Sheppard et al.: Photometric Lightcurves 129 Photometric Lightcurves of Transneptunian Objects and Centaurs: Rotations, Shapes, and Densities Scott S. Sheppard Carnegie Institution of Washington Pedro Lacerda Grupo de Astrofisica da Universidade de Coimbra Jose L. Ortiz Instituto de Astrofisica de Andalucia We discuss the transneptunian objects and Centaur rotations, shapes, and densities as deter- mined through analyzing observations of their short-term photometric lightcurves. The light- curves are found to be produced by various different mechanisms including rotational albedo variations, elongation from extremely high angular momentum, as well as possible eclipsing or contact binaries. The known rotations are from a few hours to several days with the vast majority having periods around 8.5 h, which appears to be significantly slower than the main- belt asteroids of similar size. The photometric ranges have been found to be near zero to over 1.1 mag. Assuming the elongated, high-angular-momentum objects are relatively strengthless, we find most Kuiper belt objects appear to have very low densities (<1000 kg m–3) indicating high volatile content with significant porosity. The smaller objects appear to be more elongated, which is evidence of material strength becoming more important than self-compression. The large amount of angular momentum observed in the Kuiper belt suggests a much more numer- ous population of large objects in the distant past. In addition we review the various methods for determining periods from lightcurve datasets, including phase dispersion minimization (PDM), the Lomb periodogram, the Window CLEAN algorithm, the String method, and the Harris Fourier analysis method. 1. INTRODUCTION show the primordial distribution of angular momenta ob- tained through the accretion process while the smaller ob- The transneptunian objects (TNOs) are a remnant from jects may allow us to understand collisional breakup of the original protoplanetary disk. Even though TNOs may TNOs through their rotations and shapes. be relatively primitive, their spins, shapes, and sizes from Like the rotations, the shape distribution of TNOs is the accretion epoch have been collisionally altered over the probably also a function of their size. The largest TNOs age of the solar system. The rotational distribution of the should be dominated by their gravity with shapes near their TNOs is likely a function of their size. In the current Kuiper hydrostatic equilibrium point. The smaller TNOs are prob- belt the smallest TNOs (radii r < 50 km) are susceptible to ably collisional fragments with self-gravity being less im- erosion and are probably collisionally produced fragments portant, allowing elongation of the objects to dominate their (Farinella and Davis, 1996; Davis and Farinella, 1997; lightcurves. Bernstein et al., 2004). These fragments may have been The main technique in determining the rotations and disrupted several times over the age of the solar system, shapes of TNOs is through observing their photometric var- which would have highly modified their rotational states iability (Sheppard and Jewitt, 2002; Ortiz et al., 2006; La- (Catullo et al., 1984). Intermediate-sized TNOs have prob- cerda and Luu, 2006). For the largest TNOs most photo- ably been gravitationally stable to catastrophic breakup but metric variations with rotation can be explained by slightly are likely to have had their primordial rotations highly in- nonuniform surfaces. Two other distinct types of lightcurves fluenced through relatively recent collisions. The larger stand out in rotation period and photometric range space TNOs (r > 100 km) have disruption lifetimes longer than for the largest TNOs (r > 100 km). The first type of light- the age of the solar system and probably have angular curve [examples are (20000) Varuna and (136108) 2003 momentum and thus spins that were imparted during the EL61] have large amplitudes and short periods that are in- formation era of the Kuiper belt. Thus the largest TNOs may dicative of rotationally elongated objects near hydrostatic 129 130 The Solar System Beyond Neptune equilibrium (Jewitt and Sheppard, 2002; Rabinowitz et al., and 2006). The second type, of which only 2001 QG298 is a M (n – 1)s2 member to date, show extremely large amplitude and slow ∑ = j j 2 j 1 rotations and are best described as contact binaries with sim- s = M ∑ (nj – M) ilar-sized components (Sheppard and Jewitt, 2004). j = 1 Two objects, (19308) 1996 TO66 and (24835) 1995 SM55, may have variable amplitude lightcurves, which may result N is the number of observations, xi are the measurements, from complex rotation, a satellite, cometary effects, a recent x is the mean of the measurements and sj are the variances collision, or most likely phase-angle effects (Hainaut et al., of M distinct samples. The samples are taken such that all φ φ 2000; Sheppard and Jewitt, 2003; Belskaya et al., 2006; see the members have a similar i, where i is the phase corre- also chapter by Belskaya et al.). sponding to a trial period. Usually the phase interval (0,1) is This chapter is organized as follows: In section 2 we dis- divided into bins of fixed size, but the samples can be cho- cuss how rotation periods are determined from lightcurve sen in any other way that satisfies the criterion mentioned observations and the possible biases involved. In section 3 above. the possible causes of the detected lightcurves are consid- The Lomb method (Lomb, 1976) is essentially a modified ered. In section 4 we mention what the measured lightcurves version of the well-known Fourier spectral analysis, but the may tell us about the density and composition of the TNOs. Lomb technique takes into account the fact that the data are In section 5 we look at what the shape distribution of the unevenly spaced and therefore the spectral power is “nor- TNOs looks like when assuming that elongation is the rea- malized” so that it weights the data in a “per point” basis son for the larger lightcurve amplitudes and double-peaked instead of on a “per time” interval basis. The Lomb-normal- ω rotation periods. In section 6 we discuss what the observed ized spectral power as a function of frequency PN( ) is angular momentum of the ensemble of TNOs may tell us 2 about the Kuiper belt’s past environment. Finally, section 7 – ω τ ∑ (hj – h)cos (tj – ) examines possible correlations between spin periods and ω 1 j + PN( ) = σ2 2ω τ amplitudes, and the dynamical and physical properties of 2 ∑ cos (tj – ) the TNOs. j (2) 2 – ω τ ∑ (hj – h)sin (tj – ) 2. ANALYZING LIGHTCURVES j 2ω τ ∑ sin (tj – ) 2.1. Period-Detection Techniques j where ω is angular frequency (2πf), σ2 is the variance of the – There are currently several period-detection techniques data, h is the mean of the measurements, hj and tj are the such as phase dispersion minimization (PDM) (Stellingwerf, measurements and their times, and τ is a kind of offset that ω 1978), the Lomb periodogram (Lomb, 1976), the Window makes PN( ) independent of shifting all the ti by any con- CLEAN algorithm (Belton and Gandhi, 1988), the String stant. Quantitatively, τ is such that method (Dworetsky, 1983) and a Fourier analysis method ωτ ωτ ωτ (Harris et al., 1989) that can be efficiently used to fit as- tan(2 ) =∑ sin2 j/∑ cos2 j j j teroid lightcurves. All these techniques are suitable to data that are irregularly spaced in time. Although the photometry In this method, the best period is the one that maximizes data of TNOs are usually unevenly sampled, the sampling the Lomb-normalized spectral power. times are not random. This results in what is usually called The String method (e.g., Dworetsky, 1983) finds the best “aliasing problems” (see section 2.3). period by searching for the period that minimizes a parame- The PDM method (Stellingwerf, 1978) is especially ter that can be regarded as a length suited to detect periodic signals regardless of the lightcurve n shape. The PDM method searches for the best period that L = ∑[(m – m )2 + (Φ – Φ )2]1/2 + Θ i i – 1 i i – 1 minimizes a specific parameter . This parameter measures i = 1 (3) the dispersion (variance) of the data phased to a specific [(m – m )2 + (Φ – Φ + 1)2]1/2 period divided by the variance of the unphased data. There- i n i n Φ fore, the best period is the one that minimizes the disper- where i are the phases from a trial period, mi are the exper- sion of the phased lightcurve. imental values, and n is the number of observations. The Window CLEAN algorithm (Belton and Gandhi, Θ = s2/σ2 (1) 1988) is a special application of the CLEAN algorithm (well known to radio astronomers, as it is widely used to synthe- where size images when dealing with synthetic aperture data). In N the case of time series analysis, its application is made by σ2 x 2 = ∑ (xi – ) /(N – 1) computing a window function (which takes into account the i = 1 observing times having zero value at the times when no data Sheppard et al.: Photometric Lightcurves 131 were taken and 1 for the rest). This window function is used etry values within the range observed at exactly the same to deconvolve the true signal by means of the CLEAN al- times as each data point was taken. The simulated datasets gorithm. In other words, the window function is used to can be analyzed with the particular technique that the au- generate sort of a filter in the frequency domain to be ap- thor has chosen and one can generate a distribution of val- plied to the regular Fourier spectrum in order to smooth out ues for, e.g., maximum Lomb-normalized spectral power the spectral power of the signals that arise from the sam- or a distribution of values of minimum Θ or a distribution pling pattern.