Download 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:
T 2  cr 3
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:
4 2 r 3
T 
Gm
2
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):
4 2 r13
4 2 r23
2
2
T1 
T2 
Gm
Gm
Divide one equation by the other and then solve for the mean orbiting radius of planet 1:
T12 r13

T22 r23
T
 r1  r2  1
 T2



2/3
Use the preceding equation to determine the mean orbiting radius of the planets:
Planet
Period (T1/T2)
Mean radius (r1)
Mean radius (r1)
[years]
[AU]
[m]
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
0.241
0.615
1
1.88
11.9
29.5
84
165
1
15 x 1010
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:
What does the plot tell you about the relationship between period and mean orbiting radius?