Fourier analysis of the light curves of eclipsing variables,V

The methods of analysis of the light changes of eclipsing variables in the frequency-domain, developed in our previous papers (Kopal 1975a, b, c, d) for an interpretation of mutual eclipses in systems consisting of spherical stars, have now been extended to analyse the light variations — between minima as well as within eclipses — ofclose binaries whose components are distorted by axial rotation and mutual tidal action. Following a brief introduction (Section 1) in which the need of this new approach will be expounded, in Sections 2 and 3 we shall deduce the theoretical changes of close eclipsing systems between minima (Section 2) as well as within eclipses (Section 3), which in Sections 4 and 5 will be analysed in the frequency-domain; and explicit formulae obtained which should enable us to separate the photometric proximity and eclipse effects directly from the observed data as they stand-without the need of any preliminary ‘rectification’. Section 6 will contain the explicit forms of the expressions for photometric perturbations in the frequency-domain, due to rotational and tidal distortion of both stars; and the concluding Section 7 will then be concerned with practical aspects of the application of these new methods to an analysis of the observed light changes of close eclipsing systems — in which the proximity and eclipse effects cannot be distinguished from each other by mere inspection.     

On the evaluation of the photometric proximity effects of close eclipsing systems

The aim of the present paper has been to generalize the method, developed by Kopal (1976), for the evaluation of the photometric proximity effects (between minima as well as within eclipses) of distorted eclipsing systems. The method is extended to quantities of second order (n-7) is surficial distortion. Analytical expressions of the polynomials to be used for modulation of the light curve are also given.     

Fourier analysis of the light curves of eclipsing variables XVII: The photometric perturbations

Values of the photometric perturbations required in the frequency-domain analysis for the accurate determination of parameters characterizing stars in eclipsing binary systems when some degree of proximity effect complications are present in the light curves are tabulated, corresponding to a wide range of such parameters. The relative smallness of the perturbation values permit a rapidly convergent iterative approach to the final parameter specification. Also, in view of the relative smallness, linear interpolation from the table over a wider range of nontabulated parameters should be effective in assessing the effects of the perturbations in a general case. The illustrative application to the light curve of RT Per appears to confirm this.     

An analysis of the light changes of the eclipsing system AK Herculis in the frequency domain

The aim of the present paper has been to analyse the light changes of the close eclipsing system AK Herculis in the frequency domain. This analysis is based on Kopal's new theory recently developed for the study of light variations—between minima as well as within eclipses—of close eclipsing binaries whose components are distorted by axial rotation and mutual tidal action.A new method for the separation of the photometric proximity and eclipse effects directly from the observed data is also presented. In this method no rectification is needed. New elements taking into account the photometric perturbations are also given.     

Fourier analysis of the light curves of eclipsing variables,XI

The aim of the present paper will be to introduce a new definition of the loss of light suffered by mutual eclipses of the components of close binary systems: namely, as across-correlation of two apertures representing the eclipsing and eclipsed discs.The advantages of such a strategy over the more conventional (geometrical) approach are (a) greater symmetry of the respective expressions; (b) greater affinity of expressions arising from distortion with those expressing the light changes due to eclipses of spherical stars; and (c) greater freedom in dealing with the effects of particular distribution of brightness over the disc of the star undergoing eclipse (generalized limb-darkening), as well as of possible semi-transparency of the eclipsing component (Wolf-Rayet stars!). In point of fact, none of these tasks could be handled with equal ease by any other technique; nor could the corresponding loss of light be so automated by any other approach.In Section 2 which follows brief introductory remarks we shall evaluate the loss of light arising from distribution of brightness within the aperture undergoing eclipse, and appropriate opacity of the occulting disc. In Section 3 we shall take advantage of these new forms of our results to deduce a number of new properties of the eclipse functions — both algebraic and differential — which have so far escaped attention and which are of considerable practical interest. Lastly, in Section 4 we shall generalize the same concepts to the modification of the light changes caused by the departures of the respective apertures from circular forms.It will be shown that all these phenomena can be most naturally described in terms of Hankel transforms of the products of two Bessel functions with orders depending on the physical characteristics (distribution of brightness; opacity) of the two components; while the geometry of the system (i.e., the fractional radiir _1,2 of the two stars; or the inclinationi of their orbit) enter only through their arguments. Such formulation of our problem should bring a theory of the light changes of eclipsing variables in much closer contact with the adjacent parts of physical optics.     

Analysis of the light curves of PV Cassiopeiae by use of automatized fourier techniques

UBV Light Curves of the eclipsing binary system PV Cassiopeiae have been investigated using recently developed frequency-domain techniques. This analysis is based on Kopal's new theory for the study of the light variations, between minima as well as within eclipses, of eclipsing binaries whose components undistorted or distorted by axial rotation and mutual tidal action.A method for the distinguishing of the photometric proximity and eclipse effects directly from the observed data is also presented. In this method no rectification is needed. The automated method has been tested successfully on the light curves of PV Cassiopeiae. Finally, a comparative discussion is given of Kopal's and Kitamura's methods of the light curves analysis.     

Analysis of the light curves of SX Aurigae,by automated fourier techniques

The aim of the present paper has beento analyse the light changes of the close eclipsing system SX Aurigae in the frequency-domain. This analysis is based on Kopal's new theory recently developed for the study of light variations, between minima as well as within eclipses, of close eclipsing binaries whose components are distorted by axial rotation and mutual tidal action.A method for the separation of the photometric proximityand eclipse effects directly from the observed data is also presented. In this method no rectification is needed. The automated method has been tested on the light curves of SX Aurigae. Finally, a comparative discussion is given of Kopal and Kitamura methods of thelight curves analysis.     

Observations and models of some neglected southern eclipsing binaries

Results of a photometric investigation of some photoelectrically neglected, southern eclipsing binaries, are presented for GW Car, X Car, and RS Sct. Light curve solutions obtained by the Wilson-Devinney and Wood synthetic light curve techniques are described.Paper presented at the Lembang-Bamberg IAU Colloquium No. 80 Double Stars: Physical Properties and Generic Relations, held at Bandung, Indonesia, 3–7 June, 1983.     

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,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,XVIII

The aim of the present paper has been to establish explicit expressions for the photometric perturbations in the light changes of close eclipsing systems, arising from the mutual distortion of the components, for any type of eclipses — be these occultations or transits; partial, total, or annular — and exhibiting arbitrary distribution of brightness (limb- or gravity-darkening) over the apparent disc of the eclipsed star.These perturbations have been expressed in terms of certain general types of series that can be easily programmed for automatic computation. They represent a generalization of results previously obtained by Kopal (1975) or Livaniou (1977, 1978) in so far as the expansions derived in this paper hold good for any real (not necessarily integral) value ofm>0. As such, they can be used to free from the photometric proximity effects within eclipses the empirical momentsA _2m of the light curves of non-integral orders, and the task performed within seconds of real time on high-speed automatic computers now available. Closed-form expressions for such perturbations, obtaining in the case of total eclipses, are given correctly to terms of first order in quantities which represent the distortion of each component.     

Fourier analysis of the light curves of eclipsing variables,XIV

The aim of the present paper will be to utilize the results obtained in the preceding papers of this series for the development of practical procedures for obtaining the elements of any eclipsing system from the observed photometric data by their analysis in the frequency-domain, for any type of eclipses, any proximity of the two components, and any degree of the law of limbdarkening of the eclipsed star.In Section 2, which follows a brief introduction to the subject, procedures will be developed which should permit us to perform such an analysis — by hand or automatic machine computation — for the case of mutual eclipses in binary systems the components of which can be regarded as spheres; and whose apparent discs are characterized by an arbitrary radially symmetrical distribution of surface brightness. In Section 3 we shall generalize these procedures to systems consisting of arbitrarily distorted stars.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May, 1978.     

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.     

Analysis of the light curves of RT Persei by automatized fourier techniques

A new method has been developed by Kopal (1977c, Paper XII) to make use of expressions for the fractional loss of light ₀ ^l of the arbitrarily limb-darkened stars in the form of Hankel transforms of zero-order, in order to evaluate the explicit forms of the ₀ ^l s for different types of eclipse, as well as of the momentsA _2m of the respective light curves in a closed form. The automated method has been tested successfully on the light curves of RT Persei. Also, a photometric curve fit of RT Persei is investigated by application of numerical quadratures to determine the theoretical light curve appropriate for the Roche model. Finally a comparative discussion is given of various methods of light curve analaysis.On leave from Department of Physics, University of Ferdowsi, Mashad, Iran.     

A generalized approach to the evaluation of reflection effects in close binary systems

A convenient integral formulation (-integrals) proves useful in describing reflection effects where there is an axisymmetric illumination, and the form of the emergent intensity distribution of the reflected component can be approximated by a cosine series. The formulation is applied to close binary systems where theoretical curves for the reflected light variation are considered. The treatment proceeds to the fifth power in the fractional dimensions — i.e. including combination of the leading terms of reflection with physical distortion of the components — and is therefore consistent with the usual first order theory of photometric proximity effects in close binary systems.     

Chromospheric variability in late-type dwarfs from high-resolution spectra of Caii H

Precise photometric observations of stars in the double cluster h and Persei reveal a large spread in the colours and magnitudes of the upper Main-Sequence; half of the stars are variables that are Be stars or related stars. The reported age difference between both clusters is found to be spurious. Rotation apparently affects both the intrinsic and the observed colours of the early-type stars in h and Persei. This result questions the validity of photometric calibrations that heavily rely on h and Persei or similar clusters.Paper presented at the 11th European Regional Astronomical Meeting of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

Calculations of proximity effects in close binary systems

The proximity effects in the light curves of close binary systems are investigated with the aid of facilities provided by a high-speed, digital computer. The treatment follows, to a large extent, that summarized by Kopal (1959) and makes extensive use of the alpha-functions and related integrals provided by that author. The ellipticity and reflection effects are studied individually and different expressions are checked and compared. Test data are drawn continually from the well known system SZ Cam for which a rectified (in a more realistic sense) light curve is ultimately produced.     

Light curve variations of the RS CVn-type eclipsing binary ER Vulpeculae

Photoelectric observations of the double-lined spectroscopic and eclipsing binary ER Vul were obtained inB andV colours of Johnson'sUBV system, between 1981 and 1986. The light curves show that the system is sometimes very active and sometimes very quiet and have changes in short time intervals. The were-like distortion, which is superimposed on the light curves, has been obtained. Moreover, small-amplitude light fluctuations in the light curves are noticeable.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

Spectrophotometric observations of delta Capricorni

Spectrophotometric observations of the eclipsing binary system Capricorni, covering the wavelength interval 3300–7300 Å, have been presented. Comparison of the standard spectral scans of Cap with the spectral scans of the stars of known spectral types and luminosity classes taken from the Breger (1976) catalogue shows that, near the phase of secondary eclipse, Cap shows the spectral-luminosity type as A8-9III. This is in agreement with the photometric findings of Srivastava (1987b). H emission is not visible convincingly, however, some irregularities are apparent, at least in two scans, around the H region.     

Gravity darkening of the secondary component of Algol

An analysis of the infrared light curve of the secondary minimum of the Algol system — when the contact subgiant component of spectral class K0 undergoes eclipse by its nearly spherical and early-type mate — for the gravity-darkening of the distorted contact star has disclosed that the monochromatic coefficient of this darkening at =1.6 is some three to four timeslarger than that resulting from von Zeipel's theory. This is in agreement with recent photometric studies of gravity darkening in WUMa-type systems, but in flagrant contrast with the consequences of theories requiring deep sub-surface convection zones for subgiants of advanced spectral types. The origin of this discrepancy remains unresolved, and underlines the need of further observations of the secondary minima in semi-detached eclipsing systems in the infrared.