# Hodographs for Kepler Orbits

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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.

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Contributed by: S. M. Blinder (October 2019)

Open content licensed under CC BY-NC-SA

## 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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