An Independent Analysis of the Spitzer/IRAC Phase Curves of WASP43 b

In our model the stellar flux is constant in time, and normalized to 1. The exoplanetary flux is given by

$$\begin{eqnarray}&&\begin{array}{l}{c}_{0}+{c}_{1}\cos [2\pi ({\rm{\Phi }}-{\rm{\Delta }}{\rm{\Phi }}-{c}_{2})]\\ \quad +\,{c}_{3}\cos [4\pi ({\rm{\Phi }}-{\rm{\Delta }}{\rm{\Phi }}-{c}_{4})],\end{array}\end{eqnarray} \tag{ 1 }$$

where Φ is the so-called orbital phase, i.e., the time from the reference epoch of transit (E.T.) in units of the orbital period (P), ΔΦ is the mid-transit phase offset, and c₀–c₄ are free parameters used to model the phase curve modulations. Equation (1) is equivalent to the formula adopted by S17 and M18. We used the formalism of Mandel & Agol (2002) for modeling the exoplanetary transit and eclipses.

We calculated multiple sets of four limb-darkening coefficients (Claret 2000) hereafter claret-4, for the WASP43 star in the 3.6 and 4.5 μm Spitzer/IRAC passbands, using the code provided by Espinoza & Jordán (2015) at GitHub.³ The code adopts two grids of stellar-atmosphere intensity models, i.e., ATLAS9⁴ (Kurucz 1979) and PHOENIX (Husser et al. 2013). The intensities in the models are given as a function of $\mu =\cos \theta$, where θ is the angle between the surface normal and the line of sight. The ATLAS models adopt a plane-parallel approximation for the stellar atmosphere, while the PHOENIX models use spherical geometry. As a consequence, the PHOENIX models show a characteristic steep drop-off in intensity at small, but finite μ values, which is not well approximated by any of the standard parametric laws (Claret et al. 2012, 2013; Morello et al. 2017). The limb-darkening coefficients also depend on the sampling of the intensities (Howarth 2011; Neilson & Lester 2013a, 2013b; Espinoza & Jordán 2015). We tested the following fitting options:

• 1.  
  A17, i.e., least-squares fit to the ATLAS model intensities calculated at 17 angles;
• 2.  
  A100, i.e., least-squares fit to the ATLAS intensities interpolated at 100 angles, uniformly sampled in μ, with a cubic spline;
• 3.  
  P100, i.e., least-squares fit to the PHOENIX intensities interpolated at 100 angles, uniformly sampled in μ, with a cubic spline;
• 4.  
  PQS, i.e., least-squares fit to the PHOENIX model intensities with μ ≥ 0.1 (quasi-spherical models, as defined by Claret et al. 2012).

We discarded the least-squares fit to all the PHOENIX model intensities, because it led to anomalous (non-monotonic) limb-darkening profiles. The most likely cause of the anomalous results was that the PHOENIX model intensities are more finely sampled near the steep drop-off, which is then overweighted in the fit. We interpolated the limb-darkening coefficients in T_eff and $\mathrm{log}g$ to the WASP43 parameter values reported in Table 1. Table 3 reports the four sets of claret-4 limb-darkening coefficients obtained with the different fitting options. Figure 1 shows the corresponding intensity profiles. We note that the ATLAS limb-darkening profiles, A17 and A100, overlap in the plot. The PHOENIX profiles, P100 and PQS, indicate stronger limb-darkening than the ATLAS profiles. The P100 profiles reach zero intensity at the stellar limb, while the PQS profiles remain significantly above zero. Note that the PQS profiles are not accurate at the stellar limb, given that their behavior is extrapolated from the model intensities with μ ≥ 0.1.

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For our analysis we used the basic calibrated data (BCD) provided by the Spitzer Heritage Archive (Wu et al. 2010). BCD are flat-fielded and flux-calibrated frames (Fazio et al. 2004; IRAC Instrument & Instrument Support Teams 2015). We extracted the individual pixel time series from a 5 × 5 array having the stellar centroid at its center, and computed the sum-of-pixel time series, here referred to as raw light curves. We binned the time series by a factor of 8, i.e., temporal bin size of 16 s, in order to reduce the computational time for the data analysis. The chosen bin size is smaller than the time scales of interest, e.g., it is ∼1/63 of the transit ingress duration.

Then, we applied the wavelet pixel-ICA technique (Morello et al. 2016) to simultaneously fit the phase curve model and the instrumental effects. We also tested the time pixel-ICA technique (Morello 2015), obtaining similar or less robust results which we report in Appendix B, together with a detailed comparison of both techniques. Both algorithms rely on ICA, i.e., a blind source separation technique, to extract the instrumental components from the light curves. Such blind approaches have proven to perform as well as or better than other state-of-the-art pipelines to detrend Spitzer/IRAC observations of exoplanetary transits and eclipses (Morello et al. 2015; Ingalls et al. 2016).

In this work, the pixel-ICA pipelines were applied to full phase curve observations, which may be affected by detector systematics with longer time scales compared to the transit-only and eclipse-only observations. We checked for residual long-trend systematics by adding a linear or quadratic function of time in the light curve fits, and by comparing the differences in the Bayesian information criterion (BIC; Schwarz 1978) obtained with these various ramp models (constant, linear, or quadratic), as suggested by S17. Then, following the Occam's Razor principle, we confirmed the solution obtained with the constant ramp, if it had the lowest BIC. In an alternative case, the model selection was based on a number of considerations that will be explained in the following sections.

Figure 5 shows the best-fit phase curve models. Figure 6 reports the corresponding estimates of the planet dayside maximum and nightside minimum flux normalized to the stellar flux (${F}_{{\rm{day}}}^{{\rm{MAX}}}$ and ${F}_{{\rm{night}}}^{\mathrm{MIN}}$), and their offsets relative to the mid-eclipse and mid-transit times (${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{day}}}^{{\rm{MAX}}}$ and ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{night}}}^{\mathrm{MIN}}$), respectively.

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The results obtained for the second 3.6 μm data set with the quadratic ramp parameterization appear to be unphysical, yielding negative nightside fluxes, but still consistent with zero at the 1σ level. The results obtained with the linear ramp parameterization are more plausible, because they can be explained by a simpler physical model (see Section 5.2). We present here the two sets of results for the second 3.6 μm data set, together with the selected results for the other data sets. A more detailed discussion about the model selection criteria is reported in Section 5.1.

The (normalized) planet dayside flux at 4.5 μm is (3.90 ± 0.12) × 10⁻³. The nightside flux is (3.0 ± 1.5) ×10⁻⁴. The maximum dayside flux occurs 37 ± 7 minutes prior to the mid-eclipse time, which corresponds to a shift of 113 ± 21 east of the substellar point. The minimum nightside flux occurs within the interval −10 ± 15 minutes relative to the mid-transit time, i.e., between 72 east and 15 west of the anti-stellar point.

The 3.6 μm phase curve models have remarkably different amplitudes and shapes, but with similar dayside fluxes: (3.43 ± 0.11) × 10⁻³ for the first visit, and (3.34 ± 0.10) ×10⁻³ (quadratic ramp) or (3.32 ± 0.10) × 10⁻³ (linear ramp) for the second visit. The three estimates are mutually consistent within 0.5σ and smaller than the 4.5 μm maximum with 2–2.5σ significance level. The planet flux minima are (6.9 ± 1.6) × 10⁻⁴ for the first 3.6 μm visit, and (−1.6 ± 1.9) × 10⁻⁴ (quadratic ramp) or (3.0 ± 1.5) × 10⁻⁴ (linear ramp) for the second visit.

For the second 3.6 μm visit, the phase curve maximum occurs 14 ± 7 minutes (quadratic ramp), or 18 ± 9 minutes (linear ramp), earlier than the mid-eclipse time. These offsets correspond to hotspot shifts of 44 ± 23 and 56 ± 27 east of the substellar point. The phase curve minimum occurs at +6 ± 14 minutes (quadratic ramp), or ${0}_{-14}^{+16}$ minutes (linear ramp), relative to the mid-transit time. These offsets correspond to shifts of 2° ± 4° and ${0}_{-4}^{+5}$ ° west of the anti-stellar point.

The first 3.6 μm phase curve model is strongly asymmetric, with peaks occurring 64 ± 13 (maximum) and 103 ± 18 (minimum) minutes earlier than the mid-eclipse and mid-transit time, or, equivalently, 20° ± 4° and 32° ± 6° east of the substellar and anti-stellar points.

However, the tests reported in the Appendix C suggest that the true uncertainties in the peak offsets estimated for the 3.6 μm observations may be larger than the nominal error bars. For example, our estimate of the dayside peak offset for the first 3.6 μm visit becomes 18 ± 9 minutes before mid-eclipse when considering only the first two out of three AORs, which is identical to the estimate from the second visit (linear ramp). The corresponding dayside and nightside fluxes are consistent with those obtained from the full data set analysis within 1σ. The same tests confirm the robustness of the parameter estimates for the 4.5 μm observation within their nominal error bars.

Table 4 reports our final measurements of the day and nightside fluxes and peak offsets at 3.6 and 4.5 μm. The results at 3.6 μm are the weighted averages between those obtained for the two visits, with inflated error bars for those parameters which were not consistent within 1σ. We discarded the (unphysical) results obtained for the second 3.6 μm visit with a quadratic ramp, for reasons that will be further elaborated in Sections 5.1 and 5.2.

Table 4.  Phase Curve Parameters of WASP43 b

Parameter	3.6 μm	4.5 μm
${F}_{{\rm{day}}}^{{\rm{MAX}}}$	(3.37 ± 0.07) × 10−3	(3.90 ± 0.12) × 10−3
${F}_{{\rm{night}}}^{{\rm{MIN}}}$	(4.8 ± 1.4) × 10−4	(3.0 ± 1.5) × 10−4
${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{day}}}^{{\rm{MAX}}}$	−0.028 ± 0.013	−0.031 ± 0.006
${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{night}}}^{{\rm{MIN}}}$	−0.032 ± 0.030	−0.009 ± 0.013

Note. The 3.6 μm parameters are the weighted mean values over the two visits (using the linear ramp for the second visit); the error bars on ${F}_{{\rm{night}}}^{{\rm{MIN}}}$, ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{day}}}^{{\rm{MAX}}}$, and ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{night}}}^{{\rm{MIN}}}$ are inflated by the difference between the individual estimates in units of σ (factors of 1.26, 2.15, and 3.05, respectively).

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Figure 7 reports the best-fit transit depth (p²), impact parameter (b), and transit duration (T₀) obtained with the four sets of limb-darkening coefficients reported in Table 3 (see Section 3.2). There appear to be systematic offsets between the parameters obtained using the ATLAS and PHOENIX sets of coefficients. In particular, the PHOENIX models lead to smaller transit depths by ∼400–700 parts per million (ppm), smaller impact parameters by ∼0.02–0.04, and longer transit durations by 55–70 s, at 3.6 and 4.5 μm, respectively. These differences correspond to two to five times the respective parameter error bars.

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We did not find strong evidence in favor of one specific model (see Appendix D). Table 5 reports the arithmetic and weighted mean values of the geometric parameters, b and T₀, across the three visits for each limb-darkening model, and the mean transit depths at 3.6 and 4.5 μm. Table 5 also reports the global mean values over all the different limb-darkening models. While the absolute transit depths are model-dependent, the difference between the 3.6 and 4.5 μm transit depths is always consistent with zero within 1σ.

The minimum BIC solution for the second 3.6 μm visit (quadratic ramp) includes negative nightside flux values, which are unphysical. Even if the minimum nightside flux is consistent with being positive within 1σ, the low upper limit poses a challenge for the modeling of exoplanetary atmospheres (e.g., Kataria et al. 2015). The solution obtained using a linear ramp parameterization, instead of quadratic, appears to be less problematic, as it is discussed in Section 5.2.

We tested model selection tools other than the BIC, which all agreed on the choice of the quadratic ramp model, although with different strengths of evidence. In particular, the ΔBIC = 8.9 between the linear and quadratic parameterizations (see Table 6) denotes a strong, but not conclusive, preference for the latter according to Raftery (1995). The Akaike information criterion (AIC; Akaike 1974) and the consistent Akaike information criterion (CAIC; Bozdogan 1987) favor the quadratic ramp model more/less strongly than BIC (ΔAIC = 15.6, ΔCAIC = 7.9), because of a smaller/larger penalty for the number of model parameters. The deviance information criterion (Spiegelhalter et al. 2002) gives results similar to the AIC, while the Bayesian evidences calculated with MultiNest (Buchner et al. 2014) are consistent with the BIC estimates. Therefore, any information criterion weighted average of the alternative models, e.g., the marginalization method proposed by De Wit et al. (2016), would be driven by the quadratic ramp. A more sophisticated approach consists of marginalization over the hyperparameters of a Gaussian process (GP; Gibson 2014; Evans et al. 2015). In this paper, we did not pursue the GP method, because of its high computational cost and unclear performances in previous analyses of the Spitzer/IRAC data (Ingalls et al. 2016).

All of the tests discussed above rely on the relative amplitudes of the residuals. However, it is not guaranteed that smaller residuals correspond to more reliable parameter estimates. The potential errors in the instrumental systematics model may be compensated in part by biasing the retrieved astrophysical parameters, especially if the two sets of parameters (instrumental systematics and astrophysical) are correlated. We observed that, in the second 3.6 μm data set, the minimum nightside flux is strongly correlated with the two quadratic ramp coefficients, as measured by the absolute value of the Pearson correlation coefficients (PCCs ∼0.6). The correlation with the linear ramp coefficient is much smaller (PCC ∼ 0.1).

In conclusion, simple statistical criteria based on the amplitude of the best-fit residuals can provide useful guidelines to model selection, but they should not be considered alone. Physical plausibility may pose important constraints to the model selection, especially when the competing models have similar scores (Ingalls et al. 2016). We highly recommend to perform some self-consistency tests on the data, e.g., checking that the best-fit parameters do not vary dramatically if analyzing smaller portions rather than the whole data set (see Appendix C).

For the Spitzer/IRAC 3.6 and 4.5 μm channels, the exoplanet flux contribution is predominantly thermal emission. By neglecting the nonthermal contributions, the observed planet-to-star flux ratio in a given passband is (Charbonneau et al. 2005)

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{p,\lambda }}{{F}_{* ,\lambda }}={\left(\displaystyle \frac{{R}_{p}}{{R}_{* }}\right)}^{2}\displaystyle \frac{{B}_{\lambda }({T}_{p})}{{B}_{\lambda }({T}_{* })},\end{eqnarray} \tag{ 2 }$$

where ${({R}_{p}/{R}_{* })}^{2}$ is the sky-projected planet-to-star area ratio, T_p is the phase-dependent exoplanet brightness temperature, and T_* is the star brightness temperature. We computed the brightness temperatures for a star with T_eff = 4400 K, $\mathrm{log}g=4.65$ (see Table 1), in the two IRAC passbands, by interpolating on a grid of PHOENIX stellar-atmosphere models (Husser et al. 2013). By inverting Equation (2), we calculated the exoplanet maximum dayside temperature, ${T}_{{\rm{day}}}^{{\rm{MAX}}}$, to be 1660 ± 21 K for the first 3.6 μm visit, 1643 ± 19 K (quadratic ramp) or 1639 ± 19 K (linear ramp) for the second 3.6 μm visit, and 1502 ± 18 K at 4.5 μm. The corresponding minimum nightside temperatures, ${T}_{{\rm{night}}}^{{\rm{MIN}}}$, are ${1016}_{-{\rm{63}}}^{+{\rm{56}}}$ K, <568 K, ${837}_{-{\rm{103}}}^{+{\rm{79}}}$ K, and ${700}_{-{\rm{93}}}^{+{\rm{68}}}$ K. These temperature estimates can be visualized in Figure 8.

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The choice of a linear or a quadratic ramp parameterization for the second 3.6 μm data set changes dramatically the inferred astrophysical scenario. The former leads to consistent measurements between the two 3.6 μm visits within 1.5σ, and lower brightness temperatures at 4.5 μm. The lower brightness temperatures suggest higher absorption/scatter at 4.5 μm within the WASP43 b atmosphere, assuming a non-inverted thermal profile (Blecic et al. 2014).

The quadratic ramp model implies lower nightside flux, and brightness temperature (upper limit), for the second 3.6 μm visit, suggesting some variability with the 2.5σ significance level. The wavelength–temperature trend is inverted between the exoplanet dayside and nightside, which indicates different atmospheric opacities between the two sides. These results might be explained with the appearance of high-altitude clouds in the exoplanet atmosphere during the second 3.6 μm visit. In conclusion, we cannot rule out this solution as physically impossible, but it is most likely biased by the strong parameter correlations, as mentioned in Section 5.1. This idea is reinforced by the simpler physical interpretation (and smaller parameter correlations) associated with the alternative solution using a linear ramp model.

We used the 2D-ATMO code (Tremblin et al. 2017) to compute a grid of phase curve models for WASP43 b. The 2D-ATMO is an extension of 1D-ATMO (Tremblin et al. 2015) that takes into account the circulation induced by the irradiation from the host star at the equator of the planet. The atmospheric model for WASP43 b has been computed as part of a model comparison performed for future JWST observations (O. Venot et al. 2019, in preparation). The magnitude of the zonal wind is imposed at the substellar point at 4 km s⁻¹ and is computed accordingly to the momentum conservation law in the rest of the equatorial plane. The vertical mass flux is assumed to be proportional to the meridional mass flux with a proportionality constant 1/α; the wind is therefore purely longitudinal and meridional if $\alpha \to \infty$ or purely longitudinal and vertical for $\alpha \to 0$. As in Tremblin et al. (2017), a relatively low value of α drives the vertical advection of entropy/potential temperature in the deep atmosphere that can produce a hot interior, which can explain the inflated radii of hot Jupiters. A high value of α will produce a cold deep interior as in the standard 1D models. In this study, we used a simulation with α = 10⁴ that should be representative of WASP43 b since the planet is not highly inflated. In order to reproduce the low fluxes on the nightside of the planet, we added a simple cloud model consisting of a gray absorbing cloud deck with an absorption of 2.5 m² kg⁻¹ with a fixed bottom pressure of 0.1 bar. We explored different metallicities (1×, 3×, and 10× solar) and different top pressure levels for the cloud deck (0.1 bar, i.e., no clouds, 0.02, 0.01, 10⁻³, 10⁻⁴, and 10⁻⁷ bar).

Figure 9 compares the measured day and nightside fluxes and peak offsets (from Table 4) with those predicted by the atmospheric models. Figure 10 compares the whole phase curve profiles obtained from the data with the best matching profiles from the atmospheric models.

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A number of atmospheric phase curve models are in excellent agreement with the observed profile at 4.5 μm. The best matches are the models with 3× or 1× solar metallicity and cloud top pressure of 10⁻³ bar. In both cases, the discrepancies between the fitted and the model profiles are smaller than 200 ppm with rms amplitudes of ∼100 ppm. The corresponding model phase curve parameters are all consistent with the measured values within 1σ. In general, all of the models with cloud top pressure lower than 10⁻² bar are in good agreement with the 4.5 μm observation, but the models with 10× solar metallicity tend to predict a smaller dayside peak offset. The models with no clouds can be ruled out at the 4–8σ level in the nightside flux, and they also tend to predict significantly larger peak offsets, depending on the metallicity.

The results at 3.6 μm are more problematic, as the measured dayside flux is higher than predicted by the atmospheric models. The best match to the observed profile (average of the two observations) is the model with 10× solar metallicity and cloud top pressure of 2 × 10⁻² bar. In this case, the discrepancies between the fitted and the model profiles are within ∼400 ppm with rms amplitudes below 300 ppm, which are larger than the error bars of the best-fit profile at certain orbital phases. The corresponding model phase curve parameters are consistent with the measured values within 1.5σ. The models with 3× and 1× solar metallicity predict smaller than measured dayside fluxes by ∼150 ppm (2σ) and 300 ppm (4σ), respectively.

The observations at 3.6 and 4.5 μm are best described by atmospheric models with different metallicity and cloud top pressure, although a range of models with metallicity higher than solar and cloud top pressure of ∼10⁻² bar can reproduce all of the measured phase curve parameters within less than 2σ. It is likely that a chemical composition different than scaled solar abundances could provide a better match to the data, without the need to find a compromise at the edges of the acceptable parameter ranges for the observations at the two wavelengths. In this paper, we refrain from speculating about the possible nature of the non-standard chemistry in the atmosphere of WASP43 b, which cannot be probed with the current data.

As the observations were not taken simultaneously, we cannot exclude some variability of the nightside clouds over the different visits. In Section 4.1 we noted that the parameter results derived from the individual 3.6 μm visits were not fully consistent at the 1σ level, but the apparent discrepancies might be caused by correlated noise in the fitting residuals. In general, the observations with 4.5 μm channel are much less affected by correlated noise (Krick et al. 2016), which is also confirmed by the analyses in this paper (see Section 4 and Appendix C). Therefore, new observations at 4.5 μm would help to assess the level of variability in the atmosphere of WASP43 b.

The rms amplitudes of the light-curve fitting residuals obtained with our wavelet pixel-ICA are within 1% of those reported by M18, using an extension of the BLISS mapping algorithm (Stevenson et al. 2012).

Figure 11 reports the phase curve peak-to-peak amplitudes that we computed to compare with those reported in the previous literature. When taking the linear ramp model for the second 3.6 μm visit, our best-fit amplitudes are consistent with those reported by M18 within 1σ, though our central values are ∼300 ppm larger. S17 obtained larger peak-to-peak amplitudes for the second 3.6 and 4.5 μm visits. When taking the quadratic ramp model for the second 3.6 μm visit, we also obtain a larger peak-to-peak amplitude, in agreement with S17. Overall, the different estimates for each visit are consistent at the 2σ level.

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We observed that, in the analyses discussed here, a smaller peak-to-peak amplitude corresponds to a higher nightside flux (at a given wavelength). Figure 8 compares the brightness temperatures obtained in this work with those reported by S17. The dayside temperatures are consistent within 0.5σ. Our estimates of the nightside temperatures are higher than the 2σ upper limits reported by S17. Figure 6 shows that we obtained a significantly smaller dayside peak offset compared to S17 at 4.5 μm.

S17 could not find adequate approximations to the observed phase curve profiles with the cloud-free atmospheric models of Kataria et al. (2015). M18 computed new global circulation models with THOR (Mendonça et al. 2016); their best match to the Spitzer/IRAC data is a model with a nightside cloud deck with a top pressure of 10⁻² bar and enhanced carbon dioxide (CO₂) with a longitudinal gradient. A visual inspection of the Figure 6 in M18 reveals that the maximum discrepancies between the fitted and the best model phase curves are ∼400–500 ppm, i.e., equal or larger than those obtained in this study (see Section 5.3). Furthermore, the non-equilibrium CO₂ was introduced ad hoc by M18 to reproduce the low nightside flux observed at 4.5 μm, with a lower limit for the cloud top pressure of 10⁻² bar (from Kreidberg et al. 2014). Mendonça et al. (2018b) could not physically explain that chemical disequilibrium.

Figure 12 reports our final estimates of the geometric parameters averaged over the three observations, and the analogous parameters derived from those reported by Stevenson et al. (2014) using HST/WFC3 data. Our estimates of the impact parameter using ATLAS limb-darkening coefficients are marginally consistent within 1σ with the value reported by Stevenson et al. (2014). We found longer transit duration than Stevenson et al. (2014) by 42 s and 101–106 s using ATLAS and PHOENIX limb-darkening coefficients, respectively. These discrepancies are above 3σ. Note that the small uncertainties in the parameters obtained from Stevenson et al. (2014) do not account for the degeneracy with the stellar limb-darkening.

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Figure 13 compares the 3.6 and 4.5 μm transit depths with those obtained by S17. Unsurprisingly, our estimates using ATLAS (claret-4) limb-darkening coefficients best match the results obtained by S17, which also adopted ATLAS (quadratic) limb-darkening coefficients.

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Some authors suggested the existence of a simple relationship between the irradiation temperature and circulation efficiency of the exoplanetary atmospheres (Cowan & Agol 2011; Perez-Becker & Showman 2013; Schwartz & Cowan 2015; Komacek & Showman 2016; Keating & Cowan 2017). The S17 claim of zero circulation efficiency for the WASP43 b atmosphere injected an apparent outlier to the expected trend. WASP43 b and HD 209458 b have similar irradiation temperatures of ∼2000–2100 K. Schwartz et al. (2017) reported $\varepsilon ={0.49}_{-0.14}^{0.15}$ for HD 209458 b, based on visible-to-infrared observations, often limited to the secondary eclipses.

In this work, we obtained significantly higher nightside temperatures than the previous estimates by S17 for WASP43 b in the Spitzer/IRAC passbands. Assuming a blackbody-like emission, the circulation efficiency goes up to ε ∼ 0.1–0.3. We emphasize that this is just a broad estimate of the circulation efficiency in the WASP43 b atmosphere. In fact, the blackbody assumption is not valid, as revealed by the >4.5σ difference between the 3.6 and 4.5 μm dayside temperatures (see Section 5.2). We propose a more direct comparison between the 4.5 μm phase curves of WASP43 b and HD 209458 b. Zellem et al. (2014) reported T_day = 1499 ± 15 K and T_night = 972 ± 44 K for HD 209458 b at 4.5 μm. We obtained the same dayside temperature (within 0.2σ) and ∼200–300 K lower nightside temperature for WASP43 b at the same wavelength. These comparisons suggest the WASP43 b atmosphere may have a lower circulation efficiency than HD 209458 b, but the difference appears to be significantly smaller than from the original estimates reported in the literature. Furthermore, there are some hints of variability in the nightside cloud deck of WASP43 b (see Section 5.3), that would affect the temperature measurements. A new set of observations is desirable to test this hypothesis.