## Ellipse

Previous: Escape Velocity

An ellipse is one of four possible types of conic section. The others are the circle, the parabola, and the hyperbola.

conic section is a shape obtained from cutting a cone.

Types of conic sections (from wikipedia, modified)

There are two parameters controlling the shape of these guys. Let’s focus on the ellipse, which is the one interesting regarding planetary motion.

Ellipse

The two parameters that control the shape of the elipse are a (semimajor axis, which can have any positive value) and e (eccentricity, which for an ellipse 0 < e < 1). So we can calculate the other stuff as a function of a and e. For example:

Kepler suggested that the planets orbit in ellipses, with the Sun in one of the focus, the principal focus, while the other focus is empty.

Elliptic trajectory of planet around the Sun according to the Kepler model

It is useful to express the position of a planet as a function of θ and r (instead of cartesian coordinates).

The point where the planet is closest to the Sun is for θ = 0º (perihelion). The point where the planet is farthest from the Sun is for θ = 180º (aphelion).

The other conic sections have different ranges for e (a simply indicates the size):

Circle: e = 0
Ellipse: 0 < e < 1
Parabola: e = 1
Hyperbola: e > 1

Next: Kepler’s Laws Revisited