The perihelion of Mercury advance calculated in Newton's theory

TL;DR

This paper derives a Newtonian formula for Mercury's perihelion advance by analyzing different radii associated with a circle in curved space and their roles in gravitational and centrifugal forces.

Contribution

It introduces a novel approach to calculating Mercury's perihelion shift within Newtonian mechanics using geometric radii in curved space.

Findings

Derived a Newtonian formula for Mercury's perihelion advance.

Clarified the roles of different circle radii in Newtonian gravity.

Provided insights into geometric interpretations of gravitational forces.

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.