Chaos and resonances in the Edgeworth-Kuiper-belt

我们将研究Edgeworth-Kuiper带中选定的共振区，特别是4：7、3：5和1：2的共振区。将使用不同的数值工具来研究它们的特性、稳定性和共振宽度。对小行星动力学问题实施一种相当新的混沌指标(MEGNO)将具有特别重要的意义。有了这个指示剂，就有可能分辨出相空间的精细结构，使得更高阶的共振、Kozai共振等也是可见的。为了理解通过数值积分找到的各种特征，将开发一些分析模型。借助于摄动理论，我们将大大降低问题的自由度，使其能够得到解析解。这种复杂性的降低包括对短周期扰动的哈密顿量求平均、围绕精确共振点的展开和变量的正则变换。将修改数值程序，并引入额外的分析模型，以便也可以调查主要行星由于与小行星相互作用而发生的迁移及其对小行星动力学的影响。

Selected resonant areas in the Edgeworth-Kuiper-belt will be investigated, especially the 4:7, 3:5 and the 1:2 resonances. Different numerical tools will be employed to study their characteristics, stability and resonance widths. Of special importance will be the implementation of a fairly new chaos indicator (MEGNO) to the problem of asteroid dynamics. With this indicator it will be possible to resolve the fine structure of phase-space such that also resonances of higher order, Kozai resonances etc. become visible. To understand the various signatures found by means of numerical integrations some analytical models will be developed. By means of perturbation theory we will reduce the degrees of freedom of a problem significantly so that it can be solved analytically. This reduction of complexity involves averaging of the Hamiltonian over short periodic perturbations, expansions around an exact resonance point and canonical transformations of variables. The numerical routines will be modified and additional analytical models will be introduced so that also the migration of major planets due to their interactions with planetesimals and its impact onto asteroid dynamics can be investigated.