Dynamical tides in close binary systems

The aim of the present paper will be to investigate the circumstances under which an irreversible dissipation of the kinetic energy into heat is generated by the dynamical tides in close binary systems if (a) their orbit is eccentric; (b) the axial rotation of the components is not synchronized with the revolution; or (c) the equatorial planes are inclined to that of the orbit.In Section 2 the explicit form of the viscous dissipation function will be set up in terms of the velocity-components of spheroidal deformation arising from the tides; in Section 3, the principal partial tides contributing to the dissipation will be detailed; Section 4 will be devoted to a determination of the extent of stellar viscosity — both gas and radiative; while in the concluding Section 5 quantitative estimates will be given of the actual rate at which the kinetic energy of dynamical tides gets dissipated into heat by viscous friction in stellar plasma.The results disclose that the amount of heat produced per unit time by tidal interaction between components of actual close binaries equals only about 10^–10th part of their nuclear energy production; and cannot, therefore, affect the internal structure of evolution of the constituent stars to any appreciable extent. Moreover, it is shown that the kinetic energy of their axial rotation can be influenced by tidal friction only on a nuclear, rather than gravitational (Kelvin) time-scale — as long as plasma or radiative viscosity constitute the sole sources of dissipation. However, the emergence of turbulent viscosity in secondary components of late spectral types, which have evolved away from the Main Sequence, can accelerate the dissipation 10⁵–10⁶ times, and thus give rise to appreciable changes in the elements of the system (particularly, in the orbital periods) over time intervals of the order of 10⁵–10⁶ years. Lastly, it is pointed out that, in close binary systems consisting of a pair of white dwarfs, a dissipation of the kinetic energy through viscous tides in degenerate fermion-gas could produce enough heat to account, by itself, for the observed luminosity of such objects.     

Digital mapping of coastal boulders – high-resolution data acquisition to infer past and recent transport dynamics

Coastal boulder fields provide clues to long-term frequency-magnitude patterns of coastal flooding events and have the potential to play an important role in coastal hazard assessment. Mapping boulders in the field is time and labour-intensive, and work on intertidal reef platforms, as in the present study, is physically challenging. By addressing coastal scientists who are not specialists in remote sensing, this contribution reports on the possibilities and limitations of digital applications in boulder mapping in Eastern Samar, Philippines, where recent supertyphoons Haiyan and Hagupit induced high waves, coastal flooding and boulder transport. It is demonstrated how satellite imagery of sub-metre resolution (from Pléiades and WorldView-3 imagery) enables efficient analysis of transport vectors and distances of larger boulders, reflecting variation in latitudes of both typhoon tracks and approaching angles of typhoon-generated waves. During the investigated events, boulders with a-axes of up to 8 m were clearly identified to have been shifted for up to 32 m, mostly along the seaward margin of the boulder field. It is, however, hard to keep track of smaller boulders, and the length of a-axes and b-axes including their orientation is often impossible to map with sufficient accuracy. Orthophotographs and digital surface models created through the application of an unmanned aerial vehicle and the ‘Structure from Motion’ technique provide ultra-high-resolution data, and have the potential to not only improve the results of satellite image analysis, but also those from field mapping and may significantly reduce overall time in the field. Orthophotographs permit unequivocal mapping of a-axes and b-axes including their orientation, while precise values for c-axes can be derived from the respective digital surface models. Volume of boulders is best inferred from boulder-specific Structure from Motion-based three-dimensional models. Battery power, flight speed and altitude determine the limits of the area covered, while patches shielded by the boulders are difficult to resolve. For some tasks, field mapping remains mandatory and cannot be replaced by currently available remote sensing tools: for example, sampling for rock type, density and age dating, recording of lithological separation of boulders from the underlying geological unit and of geomorphic features on a millimetre to decimetre-scale, or documentation of fine-grained sediment transport in between the boulders in supratidal settings. In terms of future events, the digital products presented here will provide a valuable reference to track boulder transport on a centimetre to decimetre-scale and to better understand the hydrodynamics of extreme-wave events on a fringing reef coastline.     

The precession and nutation of deformable bodies,III

In preceding papers of this series (Kopal, 1968; 1969) the Eulerian equations have been set up which govern the precession and nutation of self-gravitating fluid globes of arbitrary structures in inertial coordinates (space-axes) as well as with respect to the rotating body axes; with due account being taken of the effects arising from equilibrium as well as dynamical tides.In Section 1 of the present paper, the explicit form of these equations is recapitulated for subsequent solations. Section 2 contains then a detailed discussion of the coplanar case (in which the equation of the rotating configuration and the plane of its orbit coincide with the invariable plane of the system); and small fluctuations in the angular velocity of axial rotation arising from the tidal breathing in eccentric binary systems are investigated.In Section 3, we consider the angular velocity of rotation about theZ-axis to be constant, but allow for finite inclination of the equator to the orbital plane. The differential equations governing such a problem are set up exactly in terms of the time-dependent Eulerian angles and , and their coefficients averaged over a cycle. In Section 4, these equations are linearized by the assumption that the inclinations of the equator and the orbit to the invariable plane of the system are small enough for their squares to be negligible; and the equations of motion reduced to their canonical form.The solution of these equations — giving the periods of precession and nutation of rotating components of close binary systems, as well as the rate of nodal regression which is synchronised with precession — are expressed in terms of the physical properties of the respective system and of its constituent components; while the concluding Section 6 contains a discussion of the results, in which the differences between the precession and nutation of rigid and fluid bodies are pointed out.     

Possibilities of GPS kinematic measurements for geological survey in East Slovakia

This paper deals with transformation procedures for observed GPS data from the world geodetic system WGS-84 into the national geodetic grid datum S-UTCN (system of united trigonometric cadaster network) and Baa (the Baltic Sea after adjustment). Transformation from WGS-84 into SUTCN is performed most frequently by means of the 7-element Helmert transformation with three identical points. Geodetic network was adjusted by two ways.     

Manifestations of the North – South Asymmetry in the Photosphere and in the Green Line Corona

The north – south asymmetries (NSA) of three solar activity indices are derived and mutually compared over a period of more than five solar cycles (1945 – 2001). A catalogue of the hemispheric sunspot numbers, the data set of the coronal green line brightness developed by us, and the magnetic flux derived from the NSO/KP data (1975 – 2001) are treated separately within the discrete low- and mid-latitude zones (5° – 30°, 35° – 60°). The calculated autocorrelations, cross-correlations, and regressions between the long-term NSA data sets reveal regularities in the solar activity phenomenon. Namely, the appearance of a distinct quasi-biennial oscillation (QBO) is evident in all selected activity indices. Nevertheless, a smooth behavior of QBO is derived only when sufficient temporal averaging is performed over solar cycles. The variation in the significance and periodicity of QBO allows us to conclude that the QBO is not persistent over the whole solar cycle. A similarity in the photospheric and coronal manifestations of the NSA implies that their mutual relation will also show the QBO. A roughly two-year periodicity is actually obtained, but again only after significant averaging over solar cycles. The derived cross-correlations are in fact variable in degree of correlation as well as in changing periodicity. A clear and significant temporal shift of 1 – 2 months in the coronal manifestation of the magnetic flux asymmetry relative to the photospheric manifestation is revealed as a main property of their mutual correlation. This shift can be explained by the delayed large-scale coronal manifestation in responding to the emergence of the magnetic flux in the photosphere. The reliability of the derived results was confirmed by numerical tests performed by selecting different numerical values of the used parameters.     

Sixty Years in Solar Physics

I recount my career in solar physics beginning at Ondřejov Observatory in 1948 and ending with my ∼30 year stay at the Laboratory of Space Research in Utrecht.     

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.     

Vibrational stability of double-star Roche model

The aim of the present paper will be to extend the methods of our previous investigations (Kopal, 1980, 1987) by employing the Clairaut coordinates (in which the radial component is identified with the total potential) to analyze the nature of small oscillations about the equilibrium form of Roche double-star model (identical, in fact, with zero-velocity surfaces of the restricted problem of three bodies).Linearized equations of this problem have been set up in Clairaut coordinates, and solved in a closed form. This solution turns out to be closely analogous to that obtained already for the rotating single-star Roche model, and discloses that (like in the preceding case) the terms secular in time appear already in the linear approximation. However, whether or not a retention of nonlinear terms in the equations of motion can regain secular stability of the respective configurations remains yet to be clarified by future investigations.     

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.