DetectabilityOccultationsKBO Nihei

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DETECTABILITY OF OCCULTATIONS OF STARS BY OBJECTS IN THE KUIPER BELT AND OORT CLOUD

T. C. Nihei,1,2

M. J. Lehner,2

F. B. Bianco,1,2

S.-K. King,3

J. M. Giammarco,4

and C. Alcock2

Received 2007 March 15; accepted 2007 June 30

ABSTRACTThe serendipitous detection of stellar occultations by outer solar system objects is a powerful method for ascertain-

ing the small end (rP15 km) of the size distribution of Kuiper Belt objects and may potentially allow the explorationof objects as far out as the Oort Cloud. The design and implementation of an occultation survey is aided by a detailedunderstanding of how diffraction and observational parameters affect the detection of occultation events. In this study,stellar occultations are simulated, accounting for diffraction effects, finite source sizes, finite bandwidths, stellar spec-tra, sampling, and signal-to-noise ratios. Finally, the possibility of detecting small outer solar system objects from theKuiper Belt all the way out to the Oort Cloud is explored for three photometric systems: a proposed space telescope,Whipple, the Taiwanese-American Occultation Survey, and the MMT.

Key words: comets: general Kuiper Belt Oort Cloud solar system: formation

1. INTRODUCTIONStellar occultation, the dimming of a background star by a fore-

ground object passing through the line of sight, is used in a varietyof scientific studies to probe the properties of foreground objects.With stellar occultations, it has been possible to discover and studyplanetary rings (e.g., Bosh et al. 2002 and references therein) andthe atmospheres of planets and their satellites (e.g., Elliot et al.2003a, 2003b; Gulbis et al. 2006; Pasachoff et al. 2005; Sicardyet al. 2006).

Studies suggest that by searching for serendipitous occulta-tions in monitored stars it may be possible to detect outer solarsystem objects in the Kuiper Belt (Dyson 1992; Axelrod et al.1992) and objects as far out as the inner edge of the Oort Cloud(Bailey 1976). This provides a novel way to ascertain the size

distribution of small-object populations in the Kuiper Belt andthe inner Oort Cloud.

The standard planet-formation scenario begins with a disk ofsmall planetesimals (with radii r< 0:5 km) surrounding a proto-star. These planetesimals collide with one another and merge tobecome larger objects. When sufficiently massive, gravitationalfocusing leads to runaway accretion. Largerobjects dominate theaccretion process and eventually go on to become planets. Re-maining planetesimals, by a variety of mechanisms, are clearedfrom the planet region of the disk. Some objects are ejected todistant orbits by perturbations from the giant planets. At the outeredge of the protostellar disk, a slow rate of collision fails to pro-duce large enough objects and planet formation fails to occur. Farenough removed from the massive planets, these smaller plane-tesimals are invulnerable to perturbations and remain in the disk.

In our own solar system, the Kuiper Belt is a remnant of theouter protostellar disk that failed to form planets. The populationof planetesimals in this region has most likely been perturbed byNeptune and perhaps other massive bodies; therefore, the Kuiper

Belts size distribution, spatial distribution, and mass are impor-tant keys to understanding the evolution of planetary disks.Models and observations suggest the differential size distri-

bution of the Kuiper Belt follows a broken power law N / rq,where q takes on different values in the lower and upper size re-gions. The location of the breakrb in this power law depends onthe initial mass, sizespectrum, and bulk properties of KuiperBeltobjects ( KBOs), as well as Neptunes orbital evolution, amongother parameters (Kenyon & Bromley 2004; Kenyon 2002; Pan& Sari 2005; Stern 1996). Observations have constrained the sizespectrum forlargeobjects (rk100 km) to an index q $ 4 (Bernsteinet al. 2004; Trujillo et al. 2001). Bernstein et al. (2004) has foundevidence for a break in the size distribution near rb $ 30Y50 kmby a faint-object survey mounted on the Advanced Camera for

Surveys on the Hubble Space Telescope. Previous models pre-dicted the location of the break radius at 0:05 km < rb < 5 km(Stern 1996; Kenyon & Luu 1999; Kenyon 2002). Kenyon &Bromley (2004) and Pan & Sari (2005) have since modeled theKuiper Belt size distribution, revisiting assumptions made aboutthe relative gravitational and tensile energies of KBOs. Pan &Sari (2005) concluded that bulk strength plays little role in thefragmentation of small objects and found reasonable agreementwith the suggested break location evidenced by Bernstein et al.(2004). Kenyon & Bromley (2004) estimated the break radiusat 0:5 km < rb < 15 km. In addition they suggested that a dipshould be present at 5 km < r< 35 km due to the removal ofsmall objects due to collisional erosion. These two models differin the small size range where direct observations are difficult due

to the dim surface brightness of KBOs smaller than r 15 km.Surveys monitoring background stars for occultations due to

KBOs have the capability to determine the small end of the KBOsize distribution ( Roques & Moncuquet 2000; Cooray & Farmer2003). Various groups are now attempting to implement suchsurveys (e.g., Roques et al. 2003; Lehner et al. 2006; Chang et al.2006). Roques et al. (2003) have implemented a survey at thePic du Midi Observatory with sampling at 20 Hz frequency. Theyreported a candidate event at 3 , which may be ascribed to anoccultation by a r $ 0:15 km KBO. Three candidate occultationevents were also reported by a later survey conducted with aframe-transfer camera mounted on the 4.2 m William HerschelTelescope at La Palma (observing at 46 Hz; Roques et al. 2006).However, the survey team ruled out the detection of any KBOs

1 Department of Physics and Astronomy, University of Pennsylvania, 209South 33rd Street, Philadelphia, PA 19104, USA.

2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cam-

bridge, MA 02138, USA.3

Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box23-141, Taipei 10617, Taiwan.

4 Department of Physics, Temple University, Barton Hall, Philadelphia, PA19122, USA.

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The Astronomical Journal, 134:1596Y1612, 2007 October

# 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.


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in the30Y50 AU range. S. J. Bickerton (2006, private communi-cation) has also followed suit by implementing a high-speed CCDcameradesign with 40 Hz cadence andmounting a searchwiththe72 inch (1.8 m) Plaskett telescope in Victoria, British Columbia.In addition, King et al. (2002) and Lehner et al. (2006) have de-scribed a dedicated occultation survey known as the Taiwanese-American Occultation Survey (TAOS), which uses three wide-fieldrobotic telescopes to monitor as many as 2000 stars for chance

occultations by KBOs.Recently, candidate occultation events at millisecond timescaleswere observedin X-ray light curves of Scorpius X-1 (Chang et al.2006). These events were claimed to be compatible with KBOs inthe size range 5 m < r< 100 m. The reported event rate is muchhigher than expected from models of KBO formation. Jones et al.(2006) have argued that the candidate events observed by Changet al.(2006) canbe attributed to dead time response due to chargedparticle scintillation. Subsequently, Chang et al. (2007) adjustedtheir number of candidate occultation events to account for thiseffect. To date, occultation surveys have been unable to providedefinite constraints on the small-size distribution of objects in theKuiper Belt.

Beyond the Kuiper Belt, there remain many questions that

could be addressed by stellar occultation surveys. The structureof the outer solar system is thought to extend from the KuiperBelt out to distances as large as $100,000 AU. The nature of thisvast region holds important clues to the complex evolution of oursolar system.

The Oort Cloud is a population of planetesimals that is thoughtto have been scattered from the planetary disk roughly $4.5 Gyrago and is predicted to have a wide range of orbits, with semi-major axes spanning distances of 1000 AU < a < 100;000 AU.Its existence was proposed as a probable reservoir of long-periodcomets (Oort 1950).

Models suggest that these objects formed in the region of thegiant planets ($4Y40 AU) and were perturbed by the giant plan-ets to large orbits with relatively unchanged perihelia. Interac-

tions with nearbypassing stars, giant molecular clouds, and othermaterial in the solar neighborhood would have increased theirperihelia to larger distances, increased the inclinations, and placedthem in large orbits to form the Oort Cloud (Dones et al. 2004).Dynamical simulations combined with the flux of long-periodcomets into the planetary region of the solar system lead to esti-matesof the number of cometsin theOort Cloud to be $1011Y1012.However, because the Oort Cloud is at such large distances andits comets are expected to be the size of observed long-periodcomet nuclei (r $ 3 km), direct observations of the Oort Cloudare unlikely.

The discovery of Sedna (2003 VB12), a r 830 km object onan eccentric orbit at heliocentric semimajor axis ofa 490 AU,came as a surprise ( Brown et al. 2004). An object of substantial

size at such a distant orbit was unexpected. Sednas origins re-main unclear, but its discoverers (Brown et al. 2004) have sug-gested that Sedna may be part of the Oort Cloud. This impliesthat Sedna amassed size in the planetary regionand was perturbedto its current orbit ( Morbidelli & Levison 2004; Matese et al.2005). Stern (2005) has pointed out that Sednas eccentric andinclined orbit does not preclude its formation in situ. If this istrue, it is possible that Sedna is part of a population of objects thatlie in a proposed annular region beyond the Kuiper Belt (Brasseret al. 2006) that is referred to in this paper as the extended disk(a $ 50Y1000 AU).

Using stellar occultations to search for extended disk objectsis also a possibility. To date, no known occultation surveys arededicated to this purpose. However, Roques et al. (2006) showed

that this may be a real possibility, as they identified two candidateoccultation events by 300 m radius objects beyond 100 AU viathis method.

The potential for occultation surveys to detect objects in theKuiper Belt is well discussed in the literature. However, the ca-pability of a given occultation survey to detect objects out to theOort Cloud still requires careful consideration, as it depends onseveral factors. In particular, Roques & Moncuquet (2000) have

shown that starlight diffraction must be taken into account forsurveys looking to detect stellar occultations by small objects inthe Kuiper Belt. In this paper, the effects of stellar types, finitebandwidth, and sampling on a surveys ability to detect occulta-tion events in the outer solar system are studied. Occultation eventsfor three photometric systems are simulated in order to guide theselection of observational parameters for occultation surveys. Inx 2 a brief introduction to stellar occultation and diffraction ef-fects is given. The effects of finite bandwidth, stellar spectra,and finite source sizes are considered later in x 3. The effects ofsampling is discussed in x 4. Noise for three photometric sys-tems as mentioned above is included in x 5. Finally, the de-tectability of an occultation event for photometric systems andthe conclusions of this study are discussed in xx 6Y8.

2. STELLAR OCCULTATIONS AND DIFFRACTION

The diffraction pattern created by the stellar occultation of adistant star by a foreground spherical object is described usingLommel functions,

Un(; ) X1k0

1 k

n2kJn2k();

where Jn is a Bessel function of ordern. For the case of an oc-cultation by an object of radius r at a distance a, the measuredintensity of a star at wavelength k is described (Roques et al.1987 and references therein) by

I()

U20 (; ) U2

1 (; ); ;

1 U21 (; ) U2

2 (; )

2U1(; ) sin

22 2

2U2(; ) cos

22 2

; ! ;

8>>>>>>>>>>>:

1

where r/F and x/F are the radius and distance from theline of sight in units of the Fresnel scale F (ka/2)1

=2. The sub-script on I indicates that the occultation pattern depends solelyon the dimensionless parameter.

Figure 1 shows a projected occultation pattern computed from

the above expression for 1:0. Also shown are four trajec-tories through the occultation pattern at four different values ofthe impact parameterb, given in terms of the diameter of the firstAiry ring (this is an approximate definition as will be discussedfurther), which we call . The corresponding intensity profilecurves for each of these trajectories are shown in Figure 2. Anobject crossing a source with a finite impact parameter leads toseveral possible intensity curves for a given occultation pattern.From the plots shown in Figure 2, it can be seen that for impactparameters b ! 0:5 the occultation event depth is quite small,and such events will be very difficult to detect. We thus define anoccultation eventas an object crossing the line of sight to a starwith an impact parameter b 0:5, and we define the eventwidth as the first Airy ring diameter. Finite impact parameters

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complicate the characterization of an occultation event becauseof the multitude of possible intensity curves that may arisefrom one occultation pattern. In order to simplify the discus-sion, only the case of b 0 is considered throughout the re-mainder of this paper. This assumption does not significantlyalter the conclusions.

Notethatat center of the diffractionpattern in Figure 1,I(0) 1;this is the Poisson spot and is a consequence of diffraction presentfor circular objects passing before a point source.

The parameter defines the strength of diffraction effects, inparticular the width and depth of the occultation.Assuming a point-source background star, the occultation width (see Fig. 2) takeson two values in the limiting cases of large and small . ForT1,the occultation pattern is completely dominated by diffraction, and

thediameter of the Airyring isgivenby 2ffiffiffi

3p in dimensionlessFresnelscale units. In the case where31, the diffraction effects becomenegligible, and the width approaches the limiting case of 2.Anem-pirical approximation over the entire region of interest is given by

% 2ffiffiffi

3p 3=2

3=2 ! 2=3

: 2

In physical units, the width W is

W % 2ffiffiffi

3p

F 3=2

r3=2 ! 2=3

: 3

Figure3 shows the measured width (solid line) against the approx-

imation in equation (2) (dotted line). The two limiting expressionsfor large and small are indicated by dotted lines. The jumps inoccultation width for 0:3PP 9 are the result of the gradualshifting of diffraction fringes near the shadow edge. Figure 4 de-picts three light curves near $ 1 ( 0:98, 1.00, and 1.02). De-termining the maximum peak is complicated by diffractionfringesthat span theshadow edge.The jumps in the event width are due toone of twolocal maxima becoming larger than the other. Themea-sured widths for each of the curves are depicted in Figure 4 byvertical lines drawn to match the appropriate curve.

For a source of finite angular radius , the projected radius inthe plane of the occulting object is r a. This finite radius

Fig. 2.Intensitycurves for an occultation patternfor 1 traversedat various impactparameters b. Impactparameter valuesare labeledat thebottom of each plotand correspond to their labeled counterparts in Fig. 1.

Fig. 1.Projected occultation pattern for 1. The scale of the pattern isfive Fresnel units on a side. Four trajectories crossing the occultation patternwithimpactparameters b are shown withcorresponding intensitycurves in Fig.2.Values for the impact parameters are expressed in terms of the occultation width shown in Fig. 2.

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extends the occultation width by the projected diameter of thestar such that

% 2

ffiffiffi3

p 3=2 3=2

! 2=3 2; 4

where r /F, and the asterisk in the superscript of indi-cates that a finite source disk is accounted for in the occultationwidth. The width in physical units is thus

W % 2ffiffiffi

3p

F 3=2

r3=2 !2=3

2r: 5

The occultation depth is defined as the magnitude of themaximum downward deviation of an occultation pattern, as drawninFigure 2. Figure 5 is a plot ofthe depth as a function of. Thereare two limiting cases of the occultation depth as a function of .An empirical fit to() forT1 shows that the depth roughlyfollows a power law % 32. In the limit where 31, the ob-ject disk completely extinguishes the background source; hence,

the depth is constantat 1. By an empirical fit to the measured

depth, we arrive at the expression

1 32 3=2h i2=3

: 6

Three regimes are apparent in Figures 3 and 5: (1) the far-fieldor Fraunhofer regime forT1, (2) the near-field or Fresnel re-gime for $ 1, and (3) the geometric regime for31. In theFraunhofer regime the occultation width 2

ffiffiffi3

pis independent

of, but the depth varies as % 32. Example diffraction pat-terns in this regime are shown in Figure 6 for the cases 0:03,0.01, and 0.3. It is clearly evident in this figure that as decreases,the width remains constant and the depth decreases. In the Fresnelregime the depth and width of the occultation event are bothdependent on ; hence, as increases both parameters will in-

crease as seen in Figure 6, where 1:0 and 3.0. Finally, geo-metric patterns have a width that approaches the geometric shadow! 2, and complete extinction of the background source isapparent such that 1. Figure 6 shows an occultation eventapproaching the geometric regime in the panel for 9:0.

3. SPECTRAL TYPE AND FINITE SIZE OF SOURCE STAR

Occultation patterns described in x 2 depend on the observa-tion wavelength k. In reality, a survey will monitor stars througha finite bandwidth, and this can affect the diffraction features ob-served in an occultation pattern. If the bandwidth is narrow, amonochromatic pattern is a reasonable approximation for anobserved occultation. For broader bandwidths like that of TAOS,we describe the intensity pattern as

Ir; a(x)

Z11

s(k) f(k)Ir;a(x; k) dk; 7

where

Ir;a(x; k) I()

Ir=F(a;k) x=F(a; k) ;

and the wavelength-dependent filter transmission and stellar spec-trum are represented by f(k) and s(k), respectively.

To determine broad bandwidth effects, Ir;a(x; k) for severalr and a were integrated against a flat spectrum with a series of

Fig. 3.Occultation width (see Fig. 2) vs. the object radius . The solidline shows the measured width, the dotted line shows the approximation givenin eq. (2), and the dashed lines indicate the asymptotic behavior for in the twolimiting cases for.

Fig. 4.Occultation patterns for 0:98, 1.00, and 1.02. Vertical linesindicate the measured widths for each pattern. Increases in shift the absolutemaximum peak by a significant amount.

Fig. 5.Occultation event depth (see Fig. 2) as a function of. The solidline shows the measured depth, the dashed line shows the empirical approx-imation in eq. (6), and the dotted lines show the two limiting cases.

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top-hat filters. Stellar spectra were accounted for using the UVILIBlibrary, a compiled stellar flux library spanning a total wavelengthrange of 1150Y10620 8 (Pickles 1998), and were incorporatedinto occultation curve simulations. We found that for point sourcesbroader bandwidths will smooth and dampen diffraction fringes,while broadening of the occultation width is minimally seen. Inaddition, stellar spectra appear to have little effect on the inten-sity profile of a diffraction pattern for a background point source.

Throughout the remainder of this paper, the Fresnel scale of anevent with a broadband filter is redefined as

F

ffiffiffiffiffiffika

2

s;

where k is the median wavelength of the filter of interest.

While the stellar spectra themselves have a minimal effect, fi-nite source sizes will significantly affect the occultation intensitypatterns, specifically when k1. The source size of a given staris determined by a combination of its stellar class and apparentbrightness. To address finite source size effects, stellar radii fromtabulated values (Cox 2000; Lang 1992) were incorporated intosimulations of occultation events. (Limb-darkening effects alsowere considered using a solar limb-darkening model [Cox 2000];however, the effects were found to beP1%, hence they are ig-nored throughout the rest of this discussion.)

The intensity profile from a finite source disk of projected ra-dius r is expressed as an integral over the projected source disk,

Ir; a(x) 1

r2

Z20

Zr0

r0 dr0 d

; Ir; a(x2 r02 2x r

0 cos )

:

Here r0 and are the distance from the center of the source diskand the polar angle, respectively. The asterisk in the superscriptI denotes that the occultation pattern was calculated accountingfor the finite source size.

Occultation profiles for four stellar types each with magnitude

V 12 forr 0:5 and 1.5 km objects are plotted in Figures 7and 8. Both illustrate that the finite source size of a stellar diskcan significantly broaden the occultation width and smooth thediffraction fringes. In cases where the source size is relativelylarge such as an M2 V V 12 star, the smoothing is so signifi-cant that diffraction fringes with variations as large as $35% arereduced to variations of$5%.

The smoothing and broadening effects at distances well be-yond the Kuiper Belt are more drastic, as is expected due to thedependence of the projected source size on the distance a. Oc-cultation patterns were calculated for four V 12 backgroundsource stars for a r 5 km extended disk object ata 1000 AU(Fig. 9) and for a r 10 km Oort Cloud objecta 10;000 AU(Fig. 10). These objects are detectable with the A0 V star, but

Fig. 6.Occultation intensity profiles measured across the center of the pattern for several values of .

Fig. 7.Diffraction profiles for a KBO for different V 12 backgroundstars: A0 V, G5 V, K5 III, and M2 V. The occulting object for this set of curveshas a radius of 0.5 km and is at a distance of 40 AU. Projected stellar radii at40 AU are 0.2, 0.5, 1.2, and 2.6 km, respectively.



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they would be extremely difficult to detect with the K5 III or

M2 V stars because of the very small occultation depth due to thelarge projected radius of the source star.

The angular size of a star depends not only on its spectral type,but on its magnitude as well. Figure 11 shows the smoothing ef-fects due to various stellar brightnesses (V 8, 10, 12, and 14)for an A0 V star occulted by an r 1:5 km object at 40 AU. Thestellar radii projected into the plane of the object are 1.1, 0.44,0.17, and 0.069 km, respectively. The change in radius from a 14to 10 mag A0 V star will dampen the Poisson peak by $30%.However, overall smoothing due to the increase in stellar ra-dius with brightness is minimal for this range. In the case of aV 8 A0 V star that has a projected source radius comparableto both the Fresnel scale and object radius, the smoothing is sig-nificant enough to eliminate any trace of the Poisson peak and

most of the other diffraction features.Occultation patterns for stars of various types and brightnesses

and for objects of various sizes and distances exhibit a wide rangein morphologies, as can be seen from the previously discussedfigures. A summary of the expected occultation pattern morpholo-gies is presented in a plot of object radius r versus distance a

(Fig. 12) with accompanying example occultation patterns (Fig. 13).

As described above, the occultation pattern depends on the pro-jected source size r relative to the Fresnel scale F. The mor-phologies are then split into five regions labeled A, B, C, D, andE. Regions A and D lie inthe distance range a P2000 AU wherethe projected source radius < 1. Conversely, regions B and Clie in the distance range a k 2000 AU where > 1. Region Elies in the parameter space $ 1.

Recall that when 31, the occultation pattern for a pointsource can be approximated by a geometric shadow. The A andB regions therefore represent the area of the parameter space forgeometric occultation patterns. The difference between the twoarises from the source size . At large distances where k1(region B), significant smoothing by the stellar source disk washesout all diffraction details including the Poisson peak. To contrast,

diffraction features remain in the geometric occultation patternwhen P1 (region A).

Occultation patterns that fall into regions C and D in Figure 12fall within the Fresnel and Fraunhofer regimes (P1). They bothexhibit shallower occultation depths, but again, depending on thesource size , the occultation will exhibit differing occultationwidth broadening and smoothing. In region C, the Fresnel scale

Fig. 9.Diffraction profiles for an extended disk object for different V 12 backgroundstars: A0 V, G5 V, K5 III, and M2 V. The occulting objectfor thisset ofcurveshas a radiusof 5 kmandis ata distanceof 1000AU.Projectedstel-lar radii at 1000 AU are 4.3, 12.9, 30.6, and 64.1 km, respectively.

Fig. 10.Diffraction profiles for an Oort Cloud object for different V 12 background stars:A0 V, G5 V, K5 III, and M2 V. The occulting object for thisset of curves has a radius of 10 km and is at a distance of 10,000 AU. Projectedstellar radii at 10,000 AU are 43.5, 129.4, 306.3, and 641.4 km, respectively.

Fig. 11.Diffraction profiles for background A0 V stars of varying mag-nitudes: V 8, 10, 12, and 14. The occulting object for this set of curves has aradius of 1.5 km and is at a distance of 40 AU.

Fig. 8.Diffraction profiles for a KBO for different V 12 backgroundstars: A0 V, G5 V, K5 III, and M2 V. The occulting object for this set of curveshas a radius of 1.5 km and is at a distance of 40 AU. Projected stellar radii at40 AU are 0.2, 0.5, 1.2, and 2.6 km, respectively.



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is smaller than the projected source size, and therefore only afraction of the stellar flux is diffracted. These events are wide butextremely shallow. Occultation patterns in region D simply re-semble the Fraunhofer patterns we discussed previously becausethe small source sizes here have little effect. Finally, patterns inregion E, where $ 1 and $ 1, show a significant depth with-out complete extinction of the source and are devoid of signifi-cant diffraction features.

From the above discussion, it follows that in order to minimizesmoothing and maximize the occultation depths, selected targetstars should be relatively dim and blue, since brighter and redderstars have larger apparent disks. With this choice a survey wouldhave to contend with a reduced signal-to-noise ratio. With bet-ter photometry, an improved signal-to-noise ratio augmented bybrighter background sources may slightly increase the detectionof smaller objects as the occultation event widths are extendedby the stellar disk (see eqs. [4] and [5]). This approach, however,would sacrifice diffraction details and diminish the depths ofoccultation events. Background source selection is furthermorecomplicated by the linear dependence of the projected source ra-dius r on a. Effective selection of target stars therefore dependson considerations such as the target population, observational pa-

rameters, and expected event rates, as well as light-curve shape.4. RELATIVE VELOCITY AND SAMPLING

Rather than observing stationary occultation patterns like theones that have so far been discussed, real surveys will observe

Fig. 12.Plot summarizing resulting light-curve classes for object radii r

and distances a passing before a V 12 A0 V star. Each class is associated witha region demarcated by the drawn lines and is labeled accordingly (A, B, C, D,and E). Characteristic curve shapes are dependent on stellar disk radius r (solidline) andthe Fresnel scaleF(long-dashedYshort-dashed line). Exemplarycurvesfor each of the listed regions are shown in Fig. 13.

Fig. 13.Plots of example light curves for each of the five morphologies shown in Fig. 12. Each curve is labeled with its corresponding region.



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these patterns in time due to the relative velocity of the objecttransverse to the line of sight. The main source of the transversemotion is the orbital motion of the Earth, with a small contribu-tion from the velocity of the occulting object itself. Assumingthat the occulting object lies in the ecliptic plane on a circularorbit, the transverse velocity expression is

vT vE cos

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiaE

a1

a 2Ea 2

sin2

s" #; 8

where is the angle of opposition, the angle between the objectand the opposing direction of the Sun, vE 29:8 km s

1 is theorbital speed of Earth, and aE 1 AU is the distance from theSun to Earth. At opposition ( 0), where the transverse veloc-ity reaches a maximum, the expression forvT simplifies to

vT vE 1

ffiffiffiffiffiffiffiffiffiffiffiffi1 AU

a

r !: 9

A typical KBO at a distance ofa 40 AU will have a transversevelocity vT % 2 5 k m s

1. The occultation profile is then measuredasa light curve intime t x/vT. Given the occultation width (seeeq. [5]), the occultation has a duration

T W

vT

: 10

A true survey will collect images by temporally integratinginfinitesimally sampled light curves over a finite exposure timet. Shutter speed, frame transfer, and other photometric delays

will lead to time lags between images. However, here we assumethat any delays that arise from the photometric system are mini-mal and that the exposure time is equal to the inverse of the sam-pling frequency f 1/t. The discrete light curve is then givenby points integrated in time,

I(ti)

Ztit=2

tit=2

I(t) dt: 11

To give an example of how an occultation pattern is measured,accounting for finite exposure, consider observations made atopposition with relatively fastf 20 Hz sampling. This wouldcorrespond to an exposuretimet % 50ms.ForaKBOat40AU,intensity values willbe integrated overa distance ofx % 1:3 km.This is near the target KBO size of TAOS, and near the Fresnelscale for KBOs and objects in the extended disk. At Oort Clouddistances the integration distancex is roughly 10 times smallerthan the Fresnel scale.

The sampling rate is a critical parameter for an occultation sur-

vey. In order to resolve diffraction effects, the sampling rate of asurvey must be high enough such thatxTW. In cases wherex $ W, events will still be detectable; however, diffraction ef-fects will not be evident in the light curve because the entire eventwill appear in only one or twosamples. However, whenx3W

events become difficult to detect, because all the power of thediffraction occurs within a small fraction of a single sample timeand is therefore averaged out. This is illustrated in Figures 14Y16.

Figure 14 depicts occultation pattern profiles forr 0:5, 1.5,and 5 km objects at 40 AU occulting an A0 V V 12 star sam-pled at frequencies of 5, 20, and 40 Hz. A sampling of 5 Hzcorresponds to x % 5 km at 40 AU, which is comparable tothe minimum event width in the Fraunhofer regime. It can be

Fig. 14.Diffraction profiles observed at opposition for a V 12 A0 V background star occulted by r 0:5, 1.5, and 5 km objects at 40 AU. Each curve iscontinuously sampled at one of three frequencies: 5, 20, and 40 Hz. The unsampled light curve for each object is shown in the top panels.



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seen that for the r 0:5 and 1.5 km objects, only one data pointshows significant deviation from the otherwise flat light curve.On the other hand, for the r 5 km object, the width is slightlylarger than the sampling size, and thus three data points are af-fected. In all three cases, diffraction effects cannot be discerned

in the light curves. In the case of 20 Hz sampling, the widths ofthe r 0:5 and 1.5 km events are slightly larger than the sam-pling distance x, and thus typically three data points are af-fected. For the r 5 km event, the sample size is small enoughrelative to the event width that diffraction effects are slightly

Fig. 15.Diffraction profiles observed at opposition for a V 12 A0 V background star occulted by r 2:5, 5, and 10 km objects at 1000 AU. Each curve iscontinuously sampled at one of three frequencies: 5, 10, and 20 Hz. The unsampled light curve for each object is shown in the top panels.

Fig. 16.Diffraction profiles observed at opposition for a V 12 A0 V background star occulted by r 5, 10, and 20 km objects at 10,000 AU. Each curve iscontinuously sampled at one of three frequencies: 1, 5, and 10 Hz. The unsampled light curve for each object is shown in the top panels.



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visible. Finally, in the case of the 40 Hz sampling, xTW,diffraction effects can be seen for all three objects.

Figures 15 and 16 show that at larger distances the occultationwidth is augmented by the size of the Fresnel scale and size ofthe source star, and the sampling requirements are therefore lesssevere. Due to the large relative size of r most of the diffraction

effects are washed out anyway, so no gain can be obtained fromhigher sampling. In fact, because the occultation depths are sig-nificantly shallower at larger distances, signal-to-noise ratio con-siderations will require the lowest sampling rate possible thatwill still allow the event to be resolved.

Note that observations could be made away from opposi-tion, which would have the effect of increasing the duration of anevent. This would allow for longer sample times and increasedsignal-to-noise ratio for each measurement. However, this comesat a cost to the overall event rate (Cooray & Farmer 2003). Tosimplify the discussions throughout the remainder of this pa-per, it is assumed that all measurements are made at opposition.The effects of observing away from opposition is revisited in

xx6 and 7.

5. NOISE

In order to determine the observable target population of anoccultation survey, it is necessary to consider the effects of noise.Three photometric systems, TAOS, Whipple, and the MMT, aredescribed here, and estimates of signal-to-noise ratio for eachsystem are used to simulate occultation curves. In the discussionthat follows, S is used to indicate the signal-to-noise ratio of aV 12 A0 V star.

TAOS uses three to four 50 cm wide-field telescopes installedat the Lu-Lin Observatory in central Taiwan. Details of the sur-vey can be found in several references (e.g., Lehner et al. 2006;King et al. 2002). Simultaneous monitoring for occultations onmultiple telescopes allow for the rejection of false positive events.

Each telescope is equipped with a 2048 ; 2048 CCD array with a3 deg2 field of view. The TAOS cameras read out at 5 Hz and mon-itor anywhere from several hundred to a few thousand stars in a tar-get field. The reported signal-to-noise ratio for an A0 V V 12 staris STAOS $ 8 (M. J. Lehner 2007, private communication).

A campaign on the 6.5 m MMT, located at the Fred Lawrence

Whipple Observatory on the summit of Mount Hopkins, Arizona,has been proposed (F. B. Bianco 2007, private communication).The survey would use Megacam, an array of 36 edge-butting1024 ; 2304 pixel CCDs. Megacam would be used in continuousreadout mode, achieving a sampling rate of 200 Hz (in this paperwe assume a sampling rate of 20 Hz as we expect the observa-tions to be binned) on 0.16deg2 fields. For this system the mon-itoring of several hundred bright stars ($500 with magnitudeV 15) is possible, and test images show a signal-to-noise ratioof SMMT ! 70 for stars with magnitudes V 13 can be achieved.This would lead to the collection of$104 star hours over threenights.

A dedicated telescope system like that of TAOS, with theability to sample light curves with a high signal-to-noise ratio and

fast sampling rates as the previously described campaign on theMMT, is the ideal system. Whipple, a proposed space telescopededicated to the detection of occultations by outer solar systemobjects (Kaplan et al. 2003), is an attempt to match these require-ments. It is based on the design of theKeplermission (Koch et al.2005), with the same 100 deg 2 field of view, but has a modifiedfocal plane designed to allow readout cadences of up to 40 Hz.For the purposes of this discussion a signal-to-noise ratio ofSWhipple $ 40 is assumed. This accounts for both Poisson statisticsand read noise (M. J. Lehner 2007, private communication). Adedicated space telescope has several advantages over ground-based campaigns such as those mentioned above. Dedicated mon-itoring in space means that continuous monitoring of fields are apossibility with larger fields of view. It is estimated thatWhipple

Fig. 17.Diffraction profiles with added noise observed at opposition for a V 12 A0 V star occulted by objects with r 0:5, 1.5, and 5 km at 40 AU. MMT andWhipple light curves are simulated with signal-to-noise ratios of 70 and 40, respectively, and TAOS light curves are generated with a signal-to-noise ratio of 8.



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will observe up to $140,000 stars in a given field. Hence, thelarger number of monitored stars increases the rate of detection,which would provide more sound statistics on the surface densityof small objects. A space-based telescope also has the advantagethat it is not affected by atmospheric scintillation, which would

allow the survey to be sensitive to objects smaller and more dis-tant than those that could be observed from the ground.

For the Kuiper Belt, TAOSs modest 5 Hz sampling and signal-to-noise ratio of 8 is sufficient for the detection of objects withrk1:5 km, but provide none of the diffraction details (Fig. 17).

Fig. 18.Diffraction profiles withadded noiseobserved at opposition for a V 12 A0 V background star occulted by objects with r 2:5,5,and10km at1000AU.MMTand Whipple light curves are simulated with signal-to-noise ratios of 70 and 40, respectively, and TAOS light curves are generated with a signal-to-noise ratio of 8.

Fig. 19.Diffractionprofileswith added noiseobserved at opposition for a V 12A0 V backgroundstar occultedby objects with r 5,10,and20 kmat 10,000AU.MMTand Whipple light curves are simulated with signal-to-noise ratios of 70 and 40, respectively, and TAOS light curves are generated with a signal-to-noise ratio of 8.



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The MMT and Whipple photometry have greater signal-to-noiseratio and much higher sampling, and therefore are not only ca-pable of seeing objects smaller than r 1:5 km, but are also ableto detect diffraction effects for these smaller objects.

Simulations for extended disk objects in the region intermedi-ate between the Kuiper Belt and Oort Cloud ($1000AU) leadtoa similar conclusion (see Fig. 18). Although not the stated focusof TAOS, it turns out that TAOS could potentially provideconstraints on the number and size of Sednas smaller cousins, asobjects with r $ 5 km are potentially detectable with the TAOSphotometry. Objects much smaller in size would unlikely be seenby TAOS. The higher signal-to-noise ratio of an MMT-basedsurvey and the proposed Whipple space telescope coupled with

rapid sampling make the detection of the weak diffraction effectspossible.

The top panels of Figure 19 indicate that small objects (r $10 km) in the Oort Cloud region will not provide enough signalfor TAOS. Events from a r 10 km object at 10,000 AU is barelydetectable by the MMT and Whipple surveys, and larger objectsupwards ofr 20 km in size are a possibility for all three sur-veys. At distances ak1000 AU, the difference between 20 and40 Hz sampling is minimal. Both of these sampling frequencies,at large distances where the event durations are longer, are suffi-cient to determine the shape of the occultation profile. However,it is clear from Figure 17 that there is a clear benefit of samplingat 40 Hz versus 20 Hz when searching for events from KBOs at40 AU.

6. DETECTABILITY

It would be useful to describe an occultation event by a singleparameter that can be used in conjunction with the parameters ofa survey to determine whether or not the occultation event hassufficient signal to be detected above the underlying noise of ameasured light curve. For ideal curves parameterized by the Fresneldistance scale and radius , we thus define the detectability pa-rameter as

XFSU

Z11

I() 1 2

d: 12

When curves are discretely sampled, the detectability is a sumover a discrete set of observations I(i). If m samples are takenover intervals ofcorresponding to a sampling frequency 1/,then

XFSU Xmi1

I(i) 1 2

: 13

Notethatfor a geometric occultation where 31, completeextinc-tion ( 1) of the background source throughout the duration

Fig. 20.Dimensionless detectability (for a point background source) plot-tedvs. theoccultingobject radiusin Fresnel scale units at various sampling rates1/ listed in the left key. Low sampling rates result in lower detectability atsmaller values of.

Fig. 21.Detectability contour plot of objects as a function of diameter anddistance for a point source. An A0 V spectrum was assumed.

Fig. 22.Detectability contour plot of objects as a function of diameter anddistance for an A0 V V 12 star. The finite size of the stellar disk was accountedfor. Also shown are the limits where the object radius is equivalent to the Fresnelscale F(dash-dotted line) and equivalent to the stellar disk (straight solid line).



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of the occultation event means that the detectability of a geo-metric occultation is roughly equal to the occultation width(XFSU $ ). Light curves in this regime clearly have the largestdetectability, while the shallow depths for Fraunhofer occultationevents makes such events the least detectable of the three regimes.

Figure 20 is a plot ofXFSU versus . The solid line traces thedetectability for the ideal continuous occultation intensity profile

as expressed in equation (12). Atk10, the detectability asymp-totes to the occultation width 2. The effect of finite sampling isalso shown in this plot for three sampling frequencies 1/10, 1, and 0.1. Lower sampling rates result in a decreased abilityto detect occultation events. For larger values of, this effect isdiminished as occultation event widths increase. This is becauselarger objects produce wide geometric occultation patterns thatcompletely extinguish the background source, and therefore de-tection of objects with larger is not greatly improved with in-creases in the sampling frequency.

The scaled detectability XFSU shows the general trend for alloccultation patterns scaled to the Fresnel scale. However, it pro-vides little intuition for event detection in a real survey. Theabove discussion is extended to the physical parameter space of

a and rby introducing the corresponding expression in physicalunits to equation (12):

X

Z11

Ir; a(t) 1 2

dt: 14

Note thatX now has the dimension oftime and is related toXFSUas

X XFSUF=vT:

Figure 21 is a contour plot of the detectability for occultationsofan A0V point source asa functionofa and r. The detectabilityincreases for objects at closer distances and decreases at largerdistances. It can also be seen that larger objects show less drasticchanges in the detectability as their distance is increased or de-creased. Similarly, Figure 22 shows a detectability contour plotfor occultations of a finite size V 12 A0 V star by objectsat various distances a and with different radii r. Note that the

Fig. 23.Contour plots of the detectability as a function of radius rand distance a for occultation curves sampled at frequencies of 1, 5, 20, and 40 Hz. An A0 VV 12 star is assumed.



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two plots are similar at lower values ofa and larger values ofr, but for smaller objects at larger distances, when the projectedsource size becomes larger than both the Fresnel scale and

the object size, the detectability of the event drops. The up-turn in the contours of Figure 22 roughly correspond to the dis-tance a, where the projected source size is equal to the Fresnelscale. Note that for dimmer stars with smaller radii this up-turn in the contours will occur at larger distances, and for anA0 V star with magnitude 16 or greater the detectability con-tour plot would be virtually identical to the plot shown inFigure 21.

As before, a continuously sampled curve at a frequency fandcorresponding exposuretis a set of discrete observations I(ti).In this case, equation (14) becomes

X Xm

i1

t I(ti) 1 2; 15

where m is set to be large enough to accommodate the entirewidth of the event. The effects of sampling can be seen in Fig-ure 23, which shows contour detectability plots for a finite-sized A0V V 12 star for various occultation curves sampled atfrequencies of 1, 5, 20, and 40 Hz. Just as in Figure 20, the de-tectability changes very little for geometric occultation light curvesirrespective of the sampling. These curves correspond roughly todetectabilities that are of order 1 or greater. For regions of the plotwhere the object radius is either of order or less than the Fresnelscale, sampling rates will change the detectability more dras-tically. As an example, recall from Figure 14 that an r 0:5 kmobject ata 40 AU generates a fluctuation of$20%. This event

sampled at a rate of 5 Hz has a detectability of order 103. Thedetectability of this event, however, is improved 10-fold whenobserved at 40 Hz.

As discussed in x 4, observations can be made away from op-position, reducing the relative velocity of the target object. Thisaffects the detectability as well. To illustrate, consider the case ofa KBO occultation event measured at 75

from opposition. From

equation (8), it can be shown that the relative velocity vT is re-

duced by a factor of about 8. If the sampling rate is high enoughthat the diffraction effects can be resolved, the event duration T,and thus the detectability X, are subsequently increased by a fac-tor of 8. As mentioned in x 4, the event rate for such events is de-creased by a factor of 8 as well. In cases where vTt3W

andthe event is averaged out with the nominal flux of the star dueto the large sample time, the detectability X is close to 0. How-ever, the detectability could possibly be dramatically increasedby moving away from opposition such thatvTt $ W

, allow-ing the event to be detected. This is illustrated in Figure 24, whichshows light curves for an r 0:5 km KBO at 40 AU measuredat opposition and at an opposition angle of 75

. At opposition,

the detectability of the event is X 1:2 ; 104, but moving to 75 increases the detectability toX 5:1 ; 103, indicating

that such an event is much more likely to be detected away fromopposition.

7. DETECTION SENSITIVITY LIMITSFOR PHOTOMETRIC SYSTEMS

The detectability parameter described in x 6 can be used inconjunction with the signal-to-noise ratio of a given survey todefine the sensitivity of the survey to objects of various sizes anddistances. Consider the measured light curve consisting of a setof measured photon counts at a detector Ni ntI(ti), where nis the photon rate of the background star. [Variables with a tildeindicate that they are measured and contain an error term, e.g.,I(ti) I(ti) ei , where ei is a random variable with zero meanand variance 2.]

Fig. 24.Light curves for an r 0:5 km KBO at 40 AU measured at oppo-sition and at an opposition angleof 75. Moving to 75 reduces the relativevelocity vT by roughly a factor of 8, significantly increasing the detectability ofthe event.

Fig. 25.Plot ofP(2 > h2i) 1011 for each of three telescope systemsindicating the lower limit sensitivity to occultation events in the a-r plane.TAOS is indicated by the solid line. The dashed and dotted lines indicate thelimits for the MMT and Whipple systems, respectively.



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If the null hypothesis of a constant light curve in which no oc-cultation event is present is selected, then the 2 value over a set

ofm points is

2 Xmi1

1

2Ni nt

2: 16

The sensitivity of a given survey is determined by calculatingthe expected 2 value h 2i and its corresponding probabilityP(2 > h2i) and asserting that a detection of an event requiresthat the null hypothesis is rejected at a confidence level p. To doso Ni is replaced with its definition given above, and the expec-tation value is taken. The expected value of2 for an observedsignal then becomes

h2i n 2t2

2Xmi1

I(ti) 1 2n o

m; 17

where I(ti) is the observed normalized light curve. This can beexpressed in terms ofX using equation (15):

h 2i n2t

2X m: 18

Note that the expectation value of this calculation with no eventis given by

h2i m; 19

where m, the number of degrees of freedom, represents the un-derlying variance due to noise. Therefore, we claim an occulta-

tion event is detectable ifh2i is sufficiently larger than m, suchthatP(2 > h 2i) < p.

For pure Poisson statistics, 2 nt, and therefore equa-tion (18) simplifies to

h 2i nX m: 20

This expression represents the best-case scenario for any surveyin detecting an occultation curve.

Carefully considering the number of points m over which 2

is calculated will help maximize h 2i relative to m. Note thatfrom the definition ofX in equation (15), the summand does notsignificantly contribute to the sum beyond the occultation eventduration (eq. [10]) where I(ti) % 1. Therefore, the number of

points needed to sufficiently sample an occultation event shouldspan the occultation event such that

m %W

vTt: 21

The optimal value for p depends on the survey statistics of agiven star and the target population, butit should be large enoughto allow sensitivityto as many events as possible butsmall enoughto minimize the false positive rate. A reasonable value shouldroughly be of order 1/M, where Mis the total number of obser-vations of a background source. A reasonable value of the thresh-old for the TAOS survey is p 1011 (Lehner et al. 2006), andwe adopt this value for the examples that follow.

Fig. 26.Light curves for an r 3 km Oort Cloud object at 10,000 AU measured at opposition and at an opposition angle of 75. The light curve at opposition issampled at 5 Hz, while the light curve at 75 is sampled at 1.25 Hz. The panels on the left show the theoretical light curve, while the panels on the right show lightcurves with simulated noise added for the proposed Whipple survey.



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Equation (18) can be written in terms of the signal-to-noiseratio S by noting that for a constant source S nt/. Giventhis, and the above approximation form, we can compute h2ifor a given photometric system:

h2i S2

tX

W

vTt: 22

Sensitivity limits for the three systems (TAOS, Whipple, and theMMT) discussed in x 5 are plotted in Figure 25, given p 1011.This plot indicates that TAOS is sensitive to Fresnel occultationevents in the Kuiper Belt down to objects of r $ 2 km, as wellas geometric events in the more distant extended disk and OortCloud. Looking at the photometric sensitivity limits plotted forthe MMTand Whipple systems, we see very little difference betweena ground-based survey on a high signal-to-noise ratio telescopelike the MMT versus Whipple. However, photometric perfor-manceis only one requirement of a survey. The proposed Whipplesystem would follow more than 140,000 stars simultaneously,compared to fewer than 400 at the MMT. The dedicated, 24 hourusage of Whipple would result in a vastly greater number ofdetections.

As discussed in x 6, moving away from opposition can signifi-cantly increase the detectability of an event ifWTvTt. How-ever, even ifW3 vTt, the detectability increases simply due tothe larger event duration T. In this case, the signal-to-noise ratio Scan be increased as well by increasing the sample time accord-ingly. This is illustrated in Figure 26, which shows simulatedlight curves for the Whipple survey for an r 3 km Oort Cloudobject at 10,000 AU measured at opposition and at an opposi-tion angle of 75

. The relative velocity at 75 is a factor of 4

lower than at 0, and therefore both Tand X are increasedby a factor of 4. At 75, we have reduced the sampling fre-quency by a factor of 4, which increases the signal-to-noise ratioSby a factor of about 3.2 while holding m constant. We can thencalculate

h 2

i 40:5 at opposition and

h 2

i 125:3 at 75.

Given the above considerations, it is clear that the first term inequation (22) increases by a factor of 3:2 2 (both X and t in-crease by a factor of 4, so those factors cancel), while the secondterm (the degrees of freedom) remains constant (vTt does notchange). Thus the value ofh 2i increases relative to the numberof degrees offreedom, making it more likely that the event can bedetected. In the example shown in Figure 26, the measured val-ues of2 are 54.36 and 124.58 for 0 and 75, respectively,and with m 31 degrees of freedom, these values correspond toP 0:0058 and 3.7;1013. With a threshold valueofp 1011,the object could be detected at 75, but not at opposition.

8. DISCUSSION

In this paper we have simulated occultation events in theKuiper Belt out to the Oort Cloud for three photometric systemsincorporating the effects of finite bandwidths and stellar spectra,finite source sizes, and sampling. Via these simulated occultation

events, we have quantitatively parameterized observed occulta-tion events with the use of the Fresnel scale, width W, depth D,and the detectabilityX, and we have shown how the detectabilityX can be used to calculate the sensitivity of a survey using theh 2i parameter for an event.

In x 3 light-curve smoothing and occultation width broaden-ing due to finite source-size effects were shown to be significantfactors in the detection of an event. In Figures 21 and 22 it was

shown that in spite of the augmentation of the occultation widthby a finite source radius, dampening of occultation event varia-tion due to background-source smoothing lowers the detectabil-ity of events at larger distances beyond the point of intersectionbetween the background source radius and the Fresnel scale.Therefore, monitored background stars significantly minimizethe detectability of an occultation event when r > F. Selectionof background target stars should aim to minimize the stellar sizerelative to the Fresnel scale while balancing the need for a decentsignal-to-noise ratio. This becomes a greater challenge at largerdistances a because of the linear dependence of the projectedstellar radius r on a. Even a relatively blue star of moderatebrightness like an A0 V V 12 star will lower the detectabilityat larger distances in the outer solar system. For the distant

regions of the solar system, the ability to detect smaller objectswill rely on a photometry systems limiting magnitude beingrelatively high.

In xx 6 and 7 a method of determining a surveys sensitivity toobjects in the outer solar system regions by comparing the var-iation due to occultation events X and the variation due to instru-mental and photon-count fluctuations was presented. In Figure 25it was shown that even a ground survey of modest signal-to-noiseratio is capable of viewing inner Oort Cloud objects as smallas r $ 10 km and perhaps even outer Oort Cloud objects withr $ 100 km using the occultation method.

The benefits of using larger telescopes with higher signal-to-noise ratios, and the ability to sample at higher rates such as thedescribed MMT and Whipple campaigns, are evident. The increase

in detectability gained in such cases would allow the surveys topush the lower limit of detection for small objects in the KuiperBelt even further almost by an order of magnitude to objects withradii that are a few hundreds of meters. Similarly, the detection ofsmaller objects farther out in the solar system is improved.

What can be detected bya survey in order to further aid the se-lection of survey parameters can be further explored using ourdescribed method. Further work on incorporating the statistics ofthe occultation event rates to determine the amount of necessaryobservation time to significantly determine the population sizedistribution of a set of outer solar system objects would con-tribute greatly to the design and implementation of occultationsurveys.

The efforts at Harvard University are supported by NASAgrants NNG05GA28G and NNG05GO66G.

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