ANALYSIS OF THE ROTATIONAL PROPERTIES OF KUIPER BELT OBJECTS

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1 The Astronomial Journal, 131: , 2006 April # The Amerian Astronomial Soiety. All rights reserved. Printed in U.S.A. ANALYSIS OF THE ROTATIONAL PROPERTIES OF KUIPER BELT OBJECTS Pedro Laerda 1, 2 Grupo de Astrofísia da Universidade de Coimbra, Departamento de Matemátia, Largo D. Dinis, 3000 Coimbra, Portugal; plaerda@strw.leidenuniv.nl and Jane Luu Linoln Laboratory, Massahusetts Institute of Tehnology, 244 Wood Street, Lexington, MA Reeived 2005 Otober 4; aepted 2006 January 6 ABSTRACT We use optial data on 10 Kuiper Belt objets (KBOs) to investigate their rotational properties. Of the 10, 3 (30%) exhibit light variations with amplitude m 0:15 mag, and 1 out of 10 (10%) has m 0:40 mag, whih is in good agreement with previous surveys. These data, in ombination with the existing database, are used to disuss the rotational periods, shapes, and densities of KBOs. We find that, in the sampled size range, KBOs have a higher fration of low-amplitude light urves and rotate slower than main-belt asteroids. The data also show that the rotational properties and the shapes of KBOs depend on size. If we split the database of KBO rotational properties into two size ranges with diameters larger and smaller than 400 km, we find that (1) the mean light-urve amplitudes of the two groups are different with 98.5% onfidene, (2) the orresponding power-law shape distributions seem to be different, although the existing data are too sparse to render this differene signifiant, and (3) the two groups oupy different regions on a spin period versus light-urve amplitude diagram. These differenes are interpreted in the ontext of KBO ollisional evolution. Key words: Kuiper Belt minor planets, asteroids solar system: general 1. INTRODUCTION The Kuiper Belt (KB) is an assembly of mostly small iy objets, orbiting the Sun beyond Neptune. Kuiper Belt objets (KBOs) are likely remnants of outer solar system planetesimals (Jewitt & Luu 1993). Their physial, hemial, and dynamial properties should therefore provide valuable information regarding both the environment and the physial proesses responsible for planet formation. At the time of writing, roughly 1000 KBOs are known, half of whih have been followed for more than one opposition. A total of 10 5 objets larger than 50 km are thought to orbit the Sun beyond Neptune (Jewitt & Luu 2000). Studies of KB orbits have revealed an intriate dynamial struture, with signatures of interations with Neptune ( Malhotra 1995). The size distribution follows a differential power law of index q ¼ 4 for bodies k50 km ( Trujillo et al. 2001a), beoming slightly shallower at smaller sizes ( Bernstein et al. 2004). KBO olors show a large diversity, from slightly blue to very red (Luu & Jewitt 1996; Tegler & Romanishin 2000; Jewitt & Luu 2001), and seem to orrelate with inlination and/or perihelion distane (e.g., Jewitt & Luu 2001; Doressoundiram et al. 2002; Trujillo & Brown 2002). The few low-resolution optial and near-ir KBO spetra are mostly featureless, with the exeption of a weak 2 m water ie absorption line present in some of them (Brown et al. 1999; Jewitt & Luu 2001) and strong methane absorption on 2003 UB 313 (Brown et al. 2005). About 4% of known KBOs are binaries with separations larger than 0B15 (Noll et al. 2002). All the observed binaries have primary-to-seondary mass ratios 1. Two binary reation 1 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI Leiden Observatory, Postbus 9513, NL-2300 RA, Leiden, Netherlands models have been proposed. Weidenshilling (2002) favors the idea that binaries form in three-body enounters. This model requires a 100 times more dense Kuiper Belt at the epoh of binary formation and predits a higher abundane of large-separation binaries. An alternative senario (Goldreih et al. 2002), in whih the energy needed to bind the orbits of two approahing bodies is drawn from the surrounding swarm of smaller objets, also requires a muh higher density of KBOs than the present, but it predits a larger fration of lose binaries. Reently, Sheppard & Jewitt (2004) have shown evidene that 2001 QG 298 ould be a lose or ontat binary KBO and estimated the fration of similar objets in the Belt to be 10% 20%. Other physial properties of KBOs, suh as their shapes, densities, and albedos, are still poorly onstrained. This is mainly beause KBOs are extremely faint, with mean apparent red magnitude 23 (Trujillo et al. 2001b). The study of KBO rotational properties through time-series broadband optial photometry has proved to be the most suessful tehnique to date for investigating some of these physial properties. Light variations of KBOs are believed to be aused mainly by their aspherial shape; as KBOs rotate in spae, their projeted ross setions hange, resulting in periodi brightness variations. One of the best examples to date of a KBO light urve and what an be learned from it is that of (20000) Varuna (Jewitt & Sheppard 2002). The authors explained the light urve of (20000) Varuna as a onsequene of its elongated shape (axis ratio, a/b 1:5). They further argued that the objet is entripetally deformed by rotation beause of its low-density rubble pile struture. The term rubble pile is generally used to refer to gravitationally bound aggregates of smaller fragments. The existene of rubble piles is thought to be due to ontinuing mutual ollisions throughout the age of the solar system, whih gradually frature the interiors of objets. Rotating rubble piles an adjust their shapes

2 ANALYSIS OF ROTATIONAL PROPERTIES OF KBOs 2315 TABLE 1 Observing Conditions and Geometry Objet Obs. Date a Telesope Seeing b (arse) Mvt. Rate (arse hr 1 ) Int. Time d (s) R.A. Del. R e (AU) f (AU) g (deg) (19308) 1996 TO Ot 1 WHT TS Sep 30 WHT , TS Ot 1 WHT TS Ot 2 WHT , (35671) 1998 SN Sep 29 WHT , (35671) 1998 SN Sep 30 WHT (19521) Chaos Ot 1 WHT , 400, (19521) Chaos Ot 2 WHT , (79983) 1999 DF Feb 13 WHT (79983) 1999 DF Feb 14 WHT (79983) 1999 DF Feb 15 WHT (80806) 2000 CM Feb 11 WHT , (80806) 2000 CM Feb 13 WHT (80806) 2000 CM Feb 14 WHT (66652) 1999 RZ Sep 11 INT (66652) 1999 RZ Sep 12 INT (66652) 1999 RZ Sep 13 INT (47171) 1999 TC Sep 11 INT (47171) 1999 TC Sep 12 INT (47171) 1999 TC Sep 13 INT (38628) Huya Feb 28 INT (38628) Huya Mar 1 INT (38628) Huya Mar 3 INT CZ Mar 1 INT , CZ Mar 3 INT , Note. Units of right asension are hours, minutes, and seonds, and units of delination are degrees, arminutes, and arseonds. a UT date of observation. b Average seeing of the data. Average rate of motion of KBO. d Integration times used. e KBO-Sun distane. f KBO-Earth distane. g Phase angle (Sun-objet-Earth angle) of observation. to balane entripetal aeleration and self-gravity. The resulting equilibrium shapes have been studied in the extreme ase of fluid bodies and depend on the body s density and spin rate (Chandrasekhar 1969). Laerda & Luu (2003, hereafter LL03a) showed that under reasonable assumptions the fration of KBOs with detetable light urves an be used to onstrain the shape distribution of these objets. A follow-up (Luu & Laerda 2003, hereafter LL03b) on this work, using a database of light-urve properties of 33 KBOs (Sheppard & Jewitt 2002 [ hereafter SJ02], 2003), showed that although most Kuiper Belt objets (85%) have shapes that are lose to spherial (a/b 1:5), there is a signifiant fration (12%) with highly aspherial shapes (a/b 1:7). In this paper we use optial data on 10 KBOs to investigate the amplitudes and periods of their light urves. These data are used in ombination with the existing database to investigate the distributions of KBO spin periods and shapes. We disuss their impliations for the inner struture and ollisional evolution of objets in the Kuiper Belt. 2. OBSERVATIONS AND PHOTOMETRY We olleted time-series optial data on 10 KBOs at the Isaa Newton 2.5 m (INT) and William Hershel 4 m (WHT) telesopes. The INT Wide Field Camera (WFC) is a mosai of four EEV 2048 ; 4096 CCDs, eah with a pixel sale of 0B33 pixel 1 and spanning approximately 11A3 ; 22A5 in the plane of the sky. The targets are imaged through a Johnson R filter. The WHT prime fous amera onsists of two EEV 2048 ; 4096 CCDs with a pixel sale of 0B24 pixel 1 and overs a sky-projeted area of 2 ; 8A2 ; 16A4. With this amera we used a Harris R filter. The seeing for the whole set of observations ranged from 1B0to 1B9 FWHM. We traked both telesopes at sidereal rate and kept integration times for eah objet suffiiently short to avoid errors in the photometry due to trailing effets (see Table 1). No lighttravel time orretions have been made. We redued the data using standard tehniques. The sky bakground in the flat-fielded images shows variations of less than 1% aross the hip. Bakground variations between onseutive nights were less than 5% for most of the data. Cosmi rays were removed with the pakage LA-Cosmi (van Dokkum 2001). We performed aperture photometry on all objets in the field using the SExtrator software pakage ( Bertin & Arnouts 1996). This software performs irular aperture measurements on eah objet in a frame and puts out a atalog of both the magnitudes and the assoiated errors. Below we desribe how we obtained a better estimate of the errors. We used apertures ranging from 1.5 to 2.0 times the FWHM for eah frame and seleted the aperture that maximized the ratio of signal-to-noise. An extra aperture of five FWHMs was used to look for possible seeing-dependent trends in our photometry. The atalogs were mathed by seleting only the soures that are present in all frames. The slow movement of KBOs from night to night allows us to suessfully math a large number of soures in onseutive nights. We disarded all saturated soures, as well as those identified to be galaxies.

3 2316 LACERDA & LUU Vol. 131 TABLE 3 Photometri Measurements of (19308) 1996 TO 66 UT Date a Julian Date b (mag) 1999 Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, a UT date at the start of the exposure. b Julian Date at the start of the exposure. Apparent red magnitude; errors inlude unertainties in relative and absolute photometry added quadratially. Fig. 1. Frame-to-frame photometri varianes (in magnitudes) of all stars (irles, rosses) in the (a) (35671) 1998 SN 165 and (b) (38628) Huya fields, plotted against their relative magnitude. The trend of inreasing photometri variability with inreasing magnitude is lear. The intrinsially variable stars learly do not follow this trend and are loated toward the upper left region of the plot. The KBOs are shown as squares. (35671) 1998 SN 165 (top) shows a muh larger variability than the omparison stars (rosses; see x 3.1), while (38628) Huya (bottom) is well within the expeted variane range, given its magnitude. The KBO light urves were obtained from differential photometry with respet to the brightest nonvariable field stars. An average of the magnitudes of the brightest stars (the referene stars) provides a referene for differential photometry in eah frame. This method allows for small-amplitude brightness variations to be deteted even under nonphotometri onditions. The unertainty in the relative photometry was alulated from the satter in the photometry of field stars that are similar to the KBOs in brightness (the omparison stars; see Fig. 1). TABLE 2 Properties of Observed KBOs (mag) Objet Class a H b i a e (deg) e d (AU) (19308) 1996 TO C TS C (35671) 1998 SN C f (19521) Chaos... C (79983) 1999 DF 9... C (80806) 2000 CM C (66652) 1999 RZ C (47171) 1999 TC Pb (38628) Huya... P CZ C a Dynamial lass (C = Classial KBO, P = Plutino, b = binary KBO). b Absolute magnitude. Orbital inlination. d Orbital eentriity. e Semimajor axis. f This objet has a lassial-type inlination and eentriity, but its semimajor axis is muh smaller than that of other lassial KBOs. This error estimate is more robust than the errors provided by SExtrator (see below), and was used to verify the auray of the latter. This proedure resulted in onsistent time series brightness data for 100 objets (KBOs and field stars) in a time span of two to three onseutive nights. We observed Landolt standard stars whenever onditions were photometri, and used them to alibrate the zero point of the magnitude sale. The extintion oeffiient was obtained from the referene stars. Sine not all nights were photometri the light urves are presented as variations with respet to the mean brightness. These yield the orret amplitudes and periods of the light urves but do not provide their absolute magnitudes. The orbital parameters and other properties of the observed KBOs are given in Table 2. Tables 3, 4, 5, and 6 list the absolute R-magnitude photometri measurements obtained for (19308) 1996 TO 66, 1996 TS 66, (35671) 1998 SN 165, and (19521) Chaos, respetively. Tables 7 and 8 list the mean-subtrated R-band data for (79983) 1999 DF 9 and 2001 CZ LIGHT-CURVE ANALYSIS The results in this paper depend solely on the amplitude and period of the KBO light urves. It is therefore important to aurately determine these parameters and the assoiated unertainties Can We Detet the KBO Brightness Variation? We begin by investigating whether the observed brightness variations are intrinsi to the KBO, i.e., if the KBO s intrinsi brightness variations are detetable given our unertainties. This was done by omparing the frame-to-frame satter in the KBO optial data with that of (10 20) omparison stars. To visually ompare the satter in the magnitudes of the referene stars (see x 2), omparison stars, and KBOs, we plot a histogram of their frame-to-frame varianes (see Fig. 2). In general suh a histogram should show the referene stars lustered at the lowest varianes, followed by the omparison stars spread over larger varianes. If the KBO appears isolated at muh higher varianes than both groups of stars (e.g., Fig. 2j), then its brightness modulations are signifiant. Conversely, if the KBO is lustered with the stars (e.g., Fig. 2f ), any periodi brightness variations would be below the detetion threshold. Figure 1 shows the dependene of the unertainties on magnitude. Objets that do not fall on the rising urve traed out by

4 No. 4, 2006 ANALYSIS OF ROTATIONAL PROPERTIES OF KBOs 2317 TABLE 4 Photometri Measurements of 1996 TS 66 UT Date a Julian Date b (mag) 1999 Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, a UT date at the start of the exposure. b Julian Date at the start of the exposure. Apparent red magnitude; errors inlude unertainties in relative and absolute photometry added quadratially. the stars must have intrinsi brightness variations. By alulating the mean and spread of the variane for the omparison stars (rosses) we an alulate our photometri unertainties and thus determine whether the KBO brightness variations are signifiant (3 ) Period Determination In the ases in whih signifiant brightness variations (see x 3.1) were deteted in the light urves, the phase dispersion minimization method was used (PDM; Stellingwerf 1978) to look for periodiities in the data. For eah test period, PDM omputes a statistial parameter that ompares the spread of data points in phase bins with the overall spread of the data. The period that best fits the data is the one that minimizes. The advantages of PDM are that it is nonparametri, i.e., it does not assume a shape for the light urve, and it an handle unevenly spaed data. Eah data set was tested for periods ranging from 2 to 24 hr, in steps of 0.01 hr. To assess the uniqueness of the PDM solution, we generated 100 Monte Carlo realizations of eah light urve, keeping the original data times and randomizing the magnitudes within the error bars. We ran PDM on eah generated data set to obtain a distribution of best-fit periods. TABLE 5 Photometri Measurements of (35671) 1998 SN 165 UT Date a Julian Date b (mag) 1999 Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Sep ,451, Ot ,451, Ot ,451, a UT date at the start of the exposure. b Julian Date at the start of the exposure. Apparent red magnitude; errors inlude unertainties in relative and absolute photometry added quadratially Amplitude Determination We used a Monte Carlo experiment to determine the amplitude of the light urves for whih a period was found. We generated several artifiial data sets by randomizing eah point within the error bar. Eah artifiial data set was fitted with a Fourier series, using the best-fit period, and the mode and entral 68% of

5 2318 LACERDA & LUU Vol. 131 TABLE 6 Photometri Measurements of (19521) Chaos UT Date a Julian Date b (mag) 1999 Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, Ot ,451, a UT date at the start of the exposure. b Julian Date at the start of the exposure. Apparent red magnitude; errors inlude unertainties in relative and absolute photometry added quadratially. the distribution of amplitudes were taken as the light-urve amplitude and 1 unertainty, respetively. For the null light urves, i.e., those in whih no signifiant variation was deteted, we subtrated the typial error bar size from the total amplitude of the data to obtain an upper limit to the amplitude of the KBO photometri variation. 4. RESULTS In this setion we present the results of the light-urve analysis for eah of the observed KBOs. We found signifiant brightness variations (m > 0:15 mag) for 3 out of 10 KBOs (30%), and m 0:40 mag for 1 out of 10 (10%). This is onsistent with previously published results: SJ02 found a fration of 31% with m > 0:15 mag and 23% with m > 0:40 mag, both onsistent with our results. The other seven KBOs do not show detetable variations. The results are summarized in Table SN 165 TABLE 7 Relative Photometry Measurements of (79983) 1999 DF 9 UT Date a Julian Date b (mag) 2001 Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, a UT date at the start of the exposure. b Julian Date at the start of the exposure. Mean-subtrated apparent red magnitude; errors inlude unertainties in relative and absolute photometry added quadratially. The brightness of (35671) 1998 SN 165 varies signifiantly ( >5 ) more than that of the omparison stars (see Figs. 1 and 2). The periodogram for this KBO shows a very broad minimum around P ¼ 9hr(Fig.3a). The degeneray implied by the broad minimum would only be resolved with additional data. A slight weaker minimum is seen at P ¼ 6:5 hr, whih is lose to a 24 hr alias of P ¼ 9hr. Peixinho et al. (2002, hereafter PDR02) observed this objet in 2000 September, but having only one night s worth of data, they did not sueed in determining this objet s rotational period unambiguously. We used their data to solve the degeneray in our PDM result. The PDR02 data have not been absolutely alibrated, and the magnitudes are given relative to a bright field star. To be able to ombine it with our own data we had to subtrat the mean magnitudes. Our periodogram of (35671) 1998 SN 165 (entered on the broad minimum) is shown in Figure 3b and an be ompared with the revised periodogram obtained with our data ombined with the PDR02 data (Fig. 3). The minima beome muh learer with the additional data, but beause of the 1 yr time differene between the two observational ampaigns, many lose aliases appear in the periodogram. The absolute minimum, at P ¼ 8:84 hr, orresponds to a double-peaked light urve (see Fig. 4). The seond-best fit, P ¼ 8:7 hr, produes a more sattered phase plot, in whih the peak in the PDR02 data oinides with our night 2. Period P ¼ 8:84 hr was also favored by the Monte Carlo method desribed in x 3.2, being identified as the best fit in 55% of the ases versus 35% for P ¼ 8:7 hr. The large size of the error bars ompared to the amplitude is responsible for the ambiguity in the result. We use P ¼ 8:84 hr in the rest of the paper beause it was onsistently seleted as the best fit.

6 No. 4, 2006 ANALYSIS OF ROTATIONAL PROPERTIES OF KBOs 2319 TABLE 8 Relative Photometry Measurements of 2001 CZ 31 UT Date a Julian Date b (mag) 2001 Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Feb ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, Mar ,451, a UT date at the start of the exposure. b Julian Date at the start of the exposure. Mean-subtrated apparent red magnitude; errors inlude unertainties in relative and absolute photometry added quadratially. The amplitude obtained using the Monte Carlo method desribed in x 3.3 is m ¼ 0:16 0:01 mag. This value was alulated using only our data, but it did not hange when realulated adding the PDR02 data. Fig. 2. Staked histograms of the frame-to-frame variane (in magnitudes) in the optial data on the referene stars (white), omparison stars ( gray), and the KBO (blak). In (), (e), and ( j) the KBO shows signifiantly more variability than the omparison stars, whereas in all other ases it falls well within the range of photometri unertainties of the stars of similar brightness DF 9 KBO (79983) 1999 DF 9 shows large-amplitude photometri variations ( 0:4 mag). The PDM periodogram for (79983) 1999 DF 9 is shown in Figure 5. The best-fit period is P ¼ 6:65 hr, whih orresponds to a double-peak light urve (Fig. 6). Other PDM minima are found lose to P/2 3:3 hr and 9.2 hr, a 24 hr alias of the best period. Phasing the data with P/2 results in a worse fit beause the two minima of the double-peaked light urve exhibit signifiantly different morphologies ( Fig. 6); the peuliar sharp feature superimposed on the brighter minimum, whih is reprodued on two different nights, may be aused by a nononvex feature on the surfae of the KBO (Torppa et al. 2003). Period P ¼ 6:65hrwasseletedin65ofthe100Monte Carlo repliations of the data set (see x 3.2). The seond most seleted solution (15%) was at P ¼ 9hr.WeuseP ¼ 6:65 hr for the rest of the paper. The amplitude of the light urve, estimated as desribed in x 3.3, is ¼ 0:40 0:02 mag CZ 31 This objet shows substantial brightness variations (4.5 above the omparison stars) in a systemati manner. The first night of data seems to sample nearly one omplete rotational phase. As for (35671) 1998 SN 165, the 2001 CZ 31 data span only two nights of observations. In this ase, however, the PDM minima (see Figs. 7a and 7b) are very narrow, and only two orrespond to independent periods, P ¼ 4:69 hr (the minimum at 5.82 is a 24 hr alias of 4.69 hr), and P ¼ 5:23 hr. Objet TABLE 9 Light-Curve Properties of Observed KBOs a (mag) Nights b (mag) P d (hr) (35671) 1998 SN :20 0:05 2 (1) 0:16 0: (8.70) (79983) 1999 DF :40 0: (9.00) 2001 CZ (1) 0:21 0: (5.23) (19308) 1996 TO :26 0:06 1? TS :76 0:05 3 < (19521) Chaos... 20:74 0:06 2 < (80806) 2000 CM < (66652) 1999 RZ < (47171) 1999 TC < (38628) Huya < a Mean red magnitude. Errors inlude unertainties in relative and absolute photometry added quadratially. b Number of nights with useful data. Numbers in parentheses indiate number of nights of data from other observers used for period determination. Data for (35671) 1998 SN 165 are taken from PDR02, and data for 2001 CZ 31 are taken from SJ02. Light-urve amplitude. d Light-urve period (values in parentheses indiate less likely solutions not entirely ruled out by the data).

7 2320 LACERDA & LUU Vol. 131 Fig. 5. Periodogram for the (79983) 1999 DF 9 data. The minimum orresponds to a light-urve period P ¼ 6:65 hr. sattered light urve, whih onfirms the PDM result. The Monte Carlo test for uniqueness yielded P ¼ 4:71 hr as the best-fit period in 57% of the ases, followed by P ¼ 5:23 hr in 21%, and a few other solutions, all below 10%, between P ¼ 5 and 6 hr. We use P ¼ 4:71 hr throughout the rest of the paper. We measured a light-urve amplitude of m ¼ 0:21 0:02 mag. If we use both ours and SJ02 s first night data, m rises to 0.22 mag. Fig. 3. Periodogram for the data on (35671) 1998 SN 165.Panelb shows an enlarged setion near the minimum, alulated using only the data published in this paper, and panel shows the same region realulated after adding the PDR02 data. KBO 2001 CZ 31 had also been observed by SJ02 in 2001 February and April, with inonlusive results. We used their data to try to rule out one (or both) of the two periods we found. We mean-subtrated the SJ02 data in order to ombine them with our unalibrated observations. Figure 7 shows the setion of the periodogram around P ¼ 5 hr, realulated using SJ02 s first night plus our own data. The aliases are due to the 1 month time differene between the two observing runs. The new PDM minimum is at P ¼ 4:71 hr very lose to the P ¼ 4:69 hr determined from our data alone. Visual inspetion of the ombined data set phased with P ¼ 4:71 hr shows a very good math between SJ02 s first night (2001 February 20) and our own data (see Fig. 8). SJ02 s seond and third nights show very large satter and were not inluded in our analysis. Phasing the data with P ¼ 5:23 hr yields a more 4.4. Flat Light Curves The flutuations deteted in the optial data on KBOs (19308) 1996 TO 66, 1996 TS 66, (47171) 1999 TC 36, (66652) 1999 RZ 253, (80806) 2000 CM 105, and (38628) Huya are well within the unertainties. The KBO (19521) Chaos shows some variations with respet to the omparison stars but no period was found to fit all the data. See Table 9 and Figure 9 for a summary of the results Other Light-Curve Measurements The KBO light-urve database has inreased onsiderably in the last few years, largely due to the observational ampaign of SJ02, with reent updates in Sheppard & Jewitt (2003, 2004). These authors have published observations and rotational data for a total of 30 KBOs (their SJ02 paper inludes data for three other previously published light urves in the analysis). Other reently published KBO light urves inlude those for (50000) Quaoar (Ortiz et al. 2003) and the sattered KBO (29981) 1999 TD 10 (Rousselot et al. 2003). Of the 10 KBO light urves presented in this paper, 6 are new to the database, bringing the total Fig. 4. Light urve of (35671) 1998 SN 165. The figure represents the data phased with the best-fit period P ¼ 8:84 hr. Different symbols orrespond to different nights of observation. The gray line is a seond-order Fourier series fit to the data. The PDR02 data are shown as rosses. Fig. 6. Same as Fig. 4, but for KBO (79983) 1999 DF 9. The best-fit period is P ¼ 6:65 hr. The lines represent seond-order (solid line) and fifth-order (dashed line) Fourier series fits to the data. The normalized 2 values of the fits are 2.8 and 1.3, respetively.

8 No. 4, 2006 ANALYSIS OF ROTATIONAL PROPERTIES OF KBOs 2321 Fig. 7. Periodogram for the 2001 CZ 31 data. Panel b shows an enlarged setion near the minimum, alulated using only the data published in this paper, and panel shows the same region realulated after adding the SJ02 data. to 41. Table 10 lists the rotational data on the 41 KBOs that are analyzed in the rest of the paper. 5. ANALYSIS In this setion we examine the light-urve properties of KBOs and ompare them with those of main-belt asteroids ( MBAs). The light-urve data for these two families of objets over different size ranges. MBAs, being loser to Earth, an be observed down to muh smaller sizes than KBOs; in general it is very diffiult to obtain good-quality light urves for KBOs with diameters D < 50 km. Furthermore, some KBOs surpass the 1000 km barrier, whereas the largest asteroid, Ceres, does not reah 900 km. This is taken into aount in the analysis. The light-urve data for asteroids were taken from the Harris Light Curve Catalog, 3 version 5, while the diameter data were 3 See Fig. 8. Same as Fig. 4, but for KBO 2001 CZ 31. The data are phased with period P ¼ 4:71 hr. The points represented by rosses are taken from SJ02. Fig. 9. Flat light urves. The respetive amplitudes are within the photometri unertainties. obtained from the Lowell Observatory database of asteroid orbital elements. 4 The sizes of most KBOs were alulated from their absolute magnitude assuming an albedo of The exeptions are (47171) 1999 TC 36, (38638) Huya, (28978) Ixion, (55636) 2002 TX 36, (66652) 1999 RZ 36, (26308) 1998 SM 165, and (20000) Varuna, for whih the albedo has been shown to be inonsistent with the value 0.04 (Grundy et al. 2005). For example, in the ase of (20000) Varuna simultaneous thermal and optial observations have yielded a red geometri albedo of 0:070 þ0:030 0:017 (Jewitt et al. 2001) Spin Period Statistis As Figure 10 shows, the spin period distributions of KBOs and MBAs are signifiantly different. Beause the sample of KBOs of small size or large periods is poor, to avoid bias in our omparison we onsider only KBOs and MBAs with diameter larger than 200 km and with periods below 20 hr. In this range the mean rotational periods of KBOs and MBAs are 9.23 and 6.48 hr, respetively, and the two are different with a 98.5% onfidene aording to Student s t-test. However, the different means do not rule out that the underlying distributions are the same, and ould simply mean that the two sets of data sample the same 4 Available at ftp://ftp.lowell.edu/pub/elgb/astorb.html.

9 2322 LACERDA & LUU Vol. 131 TABLE 10 Database of KBO Light-Curve Properties Objet Class a ( km) Size b P ( hr) d (mag) Referenes KBOs Considered to Have m < 0:15 mag (15789) 1993 SC... P , 7 (15820) 1994 TB... P < (26181) 1996 GQ S < (15874) 1996 TL S , 7 (15875) 1996 TP P , 7 (79360) 1997 CS C < (91133) 1998 HK P < (33340) 1998 VG P < (19521) Chaos... C < , 13 (26375) 1999 DE 9... S < (47171) 1999 TC Pb < , 13 (38628) Huya... P < , 13 (82075) 2000 YW S < (82158) 2001 FP S < (82155) 2001 FZ S < KD P < (28978) Ixion... P <0.10 5, QF P < (42301) 2001 UR S < (42355) 2002 CR S < (55636) 2002 TX C :08 0:02 11 (55637) 2002 UX C < (55638) 2002 VE P < (80806) 2000 CM C < (66652) 1999 RZ Cb < TS C < KBOs Considered to Have m 0:15 mag (32929) 1995 QY 9... P :60 0:04 7, 10 (24835) 1995 SM C :19 0:05 11 (19308) 1996 TO C :26 0:03 3, 11 (26308) 1998 SM R :45 0:03 8, 10 (33128) 1998 BU S :68 0:04 8, 10 (40314) 1999 KR C :18 0:04 10 (47932) 2000 GN P :61 0:03 10 (20000) Varuna... C :42 0: AZ P :14 0: QG Pb :14 0:04 12 (50000) Quaoar... C :17 0:02 6 (29981) 1999 TD S :53 0:03 9 (35671) 1998 SN C :16 0:01 13 (79983) 1999 DF 9... C :40 0: CZ C :21 0:06 13 a Dynamial lass (C = lassial KBO, P = Plutino, R = 2:1 resonant, S = sattered KBO, b = binary KBO, = ontat binary). b Diameter assuming an albedo of 0.04 exept when measured (see text). Period of the light urve. For KBOs with both single- and double-peaked possible light urves the doublepeaked period is listed. d Light-urve amplitude. Referenes. (1) Collander-Brown et al. 1999; (2) Davies et al. 1997; (3) Hainaut et al. 2000; (4) Luu & Jewitt 1998; (5) Ortiz et al. 2001; (6) Ortiz et al. 2003; (7) Romanishin & Tegler 1999; (8) Romanishin et al. 2001; (9) Rousselot et al. 2003; (10) SJ02; (11) Sheppard & Jewitt 2003; (12) Sheppard & Jewitt 2004; (13) this work. distribution differently. This is not the ase, however, aording to the Kolmogorov-Smirnov ( K-S) test, whih gives a probability that the periods of KBOs and MBAs are drawn from the same parent distribution of 0.7%. Although it is lear that KBOs spin slower than asteroids, it is not lear why this is so. If ollisions have ontributed as signifiantly to the angular momentum of KBOs as they did for MBAs (Farinella et al. 1982; Catullo et al. 1984), then the observed differene should be related to how these two families reat to ollision events. We will address the question of the ollisional evolution of KBO spin rates in a future paper Light-Curve Amplitudes and the Shapes of KBOs The umulative distribution of KBO light-urve amplitudes is shown in Figure 11. It rises very steeply in the low amplitude range (m < 0:15 mag) and then beomes shallower reahing

10 No. 4, 2006 ANALYSIS OF ROTATIONAL PROPERTIES OF KBOs 2323 Fig. 12. Light-urve amplitudes of KBOs (irles) and MBAs(rosses) plotted against objet size. Fig. 10. Histograms of the spin periods of KBOs (top) and MBAs (bottom) satisfying D > 200 km, m 0:15 mag, and P < 20 hr. The mean rotational periods of KBOs and MBAs are 9.23 and 6.48 hr, respetively. The y-axis in both ases indiates the number of objets in eah range of spin periods. large amplitudes. In quantitative terms, 70% of the KBOs possess m < 0:15 mag, while 12% possess m 0:40 mag, with the maximum value being m ¼ 0:68 mag. (Fig. 11 does not inlude the KBO 2001 QG 298, whih has a light-urve amplitude m ¼ 1:14 0:04 mag and would further extend the range of amplitudes. We do not inlude 2001 QG 298 in our analysis beause it is thought to be a ontat binary [Sheppard & Jewitt 2004].) Figure 11 also ompares the KBO distribution to that of MBAs. The distributions of the two populations are learly distint; there is a larger fration of KBOs in the low-amplitude range (m < 0:15 mag) than in the ase of MBAs, and the KBO distribution extends to larger values of m. Figure 12 shows the light-urve amplitude of KBOs and MBAs plotted against size. KBOs with diameters larger than D ¼ 400 km seem to have lower light-urve amplitudes than KBOs with diameters smaller than D ¼ 400 km. Student s t-test onfirms that the mean amplitudes in eah of these two size ranges are different at the 98.5% onfidene level. For MBAs the transition is less sharp and seems to our at a smaller size (D 200 km). In the ase of asteroids, the aepted explanation is that small bodies (D P 100 km) are fragments of high-veloity impats, whereas their larger ounterparts (D > 200 km) generally are not (Catullo et al. 1984). The light-urve data on small KBOs are still too sparse to permit a similar analysis. In order to redue the effets of bias related to body size, we an onsider only those KBOs and MBAs with diameters larger than 200 km. In this size range, 25 of 37 KBOs (69%) and 10 of 27 MBAs (37%) have light-urve amplitudes below 0.15 mag. We used the Fisher exat test to alulate the probability that suh a ontingeny table would arise if the light-urve amplitude distributions of KBOs and MBAs were the same: the resulting probability is 0.8%. The distribution of light-urve amplitudes an be used to infer the shapes of KBOs, if ertain reasonable assumptions are made (see, e.g., LL03). Generally, objets with elongated shapes produe large brightness variations due to their hanging projeted ross setion as they rotate. Conversely, round objets, or those with the spin axis aligned with the line of sight, produe little or no brightness variations, resulting in flat light urves. Figure 12 shows that the light-urve amplitudes of KBOs with diameters smaller and larger than D ¼ 400 km are signifiantly different. Does this mean that the shapes of KBOs are also different in these two size ranges? To investigate this possibility of a size dependene among KBO shapes we onsider KBOs with diameter smaller and larger than 400 km separately. We loosely refer to objets with diameter D > 400 km and D 400 km as larger and smaller KBOs, respetively. We approximate the shapes of KBOs by triaxial ellipsoids with semiaxes a > b >. For simpliity we onsider the ase in whih b ¼ and use the axis ratio ã ¼ a/b to haraterize the shape of an objet. The orientation of the spin axis is parameterized by the aspet angle, defined as the smallest angular distane between the line of sight and the spin vetor. On this basis the light-urve amplitude m is related to ã and via the relation (eq. [2] of LL03a with ¼ 1) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ã 2 m ¼ 2:5log 1 þ ã 2 þ ã 2 : ð1þ ð 1Þos (2) Following LL03b we model the shape distribution by a power law of the form Fig. 11. Cumulative distribution of light-urve amplitude for KBOs (irles)andasteroids(rosses) larger than 200 km. We plot only KBOs for whih a period has been determined. KBO 2001 QG 298, thought to be a ontat binary (Sheppard & Jewitt 2004), is not plotted, although it may be onsidered an extreme ase of elongation. f (ã) dã / ã q dã; where f (ã)dã represents the fration of objets with shapes between ã and ã þ dã. We use the measured light-urve amplitudes ð2þ

11 2324 LACERDA & LUU Vol. 131 TABLE 11 Best-Fit Parameter to the KBO Shape Distribution Method b Size Range a LL03 MC D 400 km... 4:3 þ2:0 1:6 3:8 0:8 D > 400 km... 7:4 þ3:1 2:4 8:0 1:4 All sizes... 5:7 þ1:6 1:3 5:3 0:8 a Range of KBO diameters, onsidered in eah ase. b LL03 is the method desribed in LL03a, and MC is a Monte Carlo fit of the light-urve amplitude distribution. to estimate the value of q by employing both the method desribed in LL03a and by Monte Carlo fitting the observed amplitude distribution (SJ02; LL03b). The latter onsists of generating artifiial distributions of m (eq. [1]) with values of ã drawn from distributions haraterized by different q-values (eq. [2]), and seleting the one that best fits the observed umulative amplitude distribution ( Fig. 11). The values of are generated assuming random spin axis orientations. We use the K-S test to ompare the different fits. The errors are derived by bootstrap resampling the original data set ( Efron 1979) and measuring the dispersion in the distribution of best-fit power-law indies, q i,foundforeahbootstrap repliation. Following the LL03a method we alulate the probability of finding a KBO with m 0:15 mag: p(m 0:15) Z ãmax pffiffiffi K sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ã 2 K f (ã) (ã 2 dã; 1)K where K ¼ 10 0:8 ; 0:15, f (ã) ¼ C ã q,andcis a normalization onstant. This probability is alulated for a range of q-values to determine the one that best mathes the observed fration of light urves with amplitude larger than 0.15 mag. These frations are f (m 0:15 mag; D 400 km) ¼ 8/19, f (m 0:15 mag; D > 400 km) ¼ 5/21, and f (m 0:15 mag) ¼ 13/40 for the omplete set of data. The results are summarized in Table 11 and shown in Figure 13. The unertainties in the values of q obtained using the LL03a method (q ¼ 4:3 þ2:0 1:6 for KBOs with D 400 km and q ¼ 7:4þ3:1 2:4 for KBOs with D > 400 km; see Table 11) do not rule out similar shape distributions for smaller and larger KBOs. This is not the ase for the Monte Carlo method. The reason for this is that the LL03a method relies on a single, more robust parameter: the fration of light urves with detetable variations. The sizeable error bar is indiative that a larger data set is needed to better onstrain the values of q. In any ase, it is reassuring that both methods yield steeper shape distributions for larger KBOs, implying more spherial shapes in this size range. A distribution with q 8 predits that 75% of the large KBOs have a/b < 1:2. For the smaller objets we find a shallower distribution, q 4, whih implies a signifiant fration of very elongated objets: 20% have a/b > 1:7. Although based on small numbers, the shape distribution of large KBOs is well fitted by a simple power law (the K-S rejetion probability is 0.6%). This is not the ase for smaller KBOs, for whih the fit is poorer (K-S rejetion probability is 20%; see Fig. 13). Our results are in agreement with previous studies of the overall KBO shape distribution, whih had already shown that a simple power law does not explain the shapes of KBOs as a whole (LL03b; SJ02). The results presented in this setion suggest that the shape distributions of smaller and larger KBOs are different. However, ð3þ Fig. 13. Observed umulative light-urve amplitude distributions of KBOs (irles) with diameter smaller than 400 km (top left) and larger than 400 km (bottom left), and of all the sample (bottom right). Error bars are Poissonian. The best-fit power-law shape distributions of the form f (ã) dã ¼ ã q dã were onverted to amplitude distributions using a Monte Carlo tehnique (see text for details) and are shown as solid lines. The best-fit q-values are listed in Table 11. the existing number of light urves is not enough to make this differene statistially signifiant. When ompared to asteroids, KBOs show a preponderane of low-amplitude light urves, possibly a onsequene of their possessing a larger fration of nearly spherial objets. It should be noted that most of our analysis assumes that the light-urve sample used is homogeneous and unbiased; this is probably not true. Different observing onditions, instrumentation, and data analysis methods introdue systemati unertainties in the data set. However, the most likely soure of bias in the sample is that some flat light urves may not have been published. If this is the ase, our onlusion that the amplitude distributions of KBOs and MBAs are different would be strengthened. On the other hand, if most unreported nondetetions orrespond to smaller KBOs then the inferred ontrast in the shape distributions of different-sized KBOs would be less signifiant. Clearly, better observational onstraints, partiularly of smaller KBOs, are neessary to onstrain the KBO shape distribution and understand its origin The Inner Struture of KBOs In this setion we wish to investigate if the rotational properties of KBOs show any evidene that they have a rubble pile struture; a possible dependene on objet size is also investigated. As in the ase of asteroids, ollisional evolution may have played an important role in modifying the inner struture of KBOs. Large asteroids (D k 200 km) have in priniple survived ollisional destrution for the age of the solar system, but may nonetheless have been onverted to rubble piles by repeated impats. As a result of multiple ollisions, the loose piees of the larger asteroids would have reassembled into shapes lose to triaxial equilibrium ellipsoids ( Farinella et al. 1981). Instead, the shapes of the smaller asteroids (D 100 km) are onsistent with ollisional fragments (Catullo et al. 1984), indiating that they are most likely by-produts of disruptive ollisions.