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 M_(KB) ～ 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 ～10~7 to ～5 x 10~8 yr, with the waves being shut off between the time when the large R approx> 100 km Kuiper Belt objects first formed and when the belt was subsequently eroded and stirred up to its present configuration.
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机译：开发了一种可快速计算引力盘行星系统长期演变的模型。圆盘被看作是一组引力环的嵌套，圆环/行星的时间演化受经典的拉普拉斯-拉格朗日解的长期演化控制，但经过修改以考虑圆盘的有限厚度h。该系统的拉格朗日行星方程产生一类特殊的螺旋波解，通常称为两极密度波和节点弯曲波。劈波有两种：长波和短波。行星通常在圆盘的近边缘或圆盘的长期共振中发射长密度波，这些波最终在更远的圆盘边缘或圆盘的Q壁垒处向下游反射，随后它们以短密度波返回。行星也发动节点弯曲波，它们具有有趣的性质，它们可以在圆盘中失速，即，当到达圆盘中太厚而无法支撑弯曲波进一步传播的区域时，它们的群速度骤降至零。环模型用于计算具有各种质量的柯伊伯带的长期演化，结果表明，早期大质量带非常容易受到巨型行星发射的低振幅两极波和节点波的传播。例如，这些波通常在质量为M_（KB）〜30地球质量的原始柯伊伯带中激发到e〜sin i〜0.01的轨道。尽管这些轨道扰动很小，但是由于短密度波而导致的磁盘表面密度的分数变化通常很大，通常为一阶。快波和节波传播的这个时期大概持续了整个柯伊伯带的最初〜10〜7至〜5 x 10〜8年，在大的R大约> 100 km的柯伊伯带物体首次形成的时间之间就将这些波关闭了。当皮带随后被腐蚀并搅拌到现在的状态时。
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