Geoscience Reference
In-Depth Information
where
()
B
l
is the
spectral radiance
, defined as the rate at which energy is
emitted per square meter per unit wavelength per steradian (the unit of solid
angle). The units of
()
B
l
are W/(m
2
sr mm) when wavelength is in microns
(1 micron 1 mm 10
−6
m). In Eq. 4.3,
h
is Planck's constant (6.63 10
−34
J s),
c
is the speed of light (3.00 10
8
m/s), and
k
is Boltzmann's constant
(1.38 10
−23
J/K).
Note the following attributes of blackbody radiation:
• For a blackbody with temperature
T
,
T
T4s is the rate of energy emission per
unit area (W/m
2
), integrated over all wavelengths and over the hemisphere
into which the surface radiates:
1
3
#
.
(4.4)
σ
T
4
=
π
B d
λ
λ
0
• As the temperature of the emitting body increases, the wavelength of
maximum emission, l
MAX
, decreases according to Wein's displacement law:
10
3
−
2 898
.
#
,
(4.5)
λ
=
MAX
T
where
T
is in Kelvin and l
MAX
is in mm.
• According to Kirchhoff's law, the emissivity, , of a blackbody at any given
wavelength is equal to its
absorptivity
,
a
, for that same wavelength. A strong
absorber is an equally strong emitter. Kirchhoff's law holds for any object at
a constant temperature.
For a given temperature, Eq. 4.3 can be used to calculate energy emitted as a
function of wavelength. The resulting plots are called
Planck curves
.
4.2 APPLICATION OF BLACKBODY
THEORY TO THE EARTH SYSTEM
Imagine the earth as a spherical blackbody in empty space. Now, switch on the
sun and let solar radiation fall onto the surface of the sphere. Because it is black,
the sphere will absorb all the incident solar radiation. It will begin to heat up
and emit energy in the form of radiation at a rate,
E
in W/m
2
, that is dependent
on its temperature,
T
, according to the Stefan-Boltzmann law (Eq. 4.1). The
temperature of this model earth will continue to increase until the rate at which
it is emitting energy is equal to the rate at which it is absorbing energy. This final
state is called
radiative equilibrium
, and the temperature at which this occurs is
called the
radiative equilibrium temperature
,
T
E
.
For this idealized example and the calculations that follow, consider the
long-time average over the diurnal and seasonal cycles and over many years.
This is equivalent to imagining that the energy incident on the earth is distrib-
uted uniformly over the entire sphere, and isothermal conditions prevail.
1
The details of this integration are not presented. The reader is referred to textbooks on radia-
tion for guidance on the integration by parts.
