"""
Blackbody - Planckian Radiator
==============================
Define objects to compute the spectral radiance of a planckian radiator
and its spectral distribution.
References
----------
- :cite:`CIETC1-482004i` : CIE TC 1-48. (2004). APPENDIX E. INFORMATION ON
THE USE OF PLANCK'S EQUATION FOR STANDARD AIR. In CIE 015:2004 Colorimetry,
3rd Edition (pp. 77-82). ISBN:978-3-901906-33-6
- :cite:`Wikipedia2003f` : Wikipedia. (2003). Rayleigh-Jeans law. Retrieved
February 12, 2022, from https://en.wikipedia.org/wiki/Rayleigh-Jeans_law
"""
from __future__ import annotations
import typing
import numpy as np
from colour.colorimetry import (
SPECTRAL_SHAPE_DEFAULT,
SpectralDistribution,
SpectralShape,
)
from colour.constants import CONSTANT_BOLTZMANN, CONSTANT_LIGHT_SPEED
if typing.TYPE_CHECKING:
from colour.hints import ArrayLike, NDArrayFloat
from colour.utilities import as_float, as_float_array
from colour.utilities.common import attest
__author__ = "Colour Developers"
__copyright__ = "Copyright 2013 Colour Developers"
__license__ = "BSD-3-Clause - https://opensource.org/licenses/BSD-3-Clause"
__maintainer__ = "Colour Developers"
__email__ = "colour-developers@colour-science.org"
__status__ = "Production"
__all__ = [
"CONSTANT_C1",
"CONSTANT_C2",
"CONSTANT_N",
"planck_law",
"blackbody_spectral_radiance",
"sd_blackbody",
"rayleigh_jeans_law",
"sd_rayleigh_jeans",
]
# 2 * math.pi * CONSTANT_PLANCK * CONSTANT_LIGHT_SPEED ** 2
CONSTANT_C1: float = 3.741771e-16
# CONSTANT_PLANCK * CONSTANT_LIGHT_SPEED / CONSTANT_BOLTZMANN
CONSTANT_C2: float = 1.4388e-2
CONSTANT_N: float = 1
[docs]
def planck_law(
wavelength: ArrayLike,
temperature: ArrayLike,
c1: float = CONSTANT_C1,
c2: float = CONSTANT_C2,
n: float = CONSTANT_N,
) -> NDArrayFloat:
"""
Compute the spectral radiance of a blackbody as a function of
wavelength at specified thermodynamic temperature :math:`T[K]` in a
medium with index of refraction :math:`n`.
Parameters
----------
wavelength
Wavelength in meters.
temperature
Temperature :math:`T[K]` in kelvin degrees.
c1
The official value of :math:`c1` is provided by the Committee on
Data for Science and Technology (CODATA) and is
:math:`c1=3.741771 \\times 10^{-16}\\ \\mathrm{W/m^2}` *(Mohr and
Taylor, 2000)*.
c2
Since :math:`T` is measured on the International Temperature
Scale, the value of :math:`c2` used in colorimetry should follow
that adopted in the current International Temperature Scale
(ITS-90) *(Preston-Thomas, 1990; Mielenz et al., 1991)*, namely
:math:`c2=1.4388 \\times 10^{-2}\\ \\mathrm{m \\cdot K}`.
n
Medium index of refraction. For dry air at 15°C and 101 325 Pa,
containing 0.03 percent by volume of carbon dioxide, it is
approximately 1.00028 throughout the visible region although
*CIE 15:2004* recommends using :math:`n=1`.
Returns
-------
:class:`numpy.ndarray`
Radiance in *watts per steradian per square metre*
(:math:`\\mathrm{W \\cdot sr^{-1} \\cdot m^{-2}}`).
Warnings
--------
The :func:`colour.colorimetry.planck_law` definition behaviour with
n-dimensional arrays is unusual: The ``wavelength`` and
``temperature`` parameters are first raveled using
:func:`numpy.ravel`. Then, they are *broadcasted* together by
transposing the ``temperature`` parameter. Finally, and for
convenience, the return value is squeezed using :func:`numpy.squeeze`.
Notes
-----
- The following implementation is expressed in terms of wavelength.
- The SI unit of radiance is *watts per steradian per square
metre* (:math:`\\mathrm{W \\cdot sr^{-1} \\cdot m^{-2}}`).
References
----------
:cite:`CIETC1-482004i`
Examples
--------
>>> planck_law(500 * 1e-9, 5500) # doctest: +ELLIPSIS
np.float64(20472701909806.5...)
>>> planck_law(500 * 1e-9, [5000, 5500, 6000]) # doctest: +ELLIPSIS
array([1.2106064...e+13, 2.0472701...e+13, 3.1754431...e+13])
"""
l = as_float_array(wavelength) # noqa: E741
t = as_float_array(temperature)
attest(np.all(l > 0), "Wavelengths must be positive real numbers!")
l = np.ravel(l)[..., None] # noqa: E741
t = np.ravel(t)[None, ...]
d = 1 / np.expm1(c2 / (n * l * t))
p = ((c1 * n**-2 * l**-5) / np.pi) * d
return as_float(np.squeeze(p))
blackbody_spectral_radiance = planck_law
[docs]
def sd_blackbody(
temperature: float,
shape: SpectralShape = SPECTRAL_SHAPE_DEFAULT,
c1: float = CONSTANT_C1,
c2: float = CONSTANT_C2,
n: float = CONSTANT_N,
) -> SpectralDistribution:
"""
Generate the spectral distribution of the planckian radiator for the
specified temperature :math:`T[K]` with values in *watts per steradian
per square metre per nanometre* (:math:`W/sr/m^2/nm`).
Parameters
----------
temperature
Temperature :math:`T[K]` in kelvins.
shape
Spectral shape used to create the spectral distribution of the
planckian radiator.
c1
The official value of :math:`c_1` is provided by the Committee on
Data for Science and Technology (CODATA) and is
:math:`c_1=3.741771 \\times 10^{16}\\ W/m^2` *(Mohr and Taylor,
2000)*.
c2
Since :math:`T` is measured on the International Temperature
Scale, the value of :math:`c_2` used in colorimetry should follow
that adopted in the current International Temperature Scale
(ITS-90) *(Preston-Thomas, 1990; Mielenz et al., 1991)*, namely
:math:`c_2=1.4388 \\times 10^{-2}\\ m \\cdot K`.
n
Medium index of refraction. For dry air at 15°C and 101 325 Pa,
containing 0.03 percent by volume of carbon dioxide, it is
approximately 1.00028 throughout the visible region although
*CIE 15:2004* recommends using :math:`n=1`.
Returns
-------
:class:`colour.SpectralDistribution`
Blackbody spectral distribution with values in *watts per
steradian per square metre per nanometre* (:math:`W/sr/m^2/nm`).
Examples
--------
>>> from colour.utilities import numpy_print_options
>>> with numpy_print_options(suppress=True):
... sd_blackbody(5000, shape=SpectralShape(400, 700, 20))
... # doctest: +ELLIPSIS
SpectralDistribution([[ 400. , 8742.5713329...],
[ 420. , 9651.6810212...],
[ 440. , 10447.3423137...],
[ 460. , 11121.8597759...],
[ 480. , 11673.7121534...],
[ 500. , 12106.0645344...],
[ 520. , 12425.4166118...],
[ 540. , 12640.4550541...],
[ 560. , 12761.1284859...],
[ 580. , 12797.9345572...],
[ 600. , 12761.3938171...],
[ 620. , 12661.6795247...],
[ 640. , 12508.3723863...],
[ 660. , 12310.3119640...],
[ 680. , 12075.5205176...],
[ 700. , 11811.1793602...]],
SpragueInterpolator,
{},
Extrapolator,
{'method': 'Constant', 'left': None, 'right': None})
"""
return SpectralDistribution(
planck_law(shape.wavelengths * 1e-9, temperature, c1, c2, n) * 1e-9,
shape.wavelengths,
name=f"{temperature}K Blackbody",
)
[docs]
def rayleigh_jeans_law(wavelength: ArrayLike, temperature: ArrayLike) -> NDArrayFloat:
"""
Approximate the spectral radiance of a blackbody as a function of
wavelength at specified thermodynamic temperature :math:`T[K]` according
to the *Rayleigh-Jeans* law.
Parameters
----------
wavelength
Wavelength in meters.
temperature
Temperature :math:`T[K]` in kelvin degrees.
Returns
-------
:class:`numpy.ndarray`
Radiance in *watts per steradian per square metre*
(:math:`W/sr/m^2`).
Warnings
--------
The :func:`colour.colorimetry.rayleigh_jeans_law` definition behaviour
with n-dimensional arrays is unusual: The ``wavelength`` and
``temperature`` parameters are first raveled using :func:`numpy.ravel`.
Then, they are *broadcasted* together by transposing the
``temperature`` parameter. Finally, and for convenience, the return
value is squeezed using :func:`numpy.squeeze`.
Notes
-----
- The *Rayleigh-Jeans* law agrees with experimental results at large
wavelengths (low frequencies) but strongly disagrees at short
wavelengths (high frequencies). This inconsistency between
observations and the predictions of classical physics is commonly
known as the *ultraviolet catastrophe*.
- The following implementation is expressed in terms of wavelength.
- The SI unit of radiance is *watts per steradian per square metre*
(:math:`W/sr/m^2`).
References
----------
:cite:`Wikipedia2003f`
Examples
--------
>>> rayleigh_jeans_law(500 * 1e-9, 5500) # doctest: +ELLIPSIS
np.float64(728478884562351.5...)
>>> rayleigh_jeans_law(500 * 1e-9, [5000, 5500, 6000])
... # doctest: +ELLIPSIS
array([6.6225353...e+14, 7.2847888...e+14, 7.9470423...e+14])
"""
l = as_float_array(wavelength) # noqa: E741
t = as_float_array(temperature)
l = np.ravel(l)[..., None] # noqa: E741
t = np.ravel(t)[None, ...]
c = CONSTANT_LIGHT_SPEED
k_B = CONSTANT_BOLTZMANN
B = (2 * c * k_B * t) / (l**4)
return as_float(np.squeeze(B))
[docs]
def sd_rayleigh_jeans(
temperature: float,
shape: SpectralShape = SPECTRAL_SHAPE_DEFAULT,
) -> SpectralDistribution:
"""
Generate the spectral distribution of the planckian radiator for the
specified temperature :math:`T[K]` with values in *watts per steradian
per square metre per nanometre* (:math:`W/sr/m^2/nm`) according to the
*Rayleigh-Jeans* law.
Parameters
----------
temperature
Temperature :math:`T[K]` in kelvins.
shape
Spectral shape used to create the spectral distribution of the
planckian radiator.
Returns
-------
:class:`colour.SpectralDistribution`
Blackbody spectral distribution with values in *watts per steradian
per square metre per nanometre* (:math:`W/sr/m^2/nm`).
Notes
-----
- The *Rayleigh-Jeans* law agrees with experimental results at large
wavelengths (low frequencies) but strongly disagrees at short
wavelengths (high frequencies). This inconsistency between
observations and the predictions of classical physics is commonly
known as the *ultraviolet catastrophe*.
Examples
--------
>>> from colour.utilities import numpy_print_options
>>> with numpy_print_options(suppress=True):
... sd_rayleigh_jeans(5000, shape=SpectralShape(400, 700, 20))
... # doctest: +ELLIPSIS
SpectralDistribution([[ 400. , 1616829.9106941...],
[ 420. , 1330169.9688456...],
[ 440. , 1104316.5840408...],
[ 460. , 924427.7490112...],
[ 480. , 779721.2146480...],
[ 500. , 662253.5314203...],
[ 520. , 566097.0941823...],
[ 540. , 486776.1157138...],
[ 560. , 420874.0917050...],
[ 580. , 365756.7299433...],
[ 600. , 319373.8095198...],
[ 620. , 280115.7588306...],
[ 640. , 246708.6655722...],
[ 660. , 218136.6091932...],
[ 680. , 193583.6389284...],
[ 700. , 172390.0279623...]],
SpragueInterpolator,
{},
Extrapolator,
{'method': 'Constant', 'left': None, 'right': None})
"""
return SpectralDistribution(
rayleigh_jeans_law(shape.wavelengths * 1e-9, temperature) * 1e-9,
shape.wavelengths,
name=f"{temperature}K Rayleigh-Jeans",
)
