Particle Dynamics Through the Earth: A Study of Non-Zero Tunnel Diameters

A standard problem in undergraduate mechanics derives the trajectory of a particle that has been dropped through a gravity tunnel passing through the center of the Earth as simple harmonic oscillation, assuming constant earth density and negligible tunnel diameter. We show that when the more accurate PREM density profile and non-negligible tunnel diameters are taken into consideration, significant deviations from harmonic oscillations are observed along with reduced particle speeds. Furthermore, we report that a particle traveling perpendicular to the axis of the gravity tunnel is unable to reach the center of the earth beyond a critical tunnel diameter of 71.36% of the earth's diameter under a constant density, or 72.36% of the diameter under the PREM density model. When the rotation of the earth is incorporated, the particle trajectories occasionally extend beyond the surface of the earth at large tunnel diameters, owing to tunnel-induced non-uniformity of the earth's potential energy surface. In addition, we numerically calculate the brachistochrones connecting two points on the surface for various tunnel diameters using the PREM density profile. Our results highlight the effect of non-negligible tunnel diameters on the overall potential energy surface and particle dynamics.