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Frey lab manual Expt 7 chapter.

Kepler?s law exercise

JOHANNES KEPLER; (Germany, 1620)

Kepler had bad eyesight and was not an observational astronomer, but he was good at math. Tycho Brahe met Kepler and hired him to work in Brahe?s observatory. Tycho Brahe was one of the best observational astronomer of the time, and he made the most quantitatively accurate measurements of the locations/motions of the five wanderers that existed up to then. Brahe died and Kepler inherited all his data. Kepler noticed that there were discrepancies between the predictions of the Copernican model and the actual positions recorded by Brahe. So Kepler set out to see if he could modify the Copernican model to get better agreement with the actual observations (this is part of the ?Scientific Method?). He succeeded, after 25 yrs of tedious work. His results are encapsulated in what we today call ?Kepler?s 3 rules of Planetary Motion? . (Your book calls them ?laws? instead of ?rules?.) We could have written ?Kepler?s 3 rules of Wanderer Motion?. Whenever you say the word ?planet ?, you are using the Greek language for the English word ?wanderer?.

His 1st rule asserts that the orbits of the wanderers are NOT perfect circles, as everyone before Kepler had thought. They are ellipses, with the Sun at one focus. To remind you from High School, here is how to construct an ellipse: (Links to an external site.)

You nail two pegs in a board, take a loop of string, use your pencil to pull the loop tight. You can construct a new ellipse by changing the distance between the two foci, keeping the length of the loop the same. Or, by changing the length of the loop of string, keeping the distance between the two foci the same. The straight line connecting the two foci is called the major axis, it is the fattest part of the ellipse. (the ?Minor axis? is perpendicular to the major axis.) Notice what you get when the distance between the two foci is zero. This special case is called an ellipse with zero eccentricity. Notice that the ellipse is symmetric about the major axis. You can fold the ellipse on top of itself along the major axis. The Sun is at one focus, and nothing is at the other.

Kepler?s second rule tells us something about the speed of the wanderer. His 2nd rule says the motions of the 5 wanderers are NOT constant speeds, as everyone before him had thought. Instead, their speed changes smoothly, governed by this requirement: the imaginary line connecting the Sun to the planet must sweep out equal areas in equal time intervals. Look at this: (Links to an external site.)

You can see from this that the speed is fastest when the planet is nearest the Sun, and slowest at the other end of the major axis, where the planet is farthest from the Sun.

Notice what this rule says about the speed for the special case of an ellipse with zero eccentricity, ie., when the distance between the 2 foci is zero.

Kepler did not know it at the time, but Newton (later in 1600?s ) showed that the second law is connected to an important fact called the ?Law of conservation of Angular Momentum?.

Kepler?s 3rd rule gives the quantitatively precise connection between 3 properties of the orbit: the orbit size ( a ), the time to complete an orbit (P), and the Mass of the Sun (M). In our course, we will use only the following form of the 3rd law:

P^2 = a^3 / M ;

This is the form of the 3rd law that we will use and that you should write on your cheat sheet ( if we were to have an in-class exam).

Definitions of symbols: P is the orbital period of the planet, i.e. the amount of time it takes for the planet to make one complete trip around the Sun, and it must be in units of Earth-years (not seconds). Lower-case ?a ? is a measure of the size of the ellipse, namely it is one-half of the length of the major axis of the ellipse, and it must be in units of A.U. (astronomical unit, see pg 12in Freedman, pg 30 in Palen. the Palen book uses a capital A for my lower case a.),. M is the mass of the Sun, not in units of pounds but in ?solar masses?. Our Sun has a mass of ONE solar mass. There exist some stars that weigh twice as much as our Sun. They have Two solar masses. Polaris has a mass 6.5 times that of our Sun, M=6.5 for planets travelling around Polaris. If God were to magically instantaneously double the mass of our Sun, it would take 1/sqrt(2) = 0.707 earth-years to orbit that Sun. As another example, pretend you discover a new planet orbiting our Sun, and you measure the size of its orbit to have a major axis = 200 AU. Then Kepler?s 3rd law says it must take _____________ to complete one orbit.

Notice the mass of the planet does not appear in Kepler?s 3rd law. Therefore, you can predict that if you were to magically move Jupiter to the same distance from the Sun as is Earth, it would take __________ Earth-years for Jupiter to complete one orbit around the Sun.

(Answers = 1000 earth-years.; One earth-year)

We would not bother to spend this much time explaining Kepler?s 3 rules were it not for the fact that they are still useful approximations TODAY. And they are a little bit more general than Kepler himself realized. For example, they apply also to the orbits of the 4 Galilean moons around Jupiter. This is how Jupiter was first ?weighed?! They also apply to the orbits of all telephone/spy satellites around Earth (except of course M is not the mass of the Sun but instead the Mass of Earth). So, although Kepler?s rules are not rigorous, they often are an adequate approximation. Later in the 1600?s, I. Newton came along and developed his 3 laws of motion and his law of gravity. Newton?s laws are much more general and fundamental than Kepler?s, and Newton proved that Keplers rules can be DERIVED FROM Newton?s laws. To put it one way, Newton?s laws explain why Kepler?s rules work. Kepler did not know about the existence of the gravity force. If one needs the most accuracy ( such as the NASA engineers who send spacecraft to Mars), you use Newton?s laws instead of Kepler?s.

You can see a planetary orbit simulation using Kepler?s laws at (Links to an external site.)

but I think you have to use Internet Explorer.

Why did no one before Kepler in the 1600?s notice that the motions of the 5 wanderers are NOT perfect circles and not constant speeds? (Everybody for the previous couple thousand years, including Copernicus, thought all 5 orbits were perfect circles and const. speeds, though different sized orbits and different const. speeds for each of the 5.) Good question, and it has a straightforward answer. Because the 5 planetary orbits happen to be ALMOST perfect circles; they are only mildly eccentric, and so by Kepler?s 2nd law, they are ALMOST constant speeds. (Eg, Jupiter?s eccentricity is only a few %). And so it took the high quality data of Tycho Brahe (1580?s) to pick out the difference between a perfect circle, and a slightly noncircular ellipse. Prior to Brahe, the observations were not accurate enough to distinguish between perfect circles and mildly eccentric ellipses (this also is a part of the ?Scientific Method?: you always want the most accurate measurements).

An important quantity in physics that Newton discovered (Kepler did not know about it) is called the angular momentum of a moving object. The angular momentum of a particle or planet is defined as the product of its mass, its speed, and its distance from the Sun, mvr. Newton proved that under fairly general conditions (that we won?t discuss), angular momentum obeys a law of conservation, remaining constant in value. And that Kepler?s second law is actually a direct consequence of this law of conservation of angular momentum (but Kepler did not know that).

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