reference transit epoch T0 = 2454697.798311+/-0.000046 (BJD). Analysis of the new transit mid-times in combination with previous measurements shows no sign of a TTV signal greater than 20s. We perform simplified numerical simulations to place upper-mass limits of a hypothetical perturber in the WASP-4b system.

Both ground and space-based transit observatories are poised to significantly increase the number of known transiting planets and the number of precisely measured transit times. The variation in a planet's transit times may be used to infer the presence of additional planets. Deducing the masses and orbital parameters of such planets from transit time variations (TTVs) alone is a rich and increasingly relevant dynamical problem. In this work, we evaluate the extent of the degeneracies in this process, systematically explore the dependence of TTV signals on several parameters and provide phase space plots that could aid observers in planning future observations. Our explorations are focused on a likely-to-be prevalent situation: a known transiting short-period Neptune or Jupiter-sized planet and a suspected external low-mass perturber on a nearly-coplanar orbit. Through approximately 10^7 N-body simulations, we demonstrate how TTV signal amplitudes may vary by orders of magnitude due to slight variations in any one orbital parameter (0.001 AU in semimajor axis, 0.005 in eccentricity, or a few degrees in orbital angles), and quantify the number of consecutive transit observations necessary in order to obtain a reasonable opportunity to characterize the unseen planet (approximately greater or equal to 50 observations). Planets in or near period commensurabilities of the form p:q, where p < 21 and q < 4, produce distinct TTV signatures, regardless of whether the planets are actually locked in a mean motion resonance. We distinguish these systems from the secular systems in our explorations. Additionally, we find that computing the autocorrelation function of a TTV signal can provide a useful diagnostic for identifying possible orbits for additional planets and suggest that this method could aid integration of TTV signals in future studies of particular exosystems.

Aims. Our aim in this paper is to refine the orbital and physical parameters of the HATS-2 planetary system and study transit timing variations and atmospheric composition thanks to transit observations that span more than 10 yr and that were collected using different instruments and pass-band filters. We also investigate the orbital alignment of the system by studying the anomalies in the transit light curves induced by starspots on the photosphere of the parent star. Methods. We analysed new transit events from both ground-based telescopes and NASA’s TESS mission. Anomalies were detected in most of the light curves and modelled as starspots occulted by the planet during transit events. We fitted the clean and symmetric light curves with the JKTEBOP code and those affected by anomalies with the PRISM+GEMC codes to simultaneously model the photometric parameters of the transits and the position, size, and contrast of each starspot. Results. We found consistency between the values we found for the physical and orbital parameters and those from the discovery paper and ATLAS9 stellar atmospherical models. We identified different sets of consecutive starspot-crossing events that temporally occurred in less than five days. Under the hypothesis that we are dealing with the same starspots, occulted twice by the planet during two consecutive transits, we estimated the rotational period of the parent star and, in turn the projected and the true orbital obliquity of the planet. We find that the system is well aligned. We identified the possible presence of transit timing variations in the system, which can be caused by tidal orbital decay, and we derived a low-resolution transmission spectrum.

Only a few exoplanets are known to orbit around fast rotating stars. One of them is XO-6b, which orbits an F5V-type star. Shortly after the discovery, we started multicolor photometric and radial-velocity follow-up observations of XO-6b, using the telescopes of Astronomical Institute of the Slovak Academy of Sciences. Our main scientific goals were to better characterize the planetary system and to search for transit timing variations. We refined several planetary and orbital parameters. Based on our measurements, the planet XO-6b seems to be about 10% larger, which is, however, only about 2σ difference, but its orbit inclination angle, with respect to the plane of the sky, seems to be significantly smaller, than it was determined originally by the discoverers. In this case we found about 9.5σ difference. Moreover, we observed periodic transit timing variations of XO-6b with a semi-amplitude of about 14 min and with a period of about 450 days. There are two plausible explanations of such transit timing variations: (1) a third object in the system XO-6 causing light-time effect, or (2) resonant perturbations between the transiting planet XO-6b and another unknown low-mass planet in this system. From the O-C diagram we derived that the assumed third object in the system should have a stellar mass, therefore significant variations are expected in the radial-velocity measurements of XO-6. Since this is not the case, and since all attempts to fit radial velocities and O-C data simultaneously failed to provide a consistent solution, more realistic is the second explanation.