Exact Results On the Number of Gravitons Radiated During Binary Inspiral

We derive an exact formula $F(e)$ which provides a concrete estimate for the total number and angular momentum of gravitons emitted during the nonrelativistic inspiral of two black holes. We show that the function $F(e)$ is a slowly growing monotonic function of the eccentricity $0 \le e \le 1$ and $F(1) = 1.0128 \cdots$. We confirm and extend the results obtained by Page for the function $F(e)$. We also get an exact result for the ratio $\nu (e_i) = \frac{2\hbar N(L_i, e_i)}{L_i}$ where the numerator $2\hbar N(L_i, e_i)$ is the sum of the spin angular momentum magnitudes of the gravitons emitted and $N(L_i, e_i)$ is the total number of gravitons emitted in the gravitational waves during nonrelativistic inspiral from an initial eccentricity $e_i$ down to a final eccentricity $e = 0$ and the denominator $L_i$ is the magnitude of the initial orbital angular momentum. If the orbit starts off with unit eccentricity $e_i=1$, we get the value $\nu(1) = 1.002\, 268\, 666\, 2 \pm 10^{-10}$ which confirms the Page's conjecture that the true value of $\nu(1)$ will lie between $1.001\cdots$ and $1.003\cdots$. We also show that the formula $F(e)$ for gravitons emitted, originally expressed as an infinite series, can be represented by a single function through an integral representation.