ESLAB #53: The Gaia Universe Edited by J.H.J. de Bruijne Asteroid proper phase curves from Gaia photometry Karri Muinonen1;2, Antti Penttilä1, Julia Grön1, Alberto Cellino3, Xiaobin Wang4, Paolo Tanga5 1 Department of Physics, University of Helsinki, Finland 2 Finnish Geospatial Research Institute FGI, Masala, Finland 3 INAF, Osservatorio Astrosico di Torino, Pino Torinese, Italy 4 Yunnan Observatories, Chinese Academy of Sciences, Kunming, China 5 Observatoire de la Cote d’Azur, Nice, France Abstract The photometric phase curve and lightcurve of an asteroid refer to the variation of the asteroid’s disk-integrated brightness as a function of the phase angle (the Sun-Object-Observer angle) and of time, respectively. They depend on the shape and rotation state of the asteroid, as well as the scattering properties of the asteroid’s surface. It follows that the shape, rotation state, and scattering properties can be estimated from the observations, to an extent allowed by the data available. We study the feasibility of retrieving phase curve parameters via statistical lightcurve inversion from the sparse asteroid photometry available from Gaia Data Release 2. With the help of the H, G1, G2 photometric function, we devise a surface reection coecient that gives rise to realistic disk-integrated phase curves. This allows us to derive phase curve parameters that are independent of the illumination and observation geometry. These proper phase curve parameters may allow for photometric classication of asteroids. 1 Introduction Asteroids are nonspherical Solar System bodies typically rotating about their axes of maximum inertia. Their surfaces are assumed to be covered by regoliths, layers of particles in the size scales of microns to meters. Asteroids oer insight into the evolution of the Solar System. They provide valuable future space resources and cause a signicant natural impact risk for the human kind. Photometry is the most common source of data for the characterization of asteroid physical properties. The lightcurve, i.e., the observed brightness as a function of time, depends on the shape and spin state of the asteroid, as well as its scattering properties. We are developing statistical in- verse methods for the retrieval of rotation periods, pole ori- entations, and scattering properties from the photometric Figure 1: Asteroid (2867) Steins [11, 12]. We show the north observations. This entails a complete Markov chain Monte polar view (top left), south polar view (bottom left), the Carlo (MCMC) assessment of the uncertainties in the physi- equatorial views at longitudes 0◦ (top middle) and 180◦ (top cal parameters. The inverse methods are based on Lommel- right), as well as the equatorial views at longitudes 90◦ (bot- Seeliger ellipsoids (LS ellipsoids) and convex shape models tom middle) and 270◦ (bottom right). [1-4] suitable for the analyses of sparse data. The up-to- date inverse methods are available through a web-based Gaia Added Value Interface [5]. In the present work, we consider sparse photometric data 2 Photometric model from the ESA Gaia mission made available by Gaia Data Re- lease 2 (Gaia DR2, [6]). Our focus is on asteroid (2867) Steins, Consider an asteroid in principal-axis rotation about its which has played an important role in the scientic valida- axis of maximum inertia, and denote its rotation period by P , tion of Gaia DR2 data [6]. In Sect. 2, we describe the pho- pole orientation in ecliptic longitude and latitude by (λ, β), tometric modeling that gives rise to the denition of proper and rotational phase at a given epoch t0 by '0. phase curves, phase curves that are independent of the as- The reection coecient R of a surface element on the as- teroid’s orientation during the observations. Section 3 shows teroid relates the incident ux density πF0 and the emergent the validation of the Gaia DR2 data for Steins, followed by an intensity I as assessment of the proper phase curve parameters of Steins using least squares ts and MCMC methods. Conclusions I(µ, φ; µ0; φ0) = µ0R(µ, φ; µ0; φ0)F0; and future prospects close the article in Sect. 4. µ0 = cos ι, µ = cos , (1) 1 Karri Muinonen et al. There are thus good reasons to expect that the model Table 1: The photometric G1 and G2 parameters for dierent asteroid taxonomic classes in the so-called single-parameter 1 ΦHG1G2 (α) phase function by Penttilä et al. [9] based on the photometric !P~ 11(α) = p ; (6) observations reported by Shevchenko et al. [10]. 8 Φ0(α) Class G1 G2 that results in E 0.1505 0.6005 Φ (α) 1 R (µ, µ ; φ) = 2p HG1G2 ; (7) S/M 0.2588 0.3721 LS 0 Φ (α) µ + µ C 0.8228 0.0193 0 0 P 0.8343 0.0489 can work well in asteroid phase-curve analyses. The result- D 0.9617 0.0165 ing disk-integrated brightness phase curves tend to mimic those observed for asteroids. For example, for a ctitious spherical asteroid, the phase curves now coincide with those where ι and are the angles of incidence and emergence as described by the H;G1;G2 phase function. measured from the outward normal vector of the element, It is important to note that the G1;G2 parameters in Eq. 7 refer to the multiple scattering properties of the surface ele- and φ0 and φ are the corresponding azimuthal angles. We measure φ so that the backscattering direction (or the direc- ment of an asteroid. They are thus dierent from the G1;G2 tion of the Sun) is with φ = 0◦. With the common assump- parameters derived from the direct ts of the H;G1;G2 phase function to the disk-integrated photometric data (e.g., [10]). tion of a geometrically isotropic surface, specifying φ0 is un- necessary. In the present preliminary work, nevertheless, we do not yet The Lommel-Seeliger reection coecient (subscript LS) introduce dierent mathematical symbols for the G1;G2 pa- is (e.g., Wilkman et al. [7]), rameters in Eq. 7. However, we choose to term them the proper G1;G2 parameters. 1 1 Table 1 shows the G1;G2 parameters for the asteroid tax- RLS(µ, µ0; φ) = !P~ 11(α) ; (2) 4 µ + µ0 onomic classes E, S/M, C, P, and D as derived by Penttilä et al. [9]. Here the classes E and S/M constitute a photometric where !~ is the single-scattering albedo, P11 is the single- class of shallow phase curves, whereas the classes C, P, and scattering phase function, and α is the phase angle, the an- D constitute a class of steep phase curves. gle between the Sun and the observer as seen from the ob- ject. The Lommel-Seeliger reection coecient, the rst- order multiple-scattering approximation from the radiative 3 Results and discussion transfer equation, is applicable to dark scattering media. The Gaia DR2 provides seven sparsely distributed pho- The disk-integrated brightness L equals the surface inte- tometric points for the E-class asteroid (2867) Steins (Fig. gral 2). In the literature [11,12], the rotation period is P = Z 6:04681 h and there are two possible pole orientations with the same period: (λ, β) = (94◦; −85◦) (Pole 1) and (λ, β) = L(α) = dA µI(µ, µ0; α) ◦ ◦ A+ (142 ; −83 ) (Pole 2). In what follows, we start by xing Z G1 = 0:1505 and G2 = 0:6005 in agreement with Penttilä = dA µµ0R(µ, µ0; α)F0; (3) et al. [9]. A+ We start with the Lommel-Seeliger (LS) ellipsoid inver- sion [1-3] by optimizing the rotational phase and the ellip- where A+ species the part of the asteroid’s surface that is both illuminated and visible. For a nonspherical asteroid, L soid axial ratios b=a and c=a and keeping the remaining pa- rameters xed. For Pole 1, the least-squares t is success- can depend strongly on the orientation of the asteroid with ◦ respect to the scattering plane (the Sun-Object-Observer ful with an rms-value of 0.017 mag and with '0 = 58 plane), where L is measured, giving rise to a lightcurve when (t0 = 51544:5 MJD), b=a = 0:83, and c=a = 0:46. For Pole 2, the rms-value is minutely larger at 0.019 mag, with the asteroid rotates. ◦ For a spherical asteroid with diameter D, the disk- '0 = 106 , b=a = 0:83, and c=a = 0:40. Based on the LS- integrated brightness equals ellipsoid analysis, Pole 1 emerges as a favored solution due to the more spherical shape and better rms-value. 1 Let us next incorporate the high-resolution shape model L (α) = πF D2!P~ (α)Φ (α); 0 32 0 11 0 of Steins available from a shape analysis based on the 1 1 1 disk-resolved imaging during the Rosetta mission yby Φ (α) = 1 − sin α tan α ln(cot α); (4) 0 2 2 4 and disk-integrated ground-based lightcurve observations [11,12] (Fig. 1). With the aforedescribed parameter values but where we have dened an auxiliary phase function Φ0(α) varying the rotational phase systematically with 0.1-degree normalized to unity at zero phase angle. resolution over its full range of [0◦; 360◦[, we compute the Empirically, the disk-integrated brightness phase curve of rms-value between the Gaia data and the disk-integrated an asteroid can be described by the H;G1;G2 phase function brightness from the high-resolution model. For Pole 1, we ◦ ΦHG1G2 (Muinonen et al. [8], Penttilä et al. [9], Shevchenko obtain the minimum rms-value of 0.026 mag at '0 = 244:2 ◦ et al.