Shell and Body Should not be Difficult to Grasp

It really puzzles me why this concept is so hard for some to grasp.  A gray body has two components, a shell and a body.  The shell by definition is isothermal, there cannot be any perpendicular transfer of energy, only in/out transfer.  To estimate the energy of a gray or black body from space, you use the simple relationship TSI*(1-albedo)/4 to determine Ein which since the definition of the body limits internal heat transfer will always equal Eout.

The shell is the part of the gray body that TSI*(1-albedo)/4 applies to, not the true "body" of the object.    Since the "body" has to have some degree of perpendicular heat flow, the average energy available at that surface is TSI*cos(lat)/pi times the Cos(lon relative to local solar noon)  Then allowance can be made for limits in transmittance to that point on the spherical body.

Anyone that has any interest in designing a solar pond is aware of the "average" insolation and the "peak" insolation for the pond location which are considered along with the thermal capacity and layer gradients to determine the "operating" temperature of the solar pond.

Once you move beyond the simplistic ideal approximations you can begin to actually understand the minimal gray body considerations that need to be made.