Consider a planet of size Rp at a distance D from the Sun. Define the albedo (a) of a planet to be the fraction of incoming energy that is reflected off the planet (albedo can be as small as 0, or as large as 1). Question: How does albedo relate to the amount of energy absorbed by a planet?
With this definition, the amount of energy a planet receives from the Sun every second is given by
Now, how much energy is the planet giving off each second (not reflecting, but actually emitting)?
In thermal equilibrium, these rates are in balance:
Equating these two rates and doing some algebra, we find that
Note the dependencies, and that Tp does not depend on Rp. Why??
Let's try this for Venus. Putting in the numbers (a=0.6, Tsun=5770K, Rsun=7x105 km, D=0.72 AU) we get the equilibrium temperature of Venus = 260 K.
The surface temperature of Venus = 740 K. Whoa! Where'd we mess up??
For that matter, the equilibrium
temperature of the Earth is 255 K (or ~ -1 F). Something's