for ε = 1, S = 1365 W/m2, and α = 0.3. The factor of four comes from accounting for the spherical, spinning Earth (see any climate science textbook, chapter 1 or 2).
So this is interesting:
"The spherical Earth assumption gives the well-known So/4 expression for mean solar irradiance, where So is the instantaneous solar irradiance at the TOA. When a more careful calculation is made by assuming the Earth is an oblate spheroid instead of a sphere, and the annual cycle in the Earth's declination angle and the Earth-sun distance are taken into account, the division factor becomes 4.0034 instead of 4."https://ceres.larc.nasa.gov/documents/DQ_summaries/CERES_EBAF_Ed2.8_DQS.pdf, pg 7.
It'd be fun to calculate this, someday, when I have the time. But unfortunately I don't have it now.
When I was an undergraduate, I took undergraduate Classical Mechanics in my junior year, from a really great professor at UNM. Dr Finley taught many of the upper division undergraduate classes I took, and I learned more from him than any other teacher in my life, including in his graduate-level special relativity class my senior year, where he introduced us to four-vectors and tensors and their notations.
Dr Finley was fantastic. One of the most memorable things he did was, in junior year classical mechanics, introduce us to perturbation theory (and special functions) by calculating the gravitational field for a nonspherical Earth. First we did the oblate spheroid, but even better was for the pear-shaped Earth, a more realistic model of our planet.
These were the same calculations NASA had to do to launch rockets or send one to the Moon.
The UNM classroom we always used had chalkboards on all four sides of the classroom, and chairs/desks that swiveled. He'd start over on the far one side of the classroom, and by its end we'd all swiveled our desks 360 degrees to follow what him had calculated, all around the classroom.
I don't remember now exactly what special functions these perturbative calculations required -- Legendre polynomials, I think. But it was a wonderful introduction to not just realistic classical mechanics, but perturbation theory, which then was very handy later when learning quantum field theory, where scattering cross sections are calculated (for QED) one order of perturbation, in α (≈ 1/137), at a time.
When the department ordered a new computer -- this was 1981 -- and put it in his office, he let a good friend and I unpack it and set it up. Then he let us play with it. This was the day before Thanksgiving, and my friend Norman and I sat there for about 18 hours, figuring it out and its programming. IIRC, we calculated the scattering of electrons from various crystal types, ending up with a 2-D surface where the electrons had landed after scattering. When Dr Finley came back in the next morning, Thanksgiving morning, to pick something up, around 10 am, we were still there programming, having been up all night. He encouraged us to go home. I rode my bike the 10 miles back to my parents' house, and I think I slept all the way through Thanksgiving dinner.
Sometimes you only recognize when you were happy much after the fact. Or maybe you just forget all the problems you had then. Does the difference really matter, decades afterwards?