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The software MAVKA is described, which was elaborated for statistically optimal determination of the characteristics of the extrema of 1000+ variable stars of different types, mainly eclipsing and pulsating. The approximations are phenomenological, but not physical. As often, the discovery of a new variable star is made on time series of a single-filter (single-channel) data, and there is no possibility to determine parameters needed for physical modelling (e.g. temperature, radial velocities, mass ratio of binaries). Besides classical polynomial approximation "AP" (we limited the degree of the polynomial from 2 to 9), there are realized symmetrical approximations (symmetrical polynomials "SP", "wall-supported" horizontal line "WSL" and parabola "WSP", restricted polynomials of non-integer order based on approximations of the functions proposed by Andronov (2012) and Mikulasek (2015) and generally asymmetric functions (asymptotic parabola "AP", parabolic spline "PS", generalized hyperbolic secant function "SECH" and "log-normal-like" "BSK"). This software is a successor of the "Observation Obscurer" with some features for the variable star research, including a block for "running parabola" "RP" scalegram and approximation. Whereas the RP is oriented on approximation of the complete data set. MAVKA is pointed to parts of the light curve close to extrema (including total eclipses and transits of stars and exoplanets). The functions for wider intervals, covering the eclipse totally, were discussed in Andronov (2017). Global and local approximations are reviewed in Andronov (2020). The software is available at http://uavso.org.ua/mavka and https://katerynaandrych.wixsite.com/mavka. We have analyzed the data from own observations, as well as from monitoring obtained at ground-based and space (currently, mainly, TESS) observatories. It may be used for signals of any nature.

The methods for determination of the characteristics of the extrema are discussed with an application to irregularly spaced data, which are characteristic for photometrical observations of variable stars. We introduce new special functions, which were named as the "Wall-Supported Polynomial" (WSP) of different orders. It is a parabola (WSP), constant line (WSL) or an "asymptotic" parabola (WSAP) with "walls" corresponding to more inclined descending and ascending branches of the light curve. As the interval is split generally into 3 parts, the approximations may be classified as a "non-polynomial spline". These approximations extend a parabolic/linear fit by adding the "walls" with a shape, which asymptotically corresponds to the brightness variations near phases of the inner contact. The fits are compared to that proposed by Andronov (2010, 2012) and Mikulasek (2015) and modified for the case of data near the bottom of eclipses instead of wider intervals of the light curve. The WSL method is preferred for total eclipses showing a brightness standstill. The WSP and WSAP may be generally recommended in a case of transit eclipses, especially by exoplanets. Other two methods, as well as the symmetrical polynomials of statistically optimal order, may be recommended in a general case of non-total eclipses. The method was illustrated by application to observations of a newly discovered eclipsing binary GSC 3692-00624 = 2MASS J01560160+5744488, for which the WSL method provides 12 times better accuracy., Odessa Astronomical Publications, vol. 30 (2017)

A comparative analysis of the special shapes (patterns, profiles) of the eclipses applied for the phenomenological modeling of the light curves of eclipsing binary stars is conducted. Families of functions are considered, generalizing local approximations (Andronov, 2010, 2012) and the functions theoretically unlimited in a width, based on a Gaussian (Mikulasek, 2015). For an analysis, the light curve of the star V0882 Car = 2MASS J11080308 - 6145589 of the classic Algol - subtype (\beta Persei) is used. By analyzing dozens of modified functions with additional parameters, it was chosen the 14 best ones according to the criterion of the least sum of squares of deviations. The best are the functions with an additional parameter, describing profiles, which are limited in phase., Comment: 15 pages, 3 figures

In this paper, analysis results of the photometric data of DO Dra will be presented. DO Dra had been observed with 1 m LOAO telescope and 0.6 m CBNUO telescope from 2005 through 2014. The data shows kind of periodic oscillation behavior in the orbital period and also in the spin period. It has been found that these QPOs are not observed always and that the periods vary from 30 min to 80 min. We also found that the period variation seems to repeat itself with the period of 13.5 days. It is essential to monitor this object in the future as well as to carry out model calculation in order to have better understanding of these QPO phenomena.

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., Comment: Journal of Physical Studies, 2020, v. 24 (Paper in Ukrainian)

Phenomenological characteristics of the sample of the Algol-type stars are revised using a recently developed NAV ("New Algol Variable") algorithm (2012Ap.....55..536A, 2012arXiv 1212.6707A) and compared to that obtained using common methods of Trigonometric Polynomial Fit (TP) or local Algebraic Polynomial (A) fit of a fixed or (alternately) statistically optimal degree (1994OAP.....7...49A, 2003ASPC..292..391A). The computer program NAV is introduced, which allows to determine the best fit with 7 "linear" and 5 "non-linear" parameters and their error estimates. The number of parameters is much smaller than for the TP fit (typically 20-40, depending on the width of the eclipse, and is much smaller (5-20) for the W UMa and beta Lyrae - type stars. This causes more smooth approximation taking into account the reflection and ellipsoidal effects (TP2) and generally different shapes of the primary and secondary eclipses. An application of the method to two-color CCD photometry to the recently discovered eclipsing variable 2MASS J18024395 + 4003309 = VSX J180243.9 +400331 (2015JASS...32..101A) allowed to make estimates of the physical parameters of the binary system based on the phenomenological parameters of the light curve. The phenomenological parameters of the light curves were determined for the sample of newly discovered EA and EW - type stars (VSX J223429.3+552903, VSX J223421.4+553013, VSX J223416.2+553424, US-NO-B1.0 1347-0483658, UCAC3-191-085589, VSX J180755.6+074711= UCAC3 196-166827). Despite we have used original observations published by the discoverers, the accuracy estimates of the period using the NAV method are typically better than the original ones., Comment: 5 pages, 6 figures, "Odessa Astronomical Publications", 2015, 28 (in press)

We present a by-product of our long term photometric monitoring of cataclysmic variables. 2MASS J18024395 +4003309 = VSX J180243.9 +400331 was discovered in the field of the intermediate polar V1323 Her observed using the Korean 1-m telescope located at Mt. Lemmon, USA. An analysis of the two-color VR CCD observations of this variable covers all the phase intervals for the first time. The light curves show this object can be classified as an Algol-type variable with tidally distorted components, and an asymmetry of the maxima (the O’Connell effect). The periodogram analysis confirms the cycle numbering of Andronov et al. (2012) and for the initial approximation, the ephemeris is used as follows: Min I. BJD = 2456074.4904+0.3348837E . For phenomenological modeling, we used the trigonometric polynomial approximation of statistically optimal degree, and a recent method “NAV” (“New Algol Variable”) using local specific shapes for the eclipse. Methodological aspects and estimates of the physical parameters based on analysis of phenomenological parameters are presented. As results of our phenomenological model, we obtained for the inclination i=90°, M1=0.745M◉, M2=0.854M◉, M=M1+M2=1.599M◉, the orbital separation a=1.65·109m=2.37R◉ and relative radii r1=R1/a=0.314 and r2=R2/a=0.360. These estimates may be used as preliminary starting values for further modeling using extended physical models based on the Wilson & Devinney (1971) code and it's extensions

This paper reviews the most important components of the national project of the Ukrainian Virtual Observatory (UkrVO). Among them, there is the establishment of a Joint Digital Archive (JDA) of observational data, which has been obtained at Ukrainian observatories since the 1890s, including an astronegative JDA (more than 200 thousand plates). Since this task requires VO-oriented software, such issues as content verification software, integrity and administration of the JDA, compliance of image formats with the IVOA standards, and photometric and astrometric calibration of images are considered as the most important directions of software development, which carried out by members of the UkrVO. The scientific projects using local data archives of the UkrVO are discussed, namely: an analysis of a long observational series of active galactic nuclei, the study of solar flares and solar active regions based on spectral observational archives, research and discovery of variable stars, and the study of stellar fields in the vicinity of gamma-ray bursts. Particular attention is devoted to the CoLiTec Program that permits us to increase a number of observed solar system bodies and allows us to discover new bodies; for example, the C/2010 (Elenin) and P/2011 N01 comets were discovered using this program at the ISON-NM Observatory. The paper notes the creation of the UkrVO JDA prototype that provides access to the databases of the Main Astronomical Observatory, National Academy of Sciences of Ukraine (MAO NAS of Ukraine); Nikolaev Astronomical Observatory (NAO); and Lvov Astronomical Observatory.

We present results of two-color VR photometry of the intermediate polar RXS J1803. The data were aquired using the Korean 1-m telescope located at Mt. Lemmon, USA. Different “high” and “low” luminosity states, similar to other intermediate polars, were discovered. No statistically significant variability of the color index with varying luminosity was detected. The orbital variability was found to be not statistically significant. Spin maxima timings were determined, as well as the photometric ephemeris for the time interval of our observations. The spin period variations, caused by interaction of the accretion structure with the rotating magnetic white dwarf, were also detected. These variations are of complicated character, and their study requires further observations. We determine the color transformation coefficients for our photometric systems, and improve on the secondary photometric standards.

We study the variability of the cataclysmic variable DO Dra, on time-scales of between minutes and decades. The characteristic decay time dt/dm=0.902(3) days/mag was estimated from our 3 nights of CCD R observations. The quiescent data show a photometric wave with a cycle about 303(15)d. We analyzed the profile of the composite (or mean) outburst. We discovered however, that a variety of different outburst heights and durations had occurred, contrary to theoretical predictions. With increasing maximum brightness, we find that the decay time also increases; this is in contrast to the model predictions, which indicate that outbursts should have a constant shape. This is interpreted as representing the presence of outburst-to-outburst variability of the magnetospheric radius. A presence of a number of missed weak narrow outbursts is predicted from this statistical relationship. A new type of variability is detected, during 3 subsequent nights in 2007: periodic (during one nightly run) oscillations with rapidly-decreasing frequency from 86 to 47 cycles/day and a semi-amplitude increasing from 0.06 to 0. 10, during a monotonic brightness increase from 14. 27 to 14. 13. This phenomenon was observed only during an unusually prolonged event of about 1 mag brightening in 2007 (lasting till autumn), during which no (expected) outburst was detected. We refer to this behaviour as to the transient periodic oscillations (TPO). To study this new and interesting phenomenon, new regular photometric and spectral (in a target of opportunity mode) observations are required., 12pages, 8figures, accepted in Astronomy and Astrophysics

A computer program is introduced, which allows to determine statistically optimal approxi-mation using the "Asymptotic Parabola" fit, or, in other words, the spline consisting of polynomials of order 1,2,1, or two lines ("asymptotes") connected with a parabola. The function itself and its derivative is continuous. There are 5 parameters: two points, where a line switches to a parabola and vice versa, the slopes of the line and the curvature of the parabola. Extreme cases are either the parabola without lines (i.e.the parabola of width of the whole interval), or lines without a parabola (zero width of the parabola), or "line+parabola" without a second line. Such an approximation is especially effective for pulsating variables, for which the slopes of the ascending and descending branches are generally different, so the maxima and minima have asymmetric shapes. The method was initially introduced by Marsakova and Andronov (1996OAP.....9..127M) and realized as a computer program written in QBasic under DOS. It was used for dozens of variable stars, particularly, for the catalogs of the individual characteristics of pulsations of the Mira (1998OAP....11...79M) and semi-regular (200OAP....13..116C) pulsating variables. For the eclipsing variables with nearly symmetric shapes of the minima, we use a "symmetric" version of the "Asymptotic parabola". Here we introduce a Windows-based program, which does not have DOS limitation for the memory (number of observations) and screen resolution. The program has an user-friendly interface and is illustrated by an application to the test signal and to the pulsating variable AC Her., Comment: 3 pages, 3 figures, "Odessa Astronomical Publications", 2015, vol. 28 (accepted)

Results of 7 nights of CCD VR photometry of the intermediate polar 1RXS J062518.2+733433 obtained at the Korean 1.8m telescope are reported. The corrected ephemeris for the orbital minimum is BJD (Orb.min)

The highlights of the recent study of variable stars of different types are discussed. We present some results (out of more than 500 papers) and plans for the further studies within bi- or multi-la...

This article contains the highlights of complex studies of long-period variable stars such as Miras: semiregular, symbiotic (particularly, pulsating symbiotic), as well as RV Tauri-type stars. In the course of these studies, important characteristics of mean light curves were determined. In the case of multi-component variability, additional periods were found. Correlations between parameters of mean light curves were investigated. The cycle-to-cycle changes of light curve parameters were analysed using various mathematical methods. Classification criteria of variable stars and effects of variability were proposed based on this research. The study of observational parameters and the correlations between them can be used to estimate the age, mass, and other physical characteristics of AGB stars.

Observations of TT Ari obtained at 11 observatories (campaign TT Ari–94) during 258 hr were carried out to study optical variability on timescales from minutes to weeks. The best-fit primary photometric period determined from 16 nights of data obtained at the Dushak-Eregdag station of the Odessa State University is P = 0133160 ± 0000004 with a mean amplitude of 0.0513 ± 0.0008 mag. This new primary photometric period is larger than that obtained during the TT Ari–88 campaign and is well outside the range of estimates published since 1961. Contrary to previous findings, the "5–7 hr" secondary photometric period is not seen. Our observations do show evidence for periods of 2916 and 03040 with amplitudes of 43 and 25 mmag, respectively. The beat period between the spectroscopic and photometric periods is not seen. No coherent oscillations in the range f = 10–2500 cycles day-1 are found. The highest peaks in the power spectrum cover the wide range of 24–139 cycles day-1. In the mean periodogram, the highest peak corresponds to 21 and 30 minutes for the largest sets of observations, i.e., those obtained at Odessa and Krakow Universities, respectively. In the instrumental B system, variations with an amplitude exceeding 0.011 mag occur 8 times (from 33 runs) at 24 minutes. We conclude that quasi-periodic variations occur at a few preferred timescales rather than at a relatively stable period with a secular decrease. In the frequency range 90–900 cycles day-1, the power spectrum obeys a power law with a slope ranging from γ = 1.3 to 2.6 for different runs.

The project of time series analysis for the photometric observations of a representative group of RV Tauri type stars is described. This class of variables consists of about 100 supergiants in the Milky Way. Their light curves show in general two unequal minima with several individual features which are not explained by now. The complementary methods of periodogram, scalegram and wavelet analysis as well as statistically optimal ts have been applied. This research will result in an atlas of the light curves within the time and phase domains. In addition, a catalog of the observational characteristics of variability including the time and magnitude of the extrema for the program stars will be computed. Statistical correlations between these characteristics will give a strict observational test for the theoretical models explaining the RV Tauri phenomenon.

Results of a multi-color study of the variability of the magnetic cataclysmic variable BY Cam are presented. The observations were obtained at the Korean 1.8m and Ukrainian 2.6m, 1.2m and 38-cm telescopes in 2003-2005, 56 observational runs cover 189 hours. The variations of the mean brightness in different colors are correlated with a slope dR/dV=1.29(4), where the number in brackets denotes the error estimates in the last digits. For individual runs, this slope is much smaller ranging from 0.98(3) to 1.24(3), with a mean value of 1.11(1). Near the maximum, the slope becomes smaller for some nights, indicating more blue spectral energy distribution, whereas the night-to-night variability has an infrared character. For the simultaneous UBVRI photometry, the slopes increase with wavelength from dU/dR=0.23(1) to dI/dR=1.18(1). Such wavelength dependence is opposite to that observed in non-magnetic cataclysmic variables, in an agreement to the model of cyclotron emission. The principal component analysis shows two (with a third at the limit of detection) components of variablitity with different spectral energy distribution, which possibly correspond to different regions of emission. The scalegram analysis shows a highest peak corresponding to the 200-min spin variability, its quarter and to the 30-min and 8-min QPOs. The amplitudes of all these components are dependent on wavelength and luminosity state. The light curves were fitted by a statistically optimal trigonometrical polynomial (up to 4-th order) to take into account a 4-hump structure. The dependences of these parameters on the phase of the beat period and on mean brightness are discussed. The amplitude of spin variations increases with an increasing wavelength and with decreasing brightness, Comment: 30pages, 11figures, accepted in Cent.Eur.J.Phys

Time series analysis of the brightness variations of R Aqr is based on 1521 visual observations from the AFOEV database. The statistically significant number of harmonics for the mean phase curve is m = 5. The cycle-to-cycle variations of the phase curve are present. The most stable in phase is the maximum, for which no secular period variations detected, but a secondary cycle of P 2 = 3955 ± 50 d and amplitude (0.032 ± 0.005)P may be suggested.

We report photometric results from 44 runs at 11 observatories during the international campaign ‘TT Ari-94’. No coherent oscillations in the frequency range 10...900 cycle d-1 are found. The highest peaks in the power spectrum cover the wide range of 28...139 cycle d-1. Variations occur at a few preferred time-scales rather than at one cycle length, with a possible secular decrease. In the frequency range 90...900 cycle d-1 the power spectrum obeys a power law with slope γ ranging from 0.8 to 2.6 for different runs.