Fractal analysis applied to light curves of δ Scuti stars★

Abstract

Fractal behaviour, i.e. scale invariance in spatio-temporal dynamics, have been found to describe and model many systems in nature, in particular fluid mechanics and geophysical related geometrical objects, like the convective boundary layer of cumulus cloud fields, topographic landscapes, solar granulation patterns, and observational astrophysical time series, like light curves of pulsating stars.The main interest in the study of fractal properties in such physical phenomena lies in the close relationships they have with chaotic and turbulent dynamic.In this work we introduce some statistical tools for fractal analysis of light curves: Rescaled Range Analysis (R/S), Multifractal Spectra Analysis, and Coarse Graining Spectral Analysis (CGSA), an FFT based algorithm, which can discriminate in a time series the stochastic fractal power spectra from the harmonic one.An interesting application of fractal analysis in asteroseismology concerns the joint use of all these tools in order to develop classification criteria and algorithms for δ-Scuti pulsating stars. In fact from the fractal and multi-fractal fingerprints in background noise of light curves we could infer on different mechanism of stellar dynamic, among them rotation, modes excitation and magnetic activity.

Toward a New Paradigm in the Analysis of Asteroseismic Lightcurves

“…We considered that most of the physical phenomena proposed to explain the complex oscillation spectrum of A-F type stars (see section 1) might be related with a specific fractal fingerprint. This hypothesis was confirmed by our study on the lightcurves of 15 CoRoT δ Sct stars (De Franciscis et al, 2018), for which different fractal fingerprints were found. Those fingerprints might provide clues for a better description of non-harmonic components of the lightcurve, such as turbulence, convection, or magnetic activity.…”

“…One of the first attempts to define fractals as mathematical objects was by mean of recursive continuous but non-differentiable functions, like the Weierstrass function studied by Pascual-Granado (2011). In this context, the fractal analysis, i.e., the study of them by mean of tools derived from the self-affine functions framework (see details in De Franciscis et al, 2018, and references therein), of time series comes up naturally. Indeed, it has also been used to study the rotational periods and flicker noise amplitudes of solar-like stars (De Freitas et al, 2013;de Freitas et al, 2016), and the magnetic activity of M dwarfs (de Freitas et al, 2017), although it is the first time that such an analysis was performed to classical pulsators, for which nonfractal fingerprints was expected to be found.…”

The potential of Shannon entropy to find the large separation of δ Scuti stars: The entropy spectrum

“…As for HD 49993, the light curves of both δ Scuti stars were observed by the CoRoT satellite (Seismo Field). Their oscillation spectra were obtained from the fractal analysis of the pre-whitening process (see De Franciscis et al, 2018;De Franciscis et al, 2019, for more details). That method allowed us to find the harmonic content of the light curve without imposing any statistical criterion about the significance, thereby minimizing the presence of spurious peaks, which is key to avoid any distortion in the échelle diagrams and hence on the computation of HSpec.…”

Analysis of pulsating variable stars using the visibility graph algorithm

“…The first two pulsate only in radial modes; however, δ Scuti stars pulsate in both radial and non-radial modes, making their light curve difficult to understand. For this reason, several studies have attempted to understand their pulsation mechanism [ 18 , 19 ], including its possible fractal behavior [ 20 ].…”