I accidently posted this before it was finished.
I came across a relatively simple formula for estimating
the emitting surface temperatures of planets based on the
Stefan–Boltzmann law, which states that total thermal
radiation is proportional to the fourth power of the absolute
temperature.
R1 is the radius of a star.
R2 is the radius of a planet's orbit.
A is the planet's albedo or reflectance.
T1 is the absolute temperature of the star's surface.
T2 is the average absolute temperature of the planet's
emitting surface, where thermal radiation from the planet
can escape into space. For a planet with an atmosphere
this is often referred to as the "optical surface".
T2= (the fourth root of ((((R1/R2) squared) (1-A)) x.25)) x (T1)
In the case of Earth:
R1, the Sun's radius is 696,000 km.
R2, the Earth's orbital radius is 149,600,000 km.
A, the Earth's albedo is .296
T1, the absolute temperature of the Sun's surface is 5,778 K
Putting these values into the formula, R1/R2 = .0047
Squaring this gives .0000216, the ratio of the radiation intensity
at the Sun's surface to the radiation intensity at the Earth's orbit.
Dividing this by 4 corrects for the fact that the Earth absorbs
solar radiation over its circular cross section but emits infrared
radiation over its entire surface area,
.0000216/4 = .00000541
Multiplying this by (1 - A) corrects for the fraction of solar
radiation that is reflected back into space as light and does
not contribute to warming the planet:
.00000541 x (1 - .296) = .00000381
This is the ratio of the intensity of the infrared radiation
emitted from the Earth's surface to the radiation intensity
at the Sun's surface.
Taking the fourth root of this ratio gives the ratio of the
absolute temperature of the Earth's emitting surface,
or optical surface, to the temperature of the Sun's surface:
The fourth root of .00000381 = .0442
.0442 x 5,778 K = 255 K, which is about 18 degrees below zero
Celsius, or zero degrees Fahrenheit. 255 K is in fact a commonly
accepted value for the emission temperature of the Earth. As
they say, that's close enough for government work. Of course,
temperatures further down in the atmosphere average higher
than this due to the atmospheric greenhouse effect.
This presumes a near-circular orbit; a planet with a notably
eccentric orbit would be subject to thermal lag. Hal Clement
had to take this into account in creating the planet Mesklin
for his novel _Mission of Gravity_.