On the detectability of lightcurves of Kuiper belt objects

Pedro Lacerdaa,* and Jane Luub

a Leiden Observatory, University of Leiden, Postbus 9513, NL-2300 RA Leiden, Netherlandsb MIT Lincoln Laboratory, 244 Wood Street, Lexington, MA 02420, USA

Received 23 April 2002; revised 21 August 2002

Abstract

We present a statistical study of the detectability of lightcurves of Kuiper belt objects (KBOs). Some Kuiper belt objects displaylightcurves that appear “flat”; i.e., there are no significant brightness variations within the photometric uncertainties. Under the assumptionthat KBO lightcurves are mainly due to shape, the lack of brightness variations may be due to (1) the objects having very nearly sphericalshapes or (2) their rotation axes coinciding with the line of sight. We investigate the relative importance of these two effects and relate itto the observed fraction of “flat” lightcurves. This study suggests that the fraction of KBOs with detectable brightness variations may provideclues about the shape distribution of these objects. Although the current database of rotational properties of KBOs is still insufficient to drawany statistically meaningful conclusions, we expect that, with a larger dataset, this method will provide a useful test for candidate KBO shapedistributions.© 2003 Elsevier Science (USA). All rights reserved.

Keywords: Kuiper Belt; Minor planets; Asteroids; Solar system: General

1. Introduction

The Kuiper belt holds a large population of small objectswhich are thought to be remnants of the protosolar nebula(Jewitt and Luu, 1993). The belt is also the most likelyorigin of other outer solar system objects such as Pluto–Charon, Triton, and the short-period comets; its studyshould therefore provide clues to the understanding of theprocesses that shaped our solar system. More than 650Kuiper belt objects (KBOs) are known to date and a total ofabout 105 objects larger than 50 km are thought to orbit theSun beyond Neptune (Jewitt and Luu, 2000).

One of the most fundamental ways to study physicalproperties of KBOs is through their lightcurves. Lightcurvesshow periodic brightness variations due to rotation, since, asthe KBO rotates in space, its cross-section as projected inthe plane of the sky will vary due to its nonspherical shape,resulting in periodic brightness variations (see Fig. 1). Awell-sampled lightcurve will thus yield the rotation period

of the KBO, and the lightcurve amplitude has informationon the KBO’s shape. This technique is commonly used inplanetary astronomy, and has been developed extensivelyfor the purpose of determining the shapes, internal densitystructures, rotational states, and surface properties of atmo-sphereless bodies. These properties in turn provide clues totheir formation and collisional environment.

Although lightcurves studies have been carried out rou-tinely for asteroids and planetary satellites, the number ofKBO lightcurves is still meager, with few of sufficientquality for analysis (see Table I). This is due to the fact thatmost KBOs are faint objects, with apparent red magnitudemR�23 (Trujillo et al., 2001), rendering it very difficult todetect small-amplitude changes in their brightness. One ofthe few high-quality lightcurves is that of (20000) Varuna,which shows an amplitude of�m � 0.42� 0.02 mag anda period ofProt � 6.3442� 0.0002 h (Jewitt and Sheppard,2002). Only recently have surveys started to yield signifi-cant numbers of KBOs bright enough for detailed studies(Jewitt et al., 1998).

Another difficulty associated with the measurement ofthe amplitude of a lightcurve is that of determining theperiod of the variation. If no periodicity is apparent in the

* Corresponding author.E-mail address: placerda@strw.leidenuniv.nl (P. Lacerda), luu@ll.mit.edu

(J. Luu).

R

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data, any small variations in the brightness of an object mustbe due to noise. Furthermore, a precise measurement of theamplitude of the lightcurve requires a complete coverage ofthe rotational phase. Therefore, any conclusion based onamplitudes of lightcurves must assume that their periodshave been determined and confirmed by well-sampled phaseplots of the data.

However, not all of the observed KBOs show detectablebrightness variations (the so-called “fl at” lightcurves). Thesimplest explanations for this could be due to (1) the objectbeing axisymmetric (the two axes perpendicular to the spinvector are equal), or (2) its rotation axis is nearly coincidentwith the line of sight (see Fig. 3). In other words, theundetectable variations are a consequence either of theKBO’s shape or of the observational geometry. By studyingthe relative probabilities of these two causes and relatingthem to the observed fraction of “fl at” lightcurves, we mightexpect to improve our knowledge of the intrinsic shapedistribution of KBOs. In this paper we address the followingquestion: Can we learn something about the shape distribu-tion of KBOs from the fraction of “fl at” lightcurves?

2. Definitions and assumptions

The observed brightness variations in KBO lightcurvescan be due to

● eclipsing binary KBOs● surface albedo variations● irregular shape.

In general the brightness variations will arise from somecombination of these three factors, but the preponderance ofeach effect among KBOs is still not known. In the followingcalculations we exclude the first two factors and assume thatshape is the sole origin of KBO brightness variations. Wefurther assume that KBO shapes can be approximated bytriaxial ellipsoids, and thus expect a typical KBO lightcurveto show a set of two maxima and two minima for each fullrotation (see Fig. 1). Table II summarizes the used symbolsand notations. The listed quantities are defined in the text.

The detailed assumptions of our model are as follows:

1. The KBO shape is a triaxial ellipsoid. This is theshape assumed by a rotating body in hydrostatic equi-librium (Chandrasekhar, 1969). There are reasons tobelieve that KBOs might have a “ rubble pile” struc-ture (Farinella et al., 1981), justifying the approxima-tion even further.

2. The albedo is constant over surface. Although albedovariegation can in principle explain any given light-curve (Russell, 1906), the large-scale brightness vari-ations are generally attributed to the object’s irregularshape (Burns and Tedesco, 1979).

3. All axis orientations are equally probable. Given thatwe have no knowledge of preferred spin vector ori-entation, this is the most reasonable a priori assump-tion.

4. The KBO is in a state of simple rotation around theshortest axis (the axis of maximum moment of iner-tia). This is likely since the damping timescale of acomplex rotation (e.g., precession), �103 yr, (Burnsand Safronov, 1973), (Harris, 1994) is smaller thanthe estimated time between collisions (107–1011 yr)that would re-excite such a rotational state (Stern,1995), (Davis and Farinella, 1997).

5. The KBO is observed at zero phase angle (� � 0). Ithas been shown from asteroid data that lightcurve

Fig. 1. The lightcurve of an ellipsoidal KBO observed at aspect angle � ��/2. Cross sections and lightcurve are represented for one full rotation ofthe KBO. The amplitude, �m, of the lightcurve is determined for thisparticular case. See text for the general expression.

Table 1KBOs with measured lightcurves

Name Classa Hb [mag] �mc [mag] Pd [h] Sourcee

1993 SC C 6.9 �0.04 RT19991994 TB P 7.1 0.3 6.5 RT19991996 TL66 S 5.4 �0.06 RT19991996 TP66 P 6.8 �0.12 RT19991994 VK8 C 7.0 0.42 9.0 RT19991996 TO66 C 4.5 0.1 6.25 Ha2000Varuna C 3.7 0.42 6.34 JS20021995 QY9 P 7.5 0.6 7.0 RT19991996 RQ20 C 7.0 — RT19991996 TS66 C 6.4 �0.16 RT19991996 TQ66 C 7.0 �0.22 RT19991997 CS29 C 5.2 �0.2 RT19991999 TD10 S 8.8 0.68 5.8 Co2000

a Dynamical class (C - classical KBO, P - plutino, S - scattered KBO).b Absolute magnitude.c Lightcurve amplitude.d Rotational period.e RT1999 (Romanishin and Tegler 1999), Ha2000 (Hainaut 2000),

Co2000 (Consolmagno et al., 2000), JS2002 (Jewitt and Sheppard 2002).

Table 2Symbols and notation used

Symbol Description

a � b � c axes of ellipsoidal KBOa� � b� � c� normalized axes of KBO (b� � 1)� aspect angle�mmin minimum detectable lightcurve amplitude�min aspect angle at which �m � �mmin

K 100.8�mmin

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amplitudes seem to increase linearly with phase an-gle,

A��, 0� � A��, ��/�1 � m��,

where � is the aspect angle, � is the phase angle andm is a coefficient which depends on surface compo-sition. The aspect angle is defined as the angle be-tween the line of sight and the spin axis of the KBO(see Fig. 2a), and the phase angle is the Sun–object–Earth angle. The mean values of m found for differentasteroid classes are m(S) � 0.030, m(C) � 0.015,m(M) � 0.013, where S, C, and M are asteroid classes(see Michalowsky, 1993). Since KBO are distant ob-jects the phase angle will always be small. Evenallowing m to be one order of magnitude higher thanthat of asteroids the increase in the lightcurve ampli-tude will not exceed 1%.

6. The brightness of the KBO is proportional to itscross-section area (geometric scattering law). This isa good approximation for KBOs because (1) mostKBOs are too small to hold an atmosphere and (2) thefact that they are observed at very small phase anglesreduces the influence of scattering on the lightcurveamplitude (Magnusson, 1989).

The KBOs will be represented by triaxial ellipsoids ofaxes a � b � c rotating around the short axis c (see Fig. 2b).In order to avoid any scaling factors we normalize all axesby b, thus obtaining a new set of parameters, a� , b� , and c�,given by

a� � a/b, b� � 1, c� � c/b. (1)

As defined, a� and c� can assume values 1 � a� � � and 0 �c� � 1. Note that the parameters a� and c� are dimensionless.

The orientation of the spin axis of the KBO relative to theline of sight will be defined in spherical coordinates (�, �),with the line of sight (oriented from the object to theobserver) being the z-axis, or polar axis, and the angle �being the polar angle (see Fig. 2a). The solution is indepen-dent of the azimuthal angle �, which would be measured inthe plane perpendicular to the line of sight, between anarbitrary direction and the projection of the spin axis on thesame plane. The observation geometry is parameterized bythe aspect angle, which in this coordinate system corre-sponds to �.

As the object rotates, its cross-section area S will varyperiodically between Smax and Smin (see Fig. 1). These areasare simply a function of a, b, c, and the aspect angle �.Given the assumption of geometric scattering, the ratiobetween maximum and minimum flux of reflected sunlightwill be equal to the ratio between Smax and Smin. Thelightcurve amplitude can then be calculated from the quan-tities a� , c�, and � and is given by

�m � 2.5 log �a� 2 cos2� � a� 2c� 2 sin2�

a� 2 cos2� � c� 2 sin2� � 1/ 2

. (2)

3. “Flat” lightcurves

It is clear from Eq. (2) that under certain conditions, �mwill be zero; i.e., the KBO will exhibit a flat lightcurve.These special conditions involve the shape of the object andthe observation geometry, and are described quantitatively

Fig. 2. (a) A spherical coordinate system is used to represent the observing geometry. The line of sight (oriented from the object to the observer) is the polaraxis and the azimuthal axis is arbitrary in the plane orthogonal to the polar axis. � and � are the spherical angular coordinates of the spin axis �s. In thiscoordinate system the aspect angle is given by �. The “nondetectability” cone, with semivertical angle �min, is represented in gray. If the spin axis lies withinthis cone the brightness variations due to changing cross-section will be smaller than photometric errors, rendering it impossible to detect brightnessvariations. (b) The picture represents an ellipsoidal KBO with axes a � b � c.

176 P. Lacerda, J. Luu / Icarus 161 (2003) 174–180

below. Taking into account photometric error bars willbring this “fl atness” threshold to a finite value, �mmin, aminimum detectable amplitude below which brightnessvariation cannot be ascertained.

The two factors that influence the amplitude of a KBOlightcurve are:

1. Sphericity

For a given ellipsoidal KBO of axes ratios a� and c� thelightcurve amplitude will be largest when � � �/2 andsmallest when � � 0 or �. At � � �/2, Eq. (2) becomes

�m � 2.5 log a� . (3)

Even at � � �/2, having a minimum detectable amplitude,�mmin, puts constraints on a� since if a� is too small, thelightcurve amplitude will not be detected. This constraint isthus

a� 100.4�mmin f “fl at” lightcurve. (4)

2. Observation geometry

If the rotation axis is nearly aligned with the line of sight,i.e., if the aspect angle is sufficiently small, the object’sprojected cross-section will hardly change with rotation,yielding no detectable brightness variations (see Fig. 3). Thefinite accuracy of the photometry defines a minimum aspectangle, �min, within which the lightcurve will appear flatwithin the uncertainties. This angle rotated around the lineof sight generates a “nondetectability cone” (see Fig. 2a)with the solid angle

��min� � �0

2� �0

�min

sin� d� d�. (5)

Any aspect angle � which satisfies � � �min falls within the

Fig. 3. Illustration of a rotating ellipsoid at different aspect angles. A quarter of a full rotation is represented. Rotational phase of ellipsoid increasing fromtop to bottom and � decreasing from left to right. T is the period of rotation. Axes ratios are a� � 1.2 and c� � 0.9.

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“nondetectability cone” and results in a nondetectable light-curve amplitude. Therefore, the probability that the light-curve will be flat due to the observing geometry is

pa� ,c��non-detection� �2 ��min�

4�

� 1 � cos�min (6a)

f pa� ,c��detection� � cos�min. (6b)

The factor of 2 accounts for the fact that the axis might bepointing toward or away from the observer and still give riseto the same observations, and the 4� in the denominatorrepresents all possible axis orientations.

From Eq. (2) we can write cos �min as a function of a�and c�,

cos�min � �a� , c� �

� � c� 2�a� 2 � K�

c� 2�a� 2 � K� � a� 2�K � 1�, (7)

where K � 100.8�mmin. The function (a� , c�), represented inFig. 4, is the probability of detecting brightness variationfrom a given ellipsoid of axes ratios (a� , c�). It is a geometry-weighting function. For a� in [1, �K] we have (a� , c�) � 0by definition, since in this case the KBO satisfies Eq. (4) andits lightcurve amplitude will not be detected irrespective ofthe aspect angle. It is clear from Fig. 4 that brightnessvariation from an elongated body is more likely to bedetected.

4. Detectability of lightcurves

In order to generate a “nonflat” lightcurve, the KBO hasto satisfy both the shape and observation geometry condi-

tions. Mathematically this means that the probability ofdetecting brightness variation from a KBO is a function ofthe probabilities of the KBO satisfying both the sphericityand observing geometry conditions.

We will assume that it is possible to represent the shapedistribution of KBOs by two independent probability den-sity functions, f(a�) and g(c�), defined as

p�a� 1 � a� � a� 2� � �a�1

a�2

f�a� � da� , �1

�

f�a� � da� � 1,

(8a)

p�c� 1 � c� � c� 2� � �c�1

c�2

g�c� � dc� , �0

1

g�c� � dc� � 1,

(8b)

where the integrals on the left represent the fraction ofKBOs in the given ranges of axes ratios. This allows us towrite the following expression for p(�m � �mmin), whereboth the shape and observation geometry constraints aretaken into account,

p��m � �mmin�

� �0

1 �1

�

�a� , c� � f�a� � g�c� � da� dc� . (9)

The right hand side of this equation represents the proba-bility of observing a given KBO with axis ratios between (a� ,c�) and (a� da� , c� dc�) at a large enough aspect angle,integrated for all possible axes ratios. This is also the prob-ability of detecting brightness variation for an observedKBO.

The lower limit of integration for a� in Eq. (9) can be

Fig. 4. The function (a� , c�) (Eq. (7)). This plot assumes photometric errors �mmin � 0.15 mag. The detection probability is zero when a� � 100.4�mmin �1.15.

178 P. Lacerda, J. Luu / Icarus 161 (2003) 174–180

replaced by �K, with K defined as in Eq. (7), since (a� , c�)is zero for a� in [1, �K]. In fact, this is how the sphericityconstraint is taken into account.

Provided that we know the value of p(�m � �mmin), Eq.(9) can test candidate distributions f(a�) and g(c�) for theshape distribution of KBOs. The best estimate for p(�m ��mmin) is given by the ratio of “nonflat” lightcurves (ND) tothe total number of measured lightcurves (N); i.e.,

p��m � �mmin� �ND

N. (10)

Because N is not the total number of KBOs there will be anerror associated with this estimate. Since we do not knowthe distributions f(a�) and g(c�) we will assume that theoutcome of an observation can be described by a binomialdistribution of probability p(�m � �mmin). This is a goodapproximation given that N is very small compared with thetotal number of KBOs. Strictly speaking, the hypergeomet-ric distribution should be used since we will not uninten-tionally observe the same object more than once (samplingwithout replacement). However, since the total number ofKBOs (which is not known with certainty) is much largerthan any sample of lightcurves, any effects of repeatedsampling will be negligible, thereby justifying the binomialapproximation. This simplification allows us to calculate theupper (p ) and lower (p�) limits for p(�m � �mmin) at anygiven confidence level, C. These values, known as theClopper–Pearson confidence limits, can be found by solvingthe following equations by trial and error (Barlow 1989),

�r�ND 1

N

P�r;p ��m � �mmin�, N� �C � 1

2(11a)

�r�0

ND�1

P�r;p���m � �mmin�, N� �C � 1

2(11b)

(see Table 2 for notation), where C is the desired confidencelevel and P(r;p, N) is the binomial probability of detectingr lightcurves out of N observations, each lightcurve havinga detection probability p. Using the values in Table 1 and�mmin � 0.15 mag we have ND � 5 and N � 13 whichyields p(�m � �mmin) � 0.38�0.15

0.17 at a C � 0.68 (1 )confidence level. At C � 0.997 (3 ) we have p(�m ��mmin) � 0.38�0.31

0.41. The value of p(�m � �mmin) could besmaller since some of the flat lightcurves might not havebeen published.

Note that for moderately elongated ellipsoids (small a�)the function (a� , c�) is almost insensitive to the parameter c�(see Fig. 4), in which case the axisymmetric approximationwith respect to a� can be made yielding c� � 1. Equation (9)then has only one unknown parameter, f(a�), and

p��m � �mmin� � ��K

a�max

�a� , 1� f�a� � da�

(12)� 0.38�0.31 0.41.

If we assume the function f(a�) to be gaussian, we can useEq. (12) to determine its mean � and standard deviation ,after proper normalization to satisfy Eq. (8a). The result isrepresented in Fig. 5, where we show all possible pairs of(�, ) that would satisfy a given p(�m � �mmin). Forexample, the line labeled “0.38” identifies all possible pairsof (�, ) that give rise to p(�m � �mmin) � 0.38, the linelabeled “0.56” all possible pairs of (�, ) that give rise top(�m � �mmin) � 0.56, etc.

Clearly, with the present number of lightcurves the un-

Fig. 5. Contour plot of the theoretical probabilities of detecting brightness variation in KBOs (assuming �mmin � 0.15 mag), drawn from gaussian shapedistributions parameterized by � and (respectively the mean and spread of the distributions). The solid lines represent the observed ratio of “nonflat”lightcurves (at 0.38) and 0.68 confidence limits (at 0.23 and 0.56, respectively).

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certainties are too large to draw any relevant conclusions onthe shape distribution of KBOs. With a larger dataset, thisformulation will allow us to compare the distribution ofKBO shapes with that of the main belt asteroids. The latterhas been shown to resemble, to some extent, that of frag-ments of high-velocity impacts (Catullo et al. 1984). Itdeviates at large asteroid sizes that have presumably relaxedto equilibrium figures. A comparison of f(a�) with asteroidalshapes should tell us, at the very least, whether KBO shapesare collisionally derived, as opposed to being accretionalproducts.

The usefulness of this method is that, with more data, itwould allow us to derive such quantitative parameters as themean and standard deviation of the KBO shape distribution,if we assume a priori some intrinsic form for this distribu-tion. The method’s strength is that it relies solely on thedetectability of lightcurve amplitudes, which is more robustthan other lightcurve parameters.

This paper focuses on the influence of the observationgeometry and KBO shapes in the results of lightcurve mea-surements. In which direction would our conclusionschange with the inclusion of albedo variegation and/or bi-nary KBOs?

Nonuniform albedo would cause nearly spherical KBOsto generate detectable brightness variations, depending onthe coordinates of the albedo patches on the KBO’s surface.This means that our method would overestimate the numberof elongated objects by attributing all brightness fluctua-tions to asphericity.

Binary KBOs would influence the results in differentways depending on the orientation of the binary system’sorbital plane, on the size ratio of the components, and on theindividual shapes and spin axis orientations of the primaryand secondary. For example, an elongated KBO observedequator-on would have its lightcurve flattened by a nearlyspherical moon orbiting in the plane of the sky, whereas twospherical KBOs orbiting each other would generate a light-curve if the binary would be observed edge-on.

These effects are not straightforward to quantify analyt-ically and might require a different approach. We intend toincorporate them in a future study. Also, with a largersample of lightcurves it would be useful to apply this modelto subgroups of KBOs based on dynamics, size, etc.

5. Conclusions

We derived an expression for the probability of detectingbrightness variations from an ellipsoidal KBO, as a functionof its shape and minimum detectable amplitude. This ex-pression takes into account the probability that a “fl at”lightcurve is caused by observing geometry.

Our model can yield such quantitative parameters as themean and standard deviation of the KBO shape distribution,if we assume a priori an intrinsic form for this distribution.It concerns solely the statistical probability of detecting

brightness variation from objects drawn from these distri-butions, given a minimum detectable lightcurve amplitude.The method relies on the assumption that albedo variegationand eclipsing binaries play a secondary role in the detectionof KBO lightcurves. The effect of disregarding albedo var-iegation in our model is that we might overestimate thefraction of elongated objects. Binaries in turn could influ-ence the result in both directions depending on the geometryof the problem, and on the physical properties of the con-stituents. We intend to incorporate these effects into a fu-ture, more detailed study.

Acknowledgments

We are grateful to Professor J. Rice and G. van de Venfor helpful discussion.

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