Computation of the lateral shift due to atmospheric refraction

Abstract

Context. Atmospheric refraction modifies the apparent position of objects in the sky. As a complement to the well-known angular offset, we computed the lateral translation that is to be considered for short-range applications, such as wavefront sensing and meteor trajectories. Aims. We aim to calculate the lateral shift at each altitude and study its variation according to meteorological conditions and the location of the observation site. We also pay special attention to the chromatism of this lateral shift. Moreover, we assess the relevance of the expressions present in the literature, which have been established neglecting Earth’s curvature. Methods. We extracted the variation equations of refraction from the geometric tracing of a light ray path. A numerical method and a dry atmosphere model allowed us to numerically integrate the system of coupled equations. In addition to this, based on Taylor expansions, we established three analytic approximations of the lateral shift, one of which is the one already known in the literature. We compared the three approximations to the numerical solution. All these estimators are included in a PYTHON 3.2 package, which is available online. Results. Using the numerical integration estimator, we calculated the lateral shift values for any zenith angle including low elevations. The shift is typically around 3 m at a zenith angle of 45°, 10 m at 65°, and even 300 m at 85°. Next, the study of the variability of the lateral shift as a function of wavelength shows differences of up to 2% between the visible and near infrared. Furthermore, we show that the flat Earth approximation of the lateral shift corresponds to its first-order Taylor expansion. The analysis of the errors of each approximation shows the ranges of validity of the three estimators as a function of the zenith angle. The ‘flat Earth’ estimator achieves a relative error of less than 1% up to 55°, while the new extended second-order estimators improves this result up to 75°. Conclusions. The flat Earth estimator is sufficient for applications where the zenith angle is below 55° (most high-resolution applications) but a refined estimator is necessary to estimate meteor trajectories at low elevations.