REPEATABILITY OF SPITZER/IRAC EXOPLANETARY ECLIPSES WITH INDEPENDENT COMPONENT ANALYSIS

We analyze 12 eclipse observations of XO3b taken with Spitzer/IRAC in the 4.5 μm band. Ten individual eclipses were observed over six months (2012 November 11–2013 May 24), including two sets of three consecutive eclipses, and another eclipse is contained within a full-orbit observation on 2013 May 5 (PID: 90032). Each individual observation consists of 14,912 frames over 8.4 hr using IRAC's sub-array readout mode with 2.0 s integration time. In sub-array mode 64 frames are taken consecutively, the reset time is ∼1 s. We extracted 14,912 frames from the full-orbit observation to analyze the light-curve of the eclipse over a time interval similar to other observations. The last eclipse was extracted out of a 66 hr observation on 2010 April 8 (PID: 60058). Table 1 reports the dates in which the eclipses were observed.

In our model the stellar flux is constant in time, and normalized to 1. We adopt the formalism of Mandel & Agol (2002) for the eclipse model, accounting for the planet being occulted by the star. We approximate the planet's phase curve in the region of the eclipse as a linear function of the time, the slope is called "phase constant," adopting the same terminology of Wong et al. (2014). The slope is only due to planet's flux variations. While the planet is completely occulted by the star, the flux is constantly 1. The eclipse depth is defined as the (unseen) planet's flux at the center of eclipse, in units of stellar flux (see Figure 1). When fitting the eclipse models, the orbital parameters are fixed to the values reported in Table 2, taken from Wong et al. (2014), while the center of eclipse, eclipse depth, and phase constant are free parameters to determine. We also considered models with zero phase curve's slope (see Appendix A.5).

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2.3.1. Continuous Wavelet Transform (CWT)

The wavelet transform (WT) decomposes a given signal, x(t), into its frequency components. This is done by convolving the time signal with a basis of highly localized impulses or "wavelets." To fix the ideas, we assume that x(t) is a time series, although this is not necessary, as the Discrete Wavelet Transform (DWT) can be applied to different kinds of signals. The individual wavelet functions are derived from a single "mother wavelet," ψ(t), through translation and dilation of the mother wavelet. The mathematical definition of the CWT is

$$\begin{eqnarray}&&{c}_{\tau ,\varphi }={\displaystyle \int }_{{\mathbb{R}}}x(t){\psi }_{\tau ,\varphi }(t)\ dt\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\psi }_{\tau ,\varphi }(t)=\displaystyle \frac{1}{\sqrt{2}}\ \psi \left(\displaystyle \frac{t-\tau }{\varphi }\right)\end{eqnarray} \tag{ 2 }$$

where ψ_τ,φ is the mother wavelet for a given scaling φ and translation τ, and c_τ,φ is the wavelet coefficient with respect to τ and φ.

If the wavelet basis is orthogonal, the inverse wavelet transform can be used to reconstruct the original time series:

$$\begin{eqnarray}&&x(t)=\displaystyle \sum _{\varphi \in {\mathbb{Z}}}\displaystyle \sum _{\tau \in {\mathbb{Z}}}{c}_{\tau ,\varphi }{\psi }_{\tau ,\varphi }(t).\end{eqnarray} \tag{ 3 }$$

The mother wavelet can be chosen among a variety of wavelet families with different properties. For more details we refer to the relevant literature, e.g., Daubechies (1992) and Percival & Walden (2000).

2.3.2. Discrete Wavelet Transform (DWT)

Astronomical data are usually in the form of discrete time series. For the DWT the mother wavelet is denoted by h(t), and the scaling function, also called "father wavelet," is denoted by g(t). The mother and father wavelets act as high-pass and low-pass frequency filters, respectively. They are related by

$$\begin{eqnarray}&&g(L-1-t)={(-1)}^{t}\ h(t),\end{eqnarray} \tag{ 4 }$$

where L is the filter length and corresponds to the number of points in the time series x(t).

The one-level DWT is defined by

$$\begin{eqnarray}{{cA}}_{1}(\tau ) & = & (x\ast g)(t)\downarrow 2\\ & = & \displaystyle \sum _{t=-\infty }^{+\infty }x(t)\ g(2\tau -t)\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}{{cD}}_{1}(\tau ) & = & (x\ast h)(t)\downarrow 2\\ & = & \displaystyle \sum _{t=-\infty }^{+\infty }x(t)\ h(2\tau -t).\end{eqnarray} \tag{ 6 }$$

The cA₁ time series approximates the underlying low-frequency trend of x(t) (average coefficients), while the cD₁ time series represents a higher frequency component (detail coefficients). They are down-sampled by a factor of two ("$\downarrow 2$" in Equations (5) and (6)) with respect to the original time series, because of the Nyquist theorem.

It is possible to apply the g and h filters to the cA₁ time series, then obtaining new sets of coefficients, cA₂ and cD₂, and iterate the process. The n-level DWT includes the cA_n series of average coefficients, down-sampled by a factor of 2ⁿ, and n series cD₁–cD_n of detail coefficients, representing bands of higher frequencies. The original data can be reconstructed by reversing the process:

$$\begin{eqnarray}x(t) & = & {{cA}}_{n}(\tau )\ g(-t+2\tau )\\ & & +\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{\tau =-\infty }^{+\infty }({{cD}}_{i}\ h(-t+2\tau )).\end{eqnarray} \tag{ 7 }$$

2.3.3. Wavelet ICA

In this section we describe the wavelet ICA algorithm, which is used in a variety of contexts, such as medical sciences (e.g., Inuso et al. 2007; La Foresta et al. 2012; Mammone et al. 2012), engineering (e.g., Lin & Zhang 2005), acoustic (e.g., Moussaoui et al. 2006; Zhao et al. 2006), image denoising (e.g., Karande & Talbar 2014), and astrophysics (e.g., Waldmann 2014).

Be ${\boldsymbol{x}}={({x}_{1},{x}_{2},...,{x}_{m})}^{T}$ the column vector of observed signals. In this paper, ${{\boldsymbol{x}}}_{k}$ are individual pixel time series, so-called pixel light curves, which are mixtures of different source signals, astrophysical or instrumental in nature, and Gaussian noise. The formalism adopted in this subsection is valid in a more general context, where the x_k can be any kind of mixed signals. ICA is a linear transformation of the observed (mixed) signals, which minimizes the mutual information to decorrelate the independent components:

$$\begin{eqnarray}&&{\boldsymbol{x}}={\boldsymbol{As}},{\boldsymbol{s}}={\boldsymbol{Wx}}\end{eqnarray} \tag{ 8 }$$

where ${\boldsymbol{s}}={({s}_{1},{s}_{2},...,{s}_{m})}^{T}$ is the column vector of the original source signals, ${\boldsymbol{A}}$ is the matrix of mixing coefficients, and ${\boldsymbol{W}}$ = ${A}^{-1}$. We refer the reader to Waldmann (2012) and Morello et al. (2014, 2015) for additional details.

The ability of ICA to decorrelate non-Gaussian signals is inherently limited to a low Gaussian noise amplitude relative to the non-Gaussian signals. Wavelet ICA algorithms are designed to be less sensitive to white noise compared to the simple ICA separation, described above. In wavelet ICA algorithms, the DWT is applied to the observed signals:

$$\begin{eqnarray}&&{x}_{k}(t)\to {\hat{x}}_{k}=({{cA}}_{k,n},{{cD}}_{k,n},...,{{cD}}_{k,1})\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\boldsymbol{x}}(t)\to \hat{{\boldsymbol{x}}}={({\hat{x}}_{1}(t),{\hat{x}}_{2}(t),...,{\hat{x}}_{m}(t))}^{T}\end{eqnarray} \tag{ 10 }$$

The ICA separation is performed with the series of coefficients:

$$\begin{eqnarray}&&\hat{{\boldsymbol{s}}}=\hat{{\boldsymbol{W}}}\hat{{\boldsymbol{x}}}\end{eqnarray} \tag{ 11 }$$

The independent components series of coefficients are

$$\begin{eqnarray}&&{\hat{s}}_{l}=({{cA}}_{l,n},{{cD}}_{l,n},...,{{cD}}_{l,1})\end{eqnarray} \tag{ 12 }$$

They can be converted into the time domain through inverse DWT (Equation (7)).

The DWT preliminarly separates the high-frequency components from the low-frequency trend, enhancing the ability of ICA to disentangle the low-frequency independent components. This step is particularly important in cases where the Gaussian noise is dominant. Additional processing options/variants have been proposed in the literature to further improve the ICA performances in specific contexts, e.g., coefficients' thresholding (Stein 1981; Donoho 1995), suppression of some frequency ranges (Lin & Zhang 2005), taking individual levels as input to ICA (Inuso et al. 2007; Mammone et al. 2012). In this paper, we aim to provide the most objective analysis of the data sets with minimal prior assumptions, hence those variants are not considered. The impact of those variants and other operations to the data will be carefully investigated in future studies.

In this section we list the main steps of the wavelet pixel-ICA method, followed by a more accurate description and comments:

• 1.  
  Selecting an array of pixels. The raw light-curve is the sum of the individual pixel time series within the selected array.
• 2.  
  Removing outliers.
• 3.  
  Subtracting the background from the raw light curve.
• 4.  
  Computing the wavelet transforms of the time series from the pixels within the selected array, hereafter called pixel light curves.
• 5.  
  Performing ICA decomposition of the wavelet-transformed pixel light curves.
• 6.  
  Computing the inverse wavelet transforms of the independent components.
• 7.  
  Simultaneous fitting of the components (except the eclipse one) and astrophysical model on the raw light curve.
• 8.  
  Estimating parameter error bars.

2.4.1. Selecting the Pixel-array

2.4.2. Outlier Rejection

2.4.3. Background Subtraction

2.4.4. Wavelet Transforms

2.4.5. ICA Decomposition

2.4.6. Inverse Wavelet Transforms

2.4.7. MCMC Fitting

2.4.8. Final Error Bars

The final parameter error bars are

$$\begin{eqnarray}&&{\sigma }_{\mathrm{par}}={\sigma }_{\mathrm{par},0}\sqrt{\displaystyle \frac{{\sigma }_{0}^{2}+{\sigma }_{\mathrm{ICA}}^{2}}{{\sigma }_{0}^{2}}}\end{eqnarray} \tag{ 13 }$$

${\sigma }_{0}^{2}$ is the sampled likelihood variance, approximately equal to the variance of the residuals for the best transit model; ${\sigma }_{\mathrm{ICA}}^{2}$ is a term accounting for the potential bias of the components obtained with ICA.

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ICA}}^{2}=\displaystyle \sum _{j}{o}_{j}^{2}{\mathrm{ISR}}_{j}\end{eqnarray} \tag{ 14 }$$

where ISR is the so-called Interference-to-Signal-Ratio matrix, and o_j are the coefficients of the non-eclipse components. The term relative to the goodness of the fit of the components does not appear in Equation (14), as it is automatically included in σ₀ when the components' coefficients and astrophysical parameters are fitted simultaneously.

Figure 2 shows the raw light curves for the 12 observations, the correspondent detrended eclipses and best models, obtained using the 5 × 5 array and binning over 32 frames, i.e., ∼64 s. The individual measurements of eclipse depth and phase constant are reported in Figure 3 and Table 3. The results from all epochs are consistent within the error bars, suggesting the lack of any detectable astrophysical variability for this system, and residual instrument variability. By taking the weighted means of the individual measurements, we obtain global best estimates of (1.57 ± 0.03) × 10⁻³ for the eclipse depth, and (4.4 ± 2.0) × 10⁻⁴ days⁻¹ for the phase constant.

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The underlying assumption for the weighted mean to be a valid parameter estimate is that individual measurements of that parameter are normally distributed around the same mean value with variances ${\sigma }_{i}^{2}$, and there are no systematic errors. The reduced chi-squared can be used to test, in part, this hypothesis:

$$\begin{eqnarray}&&{\chi }_{0}^{2}=\displaystyle \frac{1}{n-k}\displaystyle \sum _{i=1}^{n}\displaystyle \frac{{({x}_{i}-\bar{x})}^{2}}{{\sigma }_{i}^{2}}\end{eqnarray} \tag{ 15 }$$

where x_i ± σ_i are the individual measurements, $\bar{x}$ is the weighted mean value, n = 12 is the number of measurements, and k = 1 is the number of calculated parameters. Ideally, if the assumption is valid, we should expect ${\chi }_{0}^{2}$ ≲ 1. Conventionally, the hypotesis is rejected if ${\chi }_{0}^{2}\gt {M}_{n,k}$, where M_n,k is the critical value corresponding to a probability of less than 5% for the hypothesis to be valid. We found ${\chi }_{0}^{2}=0.42$ for the eclipse depth, and ${\chi }_{0}^{2}=1.0$ for the phase constant, confirming the non-detection of any inter-epoch variability. ${\chi }_{0}^{2}=0.42$ may suggest that the error bars for the eclipse depth are overestimated, but this is not totally surprising, given that we actively increased them to account for potential uncorrected systematics and biases in the detrending method. Note that the reduced chi-squared tests whether the actual dispersion in the measurements is consistent within their error bars, but it is not sensitive to a uniform bias for all measurements, e.g., a constant shift. Hence, it is not sufficient alone to justify the use of the weighted mean as global estimate of a parameter. Additional tests, reported in the appendices, show that the weighted mean result is very stable for the eclipse depth. The phase constant appears to be more dependent on certain detrending options, in particular background subtraction. In this case, the adopted weighted mean error bar of 2.0 × 10⁻⁴ days⁻¹ is a lower limit, valid under some caveats. In the worst-case scenario, the maximum error bar, calculated without scaling when combining multiple measurements, is 7 × 10⁻⁴ days⁻¹.

Our results are consistent within 1σ with the ones from a previous analysis reported in Wong et al. (2014) (see Figures 3 and 4). Our error bars are generally larger by a factor 0.8–1.5 for the eclipse depth (smaller in 1 case) and 1.0–2.0 for the phase constant compared to the ones in the literature. The factors for the weighted mean eclipse depth and phase constant are 0.9 and 1.4, respectively. Slightly larger error bars are a worthwhile trade-off for much higher objectivity, which derives from the lack of assumptions about the origin of instrument systematics and their functional forms in our detrending method. We also note that, despite the larger nominal error bars, the dispersions in our best parameter estimates are slightly smaller than the ones calculated from the results reported in Wong et al. (2014; see Table 4).

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Table 4.  Weighted Mean Parameter Results, Dispersions, and Reduced Chi Squared Values Obtained in this Paper and Reported in Wong et al. (2014)

Eclipse Depth	This Work	Wong et al. (2014)
Best estimate	(1.57 ± 0.03) × 10−3	${1.580}_{-0.039}^{+0.033}$ × 10−3
Dispersion	7.2 × 10−5	8.4 × 10−5
${\chi }_{0}^{2}$	0.42	0.86
Phase Constant (days−1)	This work	Wong et al. (2014)
Best estimate	(4.4 ± 2.0) × 10−4	${6.0}_{-1.6}^{+1.3}$ × 10−4
Dispersion	7.0 × 10−4	11.3 × 10−4
${\chi }_{0}^{2}$	1.0	4.3

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The reduced chi-squared values inferred from their individual parameter estimates are ${\chi }_{0}^{2}=0.86$ for the eclipse depth, and ${\chi }_{0}^{2}=4.3$ for the phase constant. While the first ${\chi }_{0}^{2}$ value is consistent with the hypothesis of a constant transit depth within the quoted error bars, the second ${\chi }_{0}^{2}$ value indicates that the analogous hypothesis for the phase constant can be rejected (less than 0.1% probability of being true). This may suggest either that they were able to detect some astrophysical variability of the phase curve's slope, or that their individual error bars are underestimated by a factor ∼2. Given that the astrophysical slope is degenerate with other instrumental trends, such as long-term position drift of the telescope and possible thermal heating, it is possible that their individual error bars do not fully accounts for these degeneracies, as the authors themselves state. If this is the case, we observe that their final error bar on the phase constant, derived by a joint fit of all eclipses, could be equally underestimated, because it is not guaranteed that residual systematic errors cancel out over multiple observations as if they were random errors. Note that the joint fit approach is theoretically valid under the same assumptions for which the weighted mean is valid, and the two approaches are expected to lead to very similar results (we checked that this happens in this case). In conclusion, our reanalysis confirms the results reported in Wong et al. (2014) for the eclipse depth, and relative inter-epoch variability, but not the 4σ detection of a non-flat phase curve's slope during the eclipse, as larger error bars are needed to account for the possible residual systematics.

Given the large number of frames (14,912) for each observation, binning data is very useful to decrease the computational time needed to run the MCMCs for the eclipse parameters and scaling coefficients for the independent components (see Section 2.4.7). Some authors suggest that an optimal choice of the binning size can be useful to reduce the noise on the timescale of interest (Deming et al. 2015; Kammer et al. 2015), provided that the theoretical curve is similarly binned as necessary to eliminate bias, and the bin size is not too long to cause significant loss of astrophysical information (Kipping 2010).

An additional choice for the pixel-ICA algorithm is whether to bin the pixel time series prior the ICA separation, or to bin the independent components extracted from unbinned pixel time series. We found that the two options are almost equivalent, as the eclipse signals obtained after removing the systematic components from the raw light-curve are identical (discrepancies of one to two orders of magnitudes smaller than the fitting residuals), except in cases for which the unbinned ICA separation fails to retrieve an eclipse component. It is worth noting that for the unbinned case, the amplitude of the total noise plus systematics is higher than the the eclipse depth. Thus we decided to bin the individual pixel light curves prior ICA retrieval.

We compare the results obtained for all the observations with bin sizes of 32 and 64 frames, i.e., 64 and 128 s, respectively. First, we test the gaussianity of fitting residuals by calculating their rms as a function of the bin size, b. If fitting residuals are white noise, the rms would scale as $1/\sqrt{b}$. Figure 5 shows that, in both cases, the rms of fitting residuals slightly deviates from the expected behavior of white noise. Those deviations are smaller for the analysis with a bin size of 32 frames. The parameter results obtained with the two binning choices are all consistent within 0.5σ, and on average within 0.16σ (see also Figures 6 and 12). Both the error bars and the overall dispersions are smaller for the cases with bin size of 32 frames by factors of ∼1.4. We consider the results obtained with bin size of 32 frames as our best results.

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The DWT of a time series is specified by the choice of a mother wavelet and the number of levels (see Section 2.3.2). We adopted a Daubechies-4 mother wavelet, and one-level decomposition. For a few observations we tested multiple choices of the mother wavelet, among the Daubechies, Biorthogonal, Symlets, and Coiflets families, and number of decomposition levels. We refer to Daubechies (1992), Percival & Walden (2000) for details about the wavelet properties. We found that different choices of the mother wavelet are not significant, e.g., discrepancies in the eclipse signals are one to two orders of magnitudes smaller than the fitting residuals, while level decompositions higher than 1 usually appear to make it impossible for ICA to retrieve the eclipse. The difficulties with higher-level DWTs may arise from sub-sampling the average coefficients, and the fact that some of the low-frequency non-Gaussian components may be smeared over higher levels of detail coefficients.

Uncorrected background may bias the normalized amplitude of the eclipse depth, as well as the phase curve's slope, if the background is not constant over time. The typical morphology of a background time series is either a constant function or a slow monotonic drift. The lack of a distinct temporal structure and non-gaussianity makes it difficult to disentangle with ICA, as well as other statistical methods. For this reason, we performed ad hoc background subtractions before ICA detrending (see Section 2.4.3). In this section, we discuss the impact of this step in the analyses, by comparing results obtained with and without background subtraction. These are also used to infer the maximum parameter errors that can be caused by an inappropriate background correction.

Figure 7 shows an example of background time series estimated for one observation. The measured mean background level slightly varies from one observation to the other, but it is always less than 0.6% of the total flux from the system, hence potentially affecting the eclipse depth by less than 10⁻⁵, well below the error bars. The background is also not constant during one observation: it has a small ramp for the first ∼20 minutes (except for the eclipses extracted from longer observations), then continues to slowly increase.

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Figure 8 shows the parameter results obtained with and without background subtraction. The greatest discrepancies are observed for the phase constant, which is systematically higher by 2–20 × 10⁻⁴ days⁻¹ for the cases without background subtraction, suggesting the presence of uncorrected systematics. It is quite remarkable that our individual error bars automatically account for those systematics, but, given their non-random nature, the weighted mean error bars for the phase constant cannot be compared. It is difficult to prove the superiority of the results obtained with background subtraction over the others, as the residuals between the full models and the relevant raw light curves (see Section 2.4.7) are very similar for the two cases, e.g., similar amplitudes and time correlations, leading to similar error bars. Slightly smaller parameter dispersions are obtained with background subtraction rather than without, in particular 7 × 10⁻⁵ versus 9 × 10⁻⁵ for the eclipse depth, and 7 × 10⁻⁴ versus 8 × 10⁻⁴ days⁻¹ for the phase constant. The higher mean value of the phase constant obtained without background subtraction, i.e., ∼13 × 10⁻⁴ days⁻¹, appears to be less likely, as it would require a higher than expected increase in the atmospheric temperature due to stellar irradiation during the eclipse, and/or strong horizontal disomogeneities either in temperature, chemical composition and/or clouds (Cowan & Agol 2011; Kataria et al. 2013; Agúndez et al. 2014). If we assume that systematics are removed with background subtraction, we can take the weigthed mean as the best estimate for the phase constant.

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The discrepancies between eclipse depth measurements with and without background subtraction are in the range 10⁻⁵–10⁻⁴, and, on average, the eclipse depth is smaller by ∼4 × 10⁻⁵ for the case without background subtraction. Although this is more than the 10⁻⁵ difference expected from the mean background level relative to the mean stellar flux (see the discussion in the previous paragraph), in this case, the two weighted means are consistent within 0.5σ. We found that the main effect of background on the eclipse depth is due to correlations between the measured eclipse depth at the eclipse center and the phase constant, in terms of Pearson correlation coefficients between the relevant MCMCs. The details of this study are beyond the scope of this paper.

In previous studies, we found that, for high-S/N transit observations, pixel-ICA performances are only slightly dependent on the choice of the array (Morello et al. 2014, 2015). The 3 × 3 array is too narrow compared to the Spitzer/IRAC Point Spread Function, and for this reason, it is not photometrically stable (although it usually leads to consistent results). Larger arrays are more photometrically stable, but they include more noise. Also, noisier pixel light curves, appear to increase the uncertainties in the ICA decomposition (σ_ICA, see Section 2.4.8), so that the smallest error bars were usually obtained using the 5 × 5 array.

In this work, we analyzed all data sets with two different choices of the pixel-array, i.e., 5 × 5 and 7 × 7. For the analyses with the 7 × 7 array, we only adopted the 64 frames bin size, for which the MCMC fitting is faster. Figure 9 compares the results obtained with the two different arrays and the same bin size. The two sets of results are consistent well within 1σ, but error bars obtained with the 7 × 7 array are 1–1.5 times larger than the ones obtained with the 5 × 5 array, despite the residuals in the fits are similar and often smaller for the 7 × 7 cases. This confirms the conclusions obtained from previous analyses, in particular:

• 1.  
  The 5 × 5 array leads to smaller error bars than other squared arrays of pixels;
• 2.  
  Parameter results from different arrays are consistent well within 1σ.

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In this case, the choice of a less optimal array increases the error bars more significantly than in our previous analyses, most likely because of the lower S/N of the observations analyzed here.

Figure 10 compares the results obtained with and without background subtraction for the case of 7 × 7 array. The same considerations discussed in Appendix A.3 for the 5 × 5 array are valid for the 7 × 7 array.

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At the end of Appendix A.3 we revealed the existence of a correlation between the eclipse depth and phase constant parameter, which is stronger for data with a higher slope, such as the cases without background subtraction. This may affect the eclipse depth estimate in case of residual systematics with a slope, e.g., uncorrected background. In our analyses, the maximum bias on the eclipse depth due to correlations with this kind of systematics is ∼4 × 10⁻⁵, provided the systematic slope is not larger than our individual error bars.

Here we test the consequences of adopting zero phase constant while fitting for the eclipse depth. This is equivalent to the assumption that the phase curve is flat before and after the eclipse, and the slope is entirely due to instrumental effects. Compared to the cases with the phase constant as free parameter, fitting residuals with zero phase constant are not significantly larger. Figure 11 compares results for the eclipse depth with free and zero phase constant, with and without background subtraction. The weighted mean results are reported in Figure 12. We note that differences in eclipse depth measurements with and without background subtraction are smaller for the zero phase constant models, suggesting that eclipse depth measured with zero phase constant models is less affected by residual systematics with a slope. Free phase constant models are valid in a more general context, as, differently from the zero phase constant models, they approximate a planet's flux variability during the observation. The consistency between the results obtained with the two classes of models indicates that, in this case, the planet's flux variability in the proximity of the eclipse is smaller than the error bars.

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Figure 12 reports the weighted mean eclipse depth estimated for all the tests discussed in the appendices. Note that they are mutually consistent at the 0.5σ level, and the range is 6 × 10⁻⁵.