Roughness of planetary surfaces: Hapke theory and statistical multi-facet algorithm applied to dwarf planet Ceres and comet 67P data
Authors: Raponi Andrea , Ciarniello Mauro , Filacchione Gianrico , Capaccioni Fabrizio , De Sanctis Maria Cristina , Frigeri Alessandro 
Published: September 2022

&lt;p&gt;&lt;strong&gt;Introduction:&lt;/strong&gt; This study is focused on the investigation of the reflected solar light on planetary atmosphere-less bodies, as a function of the viewing geometry (direction of the solar light and direction of the ob-server with respect to the normal to the surface, identifying angles named respectively incidence and emission, and the relative angle between them, named phase). In particular, we focus on the photometric properties of large-scale roughness (hereafter referred to as roughness), which, being not spatially resolved, can be only derived by photometric models. The investigation of the photometric response as a function of roughness is mandatory to disentangle properties of the surface regolith such as albedo, porosity, and thermal inertia (the latter when thermal emission data are avail-able). Moreover, the information on the roughness can be important by itself for interpretation of the geological and physical processes in place, or for engineering use in a case of a landing site selection.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Photometric Models:&lt;/strong&gt; Among the interpretative physically-based models describing the reflection of the light from surfaces, Hapke [1] theory is among the most used in literature, thanks to its completeness in the description of the surface, and the practicality of use, being an analytical model. According to Hapke, a rough surface can be described as a collection of facets, each with a certain slope (&amp;#952;), in a way that the distribution of slopes is completely identified by a unique parameter (the Mean Slope &amp;#952;). Random Gaussian and fractal terrains are well-represented by the slope distribution described by Hapke&amp;#8217;s theory. This is the case with several minor bodies, targets of recent space missions (e.g., [2]). Hapke developed an analytic formulation to include the effect of the roughness into a general relation describing the reflectance of planetary surfaces, which is a function of several photometric parameters. However, some authors, including Hapke himself, have high-lighted some limitations of this formulation: (i) the parameter that describes roughness is derived in theory only for small angles of the slope of the facets relative to the average surface. This prevents reliable derivation of roughness for more general conditions [1]; (ii) unresolved shadows and self-illumination can affect the overall reflected signal, and if not accurately modeled, can be confused with albedo variation (e.g., [3]). In Hapke&amp;#8217;s theory, the dependence of these contributions on observation geometry is treated by means of simpli-fied approximations.&lt;/p&gt;
&lt;p&gt;An improved approach to the study of rough surfaces has been proposed to overcome the above issues [4]. It consists of a statistical approach in which N unresolved facets, each one having its own viewing geometry, are generated. The integrated resulting signal corresponding to these facets is calculated starting from the response of all facets visible to the observer. This statistical-multi-facets algorithm (SMFA) also takes into account the self-illumination and shadows projected along the observer's line of sight, so that the photometric response is not calculated through analytical ap-proximations, but with the sum of each of the N facets that describe the simulated terrain (see Fig. 1).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Roughness retrieval:&lt;/strong&gt; We investigate the properties of spatially resolved roughness of the dwarf planet Ceres and comet 67P from their shape models, derived by Dawn/NASA and Rosetta/ESA data. From this, we infer the possibility to safely apply the above-mentioned models for retrieval of non-resolved roughness. The latter is performed by the comparison of signals of the surface detected in different viewing geometries. We discuss results coming from both Hapke model and the statistical multi-facets algorithm.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;591&quot; height=&quot;393&quot; /&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Figure 1.&lt;/strong&gt; An example of the different output between SMFA and Hapke&amp;#8217;s models: S = shadowing function, &amp;#181;&lt;sub&gt;0eff&lt;/sub&gt; = effective incidence angle cosine, &amp;#181;&lt;sub&gt;eff &lt;/sub&gt;= effective emission angle cosine. All these parameters are used by the Hapke formulation, and they are functions of the Mean Slope (&amp;#952;) parameter. In this example, the incidence, emission, and phase angles are fixed respectively to 30&amp;#176;, 0&amp;#176;, 30&amp;#176; [4].&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;[1] Hapke, Theory of Reflectance and Emittance Spectroscopy, 1993&lt;/p&gt;
&lt;p&gt;[2] Davidsson et al., Icarus, 252, 2015&lt;/p&gt;
&lt;p&gt;[3] Cuzzi et al., Icarus, 289, 2017&lt;/p&gt;
&lt;p&gt;[4] Raponi et al., 14th EPSC 2020, id. EPSC2020-761&lt;/p&gt;