_{1}

^{*}

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.

All methods developed to detect gravitational waves depend more or less on the fact that the maximum variation in the particle separation occurs if the particles are located in XY-plane. The effect of gravitational waves is too small. A direct measurement of its physical properties is still lacking [^{4} Hz are the targets of several detectors like LIGO, TAMA, VIRGO, etc. Now the very low frequency is concerned 10^{−9} - 10^{−7} Hz dealt with it [^{th} order is used to obtain the second order effect.

The wave creates a field of variable accelerations of the type

F x = h 1 x + h 2 y F y = h 2 x − h 1 y F z = 0 (1)

where F_{x}, F_{y}, and F_{z} are the components of the acceleration vector of normal incident of plane gravitational wave in (x, y, z) coordinates and

h 1 = 1 2 ∂ 2 h 11 ∂ t 2 , h 2 = 1 2 ∂ 2 h 12 ∂ t 2 (2)

The two polarized components of the transverse GW are

h 11 = h + cos ( n g t + α 1 ) (3)

h 12 = h × cos ( n g t + α 2 ) (4)

where n_{g} is the frequency of the wave, α_{1} and α_{2} are the phase difference, h_{+} and h_{×} are the amplitude of the wave in the two orthogonal directions in the transverse plane. Substituting Equations (3) and (4) into (2) and then into (1), therefore the acceleration components are

F x = β cos ( n g t + α 1 ) x + γ cos ( n g t + α 2 ) y (5)

F y = γ cos ( n g t + α 2 ) x − β cos ( n g t + α 1 ) y (6)

F z = 0 (7)

where

β = − 1 2 n g 2 h + and γ = − 1 2 n g 2 h × (8)

Regarding the estimates of the values of the frequencies and amplitudes of gravitational waves from different sources, we can fairly assume that β and γ are of order the eccentricity of the elliptic orbit (e). Now we express the components of the acceleration in the directions S, T, W along the unit vectors P ^ , Q ^ and W ^ in the direction of r, normal to r in the orbital plane and normal to the orbital plane respectively as shown in

Therefore we have

S = r { A 1 + A 2 cos 2 ( f + ω ) + A 3 sin 2 ( f + ω ) } (9)

T = r { − A 2 sin 2 ( f + ω ) + A 3 cos 2 ( f + ω ) } (10)

W = r { B 1 cos ( f + ω ) + B 2 sin ( f + ω ) } sin i (11)

where

A 1 = 1 2 sin 2 i { h 1 cos 2 Ω + h 2 sin 2 Ω } (12)

A 2 = 1 + cos 2 i 2 { h 1 cos 2 Ω + h 2 sin 2 Ω } (13)

A 3 = cos i { − h 1 sin 2 Ω + h 2 cos 2 Ω } (14)

B 1 = h 1 sin 2 Ω − h 2 cos 2 Ω (15)

B 2 = cos i { h 1 cos 2 Ω + h 2 sin 2 Ω } (16)

i, ω, and f are the inclination, longitude of node, argument of perigee and the true anomaly of an orbit respectively. The Gauss form of Lagrange’s planetary equations is (Roy, 1965) [

d a d t = 2 e sin f n 1 − e 2 S + 2 a 1 − e 2 n r T (17)

d e d t = 1 − e 2 sin f n a S + 1 − e 2 n a 2 e a n − r 2 r T (18)

d i d t = r cos ( f + ω ) n a 2 1 − e 2 W (19)

d Ω d t = r sin ( f + ω ) n a 2 1 − e 2 sin i W (20)

d ω d t = − 1 − e 2 cos f n a e S + ( 1 + r p ) ( 1 − e 2 n a e sin f T − r sin ( f + ω ) n a 2 1 − e 2 cot i W (21)

d M d t = n − 1 − e 2 { − 1 − e 2 cos f n a e S + ( 1 + r p ) 1 − e 2 n a e sin f T } − 2 r n a 2 S (22)

Substituting Equations (9), (10) and (11) into Equations (17) to (22). We solve these equations numerically using Runge-Kutta four order methods, the mathematical program written by language of MATHEMATICA V10. Considering the commensurability between the gravitational wave and the mean motion of Mercury and of Pluto (i.e. we are studying the effect of GW when the frequency of GW equal to the mean motion of Mercury and when the frequency of GW equal to the mean motion of Pluto).

We now describe a perturbation approach to solve the above equations to yield the variation in the elements during any interval of time. The amount of perturbations depends on the orders of the disturbing forces such that when the perturbing force is small compared to μ r 2 we will not find large changes in the osculating elements like the force of gravitational waves, but this change may be not ignored for studying the gravitational waves effects in future. We describe a procedure to calculate the perturbations as numerical integration for the set of differential equations in the form

d X d t = є f ( x , t ) (23)

where є is a small parameter, then the solution will be in the form

F K = ∑ j = 1 k ζ j + B (24)

B is a constant n-vector and

ζ 1 = є [ ∫ f ( x , t ) d t + C ( x ) ] (25)

ζ j + 1 = − є ∫ ζ 1 ⋅ f ( x , t ) d t , j = 1 , 2 (26)

where ζ x j is the Jacobian matrix of the set ζ j with respect to the set x, x is kept constant during the integration and C ( x ) is an arbitrary function of x. The secular effects will obtain from (26), representing the second order effect. Using the elliptic orbit relations

cos f = cos E − e 1 − e cos E

sin f = 1 − e 2 sin E 1 − e cos E

r cos f = a ( cos E − e )

r sin f = a 1 − e 2 sin E

E is the eccentric anomaly and related to the mean anomaly M through the Kepler’s equation

E − e sin E = M

Changing the independent variable from the time t to the eccentric anomaly E and using the above elliptic relations in the equations of motion (17) to (22). Solving the equations numerically when the frequency of GW is equal to the frequency of Mercury and Pluto as in

Elements of the Planet | Mercury |
---|---|

The semi-major in Km (a) | 57.9 × 10^{6} |

The eccentricity (e) | 0.205627 |

Inclination in degree (i) | 7.00399 |

Longitude of node in degree Ω | 47.85714 |

Longitude of perihelion in degree ϖ | 76.83309 |

Mean daily motion in degree/day n | 4.092339 |

Elements of the Planet | Pluto |
---|---|

The semi-major in Km (a) | 5896 × 10^{6} |

The eccentricity (e) | 0.250236 |

Inclination in degree (i) | 17.1699 |

Longitude of node in degree Ω | 109.88562 |

Longitude of perihelion in degree ϖ | 224.16024 |

Mean daily motion in degree/day n | 0.003979 |

δi | δe | δa | period |
---|---|---|---|

1.735647 × 10^−24 | 3.307374 × 10^−22 | 4.947566 × 10^−15 | 0 |

1.736209 × 10^−24 | 3.307374 × 10^−22 | 4.947571 × 10^−15 | 1 |

1.736770 × 10 ^−24 | 3.307374 × 10^−22 | 4.947575 × 10^−15 | 2 |

1.737332 × 10^−24 | 3.307374 × 10^−22 | 4.947580 × 10^−15 | 3 |

1.737894 × 10^−24 | 3.307374 × 10^−22 | 4.947585 × 10^−15 | 4 |

δM | δΩ | δω | period |
---|---|---|---|

1.98933` × ^−7 | 5.0877643 × ^−26 | 5.408152 × ^−25 | 0 |

1.98933` × ^−7 | 5.0877452 × ^−26 | 5.408153 × ^−25 | 1 |

1.98933` × ^−7 | 5.0877261 × ^−26 | 5.408155` × ^−25 | 2 |

1.98933` × ^−7 | 5.0877070 × ^−26 | 5.408157 × ^−25 | 3 |

1.98933` × ^−7 | 5.0876879 × ^−26 | 5.408159 × ^−25 | 4 |

δi | δe | δa | period |
---|---|---|---|

0 | 0 | 0 | 0 |

−2.300079 × ^−17 | −5.685610 × ^−17 | −9.650559 × ^−8 | 1 |

−2.288160 × ^−17 | −5.650615 × ^−17 | −9.582255 × ^−8 | 2 |

−2.276240 × ^−17 | −5.615620 × ^−17 | −9.513952 × ^−8 | 3 |

−2.264321 × ^−17 | −5.580624 × ^−17 | −9.445648 × ^−8 | 4 |

δM | δΩ | δω | period |
---|---|---|---|

0.0000167118` | 0 | 0 | 0 |

0.0000167118` | 8.438826 × ^−24 | 7.464780 × ^−23 | 1 |

0.0000167118` | 8.439177 × ^−24 | 7.464740 × ^−23 | 2 |

0.0000167118` | 8.439528 × ^−24 | 7.464700 × ^−23 | 3 |

0.0000167118` | 8.439879 × ^−24 | 7.464660 × ^−23 | 4 |

The resulting solution for our model from Equations (17) to (22) represents the secular effects of gravitational waves on the orbital elements under the commensurability of wave’s frequency and the mean motion of the planets. The perturbations on the orbital elements of elliptic orbits are calculated numerically for five revolutions considering the time of perihelion passage is zero. The perturbation on semi-major a of Mercury increases with each revolution during the time from zero to 5π, but decreases for Pluto, and the perturbation on the eccentricity e of Mercury did not change with revolution but decreasing for Pluto and the mean anomaly M of Mercury and Pluto did not change as seen in _{p}. For low frequency n_{g}, the 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 variation in all the orbital elements will be positive (i.e. increasing).

Youssef, M.H.A. (2018) Numerical Studies of Resonance and Secular Effects of Gravitational Waves. World Journal of Mechanics, 8, 182-191. https://doi.org/10.4236/wjm.2018.85013