A blackbody refers to an object that absorbs all the light and radiation that falls on it, without reflecting or transmitting any of it. This means that a blackbody appears completely black and emits radiation solely based on its temperature.
You objected that liquid nitrogen temperatures could be higher than the effective blackbody temperature of deep space, which is about 3K. (judithcurry.com)
Why not start with a pure theoritical 24 hour rotating blackbody with 1/2 recieving full spectrum 1370 watts, from an object of approximately 30 arc min. (realclimate.org)
Finally if I am not mistaken the temperature of the top layer of an atmosphere consisting of n blackbody shells should be Tg / -LRB-(n +1) **.25) not Tg / (2. (realclimate.org)