LIGHT CURVES OF 32 LARGE TRANSNEPTUNIAN OBJECTS

With discovery of the first Transneptunian object (TNO) in 1992 (Jewitt et al. 1993) a new area of study on small bodies in our solar system was opened. Most of these objects are located in the Kuiper Belt beyond the orbit of Neptune. The dynamically unstable Centaurs are found between the orbits of Jupiter and Neptune. More than 1600 of an expected 10⁵ objects (larger than ∼100 km; Petit et al. 2011) have been observed and recorded in the Minor Planet Center (MPC) database. Of these, about half have orbits with small enough uncertainties to be observed from typical ground-based telescope facilities. Within the TNO population as a whole, objects are clustered in identifiable dynamical locations with respect to their interaction, or lack of interaction, with Neptune, or in areas near the plane of the solar system (Elliot et al. 2005; Lykawka & Mukai 2007; Gladman et al. 2008). In this paper we use the classification system defined by the Deep Ecliptic Survey (DES; Elliot et al. 2005) and for statistical purposes combine objects in scattered orbits as described. In short, cold classical objects are TNOs with low eccentricity, circular orbits, inclinations less than 5°–12° (Noll et al. 2008b; Elliot et al. 2005; Peixinho et al. 2008; for the statistical analyses in this paper we use 55), and no previously traced interaction with Neptune. Resonance objects are in mean-motion resonances with Neptune. Scattered objects consist of Centaurs (objects with orbits inside of Neptune), scattered disk objects (objects with large inclinations and eccentricities, with perihelia beyond 30 AU that are not resonant or cold classical objects), and detached objects (objects with moderate to high eccentricities whose perihelia are sufficiently far (>40 AU) from Neptune so they are not influenced by Neptune). Because of our broad grouping of objects for statistical analysis, the distinctions between the three classification systems is not significant.

The photometric variability of a TNO versus time, its light curve, is a powerful tool for learning about the shapes and surface features of these distant objects. Light curve amplitude, rotation period, color dependence, and shapes of light curves are all affected by the details the TNO's physical properties. The spin state of these bodies is important because it records the history of collisional and other evolutionary processes acting in the Kuiper Belt over time and at the extremes, can provide constraints on the material properties (and interior structure) of these objects. Disruption lifetimes for the largest objects (d ⩾ 400 km) are longer than the age of the solar system, so we expect that the rotations of these objects are the results of impacts during the formation era of the Kuiper Belt (Lacerda 2005). Objects with d ∼ 200 km have probably avoided catastrophic break-up, although their rotations could be modified by more recent collisions (Davis & Farinella 1997). The smallest objects (d ⩽ 100 km) are likely fragments resultant from multiple collisions over the age of the solar system (Catullo et al. 1984; Lacerda 2005; see Campo & Benavidez 2012 for a review of the collisional environment spanning the range of TNO sizes). Likewise, TNO shape is also thought to be related to object size, with the largest objects being dominated by self-gravity and the smallest objects being collisional fragments without significant self-gravity influence.

In this paper we present light curve work carried out on 32 bright TNOs accessible from Las Campañas Observatory. Most of the objects in our target list were discovered as part of recent large area surveys conducted for bright outer solar system objects in the Southern Hemisphere (Sheppard et al. 2011; Rabinowitz et al. 2012). Other objects were selected from newly discovered TNOs in the MPC database with the requirement that they have absolute magnitudes, H_V, brighter than ∼5.6 (corresponding to a size of d ∼ 330 km assuming an albedo of 0.1). We also observed binaries bright enough for our telescope-instrument combination (m_R < 23.0). The survey includes five binaries and four Haumea family objects.

Observations were made using the Irénée du Pont 2.5 m telescope at the Las Campañas Observatory during a series of eight 3–6 night runs (in 2007 and 2011–2012, Table 1). Images were collected using the direct CCD camera, a SITe2K chip with a pixel scale of 0259 pixel⁻¹ and a field of view of 8.85 square arcmin. The data were collected in pairs or triplets for 300 s in duration per exposure using the entire array and reading the chip in "fast" mode with a readout time of ∼67 s. Observations for each TNO were collected on multiple nights within a single run with approximately 45–75 minute spacing between visits. The runs were scheduled in pairs four to eight weeks apart so that light curve periods could be de-aliased from the 24 hr observing cycle within an observing run. Light curve data were collected using both the Sloan r ' and Bessel R (Kron-Cousins equivalent) filters. Individual images have S/N of ⩾30 (uncertainties ⩽0.03 mag), depending on the TNO magnitude and observing conditions. Photometric standard star data for overlapping stars from the Landolt (1992) and Sloan (Smith et al. 2002) surveys with TNO-like colors [104_428, 113_260, 113_339, RU_149B, and RU_149F] were collected during photometric nights. We also collected colors for some of the TNOs during one photometric night of observation bracketing the Sloan g ' and Sloan i ' measurements with the r-band light curve filter (e.g., Sloan r ' – Sloan g ' – Sloan r ' – Sloan i ' – Sloan r ') to ensure that variations did not unknowingly influence the colors. The conditions for each night are listed in Table 1.

For each run, a set of master biases, twilight sky flats, and dome flats were generated to calibrate the data using standard routines from Dr. Marc Buie's IDL libraries (http://www.boulder.swri.edu/~buie/idl/). The data was flattened using sky flats for pointings with hour angles less than or equal to 1.0 hr east, and using dome flats for pointings with hour angles greater than 1 hr east; this crossover point was empirically determined to provide the most consistent sky backgrounds across the chip. Each image was evaluated and the position of the TNO was identified by eye. Likewise, astrometry was carried out on each image, and on each star in the image with a peak pixel count <50,000 (the saturation level for the detector). For each object, three astrometric measurements (the first, last, and an observation in the middle of the sequence) were submitted to the MPC. Large (∼4 times the FWHM) and small (∼FWHM) aperture photometry was carried out on all the stars identified on the image and saved to a file. Small aperture photometry was carried out on the TNO.

The star list was then culled in the following way to find the best comparison stars for each TNO: (1) the star was identified on each night of a single run, (2) the star was bright compared to the TNO (typically 3–4 mag brighter than the TNO and with a peak pixel between 20,000–30,000 counts, well below the nonlinear regime of the detector), (3) when plotted versus time the star had no significant variations (the star was differenced by a selection of other stars of similar magnitude on the field to account for airmass and sky fluxuations, if the resulting difference varied by less then 3σ it was considered useable), and (4) the star was located relatively close to the TNO on the chip on at least one night of observation. After 10–15 comparison stars were identified each was evaluated by eye to ensure that it was not contaminated by a background source or itself a galaxy, typically a few of the stars were thrown out. Since TNOs move during the night relative to the background field stars these images were also examined versus time to look for potential background contaminates. Images with background contamination due to faint galaxies, or nearby stars were eliminated. Most of the TNOs were in average density star fields, with the TNOs moving through clear areas, so few images were excluded (the fields were pre-selected as much as possible to avoid background object overlaps so this issue affects ⩽5% of the data).

The selected field stars were photometrically calibrated using the standard star observations. If the sky was photometric for multiple nights in a given run, the calibrated star magnitudes were averaged across all the nights the field was observed. The scatter in the calibrated magnitudes was comparable to the photometric precision of the data and the calibrated magnitude uncertainties were small compared to the uncertainties on the TNO measurements themselves. To minimize uncertainties on the TNO photometry, the small aperture TNO magnitude for each frame was aperture corrected (Howell 2006) with the selected small and large aperture field star magnitudes. Finally, the TNO magnitude was calibrated with a magnitude calibrated field star. For data collected in the Bessel R filter we apply a magnitude transformation of +0.202 mag as determined using the synphot routine¹ in IRAF (Laidler et al. 2008) so that all values are based in the Sloan r ' system.

In order to combine data across runs, the TNO magnitudes were geometrically corrected to unit distance from the Sun (r) and Earth (Δ) using the formalism $H_{r^{\prime}} (\alpha) = R_{\rm obs} (\alpha) - 5\log (r\Delta)$, where α is the phase angle of the observation and R_obs is the observed Sloan r ' magnitude. In principle, one would also want to correct the magnitudes to a phase angle of 0°, however, phase curves have not been measured for the majority of our objects and more than half of our objects span <03 in phase angle difference between measurements. However, for a few objects when we plotted all the data (magnitude versus time) for that object, we noticed some large-ish (0.2–0.3 mag) offsets from earlier runs which observed the same objects (primarily data from the 2012 May dataset which was collected under constantly varying conditions); some of these offsets we thought could be explained by geometry, others could not. Therefore, we period-fit our data considering three different magnitude calculations: (1) geometric correction excluding the phase coefficient as expressed earlier in this paragraph, (2) applying the Bowell et al. (1989) formalism whose equations are given in the table note of Table 2, and (3) normalizing the values on each night by the mean magnitude of the object.

4.1. Analysis

Table 2 provides detailed observational and geometric circumstances for each TNO on each night of observation; the full table is provided electronically. It records the mid-time (in both calendar format and JD) of the observations, the magnitude range of the observations, the number of observations, the duration of time (hr) covered by the observations, heliocentric distance (AU), geocentric distance (AU), phase angle (°), light time (s) and two magnitudes ($H_{r^{\prime},\,{\rm Bowell}}$ which uses the Bowell et al. 1989 formalism and $H_{r^{\prime}}$(α), the reduced magnitude uncorrected for phase angle) for each object, each night. Table 3 provides basic characteristic information about each object and a summary of our light curve conclusions from all of our observations combined. We ran the dataset for each object through a set of periodogram analyses (described in the following paragraph), for periods ranging from 3 to 20 hr based on our observing windows and aliasing considerations. Some of the resulting phasing or suggested periods were not statistically significant, and instead of making a "best guess" which might be carried forward in the literature incorrectly, we choose instead to place an upper limit on the amplitude of a possible light curve for the duration of a single night of observation (⩽8 hr) by averaging the range on all nights of observation. Individual uncertainties on the measurements are ∼0.03 mag or better. Individual measurements and plots of the data versus time on each night of observation can be found in online Figures 1.1–1.32. We include in Table 4 and Figure 1 samples of the online data table and figures which include values for all our objects, regardless if they resulted in conclusive period fits, or flat light curves.

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Table 2. du Pont Observation Details

Object	Calendar Datea	JDa	Δmb	Nobs	Δtc	r	Δ	α	L-Time	Hr',Bowelld	Hr'(α)e
		(mid-time)			(hr)	(AU)	(AU)	(°)	(s)
79360	2011 Mar 9 02.09829	2455629.58743	0.12	8	2.57	43.501	42.704	0.786	355.152	5.10	5.23
79360	2011 Mar 10 01.77201	2455630.57383	0.11	8	2.47	43.501	42.714	0.804	355.237	5.18	5.31
79360	2011 Mar 11 01.70889	2455631.57120	0.18	8	2.59	43.501	42.724	0.823	355.324	5.15	5.28
79360	2011 Mar 12 01.21948	2455632.55081	0.18	6	1.44	43.501	42.735	0.841	355.413	5.11	5.25
79360	2011 Mar 13 01.62795	2455633.56783	0.17	8	2.43	43.501	42.746	0.859	355.505	5.00	5.14
79360	2012 Mar 18 01.64508	2456004.56855	0.28	6	2.35	43.496	42.793	0.935	355.893	4.94	5.09
79360	2012 Mar 19 02.03244	2456005.58469	0.05	6	3.62	43.496	42.805	0.951	355.993	5.10	5.25
79360	2012 Mar 20 01.14035	2456006.54751	0.38	9	2.18	43.496	42.817	0.967	356.096	5.27	5.42
79360	2012 Mar 21 01.14416	2456007.54767	0.12	9	2.28	43.496	42.830	0.983	356.201	5.07	5.22
79360	2012 Mar 22 01.05568	2456008.54399	0.13	9	2.17	43.496	42.843	0.998	356.307	5.01	5.16

Notes. ^aMid-time of exposure sequence, not corrected for light time (calendar date is the same as mid-time in readable format for convenience). ^bRange of object in Sloan_r' magnitude space during observations on each night. ^cDuration of observations per night. ^d$H_{r^{\prime},\,{\rm Bowell}}$ is the absolute magnitude calculated using the formalism of Bowell et al. (1989): H = H_obs(α) − 5 log(rΔ) + 2.5 log[(1 − G)Φ₁(α) + GΦ₂(α)], where G = 0.15 ${\rm \Phi }_1 (\alpha) = e^{[ { - 3.33\, {\rm tan}({0.5\alpha })^{0.63} }]}$ and ${\rm \Phi }_2 (\alpha) = e^{[ { - 1.87\,{\rm tan}({0.5\alpha })^{1.22} }]}$ ^e$H_{r^{\prime}}$(α) is the reduced magnitude at unit distance from the Sun and the Earth, uncorrected for phase angle: $H_{r^{\prime}} (\alpha) = R_{\rm obs} (\alpha) - 5{\rm log}({r} \Delta)$.

Only a portion of this table is shown here to demonstrate its form and content. Machine-readable and Virtual Observatory (VO) versions of the full table are available.

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Table 3. Orbit Information and Light Curve Results

Number	Designation	Classa	〈i〉b	Hr',Bowell	Single/Double	Peak–Peak	Upper Limit	No. of Nights	Notes, Other
			(°)		Peaked Period	Amplitude	on Amplitude	Observed	possible periods
					(hr)	(mag)	(mag)		(hr)
79360	1997CS29	CL	3.84	5.09 ± 0.09	...	...	<0.17	10	Binary, data consistent with
									a light curve period equal
									to the mutual orbit period,
									150.12/300.24 hours
119951	2002KX14	CL	2.82	4.46 ± 0.01	...	...	<0.05	3	...
120178	2003OP32	SD	27.02	3.82 ± 0.01	4.85/9.71	0.18 ± 0.01	...	6	Haumea, 6.09/12.18
120347	2004SB60	SD	25.57	3.97 ± 0.04	...	...	<0.04	4	Binary, Haumea
145452	2005RN43	SD	19.45	3.47 ± 0.01	6.95/13.89	0.06 ± 0.01	...	4	9.73/19.46
145453	2005RR43	SD	27.03	3.86 ± 0.01	...	...	<0.06	5	Haumea
174567	2003MW12	SD	21.24	3.18 ± 0.02	...	...	<0.04	4	Binary
225088	2007OR10	10:3	34.72	1.76 ± 0.01	...	...	<0.09	7	...
278361	2007JJ43	SD	13.45	4.17 ± 0.20	6.04/12.09	0.13 ± 0.02	...	6	4.83/9.66
303712	2005PR21	CL	2.69	5.96 ± 0.02	...	...	<0.28	2	Binary
303775	2005QU182	SD	12.82	3.36 ± 0.05	9.61/19.22	0.12 ± 0.02	...	6	...
305543	2008QY40	SD	24.02	5.19 ± 0.06	...	...	<0.15	4	...
307251	2002KW14	SD	8.48	5.50 ± 0.03	6.63/13.25	0.25 ± 0.03	...	5	5.23/10.46
312645	2010EP65	2:1	19.32	5.23 ± 0.15	7.48/14.97	0.17 ± 0.03	...	7	5.77/11.54; 8.45/16.90
...	2005EF298	CL	1.60	5.73 ± 0.05	4.82/9.65	0.31 ± 0.04	...	3	Binary, 6.06/12.13
...	2006HJ123	3:2	15.62	5.92 ± 0.03	...	...	<0.13	4	...
...	2007JF43	3:2	14.90	5.24 ± 0.05	4.76/9.52	0.22 ± 0.02	...	4	...
...	2007JH43	SD	15.09	4.49 ± 0.05	...	...	<0.08	6	...
...	2009YE7	SD	28.00	4.26 ± 0.01	...	...	<0.20	4	Haumea, multiple periods
...	2010EK139	7:2	29.13	3.89 ± 0.04	3.53/7.07	0.12 ± 0.02	...	7	...
...	2010EL139	3:2	23.99	5.23 ± 0.04	3.16/6.32	0.15 ± 0.03	...	6	...
...	2010ER65	SD	21.58	4.69 ± 0.07	...	...	<0.16	4	...
...	2010ET65	SD	30.00	4.96 ± 0.11	3.94/7.88	0.13 ± 0.02	...	6	...
...	2010FX86	SD	26.60	4.34 ± 0.04	7.90/15.80	0.26 ± 0.04	...	8	...
...	2010HE79	3:2	15.04	5.06 ± 0.04	9.75/19.49	0.11 ± 0.02	...	5	...
...	2010KZ39	SD	25.08	4.03 ± 0.01	...	...	<0.17	3	...
...	2010PU75	SD	7.42	5.80 ± 0.07	6.19/12.39	0.27 ± 0.03	...	4	4.91/9.82
...	2010RF43	CN	34.52	3.54 ± 0.04	...	...	<0.08	7	...
...	2010RO64	SD	15.86	4.84 ± 0.02	...	...	<0.16	3	...
...	2010TY53	CN	21.56	5.34 ± 0.03	...	...	<0.14	4	...
...	2010VK201	SD	27.72	4.40 ± 0.07	3.79/7.59	0.30 ± 0.02	...	8	3.28/6.55
...	2010VZ98	SD	3.46	4.81 ± 0.04	...	...	<0.18	4	4.86/9.72, just below
									significance criterion

Notes. ^aUsing the formalism of Elliot et al. (2005): CL = cold classical, CN = Centaur, SD = scattered disk, M:N = Mean Motion Resonance with Neptune. ^bThe mean heliocentric inclination based on a 10 My integration of the orbit of the object; used for identifying an object as cold classical.

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Table 4. Individual Magnitudes

Object	Filter	Julian datea	mr'b	σr'b	rc	Δc	αc	L-Time	Hr',Bowelld	Hr'(α)e
					(AU)	(AU)	(°)	(s)
2005EF298	Sloan_r	2456005.54549	21.927	0.027	40.655	39.715	0.469	330.299	6.796	6.892
2005EF298	Sloan_r	2456005.59497	21.757	0.021	40.655	39.716	0.470	330.301	6.626	6.722
2005EF298	Sloan_r	2456005.59925	21.766	0.021	40.655	39.716	0.470	330.301	6.635	6.731
2005EF298	Sloan_r	2456005.60353	21.804	0.022	40.655	39.716	0.470	330.301	6.673	6.769
2005EF298	Sloan_r	2456005.67499	21.985	0.026	40.655	39.716	0.472	330.305	6.853	6.950
2005EF298	Sloan_r	2456005.67926	21.970	0.025	40.655	39.716	0.472	330.305	6.838	6.935
2005EF298	Sloan_r	2456005.68354	21.965	0.024	40.655	39.716	0.472	330.305	6.833	6.930
2005EF298	Sloan_r	2456005.74765	22.099	0.028	40.655	39.716	0.473	330.308	6.967	7.064
2005EF298	Sloan_r	2456005.75193	22.019	0.027	40.655	39.716	0.474	330.308	6.887	6.984

Notes. ^aJulian date at the beginning of the exposure, not corrected for light-time. ^bApparent Sloan r ' magnitude and corresponding uncertainty on the individual measurement. ^cHeliocentric distance, geocentric distance, phase angle, and light time, respectively. ^d$H_{r^{\prime},\, {\rm Bowell}}$ is the absolute magnitude calculated using the formalism of Bowell et al. (1989): H = H_obs(α) − 5 log (rΔ) + 2.5 log[(1 − G)Φ₁(α) + GΦ₂(α)] where G = 0.15 ${\rm \Phi }_1 (\alpha) = e^{[ { - 3.33\, {\rm tan}({0.5\alpha })^{0.63} }]}$ and ${\rm \Phi }_2 (\alpha) = e^{[ { - 1.87\,{\rm tan}({0.5 \alpha })^{1.22} }]}$ ^e$H_{r^{\prime}}$(α) is the reduced magnitude at unit distance from the Sun and the Earth, uncorrected for phase angle: $H_{r^{\prime}} (\alpha) = R_{\rm obs} (\alpha) - 5\,{\rm log}\,(r\Delta)$.

Only a portion of this table is shown here to demonstrate its form and content. Machine-readable and Virtual Observatory (VO) versions of the full table are available.

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For objects with data to which we can fit periods with statistically significant results, we present in Figure 2 the Lomb–Scargle periodogram (top; Scargle et al. 1982) and the data phased to the most significant period assuming single- (middle) and double- (bottom) peaked light curve interpretations. For each dataset we calculate the 99.9% confidence level using the methods described by Horne & Baliunas (1986) and plot the result as a dashed line on each of the periodogram plots. Very similar results are found when using the phase-dispersed minimization (PDM; Stellingwerf 1978) and Harris–Foster (Harris et al. 1989; Foster 1995) fitting techniques. For objects with multiple peaks above the confidence interval we phased the data to each possible period for evaluation, but in all cases the resulting period was the one with the highest Lomb-normalized spectral power value. We list in the notes column of Table 3 possible additional periods that could fit the data (that we were unable to completely rule out), although we run all further analyses with our preferred periods, listed in the specified period column. We fit all three sets of magnitude calculations for each object. We draw our conclusions based on the night normalized data which minimizes any offsets between runs, although it could introduce a bias against objects observed for short periods of time with long period >8 hr rotation curves. Fits on the non-phase-corrected and Bowell-corrected values were also carried out and, where offsets were not an issue, the same periods were identified. Offsets were on order 0.1–0.2 mag, which are within the range of what one would expect for phase corrections. However, we did not sample a complete range of phases for all our objects, only for some objects (see Section 6). For consistency we analyze all the light curves using the night normalized data.

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4.2. This Sample

4.3. Axis Ratio

In addition to rotation period, we can use the amplitude of variation to estimate the sphericity of our objects. Our object sample ranges from H_V = 2–6.1 (or d ∼ 1600–250 km using the formalism of Bowell et al. (1989) and assuming an albedo of 0.1), and includes objects from both the spherical and elongated groups. We expect the smaller objects to be elongated, having double-peaked light curves where nominally we can sample the long and short axes of the object twice over a full rotation. If we assume such an object to be triaxial with semi-major axes a ⩾ b ⩾ c in rotation about the c-axis, the minimum and maximum flux of the rotation curve measured in magnitudes, Δm, can be used to determine the projection of the body shape (i.e., how spherical the object is) into the plane of the sky:

$$\begin{equation} \Delta m = 2.5\log \frac{a}{b} - 1.25\log \left({\frac{{a^2 \cos ^2 \theta + c^2 \sin ^2 \theta }}{{b^2 \cos ^2 \theta + c^2 \sin ^2 \theta }}} \right), \end{equation} \tag{ 1 }$$

where θ is the angle at which the rotation axis is inclined to the line of sight (an object with θ = 90° is being viewed equatorially; Binzel et al. 1989). If we assume that we are in fact viewing the object equatorially, then this equation can be rearranged to give the axis ratio, a/b = 10^0.4Δm. The axis ratio for our sample ranges from 1.03 to 1.33 with an average of 1.16.

4.4. Previous Results

In the following discussion we include results from this work and that collected from the literature out of the references listed in the caption of Figure 3. We use absolute magnitudes H_V from the MPC so as to interpret all objects in a consistent manner relative to intrinsic brightness which can be used as a proxy for size. One method of estimating an effective diameter for these objects is to follow the formalism of Bowell et al. (1989), where $d = 10^{\left({(6.259 - 0.4 \cdot H_V - \log \rho)/2} \right)} \;{\rm km}$ and ρ is albedo. It is known that TNO albedos range from ∼0.04 to ∼0.8 (Stansberry et al. 2008). Because the objects in our sample are relatively new discoveries, they do not yet have measured albedos. For our calculations we work in H_V space, but to give a general reference point on size for Figures 3 and 5 we assume ρ = 0.1. Figure 3 shows a histogram of absolute magnitude for the entire sample of published TNO light curves sub-divided by dynamical classification, with the sample for this work extracted in a separate plot. The majority of objects sampled thus far are from the dynamically scattered population since these objects tend to be intrinsically bright and more easily observed from smaller telescopes.

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In Figure 4, we plot the cumulative number of objects with respect to light curve amplitude normalized by the total number of objects in the sample for the entire measured population and also for the sample in this work alone using our fitted amplitudes or upper limits. The solid points/black line shows the results for the full sample, the triangles/red line for the classical objects, the squares/gray line for the resonant objects and the diamonds/blue line for the scattered objects. In both samples the scattered objects typically have smaller amplitude light curves then the larger population. However, scattered objects at all sizes are measured, so size is not the sole explanation for this effect.

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In Figure 5 we plot rotation period and amplitude versus absolute magnitude, and find that objects fainter than about H_V ∼ 5 appear to have larger amplitude light curves. Using the entire sample, a Spearman rank correlation test (Table 5) between absolute magnitude and light curve amplitude indicates this to be a 3σ result. We also see an indication of correlation between absolute magnitude and single period rotation curves, although this is less than a 3σ result and there is also some ambiguity between single and double peaked light curve interpretations. We consider correlations with orbital properties, but do not find any statistically significant results. If we run the same analysis for only the binaries or Haumea family objects (discussed in Sections 4.4 and 4.5), we also do not find significant correlations. It appears that size has a greater influence over the rotational properties of an object than one particular orbital characteristic, unless there has been obvious interaction with another TNOs as in the case of tidally locked binaries.

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Figure 5 also demonstrates that, with the exception of tidally locked objects, the rotation rate for the majority of measured objects is less than 13/26 hr (single-peak/double-peak interpretations), with the mean rotation periods being 6.73 hr and 11.30 hr, respectively (excluding Pluto/Charon and Sila/Nunam). The scatter is relatively small. Modeling by Lacerda (2005) suggests that such a spin distribution is indicative of some level of anisotropic accretion in the early Kuiper Belt.

However, if we compare the rotational properties of objects to each other with respect to dynamical class, binary, or family status (Table 6) we find a 95.60% probability that the classical and scattered object amplitudes come from different distributions and a 94.64% probability that the classical and resonant object amplitudes come from different distributions. If we combine the scattered and resonant objects the significance of the difference increases to 97.05%. We also investigated if our result was significantly affected by the absolute magnitude range of the sample. We could not test for samples brighter than an absolute magnitude of 6.5 because the sample size for the classical objects is too small, however, we plot in Figure 6 the percent results versus absolute magnitude for samples from 6.5 to 12 (the faintest object) in steps of 0.5 mag. The amplitude distinction we find is strongest for all three samples (classical/resonant; classical/scattered and classical/scattered+resonant) with an absolute magnitude limit brighter than 7.0; however, it is strong in all magnitude bins and we believe that the effect is real, not size-dependent. So while one particular orbital element does not dominate, it does appear that location within the belt plays a significant role in rotation properties. Perhaps objects in more stirred-up regions have experienced greater interaction with other bodies, but the amplitudes of larger objects are disproportionately affected by smaller collisions. There are not enough objects to compare within the binary or family populations themselves. We present all the correlations we considered in Table 6 for completeness.

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5.1. Binaries

Binaries are found throughout the Transneptunian belt, though in significantly higher fractions among the cold classical population ($29_{ - 6}^{ + 7}$% versus $5.5_{ - 2}^{ + 4}$% for all other classes combined; Noll et al. 2008a, 2008b). A number of formation mechanisms for these systems have been proposed including: (1) physical collisions (Weidenschilling 2002; Canup 2005, 2011), (2) gravitational interactions (Goldreich et al. 2002; Astakhov et al. 2005; Lee et al. 2007; Funato et al. 2004), and (3) gravitational collapse (Nesvorny & Vokrouhlicky 2006). Each of these mechanisms has the potential to influence the rotational properties of these objects, in addition to the tidal and orbital interactions between the binary objects themselves. Tidal interactions can have the effect of synchronizing the rotation period of an object with the mutual binary orbit, as is well known in the case of the (134340) Pluto/Charon system (Tholen & Tedesco 1994). Grundy et al. (2012) and this work also support the interpretation that (79360) Sila/Nunam is a tidally locked system. There are currently 24 binaries with light curve measurements and/or estimates.

Our sample includes five binary objects and, as mentioned in Section 4.1, two of these objects have the largest variations and are the smallest objects in our sample. The two binaries that do not show significant variation are the brightest of our binary sample, consistent with the idea that smaller objects have larger amplitude light curves. In Figure 7 we plot the light curve characteristics of all the binaries in the literature (references can be found in the figure caption). In a statistical sense, the sample is still small, however we note that the same low amplitude characteristic of the scattered objects is seen. The cold classical objects have the largest amplitudes with the exception of the resonant object 2001 QG₂₉₈, whose large amplitude light curve is consistent with a contact binary interpretation (Sheppard & Jewitt 2004; Takahashi & Ip 2004). We find hints that the binary amplitudes as a whole may be slightly larger then the non-binary population, but overall the distributions are similar and we are hesitant to over-interpret the statistics.

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5.2. Haumea Family Objects

In the asteroid belt, much has been learned through study of asteroid families with respect to both dynamical and photometric properties. Modeling the dynamics of collision family members in many cases can trace back a timeframe for when family creation occurred (Nesvorny & Vokrouhlicky 2006). In the Koronis family, spin studies of large numbers of objects of various sizes (Slivan 2002; Slivan et al. 2008, 2009) have demonstrated markedly nonrandom alignments of spin obliquities and correlations with spin rates which in the asteroid belt are interpreted to be thermally driven (Vokrouhlický et al. 2003). The modification of these spin rates is the result of the Yarkovsky (YORP) effect which disproportionately heats non-spherical objects and has the effect of increasing rotation rates of objects; this effect is stronger the smaller the object. YORP is not effective in the Kuiper Belt since objects are too distant from the Sun.

In the Kuiper Belt one dynamical family has been identified through spectroscopic studies (Barkume et al. 2006), and confirmed with dynamical integrations (Ragozzine & Brown 2007). Rotational studies of the largest body, (136108) Haumea, found it to have a rapid rotation, 3.9154 ± 0.0002 hr (double-peaked) with an amplitude of 0.28 ± 0.04 mag, which can be explained as the result of a physical collision (Rabinowitz et al. 2006; Schlichting & Sari 2009; Leinhardt et al. 2010; Lykawka et al. 2012), although Ortiz et al. (2012) argue that such a system could also be created through rotational fission due to collisional spin-up. Studies of the rotation properties of the Haumea family may provide insight for spin properties resultant from a formation mechanism independent from modification by thermal factors. It is possible that small collisions play a roll in rotational modification, but to date this has not been demonstrated. It is believed that large collisions influence the spin properties of the target and the material ejected during such an event (Paolicchi et al. 2002). If the Haumea system is the result of a collision, one might expect the spin properties of resultant family members to be different as a group from the background TNO light curve distribution. Perhaps these objects are more elongated as a result of the collision, or spinning more rapidly (small objects) or more slowly (large objects), depending on the energy of the initial collision.

The numbers are still too small for statistics and many of the smaller objects still need to be studied; however, we present in Figure 8 the current light curve characteristics of nine objects from this work and the literature. All have rotation periods (where measured) ⩽10 hr and distinguishable light curve amplitudes. The mean single-peaked period is 5.60 hr and the mean double-peaked period is 7.39 hr inclusive of Haumea itself. The mean amplitude is 0.14 mag. These values are in comparison with the greater TNO population (Centaurs included, Pluto/Charon and Sila/Nunam excluded) which has a mean single-peaked period of 6.73 hr, a mean double-peaked period of 11.30 hr and a mean amplitude of 0.20 mag. At this point, only the brightest Haumea family objects have been observed. Since most of the amplitudes are small, it is likely that they can be interpreted as spherical objects with surface variations due to albedo features. However, both Haumea and (55636) 2002 TX₃₀₀ are known to have high albedos compared to other TNOs (Lellouch et al. 2010; Elliot et al. 2010; Mommert et al. 2012). If all Haumea family members have high albedos then the objects would be smaller and perhaps these amplitudes are due to object elongation.

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A phase curve describes the brightness of a TNO as a function of its phase angle, the angle made between the Sun, the TNO, and the observer (Earth); for TNOs, the maximum angle is ∼2°. It is linear outside of a few tenths of a degree, but studies of asteroids and moons of the giant planets find nonlinear brightening as the phase gets close to zero (Verbiscer et al. 2005). The surge can be explained by two physical mechanisms, shadow hiding and coherent backscattering; both are related to what is happening at the surface of the TNO. Shadow hiding is the result of hills, boulders, or a mix of light and dark ices on the surface of the object. At low phase angles, no shadows occur and the object appears brighter than at larger phase angles where shadows contribute to the disk-integrated photometry. Coherent backscattering occurs when multiply scattered rays bounce off the surface of the object and follow the same path back to the observer; the light rays add together and a brightening occurs. Near-zero-phase angle light paths interfere more constructively as seen by the observer than at larger phase angles (Schaefer et al. 2009).

Slightly less than one-third of our objects span a large enough region of phase angle space (⩾03) for us to estimate the linear phase coefficient, which can be expressed in flux as ϕ(α) = 10^−0.4βα, where β is the phase coefficient in magnitudes per degree at phase angles, α < 2°. Figure 9 plots our measurements and Table 7 gives the results of our fit for each object. We find an average phase coefficient of β_R = 0.23 mag deg⁻¹ for the objects we can measure (with α > 02), higher than that found by Belskaya et al. (2003), although consistent with some of the individual values listed in Rabinowitz et al. (2007) and Schaefer et al. (2009). (278361) 2007 JJ₄₃, which has the steepest slope and relatively small uncertainty, is measured near a phase angle of 02, close to the region where the opposition surge can have an effect. We do not have any measurements between the two extremes so we suggest these values be used with caution. We do not have albedo and color-phase measurements for these objects, but based on the criteria established in Schaefer et al. (2009) for the phase curve slope, we infer that if we are seeing a surge effect it is most likely due to the coherent backscattering mechanism.

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Additionally, we collected color information on one night for each object. The individual measurements can be found in Table 8 and the cumulative results are plotted in Figure 10. Our objects span the range of TNO colors and are redder than the Sun with the exception of 2009 YE₇, which is known to be a Haumea family member (Trujillo et al. 2011). None of the other objects for which we measured colors are extreme compared to other objects in the Kuiper Belt (Sheppard 2010; Benecchi et al. 2011; Fraser et al. 2012).

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We have presented light curve results for 32 large Southern Hemisphere TNOs. We can fit rotation periods with statistically significant results for 15 of these and place amplitude limits on the light curves of 17 other objects. All of the objects in our sample have amplitudes ⩽0.3 mag with periods ⩽10 hr (assuming a single-peaked interpretation) with the exception of (79360) Sila/Nunam for which our data are consistent with the mutual orbit binary period. We find an average axis ratio of 1.16 suggestive of albedo variations on these objects. Combining light curve results from the literature with our measurements we find a correlation between light curve amplitude and absolute magnitude at the 3σ level, with small objects having larger amplitudes. However, the correlation is not as statistically significant if one considers the binary or Haumea family populations on their own. We suggest that size has a greater influence over the rotational properties of an object than one particular orbital component. We also find that comparison of light curve amplitude with respect to dynamical population results in statistically different distributions between the classical/scattered and classical/resonant populations at the 95.60% and 94.64% levels, respectively, with the classical objects having higher amplitude light curves. This statistic is 97.05% if the scattered and resonant parameters are combined. Perhaps multiple factors are at work, objects in more stirred-up regions (characteristics of binaries demonstrate that the cold classical region has been significantly less perturbed than the scattered and resonant populations in particular with respect to the number of binary systems and their orbital properties; Grundy et al. 2010; Parker et al. 2011) have experienced greater interaction with other bodies and have likewise had their shapes more greatly altered. Large bodies are less susceptible to large scale changes unless collisions are of comparably sized objects. The properties of binary light curves are largely consistent with the greater TNO population except in the case of tidally locked systems. The nine Haumea family objects with measured light curves have rotation periods <10 hr; none have completely flat light curves. It is likely that all of these objects are large enough to be dominated by surface albedo variations; alternatively, if all Haumea family objects have high albedos, then the objects are actually smaller and the light curve properties would be dominated by shape.

This paper includes data gathered with the 2.5 m Irénée du Pont telescope located at Las Campañas Observatory, Chile operated by the Carnegie Institution of Washington. We wish to thank telescope operators Herman Olivares, Sergio Castellón, and Nidia Morrell and telescope support staff Patricio Pinto and Oscar Duhalde. The work for this paper was supported through a Carnegie Fellowship at the Department of Terrestrial Magnetism. We also thank an anonymous reviewer for helpful comments that improved the manuscript.