Name: ________________________ DUE 10/11 ID number: ________________________

# Homework 7. Satellites of Jupiter

Between January 7th and 15th of 1610, Galileo observed Jupiter and discovered the four `Galilean Satellites'. On the diagram at left, the small filled circles represent Galileo's seven observations, while the dotted lines show the positions of these satellites at other times. The vertical axis is time, labeled by date. The horizontal axis is angle from Jupiter, measured in degrees.

Using these and other observations, Kepler tried to determine if the Galilean Satellites obeyed his third law. The first part of this assignment is to reproduce Kepler's calculation. To do this you need to measure the semimajor axis a, and the period P of each satellite. These can be measured from the diagram.

To measure the semimajor axis a, you need to measure the satellite's maximum angle from Jupiter. For convenience, you can use units of 0.1° to record each value of a. To measure the period P you need to measure the number of days before each satellite returns to its original position. To get a good measurement, mark the 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.

Record your measurements below, and find a3 ÷ P2 for each satellite. Don't expect the result to be equal to 1; for one thing, these satellites orbit Jupiter, not the Sun! But do the Galilean Satellites obey Kepler's third law; is the result the same for all four satellites? In answering this question, keep in mind that your measurements may have some slight error.

 Name a P a3 ÷ P2 Io __________ __________ __________ Europa __________ __________ __________ Ganymede __________ __________ __________ Callisto __________ __________ __________

Is Kepler's third law obeyed? __________

The second part of this assignment is to use the generalized form of Kepler's third law to estimate the mass of Jupiter. To do this, you first need to pick any one of the satellites, and express its semimajor axis a in AU and its period P in years. Use the small-angle formula to find a in AU:

Here D is the distance from the Earth to Jupiter; at the time
Galileo made his observations, that distance was D = 4.2 AU.
The quantity  is the semimajor axis of
your chosen satellite's orbit in units of 1°; I'm using a
different symbol here to remind you to use units of 1°, not
0.1° as above.  Keep in mind that on this page, the
semimajor axis a has units of AU.  In the space below, write
the name of your chosen satellite, the value of a in AU, and
the value of P in years.

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Given a and P in the right units, compute a3 ÷ P2, which is Jupiter's mass in units of the Sun's mass. Do this calculation in the space below. If you've done everything right, you should get a value of about 0.001 -- Jupiter is only 0.1% as massive as the Sun.

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