On the stability of the Neptune Trojans

The area of stable motion for fictitious Trojan asteroids around Neptune's equilateral equilibrium points is investigated with respect to the size of the regions and their shape, subject to the inclination of the asteroid's orbit. For this task, we used the results of extensive numerical integrations of orbits for a fine grid in initial conditions around the points L
4
and L
5
and analysed the stability of the individual orbits. Our basic dynamical model was the outer Solar system (Jupiter, Saturn, Uranus and Neptune) but for comparison reasons also simpler ones were tested. We integrated in our models the equations of motion for some 5 × 10
⁵
orbits of fictitious Trojans in the vicinity of the stable equilibrium points up to 10
⁹
yr. According to the three-dimensional model, the initial inclination of the asteroids' orbit was also varied in the range 0° < i < 60°. Using on one side a fine grid of initial conditions, the semimajor axis versus perihelion of the fictitious object and, on the other side, the proper eccentricity e
p
versus the libration width D
f
, we compiled stability maps separately for L
4
and L
5
. In addition, we computed the escape-times of the individual objects and plotted the number of escapers per time-interval of 5 × 10
⁶
yr for different initial inclinations. Finally, integrations of the equations of motion in different dynamical models shed light on the reason of the asymmetry of the stability behaviour of orbits close to the two equilateral equilibrium points of Neptune. For low-inclined Trojan orbits, the stability area around L
4
and L
5
disappeared after some 10
⁸
yr, and for larger inclinations of the Trojans the stability area survived for the time-interval of integration of 10
⁹
 yr. The largest stable regions exist for Neptune Trojans with 20° < i < 50°. The somewhat interesting asymmetry in the size and the shape of the preceding and following Lagrange points, which exist for Neptune Trojans, was confirmed, and was found to be caused mostly by the couple Saturn–Uranus.