Chapter 3: Orbits and Gravity

It's not so easy to see this rule in action in the everyday world because of the many forces acting on a body at any one time.

One important force is friction, which generally slows things down. If you roll a ball along the sidewalk, it eventually comes to a stop because the sidewalk exerts a rubbing force on the ball. But in the space between the stars, where there is so little matter that friction is insignificant, objects can in fact continue to move (to coast) indefinitely.

Next, we can make ellipses of various elongations (or extended lengths) by varying the spacing of the tacks (as long as they are not farther apart than the length of the string).

The greater the eccentricity, the more elongated is the ellipse, up to a maximum eccentricity of 1.0, when the ellipse becomes "flat," the other extreme from a circle.

Figure 3.2 Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630).

(a) A stylized engraving shows Tycho Brahe using his instruments to measure the altitude of celestial objects above the horizon. The large curved instrument in the foreground allowed him to measure precise angles in the sky. Note that the scene includes hints of the grandeur of Brahe's observatory at Hven. (b) Kepler was a German mathematician and astronomer. His discovery of the basic laws that describe planetary motion placed the heliocentric cosmology of Copernicus on a firm mathematical basis.

Angular Momentum

A concept that is a bit more complex, but important for understanding many astronomical objects, is angular momentum, which is a measure of the rotation of a body as it revolves around some fixed point (an example is a planet orbiting the Sun). The angular momentum of an object is defined as the product of its mass, its velocity, and its distance from the fixed point around which it revolves.

Newton's third law is perhaps the most profound of the rules he discovered.

Basically, it is a generalization of the first law, but it also gives us a way to define mass. If we consider a system of two or more objects isolated from outside influences, Newton's first law says that the total momentum of the objects should remain constant. Therefore, any change of momentum within the system must be balanced by another change that is equal and opposite so that the momentum of the entire system is not changed.

When P (the orbital period) is measured in years, and a is expressed in a quantity known as an astronomical unit (AU), the two sides of the formula are not only proportional but equal.

One AU is the average distance between Earth and the Sun and is approximately equal to 1.5 × 10^8 kilometers. In these units, P^2 = a^3

This is the principle behind jet engines and rockets: the force that discharges the exhaust gases from the rear of the rocket is accompanied by the force that pushes the rocket forward.

The exhaust gases need not push against air or Earth; a rocket actually operates best in a vacuum (Figure 3.7).

Through his analysis of the motions of the planets, Kepler developed a series of principles, now known as Kepler's three laws, which described the behavior of planets based on their paths through space.

The first two laws of planetary motion were published in 1609 in The New Astronomy. Their discovery was a profound step in the development of modern science.

Newton also concluded that the gravitational attraction between two bodies must be proportional to their masses.

The more mass an object has, the stronger the pull of its gravitational force. The gravitational attraction between any two objects is therefore given by one of the most famous equations in all of science: Fgravity = G(M1M2 / R^2)

The shape (roundness) of an ellipse depends on how close together the two foci are, compared with the major axis.

The ratio of the distance between the foci to the length of the major axis is called the eccentricity of the ellipse.

A more obvious example of the mutual nature of forces between objects is familiar to all who have batted a baseball.

The recoil you feel as you swing your bat shows that the ball exerts a force on it during the impact, just as the bat does on the ball. Similarly, when a rifle you are bracing on your shoulder is discharged, the force pushing the bullet out of the muzzle is equal to the force pushing backward upon the gun and your shoulder.

The volume of an object is the measure of the physical space it occupies. Volume is measured in cubic units, such as cubic centimeters or liters.

The volume is the "size" of an object. A penny and an inflated balloon may both have the same mass, but they have very different volumes. The reason is that they also have very different densities, which is a measure of how much mass there is per unit volume. Specifically, density is the mass divided by the volume. Note that in everyday language we often use "heavy" and "light" as indications of density (rather than weight) as, for instance, when we say that iron is heavy or that whipped cream is light.

Figure 3.8 Conservation of Angular Momentum

When a spinning figure skater brings in her arms, their distance from her spin center is smaller, so her speed increases. When her arms are out, their distance from the spin center is greater, so she slows down.

It was the genius of Isaac Newton that found a conceptual framework that completely explained the observations and rules assembled by Galileo, Brahe, Kepler, and others.

Newton was born in Lincolnshire, England, in the year after Galileo's death (Figure 3.6). Against the advice of his mother, who wanted him to stay home and help with the family farm, he entered Trinity College at Cambridge in 1661 and eight years later was appointed professor of mathematics. Among Newton's contemporaries in England were architect Christopher Wren, authors Aphra Behn and Daniel Defoe, and composer G. F. Handel.

Interpretation of Newton's Laws

Newton's first law is a restatement of one of Galileo's discoveries, called the conservation of momentum. The law states that in the absence of any outside influence, there is a measure of a body's motion, called its momentum, that remains unchanged. You may have heard the term momentum used in everyday expressions, such as "This bill in Congress has a lot of momentum; it's going to be hard to stop."

The momentum of a body can change only under the action of an outside influence.

Newton's second law expresses force in terms of its ability to change momentum with time. A force (a push or a pull) has both size and direction. When a force is applied to a body, the momentum changes in the direction of the applied force. This means that a force is required to change either the speed or the direction of a body, or both—that is, to start it moving, to speed it up, to slow it down, to stop it, or to change its direction.

Recall that according to Newton's second law, forces cause acceleration.

Newton's universal law of gravitation says that the force acting upon (and therefore the acceleration of) an object toward Earth should be inversely proportional to the square of its distance from the center of Earth. Objects like apples at the surface of Earth, at a distance of one Earth-radius from the center of Earth, are observed to accelerate downward at 9.8 meters per second per second (9.8 m/s^2).

Figure 3.4 Drawing an Ellipse.

(a) We can construct an ellipse by pushing two tacks (the white objects) into a piece of paper on a drawing board, and then looping a string around the tacks. Each tack represents a focus of the ellipse, with one of the tacks being the Sun. Stretch the string tight using a pencil, and then move the pencil around the tacks. The length of the string remains the same, so that the sum of the distances from any point on the ellipse to the foci is always constant. (b) In this illustration, each semimajor axis is denoted by a. The distance 2a is called the major axis of the ellipse.

Why is it then, you may ask, that the astronauts aboard the Space Shuttle appear to have no gravitational forces acting on them whenweseeimagesontelevision of the astronauts and objects floating in the spacecraft?

After all, the astronauts in the shuttle are only a few hundred kilometers above the surface of Earth, which is not a significant distance compared to the size of Earth, so gravity is certainly not a great deal weaker that much farther away. The astronauts feel "weightless" (meaning that they don't feel the gravitational force acting on them) for the same reason that passengers in an elevator whose cable has broken or in an airplane whose engines no longer work feel weightless: they are falling (Figure 3.9).[2] 2 In the film Apollo 13, the scenes in which the astronauts were "weightless" were actually filmed in a falling airplane. As you might imagine, the plane fell for only short periods before the engines engaged again.

Check Your Learning Using the orbital periods and semimajor axes for Saturn and Jupiter that are provided here, calculate P^2 and a^3, and verify that they obey Kepler's third law. Saturn's orbital period is 29.46 years, and its semimajor axis is 9.54 AU. Jupiter's orbital period is 11.86 years, and its semimajor axis is 5.20 AU.

Answer: For Saturn, P^2 = 29.46 × 29.46 = 867.9 and a^3 = 9.54 × 9.54 × 9.54 = 868.3. The square of the orbital period (867.9) approximates the cube of the semimajor axis (868.3). Therefore, Saturn obeys Kepler's third law.

Check Your learning What would be the orbital period of an asteroid (a rocky chunk between Mars and Jupiter) with a semimajor axis of 3 AU?

Answer: P = square root of 3×3×3 = square root of 27 = 5.2years

Check Your Learning By what factor would a person's weight at the surface of Earth change if Earth had its present size but only one-third its present mass?

Answer: With one-third its present mass, the gravitational force at the surface would reduce by a factor of 1/3, so a person would weight only one-third as much.

Newton's Laws of Motion

As a young man in college, Newton became interested in natural philosophy, as science was then called. He worked out some of his first ideas on machines and optics during the plague years of 1665 and 1666, when students were sent home from college. Newton, a moody and often difficult man, continued to work on his ideas in private, even inventing new mathematical tools to help him deal with the complexities involved. Eventually, his friend Edmund Halley (profiled in Comets and Asteroids: Debris of the Solar System) prevailed on himto collect and publish the results of his remarkable investigations on motion and gravity. The result was a volume that set out the underlying system of the physical world, Philosophiae Naturalis Principia Mathematica. The Principia, as the book is generally known, was published at Halley's expense in 1687.

Mass, Volume, and Density

Before we go on to discuss Newton's other work, we want to take a brief look at some terms that will be important to sort out clearly. We begin with mass, whichisameasureoftheamountofmaterialwithinanobject.

If we could look down on the solar system from somewhere out in space, interpreting planetary motions would be much simpler.

But the fact is, we must observe the positions of all the other planets from our own moving planet. Scientists of the Renaissance did not know the details of Earth's motions any better than the motions of the other planets. Their problem, as we saw in Observing the Sky: The Birth of Astronomy, was that they hadtodeducethenatureofall planetary motion using only their earthbound observations of the other planets' positions in the sky. To solve this complex problem more fully, better observations and better models of the planetary system were needed.

NEWTON'S GREAT SYNTHESIS Learning Objectives

By the end of this section, you will be able to: Describe Newton's three laws of motion Explain how Newton's three laws of motion relate to momentum Define mass, volume, and density and how they differ Define angular momentum

NEWTON'S UNIVERSAL LAW OF GRAVITATION Learning Objectives

By the end of this section, you will be able to: Explain what determines the strength of gravity Describe how Newton's universal law of gravitation extends our understanding of Kepler's laws

The answer to this question required mathematical tools that had not yet been developed, but this did not deter Isaac Newton, who invented what we today call calculus to deal with this problem.

Eventually he was able to conclude that the magnitude of the force of gravity must decrease with increasing distance between the Sun and a planet (or between any two objects) in proportion to the inverse square of their separation. In other words, if a planet were twice as far from the Sun, the force would be (1/2)^2, or 1/4 as large. Put the planet three times farther away, and the force is (1/3)^2, or 1/9 as large.

In Newton's time, gravity was something associated with Earth alone.

Everyday experience shows us that Earth exerts a gravitational force upon objects at its surface. If you drop something, it accelerates toward Earth as it falls. Newton's insight was that Earth's gravity might extend as far as the Moon and produce the force required to curve the Moon's path from a straight line and keep it in its orbit. He further hypothesized that gravity is not limited to Earth, but that there is a general force of attraction between all material bodies. If so, the attractive force between the Sun and each of the planets could keep them in their orbits. (This may seem part of our everyday thinking today, but it was a remarkable insight in Newton's time.)

where Fgravity is the gravitational force between two objects, M1 and M2 are the masses of the two objects, and R is their separation.

G is a constant number known as the universal gravitational constant, and the equation itself symbolically summarizes Newton's universal law of gravitation. With such a force and the laws of motion, Newton was able to show mathematically that the only orbits permitted were exactly those described by Kepler's laws.

Densities of Common Materials

Gold 19.3 Lead 11.3 Iron 7.9 Earth (bulk) 5.5 Rock (typical) 2.5 Water 1 Wood (typical) 0.8 Insulating foam 0.1 Silica gel 0.02 Table 3.1 To sum up, mass is how much, volume is how big, and density is how tightly packed. 1 Generally we use standard metric (or SI) units in this book. The proper metric unit of density in that system is kg/m^3. But to most people, g/cm^3 provides a more meaningful unit because the density of water is exactly 1 g/cm^3, and this is useful information for comparison. Density expressed in g/cm^3 is sometimes called specific density or specific weight.

The widest diameter of the ellipse is called its major axis.

Half this distance—that is, the distance from the center of the ellipse to one end—is the semimajor axis, which is usually used to specify the size of the ellipse. For example, the semimajor axis of the orbit of Mars, which is also the planet's average distance from the Sun, is 228 million kilometers.

The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity.

Here was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an acceptable cosmos. The universe could be a bit more complex than the Greek philosophers had wanted it to be.

At Hven, Brahe made a continuous record of the positions of the Sun, Moon, and planets for almost 20 years.

His extensive and precise observations enabled him to note that the positions of the planets varied from those given in published tables, which were based on the work of Ptolemy. These data were extremely valuable, but Brahe didn't have the ability to analyze them and develop a better model than what Ptolemy had published. He was further inhibited because he was an extravagant and cantankerous fellow, and he accumulated enemies among government officials. When his patron, Frederick II, died in 1597, Brahe lost his political base and decided to leave Denmark. He took up residence in Prague, where he became court astronomer to Emperor Rudolf of Bohemia. There, in the year before his death, Brahe found a most able young mathematician, Johannes Kepler, to assist him in analyzing his extensive planetary data.

Thinking Ahead

How would you find a new planet at the outskirts of our solar system that is too dim to be seen with the unaided eye and is so far away that it moves very slowly among the stars? This was the problem confronting astronomers during the nineteenth century as they tried to pin down a full inventory of our solar system.

The units of density that will be used in this book are grams per cubic centimeter (g/cm^3).[1]

If a block of some material has a mass of 300 grams and a volume of 100 cm^3, its density is 3 g/cm^3. Familiar materials span a considerable range in density, from artificial materials such as plastic insulating foam (less than 0.1 g/cm^3) to gold (19.3 g/cm^3). Table 3.1 gives the densities of some familiar materials. In the astronomical universe, much more remarkable densities can be found, all the way from a comet's tail (10^-16 g/cm^3) to a collapsed "star corpse" called a neutron star (10^15 g/cm^3).

This means that forces in nature do not occur alone: we find that in each situation there is always a pair of forces that are equal to and opposite each other.

If a force is exerted on an object, it must be exerted by something else, and the object will exert an equal and opposite force back on that something. We can look at a simple example to demonstrate this.

The Moon is 60 Earth radii away from the center of Earth.

If gravity (and the acceleration it causes) gets weaker with distance squared, the acceleration the Moon experiences should be a lot less than for the apple. The acceleration should be (1/60)^2 = 1/3600 (or 3600 times less—about 0.00272 m/s^2. This is precisely the observed acceleration of the Moon in its orbit. (As we shall see, the Moon does not fall to Earth with this acceleration, but falls around Earth.) Imagine the thrill Newton must have felt to realize he had discovered, and verified, a law that holds for Earth, apples, the Moon, and, as far as he knew, everything in the universe.

You might recall from math classes that in a circle, the center is a special point. The distance from the center to anywhere on the circle is exactly the same.

In an ellipse, the sum of the distance from two special points inside the ellipse to any point on the ellipse is always the same. These two points inside the ellipse are called its foci (singular: focus), a word invented for this purpose by Kepler.

How did Kepler miss this factor?

In units of the Sun's mass, the mass of the Sun is 1, and in units of the Sun's mass, the mass of a typical planet is a negligibly small factor. This means that the sum of the Sun's mass and a planet's mass, (M1 + M2), is very, very close to 1. This makes Newton's formula appear almost the same as Kepler's; the tiny mass of the planets compared to the Sun is the reason that Kepler did not realize that both masses had to be included in the calculation. There are many situations in astronomy, however, in which we do need to include the two mass terms—for example, when two stars or two galaxies orbit each other.

Figure 3.6 Isaac Newton (1643-1727), 1689 Portrait by Sir Godfrey Kneller.

Isaac Newton's work on the laws of motion, gravity, optics, and mathematics laid the foundations for much of physical science.

Newton's law also implies that gravity never becomes zero.

It quickly gets weaker with distance, but it continues to act to some degree no matter how far away you get. The pull of the Sun is stronger at Mercury than at Pluto, but it can be felt far beyond Pluto, where astronomers have good evidence that it continuously makes enormous numbers of smaller icy bodies move around huge orbits. And the Sun's gravitational pull joins with the pull of billions of others stars to create the gravitational pull of our Milky Way Galaxy. That force, in turn, can make other smaller galaxies orbit around the Milky Way, and so on.

Johannes Kepler

Johannes Kepler was born into a poor family in the German province of Württemberg and lived much of his life amid the turmoil of the Thirty Years' War (see Figure 3.2). He attended university at Tubingen and studied for a theological career. There, he learned the principles of the Copernican system and became converted to the heliocentric hypothesis. Eventually, Kepler went to Prague to serve as an assistant to Brahe, who set him to work trying to find a satisfactory theory of planetary motion—one that was compatible with the long series of observations made at Hven. Brahe was reluctant to provide Kepler with much material at any one time for fear that Kepler would discover the secrets of the universal motion by himself, thereby robbing Brahe of some of the glory. Only after Brahe's death in 1601 did Kepler get full possession of the priceless records. Their study occupied most of Kepler's time for more than 20 years.

Kepler's Third Law

Kepler's first two laws of planetary motion describe the shape of a planet's orbit and allow us to calculate the speed of its motion at any point in the orbit. Kepler was pleased to have discovered such fundamental rules, but they did not satisfy his quest to fully understand planetary motions. He wanted to know why the orbits of the planets were spaced as they are and to find a mathematical pattern in their movements—a "harmony of the spheres" as he called it. For many years he worked to discover mathematical relationships governing planetary spacing and the time each planet took to go around the Sun.

If these three quantities remain constant—that is, if the motion of a particular object takes place at a constant velocity at a fixed distance from the spin center—then the angular momentum is also a constant.

Kepler's second law is a consequence of the conservation of angular momentum. As a planet approaches the Sun on its elliptical orbit and the distance to the spin center decreases, the planet speeds up to conserve the angular momentum. Similarly, when the planet is farther from the Sun, it moves more slowly.

Kepler's laws describe the orbits of the objects whose motions are described by Newton's laws of motion and the law of gravity.

Knowing that gravity is the force that attracts planets toward the Sun, however, allowed Newtontorethink Kepler's third law. Recall that Kepler had found a relationship between the orbital period of a planet's revolution and its distance from the Sun. But Newton's formulation introduces the additional factor of the masses of the Sun (M1) and the planet (M2), both expressed in units of the Sun's mass. Newton's universal law of gravitation can be used to show mathematically that this relationship is actually a^3 = (M1 + M2)× P^2 where a is the semimajor axis and P is the orbital period.

THE LAWS OF PLANETARY MOTION

Learning Objectives By the end of this section, you will be able to: Describe how Tycho Brahe and Johannes Kepler contributed to our understanding of how planets move around the Sun Explain Kepler's three laws of planetary motion

Kepler's third law applies to all objects orbiting the Sun, including Earth, and provides a means for calculating their relative distances from the Sun from the time they take to orbit.

Let's look at a specific example to illustrate how useful Kepler's third law is.

For instance, suppose youtimehowlongMarstakestogoaroundtheSun(inEarthyears).Kepler'sthirdlawcan then be used to calculate Mars' average distance from the Sun.

Mars' orbital period (1.88 Earth years) squared, or P^2, is 1.88^2 = 3.53, and according to the equation for Kepler's third law, this equals the cube of its semimajor axis, or a^3. So what number must be cubed to give 3.53?The answer is 1.52 (since 1.52 × 1.52 ×1.52 =3.53). Thus, Mars' semimajor axis in astronomical units must be 1.52 AU. In other words, to go around the Sun in a little less than two years, Mars must be about 50% (half again) as far from the Sun as Earth is.

Let's define the precise meaning of momentum—it depends on three factors: (1) speed—how fast a body moves (zero if it is stationary), (2) the direction of its motion, and (3) its mass—a measure of the amount of matter in abody,whichwewilldiscusslater.

Scientists use the term velocity to describe the speed and direction of motion. For example, 20 kilometers per hour due south is velocity, whereas 20 kilometers per hour just by itself is speed. Momentum then can be defined as an object's mass times its velocity.

EXAMPLE 3.2 Applying Kepler's Third Law Using the orbital periods and semimajor axes for Venus and Earth that are provided here, calculate P^2 and a^3, and verify that they obey Kepler's third law. Venus' orbital period is 0.62 year, and its semimajor axis is 0.72 AU. Earth's orbital period is 1.00 year, and its semimajor axis is 1.00 AU.

Solution We can use the equation for Kepler's third law, P^2 ∝ a^3. For Venus, P^2 = 0.62 × 0.62 = 0.38 and a^3 = 0.72 × 0.72 × 0.72 = 0.37 (rounding numbers sometimes causes minor discrepancies like this). The square of the orbital period (0.38) approximates the cube of the semimajor axis (0.37). Therefore, Venus obeys Kepler's third law. For Earth, P^2 = 1.00 × 1.00 = 1.00 and a^3 = 1.00 × 1.00 × 1.00 = 1.00. The square of the orbital period (1.00) approximates (in this case, equals) the cube of the semimajor axis (1.00). Therefore, Earth obeys Kepler's third law.

EXAMPLE 3.3 Calculating Weight By what factor would a person's weight at the surface of Earth change if Earth had its present mass but eight times its present volume?

Solution With eight times the volume, Earth's radius would double. This means the gravitational force at the surface would reduce by a factor of (1/2)^2 = 1/4, so a person would weigh only one-fourth as much.

Calculating Periods Imagine an object is traveling around the Sun. What would be the orbital period of the object if its orbit has a semimajor axis of 50 AU?

Solution From Kepler's third law, we know that (when we use units of years and AU) P^2 = a^3 If the object's orbit has a semimajor axis of 50 AU (a = 50), we can cube 50 and then take the square root of the result to get P: P = square root of a^3 square root of P = 50×50×50 = square root of 125,000 = 353.6 years Therefore, the orbital period of the object is about 350 years. This would place our hypothetical object beyond the orbit of Pluto.

Figure 3.7 Demonstrating Newton's Third Law.

The U.S. Space Shuttle (here launching Discovery), powered by three fuel engines burning liquid oxygen and liquid hydrogen, with two solid fuel boosters, demonstrates Newton's third law. (credit: modification of work by NASA)

Figure 3.3 Conic Sections.

The circle, ellipse, parabola, and hyperbola are all formed by the intersection of a plane with a cone. This is why such curves are called conic sections.

Suppose that a daredevil astronomy student—and avid skateboarder—wants to jump from his second-story dorm window onto his board below (we don't recommend trying this!).

The force pulling him down after jumping (as we will see in the next section) is the force of gravity between him and Earth. Both he and Earth must experience the same total change of momentum because of the influence of these mutual forces. So, both the student and Earth are accelerated by each other's pull. However, the student does much more of the moving. Because Earth has enormously greater mass, it can experience the same change of momentum by accelerating only a very small amount. Things fall toward Earth all the time, but the acceleration of our planet as a result is far too small to be measured.

Newton's universal law of gravitation works for the planets, but is it really universal?

The gravitational theory should also predict the observed acceleration of the Moon toward Earth as it orbits Earth, as well as of any object (say, an apple) dropped near Earth's surface. The falling of an apple is something we can measure quite easily, but can we use it to predict the motions of the Moon?

Figure 3.5 Kepler's Second Law: The Law of Equal Areas.

The orbital speed of a planet traveling around the Sun (the circular object inside the ellipse) varies in such a way that in equal intervals of time (t), a line between the Sun and a planet sweeps out equal areas (A and B). Note that the eccentricities of the planets' orbits in our solar system are substantially less than shown here.

The First Two Laws of Planetary Motion

The path of an object through space is called its orbit. Kepler initially assumed that the orbits of planets were circles, but doing so did not allow him to find orbits that were consistent with Brahe's observations. Working with the data for Mars, he eventually discovered that the orbit of that planet had the shape of a somewhat flattened circle, or ellipse. Next to the circle, the ellipse is the simplest kind of closed curve, belonging to a family of curves known as conic sections (Figure 3.3).

Once Newton boldly hypothesized that there was a universal attraction among all bodies everywhere in space, he had to determine the exact nature of the attraction.

The precise mathematical description of that gravitational force had to dictate that the planets move exactly as Kepler had described them to (as expressed in Kepler's three laws). Also, that gravitational force had to predict the correct behavior of falling bodies on Earth, as observed by Galileo. How must the force of gravity depend on distance in order for these conditions to be met?

When falling, they are in free fall and accelerate at the same rate as everything around them, including their spacecraft or a camera with which they are taking photographs of Earth.

When doing so, astronauts experience no additional forces and therefore feel "weightless." Unlike the falling elevator passengers, however, the astronauts are falling around Earth, not to Earth; as a result they will continue to fall and are said to be "in orbit" around Earth (see the next section for more about orbits).

At about the time that Galileo was beginning his experiments with falling bodies, the efforts of two other scientists dramatically advanced our understanding of the motions of the planets.

These two astronomers were the observer Tycho Brahe and the mathematician Johannes Kepler. Together, they placed the speculations of Copernicus on a soundmathematical basis and paved the way for the work of Isaac Newton in the next century.

If the foci (or tacks) are moved to the same location, then the distance between the foci would be zero.

This means that the eccentricity is zero and the ellipse is just a circle; thus, a circle can be called an ellipse of zero eccentricity. In a circle, the semimajor axis would be the radius.

Tycho Brahe's Observatory

Three years after the publication of Copernicus' De Revolutionibus, Tycho Brahe was born to a family of Danish nobility. He developed an early interest in astronomy and, as a young man, made significant astronomical observations. Among these was a careful study of what we now know was an exploding star that flared up to great brilliance in the night sky. His growing reputation gained him the patronage of the Danish King Frederick II, and at the age of 30, Brahe was able to establish a fine astronomical observatory on the North Sea island of Hven (Figure 3.2). Brahe was the last and greatest of the pre-telescopic observers in Europe.

Newton's laws of motion show that objects at rest will stay at rest and those in motion will continue moving uniformly in a straight line unless acted upon by a force.

Thus, it is the straight line that defines the most natural state of motion. But the planets move in ellipses, not straight lines; therefore, some force must be bending their paths. That force, Newton proposed, was gravity.

It is this force of gravity on the surface of Earth that gives us our sense of weight.

Unlike your mass, which would remain the same on any planet or moon, your weight depends onthe local force of gravity. So you would weigh less on Mars and the Moon than on Earth, even though there is no change in your mass. (Which means you would still have to go easy on the desserts in the college cafeteria when you got back!)

In 1619, Kepler discovered a basic relationship to relate the planets' orbits to their relative distances from the Sun.

We define a planet's orbital period, (P), as the time it takes a planet to travel once around the Sun. Also, recall that a planet's semimajor axis, a, is equal to its average distance from the Sun. The relationship, now known as Kepler's third law, says that a planet's orbital period squared is proportional to the semimajor axis of its orbit cubed, or P^2 ∝ a^3

This property suggests a simple way to draw anellipse (Figure 3.4).

We wrap the ends of a loop of string around two tacks pushed through a sheet of paper into a drawing board, so that the string is slack. If we push a pencil against the string, making the string taut, and then slide the pencil against the string all around the tacks, the curve that results is an ellipse. At any point where the pencil may be, the sum of the distances from the pencil to the two tacks is a constant length—the length of the string. The tacks are at the two foci of the ellipse.

Gravity is a "built-in" property of mass.

Whenever there are masses in the universe, they will interact via the force of gravitational attraction. The more mass there is, the greater the force of attraction. Here on Earth, the largest concentration of mass is, of course, the planet we stand on, and its pull dominates the gravitational interactions we experience. But everything with mass attracts everything else with mass anywhere in the universe.

Figure 3.9 Astronauts in Free Fall.

While in space, astronauts are falling freely, so they experience "weightlessness." Clockwise from top left: Tracy Caldwell Dyson (NASA), Naoko Yamzaki (JAXA), Dorothy Metcalf-Lindenburger (NASA), and Stephanie Wilson (NASA). (credit: NASA)

Kepler's three laws provide a precise geometric description of planetary motion within the framework of the Copernican system.

With these tools, it was possible to calculate planetary positions with greatly improved precision. Still, Kepler's laws are purely descriptive: they do not help us understand what forces of nature constrain the planets to follow this particular set of rules. That step was left to Isaac Newton.

The conservation of angular momentum is illustrated by figure skaters, who bring their arms and legs in to spin more rapidly, and extend their arms and legs to slow down (Figure 3.8).

You can duplicate this yourself on a well-oiled swivel stool by starting yourself spinning slowly with your arms extended and then pulling your arms in. Another example of the conservation of angular momentum is a shrinking cloud of dust or a star collapsing on itself (both are situations that you will learn about as you read on). As material moves to a lesser distance from the spin center, the speed of the material increases to conserve angular momentum.

If a planet moves in a circular orbit, the elastic line is always stretched the same amount and the planet moves at a constant speed around its orbit. But,

as Kepler discovered, in most orbits that speed of a planet orbiting its star (or moon orbiting its planet) tends to vary because the orbit is elliptical.

Kepler's three laws of planetary motion can be summarized as follows:

• Kepler's first law: Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse. • Kepler's second law: The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time. • Kepler's second law: The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time. • Kepler's third law: The square of a planet's orbital period is directly proportional to the cube of the semimajor axis of its orbit.

At the very beginning of the Principia, Newton proposes three laws that would govern the motions of all objects:

• Newton's first law: Every object will continue to be in a state of rest or move at a constant speed in a straight line unless it is compelled to change by an outside force. • Newton's second law: The change of motion of a body is proportional to and in the direction of the force acting on it. • Newton's third law: For every action there is an equal and opposite reaction (or: the mutual actions of two bodies upon each other are always equal and act in opposite directions). In the original Latin, the three laws contain only 59 words, but those few words set the stage for modern science. Let us examine them more carefully.