Kepler’s 1st and 2nd Laws: Examining Ellipses
Objective:
Students will use string and tacks to explore the meaning of Kepler’s 1st Law of
planetary motion.
Procedure:
Cut a length of string that can be tied in a circle with an average diameter of
approximately 10 cm. Using a piece of paper, two thumb tacks, and a corkboard, trace
through 4 ellipses as given below, make sure the distance between foci is measured
horizontally on your paper.
I.
The distance between the foci is 0 cm
II.
The distance between the foci is 2 cm
III.
The distance between the foci is 4 cm
IV.
The distance between the foci is 7 cm
Analysis:
Indicate four or more points at different places on each of your ellipses (these points
can be wherever you’d like, just make sure there is at least one point in each quartile of
the orbital ellipse, and that you have points at the perihelion and the aphelion). These
points will mark the position of a moon orbiting around a central body located at the left
focus.
For each of your four ellipses:
a) sketch vectors whose apparent lengths correspond to the strength of the
gravitational force between the two bodies;
b) indicate which of your points is located where the moon’s tangential velocity is
the greatest; and
c) indicate which of your points is located where the moon’s tangential velocity is
the smallest.
Kepler’s 3rd Law: Mass of Saturn Investigation
Objective:
Students will use period-radius data for the moons of Saturn to determine the mass of
Saturn, using Newton’s applied version of Kepler’s 3rd Law (the “Law of Harmonies”).
Procedure:
Given data for the period of the moons orbiting Saturn and their mean distance from
Saturn, analyze the data in order to determine the average value of 𝑇 ! /𝑅! for all of the
moons. Once the average value of this ratio is determined, derive and use Newton’s
applied version of Kepler’s 3rd Law.
Given Data:
Blank column has been left for your data analysis.
Satellite
Mean Distance (km)
Period (days)
Pan
1.34×10!
0.58
Prometheus
1.39×10!
0.61
Pandora
1.42×10!
0.63
Janus
1.51×10!
0.69
Mimas
1.86×10!
0.94
Enceladus
2.38×10!
1.37
Calypso
2.95×10!
1.89
Helene
3.77×10!
2.74
Rhea
5.27×10!
4.52
Titan
1.222×10!
15.95
Hyperion
1.481×10!
21.28
Iapetus
3.561×10!
79.33
Average à
Analysis:
1) Show a complete derivation for Newton’s applied version of Kepler’s 3rd Law:
!!
!!
!!!
= !∙!
!"#$%&
.
2) Show your calculation of the Mass of Saturn.