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

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Abstract
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, 1Gyr 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 180deg. We find useful limits on the elements of the innermost planet A which are otherwise unconstrained by the observations. We also perform a preliminary study of the short-term dynamics of massless particles in the system.