Task 1 of 8 (optional task) - Ellipses

Note: It is highly recommended that your read the Introduction before you start at this exercise.

The first law states that planets move in ellipses. An ellipse with centre (0,0) is the curve that goes through the points (x,y) which have the relation:

(x/a)² + (y/b)² = 1    (1)

Take a = 10 and b = 5 and compute various points (x,y) using formula (1). Connect the points to draw an ellipse. Repeat this, but now for a = 10 and b = 2.

Which features do the values a and b correspond to in the ellipse? Finally, what kind of figure does the ellipse become when a = 10 and b = 10?

Figure 1 shows an ellipse. The lines of length a and b are called the semi-major axis and semiminor axis, respectively. An ellipse has two focal points. The total distance from one focal point through any point on the ellipse to the other focal point is a constant 2a. This fact is used by gardeners to make flowerbeds with an elliptical shape. Put two pins on a sheet of paper. Connect the pins with a loose loop of thread. Now put a pencil on the paper inside the loop and move it around while keeping the thread stretched. An ellipse will appear with focal points at the location of the pins. As the distance between the pins decreases the ellipse will become rounder, and eventually form a circle in the extreme case when the distance between the pins is reduced to zero and they are placed at the same point.

Figure 1: An ellipse, where the semi-major and minor axes and the focal points (black dots) are shown. Planets orbit in ellipses around the Sun, which is at one focal point, as indicated.

The second law is illustrated in Figure 2.

Figure 2: Kepler’s second law says that a planet (the small dot on the ellipse) moves around the Sun (indicated by the encircled dot) in such a way that in equal amounts of time .t the line joining planet and Sun traverses equal amounts of area. The equal areas each have their own shading. The orbital velocity is higher when the planet is close to the Sun.

The orbital period of a planet is the time it takes it to make one full ellipse around the Sun. Kepler’s third law says that if you know two of the following three quantities: the period (P), the semi-major axis (a) and the total mass (m₁ + m₂) of Sun and planet together, you can compute the unknown one.