Analytic determination of the eight-and-a-half post-Newtonian self-force contributions to the two-body gravitational interaction potential

We analytically compute, to the eight-and-a-half post-Newtonian order, and to linear order in the mass ratio, the radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies, thereby extending previous analytic results. These results are obtained by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. We emphasize the increase in ``transcendentality'' of the numbers entering the post-Newtonian expansion coefficients as the order increases, in particular we note the appearance of (3) (as well as the square of Euler's constant) starting at the seventh post-Newtonian order. We study the convergence of the post-Newtonian expansion as the expansion parameter u=GM/ (c^2r) leaves the weak-field domain u1 to enter the strong field domain u=O (1).