Blackbody radiation specific intensity (i.e., energy density and spectrum shape) is completely specified the formula Planck's law (AKA blackbody spectrum) which we present below.
Image 1 shows blackbody radiation spectra for a range of temperatures as aforesaid above.
Note, the name blackbody radiation is historical for reasons discussed in Blackbody file: hohlraum.html. However, it is not a very good name since blackbody radiation can look quite bright.
Thermodynamic-equilibrium radiation would be the best name, but it is not used.
All closed systems reach thermodynamic equilibrium (i.e., the state of maximum entropy) as dictated by the 2nd law of thermodynamics as long as random processes exist that redistribute the system energy among microscopic states.
Recall, thermodynamic equilibrium is a timeless state where nothing is changing at the macroscopic scale.
However, the concept of thermodynamic equilibrium can be extended to systems that evolve in time slowly enough that at any instant they can be regarded as in thermodynamic equilibrium. The slow evolutionary processes are called quasistatic processes. Also complex systems can often be modeled as consisting of subsystems that are their own systems in thermodynamic equilibrium and that interact with each other by quasistatic processes.
So the concept of thermodynamic equilibrium does have broad application. Of course, there are many systems that cannot be treated using thermodynamic equilibrium.
But this is NOT true.
Local regions in any system that are optically thick in absorption opacity and are at least approximately at one temperature will have a local radiation field that is a blackbody radiation to some degree of exactness. The last statement has an exception: crystalline solid emission which we briefly discuss below. It is not important in astrophysics usually.
Note, opacity is the property of matter (or radiation itself for very high temperatures) of stopping the flow of electromagnetic radiation (EMR). Absorption opacity is opacity that destroys photons and converts their energy into local heat energy. The term "optically thick" is explicated below.
We discuss examples of cases with nearly exact blackbody radiation in the items below.
A perfectly reflecting hohlraum is rather hard to build, but the observable universe is effectively that.
To explicate: The cosmic microwave background (CMB) is a blackbody radiation field that permeates the whole observable universe. It formed when the observable universe became transparent to radiation at what is called the recombination era at cosmic time t = 377,770(3200) Jyr = 1.192*10**13 s; cosmic redshift z = 1089.80(21), where time zero is the Big Bang. Initially, the CMB had temperature ≅ 3000 K, but the expansion of the universe has caused it to cool to CMB temperature = 2.72548(57) K (Fixsen 2009) at cosmic present = to the age of the observable universe = 13.797(23) Gyr (Planck 2018).
The CMB is in fact very nearly perfect blackbody radiation field. It formed that way and it has stayed that way since it has almost not interacted with matter since the recombination era. Occasionally, CMB photons scatter off free electrons or get absorbed by a star, planet, or an observational detector, but ∼ 95 % of them have NOT been in thermal contact with matter since the recombination era (Wikipedia: Λ-CDM model: Parameters). There actually is gravitational modulation of the CMB called the Sachs-Wolfe effect which is an important diagnostic of the evolution of the observable universe, but it is beyond the scope of this figure to discuss it.
Note, the temperatures of matter and non-CMB electromagnetic radiation (EMR) (if they have a definable temperature at all) are extremely diverse.
You may wonder where is the container for the universe hohlraum. It may not exist if the universe is unbounded or may be so far off beyond the radius of the observable that it has no detectable effect on the radiation field in the observable universe.
The CMB is an extremely important fundamental fact of modern cosmology. It is one of the most important evidences for the Big Bang theory.
What is rare are bodies radiating blackbody radiation from their optically thick surfaces (which have uniform temperatures) into transparent surroundings with radiation fields that are much lower in energy density than blackbody radiation at said surface temperature.
Such bodies are what is usually meant by the term blackbody radiators.
Blackbody radiators and their much more common relatives the modulated dilute blackbody radiators are discussed in some detail in File: Blackbody file: hohlraum.html.
However, one of the key reasons is that main-sequence stars (theoretically understood to be those stars nuclearly burning hydrogen (H) to helium (He) in their cores) approximate blackbody radiatiors. To be more exact, stellar photospheres (i.e., the layers from which photons can escape to infinity) approximate blackbody radiators. (See, e.g., Hu-21--22 for more on stellar photospheres.) Actually, stars that are main-sequence stars also emit radiation that approximates blackbody radiation, but they do not have well defined photospheres to describe as blackbody radiators.
That this is so allows us to estimate the effective temperature (a sort of average photospheric temperature) using Wien's law. We discuss Wien's law below.
This kind of EMR
is thermal radiation.
Reflection is a process
in which the
EMR does NOT
strongly interact with a body and only briefly and temporarily changes its surface state in the
ideal case.
In reflection,
EMR just "bounces off"
the body surface.
Scattering is sort of reflection at the
microscopic scale.
In scattering, photons interact
with a particle and are redirected somehow, but do not change their energy significantly
nor the heat energy of the particle significantly.
Of course, bodies usually emit
thermal radiation
AND reflect
and/or scatter
EMR
at the same time.
Recall continuum means there are
NO "missing points". The
real numbers are
a continuum of
numbers.
Integers
and rational numbers are
NOT continuums
of numbers.
Optically thick in absorption opacity
means that photons
cannot travel far in a body without being absorbed into its heat energy.
If a body is optically thin body in absorption opacity and is emitting EMR
from its own energy sources, then it generates
a emission line spectrum
(see Wikipedia: Emission line spectrum).
The emission lines are very narrow emission bands.
In fact, there is always a low background emission continuum as well as on which the
emission line spectrum is superimposed.
Bodies of dilute gases will be optically thick or thin depending on how
big they are.
If they are optically thin, then no matter what their size they will radiate
an emission line spectrum.
Laboratory samples of dilute gases are almost always optically thin,
and so radiate emission line spectra.
However, if the temperature of
blackbody radiation
is sufficiently low,
then yes blackbody radiation
will look black to the
human eye.
As temperature
increases,
a blackbody spectrum
increases at all wavelengths,
except at zero and
infinite
wavelength.
Thus, blackbody spectra
of different temperatures
only intersect at
zero
and
infinite
wavelength.
A blackbody spectrum
has single maximum
and the maximum shifts
blueward/redward
in wavelength
with increasing/decreasing
temperature.
Actually,
blackbody radiation and
thermal radiation
are often used loosely as
synonyms
including by yours truly.
Thermal radiation
that is NOT
blackbody radiation just
being thought of as mixed-temperature
blackbody radiation (dilute or not).
The temperature
of a spherical exact blackbody radiator
that gives the same emitted
radiant flux
integrated over all wavelength
as an actual main-sequence star
(which have geometrically thin photospheres which means there is a well defined photospheric radius
to use for the temperature calculation of the blackbody radiator)
is called the
star's
effective temperature.
A main-sequence star's
effective temperature
is usually a useful average
photospheric temperature.
Pre-main-sequence stars
and post-main-sequence stars
usually do NOT well approximate
blackbody radiators
(Hu-21--22).
And, of course, many
astro-bodies do NOT
radiate at all like blackbody radiators
overall:
e.g., planets which usually
emit mostly reflected
light though they usually
have fraction of their emission which has a
blackbody spectrum (diluted or not).
Below we give some blackbody radiation formulae and some explication of them.
where the
Planck constant h = 6.62607015*10**(-34) J·s (exactly),
the vacuum light speed c = 2.99792458*10**8 m/s
(exactly) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns,
and the x = hc/(kTλ),
where the
Boltzmann contant k = 1.380649*10**(-23) J/K (exactly)
and T is temperature
(in kelvins (K)),
and λ is wavelength.
For reference for the fundamental physical
constants,
see also NIST:
Fundamental Physical Constants.
Note, Planck's law
can be regarded as a function
of wavelength λ
with temperature T as
a parameter or
as a function
of both wavelength λ and
temperature T.
Rayleigh-Jeans law
asymptotically agrees with
the correct Planck's law in the
limit that wavelength λ goes to
infinity.
But as λ goes to zero, the
Rayleigh-Jeans law predicts
infinite
energy.
This is the famous
ultraviolet catastrophe which,
noticed immediately in 1900,
was a clue that
classical physics progressively fails
as you move away from what we now call the
classical limit.
In fact, Planck's law,
which gives the correct
formula for
blackbody radiation, was
also published in 1900
(Wikipedia:
Planck's law: Finding the empirical law), but
its full understanding from quantum mechanics
developed later.
The blackbody spectrum for
T = 5777 K approximates the
solar spectrum
as seen above the
Earth's atmosphere
since T = 5777 K approximates the
solar photosphere effective temperature = 5772 K (current best value).
Effective temperature
of a spherically-symmetric body
(e.g., star or
planet)
is the temperature
an exact blackbody radiator
(with a radius equal to the
defined radius of the body)
that radiates exactly as much
radiant flux
integrated over all wavelength as the body.
Effective temperatures
for stars are usually
good average temperatures
for their stellar photospheres.
The blackbody spectrum for
T = 300 K approximates the
average Earth atmospheric temperature
≅ 14 C ≅ 287 K and
the average human body temperature
≅ 37 C ≅ 310 K.
However, to give the short explication of Wien's law
for a blackbody spectrum,
Wien's law states
that the maximizing
wavelength
λ_max = hc/(x_max*kT) = (2897.771955185172661 ... μm-K)/T,
and so is
inversely proportional
to temperature T: i.e.,
λ_max ∝ 1/T.
Note, x_max = hc/(kTλ_max) = (4.965114231744276303 ...)
and
temperature as
a function of
maximizing
wavelength obeys
the inverse Wien's law
T = hc/(x_max*λ_max*k)=(2897.771955185172661 ... μm-K)/λ_max.
If we rewrite
Planck's law
as a function of λ and x, then x becomes the parameter rather than T.
Then if we specialize to x = x_max, we obtain
where we have suppressed the subscript max on the wavelengths for simplicity.
On the log-log plot in Image 2, the
B_λ(x = x_max) curve will intersect the B_λ(T) curves just
at their maxima because Wien's law
λ_max = hc/(x_max*kT) = (2897.771955185172661 ... μm-K)/T implies
as λ_max changes for fixed x_max as the temperature T of the
blackbody radiation field changes.
Note, the B_λ(x = x_max) curve is a
power law
with power p = -5,
and so would give if plotted
a straight line
on the log-log plot in
Image 2
with slope = -5.
A rough calculation from the
log-log plot values gives
We need to remark that location of
the maximum
of a spectrum of any kind
is actually representation dependent.
For example, for the
solar spectrum,
the maximum
in the
wavelength representation is 0.502 μm,
in the logarithmic representation is 0.635 μm,
and
in the frequency representation is 0.883 μm
(Wikimedia Commons:
File:Spectral Distribution of Sunlight.svg;
Wikipedia:
Sunlight: Composition and power).
For more on spectrum representations,
see
Blackbody file:
representation.html
and
J. Marr
2012, "A Better Presentation of Planck's Radiation Law"
(also at J. Marr
2012, "A Better Presentation of Planck's Radiation Law").
What color does the
human eye
with its peculiar
psychophysical response
see for a blackbody spectrum?
Well, if the
blackbody spectrum
has sufficiently low
temperature
(which means its
maximum in any representation
is far redward
from the
visible band (fiducial range 0.400--0.700 μm = 400--700 nm)),
the human eye sees nothing.
(Recall, for spectrum representation,
see Blackbody file:
representation.html.)
However, above "sufficiently low temperature",
the psychophysical response
to blackbody spectrum
is given below in
Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation.
In fact, from Image 2, it is clear that the
maximum
in the wavelength representation for
blackbody spectra
for temperature ⪅ 3000 K is
always in the
infrared band (fiducial range 0.7 μm -- 0.1 cm)
or redward.
Thus, even
blackbody spectra
that look whitish hot
at temperature ⪅ 1500 K
are, in fact, dominated by
light
from the redward end of the
visible band (fiducial range 0.400--0.700 μm = 400--700 nm).
Note there is NO "green hot" or "violet hot" in
Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation.
Of course, we see
green and
violet,
but that is NEVER our
psychophysical response
to the mixture of
colors in
blackbody spectra (diluted or not).
The lack of "green hot" is particularly striking
since the
psychophysical luminosity function
(photopic peak 0.555 μm, green band, scotopic peak 0.498 μm, green band)
(i.e., human eye
response to visible band light)
peaks for both
photopic vision (bright light) and
scotopic vision (dim light) in the
green band (fiducial range 0.495--0.570 μm).
At least for yours truly at this moment in
2025, the
colors
of stars
from both
ground-based astronomy
and space-based astronomy
is an unclear subject.
Explication of the origins of the unclearness in points:
The explication of the origins of unclearness given above probably explains why the
star
colors
for given
effective temperatures
do NOT agree exactly with
the blackbody radiator
colors for given
blackbody radiation
temperatures in
Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation.
See
blackbody keywords
below
(local link /
general link: blackbody_keywords.html).
2hc**2 1 2hc**2 1
B_λ(T) = ------ ---------- = ------ ----------------- ,
λ**5 [exp(x)-1] λ**5 [exp(hc/(kTλ))-1]
2hc**2 1
B_λ(x = x_max)= ------ -------------- ∝ λ**(-5) ,
λ**5 [exp(x_max)-1]
slope = (-1-9)/(1.5-0.5) = -5
using the fact that log(0.3) ≅ -1/2 and log(30) ≅ 3/2.
We get the exact result -5
fortuitously.
Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation
NOTE: The table is NOT an authoritative reference. It was compiled and conflated from
the references given below which were NOT entirely consistent and do
NOT give exact specifications.
T (K) T (C) Psychophysical Response Color
⪅ 750 ⪅ 480 invisibile
753 480 faint red glow (red hot)
853 580 dark red (red hot)
1003 730 bright red, slightly orange (orange red hot)
1203 930 bright orange (orange hot)
1373 1100 pale yellowish orange (yellow orange hot)
1573 1300 yellowish white (yellow hot)
1673 1400 white (yellowish if seen from a distance through atmosphere, (white hot)
6500 white (white hot)
8000 pale blue (white-pale-blue hot)
9000 pale blue (white-pale-blue hot)
>10000 blue-white (white-blue hot)
References:
The upshot of this discussion is that provisionally
yours truly accepts that
apparent star
colors in
ground-based astronomy
obey the following results with
the cited temperatures
probably being
effective temperatures:
Reference: Star Colors Explained,
Brian Ventrudo, 2008 December 23.
