We study the pseudo-synchronous orbital motion of a binary system on the main sequence. The equations of the pseudo-synchronous orbit are derived up to O(e4) where e is the eccentricy of the orbit. We integrate the equations to present their solutions. The theoretical results are applied to the evolution of the orbit and spin of the binary star Y Cygni, which has a current eccentricity of e0 = 0.142. We tabulate our numerical results for the evolution of the orbit and spin per century. The numerical results for the semi-major axes and rotational angular velocities in the evolutional time scales of three stages (synchronization, circularization, and collapse time scale) are also tabulated. Synchronization is achieved in about 5 × 103 years followed by circularization lasting about 1 × 105 years before decaying in 2 × 105 years.

In this study, I calculate the past and future dynamical states of the Earth-Moon system by using modified Lambeck’s formulae. I find that the ocean tidal effect must have been smaller in the past compared to its present amount. Even though the Moon is already in the spin-orbit synchronous rotational state, my calculation suggest that it will not be in geostationary rotational state in the next billion years or so. This is due to the associated Earth’s obliquity increase and slow retardation of Earth’s spin and lunar orbital angular velocities. I also attempt to calculate the precessional period of the Earth in the future. To avoid uncertainties in the time scale, the future state is described by using the Earth-Moon distance ratio as independent parameter. Effects due to solar tidal dissipation are included in all calculations.

This paper intends to highlight the effect of the third-body in an inclined orbit on a spacecraft orbiting the primary mass. To achieve this goal, a new origin of coordinate is introduced in the primary and the X-axis toward the node of the spacecraft. The disturbing function is expanded up to the second order using Legendre polynomials. A double-averaged analytical model is exploited to produce the evolutions of mean orbital elements as smooth curves.