We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution of canonical equations of long-period and short-period terms of the perturbed Hamiltonian of gravitational waves. We consider normal incident of plane gravitational wave and characteristic size of bound-two body system (earth's satellite or planet) is much smaller than the wavelength of the wave and the wave's frequency w n is much smaller than the particle's orbital p n . We construct the Hamiltonian of the gravitational waves in terms of the canonical variables ( ) , , , , , l g h L G H and we solve the canonical equations numerically using Runge-Kutta fourth order method using language MATHEMATICA V10. Taking Jupiter as practical example we found that there are long period perturbations on , ω Ω and i and not changing with revolution and the short period perturbations on , a e and M changing with revolution during the interval of time ( 0 t t − ) which is changing from 0 4π → .

For certain values of semi-major axis and eccentricity, orbit plane precession caused by Earth oblate is synchronous with the mean orbital motion of the apparent Sun (a sun-synchronism). Many forces cause slow changes in the inclination and ascending node of sun-synchronous orbits. In this work, we investigate the analytical perturbations due to the direct solar radiation pressure and gravitational waves effects. A full analytical solution is obtained using technique of canonical Lie-transformation up to the order three in 2 J (the oblateness of the Earth). The solar radiation pressure and gravitational waves perturbations cause second order effects on all the elements of the elliptic orbit (the eccentricity, inclination, ascending node, argument of perigee, and semi-major axis) consequently these perturbations will cause disturbance in the sun-synchronism. Also we found that the perturbation or the behavior of gravitational waves almost the same as the perturbation or the behavior of solar radiation pressure and their coupling will incorporate the sun-synchronism through the secular rate of the ascending node precession.

The optimizing total velocity increment v ∆ needed for orbital maneuver between two elliptic orbits with plane change is investigated. Two-impulse orbital transfer is used based on a changing of transfer velocities concept due to the changing in the energy. The transferring has been made between two elliptic orbits having a common centre of attraction with changing in their planes in standard Hohmann transfer with the terminal orbit which is elliptic orbit and not circular. We develop a treatment based on the elements of elliptic orbits 1 1 , a e , 2 2 , a e and , programming code of MATHEMATICA V10, with no condition on the eccentricity or the semi-major axis of the initial, transformed, and the final orbits. We find that there are constrains on the transfer angles 1 θ and α . For α it must be between 40˚ and 160˚, and there is no solution if α is less than 40˚ and bigger than 160˚ and 1 θ takes the values less than 40˚. The minimum total velocity increments obtained at the value of 1 θ less than 25 o and α equal to 160˚. This is an interesting result in orbital transfer problem in which the change of orbital plane is necessary for the transferring.

In this work we study the perturbation and the change in the orbital elements due to the earth's magnetic field and the gravitational waves. The acceleration components are derived in the radial, transverse to it and normal to the orbital plane. The equation for the rates of variation of the elements is formed and solved to find the secular variation in the element for polar and equatorial satellites.

This work deals with the numerical solution of the gravitational waves effects on the orbital elements of the planets in case of commensurability between the wave's frequency n g and the planet's mean motion n p. Taking Mercury and Pluto as practical examples for low frequency and high frequency, the variations of the orbital elements of Mercury due to resonance of gravitational wave are different and small than the perturbation on Pluto. The amount of changing in the orbital elements under the effects of gravitational waves is different from planet to planet according to the planet's mean motion n p. For low frequency n g , the secular variation in orbital elements will be negative (i.e. decreasing) in the inclination, semi-major axis and the eccentricity (i, a, e) like as Pluto. For high frequency n g like Mercury, the secular variation in all the orbital elements will be positive (i.e. increasing). The perturbation on all the orbital elements of two planets is changing during each revolution except the eccentricity e of Mercury and the mean anomaly M of Mercury and Pluto during the time.