n-body Problem - Two-body Problem

Two-body Problem

If the common center of mass of the two bodies is considered to be at rest, each body travels along a conic section which has a focus at the center of mass of the system (in the case of a hyperbola: the branch at the side of that focus). The two conics will be in the same plane. The type of conic (circle, ellipse, parabola or hyperbola) is determined by finding the sum of the combined kinetic energy of two bodies and the potential energy when the bodies are far apart. (This potential energy is always a negative value; energy of rotation of the bodies about their axes is not counted here).

• If the sum of the energies is negative, then they both trace out ellipses.
• If the sum of both energies is zero, then they both trace out parabolas. As the distance between the bodies tends to infinity, their relative speed tends to zero.
• If the sum of both energies is positive, then they both trace out hyperbolas. As the distance between the bodies tends to infinity, their relative speed tends to some positive number.

Note: The fact that a parabolic orbit has zero energy arises from the assumption that the gravitational potential energy goes to zero as the bodies get infinitely far apart. One could assign any value to the potential energy in the state of infinite separation. That state is assumed to have zero potential energy (i.e. 0 joules) by convention.

See also Kepler's first law of planetary motion.