A Semi-Analytical Model of Visible-Wavelength Phase Curves of Exoplanets and Applications to Kepler-7 b and Kepler-10 b

1.2 MB·English

by  Renyu Hu

DRAFTVERSIONJANUARY19,2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ASEMI-ANALYTICALMODELOFVISIBLE-WAVELENGTHPHASECURVESOFEXOPLANETSAND
APPLICATIONSTOKEPLER-7BANDKEPLER-10B
RENYUHU1,2,∗,BRICE-OLIVIERDEMORY3,SARASEAGER4,5,NIKOLELEWIS4,+,ANDADAMP.SHOWMAN6
1JetPropulsionLaboratory,CaliforniaInstituteofTechnology,Pasadena,CA91109
2DivisionofGeologicalandPlanetarySciences,CaliforniaInstituteofTechnology,Pasadena,CA91125
3AstrophysicsGroup,CavendishLaboratory,J.J.ThomsonAvenue,CambridgeCB30HE,UK.
4DepartmentofEarth,AtmosphericandPlanetarySciences,MassachusettsInstituteofTechnology,Cambridge,MA02139
5 5DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139and
1 6DepartmentofPlanetarySciences,UniversityofArizona,Tucson,AZ85721
0 DraftversionJanuary19,2015
2
ABSTRACT
n
Keplerhasdetectednumerousexoplanettransitsbyprecisemeasurementsofstellarlightinasinglevisible-wavelengthband.
a
J Inadditiontodetection,theprecisephotometryprovidesphasecurvesofexoplanets,whichcanbeusedtostudythedynamicpro-
cessesontheseplanets. However,theinterpretationoftheseobservationscanbecomplicatedbythefactthatvisible-wavelength
6
phase curves can represent both thermal emission and scattering from the planets. Here we present a semi-analytical model
1
frameworkthatcanbeappliedtostudyKeplerandfuturevisible-wavelengthphasecurveobservationsofexoplanets.Themodel

] efficiently computesreflection and thermalemission componentsfor both rockyand gaseousplanets, consideringboth homo-
P geneousand inhomogeneoussurfaces or atmospheres. We analyze the phase curves of the gaseous planet Kepler-7 b and the
E rockyplanetKepler-10 b using the model. In general, we find that a hot exoplanet’svisible-wavelengthphase curvehaving a
. significantphaseoffsetcanusuallybeexplainedbytwoclassesofsolutions: oneclassrequiresathermalhotspotshiftedtoone
h
sideofthesubstellarpoint,andtheotherclassrequiresreflectivecloudsconcentratedonthesamesideofthesubstellarpoint.The
p
twosolutionswouldrequireverydifferentBondalbedostofitthesamephasecurve;atmosphericcirculationmodelsoreclipse
-
o observations at longer wavelengths can effectively rule out one class of solutions, and thus pinpoint the albedo of the planet,
r allowingdecompositionofthereflectionandthethermalemissioncomponentsinthephasecurve. ParticularlyforKepler-7b,
t
s reflective clouds located on the west side of the substellar point can best explain its phase curve. We further derive that the
a reflectivityoftheclearpartoftheatmosphereshouldbelessthan7%andthatofthecloudypartshouldbegreaterthan80%,and
[ thatthecloudboundaryislocatedat11±3degreetothewestofthesubstellarpoint. ForKepler-10b,thephasecurvedoesnot

showasignificantphaseoffset,andanymodelwithaBondalbedogreaterthan0.8wouldprovideanadequatefit. We suggest
1
single-bandphotometrysurveyscouldyieldvaluableinformationonexoplanetatmospheresandsurfaces.
v
6 Subjectheadings: radiativetransfer—atmosphericeffects—planetarysystems—techniques: photometric
7 —planets:Kepler-10b—planets:Kepler-7b
8
3
0 1. INTRODUCTION al. 2013; Heng& Demory2013; Demoryetal. 2013). The
1. A great number of exoplanets have been discovered by short-wavelengthwingofthermalemissionmayextendtothe
visible wavelengths and affect the phase curve if the planet
0 precise photometry. NASA’s Kepler spacecraft, monitor-
is hot enough (e.g., Rouan et al. 2011). It is therefore use-
5 ing 160,000 stars in the sky, discovered that more than half
fultolearnwhetheraphasecurveofcombinedreflectionand
1 of the stars should host planets smaller than Neptune (e.g.,
thermalemission canplaceconstraintsonthe atmosphereor
: Fressinetal. 2013;Howard2013).TheCHaracterizingExO-
v surfaceofanexoplanet.
PlanetSatellite(CHEOPS),theTransitExoplanetSkySurvey
i Thephasecurveatvisiblewavelengthshasbeenapowerful
X (TESS),andthePLAnetaryTransitsandOscillationsofstars
diagnostictooltocharacterizeSolar-Systembodies,amethod
(PLATO), designed to search for exoplanets around nearby
r orthogonaltoanalyzingspectralfeatures. Thecenter-to-limb
a brightstarsusingthesametechniqueasKepler,haverecently
variationofthereflectivitycontinuumandmajorspectralfea-
beenselected by ESA and NASA for launchwithinthe next
turesofJupiterhasyieldeddetailedinformationontheloca-
decade.Theprecisemeasurementsoflightcurvesofstarsina
tion and layering of its clouds (Sato & Hanson 1979). The
singlevisible-wavelengthbandwillcontinuetobeadominant
phase curve of Venus alone has provided strong constraints
wayto detectexoplanets,especiallyrockyexoplanets,in the
ontherefractiveindexofitscloudparticles(Arking&Potter
comingyears.
1968), which were later improved to effectively only allow
Beyond planet detection, Kepler and future transiting ex-
sphericalsulfateparticlesbymeasuringpolarization(Hansen
oplanetsearch missions may also providevaluable informa-
&Hovenier1974). Thephotometriclightcurvesofasteroids
tion on the nature of exoplanets via measuring the planets’
havebeenusedextensivelytostudytheirshapeandrotational
phasecurves. Thephasecurvesreveallongitudinalinforma-
properties(e.g.,Torppaetal. 2003).
tionregardingtheplanets’atmosphereorsurface. Atvisible
For exoplanets, the phase curve characterization has been
wavelengths, a phase curve could illustrate how a planet re-
mostly limited to mid-infrared wavelengths due to greater
flects stellar light and provide an effective way to study the
planet-to-star flux ratios than at visible wavelengths (e.g.,
condensed-phase particles in the planet’s atmosphere or the
Knutsonetal.,2007,2009a,b,2012;Cowanetal.2007,2012;
planet’ssurface(e.g.,Madhusudhan&Burrows2012;Huet
Crossfieldetal. 2010;Lewisetal. 2013;Maxtedetal. 2013;
∗HubbleFellow+SaganFellowEmail:[e-mail]eld..2014). Manymid-infraredphasecurvesofex-
2 Huetal.
oplanetsshowphasepeaksbeforesecondaryeclipses, which withoutatmospheres, and they differ by whether the surface
havebeeninterpretedasevidenceforeastwarddisplacedhot ishomogeneous. Inthe “homogenoussurface” scenario,the
spots driven by super-rotating planet-encircling jet streams planet is covered by a solid surface or a molten lava ocean.
in the atmospheres(e.g., Showman et al. 2008, 2009). The In the “inhomogeneous surface” scenario, the planet’s sur-
visible-wavelengthphase curveshavebeenobservedforone face contains patches of differing albedo and/or thermal in-
Jupiter-sized exoplanet before Kepler (Snellen et al. 2009), ertia. One example of the inhomogeneoussurface scenarios
andrecentlyforanumberofexoplanetswithKeplerdata(e.g., isthatonlypartsoftheplanet’sdaysidearemolten,forming
Demoryetal. 2013;Estevesetal. 2014). alavalake(e.g.,Légeretal. 2011). Themoltensurfaceand
Theinterpretationofvisible-wavelengthphasecurveswith- thesolidsurfacemayhavedifferentreflectivities. Akeyfea-
outanyspectralinformation,asisthecaseforKeplerobserva- turethatmakesthisscenariorelevantisthatthelavalakedoes
tionsandalsoexpectedforfuturespacephotometrymissions, notneedtoextendsymmetricallywithrespecttothesubstel-
arelikelytobecomplicatedbythefollowingtwofactors: (1) larpoint. Thisisan analogofa recentsimulationontheice
avisible-wavelengthphasecurvethatcontainsreflectedlight coverageoftidally-lockedoceanplanetsaroundMdwarfstars
willdependonthespatialdistributionofcloudsontheplanet; (Hu&Yang2014). Ifamoltensurfacehasalowerreflectiv-
(2) a visible-wavelength phase curve may contain both re- ity than a solid surface of the same material, an asymmetric
flection and thermal emission from the planet, if the planet lava lake may lead to a phase shift via reflection or thermal
ishighlyirradiated. emission.
To interpret the visible-wavelength phase curves of exo- The third and fourth scenarios are for planets with at-
planets,weconstructasimplesemi-analyticalmodelthatcan mospheres, and they differ by whether the atmosphere has
beappliedtobothKeplerandfuturevisible-wavelengthphase patchy clouds. The distribution of the condensate particles,
curve observationsof exoplanets. Models at differentlevels controlledby atmosphericcirculation(e.g., Parmentieret al.
of sophistication have been developed to study atmospheric 2013),maybeinhomogeneous,e.g.,asymmetricwithrespect
circulation on exoplanets and interpret their thermal phase tothesubstellarpoint.Thepatchycloudscenarioismotivated
curves(e.g.,Showman&Guillot2002;Showmanetal.2008, by a similar scenario proposed to explain a post-occultation
2009;Rauscher&Menou2010,2012;Castan&Menou2011; phase shift for Kepler-7 b (Demory et al. 2013). A cloudy
Wordsworth et al. 2011; Heng et al. 2011a, b; Cowan & patch of the atmosphere should have differentalbedo than a
Agol2011;Heng&Kopparla2012;Pernaetal. 2012;Show- clearpatchoftheatmosphere–typicallybrighter–therefore,
man & Kaspi 2013; Showman et al. 2013; Dobbs-Dixon patchyclouds can induce phase modulationat visible wave-
& Agol 2013; Parmentier et al. 2013; Mayne et al. 2014; lengths.Wedonotseparateplanetshavingthickatmospheres
Rauscher&Kempton2014;Katariaetal. 2014;Showmanet andplanetshavingthinatmospheresinthemodel,becausewe
al. 2014). However,becausetheKeplerobservationscontain useaheatredistributionefficiency(seeSection3),ratherthan
only a single band, it is impractical to constrain the compo- thesurfacepressureorthewindspeed,astheprimarymodel
sition of the planet’s atmosphere or surface. We approach parameter.
the problem based on a physically motivated parameteriza-
tion: instead of studyingthe detailedphysicalprocesses, we 3. AGENERALMODELFORANALYZINGVISIBLE-WAVELENGTH
PHASECURVES
trytoconstrainseveraloverarchingphysicalparametersfrom
theobservations.Thiswaywecancompareplanetaryscenar- Weaimtoconstructageneralmodeltodescribethephase
ios and shed light on generalquestionssuch as, whether the curve of an exoplanet observed at visible wavelengths. The
planethasanatmosphere,andwhethertheatmospherehasa phasecurvemeasuresFP/FSasafunctionoftheorbitalphase,
standingcirculationpatternand/orpatchyclouds. whereFPisthefluxoftheplanetandFSistheemissionfluxof
Thispaperisorganizedasfollows. Webrieflyoutlinepos- theparentstar. Theplanet’sfluxcomesfromdisk-integrated
sible planetary scenarios important for phase curves in § 2, reflectionandthermalemission,as
and then describe our semi-analyticalmodelfor interpreting F =F +F , (1)
P R T
the visible-wavelength phase curve of exoplanets in § 3. In
§4weapplyourinterpretationtooltostudythephasecurve whereFRisthefluxofstellarlightreflectedbytheplanetand
ofthegaseousplanetKepler-7b,andin§5weapplyourin- FT is the thermalemission flux of the planet. The reflection
terpretationtooltostudythephasecurveoftherockyplanet componentdependsonthealbedooftheplanet,andthether-
Kepler-10b.Wediscussmodeldegeneraciesandsuggesthow mal emission component depends on the efficiency of heat
they could be addressed with additional observations in § 6 redistribution,anda potentialgreenhouseeffectif the planet
andconcludein§7. hasanatmosphere.
Themodelisdesignedforplanetsthathavecircularorbits.
2. PLANETARYSCENARIOS Theplanetarysystemsforwhichthemodelwouldbeapplied
To make the modelapplicablefor bothgaseousandrocky towouldhavetheirprimaryandsecondarytransitswellmea-
exoplanets,we considerthefollowingfourplanetaryscenar- sured.Thetiminganddurationofthetransitsconstraintheor-
ios (see Figure 1 for a schematic illustration). These four bitaleccentricity(Kallrath&Milone1999;Barnes2007;Ford
scenarios are separated by whether a planet has an atmo- etal. 2008;Kipping2008; Demoryetal. 2011). Therefore,
sphere, and whether a planet has a homogeneous reflecting onecanconfirmaplanettohaveacircularorbitbeforeapply-
layer. Whether a planet has an atmosphere is important be- ingthemodel.MostKeplerplanetsthathavemeasurementof
cause the atmospherecould cause the greenhouseeffectand the secondary eclipse depths have circular orbits (Esteves et
raisetheemittingtemperatureoftheplanet.Whetheraplanet al. 2014).
hasahomogeneousreflectinglayerisalsoimportantbecause
3.1. ReflectionComponent
an inhomogeneousreflecting layer may cause a phase curve
offset. We model the reflection component in the following two
The first and second scenarios are for bare-rock planets ways. The first possibility is that the reflection component
VisiblePhaseCurvesofExoplanets 3
Homogeneous Surface Inhomogeneous Surface Homogeneous Atmosphere Patchy Cloud
Figure1. Schematicillustrationofthefourplanetaryscenariosimportantforvisible-wavelengthphasecurves. Thefigureshowstheplanet’sdayside,andthe
starmarkindicatesthelocationofthesubstellarpoint. Inthehomogeneoussurfacescenario,theplanetiscoveredbyasolidsurfaceoramoltenlavaocean,
anddoesnothaveanatmosphere. Intheinhomogeneoussurfacescenario,theplanethasasurfacecontainingpatchesofdifferingalbedoand/orthermalinertia,
anddoesnothaveanatmosphere. Oneexampleoftheinhomogeneoussurfacescenarios,asdepictedonthefigure,isthatafractionofthedaysideiscovered
bymoltenlava(darkred)andtherestiscoveredbysolidsurface(lightred). Thesolidpartcanbemorereflectivethanthemoltenpart. Inthehomogeneous
atmospherescenario,theplanethasanatmospherefreeofcloudsorfullycoveredbyclouds.Inthepatchycloudscenario,afractionofthedaysideiscloud-free
(darkgrey)andtherestiscloudy(lightgrey).Thecloudypartismorereflectivethanthecloud-freepart.
is symmetric with respect to the occultation. This corre- range having low reflectivity, and then the local longitude
spondstothe“homogeneoussurface”and“homogeneousat- ranges having high reflectivity are [- π/2,ξ ] and [ξ ,π/2].
1 2
mosphere”scenarios.Thesecondpossibilityisthatthereflec- Thereflectioncomponentis
tioncomponentisasymmetricwithrespecttotheoccultation.
Thiscorrespondstoaplanetwithdifferentreflectivitiesatdif- FA=FS(2r /3)+F RP 22r1 2 cos(α- φ)cos(φ)dφ,
ferentlongitudesduetoheterogeneoussurfaceoratmosphere R R 0 S(cid:18) a (cid:19) 3 πZ
solid
(i.e., the “inhomogeneous surface” and “patchy cloud" sce- (3)
narios). in which the first term is the Lambertian phase curve, char-
For symmetric reflection, we approximate the reflection acterizedbyr ,andthesecondtermcontainsintegrationover
0
componentby the phase curve of a Lambertian sphere (i.e., thehigh-reflectivitylongitudesvisibletoobservers,character-
all incident photons are isotropically scattered), scaled by a izedbyr . Theintegrationisexpressedintheobserver’slon-
1
uniformgeometricalbedo(Ag)oftheplanet,namely gitude(φ) suchthatthe observeris atthe directionofφ=0.
Therelationshipbetweentheobserver’slongitudeandthelo-
FS(A )=F RP 2A 1[sin|α|+(π- |α|)cos|α|], (2) callongitudeis ξ≡φ- α. Theexactexpressionforthesec-
R g S(cid:18) a (cid:19) gπ ondtermasafunctionofξ1andξ2isexplicitlycalculatedand
giveninAppendixA.
whereαisthephaseangleoftheplanetthatrangesfrom- π
toπ (α=0isatoccultationandα=π isattransit),R isthe
P 3.2. ThermalEmissionComponent
planet’sradius,andaisthesemi-majoraxis. TheLambertian
approximationissufficientforcurrentinvestigationbecauseit WecalculatethethermalemissioncomponentintheKepler
isverydifficulttoextractanon-Lambertiancomponentfrom bandbymulti-colorblackbodyemissionbasedonalongitudi-
asymmetricphasecurve(Seageretal.2000;Madhusudhan& naldistributionoftemperature.Thethermalemissionphotons
Burrows2012). Iftheplanethasahomogeneouslyreflecting canbeeitherfromtheplanet’ssurface,orfromacertainpres-
atmosphere,theshapeofitsreflectionphasecurvewillclosely surelevelintheplanet’satmosphere,whichwebroadlyrefer
resembletheLambertianphasecurve(Seageretal. 2000;Ca- toasthe“photosphere”.Thephasedependencyoftheplanet’s
hoy et al. 2010). For an airless rocky planet, we have used thermalemissioniscomputedby
theHapkeplanetaryregolithreflectionmodeltocomputethe π π
shapeofitsphasecurvebasedonthemethoddescribedinHu F =R2 2 2 B [T(α,θ,φ)]cos2θcosφdθdφ, (4)
T P K
etal.(2012),andconfirmedthattheresultingphasecurvewill Z- π Z- π
2 2
besufficientlyapproximatedbyaLambertianone,aslongas
where B is the Planck function integrated over Kepler’s
theparticlesizeoftheregolithislessthan100µm. K
bandpass,andthecoordinatesarespecifiedbyobserver’slati-
For asymmetricreflection we considersome longitudesto
tudeandlongitude(θ,φ),suchthattheobserverisatthedirec-
haveahighreflectivityandsomelongitudestohavealowre-
tionof(θ=0,φ=0).Thethermalemissioncomponentateach
flectivity.Onecouldenvisionalavalakepicture–themolten
phaseangleαiscontrolledbythecorrespondingtemperature
surface has a lower reflectivity than the solid surface. One
distributionT(θ,φ).
could also envision a “hole-in-a-cloud” picture – the atmo-
The temperature distribution is determined by interaction
sphere is clear and poorlyreflective at some longitudesnear
betweenirradiation,heatredistribution,andradiativecooling.
the substellar pointand cloudyandhighlyreflective atother
Atmospheric circulation models have been developedfor ir-
longitudes. We denotethelowreflectivityasr andthehigh
reflectivity as r +r . Here a positive value for0r represents radiated gas giants (e.g., Showman & Guillot 2002; Show-
0 1 1 manetal. 2009;Rauscher&Menou2010,2012;Hengetal.
theincreaseinreflectivitycausedbysurfacefreezingorcloud
2011a, b; Perna et al. 2012; Showman et al. 2013; Dobbs-
formation. We further define [ξ ,ξ ] as the local longitude1
1 2 Dixon&Agol2013;Mayneetal. 2014;Rauscher&Kemp-
1Thelocallongitudeisdefinedsuchthatthesubstellarmeridianisatthe ton2014;Showmanetal.2014)andsuperEarths(e.g.,Castan
longitudeofzero,thedawnterminatorisat- π/2,andtheduskterminatoris & Menou2011; Wordsworth et al. 2011; Heng & Kopparla
atπ/2.The“dawn”and“dusk”aredefinedforaplanetofprograderotation. 2012;Katariaetal. 2014). Hereweaimatfastcalculationof
4 Huetal.
thetemperaturedistributionthatwouldenableparameterex- ontheday-nighttemperaturepattern(Perez-Becker&Show-
ploration. Therefore,we adoptthe semi-analyticalmodelof man2013). Inourabovetoymodel,noneofthesedynamical
Cowan&Agol(2011),inwhichtheatmosphereismimicked processesare considered;ω shouldsimplybeviewedasa
adv
byarigidlyrotating“photosphere”subjecttoirradiationand proxyfordynamicaladjustmentoftheatmospherictempera-
radiativecooling2. Thetemperaturedistributioniswrittenas turestructure,bywhatevermechanism.
The scaling factor f in Equation (5) is included as a free
T(α,θ,φ)= fT (θ)P(ǫ,ξ), (5)
0 parameterto mimic a possible greenhouseeffectwhen there
inwhichT isthetemperatureofthesub-stellarmeridian,P is an atmosphere. f = 1 corresponds to the homogeneous
0
is the thermalphase function that only dependson the local surfaceandinhomogeneoussurfacescenarioswithoutgreen-
longitude and a heat redistribution efficiency (ǫ), and f is a houseeffects,and f >1correspondstotheplanetthathasan
scalingfactortoaccountforapossiblegreenhouseeffect.Pis atmospherewith infrared-absorbingmolecules. Such a scal-
computedbysolvingEquation(10)inCowan&Agol(2011), ingparameterisnecessarybecausetherecouldbeinfraredab-
sorbersintheatmosphere,suchasCOandCO ,whichdonot
2
dP = 1(max(cosξ,0)- P4), (6) contributesignificantlytotheopacityatvisiblewavelengths.
dξ ǫ Inotherwords, Keplercouldpotentiallyprobea deeper,and
presumablywarmer,layerontheplanetcomparedtothelayer
andthesub-stellartemperatureis
thatreflectsstellarlight.Usingauniformscalingfactortoac-
countfor the possible greenhouseeffect is of course coarse,
R 1/2
T =T S (1- A )1/4cos(θ)1/4, (7) astheradiativetransferprocessesintheatmospherecouldbe
0 S B
(cid:18) a (cid:19) quite different from the dayside to the nightside (e.g., Bur-
rows et al. 2008). However, without knowingthe details of
whereT andR are the stellar effectivetemperatureand ra-
S S the atmosphericcomposition, a scaling factor would be best
dius,respectively,andA istheBondalbedooftheplanet.
B suitedforthepurposeofthismodel.
In this toy model, the heat redistribution efficiency is de-
finedastheratiobetweentheradiativetimescaleandthead-
vectivetimescale,i.e., 3.3. LinkingReflectionandEmissionComponents
Atthispointwecanlinkthelongitudinalvariationofreflec-
ǫ=τ ω , (8)
rad adv tivitytothelongitudinalvariationoftemperature.Weassume
where τ is the radiative timescale of the photosphere, and thatthepatternforlavaorcloudsfollowsalongitudinaldistri-
rad
the advective frequency ω is ω ≡ ω - ω . butioncontrolledbythelocaltemperatureofthesurfaceorthe
adv adv photosphere orbit
ω and ω are the angular velocities of the photo- atmosphere(seeAppendixBforjustification). Weintroduce
photosphere orbit
sphereandthebulkpartoftheplanetintheinertialframeof a single, physically motivatedparameterT, called “conden-
c
reference,respectively. When|ǫ|≫1thelongitudinalvaria- sationtemperature”inthefollowing,todescribethefreezing
tionoftemperaturewillbesmallasheatredistributionismuch temperatureof the molten lava, or the condensationtemper-
moreefficientthanradiativecooling;when|ǫ|≪1theplanet ature of the condensable species in the atmosphere. When
willbeinlocalthermalequilibriumandtheday-nightcontrast T0P(ǫ,ξ)> Tc, the surface is molten, or the atmosphere is
will be large. The sign of ǫ, inheritedfromthe sign of ωadv, cloud free. When T0P(ǫ,ξ)<Tc, the surface is solid, or the
indicates the direction of the equatorial jets and the thermal atmosphereiscloudy. T P(ǫ,ξ)=T definesthe longitudinal
0 c
phaseshift:whenǫ>0thephotosphereissuper-rotatingwith boundariesofthelavalakeortheholeinthecloud,i.e.,ξ1and
respecttotheplanet’sorbit,andthethermalphaseshiftiseast- ξ . Wedropthecos(θ)1/4terminT (Equation7)todetermine
2 0
ward(i.e., the peakof thethermalphasecurveappearsprior ξ andξ , becauseanyphasecurvesignalwillbedominated
1 2
totheoccultation,asinthecaseofhotJupiterHD189733b; bythe atmosphericpropertiesnearthe equatorialregionthat
e.g., Knutson et al. 2007); when ǫ< 0 the photosphere is cos(θ)∼1.
sub-rotatingwithrespecttotheplanet’sorbit,andthethermal Puttingthesepiecestogether,wefindthatforthesymmet-
phase shift is westward (i.e., the peak of the thermal phase ricreflectionscenarios(i.e.,lavaplanetandhomogeneousat-
curveappearsaftertheoccultation).Ifthephotospherecanbe mosphere),themodelisfullyspecifiedbythreeindependent
treated as a single atmosphericlayer, the radiative timescale parameters: the Bond albedo of the planet(A ), the heat re-
B
wouldbe distributionefficiency (ǫ), and the greenhousefactor f. The
cpP geometricalbedoandtheBondalbedoarelinkedbythephase
τ = , (9)
rad gσT3 integral;andsinceweworkwiththeLambertsphereassump-
0
tion in this paper, A = 2A /3. For the asymmetric reflec-
g B
where c is the heat capacity of the atmosphere, P is the
p tionscenarios(i.e., homogeneoussurfaceandpatchycloud),
pressure of the thermalemission photosphere, g is the grav-
themodelneedstwoadditionalparameters:thecondensation
itational acceleration of the planet, and σ is the Stephan- temperature (T), and a reflectivity boosting factor (κ). The
c
Boltzmannconstant(Showman&Guillot2002).
latterisdefinedas
Note that dynamical timescales other than the horizontal
r ≡κr . (10)
advectiontimescalemaybeimportantincontrollingthetem- 1 0
perature distribution on hot Jupiters; for example, in some
A andκaretwoindependentparameters,fromwhichr and
cases, the vertical advection timescales and horizontal grav- B 0
r canbecalculatedas
itywavepropagationtimescalesexertacontrollinginfluence 1
A
2AsthethermalemissionradiationintheKeplerbandwouldalmosten- r0= 1+2Bq′κ,
tirelycomefromtheequatorialregionontheplanet,arotatingphotosphere 3
issufficienttodescribethelongitudinalvariationofthephotospheretemper- κA
ature. r = B , (11)
1 1+2q′κ
3
VisiblePhaseCurvesofExoplanets 5
whereq′ is the phaseintegralgivenin AppendixA and Fig- Total =
Symmetric Reflection+Asymmetric Reflection+Thermal Emission
ure7. Forphysicalplausibility,weverifyeachsimulationto
ensurer +r ≤1,andrejectallattemptedmodelsthatdonot Homogeneous Atmosphere Kepler-7 b
0 1 150
satisfythiscriterion.
Tosummarize,ourmodelcomputesthecombinedreflection 100
and thermal emission from an irradiated exoplanetobserved 6]
−
at visible wavelengths, based on three (assuming symmetric 10 50
reflection) or five (assuming asymmetric reflection) parame- F [S
tersthatdescribephysicalprocessesontheplanet.Ourmodel / P 0
F
coversawiderangeofpotentialscenariosforexoplanets,with
−50
orwithoutanatmosphere.
3.4. FittingtoLightCurves −100
Thesemi-analyticalmodelcanbeusedtofitobservedlight Patchy Cloud Kepler-7 b
150
curves. Typically, we do notinclude the primarytransit, as-
sumingthatithasbeenusedtoderivedkeyplanetaryandor-
100
bitalparameters,butwedoincludethesecondaryoccultation
6]
in the fit. We use the model of Mandel & Agol (2002) for −10 50
theshape ofingressandegressofthe eclipse. Thisway, the [S
F
phase curve characteristics including the occultation depth, / P 0
thephaseamplitude,andthephaseoffsetcanbedirectlyde- F
rivedfromourfittingresults. −50
Since ourmodelcomputesphase curvesveryfast, we can
−100
use the Markov-ChainMonte Carlo(MCMC) methodto ex-
plore the parameter space and determine the posterior pa- −3 −2 −1 0 1 2 3
Phase Angle
rameter distribution of the three or five parameters from the
phasecurveobservations. Inpractice,wecalculate2Markov Figure2. Thebest-fitmodelsforthephasecurveofKepler-7b. Thema-
gentalinesarethemodeledphasecurves,andtheotherlinesshowcontribu-
chains, each containing 1 million steps, for each planetary
tionofthermalemission(red),symmetricreflection(green),andasymmetric
scenario,withtheMCMCmethodimplementedasinHaario reflection(blue). Theupper-panelshowsthebest-fitmodelforaplanethav-
etal. (2006). Thisnumberof stepsis sufficientfor the con- ingahomogeneousatmosphere,andthemodeledphasecurvehassignificant
vergenceoftheMarkovchainsforallparametersinthecases contributionfromathermalemissioncomponentcharacterizedbyahotspot
shifted westward. Bydefinition inthis scenario the asymmetric reflection
studied here, and we validate the convergence by compar-
component iszero. Thelower-panel shows thebest-fitmodelforaplanet
ingthetwochainsusingthestandardGelman-Rubinstatistics havinganatmospherewithpatchyclouds,andthemodeledphasecurvehas
(R<1.01 for all parameters; Gelman & Rubin 1992). The significantcontributionfromtheasymmetricreflectioncomponentattributed
firsthalfofeachchainisconsideredthe“burn-in”periodand totheclouds.Thecloudsconcentrateonthewestsideofthesubstellarpoint,
drivenbyahotspotshiftedeastward.
removedfromthefinalresults. Thephysicallyallowedranges
and the prior distributions of the model parameters are: A
B
uniformlyrangesin[0,1];ǫuniformlyrangesin[- ∞,∞]; f effects can be effectively removed for transiting planets be-
uniformlyrangesin[1,2];T uniformlyrangesin[200,3000] causerelevantorbitalparametersareknownfromthetransits.
c
K; and κ uniformly ranges in [0,∞]. The actual ranges of Theinteractionbetweentheseeffectsandtheatmosphericsig-
parametersusedinthefitstospecificplanetaryscenariosmay natureswill be discussed in a separate paper(Shporer& Hu
benarrowerthanthesegeneralranges. 2015).
Ourmodeldoesnotrequireknowledgeofthestellarradius.
With our formulation,the planetaryphase curve(F /F ) de- 4. APPLICATIONSTOKEPLER-7B
P S
pendsonR /R anda/R ,bothofwhichareuniquelyderived Wenowapplyourmodelframeworktoanalyzethevisible-
P S S
fromtheprimarytransit(Seager&Mallen-Ornellas2003).In wavelengthphasecurvesofexoplanets. Keplerhasprovided
practice, the precision for the measurements of R /R and data to derivevisible-wavelengthphase curvesfor a number
P S
a/R is on the order of 1% for Jupiter-sized planets whose ofJupiter-sizedexoplanets,andamongtheseplanets,Kepler-
S
transits and eclipses are detected by Kepler (e.g., Esteves et 7bhasthebestsignal-to-noiseratiosfortheoccultationdepth
al. 2014), and the precisionfor the super-Earth-sizedplanet andthephaseoffset(Demoryetal.2013;Estevesetal.2014).
Kepler-10 b is better than ∼ 3% (Batalha et al. 2011). It Kepler-7bhasacircularorbitwithanorbitaleccentricityless
is therefore legitimate to not propagate the uncertainties in than 0.02, the 3- σ upper limit derived from the light curve
R /R and a/R to the fitted parameters, when the uncer- (Demoryetal. 2011).Asanexample,weapplyourmodelto
P S S
taintiesofthefittedparametersaremuchgreaterthan∼1%. analyzethephasecurveofKepler-7b.Weusethephasecurve
Themodelalso dependson the stellar effectivetemperature. dataofDemoryetal. (2013)inthisstudy.Tofocusontesting
The stellar temperatureis not cancelled out because of non- what we could learn from visible-wavelength phase curves,
lineardependencyofthePlanckfunctiononthetemperature. weonlyusethephasecurveforthefit,andthenconsiderthe
For well-characterized Kepler stars, such as Kepler-7 b and constraintsoftheoccultationdepthsmeasuredatlongerwave-
Kepler-10 b, the precision on the stellar temperature is well lengthsbyDemoryetal.(2013).WeshowourresultsinTable
within1%. Wehavetestedourmodelsandfound1%change 1andFigures2,3,and4.
inthestellartemperaturewouldproducenegligiblechangein Abimodaldistributionforthefittedandderivedparameters
thephasecurve. emerge when we apply the model framework to explain the
Inthispaperwedonotexplicitlytreatthedopplerbeaming phase curve of Kepler-7 b. The key feature of the observed
or ellipsoidaleffectsto the phasecurve, assumingthatthese phasecurveisthatthepeakofplanetarylightoccursafterthe
6 Huetal.
Table1
EstimationofparametersforthegaseousplanetKepler-7b,basedonfittingtotheobservedphasecurve.WeusethephotometryderivedbyDemoryetal.
(2013).Inthegeneralfit,allfiveparametersareallowedtovaryintheirphysicallyplausibleranges.Inthehomogeneousatmospherefit,theasymmetric
reflectioncomponentisnotincludedinthecalculation,andthenthemodelnolongerdependsonthecloudcondensationtemperatureorthecloudreflection
boostingfactor.Inthepatchycloudfit,wesetǫ>0,assumingtheplanettohavesuper-rotatingequatorialwinds.Undersuchassumption,patchycloudsare
requiredtoexplaintheobservedphasecurve(seethetext).
Parameter General HomogeneousAtmosphere PatchyCloud
FittedParameters
AB 0.30+- 00..1135 0.18±0.03 0.42±0.01
ǫ - 3.3∼62 - 3.2±0.5 20∼82
f 1.19+- 00..0156 1.23±0.02 1.08+- 00..0085
Tc(K) 1476∼2604 - 1480±10
κ 14∼82 - 13∼69
DerivedParameters
OccultationDepth(ppm) 41.5+- 33..92 44.5±2.0 39.0+- 11..86
PhaseAmplitude(ppm) 48.0±2.1 49.4±1.4 46.6±1.3
PhaseOffset(degree) 37.2+- 23..60 38.0+- 22..46 36.2±2.7
r0 0.003∼0.049 - 0.014∼0.072
r1 0.15∼0.94 - 0.92±0.04
ξ1(degree) - 13.8∼0 - - 11.2±2.7
ξ2(degree) 0∼90 - 90
FitQuality
Minimumχ2/dof 0.992 0.991 1.028
BIC 1340.8 1327.1 1387.1
secondary occultation, i.e., a post-eclipse phase offset. Our ilarbimodaldistributionisalsofoundfortheposteriorofthe
modelcangeneratethisoffsetandprovideasatisfactoryfitto phase amplitude, but the two peaks differ to a lesser extent.
theobservation(Figure2). Whenallparametersareallowed This difference identified here highlights the importance to
to vary in their physically plausible ranges (i.e., the general use appropriate models for the phase curve, even when the
fit), the posteriordistributionshowstwo clearlyseparate pa- occultationdepthbearsmost interest. A detailed lookat the
rameterspacesthatproducethefittotheobservation(Figure best-fit model phase curve (Figure 2) would reveal that the
3and4). Thebimodaldistributionisespeciallyapparentfor homogeneousatmospheremodelhas morethermalemission
the Bondalbedo,which cantake valuesaroundeither 0.2or contributionthanthepatchycloudmodel,andthentheonset
0.4. Suchabimodaldistributionindicatesthattwoclassesof oftheplanetarylightissmootherastheplanetrotatesfromthe
modelscanbeconsistentwiththeobservedphasecurve. nightsidetothe dayside. Tofit thephasecurvethatcontains
Thetwoclassesofmodelscorrespondtothehomogeneous thesecondaryoccultation,themodelautomaticallyadjustthe
atmosphere scenario and the patchy cloud scenario, respec- “zero” point of the planetary flux to seek the minimum χ2,
tively. To separate the two classes, we additionally perform whichaffectsthedeterminationoftheoccultationdepth.
twofits,eachcorrespondingtooneplanetaryscenario,bylim-
iting the rangesin whichthe fitted parameterscan vary. For
4.1. HomogeneousAtmosphere
thehomogeneousatmospherefit, weassumetheasymmetric
reflectioncomponenttobezero,whichmakesthecloudcon- The best-fit model for the homogeneous atmosphere sce-
densationtemperatureandthecloudreflectionboostingfactor nario has thermal emission as the dominant source of plan-
dummyparameters. Forthe patchycloudfit, we assume the etary light. In this scenario, the post-occultationphase shift
planet to have super-rotating equatorial winds (ǫ>0). Un- canbeexplainedbyahotspotlocatedonthewestsideofthe
dersuchassumption,homogenousatmospherecannolonger substellar point (Figure 2). All three fitting parameters (the
provideafit,andpatchycloudwouldhavetobeinvokedtoex- Bondalbedo,theheatredistributionefficiency,andthegreen-
plainthepost-eclipsephaseoffset. Table1showstheresults housefactor)aretightlyconstrainedinthisscenario(seethe
of the two additional fits, and Figure 3 compares the poste- red lines in Figure 3). With only three parameters, the sce-
rior distributions from the two separate fits with those from narioachievesasuperiorgoodnessoffitindicatedbyχ2 and
the generalfit, andshowsthatthe two separate fits correctly significantlybettervaluefortheBayesianInformationCrite-
capturethetwoclassesofsolutionssuggestedbythegeneral rion(BIC)thananyotherscenarios(Table1).
fit. Assuming a homogeneous atmosphere, the visible-
While producingthesame phase offset, thetwo classes of wavelength phase curve contains enough information to de-
modelsresultinslightlydifferentoccultationdepths,because terminethethreefittingparameters.Thephasecurvedepends
they differ in the shape of the phase curve. Table 1 shows ontheseparametersnon-linearly,andtheseparametersappear
thatthederivedoccultationdepthsdifferby2-σ betweenthe tobecorrelatedbutnotfullydegenerate(Figure4). Themost
homogenousatmospherefitandthepatchycloudfit,andFig- prominent correlation is between the greenhouse factor (f)
ure 3 shows that the general fit would give the combination andthe Bondalbedo(A ), becausethemodeledtemperature
B
of the above two fits, and therefore have much greater error distributionisproportionalto f(1- A )1/4.Thesetwoparame-
B
bars. Comparingwith Demoryetal. (2013),the occultation terstendtobecorrelated,inordertomaintainthetemperature
depthfromthepatchycloudfitis1-σconsistent,andthatfrom andthenthethermalemissioncomponent(Figure4).Wealso
thehomogeneousatmospherefitisonly2-σconsistent. Sim- find that for a greater A , the model requires a greater (i.e.,
B
VisiblePhaseCurvesofExoplanets 7
0.25 0.35 10
9
0.3
0.2 8
Density0.15 Density00.2.25 Density 67
Probability 0.1 Probability 00.1.15 Probability 345
0.05 2
0.05
1
030 35 40 45 50 55 040 42 44 46 48 50 52 54 56 00.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
EclipseDepth[ppm] PhaseAmplitude[ppm] PhaseOffset
30 100 18
16
25
10−1 14
Density20 Density Density1102
Probability 1105 Probability 10−2 Probability 68
10−3
4
5
2
00 0.05 0.1 0.15 0.2 0A.2B5 0.3 0.35 0.4 0.45 0.5 10−−420 0 Heat2 0Redistribu40tion Efficie60ncy ε 80 100 01 1.05 1.1 G1r.1e5enhou1s.2e Fact1o.2r5 f 1.3 1.35 1.4
0.04 0.04
0.035 0.035
0.03 0.03
Density0.025 Density0.025 GHPaoetmncehoryga elC nFloeituodu sF Ait tmosphere Fit
Probability 00.0.0125 Probability 00.0.0125
0.01 0.01
0.005 0.005
10000 1200 1400 C1o60n0den1s8a00tion2 0T0e0mp2e2r0a0tur2e4 0T0c 2600 2800 3000 00 1 2Clou3d Refle4ction 5Boosti6ng Fac7tor k8 9 10
Figure3. PosteriorprobabilitydistributionoftheparametersforKepler-7bfromMCMCsimulationstofitthephasecurve. Resultsfromthegeneralfit,the
homogeneousatmospherefit,andthepatchycloudfitareshownbydifferentcolors, andthecolordesignation istabulatedonthelowerright. Theposterior
probabilitydistributionresultedfromthegeneralfitshowsbimodalsolutions,andappearstobethecombinationoftheresultsfromthehomogeneousatmosphere
fitandthepatchycloudfit. Thehomogeneous atmospherefittightlyconstrainstheheatredistribution efficiencyto- 3.2±0.5,andthegreenhousefactorto
1.23±0.02.Thepatchycloudfitallowsapositiveǫ,butthephasecurveyieldsnoconstraintontheexactvalueforǫ;instead,thephasecurvewellconstrainsthe
cloudcondensationtemperature.
more negative)value for the redistributionefficiency ǫ (Fig- the substellar point has the greatest temperature, and there-
ure4). ThisisbecausewhenA isgreater,thesymmetricre- fore that air at photospheric pressures travels westward in a
B
flectioncomponentbecomesmoresignificantandthethermal synchronously rotating reference frame. This could occur
emission component becomes less significant. To keep the on a synchronously rotating planet if the photospheric-level
phaseoffsetconsistentwiththeobservation,theoffsetofthe windswerewestwardatlowlatitudes. However,todate,cir-
thermalemissioncomponentalonewouldhavetobegreater, culation modelsof highly irradiated, tidally locked exoplan-
andtherefore|ǫ|wouldhavetobegreater.Furthermore,when ets have generally predicted eastward equatorial winds and
|ǫ| is greater, the temperature becomes lower at the dayside thereforeeastwardoffsetsofhotspotsrelativetothesubstel-
andhigheratthenightside,whichinturnwouldaffecttheoc- lar point (e.g., Showman & Guillot 2002, Cooper & Show-
cultationdepthandthephaseamplitude.Inall,three“model- man2005, Showmanetal. 2008,2009; Rauscher& Menou
independent” observed quantities, the occultation depth, the 2010,2012;Hengetal. 2011;Pernaetal. 2012). Sucheast-
phaseamplitude,andthephaseoffsetcanuniquelydetermine wardjets wereexplainedin a theorypresentedby Showman
the three fitting parameters. Once their values are found, & Polvani (2011), which shows that the day-night thermal
we can separate the contributionfrom thermal emission and forcinginducesglobal-scalewavesthattransportprogradean-
thatfromreflection,solelyfromthevisible-wavelengthphase gular momentum to the equator, allowing such a so-called
curve. “superrotating”equatorialjetto emerge. Todate, nomodels
Particularly for Kepler-7 b, we find that the heat redistri- ofhighlyirradiated,synchronouslylockedplanetshavebeen
bution efficiency (ǫ) is smaller than zero by 6- σ, implying publishedthatexhibitastrongwestwardjetattheequator,as
that the advective frequency, ω ≡ω - ω must would seem to be needed to explain the westward offset in
adv photosphere orbit
have a large negative value, in order to explain the post- the Kepler-7 b phase curve in the homogeneousatmosphere
occultationphaseshift. Therefore,thewestwardoffsetofthe scenario.
bright spot would seem to suggest that the air westward of In principle, non-synchronousrotation could contributeto
8 Huetal.
5
1
0
100
ε
−5
50
ε
1 −10
0 1.35 1800
2 3
1.3
1700
−50
1.4 1.25
c1600
2 f T
1.2
1500
1.15
f 1.2
1.1 1400
0 0.1 0.2 0.3 0.3 0.4 0.5
A A
B B
1
3000
Tc2000 3
1000
100
κ 50
0
0 0.5 −50 0 50 100 1 1.2 1.4 1000 2000 3000
A ε f T
B c
Figure4. Correlations between fittedparameters forKepler-7b. Theresults fromthegeneralfitareshownasblackdots. Thefiguresshowtwoseparate
populations corresponding to the homogeneous atmosphere solution and the patchy cloud solution. Small Bond albedos (AB), negative heat redistribution
efficiencies(ǫ),andgreater-than-unitygreenhousefactors(f)correspondtothehomogeneousatmospheresolution;theothertwoparametersareunconstrained.
Thethreeconstrainedparametersarecorrelated,andtheircorrelationsarehighlightedbyinsertedpanel1and2thatprovidezoom-inviews.LargeBondalbedos,
positiveheatredistributionefficiencies,well-constrainedcondensationtemperatures(Tc)near1500K,andpositivereflectivityboostingfactors(κ)correspondto
thepatchycloudsolution.TheBondalbedoandthecondensationtemperaturearecorrelatedandtheircorrelationsarehighlightedbyinsertedpanel3.
a hot spot offset. Circulation models of non-synchronously best-fithomogeneousatmospheremodelis1820K,whichis
rotatinghotJupitershavebeencomputedbyShowmanetal. higherthanthe3-σupperlimitofthebrightnesstemperature
(2009), Showman et al. (2014), and Rauscher & Kempton measured at the Spitzer 3.6 µm band (Demory et al. 2013).
(2014),consideringrotationthatisprograde(i.e.,inthesame With the caveat that Kepler and Spitzer may probe different
direction as the orbital motion), with rotation periods both pressurelevelsandhavedifferentbrightnesstemperatures,the
shorter and longer than the orbital period. Like their syn- homogeneousatmospherescenarioappearstobeinconsistent
chronously rotating counterparts, almost all of these mod- withtheSpitzerobservations. Based onthe atmosphericcir-
els develop fast eastward equatorial jets – fast enough to culation model results and the Spitzer observations, the ho-
cause eastward motion in the synchronously rotating refer- mogeneousatmospherescenarioisunlikely,whicheffectively
enceframeregardlessoftherotationrate–andthereforeeast- makesthepatchycloudscenariotoonlyplausiblescenariofor
ward hotspot offsets. That said, one slowly rotating non- Kepler-7b.
synchronoussimulation in Rauscher & Kempton (2014) de-
velops a “westward” equatorial jet that causes the thermal 4.2. PatchyCloud
hotspot to be shifted west of the substellar point. In princi-
The best-fit model for the patchy cloud scenario has the
ple,planetaryrotationthatisretrograde(i.e.,inthedirection
asymmetric reflection componentas the dominant source of
oppositetheorbitalmotion)couldalsoleadtothermalhotspot
planetarylight (Figure 2). In this scenario, reflective clouds
offsetsofthecorrectsigntoexplaintheKepler-7bdata.How-
locatedonthewestsideofthesub-stellarmediancanbestex-
ever, given the short tidal spindown timescales for very hot
plainthepost-eclipsephaseshiftof Kepler-7b. Thisiscon-
JupiterslikeKepler-7b,itislikelythattheplanetiscloseto
sistentwiththeexplanationproposedbyDemoryetal. (2013)
synchronousrotation,inwhichcaseonewouldexpecttheex-
butouranalysisoffersmoreinformationontheatmosphere’s
istenceofasuperrotatingjetandaneastwardhotspotoffset.
properties. The Bond albedo is well constrained (Table 1),
ForKepler-7b,thedaysideequilibriumtemperatureforthe
andismuchgreaterthanthatinthehomogeneousatmosphere
VisiblePhaseCurvesofExoplanets 9
scenario. With a Bondalbedoof∼0.4, the daysideequilib- requiredhighreflectivities. Mg SiO andMgSiO arehighly
2 4 3
riumtemperatureisconsistentwiththeSpitzerobservations. reflective(Sudarskyetal. 2003),andthereforearecandidate
The asymmetric reflection component is produced by a cloud-formingmaterialsfortheatmosphereofKepler-7b.
cloud distribution, in which the east side of the sub-stellar Finally, theefficiencyofheatredistributioncannotbe suf-
median is devoidof reflective clouds. To form such a cloud ficiently constrained by the phase curve. For Kepler-7 b, a
distribution, the model requires non-zero positive values for non-zeropositivevalueforǫisrequired,butthephasecurve
theheatredistributionefficiency(ǫ)andthecloudreflectivity does not prefer a specific value for ǫ (Figure 3) - meaning
boostingfactor(κ)(Table1). Withapositivevalueforǫ,the thatthephasecurveisratherinsensitivetotheexacttemper-
eastsideishotterthanthewestside;andifthecloudconden- ature distribution as long as a hot spot offset exists. This is
sation temperature is suitable, condensation can only occur becausewhenǫislarge,thelongitudinalvariationoftemper-
onthewestsidebutnotontheeastside. Thisformsa“hole” aturewouldbesmall(Cowan&Agol2011),andaslightad-
inthecloudontheeastside. ParticularlyforKepler-7b, the justmentof T would be enoughto keep the cloudboundary
c
cloudboundarywouldhavetobelocatedat∼10◦tothewest unchanged. This is different from the homogeneous atmo-
of the substellar point to produce the observed phase offset spherescenario,inwhichthephaseoffsetdirectlydependson
(Table1). Apositivevalueforǫindicatessuper-rotatingjets the hot spot offset and ǫ can be tightly constrained. In the
thattransportheat towardseast, consistentwith atmospheric patchy cloud scenario, due to the uncertainties of the cloud
circulationtheories(Showman& Guillot2002; Showmanet condensationtemperature,thephasecurvesetsa1- σ lower
al. 2011). bound of 20 and does not yield an upper bound. In other
Also,toexplainthephaseoffset,thecloudypartoftheat- words,thephasecurveonlyrequiresa“fairlysignificant”heat
mosphere must be more reflective than the cloud-free part, redistribution,butcannotyieldquatitativeconstraintsonthis
and our analysis shows that this reflectivity contrast would parameter.
have to be quite significant to explain the phase curve of
5. APPLICATIONSTOKEPLER-10B
Kepler-7b. Wefindthatthecloud-freepartoftheatmosphere
mustbequitedark,havingareflectivitylessthan7%,andthe WederiveafullphasecurveofKepler-10bbasedonKepler
cloudypartoftheatmospheremustbequitebright,havinga observationsduringthequarters1to17.Kepler-10bisa1.4-
reflectivitygreaterthan90%,atvisiblewavelengths(Table1). R⊕,4.6-M⊕predominantlyrockyplanet(Batalhaetal. 2011;
The significant contrastin reflectivities is one of the main Fogtmann-Schulzet al. 2014),makingourresultamongthe
reasons why the modelcan constrain the Bond albedo. The first revelation of any rocky exoplanets’ phase curve signa-
Bondalbedocannotbetoosmall, otherwisetherewouldnot tures. Thisobservationismadepossiblebycontinuousmon-
beenoughplanetarylighttoexplainthephaseamplitude.Al- itoring of the system by Kepler that brings down the error
ternatively, the phase amplitude could be explained by ad- budgetofphotometry. We also benefitfromthefactthatthe
ditionalthermalemission (i.e., byincreasingthe greenhouse planetishotenoughtohavesignificantthermalemissioncon-
factor f),butthatwoulddrivetheoverallphaseshifttowards tributiontotheKeplerband(Rouanetal. 2011),andthestar
the opposite direction than the asymmetric reflection. The isintrinsicallyquiet(Batalhaetal.2011).Weapplyoursemi-
Bond albedo cannot be too large also, because the hole in analyticalmodeltoanalyzethephasecurveofKepler-10b.
cloud has to be large enough (i.e., covering a significantly
5.1. DataReduction
large part of the dayside), and dark enough for the signifi-
cantcontrastbetweenthecloudypartandthecloud-freepart. OurdatareductionissimilartotheoneforKepler-7bpre-
Inall,thephaseamplitudeandthephaseoffsettogetherputa sented in Demory et al. (2013). We use Kepler (Batalha et
tightconstraintontheBondalbedooftheplanetinthepatchy al. 2013)long-cadencesimple aperturephotometry(Jenkins
cloudscenario. et al. 2010) obtained during the quarters 1 to 17. We take
Inadditionto the Bondalbedo, fitting to the Keplerphase into accountthe crowdingmatrix correctionfactor indicated
curveconstrainsthecloudcondensationtemperatureandmay ineachFITSfileonaquarter-per-quarterbasis. Wemitigate
implythephysicalpropertiesofthecondensatespecies. The instrumentalsystematicsbyfittingthefirstfourcotrendingba-
cloud condensation temperature is correlated with the Bond sis vectors (CBV) to each quarter using the PyKE software
albedo, as T ∝(1- A )1/4 (Figure 4). This is because the (Still&Barclay2012).Wethennormalizeeachquartertothe
c B
cloud condensation temperature needs to have a value be- median. We accountforphotometrictrendslongerthanfour
tween the maximum and the minimum temperature on the times the planetary orbital period by fitting a second-order
dayside,inordertoproduceapatchyclouddistribution.Cau- polynomial to the out-of-eclipse data in the MCMC frame-
tionshouldbeexercisedwhencomparingT withtheconden- work detailed below. We estimate and include the corrected
c
sationcurvesofpotentialcloud-formingmaterials. Wecom- noise the same way as in Demory et al. (2013). We find a
pareT withtheequilibriumtemperaturesinthemodelsetup. nominallevel(lessthan10%)ofcorrelatednoisethroughout
c
However, the cloud may be located deep in the atmosphere thedataset.
atthepressureof0.1-1bars,andthetruecloudcondensation
5.2. Model-IndependentAnalysis
temperaturemaybehigherthanT. Amorerealisticrangefor
c
thecloudcondensationtemperaturewouldbebetweenT and Beforeisolatingtheplanetaryphasecurvesignalwesearch
c
T f, where f is the derived greenhouse factor. For Kepler- forallfrequenciesinthedatasettoassessanyriskofcontam-
c
7 b, we find this range corresponds to 1480-1730 K, using ination. AtypicalLomb-Scargleperiodogramisnotoptimal
the values tabulated in Table 1. The condensationcurves of inthecaseofdatasetsspanninglongobservationsasslightly
Fe,Mg SiO ,MgSiO ,andCrcrosstheinferredtemperature changing periodicities damp amplitudes in the power spec-
2 4 3
rangeatthepressureof10- 3- 1barsforthesolarabundances trum. Toquantifyhowfrequenciesandamplitudesevolvein
our dataset, we perform a wavelet transform analysis (e.g.,
(Lodders & Fegley 2006). Fe and Cr are strongly absorp-
Torrence & Compo 1998) using the weighted wavelet Z-
tive in the visible wavelengths, and thus cannot lead to the
transformalgorithmdevelopedby Foster (1996). We do not
10 Huetal.
detect any clear signature in the frequency/time spectrum, Total =
Symmetric Reflection+Asymmetric Reflection+Thermal Emission
apartfromtheplanetorbitalsignal. Kepler-10isintrinsically 30
quietandanystellaractivityremainsnominaloverthecourse Homogeneous Surface Kepler-10 b
of these Kepler observations. The frequency/time spectrum 25
doesnotrevealquarter-dependentfluctuations. 20
We then conduct a model-independent Bayesian analysis
15
of the entire datasetby employingthe MarkovChain Monte
Carlo (MCMC) implementation presented in Gillon et al. −6] 10
0
(2012). We assume a circular orbit, and set the occultation [1S 5
depth, phase-curve amplitude, phase-curve peak offset, pe- F
riod, transitduration, time of minimumlightandimpactpa- F / P 0
rameter as jump parameters. We assume a simple trapezoid
−5
function for the occultation, and a Lambertian sphere mod-
ulation for the phase curve. We further assume a quadratic −10
law for the limb-darkening (LD) and use c1 = 2u1+u2 and −15
c = u - 2u as jump parameters, where u and u are the
2 1 2 1 2 −20
quadraticcoefficients. u andu aredrawnfromthetheoret-
1 2 30
ical tables of Claret & Bloemen (2011) for the correspond- General Kepler-10 b
ing effective temperature and log g values from Batalha et 25
al. (2011). We add the two LD combinations c1 and c2 as 20
GaussianpriorsintheMCMCfit,basedonthetheoreticalta-
15
bles. WeruntwoMarkovchainsof105stepsandassesstheir
convergenceusing the statistical test from Gelman & Rubin −6] 10
0
(1992). We alsoexploretheeffectsofthedatareductionpa- 1
[S 5
rametersonderivingthe transitparameters. We increase the F
CBVvectorsto8andreduceandanalyzethedataseparately, F / P 0
bypairsofquarters. Duringalltheseanalysesvariations,our
−5
MCMCfitsresultinindividualtransitparametervalueswithin
1-σofthefinalQ1-Q17valuesstatedabove. −10
We examine the robustness of the planetary phase-curve −15
signal. We stackapproximatelythreeyearsofdata,meaning
−20
thatforstellarcontaminationtohappen,stellaractivityhasto −3 −2 −1 0 1 2 3
bephasedexactlyontheplanetaryorbitalperiodof0.8days Phase Angle
(oramultiple).Kepler-10isanevolvedstarthatisunlikelyto Figure5. Phase-foldedlightcurveofKepler-10bsystemandexamplesof
havevsiniconsistentwiththeshortorbitalperiodofKepler- modelfit. Keplerphotometryobtainedduringthequarters1to17isusedto
derive this light curve, andkeysteps ofdata analysis areprovided in§5.
10b. Kepler-10bisnotmassiveenoughtocauseanyappre-
The light curve is binned to 200 bins for clarity. The magenta lines are
ciable ellipsoidalor beaming componentsin the light curve. themodeledphasecurves,andthecoloredlinesshowcontributionofther-
Wethereforeconcludethattheorbitalphase-curveisofplan- malemission(red),symmetricreflection(green),andasymmetricreflection
etaryorigin. (blue).Theupperpanelshowsthesimplestmodelassumingahomogeneous
surface.Themodelisdominatedbythesymmetricreflectioncomponent,and
We derive an occultation depth of 7.5±1.5 ppm and a
providesagoodfittotheobservedphasecurve. Thelowerpanelshowsthe
phase-curve amplitude of 8.5±1.2 ppm (Figure 5). We do best-fitmodelofthegeneralfit,inwhichathermalemissioncomponentfrom
notdetectaphaseoffsetforKepler-10b,withthephaseoff- ahotspotshiftedwestwardhassignificantcontribution.Multiplemodelscan
set angle constrained to be 9±6 degrees. Our value of the provideadequatefittothephasecurveofKepler-10b.
occultationdepthis1- σ consistentwithpreviouslyreported
values (5.8±2.5 ppm, Batalha et al. 2011; 9.9±1.0 ppm,
etal. (2014),ouranalysisdoesnotyieldadefinitivenightside
Fogtmann-Schulzetal. 2014)
emissionflux. Notethatthisfluxconstraintissensitivetothe
Theoccultationdepthtranslatestoabrightnesstemperature
limb-darkeningparameterizationofthestar,aswellasthein-
at the Kepler band of 3220+90 K if the planet’s flux is from
- 110 terval size used in the estimate. In any case, the nightside
thermalemission,orageometricalbedoof0.55±0.11ifthe
brightnesstemperaturemustbemuchlowerthanthedayside
planet’sfluxisfromreflection.Thephasecurvemagnitude,if
brightnesstemperature.
solelyattributedtoreflection,correspondstoaneffectivegeo-
metricalbedoof0.63±0.09,oraBondalbedoof0.94±0.13
5.3. Model-AssistedAnalysis
foraLambertiansphere. Giventhatthezero-albedodayside-
averageequilibriumtemperatureoftheplanetis2570K,both We then apply our semi-analytical model to analyze the
thermalemissionandreflectioncanhaveconsiderablecontri- phase curve of Kepler-10 b. The eccentricity of the planet’s
butiontotheplanet’semergingradiation. orbit is consistent with zero (Batalha et al. 2011). We per-
The photometric measurements immediately outside the formageneralfit,inwhichallparametersareallowedtovary
transitscontaininformationabouttheplanet’snightside. The intheirphysicallyplausiblerange,andanatmosphere-lessfit,
transitsoccurfrom-20to20degree(firstcontact)forthissys- in which the greenhouse factor is force to f = 1, assuming
tem,andwetakea20-degreeintervalonbothsidesofthetran- thattheplanetdoesnotanatmosphere. Theatmosphere-less
sittoderivetheplanet’snightsideemissionflux. Theresultis fitismotivatedbythelargebulkdensityoftheplanet,andthe
afluxof- 0.7±1.2ppm,whichplacesthe1- σupperlimitof strongirradiationreceivedbytheplanet(Batalhaetal. 2011).
thenightsidetemperatureat2270K.UnlikeFogtmann-Schulz Theposteriordistributionsresultedfromthesefitsareshown
in Figure6, andexamplesofmodelfits are shown in Figure