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Taking the example of a low earth orbit (LEO) satellite, orbiting the Earth in a circular orbit, it is obvious that the influence of orbital decay due to gravitational wave radiation(GWR) on its lifetime is almost nil(please see [1]). The satellite's time of in-spiral depends upon various other dominant forces such as the neutral drag force, the coulombic drag, drag due to solar radiation pressure, just to name a few.

But the same cannot be said for comapact objects, they possess a significant fraction of the mass of the Sun, or most of the times more mass than that of the Sun. A companion(of significant mass) in a binary system is less likely to be affected by the neutral and coulombic drag forces, however its rate of orbital decay can be significantly affected due to GWR. What this would do is reduce the system's semi-major axis(please see [2] for rate calculations) and increase its eccentricity. So the GWR is source of drag that decreases the binary system's lifetime.

Hence, is there any way to formulate the drag force exercted on a binary system due to GWR? Any help is appreciated.

Calculations:

Using the information given in this paper, the equation for the rate of change of semi-major axis due to GWR is given by: $$\frac{da}{dt}=-\frac{64}{5}\frac{Mm(M+m)}{c^{5}a^{3}(1-e^{2})^{2.5}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right)$$

For a LEO satellite of mass 100kg orbiting the Earth at $7800km/s$ at a distance of 200km (e=0) from the Earth in a circular orbit, the rate of decrease of radius is: $$\frac{dr}{dt}\approx{-10^{-37}}\tag{1}$$

Now taking the example of a binary system ( mass of primary= mass of companion= 1 Solar mass), the rate is as follows: Let $a=1AU$ and e=0.5

$$\frac{da}{dt}\approx{-10^{-14}}\tag{2}$$

Edit I saw this question which talks more on the physical aspects of the problem via a gendaken experiment, while mine is to do more about the theoretical approach to be taken.

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    $\begingroup$ If you focus on the inspiralling phase, you could compute the radiated energy and then derive the drag force (which is actually a loss of angular momentum of the compact binary system). Going with post newtonian formalism for this, you could try looking it up here (this is a rather technical paper). Edit : with your formula for the change of $a$ you should be able to compute the change of angular momentum by yourself, which is probably what @BobBee mean (please correct if mistaken) $\endgroup$
    – Naptzer
    Commented Mar 14, 2017 at 3:22
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    $\begingroup$ What is your question? I looked up the paper you refer to and don't see that equation, but maybe it is there. In any event the radiation and numbers are going to be small until the orbits are close, as Naptzer says, in the in spiral phase, and relatively close or towards the end, the highly relativistic phase. Naptzer's reference may be the best one, if not you can try looking for the papers published with the GW radiation first detected by the LIGO collaboration. If you want the energy equations they are there. Note the last few orbitS were done in about 1/4 sec, highly relativistic. $\endgroup$
    – Bob Bee
    Commented Mar 14, 2017 at 3:37
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    $\begingroup$ I did have a comment - you have an answer, what's the question. That's also what Naptzer refers to. I just don't know if your equation is in the highly relativistic phase. And 1 AU will always give you negligible GWR, need them closer. $\endgroup$
    – Bob Bee
    Commented Mar 14, 2017 at 3:41

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