General Radiative Transfer Equation Research Articles

The MAMA Algorithm for Fast Computations of Upwelling Far- and Mid-Infrared Radiances in the Presence of Clouds

A methodology for the computation of spectrally resolved upwelling radiances in the presence of atmospheric diffusive layers is presented. The algorithm, called MAMA (Martinazzo–Maestri), provides fast simulations over the whole longwave spectrum, with high accuracy, particularly for optically thin scattering layers like cirrus clouds. The solution is obtained through a simplification of the multiple-scattering term in the general equation of the radiative transfer in a plane-parallel assumption. The scattering contribution is interpreted as a linear combination of the mean ambient radiances involved in the forward and back-scatter processes, which are multiplied by factors derived from the diffusive features of the layer. For this purpose, a fundamental property of the layer is introduced, named the angular back-scattering coefficient, which describes the fraction of radiation coming from a hemisphere and back-scattered into a specific direction (the observer in our case). This property, easily derived from the phase function of the particle size distribution, can be calculated from any generic single-scattering properties database, which allows for simple upgrades of the reference optical properties within the code. The paper discusses the solutions for mean upward and downward ambient radiances and their use in the simplification of the general radiative transfer equation for thermal infrared. To assess the algorithm performance, the results obtained with the MAMA code are compared with those derived with a discrete ordinate-based radiative transfer model for a large range of physical and optical properties of ice and liquid water clouds and for multiple atmospheric conditions. It is demonstrated that, for liquid water clouds, the MAMA code accuracy is mostly within 0.4 mW/(m2cm−1sr) with respect to the reference code both at far- and mid-infrared wavelengths. Ice cloud spectra are also accurately simulated at mid-infrared for all realistic cloud cases, which makes the MAMA code suitable for the analysis of any spectral measurements of current satellite infrared sounders. At far infrared, the MAMA accuracy is excellent when ice clouds with an optical depth of less than 2 are considered, which is particularly valuable since cirrus clouds are one of the main targets of the future mission FORUM (Far-infrared Outgoing Radiation Understanding and Monitoring) of the European Space Agency. In summary, the MAMA method allows computations of cloudy sky high-resolution radiances over the full longwave spectrum (4–100 μm) in less than a second (for pre-computed gas optical depths and on a standard personal computer). The algorithm exploits the fundamental properties of the scattering layers, and the code can be easily updated in relation to new scattering properties.

3D photon conserving code for time-dependent general relativistic radiative transfer: CARTOON

ABSTRACT We develop the three-dimensional general relativistic radiative transfer code: CARTOON (Calculation code of Authentic Radiative Transfer based On phOton Number conservation in curved space–time) which is improved from the two-dimensional code: ARTIST developed by Takahashi & Umemura (2017). In CARTOON, the frequency-integrated general relativistic radiative transfer equation is solved in a photon number-conserving manner, and the isotropic and coherent scattering in the zero angular momentum observers (ZAMO) frame and the fluid rest frame is incorporated. By calculating the average energy of photons, energy conservation of the radiation is also guaranteed. With the test calculations in two-dimensional and three-dimensional space, we have demonstrated that the wavefront propagation in black hole space–time can be correctly solved in CARTOON conserving photon numbers. The position of the wavefront coincides with the analytical solution and the number of photons remains constant until the wavefront reaches the event horizon. We also solve the radiative transfer equation on the geodesic reaching the observer’s screen. The time variation of the intensity map on the observer’s screen can be simultaneously and consistently calculated with the time variation of the radiation field around the black hole. In addition, the black hole shadow can be reproduced in moderately optically thin situations.

Numerical Study on the Effect of Lambda Value (Oxygen/Fuel Ratio) on Temperature Distribution and Efficiency of a Flameless Oxyfuel Combustion System

The flameless oxyfuel combustion technology has been proven to be a promising new method to reduce the fuel consumption and pollutants in industrial applications [...]

Covariant Compton scattering kernel in general relativistic radiative transfer

A covariant scattering kernel is a core component in any self-consistent general relativistic radiative transfer formulation in scattering media. An explicit closed-form expression for a covariant Compton scattering kernel with a good dynamical energy range has unfortunately not been available thus far. Such an expression is essential to obtain numerical solutions to the general relativistic radiative transfer equations in complicated astrophysical settings where strong scattering effects are coupled with highly relativistic flows and steep gravitational gradients. Moreover, this must be performed in an efficient manner. With a self-consistent covariant approach, we have derived a closed-form expression for the Compton scattering kernel for arbitrary energy range. The scattering kernel and its angular moments are expressed in terms of hypergeometric functions, and their derivations are shown explicitly in this paper. We also evaluate the kernel and its moments numerically, assessing various techniques for their calculation. Finally, we demonstrate that our closed-form expression produces the same results as previous calculations, which employ fully numerical computation methods and are applicable only in more restrictive settings.

Computational Laser Spectroscopy in a Biological Tissue

We present a numerical spectroscopic study of visible and infrared laser radiation in a biological tissue. We derive a solution of a general two-dimensional time dependent radiative transfer equation in a tissue-like medium. The used model is suitable for many situations especially when the external source is time-dependent or continuous. We use a control volume-discrete ordinate method associated with an implicit three-level second-order time differencing scheme. We consider a very thin rectangular biological-tissue-like medium submitted to a visible or a near infrared light sources. The RTE is solved for a set of different wavelength source. All sources are assumed to be monochromatic and collimated. The energetic fluence rate is computed at a set of detector points on the boundaries. According to the source type, we investigate either the steady-state or transient response of the medium. The used model is validated in the case of a heterogeneous tissue-like medium using referencing experimental results from the literature. Also, the developed model is used to study changes on transmitted light in a rat-liver tissue-like medium. Optical properties depend on the source wavelength and they are taken from the literature. In particular, light-transmission in the medium is studied for continuous wave and for short pulse.

Observational testability of a Kerr bound in the x-ray spectrum of black hole candidates

The specific angular momentum of a Kerr black hole must not be larger than its mass. The observational confirmation of this bound which we call a Kerr bound directly suggests the existence of a black hole. In order to investigate observational testability of this bound by using the x-ray energy spectrum of black hole candidates, we calculate energy spectra for a super-spinning object (or a naked singularity) which is described by a Kerr metric but whose specific angular momentum is larger than its mass, and then compare the spectra of this object with those of a black hole. We assume an optically thick and geometrically thin disk around the super-spinning object and calculate its thermal energy spectrum seen by a distant observer by solving general relativistic radiative transfer equations including usual special and general relativistic effects, such as Doppler boosting, gravitational redshift, light bending and frame-dragging. Surprisingly, for a given black hole, we can always find its super-spinning counterpart with its spin a∗ in the range whose observed spectrum is very similar to and practically indistinguishable from that of the black hole. As a result, we conclude that to confirm the Kerr bound we need more than the x-ray thermal spectrum of the black hole candidates.

A Fast-Forward Solver of Radiative Transfer Equation

In this article, we develop an efficient forward solver for steady-state or frequency-domain radiative transfer equation on 2D and 3D structured and unstructured meshes with vacuum boundary condition or reflection boundary condition. Although we emphasize applications in optical imaging, our method may be applied to more general radiative transfer equations, for example, other types of scattering function. Efficiency and accuracy of our algorithm is demonstrated by comparison with both analytical solutions and Monte Carlo solutions and with various numerical tests in optical imaging.

Energetic Electrons in Loop Top and Footpoint Microwave Sources

We have studied two microwave events with one-loop top (LT) and two-footpoint (FP) sources observed at 17 and 34 GHz by the Nobeyama Radioheliograph (NoRH). The microwave brightness peak is located near the FPs of the flare loop for one event, but near the LT for the other event. The microwave spectra of the FP sources are considerably softer (by Δα≈2.0) than that of the LT’s for both events. We assume that the microwave emission is gyro-synchrotron radiation from energetic electrons trapped in a magnetic dipole field and the energetic electron distribution is isotropic in pitch angle and power law. In the gyro-synchrotron calculations, the self-absorption and gyro-resonance absorption are taken into account simultaneously. The numerical calculations based on the general equation of radiative transfer show that the distributions of energetic electrons along a flare loop are highly inhomogeneous: accelerated electrons are concentrated in the FPs for both events. Even for the event with brightness maximum near the LT the electron number density of the LT source is still an order of magnitude lower than that of the FP sources. The emission peak near LT results mainly from the much harder spectral index of the energetic electrons in the LT source.

Simplified atmospheric radiative transfer modelling for estimating incident PAR using MODIS atmosphere products

Photosynthetically active radiation (PAR) is an important parameter in the estimation of vegetation growth. A method is presented with which instantaneous PAR can be calculated with high accuracy from Moderate Resolution Imaging Spectroradiometer (MODIS) atmosphere data products. The method, PARcalc, is based on a simplification of the general radiative transfer equations, whereby the atmosphere is modelled in a few layers depending on the opacity of the atmosphere due to clouds and aerosol load. The four major processes of atmospheric attenuation of solar radiation, Rayleigh scattering, absorption by ozone and water vapour, and aerosol scattering, are treated individually, using calibrated data from MODIS imagery. Comparing the calculations to field measurements made in Costa Rica in 2002, absolute errors in the order of 2–3% are obtained, which should be sufficiently accurate for application of the PARcalc method in models of vegetation growth as a source of incident PAR.

Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics.

The concepts of statistical electromagnetics are used to derive the general radiative transfer equation (RTE) that describes multiple scattering of polarized light by sparse discrete random media consisting of arbitrarily shaped and arbitrarily oriented particles. The derivation starts with the volume integral and Lippmann-Schwinger equations for the electric field scattered by a fixed N-particle system and proceeds to the vector form of the Foldy-Lax equations and their approximate far-field version. I then assume that particle positions are completely random and derive the vector RTE by applying the Twersky approximation to the coherent electric field and the Twersky and ladder approximations to the coherency dyad of the diffuse field in the limit N --> infinity. The concluding section discusses the physical meaning of the quantities that enter the general vector RTE and the assumptions made in its derivation.

Radiative Transfer in Weakly Polarizing Media

We study radiative transfer through a weakly polarizing medium, i.e., a medium in which the degree of polarization of the absorbed, retarded, and emitted light is always weak. In this case, the general radiative transfer equation for the Stokes parameters yields very simple formal solutions. The intensity does not depend on the polarization, and the other Stokes parameters are uncoupled from each other. It is shown how this simplified radiative transfer equation holds in many realistic cases relevant for solar and stellar magnetometry. It can be applied whenever the weak magnetic field approximation works, i.e., for weakly split lines. In addition, it handles weak spectral lines, structures with complex magnetic topology, chromospheric lines formed under non-LTE conditions, etc. The merits of the approximation, which we call the weakly polarizing medium (WPM) approximation, are illustrated by means of several LTE and non-LTE line syntheses in realistic solar model atmospheres. The WPM approximation should be useful in planning and understanding measurements based on polarization. It simplifies the relationship between the observed polarization and the physical structure that one tries to retrieve. The approximation may also be used in numerical problems requiring extensive polarized radiative transfer (inversion codes, syntheses of stellar spectra, self-consistent multilevel non-LTE Zeeman line transfer with atomic polarization, etc.).

The Extended Power Law as an Intrinsic Signature for a Black Hole

We analyze the exact general relativistic integrodifferential equation of radiative transfer describing the interaction of low-energy photons with a Maxwellian distribution of hot electrons in the gravitational field of a Schwarzschild black hole. We prove that, owing to Comptonization, an initial arbitrary spectrum of low-energy photons unavoidably results in spectra characterized by an extended power-law feature. We examine the spectral index by using both analytical and numerical methods for a variety of physical parameters as such the plasma temperature and the mass accretion rate. The presence of the event horizon as well as the behavior of the null geodesics in its vicinity largely determine the dependence of the spectral index on the flow parameters. We come to the conclusion that the bulk motion of a converging flow is more efficient in upscattering photons than thermal Comptonization, provided that the electron temperature in the flow is of order of a few kilo-electron volts or less. In this case, the spectrum observed at infinity consists of a soft component, which is produced by those input photons that escape after a few scatterings without any significant energy change, and a hard component (described by a power law), which is produced by the photons that underwent significant upscattering. The luminosity of the power-law component is relatively small compared to that of the soft component. For accretion into a black hole, the spectral energy index of the power law is always higher than 1 for plasma temperatures of order of a few kilo-electron volts. This result suggests that the bulk motion Comptonization might be responsible for the power-law spectra seen in the black hole X-ray sources.

Radiative transfer theory with time delay for the effect of pulse entrapping in a resonant random medium: General transfer equation and point-like scatterer model

A pulse propagation of a vector electromagnetic wave field in a discrete random medium under the condition of Mie resonant scattering is considered on the basis of the Bethe–Salpeter equation in the two-frequency domain in the form of an exact kinetic equation which takes into account the energy accumulation inside scatterers. The kinetic equation is simplified using the transverse field and far wave zone approximations which give a new general tensor radiative transfer equation with strong time delay by resonant scattering. This new general radiative transfer equation, being specified in terms of the low-density limit and the resonant point-like scatterer model, takes the form of a new tensor radiative transfer equation with three Lorentzian time-delay kernels by resonant scattering. In contrast to the known phenomenological scalar Sobolev equation with one Lorentzian time-delay kernel, the derived radiative transfer equation does take into account effects of (i) the radiation polarization, (ii) the energy accumulation inside scatterers, (iii) the time delay in three terms, namely in terms with the Rayleigh phase tensor, the extinction coefficient and a coefficient of the energy accumulation inside scatterers, respectively (i.e. not only in a term with the Rayleigh phase tensor). It is worth noting that the derived radiative transfer equation is coordinated with Poynting's theorem for non-stationary radiation, unlike the Sobolev equation. The derived radiative transfer equation is applied to study the Compton–Milne effect of a pulse entrapping by its diffuse reflection from the semi-infinite random medium when the pulse, while propagating in the medium, spends most of its time inside scatterers. This specific albedo problem for the derived radiative transfer equation is resolved in scalar approximation using a version of the time-dependent invariance principle. In fact, the scattering function of the diffusely reflected pulse is expressed in terms of a generalized time-dependent Chandrasekhar H-function which satisfies a governing nonlinear integral equation. Simple analytic asymptotics are obtained for the scattering function of the front and the back parts of the diffusely reflected Dirac delta function incident pulse, depending on time, the angle of reflection, the mean free time, the microscopic time delay and a parameter of the energy accumulation inside scatterers. These asymptotics show quantitatively how the rate of increase of the front part and the rate of decrease of the rear part of the diffusely reflected pulse become slower with transition from the regime of conventional radiative transfer to that of pulse entrapping in the resonant random medium.

Simulated emission spectra of samples at non-uniform temperature based on numerical solutions of the general radiative transfer equation

Infrared spectroscopic methods are increasingly being used for analysis in the food industry. A potentially attractive approach is that of infrared emission spectroscopy: in many food processes, the raw materials are subjected to heating, and once hot, act as sources of infrared radiation. If the emitted radiation is steered directly into a spectrometer with the conventional source removed, an emission spectrum may be obtained of the sample without any further light guiding requirement. However, for bulk samples, the spectra obtained are not very useful, due to the mechanism of self-absorption, whereby chemically different samples give rise to the same, featureless spectral profile. Promising strategies for overcoming this problem are Transient Infrared Emission and Transient Infrared Transmission Spectroscopies (TIRES and TIRTS), in which thermal gradients are induced at the sample surface, enabling composition-specific features to be seen in the emission spectrum. In this paper mathematical models of the effect of thermal gradients on emission spectra are presented, based on numerical solutions of the Radiative Transfer Equation (RTE) for the non-uniform temperature case, using parameters determined experimentally. This provides a method of relating the nature of thermal gradient within a sample to the information content expected in its emission spectrum. Thus, the conditions required to obtain useful emission spectra from bulk samples can be determined, and the potential of emission spectroscopy as an analysis tool in food production processes assessed.

Winds from neutron stars and strong type I X-ray bursts

view Abstract Citations (34) References (48) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Winds from Neutron Stars and Strong Type I X-Ray Bursts Nobili, Luciano ; Turolla, Roberto ; Lapidus, Iosif Abstract A model for stationary, radiatively driven winds from X--ray bursting neutron stars is presented. General relativistic hydrodynamical and radiative transfer equations are integrated from the neutron star surface outwards, taking into account for helium nuclear burning in the inner, dense, nearly hydrostatic shells. Radiative processes include both bremsstrahlung emission--absorption and Compton scattering; only the frequency--integrated transport is considered here. It is shown that each solution is characterized by just one parameter: the mass loss rate $\Mdot$, or, equivalently, the envelope mass $\Menv$. We found that, owing to the effects of Comptonization, steady, supersonic winds can exist only for $\Mdot$ larger than a limiting value $\Mdot_{min} \approx\Mdot_{E}$. Several models, covering about two decades in mass loss rate, have been computed for given neutron star parameters. We discuss how the sequence of our solutions with decreasing $\Menv$ can be used to follow the time evolution of a strong X--ray burst during the expansion/contraction phase near to the luminosity maximum. The comparison between our numerical results and the observational data of Haberl {\it et al.\/} (1987) for the bursts in 4U/MXB 1820-30 gives an estimate for both the spectral hardening factor and the accretion rate in this source. Publication: The Astrophysical Journal Pub Date: September 1994 DOI: 10.1086/174643 arXiv: arXiv:astro-ph/9405065 Bibcode: 1994ApJ...433..276N Keywords: Hydrodynamics; Neutron Stars; Radiative Transfer; Stellar Mass; Stellar Models; Stellar Radiation; Stellar Winds; X Rays; Bremsstrahlung; Bursts; Compton Effect; Mathematical Models; Stellar Mass Accretion; Astrophysics; HYDRODYNAMICS; RADIATION MECHANISMS: NONTHERMAL; RADIATIVE TRANSFER; STARS: INDIVIDUAL ALPHANUMERIC: 4U/MXB 1820-30; STARS: MASS LOSS; STARS: NEUTRON; X-RAYS: BURSTS; Astrophysics E-Print: 32 pages (10 postsript figures available on request), PlainTeX full text sources arXiv | ADS | data products SIMBAD (4)

The Differential-Discrete-Ordinate Method for Solutions of the Equation of Radiative Transfer

This paper introduces a powerful but simple methodology for solving the general equation of radiative transfer for scattering and/or absorbing one-dimensional systems. Existing methods, usually designed to handle specific boundary and energy equilibrium conditions, either provide crude estimates or involve intricate mathematical analysis coupled with numerical techniques. In contrast, the present scheme, which uses a discrete-ordinate technique to reduce the integro-differential equation to a system of ordinary differential equations, utilizes readily available software routines to solve the resulting set of coupled first-order ordinary differential equations as a two-point boundary value problem. The advantage of this approach is that the user is freed from having to understand complicated mathematical analysis and perform extensive computer programming. Additionally, the software used is state of the art, which is less prone to numerical instabilities and inaccuracies. Any degree of scattering anisotropy and albedo can be incorporated along with different conditions of energy equilibrium or specified temperature distributions and boundary conditions. Examples are presented where the radiative transfer is computed by using different quadratures such as Gaussian, Lobatto, Fiveland, Chebyshev, and Newton-Cotes. Comparison with benchmark cases shows that in a highly forward scattering medium Gaussian quadrature provides the most accurate and stable solutions.

Extending the radiosity method to include specularly reflecting and translucent materials

An extension of the radiosity method is presented that rigorously accounts for the presence of a small number of specularly reflecting surfaces in an otherwise diffuse scene, and for the presence of a small number of specular or ideal diffuse transmitters. The relationship between the extended method and earlier radiosity and ray-tracing methods is outlined. It is shown that all three methods are based on the same general equation of radiative transfer. A simple superposition of the earlier radiosity and ray-tracing methods in order to account for specular behavior is shown to be physically inconsistent, as the methods are based on different assumptions. Specular behavior is correctly included in the present method. The extended radiosity method and example images are presented.

On the fundamental solution of the radiative transfer equation

This paper outlines the general solution of the one‐dimensional, azimuthally averaged radiative transfer equation in terms of a matrix exponential. The link between this fundamental solution and those more commonly used in radiative transfer is established. The formulation is developed for a general vertically inhomogeneous atmosphere with sources. Several new concepts, based on properties of the matrix exponentials, are described in the context of radiative transfer, including the use of the commutator and product integrals. It is also demonstrated how the matrix exponential formulation provides for new insights, not only into improvements of the numerical efficiency and stability of the solution, but also into the understanding of radiative transfer through a layered atmosphere. The various concepts introduced in this paper are illustrated throughout by the two‐stream simplification of the general radiative transfer equation.

The Interpretation of Diffuse Reflectance Spectra.

Numerous treatments of the diffuse reflecting properties of scattering media have been described. Many theories give an adequate account of the reflectance for a specific set of conditions for which the model was constructed and the solution tested experimentally. Only those models which are considered to be fairly general are considered here.It is convenient to divide the theories into those based upon continuum models and those based upon statistical models. The continuum models typically describe the scattering and absorbing properties of a given medium in terms of two phenomenological constants. These models may all be regarded as varying levels of approximate solution to the general equation of radiative transfer. This provides a convenient basis for comparison of the various theories.The statistical models are based upon a summation of transmittances and reflectances from individual layers or particles. Thus, some assumptions must be made about the nature of the fundamental units, and the validity of the ultimate result will depend upon how closely these assumptions correspond with reality. Only the statistical models lead to expressions from which absolute absorptivities and scattering coefficients can be calculated and related to the actual particle characteristics.The relationship between the various models will be discussed and the features which typify the absorptivity and scattering coefficient according to each will be compared and related to the available experimental data. This leads to a consideration of the characteristics of appropriate model systems and standards.

Angular quadrature perturbations in radiative transfer theory

A new numerical method is presented for solving the general equation of radiative transfer. The approximation, which replaces the integral term over angle in the transfer equation by a quadrature sum, is studied; an estimate of the error involved is obtained and this error, which may be thought of as a further source or sink of photons (depending upon the sign), can then be used to evaluate a corection to the radiation field originally determined. This process may then be continued as a perturbation series. The method is found to give a final solution, when starting from the Eddington approximation, at least as accurate as that obtained using variable Eddington factors. Furthermore, the technique involves very little extra computing over that required using the Eddington approximation, and may be trivially generalized to any radiative transfer problem. It can also be used in conjunction with any of the existing methods for solving the equation of transfer. Examples are given in the context of spectral line formation in slab geometry.