Working model to show Kepler's 1st and 2nd laws and the equivalence of the area law to the law of equants.
What it shows:
A demonstration illustrating the equivalence of Kepler's second law, the Law of Areas, with the Law of Angles.
How it works:
In order to determine the orbit of Mars using circular orbits, Kepler had to offset the focus of Mars' orbit from the Sun to a point C (figure 1). Kepler's 2nd Law of planetary motion states that a planet's orbit around the Sun will sweep out equal areas in equal times. In Kepler's model, the elliptical path of a planet has the Sun at one focus. He then introduced the approximation that the planet would also sweep out equal angles in equal times when viewed from the empty focus. This gave the empty focus the role of the Equant from the Ptolemaic geocentric system, introduced to explain the speeding up and slowing down of a planet's motion when viewed from Earth. Ptolemy asserted that a planet's orbital speed was constant when viewed from the equant, a point displaced from the center of the circular orbit. Figure 1 shows the geometrical relationship between the areas and the angles.
figure 1. relationship between "equal angles" (shaded) and equal angles (marked α and β).
The model consists of a vertically mounted disc (figure 1) and displays a representation of the orbits of Earth and Mars. The front surface is 1/4" white Plexiglas backed with 1/2" plywood. The circumference of the disc represents the orbit of Mars. An inner circle, marked with a ring of 100 (type?) bulbs wired in parallel, represents the orbit of the Earth. The bulbs are mounted through the plywood back but below the Plexiglas, so when on they glow through the translucent plastic. The Earth ring is centered on the Sun, a 25W light bulb protruding out of the disc; the Sun is displaced by 9cm below the center of the plexi disc. The equant, 18 cm above the Sun, consists of a motor driven 1 clock hand that turns at 1 rpm. Mounted behind the disc are three switches, to turn on each of the Sun, clock and Earth orbit ring.
figure 2. the disc, displaying the orbits of Mars and of Earth.
Setting it up:
The machine is mounted on a wheeled stand, so can just be rolled into place. There is a small holder on the back of the disc for three OH pens to sketch angles etc. on the disc.
Kepler's resurrection of the abhorrent equant, as well as his abandonment of circular orbits, were two major reasons for the lack of acceptance of his theories by contemporary scientists such as Galileo.
1 Conrac type 117 motor