Simultaneous multicolour transit photometry of hot Jupiters HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b

Accurate physical parameters of exoplanet systems are essential for further exploration of planetary internal structure, atmo-spheres, and formation history. We aim to use simultaneous multicolour transit photometry to improve the estimation of transit parameters, to search for transit timing variations (TTVs), and to establish which of our targets should be prioritised for follow-up transmission spectroscopy. We performed time series photometric observations of 12 transits for the hot Jupiters HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b using the simultaneous four-colour camera MuSCAT2 on the Telescopio Carlos Sánchez. We collected 56 additional transit light curves from TESS photometry. To derive transit parameters, we modelled the MuSCAT2 light curves with Gaussian processes to account for correlated noise. To derive physical parameters, we performed EXOFASTv2 global fits to the available transit and radial velocity data sets, together with the Gaia DR3 parallax, isochrones, and spectral energy distributions. To assess the potential for atmospheric characterisation, we compared the multicolour transit depths with a flat line and a clear atmosphere model. We consistently refined the transit and physical parameters. We improved the orbital period and ephemeris estimates, and found no evidence for TTVs or orbital decay. The MuSCAT2 broadband transmission spectra of HAT-P-19b and HAT-P-65b are consistent with previously published low-resolution transmission spectra. We also found that, except for HAT-P-65b, the assumption of a planetary atmosphere can improve the fit to the MuSCAT2 data. In particular, we identified HAT-P-55b as a priority target among these four planets for further atmospheric studies using transmission spectroscopy.


INTRODUCTION
Since the discovery of the first exoplanet around a Sun-like star (Mayor & Queloz 1995), more than 5500 exoplanets have been found, three-quarters of them by transit.When an exoplanet transits its host star, part of the starlight is blocked by the planets in the line of sight of the observer, and the starlight passes through the day-night terminator of the exoplanet's atmosphere.With the flux variation of the planetary system, the transit parameters of the system can be calculated from the transit light curves, which carry information about the internal structure and formation process of the exoplanets (Seager & Mallén-Ornelas 2003;Fortney et al. 2007;Mordasini et al. 2016).Because the atmospheric opacity varies in different passbands, the properties of the planetary atmosphere can be studied through transmission spectra (Seager & Sasselov 2000;Charbonneau et al. 2002), potentially linking the atmospheric chemistry to the planet's formation history and habitability (Madhusudhan et al. 2014(Madhusudhan et al. , 2016;;Mordasini et al. 2016).
Meaningful investigations of planetary internal structure, atmospheric properties, atmospheric evolution, formation, and migration histories all require precise determination of orbital and physical parameters for the planetary systems as input.In particular, precise transit parameters, together with the latest parallaxes provided by Gaia and additional constraints from spectral energy distributions and stellar evolution models, can improve the estimates of the physical parameters.In the past decade, simultaneous multi-channel imagers, such as GROND (Greiner et al. 2008) and MuSCAT1/2/3 (Narita et al. 2015(Narita et al. , 2019(Narita et al. , 2020)), have been extensively used to conduct multicolour follow-up transit photometry.The simultaneous multicolour capability not only allows precise measurements of colour-independent transit parameters to revise physical parameters and to search for transit timing variations (TTVs), but also helps validate candidate planets orbiting faint stars, constrain starspot properties and stellar obliquity, and provide a preliminary assessment of planetary atmospheres (e.g., Mancini et al. 2014;Chen et al. 2014;Parviainen et al. 2019).
To refine orbital and physical parameters for the known hot Jupiter systems and to prioritise targets for future spectroscopic atmospheric characterisation, we initiated a multicolour transit photometry observing campaign using the four-colour simultaneous camera MuS-CAT2 (Narita et al. 2019) in the , , , and   bands.In previous studies, we have found evidence for scattering features in the atmospheres of the hot Jupiters WASP-74b (Luque et al. 2020) and WASP-104b (Chen et al. 2021a) by combining the MuSCAT2 photometric measurements with those from transit spectrophotometry.Here, we present the MuSCAT2 transit observations for four Saturnmass hot Jupiters, HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b, complemented by archival data from the Transiting Exoplanet Survey Satellite (TESS).
HAT-P-65b was discovered by Hartman et al. (2016), with a mass of 0.53  J , a radius of 1.89  J , and an equilibrium temperature of 1930 K, orbiting a  = 13.1 mag star (1.21  ⊙ , 1.86  ⊙ ,  eff = 5835 K, and [Fe/H] = 0.10) every 2.605 days.Based on two transits with OSIRIS at the GTC, Chen et al. (2021b) reported the detection of TiO and possible evidence for Na and VO in the atmosphere of HAT-P-65b.
This paper is organised as follows.In Section 2, we summarise the transit observations and data reduction procedures.In Section 3, we describe the light-curve analysis for MuSCAT2 and TESS, and present the derived transit parameters and orbital period.In Section 4, we perform the global modelling to refine the physical parameters of the planetary systems.In Section 5, we discuss the wavelength dependence of the transit depth for future atmospheric characterisation.Finally, we draw conclusions in Section 6.
We reduced the MuSCAT2 data with customised IDL scripts as detailed in Chen et al. (2021a).In brief, we corrected bias, dark and flat field from the raw images, and performed aperture photometry using the APER routine from DAOPHOT 1 .We extracted the central time of each exposure and converted it to Barycentric Julian Dates in Barycentric Dynamical Time (BJD TDB ; Eastman et al. 2010).We determined the best aperture radius by minimising the light-curve scatter among a grid of radii ranging from 3 to 32 pixels (equivalent to 1.3 ′′ -14.1 ′′ ).We also tested different combinations of reference stars to produce the best synthetic reference light curve that minimises the light-curve scatter.The final chosen aperture radii are listed in Table 1.

TESS photometry
To enlarge the mid-transit time dataset for the refinement of the orbital period and ephemeris, we made use of the archival transit observations conducted by the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015).
-For HAT-P-19b, six transits with an exposure time of 1800 s were observed in full-frame images in Sector 17 between October 8 and November 2, 2019.Another seven transits with an exposure time of 20 s were observed in target pixel files in Sector 57 between September 30 and October 29, 2022.Notes: a The first and third values refer to the airmass at the beginning and end of the observation.The second value gives the minimum airmass.
-For HAT-P-51b, six transits with an exposure time of 1800 s were observed in full-frame images in Sector 17 between October 8 and November 2, 2019.Another six transits with an exposure time of 120 s were observed in full-frame images in Sector 57 between September 30 and October 29, 2022.
-For HAT-P-55b, thirteen transits with an exposure time of 120 s were observed in target pixel files in Sector 25 and 26 between May 14 and July 4, 2020.Another thirteen transits with an exposure time of 120 s were observed in target pixel files in Sector 52 and 53 between May 25 and July 8, 2022, of which two transits were not used in the subsequent analysis due to incomplete transit coverage.
-For HAT-P-65b, ten transits of HAT-P-65b with an exposure time of 120 s were observed in target pixel files in Sector 55 in August 5 and September 1, 2022, and three transits of HAT-P-65b were not used in the subsequent analysis due to bad data quality.We used the Python package lightkurve (Lightkurve Collaboration et al. 2018) to download the observation data from the MAST data archive 2 .For HAT-P-19b and HAT-P-51b, the raw light curves in 2019 were created from the tesscut product from full-frame images, and the SPOC light curves (Jenkins et al. 2016) were used in 2022.For HAT-P-55b, all the raw light curves were created from the target pixel file from tess phot.For HAT-P-65b, the SPOC light curves were used.The adopted time windows were three times the transit duration from the expected transit centre for data with an exposure time of 1800 s, one and a half times for data with 120 s, and one times for data with 20 s.The raw light curves were normalised by the decile value of the whole TESS raw flux of all transits.

LIGHT-CURVE ANALYSIS
We modelled the raw light curves with the transit model from Mandel & Agol (2002) using the Python package batman (Kreidberg 2015): (1) The free parameters of the transit model consist of radius ratio   / ★ , mid-transit time  mid , orbital inclination , orbital semimajor axis in units of the stellar radius / ★ , the quadratic limbdarkening coefficients (LDCs)  1 and  2 .Circular orbits are adopted for HAT-P-51, HAT-P-55, and HAT-P-65.For HAT-P-19, the orbital eccentricity and argument of periastron were fixed to 0.084 and 256 deg (Hartman et al. 2011), respectively.The orbital period is fixed to literature values in the transit model.For the planetary systems which are diluted by an unresolved companion star, the transit model is revised to  * (; ,  c ) to account for the flux dilution from the companion: where (; ) is the transit model without the dilution effect,  c is the companion-to-target flux ratio.We employed Gaussian processes (GP) to account for the correlated noise present in the light curves, which was first introduced to transmission spectroscopy by Gibson et al. (2012).The onedimensional GP regression was performed by the Python package celerite (Foreman-Mackey et al. 2017), which accepts time series as the input vector.The GP mean function was described by the transit model multiplied by a linear polynomial baseline function ().The GP covariance matrix  was described by a combined kernel which consisted of an approximated 3/2-order Matern kernel for time-correlated red noise and a jitter kernel for underestimated white noise: where  = |  −   | is the distance between two data points in time,  and  2 1 are the length and variance scales of systematic noise,  2 2 is the variance of underestimated white noise.
We performed the affine invariant Markov Chain Monte Carlo (MCMC) ensemble sampler using the Python package emcee (Foreman-Mackey et al. 2013) to explore the posterior probability distributions of the free parameters.We adopted uniform priors for most of the transit parameters, polynomial coefficients for the baseline function, and log-uniform priors for the GP hyperparameters.We imposed normal priors on the LDCs  1 and  2 (see Table A1).We calculated LDCs from the ATLAS stellar atmosphere models by interpolating in the model grids using the stellar parameters (Espinoza & Jordán 2015).We ran two short chains for the burn-in phase and one long to ensure convergence chain for formal production.

MuSCAT2 light curves
For the MuSCAT2 light curves, we adopted the baseline function () in the form of: where  and  are the coordinates of the target,  is the full width at half maximum of the target's point spread function.The mean function of GP was  * (; ,  c )() for HAT-P-65 and (; )() for the others.HAT-P-65 has a background star located at 3.6 ′′ in the west according to Hartman et al. (2016).The background star could not be spatially resolved by the defocused MuSCAT2 observations.Therefore, we estimated the companion-to-target flux ratios  c within the MuSCAT2 passbands, which were 0.0086, 0.0094, 0.0098, 0.0101 for , , , and   , respectively, based on the GTC OSIRIS measurements presented in Chen et al. (2021b).We performed two runs of light-curve modelling for each target.In the first run, we aimed to derive the common transit parameters.We jointly fitted multicolour light curves on a nightly basis.Each night had the same values of , / ★ , and  mid for all light curves, and each light curve had independent values of   / ★ ,  1 , and  2 .The coefficients of the baseline function and the GP hyperparameters were always light-curve-dependent.We reported the weighted mean of , / ★ , and   / ★ of all nights as the final updated values in Table 2.The detrended MuSCAT2 light curves and best-fit residuals are shown in Fig. 1.
In the second run, we attempted to evaluate the potential variation in transit depth as a function of wavelength.We fitted each light curve individually and fixed the values of , / ★ , and  mid to those obtained in the first run.The free parameters were   / ★ ,  1 ,  2 , baseline function coefficients, and GP hyperparameters.For each passband, the weighted mean of   / ★ was taken as the final value and listed in Table 3.

TESS light curves
For the TESS light curves, we adopted the baseline function () in the form of: where  and  are the coordinates of the target.The diluted transit model  * (; ,  c ) was adopted to account for the potential dilution from nearby stars given the large pixel size of TESS.Therefore,  * (; ,  mid )() was adopted as the GP mean function for all the targets.For HAT-P-19b and HAT-P-51b, we used the supersampling feature of batman to account for the long cadence smearing effect (Kipping 2010).Since our purpose of modelling the TESS light curves was to measure the mid-transit times, we fixed the radius ratio   / ★ , the inclination , and the semi-major axis / ★ to the values obtained from the analysis of the MuSCAT2 light curves.We also fixed the limb darkening coefficients to the pre-calculated values derived from the code of Espinoza & Jordán (2015).Each light curve had an independent mid-transit time.The detrended TESS light curves and their best-fit residuals are shown in Fig. A1.

Transit parameter refinement
Based on MuSCAT2's light curve analysis of all nights and four passbands in the first run, we are able to refine the transit parameters for the four hot Jupiter systems, which are shown in Table 2 along with literature values for comparison.
For HAT-P-19b, HAT-P-51b, and HAT-P-55b, the transit parameters are derived from at least three MuSCAT2 transits, resulting in smaller uncertainties than those reported in the literature.In the case of HAT-P-65b, only two transits were observed by MuSCAT2, and only one of them covered the entire transit event.The uncertainties of  and / ★ are slightly larger than those derived from two GTC transits (Chen et al. 2021b), but still consistent with the latter.
However, the transit parameters measured in different studies are not exactly in agreement.This discrepancy is likely due to the degeneracy between  and / ★ , since the measurements from different studies show a correlation trend (i.e., larger  with larger / ★ ) consistent with the -/ ★ degeneracy.Thanks to the multiple observations and the wide wavelength coverage of MuSCAT2, the transit parameters can be tightly constrained, with colour-dependent bias being eliminated.

Orbital period determination
We derived the mid-transit time of each transit for all the MuSCAT2 and TESS observations.To investigate the transit timing variations and to improve the orbital ephemeris, we also collected other midtransit times published in the literature.All the mid-transit times have been converted to the BJD TDB standard and presented in Table A3.
• For those with raw light curves available (Hartman et al. 2011(Hartman et al. , 2015(Hartman et al. , 2016;;Chen et al. 2021b), we recalculated their mid-transit times using our light curve analysis method.
• For HAT-P-19b, we did not include the mid-transit times in Baştürk et al. (2020), which have very small error bars and show a general downward offset from the linear ephemeris derived from the other times.
• For HAT-P-51b, we discarded the mid-transit time of the 2018-11-23 transit because the computer time of that night was not properly synchronised with the Network Time Protocol server.
The mid-transit times  mid were fitted as a function of the epoch  using a linear model and a quadratic model, respectively.For the linear model, the planet was assumed to have a constant orbital period : where  0 is the mid-transit time at zero epoch.The zero epoch was optimised to give the smallest error bar for  0 .For the quadratic model, the planet was assumed to have a decaying orbital period: where d/d is the decay rate between successive transits.We used the Bayesian Information Criterion (BIC =  2 +  log ) to perform model comparison, where  is the number of free parameters and  is the number of data points.
The results of the model comparison are shown in Table 4.For the four planetary systems, the difference between the constant period model and the orbital decay model ΔBIC is −3.26, −2.64, −2.34, and −2.11, respectively.The constant period model is favored with a lower BIC value in all four planetary systems, indicating that there is no evidence for orbital decay.Therefore, our results do not support the claims of potential orbital decay in HAT-P-19b (Hagey et al. 2022) and HAT-P-51b (Yeh et al. 2024).Meanwhile, the timing residuals of the four systems with the best-fit period model show no sign of transit timing variation (Fig. 2).The refined period and reference ephemeris are given in Table 2.

PHYSICAL PROPERTIES
Except for HAT-P-19, no follow-up physical property determinations have been made for these planetary systems.To refine their physical parameters, we used the IDL package EXOFASTv2 (Eastman et al. 2019) to perform a global modelling of the MuSCAT2 transit light curves, the radial velocity (RV) measurements from the literature, the isochrones from the MESA Isochrones and Stellar Tracks (MIST; Dotter 2016), and the spectral energy distribution (SED) from broadband photometry.The use of the MIST stellar evolutionary models produces consistent models for both isochrones and SED.The latest stellar parallax from the Gaia third data release (DR3; Gaia Collaboration 2022) provides an accurate prior on the stellar distance, which places a tight constraint on the stellar radius in the SED model.The collected broadband photometry and Gaia DR3 stellar parallax are listed in Table A2.

VARIATION OF TRANSIT DEPTH WITH WAVELENGTH
Based on the MuSCAT2 data, we measured a difference between the maximum and minimum transit depths of 463 ± 293 ppm, 525 ± 424 ppm, 887 ± 370 ppm, and 1138 ± 758 ppm for HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b, respectively, corresponding to 1.8 ± 1.1, 1.8 ± 1.5, 4.8 ± 2.0, and 5.4 ± 3.6 times the transit depth variation caused by one atmospheric scale height.The atmospheric scale height,  =  B  eq /( p ), is estimated to be 0.0072, 0.0093, 0.0061, and 0.0107  p , where  B is the Boltzmann constant,  eq is   .Broadband transmission spectra of the four hot Jupiters (HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b) along with the retrieved atmospheric models.For HAT-P-19b and HAT-P-65b, the retrievals were performed on the MuSCAT2 and OSIRIS combined dataset, with the assumption of patchy clouds and equilibrium chemistry.The OSIRIS data for HAT-P-19b and HAT-P-65b were obtained from Mallonn et al. (2015) and Chen et al. (2021b), respectively.For HAT-P-51b and HAT-P-55b, due to limited data points, the retrievals adopted a simplified assumption of cloud-free solar atmosphere.
the planetary equilibrium temperature,  p is the planetary surface gravity, and  = 2.3 g mol −1 is the mean molecular weight.
The transit depths measured by MuSCAT2 in the , , ,   bands sample a broadband transmission spectrum for each planet.In particular, MuSCAT2's ability to perform simultaneous multicolour photometry eliminates the impact of stellar rotational modulation, allowing us to take a first look at the planetary atmospheres.The optical transmission spectrum is sensitive to both optical absorbers (such as alkali metals and metal oxides) and particle sizes of scattering sources.However, the broad band averages the expected spectral features resulting from the planetary atmosphere, making it difficult to unambiguously distinguish the opacity sources of origin.Instead of using the broadband transmission spectra to infer atmospheric properties, we tried to answer which targets have a higher priority for follow-up transmission spectroscopy.
Assuming that the variation in transit depth is potentially caused by the planetary atmosphere, we performed a simplified Bayesian spectral retrieval analysis on the MuSCAT2 broadband transmission spectra.Two model hypotheses were considered, including a flat model and a planetary atmosphere model.The flat model has a constant planetary radius as the only free parameter.The planetary atmosphere model assumes a clear atmosphere of solar composition (C/O = 0.55, log / ⊙ = 0) in chemical equilibrium, consisting of two free parameters, the planetary radius at 10 mbar ( 10mbar ) and the isothermal temperature ( iso ).We used petitRADTRANS (Mollière et al. 2019) to create the planetary atmosphere model, and Py-MultiNest (Buchner et al. 2014) to implement the multimodal nested sampling (Feroz & Hobson 2008;Feroz et al. 2009) for parameter estimation.
Figure 6 presents the MuSCAT2 broadband transmission spectrum along with the retrieved atmosphere models.Compared to the flat model, the atmosphere model resulted in decreasing reduced chi-square values for HAT-P-19b (from 1.18 to 1.08), HAT-P-51b (from 0.58 to 0.10), and HAT-P-55b (from 2.25 to 0.24), but an increasing value for HAT-P-65b (from 1.08 to 1.70), indicating that the atmosphere model provides a better fit than the flat model for the first three planets.We also calculated the log-evidence to compare these two models, and obtained Δ ln Z(= ln Z atmos − ln Z flat ) values of −1.2 ± 0.1, −0.2 ± 0.1, 2.5 ± 0.1, and −0.5 ± 0.1 for HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b, respectively.Therefore, HAT-P-55b is the only planet with moderate evidence in the Bayesian framework that a planetary atmosphere is required to explain the data, making it a priority target for future follow-up spectrophotometric observations.Of the four planets, two have been observed for optical transmission spectra prior to our MuSCAT2 observations.Mallonn et al. (2015) obtained a flat featureless spectrum for HAT-P-19b using the R2500R grism of GTC's OSIRIS spectrograph, while Chen et al. (2021b) reported the detection of TiO and the possible detection of Na and VO in the atmosphere of HAT-P-65b using the R1000R grism of GTC OSIRIS.Therefore, we also performed retrievals on the combined MuSCAT2 and GTC dataset for HAT-P-19b and HAT-P-65b.In this case, we adopted a more complicated planetary atmosphere model because more data points were available.The model assumes an isothermal atmosphere at a temperature of  iso in chemical equilibrium controlled by the metallicity  and the C/O ratio with a clear and a cloudy sector.The cloudy sector has a cloud fraction of , a cloud top at pressure  cloud , and a scattering amplitude  scatt times that of H 2 Rayleigh scattering.To account for the offsets introduced by different orbital parameters in deriving the transit depth and different instrumental systematics, the GTC OSIRIS spectra were allowed to have a free offset in the retrieval.
Table 6 presents the retrieved parameters based on the combined MuSCAT2 and GTC dataset for HAT-P-19b and HAT-P-65b.For HAT-P-19b, the atmospheric metallicity tends to be super solar.For HAT-P-65b, the retrieved parameters agree well with those of Chen  (2021b).Unfortunately, due to the lack of infrared wavelengths that cover sufficient molecular spectral features to characterise atmospheric chemistry and cloud altitude, it is difficult to constrain the parameters other than temperature, reference radius, and instrumental offset.Future transmission spectroscopy conducted with the James Webb Space Telescope, together with the current optical transmission spectra, should be able to place more meaningful constraints on the atmospheric metallicity, cloud properties, and relative elemental ratios, paving the way for tracing planetary formation and migration histories (e.g., Öberg et al. 2011;Madhusudhan et al. 2014;Mordasini et al. 2016;Lothringer et al. 2021;Ohno & Fortney 2023).

CONCLUSIONS
We performed simultaneous multicolour photometric observations of the transiting exoplanet systems HAT-P-19, HAT-P-51, HAT-P-55, and HAT-P-65 with the MuSCAT2 camera on the 1.52 m TCS telescope.We observed 12 transits for the four planets and obtained a total of 43 MuSCAT2 transit light curves.The transit parameters were revised based on the MuSCAT2 multicolour transit light curves.
We also collected light curves for 56 transits from the TESS photometry, and combined the TESS timings with MuSCAT2 and literature timings to improve the orbital period and ephemeris estimates.We then consistently refined the physical parameters of these planetary systems by performing EXOFASTv2 global fits to the MuSCAT2 transit data, archival RV data, Gaia DR3 parallax, isochrones, and broadband spectral energy distributions.Finally, we investigated the potential for atmospheric characterisation using the MuSCAT2 multicolour transit depths for these four hot Jupiters.Our conclusions can be summarised as follows: -We have improved the transit parameter estimates for HAT-P-19b, HAT-P-51b, and HAT-P-55b, with smaller uncertainties than previous studies.The MuSCAT2 uncertainties for HAT-P-65b are slightly larger than those derived from the very precise GTC observations.
-We have consistently refined the physical parameters for all four planetary systems based on the improved transit parameters, which were derived from MuSCAT2 and GTC for HAT-P-65b, but only from MuSCAT2 for the other three.All the stellar and planetary radii are more tightly constrained than in previous studies, with typical relative errors of less than 2%.
-We have improved the orbital period and ephemeris estimates for all four planetary systems.All of them are consistent with linear ephemeris.No significant transit timing variations or evidence of orbital decay were found.Based on our results, the typical uncertainties of the predicted mid-transit time by mid-2035 would be 33, 84, 77, and 104 seconds for HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b, respectively, which are reasonably precise even in the ARIEL era.
-We have found that planetary atmosphere models can improve the fit to the MuSCAT2 broadband transmission spectra of HAT-P-19b, HAT-P-51b, and HAT-P-55b compared to a flat line based on  2 statistics.However, in terms of Bayesian model statistical significance, only HAT-P-55b shows (moderate) evidence of the presence of a planetary atmosphere.This makes HAT-P-55b a priority target for future transmission spectroscopy.this work will be made available at the CDS (http://cdsarc.u-strasbg.fr/).Notes.a These mid-transit times have been recalculated using our light-curve analysis method.

Figure 1 .
Figure 1.MuSCAT2 multicolour transit light curves of HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b.The first and third panels show the light curves after removal of the systematics.The second and fourth panels show the best-fit residuals.The black solid lines show the best-fit model, and the navy circles show the 15-min binned points.

Figure 2 .
Figure 2. Timing residuals of four systems with the constant period model.Each data point is the difference between the observed mid-transit time and the best-fit linear model.The middle dashed line is the zero line, the other two dashed lines show the range of 1 uncertainty.The inset shows the zoomed view of the residuals of the TESS light curves.The crossed points were discarded in the period modelling.

Figure 3 .
Figure 3. Spectral energy distributions (SED) of HAT-P-19, HAT-P-51, HAT-P-55, and HAT-P-65 from broadband photometry.The red data points with error bars are the broadband photometric measurements.The blue circles are the best-fit SED model values.

Figure 6
Figure6.Broadband transmission spectra of the four hot Jupiters (HAT-P-19b, HAT-P-51b, HAT-P-55b, and HAT-P-65b) along with the retrieved atmospheric models.For HAT-P-19b and HAT-P-65b, the retrievals were performed on the MuSCAT2 and OSIRIS combined dataset, with the assumption of patchy clouds and equilibrium chemistry.The OSIRIS data for HAT-P-19b and HAT-P-65b were obtained fromMallonn et al. (2015) andChen et al. (2021b), respectively.For HAT-P-51b and HAT-P-55b, due to limited data points, the retrievals adopted a simplified assumption of cloud-free solar atmosphere.

Table 2 .
Derived transit parameters and orbital ephemeris.
a Mid-transit times in BJD UTC .

Table 3 .
Chromatic radius ratios of four hot Jupiter systems.

Table 4 .
Comparison of the constant period model and the orbital decay model.

Table A3 .
Mid-transit times of the four hot Jupiters.