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Hodographs for Kepler Orbits

​
semi-latus rectum p
1
eccentricity e
0.5
Kepler orbit
hodograph
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
L
is the orbital angular momentum,
M
is the solar mass,
m
is the planetary mass and
G
is the gravitational constant. It is assumed that
M≫
m
. The eccentricity of the orbit is given by
e=
1+
2E
2
L
2
G
2
M
3
m
,
where
E
is the energy of the planetary orbit. For an elliptical orbit,
E<0
, so that
0⩽e<1
.
For selected values of
p
and
e
, the Kepler ellipse and the corresponding hodograph are shown. A set of velocity vectors for evenly spaced values of the true anomaly
θ
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.

Details

The spherical components of velocity are converted to Cartesian coordinates using
v=
r
r
+r
θ
θ
,
r
=cosθ
x
+sinθ
y
,
θ
=-sinθ
x
+cosθ
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
1
p
in velocity space centered at
0,
e
p
.

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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