Determination of the elements of eclipsing variables,RW Gem and AY Cam,by Fourier analysis of their light changes

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.     

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.     

Cosmogony of the solar system

In Sections 1–6, we determine an approximate analytical model for the density and temperature distribution in the protoplanetary could. The rotation of the planets is discussed in Section 7 and we conclude that it cannot be determined from simple energy conservation laws.The velocity of the gas of the protoplanetary cloud is found to be smaller by about 5×10³ cm s^–1 in comparison to the Keplerian circular velocity. If the radius of the planetesimals is smaller than a certain limitr ₁, they move together with the gas. Their vertical and horizontal motion for this case is studied in Sections 8 and 9.As the planetesimals grow by accretion their radius becomes larger thanr ₁ and they move in Keplerian orbits. As long as their radius is betweenr ₁ and a certain limitr ₂ their gravitational interaction is negligible. In Section 10, we study the accretion for this case.In Section 11, we determine the change of the relative velocities due to close gravitational encounters. The principal equations governing the late stages of accretion are deduced in Section 12, In Section 13 there are obtained approximate analytical solutions.The effect of gas drag and of collisions is studied in Sections 14 and 15, respectively. Numerical results and conclusions concerning the last and principal stage of accretion are drawn in Section 16.     

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 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 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.     

Fourier analysis of the light curves of eclipsing variables,XII

The aim of the present paper will be to make use of the expressions, established in Paper XI, for the fractional loss of light _l ⁰ of arbitrarily limb-darkened stars in the form of Hankel transforms of zero order, in order to evaluate the explicit forms of the _l ⁰'s for different types of eclipses (Section 2), as well as of the momentsA _2mof the respective light curves (Section 3)-in a closed form; or in terms of expansions that converge under all circumstances envisaged. Particular attention will be directed to a connection between these expansions and other functions already available in tabular form; or to alternative forms amenable to automatic computation.     

Magnitude-Redshift Relation in Clusters,Groups and Pairs of Galaxies

Magnitude-redshift (m, z) relation within systems of galaxies is studied in detail in several kinds of systems. The main data contains 17 clusters, 64 groups, 121 pairs and 14 systems with two measured members, altogether 1043 galaxies in 162 separate systems. In addition, another sample of nearby groups and pairs, recently published data on six clusters with 121 measured members, as well as 65 compact galaxies in four groups and 23 pairs are studied. In Section 2 the data and the method are described. The numerical results for the main data are given in Table 1. There exists a significant positive (m, z)-relation in groups and pairs, but for clusters the same is valid only with a rather loose selection of members. The (m, z)-relations are calculated separately for each morphological type, but systematic differences between the types are not found. In Section 3 some properties of the velocity (redshift) dispersion σ_V are discussed. The joint dispersion decreases significantly from the early to the late types. This may point to an early dynamical state of the systems but it is also quite possible that this result is due to a selection effect. The presence of a selection effect in some commonly used samples of systems is indicated by the increase of velocity dispersion σ_V with increasing distance (Section 4). This effect which was first found for clusters and groups by SCOTT is present also in the larger sample of these systems and in the sample of pairs. Implications of this feature are discussed. As one of them, it is concluded in Section 5 that there exists no separate Canes Venatici cluster of galaxies but the galaxies supposed to form it belong to the Ursa Major cloud of galaxies. Several independent arguments supporting this conclusion are pointed out. In the Ursa Major-Canes Venatici complex of galaxies a distinct positive (m, z)-relation is found. In Section 4 the distance-dependence of the (m, z)-relations is studied and it is found that positive relations are most common for nearby systems. This is natural if the effect is an intergalactic one, the redshift being dependent on the distance of the galaxy. The (m, z)-relations is are studied as function of size of the systems in Section 6. It is found that σ_V, (m, z) regression coefficient b_m, and parameter h which measures strength of redshift within the system, are largest in the systems with smallest radii. The result is opposite to that obtained using the virial theorem. In the Dopplerian context it would mean that the systems disperse the more rapidly the more dense these are. Dependence of the results on the number of data is studied in Section 7. As expected for a real effect, the frequency of positive relations increases with increasing number. The dispersion σ_V is usually larger in the central areas of the clusters than in the outskirts (Section 8). In these areas, σ_V is systematically larger for faint galaxies than for bright ones. The reason for large σ_V for faint galaxies projected on the centre is considered, studying in particular in the Coma cluster the velocity (redshift) distribution, colour-redshirt relation and morphological features which might be used in localization of the galaxies along the line of sight. The results of these three kinds of tests point to the possibility that redshift increases along the line of sight, but the results refer to sparse data and are very uncertain. A similar effect is suggested independently by observations of the galaxies in the background of the clusters. If true, the effect must be non-Dopplerian. In combination with brightness seggregation and preponderance of measured galaxies in the near side over those in the rear, this may cause the observed negative (m, z)-relations for some clusters. In SANDAGE 's and TAMMANN 's sample of nearby groups and pairs redshift appears dependent on luminosity class. This points to intrinsic redshifts in faint galaxies (Section 9). A similar implication is valid for the positive (m, z)-relations in the case of pairs and groups of compact galaxies (Section 10). Since there are indications of physical association in the latter case, the result cannot be explained by optical members. The present results are compared with previous ones in Section 11. This includes a study of redshifts with regard to brightness and surface brightness simultaneously, leading to a new statistical definition of relative compacity of galaxies belonging to the systems. Recent observations not included in the main data are viewed in Section 12. These show a positive (m, z)-relation. Interpretation of the results is discussed in Sections 13 and 14. From the numerous ones, three main possibilities remain, i. e. projected galaxies, intrinsic redshifts in faint galaxies and non-Dopplerian integalactic redshifts. There are several arguments suggesting that chance projections are not the principal explanation of the positive (m, z)-relations. If so, intrinsic redshifts in faint galaxies give probably the main explanation for pairs and small groups and integralactic redshifts for larger systems. This is in accordance with the general view of the redshift phenomenon in other scales. However, definitive proof of this conclusion could not be obtained from the present data which, though considerably large in number, is too small regarding the complexity of the problem.     

Fourier analysis of the light curves of eclipsing variables,XX

In the present paper _n ⁰ , for occulation and transit eclipses of partial phases, are evaluated numerically by means of the Runge-Kutta methods. Section 2 contains the required differential equations of _n ⁰ with respect to the modulusX orC, and Section 3 includes the numerical method of the solutions of these differential equations. Theoretical values of ₀ ⁰ and ₁ ⁰ , with corresponding values ofC, are also added in this section.     

New photoelectric observations of VW Cephei

The aim of the present paper is to present the newB andV light curves of the eclipsing binary VW Cep, obtained with the 48-in. reflector of the National Observatory of Athens, Greece.In the introduction general information for the system is given. In Section 2 some observational and reductional details are given and the obtained light curves are represented. Section 3 deals with the period of the system which was found to continue its shortening. Finally, in Section 4, a general discussion concerning our light curves is given.     

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.     

Dynamical tides in close binary systems

The aim of the present paper will be to deduce the explicit form of differential equations which govern dynamical tides in close binary systems, with simplifications which are permissible for the mass-point model (Section 2), as well as for one exhibiting finite but high internal density concentration (Section 3). It is pointed out that, whereas the exact formulation of the problem leads to a simultaneous system of equations of sixth order (fourth in the inviscid case), this order reduces to four (or two for inviscid fluids) for the mass-point model; and to five (three for inviscid case) if the density concentration is high but finite.In the last section of this paper the coefficientsC _i,j which specify the amplitudes of the individual partial tides are explicitly formulated as functions of the time.     

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

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.     

Vibrational stability of rotating stars

The aim of the present paper will be to detail the explicit form of the equations which govern first-order oscillations of fast-rotating globes of self-gravitating fluids; with due account taken of the effects arising from the centrifugal as well as Coriolis force. As such configurations oscillate in general about distorted figures of equilibrium, the equations governing them can be conveniently expressed in terms of the Clairaut coordinates, associated with distorted spheroidal figures, and introduced in our previous paper (Kopal, 1980) for this purpose.In Section 2 which follows a brief outline of our problem, the equilibrium properties of fast-rotating configurations or arbitrary structure will be formulated. In Section 3 we shall carry out a separation of the variables in the equations of motion, and reduce the partial differential equations of the problem to an equivalent system of ordinary differential equations, by an expansion of expressions for the velocity componentsU, V, W in terms of tesseral harmonicsY _n ^m (, ). The explicit form of such a system, including the effects of all tesseral harmonics of orders up tom=n=4, will be specified in Section 3 for configurations whose equilibrium form is a sphere; while in Section 4 this latter condition will be relaxed to allow for the equilibrium configuration to become a rotational spheroid.In the concluding Section 5 we shall convert the complex form of our equations of motion into real terms, amenable to a solution-analytical or numerical-in terms of real variables; and shall establish the boundary conditions necessary for a specification of the characteristic frequencies of oscillation.     

Structure and evolutionary history of the solar system,III

Parts I and II of our analysis of the evolution of the solar system were devoted mainly to the mechanical processes. The present part (Part III) deals primarily with the plasma processes and the hydromagnetic aspects.Much confusion in the cosmogonic field is due to the treatment of the early phases of the evolution of a circumstellar medium by pre-hydromagnetic methods, or by erroneous application of magnetohydrodynamics. In order to reduce the speculative element as far as possible the present analysis tries to connect the cosmogonic processes as directly as possible to laboratory plasma physics and to space phenomena actually observed today (Section 10).Models of the Laplacian type have been made obsolete by magnetohydrodynamics. Furthermore they are in conflict with observations. A new model is suggested (Section 11).A plasma surrounding a rotating central body may attain a state of partial corotation which is determined by the balance between gravitation and the centrifugal force acting on a plasma in a dipole field. Condensation from a partially corotating plasma results in grains orbiting in ellipses withe=1/3 and finally accreting to bodies at 2/3 of the central distance of the point of condensation (Section 12).An application of the theory to the Saturnian rings and to the asteroidal belt shows that the falldown ratio 2/3 (derived from the geometry of a dipole field) is essential for the understanding of their structure. The structure of the groups of planets and satellites is also discussed but only in a preliminary way. The behavior of volatile substances is a major problem which still awaits an appropriate treatment (Section 13).     

Relativistic hierarchical cosmology

While Euclidean models with uniform matter density have a number of radio sources of flux density greater thanF at frequencyv that varies asN(>F, v)¹ F ^–3/2, hierarchical models with = ₀ r ^–2 haveN(>F,)F ^–1/2 (Section 1). Since the observed dependency isN(>F,)F ^–1.8, severe density and/or luminosity evolution must be present in a workable hierarchical cosmology (Section 2). The same argument applies (Section 3) to the number of sources of apparent luminosity greater thanl,N(>l) and (Section 4) to the number of sources within redshift distancez from the local origin. To give agreement with empirical data demandsq _o=+1 and large first and second derivatives with respect to time of the number source density (Section 5). The adoption ofq _o=+1 allows one to show (Section 6) that a Lemaitre-type hierarchical Universe with a long coasting or waiting time can give agreement with observations of the numbers of QSO's etc. if the age of the Universe is more than 10¹³ yr. The dependence of the effective Hubble parameter onk(t), (t) andR (Section 7) leads one to suggest that ak=0, =0 hierarchy with 0 might be the simplest acceptable form of model Universe. Section 8 (Conclusion) points out that further data on source count anisotropies should allow the component levels of the hierarchy to be delineated.     

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.     

30 Doradus as the active centre of the Large Magellanic Cloud

Evidence is presented for the hypothesis that the supergiantHii-complex 30 Doradus (NGC 2070) is the mildly active galactic nucleus of the Large Magellanic Cloud. For this purpose the general properties of galactic nuclei and the characteristics of active nuclei are reviewed (Section 2). Examination of 30 Doradus shows that it plasy the same exceptional role among allHii-regions of the LMC as Sgr A among those of our Galaxy, and has all the properties of a galactic nucleus (luminosity, emission spectrum, IR source, semistellar central object R 136, symmetry centre of an starting point of spiral structure). Evidence for the activity is given by the peculiar filamentary structure (Figure 1), the young spiral filaments superposed on old, broad and smooth near-circular arms (Figure 2), the splitting of the [Oiii] 5007 profile in two components corresponding to an expansion velocity of 50 km s^–1, and the strong non-thermal component (Section 3). The mass loss of 30 Dor is estimated at 0.05M /a. It is speculated that the nucleus of a galaxy may be wandering due to explosive events.