# Planck function

The Planck function, B (T), is given by
B (T) = ( 1 3)/[exp( 2 /T) - 1]
where, = wavenumber in cm-1 = 104/ , in µm.
T = temperature in degrees Kelvin 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
h is Planck's constant (6.62620 · 10-34 Joule second)
c is the speed of light (2.99793 · 108 m/second)
k is Boltzmann's constant (1.38062 · 10-23 Joule/K) is the Stefan-Boltzmann constant = 5.67 · 10-5 erg · s-1 · cm-2 · K-4 or mW · m-2 · K-4

Radiant Energy = A · · T4

The brightness temperature, Tb, of a given radiance, R , is found with the inverse of the Planck function.

Tb 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.

B (T)  1 3 exp(- 2 /T)   mW · m-2 · steradian-1/cm-1
Tb (- 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/ v v = 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

B   1/( 2 · c2 · 10-18) · f2 · T = 9.2105 · 10-9 · f2 · T   mW · m-2 · steradian-1 ·/cm-1
Tb 1.0857 · 108 R /f2 B / T 2 · c · k · 2 = 1/ 2 · 2 = 1/( 2 · c2 · 10-18) · f2

 wavelengthmm frequencyGHz B(T=300K)mW/m2/ster/cm-1 dB/dTmW/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

 wavelengthµm wavenumbercm-1 B(T=300K)mW/m2/ster/cm-1 dB/dTmW/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