  We examine the motion of a charged particle in a weak gravitational field. In
addition to the Newtonian gravity exerted by a large central body, the particle
is subjected to an electromagnetic self-force that contains both a conservative
piece and a radiation-reaction piece. This toy problem shares many of the
features of the strong-field gravitational self-force problem, and it is
sufficiently simple that it can be solved exactly with numerical methods, and
approximately with analytical methods. We submit the equations of motion to a
multi-scale analysis, and we examine the roles of the conservative and
radiation-reaction pieces of the self-force. We show that the
radiation-reaction force drives secular changes in the orbit's semilatus rectum
and eccentricity, while the conservative force drives a secular regression of
the periapsis and affects the orbital time function; neglect of the
conservative term can hence give rise to an important phasing error. We next
examine what might be required in the formulation of a reliable adiabatic
approximation for the orbital evolution; this would capture all secular changes
in the orbit and discard all irrelevant oscillations. We conclude that such an
approximation would be very difficult to formulate without prior knowledge of
the exact solution.
