The Trojan asteroids lie in Jupiter's orbit equidistant from Jupiter
and the sun. They are an example of what is known as the restricted three-body
problem: the motion of a small body, an asteroid, under the influence of two massive
bodies whose motion is not affected by the presence of
the asteroid. The gravitational forces exerted by the sun and
Jupiter combine to give the asteroid a stable orbit with Jupiter's period of
rotation. (Some trigonometry is needed to show that the Trojan site forms an
equilateral triangle with the sun and Jupiter). In the applet, the
massive bodies executing circular orbits about
their common centre of mass can have mass ratios ranging from 9 up to the
value for sun-earth system. It is convenient to view the motion of the asteroid
in a frame rotating with the massive bodies. This is a non-inertial frame,
and both centrifugal
( - m **w** x (**w** x
**r**) ) and Coriolis
( - 2 m **w** x **v** ) forces appear in
Newton's Law. Viewed in the rotating frame, the asteroid in a stable orbit
remains at rest at the point where the net gravitational attraction
balances the repulsive centrifugal force. Points with this property show up
clearly on a color plot of the effective potential in the rotating frame.
This potential is the sum of three negative terms: a centrifugal potential
that varies as the square of the distance from the centre of mass (much like
an "upside down" harmonic potential), and two gravitational wells centered on
the massive bodies. Points where the effective potential falls within a
specified range share a common color,
and the scale factor is adjusted to place the Trojan site near a color
boundary. Such plots usually use linear or logarithmic scales, but here an
inverse scale is used to give suitable numbers and widths
of the color bands (this produces color bands of constant width near
a massive body).
The plot is centred on the centre of mass of the system, with the more
massive body (*M*) on the left and the less massive body (*m*)
a distance *d* to its right. Choose a mass ratio, and press either "Tro"
for an orbit starting near a Trojan site or "Sad" for an orbit starting near
a saddle point (a point of equilibrium on the centreline).

The most prominent features of the potential plots are the red regions
surrounding the Trojan sites. As the red regions are the crests of hills,
it is not obvious that a Trojan orbit can be stable. As the mass ratio
changes, the shape of a crest changes, but it remains a crest.
It can be shown analytically that orbits for
*M/m* greater than 25 are stable, but the calculation requires more than
the trigonometry needed to find the equilateral triangle. The numerical study
of stability, on the other hand, requires only the Feynman algorithm for
velocity-dependent forces (such as the Coriolis force). Each plot shows the
path taken by the asteroid when it is released from
rest (in the rotating frame) with a *y*-displacement of *d*/400
(about 1/3 of a pixel) from the equilibrium site. For a Trojan site, the
displacement puts the asteroid on the far side of the hill from the centre
of mass, and initially it moves outward as we would expect. As soon as it
acquires a significant velocity, the Coriolis force deflects it to the right
(in a frame rotating counterclockwise). What happens then depends on the
mass ratio. For low mass ratios it spirals outward. For intermediate mass
ratios it traces out loops close to the Trojan site. At higher mass ratios
the potential crest becomes an elongated ridge, and the orbit bumps its way
around it.

On 9 Nov 02 my car radio informed me that an asteroid the size of a football field had recently been found to share the earth's orbit. In a frame rotating with the earth, it was said to have a horseshoe-shaped orbit. As I drove along, I visualized the crest I had already plotted for Jupiter turning into a ridge that circles the sun, and the asteroid bouncing along the crest in a horseshoe orbit until it turns at a low point near the earth. That evening I extended the Trojan applet to include the sun/earth mass ratio, and found the horseshoe orbit plotted here. It has a period of 160 years.

If you select the "Sad" option, the program finds the three
equilibrium points that lie on the *M/m* axis, one outside each mass,
and one between them. They are all saddle points: the region they are in is
bounded by different colors in the *x* and *y* directions. The axis
and the force are shown in white on the plot, and a bisection algorithm is
used to locate the equilibrium points. If a Trojan hill can produce a stable
orbit there is no reason to assume that a saddle point cannot. The
program tests stability using the same *y*-displacement used at a
Trojan site. The saddle points on either side of *m* are unstable for
all mass ratios, but the one to the left of *M* generates a horseshoe orbit
for the sun/Jupiter and sun/earth systems. (Note that the other two saddle
points are not plotted for these systems, and that the PostScript program
does not deal with any of the saddle points).

The Trojan and horseshoe orbits are closely-related examples of
stable orbits: closed curves that rotate about the centre of mass in sync
with *m*. Horseshoe orbits can be generated for both the sun/Jupiter and
sun/earth systems with a *d*/400 *y*-displacement from the left
saddle point. A Trojan orbit can be generated for the sun/earth system with a
*d*/1600 *y*-displacement from the Trojan site. A horseshoe orbit
can be generated for the sun/Jupiter system with a *d*/70 (2 pixel)
*y*-displacement from the Trojan site. (The size of the displacement
needed can be seen on the horseshoe orbit generated from the saddle point).