A historical method approach to teaching Kepler’s 2nd law

Abstract

Kepler’s 2nd law, the law of the areas, is usually taught in passing, between the 1^st and the 3^rd laws, to be explained “later on” as a consequence of angular momentum conservation. The 1^st and 3^rd laws receive the bulk of attention; the 1^st law because of the paradigm-shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3^rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2^nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3^rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler’s 2^nd is as revolutionary as the 1^st: as the 1^st law does away with the epicycle, the 2^nd law does away with the equant. This way of teaching the 2^nd law also formulates the “time=area” statement quantitatively, in the way of Kepler’s equation, M = E – e sin E, (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars’ next opposition. This method is partially based on Kepler’s original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.