Implications for the Formation of 2005 UD from a New Convex Shape Model

(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 ($\lambda = 116.6^{\circ}$, $\beta = -53.6^{\circ}$) and ($\lambda = 300.3^{\circ}$, $\beta = -55.4^{\circ}$), 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.


INTRODUCTION
Near-Earth asteroid (NEA) 2005 UD is a kilometerclass object and is a potential flyby target of JAXA's DESTINY + mission, 1 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 . 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. • 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. • 8 +1.1 −5.3 , −25. • 8 +5.3 −12.5 ) 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. • 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 1 https://destiny.isas.jaxa.jp/science/ 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.

DATA COLLECTION & PROCESSING
We define "light curve" as the time series of diskintegrated 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.

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 6144×6160 pixels and a square 12. 5 field of view. This instrument samples at a pixel scale of 0. 12 pixel −1 but was used in 3 × 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. 7, 1. 5, and 2. 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 × 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 2×2 binning and exposure times set at 30 s across both nights. Seeing varied typically from 1" to 2. 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 An-dor iKON-L BEX2-DD CCD camera imaging through a Cousins R filter. Additional instrument specifications include a 0. 60 pixel −1 scale and a 22 square field of view. The images were binned 2×2 with exposure times set at 120 s. Seeing for the first night was variable with median on-chip values ranging from ∼ 3" to 4. 2 with conditions improving on the second night, where values were in the range of ∼ 2. 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×2k sensor with 13.5 µm 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 × 1472 square pixels (6.8 micron pitch) with microlenses and we imaged in 2 × 2 binning mode, providing 1. 05 pixel −1 and a 19'×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: 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) 2 . 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 ± 0.2 and (r−i) = 0.11±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. 22 to 1. 94) for the NOT data and 3.26-5.37 binned pixels (1. 17 to 1. 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.

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  and .
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.

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 gradientbased algorithm, will cause the optimizer to get trapped  (Giorgini et al. 1996).
and converge to an ill-fitting solution. The spacing of these local minima ∆P (in hours) in the period-space χ 2 spectrum is given roughly by ∆P ≈ P 2 /(2T ), where P is the rotational period of the object and T is the time span of the entire data set . 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 χ 2 (see Section 3.1 in  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 χ 2 threshold of ∼ 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 LC for each remaining dense light curve quantitatively using W LC = 1/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 ∼ 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.
The period scan results within a search range of 5.225 and 5.245 hr produced a unique sidereal rotational pe- 2 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 χ 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 sid = 5.234246 ± 0.000097 and is shown in red.
riod value of P sid = 5.234246 ± 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.

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 • spacing, which is later opti- Note. * Rejected due to nonphysical shape. χ 2 and rms quantify fits to dense light curves only.
mized 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 χ 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 χ 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 ∼ 100 • in phase over a 3 yr span. 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.
Recent shape modeling efforts using the Lommel-Seeliger ellipsoid method by Huang et al. (2021) yielded two candidate pole solutions for 2005 UD: . 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 3σ 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: 3.1µm; W2: 4.6µm). Thermophysical modeling of these data was presented in Devogèle et al. (2020). The best-fit solution from the thermophysical model (λ TPM = 102 • ± 20 • and β TPM = −35 • ± 30 • ) 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 χ 2 among numerous trial runs, we adopt λ p = 116. • 6 ± 2. • 3, β p = −53. • 6 ± 5. • 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.
The convex shape ( Figure 4) from our preferred pole solution (λ p , β p ) = (116. • 6, −53. • 6) is nearly identical to our other candidate solution (λ, β) = (300. • 3, −55. • 4). This is expected because there remains a 180 • 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 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 χ 2 values of the thermophysical solutions, which are represented as stars; only retrograde solutions with χ 2 ≤ 5 are shown for clarity. The bestfitting 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.
For reference, we provide a master summary of known properties for 2005 UD in Table A  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 , and they showed that their light curve data alone could not eliminate the possibility of a sidereal period of ∼ 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.  Table 3.

Surface Colors
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 inter-  ) 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.
polated 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 1σ 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 ∼ 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 ∼ 52 • 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 • 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.

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 Siril 3 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 θ = 1. 8-3. 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 + F e − r 2 2σ 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 θ = 1. 8-3. 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 µm 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 0.1 kg s −1 estimate derived by Kasuga & Masiero (2022) using NEOWISE observations.

A COMMON ORIGIN WITH PHAETHON?
With additional data from new viewing geometries, we were able to constrain two possible spin solutions for 2005 UD: (λ p = 116. • 6, β p = −53. • 6) and (λ = 300. • 3, β = −55. • 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.

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 volatiledriven 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.

Nongravitational Influences
The Yarkovsky effect (Bottke et al. (2006)) is relevant when considering nongravitational perturbations for small (generally less than ∼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 and2005 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 ∼ 3.5 × 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 ∼ 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.
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.

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 sid = 5.234246 ± 0.000097 hr. We conclude that 2005 UD has two equally well-fitting spin solutions given as (λ p = 116. • 6, β p = −53. • 6) and (λ = 300. • 3, β = −55. • 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. • 8 +1.1 −5.3 , −25. • 8 +5.3 −12.5 ) is only consistent in latitude with the latter of our two solutions given 3σ 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 followup 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 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 presentday activity episodes reveal that they may not be intense enough Li 2010 andJewitt 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. ACKNOWLEDGMENTS 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 ( 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.