Long Term Stability and Dynamical Environment of the Psr 1257+12 Planetary System

We study the long-term dynamics of the PSR 1257+12 planetary system. Using the recently determined accurate initial condition by Konacki & Wolszczan (2003) who derived the orbital inclinations and the absolute masses of the planets B and C, we investigate the system stability by long-term, 1 Gyr direct integrations. No secular changes of the semi-major 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 2 and 3-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 the outer planets B and C. An important feature of the system that helps sustain the stability is the secular apsidal resonance (SAR) between the planets B and C with the center of libration about 180o. We also find useful limits on the elements of the innermost planet A which are otherwise unconstrained by the observations. Specifically, we find that the line of nodes of the 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 the planet A to the mean orbital plane of the 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 the planets A and B. Beyond the 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 on the inclinations of the orbits of the test particles to the mean orbital plane of the system. It is an encouraging result supporting the search for a putative dust disk or a Kuiper belt, especially with the SIRTF mission. Subject headings: celestial mechanics, stellar dynamics—methods: numerical, N-body simulations—planetary systems—stars: individual (PSR 1257+12)