Journal ArticleDOI

# 3.6 and 4.5 $\mu$m Phase Curves of the Highly-Irradiated Eccentric Hot Jupiter WASP-14b

AbstractWe present full-orbit phase curve observations of the eccentric ($e\sim 0.08$) transiting hot Jupiter WASP-14b obtained in the 3.6 and 4.5 $\mu$m bands using the \textit{Spitzer Space Telescope}. We use two different methods for removing the intrapixel sensitivity effect and compare their efficacy in decoupling the instrumental noise. Our measured secondary eclipse depths of $0.1882\%\pm 0.0048\%$ and $0.2247\%\pm 0.0086\%$ at 3.6 and 4.5 $\mu$m, respectively, are both consistent with a blackbody temperature of $2402\pm 35$ K. We place a $2\sigma$ upper limit on the nightside flux at 3.6 $\mu$m and find it to be $9\%\pm 1\%$ of the dayside flux, corresponding to a brightness temperature of 1079 K. At 4.5 $\mu$m, the minimum planet flux is $30\%\pm 5\%$ of the maximum flux, corresponding to a brightness temperature of $1380\pm 65$ K. We compare our measured phase curves to the predictions of one-dimensional radiative transfer and three-dimensional general circulation models. We find that WASP-14b's measured dayside emission is consistent with a model atmosphere with equilibrium chemistry and a moderate temperature inversion. These same models tend to over-predict the nightside emission at 3.6 $\mu$m, while under-predicting the nightside emission at 4.5 $\mu$m. We propose that this discrepancy might be explained by an enhanced global C/O ratio. In addition, we find that the phase curves of WASP-14b ($7.8 M_{\mathrm{Jup}}$) are consistent with a much lower albedo than those of other Jovian mass planets with thermal phase curve measurements, suggesting that it may be emitting detectable heat from the deep atmosphere or interior processes.

Topics:

### 1. INTRODUCTION

• Over the past two decades, observations of exoplanets have uncovered a stunning diversity of systems.
• By comparing the measured phase curves to theoretical phase curves generated by atmospheric models, the authors can constrain fundamental properties of the atmosphere, such as the efficiency of heat transport from the day side to the night side, radiative timescales, wind speeds, and compositional gradients between the day and night sides.
• Recently, the first spectroscopic phase curve was obtained for the hot Jupiter WASP-43b using the Hubble Space Telescope between 1.2 and 1.6 μm (Stevenson et al. 2014).
• After subtracting the sky background, any remaining transient “hot pixels” in each set of 64 images varying by more than 3σ from the median pixel value are replaced by the median pixel value.
• The authors calculate the flux of the stellar target in each image using circular aperture photometry.

### 3.1. Transit and Secondary Eclipse Model

• Each full-orbit observation contains one transit and two secondary eclipses.
• The authors model these events using the formalism of Mandel & Agol (2002).
• The transit light curve includes four free parameters: the scaled orbital semimajor axis a/R*, the inclination i, the center of transit time tT, and the planet–star radius ratio Rp/R*, which is the square root of the relative transit depth.
• Each secondary eclipse event is defined by a center of eclipse time tE and a relative eclipse depth d, measured with respect to the value of the phase curve at midtransit, in addition to a/R* and i, thus yielding four additional free parameters: tE1, tE2, d1, and d2.
• The authors ensure continuity between the phase curve and secondary eclipse light curves by scaling the amplitudes of eclipse ingress and egress (when the planet is partially occulted by the star) appropriately to match the out-of-eclipse phase curve values at the start and end of the eclipse.

### 3.2. Phase Curve Model

• Here, c1 and c2 are free parameters that are computed in their fits.
• The authors also experimented with other functional forms of the phase curve that included higher harmonics, but all of them resulted in higher values of the Bayesian Information Criterion (BIC) (see Section 3.4 for more information).

### 3.3. Correction for Intrapixel Sensitivity Variations

• Photometric data obtained using Spitzer/IRAC in the 3.6 and 4.5 μm bandpasses exhibit a well-studied instrumental effect due to intrapixel sensitivity variations (Charbonneau et al. 2005).
• In their analysis, the authors decorrelate this instrumental systematic in two ways.
• The authors first approach to removing the intrapixel sensitivity effect is called pixel mapping (Ballard et al. 2010; Lewis et al. 2013).
• In Figure 3, the authors compare the noise properties of each version of the 3.6 μm residual time series to the ideal n1 scaling they would expect for the case of independent (i.e., “white” noise) Gaussian measurement errors, where n is the bin size.
• In addition to having higher residual scatter than the pixel mapping solutions, the PLD fits yield eclipse depth and phase curve parameter estimates that often differ strongly from the corresponding values derived from fits with pixel mapping; these discrepancies sometimes exceed the 3σ level.

### 3.4. Exponential Ramp Correction

• Previous studies using Spitzer/IRAC have noted a shortduration ramp at the beginning of each observation, and again after downlinks (e.g., Knutson et al. 2012; Lewis et al. 2013).
• The authors find that in both bandpasses, they obtain the best results when they do not trim any data from the start of the observations.
• By minimizing the BIC, the authors select the type of ramp model that yields the smallest residuals without “over-fitting” the data.
• The residuals from the best-fit full phase curve solution, shown in Figures 4 and 5, do not appear to display any uncorrected ramp-like behaviors.

### 3.5. Parameter Fits

• The authors use a Levenberg–Marquardt least-squares algorithm to fit each full-orbit photometric series to their total model light curve, with the intrapixel sensitivity correction calculated via pixel mapping.
• The best-fit transit, secondary eclipse, and phase parameters are listed in Table 1 along with their uncertainties.
• The maximum and minimum flux offsets are measured relative to the center of secondary eclipse time and center of transit time, respectively, and are derived from the phase curve fit parameters c1 and c2.
• Comparing the results of their fits using pixel mapping and PLD, the authors see that while the residuals from the best-fit solution with pixel mapping show a significant anomalous signal similar to the one present in the global 3.6 μm phase curve fit, no residual anomaly is evident in the best-fit solution with PLD.
• The authors estimate the uncertainties in their best-fit parameters in two ways.

### 4.1. Orbital Parameters and Ephemeris

• The authors combine the two transit times calculated from their phase curve observations with all other published values (Joshi et al.
• This period is consistent with the best-fit transit period at the 0.5σ level.
• These values are consistent with those reported in Knutson et al. (2014).
• These fits provide new estimates of the orbital period (Pb) and center of transit time (Tc,b), as well as the semiamplitude of the planet’s radial velocity (Kb), the radial solutions (black lines).

### 4.2. Phase Curve Fits

• The authors combine the results of their global fits with model-generated spectra and light curves to provide constraints on the atmospheric properties of the planet.
• The resulting phase curves are shown in Figure 13 along with the corresponding ±1σ brightness bounds.
• In these models, the eastward hotspot displacement results from eastward advection due to a fast, eastward-flowing jet stream centered at the equator.
• It is interesting to compare their observed day–night flux differences with those of other planets observed to date.

### 4.3. Brightness Temperature

• The authors consider four types of atmosphere models.
• First, the authors use an interpolated PHOENIX spectrum for the host star WASP-14 (Husser et al. 2013) to calculate the brightness temperatures of WASP-14b in each band from the retrieved secondary eclipse depths.
• The authors find that the 3.6 and 4.5 μm eclipse depths are consistent with a single blackbody temperature and derive an effective temperature of Teff = 2402 ± 35 K from a simultaneous fit to both bandpasses.
• The predicted equilibrium temperature for WASP-14b assuming zero albedo is 2220 K if incident energy is re-radiated from the dayside only and 1870 K if the planet re-radiates the absorbed flux uniformly over its entire surface.
• The authors can also compare the computed brightness temperatures to the effective temperature of the dayside in the no-albedo, no-circulation limit assuming each region is a blackbody locally in equilibrium with the incident stellar flux: 2390 K (Cowan & Agol 2011).

### 4.4. Dynamical Models

• Next, the authors compare their phase curves and emission spectra to theoretical models generated from the three-dimensional Substellar and Planetary Atmospheric Radiation and Circulation model to investigate the global circulation of WASP-14b.
• The planet’s very low 3.6 μm nightside emission indicates a higher atmospheric opacity at that wavelength than is predicted by the models.
• In addition, Blecic et al. (2013) show that atmospheric models with high C/O ratios often predict reduced emission at longer wavelengths (6–8 μm) and may explain the low 8.0 μm planetary flux ratio .
• Therefore, an enhanced C/O ratio in WASP-14b’s atmosphere provides a more straightforward explanation for the observed differences between WASP-14b’s 3.6 and 4.5 μm phase curves.
• Figure 17 shows the corresponding dayside and nightside temperature–pressure profiles generated by the model.

### 5. CONCLUSIONS

• In this paper the authors present the first phase curve observations of the eccentric hot Jupiter WASP-14b in the Spitzer 3.6 and 4.5 μm bandpasses.
• The authors compare two different techniques— pixel mapping and PLD—for correcting the intrapixel sensitivity effect and find that the pixel mapping method yields lower residual scatter from the best-fit solution.
• Comparison of the measured dayside planetary emission with spectra generated from a one-dimensional radiative transfer model (Burrows et al. 2008) suggests relatively inefficient day–night heat recirculation and a moderate dayside temperature inversion.
• In the context of other planets with full-orbit thermal measurements, the authors find that WASP-14b fits the general trend that highly irradiated hot Jupiters have poor heat recirculation, while the derived Bond albedo is very small (<0.08 at 1σ) and the planet’s large mass might indicate that WASP-14b is emitting residual heat from its formation.
• This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.

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3.6 AND 4.5 μm PHASE CURVES OF THE HIGHLY IRRADIATED ECCENTRIC HOT JUPITER WASP-14b
Ian Wong
1
, Heather A. Knutson
1
, Nikole K. Lewis
2
, Tiffany Kataria
3
4
, Jonathan J. Fortney
5
,
Joel Schwartz
6
, Eric Agol
7
, Nicolas B. Cowan
8
, Drake Deming
9
, Jean-Michel Désert
10
, Benjamin J. Fulton
11,15
,
Andrew W. Howard
11
, Jonathan Langton
12
, Gregory Laughlin
5
13
, and Kamen Todorov
14
1
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA; iwong@caltech.edu
2
Space Telescope Science Institute, Baltimore, MD 21218, USA
3
Astrophysics Group, School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK
4
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
5
Department of Astronomy and Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95604, USA
6
Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA
7
Department of Astronomy, University of Washington, Seattle, WA 98195, USA
8
Department of Physics and Astronomy, Amherst College, Amherst, MA 01002, USA
9
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
10
Department of Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA
11
Institute for Astronomy, University of Hawaii, Honolulu, HI 96822, USA
12
Department of Physics, Principia College, Elsah, IL 62028, USA
13
Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA
14
Institute for Astronomy, ETH Zürich, 8093 Zürich, Switzerland
Received 2015 May 11; accepted 2015 August 28; published 2015 September 28
ABSTRACT
We present full-orbit phase curve observations of the eccentric (e 0.08) transiting hot Jupiter WASP-14b
obtained in the 3.6 and 4.5 μm bands using the Spitzer Space Telescope. We use two different methods for
removing the intrapixel sensitivity effect and compare their efcacy in decoupling the instrumental noise. Our
measured secondary eclipse depths of 0.1882% ± 0.0048% and 0.2247% ± 0.0086% at 3.6 and 4.5 μm,
respectively, are both consistent with a blackbody temperature of 2402 ± 35 K. We place a 2σ upper limit on the
nightside ux at 3.6 μm and nd it to be 9% ± 1% of the dayside ux, corresponding to a brightness temperature
of 1079 K. At 4.5 μm, the minimum planet ux is 30% ± 5% of the maximum ux, corresponding to a brightness
temperature of 1380 ± 65 K. We compare our measured phase curves to the predictions of one-dimensional
radiative transfer and three-dimensional general circulation models. We nd that WASP-14bs measured dayside
emission is consistent with a model atmosphere with equilibrium chemistry and a moderate temperature inversion.
These same models tend to overpredict the nightside emission at 3.6 μm, while underpredicting the nightside
emission at 4.5 μm. We propose that this discrepancy might be explained by an enhanced global C/O ratio. In
addition, we nd that the phase curves of WASP-14b (7.8 M
Jup
) are consistent with a much lower albedo than those
of other Jovian mass planets with thermal phase curve measurements, suggesting that it may be emitting detectable
heat from the deep atmosphere or interior processes.
Key words: binaries: eclipsing planetary systems stars: individual (WASP-14) techniques: photometric
1. INTRODUCTION
Over the past two decades, observations of exoplanets have
uncovered a stunning diversity of systems. Major improve-
ments in the capabilities of space- and ground-based telescopes
in recent years have led to the discovery and characterization of
hundreds of new exoplanets, covering a broad range of orbital
properties, interior and atmospheric compositions, and host star
types (Han et al. 2014). Meanwhile, these same technological
advances have enabled the detailed study of the atmospheric
properties of the brightest and largest planets through high-
precision photometry and spectroscopy. Most of these targets
belong to the class of gas giant planets known as hot Jupiters.
The high levels of incident ux, slow rotation, and potentially
large temperature gradients between hemispheres characteristic
of these planets allow us to test atmospheric models in a new
regime unlike any found in the solar system. In addition, hot
Jupiters are some of the most favorable targets for measuring
elemental abundances, since most material is not in a
condensed form at these high temperatures (e.g., Line
et al. 2014).
Atmospheric circulation models of hot Jupiters predict broad
super-rotating equatorial jets that circulate energy between the
day and night sides, with the precise effect of these winds on
the day-to-night temperature contrast being strongly dependent
on the particular orbital and atmospheric properties of the
planet (see Heng & Showman 2015 and references therein).As
a result of their short orbital periods, hot Jupiters have high
atmospheric temperatures and emit relatively strongly at
infrared wavelengths, allowing for direct measurement of their
atmospheric brightness as a function of orbital phase. These
phase curves can then be converted into a longitudinal
temperature prole (Cowan & Agol 2008). By comparing the
measured phase curves to theoretical phase curves generated by
atmospheric models, we can constrain fundamental properties
of the atmosphere, such as the ef ciency of heat transport from
the day side to the night side, radiative timescales, wind
speeds, and compositional gradients between the day and night
sides.
To date, well-characterized phase curve observations have
been published for 11 planets: υ And b (Harrington et al.
2006; Crosseld et al. 2010), HD 189733b (Knutson
et al. 2007, 2009a, 2012), HD 149026b (Knutson
The Astrophysical Journal, 811:122 (15pp), 2015 October 1 doi:10.1088/0004-637X/811/2/122
15
1

et al. 2009b), HD 80606b (Laughlin et al. 2009), HAT-P-7b
(Borucki et al. 2009; Welsh et al. 2010), Kepler-7b (Demory
et al. 2013), CoRoT-1b (Snellen et al. 2009), WASP-12b
(Cowan et al. 2012), WASP-18b (Maxted et al. 2013), HAT-P-
2b (Lewis et al. 2013), and HD 209458b (Crosseld
et al. 2012; Zellem et al. 2014). The majority of these
observations were carried out using the Spitzer Space
Telescope while the rest were obtained at optical wavelengths
by the CoRoT and Kepler missions. Recently, the rst
spectroscopic phase curve was obtained for the hot Jupiter
WASP-43b using the Hubble Space Telescope between 1.2 and
1.6 μm (Stevenson et al. 2014).
In this paper, we present full-orbit phase curves of
the hot Jupiter WASP-14b in the 3.6 and 4.5 μm bands
obtained with the Spitzer Space Telescope. Photometric
and radial velocity observations of WASP-14b indicate
a mass of M
p
= 7.3 ± 0.5 M
Jup
R
p
= 1.28 ± 0.08 R
Jup
, corresponding to a density of
ρ = 4.6 g cm
2
(Joshi et al. 2009). The planet lies on an
eccentric orbit (e = 0.0822 ± 0.003; Knutson et al. 2014)
around a young host star (age 0.51.0 Gyr, spectral
type F5, M
*
= 1.21 ± 0.13 M
e
, R
*
= 1.31 ± 0.07 R
e
,
T
*
= 6462 ± 75 K, and
glog =
4
.29 0.04;
, Joshi et al. 2009;
Torres et al. 2012), with a period of 2.24 days and an orbital
semimajor axis of a = 0.036 ± 0.01 AU.
The equilibrium temperature of WASP-14b is relatively high
(T
eq
= 1866 K, assuming zero albedo and reemission from the
entire surface; Joshi et al. 2009), suggesting that the thermal
evolution of WASP-14b may be signicantly affected by
Ohmic dissipation in a partially ionized atmosphere (e.g., Perna
et al. 2010; Batygin et al. 2011, 2013; but for debate see Rogers
& Komacek 2014; Rogers & Showman 2014). Blecic et al.
(2013) analyzed secondary eclipse observations in the 3.6, 4.5,
and 8.0 μm Spitzer bands and found no evidence for a thermal
inversion in the dayside atmosphere, while concluding that the
data are consistent with relatively poor energy redistribution
from the dayside to the nightside. Cowan & Agol (2011) nd
that, based on the dayside uxes, the most highly irradiated
planets have systematically less efcient day/night heat
circulation. Perez-Becker & Showman (2013) reached a similar
conclusion from a comparison of the fractional daynight ux
differences of planets for which phase curves have been
obtained. This overall trend has been explained by hydro-
dynamical models (Perna et al. 2012; Perez-Becker &
Showman 2013).
Phase curve observations at more than one wavelength
provide complementary information about the properties of the
planets atmosphere, as different wavelengths probe different
pressure levels within the atmosphere. Multiband measure-
ments of the planets brightness can also be transformed into
low-resolution dayside and nightside emission spectra, which
can reveal differences in atmospheric composition. Only ve
systems have phase curve observations at more than one
wavelength: HD 189733 (Knutson et al. 2012), WASP-12
(Cowan et al. 2012), WASP-18 (Maxted et al. 2013), HAT-P-2
(Lewis et al.
2013), and WASP-43 (Stevenson et al. 2014).
While single-wavelength phase curves can be reasonably well-
matched by standard atmospheric circulation models, none of
the multi-wavelength phase curves are satisfactorily reproduced
by these same models. This suggests that our understanding of
the physical and chemical processes that drive atmospheric
circulation is still incomplete.
The paper is organized as follows: the observations and data
reduction methodology are described in Section 2. In Section 3,
we discuss the phase curve model used in our analysis and
present the best-t parameters. We then use our results to
obtain updated orbital parameters and discuss the implications
of our phase curve ts for the planets atmospheric dynamics in
Section 4.
2. SPITZER OBSERVATIONS AND PHOTOMETRY
We observed two full orbits of WASP-14b in the 3.6 and
4.5 μm channels of the Infrared Array Camera (IRAC; Fazio
et al. 2004) on the Spitzer Space Telescope. The observation
periods were UT 2012 April 1517 and UT 2012 April 2426
for the 3.6 and 4.5 μm bandpasses, respectively. Both
observations lasted approximately 64 hr and were carried out
in subarray mode, which generated 32 × 32 pixel (39 × 39)
images with 2.0 s integration times, resulting in a total of
113,408 images in each bandpass. Due to long-term drift of the
telescope pointing, the telescope was repositioned approxi-
mately every 12 hr in order to re-center the target, leading to
four breaks in each phase curve observation with a combined
duration of about 16 minutes. The telescope repositioning
produced offsets of up to 0.2 pixels in the stars position on the
array after each break (Figures 1 and 2).
We extract photometry using methods described in previous
analyses of post-cryogenic Spitzer data (e.g., Todorov
et al. 2013;Wongetal.2014). The raw data les are rst
dark-subtracted, at-elded, linearized, and
ux-calibrated using
version S19.1.0 of the IRAC pipeline. The exported data
comprise a set of 1772 FITS les, each containing 64 images
and a UTC-based Barycentric Julian Date (BJD
UTC
) time stamp
designating the start of the rst image. For each image, we
calculate the BJD
UTC
at mid-exposure by assuming uniform
spacing and using the start and end times of each 64-image series
as dened by the AINTBEG and ATIMEEND header keywords.
When estimating the sky background in each image, we
avoid contamination from the wings of the stars point-spread
function (PSF) by excluding pixels within a radius of 15 pixels
from the center of the image, as well as the 13th16th rows and
Figure 1. Measured stellar x centroids (top panel), y centroids (upper middle
panel), and noise pixel values (lower middle panel) as a function of orbital
phase relative to transit for the 3.6 μm phase curve observation. The bottom
panel shows the raw photometric series with hot pixels excised. The data are
binned in 2-minute intervals.
2
The Astrophysical Journal, 811:122 (15pp), 2015 October 1 Wong et al.

the 14th and 15th columns, where the stellar PSF extends close
to the edge of the array. In addition, we exclude the top (32nd)
row of pixels, which have values that are consistently lower
than those from other pixels in the array. We take the remaining
set of pixels and iteratively trim values that lie more than 3σ
from the median. We then calculate the average sky back-
ground across the image by tting a Gaussian function to the
histogram of the remaining pixel values. After subtracting the
sky background, any remaining transient hot pixels in each
set of 64 images varying by more than 3σ from the median
pixel value are replaced by the median pixel value. In both
bandpasses, the average percentage of replaced pixels is less
than 0.35%.
To determine the position of the star on the array in each
image, we calculate the ux-weighted centroid for a circular
0
pixels centered on the estimated position of
the star (see, for example, Knutson et al. 2008). We then
estimate the width of the stars PSF by computing the noise
pixel parameter (Mighell 2005), which is dened in Section
2.2.2 of the Spitzer/IRAC instrument handbook as
I
I
,1
i
i
i
i
2
2
()
()
å
å
b =
where I
i
is the intensity detected in the ith pixel. We dene the
parameter r
1
to be the radius of the circular aperture used to
calculate
.b
We calculate the ux of the stellar target in each image using
circular aperture photometry. We generate two sets of
apertures: the rst set uses a xed aperture with radii ranging
from 1.5 to 3.0 pixels in 0.1-pixel steps and from 3.0 to 5.0
pixels in 0.5-pixel steps. The second set utilizes a time-varying
radius that is related to the square-root of the noise pixel
parameter
b
by either a constant scaling factor or a constant
shift (see Lewis et al. 2013, for a full discussion of the noise-
pixel-based aperture). The optimal choice of aperture photo-
metry is determined by selecting data from 8000 images
spanning the planetary transit with a total duration of 4.4 hr and
calculating the photometric series for each choice of aperture.
We then t each photometric series with our transit light curve
model (Section 3.1), compute the rms scatter in the resultant
residuals binned in ve-minute intervals, and choose the values
of r
0
and r
1
as well as aperture type that give the minimum
scatter. In these ts, we x the planets orbital parameters to the
most recent values in the literature: P = 2.2437661 days,
i = 84
32, a/R
*
= 5.93, e = 0.0822, and ω = 251 67 (Joshi
et al. 2009; Knutson et al. 2014). For the 3.6 μm data set, we
nd that a xed aperture with a radius of 1.8 pixels and
r
0
= 3.0 produce the minimum scatter. When using a xed
aperture, the noise pixel parameter is not needed, so r
1
is
undened. In the 4.5 μm bandpass, we prefer a xed aperture
with a radius of 2.9 pixels and r
0
= 3.5.
Prior to tting the selected photometric series with our full
light curve model, we use a moving median lter to iteratively
remove points with measured uxes, x positions, y positions, or
b
values that vary by more than 3σ from the corresponding
median values in the adjacent 64 frames in the time series.
Choosing a larger or smaller interval for computing the median
values does not signicantly affect the number of excised
points. The percentages of excised points are 1.8% and 1.6% in
the 3.6 and 4.5 μm bandpasses, respectively.
3. DATA ANALYSIS
3.1. Transit and Secondary Eclipse Model
Each full-orbit observation contains one transit and two
secondary eclipses. We model these events using the formalism
of Mandel & Agol (2002). The transit light curve includes four
free parameters: the scaled orbital semimajor axis a/R
*
, the
inclination i, the center of transit time t
T
, and the planetstar
p
/R
*
, which is the square root of the relative
transit depth. Each secondary eclipse event is dened by a
center of eclipse time t
E
and a relative eclipse depth d,
measured with respect to the value of the phase curve at mid-
*
and i, thus yielding four additional
free parameters: t
E1
, t
E2
, d
1
, and d
2
. We ensure continuity
between the phase curve and secondary eclipse light curves by
scaling the amplitudes of eclipse ingress and egress (when the
planet is partially occulted by the star) appropriately to match
the out-of-eclipse phase curve values at the start and end of the
eclipse. The host star WASP-14 has an effective temperature
T
*
= 6462 ± 75 K, a specic gravity
glog 4.29 0.04,=
and a metallicity of [Fe/H] = 0.13 ± 0.08 (Torres
et al. 2012). We model the limb-darkening in each bandpass
using a four-parameter nonlinear limb-darkening law with
parameter values calculated as described in Sing (2010) for a
6500 K star with
glog 4.50=
and [Fe/H] = 0.10:
c
c 0.0192, 0.7960, 0.8558, 0.2983
14
[]-=- -
at 3.6 μm and
c
c 0.0225, 0.3828, 0.2748, 0.0522
14
[
]
-= - -
at 4.5 μm.
3.2. Phase Curve Model
WASP-14b has a relatively low eccentricity of 0.08, and
therefore the variation in the planets apparent brightness
throughout an orbit can be modeled to rst order as a simple
sinusoidal function of the true anomaly f (Lewis et al. 2013),
analogous in form to a simple sine or cosine of the orbital phase
angle that is used in the case of a circular orbit (Cowan &
Figure 2. Measured star centroids, noise pixel values, and raw photometric
series for the 4.5 μm phase curve observations; see Figure 1 for a complete
description.
16
Tables of limb-darkening parameter values, calculated in the
Spitzer bandpasses for various stellar temperatures, specic gravities, and
metallicities, can be found on David Sings website: www.astro.ex.ac.uk/
people/sing
3
The Astrophysical Journal, 811:122 (15pp), 2015 October 1 Wong et al.

Agol 2008):
Ft F c ft ccos , 2
01 2
()
() () ( )=+ -
where F
0
is the stars ux (assumed to be constant and
normalized to one), c
1
is the amplitude of the phase variations,
and c
2
represents the lag between the peak of the planets
temperature and the time of maximum incident stellar ux due
to the nite atmospheric radiative timescale of the planet. Here,
c
1
and c
2
are free parameters that are computed in our ts. We
also experimented with other functional forms of the phase
curve that included higher harmonics, but all of them resulted
in higher values of the Bayesian Information Criterion ( BIC)
3.3. Correction for Intrapixel Sensitivity Variations
Photometric data obtained using Spitzer/IRAC in the 3.6
and 4.5 μm bandpasses exhibit a well-studied instrumental
effect due to intrapixel sensitivity variations (Charbonneau
et al. 2005). Small changes in the telescope pointing during
observation cause variations in the measured ux from the
target, resulting in a characteristic sawtooth pattern in the raw
extracted photometric series. In our analysis, we decorrelate
this instrumental systematic in two ways.
Our rst approach to removing the intrapixel sensitivity
effect is called pixel mapping (Ballard et al. 2010; Lewis
et al. 2013). In an image j, the location of the target on the array
is given by the measured centroid position (x
j
, y
j
), and the
sensitivity of the pixel at that location is determined by
comparing other images with measured centroid positions near
(x
j
, y
j
). The effective pixel sensitivity at a given position is
calculated as follows:
FFe e
e .3
jj
i
m
xx
yy
meas,
0
2
2
2
ij
xj
ij
yj
ij
j
2
,
2
2
,
2
2
,
2
()
()
()
()
å
´
s
s
bb s
=
--
--
--
b

Here, F
meas,j
is the ux measured in the jth image and F
j
is the
intrinsic ux; x
j
, y
j
, and
b
are the measured x position, y
position, and noise pixel parameter values. The quantities σ
x,j
,
σ
y,j
, and
j,
s
b
are the standard deviations of x, y, and
b
over
the full range in i. For each image j, the summation in
Equation (3 ) runs over the nearest m = 50 neighbors of the
stellar target, where we dene distance as
dxx yy .4
ij
ij
ij
ij
,
2
22
2
()
()() ()bb=- +- + -

This method in effect adaptively smoothes the raw pixel map,
allowing for a ner spatial scale in regions where the density of
points is high while using a coarser spatial scale in regions with
sparser sampling. We chose this number of neighbors to be
large enough to adequately map the pixel response while
maintaining a reasonably low computational overhead (Lewis
et al. 2013). Several previous studies of Spitzer phase curves
(e.g., Knutson et al. 2012; Zellem et al. 2014) do not include
the noise pixel parameter term in Equation (3);wend that for
our WASP-14b data, including the noise pixel parameter term
produces 5%10% smaller residual scatter from our best-t
light curve solution in both bandpasses.
The second approach is a recently proposed technique
known as pixel-level decorrelation (PLD; see Deming
et al. 2015, for a complete description). Unlike most other
treatments of the intrapixel sensitivity effect, PLD does not
attempt to relate the variations in the calculated position of the
target on the pixel to the apparent intensity uctuations. Rather,
it utilizes the actual measured intensities of the individual
pixels spanning the stellar PSF to provide an expression of the
total measured ux. We consider pixels lying in a 3 × 3 box
centered on the star, which have pixel intensities P
k
(t), k = 1,
...9. We divide each 3 × 3 pixel box by the summed ux over
all nine pixels in order to remove (at least to zeroth order) any
astrophysical ux variations, giving the following relation:
Pt
Pt
Pt
.5
k
k
k
k
1
9
ˆ
()
()
()
()
å
=
=
The intrapixel sensitivity effect is modeled as a linear
combination of the arrays
P
,
k
ˆ
and thus the total measured
intensity S is given by
St bP t bt Ft h,6
k
kk t
1
9
()
ˆ
() () ( )
å
=+++
=
where F(t) is the astrophysical model, comprising the phase
curve, transit, and eclipses. The parameters b
k
are the linear
coefcients that are determined through least-squares tting,
and h is a free normalization parameter that corrects for the
overall numerical offset introduced by the linear sum of pixel
intensity arrays. Following Deming et al. ( 2015), we also
include a linear ramp in time with a slope parameter b
t
.
In Deming et al. (2015), PLD was applied to tting
Spitzer secondary eclipses of several exoplanets, including
WASP-14b. PLD was found to be generally more effective in
removing time-correlated (i.e., red) noise, resulting in lower
residual scatter from the best-t solution when compared with
other decorrelation techniques. The best results were obtained
when the photometric time series were binned, pixel by pixel,
prior to PLD tting, since binning improves the precision of the
measured intensities in pixels at the edge of the stellar PSF and
can be adjusted to reduce the noise on the timescale of interest.
When tting our full-orbit WASP-14b photometric series using
the PLD technique, we experimented with tting either
unbinned or binned data, with bin sizes equal to powers of
two up to 256 (8.5 minutes). We found that larger bin sizes,
comparable to or exceeding the occultation ingress/egress
timescale of roughly 20 minutes, cause excessive loss of
temporal resolution and yielded unsatisfactory secondary
eclipse and transit light curve ts. After tting the model light
curve to the data using PLD, we subtract the best-t solution
from the raw unbinned data to produce the residual series,
which we use to evaluate the relative amount of time-correlated
noise remaining in the data.
In Figure 3, we compare the noise properties of each version
of the 3.6 μm residual time series to the ideal
n1
scaling we
would expect for the case of independent ( i.e., white noise)
Gaussian measurement errors, where n is the bin size. The
white noise trend is normalized at bin size n = 1 to the photon
noise level corresponding to the median photon count over the
observation data set (with sky background included). Compar-
ing the various PLD ts, we nd that at small bin sizes,
unbinned PLD gives the lowest residual scatter, while at larger
4
The Astrophysical Journal, 811:122 (15pp), 2015 October 1 Wong et al.

bin sizes approaching the duration of eclipse ingress or egress,
PLD with larger bins results in lower residual scatter. The same
trend is seen when tting the 4.5 μm data. Deming et al. (2015)
found that the optimal PLD performance is achieved when the
range of star positions is lower than 0.2 pixels. The range of
pixel motion in our full-orbit phase curve observations
modestly exceeds this limit, and we indeed nd that the
residual scatter from the pixel mapping ts is smaller than the
scatter from any of the PLD ts. We conclude that the pixel
mapping technique produces the lowest residual scatter for our
full-orbit observation data sets, and we therefore use this
technique in the nal version of our analysis.
In addition to having higher residual scatter than the pixel
mapping solutions, the PLD ts yield eclipse depth and phase
curve parameter estimates that often differ strongly from the
corresponding values derived from ts with pixel mapping;
these discrepancies sometimes exceed the 3σ level. The best-t
values derived from the PLD ts also display a higher level of
variation across different choices of binning, photometric
aperture, and exponential ramp type (see Section 3.4) than in
the case of pixel mapping ts. This points toward an inherent
instability in the PLD method when tting full-orbit phase
curves that may be related to the larger range in star motions
characteristic of such data sets.
3.4. Exponential Ramp Correction
Previous studies using Spitzer/IRAC have noted a short-
duration ramp at the beginning of each observation, and again
after downlinks (e.g., Knutson et al. 2012; Lewis et al. 2013).
The ramp has a characteristic asymptotic shape that typically
decays to a constant value on timescales of an hour or less in
the 3.6 and 4.5 μm bandpasses. We rst experimented with
removing the rst 30 or 60 minutes of data from each phase
curve observation, selecting the removal interval that mini-
mizes the residual rms from the best-t solution binned in ve-
minute intervals. We nd that in both bandpasses, we obtain
the best results when we do not trim any data from the start of
the observations. When examining the residual time series, we
noticed a small ramp visible at the start of the 3.6 μm
observations. We therefore considered whether or not our ts
might be further improved by the addition of an exponential
function.
We experimented with including an exponential ramp in our
phase curve model, using the formulation given in Agol et al.
(2010):
Faeae1,7
ta ta
13
24
()=
--
where t is the time since the beginning of the observation, and
a
1
a
4
are correction coefcients. To determine whether this
function is necessary and if so, how many exponential terms to
include in the ramp model, we use the BIC, dened as
kNBIC ln , 8
2
()c=+
where k is the number of free parameters in the t, and N is the
number of data points. By minimizing the BIC, we select the
type of ramp model that yields the smallest residuals without
over-tting the data. For the 3.6 μm data, we nd that using a
single exponential ramp gives a marginally lower BIC
compared to the no-ramp case, while for the 4.5 μm data, no
ramp is needed at all. The residuals from the best-t full phase
curve solution, shown in Figures 4 and 5, do not appear to
display any uncorrected ramp-like behaviors.
Figure 3. Plot comparing the binned residual rms resulting from the t of the
3.6 μm photometric series to the light curve model using various types of
instrumental noise decorrelation: pixel mapping (solid black line) and pixel
level decorrelation with no binning (solid red line), 16-point bins (solid green
line), and 256-point bins (solid blue line). For comparison, the
n
1
dependence of white noise on bin size is shown by the dashed black line;the
trend is normalized at bin size n = 1 to the photon noise level corresponding to
the median photon count over the 3.6 μm photometric series, with sky
background included.
Figure 4. Top panel: nal 3.6 μm photometric series with instrumental
variations removed, binned in ve-minute intervals (black dots). The best-t
total phase, transit, and eclipse light curve is overplotted in red. Middle panel:
the same data as the upper panel, but with an expanded y axis for a clearer view
of the phase curve. Bottom panel: the residuals from the best-t solution.
5
The Astrophysical Journal, 811:122 (15pp), 2015 October 1 Wong et al.

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