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1.
Zdeněk Kopal 《Astrophysics and Space Science》1975,35(1):171-183
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. 相似文献
2.
Zdeněk Kopal 《Astrophysics and Space Science》1975,38(1):191-241
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. 相似文献
3.
Zdeněk Kopal 《Astrophysics and Space Science》1982,81(1-2):411-451
The aim of the present paper will be to generalize the methods for computation of the elements of eclipsing binary systems in the frequency-domain, summarized in our recent Paper I (Kopal, 1981), to the case ofclose systems, in which photometric proximity effects become conspicuous and must be taken into account before the methods previously outlined in Paper I become directly applicable.Following a brief introduction to the subject given in Section 1, Section 2 summarizes (and comments upon) the difficulties previously encountered in separation of the photometric proximity and eclipse effects. In Section 3 we develop an alternative new approach to the problem by modulation of the light curves throughout the entire orbital cycle, intended to filter out proximity effects from the observed light changes and isolate those due to eclipses; while in Section 4 we shall present a numerical application of the new method to an analysis of the observed light changes of the eclipsing system W Ursae Maioris.In Section 5 we shall present a quantitative investigation of the photometric effects of distortion on the light changes of close eclipsing systems within eclipses-the most complicated part of the whole problem-with numerical application to the system of U Sagittae carried out in the concluding Section 6.Appendices 1–3 contain numerical data which should facilitate application of the methods developed and illustrated in Sections 3–4; while Appendix 4 will be reserved for a mathematical proof of certain expansions used in Section 5, which would have been too discursive for the main text. 相似文献
4.
Zdeněk Kopal 《Astrophysics and Space Science》1975,34(2):431-457
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. 相似文献
5.
Zdeněk Kopal 《Astrophysics and Space Science》1982,87(1-2):149-191
The aim of the present investigation will be to develop a theory of the light changes of the eclipsing systems in which one (or both) components oscillate(s) freely-radially or non-radially-in period(s) which may (though need not) be synchronized with that of their orbit. The light curves exhibited by systems with oscillating components will, in general, beasymmetric; and these oscillations may, in fact, be the cause of asymmetries observed in many eclipsing systems with evolved components (for which such oscillations-far from being anomalous-should be the rule rather than exception). The present paper will be concerned with an investigation of photometric phenomena arising from oscillations of theeclipsing components of close binary systems-cutting off (partly or wholly) the star (spherical or distorted) which undergoes eclipse.In Section 2, which follows a brief introductory survey aiming to place the entire subject in its historical perspective, a theory will be given of the light curves of close binary systems, in the frequency-domain, which are affected byradial oscillations of the eclipsed or eclipsing star; while Sections 3 and 4 will be devoted to a similar treatment ofnon-radial oscillation of the secondary (eclipsing) component distorted by equilibrium tides, or axial rotation with constant angular velocity. As will be shown, a frequency-domain approach will enable us to describe all these phenomena in algebraic terms; with auxiliary results required to this end relegated to three Appendices to the main text (in order not to render the latter too discursive and impede the main line of the argument).A similar treatment of photometric phenomena which may arise in such systems from oscillations of their components which undergo eclipse is being postponed for a subsequent communication. 相似文献
6.
Zdeněk Kopal 《Astrophysics and Space Science》1975,35(1):159-170
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. 相似文献
7.
Zdeněk Kopal 《Astrophysics and Space Science》1976,40(2):461-481
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. 相似文献
8.
Zdeněk Kopal 《Astrophysics and Space Science》1975,36(1):227-237
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. 相似文献
9.
Zdeněk Kopal 《Astrophysics and Space Science》1987,134(2):235-259
The aim of the first part (Sections 1 and 2) of this paper will be to extend an analysis of the light curves of totally-eclipsing binary systems to the case in which the distortion of the shadow-cone of the eclipsing component must be taken account of in an analysis of the system's light changes within minima. Like in the case of total eclipses of spherical stars, an extension of the analysis of their light changes in the frequency-domain can likewise be carried out in a closed form, without a need to resort to automatic computers which may not be at the disposal of every investigator. In the second part (Sections 3 and 4) of this paper, the same procedure will be extended (albeit to a lesser degree of precision) toany type of eclipses—be these total, annular or partial; while a proof of the underlying formulae the reader will find collected in the Appendix. Again, in this more general case, while an access to automatic computers may be of advantage to incorporate the photometric perturbation arising from the distortion of the components mutually eclipsing each other, their use represents a convenience rather than necessity for our work. 相似文献
10.
Hamid M. K. Al-Naimiy 《Astrophysics and Space Science》1977,46(2):261-284
The aim of the present paper has been to present an analysis of the light curve of two eclipsing systems RW Gem and AY Cam by Fourier analysis of the light changes in the frequency domain which was developed by Kopal (1975a, b, c, d, e; 1976).In Section 1, the subject is introduced in a general way, with the intention of laying the foundation of the light curve analysis. Section 2 contains the evaluation of the empirical values of the theoretical momentA
2m is demonstrated, with the equation of the condition given. Then the equations forA
2m in terms of the elements of the total and the annular eclipses, including partial and annular phase of transit eclipse, follow.The analysis of the light curves of the two eclipsing binaries (RW Gem and AY Cam), the results and the discussion of our solution, are outlined in Section 3. 相似文献
11.
Zdeněk Kopal 《Astrophysics and Space Science》1977,46(1):87-108
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. 相似文献
12.
Zdeněk Kopal 《Astrophysics and Space Science》1976,45(2):269-281
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. 相似文献
13.
Zdeněk Kopal 《Astrophysics and Space Science》1977,50(1):225-246
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. 相似文献
14.
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. 相似文献
15.
M. Kurutaç 《Astrophysics and Space Science》1978,57(1):71-98
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. 相似文献
16.
S. Evren 《Astrophysics and Space Science》1987,137(1):151-166
The RS CVn-type eclipsing binary MM Her was observed photoelectrically inB andV colours. The light curves obtained in 1984 and 1985 are presented. It was found that the depths of the primary minima are decreased from 1983 to 1985. However, the amplitude of the wave-like distortion outside the eclipses was detected to increase since 1976. The period of migration was determined to be about 3.57±0.08 years. 相似文献
17.
Huseyin Alkan 《Astrophysics and Space Science》1981,75(2):455-463
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. 相似文献
18.
A. G. Tsouroplis 《Astrophysics and Space Science》1977,47(2):361-373
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. 相似文献
19.
The circular elements of eclipsing binary MR Cygni in yellow and blue light are derived from both minima. The durations of eclipses are different and this is interpreted as an effect of orbital eccentricity. The elements deduced from one minimum and the depth of the other do not permit the construction of a theoretical light curve based on a spherical model which satisfies the other minimum. However, this might be feasible by introducing an appropriate value for the orbital eccentricity. New ephemeris are obtained from the photoelectric minima. 相似文献
20.
A new method of light curve analysis is introduced for systems containing one component possessing an extended atmosphere. Occultation or partial eclipses may be treated with or without the presence of transparency effects. Besides admitting an arbitrary degree of transparency to the eclipsing component, an arbitrary law of limb-darkening may also be assigned to the eclipsed star. The method is applied to the analysis of continuum, narrow band light curves of V444 Cygni obtained by Cherepashchuk and Khaliullin. Primary and secondary minima are examined separately with reasonable results. 相似文献