The Sun-Earth System

Preview (abbreviated version)

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
The PDF version includes the following chpaters:
Chapter 2: The Nature of Electromagnetic Radiation
Chapter 3: The Earth's Atmosphere
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



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?