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1.
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.  相似文献   

2.
A Laplace and a Hankel transform of the light curves of eclipsing binary star systems have been expressed in terms of the eclipse elements (L 1,r 1,2 andi) of the spherical basic model. These general expressions can be used for the solution of the eclipse elements of the systems in terms of the observed quantities.  相似文献   

3.
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.  相似文献   

4.
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   

5.
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.  相似文献   

6.
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.  相似文献   

7.
The aim of the present paper is to deduce relations between the integral transformsA 2m, B2m,andF 1,2 of the light curves of eclipsing binary systems. The integral transformsA 2m, B2m,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.  相似文献   

8.
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   

9.
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.  相似文献   

10.
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.  相似文献   

11.
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).  相似文献   

12.
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.  相似文献   

13.
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.  相似文献   

14.
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.  相似文献   

15.
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.  相似文献   

16.
The aim of the present paper will be to evaluate numerically Jacobian and other functions which have been discussed in more detail in a previous paper of this series (Edalati, 1978b, Paper XXII), and also choose the most convenient moments to obtain a good determination for the unknown eclipse parametersa andc 0. More than 12 different pairs ofg-functions for real values ofm have been investigated numerically and diagrammatically. The behaviour ofg-functions depends but very little on different combination of the moments, and related diagrams are approximately the same asg 2 andg 4 (Kopal and Demircan, 1978, Paper XIV).The behaviour of the vanishing Jacobian, arising from different pairs ofg-functions for real values ofm1 has been shown diagrammatically in terms ofa andc 0. Accordingly, we obtain the optimum combination of the moments (i.e.,A 6,A 7,A 8 andA 9) ing-functionsg 7 andg 8. It has been noted that the behaviour of theg-functions which depend on the combinations of the higher order moments (i.e.,m5) have been ruled out, because the proportional error of the momentsA 2m increases with increasing values of realm. The automated method has been tested successfully on the light curve of RT Per (Mancusoet al., 1977; Edalati, 1978a). Finally, a comparison is given of the elements of RT Per arising from two different pairs ofg-functions, i.e.,g 2,g 4 (Edalati, 1978a) andg 7,g 8 for the light curves analysis.  相似文献   

17.
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.  相似文献   

18.
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.  相似文献   

19.
R and I band CCD observations of the nova V1494 Aql during July-November 2002 are reported and the V, R, and I light curves are analyzed. The orbital light curve of this nova has an eclipse-like form with two unequal humps before and after the eclipse. The approach to the eclipse lasts twice as long as the emergence from it. The overall duration of the eclipse is about 0.45P orb. The depth of eclipse increases with wavelength and averages 0m.3 (V), 0m.5 (R), and 0m.7 (I). The secondary, shallow minimum has an average depth of 0m.1 in R and I and about 0m.03 in V. The hump at phase 0.65 is higher than the one at phase 0.17. The most probable explanation for the observed variations in the light with the phase of the orbital period may be self eclipsing of the accretion column in the magnetic exploding variable (white dwarf) together with partial eclipsing of the accretion region by the secondary component.  相似文献   

20.
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   

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