Further notes on the associated alpha-functions and related integrals in the theory of the light changes of eclipsing variables

With a view to furthering the theory of the light changes of eclipsing variables, developed before systematically by Z. Kopal, this paper presents a number of new (and computable) expressions for the associated alpha-function _n ⁰ ,(r₁,r₂,) (and also for its partial derivatives), where _n ⁰ ,(r₁,r₂,) represents the fractional loss of light suffered by an eclipse of a circular disc of fractional radiusr ₁ (and darkened at the limb to thenth degree) by an opaque disc of radiusr ₂, with their centres separated by a fractional (projected) distance , provided that the transparency of the occulting disc increases with the angle of foreshortening in the same manner as the limb-darkening of the eclipsed star (that is, when the transparency functiong(, ) of the second aperture is given by Equation (4) below). Many of the explicit expressions derived here are valid for any type of eclipse, occultation or transit, regardless of whetherr ₁>r ₂ orr ₁<r ₂. and for any degreen of the adopted law of limb-darkening. It is also pointed out how some of the results obtained in this paper are related to the various representations given earlier in the literature for the case =0.     

Solar flares: The gradual phase

One has to distinguish between two kinds of the gradual phase of flares: (1) a gradual phase during which no energy is released so that we see only cooling after the impulsive phase (a confined flare), and (2) a gradual phase during which energy release continues (a dynamic flare).The simplest case of (1) is a single-loop flare which might provide an excellent opportunity for the study of cooling processes in coronal loops. But most confined flares are far more complicated: they may consist of sets of unresolved elementary loops, of conglomerates of loops, or they form arcades the components of which may be excited sequentially. Accelerated particles as well as hot and cold plasma can be ejected from the flare site (coronal tongues, flaring arches, sprays, bright and dark surges) and these ejecta may cool more slowly than the source flare itself.However, the most important flares on the Sun are flares of type (2) in which a magnetic field opening is followed by subsequent reconnection of fieldlines that may continue for many hours after the impulsive phase. Therefore, the main attention in this review is paid to the gradual phase of this category of long-decay flares. The following items are discussed in particular: The wide energy range of dynamic flares: from eruptions of quiescent filaments to most powerful cosmic-ray flares. Energy release at the reconnection site and modelling of the reconnection process. The post-flare loops: evidence for reconnection; observations at different wavelengths; energy deposit in the chromosphere, chromospheric ablation, and velocity fields; loops in emission; shrinking loops; magnetic modelling. The gradual phase in X-rays and on radio waves. Post-flare X-ray arches: observations, interpretation, and modelling; relation to metric radio events and mass ejections, multiple-ribbon flares and anomalous events, hybrid events, possible relations between confined and dynamic flares.     

Pristine vs. contaminated trends in Nb,Ta-oxide minerals of the Jihlava Pegmatite District,Czech Republic

Summary Columbite-tantalite is widespread in the lepidolite-subtype rare-element pegmatites of the Jihlava pegmatite district, western Moravia, Czechoslovakia. In most pegmatites, fractionation of columbite-tantalite shows initial enrichment in Mn followed by increasing Ta, in accordance with the usual trend from ferrocolumbite through mangano-columbite to manganotantalite, typical of the lepidolite pegmatites. Many dikes, however, show local deviations toward Fe- and Ti-rich compositions. In extreme cases, all columbite-tantalite is strongly enriched in Fe and Ti, and is associated with ixiolite and tantalian rutile. The degree of enrichment of the Nb,Ta oxide-mineral assemblage in Fe and Ti is proportional to tectonic introduction of wallrock xenoliths of mafic pyroxene-biotite syenite into the pegmatites during late stages of their consolidation. Extensive reaction of the residual pegmatite melt with the xenoliths contaminated the near-by melt, and generated Nb,Ta oxide minerals and tourmaline of non-typical chemistries. Late fersmite probably formed after thermal equilibration of the pegmatites with their syenitic country rocks, from Ca-bearing regional interstitial fluids pervading through the solidified pegmatites.

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Zusammenfassung In Selten-Element Pegmatiten (Lepidolith-Subtyp) des Jihlava Pegmatit Distriktes, West-Mähren, Tschechien, ist Columbit-Tantalit weitverbreitet. In den meisten Pegmatiten führte die Fraktionierung von Columbit-Tantalit zur einer anflinglichen Anreicherung von Mn gefolgt von einer Zunahme in Ta, vergleichbar mit dem für Lepidolith-Pegmatit bekannten Trend Ferrocolumbit-Manganocolumbit-Manganotantalit. Viele Gänge zeigen jedoch lokale Abweichungen zu Fe- und Ti-reichen Zusammensetzungen. In extremen Fällen ist Columbit-Tantalit stark an Fe und Ti angereichert und mit Ixiolith und Tantalo-Rutil vergesellschaftet. Das Ausmaß der Fe und Ti Anreicherung in (Nb,Ta)-Oxid-Assoziationen ist proportional dem tektonisch bedingten Eindringen von maischen Pyroxen-Biotit-reichen Syenit-Nebengesteinsxenolithen in die Pegmatite gegen Ende ihrer Verfestigung. Tiefgreifende Reaktion der Pegmatit-Restschmelze mit den Xenolithen kontaminierte die Schmelze und führte zur Bildung von (Nb,Ta)-Oxid-Mineralen und Turmalin ungewöhnlicher Zusammensetzung. Möglicherweise nach der thermischen Gleichgewichtseinstellung der Pegmatite mit ihren syenitischen Nebengesteinen bildete sich Fersmit unter Beteiligung von Ca-führenden regionalen Porenlösungen, welche die verfestigten Pegmatite durchdrangen.

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With 6 Figures     

Geochemical and structural evolution of micas in the Rožná and Dobrá Voda pegmatites,Czech Republic

Summary The chemistry, structural parameters, polytypism, optical properties and Rb-Sr isotopes were examined in 11 to 60 samples of biotite, muscovite and lepidolite from the pegmatites at Roná (the type locality of lepidolite; 323 ± 4Ma) and Dobrá Voda (306 ± 9Ma) in western Moravia. At both localities, early endocontact biotite is followed inwards by muscovite and lepidolite, which is concentrated in and around the core. At Roná, a 1M lepidolite follows after 2M₁ muscovite but all later generations of lepidolite are 2M₂, close to Tri₅₀ Ply₅₀ and in part associated with muscovite 2M₁. At Dobrá Voda, all lepidolite types are 1M and free of muscovite, and the late varieties approximate Tri₃₀ Ply₇₀. At both localities, a trend of increasing HF is indicated during the progress of mica crystallization, culminating in precipitation of topaz. Polytypism of lepidolite is not correlatable with any compositional or growth feature, or their combination. Throughout the mica crystallization, Rb/Cs decreases but K/Rb becomes reversed after an initial decrease. Boron is partitioned preferentially into muscovite (up to 1.10 wt.% B₂O₃) but Be, Zn, Mn and Sc are enhanced in lepidolite. A slight increase in Fe, Ba and Cl in the last generation of lepidolite might be possibly due to mixing of residual pegmatite fluids with metamorphic pore solutions.

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Geochemische und strukturelle Entwicklung der Glimmer in den Pegmatiten von Roná und Dobrá Voda, Tschechische Republik

Zusammenfassung In 11 bis 60 Proben von Biotit, Muskovit und Lepidolith aus den Pegmatiten von Roná (Typlokalität des Lepidoliths; 323 ± 9 Ma) in Westmähren wurden Chemie, Struktur-parameter, Polytypie, optische Eigenschaften und Rb-Sr-Isotopie untersucht. An beiden Lokalitäten wird früher Biotit an Endokontakten nach Innen von Muskovit und Lepidolith gefolgt, letzterer ist in und um den Kern konzentriert. In Roná folgt 1M-Lepidolith auf 2M₁-Muskovit, aber alle späteren Lepidolithgenerationen sind 2M₂, nahe Tri₅₀Ply₅₀ und zum Teil mit 2M₁-Muskovit vergesellschaftet. In Dobrá Voda sind alle Lepidolithe vom Typ 1M und frei von Muskovit, die späten Varietäten kommen Tri₅₀Ply₅₀ nahe. An beiden Lokalitäten ist während des Fortschreitens der Glimmerkristallisation eine Tendenz von steigendem HF angezeigt, die in der Ausfällung von Topas ihren Höhepunkt findet. Die Polytypie des Lepidoliths kann nicht mit irgendeiner Eigenheit der Zusammensetzung oder des Wachstums korreliert worden, auch nicht mit einer Kombination von diesen. Während der ganzen Glimmerkristallisation nimmt Rb/Cs ab, aber die Tendenz von K/Rb ändert sich nach anfänglichem Abfall. Das Bor verteilt sich bevorzugt auf den Muskovit (bis zu 1.10 Gew. -% B₂O₃), aber die Be-, Zn-, Mn- und Sc-Gehalte sind im Lepidolith erhöht. Ein leichter Ansteig von Fe, Ba und Cl in der letzten Lepidolithgeneration könnte vielleicht durch eine Mischung von pegmatitischen Restlösungen mit metamorphen Porenlösungen verursacht sein.

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With 9 Figures     

On ‘the Solar Flare Myth’ postulated by Gosling

The Solar Flare Myth postulated by Gosling (1993) is a misunderstanding. It is true that most sources of coronal mass ejections (CMEs) cannot be classified as flares in the common old sense of that word. However, just for this reason the term eruptive flare has been introduced for all solar active phenomena in which an opening of field lines is involved and which lead to magnetic-field and mass ejections resulting in a CME. The process is essentially the same in all events, irrespective of' whether only adisparition brusque without any chromospheric brightening or a major two-ribbon flare is involved in it; the only difference is the different strength of the magnetic field in which the process was accomplished. The major two-ribbon (cosmic-ray) flares clearly represent the most energetic events of this kind, and, therefore, it is very misleading to claim that solar flares in general are phenomena with very little importance for solar-terrestrial physics.     

Infinitesimal orbits around lagrange points in the elliptic,restricted three-body problem

This study presents a method of obtaining asymptotic approximations for motions near a Lagrange point in the planar, elliptic, restricted three-body problem by using a von Zeipel-type method. The calculations are carried out for a second-order escape solution in the proximity of the equilateral Lagrange point, L ₄, where the primaries' orbital eccentricity is taken as the small parameter .     

Stability of the solar system

If the solar system is considered as a mechanical clockwork consisting of its present members which attract each other as mass-points, the extent of its present approach to secular stability (i.e., the state of minimum potential energy) — manifested by the existence of a number of nearcommensurabilities of the present orbital periods, not only of the planets, but also of their satellites —could not have been attained in a time-span of 4.6×10⁹ yr of its age by gravitational perturbations alone.The existence of such commensurabilities — striking in many instances— could then be understood only on the assumption that either (a) the solar system was actually born with the present 2-, 3- and 4-term couplings between the orbital period of the planets already built-in from the outset (which is improbable on any known grounds); or (b) that these couplings — in particular, the 25 Jupiter-Saturn commensurability — have arisen as a result of tidal interaction between proto-planetary globes of much larger dimensions than these planets possess today. For the present dimensions and mutual distances of these planets, their tidal interaction in 10⁹ yr would exert but negligible effects; and during that time neither their masses, nor the scale of the solar system underwent any essential change.Therefore, a hypothesis is proposed that the situation now obtaining had its origin in the early days of the formation of the solar system, when the planetary globes — in particular, those of Jupiter and Saturn (now in the terminal stage of Kelvin contraction) — were very much larger than they are today; and when, as a result, the tidal coupling between them operated at a much higher rate than at the present time.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

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.     

Fourier analysis of the light curves of eclipsing variables,IX

The aim of the present paper will be to detail the procedure outlined in our previous investigations (Kopal, 1975; Kopalet al., 1976) for a solution of the elements of distorted eclipsing systems by a Fourier analysis of their light changes. This procedure—which constitutes an equivalent, in the frequency-domain, of rectification hitherto practised in the time-domain — should enable us to free the observed momentsA _2m of the light curves from all photometric effects of distortion (between minima as well as within eclipses) — a feat impossible in the time-domain except under very restricted conditions — and thus to make it possible to obtain the geometrical elements of the eclipses which should be free from any obvious source of systematic errors.