# The Greenhouse Effect: How It Works

(c) 2016 by Barton Paul Levenson

## Introduction

How the greenhouse effect actually works:

1. Sunlight makes it through the atmosphere largely unimpaired and heats the ground.
2. The ground radiates infrared light back up.
3. Greenhouse gases in the air absorb infrared light.
4. The greenhouse gases warm up, and radiate.
5. Some of the radiation from the greenhouse gases goes back to the ground. Thus the ground has both sunshine and "airshine" (atmospheric back-radiation) heating it, and is warmer than it would be without the greenhouse gases.

The atmospheric back-radiation can be measured with a device called a pyrgeometer, in use since at least 1954. In 2004 and 2006, spectral definition studies of the back-radiation showed that the amount coming from greenhouse gases specifically had increased since 1970. This was the smoking gun that it was increased greenhouse gases that had been warming the Earth.

Those of you who hate math can quit now, if you like. All the essentials are in what I just said above.

## The Stefan-Boltzmann Law

To examine the greenhouse effect quantitatively, we need some elementary radiation physics. The first principle of interest is the Stefan-Boltzmann law, discovered by Joseph Stefan in the 19th century and explained by Ludwig Boltzmann later in the same century. This states that the electromagnetic radiation given off by a warm body is proportionate to the 4th power of its absolute temperature. In modern form:

F = ε σ T4

Here F is the flux density, in watts per square meter.

ε is the "emissivity," or radiative efficiency, which can vary from 0 to 1. A body with ε = 1 at all wavelengths is a perfect radiator; a "blackbody." One with ε = k at all wavelengths, where k < 1, is a "graybody."

σ is the Stefan-Boltzmann constant, with a value of 5.670373 x 10-8 Watts per square meter per kelvin to the fourth (a kelvin being a degree on the absolute scale--same size and shape as a Celsius degree). T is the absolute or Kelvin-scale temperature. On the Kelvin scale, absolute zero, where all molecular motion ceases (except for quantum effects), is at T = 0. Water melts (or freezes) at 273.15 K and boils at 373.15 K. Earth's mean global annual surface temperature is about 288 K, and human core temperature is about 310 K--more when you have a fever.

We also need the 'radiation identity.' Light passing through a medium (like air, or water, or a block of glass) can have five possible fates:

1. It can cause a chemical change (like zapping molecular oxygen to make ozone and atomic oxygen).
2. It can cause a phase change (like evaporating water).
3. It can be reflected away (in three dimensions, "scattered").
4. It can be absorbed, heating the substance (as when blacktop heats up in the sunlight).
5. It can be transmitted, like light going through a window.

For a medium that isn't changing, only the last 3 need be considered, and we have the "radiation identity:"

A + R + T = 1

where A is the fractional "absorptance," R the "reflectance," and T the "transmittance." Some examples:

An asphalt parking lot has high absorptance, but little reflectance or transmittance.
A mirror has high reflectance, but little absorptance or transmittance.
A glass window has high transmittance, but little absorptance or reflectance.

Now we can do the "glass slab" model of Earth's climate system...

## The Glass Slab Model

The "glass slab" model is a really simplistic model, but it's useful to illustrate some basic concepts. It has three components stacked in a column: Space, air beneath that, and the ground beneath that.

Now, we divide the electromagnetic spectrum into two parts: visible light (high frequency, short wavelength), and infrared (low frequency, long wavelength). Better names would be "shortwave" and "longwave," but what the heck. In each range, we have absorptance, reflectance, and transmittance, and with subscripts v for visible and i for infrared, these become Av, Rv, and Tv, versus Ai, Ri, and Ti.

The air is assumed to be transparent to visible light, but opaque to infrared. It has Av = 0, Rv = 0, Tv = 1.
On the other hand, it is opaque to infrared: Ai = 1, Ri = 0, Ti = 0.

The ground absorbs everything: Av = 1, Rv = 0, Tv = 0. Also Ai = 1, Rv = 0, Tv = 0.

Now. Some astronomy. The climate flux density, F, absorbed by the Earth system is

F = (S / 4) (1 - A)

where S is the Solar constant (about 1360 watts per square meter) and A is the planetary albedo (about 30%). This gives F = 238 watts per square meter, corresponding to Earth's "radiative equilibrium temperature" of 255 K. Remember the Stefan-Boltzmann law in the last post. F and T can be calculated from one another.

So space emits F = 239 W m-2 down.

The ground, at surface temperature Ts, emits G W m-2 up. We don't know what G is yet.

The air absorbs nothing of F, but all of G. It then radiates A both up and down, since it has both a top and a bottom. Since energy must be conserved, A must be half of G.

For space, input is A (half the output from the air), and output is F.
For air, input is G, output is 2 A.
For Earth, input is F + A (it gets sunlight, plus half the radiation from the air), and output is G. In equations:

A = F
G = 2 A
F + A = G

We know F is 238 W m-2, so A must also be 238 W m-2. From the second equation, G must be twice this, or 476 W m-2. Back-calculating from the Stefan-Boltzmann law, we find the temperature of the air to be 255 K, and the temperature of the Earth to be 302 K. The greenhouse effect is Tg = 302 - 255 = 47 K.

Earth's actual surface temperature is 288 K, for a greenhouse effect 288 - 255 = 33 K, so we overshot. As I said, this is an oversimplified model. But it gives the basic idea. With gases in the air that can absorb infrared light from the ground (and from other layers of air), those gases will heat up and radiate, and some of the radiation will go back down to the ground, heating it further. That's the greenhouse effect.