Measuring the Oblateness and Rotation of Transiting Extrasolar Giant Planets
a r X i v :a s t r o -p h /0301156v 1 9 J a n 2003
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
jbarnes@https://www.sodocs.net/doc/bd13633227.html,,jfortney@https://www.sodocs.net/doc/bd13633227.html,
ABSTRACT
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 system vary widely,re?ecting 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 lightcurve of a transiting oblate planet should di?er signi?cantly from that of a spherical one with the same cross-sectional area under identical stellar and orbital conditions.However,if the stellar and orbital parameters are not known a priori ,?tting for them allows changes in the stellar radius,planetary radius,impact parameter,and stellar limb darkening parameters to mimic the transit signature of an oblate planet,diminishing the oblateness signature.Thus even if HD209458b had an oblateness of 0.1instead of our predicted 0.003,it would introduce a detectable departure from a model spherical lightcurve at the level of only one part in 105.Planets with nonzero obliquity break 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 to degeneracy 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 0.7regardless of obliquity.Future measurements of oblateness will be challenging because the signal is near the photometric limits of 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 beyond the 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±0.060R Jup )and orbital inclination (86.68?±0.14?,from impact parameter 0.508)of the planet,which,along with radial velocity measurements,unambiguously determine the planet’s mass (0.69±0.05M Jup )and density (0.35g cm ?3;Brown et al.2001).Of the over 100ex-trasolar planets discovered so far,this large radius makes HD209458b the only one empirically determined to be a gas giant.
Knowledge of a planet’s cross-sectional area provides a zeroeth order determination of its geometry,and the HST photometry is precise enough to constrain the shape of HD209458b to be rounded to ?rst order.While a planet transits the limb of its star,the rate of decrease in appar-ent stellar brightness is related to the rate of increase in stellar surface area covered by the planet in the same time interval.We investigate whether it is possible to use this information to determine the shape of the planet to second order.
Rotation causes a planet’s shape to be ?attened,or oblate,by reducing the e?ective gravitational acceleration at the equator (as a result of centrifugal acceleration)and by redistributing mass within the planet (which changes the gravity ?eld).Oblateness,f ,is de?ned as a function of the equatorial radius (R eq )and the polar radius (R p ):
f ≡
R eq ?R p
0.0
0.2
0.40.60.8 1.0
1
10
100
Eccentricity
S p i n ?o r b i t R a t i o
Q = const
α
Q frequency ?1
Fig. 1.—Spin to orbital mean motion ratios for tidally evolved ?uid planets in equilibrium.The solid line represents 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 come to tidal equilibrium,rotate over 90times faster than its mean orbital motion!
1
100.20
0.22
0.24
0.26
0.28
0.30
Planet Mass (M Jup )
C /M R 2
no core
Earth
20 M core Fig.2.—The moment of inertia coe?cient,C ,as a func-tion of planet mass for hypothetical generic 1.0R Jup ex-trasolar giant planets.Extrasolar giant planets may or may not possess rocky cores depending on their formation mechanism,so we plot C for both no core (upper curve)and an assumed 20M ⊕core (lower curve).
2.EXOPLANETS AND ROTATION
While knowledge of a planet’s mass and radius provides information 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 una?ected by stellar tides,slightly a?ected by tides,or heavily in?uenced by tides.2.1.
Tidally Una?ected
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 su?ciently large distances from their parent stars are not signi?cantly a?ected by stellar tides,thus these objects rotate with their primordial rates altered by planetary contraction and gravitational interactions be-tween the planet,its satellites,and other planets.To the degree 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 protoplanetary disk inherit net prograde angular momentum from the ac-creting gas,resulting in rapid prograde rotation (Lissauer 1995).Planets that form in eccentric orbits receive less prograde speci?c angular momentum than planets in cir-cular orbits,and as a result
they rotate at rates varying from slow retrograde up to prograde rotations similar to those of circularly accreted planets (Lissauer et al.1997).Thus,comparing the current-day rotation rates for planets in circular and eccentric orbits may reveal whether extra-solar planets formed in eccentric orbits or acquired their eccentricity later from dynamical interactions with other planets or a disk.
The orientation
of a planet’s rotational axis relative to 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 (3.12?)has been suggested as evidence that its formation
Fig.3.—Anatomy of a transit,after BCGNB.
?2
?1
12
?2
?1012
O b l a t e ? S p h e r i c a l F l u x
Time from Mid?transit (hours)
Fig. 4.—Oversimpli?ed model of the e?ect of oblate-ness on a transit lightcurve.We plot the di?erences be-tween the lightcurve of an oblate planet and the lightcurve of a spherical planet with the same cross-sectional area,F f =0.1(t )?F f =0.0(t )while holding all other transit param-eters,R ?,R p ,b ,and c 1the same and setting these values equal to those values measured by BCGNB for HD209458b.The top panel plots the lightcurve di?erential for impact parameters b =0.0(solid line),b =0.3(dashed line),and b =0.6(dot-dashed line).An oblate planet for b <0.7encounters ?rst contact before,and second contact after,the equivalent spherical planet,resulting in the initial neg-ative turn for F f =0.1(t )?F f =0.0(t ),and subsequent positive section of the curve for these impact parameters.In the bottom panel,we plot the di?erential lightcurve for b =0.8(solid line),b =0.9(dashed line),and b =1.0(dot-dashed line).For b >0.7,under otherwise identical conditions,an oblate planet ?rst encounters the limb of the star af-ter the equivalent spherical planet,and last touches the limb later on ingress.For b =0.9,because R p >0.1R ?there is no second contact,i.e.the transit is partial,so the ?ux di?erential does not return to near zero,even at mid-transit.The middle plot shows the lightcurve di?erential at the changeover point between these two regimes,where b =0.7.
was dominated by orderly gas ?ow rather than the stochas-tic impacts of accreting planetesimals (Lissauer 1993).Tidally unevolved extrasolar planets determined to have high obliquities could be inferred either to have formed di?erently than Jupiter or to have undergone large obliq-uity changes as has been suggested for Saturn (t =26.73?;Ward &Hamilton 2002).
2.2.Tidally In?uenced
Tidal interaction between planets and their parent stars
slows the rotation of those planets with close-in orbits (Guillot et al.1996).This tidal braking continues until the net tidal torque on the planet becomes zero.Whether a 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 parameter Q ,which represents the internal tidal dissipation within the planet.The value of Q is poorly constrained even for the planets in our own solar system (Goldreich &Soter 1966).Nevertheless,measurements of extrasolar planet rotation rates could constrain Q for these planets based on the 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’s obliquity.Tidal torques reduce the component of a planet’s angular momentum perpendicular to the orbital plane faster than they reduce the component of the planet’s an-gular momentum in the orbital plane (Peale 1999).This occurs because at solstice the planet’s induced tidal bulge is not carried away from the planet’s orbital plane by plan-etary rotation.Therefore for large fractions of the year stellar tidal torques do not act to right the planet’s spin axis,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 primordial obliquity.2.3.
Tidally Dominated
The end state of tidal evolution for planets in circular orbits is a 1:1spin-orbit synchronization between the planet’s rotation and its orbital period,along with zero obliquity.However,most of the extrasolar planets discov-ered thus far are on eccentric orbits (Marcy et al.2000)(we sometimes shorten ’planets on eccentric orbits’to ’eccen-tric planets’even though the eccentricity is not inherent to the planet).Thus these eccentric planets will never reach 1:1spin-orbit coupling as a result of tidal evolution be-cause the tidal torque (Eq.2)on the planet from its star is much stronger (due to the r ?6dependence)near periapsis.
The tidal torque between a planet and star is given by (Murray&Dermott2000)
τp??=?3
Q r6
sgn(??˙φ),(2)
where k2is the planet’s Love number,R its radius,?its
rotation rate(in radians per second),˙φthe instantaneous orbital angular velocity(also in radians per second),and r
is the instantaneous distance between the planet and star.
The function sgn(x)is equal to1if x is positive and?1if x is negative.The magnitude of the stellar perturbation is
proportional to GM2?,the product of the stellar mass(M?) squared and the gravitational constant(G).The planet is
spun down by tidal torques if its rotation is faster than its
orbital motion(?>˙φ),and it is spun up if its rotation is slower than the orbital motion(?<˙φ).
If an eccentric planet were in a1:1spin-orbit state,
it would be spun up by the star when its orbital angular velocity is greater than average near periapsis,and it would
be spun down when the orbital angular velocity is low near
apoapsis.The total positive tidal torque induced while the planet is in close will exceed the negative torque while it
is far away,despite the shorter time spent near periapsis. Thus to be in rotational equilibrium with respect to stellar
tides,an eccentric,?uid planet must rotate faster than its
mean motion.
The Earth’s moon avoids this fate because it has a
nonzero component of its quadrupole moment in its orbital
plane.The torque that the Earth exerts on this permanent bulge exceeds the net tidal torque imparted on the moon
due to its eccentric orbit,keeping the Moon in synchronous
rotation(Murray&Dermott2000).Fluid planets,how-ever,have no permanent quadrupole moment and thus
have no restoring torque competing with the stellar tidal torques(Greenberg&Weidenschilling1984).Tidal evolu-
tion ceases for these bodies when the net torque per orbit
is zero,which can only be achieved by supersynchronous rotation.
The precise rotation rate necessary to balance the tidal
torques over the eccentric orbit depends on how Q varies with the tidal forcing frequency(the di?erence between
the rotation rate and instantaneous orbital angular veloc-
ity,?p?˙φp).Conventionally,Q has been assumed to be either independent of the forcing frequency or inversely
proportional to it(Goldreich&Peale1968).The resulting equilibrium spin-orbit ratios as a function of eccentricity for each of these cases are plotted in Figure1.
Probably both of these assumptions for the behavior of Q are too simple.In particular,if the behavior of Q changes under a varying tidal forcing frequency(?p?˙φp is a function of time),then the tidal equilibrium rotation rate would di?er signi?cantly from that plotted in Figure 1.Measurement of rotational rates of eccentric extrasolar planets in tidal equilibrium could,in principle,either dif-ferentiate between these two models or suggest other fre-quency dependences,shedding light on the yet unknown mechanism for tidal dissipation within giant planets.3.ROTATION AND OBLATENESS
Rotation a?ects the shape of a planet via two mecha-nisms:gravity must provide centripetal acceleration,thus the higher velocity at the equator causes the planet to bulge by transfer of mass from polar regions;and,secon-darily,the redistributed mass alters the planet’s gravita-tional?eld 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=
?2R3eq
M p R2eq
=
2
5 5f?1
1/2 (4)
where C is the planet’s moment of inertia around the
rotational 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 giant planets(C～0.25;Hubbard1984).
By combining Equation3and Equation4,we arrive at a relation for rotation rate,?,as a function of oblateness, f:
?= R3eq 52C 2+2
with a20M⊕core,and we calculate the C of Saturn to be 0.225with a core(we are unable to calculate the interior structure of Saturn without a core due to de?ciencies in our knowledge of the equation of state).Using these moments of inertia instead of the measured ones listed in Table1 yields similar rotation errors of a few percent.We apply our model to generic1.0R Jup extrasolar giant planets of varying masses and architectures in Figure2.
For HD209458b,our models calculate C to be0.218with no core and0.185with a20M⊕core.To estimate the oblateness of HD209458b assuming synchronous rotation, we rearrange Equation5to solve for f,
f=?2R3
2 1?35
?1,(6)
and then use the synchronous rotation rate?=2.066×10?5radians per second to obtain f=0.00285with no core and f=0.00256with a20M⊕core.These results imply an equator-to-pole radius di?erence of～200km, which,although small,is still comparable to the atmo-spheric scale height at1bar,～700km.
Showman&Guillot(2002)suggest that zonal winds on HD209458b may operate at speeds up to～2km s?1in the prograde direction,and Showman&Guillot(2002)go on to show that these winds might then spin up the planet’s interior,possibly to commensurate speeds of several hun-dred m s?1up to a few km s?1(though the model of Showman&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 planet at the surface,30km s?1,and is also comparable to the planet’s rotational velocity at the equator,2.0km s?1.As such,if radiatively driven winds prove to be important on HD209458b they would a?ect the planet’s oblateness.We use 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 were spinning at its synchronous rate plus2km s?1at the cloud tops,the rotational period would be halved to1.8days, with a corresponding oblateness of0.0109and0.0098for the no core and core models respectively.
During revision of this paper,Konacki et al.(2003)an-nounced the discovery of a second transiting planet.This new planet,OGLE-TR-56b,has a radius of1.3R Jup,a mass of0.9R Jup,and an extremely short orbital period of 1.2days.Although these parameters are less constrained than those for HD209458b,we proceed to calculate that the oblateness of this new object should be0.017with no core and0.016with a20M⊕core,for C of0.228and0.204 respectively.
Current ground-based transit searches detect low-luminosity objects like brown dwarfs and low-mass stars in addition to planets.However,the high surface grav-ity for brown dwarfs and lower main sequence stars leads to low values of q and very small oblatenesses for those objects.For a13M Jup brown dwarf with1.0R Jup in an HD209458b-like3.52day orbit,the expected oblateness is only0.00007.Measuring oblateness will therefore only be practical for transiting planets and not for other transiting low-luminosity bodies.
We also note that the measurement of oblateness along with an independent measurement of a planet’s rotation rate?would determine the planet’s moment of inertia. This would provide a direct constraint on the planet’s internal structure,possibly allowing inferences regarding the planet’s bulk helium fraction and/or the presence of a rocky core.
4.OBLATENESS AND TRANSIT LIGHTCUR VES 4.1.Transit Anatomy
Brown et al.(2001)(hereafter BCGNB)investigated the detailed structure of a transit lightcurve while studying the Hubble Space Telescope lightcurve of the HD209458b transit.Figure3relates transit events to corresponding features in the lightcurve,modeled after BCGNB Figure 4.The?ux from the star begins to drop at the onset of transit,known as the?rst contact.As the planet’s disk moves onto the star,the?ux drops further,until at sec-ond contact the entire planet disk blocks starlight.Third contact is the equivalent of second contact during egress, and fourth contact marks the end of the transit.Due to stellar limb darkening the planet blocks a greater fraction
of the star’s light at mid-transit than at the second and third contacts,leading to curvature at the bottom of the transit lightcurve.
For a given stellar mass,M?,the total transit dura-tion,l,is a function of the transit chord length and the orbital velocity.We assume a circular orbit,which?xes the planet’s orbital velocity.The chord length depends on the stellar radius,R?,and the impact parameter,b.The impact parameter relates to i,the inclination of the orbital plane relative to the plane of the sky,as
b=|a cos(i)|
4.2.Methods
We calculate theoretical transit lightcurves by compar-ing the amount of stellar?ux blocked by the planet to the total stellar?ux.The relative emission intensity across the disk of the star is greatest in the center and lowest along the edges as a result of limb darkening.Many parameteri-zations of limb darkening exist(see Claret2000);however, we use the one proposed by BCGNB because it is the most appropriate for planetary transits.
The emission intensity at a given point on the stel-lar disk,I,is parameterized as a function ofμ= cos(sin?1(ρ/R?)),whereρis the projected(apparent)dis-tance between the center of the star and the point in question.BCGNB de?ned a set of two limb darkening co-e?cients,which we call c1and c2,that are are equivalent to
I(μ)
2
+c2
(1?μ)μ
F0
,(11)
where x(ρ,τ)is the fraction of a ring of radiusρand width dρcovered by the planet at timeτ.In e?ect,we split the star up into in?nitesimally small rings and add up the ?uxes in Equation9,then we determine how much of each of these rings is covered by the planet in Equation10. We calculate the integrals numerically using Romberg’s method(Press et al.1992);x(r,t)is evaluated numeri-cally as well—there is no closed form general analytical solution to the intersection of an ellipse and a circle.
This algorithm is more e?cient than the raster method used by Hubbard et al.(2001)for planets treated as opaque 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 e?ect oblateness has on a tran-sit lightcurve,we calculate the di?erence between the
?2?1012?3×10?5
?2×10?5
?1×10?5
1×10?5
2×10?5
3×10
?3?5
?2×10?5
?1×10?5
1×10?5
2×10?5
3×10
?2?1012?3?5
?2×10?5
?1×10?5
1×10?5
2×10?5
3×10?5
O
b
l
a
t
e
?
S
p
h
e
r
i
c
a
l
F
l
u
x
Time from Mid?transit (hours)
Fig. 5.—Detectable di?erence between the lightcurve of an oblate(f=0.1)planet and the best-?t spherical model,?tting for R?,R p,b,and c1.Higher b combined with larger values for R?and R p simulate the lenghened ingress and egress of an oblate planet,diminishing the di?erence between oblate and spherical planets from Figure4for planets with b<0.7(upper panel:solid line,b=0.0; dashed line,b=0.3;dotted line,b=0.5).For planets near b=0.7,the length of ingress and egress cannot be simulated by higher b,and as a result the transit signal is highest for these planets(middle panel:solid line,b=0.6; dashed line,b=0.7;dotted line,b=0.8).Above the critical value,b>0.7,the oblate planet’s signal can be simulated by lowering b for the spherical planet?t,reduc-ing the detectability of oblateness(lower panel:solid line, b=0.9;dashed line,b=1.0).It is very di?cult to deter-mine the oblateness for planets involved in grazing transits (b～1.0).The magnitude of the detectability di?erence is proportional to f to?rst order,hence to estimate the de-tectability of a planet with arbitrary oblateness,multiply the di?erences plotted here by f
R2
HD209458b
.
?2
?1
12
?3×10?5
?2×10?5?1×10
?5
1×10?5
2×10?53?3×10?5?2×10?5?1×10
?5
1×10?5
2×10?53?2
?1
012
?3×10?5
?2×10?5?1×10?50
1×10
?5
2×10?5
3×10?5
O b l a t e ? S p h e r i c a l F l u x
Time from Mid?transit (hours)
Fig. 6.—Detectable di?erence between the lightcurve of an oblate (f =0.1)planet and its best-?t spherical model with 5parameters:R ?,R p ,b ,c 1,and c 2.As in Figure 5,di?ering values for R ?,R p ,and b for a spherical planet transit can allow it to mimic the transit of an oblate planet.With the addition of c 2,the ?t is much better for planets transiting at low impact parameter.(Upper panel:solid line,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 di?erences for the 5parameter,like those for the 4parameter model,vary as
f
R 2HD 209458b
.
lightcurve of an oblate,zero obliquity planet and that of a spherical planet with the same cross-sectional area.For fa-miliarity,we adopt the transit parameter values measured by BCGNB for the HD209458b transit:R p =1.347R Jup ,R ?=1.146R ⊙,and c 1=0.640.Plots of the oblate-spherical di?erential as a function of impact parameter are shown in Figure 4.
For nearly central transits,an oblate planet encounters ?rst contact before,and second contact after,the equiv-alent spherical planet.This situation causes the oblate planet’s transit lightcurve initially to dip below that of the spherical planet.However,near the time when the planet center is covering the limb of the star,each planet blocks the same apparent stellar area and the stellar ?ux di?er-ence is zero.As the two hypothetical transits approach sec-ond contact,this trend reverses and the spherical planet starts blocking more light than the oblate one until the oblate planet nearly catches up at second contact.Be-tween second and third contacts,the lightcurve di?erences are slightly nonzero because the two planets cover areas of the star with di?ering amounts of limb darkening.The lightcurves are symmetric,such that these e?ects repeat themselves in reverse upon egress.
At high impact parameters (nearly grazing planet tran-sits)the opposite occurs.First contact for the oblate planet occurs after that for the spherical planet,because the point of ?rst contact on the planet is closer to the pole than to the equator.In this scenario,the oblate planet’s transit ?ux starts higher than,becomes equivalent to,and then drops below that of the spherical planet before return-ing to near zero for the times between second and third contact.In the case of a grazing transit,this sequence is truncated because there is no second or third contact.The boundary between these two regions occurs when the local oblate planet radius at the point of ?rst contact is equal to R ea ,the radius of the equilvalent spherical planet.For planets that are small compared with the sizes of their stars (R p ?R ?)and that have low oblateness (f 0.1),this transition occurs when θ=π/4(from Figure 3)and b =
√
?0.3
?0.2
?0.10.00.1
0.2
0.3
0.00.20.40.60.81.01.21.4Oblateness
F i t t e d P a r a m e t e r s
p
R *b
c R 1Fig.7.—Best-?t parameters resulting from ?tting a simu-late
d transit lightcurv
e for a planet with oblateness
f usin
g a spherical planet transit model.Due to the degeneracy in measuring transit parameters R ?,R p ,and b ,and oblate-ness f ,the best spherical ?t to the oblate data have larger radii and higher impact parameters than the actual val-ues (see Section 5.1).Objects wit
h negative values of f are prolate,a physically unreasonable proposition that we include here for completeness.
0.20.40.60.8
0.0
0.20.40.60.8
1.0Actual Impact Parameter, b
C h a n g e i n F i t t e d P a r a m e t e r s
db df
dR df p
p R dR df R *
*
dc df
1Fig.8.—Here we plot the change of ?tted spherical model parameters with oblateness as a function of impact pa-rameter.The derivitive of impact parameter with respect to oblateness continues o?the top of this graph to 3.0at b =0.0.An oblate planet with zero obliquity transiting near the center of the star induces much higher deviations of the ?tted parameters from the actual parameters than would a planet transiting near the critical impact param-eter,b =0.707.
we ?t the HST HD209458b transit lightcurve and obtain R ?=1.142R ⊙,R p =1.343R Jup ,i =86.72o (b =0.504),c 1=0.647,and c 2=?0.065with a reduced χ2of 1.05,consistent with the values obtained by BCGNB.
In order for planetary oblateness to have a noticeable e?ect on a transit lightcurve,it must be distinguishable from the lightcurve of a spherical planet.For a spherical planet transit model,the combination of transit parame-ters that correspond to the lightcurve that best simulates the data from an actual oblate planet transit become the measured values,and these measured quantities may not be similar to the actual values.Therefore to consider the detectability of planetary oblateness,we compare oblate planet transit lightcurves to those of the best-?t spherical planet lightcurve instead of to the lightcurve of a planet that di?ers from the actual values only in the oblateness parameter (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 evolved planet,to the transit of the best-?t spherical planet in Fig-ure 5and Figure 6.The oblate planet transit signature is muted in each when compared to Figure 4due to a de-generacy between the ?tted parameters R ?,R p ,b ,and the oblateness f .This degeneracy is introduced by the uncon-strained nature of the problem:in essence,we are trying to solve for 5free parameters,R ?,R p ,b ,c 1,and f ,given just 4constraints,d ,w ,l ,and ηassuming (as BCGNB did)knowledge of the stellar mass M ?.Without assuming a value for M ?,absolute timescales for the problem vanish,yielding a similar conundrum of solving for R p /R ?,b ,c 1,and f given only d ,η,and w/l .The previous two sentences are intended only to be simpli?cations,as at higher photo-metric precision more information about the conditions of the transit is available.Hereafter we assume knowledge of M ?,though this analysis could also have been done without this assumption,or with an assumed relation between M ?and R ?as proposed by Cody &Sasselov (2002).Changes in R ?,R p ,and b mimic the signal of an oblate planet by altering the ingress and egress times while maintaining the total 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 w from Figure 3)are ?t better using a spherical model with a higher impact parameter than actual,thus lengthening the time between ?rst and second contact.Since for a given star transits at higher impact parameter have shorter overall duration,the best-?t spherical model has a larger R ?than the model used to generate the data to maintain the chord length,and thus a larger R p to maintain the overall transit depth.
Similarly,for simulated lightcurve data from a transit-ing oblate planet at high impact parameter (b >0.7),the best-?t spherical model has a lower impact parameter than the simulated planet to increase the duration of ingress and
?2
?1
1
2
?0.0002
?0.0001
0.00000.00010.0002
?0.0002?0.0001
0.00000.00010.0002?2
?1012
?0.0002?0.0001
0.00000.00010.0002Fig.9.—Detectability of oblateness and obliquity relative to the best-?t spherical model for planets with nonzero obliquity.The top panel shows the detectability di?erence for b =0.35and t =0?(dotted line),t =15?(dot-dashed line),t =30?(dashed line),and t =45?(solid line).The middle 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 di?erence is quali-tatively similar for each case.The di?erence is maximized for t =45?and b =0.7,and falls o?as the parameters near t =0?,t =90?,b =0.0,and b =1.0.Due to the symmetry of the problem,obliquities the ones shown plus 90?are the time inverse of the di?erences shown.
egress.For these planets,the ?tted spherical parameters have smaller R ?and R p than the simulated planet to main-tain the character of the rest of the lightcurve.
Oblate planets that have impact parameters near the critical value,b ～0.7,cannot be as easily ?t using a spher-ical model because changes in the impact parameter of the ?t cannot increase the duration of ingress and egress.For these planets,the di?erence between the oblate planet transit lightcurve and the best-?t spherical planet tran-sit lightcurve is maximized,providing the largest possible photometric signal with which to measure oblateness.
At present,it is necessary to ?t for R ?and stellar limb-darkening parameters because of our limited knowledge of
these values for the host stars of transiting planets.If R ?were known to su?cient accuracy (less than 1%),then with knowledge of M ?the degeneracy between R ?,R p ,and b would be broken,allowing measurement of planetary oblateness without ?tting.However,without assumptions about the stellar mass,knowledge of R ?would only help to constrain M ?.Cody &Sasselov (2002)show that,for the star HD209458,current evolutionary models combined with transit data serve to measure the stellar radius to a precision of only 10%.
In Figure 5we plot the di?erence between the transit lightcurve of a hypothetical planet with the characteristics of HD209458b but an oblateness f =0.1,and that planet’s best-?t spherical model ?tting for R ?,R p ,b ,and c 1.For low values of the impact parameter,b ≤0.5,the best-?t spherical lightcurve emulates the oblate planet’s ingress and egress while leaving a subtly di?erent transit bottom due to the planet traversing a di?erently limb-darkened chord across the star.The magnitude of the di?erence is approximately a factor of 10smaller than the non-?t di?erence from Figure 4.Near the critical impact param-eter,b =0.7,the transit lightcurve bottoms are very sim-ilar since the best-?t b is very similar to the actual oblate planet’s impact parameter;however,the ingress and egress di?er in ?ux by a few parts in 105for f =0.1.For grazing transits,b ～1.0,the best-?t spherical model’s lightcurve is indistinguishable from the oblate planet transit lightcurve even at the 10?6relative accuracy level.
Figure 6shows the same di?erences as in Figure 5,but with a second stellar limb darkening parameter,BCGNB’s c 2,also left as a free parameter in the ?t.Including c 2in the ?t allows the ?tting algorithm to match the transit bot-tom (the time between second and third contacts)better.This e?ect leads to excellent ?ts of oblate planet transits using spherical planet models and reduces the detectable di?erence to less than one part in 105for 0.00.9using HD209458stellar and planetary radii and an oblateness of f =0.1.For b =0.0and b =1.0the spheri-cal model emulates an oblate transit particularly well,with the di?erences between the two being only a few millionths of the stellar ?ux.
In both the 4-and 5-parameter zero-obliquity models,it is easiest to measure the e?ects of oblateness on the transit light curve for transits near the critical impact parameter.
For HD209458values with f=0.1,the oblateness signal then approaches3×10?5for tens of minutes during ingress and egress and peaks near b=0.8instead of the critical value of b=0.7due to the?nite radius of HD209458b relative 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)used5years of 3minute resolution SOHO spacecraft data to deduce the noise power spectrum of the Sun.These noise e?ects are close enough in magnitude to the transit signal so as to a?ect the detectability of a transiting oblate planet.
Future high-precision space-based photometry missions such as MOST and Kepler may be able to detect the ef-fect for highly oblate transiting extrasolar planets,but the S/N ratio could be so low as to make unambiguous mea-surements of oblateness di?ucult to obtain.
If an observer were to?t photometric timeseries data from a transit of an oblate planet without?tting explicitly for the oblateness f(as,for instance,if the precision is insu?cient to measure f),the planet’s oblateness will be manifest as an astrophysical source of systematic error in the determination of the other transit parameters.Figure 7shows the e?ect that oblateness has on the stellar radius, planetary radius,impact parameter,and limb darkening coe?cient for HD209458b.This variation is a strong func-tion of the planet’s impact parameter.In Figure8,we show the how this systematic variation changes as a function of the initial impact parameter for the HD209458b system. As a severe but still physical example,an HD209458b-like planet with oblateness f=0.1that transits at b=0.0 would be measured to have radii2%above actual and an impact parameter nearly0.3above the real impact param-eter.
5.1.1.HD209458b
To illustrate the robustness of the degeneracy between R?,R p,b,and f discussed in Section5.1,we?t the HST lightcurve from BCGNB using a planet with a?xed,large oblateness of0.3(!).The best-?t parameters were R?=
1.08R⊙,R p=1.26R Jup,b=0.39,and c1=0.633with
a reducedχ2=1.06—indistinguishable in signi?cance from the spherical planet model!Although unlikely,a high actual oblateness for HD209458
b could alter the measured value for the planet’s radius into better agreement with theoretical models.
The expected detectability of oblateness for HD209458b is extremely low.During ingress and egress the transit lightcurve for HD209458b should di?er from that of the best-?t spherical model by only one part in106for f=0.01 and at the level of3×10?7if f=0.003.For comparison, the BCGNB HST photometric precision is1×10?4.
Although the systematic error in the determination of transit parameters can be important for highly oblate plan-ets,it is not at all signi?cant for HD209458b.Assuming HD209458b has an oblateness of f=0.003as calculated in Section3,the?tted radii are only～0.05%above the ac-tual radii,and the?tted impact parameter is0.0006above what the actual impact parameter should be.
5.1.2.OGLE-TR-56b
Measuring the oblateness of the new transiting planet OGLE-TR-56b(Konacki et al.2003),f=0.016(see Sec-tion3),would require photometric precision down to at least4×10?6.Since the impact parameter for this object is as yet unconstrained,the above precision corresponds to b=0.7.For other transit geometries,higher precision photometry would be necessary to measure the oblateness of this object.
5.2.Nonzero Obliquity
We plot the detectability of oblateness and obliquity for planets with nonzero obliquity in Figure9.Here we de?ne the projected obliquity(which we refer to as just obliquity hereafter),or axial tilt t,as the angle between the orbit angular momentum vector and the rotational angular momentum vector projected into the sky plane,measured clockwise from the angular momentum vector(see Figure 3).The major e?ect of nonzero obliquity is to introduce an asymmetry into the transit lightcurve(Hui&Seager 2002).
The plots in Figure9are di?erence plots for planets with di?erent obliquities and impact parameters,yet all have the same shape qualitatively.In trying to?t the asym-metric ingress and egress lightcurves,the best-?t spherical planet splits the di?erence between them.For planets with 0 This general shape is the same for the asymmetric com-ponent of each transit lightcurve,and it is superimposed on top of the symmetric component studied in Section5.1. The asymmetric component is maximized near the critical impact parameter(b=0.7),because the planet crosses the stellar limb with its projected velocity vector at an angle of π/4with respect to the limb.For transits across the mid-dle of the star,b=0,the asymmetric component of the lightcurve vanishes as a new symmetry around the planet’s path is introduced,and the local angle between the veloc-ity vector and the limb isπ/2.Similarly the asymmetric planet signal formally goes to zero for grazing transits at b=1.0,but in practice the asymmetric component is still high for Jupiter-sized planets. Planets with obliquities of zero(t=0)are symmetric and have no asymmetric lightcurve component(see Section 5.1).Likewise,planets with t=π/2are also symmetric. The asymmetric component is maximized for planets with t=π/4. Transiting planets with nonzero obliquity can break the degeneracy between transit parameters discussed in Sec-tion5.1.However,when the obliquity is nonzero a de-generacy between projected obliquity and oblateness is in-troduced:a measured asymmetric lightcurve component of a given magnitude could be due to a planet with low oblateness but near the maximum detectability obliquity of t=π/4,or it could be the result of a more highly oblate planet with a lower obliquity.In this case only a lower limit to the oblateness can be determined based on the oblate-ness for an assumed obliquity ofπ/4.This degeneracy can be broken with measurements precise enough to determine the symmetric lightcurve component.In addition,transit photometry is only able to measure the projected oblate-ness and obliquity of such objects due to the unknown component of obliquity along the line-of-sight(Hui&Sea-ger2002),therefore the true oblateness is never smaller than the measured,projected oblateness. The asymmetric transit signal of an oblate planet with nonzero obliquity could also be muddled by the presence of other bodies orbiting the planet.Orbiting satellites or rings could both introduce asymmetries into the transit lightcurve that may not be easily distinguishable from the asymmetry resulting from the oblate planet.Satellites around tidally evolved planets are not stable(Barnes& O’Brien2002),and rings around these objects may prove to be di?cult to sustain as well.However,objects that are not tidally evolved,those farther away from their parent stars,may potentially retain such adornments,and their e?ects on a transit lightcurve could be di?cult to di?eren-tiate from oblateness. Orbital eccentricity can also cause a transit lightcurve to be asymmetric.Although the eccentricity of HD209458b is zero to within measurement uncertainty based on radial velocity measurements,future planets discovered solely by their 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 velocity change between ingress and egress may be high enough to emulate the oblateness asymmetry discussed in this sec-tion.However,we do not explicitly treat that situation in this paper. 6.CONCLUSIONS Examining a transiting planet’s precise lightcurve can allow 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 extrasolar giant planets is related to the planet’s oblateness by the Darwin-Radau relation.Measurements of a planet’s rota-tion rate could constrain the tidal dissipation factor Q for those planets,as well as possibly shed light on the tidal dissipation mechanism within giant planets based on the spin:orbit ratios of tidally evolved eccentric planets. The detectable e?ect of oblateness on the lightcurve of a planet with zero obliquity is at best on the level of a few parts in105.This e?ect may be discernable from space with ultra high precision photometry,but could prove to be indistinguishable from stellar noise or other confound-ing e?ects.Accurate independent measurements of stellar radius can break the degeneracy between the stellar radius, planetary radius,and impact parameter,allowing for much easier measurement of oblateness.Without such measure-ments,the primary e?ect of oblateness and zero obliquity when studying transit lightcurves will be to provide an as-trophysical source of systematic error in the measurement of transit parameters. Planets with nonzero obliquity have higher detectabil-ities(up to～10?4)than planets with no obliquity,but yield nonunique determinations of obliquity and oblate-ness.In order to obtain unique obliquities and oblatenesses for these objects,precision comparable to that needed for the zero obliquity case is needed. The detectability of oblateness for transiting planets is maximized for impact parameters near the critical impact parameter of b=0.7.Many transit searches are currently underway,yielding the potential for the discovery of many transiting 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 that transit near an impact parameter of0.7. 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