First, I thought the global warming is inevitable since only defense from Earth’s perpective would be to reflect the incoming energy back to space and if that’s not possible, the Earth would just accumulate energy from the Sun (neglecting internal nuclear Earth’s sources of energy).
Thus the question is – what is a spherical black body temperature one astronomical unit away from the Sun?
The power received from Sun at 1 astronomical unit is
By the Stefan-Boltzmann radiation law the radiated power is
Together, the equilibrium temperature is thus
Feeding this into Wolfram Alpha (reducing incoming power by Earth albedo), we obtain the equilibrium temperature of Earth $T \approx -25 °C$.
This is clearly a very naïve calculation. Earth is not in a thermodynamic equilibrium, its internal sources probably aren’t negligible and this simple model cannot explain nor predict global warming :-(
Is there an error? Send me a message (e.g. on Github or an e-mail).