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emission. Its radiance also called brightness Bν(T) is given by the Planck’s law:
B(T,ν) = Bν(T) = 2hν3
c 2
1
, (327)
exp(hν/kT)−1
where h is Planck’s constant, k is Boltzmann’s constant and c is the vacuum speed of
light. The factor 2 in the numerator accounts for both polarizations according to the usual
convention.
The related emission from a real body – often called grey body – at the same temperature
is AνBν(T) where Aν is the fraction of incident energy absorbed from a certain direction.
In the general theory, scattering into and from other directions can lead to both losses and
gains to the intensity along a given propagation direction and can be taken into account in
both the terms Sν and αν. But if we assume a local thermodynamic equilibrium, so that each
point into the elementary volume is characterized by the same temperature T(s), the strict
requirement of balance between the energy absorbed and emitted by any particular volume
element leads to Kirchoff’s law and for the S term we can suppose:
Sν(s,T) = [αν(s)(1−Aν)]Bν(T). (328)
For our application we will consider a no-scatter isotropic medium. In these hypothesis
the source term Sν(s,T) expresses only the locally generated contribution to the radiation,
and the absorption coefficient αν(s) becomes a local scalar characteristic of the medium
that describes a true loss of energy from the radiation field into the medium. Moreover, for
a perfectly reflecting or transmitting body, Aν is equal to zero and incident energy can be
assumed to be redirected or pass through the body without being absorbed. Under these
hypotheses for an upward looking radiometer we can rewrite Eq. (328) as:
Sν(s,T) αν(s)Bν(T), (329)
where Bν(T) is always given by Planck’s function Eq. (327).
Operating in the microwave frequency range ν < 300 GHz, according to the Rayleigh-
Jeans approximation of Planck’s law, by expanding the exponential term exp(hv/kT) and
truncating to the second term of Eq. (327)
exp(hv/kT) = 1+(hv/kT)+(hv/kT) 2 /2!+... ∼
= 1+(hv/kT) ,
we obtained for Bν(T) a well-known approximated linear form (see Fig. 173):
from which Eq. (329) becomes:
Bν(T) ∼ = 2kv2
T(s), (330)
c2 Sν(s) = αν(s)Bν(T) ∼ = αν(s) 2kv2
T(s) (331)
c2 Bν(T) is the surface brightness, which is the flow of energy across a unit area, per unit
frequency, from a source viewed through free space in an element of solid angle. Since the
brightness and the intensity have the same units (according to Fig. 174) the two are locally
equivalent in this case. It follows:
Iν(s) ≡ Bν(s) ∼ = 2kv2
T(s), (332)
c2 in which Iν is expressed as linear function of its relative thermodynamic temperature T(s) as
usually in the microwave frequency range formulation.
SubstitutingEq. (331)formofSν(s) in Eq.(326)inthe hypothesisofnoscattering,in local
thermodynamic equilibrium, plane horizontallystratified, perfectly transmitting (or reflecting)
atmosphere we obtain the following radiance power transfer differential equation:
dIν = −αν(s)Iν(s)ds+ 2kν2
c 2 αν(s)T(s)ds. (333)
262 DTU Wind Energy-E-Report-0029(EN)