(c) 2006 by Barton Paul Levenson
Under certain very restricted conditions (no scattering, homogeneous medium, etc.) "Beer's Law" describes what happens to a beam of radiant energy:
|dI = -I k ρ ds||<1>|
where dI is the amount the beam is diminished, I is the intensity of the beam to begin with, k is an "absorption coefficient" of the medium under discussion (say, air), ρ is the medium's density, and ds the change in distance traveled. Beer's Law is also known as Lambert's Law, Bouguer's Law, and the Beer-Lambert-Bouguer Law, but it's a heck of a lot simpler just to call it Beer's Law.
If we divide both sides by I and integrate, we get
|ln I/I0 = -k ρ s||<2>|
|I = I0 e-k ρ s||<3>|
Let's work an example. Say a beam of 240 Watts per square meter per steradian intersects a mass of carbon dioxide and travels along it for one meter (s = 1). The CO2 has an absorption coefficient (at the wavelength we're discussing; everything so far is monochromatic) of k = 2.5 square meters per kilogram. The CO2 is as dense as air (a 100% CO2 atmosphere), ρ = 1.293 kilograms per cubic meter. We then have
|I = 240 e-2.5 1.293 1||<4>|
which works out to I = 9.47 Watts per square meter per steradian. The CO2 has absorbed nearly all of the incoming light, 96% of it.
Now, there are ways to simplify equation <3> and make it more general. For one thing, the exponential term (e-k ρ s) is the "fractional transmissivity" T:
|I = I0 T||<5>|
But that kind of loses most of the details. Another way to do this is to define the absorbtion coefficient times the density times the distance (k ρ s) as the "optical thickness" τ (tau). We then have:
|I = I0 e-τ||<6>|
Optical thickness tau is a dimensionless measure of how much a given medium retards the passage of light. Intensity falls by a factor of e (2.7182818...) when τ = 1. Optical thickness is also known as optical path or optical depth, though the latter is often used to mean only the strictly vertical optical thickness. Optical depth becomes important when you're talking about vertically stratified atmospheres (or oceans).
Remember, the following three factors determine the optical thickness:
|1)||k, the absorption coefficient of the material. For infrared light in the Earth's atmosphere, the greenhouse gases have the highest absorption coefficients; while nitrogen and oxygen barely absorb infrared light at all.|
|2)||ρ, the density of the absorbing material.|
|3)||s, the distance traveled by the beam.|