Chaos and resonances in the Edgeworth-Kuiper-belt

我们将研究Edgewerk-Kuiper带中的某些共振区，特别是4：7、3：5和1：2共振区。将采用不同的数值工具来研究它们的特性、稳定性和谐振宽度。特别重要的是，将针对小行星动力学问题实施一种相当新的混沌指示器。利用该指示器，将有可能解析相空间的精细结构，使得更高阶的共振、Kozai共振等也变得可见。为了理解通过数值积分发现的各种特征，将开发一些分析模型。通过摄动理论，我们将大大减少问题的自由度，以便可以解析地解决问题。这种复杂性的降低涉及到平均的哈密顿量在短周期扰动，膨胀周围的一个确切的共振点和正则变换的变量。将修改数值程序，并将引入更多的分析模型，以便还可以研究主要行星由于与微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.