The Geometry of Orbits
Conic Sections

All ballistic trajectories are defined by simple conic sections.

A ballistic trajectory is simply the path taken by an object (planet, comet, spacecraft etc.), as it moves through space without any force acting on the object, except gravity.

A conic section is, as the name implies, a section taken from a right circular cone.

Eccentricity

Eccentricity is the term which quantifies the shape of a conic section. The eccentricity of a conic section is a number which uniquely characterizes the shape of a conic section, without reference to the actual size of the conic section. For example, all circles have an eccentricity equal to 0, regardless of their size.

Examples

1. Circular Orbit:
eccentricity = 0


Transparency Master 		The circle is simply the perimeter of a slice taken at right angles to the axis of symmetry of a right circular cone.

In a sense, it is this property that defines a right circular cone, because if the section were not a circle, then the cone would not have been a right circular cone.

2. Elliptical Orbits:
eccentricity < 1


Transparency Master 		An ellipse is defined by the perimeter of a tilted slice which intersects opposite sides of a right circular cone.

The ellipse is the most common path that bodies acquire when orbiting another object in space.

The orbits of all the planets are ellipses.

3. Parabolic Trajectories:
eccentricity = 1


Transparency Master 		A parabola is defined by the perimeter of a slice through a right circular cone such that the plane of the slice is exactly parallel to one side on the cone.

Parabolic trajectories (like perfectly circular orbits) are rare since they form the transition between the ellipse and the hyperbola.

4. Hyperbolic Trajectories:
eccentricity > 1


Transparency Master 		When a slice is taken through a right circular cone such that its slope exceeds the slope of the cone, the perimeter of the slice defines an hyperbola.

Objects launched into space with sufficient energy to exceed the gravitational energy binding the object to the Earth will follow hyperbolic trajectories.

A large number of comets which appear in the inner solar system follow hyperbolic paths. They are one-time visitors to the Sun. Their energy is enough to carry them into interstellar space, never to return to our solar system again.

The hyperbola is an interesting case. Mathematically it intersects the cone's symmetrical image, hence the hyperbola has two identical components. Nevertheless, an object's trajectory is limited to only one lobe of the complete hyperbola and can never reach the second lobe.

Student Activity

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