Radiative Transfer Problem Research Articles

Full non-LTE spectral line formation

In the present paper we consider the full nonlocal thermodynamic equilibrium (non-LTE) radiation transfer problem. This formalism allows us to account for deviation from equilibrium distribution of both the radiation field and the massive particles. In the present study, two-level atoms with broadened upper level represent the massive particles. In the absence of velocity-changing collisions, we demonstrate the analytic equivalence of the full non-LTE source function with the corresponding standard non-LTE partial frequency redistribution (PFR) model. We present an iterative method based on operator splitting techniques that can be used to numerically solve the problem at hand. We benchmark it against the standard non-LTE transfer problem for a two-level atom with PFR. We illustrate the deviation of the velocity distribution function of excited atoms from the equilibrium distribution. We also discuss the dependence of the emission profile and the velocity distribution function on elastic collisions and velocity-changing collisions.

Discontinuous Galerkin finite element method for the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes

Context. The study of the continuum radiative transfer problem inside circumstellar envelopes is both a theoretical and numerical challenge, especially in the frequency-dependent and multi-dimensional case. While approximate methods are easier to handle numerically, they often fail to accurately describe the radiation field inside complex geometries. For these cases, it is necessary to directly solve the radiative transfer equation numerically. Aims. We investigate the accuracy of the discontinuous Galerkin finite element method (DGFEM hereafter) applied to the frequency-dependent two-dimensional radiative transfer problem, and coupled with the radiative equilibrium equation. We next used this method in the context of axis-symmetric circumstellar envelopes. Methods. The DGFEM is a variant of finite element methods. It employs discontinuous elements and flux integrals along their boundaries, ensuring local flux conservation. However, as opposed to the classical finite element methods, the solution is discontinuous across element edges. We implemented this approach in a code and tested its accuracy by comparing our results with the benchmarks from the literature. Results. For all the tested cases, the temperatures profiles agree within one percent. Additionally, the emerging spectral energy distributions (SEDs) and images, obtained by ray-tracing techniques from the DGFEM emissivity, agree on average within 5% and 10%, respectively. Conclusions. We show that the DGFEM can accurately describe the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes. Consecutively the emerging SEDs and images are also well reproduced. The DGFEM provides an alternative method (other than Monte-Carlo methods for instance) for solving the radiative transfer equation, and it could be used in cases that are more difficult to handle with the other methods.

Jacobian-free Newton-Krylov method for multilevel nonlocal thermal equilibrium radiative transfer problems

Context. The calculation of the emerging radiation from a model atmosphere requires knowledge of the emissivity and absorption coefficients, which are proportional to the atomic level population densities of the levels involved in each transition. Due to the intricate interdependence of the radiation field and the physical state of the atoms, iterative methods are required in order to calculate the atomic level population densities. A variety of different methods have been proposed to solve this problem, which is known as the nonlocal thermodynamical equilibrium (NLTE) problem. Aims. Our goal is to develop an efficient and rapidly converging method to solve the NLTE problem under the assumption of statistical equilibrium. In particular, we explore whether the Jacobian-Free Newton-Krylov (JFNK) method can be used. This method does not require an explicit construction of the Jacobian matrix because it estimates the new correction with the Krylov-subspace method. Methods. We implemented an NLTE radiative transfer code with overlapping bound-bound and bound-free transitions. This solved the statistical equilibrium equations using a JFNK method, assuming a depth-stratified plane-parallel atmosphere. As a reference, we also implemented the Rybicki & Hummer (1992) method based on linearization and operator splitting. Results. Our tests with the Fontenla, Avrett and Loeser C model atmosphere (FAL-C) and two different six-level Ca II and H I atoms show that the JFNK method can converge faster than our reference case by up to a factor 2. This number is evaluated in terms of the total number of evaluations of the formal solution of the radiative transfer equation for all frequencies and directions. This method can also reach a lower residual error compared to the reference case. Conclusions. The JFNK method we developed poses a new alternative to solving the NLTE problem. Because it is not based on operator splitting with a local approximate operator, it can improve the convergence of the NLTE problem in highly scattering cases. One major advantage of this method is that it is expected to allow for a direct implementation of more complex problems, such as overlapping transitions from different active atoms, charge conservation, or a more efficient treatment of partial redistribution, without having to explicitly linearize the equations.

Characterization of the aerosol contribution to the top-of-atmosphere radiance for satellite ocean color retrievals

A method is proposed for characterization of the aerosol contribution to the top-of-atmosphere (TOA) radiance. The method is based on solving the problem of radiative transfer in the atmosphere-ocean system and expanding the solution in powers of the aerosol optical thickness τ a . We show that the linear term of the expansion is analytically expressed in terms of the bidirectional transmittance/reflectance of the aerosol-free Rayleigh atmosphere. A procedure is also proposed for successively extracting the terms of higher order in τ a from the data of the TOA radiance computation with the DISORT code. As analysis shows, the radiance expansion in τ a is not purely polynomial. Beginning from the quadratic term, the coefficients of the series expansion in powers of τ a become dependent logarithmically on τ a . The approach proposed enables us to reproduce analytically the τ a -dependence of the TOA radiance with controlled accuracy. We determine the expansion coefficients up to the cubic term inclusive and validate our results on the aerosol model embedded in NASA’s SeaDAS algorithm for aerosol loadings, representative for the Barents and Kara seas. In the visible and near-infrared spectral ranges, accounting for the terms up to a quadratic one is found to be sufficient for the atmospheric correction of satellite ocean color data typical for the Arctic region.

X-ray polarisation signatures in bombarded magnetar atmospheres

Abstract Magnetars are neutron stars that host huge, complex magnetic fields which require supporting currents to flow along the closed field lines. This makes magnetar atmospheres different from those of passively cooling neutron stars because of the heat deposited by backflowing charges impinging on the star surface layers. This particle bombardment is expected to imprint the spectral and, even more, the polarisation properties of the emitted thermal radiation. We present solutions for the radiative transfer problem for bombarded plane-parallel atmospheres in the high magnetic field regime. The temperature profile is assumed a priori, and selected in such a way to reflect the varying rate of energy deposition in the slab (from the impinging currents and/or from the cooling crust). We find that thermal X–ray emission powered entirely by the energy released in the atmosphere by the magnetospheric back–bombardment is linearly polarised and X-mode dominated, but its polarisation degree is significantly reduced (down to 10%–50%) when compared with that expected from a standard atmosphere heated only from the cooling crust below. By increasing the fraction of heat flowing in from the crust the polarisation degree of the emergent radiation increases, first at higher energies (∼10 keV) and then in the entire soft X-ray band. We use our models inside a ray-tracing code to derive the expected emission properties as measured by a distant observer and compare our results with recent IXPE observations of magnetar sources.

Impact of the numerical conversion to optical depth on the transfer of polarized radiation

Making the conversion from the geometrical spatial scale to the optical depth spatial scale is useful in obtaining numerical solutions for the radiative transfer equation. This is because it allows for the use of exponential integrators, while enforcing numerical stability. Such a conversion involves the integration of the total opacity of the medium along the considered ray path. This is usually approximated by applying a piecewise quadrature in each spatial cell of the discretized medium. However, a rigorous analysis of this numerical step is lacking. This work is aimed at clearly assessing the performance of different optical depth conversion schemes with respect to the solution of the radiative transfer problem for polarized radiation, out of the local thermodynamic equilibrium. We analyzed different optical depth conversion schemes and their combinations with common formal solvers, both in terms of the rate of convergence as a function of the number of spatial points and the accuracy of the emergent Stokes profiles. The analysis was performed in a 1D semi-empirical model of the solar atmosphere, both in the absence and in the presence of a magnetic field. We solved the transfer problem of polarized radiation in different settings: the continuum, the photospheric Sr i AA modeled under the assumption of complete frequency redistribution, and the chromospheric Ca i AA taking the partial frequency redistribution effects into account during the modeling. High-order conversion schemes generally outperform low-order methods when a sufficiently high number of spatial grid points is considered. In the synthesis of the emergent Stokes profiles, the convergence rate, as a function of the number of spatial points, is impacted by both the conversion scheme and formal solver. The use of low-order conversion schemes significantly reduces the accuracy of high-order formal solvers. In practical applications, the use of low-order optical depth conversion schemes introduces large numerical errors in the formal solution. To fully exploit high-order formal solvers and obtain accurate synthetic emergent Stokes profiles, it is necessary to use high-order optical depth conversion schemes.

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

Partial ionization of plasma in solar prominences.

The plasma in cool cores of solar prominences is partially ionized and its ionization degree, in general, depends on the actual kinetic temperature and density of the plasma and the ions considered. However, we demonstrate that under typical prominence conditions, the dominant mechanism contributing to partial ionization is photoionization by radiation diffusing through the plasma. This mechanism strongly depends on the radiation illuminating prominences from the surrounding solar atmosphere. The understanding of the partial ionization of prominence plasma thus requires a complex solution of the non-LTE radiative-transfer problem. In this paper, we discuss the ionization of hydrogen, helium and some other important species. We also demonstrate the role of non-equilibrium ionization. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.

Observation and Modeling of the Circular Polarization of the Cr i Magnetic-field-induced Transition at 533.03 nm

We study the circular polarization of the magnetic-field-induced transition (MIT) between the 3d 5(6 S)4d 7 D 2 and 3d 5(6 S)4p 7 P 4° states of Cr i at 533.03 nm (wavelength in air). The fractional circular polarization V/I of this spectral line resulting from the solution of the radiation transfer problem in a sunspot model permeated by a homogeneous magnetic field of 3 kG shows amplitudes of about 2%. Spectropolarimetric observations of two sunspots were obtained with the Zurich Imaging Polarimeter-3 at the Istituto ricerche solari Aldo e Cele Daccò observatory in Locarno, Switzerland. The observed V/I profiles show approximately antisymmetrical shapes with an amplitude of around 0.1% and 0.2% for the two sunspots. The center of this profile coincides with the wavelengths predicted for the abovementioned MIT. We apply an inversion code to the spectropolarimetric data of the Cr i-permitted lines at 532.91 and 532.98 nm, as well as to the MIT line at 533.03 nm, to infer a stratification of the emitting atmosphere. We compare the V/I profiles synthesized in the inferred atmosphere models with the observations, showing that the observed signal likely corresponds to the MIT line.

Influence of the greenhouse effect on heat transfer through building envelopes

The evaluation of heat input through fences is an important part of the task of predicting the summer thermal regime of a building under a transparent dome, the interest in the construction of which in the northern regions has increased recently. The calculation of the summer thermal regime of a building under a dome is complicated by taking into account the greenhouse effect, which requires solving the problem of radiative heat transfer in the system: opaque wall - semitransparent screen - environment. It is also necessary to take into account the convection in the dome space. А relatively simple model for calculating the characteristics of heat transfer through the wall of a building with a dome, which takes into account the greenhouse effect, is considered in this paper. Comparison of calculations with experimental data allows us to speak about the adequacy of the proposed model. It is shown that the presence of semitransparent screen due to the greenhouse effect leads to a significant raise in temperature under the dome and an increase in heat input into the building. The influence of ventilation in the dome space on heat access through the enclosing structure for the selected values of external parameters: external temperature and solar radiation was evaluated. The proposed model of heat transfer through the enclosing structure with a semitransparent screen can become part of a more complete model for calculating the thermal regime of a building under a dome.

Adjoint-based shape optimization for radiative transfer in porous structure for volumetric solar receiver

Due to their large surface areas, porous structures are often used for volumetric solar receivers for efficient utilization of the solar energy. However, the precise simulation and optimization of radiative heat transfer inside porous structures are still challenging due to their complex geometries. In the present study, an adjoint-based shape optimization framework for radiative transfer in porous structures is first proposed. Specifically, an arbitrary complex geometry of a porous structure is represented by a level-set function embedded in a Cartesian grid system. The volume penalization method combined with the discrete ordinate method is employed to simulate a forward radiative transfer problem in a porous structure. Then, a cost functional for an adjoint analysis is defined by the volume average of the divergence of the radiative heat flux, which corresponds to the total heat absorbed by the porous structure with a negative sign. The adjoint equations for minimizing the cost functional, and the corresponding shape update formula are derived and implemented in an open-source software, OpenFOAM. The developed adjoint-based shape optimization framework is applied to three types of initial porous structures, i.e., SC Kelvin, BCC Kelvin, and square honeycomb models. For all the initial porous models, the front surface commonly extends towards the direction of the incident radiation, and consequently notable improvements of the absorption efficiency can be achieved through the optimization. In addition, a systematic parametric survey for the initial shape of the SC Kelvin model is conducted to reveal the impacts of the number of unit cells, the solid emissivity, and the solid temperature on the optimal shapes and the resulting absorption efficiencies.

Modeling the scattering polarization in the solar Ca I 4227Å line with angle-dependent PRD effects and bulk velocities

Context. Modeling the scattering polarization signals of strong chromospheric lines requires solving the radiative transfer problem for polarized radiation, out of local thermodynamic equilibrium, taking partial frequency redistribution (PRD) effects into account. This problem is extremely challenging from a computational standpoint and, so far, most studies have been carried out by either modeling PRD effects under the angle-average approximation or by considering academic models of the solar atmosphere. Thanks to a new solution strategy, applicable to atomic systems that allow for a linearization of the problem, accurate solutions can now be routinely obtained in realistic 1D models, taking angle-dependent (AD) PRD effects into account. Aims. This work is aimed at assessing the suitability and performance of this new approach to handling dynamic scenarios. At the same time, it aims to explore the joint impact of magnetic fields and bulk velocities on the scattering polarization profiles of strong resonance lines, accounting for AD PRD effects and considering more realistic atmospheric models than in previous investigations. Methods. By using a two-level atomic model for neutral calcium, we synthesized the intensity and polarization profiles of the Ca I 4227 Å line. Our calculations were performed in 1D atmospheric models, both semi-empirical and extracted from 3D magnetohydrodynamic simulations, including vertical bulk velocities and magnetic fields of arbitrary strength and orientation, both constant and varying with height. Results. We obtained accurate solutions after only a few iterations across all considered scenarios. Even when formulating the problem in the observer’s reference frame, the frequency and angular grids required for accurate results were easily manageable. The calculated profiles showed the expected signatures of bulk velocities: wavelength shifts, enhancement of the line-core polarization amplitude, and prominent asymmetries in the wing signals. The results obtained in atmospheric models with complex thermal, dynamic, and magnetic structures unveiled the broad diversity of features in the emergent radiation that can be expected from realistic scenarios. Conclusions. The presented results assess the suitability of the proposed solution strategy and its parallel implementation, thus supporting its generalization to the 3D case. Our applications in increasingly realistic atmospheric models showed the difficulty related to precisely establishing the individual weight of bulk velocities and magnetic fields in the shape of the emergent profiles. This highlights the need to account for both these physical ingredients to perform reliable inversions of observed scattering polarization profiles.

A Variable Eddington Factor Model for Thermal Radiative Transfer with Closure Based on Data-Driven Shape Function

A new variable Eddington factor (VEF) model is presented for nonlinear problems of thermal radiative transfer (TRT). The VEF model is data-driven and acts on known (a-priori) radiation-diffusion solutions for material temperatures in the TRT problem. A linear auxiliary problem is constructed for the radiative transfer equation (RTE) whose emission source and opacities are evaluated at these known material temperatures. The solution to this RTE approximates the specific intensity distribution in phase-space and time. It is applied as a shape function to define the Eddington tensor for the presented VEF model. The shape function computed via the auxiliary RTE problem will capture some degree of transport effects within the TRT problem. The VEF moment equations closed with this approximate Eddington tensor will thus carry with them these captured transport effects. In this study, the temperature data comes from multigroup P 1 , P 1 / 3 , and flux-limited diffusion radiative transfer models. The proposed VEF model can be interpreted as a transport-corrected diffusion reduced-order model. Numerical results are presented on the Fleck-Cummings test problem which models a supersonic wavefront of radiation. The VEF model is shown to improve accuracy by 1–2 orders of magnitude compared to the considered radiation-diffusion model solutions to the TRT problem.

Numerical solutions to linear transfer problems of polarized radiation

Context. A relevant class of radiative transfer problems for polarized radiation is linear, or can be linearized, and can thus be reframed as linear systems once discretized. In this context, depending on the considered physical models, there are both highly coupled and computationally expensive problems, for which state-of-the-art iterative methods struggle to converge, and lightweight ones, for which solutions can be obtained efficiently. Aims. This work aims to exploit lightweight physical models as preconditioners for iterative solution strategies to obtain accurate and fast solutions for more complex problems. Methods. We considered a highly coupled linear transfer problem for polarized radiation, which we solved iteratively using a matrix-free generalized minimal residual (GMRES) method. Different preconditioners and initial guesses, designed in a physics-based framework, are proposed and analyzed. The action of preconditioners was also computed by applying GMRES. The overall approach thus consists of two nested GMRES iterations, one for the original problem and one for its lightweight version. As a benchmark, we considered the modeling of the intensity and polarization of the solar Ca I 4227 Å line, the Sr II 4077 Å line, and the Mg II h&k lines in a semi-empirical 1D atmospheric model, accounting for partial frequency redistribution effects in scattering processes and considering a general angle-dependent treatment. Results. Numerical experiments show that using tailored preconditioners based on simplified models of the considered problem has a noticeable impact, reducing the number of iterations to convergence by a factor of 10–20. Conclusions. By designing efficient preconditioners in a physics-based context, it is possible to significantly improve the convergence of iterative processes, obtaining fast and accurate numerical solutions to the considered problems. The presented approach is general, requiring only the selection of an appropriate lightweight model, and can be applied to a larger class of radiative transfer problems in combination with arbitrary iterative procedures.

SNOWTRAN: A Fast Radiative Transfer Model for Polar Hyperspectral Remote Sensing Applications

In this work, we develop a software suite for studies of atmosphere–underlying SNOW-spaceborne optical receiver light TRANsmission calculations (SNOWTRAN) with applications for the solution of forward and inverse radiative transfer problems in polar regions. Assuming that the aerosol load is extremely low, the proposed theory does not require the numerical procedures for the solution of the radiative transfer equation and is based on analytical equations for the spectral nadir reflectance and simple approximations for the local optical properties of atmosphere and snow. The developed model is validated using EnMAP and PRISMA spaceborne imaging spectroscopy data close to the Concordia research station in Antarctica. A new, fast technique for the determination of the snow grain size and assessment of the snowpack vertical inhomogeneity is then proposed and further demonstrated on EnMAP imagery over the Aviator Glacier and in the vicinity of the Concordia research station in Antarctica. The results revealed a large increase in precipitable water vapor at the Concordia research station in February 2023 that was linked to a warming event and a four times larger grain size at Aviator Glacier compared with Dome C.

Unified gas-kinetic particle method for frequency-dependent radiation transport

This paper proposes a unified gas-kinetic particle (UGKP) method for the frequency-dependent photon transport process. The photon transport is a typical multiscale process governed by the nonlinear radiative transfer equations (RTE). The flow regime of photon transport varies from the ballistic regime to the diffusive regime with respect to optical depth and photon frequency. The UGKP method is an asymptotic preserving (AP) scheme and a regime adaptive scheme. The teleportation error is significantly reduced, and the computational efficiency is remarkably improved in the diffusive regime. Distinguished from the standard multigroup treatment, the proposed UGKP method solves the frequency space in a non-discretized way. Therefore the Rosseland diffusion system can be precisely preserved in the optically thick regime. Taking advantage of the local integral solution of RTE, the distribution of the emitted photon can be constructed from its macroscopic moments. The Monte Carlo particles in the UGKP method need only be tracked before their first collision events, and a re-sampling process is performed to close the photon distribution for each time step. The high computational cost of calculating excessive scattering events can be saved, especially in the optically thick regime. The proposed UGKP method is implicit and removes the light speed constraint on the time step. The particle tracking approach combining the implicit formulation makes the proposed UGKP method an efficient solution algorithm for frequency-dependent radiative transfer problems. We demonstrate with numerical examples the capability of the proposed multi-frequency UGKP method.

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.

Iterative space-angle discontinuous Galerkin method for radiative transfer equation

The radiative transfer equation (RTE) is an integro-differential equation that describes the radiation energy absorbing, emitting, and scattering in both space and angle, which can be up to five-dimensional problems. It is difficult for a RTE solver to satisfy both accuracy and efficiency (less computational resources) for such high dimensional problems. In this paper, an iterative solver for one-dimensional cylindrical radiative transfer problems using the space-angle discontinuous Galerkin (DG) method is developed to achieve both accuracy and efficiency. The iterative solver is based on the angular decomposition (AD) scheme, which slices the spatial-angular domain into slabs and decouples the angular integration between slabs. Both Jacobi and successive over-relaxation (SOR) iterative schemes are investigated by numerical analysis and examples. The comparison of the two iterative schemes suggests that an appropriate relaxation factor for the SOR method may accelerate the convergence. Finally, the iterative scheme is more efficient than the direct solution of the system both in terms of memory usage and computational time, especially for finer meshes.

Assessment of the CRD approximation for the observer’s frame RIII redistribution matrix

Context. Approximated forms of the RII and RIII redistribution matrices are frequently applied to simplify the numerical solution of the radiative transfer problem for polarized radiation, taking partial frequency redistribution (PRD) effects into account. A widely used approximation for RIII is to consider its expression under the assumption of complete frequency redistribution (CRD) in the observer’s frame (RIII−CRD). The adequacy of this approximation for modeling the intensity profiles has been firmly established. By contrast, its suitability for modeling scattering polarization signals has only been analyzed in a few studies, considering simplified settings. Aims. In this work, we aim to quantitatively assess the impact and the range of validity of the RIII−CRD approximation in the modeling of scattering polarization. Methods. We first present an analytic comparison between RIII and RIII−CRD. We then compare the results of radiative transfer calculations, out of local thermodynamic equilibrium, performed with RIII and RIII−CRD in realistic one-dimensional atmospheric models. We focus on the chromospheric Ca I line at 4227 Å and on the photospheric Sr I line at 4607 Å. Results. The RIII−CRD approximation provides accurate results for the Ca I 4227 Å line. Only when velocities are included can some appreciable discrepancies be found, especially for lines of sight close to the disk center. The approximation performs well also for the Sr I 4607 Å line, especially in the absence of magnetic fields or when a micro-turbulent field is included. However, some appreciable errors appear when deterministic magnetic fields or bulk velocities are considered. Conclusions. Our results show that the RIII−CRD approximation is suited for the PRD modeling of the scattering polarization signals of strong chromospheric lines, both in the core and wings. With a few minor exceptions, this approximation is also suitable for photospheric lines, although PRD effects generally play a minor role in their modeling.

Spectro-spatial evolution of the CMB. Part I. Discretisation of the thermalisation Green's function

Spectral distortions of the cosmic microwave background (CMB) have been recognized as an important future probe of the early Universe. Existing theoretical studies primarily focused on describing the evolution and creation of average distortions, ignoring spatial perturbations in the plasma. One of the main reasons for this choice is that a treatment of the spectro-spatial evolution of the photon field deep into the primordial Universe requires solving a radiative transfer problem for the distortion signals, which in full detail is computationally challenging. Here we provide the first crucial step towards tackling this problem by formulating a new spectral discretisation of the underlying average thermalisation Green's function. Our approach allows us to convert the high-dimensional partial differential equation system (≃ 103–104 equations) into and set of ordinary differential equations of much lower dimension (≃ 10 equations). We demonstrate the precision of the approach and highlight how it may be further improved in the future. We also clarify the link of the observable spectral distortion parameters (e.g., μ and y) to the computational spectral basis that we use in our frequency discretisation. This reveals how several basis-dependent ambiguities can be interpreted in future CMB analysis. Even if not exact, the new Green's function discretisation can be used to formulate a generalised photon Boltzmann-hierarchy, which can then be solved with methods that are familiar from theoretical studies of the CMB temperature and polarisation anisotropies. We will carry this program out in a series of companion papers, thereby opening the path to full spectro-spatial exploration of the CMB with future CMB imagers and spectrometers.