ABSTRACT 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 ng and the planet’s mean motion np. 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 np. For low frequency ng, 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 ng 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.
Cite this paper
Youssef, M. (2018) Numerical Studies of Resonance and Secular Effects of Gravitational Waves. World Journal of Mechanics, 8, 182-191. doi: 10.4236/wjm.2018.85013.
 Fairhurst, S., Guidi, G. and Hello, P. (2010) Current Status of Gravitational Wave Observation. General Relativity and Gravitation, 43, 387. https://doi.org/10.1007/s10714-010-1009-1
 Prince, T.A. (2010) The Promise of Low-Frequency Gravitational Wave Astronomy for LISA International Science Team. The Astronomy and Astrophysics Decadal Survey, Science White Paper, No. 238.
 Flanagan, E. and Hughes, S. (2005) The Basics of Gravitational Wave Theory. New Journal of Physics, 7, 204. https://doi.org/10.1088/1367-2630/7/1/204
 Rudenko, V.N. (1975) Test Bodies under the Effect of Gravitational Radiation. Soviet Astronomy, 19, 270.
 Thorne, K.S. and Braginsky, V.B. (1976) Gravitational Wave Burst from the Nuclei of Distant Galaxies and Quasars. The Astrophysical Journal Letters, 204, L1. https://doi.org/10.1086/182042
 Mashoon, B. (1978) On Tidal Resonance. Astrophysical Journal, 223, 285. https://doi.org/10.1086/156262
 Mashoon, B. (1987) Wave Propagation in a Gravitational Field. Physics Letters A, 122, 299-304. https://doi.org/10.1016/0375-9601(87)90829-2
 Futamase, T. and Matsuda, T. (1979) Resonance between Primordial Gravitational Waves and Gravitationally Bound System. Progress of Theoretical Physics, 61, 86-93. https://doi.org/10.1143/PTP.61.86
 Turner, M.S. (1979) Influence of a Weak gravitational Wave on a Bound System of Two Point-Masses. Astrophysical Journal, 233, 685. https://doi.org/10.1086/157429
 Nelson, L.A. and Chau, W.Y. (1982) Orbital Perturbations of a Gravitationally Bound Two-Body System with the Passage of Gravitational Waves. Astrophysical Journal, 254, 735. https://doi.org/10.1086/159785
 Ivanshchenko, A.V. (1987) The Variation of the Keplerian Elements of a Planetary Orbit under the Action of a Gravitational Wave. Soviet Astronomy, 31, 76.
 Youssef, M.H. and Ahmed, M.K. (1997) Analytical Effects of Gravitational Waves on the Motion of an Artificial Satellite. Dynamics and Astrometry of Natural and Artificial Celestial Bodies, 431. https://doi.org/10.1007/978-94-011-5534-2_61
 Lorenzo, I. (2014) Orbital Effects of a Monochromatic Plane GW with Ultra-Low Frequency Incident on a Gravitationally Bound Two-Body System. ScienceOpen Research, V2, 1-13.
 Bertotti, B. (1973) Is the Solar System Gravitationally Closed? The Astrophysical Journal Letters, 14, 51.
 Youssef, M.H. (2017) Semi-Analytical Theory of the Mean Orbital Motion Due to the Effect of Gravitational Waves. European Journal of Scientific Research, 147, 342-350.
 Youssef, M.H. (2017) Short-Period and Long-Period Effects of Weak Gravitational Waves. International Journal of Antimicrobial Agents, 7, 230-237. https://doi.org/10.4236/ijaa.2017.73019
 Ismael, M.N. and Saad, N.A. (2011) The Effects of Gravitational Waves on the Orbital Elements of the Planets. The Open Astronomy Journal, 4, 1-5. https://doi.org/10.2174/1874381101104010001
 Roy, A.E. (1965) The Foundations of Astrodynamics. The Macmillan, New York.