## MOONS OF JUPITER

**Objectives of this lab:**

1. Determine the mass of Jupiter.

2. Gain a deeper understanding of Kepler's third law.

3. Learn how to gather and analyze astronomical data.

## Background

Kepler's third law for a moon orbiting a much larger body is

C = r^3/T^2, where C is a constant, r is the length of the semi-major axis of the elliptical orbit in units of the mean Earth-Sun distance, 1 A.U. (astro-nomical unit), and T is the planetary orbit in Earth years.

If the orbit is circular (as will be assumed in this lab) the semi-major axis is the same as the radius of the orbit.

Newton expanded on Kepler’s Third Law, by using his Universal Law of Gravitation to solve for the constant C and derived:

C = r^3/T^2, where C is a constant, r is the length of the semi-major axis of the elliptical orbit in units of the mean Earth-Sun distance, 1 A.U. (astro-nomical unit), and T is the planetary orbit in Earth years.

If the orbit is circular (as will be assumed in this lab) the semi-major axis is the same as the radius of the orbit.

Newton expanded on Kepler’s Third Law, by using his Universal Law of Gravitation to solve for the constant C and derived:

**where G is the gravitational constant 6.67x10^-11 Nm^2/kg^2 and M is the mass of the larger body.**## Procedure

## Data

jupsatdata1.dat |

## Data Analysis

callisto_results.bmp |

europa_results.bmp |

ganymede_results.bmp |

io_results.bmp |

jupiter_data.xls |

## Conclusion

1. Calculate the percentage error with the accepted mass of Jupiter (1.8986 × 10^27 kg).

2. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?

2. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?

**The periods of the moons beyond the orbit of Callisto will have larger periods then Callisto. This is due to the fact that in order to maintain the relationship (R^3/T^2)=(GM)/(4pi^2) as the orbital radius increases so must the period. In simpler terms, as one moves away from the center of any system the circumference increases which makes it a longer distance to travel in order to make one revolution. Farther distance means longer time.**

3. Which do you think would cause the larger error in the mass of Jupiter calculation: a ten percent error in**"T"**or a ten percent error in**"r"**? Why?**A ten percent error in "r" would yield a greater error in the mass because the value of "r" is greater than the value of "T". "T" is a term that is squared while "r" is a term that is cubed which gives it "more weight" in the calculation.**

4. Why were Galileo's observations of the orbits of Jupiter's moons an important piece of evidence supporting the*heliocentric*model of the universe (or, how were they evidence against the contemporary and officially adopted Aristotelian/Roman Catholic,*geocentric*view)?**Galileo's observations were important in supporting the heliocentric model because his observations proved Kepler's laws. Jupiter's moons orbit Jupiter just as all of the planets in the solar system orbit the sun. Galileo observed that it is the object with the largest mass that is in control, which led to the conclusion that the sun is the center of our solar system. In simpler terms: the biggest body wins.**