Equilibrium Temperatures and Directional Emissivity of Sunlit Airless Surfaces With Applications to the Moon

Abstract

Solar irradiance dominates the heat flux incident on airless planetary bodies. In thermal equilibrium, surface roughness affects the temperature distribution by changing the incidence angle local to each slope. In order to simulate temperatures and thermal emissions at different phase angles, existing thermophysical models usually employ computationally expensive techniques such as ray tracing. Here we derive the equilibrium surface temperature distribution of sunlit Gaussian rough surfaces, providing an exact solution for the Sun at the zenith and an approximate solution for the general case. We find that although the slope distribution of realistic airless surfaces is often non‐Gaussian, their temperature distribution is well modeled assuming a Gaussian slope distribution. We additionally present closed‐form expressions that describe the radiation emitted from rough surfaces at different emissions angles and employ them to radiometrically estimate the roughness of the lunar surface using measurements obtained by Lunar Reconnaissance Orbiter (LRO) Diviner. Our model may also be applied to studying the roughness of resolved and unresolved surfaces on other airless planetary bodies.

Theoretical Modeling and Analysis of Directional Spectrum Emissivity and Its Pattern for Random Rough Surfaces with a Matrix Method

“…The patterns shown in Figure 9b are reported in experimental studies [14,29]. It is shown in Figure 9 that ε F and ε have similar patterns.…”

“…Theoretical modelling studies have been carried out to predict the emissivity of rough surfaces. This was predicted considering the shadowing effect [14]. The surface was however assumed as a grey body, which limits its application.…”

The Moon at thermal infrared wavelengths: a benchmark for asteroid thermal models

“…Their data did not show any clear variation in roughness between different regions or surface units. However, Rubanenko et al (2020) used telescope (Sinto 1962) and LRO Diviner data (Bandfield et al 2015) to produce a detailed lunar surface roughness map. They list 30.2 • ± 5.9 • (Maria) and 36.8 • ± 4.4 • (Highlands) as representative values for the bidirectional rms slope at the thermal insolation scale, consistent with results presented by Bandfield et al (2015) and Rozitis & Green (2011).…”

“…If we assume that the lunar maria spectra are more relevant (darker zones are hotter and contribute more to the thermal emission at these wavelengths) then this would point to a strongly wavelength-dependent surface roughness (low roughness values at short wavelengths and high values at longer wavelengths) that is unphysical. The emission measured by HIRS originates from the very top few millimeters to centimeters of the surface (the thermal skin depth is ≈1 cm) and baselines up to several kilometers play a role for the relevant roughness properties (see also Rosenburg et al 2011;Rozitis & Green 2011;Rubanenko et al 2020). Rozitis & Green (2011) summarized lunar roughness studies on different scales, including results from radar measurements.…”

“…At positive phase angles, the HIRS measurements sample mainly "morning" illuminated slopes, whereas the negative phase angles sample more afternoon-evening slopes, which are slightly hotter than the morning ones. Additional contributions might also be related to albedo (Vasavada et al 2012) or roughness properties (Rubanenko et al 2020) that show regional variations over the lunar surface. At large negative phase angles the illuminated fraction of the surface has probably a lower mean albedo, leading to higher temperatures and a few percent higher (than predicted) fluxes and vice versa.…”