Multiperiodicity, modulations and flip-flops in variable star light curves

Abstract

Context. The light curves of variable stars are commonly described using simple trigonometric models, that make use of the assumption that the model parameters are constant in time. This assumption, however, is often violated, and consequently, time series models with components that vary slowly in time are of great interest. Aims. In this paper we introduce a class of data analysis and visualization methods which can be applied in many different contexts of variable star research, for example spotted stars, variables showing the Blazhko effect, and the spin-down of rapid rotators. The methods proposed are of explorative type, and can be of significant aid when performing a more thorough data analysis and interpretation with a more conventional method. Methods. Our methods are based on a straightforward decomposition of the input time series into a fast "clocking" periodicity and smooth modulating curves. The fast frequency, referred to as the carrier frequency, can be obtained from earlier observations (for instance in the case of photometric data the period can be obtained from independently measured radial velocities), postulated using some simple physical principles (Keplerian rotation laws in accretion disks), or estimated from the data as a certain mean frequency. The smooth modulating curves are described by trigonometric polynomials or splines. The data approximation procedures are based on standard computational packages implementing simple or constrained least-squares fit -type algorithms. Results. Using both artificially generated data sets and observed data, we demonstrate the utility of the proposed methods. Our interest is mainly focused on cases where multiperiodicity, trends or abrupt changes take place in the variable star light curves. Conclusions. The presented examples show that the proposed methods significantly enrich the traditional toolbox for variable star researchers. Applications of the methods to solve problems of astrophysical interest will be presented in the next papers of the series.

Multiperiodicity, modulations and flip-flops in variable star light curves

“…. 1.5 (see Pelt et al 2011). Next, proper estimates for the expansion coefficients were computed for every term in the series for the fixed carrier frequency ν 0 and D; this is a standard linear estimation procedure and can be implemented using any standard statistical package (see Pelt et al 2011, for a detailed description).…”

“…the review by Schwarzenberg-Czerny 2003, and references therein). Pelt et al (2011) developed a novel method for stellar light curve analysis based on the simple idea of decomposing the light curve into two separate components: a rapidly changing carrier that traces the regular part of the signal, and its slowly changing modulation. The carrier frequency can be obtained from observations (e.g.…”

“…Such very local events, for instance flaretype activity, can actually lead to local misfits in the light curve model for sparse data. For a complete description of the method we refer to Pelt et al (2011), and for another practical application for stellar light curves to Hackman et al (2013, analysis of FK Com); here we merely highlight the most important input quantities and properties of the method. We also stress that the CF procedure belongs to the class of explorative analysis methods; the qualitative results obtained from the phase diagrams should be quantified with more precise models with proper error estimates.…”

“…The next task of the analysis is to choose a suitable model for the modulators. In Pelt et al (2011) we proposed two different types of models based on either trigonometric or spline approximation. The II Peg light curve contains a fairly small amount of complicated features, making the usage of the trigonometric model adequate.…”

“…Whichever the cause or persistence of the drift, this type of problem is especially suited to be analyzed with the recently developed carrier fit (CF) analysis method (Pelt et al 2011), which is based on the idea of a continuous fitting of the time series (e.g., a light curve of a star) clocked by a certain shorter carrier period (or higher frequency) while being modulated by a longer period (or lower frequency). This method has previously been successfully applied to analyze the phase changes seen in the light curve of FK Com .…”