Eccentricity of an Ellipse

Eccentricity (e) is the ratio of the focal distance to the semi-major axis length, determining how stretched an ellipse becomes. When e equals zero, the ellipse is a perfect circle. As e increases toward one, the ellipse elongates along its major axis, creating increasingly elongated shapes seen in celestial orbits.

An ellipse can be traced by anchoring a string at two fixed points (foci) and keeping it taut with a pencil. As the pencil moves around the pins, it maintains a constant combined distance to both foci, defining the ellipse's geometric path and demonstrating its fundamental property.

The focal distance equals the semi-major axis multiplied by eccentricity (c = a × e). The semi-minor axis is then calculated using the Pythagorean theorem, which relates it to both the semi-major axis and focal distance, completing the ellipse's geometric definition.

Halley's Comet travels along an elliptical path with eccentricity of 0.967, placing the Sun at one focus. Its major axis measures approximately 35.88 astronomical units. This highly eccentric orbit causes the comet to vary dramatically in distance from the Sun, ranging from perihelion to aphelion.

The ellipse is centered at the origin with its major axis aligned along the x-axis. The semi-major axis (a) and semi-minor axis (b) values are substituted into the standard form equation to model the orbit. For Halley's Comet, a = 17.95 AU and the focal distance is calculated from eccentricity.

Perihelion (closest approach) is approximately 0.59 AU, calculated as a − c. Aphelion (farthest point) is approximately 35.29 AU, calculated as a + c. This substantial variation illustrates the pronounced eccentricity of Halley's orbit and demonstrates how polar equations of conics model celestial mechanics.

Eccentricity uniquely characterizes each conic section. For ellipses, e ranges from zero to one; when e = 0, it becomes a circle. Other conic sections like parabolas and hyperbolas have different eccentricity ranges, making e a fundamental parameter in classifying and analyzing all conic geometries.