An invariant relation in the elliptic restricted problem of three bodies. II - Trojan asteroids

A previously derived invariant relation is reduced to an approximate integral valid in the case of the sun-Jupiter-Trojan system. Nonintegrable terms neglected contain the Jupiter-sun mass ratio as a factor and are of the second order in eccentricities or the fourth order in the relative inclination. The approximate integral is then reduced to the ecliptic as a reference plane, allowing the 'Jacobi constant' J to be calculated to four decimals for any Trojan. The Trojan's orbital inclination appears to be the main factor determining the value of J, but Jupiter's orbital eccentricity has to be taken into account to achieve this accuracy. The values of J turn out to be between 2.6769 and 2.9911 for 15 well-known Trojans, and between 2.9055 and 3.0012 for 15 noncataloged Trojans, relative to the ecliptic and equinox of 1950, with an error less than + or - 0.0002. By calculating J from osculating orbital elements for the same Trojan at two or more epochs, its constancy is checked empirically in 13 cases.