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The Astrophysical Journal Letters, 711:L7L11, 2010 March 1 doi:10.1088/2041-8205/711/1/L7C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

DETECTABILITY OF OORT CLOUD OBJECTS USINGKEPLER

Eran O. Ofek1,3

and Ehud Nakar2

1 Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA2 School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel

Received 2009 December 3; accepted 2010 January 19; published 2010 February 10

ABSTRACTThe size distribution and total mass of objects in the Oort Cloud have important implications to the theoryof planet formation, including the properties of, and the processes taking place in the early solar system. Wediscuss the potential of space missions, such as Keplerand CoRoT, designed to discover transiting exoplanets,to detect Oort Cloud, Kuiper Belt, and main belt objects by occultations of background stars. Relying onpublished dynamical estimates of the content of the Oort Cloud, we find that Keplers main program is expectedto detect between 0 and100 occultation events by deca-kilometer-sized Oort Cloud objects. The occultationrate depends on the mass of the Oort Cloud, the distance to its inner edge, and the size distribution of itsobjects. In contrast, Kepleris unlikely to find occultations by Kuiper Belt or main belt asteroids, mainly due tothe fact that it is observing a high ecliptic latitude field. Occultations by solar system objects will appear as aphotometric deviation in a single measurement, implying that the information regarding the timescale and light-curve shape of each event is lost. We present statistical methods that have the potential to verify the authenticityof occultation events by solar system objects, to estimate the distance to the occulting population, and to constraintheir size distribution. Our results are useful for planning of future space-based exoplanet searches in a way

that will maximize the probability of detecting solar system objects, without hampering the main science goals.Key words: comets: general Kuiper belt: general Oort Cloud stars: variables: general techniques:photometric

Online-only material:color figure

1. INTRODUCTION

Oort (1950) postulated the existence of a cloud of cometsorbiting the Sun with typical semimajor axes, a, of104 AU.This cloud is required to explain the existence of long-periodcomets and the fact that a considerable fraction of these

comets have orbital energies concentrated in a narrow range,corresponding to 1/a 104 AU1. However, to date no directobservation of objects in the Oort Cloud exists.4

Dynamical simulations suggest that the Oort Cloud wasformed by the ejection of icy planetesimals from the JupiterNeptune region by planetary perturbations (e.g., Duncan et al.1987;Dones et al.2004). On orbital timescales, galactic tidesand passage of nearby stars raised the perihelia of these cometsabove the region of the giant planets influence. Comets withsemimajor axes above2 104 AU are loosely bound to theSun, and tides by the galactic potential (e.g., Heisler & Tremaine1986) as well as impulse by passing nearby stars, may send thesecomets to the inner solar system where they reveal themselvesas long-period comets. In contrast, Oort Cloud objects with

a 2 104

AU (inner Oort Cloud; e.g., Hills 1981; Bailey1983) have more stable orbits. We note that theinner boundaryof the Oort Cloud, rmin, is set by perturbations due to the giantplanets and the primordial stellar neighborhood of the Sun. Thisrminis estimated to be in the range of 10003000 AU (Duncanet al. 1987;Dones et al. 2004;Fernandez1997). Below rmin,the number of comets steeply falls due to planetary influence.Dynamical simulations suggest that the spatial density of OortCloud objects in the 300050,000 AU range falls as r , with= 3.5 0.5 (e.g., Duncan et al.1987).3 Einstein Fellow.4 It is possible that (90377) Sedna, 2000 CR105, and 2006 SQ372 belong tothe inner Oort Cloud.

The current flux of long-period comets calibrated by dy-namical simulations suggest that the outer Oort Cloud (i.e.,a >2 104 AU) contains 1012 comets with nuclei absoluteplanetary magnitude5 brighter than 16 mag (e.g., Heisler1990;Weissman1996). This magnitude roughly corresponds to ob-jects with a radius of 2 km (assuming 4% geometric albedo).

The dynamically more stable inner Oort Cloud contains about2100 more comets than its outer counterpart (e.g., Hills1981;Duncan et al. 1987;Fernandez & Brunini 2000;Dones et al.2004; Brasser et al.2008).

Detecting Oort Cloud objects, and estimating their sizedistribution, will improve our understanding of the mass ofthe solar system planetary accretion disk, and the dynamicalprocesses in the young solar system. Moreover, it may bearsome clues to the stellar density at the region in which the Sunwas born (e.g., Fernandez1997; Fernandez & Brunini 2000;Brasser et al. 2006). However, these objects are beyond thedirect reach of even the currently largest planned telescopes, andsome indirect methods (e.g., astrometric microlensing; Gaudi &Bloom2005).

Bailey (1976) suggested that small Kuiper Belt objects(KBOs) as well as Oort Cloud objects could be detected byoccultations of background stars. Indeed several surveys arelooking for KBO occultations (e.g., Roques et al.2006; Lehneret al. 2009; Chang et al. 2006, 2007; Bianco et al. 2009;Schlichting et al. 2009). However, to date there is only onereported occultation of a KBO (Schlichting et al.2009).

In this Letter, we discuss the potential of space missionsdesigned to search for transiting exoplanets (e.g., Kepler,CoRoT) to detect occultations of background stars by small

5 Defined as the magnitude of an object observed at opposition and at 1 AUfrom the Sun and Earth.

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Table 1Occultation Regimes and the Minimum Radius of Detectable Objects

Case Condition Rmin= (km)

A obj , F 7.9 n8 S0.0022

texp900s

1/2vrel

30kms1

B obj F 16.6

n8

S0.0022

1/2 texp900s

1/4 vrel

30kms1

1/2 r

3000AU

1/2

1.6 105

1/2

C

F

obj 14.0 n8 S0.0022

1/2

texp900s1/4

vrel30kms

1 1/2

r3000AU1/2

1.6 10

5 1/2

D F obj 4.6

n8

S0.0022

1/2 texp900s

1/4 vrel

30kms1

1/2 r

3000AU

1/4

5000

1/4

E F obj 4.6

n8

S0.0022

1/2 texp900s

1/4 vrel

30kms1

1/2 r

3000AU

1/4

5000

1/4

Notes.The normalization of the star angular radius,, corresponds toR= 35 km atr= 3000 AU. The difference between case B and case C is that incase C we replace objby 21/4obj. This small correction is required in order to take into account the increased cross section of diffractive occultations(estimated numerically). All the formulas assume that texp> tdur.

Oort Cloud objects and KBOs. We note that the potential ofKeplerto detect occultations by KBOs was discussed by Gaudi(2004).

2. STELLAR OCCULTATIONS BY SMALL SOLARSYSTEM OBJECTS

Space telescope designed to look for exoplanet transits usu-ally have exposure times, texp, longer by orders of magnitudethan the typical duration of an occultation of a background starby a small solar system object,tdur. The relative flux decrementof such an occultation , integrated over the exposure time, isdiluted by a factor tdur/texp. However, these satellites havesuperb photometric accuracy. Their noise level per exposure is= St1/2exp 104 to 105, whereSis a normalization param-eter.6 Therefore, we may be able to detect these minute occul-tations. Here, we adopt a detection criterion for an occultationof > n, wherenis the signal-to-noise ratio. Throughout the

Letter we adoptn = 8.For an occultation by objects at a given distance from Earth,

r, the maximal value of (obtained for an optimal impactparameter7 b) increases monotonically with the object size.We define Rmin as the radius of the smallest object that canbe detected, and correspondingly min= Rmin/r . Furthermore,we can define the sum of impact parameters for which objectswith a radiusRare detectable, b(R,n) 2

0 [(R, b)

n]db, where is Heaviside step function. According to thesedefinitions,b(R < Rmin, n) = 0.

The size distribution of KBOs and Oort Cloud objects perunit solid angle is parameterized by a power law,d N(R)/dR=N>1 km(1km)1(q 1)(R/1km)q , or by a broken power law.Here, N>1 km is the total number of objects with R > 1 km

per unit area andqis the size distribution power-law index. ForOort Cloud objects AN>1 km 1013, whereAis the surface areaof the celestial sphere. For KBOs larger than Rbreak 45 kmq= 4.5 (Bernstein et al.2004; Fuentes & Holman2008;Fraseret al. 2008), and q 34 for smaller KBOs (e.g., Farinella &Davis1996;Pan & Sari2005;Schlichting et al.2009). For OortCloud objects, q is unconstrained by observations (see, however,Goldreich et al.2004).

6 ForKepler(CoRoT)S= 0.0022 (0.045) for 1 s integration of a 12thmagnitude star. This sensitivity parameter depends on the star magnitudemas10+0.2(m12).7 In the case of diffraction occultation, it is not at b = 0.

The number of events in a survey that observesNstars for aduration is then Nev= N

Rmin

b(R,n)dNdR

dR, where is theangular velocity of the occulting objects on the sky as seenfrom Earth. Here all objects are at the same distance r, all stars

have the same magnitude and angular radii, and is constant.When this is not the case, it is straightforward to integrate overthe distribution of these properties.

Thefunctionalformsoftdur,,Rmin,andthus b(R)dependonthe ratios between the three angular scales in the problem. Theseare the object angular radius, obj= R/r ; the stellar angularradius8 ; and the angular Fresnel scale9 F = F /r , withF= r/2, and the wavelength of observation. The durationof the eclipse is tdur 2max/, where max= max(obj, F, ).Table 1 gives the functional form of Rmin in the differentasymptotic cases (denoted by AE), where the parametersare normalized to those of a typical Oort Cloud object and theKeplersensitivity.

The cases where obj

, F (case A) and

obj, F

(cases BC) represent geometric occultations. In these cases,we approximate b(R)= max {obj, }(11.7 q)/13 for anyR > Rmin. Here, theq-dependent factor (i.e., [11.7q]/13) is aresult of the assumed circular shape of the object. We obtainedthis approximate correction factor by numerically integratinggeometrical occultations by circular-shaped objects.10 In thediffractive regime (F , obj; Cases DE), we numericallycalculateddiffractivelightcurvesusingEquation(B1)inRoqueset al. (1987). We find thatbcan be well approximated by

b(R,n) = 2F

max{obj, }

1 n

4

texp

tdur

F

obj

2. (1)

The scaling of Equation (1) can be understood as follows. Theoccultation shadow pattern of an object can be described asa set of bright and dark concentric rings with the centralring (around the Poisson peak) being dark with an angularradius of F. The area in all the rings is constant (2F ) sothe angular width of the rings is inversely proportional to itsangular distance from the center (e.g., Figure 1 in Nihei et al.2007). The depth of the decrement in the flux in the dark rings

8 T2e 100.2Mbol , whereTeand Mbolare the effective temperature andbolometric magnitude of the star, respectively.9 Forr= 3000 AU and = 5000 ,F= 10.6 km andF= 4.9 106 arcsec.10 This integral does not have an analytical solution.

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Table 2Formulas for the Expected Number of Occultation Events in the Different Asymptotic Regimes

Case Nev

A 50 100.90(3.5q)11.7q13 q1q2 AN>1 km1013N105

3 yr

n8

S0.0022

2q texp900s

(2q)/2 vrel

30kms1

3q r

3000AU

2B 35 101.22(3.5q)11.7q13 AN>1 km1013

N105

3 yr

n8

S0.0022

(1q)/2 texp900s

(1q)/4 vrel

30kms1

(3q)/2 r

3000AU

(1+q)/2

1.6105

(3q)/2

D 350

100.66(3.5q)3.5(q1)q(q+2)

AN>1 km

1013

N

105

3 yr n8 S0.0022

q/2

texp900sq/4

vrel30kms

1 (2q)/2

r3000AU(q+4)/4

5000

(4q)/4

E 36 100.66(3.5q) 4 .5q+1

AN>1 km1013

N105

3 yr

n8

S0.0022

(1q)/2 texp900s

(1q)/4 vrel

30kms1

(3q)/2 r

3000AU

(7+q)/4

1.6105

1

5000

(5q)/4

Notes.Case C is like case B but multiplied by (1 /2)(1q)/4. A is the area of the celestial sphere. Note that these formulas are not continuous and provide a betterapproximation in the asymptotic cases. All the formulas assume that texp> tdur.

6 5.5 5 4.5 40

0.2

0.4

0.6

0.8

1

log10

(*["])

Relativefraction

F(r=4AU),F=

0.4km

F(r=40AU),F=

1.2km

F(r=300AU),F

=3.3km

F(r=1000AU),

F=6.1km

F(r=3000AU),F

=10.6km

Kepler

CoRoT

Figure 1. Approximate stellar angular radius distributions for Kepler(blackline) andCoRoT(gray line). The vertical red dashed lines represent the Fresnelradii at selected distances (see labels) where = 5000 .(A color version of this figure is available in the online journal.)

is (obj/F)2 so the value offor the central ring occultationis c(tdur/texp)(obj/F)2. This decrement remains similar inall the dark rings as long as the ring width is smaller thanmax

{, obj

}. Therefore,b(R,n c)

2F / max

{, obj

}.

For each of the five cases, we use the approximationsdescribed above and give formula (Table 2) for the expectednumber of events as a function of the parametersn,S,texp,,r,, andvrel(= r). Here,vrelis the relative velocity between theocculting object and Earth, projected on the plane perpendicularto the Earthobject direction. Since the object size distributionis steep, the integration over R is dominated by the smallestobjects, and is carried to infinity (rather than the radius at whichthe condition defining the case is violated).

3. OCCULTATION RATES FORKEPLERAND CoRoT

To estimate the observed rate of events we need to take intoaccount the radial distance distribution (discussed in Section1),

the relative velocity (vrel) probability distribution, the eclipticlatitude distribution, and the stellar angular size and magnitudeco-distributions.

3.1. Stellar Angular Size and Magnitude Distribution

In Figure1, we present the estimated angular radii distributionof target stars observed by KeplerandCoRoT. To estimate this,we queried the guide star catalog (GSC; version 2.2; Laskeret al. 2008) for all stars within 2 deg from the center of theKepler(CoRoT) field, that have R-band magnitudes between 9and 14 (12 and 16), andBRcolors between 0.59 and 3.4 mag.This color range roughly corresponds to the spectral range ofKeplertargets. We converted the color index of each star to an

effective temperature and a bolometric correction by fitting thecolor indices with those obtained from synthetic photometryof blackbody spectra with different temperatures (e.g., Ofek2008). The synthetic photometry (Poznanski et al. 2002) waspreformed using the transmission curves of the filters used inthe GSC (Moro & Munari2000).

3.2. Relative Velocity Distribution

Given the unit vector in the direction of a target r , thespeed of the observer relative to the occulting object v, andthe heliocentric velocity unit vector of the observer11 v, theapproximate relative velocity, of an observer in an Earth-trailingorbit, projected on the plane perpendicular toris

vrel= v

1 (v r)2= v

1 cos2 ( )cos2 , (2)where and are the ecliptic longitude and latitude of theobserved star, is the ecliptic longitude (as measured from theSun) of the observer, andv 29.8 km s1 is the heliocentricspeed of the observer. Finally, assuming circular orbit (i.e.,

the probability distribution of is uniform) the probabilitydistribution ofvrelis Pvrel= ddv rel .

3.3. Ecliptic Latitude Distribution

The Oort Cloud is predicted to have a roughly sphericalshape. However, the Kuiper Belt and the main belt asteroidsare concentrated toward the ecliptic plane. Using the inclinationdistribution of KBOs estimated by Elliot et al. (2005), theexpectation probability to find a KBO at an ecliptic latitudeof66is about f 6 106 times smaller than that at theecliptic. This relative probability was estimated by convolvingthe inclination probability distribution of KBOs with the eclipticlatitudedistributionforagiveninclination(i.e.,Elliotetal.2005;

Equations (40) and (34), respectively). We note, however, thatthis number is highly uncertain.

3.4. Rate Estimates

To estimate the rate of occultation events we integrate theappropriate formulas in Table 2 over vrel, r, , and S co-distributions. For each combination of parameters, we integrateover the most appropriate occultation channel (i.e., cases AE). In Table 3, we present the estimated rates predicted forKeplerand CoRoTfor the OortCloud (upperblock), Kuiper Belt(middle block), and main belt (lower block) object occultations.

11 Object velocity is neglected.

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Table 3Approximate Occultation Rates

Parameters Rates (= 3 yr)Kepler CoRoT

N>1 km rmin q f= 6 106 f= 1(deg2) (AU) N= 105, 66 N= 104, 0

2.4 108 3000 3.0 30 0.12.4

108 3000 3.5 8 4

103

2.4 108 3000 4.5 0.7 1 1052.4 108 1000 3.5 60 0.032.4 108 5000 3.5 3 2 103

1.1 104f 40 3.0a 2 104 3 1037.3 104f 40 3.5 5 104 4 1034.9 105f 40 4.0 5 103 0.0247f 4 2.3b 1 104 0.2

Notes. Thefirst block refersto Oort Cloud objectsassuming rmax= 50,000 AU,= 3.5.For rmax= 100,000AU,theeventsrateis10% smaller.The secondblock refers to KBOs assuming rmax= 40 AU. The third block refers to mainbelt asteroids assumingrmax= 4 AU. The cumulative surface density of KBOsnear the break radius, Rbreak 45 km, is 5.4 deg2 at the ecliptic (Fuentes& Holman 2008). Therefore, at the ecliptic we adopt for KBOs, N>1 km of

1.1 104, 7.3 104, and 4.9 105 deg2, for q=3, 3.5, and 4, respectively.We note that these values are consistent with the findings of Schlichting et al.(2009). N>1 kmand q forthe main belt aretakenfrom Ivezicetal. (2001).Wesetthe ecliptic latitude surface density correctionfto 6106 and 1 for 66and 0, respectively, for both KBOs and main belt asteroids. In all the cases,we assume 8detection threshold (i.e., n = 8).a For KBOs, we assume a broken power law withq= 4.5 aboveR= 45 km.

The actual rate calculation is preformed by normalizing the equations in Table2to 45 km, and using differentqforRminabove and below 45 km.b For main belt asteroids, we assume a broken power law with q= 4 above

R= 2.5 km, andq= 2.3 below this radius.

The predicted rate for Oort Cloud objects in Table3in the caseof q

= 3.5 and r

min=3000 AU is lower by about an order

of magnitude compared to the pre-factors in Table2. This ismostly because Table2 is normalized for all objects being atr= 3000 AU, while in Table3the same number of objects isspread betweenrminandrmax, with a spatial density distributionproportional tor3.5.

These rate estimates suggest that if our current best guessregarding the properties of the Oort Cloud is correct, then it maybe detectable byKepler. Regardless,Keplerwill be able to putthe best constraint (so far) on the content of the Oort Cloud. Wenote that the rate estimates in Table3are an approximation tothe actual rate. This is mainly because our formulas representasymptotic approximations, while in some cases two out of thethree angular scales,obj,F,, may be similar.

4. VALIDATION AND DEGENERACY REMOVAL

Contrary to transits and grazing transits by extrasolar plan-ets, solar system occultations will appear in a single photo-metric measurement and they happen only once. Therefore,verification of each event by itself will require a thorough un-derstanding of the noise properties of the photometer. Keplerwill have about 1.5 1010 15 minute independent photometricmeasurements. Assuming purely Gaussian noise, the 8 de-tection threshold we adopted corresponds to a probability of1015. However, even small deviations from a Gaussian noisemay impair our confidence in the astrophysical nature of theevents.

Here we present two tests that can be used to check ifthe occultation signature are due to real events or due tosome sort of uncharacterized noise (see also Gaudi 2004).In addition, these tests can be used to statistically mea-sure q and to distinguish between Oort Cloud objects andKBOs.

The occultation rate depends on the relative velocity vrel,witha somewhat different dependency in each regime. Therefore, ifa large number of occultations are detected, a simple sanitytest is to look for the dependency of the events rate on vrel.Moreover, since Kepleris observing at high ecliptic latitude,the vrel modulation will be small (Equation (2)) compared tothe maximal modulation which will be seen by an instrumentobserving the ecliptic. Moreover, in all the cases (AE), thefunction Nev depends on q. Therefore, the amplitude of themodulation is related to q and the occultation type (i.e., AE).Knowing the occultation regime (see below), we can measureq(i.e., removing the degeneracy between qandNev).

Furthermore, as shown in Table 2 the occultation rate dependsonS, which depends on the stellar magnitude and , which inturnisafunctionofthestarmagnitudeandeffectivetemperature.We expect bright and angularly small stars to have higher

chance of producing detectable occultations. Moreover, thisdependency is different for the various occultation cases. Thiscan be used to further test the nature of the occultations,by comparing the distribution of magnitudes and effectivetemperatures of the stars showing occultations to those ofthe entire observed stellar population. Most importantly, fora Kepler-like observatory, KBO occultations will be mostly inthe diffractive regime, while Oort Cloud object occultations willbe in the geometric regime (Figure1). Therefore, the magnitudeand of occulted stars by KBOs and Oort Cloud objects willfollow different co-distributions. Therefore, we can statisticallydistinguish between Oort Cloud and KBO occultation events.However, quantifying these effects and estimating the numberofoccultations needed to carry out the suggested analysis requires

a more realistic estimate of the events rate (i.e., using fullnumerical integration of light curves) and a full explorationof the parameters space.

To summarize, we show that Kepler and similar spacemissions can detect deca-kilometer objects in the Oort Cloud.We present statistical methods to verify that the occultationsare real rather than due to uncharacterized noise. In addition,we suggest that it will be possible to statistically measure theirsize distribution (q), and the dominant occultation regime, andtherefore differentiate between Oort Cloud objects and KBOs.

Considering Keplercapabilities and reasonable Oort Cloudparameters, we find that Keplermay detect 0 to102 stellaroccultations by Oort Cloud objects. We find that Kepler isunlikely to detect KBOs (mainly because of its high ecliptic

latitude pointing). Moreover,CoRoTis unlikely to detect KBOsor Oort Cloud objects. However, the exact properties of the OortCloud and the Kuiper Belt are not well known. Therefore, suchsearches are warranted.

The analytical treatment we present in this Letter is usefulin order to maximize the efficiency of existing (and future,e.g., PLATO) experiments to detect trans-Neptunian objects.However, this treatment is accurate only in the asymptotic cases,and numerical calculations are needed in order to give moreprecise predictions.

We thank Reem Sari, Hilke Schlichting, Orly Gnat, andMichael Busch for many valuable discussions. Support for

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program number HST-AR-11766.01-A was provided by NASA.E.O.O. is supported by an Einstein fellowship. E.N. is partiallysupported by an IRG grant.

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