# Implications for the Formation of (155140) 2005 UD from a New Convex Shape
Model

Jay K. Kueny Lowell Observatory, 1400 W. Mars Hill Rd., Flagstaff, AZ 86001,
USA  Steward Observatory, University of Arizona, Tucson, 933 N. Cherry Ave.,
Tucson, AZ 85721, USA  Wyant College of Optical Sciences, University of
Arizona, 1630 E. University Blvd., Tucson, AZ 85721, USA  National Science
Foundation Graduate Research Fellow  Colin Orion Chandler Department of
Astronomy and Planetary Science, Northern Arizona University, PO Box 6010,
Flagstaff, AZ 86011, USA  National Science Foundation Graduate Research Fellow
Dept. of Astronomy & the DiRAC Institute, University of Washington, 3910 15th
Ave. NE, Seattle, WA 98195, USA  Maxime Devogèle Arecibo Observatory,
University of Central Florida, HC-3 Box 53995, Arecibo, PR 00612, USA
Nicholas Moskovitz Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff, AZ
86001, USA  Petr Pravec Astronomical Institute, Academy of Sciences of the
Czech Republic, Fričova 1, CZ-25165 Ondřejov, Czech Republic  Hana Kučáková
Astronomical Institute, Academy of Sciences of the Czech Republic, Fričova 1,
CZ-25165 Ondřejov, Czech Republic  Institute of Astronomy, Faculty of
Mathematics and Physics, Charles University, V Holešovickách 2, 18000 Prague,
Czech Republic  Research Centre for Theoretical Physics and Astrophysics,
Institute of Physics, Silesian University in Opava, Bezručovo nám. 13, 74601
Opava, Czech Republic  Kamil Hornoch Astronomical Institute, Academy of
Sciences of the Czech Republic, Fričova 1, CZ-25165 Ondřejov, Czech Republic
Peter Kušnirák Astronomical Institute, Academy of Sciences of the Czech
Republic, Fričova 1, CZ-25165 Ondřejov, Czech Republic  Mikael Granvik
Department of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland
Asteroid Engineering Laboratory, Luleå University of Technology, Box 848,
SE-98128 Kiruna, Sweden  Christina Konstantopoulou Department of Astronomy,
University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland  Nordic
Optical Telescope, Apartado 474, E-38700 Santa Cruz de La Palma, Spain
Nicholas E. Jannsen Institute of Astronomy, Celestijnenlaan 200D bus 2401,
B-3001 Leuven, Belgium  Nordic Optical Telescope, Apartado 474, E-38700 Santa
Cruz de La Palma, Spain  Shane Moran Department of Physics and Astronomy,
University of Turku, Vesilinnantie 5, FI-20500, Finland  Nordic Optical
Telescope, Apartado 474, E-38700 Santa Cruz de La Palma, Spain  Lauri Siltala
Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland
Grigori Fedorets Department of Physics, P.O. Box 64, 00014 University of
Helsinki, Finland  Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, Belfast BT7 1NN, UK  Marin Ferrais Space
sciences, Technologies & Astrophysics Research (STAR) Institute University of
Liège Allée du 6 Août 19, B-4000 Liège, Belgium  Aix Marseille Univ, CNRS,
LAM, Laboratoire d’Astrophysique de Marseille, Marseille, France  Emmanuel
Jehin Space sciences, Technologies & Astrophysics Research (STAR) Institute
University of Liège Allée du 6 Août 19, 4000 Liège, Belgium  Theodore Kareta
Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff, AZ 86001, USA  Josef Hanuš
Institute of Astronomy, Faculty of Mathematics and Physics, Charles
University, V Holešovickách 2, 18000 Prague, Czech Republic

###### Abstract

(155140) 2005 UD has a similar orbit to (3200) Phaethon, an active asteroid in
a highly eccentric orbit thought to be the source of the Geminid meteor
shower. Evidence points to a genetic relationship between these two objects,
but we have yet to fully understand how 2005 UD and Phaethon could have
separated into this associated pair. Presented herein are new observations of
2005 UD from five observatories that were carried out during the 2018, 2019,
and 2021 apparitions. We implemented light curve inversion using our new data,
as well as dense and sparse archival data from epochs in 2005–2021 to better
constrain the rotational period and derive a convex shape model of 2005 UD. We
discuss two equally well-fitting pole solutions
($\lambda=116\hbox{$.\\!\\!^{\circ}$}6$, $\beta=-53\hbox{$.\\!\\!^{\circ}$}6$)
and ($\lambda=300\hbox{$.\\!\\!^{\circ}$}3$,
$\beta=-55\hbox{$.\\!\\!^{\circ}$}4$), the former largely in agreement with
previous thermophysical analyses and the latter interesting due to its
proximity to Phaethon’s pole orientation. We also present a refined sidereal
period of $P_{\text{sid}}=5.234246\pm 0.000097$ hr. A search for surface color
heterogeneity showed no significant rotational variation. An activity search
using the deepest stacked image available of 2005 UD near aphelion did not
reveal a coma or tail but allowed modeling of an upper limit of 0.04 to 0.37
kg s-1 for dust production. We then leveraged our spin solutions to help limit
the range of formation scenarios and the link to Phaethon in the context of
nongravitational forces and timescales associated with the physical evolution
of the system.

Near-Earth objects; CCD observation; Astronomical Techniques; Photometry;

##  1 Introduction

Near-Earth asteroid (NEA) 2005 UD is a kilometer-class object and is a
potential flyby target of JAXA’s DESTINY+
mission,111https://destiny.isas.jaxa.jp/science/ scheduled to launch within
the next decade. It was discovered in 2005 by the Catalina Sky Survey
(Christensen et al., 2005) and was revealed to have an orbit similar to (3200)
Phaethon and the Geminid meteor stream. A subsequent observational campaign
revealed surface color variations as a function of rotational phase (Kinoshita
et al., 2007). Previous studies on the visible reflectance spectrum suggest
that 2005 UD is a B-type asteroid (Jewitt & Hsieh 2006; Devogèle et al. 2020)
though recent findings regarding the near-infrared spectrum by Kareta et al.
(2021) and a phase curve analysis by Huang et al. (2021) contest this. It is
in the Apollo dynamical class with a semimajor axis of 1.275 au, an
eccentricity of 0.87, and an orbital inclination of
$28\hbox{$.\\!\\!^{\circ}$}7$ (see Appendix A for a comprehensive reference
table). Light curve inversion by Huang et al. (2021) using the Lommel–Seeliger
ellipsoid method yielded a 2005 UD spin pole solution of
$(285\hbox{$.\\!\\!^{\circ}$}8\begin{subarray}{c}+1.1\\\
-5.3\end{subarray},-25\hbox{$.\\!\\!^{\circ}$}8\begin{subarray}{c}+5.3\\\
-12.5\end{subarray})$ which is comparable to that of Phaethon (Hanuš et al.
2018; Kim et al. 2018). A common origin with Phaethon continues to be
extensively investigated (see, e.g., Devogèle et al. 2020; Kareta et al. 2021;
MacLennan et al. 2021).

Asteroid (3200) Phaethon is a B-type NEA (Licandro et al., 2007) and exhibits
short bursts of activity at perihelion suspected to be caused by thermal
fracturing (Jewitt & Li 2010; Jewitt, D. 2012). It has a semimajor axis of
1.271 au, an eccentricity of 0.89, and an orbital inclination of
$22\hbox{$.\\!\\!^{\circ}$}3$. Phaethon is thought to be the parent body of
the annual Geminid meteor shower (Whipple, 1983) although predicted upper
limits to its current dust production rates cannot explain the inferred mass
contained within the Geminid meteor stream (Ryabova 2017; Kasuga & Masiero
2022).

Based on dynamical arguments (Ohtsuka et al. 2006; Hanuš et al. 2016;
MacLennan et al. 2021) 2005 UD and Phaethon could have split from a larger
precursor body at some point in the past which may explain the aforementioned
mass discrepancy with the Geminids. A recent study of the Daytime Sextantids
meteor shower (part of the Phaethon–Geminid stream complex) by Kipreos et al.
(2022) reinforces this by suggesting that this meteor stream, 2005 UD, and
Phaethon were created from a mutual breakup event. This common origin theory
is further supported since B-type near-Earth objects are uncommon (Jewitt &
Hsieh, 2006), and the previously mentioned color variability discovered on
2005 UD begs interesting implications for fresh surface material perhaps due
to recent detachment. Analyses of 2005 UD and Phaethon’s polarimetric phase
curve by Devogèle et al. (2020) and Ishiguro et al. (2022) reveal similarities
over broad phase-angle coverage, again hinting at a genetic relation between
the pair. However, counterpoints to the common origin narrative were made by
Kareta et al. (2021), who found that these two objects have distinctly
different spectral features in the near-infrared and by Ryabova et al. (2019)
based on dynamical tests probing the past 5000 yr.

In this work, we present further constraints on 2005 UD’s rotation period,
pole orientation, and shape model through light curve inversion using data
from new 2018, 2019, and 2021 observations and archival data. We introduce the
observations, data reduction, and shape modeling procedures in the next
section. In Section 3, we discuss the refined sidereal period and spin axis
orientation for 2005 UD as well as comment on the current state of a convex
shape model. In Section 4 we infer the most likely pole solution for 2005 UD
and present our search for surface color heterogeneity and activity. Section 5
leverages our spin solutions to inform possible formation scenarios for 2005
UD. We then conclude with Section 6 and encourage avenues for future work.

##  2 Data Collection & Processing

We define “light curve” as the time series of disk-integrated brightness of
the asteroid collected at a single site in a single filter. We adopt the
terms“dense” and “sparse” to describe the two modalities of light curves used
in our shape modeling process. Dense light curves feature photometric data
points spaced closely in time relative to the rotational period of the object
while sparse light curves typically contain interspersed points and light
curve subsections fewer than about seven points per night, spanning greater
than 30 days, and are usually the product of nightly astronomical surveys. In
total, we used 79 dense and 5 sparse light curves of 2005 UD from apparitions
in 2005–2021 for our investigation. Of the set of dense light curves, we
included 36 from Devogèle et al. (2020), 10 from Warner & Stephens (2019), 4
from Jewitt & Hsieh (2006), 4 from Kinoshita et al. (2007), and the remaining
from our own observations conducted in 2018, 2019, and 2021.

###  2.1 Observations & Photometry

We present photometric observations of 2005 UD obtained using the following
telescopes: the Ondřejov Observatory 0.65 m telescope, the Danish 1.54 m
Telescope, North 0.6 m TRAnsiting Planets and PlanetesImals Small Telescopes
(TRAPPIST-N), the 4.3 m Lowell Discovery Telescope (LDT), and the 2.6 m Nordic
Optical Telescope (NOT). Our analysis also includes sparse data sets from
various surveys (discussed below). Appendix B provides details about the
observing circumstances.

We used the 4.3 m Lowell Discovery Telescope (LDT, located in Happy Jack,
Arizona, USA) on the nights of UT 2019 October 19, 2019 November 18 and 2021
November 3. Images were captured using a broadband VR filter (approximately
encompassing the Johnson–Cousins $V$ and $R$ bands) and the Large Monolithic
Imager, which features $6144\times 6160$ pixels and a square
$12\hbox{$.\\!\\!^{\prime}$}5$ field of view. This instrument samples at a
pixel scale of $0\hbox{$.\\!\\!^{\prime\prime}$}12$ pixel-1 but was used in
$3\times 3$ binning mode. Exposure times ranged from 60 to 120 s for the first
night, from 14 to 20 s for the second, and from 30 to 35 s for the last.
Seeing conditions were very stable for the first and last night and stable at
the 25% level for the second night. We recorded median FWHM values of on-chip
point sources of about $2\hbox{$.\\!\\!^{\prime\prime}$}7$,
$1\hbox{$.\\!\\!^{\prime\prime}$}5$, and $2\hbox{$.\\!\\!^{\prime\prime}$}3$
for the first, middle, and last night, respectively.

The 2.6 m NOT is located at the Spanish Observatorio del Roque de los
Muchachos, La Palma, Canarias, Spain. For these data the NOT imaged with the
Alhambra Faint Object Spectrograph and Camera (ALFOSC), which equips a nearly
square $2048\times 2064$ pixel detector sampled at $0.21"$ pixel-1.
Observations were carried out on the nights of UT 4 November 2019 and 18
November 2019 in the Sloan Digital Sky Survey (SDSS) $r$ filter and with a
SDSS $g$-$r$-$i$ sequence, respectively. The images were subject to $2\times
2$ binning and exposure times set at 30 s across both nights. Seeing varied
typically from $1"$ to $2\hbox{$.\\!\\!^{\prime\prime}$}5$ across both nights
with conditions improving during the later half of the night for both runs.

Additional observations were collected by the robotic 0.6 m TRAPPIST-North
(Jehin et al., 2011) on the nights of UT 2019 November 24-26. TRAPPIST-N is
located at the Oukaïmeden Observatory in the Atlas Mountains in Morocco. This
telescope features an Andor iKON-L BEX2-DD CCD camera imaging through a
Cousins $R$ filter. Additional instrument specifications include a
$0\hbox{$.\\!\\!^{\prime\prime}$}60$ pixel-1 scale and a $22^{\prime}$ square
field of view. The images were binned $2\times 2$ with exposure times set at
120 s. Seeing for the first night was variable with median on-chip values
ranging from $\sim 3"$ to $4\hbox{$.\\!\\!^{\prime\prime}$}2$ with conditions
improving on the second night, where values were in the range of $\sim
2\hbox{$.\\!\\!^{\prime\prime}$}7$ to 4”. The final night unfortunately
presented poor observing conditions, so we refrained from using data from this
night in our analysis.

Observations in 2021 corresponding to nights UT October 27-30, as well as one
night in 2018 on UT November 4, were carried out at La Silla Observatory using
the Danish 1.54 m telescope (labeled ”Danish” in Appendix B). The DFOSC
instrument on this telescope has a deep depleted BI 2k$\times$2k sensor with
13.5 $\mu$m square pixels, and we used it unbinned. Integration times were
between $60$ and $140$ s, and the telescope was tracked at half the apparent
rate of the asteroid, providing star and asteroid source profiles in one
frame.

For the remaining observations from the 2018 apparition we used the Ondřejov
Observatory 0.65 m telescope (labeled ”Ondřejov” in Appendix B) The 0.65 m is
a reflecting telescope operated jointly by the Astronomical Institute of ASCR
and the Astronomical Institute of the Charles University of Prague, Czech
Republic. It uses a Moravian Instruments G2-3200 MkII CCD camera (with a Kodak
KAF-3200ME sensor and standard BVRI photometric filters) mounted at the prime
focus. The CCD sensor has $2184\times 1472$ square pixels (6.8 micron pitch)
with microlenses and we imaged in $2\times 2$ binning mode, providing
$1\hbox{$.\\!\\!^{\prime\prime}$}05$ pixel-1 and a 19’$\times$12.8’ field of
view. Integration times were between 30 and 100 s, and we set the tracking at
half-apparent rate of the asteroid.

We sourced the sparse data from the following:

  1. 1.

observations from the Catalina Sky Survey (CSS, Larson et al. 2003);

  2. 2.

observations from the Pan-STARRS project (Chambers et al., 2016);

  3. 3.

images from the ZTF project (Bellm et al., 2019) downloaded from the IRSA
server (https://irsa.ipac.caltech.edu/applications/ztf/); and

  4. 4.

observations from the ATLAS project (Tonry et al., 2018).

For all but the ZTF sparse data we utilized the calibrated chip-stage
photometry (unpublished) and associated Julian dates reported on the Minor
Planet Center (MPC)222https://minorplanetcenter.net/db_search. Processing of
the ZTF images is described in the following paragraph.

We bias- and flat-field-corrected data from our new observations using
standard techniques. We used the Python-based PhotometryPipeline (Mommert,
2017) to measure the photometry of these new data, as well as the ZTF sparse
data. We note that neither the field stars nor 2005 UD were trailed, and thus
irregular photometry was not required for these new observations. A high-level
overview of the pipeline workflow is as follows: astrometry using SCAMP
(Bertin, 2006), which utilizes the VizieR catalog service (Ochsenbein et al.,
2000) to perform image registration via the Gaia Data Release 2 catalog (Gaia
Collaboration et al., 2018); point-source extraction using Source Extractor
(Bertin & Arnouts, 1996); photometric zero-point calibration using the Pan-
STARRS DR1 catalog (Flewelling et al., 2020); and distilling the calibrated
photometry by using the object’s position in the frame as returned through a
query to the JPL Horizons system (Giorgini et al., 1996). Additionally, the
pipeline executed photometric calibration using stars with solar-like colors
in the same frame from the SDSS DR9 catalog (Ahn et al., 2012), where the
color thresholds were set at $(g-r)=0.44\pm 0.2$ and $(r-i)=0.11\pm 0.2$.
Using the curve-of-growth procedure outlined in Mommert (2017), the pipeline
determined a best-fit aperture radius of 2.84–4.53 binned pixels
($1\hbox{$.\\!\\!^{\prime\prime}$}22$ to $1\hbox{$.\\!\\!^{\prime\prime}$}94$)
for the NOT data and 3.26–5.37 binned pixels
($1\hbox{$.\\!\\!^{\prime\prime}$}17$ to $1\hbox{$.\\!\\!^{\prime\prime}$}93$)
for the LDT data. The TRAPPIST data were the only exception to this where we
manually set apertures of 5-pixel radius (3”).

Data from the 2018 and 2021 apparition that were taken with the Ondřejov
Observatory 0.65 m and Danish 1.54 m telescopes were subject to a custom
aperture photometry program Aphot + Redlink developed by Petr Pravec and
Miroslav Velen. In short, the software performs a semiautomated routine to
select optimal apertures for the photometry. Star-like sources in the science
frames are calibrated in the Johnson–Cousins V-R system with standard stars
from Landolt (1992) facilitating 0.01 mag precision in photometric conditions.

Appendix B includes further details of all light curves used in our analysis.
Figure 1 shows ecliptic coordinates corresponding to our new observations, as
well as future apparitions.

Figure 1: Geocentric ecliptic coordinates at the time observations of 2005 UD
were obtained. The small blue circles indicate the ecliptic coordinate of 2005
UD from 2005 September 5 to 2024 December 31 with a resolution of 1 day. The
2005, 2018, 2019, and 2021 dense light curves included in our analysis are
shown as the orange squares, green stars, blue triangles, and magenta circles,
respectively. A past viewing opportunity in 2020 November is represented as
the red vertical hash region. Additional viewing opportunities of 2005 UD in
late 2023 and late 2024 are represented as the black horizontal ticks and
yellow cross regions, respectively. Ecliptic coordinates of 2005 UD were
obtained from JPL Horizons (Giorgini et al., 1996).

##  3 Light curve Inversion

Light curve inversion has been used to ascertain spin states of NEAs; see
models for, e.g., Phaethon (Hanuš et al. 2016; Hanuš et al. 2018), Cuyo (Rożek
et al., 2019), and Apollo (Kaasalainen et al. 2007; Ďurech et al. 2008). In
addition to the necessity of good quality data (i.e., high signal-to-noise
ratio), a unique solution requires data obtained across a broad range of
viewing geometries (i.e., sampling reflectance data from as much of the
surface of the object as possible). To determine the shape (expressed as a
convex polyhedron) and pole solution of 2005 UD as well as refine its sidereal
rotation period, we used the convexinv program, described in Kaasalainen &
Torppa (2001) and Kaasalainen et al. (2001).

Prior to carrying out light curve inversion, we formatted the data (see the
convexinv documentation) into standard “blocks”. We employed the astropy-
affiliated (Tardioli et al., 2017) astroquery tool to obtain Sun and Earth
$xyz$ vector components, centered on 2005 UD and expressed in au, for each
data point via the JPL Horizons service (Giorgini et al., 1996). Additionally,
all light curves were normalized to unity and converted to flux units before
correcting observation times for light-travel time. As a final step, all flux
values were range-corrected to 1 au from the Earth and Sun.

Of the original light curve set from Devogèle et al. (2020), we discarded five
light curves obtained at the Lowell Observatory 0.79 m National Undergraduate
Research Observatory telescope (31in henceforth) and one light curve from
TRAPPIST-N due to high photometric noise or having temporally overlapping data
from a superior instrument (although our model light curves were able to
reproduce these data). These six light curves correspond to observations on UT
2018 September 27, October 6, October 10, October 15, October 16, and October
17. Additionally, we exclude the first half of one LDT light curve from our
new data taken on UT 2019 October 18 from the shape modeling procedure because
light curve predictions from our best-fit shape models did not agree with the
uncharacteristic 0.5 mag amplitude; the cause for the discrepancy in this
light curve with our models is unknown. Preparing the sparse data sets
included rejecting any data points taken near the magnitude limit of the
specific instrument and plotting the observed intensities vs. the associated
phase angles and performing a sigma-clipping routine to eliminate outliers.

###  3.1 Rotational Period

To get a reliable shape solution, it is imperative to constrain the rotational
period of the object. The parameter space is riddled with local minima due to
the rotational period, which, since convexinv is a gradient-based algorithm,
will cause the optimizer to get trapped and converge to an ill-fitting
solution. The spacing of these local minima $\Delta P$ (in hours) in the
period-space $\chi^{2}$ spectrum is given roughly by $\Delta P\approx
P^{2}/(2T)$, where $P$ is the rotational period of the object and $T$ is the
time span of the entire data set (Kaasalainen et al., 2001). We ensured that
the step size of the period search did not increase significantly above this
value to prevent missing the correct period. We define a unique period or spin
solution if its $\chi^{2}$ (see Section 3.1 in Kaasalainen & Torppa 2001 for
how this test statistic is calculated) is lower than that of all other
candidate solutions by at least 10% (also used in Hanuš et al. 2011). Note
that while the criterion employed in Hanuš et al. (2018) suggests that a
$\chi^{2}$ threshold of $\sim 5\%$ above the minimum is valid for our dataset,
we opt for the more rigorous 10% threshold to make the global minimum more
distinct.

Prior to running period_scan we specified a narrow period search window
centered on the literature value and assigned weights to each light curve to
improve the goodness of fit; noisy and sparse light curves were assigned lower
weights. We optimized the individual weights $W_{\text{LC}}$ for each
remaining dense light curve quantitatively using
$W_{\text{LC}}=1/{\text{rms}^{2}}$, where the rms error is obtained by fitting
a Fourier series to each light curve. Next, all nonzero weights for dense
light curves were then multiplied by a scale factor to bring the sum of the
dense light curve weights equal to the number of nonrejected dense light
curves (in our case 78). For the 2005, 2019, and 2021 apparitions we fit a
Fourier series up to the seventh order to each light curve. For the 2018
apparition, we limited the fitted Fourier series to second order as a form of
regularization since these 2018 data compose $\sim 80\%$ of our dense light
curves. Performing various shape modeling trials showed that fully optimizing
the weights for all dense light curves biased the model heavily to the 2018
apparition by suppressing the weights of the light curves from the other
apparitions. This is likely because data from the other apparitions,
particularly from 2005 and 2021, are quite limited in both quantity and
quality. Further, because of 2005 UD’s relatively smooth and invariant
sinusoidal light curve (see Figure 5 for an example), we determined that a
second order Fourier series was sufficient in penalizing lower-quality light
curves without significantly underestimating the weights of the high-quality
ones. For the sparse light curves, we assign weights at the 10% or 20% level
(see Section 3.2) by taking the average of the lowest 10% or 20% of weights of
the dense light curves.

Table 1: Comparison of Previous Rotational Period Estimates for 2005 UD with This Work. Period | Uncertainty | Source
---|---|---
(hr) | (hr) |
5.23 |  | Jewitt & Hsieh (2006)
5.2492 |  | Kinoshita et al. (2007)
5.231 | $\pm 0.034$ | Sonka et al. (2019)
5.235 | $\pm 0.005$ | Devogèle et al. (2020)
5.2340 | $\begin{subarray}{c}+0.00004\\\ -0.00001\end{subarray}$ | Huang et al. (2021)
5.234246 | $\pm 0.000097$ | This work
Figure 2: The period_scan results from our set of 84 light curves with
optimized weights. The black points represent local minima in the sidereal
period, pole orientation, and shape parameter space. The blue horizontal line
shows the $\chi^{2}$ value 10% higher than the global minimum to which we
consider points underneath to be viable solutions. The only solution that
satisfies this criterion for the rotational sidereal period with our dataset
is $P_{\text{sid}}=5.234246\pm 0.000097$ and is shown in red.

The period_scan results within a search range of 5.225 and 5.245 hr produced a
unique sidereal rotational period value of $P_{\text{sid}}=5.234246\pm
0.000097$ hr (Figure 2). A comparison of our period solution with those from
previous studies is presented in Table 1. A previous period search for 2005 UD
conducted in Devogèle et al. (2020) suggested the possibility of a three-
peaked light curve corresponding to a rotation period around 7.85 hr. With the
addition of our new data, we note significant reduction in the goodness of fit
for periods in this vicinity. As such, we can now exclude sidereal period
solutions in this range and surmise that this is an alias.

###  3.2 Spin and shape solutions

Using our period solution, we followed the standard convexinv recipe to derive
a shape model. We specified 48 initial pole orientations isotropically
distributed on the celestial sphere at $30\hbox{${}^{\circ}$}$ spacing, which
is later optimized during the inversion procedure. The uncertainties for the
sparse data from the MPC (all but the ZTF data) are not reported. Thus, the
contribution of each sparse dataset toward convergence was assessed by
performing a coarse grid search across all sparse data combinations, where
each individual sparse light curve was assigned a 0%, 10%, or 20% weight. By
scanning through these weights, we aimed to minimize the relative $\chi^{2}$
value for the dense light curves only. Following this test, we assigned a
weight of 10% to all but the ZTF data (assigned 20%) as this specific
combination produced the lowest $\chi^{2}$. This combination is not surprising
since we reduced the ZTF sparse data ourselves and thus had the knowledge to
reject individual data points based on error, source blending, etc. The ZTF
data also had the largest phase-angle coverage (i.e., more viewing geometries)
of the sparse data sets, covering $\sim 100\hbox{${}^{\circ}$}$ in phase over
a 3 yr span.

Table 2: Candidate Spin Solutions after Performing Light Curve Inversion.

Solutions | $\lambda$ | $\beta$ | $\chi^{2}$ | $\mathrm{rms}$
---|---|---|---|---
| (deg) | (deg) |  |
1* | 144.0 | -31.8 | 3.0542 | 0.0202
2 | 116.6 | -53.6 | 3.1037 | 0.0203
3 | 300.3 | -55.4 | 3.1190 | 0.0204
4* | 62.5 | -44.2 | 3.2048 | 0.0207

Note. ∗ Rejected due to nonphysical shape. $\chi^{2}$ and $\mathrm{rms}$
quantify fits to dense light curves only.

Following this, we were left with four probable spin solutions, which are
listed in Table 2. All four shape models had a small dark facet area (below 1%
of the total facet area), which is needed to preserve convexity, so we were
not able to eliminate any of these candidate solutions using this metric.
Further, this suggests that nonconvex features (e.g., concavities) do not
occupy a significant area on 2005 UD’s surface. Following this, we computed
the inertia tensors (see Dobrovolskis 1996 for a full description) of the
shape models to discover that Solution 1 and Solution 4 had an inertial axis
significantly misaligned from the z-axis, so we rejected these solutions on
the basis of being nonphysical (i.e. there is no evidence that 2005 UD is in
nonprincipal axis rotation). However, one caveat is that if 2005 UD is not
uniform in color (see Section 4) or density, then our convex approximation may
contain systematic errors which could manifest as the aforementioned inertial
axis misalignment.

This left us with Solution 2,
$(\lambda_{2},\beta_{2})=(116\hbox{$.\\!\\!^{\circ}$}6\pm
2\hbox{$.\\!\\!^{\circ}$}2,-53\hbox{$.\\!\\!^{\circ}$}6\pm
4\hbox{$.\\!\\!^{\circ}$}7$), and Solution 3,
($\lambda_{3},\beta_{3})=(300\hbox{$.\\!\\!^{\circ}$}3\pm
2\hbox{$.\\!\\!^{\circ}$}5-55\hbox{$.\\!\\!^{\circ}$}4\pm
2\hbox{$.\\!\\!^{\circ}$}2$) ($1\sigma$ errors), which are equally well
fitting. Errors were estimated by agitating various parameters during dozens
of trial runs and investigating the effects on $\chi^{2}$ values. Solution 3
is interesting because constraints on the pole orientation of (3200) Phaethon
place it in the range of $308\hbox{${}^{\circ}$}\lesssim\lambda\lesssim
322\hbox{${}^{\circ}$}$ and
$-40\hbox{${}^{\circ}$}\lesssim\beta\lesssim-52\hbox{${}^{\circ}$}$ (Kim et
al. 2018 and Hanuš et al. 2018), which is within our $3\sigma$ error. Further
discussion on the implications of these pole solutions for the formation of
2005 UD is given in Section 5.

Recent shape modeling efforts using the Lommel–Seeliger ellipsoid method by
Huang et al. (2021) yielded two candidate pole solutions for 2005 UD: (1)
$(72\hbox{$.\\!\\!^{\circ}$}6\begin{subarray}{c}+4.2\\\
-7.3\end{subarray},-84\hbox{$.\\!\\!^{\circ}$}6\begin{subarray}{c}+6.2\\\
-2.1\end{subarray})$ and (2)
$(285\hbox{$.\\!\\!^{\circ}$}8\begin{subarray}{c}+1.1\\\
-5.3\end{subarray},-25\hbox{$.\\!\\!^{\circ}$}8\begin{subarray}{c}+5.3\\\
-12.5\end{subarray})$. Pole 2 is favored as their preferred solution and is
both comparable to Phaethon’s and largely in agreement with our Solution 3
with overlap in longitude within $3\sigma$ errors.

Thermal data of 2005 UD from two different epochs were obtained during the
Near-Earth Object Wide Infrared Explorer (NEOWISE) reactivation mission
(Mainzer et al., 2014) using the two shortwave filters (W1: $3.1\mu$m; W2:
$4.6\mu$m). Thermophysical modeling of these data was presented in Devogèle et
al. (2020). The best-fit solution from the thermophysical model
($\lambda_{\mathrm{TPM}}=102\hbox{${}^{\circ}$}\pm 20\hbox{${}^{\circ}$}$ and
$\beta_{\mathrm{TPM}}=-35\hbox{${}^{\circ}$}\pm 30\hbox{${}^{\circ}$}$) is
consistent with Solution 2. Given that the other pole solution candidates
(Solutions 1, 3, and 4) were inconsistent with the predicted longitude from
the thermophysical analysis and Solution 2 consistently attained the lower
$\chi^{2}$ among numerous trial runs, we adopt
$\lambda_{p}=116\hbox{$.\\!\\!^{\circ}$}6\pm 2\hbox{$.\\!\\!^{\circ}$}3$,
$\beta_{p}=-53\hbox{$.\\!\\!^{\circ}$}6\pm 5\hbox{$.\\!\\!^{\circ}$}4$ as our
preferred solution. A comparison between our pole solutions from light curve
inversion, solutions from the above thermophysical modeling, and Phaethon’s
pole is illustrated in Figure 3.

Figure 3: Polar projection map comparing our four pole solutions from light curve inversion, candidate thermophysical pole solutions from Devogèle et al. (2020), and Phaethon’s spin pole region from Hanuš et al. (2018) and Kim et al. (2018). The map is centered on the south ecliptic pole. The green circless show accepted light curve inversion solutions, while the crossed-out red circles represent our rejected solutions. The color bar represents $\chi^{2}$ values of the thermophysical solutions, which are represented as stars; only retrograde solutions with $\chi^{2}\leq 5$ are shown for clarity. The best-fitting thermophysical solution is represented by the large black star. Phaethon’s spin axis orientation with approximate uncertainty is marked as the letter $P$ surrounded by the blue extended region. |
---|---
|
Figure 4: Convex shape model of 2005 UD using our preferred pole solution displayed in a variety of viewing geometries. View descriptions in left to right progression: positive isometric, negative isometric, along y-axis, and along x-axis. The scale values are unitless and show the relative sizes of the $xyz$-axes of our convex approximation. The equator of 2005 UD is parallel to the $x$-$y$ plane. The light source was arbitrarily placed and does not necessarily reflect the location of the Sun for any given observation. |  |
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|  |
|  |
Figure 5: Suite of 2005 UD light curves compared to model fits. Each plot
features a single light curve from assorted epochs in 2005, 2018, 2019, and
2021. Overlaid are light curve predictions of our preferred model from the
convex inversion procedure represented as a solid blue line. UT dates marking
the start of observations for each light curve are present as plot titles. The
data behind this figure are available and include all 84 individual
observations; see Tables B.1, B.2, and B.3. The data is in the form of a DAMIT
input file. User can run the code to get the model values or extract the
observed, relative brightness from the input file.

The convex shape (Figure 4) from our preferred pole solution
$(\lambda_{p},\beta_{p})=(116\hbox{$.\\!\\!^{\circ}$}6,-53\hbox{$.\\!\\!^{\circ}$}6)$
is nearly identical to our other candidate solution
$(\lambda,\beta)=(300\hbox{$.\\!\\!^{\circ}$}3,-55\hbox{$.\\!\\!^{\circ}$}4$).
This is expected because there remains a $180\hbox{${}^{\circ}$}$ ambiguity in
ecliptic longitude for the two solutions for roughly the same pole latitude.
We thus expect features to be mirrored across the xz-plane. As an additional
test in the validity of these two solutions, we computed light curve
predictions of these two shapes and analyzed them by eye to find equally good
agreement with all of our light curves. A suite of twelve plots illustrating
the fits between synthetic and real light curves from epochs in 2005, 2018,
2019, and 2021 using our preferred solution is shown in Figure 5.

##  4 Current State of Knowledge of 2005 UD Physical Properties

For reference, we provide a master summary of known properties for 2005 UD in
Table A.1.

|
---|---
Figure 6: Left: ($g-r$) color as a function of rotation. The data points were
computed by subtracting linearly interpolated $r$ from measured $g$ for a
given $r$-$g$-$r$ sequence. The calculated mean of these residuals is shown as
the dashed horizontal black line. We represent one standard deviation from the
mean as the dotted horizontal red lines, which confirm no signs of color
variation outside of this limit. The ($g-r$) color of the Sun is underlaid as
the green horizontal line with surrounding error for reference. Right: Same as
the left but with ($r-i$) color.

###  4.1 Albedo, Size, and Shape

Thermal emission observations, coupled with photometry, are of particular
value for albedo and size determination. A thermophysical study of 2005 UD by
Masiero et al. (2019) presented a value of $0.14\pm 0.09$ for the geometric
albedo and $1.2\pm 0.4$ km for the effective diameter. However, Masiero et al.
(2019) note that the modeling was only based on data from two observational
epochs from the NEOWISE survey.

The most recent values to be reported come from Devogèle et al. (2020), with
an effective diameter of $1.3\pm 0.2$ km and geometric albedo of $0.10\pm
0.02$ based on photometry and an albedo-polarimetry relation (Cellino et al.,
2015). Devogèle et al. (2020) also included thermophysical modeling (using a
different technique from that used in Masiero et al. 2019) to derive a
geometric albedo of $0.14\pm 0.07$ and effective diameter of
$1.12^{+0.49}_{-0.21}$ from the same limited NEOWISE data set. Despite the
slight differences in these nominal values, they are all consistent and
suggest that the albedo and diameter of 2005 UD are well constrained.

Our shape models of 2005 UD reveal that the general shape can be inferred as a
triaxial ellipsoid with $a/b=1.41$ axis ratio that produces symmetric
sinusoidal light curves with amplitude of about 0.4 mag. We notice prominent
flat areas at the north and south poles on our convex shape approximation,
which is expected, as these areas are the least constrained with unresolved
disk-integrated photometry. Devogèle et al. (2020) is the only other work to
have attempted shape modeling of 2005 UD using our methods (as presented in
Kaasalainen et al. 2001), and they showed that their light curve data alone
could not eliminate the possibility of a sidereal period of $\sim 7.85$ hr.
With the addition of our new data, the possibility of a three-peaked light
curve and consequently a shape more akin to a tetrahedron is nonviable.

###  4.2 Surface Colors

Table 3: Mean Surface Colors of 2005 UD and Phaethon.

Body | $B-V$ | $V-R$ | $R-I$ | Ref.
---|---|---|---|---
2005 UD | $0.65\pm 0.02$ | $0.32\pm 0.01$ | $0.32\pm 0.02$ | 1
2005 UD | $0.63\pm 0.01$ | $0.34\pm 0.01$ | $0.30\pm 0.01$ | 2
2005 UD | $0.66\pm 0.02$ | $0.35\pm 0.02$ | $0.33\pm 0.02$ | 3
Phaethon | $0.64\pm 0.02$ | $0.31\pm 0.02$ | $0.31\pm 0.03$ | 4
Phaethon | $0.61\pm 0.01$ | $0.34\pm 0.03$ | $0.31\pm 0.03$ | 5
Phaethon | $0.59\pm 0.01$ | $0.35\pm 0.01$ | $0.32\pm 0.01$ | 6
Phaethon | - | 0.34 | - | 7

References. (1) this work; (2) Kinoshita et al. 2007; (3) Jewitt & Hsieh 2006;
(4) Lee et al. 2019; (5) Kasuga & Jewitt 2008; (6) Dundon 2005; (7) Skiff et
al. 1996.

We analyzed a time series of photometric colors of 2005 UD derived from
$341\times 30$ s exposures taken in a $g$-$r$-$i$ sequence at the NOT (Section
2.1) to investigate previously reported variations in surface color (Kinoshita
et al., 2007). These multifilter light curves span one complete rotation of
the body. Calibrated photometry using the Aperture Photometry Tool (APT; Laher
et al. 2012) shows the following apparent magnitudes: $g=19.34\pm 0.01$,
$r=18.87\pm 0.01$, and $i=18.80\pm 0.01$. We thus report colors as
$(g-r)=0.47\pm 0.01$ and $(r-i)=0.07\pm 0.01$. To compare with Johnson–Cousins
colors of 2005 UD from prior analyses, we performed color transformations
using $(B-g)$, $(V-g)$, $(R-r)$, and $(R-I)$ equations from Jordi et al.
(2006). Our results are consistent with previous works and are displayed in
Table 3.

To check for rotational color variability, we computed color as a function of
rotation by linearly interpolating between adjacent points, e.g. measured $g$
minus interpolated $r$ (Figure 6). Within the signal-to-noise ratio of our
data we did not detect any systematic color variations that are more
significant than $1\sigma$ away from the mean across the region of the body
visible during this observation period. This null detection is consistent with
the results of (1) Devogèle et al. (2020) who used separate spectroscopic and
polarimetric methods to probe for surface heterogeneity, and (2) Kareta et al.
(2021), who saw no color variations in the near-infrared.

Our above-mentioned results are in contrast to Kinoshita et al. (2007), who
reported $\sim 0.2$ mag $R-I$ color variation. One explanation for this
discrepancy is that 2005 UD’s surface color differs as a function of latitude,
as there was a $\sim 52\hbox{${}^{\circ}$}$ difference in the subobserver
ecliptic latitudes accessed by these two color data sets. This modest
difference in subobserver latitude suggests that any color heterogeneity on
the surface would have to be confined to small ($<50\hbox{${}^{\circ}$}$ in
latitude) yet highly contrasting spots in order to have an influence on the
hemispherical averages represented by unresolved, ground-based photometry.
Photometric modeling of such a spotted surface could provide insight on
whether this interpretation is physically plausible, but this is beyond the
scope of this work. Notably, a recent study by MacLennan et al. (2022) found
evidence that Phethon’s surface is heterogeneous as a function of latitude,
which adds merit to the above hypothesis. Future multicolor photometric
observations could provide further insight into 2005 UD’s surface properties.

Figure 7: Left: radial surface brightness profiles of 2005 UD and a nearby
field star in an LDT stacked image from UT 2019 November 18. We see no
indication of extended emission or faint activity. The stacked image of the
asteroid corresponds to $\sim$10,960 s (about 3 hr) of integration time and
reaches a limiting magnitude of about 26.8 mag arcsec-2. The stellar profile
we measured was from a single 16 s exposure, and we normalized it to the peak
of stacked asteroid profile to facilitate comparison. Right: similar plot from
Chandler et al. (2020) featuring active Centaur C/2014 OG392 (Pan-STARRS) to
demonstrate the profile structure for an active object (reproduced with
permission from the authors).

###  4.3 Activity and Dust Production Limits

Given the detection of activity associated with Phaethon, we conducted a
search for indications of a faint coma or dust tail around 2005 UD. We
utilized the image processing tool Siril333https://www.siril.org/ to stack LDT
images from the night of 2019 November 18 using an average combination
procedure with no rejection (Figure 7). The stacked image corresponds to
10,910 s (about 3 hr) of integration time with a calculated depth of 26.8 mag
arcsec-2 within an annulus extending
$\theta=1\hbox{$.\\!\\!^{\prime\prime}$}8$–$3\hbox{$.\\!\\!^{\prime\prime}$}6$
from the object center; this is the deepest existing stacked image of 2005 UD
at the time of writing. We used the APT (Laher et al., 2012) to measure a
radial surface brightness profile to compare this stacked image of 2005 UD
with a close field star to search for indications of a faint coma. We then fit
the model

$S(r)=A+Br+Cr^{2}+Dr^{3}+Er^{4}+Fe^{-\frac{r^{2}}{2\sigma}}$

which is explained in Laher et al. (2012). Next, we subtracted the background
levels from the profiles and associated models before normalizing the field
star radial profile to the peak flux of the stacked 2005 UD profile. The
results of these efforts are showcased in Figure 7 alongside a similar
analysis performed on the active Centaur 2014 OG392 from Chandler et al.
(2020) for comparison.

An additional quantitative search for a faint dust stream or tail was
performed by analyzing summed pixel values in annular slices around the
stacked asteroid (as demonstrated in Chandler et al. 2021) to search for
excess flux in the anti-solar and anti-motion direction. Along with visual
scrutiny, no indications of activity from 2005 UD were detected. Importantly,
if 2005 UD exhibits activity via thermal fracturing of the surface regolith
(akin to Phaethon), we would expect difficult detection circumstances (Ye et
al., 2021) because 2005 UD was close to aphelion (heliocentric distance of 1.7
au) at the time these data were gathered. Targeted observations of 2005 UD
near perihelion might offer a better chance at detecting activity.

One way to place an upper limit on mass loss is by using a simple model with
knowledge of the limiting magnitude within a projected annulus. We again used
the stacked image from the LDT for this procedure using an annulus of size
$\theta=1\hbox{$.\\!\\!^{\prime\prime}$}8$–$3\hbox{$.\\!\\!^{\prime\prime}$}6$.
Using the procedure outlined in Section 3.2 of Jewitt (2013), we derive an
upper limit of either $0.37$ kg s-1 or $0.04$ kg s-1 based on assumed grain
radii of $10$ mm (the upper limit of surface grain size from Devogèle et al.
2020) and 1 $\mu$m respectively. Additionally, we assume a material density of
$1500$ kg m-3. Following Jewitt (2013) we use particle ejection velocities of
1 m s-1 for millimeter-sized grains and 1 km s-1 for micron-sized grains.
These upper limit results are largely consistent with the $\lesssim 0.1$ kg
s-1 estimate derived by Kasuga & Masiero (2022) using NEOWISE observations.

##  5 A Common Origin with Phaethon?

With additional data from new viewing geometries, we were able to constrain
two possible spin solutions for 2005 UD:
$(\lambda_{p}=116\hbox{$.\\!\\!^{\circ}$}6,\beta_{p}=-53\hbox{$.\\!\\!^{\circ}$}6)$
and
$(\lambda=~{}300\hbox{$.\\!\\!^{\circ}$}3,\beta=-55\hbox{$.\\!\\!^{\circ}$}4)$.
Whether the presence of a mirror solution is a consequence of 2005 UD’s
unusual orbit or by coincidence is not exactly known. Observations of 2005 UD
during upcoming apparitions may yield further information from more viewing
geometries so as to further constrain a single pole solution; Figure 1
displays both past and upcoming viewing geometries that will be accessible in
the next few years. The plausibility and implications for formation for each
pole solution are discussed in the next section.

###  5.1 Comparing Pole Solutions

An intriguing result to come out of our analysis is a well-fitting pole
solution that is consistent with Phaethon’s in both longitude and latitude
(see Section 3.2). This encourages a formation scenario where 2005 UD
separates from parent body Phaethon under a mechanism that does not disturb
the spin vector. Formation via Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP)
spin-up (Rubincam, 2000) is one such possible formation mechanism for the
Phaethon-2005 UD pair. Certainly this is a common mechanism for the formation
of most known asteroid pair systems (Pravec et al., 2010), and interestingly,
both the convex approximation of Phaethon from Hanuš et al. (2018) and radar
shape model from MacLennan et al. (2022) suggest the existence of an
equatorial ridge and top-like shape, characteristic of YORPoids (Ostro et al.
2006; Busch et al. 2011; Naidu et al. 2015). Under a YORP-induced, post-
fission scenario, 2005 UD and Phaethon would have engaged in a protobinary
state before disruption, and eventually evolving into the unbound asteroid
pair seen today (Jacobson & Scheeres, 2011). Due to the complexity of this
process and uncertainties regarding the mass distribution and morphology of
both Phaethon and 2005 UD, it remains unclear whether aligned poles are
expected and/or consistent with a YORP spin-up scenario. Conversely, Pravec et
al. 2019 found that current pole orientations of asteroids in a paired system
generally do not reflect the original orientations at the time of separation,
possibly due to solar torques (Breiter et al., 2005), YORP effects, and/or
planetary encounters. For Phaethon (and possibly 2005 UD), the added dynamical
effects of de-volatilization add more uncertainty on how pole orientations
would be affected (discussed more below). In the case of our preferred
solution, which only shares a common ecliptic latitude with Phaethon, we do
not consider pole realignment due to planetary flybys a likely scenario
because the minimum orbital intersection distance (MOID) value for both bodies
is too high for all planets. YORP ultimately may also be invalid here due to
the proposed recent separation of less than 100 kyr (Hanuš et al. 2016;
MacLennan et al. 2021), which is much shorter than the tens of millions of
years required for significant changes due to YORP (Rubincam, 2000). To probe
the extent to which current-day spin poles for 2005 UD and Phaethon line up or
differ as a result of any of these physical process would require detailed
modeling beyond the scope of this paper.

Due to Phaethon’s rare status as an active asteroid, volatile-driven
separation is another possible formation mechanism for this pair. This
conjecture, however, does not rule out rotational fission entirely since
induced torques due to outgassing cause spin changes in comets (Steckloff &
Jacobson, 2016). Akin to similar processes responsible for cometary splitting
(see Jewitt, D. 2021 and references therein), episodes of activity in an
ancestor body could have worked, perhaps in concert with YORP, to accelerate
the body to its critical spin rate, resulting in fission. However, the fact
that 2005 UD is clearly in a principal axis rotation state (i.e. it is not
tumbling) may speak to a more ordered scenario of formation that did not
involve processes, like volatile-driven outgassing, that can produce
nonprincipal axis rotation. Damping time estimates using Eq. 11 from Pravec et
al. (2014) of a nonprincipal axis rotator of 2005 UD’s size and rotation
exceed the proposed age of this system by millions of years, further
suggesting that a chaotic, purely volatile-driven splitting event was
unlikely.

Based on our results and arguments presented above, we lean toward a common
origin due to a YORP fission scenario being the most likely progenitor for the
2005 UD Phaethon cluster. This interpretation is consistent with that made by
Huang et al. (2021), who also presented a Phaethon-like pole solution for 2005
UD using a different method. We encourage modeling of this specific sequence
of events to probe the effects on spin pole in such a scenario.

###  5.2 Nongravitational Influences

The Yarkovsky effect (Bottke et al. (2006)) is relevant when considering
nongravitational perturbations for small (generally less than $\sim$10 km
diameter) bodies. In general, a secular decrease in semimajor axis is expected
for retrograde rotators (as opposed to outward drift for prograde rotators)
such as Phaethon and 2005 UD. 2005 UD is approximately one-fifth the size of
Phaethon, which would subject it to more rapid inward drift in semi-major
axis. However, the semi-major axis of 2005 UD is currently greater by $\sim
3.5\times 10^{-3}$ au than that of Phaethon and dynamical integrations
(Ohtsuka et al., 2006) suggest that that has been the case for thousands of
years. Furthermore, the presumed recent ($<100$ kyr ago) separation of this
pair means that the Yarkovsky effect has not had enough time to significantly
alter these object’s orbits. Typical Yarkovsky drift rates of $10^{-4}$ au
Myr-1 for kilometer-scale bodies suggest that $\sim 10$ Myr would be needed to
explain their current difference in semi-major axis, and that does not account
for any time associated with a necessary change in pole orientation (e.g., by
the YORP effect) that would allow for Yarkovsky drift to be in the right
direction. It is additionally unlikely that the current low levels of activity
seen for Phaethon play any significant role in the orbital dynamics of the
system, though prior epochs with higher levels of activity may have
contributed to the present-day separation.

If Phaethon and 2005 UD did in fact separate recently, it is suggested
(MacLennan et al., 2021) that orbital dynamics governed by interactions with
the terrestrial planets remain the most plausible dynamical pathway to produce
the configuration of orbital elements we see today. However, a definitive time
line associated with a separation event between Phaethon and 2005 UD has yet
to come to fruition. We hope that improved orbit solutions and physical
evidence of a fission-type disruption event following the DESTINY+ flyby could
provide more insight into this issue.

Lastly, it is worth mentioning that (225416) 1999 YC is a putative third
component in the 2005 UD-Phaethon system (Kasuga & Jewitt 2008; Ohtsuka et al.
2008). The addition of this body to the system supports a rotational fission
formation scenario (Hanuš et al., 2018), but a spectral type inconsistent with
the other two bodies (Kasuga & Jewitt, 2008) and a significantly larger semi-
major axis are hard to reconcile with any common origin theory.

Given these considerations, it seems unlikely that nongravitational forces
would have played an important role in any evolution of 2005 UD relative to
Phaethon. This does not necessarily influence arguments in favor or against
these two objects sharing a common origin.

##  6 Summary and Future Work

We conducted observations of 2005 UD during its recent apparitions in 2018,
2019, and 2021 using six different telescopes (4.3 m LDT, 2.6 m NOT, Danish
1.54 m, Ondřejov 0.65 m, and 0.6 m TRAPPIST-North and TRAPPIST-South)
motivated by a fortuitous opportunity to collect data at new aspect angles.
Our goals for this analysis included finding a unique sidereal rotational
period, pole solution, and thus an accurate shape for 2005 UD by means of
light curve inversion. We supplemented these data with archival dense and
sparse light curves from epochs in 2005–2021 prior to performing light curve
inversion. We presented a refined sidereal rotational period of
$P_{\text{sid}}=5.234246\pm 0.000097$ hr. We conclude that 2005 UD has two
equally well-fitting spin solutions given as
($\lambda_{p}=116\hbox{$.\\!\\!^{\circ}$}6$,
$\beta_{p}=-53\hbox{$.\\!\\!^{\circ}$}6$) and
($\lambda=300\hbox{$.\\!\\!^{\circ}$}3$,
$\beta=-55\hbox{$.\\!\\!^{\circ}$}4$), the former having preference
considering the proximity of all other candidate solutions from our extensive
light curve inversion trials and thermophysical modeling results from Devogèle
et al. 2020. Furthermore, the preferred solution from Huang et al. (2021) of
$(285\hbox{$.\\!\\!^{\circ}$}8\begin{subarray}{c}+1.1\\\
-5.3\end{subarray},-25\hbox{$.\\!\\!^{\circ}$}8\begin{subarray}{c}+5.3\\\
-12.5\end{subarray})$ is only consistent in latitude with the latter of our
two solutions given $3\sigma$ errors. Additional light curve data of 2005 UD
from more viewing geometries may be needed to further constrain a unique spin
solution and/or resolve inconsistencies with earlier studies. We thus
encourage follow-up observations in upcoming apparitions to either help solve
this problem or confirm our results (see Figure 1).

An activity search using the deepest stacked image of 2005 UD at the time of
writing revealed no presence of dust production at orbital longitudes near
aphelion. A simple model was used to infer an upper limit to mass loss of
around 0.04–0.37 kg s-1 depending on assumed grain size. This analysis should
be applied to images of 2005 UD taken at or very close to perihelion to
provide further constraints on the possibility of mass loss when surface
temperatures are highest.

We analyzed time-series color data of 2005 UD over a full rotation and found
no significant indications of surface color heterogeneity in Sloan $g$-$r$-$i$
bands. Understanding of the link between 2005 UD and Phaethon could be
enhanced with additional searches for color heterogeneity, as the results
presented in Section 4 are in contrast to those presented in Kinoshita et al.
(2007), but are consistent with two other studies (Devogèle et al. 2020 and
Kareta et al. 2021). Additionally, there would be value to investigating the
size of a patch of surface material that would be needed to result in
detectable color variation on a body the size and shape of 2005 UD. This could
be done, for example, through radiative transfer models or laboratory
experiments.

Our secondary spin solution is aligned with Phaethon’s within error, which
strengthens certain common origin scenarios. One such possibility is recent
separation of 2005 UD and Phaethon in a YORP-induced fission event with
conservation of angular momentum keeping the poles more or less aligned.
However, the extent to which a parent body’s spin pole orientation would be
preserved between separated pieces in this circumstance is currently a
question that’s unable to be definitively answered. Phaethon’s perihelion-
driven activity complicates this thought experiment, but present-day activity
episodes reveal that they may not be intense enough (Jewitt & Li 2010 and Li &
Jewitt 2013) to have any real effect on spin axis. As such, we encourage
efforts to model such scenarios.

Our spin solutions for 2005 UD spark interesting implications for orbital
evolution via Yarkovsky forces. Current estimates for the age of the Phaethon
cluster (assuming they are genetically related) suggest that the Yarkovsky
effect alone could not have resulted in the currently observed separation of
members. While Ohtsuka et al. (2006) suggest dynamics consistent with a common
origin, most recently Ryabova et al. (2019) performed backward orbital
integrations spanning 5000 yr and concluded that 2005 UD and Phaethon do not
share a common origin. It is, however, worth noting that dynamical analyses
for 2005 UD have thus far neglected consideration of Yarkovsky and cometary
forces which could have a significant effect on the system dynamics. Although
the presented spin solutions of 2005 UD may help to offer clues on its
ancestry, more data and a better understanding of its detailed orbital
dynamics are required to fully understand the origin of this unusual object.

We would like to thank the anonymous reviewers, whose feedback improved the
quality of this manuscript markedly. Support from Michael Mommert on the
PhotometryPipeline simplified data processing steps during critical phases of
this project. Matthew Knight offered helpful guidance on the mass-loss
modeling work in this project. We would like to give thanks to Northern
Arizona University’s HABLab research group, Prof. Tyler Robinson, and Prof.
Chadwick Trujillo for comments and suggestions that improved this work
greatly. The data presented here were obtained (in part) with ALFOSC, which is
provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint
agreement with the University of Copenhagen and NOT. This work is based on
observations made with the Nordic Optical Telescope, owned in collaboration by
the University of Turku and Aarhus University and operated jointly by Aarhus
University, the University of Turku, and the University of Oslo, representing
Denmark, Finland, and Norway, respectively; the University of Iceland; and
Stockholm University, at the Observatorio del Roque de los Muchachos, La
Palma, Spain, of the Instituto de Astrofisica de Canarias. The work at
Ondřejov Observatory, the work at the Danish 1.54 m telescope on the ESO La
Silla station, and the work of J.H. were supported by the Grant Agency of the
Czech Republic, grant 20-04431S. TRAPPIST is funded by the Belgian Fund for
Scientific Research (Fond National de la Recherche Scientifique, FNRS) under
the grant PDR T.0120.21. TRAPPIST-North is a project funded by the University
of Liège, in collaboration with the Cadi Ayyad University of Marrakech
(Morocco). E.J. is F.R.S.-FNRS Senior Research Associate. This material is
based on work supported by the National Science Foundation Graduate Research
Fellowship Program under grant Nos. 2020303693 and 2018258765. Any opinions,
findings, and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the
National Science Foundation. J.K. and N.M. acknowledge support from NASA
Hayabusa2 participating scientist grant NNX16AK68G. J.K., N.M., and M.D.
acknowledge support from NASA NEOO grant NNX17AH06G awarded in support of the
Mission Accessible Near-Earth Object Survey (MANOS). M.G., L.S., and G.F.
acknowledge support from the Academy of Finland. C.K. acknowledges support by
the Swiss National Science Foundation under grant 185692. S.M. acknowledges
support from the Magnus Ehrnrooth Foundation and the Vilho, Yrjö, and Kalle
Väisälä Foundation. This research has made use of data and/or services
provided by the International Astronomical Union’s Minor Planet Center. This
research has made use of NASA’s Astrophysics Data System. This research has
made use of SAO Image DS9, developed by Smithsonian Astrophysical Observatory
(Joye & Mandel, 2003). This work made use of the Lowell Observatory Asteroid
Orbit Database astorbDB (Moskovitz et al., 2022). This work made use of the
astropy (Astropy Collaboration et al., 2013) and SciPy (SciPy 1.0 Contributors
et al., 2020) software packages.

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##  Appendix A Master summary table for 2005 UD

Table A.1 encompasses known physical parameters for 2005 UD which we include
for reference.

Table A.1: 2005 UD Physical Parameters Parameter | Value | Reference
---|---|---
Discovery date | 2005-Oct-22 | MPEC 2005-U22
Discovery observer | E. J. Christensen | MPEC 2005-U22
Discovery survey | Catalina Sky Survey | MPEC 2005-U22
| Orbital Elements |
Orbit type | NEA | NASA JPL Horizons
Family | Apollo | NASA JPL Horizons
Perihelion | 0.1627 au | NASA JPL Horizons
Aphelion | 2.3867 au | NASA JPL Horizons
Semimajor axis | 1.2747 au | NASA JPL Horizons
Orbital inclination | $28\hbox{$.\\!\\!^{\circ}$}6677$ | NASA JPL Horizons
Mean anomaly | $90\hbox{$.\\!\\!^{\circ}$}7904$ | NASA JPL Horizons
Argument peri. | $207\hbox{$.\\!\\!^{\circ}$}597$ | NASA JPL Horizons
Eccentricity | 0.8723 | NASA JPL Horizons
Long. ascending | $19\hbox{$.\\!\\!^{\circ}$}7247$ | NASA JPL Horizons
T_J | 4.507 | NASA JPL Horizons
Orbital period | 1.44 yr | NASA JPL Horizons
| Spectroscopy |
Taxonomy (Bus-DeMeo) | B-type | Devogèle et al. (2020)
Spectral Slope (Optical) |  $20\%\pm 10\%$ $\mu$m-1 | Devogèle et al. (2020)
Spectral Slope (NIR) |  $6\%\pm 1\%$ $\mu$m-1 | Kareta et al. (2021)
| Thermophysics |
Thermal inertia |  $300\begin{subarray}{c}+120\\\ -110\end{subarray}$ J m-2 K-1 s-1/2 | Devogèle et al. (2020)
Grain size | 0.9-10 mm | Devogèle et al. (2020)

Table A.1: 2005 UD Physical Parameters

(Continued)  Parameter Value Reference Mass loss rate 0.04–0.37 kg s-1 This
work (Section 4.3) Photometry Slope Parameter $G_{1}=0.61\pm 0.02$ Huang et
al. (2021) Slope Parameter $G_{2}=-0.006\pm 0.006$ Huang et al. (2021)
Effective diameter 1.32 $\pm 0.06$ km Ishiguro et al. (2022) Absolute mag ($V$
filter) $17.54\pm 0.02$ Ishiguro et al. (2022) Sidereal period $5.234246\pm
0.000097$ hr This work (Section 3.1) Bond albedo 0.052 Devogèle et al. (2020)
Geometric Albedo 0.10 $\pm 0.02$ Devogèle et al. (2020) Color $(g-r)$
$0.472\pm 0.05$ This work (Section 4) Color $(r-i)$ $0.065\pm 0.05$ This work
(Section 4) Shape aspect ratio 1.45 This work (Section 3.2) Spin axis
(preferred) $\lambda_{p}=116\hbox{$.\\!\\!^{\circ}$}6$,
$\beta_{p}=-53\hbox{$.\\!\\!^{\circ}$}6$ This work (Section 3.2) Spin axis
$\lambda=300\hbox{$.\\!\\!^{\circ}$}3$, $\beta=-55\hbox{$.\\!\\!^{\circ}$}4$
This work (Section 3.2) Critical density 570 kg m-3 Jewitt & Hsieh (2006)
Photometric range 0.4 Jewitt & Hsieh (2006) Mass loss rate $\lesssim 0.1$ kg
m-1 Kasuga & Masiero (2022) Polarimetry† Max degree of neg. polarization
$P_{min}=-1.2\%\pm 0.1\%$ Devogèle et al. (2020) Phase where $P_{min}$ occurs
$\alpha_{min}=9\hbox{$.\\!\\!^{\circ}$}5\pm 0\hbox{$.\\!\\!^{\circ}$}2$
Devogèle et al. (2020) Inversion angle
$\alpha_{inv}=20\hbox{$.\\!\\!^{\circ}$}2\pm 0\hbox{$.\\!\\!^{\circ}$}2$
Devogèle et al. (2020) Slope at $\alpha_{inv}$ $h=0.22\%\pm 0.01\%$ Devogèle
et al. (2020) Geometric albedo $p_{\mathrm{R}}=0.1$ Ishiguro et al. (2022)

Note. † Additional polarimetric results are contained in Ishiguro et al.
(2022) that are too broad for this table.

##  Appendix B 2005 UD Observations

Tables B.1, B.2, and B.3 describe all observations of 2005 UD used in this
work including those rejected from the shape modeling procedure.

Table B.1: Observational Circumstances

# | Date/Time | Span | Telescope | Filter | $N_{p}$ | $m_{V}$ | $r$ | $\Delta$ | $\angle_{\mathrm{STO}}$ |  | Reference |
---|---|---|---|---|---|---|---|---|---|---|---|---
| (UT) | (h) |  |  |  |  | (au) | (au) | (∘) |  |  |
1 | 2005-Nov-02 11:54:32 | 3.26 | LOT | $R$ | 6 | 18.3 | 1.37 | 0.55 | 36.7 |  | 1 |
2 | 2005-Nov-03 12:11:24 | 3.22 | LOT | $R$ | 5 | 18.4 | 1.39 | 0.57 | 36.7 |  | 1 |
3 | 2005-Nov-04 12:05:20 | 3.00 | LOT | $R$ | 13 | 18.5 | 1.4 | 0.59 | 36.7 |  | 1 |
4 | 2005-Nov-05 11:52:45 | 1.78 | LOT | $R$ | 16 | 18.6 | 1.41 | 0.61 | 36.6 |  | 1 |
5 | 2005-Nov-19 08:50:19 | 0.56 | UH88 | $R$ | 15 | 19.6 | 1.57 | 0.91 | 35.9 |  | 2 |
6 | 2005-Nov-20 07:23:39 | 0.79 | UH88 | $R$ | 16 | 19.7 | 1.58 | 0.93 | 35.9 |  | 2 |
7 | 2005-Nov-21 05:48:28 | 3.19 | UH88 | $R$ | 17 | 19.8 | 1.59 | 0.95 | 35.8 |  | 2 |
8 | 2005-Nov-22 05:31:34 | 3.53 | UH88 | $R$ | 9 | 19.9 | 1.61 | 0.99 | 35.7 |  | 2 |
9 | 2018-Sep-27 00:49:29 | 1.97 | LCO-fl16 | $r$ | 92 | 16.6 | 1.09 | 0.23 | 62.5 |  | 4 |
10* | 2018-Sep-27 07:09:27 | 5.23 | 31in | - | 87 | 16.5 | 1.1 | 0.22 | 58.2 |  | 4 |
11 | 2018-Sep-27 16:35:01 | 1.97 | LCO-fl11 | $r$ | 114 | 16.5 | 1.1 | 0.22 | 58.2 |  | 4 |
12 | 2018-Oct-01 15:55:45 | 2.47 | LCO-fl11 | $r$ | 115 | 16.1 | 1.16 | 0.23 | 40.6 |  | 4 |
13 | 2018-Oct-01 23:39:44 | 2.45 | LCO-fl16 | $r$ | 122 | 16.1 | 1.16 | 0.23 | 40.6 |  | 4 |
14 | 2018-Oct-03 05:41:52 | 2.45 | LCO-fa15 | $r$ | 142 | 16 | 1.18 | 0.23 | 36.3 |  | 4 |
15 | 2018-Oct-03 21:57:50 | 0.29 | Ondřejov | $R$ | 17 | 15.9 | 1.2 | 0.24 | 28.0 |  | 5 |
16 | 2018-Oct-03 23:22:49 | 1.46 | Ondřejov | $R$ | 134 | 15.9 | 1.2 | 0.24 | 28.0 |  | 5 |
17 | 2018-Oct-03 23:50:50 | 2.98 | LCO-fl16 | $r$ | 146 | 15.9 | 1.2 | 0.24 | 28.0 |  | 4 |
18 | 2018-Oct-04 00:51:58 | 0.96 | Ondřejov | $R$ | 86 | 15.9 | 1.2 | 0.24 | 28.0 |  | 5 |
19 | 2018-Oct-04 02:20:05 | 1.63 | Ondřejov | $R$ | 110 | 15.9 | 1.2 | 0.24 | 28.0 |  | 5 |
20 | 2018-Oct-05 21:17:57 | 2.18 | Ondřejov | $R$ | 158 | 15.9 | 1.22 | 0.24 | 24.1 |  | 5 |
21 | 2018-Oct-05 23:32:08 | 1.97 | Ondřejov | $R$ | 140 | 15.9 | 1.22 | 0.24 | 24.1 |  | 5 |
22 | 2018-Oct-05 01:31:54 | 2.23 | Ondřejov | $R$ | 160 | 15.9 | 1.22 | 0.24 | 24.1 |  | 5 |
23* | 2018-Oct-06 04:07:48 | 8.38 | 31in | $r$ | 32 | 15.9 | 1.23 | 0.25 | 20.4 |  | 4 |
24 | 2018-Oct-06 02:59:19 | 6.29 | TRAPPIST-S | $R$ | 348 | 15.9 | 1.23 | 0.25 | 20.4 |  | 4 |
25 | 2018-Oct-09 20:40:28 | 1.30 | Ondřejov | $R$ | 80 | 15.8 | 1.27 | 0.28 | 10.6 |  | 5 |
26 | 2018-Oct-09 22:00:18 | 2.26 | Ondřejov | $R$ | 154 | 15.8 | 1.27 | 0.28 | 10.6 |  | 5 |
27 | 2018-Oct-09 00:17:27 | 1.92 | Ondřejov | $R$ | 125 | 15.8 | 1.27 | 0.28 | 10.6 |  | 5 |
28 | 2018-Oct-09 02:17:16 | 0.46 | Ondřejov | $R$ | 28 | 15.8 | 1.27 | 0.28 | 10.6 |  | 5 |
29 | 2018-Oct-09 13:45:09 | 3.46 | LCO-fl11 | $r$ | 155 | 15.8 | 1.27 | 0.28 | 10.6 |  | 4 |
30 | 2018-Oct-09 20:30:22 | 3.55 | LCO-fl06 | $r$ | 168 | 15.8 | 1.27 | 0.28 | 10.6 |  | 4 |

Notes. Date/Time: start of observations; Span: duration of observations;
$N_{P}$: number of points; $m_{V}$: 2005 UD apparent $V-$band magnitude; $r$:
Sun-target distance; $\Delta$: Earth—target distance; $\angle_{\mathrm{STO}}$:
Sun–target–observer (phase) angle; LOT: Lulin One-meter Telescope; UH88:
University of Hawaii 88-in Telescope; LCO: Las Cumbres Observatory; 31in:
Lowell Observatory NURO 31-inch Telescope; Ondřejov: Ondřejov Observatory 0.65
m Telescope; TRAPPIST-S: South TRAnsiting Planets and PlanetesImals Small
Telescope.
∗ Rejected from the inversion process due to excessive photometric noise or
temporal overlap with other data. References. (1) Kinoshita et al. (2007); (2)
Jewitt & Hsieh (2006); (3) Warner & Stephens (2019); (4) Devogèle et al.
(2020); (5) this work.

Table B.2: Observational Circumstances

# | Date/Time | Span | Telescope | Filter | $N_{p}$ | $m_{V}$ | $r$ | $\Delta$ | $\angle_{\mathrm{STO}}$ |  | Reference |
---|---|---|---|---|---|---|---|---|---|---|---|---
| (UT) | (h) |  |  |  |  | (au) | (au) | (∘) |  |  |
31 | 2018-Oct-10 00:04:14 | 5.45 | TRAPPIST-N | $R$ | 285 | 15.8 | 1.29 | 0.29 | 7.7 |  | 4 |
32 | 2018-Oct-10 00:30:22 | 2.69 | Ondřejov | $R$ | 154 | 15.8 | 1.29 | 0.29 | 7.7 |  | 5 |
33* | 2018-Oct-10 03:09:27 | 9.02 | 31in | - | 57 | 15.8 | 1.29 | 0.29 | 7.7 |  | 4 |
34 | 2018-Oct-10 04:59:36 | 2.28 | LCO-fa15 | $r$ | 112 | 15.8 | 1.29 | 0.29 | 7.7 |  | 4 |
35 | 2018-Oct-12 20:38:08 | 3.17 | Ondřejov | $R$ | 107 | 15.8 | 1.31 | 0.31 | 2.6 |  | 5 |
36 | 2018-Oct-12 23:49:46 | 2.09 | Ondřejov | $R$ | 114 | 15.8 | 1.31 | 0.31 | 2.6 |  | 5 |
37 | 2018-Oct-12 06:00:36 | 1.34 | CS3 | $V$ | 26 | 15.8 | 1.31 | 0.31 | 2.6 |  | 3 |
38 | 2018-Oct-12 08:15:57 | 2.54 | CS3 | $V$ | 26 | 15.8 | 1.31 | 0.31 | 2.6 |  | 3 |
39 | 2018-Oct-13 05:10:00 | 1.90 | LCO-fa15 | $r$ | 88 | 15.7 | 1.33 | 0.33 | 0.6 |  | 4 |
40 | 2018-Oct-13 12:04:43 | 3.36 | LCO-fl11 | $r$ | 153 | 15.7 | 1.33 | 0.33 | 0.6 |  | 4 |
41 | 2018-Oct-14 19:55:12 | 0.46 | Ondřejov | $R$ | 19 | 15.9 | 1.34 | 0.34 | 2.1 |  | 5 |
42 | 2018-Oct-14 12:13:49 | 2.50 | LCO-fl11 | $r$ | 122 | 15.9 | 1.34 | 0.34 | 2.1 |  | 4 |
43 | 2018-Oct-15 18:47:03 | 2.69 | Ondřejov | $R$ | 120 | 16.2 | 1.35 | 0.35 | 4.1 |  | 5 |
44 | 2018-Oct-15 21:29:53 | 2.54 | Ondřejov | $R$ | 121 | 16.2 | 1.35 | 0.35 | 4.1 |  | 5 |
45* | 2018-Oct-15 02:26:25 | 9.10 | 31in | $r$ | 48 | 16.2 | 1.35 | 0.35 | 4.1 |  | 4 |
46 | 2018-Oct-15 03:32:17 | 1.49 | CS3 | $V$ | 27 | 16.2 | 1.35 | 0.35 | 4.1 |  | 3 |
47 | 2018-Oct-15 05:07:32 | 1.68 | CS3 | $V$ | 37 | 16.2 | 1.35 | 0.35 | 4.1 |  | 3 |
48 | 2018-Oct-15 06:56:53 | 1.80 | CS3 | $V$ | 31 | 16.2 | 1.35 | 0.35 | 4.1 |  | 3 |
49 | 2018-Oct-15 09:02:17 | 1.90 | CS3 | $V$ | 37 | 16.2 | 1.35 | 0.35 | 4.1 |  | 3 |
50 | 2018-Oct-15 19:22:50 | 1.75 | LCO-fl16 | $r$ | 60 | 16.2 | 1.35 | 0.35 | 4.1 |  | 4 |
51 | 2018-Oct-16 01:59:55 | 3.98 | LCO-fa15 | $r$ | 171 | 16.4 | 1.36 | 0.37 | 5.9 |  | 4 |
52* | 2018-Oct-16 02:24:27 | 3.17 | 31in | $r$ | 14 | 16.4 | 1.36 | 0.37 | 5.9 |  | 4 |
53 | 2018-Oct-16 03:13:13 | 1.75 | CS3 | $V$ | 22 | 16.4 | 1.36 | 0.37 | 5.9 |  | 3 |
54 | 2018-Oct-16 05:00:16 | 1.54 | CS3 | $V$ | 27 | 16.4 | 1.36 | 0.37 | 5.9 |  | 3 |
55 | 2018-Oct-16 06:42:44 | 3.26 | CS3 | $V$ | 52 | 16.4 | 1.36 | 0.37 | 5.9 |  | 3 |
56 | 2018-Oct-16 09:59:22 | 0.74 | CS3 | $V$ | 15 | 16.4 | 1.36 | 0.37 | 5.9 |  | 3 |
57 | 2018-Oct-16 12:15:49 | 0.36 | LCO-fl12 | $r$ | 14 | 16.4 | 1.36 | 0.37 | 5.9 |  | 4 |
58 | 2018-Oct-16 19:59:50 | 3.48 | LCO-fl16 | $r$ | 132 | 16.4 | 1.36 | 0.37 | 5.9 |  | 4 |
59 | 2018-Oct-16 21:42:10 | 2.71 | Ondřejov | $R$ | 23 | 16.6 | 1.38 | 0.38 | 7.6 |  | 5 |
60 | 2018-Oct-17 02:59:36 | 3.46 | LCO-fa15 | $r$ | 150 | 16.6 | 1.38 | 0.38 | 7.6 |  | 4 |

Notes. Date/Time: start of observations; Span: duration of observations;
$N_{P}$: number of points; $m_{V}$: 2005 UD apparent $V-$band magnitude; $r$:
Sun-target distance; $\Delta$: Earth—target distance; $\angle_{\mathrm{STO}}$:
Sun–target–observer (phase) angle; TRAPPIST-N: North TRAnsiting Planets and
PlanetesImals Small Telescope; Ondřejov: Ondřejov Observatory 0.65 m
Telescope; Lowell Observatory NURO 31-inch Telescope; LCO: Las Cumbres
Observatory; CS3: Center for Solar System Studies.
∗ Rejected from the inversion process due to excessive photometric noise or
temporal overlap with other data.
References. (1) Kinoshita et al. (2007); (2) Jewitt & Hsieh (2006); (3) Warner
& Stephens (2019); (4) Devogèle et al. (2020); (5) this work.

Table B.3: Observational Circumstances

# | Date/Time | Span | Telescope | Filter | $N_{p}$ | $m_{V}$ | $r$ | $\Delta$ | $\angle_{\mathrm{STO}}$ |  | Reference |
---|---|---|---|---|---|---|---|---|---|---|---|---
| (UT) | (h) |  |  |  |  | (au) | (au) | (∘) |  |  |
61 | 2018-Oct-17 04:20:39 | 3.12 | TRAPPIST-N | $R$ | 179 | 16.6 | 1.38 | 0.38 | 7.6 |  | 4 |
62 | 2018-Oct-17 18:52:22 | 2.30 | Ondřejov | $R$ | 76 | 16.7 | 1.39 | 0.4 | 9.3 |  | 5 |
63 | 2018-Oct-17 21:11:36 | 2.69 | Ondřejov | $R$ | 91 | 16.7 | 1.39 | 0.4 | 9.3 |  | 5 |
64 | 2018-Oct-20 23:58:39 | 2.28 | TRAPPIST-N | $r$ | 133 | 17 | 1.41 | 0.43 | 12.1 |  | 4 |
65 | 2018-Oct-20 01:41:08 | 2.30 | LCO-fa15 | $r$ | 91 | 17 | 1.41 | 0.43 | 12.1 |  | 4 |
66 | 2018-Oct-25 10:28:53 | 3.77 | LCO-fl12 | $r$ | 75 | 17.7 | 1.47 | 0.51 | 17.8 |  | 4 |
67 | 2018-Oct-27 01:43:35 | 2.98 | LCO-fa15 | $r$ | 69 | 17.9 | 1.5 | 0.55 | 19.6 |  | 4 |
68 | 2018-Oct-28 00:08:31 | 3.43 | LCO-fa03 | $r$ | 84 | 18.1 | 1.5 | 0.56 | 20.4 |  | 4 |
69 | 2018-Oct-30 10:38:14 | 3.46 | LCO-fl11 | $r$ | 73 | 18.3 | 1.53 | 0.6 | 21.9 |  | 4 |
70 | 2018-Oct-31 10:24:58 | 3.02 | LCO-fl11 | $r$ | 60 | 18.4 | 1.54 | 0.62 | 22.6 |  | 4 |
71 | 2018-Nov-01 01:42:22 | 2.93 | LCO-fa15 | $r$ | 62 | 18.5 | 1.55 | 0.64 | 23.2 |  | 4 |
72 | 2018-Nov-02 02:43:58 | 1.90 | LCO-fa15 | $r$ | 43 | 18.6 | 1.56 | 0.67 | 23.8 |  | 4 |
73 | 2018-Nov-04 00:24:18 | 4.06 | Danish | $R$ | 20 | 18.8 | 1.58 | 0.69 | 24.8 |  | 5 |
$74^{+}$ | 2019-Oct-18 09:19:18 | 3.24 | LDT | $VR$ | 83 | 20.3 | 1.66 | 1.16 | 36.3 |  | 5 |
75 | 2019-Nov-03 01:20:13 | 5.16 | NOT | $r$ | 329 | 19.6 | 1.49 | 0.87 | 39.6 |  | 5 |
76 | 2019-Nov-18 07:24:19 | 5.81 | LDT | $VR$ | 666 | 18.8 | 1.32 | 0.66 | 45.2 |  | 5 |
77 | 2019-Nov-18 01:15:07 | 5.47 | NOT | $gri$ | 254 | 18.8 | 1.33 | 0.66 | 45.8 |  | 5 |
78 | 2019-Nov-23 00:25:28 | 5.52 | TRAPPIST-N | $Rc$ | 130 | 18.6 | 1.25 | 0.58 | 50.7 |  | 5 |
79 | 2019-Nov-24 02:10:49 | 2.57 | TRAPPIST-N | $Rc$ | 62 | 18.5 | 1.23 | 0.56 | 51.7 |  | 5 |
80* | 2019-Nov-25 00:34:05 | 5.57 | TRAPPIST-N | $Rc$ | 115 | 18.5 | 1.22 | 0.55 | 52.7 |  | 5 |
81 | 2021-Oct-27 03:30:14 | 0.65 | Danish | $R$ | 8 | 19.2 | 1.90 | 0.92 | 7.2 |  | 5 |
82 | 2021-Oct-28 03:11:31 | 1.27 | Danish | $R$ | 13 | 19.3 | 1.91 | 0.93 | 7.7 |  | 5 |
83 | 2021-Oct-29 00:05:46 | 3.58 | Danish | $R$ | 19 | 19.3 | 1.91 | 0.94 | 8.2 |  | 5 |
84 | 2021-Oct-30 00:02:53 | 3.56 | Danish | $R$ | 25 | 19.4 | 1.92 | 0.95 | 8.7 |  | 5 |
85 | 2021-Nov-03 01:29:31 | 2.47 | LDT | $VR$ | 115 | 19.6 | 1.95 | 1.00 | 11.2 |  | 5 |
86 | 2005—2011 SPARSE | — | CSS | $V$ | 19 | — | — | — | — |  | 6 |
87 | 2014—2021 SPARSE | — | PS | $w$ | 31 | — | — | — | — |  | 7 |
88 | 2018—2021 SPARSE | — | ZTF | $V$ | 71 | — | — | — | — |  | 8 |
89 | 2005—2011 SPARSE | — | ATLAS | $o$ | 41 | — | — | — | — |  | 9 |
90 | 2005—2011 SPARSE | — | CSS | $G$ | 44 | — | — | — | — |  | 6 |

Notes. Date/Time: start of observations; Span: duration of observations;
$N_{P}$: number of points; $m_{V}$: 2005 UD apparent $V-$band magnitude; $r$:
Sun-target distance; $\Delta$: Earth—target distance; $\angle_{\mathrm{STO}}$:
Sun–target–observer (phase) angle; TRAPPIST-N: North TRAnsiting Planets and
PlanetesImals Small Telescope; Ondřejov: Ondřejov Observatory 0.65 m
Telescope; LCO: Las Cumbres Observatory; LDT: Lowell Discovery Telescope; NOT:
Nordic Optical Telescope; Danish: La Silla Observatory 1.54 m Telescope; CSS:
Catalina Sky Survey; PS: Pan-STARRS; ZTF: Zwicky Transient Facility; ATLAS:
Asteroid Terrestrial-impact Last Alert System.
∗ Rejected from the inversion process due to excessive photometric noise or
temporal overlap with other data.
+ First 1.5 hr rejected from the inversion process due to being inconsistent with our best-fit models.
References. (1) Kinoshita et al. (2007); (2) Jewitt & Hsieh (2006); (3) Warner
& Stephens (2019); (4) Devogèle et al. (2020); (5) This work; (6) Larson et
al. (2003); (7) (Chambers et al., 2016); (8) Bellm et al. (2019); (9) Tonry et
al. (2018).