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.     

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

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

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

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

Fourier analysis of the light curves of eclipsing variables,II

The aim of the present paper will be to extend the Fourier methods of analysis of the light curves of eclipsing binaries, outlined in our previous communication (Kopal, 1975) in connection with systems whose components would appear as uniformly bright discs, to systems whose components exhibit discs characterized by an arbitrary radially-symmetrical distribution of brightness —i.e., an arbitrary law of darkening towards the limb — be it linear or nonlinear.In Section 2 which follows a few brief introductory remarks, fundamental equations will be set up which govern the light changes arising from the mutual eclipses of limb-darkened stars — be such eclipses total, partial or annular; and Section 3 will contain a closed algebraic solution for the elements of the occulation eclipses terminating in total phase. Such a solution proves to be no more complicated than it turned out to be for uniformly bright discs in our previous paper; and calls for no special functions for the purpose — as will be put in proper perspective in the concluding Section 4.The cases of transit eclipses terminating in an annular phase, of partial eclipses of occulation or transit type, will be similarly dealt with by Fourier methods in the next paper of the present series.     

Grativy-darkening in the Algol system

Infrared observations of the secondary minimum of the eclipsing system of Algol, secured recently by Nadeauet al. (1978) with the 200 in and 60 in reflectors of Mount Wilson and Palomar Observatories at the effective wavelength of 10 m, show its light curve to be distinctlydish-shaped-i.e., the light diminishes relatively fast in the early stages of the eclipse, and its rate of decline slows down in advanced partial phases. This fact indicates convincingly that the light distribution over the apparent disc of Algol's late-type (contact) component is akin to that produced by the phenomenon of gravity-darkening to a very pronounced degree.Past observational evidence on close binary systems has indicated that gravity-darkening consistent with the equilibrium structure of the stars was adequate to account for the light variations of such systems exhibited between minima, as long as their components remain on (or do not depart much from) the Main Sequence (Kopal, 1968b). However, for components close to (or at) their Roche limits, the photometric effects of gravity-darkening appear to be greatly exaggerated (Kopal, 1968a)—a result confirmed by subsequent investigators (cf. Parthasarathy, 1972; or Niarchos, 1978).An analysis of Algol's infrared light curve during the secondary minimum (when its contact component undergoes eclipse by its nearly spherical mate) observed at an effective wavelength of 10 m, discloses now that the (monochromatic) coefficient of the linear law of gravity-darkening, characterizing the distribution of brightness over the apparent disc of the contact star comes out again at least twice as large as one which would correspond to a purely radiative energy transfer of total light in the far interior of this star. No physical theory can be advanced to explain this fact—except, possibly, a hypothesis that the observed enhancement of the monochromatic coefficient of gravity-darkening over that appropriate for total radiation may be caused by a very wide departure of the outer layers of the respective stars from thermodynamic equilibrium.     

Radiative energy transfer in extended photospheres of the components of close binary systems

The aim of the present paper will be to set up, and solve, the equations governing transfer of radiation in semi-transparent envelopes of the stars; and, in order to do so, to employ a system of curvilinear (non-orthogonal) three-dimensional coordinates in which the radial coordinate has been identified with equipotential surfaces. Such coordinates are particularly suitable to a treatment of the problems arising in close binary systems, which render the outcome more than any other amenable to observable tests, but which has so far received but very scant attention.The introductory section of this paper will contain a statement of the problem; and its mathematical formulation in terms of Clairaut coordinates (cf. Kopal, 1980, 1989, Chapter V) will be outlined in Section 2; their methods in Section 3. Section 4 will then contain an application to the problem of distribution of surface brightness (limb-darkening) over the apparent discs of distorted components of close binary systems; while in Section 5 we shall do the same for radiative flux of distorted stars as a function of the phase (gravity darkening).The concluding Section 6 will then contain an outline of additional problems arising in this connection, to which we shall turn in successive parts of this series.     

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

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

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

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

The radio radius of the Sun at millimeter and centimeter wavelengths

The radio radius of the Sun is determined from an analysis of the radio contact times of the 7 March, 1970 and 10 July, 1972 solar eclipses from = 3 mm to = 31 cm. Agreement with other eclipse measurements is good. A best fit curve through the several points gives the radio radius to within approximately ±0.01 of the photosheric radius below -5 cm.     

The eclipses of starspots

Formulae are set out from which the light loss may be derived in a close binary system in which a component affected by a starspot, of the type considered previously (Budding, 1977) is also in eclipse, in such a way that the starspot itself is undergoing eclipse. In this way the hitherto postponed problem of thecombination of eclipse and maculation effects — which are sometimes believed to be present, e.g., in RS CVn-stars — is treated, and shown to be reducible to expressions involving algebraic or elliptic integrals.A more practical approach using a small-spot approximation is given, and demonstrated on an example. Such an approach should be easy to manage by computer, and is expected to find a ready application in the analysis of active cool, close binaries.     

Disc system delta Capricorni

FirstUBV photoelectric photometry of the eclipsing binary system Cap has been presented. An improved period of 1^d.022766 has been given. The duration of primary eclipse comes out to be more than double the duration given earlier. The depth has also been found to have increased. The light changes during eclipses show slight asymmetry. Eccentricity appears to be present in the system. Light and colour curves show variations. Primary component appears to be surrounded by a disc, the size of which is comparable to the size of the primary component. Two dips are seen around phases 0.20 and 0.67, the first appears more definitive, and is attributed to the wave-like distortion, like the one found in RS CVn binaries. The discussion reveals that Cap is a very complicated system.     

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.     

Fourier analysis of the light curves of eclipsing variables,XVIII

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

Distribution of polarization and intensity of radiation across the stellar disk and numberical values of atmospheric characteristics governing this distribution

Computations of polarization and intensity of radiation from a unit stellar surface area are presented, as well as a study of the numerical characteristics of atmospheres — single-scattering albedo and the initial source function(), which define the polarization behaviour of atmospheres. The radiatively stable models of stellar atmospheres presented by Kuruczet al. (1974) and Kurucz (1979) have been used for calculations. Since the versus optical depth dependence is rather weak, it has been assumed that (=cost. With a fixed effective temperatureT _eff maximum values of are characteristic of stars featuring the lowest surface gravity accelerationg. Among stars with radiatively stable atmospheres, maximum values of (=5000 Å) 0.4–0.6 are exhibited by supergiants withT _eff=8000–20 000 K. The plot of () is characterized by discontinuities at the boundaries of spectral series for hydrogen and, sometimes, for helium. Maximum are attained in the Lyman region of =912–1200 Å, where can reach the value 0.7–0.9 for supergiants, this value being 0.3 for Main-Sequence stars. For stars withT _eff 35 000 K, high values of also are attained for <912 Å. Within the infrared region, is always small because of bremsstrahlung absorption.A rapid growth of the source functionB with < typical for ultraviolet range (within the Wien part of spectrum), together with high values of results in the strong polarization of emission from a unit stellar surface element, sometimes exceeding the values for the case of a pure electron scattering. For longer wavelengths, where the limb-darkening coefficient is smaller, the plane of polarization abruptly turns 90° in the central parts of the visible stellar disk.     

Rosat X-ray light curves of early-type stars

We have investigated ROSAT X-ray data of OB stars to search for evidence of time variability in the X-ray emission from early-type stars. As an example for such studies we present a detailed variability analysis for our two program stars Ori and Ori which have been multiply observed with ROSAT. The long-term analysis of both stars now covers a time range of 2.5 years and includes six pointed PSPC observations, an additional pointed HRI observation of Ori and the ROSAT all-sky survey data of both stars. Over a long time range the X-ray light curves of Ori and Ori show no evidence for variability. In the case of Ori we detected a moderate increase in X-ray count rate during a period of 2 days which can be explained as a strong shock propagating in the wind of an O-type star.     

Correlations between statistical population indices and spectral characteristics of carbon stars

The coefficients of correlation between spectroscopic data published by Yamashita (1967) and others for carbon stars and the statistical population indices calculated for these stars at the Toru Observatory are calculated (Table II). The intensity estimates of Cai 4227 Å, Nai D lines, the C¹³/C¹² ratio, then=n(Li/Ca) index, as well as CN and probably C₂ bands are higher in population I carbon stars. The CH(G) band and probably hydrogen (H, H, H) lines as well as Baii 4554 and 4934 Å lines are stronger in population II carbon stars. The photoelectric colour indices, corrected for interstellar reddening do not show significant population effects. They can be used as spectral type equivalents. For a population criterion the CH/CN intensity ratio is proposed.     

Narrow-band photometry of Wolf-Rayet stars and planetary nebulae

We present observational data in the (16)(9)(25)-photometric system for 11 Wolf-Rayet stars and 7 planetary nebulae. The results show anomalous (16)(9)-indices for these emission objects. Six W-R stars show possible variations in the strength of the HeII line at 6560 Å, perhaps none in the strength of the HeI line at 10830 Å. In spite of these misconstrued (16) and (9)-indices, most WC stars are separated from WN stars in the (16)(9)-array; and PN, in this diagram lie far apart from all kind of stellar objects. The results, yielded by the (25)-index, indicate that the HeI and HeII lines at 10830 Å and 10124 Å, respectively, are probably well suited for photometric examination in W-R stars, PN and other peculiar objects.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

The spectrum of the Bp star HD 36916

The spectrum of the peculiar Bp star HD 36916 has been studied on plates with dispersion 9.7 A/mm and compared with those of Aur, B3V andi And, B8V. A comparison of the observed contours of H, H and of the Balmer discontinuity D with those computed by Mihalas gives the following results: HD 36916: _e=0.347, logg=3.8;i And: _e=0.383, logg=3.2; Aur: _e=0.302, logg=3.6. HD 36916 presents the characteristics of the Si-4200 stars: strong deficiency in helium and probably also in oxygen and nitrogen, strong excess of silicon and strontium; 3984 Hgii is present. Moreover this star also has characteristics which are not common to Si-4200 stars but rather to Mn stars: excess by a factor of ten of the elements of the iron group and an excess larger, probably of the order of 100, for manganese. The star is a member of the Sword subgroup of the Orion association.     

Ionization equilibrium and spectrum formation in QSOs' absorption regions

The behaviour of the primary and secondary components in a sample of double-lined Algol-type eclipsing binaries in the logg-logT _e diagram is analyzed. Our results indicate that the hotter components behave like normal Main-Sequence stars while the effect of irradiation may partly explain the overluminosity of the cool components.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

Hα Doppler brightening and Lyman-α Doppler dimming in moving Hα prominences

We consider the effect that coherent motion has on the observed brightness of moving clouds above the photosphere. We find that steady state clouds (constant N _e and T _e ) that are moving perpendicular to the line of sight will appear brighter in H for speeds between 8 and 100 km/sec and dimmer for speeds greater than 135 km/sec. The brightening and dimming are due to apparent Doppler shifts of the respective H absorption and the Lyman- emission profiles seen by the absorption profile of the moving cloud.We apply this analysis, along with optical depth and geometrical considerations, to the observed brightness variations of the 1 March 1969 limb eruptive prominence. We find that all of the observed brightening and dimming can be explained by the motions, and that no significant change in the prominence N _e or T _e was necessary during the observed H event. This conclusion is significant in interpreting an X-ray burst that began as the prominence velocity increased abruptly at the time of maximum H intensity. The thermal X-ray peak occurred 150 sec later when the prominence had become faint again. There was no associated flare that was visible in H. We discuss the relative brightness of H and D ₃ in a specific moving prominence knot.We note that the observed range of limb speeds (30–150 km/sec) may be due to the combined H Doppler brightening and Lyman- dimming effects. We also discuss generally the H brightness of disk surges (bright and dark) and flares, and sprays and puffs that occur at or near the limb.Now at the Dept. of Physics and Astrophysics, University of Colorado, and High Altitude Observatory (NCAR) Boulder, Colo., U.S.A.