How to calculate the expected surface temperature of a planet

The formula

$$4 pi R ^ 2 ơ T ^ 4 = frac{pi R ^ 2 L_{sun}(1 - a)}{4 pi d ^ 2}$$

is correct, if you want to calculate the radiative equilibrium temperature. You only need to use the right units. We can further simplify the formula to

$$T ^ 4 = frac{ L_{sun}(1 - a)}{16 pi d ^ 2 ơ};.$$

You should input the luminosity in watts, the distance to the star in meters and the Stefan-Boltzmann constant as

$$σ = 5.670373 × 10^{−8} ;mathrm{W}; mathrm{m}^{−2}; mathrm{K}^{−4}.$$

The albedo is dimensionless. The resulting temperature will be in Kelvins. Let me make an example for Earth:

$d = 149,000,000,000 ;mathrm{m}$

$L = 3.846×10^{26} ;mathrm{W}$

Albedo of Earth is 0.29. (The Bond albedo should be used.) You will get

$$T ^ 4 = frac{ 3.846×10^{26}(1 - 0.29)}{16 pi times (149,000,000,000) ^ 2 times (5.670373 × 10^{−8})}=4,315,325,985 ;mathrm{K}^4;.$$

After powering this number to 1/4, we obtain temperature 256 K, which is -17° C. This looks reasonable. The real average temperature on Earth is closer to 15° C, but the greenhouse effect is responsible for the difference.

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