In recent years the author has derived and studied extensively exact solutions of Kerr geodesics in [Class. Quantum Grav. 21 4743-69 (2004; Zbl 1060.83006) and 22 4391–4424 (2005; Zbl 1097.83010)]. However, some important classes of possible orbits were left out in these works. This lack of results is filled in the present paper by studying the motion of a test particle in both, a non-spherical polar orbit and an equatorial non-circular timelike orbit in the presence of the cosmological constant \(\Lambda\).
In the first case the solution is applied to derive novel expressions for the Lense-Thirring effect (frame dragging), periapsis advance and the orbital period. The resulting formulae are expressed in terms of Appell’s first hypergeometric function \(F_{1}\), Jacobi’s amplitude function and Appell’s \(F_{1}\) and Gauss hypergeometric function, resp. Then, by assuming that the galactic centre is a rotating Kerr black hole (as pointed out by recent observations), the author applies these expressions in two situations: (a) for the particle stellar orbits near the outer horizon of the galactic centre black hole (strong field regime) (b) for the more distant (with respect to the outer horizon of the Kerr galactic black hole) observed orbits of S-stars in the central arcsecond of our galaxy (weak field regime).
In the second case, the solution is applied to obtain an exact closed form formula for the orbital period of a test particle in a non-circular equatorial motion around a Kerr black hole. Exact expressions for the periapsis advance and the orbital period are obtained in terms of Lauricella’s fourth hypergeometric function \(F_{D}\). As remarked by the author, these exact formulae in closed analytic form are important for the precise determination of the combined effects of the cosmological constant and the rotation of the central mass on the orbits of test particles.