CDS&E: Multiscale Integrations of Exoplanetary Systems

近年来，随着太阳系外数千颗行星的发现，人们越来越需要从大量观测数据中提取科学知识的工具。该项目的目的是开发一种工具，用于了解相互作用的系外行星的长期轨道演变。需要创新工具，因为许多系外行星系统的配置与我们太阳系的配置不同，导致现有方法不足。该研究项目开发的工具将有助于探测系外行星并确定其参数，帮助解释观测到的系统如何演变成目前的结构，并帮助确定观测到的系统是否适合普通生命形式居住。这项研究不仅有助于数学和天文学，而且还将向公众介绍科学思想和发现。例如，它的实现将适用于带有社交媒体界面的免费屏幕保护程序，这将允许学术界以外的人执行和享受系外行星探测和演化预测。计算效率是这个项目的核心问题。由非线性行星-行星相互作用引起的行星轨道变化通常是缓慢的，需要模拟数十亿轨道的轨迹。此外，上述许多科学研究需要大量的轨迹，因此直接模拟太耗时和存储。幸运的是，行星系统包含行星/星星质量比的小参数，这使得它们几乎是可积的，并且在分离的时间尺度上表现出不同的动力学。本研究项目利用这一事实并实现加速模拟。几个突出的挑战是缺乏平均积分的解析表达式，近似误差由于小，但不是无穷小的质量比，平均近通道通过共振的不准确性。这些挑战是由两个数学贡献。一种是多尺度方法，它通过对快尺度非线性振荡进行平均来计算长时间轨道演化。该方法基于近恒等变换得到的同调偏微分方程的数值解。另一个是一个自适应程序，允许通过瞬态共振，这是通过匹配的渐近展开与适当的近共振重新缩放的准确集成。

With thousands of planets outside our solar system detected in recent years, there is an emerging need for tools that help extract scientific knowledge from the large amount of observational data. The purpose of this project is to develop a tool for understanding the long-time orbital evolution of interacting exoplanets. Innovative tools are needed because many exoplanet systems exhibit configurations distinct from that of our solar system, rendering existing approaches insufficient. The tools developed in this research project will facilitate the detection of exoplanets and the identification of their parameters, help explain how observed systems evolved to their current configuration, and help determine whether an observed system is habitable by common life forms. The research will contribute not only to mathematics and astronomy, but will also introduce the general public to scientific thinking and discovery. For instance, its implementation will be adapted to a free screen saver with social media interface, which will allow those outside academia to perform and enjoy exoplanet detection and evolution prediction. Computational efficiency is a central concern in this project. Changes in planet orbits induced by nonlinear planet-planet interactions are oftentimes slow, requiring the simulation of a trajectory for billions of orbits. Moreover, many of the aforementioned scientific investigations require large numbers of trajectories, and hence direct simulations are too time- and storage-consuming. Fortunately, planetary systems contain a small parameter of planet/star mass ratio, which makes them nearly-integrable and exhibiting different dynamics over well-separated timescales. This research project utilizes this fact and achieves accelerated simulation. Several outstanding challenges are the lack of analytical expressions for averaging integrals, approximation errors due to small but not infinitesimal mass ratio, and the inaccuracy of averaging near passage through resonance. These challenges are addressed by two mathematical contributions. One is a multiscale method that allows the computation of long time orbital evolution by averaging over fast scale nonlinear oscillations. This method is based on numerical resolution of homological PDEs derived from near identity transformations. The other is an adaptive procedure that allows accurate integration through transient resonances, which is accomplished by matched asymptotic expansions with an appropriate near-resonance rescaling.