A NEW LOOK AT SPITZER PRIMARY TRANSIT OBSERVATIONS OF THE EXOPLANET HD 189733b

The two Spitzer observations of HD 189733b discussed here were performed on 2006 October 31 (ID 30590) and 2007 November 25 (ID 40732).

The first observation consists of 1936 exposures using IRAC's stellar mode (full-array), taken over 4.5 hr: 1.8 hr on the primary transit of the planet, 1.6 hr before, and 1.1 hr after transit. The reset time is 8.4 s. During the observation, the centroid of the star HD 189733 was stable to within one pixel.

The second observation was of 1920 exposures using IRAC's sub-array mode, over 4.5 hr; 1.8 hr were spent on the primary transit of the planet, 1.7 hr before, and 1 hr after transit. The interval between consecutive exposures is 8.4 s. Each exposure consists of 64 reads at high speed cadence of 0.1 s. Only for the ObsID 40732, we replaced the 64 reads of each exposure with their mean values, in order to have a manageable number of data points, to reduce the random scatter, and to have the same sampling of the ObsID 30590. During the observation, the centroid of the star HD 189733 was again stable to within one pixel.

ICA consists of a transformation from a set of recorded signals to an equivalent set of maximally independent components. The underlying assumptions are that:

• 1.  
  each recorded signal is a linear combination of the same source signals;
• 2.  
  the source signals are mutually independent.

We can express this model as

$$\begin{equation} \begin{array}{c}x_1 = a_{1,1} s_1 + a_{1,2} s_2 +... + a_{1,n} s_n \\ x_2 = a_{2,1} s_1 + a_{2,2} s_2 +... + a_{2,n} s_n \\ \vdots \\ x_n = a_{n,1} s_1 + a_{n,2} s_2 +... + a_{n,n} s_n, \end{array} \end{equation} \tag{ 1 }$$

where x_i, i = 1...n, are the recorded signals, s_j, j = 1...n, are the source signals, and a_{i, j} are numerical coefficients. Equation (1) can be written in matrix form as

$$\begin{equation} {\bf x} = {\bf A} {\bf s}, \end{equation} \tag{ 2 }$$

where x is the column vector containing the recorded signals, s is the column vector containing the source signals, and A is the matrix of the coefficients, the so-called mixing matrix.

The aim of ICA is the "blind" separation of the source signals from the observations, i.e., without any additional information (except the assumed mutual independence of the source signals). In other words, the ICA algorithms search for the matrix W that transforms the recorded signals such that the mutual statistical independence is maximized:

$$\begin{equation} {\bf W} {\bf x} = {\bf W} {\bf A} {\bf s} \end{equation} \tag{ 3 }$$

If the assumptions are valid, then WA = D, where D is a diagonal matrix, so that

$$\begin{equation} {\bf W} {\bf x} = {\bf D} {\bf s}\protect. \protect \end{equation} \tag{ 4 }$$

The diagonal matrix D means that the extracted signals can be rescaled without changing the mutual independence.

To maximize the said independence, several approaches and implementations have been proposed (Hyvärinen et al. 2001; Tichavský et al. 2008). We used the MULTICOMBI algorithm (Tichavský et al. 2008), which optimally mixes EFICA and WASOBI, based on maximizing the non-Gaussianity of the extracted signals (Koldovský et al. 2006) and their temporal decorrelations (Yeredor 2000), respectively.

In this work, the observed signals are light curves of a star, recorded for a time interval that includes an exoplanetary transit event. These light curves contain at least three independent contributing signals:

• 1.  
  the astrophysical signal,
• 2.  
  the signal of instrumental systematics, and
• 3.  
  stochastic noise.

It is possible, in principle, to decompose further the astrophysical and instrumental systematics signals. The former is the sum of the transit signal, the astrophysical background, possible stellar activity signals, etc., the latter is the sum of different effects from different parts of the instrumentation. All these signals are expected to be independent from each other as they have different origins. By contrast, their linear combinations (i.e., the observed light curves) are clearly not mutually independent. It is worth stressing that to disentangle effectively all these signals, we need, at least, the number of available light curves to be equal to the number of signals. Therefore, we need light curves recorded with the same instrument (since light curves recorded with different instruments have different systematics plus the astrophysical signals, so that the number of source signals is greater than the number of light curves). In principle, using light curves recorded at different times with the same instruments should not work; although the systematics have the same origins, the relevant signals are not necessarily in phase, and so may differ by more than a simple scaling factor. Additionally, further differences might be present due to stellar variability. However, the transit signal, being common to all the light curves, is potentially detrendable. A successful extraction of a transit signal from a time series spanning several exoplanetary transit events, conveniently split into sub-light curves, is described in Waldmann (2012).

The advantage of spectroscopic observations over photometry is the provision of simultaneous light curves at different wavelengths with largely common instrumental systematics. The transit signals at each wavelength can be obtained by subtracting proper systematics models from the light curves (an accurate direct extraction of the transit is impossible due to the limb darkening effect). By using this technique, Waldmann et al. (2013) extracted an infrared transmission spectrum of HD 189733b between 1.51  μm and 2.43  μm from a Hubble/NICMOS data set.

The main novelty of the algorithms we use here is their ability to detrend the transit signal from a single photometric observation of just one primary transit. This is possible because, even if stars can be approximated by point sources, the instrument is purposely de-focused to spread the PSF over several detector pixels, and the position of the target star on the detectors is stable to within one pixel. During an observation, there are several pixels detecting the same astrophysical signals at any time, but with different scaling factors, depending on their received flux, their quantum efficiency, and the instrument PSF.

We performed an ICA decomposition over several pixel-light curves, i.e., the time series from individual pixels, in order to extract the transit signal and other independent signal components (stellar or instrumental in nature).

Once a set of independent components has been obtained from a selected set of pixel-light curves, different approaches to obtain the transit signal can be considered.

Method 1: direct identification of the transit component. In principle, if one of the independent components extracted has the morphology of the transit signal, we assume that one to be the transit signal, multiplied by an undetermined scaling factor. We renormalize the signal by the mean value calculated on the out-of-transit part, so that the out-of-transit level is unity.

Method 1 is not applicable to the extraction of accurate transit signals from spectroscopically resolved observations of a primary transit at different wavelengths because of the wavelength dependence of stellar limb darkening. This is not a problem in our case because all the pixels record the same wavelengths.

Method 2: non-transit-components subtraction. Another approach to estimating the transit signal is to remove all the other effects from an observed light curve, i.e., by subtracting all the components other than the transit one, properly scaled. The scaling factors can be determined by fitting a linear combination of the components, plus a constant term, to the out-of-transit part of the light curve.⁵ The coefficients of the linear combination and the constant are the free parameters to fit.

Instead of fitting the non-transit-components on the pixel-light curves and then subtracting, we performed these processes on the spatially integrated light curves, obtained by summing all the individual pixel-light curves. The integrated light curves are much less noisy than the individual pixel curves.

After the extractions of the detrended and normalized transit time series, we modeled them using the Mandel & Agol (2002) analytical formulae. We can compute the observed flux as a function F(p, z), where p = r_p/R_s is the ratio between the planetary and the stellar radii, and z = d/R_s is the distance between the centers of their disks projected onto the sky divided by the stellar radius. The relative distance z is a function of time, determined by the orbital parameters.

We assumed the orbital period P, zero eccentricity, and a quadratic limb darkening model (Howarth 2011). The values of the fixed parameters are reported in Table 1.

We first determined the centers of the transit ephemeris by fitting some symmetric models with all other parameters fixed. Recent papers (Collier Cameron et al. 2010; Triaud et al. 2010) report a small but non-zero eccentricity (e ≃ 4 × 10⁻³), but we verified that this would affect our estimates of the other parameters by a negligible fraction of their error bars.

We then performed a fit with three free parameters:

• 1.  
  the ratio of planetary to stellar radii, p = r_p/R_s,
• 2.  
  the orbital semimajor axis (in units of the stellar radius), a₀ = a/R_s, and
• 3.  
  the orbital inclination, i.

We chose these as free parameters, because

• 1.  
  there is a large range of values published in the literature,
• 2.  
  they largely determine the shape of the transit signal, and
• 3.  
  they do not show strong cross-correlations.

For completeness, and for comparisons with the literature, in the final results we also report the transit depth p², the impact parameter b, and the duration of the transit T, where

$$\begin{equation} b = a_0 \cos {i} \end{equation} \tag{ 5 }$$

$$\begin{equation} T = \frac{P \sqrt{1-b^2}}{ \pi a_0}. \end{equation} \tag{ 6 }$$

We used a Nelder–Mead optimization algorithm (Lagarias et al. 1998) to obtain first estimates of the parameters of a model. To confirm/improve these estimates and to determine error bars, we ran an Adaptive Metropolis algorithm with delayed rejection (Haario et al. 2006) for 20,000 iterations, starting from the optimal values initially determined, in order to sample the probability distributions of the fitted parameters. The updated best estimates and error bars of the parameters are the means and the standard deviations of the sampled distributions (approximately Gaussian), respectively. No burn-in is required because of the optimal starting points of the chains.

The variance of the likelihood function is initialized as the variance of the residuals obtained for the first model and then sampled together with the other free parameters ($\sigma _0^2$). In this way, we take into account both white and the autocorrelated noise present in the detrended time series, but we ignore possible systematic errors due to the preliminary ICA deconvolution. The ICA errors can be represented as an additional uncertainty, σ_ICA, on each point in the time series. The likelihood's variance, $\sigma _{{\rm like}}^2$, becomes

$$\begin{equation} \sigma _{{\rm like}}^2 = \sigma _0^2 + \sigma _{{\rm ICA}}^2. \end{equation} \tag{ 7 }$$

We tested that resampling the parameters' chains with $\sigma _{{\rm like}}^2$ does not affect their best values, while the total error bars of the single parameters, σ_par, increase with respect to the previous estimates (without the ICA errors), σ_{par, 0}, as

$$\begin{equation} \sigma _{{\rm par}} = \sigma _{{\rm par},0} \frac{ \sigma _{{\rm like}}}{ \sigma _{0}} = \sigma _{{\rm par},0} \sqrt{ \frac{ \sigma _{0}^2 + \sigma _{{\rm ICA}}^2 }{ \sigma _{0}^2}}. \end{equation} \tag{ 8 }$$

A measure of the uncertainties on the independent components extracted by ICA is given by the interference-to-signal ratio matrix, ISR, i.e., an n × n matrix, where n is the number of signals. The ISR_ij element estimates the relative remaining presence of the j^th component in the i^th one. Then,

$$\begin{equation} {\rm ISR}_i = \sum _{j=1, \ j \ne i}^{n} {\rm ISR}_{ij} \end{equation} \tag{ 9 }$$

estimates the relative remaining presence of all the other components in the i^th one.

If the i^th component represents the transit signal, and if we estimate the transit signal through method 1, we can identify

$$\begin{equation} \sigma _{{\rm ICA}}^2 = f^2 {\rm ISR}_i, \end{equation} \tag{ 10 }$$

with f the scaling factor used.

If the i^th component represents the transit signal, but we estimate it through method 2, σ_ICA has to contain a weighted sum of the ISRs of the non-transit components removed, plus the discrepancies of the fit to the out-of-transit phases:

$$\begin{equation} \sigma _{{\rm ICA}}^2 = f^2 \left(\sum _{j=1}^{m} o_j^2 {\rm ISR}_j + \sigma _{{\rm ntc-fit}}^2 \right), \end{equation} \tag{ 11 }$$

o_j being the coefficients of the non-transit components, m the number of components considered, σ_ntc-fit the standard deviation of the residuals from the reference light curve (out of the transit), and f the normalizing factor for the model-subtracted light curve.

The MULTICOMBI code produces two ISR matrices, ISR^EF, associated with the algorithm EFICA, and the ISR^WA, associated with the algorithm WASOBI. We estimated the global ISR as their average:

$$\begin{equation} {\rm ISR} = \frac{ {\rm ISR}^{{\rm EF}} + {\rm ISR}^{{\rm WA}}}{2}. \end{equation} \tag{ 12 }$$

This is a conservative estimate, given that, according to Tichavský et al. (2008), the MULTICOMBI ISR slightly outperforms the best of ISR^EF and ISR^WA (then it could be smaller), but these estimates are entirely reliable only under certain assumptions on the signals which may not be satisfied in these cases. Here we take them as worst-case estimates.

Here, we describe the main steps of the analyses performed on each of the two observations (ID 30590 and ID 40732), which include some tests of robustness. We now discuss only results obtained with method 2, as they are much more stable; results obtained with method 1 are reported in Appendix C, along with a critical comparison of the two methods.

2.5.1. Choice of the Pixels

The first step in the analysis is the choice of the pixel-light curves to analyze. This is determined by

• 1.  
  the instrument point response function (PRF), i.e., the measured intensity profile of the star on the detector⁶,
• 2.  
  the noise level of the detector, and
• 3.  
  the effective number of significant components to disentangle.

The number of significant components is not known a priori. The ICA code extracts a number of components equal to the number of light curves that it receives as input. Apart from the collective behavior common to all the pixel-light curves, each pixel introduces an individual signature. Only if the individual signatures are negligible compared to the collective behavior are we able to select enough light curves to disentangle the significant components. The PRF and the noise level of the detector limit the number of pixels containing potentially useful astrophysical information.

In practice, we considered several arrays of pixels with the stellar centroid at their centers, having dimensions 3 × 3, 5 × 5, 7 × 7, 9 × 9, and 11 × 11 pixels. Figure 1 shows the "integral light curves," obtained by summing the contributions from the various pixels. We looked for outliers in the time series, i.e., points discrepant more than 5σ from a first transit-light curve model (fitted on the original data), and we replaced those outliers with the averages of the points immediately before and after. We found only one outlier in ObsID 30590, and nine in ID 40732. Although the observed light curves are two primary transits of the same exoplanet, observed at the same wavelength through the same instrument, they appear very different, mainly because of the different observing strategies. In particular, ObsID 40732 seems to be much less affected by systematics, and less noisy. The integral light curves from the various arrays of pixels look very similar in shape, but have different absolute intensities, as expected. The mean intensities of the integral 3 × 3, 5 × 5, 7 × 7, and 9 × 9 light curves are, respectively, ∼83%, ∼92%, ∼96%, and ∼98% of the mean intensity of the integral 11 × 11 light curve. We are not interested in absolute photometry, but only in relative variations of the intensity, therefore it is not important whether the PRF is totally contained in the square used for the analysis or not, provided it contains enough information to detrend the transit signal. Larger arrays include pixels which add noise with little or no astrophysical information. We concluded that the 3 × 3 or the 5 × 5 arrays were the optimal choices. However, we tested all the pixel arrays to assess the robustness of the results.

Zoom In Zoom Out Reset image size

We binned the transit time series by replacing groups of nine consecutive points with their mean values in order to reduce the computational time required to sample the parameters' distributions in the Mandel & Agol (2002) model (see Section 2.4). We checked that in select cases this approach does not affect the parameter estimates.

The best values of p, a₀, and i are stable, within the error bars, with respect to the choice of the set of pixel-light curves used to detrend the signals. The discrepancies between the extracted signals and the relative fits are the biggest for the 3 × 3 array; for larger arrays they are smaller, and are either all at the same level (ID 30590) or slightly decrease with the size of the array (ID 40732). Our interpretation of this is that the 5 × 5 and larger arrays contain the same amount of useful information, while in the 3 × 3 array something is missed. The ICA errors confirm this hypothesis, being the smallest for the 7 × 7 (ID 30590) and 5 × 5 (ID 40732) arrays. Higher values for larger arrays were expected, but do not differ significantly. We conclude that the choice of the array size is not crucial.

2.5.2. Choice of the Components

In Section 2.5.1, we corrected the observed light curves by subtracting all the non-transit components (see Section 2.3). Here, we show how to identify the most significant components, and how many should be considered. We generally expect that some components are related to collective behaviors common to all the pixels, and others to individual pixels' signatures and/or noisy mixtures of the sources. By inspection, a few of the components clearly present time structures, while others have randomly scattered time series.

We report results from the 5 × 5 array only, as it is the smallest array containing all the astrophysical and instrumental information.

To evaluate the impact of each component in the out-of-transit data, we found the best fits of the single components (plus additive constants) to that part of the integral light curve, and calculated the means and standard deviations of the residuals. In this way, we established a ranking of importance of the components, based on the minimization of the discrepancies between their fits and the integral light curve, out of transit. We then computed other best fits by using the n most important components, according to that ranking, with n from 1 to 24. The best coefficients for the components and the additive constants were determined through the Nelder–Mead optimization algorithm.

Figure 2 reports the standard deviations of the residuals of the single-component fits, normalized to the out-of-transit level and Figure 3 reports analogous fits obtained using more components. Note that figures related to different observations are not reported with the same scale because they have very different accuracies. Most systematics are contained in one major component, but the use of multiple components increases the detrending accuracy.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We computed 24 estimates of the transit signal by removing the n most significant non-transit components from the integral light curve. We binned these over nine points, as in Section 2.5.1, and fitted Mandel & Agol (2002) models to these curves. The standard deviations of the residuals between each curve and the corresponding best model of Mandel & Agol (2002) are reported in Figure 4, and confirm that the use of multiple components for detrending improves the results.

Zoom In Zoom Out Reset image size

ICA separation errors are plotted in Figure 5, showing the same trends.

Zoom In Zoom Out Reset image size

Given these tests, we expect to have good estimates of the transit signals by removing the first few most significant components, but some improvements can be made by removing more components, up to a saturation point. The best values of the parameters p, a₀, and i for each estimated transit signal are shown in Figure 12.

The dispersions in the parameters are fully contained in the intervals previously estimated by using the signals with all non-transit components removed (see Appendix A, Table 6, Column 2), except for values from the transit signal from ObsID 40732 with only one component removed.

The parameter estimates determined from ObsID 40732 are much more accurate than those from ID 30590. Assuming that the orbital parameters were the same along the two observations, as expected because of the stability of the planetary orbit, we computed a chain for ID 30590, for p only, with a₀ and i fixed to the best values estimated from ID 40732. In this way, we can make a direct comparison of p between the two observations, and avoid possible correlations with the other parameters. Even if a₀ and i were badly determined due to an inaccurate stellar model being assumed (e.g., wrong limb darkening coefficients, star spots, or faculae), they would introduce a systematic error on p that would be equal for both observations. Thus variations of p (or p²), obtained while keeping all other parameters fixed, are a more objective measurement of the stellar variations. Results are reported in Table 4; note that σ₀ is unchanged. Figure 8 shows the discrepancies between the detrended signal and the model. Including the ICA errors, we found

$$\begin{equation} \begin{array}{c}p = 0.1551 \pm 0.0004 \\ p^2 = 0.02405 \pm 0.00014 \end{array} \end{equation} \tag{ 13 }$$

Figure 9 compares the original estimates for p and p² with those obtained by keeping a₀ and i fixed. We note that

• 1.  
  the best value from ID 30590 with a₀ and i fixed agrees better with the result from ID 40732 and
• 2.  
  the new estimate from ID 30590 is consistent with the previous one, but with a (slightly) smaller error bar.

Zoom In Zoom Out Reset image size

Table 4. Estimated Best Values and Standard Deviations of p, from ObsID 30590, with a₀ = 8.92 and i = 85.78, without Including the ICA Error

p	0.15507
$\sigma _p^0$	2.7 × 10−4
σ0	6.5 × 10−4

Note. Discrepancies between the signals and the related best model fits (see Section 2.4).

Download table as:  ASCIITypeset image

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The planetary, orbital, and stellar parameters derived separately from the two observations are all consistent within 1σ. In particular, the duration of the transit is extremely stable between the two observations. This is not surprising, because its measure is almost insensitive to calibration errors and stellar activity; all other parameters are much more affected by these sources of noise. Furthermore, these other parameters are strongly correlated; e.g., a non-optimal estimate of the impact parameter b will result in an imprecise transit depth p, etc. Figure 10 shows the differences between the transit signals extracted for the two observations. The standard deviation of the differences is ∼6.8 × 10⁻⁴, which is comparable with the standard deviation of the discrepancies between the signal from ObsID 30590 and the relative model fit ($\sigma _{0}^{{\rm ID}\, 30590} = \left(6.5 \pm 0.3 \right) \times 10^{-4}$); the discrepancies between the signal from ObsID 40732 and the corresponding model fit are negligible. Hence, there is no evidence of physical variations in the transit signal from one observation to the other one.

The results of Section 3.1 reinforce our claim of non-detectable stellar activity variations.

We conclude that the two observations lead to consistent results, but the second constrains the orbital and stellar parameters much better and allows the estimate of the transit depth for the first one to be refined.

Agol et al. (2010) report a detailed study of seven primary transits and seven secondary eclipses of HD 189733b, observed with Spitzer/IRAC at 8 μm (channel 4 of IRAC). Their measured orbital parameters differ from ours by less than the joint 1σ uncertainties. Figure 11 includes a comparison of the parameters a₀, i, and b, obtained in this paper with their values. Given the number of primary transits and secondary eclipses they analyzed, and the small impact of the limb darkening effect at 8 μm, this is a robust confirmation of the validity of our results at 3.6 μm. We suggest the use of these parameters for future observations at other wavelengths.

Zoom In Zoom Out Reset image size

Agol et al. (2010) found variations in the transit depth with a range of ∼2 × 10⁻⁴ on p². We could not detect such a difference between the two observations analyzed at 3.6 μm, as it is comparable with the first error bar.

Our results are consistent, at the 1σ level, with those of Beaulieu et al. (2008) and Désert et al. (2009) for ID 30590, and Désert et al. (2011) for ID 40732. However, our results afford a substantial agreement (within 1σ) between the transit parameters determined from the two observations, while previous analyses by Désert et al. (2009) and Désert et al. (2011) claimed significant variations of all parameters (e.g., discrepancy >4σ for transit depths). Désert et al. (2011) suggested stellar activity as a possible explanation for those differences. Our results do not support such conclusions, and we find that any possible stellar activity variations are within the error bars. Our error bars from ObsID 40732 are of the same order as (for transit depth) or even smaller (for orbital parameters) than in Désert et al. (2011), while those from ObsID 30590 are larger by a factor ∼1.6 with respect to the error bars in Désert et al. (2009). The factor ∼1.6 comes from adding the ICA errors to the parameter error bars derived from the extracted signals. Désert et al. (2009, 2011) applied parametric corrections to detrend the transit signals from other disturbances, without attributing any uncertainties to the detrending processes. The fact that we obtained smaller error bars from ObsID 40732, even including the contributions from the detrending process, indicates that, in that context, our blind extraction performed better than their parameterization. Orbital parameters determined by Beaulieu et al. (2008) and Désert et al. (2009) for ObsID 30590 are not consistent with those for ObsID 40732 obtained by Désert et al. (2011), the results presented here, or the 8 μm observations by Agol et al. (2010). Given that the second measurement was superior in quality, and given the agreement with observations at another wavelength, we conclude that the parameters presented in this paper are more robust than those reported by Beaulieu et al. (2008), or by Désert et al. (2009) using the same data.

We note that Beaulieu et al. (2008) used the same impact parameter at 3.6 μm and 5.8 μm, while Désert et al. (2009) used similar, but not identical, values. Given the conclusions obtained in this paper on the orbital parameters, we suggest that the transit depth at 5.8 μm be recalculated accordingly. A re-analysis of the observation at 5.8 μm and then of the differences between the transit depths at the two wavelengths, which were used to infer about the atmosphere of the planet, should not be strongly affected by this bias, at least in the first case. However, because their conclusions were controversial, a re-analysis of the observation at 5.8 μm, with more precise orbital parameters and possibly non-parametric technique, as done here, is needed.

Table 5 reports the best values of the parameters for the transit signals extracted from different arrays of pixels, the standard deviations of the residuals between the signals and the best transit models, and the standard deviations attributed to the ICA separation.

Table 5. Best Values of p, a₀, and i for the Transit Signals Extracted from Different Arrays of Pixels, Through Method 2, Considering All the Independent Components, Binned by Nine Points

	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
p	0.1549	0.1547	0.1546	0.1547	0.1547
a0	9.02	9.05	9.07	9.06	9.07
i	85.90	85.93	85.95	85.94	85.95
σ0	7.1 × 10−4	6.5 × 10−4	6.5 × 10−4	6.5 × 10−4	6.5 × 10−4
σICA	7.8 × 10−4	7.6 × 10−4	7.4 × 10−4	8.2 × 10−4	8.3 × 10−4
σp	0.00058	0.00055	0.00054	0.00057	0.00058
$\sigma _{a_0}$	0.17	0.16	0.16	0.17	0.17
σi	0.15	0.15	0.14	0.15	0.15

Notes. Correspondents σ₀ (computed from the residuals between the signals and the best models), and σ_ICA. Derived total standard deviations of the parameters' distributions (ObsID 30590).

Download table as:  ASCIITypeset image

Figure 12 shows the best values of the parameters p, a₀, and i, respectively, for the transit signals extracted removing the n most significant components from the integral 5 × 5 light curve, binned by nine points.

Zoom In Zoom Out Reset image size

Table 6 reports the best values of the parameters for the transit signals extracted from different arrays of pixels, the standard deviations of the residuals between the signals and the best transit models, and the standard deviations attributed to the ICA separation.

Table 6. Best Values of p, a₀, and i for the Transit Signals Extracted from Different Arrays of Pixels, Through Method 2, Considering All the Independent Components, Binned by Nine Points

	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
p	0.15546	0.15534	0.15533	0.15533	0.15528
a0	8.93	8.92	8.93	8.93	8.94
i	85.79	85.78	85.79	85.79	85.79
σ0	1.62 × 10−4	1.46 × 10−4	1.45 × 10−4	1.45 × 10−4	1.41 × 10−4
σICA	1.70 × 10−4	1.45 × 10−4	1.60 × 10−4	1.53 × 10−4	1.65 × 10−4
σp	0.00013	0.00011	0.00012	0.00011	0.00012
$\sigma _{a_0}$	0.03	0.03	0.03	0.03	0.03
σi	0.03	0.03	0.03	0.03	0.03

Notes. Correspondents σ₀ (computed from the residuals between the signals and the best models), and σ_ICA. Derived total standard deviations of the parameters' distributions (ObsID 40732).

Download table as:  ASCIITypeset image

Figure 13 reports the best values of the parameters p, a₀, and i, respectively, for the transit signals extracted by removing the n most significant components from the integral 5 × 5 light curve, binned by nine points.

Zoom In Zoom Out Reset image size

We considered 28 sub-data sets, obtained by combining seven different starting and four ending times, disposed with regular cadence of ∼14 minutes (see Figure 14).

Zoom In Zoom Out Reset image size

As before, we used the 5 × 5 array, and we applied method 2, by removing all the independent components from the integral light curve. Figure 15 shows the best values of the parameters p, a₀, and i, estimated using each sub-data set.

Zoom In Zoom Out Reset image size

We can point out some correlations between the best values and both the start and the end points of the sub-data sets. The overall scatters are compatible with the ranges determined before. Table 7 reports the estimated ranges of the parameters with the scatters observed by the sub-data sets, either by including or by rejecting the two shortest sub-data sets.

We considered 32 sub-data sets, obtained by combining eight different starting times and four ending times, disposed with regular cadence of ∼14 minutes (see Figure 16).

Zoom In Zoom Out Reset image size

As usual, we used the 5 × 5 array, and we applied method 2, and removed all the independent components from the integral light curve. Figure 17 shows the best values of the parameters p, a₀, and i, estimated using each sub-data set.

Zoom In Zoom Out Reset image size

Again, there are some correlations between the best values and the extremes of the sub-data sets, but the overall scatters are compatible with the ranges previously estimated. Table 8 reports the estimated ranges of the parameters with the scatters observed by the sub-data sets.