Characterization of Kepler ’s Transiting Extrasolar Giant Planets
Jason W. Barnes
1 Statement of Problem
I propose to help characterize the transiting extrasolar planets detected by NASA’s Kepler
mission by reﬁning and applying transit lightcurve ﬁtting algorithms.
Unlike HD209458, the parent star of the best-known transiting planet, the stars that
Kepler will monitor do not have accurately measured parallaxes. Therefore the radii of
these stars is unknown. This uncertainty in stellar radius directly translates to uncertainty
in the radius of any transiting planets, which is the most important measurement that Kepler
will make. Hence it is critical that Kepler employ a reliable method for estimating stellar
Previous investigations into transit lightcurves (Seager and Mall´n-Ornelas, 2003i.e.)
have noted the advantages of minimizing stellar limb darkening when analyzing low-precision
photometric data. I suggest that in circumstances where the photometric precision is high,
stellar limb darkening can be used to an transit observer’s advantage to potentially constrain
the stellar radius without any previous knowledge of the star’s mass or radius. Since the
technique only requires the transit lightcurve, no additional observations are needed. Essen-
tially, if the nature of the stellar limb darkening itself is regular, or predictable, between stars
in Kepler ’s spectral ﬁlter, then it may be used as a constraint that will set the transit impact
parameter, constraining the relationship between M∗ , the stellar mass, and R∗ , the stellar
radius. By then applying stellar models similar to those used by Cody and Sasselov (2002),
the M∗ - R∗ degeneracy can be broken, establishing both the star’s radius and its mass. This
procedure will be particularly useful for planets in the middle of Kepler ’s size detectability
range: those whose ingress and egress are not temporally resolved but for which the limb
darkening is photometrically resolved. I would like to investigate this idea quantitatively
during my NRC Postdoctoral Research Associateship.
Transits can also reveal moons (Sartoretti and Schneider, 1999). Brown et al. (2001)
detected no moons orbiting HD209458b, consistent with theoretical expectation (Barnes
and O’Brien, 2002). Kepler is capable of detecting large moons of extrasolar planets. I will
extend the work of Sartoretti and Schneider (1999), incorporating modern lightcurve-ﬁtting
techniques to the direct transit of extrasolar moons. The resulting algorithm will be used to
search for moons around planets detected by Kepler.
Subtle (1 part in 105 for zero obliquity, up to 1 in 104 for nonzero obliquity) deviations
during a planet’s transit ingress and egress can reveal the planet’s oblateness (Barnes and
Fortney, 2003), and from that the planet’s rotation rate. For Kepler this precision level
will require the coaddition of many separate transit events in order to achieve the necessary
signal-to-noise ratio, and will still be diﬃcult. However, for low-density, rapidly rotating
planets at intermediate semimajor axis (0.2 < a < 1.6) it may be possible to measure
rotation from the Kepler transit lightcurve.
In Barnes and Fortney (2004) I showed that large, Saturn-like ring systems, in the right
viewing geometry, can show whopping transit lightcurve detectabilities on the order of 1 ×
10−3 . The potentially intricate complexity of such ring systems, like that of Saturn, make
ﬁtting a lightcurve for ring sizes and optical depths computationally intensive. In order to
ﬁt potentially anomalous Kepler transit lightcurves, I propose to develop and reﬁne ringed
planet transit ﬁtting algorithms until they are capable of ﬁtting Kepler data on a reasonable
timescale. I will then apply the ﬁtting scheme to the Kepler transiting planets.
2 Background and Relevance
Although over 150 extrasolar planets have been discovered (see http://c3po.barnesos.net/egpdb),
characterization has been possible only for those 9 that transit. Radial velocity monitoring
alone measures planets’ minimum masses and orbital characteristics (Marcy et al., 2000,
e.g.). Precision transit photometry reveals a planet’s orbital inclination and radius. To-
gether, the two techniques are able to nail down the planet’s actual mass and therefore its
This analysis was ﬁrst done for transiting planet HD209458b (Charbonneau et al., 2000;
Henry et al., 2000). The edge-on inclination of HD209458b’s orbit removed the sin i degen-
eracy from the radial-velocity results, pinning the planet’s mass at 0.69 Jupiter masses (M J ).
Subsequent Hubble Space Telescope photometry of the transit showed no evidence for moons
or rings (Brown et al., 2001), consistent with theoretical constraints placed on extrasolar
moons (Barnes and O’Brien, 2002). Though its parallax was measured by Hipparcos, the
leading source of error in the determination of HD209458b’s radius remained the uncertainty
in the radius of its parent star, HD209458. The current best estimate for HD209458b’s ra-
dius, 1.42+0.10 RJ , was calculated by Cody and Sasselov (2002) by coupling the HST transit
lightcurve ﬁt to a stellar structure model. However, this sophisticated treatment requires
knowledge of the star’s luminosity, which cannot be calculated without knowledge of the
The transits of HD209458b allow additional follow-up observations that have made HD209458b
the best-characterized extrasolar planet to date. Charbonneau et al. (2002) detected sodium
in HD209458b’s outer atmosphere from HST spectrophotometry. Hydrogen, carbon, and
oxygen were all detected in the process of hydrodynamic escape from HD209458b’s atmo-
sphere (Vidal-Madjar et al., 2003; ?). The Spitzer Space Telescope detected infrared emission
from HD209458b’s dayside during its secondary eclipse, directly measuring its day temper-
ature to be 1100K (?). Very recently, ?) used transit observations to measure the relative
angle between the planet’s orbital plane and the stellar equatorial plane to be 4.4◦ .
Since 1999, 8 additional transiting planets have been discovered. Like HD209458b, two
others were ﬁrst found using radial velocity techniques but were later found to transit. Both
were discovered in the last 6 months. HD149026b (Mp = 0.36MJ , Rp = 0.73RJ ) (?) and
HD189733b (Mp = 1.15MJ , Rp = 1.26RJ ) (?) are both close-in hot Jupiters. HD149026b in
particular is important because although its mass is similar to that of Saturn, its radius is
lower despite high insolation, implying that the planet has a large core (?). Both of these
planets are around stars bright enough for signiﬁcant follow-up observations.
Six extrasolar planets have been discovered using the transit technique so far. TrES-1 (?),
an 0.75 MJ , 1.08 RJ planet in a 3 day orbit, was found from its transits of a relatively bright
(V=11.79) star based on a wide-ﬁeld transit search. As such, TrES-1 is bright enough for
follow-up observations which have included the detection of dayside infrared emissions using
the Spitzer Space Telescope (Charbonneau et al., 2005). The Optical Gravitational Lensing
Experiment (OGLE) team has spent one month per year on a narrow-ﬁeld, deep search
for planetary transits (Udalski et al., 2002a,b). Weeding out false positives from possible
transits is the hardest part of the process – less than 10% of prospective transits have yielded
planets. There are currently 5 generally-accepted OGLE planets: OGLE-TR-56b (Konacki
et al., 2003), OGLE-TR-113b, OGLE-TR-132b (Bouchy et al., 2004), OGLE-TR-111b (?),
and OGLE-TR-10b (?). The OGLE planets are all far away and their parent stars are faint.
Ground-based follow-up for them is challenging.
The prospects for characterizing extrasolar giant planets and their environs are very
favorable with the 2008 launch of NASA’s Kepler mission(?). Kepler is a 0.95-m aperture
photometer designed to continuously monitor 100,000 stars for 4 years searching for transiting
planets. Although designed to ﬁnd Earth-sized worlds, Kepler is also expected to discover
30 transiting giant planets in large (¿ 1.6 AU) orbits, 35 in moderate (0.2 ¡ a ¡ 1.6 AU)
orbits, and 100 in close-in orbits. Any Neptune-mass objects that Kepler ﬁnds will also be
amenable to characterization. Furthermore, Kepler ’s high photometric precision (∼ 1 × 10 −4
or better for 15 minute data points, depending on the target) will result in lightcurves of
comparable or superior quality to Hubble Space Telescope observations of HD209458 (Brown
et al., 2001) without the need for targeted follow-up.
3 Methodology, Procedures, and Techniques
Transit lightcurve ﬁtting algorithms measure just 4 parameters: b, the transit impact pa-
rameter, related to the planet’s orbital inclination; R∗ , the stellar radius; Rp , the planetary
radius; and c1, a composite limb-darkening coeﬃcient (after Brown et al. (2001)). Roughly
speaking, the depth (fraction of the star’s light blocked in mid-transit) of the transit sets the
ratio R∗ . The curvature of the transit bottom constrains c1. The duration of ingress and
egress establishes b, and the total transit duration sets the length of the transit chord which
then yields R∗ . These constraints rely on previous knowledge of the star’s mass, M∗ .
Seager and Mall´n-Ornelas (2003) showed that, in cases where limb darkening is negligi-
ble, the length of ingress and egress can be directly measured, and an analytical solution to
the mappings discussed in the previous paragraph exists. However, Kepler operates in visi-
ble wavelengths where limb darkening is not negligible. For planets with Rp ∼ RJ Kepler ’s
excellent photometric precision will still resolve the end of ingress and beginning of egress
suﬃciently for a model-based ﬁtting routine to measure the 4 basic transit parameters.
However, as the duration of ingress and egress depends on the radius of the transiting
planet, for planets with Rp < 0.3RJ or so the ingress and egress will not be temporally
resolved. In this case, the stellar limb darkening may be capable of constraining the transit
impact parameter, thus taking the place of the ingress and egress duration in model ﬁts.
To test this hypothesis, I will alter my transit-ﬁtting algorithm developed for Barnes and
Fortney (2003) and Barnes and Fortney (2004) to accomodate the new procedure. I will then
generate synthetic lightcurves simulating Kepler transit photometry and ﬁt those data using
both ﬁxed and variable limb darkening parameters. The process may require prior knowledge
of the precise value of c1, or it may be that the mere knowledge that the star’s limb darkening
structure can be wholly determined by the single parameter c1 may be suﬃcient.
Either way, this technique will only establish a relationship between M∗ and R∗ , and will
not by itself unambiguously determine the stellar radius. The application of stellar modelling,
of a type similar to that pioneered for use with planetary transits by Cody and Sasselov
(2002), may resolve the degeneracy. Along with ground-based astrometric measurements of
stellar parallax, application of the star’s spectral type should be able to limit the allowable
range of M∗ and R∗ as set by the lightcurve ﬁt. I do not yet know what level of uncertainty
in R∗ will remain after this type of analysis, but determination of the expected residual
uncertainty will be one of the research goals.
A possible back-door method for determining M∗ involves moons. If a moon were detected
in transit, its period, along with multiple determinations of its distance from the parent
planet, would independently and unambiguously determine Mp . The amplitude and period
of later-measured planet-induced radial velocity variations of the star could then set the
stellar mass, and thereby allow easy determination of the stellar radius from the planet’s
transit lightcurve. In Barnes and O’Brien (2002) I showed that large, detectable moons
should not exist around close-in giant planets. However, moons larger than 1R⊕ can exist
around giant planets with semimajor axes larger than ∼ 0.2 AU, and can serve to measure
the planet’s mass.
Direct measurement of the semimajor axis of an extrasolar moon may be challenging. I
will ﬁrst develop the capability to ﬁt the transit lightcurve of an extrasolar moon incorpo-
rating orbital motion on a variably inclined orbital plane, and then apply the technique to
synthetic Kepler lightcurves to evaluate its feasibility and limitations. The resulting program
will be used to ﬁt anomalous lightcurves from Kepler after its launch in 2008.
The algorithm for ﬁtting planetary oblateness has already been written (Barnes and
Fortney, 2003), and can be used on Kepler lightcurves with little modiﬁcation.
However, no algorithm exists for ﬁtting the transits of ringed planets. The reduced
symmetry of the problem makes a brute-force ﬁtting approach glacially slow – ﬁts would
take many months. I will make optimizing improvements in the ﬁtting algorithm in order to
ﬁt transiting ringed planets on a reasonable timescale. The technique will assume a single
ring of uniform optical depth (to be ﬁt) and an interior and exterior radius to be ﬁt, along
with the planetary obliquity.
4 Expected Results and Signiﬁcance
The Kepler mission will establish the types and distribution of terrestrial planets and plan-
etary systems in the Sun’s vicinity. It will also discover and characterize hundreds of giant
My proposed contribution to the project will be to develop a sophisticated transit lightcurve
ﬁtting algorithm for eventual application to Kepler data. In the meantime, I will publish
my studies of the usefulness and limitations of the new techniques.
Planetary radius is the fundamental measurable quantity that Kepler will provide, and
its measurement of that value will depend on the stellar radius. My algorithm and analysis
will help to constrain stellar radii throughout the Kepler project.
Moons of extrasolar planets are exciting in their own right and as possible habitable
worlds. Moons also might be used to independently measure the mass of an extrasolar
planet. The types and distribution of extrasolar moons will help to constrain the formation
and evolution of planets and planetary systems.
I also expect to be able to measure the rotation rate of perhaps a handful of extrasolar
giant planets based on a determination of oblateness from a transit lightcurve (Barnes and
Fortney, 2003). This analysis can also constrain the planet’s obliquity (Seager and Hui,
2002). These measurements also constrain formation and evolution histories, and may also
help to reveal the yet-unknown tidal dissipation mechanism within giant planets (Barnes
and Fortney, 2003).
In our own solar system, the origin, age, and longevity of Saturn’s spectacular ring
system remain unresolved. If they exist, Kepler will ﬁnd large, Saturn-like ring systems
around extrasolar giant planets. The distribution of these extrasolar ring systems will shed
light on the prevalence of rings in general, as well as provide fuel for a new understanding
of the workings of rings.
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