Periodic Variable Stars Modulated by Time-Varying Parameters

Abstract

Many astrophysical phenomena are time-varying, in the sense that their brightness change over time. In the case of periodic stars, previous approaches assumed that changes in period, amplitude, and phase are well described by either parametric or piecewise-constant functions. With this paper, we introduce a new mathematical model for the description of the so-called modulated light curves, as found in periodic variable stars that exhibit smoothly time-varying parameters such as amplitude, frequency, and/or phase. Our model accounts for a smoothly time-varying trend, and a harmonic sum with smoothly time-varying weights. In this sense, our approach is flexible because it avoids restrictive assumptions (parametric or piecewise-constant) about the functional form of trend and amplitudes. We apply our methodology to the light curve of a pulsating RR Lyrae star characterised by the Blazhko effect. To estimate the time-varying parameters of our model, we develop a semi-parametric method for unequally spaced time series. The estimation of our time-varying curves translates into the estimation of time-invariant parameters that can be performed by ordinary least-squares, with the following two advantages: modeling and forecasting can be implemented in a parametric fashion, and we are able to cope with missing observations. To detect serial correlation in the residuals of our fitted model, we derive the mathematical definition of the spectral density for unequally spaced time series. The proposed method is designed to estimate smoothly time-varying trend and amplitudes, as well as the spectral density function of the errors. We provide simulation results and applications to real data.

arXiv comment26 pages, 6 figures, to be published in The Astrophysical Journal

References in corpus (14)

• Kepler Mission Design, Realized Photometric Performance, and Early Science
• On Machine-Learned Classification of Variable Stars with Sparse and Noisy Time-Series Data
• Flexible and Scalable Methods for Quantifying Stochastic Variability in the Era of Massive Time-Domain Astronomical Data Sets
• Flavours of variability: 29 RR Lyrae stars observed with Kepler
• On the Blazhko Effect in RR Lyrae Stars
• Autoregressive Times Series Methods for Time Domain Astronomy
• Long time-scale behavior of the Blazhko effect from rectified Kepler data
• Blazhko RR Lyrae light curves as modulated signals
• Extended Aperture Photometry of K2 RR Lyrae stars
• Blazhko Effect in the first overtone RR Lyrae stars of the OGLE Galactic bulge collection
• Dynamical structure of the pulsating atmosphere of RR Lyr - I. A typical pulsation cycle
• The light-curve modulation of XY And and UZ Vir - Two Blazhko RR Lyrae stars with additional frequencies
• Change point detection and image segmentation for time series of astrophysical images
• On the connection between almost periodic functions and Blazhko light curves