A unified presentation of the Voigt functions

In the present paper, we have given a generalization of a unified study of the Voigt functionsK(x, y) andL(x, y) obtained by Srivastava and Miller (1987; Vol. 135, pp. 111–118) which play an important role in several diverse fields of physics-such as astrophysical spectroscopy and the theory of neutron reactions. Explicit expressions for these functions are given in terms of relatively more familiar special functions of one and two variables; indeed, each of these representations will naturally lead to various other needed properties of the Voigt functions.     

Astrophysical spectroscopy and neutron reactions: Integral transforms and Voigt functions

The Voigt functionsK(x, y) andL(x, y) which play an essential role in astrophysical spectroscopy and neutron physics are investigated and generalized from the viewpoint of integral operators. Unified representations and series expansions involving classical functions of mathematical physics and multivariable hypergeometric functions are established. From the delicate asymptotic analysis of Laplace and Hankel integral transforms we extract complete and rigorous asymptotic expansions of the generalized Voigt functions for large values of the variablesx andy which are of great value in the theory of spectral line profiles.     

Some unified presentations of the Voigt functions

The principal object of this note is to provide a natural further step toward the unified presentations of the Voigh functionsK(x, y) andL(x, y) which play a rather important role in such diverse fields of physics as astrophysical spectroscopy and the theory of neutron reactions. Explicit representations for these functions, given in terms of some relatively more familiar special functions of one and two variables, are potentially useful in finding many other needed (numerical or analytical) properties of the Voigt functions. Several erroneous recent contributions to the theory of Voigt functions, including (for example) the main result of A. Siddiqui (1990), are also corrected here.     

Spectral line profiles and neutron cross sections: New results concerning the analysis of Voigt functions

The Voigt functions, so important in spectroscopy and neutron physics, are represented as generalized hypergeometric functions (G-functions) of two real variables. A system of partial differential equations for the Voigt functions is derived. By applying Hölder's inequality to an integral representation of the Voigt functions apparently not known in the literature until now, lower and upper bounds are obtained. Moreover, from this representation an asymptotic expansion of Voigt functions for large values of one variable is extracted.     

The angle-dependent redistribution functionsR III andR IV: Line absorption probability distribution functions and reemission coefficients

The line absorption probability distribution functions and the reemission coefficients are derived for the non-coherent scattering functionsR _III andR _IV. The appropriate line profile function forR _III is shown to be a simple Voigt function, while forR _IV, the line absorption probability distribution function is more complex involving a linear combination of two Voigt functions and another more complex probability distribution. The structure of the reemission coefficients forR _III andR _IV is then discussed.     

Solution of the three-dimensional inverse problem

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c ₁,g(x, y, z)=c ₂, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.     

Boundary curves for families of planar orbits

For monoparametric familiesf(x,y)=c of planar orbits, created by a planar potentialV(x,y), we introduce the notion of the family boundary curves (FBC). All members of the familyf(x,y)=c are traced in an allowable region of thexy plane, defined by the corresponding FBC, with total energyE=E(c) varying along the family. Family boundary curves are also found for two-parametric familiesf(x,y,b)=c. The relation of equilibrium points and asymptotic orbits, possibly possessed by the potentialV(x,y), to be FBC is studied.     

Inverse problem with two-parametric families of planar orbits

As a possible extension of recent work we study the following version of the inverse problem in dynamics: Given a two-parametric familyf(x, y, b)=c of plane curves, find an autonomous dynamical system for which these curves are orbits.We derive a new linear partial differential equation of the first order for the force componentsX(x, y) andY(x, y) corresponding to the given family. With the aid of this equation we find that, depending on the given functionf, the problem may or may not have a solution. Based on given criteria, we present a full classification of the various cases which may arise.     

The gravitational field of a disk

This note gives the gravitational potential of the disk {(x, y, z):x ² +y ² p ² , z=0} and the gravitational field at the point (x, y, z). Formulas for a ring can be obtained as the difference of our results for two different values ofp. Results are obtained in terms of elliptic integrals and we indicate how these functions can be computed efficiently. Formulas necessary for the computation of partial derivatives are also given.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract NAS7-100, sponsored by the National Aeronautics and Space Administration.     

Compatibility conditions for a non-quadratic integral of motion

Conditions are found which are satisfied by the coefficients of the expression being a second integral of the motion of an autonomous dynamical system with two degrees of freedom. The coefficientsA, B. , ,E are differentiable functions of the cartesian position coordinatesx, y. The velocity components are denoted by . It is shown that must be constant andB must be of the formB =f(x+y) +g(x-y) wheref, g are arbitrary.Given andB one can always find the remaining coefficientsA, E and also the corresponding potential and second integral. Depending on the specifica case at hand a certain number of arbitrary constants (or arbitrary functions) enter into the potential and the second integral. To each potential (which may be of the separable or nonseparable type in the coordinatesx andy)there corresponds one integral of the above form.     

Intrinsic stability of periodic orbits

Families of orbits of a conservative, two degree-of-freedom system are represented by an unsteady velocity field with componentsu(x, y, t) andv(x, y, t). Intrinsic stability properties depend on velocity field divergence and curl, whose dynamical evolution is determined by a matrix Riccati equation. Near equilibrium, divergence-free or irrotational fields are dynamically compatible with the conservative force field. It is shown that a necessary condition for stable periodic orbits is satisfied when the orbitaveraged divergence is zero, which results in bounded normal variations. A sufficient condition for stability is derived from the requirement that tangential variations do not exhibit secular growth.In a steady, divergence-free field, velocity component functionsu(x, y) andv(x, y) may be continuedanalytically from any initial condition, except when velocity is parallel to U or at equilibria. In an unsteady field, the orbit-averaged divergence is zero when the vorticity function is periodic. When such a field exists, initial conditions for stable periodic orbits (i.e., characteristic loci) may be determinedanalytically.     

Generalization of Szebehely's equation

Szebehely's partial differential equation for the force functionU=U(x,y) which gives rise to a given family of planar orbitsf(x,y)=Constant is generalized to account for velocity-dependent potentials V^*=V^*(x,y, ). The new partial differential equation is quasi-linear and of the first order. An example is given and a comparison is made of the two equations.     

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.     

The influence of a resisting medium on 3-dimensional periodic orbits of the restricted 3-body problem

Summary The evolution of the extreme values of the functionsx(t),y(t) andz(t) which are the coordinates of the third bodyP in the barycentric rotating frame of reference, when friction is present, is discussed. These values, which are constant for periodic orbits, change due to the presence of the resisting medium. It is shown that either the orbit tends to become circular and coplanar with the two primaries or to collide with one of the primaries.     

A systematic method for the analysis of high-resolution fraunhofer line profiles

A fraunhofer line profile depends on various parameters, partly related to the photospheric structure (T, P _g, P _e, v _conv, v _turb), partly to the atom or ion involved (such as oscillator strength, energy levels), partly also resulting from the interaction of the relevant kind of particles with the photosphere, and the photospheric radiation field. In this paper we shall mainly pay attention to the determination of: the macroturbulent (convective) velocities, v _conv (); the damping constant (); the abundance, A _el; the distribution function (v _conv, ) of the convective velocities at each depth ; the source function, S (); the microturbulent velocities, v _turb ().The particular difficulty with these unknowns is that they are, as a rule, coupled in the resulting line profiles, that is: the shapes and intensities in these profiles are determined by the combined influence of these unknowns (together with the other above-given parameters).In this paper we describe a method to determine these six unknowns empirically by separating them, in analysing accurate high-resolution observations of line profiles of a multiplet. The unknown functions and quantities are consecutively determined in the above given succession. For each determination another, appropriate part of the line profile is used. In some cases the influence of the mutual coupling of the various parameters cannot be completely eliminated, and an iterative method has to be used.The method is summarized in Table II and section 2, and is further explained in sections 3 to 8. It is applied to an infrared Ci multiplet. The main results are the following:     

Theory of relativity and super-luminal speeds

It is possible that the Finsler space-timeF( _{x, y} ) may be endowed with a catastrophic nature. In particular, the horizon of the field of the general relativity is just a catastrophic set. If so, a particle with the super-luminal speeds could be projected near the horizon of these fields, and the particle will move on the space-like curves. It is very interesting that, in the Schwarzschild fields, the theoretical calculation as the space-like curves should be in agreement with the data of the superluminal expansion of extragalactic radio sources observed year after year.The project has been supported by the National Natural Science Foundation of China.     

OnH-functions of radiative transfer

Chandrasekhar'sH-functionH(z) corresponding to the dispersion functionT(z)=| _rs –frs(z)|, where [f _rs (z)] is of rank 1, is obtained in terms of a Cauchy integral whose density functionQ(x, ₁, ₂,...) can be approximated by approximating polynomials (uniformly converging toQ(x)) having their coefficients expressed as known functions of the parameters _r 's. A closed form approximation ofH(z) to a sufficiently high degree of accuracy is then readily available by term by term integration.     

A theory of libration

It is shown here that many problems of libration in celestial mechanics can be reduced to a perturbation of anintermediary defined by the Hamiltonian $$F = B\left( y \right) + 2\mu ^2 A\left( y \right)f\left( x \right).$$ This generalization of the Ideal Resonance Problem, with a periodic functionf(x) replacing sin² x, is solved here toO(μ ²) by an algorithm that is essentially the same as the one used in the original formulation. The solution is of the formx=x(u), u=u(t), y=y(x), with the functionx(u) commonly involving the inversion of a hyperelliptic integralu(x), evaluated by quadrature. Libration may be simple or multiple, depending on the nature of the functionf(x) and on the initial conditions. Double libration is illustrated here by the horseshoe-shaped orbits enclosing two libration centers.     

Lie transforms and the Hamiltonization of non-Hamiltonian systems

To develop the perturbation solution of the non-Hamiltonian system of differential equationsy=g(y, t; ), it is sufficient to obtain the perturbation solution of a Hamiltonian system represented by the HamiltonianK=Y·g(y, t; ) which is linear in the adjoint vectorY. This Hamiltonization allows the direct use of the perturbation methods already established for Hamiltonian systems. To demonstrate this fact, a Hamiltonian algorithm developed by this author and based on the Lie-Deprit transform is applied to the Hamiltonized system and is shown to be equivalent to the application of the non-Hamiltonian form of this same algorithm to the original non-Hamiltonian system.