Two small satellites of Pluto , S/2005 P1 ( hereafter P1 ) and S/2005 P2 ( hereafter P2 ) , have recently been discovered outside the orbit of Charon , and their orbits are nearly circular and nearly coplanar with that of Charon .
Because the mass ratio of Charon-Pluto is \sim 0.1 , the orbits of P2 and P1 are significantly non-Keplerian even if P2 and P1 have negligible masses .
We present an analytic theory , with P2 and P1 treated as test particles , which shows that the motion can be represented by the superposition of the circular motion of a guiding center , the forced oscillations due to the non-axisymmetric components of the potential rotating at the mean motion of Pluto-Charon , the epicyclic motion , and the vertical motion .
The analytic theory shows that the azimuthal periods of P2 and P1 are shorter than the Keplerian orbital periods , and this deviation from Kepler ’ s third law is already detected in the unperturbed Keplerian fit of Buie and coworkers .
In this analytic theory , the periapse and ascending node of each of the small satellites precess at nearly equal rates in opposite directions .

From direct numerical orbit integrations , we show the increasing influence of the proximity of P2 and P1 to the 3:2 mean-motion commensurability on their orbital motion as their masses increase within the ranges allowed by the albedo uncertainties .
If the geometric albedos of P2 and P1 are high and of order of that of Charon , the masses of P2 and P1 are sufficiently low that their orbits are well described by the analytic theory .
The variation in the orbital radius of P2 due to the forced oscillations is comparable in magnitude to that due to the best-fit Keplerian eccentricity , and there is at present no evidence that P2 has any significant epicyclic eccentricity .
However , the orbit of P1 has a significant epicyclic eccentricity , and the prograde precession of its longitude of periapse with a period of 5300 days should be easily detectable .
If the albedos of P2 and P1 are as low as that of comets , the large inferred masses induce significant short-term variations in the epicyclic eccentricities and/or periapse longitudes on the 400–500-day timescales due to the proximity to the 3:2 commensurability .
In fact , for the maximum inferred masses , P2 and P1 may be in the 3:2 mean-motion resonance , with the resonance variable involving the periapse longitude of P1 librating .
Observations that sample the orbits of P2 and P1 well on the 400–500-day timescales should provide strong constraints on the masses of P2 and P1 in the near future .