Multidimensional Radiative Transfer Research Articles

Type Ia Supernovae Can Arise from the Detonations of Both Stars in a Double Degenerate Binary

The precise origin of Type Ia supernovae (SNe Ia) is unknown despite their value to numerous areas in astronomy. While it is a long-standing consensus that they arise from the explosion of a carbon/oxygen white dwarf, the exact progenitor configurations and explosion mechanisms that lead to SNe Ia are still debated. One popular theory is the double detonation, in which a helium layer, accreted from a binary companion, detonates on the surface of the primary star, leading to a converging shock-induced detonation of the underlying core. It has recently been seen in simulations that a helium-rich degenerate companion may undergo its own explosion triggered by the impact from the ejecta of the primary star. We show 2D simulations that approximate a white dwarf undergoing a double detonation, which triggers the explosion of the degenerate companion, leading to either a triple or quadruple detonation. We also present the first multidimensional radiative transfer results from the triple and quadruple detonation scenario. We find that within a range of mass configurations of the degenerate binary, the synthetic light curves and spectra of these events match observations as well as theoretical models of isolated double detonations do. Notably, double and quadruple detonations that are spectrally similar and reach the same peak brightnesses have drastically different ejecta masses and produce different amounts of Si- and Fe-group elements. Further understanding of this scenario is needed in order to determine if at least some observed SNe Ia actually originate from two stars exploding.

Lattice Boltzmann simulations on transient radiative transfer problems in irregular geometries with constant or graded refractive index

This work is concerned with the transient radiative transport in inhomogeneous two- or three-dimensional semitransparent graded-index media with irregular geometry. The lattice Boltzmann method is applied to the graded-index media mapped with non-orthogonal grids to solve the radiative transfer while the finite volume method expresses the angular processing. The solution of the transient radiation from the methodology developed is firstly validated with good accuracy by considering benchmark solutions available in the literature for comparisons for various forms of geometry including a cube, elliptic cylinder as well as irregular shapes. Afterwards, the solver is used to analyze the radiative transfer in graded-index media including regular and irregular geometries, and various governing parameters such as laser and optical properties, scattering laws, incident angle of the incoming beam, refractive index, and temperature distributions. Suitable calculations of the transient radiometric heat flux and incident energy help to evaluate the time-resolved reflectance and transmittance. The results from this study indicate that the present methodology and boundary treatments are simple and suitable to deal with multidimensional radiative transfer in irregular domains.

The He I 10 830 Å line: Radiative transfer and differential illumination effects

We study the formation of the Stokes profiles of the He I multiplet at 10 830 Å when relaxing two of the approximations that are typically considered in the modeling of this multiplet. Specifically, these are the lack of self-consistent radiation transfer and the assumption of equal illumination of the individual multiplet components. This He I multiplet is among the most important for the diagnostics of the outer solar atmosphere from spectropolarimetric observations, especially in prominences, filaments, and spicules. However, the aptness of these approximations is yet to be assessed, especially in situations where the optical thickness is on the order of one (or greater) and the radiation transfer has a significant impact in the local anisotropy as well as the ensuing spectral line polarization. This issue becomes particularly relevant in the ongoing development of new inversion tools that take into account multi-dimensional radiation transfer effects. To relax these approximations, we generalized the multi-term equations for the atomic statistical equilibrium to allow for a differential illumination of the multiplet components and implement them in a one-dimensional (1D) radiative transfer code. We find that even for this simple geometry and relatively small optical thickness, both radiation transfer and differential illumination effects have a significant impact on the emergent polarization profiles. These effects should be taken into account in order to avoid potentially significant errors when inferring the magnetic field vector.

Generalized lattice Boltzmann method for radiative transfer problem in slab and irregular graded-index media.

The lattice Boltzmann method (LBM) has been developed as a powerful solution method in computational fluid dynamics and heat transfer. However, the development of the LBM for solving radiative transfer problems has been far from perfect. This paper proposes a generalized form of the lattice Boltzmann model for the multidimensional radiative transfer equation (RTE) in irregular geometry with a graded index based on body-fitted coordinates. The macroscopic RTE is recovered from Chapman-Enskog analysis, which provides two possible procedures to formulate the Boltzmann equation in graded-index media and irregular geometries. These proposed models have been tested by considering one-and two-dimensional problems of the RTE, and the benchmark solutions reported in the literature were used for comparisons. Afterwards, the LBM is used to analyze the radiation transport in graded-index media for various forms of scattering law, refractive index, boundary reflection, laser and optical properties, and temperatures. The graded-index function and the geometry type have a significant effect on radiative transport in cases in which the refractive index matches or mismatches the boundary. It is also apparent that the developed LBM is an efficient, powerful, robust, and accurate solver for radiative transport in inhomogeneous media with a graded-index function and irregular geometries.

Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor

A new group of structure and asymptotic preserving reduced-order models (ROMs) for multidimensional nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. Approximation of the Eddington tensor that provides exact closure for the system of moment equations is found with projection-based data-driven methodologies. These include the (i) proper orthogonal decomposition (POD), (ii) dynamic mode decomposition (DMD) and (iii) a variant of the DMD. A parameterization is derived for this ROM for the temperature of radiation incoming to the problem domain (the radiation drive temperature). This parameterization is informed from results of a dimensionless study of the TRT problem. Analysis of the ROMs is performed on the classical Fleck-Cummings TRT multigroup test problem in 2D geometry with a radiation-driven Marshak wave. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be sufficiently accurate for practical computations with rather low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.

A fundamental assessment of the concept of mean beam length for application for multi-dimensional non-gray radiative heat transfer

The traditional concept of mean beam length (MBL) and its recommended empirical expression is demonstrated to be inaccurate for general application for the evaluation of multi-dimensional non-gray radiative heat transfer. A new concept, namely Point Mean Beam Length (PMBL), is proposed and the formulation of PMBL is provided. The mathematical properties of PMBL for three common geometries including sphere, cylinder, and slab are presented. Results show that PMBL is more effective in generating an accurate evaluation of radiative heat transfer from a differential area to any finite area at the boundary of an enclosure with an isothermal absorbing/emitting medium. The deficiency of the traditional mean beam length empirical expression is illustrated. A concept of “optimal” point mean beam length (OPMBL) is demonstrated to be a more accurate length scale for practical applications. In contrast to the traditional MBL, a single value of OPMBL is applicable for all gas absorption bands, independent of the strength and shape of the absorption bands. The proposed work provides a mathematically validated approach to efficiently and accurately evaluate the radiation heat transfer within an isothermal, non-gray, multi-dimensional medium.

Multi-frequency implicit semi-analog Monte-Carlo (ISMC) radiative transfer solver in two-dimensions (without teleportation)

We study the multi-dimensional radiative transfer phenomena using the ISMC scheme, in both gray and multi-frequency problems. Implicit Monte-Carlo (IMC) schemes have been in use for five decades. The basic algorithm yields teleportation errors, where photons propagate faster than the correct heat front velocity. Recently (Poëtte and Valentin, 2020 [22]), a new implicit scheme based on the semi-analog scheme was presented and tested in several one-dimensional gray problems. In this scheme, the material energy of the cell is carried by material-particles, and the photons are produced only from existing material particles. As a result, the teleportation errors vanish, due to the infinite discrete spatial accuracy of the scheme. We examine the validity of the new scheme in two-dimensional problems, both in Cartesian and Cylindrical geometries. Additionally, we introduce an expansion of the new scheme for multi-frequency problems. We show that the ISMC scheme presents excellent results without teleportation errors in a large number of benchmarks, especially against the slow classic IMC convergence.

Multidimensional Radiative Transfer Calculations of Double Detonations of Sub-Chandrasekhar-mass White Dwarfs

Abstract Study of the double-detonation Type Ia supernova scenario, in which a helium-shell detonation triggers a carbon-core detonation in a sub-Chandrasekhar-mass white dwarf (WD), has experienced a resurgence in the past decade. New evolutionary scenarios and a better understanding of which nuclear reactions are essential have allowed for successful explosions in WDs with much thinner helium shells than in the original, decades-old incarnation of the double-detonation scenario. In this paper, we present the first suite of light curves and spectra from multidimensional radiative transfer calculations of thin-shell double-detonation models, exploring a range of WD and helium-shell masses. We find broad agreement with the observed light curves and spectra of nonpeculiar Type Ia supernovae, from subluminous to overluminous subtypes, providing evidence that double detonations of sub-Chandrasekhar-mass WDs produce the bulk of observed Type Ia supernovae. Some discrepancies in spectral velocities and colors persist, but these may be brought into agreement by future calculations that include more accurate initial conditions and radiation transport physics.

Lattice Boltzmann model for multidimensional polarized radiative transfer: theory and application

Polarimetric observations of planets are providing increasing details of the three-dimensional (3D) atmospheric structure. The one-dimensional plane-parallel approximation model neglects horizontally polarized radiative transfer. Multidimensional polarized radiative transfer models, especially 3D models, are required to contain the horizontal polarization mechanism. Here, we propose a lattice Boltzmann (LB) model for multidimensional polarized radiative transfer, which enables a simple solution of the multidimensional vector radiative transfer equation (VRTE) by performing collision and streaming processes. Through the Chapman–Enskog analysis, we rigorously derive the multi-dimensional VRTE from the proposed LB model. 2D and 3D numerical tests demonstrate that the proposed LB model is effective and accurate for simulating multidimensional polarized radiative transfer. Furthermore, we apply the proposed LB model to investigate the effects of multiple scattering on radiation intensity and degree of polarization in a 3D case and find that multiple scattering enhances the radiation intensity but dampens the degree of polarization throughout almost the whole angular space in multidimensional polarized radiative transfer. This work is expected to provide a simple and effective mesoscopic tool for multidimensional polarized radiative transfer.

Point mean beam length, a new concept to enhance the computational efficiency of multidimensional, non-gray radiative heat transfer

A new concept of point mean beam length (PMBL) is introduced. For enclosures with simple geometry, this concept provides a fundamental self-consistent interpretation on the various different definition of the conventional mean beam length. The concept is further demonstrated to be effective in enhancing the computational efficiency for multidimensional radiative heat transfer in non-gray media. In the evaluation of radiative exchange between two perpendicular areas with a common edge, the use of PMBL leads to a factor of 100 to 400 reduction in computational effort compared to the direct integration approach. For practical applications, PMBL is combined with RADNNET (a neural network correlation for a one-dimensional CO2/H2O/soot combustion mixture) to generate two highly efficient and accurate solvers for the evaluation of exchange factors between two parallel or perpendicular rectangular areas of arbitrary dimensions with an intervening combustion mixture.

Solving steady and transient radiative transfer problems with strong inhomogeneity via a lattice Boltzmann method

A complete lattice Boltzmann method (LBM) is developed to solve the steady and transient radiative transfer problems with strong inhomogeneity. The transient radiative transfer equation (RTE) is regarded as a hyperbolic conservation law equation and then solved by the LBM. A forward difference scheme is applied to the time derivative of the source term in the evolution equation, which makes the LBM accurate and stable for radiative transfer problems with strong inhomogeneity. The correctness and accuracy of the proposed LBM is verified first. Afterward, the LBM is applied to solve the one-dimensional radiative transfer problems involving strong inhomogeneous radiative intensity and discontinuous medium system, including a planar slab subjected to Gaussian-shaped source, an infinite slab with an inclusion layer, and a vacuum gapped medium system. The time-resolved radiation is presented and analyzed as well as the steady results. Finally, the present LBM is extended to multidimensional radiative transfer problems.

An Approximation Method for Time-Dependent Problems in High Energy Density Thermal Radiative Transfer

We analyze a computational method for solving multidimensional thermal radiative transfer (TRT) problems based on the radiative transfer equation (RTE) with approximate time evolution operator. The variation in the specific intensity of radiation over each time interval is approximated by an exponential function. This approximation reduces the RTE to an equation of steady-state form with time-dependent coefficients and a modified collision rate density term. This enables one to avoid storage of the high-dimensional solution from the previous time level. The numerical method for TRT problems applies the RTE with the time-dependent approximation as a high-order problem in multilevel system of low-order quasidiffusion (LOQD) equations. The change rate in the intensity is evaluated by the solution of the LOQD equations for the moments of the specific intensity. We study the accuracy of the approximate method in modeling the evolution of temperature and radiation waves in TRT problems.

Impact of ejecta morphology and composition on the electromagnetic signatures of neutron star mergers

The electromagnetic transients accompanying compact binary mergers ($\gamma$-ray bursts, afterglows and 'macronovae') are crucial to pinpoint the sky location of gravitational wave sources. Macronovae are caused by the radioactivity from freshly synthesised heavy elements, e.g. from dynamic ejecta and various types of winds. We study macronova signatures by using multi-dimensional radiative transfer calculations. We employ the radiative transfer code SuperNu and state-of-the-art LTE opacities for a few representative elements from the wind and dynamical ejecta (Cr, Pd, Se, Te, Br, Zr, Sm, Ce, Nd, U) to calculate synthetic light curves and spectra for a range of ejecta morphologies. The radioactive power of the resulting macronova is calculated with the detailed input of decay products. We assess the detection prospects for our most complex models, based on the portion of viewing angles that are sufficiently bright, at different cosmological redshifts ($z$). The brighter emission from the wind is unobscured by the lanthanides (or actinides) in some of the models, permitting non-zero detection probabilities for redshifts up to $z=0.07$. We also find the nuclear mass model and the resulting radioactive heating rate are crucial for the detectability. While for the most pessimistic heating rate (from the FRDM model) no reasonable increase in the ejecta mass or velocity, or wind mass or velocity, can possibly make the light curves agree with the observed nIR excess after GRB130603B, a more optimistic heating rate (from the Duflo-Zuker model) leads to good agreement. We conclude that future reliable macronova observations would constrain nuclear heating rates, and consequently help constrain nuclear mass models.

Multidimensional Models of Type Ia Supernova Nebular Spectra: Strong Emission Lines from Stripped Companion Gas Rule Out Classic Single-degenerate Systems

Abstract The classic single-degenerate model for the progenitors of Type Ia supernova (SN Ia) predicts that the supernova ejecta should be enriched with solar-like abundance material stripped from the companion star. Spectroscopic observations of normal SNe Ia at late times, however, have not resulted in definite detection of hydrogen. In this Letter, we study line formation in SNe Ia at nebular times using non-LTE spectral modeling. We present, for the first time, multidimensional radiative transfer calculations of SNe Ia with stripped material mixed in the ejecta core, based on hydrodynamical simulations of ejecta–companion interaction. We find that interaction models with main-sequence companions produce significant Hα emission at late times, ruling out these types of binaries being viable progenitors of SNe Ia. We also predict significant He i line emission at optical and near-infrared wavelengths for both hydrogen-rich or helium-rich material, providing an additional observational probe of stripped ejecta. We produce models with reduced stripped masses and find a more stringent mass limit of M st ≲ 1 × 10−4 M ⊙ of stripped companion material for SN 2011fe.

Infrared Emission from Kilonovae: The Case of the Nearby Short Hard Burst GRB 160821B

Abstract We present constraints on Ks-band emission from one of the nearest short hard gamma-ray bursts, GRB 160821B, at z = 0.16, at three epochs. We detect a red relativistic afterglow from the jetted emission in the first epoch but do not detect any excess kilonova emission in the second two epochs. We compare upper limits obtained with Keck I/MOSFIRE to multi-dimensional radiative transfer models of kilonovae, that employ composition-dependent nuclear heating and LTE opacities of heavy elements. We discuss eight models that combine toroidal dynamical ejecta and two types of wind and one model with dynamical ejecta only. We also discuss simple, empirical scaling laws of predicted emission as a function of ejecta mass and ejecta velocity. Our limits for GRB 160821B constrain the ejecta mass to be lower than 0.03 M ⊙ for velocities greater than 0.1 c. At the distance sensitivity range of advanced LIGO, similar ground-based observations would be sufficiently sensitive to the full range of predicted model emission including models with only dynamical ejecta. The color evolution of these models shows that I–K color spans 7–16 mag, which suggests that even relatively shallow infrared searches for kilonovae could be as constraining as optical searches.

Solution of multi-dimensional radiative heat transfer in graded index media using the discrete transfer method

The discrete transfer method is extended to solve the radiative transfer equation in two- and three-dimensional semi-transparent media with variable refractive index. The radiative transfer equation and other related equations are organized based on the refracted quantities. A cell-based ray tracing approach is used to track the curve paths of radiative rays in the medium. The complex geometries in multi-dimensional media are simulated by a simple blocked-off strategy. The present method is verified by comparing its results with benchmark solutions, and the performance of the method is examined by some numerical experiments.

The effect of BRDF surface on radiative transfer within a two-dimensional graded index medium

The traditional diffuse or specular surface assumption is insufficient to accurately characterize the reflection characteristics of a practical surface under some circumstances, which could cause distinct deviation in radiative heat transfer analysis and temperature measurements. The anisotropic reflection of a real surface can be well featured by the Bidirectional Reflectance Distribution Function (BRDF). This study aims to solve the radiative heat transfer in a two-dimensional non-uniform refractive index and anisotropic scattering medium coupled with the BRDF surface. The Distribution of Ratios of Energy Scattered by the medium Or Reflected by the boundary surface (DRESOR) method is extended to deal with this issue. The effect of BRDF surface on the radiative heat flux and radiative intensity is investigated varying with optical thickness and scattering albedo of medium by compared to the corresponding diffuse surface. It is found that the more deviation from the diffuse characteristics the BRDF surface has, the larger difference of the radiative heat flux and intensity between the BRDF and diffuse surface there exits. The increased optical thickness and scattering albedo both decrease the difference of radiative flux on the wall between the BRDF and corresponding diffuse surface. And the influence of the reflection of the BRDF boundaries can be alleviated with thick optical thickness or large scattering capacity in the medium. The results demonstrate that the BRDF surface can impose remarkable impact on the radiative heat transfer, and an appropriate BRDF model should be introduced to be coupled with radiative transfer procedure in the accurate analysis of multi-dimensional radiative transfer problem if the boundary possesses the anisotropic reflection properties.

Lattice Boltzmann model for a steady radiative transfer equation.

A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed.

TIME-DEPENDENT MULTI-GROUP MULTI-DIMENSIONAL RELATIVISTIC RADIATIVE TRANSFER CODE BASED ON SPHERICAL HARMONIC DISCRETE ORDINATE METHOD

We develop a time-dependent multi-group multidimensional relativistic radiative transfer code, which is required to numerically investigate radiation from relativistic fluids involved in, e.g., gamma-ray bursts and active galactic nuclei. The code is based on the spherical harmonic discrete ordinate method (SHDOM) that evaluates a source function including anisotropic scattering in spherical harmonics and implicitly solves the static radiative transfer equation with a ray tracing in discrete ordinates. We implement treatments of time dependence, multi-frequency bins, Lorentz transformation, and elastic Thomson and inelastic Compton scattering to the publicly available SHDOM code. Our code adopts a mixed frame approach; the source function is evaluated in the comoving frame whereas the radiative transfer equation is solved in the laboratory frame. This implementation is validated with various test problems and comparisons with results of a relativistic Monte Carlo code. These validations confirm that the code correctly calculates intensity and its evolution in the computational domain. The code enables us to obtain an Eddington tensor that relates first and third moments of intensity (energy density and radiation pressure) and is frequently used as a closure relation in radiation hydrodynamics calculations.

Spatial and angular finite element method for radiative transfer in participating media

A computational approach for the modeling of multi-dimensional radiative transfer in participating media, including scattering, is presented. The approach is based on the sequential use of angular and spatial Finite Element Methods for the discretization of the Radiative Transfer Equation (RTE). The angular discretization is developed with an Angular Finite Element Method (AFEM) based on the Galerkin approach. The AFEM leads to a counterpart of the RTE consisting of a coupled set of transient-advective–reactive equations that are continuously dependent on space and time. The AFEM is ideally suited for so-called h- and/or p-refinement for the discretization of the angular domain: h-refinement is obtained by increasing the number of angular elements and p-refinement by increasing the order of the angular interpolating functions. The spatial discretization of the system of equations obtained after the angular discretization is based on a Variational Multi-Scale Finite Element Method (VMS-FEM) suitable for the solution of generic transport problems. The angularly and spatially discretized system is solved with a second-order accurate implicit predictor multi-corrector time stepper together with a globalized inexact Newton–Krylov nonlinear solver. The overall approach is designed and implemented to allow the seamless inclusion of other governing equations necessary to solve coupled fluid–radiative systems, such as those in combustion, high-temperature chemically reactive, and plasma flow models. The combined AFEM and VMS-FEM for the solution of the RTE is validated with two- and three-dimensional benchmark problems, each solved for 3 levels of angular partitioning (h-refinement) and for 2 orders of angular basis functions (p-refinement), i.e. piecewise constant (P0) and piecewise linear (P1) basis over spherical triangles. The overall approach is also applied to the simulation of radiative transfer in a parabolic concentrator with participating media, as encountered in solar thermochemical applications, for different values of absorption and scattering coefficients, and for different angles of inclination of the incident radiation.