The steady-state temperature across a plate placed on the surface of a planet is directly proportional to the heat coming out of the planet. With a judicious choice of plate thermal conductivity and surface radiative properties, the temperature across the plate can be maximized for the environmental conditions, and therefore the systematic errors minimized.
Here, I solve the 1-D heat equation to obtain the steady state and transient solutions. The lower boundary condition is assumed to be in perfect contact with the surface, which is at temperature Ts. The upper boundary condition is convective and radiative heat flow due to the temperature difference between the plate and the Venus environment, which is also at temperature Ts.
The temperature field is u(z,t) where the surface of the planet is at z=0. The depth of the plate is a, and the thermal diffusivity is k. The convection coefficient at the top of the plate is h, and the quantity I am solving for is Fg, the geothermal heat flux. The heat equation and boundary conditions are therefore:
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