Asteroid phase curves from ATLAS dual-band photometry
AA steroid phase curves from ATLAS dual - band photometry P reprint Max Mahlke , Benoit Carry , and Larry Denneau Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, France CAB (INTA-CSIC), Campus ESAC (ESA), Villanueva de la Ca˜nada, Madrid, Spain University of Hawaii, 2680 Woodlawn Dr., Honolulu HI 96822 USA A bstract Asteroid phase curves are used to derive fundamental physical properties through the determination of the abso-lute magnitude H . The upcoming visible Legacy Survey of Space and Time (LSST) and mid-infrared
Near-EarthObject Surveillance Mission (NEOSM) surveys rely on these absolute magnitudes to derive the colours and albe-dos of millions of asteroids. Furthermore, the shape of the phase curves reflects their surface compositions,allowing for conclusions on their taxonomy. We derive asteroid phase curves from dual-band photometry ac-quired by the
Asteroid Terrestrial-impact Last Alert System telescopes. Using Bayesian parameter inference,we retrieve the absolute magnitudes and slope parameters of 127,012 phase curves of 94,777 asteroids in thephotometric H , G , G - and H , G ∗ -systems. The taxonomic complexes of asteroids separate in the observed G , G -distributions, correlating with their mean visual albedo. This allows for di ff erentiating the X-complexinto the P-, M-, and E-complexes using the slope parameters as alternative to albedo measurements. Further,taxonomic misclassifications from spectrophotometric datasets as well as interlopers in dynamical families ofasteroids reveal themselves in G , G -space. The H , G ∗ -model applied to the serendipitous observations is un-able to resolve target taxonomy. The G , G phase coe ffi cients show wavelength-dependency for the majority oftaxonomic complexes. Their values allow for estimating the degree of phase reddening of the spectral slope. Theuncertainty of the phase coe ffi cients and the derived absolute magnitude is dominated by the observational cov-erage of the opposition e ff ect rather than the magnitude dispersion induced by the asteroids’ irregular shapes andorientations. Serendipitous asteroid observations allow for reliable phase curve determination for a large numberof asteroids. To ensure that the acquired absolute magnitudes are suited for colour computations, it is imperativethat future surveys densely cover the opposition e ff ects of the phase curves, minimizing the uncertainty on H .The phase curve slope parameters o ff er an accessible dimension for taxonomic classification, correlating with thealbedo and complimentary to the spectral dimension. ntroduction The absolute magnitude H of asteroids is defined as their ap-parent Johnson V -band magnitude observed at zero degree solarphase angle and reduced to 1 AU distance from both the Sunand the Earth, averaged over a full period of their rotation. Thephase angle α is the angle between the Sun, the asteroid, and theobserver. The reduced magnitude V ( α ) is calculated from theobserved apparent magnitude m as V ( α ) = m + r ∆ ) , (1)where r is the distance between the asteroid and the Sun at theepoch of observation and ∆ the respective distance between theasteroid and Earth. V ( α ) is referred to as the phase curve, and,by definition, H = V (0).The inference of principal physical parameters of minor bodiesrequires accurate knowledge of their absolute magnitudes. Theirdiameter D and visual geometric albedo p V are related to H by(Harris & Lagerros, 2002)log D = . − . H − . p V . (2)Any uncertainty in H enters logarithmically in the derivation ofthe physical properties. The diameters and visual albedos ofmore than 100,000 Main Belt asteroids observed with NASA’s Wide-field Infrared Survey Explorer (WISE) carry 20% and 40%accuracy, under the assumption that the referenced absolute mag-nitudes are accurate (Masiero et al., 2011; Nugent et al., 2015).NASA’s planned
Near- Earth Object Surveillance Mission (NEOSM, previously NEOCam, Grav et al., 2019) aims to ex-tend this catalogue by an order of magnitude, thereby vastly in-creasing the demand for accurate determinations of H .Deriving H in di ff erent wavelength bands further o ff ers the con-solidation of asteroid photometry obtained at di ff erent epochsfor colour computation and subsequent taxonomic classification.This is vital for the upcoming Legacy Survey of Space and Time (LSST) executed at the Vera C. Rubin Observatory. The LSSTaims to provide catalogues of photometric variability and coloursfor millions of minor bodies (Jones et al., 2009). The latter ne-cessitates either quasi-simultaneous multi-band observations ofa single target or reduction of the observed magnitudes to zerophase angle (Szab´o et al., 2004; Warner et al., 2009). Given thenumerous competing science cases of the LSST and its plannedoperations with two filters per night at most, the Solar Systemscience community cannot rely on the realization of the requiredobservation cadence alone. Instead, H must be derived in eachband by fitting the observed phase curves to obtain the colours.The definition of H requires asteroid magnitudes at zero degreephase angle. This is practically di ffi cult to achieve, hence H is instead extrapolated from photometric observations acquiredclose to opposition, but at non-zero phase angles, by means ofphase curve modelling. We summarize here the most basic mod-elling advances and refer to Muinonen et al. (2002) and Li et al.(2015) for detailed reviews.In first order, an asteroid’s apparent brightness increases linearlywith decreasing phase angle. The slope of the phase curve is a r X i v : . [ a s t r o - ph . E P ] S e p reprint – A steroid phase curves from ATLAS dual - band photometry opposition ef-fect (Gehrels, 1956). In 1985, the International AstronomicalUnion (IAU) adopted the H , G -magnitude system, where G de-scribes the overall slope of the phase curve (Bowell et al., 1989).The H , G -system successfully describes phase curves in largeranges of the phase space, however, it fails to reproduce the op-position e ff ect, especially for exceptionally dark or bright ob-jects (Belskaya & Shevchenko, 2000). In 2010, the H , G systemwas extended by Muinonen et al. (2010) to the three-parameter H , G , G -system, V ( α ) = H − . [ G Φ ( α ) + G Φ ( α ) + (1 − G − G ) Φ ( α )] , (3)where the Φ i are basis functions describing the linear part (sub-scripts 1 and 2) and the opposition e ff ect (subscript 3). For low-accuracy and sparsely-sampled phase curves, the authors pro-pose the H , G -system, later refined by Penttil¨a et al. (2016) tothe H , G ∗ -system, where( G , G ) = (cid:32) . (cid:33) + G ∗ (cid:32) . − . (cid:33) . (4)Taking into account the physical constraint that asteroids getfainter with increasing phase angle, we confine the G , G -spaceusing Equation 3 to G , G ≥ , (5a)1 − G − G ≥ . (5b)We gain physical interpretability of the phase coe ffi cients by ex-pressing the photometric slope k between 0 deg and 7.5 deg fol-lowing Muinonen et al. (2010) as k = − π G + G G + G , (6)and the size of the opposition e ff ect ζ − ζ − = − G − G G + G , (7)where ζ is the ratio of the amplitude of the opposition e ff ectand the background intensity. k is in units of mag / rad, while ζ − ff ect to the abso-lute magnitude in units of mag. Belskaya & Shevchenko (2000)showed that the opposition e ff ect and the photometric slope cor-relate with the albedo. The former peaks for moderate-albedoasteroids, while minor bodies with high- and low-albedo aster-oids display smaller opposition e ff ects. k is proportional to thealbedo, with dark asteroids exhibiting steeper phase curves thanbright minor bodies.The derivation of accurate phase curve parameters requires mul-tiple observational campaigns at di ff erent solar elongations of asingle target. The observations need to account for the modula-tion of the apparent magnitude by the asteroid’s irregular shapeand rotation, in addition to possible o ff sets due to varying aspectangles when combining data from distinct apparitions. Exam-ples of targeted campaigns can be found in Shevchenko et al.(1997, 2002, 2008, 2016). The number of asteroids with ac-curate and reduced phase curves available remains in the lower hundreds due to the requirements of extensive telescope time andasteroid shape models.To obtain catalogues of phase curve parameters in the order ofmagnitude required for future large-scale surveys, serendipitousasteroid observations need to be exploited. Oszkiewicz et al.(2011) determined the H , G , G - and H , G -model values ofmore than 500,000 asteroids by combining observations fromdi ff erent observatories. Since the publication of this catalogue,the number of known minor planets has increased almost two-fold. We aim to extend this e ff ort while taking note of twocaveats of the analysis. First, the fitted H , G , G - and H , G -model were not constrained as in Equation 5b, resulting in 52%of the reported slope parameters lying outside the physical range.Furthermore, the authors combined observations from di ff erentwavebands, applying average asteroid colour-indices to unify thedata. However, the slopes and band widths of asteroid spectra in-crease with increasing phase angle (e.g., Shkuratov et al., 2002;Sanchez et al., 2012), resulting in wavelength-dependent phasecurves. Therefore, we refrain from combining observations ac-quired in di ff erent wavebands.Utilising serendipitous observations o ff ers the advantage of largecatalogues, however, the derived phase curves are subject to sev-eral undesirable e ff ects. The majority of observations reportedto the Minor Planet Centre (MPC) is collected by large-scalesurveys aiming to monitor the near-Earth environment. To iden-tify asteroids on collision trajectories with Earth, these surveysfavour observing asteroids in quadrature rather than opposition.This introduces a bias towards observations at the maximum ob-servable phase angle for asteroid populations with superior or-bits to that of Earth. In addition, the light curve modulation intro-duced by rotation and apparition e ff ects can be reduced using ac-curate targeted observations, e.g. by means of a Fourier analysisto derive the shape of the light curve and by treating observationsfrom multiple oppositions separately. For non-targeted observa-tions, the comparatively large photometric uncertainty inhibitssuch a reduction. Furthermore, the corresponding increase inrequired observations would decrease the size of the availablesample, diminishing the statistical significance of the resultingcatalogue. As a consequence, serendipitously observed phasecurves exhibit stochastic fluctuations, translating into larger un-certainties on the fitted phase coe ffi cients.In this work, we derive the phase curve parameters of serendip-itously observed asteroids. In Section 2, we describe the obser-vations at hand and the Bayesian parameter inference approach.The fitted phase curve parameters are summarized in Section 3.The taxonomic interpretability of the G , G -parameters and theirwavelength-dependency are outlined in Section 4. We illustratethese results with the taxonomy of asteroid families in Section 5.In Section 6, we quantify the e ff ect of various sources of uncer-tainties and limited phase curve coverage at opposition on thederived phase curve parameters. The conclusions are presentedin Section 7. https://wiki.helsinki.fi/display/PSR/Asteroid+absolute+magnitude+and+slope https://minorplanetcenter.net reprint – A steroid phase curves from ATLAS dual - band photometry ethodology The MPC observations database contains 246 million asteroidobservations as of March 2020. We aim to acquire densely sam-pled phase curves for a large, unbiased corpus of asteroids. Atthe same time, we seek to quantify the inherent e ff ects of theasteroids’ shape-induced light curve modulation on the phasecurve parameters. We therefore attempt to exclude possiblesources of systematic e ff ects rigorously. These derive foremostfrom non-homogeneous photometry between di ff erent observa-tories. Di ff erences in the filter transmission, reduction pipeline,or stellar catalogues introduce discrepancies in the reported mag-nitudes of asteroids.Instead, we choose to utilise observations from a single obser-vatory, maximising the likelihood of consistent data treatment.In recent years, both the Panoramic Survey Telescope and RapidResponse System (Pan-STARRS, Hodapp et al. (2004)) and the
Asteroid Terrestrial-impact Last Alert System (ATLAS, Tonryet al. (2018)) have placed among the top five contributors tothe MPC in terms of number of observations. Comparing theephemerides at the epoch of observation of several thousand as-teroids observed by the surveys, we find that the bias towards ob-servation at asteroid quadrature is less pronounced in the ATLAScatalogues. In addition, ATLAS has acquired dual-band pho-tometry of a large number of asteroids at comparable phase an-gles, o ff ering an excellent dataset to investigate the wavelength-dependency of the phase curves. Hence, we make use of ob-servations by ATLAS, referring the reader to Vereˇs et al. (2015)for a derivation of H , G -parameters using Pan-STARRS obser-vations. ATLAS is a NASA-funded sky-survey aiming to observe near-Earth asteroids (NEAs) on impactor trajectories with the Earth.It was designed with a focus on a high survey speed per unitcost (Tonry, 2011). Two independent 0.5 m telescopes located atHaleakala and Mauna Loa in Hawaii are in operation since 2015and 2017 respectively, achieving multiple scans of the northernsky every night. Each telescope observes a 30 deg field-of-view.By March 2020, ATLAS has discovered 426 NEAs, including44 potentially hazardous ones. Standard observations are car-ried out in two filters, a bandpass between 420 - 650 nm termed cyan and a bandpass between 560 - 820 nm termed orange . Thetransmission curves of these filters are depicted in Figure 1. Theobserved asteroid astrometry and photometry are reported to theMPC.We received dual-band photometry of 180,025 distinct asteroidsfrom the ATLAS collaboration. A third of the objects was ob-served at phase angles below 1 deg. The observations were ac-quired between June 2015 and December 2018. We extend thisdatabase by including ATLAS observations from 2019 reportedto the MPC.The original database contained 26.8 million observations, towhich we add 8 million using the MPC database. The requiredephemerides are retrieved using the IMCCE’s Miriade tool http://atlas.fallingstar.com http://vo.imcce.fr/webservices/miriade/
400 500 600 700 800 λ / nm . . . . . . T r an s m i ss i on cyan orange Figure 1: Transmission curves of the cyan and orange filtersused by the ATLAS survey (Tonry et al., 2018). Data from theFilter Profile Service of the Spanish Virtual Observatory (Ro-drigo et al., 2012). (Berthier et al., 2008). All 180,025 asteroids were observed in orange , while 179,719 were observed in cyan as well. A smallfraction of visually inspected phase curves showed large outliermagnitudes likely caused by blended sources in the images. Weremove these detections by rejecting observations where the dif-ference between the predicted and the observed apparent mag-nitude was larger than 1 mag. This cut is well above the am-plitude modulation of asteroid light curves induced by the spin(Marciniak et al., 2015; Carry, 2018). Fitting scattering model functions to phase curves is notoriouslyambiguous and the results do not necessarily describe the ob-served surface, especially in the case of observations where theshape-induced light curve modulation has not been subtracted(Karttunen & Bowell, 1989; Kaasalainen et al., 2003). We choosea computationally expensive Bayesian parameter inference with
Markov chain Monte Carlo (MCMC) simulations to fit the H , G , G - and H , G ∗ -models, allowing to examine the poste-rior distributions of the phase curve parameters. To di ff erentiatebetween the absolute magnitudes obtained with these two mod-els, we use the subscript H for H , G ∗ .For both absolute magnitudes H and H , we choose a weaklyinformative, normally distributed prior, p ( H ) , p ( H ) = N ( µ = , σ = , (8)where N ( µ, σ ) describes the Gaussian normal distribution withmean µ and standard deviation σ . The mean and standard de-viation are set to approximate a uniform distribution over therelevant absolute magnitude parameter space. Alternatively, in-formative prior distributions could be derived from the distri-bution of the absolute magnitude of Main-Belt asteroids, up tothe limiting magnitude of ATLAS (m ∼
19, Tonry et al., 2018),or from computing least-squares fits of the
H G -model to eachphase curve and using the acquired H and its uncertainty as mo-ments of the Gaussian distribution.To quantify the e ff ect of the prior choice, we computed the H , G , G - and H , G ∗ -model fits for 100 randomly chosen phase curves http://svo2.cab.inta-csic.es/svo/theory/fps3/ reprint – A steroid phase curves from ATLAS dual - band photometry G G Phase Angle / deg . . . . . R edu c ed M agn i t ude (20) Massalia
Gehrels (1956) H , G , G
95% HDI ( G , G ) H , G ∗
95% HDI G ∗ Figure 2: The phase curve of (20)
Massalia , as observed byGehrels (1956), fitted with the H , G , G -model (solid black).The black dashed curves are plotted using the 95% highest den-sity interval (HDI) values of the three fit parameters. The gray,dash-dotted line represents the H , G ∗ -model fit with the graydotted line representing the uncertainty envelope. The measure-ment uncertainties of 0.01 mag are smaller than the marker size.The inset shows the 1 D- and 2 D-distributions of the G and G Markov chain Monte Carlo samples.using the three outlined priors. The distribution of H for Main-Belt objects is approximated with a Gaussian distribution with µ = . σ = .
6. The resulting distributions of the modelparameters H and H show negligible variation with averageddi ff erences below 0.01, only the prior based on the H G -modelyields larger H -values with an averaged di ff erence of 0.06 asit limits the size of the opposition e ff ect. The quantificationsupports the choice of the weakly informative choice, thoughthe prior based on the Main-Belt magnitudes would have beenequally acceptable.Following the slope parameter constraints in Equation 5, wechoose uniform distributions between 0 and 1 as prior proba-bilities for G , G , and G ∗ , p ( G ) , p ( G ) , p ( G ∗ ) = U [0 , . (9)Note that this choice in priors does not necessarily lead to G and G satisfying constraint 5b. To accommodate for this, weremove solutions where 1 − G − G < Θ .We define the likelihood function by assuming that the observedapparent magnitudes m α at phase angle α i follow a normal distri-bution with the true apparent magnitude as mean and a standarddistribution σ α, i dictated by the asteroid’s light curve modulationand the observation accuracy, p ( m α | Θ ) = N ( µ = m α, i , σ = σ α, i ) . (10)With the given prior probabilities, likelihood function and data,the posterior probability distribution p ( Θ | m ) is defined. How-ever, we cannot derive it analytically and need to approximate itby means of MCMC simulations.We use the pymc3 python package (Salvatier et al., 2016) to https://docs.pymc.io/ G G Phase Angle / deg . . . . . R edu c ed M agn i t ude (442) Eichsfeldia
ATLAS in cyan H , G , G
95% HDI ( G , G ) H , G ∗
95% HDI G ∗ Figure 3: Like Figure 2, using the non-targeted ATLAS obser-vations of (442)
Eichsfeldia in cyan instead. The H , G ∗ -modeldeviates towards the opposition e ff ect as the G , G -parametersof the asteroid are outside the definition of G ∗ . This furtherleads to the unreasonably small 95% highest density interval ofthe G ∗ parameter.perform these simulations. The photometric models are imple-mented in the sbpy package (Mommert et al., 2019) . As best-fit parameters, we use the mean values of the respective param-eter’s posterior probability distribution. The uncertainties aregiven by the bounds of the 95% highest density interval (HDI)of the posterior distributions.In Figure 2, we depict the parameter inference for the phasecurve of (20) Massalia , as observed by Gehrels (1956), who firstnoted the opposition e ff ect on the surface of an asteroid usingthese targeted observations. The resulting H , G , G - and H , G ∗ -model fits are displayed, including the 1 D- and 2 D distributionsof the G and G MCMC samples. The joint distribution of G , G illustrates that the uncertainty in the fit derives primar-ily from the photometric slope k as opposed to the size of theopposition e ff ect, as can be seen in the uncertainty profile of thefitted phase curve (dashed lines). Nevertheless, the posterior dis-tributions of G and G are Gaussians with comparatively smallstandard deviations due to the targeted nature of the observa-tions.In comparison, we show the e ff ect of serendipitous observationson the parameter inference in Figure 3, which depicts the H , G , G - and H , G ∗ -model fits to observations of (442) Eichsfeldia by ATLAS in cyan . The same range of reduced magnitudes isgiven on the y-axis as in Figure 2. The light curve modulationyields a large dispersion in reduced magnitude at each phaseangle, which is reflected in the dispersion of MCMC G , G -samples, depicted in the plot inset. We further observe thatthe posterior distribution tends towards the unphysical G < ff ect and the inferredabsolute magnitudes vary considerably ( H = . H = . H , G ∗ -model; the G , G -parameters of (442) Eichsfeldia in cyan are (0.64, 0.05), https://sbpy.readthedocs.io/ reprint – A steroid phase curves from ATLAS dual - band photometry G , G -relation of G ∗ . Hence,the H , G ∗ -model cannot adequately describe the opposition ef-fect of the phase curve. esults In this section, we present the phase curve parameters acquiredby applying the H , G , G - and H , G ∗ -models to the serendipi-tous ATLAS observations. Erasmus et al. (2020) investigate thetaxonomic interpretability of the cyan - orange colour to identifyasteroid family members. We highlight the di ff erences in theabsolute magnitudes derived with the H , G , G - and H , G ∗ -model. From the 180,025 asteroids observed by ATLAS, we select theones with at least one observation at a phase angle α below threedegree, α min ≤ ff ect. This decreases the sample size to 124,072asteroids, rejecting almost a third of the available sample. Wechoose this limit based on the significant importance of the oppo-sition e ff ect on the phase curve parameters, and after simulatingdi ff erent degrees of incomplete phase curve coverage towardsopposition, refer to Section 6. We further apply lower limits onthe number of observations, N ≥
50, and the maximum phase an-gle of observation, α max ≥
10 deg, to remove sparsely-sampledphase curves. The final sample consists of 94,777 unique aster-oids, 36,441 observed in cyan and 90,571 observed in orange .We provide the H , G , G - and H , G ∗ -model parameters forall 127,012 fitted phase curves in an online catalogue publiclyavailable at the Centre de Donn´ees astronomiques de Strasbourg (CDS). The format of the catalogue is described in A. For 43phase curves, the H , G , G -model failed to fit the phase curve,meaning that not a single of the 48,000 MCMC samples satisfiedEquation 5. By visually inspection, we found that large magni-tude dispersions, insu ffi cient sampling, or strong apparitions ef-fects lead to unphysical shapes of the phase curves, where themagnitude decreased with increasing phase angle. An exam-ple is given in Figure 10 in Section 6 for the phase curve of(250) Bettina , exhibiting a particularly strong apparition e ff ect. We display the absolute magnitudes H and H in cyan and or-ange derived from the model fits on the left hand side of Figure 4.It is apparent that the absolute magnitudes from the H , G ∗ -model are lower on average. This is highlighted in the right handside part of Figure 4, where we display the histograms of the dif-ference H − H for all objects. Both in cyan and in orange , thedistributions peak around 0.1 mag and extend up to 1 mag abso-lute di ff erence.The origin of these discrepancies can be seen in Figure 5. On theleft hand side, we give the 2 D kernel density estimator (KDE)distribution fitted to the G , G -pairs of the whole sample in cyan and orange using a Gaussian kernel. The black 1 σ -contour The catalogue is available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr ( ) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?VII/288 http://cdsweb.u-strasbg.fr/
12 14 16 180 . . . . . . cyan N = , HH − . − . . . . . . . . . H − H
12 14 16 18
Absolute Magnitude . . . . . . orange N = , HH − . − . . . . ∆ H . . . . . H − H Figure 4:
Left:
The distribution of absolute magnitudes H (black) and H (white) derived from ATLAS phase curves of94,777 asteroids using the H , G , G - and H , G ∗ -models re-spectively, for phase curves observed in cyan (top) and orange (bottom). For readability, magnitudes below 10 (0.8% of thesample) are not shown. Right:
The di ff erence in the absolutemagnitude derived with the two models, for phase curves in cyan (top) and orange (bottom).gives the KDE level at which 68% of the summed probabilitiesis contained in the area, resembling the 1 σ -level of a Gaussiandistribution. The G ∗ -parameter space is superimposed as white,dashed line (refer to Equation 4). We observe a clustering to-wards low G , medium G values in both wavebands, centeredaround the region where we expect S-type asteroids to be lo-cated (refer to Section 4). S-types dominate the inner and mid-dle Main Belt in terms of absolute number (DeMeo & Carry,2013). The distributions further extend towards larger photo-metric slopes to the region of low-albedo complexes such as theC-types, with a larger fraction of low-albedo asteroids visiblein the cyan -band. Further noticeable is an extension of distri-butions towards G =
1, i.e. negligible photometric slopes andopposition e ff ects, indicative of high-albedo complexes.In both wavebands, the majority of asteroids exhibits G , G -values above the G ∗ -definition in G , G -space. The H , G ∗ -model therefore fails at describing these phase curves, particu-larly the size of the opposition e ff ect will be overestimated forobjects above the G ∗ -line. ζ − G , G -space, leading to the large tail towards neg-ative di ff erences of the distributions on the right hand side ofFigure 4.The G ∗ -parameters of the phase curves are depicted in the his-tograms on the right hand side of Figure 5. 42% of the samplein cyan and 50% of the sample in orange are below 0.1, resem-bling closely the distribution in G , G -space. In both bands, weobserve a decline of the number of objects towards larger G ∗ reprint – A steroid phase curves from ATLAS dual - band photometry . . . G . . G cyan N = , u ( G ∗ ) σ σ . . G . . . . . · cyan N = , . . . G . . G orange N = , u ( G ∗ ) σ σ . . G . . . . · orange N = , Figure 5:
Left:
The 2 D-KDE distribution fitted to the G , G parameters of the phase curves observed in cyan (top) and or-ange (bottom). The black contours outline the 1 σ -levels of theKDE distributions. Right:
The histogram of the G ∗ -parameterderived from the same sample of phase curves, aligned in thesame order.values up to about 0.9, where the number rises again. These ten-dencies of G ∗ towards the limiting 0 and 1 values indicate thata large number of phase curve lies outside the defining relation,hence, they cannot be represented appropriately by the model.We point out that we observe phase curves fitted with H , G , G on the edges of G , G -space as well, though in a much smallerratio. As mentioned in Section 2, we attribute these to stochas-tic magnitude variations leading to unphysical shapes of the ob-served phase curves. , G ∗ for taxonomic classification usingnon-targeted observations The di ff erent results between H , G , G and H , G ∗ are expectedas the H , G , G is more flexible due to the third photometricparameter. Muinonen et al. (2010) stress that the main advan-tage of the H , G -model with its reduced parameter space isits predictive power when utilized with sparsely-sampled phasecurves. Indeed, giving non-targeted, sparse observations, and aprior knowledge on the target taxonomy, Penttil¨a et al. (2016)show that the absolute magnitude can be estimated using class-specific fixed slope parameters in the fitting procedure.However, regarding a taxonomic classification based on the pa-rameters of the H , G ∗ -model, we conclude here that neither theabsolute magnitudes nor the slope parameter are su ffi ciently reli-able. The discrepancy between H and H prevents classificationbased on the absolute magnitude. To compare, we compute thecolours of the asteroid taxonomic classes in the Sloan Digital Sky Survey (SDSS) using the spectral templates of the classes fromDeMeo & Carry (2013). For each colour, we compute the av-erage di ff erence between the complexes, resulting in 0.03 mag( u - g ), -0.03 mag ( g - r ), -0.04 mag ( r - i ), and 0.02 mag ( i - z ). Theinaccuracies introduced by the H , G ∗ - model are on averagegreater than these di ff erences, preventing taxonomic classifica-tion. As outlined in Subsection 3.2, we regard the G ∗ -parameterinsu ffi cient for any conclusion on the surface composition aswell. axonomy In the following, we evaluate the taxonomic information contentof the phase curve parameters, focusing on G , G -values de-rived from the serendipitous phase curves. We illustrate the dis-tributions of the asteroid complexes and quantify their similari-ties in cyan and orange . Further evaluated are their wavelength-dependency and the ability to solve degeneracies of asteroid spec-tra using phase curve parameters.The G , G - and G -distributions of di ff erent complexes havebeen studied by Oszkiewicz et al. (2011) andShevchenko et al.(2016). We do not further explore the G ∗ -parameter followingthe conclusion of Section 3. ff ect In a first iteration, we performed the following analysis on allacquired phase curves, using the limits on α min , α max , and N as outlined in Subsection 3.1. However, we noticed large dis-persions in the arising G , G -distributions of the complexes,which showed a clear trend with respect to the number N ofobservations in each phase curve. G , G -parameters derivedfrom phase curves with low N dispersed more from the center ofthe distributions than the ones from more densely covered phasecurves.The vital role of the opposition e ff ect both for determining theabsolute magnitude H and the taxonomic interpretation of thephase curve has been pointed out in the previous sections. Itsnon-linear dependence on the phase angle and the inherent mag-nitude dispersion of the serendipitous observations (refer to Sec-tion 6) require a dense coverage of observations to accuratelydescribe the brightness surge. As ATLAS aims to observe aster-oids on impact trajectory, only 7.3% of the 24 million observa-tions analysed here have been acquired of asteroids at α ≤ N =
50, this cor-responds to 3-4 observations covering the most important partof the phase curve. We therefore evaluated the trade o ff be-tween dispersion introduced in G , G -space by phase curveswith insu ffi cient sampling of the opposition e ff ect and by smallsample numbers in less common asteroid taxa. Through visualinspection of the resulting complex distributions, we settled on N =
125 as limit for the following analyses, decreasing the ini-tial sample size of 127,012 by more than half, down to 61,184.We stress that this large number of required observations stemsfrom the science goal of the observatory providing the data; fu-ture large scale surveys like LSST can derive accurate phase The template spectra are retrieved from http://smass.mit.edu/busdemeoclass.html . reprint – A steroid phase curves from ATLAS dual - band photometry ff ectis in the focus of the observation schedule. We retrieve previous taxonomic classifications from various ref-erences for 19,708 objects, in addition to reference albedo valuesfor 14,384 of these classified asteroids. The albedos are em-ployed to identify misclassifications and to separate classes intodi ff erent complexes as outlined below. We collected these valuesfrom numerous sources and refer the reader to A and the onlinecatalogue of the phase curve parameters for details.The majority of classifications follow the Bus- or Bus-DeMeo-schemes (Bus & Binzel, 2002; DeMeo et al., 2009), which areperformed on low-resolution asteroid reflectance spectra. As thephase curve parameters a priori contain less taxonomic informa-tion than spectra and to increase the size of the subsamples, wemap the classes into broader taxonomic complexes. In the Bus-DeMeo taxonomy, there are 25 classes spanning a space of 13complexes which are designated by unique letters of the alphabet(Binzel et al., 2019). We map the asteroids onto these complexesbased on their previous classifications. For classes which havebeen defined in previous taxonomies but are no longer present inthe Bus-DeMeo one, we choose the current complex resemblingthe previous class the most. As an example, the F-type definedin Tholen (1984) is mapped onto the B-complex.Previous taxonomies like Tedesco et al. (1989) di ff erentiate theX-type asteroids into low-albedo P-types, medium-albedo M-types, and high-albedo E-types. Asteroids with the same spectralshape but lacking albedo measurement are grouped into the X-types. As the albedo was dropped in subsequent taxonomies, sowas the di ff erentiation of the X-type classes. Given the corre-lation of the phase curve parameters with the albedo (e.g., Bel-skaya & Shevchenko, 2000; Penttil¨a et al., 2016; Belskaya &Shevchenko, 2018), we expect the X-type asteroids to separatein G , G -space. Therefore, we map asteroids classified in theX-complex into the P- ( p V ≤ . . < p V < . p V ≥ . µ m. TheATLAS orange filter covers the 0.7 µ m band, therefore, theseclasses may separate in phase-parameter space. We split theCgh- and Ch-types from the C-complex to investigate whether G , G reveal the hydration.The final mapping of classes to complexes is given in Table 1.Due to the low number of O-type asteroids, we exclude the com-plex from the analysis. We further rejected several ambigu-ous class assignments such as DS, CQ, SA, CS, XS from Car-vano et al. (2010), which were performed on low-resolution vis-ible photometry from the SDSS and given to objects which pre-sented photospectra with di ff erent features in di ff erent observa-tions. Further, the D-complex contains more than 200 objectswith albedos between 0.1 and 0.5, indicating that they are mis-classified. We therefore introduce an upper limit of 0.1 albedoon the D-type complex.Finally, we exclude the Ad, Bk, Ds, and Kl classes from Popescuet al. (2018) temporarily. These classes are assigned based on Table 1: The applied mapping of asteroid taxa to complexes.The previous classifications are mapped to the complex denotedunder Σ . N refers to the number of asteroids in each complex. ¯ p V and σ p V give the mean visual albedo and its standard deviationrespectively of all asteroids in the complex. The X-complex doescontain asteroids with albedo measurements by definition.Class Σ N ¯ p V σ p V P, PC, PD, X, XC,XD, XL, Xc, Xe, Xk, Xt → P 593 0.05 0.02D, DP → D 425 0.06 0.02Cgh, Ch → Ch 266 0.06 0.06B, F, FC → B 523 0.08 0.06C, CB, CD, CF, CG,CL, CO, Cb, Cg, Cgx,Co → C 3,670 0.09 0.09T → T 62 0.12 0.06M, X, XD, XL, Xc,Xe, Xk, Xt → M 660 0.15 0.05K → K 586 0.18 0.09L, LQ, Ld → L 776 0.19 0.09O → O 5 0.21 0.10S, SQ, SV, Sa, Sk,Sl, Sp, Sq, Sqw, Sr,Srw, Sv, Sw → S 8,875 0.26 0.08A, AQ → A 69 0.28 0.09Q, QO, QV → Q 185 0.28 0.11V, Vw → V 1,412 0.36 0.11E, X, XD, Xc, Xe, Xn, Xt → E 46 0.46 0.16X, XD, XL, Xe,Xk, Xt → X 202 - -near-infrared spectrophotometry using the VISTA-MOVIS cat-alogue (Popescu et al., 2016). The spectra of these types aredegenerate in the regarded wavelength regime, therefore, the ob-jects are classed together. The authors note that these classesare likely made up objects belonging to the denominating com-plexes (i.e. Ad is made up of A- and D-type asteroids). In asubsequent analysis step, we investigate the class ratios in thesecombinations using the phase curve parameters. We choose theVISTA-MOVIS sample rather than the SDSS sample by Car-vano et al. (2010) as the degeneracy in near-infrared cannot beresolved without additional information such as the phase curveparameters. For the ambiguous SDSS results, additional obser-vations in the visible could su ffi ce to resolve the classifications. , G –space The 2 D KDE distributions fitted to the G , G -parameters of thephase curves of the 15 complexes are shown in Figure 6, both in cyan and orange , with a black contour marking the KDE levelat which 68% of the summed probability is encompassed. Thecomplexes are depicted in increasing order of their average vi-sual albedo. It is readily apparent that the albedo-dependenceof the opposition e ff ect and photometric slope as described byBelskaya & Shevchenko (2000) is present in the ATLAS ob-servations as well; with increasing average visual albedo, thedistribution centers shift from large G -values towards medium-and finally large G values, i.e. towards flatter phase curves andsmaller opposition e ff ects. We further find good agreement with reprint – A steroid phase curves from ATLAS dual - band photometry . . . . . G cyan N = P σ . . . . . cyan N = D σ σ . . . . . cyan N = Ch σ . . . . . cyan N = B σ . . . . . cyan N = C σ . . . . . G P σ σ orange N = p = .
00 0 . . . . . D σ σ orange N = p = .
00 0 . . . . . Ch σ orange N = p = .
02 0 . . . . . B σ σ orange N = p = .
00 0 . . . . . C σ σ orange N = p = . . . . . . G cyan N = T σ σ . . . . . cyan N = M σ σ . . . . . cyan N = K σ . . . . . cyan N = L σ . . . . . cyan N = , S σ . . . . . G T σ σ orange N = p = .
66 0 . . . . . M σ orange N = p = .
00 0 . . . . . K σ orange N = p = .
00 0 . . . . . L σ orange N = p = .
00 0 . . . . . S σ orange N = p = . . . . . . G cyan N = A σ . . . . . cyan N = Q σ . . . . . cyan N = V σ . . . . . cyan N = E σ . . . . . cyan N = X σ . . . G . . G A σ orange N = p = .
22 0 . . . G . . Q σ orange N = p = .
02 0 . . . G . . V σ orange N = p = .
00 0 . . . G . . E σ σ orange N = p = .
23 0 . . . G . . X σ orange N = p = . Figure 6: Depicted are the G , G -distributions for several taxonomic asteroid complexes comprising 19,708 objects, derived fromserendipitous phase curves observed by ATLAS in cyan and orange . The complexes are sorted in increasing order of their averagevisual albedo. The distributions are represented by 2 D Gaussian kernel density estimators (KDE) fitted to the G , G -pairs. Theblack contours give the KDE level at which 68% of the summed probabilities is encompassed, resembling the 1 σ -level of aGaussian distribution. Further given are the number of asteroids N in each complex and waveband as well as the two-sample 2 DKolmogorov-Smirnov p -values computed between the distributions in cyan and orange for each complex. reprint – A steroid phase curves from ATLAS dual - band photometry N of analysed phase curves as well as the geometric center C and area A of the95%-probability contour in cyan (subscript c ) and orange (subscript o ). The areas are multiplied by 1,000 for notation purposes.Further given are the photometric slope parameter k , the size of the opposition e ff ect ζ −
1, and the Kolmogorov-Smirnov p -valuesfor each complex in this study. k and ζ − G , G -pairs of the geometric centers, following Equation 6 andEquation 7. Σ N c N o C c C o A c A o k c k o ζ − c ζ − o p P 255 576 (0.80, 0.05) (0.83, 0.06) 4.0 6.4 -1.82 -1.81 0.16 0.12 0.00D 179 419 (0.77, 0.17) (0.72, 0.20) 8.5 10.6 -1.67 -1.62 0.06 0.09 0.00Ch 125 255 (0.77, 0.05) (0.76, 0.07) 4.1 5.2 -1.84 -1.80 0.22 0.21 0.02B 172 519 (0.82, 0.06) (0.77, 0.08) 4.5 8.0 -1.82 -1.79 0.14 0.17 0.00C 965 3,609 (0.82, 0.06) (0.83, 0.06) 6.2 5.0 -1.81 -1.82 0.13 0.13 0.00T 30 62 (0.65, 0.19) (0.53, 0.24) 6.3 7.5 -1.61 -1.49 0.18 0.29 0.66M 203 642 (0.19, 0.34) (0.07, 0.42) 9.0 7.5 -1.05 -0.77 0.92 1.02 0.00K 147 566 (0.18, 0.40) (0.06, 0.48) 8.6 6.6 -0.99 -0.72 0.71 0.87 0.00L 176 758 (0.16, 0.37) (0.06, 0.47) 9.0 6.7 -0.96 -0.73 0.89 0.89 0.00S 2,076 8,702 (0.08, 0.46) (0.04, 0.51) 6.4 3.5 -0.76 -0.67 0.87 0.81 0.00A 17 68 (0.30, 0.39) (0.05, 0.57) 7.5 6.2 -1.16 -0.68 0.46 0.60 0.22Q 14 184 (0.36, 0.44) (0.05, 0.52) 9.2 4.6 -1.18 -0.70 0.25 0.74 0.02V 254 1,371 (0.10, 0.56) (0.04, 0.58) 6.5 3.2 -0.78 -0.67 0.50 0.60 0.00E 19 43 (0.33, 0.45) (0.06, 0.48) 8.0 8.8 -1.14 -0.73 0.29 0.86 0.23X 31 200 (0.11, 0.45) (0.06, 0.52) 9.0 5.6 -0.83 -0.70 0.81 0.73 0.04the G , G - parameters extracted from targeted campaigns byShevchenko et al. (2016) and Penttil¨a et al. (2016). The medium-and high-albedo S-, M-, and E-types populate regions of smallphotometric slopes, while low-albedo B-, C-, D-, and P-typespresent much larger slopes. Overall, the intermittent region around G = cyan presentlarge probabilities there. We summarize the distributions in Ta-ble 2, giving the G , G -coordinates of the geometric center ofthe 95%-probability-level contour for each complex. Furtherstated are the sizes of the areas encompassed by the 95% - prob-ability contours, approximating the dispersion of the complexesin G , G -space after outlier rejection.The strong disparity in the distributions of the E- and P-complexesshows that the phase coe ffi cients present a reliable distinctionbetween members of the X-complex, independent on referencealbedo measurements. For the complexes where we discrimi-nate based on on albedo, i.e. the P-, M-, E-, and D-type, wesee large tails in the 1 σ -distributions, which we attribute to re-maining misclassifications. Asteroid albedo measurements carryuncertainties around 17.5% (Masiero et al., 2018), suggestingthat the P-, M-, and E-complexes are overlapping due to theseinterlopers. The C- and D-types present broad distributions,specifically in the orange samples. This indicates a substantialfraction of misclassifications in the literature. The majority ofclassifications is retrieved from visible photometry based on theSDSS. As noted in Subsection 4.2 and Carvano et al. (2010), as-teroids can display ambiguous spectral features of several tax-onomies, leading to mixing of high- and low-albedo classifi-cations (misclassification of X- to C-types and S- to D-types).This hypothesis is further supported by the distribution of theCh-complex. The classification of hydrated C-types is subjectto more scrutiny than the more general C-types, hence we ex-pect a much smaller fraction of misclassifications. Indeed, weobserve less dispersed G , G -distributions in the lower-albedoregime for the Ch-complex. Finally, the contamination of theC-complex prevents a conclusion on the ability to observe hy- dration in slope parameter space.We conclude that the parameters of phase curves carry substan-tial taxonomic information, even for serendipitously acquiredobservations. Several observational requirements need to be ful-filled, such as a dense coverage of the opposition e ff ect. Never-theless, this promises a classification dimension as insightful asthe albedo while being more accessible to the observer. Phase reddening describes the steepening of the spectral slopeand a change in the bandwidths of asteroid spectra with increas-ing phase angle. The e ff ect is non-linear, see Sanchez et al.(2012). As the asteroid spectra are phase-angle dependent, it fol-lows that their phase curves in turn are wavelength-dependent,resulting in varying G , G -parameters. Carvano & Davalos(2015) investigate the phase-angle dependency of taxonomic clas-sifications of asteroids in the visible wavelength-regime. Theyfind that the taxonomic complexes are a ff ected to di ff erent de-grees; objects presenting the 1 µ m-olivine / pyroxene-band showstronger correlations between spectral slope and phase angle thanasteroids lacking the absorption feature.The wavelength-dependency of the G , G -parameters is under-lying to the question of whether it is admissible to combine ob-servations acquired in di ff erent wavelength-regimes to overcomeincomplete phase curve coverage. Though the overlap of the AT-LAS cyan and orange filters decreases the apparent wavelength-dependency (refer to Figure 1), the dataset at hand o ff ers a primeopportunity to investigate the dependency using similar asteroidsamples, phase curve coverages, and apparent magnitude reduc-tion pipelines.We regard the G , G -distributions acquired in cyan and orange as two independent samples. The Kolmogorov-Smirnov (KS) p -value statistic evaluates the probability of the null hypothe-sis that the underlying distribution of the two compared sam-ples is identical (Peacock, 1983). In general, provided the two reprint – A steroid phase curves from ATLAS dual - band photometry P D Ch B C T M K L S A Q V E XPDChBCTMKLSAQVEX nan nan 0.02 nan nan nan nan nan nan nan nan nan nan nannan nan nan nan nan nan nan nan nan nan nan nan nan nannan nan nan nan nan nan nan nan nan nan nan nan nan nan0.61 nan nan nan nan nan nan nan nan nan nan nan nan nannan nan nan 0.06 nan nan nan nan nan nan nan nan nan nannan nan 0.03 0.01 0.04 nan nan nan nan nan nan nan nan nannan nan nan nan nan 0.03 nan 0.01 nan nan nan nan nan nannan nan nan nan nan 0.01 0.31 0.57 nan 0.01 nan nan nan nannan nan nan nan nan 0.04 0.20 0.27 nan 0.01 nan nan nan nannan nan nan nan nan nan nan nan nan 0.03 nan nan 0.01 0.04nan nan 0.01 nan nan nan 0.01 0.02 0.01 0.37 0.09 0.02 0.22 0.11nan nan 0.10 nan nan nan 0.04 0.06 0.03 0.03 0.34 nan 0.14 0.01nan nan nan nan nan nan nan nan nan nan 0.09 0.08 0.02 nannan nan nan nan nan nan nan nan nan 0.03 0.73 0.63 0.13 0.03nan nan 0.04 nan nan nan 0.04 0.22 0.05 0.04 0.07 0.14 nan 0.06 cy ano r ange Figure 7: The two-sample 2 D Kolmogorov-Smirnov p -valuesestimating the similarity between the observed G , G -pair dis-tributions of the 15 asteroid taxonomic complexes in cyan (upperleft) and orange (lower right). Values above 0.2 indicate that thetwo paired complexes may have the same underlying distributionin G , G -space. Values below 0.01 are not shown for readabil-ity.compared samples are su ffi ciently large, p -values above 0.2 in-dicate a strong similarity, while values below 0.2 reject the nullhypothesis. The results are given in Figure 6 above the G , G -distribution orange of each complex as well as in Table 2. Mostcomplexes present p -values equal or close to zero, i.e. they showwavelength-dependency. The A-, E-, and T-complex are abovethe 0.2-threshold. They are the three smallest samples, however,and their G , G -distributions are noticeably di ff erent. We there-fore conclude that combining observations acquired in di ff erentwavebands should be strictly avoided. Additional support for thewavelength dependency of the phase curves can be derived fromthe Euclidean distance of the G , G -pairs in cyan and orange for objects observed in both bands. Computing the distancesyields a distribution with mode at 0.127, in good agreement withthe displacements of the complex centroid centers between thetwo wavebands given in Table 2.Di ff erences in the slopes of the phase curves observed at dif-ferent e ff ective wavelengths lead to spectral reddening which isproportional to the phase angle of observation. This is of partic-ular importance for the ESA Gaia mission, which is scheduledto release asteroid spectra obtained at large solar elongation inits third data release in 2021 (Delbo et al., 2012). The acquired G , G -distributions describing the shapes of the phase curvesallows us to quantify the amount of spectral reddening per de-gree phase angle for each taxonomic complex between the ef-fective wavelengths of the cyan and orange bands. The spectral slope in units of % /
100 nm is given by S S = f o − f c λ o − λ c · , (11)where f c and f o are the observed reflectance in cyan and orange ,and λ c =
518 nm and λ o =
663 nm are the e ff ective wavelengths.By relating the reflectances to the apparent magnitudes using thePogson scale, we can express the spectral slope as S S = f c (10 − . m o − m c ) − λ o − λ c · . (12)Normalizing the reflectance at λ c gives f c =
1, and the remainingvariable is the di ff erence m o − m c , which we can derive using thephase curves m c ( α ) and m o ( α ), ∆ m = m o ( α, H o , G , o , G , o ) − m c ( α, H c , G , c , G , c ) . (13) , G The G , G -parameters o ff er an additional dimension to taxo-nomic classification, which is predominantly done in spectralspace. The combination of both dimensions allows to identifyinterlopers and misclassifications.Using the 2 D- KS statistic, we compute the p -values to quan-tify the resemblance of the asteroid taxa in G , G -space. InFigure 7, we display the heatmap of the two-sample 2 D KS p -values quantifying the similarity of the distributions. The inter-sections on the upper left hand side compare the distributions in cyan , while the orange waveband comparison is depicted on thelower right hand side. The average visual albedo increases to-wards the upper right. Complex-combinations yielding p -valuesbelow 0.01 are left blank for readability.Two trends are visible in the heatmap. First, high-albedo com-plexes tend to show more resemblance to each other than low-albedo complexes, where only the P- and B-complexes in cyan show strong likeness. Second, the complexes present larger p -values in cyan , where nine pairings cross the 0.2-threshold, pro-hibiting the rejection of the null hypothesis, as opposed to twopairs in orange . Both pairs, the K-, L- and the A-, E-complexes,cannot be distinguished in either waveband.The degeneracies in phase curve parameter space appears re-versed to the degeneracies in spectral feature space. High-albedoobjects depict distinct absorption band properties in band depth,width, and wavelength, which allows for di ff erentiation even inlow-resolution data. Low-albedo types are separated based ontheir spectral slopes, which is in general less certain (Marssetet al., 2020). The phase parameters o ff er a complimentary clas-sification space.We apply this conclusion to four classes reported by Popescuet al. (2018) in the VISTA-MOVIS based classification. As out-lined in subsection 4.2, the near-infrared photometry presentsseveral degenerate classes, of which we show the G , G -pairs inFigure 8. To estimate the ratios of the di ff erent taxa, we computethe distance in G , G -space for each object to the center coor-dinates of the complexes and assign the object to the complex itis closer to. This is a simple test and proper interloper identifica-tion should be performed accounting for the complete complexdistributions; nevertheless, it is used as a proof of concept here.As we are working with center coordinates derived from statisti-cal ensembles, we may misclassify single objects. However, thederived probabilities should hold for the entire samples. reprint – A steroid phase curves from ATLAS dual - band photometry . . . . . G A
75 : 25 D cyan N = Ad σ σ . . . . . B
48 : 52 K cyan N = Bk σ . . . . . D
38 : 62 S cyan N = Ds σ σ . . . . . K
70 : 30 L cyan N = Kl σ . . . G . . G A
75 : 25 D Ad σ orange N =
65 0 . . . G . . B
49 : 51 K Bk σ orange N =
71 0 . . . G . . D
37 : 63 S Ds σ orange N =
219 0 . . . G . . K
28 : 72 L Kl σ orange N = Figure 8: Depicted are the G , G -distributions for four spectral classes from Popescu et al. (2018) containing two distinct asteroidtaxa. The 2 D kernel density estimates of their G , G -distributions are shown for observations in cyan (blue) and orange (red).The black contour gives the 1 σ -level. The white letters denote the position of the asteroid complexes in G , G -space. N gives thenumber of asteroids in each sample, while the derived ratios of the principal classes are given below.The resolution of degeneracies is e ff ective for the classes at op-posite ends of the albedo spectrum, which here are the A-D- andD-S combinations. We retrieve the same ratios for both wave-bands in these combinations, three-quarters of A-types in theformer and about two-thirds of S-types in the latter.For the Bk superposition, we observe almost identical ratios aswell, while we note that the observed G , G -distribution peaksbetween the two complex centers. Properly accounting for thedispersion of the complexes in G , G -space might change theretrieved ratios considerably. The Kl class cannot be resolved asexpected following Figure 7.Thus, we conclude that G , G -values derived from serendipi-tous observations are su ffi cient to untangle degeneracies arisingin spectral feature space if the classes separate in albedo-space.Lower-albedo classes may even be separated from one anotherprovided a reliable observation of the size of the opposition ef-fect, which is the principal distinction between the B-, C-, D-,and P-types in G , G -space. ∗ –space Oszkiewicz et al. (2012) found the S-, C-, and X-types followGaussian distributions in G -space (rather than G ∗ -space). Fol-lowing the discussion in Section 3, we do not expect any reliabletaxonomic information in the G ∗ -distributions. However, forcompleteness, we show them analogously to Figure 6 in Fig-ure C.2. Note that by definition, the D-, E-, and P-complexescannot be modelled with H , G ∗ , hence, we use a dashed linestylefor their distributions. steroid families in phase space In the following, we illustrate the use of the G , G phase curvecoe ffi cients as an extension of the physical parameter space of families. We intend this as a proof-of-concept of the results inSection 4 rather than a full analysis of the implications.The identification of asteroid families requires accurate parame-ter derivation and large number statistics to discern their mem-bers from the background of minor bodies (Milani et al., 2014).It is an interplay of their dynamical parameter space, specificallythe proper orbital elements, and their physical parameters suchas albedos and colours (Ivezi´c et al., 2002; Parker et al., 2008;Masiero et al., 2013)The phase curve coe ffi cients represent a large corpus of physicalquantities when derived from serendipitous observations. Os-zkiewicz et al. (2011) compute family-specific phase curves byfitting family members with constant G , G -parameters, mini-mizing a global χ in a grid search and describing the qualityof all fits simultaneously to arrive at the best fit for the fam-ily collective. The resulting G , G -values are concentrated to-wards medium photometric slopes and opposition e ff ect sizesfor all 17 families in the study, among which is the high-albedo(4) Vesta -family. We interpret this as indication that the simul-taneous treatment of all family members suppresses the inherentinformation on family taxonomy and fraction of interlopers.The distribution of family members in G , G -space can yieldinsights on the nature of their parent body or bodies. Unimodaldistributions suggest a homogeneous taxonomy, e.g., from a ho-mogeneous single parent body or from compositionally similarparent bodies of overlapping families. A heterogeneous taxon-omy in either case would give rise to multimodal distributions in G , G -space, as would the presence of a considerable fractionof interlopers. Finally, the superposition of distinct families inorbital space could be reflected in their phase coe ffi cients in thecase of di ff erent taxonomic nature.We retrieve the proper orbital elements (semi-major axis a p , ec-centricity e p , and orbital inclination angle i p , refer to Milani & reprint – A steroid phase curves from ATLAS dual - band photometry . . . . . Proper Semi-Major Axis . . . P r ope r E cc en t r i c i t y Vesta Astraea HygieaEunomia ThemisMinervaHertha KoronisMaria Eos . . . . . Proper Semi-Major Axis / AU P r ope r I n c li na t i on / deg Vesta Astraea HygieaEunomia ThemisMinervaHertha KoronisMaria Eos G1 G Figure 9: Illustrated are the G , G -parameters of several asteroid families, plotted in proper orbital elements space as semi-major-axis versus eccentricity (top) and versus inclination angle (bottom). The phase curves were observed by ATLAS in orange . Theproper orbital elements and family memberships are provided by AstDyS-2 (Milani et al., 2014). The colour-coding of the G , G -space for both figures is given in the inset of the right-hand plot.Table 3: Asteroid families with the number N of members, the geometric centers C of the 95% probability-level contour, the area A of the 1 σ -contour, as observed by ATLAS in cyan (subscript c ) and orange (subscript o ). The areas A are multiplied by 1,000 fornotation purposes. Further given are the taxonomic classifications of the families and their references.Family N c N o C c C o A c A o Class Reference(4)
Vesta
229 1647 (0.07, 0.50) (0.04, 0.55) 162 106 V Zappal`a et al. (1990)(5)
Astraea
59 524 (0.11, 0.48) (0.07, 0.48) 197 156 S Huaman et al. (2017)(10)
Hygiea
101 473 (0.75, 0.11) (0.08, 0.44) 191 184 C Carruba (2013)(15)
Eunomia
383 1647 (0.11, 0.43) (0.06, 0.49) 183 170 S Nathues (2010)(24)
Themis
528 1218 (0.80, 0.05) (0.73, 0.08) 96 151 C Moth´e-Diniz et al. (2005)(93)
Minerva
114 539 (0.08, 0.49) (0.07, 0.49) 159 170 S Moth´e-Diniz et al. (2005)(135)
Hertha
264 1777 (0.36, 0.34) (0.07, 0.49) 172 131 S Dykhuis & Greenberg (2015)(158)
Koronis
502 1333 (0.06, 0.46) (0.03, 0.52) 122 71 S Tholen (1984)(170)
Maria
100 472 (0.19, 0.41) (0.05, 0.47) 228 169 S Zappal`a et al. (1997)(221)
Eos
697 2732 (0.13, 0.36) (0.04, 0.44) 174 134 K Masiero et al. (2014)Kneˇzevi´c (1992); Kneˇzevi´c & Milani (2000)) for 93,200 aster-oids observed by ATLAS as well as their family membershipsfrom the
Asteroids - Dynamic Site 2 ( AstDyS-2 ) (Milani et al.,2014). We select all families of which more than 500 mem-bers have been observed by ATLAS, either in cyan or in or-ange , after applying the limit of N ≥
150 on the sample fromSection 3. 10 families pass the required number of observedmembers: (4)
Vesta , (5)
Astraea , (10)
Hygiea , (15)
Eunomia ,(24)
Themis , (93)
Minerva , (135)
Hertha , (158)
Koronis ,(170)
Maria , and (221)
Eos . The distributions of the familiesin a p − e p - and a p − i p -space are shown in Figure 9. Each dot rep-resents an asteroid, colour-coded by its G , G -values. The il-lustrated sample is restricted to phase curves observed in orange to show the larger subsample while eliminating the wavelength-dependency. We quantify the G , G -distributions for the fam- https://newton.spacedys.com/astdys2/ Note that the (8)
Flora family is not present in
AstDyS-2 as it doesnot di ff erentiate su ffi ciently from the background in the hierarchicalclustering method used, see Milani et al. (2014). ilies as done in Section 4 for the taxonomic complexes. The2 D KDEs are depicted in Figure D.3, split into cyan and or-ange . We summarize them in Table 3, giving the area of the 1 σ -contour and the geometric center of the 95% probability-levelcontour. The former is indicative of the fraction of interlopers orthe taxonomic heterogeneity of the parent bodies, while the lat-ter characterizes the G , G -values of the core family members.In addition, we state reference taxonomic classifications of thefamilies.Three families show strong uniformity, both visually in Figure 9and in their small area sizes in Table 3. (4) Vesta is the archetypeof the high-albedo taxonomic class, the V-types (Zappal`a et al.,1990), in agreement with the large G values of its core membercenter positions.(24) Themis is a C-type family with known low-albedo inter-lopers such as the B-type subfamily (656)
Beagle (Moth´e-Dinizet al., 2005; Fornasier et al., 2016). While in cyan these com-plexes appear indistinguishable, the blue B-types separate from reprint – A steroid phase curves from
ATLAS dual - band photometry orange wavelength-regime, refer toTable 2. We are not able to resolve this shift using the classifiedB- and C-types in the (24) Themis family subsample, neverthe-less, phase curves from targeted observations might detect thisdi ff erence (Shevchenko et al., 2016).(158) Koronis is one of the largest families in terms of numberand we confirm here its homogeneous S-type taxonomy (Tholen,1984).The C-type family (10)
Hygiea shows a considerable fraction ofobjects with high albedos in cyan , as well as objects with highalbedos in orange . Carruba (2013) have identified S- and X-typeinterlopers in the family. We further attribute this partially toremaining phase curves with insu ffi cient opposition e ff ect cov-erage, as the distribution shifts towards C-type objects with in-creasing N .The C-type asteroid (93) Minerva is the namesake of an S-typefamily (Moth´e-Diniz et al., 2005).We note that the family centers given in Table 3 are not compat-ible with the results of Oszkiewicz et al. (2011); the geometriccenters of the families extend more towards the upper G - and G -values as seen by Penttil¨a et al. (2016). We attribute this toour treatment of each family member separately, allowing fora di ff erentiated look into the G , G -distributions, specificallyseparating the core family members and potential interlopers.As in Subsection 4.5, we conclude here that the G , G -space iswell suited for interloper detection, adding a physical parameterspace to asteroid families that can confirm dynamical identifi-cation and strengthen the definition of families. This, in turn,improves their age estimates (Spoto et al., 2015). rror sources in serendipitous phase curves Reduced phase curves from targeted campaigns are available forin the order of 100 asteroids. To increase the number of aster-oids with available phase coe ffi cients, exploiting serendipitousasteroid observations is necessary. The uncertainty σ of the re-duced magnitudes in serendipitous phase curves is a propagationof uncertainties arising from their 3 D-shape and from the obser-vational parameters themselves, σ ∝ (cid:113) σ + σ + σ + σ + σ , (14)where σ PHOT is the photometric uncertainty of a single obser-vation, σ PREC refers to the loss in precision when magnitudesare reported in a truncated format, σ SYS is introduced by vary-ing photometric systems used either in di ff erent observatories orby an observatory over time, and σ ROT and σ APP are magnitudemodulations introduced by the asteroid’s shape, specifically theasteroid’s rotation and the change in aspect angle over di ff erentapparitions respectively. These uncertainties, disperse the ob-served reduced magnitudes, leading to broader posterior distri-butions (i.e. uncertainties) of the phase curve parameters as seenin Figure 3. Equation 14 is a non-exhaustive list, though thedominating error sources are encompassed.Following the discussions in Section 3 and Section 4, the cover-age of the opposition e ff ect further a ff ects the derived H , G , G -parameters. As opposed to the factors in Equation 14, its impactcan be minimized by a carefully set observation schedule. Phase Angle / deg . . . . R edu c ed M agn i t ude (250) Bettina
ATLAS in orange
Epoch / MJD
Figure 10: The phase curve of (250)
Bettina as observed byATLAS in orange . The observations are colour-coded by theirepoch, highlighting the four di ff erent apparitions that were cap-tured. The triaxial ellipsoid ratios of (250) Bettina are 1.4:1:1(Viikinkoski et al., 2017). The gray lines show the H , G , G -model fits to the apparitions, split into pairs of two.In the following, we examine the order of magnitude of eachuncertainty listed in Equation 14 and di ff erent degrees of cover-age of the opposition e ff ect, intending to identify the dominatinguncertainty and means to minimize it. In the most basic form, any observed magnitude carries an uncer-tainty e.g. due to random photon noise. These cannot be avoidedand are present also in targeted campaigns. In the ATLAS obser-vations, the mean photometric error is 0.14 mag, with a standarddeviation of 0.08 mag depending largely on the apparent mag-nitude of the target. The LSST aims at σ PHOT ∼ .
01 mag forsingle exposures of objects with a magnitude in r of 21 (LSST,2009).When working with serendipitous observations, however, thisprecision is frequently truncated to 0.1 mag either when the ob-servations are reported to or retrieved from the MPC. This addsan uncertainty of σ PREC = . / √
12 mag on top of σ PHOT , wherethe √
12 divisor comes from the standard deviation of the uni-form distribution. Once the new data pipeline accepting the up-dated observation report format has been put into place by theMPC, this source of error will be removed (Chesley et al., 2017).
There are systematic magnitude o ff sets which have to be takeninto account when combining magnitudes observed by di ff er-ent observatories, or even data from a single observatory whichunderwent recalibration. A more in-depth look is done by Os-zkiewicz et al. (2011). The di ff erences in the photometric sys-tems are not always apparent, e.g. when retrieving observationsfrom the MPC.As a baseline estimation of σ SYS , we use the example of theSDSS and Pan-STARRS systems. Both share the g , r , i , and z filters, though there are slight di ff erences in their throughputs.Computing the average di ff erences in apparent magnitude of the reprint – A steroid phase curves from ATLAS dual - band photometry . . . . . . ∆ m σ APP σ ROT
Figure 11: The amplitude of magnitude dispersion due to theasteroids’ rotation (black) and change in aspect angle betweendi ff erent apparitions (white). The values are derived using theDAMIT shape models from Durech et al. (2010).mean spectra for the 24 taxonomic classes in (DeMeo et al.,2009) gives σ SYS values of 0.09 mag, 0.01 mag, 0.02 mag, and0.08 mag for the four filters respectively. For bright asteroids,this is in the order of σ PHOT .In Subsection 4.1, we highlight the importance of densely sam-ple phase curves. We see here that achieving a large number ofobservations by combining data from di ff erent photometric sys-tems is a trade-o ff between increased phase curve coverage andintroduced dispersion in apparent magnitude. ff ect The light curve of an asteroid is modulated by to its 3 D-shape ro-tating around its spin axis. The rotation imprints a periodic mod-ulation of the apparent magnitude over the rotation period, whichis typically in the order of a few hours (Warner et al., 2009). In addition, the varying aspect angles over di ff erent apparitionsof the asteroid introduce o ff sets in the observed magnitude, ef-fectively shifting the whole phase curve along the y -axis, hencebiasing the determination of the absolute magnitude H .Figure 10 shows an example of both e ff ects a ff ecting the phasecurve of (250) Bettina , as observed by ATLAS in orange . Theepoch of observation is colour-coded. Four distinct apparitionscan be seen in the observations, leading to shifts in the reducedmagnitudes which give the impression of two superimposed phasecurves being displayed. On top of the apparition e ff ect, we seethe magnitude dispersion within the apparitions, introduced inpart by the asteroid’s spin.The strength of the magnitude modulation due to rotation andchange in aspect angle depend on the shape of the asteroidsand the viewing geometry. The Database of Asteroid Modelsfrom Inversion Techniques (DAMIT) provides shape modelsfor 2,407 asteroids (Durech et al., 2010). We use these shapemodels to quantify the e ff ect of the rotational modulation ( σ ROT )and of the varying aspect angle ( σ APP ). http://alcdef.org/ http://astro.troja.mff.cuni.cz/projects/damit For each asteroid, we compute its triaxial dimensions ( a > b > c )and assimilate its shape to a smooth ellipsoid in the following.Under this assumption, the modulation of the apparent magni-tude due to spin and 3D shape writes m = − . (cid:32) π abc · (cid:32) (cid:18) cos β cos λ b (cid:19) + (cid:32) cos β sin λ a (cid:33) + (cid:32) sin β c (cid:33) (cid:33) . (cid:33) , (15)with λ , β being the longitude and latitude of the subobserverpoint (Surdej & Surdej, 1978; Ostro & Connelly, 1984).We generate a full-rotation synthetic light curve every 10 daysover an entire orbital revolution around the Sun, e ff ectively prob-ing the range of Sun-target-observer geometries. As serendipi-tous observations randomly occur over the rotation period, themeasured magnitude is o ff set from the average value. We esti-mate this o ff set σ ROT by computing the root mean square resid-uals of each light curve to its average.The influence of the varying aspect angle is computed from thedi ff erence σ APP between the average magnitude of each lightcurve and a light curve taken while the observer is located withinthe equatorial plane of the asteroid.The distributions of the changes in apparent magnitude for bothe ff ects are shown in Figure 11. σ ROT is in general larger than σ APP , with a median value of 0.11 mag compared to 0.07 magrespectively. For σ APP , the situation is analogous to σ SYS ; itcan be removed by avoiding the combination of observationsfrom di ff erent apparitions of the target. Nevertheless, the ben-efits of adding more samples of the phase curve may outweighthe downsides of increased magnitude dispersion. In Figure 10,the parameter inference failed when applied to all observations,while we could retrieve two phase curves of (250) Bettina aftersplitting the apparitions in pairs of two. ff ect of magnitude dispersion on H , G , G To quantify how the magnitude dispersion due to the uncertain-ties listed in Equation 14 a ff ect the H , G , G -model parameters,we simulate the (20) Massalia phase curve by Gehrels (1956) inFigure 2 as serendipitous observations. Using the order of mag-nitudes for the uncertainties derived above ( σ PHOT = . σ PREC = . / √
12 mag, σ SYS = .
05 mag, σ ROT = .
11 mag,and σ APP = .
07 mag), we compute the propagated uncertaintyand simulate 100 phase curves of (20)
Massalia with N = σ . Forthese 100 phase curves, we compute the H , G , G -parametersand the di ff erences to the parameters of the original phase curve, ∆ Θ = Θ i − Θ , i ∈ { , . . . , } , (16)where Θ refers to the H , G , G -parameters. The mean di ff er-ence and the median value of the absolute di ff erence of the re-sulting distributions are given in the first row of Table 4. Theformer indicates systematic parameter shifts with respect to theoriginal values, while the latter indicates the spread of the di ff er-ences. We choose the median rather than the standard deviationto be able to compare the results to the non-Gaussian distribu-tions we obtain in the next subsection. reprint – A steroid phase curves from ATLAS dual - band photometry ff erences between the 100phase curves with simulated noise ( σ ) and the H , G , G -parameters of the targeted (20) Massalia observations byGehrels (1956). The same is given for the truncated ATLASphase curves, with i deg describing the dropout degree. µ refersto the mean values of the distributions, while σ gives the medianof the absolute di ff erences. µ ∆ H σ ∆ H µ ∆ G σ ∆ G µ ∆ G σ ∆ G σ H and G ,i.e. they are steeper while depicting smaller opposition e ff ect sizes.While the opposition e ff ect in the original observations is sam-pled su ffi ciently, the stochastic nature of the simulated obser-vations is reflected in the large median values of the absolutedi ff erences of all three parameters. This highlights the need fordense sampling to allow for restricting the parameter space; theo ff set in µ ∆ H in the simulated phase curves renders taxonomicclassification from the computed colours inconclusive. ff ect coverage When relying on serendipitous asteroid observations, the cov-erage of the opposition e ff ect is pre-determined by the surveyfootprint. This dependence of the scientific yield in terms ofasteroid phase curves, colours, or taxonomies on the solar elon-gation coverage should be taken into account early on.To quantify the influence of insu ffi cient phase angle coverage,we select all ATLAS phase curves in cyan and orange with α min ≤ N ≥
50 of asteroids with shape models present inDAMIT. 917 phase curves of 720 asteroids fulfill these crite-ria. Next, we reduce the spin- and apparition-induced magni-tude dispersion in the phase curves using shape models and thelight curve generation software provided by DAMIT. This in-creases the probability that in the following simulations, we ob-serve the influence of the opposition e ff ect coverage rather thanthe magnitude dispersion on the H , G , G -parameters. How-ever, the photometric noise cannot be removed by essence andsome residuals arise from the non-ideal fit of the photometry bythe shape-induced light curve.The mean minimum phase angle of the reduced subset is 0.6 deg.We remove observations below {
1, 2, 3, 4, 5 } deg phase angle andcompute the H , G , G -model fit. The relevance of the opposi-tion e ff ect can then be quantified by looking at the di ff erenceof the H , G , G -parameters of the complete and the truncatedphase curves. In Table 4, we display the di ff erence in the pa-rameters of the truncated phase curves and the complete phasecurves, defined analogously to Equation 16 with i referring tothe truncation angle. With diminishing coverage of the opposi-tion e ff ect, the error on H increases up to 0.2 mag in these simu- https://astro.troja.mff.cuni.cz/projects/damit/pages/software_download . . . G . . G C orange N =
112 0 . . . G . . G S orange N = < Figure 12: The e ff ect of insu ffi cient phase curve coverage at lowangles on the G , G -parameters. Shown are the 2 D fitted ker-nel density estimators of the G , G -distribution for C-type andS-type asteroids (red). The solid white line displays the 1 σ -levelof the distribution using the complete ATLAS phase curves, theblack solid, dash-dotted, and dotted contours the distributionsusing the phase curves truncated at 1 deg, 3 deg, and 5 deg re-spectively.lations. Following Equation 2, this translates to an error of 17%on the derived asteroid albedo. Further, we observe systematicshifts of the G , G parameters. We display this in Figure 12, de-picting the G , G -distributions for 112 C-types and 218 S-typesobserved in orange . The contours depict the 1 σ -outlines for thecomplete phase curves (white), and increasing truncation angle(black). As the angle increases, the taxonomic information getslost. onclusion We perform phase curve parameter inference using serendipitousasteroid observations for a large number of minor bodies. TheATLAS observatory provided us with dual-band photometry formore than 180,000 objects, of which we selected about 95,000based on the sampling statistics of their phase curves. As AT-LAS continues to survey the night sky, we will be able to add anincreasing number of asteroids to this analysis.Our results show that the H , G , G -model parameters containsignificant taxonomic information of the target surface, providedthe opposition e ff ect is densely sampled. The close correla-tion between the G , G -parameters and the albedo allows touse serendipitously observed phase curves as accessible albedoproxy for hundreds of thousands of asteroids with the upcomingLSST and NEOSM surveys. The taxonomic complexes separatesu ffi ciently in the phase-coe ffi cient space to study ensembles ofasteroids such as asteroid families, while the large tails of thedistributions prevent classification of single objects from G , G alone.We find evidence for a wavelength-dependency of the phase co-e ffi cients. Provided the taxonomy is known, the derived slopeparameters of the complexes allow for estimating the degree ofphase reddening in the slopes of asteroid spectra.We quantified the sources of uncertainties of serendipitously ac- reprint – A steroid phase curves from ATLAS dual - band photometry ff ect on the G , G parameters.By simulating incomplete phase curves at low phase angles, wehighlight the importance of observations close to opposition ( ≤ H , usedto derived properties such as albedo, colours, and taxonomicclass. A cknowledgements This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. ATLAS is primarilyfunded to search for near earth asteroids through NASA grantsNN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byprod-ucts of the NEO search include images and catalogs from thesurvey area. The ATLAS science products have been made pos-sible through the contributions of the University of Hawaii In-stitute for Astronomy, the Queen’s University Belfast, the SpaceTelescope Science Institute, and the South African AstronomicalObservatory.This research has made use of IMCCE’s Miriade VO tool.This research has made use of the SVO Filter Profile Servicesupported from the Spanish MINECO through grant AYA2017-84089. 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ATLAS dual - band photometry reprint – A steroid phase curves from ATLAS dual - band photometry A O nline catalogue
We briefly describe the structure of the online catalogue submitted to the CDS of phase curve parameters. The results are providedin a CSV-formatted table consisting of 127,012 rows, with each row providing the parameters of a single asteroid in a singleATLAS observation band. About a third of the asteroids appear in twice, once for cyan and once for orange . The column namesand brief descriptions are given in Table A.1. For 43 H , G , G -model fits, no MCMC sample was withing the physical range ofthe G , G -parameters. These fits are considered failed and the parameters are empty in the catalogue provided online.Table A.1: Structure of the online catalogue providing the phase curve parameters. Column
Description
Column
Description number
Asteroid number rms12
RMS of H , G ∗ -model fit name Asteroid name or designation h up
Upper 95% HDI value of H band ATLAS observation band, either cyan or orange h low Lower 95% HDI value of H class Reference taxonomic classification for g1 up
Upper 95% HDI value of G scheme Taxonomic scheme of reference g1 low
Lower 95% HDI value of G ref tax Code to identify taxonomy reference g2 up
Upper 95% HDI value of G ap Proper semi-major-axis from
AstDyS-2 g2 low
Lower 95% HDI value of G ep Proper eccentricity from
AstDyS-2 h12 up
Upper 95% HDI value of H ip Proper inclination from
AstDyS-2 h12 low
Lower 95% HDI value of H N Total number of observations g12 up
Upper 95% HDI value of G ∗ phmin Minimum phase angle of observations g12 low
Lower 95% HDI value of G ∗ phmax Maximum phase angle of observations albedo
Reference albedo h Fitted H of H , G , G -model err albedo Uncertainty of reference albedo g1 Fitted G of H , G , G -model ref albedo Code to identify albedo reference g2 Fitted G of H , G , G -model family number Family number from
AstDyS-2rms
RMS of H , G , G -model fit family name Family name from
AstDyS-2h12
Fitted H of H , G ∗ -model family status Family status from
AstDyS-2g12
Fitted G ∗ of H , G ∗ -model B O bservation bias towards large phase angles for impactor detection α / deg . . . . . α obs . . . . . α max Figure B.1: The black line shows the kernel density estimation of the distribution of asteroid phase angles at the epoch of ob-servations, for all 20.7 million ATLAS observations analysed in this work. We further show the kernel density estimation of thedistribution of the maximum observable phase angles of all asteroids in the sample, derived from their proper semi-major axis a p as α max = / sin − (1 / (2 a p )). reprint – A steroid phase curves from ATLAS dual - band photometry C G ∗ of taxonomic complexes . . . cyan N = P . . . cyan N = D . . . cyan N = Ch . . . cyan N = B . . . cyan N = C . . . orange N = p = . P . . . orange N = p = . D . . . orange N = p = . Ch . . . orange N = p = . B . . . orange N = p = . C . . . cyan N = T . . . cyan N = M . . . cyan N = K . . . cyan N = L . . . cyan N = S . . . orange N = p = . T . . . orange N = p = . M . . . orange N = p = . K . . . orange N = p = . L . . . orange N = p = . S . . . cyan N = A . . . cyan N = Q . . . cyan N = V . . . cyan N = E . . . cyan N = X . . . G orange N = p = . A . . . G orange N = p = . Q . . . G orange N = p = . V . . . G orange N = p = . E . . . G orange N = p = . X Figure C.2: The G ∗ -distributions for several taxonomic complexes of asteroids, derived from serendipitous observations by ATLASin cyan and in orange . We give the sample sizes N and the two-sample Kolmogorov-Smirnov p-value between the cyan and orangeG ∗ -distributions of the complexes. The H , G ∗ -model is not suited for the D-, E-, and P-complexes, hence they are displayed witha dotted linestyle. reprint – A steroid phase curves from ATLAS dual - band photometry D G , G of families . . . . . G cyan N = Themis σ . . . . . cyan N = Hygiea σ . . . . . cyan N = Eos σ . . . . . cyan N = Hertha σ σ . . . . . cyan N = Koronis σ . . . . . G Themis σ σ orange N = ,
218 0 . . . . . Hygiea σ orange N =
473 0 . . . . . Eos σ orange N = ,
732 0 . . . . . Hertha σ orange N = ,
777 0 . . . . . Koronis σ orange N = , . . . . . G cyan N = Astraea σ σ . . . . . cyan N = Maria σ σ . . . . . cyan N = Minerva σ . . . . . cyan N = Eunomia σ . . . . . cyan N = Vesta σ . . . G . . G Astraea σ orange N =
524 0 . . . G . . Maria σ orange N =
472 0 . . . G . . Minerva σ orange N =
539 0 . . . G . . Eunomia σ orange N = ,