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.     

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.     

Computation of the elements of eclipsing binaries from a part of their light curves

All methods existing today for computation of the elements of eclipsing binaries require photometric observations to cover the complete revolution of binaries, outside as well as within the eclipses. The maximum light and the depth of the eclipses are to be determined prior to any analysis, and the light curves are to be expressed in terms of the total light at quadratures. A method has been introduced in the present paper to enable us to obtain the elements of eclipsing binaries by an analysis of, not the whole, but a part of their eclipse light curves. In the new method no empirical knowledge of the maximum light or depth of eclipes is needed, and no normalization is necessary. The method can be used in any models (ellipsoidal, Roche, etc.). However, the larger the phase coverage of the light changes considered for the computation of the elements, the better the determinacy (or smaller the errors) is obtained.     

Determinacy of the eclipse elements from a part of the light curves of eclipsing binaries

A part of the light curves of two detached (CD Tau and V909 Cyg) and one semi-detached (Algol) binaries have been analysed with the emphasis on the determinacy of the parameters of different systems from a part of their eclipse curves. A model light curve for whichr ₁=0.050,r ₂=0.150,i=90°, andL ₁=0.800 were chosen, has also been considered for a further clarification of the determinacy problem of the eclipse parameters. Some insight into the determinacy is gained by varying one parameter and adjusting the others for the best fit to the light curves in the considered eclipse phases. It was realised that the determinacy problem of the unknown parameters is stored in the determinacy of the type of the eclipse minima. The attempts of the solutions with a false choice of initial elements result in finding (i) no minimum of ² to give any acceptable solution, (ii) rapid departure from the false start and convergence on the correct solution, or (iii) a false solution mostly with a true inclinationi, but wrong sense of the ratio of the radii.     

Automated analysis of eclipsing binary light curves – II. Statistical analysis of OGLE LMC eclipsing binaries

In the first paper of this series, we presented EBAS – Eclipsing Binary Automated Solver, a new fully automated algorithm to analyse the light curves of eclipsing binaries, based on the ebop code. Here, we apply the new algorithm to the whole sample of 2580 binaries found in the Optical Gravitational Lensing Experiment (OGLE) Large Magellanic Cloud (LMC) photometric survey and derive the orbital elements for 1931 systems. To obtain the statistical properties of the short-period binaries of the LMC, we construct a well-defined subsample of 938 eclipsing binaries with main-sequence B-type primaries. Correcting for observational selection effects, we derive the distributions of the fractional radii of the two components and their sum, the brightness ratios and the periods of the short-period binaries. Somewhat surprisingly, the results are consistent with a flat distribution in log P between 2 and 10 d. We also estimate the total number of binaries in the LMC with the same characteristics, and not only the eclipsing binaries, to be about 5000. This figure leads us to suggest that  (0.7 ± 0.4)  per cent of the main-sequence B-type stars in the LMC are found in binaries with periods shorter than 10 d. This frequency is substantially smaller than the fraction of binaries found by small Galactic radial-velocity surveys of B stars. On the other hand, the binary frequency found by Hubble Space Telescope ( HST ) photometric searches within the late main-sequence stars of 47 Tuc is only slightly higher and still consistent with the frequency we deduced for the B stars in the LMC.     

Evaluation of the light changes for distorted eclipsing binaries

This paper discusses a method for improving on the numerical evaluation of the light changes exhibited by a distorted eclipsing binary system.In the theory formulated by Kopal (1959), certain boundary integrals due to the distortion of both components have been calculated in terms of the Appell hypergeometric series of the first kind. The values of the four parameters appearing in these series differ according as to whether one is dealing with a partial or an annular eclipse.To accelerate the numerical evaluation of the light changes one should avoid recomputing such infinite series for contiguous values of the parameters. This can be achieved by making use of certain recursion formulae which hold for the foregoing series.We have provided here a procedure that yields forty-eight recursion formulae for the Appell hypergeometric series and have specifically calculated four new independent recursion formulae relevant to the astrophysical problem.     

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.     

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.     

Photometric light curve solutions of three ultra-short period eclipsing binaries

We present the results of our study of the eclipsing binary systems CSS J112237.1+395219,LINEAR 1286561 and LINEAR 2602707 based on new CCD B, V, Rcand Iccomplete light curves. The ultra-short period nature of these stars, as reported by Drake et al., is confirmed and the system's periods are revised. The light curves were modeled using the 2003 version of the Wilson-Devinney code. When necessary, cool spots on the surface of the primary component were introduced to account for asymmetries in the light curves. As a result, we found that CSS J112237.1+395219 is a W UMa type contact binary system belonging to W subclass with a mass ratio of q = 1.61 and a shallow degree of contact of 14.8%where the primary component is hotter than the secondary one by 500 K. LINEAR 1286561 and LINEAR2602707 are detached binary systems with mass ratios q = 3.467 and q = 0.987 respectively. These detached systems are low-mass M-type eclipsing binaries with similar temperatures. The marginal contact,fill-out factor and temperature difference between components of CSS J112237.1+395219 suggest that this system may be at a key evolutionary state predicted by thermal relaxation oscillation(TRO) theory. From the estimated absolute parameters, we conclude that our systems share common properties with other ultrashort period binaries.     

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 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.     

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.     

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,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.     

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,II

The aim of the present paper will be to extend the Fourier methods of analysis of the light curves of eclipsing binaries, outlined in our previous communication (Kopal, 1975) in connection with systems whose components would appear as uniformly bright discs, to systems whose components exhibit discs characterized by an arbitrary radially-symmetrical distribution of brightness —i.e., an arbitrary law of darkening towards the limb — be it linear or nonlinear.In Section 2 which follows a few brief introductory remarks, fundamental equations will be set up which govern the light changes arising from the mutual eclipses of limb-darkened stars — be such eclipses total, partial or annular; and Section 3 will contain a closed algebraic solution for the elements of the occulation eclipses terminating in total phase. Such a solution proves to be no more complicated than it turned out to be for uniformly bright discs in our previous paper; and calls for no special functions for the purpose — as will be put in proper perspective in the concluding Section 4.The cases of transit eclipses terminating in an annular phase, of partial eclipses of occulation or transit type, will be similarly dealt with by Fourier methods in the next paper of the present series.     

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).