2005ApJ...619.1084G

We study the long-term dynamics of the PSR 1257+12 planetary system. Using the recently determined accurate initial condition by Konacki & Wolszczan, who derived the orbital inclinations and the absolute masses of planets B and C, we investigate the system stability by long-term, 1 Gyr direct integrations. No secular changes of the semimajor axes, eccentricities, and inclinations appear during such an interval. This stable behavior is confirmed with the fast indicator MEGNO. The analysis of the orbital stability in the neighborhood of the nominal initial condition reveals that the PSR 1257+12 system is localized in a wide stable region of the phase space but close to a few weak two- and three-body mean motion resonances. The long-term stability is additionally confirmed by a negligible exchange of the angular momentum deficit between the innermost planet A and the pair of outer planets B and C. An important feature of the system that helps sustain the stability is the secular apsidal resonance between planets B and C with the center of libration about 180°. We also find useful limits on the elements of the innermost planet A that are otherwise unconstrained by the observations. Specifically, we find that the line of nodes of planet A cannot be separated by more than about ±60° from the nodes of the bigger companions B and C. This limits the relative inclination of the orbit of planet A to the mean orbital plane of planets B and C to moderate values. We also perform a preliminary study of the short-term dynamics of massless particles in the system. We find that a relatively extended stable zone exists between planets A and B. Beyond planet C, the stable zone appears already at distances 0.5 AU from the parent star. For moderately low eccentricities beyond 1 AU, the motion of massless particles does not suffer from strong instabilities and this zone is basically stable, independent of the inclinations of the orbits of the test particles to the mean orbital plane of the system. This is an encouraging result supporting the search for a putative dust disk or a Kuiper Belt, especially with the Spitzer Space Telescope.