This chapter covers the revolutionary advancements due to probably the most brilliant scientist who ever lived: Isaac Newton (lived 1641--1727). His greatest contributions were in all branches of physics. Kepler's discoveries about elliptical orbits and the planets' non-uniform speeds made it impossible to maintain the idea of planetary motion as a natural one requiring no explanation. Newton had to answer some basic questions: What keeps the planets in their elliptical orbits? On our spinning Earth what prevents objects from flying away when they are thrown in the air? What keeps you from being hurled off the spinning Earth? Newton's answer was that a fundamental force called "gravity" operating between all objects made them move the way they do.
Newton developed some basic rules governing the motion of all objects. He used these laws and Kepler's laws to derive his unifying Law of Gravity. I will first discuss his three laws of motion and then discuss gravity. Finally, several applications in astronomy will be given. This chapter uses several math concepts that are reviewed in the mathematics review appendix. If your math skills are rusty, study the mathematics review appendix and don't hesitate to ask your astronomy instructor for help.
I include images of world atlases from different time periods in this chapter and the previous one as another way to illustrate the advances in our understanding of our world and the universe. Links to the sites from which the photographs came are embedded in the images. Select the picture to go to the site.
Nova orbis tabula [De Wit 1688]. Select the image to go to the Hargrett Library at the Univ. of Georgia from which this picture came. Note the curved path of the Sun between the Tropic of Cancer (at latitude 23.5° N) and the Tropic of Capricorn (at latitude 23.5° S).
In order to accurately describe how things move, you need to be careful in how you describe the motion and the terms you use. Scientists are usually very careful about the words they use to explain something because they want to accurately represent nature. Language can often be imprecise and as you know, statements can often be misinterpreted. Because the goal of science is to find the single true nature of the universe, scientists try to carefully choose their words to accurately represent what they see. That is why scientific papers can look so "technical" (and even, introductory astronomy textbooks!)
When you think of motion, you may first think of something moving at a uniform speed. The speed = (the distance travelled)/(the time it takes). Because the distance is in the top of the fraction, there is a direct relation between the speed and the distance: the greater the distance travelled in a given time, the greater is the speed. However, there is an inverse relation between time and speed (time is in the bottom of the fraction): the smaller the time it takes to cover a given distance, the greater the speed must be.
To more completely describe all kinds of changes in motion, you also need to consider the direction along with the speed. For example, a ball thrown upward at the same speed as a ball thrown downward has a different motion. This inclusion of direction will be particularly important when you look at an object orbiting a planet or star. They may be moving at a uniform speed while their direction is constantly changing. The generalization of speed to include direction is called velocity. The term velocity includes both the numerical value of the speed and the direction something is moving.
Galileo conducted several experiments to understand how something's velocity can be changed. He found that an object's velocity can be changed only if a force acts on the object. The philosopher René Descartes (lived 1596--1650, picture at left) used the idea of a greater God and an infinite universe with no special or privileged place to articulate the concept of inertia: a body at rest remains at rest, and one moving in a straight line maintains a constant speed and same direction unless it is deflected by a "force". Newton took this as the beginning of his description of how things move, so this is now known as Newton's 1st law of motion. A force causes a change in something's velocity (an acceleration).
An acceleration is a change in the speed and/or direction of motion in a given amount of time: acceleration= (the velocity change)/(the time interval of the change). Something at rest is not accelerating and something moving at constant speed in a straight line is not accelerating. In common usage, acceleration usually means just a change in speed, but a satellite orbiting a planet is constantly being accelerated even if its speed is constant because its direction is constantly being deflected. The satellite must be experiencing a force since it is accelerating. That force turns out to be gravity. If the force (gravity) were to suddenly disappear, the satellite would move off in a straight line along a path tangent to the original circular orbit.
A rock in your hand is moving horizontally as it spins around the center of the Earth, just like you and the rest of the things on the surface are. If you throw the rock straight up, there is no change in its horizontal motion because of its inertia. You changed the rock's vertical motion because you applied a vertical force on it. The rock falls straight down because the Earth's gravity acts on only the rock's vertical motion. If the rock is thrown straight up, it does not fall behind you as the Earth rotates. Inertia and gravity also explain why you do not feel a strong wind as the Earth spins---as a whole, the atmosphere is spinning with the Earth.
Newton's first law of motion is a qualitative one---it tells you when something will accelerate. Newton went on to quantify the amount of the change that would be observed from the application of a given force. In Newton's second law of motion, he said that the force applied = mass of an object · acceleration. Mass is the amount of material an object has and is a way of measuring how much inertia the object has. For a given amount of force, more massive objects will have a smaller acceleration than less massive objects (a push needed to even budge a car would send a pillow flying!). For a given amount of acceleration, the more massive object requires a larger force than a less massive object.
Newton also found that for every action force ON an object, there is an equal but opposite force BY the object (Newton's third law of motion). For example, if Andre the Giant is stuck on the ice with Tom Thumb and he pushes Tom Thumb to the right, Andre will feel an equal force from Tom pushing him to the left. Tom will slide to the right with great speed and Andre will slide to the left with smaller speed since Andre's mass is larger than Tom's.
Another example: an apple falls to the Earth because it is pulled by the force of the Earth's gravity on the apple and the acceleration of the apple is large. The apple also exerts a gravitational force on the Earth of the same amount. However, the acceleration the Earth experiences is vastly smaller than the apple's acceleration since the Earth's mass is vastly larger than the apple's---you will ordinarily refer to the apple falling to the Earth, rather than the Earth moving toward the apple or that they are falling toward each other.
Formulae
Newton's 2nd law: Force = mass · acceleration: F = m · a
If you give a bowling ball a push FAR away from any gravitational effects, what will it do? If you throw a feather (again far out in space) at the same speed as the bowling ball, how will its speed compare to the bowling ball after 5 minutes?
Let's say you're twirling a ball on a string and the string breaks. What path does the ball take and why is that?
How do you know gravity acts on an orbiting satellite?
How does a force exerted on an object relate to the object's mass or acceleration? Given the same force will a boulder accelerate more than a regular marble? Why?
Why would you need to apply more force to a bowling ball than a feather (far out in space) so that they would be travelling at the same speed after 10 minutes?
Using Kepler's third law and his own second law, Newton found that the amount of the attractive force, called gravity, between a planet and Sun a distance d apart is Force = kp · (planet mass) / (d)2, where kp is a number that is the same for all the planets. In the same way he found that the amount of the gravity between the Sun and a planet is Force = ks · (Sun mass) / (d)2. Using his third law of motion, Newton reasoned that these forces must be the same (but acting in the opposite directions). He derived his Law of Gravity: the force of gravity = G · (mass #1) · (mass #2) / (distance between them)2 and this force is directed toward each object, so it is always attractive. The term G is a universal constant of nature. If you use the units of kilograms (kg) for mass and meters (m) for distance, G = 6.672 · 10-11 m3 /(kg sec2). If you need a refresher on exponents, square & cube roots, and scientific notation, then please study the math review appendix.
Spherically symmetric objects (eg., planets, stars, moons, etc.) behave as if all of their mass is concentrated at their centers. So when you use Newton's Law of Gravity, you measure the distance between the centers of the objects.
In a bold, revolutionary step, Newton stated that his gravity law worked for any two objects with mass---it applies for any motions on the Earth, as well as, any motions in space. He unified celestial and terrestrial physics and completed the process started by Copernicus of removing the Earth from a unique position or situation in the universe. His law of gravity also explained Kepler's 1st and 2nd laws.
Newton's Law of Gravity says a lot about this force in a very compact, elegant way. It says that any piece of matter will feel it whether it is charged or not (this sets it apart from electrical and magnetic forces that affect only charged objects). Gravity depends only the masses of the two attracting objects and their distance from each other. It does not depend on their chemical composition or density. A glob of peanut butter the mass of the Sun will have the same gravitational effect on the Earth as the Sun does. Gravity is always attractive, never repulsive (this is another way it is different from electrical and magnetic forces).
Because the masses are in the top of the fraction, more mass creates more gravity force. This also means that more massive objects produce greater accelerations than less massive objects. Since distance is in the bottom of the fraction, gravity has an inverse relation with distance: as distance increases, gravity decreases. However, gravity never goes to zero---it has an infinite range (in this respect it is like the electrical and magnetic forces). Stars feel the gravity from other stars, galaxies feel gravity from other galaxies, galaxy clusters feel gravity from other galaxies, etc. The always attractive gravity can act over the largest distances in the universe.
There is no way to get rid of the force of gravity. If you want to prevent a body from producing a gravitational acceleration on an object, you need to use a second body, with the same amount of gravity pull as the first body, in a way that its gravity pulling on the object is in the opposite direction. The resulting accelerations due to the forces from the two bodies will cancel each other out.
What basic fundamental assumption did Newton make about the laws of nature on the Earth and in space?
Why is gravity often the most important force in astronomical interactions?
What things does gravity depend on?
How does gravity vary with distance between objects and with respect to what do you measure the distances?
What would happen to the orbit of Io (one of Jupiter's moons) if all of the Hydrogen and Helium in Jupiter were converted to Silicon and Oxygen? Explain your answer.
What would happen to the Earth's orbit if the Sun suddenly turned into a black hole (of the same mass)? Why?
How would antimatter respond to gravity? (Hint: antimatter has mass just like ordinary matter.)
What important laws of planet motion can be derived from Newton's law of gravity?
Though the terms weight and mass are used interchangeably in common language, in science there is distinct difference between the two terms. The weight of an object = force of gravity felt by that object but the mass of an object is the amount of matter the object has. Mass is a measure of the object's resistance to acceleration: a push on a skateboard will make it roll away quickly but the same push on a more massive car will barely budge it.
An object's weight depends on the pull of the gravitating object but the object's mass is independent of the gravity. For example, Joe Average weighs himself on the Earth's surface and then on the Moon's surface. His weight on the Moon will be about six times less than on the Earth but the number of atoms in his body has not changed so his mass is the same at the two places. In the old English unit system, there is a "pound" of force and "pound" of mass. On only the Earth's surface, an object's pound of mass = the number of pounds of force felt by the object due to the Earth's gravity.
In the metric system there is no confusion of terms. A kilogram is a quantity of mass and a newton is a quantity of force. One kilogram (kg) = 2.205 pounds of mass and 4.45 newtons (N) = 1 pound of force. If someone uses "pounds", be sure you understand if s/he means force or mass!
How do you do that?
To find something's weight in newtons, you multiply the mass in kilograms by the acceleration of gravity in the units of meters/seconds2. For example: Joe Average has a mass of 63.5 kg and he feels a force of gravity on the Earth = 63.5 kg · 9.8 m/s2 = 623 kg m/s2 = 623 N. His weight is 623 N. The other value in the preceding equation, 9.8 m/s2, is the acceleration due to gravity close to the Earth's surface. Joe Average's weight at other places in the universe will be different but his mass will remain the same.
When the astronauts landed on the Moon, how were they able to stay on the ground?
On the Moon, the astronauts weighed about six times less than they did on the Earth. Compare the amount of gravity on the Moon's surface with that on the Earth's surface. If objects fall with an acceleration of about 10 m/s2 on the Earth, how much would the acceleration be on the Moon's surface? Explain your answer.
If Joe Astronaut has a mass of 40 kilograms on the Earth, how much mass would he have on an asteroid with 10 times less surface gravity than the Earth's surface gravity? Explain your answer.
Newton's law of gravity describes a force that decreases with the SQUARE of the distance. For every factor of 2 the distance increases, the gravitational attraction decreases by a factor of 2 · 2 = 4; for every factor of 3 increase in distance, the gravity decreases by a factor of 3 · 3 = 9 (not by 3 + 3 = 6!); for every factor of 4 increase in distance, the gravity decreases by a factor of 4 · 4 = 16 (not by 4 + 4 = 8!), etc. See the mathematics review appendix for a review of "factor" and "times". Some more examples are given in the table below. Notice how quickly an inverse square law gets very small.
A comparison of inverse and inverse square relations
distance
inverse
inverse square
1
1/1 = 1
1/12 = 1
2
1/2 = 0.5
1/22 = 1/4 = 0.25
3
1/3 = 0.33
1/32 = 1/9 = 0.11
4
1/4 = 0.25
1/42 = 1/16 = 0.0625
7
1/7 = 0.14
1/72 = 1/49 = 0.02
10
1/10 = 0.1
1/102 = 1/100 = 0.01
100
1/100 = 0.01
1/1002 = 1/10,000 = 0.0001
Example: Joe Average has a mass of 63.5 kilograms, so he weighs 623 newtons (=140 pounds) on the Earth's surface. If he moves up 1 Earth radius (= 6378 kilometers) above the surface, he will be two times farther away from the Earth's center (remember that distances are measured from center-to-center!), so his weight will be four times less, or 623/4 newtons = 155.8 newtons (= 140/4 pounds); NOT two times less, or 623/2 newtons = 311.5 newtons. If he moves up another Earth radius above the surface, he will be three times farther away than he was at the start, so his weight will drop by a factor of nine times, NOT 3 times. His weight will be 623/9 newtons = 69.22 newtons (= 140/9 pounds); NOT 623/3 newtons = 207.7 newtons. His mass will still be 63.5 kilograms. Figure below illustrates this.
Let us generalize this for any situation where the masses do not change: the force of gravity at distance A = (the force of gravity at distance B) · (distance B / distance A)2. Notice which distance is in the top of the fraction! To use this relation, have the gravity at distance A represent the unknown gravity force you are trying to find and the gravity at distance B represent the reference gravity force felt at the reference distance B.
How do you do that?
Let's find where the weight values in the inverse square law figure come from. For Mr. Average's case the reference weight is his weight on the surface of the Earth = 623 N. His weight at 6378 kilometers above the surface is gravity at A = 623 · [6378/(2 · 6378)]2 = 623 · 1/22 = 623 · 1/4 = 155.8 N. When he is at two Earth radii above the surface, the gravity at A = 623 · [6378/(3 · 6378)]2 = 623 · 1/32 = 623 · 1/9 = 69.22 N.
Formulae
Inverse Square Law: Gravity at A = gravity at B · (distance B / distance A)2.
What is the difference between a simple inverse relation and an inverse square relation?
If the Earth was 3 A.U. from the Sun (instead of 1 A.U.), would the gravity force between the Earth and the Sun be less or more than it is now? By how many times?
If Mercury was 0.2 A.U. from the Sun (instead of 0.4 A.U.), would the gravity force between Mercury and the Sun be less or more than it is now? By how many times?
Galileo found that the acceleration due to gravity (called "g") depends only on the mass of the gravitating object and the distance from it. It does not depend on the mass of the object being pulled. In the absence of air drag, a huge boulder will fall at the same rate as a small marble dropped from the same height as the boulder. A tiny satellite at the same distance from the Sun as Jupiter's orbit from the Sun feels the same acceleration from the Sun as the large planet Jupiter does from the Sun. How is this possible? Most people would agree with Aristotle that the bigger object should fall faster than the smaller object, but experiments show they would be wrong.
A boulder falling toward the Earth is pulled by a stronger gravity force than the marble, since the boulder's mass is greater than the marble, but the boulder also has greater resistance to a change in its motion because of its larger mass. The effects cancel each other out, so the boulder accelerates at the same rate as the marble. The same line of reasoning explains the equal acceleration experienced by Jupiter and the satellite.
You can use Newton's second law of motion F = m · a (which relates the acceleration, a, felt by a object with mass m when acted on by a force F) to derive the acceleration due to gravity (here replace a with g) from a massive object:
The force of gravity =
(G M m)
d2
= m g
so
g =
(G M)
d2
.
The gravitational acceleration depends on only the mass of the gravitating object M and the distance d from it. Notice that the mass of the falling object m has been cancelled out. This explains why astronauts orbiting the Earth feel "weightless". In orbit they are continually "falling" toward the Earth because of gravity (the Earth's surface curves away from them at the same rate they are moving forward). If Jane Astronaut drops a pen in the space shuttle, it accelerates toward the Earth, but she accelerates by the same amount so the pen remains at the same position relative to her. In fact the entire shuttle and its contents are accelerating toward the Earth at the same rate, so Jane and her companions "float" around inside! This is because all of them are at very nearly the same distance from the Earth.
The acceleration decreases with the SQUARE of the distance (inverse square law). To compare gravity accelerations due to the same object at different distances, you use the gravity acceleration g at distance A = (the gravity acceleration g at distance B) · (distance B / distance A)2. Notice which distance is in the top of the fraction. An example of using the inverse square law is given in the "How do you do that?" box below.
How do you do that?
Find how many times more gravitational acceleration the Galileo atmosphere probe felt at 100,000 miles from Jupiter's center than the orbiter felt at 300,000 miles. You have
probe's g
= orbiter's g · (300,000/100,000)2 = orbiter's g · (3/1)2
= orbiter's g · 9.
The probe accelerated by an amount nine times greater than the orbiter.
Measuring the acceleration of an object dropped to the ground enables you to find the mass of the Earth. You can rearrange the gravity acceleration relation to solve for the mass M to find M = g d2/G. Close to the Earth's surface at a distance of 6.4 · 106 meters from the center, g = 9.8 meters/second2. The distance is given in meters to match the units of the gravity acceleration---when you do a calculation, you must be sure you check that your units match up or you will get nonsense answers. The big G is the universal gravitational constant, approximately 6.7·10-11 m3/(kg sec2). Plugging in the values, you will find the Earth's mass = 9.8 · (6.4·106)2 / (6.7 · 10-11) kilograms = 6.0 · 1024 kilograms. If you are unsure of how to work with scientific notation, read the scientific notation section in the mathematics review appendix (pay close attention to the part describing how to enter scientific notation on your calculator!).
You can determine masses of stars and planets in a similar way: by measuring the acceleration of objects orbiting them and the distance between the star or planet and the object. A small object falling to the Earth has mass and, therefore, has a gravitational acceleration associated with it: the Earth is accelerated toward the falling object (an example of Newton's third law)! However, if you plug some typical masses of terrestrial objects (less than, say, 1000 kilograms) into the acceleration formula, you will see that the amount the Earth is accelerated is vastly smaller than the falling object's acceleration. You can ignore the Earth's acceleration.
A side note: determining the mass of the Earth also depends on knowing the value of the gravitational constant G. The constant was first measured by Henry Cavendish in 1798. After discussing his experimental results, he then applied his measurement to the subject of his paper's title: "Weighing the Earth."
Formulae
Gravitational Acceleration: g = (G · Mass)/(distance from the center)2.
Comparing gravitational accelerations: acceleration at position A = acceleration at position B · (distance B/distance A)2.
What did Galileo discover about how objects of different masses fall to the Earth?
If you dropped a hammer and feather from the same height above the Earth's surface, which would actually hit the ground first? Why would it be different than what Galileo said about falling objects? Explain why if you let the feather fall quill end first, the result is closer to what Galileo said.
If you dropped a hammer and feather from the same height above the airless Moon's surface, which would actually hit the ground first? Explain why your answer is different than for the previous question.
How many times less/more gravity acceleration due to the Sun does the Ulysses spacecraft feel at 2.3 A.U. above the Sun than the solar gravity acceleration it felt at Jupiter (5.2 A.U. from the Sun)? Is it accelerated more or less at 2.3 A.U. than when it was at 5.2 A.U.?
Why do astronauts in orbit around the Earth feel "weightless" even though the Earth's gravity is still very much present?
Put the following in order of their acceleration around the Earth: a 200-ton space station 6580 kilometers from the center, a 60-kilogram astronaut 6580 kilometers from the center, a 1-ton satellite 418,000 kilometers from the center, and the 7.4·1019-ton Moon 384,000 kilometers from the center. Explain your answer.
How can you find the mass of the Earth using ordinary objects in your house?
Newton found that his gravity law is obeyed everywhere in the universe and could explain Kepler's three laws of orbital motion. Newton's development of the unifying law of gravity was also the culmination of a process of Occam's Razor in action. From Ptolemy to Newton, the theories of how the planets move got simpler and more powerful as time went on. Ptolemy's model had become extremely complicated by the time of the Renaissance and Copernicus reduced the number of circular motions to around 50 so it was simpler to use. Kepler vastly simplified the theory of planet motion by reducing the number of essential parts to just three laws. Newton unifed all of those laws to the ONE unifying law of gravity. This law was so simple and elegant that it could also explain motions on the Earth.
But what is gravity? Newton understood how the gravity force affected the motion of objects but not why gravity worked the way it did. Recognizing the limits of his knowledge, he adopted an instrumentalist view: the scientist's job is to capture observations in precise mathematical equations; explain the "how" not the "why". Only things verified by our experience of the world are admissible into science. Though the "why" question is intriguing and a few scientists will spend years trying to answer it, most scientists share Newton's instrumentalist view.
With Newton, there was no longer a hierarchical-teleological universe (one designed by God for some purpose with man playing a crucial role in the plan). The universe was now a perfect machine, based on mathematics, set in motion by God long ago. God is the reference point for absolute space and time. Newtonian mechanics requires an absolute coordinate system to keep things sensible (according to Newton this also gave God something to do).
With the success of Newton's ideas, a major change occurred in how people viewed the world around them. Reality was completely reduced to material objects. Ideas, thought, feelings, and values were secondary. Newtonism undercut the role of God and religion and the validity of science: science became just a subjective perspective of the machine universe.
Descartes saw the need to rescue thoughts, ideas and values. He developed a mind-body dualism: a world of thought and spirit exists independent of, but parallel to, the material world. There is a correspondence between the God-inaugurated, mathematical thoughts of scientists and the motions in the physical world. Descartes said that mathematical ideas work so well because there is a pre-established parallelism between the physical world and the human mind. What is real does NOT depend on us---this is probably the actual completion of the Copernican revolution and was soon so widely accepted that it became "common sense" (how about that for a paradigm shift!).
This chapter covers the revolutionary advancements due to probably the most brilliant scientist who ever lived: Isaac Newton (lived 1641--1727). His greatest contributions were in all branches of physics. Kepler's discoveries about elliptical orbits and the planets' non-uniform speeds made it impossible to maintain the idea of planetary motion as a natural one requiring no explanation. Newton had to answer some basic questions: What keeps the planets in their elliptical orbits? On our spinning Earth what prevents objects from flying away when they are thrown in the air? What keeps you from being hurled off the spinning Earth? Newton's answer was that a fundamental force called "gravity" operating between all objects made them move the way they do. 
Orbits and Tides
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