Gravity and the Orbits of the Planets II

Universal Gravitation and Kepler's Laws

• Introduction
• Newton, Gravity, and Planetary Motion
• Practice with Concepts
• Discussion Questions
• Optional Website Reading

Introduction

We come now to the physical explanation for Copernicus' model of the planetary system, and we need to establish some concepts clearly.

• A force, however it arises, is a push or pull that changes the speed or direction of motion fo some object. We call this change acceleration, and by Newton's laws of motion, if a net outside force acts on a body, it will accelerate. Conversely, if we see something accelerating, we can be confident something is exerting a force on the object.
• Energy isn't the same thing as force, although the two concepts are related. When a force moves a body through a distance, it performs work, and that work is energy. When we do work, we put energy into a system, or store it in the system. Energy can't be destroyed or created, so the energy for the work we do has to come from outside the system, and we can get it back out of the system in different forms: light, heat (which is energy of motion or kinetic energy, chemical bonds, electricity.
• Matter has certain characteristics. All matter attracts other matter (gravitational force), and some matter has electrical charge which can attract or repel other charged matter, depending on the type of charge.

Newton, Gravity, and Planetary Motion

While Kepler was able to put together laws of planetary motion and Galileo was able to observe similar movements in the Jovian moons, neither was able to explain these motions in terms of forces driving them. Galileo postulated that the sun had some native force that allowed it to capture planets, and Kepler granted the sun a mystical force as well, but neither could explain what this force was or how it worked. Galileo did, however, show through a long series of experiments that all matter resisted changes to motion.

The rules Galileo and Kepler put forward were strictly empirical laws: summaries of observational data without causal explanation. The missing key was provided by Isaac Newton, who formulated three basic laws of motion, and a general law about the force common to all matter:

• Inertia. Bodies remain in their current state of motion unless acted on by a net outside force, where "current state" is linear motion at constant speed. Thus a body will remain in straight-line motion forever without forces acting on it, or if the forces acting on it balance each other out (so there is no net force). It takes force to slow the body down and force to speed it up and force to cause it to change direction, so some force is required to keep bodies moving in circles, since they are constantly changing direction. Bodies at rest stay at rest. This is actually a generalization of a law of inertia that Galileo formulated from his falling body experiments. Any change in motion is called acceleration, whether it involves change in speed, change in direction, or both.
  
  
• The amount of force is directly related to the amount of acceleration and the mass accelerated. Newton defined a force as that which causes acceleration, or any change in the direction or speed of a body. Force is always in the same direction as the change (don't confuse this with the actual direction of motion; deceleration is acceleration in the direction opposite to the current velocity). The force required to change a body's motion is equal to the mass of the object times the change in velocity, or the acceleration: F = m/a. (a is just change in velocity over time: v/t, and velocity is change in position over time: d/t).
  
  
• Action and reaction. Newton's third law of motion explains that when one body exerts a force on another, the second exerts an equal and opposite force on the first. The results may be different depending on the properties and initial state of the objects: whether the stone hits the glass pitcher of lemonade, or the pitcher hits the stone, it is going to be bad for the pitcher. Both the pitcher and the stone receive the same magnitude of force. Pay attention to the fact that the equal but opposite forces act on different objects! This behavior is absolutely necessary and is driven by the fact that solid bodies have coherence — their atoms attract one another very strongly. If matter behaved different and didn't push back, gravity would pull you right through the floor.
  
  

Newton continued to think about this problem of force, and gradually perceived that the same force that pulls the apple to the ground when it falls from the tree also keeps the moon in its orbit. The attraction of matter to all other matter is a universal and constant force that depends only on the amount of matter and the distance between any two chunks of it. If we have any two masses M and m, separated by a distance R, then the force of mutual attraction or gravity between them is

$$F=G(\frac{{m_{1}m}_2}{r^2})$$

G is a numerical constant that depends on the units we use for mass and distance and force. We say that force varies directly[in the same way] as the masses involved. So if the masses increase, the force increases in the same proportion. We also note that the force varies inversely as the square of the distance between the two objects. So if the distance increases, the force decreases by the square amount

This relationship is true for all masses in the universe: as you sit reading this, you attract the computer, the computer attracts you, the earth attracts you both, the sun attracts the earth, and so on....but remember that the attraction is mutual in all cases.

This universal force of attraction or gravity is the force that keeps planets in circular orbits (remember that circular motion involves a change of direction, so there MUST be a force in there somewhere). It also explains Kepler's law...but I won't make you learn the derivation for this class.

Kepler's First Law: Planets move in ellipses with the sun at one foci

In a perfect world with only two objects, a gravitational force between them and equal to mv2/r would create perfectly circular orbits around their common center of gravity. But even the smallest perturbation will send the objects into elliptical orbits.

Assume that we have a disproportionate system: one mass is large, the other small, so that for all intents, the small object orbits a center of gravity inside the mass of the larger one. For a perfectly circular orbit, the force of gravitation F_g = G Mm/r² provides exactly the centripetal force F_c = mv²/r. If the orbit is disturbed in such a way that the object m moves towards its primary M, then in order to conserve angular momentum, it speeds up. This forces it to enlarge its orbit. If it is displaced outward, it slows down and "falls back" toward the primary. The oscillation between the two states produces an elliptical orbit.

Kepler's Second Law: The law of areas

Kepler observed that planets speed up when they are closer to the sun and move more slowly when they are further from the sun. This is also true of comets. The relationship is actually a precise one: planets sweep out equal areas in equal times. While we won't go into the details here, this phenomenon is a result of the conservation of energy law that was not discovered until the late nineteenth century. The "area" represents the work done to move the planet during a set period of time under varying force as the distance changes. While Kepler was able to show this graphically, precise calculations require integral calculus.

Kepler's Third Law: The law of periods

Kepler observed that if he used the Earth's period (1 year) and distance H from the sun (1 AU), as his time and distance units, the periods and distances of the other planets produced a pattern: within his ability to measure distances and times accurately, the square of the period equaled the cube of the distance:

Planet	Distance from Sun (AU)	Period (Yr)	a3	P2
Mercury	0.387	0.246	0.058	0.061
Venus	0.723	0.615	0.378	0.378
Earth	1.000	1.000	1.000	1.000
Mars	1.524	1.881	3.540	3.534
Jupiter	5.203	11.860	140.85	140.66
Saturn	9.554	29.460	872.08	867.89

This is an empirical law, a pattern based on direct observation. Kepler had no cause-and-effect relationship to explain this pattern, but it worked for all the known planets. It wasn't until Newton formulated his Universal Law of Gravitation that anyone could explain the relationship between distance from the sun and speed around the sun (which determines period) in terms of physical forces producing a predictable result.

To see how Newton's law of gravity will result in Kepler's third law of planetary, open the derivation movie. To move through the frames and animate the objects, click with your mouse or use the space bar. Do not click too fast!

Practice with the Concepts

Two masses start at a distance of 1 unit and are moved to a distance 3 units apart. What is the change in the force of attraction between them?

Discussion Questions

• How does Newton's concept of force differ from previous ideas of force? Can gravitational force operate across a vacuum? Can energy be transmitted across a vacuum?
• Gravitational force is the weakest of the four fundamental forces (gravity, electromagnetic force, strong nuclear force, weak nuclear force). How is it then able to affect the movement of galaxies separated by enormous distances? Why wouldn't electrical forces, like the charged ions in stellar atmospheres, affect stellar and planetary motions more than gravity?

Optional Readings

• The Gravity at NASA's SPACE PLACE site gives a very elementary but useful summary of the affects of gravity and the GRACE mission to detect changes in the Earth's gravitational field.
• The Gravity Simulator lets you initialize and change the gravitational parameters of a small solar system to see how gravity affects planetary motion.

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