Radiative Transfer in Planetary Subsurfaces

The formal solutions of the equations of heat transfer in solids and radiative transfer depend on the following properties of the near-surface material the complex dielectric constant (X), thermal conductivity k (ergs per centimeter per second per Kelvin), specific heat c (ergs per gram per Kelvin), and density p (grams per cubic centimeter). The analytic theory of heat transfer at planetary surfaces begins by assuming that the temperature at any point on the surface can be expanded in a Fourier series in time  

Equation (21) represents a series of thermal waves propagating into the surface and attenuating with distance. The higher harmonics are attenuated more rapidly than the lower harmonics since increases as the square root of the harmonic number n. The attenuation and phase of each harmonic depend on the quantity Pi, which is termed the thermal absorption coefficient of the planetary material. The reciprocal of(Lt = l/ i) is termed the thermal skin 

Given the thermal absorption coefficient and the boundary conditions on the heating, it is possible to determine the constants of temperature (To and T ) and phase ( ) in Eq. (21). The inverse problem is faced by the radio astronomer, namely to determine the thermal absorption coefficient from measurements of the thermal emission. This is done in the following manner. The temperature distribution given by Eq. (21) is used in the equation of radiative transfer 

The parameter Si can be written in terms of the physical characteristics of the surface material as follows  

Radio observations by themselves do not permit separation of the physical parameters contained in 5i and 4 i. Nevertheless, the radio data, when combined with infrared data, radar data, and laboratory data for real materials, constrain the material properties and in some cases allow one to exclude certain classes of materials in favor of others. 

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