Elliptical Orbit ⇒ 1/r2 Force

Abstract

Newton's proof of the connection between elliptical orbits and inverse-square forces ranks among the “top ten” calculations in the history of science. This time-honored calculation is a highlight in an upper-level mechanics course. It would be worthwhile if students in introductory physics could prove the relation elliptical orbit ⇒ 1/r2 force without having to rely on upper-level mathematics. We introduce a simple procedure—Newton's Recipe—that allows students to readily and accurately deduce the algebraic form of force laws from a geometric analysis of orbit shapes.

Weighing the Sun with five photographs

“…Therefore, we have proved the Kepler's first law. This law establishes that elliptical orbits are direct consequence of the Newton central gravitational force (that is inverse to the square of the distance) owe to the central star [7,8].…”

“…In our case, when the diameter of the Sun is assumed with a value given in equation (3), we can readily calculate distance r by using photographs and equation (9). In table 1, we tabulate the different photographs (ordered by date) with their corresponding values of r o and r aS measured directly from the photographs (see figure 2), together with their calculated values of f aS and r obtained from equations (8) and (9).…”

“…Estimation of distances r to the Sun, given in equation(9), at different dates using telescope and image measurements. , let us then assume that θ=2π in one year.…”