Download Kirchhoff law

Black Body radiation
Basic radiation laws
Radiation and Climate Change FS 2016 Martin Wild
Black Body radiation
Concept introduced by Gustav Kirchhoff (1860),
German Physicist, 1824-1887
Professor of Physics, Heidelberg/Berlin
Definition: A blackbody is an idealized physical body that absorbs all
incident electromagnetic radiation
Emission of a blackbody is only a function of temperature at a given
wavelenght
The concept of the black body is an idealization, as perfect black bodies
do not exist in nature (=> graybody: part of the radiation is reflected)
Radiation and Climate Change FS 2016 Martin Wild
Basic radiation laws: Kirchhoff law
Thermal Equilibrium: Tblackbody = Tgreybody (2nd law of thermodynamics)
rλ reflectivity, aλ absorbtivity, Eλ Greybody emission, Bλ Blackbody emission
Black coated cavity:
(Hohlraum)
aλ = 1
aλ = 1 − rλ
Between the bodies, the radiation is exactly balanced:
Cavity effect:
Multiple reflections before
beam can leave the cavity
Radiation and Climate Change FS 2016 Martin Wild
Bλ = Eλ + rλ Bλ
or
Eλ
= 1− rλ ≡ aλ
Bλ
⇒ Eλ = aλ Bλ
where aλ is the spectral absorptivity (or emissivity) of the grey body
(Kirchhoff Law)
Radiation and Climate Change FS 2016 Martin Wild
Basic radiation laws: Kirchhoff law
Basic radiation laws: Kirchhoff law
1860: Kirchhoff law:
Eλ
= Bλ ≡ Fλ (T )
aλ
aλ : absorptivity (coefficient of absorption): fraction of incident power that is
absorbed by the body
Eλ : Emissive power of the body
aλ= 1/3
aλ= 2/3
A good absorber is a
good emitter
A weak absorber is a
weak emitter
For a body of any arbitrary material in thermodynamic equilibrium, the ratio of
its emissive power Eλ to its dimensionless coefficient of absorption aλ is at
a given wavelength equal to a universal function only of temperature
In case of a blackbody
Eλ = aλ Bλ
Radiation and Climate Change FS 2016 Martin Wild
⇒ aλ = 1, Eλ = Fλ (T ) ≡ Bλ (T )
Radiation and Climate Change FS 2016 Martin Wild
Max Planck
Planck Law
The Planck Law describes the monochromatic radiance of the emitted
radiation Bλ of a black body as a function of its surface temperature T:
Bλ (T ) =
2hc 2
⎛
⎛ hc ⎞ ⎞
λ5 ⎜ exp⎜
⎟ − 1⎟
⎝ λkT ⎠ ⎠
⎝
where c is the speed of light, λ the wavelength in µm , k the Boltzmann
constant (1.381*10-23 JK-1) and h the Planck constant (6.626*10-34 Js).
Unit: Wm-2 per µm per sterad (Wm-2µm-1sr-1)
German Physicist 1858-1947
Theoretically determined radiation emitted by a black body Bλ(T)
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
Planck Law
Wien’s displacement law
The determination of the maximum of Bλ by differentiation with respect to λ
2hc 2
Bλ (T) =
# # hc & &
λ5 %exp%
( −1(
$ $ λkT ' '
dBλ (T)
=0
dλ
yields Wien’s displacement law:
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Increasing the temperature increases the intensity at all wavelengths
Total area under the curve increases as temperature increases, corresponding to
increased total emission as the object becomes hotter.
Maximum intensity shifts to shorter wavelengths with increasing temperatures
Function defined over entire electromagnetic spectrum but significant values are
only obtained in a limited spectral range.
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•  the higher the temperature, the shorter the wavelength with peak emission
•  dominant wavelength of radiation emitted by a blackbody is inversely
proportional to its temperature.
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
Wien’s displacement law
Wien’s displacement law
The determination of the maximum of Bλ by differentiation with respect to λ
dBλ (T)
=0
dλ
yields Wien’s displacement law:
Application to Sun and Earth:
Sun:
effective temperature 5777 K
=> peak emission:
2.8978 *10-3 mK/5777K
= 0.50 *10-6m=0.5µm
Earth surface:
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effective temperature 290 K
•  the higher the temperature, the shorter the wavelength with peak emission
•  dominant wavelength of radiation emitted by a blackbody is inversely
proportional to its temperature.
Radiation and Climate Change FS 2016 Martin Wild
=>peak emission:
2.8978 *10-3 mK/290K
=0.10*10-4m=10 µm
Radiation and Climate Change FS 2016 Martin Wild
Implications Planck and Wien Law (I)
Earth receives energy in the shorter wavelength portions of the
spectrum, while it loses its energy in the longwave portions of
the spectrum > clear separation possible around 4 µm.
Ø Distinction between
incoming and outgoing
energy is made easy for
planets such as Earth
Implications Planck and Wien Law (II)
If sun had a lower temperature > strongest intensity would no
longer be in visible range, but in near infrared
> human eye developed during evolution to profit maximally
from sunlight
Ø This would not be the case
on very hot planets that could
radiate at some several
thousand degrees K.
Radiation and Climate Change FS 2016 Martin Wild
Spectral sensitivity of the human eye
Radiation and Climate Change FS 2016 Martin Wild
Implications Planck and Wien Law (III)
Practical evidence
eye
“Photometry” is the
measurement of visible-band
light, weighted by the spectral
response function of the human
eye.
light-adapted state, known as photopic
dark-adapted state, known as scotopic
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If you turn on a heating plate it will first emit in the infrared (felt as heat), but
as it gets hotter the wavelength of its radiation shifts to shorter wavelengths
that will eventually be visible in red.
⇒  only very hot objects emit
radiation that we can actually see
⇒  most objects we encounter in our
daily life are much too cold to emit in
the visible.
“Radiometry” is the
measurement of the entire
climate relevant spectrum
(0.1µm-100µm).
Radiation and Climate Change FS 2016 Martin Wild
Stefan-Boltzmann law
Stefan-Boltzmann law
Integration of Planck law
Already experimentally derived by Stefan in 1879, more than 20 years
before Planck developed his theory
2hc 2
Bλ (T) =
# # hc & &
λ5 %exp%
( −1(
$ $ λkT ' '
Over the entire spectrum yields the total radiance:
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λ =∞
B (T)= ∫ B (T)dλ =
TOT
λ
λ =0
2k 4 π 4 4
T
15c 2 h 3
Jožef Stefan
1835-1893
F = σT 4
B
Ludwig
Boltzmann
1844-1906
Unit Wm-2
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The total irradiance (radiant emittance) FB at the surface of a blackbody thus
becomes (multiplication with π, cf. Exercice 1)
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F=
B
2k 4 π 4 4 2k 4 π 5 4
T π = 2 3 T = σT 4
2 3
15c h
15c h
Emission increases non-linearly with increasing temperature: Doubling
temperature leads to 16x higher emission
Stefan-Boltzman Law
F2T σ (2T) 4 16σT 4
=
=
FT
σT 4
σT 4
where σ=5.67*10-8 Wm-2K-4
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Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
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Stefan-Boltzmann law
Stefan-Boltzmann law
Application: Emission from Sun versus Emission from Earth
Sensitivity of Emission to temperature changes:
Comparison of emissions from Sun and Earth surfaces per unit area:
4
F = σT 4
4
FSun
σTSun
σ (6000)
=
= (20) 4 = 160'000
4 =
4
FEarth σTEarth
σ (300)
B
(plus surface of Sun is more than 10‘000 x larger than surface of Earth)
Radiation and Climate Change FS 2016 Martin Wild
B
Sensitivities of emission to temperature changes for typical Earth surface
conditions:
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Sun radiates about 160‘000 times more energy per square meter than
Earth
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dF
= 4σT 3
dT
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288 K (global mean)
=> dF/dT = 4 * 5.67*10-8 Wm-2K-4 *2883K3= 5.4 Wm-2/K
300 K (tropics)
=> dF/dT = 4 * 5.67*10-8 Wm-2K-4 *3003 K3= 6.1 Wm-2/K
273 K (0°C)
=> dF/dT = 4 * 5.67*10-8 Wm-2K-4 *2733 K3= 4.6 Wm-2/K
250 K (poles)
=> dF/dT = 4 * 5.67*10-8 Wm-2K-4 *2503 K3= 3.5 Wm-2/K
Radiation and Climate Change FS 2016 Martin Wild
Stefan-Boltzmann law
Application: Temperature response to Volcanic Eruption
Assume a volcanic eruption that decreases solar radiation incident at the
surface by 4 Wm-2 (typical value for an eruption like Mt. Pinatubo in 1991).
How will the surface temperature react in equilibrium?
Tropics: dF/dT=6 Wm-2K-1
dT = dF / 6 Wm-2K-1, with dF =4 Wm-2 from volcano
=> dT = 4 Wm-2 / 6 Wm-2K-1 = 0.66K required to reduce surface emission
by 4 Wm-2 to balance lower insolation.
Stefan-Boltzmann law
Application: Temperature response to increasing CO2
A doubling of CO2 (300 ppm > 600 ppm) (without any further feedbacks)
would lead to an increase in downward thermal radiation by 1.2 Wm-2
(Ramanathan 1982). What would be the surface temperature response to
equilibrate this additional energy a) in the tropics b)at the poles?
a) Tropics: dF/dT= 6 Wm-2K-1
dT= dF/6 Wm-2K-1 , with dF=1.2 Wm-2 => dT= 0.20 K required to
compensate for the additional greenhouse gas forcing
b) Poles: dF/dT=3.5 Wm-2K-1
dT= dF/3.5 Wm-2K-1 , with dF=1.2 Wm-2 => dT= 0.35 K required to
compensate for the additional greenhouse gas forcing
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
Stefan-Boltzmann law
Stefan-Boltzmann law
In case of a grey body:
Radiation temperature:
F = εσT 4
Temperature that a blackbody requires to match with its emission a
given radiation field FB, even if that does not stem from a blackbody
(“blackbody equivalent temperature” of a greybody)
B
where ε = Emissivity
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F =σT ⇒ T =
0<ε<1
B
4
4
F
B
σ
Earth surface: ε approx. 0.98
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
2. Sun Earth-Relationships
2.1 Sun as central energy source
2.1 Sun as central energy source
2.2 Celestial mechanics
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
The Sun
Solar Factoids (I)
•  The sun, a medium-size star in the milky way galaxy, consisting of
about 300 billion stars.
•  A gaseous sphere of radius about 695‘500 km (about 109 times of
Earth radius) => by far the largest object in the solar system
•  Mass: 1.989 * 1030kg (99.8% of total mass of solar system)
The Sun
Solar Factoids (II)
•  Sun consists of 3 parts of hydrogen, one part of helium. Proportion
changes over time.
•  Sun‘s energy output is produced in the core of the sun by nuclear
reactions (fusion of four hydrogen (H) atoms into one helium (He)
atom).
•  Sun is about 4.5 billion years old. Since its birth it has used up about
half of the hydrogen in its core.
Our Sun
•  Sufficient fuel remains for the Sun to continue radiating "peacefully"
for another 5 billion years (although its luminosity will approximately
double over that period), but eventually it will run out of hydrogen
fuel.
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
The Sun
The Sun
Solar Factoids (III)
•  The Sun's energy output is 3.84 * 1017Gigawatts:
(a typical nuclear power plant produces 1 Gigawatt)
•  The outer 500 km of the sun (“photosphere“) emits most of radiation
received on Earth
•  Radiation emitted by the photosphere closely approximates that of a
blackbody of 5777K
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild
Emission of Sun
Solar fusion
Binding energy per nucleon in He core: 1.1*10-12 J
Effective surface temperature of the sun: 5777 K
=> Emission Bs (per m2) at the sun surface (Stefan-Boltzman law):
Bs = σ T4=5.67 10-8 Wm-2K-4*(5777 K)4= 6.32*107 Wm-2
⇒  Energy generated by one fusion reaction combining 4 H nuclei into one He
core: 4 *1.1 10-12 J = 4.4 10-12J
Total energy per second emitted by sun: ETOT=3.84*1026W (Js-1)
=> Total emission of Sun ETOT:
ETOT =4 π
rs2 Bs
⇒  Number of fusion reactions per second required:
with rs=6.955
*108m=
radius of the sun:
4 * 3.14* (6.955 108m)2*6.32 107 Wm-2 =3.84 1026W= 3.84 1014 TerraW
cf. World‘s energy consumption: 1.5
1013
ETOT/ energy generated per fusion = 3.84 1026Js-1/ 4.4 10-12 J = 0.9 1038s-1
1 proton mass= 1.67*10-27kg
=> Per single fusion reaction 4*1.67*10-27 kg of H is consumed.
W = 15 TerraW
Area on Sun surface required to cover world‘s energy cosumption:
Total amount of H consumed in the Sun per second:
1.5 1013 W / Bs = 1.5 1013 W / 6.3 107 Wm-2=2.5 105m2=0.25 km2.
= number of fusion reactions * amount of H consumed per reaction =
=>if we could harvest energy directly on the sun surface, 0.25
would be sufficient to cover world‘s energy demands.
Radiation and Climate Change FS 2016 Martin Wild
km2
0.9*1038s-1 * 4*1.67e-27 kg = 6 *1011kg= 600 Mio Tons
=> Every second 600 Mio Tons of H are transformed to He
Radiation and Climate Change FS 2016 Martin Wild
Solar radiation reaching planet Earth
S=1366Wm-2
Total emission ETOT of Sun:
More generally, if a planet is at distance rp from the sun, then the solar
irradiance Sp (in Wm-2) onto the planet is:
ETOT = 4 π rs2 * Bs
⇒  Total Emission of Sun (in W) spread out
over a sphere (in m2) with radius a, where
a= Earth-Sun Distance (semi major axis of
Earth’s orbit, 149.6 * 109m), determines the
Solar irradiance S per m2 at the Top of
the Earth’s atmosphere (Solar Constant)
at distance a :
Solar radiation: inverse square law
SP =
a
rs
4πrs2 Bs c
= 2 with c = rs2 Bs
2
4πrp
rp
Intensity of solar irradiance decreases with distance according to
inverse square law.
S = 4 π rs2 Bs / (4 π a2) = (rs/a)2 Bs
=(6.955*108m / 149.6*109m)2*6.32*107 Wm-2 = 1366 Wm-2
Current best estimate from measurements: 1361 Wm-2
5 Wm-2 deviation may due to difference from ideal black body and
measurement uncertainties
Radiation and Climate Change FS 2016 Martin Wild
Solar radiation: inverse square law
Application to other planets:
SP =
4πrs2 Bs c
= 2 with c = rs2 Bs
2
4πrp
rp
Sp = Solar constant of Planet P at distance rp from the Sun
rs =6.955 *108m radius of Sun
Bs=6.32*107 Wm-2 Emission at Sun’s surface
=>c=3.057* 1025W
Planet
Distance from
Sun (109 m)
Intensity of solar
radiation (Wm-2)
Venus
108
2620
Earth
149.6
1366
Mars
228
558
Radiation and Climate Change FS 2016 Martin Wild
Radiation and Climate Change FS 2016 Martin Wild