Theory of the light curves of eclipsing systems with oscillating components,II

In continuation of our preceding investigation (Kopal, 1982c) of photometric consequences of free oscillations — radial or nonradial — of the eclipsing component on the light changes of close binary systems, the aim of the present paper will be to extend the scope of our inquiry to include the effects of similar oscillations of the component which undergoes eclipse, with arbitrary period and phase.In Sections 2 and 3 which follow a brief introduction to the particular aspect of the problem treated in this paper, a theory will be given of photometric effects caused by arbitrary harmonic oscillations of the components of eclipsing binary systems distorted by equilibrium tides, or by axial rotation with constant angular velocity. A translation of these effects into the frequency domain constitutes a problem which is solvable in a closed form only if the eclipse in question ends in totality. In any other case, the desired effects can be mathematically described only in terms of infinite series (of satisfactory asymptotic properties); and a construction of their explicit forms will be given in the Appendix.     

Note on the analysis of light curves of eclipsing binaries with dissimilar components

The term l _per giving a correction needed to reduce the light curves of eclipsing binaries with dissimilar components to those of rectifiable models with similar components is discussed in view of a practical definition of the luminosities of the components. A new form of the term is proposed. Furthermore, a way is shown for applying this term also when the light curve has already been rectified in the usual manner.Mitteilung des Astronomischen Instituts der Universität Tübingen Nr. 108.     

Optimization analysis of the light curves of eccentric eclipsing binary systems

The iterative optimization procedures described in a previous paper by the author have been extended to cover unrectified light curves of eccentric eclipsing binary systems. Two stars have been studied in detail: namely, Coronae Borealis and V477 Cygni. In both cases improved element sets have been obtained. Moreover, in the case of V477 Cygni the obtained values of the longitude of periastron for three different light curves are found to provide a useful basis for discussion of the apsidal motion of this system.     

Fourier analysis of the light curves of eclipsing variables,I

The aim of the present paper will be to pioneer a new approach to the analysis of the light changes of eclipsing binary systems in the frequency domain, and to point out its merits in comparison with a conventional treatment of the same problem in the time-domain which has been developed so far. Following an introductory section in which the broad features of our problem will be set forth, Section 2 will contain an outline, and critique, of the time-domain approach. Section 3 will give an explicit treatment of the light changes arising from total and annular eclipses in the frequency domain — a problem which we succeeded in solving in close algebraic form. Section 4 will extend this treatment to the case of partial eclipses; and in the concluding Section 5 the relative merits of our new results will be discussed in broader perspective. Sections 3 and 4 contain explicit results pertaining to mutual eclipses of spherical stars exhibiting uniformly bright discs. An extension of these results to the case of arbitrary limb-darkening, and taking account of mutual distortion of both components, will be given in subsequent communications.     

Fourier techniques for an analysis of eclipsing binary light curves: I

The Fourier techniques developed so far for an analysis of eclipsing binary light curves have been re-discussed. The Fourier coefficients for the analysis have been derived in a simple form of series expansions, in terms of eclipse elements, valid for any type of eclipse (regardless of whetherr ₁r ₂).These coefficients may be utilized to solve the eclipse elements in terms of the observed characteristics of the light curves. A general relation between the observed quantitiesl and , and the eclipse elementsr _1,2,i andL ₁ has also been given in the form of series expansions which can be used for the synthesis of the light curves.     

Fourier analysis of the light curves of eclipsing variables

The aim of the present paper has been to construct analytic expressions for incomplete Fourier transforms underlying Kitamura's method for an analysis of the light curves of eclipsing binary systems (Kitamura, 1965).The expansions established for Kitamura's coefficientsc _n ands _n have been used as a basis for checking numerical accuracy of these coefficients tabulated by Kitamura in 1967; and the outcome bespeaks a high quality of his tables constructed by numerical quadratures.     

Two new light curves of the eclipsing variableu Herculis

This paper presents two new light curves (inB andV) of the eclipsing variable staru Her which have been derived from photoelectric observations made at the National Observatory of Athens. General information of the system is given in the introduction, while Section 2 deals with some observational and reductional details. Section 3 gives the residuals O-C and Section 4 the normal points.     

Linear analysis of the light curves of eclipsing variables

In this paper, general model analysis of the form c _{i i} () for the fractional loss of lightf() exhibited by a close binary system will be considered in the sense of the least-squares criterion. A general recursive method will be established for constructing the normal equations for the most useful functions _i (), and an economical storage of the method on a digital computer, with its computational steps, will also be given. Moreover, a full recursive computational algorithm for the least-squares approximation will also be established. By means of this algorithm, all the solution vectors, the variance for different orders of fit and the corresponding variance-covariance matrix could be computed once and for all and, moreover, recursively. The economical storage of the algorithm and its computational steps will be given. Finally, some of the practical difficulties encountered in the application of the least-squares criterion will be analysed, and some techniques for detecting and controlling these difficulties are also given. Numerical examples on the Algol system using a Fourier cosine series of the form c _i cos [(j - 1) /] will be given for illustration.     

Fourier analysis of the light curves of eclipsing variables. XIXA:

The radiation of a charge rotating in a circle with the constant velocity (in the external magnetic field) in the isotropic plasma with random inhomogeneities of the electron density has been considered. A general expression is obtained for the radiation intensity at thenth harmonic, which is a generalization of the known Shott formula. In the ultra-relativistic case the conditions are clarified under which the inhomogeneity effect on the form of the spectrum of radiation from a particle is essential. An asymptotic formula is derived for the spectral intensity in the region of sufficiently low frequencies. The mechanism of transient radiation in this case is shown to prevail over the synchrotron one.     

Fourier analysis of the light curves of eclipsing variable stars

The aim of the present paper is to find the eclipse perturbations, in the frequency-domain, of close eclipsing systems exhibiting partial eclipses.After a brief introduction, in Section 2 we shall deal with the evaluation of thea _n ^(l) integrals for partial eclipses and give them in terms ofa ₀ ⁰ ,a ₀ ⁰ (of the associated -functions) and integrals; while Section 3 gives the eclipse perturbations arising from the tidal and rotational distortion of the two components. The are given for uniformly bright discs (h=1) as well as for linear and quadratic limb-darkening (h=2 and 3, respectively).Finally, Section 4 gives a brief discussion of the results and the way in which they can be applied to practical cases.     

Light curves of the eclipsing systems of RS CVn-type

This work presents the light curves for rotating spotted stars generated by a computational code developed by us. This code is based in a model of the system and its analytical solution. The work also present an analysis of the photometric curve variations with the changes in the model's parameters.As a first result we have simulated a temporal evolution of the light curve when the photospheric spot varies its size and position and we reproduced some observational light curves of the II Peg star.Work supported by Dirección de Investigación, Universidad de La Serena, Chile.     

Fourier analysis of the light curves of eclipsing variable stars

The aim of the present paper is to establish the explicit forms of the photometric perturbations, in the frequency-domain, of close binaries, whose components are distorted by axial rotation and mutual tidal action.Following a brief introduction, Section 2 describes the light changes and the photometric perturbations within eclipses in the frequency-domain. In Section 3 the explicit forms of the perturbations for occultation eclipses terminating in totality are given; while in Section 4 analogous results are established for transit eclipses terminating in annular phases. In this latter case the results can be expressed in terms of the photometric perturbations for total eclipses and in terms of some series. To facilitate applications to actual stars these series have been computed and their results are represented in Table I and by the Graphs. Finally, Section 5 gives a discussion of the results.An extension of the photometric perturbations to the case of partial eclipses will be given in a subsequent communication.     

Synthetic light curves of two eclipsing binary systems,U Sge and AW UMa

Photometric curve fits have been investigated by means of numerical quadratures to develop theoretical light curves appropriate to stars built up in accordance with the Roche model. The method has been applied previously to β Per (Al-Naimiy and Budding, 1977) on the basis of available observations in red and infrared, while presently applied to two systems with contact components.

1. U Sge, spherical primary totally eclipsed by a contact component secondary. Improved photometric elements of the system have been found, and compared with those obtained by Kopal's method in the frequency domain. The outcome of the curve fitting corresponds well with the results of an analysis in the frequency domain.
2. AW UMa, exhibiting the shallowest minima known for totally eclipsing W UMa systems. The physical and geometrical elements of the system have been found, and the contact nature of the two components confirmed.

     

Fourier analysis of light curves of eclipsing variable stars

The photometric perturbationsB _h ^(l) arising from both tidal and rotational distortion of a close eclipsing binary have been given in two previous papers (Livaniou, 1977; Rovithis-Livaniou, 1977). The aim of the present paper will be to find the eclipse perturbationsB _2m =B _{2m, tid} +B _{2m, rot} of a close binary exhibiting partial eclipses. This will be done giving the suitable combinations of theB _h ^(l) 's and will make easier the application to real stars. After a very brief introduction, Section 2 gives both theB _{2m, tid} andB _{2m, rot} for uniformly bright discs; while in Sections 3 and 4 they are given for linear and quadratic limb-darkening, respectively. Finally, Section 5 gives a brief discussion of the results.     

Fourier analysis of the light curves of eclipsing variables,XVI

The practical procedures for the solutions of the elements of any eclipsing system in the frequency-domain have been described in a previous paper of this series (Kopal and Demircan, 1978, Paper XIV). The fundamental quantities from which we depart in quest of our solution are twog-functions defining by the momentsA _2m (see Equations (2.13)–(2.16) in Paper XIV, or Equations (3.2)–(3.6) in Paper XV: Demircan, 1978b). If we establish the observational values for these functions, they constitute two independent relations between the unknown parametersa andc _o, and can be numerically solved for them with the aid of the general expressions for the respective moments. However, the determinacy of these parameters depends on not only the accuracy of observations but also the employedg-functions. For better understanding of the geometrical determinacy of the eclipse parametersa andc _o, different combinations of the momentsA _2m have been worked out asg-functions. For the index 2m, the values between 0 and 6 were applied. It has been noted that the behaviour of these functions vary but very little with applied different combinations of the moments. A choice of the most convenient moments to obtain a good determinacy for the eclipse elements were discussed. In this connection, (i) them-dependence of the moments, and the errors in their observational values have been considered, (ii) different practical procedures for the solution of eclipse elements were introduced, and (iii) different type of moments were tested.     

Fourier analysis of the light curves of eclipsing variables. XIII

New expressions for the fractional loss of light _l ⁰ have been derived in the simple forms of rapidly converging expansions to the series of Chebyshev polynomials, Jacobi polynomials, and Kopal'sJ-integrals. In these expansions, which are a supplement to those given by Kopal (1977b), variablesk andh occur in different products that simplify the numerical computation. The treatment follows the new definition of _l ⁰ which has been recently developed by Kopal (1977a).     

Fourier analysis of the light curves of eclipsing variables,IX

The aim of the present paper will be to detail the procedure outlined in our previous investigations (Kopal, 1975; Kopalet al., 1976) for a solution of the elements of distorted eclipsing systems by a Fourier analysis of their light changes. This procedure—which constitutes an equivalent, in the frequency-domain, of rectification hitherto practised in the time-domain — should enable us to free the observed momentsA _2m of the light curves from all photometric effects of distortion (between minima as well as within eclipses) — a feat impossible in the time-domain except under very restricted conditions — and thus to make it possible to obtain the geometrical elements of the eclipses which should be free from any obvious source of systematic errors.     

Application of the Kalman filter to light curves of eclipsing binaries

Recently Kopal (1975a, b, c, d) initiated a new approach to the analysis of the light curves of eclipsing binary systems in which the solution is based on transforming the problem from the conventional time-domain into the frequency-domain. Irrespective of the type of eclipse, the present formulation of the frequency approach requires that a set of quantitiesA _2m, called moments, be determined from the observations.It is the purpose of the present paper to describe a data interpolation and smoothing technique based on a version of the Kalman filter to pre-process observations and to determine the quantitiesA _2m in an optimal sense.