Because that is the form of the SB law used for a radiator that is less than an ideal black body but is still radiating into a vacuum...you don't seem to grasp that different forms of the equation exist for different circumstances...if you don't see a Tc term, then the equation implies that the black body is radiating into a vacuum. If you don't see an e term, then the equation implies that you are talking about a theoretical ideal black body..
Generally there is no explanation of what the equation is saying, or implying because it is assumed that you know..
Of course we're radiating into space (technically not a vacuum) ... where did you think we were radiating to? ... chocolate pudding, the persistent rash on my ankle, some magically charged æther? ... are you trying to put two black bodies together, without any space in between? ... that's about as far removed from what Stefan-Blotzmann is addressing as we can get ...
I'm sorry the Alarmists have convinced you that we can treat the atmosphere as a black body, we can't, we treat it as a fluid ... whatever's going on, we can assign a value of emissivity and use this in our equation without bothering with all the details of NS ... emissivity is a synonym of the greenhouse effect ... have you even bothered to solve the equation for this value of emissivity? ... didn't think so ...
What you don't seem to grasp is all forms of SB are algebraically identical ... if you'll remember from high school algebra, if we include a Tc term on one side of the equals sign, we have to include it on the other side ... P in this context is irradiance, somehow subtracting Tc from this makes it something
other than irradiance ... therefore yours is something
other than SB ... get it, the Earth radiates into space, no where else ...
I checked yesterday ... weather satellites are still showing IR radiation at local noon ... apparently the Earth does radiate towards the Sun ... care to explain how this is possible if it violates your version of the 2nd Law? ... maybe the dimmer switch is broken again ...