Jupiter and Kepler’s Laws



Jupiter and Kepler’s Laws

Background: Kepler discovered the laws of planetary motion in 1609 by analyzing Tycho Brahe’s extensive data of planetary positions. He discovered that planets revolve in elliptical orbits (not circular) and that the period squared is proportional to the mean radius (distance from the Sun) cubed, such that:

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Where c is a constant for the orbiting system. In the same year, Galileo used a telescope and discovered that the moons of Jupiter obeyed this law (each with a different constant of proportionality, c). This set the stage for Newton’s discovery of the laws of mechanics and the law of universal gravitation some 50 years later, after which it was found that:

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where G is a universal constant (G = 6.673 x 10-11 N m2/kg2) and m is the mass of the object being orbited (assumed to be much greater than that of the orbiting satellite).

Objective: In this activity, you are going to estimate the mass of Jupiter using Kepler’s Laws and the observation of its moons over several days.

Procedure:

1) Observe the motion of Jupiter’s moons over several days (movie)

2) Estimate the orbiting period (T) of Moon I or II

T =

3) Estimate the mean radius (r) of the moon’s orbit by comparing it to the diameter of Jupiter (142984 km)

r =

4) Solve for the mass of Jupiter using Kepler’s 3rd law (use consistent units!)

5) Compare your estimate with the actual mass of Jupiter (pg 250), what is the percentage error? Does it support Kepler’s 3rd Law?

The Solar System

How far away are the planets from the Sun compared to Earth? This question can be answered by knowing their periods (through observation) and Kepler’s 3rd law.

For two planets orbiting the same body (ex: 1 is a planet, 2 is Earth):

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Divide one equation by the other and then solve for the mean orbiting radius of planet 1:

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Use the preceding equation to determine the mean orbiting radius of the planets:

|Planet |Period (T1/T2) [years] |Mean radius (r1) |Mean radius (r1) |

| | |[AU] |[m] |

|Mercury |0.241 | | |

|Venus |0.615 | | |

|Earth |1 |1 |15 x 1010 |

|Mars |1.88 | | |

|Jupiter |11.9 | | |

|Saturn |29.5 | | |

|Uranus |84 | | |

|Neptune |165 | | |

Compare your answers to the actual mean distances from the Sun in pg. 250, where your estimates correct?

Plot the period squared (years2) vs. mean radius cubed (AU3) in the following log-log axis:

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What does the plot tell you about the relationship between period and mean orbiting radius?

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