Presentation: the lawgivers

Throughout the history of astronomy chapter in your textbook we have tracked how astronomy became the first science, in order to ultimately explain the motions of planets.

This presentation will touch upon the laws developed near the end of this story. Not laws that are written down and should subsequently be obeyed, but in science, laws are back-engineered--things that are observed to always happen a certain way must be obeying certain laws.

Kepler and Newton are the lawgivers of astronomy. We'll see that the characters of their laws are quite different, even though they both concern themselves with the motions of planets. You should be able to differentiate between Kepler's three laws and Newton's three laws, but most importantly understand the distinct difference between describing how things happen, and explaining why things happen.

First, Kepler's laws describe how planets move, which we will also demonstrate with an online simulation.

Kepler's first law observes from careful plotting of positions that planets don't quite move around the sun in perfect circles, but along elliptical orbits (with the sun located off-center). Circles are merely a special case of ellipses that have the same length and width, such that the sun is located at the center. Note that this law doesn't explain why planet orbits are elliptical, it just describes how elliptical they are.

To illustrate Kepler's first law, we compare the actual orbital shapes of each of the planets known to Kepler--Mercury, Venus, Earth, Mars, Jupiter, and Saturn.

Which planet has the most nearly circular orbit? For this planet, where is the sun located with respect to the center?

Which planet has the most elliptical orbit? For this planet, where is the sun located with respect to the center?

Kepler's second law observes from careful tracking of motion along elliptical orbits that planets move fastest when nearest to the sun, and move slowest when farthest from the sun. Notice in the garden above that the circular stone bricks are off-center, but each section of flowers has the same area, because the section where the circular stone bricks are closest to the edge is wider, and the sections where the circular stone bricks are farthest from the edge are narrower. (This is somewhat hard to see because of the perspective in this photograph, but count the flowers in each section, which should be approximately the same.) The original statement is that "equal areas are swept in equal times," which should make more sense in the animation below. Again, this law doesn't explain why a planet slows down and speeds up, it just describes how it slows down and speeds up.

To illustrate Kepler's second law, we observe how a planet moves along a circular orbit, and compare it to how it would move along an elliptical orbit. Here the length of the arrow shows how fast (or slow) the planet moves along its orbit. Also note the amount of time a planet "sweeps" out each of the colored areas is always the same (despite moving faster or slower along different parts of the orbit).

For the planet moving along a circular orbit, notice that its speed (the length of its arrow) is always constant, and that it spends an equal amount of time in each colored section, such that each colored section has the same area. (This is how you would fairly cut a pie with equal slices, right?)

For the planet moving along an elliptical orbit, where along its orbit does the planet move slowest? Where does it move the quickest?

Are there colored sections that contain more area than others? Which section(s) contain the most area? The least area? Or do they all contain the same amount of area?
(If you had an elliptical pie, would this be a fair way to cut the slices?)

Kepler's third law is something that Copernicus had conjectured, but Kepler defined a strict mathematical equation for--a planet in an inner orbit takes less time to complete an orbit, while a planet in an outer orbit takes more time to complete an orbit. Yet again, this law doesn't explain why inner and outer planets have different periods (the time to make one complete orbit), it just describes how their periods happen to be different.

To illustrate Kepler's third law, we observe how Earth (in white) orbits the sun compared to the inner planets Mercury and Venus, and Mars in the outermost orbit in this view. Which planet completes its orbit with the shortest period? The longest period? Which planet moves along its orbit with the fastest speed? The slowest speed?

In contrast to Kepler's laws, which are merely a set of rules that planets seem to follow, Newton's laws explain why planets move the way they do.

Newton's first law states what happens if there were no force (such as gravity) exerted on an object (such as a planet)--the motion of the object would be unchanged. If the object is stationary, then it remains stationary; if the object is moving steadily in a straight line, it keeps doing that.

Newton's second law is much more interesting--exerting a force (such as gravity) on an object (such as a planet) will change its motion. The more force you exert, then the object's motion will change more. This is a cause-and-effect statement, and surprisingly it is the very first such statement in the history of science. This is also the key to understanding why planets move the way that they do--they are forced to orbit the sun because of the sun's gravitational force.

Newton's third law, for the sake of completeness, notes that forces are two-way streets. Earth exerts a gravitational force on the moon, causing its motion to orbit us, while at the same time the moon does exert a gravitational force on Earth. Wait, how do we know this is true?

Yes, there's that apple thing people associate with Newton. But we will better understand how Newton's laws explain why planets move the way that they do with a more elaborate thought experiment than his falling apple.

Newton's cannon, which we will demonstrate with another online simulation.

Newton's cannon is not something that can ever be built, but assuming that such a powerful cannon were constructed at the North Pole, and that we could neglect the drag of the atmosphere on its projectile, then some interesting things can be observed. If the projectile is fired horizontally, but with relatively slow(!) speeds, then the projectile will eventually be pull by Earth's gravitational force to fall down and hit the ground. Firing the projectile with slightly faster speeds will allow it to travel farther along before eventually hitting the ground.

If Earth were flat, then firing the projectile with faster and faster speeds would cause it to travel farther and farther before eventually hitting the ground. However, because Earth is round, then eventually firing the projectile with just enough speed will allow it to travel far enough before "falling down," such that when it does arc downwards, Earth's surface is rounded enough that the projectile will never hit the ground, and just continue to orbit around the planet! Do you see Kepler's first law being illustrated here? Kepler's second law?

There is a certain speed that can be given to the projectile such that Earth's gravitational force will pull downwards (inwards) on it that the projectile will continue in a circular orbit.

Firing the projectile with an even faster speed would allow it to travel out farther, and another elliptical orbit results. Do you see Kepler's first law being illustrated here? Kepler's second law? How about Kepler's third law? This illustrates the cause-and-effect nature of Newton's second law on the motion of the projectile--cause: Earth's gravitational force; effect: projectile's sideways motion pulled into an arc or orbit (otherwise Newton's first law says that if were no gravitational force from Earth, the projectile would continue in a straight line).

And Dr. Alex Filippenko from the University of California-Berkeley, twirling a donut on a string. The donut represents a planet that has an initial (sideways) speed, and the string represents the gravitational force that pulls on the donut to keep it in its orbit. Cause: string force. Effect: sideways motion pulled into circular path (otherwise Newton's first law says that if the string breaks, the donut would fly off sideways, as shown later in this video). (Video link: "[Alex] Filippenko and the moon’s orbit demonstration.")

No comments: