Priciples of Active and Passive Remote Measurements ...
101
5.2.2 Planck's law
Bv(T) is the Planck function which gives the dependence of black body emission with temperature and wavenumber.
c'v 3
B (T) - ---,--1;;;;-_
v
- ec',vfT _ 1
v is in cm- 1
Bv(T) in Wm- 2 ster- 1 (cm- 1 )-1
C~ = 2hc 2 = 1.19110- 8 Wm- 2 ster- 1 (cm- 1 )-4
c~ = hc/k = 1.439 K( cm- I )-1
5.2.3 Stefan's law
Cl A- 5
B>.(T) =
f>.T
e C '
-
1
A is in J.lm
B>.(T) in Wm- 2 ster- I J.lm- 1
CI = 1.19110 8 Wm- 2 ster- 1 J.lm 4
C2 = 1.439 10 4 K ( J.lm )
Since B>.(T) is isotropic, the angular integration of the Plank function over the half space gives
the exitance,
M(A) = ( B>.(T)cosOdw = 1fB>.(T)
J27r
Integrating over all wavenumbers gives
hence
M= {(X) { B>.(T)cosOdwdA=(7T4
10 J2rr
{(X) B>.(T)dA = (7T
4
Jo
1f
with the Stephan's constant (7 = 5,6710- 8 [Wm- 2 K-4].
5.2.4 Wien's law
(5.18)
(5.19)
(5.20)
The maximum of the Planck's function is 'tten by the Wien's law. Neglecting 1 in the denominator of B>.(T), it is easy to check that a~ = 0 for
Examples:
C2
4
T Am ~ "5 = 0.29.10 , or Vm = 1.96 T
Sun
Earth
South pole
Venus
(T ~ 5700K)
(T ~ 220K)
(T ~ 270K)
(T ~ 200K)
(T ~ 753K)
Am ~ 0.505 J.lm
Am ~ 13 J.lm
Am ~ 9 J.lm
Am ~ 14 J.lm
Am ~ 3.8 pm
(5.21 )
There is thus a very clear distinction between the wavelength domains corresponding to solar
and terrestrial radiation.