black body radiation basic radiation laws: kirchhoff …...basic radiation laws: kirchhoff law...

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)

Black Body radiation


Cavity effect: Multiple reflections before beam can leave the cavity

Black coated cavity: (Hohlraum)

Between the bodies, the radiation is exactly balanced: where aλ is the spectral absorptivity (or emissivity) of the grey body (Kirchhoff Law) Radiation and Climate Change FS 2016 Martin Wild

Thermal Equilibrium: Tblackbody = Tgreybody (2nd law of thermodynamics) rλ reflectivity, aλ absorbtivity, Eλ Greybody emission, Bλ Blackbody emission

Basic radiation laws: Kirchhoff law

Bλ = Eλ + rλBλ or Eλ

Bλ=1− rλ ≡ aλ ⇒ Eλ = aλBλ

aλ =1 aλ =1− rλ



A good absorber is a good emitter

A weak absorber is a weak emitter

Eλ = aλBλ

aλ= 2/3 aλ= 1/3


1860: Kirchhoff law:

aλ : absorptivity (coefficient of absorption): fraction of incident power that is

absorbed by the body

Eλ : Emissive power of the body

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 ≡ Fλ (T )

⇒ aλ =1, Eλ = Fλ (T ) ≡ Bλ (T )

Max Planck


German Physicist 1858-1947 Theoretically determined radiation emitted by a black body Bλ(T)

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:

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)

⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛=

1exp

2)(

5

2

kThchcTB

λλ

λ

Planck Law


€

Bλ (T) =2hc2

λ5 exp hcλkT#

$ %

&

' ( −1

#

$ %

&

' (

•  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.

Wien’s displacement law

The determination of the maximum of Bλ by differentiation with respect to λ yields Wien’s displacement law:

•  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

€

dBλ (T)dλ

= 0


The determination of the maximum of Bλ by differentiation with respect to λ yields Wien’s displacement law:

•  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

€

dBλ (T)dλ

= 0



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:

effective temperature 290 K

=>peak emission:

2.8978 *10-3 mK/290K

=0.10*10-4m=10 µm


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

Ø This would not be the case on very hot planets that could radiate at some several thousand degrees K.

Implications Planck and Wien Law (I)


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

Implications Planck and Wien Law (II)

15

Spectral sensitivity of the human eye

eye

“Photometry” is the measurement of visible-band light, weighted by the spectral

response function of the human eye.

“Radiometry” is the

measurement of the entire climate relevant spectrum

(0.1µm-100µm). light-adapted state, known as photopic

dark-adapted state, known as scotopic

Implications Planck and Wien Law (III)


Practical evidence 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.

Stefan-Boltzmann law Integration of Planck law

Over the entire spectrum yields the total radiance:

The total irradiance (radiant emittance) FB at the surface of a blackbody thus becomes (multiplication with π, cf. Exercice 1)

Stefan-Boltzman Law

where σ=5.67*10-8 Wm-2K-4


€

BTOT (T )= Bλ

λ=0

λ=∞

∫ (T)dλ = 2k4π 4

15c2h3T4€

Bλ (T) =2hc2

λ5 exp hcλkT#

$ %

&

' ( −1

#

$ %

&

' (

€

FB = 2k4π 4

15c2h3T4π = 2k

4π 5

15c2h3T4 =σT4

Jožef Stefan 1835-1893

Ludwig Boltzmann 1844-1906

Already experimentally derived by Stefan in 1879, more than 20 years before Planck developed his theory

Emission increases non-linearly with increasing temperature: Doubling temperature leads to 16x higher emission

Stefan-Boltzmann law


€

FB =σT4

€

F2TFT

=σ (2T)4

σT 4 =16σT 4

σT 4

Unit Wm-2

Application: Emission from Sun versus Emission from Earth Comparison of emissions from Sun and Earth surfaces per unit area:

Sun radiates about 160‘000 times more energy per square meter than

Earth

(plus surface of Sun is more than 10‘000 x larger than surface of Earth)


€

FSun

FEarth

=σTSun

4

σTEarth4 =

σ (6000)4

σ (300)4= (20)4 =160'000

Stefan-Boltzmann law Stefan-Boltzmann law

Sensitivity of Emission to temperature changes: Sensitivities of emission to temperature changes for typical Earth surface

conditions: 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


€

FB =σT4

€

dFB

dT= 4σT3

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.



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




In case of a grey body: where ε = Emissivity 0 < ε < 1 Earth surface: ε approx. 0.98


€

FB =εσT4



44

σσ B

FTTFB =⇒=

Radiation temperature: 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)

2. Sun Earth-Relationships 2.1 Sun as central energy source 2.2 Celestial mechanics


2.1 Sun as central energy source


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)

Our Sun

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.

•  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.

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


The Sun

Emission of Sun

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

=> Total emission of Sun ETOT:

ETOT =4 π rs2 Bs 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 W = 15 TerraW

Area on Sun surface required to cover world‘s energy cosumption: 1.5 1013 W / Bs = 1.5 1013 W / 6.3 107 Wm-2=2.5 105m2=0.25 km2.

=>if we could harvest energy directly on the sun surface, 0.25 km2 would be sufficient to cover world‘s energy demands.


Binding energy per nucleon in He core: 1.1*10-12 J ⇒  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) ⇒  Number of fusion reactions per second required: 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.

Total amount of H consumed in the Sun per second: = number of fusion reactions * amount of H consumed per reaction = 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


Solar fusion

Total emission ETOT of Sun:

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 :


a

S=1366Wm-2

Solar radiation reaching planet Earth

rs

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

More generally, if a planet is at distance rp from the sun, then the solar irradiance Sp (in Wm-2) onto the planet is:

Intensity of solar irradiance decreases with distance according to inverse square law.


sspp

ssP Brcwith

rc

rBrS 2

22

2

44

===ππ

Solar radiation: inverse square law

Application to other planets:

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

sspp

ssP Brcwith

rc

rBrS 2

22

2

44

===ππ

Solar radiation: inverse square law

black body radiation basic radiation laws: kirchhoff …...basic radiation laws: kirchhoff law...

Documents