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.     

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.     

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.     

Photoelectric observations of BV Draconis

In this paper the twoB andV light curves of the close eclipsing binaryBV Draconis, obtained with the 48-in reflector of the National Observatory of Athens, are represented and discussed. Also, new period of the system is given.     

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.     

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.     

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.     

Determination of the elements of eclipsing variables RW tauri and U sagittae by an analysis of the light changes in the frequency domain

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.     

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.     

Accretion and electrostatic interaction of interstellar dust grains; Interstellar grit

This work is divided into 13 sections and 2 appendices, and aims to elucidate the accretion mechanism, which operates via image-theory forces, whenever two interstellar dust grains come close together. Section 1 is an introduction. Section 2 proposes that the distribution of interstellar grains be taken asn(r) r ^–4 to avoid distortion of the 3K microwave background by radiation from spinning grains. Section 3 examines each of three types of image force accretion processes, finding them to be dominant compared to radiation or gravitational forces by at least a factor of 10¹⁹. Section 4 states that only grains made of conducting material (e.g., graphite, ice, iron) are involved in image theory. Section 5 presents reasons for believing that two grains should coalesce on impact. Section 6 examines the motion of charged interstellar grains in Hi and Hii regions. Section 7 demonstrates, by way of four examples involving dust grains ofr=10^–7 cm up tor=10^–4 cm, that the image effects on conducting grains are not trivial, and that the dynamics involved is not to be compared at all with elementary Coulomb interaction of two changes. Section 8 concludes that accretion with not take place in Hi clouds if thermal (equipartition) velocities prevail among the dust particles. section 9 examines grain interactions in Hii regions: here, following an argument due to Spitzer, consideration is given to the case of a population of dust grains all streaming in the direction of the local magnetic field B at velocities of order 0.1 km s^–1. It is shown that accretion takes place effectively, leading to the formation of interstellar grit, meaning grains of mass 10^–8 to 10^–7 gm, radius 0.1 mm; and leaving also a population ofr10^–6 cm grains, which are observed in polarization and extinction measurements. The existence of the latter is now a deduction and not an ad hoc postulate, as previously, and implies a distribution of the general formn(r) r _mean ^–3 , in approximate agreement with that of Section 2. Section 10 considers the accretion mechanism as a cascade process. Section 11 shows that the existence of grains in space ofr 10^–6 cm rules out an origin in supernova or galactic explosions, and supports a passive origin, perhaps in red giants or Mira variables. Section 12 discusses the implications of the results found for polarization observations and cosmogony, the latter being given a new foundation in which planets of different composition form automatically from a solar nebula. Section 13 is a conclusion.     

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 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.     

UBV photometry of the WR binary CQ Cephei

Photoelectric observations of the WR binary CQ Cephei (WN6+O9) are presented. the depths of the eclipses in the light curves are best represented by an inclination of the orbit i = (68°.8±0.6) and the width of the very asymmetric eclipse curves can be represented by only an overcontact configuration (Ω₁ = Ω₂ = 3.65 ± 0.05, and f = 27%). Simultaneous solution of the light and radial velocity curves strongly supports CQ Cep's membership of the Cep OB1 association. By considering this membership we obtained absolute dimensions of the system, which lead to a consistent physical model for CQ Cephei. The more luminous WR primary turns out to be the hotter but slightly less massive component: M_WR = 20.8 M⊙, R_WR = 8.2R⊙, T_eff(WR) = 43600 K, and M_o = 21.4 M⊙, R_o = 8.3 R⊙, T_eff(O) = 37000 K.     

Expanding clusters of galaxies as components of a relativistic hierarchical cosmology

Approximate metries for clusters embedded in an expanding Robertson/Walker background (Section 1) indicate that in general clusters of galaxies cannot remain uninfluenced by the expansion. The potential for an expanding cluster (Section 2) is seen to suggest that clusters as presently observed are not in static equilibrium and that the missing mass problem can be elucidated via use of the virial theorem (Section 3) given certain density conditions in compact clusters. Non-compact clusters can be treated similarly, and it is found that the behaviour of a low-density cluster is equivalent to that of a domain of a Friedmann (k=0) model Universe (Section 4). The treatment of compact clusters (Section 5) is based on approximate metrics for clusters embedded in a Robertson/Walker background. These comparisons lead to information on the evolution of non-compact clusters and compact clusters (Section 7), and to observable consequences that seem to be borne out successfully (Section 8). Data on clusters themselves tend to show that there exists an expanding supercluster and/ or that >0 (Section 9). Several tests of the hypothesis of expansion of clusters (Section 10) are proposed.     

The roche coordinates and their use in hydrodynamics or celestial mechanics

The aim of the present paper will be to introduce a new system of curvilinear coordinateshereafter referred to as Roche coordinates-in which spheres of constant radius are replaced by equipotential surfaces of a rotating gravitational dipole (which consists of two discrete points of finite mass, revolving around their common center of gravity); while the remaining coordinates are orthogonal to the equipotentials. It will be shown that the use of such coordinates offers a new method of approach to the solution of certain problems of particle dynamics (such as, for instance, the construction of certain types of trajectories in the restricted problem of three bodies); as well as of the hydrodynamics of gas streams in close binary systems, in which the equipotential surfaces of their components distorted by axial rotation and mutual tidal interaction constitute essential boundary conditions.Following a general outline of the problem in Section 1, the Roche coordinates associated with the equipotentials of a rotating gravitational dipole will be constructed in the plane case (Section 2), and their geometrical properties discussed. In Section 3, we shall transform the fundamental equations of hydrodynamics to their forms appropriate in the curvilinear Roche coordinates. The metric coefficients of this transformation will be formulated in a closed form in Section 4 in terms of the respective partial derivatives of the potential; while in Section 5 analytic expressions for the Roche coordinates will be given in the orbital plane of the dipole, which are exact as far as the distortion of the equipotential curves from circular form can be described by the second, third and, fourth harmonics.The concluding Section 6 will be devoted to a formulation of the equations of a mass-point in the restricted problem of three bodies in the Roche coordinates. Three special cases will be considered: (a) motion in the neighborhood of the equipotential curves; (b) motion in the direction normal to such curves; and (c) motion in the neighbourhood of the Lagrangian points. It will be shown that motion in one coordinate is possible only in limiting cases which will be enumerated; but twodimensional motions in which one velocity component is very much smaller than the other invite further study.A generalization of the plane Roche coordinates to three dimensions, with application to additional classes of problems, is being postponed for a subsequent paper.     

Theory of the light curves of eclipsing systems with oscillating components: I

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.     

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.     

New photoelectric observations and elements of the eclipsing W UMa-type system V839 Oph

The W UMa-type system V839 Oph was observed photoelectrically in two wavebands (B andV) during 1985 at the Kryonerion Astronomical Station of the National Observatory of Greece. The light curves of the system are analysed using the frequency-domain techniques and new geometric and photometric elements are given. The light variations of the system are also discussed.     

Inverse problem with two-parametric families of planar orbits

As a possible extension of recent work we study the following version of the inverse problem in dynamics: Given a two-parametric familyf(x, y, b)=c of plane curves, find an autonomous dynamical system for which these curves are orbits.We derive a new linear partial differential equation of the first order for the force componentsX(x, y) andY(x, y) corresponding to the given family. With the aid of this equation we find that, depending on the given functionf, the problem may or may not have a solution. Based on given criteria, we present a full classification of the various cases which may arise.