From dawn to dusk

Extract

It used to be the case in some jurisdictions that a rhythmic knell of the Angelus bell would mark the onset of dawn, noon and the passing of a day at dusk. Between these times there is daylight whose duration varies from place to place and from day to day and which can be predicted either exactly or approximately. An illuminating problem concerning the number of daylight hours at a winter solstice in London was posed recently and answered in the Problem Corner of The Mathematical Gazette [1]. It was shown, for example, that a calculation of daylight hours rested strictly upon the numerical solution to a transcendental trigonometric equation. Related references to earlier works in the Gazette involving a point source Sun were also given.

The purpose of this note is multifold. First, it is to point out that the above-cited equation might be viewed also as a “Sun-ray-to-Earth tangency condition”. Such a condition was developed earlier by the author in a publication that is now defunct [2], and so for completeness the steps necessary to establish the condition will be produced again here. Second, it will be evident from the manner of its derivation that the governing equation is valid at all orbit points, not just at a given solstice.