New approximate forms for theH-function for isotropic scattering

Three new approximate forms for theH-function for isotropic scattering have been developed. Each of the forms involve three unknown functions of albedo .     

Investigation of numerical properties of Hovenier's Exit Function equation for multiple scattering of light

We show that Hovenier's Exit Function equation describing reflection and transmission by a plane-parallel layer can be obtained from the Invariant Imbedding equations. As an immediate extension we obtain a similar equation for an Exit Function defined in terms of reflection and transmission functions for successive orders of scattering. These equations allow the reflection and transmission functions of a homogeneous atmosphere of arbitrary optical thickness to be obtained from angle integrations of only one function.A technique based on successive iterations is developed to solve Hovenier's equation. The numerical behavior of this equation is then investigated employing a few representative (i.e., isotropic, Rayleigh, and Henyey-Greenstein) phase functions with the following conclusions. (i) As long as the deviation from isotropy is small (cos 0.15), the Exit Function equation can be numerically solved with an efficiency comparable to that of the standard Doubling technique, which is one of the fastest algorithms available. (ii) The reflection function generated from the Exit Function is usually more accurate than the corresponding transmission function, particularly in the case of large optical thickness. (iii) As the degree of anisotropy increases, so does the difficulty in obtaining the numerical solution for the Exit Function. The solution of the equation depends sensitively on the treatment of the numerical singularities which arise from the integrands and also on the initial approximation employed for the iteration. An improved scheme is required for numerically obtaining the Exit Function in order for this method to yield accurate reflection and transmission functions for strongly anisotropic scattering.     

Photon path-length distribution function (I)

The determination of the photon path-length distribution function (PLDF) for the case of linearly anisotropic (Rocard) scattering in a semi-infinite plane-parallel homogeneous atmosphere using the Piessens-Huysmans method is described in detail. It has been shown that in this case the PLDF may have a minimum at small path-lengths — a feature which is never encountered in isotropic scattering. The respective regions with minima in the (µ, )-plane have been sorted out. As a rule, the average path-length in the case of the forward/backward Rocard scattering is larger/smaller than that in the isotropic case. The precise average path-lengths for a number of parameters are shown in Table I.     

A simple approximation forJ v (σ v ,κ v ,τ v,B v ), and its application for temperature-correction procedures

For the case of isotropic coherent scattering plus absorption a simple expression is given (Equation 12) to compute the mean intensity of the radiationJ _v (as a function of optical depth _v ) if the scattering coefficient _v , the absorption coefficientK _v and the Planck functionB _v are given as a function of depth. In general the accuracy of this approximation is of the order of a few percent.A fairly simple temperature-correction procedure for the case when scattering is important is described.     

Light scattering in cometary dust comae

Detailed single and multiple scattering calculations were carried out for a spherically symmetric cometary atmosphere irradiated by a plane parallel source. Using simplifying assumptions in the single scattering approximation, analytical expressions were derived for the total flux impinging the cometary nucleus, which was shown to be a decreasing function of the coma opacity. Moreover, while highly anisotropic phase functions resulted in more light reaching the nucleus than was the case for isotropic phase functions, the net energy flux at the nucleus surface was still found to be smaller in the presence of a coma than in the no coma case. This increased flux due to the anisotropic phase functions was attributed mostly to the effect of directional scattering in the forward Sun-comet axis. The isotropic multiply scattered flux at the surface was found tobe an increasing function of the opacity, , for 2.5. At larger values of , the maximum in the downward directed scattered flux was still seen to increase, but occurred at a height of several radii above the nucleus, resulting in a reduction at the surface. On the other hand, the total flux at the surface was again shown to be a decreasing function of and always less than in the no coma case. Finally, on comparing the multiply scattered flux with that obtained in the plane parallel approximation, it was quite apparent that except in the vicinity of the Sun-comet axis, the plane parallel geometry tends to underestimate the degree of scattering.NRC Resident Research Associate.     

15-digit accuracy calculations of Ambartsumian-Chandrasekhar’s $$-functions for four-term phase functions with the double-exponential formula

We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar’s \(H\)-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (Astrophys. Space Sci. 358:32, 2015) and the double-exponential integration formula (DE-formula) of Takahashi and Mori (Publ. Res. Inst. Math. Sci. Kyoto Univ. 9:721, 1974), which proved highly effective in Kawabata (Astrophys. Space Sci. 361:373, 2016). Actual calculations of the \(H\)-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo \(\varpi_{0}\), and 36 values of an angular variable \(\mu(= \cos\theta)\), with \(\theta\) being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. They indicate that it is important to simultaneously verify accuracy of the numerical values of the \(H\)-functions for \(\mu<0.05\), the domain often neglected in tabulation. As a sample application of the isotropic scattering \(H\)-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (J. Quant. Spectrosc. Radiat. Transf. 183:128, 2016), who employed their analytical representations for these quantities and the single-term and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy. While our results are in satisfactory agreement with theirs, our procedure is in need of a faster algorithm to routinely deal with problems involving highly anisotropic phase functions giving rise to near-conservative scattering.     

A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere

A discrete spherical harmonics method is developed for the radiative transfer problem in inhomogeneous polarized planar atmosphere illuminated at the top by a collimated sunlight while the bottom reflects the radiation. The method expands both the Stokes vector and the phase matrix in a finite series of generalized spherical functions and the resulting vector radiative transfer equation is expressed in a set of polar directions. Hence, the polarized characteristics of the radiance within the atmosphere at any polar direction and azimuthal angle can be determined without linearization and/or interpolations. The spatial dependent of the problem is solved using the spectral Chebyshev method. The emergent and transmitted radiative intensity and the degree of polarization are predicted for both Rayleigh and Mie scattering. The discrete spherical harmonics method predictions for optical thin atmosphere using 36 streams are found in good agreement with benchmark literature results. The maximum deviation between the proposed method and literature results and for polar directions \(\vert \mu \vert \geq0.1 \) is less than 0.5% and 0.9% for the Rayleigh and Mie scattering, respectively. These deviations for directions close to zero are about 3% and 10% for Rayleigh and Mie scattering, respectively.     

The complexH-function

The properties of theH -function for the complex albedo of single scattering are studied. It is shown that basically all the formulae derived for the real albedo of single scattering can be transferred into the complex-plane without alteration.An iterational procedure to find the numerical values of the complexH - function is set up. Some examples of the numerical results are given in the figures.     

Effects of an ambient medium on the emission,absorption and scattering of waves by atoms and molecules

A theory describing the interaction between atoms or molecules (or other systems with discrete energy eigenvalues) and waves in an arbitrary mode in an arbitrary ambient medium is developed. Rules for generalizing formulae describing processes for waves in vacuo to include the effects of a medium are stated and the illustrative examples of multipole radiation, the photo-electric effect and Rayleigh and Raman scattering are given.The following specific results are discussed: (1) In an isotropic medium with refractive indexn(), the rate of transitions with frequency isn(),n ³(), ... times that in vacuo for electric dipole, magnetic dipole or electric quadrupole, ..., transitions. (2) The conventional multipole expansion is inadequate when waves with a longitudinal component of polarization exist, but this does not affect the theory of electric and magnetic dipole transitions.A possible astrophysical application of resonant scattering by molecules of electron plasma waves is discussed briefly.     

Light scattering calculations by the Fredholm integral equation method

The Fredholm integral equation method (FIM), originally introduced by Holtet al. to solve the light scattering problem for ellipsoidal particles, is reinvestigated by taking into account a recent great progress in numerical computers. A numerical code optimized for vector-processing computers is developed, and is applied to the light scattering by spherical and spheroidal particles. The results for these particles are compared with those by the Mie theory and by Asano and Yamamoto, respectively, and it is confirmed that the agreement with both of them is satisfactory. Sample calculations are also performed for the oblique incidence, in which the direction of incidence is not parallel nor perpendicular to the symmetry axis of the particle. No difficulties in the computation are found compared with the calculations for the parallel or perpendicular incidence. We study the efficiency factor for polarization (Q _pol) in general direction of incidence for spheroidal particles, and discuss the deviation from the Rayleigh approximation.     

Gas heating inside radio sources to mildly relativistic temperatures via induced compton scattering

The measured brightness temperatures of the low-frequency synchrotron radiation from intense extragalactic sources reach 10¹¹–10¹² K. If there is some amount of nonrelativistic ionized gas within such sources, it must be heated through induced Compton scattering of the radiation. If cooling via inverse Compton scattering of the same radio radiation counteracts this heating, then the plasma can be heated up to mildly relativistic temperatures kT～10–100 keV. In this case, the stationary electron velocity distribution can be either relativistic Maxwellian or quasi-Maxwellian (with the high-velocity tail suppressed), depending on the efficiency of Coulomb collisions and other relaxation processes. We derive several simple approximate expressions for the induced Compton heating rate of mildly relativistic electrons in an isotropic radiation field, as well as for the stationary electron distribution function and temperature. We give analytic expressions for the kernel of the integral kinetic equation (one as a function of the scattering angle, and the other for an isotropic radiation field), which describes the photon redistribution in frequency through induced Compton scattering in thermal plasma. These expressions can be used in the parameter range [in contrast to the formulas written out previously in Sazonov and Sunyaev (2000), which are less accurate].     

Rayleigh-Cabannes scattering in planetary atmospheres III. The Milne problem in conservative atmospheres

The discrete ordinale method by Chandrasekhar is used to solve the conservative Milne problem in a homogeneous plane-parallel atmosphere which scatters the radiation according to the Rayleigh-Cabannes law.The approximate solution which is supposed to converge uniformly to an exact one when increasing the order of approximation is obtained explicitly. In addition to a tabulation of the Hopf vector for different factors of depolarization, the extrapolation distance, the values of c, q and the Rubenson degrees of polarization at the limb are given.     

Time-dependentX- andY-functions by integral equation method

The time-dependent equation of radiative transfer for isotropic scattering has been solved by integral equation technique in terms ofX- andY-functions appropriate for the problem. It is seen thatX- andY-functions are reducible to the corresponding function for steady-state problems by simply changing the Laplace transform parameters-i.e., byS0.     

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.     

Time-dependent radiation transfer and a possible explanation of the interpulse in CP 0950

The time-dependent equation of radiative transfer is solved exactly and in then-th Gaussian approximation.The atmosphere is plane-parallel and semi-infinite; isotropic scattering is assumed, but the boundary condition at =0 is arbitrary.The results are used to investigate a suggested mechanism for the origin of the secondary pulses in CP 0950; it is found that a binary system of neutron stars can indeed explain formation, time delay and intensity of the observed interpulse.     

Numerical methods in polarized radiative transfer

This paper looks at three aspects of numerical methods for solving polarized radiative transfer problems associated with spectral line formation in the presence of a magnetic field. First we prove Murphy's law for Stokes evolution operators which is the basis of the efficient algorithm used in the SPSR software package to compute the Stokes line depression contribution functions. Then we use a two-stream model to explain the efficacy of the field-free method in which the non-LTE line source function in a uniform magnetic field is approximated by the source function neglecting the magnetic field. Finally we introduce a totally new and computationally efficient approach to solving non-LTE problems based on a method of sparsely representing integral operators using wavelets. As an illustration, the wavelet method is used to solve the source function integral equation for a two-level atomic model in a finite atmosphere with coherent scattering, ignoring polarization.     

Average photon path-length of radiation emerging from finite atmospheres

The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching 6.9% in the reflected radiation.     

An exact solution of the equation of transfer with three-term scattering indicatrix in an exponential atmosphere

The general equation for radiative transfer in the Milne-Eddington model is considered here. The scattering function is assumed to be quadratically anisotropic in the cosine of the scattering angle and Planck's intensity function is assumed for thermal emission. Here we have taken Planck's function as a nonlinear function of optical depth, viz.,B _v(T)=b _o+b ₁ e ^–. The exact solution for emergent intensity from the bounding face is obtained by the method of the Laplace transform in combination with the Wiener-Hopf technique.     

An element formulation for perturbed motion about the center of mass

The perturbed motion of a rigid body about its center of mass, is formulated in terms of the six elements:l, the magnitude of the angular momentum vector;h, the total energy; and , two linear functions of the independent variable; and ₁ and ₁, two Euler angles that orientate the inertial frame with respect to the unperturbed solution. Solutions from the element formulation and the original Euler equations are numerically compared using shuttle-type data. For applied torques smaller than a given magnitude, the element formulation produced the following results: (1) larger step sizes in the numerical integration of the differential equations, resulting in an overall computational time-saving, and (2) more significant figures of accuracy in the computation of the variables describing the state of the rigid body.     

Polarization of intrinsic radiation of tidally distorted stars with electron-scattering atmospheres

Linear polarization of radiation emitted by tidally distorted stars as a function of the binary system phase is computed, taking into account true absorption and the scattering of light on free and bound electrons within hot stellar atmospheres. Computations are made both for the linear distribution of true sources across the atmospheres and for radiative-stable model atmospheres presented by Kurcuzet al. (1974) and Kurucz (1979). Polarization variability was investigated as a function of wavelength . In a number of cases, polarization variability was found to be at an observable level. The most marked variability was expected in the ultraviolet range adjacent to the boundaries of the spectral series for H and He. Near the Lyman limit of approximately =912 Å for stars with an effective temperatureT _eff35 000 K and near the ionization boundary for HeII 226 Å for stars withT _eff>35 000 K, the amplitude of polarization variability is greater than in the case of pure electron atmospheres, sometimes reaching the level of 0.5–1%. For fairly long waves where the limb-darkening coefficient falls below a certain critical valueu _cr0.5, the plane of polarization is found to be turned by 90° as compared to the case of a pure electron atmosphere. For limb-darkening coefficients far from the value ofu _cr; the form of the polarization phase curves, as well as dependence on the parameters of a binary system, remain approximately the same as those in the case of pure electron scattering.