Asteroids in the inner solar system - I. Existence

Ensembles of in-plane and inclined orbits in the vicinity of the Lagrange p oints of the terrestrial planets are integrated for up to 100 Myr. The inte grations incorporate the gravitational effects of the Sun and the eight pla nets (Pluto is neglected). Mercury is the least promising planet, as it is unable to retain tadpole orbits over 100-Myr time-scales. Mercurian Trojans probably do not exist, although there is evidence for long-lived, corotati ng horseshoe orbits with small inclinations. Both Venus and the Earth are m uch more promising, as they possess rich families of stable tadpole and hor seshoe orbits. Our survey of Trojans in the orbital plane of Venus is under taken for 25 Myr. Some 40 per cent of the survivors are on tadpole orbits. For the Earth, the integrations are pursued for 50 Myr. The stable zones in the orbital plane are larger for the Earth than for Venus, but fewer of th e survivors (similar to 20 per cent) are tadpoles. Both Venus and the Earth also have regions in which inclined test particles can endure near the Lag range points. For Venus, only test particles close to the orbital plane (i less than or similar to 16 degrees) are stable. For the Earth, there are tw o bands of stability, one at low inclinations (i less than or similar to 16 degrees) and one at moderate inclinations (24 degrees less than or similar to i less than or similar to 34 degrees). The inclined test particles that evade close encounters are primarily moving on tadpole orbits. Two Martian Trojans (5261 Eureka and 1998 VF31) have been discovered over the last dec ade and both have orbits moderately inclined to the ecliptic (20 degrees .3 and 31 degrees .3 respectively). Our survey of in-plane test particles nea r the Martian Lagrange points shows no survivors after 60 Myr. Low-inclinat ion test particles do not persist, as their inclinations are quickly increa sed until the effects of a secular resonance with Jupiter cause destabiliza tion. Numerical integrations of inclined test particles for time-spans of 2 5 Myr show stable zones for inclinations between 14 degrees and 40 degrees. However, there is a strong linear resonance with Jupiter that destabilizes a narrow band of inclinations at similar to 29 degrees. Both 5261 Eureka a nd 1998 VF31 lie deep within the stable zones, which suggests that they may be of primordial origin.