In this work we aim to measure \(A_g\) and \(\epsilon\) for the planet WASP-178 b, a highly irradiated giant planet with an estimated equilibrium temperature of 2450 K. We analyzed archival spectra and the light curves collected by CHEOPSand TESSto characterize the host WASP-178, refine the ephemeris of the system and measure the eclipse depth in the passbands of the two respective telescopes. We measured a marginally significant eclipse depth of 70\(\pm\)​40 ppm in the TESSpassband and statistically significant depth of 70\(\pm\)​20 ppm in the CHEOPSpassband. Combining the eclipse depth measurement in the CHEOPS and TESS passbands we constrained the dayside brightness temperature of WASP-178 bin the 2250-2800 K interval. The geometric albedo 0.1<\(\rm A_g\)<0.35 is in general agreement with the picture of poorly reflective giant planets, while the recirculation efficiency \(\epsilon>\)​0.7 makes WASP-178 ban interesting laboratory to test the current heat recirculation models.

1 Introduction↩︎

In the last few years we have gained access to a detailed characterization of exoplanets. Ground-based and space-born instrumentation have progressed such to allow the analysis of the atmosphere of exoplanets, in terms of thermodynamic state and chemical composition [1], [2]. In particular, current photometric facilities allow to observe the secondary eclipse of giant exoplanets in close orbits [3]–[6]. In this research area, CHEOPS[7] is bringing a valuable contribution given its ultra-high photometric accuracy capabilities [4], [8]–[13].

The depth of the eclipse quantifies the brightness of the planetary dayside with respect to its parent star. Depending on the temperature of the planet and the photometric band used for the observations, the eclipse depth provides insight into the reflectivity and energy redistribution of the atmosphere.

WASP-178 b(HD 134004 b) is a hot Jupiter discovered by [14] and independently announced by [15] as KELT-26 b. It orbits an A1 IV-V dwarf star (Table 1) at a distance of 7 stellar radii: these features place WASP-178 bamong the planets that receive the highest energy budget from their respective host stars. It is thus an interesting laboratory to test atmospheric models in the presence of extreme irradiation.

In this paper we analyze the eclipse depths measured using the data collected by the CHEOPSand TESSspace telescopes. It is organized as follows. Sect. 2 describes the data acquisition and reduction, while in Sect. 3 we describe how we derive the stellar radius, mass and age. In Sect. 4 we update the orbital solution of WASP-178 b, put an upper limit on the eclipse depth using TESSdata and get a significant detection using CHEOPSphotometry. Finally, in Sect. 5 we discuss the implication of the extracted eclipse signal in terms of geometric albedo and atmospheric recirculation efficiency.

2 Observations and data reduction↩︎

2.1 TESS observations↩︎

TESS[16] observed the WASP-178system in sector 11 (from 2019 April 23 to 2019 May 20) with a 30 min cadence and in sector 38 (from 2020 April 29 to 2020 May 26) with a 2 min cadence. [15] claimed a modulation with period of 0.369526 days in the photometry of sector 11, interpreting it as a \(\delta\) Scuti pulsation mode. Later, [17] found out that the TESSphotometry is heavily contaminated by the background eclipsing binary ASASSN-V J150908.07-424253.6, located at a projected distance of 50.4from the target, whose orbital period matches the periodicity in the photometry of WASP-178. In order to optimize the photometric extraction and avoid the background variable contamination, we extracted the LC corresponding to each pixel in the aperture mask defined by the TESSpipeline, and we excluded the pixels for which the periodogram shows a peak at the same orbital period of the binary system. We thus re-extracted the photometry by retrieving the calibrated FFIs and TPFs for sector 11 and 38 respectively. We used a custom extraction pipeline combined with the default quality bitmask. The extracted LCs were then background-corrected after determining the sky level using custom background masks on the FFIs and TPFs. A principal component analysis was then conducted on the pixels in these background masks across all frames in order to measure the flux contribution of scattered light in the TESScameras. We detrended the data using these principal components as a linear model. Lastly, we further corrected for any photometric trends due to spacecraft pointing jitter by retrieving the CBVs and two-second cadence engineering quaternion measurements for the specific cameras WASP-178was observed in. For each sector we computed the mean of the quaternions over the length of a science observation, i.e. 30 min for the FFIs and 2 min for the TPFs. We subsequently used these vectors along with the CBVs to detrend the TESSphotometry in a similar manner as was done in [18].

We clip out the photometry taken before BJD 2459334.7 and in the windows 2458610–2458614.5 and 2459346–2459348: these data present artificial trends due to the momentum dumps of the telescope. We also visually identified and excluded a short bump in the LC between BJD 2459337.6 and 2459337.9, most likely an instrumental artifact or some short term photometric variability feature. The final LCs cover 7 and 8 transits for sector 11 and 38 respectively.

2.2 CHEOPS observations↩︎

CHEOPS[7] observed the WASP-178system during six secondary eclipses of the planet WASP-178 bwith a cadence of 60 s. The aim is the measurement of the eclipse depth and derivation of the brightness of the planet in the CHEOPSpassband (3500–11000 Å). Each visit is \(\sim\)​12 hr long, scheduled in order to bracket the eclipse and equally long pre- and post-eclipse photometry. The logbook of the observations, which are part of the CHEOPSGuaranteed Time Observation (GTO) program, is summarized in Table 2.

The data were reduced using version 13 of the CHEOPSDRP [19]. This pipeline performs the standard calibration steps (bias, gain, non-linearity, dark current and flat fielding) and corrects for environmental effects (cosmic rays, smearing trails from nearby stars, and background) before the photometric extraction.

As for the case of TESSLCs, the aperture photometry is significantly contaminated by ASASSN-V J150908.07-424253. To decontaminate the LC of WASP-178we performed the photometric extraction using a modified version of the package PIPE³ [10], [20], [21], upgraded in order to compute the simultaneous PSF photometry of the target and the background contaminant.

The extracted LCs present gaps due to Earth occultations which cover \(\sim\)​40% of the visits. The exposures close to the gaps are characterized by high value of the background flux, due to stray-light from Earth. The corresponding flux measurements are thus affected by a larger photometric scatter. To avoid these low quality data, we applied a 5\(\sigma\) clipping to the background measurements: this selection criterion removes less than 20% of the data with the highest background counts. Finally, for a better outlier rejection, we smoothed the data with a Savitzky-Golay filter, computed the residuals with respect to the filtered LC and 5\(\sigma\)-clipped the outliers. This last rejection criterion excludes a handful of data points in each LC.

Finally, we normalize the unsmoothed LCs by the median value of the photometry. These normalized LCs are publicly available at CDS PUT LINK.

3 Stellar radius, mass and age↩︎

We use a IRFM in a MCMC approach to determine the stellar radius of WASP-178[22], [23]. We downloaded the broadband fluxes and uncertainties from the most recent data releases for the following bandpasses: Gaia G, G\(_{\rm BP}\), and G\(_{\rm RP}\), 2MASS J, H, and K, and WISEW1 and W2 [24]–[26]. Then we matched the observed photometry with synthetic photometry computed in the same bandpasses by using the theoretical stellar SEDs corresponding to stellar atmospheric parameters (Table 1). The fit is performed in a Bayesian framework and, to account for uncertainties in stellar atmospheric modeling, we averaged the atlas [27], [28] and phoenix [29] catalogs to produce weighted averaged posterior distributions. This process yields a \(R_{\star}=1.722\pm0.020\, R_{\odot}\).

Assuming the \(T_{\mathrm{eff}}\), [Fe/H], and \(R_{\star}\) listed in Table 1 as input parameters, we also computed the stellar mass \(M_{\star}\) and age \(t_{\star}\) by using two different sets of stellar evolutionary models. In detail, we employed the isochrone placement algorithm [30], [31] and its capability of interpolating within pre-computed grids of PARSEC⁴ v1.2S [32] for retrieving a first pair of mass and age estimates. A second pair of mass and age values, instead, was computed by CLES [33], which generates the best-fit evolutionary track of the star by entering the input parameters into the Levenberg-Marquadt minimisation scheme as described in [34]. After carefully checking the mutual consistency of the two respective pairs of estimates through the \(\chi^2\)-based criterion broadly discussed in [35], we finally merged the outcome distributions and we obtained \(M_{\star}=2.169_{-0.089}^{+0.083}\,M_{\odot}\) and \(t_{\star}=50_{-50}^{+60}\) Myr.

4 Light curve analysis↩︎

4.1 TESSphotometry↩︎

To compare in a homogeneous way the LCs of sectors 11 and 38, we rebinned the photometry of sector 38 to 30 min. The standard deviation of the TESSLCs, after transits and secondary eclipses are clipped, is of \(\sim\)​300 ppm for both sectors, and in both cases the photometric uncertainty can account for only \(\sim\)​70% of the variance. This indicates that there is some noise in the LCs due to astrophysical signals and/or instrumental leftovers. To investigate if the unexplained variance is related to periodic signals, for both sectors we compute the GLS periodogram [36] of the out-of-transit and out-of-eclipse photometry (Fig. 1). For sector 11 we did not find any significant periodic signal, while for sector 38 there is a strong peak at frequency \(\rm \nu=0.304\pm0.003~d^{-1}\) with FAP lower than 0.1%. The amplitude of the corresponding sinusoidal signal is \(90\pm10\) ppm.

The periodicity detected in sector 38 is consistent within uncertainties with the planetary orbital period. Nonetheless, we exclude that this signal is of planetary origin because it remained undetected in sector 11 and is not coherent from orbit to orbit in sector 38. This signal might be due to stellar rotation, and is consistent with the typical amplitude reported by [37] for A-type stars. Its periodicity would correspond to the \(v \sin i_\star\)in Table 1 if the inclination of the stellar rotation axis is \(\sim15^{\circ}\). This speculation supports the hypothesis of [15] that the star is seen nearly pole-on. A significant misalignment between the stellar rotation axis and the planetary orbit axis is not surprising, both because of its young age (no time for realignment to occur), and the stellar temperature. As a matter of fact, hot stars (T\(_{\rm eff}\)>6200 K) are observed to often have large misalignments, which is thought to be due to their lack of a convective zone [38], needed to tidally align the planetary orbit with the stellar spin axis. Furthermore, [39] found that misaligned orbits are most often polar, or close to polar, as seems to be the case of WASP-178 b. We expand the discussion on the photometric variability and the orientation of the stellar rotation axis in Appendix 8.

The fact that the period of the correlated noise is similar to the orbital period of WASP-178 bmakes it difficult to extract the complete planetary PC signal. We thus first tried a simpler and more robust approach to analyze the planetary transits and eclipses (Sect. 4.1.1), then we attempted a more complex analysis framework aimed at retrieving the full PC of WASP-178 b(Sect. 4.1.2).

4.1.1 Fit of transits and eclipses↩︎

We computed the ephemeris of the planet by trimming segments of the LCs centered on the transit events (7 transits in sector 11 and 8 transits in sector 8) and as wide as 3 times the transit duration. To further constrain the ephemeris of WASP-178 b, we included in our analysis the WASP-South photometry (2006 May – 2014 Aug) and EulerCAM I-band photometry (2018 Mar 26) presented in [14]. Also for the case of the WASP-South photometry, we trimmed the LC around the transits and kept the 24 intervals containing more than 20 data points.

We fit the data using the same Bayesian approach described in [12]. In summary, it consists in the fit of a model - in a likelihood-maximization framework - which includes the transits, a linear term for each transit to detrend against stellar/instrumental systematics and a jitter term to fit the white noise not included in the photometric uncertainties. The transit profile is formalized using the quadratic limb darkening (LD) law indicated by [40] with the reparametrization of the LD coefficients suggested by [41]. For TESSsector 11 the model is rebinned to the same 30 min cadence of the data. The likelihood maximization is performed with MCMC using the python emcee package version 3.1.3 [42], using a number of samplings long enough to ensure convergence. We used flat priors for all the fitting parameters but the stellar density, for which we used the Gaussian prior \(N(0.43,0.02)\) given by the stellar mass and radius in Table 1. We also used the Gaussian priors for the LD coefficients given by the LDTk package ⁵. For simplicity, we use the same LD coefficients for the three datasets. This is motivated by the fact that the WASP-South photometry is not accurate enough to constrain the LD profile, and that the TESSpassband is basically centered on the standard I band, thus the same LD profile is expected for the TESSand EulerCAM LCs. Since the model fitting is computationally demanding, we ran the code in the HOTCAT computing infrastructure [43], [44].

The result of the model fitting is listed in Table 3. The orbital solution we derived is consistent with previous studies [14], [15] within 5\(\sigma\). The best fit model, corresponding to the MAP parameters, is over-plotted to the phase-folded data in Fig 2, where we rebinned the model and the LC of sector 38 to the same 30 min cadence of sector 11 (we do not show the WASP-South and EulerCAM photometry not to clutter the plot). In Fig. 9 we show the corner plot of the system’s parameters from the fit of the transit LCs.

Since the TESSLCs of sector 11 and 38 show different levels of variability, we investigated any seasonal dependence of the apparent planet-to-star radius ratio. We used the same Bayesian framework as above, where we fix the ephemeris of WASP-178. The planet-to-star radius ratio we derived is 0.1108\(\pm\)​0.0004 for sector 11 and 0.1141\(\pm\)​0.0004 for sector 38. The difference is thus 0.0033\(\pm\)​0.0006, that is we found different transit depths with a 5.5\(\sigma\) significance. In particular, we remark that the planet looks larger in sector 38, where the rotation signal is stronger. We thus speculate that WASP-178was in a low variability state during sector 11, while one year later (\(\sim\)​80 stellar rotations) during sector 38 the stellar surface hosted dark spots in co-rotation with the star. This hypothesis is consistent with [45], according to which magnetic chemically peculiar stars may show complex photometric patterns due to surface inhomogeneities.

We used a similar framework to extract the secondary eclipse signal of WASP-178: we simultaneously fit segments of the TESSLCs centered on the planetary eclipses keeping fixed the ephemeris of WASP-178to the values listed in Table 3. We did not attempt the disjoint analysis of the two sectors as the expected eclipse depth is of the order of \(\sim\)​100 ppm (see below), and the photometric precision of the LCs is not good enough to appreciate differences in the eclipse depth with enough statistical evidence. The only free planetary parameter of the model is thus the eclipse depth, which turned out to be \(\delta_{\rm ecl}=70\pm40\) ppm (MAP: 70 ppm). The detrended phase-folded eclipses are shown in Fig 2 together with the MAP model.

4.1.2 Fit of the phase curve↩︎

As a more advanced data analysis, we jointly fit the two TESSsectors to extract the full planetary PC (for a homogeneous analysis, we rebinned sector 38 to the same 30 min cadence of sector 11). The fitting model now includes the transits, secondary eclipses, the planetary PC, a GP to fit the correlated noise in the data and, for each TESSsector, a long-term linear trend and a jitter term. Given the system parameters listed in Table 1, the expected amplitude of ellipsoidal variations [46] and Doppler boosting [47] in the planetary PC are 2.5 ppm and 1 ppm respectively, out of reach for the TESSphotometry. Hence, for the sake of simplicity, we do not include them in the model fitting.

We jointly fit the two TESSsectors, starting with the simplest model where the out of transit PC is flat (that is, we assumed that the planetary PC is not detectable) and the putative stellar rotation is modeled as GP with the SHO kernel⁶, indicated for quasi-periodic signals. Checking the residuals of the fit we notice that some aperiodic correlated noise is left. We thus increased the complexity of the GP model by adding a Matérn3/2 kernel to capture the remaining correlated noise adding only two free parameters to the model. This composite GP model turned out to ensure better convergence to the fit of the model.

Finally, we also included the planetary PC. Assuming that the planet is basically composed of two homogeneous dayside and nightside, the PC is in principle the combination of three components: the PC due to reflected light (with amplitude \(A_{\rm refl}\)), the PC of the planetary night side (with amplitude \(A_{\rm n}\)) and the PC of the planetary day side, whose amplitude \(A_{\rm d}\) is parameterized such to be larger than \(A_{\rm n}\) by an increment \(\delta_{\rm d}\) (this parametrization avoids the nonphysical case of a planetary night side brighter than the dayside). The three terms are modeled assuming Lambert’s cosine law. For the reflected PC, the amplitude can be expressed as: \[A_{\rm refl}=A_{\rm g}\left(\frac{R_{\rm p}}{a}\right)^2,\label{eq:albedo}\tag{1}\] where \(A_{\rm g}\)is the planetary geometric albedo [50]. Assuming a Lambertian reflective planetary surface, the geometric albedo is fixed by the Bond albedo \(A_{\rm B}\) by \(A_{\rm g}\)=\(\frac{2}{3}A_{\rm B}\). In the optimistic case of a perfectly reflecting body (\(A_{\rm B}\)=1), the maximum amplitude for the reflected PC is thus \(A_{\rm refl}=167\pm5\) ppm, where we have used the system parameters in Table 3.

With a slight re-adaptation of the formalism in [50], the amplitude thermal component \(A_{\rm therm}\) can be expressed as: \[A_{\rm therm}=\left(\frac{R_{\rm p}}{R_\star}\right)^2\frac{\int\eta(\lambda) B(\lambda,T)d\lambda}{\int\eta(\lambda) I_\star(\lambda,T_{\rm eff})d\lambda},\label{eq:thermal}\tag{2}\] where \(\eta(\lambda)\) is the optical throughput of the telescope and \(I_\star(\lambda,T_{\rm eff})\) is the expected stellar intensity computed with the NextGen model [51] corresponding to the effective temperature \(T_{\rm eff}\). For the planetary thermal emission we lack the infrared information which can constrain the emission spectrum. We thus assume a black body spectrum \(B(\lambda,T)\) at temperature \(T\).

The amplitude of the thermal emission from the dayside and the nightside depends on the respective temperatures, which we estimated following [52]. The expected temperature of the substellar point of WASP-178 bis: \[T_{\rm 0}=T_{\rm eff}\sqrt{\frac{R_\star}{a}}=3530\pm60~{\rm K},\] where we used the stellar effective temperature in Table 1 and the scaled semi-major axis in Table 3.

The nightside temperature \(T_{\rm n}\) and the dayside temperature \(T_{\rm d}\) depend on the Bond albedo \(A_B\) and heat recirculation coefficient \(\epsilon\): \[\begin{align} T_{\rm n}&=T_{\rm 0}\left(1-A_{\rm B}\right)^{1/4}\left(\frac{\epsilon}{4}\right)^{1/4}\tag{3}\\ T_{\rm d}&=T_{\rm 0}\left(1-A_{\rm B}\right)^{1/4}\left(\frac{2}{3}-\frac{5}{12}\epsilon\right)^{1/4}\tag{4} \end{align}\]

The highest dayside temperature, corresponding to \(A_{\rm B}\)=0 and no heat recirculation (\(\epsilon=0\)), is: \[T_{\rm d}=T_{\rm 0}\left(\frac{2}{3}\right)^{1/4}=3190\pm60~{\rm K},\] while the maximum nightside temperature, obtained in the case of full energy recirculation (\(\epsilon=1\)), is: \[T_{\rm n}=T_{\rm 0}\left(\frac{1}{4}\right)^{1/4}=2500\pm40~{\rm K}.\] These maximum temperatures lead to the upper limit on the dayside (\(A_{\rm d}\)) and nightside (\(A_{\rm n}\)) thermal emission amplitude of: \[A_{\rm d}=225\pm27~{\rm ppm}; A_{\rm n}=48\pm7~{\rm ppm}.\]

In Fig. 3 we plot the upper limits for the three PCs discussed so far. The figure shows that the three components may reach amplitudes of comparable orders of magnitude, meaning that none of them can in principle be neglected in the extraction of the planetary PC. We note that the two components belonging to the dayside (reflection and thermal emission) have the same shape, thus it is not possible to disentangle them. To simplify the model and avoid the degeneracy between dayside reflection and emission, we thus artificially set \(A_{\rm refl}=0\) and let \(A_{\rm d}\) absorb the whole signal belonging to the planetary dayside.

The Akaike Information Criterion [53] favors the model that includes the planetary PCs. The simpler model with flat out-of-transit PC has a relative likelihood of 55% and cannot be rejected. The posterior distributions of the common parameters (system ephemeris and planet-to-star radii ratio) between the two models do not differ significantly.

For both models, the SHO GP has a frequency \(\nu\sim\)​0.30 d\(^{-1}\), an amplitude \(\sim\)​70 ppm and a timescale \(\sim\)​11 d. Frequency \(\nu\) and the amplitude of the quasi-periodic GP are consistent with the frequency and amplitude of the periodic signal found in the periodogram (Sect. 4.1). While the correspondence between the orbital period of WASP-178 band the periodicity of the GP suggests that the red noise in the data is of planetary origin, we do not have other strong evidence that support this hypothesis. On the contrary, the planetary origin seems odd for the reasons discussed in Sect. 4.1. The aperiodic red noise fitted as a Matérn3/2 GP has an amplitude \(\sim\)​150 ppm and a timescale of \(\sim\)​50 min. Despite the aperiodic red noise evolves on timescales shorter than the transit duration (\(T_{\rm 14}\)=3.53 hr), the retrieved posterior distributions of the orbital parameters are similar to the ones obtained in Sect. 4.1.1.

Regarding the retrieval of the planetary PCs, we derived an upper limit of \(A_{\rm n}<\)​174 ppm at the 99.9% confidence level, while for the dayside we derived a 1\(\sigma\) confidence band of 110\(\pm\)​40 ppm which includes both the reflection and thermal emission components. We remark that the \(A_{\rm d}\) parameter corresponds in our model to the eclipse depth \(\delta_{\rm ecl}\) and is consistent within uncertainties with the estimate reported in Table 3.

The fit of the full TESSLC thus confirms the results previously obtained in Sect. 4.1.1. Given the similarities between the results obtained with the two approaches, we preferred the former one as it is less prone to interference between the data detrending and the extraction of the orbital parameters and the eclipse depth.

4.2 CHEOPSphotometry↩︎

To analyze the CHEOPSLCs we use the same approach as for the fit of the eclipses observed by TESS(Sect. 4.1.1), with an additional module in the fitting model that takes into account the systematics in the CHEOPSdata. As a matter of fact, CHEOPSphotometry is affected by variable contamination from background stars in the field of view [4], [8], [9], [12], [54]. This variability is due to the interplay of the asymmetric PSF with the rotation of the field of view [7]. PIPE by design uses nominal magnitudes and coordinates of the stars in the field to fit the stellar PSFs, but residual correlated noise is present in the LCs due to inaccuracies in the assumptions. This signal is phased with the roll angle of the CHEOPSsatellite and, in the case of WASP-178, is clearly visible in the GLS periodogram of the data together with its harmonics. To remove this signal, we included in our algorithm a module which fits independently for each visit the harmonic expansions of the telecope’s orbital period and its harmonics. A posteriori, we found that the roll angle phased modulation is adequately suppressed (that is, there is no peak in the periodograms at the frequencies in the harmonic series) if we include in the model the fundamental harmonic and its first two harmonics.

We also noticed that the photometry is significantly correlated with the coordinates of the centroid of the PSF on the detector. To decorrelate against this instrumental jitter we thus included in the model a bi-linear function of the centroid coordinates.

Finally, to take into account a white noise not included in the formal photometric uncertainties, we add to our model a diagonal GP kernel of the form: \[k(t_i,t_j)=j_v^2\delta_{i,j},\label{eq:kernelCHEOPS}\tag{5}\] with an independent jitter term \(j_v\) for each CHEOPS visit.

As for the fit of the TESSdata (Sect. 4.1.1), we search the bestfit parameters through likelihood-maximization in a MCMC framework, and we find \(\rm \delta_{ecl}=70\pm20\) ppm. The ephemeris of the planet were locked to the MAP values listed in Table 3, which guarantee an uncertainty on the eclipse time less than 23 s for each CHEOPS visit. The phase-folded data detrended against stellar and instrumental correlated noise are shown in Fig. 4.

5 Discussion↩︎

In the previous sections we have analyzed the TESSand CHEOPSPCs of WASP-178 bin order to extract its planetary eclipse depth \(\delta_{\rm ecl}\), obtaining respectively \(70\pm40\) ppm and \(70\pm20\) ppm. These two measurements take into account both the reflection from the planetary dayside and its thermal emission. With the assumptions discussed below, the combined information on the eclipse depth from two different instruments allows to disentangle these two contributions. For a given telescope with passband \(\eta(\lambda)\) we can combine Eq. 1 and Eq. 2 obtaining: \[\delta_{\rm ecl}=A_{\rm g}\left(\frac{R_{\rm p}}{a}\right)^2+\left(\frac{R_{\rm p}}{R_\star}\right)^2\frac{\int\eta(\lambda) B(\lambda,T_{\rm d})d\lambda}{\int\eta(\lambda) I_\star(\lambda,T_{\rm eff})d\lambda},\label{eq:decl}\tag{6}\] where \(T_{\rm d}\) is the planetary dayside temperature.

Solving for \(A_{\rm g}\), and indicating with the subscripts \(C\) (for CHEOPS) and \(T\) (for TESS) the wavelength-dependent quantities, we obtain: \[\begin{cases} A_{\rm g}^C=\left(\frac{a}{R_\star}\right)^2\left[\frac{\delta_{\rm ecl}^C}{\left(\frac{R_{\rm p}}{R_\star}\right)^2}-\frac{\int\eta^C(\lambda) B(\lambda,T_{\rm d})d\lambda}{\int\eta^C(\lambda) I_\star(\lambda,T_{\rm eff})d\lambda}\right],\\ A_{\rm g}^T=\left(\frac{a}{R_\star}\right)^2\left[\frac{\delta_{\rm ecl}^T}{\left(\frac{R_{\rm p}}{R_\star}\right)^2}-\frac{\int\eta^T(\lambda) B(\lambda,T_{\rm d})d\lambda}{\int\eta^T(\lambda) I_\star(\lambda,T_{\rm eff})d\lambda}\right].\label{eq:ag} \end{cases}\tag{7}\]

In the most general case, \(A_g^C\) and \(A_g^T\) differ according to the planetary reflection spectrum. Since we do not have any spectroscopic analysis of the dayside of WASP-178 b, we define the proportionality coefficient \(\alpha=A_g^T/A_g^C\) to account for differences between the two passbands. Solving Eq. 7 for \(T_{\rm d}\) we thus obtain the implicit function: \[\frac{\alpha\delta_{\rm ecl}^C-\delta_{\rm ecl}^T}{\left(\frac{R_{\rm p}}{R_\star}\right)^2}+\frac{\int\eta^T(\lambda) B(\lambda,T_{\rm d})d\lambda}{\int\eta^T(\lambda) I_\star(\lambda)d\lambda}-\alpha \frac{\int\eta^C(\lambda) B(\lambda,T_{\rm d})d\lambda}{\int\eta^C(\lambda) I_\star(\lambda)d\lambda}=0.\label{eq:tdRoot}\tag{8}\]

We initially assumed the simplest scenario of a gray albedo spectrum (\(\alpha\)=1) in the spectral range covered by CHEOPSand TESS(the respective bandpasses are shown in Fig. 6). In this scenario, we expect that the contribution of reflection to the eclipse depth is the same in the CHEOPSand TESSpassbands. We also expect that the thermal emission from a \(T\simeq\)​3000 K black body has a larger contribution in the TESSpassband, because it favors redder wavelengths compared with CHEOPS(see Fig. 6).

To estimate the dayside temperature \(T_{\rm d}\) that best explains our best observations together with its corresponding uncertainty, we randomly extracted 10 000 steps from the Monte Carlo chains obtained in the fit of the transit and eclipse LCs (Sect. 4.1.1 and 4.2). We also generated 10 000 samples of the stellar effective temperature using a normal distribution with mean and standard deviations as reported in Table 1. For each of the 10 000 samples we thus numerically solved Eq. 8 using the fsolve method of the scipy Python package, thus obtaining 10 000 samples of \(T_{\rm d}\). Then, we plugged all the samples back into Eq. 7 to derive \(A_g^C\)(=\(A_g^T\)).

The root finding algorithm failed for \(\sim\)​50% of the samples. These failures correspond to the combination of parameters that prevent Eq. 8 from having a \(T_{\rm d}\) root in the domain of real numbers. This happens in particular for the samples where \(\delta_{\rm ecl}^T<\delta_{\rm ecl}^C\) that, as discussed above, are not consistent with the gray albedo hypothesis. We thus reject these samples and show in Fig. 7 the \(A_g^C\)–\(T_{\rm d}\) density map of the remaining ones. Our results indicate \(A_g^C=0.2\pm0.1\) and \(T_{\rm d}=2400\pm300\) K, consistent with the temperature-pressure profile derived by [17] at pressure higher that 1\(\rm\mu\)bar by means of ultraviolet transmission spectroscopy. Our result also follows the general trend of \(A_{\rm g}\)increasing with \(T_{\rm d}\) indicated by [55] (Fig. 5).

By definition, the geometric albedo \(A_{\rm g}\)refers to the incident light reflected back to the star at a given wavelength (or bandpass). Integrating at all angles, the spherical albedo \(A_{\rm S}\)is related to \(A_{\rm g}\)by the phase integral \(q\): \(A_{\rm S}\)=\(q\) \(A_{\rm g}\)[50]. Depending on the scattering law, exoplanetary atmospheres have 1<\(q\)<1.5 [56], [57]. Unfortunately, for the reasons explained in Sect. 4.1.2, we could not extract a robust phase curve for WASP-178 b, hence we could not place any better constraint on \(q\). In the following we thus consider the two limiting scenarios \(A_S^{\rm min}=A_g\) and \(A_S^{\rm max}=1.5A_g\).

The Bond albedo \(A_{\rm B}\)is computed as the average of \(A_{\rm S}\)weighted over the incident stellar spectrum: \[A_{\rm B}=\frac{\int_0^\infty A_{\rm S}(\lambda)I_\star(\lambda)d\lambda}{\int_0^\infty I_\star(\lambda)d\lambda}.\label{eq:abond}\tag{9}\] The conversion into \(A_{\rm B}\)thus relies on the measurement of \(A_{\rm S}\)across the stellar spectrum. In this study we only covered the optical and near-infrared part of the spectrum, as shown in Fig. 6, and we lack the necessary information in the UV, mid, and far infrared domains. Following the approach of [58], we explore the scenario of minimum Bond albedo \(A_{\rm B}^{\rm min}\) obtained through Eq. 9 assuming \(A_{\rm S}\)=\(A_{\rm S}^{\rm min}\) in the spectral range covered by CHEOPSand TESSand \(A_{\rm S}\)=0 otherwise. The opposite limiting case assumes \(A_{\rm S}\) \((\lambda)=A_{\rm S}^{\rm max}\) at all wavelength, which leads to \(A_{\rm B}^{\rm max}=A_{\rm S}^{\rm max}\). To compute the integrals in Eq. 9 , we used the synthetic spectrum in the BT-Settl library corresponding to the parameters of WASP-178[59].

Under the hypothesis that \(\alpha=1\), we obtained samples of \(A_{\rm g}^C=A_{\rm g}^T=\) \(A_{\rm g}\), which now translate into samples of \(A_{\rm B}^{\rm min}\) and \(A_{\rm B}^{\rm max}\) with the assumptions discussed above. Moreover, inverting Eq. 4 yields to: \[\epsilon=\frac{1}{5}\left[8-\left(\frac{T_{\rm d}}{T_{\rm eff}}\right)^4\left(\frac{a}{R_\star}\right)^2\frac{12}{1-A_{\rm B}}\right].\label{eq:epsilon}\tag{10}\] Equation 10 indicates that \(\epsilon\) is a decreasing function of \(A_{\rm B}\). Plugging in the samples of \(T_{\rm d}\), T\(_{\rm eff}\), \(a/R_{\star}\) and alternatively \(A_{\rm B}^{\rm min}\) and \(A_{\rm B}^{\rm max}\), we obtained the corresponding samples \(\epsilon^{\rm max}\) and \(\epsilon^{\rm min}\).

The albedo–recirculation density maps of the two scenarios are shown in the right panel of Fig. 7. Unsurprisingly, we find that the upper limit on \(A_{\rm B}\)is larger (\(\sim\)​0.6) in the scenario where all the factors concur to push up the reflectivity of the atmosphere (high \(q\) and maximum \(A_{\rm S}\)), and it decreases to \(\sim\)​0.3 in the opposite scenario of minimum reflectivity.

The recirculation coefficient \(\epsilon\) tends towards large values, eventually exceeding 1 due to measurement uncertainties, in both scenarios. While \(\epsilon>1\) is physically meaningless, still the posterior distribution indicates a high level of atmospheric energy recirculation in both cases of minimum and maximum albedo (\(\epsilon^{\rm max}=1.0\pm0.3\) and \(\epsilon^{\rm min}=0.8\pm0.3\) respectively). Following different atmospheric models [60]–[63], WASP-178 bis in a temperature regime where heat recirculation of zonal winds is suppressed and recirculation efficiency is expected to be low.

Nonetheless, [64] collected observational evidence that HJs with irradiation temperatures similar to the one of WASP-178 bare characterized by efficient day-to-night recirculation. Using the global circulation models of [65], they ascribed this efficiency to the presence of zonal winds, that also explain the eastward offset of the PC of the same planets. While the theoretical predictions on the offset (and the recirculation efficiency correspondingly) likely overestimate the truth [64], still there is an indication that zonal winds might indeed explain the energy transfer from the dayside to the nightside of WASP-178 b.

Unfortunately, as discussed in Sect.4.1.2, it is difficult to disentangle the planetary phase curve and the stellar variability. By consequence, any measurement of the phase curve offset is precluded. Another possibility is that ionized winds flow from the dayside, where temperatures higher that 2500 K leads to complete dissociation of H\(_2\), to the nightside, where lower temperatures allow the molecular recombination and the consequent energy release. This scenario is supported by several recent studies [66]–[70].

According to the synthetic models computed by [71], the albedo spectrum of highly irradiated giant planets is expected to show molecular absorption bands by H\(_2\)O at around 1 \(\mu m\) and longer wavelength. Since the passband of TESSis more sensitive in the near-infrared than CHEOPS, it is plausible to assume that the geometric albedo in the TESSpassband is lower than for CHEOPS. In order to assess how the assumption on \(\alpha\) affects our results, we re-run our analysis assuming an extreme \(\alpha=0.5\). The results are shown in Fig. 8: we find almost the same posterior distribution for dayside temperature, albedo and recirculation, indicating that the most important source of uncertainties are not the assumptions we make but the measurement uncertainties on the eclipse depth extracted from the CHEOPSand TESSLCs.

6 Summary and conclusion↩︎

In this work we have analyzed the space-borne photometry of the WASP-178system obtained with TESSand CHEOPS. For both telescopes we have tailored the photometric extraction in order to avoid the strong contamination by a background eclipsing binary. We have found evidence that the stellar host is rotating with a period of \(\sim\)​3 days, consistent with the orbital period of its HJ, and that it is seen in a pole-on geometry.

The similarity between the stellar rotation period and the orbital period of the planet, coupled with the quality of the TESSdata, does not allow a robust analysis of the full PC of the planet, nor we could put strong constraints on the planetary nightside emission. Nonetheless, focusing on the transit and eclipse events, we could update the ephemeris of the planet and measure an eclipse depth of 70\(\pm\)​40 ppm and 70\(\pm\)​20 ppm in the TESSand CHEOPSpassband respectively.

The joint analysis of the eclipse depth measured in the two passbands allowed us to constrain the temperature in the 2250–2750 K range, consistent with the temperature-pressure profile derived by [17] at pressure higher that 1\(\rm\mu\)bar. Moreover, we could also constrain the optical geometric albedo \(A_{\rm g}\)<0.4, which fits the general increasing albedo with stellar irradiation indicated by [55].

Finally, we found indication of an efficient atmospheric heat recirculation of WASP-178 b. If confirmed, this evidence challenges the models that predict a decreasing heat transport by zonal winds as the equilibrium temperature increases [60]–[63]. Conversely, WASP-178 bis an interesting target to test atmospheric models where heat transport is granted by ionized winds from the dayside to the nightside, where the recombination of H\(_2\) takes place [68]–[70]. Additional observations of WASP-178 bare needed to better constrain its atmospheric recirculation and rank current competing models. The WASP-178system is going to be observed again by TESSin May 2023, but this is not expected to add much to the two sectors analyzed in this work. Conversely, dedicated observations from larger space observatories might provide helpful insights of the physical and chemical equilibrium of the atmosphere of WASP-178 b.

References↩︎