From the Sun, the Earth looks like a flat disk with radius Re. The area of that disk is:
pRe2
Using the solar constant(1.35Kw/m2) and assuming that the Earth's albedo(reflectivity) is 30%, the available power at the Earth's orbit is:
Pabsorbed = (1 - 0.3) (1.35 kW/m2) (pRe2)
To be in equilibrium, so that the temperature is stable, the Earth must radiate the same power as it absorbs. The temperature of a blackbody that radiates at this power level can be derived from the Stefan-Boltzmann law:
Pradiated = (5.67 x 10-8 W/m2K4) T4 (4pRe2)
Note that the power radiated is related to the fourth power of temperature, and that this power is radiated over a spherical surface representing the Earth.
Solving for T4 gives:
T4 = ( 0.7 1350W/m2 ) / (4 x 5.67 x 10-8 W/m2K4) = 4.17 x 109 K4
Or,
T = 254 K
Note that the temperature at the top of the Earth's atmosphere is 255 K, while it averages 287 K at the Earth's surface.