THE EMERGENT 1.1–1.7 μm SPECTRUM OF THE EXOPLANET COROT-2B AS MEASURED USING THE HUBBLE SPACE TELESCOPE

Bad pixel correction due to energetic particle hits is part of the calwf3 processing used for our α analysis. Additional pixels not identified by calwf3 may still be erroneously high or low in value and need correction. For both α and β analyses, we identify and correct bad pixels immediately prior to the spectral extraction (i.e., before summing the box). Our α analysis inspects pixels in each column of the spectral box (i.e., a single wavelength) that deviate significantly from a Gaussian profile of the spectral trace. Such deviations are virtually always characterized by much higher intensity levels. Those pixels that are more than 10 times greater than the fitted Gaussian value are replaced by a 7 pixel median in the vertical direction (perpendicular to the dispersion) at that wavelength.

Our β analysis must be more sophisticated as regards bad pixels, since these data have not been processed by calwf3. We examine the ratio of a given pixel to the total of all pixels in that row, i.e., the ratio of a single pixel to the sum over wavelength at each spatial position. Because of spatial pointing jitter, pixel intensities can vary with time in an absolute sense, but their relative variation should be similar at all wavelengths. We examine the ratio as a function of time (i.e., for each exposure) and we identify instances where a given pixel does not scale with its row sum. We identify >4σ outliers, and correct them using a five-frame median value of the ratio at that time.

For both the α and β analyses, we calculate the background individually for each exposure by using pixels outside of the spectral box. Specifically, the pixels used are those that lie directly below the spectrum on the subarray, which is the section of the subarray corresponding to the width of the spectrum and extending from the bottom edge of the spectral box to the bottom edge of the subarray. We construct a histogram of intensity values in these pixels and fit a Gaussian to the histogram. The adopted background value is the intensity corresponding to the central value of the fitted Gaussian, and we do assume that it is independent of wavelength to the limit of our precision. This is typically a few tens of electrons per pixel, several orders of magnitude less than the signal in the stellar spectrum, and the sum is thus also significantly lower when calculating the white-light curve and its corresponding background. Background subtraction therefore has a relatively minor effect on our analysis.

Wavelength calibration utilizes both the direct image and the spectral image, as the wavelength of a given pixel depends upon its location on the detector relative to the direct image. Kuntschner et al. (2009) outline the procedure for wavelength calibration in an STScI calibration report. The equations governing the wavelength for a pixel at a given x-position in the first-order spectrum are

$$\begin{equation} \lambda (x) = dldp_{0} + dldp_{1}\Delta x, \end{equation} \tag{ 1 }$$

where

$$\begin{equation*} dldp_{0} = a^{0}_{0} + a^{1}_{0}x_{{\rm center}} \end{equation*}$$

$$\begin{equation*} dldp_{1} = a^{0}_{1} + a^{1}_{1}x_{{\rm center}} + a^{2}_{1}y_{{\rm center}} + a^{3}_{1}x_{{\rm center}}^{2}+ \end{equation*}$$

$$\begin{equation*} a^{4}_{1}x_{{\rm center}}y_{{\rm center}} + a^{5}_{1}y_{{\rm center}}^{2} \end{equation*}$$

$$\begin{equation*} \Delta x = x -x_{{\rm center.}} \end{equation*}$$

The terms x_center and y_center are the central coordinates of the direct image. The coefficients ($a^{0}_{0}$, $a^{1}_{0}$, etc.) are calculated in Kuntschner et al. (2009).

In performing this calibration, we found that the calibrated grism response (sensitivity) curve did not line up precisely in wavelength space with the observed response (see Figure 1). We therefore adjusted the coefficients empirically to obtain optimal agreement with the observed grism response curve and with the wavelengths of two stellar hydrogen lines (Pa-β at 1.282 μm and Br-12 at 1.646 μm). These adjustments yielded

$$\begin{equation*} a^{0}_{0} \rightarrow 0.997 \times a^{0}_{0} \end{equation*}$$

$$\begin{equation*} a^{1}_{0} \rightarrow 0.90 \times a^{1}_{0} \end{equation*}$$

$$\begin{equation*} a^{0}_{1} \rightarrow 1.029 \times a^{0}_{1}. \end{equation*}$$

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We used these adjusted values in the calibration presented in this work and also successfully applied them to other data sets in this HST program. Therefore, this empirical correction is not specific to this target or these visits, and we in fact used another object in the program (TrES-2) to find the correction, as it was observed on the larger subarray, and thus the observations include all 150 pixels of the spectrum.

The flat field and sensitivity curve of the G141 grism on the WFC3 detector are the two components of flux calibration, and both are wavelength-dependent.

For imaging observations, calwf3 applies the flat field to the data, but flat-fielding of grism data must be done by the observer. STScI provides a flat-field cube for the G141 grism. This cube is a four-extension FITS file, and each extension is the size of the full WFC3 IR array. For a given pixel on the data image with a given wavelength, the flat-field value for that pixel is given by a polynomial function with coefficients defined by the values of the flat-field cube extensions at the pixel's location.

This method is described in the aXe handbook (Kümmel et al. 2011) and laid out with the equations that follow. For a pixel at position (i, j), they define a normalized wavelength coordinate, x:

$$\begin{equation*} x = \frac{\lambda -\lambda _{{\rm min}}}{\lambda _{{\rm max}} - \lambda _{{\rm min}}}\,. \end{equation*}$$

The parameters λ_min and λ_max are constants found in the flat-field cube header. The flat field value of a pixel (i, j) with normalized wavelength coordinate x is then a polynomial function in x:

$$\begin{equation} f(i,j,x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3}, \end{equation} \tag{ 2 }$$

where a₀, a₁, a₂, and a₃ are the values at (i, j) in the zeroth, first, second, and third extension arrays in the flat-field cube file, respectively. For both our α and β analyses, we apply the flat-field correction to the spectral box by dividing by the corresponding flat-field "box," generated pixel-by-pixel from the method above.

STScI also provides the wavelength-dependent sensitivity of the G141 grism. In Figure 1, we plot the scaled sensitivity curve over a flat-fielded spectrum from a single exposure of TRES-2, along with the two hydrogen lines used as reference points in adjusting the wavelength calibration coefficients.

CoRoT-2 has a companion star, so our analysis must remove or correct for this second source. The direct image of CoRoT-2 appears in Figure 2, where the second, fainter source is evident. The proximity of the second source in the image depends on the orientation angle of the telescope, and varies between the three visits, but it is close enough to be of concern for source contamination. The spectra overlap minimally in visits A and C, but there is significant overlap in visit B, which has the lowest orientation angle and thus the smallest distance between the two spectra of the three observations. The orientation angles, which only vary a few degrees from each other, are reported in Table 1.

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3.5.1. Characterization

3.5.2. Removal

One potential cause of the hook effect is detector persistence, the phenomenon in which trapped charge in an exposure is slowly released in the following exposure(s) to produce a falsely increased signal detection (Smith et al. 2008). STScI publishes persistence models and even predictions of persistence for a given exposure based on the exposures prior to it. The predictions are for an additive effect, and the data product for a given exposure is an image array the size of the original exposure, but with each pixel value equal to the predicted persistence, so the correction is simply to subtract the corresponding pixel values. The persistence is low for the first exposure, but jumps up quickly and remains at a higher value until the time of the data transfer, when it, too, resets. The additive correction as given by STScI do decrease the severity of the hook, but they do not entirely remove the hook, and we conclude that the hook is a combination of a additive and multiplicative effect. This will justify our methods of correction outlined and examined in the sections that follow. We have made the STScI persistence correction in our α analysis. Our β analysis ignores additive persistence, as do most WFC3 exoplanetary investigations published to date.

Berta et al. (2012) demonstrated that the hook is more prominent at high exposure levels. We have investigated the amplitude of the hook as a function of the per-pixel exposure level, and other parameters, and we seek quantitative relationships. For each pixel, we average the change in signal level over the multiple iterations of the pattern within one orbit, and then examine the change as a function of time within the pattern, flux of the pixel, and location of the pixel on the detector.

The average shape of the hook for two objects in the program can be seen in Figure 5. The normalized signal is shown against the exposure number within the pattern. For each visit, the pixels have been split between those with flux below the mean and those with flux above the mean. This is done to confirm that the existence of the hook does indeed depend upon the flux of the pixel.

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Figure 6 shows this dependence of the additive change on the flux of the pixels, where every pixel has been plotted by its initial flux and "jump" in electrons between the first exposure and the last exposure in the pattern. The jump is statistically insignificant below a certain original pixel value, but shows a reliable parabolic rise starting around 30,000 electrons. The scatter is nevertheless remarkably large, which ultimately means that we cannot depend on a unique, quantitative relation to correct this effect.

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In principle, the hook could be removed by using Figure 6 to predict the magnitude of the jump for a pixel given its initial flux in the first exposure of the pattern, and thereby correct each pixel in each image. We attempted such a correction, and it does remove the obvious appearance of the hook pattern, but it leaves the data with much more scatter than is acceptable, due to the wide variations seen in Figure 6.

We also examined the amplitude of the hook as a function of position on the detector. We find no correlation in column (wavelength) space, but some correlation with the slope of the hook pattern and the row on the detector, i.e., how far a given row is from the spatial center of the spectrum. This correlation does seem to strongly depend on which subarray we used. Especially in the case of the 128 × 128 subarray, the slope of the trend is more positive for the rows of pixels below the central peak of the spectrum (in the direction perpendicular to dispersion), while the slope is less positive for those rows above the central peak. This correlation is weaker for the 512 × 512 subarray, but still discernible. Figure 7 shows the correlation for the smaller subarray by demonstrating the shift in the spatial center of the spectrum between the starting and ending frames of the hook. Our finding that the nature of the hook depends on the row of the spectrum may be a significant clue to the nature of this effect. Reading the detector involves addressing the pixels by row, and it is conceivable that the hook is related to the manner in which the detector is addressed and sampled. We conclude that the effect in Figure 7 cannot be explained by anything like telescope drift. The trend featured in Figure 7 is correlated with the hook and therefore the transfer of the detector buffer, a task performed with no relation to telescope motion.

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We observe CoRoT-2 in four orbits per visit, but each visit contains, at most, one orbit that is completely in-eclipse (when the planet does not contribute), and each visit contains the virtually unusable first orbit. For this reason and due to our significantly lower signal-to-noise ratio than for the Berta et al. (2012) planet's observations—the GJ1214b transit depth—is two orders of magnitude larger than the depth of the CoRoT-2b secondary eclipse in the same waveband and on the same grism—our CoRoT-2 data are not well-suited for the divide-oot method per se. Another issue with CoRoT-2 is the severity of the visit-long ramp, which causes trouble when trying to average pattern shapes before removing the ramp. Therefore, instead of dividing by an average orbit, we elect to divide by an average pattern, defined both by the occurrence of a buffer dump and through visual assessment, and we proceed as follows.

• 1.  
  Identify the patterns that are out-of-eclipse. Divide the entire white-light curve by the median of the out-of-eclipse exposures from a single, early orbit (usually orbit 2). This normalizes the curve to unity.
• 2.  
  Fit a line to the out-of-eclipse patterns, but exclude all points below intensity level 0.997. These outliers are due to the hook effect, and would bias the visit-long slope correction.
• 3.  
  Divide by the fitted curve to re-normalize to unity.
• 4.  
  Create an average pattern by averaging the out-of-eclipse patterns.
• 5.  
  Divide each occurrence of the pattern in the entire white-light curve by the average pattern.

This creates a vast improvement in the data, with a significant reduction in the presence of systematic effects. We are also able to utilize the later patterns of the first orbit, rather than discarding it completely, as the problems presented by settling or other effects of unknown origin diminish significantly after one to two iterations of the pattern. An average pattern is plotted in the inset of Figure 8, and the corrected data are shown in comparison to our best-fit eclipse curve in Figure 9.

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With the corrected data in hand after applying our modified divide-oot procedure, we fit an eclipse curve using the data from our α analysis. We calculate the shape of the theoretical eclipse curve using expressions from Mandel & Agol (2002), with orbital parameters from Alonso et al. (2009), except for the orbital period where we adopt the updated value from Sada et al. (2012). In fitting the data, we vary only the central phase and amplitude of the eclipse, the latter by scaling the amplitude of the theoretical curve. We perform the fit using two χ²-minimization methods. First, we implement a Levenberg–Marquardt algorithm to vary the eclipse amplitude and central phase simultaneously, to find the global minimum in χ². Second, we vary the central phase incrementally from 0.49 to 0.51 in steps of 10⁻⁵. At each step, we calculate the best-fit eclipse amplitude at that phase in closed form, using linear least-squares. Cycling through the range of trial central phases, we again find the global minimum χ². Results from the two methods were in excellent agreement.

We find a best-fit eclipse depth of $395^{+69}_{-45}$ ppm (parts per million); the fit is shown in Figure 9. The reduced $\chi ^{2}_{{\rm red}} = 6.60$; as it was calculated estimating the error to be Poissonian, the ideal scenario, this $\chi ^{2}_{{\rm red}}$ value indicates that the achieved per-point scatter is 2.6 times the photon noise. The error level, and the appearance of Figure 9, suggests that red noise remains in the data, in spite of our modified divide-oot procedure. To verify the presence of red noise, we binned the residuals from the best-fit eclipse over N points per bin, and calculated the standard deviation of the binned points, σ_N. We solve for the slope of the relation between log (N) and log (σ_N) using linear least-squares. Poisson noise will produce a slope of −0.5, whereas we find a slope of −0.33 ± 0.03 for the Figure 9 data, confirming the presence of red noise.

Given the presence of red noise in the white-light eclipse data, we assign errors to the best-fit eclipse parameters (eclipse amplitude and central phase) using the residual permutation ("prayer-bead") method (Bouchy et al. 2005; Gillon et al. 2007). Figure 10 shows histograms of the results for the best-fit amplitude and central phase, based on the residual permutation fits. For reference, we fit Gaussians to these histograms. A Gaussian is a reasonable approximation to the central phase histogram, but the eclipse amplitude histogram has a higher central peak, and lower wings, than does a Gaussian. Our adopted errors are equivalent to the ±1σ points in the histograms, in the sense that 15.8% of the histogram area lies beyond each quoted 1σ value (31.6% considering both ends of the range).

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Our best-fit eclipse is centered at a phase of 0.4998 ± 0.0030. The light-travel time across the orbit is 28 s. The central phase for a circular orbit would be 0.50019, consistent with our result, within our errors. Gillon et al. (2010) found the eclipse to occur slightly earlier than expected for a circular orbit, at phase 0.4981 ± 0.0004. (Deming et al. 2011) found a central phase of 0.4994 ± 0.0007, weakly supporting the result from Gillon et al. (2010). The low signal-to-noise—due to the shallower secondary eclipse at shorter wavelengths—of the eclipse in the WFC3 bandpass contributes to a relatively large error level for the central phase (approximately four to eight times larger than the Spitzer errors). Although we find good agreement with a circular orbit, we cannot exclude the result of Gillon et al. (2010), who concluded that the orbit is slightly eccentric.

The differential method is intended to exploit the characteristics of the systematic hook pattern in order to cancel it, while also correcting for the effects of jitter in wavelength over time. The amplitude of the hook is a function of the flux level in the affected pixels (Section 4). The procedure of the differential method, in its simplest form, is to therefore extract the intensity in each column of each grism image, and divide that intensity by the wavelength-integrated intensity in the entire spectrum observed at that time. In other words, ratio the intensity in a given column on the detector (after subtracting the background, and integrating over rows) to the sum of all columns, and we repeat this process for the grism image at each orbital phase ϕ. This ratio adds minimal noise, because the precision of the wavelength-integrated spectrum is much better than the precision of a single wavelength. Moreover, the ratio should be effective in removing the hook, as long as the wavelength used in the numerator is not too close to the edges of the grism response, where the intensity rolls-off to much smaller values, as does the hook (Section 4). The observed grism spectral intensity varies only modestly (Figure 1) over the 1.1–1.7 μm range of our analysis. Thus, dividing a single wavelength by the sum of all wavelengths is a comparison of similar intensity levels, so we expect much of the hook pattern to cancel, and this expectation is met by the actual data (see below).

The differential method also removes the white-light eclipse. Specifically, the eclipse shown on Figure 9, by summing over wavelengths, will identically cancel. However, wavelength-to-wavelength variations in the eclipse depth will be preserved. We call these differences differential depths and we derive them either positive or negative, by fitting to the wavelength-ratioed data. We then add the depth of the white-light eclipse, reconstructing the full emergent spectrum of the planet at eclipse.

In actual practice, the implementation of this differential method is more complex than the simple division implied above. We do not explicitly divide by a wavelength integral; we use an equivalent, but more subtle procedure that we now describe.

We must account for possible wavelength shifts in each grism spectrum. Wavelength shifts have two effects. First, a shift of the spectrum changes the intensity in a given column because the grism response varies with wavelength. Second, a shift in the spectrum changes the range of wavelengths sampled by a given column of the detector. We find that the wavelength shifts are of order 0.02 pixels, and they vary within an orbit, but tend to reset and exhibit a similar pattern in subsequent orbits. Given this magnitude of shifts, the second effect mentioned above—a perturbation to the wavelength assigned to a given column—has negligible effect. We therefore ignore the wavelength perturbations per se, and we use the wavelength scale from the calibration described in Section 3.3. However, the first effect (changes in grism response with wavelength) is important, and we deal with it as follows.

• 1.  
  For each visit, form a "template" spectrum of the star alone by summing the in-eclipse (planet hidden) spectra. Denote this spectrum by S_x, where x is the column coordinate on the detector.
• 2.  
  Fit the template to each individual spectrum by re-sampling, shifting (in steps of 10⁻⁴ pixels), and scaling S_x in intensity using linear least-squares. Perform this least-squares fit over a large range of shift values (±0.1 pixels) and choose the shift that exhibits the best fit as judged by the standard deviation of the ratio.
• 3.  
  Each individual spectrum, P_x at orbital phase ϕ, matches a version of S_x with a scaling factor a: $aS^{^{\prime }}_{x}+b$. The prime marks the change in intensity due to the shift in x, and the zero-point constant b is negligibly small.
• 4.  
  Form the ratio $R^{\phi }_{x} = ({ P_{x}}/{ aS^{^{\prime }}_{x}+b })$.

An example of this basic process of shifting and fitting the template spectrum, for a randomly selected spectrum in visit C, is illustrated in Figure 11. However, our actual analysis adds an additional step in order to deal with the undersampling of the stellar spectrum as discussed by Deming et al. (2013). Between steps 3 and 4 above:

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3.5. Smooth all of the spectra using a Gaussian kernel with FWHM = 4 pixels.

The choice of pixels (columns in wavelength) is dictated by the tradeoff between suppressing the undersampling and preserving the spectral resolution.

The wavelength integrals of P_x and $aS^{^{\prime }}_{x}+b$ are closely equal because of the fitting process that matches them. Moreover, the shape of S_x is constant over a visit, i.e., its value at any single wavelength, relative to its wavelength integral, is constant. Hence, the point-by-point division described above is conceptually equivalent to dividing a single wavelength (equivalently, x-value) in P_x by the wavelength integral of P_x. However, our procedure has the advantage that we do not have to re-sample any spectra wherein the potential signal is present, or where the reference stellar spectrum is changing. Hence, we introduce no extra noise in this process, while also correcting for wavelength jitter in the spectrum.

6.1.1. The Spectrum of CoRoT-2b Using the Differential Method

Performing the procedure described above yields a set of ratio values $R^{\phi }_{x}$ for each visit. We now combine visits as follows.

• 1.  
  For each column of the detector x, fit a straight line to the $R^{\phi }_{x}$, where the independent variable in the linear fit is phase ϕ, and then divide by that line.Dividing each visit by the linear fit removes any slight slopes that are present in each visit (as described by Berta et al. 2012 and Section 4) and places all three visits on a common scale.
• 2.  
  Fit an eclipse curve to the combined $R^{\phi }_{x}$ at each x, holding the central phase fixed at 0.5 for the eclipse fit, solving only for the depth.
• 3.  
  Use the wavelength calibration to associate a wavelength with each column x; $R^{\phi }_{x}$ becomes $R^{\phi }_{\lambda }$.The wavelength scale is sufficiently similar for each of the visits that we associate visit-averaged wavelengths with each x. The upper panel of Figure 12 shows the result of fitting an eclipse curve to the visit-combined $R^{\phi }_{\lambda }$ at a randomly selected wavelength. Because the white-light eclipse has been removed by the process used to generate the $R^{\phi }_{\lambda }$, the differential eclipse depth at individual wavelengths can be either positive or negative depending on whether the intensity of the exoplanetary spectrum is greater or less at that wavelength compared to the average over the band defined by the grism response. Note that the scatter in the individual points on Figure 12 is large compared to these differential eclipse depths. However, the precision of the differential eclipse depths is much better than the single-point scatter in $R^{\phi }_{\lambda }$, and we also average adjacent wavelengths to derive spectral structure in the exoplanetary spectrum (see below).As the final step,
• 4.  
  We add the white-light eclipse depth (0.000495, Section 5.2) to the differential eclipse depths, and thereby derive the planet-to-star contrast versus wavelength.

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This emergent spectrum of the planet is illustrated on Figure 13, from both our α- and β-analyses. The upper panel shows the values for individual wavelengths, i.e., single columns of the detector, and the lower panel bins the results in bins of width 0.05 μm (4 columns).

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6.1.2. Errors

The lower panel of Figure 13 includes the contrast produced by a best-fit blackbody for the planet compared to the results from our α and β analyses, and Figure 14 plots that blackbody in comparison to the totality of existing eclipse data. We adopt a Kurucz model for the star (Teff = 5750, log(g) = 4.5), yielding a best-fit blackbody temperature of 1788 ± 18 K for the planet in our WFC3 band, from our α-analysis. This blackbody temperature gives acceptable agreement with the infrared eclipse results at longer wavelength (Figure 14). The 1788 K blackbody—derived from the WFC3 data alone—misses the ground-Spitzer eclipse amplitudes by an average of about 1.8σ. However, a blackbody spectrum for the planet does not produce the best slope over the WFC3 band, as we now discuss.

Our observed WFC3 spectrum for CoRoT-2b has two striking features: (1) it slopes slightly upward with increasing wavelength, and (2) it shows little to no evidence for water absorption or emission in the 1.4 μm band. Statistically, the first question to resolve is whether the simplest possible fitting function can account for our spectrum. The simplest function is a single value in contrast, i.e., a flat line at the average contrast level. For our α analysis spectrum (red points on Figure 13) the χ² of the best-fit flat line is 12.8 for 10 degrees of freedom, so our α analysis accepts a flat line as representing the planet's contrast across the WFC3 band. For our β analysis (blue points on Figure 12), the flat line χ² is 28.6, rejecting the flat line at >99% confidence. So our β analysis indicates a stronger and more significant upward slope than does our α analysis. That is the single largest difference between our α and β analyses, that are otherwise very consistent, with all points overlapping within their error bars (Figure 13). Both of our WFC3 analyses reject the best-fit blackbody slope for the planet, but only at about the 93% confidence level. The χ² values are 17.0 and 17.8 (10 dof) for our α and β spectra, respectively. On the other hand, the blackbody is obviously consistent with the weakness of water absorption in the WFC3 band.

We checked that our results are not affected by inadequate corrections for detector nonlinearity at the high fluence levels of our data. We repeated the β analysis, omitting the last (fourth) sample of the exposure, and using only the first three samples, where the fluence level (∼47,000 electrons) is well within the linear regime. That modified version of the β analysis shows little difference from the β spectrum shown on Figures 13 and 14 (but, with larger errors due to the lower fluence levels).

The slope of the planet's spectrum across the WFC3 band is relevant to the interpretation of the eclipse amplitude observed in the optical by CoRoT (Alonso et al. 2009; Snellen et al. 2010). If a 1788 K blackbody agreed with the slope of our observed spectrum, it would be reasonable to extrapolate that blackbody to judge the magnitude of the thermal emission from the planet at optical wavelengths. A blackbody of 1788 K would produce negligible thermal emission in the optical, and we would conclude that the optical eclipses are due to reflected light. However, given that the observed slope across the WFC3 band does not decline as strongly as a 1788 K blackbody, it remains possible that the optical eclipses are due to thermal emission. That could happen, for example, if temperatures on the star-facing hemisphere of the planet were spatially inhomogeneous. Hotter regions having a small filling factor, combined with cooler regions of larger filling factor, could in principle produce the observed slope across the WFC3 band, and account for the optical eclipses, while still remaining consistent with the observed contrast at wavelengths exceeding 2 μm.

In order to probe the viability of our speculation concerning temperature inhomogeneities, we performed exploratory fits (not illustrated) using two different blackbody temperatures and filling factors on the star-facing hemisphere of the planet. We find a good fit to our WFC3 and the CoRoT data using T₁ = 1500 K and T₂ = 3600 K, with filling factors of 0.96 and 0.04, respectively. This combination matches the level of the contrast in the CoRoT bands as well as the contrast level and wavelength dependence of our WFC3 results, but it significantly underestimates the contrast in the Spitzer bands (by about 0.001). Recent hydrodynamic simulations of hot Jupiter atmospheres show brightness temperature variations as large as a factor of two on the star-facing hemisphere of HD189733b (Dobbs-Dixon & Agol 2013). Since that planet is less strongly irradiated than CoRoT-2, the temperatures found by our exploratory fits seem plausible. Nevertheless, here we do not attempt to model the atmosphere of CoRoT-2 using a self-consistent three-dimensional approach (temperature varying with depth and with horizontal coordinate). Higher quality data, such as we anticipate from the James Webb Space Telescope, may justify such an approach in the future.

Both our α and β spectra agree that a straight line (contrast increasing linearly with wavelength) gives a good account of our results across the WFC3 band: the χ² values for a linear fit (9 dof) are 6.1 and 13.7 for our α and β spectra, respectively. These values leave little room for absorption or emission by water vapor at 1.4 μm. In order to specify a limit on the degree of water absorption or emission, we scale and fit a Burrows model to the data, using the model shown in blue on Figure 14. In order to make the limit responsive to the modulation caused by the actual water absorption (as opposed to the slope of the continuum), we allow for a linear baseline difference as a function of wavelength. We construct 10,000 trial data sets, adopting the error at each binned wavelength from our β-analysis, and we fit the model plus a linear baseline to each trial data set using linear regression. Based on the distribution of fitted amplitudes, we find an 85 ppm 3σ limit on the amplitude of water absorption or emission, measured at the bandhead at 1.38 μm. This limit assumes that the shape of the water absorption is the same as in the Burrows model. The 3σ limit of 85 ppm is significantly less than the already weak water absorption seen during transmission spectroscopy of the giant planets XO-1b and HD 209458b (Deming et al. 2013), WASP-19 (Huitson et al. 2013; Mandell et al. 2013), and HAT-P-1b (Wakeford et al. 2013). This conclusion is significant, as can been seen by reference to one conventional solar abundance Burrows model (Burrows et al. 2001, 2008a, 2008b, 2010) illustrated as the dark blue line on Figure 14. This model is not intended as a fit to the WFC3 data, but it was invoked by Deming et al. (2011) in an attempt to account for the Spitzer observations. Although it misses the 4.5 μm Spitzer point, Deming et al. (2011) discussed the possibility of circumplanetary carbon monoxide emission in that band, due to tidal stripping by the star. However, this model produces a much larger spectral modulation in the WFC3 band than is seen in our observed spectra.

CoRoT-2b is an unusual planet, and the Spitzer data have been particularly difficult to understand, as discussed by Deming et al. (2011; however, see Madhusudhan 2012). The relatively high contrast at 3.6 and 4.5 μm seems to require a hot continuum, allowing little if any molecular (principally water) absorption. Simultaneously, the lower contrast at 8 μm requires absorption to a significant degree. Here we explore the potential for conventional solar abundance model planetary atmospheres to account for the totality of the CoRoT-2b eclipse data.

The weakness of absorption features can be produced in a solar abundance model by adding continuous opacity by small particle scattering and/or absorption. That could dampen features in the emergent spectrum at short wavelengths, but a reduced scattering cross-section with increasing wavelength could allow greater spectral contrast at 8 μm (mentioned by Deming et al. 2011). If the temperature remains nearly constant as a function of pressure/altitude in the planet's atmosphere, that would also suppress any absorption or emission features in the emergent spectrum. Figure 14 shows the contrast from a Burrows model (Burrows et al. 2001, 2008a, 2010) having three additional sources of opacity not present in a clear atmosphere. This model is shown in green on Figure 14, and has redistribution parameter P_n = 0.1 (Burrows et al. 2008a). The additional opacity sources are first, a high altitude optical (0.4–1.0 μm) absorber of opacity 0.2 cm² g⁻¹. Second, an absorbing haze opacity of 0.04 cm² g⁻¹ uniformly distributed at all pressures and wavelengths, and third, a scattering opacity of 0.08 cm² g⁻¹ also uniformly distributed at all pressures and wavelengths. The scattering opacity acts to increase the reflected light, but not increasing the thermal emission. Note that, in principle, we could include a wavelength dependence to the opacity of the broadly distributed hazes, but we prefer to keep this ad hoc opacity as simple as possible.

The uniformly distributed hazes dampen the spectral modulation in the WFC3 bandpass to an acceptable degree, but the model misses the overall WFC3 contrast level, being too high by 161 ppm. Like all single-spatial-component models, its slope across the WFC3 band is larger than our data. Given the error level of our white-light eclipse ($395^{+69}_{-45}$ ppm), the overall contrast difference is significant at 2.3σ, which is the single largest problem with this model. On the other hand, the scattering opacity increases the contrast in the optical to the point where it underestimates the CoRoT eclipse amplitude by less than 2σ. Also, among the models we have tested, it does the best job of reproducing the long wavelength eclipse amplitudes (1.5σ on average).

The aggregate eclipse data are ambiguous concerning the possibility of a thermal inversion in the dayside atmosphere of CoRoT-2b. As discussed in Madhusudhan (2012), the lower brightness temperature in the 8 μm Spitzer bandpass compared to the brightness temperatures in the shorter wavelength channels (except 4.5 μm) suggests a temperature profile decreasing outward in the atmosphere. If that gradient is flatter than radiative equilibrium models predict, it will help to account for the lack of strong spectral features. On the other hand, the solar abundance radiative equilibrium model (green line on Figure 14) achieves good agreement with the 4.5 μm Spitzer eclipse depth by incorporating 0.2 cm² g⁻¹ of extra optical-wavelength opacity at low pressures (∼1 mbar) close to where radiation in the 4.5 μm band is formed (Burrows et al. 2007). Indeed, that model shows a temperature rise of about 75 K, near 0.2 mbars. However, due to the more widely distributed absorbing haze, the temperatures in this model at high altitude are already hundreds of Kelvins over the values they would have in a clear atmosphere. To the extent that this model is preferred, or that a flattened temperature gradient counts as a weakly inverted atmosphere, then CoRoT-2b could be claimed to have a temperature inversion. However, this evidence for an inversion is weaker than for HD 209458b (Burrows et al. 2007; Knutson et al. 2008), and is ambiguous in the sense that the atmosphere could be heated without satisfying a strict definition of inversion (temperature increasing with height). Knutson et al. (2010) hypothesized that planets orbiting active stars will not have strong atmospheric temperature inversions, because the absorbing species that causes the inversion (e.g., Hubeny et al. 2003; Fortney 2008) may be destroyed by the enhanced UV flux from stellar activity. CoRoT-2a is an active star (Guillot & Havel 2011), and lack of a strong thermal inversion in CoRoT-2b would support the Knutson et al. (2010) hypothesis.

An alternate way to reduce the spectral modulation by water vapor in the WFC3 bandpass is to reduce the equilibrium water vapor mixing ratio, for example, by increasing the carbon abundance relative to oxygen. This also helps to decrease absorption in the 3.6- and 4.5 μm bands (although methane does contribute some absorption at 3.6 μm), while preserving absorption at 8 μm via the 7.8 μm methane band. We used the methodology described by Madhusudhan (2009) and Madhusudhan & Seager (2010) to find a possible carbon-rich match to the aggregate data for this planet (except for the optical eclipses). Madhusudhan (2012) discussed CoRoT-2b and was able to fit the pre-WFC3 data by varying the C/O ratio to various degrees. The magenta line on Figure 14 is a model with an enhanced carbon abundance (C/O = 1), and having a non-inverted atmosphere with modest thermal contrast (700 K increase in temperature from upper boundary to the optically thick photosphere). The enhanced carbon abundance weakens the water absorption, but allows sufficient absorption near 8 μm to account for that Spitzer point to within ∼1σ. The average agreement with the ground Spitzer eclipses is 1.9σ, not quite as good as the blackbody and the solar abundance model. On the other hand, the carbon-rich model does the best job —of the atmospheric models, i.e., beyond just a linear fit— of reproducing the WFC3 spectrum (χ² = 16.1 for 10 dof), and in particular it agrees essentially perfectly with the amplitude of the WFC3 white-light eclipse. It does not require additional haze opacity.

Here we summarize the main conclusions from comparing the aggregate eclipse data for this planet with emergent spectra from different models. We tested a blackbody as well as more sophisticated solar abundance and carbon-rich models. No model fits all of the data. The limit on spectral modulation due to water absorption in the WFC3 band is our main observational result. Given the lack of clear and unequivocal molecular absorption features in the WFC3 and other bands, emergent spectra more sophisticated than a blackbody are unproven. A blackbody spectrum gives an acceptable fit to the data except for the optical eclipses as seen by CoRoT. A blackbody spectrum fits the slope over the WFC3 band poorly, but multi-component blackbodies due to spatial inhomogeneities on the star-facing hemisphere have the potential to help account for the observations, including the optical eclipses as seen by CoRoT, especially if extra scattering opacity increases toward short wavelengths. Note that the absorbing and scattering hazes invoked in our solar abundance model are qualitatively similar to extra absorption and scattering opacity inferred for the archetype planet HD 189733b (Pont et al. 2013; Evans et al. 2013).

Although a blackbody spectrum reasonably accounts for the infrared eclipse data, it is not a model of the planet's atmosphere per se. Instead, the planetary atmosphere can mimic a blackbody via the presence of extra continuous opacity that damps the observed thermal contrast, or due to a high carbon abundance that weakens the bands of the principal molecular absorber (water vapor). In either case, extra scattering opacity at optical wavelengths could help to account for the amplitude of the optical eclipses. We find only weak evidence for a strong temperature inversion, but extra absorbing opacity in the solar abundance model would perturb the temperature profile in a manner similar to a temperature inversion, but less extreme.