Dept. of Geosciences Colloquium: Survival and extinction of mean-motion resonances in the three-body problem and in planetary systems

Shuki koriski, TAU (PhD Thesis defence)

12 April 2021, 11:00 
Dept. of Geosciences Colloquium




We study the mean-motion resonance (hereafter MMR) in the three-body problem, trying to answer the question whether MMRs are destined to suffer dynamical disruption and if so, whether the reason for this may be due only to the chaotic nature of the problem. The study carries us from the vastness of extrasolar planetary systems to the simple driven pendulum model that mimics the dynamic behavior of three-body system in resonance. 


The thesis contains three parts: 

- In the first part the goal is to determine if MMRs are eternal phenomena or if MMR configurations tend to break down over time, as suggested in several studies. To answer this, we compare the average age of systems in MMR against the average age of systems that are not in MMR. If MMR tends to
vanish, we would expect to see a shortage of MMRs around older stars. We show that the 2/1 MMR, which is quite common among planetary systems, tends to be disrupted after a few Gyr. Therefore at least, the 2/1 MMRs are not eternal.

We go further to explore whether this finite nature is inherent to the pure three-body problem (of point particles), or it has to arise because of other intervening factors, like tidal effects or other bodies. As a research hypothesis we suggest that the chaotic nature of threebody systems is a possible cause for MMR disruption.

- In the second part we intend to better understand the impact of chaos upon the stability of MMRs in the three-body problem. We simplify the problem by the commonly used toy model of a periodically driven pendulum to approximate the behavior of the threebody system near resonance. We study the statistics of energy time series of the pendulum during its irregular and chaotic motion near the separatrix and find typical patterns which may serve as indicators of chaotic behavior. Familiarity with these patterns helps us to distinguish between regular and chaotic behavior in the next part of our work.

- In the third part, we use the energy time series method to study the typical times and rates over which a driven pendulum, initially in stable motion, crosses for the firsttime specific energy levels (hereafter, disruptive energies). The disruptive energies were chosen from within the energy zone in which the pendulum motion is significantly chaotic and thus unstable. In this model, an unstable motion of the pendulum is analogous to the behavior of the resonance argument in the three-body problem when the system is not in resonance. To improve the similarity to the threebody problem, we subject the pendulum to a parametric driving force that mathematically resembles the disturbing function of the three-body problem when expanded in Legendre polynomials. We perform thousands of numerical integrations for various parameters of the driving force and record the exact times at which the
instantaneous pendulum energy crosses initially each of the disruptive energies. The results show that, for a wide range of driver parameters, the set of energy crossing times for each of the disruptive energies can be assumed to obey an exponential distribution. Thus, the events that cause the system to become unstable occur randomly but at a constant mean rate. 


We show that within a family of driven pendulums that differ in their driver parameters and are initially in a stable librating motion, some will maintain their stable motion eternally and some will eventually cross the typical disruptive energies and, according to our definitions, will become unstable. In addition, we formulate the dependence between the parameters that define the driving force and the typical time on which a stable motion becomes unstable. This dependence provides a simple condition that can be used as a tool to distinguish between stable and unstable pendulums.


In the thesis conclusion we suggest that the method of disturbing energy, used in the pendulum case, may be applied to the three-body problem as well. That is, for every threebody system there is a typical energy threshold, dependent on the physical parameters of the system. For some parameter values, the system will never attain the energy threshold and will stay forever in MMR. In contrast, systems with different combination of parameters will eventually lose their MMR state and become unstable. This instability can change the system configuration and may practically cause its destruction. As for the pendulum case, we hypothetically suggest for the three-body problem a simple relation, based on the masses and the trajectories of the rotating bodies, to serve as rough indicator for stability of MMRs.


If our assumptions are correct, this work supports the hypothesis that disruption of MMRs in planetary systems may be an evolutionary process that stems only from the chaotic nature of the 3-body problem, and does not necessarily rely on cataclysmic events or forces external to the three-body system of point masses.




Event Organizers: Dr. Roy Barkan and Dr. Asaf Inbal



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