We investigate the non-resonant , 3-D ( spatial ) model of the hierarchical system composed of point-mass stellar ( or sub-stellar ) binary and a low-mass companion ( a circumbinary planet or a brown dwarf ) .
We take into account the leading relativistic corrections to the Newtonian gravity . The secular model of the system relies on the expansion of the perturbing Hamiltonian in terms of the ratio of semi-major axes \alpha , averaged over the mean anomalies . We found that the low-mass object in a distant orbit may excite large eccentricity of the inner binary when the mutual inclination of the orbits is larger than about of 60 deg . This is related to strong instability caused by a phenomenon which acts similarly to the Lidov-Kozai resonance ( LKR ) . The secular system may be strongly chaotic and its dynamics unpredictable over the long-term time scale .
Our study shows that in the Jupiter– or brown dwarf– mass regime of the low-massive companion , the restricted model does not properly describe the long-term dynamics in the vicinity of the LKR .
The relativistic correction is significant for the parametric structure of a few families of stationary solutions in this problem , in particular , for the direct orbits configurations ( with the mutual inclination less than 90 degrees ) .
We found that the dynamics of hierarchical systems with small \alpha \sim 0.01 may be qualitatively different in the realm of the Newtonian ( classic ) and relativistic models .
This holds true even for relatively large masses of the secondaries .