Orbital Precession in the Restricted Three-Body Problem: Exact Representations

Analytical representations of the rate of apsidal precession in the planar elliptical restricted three-body problem are considered in the case when the orbit of the disturbing body is external with respect to the orbit of the test particle. The analytical expressions are compared with the numerical data obtained for the apsidal precession rate in the form of a power series with a parameter equal to the ratio of the semi-major axis of the orbit of the test particle to that of the disturbing planet. It is shown that the analytical expressions for the rate of apsidal precession of the particle are reliable only at distances not close to the instability zone near the orbit of the disturbing planet. Near the Wisdom gap, the linear secular theory is no more valid. An empirical correction formula is proposed to calculate the apsidal procession rate at relatively high (however less than 0.5) eccentricities of the particle and disturbing planet. The proposed formulas are applied to describe the precession of orbits in real exoplanetary systems.