Asteroid Photometric Phase Functions From Bayesian Lightcurve Inversion

Asteroid Photometric Phase Functions From Bayesian Lightcurve Inversion Karri Muinonen 1 , 2 *, Elizaveta Uvarova 1 , Julia Martikainen 3 , Antti Penttilä 1 , Alberto Cellino 4 and Xiaobin Wang 5 , 6 1 Department of Physics, University of Helsinki, Helsinki, Finland, 2 Finnish Geospatial Research Institute FGI, Masala, Finland, 3 Instituto de Astrofísica de Andalucía, CSIC, Granada, Spain, 4 INAF, Osservatorio Astrofisico di Torino, Pino Torinese, Italy, 5 Yunnan Observatories, Chinese Academy of Sciences, Kunming, China, 6 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing, China Photometry is an important tool for characterizing the physical properties of asteroids. An asteroid’s photometric lightcurve and phase curve refer to the variation of the asteroid’s disk-integrated brightness in time and in phase angle (the Sun-asteroid-observer angle), respectively. They depend on the asteroid’s shape, rotation, and surface scattering properties, and the geometry of illumination and observation. We present Bayesian lightcurve inversion methods for the retrieval of the asteroid’s phase function, the unambiguous phase curve of a spherical object with surface scattering properties equal to those of the asteroid. A collection of such phase functions can give rise to a photometric taxonomy for asteroids. In the inverse problem, first, there are four classes of lightcurves that require individual error models. The photometric observations can be absolute or relative and they can have dense or sparse cadence in comparison to the rotation period of the asteroid. Second, the observations extend over varying phase angle ranges, requiring different phase function models. Asteroid photometry from the European Space Agency Gaia space mission extends, typically, over a range of phase angles, where the phase curve tends to be linear on the magnitude scale. Photometry from groundbased observing programs can reach small phase angles, where the asteroids show an opposition effect, a nonlinear increase of brightness on the magnitude scale towards zero phase angle. We provide error models for all four classes of lightcurves and make use of linear or linear-exponential phase functions for phase angles below 50°. We apply the inverse methods to sparse absolute Gaia and dense relative ground-based lightcurves and obtain absolute magnitudes and phase functions, with uncertainties, for ~500 asteroids. Finally, we assess the lightcurve inversion problem for dense absolute photometry with the help of a numerical simulation for a Gaussian-random-sphere asteroid. Keywords: asteroid, lightcurve, phase curve, phase function, absolute magnitude, convex inversion, asteroid taxonomy, Gaia mission Edited by: Aaron Golden, National University of Ireland Galway, Ireland Reviewed by: Josep M. Trigo-Rodríguez, Institute of Space Sciences (CSIC), Spain Apostolos Christou, Armagh Observatory, United Kingdom *Correspondence: Karri Muinonen karri.muinonen@helsinki.fi Specialty section: This article was submitted to Astrostatistics, a section of the journal Frontiers in Astronomy and Space Sciences Received: 23 November 2021 Accepted: 23 May 2022 Published: 17 June 2022 Citation: Muinonen K, Uvarova E, Martikainen J, Penttilä A, Cellino A and Wang X (2022) Asteroid Photometric Phase Functions From Bayesian Lightcurve Inversion. Front. Astron. Space Sci. 9:821125. doi: 10.3389/fspas.2022.821125 Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211251 ORIGINAL RESEARCH published: 17 June 2022 doi: 10.3389/fspas.2022.821125 1 INTRODUCTION Asteroids are small Solar System objects that originate from times preceding planet formation. Typically, when rotating about its principal axis of inertia, an asteroid exhibits a periodic change of brightness caused by the varying part of its surface being both illuminated by the Sun and visible to the observer. On one hand, a photometric lightcurve is the result of photometric observations extending over a time span covering a substantial part of the rotation period or more. On the other hand, a photometric phase curve is the result of observations of the change in an asteroid’s apparent brightness obtained at different epochs during a single apparition, corresponding to a slowly changing Sun-observerobject geometry. Consequently, photometric lightcurves provide immediate information about the rotation period. Historically, asteroid lightcurve observations have been the first tool intensively applied to derive physical properties of these objects. In addition to the retrieval of the rotation period, lightcurve amplitudes have been used to derive rough characteristics of the shape (elongated shapes generally producing lightcurves with larger amplitudes) and, when lightcurves obtained at different apparitions were available, to derive estimates of the direction of the asteroid’s pole orientation (Taylor, 1979;Magnusson et al., 1989;Kaasalainen et al., 2002;Durech et al., 2015). Some early statistical analyses of lightcurve amplitudes and periods led also to the discovery of the existence of equilibrium shapes among large asteroids (Farinella et al., 1981), and opened the way to the discovery of the importance of the general phenomenon of asteroid collisional evolution (Farinella et al., 1982). Photometric phase curves provide information about the intrinsic light-scattering properties of the surface that are intimately related to the regolith composition and structure. Since the composition and structure determine also the reflectance properties observed at different wavelengths, phase curves are strictly related to the taxonomic classification of asteroids. Phase curves are often represented with the help of a few parameters. The classical two-parameter H,Gmagnitude system has been used for a long time (Bowell et al., 1989). His the asteroid’s absolute magnitude, namely the magnitude (corresponding to unit distance from the Sun and the observer) measured at zero phase angle, and Gis a parameter describing the overall variation of magnitude at different phase angles, including a nonlinear brightness surge at phase angles smaller than 10°(opposition effect). In recent years, it has been replaced by the so-called H,G 1 ,G 2 magnitude system, a threeparameter model developed by Muinonen et al. (2010) to remove the caveats of the H,Gsystem in the case of low-albedo and highalbedo asteroids. Further refinements and applications of the H, G 1 ,G 2 system have been published, among others, by Penttilä et al. (2016) and Shevchenko et al. (2016). Gaia Data Release 3 (DR3) is imminent (13 June 2022), including extensive astrometric, photometric, and spectroscopic observations of small Solar System objects, primarily asteroids. It is important to prepare to analyze such a large amount of new data. In the field of asteroid photometry, in particular, significant progress has been made recently in what concerns the capability of obtaining efficient processing and reliable interpretation of sparse photometric data. Martikainen et al. (2021) carried out an analysis of photometric data combined from ground-based lightcurves and observations published in Gaia Data Release 2 (DR2) in order to invert the data for reliable estimates of shape, rotational properties, and phase curves for a large number of objects belonging to a variety of asteroid taxonomic classes. They derived photometric phase curve slopes, rates of brightness change on the magnitude scale, for more than 300 asteroids in the so-called reference geometry of equatorial illumination and observation (Kaasalainen et al., 2001). Expanding on the study in Muinonen et al. (2020), Martikainen et al. (2021) provided unequivocal proof that the projection to similar illumination and observation conditions was needed to enable unbiased comparative studies of asteroid phase curves. It is possible to strive towards minimizing the biases by incorporating all practical geometries of illumination and observation (Oszkiewicz et al., 2011). However, for asteroids with their individual pole orientations and orbits, different illumination and observation geometries are sampled, and biases remain. In spite of these biases, Oszkiewicz et al. (2011) and Mahlke et al. (2021) have successfully related phase curve parameters from massive observing programs to the taxonomical classes. We extend the work by Martikainen et al. (2021) by providing computational tools for the derivation of asteroid phase functions using fictitious spherical asteroids with equal surface properties. In earlier works, we have described these phase functions as being proper phase functions that describe the intrinsic properties of the surfaces. We apply the methods to the asteroids studied by Martikainen et al. (2021), by starting from their results of Markov-chain Monte Carlo (MCMC) lightcurve inversion for some 500 asteroids. Furthermore, we generalize the studies by Martikainen et al. (2021) and Muinonen et al. (2020) by incorporating a combined linear-exponential model of phase functions on the magnitude scale. In earlier work (Muinonen et al., 2020), models of observational uncertainties were developed for dense relative photometry and sparse relative photometry. The former entailed ground-based lightcurves that were treated, in lightcurve inversion, on a relative magnitude scale. The latter comprised lightcurves of sparse Gaia photometry that were incorporated on a relative magnitude scale, too. Martikainen et al. (2021) then treated the Gaia photometry in the absolute sense, deriving absolute magnitudes for a large number of asteroids. In the present work, we provide a complete set of four models for observational uncertainties, including models for dense relative, sparse relative, dense absolute, and sparse absolute lightcurves. The paper is organized as follows. Section 2 describes the theoretical framework in asteroid lightcurve inversion. Particular attention is paid to the different surface scattering models utilized in the forward and inverse problems. In Sect. 3, error models are presented for dense and sparse lightcurves (relative or absolute), and the retrieval of absolute magnitudes and phase functions is outlined. Section 4 first provides the application of the methods to ~500 asteroids with both sparse absolute Gaia photometry and Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211252 Muinonen et al. Asteroid Photometric Phase Functions dense relative ground-based photometry. Thereafter, a numerical forward simulation of dense absolute photometry with a Gaussian-random-sphere asteroid model (GS-asteroid) is described, followed by inversion results and their discussion. Conclusions and future prospects are offered in Sect. 5. 2 LIGHTCURVE INVERSION 2.1 Asteroid Modeling Consider an asteroid in principal-axis rotation. We denote the asteroid’s rotation period by P, the pole orientation in ecliptic longitude and latitude by (λ,β) T (J2000.0, Tstands for transpose), and the rotational phase at a given epoch t 0 by φ 0 . As to the asteroid’s shape, in the inverse problem, we consider both ellipsoidal shapes and general convex shapes. In the forward problem, we incorporate star-like shapes described by the Gaussian-random-sphere geometry. The apparent V-band magnitude (in mag) of a fictitious spherical asteroid (diameter Din km and geometric albedo pV) at a given phase angle αis mVα () H−2.5lgΦα () +5lg rΔ () , H2.56.259 −2lgD−lgpV  ,(1) where His the absolute magnitude (in mag), rand Δare the asteroid’s heliocentric and topocentric distances (in au), Φis the phase function (Φ(0°) = 1), and lg denotes the 10-based logarithm. The corresponding reduced V-band magnitude is projected to r=Δ= 1 au: Vα () H−2.5lgΦα () .(2) Scattering of light from a surface element on an asteroid is described by the diffuse reflection coefficient Rthat relates the incident solar flux density πF 0 to the emergent, scattered intensity I: Iμ,ϕ;μ0,ϕ0  μ0Rμ,ϕ;μ0,ϕ0  F0, μ0cos ι,μcos ϵ.(3) Here ιand ϵare the angles of incidence and emergence as measured from the outward normal vector of the element, and ϕ 0 and ϕdenote the respective azimuthal angles. For a geometrically isotropic surface, it is unnecessary to specify ϕ 0 andwesetthecoordinatesystemso that ϕ 0 =0 °. Consequently, the backscattering direction, the direction for the source of light, is with ϕ=0 °. The Lommel-Seeliger surface reflection coefficient (subscript LS) derives from radiative transfer (e.g., Lumme and Bowell, 1981): RLS μ,μ0,ϕ  2pΦ11 α () 1 μ+μ0 ,(4) where pis the geometric albedo for the wavelength band considered. The diffuse reflection coefficient for dark particulate media (subscript PM)—such as planetary regoliths of low-albedo asteroids—can be expressed in the form (Muinonen et al., 2011;Wilkman et al., 2015) RPM μ,μ0,ϕ  2pΦ11 α () ΦSμ,μ0,ϕ  1 μ+μ0 , Φ11 P11 α () P11 0° () , p1 8~ ωP11 0° () , (5) where pis the geometric albedo, αis the phase angle, and ~ ωand P 11 are the single-scattering albedo and single-scattering phase function (different from the phase function of the asteroid). The function Φ S represents the corrections to radiative transfer due to the dense packing of the particulate medium, for example, the corrections due to shadowing among the particles typically much larger than wavelength. Φ 11 and Φ S , as all functions Φof the present study, are normalized to unity at backscattering: Φ11 0° () 1, ΦSμ0,μ0,0°  1. (6) The reflection coefficient in Eq. 5 belongs to a class of photometric models consisting of a Lommel-Seeliger-type volume-element part and a part describing scattering among volume elements in a particulate medium (e.g., Lumme and Bowell, 1981;Muinonen and Lumme, 1991). Wilkman et al. (2015) provide an extensive set of numerical computations of the dense-packing correction Φ S using particulate media of opaque spherical particles. They complete the modeling with the help of a fractional-Brownian-motion model for the roughness of interface between the medium and free space. There are altogether three parameters: the packing density of the particles v, the fractal Hurst exponent H fBm , and the amplitude σ fBm . Smaller H fBm and higher σ fBm imply rougher interfaces with stronger effects of interface roughness. The function Φ S is an unknown function for asteroid surfaces. In what follows, as in Muinonen et al. (2015), we utilize Φ S estimated for the lunar mare regolith by Wilkman et al. (2014). Consider next the function Φ 11 in Eqs 4,5and the large numbers of existing phase curve observations of asteroids. Indeed, we may consider that the principal form for the phase function of a fictitious spherical asteroid is already known to be described by, for example, the H,G 1 ,G 2 phase function. Consequently, we introduce denominators in Eqs 4,5that cancel the inherent phase function that would result from the assumption of Φ 11 = 1 (isotropic scattering): RLS μ,μ0,ϕ  2pΦα () ΦLS α () 1 μ+μ0 , RPM μ,μ0,ϕ  2pΦα () ΦPM α () ΦSμ,μ0,ϕ  1 μ+μ0 , (7) where Φrepresents an unknown function that should nevertheless be close to the empirically known phase functions of asteroids. In other words, the functions Φ LS and Φ PM are the phase functions for a spherical asteroid with reflection coefficients R LS and R PM assuming Φ 11 = 1, respectively. Figure 1 depicts the lunar disk-integrated phase function with the help of the H,G 1 ,G 2 phase function as well as the phase function Φ PM derived for the lunar mare regions by Wilkman Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211253 Muinonen et al. Asteroid Photometric Phase Functions et al. (2015). Their ratio is also given. By approximating that Φ PM can be valid for the higher-albedo lunar highland regions, too, the ratio points to the part of the lunar opposition effect at small phase angles unexplained by shadowing and rather explained by the coherent backscattering mechanism (Muinonen et al., 2002). Significantly different single-scattering phase functions Φ/Φ LS or Φ/Φ PM result from utilizing R LS or R PM :asR PM assigns a significant part of phase curve steepness to shadowing, the resulting single-scattering phase function is shallower. In order to model the function Φin Eq. 7 for lightcurve inversion, we make use of a linear-exponential model on the magnitude scale: −2.5lgΦα () −m0exp −α α0  +m0+β0α,0°≤α≤50°, (8) where m 0 and α 0 are the amplitude and angular width of the opposition effect, respectively, and β 0 is a slope parameter. In the linear-exponential model of Eq. 8, the photometric slope at α= 20°(the phase angle chosen for comparative studies) equals βS20° () β0+m0 α0 exp −20° α0  .(9) For phase angles outside the angular regime of the opposition effect, 10°≤α≤50°, a linear model can be utilized: −2.5lgΦα () β0α,10°≤α≤50°, βS20° () β0.(10) The H,G 12 phase function is a two-parameter phase function developed for scarce photometric data (Muinonen et al., 2010). As in Martikainen et al. (2021), we rule out increasing brightness with increasing phase angle and enforce the absence of an opposition effect for G 12 values that would result in negative weights 1 −G 1 −G 2 : G2 10 3G1,G 12 ≤−0.53784, 1−G1,G 12 ≥1.23728. ⎧ ⎪ ⎨ ⎪ ⎩(11) The extension does not affect the H,G 12 phase function within its approximate nominal range of 0 ≤G 12 ≤1. FIGURE 1 | Phase functions and their ratios on the magnitude scale. The red curves (top left and right) correspond to the lunar phase function Φ(lowermost solid line), the ratio Φ/Φ LS (middle dashed line), and the ratio Φ/Φ PM (uppermost solid line; see Eq. 7). The blue curves (bottom left and right) denote Φ PM (lowermost solid line), Φ LS (at α= 120°, middle dashed line), and Φ PM /Φ LS (at α= 120°, uppermost solid line). The raggedness at small phase angles is an artefact deriving from the interpolation of Φ PM . Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211254 Muinonen et al. Asteroid Photometric Phase Functions 2.2 Inverse Problem Consider the free parameters (unknowns) of the asteroid forward model. In the case of the ellipsoid shape model, the parameters are described by the vector PP, λ,β,φ0,b a, ca, m0,α0,β0  T,(12) where the N P = 9 free parameters are, respectively, the rotation period, ecliptic pole longitude, ecliptic pole latitude, rotational phase at a given epoch t 0 , two ellipsoid axial ratios (with a>b>c denoting the semiaxes), and three parameters of the linearexponential phase function. In the case of the convex shape models, the parameters are PP, λ,β,φ0,s 00,...,s lmaxlmax,m 0,α0,β0  T,(13) where the shape is described by the (lmax +1)2−l2 min spherical harmonics coefficients s00,...,s lmaxlmax for the Gaussian surface density. Here φ 0 and the real-valued s 00 can be fixed as the rotational phase is immersed in the spherical harmonics coefficients and the asteroid size information is omitted. Thus, the total number of parameters equals NPlmax +1 () 2−l2 min +5. (14) Let p p be the a posteriori probability density function (p.d.f.) for the parameters. Within the Bayesian framework (cf. Muinonen et al., 2020), p p is proportional to the a priori and observational uncertainty p.d.f.s p pr and p ϵ+υ ,ϵand υreferring to random and systematic uncertainties. p ϵ+υ is evaluated for the “Observed-Computed”(O-C) residual magnitudes ΔM(P), ppP ()∝ppr P () pϵ+υΔMP ()() , ΔMP () Mobs −MP () .(15) Even though p ϵ+υ is here related to the observations, it can also describe the uncertainties deriving from the shortcomings in the physical model. It is currently assumed that p ϵ+υ is Gaussian and that p pr will describe, for example, the regularization needed in convex inversion. The final a posteriori p.d.f. is thus ppP ()∝ppr P () exp −1 2χ2P () , χ2P () ΔMTP () Λ−1 ϵ+υΔMP () , (16) where χ 2 measures the O-C distance in terms of the model for the uncertainties. The observation vector is composed of a number of lightcurves with their varying numbers of magnitudes, and the uncertainties are assumed to be uncorrelated between the lightcurves. We may thus rephrase χ 2 (P)as χ2P ()  K k1 ΔMT kP () Λ−1 ϵ+υ,kΔMkP () , ΔMkP () Mobs,k −MkP () , (17) where M obs,k ,M k (P), and Λ ϵ+υ,k pertain to the observations, computations, and the covariance matrix for the uncertainties in lightcurve k, the total number of lightcurves being K. In detail, we simplify the χ 2 -value in Eq. 17 to the form χ2P ()  K k1 1 σ2 ϵ,k  Nk j1 Mobs,kj −Mkj P () 2,(18) where the σ ϵ,k values describe the uncertainty (and weight) of the N k observations in lightcurve k, and M obs,kj and M kj (P) are the observed and computed magnitudes. For small relative uncertainties in brightness, the χ 2 -value can be approximated by (Muinonen et al., 2020) χ2P () ≈ K k1 2.5lg e  2 σ2 ϵ,k  Nk j1 ℓobs,kj −ℓkj P () 100.4ΔMk0P () ℓobs,kj  2 ,(19) where ΔM k0 (P) denote the O-C difference of the mean magnitudes in lightcurve k, and ℓ obs,kj and ℓ kj (P) are the observed and computed brightnesses relative to the brightnesses corresponding to the lightcurve mean magnitude lg~e 0.43429. Thus, in Eq. 19, the χ 2 -value is computed using the differences in the observed and computed relative brightnesses, relative to the observed relative brightnesses. 3 NUMERICAL METHODS The MCMC inverse methods are based on proposal probability densities characterized with the help of the so-called virtual leastsquares solutions and are described in detail by Muinonen et al. (2020). In what follows, we introduce the error models for the FIGURE 2 | Example photometric lightcurves for asteroid (167) Urda (blue circles) on the magnitude scale. We show the ground-based dense lightcurve #8 (left) and the Gaia sparse lightcurve (middle and right). Red crosses stand for the best-fit convex model lightcurves with the 1-σuncertainties (red bars), and the red points are the uncertainty envelopes using 5,000 MCMC sample solutions. The symbols t,α,andkdenote the time, phase angle, and lightcurve observation counter. Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211255 Muinonen et al. Asteroid Photometric Phase Functions observations and the algorithm for the retrieval of absolute magnitudes and phase functions from DR2 photometry. 3.1 Error Models for Observations We define dense lightcurves as having photometric points timed, on average, less than the asteroid’s estimated rotation period apart. For dense relative lightcurves, the points are calibrated against each other but not against absolute standards. We set (Muinonen et al., 2020) σϵ,k  Nk Nk,eff max σ0,k,σpr,k  , Nk,eff ΔT~ k ΔTk ,k1,...,K, (20) where σ 0,k denotes the rms-value and σ pr,k denotes the a priori threshold value for the uncertainty in the lightcurve k, obtainable, for example, from cubic spline fits to the lightcurves. Furthermore, ΔT k is the mean sampling time interval of the lightcurve k, ΔTkTk Nk−1, ΔT~ kmaxk1,...,K ΔTk, (21) where ~ kmarks the lightcurve with the lowest sampling rate in time and T k stands for the time span of the lightcurve k.Equation 20 can be interpreted in the following way. The lightcurve ~ k contributes a χ 2 -value of unity in the inverse problem, whereas each other lightcurve kis considered to split into approximately N k,eff independent lightcurves. For dense absolute lightcurves composed of Klightcurves, the points are calibrated both against each other and absolute standards. We start by setting σϵ,k  Nk Nk,eff max σ0,k,σpr,k  , k1,...,K, (22) where σ 0,k and σ pr,k are defined as for Eq. 20. The definition of N k,eff is the key point of the error model. For simplicity, consider the case σ 0,k ≥σ pr,k . First, if we set N k,eff = 1, each lightcurve obtains an equal weight of unity. This is the most conservative model with systematic errors dominating over the random errors. Second, we may set N k,eff = 1 for the lightcurve ~ kthat has the minimum number of observations and consider the other lightcurves as composed of an effective number of Nk,eff Nk N~ k ,(23) lightcurves like lightcurve ~ k, balancing the weights of lightcurves with drastically different numbers of observations. We recall that the systematic errors include, in addition to the observational errors, the effects of simplified forward modeling, in particular, for the phase function in Eqs 8–10. We define sparse lightcurves as having photometric points timed, on average, more than the estimated rotation period apart. For sparse relative lightcurves, as for dense relative lightcurves, the points are calibrated against each other but not against absolute standards. In accordance with Muinonen et al. (2020), we set Nk,eff N~ kin an error model coinciding with the form in Eq. 22, the index ~ know denoting the sparse lightcurve with the smallest number of observations. Finally, for sparse absolute lightcurves, we introduce a model that coincides with the FIGURE 3 | Histograms of the derived β S (20°)andβ ref values (both in mag/rad), obtained using 358 asteroids. FIGURE 4 | Comparison of the derived β S (20°)andβ ref values using 358 asteroids. The blue line represents perfect correlation. Two noticeable outliers (5902) Talima and (15172) 3086 P-L are marked with blue crosses. Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211256 Muinonen et al. Asteroid Photometric Phase Functions one for the sparse relative lightcurves. Now the observations are, however, calibrated against absolute standards. Thecompleteaposteriorip.d.f.istheproductofthep.d.f.sforthe four classes so that the corresponding χ 2 -values are added together: χ2P ()  4 i1 Ki k1 1 σ2 ϵ,ik  Nik j1 Mobs,ikj −Mikj P () 2(24) or FIGURE 5 | The topmost graphs depict σβS(20°)as a function of β S (20°) using 358 asteroids (left) and highlighting 13 new asteroids presently processed (right). The middle graphs show σβS (20°)as a function of β S (20°) for 97 asteroids with known Tholen classes (left), highlighting six new asteroids (right). The bottom graphs provide a zoom-in of the middle graphs. FIGURE 6 | Comparison of the derived equal-sphere G-band absolute magnitudes G(1,0) and the V-band absolute magnitudes Hbased on the Jet Propulsion Laboratory Small-Body Database for 358 asteroids (left), the right panel highlighting the 13 new asteroids. The blue line represents perfect correlation of the absolute magnitudes. Two noticeable outliers (446) Aeternitas and (1368) Numidia are marked with blue crosses. Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211257 Muinonen et al. Asteroid Photometric Phase Functions χ2P () ≈4 i1 Ki k1 2.5lge  2 σ2 ϵ,ik  Nik j1 ℓobs,ikj −ℓikj P () 100.4ΔMik0P () ℓobs,ikj  2 , (25) where the index idescribes the dense relative (i= 1), sparse relative (i= 2), dense absolute (i= 3), and sparse absolute photometry (i= 4). For dense and sparse relative lightcurves, the relative brightnesses ℓ obs,ikj and ℓ ikj (P) are computed using the mean-magnitude brightnesses of each lightcurve. However, for dense and sparse absolute lightcurves, they are computed using the mean magnitude of the entire absolute lightcurve data. Note that the sparse relative lightcurves are weighted equally to the absolute photometric data but they enter the inverse problem separately with no regard to the absolute level of brightness. 3.2 Absolute Magnitudes and Phase Functions We refine the algorithm in Martikainen et al. (2021) for the derivation of absolute magnitudes and phase functions from Gaia photometry. As in their study, first, we start from the phase function slope parameter β S retrieved from MCMC inversion, recalling that β S describes the intrinsic surface-element properties of an asteroid. Second, using the full asteroid modelavailablefromtheinversion,wemovetothereference TABLE 1 | Example asteroids with photometry in Gaia Data Release 2. Tholen stands for the Tholen taxonomic class, B-DM stands for the Bus-DeMeo taxonomic class, Nis the total number of Gaia and ground-based observations, K−1 is the number of observed ground-based lightcurves, N Gaia is the number of observed Gaia points, and αdenotes the phase angle coverage for each asteroid using Gaia data. Asteroid Tholen B-DM NK−1N Gaia α(°) (55) Pandora M Xk 1,126 37 18 14.2–24.1 (95) Arethusa C Ch 180 5 30 11.6–16.6 (97) Klotho M Xc 429 26 9 17.0–26.1 (122) Gerda ST L 1,485 18 13 14.5–18.1 (245) Vera S S 121 4 16 12.5–21.5 (246) Asporina A A 81 7 15 14.4–22.2 (376) Geometria S Sl 870 40 13 17.5–24.2 (377) Campania PD Ch 1,088 35 11 16.9–21.6 (404) Arsinoe C Ch 2,332 50 9 19.6–28.7 (596) Scheila PCD T 358 8 26 11.9–18.8 (731) Sorga CD Xe 620 10 16 11.9–18.7 (1251) Hedera E X 343 11 10 16.7–24.1 TABLE 2 | The photometric slope β S (mag/rad) of the phase function retrieved using convex inversion by Martikainen et al. (2021) (CXI), together with the mean-magnitude reference phase curve slope β ref (mag/rad) and the slope β S (mag/rad), both computed for the phase angle of 20°. All the slope parameters represent the means from MCMC sampling. The uncertainties are given in units of the last digit shown. Asteroid Tholen β S (CXI) β ref (20°)β S (20°) (55) Pandora M 1.434 (86) 1.456 (76) 1.446 (77) (95) Arethusa C 2.10 (10) 1.920 (69) 1.910 (69) (97) Klotho M 1.957 (80) 1.988 (86) 1.982 (86) (122) Gerda ST 1.54 (15) 1.54 (11) 1.53 (11) (245) Vera S 1.683 (56) 1.646 (41) 1.643 (41) (246) Asporina A 1.425 (67) 1.454 (54) 1.442 (53) (376) Geometria S 1.572 (66) 1.574 (69) 1.572 (69) (377) Campania PD 2.398 (81) 2.344 (78) 2.340 (78) (404) Arsinoe C 2.096 (61) 2.258 (80) 2.247 (80) (596) Scheila PCD 1.995 (69) 1.833 (54) 1.831 (54) (731) Sorga CD 1.754 (69) 1.739 (49) 1.709 (49) (1251) Hedera E 1.635 (87) 1.666 (88) 1.637 (87) FIGURE 7 | Photometric phase functions with H,G 1 ,G 2 -fits for asteroids (55) Pandora (95) Arethusa (245) Vera (596) Scheila (731) Sorga, and (1251) Hedera. For illustration, the phase functions are presented on a relative magnitude scale with 0.5-mag offsets (left) and normalized at 20°(right). Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211258 Muinonen et al. Asteroid Photometric Phase Functions geometry of equatorial illumination and observation at the epochs and phase angles of the individual photometric points. Third, by computing the asteroid model brightnesses over one full rotation for each epoch, we determine the magnitudes of lightcurve brightness maxima. Martikainen et al. (2021) then carried out linear least-squares fitting to determinetheirphasecurveslopeparameterβ max ,from which they derived the full H,G 12 phase function. As a small TABLE 3 | The Tholen classes, the derived G-band absolute magnitudes G RG (1,0) and G EQ (1,0) for the reference geometry and equal-sphere case, respectively, the derived absolute magnitudes H G using the H,G 1 ,G 2 fit, and the derived G 1 and G 2 parameters for the equal-sphere photometric phase functions. The uncertainties are given in units of the last digit shown. Asteroid Tholen G RG (1,0) G EQ (1,0) H G G 1 G 2 (55) Pandora M 7.976 (67) 7.762 (69) 7.768 (69) 0.192 (54) 0.512 (73) (95) Arethusa C 7.9151 (58) 7.892 (19) 7.899 (19) 0.626 (71) 0.159 (45) (97) Klotho M 7.639 (18) 7.466 (23) 7.475 (24) 0.674 (85) 0.123 (54) (122) Gerda ST 7.634 (80) 7.590 (76) 7.597 (76) 0.276 (88) 0.420 (98) (245) Vera S 7.644 (29) 7.592 (17) 7.597 (18) 0.346 (40) 0.345 (28) (246) Asporina A 8.377 (53) 8.410 (48) 8.412 (48) 0.163 (47) 0.532 (60) (376) Geometria S 9.371 (45) 9.292 (47) 9.295 (47) 0.264 (48) 0.408 (52) (377) Campania PD 8.8774 (98) 8.772 (38) 8.776 (38) 1.046 (64) −0.105 (41) (404) Arsinoe C 8.967 (19) 8.721 (55) 8.733 (55) 0.958 (79) −0.050 (51) (596) Scheila PCD 8.712 (29) 8.684 (29) 8.686 (28) 0.528 (63) 0.223 (40) (731) Sorga CD 9.7093 (93) 9.5560 (92) 9.5681 (95) 0.432 (46) 0.282 (31) (1251) Hedera E 10.732 (71) 10.773 (69) 10.788 (69) 0.357 (56) 0.373 (74) FIGURE 8 | The distributions of the different Tholen classes (black dots) and their range (black crosses) in the G 1 and G 2 parameter space based on the equal-sphere phase functions. The grey dots represent single asteroids and the green line shows how the G 12 parameter maps into G 1 and G 2 in the H,G 12 magnitude system. TABLE 5 | Lightcurve characteristics for the simulated GS-asteroid. First, we give the lightcurve identifier (k), time span (T k ), mean sampling time interval (ΔT k ), and number of observations (N k ). Second, we give the number of nodes for the cubic spline fit based on the Bayesian information criterion (N BIC ) and the rmsvalue of the spline fit in relative magnitude (rms(m)). In the third column, for example, 0.9674 (−3) stands for 0.9674 × 10 –3 . kT k ΔT k N k N BIC rms(m) (d) (d) (mag, ini) 1 0.3560 0.9674 (−3) 369 22 0.0096 2 0.3799 0.1076 (−2) 354 25 0.0101 3 0.3796 0.1078 (−2) 353 26 0.0102 4 0.4232 0.1096 (−2) 387 24 0.0098 5 0.4307 0.1297 (−2) 333 19 0.0102 6 0.2521 0.9233 (−3) 274 16 0.0101 7 0.3741 0.1140 (−2) 329 21 0.0104 8 0.3229 0.1228 (−2) 264 25 0.0094 9 0.2363 0.1041 (−−2) 228 9 0.0091 10 0.2303 0.1061 (−2) 218 10 0.0095 11 0.1995 0.1187 (−2) 169 13 0.0101 12 0.3140 0.5925 (−2) 54 18 0.0067 13 0.2461 0.4171 (−2) 60 12 0.0094 14 0.2083 0.1894 (−2) 111 12 0.0104 15 0.3135 0.1479 (−2) 213 20 0.0088 16 0.3181 0.8552 (−3) 373 14 0.0100 17 0.3178 0.9184 (−3) 347 13 0.0097 18 0.2375 0.1484 (−2) 161 16 0.0081 19 0.3321 0.8260 (−3) 403 15 0.0106 20 0.2375 0.9063 (−3) 263 11 0.0099 21 0.2836 0.8103 (−3) 351 14 0.0098 22 0.2797 0.8250 (−3) 340 12 0.0104 23 0.2170 0.2973 (−2) 74 10 0.0094 24 0.1456 0.3033 (−2) 49 7 0.0118 25 0.0169 0.7046 (−3) 25 4 0.0084 26 0.0851 0.1576 (−2) 55 8 0.0089 27 0.1154 0.1538 (−2) 76 5 0.0110 28 0.3833 0.1645 (−2) 234 15 0.0100 29 0.3757 0.1917 (−2) 197 11 0.0102 30 542.7794 0.1428 (2) 39 —0.0100 TABLE 4 | Lightcurve characteristics for the simulated Gaussian-sphere asteroid (GS-asteroid). “Class”denotes the Tholen taxonomical class, Nand Kdenote the numbers of observations and lightcurves, respectively, and T obs is the time span of the observations. The uppermost numbers refer to the simulated dense ground-based observations, the ones in the middle to the simulated sparse observations, and the lowermost ones refer to the combined observations. Asteroid Class NKT obs (d) T obs (a) GS-asteroid S 6,664 29 17,609.83 48.21 39 1 542.78 1.49 6,703 30 18,694.55 51.18 Frontiers in Astronomy and Space Sciences | www.frontiersin.org June 2022 | Volume 9 | Article 8211259 Muinonen et al. Asteroid Photometric Phase Functions helped in the simulations and analysis. All have offered comments on the article. FUNDING Research supported by the Academy of Finland grants No. 1325805, No. 1336546, and No. 1345115, and the Chinese Academy of Sciences President’s International Fellowship Initiative (PIFI) Grant No. 2021VMA0017. This work presents results from the European Space Agency (ESA) space mission Gaia.Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). 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An Analysis of the Amplitude-Phase Relationship Among Asteroids. Astron. Astrophys. 231, 548–560. Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer AC declared a past collaboration with the author AC to the handling editor. Publisher’s Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. Copyright © 2022 Muinonen, Uvarova, Martikainen, Penttilä, Cellino and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). 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