Phase-curve Pollution of Exoplanet Transit Depths

Marine Martin-Lagarde1 , Giuseppe Morello1,2 , Pierre-Olivier Lagage1, René Gastaud1, and Christophe Cossou1,31 AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, F-91191 Gif-sur-Yvette, France; marine.martin-lagarde@cea.fr

2 INAF—Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, I-90134 Palermo, Italy3 Institut d’Astrophysique Spatiale, CNRS, Université Paris-Saclay, Bât. 121, F-91405, Orsay Cedex, France

Received 2020 April 11; revised 2020 July 25; accepted 2020 July 28; published 2020 October 7

Abstract

The next generation of space telescopes will enable transformative science to understand the nature and origin ofexoplanets. In particular, transit spectroscopy will reveal the chemical composition of the exoplanet atmosphereswith unprecedented detail thanks to precise measurements of the visible-to-infrared transit depths down to 10 partsper million. Such a level of instrumental precision raises the challenge to obtain even more precise astrophysicalmodels so as not to significantly influence the interpretation of the observed data. We must therefore criticallyrevisit some of the commonly accepted assumptions that were adequate for analyzing past and currentobservations. A common approximation in the analysis of exoplanetary primary transits is that the planet does notcontribute to the recorded flux, so-called dark planet hypothesis. In this paper, we investigate the impact of thedark planet hypothesis on the parameters obtained from the analysis of transits with particular attention to thetransit depth. We develop mathematical formulae and release new software to estimate the magnitude of thepotential bias. These tools will be useful in the preparation of observing proposals, as well as within the scientificconsortia of the James Webb Space Telescope (JWST) and the Atmospheric Remote-sensing Infrared ExoplanetLarge-survey (ARIEL) missions. We probe the accuracy of the mathematical formulae through the analysis ofsynthetic observations with the JWST Mid-InfraRed Instrument. We find that self-blending from nightsideemission attenuates the transit depth by >3σ for some of the known exoplanet systems, in agreement with previouswork. An additional unreported effect caused by the nightside rotating into view can also impart a significanteffect, but in the opposite direction (increasing the transit depth); this effect can largely be removed withconventional detrending practices, at the expense of a slight increase in noise, and mixing astrophysical variationsand instrumental drifts.

Unified Astronomy Thesaurus concepts: Exoplanet systems (484); Transit photometry (1709); Exoplanetatmospheres (487); High time resolution astrophysics (740); Observational astronomy (1145); Exoplanets (498);Astronomical instrumentation (799); Infrared photometry (792)

1. Introduction

The discovery of the first exoplanet transiting in front of itshost star (Charbonneau et al. 2000; Henry et al. 2000)represents a breakthrough for the detection and characterizationof exoplanets. Transit light-curves enable measurements of theplanet radius and orbital parameters, as well as of the planetmass, if combined with radial velocity observations. Transitspectroscopy is one of the most successful techniques to obtaininformation on the exoplanet atmospheres. It relies on precisemeasurements of the transit depth (i.e., the squared planet-to-star radii ratio) at multiple wavelengths, which constitute theso-called transmission spectrum of the exoplanet atmosphere.The transmission spectrum can reveal the presence ofabsorbing species, particle scattering, and clouds in theatmosphere through features on the order of tens to hundredsof parts per million (ppm) in amplitude relative to the stellarflux (Brown 2001).Typically, the transit light-curves are modeled using a

parametric function such as that introduced by Mandel & Agol(2002). Two common approximations are that (1) the stellarflux is constant, and (2) the planetary flux is zero. We refer tothe second approximation as the dark planet hypothesis.Kipping & Tinetti (2010) considered the effects of emissionfrom the nightside of the exoplanet. In their study, theyassumed a constant planetary flux in proximity of the transitevent, i.e., a flat out-of-transit hypothesis. The main conse-quence of a nonzero, but constant, planetary flux is a dilution of

the apparent transit depth, to which the authors referred to asthe self-blend effect. They showed that such dilution can be ofthe order of∼100 ppm for some exoplanets observed in themid-infrared; therefore, it is not negligible at the precision levelof some of the current and future observations.In this paper, we take a step forward by including the phase-

curve variability. We identify a phase-blend effect in additionto self-blend, that tends to increase rather than dilute theapparent transit depth. We explore the impact of both effects onsimulated observations of a sample of exoplanet systems withthe Mid-InfraRed Instrument (MIRI) on board the James WebbSpace Telescope (JWST). In particular, this first study focuseson the white light-curves from the MIRI-Low ResolutionSpectrometer (LRS). Errors in the white light-curve analysiscan cause an offset and a trend on the measured transmissionspectrum of the exoplanet atmosphere, if the white light-curveis used to correct the instrument systematic effects and/or toderive parameters common to the spectral bins (e.g., Tsiaraset al. 2018). Future studies will investigate the bias in thespectrum due to the wavelength-dependent phase-curve effects.Structure of the paper—Section 2 introduces the ExoNoo-

dle package, which we used to generate the syntheticobservations. Section 3 reports the information about thesimulated light-curves and their analysis. Appendix Adiscusses technical details about our method of analysis.Section 4 describes the mathematical formulae accounting forthe effects of the exoplanet phase curve in transit observations

The Astronomical Journal, 160:197 (13pp), 2020 November https://doi.org/10.3847/1538-3881/abac09© 2020. The American Astronomical Society. All rights reserved.

1

Software reviewed by the Journal of Open Source Software

and discusses their correctability. Section 5 shows the results ofthe synthetic data analysis in terms of parameter bias andstatistical significance, along with the predictions from themathematical formulae; a correction of the phase-blend effectbased on polynomial fitting is also discussed in this section.Section 6 discusses the implications and usage of our findings.Finally, Section 7 summarizes the key points discussed in thispaper.

2. The ExoNoodle Package

ExoNoodle is a package that generates spectroscopic light-curves of exoplanetary systems (Martin-Lagarde et al. 2020).Starting with the IDL routine of Mandel & Agol (2002), a longlist of software has been published online for the calculation oftransit light-curves, such as JKTEBOP (Southworth et al.2004), TAP (Gazak et al. 2012), EXOFAST (Eastman et al.2013), PYTRANSIT (Parviainen 2015), BATMAN (Kreidberg2015), PYLIGHTCURVE (Tsiaras et al. 2016), ExoTETHyS.TRIP (Morello et al. 2020), and Limbdark (Agol et al. 2020).Recently, some routines have appeared to compute theexoplanet phase-curves (but neglecting the occultations), suchas SPIDERMAN (Louden & Kreidberg 2018) and BELL-EBM(Bell & Cowan 2018). ExoNoodle covers both aspects, i.e., itcan generate the complete exoplanet phase-curves includingthe transit and eclipse events, with a spectral dimension. Ittakes model spectra of the planet and star and computes thecombined spectrum of the system (in Jansky) over timeaccounting for the orbital motions. The code is modular toallow a high degree of flexibility for the input information (e.g.,detailed spectra or blackbody temperatures). The spectracreated with ExoNoodle are noiseless and without anysystematics or distortion from the instrument. ExoNoodle isdesigned to guarantee a precision of <1 ppm in the computedtime-series.

The spectroscopic light-curves can be weighted by aninstrumental response and photon noise can be added togenerate synthetic observations. The input spectra can have anywavelength range and resolution; therefore, it is possible tocreate time series for any instrument. Such synthetic data setscan be used for feasibility studies when preparing observingproposals and/or testing new science cases within missionconsortia. These synthetic observations are complementary tothose obtained with typical noise simulators such as pysyn-phot (Lim et al. 2015) and pandexo (Batalha et al. 2017).Additionally, ExoNoodle can be coupled with more completeinstrument simulators, for example, including systematics. It iscurrently being used coupled with MIRISim (P. Klaassen et al.2020, in preparation) in the frame of preparing the JWST-MIRIobservations of the Early Release Science (ERS) program.

ExoNoodle is an open-source project. Instructions and adetailed user manual are available in the GitLab repository.4

2.1. Overview of the Algorithm

In ExoNoodle, computations are conducted with absoluteunits. The final product of an ExoNoodle run is the spectrumof the system at a given time. The system spectrum is the sumof three terms:

( )★a b g= + +F F F F , 1system day night

where F★, Fday, and Fnight are template spectra for the star,exoplanet dayside and nightside, respectively. The coefficientsα, β, and γ denote the flux fractions directed to the observer,hereinafter referred to as visibility coefficients. These can beexpressed as

( ) ( )★a = - L1 , 2

( ) ( )b y= - L1 , 3day

( )( ) ( )g y= - L -1 1 . 4night

Here ★L denotes the fraction of stellar flux occulted by theplanet, which depends on the system parameters, the orbitalphase, and the stellar limb-darkening (see Sections 2.2 and2.3). Similarly, Λday and Λnight denote the fractions of planetdayside and nightside flux occulted by the star. Finally, ψ is thesurface fraction of planet projected disk showing the dayside,and 1−ψ is the complementary fraction showing the night-side. Assuming that the planet lies on a circular, tidally lockedorbit, the dayside fraction is

( ) ( )yj j

=- + D1 cos

2, 5

where j is the orbital phase angle from mid-transit, and Δj canmimic the hotspot displacement from the substellar point, which isa possible effect induced by the atmospheric circulation (Burrowset al. 2010).The dayside and nightside fluxes can be decomposed as

( )= +F F F , 6day dayemission

dayreflection

( )=F F , 7night nightemission

where the superscript “emission” refers to the flux emitted bythe planet, while “reflection” refers to the reflected stellar light.The reflection component is given by

( )★⎛⎝⎜

⎞⎠⎟= ´ ´F A

R

aF

2, 8

pdayreflection

2

where ( )R

a2

2p is the geometric scaling factor (a is the orbital

semimajor axis, and Rp is the radius of the planet), and A is theplanetary albedo. Note that Rp and A can be wavelength dependentbecause of the exoplanet atmosphere (Brown 2001).

2.2. Phase Configurations

The compilation of the system spectrum depends on thegeometric configuration at the given orbital phase. There arethree cases:

1. TRANSIT: The transit occurs when the planet occults partof the star. The transit time interval is delimited by theexternal contacts, i.e., when the planet and the starprojected disks are externally tangent (Seager & Mallén-Ornelas 2003; Kipping 2008). We adopt the symbol T1−4

to denote the duration of this interval.During the transit interval, the planet is entirely

visible (Λday=Λnight=0), while the star is partiallyocculted ( ★L > 0). If the star is represented by auniformly emitting disk, ★L is the fraction of areaocculted by the planet, given by the lune (the shapeformed by two intersecting circles) expression from4 https://gitlab.com/mmartin-lagarde/exonoodle-exoplanets

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The Astronomical Journal, 160:197 (13pp), 2020 November Martin-Lagarde et al.

Mandel & Agol (2002). In this case, ( )★★

L = R

R

2p when

the full planet disk lies over the star disk. In mostapplications, the star is represented by a limb-darkeneddisk, for which we perform a numerical integration (seeSection 2.3).

2. ECLIPSE: The eclipse occurs when the planet passesbehind the star. Likewise, the eclipse time interval isdelimited by the analogous external contacts.

During the eclipse interval, the star is entirely visible( ★L = 0), while the planet is partially or totally occulted(Λday, Λnight>0). The ingress and egress phases, whenthe planet is only partially behind the star are computedwith the lune expression from Mandel & Agol (2002).When the planet disk is entirely behind the star Λday=Λnight=1.

3. TRANSITION PHASE: If the two disks are not overlapping,we are neither in transit or eclipse. During this transitionphase, both the star and the planet are entirelyvisible ( ★L = L = L = 0day night ).

2.3. Limb-darkening and Meshing

The stellar limb-darkening is the radial decrease of the lightintensity from the center to the edge of the disk. Because of thiseffect, the fraction of occulted stellar flux varies while theplanet moves across the stellar disk. The intensity profile isoften parameterized by analytical functions Iλ(μ), the so-calledlimb-darkening laws, where m g= cos and γ is the anglebetween the line of sight and the normal to the stellar surface(Claret 2000; Espinoza & Jordán 2016; Morello et al. 2017).ExoNoodle can take a set of limb-darkening coefficients ateach wavelength, or a unique set of coefficients. During thetransit, the occulted stellar flux is computed by numericalintegration using a radial mesh, as described hereafter.

First, the algorithm calculates the projected planet–starseparation ( ★-d p), based on the orbital parameters andreference mid-transit time (Winn 2010). The overlapping areais radially meshed in 3000 parts, this number being fixed toguarantee the 1 ppm precision. The area of each mesh (Sn) isobtained analytically from the difference of two lunes, asshown in Figure 1. The mesh flux is approximated by

( ) ´lI r Sn n. Finally, the occulted stellar flux is the sum of the

mesh fluxes. Therefore, we can write

( ( ) ) ( )★★åL = ´l=F

I r S1

. 9n

n n1

3000

3. Simulated Light-curves

3.1. Selection of the Planets

We selected a sample of 17 planets covering the range0.7–8.5 days for the orbital period, and corresponding to asemimajor axis of 0.014–0.077 au. These ranges include 80%and 90% of the known population of transiting exoplanets,respectively.5 The chosen planets are mostly hot Jupiters,although we also included a hot and a warm super-Earth(55Cnc-e and GJ1214-b). The detailed stellar, planetary,and orbital parameters are reported in Tables B1–B3 inAppendix B.We used ExoNoodle to compute the synthetic light-curves

(see Section 2). We adopted a library of PHOENIX spectra(Baraffe et al. 2015) for the stars. These spectra are identifiedby the effective temperature (Teff), the surface gravity (log(g)),and the metallicity ([ ]Fe H ). The spectra seen from Earth arecalculated with the scaling factor ( )★R d 2, where d is thedistance from Earth. We modeled the exoplanets as two half-spheres, with blackbody emission spectra defined by thedayside and nightside temperatures. The majority of transitingplanets currently detected are thought to be tidally locked,therefore having constant daysides and nightsides withdifferent average temperatures (Bolmont & Mathis 2016;Barnes 2017; Parmentier & Crossfield 2018). Comparing ourmodel with spiderman (Louden & Kreidberg 2018) modelsthat include a temperature gradient based on sphericalharmonics, we find some configurations with nearly identicalphase-curves (within 0.5 ppm).The dayside and nightside temperatures are given by Cowan

& Agol (2011):

( ) ( )⎜ ⎟⎛⎝

⎞⎠e= - -T T A1

2

3

5

12, 10bday irr

14

14

( ) ( )⎜ ⎟⎛⎝

⎞⎠

e= -T T A1

4, 11bnight irr

14

14

where ★= ´T T R

airr eff is the irradiation temperature, ε is thecirculation efficiency, and Ab is the bond albedo. Usually, thelast two parameters are poorly constrained due to the lack ofobservational data. For our simulations we set Ab=0.3, whichis a typical value for hot Jupiters (Mallonn et al. 2019), andε=0.5, which is the central value of the allowed range [0; 1].We calculated the spectroscopic time-series over a time

window of Δtobs=2.5×T1–4 with a time sampling of12.35 s. For modeling the transit, the limb-darkening coeffi-cients are computed using ExoTETHyS (Morello et al. 2020).All the spectra are multiplied by the MIRI LRS photon–electron conversion (Kendrew et al. 2015) and integrated overthe instrument wavelengths and ramp time. The integratedspectra are multiplied by the collecting area and throughput ofJWST. These operations return a white light-curve in numberof electrons, to which we add the photon noise as a Poisson

Figure 1. Sketch of an elementary mesh for computing the flux occulted by theplanet (in red), shown here as the substraction of two lunes (dark and clearblue), defined with their radius relative to the center of the star.

5 From www.exoplanet.eu.

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The Astronomical Journal, 160:197 (13pp), 2020 November Martin-Lagarde et al.

distribution. The white light-curves with noise are the syntheticobservations analyzed in this study.

3.2. Fitting Method

We fitted the synthetic observations using transit light-curvesgenerated with pylightcurve. This model does not accountfor the exoplanet emission and reflection, as is usual in thestudy of exoplanetary transit. The free parameters in the light-curve fits were

1. the transit depth

( )★

⎛⎝⎜

⎞⎠⎟=p

R

R; 12

p22

2. the impact parameter

( ) ( )★

=ba

Ricos ; 13

3. the “central” transit duration (Kipping 2010)

( )( )

★

⎛

⎝⎜⎜

⎞

⎠⎟⎟p

=-

TP b

iarcsin

1

sin; 14a

R

0

2

4. the mid-transit shift (that accounts for potential timingvariation or errors in the ephemerides); and

5. the normalization factor.

The choice of the set of free parameters is equivalent to thosethat are typically adopted in the analysis of transit photometric(e.g., Beaulieu et al. 2011; Morello 2018; Vanderspek et al.2019; Shporer et al. 2020) or white light-curve observations(e.g., Sing et al. 2016; Tsiaras et al. 2018).

Note that we assume a perfect knowledge of the limb-darkening coefficients to prevent more parameter degeneracies.This assumption is realistic for observations in the mid-IR, asthe JWST-MIRI simulated observations of this study (Magicet al. 2015; Morello et al. 2017). We also performed fits withtransit depth and normalization factor as the only freeparameters, while fixing the other parameters to their inputvalues.

In order to explore the parameter space efficiently, first werun a simple least-squares minimization function to obtainpreliminary estimates of the parameter values. Then we executeemcee (Foreman-Mackey et al. 2013) with 2×105 iterationsand 40 walkers. Each walker is initialized at a random positionclose to the least-squares solution. The results of the fit are themedians of the five parameter chains. The upper and lowererror bars are defined by the differences of the 84th and 16thpercentiles from the median.

In order to eliminate the random biases associated with aspecific noise realization, we applied the noise twice: one timeby adding the noise to the curve, the second by subtracting it.We take the mean of the two estimates to obtain the purelybiased parameter values independent of the specific noiserealization. The validity of this approach is demonstrated inAppendix A.

4. Mathematical Derivation

4.1. Analytical Estimate of the Bias

We derive here an analytical formula to estimate the biasin transit depth due to neglecting the thermal and reflection

phase-curve of the planet. The phase-curve modulation isapproximated by a step function with three intervals, theconstant values being the flux averaged before, during, andafter transit (see Figure 2). Based on Equations (1)–(5),considering that Δj=0 in this study, we define the followingintegral functions:

( ) ( )

( ) ( )( )

⎛⎝⎜

⎞⎠⎟

òj jj j

jj

j jj j

=-

-

= ---

j

j

Fd

F

,1 cos

2

21

sin sin, 15

day 1 2day

2 1

day 2 1

2 1

1

2

( ) ( )

( ) ( )( )

⎛⎝⎜

⎞⎠⎟

òj jj j

jj

j jj j

=-

+

= +--

j

j

Fd

F

,1 cos

2

21

sin sin. 16

night 1 2night

2 1

night 2 1

2 1

1

2

Using again the phase-curve symmetry (Δj=0), the averageplanet flux in and out of the transit can be expressed as

( )⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠=

F+

F- - F 0,2

0,2

, 17pin

day1 4

night1 4

( )⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠=

FF +

FF-

--

- F m m2

,2

, , 18pout

day1 4

1 4 night1 4

1 4

where ( )pF =- -T P21 4 1 4 is the phase angle associated withthe transit duration (P is the orbital period), and m=5/4 in oursimulated light-curves. Explaining the integral means throughEquations (15) and (16),

( )( )=

+-

-F

F

-

-F

F F F F

2 2

sin, 19p

in day night day night 2

2

1 4

1 4

( )( )

( )( )

=+

--

´F -

- F

-F

-

-

FF F F F

m

m

2 2

sin sin. 20

pout day night day night

1 4 2

1

2 1 4

1 4

Note that the above equations give at first order theapproximation of Kipping & Tinetti (2010):

( )» »F F F . 21p pin out

night

We provide here an approximation for the flux difference-F Fp p

out in by replacing the sine functions with their third-orderMaclaurin expansion:

( ) ( ) ( )- » -+

F -F F F Fm m2 1

24. 22p p

out inday night 1 4

2

This approximation will be used in Section 4.2.The simplified step function describing the planet flux can be

reformulated as

( ) ( ( )) ( )j j= - GF F 1 , 23p pout

with

( ) ( )

⎧⎨⎪

⎩⎪jG =

-F F

Fin transit

0 out of transit

. 24p p

p

out in

out

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The Astronomical Journal, 160:197 (13pp), 2020 November Martin-Lagarde et al.

The total system flux is

( ( )) ( ( )) ( )★ ★ j j= - L + - GF F F1 1 , 25psystemout

which can be normalized as

˜ ( )

( ) ( )

★

★

★★

★

j

j

=+

= -+

L

-+

G

FF

F F

F

F F

F

F F

1

. 26

p p

p

p

systemsystem

out out

out

out

The normalized flux can be reformulated as

˜ ( ) ( )j= - LF 1 , 27system system

where

( ) ( ) ( ) ( )★

★★

★j j jL =

+L +

+G

F

F F

F

F F28

p

p

psystem out

out

out

is the analogous occultation fraction for the system flux. Notethat ★L = Lsystem if the planet flux is zero. We conclude that,in a real case, the dark planet approximation leads to a biasedtransit depth (based on Equations (28) and (24))

( )★

★ ★=

++

-

+p

F

F Fp

F F

F F. 29

p

p p

pbias2

out2

out in

out

We separate the two contributions to the bias:

( ) ( )‐★

D = -+

pF

F Fp , 30

p

p

2self blend

out

out2

( ) ( )‐★

D =-

+p

F F

F F. 31

p p

p

2phase blend

out in

out

The first term is the planet self-blend that was introduced byKipping & Tinetti (2010). The transit depth tends to beunderestimated because of self-blending, when using the darkplanet approximation. The second term comes from the planet

flux variation with the orbital phase, whose contribution to thebias has not been taken into account in the previous literature.We name this term as the phase-blend effect. The transit depthtends to be overestimated because of the phase-blend effect,when using a dark planet approximation. These two effects actin opposite directions and tend to cancel each other.

4.2. Dependence of the Bias on the System Parameters

In this section, we provide a simple way to estimate thepotential bias in transit depth for a given exoplanet systembased on its physical parameters. We assume that star andplanet fluxes can be approximated by blackbody emissions,

( )µ lF R B T2 , where λ is the effective wavelength of theinstrument passband, and the reflected stellar flux is negligible.By replacing the approximate formulae for Fp

out (fromEquation (21)) and -F Fp p

out in (from Equation (22)) into theEquations (30) and (31), and considering that ★F Fp

out , weobtain

( )( )( )

( )‐★

⎛⎝⎜

⎞⎠⎟D » - l

lp

B T

B T

R

R, 32

p2self blend

night

eff

4

( ) ( ) ( ) ( )( )

( )

‐

★⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

D »+ -

´

l l

l

-

pm m B T B T

B T

R

R

T

P

2 1

24

. 33p

2phase blend

day night

eff2

1 42

For the mid-infrared observations, the spectral radiance can beapproximated by the Rayleigh–Jeans formula, ( ) µlB T T . FromEquations (10) and (11), we can infer that µT T T,day night irr. Weconclude that

( ) ( )‐★

★

⎛⎝⎜

⎞⎠⎟D » -p C

R

a

R

R, 34

p2self blend night

4

Figure 2. Sketch illustrating different transit light-curve models. The gray curve is the flux that would be observed over time in the case of a dark planet, i.e., with aflat out-of-transit line at F=Få. The black curve is from the ExoNoodle model, accounting for the planetary flux and its variation due to the orbital rotation. Thecorresponding planetary phases (not representative) are illustrated above the curves. The average flux in transit ★ +F Fp

in (with the occultation removed) and out oftransit ★ +F Fp

out are shown in blue and red, respectively. The times A and B are the transit boundaries (tB−tA=T1–4).

5

The Astronomical Journal, 160:197 (13pp), 2020 November Martin-Lagarde et al.

( ) ( )‐★

★⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠D » -p C C

R

a

R

R

T

P, 35m

p2phase blend day

21 4

2

with

( ) ( )⎜ ⎟⎛⎝

⎞⎠

e= -C A1

4, 36bnight

14

14

( ) ( )⎜ ⎟ ⎜ ⎟⎡⎣⎢⎢⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥⎥e

e= - - -C A1

2

3

5

12 4, 37bday

14

14

14

( ) ( )=+

Cm m2 1

24. 38m

4.3. Morphological Degeneracy and Correctability of the Bias

The self-blend effect acts as a contraction of the normalizedtransit light-curve along its vertical axis. While it is notequivalent to a simple rescaling of the planet-to-star radii ratio,the resulting light curve can be very well approximated by thatof a system with slightly different geometric parameterswithout self-blend. Therefore the self-blend effect cannot becorrected based solely on the transit observation, as stated byKipping & Tinetti (2010).

The phase-blend effect is associated with a continuous signalthat affects the overall light-curve shape, e.g., a portion of sinusoidin our simulations. If significant, this signal can cause time-correlated noise in the light-curve residuals. The phase-curve signalcan be confused with other long-term instrumental systematiceffects that can be approximated with polynomial, exponential, orother parametric functions. It is desirable to separate the phasecurve from instrumental systematic signals to maximize theastrophysical information on the specific system. This separation isalso important for a good characterization of the instrument itself,e.g., to improve the precision of the data-processing methods, andto avoid injecting false instrument variability. However, aparametric model that mimics the phase curve, albeit incorrect,can still decrease the phase-blend bias on the transit depth. In other

words, the act of detrending can actually remove a large fraction ofthe phase-blend, but would slightly increase the noise.

5. Results

5.1. Comparison between Simulations and AnalyticalFormulae

Figure 3 shows the differences between the best-fit and inputtransit depths for the simulated exoplanet systems. We detectedsignificant biases (3σ) in transit depth for five planets. Inthree cases (WASP12-b, WASP18-b, and WASP19-b), thefitted transit depth is overestimated, suggesting that the phase-blend effect is dominant over the self-blend. Note that theseplanets are the ones with the largest irradiation temperature,hence the largest day–night temperature contrast. They alsohave an orbital period shorter than 1 day. In the other two cases(HD189733-b and HD209458-b), the transit depth is under-estimated, suggesting the prevalence of the self-blend effect.These two planets are average hot Jupiters with orbital periodsof 2.2 and 3.5 days, respectively. We do not report significantdifferences between the best-fit values of the transit depthobtained with five or two free parameters (see Section 3.2). Theonly noticeable differences are found for WASP37-b (17 ppm)and WASP43-b (9 ppm), which are however less than half oftheir respective error bars. The error bars are 6%–17% smallerwhen fitting two parameters only, owing to the reducedparameter degeneracies.The analytical estimates obtained from Equation (29) are in

excellent agreement with the fitted values, being always consistentto within 0.65σ. The median difference between the results of thefit and the analytical estimates is below 1 ppm. For the two planetswith the largest transit depth bias, WASP12-b and WASP19-b,the predicted biases are 8–15 ppm below the corresponding best-fit values. In the other cases the discrepancies are below 5 ppm,except for the five-parameter fit of WASP37-b and the two-parameter fit of WASP43-b.Figure 4 shows the differences between the best-fit and

input values for the other physical parameters. For some of the

Figure 3. Differences between the best-fit and input transit depths for the simulated exoplanet systems obtained with five (dark green, open circles) and two (lightgreen, open circle) free parameters, as described in Section 3.2. The error bars are the statistical 1σ obtained from the Markov Chain Monte Carlo fits. The analyticalpredictions by Equation (29) are also reported (dark green, filled circles).

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Figure 4. Top panel: differences between the best-fit and input impact parameter for the simulated exoplanet systems obtained with five free parameters, as describedin Section 3.2. Bottom panel: analogous plot for the central transit duration.

Figure 5. Analytical estimates of the transit depth bias with different values of the circulation efficiency: ε=0 (blue squares), ε=0.5 (green circles), and ε=1 (redtriangles). The light gray area represents the predicted 3σ error.

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planets with a significant bias in transit depth, the impactparameter and transit duration are also biased, but withsignificance 2σ. If the bias in transit depth is not significant,the other parameters are unbiased as well.

5.2. Exploration of the ε-dependence

Figure 5 shows the expected transit depth bias with differentvalues of the circulation efficiency. For ε=1 the thermalenergy is efficiently redistributed over the planet, so that thereis no day–night temperature difference. In this case, only theself-blend is effective, leading to an underestimated transitdepth. On the contrary, for ε=0 the thermal energy stays onthe dayside; therefore, the day–night temperature contrast is themaximum. In this case, the phase-blend dominates over theself-blend effect, leading to an overestimated transit depth.The total bias in transit depth is a monotonic function of thecirculation efficiency, as can be demonstrated starting fromEquations (10) and (11). Figure 6 shows the transit depth biasversus circulation efficiency for the WASP12 system, whichhas the largest variation between the two extreme cases.

We can see in Figure 5 that the exoplanet systems do nothave the same sensitivity to that parameter, given the verydifferent intervals delimited by the two extreme cases. We canidentify seven planets with statistically significant biases (>3σ)when ε=0 or ε=1; these are in descending order of theabsolute value of the maximum bias: WASP12-b, WASP19-b,WASP18-b, WASP43-b, HD189733-b, WASP79-b, andHD209458-b. Five of these planets were already identified ashaving detectable bias even if ε=0.5. The top three planetspresent very asymmetric intervals with a maximum positivebias of 230–550 ppm for ε=0, and a negative bias of20–100 ppm for ε=1. Instead, WASP43-b presents a nearlysymmetric interval, the potential bias being between −110 ppmand +150 ppm. HD189733-b is more likely to have a negative

bias, the predicted interval being between −60 ppm and+30 ppm.

5.3. Correlations with the System Parameters

Figure 7 shows how the theoretical bias in transit depth withε=0 and ε=1 are correlated with the system parametersaccording to Equations (35) and (34), respectively. Thecorrelations are striking and enabling good order-of-magnitudeestimates of the possible bias. The proportionality factors are∼5 (Equation (35)) and ∼0.34 (Equation (34)) for observationsin the JWST-MIRI passband. The points that deviate moresignificantly from the linear correlations correspond to thecoldest planets, for which the Rayleigh–Jeans approximation istoo crude. However the predicted bias for these planets isbelow the detection level. The point below 10−7 is notmeaningful because it is comparable with the numericalprecision of ExoNoodle.

5.4. Polynomial Baseline Correction

We investigated whether the phase-blend effect leaves adetectable trace in the light-curve residuals of the simulatedobservations. We also experimented with the use of apolynomial instead of flat baseline multiplied by the transitsignal. This approach is equivalent to modeling the exoplanetphase-curve as if it were due to a modulation of the detectorsensitivity (Todorov et al. 2014; Morello 2018; Teachey &Kipping 2018; Teachey et al. 2020).In some cases, the variance of the residuals obtained with the

flat baseline can exceed the variance of the input noise by up to6%. Note that it is not rare to obtain even more excess red noisefrom the analysis of transits observed with the Spitzer SpaceTelescope and Hubble Space Telescope (HST), most likelydue to the imperfect removal of instrumental systematic effects(Ingalls et al. 2016; Tsiaras et al. 2018).

Figure 6. Predicted transit depth bias vs. ε for WASP12-b. The gray areas around 0 ppm denote the 1σ and the 3σ errors.

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The variance of the residuals obtained with the quadraticbaseline is less than or equal to that obtained with the flatbaseline, thanks to the two additional parameters. The Bayesianinformation criterion (BIC; Schwarz 1978) is commonlyadopted to select the best model to fit to the data:

( )s= +n k nBIC log log , 39res2

where s res2 is the variance of the residuals, n is the number of

data points, and k is the number of free model parameters. Thequadratic baseline has to be preferred only if it leads to a lowerBIC. Note that the second term in Equation (39) is a penalty forthe higher number of free model parameters.

Table 1 compares the results obtained with the flat andquadratic baseline for four simulations. The value of ΔBIC ispositive when the quadratic fit is preferred. It appears that thequadratic baseline significantly reduces the phase-blend bias, sothat the measured transit depth is mostly affected by the self-blendterm. The corresponding error bars are ∼50% larger than thoseobtained with the flat baseline, owing to parameter degeneracies.For the two cases with the largest phase-curve signature(WASP12-b and WASP19-b), the solution with the quadraticbaseline is favored by the BIC. Interestingly for WASP43-b thesolution with the flat baseline is preferred according to the BIC,despite it leading to a significant phase-blend bias at the level of

∼2.5σ. Also note that, even if the phase-blend bias is below1σ with a single visit, its significance may increase when takingthe weighted average of the best-fit results from multipleobservations.

6. Discussion

The Spitzer Space Telescope has observed a few dozens ofexoplanet phase-curves. The majority have been observed withthe InfraRed Array Camera (IRAC) using two broadband filtersat 3.6 and m4.5 m. The phase curves of three exoplanets havebeen observed with Spitzer/IRAC at m8.0 m. The HST hasobserved the spectroscopic phase-curves of at least threeexoplanets using the Wide Field Camera 3 (WFC3) in thewavelength range 1.1–1.7μm.Morello et al. (2019) report the first tentative detection of the

phase-blend effect in the infrared transits of WASP43-b. Inparticular, they measured systematically higher transit depthswhen analyzing transit-only observations extracted from threeSpitzer/IRAC phase-curves. In all cases, the transit depth valuewas increasing with the length of the out-of-transit baseline, asexpected in presence of a phase-blend. The detection was notconclusive as the offsets were comparable to the 1σ error bars.Based on the infrared phase-curves, the inferred circulation

efficiency of the exoplanet atmospheres covers a wide range of

Figure 7. Left panel: predicted transit depth bias with ε=0 vs. the phase-blend correlation factor. For the highlighted points (darker blue) the predicted bias is above3σ. The regression line with zero intercept is overplotted. Right panel: analogous plot for the transit depth bias with ε=1 vs. the self-blend correlation factor.

Table 1Comparison between Model Fits with Flat and Quadratic Baselines for Selected Simulations: Measured Transit Depth Bias and 1σ Error (in ppm) for the Two

Configurations, and D = -BIC BIC BICflat quadratic

Name Flat Quadratic ΔBIC Predicted

Bias 1σ Bias 1σ Self Phase Total

WASP12-b +164 19 −39 29 +58/+102 −47 +200 +152

WASP19-b +77 33 −73 47 +15/+7 −80 +172 +92

WASP43-b −11 25 −74 34 −8/−6 −79 +64 −15

WASP38-b −1 6 −3 9 −16/−17 −3 +2 −1

Note. The two ΔBIC values refer to the simulations with the noise added and subtracted, respectively (see Appendix A). The predicted self-blend, phase-blend, andtotal bias are also reported (see Equations (30) and (31)).

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values. For example, Schwartz et al. (2017) report valuesbetween ε=0.01 for the atmosphere of WASP18-b, andε=0.69 for that of HD189733-b. Schwartz & Cowan (2015)also report higher efficiencies up to ε=1 for some targets,solely based on their infrared eclipses and optical phase-curves.

Given the lack of a priori knowledge about the circulationefficiency in the exoplanet atmospheres, we recommendestimating the transit depth bias in the two extreme cases toevaluate whether it could be significant for a specific target.There are at least two circumstances in which this check can beimportant:

1. During the preparation of observing proposals, e.g., toavoid observations that contain only the transit if ignoringthe phase curve can introduce a significant bias.

2. In the analysis of transit photometric and spectroscopicobservations, e.g., to assess the risk of making wronginferences from the data.

In future work, we will explore the possible strategies tomitigate the bias when some information is available, e.g., ifthe eclipse has been observed but not necessarily the full phase-curve.

7. Conclusions

In this paper, we study the impact of the common dark planetapproximation in the analysis of transit light-curves. Ourapproach is twofold, as it is based on both mathematicalderivations and simulated observations.

We identify two terms that tend to bias the measured transitdepth in opposite directions due to neglecting the planetaryflux. The first term is associated with the dilution of theoccultation signal due to the presence of the additional fluxfrom the planet that acts as a blend, therefore named the self-blend effect by Kipping & Tinetti (2010). The second term isassociated with the variation of the planetary flux with theorbital phase during the observation, therefore named thephase-blend effect in this paper. The phase-blend effect was nottaken into account in the previous literature. However, we findthat both terms have the same order of magnitude, and thephase-blend effect can be much larger than the self-blend effectfor the most compact systems. However, the phase-blend signalcan be modeled with a smooth function of time, analogous tothe long-term instrumental systematic effects. In this way, thephase-blend bias in transit depth can be significantly reduced,the trade-off being a larger error bar owing to strongerparameter degeneracies. Note that the self-blend bias in transitdepth cannot be corrected solely based on the transitobservation.

We also introduce ExoNoodle, a python package thatgenerates time-series spectra of exoplanetary systems. We used

ExoNoodle to generate the synthetic light-curves of 17transiting exoplanets, integrated over the JWST-MIRI LRSpassband. The results of the synthetic data analysis were inexcellent agreement with the theoretical formulae derived inthis paper. We show that several of the known exoplanetsystems are likely to be affected by the self-blend and phase-blend effects in the future JWST-MIRI observations. Knowl-edge of the phase curve is necessary, in some cases, to avoidbiases >3σ in transit depth.We provide simple formulae to obtain order-of-magnitude

estimates of the self-blend and phase-blend effects based thesystem parameters. Such formulae are particularly accurate inthe mid-infrared, being optimized for the JWST-MIRI LRSpassband. We also developed more sophisticated tools that canprovide precise estimates of these effects at all wavelengths andfor any instrument passband. These tools will be available soonwithin the ExoTETHyS package.6 We expect these tools to beused both in the preparation of observing proposals and in thetransit data analysis, especially with the next-generationinstruments on board JWST and the Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL). Thecorrection of these effects is crucial to achieve the10–100 ppm in transit spectroscopy, also depending on thesystem architecture.

The authors would like to thank Camilla Danielski forproviding the PHOENIX stellar spectra used to generate thesynthetic observations. We would also like to thank the referee,David Kipping, for his useful comments that have improved themanuscript. The research leading to these results has receivedfunding from the European Unionʼs Horizon 2020 Researchand Innovation program, under Grant Agreement 776403, andthe LabEx P2IO, the French ANR contract 05-BLANNT09-573739. M.M.-L. is partly funded with a CNES scholarship.

Appendix ADemonstration of the Method to Eliminate the Noise

Randomness

In this appendix, we demonstrate the validity of ourapproach to average between two simulations with the addedand subtracted noise realization. Figure A1 shows the fittingresults to the light curve of WASP19 with 10 noise realizations,added and subtracted, leading to 20 configurations in total. Theindividual best-fit parameters for each configuration are spreadover large intervals comparable with their statistical error bars.Instead, the pair-averaged parameter results are essentiallycoincident, their spread being a small fraction of the respectiveerror bars. The average mid-transit shift is zero, as expected forthe light-curve symmetry.

6 https://github.com/ucl-exoplanets/ExoTETHyS/

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Appendix BParameters of the Planet Sample

This appendix contains the tables with all the parametersused to generate the simulated light-curves with ExoNoodle.Table B1 reports the planetary and orbital parameters. Table B2

reports the stellar parameters. Finally, Table B3 reports thecalculated dayside and nightside temperatures. The referencesof the values are reported with a number. All the figures that donot have any reference are calculated using these referencedvalues.

Figure A1. Best-fit parameters to the WASP19 light curves with 10 noise realizations, added and subtracted. The results from each pair are drawn with the same color.The thick black line is the mean of the 10 pair-averaged results (see Section 3.2); the gray area around this line contains all the pair-averaged results. The mean 1σerror bars of all the noise realizations are overplotted for comparison, as typical sigma.

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Table B1List of the Planets Studied in This Work, Along with the Parameters Used for the Planet

Planet

Name Mass (Mjup) Radius (Rjup) Period (day) SMA (au) Inclination (deg) Impact Param

55Cnc-e 0.0251 (12) 0.17 (12) 0.7365474 (12) 0.01544 83.59 (12) 0.393

GJ1214-b 0.0197 (13) 0.254 (13) 1.58040456 (13) 0.01411 88.17 (13) 0.449

HD189733-b 1.15 (1) 1.151 (1) 2.21857578 (1) 0.03141 85.78 (1) 0.661

HD209458-b 0.714 (1) 1.38 (1) 3.52474859 (1) 0.04745 86.59 (1) 0.522

WASP10-b 3.16 (1) 1.067 (1) 3.0927616 (1) 0.03777 88.81 (1) 0.240

WASP12-b 1.47 (6) 1.9 (6) 1.0914203 (6) 0.02339 83.37 (6) 0.351

WASP15-b 0.592 (11) 1.408 (11) 3.75209748 (11) 0.05163 85.74 (11) 0.542

WASP18-b 10.29 (1) 1.158 (1) 0.94145181 (1) 0.02028 85 (1) 0.311

WASP19-b 1.069 (7) 1.392 (7) 0.788838989 (7) 0.01615 78.78 (7) 0.673

WASP37-b 1.8 (5) 1.16 (5) 3.577469 (5) 0.04460 88.82 (5) 0.197

WASP38-b 2.691 (2) 1.094 (2) 6.871815 (2) 0.07522 88.69 (2) 0.278

WASP43-b 1.78 (3) 0.93 (3) 0.813475 (3) 0.01438 82.6 (3) 0.687

WASP73-b 1.88 (4) 1.16 (3) 4.08722 (3) 0.05515 87.4 (3) 0.260

WASP79-b 0.9 (8) 2.09 (8) 3.6623866 (8) 0.05346 83.3 (8) 0.702

WASP84-b 0.694 (10) 0.942 (10) 8.5234865 (10) 0.07710 88.368 (10) 0.631

WASP86-b 0.95 (9) 1.78 (9) 5.031623 (9) 0.06708 84.47 (9) 0.587

XO3-b 11.83 (1) 1.248 (1) 3.1915289 (1) 0.04515 84.89 (1) 0.614

Table B2List of the Planets Studied in This Work, Along with the Parameters Used for the Host Stars

Star

Name Distance (pc) Mass (MSun) Radius (RSun) Teff (K) Fe/H log(G) Tirr (K)

55Cnc-e 12.5901 (17) 0.905 (12) 0.943 (12) 5172 (12) 0.35 (12) 4.43 (12) 2756

GJ1214-b 14.6487 (17) 0.15 (13) 0.216 (13) 3026 (13) 0.39 4.94 (13) 807

HD189733-b 19.7752 (17) 0.84 (1) 0.752 (1) 5050 (1) −0.03 (1) 4.61 (1) 1685

HD209458-b 48.3688 (17) 1.148 (1) 1.162 (1) 6117 (1) 0.02 (1) 4.37 (1) 2064

WASP10-b 141.571 (17) 0.752 (1) 0.703 (1) 4675 (1) 0.03 (1) 4.62 (1) 1375

WASP12-b 432.589 (17) 1.434 (6) 1.657 (6) 6360 (6) 0.33 4.16 (6) 3650

WASP15-b 284.712 (17) 1.305 (11) 1.522 (11) 6405 (11) 0.00 (11) 4.19 (11) 2371

WASP18-b 123.925 (17) 1.256 (1) 1.222 (1) 6400 (1) 0.00 (1) 4.36 (1) 3387

WASP19-b 270.409 (17) 0.904 (7) 1.004 (7) 5568 (7) 0.15 (7) 4.45 (7) 2993

WASP37-b 397.093 (17) 0.925 (5) 1.003 (5) 5917 (16) −0.29 (16) 4.45 (16) 1913

WASP38-b 136.771 (17) 1.203 (2) 1.331 (2) 6436 (14) 0.06 (14) 4.80 (14) 1846

WASP43-b 86.9633 (17) 0.6 (3) 0.58 (3) 4798 (15) −0.13 (15) 4.55 (15) 2078

WASP73-b 319.57 (17) 1.34 (4) 2.07 (4) 6036 (4) 0.14 (4) 3.93 (4) 2521

WASP79-b 248.447 (17) 1.52 (8) 1.91 (8) 7002 (14) 0.03 (14) 4.20 (14) 2854

WASP84-b 100.878 (17) 0.842 (10) 0.748 (10) 5314 (10) 0.00 (10) 4.40 (10) 1129

WASP86-b 371.960 (17) 1.591 (9) 2.37 (9) 6279 (9) 0.19 (9) 3.89 (9) 2545

XO3-b 214.316 (17) 1.206 (1) 1.409 (1) 6429 (1) −0.18 (1) 4.22 (1) 2449

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(1) Southworth (2010), (2) Barros et al. (2011), (3) Hellieret al. (2011), (4) Delrez et al. (2014), (5) Simpson et al. (2010),(6) Collins et al. (2017), (7) Wong et al. (2016), (8) Smalleyet al. (2012), (9) Stevens et al. (2017), (10) Anderson et al.(2014), (11) Southworth et al. (2013), (12) Bourrier et al.(2018), (13) Harpsøe et al. (2012), (14) Mortier et al. (2013),(15) Sousa et al. (2018), (16) Andreasen et al. (2017), (17) GaiaCollaboration et al. (2018).

ORCID iDs

Marine Martin-Lagarde https://orcid.org/0000-0003-0523-7683Giuseppe Morello https://orcid.org/0000-0002-4262-5661Christophe Cossou https://orcid.org/0000-0001-5350-4796

References

Agol, E., Luger, R., & Foreman-Mackey, D. 2020, AJ, 159, 123Anderson, D. R., Cameron, A. C., Delrez, L., et al. 2014, MNRAS, 445, 1114Andreasen, D. T., Sousa, S. G., Tsantaki, M., et al. 2017, A&A, 600, A69Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, A&A, 577, A42Barnes, R. 2017, CeMDA, 129, 509Barros, S. C. C., Faedi, F., Collier Cameron, A., et al. 2011, A&A, 525, A54Batalha, N., Mandell, A., Pontoppidan, K., et al. 2017, PASP, 129, 064501Beaulieu, J. P., Tinetti, G., Kipping, D. M., et al. 2011, ApJ, 731, 16

Bell, T. J., & Cowan, N. B. 2018, ApJL, 857, L20Bolmont, E., & Mathis, S. 2016, CeMDA, 126, 275Bourrier, V., Dumusque, X., Dorn, C., et al. 2018, A&A, 619, A1Brown, T. M. 2001, ApJ, 553, 1006Burrows, A., Rauscher, E., Spiegel, D. S., & Menou, K. 2010, ApJ, 719, 341Charbonneau, D., Brown, T. M., Latham, D. W., & Mayor, M. 2000, ApJL,

529, L45Claret, A. 2000, A&A, 363, 1081Collins, K. A., Kielkopf, J. F., & Stassun, K. G. 2017, AJ, 153, 78Cowan, N., & Agol, E. 2011, ApJ, 729, 54Delrez, L., Van Grootel, V., Anderson, D. R., et al. 2014, A&A, 563, A143Eastman, J., Gaudi, B. S., & Agol, E. 2013, PASP, 125, 83Espinoza, N., & Jordán, A. 2016, MNRAS, 457, 3573Foreman-Mackey, D., Hogg, D., Lang, D., & Goodman, J. 2013, PASP,

125, 306Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1Gazak, J. Z., Johnson, J. A., Tonry, J., et al. 2012, AdAst, 2012, 697967Harpsøe, K. B. W., Hardis, S., Hinse, T. C., et al. 2012, A&A, 549, A10Hellier, C., Anderson, D. R., Collier Cameron, A., et al. 2011, A&A, 535, L7Henry, G. W., Marcy, G. W., Butler, R. P., & Vogt, S. S. 2000, ApJL, 529, L41Ingalls, J. G., Krick, J. E., Carey, S. J., et al. 2016, AJ, 152, 44Kendrew, S., Scheithauer, S., Bouchet, P., et al. 2015, PASP, 127, 623Kipping, D., & Tinetti, G. 2010, MNRAS, 407, 2589Kipping, D. M. 2008, MNRAS, 389, 1383Kipping, D. M. 2010, MNRAS, 407, 301Kreidberg, L. 2015, PASP, 127, 1161Lim, P. L., Diaz, R. I., & Laidler, V. 2015, PySynphot Userʼs Guide

(Baltimore, MD: STScI)Louden, T., & Kreidberg, L. 2018, MNRAS, 477, 2613Magic, Z., Chiavassa, A., Collet, R., & Asplund, M. 2015, A&A, 573, A90Mallonn, M., Köhler, J., Alexoudi, X., et al. 2019, A&A, 624, A62Mandel, E., & Agol, E. 2002, ApJL, 580, L171Martin-Lagarde, M., Cossou, C., & Gastaud, R. 2020, JOSS, doi:10.21105/

joss.02287Morello, G. 2018, AJ, 156, 175Morello, G., Claret, A., Martin-Lagarde, M., et al. 2020, AJ, 159, 75Morello, G., Danielski, C., Dickens, D., Tremblin, P., & Lagage, P. O. 2019,

AJ, 157, 205Morello, G., Tsiaras, A., Howarth, I. D., & Homeier, D. 2017, AJ, 154, 111Mortier, A., Santos, N. C., Sousa, S. G., et al. 2013, A&A, 558, A106Parmentier, V., & Crossfield, I. J. M. 2018, in Handbook of Exoplanets, ed.

H. Deeg & J. Belmonte (Cham: Springer), 116Parviainen, H. 2015, MNRAS, 450, 3233Schwartz, J. C., & Cowan, N. B. 2015, MNRAS, 449, 4192Schwartz, J. C., Kashner, Z., Jovmir, D., & Cowan, N. B. 2017, ApJ, 850, 154Schwarz, G. 1978, AnSta, 6, 461Seager, S., & Mallén-Ornelas, G. 2003, ApJ, 585, 1038Shporer, A., Collins, K. A., Astudillo-Defru, N., et al. 2020, ApJL, 890, L7Simpson, E. K., Faedi, F., Barros, S. C. C., et al. 2010, AJ, 141, 8Sing, D. K., Fortney, J. J., Nikolov, N., et al. 2016, Natur, 529, 59Smalley, B., Anderson, D. R., Collier-Cameron, A., et al. 2012, A&A,

547, A61Sousa, S. G., Adibekyan, V., Delgado-Mena, E., et al. 2018, A&A, 620,

A58Southworth, J. 2010, MNRAS, 408, 1689Southworth, J., Mancini, L., Browne, P., et al. 2013, MNRAS, 434, 1300Southworth, J., Maxted, P. F. L., & Smalley, B. 2004, MNRAS, 351, 1277Stevens, D. J., Collins, K. A., Gaudi, B. S., et al. 2017, AJ, 153, 178Teachey, A., Kipping, D., Burke, C. J., Angus, R., & Howard, A. W. 2020, AJ,

159, 142Teachey, A., & Kipping, D. M. 2018, SciA, 4, eaav1784Todorov, K. O., Deming, D., Burrows, A., & Grillmair, C. J. 2014, ApJ,

796, 100Tsiaras, A., Rocchetto, M., Waldmann, I. P., et al. 2016, ApJ, 820, 99Tsiaras, A., Waldmann, I. P., Zingales, T., et al. 2018, AJ, 155, 156Vanderspek, R., Huang, C. X., Vanderburg, A., et al. 2019, ApJL, 871, L24Winn, J. N. 2010, in Exoplanets, ed. S. Seager (Tucson, AZ: Univ. Arizona

Press), 55Wong, I., Knutson, H. A., Kataria, T., et al. 2016, ApJ, 823, 122

Table B3List of the Planets Studied in This Work, Along with the Parameters used for

the Planets Thermal Properties

Other Properties

Name Albedo Circulation Efficiency Tday (K) Tnight (K)

55Cnc-e 0.30 0.50 2074 1499

GJ1214-b 0.30 0.50 608 439

HD189733-b 0.30 0.50 1268 916

HD209458-b 0.30 0.50 1553 1123

WASP10-b 0.30 0.50 1035 748

WASP12-b 0.30 0.50 2747 1985

WASP15-b 0.30 0.50 1785 1290

WASP18-b 0.30 0.50 2549 1842

WASP19-b 0.30 0.50 2253 1628

WASP37-b 0.30 0.50 1440 1041

WASP38-b 0.30 0.50 1389 1004

WASP43-b 0.30 0.50 1564 1130

WASP73-b 0.30 0.50 1898 1371

WASP79-b 0.30 0.50 2148 1552

WASP84-b 0.30 0.50 849 614

WASP86-b 0.30 0.50 1915 1384

XO3-b 0.30 0.50 1843 1332

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