A&A 604, A27 (2017) Astronomy DOI: 10.1051/0004-6361/201730868 & c ESO 2017 Astrophysics Asteroid shapes and thermal properties from combined optical and mid-infrared photometry inversion J. Durechˇ 1, M. Delbo’2, B. Carry2, J. Hanuš1, and V. Alí-Lagoa2; 3 1 Astronomical Institute, Faculty of Mathematics and Physics, Charles University, V Holešovickáchˇ 2, 180 00 Prague 8, Czech Republic e-mail: [email protected] 2 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, 06304 Nice, France 3 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstraße, Postfach 1312, 85741 Garching, Germany Received 24 March 2017 / Accepted 1 June 2017 ABSTRACT Context. Optical light-curves can be used to reconstruct the shape and spin of asteroids. Because the albedo is unknown, these models are scale free. When thermal infrared data are available, they can be used to scale the shape models and derive the thermophysical properties of the surface by applying a thermophysical model. Aims. We introduce a new method for simultaneously inverting optical and thermal infrared data that allows the size of an asteroid to be derived along with its shape and spin state. Methods. The method optimizes all relevant parameters (shape and size, spin state, light-scattering properties, thermal inertia, and surface roughness) by gradient-based optimization. The thermal emission is computed by solving the 1D heat diffusion equation. Calibrated optical photometry and thermal fluxes at different wavelengths are needed as input data. Results. We demonstrate the reliability and test the accuracy of the method on selected targets with different amounts and quality of data. Our results in general agree with those obtained by independent methods. Conclusions. Combining optical and thermal data into one inversion method opens a new possibility for processing photometry from large optical sky surveys with the data from WISE. It also provides more realistic error estimates of thermophysical parameters. Key words. minor planets, asteroids: general – radiation mechanisms: thermal – techniques: photometric 1. Introduction (Masiero et al. 2011), which means that the size estimation from the visible brightness has a very large uncertainty. The models Optical and thermal infrared disk-integrated radiation of as- can be set to scale by disk-resolved data (Hanuš et al. 2013a), teroids is routinely used for the determination of their physi- stellar occultation silhouettes (Durechˇ et al. 2011), or thermal in- cal properties (Kaasalainen et al. 2002; Harris & Lagerros 2002; frared data (Hanuš et al. 2015). Durechˇ et al. 2015; Delbo et al. 2015, and references therein). The transition from purely reflected light to purely thermal The reflected sunlight at optical wavelengths serves as the first emission is continuous, and the transition zone where the de- guess for the size of the asteroids (from the brightness) and tected flux is a mixture of reflected solar radiation and thermal taxonomic type (color information). The periodic variations emission depends on the heliocentric distance, albedo, and ther- in brightness that are due to rotation carry information about mal properties of the surface. For asteroids in the main belt it is the shape and spin state of the asteroid. Kaasalainen & Torppa around 3–5 µm (Delbó & Harris 2002; Harris & Lagerros 2002). (2001) and Kaasalainen et al.(2001) developed a method for For longer wavelengths, the flux can be treated as pure thermal reconstructing the asteroid shape and spin state from disk- emission. Measurements of this flux can be used for a direct es- integrated reflected light. The method provides reliable results timation of the size, and sophisticated thermal models have been and is mathematically robust, which means it is not sensitive developed to reveal the thermal properties of the surface from to the noise in data. It has been successfully applied to hun- measurements of thermal emission at different wavelengths. dreds of asteroids (Durechˇ et al. 2009, 2016a; Hanuš et al. 2011, The simplest approach to thermal modeling that is often 2016; Marciniak et al. 2011, for example), and its results were used when no information about the shape is available is to confirmed by independent methods (Kaasalainen et al. 2005; assume that the shape is a sphere. Then the standard thermal Marchis et al. 2006; Keller et al. 2010; Durechˇ et al. 2011). Be- model (STM, Lebofsky et al. 1986) or the near-Earth asteroid cause the general inverse problem in case of disk-integrated data thermal model (NEATM, Harris 1998) are used. When interpret- is ill-posed, the shape is usually modeled as convex. This ap- ing thermal data by a thermophysical model (TPM), the shape proach guarantees uniqueness of the model. The output of the and spin state of the asteroid has to be known to be able to light-curve inversion method is a shape model represented by a compute the viewing and illumination geometry for each facet convex polyhedron that approximates the real nonconvex shape on the surface (and shadowing in case of a nonconvex model). of an asteroid. Because of the ambiguity between size D and ge- Usually, the 1D heat diffusion problem is solved in the subsur- ometric albedo pV, the models are scale free. The visible flux face layers. There are many different TPM codes, which differ 2 is proportional to D pV and in principle 0:01 . pV . 1:0 mainly in the way they deal with surface roughness (for a review, Article published by EDP Sciences A27, page 1 of8 A&A 604, A27 (2017) see Delbo et al. 2015). Shape models are usually reconstructed However, the Bond albedo AB as well as the geometric visi- from other data sources such as photometry (Kaasalainen et al. ble albedo pV are not material properties, and they are unam- 2002), radar echos (Benner et al. 2015), or high angular reso- biguously defined only for a sphere. Instead of this traditional lution imaging (Carry et al. 2010a). Although this approach in approach, we therefore use a self-consistent model, where the set general works and provides thermophysical parameters, the main of Hapke’s parameters is also used to compute the total amount caveat here is that the shape and spin state are taken as a priori of light scattered into the upper hemisphere, which defines the assumptions and their uncertainties are not taken into account. hemispherical albedo Ah (the ratio of power scattered into the This may lead to an underestimation of errors of the derived pa- upper hemisphere by a unit area of the surface to the collimated rameters or even to erroneous results (see Rozitis & Green 2014; power incident on the unit surface area, Hapke 2012) that is Hanuš et al. 2015, for example). needed to compute the energy balance between incoming, emit- To overcome the limitation of a two-step approach where ted, and reflected flux. This hemispherical albedo is dependent first the shape and spin model is created and then thermal data on the angle of incidence (it is different for each surface ele- are fitted, we have developed a new algorithm that allows for ment), and it is at each step computed by numerically evaluating simultaneous optimization of all relevant parameters. We call it the integral convex inversion TPM (CITPM), and we describe the algorithm 1 Z in Sect.2 and show how this method works for some test aster- Ah(i) = r(i; e; α) µ dΩ; (2) µ oids in Sect.3. 0 Ω where µ = cos e, µ0 = cos i, and the integration region Ω is over the upper hemisphere. Because the above defined quanti- ties are in general dependent on the wavelength λ, we need the 2. Combined inversion of optical and thermal bolometric hemispherical albedo A , that is, the average of the infrared data bol wavelength-dependent hemispherical albedo Ah(λ) weighted by the spectral irradiance of the Sun J (λ): Our new code joins two widely used and well-tested methods: S R 1 (i) the light-curve inversion of Kaasalainen et al.(2001), and A λ J λ λ 0 h( ) s( ) d (ii) the thermophysical model of Lagerros(1996, 1997, 1998). Abol = R 1 · (3) J λ λ We use the convex approach, which enables us to work in the 0 s( ) d Gaussian image representation: the convex shape is represented by areas of surface facets and corresponding normals. The nor- In practice, we approximate the integrals by sums. The depen- mals are fixed while the areas are optimized to achieve the dence of Ah on λ is not known from either theory or measure- best agreement between the visible light and the thermal in- ments, therefore we assume that it is the same as the dependence frared fluxes calculated by the model and the observed fluxes. of the physical (geometric) albedo p(λ) that can be estimated Moreover, the distribution of individual areas is parametrized from reflectance curves. If the asteroid taxonomy is known, we by spherical harmonics (usually on the order and degree of six take the spectrum of that class in the Bus-DeMeo taxonomy to eight). The polyhedral representation of the shape is then re- (DeMeo et al. 2009) and extrapolate the unknown part outside constructed by the Minkowski iteration (Kaasalainen & Torppa the 0.45–2.45 µm interval with flat reflectance. We then multiply 2001). The spin vector is parametrized by the direction of the this spectrum by the spectrum of the Sun, taken from Gueymard spin axis in ecliptic coordinates (λ, β) and the sidereal rotation (2004). If the taxonomic class is unknown, we assume flat re- flectance, that is, Ah(λ) = const. period P. Together with the initial orientation '0 at epoch JD0, these parameters uniquely define the orientation of the asteroid The Bond and geometric albedos can both be computed from Hapke’s parameters, but they are not directly used in our in the ecliptic coordinate frame (Durechˇ et al.