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

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.