Luca Cimbaro has a lifelong passion for studying matter, energy and mathematics. In 2015, Luca received an MSc in Physics from Imperial College London. He is completing his PhD in Condensed Matter Theory at Imperial College London nowadays. His research interests concern the mechanical properties of matter, such as material toughness. Luca’s other interests are varied. He is an avid reader and appreciates creative, original and novel ideas. He is enthusiastic about sports and traveling. He enjoys playing chess, live events, visiting museums and expositions, and sunny days.
Computational Essay: The Golden Ratio
Project: The geographic distance on asteroids
Asteroids formed five billion years ago in the solar system. Having failed to integrate with planets, they are the largest space rocks orbiting the Sun. Their size ranges from a few meters to hundreds of kilometers across, and their weight is much less than the moon. As a result of their formation and impacts, they have irregular shapes. Their heterogeneous surfaces are also heavily cratered. What is the shortest path between two distinct points on the surface of the asteroid?
Summary of Results
The geodesic is calculated using differential geometry. The asteroid’s shape is imported from the NASA databases. Because in this data the asteroid’s surface is discrete, a linear superposition of spherical harmonics is used to calculate a smooth two-dimensional surface. Arbitrary precision can be achieved for larger sets of spherical harmonics. The smooth surface is then used to solve the nonlinear second-order differential equations for the geodesic. Boundary conditions on the differential equation can be used to define the initial and the final points of the geodesic. For large numbers of geodesic, the functional representation of the Christoffel symbols will reduce the time needed to compute the differential equation. Because the model does not assume any constraint on the shape of the asteroid’s surface, it can be used to calculate the geodesic on any irregular surface.
What is the length of the geodesic? This length can be calculated using the metric tensor. A limitation of the current geodesic is that the polar and the azimuthal angle cannot assume values outside their domain. How does the solution of the nonlinear second-order differential equation change for arbitrary angles? Finally, for simplicity, the present project uses the boundary conditions to define the initial point of the geodesic and its rate of change at this point. What is the fastest method to solve the differential equation with boundary conditions at the initial and final points of the geodesic?