Thermophysical Modeling of Asteroid Surfaces using Ellipsoid Shape Models Eric M. MacLennana,∗, Joshua P. Emerya aEarth and Planetary Sciences Department, Planetary Geosciences Institute, The University of Tennessee, Knoxville, TN 37996 Abstract Thermophysical Models (TPMs), which have proven to be a powerful tool in the interpretation of the infrared emission of asteroid surfaces, typically make use of a priori obtained shape models and spin axes for use as input boundary conditions. We test then employ a TPM approach - under an assumption of an ellipsoidal shape - that exploits the combination of thermal multi-wavelength observations obtained at pre- and post-opposition. Thermal infrared data, when available, at these observing circumstances are inherently advantageous in constraining thermal inertia and sense of spin, among other physical traits. We show that, despite the lack of a priori knowledge mentioned above, the size, albedo, and thermal inertia of an object are well-constrained with precision comparable to that of previous techniques. Useful estimates of the surface roughness, shape, and spin direction can also be made, to varying degrees of success. Applying the method to WISE observations, we present best-fit size, albedo, thermal inertia, surface roughness, shape elongation and sense of spin direction for 21 asteroids. We explore the thermal inertia's correlation with asteroid diameter, after accounting for its dependence on the heliocentric distance. Keywords: Asteroids, Thermophysical Modeling, Thermal Inertia 1. Introduction Multi-wavelength, photometric infrared observations of asteroids provide essential information about the thermophyiscal properties of their surfaces. A handful of both simple thermal models (Harris, 1998; Lebofsky et al., 1978, 1986; Wolters and Green, 2009; Myhrvold, 2016) and more sophisticated themophysical models (TPMs; Spencer et al., 1989; Spencer, 1990; Lagerros, 1996; Delbo' et al., 2007; Mueller, 2007; Rozitis and Green, 2011) have been established as effective means of modeling thermal infrared observations of asteroids. The general purpose of a thermal model is to compute surface temperatures for an object, which are in turn used to calculate the emitted flux at the desired wavelengths (see Delbo' et al., 2015, for a recent review). Thermophysical models have proved to be a powerful tool in providing meaningful estimates of an object's size and albedo and provide insight into thermophysical characteristics of asteroid regoliths (Emery et al., 2014; Hanu˘set al., 2015, 2018; Landsman et al., 2018; Rozitis and Green, 2014; Rozitis et al., 2014, 2018). Most simple thermal models assume an idealized (often spherical) object shape, instead of including a priori knowledge of the shape, in order to estimate the diameter and albedo of an object. However, unlike arXiv:1811.02849v1 [astro-ph.EP] 7 Nov 2018 TPMs, simple thermal models lack the ability to estimate geologically-relevant thermophysical properties such as the thermal inertia. TPMs require a spin vector - typically sourced from the Database of Asteroid Models from Inversion Techniques (DAMIT1; Durech˘ , 2010) - and object shape model as input (e.g., Hanu˘s et al., 2018). ∗corresponding author Email address: [email protected] (Eric M. MacLennan) URL: web.utk.edu/~emaclenn/home (Eric M. MacLennan) 1http://astro.troja.mff.cuni.cz/projects/asteroids3D/web.php Preprint submitted to Astronomical Journal November 8, 2018 With the recent surge in thermal infrared observations from large-scale surveys, an ever-increasing num- ber of asteroids are being observed across several epochs, building up sets of observations that span both pre- and post-opposition and often at large solar phase angles (Mainzer et al., 2011a). Such observations provide additional constraints and/or allow for more free parameters in data-modeling inversion techniques. However, the rate of thermal infrared observations significantly outpace the efforts to characterize the shapes of individual asteroids. Advantages of multi-epoch observations have previously been noted (e.g. Spencer, 1990) and been used recently to derive thermophysical and spin properties of objects (M¨ulleret al., 2011, 2014, 2017; Durech˘ et al., 2017). These studies serve as the foundation on which we build a methodologi- cal approach aimed at extracting important thermophysical properties from the large number of asteroids observed at pre- and post-opposition at multiple thermal wavelengths often acquired from thermal infrared surveys. In this paper, we outline and test a TPM approach that utilizes pre- and post-opposition multi-wavelength thermal observations when information about an object is limited. To do so, we use various simple shapes (both spherical and prolate ellipsoids) of varying spin vectors. The goal of this work is to demonstrate and establish the effectiveness of this modeling approach and compare it to previous studies. In section 2 we briefly review established thermal modeling techniques and their ability to constrain diameter, albedo, thermal inertia, surface roughness, shape and spin direction - highlighting the use of pre-/post- opposition observing geometries. We describe our thermophysical model and its implementation in section 3. In section 4 we implement and analyze the effectiveness of our approach to that of the previously reviewed works. We then apply our modeling approach and present results for 21 asteroids in section 5. In a follow-up paper, we will supplement the number of objects analyzed with this method by an order of magnitude. 2. Thermal Modeling Background and Motivation A few simple thermal models and TPMs have been developed to estimate various physical and thermo- phyphysical properties of asteroids. Many thermal models attempt to match a disk-integrated flux to a set of telescopic observational data by calculating surface temperatures for a given shape. Simple thermal models model surface temperatures by using closed-form equations (which can be evaluated in a finite number of operations) based off of the equilibrium surface temperature (Teq), which is calculated via the energy balance between the amount of absorbed insolation and emitted thermal energy: S (1 − A) 4 2 − "BσTeq = 0: (1) RAU −2 In equation (1), S is the solar constant at 1 AU (1367 Wm ; Fr¨ohlich, 2009), A is the bolometric Bond albedo, RAU is the heliocentric distance in Astronomical Units, "B is the bolometric emissivity, and σ is the Stefan-Boltzmann constant. The \beaming parameter", η, modifies the energy balance in order to account for various thermophysical effects that cause temperatures to depart from thermal equilibrium. On the other hand, TPMs compute surface temperatures using Fourier's Law of heat-diffusion (evaluated numerically). Well established simple thermal models include the Standard Thermal Model (STM; Lebofsky et al., 1986, and references within), Near-Earth Asteroid Thermal Model (NEATM; Harris, 1998), and the Fast Rotating Model (FRM; Lebofsky et al., 1978) More recently, the Night Emission Simulated Thermal Model (NESTM; Wolters and Green, 2009)) and Generalized FRM (GFRM; Myhrvold, 2016) have been created by modifying the NEATM and FRM, respectively. For reasons of applicability and flexibility, especially when analyzing large datasets, the NEATM has been used most often (Trilling et al., 2010; Mainzer et al., 2011b; Masiero et al., 2011). A brief description and history of the STM, FRM, and NEATM can be found in Harris and Lagerros(2002). The NEATM has been the preferred simple thermal model for multi-wavelength thermal surveys; due to the ease of application to single epoch observations of objects for which no shape or spin information exists. But, a TPM is the preferred tool for interpreting the multi- wavelength thermal observations of objects with shape and spin vector information. 2 2.1. Thermophysical Models Many versions of TPMs, with varying levels of complexity, have been developed throughout the past few decades. Such models explicitly account for the effects that physical attributes of the surface have on the thermal emission (Spencer et al., 1989). Subsurface heat conduction, self-shadowing from direct insolation (incoming solar radiation), multiply scattered insolation, and re-absorbed thermal radiation (i.e. self-heating) are implemented in a multitude of ways. The energy conservation expressed by equation (1) is amended to include additional terms that account for these effects: S (1 − A) solar therm dT 4 2 cos(i)(1 − s) + Escat + Eabs + k − "BσTsurf = 0: (2) RAU dx surf Where, k is the effective thermal conductivity, i is the angle between a surface facet's local zenith and sun-direction (solar incidence angle), and s is a binary factor indicating whether an element is shadowed by solar therm another facet (s = 1), or not (s = 0). The terms Escat and Eabs encompass the contributed energy input from scattered solar radiation and re-radiated thermal photons, respectively, from other surface elements. Rough topography has been modeled as spherical section craters (Spencer, 1990; Emery et al., 1998), random Gaussian surface (Rozitis and Green, 2011), and using fractal geometry (Davidsson and Rickman, 2014). The mathematics and numerical implementations of these features differ somewhat from one another. For further details, we refer the reader to the primary papers and to Davidsson et al.(2015) for a comparison of the different implementations. TPMs often use the time-dependent, one-dimensional heat diffusion equation, @T (x; t) k @2T (x; t) = ; (3) @t ρc