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THE
SUN-EARTH
SYSTEM
THE
SUN-EARTH
SYSTEM
John Streete
Physics Department
Rhodes College
Memphis, Tennessee
UNIVERSITY SCIENCE BOOKS
SAUSALITO, CALIFORNIA
University Science Books
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Managing Editor: Lucy Warner
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This book is printed on acid-free paper.
Copyright © 1996 by University Corporation for Atmospheric
Research. All rights reserved.
Reproduction or translation of any part of this work beyond
that permitted by Section 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is
unlawful. Requests for permission or further information should
be addressed to UCAR Communications, Box 3000, Boulder, CO
80307-3000.
Library of Congress Catalog Number: 95-061059
ISBN: 0-935702-86-5
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
A Note on the Global Change Instruction Program
This series has been designed by college professors to fill an
urgent need for interdisciplinary materials on the emerging
science of global change. These materials are aimed at
undergraduate students not majoring in science. The modular
materials can be integrated into a number of existing courses
—in earth sciences, biology, physics, astronomy, chemistry,
meteorology, and the social sciences. They are written to capture
the interest of the student who has little grounding in math
and the technical aspects of science but whose intellectual
curiosity is piqued by concern for the environment. The material
presented here should occupy about two weeks of classroom
time.
For a complete list of modules available in the Global Change
Instruction Program, contact University Science Books, Sausalito,
California, fax (415) 332-5393. Information about the Global
Change Instruction Program is also available on the World Wide
Web at http://home.ucar.edu/ucargen/education/gcmod/
contents.html.
Contents
Preface ix
I.
The Sun and Solar Energy 1
Formation of the Sun 1
Fusion 2
Energy Transport 5
Problems 6
II. The Nature of Electromagnetic Radiation
7
Energy Levels of Atoms 7
Absorption and Emission of Electromagnetic Radiation
Blackbody Radiation 9
The Electromagnetic Spectrum 9
Problems 11
8
III. The Earth’s Atmosphere 12
Composition and Distribution of the Atmosphere 12
The Atmosphere’s Interaction with Solar Radiation 12
Problems 21
Appendix I. Scientific Notation 22
Appendix II. Units and Dimensional Analysis 24
Appendix III. Physical Constants and Data for the Sun and Earth 26
Appendix IV. Measuring the Solar Constant 27
Glossary 30
Additional Reading Material
Index 34
33
Preface
The information in this module on the Sun-Earth system is
necessary to comprehend such physical phenomena as the
greenhouse effect and global warming. It can be used as one
element of a course on global change or can be integrated into
more traditional introductory classes in astronomy, physics, or
Earth sciences. I hope to show students that they can understand
the basic physical principles behind many of the global changes
they read about in the paper and weekly news magazines.
The material is aimed at first- or second-year students. The
mathematics should be accessible to any student who has
studied introductory algebra in high school.
There is a review of scientific notation in Appendix I, and
Appendix II contains a brief discussion of units and dimensional
analysis. One of the best ways to learn new concepts in science is
through problem-solving. The problems at the end of each
section are intended to give the students practice in using
scientific notation and require them to make use of the
information discussed in each section. Appendix III lists values
for the quantities that will be needed to work the problems.
For those students who would like hands-on experience with
scientific measurements, Appendix IV describes an experiment
to measure the solar constant with easily obtainable items. This
experiment is fun, and the results can be surprisingly accurate if
sufficient care is taken with the measurements.
John Streete
Rhodes College
ix
Acknowledgments
This instructional module has been produced by the the Global
Change Instruction Program of the Advanced Study Program of
the National Center for Atmospheric Research, with support
from the National Science Foundation. Any opinions, findings,
conclusions, or recommendations expressed in this publication
are those of the author and do not necessarily reflect the views of
the National Science Foundation.
Executive Editors: John W. Firor, John W. Winchester
Global Change Working Group
Louise Carroll, University Corporation for Atmospheric Research
Arthur A. Few, Rice University
John W. Firor, National Center for Atmospheric Research
David W. Fulker, University Corporation for Atmospheric Research
Judith Jacobsen, University of Denver
Lee Kump, Pennsylvania State University
Edward Laws, University of Hawaii
Nancy H. Marcus, Florida State University
Barbara McDonald, National Center for Atmospheric Research
Sharon E. Nicholson, Florida State University
J. Kenneth Osmond, Florida State University
Jozef Pacyna, Norwegian Institute for Air Research
William C. Parker, Florida State University
Glenn E. Shaw, University of Alaska
John L. Streete, Rhodes College
Stanley C. Tyler, University of California, Irvine
Lucy Warner, University Corporation for Atmospheric Research
John W. Winchester, Florida State University
This project was supported, in part, by the
National Science Foundation
Opinions expressed are those of the authors
and not necessarily those of the Foundation
xi
THE
SUN-EARTH
SYSTEM
I
The Sun and Solar Energy
there contained the other known elements in an
abundance generally decreasing with the mass
of the element. These additional elements were
produced inside a very massive star (or stars)
and then hurled out into space in the cataclysmic explosion of a star known as a supernova.
One of the most important forces behind global
change on Earth is over 90 million miles distant
from the planet. The Sun is the ultimate,
original source of the energy that drives the
Earth’s climate system and nurtures life itself. It
provides essentially all the energy the Earth
and its atmosphere receive. If we are to understand global warming and climate change we
should examine the source of the energy that is
responsible for producing the environment we
enjoy on Earth, how this solar energy interacts
with the Earth and its atmosphere, and how the
composition of the atmosphere determines the
ultimate temperature of Earth.
Gravity
Space is not a total vacuum; even in the vast
reaches between stars there are tenuous wisps
of gas. Stars are born where the interstellar
matter is denser than average. The closer together particles are, the greater the gravitational
force between them. Gravity, the same familiar
force that keeps the Earth in orbit around the
Sun and holds us on the surface of the Earth, is
one of the major forces in the universe. It is
always attractive, tending to draw things together, and the attraction between two objects
increases faster than the distance between them
decreases: as the distance halves, the force of
gravity increases by a factor of four.1
Formation of the Sun
Throughout the universe, for more than 10
billion years, stars have been forming continuously. Galaxies—huge aggregates containing
stars, groups of stars, and interstellar (literally,
“between-star”) matter—may contain hundreds
of billions of stars, and there are billions of
galaxies.
Our Sun, a fairly ordinary star, was born
over 4.5 billion years ago in an outlying region
of the Milky Way galaxy. The Sun formed
where there was a higher than average concentration of hydrogen and helium, the two elements that have been present since early in the
evolution of the universe. The interstellar space
Kinetic Energy
Counteracting the tendency for particles to
be pulled together—in interstellar space or anywhere in the universe—is their random motion.
The energy associated with this motion is called
kinetic energy, and it depends on the temperature of the particles: the higher the temperature,
1
Mathematically, the gravitational force between two particles is proportional to one over the square of the distance between
them. This is an instance of the “inverse-square law” (see p. 5).
1
THE SUN-EARTH SYSTEM
particles of opposite charge attract. Two protons—subatomic particles with a positive
charge—repel each other with a greater and
greater force as they get closer together.
The electrical force is also much stronger
than the gravitational force—1038 times as great
for two protons a given distance apart. The
nuclear strong force is even stronger than the
electrical force (at close range, anyway), and it
is always attractive. The nuclear force does not
depend on charge, and although its exact relation to the distance between particles is not
known, it is known that the force only operates
within a distance of 3 x 10-15 meters (3 femtometers, or fm).
the greater the kinetic energy. If the particles
become close enough or their temperature low
enough, gravity can overcome the energy of
motion. When the number density of particles
in a particular region of space becomes great
enough, and when the gases in this region are
compressed even more, perhaps by the shock
wave from a supernova, the particles collapse
on themselves and are on their way to becoming a star.
If a forming star, or protostar, is to become
stable, however, something must happen to halt
this collapse. What happens is that the compression raises the temperature of the protostar
by the same physical process that heats the
gases in the cylinder during the compression
phase in a combustion engine. Eventually the
temperature in the central region, or core, of the
protostar becomes so high that nuclear fusion
begins.
The Proton-Proton Chain
Imagine two protons moving toward each
other. The closer they approach, the greater the
electrical force trying to push them apart. But if
they are able to move close enough, the nuclear
force takes hold and rapidly overwhelms the
electrical force. What determines whether the
protons can get close enough for the strong
force to predominate? Their speed. It’s as if you
were trying to kick a soccer ball into a hole at
the top of a steep hill. If you don’t kick the ball
hard enough it will go only part way up the hill
and then roll back down. The harder you kick,
the higher it goes, and if you kick it hard
enough, giving it enough initial speed (and if
your aim is good), you can sink it into the hole
at the top.
Protons, of course, aren’t being kicked. Temperature determines how fast they move. The
higher the temperature, the faster, on the average, they go. As the core of a star is compressed
by its collapse, its temperature rises. If it gets up
to about 10 million kelvins (kelvin, K, is a unit
of temperature equal to Celsius plus 273), the
particles are moving fast enough for protons to
collide and bond together to produce three
other subatomic particles, a deuteron, a
positron, and a neutrino. This is the first step of
the proton-proton chain, (see Table 1).
Fusion
The energy released by nuclear fusion in the
core of the protostar produces an outward pressure that eventually equals the inward pressure
from gravity. When a balance between outward
and inward pressure is reached, the protostar
becomes a star. This is how our Sun formed.
Nuclear fusion not only provided the energy to
halt its collapse, it also provides almost all the
energy the Earth receives.
Electrical and Nuclear Strong Forces
Two forces are involved in fusion: the electrical and nuclear strong forces. The electrical
force, like the gravitational force, increases as
the distance between two particles decreases.
But unlike gravity, it is not always attractive; it
may be repulsive, tending to separate the particles, depending on their charge. Charge is a
basic property of elementary particles of matter.
It may be positive, negative, or zero. Two particles with the same charge repel each other;
2
THE SUN AND SOLAR ENERGY
Table 1
Particles Involved in the Proton-Proton Chain
Step 1 (The particles’ charges are noted
above their symbols.)
+ +
+ + 0
p + p ———> d + e+ + ν
Proton. Symbol: p; charge: +. Subatomic,
positively charged particle that is one of
the two principal particles comprising the
nucleus.
Step 2. The deuteron is bound to a proton to
produce helium 3 and a high-energy photon, g.
+ +
++
0
3
d + p ———> He + γ
Neutron. Symbol: n; charge: 0. Subatomic,
uncharged particle found as the other
principal particle making up the nucleus.
Steps 1 and 2 occur again, so that there are
available two He3 nuclei. Then, in step 3, two
He3 nuclei collide to produce He4 and two protons.
Deuteron. Symbol: d; charge: +. One form
of “heavy hydrogen.” The nucleus of an
ordinary hydrogen atom consists of a
single proton. Deuteron contains a neutron in addition to the proton.
++ ++
++ ++
He3 + He3 ———> He4 + 2p
When the chain is complete, the two protons
are free to begin again. The complete chain,
steps 1 and 2 occurring twice and step 3 once,
converts a total of four protons (six protons are
used and two are returned) into one nucleus of
He4. Note that the charge is conserved at each
step; it is the same before the collision (left side
of the arrow) and after it (right side). Figure 1
illustrates the processes involved in the protonproton chain.
Electron. Symbol: e; charge: –. Subatomic
particle that orbits around the nucleus.
Positron. Symbol: e+; charge: +. An
“antielectron,” the same as an electron but
with a positive rather than negative
charge.
Helium 3 nucleus. Symbol: He3; charge
++. An isotope of ordinary helium containing two protons and only one neutron
instead of two.
Helium 4 nucleus. Symbol: He4; charge:
++. “Ordinary” helium nucleus containing
two protons and two neutrons.
Neutrino. Symbol: n; charge: 0. A
chargeless particle having little, if any,
mass. Important in many nuclear processes.
Photon (gamma ray). Symbol: g; charge: 0.
A chargeless “particle,” or packet, of
electromagnetic energy.
Figure 1. In the proton-proton chain, four protons combine
to form one helium nucleus and emit energy.
3
THE SUN-EARTH SYSTEM
INVERSE-SQUARE LAW AND THE SOLAR CONSTANT
Because we know how much energy leaves
the Sun each second, we can calculate how
much energy the Earth receives. The
amount of radiative energy that reaches
the top of the Earth’s atmosphere when the
Earth is at its average distance from the Sun
is called the solar constant. It is also the
amount that would strike each square
meter of the surface if there were no
atmosphere.
The amount of radiative energy per unit
area arriving at a particular point each second from a source like a star decreases as
one over the square of the distance from the
source. This is an instance of the inversesquare law, and is illustrated in Figure 2.
Assume the energy leaving the star moves
out uniformly in all directions and the total
amount leaving per second is a quantity L.
Suppose we are at a distance R from the
star. The amount of energy reaching us
each second can be determined by considering a hollow sphere whose center is the
star’s center and whose radius—the distance from the center to the outer shell—is
R, our distance. The total amount of energy
passing through the shell each second has
to be the same amount that leaves the star
each second. To find out how much energy
falls on a square meter, calculate the surface
area of the imaginary spherical shell. The
formula for the surface area of a sphere is
When the source is the Sun and the distance is the distance from the Sun to the
Earth, E is called the solar constant.
Appendix IV describes a simple experiment you may do to measure the solar constant using water, black ink, a thermometer,
and a few other easily obtainable items.
We know that the amount of energy
leaving the Sun is 3.83 x 1026 joules per second, and the average distance between the
Earth and the Sun is about 1.496 x 1011
meters (this distance is defined as one astronomical unit, 1 AU), so we can calculate the
actual value of the solar constant to be 1.36
x 103 joules per second per square meter.
4πR2
where R is the radius and π is a constant.
The amount of energy per second falling on
a unit area of a surface is called the irradiance (E); we derive it by dividing the total
amount of energy leaving the Sun each second (L, the total amount passing through
the shell) by the surface area of the shell:
Figure 2. Illustration of the inverse-square law. L is
the total energy per second leaving the distant body
equally in all directions. R is the distance from the
body to the point of measurement, and E is the
amount of energy per second striking each unit area
at a distance R from the body. From the text, E = L/
4πR2.
E = L/4πR2
4
THE SUN AND SOLAR ENERGY
Figure 3 is a cross section of the Sun. The
temperature of the Sun decreases drastically
and rapidly from the core outward. The surface
temperature, the temperature we measure from
Earth, is only about 5,800 K. About 70% of the
way from the center to the surface, the temperature becomes low enough for atoms to exist.
Atoms are very effective in absorbing radiation,
so they themselves take over the job of energy
transport. Heated by absorbing the radiation
from below, they begin to rise in the same way
that warm air in a room rises. When the atoms
reach the surface of the Sun, the photosphere,
their energy is radiated to space, they cool, and
begin to fall. This is energy transport by the
actual movement of matter, or convection.
Energy moves outward from the surface of
the Sun, once again in the form of electromagnetic radiation, travelling unimpeded and uniformly in all directions. Moving at the speed of
light, it strikes the Earth about eight minutes
later. It is this energy, coupled with the Earth’s
rotation, that drives our weather and establishes the Earth’s climate.
At each step a small amount of mass is converted into energy. Einstein’s law of mass-energy equivalence says that E = mc2. The m in
this case is the difference in mass before and
after the collision. E is the energy produced,
and c is the speed of light, which is a constant:
about 3 x 108 meters per second.
Each chain produces only a tiny amount of
energy, about 4.4 x 10-12 joules. (A joule is a unit
of energy.) But in the solar core, each second
there are enough chains to generate the enormous total of 3.9 x 1026 joules. About 0.7% of the
mass of the four protons is converted to energy.
This means that when 1,000 kilograms of hydrogen undergo fusion, 993 show up as helium
and 7 as energy.
It is obvious from all this that the number of
protons in the Sun’s core is steadily decreasing.
Each second about 600 billion (6 x 1011) kilograms of hydrogen are converted to helium.
When it’s all gone, in a mere 4 billion years, the
Sun will die.
Energy Transport
All this energy is produced in the Sun’s core.
Before it can be radiated to Earth, it has to get to
the surface. It makes the first part of the journey
from the core to the surface in the form of electromagnetic radiation, radiant energy that travels through space and matter.
The heat that we feel when when we hold a
hand over an electric light bulb or lie on a beach
on a hot, sunny day is produced by electromagnetic radiation. In the Sun’s core the temperature is around 15 million K. Atoms cannot exist
at the extremely high temperatures in the inner
regions of the Sun. They are moving so fast
that, if they form, collisions between them immediately break them apart again. Subatomic
particles such as protons and electrons deflect,
or scatter, electromagnetic radiation, but do not
remove much of its energy. So most of the energy created by fusion moves outward from the
core in this form.
Figure 3. Cross section of the sun’s interior, showing the
radiative and convective zones. Also shown are the sun’s
atmospheric regions, called the chromosphere and corona.
From Robert Jastrow, Astronomy: Fundamentals and
Frontiers. John Wiley & Sons. Reprinted by permission.
5
THE SUN-EARTH SYSTEM
Problems
4. When Voyager II encountered Neptune on
August 24, 1989, what was the ratio of the
tug of the Sun’s gravity on it compared to
when it was launched? (Neptune is about 30
AU from the Sun.)
1. Calculate how long it takes light leaving the
surface of the sun to reach the Earth.
2. How much mass has the Sun lost, in terms of
equivalent Earth masses, in its 4.5-billionyear life?
5. We learned how much energy is produced
each second in the solar core and how much
energy each p–p chain produces. How many
p–p chains are occurring each second in the
Sun’s core?
3. What is the average mass density (mass per
unit volume) of the Sun?
6
II
The Nature of
Electromagnetic Radiation
The Sun’s energy has traveled across space as
electromagnetic radiation, and that is the form
in which it arrives on Earth. It is this radiation
that determines the effect of the Sun’s energy
on the Earth and its climate. Infrared radiation,
radio waves, visible light, and ultraviolet rays
are all forms of electromagnetic radiation. One
of the best ways to understand the production
of this type of energy is to consider how it is
emitted by atoms, in particular the hydrogen
atom.
Energy of Electromagnetic Radiation
In many situations electromagnetic radiation
may be described as having a wave-like nature.
Three important features of waves of any sort
are the wavelength (the distance between adjacent crests), the frequency (how fast the crests
move up and down), and the speed (how fast
the crests move forward). There is a basic relationship between these features. If we multiply
the wavelength (symbolized by λ, the Greek
letter lambda) by the frequency (f), we obtain
the speed of the wave, v. The mathematical
formula is
Energy Levels of Atoms
λf = v
Allowed Energies of the Electron
When the electromagnetic radiation is moving
through space or another vacuum, regardless of
its wavelength or frequency, it travels at the
speed of light, c. Because c is constant, the
product of λ and f is always the same, so if one
gets larger, the other gets smaller.
Electromagnetic radiation also, under certain
conditions, exhibits a particle-like nature. The
particles are called photons, and it is helpful to
think of them as energy packets having a welldefined wavelength and frequency. In the early
part of this century, Albert Einstein demonstrated that the energy of these photons, E, is
directly proportional to their frequency:
Since before the turn of the century, it has
been known that an individual atom is made up
of a nucleus (composed of protons and neutrons) and electrons bound to the nucleus, and
that the electrons (and hence the atom) have
very well defined, discrete amounts of energy.
The simplest atom, hydrogen, is composed of a
proton (its nucleus) and an electron bound to it
by the electrical force of attraction. (Electrons
have a negative charge and protons a positive
one.) The electron may have only certain values
of energy when it is bound to the nucleus in
this way. The lowest energy level, in which the
electron is closest to the nucleus, is called the
ground level. The next level is the first excited
level, and so on (see Figure 4). There are various ways the electron may be moved to higher
levels, and one of those ways is by receiving
energy from electromagnetic radiation. What
determines the amount of energy electromagnetic radiation may have?
E = hf
where h is a constant called Planck’s constant
(see Appendix III). The frequency of electromagnetic radiation is inversely proportional to
its wavelength, so its energy is, too. Radiation
with a long wavelength has less energy than
short-wavelength radiation.
7
THE SUN-EARTH SYSTEM
Absorption and Emission of
Electromagnetic Radiation
radiation over a particular set of wavelengths.
The pattern of wavelengths absorbed is called
the absorption spectrum of the atom or molecule. The radiation has vanished (been absorbed), but the total energy of the radiation
and the atom energy is conserved. The atom
now has more energy than it did.
Absorption
Suppose that electromagnetic radiation of a
given frequency strikes a hydrogen atom and
that the frequency is such that the energy of the
radiation equals the difference in the energy of
the ground and first excited levels of the atom.
Then an electron in the ground level may be
raised to the first excited level. This process is
called absorption. Because the atom has a
unique set of energy levels, each will absorb
Emission
Line Emission
Just as a ball kicked up a hill will roll back
down to the bottom, the electron very quickly
returns to its lowest possible energy level (usually within a hundred-millionth of a second).
On the way back to its ground level it must
release energy, and it emits the energy as radiation that has the same wavelength as the radiation that first hit the atom. So if the atom is
being given energy by some means (such as
radiation or collisions with other particles), we
can expect to find it emitting electromagnetic
radiation at wavelengths governed by the difference in these energy levels.
This kind of radiation is called line emission,
because when the individual wavelengths are
measured with an instrument called a spectrograph, the results show up as lines on a photographic plate.
Figure 4. Energy-level diagram for hydrogen. Arrows
indicate direction of electron transitions. When the
electron moves from a higher to a lower energy level, a
photon is emitted; the atom emits energy. When a photon
of the right energy strikes the hydrogen atom, the electron
moves from a lower to a higher energy level. The atom
absorbs energy.
Figure 5. Relative amounts of energy emitted by the sun at
different wavelengths. Note that most of this energy occurs
in the visible region, which extends from about 400 nm to
750 nm.
8
THE NATURE OF ELECTROMAGNETIC RADIATION
λ maxT = 2.898 x 106
Continuous Emission
where wavelength is in nanometers and temperature is in kelvins. (Note that the constant is
very close to 3.0 x 106, which is useful for rough
calculations.) Wien’s law and the StefanBoltzmann law are central to understanding the
greenhouse effect, discussed in Section III.
When the density of atoms in a given area is
sufficiently high, the radiation that ultimately
leaves the area is smeared into a continuous
distribution of wavelengths made up of the
many separate wavelengths that the individual
atoms emit. This is called continuous emission.
The radiation we receive from the Sun is
continuous radiation. Figure 5 shows a graph of
the relative amount of energy of different wavelengths that the Earth receives from the Sun.
The Electromagnetic Spectrum
The continuous emission “spectrum” an
object radiates is a display of the amount of
energy it emits at all wavelengths. The entire
electromagnetic spectrum covers an enormous
range of wavelengths, divided into regions.
Going from the shortest wavelengths to the
longest, there are: gamma rays, x-rays, ultraviolet radiation, visible light, infrared radiation,
and radio waves. Figure 6 shows the various
regions of the spectrum and their approximate
wavelength ranges. The visible region occupies
only a small portion of the entire spectrum.
As Figure 5 shows, the vast majority of our
Sun’s energy is emitted in the visible, ultraviolet, and infrared ranges. In fact, about 41% of
the energy emitted from the Sun lies in the visible bands alone, between 390 nm and 750 nm.
Since the surface temperature of the Sun is
about 5,800 K, we can calculate from Wien’s
displacement law that the maximum amount
of energy is emitted at about 500 nm, right in
the middle of the visible spectrum. About 50%
of the energy the Sun emits lies in the infrared
and radio regions, above 750 nm, and only
about 9% is in the ultraviolet, x-ray, and
gamma-ray regions, below 390 nm.
It is this electromagnetic radiation, of all
wavelengths and frequencies, that determines
what effect the Sun’s energy has on the Earth
and its climate. We have seen that one of the
ways electromagnetic radiation interacts with
atoms (and molecules) is by being absorbed by
them, and that very soon afterward it is emitted
again. The energy is taken away but then given
back. Does this mean, then, that there is no net
Blackbody Radiation
The type of continuous radiation the Sun
emits is often called blackbody radiation. There
are many special features of this type of radiation that allow us to determine various properties of the objects emitting it. One is that the
total amount of energy the blackbody emits is
determined solely by its temperature. Specifically, the amount of energy that a body emits
increases as the fourth power of the temperature. The mathematical expression for this is the
Stefan-Boltzmann law:
E = σ T4
where s is a constant (the Stefan-Boltzmann
constant; see Appendix III). When the temperature of a blackbody doubles, for example, the
amount of energy emitted increases by a factor
of two to the fourth power, or 16 (two multiplied by itself four times: 2x2x2x2). A blackbody whose temperature is 4,000 K emits 16
times as much energy as one at 2,000 K.
The energy emitted by a blackbody always
peaks at some wavelength and decreases toward longer and shorter wavelengths. Figure 5
shows the energy curve for an object at 5,800 K,
the approximate temperature of the surface of
the Sun. In fact, there is a simple equation,
called Wien’s displacement law, to determine
the temperature of an object by measuring this
peak in the energy curve. The equation is:
9
THE SUN-EARTH SYSTEM
effect on the amount of radiation traversing the
atmosphere? Indeed, it does not. The radiation
being absorbed by an atom is moving in a given
direction (for example, from the Sun to the
Earth). The radiation that the atom emits may
go in any direction with about equal probability. So at the wavelength being absorbed, only
an insignificant fraction of the original radiation
will continue moving in the original direction,
and there will be a net loss of energy in the
direction the radiation was originally traveling.
Therefore, how much energy, at various wavelengths, is lost from the beam of radiation
depends on what atoms and molecules (and
how many of them per unit volume) are in
its path.
So it is the makeup of our atmosphere that
determines how much of the Sun’s energy gets
to the Earth’s surface (as well as how much
leaves the Earth).
Figure 6. The electromagnetic spectrum consists of many
regions. Each differs in wavelength, frequency, and
energy.
10
THE NATURE OF ELECTROMAGNETIC RADIATION
THE STEFAN-BOLTZMANN LAW AND THE EARTH’S TEMPERATURE
multiply this fraction by the total energy
emitted by the Sun, we have the amount of
energy per second the Earth receives from
the Sun.
Assume the Earth absorbs all of this
energy. It must lose the same amount as
it gains each second if it is to remain at a
constant temperature. We know from the
above calculation how much energy the
Earth receives and loses. Now we can use
the Stefan-Boltzmann law to calculate what
temperature the Earth must be to radiate
away that amount of energy. It works out
at 279 K. To get that number, we simplified
some things; we assumed that all the Sun’s
radiation made it to the Earth’s surface
and that all the Earth’s radiation left unimpeded. The actual average temperature of
the Earth is about 288 K. To obtain this
number we must take into account the
natural greenhouse warming and the fact
that about 30% of the Sun’s radiation is
reflected away and thus does not heat the
Earth. These effects will be discussed later.
Using the Stefan-Boltzmann law, we should
be able to determine the approximate
average temperature of the Earth, if we
assume that the Earth emits the same
amount of energy each second that it
absorbs from the Sun; that is, that the Earth,
on average, is neither heating up nor
cooling down.
The Stefan-Boltzmann law tells us how
much radiant energy per second per unit
area an object at a given temperature emits.
We know the Sun’s temperature, so we can
calculate this value for the sun. Multiplying
this number by the Sun’s surface area gives
us the total energy the Sun emits each second. To figure out what fraction of this the
Earth receives, imagine, as we did in earlier,
a huge sphere with a radius equal to the
distance between the Sun and Earth. Imagine a round spot the size of a cross section
of Earth pasted on the surface of that
sphere. If we divide the cross sectional area
of the Earth (the size of the spot) by the
surface area of the imaginary sphere, and
Problems
levels of hydrogen that absorb radiation at
656.3 nm? This line is called the Hα line and
is one of the most important spectral lines in
astrophysics. It is the first line of the “Balmer
series,” and the two energy levels involved
are the first and second excited levels.
1. In calculating the temperature of the Earth,
we assumed that all of the Sun’s energy was
absorbed by the Earth and its atmosphere. In
fact, only about 70% of this energy is absorbed. The rest is reflected and scattered by
the atmosphere and Earth’s surface. (This is
another way of saying that the albedo of the
Earth is about 30%.) What would the Earth’s
temperature be if its albedo were only 10%?
4. At what wavelength does the maximum
radiation occur for an “O-type” star with a
temperature of 30,000 K? This type of star
appears blue, since much more radiation is
emitted at shorter, blue, than longer, red,
wavelengths.
2. How much solar energy would be collected
by a 3-foot by 5-foot solar collector on a clear
day if the collector absorbs 95% of this energy, and if 70% of the solar radiation penetrates the atmosphere?
5. How much more energy per second per
square meter does the star in Problem 4 emit
than the Sun?
6. What is the temperature of an “M-type” star
whose maximum radiation is at 1,000 nm?
3. What is the energy difference in the two
11
THE SUN-EARTH SYSTEM
III
The Earth’s Atmosphere
Composition and Distribution
of the Atmosphere
troposphere is heated by energy from the Earth,
so its temperature decreases with altitude: from
about 288 K at sea level to 216 K at its upper
boundary.
Above the troposphere, from about 11 to 50
km altitude, is the stratosphere. This level contains the ozone layer. Ozone, O3, is composed of
three oxygen atoms. This molecule strongly
absorbs certain ultraviolet wavelengths, and the
absorption of this radiation from the Sun heats
the air. So temperature increases with height in
the stratosphere, to about 271 K at its top.
Moving upward, we come to a region in
which the temperature again falls with altitude,
because it has hardly any ozone to absorb radiation. This region, the mesosphere, extends to 90
km or so, where the temperature has fallen to
about 183 K.
The top layer is the thermosphere, where
oxygen molecules, O2, absorb solar radiation at
wavelengths below about 200 nm. The absorption of this energy increases the energy of the
atoms, raising their temperature, which climbs
rapidly with increasing altitude.
The composition of the atmosphere and the
way its gases interact with electromagnetic
radiation determine the atmosphere’s effect on
energy from the Sun—and vice versa. Table 2
lists the major components of our atmosphere
and their relative concentrations.
The atmosphere may conveniently, if rather
arbitrarily, be broken into four layers, depending on whether the temperature is falling or
rising with increasing height above the Earth’s
surface (Figure 7). The lowest layer, the troposphere, extends up to about 11 km. It contains
about 75% of the mass of the atmosphere and is
the region in which weather takes place. The
Table 2
Major Components of the Earth’s Atmosphere
Gas
Concentration
Nitrogen, N2
Oxygen, O2
Argon, A
Carbon dioxide, CO2
Water vapor, H2O
Ozone, O3
Methane, CH4
Carbon monoxide, CO
Nitrous oxide, N2O
78.1% by volume
20.9% by volume
0.9% by volume
350 ppm
0–4%, variable
4–65 ppb
1,750 ppb
150 ppb
280 ppb
The Atmosphere’s Interaction with
Solar Radiation
The Earth’s atmosphere is like an obstacle
course for incoming solar radiation. About 30%
of it is reflected away by the atmosphere,
clouds, and surface. (The percentage of solar
radiation a planet reflects back into space, and
that therefore does not contribute to the planet’s
warming, is called its albedo. So the albedo
of the Earth is about 30%.) Another 25% is
Note: ppm means parts per million; ppb means
parts per billion
12
THE EARTH’S ATMOSPHERE
Figure 7. Atmospheric layers are determined by the way the temperature
changes with increasing height. From J. R. Eagleman, Meteorology: The
Atmosphere in Action. Wadsworth. Reprinted by permission.
13
THE SUN-EARTH SYSTEM
absorbed by the atmosphere and clouds, leaving only 45% to be absorbed by the Earth’s
surface.
vibrational and rotational energy levels (E) of
molecules, however, is less than that of electrons in atoms. We know, from Section II, that
electrons can be bumped up to higher levels
when they absorb radiation of the proper wavelength. We also know that wavelength is inversely proportional to the amount of energy
the wave has. Short-wavelength radiation has
more energy than long-wavelength radiation.
So, if E is less for molecules than for atoms, the
wavelengths emitted or absorbed by molecules
are longer—mainly in the infrared part of the
spectrum. These molecules absorb the Sun’s
radiation almost completely over most of the
infrared spectrum, but there are certain wavelengths, called “windows,” in which much of
the radiation does reach the Earth. These
Absorption of Solar Radiation
by the Atmosphere
Different gases in the atmosphere absorb
radiation at different wavelengths. Figure 8
is a schematic representation of how much of
each segment of the electromagnetic spectrum
reaches the Earth’s surface. Only the visible
radiation and parts of the infrared and radio
regions penetrate the atmosphere completely.
None of the very short wavelength, high-energy
gamma rays and x-rays make it through. Atmospheric nitrogen and oxygen remove them and
short-wavelength (up to about 210 nm) ultraviolet radiation as well. Stratospheric ozone
eliminates another section of the ultraviolet
band, between about 210 and 310 nm.
Most of the radiation between 310 nm and
the visible makes it to ground level. It is this
radiation that produces suntans (and sunburns,
skin cancer, and eye damage, if we are not careful) and that is blocked by effective suntan lotions and sunglasses. A slight increase in the
amount of short-wavelength UV radiation that
reaches the Earth could have devastating effects, and it is the ozone layer that controls how
much of this radiation reaches us.
As we move from the visible toward longer
wavelengths, we see, from Figure 8, that only a
portion of the infrared radiation gets through
the atmosphere. The main constituents responsible for absorbing it are water vapor (H2O) and
carbon dioxide (CO2). Remember that electrons
in atoms have certain energy levels they may
occupy. This is true, too, of molecules such as
H2O and CO2. Molecules are able to vibrate or
rotate, and different vibrational and rotational
modes are associated with different amounts of
energy. Figure 9 is a schematic representation of
these motions. These modes, like the energy
levels of electrons, represent only certain discrete amounts of energy. The difference in the
Figure 8. Solar radiation reaching the Earth’s surface. At
the top of the atmosphere most of the solar radiation is still
present. By the time the radiation reaches the Earth’s
surface, radiation in most spectral regions has been
removed by the Earth’s atmosphere.
14
THE EARTH’S ATMOSPHERE
windows lie in the part of the infrared region
closest to the visible, called the “near” infrared.
Between the near infrared and the radio region, no radiation reaches the Earth. The radio
region has a wide transparent window from
slightly less than 2 cm to about 30 m. Another
part of the atmosphere, the ionosphere, blocks
out wavelengths longer than 30 m. The ionosphere, lying between about 90 and 300 km
above the Earth, contains fairly large concentrations of ions and free electrons. (An ion is an
atom or molecule with either an excess or deficiency of electrons.) Free electrons (electrons
that are unattached to atoms) are primarily
responsible for reflecting the long radio waves.
This has the highly useful side-effect of allowing long-range radio communication by reflecting low-frequency radio waves transmitted
from the ground back toward the Earth.
We see, then, that radiation in most of the
spectral regions does not reach the Earth. However, the majority of radiation from the Sun lies
in spectral regions for which the Earth’s atmosphere is transparent or partially transparent:
the visible, near ultraviolet, and near infrared.
This is the radiation that is critical to the energy
balance of the Earth and its atmosphere.
Absorption of Earth’s Radiation by the
Atmosphere: The Greenhouse Effect
The presence of gases in the atmosphere that
absorb and reradiate infrared radiation, combined with the fact that the Sun and Earth emit
radiation in different wavelength regions, is the
principal cause of the phenomenon called the
greenhouse effect.
When the Earth, or any body, absorbs radiation, its temperature rises. We know this if we
have ever tried walking barefoot on black pavement on a hot, sunny day. We also know that,
under the same sunlight, some things will get
hotter than others. The sand on the beach will
WHY THE SKY IS BLUE
Practically none of the Sun’s visible radiation is absorbed by the atmosphere, but
some of it is scattered. The molecules and
tiny dust particles in the atmosphere scatter the shorter-wavelength blue light more
effectively than the longer-wavelength red
light, a phenomenon known as Rayleigh
scattering. What we see when we look at
the sky is light from the Sun that has been
scattered in our direction. Since blue is
scattered most, the sky appears blue.
This same process gives us our red Sun
at sunset and sunrise. When we look directly at the Sun, we see only the light
from it that has not been scattered out of
the beam. At sunset or sunrise (which are
the only times we may safely look directly
at the Sun), the Sun’s radiation must go
through the maximum thickness of atmosphere. The greater the thickness, the
more the scattering. Since the shorter
wavelengths (violet and blue) are scattered most, what is left when the light
reaches our eyes is light of longer wavelengths (red and orange). Even at noon, if
we could look at the Sun we would see it
as slightly more yellow than it actually is.
Figure 9. A two-atom molecule may vibrate and rotate.
15
THE SUN-EARTH SYSTEM
by volume, do not absorb energy in the near
infrared, so relatively minor atmospheric constituents turn out to have an importance far
greater than their numbers would indicate.
Figure 11 shows a summary of the radiation
from the Sun arriving at the top of the atmosphere and the radiation leaving the surface of
the Earth. Notice the very different wavelength
regions of the two, a major factor in the greenhouse effect. The figure also shows, roughly,
the spectral regions absorbed and transmitted
by the Earth’s atmosphere, listing some of the
gases responsible for the absorption.
Now let us assume that the Earth is in equilibrium, so that the amount of energy coming in
equals the amount leaving. One thing to keep in
mind is that the Earth may lose energy only in
the upward (or outward) direction, while the
atmosphere and clouds lose part of their energy
in an upward direction and part downward.
The Earth’s surface loses energy not only by
be warmer than the water; a dark shirt gets
hotter than a white one. But if we were to measure the temperature of everything all over the
Earth, we would find a relatively constant longtime average temperature: 288 K. Solid objects
at any temperature emit radiation at all wavelengths, and Wien’s law tells us where, for a
given temperature, the maximum amount of
radiation is emitted. For an object whose temperature is 288 K, this peak is at about 10 micrometers. (A micrometer is a millionth of a
meter.) A more involved calculation would
show that about 60% of the radiation from an
object at 288 K lies between 6 and 17 micrometers, the infrared region of the spectrum. So the
majority of the radiation being emitted by the
Earth lies in the near and intermediate infrared,
the very region where, as Figure 10 shows, water vapor, carbon dioxide, and ozone are
strongly absorbing. Nitrogen and oxygen,
which comprise over 99% of the atmosphere
Figure 10. Radiated Earth energy showing atmospheric regions. Note that the
visible region is off the figure to the left.
16
THE EARTH’S ATMOSPHERE
From Figure 12, and defining the amount of
energy from the Sun striking the top of the atmosphere as 100 units, we can get the following
energy balances:
radiating it away but also by conduction, convection, and evaporation. The atmosphere,
including clouds, loses energy by radiation
alone.
Figure 12 shows that 70% of the incoming
solar radiation is absorbed by, and heats, the
Earth’s atmosphere and clouds (25%), and surface (45%). The atmosphere also absorbs about
96% (100/104) of the energy that the Earth radiates. (The rest is lost to space.) The gases in the
atmosphere responsible for this absorption—
such as water vapor, carbon dioxide, methane,
and ozone—are the so-called greenhouse gases.
The greenhouse gases reradiate the long-wavelength radiation back to the Earth’s surface and
to space. They absorb radiation from the Sun
and Earth and emit radiation to the Earth and
space.
Atmosphere and Clouds
Energy Gained = 100 + 29 + 25 = 154 units
Energy Lost = 66 + 88 = 154 units
Net Energy = 0
Surface (Land and Oceans)
Energy Gained = 45 + 88 = 133 units
Energy Lost = 104 + 29 = 133 units
Net Energy = 0
Space
Energy Lost = 25 + 5 +66+ 4 = 100 units
Energy Gained = 100 units
Net Energy = 0
Figure 11. Comparison of solar and Earth radiation. The solar curve is scaled to
equal the Earth’s curve at the peak.
17
THE SUN-EARTH SYSTEM
of the atmosphere remain at a fixed concentration, but that the amount of CO2 increases due
to, for example, continued burning of fossil
fuels. This means that more of the Earth’s radiation will be absorbed and reemitted back toward the surface, increasing the net amount of
energy striking the Earth. This net increase in
energy absorbed by the Earth will result in a
temperature increase.
Some greenhouse gases are much more effective in trapping the Earth’s radiative energy
than others. Figure 13 shows the way different
atmospheric gases absorb radiation at different
wavelengths. These atmospheric gases account
for the overall absorption of radiation by the
atmosphere. Although these gases (except for
water vapor) are present in the atmosphere in
only trace amounts (parts per billion to parts
The net energy is zero at each level—the top
of the atmosphere, inside the atmosphere, and
at the Earth’s surface. We can see immediately
that the Earth is warmer due to the greenhouse
gases in its atmosphere than it would be without them. It gains 133 units with the atmosphere present, only 90 units without it
(assuming that if there were no atmosphere the
Earth’s surface would reflect 10% of the incident energy). The process is referred to as the
greenhouse effect.
One way of looking at the greenhouse effect
is to think of the carbon dioxide or other greenhouse gases as a one-way filter; their increase in
the atmosphere only negligibly affects the
amount of radiation reaching the Earth, but it
significantly affects the amount leaving it.
Now assume that all of the other components
Figure 12. An illustration of how the atmosphere and clouds act to trap energy
from the Earth’s surface and reradiate most of it back to the Earth. This raises
the surface temperature by about 33 K above what it would be without the
atmosphere. From S.H. Schneider, “The Changing Climate,” Scientific
American 261 (3), p. 72. Reprinted by permission.
18
THE EARTH’S ATMOSPHERE
per million), they are producing essentially all
of the absorption of radiation, so a slight change
in their concentrations will produce a large
change in the amount of radiation absorbed.
Also notice that water vapor (H2O) and carbon
dioxide (CO2) are absorbing essentially 100% of
the radiation in their absorption regions,
whereas methane (CH4) and nitrous oxide
(N2O) are not. It is not as easy for carbon dioxide and water vapor to absorb additional radiative energy as it is for methane and nitrous
oxide. Also, methane and nitrous oxide have
absorption regions closer to the peak of the
Earth’s radiation spectrum. The combination of
these two facts means that, for example, an
increase in methane is 25 times as effective at
absorbing radiation from the Earth as is the
same relative increase in carbon dioxide, even
though methane’s concentration is only about
half of 1% that of carbon dioxide. Chlorofluorocarbons are also trace gases that are very proficient at absorbing terrestrial radiation.
While it is true that the Earth is warmer due
to its atmosphere than it would be otherwise, as
of this writing there is no definitive answer to
the question of how rapidly and severely an
increase in atmospheric concentrations of CO2
and other greenhouse gases will increase the
Earth’s surface temperature. This is because
other effects occur when the Earth’s surface
begins to warm. There might, for instance, be
an increase in cloud cover, which would reduce
the rate at which the temperature increases.
The question of what effect changes in atmospheric composition will actually and ultimately have is enormously complex because
the Earth’s climate system is complex and we
have yet to understand, or even define, all of
its components.
What is clear is that gases capable of
warming the Earth are building up in the atmosphere. See Figures 14 and 15 as examples of the
measured increase in concentration of CO2 and
CH4 in our atmosphere.
By now you should have an understanding
WHY DO WE CALL IT THE GREENHOUSE EFFECT?
The term greenhouse effect, strictly speaking, is a misnomer. It developed from the
assumption that the glass panes on a
greenhouse serve the same purpose as the
carbon dioxide and water vapor in the
atmosphere: they let visible solar radiation
in but trap much of the long-wavelength,
infrared radiation. (In the case of the
greenhouse, infrared radiation is emitted
by the soil, plants, etc., inside.) Studies
have shown, however, that the glass panes
of a greenhouse contribute to its heating
more by keeping the air inside from mixing with the outside air than by absorbing
the outgoing radiation. Still, the term continues to be used in describing the warming effect of certain gases in the
atmosphere.
Figure 13. Absorption of different wavelengths by trace
gases and H2O in the atmosphere. The scale at left shows
the percent of absorption; along the bottom is the
wavelength. For example, carbon dioxide absorbs almost
all the incoming radiation at 2.6 and 4.3 micrometers.
19
THE SUN-EARTH SYSTEM
of how and where the Earth’s external energy is
produced, the nature of this electromagnetic
energy, and how it interacts with the Earth and
its atmosphere to produce an environment that
is, so far, relatively comfortable to the Earth’s
inhabitants.
Figure 14. The concentration of carbon dioxide in the air as measured at the
Mauna Loa Observatory from 1958 to 1986. Units of concentration are parts
per million by volume. From John Firor, The Changing Atmosphere: A
Global Challenge. Yale University Press, 1990. Reprinted by permission.
20
THE EARTH’S ATMOSPHERE
Problems
1. Quantitatively, the wavelength dependence
of Rayleigh scattering is 1/λ4. How much
more strongly is blue light (say, 400 nm)
scattered by the atmosphere than red light
(use 700 nm)?
2. What color would the sky be if the atmospheric scattering were the same average
strength as at present, but not dependent
on wavelength?
3. The energy level separation that produces
the hydrogen red line at 656.3 nm is 3.03 x
10–19 joules. What is the difference in energies
of the two levels of CO2 that produces its
absorption at 4,300 nm?
Figure 15. The concentration of methane in the
atmosphere at various times in the past as deduced from
measurements of air trapped in ice cores (1480–1950) and
from direct measurements of air samples (after 1950). The
concentration is plotted in parts per billion by volume.
From John Firor, The Changing Atmosphere: A Global
Challenge. Yale University Press, 1990. Reprinted by
permission.
4. In what part of the spectrum does the CO2
absorption in the above problem occur?
5. If the atmosphere did not absorb in the infrared, would the Earth’s surface be warmer or
cooler?
21
THE SUN-EARTH SYSTEM
APPENDIX I
Scientific Notation
Science deals with very large and very small
numbers, numbers that are inconvenient to
write in the everyday “long form.” If we need
to write a number like one hundred fifty, it’s
easy and quick—150. But suppose, instead, we
are dealing with a number like one hundred
fifty million, the number of kilometers between
the Sun and the Earth. That is a little harder—
150,000,000. In fields of science like astronomy,
even this is a relatively small number. Numbers
like a million million million million are not
uncommon. We see, then, that it would be
helpful to have a “shorthand” way to write
very large (and very small) numbers. Scientific
notation is a convenient way to write such
numbers. It involves expressing numbers in
powers of ten, using a superscript or exponent.
The exponent gives the number of zeros to add
after 1. For example,
such as 1/1,000,000. We may write this as
0.000001. In scientific notation this number is
written as 1.0 x 10-6. When the exponent is negative, we move the decimal point to the left instead of the right.
Multiplying and dividing numbers written in
scientific notation is especially easy, when you
get the hang of it. Simply multiply or divide the
numbers in front of the tens and add or subtract, respectively, the exponents. As an example, let us multiply 4.0 x 108 and 2.0 x 10-4.
(4.0 x 108) x (2.0 x 10-4) = (4.0 x 2.0) x (108+ (-4))
= (8.0) x (104)
= 8.0 x 104.
Now let us divide the same numbers.
(4.0 x 108)/(2.0 x 10-4) = (4.0/2.0) x (108-(-4))
= (2.0) x (108+4)
= 2.0 x 1012.
101 = 10
102 = 100
103 = 1,000
1012 = 1,000,000,000,000.
You might want to check these results by
writing the numbers out in the long form. If
you do you will see how much easier and
quicker using scientific notation can be.
If you need to add or subtract numbers written in scientific notation, you must first write
the numbers so that they are given in the same
power of ten. Then just add or subtract as you
normally would. For example, to add 2.46 x 103
to 5.23 x 104 we would do the following:
This system can be applied to any number.
Suppose, for example, we wanted to write the
number 5,280, the number of feet in a mile.
Since 5,280 is 5.280 times 1,000, we may write it
as 5.280 x 103. (Always place the decimal point
between the first and second number when
using scientific notation.) The exponent tells us
how many places to move the decimal point to
the right to express the number in the long
form.
We may also need to write fractions, numbers less than one, that are extremely small,
2.46 x 103 + 5.23 x 104 = 2.46 x 103 + 52.3 x 103.
Now we have expressed both numbers to the
same power of ten, namely 3.
22
APPENDIX I
The result is 54.76 x 103. We would then
round this number to 54.8 x 103 and, to follow
our convention, move the decimal point to between the first and second number and raise the
power of 10 by 1, that is, to 4. This would finally
give us the answer 5.48 x 104. The complete
process looks like this:
Scientists also use prefixes to indicate powers
of ten in describing large and small numbers.
For instance, we use nanometer in talking about
wavelengths. A nanometer is a meter times 10-9.
The following table lists some common prefixes
and a few familiar examples.
2.46 x 103 + 5.23 x 104 = 2.46 x 103 + 52.3 x 103
= 54.76 x 103
= 54.8 x 103
= 5.48 x 104.
Table 3
Scientific Prefixes for Common Powers of Ten
Factor
Prefix
Symbol
1012
109
tera
giga
T
G
106
103
102
101
10–1
10-2
10-3
10-6
10-9
10–12
10–15
10–18
mega
kilo
hecto
decka
deci
centi
milli
micro
nano
pico
femto
atto
M
k
h
da
d
c
m
µ
n
p
f
a
Example
GHz – gigahertz (Hz, hertz, means cycles
per second)
MHz – megahertz
km – kilometer
cm – centimeter
mm – millimeter
mm – micrometer
nm – nanometer
23
THE SUN-EARTH SYSTEM
APPENDIX II
Units and Dimensional Analysis
Units
The principle of conservation of energy says
that the amount of energy (of all types) around
before an event takes place exactly equals the
amount after the event. Let us assume that the
amount of energy (called potential energy, PE) a
ball of mass m has when raised to a height h is
mgh, where g is the acceleration due to gravity
(PE = mgh). We know the ball has energy, because, for example, if we dropped it on a drumhead it would not only bounce back up some
distance but would also create sound waves, a
form of energy. Before we drop the ball, its
speed is zero and its height is h. We know that
it will have a certain speed, v, when it reaches
the drumhead after falling. Since at the drumhead its height will be zero (we are measuring
height from the drum), mgh will be zero. But
energy must be conserved, so the energy of the
ball right at the drumhead must depend on its
speed. It now has some speed, but no height, so
the potential energy must have been converted
into some other form of energy that has to do
with its speed, v. How does this “energy of
motion” (called kinetic energy, KE) depend on
the ball’s speed and possibly its mass?
Scientists use an International System of
Units (SI) to describe and measure various
quantities. The three basic units from which all
other quantities may be derived are length,
mass, and time. In the SI system, length is measured in meters and mass in kilograms. Time is
measured in seconds. In the SI system, the basic
quantities are written as follows:
• Length in meters, m (approximately
1.1 yard)
• Mass in kilograms, kg (approximately
2.2 pounds)
• Time in seconds, s
• Temperature in kelvins, K
All other quantities are derived from these
units. For example, we express energy in the SI
system as joules, J. In terms of the basic units a
joule is a kg m2s–2 (s–2 means per second
squared, or per second per second), and is
about equal to the amount of energy imparted
to the floor by dropping a 2.25 kg (5 pound) bag
of sugar a distance of 5 cm (2 inches). The familiar measure of radiant power, watt (W), is a
joule per second.
In science, the units on each side of an equation must be the same. This is another way of
saying that we must compare apples with
apples and not oranges. This property can help
scientists find a correct relationship between
quantities even when the theory behind the
relationship is not completely understood.
Dimensional Analysis
Dimensional analysis or “keeping the units
the same on both sides of the equation” can
help. We know that the kinetic energy must in
some way depend on the speed and mass,
maybe each raised to some power. Let us set up
an equation with the total energy at height h
24
APPENDIX II
becomes LT–2. On the right side, v has dimensions of length per time, so v becomes LT–1.
Using these notations for mass, length, and
time, we may write mgh = mavb as
equal to the total energy at height zero, so that
energy is conserved.
Energy at height h + Energy with speed zero =
Energy at height zero + Energy with speed v
(1)
In symbols this may be written:
mgh + 0 = 0 + KE
mgh = KE
(M)(LT–2)(L) = (Ma)(LT–1)b
ML2T–2 = Ma(LT–1)b
ML2T–2 = MaLbT–b
(2)
(3)
Since the powers of like units on each side of
the equation must be the same, we see from
inspection that a = 1 and b = 2. Going back to
Equation 4, since v = LT–1, we may write KE as
As we reasoned earlier, KE must also equal
m times v, each raised to some power. We will
raise m to the power “a” and v to the power
“b,” realizing that these exponents could be
positive, negative, or zero.
KE = mavb
KE = mv2.
Actually, the correct expression is KE =
(1/2)mv2, a fact we could determine by measuring the mass of the ball and the velocity after it
had fallen a distance h. But the point is that the
dependence of KE on mass and speed are right,
and we obtained the relationship solely by
some intuition and dimensional analysis.
(4)
So, because they both equal KE, mgh = mavb.
In order to keep tabs on the basic units we will
write mass as M, length as L, and time as T.
Now, g is the acceleration due to gravity, and
its units are length per time squared, so g
25
THE SUN-EARTH SYSTEM
APPENDIX III
Physical Constants and
Data for the Sun and Earth
Quantity
Symbol
Value
Units
Speed of light in vacuum
c
2.998 x 108
m s-1
Planck’s constant
h
6.626 x 10-34
Js
Stefan-Boltzmann constant
σ
5.670 x 10-8
W m-2 K-4
Acceleration of gravity near Earth
g
9.80
m s-2
Radius of Sun
RSun
6.96 x 105
km
Radius of Earth
REarth
6.37 x 103
km
Mass of Sun
MSun
1.989 x 1030
kg
Mass of Earth
MEarth
5.974 x 1024
kg
Temperature of Sun
TSun
5,800
K
Temperature of Earth
TEarth
290
K
Luminosity of Sun
LSun
3.83 x 1026
J s-1
Albedo of Earth
α
0.30
dimensionless
Earth–Sun distance
1 AU
1.495 x 108
km
Solar constant
SC
1,376
W m-2
26
APPENDIX IV
Measuring the Solar Constant
Supplies needed
through the (plastic) wall of the bottle in order
to raise the temperature of the water inside.
Plastic transmits very little radiation in the ultraviolet and infrared, so let us assume that the
water will be detecting about 50% of the energy
reaching the bottle (since 44% is in the visible,
where the plastic obviously transmits very
well). Therefore, we should detect 757/2 or
379 J s–1 m–2 with our setup. You see that we
are making a number of approximations, but
at least we are aware of the aproximations that
are being made. If, after making your measurement, you multiply the value obtained by 2
(1/0.50) and then by 1.82 (1/0.55), you should
get a number close to 1,376 J s-1 m-2.
Water is a good detector of energy in this
experiment, since at normal temperatures it
takes 4,186 J per kg to raise its temperature 1
kelvin (or Celsius). The reason for the black ink
(a few drops of which is placed into the water)
is to allow the water-ink solution to absorb all
wavelengths transmitted by the plastic approximately equally well.
The experiment is done by pouring a measured amount of water (along with a few drops
of India ink) into the bottle and then placing
the bottle in the direct sun. The flat face of the
bottle, in which the thermometer and cork have
been placed, is kept perpendicular to the direction of the sun for a measured amount of time.
The temperature rise is then noted. From these
measured values, we may calculate the amount
of energy (Joules) the water received per second. We can then measure the area of the surface of the bottle, and after dividing the above
value by this area (in square meters) determine
the solar constant.
• Small (~250 ml), flat-sided, bottle [Note: a
T-75 tissue-culture flask works well here]
• Cork stopper for bottle bored for the
thermometer
• Black India ink
• Thermometer (range: ambient to 10°C
above)
• Stopwatch
• Metric measuring cup
Discussion of Experiment
In this experiment we let radiation from
the Sun raise the temperature of a measured
amount of water. Then, by making careful measurements of the temperature rise andthe water
volume (from which we may obtain its mass)
we will be able to say how much energy the
water has received. Knowing the amount of
time over which the radiation energy was
collected and the cross-sectional area of the
container, we will finally be in a position to
calculate the ground-level solar constant.
We have learned that the amount of radiant
energy each square meter of the Earth receives
each second from the Sun (at the top of the atmosphere) is about 1,376 J s–1 m–2. Also, we
have seen that only about 55% of this energy
reaches the ground. In addition, this energy is
spread out over the electromagnetic spectrum,
with about 44% being in the visible, 7% in the
ultraviolet, and 49% in the infrared and longer
wavelengths.
In our experiment, the solar energy must go
27
THE SUN-EARTH SYSTEM
Procedure
Cool the water solution to ambient by placing it under running water, and repeat the experiment. Do the experiment three times total.
Now let's do the experiment. It should be
done as close to noon as practical and on a
clear, cloudless day. Your instructor will supply
you with the items listed above. Measure 200
ml of cool water (close to ambient air temperature) and pour it into the bottle. Then add a few
drops of India ink to the water until it is fairly
black.
Place the thermometer and cork in the bottle
and put the bottle in the shade close to where
you will do the experiment. This is to let the
water and thermometer come to ambient air
temperature. After several checks, a few minutes apart, indicate that the temperature of the
water is neither rising nor falling (record this
temperature), you may prepare to place the
bottle in the direct sun.
The bottle should be propped so that its face
is as close to perpendicular to the sun's direction as possible. You can assure this by noting
the shadow on a white card placed behind the
bottle. Until you are ready to start timing, have
a card or book or something opaque in front of
the bottle to block the direct rays from the sun.
(It will help to have a lab partner for this experiment.) Remove the sun block and begin timing.
Allow the sun’s rays to strike the bottle for
about 20 minutes. This will be enough time to
cause the temperature of the water to rise several degrees Celsius (See example.) Record the
temperature and elasped time.
Example: Calculation of Temperature
Rise of the Water
Remember we are expecting the “effective”
solar constant to be about 379 J s–1 m–2. The
density of water is 1 kg per liter. The amount of
water in the bottle is 200 ml, or 0.20 liter. Therefore, the mass of the water is 0.20 kg. Now, the
specific heat capacity of water is 4,186 J kg–1K–1,
and since we have only 0.20 kg, we should get
1 degree temperature rise for every 0.20 x 4,186
joules of energy input. That is, every 837 joules
of energy input should produce 1 degree rise in
temperature.
Now we need to determine the energy input
expected from the solar radiation. The face of
the bottle is about 8 cm by 9 cm, or it has an
area of 72 cm2. When we convert this to meters,
we get 7.2 x 10–3m2. Therefore we may expect to
collect 379 x 7.2 x 10–3 = 2.73 Js–1. This comes out
to be about 164 J per minute. Since it will take
837 J to produce a 1-degree rise in temperature,
we will have to wait 837/164 minutes, or 5.1
minutes, for the temperature to increase by 1
degree. In order to read the temperature rise of
the thermometer reasonably accurately, we will
want it to change by 3 or 4 degrees. This means
that we should expect to wait 15 to 20 minutes
or so before recording the temperature rise.
28
APPENDIX IV
DATA SHEET
Name:
Date:
cm3
Volume of water:
liter (1000 cm3 = 1 liter of water)
Mass of water:
kg (1 liter of water weighs 1 kg)
m2
Surface area of bottle:
Trial No.
Initial Temperature
(o C)
(1 cm2 = 1 x 10-4 m2)
Final Temperature
(o C)
Average change in temperature: ∆T =
Elapsed Time
(sec)
(o C s-1)
(Energy needed to raise 1 kg of water 1 °C is 4,186 joules, so energy needed to raise m kg of water
[where m is the mass of water you got above] ∆T °C is m times ∆T times 4,186 joules.)
The energy your detector absorbed is 4,186 times
the mass, m, of your water sample times ∆T, which is:
joules.
Now your detector received this amount of energy in, say, 20 minutes (1200 seconds), and the
area of the water receiving the energy was 7.2 x 10-3 m2.
(J s-1)
Average amount of energy absorbed per second:
Average amount of energy absorbed per second per square meter:
(J m-2 s-1).
This value is your measurement of the “ground level,” uncorrected solar constant. As discussed
above, if you now multiply this value by 2 and then 1.82, you should get a “corrected value” for
the solar constant.
“Corrected” solar constant:
(J m-2 s-1)
Accepted value for the solar constant:
(J m-2 s-1)
Difference:
(J m-2 s-1)
% error
29
THE SUN-EARTH SYSTEM
GLOSSARY
absolute zero—zero degrees kelvin, the lowest
temperature on the scientific temperature
scale.
continuous emission—radiation that is emitted
in the form of a continuous distribution of
wavelengths.
absorption—the process by which electromagnetic energy is given up to an object.
convection—energy transfer by the movement
of matter.
absorption spectrum—the distinctive pattern
of wavelengths of electromagnetic radiation
that an atom absorbs.
corona—the outer layer of the Sun’s
atmosphere.
albedo—the percentage of incident radiation
that a surface reflects and that thus does not
contribute to its heating.
Einstein’s law of mass-energy equivalence,
E=mc2— The law of physics stating that mass
may be converted to energy and vice versa.
astronomical unit (AU)—the average distance
between the Sun and the Earth.
electrical force—the force between two charged
particles. It is proportional to the charge of
each particle and to the inverse of the square
of the distance between them. The electrical
force may be attractive or repulsive.
atom—the smallest unit of an element, consisting of a nucleus (composed of protons and
neutrons) and electrons bound to the
nucleus.
electromagnetic radiation—energy, such as
that emitted from the Sun or an electric light
bulb, that is transmitted in the form of oscillating electric and magnetic fields.
blackbody radiation—continuous electromagnetic radiation emitted from an object that
absorbs all the radiation it receives.
element—a substance that contains only one
kind of atom.
chromosphere—the layer of the sun immediately above the photosphere.
emission—a process by which electromagnetic
radiation is given off by an object, an atom,
or a molecule. The emitting particle then
goes from a higher to a lower energy level.
conduction—the passage of energy through a
substance, not involving net motion of the
particles of the substance.
conservation of energy—a fundamental law
of physics stating that the total amount of
energy in all its forms after an event equals
the total amount before the event.
energy level—a possible (or allowed) value of
energy that an electron may have within an
atom.
30
GLOSSARY
evaporation—the process by which a liquid
becomes a gas; the vapor removes energy
from the parent substance.
isotope—one of two or more atoms whose nuclei have the same number of protons but a
different number of neutrons.
free electrons—electrons that move independently, without being bound to any particular atom.
joule—a unit of energy in the International
System.
kelvin—the absolute unit of temperature equal
to degrees Celsius plus 273, which, on this
scale, is the freezing point of water.
frequency—for a wave, the number of whole
wavelengths that pass a given point in one
second.
kinetic energy—the energy an object has due to
its motion.
galaxy—a collection of billions of stars that
evolved from a common source.
line emission—the emission of a specific electromagnetic wavelength by an element.
gravitational force—the force of attraction that
one mass exerts on another. It is proportional
to the masses of the two objects and inversely proportional to the square of the
distance between them.
mesosphere—the layer of the atmosphere
above the stratosphere, in which the temperature decreases with height. The mesosphere extends from about 50 km (the top
of the stratosphere) to 90 km.
greenhouse effect—the planetary warming
produced by the trapping of infrared radiation from the planet’s surface by its atmosphere.
molecule—an atomic group composed of at
least two atoms. Molecules are the building
blocks from which complex substances are
formed.
ion—an atom or molecule that has lost or
gained one or more electrons.
nuclear fusion—the process by which two
lighter nuclear masses join to form a heavier
nucleus and release energy.
ionosphere—a layer of the atmosphere, between about 90 and 300 km altitude, containing large concentrations of ions and free
electrons.
nuclear strong force—the short-range attractive
force that holds the atomic nucleus together
against the repulsive electrical force.
interstellar matter—the matter between stars in
a galaxy.
ozone layer—the region of the stratosphere and
lower mesosphere containing ozone, which
strongly absorbs solar ultraviolet radiation.
inverse-square law—a law of nature by which
a physical quantity varies with distance from
a source inversely as the square of the distance.
photosphere—the visible layer of the Sun’s
atmosphere.
proton-proton chain—a nuclear fusion process,
the net results of which are the conversion of
four protons into one nucleus of helium and
the release of energy.
irradiance—the amount of electromagnetic
energy received by a surface per second per
unit area.
31
THE SUN-EARTH SYSTEM
radiant exitance—the radiant energy emitted
by an object, measured in energy per second
per unit area.
to the absorption of solar radiation by
ozone.
supernova—a massive star that explodes, becoming extremely bright and emitting vast
amounts of energy.
Rayleigh scattering—scattering of radiant energy by small particles, such as molecules in
the atmosphere.
thermosphere—the uppermost layer of the
Earth’s atmosphere, from about 90 km up, in
which the temperature increases with height
due to absorption of high-energy radiation
from the Sun.
solar constant—the amount of solar radiant
energy received at the top of the Earth’s atmosphere per second per unit area when the
Earth is at its average distance from the Sun.
spectrograph—an instrument that photographs
the electromagnetic spectrum. It is used to
analyze the chemical composition of the
atmosphere.
troposphere—the lowest layer of the Earth’s
atmosphere, from ground level to 11 km, in
which our weather takes place.
visible spectrum—the region of the electromagnetic spectrum, ranging approximately
from 390 nm to 780 nm, that we can see with
our eyes.
spectrum—the distribution of electromagnetic
energy an object radiates over all wavelengths.
speed (of wave)—how fast the crests of a wave
move forward.
wavelength—the distance between adjacent
crests of a wave.
Stefan-Boltzmann law—a law stating that the
amount of energy per second per unit area
emitted by a blackbody is proportional to the
fourth power of the body’s temperature.
Wien’s displacement law—the radiation law
that relates the wavelength of maximum
energy output of a blackbody to its temperature.
stratosphere—the layer of the Earth’s atmosphere above the troposphere, lying between
11 and 50 km altitude, in which the temperature increases with height, due primarily
window—a region of the electromagnetic spectrum that is not absorbed by the atmosphere
and thus reaches the Earth.
32
ADDITIONAL READING MATERIAL
Periodicals
Friedman, H. Sun and Earth. New York: Scientific American Books, 1986.
Astronomy, published monthly by Astromedia
Corporation, 441 Mason Street, P. O. Box
92788, Milwaukee, WI 53202.
Frazier, K. Our Turbulent Sun. Englewood
Cliffs, N.J.: Prentice-Hall, 1982.
Scientific American, published monthly by Scientific American, Inc., 415 Madison Ave., New
York, NY 10017.
Kellogg, W.W., and M. Mead, Eds. The Atmosphere: Endangered and Endangering. Washington, D.C.: Government Printing Office,
1977.
Sky and Telescope, published monthly by Sky
Publishing Corporation, 49 Bay State Road,
Cambridge, MA 02139.
Mitton, S. Daytime Star: The Story of Our Sun.
New York: Scribners, 1983.
Science News, published weekly by Science Service, Inc., 1719 N Street N.W., Washington,
D.C. 20036.
Washburn, M. In the Light of the Sun. New
York: Harcourt Brace Jovanovich, 1981.
Books
Wilson, J.D. Physics: A Practical and Conceptual
Approach, second ed. New York: Saunders
College Publishing, 1989.
Eddy, J. The New Sun, NASA SP-402. Washington, D.C.: Government Printing Office, 1979.
33
THE SUN-EARTH SYSTEM
INDEX
argon, 12
atom,
energy level of, 7
atmosphere, Earth’s, 12
absorption of radiation, 15–20
appearance of, 15
interaction with solar radiation, 12–15
layers of, 13
major components, 12
ion, 15
inverse-square law, 4
kinetic energy, 1, 2
methane, 12, 17, 19, 21
neutrino, 3
neutron, 3, 7
nitrogen, 12
nitrous oxide, 12
nuclear strong forces, 2
nuclear fusion, 2
blackbody radiation, 9
carbon dioxide, 12, 14, 17, 19, 20
carbon monoxide, 12
chlorofluorocarbons, 19
ozone, 12
photon, 3, 7
Planck’s constant, 7
positron, 3
proton-proton chain, 2–3
deutron, 3
electrical force, 2
electron, 3, 7
free, 15
Einstein, 5, 7
law of mass-energy equivalence, 5
electromagnetic radiation, 5, 7–11
absorption of, 8
emission of, 8, 9
electromagnetic spectrum, 9, 10, 14
energy transport
conduction, 5
convection, 5
radiation, 5
scientific notation, 22–26
dimensional analysis, 24, 25
physical constants, 26
units, 24
star formation, 1–2
Stefan-Boltzman law, 9–11
sun,
age of, 1
appearance of, 15
core of, 5
elemental constituents of, 1
energy of, 4, 9–11
formation of the, 1
interior of, 5
solar constant, 27
fusion, see nuclear fusion
galaxy, 1
gravitational force, 1
greenhouse
effect, 15–20
gases, 15–20
temperature,
of Earth, 15–16, 18–19
of sun’s surface, 5, 9
helium
3 nucleus, 3
4 nucleus, 3
water vapor, 12, 14, 16–18
Wien’s displacement law, 9
34