A blackbody or planckian radiator is an ideal thermal radiator that absorbs completely all incident radiation, whatever the wavelength, the direction of incidence or the polarization. [1]
A blackbody in thermal equilibrium emits electromagnetic radiation called blackbody radiation.
The spectral radiance of a blackbody at thermodynamic temperature $T [K]$ in a medium having index of refraction $n$ is given by the Planck's law equation: [2]
$$ \begin{equation} L_{e\lambda}(\lambda,T)=\cfrac{C_1n^{-2}\lambda^{-5}}{\pi}{\Biggl[\exp\biggl(\cfrac{C_2}{n\lambda T}\biggr)-1\Biggr]^{-1}} \end{equation} $$where $$ \begin{equation} \begin{aligned} C_1&=2\pi hc^2\\ C_2&=\cfrac{hc}{k} \end{aligned} \end{equation} $$
$h$ is Planck's constant, $c$ is the speed of light in vacuum, $k$ is the Boltzmann constant and $\lambda$ is the wavelength.
As per CIE 015:2004 Colorimetry, 3rd Edition recommendation $C_2$ value when used in colorimetry should be $C_2= 1,4388x10^{-2}mK$ as defined by the International Temperature Scale (ITS-90).
$C_1$ value is given by the Committee on Data for Science and Technology (CODATA) and should be $C_1=3,741771x10^{16}Wm^2$.
In the current CIE 015:2004 Colorimetry, 3rd Edition recommendation, colour temperature and correlated colour temperature are calculated with $n=1$.
Colour implements some blackbody related objects available in colour.colorimetry sub-package:
In [1]:
import colour.colorimetry
Note: colour.colorimetry package public API is also available from the colour namespace.
The Planck's law is called using either the colour.planck_law or colour.blackbody_spectral_radiance definitions, they are expecting the wavelength $\lambda$ to be in nanometers and the temperature $T$ to be in degree kelvin:
In [2]:
%matplotlib inline
In [3]:
import colour
colour.utilities.filter_warnings(True, False)
colour.colorimetry.planck_law(500 * 1e-9, 5500)
Out[3]:
Generating the spectral power distribution of a blackbody is done using the colour.blackbody_spectral_power_distribution definition:
In [4]:
colour.blackbody_spd(6500, colour.SpectralShape(0, 10000, 10))
Out[4]:
With its temperature lowering, the blackbody peak shifts to longer wavelengths while its intensity also decreases:
In [5]:
from colour.plotting import *
colour_plotting_defaults()
# Plotting various *blackbodies* spectral power distributions.
blackbodies_spds = [colour.blackbody_spd(i, colour.SpectralShape(0, 10000, 10))
for i in range(1000, 15000, 1000)]
multi_spd_plot(blackbodies_spds,
y_label='W / (sr m$^2$) / m',
use_spds_colours=True,
normalise_spds_colours=True,
legend_location='upper right',
bounding_box=[0, 1000, 0, 2.25e15])
Let's plot the blackbody colours from temperature in domain [150, 12500, 50]:
In [6]:
blackbody_colours_plot(colour.SpectralShape(500, 12500, 50))
Let's compare the extraterrestrial solar spectral irradiance to the blackbody spectral radiance of a thermal radiator with a temperature of 5778 K:
In [7]:
# Comparing theoretical and measured *Sun* spectral distributions.
# Arbitrary ASTM_G_173_ETR scaling factor calculated with
# :def:`colour.spectral_to_XYZ` definition.
ASTM_G_173_spd = ASTM_G_173_ETR.copy() * 1.37905559e+13
blackbody_spd = colour.blackbody_spd(
5778,
ASTM_G_173_spd.shape)
blackbody_spd.name = 'The Sun - 5778K'
multi_spd_plot([ASTM_G_173_spd, blackbody_spd],
y_label='W / (sr m$^2$) / m',
legend_location='upper right')
As you can see the Sun spectral power distribution is very close to the one from a blackbody at similar temperature $T$.
Calculating theoritical colour of any star is possible, like for instance the VY Canis Majoris red hypergiant in the constellation Canis Major.
In [8]:
blackbody_spectral_radiance_plot(temperature=3500, blackbody='VY Canis Majoris')
Or Rigel the brightest star in the constellation Orion and the seventh brightest star in the night sky.
In [9]:
blackbody_spectral_radiance_plot(temperature=12130, blackbody='Rigel')
And finally the Sun, our star:
In [10]:
blackbody_spectral_radiance_plot(temperature=5778, blackbody='The Sun')