Our model for Phoebe's orbit is a numerical integration of its equation of motion (Peters 1981) which includes the effects of an oblate Saturn, perturbations from seven of the eight major Saturnian satellites (Hyperion, whose mass is unknown but presumed quite small, is neglected), and perturbations from the Sun, Jupiter, and Uranus. The formulation is in Cartesian coordinates centered at the Saturnian system barycenter and referenced to the Earth mean equator and equinox of J2000 system. Because it is small and its mass unknown, Phoebe, like Hyperion, is assumed to be massless, hence the barycenter location depends only upon the planet and perturbing satellites. JPL planetary ephemeris DE403 (Standish et al. 1995) provides the masses and positions of the Sun and perturbing planets, and JPL satellite ephemeris SAT077 (Jacobson 1996a) provides those of the perturbing satellites.
The model is far more complete than necessary to fit the observations. Rose's original work included only solar perturbations, that of Bykova and Shikhalev accounted for the Sun and Jupiter, and that of Bec-Borsenberger and Rocher took into account the effects of Titan, Jupiter, and the Sun. All three of these previous investigations obtained reasonable fits to the observations. We selected the model in this work primarily for consistency with the integrations of the other Saturnian satellites being performed in preparation for support of the Cassini mission (Jacobson 1996b). Table 1 (click here) gives the ratio of the maximum perturbing accelerations encountered in our integrations to the average point mass central body acceleration. The table entries show that Phoebe's acceleration due to Jupiter is about 0.47% that of the Sun and due to Titan is about 0.13%. Interestingly, the effect of Uranus is greater than that of any of the remaining major satellites. We also investigated a simplified model which replaces the perturbing satellites, except for Titan, with uniform circular equatorial rings retaining only their quadrupole effect (see Roy et al. 1988 for a discussion of this technique). In addition, the simplified model obtains its Titan positions from a precessing ellipse approximate representation of the Titan ephemeris (the elements for the ellipse can be found in Appendix A). Integrations with the simplified model differ from those with the complete model by less than 50 km over a 50 year period. In fact, because such differences are well below the accuracy of the pre-1966 observations (the integration epoch is November 11, 1966), we processed those early observations with only the simplified model.
| Accel. | Magnitude | Accel. | Magnitude |
| Sun | 3.2 | Dione | 2.9 |
| Jupiter | 1.5 | Tethys | 1.6 |
| Titan | 4.1 | Harmonics | 1.0 |
| Uranus | 1.3 | Enceladus | 1.9 |
| Iapetus | 8.5 | Mimas | 9.7 |
| Rhea | 6.3 | ||
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The integration was carried out with a variable step size, variable order, Gauss-Jackson method. An absolute truncation error limit of 10-10 km s-1 imposed on the velocity controlled the integration step. The average step size was 25896 seconds and the maximum order was 15.