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Register a new userMAVKA: Program Of Statistically Optimal Determination Of Phenomenological Parameters Of Extrema. Parabolic Spline Algorithm and Analysis of Variability of the Semi-Regular Star Z UMa
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Advanced MAVKA software for the approximation of extrema observations is used to analyze the variability of the brightness of pulsating and eclipsing stars, but may be useful in analyzing signals of any nature. A new algorithm using a parabolic (quadratic) spline is proposed. In contrast to the traditional definition of a spline as a piecewise-defined function at fixed intervals, a spline is proposed to be divided into three intervals, but the positions of the boundaries between the intervals are additional parameters. The spline defect is 1, that is, the function and its first derivative are continuous and the second derivative can be discontinuous at the boundaries. Such a function is an enhancement of the asymptotic parabola (Marsakova and Andronov 1996). The dependence of the fixed signal approximation accuracy on the location of the boundaries of the interval is considered. The parameter accuracy estimates using the least squares method and bootstrap are compared. The variability of the semi-regular pulsating star Z UMa is analyzed. The presence of multicomponent variability of an object, including, four periodic oscillations and significant variability of the amplitudes and phases of individual oscillations is shown.
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We introduce the program MAVKA for determination of characteristics of extrema using observations in the adjacent data intervals, with intended applications to variable stars, but it may be used for signals of arbitrary nature. We have used a dozen of basic functions, some of them use the interval near extremum without splitting the interval (algebraic polynomial in general form, Symmetrical algebraic polynomial using only even degrees of time (phase) deviation from the position of symmetry argument), others split the interval into 2 subintervals (a Taylor series of the New Algol Variable, the function of Prof. Z. Mikulav{s}ek), or even 3 parts (Asymptotic Parabola, Wall-Supported Parabola, Wall-Supported Line, Wall-Supported Asymptotic Parabola, Parabolic Spline of defect 1). The variety of methods allows to choose the best (statistically optimal) approximation for a given data sample. As the criterion, we use the accuracy of determination of the extremum. For all parameters, the statistical errors are determined. The methods are illustrated by applications to observations of pulsating and eclipsing variable stars, as well as to the exoplanet transits. They are used for the international campaigns Inter-Longitude Astronomy, Virtual Observatory and AstroInformatics. The program may be used for studies of individual objects, also using ground-based (NSVS, ASAS, WASP, CRTS et al.) and space (GAIA, KEPLER, HIPPARCOS/TYCHO, WISE et al.) surveys.
Multiple algorithms of time series analysis are briefly reviewed and partially illustrated by application to the visual observations of the semi-regular variable DY Per from the AFOEV database. These algorithms were implemented in the software MCV (Andronov and Baklanov, 2004), MAVKA (Andrych and Andronov, 2019; Andrych et al., 2019). Contrary to the methods of physical modeling, which need to use too many parameters, many of which may not be determined from pure photometry (like temperature/spectral class, radial velocities, mass ratio), phenomenological algorithms use smaller number of parameters. Beyond the classical algebraic polynomials, in the software MAVKA are implemented other algorithms, totally 21 approximations from 11 classes. Photometric observations of DY Per from the AFOEV international database were analyzed. The photometric period has switched from P=851.1d(4.1) to P=780.5d(2.7) after JD 2454187(9)d. A parameter of sinusoidality is introduced, which is equal to the ratio of effective semi-amplitudes of the signal determined from a sine fit and the running parabola scalegram.
Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons are spectrally determined in the class of convex piecewise smooth domains. Specifically, we show that if $Omega$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P_n$, then $Omega$ is congruent to $P_n$.
The structural and dynamical properties of star clusters are generally derived by means of the comparison between steady-state analytic models and the available observables. With the aim of studying the biases of this approach, we fitted different analytic models to simulated observations obtained from a suite of direct N-body simulations of star clusters in different stages of their evolution and under different levels of tidal stress to derive mass, mass function and degree of anisotropy. We find that masses can be under/over-estimated up to 50% depending on the degree of relaxation reached by the cluster, the available range of observed masses and distances of radial velocity measures from the cluster center and the strength of the tidal field. The mass function slope appears to be better constrainable and less sensitive to model inadequacies unless strongly dynamically evolved clusters and a non-optimal location of the measured luminosity function are considered. The degree and the characteristics of the anisotropy developed in the N-body simulations are not adequately reproduced by popular analytic models and can be detected only if accurate proper motions are available. We show how to reduce the uncertainties in the mass, mass-function and anisotropy estimation and provide predictions for the improvements expected when Gaia proper motions will be available in the near future.
SU UMa stars are characterized by superoutbursts which are brighter at maximum light and which last much longer than the more frequent ordinary outbursts of these dwarf novae. Although there are now more than 1180 SU UMa type dwarf novae catalogued, our knowledge on their superoutburst cycle length Cso was hitherto limited to about 6$%$ of the entire sample of known SU UMa stars. Using public data bases we have determined new Cso values for a total of 206 additional SU UMa stars in the range 17 d $<$ Cso $<$ 4590 d (including some ER UMa and WZ Sge type representants) within total time intervals between 2 and 57 years, and with an estimated uncertainty of $pm$11$%$. This way, we are increasing our present knowledge of Cso values by a factor $sim$3.8. Its distribution is characterized by a broad maximum around Cso $approx$ 270 days, and slowly decreasing numbers till Cso $approx$ 800 d. The domain Cso $>$ 450 d was unexplored until now; we add here 106 cases ($sim$51$%$ of our total sample) in this range of long cycles, implying a better statistical basis for future studies of their distribution. Our sample contains 16 known WZ Sge stars, and we propose WZ Sge membership for 5 others hitherto classified as ordinary SU UMa stars. Individual superoutburst timings deviate in average about $pm$7$%$ of the cycle length from their overall linear ephemeris, conrming a pronounced quasi-periodic repeatability of superoutbursts. All relevant parameters are listed with their errors, and a table with individual superoutburst epochs of our targets is given, enabling future researchers to combine our results with other (past or future) observations.
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