## Radiative Transfer

##
Planetary Equilibrium Temperature:

For the earth, the atmospheric albedo, which is entirely produced by
reflections from cloud tops, is approximately 0.3.

What we wish to do is to derive a simple
expression for how the atmosphere modifies the tempearature of a
planetary surface

Before doing this, we need to make some assumptions about our
atmosphere.

- The atmosphere is Thin

- The atmosphere is supported by pressure equilibrium (hydrostatic
equilibrium)

- The atmosphere is isothermal

- The equation of state is the Ideal Gas Law

These assumptions allow us to treat the atmosphere is a thin, uniform
slab of material at constant density and temperature.

First some constants:

- Solar Constant:

- F
_{o} =
Flux at the top of the Earth's atmosphere =
1370 watts per square meter (W/m^{2})

- Stefan-Boltzmann constant s = 5.67 x 10
^{-8}

Going back to the planetary equilibruim temperature we see that

4σT^{4} =F_{o}(1-A)
or

T^{4} = ((1/4*1370)/σ)(1-A)

T = 278*(1-A)^{1/4}

For A = 0.3 one gets T = 254K

*One Layer Model* (e.g. our atmosphere)
We assume the following:

- The temperature of the surface of the planet is constant
- The atmosphere is a shell of negligible thickness
- The atmosphere is isothermal
- All the shortwave radiation that is not reflected reaches the planet's surface
- All the longwave radiation emitted by the planet is absorbed by the atmosphere

Energy Balance in the One Zone Atmosphere

F_{o} = incident flux

T_{s} = transmission percentage of short wavelength incoming radiation

T_{t} = transmission percentage of outgiong long wavelength radiation

F_{g} = Flux from ground

F_{a} = Flux from the atmosphere.