# Elliptical orbits

• I
How is it mathematically proven that gravitational orbits are elliptical?

phyzguy
Newton showed in the Principia in 1687 that bodies orbiting under the influence of a 1/r^2 force will move in elliptical orbits. This Wikipedia page describes the argument, both Newton's derivation and a more modern derivation.

gneill
Mentor
Note that in general any conic section is a solution to the differential equation of motion in an inverse square field. Thus "orbits? can be circular, elliptical, hyperbolic, parabolic, and even straight lines (a so-called "degenerate orbit" where the object just drops straight into the Sun).

Orodruin
Staff Emeritus