Phase Plane Analysis of the Photometrical Variations of Long-Period Variables

TL;DR

This paper applies phase plane analysis to long-period variable stars, using historical observational data to characterize their limit cycles and photometric variability patterns.

Contribution

It introduces a method to analyze the photometrical variations of Mira-type and semi-regular variables through phase plane diagrams and limit cycle characterization.

Findings

Limit cycles correspond to the stars' auto-oscillation processes.

Simple sine-like light curves produce elliptical limit cycles.

Complex light curves show deviations from simple elliptical limit cycles.

Abstract

Using the phase plane diagrams, the phase light curves of a group of the Mira-type stars and semi-regular variables are analyzed. As generalized coordinates x and $\dot{x}$, we have used m - the brightness of the star and its phase derivative. We have used mean phase light curves using observations of various authors. The data typically span a large time interval (nearly a century). They were compiled from the databases of AAVSO, AFOEV, VSOLJ, ASAS and approximated using a trigonometric polynomial of statistically optimal degree. As the resulting approximation characterizes the auto-oscillation process, which leads to a photometrical variability, the phase diagram corresponds to a limit cycle. For all stars studied, the limit cycles were computed. For a simple sine-like light curve, in e.g., L$_2$ Pup, the limit cycle is a simple ellipse. In a case of more complicated light curve, in which harmonics are statistically significant, the limit cycle has deviations from the ellipse. In an addition to a classical analysis, we use the error estimates of the smoothing function and its derivative to constrain an "error corridor" in the phase plane.