INFERENCE OF INHOMOGENEOUS CLOUDS IN AN EXOPLANET ATMOSPHERE

We observed two occultations of Kepler-7b with Spitzer (Werner et al. 2004) in IRAC (Fazio et al. 2004) 3.6 μm channel as well as two other in IRAC 4.5 μm channel between 2011 August and November. All Astronomical Observation Requests (AORs) were obtained as part of program 80219 (PI: H. Knutson). Datasets are ∼9 hr long and were obtained in full-array mode with an individual exposure time of 10.4 s. A total of 2440 frames was collected for each AOR. We perform a data reduction of all AORs similar to Demory et al. (2011a), using as input the Basic Calibrated Data files produced by the Spitzer pipeline version 18.18.0. In a first step, we test 12 apertures ranging from 1.8 to 4.5 pixels and find the lowest rms using 2.6 and 2.8 pixel apertures at 3.6 and 4.5 μm respectively. We obtain an rms of 6380 and 6710 ppm for the two 4.5 μm AORs with a moderate contribution from correlated noise of less than 20%. Our analysis of the 3.6 μm data resulted in significant correlated noise in both time-series (>40%). Because of the long occultation duration of Kepler-7b (5.3 hr), the remaining out-of-transit photometry is small on each side of the eclipse, making the occultation parameters retrieval delicate in the presence of correlated noise. In a second step, we apply the noise-pixel variable aperture technique (Lewis et al. 2013) to all AORs. We find this method mitigates systematics found at 3.6 μm. We report corresponding rms of 4900 and 4750 ppm for both AORs in this channel using this technique, with a reduced correlated noise contribution of ∼15%. We do not notice any improvement using noise-pixel aperture over the classical fixed-aperture photometry reduction at 4.5 μm.

In order to model these data, we use the Markov Chain Monte Carlo (MCMC) implementation presented in Gillon et al. (2012). We assume a circular orbit (D11), set the occultation depth as a jump parameter and impose priors on the orbital period P, transit duration W, time of minimum light T₀ and impact parameter b = acos i/R_⋆ based on D11. For each MCMC fit (at 3.6 and 4.5 μm), we run two chains of 10⁵ steps and assess their convergence using the statistical test from Gelman & Rubin (1992).

We use the Bayesian Information Criterion (BIC) to select the optimal baseline model for our 4.5 μm observations. We find the most adequate model based on a classical second order x–y polynomial (Demory et al. 2011a; Equation (1)) to correct the "pixel-phase" effect, added to a time-dependent linear trend. The baseline model for our 3.6 μm data consists of the noise-pixel parameter alone. We discard the first ∼25–35 minutes of all AORs that are affected by a noticeable detector ramp and/or increased noise, already noticed in warm-Spitzer photometry (e.g., Deming et al. 2011). Our Spitzer/IRAC raw lightcurves are shown on Figure 1 (left).

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We repeat the same MCMC fits for both channels setting the occultation depth to zero, to compare the BIC between a model that includes the occultation and a model that does not. The MCMC fits including the occultation model yield an occultation depth of 164 ± 150 ppm at 3.6 μm and 367 ± 221 ppm at 4.5 μm. We compare the BIC of these runs to the MCMC fits that do not include the occultation model. The odds ratio between both models is ∼180 and ∼100 in favor of the model without occultation at 3.6 and 4.5 μm respectively. Based on our dataset, the occultation is detected in none of the channels. We derive corresponding 3σ upper limits of 615 and 1010 ppm at 3.6 and 4.5 μm. We employ a PHOENIX (Hauschildt et al. 1999) model of Kepler-7 using the D11 stellar parameters to convert these occultation depth upper-limits into brightness temperatures. We find these 3σ upper-limits to be 1700 and 1840 K at 3.6 and 4.5 μm. Our final phase-folded occultation lightcurves are shown on Figure 1 (right).

We base our analysis on Kepler (Batalha et al. 2013) quarters 1–14 long-cadence simple aperture photometry (Jenkins et al. 2010) that span more than 1200 days of quasi-continuous observations, which is three times more data than in D11. We mitigate instrumental systematics by fitting the first four cotrending basis vectors (CBVs) to each quarter using the PyKE software (Still & Barclay 2012). We find that outliers represent less than ∼0.5% of the dataset. We then normalize each quarter to the median. In total, 56,000 datapoints are collected. We employ our MCMC framework presented in Section 2 to account for photometric trends longer than twice the planetary orbital period by fitting a second-order polynomial to the out-of-eclipse data.

We then evaluate the contribution from correlated noise on timescales corresponding to the orbital period. We cut the whole data into 5 day duration segments and compute a scaling factor β based on the standard deviation of the binned residuals for each light curve using different time-bins (Gillon et al. 2010). We keep the largest β value as a criterion to discard data segments affected by significant correlated noise. We obtain a mean β = 1.19 over the whole data set and discard those with threshold β > 2.1, which account for ∼5% of the complete dataset. All data discarded affect predominantly quarters 12–14, when increased solar activity and coronal mass ejections resulted in a decrease of Kepler's pointing accuracy and thus an increase in systematic noise. We finally note that in contrary to pre-whitening techniques (as employed in D11), the data-reduction method presented here preserves all phase-curve properties.

To assess the robustness of the phase-curve properties, we repeat the analysis presented above several times, by increasing the number of CBV components up to 8, by decreasing the threshold β values and by using linear or third-order polynomials to account for the long-term trends. We find the phase amplitude, peak-offset and occultation depth values to remain consistent within 1σ uncertainties (see Section 3.3). The phase-curve signal is therefore not due to (nor affected by) the detrending. Two of us (B.O.D., T.B.) performed independent analyses of the dataset and obtained results in excellent agreement.

Figures 2 and 3 demonstrate the stability of the phase-curve signal across Q1–Q14. This would not be the case if the phase curve was of instrumental origin as while Kepler systematics can be consistent in amplitude across quarters, they are definitely not consistent in phase (e.g., Kinemuchi et al. 2012). Any signal due to Kepler systematics would thus average out across quarters. This strongly favors the phase-curve being of astrophysical origin.

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We search for all frequencies in the dataset to assess any risk of contamination of the planetary phase curve. To quantify how frequencies and amplitudes evolve with time, we perform a wavelet transform analysis using the weighted wavelet Z-transform algorithm developed by Foster (1996). We do not detect any clear signature, apart from the planet orbital signal. Kepler-7 is intrinsically quiet and any stellar activity remains nominal over Q1–Q14 observations, with no quarter-dependent fluctuations. We notice a barely detectable periodicity at ∼16.7 days that could correspond to the rotational period of the star, which translates to an equatorial velocity of V_eq ∼ 6 km s⁻¹ assuming R_⋆ = 2.02 R_☉ (D11). This is broadly consistent with Kepler-7's stellar projected rotation $v\,{\rm \sin }\,i=4.2$ km s⁻¹ (Latham et al. 2010).

The host star is unlikely to contaminate our phase-curve for several reasons. As we phase-fold data over more 3.5 yr, only stellar variability exactly phased on the planetary orbital period (or a multiple) and consistent over the duration of the observations could affect the phase-curve shape. First, the stellar rotational velocity suggests that the star is not tidally locked to the planet, as the planetary orbital period is only ∼4.89 days. The stellar rotation and planetary orbital periods are different by a non-integer factor of ∼3.4. Second, stellar pulsations with a period of ∼5 days are unlikely for a sub-giant star and would have been visible in the data. Third, as we do not clearly detect stellar variability in the photometry, only small starspots could be present, but those starspots would have a short lifetime (e.g., Strassmeier 2009). Even in the case of starspots that are stable over more than 3 yr, differential rotation would cause distortions in the lightcurve across quarters that are not observed (Figure 2). Furthermore, spots or group of spots do not usually produce sinusoidal lightcurves but rather sequences of flat and V-shaped lightcurves (e.g., Harrison et al. 2012). Finally, we do not detect interactions between the star and the planet in the form of ellipsoidal or beaming components in the phase curve.

We finally take into account a faint stellar companion located 19 east of Kepler-7 with a Δmag =4.0 both in J and Ks bands (Adams et al. 2012). These flux ratios suggest a similar spectral type and discard the possibility of a cool star. In order to detect a significant contamination from the companion star with a period commensurate with Kepler-7b's orbital period, we split the full dataset in segments of duration equal to a quarter. Each quarter has a specific aperture with a different contribution from the fainter companion star. The reported consistency at the 1σ level of the phase curve properties (amplitude, phase-peak offset) across quarters suggest a negligible contamination from the stellar neighbor.

We therefore conclude that the phase curve is of planetary origin.

Kepler-7b's phase curve deviates from a pure Lambert-law phase-dependent behavior (e.g., Sobolev 1975) expected for isotropic scattering alone (Figure 3, green). The main feature of Kepler-7b's phase curve is a delay of 13 ± 3.5 hr of the phase-curve's peak from the occultation center. This delay implies that the hemisphere-integrated flux is maximum to the west of Kepler-7b's substellar point. We further measure a phase-curve amplitude of 50 ± 2 ppm and an occultation depth of 48 ± 3 ppm, corresponding to a geometric albedo A_g = 0.35 ± 0.02. This occultation depth translates to a brightness temperature of $2645^{+20}_{-30}$ K in the Kepler bandpass, which is 1000 K and 800 K larger than the infrared brightness temperatures upper limits measured at 3.6 and 4.5 μm respectively (see Section 2). We found our phase-curve amplitude and occultation depth to be in agreement with previous analyses (D11; Kipping & Bakos 2011; Coughlin & López-Morales 2012).

The key features of Kepler-7b's phase-curve translate directly into constraints on maps (Cowan & Agol 2008) assuming a tidally locked planet on a circular orbit. A planetary phase-curve F_p/F_⋆ measures the planetary hemisphere-averaged relative brightness 〈I_p〉/〈I_⋆〉 as follows:

$$\begin{equation} \frac{F_p}{F_{\star }}(\alpha)=\frac{\langle I_p\rangle (\alpha)}{\langle I_{\star }\rangle }\left(\frac{R_p}{R_{\star }}\right)^2, \end{equation} \tag{ 1 }$$

where α is the orbital phase.

We first notice that Kepler-7b's planetary flux contribution starts from phase 0.18 ± 0.03, when the meridian centered 25 ± 12° east of the substellar point appears. Second, the phase-curve's maximum is located at phase 0.61 ± 0.03, implying that the brightest hemisphere is centered on the meridian located 41 ± 12° west of the substellar point. Third, the planetary flux contribution vanishes around the transit, implying that the "bright" area extends up to the western terminator, while its extension to the east of the substellar point is nominal. We finally note that the phase-curve's amplitude of 50 ± 2 ppm converts into an hemisphere-averaged relative brightness 74 ± 2 × 10⁻⁴ (Equation (1)).

We longitudinally map Kepler-7b using the MCMC implementation presented in de Wit et al. (2012). This method has been developed to map exoplanets and to mitigate the degeneracy between the planetary brightness distribution and the system parameters. We use two model families similar to the "beach-ball models" introduced by Cowan et al. (2009): one using n longitudinal bands with fixed positions on the dayside and another using longitudinal bands whose positions and widths are jump parameters in the MCMC fit. We choose the two simplest models from these families: a 3-fixed-band model and 1-free-band model so as to extract Kepler-7b's longitudinal dependence of the dayside brightness as well as the extent of the "bright" area. For both models, we compute each band's amplitude from their simulated lightcurve by using a perturbed singular value decomposition method. The corresponding median brightness maps are shown on Figure 4. The 1-free-band model (Figure 3, blue) finds a uniformly bright longitudinal area extending from 105 ± 12° west to 30 ± 12° east with a relative brightness 78 ± 4 × 10⁻⁴ (Figure 4, left). The 3-fixed-band model (Figure 3, red) finds bands of relative brightness decreasing from the west to the east with the following values: 100 to 68 and 3 ± 6 × 10⁻⁴ (Figure 4, right). We finally note that the 1-free-band model finds a bright sector extending to the night side, due to the sharp flux increase observed around transit (Figure 3).

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