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Planck function

The Planck function, Bnu(T), is given by
Bnu(T) = (alpha1 nu3)/[exp(alpha2 nu/T) - 1]
where,
nu = wavenumber in cm-1 = 104/lambda, lambda in µm.
T = temperature in degrees Kelvin
alpha1 = 2 h c2 = 1.191066 · 10-5 for radiance in units of mW · m-2 · steradian-1/cm-1.
alpha2 = 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)
sigma 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 · sigma · T4

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

Tb ident B-1nu(Rnu) = (alpha2 · nu)/ loge [ 1 + (alpha1 nu3/Rnu) ]

The derivative of the Planck function is given by

partial Bnu/partial T = (alpha1 alpha2 nu4/T2) · exp(alpha2 nu/T)/[exp(alpha2 nu/T) - 1]2

Infrared approximation: for nu geq 600 cm-1 and T leq 300 K.

Bnu(T) appeq alpha1 nu3 exp(-alpha2 nu/T)   mW · m-2 · steradian-1/cm-1
Tb appeq (-alpha2 · nu)/ loge [ (alpha1 nu3/Rnu) ]
partial Bnu/partial T approx (alpha1 alpha2 nu4)/T2 · exp[- alpha2 nu /T]

Microwave Rayleigh Jeans approximation: lambda(mm) = 300/f, nu = f/30 cm-1, f is in GHz

Wavenumbers, nu, 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, lambdaa or wavelength in vacuum, lambdav.

nu ident 10000/lambdav
lambdav = na · lambdaa
nu = 10000/(na · lambdaa)     na = 1.00027

fcm-1µmmm
0.66 GHz0.02454545454.55P band (SAR)
1.25 GHz0.04240000240.00C band (SAR)
5.33 GHz0.185628556.29L band (SAR)
6.6 GHz0.224545445.45MIMR (surface)
23 GHz0.771304313.04AMSU-A
50 GHz1.6760006.00AMSU-A
60 GHz2.0050005.00AMSU-A
89 GHz2.9733713.37AMSU-C
118 GHz3.9325422.54MHS-X
183 GHz6.1016391.64AMSU-B

Bnu appeq alpha1/(alpha2 · c2 · 10-18) · f2 · T = 9.2105 · 10-9 · f2 · T   mW · m-2 · steradian-1 ·/cm-1
Tb appeq 1.0857 · 108 Rnu/f2
partial Bnu/partial T approx 2 · c · k · nu2 = alpha1/alpha2 · nu2 = alpha1/(alpha2 · 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
6.0500.0070.0000230.335
3.01000.0270.0000920.336
2.01500.0610.0002070.337
1.52000.1090.0003680.338

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
16.7600153.381.5591.0
9.1110081.491.4411.8
6.2160022.690.5812.6
4.323002.350.0863.7
3.727000.560.0244.3
3.330000.180.0094.8