EXTRASOLAR STORMS: PRESSURE-DEPENDENT CHANGES IN LIGHT-CURVE PHASE IN BROWN DWARFS FROM SIMULTANEOUS HST AND SPITZER OBSERVATIONS

The L5 dwarf 2MASS J15074769-1627386 (hereafter 2M1507) was discovered by Reid et al. (2000), and the L5 dwarf 2MASS J18212815+1414010 (hereafter 2M1821) was discovered by Looper et al. (2008). Spectroscopic analysis by Gagné et al. (2014) identified 2M1821 as a young field dwarf showing signs of low surface gravity. Sahlmann et al. (2015) measured the parallax of 2M1821 to be 106.15 ± 0.18 mas, corresponding to a distance of 9.38 ± 0.03 pc. Based on the J − K color of 1.78, weak water bands, triangular H-band continuum, and strong 9–11 μm silicate absorption, Looper et al. (2008) suggested that the atmosphere of 2M1821 might have unusually thick dust clouds. For 2M1507, weak silicate absorption between 9 and 11 μm was found (Cushing et al. 2006), indicating different cloud thickness than 2M1821.

With the HST/WFC3 G141 grism spectra of 2M1507 and 2M1821, Yang et al. (2015) studied the wavelength dependence of their flux density variation between 1.1 and 1.7 μm. By taking the ratio of the brightest and faintest spectra in an HST visit, Yang et al. (2015) found that for the two L5 dwarfs the flux density of the 1.4 μm water band varies at a similar rate to that of the J and H bands. This is significantly different from the reduced variation amplitude in the 1.4 μm water band observed for the two T2 dwarfs (SIMP J013656.5+093347.3 and 2MASS J21392676+0220226) reported by Apai et al. (2013). Yang et al. (2015) showed that models of an L5 dwarf with a haze layer high (<50 mbar) in the atmosphere could explain the similar flux variation amplitude in and out of the 1.4 μm water band observed for 2M1507 and 2M1821.

The T2 dwarf SIMP0136 was discovered by Artigau et al. (2006) in a proper-motion survey. At an estimated photometric distance of 6.4 ± 0.3 pc, it is the brightest T dwarf in the northern hemisphere. Ground-based photometric monitoring by Artigau et al. (2009) revealed that SIMP0136 is variable with a peak-to-peak amplitude of ∼50 mmag in J band, marking the first detection of high-amplitude IR flux variations for a T dwarf. Kao et al. (2015) detected circularly polarized radio emission with two pulses in the 4–8 GHz band and estimated a magnetic field strength of at least 2.5 kG.

The T2 dwarf 2M2139 was discovered by Reid et al. (2008). Radigan et al. (2012) reported periodic flux variation of 2M2139 in the J band with a peak-to-peak amplitude as large as 26%, making 2M2139 the most variable brown dwarf discovered to date.

HST time-resolved spectroscopy by Apai et al. (2013) provided high-quality spectral series and light curves for both SIMP0136 and 2M2139. Based on these data and comparison to state-of-the-art atmospheric models, the authors concluded that the variations in both T2 dwarfs are caused by thickness variations in the silicate cloud cover (warm thin and cooler thicker clouds). Light-curve modeling through a Genetic Algorithm-optimized ray tracer by Apai et al. (2013) and a Markov Chain Monte Carlo (MCMC)-optimized pixelized atmosphere model by Karalidi et al. (2015) both found that at least three elliptical features are required to fit the light curve of SIMP0136 and 2M2139. Principal component analysis of the HST spectral series showed that >97% of the spectral variations can be reproduced with only a single principal component on top of a mean spectrum, arguing for a single type of cloud feature, consistent with the result of the atmospheric modeling.

The T2 dwarf 2MASS J13243553+6358281 (hereafter 2M1324) was discovered independently by Looper et al. (2007) and Metchev et al. (2008). It has an usually red spectral energy distribution and also exhibits peculiar IR colors (Looper et al. 2007; Metchev et al. 2008; Faherty et al. 2009). Looper et al. (2007), Burgasser et al. (2010), and Geißler et al. (2011) discussed the possibility of 2M1324 being a close binary (L9 + T2 or L8 + T3.5).

The T6 dwarf 2M2228 was discovered by Burgasser et al. (2003). In a ground-based monitoring campaign searching for variability, Clarke et al. (2008) detected a peak-to-peak amplitude of 15.4 ± 1.4 mmag for 2M2228.

Buenzli et al. (2012) analyzed HST/WFC3 G141 grism spectra and partially simultaneous Spitzer Ch2 photometry of 2M2228 and for the first time found phase shifts among light curves of five narrow spectral bands between 1.1 and 1.7 μm as well as the broad spectral band at 4.5 μm. The fluxes in these spectral bands probe different depths in the atmosphere, and the measured phase shifts of 2M2228 were found to correlate with the characteristic pressure levels derived from atmospheric models. For lower pressure levels, the phase shift is larger. Such findings by Buenzli et al. (2012) revealed atmospheric structures in the vertical direction, adding a new dimension in the studies of the ultracool atmospheres.

We found that the light curves in both Spitzer/IRAC channels exhibit a variety of shapes, as shown in Figure 4. In some cases, the light-curve shapes could evolve over a timescale as short as one rotation period, and the shape and amplitude of the light curve observed at different wavelengths can be substantially different.

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While our observations reveal differences in the light-curve shape as a function of time and, sometimes, as a function of the wavelength of the observations, in 20 out of 48 Spitzer visits the differences between the light curves of consecutive rotations are relatively small. In these cases, the light-curve shape observed at different wavelengths is either identical or very similar, but potentially delayed ("phase-shifted") and may have different amplitudes.

In this paper, we performed analyses to quantify phase shifts of only the data sets where the light curves are slowly evolving and the features are similar between different wavelengths, so that simple, robust approaches can be applied to measure phase shifts. A separate publication (D. Apai et al. 2016, in preparation) will fully explore the evolution of the light curves.

Here we focus on the information available in the wavelength-dependent phase shifts. As different wavelengths of observation probe different pressure levels in the atmosphere, the phase shifts measured here can be used to explore the vertical-longitudinal structure of the atmosphere. This approach has been demonstrated in Buenzli et al. (2012) and Apai et al. (2013) observationally, and models by Zhang & Showman (2014) and Robinson & Marley (2014) have predicted characteristic observational signatures contained in such pressure-dependent phase shifts.

With the Storms data set, we adopted different approaches to determine phase shifts for two different data subsets: nonsimultaneous Spitzer Ch1 and Ch2 observations and simultaneous HST and Spitzer observations. The Spitzer data allow analysis of continuous temporal variations and constrain longitudinal heterogeneities in the atmospheres, while the simultaneous multiwavelength observations yield pressure-dependent phase shifts and probe atmospheric structures in the vertical direction.

One of the objectives of our Spitzer observations is to measure phase shifts between Ch1 and Ch2 light curves. In a typical Spitzer visit, as shown in Figure 4, we first observe two rotations in Ch1, followed by one rotation in Ch2 and then another rotation in Ch1.

We employed two methods to determine potential phase shifts between Ch1 and Ch2 observations. The first method derives the phase shift from the times at which the target is at maximum or mininum flux levels, T_max or T_min. For visits where we can unambiguously identify the flux maxima and minima in Rotations 2, 3, and 4 (R2, R3, and R4, as marked, e.g., in Figure 5), we first found the midpoint in T_max or T_min for R2 and R4, which are both observed in Ch1. This was achieved by fitting a third-order Fourier function (see Equation (2)) and then calculating the maxima and minima from the fits. Next, we calculated the time difference between the T_max or T_min in R3 (observed in Ch2) and the midpoint time of R2 and R4. The time difference is expressed as a phase shift in degrees using the rotation period given in Table 1. Both the time differences in hours and the phase shifts in degrees are given in Table 3. We excluded the visits where T_max or T_min cannot be well determined due to light-curve evolution and/or incomplete phase coverage, such as Visit 6 of 2M1324 (see Figure 4).

To estimate the uncertainty in the phase shift measurements, we created 1000 synthetic light curves by adding random noise to the Fourier fits. The random noise is drawn from a normal distribution derived from the residuals of the Fourier fit. We performed the same analysis described above on the synthetic light curves, and the standard deviations of the 1000 phase shift measurements were adopted as the uncertainties.

The second method we used to measure phase shifts betweeen light curves taken at different wavelengths was to directly cross-correlate the light curves in each visit between the two channels. For each visit, we first measured a rotation period by cross-correlating the two rotations observed in Ch1, R2 and R4. The shift in time at which the cross-correlation function reaches maximum plus the time difference between the end times of R2 and R4 are equal to two rotation periods. Then we cross-correlated R2 (in Ch1) and R3 (in Ch2) and also determined the shift in time corresponding to the peak of the cross-correlation function. The phase shift was calculated from the shift in time and the rotation period measured from R2 and R4. The same analysis was applied to R3 (in Ch2) and R4 (in Ch1). We also fit a Gaussian to the cross-correlation function, and the uncertainties in the center location of the fitted Gaussian expressed in phase were taken as the uncertainties of the measured phase shifts. Note that this method does not work well when the light curve evolves significantly between consecutive rotations, such as Visit 7 of 2M2139 (see Figure 4). We list the results from the visits that show little evolution of light-curve shape in Table 4.

Table 4.  Phase Shifts between Spitzer Ch1 and Ch2 Light Curves Measured from Cross-correlation

Target	Pa (hr)	Visit	PVisitb (hr)	Phase Shift
				(deg)	(deg)
SIMP0136	2.414	1	2.628	−0.4 ± 4.8	0.1 ± 4.6
		2	2.623	7.3 ± 7.1	−7.5 ± 6.6
		3	2.617	−5.7 ± 4.0	−0.3 ± 4.0
		6	2.611	10.7 ± 6.1	−14.8 ± 7.2
		7	2.531	12.8 ± 6.8	−16.9 ± 7.0
2M1324	13.0	1	12.339	13.8 ± 27.5	−11.8 ± 27.5
		4	12.346	−2.6 ± 1.6	3.8 ± 1.8
2M1821	4.2	1	4.177	−1.8 ± 9.5	−5.9 ± 9.1
		4	4.262	−8.8 ± 5.6	1.5 ± 6.7
		5	4.289	4.1 ± 5.8	−3.9 ± 5.4
		8	4.439	−2.9 ± 6.0	−8.5 ± 6.5
2M2139	7.614	5	7.790	−1.0 ± 9.0	0.2 ± 2.2
		6	7.775	−16.3 ± 21.5	18.1 ± 22.3
		7	7.654	−2.8 ± 1.6	3.2 ± 1.7
2M2228	1.369	3	1.516	8.6 ± 10.6	−5.6 ± 9.2

Notes. Visits that show strong light-curve evolution are excluded.

^aRotation period measured from cross-correlating all eight visits of data. ^bRotation period measured in the specific visit from cross-correlating the Ch1 light curves of Rotation 2 and Rotation 4.

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For the four objects simultaneously observed with HST and Spitzer, the HST observations have incomplete phase coverages. Therefore, HST–Spitzer phase shift measurement requires a different approach than the nonsimultaneous but longer and better-sampled Ch1-Ch2 phase shift comparisons.

We compare HST J-band light curves with Spitzer light curves of the four objects in Figures 5– 8. The J-band light curves were calculated by integrating the HST/WFC3 grism spectra over the Two Micron All Sky Survey (2MASS) J-band relative spectral response curve (Cohen et al. 2003). To be consistent with the time stamps of the Spitzer observations, all HST time stamps were converted from Modified Julian Dates to barycentric Modified Julian Dates, using the IDL routine barycen.pro. The mid-time of each HST exposure is used as the time of observation. The light curves from the two instruments are normalized by their respective median values.

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To quantify the phase shifts, we fit separate third-order Fourier functions (see Equation (2)) to the simultaneous HST J-band and Spitzer Ch1 or Ch2 light curves. The rotation periods in the Fourier functions were fixed to the values listed in Table 1. The residuals of the Fourier fits are shown in Figure 9. We used the best-fit parameters of the Fourier fits to calculate the phases of the first- and second-order Fourier components and then, from their difference, the corresponding phase shifts.

Given the high precision of the HST data, the main source of uncertainty in the measured phase shifts is from the Fourier fits to the Spitzer light curves. We estimated the amplitude of noise by fitting a Gaussian to the residuals of the Fourier fit and created a synthetic light curve by adding random noise drawn from the fitted Gaussian distribution to the Fourier fit. We then measured the phase shift in the synthetic light curve in the same fashion as the observed light curve. This procedure was repeated 1000 times for each segment of Spitzer light curve analyzed, and the standard deviations of the 1000 measurements were taken as the uncertainties of the phase shift. We report the phase shifts and associated uncertainties for first- and second-order Fourier components in Table 5. In the following discussion, we excluded measurements from two visits, Spitzer Visit 7 of SIMP0136 and Visit 1 of 2M1507, because for Visit 7 of SIMP0136 (bottom panel of Figure 7), light-curve evolution makes the measurement ambiguous, and for Visit 1 of 2M1507 (top panel of Figure 5), the Fourier fit to the HST data is poor.

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We found that all six sources in our sample are variable in both IRAC channels in all Spitzer visits. Most light curves are not strictly sinusoidal and can evolve on timescales as short as one rotation period, as shown in Figures 4– 8. The simultaneous HST observations of four short-period sources also exhibit flux variations in the integrated light curves of different narrow spectral bandpasses (Figures 10– 13).

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For most of our targets our Spitzer data represent the longest continuous, high-precision light curves. As such, these data can place powerful constraints on the rotational period of the objects. In order to examine periodicity and estimate the rotational period, we calculated the autocorrelation functions for each visit for the Ch1 observations (approximately three rotations of the targets per visit sampling a time interval corresponding to approximately four rotation periods; see Figure 1). We used the standard IDL script for the autocorrelation analysis and identified the first peak (corresponding to nonzero shifts) in the autocorrelation function for each visit.

For three of our sources (SIMP0136, 2M2139, and 2M2228), the autocorrelation function was well defined in most visits and the peaks identified were consistent. Examples are shown in Figure 14. For these objects we adopted, as rotation periods, the mean of the peak autocorrelation values and, as the uncertainties of the rotation periods, the standard deviation of the peak locations in the autocorrelation functions.

For the other three targets, however, the light-curve evolution was so significant that no peak in the autocorrelation function emerged consistently among the eight visits. For these three objects, therefore, the autocorrelation analysis did not yield reliable rotation period estimates. For these sources we adopted previous rotation estimates by Metchev et al. (2015), primarily based on Fourier analyses, which are qualitatively consistent with our light curves. The measured and adopted rotation periods are listed in Table 1.

We utilized two different methods to measure phase shifts between light curves observed in the two IRAC channels. While both methods are somewhat limited by light-curve evolution on short timescales, our measurements showed that there is no detectable phase shift between the Ch1 and Ch2 light curves. As listed in Tables 3 and 4, the majority of measurements are within the 1σ or 2σ limit of 0°, with a typical upper limit of 15°.

Simultaneous HST and Spitzer observations of the four targets from the Storms program provide the most comprehensive brown dwarf monitoring data set, with each object covered for at least six HST orbits. Previously, only Buenzli et al. (2012) have studied 2M2228 with simultaneous HST and Spitzer observations, which overlapped for two HST orbits. With our unique data set, we found that SIMP0136 (Figure 7) shows a phase shift of ∼30° between HST J-band and Spitzer light curves, while the other three objects (2M1507, 2M1821, and 2M2228) all show substantial phase shifts within 10°–20° of 180° (Table 5).

Buenzli et al. (2012) discovered phase shifts among several integrated light curves of characteristic wavelength regions in the HST and Spitzer observations of 2M2228 obtained in 2011 July. The selected bandpasses, including the J- and H-band windows, as well as water and methane absorption, probe different pressure levels in the atmosphere, and Buenzli et al. (2012) found that the phase shifts increase monotonically with decreasing pressure level, indicating vertical atmospheric structures.

We applied the same analysis procedure as described in Buenzli et al. (2012) to the Storms observations of 2M2228 and found that the light curves of various narrow bandpasses still display phase shifts even after 2 yr, corresponding to over 12,000 rotations. We fit sine waves to the narrowband light curves and compared the phases of the sine waves. As shown in Figure 13, with respect to the J- and H-band light curves, the water and methane bands show phase shifts close to 180° in both HST visits. We extended this analysis to all four Storms targets observed with HST. The T2 dwarf SIMP0136 (Figure 12) shows little phase shift, typically 4° ± 2°. We discuss this finding in the next section.

The greatest advantage of multiband monitoring of brown dwarfs is that emergent fluxes at different wavelengths probe different depths in the atmospheres. To investigate the characteristic pressure levels that different wavelength regions probe, we employ state-of-the-art radiative transfer and atmospheric chemistry models and calculate relative flux contributions from each model pressure level.

We first performed least-squares fits to available HST grism spectra to find the best-fit atmospheric model for each target. The atmosphere models we use are the most up-to-date versions of those published in Saumon & Marley (2008). Besides the four targets observed with the Storms program, we also used HST/WFC3 G141 spectra of 2M2139 from Apai et al. (2013). The observations were resampled according to the model spectral grid. The model spectra were normalized to match with the flux peak in the J band between 1.25 and 1.28 μm. The observed and the best-fit model spectra are shown in Figure 15, and the best-fit model parameters are given in Table 6. The observed spectrum beyond 1.55 μm for 2M1821 is contaminated by a background object and thus not well fitted by the model. As 2M1324 has not been observed with the WFC3 G141 grism, we use the best-fit model for the other T2 dwarfs.

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Table 6.  Best-fit Atmospheric Model Parameters for the Targets

Target		2M1507	2M2139, SIMP0136,	2M2228
		and 2M1821	and 2M1324a
Spectral Type		L5	T2	T6
Best-fit model parameters	Teff(K)	1700	1400	950
		5.0	5.0	4.5
	fsed	3	5	5
Model pressure (bar)	1.12–1.17 μm (H2O and alkali)	6.55	7.60	6.99
	1.21–1.32 μm (narrow J)	6.50	8.12	7.59
	2MASS J-band	6.48	7.29	6.93
	1.35–1.43 μm (H2O)	4.33	4.13	1.91
	1.54–1.60 μm (narrow H)	5.94	7.08	6.04
	1.62–1.69 μm (CH4 and H2O)	6.02	7.18	4.26
	Spitzer Ch1	2.75	0.37	0.24
	Spitzer Ch2	1.59	1.00	0.48

Note.

^aDue to the absence of a high-precision HST/WFC3 spectrum of 2M1324, we use the best-fit parameters from fits of SIMP0136, which has the same spectral type as 2M1324.

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To compute the contribution functions, we first converged a standard radiative-convective equilibrium atmosphere thermal structure model following the approach of Saumon & Marley (2008). Since the time of that model description, there have been substantial updates to the opacity database employed, which were utilized in the present analysis and will be described more fully in an upcoming paper (M. Marley et al. 2016, in preparation). Once the model converged, a temperature perturbation was iteratively applied to quarter-scale-height subregions of the atmosphere. Given this new, artificial temperature profile, a new emergent spectrum was computed. The perturbation was then removed, a new perturbation applied to the next overlying region, and the process repeated. By computing the ratio of each perturbed thermal emission spectrum to the baseline case, the sensitivity of each spectral region to temperature perturbations at depth could be computed.

We repeated this procedure for 37 model pressure levels that cover pressures from 1.8 × 10⁻⁴ bars to ∼23 bars, and we essentially obtained the relative flux contributions from a range of pressures at the wavelengths covered by our HST and Spitzer observations (left panels of Figures 16–18).

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To find the characteristic pressure level from where most of the flux emerges for a spectral bandpass of interest, we integrated the relative flux contributions over the wavelength bandpass and calculated a cumulative flux contribution function starting from the top of the atmosphere. We identified the two pressure levels between which the cumulative flux reaches 80% of the total flux and determined the exact pressure level by linearly interpolating between the two model pressure levels. The 80% cumulative flux pressure levels were calculated for eight spectral bandpasses, including five HST narrow bands, as well as 2MASS J band and Spitzer Ch1 and Ch2. The results are listed in Table 6 and also shown in the right panels of Figures 16–18.

We stress that the approach adopted here, although based on state-of-the-art atmosphere models, offers only a limited tool for identifying the specific pressure levels where modulations are introduced: In our approach of modifying the temperature but not the cloud opacity, the results strictly apply only to changes in atmospheric temperature alone. However, our model does make clear the atmospheric region to which the spectra at a given wavelength are most sensitive. Changes to the cloud opacity at or above the pressure levels we identify for each observation are likely to substantially alter the emergent flux, whereas changes to the underlying cloud opacity (below the 80% contribution levels) are less likely to be as significant.

A good example of how high-altitude clouds or haze layer can dramatically modulate the pressure levels probed in the infrared is provided by Cassini observations of hot spots in Jupiter's equatorial regions. Choi et al. (2013) present multiband Cassini imaging monitoring of the hot spots and plumes that are seen to coevolve near the Jovian equator. The hot spots are interpreted as cloud clearing in the ammonia cloud deck, which may be rapidly obscured or revealed by high-altitude clouds. In this case, for example, the infrared observations probe deep in the atmosphere when cloud opacity is absent, but are limited to the top of the atmosphere when opacity is introduced by high-level clouds.

All six sources in our sample exhibit flux variations in both Spitzer channels during all eight visits, regardless of spectral type and rotation period. We found that the derived characteristic model pressures above which 80% of the fluxes emerge from at Ch1 and Ch2 are less than 2 bars for the two L5 dwarfs and less than 1 bar for the T dwarfs, indicating heterogeneity in the upper atmospheres of all our targets.

Interestingly, the relative flux contributions (Figures 16–18) show that in the model atmosphere of a T2 dwarf most of the Spitzer Ch1 flux is from lower pressure levels than the Ch2 flux, while the case is reversed in the L5 and T6 models.

For 2M2228 (T6), shown in top left panel of Figure 18, flux between 1.1 and 1.7 μm emerges from a wide range of pressures levels. For example, the narrow J-band flux is mostly from ∼7.5 bars, while the flux of the 1.35–1.43 μm water band primarily comes from ∼2 bars. In contrast, the L5 and T2 dwarfs emit most of the flux between 1.1 and 1.7 μm from a smaller range of pressures, which are around 6.6–4.3 bars for the L5 targets and 8.1–4.1 bars for the T2 targets.

Between Spitzer Ch1 and Ch2 light curves, we have found no detectable phase shifts from analyses using two different methods. According to the relative flux contributions derived from the models (Figures 16–18), most of the observed flux in both Spitzer channels comes from a relatively narrow range of pressure levels in the upper atmospheres, e.g., between 1 and 2 bars in Ch1 for the L5 model (Figure 16), and between 0 and 0.3 bars in Ch2 for the T2 model (Figure 17).

For all four targets that have simultaneous HST and Spitzer observations, we have detected phase shifts between J-band and Ch1/Ch2 light curves (Table 4). The rotation rates of our objects range from 1.37 hr (2M2228) to 4.2 hr (2M1821). With the small sample size, the phase shifts display no obvious correlation with rotation periods. For example, SIMP0136 (T2) and 2M1507 (L5) have similar rotation periods, but their phase shifts are very different. SIMP0136 (T2) exhibits ∼30° phase shift, while the phase shifts of the mid-L and late-T dwarfs are all centered around 180°. This might hint that L/T transition objects have peculiarities in their atmospheric properties with respect to regular L and T dwarfs, resulting in different phase shifts between the HST and Spitzer light curves.

Simultaneous observations in multiple bandpasses probe different pressure levels in the atmosphere, providing vital information on the atmospheric properties in the vertical direction. With the Storms data set, we measured phase shifts between light curves of four narrow bands between 1.1 and 1.7 μm and two Spitzer broad bands centered at 3.6 and 4.5 μm. Here we explore the correlation between phase shifts of the six bandpasses and the characteristic model pressure levels from which most flux originates for those bandpasses.

Buenzli et al. (2012) first detected pressure-dependent phase shifts in 2M2228, with lower pressure levels showing larger phase shifts. In Figure 19, we examine again the phase shifts as a function of model pressures for 2M2228 observed 2 yr later. We find that the phase shifts measured among HST narrow bandpasses in this work are consistent with those reported by Buenzli et al. (2012). Our measured phase shifts between the J- and H-band light curves are −5° ± 2° and −8° ± 2° for two visits, respectively (Figure 13), while Buenzli et al. (2012) found it to be 15° ± 2°. The sine-wave fits to the 1.35–1.43 μm (water) and 1.62–1.69 μm (water and methane) bandpasses are not as good as J and H bands, and the phase shifts of those two bandpasses with respect to the narrow J band are generally close to 180°, consistent within errors with the results of Buenzli et al. (2012). (Note that a 180° phase shift could be equivalent to an anticorrelation of fluxes at different wavelengths with no phase shift.) The main difference between the results of Buenzli et al. (2012) and this work lies in the phase shift between the narrow J band and the Spitzer channels, which Buenzli et al. (2012) measured to be 118° ± 7°, compared to our measurement of around 160°.

Our Fourier-based phase shift measurements between different wavelength bands are separated into two distinctive groups in terms of the pressure levels probed by the bandpasses. For each target (left panels of Figures 19–22), the light curves probing deeper (≳4 bars) in the atmosphere are generally in phase, while the light curves probing the upper atmospheres (≲4 bars) display similar phase offsets. For the L and the L/T dwarfs, the two groups of light curves appear to probe pressure levels separated approximately by the radiative-convective boundary calculated from the model atmospheres. For the T6 object, the radiative-convective boundary is deeper in the atmosphere than where our available wavelength bands can probe.

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Compared with the locations of the condensate clouds in the models (right panels of Figures 19–22), the higher-pressure region (≳4 bars) probed by our observations generally coincides with the dense part of the cloud layers. Variable cloud thickness could explain the observed flux variations in the light curves that probe higher pressure levels in the atmosphere. On the other hand, the pressure levels that the Spitzer channels probe have very low condensate density. Along with the phase differences, this might indicate that different sources of modulation are in play in the two regions. The transition between the two groups of light curves is very abrupt in pressure, indicating a break or discontinuity between the two pressure regions. A separate patchy cloud deck at the lower pressure levels could cause the flux variations probed by the Spitzer bands.