1. Gap formation in planetary ring systems

In the simulation below a planetary ring system is modeled by 3,000 non-interacting ring particles orbiting a central body/planet. In addition to the ring system, an inner moon is introduced (a small disk between the planet and the ring). The moon interacts with the planet and with each particle from the ring. The strength of the moon-particles interaction is about 2,000 weaker than the strength of the moon-planet and particle-planet interactions. Due to the resonance between the inner moon and the ring particles, which happens at very specific distances in the ring, a gap in the ring is formed (see, for example, "Rings of Saturn"). After a certain time the gap starts to expose itself and it can be clearly observed in the ring itself (the left-side window) and in the distributions shown in the right-side window: the orbital radius as a function of the angle (the top-right graph) and the ring particle distribution as a function of radius (the bottom-right graph). The red lines show the theoretical prediction for the position of the gap.

The students involved in the project: Peter Andonov and Amaru Alzogaray.

The simulation was done in C++ with SDL2 graphics library.

2. 2-D Pendulum

This project models the motion of a 2-D physical pendulum. The two pendula shown below have identical physical parameters. They start motion with the same initial conditions and should ideally have identical trajectories. The only difference between them is that the left-side pendulum (blue) employs a more accurate algorithm to compute its trajectory. The algorithm's accuracy for each pendulum is presented in the right-side window. It can be related to the deviation of the blue and green curves from the horizontal middle line. The curves represent the mechanical energies of the pendula, which are supposed to stay constant during the motion. The greater deviation from the horizontal line, the less accurate the algorithm and the trajectory. The simulation shows that the less accurate algorithm fails to describe the motion at long time intervals, and the corresponding pendulum starts to deviate from the constant energy surface rapidly.

The students involved in the project: Edesa Montgomery, Tasmia Silmi, Sang Won Park, Adrian Echeverria.

The simulation was done in C++ with SDL2 graphics library.

3. Diffusion

The simulation below demonstrates a diffusion process of a gas or a fluid modeled by 20,000 non-interacting particles. Initially the entire gas is concentrated at the origin. Then the gas particles start to randomly "walk" with equal probabilities with steps left/right and up/down. The graphs in the right-side window show the distributions of the particles in the gas along the horizontal direction (the top graph) and along the radius (the bottom graph). Both distributions are in good agreement with the corresponding theoretical predictions, which have a pure Gaussian form. The simulated distributions are presented by the histograms and the theoretical distributions by the solid blue and red curves. With time the gas spreads in both directions and its average position stays at the origin, and the width of the gas distribution increases with time as \( \sigma \sim \sqrt{t}\).

The students involved in the project: Gabriel Diaz Espinoza, Khendo Lama, Saima Tarannum, Tamim Khan.

The simulation was done in C++ with SDL2 graphics library.

4. Brownian motion

This project started with modeling elastic collisions of two rigid balls and then continued with simulating billiard-like systems in which the balls are confined by a frame. This simulation is perfect for studying thermalization and other statistical properties of chaotic many-body systems such as gases and fluids. Below is an example of how it can be used to visualize the Brownian motion - motion of a heavy particle suspended in a gas or liquid. The graph in the right-side window shows the energy distribution of the gas particles: the black histogram represents the modeled distribution and the red curve corresponds to the theoretical prediction when the system has reached its thermal equilibrium called the Boltzmann distribution. The simulation shows that the system reaches the equilibrium very fast. Surprisingly only several collisions is enough to reach the equilibrium.

The students involved in the project: Silvio Molina, Rafael Paulino, Keven Juela, Paul Bravo, Javier Reyes.

The simulation was done in C++ with SDL2 graphics library.

5. Perihelion precession of planetary orbits

In this project we studied the precession of planetary orbits due to non-Keplerian central forces. Such perturbing forces may originate from variety of sources – from dark matter to the Einstein’s general theory of relativity, and usually result in the slow rotation of planetary orbits relative to the center of mass of the system. The simulation shown below demonstrates an example of such a precession due to an additional force proportional to \( F \sim 1/r^4 \), which is a standard general relativity correction (see, for example, "Tests of general relativity") to the Newton’s law of gravitation. The graph in the right-side window shows how the angle of the planet’s perihelion (the closest approach to the hosting star) changes from one complete orbit to another. The initial angle is close to 180°, then it starts to slowly change to 0°, and continues to decrease to -180°. The theory prediction is shown by the dashed red line. There is certain deviation between the computed precession and the theory prediction, and we believe that this deviation is due to inaccuracy of the simulation algorithm.

The students involved in the project: Rudra Chaudhary, Suyasha Tamang, Daniel Paulino, Kushal Shrestha, Orlando Flores.

The simulation was done in C++ with SDL2 graphics library.