Hodographs for Kepler Orbits

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
,
where 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
,
where is the energy of the planetary orbit. For an elliptical orbit, , so that .
For selected values of and , 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.

SNAPSHOTS

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DETAILS

The spherical components of velocity are converted to Cartesian coordinates using
,
,
.
Taking the time derivative of the orbital formula above and using the angular-momentum definition
,
we obtain the Cartesian velocity components
,
.
The relation
shows that the velocity vector traces out a circle of radius in velocity space centered at .
References
[1] E. I. Butikov, "The Velocity Hodograph for an Arbitrary Keplerian Motion," European Journal of Physics, 21(4), 2000 pp. 297–302. doi:10.1088%2F0143-0807%2F21%2F4%2F303.
[2] ThatsMaths. "Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph." (Oct 4, 2019) thatsmaths.com/2019/05/02/keplers-vanishing-circles-hidden-in-hamiltons-hodograph.
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