Abstrakt
The work presents a study of the 1:1 resonance case in a hierarchical quadruple stellar system of the 2+2 type. The resonance appears if orbital periods of both binaries are approximately equal.
It is assumed that both periods are significantly shorter than the period of principal orbit of one binary with respect to the other. In these circumstances, the problem can be treated as three independent Kepler problems perturbed by mutual gravitational interactions.
By means of canonical perturbation methods, the planar problem is reduced to a secular system with 1 degree of freedom involving a resonance angle (the difference of mean longitudes of the binaries) and its conjugate momentum (involving the ratio of orbital period in one binary to the period of principal orbit). The resonant model is supplemented with short periodic perturbations expressions, and verified by the comparison with numerical integration of the original equations of motion.
Estimates of the binaries periods variations indicate that the effect is rather weak, but possibly detectible if it occurs in a moderately compact system. However, the analysis of resonance capture scenarios implies that the 1:1 resonance should be exceptional amongst the 2+2 quadruples.