A Barometric Exponential Model of the Atmosphere's Refractive Index: Zenith Angles and Second Order Aberration in the Entrance Pupil

TL;DR

This paper develops a mathematical model of atmospheric refraction using an exponential decay profile, analyzing zenith angle deviations and optical path differences for telescopic observations.

Contribution

It introduces a novel exponential model for atmospheric refractive index and derives analytical expressions for zenith angle correction and path length distribution.

Findings

Quantifies differential zenith angles due to atmospheric refraction.

Provides analytical formulas for optical path length variations.

Enhances understanding of second order aberrations in telescopic imaging.

Abstract

This report models the refractive index above a telescope site by an atmosphere with exponential decay of the refractive index (susceptibility) as a function of altitude. The air is represented as a spherical hull around the surface. We compute (i) the differential zenith angle -- the difference between the actual zenith angle at arrival of rays at the telescope and a hypothetical zenith angle without the atmosphere -- and (ii) the optical path length distribution of rays at arrival in the entrance pupil as a function of the distance to the pupil center. The key technique in this work is to expand some integrals -- that depend on the refractive index profile along the curved path of each ray from some virtual plane in the direction of the star up to the entrance pupil -- in power series of small parameters.