WARM SPITZER AND PALOMAR NEAR-IR SECONDARY ECLIPSE PHOTOMETRY OF TWO HOT JUPITERS: WASP-48b AND HAT-P-23b

We observed both planets in secondary eclipse using the IRAC on board the Spitzer Space Telescope. For all observations, we used the IRAC full array mode, which produced images of 256 × 256 pixels (52 × 52), with 12.0 s integration times over 7.9 hr. For WASP-48b, we obtained 2167 images for both the 3.6 and 4.5 μm bands on UT 2011 August 9 and UT 2011 September 7, respectively. For HAT-P-23b, we obtained 2166 images in the same bands on UT 2011 November 30 and UT 2011 December 1. WASP-48 and HAT-P-23 have K_S band magnitudes of 10.4 and 10.8, respectively.

We extract photometry from the basic calibrated data files produced by versions S.18.18.0 (for WASP-48b, 3.6 μm) and S.19.0.0 (for HAT-P-23b, 3.6 and 4.5 μm, and WASP-48b, 4.5 μm) of the IRAC pipeline. The pipeline dark-subtracted, flat-fielded, linearized, and flux-calibrated the images. For each image, we extract the UTC-based Barycentric Julian Date (BJD_UTC) from the FITS header (keyword BMJD_OBS). We transform these time stamps into the Barycentric Julian Date based on the Terrestrial Time standard (BJD_TT) using the conversion at the time of our observations, BJD_TT ≈ BJD_UTC + 66.184 s (Eastman et al. 2010). We prefer the continuous BJD_TT standard because leap seconds are occasionally added to the BJD_UTC standard.

We correct for transient "hot pixels" within a 20×20 pixel box centered on our target star. In each frame, we compare the intensity of each individual pixel within the box to its median value in the 10 preceding and 10 following frames. If the intensity of a pixel varies by >3σ from this local median value, we replace it with that value. We corrected 0.18% and 0.25% of pixels in the 3.6 and 4.5 μm bands, respectively, for WASP-48b, and 0.16% and 0.20% of pixels for HAT-P-23b.

To find the center of the stellar point-spread function (PSF), we use the flux-weighted centroid method. That is, we first calculate the flux-weighted centroid within 4.0 pixels of the approximate position of the target star. Then, we re-calculate the centroid, using the first centroid as our new approximate position. We iterate this procedure a total of 10 times and use the final centroid as the stellar position. Alternatively, we could fit a two-dimensional Gaussian function to the stellar image, but prior experience with extracting time series photometry from Spitzer data shows that the flux-weighted centroid method is either equivalent or superior (e.g., Beerer et al. 2011; Knutson et al. 2012; Lewis et al. 2013). The x and y coordinates of our target stars changed by 0.26 pixels or less during all of our Spitzer observations.

We extract photometry using two methods. First, we perform aperture photometry with the IDL routine aper, using circular apertures with radii ranging from 2.5 to 5.0 pixels in 0.1 pixel increments. We choose aperture radii that yield the lowest scatter in the residuals of the eventual light curves, but we also confirm that eclipse depths and timings remain consistent for the entire range of aperture settings. Second, we use a time-varying aperture radius based on the noise-pixel parameter, which is proportional to the square of the FWHM of the stellar PSF (Mighell 2005; Knutson et al. 2012; Lewis et al. 2013). According to Section 2.2.2 of the Spitzer/IRAC instrument handbook, the noise-pixel parameter is defined as:

$$\begin{equation} \bar{\beta }= \frac{(\sum _iI_i)^2}{\sum _iI_i^2}, \end{equation} \tag{ 1 }$$

where I_i is the intensity recorded by the ith pixel. We include all pixels that are located at least partially within a 4.0 pixel radius of the stellar centroid, which we found minimizes the scatter in the residuals from our best-fit solution. We extract photometry from each image using a circular aperture of radius

$$\begin{equation} r = a_0\sqrt{\bar{\beta }} + a_1, \end{equation} \tag{ 2 }$$

where a₀ and a₁ are constants that we vary in steps of 0.1. We found that photometry based on the noise pixel decreased the scatter in the residuals of our ultimate light curve by ∼5% for the secondary eclipse of WASP-48b in the 3.6 μm band, with a median radius of ∼2.5 pixels for the photometric aperture, although we obtained consistent eclipse depths and timings using fixed radii. This time-varying approach did not improve the fits for the other secondary eclipses, for which we used circular apertures with fixed radii of 2.8 pixels (WASP-48b, 4.5 μm) and 3.0 pixels (HAT-P-23b, 3.6 μm and 4.5 μm). We estimate the background using the 3σ clipped mean within circular sky annuli with inner and outer radii, respectively, of 20.0 and 30.0 pixels (WASP-48b) and 30.0 and 50.0 pixels (HAT-P-23b). There were no visible bright stars within these annuli. Using different annuli yields consistent results, but with higher scatter in the residuals of the eventual light curves. To estimate the Poisson noise limit for the photometry, we converted the pixel intensities to electron counts from MJy/sr using the conversion factors in the FITS headers.

We observed secondary eclipses of HAT-P-23b in the K_S band and WASP-48b in the H and K_S bands using the Wide field IR Camera (WIRC) instrument on the Palomar 200 inch Hale telescope. The WIRC instrument has a 2048 × 2048 detector with a field of view of 87 × 87 and a pixel scale of 02487. The seeing was variable during each night but generally ≲ 15.

To avoid saturating the detector and to minimize systematic errors resulting from variations in intra-pixel sensitivity, we defocused the telescope to a FWHM of ∼25–3''. We used the new active guiding scheme for WIRC (Zhao et al. 2012a) to improve the tracking of the telescope and thus mitigate the effects of imperfect flat-fielding and inter-pixel variations in the detector. To minimize these systematics, we also refrained from dithering during all observations. We positioned the telescope so that the target and calibration stars were located as far as possible from bad pixels. The x and y coordinates of the target star varied smoothly throughout the night by less than 3.7 and 2.5 pixels in the K_S band and 5.4 and 5.3 pixels in the H band (WASP-48b), respectively, and 5.9 and 4.5 pixels in the K_S band (HAT-P-23b), which is ∼3–5 times better than the pointing stability before the work of Zhao et al. (2012a).

Our images of WASP-48b in the K_S band were obtained on UT 2012 June 22 with a total duration of 6.6 hr, yielding 1094 images with an exposure time of 8 s. Our observations of WASP-48b in the H band were obtained on UT 2012 July 07 with a total duration of 6.9 hr, yielding 1064 images with an exposure time of 10 s. Our images of HAT-P-23b in the K_S band were obtained on UT 2012 July 09 with a total duration of 6.7 hours, yielding 1037 images with an exposure time of 9 s. The airmass ranged from 1.08–1.35, 1.08–1.47, and 1.04–1.53 during the nights, respectively.

We extracted time-series photometry for the target stars and calibration stars in the collected images. First, we subtracted averaged dark frames from all images. We combined twilight flats into master flat fields by normalizing each individual flat to unity and then taking the median for each night. For the K_S and H band observations of WASP-48b, respectively, we selected three calibration stars with median fluxes ranging from ∼0.8–1.1 and ∼0.2–0.9 times that of WASP-48. For HAT-P-23b, we selected four calibration stars with median fluxes ranging from ∼0.4–0.8 times that of HAT-P-23. Brighter stars in the fields were excluded because their fluxes exceeded the linearity regime or saturated the detector. Faint stars in the fields had insufficient signal-to-noise and were thus ignored. Still other stars were neglected if their inclusion into the analysis resulted in an increased scatter in the best-fit residuals. We corrected for transient "hot pixels" around each star as for the warm Spitzer data.

To determine the position of each star, we calculated the flux-weighted centroid within 20.0 pixels of the approximate center of the star. We allowed each stellar position to vary independently. For these data, we cannot obtain accurate position estimates using a two-dimensional Gaussian fit because defocusing the telescope makes the stellar PSFs highly irregular, with elliptical shapes and uneven distributions of flux. We corrected the time stamp of each image to the mid-exposure time and then converted the time into the BJD_TT standard using the routines of Eastman et al. (2010). We extract conversion factors from the FITS headers to estimate the Poisson noise limits for the photometry.

We performed aperture photometry on each star using circular apertures with fixed radii ranging from 8.0–20.0 pixels in increments of 0.5 pixels. Radii of 11.5 pixels (WASP-48b, K_S band), 15.0 pixels (WASP-48b, H band), and 11.0 pixels (HAT-P-23b, K_S band) ultimately produced the best fits. We used the same photometric apertures for the target and calibration stars in each set of observations; varying the aperture size from star to star during the analysis degraded the quality of the resulting light curves because the shapes of the stellar PSFs were not constant during the night. We used sky annuli with inner and outer radii of 30.0 and 55.0 pixels, respectively. We again used the 3σ clipped mean as our background value and checked to make sure that there were not any bright stars within the annuli. We selected this annulus size and range to minimize the rms scatter in the residuals residuals from the final fits, but different annuli yield consistent results for the eclipse depths and times. Figures 1 and 2 show the raw photometry that we extracted for WASP-48b, HAT-P-23b, and calibration stars during all three nights.

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The data analysis routines require precise ephemerides for the systems. In particular, we need to know the orbital period, the ratio of the stellar and planetary radii, the orbital inclination, the orbital semi-major axis, and the timing of at least one eclipse or transit. Accounting for the orbit-crossing time of light and assuming circular orbits, secondary eclipses would occur at a phase of 0.5002 for both planets. We use the ephemerides from the discovery papers for HAT-P-23b (Bakos et al. 2011) and WASP-48b (Enoch et al. 2011). Sada et al. (2012) observed a more recent transit of WASP-48b and calculated different values for its inclination and semi-major axis, which are tightly correlated. We obtain consistent results with both sets of ephemerides. However, we use the original ephemerides from Enoch et al. (2011) because they ultimately yield lower formal errors.

We use standard methods to analyze the data from warm Spitzer (e.g., Beerer et al. 2011; Knutson et al. 2012; Lewis et al. 2013; Baskin et al. 2013; Todorov et al. 2013). We initially discard points in our light curve that suffered uncorrected particle hits within our science aperture or significant spatial displacement. Specifically, we discard points in which the measured flux or the x or y position vary by >3σ from the median values in the adjacent 20 frames in the time series. In total, we excised 23 (1.1%) and 39 (1.8%) frames from the 3.6 and 4.5 μm observations for WASP-48b, respectively, and 23 (1.1%) and 30 (1.4%) frames for HAT-P-23b.

The background fluxes for these data sets exhibit a ramp-like behavior and intermittent variation between three distinct levels, particularly in the 3.6 μm band (e.g., Beerer et al. 2011; Deming et al. 2011). Also, the position of the target star on the detector does not stabilize for ∼15–30 minutes. To counter these effects, we trim the number of frames from the beginning of each time series that minimizes the rms of the residuals in our best-fit solution: 126 (HAT-P-23b, 3.6 μm), 131 (HAT-P-23b, 4.5 μm), 69 (WASP-48b, 3.6 μm), and 60 (WASP-48b, 4.5 μm).

The total flux measured within our photometric aperture varies due to a well-known intrapixel sensitivity effect (e.g., Charbonneau et al. 2005, 2008). We first normalize our light curve so that the median of the out-of-eclipse flux values is equal to unity. Then, to correct for the systematic effects, we fit the data to "decorrelation functions" involving the measured x and y positions of the star, along with a term that is linear in time to reduce correlated noise. Our most general decorrelation function is

$$\begin{equation} F(\lbrace c_i\rbrace ,\bar{x},\bar{y},t)=c_0 + c_1\bar{x} + c_2 \bar{y} + c_3 \bar{x}^2 + c_4 \bar{y}^2 + c_5 t, \end{equation} \tag{ 3 }$$

where t is the predicted time from the center of the secondary eclipse, $\bar{x}$ and $\bar{y}$ are the x and y positions minus their median values over the time series, and the {c_i} are free parameters. Including all of the terms in Equation (3) for the 3.6 μm band data improved the fits for both targets, relative to fits in which the quadratic terms were excluded, where improvement is defined as a decrease in χ² (by nine for WASP-48b and two for HAT-P-23b). We assumed that the measurement error for each of our points was equal to the rms scatter in the residuals between the photometry and the best-fit eclipse model for each fit, although this is likely an underestimate, as discussed below. Including the $\bar{x}^2$ and $\bar{y}^2$ terms in the fits for the 4.5 μm band data decreased χ² by ⩽1 for both targets, so we set c₃ = c₄ = 0 in our analysis of these data. We tried including a $\bar{x}\bar{y}$ term (Désert et al. 2009) and obtained consistent results, but the χ² for our fits remained constant or increased. Figure 3 shows the raw photometry along with the best-fit decorrelation functions.

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The Bayesian information criterion (BIC) is a popular method of model selection (Liddle 2007). We may write

$$\begin{equation} {\rm BIC} = \chi ^2 + k \ln (n), \end{equation} \tag{ 4 }$$

where k is the number of free parameters in the model and n is the number of observations. Models with a lower BIC are usually preferred. That is, the improvement in the fit achieved by adding an additional parameter to the model must be larger than the penalty imposed by the rightmost term in Equation (4). For our observations, ln (n) ∼ 8. Although we desire a quantitative criterion for which terms to exclude in Equation (3), the BIC is ill-suited to this purpose (Todorov et al. 2013). In particular, our calculation of χ² depends on the intrinsic error in the light curve, σ, which is certainly larger than the Poisson noise limit because of systematic effects but otherwise undetermined. In any case, we obtain consistent results for the eclipse depths and timing offsets whether or not we include the quadratic terms in Equation (3).

We simultaneously fit a model for the eclipse, G(d, Δt), with two free parameters: the eclipse depth d and a timing offset from the predicted center of eclipse Δt (Mandel & Agol 2002). Initially, we fit these functions using the Levenberg–Marquardt algorithm (Markwardt 2009). We fit all free parameters simultaneously, but we also tried fitting the decorrelation function and then the eclipse light curve in series; we report the simultaneous fits as our final results, but both methods produce consistent results. Figures 4 and 5 show the decorrelated photometry and the best-fit eclipse models for WASP-48b and HAT-P-23b. The rms scatters in the residuals are, respectively, 0.390% and 0.264% in the 4.5 and 3.6 μm bands for WASP-48b and 0.442% and 0.331% in the same bands for HAT-P-23b. These residual errors are 24.0% and 14.4% (WASP-48b) and 14.2% and 15.0% (HAT-P-23b) above the Poisson noise limits.

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We used two methods to estimate the uncertainties in our calculated eclipse depth and time offsets. First, we used the Markov chain Monte Carlo (MCMC) method (Ford 2005; Winn et al. 2009) with 10⁷ steps to fit the parameters, using our first fit as our initial state. We ran independent Markov chains with different initial states and obtained consistent results. The distributions for all parameters were roughly Gaussian, and the eclipse depths and time offsets were neither correlated with each other nor with any parameter in the decorrelation functions. Second, we estimated the contribution to uncertainty from time-correlated noise with the "prayer-bead" (PB) method (Gillon et al. 2009; Carter & Winn 2009). That is, we removed the residuals from our best-fit solution, then shifted them in increments of one time step and added them back into the best-fit solution. At each time step, we re-fit the eclipse light curve using the Levenberg–Marquardt algorithm.

We compared the prayer-bead distribution of eclipse depths and time offsets to the 68% symmetric confidence intervals from the Markov chains and reported the larger of the two as our formal errors. Both methods returned roughly consistent results, which are summarized in Table 1. Specifically, for WASP-48b, the MCMC error for the eclipse depth (0.020%) was larger than the PB error (0.009%) in the 4.5 μm band, whereas the PB and MCMC errors were identical (0.013%) for the 3.6 μm band. For HAT-P-23b, the MCMC errors were larger in both the 3.6 μm band (0.019% versus 0.013%) and the 4.5 μm band (0.026% versus 0.020%). The PB errors in the eclipse timing were larger for every observation except the 3.6 μm band observation of HAT-P-23b, for which the MCMC error was 0.1 minute larger.

We measured the timing of the secondary eclipses to within 4.3 and 7.8 minutes (WASP-48b) and 2.2 and 2.6 minutes (HAT-P-23b) in the 4.5 and 3.6 μm bands, respectively. We calculated the predicted timing of each center of eclipse using the orbital period and transit epochs from the discovery papers (Enoch et al. 2011; Bakos et al. 2011). We added the uncertainties in the ephemerides in quadrature with our measurement error to obtain our formal errors on the eclipse offset, which are also summarized in Table 1. Overall, we have measured mean timing offsets of 2.6 ± 3.9 minutes for WASP-48b and 4.0 ± 2.4 minutes for HAT-P-23b.

We discarded the final 37 and 21 images from the WASP-48b K_S and H band data, respectively, because the sky background began to increase dramatically near sunrise. We also discarded images if any pixel counts in the photometric apertures exceeded the linearity regime of the detector, or if the total flux for any star varied by >3σ from the median values in the adjacent 20 frames in the time series. In total, we excised 39, 26, and 13 images for the WASP-48b K_S and H band and the HAT-P-23b K_S band data, respectively, under these two criteria.

Most of the variations in the raw light curves from the ground-based data result from correlated systematics, including changes in seeing and airmass, atmospheric transition, and thermal background (e.g., Zhao et al. 2012a). Fortunately, these systematics have similar effects on the target and calibration stars. For each set of observations, we take the mean of the normalized light curves of the calibration stars to form a single reference light curve. Combining the calibration light curves using the median or flux-weighted mean produced consistent, but inferior, results in the ultimate fits. We also attempted to combine the calibration stars using a weighted average, where the relative weights for each star varied as free parameters in the fit, but this did not improve our ultimate best-fit solution.

We use our averaged calibrator star light curve as part of a decorrelation function to correct for these systematics. We fit the following decorrelation function to the light curve for HAT-P-23b in the K_S band:

$$\begin{equation} F(\lbrace c_i\rbrace ,t)=c_0 + c_1t + c_2L_R, \end{equation} \tag{ 5 }$$

where L_R is the reference light curve. For the observations of WASP-48b in both the K_S and H wavebands, we obtained consistent results when we simply divided the target light curve by the reference light curve and then fit a decorrelation function that was only linear in time, i.e., with c₂ = 0 in Equation (5). Doing the same for the K_S band observation of HAT-P-23b, however, implausibly caused the eclipse depth to increase by >3σ, along with a ∼10% increase in the residual scatter, so we elected to use the full decorrelation function in Equation (5).

As with the Spitzer data, we simultaneously fit a model for the secondary eclipse from Mandel & Agol (2002). However, we elect to fix the time offset to the best-fit value from the analysis of the Spitzer data to reduce the number of free parameters in our fit, since our ground-based data are of comparatively lower quality. That is, Δt = 0.4 and 9.5 minutes for WASP-48b and HAT-P-23b, respectively. Changing the eclipse offset by ±1σ yields consistent eclipse depths to within ±1σ. In fact, the best-fit eclipse depth changes by ⩽0.002% for both observations of WASP-48b. Moreover, we obtain consistent eclipse depths when we constrain the eclipses to have no timing offsets, as predicted for circular orbits. We again first calculate the best-fit values for each parameter using the Levenberg–Marquardt algorithm. We then run Markov chains with 10⁷ and 5 × 10⁶ steps, respectively, for HAT-P-23b and WASP-48b.

We report the best-fit solutions with minimized χ² from the Levenberg–Marquardt algorithm in Table 1, but the medians of the distributions from the Markov chains are consistent within ±1σ. No parameters are strongly correlated. We run our prayer-bead routine on the best-fit solution to provide another measure of uncertainty. The prayer-bead errors for HAT-P-23b are larger than the MCMC errors (0.046% versus 0.036%), which is expected given the large systematic effects seen in Figure 5, but the MCMC errors are larger for the WASP-48b eclipse depths (0.016% versus 0.006% in the H band and 0.027% versus 0.011% in the K_S band).

Our results are summarized in Table 1. Figures 4 and 5 show our best-fit light curves. The RMS scatters in the best-fit residuals are 0.380% (WASP-48b, K_S band), 0.222% (WASP-48b, H band), and 0.393% (HAT-P-23b, K_S band). These are factors of 2.8, 3.2, and 2.2 above the respective Poisson noise limits. The limiting source of noise in our measurements is not photon statistics, but systematic instrumental and atmospheric effects. Our noise levels are similar to those achieved by other ground-based eclipse observations (e.g., Zhao et al. 2012a, also with WIRC).

Both K_S band datasets appear to have >3σ outliers when binned in 12 minute intervals and thus large formal errors. These outliers are not obviously correlated with the seeing conditions, stellar positions, or other easily measured factors. Systematics related to the unstable, highly irregular stellar PSFs, rather than anomalous flux measurements from single images, likely produce these outliers; obtaining a more stable PSF would increase the precision of our eclipse depth measurements.

We predicted the timing of the secondary eclipses under the assumption that the planetary orbits are circular, using ephemerides derived from radial velocity and transit observations. The observed time of secondary eclipse may be offset if the brightness distribution on the planet's dayside is non-uniform, as has been observed (e.g., Knutson et al. 2007; Cowan & Agol 2011a), but the expected time offsets are less than one minute for most hot Jupiters, which is less than the precision of our measurements. A non-zero orbital eccentricity would also alter the timing of the secondary eclipse. From Charbonneau et al. (2005),

$$\begin{equation} |e \cos \omega | \simeq \frac{\pi \Delta t}{2P}, \end{equation} \tag{ 6 }$$

where e is orbital eccentricity, ω is the argument of periastron, and P is the orbital period. For HAT-P-23b, Δt = 4.0 ± 2.4 minutes translates into |ecos ω| ≃ 0.0036, constrained to 0.0080 within 2σ, which is consistent with the estimate of ecos ω ≃ −0.048 ± 0.023 from Bakos et al. (2011). Likewise, for WASP-48b, Δt = 2.6 ± 3.9 minutes means |ecos ω| ≃ 0.0013, constrained to 0.0053 within 2σ, consistent with the circular orbit assumed in Enoch et al. (2011). We therefore conclude that, unless the semi-major axes of these orbits are coincidentally aligned with our line of sight, these planets must have effectively circular orbits.

In this section, we combine our secondary eclipse measurements to provide constraints on the shape of the dayside emission spectrum for each planet. For very high signal-to-noise data with broad wavelength coverage, we could, in theory, retrieve the temperature profile for the planetary atmosphere, along with the abundances of chemical species such as CO, CO₂, H₂O, and CH₄ (e.g., Madhusudhan & Seager 2009; Line et al. 2012; Benneke & Seager 2012; Lee et al. 2012). Unfortunately, the information content of our secondary eclipse measurements is not high enough to allow for a full retrieval approach (Line et al. 2013).

We instead consider three types of atmospheric models. First, we model the planetary emission spectra as blackbodies. In Table 1, we report the brightness temperatures, T_b, at which the observed eclipse depths equal the planet/star flux ratios averaged over each waveband, where we used synthetic spectra from the new PHOENIX library (Husser et al. 2013) for the stars. All of the eclipse depths for WASP-48b and HAT-P-23b can be described by isothermal blackbody emission spectra. Next, we consider two sets of more complicated atmospheric models. We do not attempt to obtain quantitative best fits using either of the following models, because neither was intended for use in a fitting algorithm and because we cannot tightly constrain the various free parameters (cf., Todorov et al. 2013). Instead, we consider a range of models for each planet and qualitatively discuss those that best match our data.

We compare our data to spectra generated using the methods of Fortney et al. (2008), which assume a one-dimensional, plane-parallel atmosphere, local thermodynamic equilibrium, solar composition, and equilibrium abundances of chemical species. The absorber TiO is added in equilibrium abundance to the upper atmosphere in some models and excluded from others in order to account for its potential loss due to settling or cold traps. The magnitude of stellar flux incident at the top of the planetary atmosphere is multiplied by a scaling factor to account for the redistribution of energy to the planet's nightside. Specifically, f = 0.5 signifies even distribution of incident energy over the dayside, but no circulation to the nightside, whereas f = 0.25 corresponds to total redistribution of energy across the entire planet.

Finally, we consider spectra generated using the methods of Burrows et al. (2008), which similarly assume local thermodynamic equilibrium, solar composition, and a plane-parallel atmosphere. These models incorporate a generalized gray absorber in the stratosphere, which is parameterized with an absorption coefficient, κ_e (units of cm² g⁻¹), along with a heat sink at a fixed pressure level between 0.01–0.1 bars. The presence of the absorber raises the local atmospheric temperature, producing a high-altitude temperature inversion. A dimensionless parameter P_n represents the efficiency of energy redistribution, with P_n = 0.0 signifying redistribution on the dayside only and P_n = 0.5 representing complete redistribution.

4.2.1. WASP-48b

The predicted equilibrium temperature for WASP-48b is 2400 K if incident energy is re-radiated from the dayside alone and 2000 K if the entire planet radiates evenly, assuming zero albedo. To calculate the brightness temperatures corresponding to our secondary eclipse depths, we used a PHOENIX spectrum for WASP-48 with T_eff = 6000 K, log g = 4.0, and [Fe/H] = 0.0. We find that the best-fit blackbody emission spectrum has T_eff = 2158 ± 100 K, with χ² = 0.88 (all our fits for effective temperature have one degree of freedom), as shown in Figures 6 and 7.

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Figure 6 compares our secondary eclipse depths to four models of the dayside emission spectrum of WASP-48b following Fortney et al. (2008). All four eclipse depths are consistent to within ∼1σ with the model without TiO in the atmosphere and with f = 0.50, indicating little redistribution of energy from the dayside to the nightside. The Spitzer measurements provide the strongest constraints because these atmospheric models are most divergent at Spitzer wavelengths. The ground-based data are also well-matched by the nominal model preferred by the Spitzer data. The model with TiO and f = 0.25 matches the 4.5 μm and H band measurements and might be consistent with the other two eclipse depths if the amount of recirculation were reduced.

Figure 7 compares four models generated using the methods of Burrows et al. (2008) to our data. Unlike the models of Fortney et al. (2008), the band-averaged flux ratios are nearly identical among the different models in the H, K_S, and 3.6 μm bands. The eclipse depth at 4.5 μm, however, is only consistent with two models. In the first model, WASP-48b has a non-zero, but low, abundance of a gray absorber and little redistribution of energy. In the second model, WASP-48b has an intermediate amount of recirculation to the night side and a weak temperature inversion.

Both sets of models indicate that WASP-48b likely has a weak or absent temperature inversion, but the inferred magnitude of energy recirculation depends on the assumed model. Like the simple blackbody fit, the models of Burrows et al. (2008) suggest an intermediate amount of circulation. The best-fit model following Fortney et al. (2008) implies very little recirculation to the planet's nightside, but the gold model with TiO and more efficient day-night circulation also provides a reasonably close match to the data.

4.2.2. HAT-P-23b

The predicted equilibrium temperature for HAT-P-23b is 2450 K if incident energy is re-radiated from the dayside alone and 2050 K if the entire planet radiates evenly, assuming zero albedo. To calculate the brightness temperatures corresponding to our secondary eclipse depths, we interpolated spectra in the PHOENIX library to produce one with T_eff = 5900 K, log g = 4.3, and [Fe/H] = 0.0 for HAT-P-23. The best-fit blackbody emission spectra has T_eff = 2154 ± 90 K (χ² = 6.11), as shown in Figures 8 and 9. Excluding the secondary eclipse in the K_S band dramatically improves the fit (χ² = 0.01), but only lowers T_eff to 2123 ± 81 K.

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Figure 8 compares our secondary eclipse depths to four models of the dayside emission spectrum of HAT-P-23b following Fortney et al. (2008). Again, the Spitzer eclipse depths are consistent to within ∼1σ with the model with f = 0.50 and without TiO in the atmosphere. The K_S band eclipse depth is higher than all of the models, but this band suffers from a higher level of time-correlated noise and the errors are correspondingly large.

Figure 9 shows three models following Burrows et al. (2008) compared to our secondary eclipse depths. Within ∼2σ, the K_S band eclipse depth is consistent with all three models. The Spitzer eclipse depths, particularly the 4.5 μm band, favor the model with a weak temperature inversion and moderate redistribution of energy, comparable to the preferred model for WASP-48b.

4.2.3. Stellar Activity and Temperature Inversions