Constraints to the semimajor axis of outer particles with nodal librations by general relativity effects

Abstract

We study nodal librations of outer particles in the framework of the elliptical restricted three-body problem including generalrelativity (GR) effects. From an analytical treatment based on secular interactions up to quadrupole level, we derive equationsthat define the nodal libration region of an outer test particle, which depends on the physical and orbital parameters of the bodiesof the system. From this, we analyze how the GR constrains the semimajor axis of outer test particles that experience nodal librations under the effects of an inner planet around a single stellar component. Such an upper limit of the semimajor axis, which is called 𝑎2,lim , depends on the mass of the star 𝑚s, the mass 𝑚1, the semimajor axis 𝑎1, and the eccentricity 𝑒1 of the inner planet, and the eccentricity 𝑒2 of the outer test particle. On the one hand, our results show that the greater 𝑚1,𝑎1 , and 𝑒2 and the smaller 𝑚s, the greater the value of 𝑎2,lim. On the other hand, for fixed 𝑚s, 𝑚1, 𝑎1, and 𝑒2, 𝑎2,lim does not strongly depend on𝑒1, except for large values of such an orbital parameter. We remark that N-body experiments of particular scenarios that include GR show results consistent with the analytical criteria derived in the present research. Moreover, the study of hypothetical small body populations of real systems composed of a single star and an inner planetary-mass companion show that the GR effects can play a very important role in their global dynamics.