Impact of Orbital Eccentricity on the Detection of Transiting Extrasolar Planets

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by  Christopher J. Burke

Submittedforpublicationin theAstrophysical Journal
PreprinttypesetusingLATEXstyleemulateapjv.03/07/07
IMPACT OF ORBITAL ECCENTRICITY ON THE DETECTION OF TRANSITING EXTRASOLAR PLANETS
Christopher J. Burke
SpaceTelescopeScienceInstitute, 3700SanMartinDr.,Baltimore,MD,21218
(Received 2007 November09)
Submitted for publication in the Astrophysical Journal
ABSTRACT
For extrasolar planets with orbital periods, P>10 days, radial velocity surveys find non-circular
orbital eccentricities are common, e 0.3. Future surveys for extrasolar planets using the transit
h i ∼
8 technique will also have sensitivity to detect these longer period planets. Orbital eccentricity affects
0 the detectionofextrasolarplanetsusing the transittechnique intwoopposingways: anenhancement
0 in the probability for the planet to transit near pericenter and a reduction in the detectability of
2 the transit due to a shorter transit duration. For an eccentricity distribution matching the currently
n known extrasolar planets with P>10 day, the probability for the planet to transit is 1.25 times
∼
a higherthanthe equivalentcircularorbitandthe averagetransitdurationis 0.88timesshorterthan
∼
J the equivalent circular orbit. These two opposing effects nearly cancel for an idealized field transit
6 survey with independent photometric measurements that are dominated by Poisson noise. The net
1 effect is a modest 4% increase in the transiting planet yield comparedto assuming all planets have
∼
circular orbits. When intrinsic variability of the star or correlatedphotometric measurements are the

] dominantsource of noise,the transitdetectability is independent ofthe transitduration. In this case
h
thetransityieldis 25%higherthanthatpredictedundertheassumptionofcircularorbits. Sincethe
p ∼
Kepler searchfor Earth-sizedplanets in the habitable zone of a Solar-type star is limited by intrinsic
-
o variability, the Kepler mission is expected to have a 25% higher planet yield than that predicted
∼
r for circular orbits if the Earth-sized planets have an orbital eccentricity distribution similar to the
t
s currently known Jupiter-mass planets.
a Subject headings: eclipses — planetary systems — techniques: photometric
[

1
1. INTRODUCTION how the transit duration is affected by orbital eccentric-
v
ity, but they do not quantify the impact this will have
9 The known extrasolar planets possess a broad distri-
on transit surveys. Recently, Barnes (2007) derives the
7 bution of orbital eccentricity (Butler et al. 2006) (see
5 Figure 1). The short period, Hot Jupiter (P<10 day) probability for a planet on an eccentric orbit to transit
2 planets predominately have circular orbits. However, at and conclude that the photometric precision of current
1. longer orbital periods circular orbits become a minority surveys and future surveys, such as Kepler, is insuffi-
cient to determine the orbital eccentricity solely from
0 andthemedianeccentricityforextrasolarplanetse 0.3.
∼ the light curve. Barnes (2007) concludes that without
8 At the extreme eccentricity end, there are three planets,
knowledge of the eccentricity from radial velocity data
0 HD 80606b (Naef et al. 2001), HD 20782b (Jones et al.
or independent measurement of the stellar host radius,
: 2006),andHD4113b(Tamuz et al.2007),havinge>0.9.
v the habitability of planets detected with Kepler will re-
HD80606bcomesclosertoitsstellarhost(a=0.033AU)
Xi than many of the circular orbit Hot Jupiter planets. main unknown.
Neglecting the impact eccentricity has on transit de-
r Ford & Rasio(2007)recentlyreviewedthevariousmech-
a anisms invoked to explain the distribution of eccentrici- tections is justified for the current sample of transit-
ing planets given the predominance of circular orbits
ties for the known extrasolar planets. Interactions with
for the Hot Jupiters and the strong bias of transit sur-
a stellar companion, planetary companion, passing star,
veys against finding long period planets on circular or-
gaseous disk, planetesimal disk, and stellar jets have all
bits (Gaudi et al. 2005). The announced transit for the
been proposed to modify the orbital eccentricity of ex-
planet orbiting HD 171561 with a 21 day period and ec-
trasolar planets.
centric orbit (Barbieri et al. 2007) is a precursor for the
Limiteddiscussionsintheliteraturehavebeengivento
kinds ofplanets detectable in transitsurveys. As transit
theimpactorbitaleccentricityhasonatransitsurveyfor
surveys continue, longer period transiting planets may
extrasolarplanets. Tingley & Sackett(2005) discuss the
be discovered. More importantly, the recently launched
impact of eccentricity on their η parameter (η is the ra-
COROT mission will surpass current surveys for sen-
tioofthe observedtransitdurationtoanestimateofthe
sitivity to longer period planets (P several months;
transit duration). As expected, they find transits occur
∼
Bord´e et al. 2003). Also, the Kepler mission, scheduled
near pericenter (apocenter) are shorter (longer) in dura-
for launch in 2009, has a goal to find Earth-sized ob-
tionthan the circularorbitcase,andthey showthat the
jects at 1 AU from their host star (Borucki et al. 2004).
transit duration of an eccentric is typically shorter than
Themainpurposeofthispaperistoshowthatgiventhe
a circular orbit of the same period. However, their dis-
distribution of eccentricities for the currently known ex-
cussionwasfocused onthe impact oforbitaleccentricity
on their η parameter. Moutou et al. (2006) also discuss
1 First detected by the radial velocity technique (Fischeretal.
2007)
Electronicaddress: [email protected]
2 Christopher J. Burke
trasolarplanets,eccentricityshouldnotbeignoredinas-
sessingthe detectability oftransitinggiantplanets when
thetransitsurveyissensitivetoplanetswithP>10day.
In addition to longer period planets, the COROT
and Kepler missions also will detect transiting plan-
ets with small, Earth-sized radii. The eccentricity
distribution for planets less massive than the cur-
rently known Jupiter-mass planets is beginning to
be explored. Ribas & Miralda-Escud´e (2007) and
Ford & Rasio (2007) provide tentative evidence for the
tendency of lower mass planets to have lower eccentric-
ities in the current sample of radial velocity planets.
Thereistheoreticalagreementthataproto-planetarygas
disk strongly damps the eccentricity of non-gap open-
ing embedded low-mass planets (Goldreich & Tremaine
1980; Cresswell et al. 2007). However, the theoretical
modelsshowthatoncethegasdiskdissipates,dynamical
interactions amongst the planets results in a random- Fig. 1.— Normalized histogram of the eccentricity distribution
walk diffusion that leads to an increasing eccentricity forknownextrasolarplanetswithP>10day. Theeccentricitydis-
tributionisgiven forthe known extrasolar planets attwo epochs:
that takes on a Rayleigh-distribution similar to what
November2006(TopPanel)andSeptember2007(BottomPanel).
is observed (dotted line in Figure 1; Juric & Tremaine For this study a two-piece model parameterizes the eccentricity
2007; Zhou et al. 2007). Opposing the increases in ec- distribution (solid line). The model has 2 parameters: ecrit and
centricity fromplanet-planet scattering,late stage inter- emax. Themodel distributionisuniformatloweccentricity upto
actions with planetesimals can preferentially damp the etocrizte;roa-nvdaltuoewaatrdemhaixgh.eArlesoccsehnotwricnitiyn,tthheetmopodpealnleilneisartlhyedReacyreleaisgehs
eccentricities of lower mass planets enabling theoretical distribution that describes the eccentricity distribution from the
models to achieve the low eccentricities of the terrestrial planet-planet scattering calculations of (Juric&Tremaine 2007)
planets of the Solar System (Raymond et al. 2006) and (dottedline). Thedottedhistograminthebottompanelshowsthe
eccentricity distribution from the latest epoch (September 2007)
possibly explain the current trend of lower eccentricities thatexcludes planets withP<20day.
for lower mass planets (Ford & Rasio 2007). Given the
number of potential physical processes that can affect certainty in the underlying orbital eccentricity distribu-
orbital eccentricity, it is premature to assume that the tion affects the results. No attempt was made to select
typical Earth-like planet has an eccentricity near zero planets with well determined eccentricities, avoidance
like the Solar System. of multiple planet or multiple star systems, correction
Inthisstudy, 2reviewstheobservedeccentricitydis- for observationalbiases againsthigh eccentricity planets
§
tributionoftheknownradialvelocityextrasolarplanets. (Cumming 2004), orevolutionoforbitalelements due to
The broad distribution of orbital eccentricity has two tidal circularization. Planets with P < 10 day are not
main effects on the sensitivity of a transit survey. First, included in order to examine the potential issues that
theplanet-hostseparationvariesalonganeccentricorbit, future transit surveys sensitive to longer period planets
enhancing the probability to transit when the planet is will encounter. In this period regime (P>10 day) high
relativelyclosertothe stellarhost. 3quantifiesthe net eccentricity planets are quite common ( e = 0.3 and
§ h i
affectonthetransitprobabilityforapopulationofplan- σ =0.2).
e
ets with non-circular orbits. Second, the planet velocity A two-piece model parameterizes the eccentricity dis-
varies along and eccentric orbit, resulting in a reduction tribution from the earlier epoch (top panel of Figure 1).
or lengthening of the transit duration. 4 quantifies the The first piece at low eccentricity is flat up to e . The
crit
§
distributionoftransitdurationsresultingfromapopula- second piece matches the first piece at e , linearly de-
crit
tionof planets on eccentric orbits. 5 describes how the creases toward higher eccentricities, and is zero at e .
max
§
transit duration affects transit detection for transit sur- Thenormalizedtwo-piecemodelisgivenbytheequation,
veys in the limit of various noise sources. 6 concludes
§
by quantifying the net result of the two aforementioned
2 0 e e
effects, the enhanced probability to transit and the re- ecrit+emax ≤ ≤ crit
duced detectability, on the yield from transit surveys. P(e)de= ecrit+2emax(e(cer−it−emeamxa)x) ecrit <e≤emax .
0 emax <e<1
2. ECCENTRICITYDISTRIBUTION (1)
The solid histograms in Figure 1 show the normal- A χ2 minimization yields the best model parameters
ized distribution of eccentricities for the known extra- e = 0.25 and e = 0.92. The χ2 minimization
crit max
solar planets with P > 10 day (Butler et al. 2006). The applied to the more recent epoch eccentricity distribu-
top panel shows the eccentricity distribution from plan- tion (bottom panel of Figure 1), yields e = 0.0 and
crit
etsreportedbeforeNovember2006andthebottompanel e =0.91. The relative increase in the number of low
max
showstheeccentricitydistributionincludingmorerecent eccentricity planets discovered in the last year results in
discoveries (September 2007). The more recent radial bestparametersthatareeffectivelyasingle-piecemodel.
velocity discoveries over the last year have increased the Juric & Tremaine (2007) and Zhou et al. (2007) pre-
sample of low eccentricity systems. Results from this dict that the eccentricity distribution of dynami-
study are given for both epochs of the observed eccen- cally active planetary systems approaches a Rayleigh-
tricity distribution in order to characterize how the un- distribution due to planet-planet scattering, and they
Eccentricity Affects Detection of Transiting Planets 3
showthe Rayleigh-distributionissimilartothe observed
planeteccentricitydistribution. ThedottedcurveinFig-
ure 1 shows the Rayleigh-distribution with σ = 0.3
e
that best describes the outcome of the planet-planet
scattering calculations from Juric & Tremaine (2007).
The Rayleigh-distribution under represents the low ec-
centricity systems in the most recent eccentricity data.
Juric & Tremaine (2007) based their study on the sam-
ple of planets known at the time (April 2006). Subse-
quently, relatively more planets with low eccentricities
have been announced. In addition, Juric & Tremaine
(2007) included systems with P>20 day whereas this
samplehasP>10day. Thedottedhistograminthelower
panel of Figure 1 shows the eccentricity distribution at
the most recent epoch (September 2007), but with sys-
tems with P<20 day removed; apparently the difference
is slight and will hereafter be ignored. Although this
study concentrates on the two-piece model to describe
the eccentricity distribution, results are also given for
the Rayleigh-distribution with σ = 0.3. To provide re-
e
sults with the Rayleigh-distribution, the distribution is
set to zero for e>0.95. Fig. 2.—Averagetransitprobabilityasafunctionoftheobserved
eccentricitydistributionmodelparameters. Thetransitprobability
3. TRANSITPROBABILITY is scaled to the equivalent transit probability for a circular orbit.
Theabscissaindicatesemax,andthecurvesareforselectedvalues
Barnes (2007) derives the impact of orbital eccentric- of ecrit as labeled. For the model parameters shown in Figure 1,
ity on the probability for a planet to transit its stellar hProbTei ∼ 25% higher than assuming all planets have circular
host (see also Seagroveset al. 2003). The probability orbits(dotted lines).
to transit depends on the planet-star separation during larger than expectations that assume circular orbits for
transit. Foraneccentricorbit,atransitoccurringduring all planets. This section discusses only the probability
pericenter enhances the transitprobability and a transit to transit, and it does not address whether the planet
occurring during apocenter decreases the transit prob- has a transit signal that is detectable. Thus, the re-
ability. When averaged over observing angles, the net sults fromthis sectionassumesthe enhancedprobability
result is an enhancement of the probability to transit for the planet to transit does not affect its detectability,
over the circular orbit case, which is the topic of the next section.
ProbTo 4. TRANSITDURATION
Prob = , (2)
Te (1 e2) 4.1. Edge-on Transit
−
where Prob is the transit probability of the circular In addition to enhancing the probability for a transit
To
orbit case (Barnes 2007). A planet with e=0.6 is 1.5 to occur, orbital eccentricity results in the transit dura-
times more likely to transit than a planet on a circ∼ular tion varying according to the planet’s longitude of peri-
orbit. center during transit. The transit duration is shortest
With the observed eccentricity distribution from 2, (longest)if the transitoccurswhen the planet is atperi-
itispossibletocalculatetheaverageenhancementin§the center (apocenter) when transiting its stellar host. A
transit probability over the circular orbit case, shorttransitdurationmayreducethedetectabilityofthe
transit event. Assuming constant velocity during tran-
1
sit, the transit duration scaled to the edge-on (i = 90◦)
Prob = Prob P(e)de. (3)
h Tei Te circular orbit case has extrema of
Z0
TheresultingintegralusingEquation1fortheeccentric- tp (1 e)
ity distribution is, τp = a = ∓ 1 e+e2/2, (5)
a to s(1 e) ∼ ∓
±
hProbTei= (e2mPaxro−bTeo2crit)[2(emax−ecrit)arctanh(ecrit) where the sign on top and bottom corresponds to the
+ln ((11−−ee2m2craitx))!+emaxln„((11+−eeccrriitt))((11+−eemmaaxx))«#. (4) p(aBperoairccneenenstteer2r0ai0sn7d)t.waipcWeochtehennetercei=rtcr0au.nl6as,rittohdreubritatrtaicnoanssiset,.dreuGsrpiaveteciontinvtehalyet
Using the numerical values for e and e from range of eccentricities observed for extrasolar planets
crit max
2, Prob =1.26 and 1.23 for the earlier and cur- with P>10 day, we expect significant variations in the
Te
§ h i
rent epochs of observed eccentricities, respectively. The transit duration compared to that of a circular orbit.
Rayleighdistributionresultsin Prob =1.31. Figure2 Tingley & Sackett (2005) (their Equation 7) provides
Te
h i
illustrates Prob for other choices of the eccentricity a simplified form of the transit duration, including lon-
Te
h i
distribution model parameters. Figure 2 is a function of gitude of pericenter, ̟, and orbital inclination, i. In
e and the curves are for selected values of e as la- deriving the transit duration, (Tingley & Sackett 2005)
max crit
beled. Thus, it is expected that the yield from a transit assume constant orbital velocity and planet-star separa-
surveythatissensitivetoP>10dayplanetswillbe 25% tionduringtransitandtheplanetcrossesthestellardisk
∼
4 Christopher J. Burke
resulting in
√1 e2dτ
p(τ)dτ = − . (9)
πτ (eτ)2 (√1 e2 τ)2
− − −
q
TherightpanelofFigure3showstheprobabilitydensity
for transit duration with i = 90◦ for various orbital ec-
centricities. As expectedthe probability density is heav-
ilyweightedtowardtheextrema. Thesingularitiesinthe
probabilitydensityattheextremaareintegrable,making
the probability density normalizable.
Fig. 3.—Left: Edge-ontransitduration,τ,scaledtotheedge-on
circularorbitcase as afunction of the longitude of pericenter, ̟, 4.2. Affect of Varying Inclination Angle
forseveraleccentricitiesaslabeled. Right: Forauniformdistribu- The previous section shows the expected transit du-
tion of ̟, solid curves show the distribution for transit duration
ration for fixed orbital eccentricity when the inclination
scaled to the edge-on circular orbit case assuming all orbits are
edge-on(i.e. i=90◦). angle,i=90◦. Varyingtheinclinationimpactsthe tran-
sitdurationintwoimportantways. First,astheinclina-
tion decreases, the path of the planet across the stellar
along a straight path (a R ). An exact calculation discshortens,resultinginareductionofthetransitdura-
⋆
(of a numerical nature giv≫en the need to solve Kepler’s tion. Second, for an eccentric orbit, the planet is closer
Equation) of the transit duration finds Equation 7 of to the star at pericenter and farther away at apocen-
Tingley & Sackett (2005) is accurate to better than 5% ter. The smaller separation at pericenter increases the
fore=0.9andorbitalseparationa 0.1AU(P 10day). range of inclination angles that result in a transit, and
Theequationisaccuratetobetter≥than1%fore∼=0.9and conversely, a planet at apocenter will have a reduction
a 1.0 AU. in the range of inclination angles for a transit to occur.
≥Afterscalingtothetransitdurationoftheedge-on(i= Thus, long transit duration events when the planet is
90◦),circularorbitwiththesameperiodandseparation, at apocenter become increasingly rare as the eccentric-
Equation 7 of Tingley & Sackett (2005) becomes, ity increases. This section quantifies these effects on the
transit duration distribution.
(1 e2) (1 e2) 2 Beginning with a simple joint distribution that is uni-
τ = − 1 ρ2 − cos(i)2, form in cos(i) and uniform in ̟ enables the more com-
(1+ecos(̟))s − (1+ecos(̟))
p (cid:20) (cid:21) plicated distribution for transit duration to be derived.
(6)
For this section, e remains fixed. The beginning joint
where ρ=a/(R +R ), R is the radius of the star and
⋆ p ⋆ distribution is,
R is the radius of the planet. The definition for ̟ is
p
with respect to the line of sight. Thus, ̟ = 0 means P(̟,0 cos(i) cos(imin)e)d(cos(i))d̟ =Ad(cos(i))d̟,
≤ ≤ |
the pericenter is aligned with the observersline of sight, (10)
and ̟ = 180◦ means the apocenter is aligned with the wherecos(imin)iscosineoftheminimuminclinationan-
observers line of sight. This varies from the definition glenecessaryfora transittooccur,andAisthe normal-
of the argument of pericenter, ω, which is defined with izationconstant. Settingτ =0inEquation6andsolving
respectto the line ofnodes onthe plane ofthe sky(̟ = forcos(i)yieldscos(imin)=(1+ecos(̟))/ρ(1 e2). In-
−
ω 90◦). tegrating Equation 10 over the range of variables yields
−Thei=90◦caseillustratesthefirstorderimpactofor- the normalization constant, A=ρ(1 e2)/π.
−
bitaleccentricityontransitduration. Inthiscase,Equa- The transformation law of probabilities for multiple
tion 6 simplifies to dimensions(Equation7.2.4inPress et al.1992)provides
the joint probability density in terms of new variables τ
(1 e2) and ̟ when starting with the probability density for
τ90◦ = (1+ec−os(̟)). (7) cos(i) and ̟ .
p ′
The left panel of Figure 3 shows the transit duration ∂cos(i) ∂cos(i)
P(τ,̟ e)dτd̟ = ∂τ ∂̟ P(cos(i),̟ )dτd̟,
with respect to the edge-on circular orbit case for a va- | ∂̟′ ∂̟′ ′
(cid:13) ∂τ ∂̟ (cid:13)
riety of orbital eccentricities as a function of ̟. For a (cid:13) (cid:13) (11)
uniform distribution in ̟, the shallow slope in transit (cid:13) (cid:13)
In Equation 11, ∂(cid:13)̟/∂τ = 0 and ∂(cid:13)̟ /∂̟ = 1. Thus,
duration at pericenter and apocenter results in a dis- ′
only ∂cos(i)/∂τ remains, and Equation 11 simplifies to
tribution of transit durations peaked at these extrema
(given by Equation 5) with low probability for a transit ∂cos(i)
P(τ,̟ e)dτd̟ = ρ(1 e2)/πdτd̟. (12)
duration in between these extrema. | ∂τ −
Assuming a uniform distribution of ̟, p(̟)d̟ = (cid:12) (cid:12)
(cid:12) (cid:12)
d̟/π, the transformationlaw ofprobabilities (Equation The requisite deriva(cid:12)tive nee(cid:12)ded in Equation 12 is ob-
(cid:12) (cid:12)
7.2.4inPress et al.1992)yields the distributionoftran- tained by solving Equation 6 for cos(i) yielding
sit durations,
(1+ecos(̟))
cos(i)= (1 e2) τ2(1+ecos(̟))2,
∂̟ ρ(1 e2)3/2 − −
p(τ)dτ = p(̟)dτ, (8) −
∂τ p (13)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Eccentricity Affects Detection of Transiting Planets 5
as the solid line. The function,
π
τ = 1 e2, (18)
e
h i 4 −
p
fits the relation to better than 10−6 (similar to the nu-
merical integration precision), which very strongly sug-
gests this is the analytical solution to the integral in
Equation 17. Tingley & Sackett (2005) find the same
resultintermsoftheaveragevaluefortheirη parameter
at fixed eccentricity (see their Equation 18).
4.3. Transit Duration Distribution For Observed
Fig. 4.—Left: Transitdurationdistributionscaledtotheedge- Eccentricity Distribution
oncircularorbitcaseforauniformdistributionofcos(i)andfixed
eccentricity. The cases e=0.2, 0.4, 0.6, and 0.8 are shown. Right: The previous section derives the transit duration dis-
Averagetransitduration scaledtothe average transitdurationof tribution at fixed eccentricity. This section derives the
acircularorbitasafunctionoforbitaleccentricity(solid line). transitdurationdistributionassumingplanetsfollowthe
and taking the partial derivative with respect to τ. Per- observed eccentricity distribution. As in the previous
forming this operation yields section,derivingthe transitdurationdistributionbegins
P(0 τ τ ,̟ ̟ π e)dτd̟ withthesimpledistributionthatisuniformincos(i),uni-
a min
≤ ≤ ≤ ≤ | formin̟, andthe distributionofefollowsthe observed
τ(1+ecos(̟))3 (14)
= dτd̟, distribution as given in 2. The transformation law of
π√1 e2 (1 e2) τ2(1+ecos(̟))2 probabilities enables tra§nsforming the simple distribu-
− − −
tion in cos(i), ̟ , and e into the distribution expressed
wheretheloweprlimitto̟ isnecessarytoavoidanimag- in τ, ̟, and e. ′ ′
inary result and is given by
The initial distribution is given by,
̟min =cos−1 MIN 1.0,√1−e2−τ . (15) P(0≤cos(i)≤cos(imin),0≤̟′≤π,0≤e′≤emax)
" τe !# dcos(i)d̟ de =AP(e)d(cos(i))d̟ de,
′ ′ ′ ′ ′
(19)
For τ τ , ̟ = 0, but as τ > τ , ̟ > 0
peri min peri min
≤
sinceonlytransitsoccurringclosertoapocenterresultin where the normalization constant A=ργ/π, where
a transit duration long enough.
Finally, the conditional probability density for transit γ =(e2 e2 )[2(e e )arctanh(e )
max− crit max− crit crit
duration alone is obtained by integrating over ̟, (1 e2 ) (1 e )(1+e ) −1
π +ln − max +e ln − crit max .
P(τ|e)dτ =Z̟minP(τ,̟|e)dτd̟. (16) (cid:18)(1−e2crit)(cid:19) max (cid:18)(1+ecrit)(1−emax)(cid:19)(2(cid:21)0)
A solution to the above integral is possible in terms of TheJacobiantransformationmatrixsimplifiesasbefore,
a summation of the incomplete elliptic integrals of the such that the transit duration distribution scaled to the
first, second, and third kind (Byrd & Friedman 1954). edge-on, circular orbit transit duration is given by,
In practice, I choose to solve the integral numerically,
which is readily solved using the Romberg open ended P(0 τ τa,̟min ̟ π,emin e emax)dτd̟de
≤ ≤ ≤ ≤ ≤ ≤
algorithm which takes into account the singularity at ∂cos(i)
the lower limit of integration at ̟ = ̟ (Press et al. = P(cos(i),̟ ,e)dτd̟de,
min ∂τ ′ ′
1992). Tests of convergenceshow Equation16 has a sin- (cid:12) (cid:12)
(cid:12) (cid:12) (21)
gularity at τ = τperi, but it is integrable elsewhere over (cid:12) (cid:12)
(cid:12) (cid:12)
the range 0≤τ ≤τa. where the lower limit to the eccentricity, emin, becomes
The left panel in Figure 4 shows the probability den- necessary for τ > 1, when too small of an eccentricity
sity for transit duration scaled to the edge-on circular cannotproduceatransitdurationaslongasτ. Solvingτ
a
orbit case at several values of eccentricity. By compari- for e yields e =MAX[0.0,(τ2 1)/(τ2+1)]. Overall,
min
sonto Figure3, the mainimpact oforbitalinclinationis the joint distribution is given by−
to strongly enhance the probability of observing a short
P(τ,̟,e)dτd̟de=
duration event at pericenter relative to a long duration
event at apocenter. The probability density also allows γP(e) τ(1+ecos(̟))3
arbitrarily short events due to the potential for grazing dτd̟de.
π (1 e2)3/2 1 e2 τ2(1+ecos(̟))2
events. − − −
(22)
The probability density for transit duration can be p
summarized by finding the average transit duration at Integrating over ̟ and e provides the final probabil-
fixed eccentricity, ity density for τ for the assumed distribution of orbital
τa eccentricities. Given the additional complication of in-
τe = τP(τ e)dτ. (17) tegration over two variables, an analytical solution was
h i |
Z0 not forthcoming. As in 4.2, the singularities in the
§
TherightpanelinFigure4shows τ scaledtotheaver- integrand are integrable, and the Romberg open ended
e
h i
agetransitdurationofthecircularorbitcase, τ =π/4, algorithmwhichtakesintoaccountthesingularityatthe
0
h i
6 Christopher J. Burke
the top panel of Figure 1 τ / τ = 0.88 and the bot-
e o
h i h i
tom panel τ / τ = 0.89. The Rayleigh distribution
e o
h i h i
results in τ / τ =0.86
e o
h i h i
5. DISCUSSION:APPLICATIONTOTRANSITSURVEYS
The results from 4 quantify the impact orbital ec-
§
centricity has on the transit duration. Overall, orbital
eccentricity results in shorter transit durations than the
circularorbitcase,andtheshorttransitdurationreduces
the transit detectability. This section quantifies the re-
duction in transit detectability for various noise models
of transit surveys.
Fig. 5.—Left: Transitdurationdistributionscaledtotheedge- Inatransitsurveywithindependentphotometricmea-
on circular orbit case for a uniform distribution of cos(i) and the surements, the transit signal to noise ratio is,
observed distribution of e as shown in the top panel (solid line)
and bottom panel (long dashed line) of Figure 1. Given the bias ∆F
SNR= N , (23)
against detecting high eccentricity planets in radial velocity sur- obs
σ
veys, a uniform distribution of orbital eccentricities up to high
eccentricity (ecrit = 0.9 and emax = 0.95) will be highly skewed where∆F isthetransitdepth(pthetransitismodeledasa
towards τ ∼ 0.3 (short dashed line). A Rayleigh distribution of box-carshape),σ istheerroronaphotometricmeasure-
orbital eccentricities results in relatively fewer transit durations
ment, and N is the number of measurements during
τ ∼ 1 due to the relatively fewer objects on circular orbits (dot- obs
ted line). Right: Average transit duration scaled to the average one or more transit(s). A shorter transit duration re-
transit duration of a circular orbit as a function of the observed duces N τ, thus, SNR √τ. The observed eccen-
obs
eccentricitydistributionmodelparameters. Theabscissaindicates tricity distri∝bution (Figure 1∝) results in τ˜ = τ / τ =
emax,andthecurvesareforselectedvaluesofecrit aslabeled. For h ei h oi
themodelparametersshowninFigure1,hτi∼0.88timesshorter 0.88,andonaveragethereducedtransitdurationresults
thanassumingallplanets havecircularorbits(dotted line). in an effectively smaller SNR =√τ˜=0.94 per transit
eff
than assuming all planets are on circular orbits.
The above impact on the transit signal SNR due to
lowerlimitsofintegration(Press et al.1992)providesthe
a shorter transit duration is for an individual star in a
solution.
survey. However, the reduced SNR will have a larger
The solid and long-dashed lines in Figure 5 shows the
impact on the overall transit detectability in an ideal
distributionoftransitdurationscaledtotheedge-oncir-
field transit survey. As described in Gaudi et al. (2005)
cular orbit case for a population of extrasolar planets
and Gaudi (2007), a specified SNR criteria for transit
that follows the observed orbital eccentricity distribu-
detection, SNR , in a field transit survey corresponds
tionsshownintopandbottompanelsofFigure1,respec- min
tively. The probability density has a broadflat top with to a maximum distance, ℓmax ∝SNR−m1in, out to which a
τ = 0.4 just as likely to occur as τ = 1.0. For compar- planetis detectable. This proportionalityassumes white
ison, the transit duration distribution that would result noise and the dominant source of photometric error is
fromanearlyuniformdistributionoforbitaleccentricity Poisson noise. In the studies of Gaudi et al. (2005) and
up to high eccentricities (ecrit = 0.9 and emax = 0.95) Gaudi(2007),ℓmax isafunctionoftheplanetradiusand
is shown as the short-dashedline in Figure 5. Given the stellar host spectral type (i.e. ℓmax is a smaller distance
biasagainstdetectinge>0.6extrasolarplanetsinradial forasmallerradiusplanetorlargerradiusstar). Forthis
velocity studies (Cumming 2004), a large population of study only the dependence ofℓmax ontransitdurationis
e 0.9 planets cannot be ruled out. The transit dura- of interest.
tio∼ndistributionresultingfromtheRayleighdistribution Overall, the yield from a transit survey is propor-
of orbital eccentricity (dotted line in Figure 5) has rela- tionalto the number of objects in the survey that meets
tciivrceulylafreworebriptsla.netswithτ ∼1duetoitsrelativelyfewer SfoNrmRmlyini,nwthhiechsuisrvNeyobvjo∝luℓm3meaxasfoarpsptarorsprdiaistteribfourtendeaurnbiy-
Inthefuture,ifastatisticallylargesampleoftransiting stars. The effective SNRmin,eff =SNRmin√τ˜ larger than
planets with orbital P > 10 days is available with accu- assuming all planets are on circular orbits. Thus, the
rate stellar parameters, histograms of observed transit number of objects in an idealized transit survey where a
duration scaled to the edge-on circular orbit case may transit is detectable is N = τ˜3/2N = 0.82N
obj,e obj,o obj,o
help characterize the underlying eccentricity distribu- times smaller than the case where the detectability of
tion. After accounting for the selection effects, a large a transit is based on assuming all planets have circular
number of τ 0.3 detections relative to τ 0.8, as il- orbits, N . Despite the reduced detectability of tran-
obj,o
∼ ∼
lustrated in the right panel of Figure 5, would indicate sits,this isoffsetbythe higherprobabilityforthe planet
e = 0.9 planets are as common as circular orbits. Work to transit in the case of significant eccentricity 3. The
§
toward understanding the sensitivity of Kepler for con- overallyieldoftheidealizedtransitsurveyisdiscussedin
straining the underlying eccentricity distribution is un- 6 taking both the reduced detectability and enhanced
§
derway (E. Ford, private communication). probability to transit into account.
The right panel of Figure 5 summarizes the transit In practice, transit surveys typically are affected by
durationdistributionbyshowing τ intermsofthe pa- correlated measurements (Pont et al. 2006). In this
e
h i
rameterscharacterizingtheeccentricitydistribution,e regime, the correlation time scale is similar to the tran-
crit
and e . Each line corresponds to a fixed value of e sit duration and repeated measurements do not add
max crit
as labeled, and the abscissa indicates e of the eccen- independent information. When correlated measure-
max
tricity distribution. For the eccentricity distribution in ments dominate the photometric error, the SNR =
Eccentricity Affects Detection of Transiting Planets 7
(∆F/σ )√N , where N is the number of transits de-
cor tr tr
tected, and the correlated measurement error, σ , no
cor
longer depends on the stellar luminosity, but is constant
(Gaudi 2007). When dominated by correlated measure-
ments, the SNR is independent of τ and the only re-
quirement is to have a short enough sampling cadence
to detect the shortest transit duration expected. Thus,
a shorter τ due to orbital eccentricity has no impact on
the transit detectability in a survey which is dominated
by correlatedmeasurements (i.e. N =N ). How-
obj,e obj,o
ever, the Poissonlimited, white noise transit survey will
have a higher planet yield than a survey dominated by
correlated measurements (Pont et al. 2006), but in the
latter case, orbital eccentricity does not impose any ad-
ditional reduction in the transit detectability.
TheKeplerspace-basedtransitsurveywhosegoalisto
detect several Earth-sized planets (Borucki et al. 2004)
contends with stellar intrinsic variability as the domi-
nantsourceofnoise(Jenkins2002). Usingintegratedflux
measurementsoftheSuntomodelintrinsicvariabilityof
stars, Jenkins (2002) characterizes the detectability of
Earth-sizedtransiting planets with Kepler. The Sun has Fig. 6.— The overall yield from an idealized transit survey
low noise on7 hr time scales typical of transiting Earth- asfunctionoftheobserveeccentricitydistributionmodelparame-
sized planets in 1 yr orbital periods. However, the solar tersscaledtotheassumptionthatallplanetshavecircularorbits.
The transit survey is assumed to have independent photometric
noise increases by 4 orders of magnitude by 10 day
∼ measurements that aredominated byPoissonnoise. Theabscissa
time scales (see Figure 2 of Jenkins 2002). The rapidly indicates emax, and the curves are for selected values of ecrit as
increasing intrinsic noise toward longer time scales can- labeled. Forthe model parameters showninFigure1, Ndet ∼4%
celsanybenefitofincreasedtransitsignalSNRfortransit higherthanassumingallplanetshavecircularorbits(dotted line).
durations >4 hr for the Kepler mission (see Figure 9 of
Jenkins 2002). Thus, as long as the transit duration re-
mainsabove4hr,orbitaleccentricitywillnotreducethe pendent of the transit duration, in which case N is
obj
detectability of Earth-sized planets with Kepler. independent of the orbital eccentricity. For these cases,
the transit survey will have higher returns by a factor
6. CONCLUSION of Prob (Equation 4) than estimated by assuming
Te
h i
Orbital eccentricity results in an enhanced probabil- all planets are on circular orbits. However, the reduced
ity for a planet to transit and potentially a reduction planet yield in a transit survey due to intrinsic stellar
in the transit detectability. The overall yield from a variability or correlatedmeasurements must be properly
transit survey is given by N Prob N . The accounted for. If every dwarf star has an Earth-sized
det T obj
∝ ×
results from this study can be used to scale the over- planetorbiting inthe habitable zone,then assuming cir-
all yield from a transit survey based on assuming all cular orbits, the Kepler mission expects to detect 100
planets are on circular orbits for an assumed distribu- Earth-sizedplanets in the habitable zone (Borucki et al.
tionoforbitaleccentricity. The enhancedprobabilityfor 2004). The work presented here indicates Kepler will
a planet to transit, Prob with a distribution of orbital have Prob 25% higher yield if Earth-sized planets
T Te
h i∼
eccentricity scaled to the circular orbit case is given by inthe habitable zonehaveaplaneteccentricitydistribu-
Equation 4. The reduced number of transiting planets tionsimilartothecurrentlyknownsampleofgiantplan-
detectable N scaled to the circular orbit case for an ets from radial velocity surveys. However,if Earth-sized
obj
idealtransitsurveywherethe photometricnoiseiswhite planetswithe 0.9areascommonase 0then,theyield
∼ ∼
anddominatedbyPoissonerrorisgivenin 5. Multiply- ofEarth-sizedplanets could be 80%higherfromthe Ke-
§
ingthesetwofactorsprovidestheoverallyieldofanideal pler mission assuming the high-e, short transit duration
transitsurveyscaledtoassumingallplanetsareoncircu- planets are still detectable.
lar orbits. Figure 6 showthat these two opposing effects The dependence of the transit survey yield on the un-
nearly cancel over the parameters that characterize the certain underlying orbital eccentricity distribution im-
eccentricity distribution. Thus, for an idealized transit plies an uncertainty in measuring the frequency of ter-
survey with the currently observed orbital eccentricity restrial planets in the habitable zone (a major goal of
distribution, the overall yield will be 4% greater than the Kepler mission Borucki et al. 2004) . An analysis
assuming all planets are on circular orbits. The result of the transit yield from a transit survey that assumes
for an idealized transit survey with a Rayleigh distribu- all planets are on circular orbits will overestimate the
tion of orbital eccentricities gives a similar enhancement frequency of habitable planets if high eccentricities are
(4%). common and not taken into account. In practice a va-
In ground-based transit surveys, correlated measure- riety of noise regimes affect a transit survey and accu-
ments limit transit detectability (Pont et al. 2006) and rate yields necessitate an accurate understanding of the
intrinsicvariabilityofthestarwilllimitthedetectability photometric noise, stellar sample, and underlying eccen-
for Earth-sized planets for the Kepler mission (Jenkins tricitydistribution(Burke et al.2006;Gould et al.2006;
2002). In both cases the transit detectability is inde- Fressin et al. 2007).
8 Christopher J. Burke
This paper benefited from discussions with Scott Ford. This work is funded by NASA Origins grant
Gaudi, Will Clarkson, Peter McCullough, and Eric NNG06GG92G.
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