"""
Blackbody - Planckian Radiator
==============================
Defines the 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 numpy as np
from colour.colorimetry import (
SPECTRAL_SHAPE_DEFAULT,
SpectralDistribution,
SpectralShape,
)
from colour.constants import CONSTANT_BOLTZMANN, CONSTANT_LIGHT_SPEED
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:
"""
Return the spectral radiance of a blackbody as a function of wavelength at
thermodynamic temperature :math:`T[K]` in a medium having 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=3,741771x10.16\\ 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 aI., 1991)*, namely
:math:`c2=1,4388x10.2\\ m/K`.
n
Medium index of refraction. For dry air at 15C 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:`W/sr/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:`W/sr/m^2`).
References
----------
:cite:`CIETC1-482004i`
Examples
--------
>>> planck_law(500 * 1e-9, 5500) # doctest: +ELLIPSIS
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:
"""
Return the spectral distribution of the planckian radiator for given
temperature :math:`T[K]` with values in
*watts per steradian per square metre per nanometer* (:math:`W/sr/m^2/nm`).
Parameters
----------
temperature
Temperature :math:`T[K]` in kelvin degrees.
shape
Spectral shape used to create the spectral distribution of the
planckian radiator.
c1
The official value of :math:`c1` is provided by the Committee on Data
for Science and Technology (CODATA) and is
:math:`c1=3,741771x10.16\\ 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 aI., 1991)*, namely
:math:`c2=1,4388x10.2\\ m/K`.
n
Medium index of refraction. For dry air at 15C 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 nanometer*
(: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:
"""
Return the approximation of the spectral radiance of a blackbody as a
function of wavelength at thermodynamic temperature :math:`T[K]` according
to *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
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:
"""
Return the spectral distribution of the planckian radiator for given
temperature :math:`T[K]` with values in
*watts per steradian per square metre per nanometer* (:math:`W/sr/m^2/nm`)
according to *Rayleigh-Jeans* law.
Parameters
----------
temperature
Temperature :math:`T[K]` in kelvin degrees.
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 nanometer*
(: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",
)