THE WARPED PLANE OF THE CLASSICAL KUIPER BELT

Abstract

By numerically integrating the orbits of the giant planets and of test particles over a period of four billion years, we follow the evolution of the location of the midplane of the Kuiper belt. The Classical Kuiper belt conforms to a warped sheet that precesses with a 1.9 Myr period. The present-day location of the Kuiper belt plane can be computed using linear secular perturbation theory: the local normal to the plane is given by the theory's forced inclination vector, which is specific to every semimajor axis. The Kuiper belt plane does not coincide with the invariable plane, but deviates from it by up to a few degrees in stable zones. For example, at a semimajor axis of 38 AU, the local Kuiper belt plane has an inclination of 1.9 degrees and a longitude of ascending node of 149.9 degrees when referred to the mean ecliptic and equinox of J2000. At a semimajor axis of 43 AU, the local plane has an inclination of 1.9 degrees and a nodal longitude of 78.3 degrees. Only at infinite semimajor axis does the Kuiper belt plane merge with the invariable plane, whose inclination is 1.6 degrees and nodal longitude is 107.7 degrees. A Classical Kuiper belt object keeps its inclination relative to the Kuiper belt plane nearly constant, even while the plane departs from the trajectory predicted by linear theory. The constancy of relative inclination reflects the undamped amplitude of free oscillation; that is, the homogeneous solution to the forced harmonic oscillator equation retains constant amplitude, even while the inhomogeneous solution cannot be written down accurately because the planetary forcing terms are chaotic. Current observations of Classical Kuiper belt objects are consistent with the plane being warped by the giant planets alone, but the sample size will need to increase by a few times before confirmation exceeds 3σ in confidence. In principle, differences between the theoretically expected plane and the observed plane could be used to infer as yet unseen masses orbiting the Sun, but carrying out such a program would be challenging.