The Sun-Earth System

(Full 34-page Text in HTML)

John Streete
Physics Department
Rhodes College
Memphis, Tennessee


 

Contents

Preface
Chapter 1: The Sun and Solar Energy
   Formation of the Sun
   Fusion
   Energy Transport
   Problems
Chapter 2: The Nature of Electromagnetic Radiation
   Energy Levels of Atoms
   Absorption and Emission of Electromagnetic Radiation
   Blackbody Radiation
   The Electromagnetic Spectrum
   Problems
Chapter 3: The Earth's Atmosphere
   Composition and Distribution of the Atmosphere
   The Atmosphere's Interaction with Solar Radiation
   Problems
Appendix I. Scientific Notation
Appendix II. Units and Dimensional Analysis
Appendix III. Physical Constants and Data for the Sun and Earth
Appendix IV. Measuring the Solar Constant
Glossary
Additional Reading Material
Index
Bibliographic Information

 

Index of Items
*Please note that the following links take you to the item's instance in the text
Figures
figure 1 Illustration of the Proton-Proton Chain
figure 2 Illustration of the Inverse-Square Law
figure 3 Cross Section of the Sun's Interior
figure 4 Energy-Level Diagram for Hydrogen
figure 5 Relative amounts of energy emitted by the sun at different wavelengths
figure 6 Regions of the Electromagnetic Spectrum
figure 7 Atmospheric Layers
figure 8 Solar Radiation Reaching the Earth's Surface
figure 9 Representation of Motion in a Molecule
figure 10 Radiated Earth Energy Showing Atmospheric Regions
figure 11 Comparison of Solar and Earth Radiation
figure 12 Earth's Atmosphere and Clouds Trap Energy
figure 13 Absorption of Different Wavelengths by Trace Gases and H2O in the Atmosphere
figure 14 Concentration of Carbon Dioxide in the Air
figure 15 Concentration of Methane in the Atmosphere
Tables
table 1 Particles Involved in the Proton-Proton Chain
table 2 Major Components of the Earth's Atmosphere
table 3 Scientific Prefixes for Common Powers of Ten
table 4 Physical Constants and Data for the Sun and Earth



Preface

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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




Chapter 1: The Sun and Solar Energy

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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.

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 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.

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. [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".]

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, 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.

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; 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 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).

Table 1. Particles Involved in the Proton-Proton Chain
Name Symbol Charge Description
proton p + Subatomic, positively charged particle that is one of the two principal particles comprising the nucleus.
neutron n 0 Subatomic, uncharged particle found as the other principal particle making up the nucleus.
deuteron d + 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.
electron e - Subatomic particle that orbits around the nucleus.
Positron e+ + An antielectron, the same as an electron but with a positive rather than negative charge.
Helium 3 nucleus He3 ++ An isotope of ordinary helium containing two protons and only one neutron instead of two.
Helium 4 nucleus He4 ++ "Ordinary" helium nucleus containing two protons and two neutrons.
Neutrino n 0 A chargeless particle having little, if any, mass. Important in many nuclear processes.
Photon (gamma ray) g 0 A chargeless "particle", or packet, of electromagnetic energy.

Step 1. (The particles' charges are noted above their symbols.)

+    +     +    +      0
p + p -> d + e+ + ν

Step 2. The deuteron is bound to a proton to produce helium 3 and a high-energy photon, g.

+    +     ++      0
d + p -> He3 + γ

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.

++      ++       ++      ++
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 proton-proton chain.

Figure 1. Illustration of the Proton-Proton Chain
In the proton-proton chain, four protons combine to form one helium nucleus and emit energy.

 

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 inverse-square 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

4πR2

where R is the radius and 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:

E = L/ 4πR2

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.

 

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.

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 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, traveling 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.

Figure 3. Cross Section of the Sun's Interior
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.

 

Problems

  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-billion-year life?

  3. What is the average mass density (mass per unit volume) of the Sun?

  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.)

  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 PP chains are occurring each second in the Sun's core?

 



Chapter 2: The Nature of Electromagnetic Radiation

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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 Levels of Atoms

Allowed Energies of the Electron

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?

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.

Energy of Electromagnetic Radiation

In many situations electromagnetic radiation may be described as having a wavelike 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

λf = v

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 well-defined 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:

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.

Absorption and Emission of Electromagnetic Radiation

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 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.

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.

Continuous Emission

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.

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.

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 = σT4

where σ 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:

λmax T = 2.898 x 106

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 Stefan-Boltzmann law are central to understanding the greenhouse effect, discussed in chapter 3.

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.

Figure 6. Regions of the Electromagnetic Spectrum
The electromagnetic spectrum consists of many regions. Each differs in wavelength, frequency, and energy.

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 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).

The Stefan-Boltzmann Law and The Earth's Temperature

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 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.

Problems

  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%?

  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?

  3. What is the energy difference in the two levels of hydrogen that absorb radiation at 656.3 nm? This line is called the H 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.

  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.

  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?

 



Chapter 3: The Earth's Atmosphere

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Composition and Distribution of the Atmosphere

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.

Table 2. Major Components of the Earth's Atmosphere
Gas   Concentration
Nitrogen N2 78.1% by volume
Oxygen O2 20.9% by volume
Argon A 0.9% by volume
Carbon dioxide CO2 350 ppm
Water vapor H2O 0-4%, variable
Ozone O3 4-65 ppb
Methane CH4 1,750 ppb
Carbon monoxide CO 150 ppb
Nitrous oxide N2O 280 ppb
Note: ppm means parts per million; ppb means parts per billion

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 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.

Figure 7. Atmospheric Layers
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.

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 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 absorbed by the atmosphere and clouds, leaving only 45% to be absorbed by the Earth's surface.

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.

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.

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 vibrational and rotational energy levels (E) of molecules, however, is less than that of electrons in atoms. We know, from chapter 2, 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 windows lie in the part of the infrared region closest to the visible, called the near infrared.

Figure 9. Representation of Motion in a Molecule
A two-atom molecule may vibrate and rotate.

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.

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.

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 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 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 10. Radiated Earth Energy Showing Atmospheric Regions
Note that the visible region is off the figure to the left.

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.

Figure 11. Comparison of Solar and Earth Radiation
The solar curve is scaled to equal the Earth's curve at the peak.

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 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.

Figure 12. Earth's Atmosphere and Clouds Trap Energy
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.

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:

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

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.

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.

Now assume that all of the other components 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 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.

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.

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 Figure 14 and Figure 15 as examples of the measured increase in concentration of CO2 and CH4 in our atmosphere.

Figure 14. Concentration of Carbon Dioxide in the Air
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.

By now you should have an understanding 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 15. Concentration of Methane in the Atmosphere
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.

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 1019 joules. What is the difference in energies of the two levels of CO2 that produces its absorption at 4,300 nm?

  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?

 



Appendix I: Scientific Notation

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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,

101 = 10
102 = 100
103 = 1,000
1012 = 1,000,000,000,000.

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, 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

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:

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.

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:

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

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.

Table 3. Scientific Prefixes for Common Powers of Ten
Factor Prefix Symbol Example
1012 tera T  
109 giga G GHz - gigahertz
(Hz, hertz - cycles per second)
106 mega M MHz - megahertz
103 kilo k km - kilometer
102 hecto h  
101 decka da  
10-1 deci d  
10-2 centi c cm - centimeter
10-3 milli m mm - millimeter
10-6 micro µ µm - micrometer
10-9 nano n nm - nanometer
10-12 pico p  
10-15 femto f  
10-18 atto a  

 



Appendix II: Units and Dimensional Analysis

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Units

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:

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.

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?

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 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     (2)
mgh = KE     (3)

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     (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 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 = m a v b as

(M)(LT-2)( L) = ( Ma)(LT-1)b
ML2T-2 = Ma(LT-1)b
ML2T-2= MaLbT-b

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

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.

 



Appendix III: Physical Constants and Data for the Sun and Earth

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Table 4. Physical Constants and Data for the Sun and Earth
Quantity Symbol Value Units
Speed of light in a vacuum c 2.998 x 108 m s-1
Planck's constant h 6.626 x 10-34 J s
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 m2

 



Appendix IV: Measuring the Solar Constant

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Supplies needed

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 and the 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-1m-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 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-1m-2 with our setup. You see that we are making a number of approximations, but at least we are aware of the approximations 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-1m-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.

Procedure

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 elapsed time.

Cool the water solution to ambient by placing it under running water, and repeat the experiment. Do the experiment three times total.

Example: Calculation of Temperature Rise of the Water

Remember we are expecting the "effective" solar constant to be about 379 J s-1m-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.

DATA SHEET

Name:

Date:

Volume of water: ____cm3 ____liter (1000 cm3 = 1 liter of water)
Mass of water: ____kg (1 liter of water weighs 1 kg)
Surface area of bottle: ____m2 (1 cm2 = 1 x 10-4m2)

Trial No. Initial Temperature (°C) Final Temperature (°C) Elapsed Time (sec)
       
       
       
       

Average change in temperature: ΔT = ____(°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] ΔT °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
ΔT, 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.

Average amount of energy absorbed per second: ____( J s-1)

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 ____

 



Glossary

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absolute zero - zero degrees Kelvin, the lowest temperature on the scientific temperature scale.

absorption - the process by which electromagnetic energy is given up to an object.

absorption spectrum - the distinctive pattern of wavelengths of electromagnetic radiation that an atom absorbs.

albedo - the percentage of incident radiation that a surface reflects and that thus does not contribute to its heating.

astronomical unit (AU) - the average distance between the Sun and the Earth.

atom - the smallest unit of an element, consisting of a nucleus (composed of protons and neutrons) and electrons bound to the nucleus.

blackbody radiation - continuous electromagnetic radiation emitted from an object that absorbs all the radiation it receives.

chromosphere - the layer of the sun immediately above the photosphere.

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.

continuous emission - radiation that is emitted in the form of a continuous distribution of wavelengths.

convection - energy transfer by the movement of matter.

corona - the outer layer of the Sun's atmosphere.

Einstein's law of mass-energy equivalence, E= mc2 - The law of physics stating that mass may be converted to energy and vice versa.

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.

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 field.

element - a substance that contains only one kind of atom.

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.

energy level - a possible (or allowed) value of energy that an electron may have within an
atom.

evaporation - the process by which a liquid becomes a gas; the vapor removes energy from the parent substance.

free electrons - electrons that move independently, without being bound to any particular atom.

frequency - for a wave, the number of whole wavelengths that pass a given point in one second.

galaxy - a collection of billions of stars that evolved from a common source.

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.

greenhouse effect - the planetary warming produced by the trapping of infrared radiation from the planet s surface by its atmosphere.

ion - an atom or molecule that has lost or gained one or more electrons.

ionosphere - a layer of the atmosphere, between about 90 and 300 km altitude, containing large concentrations of ions and free electrons.

interstellar matter - the matter between stars in a galaxy.

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.

irradiance - the amount of electromagnetic energy received by a surface per second per unit area.

isotope - one of two or more atoms whose nuclei have the same number of protons but a different number of neutrons.

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.

kinetic energy - the energy an object has due to its motion.

line emission - the emission of a specific electromagnetic wavelength by an element.

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.

molecule - an atomic group composed of at least two atoms. Molecules are the building blocks from which complex substances are formed.

nuclear fusion - the process by which two lighter nuclear masses join to form a heavier nucleus and release energy.

nuclear strong force - the short-range attractive force that holds the atomic nucleus together against the repulsive electrical force.

ozone layer - the region of the stratosphere and lower mesosphere containing ozone, which strongly absorbs solar ultraviolet radiation.

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.

radiant exitance - the radiant energy emitted by an object, measured in energy per second per unit area.

Rayleigh scattering - scattering of radiant energy by small particles, such as molecules in the atmosphere.

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.

spectrum - the distribution of electromagnetic energy an object radiates over all wavelengths.

speed (of wave) - how fast the crests of a wave move forward.

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.

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 to the absorption of solar radiation by ozone.

supernova - a massive star that explodes, becoming extremely bright and emitting vast amounts of energy.

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.

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.

wavelength - the distance between adjacent crests of a wave.

Wien's displacement law - the radiation law that relates the wavelength of maximum energy output of a blackbody to its temperature.

window - a region of the electromagnetic spectrum that is not absorbed by the atmosphere and thus reaches the Earth.

 



Additional Reading Material

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Periodicals

Astronomy, published monthly by Astromedia Corporation, 441 Mason Street, P. O. Box 92788, Milwaukee, WI 53202.
 
Scientific American, published monthly by Scientific American, Inc. , 415 Madison Ave. , New York, NY 10017.
 
Sky and Telescope, published monthly by Sky Publishing Corporation, 49 Bay State Road, Cambridge, MA 02139.
 
Science News, published weekly by Science Service, Inc. , 1719 N Street N. W. , Washington, D. C. 20036.
 

Books

Eddy, J. The New Sun, NASA SP-402. Washington, D. C. : Government Printing Office, 1979.
 
Friedman, H. Sun and Earth. New York: Scientific American Books, 1986.
 
Frazier, K. Our Turbulent Sun. Englewood Cliffs, N. J. : Prentice-Hall, 1982.
 
Kellogg, W. W. , and M. Mead, Eds. The Atmosphere: Endangered and Endangering. Washington, DC: Government Printing Office, 1977.
 
Mitton, S. Daytime Star: The Story of Our Sun. New York: Scribners, 1983.
 
Washburn, M. In the Light of the Sun. New York: Harcourt Brace Jovanovich, 1981.
 
Wilson, J. D. Physics: A Practical and Conceptual Approach, second ed. New York: Saunders
College Publishing, 1989.

 



Index

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*Use your browser's Find feature to find these topics in the module.
A
argon
atom
   energy level of
atmosphere, Earth's
   absorption of radiation
   appearance of
   interaction with solar radiation
   layers of
   major components
B
blackbody radiation
C
carbon dioxide
carbon monoxide
chlorofluorocarbons
D
deutron
E
electrical force
electron
   free
Einstein
   law of mass-energy equivalence
electromagnetic radiation
   absorption of
   emission of
electromagnetic spectrum
energy transport
   conduction
   convection
   radiation
F
fusion (try also nuclear fusion)
G
galaxy
gravitational force
greenhouse
   effect
   gases
H
helium
   3 nucleus
   4 nucleus
I
ion
inverse-square law
K
kinetic energy
M
methane
N
neutrino
neutron
nitrogen
nitrous oxide
nuclear strong forces
nuclear fusion
O
ozone
P
photon
Planck's constant
positron
proton-proton chain
S
scientific notation
   dimensional analysis
   physical constants
   units
star formation
Stefan-Boltzman law
sun
   age of
   appearance of
   core of
   elemental constituents of
   energy of
   formation of the
   interior of
solar constant
T
temperature
   of Earth
   of sun's surface
W
water vapor
Wien's displacement law

 
 



Bibliographic Information

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University Science Books
55D Gate Five Road
Sausalito, CA 94965
Fax: (415) 332-5393
http://www.uscibooks.com/

Managing editor: Lucy Warner
Editor: Louise Carroll
NCAR Graphics Team: Justin Kitsutaka, Lee Fortier, Wil Garcia, Barbara Mericle, David McNutt, and Michael Shibao
Cover Design and Photography: Irene Imfeld
Compositor: Archetype Typography, Berkeley, California

Copyright © 1996 by the 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 Cataloging-in-Publication Data

Library of Congress Catalog Number: 95-061059
ISBN: 0-935702-86-5


Acknowledgements

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.



A note on this series

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, univscibks@igc.org. Information about the Global Change Instruction Program is also available on the World Wide Web at http://www.uscibooks.com/globdir.htm or http://www.ucar.edu/communications/gcip/

 

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THE SUN-EARTH SYSTEM

How is solar energy produced? How does the Earth's atmosphere interact with solar radiation? The answers to these questions will help students understand such timely physical phenomena as the greenhouse effect and global warming. This module, which is accessible to any student who has studied introductory high school algebra, includes problems at the end of each section and a hands-on experiment designed to measure the solar constant with easily obtainable items.

GLOBAL CHANGE INSTRUCTION PROGRAM