Convex Shape and Rotation Model of Lucy Target (11351) Leucus from Lightcurves and Occultations

The photometric lightcurves presented in the previous section and in Buie et al. (2018), and the results from the stellar occultation campaigns reported in Buie et al. (2020) constitute the bulk of observational data used for our modeling work. In addition, we made use of dense lightcurve photometry by French et al. (2013; retrieved through the ALCDEF service (http://alcdef.org/)) and sparse photometry from the following sources:

• 1.  
  The Gaia DR2 database (Gaia Collaboration et al. 2018) retrieved through the VizieR server (https://vizier.u-strasbg.fr/),
• 2.  
  The ZTF project (Bellm et al. 2019) retrieved through the IRSA server (https://irsa.ipac.caltech.edu/applications/ztf/) and from the nightly transient archive at https://ztf.uw.edu,
• 3.  
  The PAN-STARRS-1 DR2 database (Flewelling et al. 2016) retrieved through a query to the MAST archive (https://catalogs.mast.stsci.edu/), and
• 4.  
  The ATLAS project (Tonry et al. 2018) retrieved through the AstDys database (https://newton.spacedys.com/astdys/).

For all but the ATLAS observations, photometric uncertainties were retrieved along with the magnitude data. In the case of the ATLAS observations, we estimated nominal photometric uncertainties by compiling the data into composites for the individual oppositions and computing their residuals. Since the survey observations were acquired in a variety of different photometric systems, for which transformations to the Johnson system are not accurately established, they were treated as relative photometry. Similarly to the dense data set, the times were light-time corrected, and the magnitudes reduced to 1 au. Although these additional data sets provide varying degrees of accuracy, they proved to be very useful to extend the coverage and baseline of the observations, which combined, cover a period of about 6 years.

In this paper we apply the convex shape inversion approach described in Kaasalainen et al. (2002) and references therein to the photometric time series to simultaneously solve for the sidereal period, the spin-axis orientation, the photometric function, and a convex, polyhedral approximation of the shape. The occultation data are used as a constraint to resolve the spin-axis ambiguity, to determine the scale of the object—and hence its albedo—and to refine the orientation of the spin axis. Although it is likely that Leucus does contain some degree of global-scale concavity, we did not feel that the available data—both because of coverage and photometric accuracy, in the case of lightcurve data, and because of limited unique observation geometries in the case of occultation data—would allow addressing the intrinsic nonuniqueness of the nonconvex problem. On the other hand, the convex inversion scheme offers the advantage of a provably convergent method that results in a unique solution in the case of a convex shape (Kaasalainen & Lamberg 2006) and gracefully degrades in the case of moderate concavities. Nonconvex modeling of Leucus will be the subject of future work as soon as more data—especially at further occultation geometries—become available. Although a working implementation of the convex inversion algorithm (convexinv) is publicly available (Durech et al. 2010) we decided to develop our own implementation based on the original publications (Kaasalainen & Torppa 2001; Kaasalainen et al. 2001). This approach has allowed us to overcome some limitations of the code (described later) to expand its functionality and to correct a minor bug in the computation of the χ² metric that is present in the current convexinv version and that has been promptly communicated to its author. Although we did not make use of any code from convexinv, we did use that software to validate the results of our own code on a few test cases.

The surface brightness of the object is described through its photometric function, which, for the purpose of this work, is assumed to be separable into a disk function and a surface phase function (see, e.g., Schröder et al. 2013). Following Kaasalainen et al. (2001), we adopt Lommel-Seelinger-Lambert (LSL) scattering as a disk function and a three-parameter exponential-linear combination as a surface phase function, as described in the Appendix. The LSL disk function is the linear combination of the Lommel-Seeliger and Lambert scattering functions through a partition parameter c and is equivalent to the Lunar-Lambert disk function (Li et al. 2015 and references therein) when the latter is used with a partition function independent of the phase angle. We fixed c to a constant value of c = 0.1, appropriate for dark asteroids (Kaasalainen et al. 2005). For this work we did not attempt to use more complex photometric models—as, e.g., the Hapke function (Hapke 2012 and references therein), because due to the small phase-angle range in which Trojans can be observed from Earth, no meaningful retrieval of the model parameters is achievable.

The brightness of the object depends on the product of its size and its albedo. From unresolved photometric measurements alone, it is not possible to retrieve independently those two quantities. Independent measurements such as thermal radiometry, stellar occultations, or direct imaging, however, offer the possibility to disentangle the two quantities. In the absence of more detailed information, we assume that the photometric properties of the target—in particular its albedo—are uniform over its surface. This is quite a reasonable assumption, because, although albedo variations on asteroids do occur, the observed lightcurve variations tend to be dominated by the changing cross-section of a nonspherical shape.

The convex inversion scheme models the brightness of the target in the space of its Extended Gaussian Image (EGI; Kaasalainen & Torppa 2001), as opposed to the 3D object space. The EGI represents the discrete equivalent of the inverse of the Gaussian Surface Density and is used to represent the area of the facets oriented toward a particular direction. For the Leucus model we parameterize the EGI with a spherical harmonic expansion of rank and order eight, which results in a total of 81 shape parameters, one of which represents its scale. An exponential-function representation of the EGI is used to guarantee positive surface areas. We integrate the EGI over the unit sphere by sampling the spherical harmonics expansion at discrete points by following a Lebedev quadrature (Lebedev & Laikov 1999), which offers greater efficiency compared to a quadrature based, e.g., on a uniform or random sampling (Kaasalainen et al. 2012). In the case of Leucus, we applied a Lebedev rule of order 302, which results in an EGI with the same number of facets.

Contrary to Kaasalainen & Torppa (2001)—and to the convexinv implementation—we minimize a weighted metric for solving the least-squares problem in order to properly account for the varying degree of accuracy of the different data sets. In the case of absolute observations, the optimization is performed by minimizing the reduced ${\chi }_{\mathrm{red}}^{2}$ metric defined as

$$\begin{eqnarray}&&{\chi }_{\mathrm{red}}^{2}=\displaystyle \frac{1}{N}\displaystyle \sum _{i}\displaystyle \frac{{\left({L}_{i}^{\mathrm{obs}}-{L}_{i}^{\mathrm{mod}}\right)}^{2}}{{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 1 }$$

where L_i^obs and L_i^mod are the observed and model intensities, respectively, σ_i is the intensity uncertainty, and N is the number of degrees of freedom for errors. The index i runs over all of the n photometric data points.

In the case of relative observations, we minimize the deviations of the intensities relative to the average of the respective lightcurve:

$$\begin{eqnarray}&&{\chi }_{\mathrm{rel}}^{2}=\displaystyle \frac{1}{N}\displaystyle \sum _{k,j}{\left(\displaystyle \frac{{\bar{L}}_{j}^{\mathrm{obs}}}{{\sigma }_{k,j}}\right)}^{2}{\left(\displaystyle \frac{{L}_{k,j}^{\mathrm{obs}}}{{\bar{L}}_{j}^{\mathrm{obs}}}-\displaystyle \frac{{L}_{k,j}^{\mathrm{mod}}}{{\bar{L}}_{j}^{\mathrm{mod}}}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

where the index k runs over the data points of the lightcurve j and ${\bar{L}}_{j}^{\mathrm{obs}}$ and ${\bar{L}}_{j}^{\mathrm{mod}}$ are the average intensities of the observed and modeled lightcurve j, respectively. The term ${\sigma }_{k,j}$ represents the uncertainty of the kth data point of lightcurve j. If absolute observations were performed during the same apparition in two photometric bands, then a color index term is introduced to tie the two photometric systems. The color term is also optimized in the procedure. If, on the other hand, one photometric band is never used together with another photometric system at least during one apparition, then these series of observations are treated as relative photometry in order to avoid a possible parameter coupling between the color index and the spin-axis orientation. The nonlinear optimization is performed with the Levenberg–Marquardt algorithm (Press et al. 1992).

A unique transformation from the EGI to a polyhedron in 3D space is guaranteed by a Minkowski theorem (Minkowski 1897). For this transformation we use the iterative scheme proposed by Lamberg (1993). The body reference system is defined such that the Z-axis coincides with the spin axis. The plane that contains the body center of mass and that is perpendicular to the Z-axis defines the XY plane in the body system. The direction of the X-axis is chosen such that it coincides with the projection of the principal axis of smallest inertia onto the XY plane. The body X-axis also defines the location of the prime meridian and thus the zero longitude.

The rotation period strongly modulates the spectrum of the χ² of the fit with periodic local minima at a minimum spacing ΔP ≈ P²/(2T) (Kaasalainen et al. 2001), where P is the rotation period and T is the total baseline of the lightcurve coverage. It is therefore important that the optimization be started in the vicinity of the correct period in order to avoid the optimizer becoming trapped in the local minimum of an alias period. For this reason, the search of the correct sidereal period is the first step in the convex inversion scheme. The search is performed by running the inversion procedure using all trial periods in a relevant range as starting conditions with a step sufficiently smaller than the minima separation. For each trial period we use 12 different starting pole directions. Figure 2 shows the results of the period scan for Leucus.

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In order to identify the coarse direction of the spin axis we run the optimization procedure using the best-fit sidereal period derived in the previous paragraph as a starting value and by fixing the spin-axis orientation to each of about 20,000 trial directions equally spaced on the celestial sphere. The shape of the object, the sidereal period, and the photometric parameters (but not the pole coordinates) are simultaneously optimized for each trial pole. The resulting χ² values for each solution are mapped on the celestial sphere via a polar azimuthal equidistant projection and are shown in Figure 3. As expected, two equally significant loci for the best solution are identified. This is the consequence of the ambiguity theorem (Kaasalainen & Lamberg 2006) that states that if disk-integrated photometric observations are always carried out in the same photometric plane—as is the case for low-inclination objects observed from Earth—then two indistinguishable solutions exist that satisfy the observations and that are separated by about 180° in ecliptic longitude. The shapes corresponding to the two solutions are approximately mirrored shapes of one another around the body XY-plane. The graph also shows that both solutions are prograde and, as already inferred by Buie et al. (2018), the obliquity of the spin axis is low.

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By providing disk-resolved information, occultation data can resolve the pole ambiguity, fix the absolute scale of the shape model, and, together with the determined H_V-value, measure its geometric albedo.

In principle, occultation data can also be used to derive nonconvex shape models provided sufficient, densely sampled silhouettes are available at multiple rotation phases. Much work has been recently done concerning the optimal fusion of data coming from disk-integrated photometry and disk-resolved techniques as stellar occultations, adaptive-optics direct imaging, and interferometry (e.g., Kaasalainen & Viikinkoski 2012; Viikinkoski et al. 2015). In the case of Leucus, however, both the limited lightcurve coverage and the sparse silhouette sampling would not allow a reliable nonconvex model to be derived. For this reason we decided to adopt an approach similar to Durech et al. (2011) and favor the advantages of the uniqueness and stability of a convex solution.

For each of the two best-candidate solutions from the previous section we project the vertices of the shape model onto the plane of sky at the time of each occultation event and then compute the 2D convex hull of the projected points. Applying this procedure to the two complementary solutions visible in Figure 3 allowed us to unambiguously identify the correct solution as the one centered at an ecliptic longitude of around 210°. However, it also became apparent that the best solution had a slight systematic deviation in the orientation of the projection with respect to the occultation data that could be explained by a slight offset (≈5°) in the direction of the spin-axis orientation of the model. Such a small mismatch was not unexpected, as the loci of the solutions in Figure 3 are quite broad and shallow, and a small shift in the spin axis direction of the model would produce fits to the lightcurves with similar χ². On the other hand, disk-resolved data such as stellar occultations are much more sensitive to a pole misalignment. For this reason we decided to use the occultation data for the refinement of the solution.

As a goodness of fit for the occultation data we define a metric

$$\begin{eqnarray}&&{\chi }_{\mathrm{occ}}^{2}=\displaystyle \sum _{i,j}\displaystyle \frac{{\left({D}_{{ij}}\right)}^{2}}{{N}_{\mathrm{op}}},\end{eqnarray} \tag{ 3 }$$

where D_ij represents the minimum Euclidean distance between the occultation transition point j (either ingress or egress) of the event i and the model 2D convex hull of the event i. N_op is the total number of the observed transition points. This metric is different from the one chosen by Durech et al. (2011), who prefer to use the distance of the occultation points to the occultation limb measured in the direction of the asteroid ground track. Their choice is justified by the fact that the largest contribution to their occultation data is given by timing errors and observer reaction times, which act along track. In our case, on the other hand, we estimate the largest errors to be due to the convex-shape model, which are not expected to have a preferred direction.

${\chi }_{\mathrm{occ}}^{2}$ is minimized by optimizing the global scale and the Cartesian coordinates of the centers of the projections for each occultation epoch. The latter is necessary to compensate both for inaccuracies in the position of the occulted stars and for the uncertainties of the target ephemeris at the epochs of the occultations. In practice the minimization is performed by varying the projection centers and the A_LSL albedo (see the Appendix)—which constrains the scale—in an adaptive grid search fashion.

One practical problem arises from the fact that the convex-shape optimization is performed in the EGI space, while the occultation profile optimization is done in the space of the projected shape. It is therefore impractical to perform a simultaneous, combined optimization. For this reason we used the ${\chi }_{\mathrm{occ}}^{2}$ as a mild penalty function for a combined ${\chi }_{\mathrm{tot}}^{2}$ of the form

$$\begin{eqnarray}&&{\chi }_{\mathrm{tot}}^{2}={\chi }_{\mathrm{conv}}^{2}+\lambda {\chi }_{\mathrm{occ}}^{2},\end{eqnarray} \tag{ 4 }$$

where ${\chi }_{\mathrm{conv}}^{2}$ is the term coming from the convex inversion and λ is a small weight. The value of λ has been determined by experimentation by adopting a value of λ that minimized ${\chi }_{\mathrm{occ}}^{2}$ without significantly increasing ${\chi }_{\mathrm{conv}}^{2}$. We have then computed the quantity ${\chi }_{\mathrm{tot}}^{2}$ for one hundred discrete pole directions within a radius of 10° of the best solution (and of the complementary one).

The parameters for the best-fit solution are reported in Table 2. The errors quoted for the different quantities have been determined by computing perturbed solutions and investigating the effects on the χ². It must be noted that this method produces an evaluation of the statistical component of the uncertainties only. The largest contribution to the errors is thought to be due to the violations of the assumptions, as the assumption of the functional form of the photometric model and the convexity assumption. Those contributions, however, are virtually impossible to formally quantify. For this reason we refrained from using more sophisticated statistical error models, such as the Markov Chain Monte Carlo (MCMC) method, because they also only address the stochastic component of the uncertainty.

Table 2.  Results

Sidereal Period (h)	445.683 ± 0.007
Pole J2000 Ecl. Longitude (°)	208
Pole J2000 Ecl. Latitude (°)	+77
Pole J2000 R.A. (°)	248
Pole J2000 Decl. (°)	+58
Radius of pole uncertainty (°, 1σ)	3
Ecliptic obliquity of pole (°)	13
Orbital obliquity of pole (°)	10
T0 (JDTDB)	2 456 378.0
Φ0 (°)	−76.129
W0 (°)	60.014
$\dot{W}$ (°day−1)	19.385 96 ± 0.00030
pV	0.043 ± 0.002
ALSL	0.061 ± 0.002
A0	0.23 ± 0.09
D (rad)	0.075 ± 0.015
k (rad−1)	−1.07 ± 0.23
c (fixed)	0.1
${H}_{{\rm{V}}-\mathrm{LSL}}$ (sph. int.)	10.979 ± 0.037
${H}_{{\rm{V}}-\mathrm{lin}}$ (sph. int.)	11.034 ± 0.035
β ($\mathrm{mag}/^\circ$)	0.039 5 ± 0.0005
${H}_{{\rm{V}}-\mathrm{HG}}$ (sph. int.)	10.894 ± 0.004
G	0.34 ± 0.02
${H}_{{\rm{V}}-{\mathrm{HG}}_{1}{{\rm{G}}}_{2}}$ (sph. int.)	10.95 ± 0.01
G1	0.63 ± 0.04
G2	0.23 ± 0.02
V − R	0.464 ± 0.015
$V-{\text{}}r^{\prime}$	0.313 ± 0.021
${L}_{X}(\mathrm{km})$	60.8
${L}_{Y}(\mathrm{km})$	39.2
${L}_{Z}(\mathrm{km})$	27.8
Surface-equivalent spherical diam. $(\mathrm{km})$	41.0 ± 0.7
Photometric surface $({\mathrm{km}}^{2})$	5288 ± 180
Volume $({\mathrm{km}}^{3})$	≤3.0 × 104

Note. Please refer to the text for the definition of the respective quantities.

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Figure 4 shows the synthetic lightcurves for the corresponding shape model for the epochs covered by the dense observations reported in this paper. The lightcurve intensities are corrected for changing heliocentric and topocentric range and are normalized to unity at the maximum of the respective observation window. Besides the rotational variation, the intensity variation due to the changing phase angle is visible.

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Figure 5 shows the resulting best-fit model overplot onto the occultation chords by Buie et al. (2020). The complementary (wrong) solution is also plotted in light gray, showing how occultation data can help identify the correct solution. Please note that, for comparison purposes, we plot the data following Buie's convention of projecting the shape onto the plane of sky (Green 1985), with the η coordinate increasing toward celestial north and the ξ coordinate increasing toward east. This is a different convention than in Durech et al. (2011) who project the shape onto the fundamental plane, with the η coordinate increasing toward west and the ξ coordinate increasing toward north.

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We note that the model shape well reproduces the occultation profiles and captures the general shape of the object within the limits of a convex representation. In particular, it reproduces well the flat sides visible during the events LE20181118 and LE20191002 and the polygonal appearance of event LE20181118. It is also important to note that the occultation data are not directly used to derive the convex shape. The shape is influenced by the occultation data only indirectly through the refinement of the spin-axis orientation. In this light, the match of the convex shape model to the occultation data appears even more convincing.

Figure 6 shows six orthogonal views of the best-fit Leucus shape model. As already hinted by the occultation silhouettes, Leucus' shape considerably deviates from an ellipsoid and is characterized by a comparatively flat northern hemisphere. The convexity residual of the shape model resulting from the Minkowski transformation (Kaasalainen & Torppa 2001) is a mere 0.14%, which, taken at face value, would suggest negligible global-scale concavities. However, the small phase-angle range at which observations are available reduces the diagnostic value of this parameter.

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As a sanity check, we computed the model's inertia tensor with the method of Dobrovolskis (1996)—assuming a uniform bulk density—and established that the principal inertia axis is misaligned with respect to the body's rotation axis by about 8°. To this end, we have to recall that the derived shape model represents a convex approximation of the shape and ignoring concavities can contribute to shift the direction of the inertia axes. Also the assumption of uniform bulk density, if violated, would contribute to shifting the direction of the principal axis of inertia. The observed misalignment is therefore not necessarily a hint that the object is not in a principal rotation state. Rather, it is an expression of the fact that the convex shape represents a photometric shape, which can locally differ from the physical shape.

The maximum extent of a complex shape can be defined in several different ways (see, e.g., Torppa et al. 2008). For our model we define the maximum dimensions as

$$\begin{eqnarray}&&\begin{array}{l}{L}_{X}=\max ({x}_{1},...,{x}_{i})-\min ({x}_{1},...,{x}_{i})\\ {L}_{Y}=\max ({y}_{1},...,{y}_{i})-\min ({y}_{1},...,{y}_{i})\\ {L}_{Z}=\max ({z}_{1},...,{z}_{i})-\min ({z}_{1},...,{z}_{i}),\end{array}\end{eqnarray} \tag{ 5 }$$

where (x_i, y_i, z_i) represent the Cartesian coordinates of the ith vertex of the shape model and the X, Y, and Z body axes are defined as per Section 4.2. This definition produces similar—but not identical—extents as the overall dimensions (OD) definition of Torppa et al. (2008; see their Figure 1). In particular, with our definition, the largest extent is computed in the general direction of the principal axis of the smallest inertia, which represents a natural axis of the body. In the case of the OD definition by Torppa et al. (2008), on the other hand, the maximum dimension is the largest extent that occurs anywhere in the XY plane. As an example, for a hypothetical body with a rectangular equatorial cross-section, with our definition, the maximum extent would be represented by the longest side of the rectangle, while, according to the definition by Torppa, the maximum extent would be represented by the diagonal of the rectangle.

With our definition, the maximum dimensions for Leucus are L_X = 60.8 km, L_Y = 39.2 km, and L_Z = 27.8 km (see Table 2). Those compare with the axes 63.8, 36.6, and 29.6 km of the ellipsoidal approximation by Buie et al. (2020) which were derived under the assumption of a strictly equatorial aspect during the occultation events. In the same table, the orientation of the model is described both by reporting the ecliptic J2000 coordinates of the spin axis and initial angle Φ₀ at the epoch T₀ using the formalism by Kaasalainen et al. (2001) as well as using the IAU convention of reporting the ICRF equatorial coordinates of the spin axis and the W₀ angle at the standard epoch J2000 (Archinal et al. 2018, 2019).

The surface-equivalent spherical diameter of the convex model is D = 41.0 ± 0.7 km whereas its volume is (30 ± 2) × 10³  km³. It should be noted, however, that due to the likely presence of concavities, the quoted value for the volume rather represents an upper bound.

The disk-integrated phase curve of an object condenses the often complex parameter space of a photometric function into a two-dimensional space. As such, the phase curve is a useful phenomenological tool to compare and classify different objects and to infer the presence of physical phenomena, such as, e.g., coherent backscattering. A practical difficulty in deriving phase curves, however, is that during a single apparition—and even more so across multiple apparitions—in addition to the phase angle, other viewing and illumination angles also change (i.e., aspect angle and photometric obliquity), which affects the phase curves. This effect is more pronounced the more the object deviates from a sphere. Having determined through modeling the surface phase function, however, it is possible to compute the phase curve that the object would display if it were a sphere, freeing it from any dependence on the shape and thus being only the expression of the photometric properties of the surface regolith. This is achieved by integrating the photometric function over a sphere in a range of phase angles, as detailed in the Appendix. This representation, which we refer to as the sphere-integrated phase curve corresponds to the reference phase curves defined by Kaasalainen et al. (2001) in the particular case of a sphere. Figure 7 shows the sphere-integrated phase curve for the LSL photometric function for Leucus (red line) corresponding to the best-fit photometric parameters derived in Section 4.4 and listed in Table 2. For the purpose of comparison, fits are also shown for (1) the best-fit linear phase function (β = 0.0395 mag/°), (2) the IAU HG system (Bowell et al. 1989), and (3) the more recently adopted IAU ${\mathrm{HG}}_{1}{{\rm{G}}}_{2}$ system (Muinonen et al. 2010). The latter was computed with the online tool described in Penttilä et al. (2016).

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As already apparent during the 2016 apparition (Buie et al. 2018), and as observed for several other Trojans (see, e.g., Shevchenko et al. 2012), Leucus has a very subtle—if at all—opposition effect. Within the observed phase angle range, all of the phase curves except for the HG curve—which shows systematic variations both at the small and at the large end of the phase-angle range—provide a good fit to the data. The extrapolation at zero phase produces ${H}_{{\rm{V}}-\mathrm{lin}}$ and ${H}_{{\rm{V}}-{\mathrm{HG}}_{1}{{\rm{G}}}_{2}}$ values that differ from the LSL solution (${H}_{{\rm{V}}-\mathrm{LSL}}$) by about 0.05 mag and 0.03 mag, respectively. These small deviations are partly due to the fact that no calibrated data were available in the V band below the phase angle of about 16 to constrain the fit. Buie et al. (2018) did observe Leucus in the r' band at phase angles as low as 0125 during the 2016 apparition. However, there appear to be calibration inconsistencies between the 2016 observations and those acquired at the LCOGT in 2017 that are not fully understood. For this reason the 2016 observations were used as a relative data set and do not contribute to our phase curve. It has long been realized (see, e.g., Oszkiewicz et al. 2012; Shevchenko et al. 2016 and references therein) that a correlation exists between an asteroid's taxonomic class and the shape of its phase curve. The above-mentioned tool by Penttilä et al. (2016) also performs an unsupervised taxonomic classification based on the derived ${\mathrm{HG}}_{1}$,${{\rm{G}}}_{2}$ solution. Interestingly, the software classifies Leucus as a D type, solely based on its phase curve. This further supports the tentative classification by Levison & Lucy Science Team (2016) that was based on spectral information.

It is also important to recall that since Leucus has a considerably larger equatorial cross-section than the polar one, oppositions with a pole-on aspect would result in measured phase curves that are brighter than the sphere-integrated phase curve and conversely, apparitions with an equator-on aspect would appear fainter than the spherical model. Given the low obliquity of Leucus' spin axis, however, all apparitions tend to be at a near-equatorial aspect, as observed from Earth, and therefore Leucus never exposes its largest cross-section to the observer. This effect must be considered, e.g., when computing spherical-equivalent diameters from observations.

The geometric albedo is quite an elusive quantity to measure. First, it is defined for an observation geometry (0° solar phase angle) that is rarely observable from Earth and in which the photometric behavior of different planetary surfaces can wildly vary. Second, it is the result of an indirect measurement that requires the brightness at zero phase and a further measurement as a thermal flux or a geometric cross-section, which adds to the total error budget. Third, the albedo depends in principle on the shape of the object, although this issue is less of a problem for dark objects than for Trojans. The error on the brightness at zero phase directly translates into the same relative error for the geometric albedo (i.e., a 10% error in the brightness would cause a 10% error in the albedo). In our case the geometric albedo is derived through the simultaneous fit of the photometric lightcurves and the occultation data and by applying Equation A3, which results in an accurate geometric albedo determination $\,{p}_{{\rm{V}}}=0.043\pm 0.002$.

Grav et al. (2012) report for Leucus a much higher geometric albedo of ${p}_{{\rm{V}}}=0.079\pm 0.013$ and a spherical-equivalent diameter D = 34.155 ± 0.646 km derived from WISE observations. Those values are clearly incompatible with the occultation footprints. Part of the reason for their overestimation of the albedo is that they used an inaccurate H_V = 10.70, retrieved from the Minor Planet Center database. Very often, those photometric measurements are acquired in nonstandard photometric systems, are subject to inaccurate calibrations, and are therefore affected by large uncertainties. A further reason for the discrepancy is that the WISE observations happen to have occurred in the vicinity of the lightcurve minimum (see Figure 8), thereby underestimating the average thermal flux of Leucus. If we use our value ${H}_{{\rm{V}}-\mathrm{LSL}}=10.979$ to correct their determination by using the method proposed by Harris & Harris (1997) and account for the apparent visible cross-section at the time of the WISE observations, we obtain a corrected geometric albedo p_V = 0.048 ± 0.014 and a diameter D = 39 ± 1 km. With these corrections applied, the WISE determinations are compatible with our own within the respective uncertainties.

The IRAS albedo and size determinations by Tedesco et al. (2004; 0.063±0.014 and 42.2 ± 4.0 km, respectively) were based on an inaccurate H value of 10.5. By using our ${H}_{{\rm{V}}-\mathrm{LSL}}$ value we update their determinations to p_V = 0.041 ± 0.014 and D = 41.9 ± 4.0, which are also in agreement with our own determinations.

Buie et al. (2020) determined geometric albedo values for the four occultation events in 2018 and 2019 by estimating the object cross-sectional area from best-fit ellipses and by using the absolute photometry reported in this paper. They derived geometric albedo values ranging from 0.035 to 0.043 for the different occultation events with the scatter of the measurements probably reflecting the uncertainty in the different elliptical approximations of the occultation profiles.