frudawski

planck

The planck function calculates the spectral radiant exitance M of a Planck radiator for a given black body temperature T in K, as in equation 1:

(1) \qquad M(\lambda,T) = \frac{c_1}{\lambda^5}\frac{1}{\exp\left({\frac{c_2}{T\cdot\lambda}}\right)-1}

With:

c_1 = 3.741771852 \cdot 10^{-16}~\mathrm{W}\cdot\textrm{m}^2

c_2 = 1.438776877 \cdot 10^{-2}~\mathrm{m}\cdot\textrm{K}

Note: Since the redefinition of the Boltzmann constant in 2018 the constants c_1 and c_2 are now exact values.

Usage:

[M,x,y,u,v] = planck(T,lam,mode)

Where:

ParameterDescription
MIs the planck radiator’s spectral radiant exitance in \textrm{W} \textrm{m}^{-2} \textrm{nm}^{-1}.
x,yAre the colour coordinates in CIE 1931 chromaticity of M.
u,vAre the colour coordinates in CIE 1960 chromaticity of M.
TIs the planck radiator’s temperature in K.
lam
(optional)
Specifies the wavelengths in nm, default: 360:830
mode
(optional)
Specifies the constants c_1 and c_2:
‘exact’ using the exact values as from NIST
\qquad c_1 = 3.741771852 \cdot 10^{-16}~\mathrm{W}\cdot\textrm{m}^2
\qquad c_2 = 1.438776877 \cdot 10^{-2}~\mathrm{m}\cdot\textrm{K}
‘CIE’ using the values specified by the CIE e.g. in CIE TR 224
\qquad c_1 = 3.74183 \cdot 10^{-16}~\mathrm{W}\cdot\textrm{m}^2
\qquad c_2 = 1.4388 \cdot 10^{-2}~\mathrm{m}\cdot\textrm{K}

Examples

Spectral radiant exitance for a block body temperature of 5000 K:

M = planck(5000);
plotspec(360:830,M)
ylabel('spectral radiant exitance in W m^{-2} nm^{-1}')

See also: plotspec

Result:

Spectral radiant exitance for several black body temperatures:

lam = 0:10:1000;
T = 4000:1000:9000;
M = planck(T,lam);
plot(lam,M)
xlabel('\lambda in nm')
ylabel('spectral radiant exitance in W m^{-2} nm^{-1}')
legend('4000 K','5000 K','6000 K','7000 K','8000 K','9000 K')

Result:

Colour coordinates for a planck radiator of 9000 K:

[~,x,y,u,v] = planck(9000)

Result:

x = 0.2869
y = 0.2956
u = 0.1921
v = 0.2969

Reference

Günther Wyszecki, W. S. Stiles: Colour Science - Concepts and Methods, Quantitative Data and Formulae, 2nd Edition. John Wiley & Sons, Inc., 2000, ISBN: 978-0-471-39918-6.

The NIST Reference on Constants, Units and Uncertainty, first radiation constant

The NIST Reference on Constants, Units and Uncertainty, second radiation constant

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