Measuring the Oblateness and Rotation of Transiting Extrasolar Giant Planets

13 Pages·0.32 MB·English

by  Jason W. Barnes

Measuring the Oblateness and Rotation of Transiting Extrasolar Giant
Planets
Jason W. Barnes and Jonathan J. Fortney
Department of Planetary Sciences, University of Arizona, Tucson, AZ, 85721
[email protected], [email protected]
3
0 ABSTRACT
0
2 We investigate the prospects for characterizing extrasolar giant planets by measuring planetary oblateness
from transit photometry and inferring planetary rotational periods. The rotation rates of planets in the solar
n
systemvarywidely,reflectingthe planets’diverseformationalandevolutionaryhistories. Ameasuredoblateness,
a
J assumedcomposition,andequationofstateyieldsarotationratefromtheDarwin-Radaurelation. Thelightcurve

9 of a transiting oblate planet should differ significantly from that of a spherical one with the same cross-sectional
areaunderidenticalstellarandorbitalconditions. However,ifthe stellarandorbitalparametersarenotknowna

1 priori, fitting for them allows changes in the stellar radius, planetary radius, impact parameter, and stellar limb
v
darkening parameters to mimic the transit signature of an oblate planet, diminishing the oblateness signature.
6
Thus even if HD209458bhad an oblateness of 0.1 instead of our predicted 0.003, it would introduce a detectable
5
1 departure from a model spherical lightcurve at the level of only one part in 105. Planets with nonzero obliquity
1 break this degeneracy because their ingress lightcurve is asymmetric relative to that from egress, and their best-
0 case detectability is of order 10−4. However, the measured rotation rate for these objects is non-unique due to
3
degeneracy between obliquity and oblateness and the unknown component of obliquity along the line of sight.
0
Detectability of oblateness is maximized for planets transiting near an impact parameter of 0.7 regardless of
/
h obliquity. Futuremeasurementsofoblatenesswillbechallengingbecausethe signalisnearthephotometriclimits
p
of current hardware and inherent stellar noise levels.
-
o
Subject headings: occultations — planets and satellites: general — planets and satellites: individual(HD209458b)
r
t
s
a 1. INTRODUCTION Rotation causes a planet’s shape to be flattened, or
: oblate, by reducing the effective gravitationalacceleration
v
i Discovery of a transiting extrasolar planet, HD209458b at the equator (as a result of centrifugalacceleration)and
X
(Charbonneau et al. 2000; Henry et al. 2000), has pro- by redistributing mass within the planet (which changes
r vided one mechanism for researchers to move beyond the gravity field). Oblateness, f, is defined as a function
a
the discovery and into the characterization of extraso- of the equatorial radius (Req) and the polar radius (Rp):
lar planets. Precise Hubble Space Telescope measure-
R R
ments of HD209458b’s transit lightcurve revealed the ra- f eq− p. (1)
dius (1.347 0.060RJup) and orbital inclination (86.68◦ ≡ Req
0.14◦, from±impact parameter 0.508) of the planet, which±,
ForJupiter andSaturn,highrotationratesandlowdensi-
along with radial velocity measurements, unambiguously
ties resultinhighlyoblate planets: f =0.06487and
determinetheplanet’smass(0.69 0.05M )anddensity Jupiter
(0.35g cm−3; Brown et al. 2001±). Of thJuepover 100 ex- fSaturn =0.09796,compared to fEarth =0.00335.
trasolar planets discovered so far, this large radius makes Anearlierinvestigationofthedetectabilityofoblateness
HD209458b the only one empirically determined to be a intransitingextrasolarplanets waspublishedbySeager&
gas giant. Hui (2002). Our work represents an improvement over
Seager & Hui (2002) in the use of model fits to compare
Knowledge of a planet’s cross-sectional area provides a
oblateandsphericalplanettransits,theuseoftheDarwin-
zeroeth order determination of its geometry,and the HST
Radau relation to associate oblateness and rotation, and
photometry is precise enough to constrain the shape of
a thorough investigation of the degeneracies involved in
HD209458b to be rounded to first order. While a planet
fitting a transit lightcurve.
transits the limb of its star, the rate of decrease in appar-
ent stellar brightness is related to the rate of increase in In this paper, we investigate the reasons for measuring
stellarsurfaceareacoveredbythe planetinthe same time planetaryrotationrates,therelationshipbetweenrotation
interval. We investigate whether it is possible to use this rate and oblateness for extrasolar giant planets, the effect
informationtodeterminetheshapeoftheplanettosecond of oblateness on transit lightcurves, and the prospects for
order. determinating the oblateness ofa planetfromtransitpho-
tometry.
2. EXOPLANETS AND ROTATION
100
Whileknowledgeofaplanet’smassandradiusprovides
informationregardingcompositionandthermal evolution,
measurements of rotation and obliquity promise to con-
straintheplanet’sformation,tidalevolution,andtidaldis-
o sipation. Whatthesedatamightrevealaboutaplanetde-
ati pends on whether the planet is unaffected by stellar tides,
R
bit 10 slightly affected by tides, or heavily influenced by tides.
r
o
− 2.1. Tidally Unaffected
n
pi
S Q = const The present-day rotation rate of a planet is the prod-
uct of both the planet’s formationandits subsequentevo-
Q α frequency−1
lution. Planets at sufficiently large distances from their
parent stars are not significantly affected by stellar tides,
thustheseobjectsrotatewiththeirprimordialratesaltered
1 byplanetarycontractionandgravitationalinteractionsbe-
0.0 0.2 0.4 0.6 0.8 1.0 tween the planet, its satellites, and other planets. To the
Eccentricity degreethataplanet’srotationalangularmomentumispri-
mordial, it may be diagnostic of the planet’s formation.
Fig. 1.— Spin to orbital mean motion ratios for tidally Planets formed in circular orbits from a protoplanetary
evolved fluid planets in equilibrium. The solid line disk inherit net progradeangularmomentum fromthe ac-
represents the ratio as calculated under the assump- creting gas, resulting in rapid prograde rotation (Lissauer
tion of a frequency-indenpendant tidal dissipation fac- 1995). Planets that form in eccentric orbits receive less
tor Q, and the dashed line is calculated assuming Q prograde specific angular momentum than planets in cir-
frequency−1. Under these assumptions, extremely eccen∝- cular orbits, and as a result they rotate at rates varying
tricplanetHD80606b(e=0.93)would,ifallowedto come from slow retrograde up to prograde rotations similar to
to tidal equilibrium, rotate over 90 times faster than its those of circularly accreted planets (Lissauer et al. 1997).
mean orbital motion! Thus,comparingthecurrent-dayrotationratesforplanets
in circular and eccentric orbits may reveal whether extra-
solar planets formed in eccentric orbits or acquired their
0.30
eccentricity later from dynamical interactions with other
planets or a disk.
The orientation of a planet’s rotational axis relative to
0.28
no core the vector perpendicular to the orbital plane, the planet’s
obliquity or axial tilt t, can also provide insight into the
planet’s formation mechanism. Jupiter’s low obliquity
0.26 (3.12◦) has been suggested as evidence that its formation
2
R 20 M core
M Earth
C/
0.24
t
0.22 θ
ρ b
1st 2nd 3rd 4th
0.20
1 10 w
Planet Mass (M )
Jup
d
Fig. 2.— The moment of inertia coefficient, C, as a func-
tion of planet mass for hypothetical generic 1.0 R ex-
Jup
trasolar giant planets. Extrasolar giant planets may or η
may not possess rocky coresdepending on their formation l
mechanism, so we plot C for both no core (upper curve)
and an assumed 20 M⊕ core (lower curve). Fig. 3.— Anatomy of a transit, after BCGNB.
wasdominatedbyorderlygasflowratherthanthestochas-
tic impacts of accreting planetesimals (Lissauer 1993).
Tidally unevolved extrasolar planets determined to have
high obliquities could be inferred either to have formed
differently than Jupiter or to have undergone large obliq-
uity changesashasbeen suggestedforSaturn(t=26.73◦;
Ward & Hamilton 2002).
−2 −1 0 1 2
0.0002
2.2. Tidally Influenced
0.0001
Tidalinteractionbetweenplanetsandtheirparentstars
0.0000
slows the rotation of those planets with close-in orbits
−0.0001 b = 0.0, 0.3, 0.6 (Guillot et al. 1996). This tidal braking continues until
−2 −1 0 1 2
the net tidal torque on the planet becomes zero. Whether
−00..00000022
x a planet reaches this end state depends on its age, radius,
u
pherical Fl 00..00000001 semTihmearjaotreaoxfist,idaanldbrtahkeinpglaanlesto:destpaernmdsaossnrtahteiop.arameter
S
Oblate − −−000...000000000221 b = 0.7 Qth,ewphlainchet.reTprheesevnatlsuethoef Qintiesrnpaolortliydaclondsistsriapianteidonevwenithfoinr
the planets in our own solar system (Goldreich & Soter
0.0001 1966). Nevertheless, measurements of extrasolar planet
0.0000 rotationratescouldconstrainQfortheseplanetsbasedon
the degree of tidal evolution that has taken place (Seager
−0.0001
b = 0.8, 0.9, 1.0 & Hui 2002).
−0.0002
−2 −1 0 1 2 Tidal braking for objects with nonzero obliquity can,
Time from Mid−transit (hours) somewhat counterintuitively, act to increase an object’s
obliquity. Tidaltorquesreducethecomponentofaplanet’s
Fig. 4.— Oversimplified model of the effect of oblate-
angular momentum perpendicular to the orbital plane
ness on a transit lightcurve. We plot the differences be-
faster than they reduce the component of the planet’s an-
tweenthelightcurveofanoblateplanetandthelightcurve gular momentum in the orbital plane (Peale 1999). This
of a spherical planet with the same cross-sectional area,
occurs because at solstice the planet’s induced tidal bulge
Ff=0.1(t) Ff=0.0(t)whileholdingallothertransitparam- isnotcarriedawayfromtheplanet’sorbitalplanebyplan-
−
eters, R∗, Rp, b, and c1 the same and setting these values etary rotation. Therefore for large fractions of the year
equaltothosevaluesmeasuredbyBCGNBforHD209458b.
stellar tidal torques do not act to right the planet’s spin
The top panel plots the lightcurve differential for impact axis,whiletorquesthatreducetheangularmomentumper-
parameters b =0.0 (solid line), b =0.3 (dashed line), and
pendiculartotheorbitalplaneactyear-round. Asaresult,
b = 0.6 (dot-dashed line). An oblate planet for b < 0.7
planetsthathaveundergonepartialtidalevolutioncanex-
encounters first contact before, and second contact after, hibit temporarily increased obliquity as the planet’s rota-
theequivalentsphericalplanet,resultingintheinitialneg-
tionratedecreases. Eventually,suchaplanetreachesmax-
ativeturnforFf=0.1(t) Ff=0.0(t),andsubsequentpositive imumobliquityandthereafterapproachessynchronousro-
−
section of the curve for these impact parameters. In the
tation and zero obliquity simnultaneously. Planets under-
bottompanel,weplotthedifferentiallightcurveforb=0.8
goingtidalevolutionmaybeexpectedtohavehigherobliq-
(solid line), b=0.9 (dashed line), andb=1.0 (dot-dashed uities on average than planets retaining their primordial
line). For b > 0.7, under otherwise identical conditions,
obliquity.
an oblate planet first encounters the limb of the star af-
ter the equivalent spherical planet, and last touches the
2.3. Tidally Dominated
limb later on ingress. For b = 0.9, because Rp > 0.1 R∗
thereisnosecondcontact,i.e. thetransitispartial,sothe
The end state of tidal evolution for planets in circular
flux differential does not return to near zero,even at mid-
orbits is a 1 : 1 spin-orbit synchronization between the
transit. The middle plot shows the lightcurve differential
planet’s rotation and its orbital period, along with zero
at the changeoverpoint between these two regimes, where
obliquity. However, most of the extrasolar planets discov-
b=0.7.
eredthusfarareoneccentricorbits(Marcyetal.2000)(we
sometimes shorten ’planets on eccentric orbits’ to ’eccen-
tricplanets’eventhoughtheeccentricityisnotinherentto
the planet). Thus these eccentric planets will never reach
1 : 1 spin-orbit coupling as a result of tidal evolution be-
causethetidaltorque(Eq. 2)ontheplanetfromitsstaris
muchstronger(duetother−6 dependence)nearperiapsis.
The tidal torque between a planet and star is given by 3. ROTATION AND OBLATENESS
(Murray & Dermott 2000)
Rotation affects the shape of a planet via two mecha-
3k G M2 R5
τp−∗ = − 2 2 Q r6∗ sgn(Ω−φ˙) , (2) nthisemhsi:gghrearvivteylomciutsytaptrotvhiedeecqeunattroiprectaaulsaecscetlhereatpiloann,etthutos
bulge by transfer of mass from polar regions; and, secon-
where k is the planet’s Love number, R its radius, Ω its
2 darily, the redistributed mass alters the planet’s gravita-
rotation rate (in radians per second), φ˙ the instantaneous
tional field and attracts even more mass towardthe equa-
orbitalangularvelocity(alsoinradianspersecond),andr
torial plane. The ratio of the required centripetal accel-
is the instantaneousdistance betweenthe planetand star.
eration at the equator to the gravitationalacceleration, q,
The function sgn(x) is equal to 1 if x is positive and 1 if
− representsthe relativeimportanceofthe centripetalaccel-
xisnegative. Themagnitudeofthestellarperturbationis
eration term:
proportionaltoGM∗2,theproductofthestellarmass(M∗) Ω2R3
eq
squared and the gravitationalconstant (G). The planet is q = , (3)
GM
spun downby tidal torquesif its rotationis fasterthan its p
orbital motion (Ω>φ˙), and it is spun up if its rotation is where Ω is the rotation rate in radians per second, Mp is
slower than the orbital motion (Ω<φ˙). the mass of the planet, and Req is the planet’s equatorial
radius (Murray & Dermott 2000).
If an eccentric planet were in a 1 : 1 spin-orbit state,
it would be spun up by the star when its orbital angular We use the Darwin-Radau relation to relate rotation
velocityisgreaterthanaveragenearperiapsis,anditwould and oblateness accounting for the gravitationalpull of the
bespundownwhentheorbitalangularvelocityislownear shifted mass: (Murray & Dermott 2000):
apoapsis. Thetotalpositivetidaltorqueinducedwhilethe
planet is in close will exceed the negative torque while it C 2 2 5q 1/2
C = 1 1 (4)
is far away, despite the shorter time spent near periapsis. ≡ MpRe2q 3" − 5(cid:18)2f − (cid:19) #
Thustobeinrotationalequilibriumwithrespecttostellar
tides, an eccentric, fluid planet must rotate faster than its where C is the planet’s moment of inertia around the
mean motion. rotational axis and C is shorthand for CM−1R−2. The
p eq
Darwin-Radau relation is exact for uniform density bod-
The Earth’s moon avoids this fate because it has a ies (C = 0.4), but is only an approximation for gas giant
nonzerocomponentofitsquadrupolemomentinitsorbital planets (C 0.25; Hubbard 1984).
plane. ThetorquethattheEarthexertsonthispermanent ∼
bulge exceeds the net tidal torque imparted on the moon By combining Equation 3 and Equation 4, we arrive at
duetoitseccentricorbit,keepingtheMooninsynchronous a relation for rotation rate, Ω, as a function of oblateness,
rotation (Murray & Dermott 2000). Fluid planets, how- f:
ever, have no permanent quadrupole moment and thus 2
fGM 5 3 2
have no restoring torque competing with the stellar tidal Ω= p 1 C + . (5)
torques (Greenberg & Weidenschilling 1984). Tidal evolu- vu Re3q "2(cid:18) − 2 (cid:19) 5#
u
tion ceases for these bodies when the net torque per orbit t
For our solar system, the Darwin-Radau relation yields
is zero, which can only be achieved by supersynchronous
rotation periods accurate to within a few percent (Table
rotation.
1) using model-derived moments of inertia from Hubbard
The precise rotation rate necessary to balance the tidal & Marley (1989).
torques over the eccentric orbit depends on how Q varies
Extending the Darwin-Radau relation to extrasolar
with the tidal forcing frequency (the difference between
planets requires estimation of the appropriate moment
the rotation rate and instantaneous orbital angular veloc-
of inertia coefficients, C, for those planets. For transit-
ity, Ω φ˙ ). Conventionally, Q has been assumed to be
p− p ing planets whose masses and radii are known, we use a
either independent of the forcing frequency or inversely
self-consistent hydrodynamic model of the planet and as-
proportionaltoit(Goldreich&Peale1968). Theresulting
sumptions about its composition to estimate C following
equilibrium spin-orbit ratios as a function of eccentricity
Fortney & Hubbard (2002). To first order, C is indepen-
for each of these cases are plotted in Figure 1.
dent of oblateness due to similar symmetry around the
Probably both of these assumptions for the behavior rotationalaxis,so ourhydrodynamic modeldoes notneed
of Q are too simple. In particular, if the behavior of Q to explicitly incorporate oblateness. The Darwin-Radau
changes under a varying tidal forcing frequency (Ω φ˙ relation and spherically symmetric hydrodynamic models
p p
−
is a function of time), then the tidal equilibrium rotation provide sufficient precision for the current work; however,
rate would differ significantly from that plotted in Figure to estimate the oblateness as a function of rotation more
1. Measurement of rotational rates of eccentric extrasolar robustly, a two-dimensional interior model involving both
planets in tidal equilibrium could, in principle, either dif- rotational and gravitational forces should be used (e.g.,
ferentiate between these two models or suggest other fre- Hubbard & Marley 1989).
quency dependences, shedding light on the yet unknown
Under the spherically symmetric assumption, we calcu-
mechanism for tidal dissipation within giant planets.
late the C of Jupiter to be 0.277 with no core and 0.253
witha20M⊕ core,andwecalculatethe CofSaturnto be only 0.00007. Measuring oblateness will therefore only be
0.225 with a core (we are unable to calculate the interior practicalfortransitingplanetsandnotforothertransiting
structureofSaturnwithoutacoreduetodeficienciesinour low-luminosity bodies.
knowledge of the equationof state). Using these moments
We also note that the measurement of oblateness along
of inertia instead of the measured ones listed in Table 1
with an independent measurement of a planet’s rotation
yields similar rotation errors of a few percent. We apply
rate Ω would determine the planet’s moment of inertia.
our model to generic 1.0 R extrasolar giant planets of
Jup
This would provide a direct constraint on the planet’s
varying masses and architectures in Figure 2.
internal structure, possibly allowing inferences regarding
ForHD209458b,ourmodelscalculateCtobe0.218with the planet’s bulk helium fraction and/or the presence of a
no core and 0.185 with a 20 M⊕ core. To estimate the rocky core.
oblateness of HD209458b assuming synchronous rotation,
we rearrange Equation 5 to solve for f,
4. OBLATENESSANDTRANSITLIGHTCURVES
−1
Ω2 R3 5 3 2 2
f = 1 C + , (6) 4.1. Transit Anatomy
G Mp "2(cid:18) − 2 (cid:19) 5#
Brownetal.(2001)(hereafterBCGNB)investigatedthe
and then use the synchronous rotation rate Ω = 2.066
10−5 radians per second to obtain f = 0.00285 with n×o detailed structure of a transit lightcurve while studying
the Hubble Space Telescope lightcurve of the HD209458b
core and f = 0.00256 with a 20 M⊕ core. These results transit. Figure 3 relates transit events to corresponding
imply an equator-to-pole radius difference of 200 km,
∼ features in the lightcurve, modeled after BCGNB Figure
which, although small, is still comparable to the atmo-
4. The flux from the star begins to drop at the onset of
spheric scale height at 1 bar, 700 km.
∼ transit, known as the first contact. As the planet’s disk
Showman& Guillot(2002)suggestthat zonalwinds on moves onto the star, the flux drops further, until at sec-
HD209458bmayoperateatspeedsupto 2km s−1inthe ond contact the entire planet disk blocks starlight. Third
prograde direction, and Showman & Gu∼illot (2002) go on contact is the equivalent of second contact during egress,
to show that these winds might then spin up the planet’s and fourth contact marks the end of the transit. Due to
interior, possibly to commensurate speeds of several hun- stellar limb darkening the planet blocks a greater fraction
dred m s−1 up to a few km s−1 (though the model of of the star’s light at mid-transit than at the second and
Showman & Guillot (2002) does not treat the outer lay- third contacts, leading to curvature at the bottom of the
ers and interior self-consistently). This speed is a non- transit lightcurve.
negligiblefractionoftheorbitalvelocityaroundtheplanet
at the surface, 30 km s−1, and is also comparable to the For a given stellar mass, M∗, the total transit dura-
tion, l, is a function of the transit chord length and the
planet’s rotationalvelocityatthe equator,2.0km s−1. As
orbital velocity. We assume a circular orbit, which fixes
such, if radiatively driven winds prove to be important on
the planet’s orbitalvelocity. The chordlength depends on
HD209458b they would affect the planet’s oblateness. We
the stellar radius, R∗, and the impact parameter, b. The
use the rotation rate implied by the Showman & Guillot
impactparameterrelatestoi,theinclinationoftheorbital
(2002) calculations to provide an upper limit for the ex-
plane relative to the plane of the sky, as
pected oblateness ofHD209458b. If the entire planet were
spinningatitssynchronousrateplus2km s−1atthecloud acos(i)
b= | | , (7)
tops, the rotational period would be halved to 1.8 days, R∗
with a corresponding oblateness of 0.0109 and 0.0098 for
the no core and core models respectively. where a is the semimajor axis of the planet’s orbit.
During revisionof this paper, Konackiet al. (2003) an- The duration of ingress and egress, w, is the time be-
nounced the discovery of a second transiting planet. This tween first and second contact, and is a function of Rp,
new planet, OGLE-TR-56b, has a radius of 1.3 RJup, a R∗, and b (or i). Transits with b∼0 have smaller w than
mass of0.9 R , andan extremelyshortorbitalperiodof transitswithhigherb. Fortransitswithb 1,calledgraz-
Jup ∼
1.2 days. Although these parameters are less constrained ing transits, w is undefined because there is no second or
than those for HD209458b, we proceed to calculate that third contact.
the oblateness of this new object should be 0.017 with no
coreand0.016witha20M⊕ core,forCof0.228and0.204 R Tishtehteortaalditurasnosfitthdeepptlahn,edt,(fiexxecsepthteinrathtieocRaspe/Rof∗g,rwahzienrge
respectively. p
transits).
Current ground-based transit searches detect low-
The magnitude of the curvature at the bottom of the
luminosity objects like brown dwarfs and low-mass stars
transitlightcurve,η,determinesthestellarlimbdarkening.
in addition to planets. However, the high surface grav-
We use limb darkening parameters u and u , or c and
1 2 1
ity for brown dwarfs and lower main sequence stars leads
c , which we define mathematically in Section 4.2.
2
to low values of q and very small oblatenesses for those
objects. For a 13M brown dwarf with 1.0R in an
Jup Jup
HD209458b-like 3.52 day orbit, the expected oblateness is
4.2. Methods
We calculate theoretical transit lightcurves by compar-
ing the amountof stellar flux blockedby the planet to the
totalstellarflux. Therelativeemissionintensityacrossthe
disk of the star is greatest in the center and lowest along
the edges as a resultof limbdarkening. Many parameteri-
zationsoflimbdarkeningexist(seeClaret2000);however,
weusetheoneproposedbyBCGNBbecauseitisthemost
appropriate for planetary transits. −2 −1 0 1 2
3×10−5
The emission intensity at a given point on the stel- 2×10−5 b = 0.0, 0.3, 0.5
lar disk, I, is parameterized as a function of µ = 1×10−5
cos(sin−1(ρ/R∗)), where ρ is the projected(apparent)dis- 0
−1×10−5
tance between the center of the star and the point in
question. BCGNB defined a set of two limb darkening co- −2×10−5−2 −1 0 1 2
−33××1100−−55
efficients, which we call c and c , that are are equivalent
1 2 ux 2×10−5 b = 0.6, 0.7, 0.8
to Fl
II((µ1)) =1−c1(1−µ)2(2−µ) +c2(1−2µ)µ . (8) pherical 1×10−05
− S−1×10−5
Tdehsecraibdevsanthtaegmeaogfntihtuisdelimofbthdeadrkaernkeinnginfgu,nwchtiiolenci2sitshaactocr1- Oblate −−323×××111000−−−555
rectionforcurvature. This makesthe BCGNB coefficients 2×10−5 b = 0.9, 1.0
particularly useful for fitting transit observations because 1×10−5
a good fit to data can be achieved by fitting only for the 0
c1 coefficient. −1×10−5
−2×10−5
Our algorithm for calculating the lightcurve takes ad-
−3×10−5
vantage of the symmetry inherent in the problem: that −2 −1 0 1 2
I(µ) depends only on ρ and not on the angle around the Time from Mid−transit (hours)
star’s center, θ. We evaluate the apparent stellar flux at
Fig. 5.— Detectable difference between the lightcurve of
timeτ, F(τ), relativetothe out-of-transitfluxF ,bysub-
0
anoblate(f =0.1)planetandthebest-fitsphericalmodel,
tracting the amount of stellar flux blocked by the planet
from F : fitting for R∗, Rp, b, and c1. Higher b combined with
0
R∗ largervaluesforR∗ andRp simulatethelenghenedingress
F0 = 2πI(ρ) dρ , (9) and egress of an oblate planet, diminishing the difference
Z0 between oblate and spherical planets from Figure 4 for
R∗
planets with b < 0.7 (upper panel: solid line, b = 0.0;
F = 2πI(ρ) x(ρ,τ) dρ , (10)
blocked
Z0 dashed line, b = 0.3; dotted line, b = 0.5). For planets
and near b = 0.7, the length of ingress and egress cannot be
F F
F(τ)= 0− blocked , (11) simulated by higher b, and as a result the transit signal is
F0 highestforthese planets(middle panel: solidline, b=0.6;
wherex(ρ,τ)isthefractionofaringofradiusρandwidth dashed line, b = 0.7; dotted line, b = 0.8). Above the
dρ covered by the planet at time τ. In effect, we split critical value, b > 0.7, the oblate planet’s signal can be
the star up into infinitesimally small rings and add up the simulated by lowering b for the spherical planet fit, reduc-
fluxes inEquation9,thenwe determine howmuchofeach ing the detectability of oblateness (lower panel: solid line,
of these rings is covered by the planet in Equation 10. b =0.9; dashed line, b =1.0). It is very difficult to deter-
We calculate the integrals numerically using Romberg’s minetheoblatenessforplanetsinvolvedingrazingtransits
method (Press et al. 1992); x(r,t) is evaluated numeri- (b 1.0). The magnitude of the detectability difference is
∼
cally as well — there is no closed form general analytical proportional to f to first order, hence to estimate the de-
solution to the intersection of an ellipse and a circle. tectability of a planet with arbitrary oblateness, multiply
This algorithmis more efficient than the raster method the differences plotted here by 0f.1R2HDR202p9458b.
usedbyHubbardetal.(2001)forplanetstreatedasopaque
disks because the use of symmetry and Romberg integra-
tion minimize the number of computations of the stellar
intensity, I(µ).
4.3. Results
To illustrate the effect oblateness has on a tran-
sit lightcurve, we calculate the difference between the
lightcurveofanoblate, zeroobliquity planetand thatofa
sphericalplanetwiththesamecross-sectionalarea. Forfa-
miliarity, we adopt the transit parameter values measured
by BCGNB for the HD209458b transit: R = 1.347R ,
p Jup
R∗ = 1.146R⊙, and c1 = 0.640. Plots of the oblate-
spherical differential as a function of impact parameter
are shown in Figure 4.
For nearly central transits, an oblate planet encounters
first contact before, and second contact after, the equiv-
alent spherical planet. This situation causes the oblate
planet’stransitlightcurveinitiallytodipbelowthatofthe
spherical planet. However, near the time when the planet
center is covering the limb of the star, each planet blocks
−2 −1 0 1 2
3×10−5 the same apparent stellar area and the stellar flux differ-
2×10−5 b = 0.1, 0.3, 0.5 enceiszero. Asthetwohypotheticaltransitsapproachsec-
1×10−5 ond contact, this trend reverses and the spherical planet
0 starts blocking more light than the oblate one until the
−1×10−5 oblate planet nearly catches up at second contact. Be-
−2×10−5−2 −1 0 1 2 tweensecondandthirdcontacts,thelightcurvedifferences
−33××1100−−55 are slightly nonzero because the two planets cover areas
Flux 2×10−5 b = 0.6, 0.7, 0.8 of the star with differing amounts of limb darkening. The
pherical 1×10−05 ltihgehmtcsuerlvveessianreresvyemrsmeeutproicn, esgurcehsst.hat these effects repeat
− S−1×10−5
blate −2×10−5 Athighimpactparameters(nearlygrazingplanettran-
O−33××1100−−55 sits) the opposite occurs. First contact for the oblate
2×10−5 b = 0.9, 1.0 planet occurs after that for the spherical planet, because
1×10−5
the pointoffirstcontactonthe planetisclosertothe pole
0
than to the equator. In this scenario, the oblate planet’s
−1×10−5
transitflux starts higher than,becomes equivalent to,and
−2×10−5
thendropsbelowthatofthesphericalplanetbeforereturn-
−3×10−5
−2 −1 0 1 2 ing to near zero for the times between second and third
Time from Mid−transit (hours) contact. In the case of a grazing transit, this sequence is
truncated because there is no second or third contact.
Fig. 6.— Detectable difference between the lightcurve of
an oblate (f =0.1) planet and its best-fit spherical model The boundary between these two regions occurs when
thelocaloblateplanetradiusatthepointoffirstcontactis
with 5 parameters: R∗, Rp, b, c1, and c2. As in Figure
5, differing values for R∗, Rp, and b for a spherical planet equaltoRea,theradiusoftheequilvalentsphericalplanet.
Forplanetsthataresmallcomparedwiththesizesoftheir
transitcanallowittomimicthetransitofanoblateplanet.
With the addition of c2, the fit is much better for planets stars (Rp ≪ R∗) and that have low oblateness (f . 0.1),
transiting at low impact parameter. (Upper panel: solid this transition occurs when θ = π/4 (from Figure 3) and
b=√2/2=0.707. The flux difference between transits of
line, b = 0.0; dashed line, b = 0.3; dotted line, b = 0.5.
oblateplanetsandthatforsphericalplanets,allelsebeing
Middle panel: solid line, b = 0.6; dashed line, b = 0.7;
equal, is at a minimum at this point and deviates from
dotted line, b = 0.8. Lower panel: solid line, b = 0.9;
zero because the rate of change in stellar area covered is
dashed line, b=1.0.) The differences for the 5 parameter,
likethoseforthe4parametermodel,varyas f R2p . different for the two planets. First, second, third, and
0.1R2HD209458b fourth contacts all occur at nearly the same time for each
planet. However, if all else is not equal, as is usually the
case since the stellar parameters are poorly constrained,
then this result is misleading and does not represent the
detectability of planetary oblatness.
5. TRANSITLIGHTCURVESANDEXOPLAN-
ETS
To test whether these flux differences are detectable,
we use a Levenberg-Marquardt curve-fitting algorithm
adapted from Press et al. (1992) to fit model transits
to both the HST HD209458b lightcurve and hypothetical
model-generated lightcurves by minimizing χ2. As a test,
we fit the HST HD209458b transit lightcurve and obtain
1.4 R R∗ = 1.142R⊙, Rp = 1.343RJup, i = 86.72o (b = 0.504),
* c = 0.647, and c = 0.065 with a reduced χ2 of 1.05,
1 2
−
consistent with the values obtained by BCGNB.
1.2 Rp
In order for planetary oblateness to have a noticeable
rs 1.0 effect on a transit lightcurve, it must be distinguishable
e
et from the lightcurve of a spherical planet. For a spherical
m
a 0.8 planet transit model, the combination of transit parame-
r
a ters that correspond to the lightcurve that best simulates
P c
1
d 0.6 the data from an actual oblate planet transit become the
e
Fitt b mbeeassimurieladrvtaolutehse,aacntduatlhveaselumese.aTsuhreerdefqoureanttoitcieosnsmidaeyrnthoet
0.4
detectability of planetary oblateness, we compare oblate
planet transit lightcurves to those of the best-fit spherical
0.2
planet lightcurve instead of to the lightcurve of a planet
that differs from the actual values only in the oblateness
0.0
−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 parameter (as we did in Section 4.3).
Oblateness
5.1. Zero Obliquity
Fig. 7.—Best-fitparametersresultingfromfittingasimu-
latedtransitlightcurveforaplanetwithoblatenessf using We compare a model transit of an oblate planet (f =
asphericalplanettransitmodel. Due tothe degeneracyin 0.1)with zeroobliquity, as is the casefor a tidally evolved
measuring transit parameters R∗, Rp, and b, and oblate- planet,tothetransitofthebest-fit sphericalplanetinFig-
nessf,the bestsphericalfittothe oblatedata havelarger ure 5 and Figure 6. The oblate planet transit signature
radii and higher impact parameters than the actual val- is muted in each when compared to Figure 4 due to a de-
ues (see Section 5.1). Objects with negative values of f generacybetweenthe fittedparametersR∗, Rp,b,andthe
are prolate, a physically unreasonableproposition that we oblatenessf. Thisdegeneracyisintroducedbythe uncon-
include here for completeness. strained nature of the problem: in essence, we are trying
to solve for 5 free parameters, R∗, Rp, b, c1, and f, given
just 4 constraints, d, w, l, and η assuming (as BCGNB
1.0 did) knowledge ofthe stellarmass M∗. Without assuming
avalueforM∗,absolutetimescalesfortheproblemvanish,
yielding a similar conundrum of solving for Rp/R∗, b, c1,
andf givenonlyd,η,andw/l. Theprevioustwosentences
s 0.8
er areintendedonlytobesimplifications,asathigherphoto-
et db metric precision more information about the conditions of
m df
a 0.6 the transitis available. Hereafter we assume knowledge of
r
Pa M∗,thoughthisanalysiscouldalsohavebeendonewithout
d this assumption, or with an assumed relation between M∗
e in Fitte 0.4 ddRfp Rp iaanlntdeRrR∗in,∗gRatphs,epairnnodgprobessemsdaimbnydicCetgohrdeesyssi&tginmSaealssoswfelhaoinvleo(m2b0laa0it2ne)t.apiClnaihnnaegnttgbheyes
g total transit duration by keeping the chord length con-
n 0.2 dR
ha df*R* stant.
C
For planets transiting at low impact parameter (b <
0.0 0.7),anoblateplanet’slongeringressandegress(higherw
0.2 0.4 0.6 0.8 from Figure 3) are fit better using a spherical model with
dc
1 Actual Impact Parameter, b a higher impact parameter than actual, thus lengthening
df
the time between first and second contact. Since for a
Fig. 8.—Hereweplotthechangeoffittedsphericalmodel givenstartransitsathigherimpactparameterhaveshorter
parameters with oblateness as a function of impact pa- overall duration, the best-fit spherical model has a larger
rameter. The derivitive of impact parameter with respect R∗ than the model used to generate the data to maintain
to oblateness continues off the top of this graph to 3.0 at the chord length, and thus a larger R to maintain the
p
b = 0.0. An oblate planet with zero obliquity transiting overalltransit depth.
near the center of the star induces much higher deviations
Similarly, for simulated lightcurve data from a transit-
of the fitted parameters from the actual parameters than
ing oblate planet at high impact parameter (b > 0.7), the
would a planet transiting near the critical impact param-
best-fitsphericalmodelhasalowerimpactparameterthan
eter, b=0.707.
thesimulatedplanettoincreasethedurationofingressand
egress. For these planets, the fitted spherical parameters
havesmallerR∗ andRpthanthesimulatedplanettomain-
tain the character of the rest of the lightcurve.
Oblate planets that have impact parameters near the
criticalvalue,b 0.7,cannotbeaseasilyfitusingaspher-
∼
ical model because changes in the impact parameter of
the fit cannot increase the duration of ingress and egress.
Fortheseplanets,the difference betweenthe oblateplanet
transit lightcurve and the best-fit spherical planet tran-
sit lightcurve is maximized, providing the largest possible
photometric signal with which to measure oblateness.
Atpresent,itis necessaryto fitfor R∗ andstellarlimb-
−2 −1 0 1 2
darkeningparameters because of our limited knowledgeof
0.0002
these values for the host stars of transiting planets. If
0.0001 R∗ were knownto sufficient accuracy(less than 1%), then
0.0000 with knowledge of M∗ the degeneracy between R∗, Rp,
andbwouldbebroken,allowingmeasurementofplanetary
−0.0001 oblateness without fitting. However, without assumptions
−2 b=0.35, t = 0, 1−51, 30, 45 degrees 0 1 2 about the stellar mass, knowledge of R∗ would only help
−00..00000022
to constrain M∗. Cody & Sasselov (2002) show that, for
0.0001 thestarHD209458,currentevolutionarymodelscombined
with transit data serve to measure the stellar radius to a
0.0000
precision of only 10%.
−0.0001
In Figure 5 we plot the difference between the transit
b = 0.7, t = 0, 15, 30, 45 degrees
−00..00000022 lightcurveofahypotheticalplanetwiththecharacteristics
ofHD209458bbutanoblatenessf =0.1,andthatplanet’s
0.0001
best-fit spherical model fitting for R∗, Rp, b, and c1. For
0.0000 low values of the impact parameter, b 0.5, the best-fit
≤
spherical lightcurve emulates the oblate planet’s ingress
−0.0001
b = 0.83, q = 0, 15, 30, 45 degrees and egress while leaving a subtly different transit bottom
−0.0002 due to the planet traversing a differently limb-darkened
−2 −1 0 1 2 chord across the star. The magnitude of the difference
is approximately a factor of 10 smaller than the non-fit
Fig. 9.—Detectabilityofoblatenessandobliquityrelative
difference from Figure 4. Near the critical impact param-
to the best-fit spherical model for planets with nonzero
eter, b=0.7, the transit lightcurve bottoms are very sim-
obliquity. Thetoppanelshowsthe detectability difference
ilar since the best-fit b is very similar to the actual oblate
for b=0.35 and t=0◦ (dotted line), t =15◦ (dot-dashed
planet’simpactparameter;however,theingressandegress
line), t = 30◦ (dashed line), and t = 45◦ (solid line). The differ influx byafew partsin105 forf =0.1. Forgrazing
middle panel represents the same varying obliquities cal-
transits,b 1.0,thebest-fitsphericalmodel’slightcurveis
culated for b = 0.7, and the bottom panel shows the dif- ∼
indistinguishable from the oblate planet transit lightcurve
ferences for b=0.83. The shape of the difference is quali- even at the 10−6 relative accuracy level.
tatively similar for eachcase. The difference is maximized
for t = 45◦ and b = 0.7, and falls off as the parameters Figure 6 shows the same differences as in Figure 5, but
near t = 0◦, t = 90◦, b = 0.0, and b = 1.0. Due to the withasecondstellarlimbdarkeningparameter,BCGNB’s
symmetry of the problem, obliquities the ones shown plus c , also left as a free parameter in the fit. Including c in
2 2
90◦ are the time inverse of the differences shown. thefitallowsthefittingalgorithmtomatchthetransitbot-
tom (the time between second and third contacts) better.
This effect leads to excellent fits of oblate planet transits
using spherical planet models and reduces the detectable
differencetolessthanonepartin105for0.0<b<0.5and
b>0.9usingHD209458stellarandplanetaryradiiandan
oblateness of f = 0.1. For b =0.0 and b= 1.0 the spheri-
calmodelemulatesanoblatetransitparticularlywell,with
thedifferencesbetweenthetwobeingonlyafewmillionths
of the stellar flux.
Inboththe4-and5-parameterzero-obliquitymodels,it
iseasiesttomeasuretheeffectsofoblatenessonthetransit
light curvefor transits nearthe criticalimpactparameter.
For HD209458 values with f = 0.1, the oblateness signal Section 3, the fitted radii are only 0.05% above the ac-
thenapproaches3 10−5fortensofminutesduringingress tualradii,andthefittedimpactpar∼ameteris0.0006above
×
and egress and peaks near b = 0.8 instead of the critical what the actual impact parameter should be.
value of b = 0.7 due to the finite radius of HD209458b
relative to its parent star.
5.1.2. OGLE-TR-56b
Basedon observations of the Sun, Boruckiet al. (1997)
Measuring the oblateness of the new transiting planet
expecttheintrinsicstellarphotometricvariabilityontran-
sittimescalestobe 10−5. Jenkins(2002)used5yearsof OGLE-TR-56b (Konacki et al. 2003), f = 0.016 (see Sec-
∼ tion 3), would require photometric precision down to at
3 minute resolution SOHO spacecraft data to deduce the
least 4 10−6. Since the impact parameter for this object
noise power spectrum of the Sun. These noise effects are ×
is as yet unconstrained, the above precision corresponds
close enough in magnitude to the transit signal so as to
to b = 0.7. For other transit geometries, higher precision
affect the detectability of a transiting oblate planet.
photometry would be necessary to measure the oblateness
Future high-precisionspace-basedphotometry missions of this object.
such as MOST and Kepler may be able to detect the ef-
fectforhighlyoblatetransitingextrasolarplanets,butthe
5.2. Nonzero Obliquity
S/N ratio could be so low as to make unambiguous mea-
surements of oblateness diffucult to obtain.
We plot the detectability of oblateness and obliquity
for planets with nonzero obliquity in Figure 9. Here we
If an observer were to fit photometric timeseries data
define the projected obliquity (which we refer to as just
fromatransitofanoblateplanetwithoutfittingexplicitly
obliquityhereafter),oraxialtiltt,astheanglebetweenthe
for the oblateness f (as, for instance, if the precision is
orbitangularmomentumvectorandtherotationalangular
insufficient to measure f), the planet’s oblateness will be
momentum vector projected into the sky plane, measured
manifest as an astrophysical source of systematic error in
clockwise from the angular momentum vector (see Figure
the determination of the other transit parameters. Figure
3). The major effect of nonzero obliquity is to introduce
7showstheeffectthatoblatenesshasonthestellarradius,
an asymmetry into the transit lightcurve (Hui & Seager
planetary radius, impact parameter, and limb darkening
2002).
coefficientforHD209458b. This variationisa strongfunc-
tionoftheplanet’simpactparameter. InFigure8,weshow
TheplotsinFigure9aredifferenceplotsforplanetswith
the how this systematic variation changes as a function of
different obliquities and impact parameters, yet all have
the initial impact parameter for the HD209458b system.
the same shape qualitatively. In trying to fit the asym-
As a severe but still physical example, an HD209458b-like
metricingressandegresslightcurves,the best-fitspherical
planet with oblateness f = 0.1 that transits at b = 0.0
planetsplitsthedifferencebetweenthem. Forplanetswith
would be measured to have radii 2% above actual and an
0 < t < π/2, this process yields a difference curve that
impactparameternearly0.3abovetherealimpactparam-
initially increases as the spherical planet covers the star
eter.
at a faster rate than does the oblate planet. Because of
theasymmetricnatureoftheproblem,however,theegress
5.1.1. HD209458b of the oblate planet takes longer than the spherical one,
causing an initial upturn near third contact. The transit
To illustrate the robustness of the degeneracy between lightcurvedifferenceforplanetswith0>t> π/2isequal
−
R∗, Rp, b, and f discussed in Section 5.1, we fit the HST to the one for 0<t<π/2, but reversed in time.
lightcurve from BCGNB using a planet with a fixed, large
Thisgeneralshapeisthesamefortheasymmetriccom-
oblateness of 0.3(!). The best-fit parameters were R∗ =
ponent of each transit lightcurve, and it is superimposed
1.08 R⊙, Rp = 1.26 RJup, b = 0.39, and c1 = 0.633 with
a reduced χ2 = 1.06 — indistinguishable in significance ontopofthesymmetriccomponentstudiedinSection5.1.
The asymmetriccomponentis maximizednearthe critical
fromthesphericalplanetmodel! Althoughunlikely,ahigh
impactparameter(b=0.7),becausetheplanetcrossesthe
actualoblatenessforHD209458bcouldalterthe measured
stellarlimbwithitsprojectedvelocityvectoratanangleof
value for the planet’s radius into better agreement with
π/4 with respect to the limb. For transits across the mid-
theoretical models.
dle of the star, b = 0, the asymmetric component of the
TheexpecteddetectabilityofoblatenessforHD209458b lightcurvevanishesasanewsymmetryaroundtheplanet’s
is extremely low. During ingress and egress the transit path is introduced, and the local angle between the veloc-
lightcurve for HD209458b should differ from that of the ity vector and the limb is π/2. Similarly the asymmetric
best-fitsphericalmodelbyonlyonepartin106forf =0.01 planet signal formally goes to zero for grazing transits at
and at the level of 3 10−7 if f =0.003. For comparison, b = 1.0,butinpracticetheasymmetriccomponentisstill
the BCGNB HST ph×otometric precision is 1 10−4. high for Jupiter-sized planets.
×
Although the systematic error in the determination of Planets with obliquities of zero (t = 0) are symmetric
transitparameterscanbeimportantforhighlyoblateplan- andhavenoasymmetriclightcurvecomponent(seeSection
ets, it is not at all significant for HD209458b. Assuming 5.1). Likewise, planets with t = π/2 are also symmetric.
HD209458bhasanoblatenessoff =0.003ascalculatedin The asymmetric componentis maximizedfor planets with