General formulation of optical paths for large zenith angles in the earth's curved atmosphere

Formulas that can be used to determine the optical path between two points along an atmospheric ray path are derived for the case when the local zenith angle of the ray path is larger than 70 deg. For angles less than 70 deg, these formulas reduce to the airmass function; viz., the secant of the zenith angle. The formulation presented in this paper is general enough to be applicable to a wide variety of atmospheric conditions, such as spherical and nonspherical atmospheres, and vertically and horizontally homogeneous as well as inhomogeneous atmospheres. A formulation for the case when atmospheric refraction is important also is presented here.