Observer
E
W
January 1
July 1
October 1 April 1
At an easterly elongation.
At a westerly elongation.
MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 3: Kepler's Laws of Planetary Motion
Prior to Johannes Kepler's (1571-1630) statement of his three Laws of Planetary Motion:
1. A planet revolves around the Sun in an elliptical orbit with the Sun at one focus.
2. The line joining the Sun to a planet sweeps out equal areas in equal times.
3. The square of the period of revolution of a planet is proportional to the cube of the length of the major
axis of its orbit.
it was thought that the orbits of the planets were circles with the Sun at the center of each.
Kepler's approach to his his three laws was actually
quite geometric. (It was Sir Isaac Newton who
first showed how Kepler's laws could be deduced
from two of his own laws -- his Second Law of
Motion and his Law of Universal Gravitation -- in
his Principia Mathematica in 1687.) It (Kepler's
approach) was based on the fact that if an object
travels around a smooth, closed, planar curve
(such as a circle or an ellipse) then, to an observer
located at a point in the plane of the objects motion
but outside the curve, the object will appear
to travel to the right, to stop, to travel to the left,
Figure 1
to stop, and to travel to the right again, after which the (apparent) motion repeats (see Figure 1). A point
such as E at which the object appears to stop after traveling to the right is called an easterly elongation,
and a point such as W at which the object appears to stop after traveling to the right is called a westerly
elongation.
Kepler assumed that the orbit of the earth was
a circle with the Sun at its center. He asscoiated
to each point of the circle a date of the year (see
Figure 2). He then used Tycho Brahe's observations
of the easterly and westerly elongations of
the planet Mercury -- measured as angles as seen
from the earth, from the Sun to Mercury -- to
draw a set of lines to which he concluded the orbit
of Mercury must be tangent.
Figure 2
Obtain data from the Observers' Handbook, and repeat Kepler's construction for yourself (using as much
data as possible). Can you inscribe a circle in the resulting set of lines in such a manner that the circle is
tangent to each line? If not, what kind of curve can you inscribe? This should give you some insight into
why Kepler concluded his first law; can you see how he might have arrived at the other two?
The following Maple code may help: it generates a circle with 12 equally spaced tic-marks:
> # CircleWithPoints
> restart: with(plots):
> Circle := implicitplot(x^2+y^2=1,x=-1..1,y=-1..1,scaling=constrained,axes=none):
> P := [0,0];
> N := 12;
> for i from 1 to N do P := P,[cos(2*Pi*i/N),sin(2*Pi*i/N)]: od:
> Points := pointplot([P],color=blue,symbol=circle):
> display(Circle,[Points]);