The second post-Newtonian motion in Reissner-Nordstr\"om spacetime

We derived the second post-Newtonian solution for the quasi-Keplerian motion of a charged test particle in the Reissner-Nordstr\"om spacetime under the harmonic coordinates. The solution is formulated in terms of the test particle's orbital energy and angular momentum, both of which are constants at the second post-Newtonian order. The charge effects on the test particle's motion including the orbital period and perihelion precession are displayed explicitly. Our results can be applied to the cases in which the test particle has small charge-to-mass ratio, or the test particle has arbitrary charge-to-mass ratio but the multiplication of the test particle and the gravitational source's charge-to-mass-ratios is much smaller than 1.