Influence of a weak gratitational wave on a bound system of two point-masses

We consider the effect of a weak gravitational wave (dimensionless amplitude h/sup TT/very-much-less-than1) on a nonrelativistic (orbital velocity upsilonvery-much-less-thanc) system of two point-masses by expanding the geodesic equation for each mass in terms of two parameters, (upsilon/c) and H/sup TT/. The terms linear in h/sup TT/ through order (upsilon/c) h/sup TT/upsilon2/a (aapprox. orbital size) are used to calculate the effect of the gravitational wave on the motion of the masses. The perturbation due to the wave in the orbital equation of motion is at lowest order h/sup TT/upsilon2/a and produces secular variations in the orbital elements for ω=nω0 (ωw=frequency of the wave, ω0=orbital frequency). In addition, for any ω there are periodic variations in the proper separation of the masses [deltaaapprox.ah/sup TT/(t)]. The perturbation in the center of-mass (CM) equation of motion is at lowest order (upsilon/c) h/sup TT/upsilon2/a and produces a secular CM velocity if ω=nω0. Both secular effects are gauge-invariant to linear order in h/sup T/T. The perturbed orbital equation of motion at order h/sup TT/upsilon2/a is different from the one found by Mashhoon, but is related by a simple coordinate transformation