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).     

Extrasolar planetary systems

The article deals with the occurrence of planetary systems in the Universe. In Section I, the terms “planet” and “planet-like objects” are defined. Two definitions proposed for the term “planetary system” are examined from the point of view (1) of the relation between planetary systems and binary and multiple star systems and (2) of planetary systems as abodes of intelligent beings. In Section II, the observational search for extrasolar planetary systems is described, as performable by earthbound optical telescopes, by space probes, by long baseline radio interferometry, and finally by inference from the reception of signals sent by intelligent beings in other worlds.In Section III we show that any planetary system must be preceded by a rotating disk of gas and dust around a central mass. Both observational evidence and theoretical reasons indicate the ease of formation of such disk structures in the cosmos. The time scale of collapse of a gaseous medium into a disk and that of the latter's dissipation are examined. This provides us with a new empirical approach and leads us to consider the problem of the frequency of occurrence of planetary systems to be ripe for scientific study. In Section IV, a brief review of theories of the formation of the solar system is given along with a proposed scheme for classification of these theories. In Section V, the evidence for magnetic activity in the early stages of stellar evolution is presented, as developed from six independent clues: the nuclear abundance of light elements, the behavior of flare stars, the intensities of H and K emission in stars, the nonthermal radiation of premain sequence stars, the properties of meteorites, and finally the existence of contact binaries. The magnetic braking theories of solar and stellar rotation are discussed in Section VI, thereby introducing the idea of formation of a rotating disk of gas and dust around stars in Section VII. From this disk a planetary system emerges.Section VIII gives an estimate for the frequency of occurrence of planetary systems in the Universe. It is based on the rotational behavior of main-sequence stars, and concludes that planetary systems have a far greater chance to appear around single main-sequence stars of spectral types later than F5 than around any other kind of star. The combined probability distribution of sizes and masses could be obtained. From physical considerations, it appears that sizes of planetary systems around stars of any given spectral type may not vary greatly from one to another.     

Vortex flow of matter around a singularity and a galactic hypothesis of Jeans

When compact objects or black holes move through a fluid medium, or when turbulent plasma and magnetic fields so conspire, a gas flow is set up which closely resembles the flow of water down a plug-hole (Section 1). A similar hypothesis, but in reverse, was suggested by Jeans in 1928, and would nowadays be referred to as the white hole concept. The dynamics of the flow (Section 2) lead to expressions for the rotational velocity of the fluid far away from (2.1) and near to (2.2) the origin of the vorticity. Rotation curves derived from the model (Section 3) are closely akin to actual galactic rotation curves, but observational data on the latter are not precise enough to permit a delineation to be made between (i) flow around a singularity and (ii) flow around a non-singular sink or source. The other acceptable model, that of (iii) a spreading line vortex, is ruled out by comparison with astrophysical observations (Section 4). The basic analysis for all the models shows that the old problem of the winding-up of spiral arms can be avoided, since the galactic flow system is in a steady state. Section 5 identifies Jeans' speculation as being a hypothesis compatible with singular vortex flow and so with observation, but perhaps not with the usual interpretation of general relativity metrics, even though the requisite dual space does complete the topology in a mathematically satsifying manner. Section 6 concludes that the predictions of the hypothesis of vortex flow agree with the shape, dynamics and structure of galaxies.     

Rotational dynamics of a deformable medium

The aim of the present investigation has been to derive from the fundamental Cauchy's first law of continuum mechanics the explicit form of the Eulerian general equation which governs the three-axial generalized rotation about the centre of mass of a self-gravitating deformable finite material continuum, viscolinear (i.e., Newtonian) or not, consisting of compressible fluid of arbitrary viscosity, in an external field of force. The generalized rotation is a superposition of the so-called rigid-body (i.e., time dependent only) rotation of the continuum plus a nonrigidbody (i.e., position-time dependent) rotation of its configurations.In Section 2, which follows brief introductory remarks outlining the problem, we develop a mathematical theory which describes the whole phenomenon in terms of two rotation tensors corresponding, respectively, to the rigid-body and nonrigid-body rotation modes. In Section 3, we derive the differmation vectors of velocity and acceleration. The equations we have obtained are a very general version of Navier Stokes' equations, which were not given in previous investigations. In Section 4, we perform integration of the left-hand side of Cauchy's first law, cross-multiplied by the position vector, without any restriction. In Section 6, integration of the right-hand side of the same law, cross-multiplied by the position vector, is carried out, by taking account of actually simplifying assumptions stated in Section 5. All the integral terms occurring in both sides are expressed explicitly by quantities evaluated in terms of components of properly defined moments.Finally, in Section 7, the system of the general Eulerian equations is set up; and some easy modifications are given, which describe nicely physical models of special interest; while the concluding Section 8 contains a general discussion of the results.     

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.     

The Friedmann universe and the world potential

In Section 1 of the paper the energy equation of the Friedmann universe, when matter dominates over radiation, is discussed. It is known that the value of the world potential is constant everywhere in the Universe, despite the pulsation motion of the Universe or a possible transformation of pulsation energy into matter or vice versa. The condition for the Universe being closed is deduced. Furthermore, the possibility to define the mass-energy of the Universe is discussed; and the conclusion is arrived at that the mass-energy of the Universe relative to an observer in the non-metric space outside the Universe is equal to zero; i.e. the Universe originated as a vacuum fluctuation. Finally, the view-point of an external observer is described. Such an observer can claim that our closed Universe is a black hole in a non-metric empty space. Besides, the differences between such a black hole and the astrophysical black holes are indicated.In Section 2 the origin of the gravitational force retarding the expansion is discussed, using the properties of the relativistic gravitational potential. In contradiction to Section 1, the view-point of an inner observer (inside the Universe) is used here. It is concluded that the boundary of the closed Universe is an unlocalizable potential barrier.In Section 3 of the paper the apparent discrepancy between Mach's principle and the general theory of relativity is resolved. The solution is based on the fact that, for the Euclidean open universe, the concept of mass is related to the potential of the background equal to –1, but the concept of the mass-energy is related to the zero-potential of the non-metric background. Because the universe is open and a potential barrier (a boundary of the universe) can be localized-i.e. is geometrically existing — by solution of the field equation, we have to refer to the background with zero-potential. The principal idea of the solution is then that the zero-density means the density of mass-energy, when simultaneously the mass density is equal to the critical value for which the Robertson-Walker metric becomes the Euclidean metric of the Minkowski (i.e., flat) space-time. Further a generalization of Newton's law of inertia is formulated, and the properties of nullgeodesics are touched upon. As a conclusion it is stated that this paper and the two previous ones (see Voráek, 1979a, b)de facto express Mach's principle.     

The Roche Coordinates in three dimensions and their application to problems of double-star astronomy

In a preceding paper (Kopal, 1969; in what follows referred to as Paper I) we introduced a new system of curvilinear coordinates-hereafter referred to as Roche Coordinates — in which spheres of constant radius in spherical polars have been replaced by surfaces of constant potential of a rotating gravitational dipole; while the angular coordinates are orthogonal to the equipotentials. In Paper I we established an explicit form of such a transformation, and related the Roche coordinates with polar coordinates (with which they coalesce in the immediate neighbourhood of each one of the two finite mass-points) in the plane case. The aim of the present investigation will be to generalize the definition of the Roche coordinates to three dimensions.The opening Section 1 of this paper will contain a general outline of the proposed three-dimensional transformation; and in Section 2 details of this transformation will be explicitly worked out correctly to quantities of first order in superficial distortion — an approximation which should prove adequate in regions surrounding the two finite masses; while in Section 3 we shall evaluate (to this degree of accuracy) the metric coefficients of the respective transformation, and its direction cosines, in both polar and curvilinear coordinates. Section 4 will then contain a formulation of the fundamental equations of hydrodynamics in terms of the three-dimensional Roche coordinates; and their advantages for a treatment of certain classes of dynamical problems encountered in doublestar astronomy will be illustrated in the concluding Section 5 by an investigation of the vibrational stability of the Roche model. We shall show that this model is capable of performing free radial oscillations which remain barotropic only if its equilibrium form is spherical (i.e., in the absence of any external mass in the neighbourhood); but not if it is distorted to any extent by rotation or tides.     

Dissipation of energy through tidal deformation

The aim of the present paper will be to derive an equation of dissipation of energy for a rotating body of arbitrary viscosity distorted by tides, which arise from the gravitational field of its companion in a close pair of such bodies.By a transformation of the fundamental equation of energy dissipation in terms of velocity of tidal deformation (Section 2), the dissipation function is constructed for a tidally-distorted body (Section 3). From this equation, the rate of dissipation of tidal energy is formulated for a nearly-spherical rotating body distorted by second harmonic longitudinal tides (Section 4); the coefficients of viscosity (or the bulk modulus) are treated as arbitrary functions of spatial coordinates. Finally (Section 5), expressions for the total energy dissipation within the orbital cycle are given for axial rotation of the distorted body, provided its angular velocity is constant (for example, with the Keplerian angular velocity).Research financed in part by the Division of Scientific Research and Development of Ministry of Sciences and Culture of Greece.     

Effects of rotation on internal structure of the stars

The aim of the present paper will be to establish the explicit form of the equations which govern the internal structure of stars rotating with constant angular velocity formulated in terms of Clairaut coordinates (cf. Kopal, 1980) in which the radial coordinate is replaced by the total potential, which for equilibrium configurations remains constant over distorted level surfaces. The introductory Section 1 contains an account of previous work on rotating stars, commencing with Milne (1923), von Zeipel (1924) and Chandrasekhar (1933), who all employed orthogonal coordinates for their analysis. In Section 2 we shall apply to this end the curvilinear Clairaut coordinates introduced already in our previous work (cf. Kopal, 1980, 1981); and although these are not orthogonal, this disadvantage is more than offset by the fact that, in their terms, the fundamental equation of our problem will assume the form of ordinary differential equations, subject to very simple boundary conditions. The explicit form of these equations — exact to terms of fourth order in surficial distortion caused by centrifugal force—will be obtained in Section 3; while in the concluding Section 4 these will be particularized (for the sake of comparison with work of previous investigators) to stars of initially polytropic structure. These will prove to be much simpler in Clairaut coordinates than they were in any previously used frame of reference. Lastly, in Appendix A we shall present the explicit forms, in Clairaut coordinates, of the differential operators which were needed to establish the results given in Sections 3–4; while Appendix B will summarize other auxiliary algebraic relations of which use was made to formulate our fourth-order theory developed in Section 3.     

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.     

A non-linear model for time

Some recent astronomical observations [4], and a number of experiments in particle physics, seem to cast doubt on the validity of the standard linear model for time. These results raise two (at least) questions: (1) If time is not a linear continuum (i.e., if the standard model is incorrect), then why does this model work so well in so many areas of science? (2) Whatever the “true nature” of time is, are there any advantages, to science, in replacing the standard model with a more complicated one? The purpose of this paper is to present a non-linear, mathematical model for time that enables us to answer question (1), and to partially answer question (2). Our discussion of question (2) is incomplete, but our results are intriguing. They also show promise of helping us understand some of the observations mentioned above. A rather natural extension of our model brings it into close contact with one that has been used in quantum theory (“Stochastically branching spacetime topology” by Roy Douglas [2]). These points of contact will also be discussed.     

Physical time and astronomical “time”

This sketch is based on the following line of thought. If we understand by (physical or dynamical) time the independent variable in the equations of motion, only quantities which are in a strictly linear relationship with this mechanical — physical time should properly be called time. This excludes therefore such concepts as e.g., sidereal time; in general that Earth Rotation Parameter which is one of the angles which define the orientation in space of a coordinate system fixed in a rigid model Earth. The relationships between these Eulerian angles and the ERP which follow from the theory of motion are discussed, also for the case that the reference system is an arbitrary noninertial system whose relationship (as a function of time) with an inertial system is known.     

Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle

We present the results of a statistical study of the solar cycle based on the analysis of the superficial toroidal magnetic field component phase space. The magnetic field component used to create the embedded phase space was constructed from monthly sunspot number observations since 1750. The phase space was split into 32 sections (or time instants) and the average values of the orbits on this phase space were calculated (giving the most probable cycle). In this phase space it is shown that the magnetic field on the Sun’s surface evolves through a set of orbits that go around a mean orbit (i.e., the most probable magnetic cycle that we interpret as the equilibrium solution). It follows that the most probable cycle is well represented by a van der Pol oscillator limit curve (equilibrium solution), as can be derived from mean-field dynamo theory. This analysis also retrieves the empirical Gnevyshev – Ohl’s rule between the first and second parts of the solar magnetic cycle. The sunspot number evolution corresponding to the most probable cycle (in phase space) is presented.     

The neutral atmospheres of comets

In this paper we have endeavored to critically evaluate our present understanding of cometary atmospheres. Following a brief introduction of the significance of the study of cometary atmospheres (Section 1), the relevant photometric and spectroscopic observations are summarized in Section 2.The interaction with the solar radiation, with regard to both the excitation of the observed species as well as the dissociation of stable molecules evaporating from the nucleus, is considered in Sections 3 and 4. The gas phase chemistry likely to take place in the dense inner coma is next considered in Section 5.The exospheric and hydrodynamic models of the expanding cometary atmosphere are considered in detail in Section 6, and both their limitations as well as possible improvements are discussed.The observed chemical composition of the neutral atmosphere and the inferred chemical composition of the volatile component of the nucleus, together with possible variations between different classes of comets is next considered in Section 7, and their possible cosmogonic significance is discussed.In conclusion, some of the important directions in which future research should progress, in order to provide more complete and secure knowledge of cometary atmospheres, are stressed (Section 8).Astrophysics and Space Science Review Paper.     

Sociological Profile of Astronomers in Spain

In this paper the main findings are presented of a recent study made by a team of sociologists from the University of Granada on the professional astronomers currently working in Spain. Despite the peculiarities of this group – its youth, twentyfold increase in size over the last 20 years, and extremely high rate of specialization abroad – in comparison with other Spanish professionals, this is the first time that the sociological characteristics of the group have been studied discretely. The most significant results of the study are presented in the following sections. Section 1 gives a brief historical background of the development of Astronomy in Spain. Section 2 analyzes the socio-demographic profile of Spanish Astronomy professionals (sex, age, marital status, etc.). Sections 3–5 are devoted to the college education and study programs followed by Spanish astronomers, focusing on the features and evaluations of the training received, and pre- and postdoctoral study trips made to research centers abroad. The results for the latter clearly show the importance that Spanish astronomers place on having experience abroad. Special attention is paid to scientific papers published as a result of joint research projects carried out with colleagues from centers abroad as a result of these study trips. Section 6 describes the situation of Astronomy professionals within the Spanish job market, the different positions available and the time taken to find a job after graduation. Section 7 examines Astronomy as a discipline in Spain, including the astronomers' own opinions of the social status of the discipline within Spanish society. Particular attention is paid to how Spanish astronomers view the status of Astronomy in Spain in comparison with that of other European countries. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mars atmospheric escape and evolution; interaction with the solar wind

This tutorial deals with the question of atmospheric escape on Mars. After a brief introduction describing the general context of Mars escape studies, we will present in Section 2 a simplified theory of thermal escape, of both Jeans and hydrodynamic types. The phenomenon of hydrodynamic escape, still hypothetical and not proved to have ever existed on terrestrial planets, will be treated with the help of two well known examples: (i) the isotopic fractionation of xenon in Mars and Earth atmospheres, (ii) the paradox of missing oxygen in Venus atmosphere. In Section 3, a simplified approach of non-thermal escape will be developed, treating in a specific way the different kinds of escape (photochemical escape, ion sputtering, ion escape and ionospheric outflow). As a matter of illustration, some calculations of the relative contributions of these mechanisms, and of their time evolutions, will be given, and the magnitude of the total amount of atmosphere lost by non-thermal escape will be estimated. Section 4 will present the state of knowledge concerning the constraints derived from Mars isotopic geochemistry in terms of past escape and evolution. Finally, a few conclusions, which are more interrogations, will be proposed.     

Bianchi type-V bulk viscous string cosmological model in scale-covariant theory of gravitation

A spatially homogeneous and anisotropic Bianchi type-V space–time is considered in the frame work of a scale covariant theory of gravitation proposed by Canuto et al. (Phys. Rev. Lett. 39:429, 1977) when the matter sources is a bulk viscous fluid containing one dimensional cosmic strings. Using some physically plausible conditions, we have obtained a determinate solution of the field equations of the theory which represents a Bianchi type-V bulk viscous string cosmological model in this theory. Some physical and kinematical properties of the model are also discussed.     

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.     

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.