Useful relations for the analysis of stellar scintillation at the entrance pupil of a telescope

TL;DR

This paper introduces new theoretical tools and approximations for analyzing stellar scintillation caused by atmospheric turbulence, facilitating more accurate characterization of optical turbulence in telescopic observations.

Contribution

It presents a dimensionless Fresnel filter for approximating polychromatic weighting functions and derives analytical expressions for specific aperture geometries, advancing scintillation analysis methods.

Findings

The dimensionless WF can be scaled for any aperture size.

Analytical WF expression for circular aperture with monochromatic radiation.

The WF for square and circular apertures match within 1% when the diameter is 1.15 times the square side.

Abstract

The development of new techniques for characterizing atmospheric optical turbulence (OT) has become an active topic of research again in recent years. In order to facilitate these studies, we reconsidered known theoretical results and obtained some new practically useful conclusions. We introduce a dimensionless Fresnel filter, which allows us to approximate a polychromatic weighting function (WF) by a monochromatic one with a typical precision of several percent. A so-called dimensionless WF can be easily scaled for a receiving aperture of any size. For the case of a circular aperture and monochromatic radiation, an analytical expression for the WF was found. The WFs for a square aperture and for a circular aperture match with relative difference less than 0.01 if the circular aperture diameter is 1.15 times larger than the square aperture side. A linear digital filter can be applied to the scintillation signal from an image detector. As an example of digital filtering, we considered the power law filter $\propto f^{5/3}$ with the WF being constant in a wide range of altitudes. We discuss the main limitations of this approach for measuring OT integral: finite pixel size, aliasing, and finite image detector size.