[deleted fancy math and explaination that doesn’t apply]
So, for a given A and S0, we can find the effective temperature. In the case of the Earth, the albedo (A) is about 0.3, so 1-A is 0.7, which magically explains where that factor comes from that Gerlich and Tscheuschner couldn’t explain. The factor of 4 is just a consequence of the fact that the Earth can only absorb radiation on the side facing the sun, but emits in all directions. When the values are plugged in, we (and Gerlich and Tscheuschner) get a value of -18 C.