Free falling inside flattened spheroids: Gravity tunnels with no exit

Abstract

A “gravity tunnel” is the name given to a fictitious deep shaft drilled inside the Earth so that objects dropped from the surface of the Earth would free fall without ever touching the walls. It is well known that because of the rotation of the Earth, such tunnels are not straight lines but instead they emerge westward of the antipodal point, when the Earth is approximated as a rotating sphere. In this article, we determine the shape of gravity tunnels by taking into account the polar flattening of the Earth resulting from its rotation. The Earth is described as a McLaurin spheroid, an exact equilibrium shape for a rotating homogeneous deformable body that provides a fair description of the actual shape of the Earth. It turns out that the gravitational force acting on an object located inside the spheroid has a simple form (it is harmonic), so that it is straightforward to compute analytically the free fall trajectories. This study follows a procedure presented several times in this journal and elsewhere, i.e., the trajectory is first computed in the geocentric (non-rotating) frame, and it is then analysed in the terrestrial (rotating) frame. We find that when the flattening of the Earth is taken into account, gravity tunnels have no exit: an object dropped from a point of the surface (other than on the equator or a pole) never reaches the surface again, unless the flattening has very specific (and unnatural) values. We also compute the deviations from the vertical for short falls and compare them to standard eastwards and southwards deviation expressions obtained with other modelizations of the gravity of the rotating Earth, in particular, for a rotating spherical body.