Re 392 Chris Dudley — I don't understand what you mean by R ^ 2T ^ 4 — and there should be something about how optical depth is proportional to R, and also, if you're going a significant distance toward the center of such an object, there is the issue of spherical geometry; if the optical thickness is large enough across small changes in radius, then you don't need to account for the spherical geometry in the calculation of the flux per unit area as a function of the temperature profile and optical thickness; however, the flux per unit area outward will
drop as an inverse square, except of course within the layers that are being heated through a different process (SW heating for a planet, radioactivity, latent and sensible heat loss associated with a cooling interior, gravitational potential energy conversion to enthalpy
via compression (adiabatic warming) and settling of denser material under
gravity (the later both leads to compression
via increased pressure
via increased
gravity within the interior, and also is a source of kinetic energy which can be converted to heat)...