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.     

Monte Carlo calculation of comptonization effects in spherical accretion onto black holes

Spherical accretion onto black holes at high accretion rates leads to temperatures and optical depths for which Comptonization of the emerging radiation plays an important role, altering both the plasma temperature and the emergent spectrum. In this paper the nonlocal effects of Comptonization are accounted for by numerically modeling the scattering process with Monte Carlo techniques. Calculations are performed for black hole masses ranging from 10¹–10⁹ M and are compared with the local scattering model of Ipser and Price (1983). It is found that the local scattering approximation underestimates the energy in the X-ray portion of the spectrum. Monte Carlo calculations shows that the spectrum flattens and the spectral index decreases to values about 20% below those obtained by using the local scattering approximation.     

The range of validity of the Eddington approximation

The Eddington approximation is often assumed to be useful only for optically thick media having a single-scattering albedo near unity. We present detailed evidence in this paper that, for homogeneous layers illuminated by a beam of radiation, the Eddington approximation predicts albedo and absorptivity reasonably well for all values of optical depth and single-scattering albedo, for several scattering phase functions (Rayleigh, Henyey-Greenstein, and Mie) having asymmetry factors less than or equal to $\text{1}\text{2}$. The worst errors are in the neighborhood of optical depth unity and single-scattering albedo 0.5. The Eddington approximation is further found to maintain good accuracy over almost the full range of incident beam directions and surface albedos. It is least accurate for the Mie phase function example, where one can obtain a dramatic improvement in accuracy by going over to the δ-Eddington approximation; this shows that the forward peak of the Mie phase function, and not its detailed shape, is the primary cause of diminished accuracy in the Eddington approximation.     

The effects of scattering on quiet Sun emission at frequencies less than 200 MHz

The problem of predicting the radio emission from the quiet Sun for meter and decameter waves may be formulated in terms of a standard radiative transfer problem with conservative scattering. One may therefore avoid the numerical complications involved in using a ray-tracing approach which incorporates a Monte-Carlo routine for representing scattering. The brightness and extent of the radio Sun are calculated for several values of the electron temperature and scattering parameter of Steinberg et al. (1971). It is concluded that the temperature and density model of Newkirk (1967) for solar maximum reasonably represents the observations. However, some observations appear to be inconsistent with scattering models and further observations are needed. It is shown (in Appendix B) that the standard ray-tracing technique incorporating a Monte-Carlo routine for the scattering may be replaced by a diffusion approximation.     

Average photon path-length in isotropically scattering finite atmospheres

The determination of the average path-length of photons in a finite isotropically scattering plane-parallel homogeneous atmosphere is discussed. To solve this problem we have used the kernel approximation method which easily allows us to find the derivatives of the intensity with respect to optical depth, optical thickness and albedo of single scattering.In order to check the results we have used another approach by exploiting the set of integrodifferential equations of Chandrasekhar for theX- andY-functions. This approach allows us to find the average path length only at the boundaries of the atmosphere but on the other hand it gives also the dispersion of the path-length distribution function, thus generating the input parameters for determining the approximate path-length distribution function. It occurred that the set so obtained is stable and the results are highly accurate.As a by-product we obtain the first two derivatives of theX- andY-functions with respect to the albedo of single scattering and optical thickness, and the mixed derivative.     

Bidirectional Reflectance Spectroscopy: 5. The Coherent Backscatter Opposition Effect and Anisotropic Scattering

A model published previously by the author that describes light scattering from particulate media is modified to include several improvements: (1) a better approximation to the Ambartsumian-Chandrasekhar H-functions that is especially important for particles with single scattering albedos close to 1.00, (2) increased accuracy for anisotropically scattering particles, and (3) incorporation of coherent backscattering. The goal of the original model of being analytic and mathematically tractable is preserved. No new parameters are introduced by the first and second modifications; however, the third unavoidably adds two new parameters: the amplitude and width of the coherent backscatter opposition effect. Several examples are given in which the results of calculations using the original and new models are compared with exact numerical computations.It is shown that a medium composed of complex particles that are large compared to the wavelength can have a coherent backscatter opposition effect (CBOE) that is broad enough to be readily observable. The CBOE multiplies the entire reflectance and not just the multiply scattered component, so that a low-albedo medium, such as lunar regolith, can have a strong CBOE.     

An efficient method to calculate Ambarzumian-Chandrasekhar's and hopf's functions

An efficient method is proposed to calculate scalar Ambarzumian-Chandrasekhar's and Hopf's functions. This method is based on the approximation of Sobolev's resolvent function using exponent series, the coefficients of which are readily found from approximate characteristic equation and from a system of linear algebraic equations.The approximate expressions for the above functions are given. For checking purposes the calculations were carried out in single, double, and quadruple precision. For isotropic, Rocard, and Rayleigh scattering we present a sample of results in 14 significant figures.The Hopf function for isotropic and Rayleigh scattering is presented in 18 significant figures and the well-known Hopf constantq() is found in 59 significant figures.     

Angular distribution of radiation in an isotropically scattering slab

The equation of radiative transfer in an isotropically scattering slab subject to general boundary conditions is solved. The Padé approximation technique is used to calculate the reflected and transmitted angular distributions. Numerical results for angular distributions through and at the boundaries of a finite slab are calculated by the Padé approximation technique. The results for a Padé approximation of [0/1] are compared with results obtained by the Galerkin method.     

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.     

A fast,exact code for scattered thermal radiation compared with a two-stream approximation

For radiative transfer in plane-parallel emitting, absorbing, and scattering media, the two-stream approximation, and its various modifications or related methods, is probably mathematically the most simple to use. Unfortunately this physical approximation produces errors that are neither analytically known nor controllable. For externally (Sun) driven problems, many error studies exist for reflectivity, transmissivity, and certain defined albedos. A two-stream accuracy study for internally (thermal) driven problems is presented in this paper by comparison with a recently developed “exact” adding/doubling method. The resulting errors in external (or boundary) radiative intensity and flux are usually larger than those for the externally driven problems and vary substantially with the radiative parameters. Error predictions for a specific problem are difficult. An unexpected result was that the exact method is computationally as fast as the two-stream approximation for nonisothermal media. Although the adding/doubling method is mathematically and conceptually more complex, it may be used as a developed code with no essential sacrifice in computing time.     

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.     

Radiative transfer in disc galaxies – I. A comparison of four methods to solve the transfer equation in plane-parallel geometry

Accurate photometric and kinematic modelling of disc galaxies requires the inclusion of radiative transfer models. Because of the complexity of the radiative transfer equation (RTE), sophisticated techniques are required. Various techniques have been employed for the attenuation in disc galaxies, but a quantitative comparison of them is difficult, because of the differing assumptions, approximations and accuracy requirements that are adopted in the literature. In this paper, we present an unbiased comparison of four methods to solve the RTE, in terms of accuracy, efficiency and flexibility. We apply them all to one problem that can serve as a first approximation of large portions of disc galaxies: a one-dimensional plane-parallel geometry, with both absorption and multiple scattering taken into account, with arbitrary vertical distributions of stars and dust and an arbitrary angular redistribution of the scattering. We find that the spherical harmonics method is by far the most efficient way to solve the RTE, whereas both Monte Carlo simulations and the iteration method, which are straightforward to extend to more complex geometries, have a cost that is about 170 times larger.     

Spektrum und Anisotropiegrad der 3 °K-Strahlung bei anisotroper kosmologischer Expansion

Two theorems concerning the propagation of electromagnetic radiation which interacts with matter through elastic electron scattering in arbitrary gravitational fields allow to draw conclusions relevant to the thermal history of the 3 °K microwave background: Provided, a thermalization at some instant t = t₁ in the past has established at this time a Planckian photon distribution function (possibly with the temperature T depending on the propagation direction and on space coordinates), then (i) the Planckian distribution is preserved in a first approximation, if the deviation of the actual distribution from an isotropic Planckian remains always small; and (ii) in the Rayleigh-Jeans domain the Planckian shape is preserved independent of the degree of radiation anisotropy. Due to thermalizing action of electron scattering in ionized intergalactic and pre-galactic matter the observed degree of anisotropy in the 3 °K microwave background may be smaller as is expected for collisionless radiation propagating in a given geometry. To study this effect, the equation of radiative transfer with an electron scattering term is integrated in an anisotropic universe of the Heckmann-Schücking type (Bianchi type I), starting from an early optically thick epoch (corresponding to a radiation temperature T = 5000 °K) prior to plasma recombination. With the quadrupole anisotropy of the order of 2 · 10⁻³ found by Partridge and Wilkinson in the Rayleigh-Jeans domain, metric anisotropy parameters ranging from a = 380 to 900 years are derived, if the re-ionization of intergalactic matter sets in at redshift values ranging from z_r = 0 to z_r = 8.     

Geant4 simulations of soft proton scattering in X-ray optics

Low energy protons (< 300 keV) can enter the field of view of X-ray telescopes, scatter on their mirror surfaces at small incident angles, and deposit energy on the detector. This phenomenon can cause intense background flares at the focal plane decreasing the mission observing time (e.g. the XMM-Newton mission) or in the most extreme cases, damaging the X-ray detector. A correct modelization of the physics process responsible for the grazing angle scattering processes is mandatory to evaluate the impact of such events on the performance (e.g. observation time, sensitivity) of future X-ray telescopes as the ESA ATHENA mission. The Remizovich model describes particles reflected by solids at glancing angles in terms of the Boltzmann transport equation using the diffuse approximation and the model of continuous slowing down in energy. For the first time this solution, in the approximation of no energy losses, is implemented, verified, and qualitatively validated on top of the Geant4 release 10.2, with the possibility to add a constant energy loss to each interaction. This implementation is verified by comparing the simulated proton distribution to both the theoretical probability distribution and with independent ray-tracing simulations. Both the new scattering physics and the Coulomb scattering already built in the official Geant4 distribution are used to reproduce the latest experimental results on grazing angle proton scattering. At 250 keV multiple scattering delivers large proton angles and it is not consistent with the observation. Among the tested models, the single scattering seems to better reproduce the scattering efficiency at the three energies but energy loss obtained at small scattering angles is significantly lower than the experimental values. In general, the energy losses obtained in the experiment are higher than what obtained by the simulation. The experimental data are not completely representative of the soft proton scattering experienced by current X-ray telescopes because of the lack of measurements at low energies (< 200 keV) and small reflection angles, so we are not able to address any of the tested models as the one that can certainly reproduce the scattering behavior of low energy protons expected for the ATHENA mission. We can, however, discard multiple scattering as the model able to reproduce soft proton funnelling, and affirm that Coulomb single scattering can represent, until further measurements at lower energies are available, the best approximation of the proton scattered angular distribution at the exit of X-ray optics.     

Determination of the temperature of the solar corona from the spectrum of the electron-scattering continuum

When the K-corona is formed by the scattering of photospheric radiation from free electrons, the Fraunhofer lines are greatly broadened by the thermal motions of the hot electrons. This paper discusses the possibility of measuring the coronal electron temperature from the residual depressions in the K-coronal spectrum. If the ratio of the intensities at 4100 Å and 3900 Å can be measured to an accuracy of ±1%, the coronal temperature can be inferred to an accuracy of ±0.2 MK. The temperature of a coronal inhomogeneity may also be measured by this method, provided the position angle is known.Now at Fraunhofer Institute, Freiburg, Germany.     

Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering

A comparison is made between the plane-parallel approximation and the more realistic spherical shell approximation for the radiance reflected from a planetary atmosphere. In this paper we have considered a planet of radius 6371 km (the Earth) with a homogeneous, conservative, Rayleigh scattering atmosphere extending to a height of 100 km. We have found significant departures from the plane-parallel approximation. Radiance versus height distributions for both single and multiple scattering are presented. Results are presented for the fractional radiance from altitudes in the atmosphere which contribute to the total unidirectional reflected radiance at the top of the atmosphere. We have referred to this as the radiance versus height distribution in the sequel. These data will be very useful for both remote sensing applications and planetary spectroscopy. We have also found that gross violations of the principle of reciprocity do occur in the spherical shell approximation.     

Neutrino chemical potential and neutrino heat conductivity with allowance for neutrino scattering

The equations of neutrino hydrodynamics are derived in two different approximations taking into consideration the neutrino scattering from stellar material. In a thermal-conductivity approximation which holds good when neutrino optical depth with respect to absorption exceeds 1, the neutrino scattering is taken into account, analogously with photon radiative conductivity, by introducing the transport cross-section in the neutrino mean free path. In a practically important case when the neutrino optical thickness with respect to scattering is high enough, whereas that concerning absorption is sufficiently low, another approximation of Comptonized neutrinos is valid. In this case, the neutrino and antineutrino chemical potentials are independent of each other. They have to be calculated from equations of continuity established for neutrino and antineutrino alongside with the diffusion equation expressing the law of lepton-charge conservation. The equations of neutrino hydrodynamics are written out both with and without inclusion of muon neutrinos and antineutrinos.The equations obtained are established to deal properly with neutrino diffusion inside collapsing stars.     

Analytic approach to auroral electron transport and energy degradation

The interaction of a beam of auroral electrons with the atmosphere is described by the linear transport equation, encompassing discrete energy loss, multiple scattering and secondary electrons. A solution to the transport equation provides the electron intensity as a function of altitude, pitch angle (with respect to the geomagnetic field) and energy. A multi-stream (discrete ordinate) approximation to the transport equation is developed. An analytic solution is obtained in this approximation. The computational scheme obtained by combining the present transport code with the energy degradation method of Swartz (1979) conserves energy identically. The theory provides a framework within which angular distributions can be easily calculated and interpreted. Thus, a detailed study of the angular distributions of “non-absorbed” electrons (i.e., electrons that have lost just a small fraction of their incident energy) reveals a systematic variation with incident angle and energy, and with penetration depth. The present approach also gives simple yet accurate solutions in low order multi-stream approximations. The accuracy of the four-stream approximation is generally within a few per cent, whereas two-stream results for backscattered mean intensities and fluxes are accurate to within 10–15%.