PEEB1
Project Earth Energy Balance
Purpose: To develop a quantitative understanding of the temperature of the Earth, the
warming effect of the atmosphere, the anthropogenic impact on the atmosphere and
energy balance, and how Earth’s temperature can be controlled.
Planck’s Radiation Law
Over billions of years, Earth has reached a steady-state relationship with its celestial
neighborhood. Earth looses some mass due to escaping hydrogen gas and nuclear reactions in
its core, and gains some mass from space dust and global warming, but the net effect is only
about 50,000 tons (loss) per year. In a discussion of temperature, the transfer of mass between
solar inhabitants can be neglected. For all intents and purposes, nothing escapes Earth’s
gravity.
Nothing, that is, except photons. Earth from space appears as a blue dot, but these are
reflected photons originating from the Sun. However, even if the Sun were removed from the
Solar Syatem, Earth could still be imaged, just not by the human eye. The foundational concept
underpinning Earth’s energy balance is that all planetary bodies emit photons, representing an
energy loss, but at the same time absorb photons from external sources, representing an
energy gain. On Earth at equilibrium, energy gained by photons arriving from the sun, peaked
in the visible, equals energy lost by photons emitted into space, peaked in the infrared.
One might think that the interior of the Earth plays a role in this energy balance. Earth’s
core is indeed very hot, estimated to be 5700°K – about the same as the surface of the Sun!
Some of this core energy is left over from planetary accretion when the Earth formed, but the
great majority is from radioactive decay of long-lived isotopes like 40K and 238U. The crust is
such a good insulator, however, that the average flux of energy from the interior is a little less
than 0.1 W·m–2. As we shall see, this a tiny fraction of the total heat flux driven by solar
radiation, even accounting for energy escape in concentrated amounts at tectonic boundaries.
In short, heat transfer from the core to the surface has a negligible effect on the surface
temperature of the planet.
However, the interior energy of the Earth can indirectly affect the surface temperature by
causing explosive volcanic eruptions that eject large quantities of sulfate-forming matter into the
stratosphere. During the 1 to 3 years required to wash this debris out of the stratosphere, the
sulfate particles nucleate clouds that act much like a sunshade to reduce the amount of sunlight
reaching the surface, producing a substantial transient cooling effect. As we shall see,
stratospheric sulfates, when properly geoengineered, can effectively cool the Earth using
quantities far less – about 1/20 of 1% - than already being emitted by natural sources.
Our starting point, then, is that all objects, including stars and planets, radiate energy to
their surroundings. The wavelength of the emitted radiation depends on the temperature of the
object. If you raise the temperature of an object high enough, some of the emitted wavelengths
are in the visible part of the spectrum and you see the object glowing, first red-hot and then, as
its temperature increases, white-hot like the filament of an incandescent light bulb. The surface
of Earth is not nearly hot enough to emit visible photons; again, Earth emits in the infrared, the
Sun in the visible.
The emission of energy as a function of wavelength from a warm object can be
approximated by Planck’s Law for emission from a black body. A black body is an object that
absorbs all wavelengths of light that fall on it, and at the same time emits the maximum possible
amount of radiation, determined by Planck’s Law, for the given temperature. Emission from a
C. Brucker
black body is closely approximated by the emission from a tiny hole in a hollow object held at a
constant temperature, measurements of which were first carried out in the 19th century.
Planck's law describes the electromagnetic radiation emitted by a black body in thermal
equilibrium at a definite temperature. Named after Max Planck, who originally proposed it in
1900, it is a pioneering result of modern physics and quantum theory:
( )
Bl T =
Where
h = 6.626068E-34
c = 299792500
k = 1.38066E-23
T
λ
B
2hc 2l -5
hc
e lkT -1
m2kg/s
m/s
m2kgs-2K-1
°K
m
Wm-3steradian-1
Planck’s constant
speed of light in vacuum
Boltzmann’s constant
temperature
wavelength
radiant power flux
HyperGrade PEEB1
In HyperGrade you will find Project PEEB1. You will be asked to write a program to
calculate the radiant power flux B for a given wavelength and temperature. Use your program
to generate two tables of B-values, one for the surface temperature of the sun (5800°K) and one
for the surface temperature of the earth (288°K), over the wavelength range 0.1 – 100 μm. This
will be the start of our energy balance analysis, which will compare energy incoming from the
Sun vs. energy outgoing from the Earth.
For example, given λ = 10μm and T = 5800°K, B = 4.23e9 Wm-3steradian-1. Express B
in scientific notation, accurate to two decimal places, as in this example. Use λ values of 0.1,
0.2, 0.3, 0.4, 0.6, 0.8, 1, 2, 3, 4, 6, 8, 10, 20, 30, 40, 60, 80, and 100μm. Be sure to convert μm
(micron) to m (meter) in your calculations. Plot your data with B on the vertical axis and λ on the
horizontal axis, with T as a parameter, and with λ in microns on a log scale. Plot both curves,
one for T = 5800°K and one for T = 288°K on the same graph. For display purposes, scale the
curve for T = 288°K by a factor of one million.
Your graph can be done by hand, but to facilitate further exploration it is best done in a
graphing utility. I use Excel, but you have many options.
Grading
This is an “extracurricular” activity which will be credited as a 15 point homework assignment: 5
points for HyperGrade and 10 points for your tables and graph. There is a bonus opportunity for
graphic display using Java.
C. Brucker