"Asymptotic Parabolic" Fit for Light Curves

A method is proposed for local approximation of light curves with nearly linear ascending and descending branches connected by relatively short phases of maximum or minimum. 1. Discussion Light curves of many pulsating stars and the minima of eclipsing variables may be satisfactorily explained by mathematical approximation as well as by the "O-C" diagrams of such stars whose periods abruptly change from one value to another. Andronov and Shakun (1990) have used hyperbolic functions to study abrupt "switches" of the outburst cycle length of some dwarf novae between two preferred values. Marsakova and Andronov (1996) proposed to use a more simple function which consists of two asymptotic lines connected with a parabola, so the function and its first derivative are continuous even at the "border points" T 1 and T 2 . The free parameters are the slopes of the asymptotes, the value at their crossing, and the arguments of the "border points." Assuming that the arguments of the data are t 1 ...t n and T 1 ≤ T 2 , one may obtain for this "asymptotic parabola" (AP) fit different functions: AP (t 1 < T 1 < T 2 < t n ); single line (T 2 ≤ t 1 or T 1 ≥ t n ); broken line (t 1 < T 1 = T 2 < t n ); ordinary parabola (T 1 ≤ t 1 < t n ≤ T 2 ). A program has been developed which allows the determination of T 1 and T 2 by using a method of differential corrections after their initial values are estimated by minimizing the r.m.s. deviation σ O-C of the residuals from the fit on a grid. The practical difficulty solved by this program is that it takes into account the variety of types of functions mentioned above which affect the equations for differential corrections. Figure 1 illustrates part of the light curve of W Lyr from the AFOEV database (Schweitzer 1993) and various fits for these 210 observations with m model parameters, i.e., the trigonometric polynomial (TP) of the first ( m = 3, 4) and second