Kepler orbits are conic sections, most notably ellipses for stable periodic motion of a planet around the Sun. A lesser-known property is the motion of the associated tangential velocity vector, which traces out a circular orbit in velocity space. Hamilton (1864) first introduced the term hodograph to denote this motion.
A Kepler orbit in plane polar coordinates is described by
Here the semi-latus rectum is given by
is the orbital angular momentum,
is the solar mass,
is the planetary mass and
is the gravitational constant. It is assumed that
. The eccentricity of the orbit is given by
is the energy of the planetary orbit. For an elliptical orbit,
, so that
For selected values of
, the Kepler ellipse and the corresponding hodograph are shown. A set of velocity vectors for evenly spaced values of the true anomaly
is shown by numbered red arrows, with corresponding values pertaining to the orbit and the hodograph.
The magnitude of the velocity is a minimum at the aphelion, numbered 1, and a maximum at the perihelion, numbered 7.