Yesterday was Pi day, 3.1413, well close enough anyway. I should have posted this yesterday, but I was wanting to do some double checking.
A blackbody is not a gray body. A black body distributes is energy uniformly over its entire surface. In order to produce a "blackbody" for experiments, the grand masters of radiant physics had to take a few liberties and just use a fraction of a surface that approximated a black body, a blackbody cavity.
The Sun is very close to an ideal blackbody and provided the energy that goes into the cavity to stoke the furnace. Instead of a slit though, Earth has pretty large oceans that allow the solar energy in and currents to slowly mix the energy. Unlike the moon and asteroids, the liquid oceans allow for more uniform distribution of the energy so Earth can behave more closely to a true blackbody.
Without the subsurface mixing inside the box, the energy from the sun would be distributed based on the total energy available and the cosine of the latitude of spherical surface receiving that energy. Since Earth has a fluid "interior", I speculated that the distribution would diffuse based on the Square root of the cosine latitude. Then the average would be TSI*cos(L)/pi for the moon and TSI*SQRT[Cos(L)/pi] for the Earth, if it had ideal internal transfer.
While I don't have very good data on the ocean thermoclines which would be the most idea black body surface, I do have sea surface temperature data by latitude and an estimate of average insolation by latitude that I posted in Ideal versus Observation.
What I neglected to show in that post was the Ideal with mixing versus observations.
Now armed with this fine tuning of the blackbody baseline, I should be able to produce a more accurate static model with meridional flux values for a future post.