Hodographs for Kepler Orbits
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
r=
p
1-ecosθ
Here the semi-latus rectum is given by
p=
2
L
GM
2
m
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
L
M
m
G
M≫
m
e=
1+
2E
2
L
2
G
2
M
3
m
where is the energy of the planetary orbit. For an elliptical orbit, , so that .
E
E<0
0⩽e<1
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.
p
e
θ
The magnitude of the velocity is a minimum at the aphelion, numbered 1, and a maximum at the perihelion, numbered 7.
Details
Details
The spherical components of velocity are converted to Cartesian coordinates using
v=+r
r
r
θ
θ
r
x
y
θ
x
y
Taking the time derivative of the orbital formula above and using the angular-momentum definition
L=m
2
r
θ
we obtain the Cartesian velocity components
v
x
sinθ
p
v
y
e-sinθ
p
The relation
2
v
x
2
v
y
e
p
1
p
shows that the velocity vector traces out a circle of radius in velocity space centered at .
1
p
0,
e
p
References
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.
Permanent Citation
Permanent Citation
S. M. Blinder
"Hodographs for Kepler Orbits" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/HodographsForKeplerOrbits/
Published: October 17, 2019