The horizontal axis is wavelength of electromagnetic radiation in microns (μm). The vertical axis is specific intensity (energy per unit time per unit steradian (sr) per unit area perpendicular to the beam path per wavelength) in appropriate units (i.e., kW/sr/m**2/nm).
Below we explicate thermal radiation and its very important special case blackbody radiation.
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.
Of course, bodies usually emit thermal radiation AND reflect 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.
Actually crystalline solids like rocks can have broad emission and absorption bands (HI-92), and so do NOT necessarily radiate a continuum. We will NOT go into that detail here.
The blackbody spectra for 3 temperatures are illustrated in Image 1.
The blackbody spectra only depend on the temperature of the emitting body, NOT on any other properties: e.g., density (as long as sufficiently high), chemical composition, and shape. This remarkable fact is a result of thermodynamics.
A blackbody spectrum has single maximum and the maximum shifts blueward/redward with increasing/decreasing temperature.
In fact, the blackbody radiation itself has the temperature of the emitter. Below, we give the formula for blackbody radiation that shows the temperature dependence of blackbody radiation.
Actually, many examples of thermal radiation approximate blackbody radiation so closely that they are just considered blackbody radiation. In fact, thermal radiation can often be considered as consisting of a mixture blackbody radiations of different temperature.
Also actually, thermal radiation and blackbody radiation are often used loosely as synonyms including by yours truly. Thermal radiation that is NOT blackbody radiation just being thought of as non-ideal blackbody radiation.
However, if the temperature of the blackbody radiator is sufficiently low and there is NO reflection, then yes a blackbody radiator will look black to the human eye.
In any case, tradition (beginning in the 19th century) has stuck us with the name blackbody radiation. Thermodynamic equilibrium thermal radiation would be a more accurately descriptive, if longwinded, name.
2hc**2 1 2hc**2 1 B_λ(T) = ------ ---------- = ------ ----------------- , λ**5 [exp(x)-1] λ**5 [exp(hc/(kTλ))-1]
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 and λ is wavelength. For reference for the fundamental physical constants, see also NIST: Fundamental Physical Constants.
Note that 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 energy. This is the famous ultraviolet catastrophe which, noticed immediately in 1900, was a clue that classical physics failed outside of what we now call the classical limit.
In fact, Planck's law, which gives correct formula for blackbody radiation, was also published in 1900 (Wikipedia: Planck's law: Finding the empirical law), but its understanding in 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 body
(e.g., star or
planet)
is the temperature
a perfect blackbody radiator
(with a radius equal to the
defined radius of the body)
that radiates exactly as much
radiant flux 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.
where we constrain x = x_max = hc/(kTλ) = (4.965114231744276303 ...) which
is just a constant.
Note that B_λ[T(λ)]max is
power law
with power p = -5,
and so gives 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;
J. Marr
2012, "A Better Presentation of Planck's Radiation Law", also
J. Marr
2012, "A Better Presentation of Planck's Radiation Law").
If blackbody radiation
and the matter it is in contact with
have the same temperature, then
the two are in
thermodynamic equilibrium
with respect to each other.
The zero point of the Kelvin scale
(i.e., T = 0 K) is absolute zero---which is
the coldest possible state where all heat energy
that can be removed has been removed.
If true a blackbody radiator
was at absolute zero
(where there is no heat energy to radiate)
and was NOT reflective, it would be
"black" at all
wavelengths.
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
(i.e., its
maximum is far
redward
from the
visible band (fiducial range 0.400--0.700 μm = 400--700 nm)),
the human eye sees nothing.
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.
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.
The lack of "green hot" is particularly strking
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 a unclear subject.
Explication of the origins of the unclearness in points:
The explication given above 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.
2hc**2 1
B_λ[T(λ)]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.
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.
Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation
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:
In fact, from Image 2, it is clear that the
maximum 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 white hot
at temperatures ∼< 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).
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.
See blackbody keywords below (local link / general link: blackbody_keywords.html).
See the
Blackbody radiation videos
below
(local link /
general link: blackbody_videos.html).
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