Numerical Tools for Orbital Dynamics of Planetary Systems

Project #1: A planet and star interacting under gravity result in a fixed elliptical orbit. With any more planets, the N-body problem is not analytically solvable. Nevertheless, several approximate models have been found that work well in certain regions of parameter space. The general idea is always to expand the gravitational potential in terms of the planets' orbital elements, isolate a small number of terms that are important, and argue that the remaining terms average out over time. This approach works well for many applications, but a very general problem in the field is that of how to choose initial conditions. In principle, one would need to translate the initial conditions of interest into *averaged* initial conditions appropriate to the averaged model. But how do you average over an infinite number of terms? This leads to a number of ad hoc approaches, including trying to average over a few of the most important terms, or inefficient searches for the initial conditions that most closely match the averaged and full models. The goal of the project to develop a general method to do this transformation to high accuracy.

Our approach will be to use splitting methods to construct the appropriate transformation. While we do not have a way to average over an infinite number of terms, we have existing integrators that can evolve the system under (i) the full equations (involving all terms) and (ii) the handful of isolated terms in the expansion. By composing sequences of such steps, e.g., integrate for a timestep according to the full equations, then integrate backward for a timestep using the analytical model with only a few terms, we obtain an approximation to the infinite number of neglected terms. Because the errors in such splitting schemes are well understood, we will be able to sequentially construct higher order sequences of steps forward and backward in time to obtain progressively higher accuracy. Such methods have already been developed for iterated maps, showing that this approach is feasible and likely to succeed. We also have an existing integrator for the full equations of motion in hand (REBOUND), as well as a package for integrating analytical models involving a subset of user-specified terms in the interaction potential, so most of the technical pieces are already available.

This is a very broad problem applicable to all analytical models in orbital dynamics, and more generally to analytical models that isolate particular terms in an expansion of a more complicated potential. The solution we seek is formulated in a way that's agnostic to the particular analytic model used, and by implementing it in our open-source celmech package, we will allow users to be able to calculate corrections to their initial conditions for any orbital dynamics problem. 

Project # 2: A central unsolved problem in planetary dynamics is predicting whether a given planetary system's orbital configuration will be long-term stable. We have developed a machine-learning model called SPOCK that uses physically motivated features (derived from the parts of the dynamics that we understand) to make long-term predictions. Our theoretical understanding has improved over the last few years, and the SPOCK model and training dataset provides a valuable testbed for confirming these theoretical insights, identifying exceptions to the rule, and identifying new directions for theoretical development. One thing that has become clear is that there are multiple dynamical mechanisms for driving chaos and instabilities. 

The project is to use the existing black-box machine-learning model and dataset, together with our physical understanding, to create a "white-box" analytical decision tree, that given a population of planetary systems, would sequentially assign them to different bins corresponding to different instability mechanisms. Ones that traversed this 'gauntlet' without being assigned to an instability mechanism would then be stable. By analyzing the systems mistakenly assigned by this analytical classifier, we will refine the selection criteria and improve our theoretical understanding.