All forms of electromagnetic radiation travel in a vacuum at the same velocity, which is approximately 3 x 1010 cm/sec and is denoted by the letter c. Electromagnetic radiation is usually quantified according to its wave-like properties, which include intensity and wavelength. For many applications it is sufficient to consider electromagnetic waves as being a continuous train of sinusoidal shapes.
If radiation has only one colour, it is said to be monochromatic. The colour of any particular kind of radiation is designated by its frequency, which is the number of waves passing a given point in one second and is represented by the letter f (with units of cycles/sec or Hertz). The length of a single wave (i.e., the distance between two successive maxima) is called the wavelength and is denoted by . (with units of microns to meters). For monochromatic radiation, the number of waves passing a fixed point in one second multiplied by the length of each wave is the distance the wave train travelled on one second. But that distance is equal in magnitude to the velocity of light c.
Since the frequency of a wave, f, is equal to the number of wavelengths in a distance of 3 x 1010 cm, it is usually a large number. Therefore, it is often much more convenient (especially in the infrared region of the spectrum) to consider the number of waves in one centimetre, called the wavenumber.
wavenumber has units of cm -1
This new unit the wavenumber is designated by &nu(cm-1) and
Conversion of wavelength units is demonstrated below:
1 Å = 10-10 m = 10-8 cm = 10-4 µm, and
1 µm = 10-6 m = 10-4 cm = 104 Å.
Conversion of frequency units follows:
1 cm-1 = 3 x 1010 Hz = 30 GHz, and
1 GHz = 109 Hz = 1/30 cm-1.
Table 2.1 summarizes the regions and units of the electromagnetic spectrum commonly encountered in satellite meteorology.
Table 2.1 Regions and Units of the Electromagnetic Spectrum
| WAVELENGTH | FREQUENCY | WAVENUMBER | ||||
| REGION | cm | µm | Å | Hz | GHz | cm-1 |
| Near Ultraviolet (UV) | 10-5 | 0.1 | 1,000 | 3x1015 | . | . |
| Visible | 4x10-5 | 0.4 | 4,000 | 7.5x1014 | . | . |
| Near Infrared (IR) | 7.5x10-5 | 0.75 | 7,500 | 4x1014 | . | 13,333 |
| Far Infrared (IR) | 2x10-3 | 20 | 2x105 | 1.5x1013 | . | 500 |
| Microwave (MW) | 0.1 | 103 | . | 3x1011 | 300 | 10 |
2.3 Definitions of Radiation
The rate of energy transfer by electromagnetic radiation is called the radiant flux, which has
units of energy per unit time. It is denoted by
F = dQ / dt
and is measured in joules per second or watts. For example, the radiant flux from the sun is about 3.90 x 1026 W.
The radiant flux per unit area is called the irradiance (or radiant flux density in some texts). It is denoted by
E = dQ / dt / dA
and is measured in watts per square metre. The irradiance of electromagnetic radiation passing
through the outermost limits of the visible disk of the sun
(which has an approximate radius of 7 x 108 m) is given by 3.90 x 10 26
E (sun sfc) = = 6.34 x 107 W m-2.
4p (7 x 108)2
The solar irradiance arriving at the earth can be calculated by realizing that the flux is a
constant, therefore
E (earth sfc)*4pRes
2 = E (sun sfc)*4pRs
2,
where Res is the mean earth to sun distance (roughly 1.5 x 1011 m) and Rs is the solar radius. This
yields
E (earth sfc) = 6.34 x 107 (7 x 108 / 1.5 x 1011)2 = 1380 W m-2.
The irradiance per unit wavelength interval at wavelength . is called the monochromatic
irradiance,
E. = dQ / dt / dA / d. ,
and has the units of watts per square metre per micrometer. With this definition, the irradiance is
readily seen to be
8
E = . E. d. .
o
In general, the irradiance upon an element of surface area may consist of contributions which come from an infinity of different directions. It is sometimes necessary to identify the part of the irradiance that is coming from directions within some specified infinitesimal arc of solid angle dO. The irradiance per unit solid angle is called the radiance,
I = dQ / dt / dA / d. / dO, and is expressed in watts per square metre per micrometer per steradian. This quantity is often also referred to as intensity and denoted by the letter B (when referring to the Planck function).
Table 2.2 summarizes these definitions. In order to express the relationship between irradiance and radiance quantitatively, it is necessary to define the zenith angle, ., as the angle between the direction of the radiation and the normal to the surface in question. The component of the radiance normal to the surface is then given by I cos .. The irradiance represents the combined effects of the normal component of the radiation coming from the whole hemisphere; that is,
E = . I cos . dO O where in spherical coordinates dO = sin . d. df .
Figure 2.1 illustrates the radiance geometry. Radiation whose radiance is independent of direction is called isotropic radiation. In this case, the integration over dO can be readily shown to be equal to p so that
E = p I . Radiance is often expressed per unit frequency, I(f), rather than per unit wavelength, I(.). Since the transformation from wavelength to frequency is given by f = c / . , it follows that
I(.) = I(f) c / .2.
I(.) must not be confused with I(f): they are different quantities. More will be said about this later in this
chapter.
2.4 Historical Development of Planck's Radiation Law It had long been observed that the surface of all bodies at a temperature greater than absolute zero (0 K) emits energy, in the form of thermal radiation. These electromagnetic waves were thought to be due to the motion of electric charges near the surface of the radiating body.
The study of radiation focused on the properties of a hypothetical black body, which is
characterized by (a) complete absorption of all incident radiation (hence the term black), and (b)
maximum possible emission in all wavelengths in all directions. In other words, it is the perfect
absorber and emitter of all radiation.
Many attempts, both empirical and theoretical in approach, were made in the years up to and
about 1900 to understand the black body spectrum. In 1879, Stefan had empirically found that the
irradiance of a black body was related to temperature by the law
E = s T4,
where
s = 5.67 x 10-8 W m-2 deg-4.
In 1884, Boltzmann produced a theoretical derivation of this equation.
Earliest accurate measurements of monochromatic irradiance are credited to Lummer and
Pringsheim in 1899. They observed the now well-known emission spectra for black bodies at several
different temperatures (shown in Figure 2.2).
Thermodynamical reasoning, while not giving a complete answer, did predict two
characteristic features of the radiation. Wien, in 1893, was able to shown that the monochromatic
radiance was related to temperature and wavelength by
I(.) = f(.T) / .5,
where the form of the single function f(.T) was not evaluated. Also, the peak emission wavelength of
a black body was shown to be inversely proportional to temperature, so that
.max = const / T .
In this derivation, he considered a cylindrical cavity with reflecting walls, one of which is a
movable piston, filled with radiation at a temperature T. Radiation was known to exert a pressure
proportional to its energy density. Taking this system through a Carnot cycle, a relation between the
work done by the radiation (expressed in terms of monochromatic radiance) and temperature was
obtained.
To evaluate the function f(.T) some of the detailed properties of a black body must be taken
into account. In 1900, Rayleigh and Jeans attempted to evaluate f(.T) by considering a cubical cavity
containing electromagnetic standing waves with nodes at the metallic surfaces and the energy of these
waves obeying the Boltzmann probability distribution. Assuming a continuum of energy states, the
average total energy of this system can be expressed as
8 8
eav = [ . e e .e / kT de ] / [ . e .e / kT de ]
o o
where the Boltzmann constant is given by
k = 1.381 x 10-23 J/deg.
2-5
This leads to
f(.T) = 2ck.T ,
which is in agreement with experiment only for long wavelengths. At short wavelengths the
monochromatic radiance becomes infinite (often referred to as the ultraviolet catastrophe). The form
of f(.T) obtained by Rayleigh and Jeans was a necessary consequence of the theories of classical
physics, yet it failed!
The discrepancy between experiment and theory was resolved by Planck in 1901, but at the
expense of some of the concepts known to classical physics. Assuming that electromagnetic harmonic
oscillations can only exist in quanta of hf (h is a constant, f is a frequency) and that oscillators emit
energy only when changing from one to another of their quantized energy states, then the average total
energy of this system is expressed as
8 8
eav = [ S nhf e .nhf / kT ] / [ S e .nhf / kT ]
n=o n=o
or
eav = hf / (e hf / kT - 1) .
This yields the proper form of f(.T),
f(.T) = 2hc2 /(e hc / .kT - 1) ,
and predicts the observed results if the Planck constant
h = 6.63 x 10-34 J*s .
And the Planck law for radiation intensity (or monochromatic radiance) can be expressed as
B(.,T) = c1 / [.5 (e c
2
/ .T - 1)]
where
c1 = 2hc2 = 1.191044 x 10-8 W/(m2*ster*cm-4)
and
c2 = hc/k = 1.438769 K*cm
and B(.,T) has units of W/(m2*ster*cm).
Written in terms of wavenumber rather than wavelength,
B(.,T) = c1.3 / (e c
2
. / T - 1)
where B(.,T) has units of W/(m2*ster*cm-1). The difference in the Planck radiance curves when
measured with respect to unit wavelength versus unit wavenumber is seen in Figure 2.3.
2-6
Thus, in the course of his successful attempt at resolving certain discrepancies between the
observed energy spectrum of thermal radiation and the predictions of the classical theory, Planck was
led to the idea that a system executing simple harmonic oscillations only can have energies which are
integral multiples of a certain finite amount of energy (1901). A closely related idea was later applied
by Einstein in explaining the photo-electric effect (1905), and by Bohr in a theory which predicted with
great accuracy many of the complex features of atomic spectra (1913). The work of these three
physicists, plus subsequent developments by de Broglie, Schroedinger, and Heisenberg (ca. 1925),
constitutes what is known as the quantum theory. Quantum theory and the theory of relativity together
comprise the two most significant features of modern physics.
2.5 Related Derivations
2.5.1 Wien's Displacement Law
The peak of Planck function curve shifts to shorter wavelengths with an increase in
temperature. The wavelength .max for which the Planck function peaks at a given temperature T can
be found from Planck’s law by differentiating it with respect to . and equating the result to zero. This
yields the nonlinear equation
x = 5 (1 - e-x)
where x = c2 / (.max T) whose solution is x = 4.965114. Therefore,
.max = .2897 / T (cm)
which is Wien’s displacement law. This law indicates that the wavelength of maximum Planck radiance
varies inversely with temperature. Note that when the Planck function is expressed in terms of
wavenumber, differentiating with respect to . and equating the result to zero yields y = 3 (1 - e- y) where
y = c2 .max/T with the solution .max = 1.95*T (cm-1). It is important to realize that .max . 1/ .max.
The Planck radiance at the Wien wavelength varies as temperature to the fifth power. To see
this more clearly consider
B(.max,T) = c1 / [.max
5 (e c
2
/ .maxT - 1)]
= c1 T5 / [ (.2897)5 (e c
2
/ .2897 - 1)]
= c3 T5
where c3 is a constant. In a similar way, it can be shown that B(.max,T) = c4T3.
The radiative temperature of the surface of the sun is roughly 5780 K. Applying Wien’s Law
at the sun’s surface temperature, one finds the maximum Planck radiance to be at 0.50 um, which is
near the centre of the visible region of the spectrum. Since the sun radiates nearly as a black body,
we can say that the solar energy reaching the earth is a maximum in the visible region.
On the other hand, the earth’s atmospheric temperature averaged vertically is around 255 K;
therefore, the maximum emitted energy of the earth’s atmosphere occurs roughly at 11 um, that is, well
into the infrared region. In fact, if the black body curves (Planck functions) for temperatures of 255 K
and 5780 K are plotted next to each other, the curves are almost entirely separate. Thus, the spectral
distribution of the incoming solar radiation is quite different from that of the outgoing terrestrial radiation
(see Figure 2.4).
Wien’s displacement law derives its name from the fact that as the temperature increases, the
point of maximum intensity of the black body curve is displaced toward the shorter wavelengths. Since
the wavelength of the maximum value determines the colour perceived while observing the complete
spectrum, we have an explanation for the transition in colour of a heated iron bar from red to a white
2-7
glow with increased heat. As the temperature is raised the longer wavelength red light becomes visible
first. Then higher temperatures make additional wavelengths visible. Finally, when the temperature
is sufficiently high, the radiation consists of a mixture of all the visible wavelengths and, hence, appears
white hot. For similar reasons, the filament of an incandescent lamp must be heated to thousands of
degrees K in order to be an efficient emitter of visible light, while infrared lamps operate at lower
temperatures.
2.5.2 Rayleigh-Jeans Radiation Law
In the microwave region of the electromagnetic spectrum, when . > 0.5 cm and T is at
terrestrial temperatures, the exponent in the Planck function is small, and hence one can make the
approximation
ec2/.T = 1 + c2/.T
that yields the following asymptotic expansion:
B(.,T) = c1T/(c2.4)
Similarly,
B(.,T) = c1.2T/c2
for sufficiently small v. These formulas represent the Rayleigh-Jeans law of radiation and the spectral
region . > 0.5 cm is called the Rayleigh-Jeans region in atmospheric physics. Note that in the
Rayleigh-Jeans region the Planck function is linear to T. For .T > 10 cm K, the Rayleigh-Jeans
approximation is within 2% or better.
2.5.3 Wien's Radiation Law
In the near infrared region and beyond into the visible and ultraviolet regions, i.e., . < 10-3 cm,
and when T is at terrestrial temperatures, the exponent in the Planck function is large and, hence
ec2 / .T >> 1.0
Consequently, the constant 1.0 can be ignored in the denominator to yield another asymptotic
expansion for the Planck function, namely,
B(.,T) = c1 .-5 e-c2/.T ,
or
B(.,T) = c1 .3 e-c2 ./T
for sufficiently large .. These are two forms of Wien’s law of radiation. The spectral region . < 10-3
cm is called the Wien region when dealing with atmospheric temperatures. In the Wien region, the
Planck function is highly nonlinear in temperature. . For .T < 0.5 cm K, the Wein approximation is
within 2% or better.
2-8
2.5.4 Stefan-Boltzmann Law
The black body irradiance is obtained by integrating the Planck function over all wavelengths
and angles,
8 8 c1 d.
E = . E. d. = p . ,
o o c2/.T
.5 [e -1]
Let x = c2/.T then
pc1T4 8 x3 dx p5c1T4
E = . = = sT4 .
c2
4 o (ex - 1) 15c2
4
Note that
8 8
. B(.,T) d. = . B(.,T) d. ,
o o
which implies for a given temperature
B(.,T) = .2 B(.,T) .
It is often useful to know the fraction of the total blackbody radiance coming from below a
specific wavelength, expressed by
. .
. E. d. / sT4 = p . B. d. / sT4 .
o o
Figure 2.5 shows the fraction of total blackbody radiance emitted below a specific wavelength as a
function of .T. Note that for . = .max the fraction is .23, or less than 25% of the radiance comes from
wavelengths shorter than the peak wavelength. Figure 2.5 illustrates that less than 1% of the solar
energy comes from wavelengths longer than 4 microns, while less than 1% of terrestrial energy from
a blackbody at 250 K comes from wavelengths shorter than 4 microns.
2.5.5 Brightness Temperature
Ultimately, one is interested in the temperature that corresponds to a particular Planck
function value B.. This temperature is determined by inverting the Planck function,
c1 c1.3
T = c2 / [. ln( + 1)] = c2. / [ln( + 1)]
.5B. B.
The temperature derived is called brightness temperature because of its historical connection
with radio astronomy; however, the terms radiance temperature or equivalent black body temperature
are also frequently used.
2-9
In the Rayleigh-Jeans regions, one can write
T = (c2/c1) .4 B. = (c2/c1) B. / .2,
where c2/c1 = 1.208021 x 105. On the other hand in the Wien region
c1 c1.3
T = c2 / [ . ln ( ) ] = c2. / [ln( )]
.5B. B.
where B(.,T) has units of W/(m2*ster*cm-1).
As we have seen from Planck.s Law, as temperature increases the radiance also increases; the percentage increase varies as a function of wavelength and temperature. The percentage change in radiance to a corresponding percentage change in temperature, called temperature sensitivity, a, of a given spectral band is defined as dB/B = a dT/T. For infrared wavelengths, we find that a ˜ c2./T = c2/.T. Thus the larger wavenumbers (shorter wavelengths) have a greater temperature sensitivity than the smaller wavenumbers (longer wavelengths). Consider the two infrared windows for example; at 300 K, the temperature sensitivity for 900 cm-1 (long-wave window at 11 um) is 4.3 and for 2500 cm- 1 (short-wave window at 4 um) it is 12. The temperature sensitivity indicates the power to which the Planck radiance depends on temperature, since B proportional to Ta satisfies the equation. Thus the radiance in the short-wave window is varying roughly as temperature to the twelfth power and in the long-wave window roughly as temperature to the fourth power. Table 2.3 indicates the temperature sensitivity of some of the other spectral regions.
Table 2.2 Definitions of Radiation
| QUANTITY | SYMBOL | UNITS |
| Energy | DQ | Joules |
| Flux | dQ/dT | Joules/sec = Watts |
| Irradiance | dQ/dT/dA | Watts/m2 |
| Monochromatic Irradiances | dQ/dT/dA/d&lambda or dQ/dT/dA/d&nu | W/m2/micron or W/m2cm2-1 |
| Radiance | dQ/dT/dA/d&lambda/d&Omega or dQ/dT/dA/d&nu/d&Omega | W/m2/micron/ster or W/m2/cm-1/ster |
Table 2.3 Temperature sensitivity of various spectral regions.
| Wavenumber Typical Scene | Temperature | Temperature Sensitivity |
| 700 | 220 | 4.58 |
| 900 | 300 | 4.32 |
| 1200 | 300 | 5.76 |
| 1600 | 240 | 9.59 |
| 2300 | 220 | 15.04 |
| 2500 | 300 | 11.99 |
Figure 2.1: The Radiance Geometry
Figure 2.2: Emission Spectra for Black Bodies with the Indicated Temperatures
Figure 2.3: Planck radiance curves at 300 K. . indicates B(.,T) in units of mW/ster/m2/cm (per unit wavelength) and + indicates B(.,T) in units of mW/ster/m2/cm-1 (per unit wavenumber). B(.,T) is shown multiplied by a factor of 10-7 (e.g., B(10 µm, 300 K) is 9.9x107) while B(.,T) is shown multiplied by a factor of 10-1 (e.g., B(1000 cm-1, 300 K) is 99.0). Note that Wiens law for B(.,T) becomes v(max) = 1.95*T (cm-1) while the more familiar form for B(.,T) is .(max) = .2897/T (cm).
Figure 2.4: Normalized black body spectra representative of the sun (left) and earth (right), plotted on a logarithmic wavelength scale. The ordinate is multiplied by wavelength so that the area under the curves is proportional to irradiance. Figure 2.5: Fraction of total blackbody radiance coming from below a given wavelength for a given temperature.