A method for stability determinations in the elliptic restricted problem

Abstract

In the ordinary restricted problem of three bodies, the first-order stability of planar periodic orbits may be determined by means of their characteristic exponents, as derived from the condition of a vanishing determinant for the coefficients of an infinite system of homogenous linear equations associated with the exponential series solutionu, v representing any initially small oscillations about the periodic solutionx, y. In the elliptic restricted problem, periodic solutions are possible only for periods which are equal to, or integral multiples of, the periodP of the elliptic motion of the two primary masses. It is shown that the infinite determinant approach to the determination of the characteristic exponents can be extended to the treatment of superposed free oscillations in the elliptic problem, and that in generaltwo exponents appear in any complete solutionu, v for eachone existing in the corresponding ordinary restricted problem. The value of each exponent depends on a series proceeding in even powers of the eccentricitye of the relative orbit of the two primaries, in addition to its basic dependence on the mass ratio μ. For stable periodic orbits, the oscillation frequenciesn1 (μ,e2),n2 (μ,e2) associated with these two exponents tend, withe→0, to certain limiting valuesn1 (μ),n2(μ), which differ from each other by the amount of the frequencyN=2π/P of the orbital motion of the primaries. One of the two frequencies, sayn1(μ), is identical with the frequency of the corresponding oscillations in the ordinary restricted problem, while the second one gives rise to oscillations only in the elliptic restricted problem, withe≠0.