Thermal motion of the charged particles in matter causes electromagnetic radiation,
called thermal radiation
from its cause, though it is no different from other electromagnetic radiation. This
implies that matter will also absorb electromagnetic radiation. In a closed system,
emission and radiation will equilibrate at some temperature T. Since the emission and
absorption properties do not depend on the arrangments, these properties will hold also
when there is no equilibrium. A sample of matter at temperature T in space will cool
gradually to 0K if there is no radiation present to absorb.

The emissivity e of
a surface is the energy radiated per unit area per unit time, with units W/m².
It is usually assumed that the radiation is Lambertian, or
diffuse. The absorptivity a of a surface is the fraction of incident radiation absorbed by the surface,
also assumed to be Lambertian.
The assumption of diffuse emission and absorption is not fundamental; it just makes the
discussion simpler. Both e and a may be functions of the frequency or wavelength of the
radiation. By considering radiation exchange in thermal equilibrium, Kirchhoff demonstrated that e/a was a universal
constant at any frequency and temperature equal to the emissivity of a perfect absorber,
or black body, with a = 1. That is, a good absorber is also a good emitter.

The spectrum of thermal radiation is illustrated at the right. The abscissa is the ratio x = hf/kT = hc/lambda by kT,
proportional to the frequency. hf is the quantum energy of radiation of frequency f, or
wavelength lambda = c/f, and kT is the average thermal energy per equivalent harmonic
oscillator (two “degrees of freedom,” one for kinetic
energy and one for potential energy). k is Boltzmann’s constant, the gas constant per
molecule. The amount of radiation decreases rapidly at both small and large frequencies
with a maximum at x = 2.82. This remarkable and useful formula was discovered by Max Planck, and was the beginning of quantum theory.

The maximum of the curve occurs at x = 2.82, or hf = 2.82kT, from which we can find
that lambda by T = 5.102 x 10⁶ nm-K. This is the peak of
the emission per unit frequency interval. Since
frequency and wavelength intervals are related by df = – dlambda/lambda², the energy density per unit wavelength
interval is proportional to x⁵/(e^x – 1), where x = hc/lambda by kT. The maximum of this curve is at x
= 4.96, which gives lambda_maxT
= 2.901 x 10⁶ nm-K. The rule that lambda_maxT = constant is called Wien’s Law, in either case. The maximum depends on which
expression for the spectrum that you are using, and really has no absolute significance by
itself, but is only useful for comparison. Although the wavelength interval result is more
often quoted, the frequency interval result may be more meaningful. For solar radiation at 5750 K, the maxima are at 505 nm and 887 nm,
respectively.

If a is independent of frequency, then the total energy radiated per unit area per unit
time is W = sT⁴,
which is Stefan’s Law. This
relation was known long before Planck’s formula, and is a consequence of classical
thermodynamics. The value of s was determined in terms of universal constants by Boltzmann. In calorie units, it is 8.14 x
10^-11 cal/cm²-min-K⁴. In SI, it is 5.6705 x 10^-8
W/m²-K⁴.

Many simplifying assumptions have been made here, such as neglecting radiative transfer
with the environment at some lower temperature T” <‘ T’, but the general mechanism
is clear. Per square metre, W = sT⁴, if we assume the
emissivity is 1. Actually, it is sufficient to assume that the emissivity is the same for
all surfaces involved, not necessarily 1. Then, W = sT’⁴, and 2W = sT⁴. When the energy leaving the glass equals that absorbed,
s(T⁴ – sT’⁴) = sT’⁴ = W, or T⁴ = 2T’⁴, which means that T = 2^1/4T’
= 1.189 T’. Since the glass must be hotter than its surroundings, the interior of the
greenhouse is still hotter. If we know W, then we can find both T’ and T.

The earth absorbs the short-wave sunlight falling on its projected disc of area pR², and radiates long-wave radiation from its surface area 4pR². The long-wave radiation per unit area must then be
1/4 the short-wave radiation incident per unit area. The incident short-wave radiation is
about 1.94 cal/cm²-min, which works out to 1353 W/m². Let’s assume
that half of this is absorbed by the earth, or 676 W/m². The flux that must be
radiated by every square metre is a quarter of this, or 169 W/m², presuming
that the earth reaches a uniform temperature. This is not nearly as bad an assumption as
would be imagined, since the earth is actually at about the same temperature all over,
thanks to the oceans and atmosphere. If the long-wave emissivity is assumed to be unity, also not a bad assumption, then 169
= sT⁴, or T = 235K or -38°C, if the
atmosphere were transparent to long-wave radiation. This is not far from the truth.

It is remarkable that our approximate estimate is close to actuality. The accurate calculation of the radiation balance of the atmosphere is extremely
difficult because of the variability of the earth’s surface, the differences in cloud
cover, and other factors. In fact, in spite of computers, only simplified models are
practical, and there is really not a good handle on this important factor, just
enthusiastic scientists waving papers in the air and shouting at each other. The general
effect has been well-known for a considerable time. Fortunately for us, it does not depend
too greatly on the concentration of the trace gases responsible for the long-wave
absorption, of which water vapor is the most important, and also quite variable.

The earth generates heat internally, and this internal source drives plate tectonics,
the magnetic field, volcanic eruptions and earthquakes. Some energy also comes in the fast
protons and electrons of the solar wind, which cause interesting ionospheric and auroral
effects, and there is even a little from cosmic rays, which are mainly very energetic protons and photons. All of these
sources of energy are inconsequential compared to the copious bath of radiation in
sunlight, which extends from 0.15 µm in the far ultraviolet to 4 µm in the infrared. We
can, therefore, safely neglect these inputs and
concentrate on the solar input. The total power received on a surface normal to the
direction of the sun outside the atmosphere at the earth’s distance is 1.94 cal/cm²-min
or 1353 W/m². The spectrum is close to a black-body spectrum for 6000K, crossed
by narrow absorption lines due to absorption in the chromosphere of the sun, the
Fraunhofer lines.

As observed at the surface, direct sunlight, which is about 27% of that incident, has
been modified by atmospheric absorption and scattering. The energy at wavelengths shorter than 0.29 µm has been cut off
due to the creation of ozone and its strong absorption. However, the atmosphere is
remarkably transparent to the remainder of the spectrum, only a few weak lines due to
oxygen (the diatomic molecule) and water vapor being
evident. Scattering is stronger at shorter wavelengths, so the scattered radiation is
noticeably blue, and makes the blue of the sky, while the longer wavelengths remain in the
direct beam. This scattering is from density fluctuations in the upper atmosphere, not
from individual molecules as is sometimes asserted.

An estimate of the steady-state energy budget of the earth is shown at the left. The
numbers represent energy in units of 10²² cal/year. The figures are rough estimates, but illustrate the relative
magnitudes of the contributions. The sun provides 130 in the form of short-wave radiation
(0.15 to 4.0µm) at the top of the atmosphere. Of this, 76 is scattered or reflected, and
19 is absorbed, leaving 35 to reach the surface as
direct radiation. The absorbed portion includes interaction with ozone, and this energy
heats the upper atmosphere. Of the scattered and reflected light, 25 also reaches the
surface, while 51 is radiated into space as short-wave radiation. This makes the albedo of
the earth to be about 0.39.

The atmosphere absorbs 141 from the surface, and re-emits 125 to the surface as
long-wave (4-120µm) radiation. The net amount, 65, is radiated to space. The atmosphere
receives 30 by condensation of water, and gives up
30 by evaporation, most of which is part of the hydrologic cycle. The surface receives 5
by turbulence, and releases the same amount of energy on the average by the same means.
Finally, 14 is directly radiated from the earth to space through the “window” in the infrared
spectrum from 8 – 12µm, and in the near infrared below 4 µm.

Things are quite different in the long-wave spectrum. Here, water has strong absorption from 4 –
8 µm, and beyond 25 µm, so that now clouds and snow are black, not white. Snow and thick
clouds radiate like black bodies at long wavelengths, so exchange of energy by radiation
is very important. Clouds may be warmed by radiation
from below, while they are cooled at their tops by radiation into space. This creates
instability (a large lapse rate) so thunderstorms can continue to boil during the night.
Liquid water, interestingly, strongly absorbs everything except the visible. It is even
blacker with long waves, but the energy does not penetrate as far as short waves will, so
absorption and radiation of infrared takes place only in superficial layers. Cooling is
rapid at the surface, which may produce the convection that keeps the surface waters well
mixed.

The discussion of the greenhouse effect showed that the absorption of long-wave
radiation in the atmosphere was important to its explanation. The major constituents of
the atmosphere, the diatomic gases nitrogen and oxygen, and the 1% of argon, do not
interact with electromagnetic radiation in the long-wave spectrum at all. The only trace
gases present in significant amounts are water vapor and carbon dioxide. Water vapor has a
permanent dipole moment, and so a strong pure
rotation spectrum beginning at about 25 µm and extending with greater and greater
absorption to longer wavelengths. It also has a vibration-rotation band for the bending
mode at around 6.3 µm, and for an asymmetric stretching mode at 2.66 µm.

The pure rotation spectrum of carbon dioxide is at much too long wavelengths to play a
role, but there is a strong bending mode band at 14.7 µm, as well as an asymmetric
stretching vibration at 4.26 µm. The 14.7 µm band is at a critical
wavelength, and has a considerable effect in spite of the low concentration of carbon
dioxide. Carbon dioxide must be considered in atmospheric radiative transfer, but it is
much less important than water.

It is usual to express absorption in terms of Beer’s Law,
which is the integrated form of dI/dx = -Iadx, I = I_oe^-ax, where a is the linear absorption coefficient. Sometimes
it is convenient to use the optical density, which is the product of the density and the
distance, d = rho by x, expressed in g/cm². Then, if a’ = a/rho, I = I_oe^-a’d.
The advantage is that a’ is not a function of the density and distribution of the
absorber, only on the number of absorber molecules present.

Beer’s Law is almost useless for expressing the absorption of
radiation in the atmosphere. The spectra consist of sharp, separated absorption lines. If
you start with a uniform radiation spectrum, first the centres of the lines are absorbed
and removed from the beam, then the wings of the lines, and in a complex way that depends
on the details of the spectrum (which for a long time were not accurately known). Attempts
to use an average absorption, as if the atmosphere were a “grey” body failed
utterly. This is a place where computers can help greatly when a detailed absorption curve
is available, and many good calculations have been carried out. We won’t go into this in
detail here, only point it out lest it be thought that things were straightforward and
easy.