Vibration Effect of Near Earthquakes at Different Depths in a Shallow Medieval Mine

The shallow medieval Jeroným Mine is located at a distance of about 25 km southeast of the Nový Kostel focal zone where the most intensive seismic activity in West Bohemia (Czech Republic) has been documented. Permanent seismological monitoring has been carried out since 2004 in this mine. During the 2011 and 2014 seismic swarms, more than 1000 triggered records comprising almost 1500 earthquakes were recorded at the permanent station in the mine. Three short-term seismological experiments were accomplished during these swarms. Several temporary seismic stations were simultaneously placed in different parts of underground spaces which enabled comparison of vibration effect caused by near earthquakes in different parts of the mine. Although the depth of the lowest parts of mine is only about 60 m, a vibration effect generated by earthquakes from the Nový Kostel focal zone is not the same for the whole underground complex.     

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.     

On the role of the cosmological constant in the formation of neutrino halos

We investigate the role of the cosmological constant in the formation of neutrino halos. Using the observational limit 10^–55 cm^–2 we find that it is relevant only for neutrinos of very light rest-mass: namely, 10^–1 eV.On leave of absence from Department of Physics, College of Mines, Ostrava, Czechoslovakia.     

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.     

Volume integrals of the products of spherical harmonics and their application to viscous dissipation phenomena in fluids

The equations governing the conversion of kinetic energy into heat in moving viscous media are formulated as volume integrals of products of spherical harmonics. Although the formulation of the fundamental equations is classical, difficulties in the integration of certain products of generalized spherical harmonics over a sphere have permitted heretofore the treatment of only two cases. The closed, form evaluation of eight fundamental types of definite integrals of the product of spherical harmonics, some of them new, or at least missing in the literature, makes possible for the first time the evaluation of these volume integrals in closed form for arbitrary order and index. Explicit details are given for the rates of energy dissipation produced by viscous motions characterized by spheroidal as well as toroidal symmetry.     

Erfassungskoeffizient des Aerosols AgJ durch einen fallenden Wassertropfen

Summary The author first derives the coefficient of trapping of the aerosol by a water droplet. Having determined the velocity field of the medium in the neighbourhood of the droplet and solved the equation of motion of the aerosol, he determined the critical Stokes number, St_k = 5/6, beginning with which the aerosol is trapped by inertia. The next section deals with the experiment. Using the apparatus, located at the observatory of the Institute of the Physics of the Atmosphere in Hradec Králové, the author carried out experiments which proved that there existed a critical Stokes number and also that it was possible, given sufficiently small values of St, to adopt with sufficient accuracy the trapping coefficient as = (2.5 ± 0.3) a/r, where a is the radius of the aerosol particle and r is the radius of the droplet.     

Theory of the light curves of eclipsing systems with oscillating components,II

In continuation of our preceding investigation (Kopal, 1982c) of photometric consequences of free oscillations — radial or nonradial — of the eclipsing component on the light changes of close binary systems, the aim of the present paper will be to extend the scope of our inquiry to include the effects of similar oscillations of the component which undergoes eclipse, with arbitrary period and phase.In Sections 2 and 3 which follow a brief introduction to the particular aspect of the problem treated in this paper, a theory will be given of photometric effects caused by arbitrary harmonic oscillations of the components of eclipsing binary systems distorted by equilibrium tides, or by axial rotation with constant angular velocity. A translation of these effects into the frequency domain constitutes a problem which is solvable in a closed form only if the eclipse in question ends in totality. In any other case, the desired effects can be mathematically described only in terms of infinite series (of satisfactory asymptotic properties); and a construction of their explicit forms will be given in the Appendix.