A certain planet of radius R is composed of a uniform material that, through radioactive decay, generates a net power P. This results in a temperature differential between the inside and outside of the planet as heat is transfered from the interior to the surface.
The rate of heat transfer is governed by the thermal conductivity. The thermal conductivity of a material is a measure of how quickly heat flows through that material in response to a temperature gradient. Specifically, consider a thin slab of material of area A and thickness \Delta x where one surface is hotter than the other by an amount \Delta T. Suppose that an amount of heat \Delta Q flows through the slab in a time \Delta t. The thermal conductivity k of the material is then
It is found that k is approximately constant for many materials; assume that it is constant for the planet.
For the following assume that the planet is in a steady state; temperature might depend on position, but does not depend on time.
a. Find an expression for the temperature of the surface of the planet assuming blackbody radiation, an emissivity of 1 , and no radiation incident on the planet surface. You may express your answer in terms of any of the above variables and the Stephan-Boltzmann constant \sigma.
b. Find an expression for the temperature difference between the surface of the planet and the center of the planet. You may express your answer in terms of any of the above variables; you do not need to answer part (a) to be able to answer this part.