### Activity 7A: Determining the Mass of Jupiter

Many years ago Sir Isaac Newton continued the work of Johannes
Kepler and discovered that the mass of any planet could be calculated
by knowing the period and orbital radius of any of its moons. The
result of this work** is contained in the following equation:

M =
4p^{2}r^{3}/T^{2}G

where M is the mass of the planet, T is the period of one of its moons
and r is the orbital radius of the same moon. The letter G represents
the universal gravitation constant which has the value 6.67 x 10

^{-11} N-m

^{2}/kg

^{2}.

The units used with this equation are of utmost importance. The
period, T, must be in seconds. The orbital radius, r, must be in
meters. The resulting mass, M, will be in kilograms.

***If you are interested in how Newton arrived at this equation
please refer to the *Library in the EDO - Orbits nexus community. The
document is called "Derivation of the formula for determining the Mass
of Jupiter".*

In Activity 6 you were able to determine the period and orbital
radius of the moon Io. Possibly you were able to do this for other
moons. You will now be able to use this information to perform a
relatively simple calculation and determine the mass of Jupiter.

**Do some math: **
- Go back to Activity 6 and locate the period and orbital radius you found for Io.
- Convert the period from hours into seconds.
- Convert the radius from kilometers to meters.
- Now insert these values into the formula above to calculate the mass of Jupiter in kilograms.

You will want to use a calculator for this purpose. Be careful with the exponents on the powers of ten.

Locate a source for the published value of the Jupiter's mass.
Compare the result of your calculation above to the published value.

### Activity 7B: Determining the mass of the Earth.

You now have had the practice of using Newton's equation to find the
mass of Jupiter. Recall that you needed to know only the period and
orbital radius of one of its moons, Io.

You can use this same process to determine the mass of the Earth by
obtaining the same orbital information about the earth's moon.

Find a source of information that will give you the moon's period
and orbital radius. Once you have found these values remember to
convert them into the units you used in Activity 7A. Then insert these
numbers into the mass equation in Activity 7A to calculate the mass of
the Earth.

Now find a source that will give you the published value for the
mass of the Earth. Compare your calculation to this accepted value. If
you find that your calculated value was not close to the published
value, go back and check your work, especially the conversion of units.

### Activity 7C: Measurement of the speed of light.

Interestingly enough, the moons of Jupiter can be used to determine the
speed of light. In the late 1600's Ole Roemer made a well known set of
measurements to give us an early value of light's speed. Although the
value that Roemer determined was quite different from the value we know
today, it gave other scientists an idea of how large it was and
suggested other methods that could be used in future measurements.

In this activity you are directed to some internet websites to
learn more about the measurement of the speed of light and how Ole
Roemer used the moons of Jupiter to make this measurement.

Use the link above to go to the EDO Project Links cybrary. Here you
will find three websites that are very useful for understanding how the
speed of light was first calculated.

The first website listed will give you some background on some early
attempts at this measurement, and then it gives information about how
Roemer went about his measurement. In contains a diagram that is
helpful in visualizing Roemer's thought process. **Please note that there is an important error on this website.**
Roemer did not get 186,000 miles/sec for his calculation. That is the
currently accepted value. He got a value that was somewhat higher.

The second website is a quick summary of his experiment, but also
contains a little more detail of the actual measurements and the
numbers Roemer used.

The final website contains a biography of Roemer's life, including his work on the speed of light.

After you have read some of these references, particularly the first
two, you might want to attempt a repeat of Roemer's measurements.

First, try to go through the numbers that Roemer actually used to
calculate the speed of light. The only equation you really need is: **speed = distance/time.**
The distance involved here is related to the radius of the earth's
orbit around the sun which is called an Astronomical Unit. You can find
this number in the third reference. The time you need is the delay of
the light coming from Io as it appears around Jupiter. You'll find that
number in the first and second references. Remember to use the correct
units, usually **meters** and **seconds,** to get the correct units for speed, usually **meters/second.**

After this you could try to find your own values for the time
delay of Io by looking for published values of Io's transit behind
Jupiter at two different times of the year. The times must be 6 months
apart: one time when the earth is moving toward Jupiter; the other time
when it is moving away from Jupiter.

Good luck!