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  On gravitational wave radiation changing orbital parameters

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Two celestial objects of mass m1 and m2 are orbiting around each other on a very elongated ellipse(ϵ1). The system is isolated, there is no other proximate celestial body. How would the parameters of the elliptical orbit change as a consequence of radiation of gravitational waves. How can the time of the process be estimated when the orbit changes from an ellipse to a circle?

Here is my work:
The power emitted by gravitational waves is given by:
PGW=c5G(GMc5l)5

Very compact binaries will lose energy rapidly by GW radiation.
If we assume that the two bodies making up the binary lie in the xy plane and their **orbits are circular** (ϵ=0), then only non-vanishing components of quadrupole tensors are:
Qxx=Qyy=12(μ)a2cos2Ωt

and 
Qxy=Qya=12(μ)a2sin2Ωt

Where Ω is the orbital velocity, μ=m1m2m is the reduced mass and where m=m1+m2

The luminosity of the system can be deduced as:
LGM=325Gc5μ2a4Ω6=325G4c5M3μ2a5

The latter part is obtained from Kepler third law: Ω2=GMa3
As the gravitating system loses energy by emitting radiation, the distance between the 2 bodies shrinks at a rate:
dadt=645G3Mμc5a3


The binary would hence colase at a time :
τ=5256c5G3a40μM4

I am having problems in converting this concept to that mentioned in the question, I.e., for a system of binaries of highly elliptical orbit(in my calculations, I have assumed a circle) and how would the elliptical parameters change with time. I also require help in calculating the time of the process when the orbit changes from an ellipse to a circle.

I have obtained the following integral for finding the decay time of the orbits( the orbits would eventually decay into a circular one):
T(a0,e0)=12(c40)19γe00e29/19[1+(121/304)e2]1181/2299(1e2)3/2de


Where γ=64G35c5m1m2(m1+m2)

For e0 close to 1 the equation becomes:
T(a0,e0)768425Tfa0(1e20)7/2

Where Tf=a404γ

Here T depends on both a0 an de e0 as the former determines the latter.
And γ has the same value as define do above.
I am stuck with (1) and (2); any help would be appreciated.
**My approach:**
I have tried to use Appell hypergeometric integration and obtained the following result:(let x=e)
>x(32+y)[1+Ax2]12+z(1x2)32dx=2xy+52F1(y2+54;32,zy2+94;x2,Ax2)2y+5

Where y=5538,z=634598,A=121304
Here F1(a,β,β,c;x,y) is appells function and I read that it can be expressed as the following integral:
>F1(a,β,β,c;x,y)=Γ(c)Γ(a)Γ(ca)10ua1(1u)ba1(1ux)β(1uy)βdu

Is this right? How am I to solve this integral? And also how am I to solve for e0 in equation (2)?

asked Oct 29, 2016 in Astronomy by Naveen (85 points) [ revision history ]
edited Oct 31, 2016 by Naveen

I have to say it is not clear to me at all how does equation (1) come about. For instance, do you take relativistic precession into account, or do you assume Newtonian trajectories at every point of the decay? (I.e., what is your approximation, Newtonian+linearized gravity waves, post-Newtonian...) The loss of eccentricity can be essentially seen from the slightly different behavior of energy and angular momentum - how would that look in your case?

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