_{1}

^{*}

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
*
n _{w}
* is much smaller than the particle’s orbital

*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

_{p}*a*,

*e*and

*M*changing with revolution during the interval of time (

*t*−

*t*

*0*) which is changing from 0 →4 π.

The tremendous penetrating capacity of gravitational waves is a reason why it is so difficult to detect them. Another reason is the very low intensities by which they are emitted. A great number of theoreticians and experimentalists work on gravitational radiation not only because of its important applications but also because of its fundamental role in science. In the domain of general relativity, the particular solution of the field equations leading to the wave solution is usually preferred in the domain of the post-Newtonian (or weak field) approximation. The reason is that the wave contributes to its own source, besides the effect of the relatively strong field of source itself; greatly complicate the detection of the wave close to its source. Hence it is better to study the simple properties of the waves by looking far into the wave zone where the fields are weak. The effects of incident plane gravitational waves on the orbital motion of gravitational bound two-body system were investigated by several authors with different approaches and various features of both orbits and the waves [

According to the weak field approximation of general relativity, there exist in empty space gravitational waves generated by different sources far in the universe, and propagated with velocity c. The field equations for empty space in the linear weak field approximation

□ 2 h i k = 0

Has the plane wave as the simplest solution. □ 2 is d’Alembertian operator is given by

□ 2 = ∂ 2 ∂ x i 2 − 1 c 2 ∂ 2 ∂ t 2 ( i = 1 , 2 , 3 )

With the Z axis chosen in the direction of propagation h i k , □ 2 becomes

□ 2 = ∂ 2 ∂ z 2 − 1 c 2 ∂ 2 ∂ t 2

and then the partial derivatives of h i k with respect to x and y vanish. Thus a plane gravitational wave is determined by two quantities h 12 and h 11 = − h 22 in other words, gravitational waves are transverse waves whose polarization is determined by a symmetric tensor of the second rank in the XY plane, the sum of whose diagonal terms h 11 + h 22 is zero. So the plane wave can be written a the sum of two components

h 11 = h + cos ( n w t + α 1 ) (1)

h 12 = h × cos ( n w t + α 2 ) (2)

where n w 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. From the equation of geodesic deviation we obtain the acceleration components of GW as in [

d 2 n i d t 2 = 1 2 ∂ 2 h i k ∂ t 2 n k (3)

From which we can obtained the potential like function U of GW in terms of the orbital elements

U = 1 2 h 1 ( y 2 − x 2 ) − h 2 x y (4)

where h 1 = 1 2 ∂ 2 h 11 ∂ t 2 and h 2 = 1 2 ∂ 2 h 12 ∂ t 2 , and we have

x = r ( cos Ω cos ( f + ω ) − cos i sin Ω sin ( f + ω ) ) (5)

y = r ( sin Ω cos ( f + ω ) + cos i cos Ω sin ( f + ω ) ) (6)

where i , Ω , ω and f are the inclination, longitude of node, argument of perigee and the true anomaly of an orbit respectively. Substituting Equations ((1), (2), (5) and (6)) into Equation (4) we obtain

U = ϵ r 2 { s 2 8 [ cos ( 2 Ω + n w t + α 1 ) − cos ( 2 Ω − n w t − α 1 ) − A g [ sin ( 2 Ω + n w t + α 2 ) + sin ( 2 Ω − n w t − α 2 ) ] ] + ( c 2 16 + c 8 − 1 16 ) [ cos ( 2 f + 2 ω + 2 Ω + n w t + α 1 ) + cos ( 2 f + 2 ω + 2 Ω − n w t − α 1 ) − A g [ sin ( 2 f + 2 ω + 2 Ω + n w t + α 2 ) + sin ( 2 f + 2 ω + 2 Ω − n w t − α 2 ) + sin ( 2 f + 2 ω − 2 Ω + n w t + α 2 ) + sin ( 2 f + 2 ω − 2 Ω − n w t − α 2 ) ] ] + ( c 2 16 − c 8 + 1 16 ) [ cos ( 2 f + 2 ω − 2 Ω + n w t + α 1 ) + cos ( 2 f + 2 ω − 2 Ω − n w t − α 1 ) ] } (7)

where

c = cos i , s = sin i , ϵ = 1 2 n w 2 h × and A g = − h × h + (8)

The Hamiltonian will be in the form

H = H 0 + ϵ H 1

H 0 is the unperturbed Hamiltonian given by

H 0 = − μ 2 2 L 2 (9)

μ = G M the gravitational constant and L = μ a . The perturbed Hamiltonian H 1 due to the gravitational wave is given by

H 1 = L 4 μ 2 r 2 a 2 { s 2 8 [ cos ( 2 Ω + n w t + α 1 ) − cos ( 2 Ω − n w t − α 1 ) − A g [ sin ( 2 Ω + n w t + α 2 ) + sin ( 2 Ω − n w t − α 2 ) ] ] + ( c 2 16 + c 8 − 1 16 ) [ cos ( 2 f + 2 ω + 2 Ω + n w t + α 1 ) + cos ( 2 f + 2 ω + 2 Ω − n w t − α 1 )

− A g [ sin ( 2 f + 2 ω + 2 Ω + n w t + α 2 ) + sin ( 2 f + 2 ω + 2 Ω − n w t − α 2 ) + sin ( 2 f + 2 ω − 2 Ω + n w t + α 2 ) + sin ( 2 f + 2 ω − 2 Ω − n w t − α 2 ) ] ] + ( c 2 16 − c 8 + 1 16 ) [ cos ( 2 f + 2 ω − 2 Ω + n w t + α 1 ) + cos ( 2 f + 2 ω − 2 Ω − n w t − α 1 ) ] } (10)

The Delaunay canonical variables ( l , g , h , L , G , H ) are defined in terms of the Keplerian elements ( a , e , i , Ω , ω , M ) by

l = M = n ( t – τ ) ; L = μ a g = ω ; G = μ a ( 1 − e 2 ) h = Ω ; H = G cos i (11)

The considered systems of differential equations in terms of canonical elements under the effect of GW are

d L i d t = − ∂ H ∂ l i ; d l i d t = ∂ H ∂ L i ( i = 1 , 2 , 3 , 4 ) (12)

Then

l · = ∂ H ∂ L ; g · = ∂ H 1 ∂ G ; h · = ∂ H 1 ∂ H (13)

L · = − ∂ H 1 ∂ M ; G · = − ∂ H 1 ∂ g ; H · = − ∂ H 1 ∂ h (14)

Using Equations ((11), thus (13) and (14)) yield

a · = 2 a μ L · (15)

e · = ( 1 − e 2 ) n a 2 e L · − ( 1 − e 2 ) n a 2 e G · (16)

i · = cot i n a 2 ( 1 − e 2 ) G · − csc i n a 2 ( 1 − e 2 ) H · (17)

Ω · = ∂ H 1 ∂ H = ∂ H 1 ∂ i ∂ i ∂ H (18)

ω · = ∂ H 1 ∂ G = ∂ H 1 ∂ e ∂ e ∂ G + ∂ H 1 ∂ i ∂ i ∂ G (19)

M · = ∂ H 0 ∂ L + ∂ H 1 ∂ L = ∂ H 0 ∂ L + ∂ H 1 ∂ a ∂ a ∂ L + ∂ H 1 ∂ e ∂ e ∂ L (20)

We solve this equations numerically using Runge-Kutta fourth order method, the mathematical program written by language of MATHEMATIC V10.

We expand the trigonometric functions of the true anomaly r 2 a 2 , r 2 a 2 cos 2 f and r 2 a 2 sin 2 f of Hamiltonian (10) in series of the trigonometric functions of

mean anomaly M up to order three in the eccentricity [

frequency 56 × 10 − 6 Hz and amplitude 15 × 10 − 22 . α 1 = 0 and α 2 = π 2 . The

variations of Jupiter’s elements due to the considering effect of gravitational waves are presented in

The resulting solution for our model from Equations (15) to (20) includes short-period as well as long-period terms for practical application it appears more expedient to calculate the perturbations on a revolution by revolution. The averaging process over the fast variable or short-period comes after the differentiated with respect to slow variable or long-period. The short-period perturbation on semi-major a, the eccentricity e, and the mean anomaly M increase with each revolution during the time interval ( t − t 0 ) from zero to 4π. The long-period perturbation on the inclination i, Ω, and ω takes long time to change with revolution in the same interval as seen in

Youssef, M.H.A. (2017) Short-Period and Long-Period Effects of Weak Gravitational Waves. International Journal of Astronomy and Astrophysics, 7, 230-237. https://doi.org/10.4236/ijaa.2017.73019