Abstract

Many astrophysical phenomena are time-varying, in the sense that their brightness changes 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 the trend and amplitudes. We apply our methodology to the light curve of a pulsating RR Lyrae star characterized 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 trends and amplitudes, as well as the spectral density function of the errors. We provide simulation results and applications to real data.