  For a Schwarzchild black hole of mass $M$, we consider a test particle
falling from rest at infinity and becoming trapped, at late time, on the
unstable circular orbit of radius $r=4GM/c^2$. When the particle is endowed
with a small mass, $\mu\ll M$, it experiences an effective gravitational
self-force, whose conservative piece shifts the critical value of the angular
momentum and the frequency of the asymptotic circular orbit away from their
geodesic values. By directly integrating the self-force along the orbit
(ignoring radiative dissipation), we numerically calculate these shifts to
$O(\mu/M)$. Our numerical values are found to be in agreement with estimates
first made within the Effective One Body formalism, and with predictions of the
first law of black-hole-binary mechanics (as applied to the asymptotic circular
orbit). Our calculation is based on a time-domain integration of the
Lorenz-gauge perturbation equations, and it is a first such calculation for an
unbound orbit. We tackle several technical difficulties specific to unbound
orbits, illustrating how these may be handled in more general cases of unbound
motion. Our method paves the way to calculations of the self-force along
hyperbolic-type scattering orbits. Such orbits can probe the two-body potential
down to the "light ring", and could thus supply strong-field calibration data
for eccentricity-dependent terms in the Effective One Body model of merging
binaries.
