Validation of selected TESS exoplanetary candidates

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Introduction
The Transiting Exoplanet Survey Satellite (TESS, Ricker et al., 2015) is a NASA space all-sky survey that continuously monitors strips of the sky for ∼ 27 continuous days (i.e.sectors) alternating between the north and south hemispheres.The cadence of observation was at first 30 min, 2 min and later 20 s for selected targets.The detector bandpass spans from 600 -1040 nm and is centred on the traditional Cousins I-band (λ eff = 786.5 nm).The red part is preferred for increased sensitivity to small planets transiting in front of cool, red stars, but lacks the colour information of standard photometric filters.
We have selected six TESS objects of interest (TOIs) that were still considered candidates for exoplanets (see Tab. 1) in 2021.The selection criteria had  (Jensen, 2013) and coordinated observers who were part of ETD (Poddaný et al., 2010), ExoClock (Kokori et al., 2022), and MuSCAT2 (Narita et al., 2019) projects (see Acknowledgements).This article is based on results of the master thesis by Mesarč (2023).

Reduction
The available TESS data of our targets covered 2-5 sectors per object during the September 2019 -December 2022 time frame.In total, 316 individual transits were identified.TESS data were reduced independently of the pipeline first by TPF (Target Pixel File) identification using Python package lightkurve (Lightkurve Collaboration et al., 2018) with custom pixel aperture for each TOI using tpfplotter (Aller et al., 2020).Each FOV was individually visually inspected for nearby transit-like variable signals in each pixel of the selected aperture and close vicinity.Individual sectors (with 20 s, 2 min, and 30 min cadence) were treated separately.Outliers were removed by Sigma clipping, transits were identified and incomplete ones were omitted for further analysis.For TOI-3856.01 we also had to correct for momentum dumps.Afterwards, our custom-mask flux was de-trended using exoplanet package (Foreman-Mackey et al., 2021).Follow-up observations took part from May 2021 to April 2022.In total, 136 transits were observed at 27 sites with the help of 33 observers in the campaign.Telescopes with diameters 150-1520 mm were used.Various observers had different photometric passbands at their disposal: Johnson BV RI, Sloan g r i z , or filter-less clear or luminance band.
For the basic photometric reduction, the HOPS software (Tsiaras, 2019) was used.This allowed for the use of multiple check stars with a circular aperture automatically determined by the stellar FWHM.In each field, we have selected 4-7 check stars so that they show no signs of variability.A standard dark and flat calibration of all frames was performed.From the check stars available in each FOV, an artificial star was constructed to do the relative aperture photometry.

Modelling of transits
The total de-trended TESS light curve (LC) was analysed with a box-fitting least-square algorithm by Kovács et al. (2002) to find period P and starting epoch T 0 for the transit ephemeris.This was used to predict follow-up groundbased observations.We have constructed the phased TESS LC and selected the region ±4 hours around the mid-transit.
Using the mean transit shape, we were able to find more precise times of mid-transit for 30-minute TESS cadence even when each transit was covered by only a handful of points.The data was used to construct an O − C diagram.
TESS transit model was calculated using the pyaneti code by Barragán et al. (2019).The code utilises the MCMC method to compute transit parameters: the planet-star radius ratio R P /R S , the ratio of the semi-major axis to the stellar radius a/R S , inclination i of the orbital plane (or the impact parameter b), duration t T .The limb-darkening coefficients were adopted using a parametrization proposed by Kipping (2013).
The ground-based transits were modelled by the HOPS package (Tsiaras et al., 2016).The necessary stellar limb-darkening coefficients were calculated with the ExoTETHys package (Morello et al., 2020) for each passband using the nonlinear law by Claret (2000) and the stellar parameters were taken from NASA Exoplanet Archive.

Results
Since the TESS detector has 21-arcsec pixel scale (see Ricker et al., 2015), we had to check if there are more light sources at the target TOIs position below this resolution.We have used publicly available speckle polarimetry from SAI 2.5-m telescope (Safonov et al., 2017), Alopeke Speckle Instrument at Gemini-North telescope (Howell et al., 2016;Howell & Furlan, 2022), and PHARO instrument at Palomar 5-m telescope (Hayward et al., 2001) of angular resolution in range of 20-89 mas.Only in the case of TOI 3604.01,there was a visible detection of another source within 0.3 arcsec off the target.Some transit parameters from independent modelling of confirmed exoplanets during this work are listed in Tab. 2 and as of date non-confirmed candidates are in Tab. 3. We have additionally revised the linear ephemerides (Tab.4) by incorporating every observed minimum.
Ground-based observations taken in different filters were analyzed separately.Parameters R P /R S , a/R S , and i were found for each passband.The relation between R P /R S and λ eff of filter shows "jumps" similar to rigorous models (see e.g.Fortney et al., 2010).TOI-1834.01 is an exception to this, which seems to have a monotonous function R P /R S ∝ λ eff but only from V R + r i passbands.It is also the only system with a periodic O −C signal, with ∼ 10-min amplitude and a period of 80.3501 ± 0.3149 days.
Our multi-colour ground-based photometry allowed us to construct a differential colour transit LC.While no sufficiently precise spectroscopy was available for us, we still wanted to limit the possibility of false detection by investigating the limb-darkening profile of the transiting body.If the second component is a star on an inclined orbit, it would produce shallow grazing eclipses.However, a star has a different limb-darkening profile than an exoplanet with a night side facing the observer.According to Tingley (2004), exoplanetary transits produce a double-horned profile in differential colour transit LC while a close binary shows no such feature.
Only a handful of our ground-based transits were observed in two or more filters during a single night.We have constructed a model LC for each colour.Then, by subtracting any two models we created an artificial colour LC in arbitrary phases to cover the full transit.The more distant the effective wavelength of bands, the higher the signal.We have used all identified transits from TESS as well as ground-based follow-up to construct the O − C diagram.Except for TOI-1834.01(Fig. 1), no other target shows a meaningful signal.The most notable part of O − C is taken during TESS sector 49 (February 26 -March 26, 2022).We have modelled this part and the whole O − C using the code OCFit by Gajdoš & Parimucha (2019).We have fitted a simple 3-body scenario with MCMC resulting in P 3 = 80.35013±0.31488d on a short (a 3 sin i 3 = 1.161±0.157au) eccentric (e 3 = 0.265 ± 0.135) orbit.It is a similar solution to that of Czavalinga et al. (2023) which was published shortly before submitting the thesis.The problem is that considering the total O − C amplitude is caused solely by LITE leads to unrealistic large mass function f (m) = 32.33M .
We have also tried to model the transit of TOI-1834.01as a grazing binary with the code RMF by Garai et al. (2020).The simplest case was to assume M 2 /M 1 = R 2 /R 1 and fit a model spectrum to the spectral energy distribution fixed around VizieR photometry.The best-fitting temperature was 5200 K corresponding to K0 spectral type or ∼ 0.79 M , ∼ 0.85 R (based on tables by Cox (2000)).We can use equations 8 and 12 from Borkovits et al. (2016) to estimate amplitudes of LTTE and dynamical effects (for M 1 = M 2 = M 3 ) as A LT T E = 78.68s and A dyn = 567.38 s, respectively.This is already enough to explain the observed 10-minute amplitude of O − C. TOI-3604.01 is the only target resolved by speckle interferometry in near-IR into two sources of similar apparent luminosity separated by 0.3 arcsec.We have thus removed a constant 50% of total TESS out-of-transit flux and constructed a new model (B in Tab. 3) from the normalized treated flux.In this case, the supposed exoplanet orbiting one of the two stars had to be ∼ 30 % larger than initially assumed, making it more likely to be a low-mass star.The parent object is also a very active red dwarf.Lambert et al. (2023) found maybe a similar (by R P /R S ) object TOI-5375.01which hosts a brown dwarf/low-mass star at hydrogen-burning limit with ∼ 83.8 ± 2.1 M J .TOI-3856.01 was at the time of this work not independently analyzed.The transit model based on ground photometry gives R P /R S ∈ [0.112, 0.128] in seven different passbands (c.f. with Tab. 3).With stellar radius R = 1.12864R (taken from the EXOFOP page), this gives the candidate radius as R P ∈ [1.26, 1.44] R J .The U-shaped single colour transits and double-horned differential colour transit suggest this object may be a hot Jupiter on ∼ 0.034 au orbit.

Conclusion
We have selected six exoplanetary candidates as indicated by the TESS pipeline.From TESS photometry, we modelled their transits and got independent solutions for TOIs 1518b, 2046b, and 2109b.The TOI-1834.01 was independently found to be most probably a hierarchical triple star system of late K-type stars.Based on the corrected flux in TESS aperture, we found TOI-3604.01 to be probably a very low-mass star close to the hydrogen-burning limit.Based on our analysis, TOI-3856.01appears to be a sub-stellar object.

Table 1 .
Preliminary transit properties of target TOIs at the start of the campaign: host star magnitudes in TESS and V bands, normalized depth of transit ∆F , transit duration tT , and period P .
to ensure observability in Central Europe (Dec > −10 deg), a higher occurrence of transits per night (P < 2.2 d, t T < 3 h), and bright host stars (V ≤ 12.5 mag) with deep transits (∆F ≥ 0.65 %) enough for sub-metre telescopes.We planned the observation with TransitFinder

Table 2 .
Comparison of our results to those models from papers published during the preparation of the master thesis.Individual papers are: C21 = Cabot et al. (2021), K22 = Kabáth et al. (2022), and W21 = Wong et al. (2021).

Table 3 .
Our results of assumed exoplanet transit models of TOIs without known paper published before May 2023.Note: for TOI-3604.01,model A treats all light as a single star in TESS pixel, while model B assumes half of the observed flux as contamination.

Table 4 .
Updated periods (P ) and reference time of primary transit (T0).