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A&A
Volume 687, July 2024
Article Number A38
Number of page(s) 12
Section Planets and planetary systems
DOI https://doi.org/10.1051/0004-6361/202449789
Published online 26 June 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

Called vermin of the sky by astronomers for a long time owing to the trails they left on photographic plates (Seares 1930), the accidental observations of asteroids in large sky surveys have seen a growing interest. While the cadence and mode of operation of most surveys are seldom optimized for moving objects (Solano et al. 2014), the tremendous amount of data acquired in modern times can provide a wealth of information on the compositional and physical properties of Solar System objects (SSOs), unattainable by dedicated observations and yet crucial to deciphering the events that sculpted our Solar System.

Spectrophotometry is required to determine the taxonomy (DeMeo & Carry 2013; Popescu et al. 2018), and hence composition, of objects and to map their distribution, relics of the timing, and place of formation, together with past dynamical events (DeMeo & Carry 2014). Photometry time series are necessary to determine the rotation period and spin coordinates (Kaasalainen et al. 2001; Kaasalainen 2004), which are critical parameters dictating the dynamical evolution of asteroids through the Yarkovsky effect (Farinella et al. 1998; Vokrouhlický et al. 2015), spreading dynamical structures over time (Bottke et al. 2001; Vokrouhlický et al. 2006).

Decades of targeted observations (e.g., Zellner et al. 1985; Xu et al. 1995; Bus & Binzel 2002; Lazzaro et al. 2004; DeMeo et al. 2014; Lucas et al. 2017; Devogèle et al. 2018; De Prá et al. 2018; Binzel et al. 2019) have, however, brought about 7 000 visible and near-infrared spectra only (see the compilation in the Mahlke et al. (2022) taxonomy). The situation is even more dramatic for physical properties, focusing here on rotation period and spin-axis coordinates. Detailed studies using stellar occultations, disk-resolved imaging, or radar echoes have characterized only a few tens of SSOs (e.g., Veverka et al. 2000; Ostro et al. 2006; Carry et al. 2010; Sierks et al. 2011; Tanga et al. 2015; Pajuelo et al. 2018; Vernazza et al. 2021). The most-productive method has been light curve inversion, mainly using the inversion algorithm of Kaasalainen et al. (2001). However, decades of patient accumulation of light curves only brought solutions for a few hundred SSOs (Kaasalainen et al. 2002; Torppa et al. 2003; Slivan et al. 2003; Ďurech et al. 2007).

The game changer has been the data mining of serendipitous observations of SSOs in large sky surveys, through dedicated software. The Sloan Digital Sky Survey (SDSS), the ESO VISTA Hemispherical Survey (VHS), and the SKYMAPPER Southern Survey (SMSS) have brought hundreds of thousands of multi-filter photometric observations of asteroids (e.g., Ivezić et al. 2001; Carry et al. 2016; Popescu et al. 2016; Sergeyev et al. 2022), resulting in the determination of the taxonomic class of about 143 000 SSOs (11% of the population, see the compilation in the SsODNet service1, Berthier et al. 2023). The ESA Gaia mission released spectroscopy for over 60,000 asteroids (Galluccio et al. 2023; Oszkiewicz et al. 2023; Galinier et al. 2023). The Catalina Sky Survey, the Lowell Observatory database, NASA’s Kepler/K2 and TESS, the Palomar Transient Factory (PTF), and ESA Gaia have brought a wealth of photometry on most targets (Hanuš et al. 2013b; Waszczak et al. 2015; Chang et al. 2015; Berthier et al. 2016; Ďurech et al. 2016, 2019; Ďurech & Hanuš 2023, 2018; Pál et al. 2020; Spoto et al. 2018; Muinonen et al. 2020; Kalup et al. 2021), albeit mainly sparse in time (i.e., the frequency of observations is much lower than the intrinsic rotation frequency). Most important is the theoretical framework to benefit from this photometry sparse in time (Kaasalainen 2004). Nevertheless, rotation periods are only available for about 33 000 asteroids (2.5% of the population) and spin coordinates for 10 000 (0.7%) only (see Berthier et al. 2023).

The upcoming LSST of the Vera C. Rubin Observatory is expected to discover around 5 million SSOs (LSST Collaboration 2009). The determination of their taxonomy and surface composition based on the provided observations is not simple. Owing to their irregular shape and constantly changing geometry, the photometry of SSOs is a degenerate combination of intrinsic color and geometry. The determination of their colors hence generally relies on photometry acquired over short timescales, typically within minutes (Popescu et al. 2016). While this condition is fulfilled by the modes of operation of the SDSS, SKYMAPPER, and the ESA Euclid mission (Carry 2018; Sergeyev & Carry 2021; Sergeyev et al. 2022), the LSST will provide sparse photometry only (Jones et al. 2009). The determination of the colors of SSOs must therefore rely on the determination of their absolute magnitude in each filter (Mahlke et al. 2021; Alvarez-Candal et al. 2022). It has been repeatedly shown, however, that the absolute magnitude and the slope of the phase function may vary from apparition to apparition (e.g., Kwiatkowski & Kryszczynska 1992; Carvano & Davalos 2015; Mahlke et al. 2021; Jackson et al. 2022). This severely increases the required number of observations per SSOs in a given apparition to determine its phase curve as observations from different apparitions cannot be combined as of today.

We introduce here a new model to benefit from the sparse photometry obtained over multiple bands from large sky surveys. We aim to provide a more accurate description of the photometry by simultaneously determining the absolute magnitude and phase coefficients in each band as well as the base geometric properties (spin coordinates and oblateness) of the SSOs. We describe this new model for multi-filter sparse photometry in Sec. 2. We present in Sec. 3 the data we used to validate the approach by comparing the new model with previous results from the community in Sec. 4. We present results for the asteroid phase function and spin orientation in Sec. 5. Finally, we provide easy and programmatic access to these results in Sec. 6.

2 Generalized model for sparse photometry

The absolute magnitude, H, of a SSO is defined as the magnitude of the object located at a heliocentric distance, Λ, of 1 au, a range to the observer, r, of 1 au, and a phase angle, γ, of 0°: H=H(r=1,Δ=1,γ=0)=m−f(r,Δ)−g(γ),$\[\begin{aligned}H & =H(r=1, \Delta=1, \gamma=0) \\& =m-f(r, \Delta)-g(\gamma),\end{aligned}\]$(1)

where f(r,Δ)=5 log10⁡(rΔ)$\[f(r, \Delta)=5 \log _{10}(r \Delta)\]$(2) g(γ)=−2.5 log 10⁡[G1ϕ1(γ) +G2ϕ2(γ) +(1−G1−G2)ϕ3(γ)],$\[\begin{aligned}g(\gamma)= & -2.5 \log _{10}\left[\mathrm{G}_1 \phi_1(\gamma)\right. \\& ~+\mathrm{G}_2 \phi_2(\gamma) \\& ~\left.+\left(1-\mathrm{G}_1-\mathrm{G}_2\right) \phi_3(\gamma)\right],\end{aligned}\]$(3)

with the conditions 0<G1, $\[0<\mathrm{G}_1 \text {, }\]$(4a) 0<G2, $\[0<\mathrm{G}_2 \text {, }\]$(4b) 0<1−G1−G2,$\[0<1-G_1-G_2,\]$(4c)

following the HG1G2 model of Muinonen et al. (2010), which was accepted in 2012 by the International Astronomical Union (IAU) as superseding the historical HG model (Bowell et al. 1989). The HG1G2 model offers a better description of the surge of brightness at low phase angles (called the opposition effect), and hence provides a more accurate determination of the absolute magnitude (Muinonen et al. 2010; Mahlke et al. 2021). Furthermore, the G1G2 parameters have been shown to be linked with albedo and taxonomic type (Shevchenko et al. 2016; Mahlke et al. 2021).

Although the HG1G2 model presents improvements over the HG model both in prediction and interpretation, its usage has remained limited so far. It has been used in a few studies, based mainly on targeted observations (Shevchenko et al. 2019, 2021, 2022) but also on sky surveys (in particular ESA Gaia, Martikainen et al. 2021; Colazo et al. 2021), including an extensive initial study by Oszkiewicz et al. (2011). However, ephemerides computation centers (Minor Planet Center2, Jet Propulsion Laboratory Solar System Dynamics3, and Lowell Observatory astorb4) report absolute magnitudes with the HG model and not HG1G2. Pragmatically, while HG always converges (especially as G is almost always fixed to the canonical value of 0.15), HG1G2 has strong requirements on phase coverage to produce meaningful results (in particular requiring observations at low phase angles, typically below 2-4°, see Mahlke et al. 2021). A two-parameter version of HG1G2 was proposed for that purpose by Muinonen et al. (2010) and refined by Penttilä et al. (2016), HG⋆12$\[\begin{aligned}& \star \\& 12\end{aligned}\]$, although the absolute magnitudes derived with this latter model present systematic discrepancies with HG1G2 (Mahlke et al. 2021) due to the restricted parameter space.

However, the parameters of HG1G2 are wavelength-dependent. First, the absolute magnitude, H, is expected to be different for each observing band owing to the intrinsic color of the asteroid (close to Solar colors as a first approximation, see IMCCE 2021). This is already the case for the HG model, in which H is implicitly HV, the absolute magnitude reported in the Johnson V band5. The advantage of measuring the color based on absolute magnitudes is to avoid biases introduced by differences in observing time and brightness variation related to the shape of the SSO (Popescu et al. 2016; Carry 2018; Alvarez-Candal et al. 2022). Second, the G1G2 are also different (Mahlke et al. 2021), something which has been reported as “phase reddening” and observed spectrally in both laboratory samples and asteroids in the sky (Sanchez et al. 2012; Binzel et al. 2019; Alvarez-Candal 2024).

Furthermore, the HG1G2 model suffers from a major limitation: it does not account for the nonspherical geometry of asteroids (Jackson et al. 2022). Owing to the changing aspect angle (the angle between the spin axis and the viewing direction) over apparitions, the absolute magnitude can differ (see Fig. 10 in Mahlke et al. 2021, for instance). Conversely, a single HG or HG1G2 fit on a multi-opposition dataset will provide an average absolute magnitude but will likely result in biased magnitude predictions.

In the shape modeling formalism introduced by Kaasalainen et al. (2001), this aspect is solved by the simultaneous modeling of the 3D shape and the phase function (with a linear-exponential model). However, it typically requires more data (either dense light curves or many photometry measurements sparse in time) than for phase-function fitting only (see the discussion in Ďurech et al. 2015; Ďurech & Hanuš 2023).

We propose here an intermediate solution, dubbed sHG1G2 for spinnedHG1G2, solving the apparition-to-apparition change in magnitude without adding too many requirements to the dataset. It is in essence the “reference phase curve” defined by Kaasalainen et al. (2001). We introduced a new term, s(α,δ), to the definition of the absolute magnitude, H (Eq. (2)), accounting for the orientation of the asteroid and function of its oblateness, R, and the equatorial coordinates of its spin axis (α0, δ0): H=m−f(r,Δ)−g(γ)−s(α,δ),$\[H=m-f(r, \Delta)-g(\gamma)-s(\alpha, \delta),]$(5)

where s(α,δ)=2.5 log 10⁡[1−(1−R)|cos⁡Λ|], $\[s(\alpha, \delta)=2.5 \log _{10}[1-(1-R)|\cos \Lambda|] \text {, }\]$(6)

with Λ the aspect angle, computed from the equatorial coordinates (α,δ) of the asteroid at the time of observation as cos⁡Λ=sin⁡δsin⁡δ0+cos⁡δcos⁡δ0cos⁡(α−α0).$\[\cos \Lambda=\sin \delta \sin \delta_0+\cos \delta \cos \delta_0 \cos \left(\alpha-\alpha_0\right).\]$(7)

Thus, an additional criterion was added to the definition of H: that it should correspond to the magnitude of the object at 1 au from both the Sun and the observer, with a 0° phase angle, seen from its equatorial plane As a corollary, the object is brighter by 2.5 log10 R magnitudes when observed from its pole. Noting that a > b > c the tri-axial diameters of the asteroid, R is the polar-to-equatorial oblateness (0 < R ≤ 1): R=c(a+b)2ab$\[R=\frac{c(a+b)}{2 a b}\]$(8)

Applying this new definition of H to observations taken over several apparitions with different Nf filters, one can simultaneously determine the absolute magnitude, H, and phase parameters G1G2 for each band and the oblateness and spin coordinates for a total of 3 × (Nf + 1) parameters. We present in Fig. 1 an illustration of the sHG1G2 model for the asteroid (22) Kalliope (we use the spin solution from Ferrais et al. 2022). The distance term, f(r, Δ), presents the expected minima and maxima at oppositions and conjunctions. The phase term, g(γ), has a periodicity that is twice as fast, with a decreasing phase angle at both oppositions and conjunctions. Both are symmetric around epochs of a minimal phase angle (as is visible on the blue and orange curves). Their sum defines the HG1G2 model (Muinonen et al. 2010). The new term s(α, δ) accounts for the slow change in geometry over time, allowing asymmetry. We present in Fig. 2 the phase and spin components (g(γ) and s(α, δ)), similarly to Fig. 5 of Jackson et al. (2022). The phase curve is bounded between two extreme cases, the pole at (Λ = {0°, 180°}) and the equator at (Λ = 90°), and practically presents a slow evolution between these boundaries.

thumbnail Fig. 1

sHG1G2 model of (22) Kalliope, with its three components – f(r, Δ), g(γ), and s(α, δ) – shown explicitly, together with their amplitude. The HG1G2 model is plotted for comparison.

3 Data

We implemented the sHG1G2 model in FINK6, (Möller et al. 2021) a broker of alerts for the LSST (LSST Collaboration 2009). Today, before the start of the LSST, FINK processes daily the public stream of alerts from the Zwicky Transient Survey (ZTF; Masci et al. 2019; Graham et al. 2019; Bellm et al. 2019; Patterson et al. 2019).

ZTF broadcasts a public stream for variable and transient events in two bands (g7 and r8, similar but not identical to the Sloan g and r filters, Bellm et al. 2019) of typically 215 000 alerts per night, of which about 70% is retained by FINK for scientific analyses, and 14% corresponds to SSOs. Between late 2019 and late 2023, FINK extracted 19 319 067 observations of SSOs from the ZTF public alert stream. FINK is one of the seven brokers of alerts being developed for the LSST. As such, the system has been designed to cope with a very large flow of data, covering all aspects of astrophysics, from the Solar System to variable stars, supernovae, and optical counterparts to gravitational wave events (e.g., Möller et al. 2021; Aivazyan et al. 2022; Leoni et al. 2022; Le Montagner et al. 2023). With the sHG1G2 model implemented within FINK, its results are already freely available for each SSO through the FINK Web interface9 and the corresponding Application Public Interface (API) SSO. We also propose a dedicated method to retrieve the parameters of the sHG1 G2 model for a large corpus of data at once: the Solar System Objects FINK Table10 (SSOFT, see Sec. 6).

We used FINK to retrieve 19 319 067 observations of 565 045 unique SSOs in g and r between November 2019 and December 2023. We decided to only retain objects with at least 50 observations across all filter bands (122 675 unique SSOs). This threshold for the number of observations was chosen arbitrarily after several tests, as a compromise between the sample size and the fraction of objects failing to converge to a solution (see Sec. 4.1). The average number of observations (together with the 25% and 75% quantiles) is 92−26+46$\[92_{-26}^{+46}\]$, for a length of observations (number of days between the first and the last observation) of 1,164−108+270$\[1,164_{-108}^{+270}\]$. We fit these observations with the HG (Bowell et al. 1989), HG1G2 (Muinonen et al. 2010), and sHG1G2 models, and compare the results in the following sections.

thumbnail Fig. 2

Effect of the changing geometry (described by s(α, δ)) on the phase curve (traditionally g(γ) only). Over 400 days from JD 2 460 000 (same as Fig. 1), (22) Kalliope is seen from an aspect angle increasing from 4° (pole on) to 78° (close to the equator) owing to motion as seen from the Earth. The inset presents the successive ecliptic (x,y) positions of Kalliope, color-coded by epoch, in a reference frame centered on Earth (the pale blue dot). Three reference epochs are drawn on both graphs (a circle, a square, and a diamond). The orientation of the spin axis of Kalliope is drawn as arrows at these three epochs, and the line of sight is drawn for the second (square) epoch, illustrating the aspect angle, Λ.

4 Validation of the sHG1G2 model

The improvement provided by sHG1 G2 is qualitatively visible in Fig. 3 for the asteroid (223) Rosa, a flyby candidate by the ESA JUICE mission (Grasset et al. 2013; Agostini et al. 2022). While the description of the opposition effect is different between HG and HG1G2, they both predict a globally symmetric behavior around opposition. sHG1G2 correctly describes the photometry as it accounts for the evolving geometry with time. We used the 122 675 SSOs to assert statistically the validity and limits of the sHG1G2 model.

4.1 Success and failure

We imposed a set of boundaries for the parameters in the fitting process to guarantee meaningful results. Above all, we enforced that the conditions in Eqs. (4a) and (4b) be fulfilled, and that the oblateness, R, be between 0.3 and 1 (encompassing all published shape models; see Sec. 4.5).

Out of the 122 675 initial SSO light curves, the fitting procedure for the sHG1G2 model converges in about 98% of cases. However, at this stage, there are suspicious solutions, where the minimization algorithm is clearly hitting the boundary conditions. Hence, in the following, we consider a solution to be fully valid if: the fitting procedure converges, the G1 and G2 values are non-singular (singular cases encompass values strictly equal to 0), the spin coordinates are non-singular (singular cases encompass α0 strictly equal to 0° or 360°, (α0,β0) equals (180°,0°)), and the condition in Eq. (4c) is fulfilled (it could not be incorporated in the fitting procedure).

Using the two filter bands of ZTF (g and r), this results in a sample of 63 092 SSOs (51% success). This means about half of the fitted parameters are of good quality in all bands simultaneously (a similar success rate as Ďurech & Hanuš 2023, on Gaia photometry). As the phase parameters are fitted per band, we also defined the sample of SSOs for which the constraints on spin parameters defined above are fulfilled, and G1 and G2 fulfil the conditions in Eq. (4) for at least one filter band. This results in a sample of 95 593 SSOs (78% success). This sample will be used for the rest of the analysis unless explicitly stated.

For the HG model, the rate of success is higher, as was expected due to its simplistic form (98% success), but the HG1G2 model has a somewhat lower success rate than the sHG1G2 model (38% success in the two filter bands simultaneously, and 70% success in at least one filter band). This gives us confidence that the spin component in the light curve model contributes to ameliorating the parameter estimation globally compared to the HG1G2 model.

Nonetheless, we have tried to understand the reasons for the failure of the HG1G2 and sHG1G2 models. The failures are not clearly correlated with the minimum phase angle, the number of observations per filter, the number of oppositions, and the range for the aspect angle values. Based on Fig. 1, we see that the contributions of the g(γ) and s(α, δ) components are relatively small compared to those of the other components, and often smaller than the typical error estimate on the photometric measurements reported in the ZTF alert packets. Hence, we might be in a noisy regime where the contributions from the phase and spin components are not always accessible. To test this hypothesis, we took the valid solutions for which the fitted procedure converges, and we randomized the magnitudes for each observation with a Gaussian distribution within (a) the reported observational errors, and (b) ten times the reported observational errors. We again fit the models’ parameters, and we restarted the procedure 500 times. In the (a) scenario, the results remain stable, within the reported standard deviations. On the contrary, the (b) scenario systematically leads to outlier values for the HG1G2 and sHG1G2 models, similar to what is observed in the initial dataset. Although this is perhaps not the only reason, we are confident that the rate of success should improve with higher signal-to-noise measurements, such as the data that will be soon collected by the Vera C. Rubin Observatory.

thumbnail Fig. 3

Comparison of the HG, HG1G2, and sHG1G2 models on the photometry of (223) Rosa from ZTF.

4.2 Fit to data

We compared the root-mean-square (RMS) residuals between the predicted photometry of the HG, HG1 G2, and sHG1G2 models and the ZTF photometry in g and r in Fig. 4. The improvement of HG1G2 over HG (Fig. 4, left) is marginal, with all SSOs close to the diagonal. The improvement of sHG1G2 over HG1G2 (Fig. 4, right) is clear, and over HG (Fig. 4, middle) it is striking, with a significant tail below the diagonal.

The bulk of the population remains close to the diagonal. This implies that the predictions by HG or HG1G2 fit equally well the data as sHG1G2, although the latter has more degrees of freedom. There are three cases in which sHG1G2 may not be required. First, if the SSO is (nearly) spherical, its orientation will not significantly change its brightness. The s(α, δ) term of sHG1G2 is therefore not required and converges toward zero. Second, if obliquity of the SSO is close to either 0° or 180° (as expected from YORP evolution, and as it is often the case, see Sec. 4.5 and Vokrouhlický et al. 2015; Ďurech et al. 2015; Ďurech & Hanuš 2023), as the orbital inclination is small the range of the aspect angle, Λ, covered remains limited around 90°. The brightness hence barely changes over apparitions, and the shape or spin term (s(α, δ)) contribution is minimal. Third, depending on the time span (mainly the number of apparitions) and number of observations, the range of the aspect angle, Λ, may again be limited. These three limiting cases are visible in Fig. 5.

Two conclusions can be drawn from this comparison of RMS. First, the sHG1G2 model indeed provides a clear improvement over the previous models. Second, this improvement is not always needed, for the reasons listed above. Pragmatically, this implies that the HG, HG1G2, and sHG1G2 models should be computed for each SSO (in FINK but more generally at any ephemerides computation center): from the most robust to the most informative. In the pathological cases listed above, HG1G2 should then be preferred over sHG1G2. Similarly, if low phase angles are not covered, the two-parameter G1G2 function, g(γ) (common to HG1G2 and sHG1G2), is not constrained (see Mahlke et al. 2021) and HG should be preferred.

thumbnail Fig. 4

Comparison of the residuals for the HG, HG1G2, and sHG1G2 models in both g and r filters.
thumbnail Fig. 5

Comparison of the residuals for the HG1G2 and sHG1G2 models as a function of the range of the aspect angle (ΔΛ), the obliquity, and the oblateness, R. Larger values of ΔRMS indicate cases where sHG1G2 improves over HG1G2.
thumbnail Fig. 6

Comparison of the absolute magnitude in sHG1G2 with HG and HG1G2, in the g and r filters.

4.3 Absolute magnitude

We compare in Fig. 6 the sHG1G2 absolute magnitude in g and r with those obtained with HG and HG1G2. The three models are in good agreement, as is shown by the mode of 0 in their difference in absolute magnitude. The dispersion between sHG1G2 and HG1G2 is much smaller than with HG, owing to the shared definition of the phase function, g(γ), between the two models that differs from HG. The two-parameter (G1G2) function, g(γ), was introduced by Muinonen et al. (2010) to better describe the opposition effect at small phase angles. It is therefore not unexpected to have differences in absolute magnitude between the two systems. There is a clear asymmetry in the absolute magnitude, as was expected from the definition of sHG1G2 (Eq. (2)): the absolute magnitude is defined in the equatorial plane of the SSOs and is hence larger on average (the projection area on the plane of the sky is maximal for objects seen pole-on).

As an additional validation, we compare in Fig. 7 the ZTF g-r obtained from the Hg and Hr absolute magnitudes with the sHG1G2 model and the g-r color from the SDSS and SKYMAPPER surveys (Sergeyev & Carry 2021; Sergeyev et al. 2022). The g and r of the three facilities differ, and the colors must therefore be corrected before any comparison. We converted all three systems to the PanSTARRS g–r color, using the following transformations: SKYMAPPER to SDSS (Sergeyev et al. 2022), SDSS to PANSTARRS (Finkbeiner et al. 2016), and ZTF to PANSTARRS (Medford et al. 2020). The distributions show a general trend toward the 1:1 relation, as was expected. The spread of values is larger for ZTF than for SDSS and SKYMAPPER, due to the fact that the ZTF magnitudes have an added uncertainty due to the computation via the phase curve, while SDSS and SKYMAPPER can compute the colors directly from the near-simultaneously acquired photometry. Two separate regions of higher density are visible, corresponding to the C and S taxonomy complexes (centered on g-r values of 0.4 and 0.55, respectively). This validates the approach of color determination from absolute magnitudes with the sHG1G2 model.

thumbnail Fig. 7

g–r color from sHG1G2 absolute magnitudes (Hg and Hr) compared with apparent SDSS and SKYMAPPER g–r colors (all converted to the PANSTARRS photometric system; see text).

4.4 Phase parameters G1G2

We compare in Fig. 8 the values of G1G2 obtained with the sHG1G2 model with those from the HG1G2 model. Both G1 and G2 show a good agreement, albeit some spread is present. The addition of the geometry term, s(α, δ), in the definition of the absolute magnitude (Eq. (2)) thus does not bias the slope parameters, G1G2. We further tested the values of these parameters in Fig. 9. For both the g and r filters, the bulk of the SSOs follow the HG⋆12$\[\begin{aligned}& \star \\& 12\end{aligned}\]$ line in the {G1, G2} space (Muinonen et al. 2010; Penttilä et al. 2016; Shevchenko et al. 2016), again confirming the validity of the sHG1G2 model.

thumbnail Fig. 8

Comparison of the G1G2 parameters in the g and r filters for the HG1G2 (x-axis) and sHG1G2 (y-axis) models.
thumbnail Fig. 9

Distribution of G1G2 parameters in g and r for the sHG1G2 model.

4.5 Spin coordinates

We compared the spin-axis coordinates (α0,δ0) derived with sHG1G2 and those available in the literature. We collected available spin solutions for the 95 593 SSOs in the FINK sample using the SsODNet service (Berthier et al. 2023). We find 6499 SSOs with previous estimates of the spin-axis coordinates, mainly from the light curve inversion technique (Kaasalainen et al. 2001) or in combination with other techniques (stellar occultations, direct imaging, and thermophysical modeling, Carry et al. 2010; Kaasalainen 2011; Ďurech et al. 2011; Viikinkoski et al. 2015; Hanuš et al. 2018a) by many authors (e.g., Hanuš et al. 2013a; Marciniak et al. 2021; Vernazza et al. 2021; Hung et al. 2022), compiled on the DAMIT11 service (Ďurech et al. 2010).

We present in Fig. 10 the distribution of angular separation between the spin-axis coordinates from the literature and sHG1G2. The distribution peaks around 20°, with half of the SSOs agreeing below 35°. This is an overall good agreement, especially considering the limited number of observations available to the sHG1G2 model (92−26+46$\[92_{-26}^{+46}\]$; see Sec. 3). We also present this distribution normalized by the spin uncertainty (computed as the quadratic sum of the uncertainties from the sHG1G2 and literature spin coordinates). About 60% of the solutions agree within one σ, and 95% at three σ, an indication that the uncertainties seem to have been properly estimated (and not over- or underestimated).

thumbnail Fig. 10

Angular distance between spin-axis coordinates from sHG1G2 and the literature for 6499 asteroids, in degrees (left) and normalized by uncertainty (right). The cumulative distributions up to 100% are also presented (in gray).
thumbnail Fig. 11

Distribution of the oblateness, R, compared with that of 16 317 shape models of 10 743 asteroids from DAMIT (Ďurech et al. 2010).

4.6 Polar oblateness

We present the distribution of the polar oblateness, R, in Fig. 11, compared with that of existing 3D shape models from the literature. For that, we downloaded the 16 317 models of 10 743 unique asteroids available on DAMIT (Ďurech et al. 2010). These models are the product of inversion techniques (mainly Kaasalainen & Torppa 2001; Kaasalainen et al. 2001; Carry et al. 2010; Viikinkoski et al. 2015) from many authors (e.g., Ďurech et al. 2016, 2018; Hanuš et al. 2018b, 2021; Viikinkoski et al. 2017). We computed the oblateness, R, of all these models from Eq. (8), computing first the tri-axial diameters (a ≥ b ≥ c) from the moments of inertia (using the formulae by Dobrovolskis 1996).

The distribution of oblateness determined here presents two main differences with the one from the 3D shape models: it peaks at rounder objects (R of 0.9 rather than 0.7) and has a long tail toward flying saucers (low R). The tail is an artifact, as most SSOs with an oblateness below 0.5 have observations covering a very limited range of aspect angles and/or an obliquity close to either 0° or 180°, limiting the viewing geometry, as is shown in Fig. 12.

The shift in the center of the distribution is more complex to interpret and could be real or artificial. While the range of absolute magnitude between the FINK sample and the shape models from DAMIT overlap, the median absolute magnitude is 14.8 against 13.6. The SSOs reported here thus have a smaller diameter (with half typically smaller than 4 km). The distribution of oblateness between the two samples could genuinely be different. On the other hand, the difference aligns with intrinsic biases of both methods at play. As is shown by Marciniak et al. (2018), there is a clear bias against slow rotations and low-amplitude light curves among asteroids with a determined spin axis and 3D shape, even more marked as objects get fainter. The light curve inversion indeed favors targets with a strong intrinsic variability, which creates a bias against round asteroids (as a > b > c, large a/b ratios imply smaller R). This is confirmed, as asteroids with a 3D shape model generally have larger residuals with sHG1G2 in ZTF data. Therefore, the distribution of oblateness for well-constrained solutions (e.g., large -enough ΔΛ, valid G1G2, low residuals) is likely representative of the real distribution of oblateness among asteroids. We plan to combine several datasets from different facilities, such as ZTF, ATLAS, PANSTARRS, Gaia, and LSST, to increase the time coverage, and hence the range of the observed aspect angle, Λ, to further constrain the obliquity and oblateness distributions.

thumbnail Fig. 12

Oblateness, R, as a function of the range of covered aspect angles (ΔΛ) and obliquity (given the symmetry of s(α,δ), we only plot obliquity in 0-90°).

5 Results

One of the main motivation to develop sHG1G2 was the determination of reliable colors from sparse photometry of SSOs, such as that provided by the LSST. We show in Sec. 4.3 the validity of the approach. The dataset we use here (ZTF) only contains two filters (g and r) but it is enough as a proof of concept. We further illustrate the approach in Fig. 13, comparing ZTF Hg–Hr with SDSS i–z, mimicking the situation for LSST observations with absolute magnitudes determined with sHG1G2 in all six LSST filters: u, g, r, i, z, and y (LSST Collaboration 2009). The different taxonomic groups are easily identified, and follow their expected location in such a “slope” (Hg–Hr) versus “one micron band” (i–z) plane (Ivezić et al. 2001; Nesvorný et al. 2005; DeMeo et al. 2009; DeMeo & Carry 2013; Sergeyev et al. 2023).

This is extremely promising for LSST: the G1G2 of the phase function are better constrained with larger phase coverage (Mahlke et al. 2021), and as a result the absolute magnitude, H, is too. Furthermore, the longer the time spanned by the observation, the more geometries observed and the better the constraints on the spin axis and polar oblateness. With a factor of four to five times as many observations (400 vs 92−26+46$\[92_{-26}^{+46}\]$) over a time interval three times longer (10 yr vs three), the colors obtained with the LSST will be much more precise (not to mention having a better intrinsic photometric accuracy).

We then present in Fig. 14 the distribution of the G1G2 parameters for the g and r filters separately, for asteroids in the taxonomic complexes A, B, C, Ch, D, E, K, L, M, P, Q, S, V, and X (i.e., spectrally similar to E/M/P without albedo information), following the work by Mahlke et al. (2021). There is a clear trend in G1G2 from low-albedo asteroids (C/P/D) occupying the high-G1 -low-G2 region to the high-albedo asteroids (V/A/E) located toward low-G1 -high-G2, as was expected (Shevchenko et al. 2016). The dispersion of each taxonomic group in Fig. 14 is more limited here than in Mahlke et al. (2021): this may be due to an intrinsically higher precision in ZTF photometry or the sHG1G2 model being more adapted, or both. In any case, the strong correlation of G1G2 with taxonomy opens up the possibility of using these parameters in the determination of asteroid taxonomic classes, especially with LSST photometry.

We compare the obliquity (computed from the spin coordinates determined with sHG1G2) with the diameter (retrieved from SsODNet, Berthier et al. 2023) in Fig. 15. Asteroids larger than 50 km appear roughly isotropic (although an excess of direct rotators has been observed, Johansen & Lacerda 2010; Visser et al. 2020). Asteroids smaller than 10-20 km cluster toward extreme values of 0° and 180°, although there are many asteroids with an intermediate obliquity. This is a clear demonstration of the Yarkovsky–O’Keefe–Radzievskii–Paddack effect (Radzievskii 1952; Paddack 1969; O’Keefe 1976), or YORP for short (Vokrouhlický et al. 2015). This obliquity-diameter distribution has been known for years (e.g., Hanuš et al. 2013a; Ďurech et al. 2015) The sample of 95 593 from ZTF/sHG1G2, however, increases the sample of known obliquity by a factor of about ten.

The next logical step would be to study the distribution of obliquities among asteroids belonging to families, as was recently done by Ďurech & Hanuš (2023). However, the sHG1G2 model is by construction symmetric across the equator of the target asteroid, and the determination of the spin-axis is thus ambiguous: the rotation can be either direct or retrograde. We hence report two spin solutions for each object in FINK: (α0,δ0) and (α0+180,-δ0).

We also study the distribution of ecliptic longitude of the pole. There is a strong correlation with the longitude of the ascending node, as was expected from a majority of SSOs with an obliquity close to either 0° or 180°: the longitude should be close to the longitude of the node minus 90°. We do not find a dependence between the distribution of longitude of the pole and the orbital inclination (Fig. 16), as opposed to Cibulková et al. (2016), who reported a more isotropic distribution of longitude for more inclined orbits. We compare the cumulative distributions of longitude for each range of inclination in Fig. 16. They are barely distinguishable from one another, which is confirmed by a Kolmogorov–Smirnov test: all the distributions are statistically similar.

thumbnail Fig. 13

Distribution of 3569 SSOs in SDSS i-z against ZTF Hg–Hr. The symbols and colors correspond to the taxonomic class retrieved from SsODNet (Berthier et al. 2023), and the letter mark the average color for each class.
thumbnail Fig. 14

Distribution of G1G2 parameters in g and r per taxonomic complex.
thumbnail Fig. 15

Obliquity as a function of the diameter.

6 Data availability

As was mentioned in Sec. 3, the data and results presented in the present study were all acquired and processed within FINK. The amount of SSOs for which a solution is determined, and the values of the parameters of the sHG1G2 model, are therefore in constant evolution. FINK processes the incoming stream of alerts from ZTF on a daily basis, and the SSO parameters are determined once a month. The data and derived parameters reported here correspond to the SSOFT of December 2023.

As an official community broker of alerts for LSST, FINK will receive in real time the flow of alerts from the Vera C. Rubin Observatory. As such, the absolute magnitude, H, and G1G2 in each of LSST filters, together with spin coordinates (α0, δ0) and polar oblateness (R), will be regularly determined and openly accessible to the scientific community for every SSO observed in the LSST.

The data and parameters can be retrieved from FINK Science Portal and API, in particular the summary table SSOFT containing all of the parameters. In the Python programming language, the data can be retrieved as follows12: 

import requests
r = requests.post(
  "https://fink-portal.org/api/v1/ssoft",
  json={
      "flavor":   "SHG1G2",
      "version":   "2023.12",
      "output-format":   "json"
  }
)

7 Conclusion

We propose a simple modification of the currently accepted HG1G2 model to properly render the photometric behavior of SSOs taken over long periods of time. From observations in Nf filters, the new sHG1G2 model simultaneously determines the absolute magnitude, H, and phase function coefficient, G1G2, in each filter, the spin coordinates (α0,δ0), and the polar oblateness, R, for a total of 3 × (Nf + 1) parameters. The determination of the absolute magnitude across multiple filters is required to determine the colors of SSOs in large sky surveys without back-to-back observations in different filters, such as the upcoming LSST of the Vera C. Rubin Observatory.

We tested the new sHG1G2 model on observations in g and r, collected over 3 yr by the ZTF. The sHG1G2 model provides a better description of the photometry, as is revealed by smaller residuals, than previous proposed models (HG and HG1G2). The parameters determined by sHG1G2 are nevertheless consistent with previous estimates. The absolute magnitude and phase parameters are in agreement with those determined with the HG1G2 model on the same dataset, and the spin coordinates agree with those determined with other methods on different datasets within typically 20-30°. The polar oblateness presents a spurious trend toward small values, and a systematic shift in its peak value compared to SSOs with the shape model, which could be genuine but is also aligned with the expected biases of the different methods at play.

The limitations of the sHG1G2 model are linked with the geometry of observations. Observations at a low phase angle (between 0° and 5°) are required to properly describe the opposition effect (affecting the G1G2 parameters), a limitation inherited from the HG1G2 model. The spin orientation and oblateness cannot be properly retrieved from observations spanning only a limited range of geometries, in particular the aspect angle. Finally, by construction of the model, there is a complete degeneracy between direct and retrograde rotation. The new term, s(α, δ), accounting for the oblateness and orientation can, however, also be introduced in the HG and sHG models. Practically, all four models – HG, sHG, HG1G2, and sHG1G2 – should be computed for each SSO. Then, following the principle of Occam’s razor, the simplest model, among those fitting the data satisfactorily, should be chosen and used.

The sHG1G2 model is fully implemented in FINK, a broker of alerts for the LSST. The results of the model are freely available on the FINK portal for all of the SSOs observed by ZTF – and by the LSST in the near future – for which the fitting procedure has been successful.

thumbnail Fig. 16

Distribution of the ecliptic longitude of the spin axis as a function of the orbital inclination.

Acknowledgements

This project has received financial support from the CNRS through the MITI interdisciplinary program. B.C. was supported by CNRS/INSU/PNP. We thank these programs for their support. This research used the Miriade (Berthier et al. 2009), SsODNet (Berthier et al. 2023), and TOPCAT (Taylor 2005) Virtual Observatory tools. It used the astropy (Astropy Collaboration 2013, 2018, 2022), sbpy (Mommert et al. 2019), and rocks (Berthier et al. 2023) python packages. This work was developed within the FINK community and made use of the FINK community broker resources. FINK is supported by LSST-France and CNRS/IN2P3. Thanks to all the developers and maintainers.

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12

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All Figures

thumbnail Fig. 1

sHG1G2 model of (22) Kalliope, with its three components – f(r, Δ), g(γ), and s(α, δ) – shown explicitly, together with their amplitude. The HG1G2 model is plotted for comparison.
In the text
thumbnail Fig. 2

Effect of the changing geometry (described by s(α, δ)) on the phase curve (traditionally g(γ) only). Over 400 days from JD 2 460 000 (same as Fig. 1), (22) Kalliope is seen from an aspect angle increasing from 4° (pole on) to 78° (close to the equator) owing to motion as seen from the Earth. The inset presents the successive ecliptic (x,y) positions of Kalliope, color-coded by epoch, in a reference frame centered on Earth (the pale blue dot). Three reference epochs are drawn on both graphs (a circle, a square, and a diamond). The orientation of the spin axis of Kalliope is drawn as arrows at these three epochs, and the line of sight is drawn for the second (square) epoch, illustrating the aspect angle, Λ.
In the text
thumbnail Fig. 3

Comparison of the HG, HG1G2, and sHG1G2 models on the photometry of (223) Rosa from ZTF.
In the text
thumbnail Fig. 4

Comparison of the residuals for the HG, HG1G2, and sHG1G2 models in both g and r filters.
In the text
thumbnail Fig. 5

Comparison of the residuals for the HG1G2 and sHG1G2 models as a function of the range of the aspect angle (ΔΛ), the obliquity, and the oblateness, R. Larger values of ΔRMS indicate cases where sHG1G2 improves over HG1G2.
In the text
thumbnail Fig. 6

Comparison of the absolute magnitude in sHG1G2 with HG and HG1G2, in the g and r filters.
In the text
thumbnail Fig. 7

g–r color from sHG1G2 absolute magnitudes (Hg and Hr) compared with apparent SDSS and SKYMAPPER g–r colors (all converted to the PANSTARRS photometric system; see text).
In the text
thumbnail Fig. 8

Comparison of the G1G2 parameters in the g and r filters for the HG1G2 (x-axis) and sHG1G2 (y-axis) models.
In the text
thumbnail Fig. 9

Distribution of G1G2 parameters in g and r for the sHG1G2 model.
In the text
thumbnail Fig. 10

Angular distance between spin-axis coordinates from sHG1G2 and the literature for 6499 asteroids, in degrees (left) and normalized by uncertainty (right). The cumulative distributions up to 100% are also presented (in gray).
In the text
thumbnail Fig. 11

Distribution of the oblateness, R, compared with that of 16 317 shape models of 10 743 asteroids from DAMIT (Ďurech et al. 2010).
In the text
thumbnail Fig. 12

Oblateness, R, as a function of the range of covered aspect angles (ΔΛ) and obliquity (given the symmetry of s(α,δ), we only plot obliquity in 0-90°).
In the text
thumbnail Fig. 13

Distribution of 3569 SSOs in SDSS i-z against ZTF Hg–Hr. The symbols and colors correspond to the taxonomic class retrieved from SsODNet (Berthier et al. 2023), and the letter mark the average color for each class.
In the text
thumbnail Fig. 14

Distribution of G1G2 parameters in g and r per taxonomic complex.
In the text
thumbnail Fig. 15

Obliquity as a function of the diameter.
In the text
thumbnail Fig. 16

Distribution of the ecliptic longitude of the spin axis as a function of the orbital inclination.
In the text

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