A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding
From: Robin Stuart
Date: 2018 Jan 10, 08:25 -0800
Chauvenet is describing Bessel’s method but unfortunately much of his derivation involves using some tricks of trigonometric substitution to solve a quadratic equation which makes it a bit harder to follow. I personally think that Bessel’s method is conceptually the simplest approach to use. Everything is projected onto the so-called fundamental plane which runs through the centre of the Earth and its normal in the direction of the star. Effectively it’s the view of events of an observer on the star. Projected on the fundamental plane the Moon is a circle moving across the plane. The occultation occurs when the moving point that represents the observer’s position touches the circle. The x-y coordinates of the Moon’s centre on the fundamental plane are Besselian elements and were precomputed in Almanacs. To sufficient accuracy, the coordinates of the observer and centre of the Moon can be approximated as linear functions of time which results in a quadratic equation for the time (GMT) of the occultation. See for example https://babel.hathitrust.org/cgi/pt?id=nyp.33433108133277 p.630.
To get your longitude it is assumed you know your latitude and the local mean time (LMT) at which the occultation occurred. Since GMT=LMT-longitude you have a quadratic equation for your longitude,