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Week 1 Update

I finally made it to Canada and just finished my first full week at the Fields Institute!

St. George Street
One of the campus dining
halls
Fields is housed on the University of Toronto's campus, and the participants of the program are given accommodations at one of the university's residence halls. Over the past week and a half, in addition to my research, I've been exploring the campus and neighborhood. I'm really enjoying getting to live in Toronto for the summer (especially because it's not as hot and humid as Atlanta right now)!

Bloor Street West
Thomas Fisher Rare Books
Library
As part of the research program, I have an office at the Fields Institute that I share with the two other members of my group the Project 8 group. Fields is great; for one, I don't think there are any whiteboards in the building - only blackboards.  There are also conferences and talks that are going on many of the days, so that means there's also free coffee available almost all the time. It's been amazing to get to interact with so many people from different countries and cultures, and I'm so excited to spend the rest of my summer with them!






Now to explain my research a bit more fully: If you'll recall, the title is "Development and Evaluation of a Hybrid Symplectic Integrator for Planetary Systems." I'll start by explaining what an integrator is: some equations involving motion or change cannot be solved exactly, so integrators are used. The integrators we've examined are the Euler method and the Leapfrog method. The Euler method was developed unsurprisingly by mathematician Leonhard Euler, and the Leapfrog method is so named because it evaluates equations by calculating intermediate values used to calculate other final values; sort of "leapfrogging" over the intermediate value.

Because integrators are essentially approximations, there's some error generated, especially when we want to evaluate things like orbits of planets. When using integrators to evaluate orbits, we need the integrators to preserve quantities like energy and momentum so they accurately represents the orbits. Consider the following plots, implemented with the programming language Python:


The first is an orbit modeled with the standard Euler integrator. Notice that the orbit starts in the middle and spirals outward. From a physical standpoint, this means that the total energy of the system is spontaneously increasing, which does not depict reality; an orbit should conserve energy. The second plot, however, is the symplectic Euler integrator. Unlike the first orbit, this one is an ellipse, which implies that the energy is in fact conserved. 

This same method is adopted in a 20-year-old integrator called the Wisdom-Holman method. The current method used to evaluate orbits is to use the Wisdom-Holman method in general but switch to a more precise but more computationally expensive integrator in certain conditions. The issue we're investigating is the transition between integrators. Our research will focus on improving this transition so it maintains important properties of the symplectic integrator.

Here's a plot of the orbits of Jupiter, Saturn, Uranus, Neptune, and Pluto as evaluated with the Wisdom-Holman method:

(I wrote the code that implements the integrators, so any inaccuracies are due to my programming, not the method)

This past week and a half has been incredible, and I'm really looking forward to the rest of the summer!

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