how
planets got their temperatures
1) The diagram shows a planet with radius R at distance d from a star
with luminosity Lstar.
what fraction of light does the planet intercept?
fraction of star's light that is intercepted
by the planet
= (p R2)/(4p
d2)
2) But not all of the light intercepted is absorbed!
A
= albedo = fraction of light reflected by the planet
(1 - A) = fraction of light that is intercepted by
the planet that
is also absorbed
fraction of star's light that is absorbed by the
planet
= (1 - A) (p R2)/(4p d2)
3) Therefore the rate at which energy is absorbed by the planet (in
other words, the energy gained per time) is
energy absorbed/time by a planet
= Lstar (1 - A) (p R2)/(4p
d2)
4) But because planets are solid or liquids, and therefore blackbodies,
we also know the energy
emitted per time; it's the same formula (the
Stefan-Boltzmann law) as for stars:
energy emitted/time by a planet
= 4 p
R2 s T4
5) If the planet is in equilibrium (constant temperature, in other
words),
the energy emitted and the energy absorbed per
time must match:
4 p R2
s T4
= Lstar (1 - A) (p R2)/(4p
d2)
or
T4
= Lstar (1 - A) / (16 s p d2)
6) Some sample calculations
planet
distance from sun
albedo
predicted
temperature actual temperature
Venus
1.1 x 1011
m
.59
260 K
733 K
Earth
1.5 x 1011
m
.39
250 K
287 K
Saturn
1.5 x 1012
m
.44
75 K
90 K
(cloudtops)