An Introduction to Kepler's Three Laws

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OBJECTIVE

By reproducing ellipses via the "string-and-pencil method," the students will draw ellipses and determine the eccentricities; by measuring the orbits of five of Jupiter's moons, the students will test Kepler's third law; and by using characteristics of Pluto's orbit, the students will confirm Kepler's second law.

PROCEDURE

Kepler's three laws are simply a mathematical way of describing motions of objects that orbit a large central mass, such as the planets which orbit around the Sun or the moons which orbit around Jupiter. This lab explores each of Kepler's three laws.

Please write down all work, calculations, and answers on a separate sheet.

1. Kepler's First Law: Orbiting objects travel in elliptical paths with the central mass at one focus. In this section you will get acquainted with ellipses by sketching one yourself.

a) Get two tacks and a piece of string. On your paper, place the two tacks a small distance apart, pinning down the ends of the string. Be sure to leave some slack in the string.

b) Using the string as a guide (i.e., place the pencil inside the string loop and pull the loop taut), draw an ellipse.

c) Now measure and write down the distance between the foci AND the length of the major axis of the ellipse.

d) Divide the distance between the foci by the length of the major axis. This quantity is known as the eccentricity, "e".

e = (distance between foci)

(major axis)

What is the eccentricity of the ellipse you drew?

e) What familiar shape is an ellipse with an eccentricity e=0.0?

f) (optional) Sketch Pluto’s orbit, e=0.25.

2. Kepler's Third Law: The periods and semi-major axes of bodies orbiting a common object are related by

P_body1² P_body2²

-------- = ---------

a_body1³ a_body2³

In this section you will verify this law for the five largest moons of Jupiter: Almathea, Io, Europa, Callisto, and Ganymede.

a) Create a table to hold the values of orbital period (P), semi-major axis (a), and P²/a³ for all five moons.

b) Measure the semi-major axes of the moons on the screen with a ruler.

c) Measure the orbital periods either by noting the times in the movie or by timing with a watch.

d) Look at your values or P²/a³: does Kepler’s third law hold? Your numbers are probably not exactly as you expected. Comment on sources of error in your measurements.

If you’re doing this part at home, you can find the movie of Jupiter’s moons at

http://www.astro.washington.edu/labs/clearinghouse/movies/images/orbits_of_jovian_moons.gif

It might be helpful to save the gif file and use QuickTime to view it so you can stop and start the movie at will.

3. Kepler's Second Law: Objects in elliptical orbits sweep out equal areas in equal times. This implies that the orbital speed of a planet around the sun is not uniform - it moves fastest at the point closest to the sun (known as the PERIHELION) and slowest at the point farthest away (known as APHELION). In this section we will calculate the difference in this speed using Pluto as an example. Pluto's orbit has an eccentricity e=0.25. Its semi-major axis is 5.9x10⁹ km.

a) Determine the distance (D_aphelion) between Pluto and the sun at aphelion. You should be able to determine this using just the semi-major axis and the eccentricity.

b) Determine the distance (D_perihelion) between Pluto and the sun at perihelion. Again, you should be able to determine this using just the semi-major axis and the eccentricity.

c) Kepler's Second Law allows us to determine the ratio between Pluto's velocity at aphelion and perihelion: v_aphelion/v_perihelion. To do this you need to find the area swept out by Pluto's orbit. This can be approximately described as a triangle with:

Area=1/2 x (Distance to sun) x (Current Velocity) x (Time)

The law states that planets sweep out 'equal areas in equal times': this means that the area swept out in a fixed time interval (say a week) is the same at perihelion as it is at aphelion. Therefore we can say:

½ x (D_perihelion) x (v_perihelion) x (time) = ½ x (D_aphelion) x (v_aphelion) x (time)

Use this equation to determine v_aphelion/v_perihelion.

d) Pluto's minimum orbital velocity is 3.7 km/sec. Determine values for v_aphelion and v_perihelion.