# Lecture 5 -- Blackbody Radiation/ Planetary

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```					Lecture 5 -- Blackbody Radiation/
Planetary Energy Balance
Abiol 574
Electromagnetic Spectrum

visible
light

1000     100    10       1             0.1   0.01

0.7 to 0.4 m

 (m)
Electromagnetic Spectrum

visible
light ultraviolet

1000     100    10       1           0.1          0.01

 (m)
Electromagnetic Spectrum

visible
infrared             light ultraviolet

1000     100          10       1           0.1          0.01

 (m)
Electromagnetic Spectrum

visible
microwaves            infrared             light ultraviolet      x-rays

1000     100          10       1           0.1          0.01

 (m)
Electromagnetic Spectrum

visible
microwaves              infrared             light ultraviolet      x-rays

1000     100          10       1           0.1          0.01

Low                                                          High
Energy                                                       Energy
 (m)

emits (or absorbs) equally well at all wavelengths
The Planck Function

• Blackbody radiation follows the Planck function

1) All objects emit radiant energy.

1) All objects emit radiant energy.

2) Hotter objects emit more energy than colder
objects.

1) All objects emit radiant energy.

2) Hotter objects emit more energy than colder
objects. The amount of energy radiated is
proportional to the temperature of the object.

1) All objects emit radiant energy.

2) Hotter objects emit more energy than colder
objects. The amount of energy radiated is
proportional to the temperature of the object
raised to the fourth power.

 This is the Stefan Boltzmann Law

F =  T4

F = flux of energy (W/m2)
T = temperature (K)
 = 5.67 x 10-8 W/m2K4 (a constant)

1) All objects emit radiant energy.

2) Hotter objects emit more energy than colder
objects (per unit area). The amount of energy
radiated is proportional to the temperature of
the object.

3) The hotter the object, the shorter the
wavelength () of emitted energy.

1) All objects emit radiant energy.

2) Hotter objects emit more energy than colder
objects (per unit area). The amount of energy
radiated is proportional to the temperature of
the object.

3) The hotter the object, the shorter the
wavelength () of emitted energy.

This is Wien’s Law

max  3000 m
T(K)
 Stefan Boltzmann Law.

F =  T4

F = flux of energy (W/m2)
T = temperature (K)
 = 5.67 x 10-8 W/m2K4 (a constant)

 Wien’s Law

max  3000 m
T(K)
We can use these equations to calculate properties
of energy radiating from the Sun and the Earth.

6,000 K       300 K
T     max   region in     F
spectrum
(K)    (m)               (W/m2)

Sun     6000

Earth   300
T     max   region in     F
spectrum
(K)    (m)               (W/m2)

Sun     6000   0.5

Earth   300    10
Electromagnetic Spectrum

visible
microwaves              infrared             light ultraviolet      x-rays

1000     100          10       1           0.1          0.01

Low                                                          High
Energy                                                       Energy
 (m)
T     max   region in     F
spectrum
(K)    (m)               (W/m2)

Sun     6000   0.5    Visible
(yellow?)

Earth   300    10     infrared
• Blue light from the Sun is removed from the beam
by Rayleigh scattering, so the Sun appears yellow
when viewed from Earth’s surface even though its
T     max   region in     F
spectrum
(K)    (m)               (W/m2)

Sun     6000   0.5    Visible
(green)

Earth   300    10     infrared
 Stefan Boltzman Law.

F =  T4

F = flux of energy (W/m2)
T = temperature (K)
 = 5.67 x 10-8 W/m2K4 (a constant)
T     max   region in     F
spectrum
(K)    (m)               (W/m2)

Sun     6000   0.5    Visible 7 x 107
(green)

Earth   300    10     infrared     460
Solar Radiation and Earth’s Energy Balance
Planetary Energy Balance
• We can use the concepts learned so far
to calculate the radiation balance of the
Earth
Some Basic Information:

Area of a circle =  r2

Area of a sphere = 4  r2
Energy Balance:

The amount of energy delivered to the Earth is
equal to the energy lost from the Earth.

Otherwise, the Earth’s temperature would
continually rise (or fall).
Energy Balance:

Incoming energy = outgoing energy

Ein = Eout

Eout

Ein
(The rest of this derivation will be done on the
board. However, I will leave these slides in here
in case anyone wants to look at them.)
How much solar energy reaches the Earth?
How much solar energy reaches the Earth?

As energy moves away from the sun, it is
spread over a greater and greater area.
How much solar energy reaches the Earth?

As energy moves away from the sun, it is
spread over a greater and greater area.

 This is the Inverse Square Law
So = L / area of sphere
So = L / (4  rs-e2) = 3.9 x 1026 W            = 1370 W/m2
4 x  x (1.5 x 1011m)2

So is the solar constant for Earth
So = L / (4  rs-e2) = 3.9 x 1026 W            = 1370 W/m2
4 x  x (1.5 x 1011m)2

So is the solar constant for Earth

It is determined by the distance between Earth (rs-e)
and the Sun and the Sun’ luminosity.
Each planet has its own solar constant…
How much solar energy reaches the Earth?

Assuming solar radiation covers the area of a circle
defined by the radius of the Earth (re)

Ein                  re
How much solar energy reaches the Earth?

Assuming solar radiation covers the area of a circle
defined by the radius of the Earth (re)

Ein = So (W/m2) x  re2 (m2)

Ein                    re
How much energy does the Earth emit?

300 K
How much energy does the Earth emit?

Eout = F x (area of the Earth)
How much energy does the Earth emit?

Eout = F x (area of the Earth)

F =  T4

Area = 4  re2
How much energy does the Earth emit?

Eout = F x (area of the Earth)

F =  T4

Area = 4  re2

Eout = ( T4) x (4  re2)
Earth         Sun

1000   100    10       1         0.1     0.01

Hotter objects emit
more energy than
 (m)        colder objects
Earth         Sun

1000   100    10       1         0.1     0.01

Hotter objects emit
more energy than
 (m)        colder objects

F =  T4
Hotter objects emit at
shorter wavelengths.

max = 3000/T         Earth         Sun

1000      100        10       1         0.1     0.01

Hotter objects emit
more energy than
 (m)        colder objects

F =  T4
How much energy does the Earth emit?

Eout = F x (area of the Earth)

Eout
How much energy does the Earth emit?

Eout = F x (area of the Earth)

F =  T4
Area = 4  re2

Eout = ( T4) x (4  re2)
Eout
How much solar energy reaches the Earth?

Ein
How much solar energy reaches the Earth?

We can assume solar radiation covers the area of a
circle defined by the radius of the Earth (re).

Ein                 re
How much solar energy reaches the Earth?

We can assume solar radiation covers the area of a
circle defined by the radius of the Earth (re).

Ein = So x (area of circle)

Ein                    re
Remember…

So = L / (4  rs-e2) = 3.9 x 1026 W            = 1370 W/m2
4 x  x (1.5 x 1011m)2

So is the solar constant for Earth

It is determined by the distance between Earth (rs-e)
and the Sun and the Sun’s luminosity.
How much solar energy reaches the Earth?

We can assume solar radiation covers the area of a
circle defined by the radius of the Earth (re).

Ein = So x (area of circle)

Ein = So (W/m2) x  re2 (m2)

Ein                     re
How much solar energy reaches the Earth?

Ein = So  re2

BUT THIS IS NOT QUITE CORRECT!

**Some energy is reflected away**

Ein                    re
How much solar energy reaches the Earth?

Albedo (A) = % energy reflected away

Ein = So  re2 (1-A)

Ein                     re
How much solar energy reaches the Earth?

Albedo (A) = % energy reflected away
A= 0.3 today

Ein = So  re2 (1-A)

Ein = So  re2 (0.7)

Ein     re
Energy Balance:

Incoming energy = outgoing energy

Ein = Eout

Eout

Ein
Energy Balance:
Ein = Eout

Ein = So  re2 (1-A)

Eout

Ein
Energy Balance:
Ein = Eout

Ein = So  re2 (1-A)

Eout =  T4(4  re2)

Eout

Ein
Energy Balance:
Ein = Eout

So  re2 (1-A) =  T4 (4  re2)

Eout

Ein
Energy Balance:
Ein = Eout

So  re2 (1-A) =  T4 (4  re2)

Eout

Ein
Energy Balance:
Ein = Eout

So (1-A) =  T4 (4)

Eout

Ein
Energy Balance:
Ein = Eout

So (1-A) =  T4 (4)

T4 = So(1-A)
4
Eout

Ein
T4 = So(1-A)
4

If we know So and A, we can calculate the
temperature of the Earth. We call this the
expected temperature (Texp). It is the
temperature we would expect if Earth behaves
like a blackbody.

This calculation can be done for any planet,
provided we know its solar constant and albedo.
T4 = So(1-A)
4

For Earth:

So = 1370 W/m2
A = 0.3
 = 5.67 x 10-8 W/m2K4
T4 = So(1-A)
4

For Earth:

So = 1370 W/m2
A = 0.3
 = 5.67 x 10-8

T4 =   (1370 W/m2)(1-0.3)
4 (5.67 x 10-8 W/m2K4)
T4 = So(1-A)
4
For Earth:

So = 1370 W/m2
A = 0.3
 = 5.67 x 10-8

T4 =   (1370 W/m2)(1-0.3)
4 (5.67 x 10-8 W/m2K4)

T4 = 4.23 x 109 (K4)

T = 255 K
Expected Temperature:

Texp = 255 K

(oC) = (K) - 273
Expected Temperature:

Texp = 255 K

(oC) = (K) - 273

So….

Texp = (255 - 273) = -18 oC

Is the Earth’s surface really -18 oC?
Is the Earth’s surface really -18 oC?

NO. The actual temperature is warmer!

The observed temperature (Tobs) is 15 oC, or
Is the Earth’s surface really -18 oC?

NO. The actual temperature is warmer!

The observed temperature (Tobs) is 15 oC, or

The difference between observed and
expected temperatures (T):

T = Tobs - Texp

T = 15 - (-18)

T = + 33 oC
T = + 33 oC

In other words, the Earth is 33 oC warmer than
expected based on black body calculations
and the known input of solar energy.
T = + 33 oC

In other words, the Earth is 33 oC warmer than
expected based on black body calculations
and the known input of solar energy.

This extra warmth is what we call the
GREENHOUSE EFFECT.
T = + 33 oC

In other words, the Earth is 33 oC warmer than
expected based on black body calculations
and the known input of solar energy.

This extra warmth is what we call the
GREENHOUSE EFFECT.

It is a result of warming of the Earth’s surface
by the absorption of radiation by molecules in
the atmosphere.
The greenhouse effect:

Heat is absorbed or “trapped”
by gases in the atmosphere.

Earth naturally has a
greenhouse effect of +33 oC.
The concern is that the amount of greenhouse warming
will increase with the rise of CO2 due to human activity.

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