Remarkable properties of astronomical refraction in a spherically symmetric atmosphere

The general theoretical properties and various approximate formulas for the angle chi(S) of astronomical refraction at S are presented, beginning with its fundamental expression by the refraction integral and immediately leading to the Simpson formula. Assuming a spherically symmetric atmosphere, many additional results show up, starting with Oriani's "theorem" and Bouguer's equation. From these, an usual form of the refraction integral is derived. A fundamental approximation follows to deduce the Laplace formula, then the series expansions that generalize it but are divergent for an isothermal atmosphere-because of the approximation made. Two results, free from any approximation, exist and are presented: Biot's and Biot-Sang-Meyer-Fraser-White's theorems. It leads to an approximate link with extinction, as found by Laplace, and an approximation of the refraction chi(Sh) for points on the astronomical horizon. We compute the chromatic effects on chi(S), chi(Sh)-and (in a document associated with this article) on various characteristics of optical ducts considered in air-and an original expression of the "local vertical angular distortion coefficient" on the horizon (in Annexe B).