PERIODIC MOTIONS AROUND PULSATING STARS

The periodic motion of a test particle (dust, grain, or a larger body) around a pulsating star with a luminosity oscillation of small amplitude (featured by a small parameter B) is being studied. The perturbations of all orbital elements are determined to first order in B, by using Delaunay-type canonical variables and a method whose bases were put forth by von Zeipel. According to the value of the ratio oscillation frequency/dynamic frequency, three possible situations are pointed out: nonresonant (NR), quasi-resonant (QR), and resonant (R). The solution of motion equations shows that only in the (QR) and (R) cases there are orbital parameters (argument of periastron and mean anomaly) affected by secular perturbations. These solutions (which indicate a secularly stable motion in a first approximation) are valid over prediction times of order B-1 in the (NR) case and B-1/2 in the (QR) and (R) cases. The theory may be applied to various astronomical situations.