CDS&E: Multiscale Integrations of Exoplanetary Systems

CDS

• 批准号: 1521667
• 负责人: Molei Tao
• 金额: $ 20.99万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1521667&HistoricalAwards=false
• 关键词: CDS& Multiscale; Integrations; Exoplanetary; Systems

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.

近年来，随着太阳系外数以千计的行星被探测到，人们越来越需要工具来帮助从大量的观测数据中提取科学知识。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化。需要创新的工具，因为许多系外行星系统展示了与我们太阳系不同的配置，使得现有的方法不够充分。在该研究项目中开发的工具将有助于探测系外行星并确定它们的参数，帮助解释观测到的系统如何进化到目前的形态，并帮助确定观测到的系统是否适合常见生命形式居住。这项研究不仅将对数学和天文学做出贡献，还将向普通公众介绍科学思维和发现。例如，它的实施将被改造成一个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。计算效率是该项目的中心关注点。由非线性行星-行星相互作用引起的行星轨道变化往往很慢，需要模拟数十亿个轨道的轨道。此外，前面提到的许多科学调查需要大量的轨迹，因此直接模拟太耗时和耗费存储。幸运的是，行星系统包含一个行星/恒星质量比的小参数，这使得它们几乎可积，并在完全分开的时间尺度上表现出不同的动力学。本研究项目利用了这一事实，实现了加速仿真。几个突出的挑战是缺乏平均积分的解析表达式，由于小的但不是无穷小的质量比而导致的近似误差，以及通过共振进行接近通过的平均的不准确性。这些挑战通过两个数学贡献得到了解决。一种是多尺度方法，它允许通过对快速尺度的非线性振荡进行平均来计算长时间轨道演化。该方法基于同调偏微分方程组的数值解，这些偏微分方程组是由近似恒等式变换得到的。另一种是自适应程序，它允许通过瞬时共振进行精确积分，这是通过匹配的渐近展开和适当的近共振重标度来完成的。