The Trojan Asteroids

Introduction

The Trojan asteroids are a special part of the minor bodies of the solar system. They orbit sixty degrees ahead and behind a planet in two of the Lagrangian points that exist between two massive bodies. Jupiter has many and Mars has one known Trojan asteroid. One of the more interesting ways of visualizing the orbits of Trojan asteroids is to display the orbit in the Sun-planet rotating reference frame. A rotating reference frame means that, in plotting the orbit, the Sun-planet line is fixed. The eccentricity of the planet (if unaccounted for) causes the planet to form a short line. The L4 region is known as the Greek camp, while the L5 region is known as the Trojan camp.

          A very simple diagram of the Sun - planet rotating frame.

                                          L4





                       Sun ----------------------- planet





                                          L5

The asteroids do not remain perfectly still at the Lagrangian points, but actually appear to "orbit" the Lagrangian point in the rotating frame. Some of the more interesting orbits are plotted in detail below and show tremendous variety and interesting shapes. Folding mythology into the system over the years, two distinct camps were formed: one Greek, the other Trojan. There are a few spies in each camp. This author also has designated neutrals which are those asteroids which have yet to be named, thus indicating their allegiance.

The Integrations

The Bulirsch-Stoer method is used to integrate the equations of motion forward in time by 1500 years. The standard heliocentric equations of motion for Cowell's method are integrated. Six times N equations must be solved, where N is the number of perburbing objects plus one object of interest (the Trojan minor planet). In this case, we used used 12 perturbing objects (Mercury, Venus, Earth+Moon, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto, Ceres, Pallas, and Vesta) plus one of interest. This gives a total of 78 equations that must be solved for each step. The constant six comes from the three heliocentric coordinates (x,y,z) and their velocities (xdot, ydot, zdot) that describe an objects basic state.

The initial conditions and beginning epoch of integration (JD 2452200.5) for all objects were taken from the Ephemerides of Minor Planets for the year 2001.

Of the 173 Trojan asteroids integrated, 37 had interesting looking orbits as viewed in the heliocentric ecliptic (x,y) plane. Images of those follow. The points where the minor planet was above the plane of the ecliptic are colored green, while when below the points are colored red. Only the region around the minor planet is plotted in order to maintain detail in the plots. The scale is in astronomical units (A.U.).

This page is http://tycho.usno.navy.mil/trojan.html
(17 June 2001)

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