Determination of the elements of eclipsing variables RW tauri and U sagittae by an analysis of the light changes in the frequency domain

The aim of the present paper has been to present an analysis of the light changes of two eclipsing systems RW Tau and U Sge in the frequency domain, which was developed by Kopal (1975a, b, c, d, e, 1976).Following a brief introduction, Section 2 contains the evaluation of the theoretical momentsA _2m. The determination of the preliminary elements and their improvement, taking into account the photometric perturbations, are given in Sections 3 and 4. A general discussion devoted to the whole analysis of the system is presented in Section 5.     

Fourier analysis of the light curves of eclipsing variables,VIII

The main aim of this paper will be to develop explicit form of the moments of the light curvesA _2m(r ₁,r ₂,i) required for the solution for the geometrical elementsr _1,2 andi of eclipsing systems exhibiting annular eclipses (Sections 2 and 3), as well as partial eclipses (Section 4).In the concluding Section 5 we shall demonstrate that — regardless of the type of eclipse and distribution of brightness on the apparent disc of the eclipsed star, or indeed of the shape of the eclipsing as well as eclipsed components — the momentsA _2m satisfy certain simple functional equations — a fact which relates them to other classes of functions previously studied in applied mathematics.     

Fourier analysis of the light curves of eclipsing variables,IV

The methods of analysis of the light changes of eclipsing variables in the frequency domain, developed in our previous papers (Kopal, 1975b, c) for total or annular eclipses of arbitrarily limbdarkened stars, have now been extended to the case of partial eclipses of occultation as well as transit type. In Section 2 which follows brief introductory remarks the even Fourier sine coefficients are formulated — in the guise of the momentsA _2m of the light curve — in terms of the elements of the eclipse; and their use for a solution for the elements is detailed in Section 3. A brief appendix containing certain auxiliary tables to facilitate this task concludes the paper. An extension of the same method to an analysis of the light changes exhibited by close eclipsing systems — in which the photometric proximity effects arising from mutual distortion can no longer be ignored — will be given in the subsequent paper of this series.     

Fourier analysis of the light curves of eclipsing variables,III

The aim of the present paper will be to extend our new methods of analysis of the light curves, of eclipsing binary systems, consisting of spherical components, by Fourier approach to eclipses oftransit type — which arise when the eclipsing component happens to be smaller of the two. Our present principal concern will be transit eclipses, terminating in annular phase, of stars characterized by arbitrary radially-symmetrical distribution of brightness over their apparent discs — a phenomenon which will cause the light of the system to vary continuously during annular phase. In the first section which follows this abstract, an outline of the problem at issue will be given. Section 2 has been devoted to an analysis of light changes arising in the course of partial phases of transit eclipses; and the concluding Section 3 will contain an analysis of the corresponding light changes, during annular phase. Unlike for occultation eclipses considered in our previous paper (cf. Kopal, 1975b), the momentsA _2m of the light curves due to eclipses of transit type can again be expressed in terms of the geometrical elements of such eclipses in a closed form for limb darkening characterized by any value ofn; but the use of such functions will require auxiliary tables (now in preparation) for applications to practical cases. A parallel treatment of partial eclipses of the occultation or transit type — eclipses which stop short of totality or annular phase — is being postponed for a subsequent communication.     

Basic Functions of the Light Curve Analysis of Eclipsing Variables in the Frequency Domain

An analytically tractable method of transforming the problem of light curve analysis of eclipsing binaries from the time domain into the frequency domain was introduced by Kopal (1975, 1979, 1990). This method uses a new general formulation of eclipse functions α, the so-called moments A _2m , and their combinations as g _2m = A _2m+2/(A _2m A _2m+4) functions for the basic spherical model. In this paper, I will review the use of these functions in the light curve analysis of eclipsing binaries.     

Fourier analysis of the light curves of eclipsing variables,XXI

The aim of the present paper is to deduce some further properties of the fundamental quantities inherent in the frequency-domain approach-such as the fractional loss of light _l ⁰ and momentsA _2m of the light curves of eclipsing variables; and also to develop an iterative method for the solution of two key eclipse parametersa andc ₀ in terms of the observed quantities. This should facilitate practical applications of the methods developed in the preceding papers of this series for the frequency-domain light curve analysis of eclipsing variables.     

Fourier analysis of the light curves of eclipsing variables,XV

A new general expression for the theoretical momentsA _2m of the light curves of eclipsing systems has been presented in the form of infinite series expansion. In this expansion, the terms have been given as the product of two different polynomials which satisfy certain three-term recursion formulae, and the coefficients diminish rapidly with increasing number of terms. Thus, the numerical values of the theoretical momentsA _2m can be generated recursively up to four significant figures for any given set of eclipse elements. This can be utilized to solve the eclipse elements in two ways: (i) with an indirect method (for the procedures see Paper XIV, Kopal and Demircan, 1978), (ii) with a direct method as minimization to the observational momentsA _2m (area fitting). The procedures given in Paper XIV for obtaining the elements of any eclipsing system consisting of spherical stars have been automated by making use of the new expression for the momentsA _2m of the light curves. The theoretical functionsf ₀,f ₂,f ₄,f ₆,g ₂ andg ₄ which are the functions ofa andc ₀, have been used to solve the eclipse elements from the observed photometric data. The closed-form expressions for the functionsf ₂,f ₄ andf ₆ have also been derived (Section 3) in terms of Kopal'sI-integrals.The automated methods for obtaining the eclipse elements from one minimum alone have been tested on the light curves of YZ (21) Cassiopeiae under the spherical model assumptions. The results of these applications will be given in Section 5 which follows a brief introduction to the procedure we followed.     

Fourier analysis of the light curves of eclipsing variables,X

The aim of the present paper will be to develop methods for computation of the Fourier transforms of the light curves of eclipsing variables — due to any type of eclipses — as a function of a continuous frequency variablev. For light curves which are symmetrical with respect to the conjunctions (but only then) these transforms prove to be real functions ofv, and expressible as rapidly convergent expansions in terms of the momentsA _2m+1 of the light curves of odd orders. The transforms are found to be strongly peaked in the low-frequency domain (attaining a maximum forv=0), and become numerically insignificant forv>3. This is even more true of their power spectra.The odd momentsA _2m+1 — not encountered so far in our previous papers — are shown in Section 3 of the present communication to be expressible as infinite series in terms of the even momentsA _2m well known to us from Papers I–IV; and polynomial expressions are developed for approximating them to any desired degree of accuracy. The numerical efficiency of such expressions will be tested in Section 4, by application to a practical case, with satisfactory results.Lastly, in Section 5, an appeal to the Wiener-Khinchin theorem (relating the power spectra with autocorrelation function of the light curves) and Parseval's theorem on Fourier series will enable us to extend our previous methods for a specification of quadratic moments of the light curves in terms of the linear ones.     

Computation of the elements of eclipsing binary systems in the frequency-domain

The aim of the present paper will be to translate the essential parts of the theory of Fourier analysis of the light changes of eclipsing variables into more practical terms; and describe procedures (illustrated by numerical examples) which should enable their users to obtain the desired results with maximum accuracy and minimum loss of information by processes which can be fully automated.In order to unfold in steps how this can be done, the scope of the present paper-the first of two-will be restricted to an exposition of the analysis of light changes caused by eclipses of spherical stars; while between minima due to this cause the light of the system should remain sensibly constant. An extension of our analysis to incorporate photometric effects arising from mutual distortion of the components of close eclipsing systems between minima as well as within eclipses is being postponed for the second communication.In developing this subject we shall single out for the user's attention only those parts of the whole theory which are of direct relevance to practical work. Their justification can be largely found in sources already published; and new developments essential for our work, not yet made public, will be relegated to several Appendices at the end of the text, in order not to render its text too discursive and deflect the reader's attention from the main theme of its narrative.After a brief outline of the subject given in Section 1, Section 2 will introduce the reader to practical aspects of the Fourier analysis of the light curves; and Section 3 will be devoted to its use to determine the numerical values of the momentsA _2m of the light curves which constitute the cornerstones for all subsequent work. Section 4 will describe an algebraization of the process of determination of the elements for the case of total (annular) eclipses; while Section 5 will do the same for partial eclipses. The concluding Section 6 will be devoted to an error analysis of our problem, and to an outline of the way by which the errors of the individual observations will compound to the uncertainty of the final results. Lastly, Appendices 1–5 concluding the paper will contain additional details of some aspects of our work, or proofs of new processes made use of to obtain our results, whose earlier inclusion would have made the main text too discursive.     

Fourier analysis of the light curves of eclipsing binaries in the case of transit eclipses,with application to the eclipsing system YZ Cassiopeiae

The aim of this paper is to extend the Fourier approach to the transit eclipses, terminating in annular phase, with an application to YZ Cassiopeiae. The results turn out to be more complicated than those obtained by Kopal for total eclipses. However, the solution can still be obtained by successive approximations without resorting to any tables of special functions.Section 1 contains an outline of the problem. In Section 2, the evaluation of the theoretical momentsA _2m for transit eclipses is given. An application of the Fourier method to the light curves of YZ Cas is presented in Section 3. Finally, in Section 4, a general discussion of the results is given.     

Analysis of the light curves of V78 in ω Centauri in the frequency domain by automatized fourier techniques

The aim of the present paper has been to analyse the light changes of the close eclipsing system V78 in Centauri in the frequency domain. In two of his recent papers, Kopal (1977b, c) has developed new methods for the analysis of light curves using Hankel transforms of zero order. He succeeded in expressing the momentsA _2m of light curves in a closed form. The expansions, in terms of which the momentsA _2m can be expressed, converge in all circumstances. Their analytical structure presents no difficulty for automatic computation. The light variations of the eclipsing system V78 in Centauri have been studied by use of the above method. New geometrical elements are also given.     

Fourier techniques for an analysis of eclipsing binary light curves. VIa

The aim of the present paper is to deduce relations between the integral transformsA _2m, B_2m,andF _1,2 of the light curves of eclipsing binary systems. The integral transformsA _2m, B_2m,andF _1,2 have been related to one another by means of finite or rapidly converging infinite summations obtained by integrations of the series expansions of trigonometric functions.     

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.     

Fourier techniques for an analysis of eclipsing binary light curves. V

An integral transform called the momentsA _2m of the light curves has been introduced by Kopal (1975) and utilized in the subsequent papers for an analysis of the light curves of eclipsing variables. The aim of the present paper is to generalize this integral transform by two distinct ways: (i) by introducing an exponential factor, and (ii) a Jacobi polynomial as multiplicative factor into the integrand of the transformA _2m. Observational values of these general transforms are likewise obtainable. They have been expressed in terms of eclipse elementsr _1,2,i andL ₁. These expressions can be used to solve the eclipse elements in terms of observed quantities. Free parameters in the expressions increase the flexibility in applications and may be utilized to improve the determinacy of the elements.     

Fourier analysis of the light curves of eclipsing variables,VII

A new method has been developed for the evaluation of the light momentsA _2m, required for a Fourier analysis of the light curves of eclipsing variables, in terms of the elements of the eclipsea method simpler and more straightforward than that previously developed in so far as it dispenses with the auxiliary coefficientsa _n ^(l) andb _n ^(l) used before at the intermediary stage. Our present method is applicable to an analysis of the eclipses of spherical stars of any type, arbitrarily darkened at the limb; and its results agree with those previously established in Papers III and IV of this series in less explicit form.     

Fourier analysis of the light curves of eclipsing variables,XIX

The aim of the present paper has been to generalize the methods previously developed for analysis of the light changes of eclipsing binary systems in the frequency-domain to cases in which the components of such systems revolve in eccentric orbits. It is shown that these methods can indeed be generalized to systems with eccentric orbits provided that the light momentsA _2m deduced from such eclipses are suitably re-defined in terms of the true, rather than mean, anomaly in the relative orbit; and that due attention is paid to the unit of length in terms of which the fractional radii of the two stars are expressed. When this is done the Fourier methods continue to be applicable to all types of eclipses exhibited by eccentric binary systems — whether these are occultations or transits; total, annular or partial.An application of these methods to practical cases has been postponed for a subsequent communication.     

Fourier analysis of the light curves of eclipsing variables,XXII

The theoretical values of the momentsA _2m for any type of eclipses, expressed in terms of the elementsL ₁,a andc ₀, have been derived in the simple forms of rapidly convergent expansions to the series of Chebyshev polynomials, Jacobi polynomials and KopalJ-integrals (Kopal, 1977c) and hold good for any real (not necessarily integral) value ofm0.The aim of the present paper has been to establish explicit expressions for the Jacobian and its fast enough computation in the light changes of close eclipsing systems, arising from the partial derivative of different pairs ofg-functions (Kopal and Demircan, 1978, Paper XIV) with respect toa andc ₀ ² , for any type of eclipses (be these occultations or transit, partial, total or annular) and for any arbitrary degreel of the adopted law of limb-darkening. The functional behaviour of this Jacobian would determine the reasonable light curve in connection with geometrical determinacy of the parametersa andc ₀. In the expansion of Jacobian, the terms consist of two polynomials which satisfy certain three-term recursion relations having the eclipse parametersa andc ₀, as their arguments.Closed form expressions forf-functions, as well as of the Jacobian (e.g.,m=1, 2, 3), obtaining in the case of total eclipses, are given for a comparative discussion with the theoretical values of Jacobian derived from partial derivative of different pairs ofg-functions.The numerical magnitude of Jacobian would determine the best combination of the momentsA _2m in the different pairs ofg-functions and definite results would follow in the subsequent paper of this series (Edalati, 1978c, Paper XXIV).     

Photometric Observations and Light Curve Studies of the Semi-Detached Eclipsing Binary R CMa

We present observations and light curve analysis of the eclipsing binary R CMa in the narrow band filters v and b. Observations were made during 1993 at Biruni Observatory and the light curves have been analyzed using the Wilson-Devinney light curve interpretation program. Assuming a semi-detached configuration for R CMa, the parameters i, Ω₁, L ₁, T ₂ and A ₂ were adjusted for the best fit between the synthesized light curves and observations. Both light curves were fitted well with a lower value of bolometric albedo than what would be expected for a normal cool star with a convective envelope. The masses of the primary and secondary components and the absolute dimensions of the stars have been calculated using the derived relative dimensions from Wilson-Devinney codes and the spectroscopic observations.     

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

Some properties of the quantitiesB _2m (Smith, 1977) inherent in the frequency-domain approach have been deduced, and a general expression for them in terms of the eclipse elementsr _1,2,i andL ₁ of the basic model has been presented (Section 2).An expansion for the loss of light (1–l) into a Fourier sine series alone have been introduced, and its coefficientsb _m presented (Section 3) in terms of the same eclipse elements. A method of increasing the rate of convergence of this series has been given in Section 4. The methods for obtaining the elements of eclipsing binaries by making use of all these quantities in the frequency-domain can likewise be generalized to cover the photometric effects of gravitational and radiative interaction between the components.