Ellipse Geometry Explained
Understanding ellipses is essential to understanding Kepler's First Law. This page covers the geometry in detail.
Definition
An ellipse is the set of all points in a plane such that the sum of the distances from any point on the ellipse to two fixed points (the foci) is constant. If you stick two pins in a board, loop a string around them, and pull the string taut with a pencil, the curve you draw is an ellipse. (See the hands-on activity.)
Key Parameters
- Semi-major axis (a): Half the longest diameter of the ellipse. This is the "average radius" of a planetary orbit and appears in Kepler's Third Law.
- Semi-minor axis (b): Half the shortest diameter.
- Foci (F1, F2): The two special interior points. For a planetary orbit, the Sun is at one focus.
- Center: The midpoint between the two foci (not where the Sun is).
- Eccentricity (e): A number between 0 and 1 measuring how elongated the ellipse is. e = 0 is a circle; e close to 1 is very elongated.
Mathematical Relationships
b² = a² - c²
e = c / a
Perihelion distance = a(1 - e)
Aphelion distance = a(1 + e)
Conic Sections
An ellipse is one of four conic sections—shapes produced by slicing a cone at different angles. The others are the circle (a special ellipse with e = 0), the parabola (e = 1), and the hyperbola (e > 1). Comets can follow parabolic or hyperbolic paths, but bound orbits (planets, moons, most satellites) are always ellipses.
Eccentricities of the Planets
| Planet | Eccentricity |
|---|---|
| Mercury | 0.2056 |
| Venus | 0.0068 |
| Earth | 0.0167 |
| Mars | 0.0934 |
| Jupiter | 0.0484 |
| Saturn | 0.0539 |
| Uranus | 0.0473 |
| Neptune | 0.0086 |
See the full solar system data table for more orbital parameters.