Reflected lightcurves of extrasolar Jupiters and Saturns (ApJ paper)

14 Pages·2005·0.87 MB·English

TheAstrophysicalJournal,618:973–986,2005January10
#2005.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A.
PHASE LIGHT CURVES FOR EXTRASOLAR JUPITERS AND SATURNS
Ulyana A. Dyudina,1,2 Penny D. Sackett,2 and Daniel D. R. Bayliss3
ResearchSchoolofAstronomyandAstrophysics,MountStromloObservatory,AustralianNationalUniversity,
CotterRoad,WestonCreek,Canberra,ACT2611,Australia
S. Seager
ObservatoriesoftheCarnegieInstitutionofWashington,5241BroadBranchRoadNW,Washington,DC20015
Carolyn C. Porco
CICLOPS,SpaceScienceInstitute,3100MarineStreet,SuiteA353,Boulder,CO80303-1058
and
Henry B. Throop and Luke Dones
SouthwestResearchInstitute,1050WalnutStreet,Suite426,Boulder,CO80302
Received2004June16;accepted2004September16
ABSTRACT
Wepredicthowaremoteobserver wouldsee thebrightnessvariationsof giantplanetssimilarto those in our
solar system as they orbit their central stars. Our models are the first to use measured anisotropic scattering
propertiesofsolarsystemgiantsandthefirsttoconsidertheeffectsofeccentricorbits.Wemodelthegeometryof
Jupiter,Saturn,andSaturn’sringsforvaryingorbitalandviewingparameters,usingscatteringpropertiesforthe
(forwardscattering)planetsand(backwardscattering)ringsasmeasuredbythePioneerandVoyagerspacecraftat
0.6–0.7 (cid:1)m. Images of the planet with and without rings are simulated and used to calculate the disk-averaged
luminosityvaryingalongtheorbit;thatis,alightcurveisgenerated.Wefindthatthedifferentscatteringproperties
ofJupiterandSaturn(withoutrings)makeasubstantialdifferenceintheshapeoftheirlightcurves.Saturn-sized
ringsincreasetheapparentluminosityofaplanetbyafactorof2–3forawiderangeofgeometries,aneffectthat
couldbeconfusedwithalargerplanetsize.Ringsproduceasymmetriclightcurvesthataredistinctfromthelight
curve that the planet would have without rings, which could resolve this confusion. If radial velocity data are
available for the planet, the effect of the ring on the light curve can be distinguished from effects due to orbital
eccentricity. Nonringed planets on eccentric orbits produce light curves with maxima shifted relative to the po-
sitionofthemaximumphaseoftheplanet.Givenradialvelocitydata,theamountoftheshiftrestrictstheplanet’s
unknownorbitalinclinationandthereforeitsmass.Acombinationofradialvelocitydataandalightcurvefora
nonringedplanetonaneccentricorbitcanalsobeusedtoconstrainthesurfacescatteringpropertiesoftheplanet
andthusdescribethecloudscoveringtheplanet.Wesummarizeourresultsforthedetectabilityofexoplanetsin
reflectedlightinachartoflight-curveamplitudesofnonringedplanetsfordifferenteccentricities,inclinations,and
azimuthal viewing angles of the observer.
Subject headingg: methods: data analysis — planetary systems— planets: rings —
planets and satellites: individual (Jupiter, Saturn) — scattering
1.INTRODUCTION been detected by transit photometry (Udalski et al. 2002a,
2002b, 2003) and then confirmed with Doppler measure-
Modernspace-basedtelescopesandinstrumentationarenow
ments(Konackietal.2003,2004;Bouchyetal.2004).Itisex-
approaching the precision at which reflected light from extra-
pected that in the next decade, as photometric techniques are
solarplanets can be detected directly (Jenkins & Doyle 2003;
improvedfrom both the ground and in space, detection of the
Walkeretal.2003;Greenetal.2003;Hatzes2003).Since1995,
reflectedlightfromexoplanetswillproveusefulnotonlyinex-
morethan100extrasolarplanets(orexoplanets)havebeende-
pandingthenumberofknownexoplanetsbutalsoindetailing
tectedindirectlybymeasuringthereflexmotionoftheirparent
theircharacteristics.4
staralongthelineofsight(radialvelocityorDopplermethod).
Reflected light from an exoplanet can be detected in two
Oneoftheseradialvelocityplanets,Gl876b,hasbeenconfirmed
ways. First, with precise, integrated photometry, one can
by measuring the parent star motion on the sky (astrometry;
searchfortemporalvariationsinthecombinedparentstarand
Benedict et al. 2002), and a second, HD 209458b, by measur-
exoplanet light curve due to the planet changing phase in re-
ingthe changein parentstarbrightnessastheplanetexecutes
flectedlightthroughoutitsorbit,inthesamewayastheMoon
a partial eclipse of its host (transit photometry; Charbonneau
changes phase. The vast majority of the light comes from the
et al. 2000; Henry et al. 2000). Three exoplanets have now
staritself,withasmallconstantthermalcontributionforgiant
planets, but if the periodic variations due to light reflected by
1 NASAGoddardInstituteforSpaceStudies,2880Broadway,NewYork, theplanetscanbeextractedfromthetotalphaselightcurve,the
NY10025.
2 Planetary ScienceInstitute,AustralianNationalUniversity,ACT0200,
Australia. 4 Reviews and references on extrasolar planets and detection techniques
3 VictoriaUniversityofWellington,NewZealand. canbefoundathttp://www.obspm.fr/encycl/encycl.html.
973
974 DYUDINA ETAL.
planetary phase as a function of time can be deduced. Fur- of our model and the detectability of the light curves by
thermore, since the planet’s light and the starlight need not modern and future instruments. Conclusions are presented in
be spatially resolved, relatively distant planetary systems and x5.
planets at small physical orbital radiican be studied.
Already, ground-based observations of known short-period 2.MODEL
exoplanets have been conducted to search for reflected light Wemodelthereflectedbrightnessofanexo-Jupiterorexo-
signaturesinhigh-resolutionspectroscopy(Charbonneauetal. Saturn by tracing how light rays from the central star are re-
1999;CollierCameronetal.2002;Leighetal.2003a,2003b). flectedbyeachpositionontheplanet.Wethenproduceimages
Todate,nosignatureofreflectedlighthasbeendetected,butbe- (mapswitharesolutionof16–200pixelsacross,dependingon
causetheorbitalphaseisknownfromradialvelocitymeasure- the acceptable error level) of the planet and its rings for dif-
ments,thesenondetectionsconstrainthegraygeometricalbedo ferentgeometries.Thelightraysfromthestarilluminatingthe
poftheplanetforanassumedphasecurve,orbitalinclination, planetareassumedtobeparallel,consistentwithbothstarand
andplanetary radius.For theassumedparameters of(cid:2) Boob, planet being negligible in size compared with the star-planet
HD75289b,andtheinnermostplanetof(cid:3)And,nondetections distance. The reflected rays collected by the observer are as-
seemtoimplythatp<0:4(CollierCameronetal.2002;Leigh sumed to be parallel, consistent with a remote observer. The
etal.2003a,2003b). model includes reflection from the planet and rings and the
Second,withsufficientspatialresolution,directimagingcan rings’ transmission and shadows but does not incorporate
resolve the planet and the star in space, so that the projected second-ordereffectssuchasringshineontheplanetorplanet
planetary orbit can be tracked simultaneously with the mea- shine on the rings. We account for the planet’s oblateness in
surement of the planet’s phases in reflected light. To be de- someofoursimulationsbyusingthe10%oblatenessofSaturn
tected,planets must be at orbital distances large enough to be as an example.
easilyresolvedfromtheirparentstarandyetcloseenoughthat The planetary reflected brightness in our model is derived
the reflected brightness is large enough to be detected against from the data as an average brightness integrated along the
thebackground.Consequently,thefirstextrasolarplanetstobe planet’s spectrum, weighted by the wavelength-dependent
detecteddirectlyinthiswayarelikelytobegiantsorbitingrel- transmissivityofthePioneerredfilter,whichisnonzerointhe
atively nearby stars (at tens of parsecs) at intermediate semi- range 0.595–0.720 (cid:1)m. Our notation matches that of most
majoraxes(1–5AU;Clampinetal.2001;Lardie`reetal.2004; observational papers on Saturn and Jupiter. To compare our
Trauger et al.2003; Krist et al.2003; Codona & Angel 2004; results with those from the models of Arnold & Schneider
Dekany et al.2004). (2004),welistbothourandtheirparameternamesinTable1.
Direct imaging can yield both the orbit and the luminosity Figure1demonstratesthegeometryofaringedplanetona
of the planet simultaneously and thus yield robust detections. circular orbit defined by the angles listed in Table 1. Besides
Although direct exoplanet imaging from either the ground or theanglesshowninFigure1,whichdefinethegeometryofthe
space may not be available for several years, our model light planetasawhole,thescatteringbrightnessofeachpositionon
curvescanbeusedforplanningfutureobservationsofspatially theplanet’sorring’ssurfacedependsonthreeangles:thephase
resolvedplanets in reflected light. angle(cid:4),theincidenceangleviaitscosine(cid:1) ,andtheemission
0
Thepotentialofexoplanetdetectioninreflectedlightandthe angleviaitscosine(cid:1),asdefinedbythesurfacescatteringphase
possibilityofdetectingplanetaryringsisdemonstratedbythe functionP((cid:1) ,(cid:1),(cid:4)).WeobtainP((cid:1) ,(cid:1),(cid:4))forJupiter,Saturn,
0 0
numberofworkspublishedinthelastfewyears.Seageretal. and the rings from the data as described in the subsections
(2000) modeled the atmospheric and cloud composition and below.
simulated the light curves for close-in giant planets with var-
ious cloud coverages. Sudarsky et al. (2003) modeled clouds 2.1.ReflectinggProperties of Jupiter, Saturn, and Ringgs
andspectraforextrasolarplanetsatdifferentdistancesfromthe
Figure 2 shows the scattering phase function for a given
star.Barnes&Fortney(2004)modeledtransitlightcurvesfora
geometry, used in our model to describe surfaces of Jupiter,
planet with rings. Arnold & Schneider (2004) simulated light
Saturn,and its rings.The normalized brightness of a point on
curves for planets with rings of different sizes, assuming
the surface isplotted versus scattering phaseangle(cid:4):
planetswithisotropicallyscattering(Lambertian)surfacesand
rings with isotropically scattering particles, and provided a
P((cid:1) ; (cid:1); (cid:4))(cid:1)I((cid:4); (cid:1) ; (cid:1))=(F(cid:1) ); ð1Þ
discussionofthepossibilityofringpresenceatdifferentstages 0 0 0
ofplanetary system evolution.
Our model, however, is the first to use the observed aniso- where F(cid:1) is the ‘‘ideal’’ reflected brightness of a white, iso-
0
tropic scatteringof Jupiter, Saturn, and Saturn’s rings and the tropically scattering (Lambertian) surface, (cid:1) is the cosine of
0
first to calculate light curves for eccentric orbits. We find that theincidenceanglemeasuredfromthelocalvertical,andF((cid:5)sr)
anisotropicscattering yieldslightcurves that are substantially isthesolarfluxattheplanet’sorbitaldistance.Inournotation,
different from those assuming Lambertian planets and iso- (cid:4) ¼0(cid:2) indicates backscattering and (cid:4) ¼180(cid:2) indicates for-
tropically scattering rings and thus is more likely to give an wardscattering.Notethelogarithmicscaleoftheordinateand
accuratedescriptionforextrasolarplanetssimilartoJupiterand the large amplitudes of the phase functions. Lambertian scat-
Saturn. tering with reflectivity 0.82 (matching the observed Saturn’s
In x 2 we describe our model. In x 3 we present our light full-diskalbedoatopposition)isshownasahorizontalsolidline
curves for nonringed exo-Saturn and exo-Jupiter planets on for comparison. We also indicate, for the same ring opacity,
an edge-on circular orbit, for variously oriented oblate exo- albedo,andgeometry,theringphasefunctionusedbyArnold
Saturnsandforaringedplanet(withSaturnianscatteringprop- & Schneider (2004), which assumes isotropically scattering
erties)atvariouslyinclinedcircularorbits.Inaddition,wemodel particles.Figure2showsreflectedlightonlyforthegeometry
a nonringed exo-Saturn, an exo-Jupiter, and a Lambertian inwhichtheSunis2(cid:2)abovethehorizon((cid:1) ¼0:035)andthe
planet on eccentric orbits. In x 4 we discuss the uncertainties observer moves from the Sun’s location ((cid:4)0 ¼0(cid:2)) across the
TABLE1
ParametersUsedinModeling
ThisWork Arnold&Schneider(2004) Quantity
A,B.......................... CoeDcffiientsoftheBackstormlaw
AHG........................... CoeDcffiientforJovianphasefunction
a,a1.......................... Semimajoraxisoftheplanet’sorbit(AU)
D ............................. Planet-stardistance(km)
P
E............................... Eccentricanomaly(polarangleparameterization)
e................................ Eccentricity
F............................... (cid:7) IntensityofawhiteLambertiansurfacea(Wm(cid:3)2sr(cid:3)1)
g,g,f...................... ParametersofHenyey-Greensteinfunction
1 2
I................................ L,I ,I ,L ,L Intensity(orbrightness,orradiance)ofthesurface(Wm(cid:3)2sr(cid:3)1)
p R T ps pr
i................................ i Inclinationoftheorbit(0(cid:2):face-on;90(cid:2):edge-on)(deg)
L .............................. Luminosityoftheplanetbinreflectedlight
P
L .............................. Luminosityofthestarb
*
M.............................. Meananomaly(timeparameterization)
p................................ Full-diskalbedo,bLP=((cid:5)R2PF)
P((cid:1)0,(cid:1),(cid:4)).............. Scatteringphasefunctionofthesurface
R............................... Star’smagnitudeinRband
R ............................. r Equatorialradiusoftheplanet(km)
P p
rpix............................ Pixelsize(km)
V............................... Star’smagnitudeinVband
(cid:4)............................... (cid:4) Phaseangle(deg)
(cid:8)t............................... Temporalshiftoflight-curvemaximumfrompericenter(days)
(cid:6)................................ Planet’sobliquity(deg)
(cid:1).............................. (cid:9)¼(cid:1)(cid:3)90(cid:2) Orbitalangle((cid:4)180(cid:2):minimumphase;0(cid:2):maximumphase)(deg)
(cid:1) ,(cid:1)......................... (cid:1) ,(cid:1) Cosinesoftheincidenceandemissionangle
0 0
!............................... Argumentofpericenter(90(cid:2):observedfrompericenter,(cid:3)90(cid:2):fromapocenter)(deg)
!.............................. Observer’sazimuthrelativetotherings(deg)
r
a F((cid:5)sr)istheincidentstellarfluxattheplanet’sorbitaldistance(whichisalsosometimescalledFbuthasWm(cid:3)2units,unlikeourintensityF
measuredperunitsolidangle).
b The‘‘red’’or‘‘blue’’opticalpropertiesaretheconvolutionoftheplanet’s(orthering’sorthesolar)spectrumwiththewavelength-dependent
transmissivityofthePioneerfilter(whichisnonzerobetween0.595and0.720(cid:1)mfortheredfilterandbetween0.390and0.500(cid:1)mfortheblue
filter).
Fig.1.—Anglesdefiningthegeometryofaringedplanetonacircularorbitobservedfromalargedistance.Dashedlinesshowthenormaltotheringplaneand
thelineofintersectionoftheeclipticwiththeringplane.Theobserver’sazimuthrelativetotherings,!,ispositivewhentheringsaretiltedfromtheecliptictoward
theobserver,isnegativewhentheringsaretiltedfromtheeclipticawayfromtheobserver,andchangesrfrom(cid:3)90(cid:2)to90(cid:2).Wedonotconsiderotherpossiblevalues
of! ¼(cid:4)(90(cid:2) 180(cid:2))becausetheseorientationsproducelightcurvessymmetrictothosewith! ¼(cid:4)(0(cid:2) 90(cid:2))! ¼(cid:4)0(cid:2).Foranonringedplanet,thegeometryis
r r r
fullydeterminedbytheinclinationiandtheorbitalangle(cid:1)(t).Foraringedplanet,thegeometryisfullydeterminedbyi,(cid:1)(t),!,andtheobliquity(cid:6).
r
976 DYUDINA ETAL. Vol. 618
Fig. 3.—Our fit of the Henyey-Greenstein function (solid line; g ¼
Fig. 2.—Our model scattering phase functions for a Lambertian surface, 0:8; g2¼(cid:3)0:38; f ¼0:9,andAHG¼2)tothePioneer10data(P10),p1ub-
Saturn,Jupiter,andSaturn’sringsandthephasefunctionforringsconsisting lishedasTablesIIaandIIbofTomaskoetal.(1978),andtothePioneer11
ofisotropicallyscatteringparticlesusedbyArnold&Schneider(2004).The data(P11),publishedasTablesIIaandIIbofSmith&Tomasko(1984).The
ipsh2as(cid:2)eafbuonvcetiothneshaorerizpolont)t,edwhfoilreathseinogblesesravmerplmeogveeosmientrtyh:e(cid:1)p0lainsefixtheadt(itnhceluSduens dseartvaerdepwrietshenthtebreeltds((0d.a5r9k5–st0r.i7p2e0s)(cid:1)amn)daznodnebslu(ebr(0ig.h3t90st–r0ip.5es0)0o(cid:1)nmJ)ufiplitteerrs.ob-
theSunandthezenith.Thewavelengthscorrespondtovisibleredlight(0.6–
0.7(cid:1)m).
between Pioneer 10 and Pioneer 11 data is not as well con-
strained as the calibration within each data set. If we accept
zenith toward the point on the horizon opposite the Sun
((cid:4) ¼178(cid:2)). We used different methods to obtain P((cid:1) , (cid:1), (cid:4)) thecalibrationsgiveninTomaskoetal.(1978)forPioneer10
0 and those given in Smith & Tomasko (1984) for Pioneer 11,
for Jupiter, Saturn, and the rings from Pioneer and Voyager
our model curve better represents the observations in the red
data.
filter(blackdatapoints)thanintheblue.ThePioneer11blue
2.1.1.Jupiter data at moderate (cid:4) in Figure 3 seem to be systematically off-
set from the Pioneer 10 points, which may be a result of rel-
For Jupiter we fitted a simple four-parameter analytical
ativecalibrationerror.Consequently,wedonotmodeltheblue
function (two-term Henyey-Greenstein function) to the pub-
wavelengths.
lished data,
InadditiontoPioneerdata,imagesofJupiterfromavariety
ofanglesweretakenbyVoyager,Galileo,andCassini,although
P((cid:1)0; (cid:1); (cid:4))(cid:5)AHG½fPHG(g1; (cid:4))þ(1(cid:3)f )PHG(g2; (cid:4))(cid:6) ð2Þ wearenotawareofanyotherpublisheddataofthescattering
phasefunctionsfortheJoviansurface.5Datafor(cid:4) >150(cid:2) do
not exist because these directions would risk pointing space-
(Tomasko et al. 1978; Tomasko & Doose 1984). The coeffi-
craftcamerastooclosetotheSun.Ourderivedlightcurvesare
cient A is fitted to match the amplitude of the observed
HG (cid:2)
notseverelyaffectedby ourextrapolationfor(cid:4) >150,how-
phase function. The individual terms are Henyey-Greenstein
ever, since when the forward scattering is important, the ob-
functions representing forward- and backward-scattering lobes,
served crescent is small, and the reflected light phase curve
respectively:
undergoesitsminimum.
1(cid:3)g2 2.1.2.Saturn
P (g; (cid:4))(cid:1) ; ð3Þ
HG (1þg2þ2gcos(cid:4))3=2 ThescatteringphasefunctionandalbedoofSaturnarerep-
resentedbytheBackstormlaw,whichwasusedbyDonesetal.
where (cid:4) is the phase angle, f 2½0;1(cid:6) is the fraction of the (1993)to fitobservations of Saturn’s scattering:
forwardversusbackwardscattering,andgisg1org2;g12½0;1(cid:6) (cid:1) (cid:2)
controls the sharpness of the forward-scattering lobe, while I A (cid:1)(cid:1) B
g22½(cid:3)1;0(cid:6) controls the sharpness of the backward-scattering F ¼ (cid:1) (cid:1)þ0(cid:1)0 ; ð4Þ
lobe.
Figure3showsourfitoftheHenyey-Greensteinfunctionto
where A and B are coefficients that depend on the phase
the data points from the Pioneer 10 and 11 images (Tomasko
angle (cid:4). The coefficients are fitted by Dones et al. (1993) to
etal.1978;Smith&Tomasko1984).Thisfittedfunctionalso
Pioneer11phasefunctiontables,whichwereproducedbythe
reproducesthefull-diskalbedosobservedbyKarkoschka(1994,
multiple-scattering model of Tomasko & Doose (1984). We
1998)atthewavelengthscorrespondingtotheredpassbandof
averaged the coefficients published by Dones et al. (1993)
Pioneer. Pioneer images taken with broadband blue (0.390–
separately for zones and belts through Pioneer’s blue and
0.500(cid:1)m)andred(0.595–0.720(cid:1)m)filtersshowsurfaceloca-
red filters. Table 2 gives the resulting coefficients; Figure 2
tionsonJupiterwithdifferentproperties.Inparticular,Tomasko
displays the red filter curve only. Figure 4 demonstrates the
etal.(1978)andSmith&Tomasko(1984)haveindicatedtwo
types of locations: the belts, usually seen as dark stripes on
Jupiter(Fig.3,crosses),andthezones,usuallyseenasbright 5 One of us, U. A. D., plans to work on obtaining the spectral phase
stripes on Jupiter (Fig. 3, plus signs). The relative calibration functionsfromCassininine-filtervisibleimagesintheimmediatefuture.
No. 2, 2005 LIGHT CURVES FOR EXTRASOLAR PLANETS 977
TABLE2
CoefficientsfortheBackstormFunctionforSaturn
PhaseAngle(cid:4) 0(cid:2) 30(cid:2) 60(cid:2) 90(cid:2) 120(cid:2) 150(cid:2) 180(cid:2)a
A(red,0.64(cid:1)m).................... 1.69 1.59 1.45 1.34 1.37 2.23 3.09
B(red,0.64(cid:1)m).................... 1.48 1.48 1.46 1.42 1.36 1.34 1.31
A(blue,0.44(cid:1)m)................. 0.63 0.59 0.56 0.56 1.69 1.86 3.03
B(blue,0.44(cid:1)m)................. 1.11 1.11 1.15 1.18 1.20 1.41 1.63
Note.—Coefficients for the Backstorm function for Saturn are averaged between belt and zone values
publishedinTableVofDonesetal.(1993).
a The coefficients at (cid:4)¼180(cid:2) are not constrained by observations and were estimated by linearly
extrapolatingthecoefficientsat120(cid:2)and150(cid:2).
difference between the blue and red phase functions for comingfromtheplanetandtherings(ifany)toobtainthefull-
Saturn.Thevaluesforthebluecurveare1.5–2timessmaller, disk (orgeometric) albedo p((cid:4)):
indicatingthatSaturnisdarkerinbluewavelengthsbecauseof P
higher absorption by photochemical hazes. I((cid:1); (cid:1) ; (cid:4))r2 =F
p((cid:4))¼ pix 0 pix ; ð5Þ
(cid:5)R2
2.1.3.Saturn’sRinggs P
The ring brightness in reflection (illuminated side) and where r is the pixel size and R is the planet’s radius. The
pix P
transmission (unilluminated side) is provided by a physical full-disk albedo is the planet’s luminosity L normalized by
P
scattering model of the ring. This model calculates multiple thereflectedluminosityofaLambertiandiskwiththeplanet’s
scatteringwithintheringsusingaray-tracingcode.Themodel radius at the planet’s orbital distance, illuminated and ob-
ring is populated with macroscopic bodies of size 1 m, with served from the normal direction:
optical depth and albedo profiles chosen to match those of
Dones et al. (1993). The model reproduces well the bright- p(cid:1)LP=((cid:5)RP2F): ð6Þ
nessesobservedbyVoyager1and2.Weusethecodetopredict
ringbrightnessatgeometriesnotobservedbyVoyager,binthe 2.3.Eccentric Orbits
output into a look-up table, and then use the table to produce
In addition to modeling light curves from ringed planets
the ring images.
traveling in circular orbits, we also model light curves of
The ring brightness at each point is a function of three
nonringed planets moving in eccentric orbits. Although most
angles((cid:4),(cid:1),and(cid:1) ),theopticaldepth,andthealbedo.Because
0 of the planets in the solar system orbit the Sun with low ec-
of data volume restrictions, we have binned output in rather
centricities, the extrasolar planets detected to date display a
large steps,whichdependonparametervalues.Thesestepsare wide range of eccentricities.6 As can be seen in Figure 5, the
clearlyseenintheringphasefunctioninFigure2(dottedline).
geometry of an ellipse introduces an additional parameter, in
additiontoinclination,intheobserver’sazimuthalperspective
2.2.Full-Disk Albedo
ofthesystem.Thisparameteristheargumentofpericenter,!,
To produce light curves of the fiducial exoplanets that we which is the angle (in the planet’s orbital plane) between the
model,imagesoftheplanetforasetoflocationsalongtheorbit ascendingnodeline7andpericenter(Murray&Dermott2001).
are generated. For each image, we integrate the total light To model the reflected light as a function of time, we cal-
culate the angular position of the planet and the planet-star
separationoveracompleteperiodusingasolutiontoKepler’s
equation: M ¼E(cid:3)esinE, where M is the mean anomaly (a
parameterizationoftime),Eistheeccentricanomaly(aparam-
eterizationofthepolarangle),andeistheeccentricity.Kepler’s
equation is transcendental and cannot be solved directly. Ap-
plyingtheNewton-Ralphsonmethod8toKepler’sequation,we
obtaintheiterativesolutionfortheplanet’sposition:
E (cid:3)esinE (cid:3)M
Eiþ1 ¼Ei(cid:3) i 1(cid:3)ecosiE i: ð7Þ
i
3.RESULTS
The shape of the phase light curve depends on many pa-
rameters: the planet’s orbit, its geometry relative to the ob-
server, the planet’s oblateness, the scattering properties of the
Fig.4.—ScatteringphasefunctionsforSaturninthered(0.595–0.720(cid:1)m) 6 Eccentricities of extrasolar planets are listed at http://exoplanets.org/
andblue(0.390–0.500(cid:1)m)passbandsadoptedfromDonesetal.(1993)for almanacframe.html.
the same scattering geometry as in Fig. 2. The optical depth at the sample 7 The reference plane is formed by the observer’s line of sight and its
pointontherings(1.7timesSaturn’sradiifromtheplanet’scenter)is2.3;the normalintheeclipticplane,whichisalsotheascendingnodeline.
albedooftheringparticlesis0.56.Theringisobservedfromtheilluminated 8 TheapplicationoftheNewton-RalphsonmethodtoKepler’sequationis
side. describedinMurray&Dermott(2001).
978 DYUDINA ETAL. Vol. 618
Fig.5.—Argumentofpericenter,!,anadditionalangleneededtodefinethegeometryofanonringedplanetonaneccentricorbit.
planet’s surface, and the presence and geometry of rings. To Lambertianplanetbecauseofthesharpbackscatteringpeakin
studywhetherthesesignaturescanbeidentifiedunambiguously Jupiter’s scattering phase function (at(cid:4) (cid:5)0(cid:2) in Fig.2).Near
inthelightcurve,wemodeledseveralgeometriesfortheplan- zerophase((cid:1)(cid:5) (cid:4)180(cid:2))inFigure6,Jupiterismoreluminous
etarysystem,i.e.,differentorbitaleccentricitiesefornonringed than the other two models because of the large forward scat-
planetsanddifferentringobliquities(cid:6)relativetotheeclipticfor tering from its surface ((cid:1)(cid:5)180(cid:2)) in Figure 2. Such differ-
ringed planets on circular orbits. For each of these cases, we ences in phase functions are commonly attributed to larger
modeledavarietyofobserverlocations,i.e.,differentorbitalin- particle size in the main cloud deck on Jupiter. For example,
clinations i as seen by the observer and different azimuths ! Tomaskoetal.(1978)suggestparticlesizeslargerthan0.6(cid:1)m
oftheobserverrelativetotheorbit’spericenterortotherings to explain the forward scattering.
(!).Inwhatfollows,wecomparelightcurvesforthesediffer-
ingr geometriesanddiscusswhetherornotdifferentgeometric 3.2.Ligght Curvve for Oblate Planets vversus Spherical Planets
effectscanbedistinguishedfromoneanother. Next we examine the effect of planet oblateness. Saturn’s
equatorial radius is 10% larger than its polar radius. This ob-
3.1.Ligght Curvves for Jupiter vversus a Ringgless Saturn
lateness of Saturn (6% for Jupiter) makes the planet appear
Wefirstcomparesphericalringlessexoplanetswithdifferent larger when looking at the pole than when looking at the
surface reflection characteristics on circular orbits. Figure 6
comparesedge-onlightcurvesforaringlessSaturn,aJupiter,
and a Lambertian planet. The reflected planet luminosity is
normalized by the incident stellar illumination to obtain the
full-disk albedo p as described in equations (5) and (6). The
model planets have identical radii; the curves differ only be-
cause of the differentsurface scattering ofthe three planets.
The full-disk albedo can be converted to the planet’s lumi-
nosityLPasafractionofthestar’sluminosityL(cid:7)*foraplanetof
equatorialradius R at an orbitaldistance D ,
P P
LP=L(cid:7) ¼(RP=DP)2p: ð8Þ
For example, for Saturn at 1 AU, ðR =D Þ2(cid:5)1:6;10(cid:3)7. The
P P
abscissaofFigure6indicatestheazimuthalangleoftheplanet
in its orbital plane (the orbital angle (cid:1) of Fig. 1), starting at
minimum planet phase ((cid:1)¼(cid:3)180(cid:2)). The plot can be trans-
formed into a time-dependent light curve simply by dividing Fig. 6.—Comparison of light curves for a spherical Jupiter, a spherical
(cid:1) by 360(cid:2) and multiplying by the planet’s orbital period. Saturn,andasphericalLambertian(isotropicallyscattering)planet,assuming
theplanets are ringless and have the same radiiR .The planets differ only
The light curve for Jupiter peaks much more sharply at P
intheirsurfacescatteringproperties.Albedosareshownforvisibleredlight
full phase ((cid:1)(cid:5)0(cid:2)) than the light curves of Saturn or the (0.6–0.7(cid:1)m).
No. 2, 2005 LIGHT CURVES FOR EXTRASOLAR PLANETS 979
(cid:2)
is not shown in Figure 7. Observing the planet at 45 latitude
produces a small asymmetry in the curve.
3.3.Ligght Curvve for a Ringged Planet
Rings have a large effect on the light curves of planets. To
describe the geometry of ringed planetary systems, two more
parameters are required. If we assume a fixed radial density
distribution for the rings, the geometry becomes sensitive to
the ring obliquity (cid:6) (the angle between the ring and ecliptic
planes)andtotheazimuthoftheobserverrelativetotherings
! (see Fig.1).
rFigure8showsanexamplelightcurvefori¼55(cid:2),(cid:6)¼27(cid:2),
and! ¼25(cid:2),which isa convenientgeometrytodemonstrate
r
differentringeffectsinonelightcurve.Thecartoonontheright
of Figure 8 displays images of Saturn at several positions on
theorbit.Theplotontheleftshowsthelightcurveforaringed
Fig.7.—Comparisonoflightcurvesforasphericalanda10%oblateplanet planet (Fig. 8, plus signs) compared with the curve for a
wrinitghleSsast.uTrnh’esosurbrfitaciseopbroseprevretidese.dPglea-noents(ih¼ave90th(cid:2)e).sTahmeepelqanueattsorairaelrraodtaiitianngd‘‘aorne ringless planet (Fig. 8, solid curve) with the same geometry.
theirsides’’((cid:6)¼90(cid:2)).Thealbedosshownareforvisibleredlight(0.6–0.7(cid:1)m). The (cid:8)10% spread of the points in the ringed light curve is
partiallyduetothelargestepsintheringreflectiontable(steps
are also seen in Fig. 2) and should be treated as a model un-
equator,whichaffectsthelightcurves.Figure7compareslight certainty.Foralargefractionoftheorbit,thepresenceofrings
curvesforasphericalplanet,a10%oblateplanetviewedatits increasestheluminosityoftheplanet((cid:1)<(cid:3)20(cid:2))becausethe
(cid:2)
equator,anda 10% oblate planet viewed at45 latitude. observerisabletoviewmorereflectingsurface.However,for
Allthreesampleplanetshavethesameequatorialradiusand 0(cid:2) <(cid:1)<100(cid:2), rings shadow part of the planet, producing a
scattering propertiesas Saturn.We display light curves for an lowerluminositythanthatforthenonringedplanet.Moreim-
edge-on orbit (i¼90(cid:2)) of a planet rotating ‘‘on its side’’ portantly,theringedlightcurveisasymmetric,whichmakesit
((cid:6)¼90(cid:2)), because this geometry emphasizes the effects of distinguishablefromanylightcurvesofanonringedplaneton
oblateness in the reflected light. The differences among the a circular orbit.
curves are created by differences in the observer’s azimuth The variety of light curves for a Saturn-like planet for dif-
relative to the planet’s equator, an angle analogous to ! but ferentgeometriesisillustratedinFigure9,fromwhichonecan
r
measured with respect to the equatorial plane rather than the learn several lessons. First, rings generally increase the am-
ring plane. Observing the planet at the equator decreases the plitude of the light curves by a factor of 2–3. An amplitude
crosssectionoftheplanetandthusdecreasestheamplitudeof increase due to rings could be partially confused with effects
the curve. Observing the planet at the pole yields a curve in- duetoalargerplanetsizeorlargeralbedo.Thisambiguitymay
distinguishablefromthesolidcurveforasphericalplanet;this beresolvedspectrallybecausethespectrumofringsshouldbe
Fig. 8.—Effect of Saturn’s rings if observed from 35(cid:2) above the orbital plane (i¼55(cid:2)). In this example, the obliquity of Saturn’s rings is (cid:6)¼27(cid:2). Right:
Observer’sazimuthontheeclipticseparatedfromtheintersectionoftheringandeclipticplanesby! ¼25(cid:2).Valuesof(cid:1)areindicatednexttoeachimage.The
r
brightnessesoftheimagesinthecartoonaregiveninunitsofI/F,asindicatedonthenonlineargray-scalebar.Left:Lightcurvesforaringlessplanet(solidcurve)
andforaringedplanet(crosses).
980 DYUDINA ETAL. Vol. 618
Fig.9.—LightcurvesforSaturnfordifferentgeometries.Differentringobliquities(cid:6)areshownoneachsubplotasblackcurvesofdifferentlinetype.Acurvefora
sphericalplanetwithoutringsisshowningray.Eachcolumncorrespondstoadifferentorbitalinclinationi.Eachrowcorrespondstoadifferentazimuth! ofthe
r
observerrelativetotherings.
ratherflat,whereastheplanetaryatmosphereisexpectedtobe mumitselfoccursinarathersmallfractionoftheplots.How-
dark at a set of prominent gaseous absorption bands. ever,sincetheentirecurveisasymmetric,theeffectwouldbe
Thesecondlessonisthatlightcurvesforaringedplanetare detectable as an overall deviation from the simple symmetric
asymmetric.Therearetwotypesofasymmetry.Thefirst,which curveslikelytobeusedtofitthefirstdetectionsofreflectedlight.
ispotentiallyeasiertodetect,istheoffsetofthecurve’smaxi- Ifradial velocity dataexistfor a planet,the exacttiming ofthe
mumrelativeto(cid:1)¼0(cid:2),themaximumofthelightcurvesfora (cid:1)¼0(cid:2) pointwouldbemeasurable.Inthiscase,ashiftedmaxi-
ringless planet (Fig. 9, vertical lines). The offset of the maxi- muminthelightcurvewouldyielddirectevidenceoftherings.
No. 2, 2005 LIGHT CURVES FOR EXTRASOLAR PLANETS 981
Theshiftedmaximumisastrongphotometricsignaturethat
was also noted in the ring simulation of Arnold & Schneider
(2004)asa‘‘(cid:9)-shift.’’Westress,however,thatotherprocesses
maybecapableofproducingsucha(cid:9)-shift.Similar,although
probably smaller, shifts may be induced by a seasonal bright-
ness variation on the planet. Note, however, that none of the
giant planets in the solar system have pronounced seasonal
brightnessvariations.The(cid:9)-shiftscouldalsobeproducedbya
globalasymmetryofthebrightnessdistributionovertheplanet
or by the planet’s oblateness (see x 3.2). Again, spectroscopy
mayhelptoresolvetheambiguitybetweenasymmetriescaused
by rings and those caused by processes on the planet surface.
Abrupt asymmetric changes in the fine structure of the light
curvegiveamorerobustindicationoftheringsthana(cid:9)-shift,
butresolvingsuchdetailwouldrequireanotherorderofmag-
nitude in instrumentsensitivity.
ThethirdlessonthatwecandrawfromFigure9isthatinthe
case offace-on orbits (right column), both radial velocity ob-
servations and precise photometry would give no signal for a
nonringedplanet.Aringedplanet,ontheotherhand,typically
produces a double brightness maximum, as the rings are illu-
minated first from the observer’s side and then from the back
side during the orbit (dotted curve). Double maxima can also
be produced in eccentric orbits, but these have different fine
structure(seex3.4).Wenote,however,thatadoublepeakmay
begeneratedbyseasonalvariationsorbyanunevenbrightness
distribution at the planet’s surface.
3.4.Ligght Curvve for a Planet on an Eccentric Orbit
Asanexampleoftheeffectorbitaleccentricitymayhaveon
the light curve of a planet in reflected light, Figure 10 shows
lightcurvesforaringlessplanetwiththeorbitalcharacteristics
Fig.10.—Lightcurvesforaplanetonaneccentricorbit(orbitalcharacter-
ofexoplanetHD108147bandtheequatorialradiusofJupiter. isticsofexoplanetHD108147b:a¼0:104AU,e¼0:498),assumingJupiter’s
HD 108147b provides an interesting test case because some equatorialradiusand10%oblateness.Theplanet’sluminosityL isnormalized
P
oftheveryfirstplanetstobedetectedinreflectedlightwillbe bythestar’sluminosityL andplottedvs.time.Theargumentofpericenteris
‘‘close-in’’ and yet on eccentric orbits (HD 108147b has e¼ !¼(cid:3)41(cid:2). The correspon*ding time of the planet’s maximum phase (cid:1)¼0(cid:2)
0:498eventhoughitssemimajoraxisissmallata¼0:104AU). (vertical line) is (cid:8)0.4 days after pericenter (which defines time zero). Top:
Comparison of edge-on light curvesfor planets with Jovian,Saturnian, and
WecompareLambertian,Jovian,andSaturnianscatteringprop- Lambertiansurfaces.Bottom:Comparisonofdifferentorbitalinclinationsfrom
ertiesinFigure10(top)todemonstratetheeffectofanisotropic face-ontoedge-onorientations(i¼1(cid:2)–89(cid:2),respectively).
scattering,buttheprimarymotivationforFigure10istoshow
the effect of orbital eccentricity.9 The orbital parameters as-
(cid:2)
sumedforFigure10arethosedeterminedforHD108147bby 41 .Withsuchageometry,thephase-inducedandeccentricity-
precise radial velocity measurements (Pepe et al. 2002). We induced maxima on the light curve amplify one another. As a
assume the planet to be 10% oblate, as is Saturn. We plot the result,theamplitudeofthecurvefortheedge-oncaseisabout
lightcurveoveronecompleteorbit,beginningatpericenter. 3:5;10(cid:3)5, nearly 5 times larger than for the face-on case, in
The light curves of Figure 10 can be rescaled easily for a whichonlytheorbitaldistancevariationmatters.
planetwiththesameorbitaleccentricitybutdifferentsemimajor Figure 11 illustrates the importance of the argument of
axis a by multiplying the luminosity by (a=a )2 and multi- pericenter in determining the light curves of a given system
1 1
plyingthetimeaxisbytheratiooftheorbitalperiods(a =a)3=2. measured by the observer. The light curves assume the same
1
Sincetheinclinationoftheorbitalplanecannotbedetermined orbitasinFigure10,buttheargumentofpericenterisnowset
from radial velocity measurements, we display in Figure 10 to !¼60(cid:2) rather than(cid:3)41(cid:2).Such a change in the observer’s
(bottom) light curves for inclinations between i¼90(cid:2) (edge- position causes a threefold decrease in the amplitude of the
on)and0(cid:2) (face-on).HereweuseJovianscatteringasourtest edge-on (i¼89(cid:2))lightcurves.
case because it produces the most prominent inclination fea- Forthe!¼60(cid:2) geometry,asecondarypeaklocatedcloseto
tures in the light curves. The argument of pericenter ! (see pericenterappearsontheedge-onJovianlightcurve.Iftheplanet
Fig.5)canbederivedfromradialvelocitymeasurements.For islessforwardscatteringthanJupiter(e.g.,SaturnoraLambertian
HD 108147b, !¼(cid:3)41(cid:2), which means that we are fortunate planet), the second peak does not appear. The i¼89(cid:2) and 75(cid:2)
to be at the azimuth at which the fullest planet phase (cid:1)¼0(cid:2) lightcurvesdisplayasharptroughinwhichtheamplitudeofre-
(Fig. 10, vertical line) is separated from pericenter by only flectedlightisreducedalmosttozero.Thistroughisduetothe
planetshowingnophase(‘‘newmoon’’)atthispointofitsorbit.
9 Wedonotexpectaclose-inplanetsuchasHD108147btohaveclouds 3.4.1.TemporalShiftofLigght-CurvveMaximum(cid:8)t
similartothoseofJupiterorSaturn,andwedonotattempttomodelindetail
Thepositionofthelight-curvemaximumrelativetotheperi-
thescatteringpropertiesofclose-inplanets.Ourscatteringmodelsaremore
appropriateforplanetscoveredbywaterorammoniaclouds. center (the temporal shift, (cid:8)t, in Fig. 11) can yield important
982 DYUDINA ETAL. Vol. 618
tering planet. For example, in Figure 11, a 2 day (cid:8)t can be
produced only by the Jovian planet.
To learn whether the observed (cid:8)t is high enough to restrict
atmosphericscattering,onecanexaminethethreecolumnsof
Figure 12, which summarize scattering from Jovian (left col-
umn),Saturnian(middlecolumn),andLambertianplanets(right
column).Asanexample,ife¼0:4,!¼60(cid:2),andtheobserved
temporal shift is 25% of the orbital period (green; (cid:8)t values),
only the left column, corresponding to Jovian scattering, dis-
playsgreenanywherealongthe!¼60(cid:2)verticalline;thus,only
stronglybackscatteringplanetslikeJupitercouldbeconsistent
with such an observation. Note, however, that this restriction
onatmosphericscatteringmaybecontaminatedbyringeffects,
large-scale bright patches on the planet’s surface, or seasonal
variationofthecloudcoverage.
3.4.2.Contrast:AmplitudeofLigght-CurvveVariations
Thecontrast,ordifferencebetweenthemaximumandmin-
imum amplitude of the light curve, gives a measure of the
degreeofvariabilitythattheplanet’sreflectedlightdisplays.It
is this variation that new-generation space- or ground-based
photometers may be able to detect if they can achieve the re-
quired levels of precision.
We plot the light-curve contrast for all possible orbital ori-
entationsinFigure13.Thesecontourmapsdisplaythedegree
of contrast for various orbital inclinations and arguments of
pericenter at given eccentricities. Note that the backscatter-
ing peak of Jupiter’s surface makes the planet much darker at
loworbitalinclinationsthanaSaturnianorLambertianplanet.
For a given orientation, the more eccentric orbits show much
highercontrastthandothecircularorbits,althoughtheamount
Fig.11.—LightcurvesforthesameeccentricorbitasinFig.10,exceptthat of contrast depends strongly on both the inclination and the
theargumentofpericenteris!¼60(cid:2).Thetemporalshift(cid:8)tismarkedforthe argument of pericenter. At high eccentricities, favorable ge-
Joviannearlyedge-on(i¼89(cid:2))lightcurve. ometries (such as i¼90(cid:2) and !¼(cid:3)90) can increase the
contrastbyapproximately5timesoverthoseoflessfavorable
constraints on the inclination of the orbit and the cloud cover orientations.
ofthe planet.First,asthe inclination increases,themaximum
movesfrompericentertowardthetimeofthemaximumphase 4.DISCUSSION
(cid:1)¼0(cid:2); i.e., (cid:8)t increases. Unlike the amplitude of the curve,
4.1.Uncertainties
whichisalsoanindicatorofinclination,(cid:8)tcannotbeproduced
by altering the planet’s size or albedo. However, this inter- Ourmodelprovidesaratheraccuratedescriptionofreflected
pretation can be confused by the unknown strength of back- lightcurvesforJupiterandSaturn.Thelargestuncertaintiesare
scattering atthe planet’s ‘‘surface’’ (see Fig.11,top). duetothelackofobservationsatlargephaseangles(forward
(cid:2)
Figure12summarizesthesensitivityof(cid:8)ttotheinclination scatteringat(cid:4) >150 )forthesurfacesofJupiter,Saturn,and
of the orbit and different atmospheric scattering properties. its rings. The extrapolations we have made to the phase
If radial velocity data exist for the planet, then the orbital ec- functionscouldresultinafactorofafewerrorinthebrightness
centricity e and argument of pericenter ! are known, and the attheseangles,atwhichthelightcurveshavetheirminimum.
shift,indicatedbycolorcodinginFigure12,isobservable.The The contrast in the highest amplitude light curves (edge-on
orbitalinclinationi(Fig.12,ordinate)isgenerallynotknown, orbits) is constrained to within a few percent by high spectral
unless transitor astrometric data are available. resolution,ground-basedobservationsofSaturnandJupiter.In
Our method of constraining the orbital inclination i using our study, the uncertainty of these maximum amplitudes is
Figure12isachievedbymatchingthemeasured(cid:8)tvalueswith 10%–20%becausethebrightnessofaplanetvarieswithwave-
those along the vertical line corresponding to the known ! length by about this amount within the 0.6–0.7 (cid:1)m spectral
ontheplotwiththeappropriateeccentricitye,asmeasuredby range (Karkoschka 1998).
radial velocity techniques. An ambiguity remains because of Reflectivityoftheringsmaybeinerrorbyasmuchas50%
theunknownscatteringpropertiesoftheplanet’ssurface,which because many geometries, especially those for face-on ring
cansometimesberesolvedbycomparingthedetailedshapeof illuminationandface-onringobservation,arenotconstrained
thelightcurves. by observations. New observations by the Cassini spacecraft
Second,(cid:8)tmayalsoservetoconstrainthesurfaceproperties willfillthisgapinthedata.Anothersourceofringerroristhe
of the planet. In some cases, a lower limit may be put on the scatteringpropertiesofthecoarsegridinthetabulatedring(see
strength of backscattering from the planet’s surface. For any x 2.1.3). This grid-induced noise is usually below 30% of the
scattering surface, the maximum possible (cid:8)t occurs on the ring’s luminosity.
edge-on orbit (i¼90(cid:2)); see examples in Figure 11 (top). The Rings around close-in, short-period exoplanets may be un-
largest (cid:8)t values are possible only from a strongly backscat- likely because of tidal disruption and in any case can consist