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1.
By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere we derive an integral equation for the emergent intensity matrix. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity matrix in terms of singularH-matrix and an unknown matrix. The unknown matrix has been determined considering the boundary condition at infinity to be identical with the asymptotic solution for the intensity matrix.  相似文献   

2.
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.  相似文献   

3.
We consider the basic vector equation of transfer for radiation in a semi-infinite atmosphere for diffuse reflection which scatters radiation in accordance with the phase matrix obtained from a combination of Rayleight and isotropic scattering. This equation will give an integral equation for emergent intensity while subjected to the Laplace transform. The integral equation will give rise to the emergent intensity matrix on application of the Wiener-Hopf technique. This is an exact method.  相似文献   

4.
The solution of the partial differential equation describing the ‘non-isentropic’ oscillations of a star in thermal imbalance has been obtained in terms of asymptotic expansions up to the first order in the parameterII/t s, whereII is the adiabatic pulsation period for the fundamental mode andt s , a secular time scale of the order of the Kelvin-Helmholtz time. Use has been made of the zeroth order ‘isentopic’ solution derived in I. The solution obtained allows one to derive unambiguously a general integral expression for the coefficient of vibrational stability for arbitrary stellar models in thermal imbalance. The physical interpretation of this stability coefficient is discussed and its generality and its simplicity are stressed. Application to some simple analytic stellar models in homologous and nonhomologous contraction enables one to recover, in a more straightforward manner, results obtained by Coxet al. (1973). Aizenman and Cox (1974) and Davey (1974). Finally, we emphasize that the inclusion of the effects of thermal imbalance in the stability calculations of realistic evolutionary sequences of stellar models, not considered up to now by the other authors, is quite easy and straightforward with the simple formula derived here.  相似文献   

5.
We derive a formula for the nodal precession frequency and the Keplerian period of a particle at an arbitrary orbital inclination (with a minimum latitudinal angle reached at the orbit) in the post-Newtonian approximation in the external field of an oblate rotating neutron star (NS). We also derive formulas for the nodal precession and periastron rotation frequencies of slightly inclined low-eccentricity orbits in the field of a rapidly rotating NS in the form of asymptotic expansions whose first terms are given by the Okazaki-Kato formulas. The NS gravitational field is described by the exact solution of the Einstein equation that includes the NS quadrupole moment induced by rapid rotation. Convenient asymptotic formulas are given for the metric coefficients of the corresponding space-time in the form of Kerr metric perturbations in Boyer-Lindquist coordinates.  相似文献   

6.
It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry, Proc Lond Math Soc ser 2, 27:151–170, 1926; Moser, Commun Pure Appl Math 9:673, 1956 and 11:257, 1958; Moser, Giorgilli, Discret Contin Dyn Syst 7:855, 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al., Phys D 29:181, 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice the study of the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain, the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as (i) the order of truncation of the series increases, and (ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.  相似文献   

7.
The circular restricted problem of three bodies is investigated analytically with respect to the problem of deriving a second integral of motion besides the well known Jacobian Integral. The second integral is searched for as a correction the angular momentum integral valid in the two body case. A partial differential equation equivalent to the problem is derived and solved approximately by an asymptotic Fourier method assuming either sufficiently small values for the dimensionless mass parameter or sufficiently large distances from the barycentre. The solution of the partial equation then leads to a function of the coordinates, velocities and time being nearly constant, which means that its variation with time is about 40–300 times less than that of the pure angular momentum. By averaging over the remaining fluctuating part of the quasi-integral we are able to integrate the first order equations using a renormalization transformation. This leads to an explicit expression for the approximate solution of the circular problem which describes the motion of the third body orbiting both primaries with nonvanishing initial eccentricity (eccentric planetary type orbits). One of the main results is an explicit formula for the frequency of the perihelion motion of the third body which depends on the mass parameter, the initial distance of the third body from the barycentre and the initial eccentricity. Finally we study orbits of the P-Type, being defined as solutions of the restricted problem with circular initial conditions (vanishing initial eccentricity).  相似文献   

8.
This paper derives asymptotic expansions of ellipsoidal coordinates in Cartesian coordinates and an expansion in spherical harmonics of the dominant term for the solution of Laplace's equation corresponding to the gravitational force function for a two-dimensional finite body.On comparing the expansion of the dominant term derived here with known expansions of the force functions of the Earth's and Moon's gravitation the author obtains values for the semimajor axes and eccentricities of the singular ellipses of these bodies in terms of the second degree harmonic coefficientsc 20 andc 22.  相似文献   

9.
S. I. Grachev 《Astrophysics》1999,42(4):376-390
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).  相似文献   

10.
This is the second paper in a series of articles devoted to one of the basic problems of celestial mechanics: the evolution of solar-type planetary systems. In the first paper (Kholshevnikov et al., 2001), we reviewed the history and the current state of the issue, outlined the scheme of the study, introduced Jacobi coordinates and related osculating elements, and indicated the form of the Hamiltonian expansion into a Poisson series in all elements. In this paper, the expansion coefficients are found according to a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. At the first stage, we restricted our analysis to the two-planetary problem (Sun–Jupiter–Saturn). The general case will be investigated in a forthcoming paper.  相似文献   

11.
S. I. Grachev 《Astrophysics》2001,44(3):369-381
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.  相似文献   

12.
In this paper of the series, we arrive at the end of the second step of our regularization approach, and in which, elliptic expansions in terms of the sectorial variables j (i) introduced by the author in Paper IV (Sharaf, 1982b) to regularize the highly oscillating perturbation force of some orbital systems will be established analytically and computationally for the thirteenth, fourteenth, fifteenth, and sixteenth categories according to our adopted scheme of presentation drawn up in Paper V (Sharaf, 1983). For each of the elliptic expansions belonging to a category, literal analytic expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, and numerical results are included to provide test examples for constructing computational algorithms. Finally, the second and the last collection of completed elliptic expansion will be given in Appendix B, such that, the materials of Appendix A of Paper VIII (Sharaf, 1985b) and those of Appendix B of the present paper provide the reader with the elliptic expansions in terms of j (i) so explored for the second step of our regularization approach.  相似文献   

13.
The application of the Wiener-Hopf technique to the coupled linear integral equation ofX- andY-equations gives rise to the Fredholm equations with simpler kernels.X-equation is expressed in terms ofY-equation and vice-versa. These are unique in representation with respect to coupled linear constraints.  相似文献   

14.
We present measurements of ratios of elements of the scattering matrix of martian analogue palagonite particles for scattering angles ranging from 3° to 174° and a wavelength of 632.8 nm. To facilitate the use of these measurements in radiative transfer calculations we have devised a method that enables us to obtain, from these measurements, a normalized synthetic scattering matrix covering the complete scattering angle range from 0° to 180°. Our method is based on employing the coefficients of the expansions of scattering matrix elements into generalized spherical functions. The synthetic scattering matrix elements and/or the expansion coefficients obtained in this way, can be used to include multiple scattering by these irregularly shaped particles in (polarized) radiative transfer calculations, such as calculations of sunlight that is scattered in the dusty martian atmosphere.  相似文献   

15.
The behavior ofp-modes of high degree and high order in the homogeneous compressilbe model is examined. The second-order differential equation of Pekeris is used to construct asymptotic expansions near the centre and near the surface, which are singular points, and near the turning point of that equation. An equation for the frequencies is obtained by requiring the continuity of the asymptotic solutions and of their first derivatives. Numerical applications are considered.  相似文献   

16.
We present a method for computing the spectrum of the integral equation for radiation transfer in a cylinder. This method, as in the previous articles in this series, is based on a Hankel transformation applied to the equation. Calculating the spectrum then reduces to solving the equation for the auxiliary function for each eigenvalue separately. The corresponding eigenfunction is then found by an additional integration. We find asymptotic expressions for the eigenvalues and the eigenfunctions for a cylinder with a large optical radius when there is scattering in a spectral line, with complete redistribution over frequency when the absorption coefficient obeys a power law. We also derive equations determining the quantities entering into these expressions. For the simplest kernel of the equation all quantities can be expressed in terms of Bessel functions and roots of a transcendental equation.Translated from Astrofizika, Vol. 38, No. 1, pp. 75–88, January–March, 1995.  相似文献   

17.
The method of computing the radiation field in an infinite circular cylinder proposed in Part I is now applied to the case of isotropic scattering with sources on the boundary and axis of the cylinder, as well as for a uniform distribution of sources inside the cylinder. For the simplest kernel we obtain exact solutions of the basic integral equation in explicit form. For scattering in a spectral line with complete frequency redistribution and a power absorption profile we develop an asymptotic theory for the case when the optical radius of the cylinder is large. We solve the asymptotic equations for the basic characteristics of the scattering in closed form for conservative scattering and find its asymptotics. We obtain estimates of the mean number of scatterings with a layered source, and also the mean and variance of the number of scatterings with a uniform source distribution.Translated fromAstrofizika, Vol. 37, No. 4, 1994.This work was carried out with the financial support of the Russian Basic Research Fund (grant 93-02-2957).  相似文献   

18.
Some considerations about the zodiacal light brightness integral from the stand point of the theory of integral equations are made. It is shown that for observation directions confined to a plane perpendicular to the ecliptic and passing through the Sun, the Z.L. brightness integral can be formally considered as a first kind integral equation of Volterra type (V.I.E.). In a second step, this equation is transformed into a V.I.E. of the second kind, from which, and under certain assumptions, the spatial distribution of dust out of the ecliptic is obtained.  相似文献   

19.
It is demonstrated that the long term variation in cosmic ray intensity I(t) can be described by an integral equation,
I(t)=I∞?∫0∞f(τ)S(t?τ) dτ
, which is derived from a generalization of Simpson's coasting solar wind model. A source function S(t?τ) is given by some appropriate solar activity index at a time t?τ(τ ? 0) and the characteristic functionf(τ)(?0 forτ ? 0) expresses the time dependence of the efficiency of the intensity depression due to solar disturbances represented by S(t ?τ) when the disturbances generated at the solar surface propagate through the modulating region with the solar wind. It is demonstrated further that the equation can be derived from the general diffusion-convection theory on some assumptions, and as a result, the source and characteristic functions can be related to diffusion coefficient and its transition in space. Assuming the sunspot number R (or two activity indices including R) as the source function, the characteristic function f(τ) [or f(τ)'s] is obtained with data of the cosmic ray intensity extended over several decades. Based on the theory, one can obtain from f(τ) the following physical quantities in space, such as the transition and life time of solar disturbances, the boundary of the modulating region, and the radial and time dependences of the diffusion coefficient, radial density gradient and modulated intensity of cosmic rays. Results deduced from the present analysis are consistent with those obtained directly or indirectly by space observations.  相似文献   

20.
In a series of papers, Saxena et al. (2002, 2004a, 2004b) derived solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which provide the extension of the work of Haubold and Mathai (1995, 2000). The subject of the present paper is to investigate the solution of a fractional reaction-diffusion equation. The results derived are of general nature and include the results reported earlier by many authors, notably by Jespersen et al. (1999) for anomalous diffusion and del-Castillo-Negrete et al. (2003) for reaction-diffusion systems with Lévy flights. The solution has been developed in terms of the H-function in a compact form with the help of Laplace and Fourier transforms. Most of the results obtained are in a form suitable for numerical computation.  相似文献   

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