Determination and survey of periodic Trojan orbits in the restricted problem of three bodies

For the plane restricted problem of three bodies, new computational and iteration methods are employed to determine, on an IBM 650 electronic computer, a series of periodic orbits of the Trojan type. The resulting periodic solutions, established to a high degree of numerical accuracy, refute 's (1951, 1952, 1959) repeated claims of their nonexistence. Harmonic analysis is used to obtain the Fourier series representation of all the orbits. The convergence of the Fourier expansions is very satisfactory up to amplitudes of the order of those of the actual Trojan planets with the largest libration amplitudes. All the orbits computed appear to be stable, even the horseshoe-shaped periodic orbit enclosing both equilateral points and the collinear libration point opposite to Jupiter. The latter class of orbits, anticipated by Brown (1911), is of particular interest, because with their increasing amplitudes but decreasing periods these orbits link the equilateral points with the satellite region of Jupiter.