The perihelion of Mercury advance calculated in Newton's theory

Abstract

Three radii are associated with a circle: the "geodesic radius" R_1 which is the distance from circle's center to its perimeter, the "circumferential radius" R_2 which is the length of the perimeter divided by 2 pi and the "curvature radius" R_3 which is circle's curvature radius in the Frenet sense. In the flat Euclidean geometry it is R_1 = R_2 = R_3, but in a curved space these three radii are different. I show that although Newton's dynamics uses Euclidean geometry, its equations that describe circular motion in spherical gravity always unambiguously refer to one particular radius of the three --- geodesic, circumferential, or curvature. For example, the gravitational force is given by F = -GMm/(R_2)^2, and the centrifugal force by mv^2/R_3. Building on this, I derive a Newtonian formula for the perihelion of Mercury advance.