Homework 7. Satellites of Jupiter

Name: ________________________ DUE 10/16 ID number: ________________________

Between January 7th and 15th 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, measured in degrees. The filled circles show Galileo's observations, while the dotted lines show the positions of the satellites at other times.

Using such observations, Kepler tried to find out if the Galilean Satellites obeyed 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 make this measurement is to mark two successive points 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

__________ __________ __________ __________

P (days)
__________ __________ __________ __________

a (AU)
__________ __________ __________ __________

P (years)
__________ __________ __________ __________

a3 ÷ P2
__________ __________ __________ __________

It's convenient 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 convenient 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 and P you have on the third and fourth lines of the table, compute the quantity 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 will not all be exactly the same 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: October 7, 2001