Measuring the Oblateness and Rotation of Transiting Extrasolar GiantPlanets

Jason W. Barnes and Jonathan J. FortneyDepartment of Planetary Sciences, University of Arizona, Tucson, AZ, 85721

jbarnes@c3po.lpl.arizona.edu, jfortney@lpl.arizona.edu

ABSTRACT

We investigate the prospects for characterizing extrasolar giant planets by measuring planetary oblatenessfrom transit photometry and inferring planetary rotational periods. The rotation rates of planets in the solarsystem vary widely, reflecting the planets’ diverse formational and evolutionary histories. A measured oblateness,assumed composition, and equation of state yields a rotation rate from the Darwin-Radau relation. The lightcurveof a transiting oblate planet should differ significantly from that of a spherical one with the same cross-sectionalarea under identical stellar and orbital conditions. However, if the stellar and orbital parameters are not known apriori, fitting for them allows changes in the stellar radius, planetary radius, impact parameter, and stellar limbdarkening parameters to mimic the transit signature of an oblate planet, diminishing the oblateness signature.Thus even if HD209458b had an oblateness of 0.1 instead of our predicted 0.003, it would introduce a detectabledeparture from a model spherical lightcurve at the level of only one part in 105. Planets with nonzero obliquitybreak this degeneracy because their ingress lightcurve is asymmetric relative to that from egress, and their best-case detectability is of order 10−4. However, the measured rotation rate for these objects is non-unique due todegeneracy between obliquity and oblateness and the unknown component of obliquity along the line of sight.Detectability of oblateness is maximized for planets transiting near an impact parameter of 0.7 regardless ofobliquity. Future measurements of oblateness will be challenging because the signal is near the photometric limitsof current hardware and inherent stellar noise levels.

Subject headings: occultations — planets and satellites: general — planets and satellites: individual (HD209458b)

1. INTRODUCTION

Discovery of a transiting extrasolar planet, HD209458b(Charbonneau et al. 2000; Henry et al. 2000), has pro-vided one mechanism for researchers to move beyondthe discovery and into the characterization of extraso-lar planets. Precise Hubble Space Telescope measure-ments of HD209458b’s transit lightcurve revealed the ra-dius (1.347± 0.060RJup) and orbital inclination (86.68 ±0.14, from impact parameter 0.508) of the planet, which,along with radial velocity measurements, unambiguouslydetermine the planet’s mass (0.69±0.05MJup) and density(?, 0.35g cm−3; )]2001ApJ...552..699B. Of the over 100 ex-trasolar planets discovered so far, this large radius makesHD209458b the only one empirically determined to be agas giant.

Knowledge of a planet’s cross-sectional area provides azeroeth order determination of its geometry, and the HSTphotometry is precise enough to constrain the shape ofHD209458b to be rounded to first order. While a planettransits the limb of its star, the rate of decrease in appar-ent stellar brightness is related to the rate of increase instellar surface area covered by the planet in the same timeinterval. We investigate whether it is possible to use thisinformation to determine the shape of the planet to secondorder.

Rotation causes a planet’s shape to be flattened, oroblate, by reducing the effective gravitational accelerationat the equator (as a result of centrifugal acceleration) andby redistributing mass within the planet (which changesthe gravity field). Oblateness, f , is defined as a functionof the equatorial radius (Req) and the polar radius (Rp):

f ≡ Req −Rp

Req. (1)

For Jupiter and Saturn, high rotation rates and low densi-ties result in highly oblate planets: fJupiter = 0.06487 andfSaturn = 0.09796, compared to fEarth = 0.00335.

An earlier investigation of the detectability of oblatenessin transiting extrasolar planets was published by Seager &Hui (2002). Our work represents an improvement overSeager & Hui (2002) in the use of model fits to compareoblate and spherical planet transits, the use of the Darwin-Radau relation to associate oblateness and rotation, anda thorough investigation of the degeneracies involved infitting a transit lightcurve.

In this paper, we investigate the reasons for measuringplanetary rotation rates, the relationship between rotationrate and oblateness for extrasolar giant planets, the effectof oblateness on transit lightcurves, and the prospects fordeterminating the oblateness of a planet from transit pho-tometry.

1

0.0 0.2 0.4 0.6 0.8 1.01

10

100

Eccentricity

Spi

n−or

bit R

atio

Q = const

αQ frequency−1

Fig. 1.— Spin to orbital mean motion ratios for tidallyevolved fluid planets in equilibrium. The solid linerepresents the ratio as calculated under the assump-tion of a frequency-indenpendant tidal dissipation fac-tor Q, and the dashed line is calculated assuming Q ∝frequency−1. Under these assumptions, extremely eccen-tric planet HD80606b (e = 0.93) would, if allowed to cometo tidal equilibrium, rotate over 90 times faster than itsmean orbital motion!

1 100.20

0.22

0.24

0.26

0.28

0.30

Planet Mass (MJup)

C/M

R2

no core

Earth20 M core

Fig. 2.— The moment of inertia coefficient, C, as a func-tion of planet mass for hypothetical generic 1.0 RJup ex-trasolar giant planets. Extrasolar giant planets may ormay not possess rocky cores depending on their formationmechanism, so we plot C for both no core (upper curve)and an assumed 20 M⊕ core (lower curve).

2. EXOPLANETS AND ROTATION

While knowledge of a planet’s mass and radius providesinformation regarding composition and thermal evolution,measurements of rotation and obliquity promise to con-strain the planet’s formation, tidal evolution, and tidal dis-sipation. What these data might reveal about a planet de-pends on whether the planet is unaffected by stellar tides,slightly affected by tides, or heavily influenced by tides.

2.1. Tidally Unaffected

The present-day rotation rate of a planet is the prod-uct of both the planet’s formation and its subsequent evo-lution. Planets at sufficiently large distances from theirparent stars are not significantly affected by stellar tides,thus these objects rotate with their primordial rates alteredby planetary contraction and gravitational interactions be-tween the planet, its satellites, and other planets. To thedegree that a planet’s rotational angular momentum is pri-mordial, it may be diagnostic of the planet’s formation.Planets formed in circular orbits from a protoplanetarydisk inherit net prograde angular momentum from the ac-creting gas, resulting in rapid prograde rotation (Lissauer1995). Planets that form in eccentric orbits receive lessprograde specific angular momentum than planets in cir-cular orbits, and as a result they rotate at rates varyingfrom slow retrograde up to prograde rotations similar tothose of circularly accreted planets (Lissauer et al. 1997).Thus, comparing the current-day rotation rates for planetsin circular and eccentric orbits may reveal whether extra-solar planets formed in eccentric orbits or acquired theireccentricity later from dynamical interactions with otherplanets or a disk.

The orientation of a planet’s rotational axis relative tothe vector perpendicular to the orbital plane, the planet’sobliquity or axial tilt t, can also provide insight intothe planet’s formation mechanism. Jupiter’s low obliq-uity (3.12) has been suggested as evidence that its for-

4th1st nd2

w

b

t

3

d

rd

η

l

ρ

θ

Fig. 3.— Anatomy of a transit, after BCGNB.

2

−2 −1 0 1 2−0.0002

−0.0001

0.0000

0.0001

0.0002

−2 −1 0 1 2

−0.0002

−0.0001

0.0000

0.0001

0.0002

−2 −1 0 1 2

−0.0002

−0.0001

0.0000

0.0001

0.0002

0.7

b = 0.0, 0.3, 0.6

b = 0.8, 0.9, 1.0

Obl

ate

− S

pher

ical

Flu

x

b =

Time from Mid−transit (hours)

Fig. 4.— Oversimplified model of the effect of oblate-ness on a transit lightcurve. We plot the differences be-tween the lightcurve of an oblate planet and the lightcurveof a spherical planet with the same cross-sectional area,Ff=0.1(t)−Ff=0.0(t) while holding all other transit param-eters, R∗, Rp, b, and c1 the same and setting these valuesequal to those values measured by BCGNB for HD209458b.The top panel plots the lightcurve differential for impactparameters b = 0.0 (solid line), b = 0.3 (dashed line), andb = 0.6 (dot-dashed line). An oblate planet for b < 0.7encounters first contact before, and second contact after,the equivalent spherical planet, resulting in the initial neg-ative turn for Ff=0.1(t)−Ff=0.0(t), and subsequent positivesection of the curve for these impact parameters. In thebottom panel, we plot the differential lightcurve for b = 0.8(solid line), b = 0.9 (dashed line), and b = 1.0 (dot-dashedline). For b > 0.7, under otherwise identical conditions,an oblate planet first encounters the limb of the star af-ter the equivalent spherical planet, and last touches thelimb later on ingress. For b = 0.9, because Rp > 0.1 R∗there is no second contact, i.e. the transit is partial, so theflux differential does not return to near zero, even at mid-transit. The middle plot shows the lightcurve differentialat the changeover point between these two regimes, whereb = 0.7.

mation was dominated by orderly gas flow rather thanthe stochastic impacts of accreting planetesimals (Lissauer1993). Tidally unevolved extrasolar planets determinedto have high obliquities could be inferred either to haveformed differently than Jupiter or to have undergone largeobliquity changes as has been suggested for Saturn (?,t = 26.73;)]2002WardDPS.

2.2. Tidally Influenced

Tidal interaction between planets and their parent starsslows the rotation of those planets with close-in orbits(Guillot et al. 1996). This tidal braking continues untilthe net tidal torque on the planet becomes zero. Whethera planet reaches this end state depends on its age, radius,semimajor axis, and the planet : star mass ratio.

The rate of tidal braking also depends on the parameterQ, which represents the internal tidal dissipation withinthe planet. The value of Q is poorly constrained even forthe planets in our own solar system (Goldreich & Soter1966). Nevertheless, measurements of extrasolar planetrotation rates could constrain Q for these planets based onthe degree of tidal evolution that has taken place (Seager& Hui 2002).

Tidal braking for objects with nonzero obliquity can,somewhat counterintuitively, act to increase an object’sobliquity. Tidal torques reduce the component of a planet’sangular momentum perpendicular to the orbital planefaster than they reduce the component of the planet’s an-gular momentum in the orbital plane (Peale 1999). Thisoccurs because at solstice the planet’s induced tidal bulgeis not carried away from the planet’s orbital plane by plan-etary rotation. Therefore for large fractions of the yearstellar tidal torques do not act to right the planet’s spinaxis, while torques that reduce the angular momentum per-pendicular to the orbital plane act year-round. As a result,planets that have undergone partial tidal evolution can ex-hibit temporarily increased obliquity as the planet’s rota-tion rate decreases. Eventually, such a planet reaches max-imum obliquity and thereafter approaches synchronous ro-tation and zero obliquity simnultaneously. Planets under-going tidal evolution may be expected to have higher obliq-uities on average than planets retaining their primordialobliquity.

2.3. Tidally Dominated

The end state of tidal evolution for planets in circularorbits is a 1 : 1 spin-orbit synchronization between theplanet’s rotation and its orbital period, along with zeroobliquity. However, most of the extrasolar planets discov-ered thus far are on eccentric orbits (Marcy et al. 2000) (wesometimes shorten ’planets on eccentric orbits’ to ’eccen-tric planets’ even though the eccentricity is not inherent tothe planet). Thus these eccentric planets will never reach1 : 1 spin-orbit coupling as a result of tidal evolution be-cause the tidal torque (Eq. 2) on the planet from its star ismuch stronger (due to the r−6 dependence) near periapsis.

3

The tidal torque between a planet and star is given by(Murray & Dermott 2000)

τp−∗ = − 32

k2 G M2∗ R5

Q r6sgn(Ω− φ) , (2)

where k2 is the planet’s Love number, R its radius, Ω itsrotation rate (in radians per second), φ the instantaneousorbital angular velocity (also in radians per second), and ris the instantaneous distance between the planet and star.The function sgn(x) is equal to 1 if x is positive and −1 ifx is negative. The magnitude of the stellar perturbation isproportional to GM2∗ , the product of the stellar mass (M∗)squared and the gravitational constant (G). The planet isspun down by tidal torques if its rotation is faster than itsorbital motion (Ω > φ), and it is spun up if its rotation isslower than the orbital motion (Ω < φ).

If an eccentric planet were in a 1 : 1 spin-orbit state,it would be spun up by the star when its orbital angularvelocity is greater than average near periapsis, and it wouldbe spun down when the orbital angular velocity is low nearapoapsis. The total positive tidal torque induced while theplanet is in close will exceed the negative torque while itis far away, despite the shorter time spent near periapsis.Thus to be in rotational equilibrium with respect to stellartides, an eccentric, fluid planet must rotate faster than itsmean motion.

The Earth’s moon avoids this fate because it has anonzero component of its quadrupole moment in its orbitalplane. The torque that the Earth exerts on this permanentbulge exceeds the net tidal torque imparted on the moondue to its eccentric orbit, keeping the Moon in synchronousrotation (Murray & Dermott 2000). Fluid planets, how-ever, have no permanent quadrupole moment and thushave no restoring torque competing with the stellar tidaltorques (Greenberg & Weidenschilling 1984). Tidal evolu-tion ceases for these bodies when the net torque per orbitis zero, which can only be achieved by supersynchronousrotation.

The precise rotation rate necessary to balance the tidaltorques over the eccentric orbit depends on how Q varieswith the tidal forcing frequency (the difference betweenthe rotation rate and instantaneous orbital angular veloc-ity, Ωp − φp). Conventionally, Q has been assumed to beeither independent of the forcing frequency or inverselyproportional to it (Goldreich & Peale 1968). The resultingequilibrium spin-orbit ratios as a function of eccentricityfor each of these cases are plotted in Figure 1.

Probably both of these assumptions for the behaviorof Q are too simple. In particular, if the behavior of Qchanges under a varying tidal forcing frequency (Ωp − φp

is a function of time), then the tidal equilibrium rotationrate would differ significantly from that plotted in Figure1. Measurement of rotational rates of eccentric extrasolarplanets in tidal equilibrium could, in principle, either dif-ferentiate between these two models or suggest other fre-quency dependences, shedding light on the yet unknownmechanism for tidal dissipation within giant planets.

3. ROTATION AND OBLATENESS

Rotation affects the shape of a planet via two mecha-nisms: gravity must provide centripetal acceleration, thusthe higher velocity at the equator causes the planet tobulge by transfer of mass from polar regions; and, secon-darily, the redistributed mass alters the planet’s gravita-tional field and attracts even more mass toward the equa-torial plane. The ratio of the required centripetal accel-eration at the equator to the gravitational acceleration, q,represents the relative importance of the centripetal accel-eration term:

q =Ω2R3

eq

GMp, (3)

where Ω is the rotation rate in radians per second, Mp isthe mass of the planet, and Req is the planet’s equatorialradius (Murray & Dermott 2000).

We use the Darwin-Radau relation to relate rotationand oblateness accounting for the gravitational pull of theshifted mass: (Murray & Dermott 2000):

C ≡ C

MpR2eq

=23

[1− 2

5

(52

q

f− 1

)1/2]

(4)

where C is the planet’s moment of inertia around therotational axis and C is shorthand for CM−1

p R−2eq . The

Darwin-Radau relation is exact for uniform density bod-ies (C = 0.4), but is only an approximation for gas giantplanets (?, C ∼ 0.25;)]HubbardsBook.

By combining Equation 3 and Equation 4, we arrive ata relation for rotation rate, Ω, as a function of oblateness,f :

Ω =

√√√√fGMp

R3eq

[52

(1− 3

2C

)2

+25

]. (5)

For our solar system, the Darwin-Radau relation yieldsrotation periods accurate to within a few percent (Table1) using model-derived moments of inertia from Hubbard& Marley (1989).

Extending the Darwin-Radau relation to extrasolarplanets requires estimation of the appropriate momentof inertia coefficients, C, for those planets. For transit-ing planets whose masses and radii are known, we use aself-consistent hydrodynamic model of the planet and as-sumptions about its composition to estimate C followingFortney & Hubbard (2002). To first order, C is indepen-dent of oblateness due to similar symmetry around therotational axis, so our hydrodynamic model does not needto explicitly incorporate oblateness. The Darwin-Radaurelation and spherically symmetric hydrodynamic modelsprovide sufficient precision for the current work; how-ever, to estimate the oblateness as a function of rotationmore robustly, a two-dimensional interior model involvingboth rotational and gravitational forces should be used (?,e.g.,)]1989Icar...78..102H.

Under the spherically symmetric assumption, we calcu-late the C of Jupiter to be 0.277 with no core and 0.253

4

with a 20 M⊕ core, and we calculate the C of Saturn to be0.225 with a core (we are unable to calculate the interiorstructure of Saturn without a core due to deficiencies in ourknowledge of the equation of state). Using these momentsof inertia instead of the measured ones listed in Table 1yields similar rotation errors of a few percent. We applyour model to generic 1.0 RJup extrasolar giant planets ofvarying masses and architectures in Figure 2.

For HD209458b, our models calculate C to be 0.218 withno core and 0.185 with a 20 M⊕ core. To estimate theoblateness of HD209458b assuming synchronous rotation,we rearrange Equation 5 to solve for f ,

f =Ω2 R3

G Mp

[52

(1− 3

2C

)2

+25

]−1

, (6)

and then use the synchronous rotation rate Ω = 2.066×10−5 radians per second to obtain f = 0.00285 with nocore and f = 0.00256 with a 20 M⊕ core. These resultsimply an equator-to-pole radius difference of ∼ 200 km,which, although small, is still comparable to the atmo-spheric scale height at 1 bar, ∼ 700 km.

Showman & Guillot (2002) suggest that zonal winds onHD209458b may operate at speeds up to ∼ 2 km s−1 in theprograde direction, and Showman & Guillot (2002) go onto show that these winds might then spin up the planet’sinterior, possibly to commensurate speeds of several hun-dred m s−1 up to a few km s−1 (though the model ofShowman & Guillot (2002) does not treat the outer lay-ers and interior self-consistently). This speed is a non-negligible fraction of the orbital velocity around the planetat the surface, 30 km s−1, and is also comparable to theplanet’s rotational velocity at the equator, 2.0 km s−1. Assuch, if radiatively driven winds prove to be important onHD209458b they would affect the planet’s oblateness. Weuse the rotation rate implied by the Showman & Guillot(2002) calculations to provide an upper limit for the ex-pected oblateness of HD209458b. If the entire planet werespinning at its synchronous rate plus 2 km s−1 at the cloudtops, the rotational period would be halved to 1.8 days,with a corresponding oblateness of 0.0109 and 0.0098 forthe no core and core models respectively.

During revision of this paper, Konacki et al. (2003) an-nounced the discovery of a second transiting planet. Thisnew planet, OGLE-TR-56b, has a radius of 1.3 RJup, amass of 0.9 RJup, and an extremely short orbital period of1.2 days. Although these parameters are less constrainedthan those for HD209458b, we proceed to calculate thatthe oblateness of this new object should be 0.017 with nocore and 0.016 with a 20 M⊕ core, for C of 0.228 and 0.204respectively.

Current ground-based transit searches detect low-luminosity objects like brown dwarfs and low-mass starsin addition to planets. However, the high surface grav-ity for brown dwarfs and lower main sequence stars leadsto low values of q and very small oblatenesses for thoseobjects. For a 13MJup brown dwarf with 1.0RJup in anHD209458b-like 3.52 day orbit, the expected oblateness is

only 0.00007. Measuring oblateness will therefore only bepractical for transiting planets and not for other transitinglow-luminosity bodies.

We also note that the measurement of oblateness alongwith an independent measurement of a planet’s rotationrate Ω would determine the planet’s moment of inertia.This would provide a direct constraint on the planet’sinternal structure, possibly allowing inferences regardingthe planet’s bulk helium fraction and/or the presence of arocky core.

4. OBLATENESS AND TRANSIT LIGHTCURVES

4.1. Transit Anatomy

Brown et al. (2001) (hereafter BCGNB) investigated thedetailed structure of a transit lightcurve while studyingthe Hubble Space Telescope lightcurve of the HD209458btransit. Figure 3 relates transit events to correspondingfeatures in the lightcurve, modeled after BCGNB Figure4. The flux from the star begins to drop at the onset oftransit, known as the first contact. As the planet’s diskmoves onto the star, the flux drops further, until at sec-ond contact the entire planet disk blocks starlight. Thirdcontact is the equivalent of second contact during egress,and fourth contact marks the end of the transit. Due tostellar limb darkening the planet blocks a greater fractionof the star’s light at mid-transit than at the second andthird contacts, leading to curvature at the bottom of thetransit lightcurve.

For a given stellar mass, M∗, the total transit dura-tion, l, is a function of the transit chord length and theorbital velocity. We assume a circular orbit, which fixesthe planet’s orbital velocity. The chord length depends onthe stellar radius, R∗, and the impact parameter, b. Theimpact parameter relates to i, the inclination of the orbitalplane relative to the plane of the sky, as

b =|a cos(i)|

R∗, (7)

where a is the semimajor axis of the planet’s orbit.

The duration of ingress and egress, w, is the time be-tween first and second contact, and is a function of Rp,R∗, and b (or i). Transits with b ∼ 0 have smaller w thantransits with higher b. For transits with b ∼ 1, called graz-ing transits, w is undefined because there is no second orthird contact.

The total transit depth, d, fixes the ratio Rp/R∗, whereRp is the radius of the planet (except in the case of grazingtransits).

The magnitude of the curvature at the bottom of thetransit lightcurve, η, determines the stellar limb darkening.We use limb darkening parameters u1 and u2, or c1 andc2, which we define mathematically in Section 4.2.

5

4.2. Methods

We calculate theoretical transit lightcurves by compar-ing the amount of stellar flux blocked by the planet to thetotal stellar flux. The relative emission intensity across thedisk of the star is greatest in the center and lowest along theedges as a result of limb darkening. Many parameteriza-tions of limb darkening exist (?, see)]2000AA...363.1081C;however, we use the one proposed by BCGNB because itis the most appropriate for planetary transits.

The emission intensity at a given point on the stel-lar disk, I, is parameterized as a function of µ =cos(sin−1(ρ/R∗)), where ρ is the projected (apparent) dis-tance between the center of the star and the point inquestion. BCGNB defined a set of two limb darkening co-efficients, which we call c1 and c2, that are are equivalentto

I(µ)I(1)

= 1− c1(1− µ)(2 − µ)

2+ c2

(1− µ)µ2

. (8)

The advantage of this limb darkening function is that c1

describes the magnitude of the darkening, while c2 is a cor-rection for curvature. This makes the BCGNB coefficientsparticularly useful for fitting transit observations becausea good fit to data can be achieved by fitting only for thec1 coefficient.

Our algorithm for calculating the lightcurve takes ad-vantage of the symmetry inherent in the problem: thatI(µ) depends only on ρ and not on the angle around thestar’s center, θ. We evaluate the apparent stellar flux attime τ , F (τ), relative to the out-of-transit flux F0, by sub-tracting the amount of stellar flux blocked by the planetfrom F0:

F0 =∫ R∗

0

2πI(ρ) dρ , (9)

Fblocked =∫ R∗

0

2πI(ρ) x(ρ, τ) dρ , (10)

andF (τ) =

F0 − Fblocked

F0, (11)

where x(ρ, τ) is the fraction of a ring of radius ρ and widthdρ covered by the planet at time τ . In effect, we splitthe star up into infinitesimally small rings and add up thefluxes in Equation 9, then we determine how much of eachof these rings is covered by the planet in Equation 10.We calculate the integrals numerically using Romberg’smethod (Press et al. 1992); x(r, t) is evaluated numeri-cally as well — there is no closed form general analyticalsolution to the intersection of an ellipse and a circle.

This algorithm is more efficient than the raster methodused by Hubbard et al. (2001) for planets treated as opaquedisks because the use of symmetry and Romberg integra-tion minimize the number of computations of the stellarintensity, I(µ).

4.3. Results

To illustrate the effect oblateness has on a tran-sit lightcurve, we calculate the difference between the

−2 −1 0 1 2−3 ×10−5

−2 ×10−5

−1 ×10−5

0

1 ×10−5

2 ×10−5

3 ×10−5

−2 −1 0 1 2

−3 ×10−5

−2 ×10−5

−1 ×10−5

0

1 ×10−5

2 ×10−5

3 ×10−5

−2 −1 0 1 2

−3 ×10−5

−2 ×10−5

−1 ×10−5

0

1 ×10−5

2 ×10−5

3 ×10−5

0.9, 1.0

b = 0.6, 0.7, 0.8

b = 0.0, 0.3, 0.5

Obl

ate

− S

pher

ical

Flu

x

b =

Time from Mid−transit (hours)

Fig. 5.— Detectable difference between the lightcurve ofan oblate (f = 0.1) planet and the best-fit spherical model,fitting for R∗, Rp, b, and c1. Higher b combined withlarger values for R∗ and Rp simulate the lenghened ingressand egress of an oblate planet, diminishing the differencebetween oblate and spherical planets from Figure 4 forplanets with b < 0.7 (upper panel: solid line, b = 0.0;dashed line, b = 0.3; dotted line, b = 0.5). For planetsnear b = 0.7, the length of ingress and egress cannot besimulated by higher b, and as a result the transit signal ishighest for these planets (middle panel: solid line, b = 0.6;dashed line, b = 0.7; dotted line, b = 0.8). Above thecritical value, b > 0.7, the oblate planet’s signal can besimulated by lowering b for the spherical planet fit, reduc-ing the detectability of oblateness (lower panel: solid line,b = 0.9; dashed line, b = 1.0). It is very difficult to deter-mine the oblateness for planets involved in grazing transits(b ∼ 1.0). The magnitude of the detectability difference isproportional to f to first order, hence to estimate the de-tectability of a planet with arbitrary oblateness, multiply

the differences plotted here by f0.1

R2p

R2HD209458b

.

6

−2 −1 0 1 2−3 ×10−5

−2 ×10−5

−1 ×10−5

0

1 ×10−5

2 ×10−5

3 ×10−5

−2 −1 0 1 2

−3 ×10−5

−2 ×10−5

−1 ×10−5

0

1 ×10−5

2 ×10−5

3 ×10−5

−2 −1 0 1 2

−3 ×10−5

−2 ×10−5

−1 ×10−5

0

1 ×10−5

2 ×10−5

3 ×10−5

0.6, 0.7, 0.8

b = 0.9, 1.0

b = 0.1, 0.3, 0.5

Obl

ate

− S

pher

ical

Flu

x

b =

Time from Mid−transit (hours)

Fig. 6.— Detectable difference between the lightcurve ofan oblate (f = 0.1) planet and its best-fit spherical modelwith 5 parameters: R∗, Rp, b, c1, and c2. As in Figure5, differing values for R∗, Rp, and b for a spherical planettransit can allow it to mimic the transit of an oblate planet.With the addition of c2, the fit is much better for planetstransiting at low impact parameter. (Upper panel: solidline, b = 0.0; dashed line, b = 0.3; dotted line, b = 0.5.Middle panel: solid line, b = 0.6; dashed line, b = 0.7;dotted line, b = 0.8. Lower panel: solid line, b = 0.9;dashed line, b = 1.0.) The differences for the 5 parameter,

like those for the 4 parameter model, vary as f0.1

R2p

R2HD209458b

.

lightcurve of an oblate, zero obliquity planet and that of aspherical planet with the same cross-sectional area. For fa-miliarity, we adopt the transit parameter values measuredby BCGNB for the HD209458b transit: Rp = 1.347RJup,R∗ = 1.146R, and c1 = 0.640. Plots of the oblate-spherical differential as a function of impact parameterare shown in Figure 4.

For nearly central transits, an oblate planet encountersfirst contact before, and second contact after, the equiv-alent spherical planet. This situation causes the oblateplanet’s transit lightcurve initially to dip below that of thespherical planet. However, near the time when the planetcenter is covering the limb of the star, each planet blocksthe same apparent stellar area and the stellar flux differ-ence is zero. As the two hypothetical transits approach sec-ond contact, this trend reverses and the spherical planetstarts blocking more light than the oblate one until theoblate planet nearly catches up at second contact. Be-tween second and third contacts, the lightcurve differencesare slightly nonzero because the two planets cover areasof the star with differing amounts of limb darkening. Thelightcurves are symmetric, such that these effects repeatthemselves in reverse upon egress.

At high impact parameters (nearly grazing planet tran-sits) the opposite occurs. First contact for the oblateplanet occurs after that for the spherical planet, becausethe point of first contact on the planet is closer to the polethan to the equator. In this scenario, the oblate planet’stransit flux starts higher than, becomes equivalent to, andthen drops below that of the spherical planet before return-ing to near zero for the times between second and thirdcontact. In the case of a grazing transit, this sequence istruncated because there is no second or third contact.

The boundary between these two regions occurs whenthe local oblate planet radius at the point of first contact isequal to Rea, the radius of the equilvalent spherical planet.For planets that are small compared with the sizes of theirstars (Rp R∗) and that have low oblateness (f . 0.1),this transition occurs when θ = π/4 (from Figure 3) andb =

√2/2 = 0.707. The flux difference between transits of

oblate planets and that for spherical planets, all else beingequal, is at a minimum at this point and deviates fromzero because the rate of change in stellar area covered isdifferent for the two planets. First, second, third, andfourth contacts all occur at nearly the same time for eachplanet. However, if all else is not equal, as is usually thecase since the stellar parameters are poorly constrained,then this result is misleading and does not represent thedetectability of planetary oblatness.

5. TRANSIT LIGHTCURVES AND EXOPLAN-ETS

To test whether these flux differences are detectable,we use a Levenberg-Marquardt curve-fitting algorithmadapted from Press et al. (1992) to fit model transitsto both the HST HD209458b lightcurve and hypotheticalmodel-generated lightcurves by minimizing χ2. As a test,

7

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.30.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Oblateness

Fitt

ed P

aram

eter

s

p

R*

b

c

R

1

Fig. 7.— Best-fit parameters resulting from fitting a simu-lated transit lightcurve for a planet with oblateness f usinga spherical planet transit model. Due to the degeneracy inmeasuring transit parameters R∗, Rp, and b, and oblate-ness f , the best spherical fit to the oblate data have largerradii and higher impact parameters than the actual val-ues (see Section 5.1). Objects with negative values of fare prolate, a physically unreasonable proposition that weinclude here for completeness.

0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

0.8

1.0

Actual Impact Parameter, b

Ch

an

ge

in F

itte

d P

ara

me

ters

dbdf

dRdf

p

pR

dRdf R

*

*

dcdf

1

Fig. 8.— Here we plot the change of fitted spherical modelparameters with oblateness as a function of impact pa-rameter. The derivitive of impact parameter with respectto oblateness continues off the top of this graph to 3.0 atb = 0.0. An oblate planet with zero obliquity transitingnear the center of the star induces much higher deviationsof the fitted parameters from the actual parameters thanwould a planet transiting near the critical impact param-eter, b = 0.707.

we fit the HST HD209458b transit lightcurve and obtainR∗ = 1.142R, Rp = 1.343RJup, i = 86.72o (b = 0.504),c1 = 0.647, and c2 = −0.065 with a reduced χ2 of 1.05,consistent with the values obtained by BCGNB.

In order for planetary oblateness to have a noticeableeffect on a transit lightcurve, it must be distinguishablefrom the lightcurve of a spherical planet. For a sphericalplanet transit model, the combination of transit parame-ters that correspond to the lightcurve that best simulatesthe data from an actual oblate planet transit become themeasured values, and these measured quantities may notbe similar to the actual values. Therefore to consider thedetectability of planetary oblateness, we compare oblateplanet transit lightcurves to those of the best-fit sphericalplanet lightcurve instead of to the lightcurve of a planetthat differs from the actual values only in the oblatenessparameter (as we did in Section 4.3).

5.1. Zero Obliquity

We compare a model transit of an oblate planet (f =0.1) with zero obliquity, as is the case for a tidally evolvedplanet, to the transit of the best-fit spherical planet in Fig-ure 5 and Figure 6. The oblate planet transit signatureis muted in each when compared to Figure 4 due to a de-generacy between the fitted parameters R∗, Rp, b, and theoblateness f . This degeneracy is introduced by the uncon-strained nature of the problem: in essence, we are tryingto solve for 5 free parameters, R∗, Rp, b, c1, and f , givenjust 4 constraints, d, w, l, and η assuming (as BCGNBdid) knowledge of the stellar mass M∗. Without assuminga value for M∗, absolute timescales for the problem vanish,yielding a similar conundrum of solving for Rp/R∗, b, c1,and f given only d, η, and w/l. The previous two sentencesare intended only to be simplifications, as at higher photo-metric precision more information about the conditions ofthe transit is available. Hereafter we assume knowledge ofM∗, though this analysis could also have been done withoutthis assumption, or with an assumed relation between M∗and R∗ as proposed by Cody & Sasselov (2002). Changesin R∗, Rp, and b mimic the signal of an oblate planet byaltering the ingress and egress times while maintaining thetotal transit duration by keeping the chord length con-stant.

For planets transiting at low impact parameter (b <0.7), an oblate planet’s longer ingress and egress (higher wfrom Figure 3) are fit better using a spherical model witha higher impact parameter than actual, thus lengtheningthe time between first and second contact. Since for agiven star transits at higher impact parameter have shorteroverall duration, the best-fit spherical model has a largerR∗ than the model used to generate the data to maintainthe chord length, and thus a larger Rp to maintain theoverall transit depth.

Similarly, for simulated lightcurve data from a transit-ing oblate planet at high impact parameter (b > 0.7), thebest-fit spherical model has a lower impact parameter thanthe simulated planet to increase the duration of ingress and

8

−2 −1 0 1 2−0.0002

−0.0001

0.0000

0.0001

0.0002

−2 −1 0 1 2

−0.0002

−0.0001

0.0000

0.0001

0.0002

−2 −1 0 1 2

−0.0002

−0.0001

0.0000

0.0001

0.0002

b = 0.83, q = 0, 15, 30, 45 degrees

b=0.35, t = 0, 15, 30, 45 degrees

b = 0.7, t = 0, 15, 30, 45 degrees

Fig. 9.— Detectability of oblateness and obliquity relativeto the best-fit spherical model for planets with nonzeroobliquity. The top panel shows the detectability differencefor b = 0.35 and t = 0 (dotted line), t = 15 (dot-dashedline), t = 30 (dashed line), and t = 45 (solid line). Themiddle panel represents the same varying obliquities cal-culated for b = 0.7, and the bottom panel shows the dif-ferences for b = 0.83. The shape of the difference is quali-tatively similar for each case. The difference is maximizedfor t = 45 and b = 0.7, and falls off as the parametersnear t = 0, t = 90, b = 0.0, and b = 1.0. Due to thesymmetry of the problem, obliquities the ones shown plus90 are the time inverse of the differences shown.

egress. For these planets, the fitted spherical parametershave smaller R∗ and Rp than the simulated planet to main-tain the character of the rest of the lightcurve.

Oblate planets that have impact parameters near thecritical value, b ∼ 0.7, cannot be as easily fit using a spher-ical model because changes in the impact parameter ofthe fit cannot increase the duration of ingress and egress.For these planets, the difference between the oblate planettransit lightcurve and the best-fit spherical planet tran-sit lightcurve is maximized, providing the largest possiblephotometric signal with which to measure oblateness.

At present, it is necessary to fit for R∗ and stellar limb-darkening parameters because of our limited knowledge ofthese values for the host stars of transiting planets. IfR∗ were known to sufficient accuracy (less than 1%), thenwith knowledge of M∗ the degeneracy between R∗, Rp,and b would be broken, allowing measurement of planetaryoblateness without fitting. However, without assumptionsabout the stellar mass, knowledge of R∗ would only helpto constrain M∗. Cody & Sasselov (2002) show that, forthe star HD209458, current evolutionary models combinedwith transit data serve to measure the stellar radius to aprecision of only 10%.

In Figure 5 we plot the difference between the transitlightcurve of a hypothetical planet with the characteristicsof HD209458b but an oblateness f = 0.1, and that planet’sbest-fit spherical model fitting for R∗, Rp, b, and c1. Forlow values of the impact parameter, b ≤ 0.5, the best-fitspherical lightcurve emulates the oblate planet’s ingressand egress while leaving a subtly different transit bottomdue to the planet traversing a differently limb-darkenedchord across the star. The magnitude of the differenceis approximately a factor of 10 smaller than the non-fitdifference from Figure 4. Near the critical impact param-eter, b = 0.7, the transit lightcurve bottoms are very sim-ilar since the best-fit b is very similar to the actual oblateplanet’s impact parameter; however, the ingress and egressdiffer in flux by a few parts in 105 for f = 0.1. For grazingtransits, b ∼ 1.0, the best-fit spherical model’s lightcurve isindistinguishable from the oblate planet transit lightcurveeven at the 10−6 relative accuracy level.

Figure 6 shows the same differences as in Figure 5, butwith a second stellar limb darkening parameter, BCGNB’sc2, also left as a free parameter in the fit. Including c2 inthe fit allows the fitting algorithm to match the transit bot-tom (the time between second and third contacts) better.This effect leads to excellent fits of oblate planet transitsusing spherical planet models and reduces the detectabledifference to less than one part in 105 for 0.0 < b < 0.5 andb > 0.9 using HD209458 stellar and planetary radii and anoblateness of f = 0.1. For b = 0.0 and b = 1.0 the spheri-cal model emulates an oblate transit particularly well, withthe differences between the two being only a few millionthsof the stellar flux.

In both the 4- and 5-parameter zero-obliquity models, itis easiest to measure the effects of oblateness on the transitlight curve for transits near the critical impact parameter.

9

For HD209458 values with f = 0.1, the oblateness signalthen approaches 3×10−5 for tens of minutes during ingressand egress and peaks near b = 0.8 instead of the criticalvalue of b = 0.7 due to the finite radius of HD209458brelative to its parent star.

Based on observations of the Sun, Borucki et al. (1997)expect the intrinsic stellar photometric variability on tran-sit timescales to be ∼ 10−5. Jenkins (2002) used 5 years of3 minute resolution SOHO spacecraft data to deduce thenoise power spectrum of the Sun. These noise effects areclose enough in magnitude to the transit signal so as toaffect the detectability of a transiting oblate planet.

Future high-precision space-based photometry missionssuch as MOST and Kepler may be able to detect the ef-fect for highly oblate transiting extrasolar planets, but theS/N ratio could be so low as to make unambiguous mea-surements of oblateness diffucult to obtain.

If an observer were to fit photometric timeseries datafrom a transit of an oblate planet without fitting explicitlyfor the oblateness f (as, for instance, if the precision isinsufficient to measure f), the planet’s oblateness will bemanifest as an astrophysical source of systematic error inthe determination of the other transit parameters. Figure7 shows the effect that oblateness has on the stellar radius,planetary radius, impact parameter, and limb darkeningcoefficient for HD209458b. This variation is a strong func-tion of the planet’s impact parameter. In Figure 8, we showthe how this systematic variation changes as a function ofthe initial impact parameter for the HD209458b system.As a severe but still physical example, an HD209458b-likeplanet with oblateness f = 0.1 that transits at b = 0.0would be measured to have radii 2% above actual and animpact parameter nearly 0.3 above the real impact param-eter.

5.1.1. HD209458b

To illustrate the robustness of the degeneracy betweenR∗, Rp, b, and f discussed in Section 5.1, we fit the HSTlightcurve from BCGNB using a planet with a fixed, largeoblateness of 0.3(!). The best-fit parameters were R∗ =1.08 R, Rp = 1.26 RJup, b = 0.39, and c1 = 0.633 witha reduced χ2 = 1.06 — indistinguishable in significancefrom the spherical planet model! Although unlikely, a highactual oblateness for HD209458b could alter the measuredvalue for the planet’s radius into better agreement withtheoretical models.

The expected detectability of oblateness for HD209458bis extremely low. During ingress and egress the transitlightcurve for HD209458b should differ from that of thebest-fit spherical model by only one part in 106 for f = 0.01and at the level of 3× 10−7 if f = 0.003. For comparison,the BCGNB HST photometric precision is 1× 10−4.

Although the systematic error in the determination oftransit parameters can be important for highly oblate plan-ets, it is not at all significant for HD209458b. AssumingHD209458b has an oblateness of f = 0.003 as calculated in

Section 3, the fitted radii are only ∼ 0.05% above the ac-tual radii, and the fitted impact parameter is 0.0006 abovewhat the actual impact parameter should be.

5.1.2. OGLE-TR-56b

Measuring the oblateness of the new transiting planetOGLE-TR-56b (Konacki et al. 2003), f = 0.016 (see Sec-tion 3), would require photometric precision down to atleast 4× 10−6. Since the impact parameter for this objectis as yet unconstrained, the above precision correspondsto b = 0.7. For other transit geometries, higher precisionphotometry would be necessary to measure the oblatenessof this object.

5.2. Nonzero Obliquity

We plot the detectability of oblateness and obliquityfor planets with nonzero obliquity in Figure 9. Here wedefine the projected obliquity (which we refer to as justobliquity hereafter), or axial tilt t, as the angle between theorbit angular momentum vector and the rotational angularmomentum vector projected into the sky plane, measuredclockwise from the angular momentum vector (see Figure3). The major effect of nonzero obliquity is to introducean asymmetry into the transit lightcurve (Hui & Seager2002).

The plots in Figure 9 are difference plots for planets withdifferent obliquities and impact parameters, yet all havethe same shape qualitatively. In trying to fit the asym-metric ingress and egress lightcurves, the best-fit sphericalplanet splits the difference between them. For planets with0 < t < π/2, this process yields a difference curve thatinitially increases as the spherical planet covers the starat a faster rate than does the oblate planet. Because ofthe asymmetric nature of the problem, however, the egressof the oblate planet takes longer than the spherical one,causing an initial upturn near third contact. The transitlightcurve difference for planets with 0 > t > −π/2 is equalto the one for 0 < t < π/2, but reversed in time.

This general shape is the same for the asymmetric com-ponent of each transit lightcurve, and it is superimposedon top of the symmetric component studied in Section 5.1.The asymmetric component is maximized near the criticalimpact parameter (b = 0.7), because the planet crosses thestellar limb with its projected velocity vector at an angle ofπ/4 with respect to the limb. For transits across the mid-dle of the star, b = 0, the asymmetric component of thelightcurve vanishes as a new symmetry around the planet’spath is introduced, and the local angle between the veloc-ity vector and the limb is π/2. Similarly the asymmetricplanet signal formally goes to zero for grazing transits atb = 1.0, but in practice the asymmetric component is stillhigh for Jupiter-sized planets.

Planets with obliquities of zero (t = 0) are symmetricand have no asymmetric lightcurve component (see Section5.1). Likewise, planets with t = π/2 are also symmetric.The asymmetric component is maximized for planets with

10

t = π/4.

Transiting planets with nonzero obliquity can break thedegeneracy between transit parameters discussed in Sec-tion 5.1. However, when the obliquity is nonzero a de-generacy between projected obliquity and oblateness is in-troduced: a measured asymmetric lightcurve componentof a given magnitude could be due to a planet with lowoblateness but near the maximum detectability obliquityof t = π/4, or it could be the result of a more highly oblateplanet with a lower obliquity. In this case only a lower limitto the oblateness can be determined based on the oblate-ness for an assumed obliquity of π/4. This degeneracy canbe broken with measurements precise enough to determinethe symmetric lightcurve component. In addition, transitphotometry is only able to measure the projected oblate-ness and obliquity of such objects due to the unknowncomponent of obliquity along the line-of-sight (Hui & Sea-ger 2002), therefore the true oblateness is never smallerthan the measured, projected oblateness.

The asymmetric transit signal of an oblate planet withnonzero obliquity could also be muddled by the presenceof other bodies orbiting the planet. Orbiting satellites orrings could both introduce asymmetries into the transitlightcurve that may not be easily distinguishable from theasymmetry resulting from the oblate planet. Satellitesaround tidally evolved planets are not stable (Barnes &O’Brien 2002), and rings around these objects may proveto be difficult to sustain as well. However, objects that arenot tidally evolved, those farther away from their parentstars, may potentially retain such adornments, and theireffects on a transit lightcurve could be difficult to differen-tiate from oblateness.

Orbital eccentricity can also cause a transit lightcurve tobe asymmetric. Although the eccentricity of HD209458bis zero to within measurement uncertainty based on radialvelocity measurements, future planets discovered solely bytheir transits may not have constrained orbital parameters.For these objects with unknown eccentricity, if the eccen-tricity is very high then under some conditions the velocitychange between ingress and egress may be high enough toemulate the oblateness asymmetry discussed in this sec-tion. However, we do not explicitly treat that situation inthis paper.

6. CONCLUSIONS

Examining a transiting planet’s precise lightcurve canallow the measurement of the planet’s oblateness and,therefore, rotation rate, beginning the process of charac-terizing extrasolar planets. To a reasonably close approxi-mation (a few percent), the rotation rate of an extrasolargiant planets is related to the planet’s oblateness by theDarwin-Radau relation. Measurements of a planet’s rota-tion rate could constrain the tidal dissipation factor Q forthose planets, as well as possibly shed light on the tidaldissipation mechanism within giant planets based on thespin : orbit ratios of tidally evolved eccentric planets.

The detectable effect of oblateness on the lightcurve ofa planet with zero obliquity is at best on the level of a fewparts in 105. This effect may be discernable from spacewith ultra high precision photometry, but could prove tobe indistinguishable from stellar noise or other confound-ing effects. Accurate independent measurements of stellarradius can break the degeneracy between the stellar radius,planetary radius, and impact parameter, allowing for mucheasier measurement of oblateness. Without such measure-ments, the primary effect of oblateness and zero obliquitywhen studying transit lightcurves will be to provide an as-trophysical source of systematic error in the measurementof transit parameters.

Planets with nonzero obliquity have higher detectabil-ities (up to ∼ 10−4) than planets with no obliquity, butyield nonunique determinations of obliquity and oblate-ness. In order to obtain unique obliquities and oblatenessesfor these objects, precision comparable to that needed forthe zero obliquity case is needed.

The detectability of oblateness for transiting planets ismaximized for impact parameters near the critical impactparameter of b = 0.7. Many transit searches are currentlyunderway, yielding the potential for the discovery of manytransiting planets in coming years. Given the opportu-nity to attempt to measure oblateness, our analysis sug-gests that the optimal observational target selection strat-egy would be to observe planets around bright stars thattransit near an impact parameter of 0.7.

The authors wish to acknowledge Bill Hubbard and FredCiesla for useful conversations; Wayne Barnes, RobertH. Brown, Christian Schaller, and Paul Withers formanuscript suggestions; and Robert H. Brown for valu-ble advice.

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Table 1

Darwin-Radau in the Solar System

Planet f C Calculated Period Actual Period Error

Jupiter 0.06487 0.264011 10.1609 hr 9.92425 hr 2.38%Saturn 0.09796 0.220371 10.8817 hr 10.6562 hr 2.12%Uranus 0.02293 0.2268 1 16.6459 hr 17.24 hr 3.45%Neptune 0.01708 0.23 ∗ 16.8656 hr 16.11 hr 4.69%Earth 0.00335 0.3308 23.8808 hr 23.9342 hr 0.223%

1Hubbard & Marley (1989)

∗Assumed

Note.— Comparison of calculated rotation rates from the Darwin-Radau ap-proximation (Equation 4) to actual rotation rates for selected planets in our solarsystem.

13