# Blackbody radiation and the greenhouse effect

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```					       Blackbody radiation and
the greenhouse effect
Keywords
2. Radiation and the Earth atmosphere
3. Greenhouse effect (simple models)
4. Heat conduction/diffusion
5. Convection
Ronald Griessen 2008
Vrije Universiteit, Amsterdam                vrije Universiteit amsterdam

I give here the background information necessary for the understanding of pages
10-13, 18-23 and 295-299

1
The Sun

Ronald Griessen 2008
Vrije Universiteit, Amsterdam                vrije Universiteit amsterdam

There are three ways to transport heat: radiation, conduction and convection. We

2
564 million tons of hydrogen
are converted into 560 million
tons of helium per second

E=mc2

3
564 million tons of hydrogen
are converted into 560 million
tons of helium per second

E=mc2
3.8 x 1026 W
63 MW/m2

4
What heats the
1
H + 1 H → 2H + e + + ν e

e+ + e− → 2γ (1.02 MeV)

2
H + 1H → 3He + γ (5.49 MeV)

1 MeV=1.6 x 10-13 J
3
He +3He → 4He + 1H + 1H + 12.86 MeV

γ Gamma ray             Proton
ν Neutrino              Neutron
Positron

5
564 million tons of hydrogen
are converted into 560 million
tons of helium per second

Power [W/m2]= 5.67x10-8 T4

Power= 5.67x10-8 (5778)4=63.17 MW/m2

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1368 W/m2

63 MW/m2

7
Fuel shares of World Total Primary Energy Supply 2005

15.1 TW               4.8 1014 MJ

http://www.iea.org/textbase/nppdf/free/2007/Key_Stats_2007.pdf

The contribution of renewable energy sources has increased from
(1.8%+10.6%+0.1%=12.5%) in 1973 to (2.2%+10%+0.5%=12.7%) in 2005. In
percentage this is not impressive. One should however keep in mind that the total
energy consumption doubled between 1973 and 2005.

8
10960 km2 give 15 TW

0.24 km2 give 15 TW

9
Solar area Sahara
Future world power
consumption
1010 persons × 5 kW/person
•   Solar constant: 1350 W/m2
•   50% reaches the Earth’s surface
•   50% is day
•   Efficiency of photovoltaics: 10%.

148 m2/person

1.48 × 106 km2

10
10960 km2 give 15 TW

0.24 km2 give 15 TW

11

Ronald Griessen 2008
Vrije Universiteit, Amsterdam                vrije Universiteit amsterdam

There are three ways to transport heat: radiation, conduction and convection. We

12

13
Hot body

1.5
Spectral energy density dE/df

1.0
300 K
600 K
0.5                        900 K
1200 K

0.0

0.0   0.5   1.0    1.5    2.0     2.5   3.0
Photon energy [eV]

There are three main properties that characterize thermal radiation:
Thermal radiation at a single temperature occurs at a wide range of frequencies.
How much of each frequency is given by Planck’s law of radiation (for idealized
materials). This is shown by the curves in the diagram for four different
temperatures, 300, 600, 900 and 1200 K The peak photon energy( or frequency or
color) of the emitted radiation increases as the temperature increases. For example,
a red hot object radiates mostly low energy photons (i.e. long wavelength light)
which is why it appears red. If it heats up further, the main frequency shifts to the
middle of the visible band, and the spread of frequencies mentioned in the first point
make it appear white. We then say the object is white hot.
The total amount of radiation, of all frequencies, goes up very fast as the
temperature rises (it grows as T4, where T is the absolute temperature of the body).
An object at the temperature of a kitchen oven (about twice room temperature in
absolute terms - 600 K vs. 300 K) radiates 16 times as much power per unit area. An
object the temperature of the filament in an incandescent bulb (roughly 3000 K, or
10 times room temperature) radiates 10,000 times as much per unit area.
Mathematically, the total power radiated rises as the fourth power of the absolute
temperature, the Stefan–Boltzmann law. In the plot, the power radiated is simply the
area under each curve in the graph.

1 eV energy is 2.417970×1014 Hz

14

This picture represents, in false colours, the radiation emitted by a man holding a
burning match. White is a high temperature and blue a low temperature.
Thermal radiation is electromagnetic radiation emitted from the surface of an object
which is due to the object's temperature. Infrared radiation from a common
household radiator or electric heater is an example of thermal radiation, as is the
light emitted by a glowing incandescent light bulb. Thermal radiation is generated
when heat from the movement of charged particles is converted to electromagnetic
temperature, and for a genuine black body is given by Planck’s law of radiation. The
Stefan–Boltzmann law in the next slide gives the total intensity of emitted radiation,
i.e. the intensity summed over all frequencies.

15

250
Spectral Irradiance (W/m /eV))                 P=5.67 x 10-8 T4 [W/m2]
200
2

Stefan–Boltzmann law

150                       300 K
600 K
100                       900 K
1200 K
50

0
0.0   0.5   1.0    1.5    2.0       2.5   3.0
Photon energy [eV]

1 eV energy is 2.417970×1014 Hz

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Some constants
Planck's constant: h     6.626 0693(11)×10-34 Js =
4.135 667 43(35)×10-15 eVs

Wien's displacement      2.897 7685(51)×10–3 mK
constant

Boltzmann constant: kB   1.380 6505(24)×10−23 J/K =
8.617 343(15)×10−5 eV/K

Stefan–Boltzmann         5.670 400(40)×10−8 W/m2/K4
constant: σ

Speed of light: c        299 792 458 m/s

19
30000

25000
2

20000
5800 K
15000

10000

5000

0
0   2           4            6   8
Photon energy [eV]

The radiation emitted by the Sun can be approximately modeled as that of a black
body with a temperature of 5800 K (pink curve). The energy E of a photon is related
to the frequency ν of light by the relation E=hν where h= 6.626 x 10-34 Js. The unit
of energy 1 eV (electron-volt) is equal to 1.602 x 10-19 J.

20
Black body radiation of the Sun
30000

2                                25000

20000
5800 K
15000

10000

5000

0
0   2           4            6   8
Photon energy [eV]

21
Low energy part of the spectrum

1000

Spectral Irradiance (W/m /eV))    100                5800 K
2

10

1
IE ∝ E2
0.1                           2π kT 2
IE =          E
h3c 2
0.01
1E-4   1E-3          0.01         0.1
Photon energy [eV]

By plotting logarithmically the low photon energy part of the spectrum on a double
logarithmic scale we conclude that the spectral energy density varies like ν2

22
1000000
100000
10000
1000
2

100
5800 K
10
1
0.1                          −E
0.01                 IE ∝ e        kT

1E-3
2π − E kT
1E-4                  IE =          e
1E-5                         h 3c 2
1E-6
0   2   4       6      8         10   12   14
Photon energy [eV]

By plotting the logarithm of the spectral energy density as a function of the photon
energy we obtain a straight line for high photon energies. This implies that the
spectral energy density decreases exponentially. The slope is equal to -
1/(5800*ln(10)*kB) where kB is the Boltzmann constant ~1.38x10-23 J

23
How do we combine these two relations ?
30000

2π E 3  1
25000                   IE = 3 2 E
h c e kT − 1
2

20000
5800 K
15000

2π kT 2                            2π − E kT
IE =
10000          E                   IE =          e
h3c 2                            h 3c 2
5000

0
0          2           4            6          8
Photon energy [eV]

24

25
Radiation in terms of E, f and λ

k =1.38 × 10-23 J/K
2π E 3  1
I E dE = 3 2 E        dE                                h = 6.63×10-34 Js.
h c e kT − 1                                   E = hf

2π hf 3    1
I f df =                    df = I E dE
c 2 ehf kT − 1                                λf =c
df    c
=− 2
dλ   λ
2π hc 2             1
Iλ d λ =                                  d λ = I E dE
λ   5
e
hc
λ kT
−1

I (T ) = σ T 4 = 5.67 ×10−8 T 4 [W / m2 ]

26
Spectral irradiance at the surface of the Sun
100000

2

60000                         5800 K

40000

20000

0
0   500     1000       1500   2000
Photon wavelength [nm]

27
1368 W/m2

63 MW/m2

28
Comparison with measurements

2.0

2
1.5
5800 K

1.0

0.5

0.0
0   500     1000       1500   2000
Photon wavelength [nm]

30
The Greenhouse effect

We have now enough ingredients for a first, and thus approximate, description of
the Greenhouse effect.

31
Incoming and reflected solar energy

100 %

Atmosphere:
6%
Clouds: 20 %

Earth’s surface: 4 %

We look now at what happens with the solar light when it hits the Earth. At the top
of the atmosphere the light intensity is 1366 W/m2 and has a spectrum with a
maximum around λ=500 nm. Approximately 6% of the light is directly reflected by
the atmosphere, 20% by the clouds and 4% by the Earth’s surface. This means that
only 70% of the light is absorbed by the Earth. The Earth is heated by this incoming
radiation to a temperature around 300 K. As all bodies it also re-emit radiation, but
now it is radiation with a very long wave length. This thermal radiation is able to
excite molecules in the atmosphere. This creates a sort of thermal blanket and is at
the origin of the Greenhouse effect.

32
Without atmosphere

5600 K                     260 K

The Sun being much hotter than the Earth emits radiation at a much lower
wavelength (typically 500 nm (nanometer)) than the Earth (typically around 10 μm
(micrometer).

33
Absorption by the atmosphere

5600 K                             260 K

The incoming sunlight is much less absorbed by the atmosphere than the outgoing thermal radiation
of the Earth. The absorption bands generated by various greenhouse gases and their impact on both
incoming solar radiation and outgoing thermal radiation from the Earth's surface are indicated in
grey. Note that a greater quantity of outgoing radiation is absorbed, which contributes to the
greenhouse effect.
Quantum mechanics provides the basis for computing the interactions between molecules and
electromagnetic radiation. Most of this interaction occurs when the frequency of the radiation closely
matches that of the spectral lines of the molecule, determined by the quantization of the modes of
vibration and rotation of the molecule. The molecules/atoms that constitute the bulk of the
atmosphere: oxygen (O2), nitrogen (N2) and argon (Ar) do not interact with infrared radiation
significantly. While the oxygen and nitrogen molecules can vibrate, because of their symmetry these
vibrations do not create any transient charge separation. Without such a transient charge separation
moment, they can neither absorb nor emit infrared radiation. In the Earth’s atmosphere, the dominant
infrared absorbing gases are water vapor, carbon dioxide, and ozone (O3). The same molecules are
also the dominant infrared emitting molecules. CO2 and O3 have "floppy" vibration motions whose
quantum states can be excited by collisions at energies encountered in the atmosphere. For example,
carbon dioxide is a linear molecule, but it has an important vibrational mode in which the molecule
bends with the carbon in the middle moving one way and the oxygens on the ends moving the other
way, creating some charge separation, a dipole moment, thus carbon dioxide molecules can absorb
long wave length infra-red (IR) radiation. Collisions will immediately transfer this energy to heating
the surrounding gas. On the other hand, other CO2 molecules will be vibrationally excited by
collisions. Roughly 5% of CO2 molecules are vibrationally excited at room temperature and it is this
5% that radiates. A substantial part of the greenhouse effect due to carbon dioxide exists because this
vibration is easily excited by infrared radiation. Water vapor has a bent shape. It has a permanent
dipole moment (the O atom end is electron rich, and the H atoms electron poor) which means that IR
radiation can be emitted and absorbed during rotational transitions and collisions. Clouds are also
very important infrared absorbers. Therefore, water has multiple effects on infrared radiation,
through its vapor phase and through its condensed phases. Other absorbers of significance include
methane, nitrous oxide and the chlorofluorocarbons.

34

In the literature solar spectra are also often plotted as a function of wavelength λ.
The wavelength enters the relation λν=c where c is the velocity of light. The
maximum intensity is seen around λ=500 nm.

36
Earth without atmosphere

ASSun           π R 2 S Sun (1 − A ) = 4π R 2 S Earth

(1 − A) SSun = 4S Earth = 4σ TEarth
4

SSun
SE

(1-A)SSun

SE

Radiation model of the Earth without atmosphere. We have, SSun=1366 W/m2 times
the area πR2 where R is the radius of the Earth is the total solar energy hitting the
top od the Earth’s atmosphere. The thermal radiation of the Earth (at long wave
lengths) is emitted in all directions (that is why nights are cooler than days). The
total emitted radiation is obtained by multiplying the flux SEarth by the area of the
Earth which is 4 πR2. We obtain then 1366(1-0.3)=4*5.67*10-8T4 gives TEarth=255
K. Without atmosphere the Earth would be at an average -18 degree Celsius. There
would be no liquid water.
The present average temperature of the Earth is 287 K. If we attributed the
calculated 32 K temperature increase solely to CO2 we could reason as follows:
There are presently 380 ppm CO2 in the atmosphere. If the CO2 content would
increase to 500 ppm (as predicted by certain models) the temperature would then
become 255+(33K/380ppm)500ppm=298 K. This is an increase of 11 K in the
average temperature. This is of course a very rough approximation. Modern climate
models estimate that with 500 ppm CO2 in the atmosphere the overall temperature of
the Earth would increase by about 2 degrees.

37
Earth with totally absorbing atmosphere:
Greenhouse effect

ASSun                  Sa
2 S a = S Earth
SSun
SE       Sa

(1-A)SSun
π R 2 S Sun (1 − A ) + 4π R 2 Sa = 4π R 2 S Earth

(1 − A) S Sun = 2S Earth = 2σ TEarth
4

SE

This model involves only thermal radiation according to Stefan-Boltzmann’s law. It
assumes furthermore that all the thermal radiation emitted by the Earth is absorbed
in the atmosphere (clouds+greenhouse gases).

38
Greenhouse effects: more realistic model

A more realistic model for the energy fluxes in the atmosphere.

From:
es_251.html

a) When we look at the radiation budget of the Earth, we can divide the system
into three parts:
1) outer space
2) our atmosphere
3) the surface of the Earth
In each part of the system the amount of energy coming in equals the amount of
the energy leaving. If this wasn't the case, one part of the system would become
either warmer and warmer or colder and colder over time and this isn't
happening. So the system is in equilibrium (balance).

b) Greenhouse gases do NOT produce energy. They help to generate an equilibrium
state where the surface layer of the atmosphere is unusually warm.

Note that the incoming Solar Radiation is indicated per m2 of Earth surface. 1366/4
≅ 342 W/m2.

44
Direct Global Warming Potentials (mass basis)
relative to carbon dioxide
GAS                       Pre-1750 Current          GWP (100- Atmospheric
-tion     concentration   horizon)  (years)
Carbon dioxide (CO2)      280 ppm   377.3 ppm       1          variable
Methane (CH4)             730 ppb   1847 ppb        23         12
Nitrous oxide (N2O)       270 ppb   319 ppb         296        114
Tropospheric ozone (O3)   25 ppb    34 ppb          n.a.       hours-days
CFC-11 (CCl3F)            zero      253 ppt         4600       45
HFC-23 (CHF3)             zero      14 ppt          12000      260
Perfluoroethane (C2F6)    zero      3 ppt           11900      10000

http://cdiac.ornl.gov/pns/current_ghg.html
Increased radiative forcing is the change in the rate at which additional energy is
made available to the earth-atmosphere system over an "average" square meter of
the earths surface due to increased concentration of a "greenhouse" gas, or group of
gases, since 1750. Energy is measured in Joules; the rate at which it is made
available is in Joules/second, or Watts; hence, radiative forcing is measured in Watts
per square meter (W/m2).

From this table one concludes that:
CO2 has a variable atmospheric lifetime, and cannot be specified precisely. Recent
work indicates that recovery from a large input of atmospheric CO2 from burning
fossil fuels will result in an effective lifetime of tens of thousands of years. Carbon
dioxide is defined to have a GWP of 1 over all time periods.
Methane has an atmospheric lifetime of 12 ± 3 years and a GWP of 62 over 20
years, 23 over 100 years and 7 over 500 years. The decrease in GWP associated
with longer times is associated with the fact that the methane is degraded to water
and CO2 by chemical reactions in the atmosphere.
Nitrous oxide has an atmospheric lifetime of 120 years and a GWP of 296 over 100
years.
CFC-11 has an atmospheric lifetime of 45 years and a GWP of 4600 over 100 years.
HCFC-23 has an atmospheric lifetime of 260 years and a GWP of 12000 over 100
years.

46
The Greenhouse effect: Part 2
Heat transfer through conduction

Ronald Griessen 2008
Vrije Universiteit, Amsterdam   vrije Universiteit amsterdam

48
Fourier’s law for thermal diffusion

Temperature profile

⎛ x⎞        ⎛x⎞
T ( x, ) = T1 ⎜1 − ⎟ + T2 ⎜ ⎟
T1                                                   ⎝ L⎠        ⎝L⎠
Heat flow

A
J=
δQ     (T − T )
= kA 1 2
δt        L
T2

0                L               x

Consider a wall with the temperature T1 on the left and T2 on the right. It
is “natural” to expect that the heat flow going through the wall, i.e. the
amount of energy which is transported from left to right :
•Is proportional to the area A of the wall
•Is inversely proportional to the thickness L of the wall
•Is proportional to the temperature difference (T1 - T2 )

With Fourier’s equation you can then easily calculate how much heat
flows from left to t right through the wall.

49
Heat transfer through convection

Ronald Griessen 2008
Vrije Universiteit, Amsterdam                  vrije Universiteit amsterdam

Hot air being less dense than cold air at the same pressure feels an upward force.
We consider here the simple situation of a volume of air that has been locally
heated at the surface of the Earth (for example above a golden wheat field before
harvest) in a cooler surrounding atmosphere. This is leading to the formation of the
fair weather cumulus clouds.

55

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