Measuring the elliptical form of the Moon’s terminator curve and its illuminated area by direct and indirect methods

Because solar light, from Earth perspective, a curve is drawn in Moon’s surface that separates its dark side from the illuminated one. This curve is known as the terminator curve. In this article we prove, using direct and indirect methods, that the terminator curve corresponds to an ellipse. This is demonstrated using mathematical concepts and photographs of the Moon that are analysed with a geometrical software. Using this information, we also show how to calculate the illuminated fraction area of the Moon depending on its day of rotation. We obtain excellent approximations regarding the values given by computational systems. We discuss the results of considering the Moon as a flat disk or like a sphere. We analyse the technical difficulties of the process and the mathematical tools needed for more precise calculations. We also put in context this demonstration of the ellipticity of the terminator curve for any interior planet illuminated by a central star of any planetary system, seen from a outsider planet, the case in which the phases are more noticeable. Finally we extend the previous calculations to analyse the illuminated percentage of the inner planets Mercury and Venus, obtaining excellent results in the case of Venus.