Two celestial objects of mass m1 and m2 are orbiting around each other on a very elongated ellipse(ϵ≈1). 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 x−y 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)19γ∫e00e29/19[1+(121/304)e2]1181/2299(1−e2)3/2de
Where
γ=64G35c5m1m2(m1+m2)
For
e0 close to
1 the equation becomes:
T(a0,e0)≈768425Tfa0(1−e20)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(1−x2)32dx=2xy+52F1(y2+54;32,−z−y2+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(c−a)∫10ua−1(1−u)b−a−1(1−ux)−β(1−uy)−β′du
Is this right? How am I to solve this integral? And also how am I to solve for
e0 in equation (2)?