PDS: The Planetary Atmospheres NodePlanck function
The Planck function, B
(T), is given by(T) = (
1 3)/[exp(2 /T) - 1]where,
= wavenumber in cm-1 = 104/
, in µm.1 = 2 h c2 = 1.191066 · 10-5 for radiance in units of
mW · m-2 · steradian-1/cm-1.2 = h c / k = 1.438833 K cm
is the Stefan-Boltzmann constant = 5.67 · 10-5 erg · s-1 · cm-2 · K-4
or mW · m-2 · K-4
· T4
The brightness temperature, Tb, of a given radiance, R,
is found with the inverse of the Planck function.
B-1(R) = (2 · )/
loge [ 1 + (1 3/R) ]
The derivative of the Planck function is given by
B/ T =
(1 2 4/T2) ·
exp(2 /T)/[exp(2 /T) - 1]2
Infrared approximation: for 
600 cm-1 and
T 
300 K.
(T) 
1 3 exp(-2 /T)
mW · m-2 · steradian-1/cm-1 (-2 · )/ loge [ (1 3/R) ] B/ T 
(1 2 4)/T2 · exp[- 2 /T]
Microwave Rayleigh Jeans approximation: (mm) =
300/f, = f/30 cm-1, f is in GHz
Wavenumbers, , are frequency units and are assumed to be in
vacuum while wavelength is the wavelength specified within the
medium. Typically, wavelengths are expressed as wavelength in air,
a or wavelength in vacuum, v.
10000/vv = na · a = 10000/(na · a) na = 1.00027
| f | cm-1 | µm | mm | |
| 0.66 GHz | 0.02 | 454545 | 454.55 | P band (SAR) |
| 1.25 GHz | 0.04 | 240000 | 240.00 | C band (SAR) |
| 5.33 GHz | 0.18 | 56285 | 56.29 | L band (SAR) |
| 6.6 GHz | 0.22 | 45454 | 45.45 | MIMR (surface) |
| 23 GHz | 0.77 | 13043 | 13.04 | AMSU-A |
| 50 GHz | 1.67 | 6000 | 6.00 | AMSU-A |
| 60 GHz | 2.00 | 5000 | 5.00 | AMSU-A |
| 89 GHz | 2.97 | 3371 | 3.37 | AMSU-C |
| 118 GHz | 3.93 | 2542 | 2.54 | MHS-X |
| 183 GHz | 6.10 | 1639 | 1.64 | AMSU-B |
1/(2 · c2 · 10-18) · f2 · T
= 9.2105 · 10-9 · f2 · T mW · m-2 · steradian-1 ·/cm-1 1.0857 · 108 R/f2 B/ T
2 · c · k · 2 = 1/2 · 2
= 1/(2 · c2 · 10-18) · f2
| wavelength mm | frequency GHz | B(T=300K) mW/m2/ster/cm-1 | dB/dT mW/m2/ster/cm-1/K | (1/B) · dB/dT %/K |
| 6.0 | 50 | 0.007 | 0.000023 | 0.335 |
| 3.0 | 100 | 0.027 | 0.000092 | 0.336 |
| 2.0 | 150 | 0.061 | 0.000207 | 0.337 |
| 1.5 | 200 | 0.109 | 0.000368 | 0.338 |
| wavelength µm | wavenumber cm-1 | B(T=300K) mW/m2/ster/cm-1 | dB/dT mW/m2/ster/cm-1/K | (1/B) · dB/dT %/K |
| 16.7 | 600 | 153.38 | 1.559 | 1.0 |
| 9.1 | 1100 | 81.49 | 1.441 | 1.8 |
| 6.2 | 1600 | 22.69 | 0.581 | 2.6 |
| 4.3 | 2300 | 2.35 | 0.086 | 3.7 |
| 3.7 | 2700 | 0.56 | 0.024 | 4.3 |
| 3.3 | 3000 | 0.18 | 0.009 | 4.8 |
