Abstract We consider the coplanar planetary four-body problem, where three planets orbit a large star without the cross of their orbits. The system is stable if there is no exchange or cross of orbits. Starting from the Sundman inequality, the equation of the kinematical boundaries is derived. We discuss a reasonable situation, where two planets with known orbits are more massive than the third one. The boundaries of possible motions are controlled by the parameter c2E. If the actual value of c2E is less than or equal to a critical value (c2E)cr, then the regions of possible motions are bounded and therefore the system is stable. The criteria obtained in special cases are applied to the Solar System and the currently known extrasolar planetary systems. Our results are checked using N-body integrator.
Keywords methods: analytical — methods: numerical — celestial mechanics — planets and satellites: dynamical evolution and stability— stars: kinematics and dynamics— galaxies: kinematics and dynamics
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