Approximation of the electron–proton mass ratio as a series in powers of $$\uppi $$ π

Abstract

Eddington in 1923, first identified four dimensionless numbers, derived from combinations of the basic physical constants, which are known as the “Eddington constants”. In formulating these dimensionless numbers, Eddington, a leading physicist of his time, claimed that they are characteristic of the structure and dynamics of the Universe at large, on the microscopical scale and at the macroscopical scale. Recently, there has been suggested a possible way of accounting for the magnitude of one of these four dimensionless constants, indicated as the “fine structure constant”, $$\upalpha $$ , that first emerged from studies of the atomic line spectrum of H. A simple power series in the product $$\hbox {e}\cdot \uppi $$ has been proposed, that fits the measured value of the fine structure constant to better than 9999 parts in 10,000. Following along these lines, the authors here propose a simple power series expansion in $$\uppi $$ that agrees with the currently accepted measurement of the value of the electron–proton mass ratio (m/M), or $$\upbeta $$ , to better than 999 parts in 1000.