ROTATIONAL PROPERTIES OF THE HAUMEA
FAMILY MEMBERS AND CANDIDATES:
SHORT-TERM VARIABILITY

Audrey Thirouin, Scott Sheppard, Keith Noll, Nicholas Moskovitz, José Luis Ortiz, Alain Doressoundiram

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To cite this version:

Audrey Thirouin, Scott Sheppard, Keith Noll, Nicholas Moskovitz, José Luis Ortiz, et al. ROTATIONAL PROPERTIES OF THE HAUMEA FAMILY MEMBERS AND CANDIDATES: SHORT-TERM VARIABILITY. The Astronomical Journal, 2016, 151 (6), pp.148. 10.3847/0004-6256/151/6/148. [hal-02461470]

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# ROTATIONAL PROPERTIES OF THE HAUMEA FAMILY MEMBERS AND CANDIDATES: SHORT-TERM VARIABILITY

AUDREY THIROUIN¹, SCOTT S. SHEPPARD², KEITH S. NOLL³, NICHOLAS A. MOSKOVITZ¹,
JOSE LUIS ORTIZ⁴, AND ALAIN DORESSOUNDIRAM⁵

¹Lowell Observatory, 1400 W Mars Hill Road, Flagstaff, AZ 86001, USA; thirouin@lowell.edu

²Department of Terrestrial Magnetism (DTM), Carnegie Institution for Science, 5241 Broad Branch Road NW, Washington, DC 20015, USA

³NASA Goddard Space Flight Center (NASA-GSFC), Greenbelt, MD 20771, USA

⁴Instituto de Astrofísica de Andalucía (IAA-CSIC), Apt 3004, E-18080, Granada, Spain

⁵Observatoire de Paris-LESIA, 5 Place Jules Janssen, Meudon Cedex F-92195, France

Received 2015 October 19; accepted 2016 March 11; published 2016 May 25

## ABSTRACT

Haumea is one of the most interesting and intriguing trans-Neptunian objects (TNOs). It is a large, bright, fast rotator, and its spectrum indicates nearly pure water ice on the surface. It has at least two satellites and a dynamically related family of more than 10 TNOs with very similar proper orbital parameters and similar surface properties. The Haumean family is the only one currently known in the trans-Neptunian belt. Various models have been proposed, but the formation of the family remains poorly understood. In this work, we have investigated the rotational properties of the family members and unconfirmed family candidates with short-term variability studies, and report the most complete review to date. We present results based on five years of observations and report the short-term variability of five family members and seven candidates. The mean rotational periods, from Maxwellian fits to the frequency distributions, are 6.27 ± 1.19 hr for the confirmed family members, 6.44 ± 1.16 hr for the candidates, and 7.65 ± 0.54 hr for other TNOs (without relation to the family). According to our study, there is a possibility that Haumea family members rotate faster than other TNOs; however, the sample of family members is still too limited for a secure conclusion. We also highlight the fast rotation of 2002 GH<sub>32</sub>. This object has a 0.36 ± 0.02 mag amplitude lightcurve and a rotational period of about 3.98 hr. Assuming 2002 GH<sub>32</sub> is a triaxial object in hydrostatic equilibrium, we derive a lower limit to the density of 2.56 g cm<sup>-3</sup>. This density is similar to Haumea's and much more dense than other small TNO densities.

Key words: Kuiper belt: general – Kuiper belt objects: individual (Haumea) – techniques: photometric

## 1. INTRODUCTION

The dwarf planet Haumea, (136108) 2003 EL<sub>61</sub>, has been well observed since its discovery and exhibits a number of interesting and unusual characteristics.

1. Bright with a visual magnitude of about 17.

2. Large with a diameter of around 1200 km and a geometric albedo of 0.70–0.75 (Lellouch et al. 2010).

3. Very elongated shape and a dark spot (Lacerda et al. 2008).

4. Fast rotational period around 3.9 hr (Rabinowitz et al. 2006; Lacerda et al. 2008; Thirouin et al. 2010).

5. High density of 2.5–3.3 g cm<sup>-3</sup> (Rabinowitz et al. 2006; Thirouin et al. 2010).

6. Pure water ice surface (Merlin et al. 2007; Tegler et al. 2007; Trujillo et al. 2007).

7. At least two satellites: Namaka and Hi'iaka (Brown et al. 2005, 2006; Brown 2005).

8. 10 objects (plus Haumea and its satellites) have similar proper orbital parameters and similar surface properties.

Noll et al. (2005) noted that three objects, (19308) 1996 TO<sub>66</sub>, (24836) 1995 SM<sub>55</sub>, and (86047) 1999 OY<sub>3</sub> had unusually blue colors in a *Hubble Space Telescope* (HST) survey of 81 trans-Neptunian objects (TNOs) using NICMOS (F110W–F160W color). The authors suggested that these unusual colors could be due to bodies covered by relatively fresh ices. Brown et al. (2007) further suggested that these three objects with several others had similar proper orbital parameters and surface properties to Haumea. This lead to the idea that Haumea, its satellites, and these other objects formed a family. Brown et al. (2007) proposed

that the proto-Haumea<fnref n="6" /> suffered a catastrophic impact that ejected a large fraction of its icy mantle, which formed the two satellites and the dynamical family. Levison et al. (2008) found that the Haumean family is likely the only collisional family in the Trans-Neptunian belt. However, Marcus et al. (2011) and Campo Bagatin & Benavidez (2012) have argued that there could be more families in this region. They estimated that a collision on a 400 km body would have produced the largest fragment, no smaller than ~300 km, and fragments in the 50–100 km size range (23–24 mag) making their identification difficult with existing surveys (e.g., Sheppard et al. 2011; Brown et al. 2015).

In this work, we report observations of short-term variability of Haumea family members and candidates, including objects not previously observed. We performed CCD photometric observations using several telescopes in Spain and the United States over a period of five years. We report the short-term variability of 12 objects. We compare the rotational properties of the Haumea family members to non-family TNOs. As a test of whether the peculiar creation and history of this group of objects may have resulted in rotational properties that are different from those of other TNOs, we examine the lightcurve amplitude distribution of the family members and candidates. Body elongation and lower limit to the density are derived from lightcurves.

This paper is divided into six sections. In the next section, we review the Haumea family and define terminology. Then, we describe the observations and the data set analyzed. In

<footnote n="6">⁶ The term “proto-Haumea” is used to refer to the object prior to the formation of the family. The name “Haumea” is used to refer to the actual object.</footnote>
Section 4, we present our main results for each object. In Section 5, we discuss our results and present a summary about the Haumea family members and candidates. Finally, Section 6 is dedicated to the conclusions of this work.

## 2. HAUMEA FAMILY MEMBERS

The Haumea family is composed of objects sharing similar proper orbital elements and similar surface properties (Brown et al. 2007). Currently, the list of confirmed Haumea family members is as follows.

1. Brown et al. (2007) identified (24835) 1995 SM$_{55}$, (19308) 1996 TO$_{66}$, (55636) 2002 TX$_{300}$, (120178) 2003 OP$_{32}$, (145453) 2005 RR$_{43}$, (136108) Haumea, Namaka, and Hi'iaka.

2. Ragozzine & Brown (2007) added (86047) 1999 OY$_{3}$ and 2003 UZ$_{117}$.

3. Schaller & Brown (2008) added (308193) 2005 CB$_{79}$.

4. Snodgrass et al. (2010) included 2003 SQ$_{317}$.<fnref n="7" />

5. Trujillo et al. (2011) confirmed the membership of 2009 YE$_{7}$.

Ragozzine & Brown (2007) published a list of possible family members (hereinafter, candidates) that have proper orbital elements consistent with the family but without any near-infrared spectra confirming the presence of water ice on their surfaces (see Ragozzine & Brown 2007 for a more complete definition of the candidate sample). Some of these candidates have been rejected by Snodgrass et al. (2010) and Carry et al. (2012) based on color studies. Recently, Sheppard and Trujillo discovered another candidate in their survey dedicated to the search of Sedna-like objects (Trujillo & Sheppard 2014; Sheppard & Trujillo 2015a). This object, 2014 FT$_{71}$, has proper orbital elements consistent with Haumea's, but color/spectroscopic observations needed to confirm family membership are lacking (Sheppard & Trujillo 2015b).

Volk & Malhotra (2012) suggested that (315530) 2008 AP$_{129}$ belongs to a new class of rockier family members. This object has proper elements consistent with being a member of the family but does not have a strong water ice signature (Brown et al. 2012). Volk & Malhotra (2012) speculated that this object could be a fragment from an inner part of a differentiated proto-Haumea. On the other hand, Cook et al. (2011) based on Desch et al.'s (2009) work suggested that the proto-Haumea was only partially differentiated. In fact, Desch et al. (2009) showed that TNOs with radii in the range of 500–1000 km are only partially differentiated with a rocky core and an icy mantle surrounded by a thick crust of rock/ice mixture. Such a crust never reached temperatures high enough to melt or differentiate. In that case, the fragments forming the Haumea family are from the icy mantle and the crust, and so one might expect a mix of icy and rocky members in the family. Because 2008 AP$_{129}$ appears as a “transition object” between the icy and rocky population, for the purpose of this work, we will consider it as a candidate. Orcus and Pluto-Charon also have water ice on their surface, but, because their

proper orbital elements are not similar to the rest of the family and candidates, they will not be considered in this work (Fornasier et al. 2004; Trujillo et al. 2005; Carry et al. 2011; Cruikshank et al. 2015).

## 3. OBSERVATIONS AND DATA REDUCTION

### 3.1. *Runs and Telescopes*

We present data obtained with Lowell Observatory's 4.3 m Discovery Channel Telescope (DCT), the 3.58 m Telescopio Nazionale Galileo (TNG), the 2.5 m Isaac Newton Telescope (INT), the 2.2 m Centro Astronómico Hispano Alemán (CAHA) telescope at Calar Alto Observatory, and the 1.5 m Sierra Nevada Observatory (OSN) telescope between 2011 and 2015.

The DCT is 40 miles southeast of Flagstaff at the Happy Jack site (Arizona, United States of America). Images were obtained using the Large Monolithic Imager (LMI), which is a 6144 × 6160 CCD. The total field of view is 12.5 × 12.5 with a pixel scale of 0.′′12 pixel$^{-1}$ (unbinned). Images were obtained using the 3 × 3 binning mode.

The TNG and INT are located at the Roque de los Muchachos Observatory (La Palma, Canary Islands, Spain). Images were obtained using the Device Optimized for the LOw RESolution instrument (DOLORES or LRS). This device has a camera and a spectrograph installed at the Nasmyth B telescope focus. We observed in imaging mode and a 2 × 2 binning mode. The camera is equipped with a 2048 × 2048 CCD with a pixel size of 13.5 µm. The field of view is 8.6 × 8.6 with a 0.′′252 pixel$^{-1}$ scale (pixel scale for a 1 × 1 binning). Our observations with the INT were obtained with the Wide Field Camera instrument. This camera consists of four thinned EEV 2154 × 4200 CCDs for a total field of view of 34′ × 34′. The pixel scale is 0.′′33 pixel$^{-1}$ (pixel scale for a 1 × 1 binning).

The 2.2 m CAHA telescope at the Calar Alto Observatory is located in the Sierra de Los Filabres (Almeria, Spain). We used the Calar Alto Faint Object Spectrograph (CAFOS) instrument located at the Cassegrain focus of the telescope. CAFOS is equipped with a 2048 × 2048 pixels CCD and the image scale is 0.′′53 pixel$^{-1}$ (pixel scale for a 1 × 1 binning). Images were obtained using 2 × 2 binning.

The 1.5 m telescope is located at the Observatory of Sierra Nevada (OSN), at Loma de Dilar in the National Park of Sierra Nevada (Granada, Spain). Observations were carried out by means of a 2k × 2k CCD, with a total field of view of 7.8 × 7.8. We used a 2 × 2 binning mode, which changes the image scale to 0.′′46 pixel$^{-1}$.

### 3.2. *Observing Strategy, Data Reduction, and Analysis*

Exposure times were chosen based on two main factors: (1) exposure time had to be long enough to achieve a signal-to-noise ratio (S/N) sufficient to study the observed object (typically, S/N > 20); (2) exposure time had to be short enough to avoid elongated images. We always tracked the telescope at sidereal speed. The drift rates of TNOs are low, typically ~2′′/hr, so exposure times around 200–900 s were used.

Observations at the OSN were performed without filter in order to maximize the S/N. Because the main goal of our study is short-term variability via relative photometry, the use of unfiltered images without absolute calibration is not a problem. The r' Sloan filter was used during our observations with the TNG. With the 2.2 m CAHA telescope, we used the Schott KG1 filter (near-infrared blocking filter), and the VR filter

<footnote n="7">⁷ Snodgrass et al. (2010) considered this object only as a *probable* member due to the lack of optical colors and the relatively large uncertainty of its (J–H$_{S}$) value (see Snodgrass et al. 2010 for more details). Therefore, the water ice detection is only preliminary. According to Lacerda et al. (2014), this object has a nearly solar surface color matching colors of the other members, but its phase function is much steeper. In order to provide the most complete study, we will consider this object a member.</footnote>
<table><caption>**Table 1**
Observational Circumstances<sup>a</sup></caption><thead><tr><th>Object</th><th>Date</th><th>Nb.</th><th>r<sub>h</sub> (au)</th><th>Δ (au)</th><th>α (°)</th><th>Filter</th><th>Tel.</th></tr></thead><tbody><tr><td rowspan="19">(24835) 1995 SM<sub>55</sub></td><td>2012 Sep 12</td><td>9</td><td>38.437</td><td>37.892</td><td>1.27</td><td>Clear</td><td>OSN</td></tr><tr><td>2012 Sep 13</td><td>20</td><td>38.436</td><td>37.878</td><td>1.26</td><td>Clear</td><td>OSN</td></tr><tr><td>2012 Sep 15</td><td>23</td><td>38.436</td><td>37.851</td><td>1.23</td><td>Clear</td><td>OSN</td></tr><tr><td>2012 Sep 16</td><td>22</td><td>38.436</td><td>37.837</td><td>1.21</td><td>Clear</td><td>OSN</td></tr><tr><td>2012 Oct 15</td><td>10</td><td>38.430</td><td>37.545</td><td>0.69</td><td>Clear</td><td>OSN</td></tr><tr><td>2012 Oct 16</td><td>61</td><td>38.430</td><td>37.538</td><td>0.67</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Nov 28</td><td>25</td><td>38.350</td><td>37.441</td><td>0.58</td><td>R</td><td>INT</td></tr><tr><td>2013 Dec 01</td><td>22</td><td>38.349</td><td>37.459</td><td>0.64</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Dec 02</td><td>13</td><td>38.349</td><td>37.466</td><td>0.66</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Dec 03</td><td>52</td><td>38.349</td><td>37.473</td><td>0.68</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Dec 05</td><td>54</td><td>38.348</td><td>37.487</td><td>0.72</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Dec 06</td><td>6</td><td>38.348</td><td>37.495</td><td>0.74</td><td>Clear</td><td>OSN</td></tr><tr><td>1999 CD<sub>158</sub></td><td>2015 Mar 24</td><td>51</td><td>46.288</td><td>46.933</td><td>0.93</td><td>VR</td><td>DCT</td></tr><tr><td rowspan="6">(86047) 1999 OY<sub>3</sub></td><td>2015 May 25</td><td>5</td><td>41.127</td><td>41.264</td><td>1.40</td><td>VR</td><td>DCT</td></tr><tr><td>2015 Jul 25</td><td>26</td><td>40.363</td><td>41.289</td><td>0.58</td><td>VR</td><td>DCT</td></tr><tr><td>2015 Aug 19</td><td>4</td><td>40.304</td><td>41.300</td><td>0.25</td><td>VR</td><td>DCT</td></tr><tr><td>2015 Aug 20</td><td>21</td><td>40.305</td><td>41.301</td><td>0.25</td><td>VR</td><td>DCT</td></tr><tr><td>2015 Aug 21</td><td>5</td><td>40.307</td><td>41.301</td><td>0.26</td><td>VR</td><td>DCT</td></tr><tr><td>2015 Aug 22</td><td>24</td><td>40.309</td><td>41.302</td><td>0.27</td><td>VR</td><td>DCT</td></tr><tr><td>2000 CG<sub>105</sub></td><td>2015 Mar 25</td><td>9</td><td>46.220</td><td>47.025</td><td>0.72</td><td>VR</td><td>DCT</td></tr><tr><td rowspan="2">2002 GH<sub>32</sub></td><td>2015 Mar 25</td><td>12</td><td>43.619</td><td>42.929</td><td>0.95</td><td>VR</td><td>DCT</td></tr><tr><td>2015 Apr 15</td><td>30</td><td>43.624</td><td>42.713</td><td>0.56</td><td>VR</td><td>DCT</td></tr><tr><td rowspan="5">(120178) 2003 OP<sub>32</sub></td><td>2011 Aug 29</td><td>10</td><td>41.652</td><td>40.682</td><td>0.39</td><td>KG1</td><td>CAHA</td></tr><tr><td>2011 Aug 30</td><td>15</td><td>41.652</td><td>40.684</td><td>0.40</td><td>KG1</td><td>CAHA</td></tr><tr><td>2011 Aug 31</td><td>5</td><td>41.652</td><td>40.866</td><td>0.41</td><td>KG1</td><td>CAHA</td></tr><tr><td>2013 Aug 09</td><td>52</td><td>41.842</td><td>40.900</td><td>0.52</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Aug 31</td><td>33</td><td>41.848</td><td>40.873</td><td>0.36</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Sep 01</td><td>47</td><td>41.848</td><td>40.875</td><td>0.37</td><td>Clear</td><td>OSN</td></tr><tr><td rowspan="3">2003 HA<sub>57</sub></td><td>2015 Mar 24</td><td>15</td><td>32.148</td><td>32.937</td><td>1.07</td><td>VR</td><td>DCT</td></tr><tr><td>2015 Mar 25</td><td>7</td><td>32.138</td><td>32.937</td><td>1.06</td><td>VR</td><td>DCT</td></tr><tr><td>2015 May 24</td><td>8</td><td>32.028</td><td>32.946</td><td>0.75</td><td>VR</td><td>DCT</td></tr><tr><td>2003 HX<sub>56</sub></td><td>2015 May 26</td><td>18</td><td>47.140</td><td>47.968</td><td>0.70</td><td>VR</td><td>DCT</td></tr><tr><td>2003 UZ<sub>117</sub></td><td>2014 Nov 28</td><td>109</td><td>38.944</td><td>38.005</td><td>0.45</td><td>VR</td><td>DCT</td></tr><tr><td>2005 GE<sub>187</sub></td><td>2015 Apr 16</td><td>26</td><td>28.887</td><td>28.106</td><td>1.26</td><td>VR</td><td>DCT</td></tr><tr><td rowspan="7">(315530) 2008 AP<sub>129</sub></td><td>2012 Jan 25</td><td>25</td><td>37.814</td><td>36.928</td><td>0.66</td><td>Clear</td><td>OSN</td></tr><tr><td>2012 Jan 26</td><td>30</td><td>37.814</td><td>36.928</td><td>0.66</td><td>Clear</td><td>OSN</td></tr><tr><td>2012 Jan 30</td><td>15</td><td>37.815</td><td>36.929</td><td>0.66</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Feb 08</td><td>20</td><td>37.930</td><td>37.051</td><td>0.69</td><td>r'</td><td>TNG</td></tr><tr><td>2013 Feb 09</td><td>37</td><td>37.930</td><td>37.054</td><td>0.69</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Feb 13</td><td>27</td><td>37.932</td><td>37.068</td><td>0.73</td><td>Clear</td><td>OSN</td></tr><tr><td>2013 Feb 14</td><td>63</td><td>37.932</td><td>37.073</td><td>0.74</td><td>Clear</td><td>OSN</td></tr><tr><td>(386723) 2009 YE<sub>7</sub></td><td>2014 Nov 27</td><td>90</td><td>50.694</td><td>49.832</td><td>0.55</td><td>VR</td><td>DCT</td></tr></tbody></table>

**Note.**

(broadband filter) at the DCT. Such filters were chosen to maximize the object S/N and to minimize the fringing that appears at longer wavelengths in these instruments.

In this work, we focused on five confirmed members of the family: 1995 SM<sub>55</sub>, 1999 OY<sub>3</sub>, 2003 OP<sub>32</sub>, 2003 UZ<sub>117</sub>, and

2009 YE<sub>7</sub>, and seven candidates: 1999 CD<sub>158</sub>, 2000 CG<sub>105</sub>, 2002 GH<sub>32</sub>, 2003 HA<sub>57</sub>, 2003 HX<sub>56</sub>, 2005 GE<sub>187</sub>, and 2008 AP<sub>129</sub>. All relevant geometric information about the observed objects at the date of observation, the number of images, and the filters are summarized in Table 1.

<footnote><sup>a</sup> UT-Dates, heliocentric ($r_h$) and geocentric ($\Delta$) distances, and phase angle ($\alpha$) of the observations are reported. We also indicate the number of images (Nb.) obtained each night, the filter used, and the telescope (Tel.) for each observational run. “DCT” stands for the Discovery Channel Telescope, “OSN” stands for the Observatory of Sierra Nevada Telescope, “TNG” stands for the Telescopio Nazionale Galileo, “INT” for the Isaac Newton Telescope, and “CAHA” stands for the Centro Astronómico Hispano Alemán telescope. Some data from others publications have been used, observational circumstances can be found in respective publications (see Section 4 for more details).</footnote>
We used the standard data calibration and reduction
techniques described in Thirouin et al. (2010, 2012, 2014).

3.3. Period-detection Methods

The time-series photometry of each target was inspected for periodicities by means of the Lomb technique (Lomb 1976) as implemented in Press et al. (1992). We also checked our results with several other time-series analysis techniques, such as Phase Dispersion Minimization (PDM; Stellingwerf 1978) and the CLEAN technique (Foster 1995). Harris et al.'s (1989) method and its improvement (Pravec et al. 1996) were also used (hereinafter, the Pravec-Harris method). Finally, in order to measure the full amplitude (or peak-to-peak amplitude) of short-term variability, a first or second order Fourier fit (depending on whether we are considering a single- or double-peaked rotational periodicity) to the data has been performed.

We must point out that to distinguish between shape and/or
albedo contribution(s) in a lightcurve is not trivial. In fact, care
has to be taken to distinguish between a single- or double-peaked
lightcurve (see Figure 1 and the discussion related in Thirouin
et al. 2014). Except for a pole-on view of an object, in which no
rotational variability can be observed, the observer will detect
rotational variability for the rest of the configurations of the spin
axis. Assuming a triaxial ellipsoid (Jacobi ellipsoid), we have to
expect a lightcurve with two maxima and two minima,
corresponding to a full rotation (a double-peaked lightcurve).
However, if the object is spherical or oblate (MacLaurin
spheroid) without any albedo variation on its surface, we have
to expect a flat lightcurve. If this spheroid presents albedo
variation on its surface, we have to expect a lightcurve with one
maximum and one minimum (i.e., a single-peaked lightcurve).

When the lightcurve amplitude is small, it is very difficult or
even impossible to distinguish if the lightcurve is single- or
double-peaked. Therefore, we have to find a criterion to
distinguish between both cases. In Thirouin et al. (2010) and
Duffard et al. (2009), we proposed a threshold at 0.15 mag to
distinguish between shape and albedo effects. This criterion is a
simplification because there may be elongated objects whose
rotational variability is smaller than 0.15 mag simply because
their rotation axes are viewed close to pole-on from Earth.
However, these objects are only a small fraction because
statistically only very few objects have spin axes near the pole-
on orientation.<fnref n="8" /> In Thirouin (2013), we determined the
lightcurve amplitude limit to distinguish between shape- and
albedo-dominated lighcurves (i.e., to distinguish between
single- and double-peaked lighcurves). We tested three light-
curve amplitude limits: (1) a threshold at 0.10 mag, (2) at
0.15 mag, and (3) at 0.20 mag, to distinguish between single-
and double-peaked lightcurves. The best fit (i.e., with the
highest confidence level) has been obtained considering a
lightcurve amplitude limit of 0.15 mag. Such a criterion with a
threshold at 0.15 mag has already been used by several
investigators as the transition from low to medium variability
(Lacerda & Luu 2006; Sheppard et al. 2008). In the asteroid
case, albedo variations are usually responsible for lightcurve
amplitudes between 0.1 and 0.2 mag (Degewij et al. 1979,
Lupishko et al. 1983; Magnusson 1991).

<figure><img src="image_4.png" /><figcaption>**Figure 1.** Lomb-normalized spectral power vs. frequency in cycles/day for 1995 SM$_{55}$: the Lomb periodogram shows one main peak located at 4.04 hr (5.94 cycles/day), and two aliases located at 4.77 hr (5.04 cycles/day) and 3.45 hr (6.94 cycles/day).</figcaption></figure>

Albedo variation in the asteroid modeling is negligible, but
not in the TNO case. The shape dominates in the asteroid case,
but not for the TNOs (Lacerda et al. 2008). Lacerda et al.
(2008) simulated a synthetic ellipsoidal object without any
mark of albedo on its surface, and the result is a perfectly
symmetric double-peaked lightcurve. However, if a spot or a
hemispheric albedo variation is present on the surface of the
same ellipsoidal object, the result is an asymmetric double-
peaked lightcurve. In other words, the double-peaked light-
curve is due to the shape of the object, and it is clear that the
asymmetry of the maxima is due to the spot/hemispheric
albedo variation. Thus, the difference between both maxima
(and both minima) gives us information about the albedo
variation on the object's surface. In the case of Haumea, such a
difference between both maxima is around 0.04 mag (or 4%),
and there is also a difference between both minima of around
0.04 mag (Lacerda et al. 2008). Therefore, the total change can
amount to 0.08 mag. Haumea is not the only TNO presenting
such a characteristic. We can cite (1) Varuna with a difference
around 0.1 mag (10%; Thirouin et al. 2010; Thirouin 2013;
J. L. Ortiz et al., in preparation), and (2) 2003 VS$_2$ also presents
an asymmetric lightcurve with a 0.04 mag difference (Ortiz
et al. 2006; Sheppard 2007; Thirouin et al. 2010). The observed
asymmetric lightcurves can be perfectly explained thanks to the
lightcurve modeling of objects with spot or hemispheric albedo
variations reported by Lacerda et al. (2008; Lellouch
et al. 2010; Snodgrass et al. 2010; Carry et al. 2012; Lockwood
et al. 2014). Besides these cases, there is an even more well-
known case: Pluto. Pluto is a MacLaurin body whose
lightcurve is exclusively due to albedo variations on its
surface. In conclusion, based on the typical hemispherically
averaged albedo, and the best Maxwellian fit distribution
(Thirouin 2013), we estimate that 0.15 mag is a good measure
of the typical variability caused by albedo features. On the
other hand, we know that the lightcurve amplitude of large and
small TNOs are significantly different (Lacerda & Luu 2003,
2006). These authors, based on numerical and observations
evidence, noticed a cut-off for objects with a diameter ~400 km
(with an albedo of 0.04). They demonstrated that large objects
have nearly spherical shapes and small objects have irregular
shapes. Based on a larger sample, and using an albedo of 0.12,

<footnote n="8">⁸ In the case of a pole-on observation, even if the object is very elongated, the lightcurve will be flat. However, the probability of such a case is low. In fact, Equation (6a) of Lacerda & Luu (2003) estimates the probability to observe an object with a pole-on orientation. Assuming an angle of $\theta = 5^\circ$, the probability to see an object with a pole-on orientation $\pm 5^\circ$ is less than 1%.</footnote>
<table><caption>**Table 2**
The Time Series Photometry of All the Objects is Provided in the 'Center of Astronomical Data of Strasbourg' (CDS)</caption><thead><tr><th>Object</th><th>JD<br>(2450000+)</th><th>Mag.<br>(mag)</th><th>Err.<br>(mag)</th><th>Fil.</th><th>α<br>(°)</th><th>r<sub>h</sub><br>(au)</th><th>Δ<br>(au)</th></tr></thead><tbody><tr><td rowspan="5">315530</td><td>5952.44390</td><td>0.007</td><td>0.022</td><td>Cle</td><td>0.66</td><td>36.928</td><td>37.813</td></tr><tr><td>5952.45564</td><td>0.000</td><td>0.017</td><td>Cle</td><td>0.66</td><td>36.928</td><td>37.813</td></tr><tr><td>5952.46149</td><td>-0.007</td><td>0.015</td><td>Cle</td><td>0.66</td><td>36.928</td><td>37.813</td></tr><tr><td>5952.55931</td><td>-0.006</td><td>0.030</td><td>Cle</td><td>0.66</td><td>36.928</td><td>37.813</td></tr><tr><td>5952.56510</td><td>-0.034</td><td>0.015</td><td>Cle</td><td>0.66</td><td>36.928</td><td>37.813</td></tr></tbody></table>

**Note.** We present our photometric results. Following the name of the object, for each image we specify the Julian Date (JD, not corrected for light time), the relative magnitude (mag in magnitudes) and the $1\sigma$ error associated (Err. in magnitude), the filter (Fil.) used during observational runs, the phase angle ($\alpha$, in degree), and topocentric ($r_h$) and heliocentric ($\Delta$) distances (both distances in astronomical units, au). “Cle” stands for clear filter.

this cut-off is closer to ~250–300 km (Vilenius et al. 2012, 2014). From observations and numerical simulations, we can conclude that objects with a diameter larger than 250–350 km (conservative cut-off) are spherical, whereas smaller objects have an elongated shape (Leinhardt et al. 2000; Lacerda & Luu 2003, 2006; Duffard et al. 2009). In the case of an asymmetric lightcurve, we always choose the double-peaked option because it clearly shows a complex shape and/or surface variations that cannot be explained with a single-peaked lightcurve.

## 4. PHOTOMETRIC RESULTS

In this section, we discuss our short-term variability results. We report new data for 11 objects. For 1 object, we present a new analysis of previously published results by our team, plus additional data we obtained. For the first time, rotational periods and lightcurve amplitude are reported for the entire family (except Haumea's satellites). To present a complete study, we also focus on several candidates. Observations of candidates are challenging because these objects are small and faint. Only a few attempts of short-term variability of faint objects have been published (e.g., Kern 2006; Kern & Elliot 2006; Trilling & Bernstein 2006). HST is the most prolific tool to study these faint objects; however, thanks to the 4 m class telescope, we reached objects with a visual magnitude up to 24 (the faintest objects observed under excellent weather and seeing conditions).

Lomb periodograms and lightcurves for all objects are provided in Figures 1–20. We plotted all lightcurves over two cycles (rotational phase from 0 to 2) for better visualization of the cyclical variation. For each lightcurve, a first or second order Fourier series is used to fit the photometric data. Error bars for the measurements are not shown on the plots for clarity, but one-sigma error bars on the relative magnitudes are reported in the supplementary material (see Table 2). We must point out that when we combined several observing runs obtained at different epochs, light time correction of the data is required (see Thirouin 2013 for more details about data combination). Our photometric results are reported in Table 4. A complete summary of the short-term variability of the Haumea family members and candidates can be found in Table 3.

### 4.1. (24835) 1995 SM$_{55}$

Sheppard & Jewitt (2003) observed this object for several nights on 2001 October and November with the University of Hawaii 2.2 m telescope. Based on the October data set, they

reported a scattered photometry and no rotational period estimation. With the additional November data set, they suggested a single-peaked rotational period of 4.04 hr or a double-peaked periodicity of 8.08 hr and an average peak-to-peak amplitude of $0.19 \pm 0.05$ mag (they only reported photometric amplitudes estimated from apparent maximum and minimum, and no lightcurve amplitude was obtained thanks to a lightcurve fit as has been done in this work). Unfortunately, in both cases, the curves were too noisy given the photometric uncertainties. Sheppard & Jewitt (2003) concluded that the amplitude of the lightcurve may be variable from night to night. Such variations could be due to the (1) presence of a companion, (2) cometary activity, or (3) complex rotational state. Sheppard & Jewitt (2003) pointed out that this object has been investigated for binarity with the HST and that no satellite with a separation $\ge 0.01$ arcsec and with a magnitude difference $\le 2.5$ was found. 1995 SM$_{55}$ is one of the bluest TNOs that could be attributed to recent excavation (due to a collision, for example) of its volatile-rich interior (Hainaut & Delsanti 2002). On the other hand, the lightcurve amplitude may be due to freshly exposed material by cometary activity (Hainaut & Delsanti 2002).

This object was observed in 2012 and 2013 to look for a possible change in the lightcurve. By merging our data with Sheppard & Jewitt (2003) data, the Lomb periodogram plotted in Figure 1 is obtained. The main peak is located at 5.94 cycles/days (4.04 hr), and there are two other peaks with a lower spectral power located at 5.04 cycles/days and at 6.94 cycles/days. The lightcurve is asymmetric with a first peak taller than the second one and a second minimum deeper than the first one, so the double-peaked lightcurve with a rotational period of 8.08 hr seems the best option. In Figure 2, the corresponding double-peaked lightcurve with a lightcurve amplitude of $0.04 \pm 0.02$ mag is plotted. A second argument in favor of the double-peaked lightcurve is the goodness of the fit based on a reduced $\chi^2$ test ($\chi^2 = 1.496$ for the double-peaked option and $\chi^2 = 1.510$ for the single-peaked one). We also tested higher harmonics, but the $\chi^2$ test discarded all of them.

In conclusion, we confirmed the rotational period obtained by Sheppard & Jewitt (2003), as well as the lightcurve amplitude. We also report no significant change for this lightcurve over 12 years. The lightcurve reported by Sheppard & Jewitt (2003), as well as our new lightcurve, are noisy despite the high data quality. Possible explanations for such a lightcurve will be studied in a future work.
<table><caption>**Table 3**
We Listed the Short-term Variability of the Confirmed Haumea Family Members and the Candidates</caption><thead><tr><th>Object</th><th>Dyn. Class</th><th>Single Peak Periodicity (hr)</th><th>Double Peak Periodicity (hr)</th><th>Amplitude (mag)</th><th>Abs mag.</th><th>Ref.</th></tr></thead><tbody><tr><td colspan="7"><strong>Confirmed members</strong></td></tr><tr><td rowspan="3">(136108) Haumea</td><td>Resonant</td><td>...</td><td>3.9154 ± 0.0002</td><td>0.28 ± 0.02</td><td>0.1</td><td>R06</td></tr><tr><td>...</td><td>...</td><td>3.9155 ± 0.0001</td><td>0.29 ± 0.02</td><td>...</td><td>L08</td></tr><tr><td>...</td><td>...</td><td>3.92</td><td>0.28 ± 0.02</td><td>...</td><td>T10</td></tr><tr><td>Namaka<sup>a</sup></td><td>Resonant</td><td>...</td><td>...</td><td>...</td><td>4.5</td><td>...</td></tr><tr><td>Hi'iaka<sup>a</sup></td><td>Resonant</td><td>...</td><td>...</td><td>...</td><td>2.9</td><td>...</td></tr><tr><td rowspan="2">(24835) 1995 SM<sub>55</sub><sup>b</sup></td><td>Classical</td><td>4.04 ± 0.03</td><td>8.08 ± 0.03</td><td>0.19 ± 0.05</td><td>4.8</td><td>SJ03</td></tr><tr><td>...</td><td>...</td><td>8.08</td><td>0.05 ± 0.02</td><td>...</td><td>TW</td></tr><tr><td rowspan="4">(19308) 1996 TO<sub>66</sub><sup>b</sup></td><td>Classical</td><td><strong>3.96 ± 0.04</strong> or 4.80 ± 0.05</td><td><strong>7.92 ± 0.04</strong> or 5.90 ± 0.05 or 9.6 ± 0.1</td><td>0.26 ± 0.03</td><td>4.5</td><td>SJ03</td></tr><tr><td>...</td><td>...</td><td>11.9</td><td>0.25 ± 0.05</td><td>...</td><td>B06</td></tr><tr><td>...</td><td>...</td><td>6.25 ± 0.03</td><td>0.12 to 0.33</td><td>...</td><td>H00</td></tr><tr><td>...</td><td>...</td><td>...</td><td>&lt;0.10</td><td>...</td><td>RT99</td></tr><tr><td>(86047) 1999 OY<sub>3</sub><sup>b</sup></td><td>Resonant</td><td>-</td><td>18.02</td><td>0.08</td><td>6.8</td><td>TW</td></tr><tr><td rowspan="5">(55636) 2002 TX<sub>300</sub></td><td>Classical</td><td>(8.12 or 12.101) ± 0.08</td><td>(16.24 or 24.20) ± 0.08</td><td>0.08 ± 0.02</td><td>3.2</td><td>SJ03</td></tr><tr><td>...</td><td>7.89 ± 0.03</td><td>15.78</td><td>0.09 ± 0.08</td><td>...</td><td>O04</td></tr><tr><td>...</td><td>...</td><td>8.16</td><td>0.04 ± 0.01</td><td>...</td><td>T10</td></tr><tr><td>...</td><td>...</td><td>8.15 or 11.7</td><td>(0.01 or 0.05) ± 0.01</td><td>...</td><td>T12</td></tr><tr><td>...</td><td>...</td><td>18.35</td><td>...</td><td>...</td><td>S12</td></tr><tr><td rowspan="3">(120178) 2003 OP<sub>32</sub><sup>b</sup></td><td>Classical</td><td>4.845 ± 0.003</td><td>...</td><td>0.26 ± 0.04</td><td>3.6</td><td>R08</td></tr><tr><td>...</td><td>4.05</td><td>...</td><td>0.13 ± 0.01</td><td>...</td><td>T10</td></tr><tr><td>...</td><td><strong>4.85</strong>/6.09</td><td><strong>9.71</strong>/12.18</td><td>0.18 ± 0.01</td><td>...</td><td>BS13</td></tr><tr><td rowspan="2">2003 SQ<sub>317</sub><sup>c</sup></td><td>Classical</td><td>3.74</td><td>...</td><td>0.14 ± 0.02</td><td>...</td><td>TW</td></tr><tr><td>...</td><td>...</td><td>1</td><td>...</td><td>6.3</td><td>S10</td></tr><tr><td rowspan="2">2003 UZ<sub>117</sub><sup>b</sup></td><td>Classical</td><td>~6</td><td>7.210 ± 0.001</td><td>0.85 ± 0.05</td><td>...</td><td>L14</td></tr><tr><td>...</td><td>...</td><td>11.29</td><td>0.09 ± 0.01</td><td>...</td><td>TW</td></tr><tr><td>(308193) 2005 CB<sub>79</sub></td><td>Classical</td><td>6.76</td><td>...</td><td>0.05 ± 0.02</td><td>4.7</td><td>T10</td></tr><tr><td rowspan="2">(145453) 2005 RR<sub>43</sub></td><td>Classical</td><td>...</td><td>5.08 ± 0.04</td><td>0.12 ± 0.03</td><td>4.0</td><td>P09</td></tr><tr><td>...</td><td>7.87</td><td>...</td><td>0.06 ± 0.01</td><td>...</td><td>T10</td></tr><tr><td rowspan="2">2009 YE<sub>7</sub></td><td>Classical</td><td>...</td><td>...</td><td>&lt;0.06</td><td>...</td><td>BS13</td></tr><tr><td>...</td><td>5.65</td><td>...</td><td>&lt;0.20</td><td>4.4</td><td>BS13</td></tr><tr><td>...</td><td>...</td><td>...</td><td>0.06 ± 0.02</td><td>...</td><td>TW</td></tr><tr><td colspan="7"><strong>Candidates</strong></td></tr><tr><td rowspan="3">(120347) Salacia</td><td>Classical</td><td>...</td><td>~17.5</td><td>0.2</td><td>4.4</td><td>S10</td></tr><tr><td>...</td><td>6.09 or 8.10</td><td>...</td><td>0.03 ± 0.01</td><td>...</td><td>T10</td></tr><tr><td>...</td><td>...</td><td>...</td><td>&lt;0.04</td><td>...</td><td>B13</td></tr><tr><td rowspan="3">(136472) Makemake<sup>d</sup></td><td>Classical</td><td>...</td><td>...</td><td>0.06 ± 0.02</td><td>...</td><td>T14</td></tr><tr><td>Scattered</td><td>11.24 ± 0.01</td><td>20.54/<strong>22.48</strong></td><td>0.03 ± 0.01</td><td>-0.3</td><td>O07</td></tr><tr><td>...</td><td>7.7710 ± 0.0030</td><td>...</td><td>0.0286 ± 0.0016</td><td>...</td><td>H09</td></tr><tr><td rowspan="2">1996 RQ<sub>20</sub><sup>b</sup></td><td>Classical</td><td>...</td><td>...</td><td>0.014 ± 0.002</td><td>...</td><td>T10</td></tr><tr><td>...</td><td>...</td><td>...</td><td>0.022 ± 0.002</td><td>...</td><td>T13</td></tr><tr><td>(20161) 1996 TR<sub>66</sub><sup>b</sup></td><td>Resonant</td><td>...</td><td>...</td><td>...</td><td>6.9</td><td>...</td></tr><tr><td>1997 RX<sub>9</sub></td><td>Classical</td><td>...</td><td>...</td><td>...</td><td>7.5</td><td>...</td></tr><tr><td>1998 HL<sub>151</sub></td><td>Classical</td><td>...</td><td>...</td><td>...</td><td>8.3</td><td>...</td></tr><tr><td>(181855) 1998 WT<sub>31</sub></td><td>Classical</td><td>...</td><td>...</td><td>...</td><td>8.1</td><td>...</td></tr><tr><td>1999 CD<sub>158</sub><sup>b</sup></td><td>Resonant</td><td>...</td><td>...</td><td>0.6</td><td>5.1</td><td>S10</td></tr><tr><td rowspan="2">(40314) 1999 KR<sub>16</sub></td><td>Detached</td><td>(5.840 or 5.929) ± 0.001</td><td>(11.680 or 11.858) ± 0.002</td><td>0.49 ± 0.03</td><td>...</td><td>TW</td></tr><tr><td>...</td><td>5.8</td><td>...</td><td>0.18 ± 0.04</td><td>5.8</td><td>SJ02</td></tr><tr><td>1999 OH<sub>4</sub></td><td>Classical</td><td>...</td><td>...</td><td>0.12 ± 0.06</td><td>...</td><td>T12</td></tr><tr><td>1999 OK<sub>4</sub></td><td>Classical</td><td>...</td><td>...</td><td>...</td><td>8.3</td><td>...</td></tr><tr><td>(86177) 1999 RY<sub>215</sub><sup>b</sup></td><td>Classical</td><td>...</td><td>...</td><td>...</td><td>7.6</td><td>...</td></tr><tr><td>2000 CG<sub>105</sub><sup>b</sup></td><td>Classical</td><td>...</td><td>...</td><td>0.45</td><td>6.6</td><td>S10</td></tr><tr><td rowspan="2">(130391) 2000 JG<sub>81</sub></td><td>Classical</td><td>...</td><td>&gt;4</td><td>&gt;0.2</td><td>...</td><td>TW</td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>8.0</td><td>...</td></tr><tr><td>2001 FU<sub>172</sub></td><td>Resonant</td><td>...</td><td>...</td><td>...</td><td>8.3</td><td>...</td></tr><tr><td>2001 QC<sub>298</sub></td><td>Classical</td><td>...</td><td>~12</td><td>0.4</td><td>6.1</td><td>S10</td></tr><tr><td rowspan="2">(55565) 2002 AW<sub>197</sub></td><td>Classical</td><td>...</td><td>3.89 ± 0.24</td><td>7.78 ± 0.48</td><td>0.30 ± 0.04</td><td>K06</td></tr><tr><td>...</td><td>8.86 ± 0.01</td><td>...</td><td>0.08 ± 0.07</td><td>3.5</td><td>O06</td></tr><tr><td rowspan="2">2002 GH<sub>32</sub></td><td>Classical</td><td>...</td><td>...</td><td>0.02 ± 0.02</td><td>...</td><td>T10</td></tr><tr><td>...</td><td>...</td><td>...</td><td>0.75</td><td>5.5</td><td>S10</td></tr><tr><td>2003 HA<sub>57</sub></td><td>Resonant</td><td>...</td><td>...</td><td>0.36 ± 0.02</td><td>...</td><td>TW</td></tr><tr><td>...</td><td>...</td><td>...</td><td>~0.5</td><td>8.1</td><td>R05</td></tr></tbody></table>
<table><caption>**Table 3**
(Continued)</caption><thead><tr><th>Object</th><th>Dyn. Class</th><th>Single Peak Periodicity (hr)</th><th>Double Peak Periodicity (hr)</th><th>Amplitude (mag)</th><th>Abs mag.</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="3">2003 HX<sub>56</sub></td><td>...</td><td>...</td><td>6.44</td><td>0.31 ± 0.03</td><td>...</td><td>TW</td></tr><tr><td>Classical</td><td>...</td><td>...</td><td>~0.4</td><td>7.1</td><td>R05</td></tr><tr><td>...</td><td>...</td><td>>10</td><td>>0.4</td><td>...</td><td>TW</td></tr><tr><td>2003 QX<sub>91</sub></td><td>Resonant</td><td>...</td><td>...</td><td>...</td><td>8.3</td><td>...</td></tr><tr><td>2003 TH<sub>58</sub></td><td>Resonant</td><td>...</td><td>...</td><td>~0.2</td><td>7.6</td><td>R05</td></tr><tr><td>2004 PT<sub>107</sub></td><td>Classical</td><td>~20</td><td>...</td><td>0.05</td><td>6.0</td><td>S10</td></tr><tr><td>2005 GE<sub>187</sub><sup>b</sup></td><td>Resonant</td><td>6.1</td><td>...</td><td>0.5</td><td>7.1</td><td>S10</td></tr><tr><td rowspan="2">(202421) 2005 UQ<sub>513</sub></td><td>...</td><td>...</td><td>11.99</td><td>0.29 ± 0.02</td><td>...</td><td>TW</td></tr><tr><td>Classical</td><td>7.03 or 10.01</td><td>...</td><td>0.06 ± 0.02</td><td>3.4</td><td>T12</td></tr><tr><td>2008 AP<sub>129</sub></td><td>Classical</td><td>9.04</td><td>...</td><td>0.12 ± 0.02</td><td>4.7</td><td>TW</td></tr><tr><td>2010 KZ<sub>39</sub></td><td>Scattered</td><td>...</td><td>-</td><td>&lt;0.17</td><td>4.0</td><td>BS13</td></tr><tr><td>2014 FT<sub>71</sub></td><td>Classical</td><td>...</td><td>...</td><td>...</td><td>4.7</td><td>...</td></tr></tbody></table>

**Notes.** In the case of multiple rotational periods, the preferred rotational period according to the authors of each study is indicated in bold. Absolute magnitudes (abs mag.) are from the minor planet center (MPC) database. The complete reference list can be found after the table. Dynamical classification (dyn. class) from Gladman et al. (2008) is also indicated.

<sup>a</sup> Haumea's moons.

<sup>b</sup> No satellite/companion detected with the Hubble Space Telescope.

<sup>c</sup> Contact binary (Lacerda et al. 2014).

<sup>d</sup> Different lighthcurve amplitudes according to the filter used, as reported in Heinze & de Lahunta (2009) and Thirouin (2013).

**References.** RT99: Romanishin & Tegler (1999), H00: Hainaut et al. (2000), SJ02: Sheppard & Jewitt (2002), SJ03: Sheppard & Jewitt (2003), R05: Rousselot et al. (2005), O04: Ortiz et al. (2004), B06: Belskaya et al. (2006), K06: Kern (2006), O06: Ortiz et al. (2006), R06: Rabinowitz et al. (2006), O07: Ortiz et al. (2007), L08: Lacerda et al. (2008), R08: Rabinowitz et al. (2008), H09: Heinze & de Lahunta (2009), P09: Perna et al. (2009), S10: Snodgrass et al. (2010), T10: Thirouin et al. (2010), S12: Sonnett et al. (2012), T12: Thirouin et al. (2012), BS13: Benecchi & Sheppard (2013), T13: Thirouin (2013), L14: Lacerda et al. (2014), T14: Thirouin et al. (2014), TW: this work.

<figure><img src="image_5.png" /><figcaption>**Figure 2.** 1995 SM<sub>55</sub> lightcurve: rotational phase curve for 1995 SM<sub>55</sub> obtained by using a rotational period of 8.08 hr. The continuous line is a Fourier fit of the photometric data. Different symbols correspond to different dates.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Lomb-normalized spectral power vs. frequency in cycles/days for 1999 OY<sub>3</sub>: the Lomb periodogram shows one main peak located at 9.01 hr (2.66 cycles/days), and other peaks at 14.37 hr (1.67 cycles/days), and 5.97 hr (4.02 cycles/days).</figcaption></figure>

## 4.2. (86047) 1999 OY<sub>3</sub>

We report the first attempt of a short-term variability study for this object. We observed 1999 OY<sub>3</sub> during several nights in 2015 with the DCT. We report two long observing blocks of about 6 hr and several shorter blocks. Techniques used to derive the object rotation favored a periodicity around 9 hr. Figure 3 showed one main peak at 2.66 cycles/days. Peaks around 4 cycles/days are consistent with the duration of our longer observing runs and so are not due to the object rotation. We obtained a single-peaked period of 9.01 hr (Figure 4, plot (a)) and a double-peaked periodicity of 18.02 hr (Figure 4, plot (b)). We favored the double-peaked option because minima/maxima are different by about 0.02 mag, and because the χ<sup>2</sup> of

the single-peaked fit is 1.633, whereas the double-peaked is 1.548. Considering the double-peaked option, we found a χ<sup>2</sup> = 1.346 for the eighth harmonic. However, due to our data quality, we are not confident about this result. Only more data will confirm the possibility of such a harmonic.

## 4.3. (120178) 2003 OP<sub>32</sub>

Rabinowitz et al. (2008) presented 78 R-band observations of 2003 OP<sub>32</sub> obtained in 2006 with the 1.3 m SMARTS telescope. They proposed a single-peaked lightcurve with a periodicity of 4.845 hr and an amplitude of 0.26 mag. Thirouin et al. (2010) also observed 2003 OP<sub>32</sub> during several runs between 2005 and 2007, and proposed a single-peaked
<figure><img src="image_7.png" /></figure>

**Figure 4.** 1999 OY$_3$ lightcurve: rotational phase curve for 1999 OY$_3$ obtained by using a rotational period of 9.01 hr (plot (a)), and a double-peaked periodicity of 18.02 hr (plot (b)). Continuous lines represent the Fourier fit of the photometric data. Different symbols correspond to different dates. Same legend for both plots.

<figure><img src="image_8.png" /></figure>

**Figure 5.** Lomb-normalized spectral power vs. frequency in cycles/days for 2003 OP$_{32}$: the Lomb periodogram shows one main peak located at 4.85 hr (4.95 cycles/days), and two aliases located at 6.07 hr (3.96 cycles/day), and 4.03 hr (5.96 cycles/days).

lightcurve with a rotational period of 4.05 hr and an amplitude peak-to-peak of 0.13 ± 0.01 mag. Benecchi & Sheppard (2013) observed 2003 OP$_{32}$, during six nights with the Irénée du Pont 2.5 m telescope at Las Campanas Observatory (Chile). They favored a single-peaked rotational period of 4.85 hr or a double-peaked rotational period of 9.71 hr. Their peak-to-peak lightcurve amplitude is 0.18 ± 0.01 mag.

2003 OP$_{32}$ was re-observed on 2011 and 2013 with the 2.2 m CAHA and the 1.5 m OSN telescopes. The Lomb periodogram of our 2005, 2007, 2011, 2013, and Benecchi & Sheppard (2013) data sets altogether shows one peak located at 4.95 cycles/days (4.85 hr) and two aliases located at 3.96 cycles/days (6.07 hr) and at 5.96 cycles/days (4.03 hr; Figure 5). All techniques confirm a periodic signature at 4.85 hr with high spectral power. In Figure 6, the corresponding single-peaked lightcurve with an amplitude of 0.14 ± 0.02 mag is plotted. In conclusion, our and Benecchi & Sheppard's (2013) results completely ruled out the possibility of the large amplitude lightcurve noted by Rabinowitz et al. (2008), and there is an agreement about the single-peaked periodicity of 4.85 hr.

<figure><img src="image_9.png" /></figure>

**Figure 6.** 2003 OP$_{32}$ lightcurve: rotational phase curve for 2003 OP$_{32}$ obtained by using a rotational period of 4.85 hr (upper plot) and a rotational period of 9.71 hr (double-peaked periodicity, lower plot). Continuous lines are Fourier fits of the photometric data. Different symbols correspond to different dates.

Though we cannot rule out the double-peaked lightcurve, we favored the single-peaked option for this object for several reasons: (1) the moderate lightcurve amplitude suggests albedo variation on the object's surface and not elongated shape behavior, (2) the lightcurve is symmetric, (3) because of the size of this object, it is more likely that it is a MacLaurin spheroidal object (see Section 3.3 for more details), and (4) $\chi^2 = 1.559$ for the single-peaked lightcurve and $\chi^2 = 1.616$ for the double-peaked.

### 4.4. 2003 UZ$_{117}$

Using the EMMI instrument installed at the New Technology Telescope (NTT) of the European Southern Observatory (ESO), Perna et al. (2009) observed this object for ~10.5 hr during two nights on 2007 December. As they mentioned, data points were not good enough to find an unambiguous rotational period. They suggested a rotational period of about 6 hr.

We observed 2003 UZ$_{117}$ during one night with the DCT on 2014 November. Based on our DCT data, we obtained a single-peaked period of 5.30 hr (Figure 7, plot (a)) and a double-peaked periodicity of 10.61 hr (Figure 7, plot (b)). Using the ESO archive<fnref n="9" />, we downloaded and re-reduced the images obtained by Perna et al. (2009). By merging both data sets, reduced and analyzed the same way, we derived a double-peaked periodicity of 11.29 hr (Figure 8). We favored the double-peaked option based on the fact that one of the minima is deeper than the other one and one of the maxima is taller than the other one (~0.01–0.02 mag). In Figure 7 (plot (c)) is plotted the corresponding lightcurve with an amplitude of 0.09 ± 0.01 mag is plotted. We must point out that the rotational period of about 5.64 hr was also an option based on our DCT data, but with a lower confidence level than the 5.30 hr option. We calculated a $\chi^2$ of 1.246 for the second harmonic and 1.104 for the sixth harmonic. However, such a higher harmonic was only favored because Perna et al. (2009) data have a higher dispersion than our data. Therefore, we discarded this harmonic.

<footnote n="9">⁹ Data can be downloaded at http://archive.eso.org.</footnote>
<figure><img src="image_10.png" /></figure>

**Figure 7.** 2003 UZ<sub>117</sub> lightcurve: rotational phase curve for 2003 UZ<sub>117</sub> obtained by using a rotational period of 5.30 hr (single-peaked lightcurve, plot (a)), using a rotational period of 10.61 hr (double-peaked lightcurve, plot (b)), and using a rotational period of 11.29 hr (double-peaked lightcurve, plot (c)). The continuous lines are a Fourier fits of the photometric data. Different symbols correspond to different dates. Same legend for all the plots.

<figure><img src="image_11.png" /></figure>

**Figure 8.** Lomb-normalized spectral power vs. frequency in cycles/day for 2003 UZ<sub>117</sub>: the Lomb periodogram shows one main peak located at 5.64 hr (4.25 cycles/day) and two other peaks with a lower significance level at 6.42 hr (3.74 cycles/day), and at 5.09 hr (4.16 cycles/day).

4.5. (386723) 2009 YE<sub>7</sub>

Benecchi & Sheppard (2013) observed 2009 YE<sub>7</sub> during four nights using the 2.5 m Irénée du Pont telescope located at Las Campanas Observatory (Chile). They concluded that this object has a lightcurve amplitude <0.2 mag and they were not able to favor a rotational period based on their data set.

We observed this object during one night with the DCT in 2014 November under variable weather conditions. Based on our data set, we derived a possible single-peaked lightcurve with a rotational period of about 5.5 hr. By merging our sample and Benecchi & Sheppard (2013) data, we obtained a single-peaked periodicity of 5.65 hr (Figure 9). The lightcurve amplitude is 0.06 ± 0.02 mag (Figure 10). The goodness of the fit is $\chi^2 = 1.759$ for the single-peaked lightcurve and $\chi^2 = 2.005$ for the double-peaked lightcurve. Based on the $\chi^2$ value, the single-peaked option is favored. Though we cannot rule out the double-peaked lightcurve, the small lightcurve amplitude is compatible with albedo variation, and so we infer

<figure><img src="image_12.png" /></figure>

**Figure 9.** Lomb-normalized spectral power vs. frequency in cycles/day for 2009 YE<sub>7</sub>: the Lomb periodogram shows one main peak located at 5.65 hr (4.25 cycles/day), and a second one located at 4.59 hr (5.23 cycles/day).

<figure><img src="image_13.png" /></figure>

**Figure 10.** 2009 YE<sub>7</sub> lightcurve: rotational phase curve for 2009 YE<sub>7</sub> obtained by using a rotational period of 5.65 hr (upper plot) and a rotational period of 11.30 hr (double-peaked, lower plot). Continuous lines are Fourier fits of the photometric data. Different symbols correspond to different dates.

a single-peaked lightcurve (Duffard et al. 2009; Thirouin et al. 2010).

4.6. (315530) 2008 AP<sub>129</sub>

2008 AP<sub>129</sub> did not previously have an observed lightcurve. We observed this object during a run in 2012 January, in poor atmospheric conditions, and during two more runs in 2013 February with the 3.58 m TNG and the 1.5 m OSN telescope. The Lomb periodogram of our 2012 and 2013 data sets (Figure 11) shows one main peak located at 9.04 hr (2.65 cycles/day) and the second one with a lower spectral power is located at 3.84 cycles/day (6.25 hr). PDM, CLEAN, and Pravec-Harris techniques confirmed these two peaks with a higher spectral power for the 9.04 hr rotational period. In Figure 12, the corresponding lightcurve using a rotational periodicity of 9.04 hr is plotted. The amplitude of the curve is 0.12 ± 0.02 mag. In summary, 9.04 hr is preferred but 6.25 hr is also a possible period. In both cases, we preferred the single-peaked periodicity because of the small lightcurve amplitude,
<figure><img src="image_14.png" /><figcaption>**Figure 11.** Lomb-normalized spectral power vs. frequency in cycles/days for 2008 AP<sub>129</sub>: the Lomb periodogram shows one main peak located at 9.04 hr (2.65 cycles/days), and a second one located at 6.25 hr (3.84 cycles/day).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 12.** 2008 AP<sub>129</sub> lightcurve: rotational phase curve for 2008 AP<sub>129</sub> obtained by using a rotational period of 9.04 hr (upper plot), and rotational period of 18.08 hr (lower plot). The continuous line is a Fourier fit of the photometric data. Different symbols correspond to different dates.</figcaption></figure>

as well as the symmetric lightcurve and the potentially large size of this object. The goodness of the fit is $\chi^2 = 1.539$ for the single-peaked lightcurve and $\chi^2 = 1.571$ in the case of the double-peaked option. Therefore, the single-peaked lightcurve seems to be the best option, but the double-peaked option is not ruled out. Based on the data quality (data obtained under poor weather conditions), we do not favor higher harmonics. Only more data will allow us to check if higher harmonics have to be considered.

## 4.7. 1999 CD<sub>158</sub>

1999 CD<sub>158</sub> has not been previously observed for short-term variability. We observed this object during one night in 2015 March with the DCT. We derived a single-peaked period of 3.55 hr and a double-peaked periodicity of 7.1 hr. Lomb periodogram and other techniques confirmed this periodicity (Figure 13). We favored the double-peaked option based on the large lightcurve amplitude. In Figure 14 (Plot (a)), the corresponding lightcurve with an amplitude of 0.51 ± 0.03 mag is plotted. Snodgrass et al. (2010) observed this object

<figure><img src="image_16.png" /><figcaption>**Figure 13.** Lomb-normalized spectral power vs. frequency in cycles/days for 1999 CD<sub>158</sub>: the Lomb periodogram suggests a rotational period of 3.44 hr (6.98 cycles/day).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 14.** 1999 CD<sub>158</sub> lightcurve: rotational phase curve for 1999 CD<sub>158</sub> obtained by using a rotational period of 7.1 hr (plot (a)), and a rotational period of 6.88 hr (plot (b)). Continuous lines are Fourier fits of the photometric data. Different symbols correspond to different data sets.</figcaption></figure>

in 2008 using the 3.6 m NTT located at the La Silla Observatory (Chile). The main purpose of their work was to derive BVRI colors, and their data are not entirely suitable for lightcurve study (see Snodgrass et al. 2010 for more details and observing circumstances). Thanks to the ESO archive, we downloaded and re-reduced images obtained by Snodgrass et al. (2010). By merging our sample and their R-data, we obtained a single-peaked periodicity of 3.44 hr (Figure 13). Based on the large lightcurve amplitude ($\Delta m = 0.49 \pm 0.03$ mag) and the asymmetric peaks, we favored the double-peaked periodicity of 6.88 hr (Figure 14, plot (b)). The best fit was obtained for the double-peaked lightcurve, and our result is confirmed.

## 4.8. 2000 CG<sub>105</sub>

2000 CG<sub>105</sub> was observed only one night in 2015 with the DCT. From about 2 hr of observations, we report a 0.2 mag amplitude variation. We searched for a rotational periodicity but, unfortunately, with only few observational hours, we are not able to propose a reliable rotational-period estimation. We try to merge our data with R-band images from Snodgrass et al.
<figure><img src="image_18.png" /><figcaption>**Figure 15.** Lomb-normalized spectral power vs. frequency in cycles/days for 2002 GH<sub>32</sub>: the Lomb periodogram favors a periodicity of 1.99 hr (12.06 cycles/days).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure 16.** 2002 GH<sub>32</sub> lightcurve: rotational phase curve for 2002 GH<sub>32</sub> obtained by using a rotational period of 3.98 hr. The continuous line is a Fourier fit of the photometric data. Different symbols correspond to different observing dates.</figcaption></figure>

(2010), but we are not able to derive a secure rotational period. However, based on Snodgrass et al. (2010) and our data, we can confirm the large variability of this object.

## 4.9. 2002 GH<sub>32</sub>

2002 GH<sub>32</sub> has not been observed for a lightcurve before. We observed this object during one night in March and one night in 2015 April with the DCT. We obtained a single-peaked period of 1.99 hr. Lomb periodogram and other techniques confirmed such a periodicity (Figure 15). However, the lightcurve is asymmetric and so we favored a double-peaked periodicity of 3.98 hr. In Figure 16, the corresponding lightcurve with a peak-to-peak amplitude of 0.36 ± 0.02 mag is plotted. Harris et al. (2014) noticed that the fourth harmonic is potentially the primary harmonic for an asteroid with a lightcurve dominated by shape and an amplitude up to ~0.38 mag. Assuming the second harmonic, the goodness of the fit is $\chi^2 = 1.310$, whereas the fourth harmonic has $\chi^2 = 1.339$. In conclusion, the second harmonic is favored.

<figure><img src="image_20.png" /><figcaption>**Figure 17.** Lomb-normalized spectral power vs. frequency in cycles/days for 2003 HA<sub>57</sub>: the Lomb periodogram shows one main peak located at 3.22 hr (7.46 cycles/days), and two other peaks at 3.83 hr (6.27 cycles/days), and at 2.79 hr (8.59 cycles/days).</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 18.** 2003 HA<sub>57</sub> lightcurve: rotational phase curve for 2003 HA<sub>57</sub> obtained by using a rotational period of 3.22 hr (single-peaked lightcurve, plot (a)), and a period of 6.44 hr (double-peaked lightcurve, plot (b)). Different symbols correspond to different dates. Continuous lines are a Fourier fit of the photometric data. Same legend for both plots.</figcaption></figure>

The first minimum of the curve is deeper than the second one by ~0.08 mag. Such a lightcurve suggests that this object has a very elongated shape. In addition to similarities with Haumea, it is also interesting to point out that this object is the second fastest rotator in the trans-Neptunian belt after Haumea.

## 4.10. 2003 HA<sub>57</sub>

We observed 2003 HA<sub>57</sub> during two consecutive nights in 2015 March and one night in 2015 May with the DCT. We derived a single-peaked lightcurve with a rotational period of 3.22 hr (Figure 17). All techniques confirmed such a periodicity. However, due to the large amplitude, such a curve is probably shape-dominated and so we have to consider a double-peaked periodicity of 6.44 hr. In Figure 18, the single- and double-peaked options are plotted (plots (a) and (b), respectively). We report an amplitude peak-to-peak of 0.31 ± 0.03 mag. The $\chi^2$ value is 1.486 for the second harmonic. We
<figure><img src="image_22.png" /><figcaption>**Figure 19.** Lomb-normalized spectral power vs. frequency in cycles/days for 2005 GE<sub>187</sub>: the Lomb periodogram shows one main peak located at 5.99 hr (4 cycles/days), and two other peaks at 7.81 hr (3.07 cycles/days), and at 4.87 hr (4.93 cycles/days).</figcaption></figure>

also tested the goodness of the fit considering higher
harmonics, but none of them are favored based on our data.

4.11. 2003 HX$_{56}$s

We observe this object ~4 hr during one night with DCT.
With only a few images, we are not able to derive a secure
rotational period. However, several constraints can be reported.
We found a lightcurve amplitude higher than 0.4 mag and a
rotational period higher than 5 hr. Based on the large
amplitude, we may have to consider a double-peaked light-
curve (rotational period higher than 10 h). 2003 HX$_{56}$ has not
been observed for short-term variability before and so more
data are needed to complete this lightcurve.

4.12. 2005 GE<sub>187</sub>

Snodgrass et al. (2010) observed 2005 GE<sub>187</sub> in 2008 using the
3.6 m NTT located at La Silla Observatory (Chile). Besides the
fact that their main goal was to obtain BVRI colors, they noted that
a single-peaked lightcurve with a period of 6.1 hr and an amplitude
of 0.5 mag seemed reasonably convincing for this object.

We observed this object during one night with the DCT on
2015 April under very poor seeing conditions. Based on our data
set, we derived a possible single-peaked lightcurve with a
rotational period of 5.57 hr and a lightcurve amplitude of
0.31 mag. We decided to take advantage of R-filter data from
Snodgrass et al. (2010) and include them in our study. Using the
ESO archive, we downloaded and re-reduced their images. By
merging our sample and Snodgrass et al. (2010) R-data, we
obtained a single-peaked periodicity of 5.99 hr (Figure 19). Based
on the large lightcurve amplitude (Δm = 0.29 ± 0.02 mag) and
the asymmetric peaks, such a curve is probably shape-dominated
and therefore we favored the double-peaked rotational period of
11.99 hr (Figure 20). For this object, we calculated a χ² of 1.697
for the double-peaked lightcurve and χ² = 1.959 for the single-
peaked option. In conclusion, based on our data set, the best
lightcurve is the double-peaked lightcurve. Higher harmonics
have been considered, but all of them have been discarded based
on our data.

<figure><img src="image_23.png" /><figcaption>**Figure 20.** 2005 GE<sub>187</sub> lightcurve: rotational phase curve for 2005 GE<sub>187</sub> obtained by using a rotational period of 11.99 hr. The continuous line is a Fourier fit of the photometric data. Different symbols correspond to different data sets.</figcaption></figure>

5. ROTATIONAL PROPERTIES AND DERIVED
PARAMETERS

Here, we report lightcurves for all the family members
(except Haumea's satellites). Despite studying the most
complete sample to date, the Haumea family is only currently
known to have 11 objects; therefore, results presented here are
preliminary.

5.1. *Rotational Period Distributions*

In the asteroid belt, members of dynamical families are
thought to be fragments of collisions that could influence the
spin properties of the family members (Paolicchi et al. 2002). If
one assumes that the Haumea family is the result of a collision,
one might expect the rotational properties of the family to be
different from other TNOs.

In Figure 21, all the TNOs with a known rotational period
are plotted and we highlighted the Haumea family members
and candidates. A running mean is also reported for all samples
considered. Other TNO and candidate samples exhibit a mostly
flat running mean; therefore, there is not a clear tendency
between size and rotational period. However, smaller family
members tend to spin slower than the larger members.

We noticed a tendency between size and rotational period for
the Haumea family members suggesting that the smaller
members of the family rotate slower than the biggest ones.
With a significance level of 98.43%, and a Spearman
coefficient of 0.638, there is strong evidence of correlation
between absolute magnitude and rotational period for the
family members (see Section 5.5). We also tested the
probability that the family members and other TNOs are from
the same distribution using the 2D Kolmogorov–Smirnov test
(KS test). The KS test estimates the maximum deviation
between the cumulative distribution of both data sets to test the
similarity (or not) between the two distributions (Df). The
significance level of the KS test is a value between 0 and 1.
Small values show that the cumulative distribution of the first
data set is significantly different from the second data set.
Considering two samples of the family members and the other
TNOs, we obtained a value of Df = 0.54 and a significance
level of 0.004, indicating that the rotational periods of the
<figure><img src="image_24.png" /></figure>

**Figure 21.** Lightcurve amplitude and rotational period vs. absolute magnitude: black circles for the confirmed Haumea family members, open circles for the candidates, and gray triangle for the other TNOs. Same legends for both plots. Plot (a): continuous black horizontal line represents the shape- or albedo-dominated lightcurve as in Thirouin et al. (2012, 2014). Plot (b): spin barrier (dash horizontal line) around 4 hr as suggested in Thirouin et al. (2010). Absolute magnitudes from the Minor Planet Center (MPC). Lightcurve amplitudes and rotational periods are from Table 3 and Thirouin (2013). In the case of multiple rotational periods proposed in the literature for the same object, we computed the mean period and the corresponding range of values. Pluto-Charon and Sila-Nunam are not plotted. Running means are also plotted: black continuous line for the Haumea family members, discontinuous black line for the candidates, and gray line for the other TNOs.

family are significantly different. In Figure 22, rotational frequency of the family members and candidates are plotted (data from this work and the literature). The Maxwellian fits<fnref n="10" /> to the family sample give a mean rotational period of 6.27 ± 1.19 hr, whereas the candidates have a mean rotational period of 6.57 ± 1.14 hr.<fnref n="11" /> In Figure 23, several samples without the family members and with/without the candidates and the binary population are plotted.<fnref n="12" /> Maxwellian fit to the other TNOs sample gives a mean rotational period of 8.98 ± 0.77 hr. In Table 5, the mean rotational periods from Maxwellian fits as well as average and median for several samples including/ excluding the family members or the candidates are reported. Standard error and standard deviation are indicated for all the samples. Without considering the error bars, average and median rotational periods of the family members indicate that the objects in relation to Haumea seem to rotate faster than the other TNOs. However, because of the large error bars regarding the family sample, such a tendency may not be true.

Regarding the candidate sample, we found no evidence of correlation between absolute magnitude and rotational period. The running mean suggests a flat distribution similar to the one for the other TNOs sample. Maxwellian fit to the candidate sample favors a mean rotational period comparable to the family (but large error bars have to be considered). The candidate sample may be “contaminated” by icy members of the family yet to be identified, as well as objects without relation with the Haumea family. Despite the fact that 2008

AP129 is not a confirmed member of the family, it follows the tendency between size and rotational period as the rest of the family.

Several families have been identified and studied in the Main Belt of asteroids. Based on the Koronis and Eos family studies, the largest fragments of the families appear to have relatively similar rotation rates (Binzel et al. 1989, p. 416). In other words, the largest fragments “remember” the spin rate of their parent body. The largest<fnref n="13" /> members of the family are Haumea, 2002 TX300, and 2003 OP32 with rotational periods between ~4 hr and ~8 hr. The lightcurve of 2002 TX300 is very flat and thus the rotational period may have significant uncertainty (Sheppard & Jewitt 2003; Ortiz et al. 2004; Thirouin et al. 2010, 2012). If 2003 OP32 has a single-peaked lightcurve, then two of the three largest members of the family Haumea and 2003 OP32 are also the fastest rotators of the family. If the largest fragments of the family remember the spin rate of their parent body, we can conclude that the proto-Haumea was also a fast rotator with an elongated shape due to its rotation. We emphasize that a catastrophic collision is not able to create a fast-spinning elongated object. In fact, Leinhardt et al. (2010) simulated a catastrophic collision with exactly the same parameters proposed by Brown et al. (2007). The results of their simulations show that the largest remnant is bigger than the current Haumea, and such a catastrophic collision produced a slow rotator with a rotational period of 28 hr, far from the 3.92 hr period of the current Haumea. Similar conclusion regarding the slow rotation has been obtained by Takeda & Ohtsuki (2009). Numerical simulations of catastrophic disruptions with enough resolution to resolve the shape of the largest remnant produce spherical objects, not fast-spinning elongated remnants (Leinhardt et al. 2000; Leinhardt & Stewart 2009).

Some experiments have focused on the rotation of fragments in catastrophic impacts for scenarios relevant to asteroid-like objects (Fujiwara & Tsukamoto 1981). They performed catastrophic destruction of basalt targets by impacts of high-velocity projectiles. The rotational periods of the ejected fragments were measured as a function of particle size. Review of catastrophic disruption experiments using a wider range of materials has been reported by Martelli et al. (1994). Both publications reported that the general tendency is that the smaller fragments have shorter rotational period than the bigger ones. Obviously, the size range of the fragments as well as the target composition are different for the Haumea case. However, if the relation noticed by Fujiwara & Tsukamoto (1981) and Martelli et al. (1994) is independent of size and composition, one could suggest that the Haumea family was not formed during a catastrophic collision. In conclusion, there is no clear explanation of such a relation between size and rotational period in the Haumea family.

## 5.2. Lightcurve Amplitude Distributions

In Figure 24, we focused on the lightcurve amplitude distributions for objects with/without relation with Haumea. Only two members of the family present a large lightcurve amplitude, Haumea and 2003 SQ317, and maybe 1996 TO66. The mean lightcurve amplitude for the confirmed members is 0.19 mag, and only 0.12 mag without the contact binary, 2003

<footnote n="10">¹⁰ We used Maxwellian fits based on the works of Salo (1987) and Binzel et al. (1989).</footnote>

<footnote n="11">¹¹ Mean rotational rate in cycles/days, $\Omega$, is $(8/\pi)^{0.5} \times \sigma$ where $\sigma$ is the width of the distribution.</footnote>

<footnote n="12">¹² We have shown that the rotational properties of the binary population are different from the non-binary one and so we removed the binary population in some samples (Thirouin et al. 2014).</footnote>

<footnote n="13">¹³ Haumea family member sizes have large uncertainty mainly because albedo is only known for two objects (see Table 7), and so we use here the absolute magnitude to report the largest fragments.</footnote>
<table><caption>**Table 4**
Summary of Results from This Work</caption><thead><tr><th>Object</th><th>Phot. per.<br>(hr)</th><th>Rot. per.<br>(hr)</th><th>Δm<br>(mag)</th><th>φ<sub>0</sub> [JD]<br>(2450000+)</th><th>Asym.<br>LC?</th><th>Bin.<br>?</th></tr></thead><tbody><tr><td>1995 SM<sub>55</sub></td><td>4.04</td><td>8.08</td><td>0.04 ± 0.02</td><td>2193.90249<sup>a</sup></td><td>Yes</td><td>No</td></tr><tr><td>1999 OY<sub>3</sub></td><td>9.01</td><td>18.02</td><td>0.08 ± 0.02</td><td>7167.88515</td><td>Yes</td><td>No</td></tr><tr><td>2003 OP<sub>32</sub></td><td>4.85</td><td>4.85</td><td>0.14 ± 0.02</td><td>3588.39312</td><td>No</td><td>No</td></tr><tr><td>2003 UZ<sub>117</sub></td><td>5.65</td><td>11.29</td><td>0.09 ± 0.01</td><td>4438.54307<sup>b</sup></td><td>Yes</td><td>No</td></tr><tr><td>2009 YE<sub>7</sub></td><td>5.65</td><td>5.65</td><td>0.06 ± 0.02</td><td>5833.71848<sup>c</sup></td><td>No</td><td>?</td></tr><tr><td>2008 AP<sub>129</sub></td><td>9.04</td><td>9.04</td><td>0.12 ± 0.02</td><td>5952.41458</td><td>No</td><td>?</td></tr><tr><td>1999 CD<sub>158</sub></td><td>3.44</td><td>6.88</td><td>0.49 ± 0.03</td><td>7105.61716</td><td>Yes</td><td>No</td></tr><tr><td>2000 CG<sub>105</sub><sup>d</sup></td><td>&gt;2</td><td>...</td><td>&gt;0.2</td><td>7106.61325</td><td>...</td><td>No</td></tr><tr><td>2002 GH<sub>32</sub></td><td>1.99</td><td>3.98</td><td>0.36 ± 0.02</td><td>7106.59003</td><td>Yes</td><td>?</td></tr><tr><td>2003 HA<sub>57</sub></td><td>3.22</td><td>6.44</td><td>0.31 ± 0.03</td><td>7105.86625</td><td>Yes</td><td>?</td></tr><tr><td>2003 HX<sub>56</sub><sup>e</sup></td><td>&gt;5</td><td>&gt;10</td><td>&gt;0.4</td><td>7168.65047</td><td>...</td><td>?</td></tr><tr><td>2005 GE<sub>187</sub></td><td>5.99</td><td>11.99</td><td>0.29 ± 0.02</td><td>4622.47907<sup>f</sup></td><td>Yes</td><td>No</td></tr></tbody></table>

**Notes.** In this table, we present the preferred rotational period (Rot. per. in hour), the preferred photometric period (Phot. per. in hour), the peak-to-peak lightcurve amplitude (Δm in magnitude), and the Julian Date (φ₀) for which the phase is zero in our lightcurves. The Julian Date is without light time correction. The preferred photometric period is the periodicity obtained thanks to the data reduction. Preferred rotational period is estimated based on our criteria to distinguish if a lightcurve is shape- or albedo-dominated and based on the asymmetry of the lightcurve (see Section 3.3 for more details). We also report if the lightcurve is asymmetric or not (Asym. LC column) and if the object is known to be a binary one (i.e., binary companion detected or not with the Hubble Space Telescope (HST), last column). Some objects have not been observed with HST to detect binarity and are indicated with a question mark.

a Zero phase from Sheppard & Jewitt (2003).

b Zero phase from Perna et al. (2009).

c Zero phase from Benecchi & Sheppard (2013).

d Amplitude variation based on ~2 hr of observations (see Section 4.8).

e Constraints for lightcurve amplitude and rotational period. Based on the large amplitude, we favor the double-peaked option.

f Zero phase from Snodgrass et al. (2010).

<figure><img src="image_25.png" /><figcaption>**Figure 22.** Number of objects vs. rotational rate in cycles/days. Two different samples are plotted: confirmed members of the family (Number of objects (N) = 11, black bars) and candidates (N = 12, gray bars). A Maxwellian fit to the confirmed family members gives a mean rotational period of 6.27 ± 1.19 hr (continuous black line). The Maxwellian fit of the sample with the candidates gives a mean rotational period of 6.44 ± 1.16 hr (continuous gray line).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 23.** Number of objects vs. rotational rate in cycles/days. Three different samples are plotted: other TNOs with candidates and 2008 AP<sub>129</sub> (i.e., all TNOs except confirmed members of the family; number of objects (N) = 90, black bars), other TNOs without candidates (N = 78, gray bars), and other TNOs without the binary population and candidates (N = 53, white bars). A Maxwellian fit to the first sample gives a mean rotational period of 8.98 ± 0.77 hr (continuous black line). The Maxwellian fit of the second sample gives a mean rotational period of 7.65 ± 0.54 hr (continuous gray line). Maxwellian fit to the third sample gives a mean rotational period of 8.98 ± 0.77 hr (discontinuous black line).</figcaption></figure>

SQ<sub>317</sub>. Thus, most of the family members have low lightcurve amplitudes and thus should be considered as spherical (or nearly spherical) objects also known as MacLaurin spheroids. On the other hand, the candidates have larger lightcurve amplitudes with a mean lightcurve amplitude of 0.20 mag. Such a mean value increases up to 0.28 mag if we only consider the smallest candidates with absolute magnitudes higher than 5. Objects with a large lightcurve amplitude have an elongated shape and are usually named Jacobi ellipsoids.

This tendency has already been reported in the other TNOs sample (Duffard et al. 2009; Thirouin et al. 2012, 2014). In fact, smaller objects tend to have higher lightcurve amplitudes (Figure 21). Thus, it seems reasonable to expect that the candidates (i.e., the smallest objects) have higher lightcurve amplitudes. However, we cannot totally discard the fact that
<caption>Table 5</caption>

"STDEV" Stands for Standard Deviation and "SE" Stands for Standard Error, and Both Are Associated with the Average and Median Rotational Periods

<table><thead><tr><th>Sample</th><th>Average (hr)</th><th>Median (hr)</th><th>SE/STDEV (hr)</th><th>Maxwellian (hr)</th></tr></thead><tbody><tr><td>Confirmed<sup>a</sup></td><td>7.90</td><td>7.21</td><td>1.24/3.92</td><td>6.27 ± 1.19</td></tr><tr><td>Candidates</td><td>8.09</td><td>6.88</td><td>1.43/4.53</td><td>6.44 ± 1.16</td></tr><tr><td>Other TNOs, and candidates</td><td>13.85</td><td>8.55</td><td>8.24/26.06</td><td>8.98 ± 0.59</td></tr><tr><td>Other TNOs, no candidates</td><td>14.64</td><td>8.84</td><td>8.82/27.89</td><td>7.65 ± 0.54</td></tr><tr><td>Other TNOs, no candidates, no binary pop</td><td>8.96</td><td>8.22</td><td>1.04/3.29</td><td>8.98 ± 0.77</td></tr></tbody></table>

**Notes.** The last column reports the mean rotational period obtained from the Maxwellian Fits. All values are in hours. We consider several sub-samples to account for the confirmed family members, the candidates, and the other TNOs. We also removed the binary population (labeled as binary pop) from the last sample because it has been shown that such a population is affected by tidal effects between the components of the system, and therefore their rotational period is not primordial (for more details see Thirouin et al. 2014).

<sup>a</sup> Average is 6.89 hr, median is 6.99 hr and SE/STDEV are, respectively, 0.68/2.14 for the confirmed family members without the slow rotator 1999 OY<sub>3</sub>.

such a tendency has some connection to the formation of the Haumea family (assuming that the candidates are members of the family).

### 5.3. Size and Density

According to Binzel et al. (1989), if we assume that TNOs are triaxial ellipsoids with axes of $a > b > c$ and rotate along the c-axis, the lightcurve amplitude ($\Delta m$) varies as a function of the observational (or viewing) angle $\xi$ as

$$ \Delta m = 2.5 \log \left( \frac{a}{b} \right) - 1.25 \log \left( \frac{a^2 \cos^2 \xi + c^2 \sin^2 \xi}{b^2 \cos^2 \xi + c^2 \sin^2 \xi} \right). \quad (1) $$

The lower limit for the object elongation ($a/b$) is obtained assuming an equatorial view ($\xi = 90^{\circ}$). For a random distribution of spin vectors, the probability of viewing an object on the angle range of [$\xi, \xi + d\xi$] is proportional to $\sin(\xi)d\xi$. The average viewing angle is $\xi = 60^{\circ}$ (Sheppard 2004). We will consider viewing angles of 60° and 90°.

According to the study of equilibrium figures for fluid bodies by Chandrasekhar (1987), one can estimate lower limits for densities from rotational periods and the elongation of objects. We want to point out that it is difficult to estimate the size transition for an object to be in hydrostatic equilibrium or not. Lacerda & Luu (2003, 2006), based on observational and numerical considerations, noticed a cut-off for objects with diameters of ~400 km (assuming an albedo of 0.04). According to their studies, small objects have irregular shapes whereas large ones have a nearly spherical shape. This cut-off can be interpreted as a transition between spherical objects in hydrostatic equilibrium and rubble-pile objects. The works of Lacerda and Luu were based on a limited sample of lightcurves. By including newer results and assuming an albedo of 0.12 (which is more appropriate based on recent results), this cut-off is closer to ~250–300 km (Vilenius et al. 2012, 2014; Thirouin 2013). On the other hand, Duffard

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /><figcaption>**Figure 24.** Number of objects vs. lightcurve amplitude. (Upper plot) Two different samples are plotted: confirmed members of the family (number of objects ($N$) = 11, black bars) and candidates ($N$ = 12, gray bars). (Lower plot) Two different samples are plotted: other TNOs (i.e., all TNOs except confirmed members of the family; number of objects ($N$) = 90, black bars), other TNOs, except the binary population, the confirmed and candidates members of the family ($N$ = 54, gray bars).</figcaption></figure>

et al. (2009), based on a simple Monte-Carlo model suggested that even small objects of about 120 km are in hydrostatic equilibrium. Finally, Lineweaver & Norman (2010) derived the potato-to-sphere transition for an icy body with a diameter of ~200 km, and ~300 km for rocky asteroids. This means that icy objects with diameters smaller than about 200 km have an irregular shape whereas the biggest objects have a spherical shape. However, they are not considering that TNOs are rubble-piles. In this work, we used a conservative cut-off of 250–350 km for the transition between spheroidal and elongated objects. In conclusion, TNOs in the 200 km and below that size range can be rubble-piles. Collisional evolution models of the trans-Neptunian region indicate that objects in this size range have received several collisions on average, so they are likely re-accumulated objects. Besides, Duffard et al. (2009) indicated that the lightcurve statistics of small objects is consistent with the figures of hydrostatic equilibrium.

Assuming that a given TNO is a triaxial ellipsoid (Jacobi ellipsoid) in hydrostatic equilibrium, one can compute a lower density limit (Figure 25 and Table 6). Most of our targets have low amplitude lightcurves (i.e., low elongation), and their
<figure><img src="image_29.png" /><figcaption>**Figure 25.** Lightcurve amplitude vs. rotational period for theoretical Jacobi ellipsoids of various densities compared with observations. Density values are indicated in the legend. Vertical lines have been computed using Chandrasekhar (1987). All objects presented in this work are shown: black circles for the confirmed Haumea family members, and open circles for the candidates. The continuous black horizontal line represents the shape- or albedo-dominated lightcurve as in Thirouin et al. (2012) and Thirouin et al. (2014). Lightcurve amplitudes and rotational periods are from Table 3.</figcaption></figure>

<table><caption>**Table 6**
Elongation and Lower Limit to the Density for All Objects Studied
Are Summarized in this Table</caption><thead><tr><th>Object</th><th>b/a<br>eq.view</th><th>b/a<br>ξ = 60°</th><th>ρ<br>eq.view<br>(g cm<sup>-3</sup>)</th><th>ρ<br>ξ = 60°<br>(g cm<sup>-3</sup>)</th></tr></thead><tbody><tr><td>1995 SM<sub>55</sub></td><td>0.95</td><td>0.83</td><td>&gt;0.60</td><td>&gt;0.60</td></tr><tr><td>1999 OY<sub>3</sub></td><td>0.93</td><td>0.81</td><td>&gt;0.12</td><td>&gt;0.12</td></tr><tr><td>2003 OP<sub>32</sub></td><td>0.88</td><td>0.76</td><td>&gt;1.66</td><td>&gt;1.70</td></tr><tr><td>2003 UZ<sub>117</sub></td><td>0.92</td><td>0.80</td><td>&gt;0.31</td><td>&gt;0.31</td></tr><tr><td>2009 YE<sub>7</sub></td><td>0.95</td><td>0.82</td><td>&gt;1.22</td><td>&gt;1.23</td></tr><tr><td>2008 AP<sub>129</sub></td><td>0.90</td><td>0.78</td><td>&gt;0.48</td><td>&gt;0.49</td></tr><tr><td>1999 CD<sub>158</sub></td><td>0.63</td><td>0.54</td><td>&gt;0.85</td><td>&gt;0.89</td></tr><tr><td>2000 CG<sub>105</sub><sup>a</sup></td><td>0.83</td><td>0.72</td><td>&gt;0.61</td><td>&gt;0.63</td></tr><tr><td>2002 GH<sub>32</sub></td><td>0.72</td><td>0.62</td><td>&gt;2.56</td><td>&gt;2.68</td></tr><tr><td>2003 HA<sub>57</sub></td><td>0.75</td><td>0.65</td><td>&gt;0.87</td><td>&gt;1.01</td></tr><tr><td>2003 HX<sub>56</sub><sup>b</sup></td><td>0.69</td><td>0.60</td><td>&gt;0.41</td><td>&gt;0.43</td></tr><tr><td>2005 GE<sub>187</sub></td><td>0.63</td><td>0.55</td><td>&gt;0.89</td><td>&gt;0.94</td></tr></tbody></table>

**Notes.** We consider two cases: (1) equatorial view (viewing angle of 90°) and (2) viewing angle of 60°. Lower limit to the density has been computed using Chandrasekhar (1987). As mentioned in the discussion, this model assumes that objects are in hydrostatic equilibrium and are triaxial (Jacobi) objects. Lower limits for maclaurin spheroids are also reported here, but one has to keep in mind that these densities are based on assumptions that do not hold for this kind of objects.

<sup>a</sup> Axis ratio derived assuming a lightcurve with a 0.2 mag as amplitude. Assuming a rotational period of 8 hr for this object (mean rotational period of the non-binary TNOs as reported in Thirouin et al. 2014), we derived a lower limit to the density ~0.6 g cm⁻³.

<sup>b</sup> Axis ratio derived assuming a lightcurve with a 0.4 mag amplitude. Assuming a rotational period of 10 hr for this object, we derived a lower limit to the density ~0.4 g cm⁻³.

lightcurves are probably due to albedo effects. In other words, they are MacLaurin spheroids and the study on lower limit densities cannot be applied. In fact, most of the observed objects are far from the theoretical curves for acceptable values for the density, which indicates that those objects are likely spheroids or are not in hydrostatic equilibrium (Thirouin et al. 2014). Only a few members have a high amplitude lightcurve

<figure><img src="image_30.png" /><figcaption>**Figure 26.** Density vs. absolute magnitude: two samples are plotted: confirmed Haumea family members (black circles) and candidates (open circles). Most of the densities are only lower limit densities derived from lightcurves assuming an equatorial view (see the discussion). In the case of 2003 SQ<sub>317</sub>, density, assuming a contact binary and reported in Lacerda et al. (2014), has been used. It seems that the smallest members have the lowest density, except in the case of the contact binary, which seems to have the highest density. Lower limits to density are reported in Table 6 for objects studied in this work. In the case of Makemake and Salacia, we used densities from Ortiz et al. (2012) and Stansberry et al. (2012). For candidates not studied in this work, but with a rotational period estimate, we derive their densities as in Section 5.3. Absolute magnitudes are from the Minor Planet Center (MPC).</figcaption></figure>

(>0.15 mag): Haumea, 2003 SQ<sub>317</sub>, and maybe 1996 TO<sub>66</sub>. Haumea has a high density, ~2.5 g cm⁻³ (Rabinowitz et al. 2006; Thirouin et al. 2010) whereas 1996 TO<sub>66</sub> seems to have a density higher than 1 g cm⁻³ (Figure 25). 2003 SQ<sub>317</sub> is a contact binary with a density of 2.7 g cm⁻³ (or 0.8 g cm⁻³ assuming a single object, but based on the lightcurve, this object seems to be a contact binary and therefore the highest density will be considered), according to Lacerda et al. (2014). All the candidates presented here have large lightcurve amplitudes and so can be considered as Jacobi objects. For the candidates, the range of densities varies from 0.9 to 2.6 g cm⁻³, and varies from 0.12 to 1.7 g cm⁻³ for the confirmed family members. Mean density for the members is 1.1 g cm⁻³ (0.7 g cm⁻³ without the contact binary, and Haumea), and is 1 g cm⁻³ for the candidates. We want to emphasize the case of 2002 GH<sub>32</sub>. This object is a very fast rotator with a double-peaked periodicity of 3.98 hr and a lightcurve amplitude of about 0.36 mag. The morphology of the 2002 GH<sub>32</sub> lightcurve is similar to Haumea's. In fact, both objects present a fast rotation as well as an asymmetric lightcurve. One possible interpretation of such a lightcurve is that 2002 GH<sub>32</sub> has an elongated shape with strong albedo variation on its surface, as is the case for Haumea (Lacerda et al. 2008). From the lightcurve, we derived a lower limit to the density of 2.6 g cm⁻³ (equatorial view) or 2.7 g cm⁻³ (viewing angle of 60°).

In Figure 26, as previously noticed in Sheppard et al. (2008) the biggest objects have higher densities than the smallest ones. Based on the family sample, an anti-correlation with ρ = -0.391, and a low significance level of 78.36% is found. Without the contact binary, 2003 SQ<sub>317</sub>, a reasonably strong anti-correlation with ρ = -0.733 and a significance level of 97.33% is noticed. As pin-pointed by Lacerda et al. (2014), the contact binary presents several atypical characteristics. To explain the lightcurve of this object, authors considered two cases: (i) ellipsoidal object (Jacobi ellipsoid), and (ii) compact
binary near hydrostatic equilibrium (Roche binary). According to their models, 2003 SQ317 may have a density between 670 and 1100 kg cm⁻³ considering a Jacobi object, or a density between 2050 and 3470 kg cm⁻³ based on a Roche model (see Lacerda et al. 2014 for more details). Assuming that the family members are from the icy mantle of the proto-Haumea, members would be expected to have an icy composition. The high density of the contact binary option suggests that it must be a fragment from the rocky core of the proto-Haumea. But, assuming an ellipsoidal object, the lower density indicates an icy composition. On the other hand, visible and near-infrared colors of this object confirm its membership, despite a much steeper phase function as the other members. In conclusion, as mentioned by Lacerda et al. (2014), both options (Jacobi ellipsoid and Roche binary) are potential options. Only more observations over the next decade can confirm one of those options. Assuming a density of 860 kg cm⁻³ (i.e., Jacobi option) for 2003 SQ317, we found an anti-correlation between density and absolute magnitude with a ρ = -0.636, and a significance level of 95.58% corresponding to a reasonably strong anti-correlation. Regarding only the candidate population, the trend between size and density is not evident (ρ = -0.251, significance level of 57.19%).

## 5.4. Total Haumea Family Mass

The mass of the family can be roughly estimated as follows. First, we computed the diameter and the mass of each confirmed member (Table 7). The diameter (D) according to Pravec & Harris (2007) can be estimated by

$$D = \frac{K}{\sqrt{p}} 10^{-0.2 H} \quad (2)$$

where p is the geometric albedo and H is the absolute magnitude. The constant K is

$$K = 2 \text{ au} \times 10^{\frac{V_{\text{Sun}}}{5}} \quad (3)$$

where V_Sun is the visual magnitude of the Sun. Assuming that the objects are spherical, the mass M is

$$M = \frac{4}{3} \pi \rho R^3 \quad (4)$$

where ρ is the density and R is the radius of the object. By combining the previous equations, one can derive the mass, M, from

$$M = \frac{\pi \rho}{6} \left( \frac{K \times 10^{-0.2 H}}{\sqrt{p}} \right)^3 \quad (5)$$

An albedo of 0.7 has been assumed for the confirmed members of the family without a known albedo. Such an estimation is reasonable because all members are believed to be from the icy mantle of the proto-Haumea. For the candidates without a known albedo, we used albedos of 0.08 and 0.30 to derive a range of possible sizes and masses. Based on the masses computed and reported in Table 7, we found a total mass of 4.06 × 10²¹ kg for the known family members (without Haumea, the total mass is 5.68 × 10¹⁹ kg ≈ 2%M_Haumea, where M_Haumea is the mass of Haumea). We did not include 2008 AP129 as a member of the family because its membership is not yet confirmed

(contribution would be very small). This mass estimation is obviously a lower limit because more small icy family members (and maybe rocky members) are expected to be found. On the other hand, as several members have no albedo reported, computed sizes indicated in Table 7 are only estimations. A more complete review will be proposed by E. Vilenius et al. (in preparation) in which size and geometric albedos of several family members as well as some candidates will be derived thanks to thermal modeling of Herschel Space Observatory data.

## 5.5. Correlation/Anti-correlation Search

We searched for correlations between physical (rotational period and lightcurve amplitude) and orbital parameters. We used the Spearman rank correlation (Spearman 1904) because this method is less sensitive to atypical/wrong values and does not assume any population probability distribution. We computed the strength of the correlations by calculating the Spearman coefficient ρ and the significance level (SL). The ρ coefficient has values between -1 and 1. If ρ > 0, there is a possible correlation, whereas ρ < 0 indicated a possible anti-correlation, and if ρ = 0, there is no correlation. We consider a correlation as (1) strong if |ρ| > 0.6, (2) weak if 0.3 < |ρ| < 0.6, and (3) non-existent if |ρ| < 0.3. The significance of the ρ parameter is measured by the SL: (1) very strong evidence of correlation if SL > 99% (i.e., 3σ), (2) strong evidence of correlation if SL > 97.5% (i.e., 2.5σ), and (3) reasonably strong evidence of correlation if SL > 95% (i.e., 2σ). Such criteria have been used in several studies of correlations/anti-correlations between colors and orbital elements, for example, in Peixinho et al. (2008, 2012, 2015).

Our search for correlations/anti-correlations is reported in Table 8. We used several samples, such as the entire family with/without the candidates, as well as other TNOs with/without the binary population (centaurs are not taken into account). As reported in Thirouin et al. (2014), the binary population seems to exhibit distinct characteristics; therefore, special care has been taken to remove this population. We also point out that the family and candidate samples are limited and so care has to be taken regarding the correlation/anti-correlation detection and interpretation. Only correlations regarding the family and candidates with a significance level higher than 95% and |ρ| ≥ 0.3 are reported in Table 8.

**Rotational period versus orbital elements:** correlations between rotational period and ascending node are only reported in several sub-samples but seem to be a specific characteristic of the candidates pool. It is also interesting to point out that such correlations are not presented in the other TNOs populations. We also report an anti-correlation between rotational period and inclination in the family. The most interesting and significant correlation is between rotational period and absolute magnitude. It seems that the biggest members of the family are rotating faster than the smallest ones. Such a correlation is only found in the family, not in the candidate nor other TNO pools. As mentioned in the previous section, this correlation, if real, is not yet understood.

**Lightrurve amplitude versus orbital elements:** a well-known correlation is the one between lightcurve amplitude and absolute magnitude (Duffard et al. 2009; Thirouin et al. 2010). However, such a correlation is not found in the family sample. In fact, it seems that such a feature is only reported in the other TNOs, and candidates. Anti-correlations between
<table><caption>**Table 7**
We Summarize the Diameter, the Mass, the Albedo, and the Absolute Magnitude for Confirmed and Candidates</caption><thead><tr><th>Object</th><th><i>H</i></th><th>Albedo<sup>a</sup></th><th>Diameter (km)</th><th>Mass<sup>b</sup> (×10<sup>18</sup> (kg))</th><th>Ref.</th></tr></thead><tbody><tr><td colspan="6"><em>Confirmed members:</em></td></tr><tr><td>1995 SM<sub>55</sub></td><td>4.8</td><td>0.70</td><td>149</td><td>1.73</td><td>TW</td></tr><tr><td>1996 TO<sub>66</sub></td><td>4.5</td><td>0.70</td><td>171</td><td>2.62</td><td>TW</td></tr><tr><td>1999 OY<sub>3</sub></td><td>6.8</td><td>0.70</td><td>59</td><td>0.11</td><td>TW</td></tr><tr><td>2002 TX<sub>300</sub></td><td>3.2</td><td>0.88<sup>+0.15</sup><sub>-0.06</sub></td><td>286 ± 10</td><td>12.25</td><td>E10</td></tr><tr><td>Haumea</td><td>0.1</td><td>0.70-0.75</td><td>1150</td><td>4006 ± 40</td><td>L10, R09</td></tr><tr><td>2003 OP<sub>32</sub></td><td>3.6</td><td>0.70</td><td>259</td><td>9.09</td><td>TW</td></tr><tr><td>2003 SQ<sub>317</sub></td><td>6.3</td><td>0.70</td><td>75</td><td>0.22</td><td>TW</td></tr><tr><td>2003 UZ<sub>117</sub></td><td>5.3</td><td>0.70</td><td>118</td><td>0.87</td><td>TW</td></tr><tr><td>Hi'iaka</td><td>2.9</td><td>~0.70</td><td>~320</td><td>17.9 ± 1.1</td><td>R09</td></tr><tr><td>Namaka</td><td>4.5</td><td>~0.70</td><td>~160</td><td>1.79 ± 1.48</td><td>R09</td></tr><tr><td>2005 CB<sub>79</sub></td><td>4.7</td><td>0.70</td><td>156</td><td>1.99</td><td>TW</td></tr><tr><td>2005 RR<sub>43</sub></td><td>4.0</td><td>0.70</td><td>215</td><td>5.23</td><td>TW</td></tr><tr><td>2009 YE<sub>7</sub></td><td>4.4</td><td>0.70</td><td>179</td><td>3.01</td><td>TW</td></tr><tr><td colspan="6"><em>Candidates:</em></td></tr><tr><td>Salacia</td><td>4.4</td><td>0.0439 ± 0.0044</td><td>901 ± 45</td><td>311.96 ± 46.71</td><td>V12</td></tr><tr><td>Makemake</td><td>-0.3</td><td>0.77 ± 0.03</td><td>1430 ± 9</td><td>2761.05 ± 161.26</td><td>O12</td></tr><tr><td>1996 RQ<sub>20</sub></td><td>6.9</td><td>0.08/0.30</td><td>168/87</td><td>2.46/0.34</td><td>TW</td></tr><tr><td>1996 TR<sub>66</sub></td><td>7.5</td><td>0.08/0.30</td><td>127/66</td><td>1.08/0.15</td><td>TW</td></tr><tr><td>1997 RX<sub>9</sub></td><td>8.3</td><td>0.08/0.30</td><td>88/45</td><td>0.36/0.05</td><td>TW</td></tr><tr><td>1998 HL<sub>151</sub></td><td>8.1</td><td>0.08/0.30</td><td>96/50</td><td>0.47/0.06</td><td>TW</td></tr><tr><td>1998 WT<sub>31</sub></td><td>7.2</td><td>0.08/0.30</td><td>146/75</td><td>1.63/0.22</td><td>TW</td></tr><tr><td>1999 CD<sub>158</sub></td><td>5.1</td><td>0.08/0.30</td><td>384/198</td><td>29.62/4.08</td><td>TW</td></tr><tr><td>1999 KR<sub>16</sub></td><td>5.8</td><td>0.204<sup>+0.07</sup><sub>-0.05</sub></td><td>254 ± 37</td><td>4.79 ± 1.95</td><td>SS12</td></tr><tr><td>1999 OH<sub>4</sub></td><td>8.3</td><td>0.08/0.30</td><td>88/45</td><td>0.36/0.05</td><td>TW</td></tr><tr><td>1999 OK<sub>4</sub></td><td>7.6</td><td>0.08/0.30</td><td>121/63</td><td>0.94/0.13</td><td>TW</td></tr><tr><td>1999 RY<sub>215</sub></td><td>7.1</td><td>0.0388<sup>+0.0122</sup><sub>-0.0065</sub></td><td>263<sup>+29</sup><sub>-37</sub></td><td>8.75 ± 2.89</td><td>V14</td></tr><tr><td>2000 CG<sub>105</sub></td><td>6.6</td><td>0.08/0.30</td><td>192/99</td><td>3.73/0.51</td><td>TW</td></tr><tr><td>2000 JG<sub>81</sub></td><td>8.0</td><td>0.08/0.30</td><td>101/52</td><td>0.54/0.07</td><td>TW</td></tr><tr><td>2001 FU<sub>172</sub></td><td>8.3</td><td>0.08/0.30</td><td>88/45</td><td>0.34/0.05</td><td>TW</td></tr><tr><td>2001 QC<sub>298</sub></td><td>6.1</td><td>0.061<sup>+0.027</sup><sub>-0.017</sub></td><td>303<sup>+27</sup><sub>-30</sub></td><td>19.72 ± 9.42</td><td>V14</td></tr><tr><td>2002 AW<sub>197</sub></td><td>3.5</td><td>0.112<sup>+0.012</sup><sub>-0.011</sub></td><td>768<sup>+39</sup><sub>-38</sub></td><td>263.87 ± 40.29</td><td>V14</td></tr><tr><td>2002 GH<sub>32</sub><sup>c</sup></td><td>5.5</td><td>>0.13</td><td>&lt;180</td><td>>13.14</td><td>V14</td></tr><tr><td>2003 HA<sub>57</sub></td><td>8.1</td><td>0.08/0.30</td><td>96/50</td><td>0.47/0.06</td><td>TW</td></tr><tr><td>2003 HX<sub>56</sub></td><td>7.1</td><td>0.08/0.30</td><td>153/79</td><td>1.87/0.26</td><td>TW</td></tr><tr><td>2003 QX<sub>91</sub></td><td>8.3</td><td>0.08/0.30</td><td>88/45</td><td>0.34/0.05</td><td>TW</td></tr><tr><td>2003 TH<sub>58</sub></td><td>7.6</td><td>0.08/0.30</td><td>121/63</td><td>0.94/0.13</td><td>TW</td></tr><tr><td>2004 PT<sub>107</sub></td><td>6.0</td><td>0.0325<sup>+0.0111</sup><sub>-0.0066</sub></td><td>400<sup>+45</sup><sub>-51</sub></td><td>53.99 ± 20.08</td><td>V14</td></tr><tr><td>2005 GE<sub>187</sub></td><td>7.1</td><td>0.08/0.30</td><td>153/79</td><td>1.87/0.26</td><td>TW</td></tr><tr><td>2005 UQ<sub>513</sub></td><td>3.4</td><td>0.202<sup>+0.084</sup><sub>-0.049</sub></td><td>498<sup>+63</sup><sub>-75</sub></td><td>130.30 ± 57.01</td><td>V14</td></tr><tr><td>2008 AP<sub>129</sub></td><td>4.7</td><td>0.08/0.30</td><td>462/238</td><td>51.48/7.09</td><td>TW</td></tr><tr><td>2010 KZ<sub>39</sub></td><td>4.0</td><td>0.08/0.30</td><td>637/329</td><td>135.41/18.65</td><td>TW</td></tr><tr><td>2014 FT<sub>71</sub></td><td>4.7</td><td>0.08/0.30</td><td>462/238</td><td>51.48/7.09</td><td>TW</td></tr></tbody></table>

**Notes.** Absolute magnitudes ($H$) are from the Minor Planet Center (MPC) database. Diameters reported in this table are only approximations. More accurate diameters and albedos for some family members will be published in E. Vilenius et al. (in preparation) using thermal modeling of *Herschel Space Observatory* data.

<sup>a</sup> Assuming an albedo of 0.70 for confirmed members of the family, except for Haumea and 2002 TX<sub>300</sub> whose albedos are known. Assuming albedos of 0.08/0.30 for the candidates when the albedo is unknown.

<sup>b</sup> Masses (except Haumea, Hi'iaka, and Namaka masses) computed assuming a density of 1 g cm<sup>-3</sup>.

<sup>c</sup> Assuming an albedo of 0.08/0.30, we derived a diameter of 319/165 km and a mass of 17.05/2.35 × 10<sup>18</sup> kg.

**References.** R09: Ragozzine & Brown (2009), E10: Elliot et al. (2010), L10: Lellouch et al. (2010), O12: Ortiz et al. (2012), SS12: Santos-Sanz et al. (2012), V12: Vilenius et al. (2012), V14: Vilenius et al. (2014), TW: this work.

lightcurve amplitude versus eccentricity and inclination are only reported in the other TNOs samples.

## 6. SUMMARY AND CONCLUSIONS

We have collected photometric data for several objects related to the Haumea family over the past five years using several facilities in Spain and United States of America. We

present a homogeneous data set composed of 12 objects: 5 family members and 7 candidates. We report rotational periods and lightcurve amplitude for most of them, but in two cases, we only report constraints. Half of the studied objects have low lightcurve amplitudes (peak-to-peak amplitude less than 0.15 mag). Of the 12 objects, 6 can be considered Jacobi ellipsoids with a high lightcurve amplitude due to the shape of the body. Some of the large amplitude lightcurves are
<table><caption>**Table 8**
Some Correlations/Anti-correlations Found Using the Lightcurve Parameters and Orbital/Physical Variables</caption><thead><tr><th>Correlated Values</th><th>Sample</th><th>ρ</th><th>SL (%)</th><th>Nb.</th></tr></thead><tbody><tr><td>Rotational period versus absolute magnitude</td><td>Family</td><td>0.638</td><td>98.43</td><td>11</td></tr><tr><td>Rotational period versus inclination</td><td>Family</td><td>-0.565</td><td>96.33</td><td>11</td></tr><tr><td>Rotational period versus ascending node</td><td>Family</td><td>0.426</td><td>95.27</td><td>11</td></tr><tr><td rowspan="2">Lightcurve amplitude versus absolute magnitude</td><td>Family, and candidates</td><td>0.358</td><td>99.28</td><td>23</td></tr><tr><td>Family, and candidates</td><td>0.473</td><td>99.32</td><td>23</td></tr></tbody></table>

**Note.** We looked into 13 samples of data. *family* stands for all confirmed members of the family, *candidates* are reported in Table 3, *Other TNOs* refers to all TNOs without relation with the family, and *no binary pop* excludes the binary population. We indicate the spearman rank correlation (*ρ*), the significance level (SL in percent), and the number of objects in each sample (Nb). Only positive correlations/anti-correlations with a spearman rank and significance level in agreement with our criterion are reported.

asymmetric and such a fact can be explained by albedo variation(s) on the surface of the objects, as is the case with the dark spot on Haumea's surface. We compared the rotational frequency distributions of the family members with/without candidates and we conclude that the family members as well as candidates seem to rotate faster than the other TNOs (objects without relations with Haumea). We also have shown that the family members rotate at different rates according to their size. In fact, smaller members of the family rotate slower than the biggest fragments. Such a tendency is yet to be understood and will be discussed in future work. Rotational periods of the Haumea family members give us information about the characteristics of the Proto-Haumea. One of the conclusions is that the Proto-Haumea was probably a fast rotator too and thus probably an elongated object (deformation due to the fast rotation).

Regarding the lightcurve amplitude distribution, Haumea family members mostly show low amplitudes, besides the case of the contact binary 2003 SQ317 and Haumea itself. However, it is interesting to point out that most of the candidates have moderate to large lightcurve amplitudes. Objects in relation with Haumea seem to follow the same tendency as objects without relation to Haumea, i.e., smaller objects have higher lightcurve amplitude thus have more irregular shapes.

From short-term variability study, we derive several physical properties such as axis ratio and lower limit to the density. We have shown that the mean density for the members as well as for the candidates is around $1 \text{ g cm}^{-3}$. In the family, bigger objects seem to have higher densities, except for the case of the contact binary, whereas such a tendency is not evident in the candidate sample.

The definition of the Haumea family is not clear. The classic definition considers that proper orbital elements and water ice detection are necessary to identify family members, but the enlarged definition suggests that rocky members without (or a small amount) water ice have to be considered too. Based on our study, it seems that the family members have rotational properties significantly different from the other TNOs. Therefore, such properties are probably useful to identify family members.

We also report the first lightcurve of 2002 GH32. This object presents an asymmetric lightcurve and a very fast rotation. Our interpretation for such a lightcurve is that 2002 GH32 is a very elongated object. 2002 GH32 is the second fastest rotator in the trans-Neptunian belt, after Haumea. We report a double-peaked periodicity of 3.98 hr and a lightcurve amplitude of 0.36 mag. Assuming this object as a triaxial ellipsoid in hydrostatic

equilibrium, we derive a lower limit to the density of $2.6 \text{ g cm}^{-3}$. Such a density is similar to Haumea's. We also want to emphasize the similarity of this lightcurve with Haumea's. In fact, both objects have similar rotational period and asymmetric lightcurve despite their size range difference.

We thank the anonymous referee for her/his careful reading of the paper and for useful comments. We thank Will Grundy for very useful discussions. This research was based on data obtained at the Observatorio de Sierra Nevada, which is operated by the Instituto de Astrofísica de Andalucía, CSIC. Other results were obtained at the Telescopio Nazionale Galileo. The Telescopio Nazionale Galileo (TNG) is operated by the Fundación Galileo Galilei of the Italian Istituto Nazionale di Astrofisica (INAF) on the island of La Palma in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Other results were obtained at the Isaac Newton Telescope (INT). The Isaac Newton Telescope is operated on the island of La Palma by the Isaac Newton Group (ING) in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofísica de Canarias (IAC). This research is also based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (IAA-CSIC). These results made use of Lowell Observatory's Discovery Channel Telescope (DCT). Lowell operates the DCT in partnership with Boston University, Northern Arizona University, the University of Maryland, and the University of Toledo. Partial support of the DCT was provided by Discovery Communications. LMI was built by Lowell Observatory using funds from the National Science Foundation (AST-1005313). We acknowledge the DCT operators H. Larson, T. Pugh, and J. Sanborn, and the OSN operators F. Aceituno, V. Casanova, and A. Sota. Special thanks to Heidi Larson who managed to fix all the technical problems we had during our DCT observing runs. We thank S. Levine, B. De Groff, L. Wasserman, P. Massey, D. Hunter, and D. Trilling for extra observing time at DCT. A. Thirouin was/is supported by AYA2008-06202-C03-01, NASA NEOO grant number NNX14AN82G, awarded to the Mission Accessible Near-Earth Object Survey (MANOS), and Lowell Observatory funding. J.L. Ortiz is supported by AYA2014-56637-C2-1-p, which is a Spanish MICINN/MEC project, by the Proyecto de Excelencia de la Junta de Andalucía, J.A. 2012-FQM1776, and by FEDER funds.
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