Photometric follow-up of the 20 Myr-old multi-planet host star V1298 Tau with CHEOPS and ground-based telescopes

Context. The 20 Myr-old star V1298Tau hosts at least four planets. Since its discovery, this system has been a target of intensive photometric and spectroscopic monitoring. The characterisation of its architecture and planets’ fundamental properties turned out to be very challenging so far. Aims. The determination of the orbital ephemeris of the outermost planet V1298Tau e remains an open question. Only two transits have been detected so far by Kepler / K 2 and TESS, allowing for a grid of reference periods to be tested with new observations, without excluding the possibility of transit timing variations. Observing a third transit would allow to better constrain the orbital period, and would also help determining an accurate radius of V1298Tau e because the former transits showed di ff erent depths. Methods. We observed V1298Tau with the CHaracterising ExOPlanet Satellite (CHEOPS) to search for a third transit of planet e within observing windows that have been selected in order to test three of the shortest predicted orbital periods. We also collected ground-based observations to verify the result found with CHEOPS. We reanalysed Kepler / K 2 and TESS light curves to test how the results derived from these data are a ff ected by alternative photometric extraction and detrending methods. Results. We report the detection with CHEOPS of a transit that could be attributed to V1298Tau e . If so, that result implies that the orbital period calculated from fitting a linear ephemeris to the three available transits is close to ∼ 45 days. Results from the ground-based follow-up marginally support this possibility. We found that i ) the transit observed by CHEOPS has a longer duration compared to that of the transits observed by Kepler / K 2 and TESS; ii ) the transit observed by TESS is > 30% deeper than that of Kepler / K 2 and CHEOPS, and deeper than the measurement previously reported in the literature, according to our reanalysis. Conclusions. If the new transit detected by CHEOPS is due to V1298Tau e , this implies that the planet experiences TTVs of a few hours –as it can be deduced from three transits– and orbital precession, which would explain the longer duration of the transit compared to the Kepler / K 2 and TESS signals. Another, a priori less likely possibility is that the newly detected transit belongs to a fifth planet with a longer orbital period than that of V1298Tau e . Planning further photometric follow-up to search for additional transits is indeed necessary to solve the conundrum, also for pinning down the radius of V1298Tau e .


Introduction
V1298 Tau (also known as K2-309) is a 20 ± 10 Myr-old K1 star bolometric luminosity L = 0.954 ± 0.040 L ⊙ ; magnitude ⋆ Based on data collected within the CHEOPS AO-3 proposal PR230032 Pinning down orbital period, transit ephemeris, and radius of the infant planet V1298 Tau e.
tems detected so far.Right after the discovery, V1298 Tau became a target of intense multi-instrument, multi-band spectroscopic and photometric follow-up (Poppenhaeger et al. 2021;Feinstein et al. 2021;Suárez Mascareño et al. 2021;Vissapragada et al. 2021;Gaidos et al. 2022;Johnson et al. 2022).The study by Suárez Mascareño et al. (2021) had the primary aim of measuring masses and bulk densities of the four planets in the V1298 Tau system through spectroscopic observations.Their results show that V1298 Tau b has a mass of 0.64 ± 0.19 M Jup and a density similar to that of giant planets in the Solar System and other known old giant exoplanets.For the outermost V1298 Tau e, for which only one transit was detected at that time, Suárez Mascareño et al. (2021) determined an orbital period P e = 40.2± 0.1 days, a mass of 1.16 ± 0.30 M Jup , and a density slightly larger, but compatible within error bars, than that of most of the older giant exoplanets.This unexpected result suggests that giant planets of such a young age might evolve and contract within the first million years after system's birth, thus challenging current models of planetary formation and evolution.Following the results of Suárez Mascareño et al. (2021), Maggio et al. (2022) investigated the current escape rates from the planetary atmospheres, and predicted the future evolution of atmospheric photo-evaporation, while Tejada Arevalo et al. ( 2022) investigated stability constraints on the system's orbital architecture.
Despite a first and demanding follow-up, the results by Suárez Mascareño et al. (2021) and their implications need to be confirmed, and an accurate characterisation of the V1298 Tau system remains an open issue and a hot topic.In September-October 2021 the Transiting Exoplanet Survey Satellite (TESS) observed V1298 Tau and detected a second transit attributed to planet e, as reported by Feinstein et al. (2022).They constrained the orbital period of the outermost planet to a grid of discrete values (P e > 44 days, not in agreement with the solution found by Suárez Mascareño et al. 2021 through spectroscopy), which is the only reasonable calculation when just two transits are detected.The discrete periods in the grid have been determined by dividing for integer numbers the time separation between the epochs of central transit observed by Kepler/K2 and TESS, and have to be interpreted as a guidance to search for future transits of planet e.They would correspond to actual values of P e if there are no substantial changes in the orbital parameters over short time scales, and without the presence of large transit timing variations (TTVs).The information currently available about the system does not allow us to reliably predict any significant orbital instability, and nowadays a transit follow-up strategy has to be necessarily based on the grid of orbital periods defined by Feinstein et al. (2022).Feinstein et al. (2022) also showed that the transit depths of V1298 Tau bcd measured from TESS data are slightly shallower, but compatible within 1-2σ, than those observed by Kepler/K2, while the transit depth of V1298 Tau e is ∼ 3σ larger.They conclude by proposing a few scenarios to qualitatively explain this discrepancy in radius measurements between Kepler/K2 and TESS, all of them involving phenomena not yet well characterised for systems with very young ages which make any interpretation very complex, going from the rapidly evolving stellar activity to haze-dominated planetary atmospheres.Sikora et al. (2023) presented a reanalysis of the spectroscopic and photometric data of V1298 Tau, including additional RVs to the dataset of Suárez Mascareño et al. (2021).Concerning V1298 Tau e, and with a reference to the grid of guidance orbital periods derived by Feinstein et al. (2022), Sikora et al. (2023) conclude that periods P e > 55.4 days can be ruled out at 3σ level, and placed a tight constraint to P e (P e = 46.768131±0.000076days assuming a circular orbit, although the actual orbital architecture of V1298 Tau e remains puzzling. In order to improve the characterisation of this system, we followed-up V1298 Tau with the CHaracterising ExOPlanet Satellite (CHEOPS) space telescope (Benz et al. 2021).Our aim was twofold1 : i) detecting additional transits of planet e, providing better constraints to its orbital period and a first identification of TTVs, with the consequent improvement of the results by Feinstein et al. (2022), and ii) getting a new measurement of its radius, possibly helping to explain the different transit depths measured from Kepler/K2 and TESS light curves (LCs).In this work, we present results from the CHEOPS monitoring conducted between 18 November and 18 December 2022, and from further ground-based photometric follow-up that was motivated by the outcomes of CHEOPS observations.We also reanalysed the TESS photometry that we extracted using an alternative pipeline to correct for contamination from nearby faint stars, and compare the results to those of Feinstein et al. (2022).

TESS
V1298 Tau was observed by TESS between 16 September and 6 November 2021 (Sectors 43 and 44), and the observations were analysed shortly after by Feinstein et al. (2022).We reextracted the LC from the TESS Full Frame Images (FFIs) with the PATHOS pipeline described by Nardiello et al. (2019Nardiello et al. ( , 2020)).First, we extracted the long-cadence LC (cadence of 10 minutes) by using six different photometric apertures (PSF-fitting, 1-px, 2-px, 3-px, 4-px, and 5-px radius aperture photometry) after subtracting from each FFI the sources within 3 arcmin from V1298 Tau.Then, we corrected the LC by applying the cotrending basis vectors (CBVs) calculated as in Nardiello et al. (2021).We selected the best photometric aperture comparing the point-to-point (P2P) rms of the different LCs, finding that the lower P2P rms is that associated with the PSF-fitting light curve.V1298 Tau is an ideal target to be analysed with the PATHOS pipeline, that was developed to extract high-precision light curves in crowded fields, like open clusters.The PSF-fitting photometry performed with empirical PSFs allow us to minimise the dilution effects due to the contamination of close-by stars (following the TIC catalogue, there are 63 sources within 3 arcmin, two of them having T ∼ 9.6 and T ∼ 7.9).Moreover, in Nardiello et al. (2022), we discussed that our CBV correction of systematic errors in the light curves is expected to perform more effectively than other pipelines in the case of active stars such as V1298 Tau (no spurious signals or transit deformations occurred in our final light curves).

CHEOPS
We were awarded a total of 112 CHEOPS orbits to test three orbital periods of V1298 Tau e with the highest probability among those calculated by Feinstein et al. (2022) from the temporal separation between the two transits seen by Kepler/K2 and TESS, namely 44.1699, 45.0033, and 46.7681 (±0.0001) days.It must be emphasised that the grid of periods proposed by Feinstein et al. (2022) originates from just two observed transits, and they can be used as a guidance to search for a third transit that would better constraint the orbital ephemeris.Our original plan was dividing the 112 orbits in six visits, two visits per test period, in order to collect two transits in the case of a successful detection, and get a robust measurement of the orbital period and planet's radius.In practice, three over six visits have been executed.We did not select P=45.8687 days as a test period from the grid because only one CHEOPS visit could be accommodated in the available time span, due to observing constraints.Concerning this period, we did not find clear evidence for a transit event in photometry collected between BJD TDB 2 459 619.2380 and 2 459 619.4459 (8 February 2022) with the Schmidt 67/92cm telescope at the INAF-Astrophysical Observatory of Asiago, when a transit was predicted with ingress, centre, and egress BJD TDB 2 459 619.24662, 2 459 619.40162 and 2 459 619.55662 respectively.This LC is shown in Fig. 1 and was extracted as described in Nardiello et al. (2015Nardiello et al. ( , 2016)).Looking at the data, we cannot claim a detection with a significance better than nearly 2σ.Also, possible large TTVs should be taken into account before discarding P=45.8687 days from the list of the possible guidance orbital periods.The TTVs cannot be reliably predicted yet because, as already mentioned, many critical information about the system are still missing.Nonetheless, we found convenient for the sake of our proposal to focus on the other three high-probability test periods that we selected.The observing windows have been centred to the predicted time of central transit assuming no TTVs, and their time span (nearly 1 or 2 days) has been defined to allocate to some extent possible, but still unknown TTVs.
We extracted the LC using the Point Spread Function (PSF) photometry package PIPE2 (Brandeker et al. 2022), which is suitable for crowded fields.This pipeline models the background stars before extracting the photometry of the target by fitting a PSF.The PSF fitting is weighted by the signal and noise of each pixel, thus the extraction is less sensitive than aperture photometry to background star contamination.We use this curve in the present analysis.A comparison with the default Data Reduction Pipeline (DRP) of CHEOPS (Hoyer et al. 2020) reveals a general 30% improvement in the overall photometric scatter.However, as a cross-check we also analysed the DRP photometry, finding no differences in the results within the uncertainties.
For convenience, we show the bandpasses of the three space telescopes in Fig. 2.

Ground-based observations
V1298 Tau was observed during the night of 31 January -01 February 2023 by ground-based facilities in Italy and Spain.This joint follow-up was aimed to support the findings of CHEOPS described in Sect.3, and the involved facilities are described in the following:  -Asiago Copernico.We gathered a photometric series through a Sloan r filter with the AFOSC instrument (Asiago Faint Object Spectrograph and Camera) mounted at the 1.82-m Copernico telescope at Cima Ekar, in northern Italy 3 .A total of 3 364 frames were collected with a constant exposure time of 5 s from 17:07 UT to 00:18 UT, when the target elevation went below the safety limit.The sky was mostly clear throughout the series with a few passing thin veils.The 81%-illuminated Moon at just 16 • from the line of sight resulted in a very high background level and a scatter larger than usual, but with no significant impact on photometric accuracy.The images, purposely defocused to ∼ 7 ′′ FWHM to improve photometric accuracy and avoid saturation, were bias-and flat-field corrected using standard procedures.A LC was then extracted with the STARSKY code (Nascimbeni et al. 2013), a photometric pipeline developed by the TASTE project (Nascimbeni et al. 2011)

Light curve modelling
We analysed the LCs of all instruments in a Bayesian framework.We used the python package SnappyKO (Parviainen & Korth 2020) to model the transits.Correlated signals due to stellar activity are modelled through a Gaussian Process (GP) regression using the S+LEAF python library (Delisle et al. 2020(Delisle et al. , 2022)).All the details of the GP modelling are provided in Sect. A. The likelihood maximisation is performed with a Markov Chain Monte Carlo (MCMC) using the package emcee version 3. 1.3 (Foreman-Mackey et al. 2013), a python implementation of the affine invariant MCMC ensemble sampler by Goodman & Weare (2010).We ran the code in the HOTCAT computing infrastructure (Bertocco et al. 2020;Taffoni et al. 2020), to speedup the computational time.

Kepler/K2 light curve
We masked the transits of planets bcd from the Kepler/K2 LC by computing the time of transits using the ephemeris of David et al. (2019b) and trimming a time window of 1.5 times the corresponding transit duration centred on the expected transit time.Then, we flattened the LC using the best-fit model of David et al. (2019b), in order to identify and reject stellar flares (using a +5σ clipping threshold above the best-fit model).
As discussed in Appendix A, we found that the best GP model is a combination of a quasi-periodic Exponentialsine periodic (ESP) kernel to trace the rotational signal, and stochastically-driven harmonic oscillator (SHO) kernel to modelling the low-frequency correlated noise, which is likely of stellar origin, as it is recovered also in the analysis of the TESS LC (see Sect. 3.2).
We fitted the data using a model containing simultaneously the transit signal of planet e (assuming a circular orbit) and the correlated stellar rotation signal.We used the priors listed in Table 1.For the orbital period P e we used a uniform prior U(35, 100) days, and for the time of central transit we used a uniform prior U(2 457 096.46, 2 457 096.79)BJD TDB centred on the epoch of transit indicated by David et al. (2019b).We used the stellar density instead of fitting directly the scaled semimajor axis a/R * , adopting a Gaussian prior based on the measurement of Suárez Mascareño et al. (2021).To fit the limb darkening (LD) coefficients (adopting a quadratic law), we used priors based on the results of David et al. (2019b).The Kepler/K2 LC has a cadence of 29.4 minutes.Following a common practice, in order to mitigate morphological distortions introduced when fitting transit light curves with such a long-cadence (see, e.g., Kipping 2010), we used an oversampling factor of 3.This means that each simulated photometric data point at the epoch of a real observation is calculated as the average value of three evenly spaced simulated data points each corresponding to a 10minute exposure.
We ran the MCMC for 100 000 steps, which turned out to be ∼150 times the auto-correlation length of the chains, estimated following Goodman & Weare (2010).This indicated successful convergence, as suggested by Sokal (1997) and adapted to parallel Monte Carlo chains (see https://dfm.io/posts/autocorr/).The best-fit model of the Kepler/K2 LC is shown in Fig. 3, while in Fig. 4 we show the detrended transit signal (black dots).We obtained a transit best-fit solution consistent with David et al. (2019b) within 1σ (Table 1).

TESS light curve
For the fit of the TESS LC we used the same approach adopted for the Kepler/K2 data, as described above.Also in this case, we found out that a mixture of a ESP and SHO kernels provides a more realistic representation of the correlated noise in the LC, and through several tests we found that using different kernels does not affect the transit parameters.Following the results of Feinstein et al. (2022), we used the prior U(2459481.63,2459481.96)BJD TDB for the time of transit, and a Gaussian prior for the LD coefficients.We used an oversampling factor of 3 to model the data with a cadence of 10 min.The best-fit model and the detrended transit signal are shown in Fig. 3  The two LCs looks very similar, except for the out of transit segments, and our GP-modelled flux is higher during the transit timespan, overall determining a deeper transit in our case.

CHEOPS light curve
We modelled the LC of all the CHEOPS visits using the same framework described above and taking advantage of some of the results from the previous analysis.First of all, we found that a simple undamped SHO kernel could effectively mitigate the correlated signal present in the LC that is due to the stellar rotation.This is expected from the fact that the Lomb-Scargle (LS) periodogram of the data shows only the rotational peak, not its harmonics.Moreover, the adoption of the SHO kernel minimises the number of free parameters compared to the ESP kernel.We assumed that the low-frequency correlated noise detected in the Kepler/K2 and TESS data could not be recovered in CHEOPS data due to the limited time span and sparse sampling.Thus, we fitted the GP correlated noise using a Gaussian prior for the stellar rotation frequency ν rot measured in the TESS data (corresponding to the rotation period of ∼ 2.9 days).For the LD coefficients we used Gaussian priors based on the results of David et al. (2019b), because Kepler and CHEOPS passbands are similar (Fig. 2, and Benz et al. 2021).The prior on the time of inferior conjunction T 0 is uniform, and spans the full extent of the CHEOPS follow-up (U(2 459 901.8, 2 459 932.5)BJD TDB ) in order to keep the search for a transit signal blind.We adopted a uniform prior U(35, 100) days for the orbital period of a possible transiting companion.
The CHEOPS LCs are affected by periodic instrumental noise due to the roll of the telescope.We modelled this systematic effect jointly with the transit and stellar rotational signals by using a harmonic expansion up to the fifth harmonic of a periodic signal phased with the roll angle of the telescope, as described in Scandariato et al. (2022).A posteriori, we also found some residual correlation with the coordinates (x PSF , y PSF ) of the centroid of the stellar PSF, which we corrected by including in the model a bi-linear detrending in x PSF and y PSF .
The results of this analysis are i) the non-detection of transitlike signals in the first and second visit.This result puts some constraints on the orbit of the planet V1298 Tau e, but it is not conclusive, and it does not allow us to exclude that the orbital period, assuming the presence of TTVs not yet verified, is close to P=44.1699 and 46.7681 days (Feinstein et al. 2022).This point is further discussed in Section 4; ii) the detection of a transit-like signal in the first half of the third CHEOPS slot (last panel in Fig. 3), where a transit of V1298 Tau e is expected to occur if the orbital period were close to 45.0033 days, assuming some TTVs.During the third visit, there were no predicted transits of any of the other three planets, well within the uncertainties, according to the ephemeris calculated by Feinstein et al. (2022).The Bayesian Information Criterion (BIC) of this model is -34064.Then, we fitted the data without including the transit signal.The corresponding solution has a BIC=-34048 (∆BIC=+16), thus this model is highly disfavoured with respect to the one that includes the transit.
We derive a transit duration T 14 ∼ 9 hr and an impact parameter b = 0.11 +0.09 −0.07 which are, respectively, longer and shorter than those measured from Kepler/K2 and TESS data (Table 2 and Fig. 4).Specifically, the discrepancies between the transit duration measured by CHEOPS and the duration observed by Kepler/K2 and TESS are (T 14, CHEOPS -T 14, K2 )/ (σ 2 T 14 , CHEOPS + σ 2 T 14 , K2 ) =∆(T 14, CHEOPS, K2 )/ Σ(σ 2 T 14, CHEOPS, K2 )=14.2, and ∆(T 14, CHEOPS, TESS )/ Σ(σ 2 T 14, CHEOPS, TESS )=10.6, respectively (in case of values with asymmetric error bars, in the calculation we used the average values of the upper and lower uncertainties).For the impact parameter, the discrepancies between the measurements are ∆(b CHEOPS, K2 )/ Σ(σ 2 b CHEOPS, K2 )=3.7, and ∆(b CHEOPS, TESS )/ Σ(σ 2 b CHEOPS, TESS )=5.2Our value of the orbital period P orb = 58 ± 7 days is consistent within 2σ with the value of 45 days expected for the third visit by CHEOPS.The derived transit depth R 2 p /R 2 ⋆ = 0.0036 ± 0.006 is in very good agreement with that measured from Kepler/K2 photometry, after the TESS observations).In principle, significant TTVs could not be ruled out, and would explain the significant discrepancy with the predicted epoch T 0 .We note that the best-fit timescale of the SHO kernel is ∼2 days (Table 2), meaning that the GP model cannot filter out features of the LC with shorter timescales, such as the planetary ingress and egress.Thus, we conclude that the GP is not responsible for altering the shape of the transit signal.We show in Fig. C.2 the original transit LCs gathered by the three telescopes that we analysed in this work, with overplotted our best-fit GP models.As a final test, we fit the CHEOPS LC but substituting the GP detrending with a fifth-degree polynomial (a polynomial for each CHEOPS visit).The polynomial detrending does not perform as effectively as the GP in this case, as it is confirmed by the presence of correlated noise in the residuals of the best-fit.This introduces biases in the estimated best-fit parameters, but the advantage is that polynomials are less prone to absorb astrophysical signals or introduce false positives than GPs.Following this alternative approach, we still found a transit signal in the first half of the third visit which is again significantly favoured by the BIC over the model without a transit.The best-fit parameters, as said above, are less reliable due to the worse quality of the detrending, nonetheless in both cases (fixed and free orbital period) the transit duration is longer than that observed by Kepler/K2 and TESS.

Ground-based follow-up
All the ground-based photometric data we gathered during the night of 31 January 2023 (Sect.2.4) are plotted in Fig. 5.We remark that all the instruments that we used could in principle significantly detect a transit with a depth of ∼ 0.5%.We globally fitted all of them with the PyORBIT code (Malavolta et al. 2016(Malavolta et al. , 2018) )    Notes.Referred to the stellar descending point.
Article number, page 6 of 15  the observations we adopted a single-planet transit model where all the stellar and planetary parameters were fixed to the values published by Feinstein et al. (2022), and only the central transit time T 0 and a linear baseline f (t) = a 0 + a 1 • t were left free to vary using a uninformative prior.After 100 000 steps with a thinning factor of 100 and a burn-in phase of 20 000 steps, the posterior distribution of T 0 regularly converged to T 0 = 2 459 976.0918 ± 0.0016 BJD TDB meaning that just the egress of planet e was captured by the Asiago Copernico and Schmidt LCs and missed by the other two observing sites, as can be seen by the maximum a-posteriori probability (MAP) model plotted as a black line in Fig. 5.This unexpected finding is in striking contrast with our initial prediction T 0 = 2 459 976.8309BJD TDB and would imply an O − C of about −12 hours.
We note that the egress-like feature falls at the very beginning of our coverage, during nautical twilight at Asiago, and this feature could be influenced by the sky conditions.On the other hand, all the instrumental diagnostics look perfectly nominal for both LCs, which agree equally well with the best-fit transit model, and in particular with the transit depth and the ingress/egress duration (T 14 ) of planet V1298 Tau e.

Discussion and Conclusions
The main result presented in this work is the detection of a transit in the LC of the young star V1298 Tau collected by CHEOPS between 17 and 18 December 2022.The signal falls within a time interval when a transit of the planet V1298 Tau e was predicted to occur, if its orbital period were close to P e = 45.0033days.This result looks not compatible with an orbital period close to P e = 46.7681days, which is the most probable value proposed by Sikora et al. (2023), because no transit would then be expected to fall within the third CHEOPS observing slot, even though this conclusion is necessarily based on a very limited amount of information that we have so far about the orbital dynamics of V1298 Tau e.
We reanalysed the individual transits observed by Kepler/K2 and TESS that have been ascribed to V1298 Tau e, and compared them with the signal that we detected in the CHEOPS LC, finding some differences which make a coherent description of this puzzling system challenging.The main observational results and findings from the comparative analysis of the three Kepler/K2, TESS, and CHEOPS transits can be summarised as follows (see Section 3.3 for details): the transit duration as seen by CHEOPS is significantly longer than that derived from Kepler/K2 and TESS observations.We note that our derived transit duration from TESS  data is longer than that observed by Kepler/K2, but compatible within ∼ 3σ; without constraining the orbital period in the MCMC analysis, we get posteriors for P orb that are in agreement within 1σ for the three telescopes, but the impact parameter derived from CHEOPS data is significantly lower (b = 0.11 +0.09 −0.07 ), as it is expected for a transit with a longer duration; the transit depth as seen by CHEOPS is consistent with that of the Kepler/K2 transit, while the transit depth measured from the TESS LC is larger.This implies a radius for the transiting companion of ∼ 0.74 R Jup (Kepler/K2 and CHEOPS), or ∼ 0.98 R Jup (TESS).We remark that Feinstein et al. ( 2022) found a radius ∼ 3σ larger in the TESS data (R p = 0.89 ± 0.04 R Jup ) than that found in the Kepler/K2 data by David et al. (2019b), while in our case this discrepancy increases to ∼ 4σ (i.e. the ratio R p /R ⋆ measured by TESS is ∼ 34% higher than for Kepler/K2).That is because our derived TESS transit is deeper than that of Feinstein et al. ( 2022), as shown in Fig. 4, implying a larger planet radius (0.98 ± 0.06 vs. 0.89 ± 0.04 R Jup ).
Regarding the comparison between our derived TESS LC and that of Feinstein et al. (2022), we investigated whether a similar difference in the transit depths is also observed for planet b, which has a transit depth comparable to that of planet e (Appendix B).In this case, we recover a transit depth slightly shallower, but well in agreement within the error bars (Fig. B.1), and conclude that apparently a significant difference with Feinstein et al. ( 2022) is only observed for planet e.
These results do not have a straightforward interpretation, and bring into question even the identity of the transiting companion(s).Nonetheless, it must be emphasised that any reasonable explanation is naturally constrained by the limited number of the observed transits.We propose two main interpretative frameworks based on the available data.
First, let us make the hypothesis that Kepler/K2, TESS, and CHEOPS indeed observed the same planet V1298 Tau e, as previously identified by David et al. (2019b) andFeinstein et al. (2022).Thanks to CHEOPS, this assumption would solve the uncertainty on the orbital period, implying that P e ∼ 45 days, and also implying that the outermost planet in the system experiences changes to the orbital parameters on a timescale of a few years.Indeed, a change in the orbital elements occurring over short timescales would explain the presence of large TTVs (Fig. 6), and transit duration variations (TDV).As an illustrative case, we performed a few N-body numerical integration of the V1298 Tau system for different random choices of the initial orbital angles of planet b and e (mean anomaly and nodal longitude).For this exercise, we adopted the masses of V1298 Tau b and e derived by Suárez Mascareño et al. (2021), and mild eccentricities (0.01-0.05).We see from Fig. C.3 that the orbital period could change significantly over a very short timescale, therefore the orbit of V1298 Tau e can be likely chaotic.Changes in the orbital parameters can determine a corresponding change in the transit parameters over a few years, which is an interval similar to the time span covered by the photometric observations.We emphasise that the result of Fig. C.3 is only suggestive because of our limited prior knowledge about the system's architecture and fundamental properties.As an alternative and illustrative case, we ran another loosely constrained set of dynamical integration over a time interval of 10 years by imposing circular orbits, and fixing the longitude of the ascending nodes to 180 deg.In this case, planet masses have been drawn from the posteriors derived by Sikora et al. (2023).The results show that there could be a significant probability that we may have missed transits in the first two CHEOPS observing slots, if the orbital period P e is close to 46.76 and 44.17 days, respectively, and TTVs are ≲1 hour.If that is the case, this would imply that the observed transit in the third CHEOPS slot belongs to a fifth companion.Any in-depth analysis of the system's dynamics can be performed only after additional transits of planet e will be observed, and must be left to future studies.We fitted the three transits by fixing P e to 45.0033 days, and tested both circular and eccentric orbits (Tab. 1 and 2).In all the cases, the BIC statistics does not strongly favour these models over those with P e treated as a free parameter.CHEOPS data are better fitted by a circular model (∆BIC = −18), but the posterior of the stellar density significantly differs from the prior.That is not surprising, because a larger stellar radius (lower stellar density) has to be assumed to explain a longer transit duration when fixing P e to 45 days.We recover a correct stellar density when we include the eccentricity as a free parameter, and we get e e = 0.12 +0.09 −0.06 .In the cases of Kepler/K2 and TESS, a circular model does not result in a similar issue.This first scenario is tentatively supported by ground-based photometry (Sect.3.4) that we collected 45 days after the transit detected by CHEOPS.The results from this follow-up do not exclude the possibility that we detected the egress of V1298 Tau e, with the transit occurring ∼ −12 hr earlier than predicted assuming the reference period from Feinstein et al. (2022).We note that a study of the V1298 Tau system focused on the determination of TTVs is not yet available in the literature.Obviously, to confirm this scenario additional and intensive photometric follow-up is necessary.Unfortunately, at the time of writing, we know that TESS is not going to observe V1298 Tau during Cycle 6 starting from Sept. 2023 7 .New observations with CHEOPS are indeed required, and they must be scheduled allowing for a sufficiently large observational window to take into account several hours of TTVs which are presently not predictable.
In a second framework, we interpret the transit detected by CHEOPS as due to a new planet-sized companion in the system, different from that observed by Kepler/K2 and TESS, which here we assume to be produced by the same planet.Assuming circular orbits, this would reconcile the different observed transit duration, and it would exclude P e = 45.0033days from the list of possible orbital periods of the Kepler/K2 and TESS companion.Future photometric follow-up is indeed necessary also to confirm this scenario, in case no additional transits corresponding to P ∼ 45 days will be detected.However, taking into account the 71day time span of almost uninterrupted observations, the chances that Kepler/K2 would have detected one transit of a planet with P orb = 58 ± 7 days are not negligible, in that it falls nearly 2σ within the duration of the Kepler/K2 observing window.
7 https://heasarc.gsfc.nasa.gov/wsgi-scripts/TESS/TESS-point_Web_Tool/TESS-point_Web_Tool/wtv_v2.0.py/Whatever the correct scenario is, we need to find an explanation also for the different transit depth observed by Kepler/K2, TESS, and CHEOPS.For a preliminary assessment, we neglect the differences in the Kepler/K2 (CHEOPS) and TESS passbands (Fig. 2).In active stars where a significant fraction of the disk is covered by spots, as it is expected in the case of V1298 Tau, unocculted starspots during a transit would produce an apparent increase in the planet's radius up to 10%, when the transits occur at very different locations in a starspot modulated LC (e.g.see Eq. 17 in Morris 2020): assuming that the planet does not cover any of the spots during its transit, when the unocculted spots occupy a smaller area of the stellar disc (i.e. at a maximum of the LC) the out-of-transit stellar brightness baseline will be higher, and the transit depth will be shallower, and the other way round.Here, the transits observed by Kepler/K2 and CHEOPS occur close to a minimum in their LCs, while the TESS transit is located close to a maximum of the stellar brightness (Fig. 3).Thus, assuming that the results of a GP model do not depend on where the transit is located on the LC, we would expect the TESS transit to be shallower than the other two, while we observe the opposite.We could reconcile the observations of the three individual transits, at least qualitatively, by assuming that the atmosphere and surface magnetic activity of V1298 Tau is faculae-dominated instead of being spot-dominated.For premain sequence stars, this is a plausible scenario, especially for fast rotators for which an anti-correlation between X-ray and simultaneous optical variability is observed (e.g.Guarcello et al. 2019).Nonetheless, that possibility has not been yet investigated for V1298 Tau.We also investigated a possible dependence of the transit depths of planet b from the specific location of the transit signals on the TESS LC.The two transits occurred at different phases of the LC, and they perfectly match (Fig. C.4).
Considering the effects of starspot occultation by a transiting planet, this might in principle explain the observed differences if we assume an uninterrupted band of dark spots extended in latitude running through all the stellar disk.When the transit chord runs within that band, depending on the fraction of the dark strip covered by the planet disk a transit depth would be more or less affected, and significantly change.This is indeed a ad hoc scenario that should be tested with additional transit observations and modelling, but it would explain the observations if we assume that the planet was covering a larger part of the dark band when observed K2/Kepler and CHEOPS.Starspots on the stellar disc may produce small variations in the apparent duration of a transit (longer or shorter), of the order of 4% as estimated by Oshagh et al. (2013).Therefore, it looks hard to invoke starspots to explain the 1.5-hour longer duration observed by CHEOPS, which is nearly 20% longer than the duration of the Kepler/K2 and TESS transits.
The detection of a new transit-like signal with interesting properties is indeed an important outcome of this study, nonetheless our result does not allow us to draw unambiguous conclusions on the identity of the transiting companion(s) detected by three different telescopes.That is not surprising in that V1298 Tau is a very challenging system to be characterised, both in terms of planet properties and architecture, and stellar activity.The more plausible interpretation may be that we observed the same transiting planet detected by Kepler/K2 and TESS with an orbital period of ∼ 45 days (from a linear ephemeris), which shows TTVs and TDVs due to dynamical instability of its orbit.After all, we detected a transit within the time window where it was expected to occur assuming one of the orbital periods in the grid of Feinstein et al. (2022) as a guidance.The hypothesis of a fifth planet in the system appears less probable with the data cur- rently available.We plan to observe V1298 Tau again with the CHEOPS telescope in the near future, with the primary aim of detecting more transit signals of planet e, and revising the results of this study.The priority will be looking for transits to confirm an orbital period of ∼45 days.At the same time, we will analyse a time series of hundreds of RVs of V1298 Tau, most of them collected with the HARPS-N spectrograph as a continuation of the work by Suárez Mascareño et al. (2021).As also shown by the studies of Sikora et al. (2023) and Blunt et al. (2023), the analysis of the RVs for such an active star is notoriously very challenging, especially if the transit ephemeris are not well constrained for all the planets in the system.Our goal will be to exploiting the synergy between CHEOPS and RV follow-up in order to improve the characterisation of the orbital and fundamental physical parameters of the planets in the V1298 Tau system.This, in turn, will allow for a well-informed analysis of the planet formation history, and of the current architecture and dynamical state of the system, improving the results of a preliminary investigation carried out by Turrini et al. (2023).
From a more quantitative perspective, we compared the PSD of the data with the typical PSD corresponding to the GPs derived by David et al. (2019b) andFeinstein et al. (2022).For each GP we generated 1000 LCs using the same timestamps of the corresponding LC (either Kepler/K2 or TESS) to take into account any aliasing effect due to the time sampling.Finally, for each set of 1000 LSs, we computed the average LS periodogram and its 1-σ uncertainty.The results are shown in Fig. A.1.The most striking feature of these average periodograms is that they recover the two peaks in the periodogram of the data, but the peaks are broader than the peaks in the periodogram of the data.This means that the quality factor Q of the oscillations in the rotation GPs underestimate the true values or, in physical units, the timescales of the oscillations are shorter than in the data.This is most critical for the TESS LC, where the Q of the second SHO GP is so low (and the corresponding timescale is so short) that the corresponding peak at ν ∼ 0.70 d −1 is completely flattened out.
A closer look at the periodograms shows a hint of a decreasing slope at low frequencies (ν < 0.3 d −1 ).This may indicate the presence of an additional correlated noise (other than the stellar rotation) with a PSD that decreases with ν.The presence of a slope at low frequencies is a challenge for the SHO GP and also for the rotation GP, as by construction their corresponding PSD is flat for frequencies lower than the peak frequencies.
We ran a few simulation tests by generating LCs using a rotation GP combined with an overdamped SHO, whose PSD monotonically decreases with ν (Foreman-Mackey et al. 2017).In order to save computing time, we fitted the LCs by means of least-square regression.We found that the presence of a neglected GP biases the fit of the rotation GP towards lower quality factors and shorter timescales.In the most critical cases, the fit of the rotation GP also diverged towards unrealistic timescales and/or amplitudes of the correlated noise.This can explain why David et al. (2019b) andFeinstein et al. (2022) use informative priors in their analysis.In the frequency domain, the fit of the simulated LCs leads to a rotation GP whose corresponding LS periodogram has broader peaks compared to the periodogram of the fitted LC.In other words, the best-fit rotation GP tries to include the spectral power due to the extra GP by broadening the rotation peaks.
In the domain of Fourier analysis, the assumption of a PSD that does not match the observations leads to a biased spectral decomposition of the signal.It is difficult to asses how this translates in the GP analysis framework, and any further detailed analysis is not the scope of this work.Nonetheless, we tested a different GP for two reasons.First of all, we aimed at a better representation of the correlated noise in the Kepler/K2 and TESS LCs in order to put our analysis on more solid ground.Secondly, we wanted to asses how the planetary parameters are affected by the choice of a different GP model.
We modelled the correlated noise using the S+LEAF python library (version 2.1.2Delisle et al. 2020Delisle et al. , 2022)).This library implements the Exponential-sine periodic (ESP) GP, an approximation of the quasi-periodic kernel, with the generic element of where σ ESP is the standard deviation of the GP, λ ESP is its timescale, ν ESP is the rotational frequency (≡ ν rot in Table 1), and η ESP is the scale of the oscillations, which sets how the spectral power is distributed among the harmonics of the periodic signal.
In the current implementation of this kernel it is necessary to set the number of harmonics to include through the nharm keyword: since the PSD of the data shows two predominant peaks (the fundamental and first harmonics), we manually set the nharm = 2.We initially tried to use this GP to fit by least-square regression the same stellar LCs discussed above, but we did not find any satisfactory results, mainly because the fit did not converge to a realistic set of GP parameters.Once again, some tests showed that a low-frequency power excess might prevent the ESP GP alone to properly model the correlated noise.We thus added an extra SHO GP in order to take into account the slope at low frequencies discussed above, obtaining a more meaningful set of GP parameters.
First of all, for the Kepler/K2 LC the ESP GP converged to σ ESP = 18 mmag, P ESP = ν −1 ESP =2.88 d and λ ESP =17 d, while for the TESS LC we obtained σ ESP = 11 mmag, P ESP =2.90 d and λ ESP =30 d.Thus, we recovered similar periodicities, timescales and amplitudes for the two LCs, small differences being likely due to differences in the bandpasses and/or different active stellar latitudes at the epoch of the observations.Most importantly, we derived a GP timescale approximately an order of magnitude longer than the rotation period.
The SHO GP converged to σ SHO =1.7 (3.3) ppm, ¶ SHO =1 (2) d and λ SHO =0.4 (0.2) d for the Kepler/K2 (TESS) LC.By means of Eqs.A.2-A.3, these parameters translate into Q=1.2 and Q=0.3 for the Kepler/K2 and TESS LC respectively.These two quality factors are close to the critical damping value of Q = 1/ √ 2, below which the PSD of the SHO GP becomes monotonically decreasing with ν.From a physical point of view, this corresponds to an overdamped GP that can be representative of low-frequencies stellar processes.As an example, the overdamped SHO GP with fixed Q = 1/2 is commonly used to model the surface granulation of convective stars (Harvey 1985;Kallinger et al. 2014).
Similarly to what we have done above, we used our bestfit composite GP to simulate 1000 LCs, using either the time sampling of Kepler/K2 or TESS.Then we computed the mean LS periodogram, obtaining the PSDs shown in Fig. A.1.Our GP model is able to better reproduce the two peaks in the PSD of the data and also provides a better match at low frequencies.For all the reasons discussed so far in this Appendix, in our combined analysis of the stellar correlated signal and the planetary transits (Sect.3.1 and 3.2) we adopted the mixture of a SHO and a ESP GP models.We also remark that we used uninformative priors on all the GP parameters (see Table 1) without any issue in the convergence of the MCMC chains.

Fig. 1 .
Fig. 1.Light curve of V1298 Tau collected on the night of 8 February 2022 with the Schmidt 67/92cm telescope at the INAF-Astrophysical Observatory of Asiago.The red dots corresponds to the binned light curve, and the green curve represents the transit model of planet e based on the ephemeris and transit depth derived by Feinstein et al. (2022).

Fig. 2 .
Fig.2.Bandpasses of Kepler/K2, TESS, and CHEOPS normalised to their peak values, as a function of wavelength.The cyan and red curves represent the spectral energy distributions of a T eff =5500 K and T eff =4500 K dwarf star, respectively (taken from https://www.cosmos.esa.int/web/cheops/performances-bandpass).
and Fig. 4 (orange dots), respectively.For comparison, we also show in Fig. 4 the TESS transit measured by Feinstein et al. (2022) (green data points), which has a lower depth.This discrepancy is due to both the different LC extraction methods and GP regression analysis, as it is evident looking at Fig. C.1.
(a)   The best-fit values are given as the median of the posterior distributions, and the uncertainties are given as the 16

Fig. 3 .
Fig. 3. Kepler/K2, TESS and CHEOPS LCs of V 1298 Tau from top to bottom row respectively.In each panel, the blue solid line shows the best-fit model.The arrows indicate the transits of the companion labelled as planet e by David et al. (2019b) and Feinstein et al. (2022), based on data from K2 and TESS.

Fig. 4 .Fig. 5 .
Fig. 4. Comparison of the detrended transit LCs discussed in this work as seen by Kepler/K2, TESS and CHEOPS.For each dataset, the best-fit transit model is shown by curves with the same colour as the data points.The green data points identify the detrended and flattened TESS light curve analysed by Feinstein et al. (2022), obtained from the jupyter notebook made publicly available by the authors at https://github.com/afeinstein20/v1298tau_tess/blob/main/notebooks/TESS_V1298Tau.ipynb

Fig. 6 .
Fig.6.Transit timing variations for the three transits observed by Kepler/K2, TESS, and CHEOPS assuming that they are all ascribable to V1298 Tau e.The observed epochs of central transit have been fitted with a linear function, resulting in a mean orbital period of 45.0 ± 0.1 days.

Fig. A. 1 .
Fig. A.1.Analysis of the PSD of the Kepler/K2 (left) and TESS (right) LCs.In each panel, the black solid line shows the PSD of the LC after masking the planetary transits.The two vertical dashed lines mark the frequencies 0.35 d −1 and 0.70 d −1 .The confidence band in orange shows the average periodogram obtained using the rotation GP, as discussed in the text.Similarly, the blue confidence band corresponds to the GP model used in this work.

Fig. B. 1 .
Fig. B.1.Portions of Kepler/K2 and TESS LCs showing a transit of planet V1298 Tau b (left and right panel respectively).The plots at the top show the undetrended LCs analysed in our work centred on the transit (black dots), and the data after removing the best-fit planetary transit (red dots) to show the GP signal during the transit timespan.The plots at the bottom show the transit signals after subtracting the best-fit GP model.The corresponding detrended and flattened TESS light curve analysed by Feinstein et al. (2022) is shown for comparison (green cross symbols).It has been obtained from the jupyter notebook made publicly available by Feinstein et al. (2022) at https://github.com/afeinstein20/v1298tau_tess/blob/main/notebooks/TESS_V1298Tau.ipynb

Fig. C. 1 .
Fig. C.1.Comparison between our extracted TESS LC (orange dots) showing the transit attributed to V1298 Tau e, and the one extracted by Feinstein et al. (2022) (blue dots), including the corresponding best-fit GP models (solid curves).

Fig
Fig. C.2. Undetrended LCs of the three transits analysed in our work and shown in Fig. 4 (Kepler/K2, TESS, and CHEOPS), with overplotted our calculated best-fit GP models.An offset has been added to each curve in order to improve the clarity of the plot.

Fig
Fig. C.3.Temporal evolution of the orbital period of V1298 Tau e, assuming that K2/Kepler, TESS, and CHEOPS observed a transit of the same planet.This illustrative result is based on a few simulated N-body integrations covering a time span of five years, as described in Section 4. Each curve in the plot represents a simulation.

Table 1 .
(using the emcee Markov Chain Monte Carlo sampler) to search for a transit of planet e that was predicted to occur by assuming the orbital period P e = 45.0003d.Except for the shallow transit of V1298 Tau c (ingress: 2459976.28146;centre: 2 459 976.37856; egress: 2 459 976.47566BJD TDB , based on the ephemeris of Feinstein et al. 2022; depth ∼ 0.1%) no other transit event was predicted in that observing window.To model A&A proofs: manuscript no.V1298Tau_CHEOPS Model parameters for the fit of the Kepler/K2 and TESS data.

Table 2 .
Model parameters for the fit of the CHEOPS data.a