Homework 5. Satellites of Jupiter

Name: ________________________

Due 9/26

ID Number: ________________________

Between January 7^th and 15^th of 1610, Galileo discovered the four `Galilean Satellites' of Jupiter; listed in order of their distance from Jupiter, these are Io, Europa, Ganymede, and Callisto. On the diagram at left, the vertical axis is the date, and the horizontal axis is the angle from Jupiter, marked in units of 0.1°. The filled circles show Galileo's observations, while the lines show the positions of the satellites at other times.

Using such observations, Kepler tried to find out if the Galilean Satellites obey his third law. This assignment involves reproducing some of Kepler's calculations. To do this you need to measure the semimajor axis a and the period P of each satellite's orbit.

To measure the semimajor axis of each satellite's orbit, you must first find its greatest angle from Jupiter; call this angle and write it on the first line of the table below. Try to make your measurement accurate to one-hundredth of a degree.

To measure the period of each satellite's orbit you must determine the length of time before it returns to its original position. One way to do this is to mark two successive times where the satellite crosses the solid vertical line (which represents Jupiter's position); the period is twice the time interval between two such crossings. Write the period on the second line of the table. Try to make your measurement accurate to one-tenth of a day.

	Io	Europa	Ganymede	Callisto
(deg.)	__________	__________	__________	__________
P (days)	__________	__________	__________	__________
a (AU)	__________	__________	__________	__________
P (years)	__________	__________	__________	__________
a³ / P²	__________	__________	__________	__________

It's useful to convert each greatest angle to a semi-major axis a measured in AU; this can be done using the small-angle formula:

Here D is the distance from the Earth to Jupiter; that distance was D = 4.2 AU at the time Galileo made his observations. Note that because D is given in units of AU, you're guaranteed that a will also be in AU. Write the semi-major axis a of each satellite on the third line of the table.

Likewise, it's useful to convert each period from days to years. Just divide the periods P listed in the second line of the table by 365.26; write the results on the fourth line of the table.

Finally, you're ready to check if Kepler's third law is obeyed by the Galilean Satellites! Using the values of a in AU and P in years that you've just found, compute a³ / P² and write it on the last line of the table.

Now the question is: do the satellites of Jupiter obey Kepler's third law? Recall that the third law says that a³ / P² has the same value for all objects orbiting a given body. But no real measurement is perfectly accurate, so even if the law is obeyed, the values of a³ / P² you get will probably not be identical for all four satellites. But are they close? In other words, do you think that the differences in your a³ / P² values are due to errors in your measurements and calculations, or do they show that Kepler's third law really doesn't apply to the Galilean Satellites? Why or why not?

Extra credit: if a is measured in AU and P is measured in years, the quantity a³ / P² is equal to the mass of the central body - in this case Jupiter - in units of the Sun's mass, M. From your results from the last line of the table, what is your best value of Jupiter's mass in units of M?

Joshua E. Barnes (barnes@ifa.hawaii.edu)

Last modified: September 18, 2006
http://www.ifa.hawaii.edu/~barnes/ast110_06/homework/hw05.html