Solution of Riemann–Hilbert problems for determination of new decoupled expressions of Chandrasekhar’s <Emphasis Type="Italic">X</Emphasis>- and <Emphasis Type="Italic">Y</Emphasis>-functions for slab geometry in radiative transfer

In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness can be expressed in terms of Chandrasekhar’s X- and Y-functions. The nonlinear nonhomogeneous coupled integral equations which the X- and Y-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann–Hilbert problems using Plemelj’s formulae. Solutions of those Riemann–Hilbert problems are obtained using the theory of linear singular integral equations. New forms of linear nonhomogeneous decoupled expressions are derived for X- and Y-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known N-function and two new unknown functions N ₁- and N ₂- in the complex plane for both nonconservative and conservative cases. The N ₁- and N ₂-functions are expressed in terms of the known N-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the H-function for a semi-infinite atmosphere are obtained as a limiting case.     

Combination Scattering by Anisotropic Langmuir Turbulence with Application to Solar Radar Experiments

We develop a theory for radar signal scattering by anisotropic Langmuir turbulence in the solar corona due to a t+l ⇄ t process. Langmuir turbulence is assumed to be generated within a cone by a narrow type III burst electron beam. Using wave-kinetic theory we obtain expressions for the frequency shift, scattering cross-section of the turbulence, coefficient of absorption (due to scattering) and optical depth. On the basis of those expressions we give some estimates for an echo spectrum. We show that the minimum radar echo frequency shift is determined by the minimal phase velocity of the Langmuir waves, the maximum shift is determined by the electron beam velocity, but in any case it can not exceed −wt/2 (decay) and wt (coalescence), where wt is the frequency of a radar signal. The angular characteristics of the scattered signal differ dramatically for the cases of coalescence and decay. The signal is scattered into a narrow cone high above the specular reflection point (wp ≪ wt), but in the vicinity of wp ∼ wt/2 the red-shifted echo is scattered isotropically, while the blue-shifted echo is scattered into a even narrower cone. We show that absorption (due to scattering) increases with increasing radar frequency. The dependence of the absorption on the local plasma frequency is strongly determined by the Langmuir turbulence spectrum. Our theory shows that the role of the nonlinear scattering process t+l ⇄ t is essential and that such process can be used for radar studies of the spectral energy density of anisotropic Langmuir turbulence.     

Optimising optimal image subtraction

Difference imaging is a technique for obtaining precise relative photometry of variable sources in crowded stellar fields and, as such, constitutes a crucial part of the data reduction pipeline in surveys for microlensing events or transiting extrasolar planets. The Optimal Image Subtraction (OIS) algorithm of Alard & Lupton (1998) permits the accurate differencing of images by determining convolution kernels which, when applied to reference images with particularly good seeing and signal‐to‐noise (S/N), provide excellent matches to the point‐spread functions (PSF) in other images of the time series to be analysed. The convolution kernels are built as linear combinations of a set of basis functions, conventionally bivariate Gaussians modulated by polynomials. The kernel parameters, mainly the widths and maximal degrees of the basis function model, must be supplied by the user. Ideally, the parameters should be matched to the PSF, pixel‐sampling, and S/N of the data set or individual images to be analysed. We have studied the dependence of the reduction outcome as a function of the kernel parameters using our new implementation of OIS within the IDL‐based TRIPP package. From the analysis of noise‐free PSF simulations of both single objects and crowded fields, as well as the test images in the ISIS OIS software package, we derive qualitative and quantitative relations between the kernel parameters and the success of the subtraction as a function of the PSF widths and sampling in reference and data images and compare the results to those of other implementations found in the literature. On the basis of these simulations, we provide recommended parameters for data sets with different S/N and sampling. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On obtaining the forward phase functions of Saturn ring features from radio occultation observations

The near-forward scattering functions of particles in Saturn ring features are related to 3.6 cm radio occultation power spectra by a Fredholm integral equation of the first kind. The equation reduces to an algebraic system of equations whose solution by usual inversion techniques (i.e., least mean squares) is precluded by the near singularity of the forward transformation matrix. The instabilities are reduced by applying a combination of constrained linear inversion and a filtering algorithm based on eigenvector decomposition of the matrix, which yields derived phase functions valid over the range of zero to about 12 mrad. These functions represent the collective forward diffraction lobe of particles greater than about 1 m in radius. Multiple scattering of the signal is a significant effect, and the measured phase functions must be adjusted to obtain the singly scattered component. This single-scattering correction is examined for two physical ring models, (a) the monolayer and (b) the classical discrete random slab, and the fraction of opacity in submeter particles for each model for specific ring features is estimated. Four representative regions of the rings approximately between 1.3 and 1.4R_s, 1.5 and 1.52R_s, 2.0 and 2.02R_s, and 2.08 and 2.16R_s have been studied in detail and single-scatter phase functions produced. Each of these features exhibits effective particle sizes in the range of 3–6 m radius. The approximate fractions of optical thickness due to the submeter particles in each of these regions are 0.58, 0.54, 0.23, and 0.0, respectively, for the many-particle-thick model, and 0.67, 0.67, 0.50, and 0.50, for the monolayer model.     

Unified structure of analytical solutions for radiative transfer problems in plane-parallel,homogeneous media

The basic concepts for developing a system of analytic solutions for the standard problems of radiative transfer theory are discussed. These solutions, which are found using Ambartsumyan’s layer addition method in Sobolev’s probabilistic interpretation for radiative diffusion problems, are maximally compact and easily used in numerical computations. New expressions are obtained for the resolvents and the resolvent functions, as well as a unified structure for the form of an integral representation for solving different radiative transfer problems in semi-infinite media and in finite layers. Block diagrams of the sequence of stages for solving these problems are provided, where the Ambartsumyan function φ(η) (more precisely, 1/φ(η)) plays a fundamental role in the case of semi-infinite media while the functions a(η, τ₀ ) and b(η, τ₀) play an analogous role for finite layers.     

On the “rotational angular momentum” of the oceans and the corresponding polar motion

Periodic polar motions caused by ocean tides are predicted. In the Liouville equations for rotational motion the complete excitation functions for the ocean tides have to be used. This does not depend on the fact that hydrodynamical ocean tide models do not consider the centrifugal acceleration. The observable polar motion of the Celestial Ephemeris Pole CEP (more exactly: the terrestrial location of the CEP) is tabulated for the ten ocean tides M₂, S₂, N₂, K₁, O₁, P₁, M f, M f′, M m, Ssa. Typical amplitudes for the largest ocean tides are 0.4 milliarcseconds. This is within the reach of geodetic VLBI and SLR observations.     

Small perturbations in flat galaxies

The aim of this series of papers is to develop straightforward methods of computing the response of flat galaxies to small perturbations. This Paper I considers steady state problems; Paper II considers time varying perturbations and the effects of resonances; and Paper III applies the methods developed in Papers I and II to a numerical study of the stability of flat galaxies.The general approach is to study the dynamics of each individual orbit. The orbits are described by their apocentric and pericentric radii,r _a andr _p , and the distribution function of an equilibrium model is a function ofr _a andr _p . The mass density and potential corresponding to a distribution function is found by means of an expansion in Hankel-Laguerre functions; the coefficients of the expansion being found by taking moments of the mass density of the individual orbits. This leads to a simple method of constructing equilibrium models.The response to a small perturbation is found by seeking the response of each orbit. When the perturbations are axisymmetric and slowly varying, the response can be easily found using adiabatic invariants. The potential is expanded in a series of Hankel-Laguerre functions, and the response operator becomes a discrete matrix. The condition that the model is stable against adiabatic radial perturbations is that the largest eigenvalue of the response matrix should be less than one.An analytic approximation to the response matrix is derived, and applied to estimate the eccentricity needed for stability against local perturbations.     

An efficient method to calculate Ambarzumian-Chandrasekhar's and hopf's functions

An efficient method is proposed to calculate scalar Ambarzumian-Chandrasekhar's and Hopf's functions. This method is based on the approximation of Sobolev's resolvent function using exponent series, the coefficients of which are readily found from approximate characteristic equation and from a system of linear algebraic equations.The approximate expressions for the above functions are given. For checking purposes the calculations were carried out in single, double, and quadruple precision. For isotropic, Rocard, and Rayleigh scattering we present a sample of results in 14 significant figures.The Hopf function for isotropic and Rayleigh scattering is presented in 18 significant figures and the well-known Hopf constantq() is found in 59 significant figures.     

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.     

Partial derivatives of eclipse functions αoc and αtr

Expressions are given for partial derivatives of eclipse functions with respect to geometrical depth,p, and the ratio of radii,k. The derivatives are evaluated for critical combinations ofp andk at which indeterminacies occur and the resulting expressions are listed. All expressions are given in a form suitable for numerical evaluation. Notation employed is that of Merrill.     

Poynting–Robertson effect in non-singular gravitational potential

We use a non-singular potential that appears in the literature under the influence of which the Poynting-Robertson effect is studied. For that, dust particles originating within the asteroid belt are used, in circular and elliptic orbits, and expressions for the semimajor axis as a function of time are obtained. The derived expressions are written in terms of the two basic dust particle parameters, namely the density and the diameter. In both cases, we obtain expressions for the time that the dust particles take to reach the orbit of Earth under the action of the non-singular potential and solar radiation. For the non-singular potential, dust particles of diameter 10⁻³ m in circular and elliptical orbits require times of the order of 4.058×10⁷ and 2.823×10⁷ y to reach the orbit of the Earth respectively. Finally, the derived expressions and numerical results are compared with those of the Newtonian potential.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

Higher-dimensional vacuum bianchi-mixmaster cosmologies

We derive some new exact 7-dimensional cosmological solutions |R⊗ I ⊗N, whereN = I, II, VI₀, VII₀, VIII and IX are the various 3-dimensional Bianchi models. The solutions given are higher-dimensional generalizations of the mixmaster cosmologies. There is a strong influence of the extra spacesN, which results in a fundamental change of the 3-dimensional cosmology.     

Galaxies in the connection machine

Many problems in galactic dynamics require computer speeds orders of magnitude larger than what can be achieved on current single-processor machines. In the near future such speeds are likely to become available through computer architectures based on large-scale, fine-grained parallelism. An example of a highly parallel computer is the Connection Machine, with up to 65,636 processors. We have benchmarked gravitationalN-body algorithms on the Connection Machine, and compared those with similar benchmarks which we have obtained on more traditional vector supercomputers. Our conclusions are: (1) The direct summation algorithm, with of orderN ² interactions forN particles, can be made to run with high efficiency on either type of computer. As a result, the Connection Machine clearly wins in speed over all supercomuters tested, with the exception of an 8-processor ETA, which shows a comparable performance. (2) A more efficient tree algorithm reduces the growth of the number of interactions fromN ² toN logN. However, the greater complexity of this algorithm causes a considerable degradation of efficiency, by a factor which is larger on the Connection Machine than on vector supercomputers. As a result, out tree code runs at comparable speeds on both types of machines, with the notable exception of the 8-processor ETA, which has an extrapolated speed for running our tree code which is higher than any of the other machines we have tested.     

A Fast Current-Field Iteration Method for Calculating Nonlinear Force-Free Fields

Existing methods for calculating nonlinear force-free magnetic fields are slow, and are likely to be inadequate for reconstructing coronal magnetic fields based on high-resolution vector magnetic field data from a new generation of spectro-polarimetric instruments. In this paper a new implementation of the current-field iteration method is presented, which is simple, fast, and accurate. The time taken by the method scales as N ⁴, for a three-dimensional grid with N ³ points. The method solves the field-updating part of the iteration by exploiting a three-dimensional Fast Fourier Transform solution of Ampere’s law with a current density field constructed to satisfy the required boundary conditions, and uses field line tracing to solve the current-updating part of the iteration. The method is demonstrated in application to a known nonlinear force-free field and to a bipolar test case.     

Fourier analysis of the light curves of eclipsing variables,X

The aim of the present paper will be to develop methods for computation of the Fourier transforms of the light curves of eclipsing variables — due to any type of eclipses — as a function of a continuous frequency variablev. For light curves which are symmetrical with respect to the conjunctions (but only then) these transforms prove to be real functions ofv, and expressible as rapidly convergent expansions in terms of the momentsA _2m+1 of the light curves of odd orders. The transforms are found to be strongly peaked in the low-frequency domain (attaining a maximum forv=0), and become numerically insignificant forv>3. This is even more true of their power spectra.The odd momentsA _2m+1 — not encountered so far in our previous papers — are shown in Section 3 of the present communication to be expressible as infinite series in terms of the even momentsA _2m well known to us from Papers I–IV; and polynomial expressions are developed for approximating them to any desired degree of accuracy. The numerical efficiency of such expressions will be tested in Section 4, by application to a practical case, with satisfactory results.Lastly, in Section 5, an appeal to the Wiener-Khinchin theorem (relating the power spectra with autocorrelation function of the light curves) and Parseval's theorem on Fourier series will enable us to extend our previous methods for a specification of quadratic moments of the light curves in terms of the linear ones.     

Evaluation of Jefferies' level population ratios,and generalization of Seaton's cascade matrix,by a Markov-chain method

Closed expressions are obtained for the conditional probabilitiesq _ij,k required in evaluating particular ratios of atomic level populations, using a Markov-chain representation of the system of levels. The total transition probability between two arbitrary levels is also evaluated and its relation to population ratios is clarified. It is shown that Seaton's cascade matrix is a subset of the total transition probability matrix.     

Analytical theory of earth's artificial satellites (A.T.E.A.S.)

The problem of A.T.E.A.S. is treated, for the zonal perturbations, in its Hamiltonian form. The method consists in eliminating angular variables from the Hamiltonian function. Nearly identity canonical transformations are used, first to remove short periodic terms, second to remove long periodic terms. The general solution, up toJ ₂ ³ , is represented by the generators of the transformations and by the mean motions of averaged variables, known up toJ ₂ ⁴ . Open expressions in the eccentricity are avoided as far as possible. It permits to obtain a closed second order theory with closed third order mean motions.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Expressions to determine temperatures and emission measures for solar X-ray events from GOES measurements

We have developed expressions which give the effective color temperatures and corresponding emission measures for solar X-ray events observed with instruments onboard any of the GOES satellites. Since 1976, these satellites have been used to monitor continuously the full-Sun X-ray emission in two broadband wavelength intervals (approximately 0.5–4 Å and 1–8 Å) with a time resolution of 3 s. To simulate the solar X-ray input at a variety of plasma temperatures, we used theoretical spectra provided by D. L. McKenzie. These spectra were folded through the wavelength dependent transfer functions for the two GOES detectors as given by Donnelly et al. (1977). The resulting detector responses and their ratio as a function of plasma temperature were then fit with simple analytic curves. Over the entire range between 5 and 30 million degrees, these fits reproduce the calculated color temperatures within 2% and the calculated emission measures within 5%. With the theoretical spectra provided by McKenzie, we can determine similar expressions for any pair of broadband X-ray detectors whose sensitivities are limited to wavelengths between 0.2 and 100 Å.     

An equation for the characteristic curve of a family of symmetric periodic orbits

Szebehely's equation for the inverse problem of Dynamics is used to obtain the equation of the characteristic curve of a familyf(x,y)=c of planar periodic orbits (crossing perpendicularly thex-axis) created by a certain potentialV(x,y). Analytic expressions for the characteristic curves are found both in sideral and synodic systems. Examples are offered for both cases. It is shown also that from a given characteristic curve, associated with a given potential, one can obtain an analytic expression for the slope of the orbit at any point.