Volterra equation for the matrix source function at resonance scattering

A matrix transfer equation for multiple resonance scattering of radiation in a spectral line in a semiinfinite atmosphere with a uniform distribution of primary radiation sources is examined. A nonlinear matrix integral is obtained for this equation as a generalization of the Rybicki two-point Q-integral. One special case of the matrix [^(Q)] {\mathbf{\hat{Q}}} -integral is the Volterra equation for the matrix source function of the problem discussed here. The Volterra equation is solved numerically for a Doppler profile of the absorption coefficient. Several polarization characteristics of the emerging radiation are obtained.     

Radiation forces and nonspherical dust particles

Action of electromagnetic radiation on nonspherical dust particles is discussed. It is stressed that the radiation pressure coefficientQ _PR cannot be considered to be a scalar quantity, as it is used in all calculations for dynamical studies of interplanetary dust particles. Also the equation <Q _PR A> = <Q _PR ><A> (A - area of the particle) holds only for perfectly absorbing convex dust particle (Q _PR = 1) and not even one of these two properties holds for interplanetary dust particles. Plane mirror is discussed in detail - all calculations can be done in this simple case.     

Transfer of Polarized Radiation: Nonlinear Integral Equations for I-Matrices in the General Case and for Resonance Scattering in a Weak Magnetic Field

General equations of the Wiener-Hopf type for a matrix source function with nonsymmetrical kernel matrices are considered in the form of continuous superpositions of exponentials. Certain problems in the transfer of polarized radiation reduce to equations of this kind. In general there are two different H-matrices in the theory (which are a generalization of the Ambartsumian-Chandrasekhar scalar H-function), generated by an initial equation of the Wiener-Hopf type and its analog, but with the kernel matrix and the unknown matrix of the source function being transposed. In addition there are two corresponding I-matrices, actually consisting of Laplace transforms of the matrix source functions, through which the Stokes vector of the escaping radiation is directly determined. In the problem of diffuse reflection from a half-space, the I-matrices are expressed in terms of a product of these two H-matrices, and for the latter there is a system of nonlinear equations which is a generalization of the corresponding Ambartsumian-Chandrasekhar scalar equation. In the problem of the emission of partially polarized radiation from a half-space containing uniformly distributed internal sources we have obtained a system of two nonlinear equations for the I-matrices directly. In the special case of a symmetrical kernel matrix, this system of two equations reduces to one equation. It is shown that in the case of resonance scattering in a weak magnetic field (the Hanle effect) in the approximation of complete frequency redistribution, the system of two nonlinear equations for the I-matrices (of dimension 6×6) also reduces to one nonlinear equation, although the kernel matrix for the main integral equation for the matrix source function () is not symmetrical. For this case we have found a matrix generalization of the so-called law, consisting of an equation of the type (0)Â ^T (0) = (where T denotes transposition) at the boundary of a half-space containing uniformly distributed primary sources of partially polarized radiation.     

On the solving of the Hill's differential equation through the method of operational calculus

Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter ² which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree 12 with respect to which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in ², ( ²)² and the coefficient of the term in ( ²)³ which depends upon the initial valuey(0) of the Hill's differential equation.     

Possible Evidence of Long-Term Oscillations in the Solar Atmosphere

We have obtained the temporal correlation function, Q(t), from time sequences of Caii K filtergrams and Dopplergrams from Antarctica, Taiwan Oscillation Network (TON) and Solar and Heliospheric Observatory (SOHO). Q(t) gives the time evolution of the pattern under examination, supergranulation in this case. It has been found that Q(t) shows oscillatory signals of both 5-min and long-term periods. The 5-min oscillations are suppressed by averaging the images over 10 min. An exponential decay curve which represents the lifetime trend of supergranules, is fitted to Q(t) and subtracted out. The Q(t) residuals thus obtained contain the oscillatory component and are then subjected to a periodogram analysis. Significant periodicities in the range of 1.4–10 hours have been noted. The causes of these oscillations are not fully known at present, but the instrumental and atmospheric factors can be ruled out, pointing to solar origin. Various possibilities are discussed. Some of the observed periodicities may be considered as probable candidates for long-term oscillations in the Sun, such as the elusive gravity modes.     

Gravitational waves as a source for the polarization of the Cosmic Microwave Background

The polarization of the Cosmic Microwave Background (CMB) induced by gravitational waves (GWs) is studied by solving in a semi-analytical way the Chandrasekhar radiative transfer equation; following the Polnarev approach, the equation is written as a second-kind Volterra integral equation and its kernel is handled by performing a series expansion of the trigonometric functions it contains. In this way, a recursive calculation of the Volterra equation gets possible and the polarizing effect of the gravitational waves can be brought out.The polarization degree of the CMB coming from this analysis shows a peak for a wavenumber corresponding to GWs re-entering the horizon at the end of the recombination epoch: the position and the size of the maximum are in agreement with the results of other works, based on a totally numerical calculation. However, a difference quite relevant can be remarked when one looks at the shape of the polarization plot: a semi-analytical calculation of the solution of the Volterra integral equation gives a sharp peak due to the fact that the contribution of each packet of GWs of fixed wavenumberk is strongly singled out when one substitutes the integrals with series and sums.As a consequence, this solution method may have some usefulness when one wants to point out the contributions really dominating in producing a polarization for the CMB.From this analysis one can also infer that the best angular scales to test in order to detect a polarization for the CMB are 2°–3°, smaller than those investigated by COBE.     

A theory of differntial rotation based on the discussion of turbulent transport of angular momentum

This paper deals with the theory of the solar rotational law. We assume the turbulence to be of the largest influence compared with the momentum flux caused by molecular viscosity and meridional circulation. Firstly we use heuristical forms for the needed cross correlations Qrφ (turbulent radial momentum flux) and Qνφ (turbulent latitudinal momentum flux): Qrφ = −α₀r ϑΩ/ϑr · sin ν + Q₀ sin ν + Q₂ sin³ ν, Qνφ = −δ₀ ϑΩ/ϑν· sin ν + P₂ sin²θ cos θ. It is shown that a radial dependence of the angular velocity Ω is given by Q₀. Furthermore, the observed equatorial acceleration occurs in the case of non-negativity of Q₂ and/or P₂. Because of the spatial dependence of the solar angular velocity the coefficients of Q and P are unfortunately not to be measured. Secondly, we determine the coefficients with a theory founded upon the hypothesis that a rotating stochastical force field — independent from Ω — maintains an anisotropic turbulence. The global fast rotation produces, indeed, finite cross correlations Q₂ and P₂. It is suggested that horizontally directed turbulent motions with not too small radial correlations lengths and time scales of about 2 weeks could be responsible for the solar differential rotation. Finally, we show that also short-living turbulent horizontal modes provide the observed equatorial acceleration if they occur preferably at the equatorial region.     

Nonlinear dust acoustic waves in electronegative dusty plasmas

The problem of nonlinear localized dust acoustic (DA) is addressed in a plasma comprising positive ions, negative ions, and mobile negatively charged dust grains. We first consider the case when the grain charge remains constant and discuss later the case when the charge variations are self-consistently included. It is found that a relative increase of the positive ion density favors the propagation of the DA solitary waves, in the sense that the domain of their admissible Mach numbers enlarges. Furthermore, electronegativity makes the dust acoustic solitary structure more spiky. When the dust grain charge Q _d is allowed to fluctuate, the latter is expressed in terms of the Lambert function and we take advantage of this transcendental function to investigate the variable charge DA solitary wave. Q _d adopts a localized profile and becomes more negative as the number of charges Z ₍₋₎ of the negative ion increases. The dust grains are found to be highly localized. This localization (accumulation) caused by a balance of the electrostatic forces acting on the dust grains becomes more effective for lower values of Z ₍₋₎. An increase of Z ₍₋₎ may lead to a local depletion of the negative ions from the region of the soliton’s localization. The results are useful to understand the salient features of localization of large amplitude dust acoustic waves in cosmic plasmas such as the ionospheric D-region and the mesosphere.     

The luminosity function of nearby galaxies (mitteilungen der universitätssternwarte zu jena nr. 147)

Using the catalogue of galaxies within 10 Mpc by Kraan-Korteweg and Tammann (1979) the local luminosity function of galaxies is derived. Possibly there exists a large population of faint elliptical systems being an important constituent in the Universe. From this luminosity function the local mass density Q = 5.1 · 10^-31 g cm^-3 = 0.11 Q_c was obtained.     

New exact solutions for power-law inflation Friedmann models

We consider the spatially flat Friedmann model For a ≈ t^p, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = R^m with p = − (m − 1) (2m − 1)/(m − 2) power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field ϕ in an exponential potential V(ϕ) = exp (μϕ) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we write The general solutions reads Q(a) = (a^b + C)^f/b with free integration constant C (C = 0 gives exact power-law inflation) and m-dependent values b and f: f = −2 + 1/p, b = (4m − 5)/(m − 1). Finally, special solutions for the closed and open Friedmann model are found.     

The line-ratio method applied to vectormagnetographic measurements

Since solar magnetic fields are inhomogeneous, the averaging of Stokes parameter I within the entrance slit of the magnetograph is different from averaging Stokes Q₀ and V, because the former contains also light from non-magnetic, while the latter only contain light from magnetic regions. If the magnetographic calibration functions are calculated for homogeneous magnetic fields, errors arise, when they are used to reduce measurements of inhomogeneous fields. Therefore, we propose to use the line-ratio method to transform magnetographic measurements into the parameters of the magnetic vector field. The Q ratios and the V ratios of two carefully selected lines are free from errors of this kind. This is also the case for the Q ratios in line core and line wings in single-line magnetographs. An iterative method is presented to calculate the magnetic field parameters using the corresponding new calibration functions. An important advantage is, that the influence of scattered light in sunspots is also eliminated in a good approximation and the filling factor in plages can be estimated. This method is now used to determine magnetic vector fields in plages and sunspots of active regions with a new double-vector magnetograph.     

Solutions of the linearized Bach-Einstein equation in the static spherically symmetric case

The Bach-Einstein equation linearized around Minkowski space-time is completely solved. The set of solutions depends on three parameters; a two-parameter subset of it becomes asymptotically flat. In that region the gravitational potential is of the type φ = -m/r + ε exp (-r/l) m and ε being small parameters. Because of the different asymptotic behaviour of both terms, it became necessary to linearize also around the Schwarzschild solution φ = -m/r. The linearized equation resulting in this case is discussed using qualitative methods. The result is that for m ≤ 2l the asymptotic behaviour is φ = -m/r + εr^−m/l exp (-r/l) u, and for m ≧ 2l φ = -m/r + εr⁻² exp (-r/l) u, where u is some bounded function; m is arbitray and ε again small. Further, the relation between the solution of the linearized and the full equation is discussed.     

Dynamical determination of the quadrupole mass moment of a white dwarf

In this paper we dynamically determine the quadrupole mass moment Q of the magnetic white dwarf WD 0137-349 by looking for deviations from the third Kepler law induced by Q in the orbital period of the recently discovered brown dwarf moving around it in a close 2-hr orbit. It turns out that a purely Newtonian model for the orbit of WD 0137-349B, assumed circular and equatorial, is adequate, given the present-day accuracy in knowing the orbital parameters of such a binary system. Our result is Q=(−1.5±0.9)×10⁴⁷ kg m² for i=35 deg. It is able to accommodate the 3-sigma significant discrepancy of (1.0±0.3)×10⁻⁸ s⁻² between the inverse square of the phenomenologically determined orbital period and the inverse square of the calculated Keplerian one. The impact of i, for which an interval Δ i of possible values close to 35 deg is considered, is investigated as well.     

On the far-ultraviolet radiation of the T Tauri-type stars

It is shown that the far-ultraviolet radiation (shorter than 2000 Å) discovered by ANS observations in the few T Tauri-type stars does not have any relation to the two-photon emission of hydrogen, as suggested by some authors. This is obtained from the observational data of the numerical values of the ratioQ ^*(2q)/Q ^*() for these stars, whereQ ^*(2q) is the complete number of the observed 2q-photons andQ ^*() is the number of observedH-photons. The observational values ofQ ^*(2q)/Q ^*() for four T Tauri-type stars turned out to be in the region of 20–90, while the theoretical value of this relation is 6. Hence, the observed fluxes in the region <2000 Å are 3–15 times larger than the theoretically expected values.The emission discovered in the region <2000 Å is of non-thermal origin, and can be identified with high probability with thetransition radiation. The latter originates as a result of the electromagnetic interaction of so-called fast electrons (E1.5 MeV) with dust particles in the gas-dust clouds surrounding these stars. The theoretical spectral curves of the transition radiation, for a few values of the plasma frequency ₀ for the dust particles, are calculated taking into account also the self-absorption effect of the radiation in the cloud and the absorption in the interstellar medium. Qualitatively, these curves (Figures 2, 3 and 4) are in good accord with the observed spectral distribution curves for the T Tauri-type stars (Figure 1). In particular, in both cases a minimum of radiation flux occurs near to 2200 Å, and a maximum near 1800 Å.The starting point of our analysis has been the concept of the identity of the processes, non-thermal and non-stationary in character, taking place at the time of the flare phenomenon of UV Cet-type stars in one case, and at the generation of continuous emission and the excitation of the emission lines in T Tauri-type stars on the other. In the latter case, the T Tauri-type stars can be regarded aspermanently flaring stars, with a very high frequency of flare events.     

A class of semi-analytical solutions for a rotating toroidal plasma

In this paper, the problem of stationary MHD flow for a rotating toroidal plasma is investigated by assuming that the entropy is a surface quantity. Then, the system of ideal MHD equations is reduced to a single second-order elliptic partial differential equation known as the modified Grad-Shafranov (or Maschke-Perrin) equation. Under the assumption that both the function,P _s andf ² are quadratic polynomials of the flux function, a class of semi-analytical solutions is obtained for a plasma contained in a perfectly conducting toroidal boundary with a rectangular cross section. The flux function, poloidal current and the generalized pressure are obtained and discussed for relevant values of the parameters.     

A calculation of structural parameters of spherical stellar systems by means of characteristic functions method. I. Theory

To calculate structural parameters of stellar systems such as an effective radius and central space (or surface) density, the method of characteristic functions is suggested. The characteristic function of the system is a Fourier image of their normalized space density profile f₃(r). In the case of spherical symmetry the probability distribution of r (Q₃(r) = (3/a³)r²f₃(r)) and its orthogonal projections have the same characteristic functions. This fact is used to calculate the effective radii of a few star cluster models (King law, Plummer model and Gausian profile). It is shown, that the characteristic function for King law clusters tends to a finite generalised function if the concentration parameter c is large. The expression for the effective radius (at c ≫ 1) is given. The formula of the effective radius in the Plummer model as well as the relation between the one-dimensional central velocity dispersion and the root mean square velocity are obtained. It is shown, that in the Gaussian model and for King law clusters the effective radius (half-mass visual radius) can differ from the effective (harmonic) radius a few times. This fact should be taken into account in estimating the mass-to-light ratio from the virial mass of such systems using the King radius.     

Secular effects of tidal friction on the planet-satellite systems of the solar system

We separate the tidal evolution of a planet-satellite system with zero eccentricity in two phases:phase 1—from the formation of the system to satellite's corotation (satellite's corotation means that its spin angular velocity equals the orbital angular velocity);phase 2—after satellite's corotation.We study the planet-satellite system during phase 1 with Darwin's graphical method and obtain an upper limit to satellite'sQ which discloses whether or not it is corotating. Moreover we obtain some qualitative information about the future evolution of the corotating satellites.The present work does not give any new result for the Earth-Moon case and for the Neptune-Triton case.     

Third-order Jupiter — Saturn planetary theory

The construction of a third order J-S theory is presented. The Hori theory of planetary perturbations is employed. No Critical J-S terms due to the 2:5 commensurabilities and its multiples exist, when we take into account the periodic terms of order 0, 1, 2 with respect to the eccentricity- inclination. In this case the Lie series transformation degenerates and is meaningless. The J-S equations of motion for secular perturbations are solved when we neglect in our treatment, the Poisson terms of degree > 2 in the Poincaré canonical variables H _u , K _u , P _u Q _u (u = 1, 2). The Jacobi-Radau referential is adopted, and the theory is expressed in terms of the canonical variables of H. Poincaré.Now at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U.S.A.