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.     

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.     

A unified presentation of the Voigt functions

This paper aims at presenting a unified study of the Voigt functionsK(x,y) andL(x,y) which play a rather 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.     

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.     

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.     

An expansion of the Voigt function

The Voigt function and its derivatives are represented by means of series in Hermite polynomials. The equations obtained can be used both for numerical calculations of these functions and for analytical research.Translated from Astrofizika, Vol. 39, No. 4, pp. 619–625, October–December, 1996.     

The Lambert <Emphasis Type="Italic">W</Emphasis> function and solutions to Kepler’s equation

This paper presents a method for the truncation of infinite Fourier–Bessel representations for functions requiring a solution to Kepler’s equation. Use is made of the Lambert W function to solve for the desired index that bounds the remainder terms of the series, within the prescribed tolerance. The enforcement of a maximum on the number of Bessel functions is also useful in truncating the Bessel functions themselves, resulting in an analytical representation of the solution to a desired tolerance, without the use of infinite series.     

The redistribution functionR IV

In this paper we address comments made in a recent paper by Hubený and Heinzel (1986). By use of an identity which they report, the photon redistribution function appropriate for scattering transitions between two radiation broadened atomic levels is derived using the simpler Voigt functions. The behaviour of the redistribution function relative to complete redistribution is then examined. It is important to note that the conclusions originally arrived at by McKenna (1984a) are reinforced by this examination. In particular, for most scattering angles, there exists a significant cohrently scattered component of the redistribution function, and the deviations from complete redistribution are large over most outgoing frequencies for a given incoming frequency.     

Voigt profile fitting to quasar absorption lines: an analytic approximation to the Voigt–Hjerting function

The Voigt–Hjerting function is fundamental in order to correctly model the profiles of absorption lines imprinted on the spectra of bright background sources by intervening absorbing systems. In this work, we present a simple analytic approximation to this function in the context of absorption-line profiles of intergalactic H  i absorbers. Using basic calculus tools, we derive an analytic expression for the Voigt–Hjerting function that contains only fourth-order polynomial and Gaussian functions. In connection with the absorption coefficient of intergalactic neutral hydrogen, this approximation is suitable for modelling Voigt profiles with an accuracy of 10⁻⁴ or better for an arbitrary wavelength baseline, for column densities up to   N _H I= 10 ²² cm⁻²  , and for damping parameters   a ≲ 10⁻⁴  , that is, the entire range of parameters characteristic to all Lyman transitions arising in a variety of H  i absorbing systems such as Lyman α (Lyα) forest clouds, Lyman limit systems and damped Lyα systems. We hence present an approximation to the Voigt–Hjerting function that is both accurate and flexible to implement in various types of programming languages and machines, and with which Voigt profiles can be calculated in a reliable and very simple manner.     

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.     

Analytic properties of Hansen coefficients

Hansen’s coefficients in the theory of elliptic motion with eccentricity e are studied as functions of the parameter η = (1 − e ²)^1/2. Their analytic behavior in the complex η plane is described and some symmetry relations are derived. In particular, for every Hansen coefficient, multiplication by suitable powers of e and η results in an entire analytic function of η. Consequently, Hansen’s coefficients can be in principle computed by means of rapidly convergent series in powers of η. A representation of Hansen’s coefficients in terms of two entire functions of e ² follows.      

Line formation in moving media: Asymptotic expansions of some special functions

Line formation in the spectrum of a moving medium with a spherical geometry is considered. In the Sobolev approximation there are some special functions that determine the source function and the force of radiation pressure in the line. The most important case is that of a small dimensionless velocity gradient (i.e., a large dimensionless Sobolev length τ) and a small ratio β of the opacity in the continuum to the opacity in the line. Until now there has been no detailed analytical information about the asymptotic behavior of these functions. For the case of a Doppler profile of the absorption coefficient, we clarify the nontrivial structure of their total asymptotic expansions for τ » 1, β « 1, and arbitrary Βτ. We give an algorithm for obtaining all the coefficients of these expansions and give explicit expressions for the first few coefficients. We also compare the asymptotic expansions with the numerical calculations of these functions available in the literature. We also briefly consider the case of a power-law decrease in the absorption coefficient in the line wing (and, in more detail, the case of Lorentz wings of the Voigt profile).     

The Fourier-Chebychev approximation for time series with a great many terms

A new type of approximation is proposed to evaluate Fourier series with a great many terms. It combines Fourier series and Chebychev polynomials to represent time series over a finite time interval. On one side, the high frequencies are approximate multiples of a basic frequency to represent short periodic terms; on the other side, the slowly variable functions of the time, on a given interval, which contain terms with long periods, are approximated by Chebychev polynomials. Application is made in case of the Lunar Theory ELP which contains about 30 000 trigonometric terms. The computing time necessary to evaluate coordinates is very much reduced when we give to series the intermediary representation inF-T form, as compared to a direct substitution of the time in the arguments.One shows also on a numerical example, that the Fourier-Chebychev approximation smoothly degrades outside its range of representation.Bureau des Longitudes. Equipe de Recherche Associée au CNRS     

Analytical evaluation of the Voigt function using binomial coefficients and incomplete gamma functions

Using the binomial expansion theorem, the simple general analytical expressions are obtained for the Voigt function arising in various fields of physical research. As we will seen, the present formulation yields compact closed-form expressions which enable the ready analytical calculation of the Voigt function. The validity of this approximation is tested by other calculation methods. The series expansion relations established in this work are accurate enough in the whole range of parameters. The convergence rate of the series is estimated and discussed. Some examples of this methodology are presented.     

The height variation of granular and oscillatory velocities

Previous observations of spatially-resolved vertical velocity variations in ten lines of Fe i spanning the height range 0 h 1000 km are re-analyzed using velocity weighting functions. The amplitudes and scale heights of granular and oscillatory velocities are determined, as well as those of the remaining unresolved velocities. I find that the optimal representation of the amplitude of the outward-decreasing granular velocities is an exponentially decreasing function of height, with a scale height of 150 km and a velocity at zero height of 1.27 km s^–1. The optimal representation of the same quantities for oscillatory velocities is an exponential increase with height, with a scale height of 1100 km and a velocity at zero height of 0.35 km s^–1. The remaining unresolved velocities decrease with height, with a scale height of 380 km and a velocity at zero height of 2.3 km s^–1.     

Theory of the isotropic scattering of radiation in a plane layer: Feasibility of obtaining a complete analytical solution of the problem

Analytical investigations of the method of linear nonsingular integral equations, originally proposed by É. Kh. Danielyan [Astrofizika 36,225 (1993)] for the solution of problems in the theory of radiative transport in a medium of finite optical thickness with isotropic scattering, are continued in the present article. It is shown that the solution of problems of the stated class reduce to the determination of only the functions u ^± (, ) in the general case with true absorption. Explicit expressions are obtained for these functions at =0. The feasibility of a complete analytical solution of the problem is newly formulated as the solution of a Fredholm integral equation on the semiaxis with a kernel that admits representation by a superposition of exponential functions [Eq. (25)]. The choice of an efficient procedure for determining the Ambartsumyan -function for a semiinfinite medium is discussed. In particular, a new equation is given for this function.Translated from Astrofizika, Vol. 37, No. 1, pp. 129–145, January–March, 1994.     

Temporal behaviour of pressure in solar coronal loops

The temporal evolution of pressure in solar coronal loops is studied using the ideal theory of magnetohydrodynamic turbulence in cylindrical geometry. The velocity and the magnetic fields are expanded in terms of the Chandrasekhar-Kendall (C-K) functions. The three-mode representation of the velocity and the magnetic fields submits to the investigation of chaos. When the initial values of the velocity and the magnetic field coefficients are very nearly equal, the system shows periodicities. For randomly chosen initial values of these parameters, the evolution of the velocity and the magnetic fields is nonlinear and chaotic. The consequent plasma pressure is determined in the linear and nonlinear regimes. The evidence for the existence of chaos is established by evaluating the invariant correlation dimension of the attractorD ₂, a fractal value of which indicates the existence of deterministic chaos.     

Synthetic high-resolution near-IR spectra of the Sun for planetary data reductions made from ATMOS/Spacelab-3 and Atlas-3 data

We have constructed synthetic solar spectra for the 2302-4800 cm⁻¹ (2.08-4.34 μm) range, a spectral range where planetary objects mainly emit reflected sunlight, using ATMOS (Atmospheric Trace Molecule Spectroscopy)/Spacelab-3 and Atlas-3 spectra, of which resolution is 0.01 cm⁻¹. We adopted Voigt line profiles for the modeling of line shapes based on an atlas of line identifications compiled by Geller [Geller, M., 1992. Key to Identification of Solar Features. A High-Resolution Atlas of the Infrared Spectrum of the Sun and the Earth Atmosphere from Space. NASA Reference Publ. 1224, vol. III. NASA, Washington, DC, pp. 1-22], who derived solar line positions and intensities from contaminated high-resolution solar spectra obtained by ATMOS/Spacelab-3. Because the ATMOS spectra in these wavelength ranges are compromised by absorption lines of molecules existing in Earth's high-altitude atmosphere and in the compartment of the spacecraft, the direct use of these high-resolution solar spectra has been inconvenient for the data reductions of planetary spectra. We compared the synthetic solar spectra with the ATMOS spectra, and obtained satisfactory fits for the majority of the solar lines with the exception of abnormal lines, which do not fit with Voigt line profiles. From the model fits, we were able to determine Voigt line parameters for the majority of solar lines; and we made a list of the abnormal lines. We also constructed telluric-line-free solar spectra by manually eliminating telluric lines from the ATMOS spectra and filling the gaps with adjacent continua. These synthetic solar spectra will be useful to eliminate solar continua from spectra of planetary objects to extract their own intrinsic spectral features.     

On the evaluation of an integral connected with the thermonuclear reaction rate in closed form

The standard expression of the reaction rate for low-energy, nonresonant nuclear reactions in nondegenerate plasma contains a parameter-dependent integral which in all previous calculations with physical or astrophysical background is considered as not capable of being evaluated in a closed form. So one usually resorts to approximation methods concerning large values of the parameter. At first we point out that CONSUL (1964) has given a series representation of the integral which was identified with a MEIJER 's G-function by MATHAI (1971). Next, in view of a physically more exact determination of the reaction rate formula, especially in connection with calculations concerning stellar energy generation, we consider a more general integral containing the mentioned one as special case and give an approximation-free representation by means of MEIJER 's G-function. The G-function so obtained may be conceived as complex-valued continuation of CONSUL 's series representation of a certain class of integrals contained in the considered one. From the series we extract a small parameter approximation of the special integral.     

Effect of continuum absorption on the polarization of resonance lines

Multiple resonance scattering of radiation in a spectrum line with absorption in the continuum is examined. It is assumed that the scattering atmosphere is semi-infinite and that there is no magnetic field or continuum emission at the frequencies of the spectrum line. The polarization characteristics of the emerging radiation are determined for unpolarized primary radiation sources distributed uniformly within an atmosphere in the case of a Voigt absorption profile. The calculations employ an iterative solution of the Ambartsumyan-Chandrasekhar matrix integral equation.