On integrating decay time of binary star's orbital evaporation.

Question

I am currently researching on binary systems and their orbital evaporation due to gravitational radiation. The binary stars have a highly elliptic orbit($\epsilon\approx1$). I have obtained the following integral for finding the decay time of the orbits( the orbits would eventually decay into a circular one): $$T(a_{0},e_{0})=\frac{12(c_{0}^4)}{19\gamma}\int_{0}^{e_0}{\frac{e^{29/19}[1+(121/304)e^2]^{1181/2299}}{(1-e^2)^{3/2}}}de\tag1$$ Where $$\gamma=\frac{64G^3}{5c^5}m_{1}m_{2}(m_{1}+m_{2})$$ For $e_{0}$ close to $1$ the equation becomes: $$T(a_{0},e_{0})\approx\frac{768}{425}T_{f}a_{0}(1-e_{0}^2)^{7/2}\tag2$$ Where $$T_{f}=\frac{a_{0}^4}{4\gamma}$$ And $\gamma$ has the same value as define do above. I am stuck with $(1)$ and $(2)$; any help would be appreciated.

Answer 1

More a comment, but I lack the reputation to add comments:

• The lhs of $(1)$ says that $T$ depends on $a_0$, but the rhs doesn't contain it.

• According to WolframAlpha, the integral in $(1)$ can be solved using the Appell hypergeometric function $F_1$.