The Secular Evolution of the Primordial Kuiper Belt

Abstract

A model that rapidly computes the secular evolution of a gravitating disk-planet system is developed. The disk is treated as a nested set of gravitating rings, with the rings'/planets' time evolution being governed by the classical Laplace-Lagrange solution for secular evolution but modified to account for the disk's finite thickness h. The Lagrange planetary equations for this system yield a particular class of spiral wave solutions, usually called apsidal density waves and nodal bending waves. There are two varieties of apsidal waves—long waves and short waves. Planets typically launch long density waves at the disk's nearer edge or else at a secular resonance in the disk, and these waves ultimately reflect downstream at a more distant disk edge or else at a Q barrier in the disk, whereupon they return as short density waves. Planets also launch nodal bending waves, and these have the interesting property that they can stall in the disk, that is, their group velocity plummets to zero upon approaching a region in the disk that is too thick to support further propagation of bending waves. The rings model is used to compute the secular evolution of a Kuiper Belt having a variety of masses, and it is shown that the early massive belt was very susceptible to the propagation of low-amplitude apsidal and nodal waves launched by the giant planets. For instance, these waves typically excited orbits to e ~ sin i ~ 0.01 in a primordial Kuiper Belt of mass MKB ~ 30 Earth masses. Although these orbital disturbances are quite small, the resulting fractional variations in the disk's surface density due to the short density waves is usually large, typically of order unity. This epoch of apsidal and nodal wave propagation probably lasted throughout the Kuiper Belt's first ~107 to ~5 × 108 yr, with the waves being shut off between the time when the large R 100 km Kuiper Belt objects first formed and when the belt was subsequently eroded and stirred up to its present configuration.