Airplane Orbits and Satellite Orbits and Orbitfall: Physics Hidden in Plain Sight

Abstract

An airplane flying at constant speed and altitude is an example of physics invisible to the pilots and passengers, but visible to remote observers and manifest in the mathematics. The optimum flight path is an arc of a Great Circle, specifically that circle which is the result of rotating the equator to intersect the origin and destination airports. In order that the velocity vector remain parallel to the surface of the spherical earth, a centripetal force is required to rotate the velocity without altering its magnitude. This force must be radially inward and thus parallel to the local vertical. The assertion that ``lift balances gravity" is only an approximation. To follow the curve of the earth, the vertical component of aerodynamic lift must be \emph{slightly weaker} than gravity so that the plane can be in ``orbitfall". That is, to follow the curvature of the earth, maintaining a constant distance $R$ from the center of the earth while flying at a constant speed $V$ and generating a vertical lift per unit mass $L$, the plane and all inside it must fall towards the center of the earth with an acceleration, the ``orbitfall acceleration", $g_{orbitfall} \equiv g-L=V^{2} / R$ where $g$ is the usual gravitational acceleration constant. If the plane travels a fraction $\digamma$ of the earth's circumference , then it must execute $\digamma$ of an outside loop. This requires that the pitch angle $\xi$ must rotate nose-down at a rate of $d \xi / dt=V/R$. The dynamic stability of a non-military aircraft makes this pitch change continuously without pilot intervention: the horizontal stabilizers act as weathervanes, suppressing small perturbations so that the longitudinal axis remains at a constant angle of attack. Neither pilot nor passenger is aware of the orbitfall or the automatic pitch changes -- physics invisible but essential.