Gravity Tunnel Dynamics at Non-zero Tunnel Diameters

Abstract

A standard exercise 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 uniform earth density and negligible tunnel radius. We show that when the more accurate Preliminary Reference Earth Model (PREM) density profile and non-negligible tunnel radii are taken into consideration, significant deviations from harmonic oscillation are observed along with increasing or decreasing average particle speeds. Furthermore, we report that a particle traveling along a diameter perpendicular to the axis of the gravity tunnel is unable to reach the center of the earth beyond a critical tunnel radius of 71.36% of the earth’s radius in the uniform density model, or 72.36% of the radius in the PREM density model. When the rotation of the earth is incorporated, the particle released from the surface is observed to form planar trajectories that occasionally extend beyond the surface of the earth at large tunnel radii, owing to tunnel-induced non-uniformity of the earth\u27s potential energy surface. At the same time, the traversal times from surface to center can be closely approximated by the assumption of constant gravity even at large tunnel radii. The results presented here demonstrate the effect on the overall potential energy surface and particle dynamics when the tunnel radius is not negligible