# TESS Revisits WASP-12: Updated Orbital Decay Rate and Constraints on
Atmospheric Variability

Ian Wong *  — NASA Postdoctoral Program Fellow  NASA Goddard Space Flight
Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA Ian Wong<EMAIL_ADDRESS>Avi Shporer *  — Department of Physics and Kavli Institute for Astrophysics
and Space Research, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  Shreyas Vissapragada *  — Division of Geological and Planetary
Sciences, California Institute of Technology, 1200 East California Blvd,
Pasadena, CA 91125, USA  Michael Greklek-McKeon *  — Division of Geological
and Planetary Sciences, California Institute of Technology, 1200 East
California Blvd, Pasadena, CA 91125, USA  Heather A. Knutson *  — Division of
Geological and Planetary Sciences, California Institute of Technology, 1200
East California Blvd, Pasadena, CA 91125, USA  Joshua N. Winn *  — Department
of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
Björn Benneke *  — Department of Physics and Institute for Research on
Exoplanets, Université de Montréal, Montréal, QC, Canada

###### Abstract

After observing WASP-12 in the second year of the primary mission, the
Transiting Exoplanet Survey Satellite (TESS) revisited the system in late 2021
during its extended mission. In this paper, we incorporate the new TESS
photometry into a reanalysis of the transits, secondary eclipses, and phase
curve. We also present a new $K_{s}$-band occultation observation of WASP-12b
obtained with the Palomar/WIRC instrument. The latest TESS photometry span
three consecutive months, quadrupling the total length of the TESS WASP-12
light curve and extending the overall time baseline by almost two years. Based
on the full set of available transit and occultation timings, we find that the
orbital period is shrinking at a rate of $-29.81\pm 0.94$ ms yr-1. The
additional data also increase the measurement precision of the transit depth,
orbital parameters, and phase-curve amplitudes. We obtain a secondary eclipse
depth of $466\pm 35$ ppm, a $2\sigma$ upper limit on the nightside brightness
of 70 ppm, and a marginal $6\overset{\circ}{.}2\pm 2\overset{\circ}{.}8$
eastward offset between the dayside hotspot and the substellar point. The
voluminous TESS dataset allows us to assess the level of atmospheric
variability on timescales of days, months, and years. We do not detect any
statistically significant modulations in the secondary eclipse depth or
day–night brightness contrast. Likewise, our measured $K_{s}$-band occultation
depth of $2810\pm 390$ ppm is consistent with most $\sim$2.2 $\mu$m
observations in the literature.

Hot Jupiters (753); Exoplanet astronomy (486); Transit photometry (1709)

††facilities: TESS, Palomar/WIRC††software:  batman (Kreidberg, 2015), emcee
(Foreman-Mackey et al., 2013), exoplanet (Foreman-Mackey et al., 2021), ExoTEP
(Benneke et al., 2019; Wong et al., 2020a).

##  1 Introduction

Since its discovery in 2009 (Hebb et al., 2009), WASP-12b has become one of
the most intensively studied exoplanets. This archetypal ultrahot Jupiter,
with a mass of 1.5 $M_{\rm Jup}$ and an inflated radius of 1.9 $R_{\rm Jup}$
(Collins et al., 2017), has a 1.09 day orbit around a 6150 K F-type star
(e.g., Stassun et al., 2019). The intense stellar irradiation and high dayside
equilibrium temperature of $\approx$2600 K have made WASP-12b a prime target
for atmospheric study. Extensive spectroscopic and photometric transit and
secondary eclipse measurements have been obtained at infrared wavelengths
(Campo et al., 2011; Croll et al., 2011; Cowan et al., 2012; Crossfield et
al., 2012; Zhao et al., 2012; Sing et al., 2013; Stevenson et al., 2014a, b;
Croll et al., 2015; Kreidberg et al., 2015; Bell et al., 2019; Arcangeli et
al., 2021) and visible wavelengths (López-Morales et al., 2010; Copperwheat et
al., 2013; Föhring et al., 2013; Sing et al., 2013; Bell et al., 2017; Hooton
et al., 2019; von Essen et al., 2019). Some highlights from these observations
include the detection of water vapor in the upper atmosphere along the
day–night terminator (Kreidberg et al., 2015) and the very low dayside
reflectivity (Bell et al., 2017). In addition, full-orbit infrared phase
curves from the Spitzer Space Telescope have provided a glimpse of the
planet’s global temperature distribution (Cowan et al., 2012; Bell et al.,
2019).

WASP-12b is notable in several respects. First, the planet is undergoing
atmospheric mass loss. Repeated observations of the planet’s transit in the
near-ultraviolet and in H-alpha have revealed a strongly asymmetric signal
with an early ingress, suggestive of a distended circumplanetary disk of
stripped material (Fossati et al., 2010; Haswell et al., 2012; Nichols et al.,
2015; Jensen et al., 2018). This scenario is supported by the recent analysis
of the Spitzer 4.5 $\mu$m phase curve, which displays a significant modulation
with a period equal to half of the orbital period that could be explained by
CO emission from escaping gas flowing from the planet into the host star (Bell
et al., 2019).

Another peculiarity of WASP-12b is the apparent variation in its dayside
atmospheric brightness. Mutually discrepant secondary eclipse depths have been
reported at various optical wavelengths (López-Morales et al., 2010; Föhring
et al., 2013; Hooton et al., 2019; von Essen et al., 2019), as well as in the
2 $\mu$m region (Zhao et al., 2012; Crossfield et al., 2012; Croll et al.,
2015). In light of other purported detections of atmospheric variability in
hot Jupiters (e.g., Armstrong et al. 2016; Jackson et al. 2019; but also see
Lally & Vanderburg 2020), the question of whether these differences are real
or are instead caused by instrumental effects and/or observing conditions has
come to the fore. Meanwhile, recent theoretical work on the atmospheric
dynamics of hot Jupiters has shown that hydrodynamic instabilities and other
transient atmospheric processes may lead to orbit-to-orbit evolution in the
global average temperature, the effects of which may be detectable with
current and near-future instruments (e.g., Tan & Komacek, 2019; Komacek &
Showman, 2020; Tan & Showman, 2020).

Lastly, WASP-12b is the only exoplanet for which there is clear evidence for a
gradually shrinking orbital period. Evidence for a change in period was first
reported by Maciejewski et al. (2016). Subsequent transit and occultation
observations solidified the detection and ruled out apsidal precession and the
Rømer effect as the explanation for the changing period (Patra et al., 2017;
Yee et al., 2020). The best explanation appears to be tidal orbital decay,
which has motivated theorists to try to understand the observed rate of decay
in terms of tidal dissipation mechanisms within the star (e.g., Weinberg et
al., 2017; Bailey & Goodman, 2019; Ma & Fuller, 2021)

During the winter of 2019–2020, the Transiting Exoplanet Survey Satellite
(TESS; Ricker et al. 2015) observed WASP-12 as part of its ongoing mission to
search the entire sky for transiting exoplanets. The light curves provided
additional transit timings, which were used to refine the estimate of the
planet’s orbital decay rate (Turner et al., 2021). Meanwhile, the planetary
phase curve was analyzed by Owens et al. (2021) and Wong et al. (2021a),
yielding a robust secondary eclipse measurement and broad constraints on the
day–night temperature contrast. No evidence for significant orbit-to-orbit
variability in the dayside brightness was found in the TESS photometry. TESS
has since revisited the WASP-12 system during its ongoing extended mission,
providing an additional two and a half months of nearly continuous photometric
observations.

In this paper, we present a follow-up study of WASP-12b’s orbital decay, phase
curve, and potential atmospheric variability, incorporating the latest TESS
light curves from 2021 as well as a new ground-based $K_{s}$-band observation
of an occultation. The relevant datasets are described in Section 2. An
overview of our light-curve modeling techniques and results is provided in
Section 3. In Section 4, we use our measurements to update the orbital
ephemeris and derive constraints on the level of atmospheric variability.
Finally, the main findings of our work are summarized in Section 5.

##  2 Observations

###  2.1 TESS Light Curves

Table 1:  List of TESS Light-curve Segments

Segment$\mathrm{a}$$\mathrm{a}$footnotemark:  |  $n_{\mathrm{raw}}$$\mathrm{b}$$\mathrm{b}$footnotemark:  |  $n_{\mathrm{trimmed}}$$\mathrm{b}$$\mathrm{b}$footnotemark:  |  $T_{\mathrm{start}}$$\mathrm{c}$$\mathrm{c}$footnotemark:  |  $T_{\mathrm{end}}$$\mathrm{c}$$\mathrm{c}$footnotemark:  | Order$\mathrm{d}$$\mathrm{d}$footnotemark:  | Comments
---|---|---|---|---|---|---
20-1-1 | 3910 | 3101 | 843.510 | 847.922 | 0 | Removed 1.00 day from start
20-1-2 | 3870 | 3788 | 847.935 | 853.298 | 0 |
20-1-3 | 1119 | 1094 | 853.310 | 854.862 | 1 |
20-2-1 | 3029 | 2640 | 858.200 | 861.944 | 1 | Removed 0.25 day from start
20-2-2 | 3959 | 3874 | 861.955 | 867.444 | 1 |
20-2-3 | 989 | 952 | 867.455 | 868.827 | 1 |
43-1-1 | 2922 | 2870 | 1474.172 | 1478.208 | 0 |
43-1-2 | 2880 | 2833 | 1478.219 | 1482.209 | 1 |
43-1-3 | 2071 | 1907 | 1482.220 | 1484.896 | 0 |
43-2-1 | 4172 | 4098 | 1487.181 | 1492.960 | 3 |
43-2-2 | 3855 | 3481 | 1492.971 | 1497.875 | 1 | Removed 0.25 day from end
44-1-1 | 2832 | 2714 | 1500.281 | 1504.107 | 2 |
44-1-2 | 2790 | 2751 | 1504.118 | 1507.982 | 0 |
44-1-3 | 2689 | 2663 | 1507.993 | 1511.727 | 1 |
44-2-1 | 2356 | 1820 | 1514.420 | 1516.983 | 1 | Removed 0.50 day from start
44-2-2 | 2698 | 2666 | 1516.994 | 1520.733 | 1 |
44-2-3 | 2664 | 2619 | 1520.744 | 1524.443 | 0 |
45-1-1 | 4264 | 3705 | 1526.584 | 1531.796 | 2 | Removed 0.50 day from start
45-1-2 | 4447 | 4390 | 1531.808 | 1537.983 | 0 |
45-2-1 | 3486 | 3286 | 1540.165 | 1544.797 | 4 |
45-2-2 | 4192 | 4126 | 1544.808 | 1550.629 | 1 |

Notes. ddfootnotetext: Order of the polynomial detrending model used in the
full-orbit phase-curve fitting.

Figure 1: Normalized and outlier-removed TESS PDC light curves of WASP-12 from
Sectors 20, 43, 44, and 45. The red dots indicate portions of the photometry
that were trimmed prior to fitting due to short-term flux ramps. The vertical
green lines denote the scheduled momentum dumps, and each data segment is
labeled with the corresponding sector, orbit, and segment numbers. The primary
transits that were included in the transit-timing analysis are labeled
sequentially in blue.

TESS first observed WASP-12 (TIC 86396382) during Sector 20 of the primary
mission, which lasted from UT 2019 Dec 24 to 2020 Jan 21. The WASP-12 data
were obtained with Camera 1 in the 2-minute cadence mode. The data were
downlinked from the spacecraft and passed through the Science Processing
Operations Center (SPOC) pipeline hosted at the NASA Ames Research Center
(Jenkins et al., 2016). The pipeline determined the optimal photometric
aperture and extracted the light curve, which was deposited in the Mikulski
Archive for Space Telescopes (MAST).

Analyses of the Sector 20 light curves of WASP-12 have been published by
several teams. Turner et al. (2021) analyzed the transits and secondary
eclipses and used the measured timings to calculate a refined orbital decay
rate for WASP-12b. The full-orbit phase curve of the WASP-12 system was
analyzed independently by Owens et al. (2021) in a standalone paper and by
Wong et al. (2021a) as part of a systematic phase-curve study of short-period
exoplanet systems observed during the second year of the TESS mission.

TESS revisited the northern ecliptic hemisphere beginning in mid-2021. Due to
the fortuitous positioning of the fields of view, WASP-12 was observed in
three consecutive sectors: Sector 43 (2021 Sep 16 to Oct 12), Sector 44 (2021
Oct 12 to Nov 6), and Sector 45 (2021 Nov 6 to Dec 2). During these three
roughly month-long observations, the target was located on Camera 4, 3, and 1,
respectively. The combined TESS dataset for WASP-12 now spans almost two
years.

In the analysis of the TESS WASP-12 photometry presented in this paper, we
utilized the pre-search data conditioning (PDC) light curves, which were
corrected for common-mode instrumental systematic trends by the SPOC pipeline
using empirically derived cotrending basis vectors (Smith et al., 2012; Stumpe
et al., 2012, 2014). Our experience with TESS light-curve fitting has
demonstrated that the PDC light curves typically exhibit lower levels of time-
correlated noise when compared to the raw simple aperture photometry (SAP)
light curves, while preserving genuine astrophysical variability, including
the orbital phase curve (see, for example, Wong et al. 2020b, d).

The preprocessing steps that we completed prior to fitting were largely
identical to the methodology described in our previous TESS light-curve papers
(e.g., Wong et al., 2020c, 2021a, 2021b). First, we removed all flagged points
and trimmed $3\sigma$ outliers from the transit-masked photometry using a
16-point-wide moving median filter. Next, we separated the light curves into
data segments that are separated by the scheduled momentum dumps and data
downlink interruptions. Lastly, we searched for significant flux ramps that
often occur at the beginning or end of data segments and removed them from the
time series. The list of TESS light-curve segments is provided in Table 1,
including the number of datapoints, time ranges, and any trimmed portions.
Figure 1 shows the PDC light curves from the four sectors of TESS
observations, with the momentum dumps and trimmed ramps marked.

###  2.2 Palomar/WIRC Secondary Eclipse

We observed a full secondary eclipse of WASP-12b on UT 2021 Feb 18 with the
Wide-field Infrared Camera (WIRC; Wilson et al. 2003) on the Hale
$200^{\prime\prime}$ Telescope at Palomar Observatory, California, USA. Images
were taken in the $K_{s}$ band ($\lambda_{\rm eff}=2.13$ $\mu$m) using a
special beam-shaping diffuser for improved duty cycle and guiding stability;
see Stefansson et al. (2017) and Vissapragada et al. (2020) for details
concerning the observing setup. Following an initial two-point dither near the
target to construct a background frame and mitigate short-cadence detector
systematics, we obtained 178 exposures with a total integration time of 42 s
(14 coadds $\times$ 3 s).

After dark-correcting, flat-fielding, and background-subtracting each image
using the methods described by Vissapragada et al. (2020), we extracted the
photometry of WASP-12 and 10 nearby companion stars using circular apertures
ranging in diameter from 10 to 25 pixels (in 1 pixel steps). For our
subsequent analysis, we chose a 17-pixel aperture, which was found to minimize
the per-point scatter in the resultant WASP-12 light curve.

##  3 Data Analysis

For our analysis of the TESS photometry, we used the Python-based ExoTEP
pipeline (e.g., Benneke et al., 2019; Wong et al., 2020a). In this section, we
provide a brief overview of our fitting methodology for the transits,
secondary eclipses, and full-orbit phase curves. We refer the reader to
previous studies, such as Shporer et al. (2019) and Wong et al. (2020c), for
more detailed descriptions of the light-curve modeling and error analysis.

###  3.1 Transits

We considered all 74 full transit events in the trimmed TESS light curves that
were not interrupted by a momentum dump: 19 in Sector 20, 18 in Sector 43, 18
in Sector 44, and 19 in Sector 45. These transits are labeled sequentially
within each sector in Figure 1. To construct each individual transit time
series, we selected all datapoints with timestamps within $0.1P$ of the
predicted mid-transit time, where $P$ is the orbital period. There were about
160 selected points on average. This procedure ensured sufficient out-of-
transit baseline, while simultaneously minimizing the effect of the orbital
phase-curve variation on the fitted transit depth.

Within the ExoTEP pipeline, transit light-curve modeling is handled by batman
(Kreidberg, 2015), which is based on the analytical formulation of Mandel &
Agol (2002). For every transit, the unconstrained free parameters were the
mid-transit time $T_{c}$, planet–star radius ratio $R_{p}/R_{*}$, impact
parameter $b$, and scaled semimajor axis $a/R_{*}$. The orbital period $P$ was
held fixed at the extrapolated value based on the quadratic transit ephemeris
reported by Turner et al. (2021). We assumed a quadratic limb-darkening law
with coefficients constrained by Gaussian priors: the median values
($u_{1}=0.24$, $u_{2}=0.31$) were calculated by interpolating the coefficients
tabulated by Claret (2017) based on the measured properties of the host star
WASP-12 (from Version 8 of the TESS Input Catalog; Stassun et al. 2019), and
the Gaussian widths were set to 0.05 to account for the uncertainties in the
stellar parameters.

Parameter estimation was carried out using the affine-invariant Markov chain
Monte Carlo (MCMC) ensemble sampler emcee (Foreman-Mackey et al., 2013). In
addition to the free parameters listed above, we included a per-point
uncertainty parameter $\sigma$ that was allowed to vary freely to ensure that
the resultant reduced $\chi^{2}$ was unity. For normalization and systematics
detrending, we multiplied the transit model by either a constant
multiplicative constant or a linear function of time. We utilized the Bayesian
information criterion (BIC) to determine the optimal systematics model on a
transit-by-transit basis; the linear trend was preferred in only one case —
the second transit of Sector 43 (S43-2).

The results of our individual transit fits are listed in Table 3 in the
Appendix. The measured transit depths show a very high level of mutual
consistency, with the largest pairwise discrepancy being $2.3\sigma$.
Likewise, the transit-shape parameters $b$ and $a/R_{*}$ show little variation
across the 74 transits, with the spread in values lying within $2.4$ and
$2.5\sigma$, respectively. The median $1\sigma$ timing uncertainty for the
TESS transits is 42 s. Compilation plots of the normalized transit light
curves and best-fit models are provided in Figures 6–9.

###  3.2 Secondary Eclipses

The individual secondary eclipse events are not detected with a sufficiently
high signal-to-noise ratio to permit reliable timing measurements. Instead, we
combined all full and uninterrupted occultation events within a TESS sector
and calculated a single effective mid-occultation time. In an analogous manner
to our transit analysis, we selected points within 10% of an orbit from the
predicted mid-occultation time. After median-normalizing each secondary
eclipse event, we phase-folded the datapoints. For each sector, we held $P$,
$R_{p}/R_{*}$, $b$, and $a/R_{*}$ fixed to the best-fit values determined from
the corresponding individual sector full-orbit phase-curve fit (Section 3.3,
Table 2).

We fit for the mid-occultation time $T_{\rm occ}$ and secondary eclipse depth
$D_{d}$, along with the uniform per-point uncertainty $\sigma$. The epoch of
$T_{\rm occ}$ for each sector was set to the secondary eclipse event closest
to the midpoint of the time series. The best-fit occultation timings for the
four TESS sectors are (in ${\rm BJD}_{\rm TDB}$)

$\displaystyle T_{\rm occ,S20}=2{,}458{,}855.5543\pm 0.0029,$ $\displaystyle
T_{\rm occ,S43}=2{,}459{,}485.3066\pm 0.0021,$ $\displaystyle T_{\rm
occ,S44}=2{,}459{,}511.4976\pm 0.0026,$ $\displaystyle T_{\rm
occ,S45}=2{,}459{,}537.6908\pm 0.0020.$ (1)

The corresponding measured secondary eclipse depths are $534\pm 54$, $497\pm
49$, $450\pm 51$, $498\pm 46$ ppm, respectively. Later in this paper, we will
present improved measurements of the occultation depths derived from the full-
orbit phase-curve fits. The improvement comes from a more robust treatment of
the time-correlated noise and self-consistently accounting for the out-of-
eclipse phase-curve modulations as well as the uncertainties in the orbital
parameters.

Figure 2: The normalized and detrended $K_{s}$-band secondary eclipse light
curve observed with Palomar/WIRC on UT 2021 Feb 18 (gray). The black
datapoints show the light curve binned in 10 minute intervals. The red curve
and shaded area denote the best-fit occultation model and $1\sigma$ confidence
region.

We fit the Palomar/WIRC $K_{s}$-band secondary eclipse light curve with the
exoplanet toolkit (Foreman-Mackey et al., 2021). For detrending vectors, we
used the extracted flux arrays from the 10 companion stars, airmass,
background flux, and the relative centroid offset of WASP-12. These vectors
were combined in linear combination with a constant multiplicative offset, and
the coefficients were included in the MCMC fit as free parameters along with
the mid-occultation time and eclipse depth. We applied Gaussian priors to the
remaining orbital parameters based on the measured values from the previously
published Sector 20 phase-curve fit in Wong et al. (2021a).

We obtained a mid-occultation time of

$T_{{\rm occ,WIRC}}=2{,}459{,}263.74611\pm 0.00090\,\,{\rm BJD}_{\rm TDB}$ (2)

and an eclipse depth of $2810\pm 390$ ppm. The precision of this timing
measurement is significantly higher than that of the TESS occultation timings.
The normalized and detrended $K_{s}$-band secondary eclipse light curve is
plotted in Figure 2, along with the best-fit occultation model and $1\sigma$
confidence region.

###  3.3 Full-orbit Phase-curve Fits

Table 2:  Results from Individual Sector Phase-curve Fits and Combined
Analysis

Parameter | Sector 20 | Sector 43 | Sector 44 | Sector 45 | All
---|---|---|---|---|---
Fitted Parameters |  |  |  |
$R_{p}/R_{*}$ | $0.11666\pm 0.00081$ | $0.1152\pm 0.0010$ | $0.11581\pm 0.00079$ | $0.11563\pm 0.00070$ | $0.11600\pm 0.00045$
$T_{c}$aaThe numbers indicate the TESS sector, spacecraft orbit (two per sector), and segment number, respectively. | $855.01067\pm 0.00013$ | $1484.75805\pm 0.00010$ | $1510.95210\pm 0.00012$ | $1537.14639\pm 0.00014$ | $1196.624116\pm 0.000072$
$P$ (days) | $1.091408\pm 0.000019$ | $1.091433\pm 0.000015$ | $1.091407\pm 0.000016$ | $1.091401\pm 0.000019$ | $1.09141685\pm 0.00000025$
$b$ | $0.311_{-0.094}^{+0.069}$ | $0.327_{-0.091}^{+0.067}$ | $0.357_{-0.077}^{+0.050}$ | $0.293_{-0.093}^{+0.071}$ | $0.344\pm 0.036$
$a/R_{*}$ | $3.086\pm 0.070$ | $3.080_{-0.070}^{+0.071}$ | $3.048_{-0.056}^{+0.068}$ | $3.107_{-0.073}^{+0.068}$ | $3.061\pm 0.034$
$\bar{f_{p}}$ (ppm) | $197\pm 74$ | $217\pm 61$ | $197\pm 75$ | $201\pm 73$ | $205\pm 31$
$A_{\mathrm{atm}}$ (ppm) | $265\pm 36$ | $305\pm 27$ | $223\pm 26$ | $249\pm 37$ | $262\pm 15$
$\delta$ (deg) | $10.3\pm 6.3$ | $5.8\pm 4.0$ | $3.5\pm 5.3$ | $6.3\pm 6.3$ | $6.2\pm 2.8$
$A_{\mathrm{ellip}}$ (ppm) | $77\pm 38$ | $102\pm 30$ | $52\pm 32$ | $30\pm 38$ | $65\pm 14$
$u_{1}$bbNumber of datapoints in each data segment before and after removing flagged points, outliers, and flux ramps. | $0.22\pm 0.03$ | $0.23\pm 0.03$ | $0.22\pm 0.03$ | $0.23\pm 0.03$ | $0.22\pm 0.02$
$u_{2}$bbThe quadratic limb-darkening coefficients were constrained by Gaussian priors based on the tabulated values in Claret (2017): $u_{1}=0.24\pm 0.05$, $u_{2}=0.31\pm 0.05$. | $0.30\pm 0.05$ | $0.32\pm 0.05$ | $0.30\pm 0.05$ | $0.28\pm 0.05$ | $0.30\pm 0.04$
Derived Parameters |  |  |  |
$D_{d}$ (ppm)ccStart and end times of each data segment ($\mathrm{BJD}_{\mathrm{TDB}}-2{,}458{,}000$). | $454\pm 80$ | $519\pm 67$ | $418\pm 83$ | $450\pm 79$ | $466\pm 35$
$D_{n}$ (ppm)cc$D_{d}$ and $D_{n}$ denote the dayside and nightside fluxes, respectively. The dayside flux is equivalent to the secondary eclipse depth. | $-64\pm 83$ | $-89\pm 69$ | $-25\pm 79$ | $-46\pm 81$ | $-55\pm 35$
$i$ (deg) | $84.2^{+1.8}_{-1.4}$ | $83.9^{+1.8}_{-1.4}$ | $83.3^{+1.6}_{-1.1}$ | $84.5^{+1.8}_{-1.5}$ | $83.54\pm 0.74$

Notes. aafootnotetext: ${\rm BJD}_{\rm TDB}-2{,}458{,}000$.

The phase curve consists of the transits, secondary eclipses, and the
synchronous brightness modulation of the entire system. The flux variability
contains contributions from the changing viewing geometry of the planet’s
longitudinal atmospheric brightness modulation as it orbits around the host
star, as well as the ellipsoidal distortion of both orbiting companions due to
the mutual gravitational interaction and the Doppler boosting of the star’s
spectrum (see the review by Shporer 2017). In tandem with previous TESS phase-
curve studies, we used the following simple sinusoidal light-curve model
(e.g., Shporer et al., 2019; Wong et al., 2020b, c, d, 2021a, 2021b):

$\displaystyle F(t)$
$\displaystyle=\frac{F_{*}(t)\lambda_{t}(t)+F_{p}(t)\lambda_{e}(t)}{1+\bar{f_{p}}},$
(3) $\displaystyle F_{p}(t)$ $\displaystyle=\bar{f_{p}}-A_{\rm
atm}\cos(\phi+\delta),$ (4) $\displaystyle F_{*}(t)$ $\displaystyle=1-A_{\rm
ellip}\cos(2\phi)+A_{\rm Dopp}\sin(\phi),$ (5) $\displaystyle\phi$
$\displaystyle\equiv 2\pi(t-T_{c}).$ (6)

Here, $\lambda_{t}$ and $\lambda_{e}$ denote the unit-normalized transit and
secondary eclipse light curves. The flux from the planet, $F_{p}$, is
parameterized by the average flux level $\bar{f_{p}}$, the semiamplitude of
the atmospheric brightness modulation $A_{\rm atm}$, and the phase shift in
the planet’s phase curve $\delta$. The secondary eclipse depth (i.e., dayside
hemispheric brightness) is derived from the aforementioned three parameters
via $D_{d}=\bar{f_{p}}-A_{\rm atm}\cos(\pi+\delta)$. Similarly, the nightside
flux is given by $D_{n}=\bar{f_{p}}-A_{\rm atm}\cos{\delta}$. The stellar flux
$F_{*}$ includes two harmonic amplitudes, $A_{\rm ellip}$ and $A_{\rm Dopp}$,
which correspond to the ellipsoidal distortion and Doppler boosting signals
(e.g., Loeb & Gaudi, 2003; Faigler & Mazeh, 2011, 2015; Shporer, 2017).

The PDC light curves are affected by low-level residual systematics in the
form of long-term temporal trends. To model these systematics, we multiplied
the astrophysical phase-curve model by polynomial functions of time, with a
separate systematics function defined for each data segment. We carried out
preliminary phase-curve fits to each data segment and determined the optimal
polynomial order using the BIC. These selected orders are listed in Table 1
and are predominantly 0 or 1.

Each sector’s light curve was analyzed separately using a two-step fitting
process. All of the orbital parameters ($T_{c}$, $P$, $b$, $a/R_{*}$) and most
transit/eclipse/phase-curve parameters ($R_{p}/R_{*}$, $\bar{f_{p}}$, $A_{\rm
atm}$, $\delta$, $A_{\rm ellip}$) were allowed to vary freely. We accounted
for the light-travel time across the system when modeling the secondary
eclipses by adding $2a/c\approx 23$ s to the calculated time of superior
conjunction. Our model neglects the gradual decrease in the orbital period
over the timescale of a TESS sector, because the expected transit-timing
deviation is only about 6 ms. The limb-darkening coefficients were constrained
by the same Gaussian priors as in our transit fits ($u_{1}=0.24\pm 0.05$,
$u_{2}=0.31\pm 0.05$). Meanwhile, given the small amplitude of the Doppler
boosting signal and its degeneracy with the planet’s phase-curve offset
$\delta$, we followed the technique used by Wong et al. (2020c) and Wong et
al. (2021a) and applied a Gaussian prior to $A_{\rm Dopp}$. The median and
width of this prior was derived from the theoretical formulation described by
Shporer (2017) and the system parameters published by Stassun et al. (2019):
$2.3\pm 0.2$ ppm.

In the first MCMC fit, we allowed the per-point uncertainty $\sigma$ to vary.
The best-fit $\sigma$ values for the four TESS sectors were 1880, 1790, 1820,
and 1780 ppm, respectively. Upon examination of the residuals, we found
discernible time-correlated noise (i.e., red noise). To propagate this red
noise contribution to the best-fit phase-curve parameters and uncertainties,
we computed the scatter of the residuals binned at intervals ranging from 20
minutes to 8 hours and calculated the average fractional deviation $\beta$
from the expected $1/\sqrt{n}$ trend for pure white noise (see Wong et al.
2020c for a full description of this methodology). We then multiplied the
best-fit $\sigma$ from the first fit by $\beta$ to obtain the red-noise-
enhanced per-point uncertainty $\sigma_{r}\equiv\beta\sigma$. For the second
phase-curve fit, we fixed the flux uncertainties to $\sigma_{r}$ and reran the
MCMC analysis to arrive at the final set of results. Across the four TESS
sectors, $\beta$ ranged from 1.01 to 1.23, illustrating the small and variable
contribution of red noise throughout the light curves.

The results of our individual sector phase-curve fits are listed in Table 2.
There is a high level of consistency in the measured astrophysical parameters
across the four sectors. The planet–star radius ratios agree at better than
the $1.1\sigma$ level, and the transit-shape parameters ($b$, $a/R_{*}$) are
mutually consistent to well within $1\sigma$. Meanwhile, the secondary eclipse
depths $D_{d}$ and atmospheric brightness modulation amplitudes $A_{\rm atm}$
lie within $1.0$ and $2.2\sigma$, respectively. Likewise, the planet’s phase-
curve offsets $\delta$ are in excellent agreement, and the nightside fluxes
are all consistent with zero. The star’s ellipsoidal distortion signal
amplitudes show a $1.3\sigma$ spread. All in all, there is no sign of
statistically significant alterations in the shape of the WASP-12 phase curve
throughout the TESS observations. Constraints on atmospheric variability are
discussed in more detail in Section 4.2.

Having established that the transits, secondary eclipses, and synchronous
photometric modulations of the WASP-12 system are comparable across the four
TESS sectors, we proceeded with a joint phase-curve fit to the combined TESS
light curve. Just as in our individual sector phase-curve analyses, we assumed
the orbital period to be constant throughout the time interval of the TESS
observations. Choosing the reference epoch to be the transit event nearest to
the midpoint of the full TESS time series, the accumulated transit-time
deviation due to the shrinking period is expected to be less than 15 s at the
first and last epochs. A deviation of 15 s is shorter than both the 120 s
cadence of the TESS observations and the average precision with which the
individual mid-transit times are measured (42 s). Prior to fitting, we divided
away the best-fit systematics models from the light curve (calculated in the
individual sector phase-curve fits) and fixed the per-point uncertainties to
the respective red-noise-enhanced $\sigma_{r}$ values that we computed
previously.

Figure 3: Top: the combined four-sector TESS phase curve, phase-folded and
binned in 10 minute intervals. The red curve shows the best-fit full-orbit
phase-curve model. Middle: a zoomed-in view of the phase-curve variability.
The solid and dashed blue curves denote the individual components of the phase
curve that correspond to atmospheric brightness modulation and ellipsoidal
distortion, respectively. Bottom: the residuals from the best-fit model.

The results of our combined phase-curve analysis are listed in the rightmost
column of Table 2. The phase-folded light curve and best-fit model are shown
in Figure 3. The fitted values yield a more than twofold improvement in
precision when compared to the previously published Sector 20 phase-curve fit
results in Wong et al. (2021a). The transit-shape parameters were tightly
constrained: $b=0.344\pm 0.036$, $a/R_{*}=3.061\pm 0.034$, and
$i=83\overset{\circ}{.}54\pm 0\overset{\circ}{.}74$. The planet–star radius
ratio of $R_{p}/R_{*}=0.11600\pm 0.00045$ is among the most precise in the
literature. Using the stellar radius $R_{*}=1.75\pm 0.09\,R_{\sun}$ published
in Stassun et al. (2019), we find $R_{p}=1.98\pm 0.10\,R_{\rm Jup}$ and
$a=0.0249\pm 0.0013$ au.

The TESS-band secondary eclipse depth of WASP-12b is $466\pm 35$ ppm, and the
nightside flux is consistent with zero at $1.6\sigma$ ($2\sigma$ upper limit
of 70 ppm). The day–night brightness contrast has a peak-to-peak amplitude of
$524\pm 30$ ppm. There is a small, marginally significant
$6\overset{\circ}{.}2\pm 2\overset{\circ}{.}8$ eastward shift in the location
of maximum dayside brightness. Given the low optical geometric albedo of
WASP-12b (Bell et al., 2017; Wong et al., 2021a), the planetary phase curve
observed in the TESS bandpass is dominated by the thermal emission from the
atmosphere. It follows that the measured phase offset suggests the presence of
superrotating equatorial winds that advect heat toward the colder nightside.

The star’s ellipsoidal distortion signal has a semiamplitude of $65\pm 14$
ppm. Using the theoretical relationship between the ellipsoidal distortion
amplitude and the planet–star mass ratio (e.g., Shporer, 2017), along with the
measured system parameters and coefficients for limb darkening and gravity
darkening interpolated from the tables in Claret (2017), we arrived at an
independent estimate of WASP-12b’s mass: $1.96\pm 0.57\,M_{\rm Jup}$. This
value is consistent with the planet mass presented in the discovery paper
($1.41\pm 0.10\,M_{\rm Jup}$; Hebb et al. 2009) to within $1\sigma$. The close
agreement indicates that the photometric modulation at the first harmonic of
the orbital frequency is adequately modeled by the classical formalism of
stellar tidal distortion.

##  4 Discussion

###  4.1 Orbital Decay

Figure 4: Plot of the transit (top) and occultation (bottom) timing deviations
from the best-fit constant-period orbital ephemeris. The black points are
previously-published data from Yee et al. (2020), while the blue points denote
the newly measured timings from our TESS light-curve analysis (Table 3 and
Section 3.2). The red datapoint is based on the $K_{s}$-band Palomar/WIRC
secondary eclipse observation. The green curves and shaded areas indicate the
best-fit orbital decay ephemeris model and corresponding $1\sigma$ confidence
regions.

Previous transit-timing analyses (Patra et al., 2017; Maciejewski et al.,
2018; Yee et al., 2020; Turner et al., 2021) uncovered strong evidence for a
shrinking orbital period. Having obtained a large number of new transit
timings from TESS Sectors 43–45 and extended the overall time baseline by
almost two years, we carried out an updated ephemeris fit to refine the
orbital decay rate estimate.

We combined the TESS transit timings listed in Table 3 and the five mid-
occultation timings presented in Equations (1) and (2) with the full list of
prior transit and occultation times compiled by Yee et al. (2020).111The
previous transit and secondary eclipse timings were contributed by Hebb et al.
(2009), Campo et al. (2011), Chan et al. (2011), Cowan et al. (2012),
Crossfield et al. (2012), Sada et al. (2012), Copperwheat et al. (2013),
Föhring et al. (2013), Maciejewski et al. (2013), Stevenson et al. (2014b),
Croll et al. (2015), Deming et al. (2015), Kreidberg et al. (2015),
Maciejewski et al. (2016), Collins et al. (2017), Patra et al. (2017),
Maciejewski et al. (2018), Hooton et al. (2019), Öztürk & Erdem (2019), von
Essen et al. (2019), Yee et al. (2020), and Patra et al. (2020). There are a
total of 213 transit timings and 24 occultation timings. The transit- and
occultation-time model for orbital decay is as follows:

$\displaystyle T_{\rm tra}(E)$
$\displaystyle=T_{0}+PE+\frac{P}{2}\frac{dP}{dE}E^{2},$ $\displaystyle T_{\rm
occ}(E)$
$\displaystyle=T_{0}+P\left(E+\frac{1}{2}\right)+\frac{P}{2}\frac{dP}{dE}E^{2}.$
(7)

The variable $dP/dE$ represents the decay rate in units of days per orbit,
$T_{0}$ is the zeroth epoch transit time, and $E$ is the transit epoch.

Before fitting the measured timings with emcee, we subtracted the light-travel
time of 23 s from the mid-occultation times. We designated the transit event
closest to the weighted average of timings as the zeroth epoch in our fit. The
results of our MCMC analysis are listed below:

$\displaystyle T_{0}$ $\displaystyle=2{,}457{,}103.283654\pm 0.000032\,\,{\rm
BJD}_{\rm TDB},$ $\displaystyle P$ $\displaystyle=1.091419370\pm
0.000000020\,\,{\rm days},$ $\displaystyle\frac{dP}{dE}$
$\displaystyle=(-1.031\pm 0.033)\times 10^{-9}\,{\rm days}~{}{\rm
orbit}^{-1},$ $\displaystyle\frac{dP}{dt}$ $\displaystyle=-29.81\pm
0.94\,\,{\rm ms}~{}{\rm yr}^{-1}.$ (8)

The updated orbital decay timescale is $\tau=P/|\dot{P}|=3.16\pm 0.10\,{\rm
Myr}$, which lies in between the published estimates of Yee et al. (2020) and
Turner et al. (2021). Figure 4 shows the timing deviations of the WASP-12b
transits and secondary eclipses relative to a constant-period ephemeris model,
with the best-fit orbital decay model and corresponding $1\sigma$ confidence
regions plotted in blue. The fit quality is excellent, with a reduced
$\chi^{2}$ of 1.05 and 234 degrees of freedom (237 datapoints and 3 free
parameters).

The additional time observations from the TESS extended mission improved the
precision of the measured orbital decay rate by almost 30% when compared to
the estimate by Turner et al. (2021), which incorporated timings from Sector
20 only. The magnitude of our derived decay rate ($29.81\pm 0.94$ ms yr-1) is
$1.5\sigma$ larger than the previously published value ($27.3\pm 1.4$ ms
yr-1).222We note that the transformation from $dP/dE$ to $dP/dt$ in Turner et
al. (2021) is erroneous: the orbital period conversion factor 1.09 days
orbit-1 was multiplied, instead of divided. To assess the influence of the
occultation timings on the measured orbital decay rate, we carried out an
analogous fit without the occultation timings and obtained $30.01\pm 0.96$ ms
yr-1. This indicates that while the occultations are crucial for
distinguishing between the orbital decay and apsidal precession scenarios (Yee
et al., 2020; Turner et al., 2021), the timings do not have a substantial
impact on the decay-rate estimate.

The measured orbital decay rate $dP/dt$ can be used to constrain the host
star’s modified333$Q^{\prime}_{*}$ is defined as $3Q_{*}/2k_{2}$, where
$Q_{*}$ is the tidal quality factor, and $k_{2}$ is the second-order Love
number. tidal quality factor $Q^{\prime}_{*}$, which in the model of Goldreich
& Soter (1966) is related to the period derivative via

$\left|\frac{dP}{dt}\right|=\frac{27\pi
q}{2Q^{\prime}_{*}}\left(\frac{R_{*}}{a}\right)^{5}.$ (9)

Using the planet–star mass ratio $q\equiv M_{p}/M_{*}$ from Collins et al.
(2017) and $a/R_{*}$ from our combined phase-curve fit (Table 2), we arrived
at $Q^{\prime}_{*}=(1.50\pm 0.11)\times 10^{5}$.

As has been discussed in the previous orbital decay studies (Yee et al., 2020;
Turner et al., 2021), the measured $Q^{\prime}_{*}$ value of WASP-12 lies at
the lower end of most other estimates in the literature that based on
theoretical calculations or less direct empirical methods. The range of tidal
quality factors inferred from analyzing the eccentricity distribution of main-
sequence stellar binaries is $10^{5}$–$10^{7}$ (e.g., Ogilvie & Lin, 2007;
Lanza et al., 2011; Meibom et al., 2015), while analogous population studies
of hot Jupiter hosts point toward $Q^{\prime}_{*}$ values in the range
$10^{5.5}$–$10^{6.5}$ (e.g., Jackson et al., 2008; Husnoo et al., 2012;
Barker, 2020). Other population-level approaches indicate a broad range of
possible tidal quality factors ($10^{5}$–$10^{8}$; Collier Cameron & Jardine
2018; Penev et al. 2018; Hamer & Schlaufman 2019).

Meanwhile, the theoretically expected values of $Q^{\prime}_{*}$ are generally
higher ($10^{7}$–$10^{10}$; see Ogilvie 2014 and references therein). Weinberg
et al. (2017) proposed that WASP-12 is a subgiant star for which more rapid
tidal dissipation might be expected, due to breaking of internal gravity waves
near the star’s core. This hypothesis was investigated further by Bailey &
Goodman (2019), who agreed with the theoretical premise but found that the
observed characteristics of WASP-12 made it more likely to be a main sequence
star. Barker (2020) also calculated tidal dissipation rates for WASP-12 and
demonstrated that the theoretical and observed decay rates could only be
brought into agreement if the star is a subgiant. Taking a different tack,
Millholland & Laughlin (2018) proposed that the orbital decay is driven by
tidal dissipation within the planet, rather than the star; this would be due
to a hypothetical nonzero planetary obliquity that is maintained by a nearby
planetary companion. This intriguing scenario was investigated further by Su &
Lai (2021), who concluded that it required too much fine tuning to be
plausible.

In short, there is not yet a completely satisfactory explanation for the rapid
orbital decay of WASP-12b. Some priorities for future progress in this realm
are better characterization of the host star and its evolutionary state, as
well as more sensitive searches for nearby planetary companions.

Orbital decay has not yet been detected for any other planet, at least not
with nearly the same high confidence level as it has been detected for
WASP-12b. However, looking at the wider population of hot Jupiters, there are
several other planets with characteristics similar to those of WASP-12b that
might be expected to show comparable rates of orbital decay. In particular, if
we assume Equation (9) is correct and that $Q^{\prime}_{*}$ is the same for
all hot Jupiter systems, then WASP-103b, KELT-16b, WASP-18b, and the recently
discovered ultrahot Jupiter TOI-2109b have larger predicted $|dP/dt|$ values
than WASP-12b (see Figure 14 in Wong et al. 2021b). Notably, the hosts of all
five aforementioned planets are F-type stars. The transits of WASP-18b have
been monitored for over 15 years, with no sign yet of a decaying orbital
period; this nondetection constrains WASP-18’s tidal quality factor to
$Q^{\prime}_{*}>(1.7\pm 0.4)\times 10^{6}$ (Patra et al., 2020). Perhaps there
is a diversity in the dissipative properties of planet-hosting stars, or a
very sensitive dependence on the forcing frequency and internal structure of
the star.

TOI-2109b is a particularly promising candidate for probing tidal orbital
decay. Due to its ultrashort 0.67 day orbital period and its large mass of
roughly $5\,M_{\rm Jup}$, this extreme hot Jupiter has a predicted orbital
decay rate that is more than an order of magnitude faster than that of
WASP-12b (assuming the same $Q^{\prime}_{*}$). Follow-up transit observations
of the TOI-2109 system over the coming years should soon allow us to determine
if the host star has a $Q^{\prime}_{*}$ value comparable to that of WASP-12,
or a higher $Q^{\prime}_{*}$ similar to WASP-18.

###  4.2 Constraints on Atmospheric Variability

Figure 5: Plot of the individual secondary eclipse depths measured from the
TESS light curve (black). The datapoints are arranged sequentially by epoch,
with occultations that coincided with momentum dumps removed. The colored
lines and shaded areas indicate the median eclipse depths and $1\sigma$
confidence regions that we derived from full-orbit phase-curve fits for each
TESS sector (see Table 2). The scatter in the individual eclipse depths is
comparable to the typical uncertainties, and the depths measured for each
sector are mutually consistent at better than the $1\sigma$ level.

As was discussed in the Introduction, several prior secondary eclipse
observations of WASP-12b have yielded discrepant depth measurements, raising
the possibility of time-varying behavior of the planet’s atmosphere. However,
most of these previous results were obtained using different ground-based
facilities, and as such it is difficult to ascertain the extent to which
variable observing conditions and photometric extraction methodologies may
have caused the formal uncertainties in the occultation depths to be
underestimated.

Wong et al. (2021a) fit the individual WASP-12b secondary eclipses from the
TESS Sector 20 light curve and found that the measured depths show no sign of
variability to within the measurement precision. We expanded this analysis to
all four TESS sectors. After dividing out the best-fit systematics model from
each sector’s light curve, we fixed the mid-eclipse times based on the best-
fit decaying transit ephemeris from Equation (8). The parameters $b$,
$a/R_{*}$, and $R_{p}/R_{*}$ were fixed to the best-fit values computed from
our combined phase-curve fit (Table 2), and the per-point photometric
uncertainties were set to the respective red-noise-enhanced $\sigma_{r}$
values. No additional systematics modeling was included.

Figure 5 shows the depths measured from the 76 secondary eclipse light curves,
arranged sequentially by sector. The occultation depths derived from the
individual sector phase-curve fits are also shown. Due to the low signal-to-
noise ratio of the eclipse light curves, the precision of the individual depth
measurements is poor. Nevertheless, we can still place broad constraints on
any orbit-to-orbit atmospheric variability that may be present. The majority
of individual eclipse depths agree with the corresponding sector-wide values
at better than the $1\sigma$ level. The largest discrepancy between a pair of
eclipse depths is $3.1\sigma$. The overall scatter in the measurements (223
ppm) is smaller than the median uncertainty (242 ppm), indicating that the
individual eclipse depths are consistent with being drawn from the same
distribution. We calculated the Lomb–Scargle periodogram for the eclipse
measurements from the three consecutive sectors (43–45) and detected no
significant periodicities. We are able to place a $2\sigma$ upper limit of
roughly 450 ppm on the orbit-to-orbit variability of WASP-12’s dayside
brightness in the TESS bandpass.

We now compare this constraint to the previously reported instances of
variable secondary eclipse depths. The most significant discrepancy in the
literature lies between two $V$-band observations in von Essen et al. (2019):
$280\pm 100$ and $1160\pm 170$ ppm. This $4.5\sigma$ difference greatly
exceeds our upper limit; moreover, given the lower planet–star contrast ratio
in $V$-band, such a discrepancy in secondary eclipse depth would translate to
an even larger absolute change in the TESS-band dayside brightness. It is
important to mention that a third $V$-band eclipse light curve from the same
instrument yielded a nondetection ($-30\pm 170$ ppm). Therefore, barring the
serendipitous observation of a hitherto unexplained astrophysical event, we
suggest that uncorrected instrumental systematics and/or observing conditions
led to significant biases in one or both of the reported $V$-band secondary
eclipse depths in von Essen et al. (2019).

The remaining instances of short-wavelength ($<$1 $\mu$m) secondary eclipse
depth discrepancies in the literature are smaller in magnitude. Hooton et al.
(2019) measured $i^{\prime}$-band depths of $440\pm 210$ and $970\pm 140$ mmag
using two different telescopes. The former value is in good agreement with our
TESS-band sector-wide depth measurements. The $2.1\sigma$ difference between
the two depths is only marginally larger than our $2\sigma$ upper limit.
Lastly, the three $z^{\prime}$-band eclipse depths published in López-Morales
et al. (2010) and Föhring et al. (2013) are $680\pm 210$, $1000\pm 230$, and
$1300\pm 130$ ppm, with a $2.5\sigma$ mutual discrepancy between the largest
and smallest values. In light of the relatively low statistical significance
of these mismatches, as well as the lack of uniformity in instrumentation and
possible $i^{\prime}$\- and $z^{\prime}$-band telluric contamination in
ground-based observations, we should be cautious about interpreting these
measurements as evidence for atmospheric variability.

The TESS light curves provide much tighter constraints on longer-term
variability of WASP-12b’s atmosphere. The phase-curve results that we obtained
for the individual sectors offer four independent snapshots of the planet’s
properties, each averaged over a roughly one-month period. As illustrated in
Figure 5, the best-fit secondary eclipse depths for the four sectors are in
excellent agreement, with a standard deviation of 37 ppm and a median
uncertainty of 80 ppm. Moreover, there is no evidence of a systematic shift in
the size of the occultation across the roughly 600 days that elapsed between
Sector 20 and Sectors 43–45. We place a $2\sigma$ upper limit of $\approx$80
ppm on the TESS-band dayside atmospheric variability of WASP-12b across both
month-long and year-long timescales. In other words, the planet’s dayside
brightness is invariable to within $\approx$17%. Likewise, there are no
significant variations in the day–night temperature contrast, with the
atmospheric brightness amplitudes ($A_{\rm atm}$) from the four TESS sectors
consistent to within $2.2\sigma$ (see Table 2).

Our Palomar/WIRC $K_{s}$-band occultation measurement ($2810\pm 390$ ppm;
Section 3.2) adds to the appreciable body of previous $\sim$2.2 $\mu$m
observations. The first $K_{s}$-band eclipse depth was published in Croll et
al. (2011): $3090\pm 130$ ppm. Zhao et al. (2012) obtained two secondary
eclipse measurements in $K_{s}$ band with the TIFKAM instrument on the Hiltner
Telescope: $2810\pm 850$ and $3160\pm 950$ ppm, with a joint depth of $2990\pm
640$ ppm. Narrowband 2.312 $\mu$m observations at the Subaru Observatory
yielded a notably deeper occultation depth of $4500\pm 600$ ppm (Crossfield et
al., 2012). Croll et al. (2015) used CFHT/WIRCam on three nights and measured
$K_{s}$-band secondary eclipse depths of $2840\pm 200$, $2890\pm 180$, and
$2590\pm 420$ ppm; the same team also observed the system in the $K_{\rm
CONT}$ band ($\sim$2.22 $\mu$m) and calculated a depth of $2640\pm 450$ ppm.
The new $K_{s}$-band measurement is consistent with all prior values, except
for the significantly larger depth reported by Crossfield et al. (2012).

##  5 Summary

In this paper, we analyzed the full TESS light curve of WASP-12, including
three sectors’ worth of photometry obtained during the spacecraft’s ongoing
extended mission. We also presented a new $K_{s}$-band secondary eclipse
observation taken with the Palomar/WIRC instrument. Below, we reprise the main
results of our work:

  * •

From our combined fit to all four sectors of TESS data (Section 3.3; Table 2),
we significantly improved the precision of the measured phase-curve
properties. We obtained a secondary eclipse depth of $466\pm 35$ ppm, a peak-
to-peak atmospheric brightness amplitude of $524\pm 30$ ppm, and a nightside
flux that is consistent with zero. A marginal phase-curve offset of
$6\overset{\circ}{.}2\pm 2\overset{\circ}{.}8$ was also detected. The observed
ellipsoidal distortion amplitude of $65\pm 14$ ppm is consistent with
theoretical predictions based on the measured planet–star mass ratio.

  * •

We fit for 74 individual transit timings (Section 3.1; Table 3) and five mid-
occultation times (four from the TESS light curves and one from the
Palomar/WIRC observation; Section 3.2). The updated orbital period is
$1.091419370\pm 0.000000020$ days, with a decay rate of $-29.81\pm 0.94$ ms
yr-1.

  * •

From the sector-wide phase-curve fits, we found close agreement in the
measured secondary eclipse depths, planetary phase-curve amplitudes, and
phase-curve offsets. We placed a tight $2\sigma$ upper limit of $\approx$80
ppm ($\approx$17%) on dayside atmospheric brightness variability in the TESS
bandpass occurring on month-long and year-long timescales.

  * •

The set of individual TESS-band secondary eclipse depths shows no signs of
variability (Figure 5). We placed a broad $2\sigma$ upper limit of 450 ppm on
the orbit-to-orbit variability of WASP-12b’s dayside brightness.

  * •

Our new $K_{s}$-band occultation depth measurement ($2810\pm 390$ ppm) agrees
with all previously published values in the literature, except for the
anomalously high value from Crossfield et al. (2012), which we consider to be
an outlier.

Funding for the TESS mission is provided by NASA’s Science Mission
directorate. This paper includes data collected by the TESS mission, which are
publicly available from the Mikulski Archive for Space Telescopes (MAST).
Resources supporting this work were provided by the NASA High-End Computing
(HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center for the production of the SPOC data products. We thank the
Palomar Observatory team, particularly Paul Nied, Carolyn Heffner, and Tom
Barlow, for enabling the $K_{s}$-band secondary eclipse observations and
facilitating remote operations on the Hale $200^{\prime\prime}$ Telescope.
I.W. is supported by an appointment to the NASA Postdoctoral Program at the
NASA Goddard Space Flight Center, administered by the Universities Space
Research Association under contract with NASA. S.V. is supported by an NSF
Graduate Research Fellowship. H.A.K. acknowledges support from NSF CAREER
grant 1555095.

##  Appendix A Results from Individual TESS Transit Fits

The individual TESS transit light curves and best-fit models are plotted in
Figures 6–9. Table 3 lists the results from our individual TESS transit light-
curve fits. Each transit is labeled by the sector and sequential transit
number; Figure 1 shows the location of the corresponding transit events. The
second–fifth columns provide the median and $1\sigma$ values for the mid-
transit time $T_{c}$, planet–star radius ratio $R_{p}/R_{*}$, impact parameter
$b$, and scaled orbital semimajor axis $a/R_{*}$. The fitted transit depths
and transit-shape parameter values are mutually consistent to within
2.3–2.5$\sigma$.

Figure 6: Compilation of the TESS Sector 20 transit light curves. The
systematics-corrected photometry is shown in black. The red curves are the
best-fit models from our individual transit fits. The corresponding residuals
from the best-fit models are shown (offset by 0.03) in each panel. Figure 7:
Same as Figure 6, but for the Sector 43 transits. Figure 8: Same as Figure 6,
but for the Sector 44 transits. Figure 9: Same as Figure 6, but for the Sector
45 transits.

Table 3:  Results of Individual Transit Fits  Transit Number |  $T_{c}$ (${\rm BJD}_{\rm TDB}$) |  $R_{p}/R_{*}$ |  $b$ |  $a/R_{*}$
---|---|---|---|---
S20-1 | $2{,}458{,}844.09668\pm 0.00046$ | $0.1168\pm 0.0024$ | $0.26\pm 0.19$ | $3.17\pm 0.20$
S20-2 | $2{,}458{,}845.18772\pm 0.00042$ | $0.1152\pm 0.0022$ | $0.31\pm 0.20$ | $3.10\pm 0.19$
S20-3 | $2{,}458{,}846.27969\pm 0.00053$ | $0.1162\pm 0.0024$ | $0.30\pm 0.21$ | $3.07\pm 0.22$
S20-4 | $2{,}458{,}847.37086\pm 0.00049$ | $0.1181\pm 0.0022$ | $0.32\pm 0.20$ | $3.08\pm 0.27$
S20-5 | $2{,}458{,}848.46246\pm 0.00047$ | $0.1188\pm 0.0019$ | $0.23\pm 0.18$ | $3.16\pm 0.19$
S20-6 | $2{,}458{,}849.55305\pm 0.00045$ | $0.1143\pm 0.0016$ | $0.17\pm 0.18$ | $3.20\pm 0.14$
S20-7 | $2{,}458{,}850.64500\pm 0.00054$ | $0.1153\pm 0.0020$ | $0.21\pm 0.18$ | $3.15\pm 0.16$
S20-8 | $2{,}458{,}851.73597\pm 0.00054$ | $0.1177\pm 0.0025$ | $0.33\pm 0.22$ | $3.04\pm 0.26$
S20-9 | $2{,}458{,}852.82816\pm 0.00044$ | $0.1174\pm 0.0014$ | $0.13\pm 0.13$ | $3.20\pm 0.07$
S20-10 | $2{,}458{,}853.91929\pm 0.00045$ | $0.1179\pm 0.0020$ | $0.30\pm 0.19$ | $3.10\pm 0.21$
S20-11 | $2{,}458{,}859.37709\pm 0.00049$ | $0.1155\pm 0.0020$ | $0.31\pm 0.19$ | $3.11\pm 0.20$
S20-12 | $2{,}458{,}860.46813\pm 0.00045$ | $0.1182\pm 0.0016$ | $0.19\pm 0.15$ | $3.14\pm 0.11$
S20-13 | $2{,}458{,}861.55807\pm 0.00047$ | $0.1168\pm 0.0018$ | $0.28\pm 0.17$ | $3.11\pm 0.17$
S20-14 | $2{,}458{,}862.65051\pm 0.00055$ | $0.1161\pm 0.0023$ | $0.27\pm 0.18$ | $3.04\pm 0.18$
S20-15 | $2{,}458{,}863.74202\pm 0.00058$ | $0.1169\pm 0.0027$ | $0.45\pm 0.29$ | $2.86\pm 0.25$
S20-16 | $2{,}458{,}864.83386\pm 0.00049$ | $0.1145\pm 0.0018$ | $0.21\pm 0.17$ | $3.18\pm 0.15$
S20-17 | $2{,}458{,}865.92430\pm 0.00047$ | $0.1174\pm 0.0019$ | $0.23\pm 0.16$ | $3.16\pm 0.16$
S20-18 | $2{,}458{,}867.01609\pm 0.00051$ | $0.1174\pm 0.0020$ | $0.33\pm 0.22$ | $3.06\pm 0.22$
S20-19 | $2{,}458{,}868.10758\pm 0.00043$ | $0.1178\pm 0.0018$ | $0.29\pm 0.21$ | $3.08\pm 0.18$
S43-1 | $2{,}459{,}474.93462\pm 0.00046$ | $0.1166\pm 0.0019$ | $0.22\pm 0.16$ | $3.20\pm 0.14$
S43-2 | $2{,}459{,}476.02746\pm 0.00056$ | $0.1135\pm 0.0019$ | $0.25\pm 0.18$ | $3.15\pm 0.19$
S43-3 | $2{,}459{,}477.11831\pm 0.00050$ | $0.1170\pm 0.0024$ | $0.38\pm 0.23$ | $3.00\pm 0.21$
S43-4 | $2{,}459{,}479.30082\pm 0.00047$ | $0.1168\pm 0.0020$ | $0.35\pm 0.17$ | $3.02\pm 0.17$
S43-5 | $2{,}459{,}480.39281\pm 0.00048$ | $0.1153\pm 0.0018$ | $0.28\pm 0.17$ | $3.09\pm 0.14$
S43-6 | $2{,}459{,}481.48348\pm 0.00051$ | $0.1145\pm 0.0021$ | $0.32\pm 0.21$ | $3.09\pm 0.21$
S43-7 | $2{,}459{,}482.57475\pm 0.00049$ | $0.1153\pm 0.0018$ | $0.21\pm 0.17$ | $3.19\pm 0.15$
S43-8 | $2{,}459{,}483.66690\pm 0.00051$ | $0.1135\pm 0.0020$ | $0.27\pm 0.18$ | $3.15\pm 0.19$
S43-9 | $2{,}459{,}484.75830\pm 0.00048$ | $0.1167\pm 0.0020$ | $0.28\pm 0.16$ | $3.14\pm 0.19$
S43-10 | $2{,}459{,}488.03344\pm 0.00046$ | $0.1155\pm 0.0016$ | $0.24\pm 0.17$ | $3.17\pm 0.16$
S43-11 | $2{,}459{,}489.12326\pm 0.00051$ | $0.1188\pm 0.0031$ | $0.51\pm 0.25$ | $2.76\pm 0.29$
S43-12 | $2{,}459{,}490.21514\pm 0.00042$ | $0.1146\pm 0.0015$ | $0.17\pm 0.14$ | $3.15\pm 0.09$
S43-13 | $2{,}459{,}491.30624\pm 0.00048$ | $0.1145\pm 0.0016$ | $0.20\pm 0.13$ | $3.12\pm 0.10$
S43-14 | $2{,}459{,}492.39827\pm 0.00051$ | $0.1170\pm 0.0022$ | $0.28\pm 0.20$ | $3.14\pm 0.22$
S43-15 | $2{,}459{,}493.48919\pm 0.00043$ | $0.1163\pm 0.0017$ | $0.25\pm 0.17$ | $3.10\pm 0.16$
S43-16 | $2{,}459{,}494.58116\pm 0.00052$ | $0.1134\pm 0.0020$ | $0.28\pm 0.20$ | $3.17\pm 0.21$
S43-17 | $2{,}459{,}495.67260\pm 0.00047$ | $0.1156\pm 0.0021$ | $0.26\pm 0.20$ | $3.16\pm 0.23$
S43-18 | $2{,}459{,}496.76348\pm 0.00049$ | $0.1165\pm 0.0026$ | $0.46\pm 0.20$ | $2.94\pm 0.21$
S44-1 | $2{,}459{,}501.12937\pm 0.00051$ | $0.1164\pm 0.0024$ | $0.45\pm 0.19$ | $2.89\pm 0.21$
S44-2 | $2{,}459{,}502.22115\pm 0.00043$ | $0.1161\pm 0.0019$ | $0.27\pm 0.18$ | $3.10\pm 0.17$
S44-3 | $2{,}459{,}503.31271\pm 0.00045$ | $0.1149\pm 0.0014$ | $0.16\pm 0.15$ | $3.16\pm 0.10$
S44-4 | $2{,}459{,}504.40350\pm 0.00047$ | $0.1158\pm 0.0022$ | $0.25\pm 0.21$ | $3.23\pm 0.23$
S44-5 | $2{,}459{,}505.49468\pm 0.00059$ | $0.1168\pm 0.0025$ | $0.48\pm 0.21$ | $2.90\pm 0.23$
S44-6 | $2{,}459{,}506.58620\pm 0.00051$ | $0.1143\pm 0.0018$ | $0.23\pm 0.16$ | $3.16\pm 0.16$
S44-7 | $2{,}459{,}507.67798\pm 0.00045$ | $0.1165\pm 0.0020$ | $0.42\pm 0.15$ | $2.98\pm 0.16$
S44-8 | $2{,}459{,}508.76993\pm 0.00048$ | $0.1140\pm 0.0016$ | $0.21\pm 0.13$ | $3.15\pm 0.09$
S44-9 | $2{,}459{,}509.86066\pm 0.00045$ | $0.1154\pm 0.0018$ | $0.23\pm 0.17$ | $3.15\pm 0.15$
S44-10 | $2{,}459{,}510.95217\pm 0.00056$ | $0.1162\pm 0.0022$ | $0.31\pm 0.15$ | $3.09\pm 0.19$
S44-11 | $2{,}459{,}515.31724\pm 0.00049$ | $0.1144\pm 0.0024$ | $0.35\pm 0.20$ | $3.02\pm 0.25$
S44-12 | $2{,}459{,}516.40891\pm 0.00053$ | $0.1150\pm 0.0022$ | $0.35\pm 0.20$ | $3.06\pm 0.24$
S44-13 | $2{,}459{,}517.50098\pm 0.00060$ | $0.1191\pm 0.0025$ | $0.48\pm 0.16$ | $2.83\pm 0.20$
S44-14 | $2{,}459{,}518.59247\pm 0.00048$ | $0.1181\pm 0.0021$ | $0.37\pm 0.21$ | $3.01\pm 0.19$
S44-15 | $2{,}459{,}519.68224\pm 0.00057$ | $0.1151\pm 0.0022$ | $0.34\pm 0.20$ | $3.07\pm 0.19$
S44-16 | $2{,}459{,}521.86571\pm 0.00042$ | $0.1153\pm 0.0018$ | $0.25\pm 0.17$ | $3.16\pm 0.17$
S44-17 | $2{,}459{,}522.95725\pm 0.00057$ | $0.1168\pm 0.0021$ | $0.27\pm 0.19$ | $3.12\pm 0.23$
S44-18 | $2{,}459{,}524.04947\pm 0.00039$ | $0.1162\pm 0.0016$ | $0.18\pm 0.16$ | $3.20\pm 0.12$
S45-1 | $2{,}459{,}527.32351\pm 0.00043$ | $0.1151\pm 0.0016$ | $0.24\pm 0.15$ | $3.11\pm 0.15$
S45-2 | $2{,}459{,}528.41517\pm 0.00054$ | $0.1179\pm 0.0024$ | $0.52\pm 0.13$ | $2.75\pm 0.19$
S45-3 | $2{,}459{,}529.50671\pm 0.00055$ | $0.1156\pm 0.0022$ | $0.44\pm 0.21$ | $2.99\pm 0.22$
S45-4 | $2{,}459{,}530.59839\pm 0.00057$ | $0.1167\pm 0.0030$ | $0.43\pm 0.28$ | $2.92\pm 0.25$
S45-5 | $2{,}459{,}531.68987\pm 0.00052$ | $0.1145\pm 0.0016$ | $0.22\pm 0.15$ | $3.19\pm 0.14$
S45-6 | $2{,}459{,}532.78034\pm 0.00048$ | $0.1170\pm 0.0024$ | $0.45\pm 0.19$ | $2.87\pm 0.20$
S45-7 | $2{,}459{,}533.87187\pm 0.00054$ | $0.1157\pm 0.0019$ | $0.32\pm 0.23$ | $3.03\pm 0.20$
S45-8 | $2{,}459{,}534.96448\pm 0.00046$ | $0.1135\pm 0.0015$ | $0.21\pm 0.16$ | $3.21\pm 0.14$
S45-9 | $2{,}459{,}536.05478\pm 0.00052$ | $0.1179\pm 0.0018$ | $0.27\pm 0.16$ | $3.14\pm 0.18$
S45-10 | $2{,}459{,}537.14683\pm 0.00055$ | $0.1165\pm 0.0016$ | $0.21\pm 0.16$ | $3.18\pm 0.14$
S45-11 | $2{,}459{,}540.42054\pm 0.00047$ | $0.1146\pm 0.0017$ | $0.17\pm 0.15$ | $3.17\pm 0.11$
S45-12 | $2{,}459{,}541.51151\pm 0.00041$ | $0.1168\pm 0.0015$ | $0.15\pm 0.12$ | $3.20\pm 0.08$
S45-13 | $2{,}459{,}542.60447\pm 0.00048$ | $0.1155\pm 0.0017$ | $0.22\pm 0.17$ | $3.11\pm 0.14$
S45-14 | $2{,}459{,}543.69504\pm 0.00046$ | $0.1171\pm 0.0020$ | $0.28\pm 0.18$ | $3.06\pm 0.16$
S45-15 | $2{,}459{,}545.87708\pm 0.00049$ | $0.1164\pm 0.0021$ | $0.28\pm 0.20$ | $3.12\pm 0.21$
S45-16 | $2{,}459{,}546.96868\pm 0.00044$ | $0.1158\pm 0.0015$ | $0.12\pm 0.12$ | $3.24\pm 0.06$
S45-17 | $2{,}459{,}548.06099\pm 0.00055$ | $0.1167\pm 0.0022$ | $0.46\pm 0.18$ | $2.90\pm 0.20$
S45-18 | $2{,}459{,}549.15196\pm 0.00049$ | $0.1160\pm 0.0017$ | $0.18\pm 0.16$ | $3.19\pm 0.13$
S45-19 | $2{,}459{,}550.24370\pm 0.00054$ | $0.1143\pm 0.0024$ | $0.41\pm 0.23$ | $2.98\pm 0.25$

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