Black-body radiation
Black-body radiation is the thermal electromagnetic radiation within
or surrounding a body in thermodynamic equilibrium with its
environment, or emitted by a black body (an opaque and non-reflective
body). It has a specific spectrum and intensity that depends only on
the body's temperature, which is assumed for the sake of calculations
and theory to be uniform and constant.[1][2][3][4]
The thermal radiation spontaneously emitted by many ordinary objects
can be approximated as black-body radiation. A perfectly insulated
enclosure that is in thermal equilibrium internally contains
black-body radiation and will emit it through a hole made in its wall,
provided the hole is small enough to have negligible effect upon the
equilibrium.
A black-body at room temperature appears black, as most of the energy
it radiates is infra-red and cannot be perceived by the human eye.
Because the human eye cannot perceive colour at very low light
intensities, a black body, viewed in the dark at the lowest just
faintly visible temperature, subjectively appears grey, even though
its objective physical spectrum peaks in the infrared range.[5] When
it becomes a little hotter, it appears dull red. As its temperature
increases further it becomes yellow, white, and ultimately blue-white.
Although planets and stars are neither in thermal equilibrium with
their surroundings nor perfect black bodies, black-body radiation is
used as a first approximation for the energy they emit.[6] Black holes
are near-perfect black bodies, in the sense that they absorb all the
radiation that falls on them. It has been proposed that they emit
black-body radiation (called Hawking radiation), with a temperature
that depends on the mass of the black hole.[7]
The term black body was introduced by
Gustav Kirchhoff
Gustav Kirchhoff in 1860.
Black-body radiation
Black-body radiation is also called thermal radiation, cavity
radiation, complete radiation or temperature radiation.
Contents
1 Spectrum
2 Explanation
3 Equations
3.1
Planck's law
Planck's law of black-body radiation
3.2 Wien's displacement law
3.3 Stefan–Boltzmann law
4 Human-body emission
5
Temperature
Temperature relation between a planet and its star
5.1 Virtual temperature of Earth
6 Cosmology
7 Doppler effect for a moving black body
8 History
8.1 Balfour Stewart
8.2 Gustav Kirchhoff
9 See also
10 References
10.1 Bibliography
11 Further reading
12 External links
Spectrum[edit]
Black-body radiation
Black-body radiation has a characteristic, continuous frequency
spectrum that depends only on the body's temperature,[8] called the
Planck spectrum or Planck's law. The spectrum is peaked at a
characteristic frequency that shifts to higher frequencies with
increasing temperature, and at room temperature most of the emission
is in the infrared region of the electromagnetic spectrum.[9][10][11]
As the temperature increases past about 500 degrees Celsius, black
bodies start to emit significant amounts of visible light. Viewed in
the dark by the human eye, the first faint glow appears as a "ghostly"
grey (the visible light is actually red, but low intensity light
activates only the eye's grey-level sensors). With rising temperature,
the glow becomes visible even when there is some background
surrounding light: first as a dull red, then yellow, and eventually a
"dazzling bluish-white" as the temperature rises.[12][13] When the
body appears white, it is emitting a substantial fraction of its
energy as ultraviolet radiation. The Sun, with an effective
temperature of approximately 5800 K,[14] is an approximate black body
with an emission spectrum peaked in the central, yellow-green part of
the visible spectrum, but with significant power in the ultraviolet as
well.
Black-body radiation
Black-body radiation provides insight into the thermodynamic
equilibrium state of cavity radiation. If each
Fourier mode
Fourier mode of the
equilibrium radiation in an otherwise empty cavity with perfectly
reflective walls is considered as a degree of freedom capable of
exchanging energy, then, according to the equipartition theorem of
classical physics, there would be an equal amount of energy in each
mode. Since there are an infinite number of modes this implies
infinite heat capacity (infinite energy at any non-zero temperature),
as well as an unphysical spectrum of emitted radiation that grows
without bound with increasing frequency, a problem known as the
ultraviolet catastrophe. Instead, in quantum theory the occupation
numbers of the modes are quantized, cutting off the spectrum at high
frequency in agreement with experimental observation and resolving the
catastrophe. The study of the laws of black bodies and the failure of
classical physics to describe them helped establish the foundations of
quantum mechanics.
Explanation[edit]
Color of a black body from 800 K to 12200 K. This range of colors
approximates the range of colors of stars of different temperatures,
as seen or photographed in the night sky.
All normal (baryonic) matter emits electromagnetic radiation when it
has a temperature above absolute zero. The radiation represents a
conversion of a body's thermal energy into electromagnetic energy, and
is therefore called thermal radiation. It is a spontaneous process of
radiative distribution of entropy.
Conversely all normal matter absorbs electromagnetic radiation to some
degree. An object that absorbs all radiation falling on it, at all
wavelengths, is called a black body. When a black body is at a uniform
temperature, its emission has a characteristic frequency distribution
that depends on the temperature. Its emission is called black-body
radiation.
The concept of the black body is an idealization, as perfect black
bodies do not exist in nature.[15]
Graphite
Graphite and lamp black, with
emissivities greater than 0.95, however, are good approximations to a
black material. Experimentally, black-body radiation may be
established best as the ultimately stable steady state equilibrium
radiation in a cavity in a rigid body, at a uniform temperature, that
is entirely opaque and is only partly reflective.[15] A closed box of
graphite walls at a constant temperature with a small hole on one side
produces a good approximation to ideal black-body radiation emanating
from the opening.[16][17]
Black-body radiation
Black-body radiation has the unique absolutely stable distribution of
radiative intensity that can persist in thermodynamic equilibrium in a
cavity.[15] In equilibrium, for each frequency the total intensity of
radiation that is emitted and reflected from a body (that is, the net
amount of radiation leaving its surface, called the spectral radiance)
is determined solely by the equilibrium temperature, and does not
depend upon the shape, material or structure of the body.[18] For a
black body (a perfect absorber) there is no reflected radiation, and
so the spectral radiance is entirely due to emission. In addition, a
black body is a diffuse emitter (its emission is independent of
direction). Consequently, black-body radiation may be viewed as the
radiation from a black body at thermal equilibrium.
Black-body radiation
Black-body radiation becomes a visible glow of light if the
temperature of the object is high enough. The
Draper point is the
temperature at which all solids glow a dim red, about
7002798000000000000♠798 K.[19] At
7003100000000000000♠1000 K, a small opening in the wall of a
large uniformly heated opaque-walled cavity (let us call it an oven),
viewed from outside, looks red; at 7003600000000000000♠6000 K,
it looks white. No matter how the oven is constructed, or of what
material, as long as it is built so that almost all light entering is
absorbed by its walls, it will contain a good approximation to
black-body radiation. The spectrum, and therefore color, of the light
that comes out will be a function of the cavity temperature alone. A
graph of the amount of energy inside the oven per unit volume and per
unit frequency interval plotted versus frequency, is called the
black-body curve. Different curves are obtained by varying the
temperature.
The temperature of a Pāhoehoe lava flow can be estimated by observing
its color. The result agrees well with other measurements of
temperatures of lava flows at about 1,000 to 1,200 °C (1,830 to
2,190 °F).
Two bodies that are at the same temperature stay in mutual thermal
equilibrium, so a body at temperature T surrounded by a cloud of light
at temperature T on average will emit as much light into the cloud as
it absorbs, following Prevost's exchange principle, which refers to
radiative equilibrium. The principle of detailed balance says that in
thermodynamic equilibrium every elementary process works equally in
its forward and backward sense.[20][21] Prevost also showed that the
emission from a body is logically determined solely by its own
internal state. The causal effect of thermodynamic absorption on
thermodynamic (spontaneous) emission is not direct, but is only
indirect as it affects the internal state of the body. This means that
at thermodynamic equilibrium the amount of every wavelength in every
direction of thermal radiation emitted by a body at temperature T,
black or not, is equal to the corresponding amount that the body
absorbs because it is surrounded by light at temperature T.[22]
When the body is black, the absorption is obvious: the amount of light
absorbed is all the light that hits the surface. For a black body much
bigger than the wavelength, the light energy absorbed at any
wavelength λ per unit time is strictly proportional to the black-body
curve. This means that the black-body curve is the amount of light
energy emitted by a black body, which justifies the name. This is the
condition for the applicability of Kirchhoff's law of thermal
radiation: the black-body curve is characteristic of thermal light,
which depends only on the temperature of the walls of the cavity,
provided that the walls of the cavity are completely opaque and are
not very reflective, and that the cavity is in thermodynamic
equilibrium.[23] When the black body is small, so that its size is
comparable to the wavelength of light, the absorption is modified,
because a small object is not an efficient absorber of light of long
wavelength, but the principle of strict equality of emission and
absorption is always upheld in a condition of thermodynamic
equilibrium.
In the laboratory, black-body radiation is approximated by the
radiation from a small hole in a large cavity, a hohlraum, in an
entirely opaque body that is only partly reflective, that is
maintained at a constant temperature. (This technique leads to the
alternative term cavity radiation.) Any light entering the hole would
have to reflect off the walls of the cavity multiple times before it
escaped, in which process it is nearly certain to be absorbed.
Absorption occurs regardless of the wavelength of the radiation
entering (as long as it is small compared to the hole). The hole,
then, is a close approximation of a theoretical black body and, if the
cavity is heated, the spectrum of the hole's radiation (i.e., the
amount of light emitted from the hole at each wavelength) will be
continuous, and will depend only on the temperature and the fact that
the walls are opaque and at least partly absorptive, but not on the
particular material of which they are built nor on the material in the
cavity (compare with emission spectrum).
Calculating the black-body curve was a major challenge in theoretical
physics during the late nineteenth century. The problem was solved in
1901 by
Max Planck
Max Planck in the formalism now known as
Planck's law
Planck's law of
black-body radiation.[24] By making changes to Wien's radiation law
(not to be confused with Wien's displacement law) consistent with
thermodynamics and electromagnetism, he found a mathematical
expression fitting the experimental data satisfactorily. Planck had to
assume that the energy of the oscillators in the cavity was quantized,
i.e., it existed in integer multiples of some quantity. Einstein built
on this idea and proposed the quantization of electromagnetic
radiation itself in 1905 to explain the photoelectric effect. These
theoretical advances eventually resulted in the superseding of
classical electromagnetism by quantum electrodynamics. These quanta
were called photons and the black-body cavity was thought of as
containing a gas of photons. In addition, it led to the development of
quantum probability distributions, called
Fermi–Dirac statistics
Fermi–Dirac statistics and
Bose–Einstein statistics, each applicable to a different class of
particles, fermions and bosons.
The wavelength at which the radiation is strongest is given by Wien's
displacement law, and the overall power emitted per unit area is given
by the Stefan–Boltzmann law. So, as temperature increases, the glow
color changes from red to yellow to white to blue. Even as the peak
wavelength moves into the ultra-violet, enough radiation continues to
be emitted in the blue wavelengths that the body will continue to
appear blue. It will never become invisible—indeed, the radiation of
visible light increases monotonically with temperature.[25] The
Stefan–Boltzmann law
Stefan–Boltzmann law also says that the total radiant heat energy
emitted from a surface is proportional to the fourth power of its
absolute temperature. The law was formulated by Josef Stefan in 1879
and later derived by Ludwig Boltzmann. The formula E = σT4 is given,
where E is the radiant heat emitted from a unit of area per unit time,
T is the absolute temperature, and σ =
6992567036700000000♠5.670367×10−8 W·m−2⋅K−4 is the
Stefan–Boltzmann constant
Stefan–Boltzmann constant .[26]
The radiance or observed intensity is not a function of direction.
Therefore, a black body is a perfect Lambertian radiator.
Real objects never behave as full-ideal black bodies, and instead the
emitted radiation at a given frequency is a fraction of what the ideal
emission would be. The emissivity of a material specifies how well a
real body radiates energy as compared with a black body. This
emissivity depends on factors such as temperature, emission angle, and
wavelength. However, it is typical in engineering to assume that a
surface's spectral emissivity and absorptivity do not depend on
wavelength, so that the emissivity is a constant. This is known as the
gray body assumption.
9-year
WMAP
WMAP image (2012) of the cosmic microwave background radiation
across the universe.[27][28]
With non-black surfaces, the deviations from ideal black-body behavior
are determined by both the surface structure, such as roughness or
granularity, and the chemical composition. On a "per wavelength"
basis, real objects in states of local thermodynamic equilibrium still
follow Kirchhoff's Law: emissivity equals absorptivity, so that an
object that does not absorb all incident light will also emit less
radiation than an ideal black body; the incomplete absorption can be
due to some of the incident light being transmitted through the body
or to some of it being reflected at the surface of the body.
In astronomy, objects such as stars are frequently regarded as black
bodies, though this is often a poor approximation. An almost perfect
black-body spectrum is exhibited by the cosmic microwave background
radiation.
Hawking radiation
Hawking radiation is the hypothetical black-body radiation
emitted by black holes, at a temperature that depends on the mass,
charge, and spin of the hole. If this prediction is correct, black
holes will very gradually shrink and evaporate over time as they lose
mass by the emission of photons and other particles.
A black body radiates energy at all frequencies, but its intensity
rapidly tends to zero at high frequencies (short wavelengths). For
example, a black body at room temperature
(7002300000000000000♠300 K) with one square meter of surface
area will emit a photon in the visible range (390–750 nm) at an
average rate of one photon every 41 seconds, meaning that for most
practical purposes, such a black body does not emit in the visible
range.[citation needed]
Equations[edit]
Planck's law
Planck's law of black-body radiation[edit]
Main article: Planck's law
Planck's law
Planck's law states that[29]
B
ν
(
T
)
=
2
h
ν
3
c
2
1
e
h
ν
k
T
−
1
,
displaystyle B_ nu (T)= frac 2hnu ^ 3 c^ 2 frac 1 e^
frac hnu kT -1 ,
where
Bν(T) is the spectral radiance (the power per unit solid angle and
per unit of area normal to the propagation) density of frequency ν
radiation per unit frequency at thermal equilibrium at temperature T.
h is the Planck constant;
c is the speed of light in a vacuum;
k is the Boltzmann constant;
ν is the frequency of the electromagnetic radiation;
T is the absolute temperature of the body.
For a black body surface the spectral radiance density (defined per
unit of area normal to the propagation) is independent of the angle
θ
displaystyle theta
of emission with respect to the normal. However, this means that,
following Lambert's cosine law,
B
ν
(
T
)
cos
θ
displaystyle B_ nu (T)cos theta
is the radiance density per unit area of emitting surface as the
surface area involved in generating the radiance is increased by a
factor
1
/
cos
θ
displaystyle 1/cos theta
with respect to an area normal to the propagation direction. At
oblique angles, the solid angle spans involved do get smaller,
resulting in lower aggregate intensities.
Wien's displacement law[edit]
Wien's displacement law
Wien's displacement law shows how the spectrum of black-body radiation
at any temperature is related to the spectrum at any other
temperature. If we know the shape of the spectrum at one temperature,
we can calculate the shape at any other temperature. Spectral
intensity can be expressed as a function of wavelength or of
frequency.
A consequence of
Wien's displacement law
Wien's displacement law is that the wavelength at
which the intensity per unit wavelength of the radiation produced by a
black body is at a maximum,
λ
max
displaystyle lambda _ max
, is a function only of the temperature:
λ
max
=
b
T
,
displaystyle lambda _ max = frac b T ,
where the constant b, known as Wien's displacement constant, is equal
to 6997289777290000000♠2.8977729(17)×10−3 K m.[30]
Planck's law
Planck's law was also stated above as a function of frequency. The
intensity maximum for this is given by
ν
max
=
T
×
1.04
×
10
11
H
z
/
K
displaystyle nu _ max =Ttimes 1.04times 10^ 11 mathrm Hz
/mathrm K
.[31]
Stefan–Boltzmann law[edit]
By integrating
B
ν
(
T
)
displaystyle B_ nu (T)
over the frequency the integrated radiance
L
displaystyle L
is
L
=
2
π
5
15
k
4
T
4
c
2
h
3
1
π
=:
σ
T
4
1
π
displaystyle L= frac 2pi ^ 5 15 frac k^ 4 T^ 4 c^ 2 h^ 3
frac 1 pi =:sigma T^ 4 frac 1 pi
by using
∫
0
∞
d
x
x
3
e
x
−
1
=
π
4
15
displaystyle int _ 0 ^ infty dx, frac x^ 3 e^ x -1 = frac
pi ^ 4 15
with
x
≡
h
ν
k
T
displaystyle xequiv frac hnu kT
and with
σ
≡
2
π
5
15
k
4
c
2
h
3
=
5.670373
×
10
−
8
W
m
2
K
4
displaystyle sigma equiv frac 2pi ^ 5 15 frac k^ 4 c^ 2
h^ 3 =5.670373times 10^ -8 frac W m^ 2 K^ 4
being the Stefan–Boltzmann constant. The radiance
L
displaystyle L
is then
σ
T
4
cos
θ
π
displaystyle sigma T^ 4 frac cos theta pi
per unit of emitting surface.
On a side note, at a distance d, the intensity
d
I
displaystyle dI
per area
d
A
displaystyle dA
of radiating surface is the useful expression
d
I
=
σ
T
4
cos
θ
π
d
2
d
A
displaystyle dI=sigma T^ 4 frac cos theta pi d^ 2 dA
when the receiving surface is perpendicular to the radiation.
By subsequently integrating over the solid angle
Ω
displaystyle Omega
(where
θ
<
π
/
2
displaystyle theta <pi /2
) the
Stefan–Boltzmann law
Stefan–Boltzmann law is calculated, stating that the power j*
emitted per unit area of the surface of a black body is directly
proportional to the fourth power of its absolute temperature:
j
⋆
=
σ
T
4
,
displaystyle j^ star =sigma T^ 4 ,
by using
∫
cos
θ
d
Ω
=
∫
0
2
π
∫
0
π
/
2
cos
θ
sin
θ
d
θ
d
ϕ
=
π
.
displaystyle int cos theta ,dOmega =int _ 0 ^ 2pi int _ 0 ^ pi
/2 cos theta sin theta ,dtheta ,dphi =pi .
Human-body emission[edit]
Much of a person's energy is radiated away in the form of infrared
light. Some materials are transparent in the infrared, but opaque to
visible light, as is the plastic bag in this infrared image (bottom).
Other materials are transparent to visible light, but opaque or
reflective in the infrared, noticeable by the darkness of the man's
glasses.
The human body radiates energy as infrared light. The net power
radiated is the difference between the power emitted and the power
absorbed:
P
net
=
P
emit
−
P
absorb
.
displaystyle P_ text net =P_ text emit -P_ text absorb .
Applying the Stefan–Boltzmann law,
P
net
=
A
σ
ε
(
T
4
−
T
0
4
)
,
displaystyle P_ text net =Asigma varepsilon left(T^ 4 -T_ 0 ^ 4
right),
where A and T are the body surface area and temperature,
ε
displaystyle varepsilon
is the emissivity, and T0 is the ambient temperature.
The total surface area of an adult is about 2 m2, and the mid- and
far-infrared emissivity of skin and most clothing is near unity, as it
is for most nonmetallic surfaces.[32][33] Skin temperature is about
33 °C,[34] but clothing reduces the surface temperature to about
28 °C when the ambient temperature is 20 °C.[35] Hence,
the net radiative heat loss is about
P
net
=
100
W
.
displaystyle P_ text net =100~ text W .
The total energy radiated in one day is about 8 MJ, or 2000 kcal (food
calories).
Basal metabolic rate
Basal metabolic rate for a 40-year-old male is about 35
kcal/(m2·h),[36] which is equivalent to 1700 kcal per day, assuming
the same 2 m2 area. However, the mean metabolic rate of sedentary
adults is about 50% to 70% greater than their basal rate.[37]
There are other important thermal loss mechanisms, including
convection and evaporation. Conduction is negligible – the Nusselt
number is much greater than unity.
Evaporation
Evaporation by perspiration is only
required if radiation and convection are insufficient to maintain a
steady-state temperature (but evaporation from the lungs occurs
regardless). Free-convection rates are comparable, albeit somewhat
lower, than radiative rates.[38] Thus, radiation accounts for about
two-thirds of thermal energy loss in cool, still air. Given the
approximate nature of many of the assumptions, this can only be taken
as a crude estimate. Ambient air motion, causing forced convection, or
evaporation reduces the relative importance of radiation as a
thermal-loss mechanism.
Application of Wien's law to human-body emission results in a peak
wavelength of
λ
peak
=
2.898
×
10
−
3
K
⋅
m
305
K
=
9.50
μ
m
.
displaystyle lambda _ text peak = frac 2.898times 10^ -3 ~ text
K cdot text m 305~ text K =9.50~mu text m .
For this reason, thermal imaging devices for human subjects are most
sensitive in the 7–14 micrometer range.
Temperature
Temperature relation between a planet and its star[edit]
Main article: Planetary equilibrium temperature
The black-body law may be used to estimate the temperature of a planet
orbiting the Sun.
Earth's longwave thermal radiation intensity, from clouds, atmosphere
and ground
The temperature of a planet depends on several factors:
Incident radiation from its star
Emitted radiation of the planet, e.g., Earth's infrared glow
The albedo effect causing a fraction of light to be reflected by the
planet
The greenhouse effect for planets with an atmosphere
Energy generated internally by a planet itself due to radioactive
decay, tidal heating, and adiabatic contraction due by cooling.
The analysis only considers the Sun's heat for a planet in a Solar
System.
The
Stefan–Boltzmann law
Stefan–Boltzmann law gives the total power (energy/second) the
Sun
Sun is emitting:
The Earth only has an absorbing area equal to a two dimensional disk,
rather than the surface of a sphere.
P
S
e
m
t
=
4
π
R
S
2
σ
T
S
4
(
1
)
displaystyle P_ rm S emt =4pi R_ rm S ^ 2 sigma T_ rm S ^ 4
qquad qquad (1)
where
σ
displaystyle sigma ,
is the Stefan–Boltzmann constant,
T
S
displaystyle T_ rm S ,
is the effective temperature of the Sun, and
R
S
displaystyle R_ rm S ,
is the radius of the Sun.
The
Sun
Sun emits that power equally in all directions. Because of this,
the planet is hit with only a tiny fraction of it. The power from the
Sun
Sun that strikes the planet (at the top of the atmosphere) is:
P
S
E
=
P
S
e
m
t
(
π
R
E
2
4
π
D
2
)
(
2
)
displaystyle P_ rm SE =P_ rm S emt left( frac pi R_ rm E ^
2 4pi D^ 2 right)qquad qquad (2)
where
R
E
displaystyle R_ rm E ,
is the radius of the planet and
D
displaystyle D,
is the distance between the
Sun
Sun and the planet.
Because of its high temperature, the
Sun
Sun emits to a large extent in
the ultraviolet and visible (UV-Vis) frequency range. In this
frequency range, the planet reflects a fraction
α
displaystyle alpha
of this energy where
α
displaystyle alpha
is the albedo or reflectance of the planet in the UV-Vis range. In
other words, the planet absorbs a fraction
1
−
α
displaystyle 1-alpha
of the Sun's light, and reflects the rest. The power absorbed by the
planet and its atmosphere is then:
P
a
b
s
=
(
1
−
α
)
P
S
E
(
3
)
displaystyle P_ rm abs =(1-alpha ),P_ rm SE qquad qquad (3)
Even though the planet only absorbs as a circular area
π
R
2
displaystyle pi R^ 2
, it emits equally in all directions as a sphere. If the planet were a
perfect black body, it would emit according to the Stefan–Boltzmann
law
P
e
m
t
b
b
=
4
π
R
E
2
σ
T
E
4
(
4
)
displaystyle P_ rm emt,bb =4pi R_ rm E ^ 2 sigma T_ rm E ^
4 qquad qquad (4)
where
T
E
displaystyle T_ rm E
is the temperature of the planet. This temperature, calculated for
the case of the planet acting as a black body by setting
P
a
b
s
=
P
e
m
t
b
b
displaystyle P_ rm abs =P_ rm emt,bb
, is known as the effective temperature. The actual temperature of the
planet will likely be different, depending on its surface and
atmospheric properties. Ignoring the atmosphere and greenhouse effect,
the planet, since it is at a much lower temperature than the Sun,
emits mostly in the infrared (IR) portion of the spectrum. In this
frequency range, it emits
ϵ
¯
displaystyle overline epsilon
of the radiation that a black body would emit where
ϵ
¯
displaystyle overline epsilon
is the average emissivity in the IR range. The power emitted by the
planet is then:
P
e
m
t
=
ϵ
¯
P
e
m
t
b
b
(
5
)
displaystyle P_ rm emt = overline epsilon ,P_ rm emt,bb
qquad qquad (5)
For a body in radiative exchange equilibrium with its surroundings,
the rate at which it emits radiant energy is equal to the rate at
which it absorbs it:[39][40]
P
a
b
s
=
P
e
m
t
(
6
)
displaystyle P_ rm abs =P_ rm emt qquad qquad (6)
Substituting the expressions for solar and planet power in equations
1–6 and simplifying yields the estimated temperature of the planet,
ignoring greenhouse effect, TP:
T
P
=
T
S
R
S
1
−
α
ε
¯
2
D
(
7
)
displaystyle T_ P =T_ S sqrt frac R_ S sqrt frac 1-alpha
overline varepsilon 2D qquad qquad (7)
In other words, given the assumptions made, the temperature of a
planet depends only on the surface temperature of the Sun, the radius
of the Sun, the distance between the planet and the Sun, the albedo
and the IR emissivity of the planet.
Notice that a gray (flat spectrum) ball where
(
1
−
α
)
=
ε
¯
displaystyle ( 1-alpha )= overline varepsilon
comes to the same temperature as a black body no matter how dark or
light gray .
Virtual temperature of Earth[edit]
Substituting the measured values for the
Sun
Sun and Earth yields:
T
S
=
5778
K
,
displaystyle T_ rm S =5778 mathrm K ,
[41]
R
S
=
6.96
×
10
8
m
,
displaystyle R_ rm S =6.96times 10^ 8 mathrm m ,
[41]
D
=
1.496
×
10
11
m
,
displaystyle D=1.496times 10^ 11 mathrm m ,
[41]
α
=
0.306
displaystyle alpha =0.306
[42]
With the average emissivity
ε
¯
displaystyle overline varepsilon
set to unity, the effective temperature of the Earth is:
T
E
=
254.356
K
displaystyle T_ rm E =254.356 mathrm K
or −18.8 °C.
This is the temperature of the Earth if it radiated as a perfect black
body in the infrared, assuming an unchanging albedo and ignoring
greenhouse effects (which can raise the surface temperature of a body
above what it would be if it were a perfect black body in all
spectrums[43]). The Earth in fact radiates not quite as a perfect
black body in the infrared which will raise the estimated temperature
a few degrees above the effective temperature. If we wish to estimate
what the temperature of the Earth would be if it had no atmosphere,
then we could take the albedo and emissivity of the Moon as a good
estimate. The albedo and emissivity of the Moon are about 0.1054[44]
and 0.95[45] respectively, yielding an estimated temperature of about
1.36 °C.
Estimates of the Earth's average albedo vary in the range 0.3–0.4,
resulting in different estimated effective temperatures. Estimates are
often based on the solar constant (total insolation power density)
rather than the temperature, size, and distance of the Sun. For
example, using 0.4 for albedo, and an insolation of 1400 W m−2, one
obtains an effective temperature of about 245 K.[46] Similarly using
albedo 0.3 and solar constant of 1372 W m−2, one obtains an
effective temperature of 255 K.[47][48][49]
Cosmology[edit]
The cosmic microwave background radiation observed today is the most
perfect black-body radiation ever observed in nature, with a
temperature of about 2.7 K.[50] It is a "snapshot" of the
radiation at the time of decoupling between matter and radiation in
the early universe. Prior to this time, most matter in the universe
was in the form of an ionized plasma in thermal, though not full
thermodynamic, equilibrium with radiation.
According to Kondepudi and Prigogine, at very high temperatures (above
1010 K; such temperatures existed in the very early universe),
where the thermal motion separates protons and neutrons in spite of
the strong nuclear forces, electron-positron pairs appear and
disappear spontaneously and are in thermal equilibrium with
electromagnetic radiation. These particles form a part of the black
body spectrum, in addition to the electromagnetic radiation.[51]
Doppler effect for a moving black body[edit]
The relativistic Doppler effect causes a shift in the frequency f of
light originating from a source that is moving in relation to the
observer, so that the wave is observed to have frequency f':
f
′
=
f
1
−
v
c
cos
θ
1
−
v
2
/
c
2
,
displaystyle f'=f frac 1- frac v c cos theta sqrt 1-v^ 2
/c^ 2 ,
where v is the velocity of the source in the observer's rest frame, θ
is the angle between the velocity vector and the observer-source
direction measured in the reference frame of the source, and c is the
speed of light.[52] This can be simplified for the special cases of
objects moving directly towards (θ = π) or away (θ = 0) from the
observer, and for speeds much less than c.
Through
Planck's law
Planck's law the temperature spectrum of a black body is
proportionally related to the frequency of light and one may
substitute the temperature (T) for the frequency in this equation.
For the case of a source moving directly towards or away from the
observer, this reduces to
T
′
=
T
c
−
v
c
+
v
.
displaystyle T'=T sqrt frac c-v c+v .
Here v > 0 indicates a receding source, and v < 0 indicates an
approaching source.
This is an important effect in astronomy, where the velocities of
stars and galaxies can reach significant fractions of c. An example is
found in the cosmic microwave background radiation, which exhibits a
dipole anisotropy from the Earth's motion relative to this black-body
radiation field.
History[edit]
Balfour Stewart[edit]
In 1858,
Balfour Stewart
Balfour Stewart described his experiments on the thermal
radiative emissive and absorptive powers of polished plates of various
substances, compared with the powers of lamp-black surfaces, at the
same temperature.[22] Stewart chose lamp-black surfaces as his
reference because of various previous experimental findings,
especially those of
Pierre Prevost
Pierre Prevost and of John Leslie. He wrote
"Lamp-black, which absorbs all the rays that fall upon it, and
therefore possesses the greatest possible absorbing power, will
possess also the greatest possible radiating power." More an
experimenter than a logician, Stewart failed to point out that his
statement presupposed an abstract general principle, that there exist
either ideally in theory or really in nature bodies or surfaces that
respectively have one and the same unique universal greatest possible
absorbing power, likewise for radiating power, for every wavelength
and equilibrium temperature.
Stewart measured radiated power with a thermo-pile and sensitive
galvanometer read with a microscope. He was concerned with selective
thermal radiation, which he investigated with plates of substances
that radiated and absorbed selectively for different qualities of
radiation rather than maximally for all qualities of radiation. He
discussed the experiments in terms of rays which could be reflected
and refracted, and which obeyed the Stokes-Helmholtz reciprocity
principle (though he did not use an eponym for it). He did not in this
paper mention that the qualities of the rays might be described by
their wavelengths, nor did he use spectrally resolving apparatus such
as prisms or diffraction gratings. His work was quantitative within
these constraints. He made his measurements in a room temperature
environment, and quickly so as to catch his bodies in a condition near
the thermal equilibrium in which they had been prepared by heating to
equilibrium with boiling water. His measurements confirmed that
substances that emit and absorb selectively respect the principle of
selective equality of emission and absorption at thermal equilibrium.
Stewart offered a theoretical proof that this should be the case
separately for every selected quality of thermal radiation, but his
mathematics was not rigorously valid.[53] He made no mention of
thermodynamics in this paper, though he did refer to conservation of
vis viva. He proposed that his measurements implied that radiation was
both absorbed and emitted by particles of matter throughout depths of
the media in which it propagated. He applied the Helmholtz reciprocity
principle to account for the material interface processes as distinct
from the processes in the interior material. He did not postulate
unrealizable perfectly black surfaces. He concluded that his
experiments showed that in a cavity in thermal equilibrium, the heat
radiated from any part of the interior bounding surface, no matter of
what material it might be composed, was the same as would have been
emitted from a surface of the same shape and position that would have
been composed of lamp-black. He did not state explicitly that the
lamp-black-coated bodies that he used as reference must have had a
unique common spectral emittance function that depended on temperature
in a unique way.
Gustav Kirchhoff[edit]
In 1859, not knowing of Stewart's work, Gustav Robert Kirchhoff
reported the coincidence of the wavelengths of spectrally resolved
lines of absorption and of emission of visible light. Importantly for
thermal physics, he also observed that bright lines or dark lines were
apparent depending on the temperature difference between emitter and
absorber.[54]
Kirchhoff then went on to consider bodies that emit and absorb heat
radiation, in an opaque enclosure or cavity, in equilibrium at
temperature T.
Here is used a notation different from Kirchhoff's. Here, the emitting
power E(T, i) denotes a dimensioned quantity, the total radiation
emitted by a body labeled by index i at temperature T. The total
absorption ratio a(T, i) of that body is dimensionless, the ratio of
absorbed to incident radiation in the cavity at temperature T . (In
contrast with Balfour Stewart's, Kirchhoff's definition of his
absorption ratio did not refer in particular to a lamp-black surface
as the source of the incident radiation.) Thus the ratio E(T, i) /
a(T, i) of emitting power to absorption ratio is a dimensioned
quantity, with the dimensions of emitting power, because a(T, i) is
dimensionless. Also here the wavelength-specific emitting power of the
body at temperature T is denoted by E(λ, T, i) and the
wavelength-specific absorption ratio by a(λ, T, i) . Again, the ratio
E(λ, T, i) / a(λ, T, i) of emitting power to absorption ratio is a
dimensioned quantity, with the dimensions of emitting power.
In a second report made in 1859, Kirchhoff announced a new general
principle or law for which he offered a theoretical and mathematical
proof, though he did not offer quantitative measurements of radiation
powers.[55] His theoretical proof was and still is considered by some
writers to be invalid.[53][56] His principle, however, has endured: it
was that for heat rays of the same wavelength, in equilibrium at a
given temperature, the wavelength-specific ratio of emitting power to
absorption ratio has one and the same common value for all bodies that
emit and absorb at that wavelength. In symbols, the law stated that
the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) has one and
the same value for all bodies, that is for all values of index i . In
this report there was no mention of black bodies.
In 1860, still not knowing of Stewart's measurements for selected
qualities of radiation, Kirchhoff pointed out that it was long
established experimentally that for total heat radiation, of
unselected quality, emitted and absorbed by a body in equilibrium, the
dimensioned total radiation ratio E(T, i) / a(T, i), has one and the
same value common to all bodies, that is, for every value of the
material index i.[57] Again without measurements of radiative powers
or other new experimental data, Kirchhoff then offered a fresh
theoretical proof of his new principle of the universality of the
value of the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) at
thermal equilibrium. His fresh theoretical proof was and still is
considered by some writers to be invalid.[53][56]
But more importantly, it relied on a new theoretical postulate of
"perfectly black bodies," which is the reason why one speaks of
Kirchhoff's law. Such black bodies showed complete absorption in their
infinitely thin most superficial surface. They correspond to Balfour
Stewart's reference bodies, with internal radiation, coated with
lamp-black. They were not the more realistic perfectly black bodies
later considered by Planck. Planck's black bodies radiated and
absorbed only by the material in their interiors; their interfaces
with contiguous media were only mathematical surfaces, capable neither
of absorption nor emission, but only of reflecting and transmitting
with refraction.[58]
Kirchhoff's proof considered an arbitrary non-ideal body labeled i as
well as various perfect black bodies labeled BB . It required that the
bodies be kept in a cavity in thermal equilibrium at temperature T .
His proof intended to show that the ratio E(λ, T, i) / a(λ, T, i)
was independent of the nature i of the non-ideal body, however partly
transparent or partly reflective it was.
His proof first argued that for wavelength λ and at temperature T, at
thermal equilibrium, all perfectly black bodies of the same size and
shape have the one and the same common value of emissive power E(λ,
T, BB), with the dimensions of power. His proof noted that the
dimensionless wavelength-specific absorption ratio a(λ, T, BB) of a
perfectly black body is by definition exactly 1. Then for a perfectly
black body, the wavelength-specific ratio of emissive power to
absorption ratio E(λ, T, BB) / a(λ, T, BB) is again just E(λ, T,
BB), with the dimensions of power. Kirchhoff considered, successively,
thermal equilibrium with the arbitrary non-ideal body, and with a
perfectly black body of the same size and shape, in place in his
cavity in equilibrium at temperature T . He argued that the flows of
heat radiation must be the same in each case. Thus he argued that at
thermal equilibrium the ratio E(λ, T, i) / a(λ, T, i) was equal to
E(λ, T, BB), which may now be denoted Bλ (λ, T), a continuous
function, dependent only on λ at fixed temperature T, and an
increasing function of T at fixed wavelength λ, at low temperatures
vanishing for visible but not for longer wavelengths, with positive
values for visible wavelengths at higher temperatures, which does not
depend on the nature i of the arbitrary non-ideal body. (Geometrical
factors, taken into detailed account by Kirchhoff, have been ignored
in the foregoing.)
Thus
Kirchhoff's law of thermal radiation
Kirchhoff's law of thermal radiation can be stated: For any
material at all, radiating and absorbing in thermodynamic equilibrium
at any given temperature T, for every wavelength λ, the ratio of
emissive power to absorptive ratio has one universal value, which is
characteristic of a perfect black body, and is an emissive power which
we here represent by Bλ (λ, T) . (For our notation Bλ (λ, T),
Kirchhoff's original notation was simply e.)[57][59][60][61][62][63]
Kirchhoff announced that the determination of the function Bλ (λ, T)
was a problem of the highest importance, though he recognized that
there would be experimental difficulties to be overcome. He supposed
that like other functions that do not depend on the properties of
individual bodies, it would be a simple function. Occasionally by
historians that function Bλ (λ, T) has been called "Kirchhoff's
(emission, universal) function,"[64][65][66][67] though its precise
mathematical form would not be known for another forty years, till it
was discovered by Planck in 1900. The theoretical proof for
Kirchhoff's universality principle was worked on and debated by
various physicists over the same time, and later.[56] Kirchhoff stated
later in 1860 that his theoretical proof was better than Balfour
Stewart's, and in some respects it was so.[53] Kirchhoff's 1860 paper
did not mention the second law of thermodynamics, and of course did
not mention the concept of entropy which had not at that time been
established. In a more considered account in a book in 1862, Kirchhoff
mentioned the connection of his law with Carnot's principle, which is
a form of the second law.[68]
According to Helge Kragh, "Quantum theory owes its origin to the study
of thermal radiation, in particular to the "black-body" radiation that
Robert Kirchhoff had first defined in 1859–1860."[69]
See also[edit]
Bolometer
Color temperature
Infrared
Infrared thermometer
Photon
Photon polarization
Planck's law
Pyrometry
Rayleigh–Jeans law
Thermography
Sakuma–Hattori equation
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Further reading[edit]
Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.).
W. H. Freeman Company. ISBN 0-7167-1088-9.
Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H.
Freeman. ISBN 0-7167-4345-0.
External links[edit]
Black-body radiation
Black-body radiation JavaScript Interactives
Black-body radiation
Black-body radiation by
Fu-Kwun Hwang and Loo Kang Wee
Calculating Black-body Radiation Interactive calculator with Doppler
Effect. Includes most systems of units.
Color-to-
Temperature
Temperature demonstration at Academo.org
Cooling Mechanisms for Human Body – From Hyperphysics
Descriptions of radiation emitted by many different objects
Black-Body Emission Applet
"Blackbody Spectrum" by Jeff Bryant, Wolfram Demonstrati