So, the final formula is:
Power = (p ^ m) * F * (c / λ) * π * r²
p - Pogson's ratio  (dimensionless)
m - magnitude  (dimensionless)
F - flux [J / (s * cm² * Hz)]
c - speed of light [cm / s]
λ - wavelength [cm]
r - Earth's radius [cm]
π - pi  (dimensionless)
Power = ( ^ ) * [J / (s * cm² * Hz)] * [cm / s] * [1 / cm] *  * [cm²]
Power = [J / (s * cm² * Hz)] * [1 / s] * [cm²]
Power = [J / (s * cm² * Hz)] * [Hz] * [cm²]
Power = [J / s] * [(Hz * cm²) / (Hz * cm²)]
Power = [J / s] = [W]
No problems here.
Using our values:
p = 2.512
m = 4
F = 3.64E-27 [J / (cm² * Hz * s)]
c = 3E10 [cm / s]
λ = 5.5E-5 [cm]
r = 6E8 [cm]
pi = 3.14
Power = (2.512 ^ 4) * 3.64E-27 * (3E10 / 5.5E-5) * 3.14 * (6E8)²
Power = 8.94E7 [W]
So… pretty close and the difference comes down mainly to rounding. Other than that, under your assumptions, I see no problems with reasoning or method. Sorry for taking so long to respond, though.
Hey, what're you up to just after January 28th of 2024? Asking for a friend.
You need to double it, that's when Centaurs would get your message.