Detailed demonstrations of the relations between coordinate systems in the celestial sphere

Abstract

Spherical Geometry is the basis for what we know as Positional Astronomy. This is one of the oldest approaches to Astronomy as a science, being used by ancient Greeks and possibly by other people before that. Concerning a more formal current mathematical description, there are several types of coordinate systems with respect to the celestial sphere. Each system differs in the choice of its referential plan. Several connections between elements of different types of coordinates are well known once it is a well-developed field. However, the reader will not find in the literature complete mathematical proofs for such formulas. Thus, this article fills that gap. We present in this text several formulas relating the coordinates Zenital Distance, Hour Angle, Azimuth, Declination and Geographic Latitude and their mathematical proofs in detail, explaining all possible steps. Our main contribution is in the form of the presentation and deduction of classical results from Positional Astronomy.