On the numerical evaluation of theJ-integrals

This is a third paper dealing with the numerical evaluation of the light changes exhibited by close binary systems; for previous communications, Lanzano, 1976a, b.TheJ-integrals which were introduced by Kopal for the purpose of this numerical evaluation can be expanded in terms of the Appell hypergeometric series of the first kind. This relationship has been instrumental in establishing a number of recursion formulae for theJ-integrals applicable to the case of an annular eclipse. It was found that some recursion formulae hold both for the case of a partial and annular eclipse.The appropriate use of these recursion formulae should facilitate any numerical computation for eclipsing binaries.     

On theJ-integrals associated with close binary systems: New recursion formulae

The integralsJ _, ^m were introduced by Kopal for the numerical evaluation of the light changes exhibited by eclipsing binaries when both the tidal and rotational distrotions are taken into account.This paper is a sequel to a previous one to appear in this journal and aims at ascertaining some recursion formulae for these integrals to alleviate the computational complexity of the problem.Using a relationship existing between theJ-integrals and the Appell hypergeometric series of the first kind, we have been able to obtain recursion formulae affecting all three parametersm , of these integrals. The present stage of development has also allowed for a complete enumeration of all independent recursion formulae applicable to the case of partial eclipses.Various recursion formulae, given here for the first time, generalize previous results by Kopal which were valid form=0 ory=0.     

Two Methods for Deriving the Recursion Formula of Inclination Function

Two kinds of recursion relations of hypergeometric series were derived, and hereby the Gooding's recursion formula of inclination function. In addition, it was demonstrated that this recursion formula can be derived also on the basis of the recursion relation of Jacobi polynomials. Comparing with the hypergeometric series, the recursion process based on the Jacobi polynomials is much simpler, indicating that the Gooding's method is the recursion of Jacobi polynomials in essence.     

Some further reduction formulas for certain classes of-generalized multiple hyper geometric series arising in physical,astrophysical, and quantum chemical applications

The multivariable hypergeometric function $$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$ considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric function_p F _q of one variable, Appell and Kampé de Fériet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.     

Closed-form evaluation and approximation considerations of the non-resonant thermonuclear reaction rate

By use of a highly efficient theory of generalized hypergeometric functions, we show that the non-resonant thermonuclear reaction rate can be evaluated in closed form. This new approach is advantageous to operate analytically with reaction rates The numerical computation of the rate may be performed with the aid of associated differential equation, a convergent series expansion for small values of the characteristic parameter and an asymptotic expansion for large values of the characteristic parameter.     

Fourier analysis of the light curves of eclipsing variables,XV

A new general expression for the theoretical momentsA _2m of the light curves of eclipsing systems has been presented in the form of infinite series expansion. In this expansion, the terms have been given as the product of two different polynomials which satisfy certain three-term recursion formulae, and the coefficients diminish rapidly with increasing number of terms. Thus, the numerical values of the theoretical momentsA _2m can be generated recursively up to four significant figures for any given set of eclipse elements. This can be utilized to solve the eclipse elements in two ways: (i) with an indirect method (for the procedures see Paper XIV, Kopal and Demircan, 1978), (ii) with a direct method as minimization to the observational momentsA _2m (area fitting). The procedures given in Paper XIV for obtaining the elements of any eclipsing system consisting of spherical stars have been automated by making use of the new expression for the momentsA _2m of the light curves. The theoretical functionsf ₀,f ₂,f ₄,f ₆,g ₂ andg ₄ which are the functions ofa andc ₀, have been used to solve the eclipse elements from the observed photometric data. The closed-form expressions for the functionsf ₂,f ₄ andf ₆ have also been derived (Section 3) in terms of Kopal'sI-integrals.The automated methods for obtaining the eclipse elements from one minimum alone have been tested on the light curves of YZ (21) Cassiopeiae under the spherical model assumptions. The results of these applications will be given in Section 5 which follows a brief introduction to the procedure we followed.     

1s2 1S-1s np 1P transitions of the helium isoelectronic sequence members up to Z = 30

Radial integrals have been calculated under the one-electron hydrogenic model. Two different values of the effective charge parameter, one for the initial state and the other for the final state, are retained in these formulae. The model is able to reasonably reproduce the existing dipole oscillator strength values with little effort. The dipole oscillator strength values are given for many ions for the first time.     

Computer-developed construction of analytic expressions for the coordinates and partial derivatives of Jupiter's Galilean satellites

In his effort to develop series expressions for the coordinates of the Galilean satellites accurate to one are second (Jovicentric), R. A. Sampson was forceda priori to adopt certain numerical values for several constants imbedded in his theory. His final numerical values for the series expressions are not amenable to adjustment of the constants of integration nor of physical constants which affect the motion of the satellites. A method which utilizes computer-based algebraic manipulation software has been developed to reconstruct Sampson's theory, to remove existing errors, to introduce neglected effects and to provide analytical expressions for the coordinates as well as for the partial derivatives with respect to orbital parameters, Jupiter and satellite masses, Jupiter's oblateness (J ₂,J ₄) and Jupiter's pole and period of rotation. The computer-based manipulations enable one to perform, for example, the approximately 10⁸ multiplications required in calculating some perturbations (and their partial derivatives) of Satellite II by Satellite III with ease, and provide algebraic expressions which can readily be adjusted to generate theories corresponding to revised constants of integration and physical parameters.     

The general analytical expression for computation of generalized relativistic Fermi-Dirac functions

In this paper, general sufficiently analytical formulae are developed for the arbitrary order generalized relativistic Fermi-Dirac (FD) functions. Analytical assessment of relativistic FD function is very important for various fields of physics especially in the theory of relativistic nondegenerate and degenerate electron gas systems. One of the more appropriate and correct approximations is based on a binomial expansion method and incomplete Gamma functions that have been used in the calculations of the generalized relativistic FD functions. Note that, the established expression in special cases of specific values of parameters becomes the evaluation formulae of other type FD functions. Calculation results of the generalized relativistic FD functions are compared with the other approximations methods and available numerical approaches and demonstrated satisfactory agreement.     

A simple method to compute spherical harmonic coefficients for the potential model of the coronal magnetic field

It is impossible to make a direct measurement of the coronal magnetic field from the ground. The coronal magnetic field is, then, usually inferred by extrapolation of the observed photospheric magnetic field. The so-called potential model has been used for this extrapolation. We have to solve the Laplacian equation of the magnetic scalar potential. This magnetic scalar potential can be expanded into a spherical harmonic series. In this paper, new simple recursion formulae are proposed to solve the Laplacian equation; that is, to determine the spherical harmonic coefficients.     

N-body simulations of tidal encounters between stellar systems

N-Body simulations have been performed to study the tidal effects of a primary stellar system on a secondary stellar system of density close to the Roche density. Two hyperbolic, one parabolic and one elliptic encounters have been simulated. The changes in energy, angular momentum, mass distribution, and shape of the secondary system have been determined in each case. The inner region containing about 40 per cent of the mass was found to be practically unchanged and the mass exterior to the tidal radius was found to escape. The intermediate region showed tidal distension. The thickness of this region decreased as we went from hyperbolic encounters to the elliptic encounter keeping the distance of closest approach constant. The numerical results for the fractional change in energy have been compared with the predictions of the available analytic formulae and the usefulness and limitations of the formulae have been discussed.     

Particle distributions and X-ray spectra in single or multiple solar current sheets

The acceleration of charged particles in a site of magnetic reconnection is analysed by detailed numerical simulations. Single or multiple encounters of the particles with Harris-type reconnecting current sheets (RCSs) are modelled as an overall stochastic process taking place within an active region. RCS physical parameters are selected in a parameter space relevant to solar flares. Initially, the charged particles form a thermal (Maxwellian) distribution corresponding to coronal temperature  ≃2 × 10⁶ K  . Our main goal is to investigate how the acceleration process changes the shape of the particles' kinetic energy distribution. The evolution of the kinetic energy distribution, calculated numerically after one encounter of the particles with a single RCS, is found to be in good agreement with our previously published analytical formulae. In the case of consecutive encounters, we find that the kinetic distribution tends to converge to a practically invariant form after a relatively small number of encounters. We construct a discrete stochastic process that reproduces the numerical distributions and we provide a theoretical interpretation of the asymptotic convergence of the energy distribution. We finally compute the theoretical X-ray spectra that would be emitted by the simulated particles in a thick target model of radiation.     

Direct solution of laplace's equation for coronal magnetic fields using line-of-sight boundary conditions

A new fast mathematical method is described for computing potential magnetic field solutions in the solar atmosphere from the observed line-of-sight component of the photospheric magnetic field. As in a standard Neumann boundary problem the orthogonality relation of the spherical harmonics is used to determine the coefficients of the harmonic expansion. This leads to a very simple set of recursion formulae that determines the harmonic coefficients successively.     

Optimizing expansions of the Earth’s gravity potential and its derivatives over ellipsoidal harmonics

A set of spherical harmonics is the most widely used representation of the Earth’s gravity potential. This series converges outside and on the surface of a reference sphere enveloping the Earth. However, the Earth’s surface is better approximated by the reference ellipsoid—a compressed ellipsoid of revolution that covers the entire Earth. The gravity potential can be expanded in a series over ellipsoidal harmonics on the surface of the reference ellipsoid and on the surface of other external confocal ellipsoids of revolution. In contrast to spherical harmonics, depending on the associated Legendre functions of the first kind, ellipsoidal harmonics depend also on the associated Legendre functions of the second kind. The latter contain the very slowly converging hypergeometric Gauss series. The number of series increases with increasing the order of their derivatives. In this work, we derived new series for the gravitational potential of the Earth and its derivatives over ellipsoidal harmonics. Starting from the first order derivative, all the series corresponding to higher order derivatives depend on the same two hypergeometric Gauss series. The latter converges considerably faster than that for the hypergeometric series previously used when computing the gravity potential and its derivatives.     

On stellar encounters and their effect on cometary orbits in the Oort cloud

We systematically investigate the encounters between the Sun and neighbouring stars and their effects on cometary orbits in the Oort cloud, including the intrinsic one with the star Gl 710 (HIP 89 825), with some implications to stellar and cometary dynamics. Our approach is principally based on the combination of a Keplerian‐rectilinear model of stellar passages and the Hipparcos Catalogue (ESA 1997). Beyond the parameters of encounter, we pay particular attention to the observational errors in parallaxes and stellar velocities, and their propagation in time. Moreover, as a special case of this problem, we consider the collision probability of a star passing very closely to the Sun, taking also into account the mutual gravitational attraction between the stars. In the part dealing with the influence of stellar encounters on the orbital elements of Oort cloud comets, we derive new simple formulae calculating the changes in the cometary orbital elements, expressed as functions of the Jeans impulse formula. These expressions are then applied to calculate numerical values of the element changes caused by close encounters of neighbouring stars with some model comets in the Oort cloud. Moreover, the general condition for an ejection of comets from the cloud effected by a single encounter is derived and discussed.     

Resonances and bifurcations in axisymmetric scale-free potentials

We investigate an analytical treatment of bifurcations of families of resonant 'thin' tubes in axisymmetric galactic potentials. We verify that the most relevant bifurcations are due to the (1:1) resonance producing the 'inclined' orbits through two different mechanisms: from the disc orbit and from the 'thin' tube associated with the vertical oscillation. The closest resonances occurring after these are the (4:3) resonance in the oblate case and the (2:1) resonance in the prolate case. The (1:1) resonances are treated in a straightforward way using a second-order truncated normal form. The higher order resonances are instead cumbersome to investigate, because the normal form has to be truncated to a high degree and the number of terms grows very rapidly. We therefore adopt a further simplification giving analytical formulae for the values of the parameters at which bifurcations ensue and compare them with selected numerical results. Thanks to the asymptotic nature of the series involved, the predictions are reliable well beyond the convergence radius of the original series.     

Scale Factor of Isotropic Homogeneous Universe

Four fundamental expressions of isotropic homogeneous universe lead to the scale factor equations. Their solution is achieved by developing the factor as a power series of time, establishing the recursion relations among its coefficients, performing the summation of this power series and obtaining the scale factor as a function of time in a closed form. This one should be used in comparison to observational data with different cosmological models, generally present within this formalism.     

Comparison of Numerical Methods for the Integration of Natural Satellite Systems

We present a new implementation of the recurrent power series (RPS) method which we have developed for the integration of the system of N satellites orbiting a point-mass planet. This implementation is proved to be more efficient than previously developed implementations of the same method. Furthermore, its comparison with two of the most popular numerical integration methods: the 10th-order Gauss–Jackson backward difference method and the Runge–Kutta–NystrRKN12(10)17M shows that the RPS method is more than one order of magnitude better in accuracy than the other two. Various test problems with one up to four satellites are used, with initial conditions obtained from ephemerides of the saturnian satellite system. For each of the three methods we find the values of the user-specified parameters (such as the method's step-size (h or tolerance (TOL)) that minimize the global error in the satellites' coordinates while keeping the computer time within reasonable limits. While the optimal values of the step-sizes for the methods GJ and RKN are all very small (less than T/100, the ones that are suitable for the RPS method are within the range: T/13<h<T/6 (T being the period of the innermost satellite of the problem). Comparing the results obtained by the three methods for these step-sizes and for the various test problems we observe the superiority of the RPS method over GJ in terms of accuracy and over RKN both in accuracy and in speed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Modelling RR Lyrae pulsation: mission (im)possible?

In order to make an in-depth comparison between theory and observations, we analyse the light and velocity curves of various hydrodynamical models simulating RRab stars. The observations are represented by empirical formulae, derived in this and our earlier papers. It is shown that the overwhelming majority of the models tested do not follow the empirical relations regarding the shape of the light curves and the physical parameters. In almost all cases the luminosities predicted from the model light curves are significantly lower than the corresponding model values. The overall discrepancy of the models is an important indication of the limitation of the applicability of the present theoretical light and velocity curves in the determination of the physical parameters of these stars. In transforming the theoretical data to the observed light curves in V colour and in computing the observed radial velocities, it is shown that both bolometric correction and tracing the line-forming regions have considerable effects on the evaluation of the observed quantities. In an effort to resolve the discrepancy between theory and observations, it is suggested that a proper evaluation of the bolometric correction and radial velocity based on complete dynamical atmosphere models may be a useful step in this direction.     

Pickup interstellar helium ions in the region of the solar gravitational cone

Our numerical analyses of the velocity and spatial distributions of pickup interstellar helium ions in the region of the solar gravitational cone in the ecliptic plane at a distance of 1 AU show that the ion density maximum must be displaced relative to the neutral helium cone axis in the direction of the Earth’s revolution around the Sun. The solar wind parameters in the numerical model correspond to their observed values during the crossing of the helium cone by the ACE spacecraft in 1998. At these parameters, the calculated angular displacement is 5°. The absence of a similar displacement in the ACE measurements is shown to stem from the fact that the spectrometer onboard ACE records and identifies only a fraction of the pickup helium ions with fairly high magnitudes and certain directions of the velocities.