Accelerated Lambda Iteration Method Research Articles

Polarized resonance line transfer in a spherically symmetric medium with angle-dependent partial frequency redistribution

ABSTRACT In a stellar atmosphere, the resonance line polarization arises from scattering of limb-darkened radiation field by atoms. This spectral line polarization gets affected particularly in the wings, when the line photons suffer scattering on electrons in thermal motion. Scattering of line photons by atoms and electrons are, respectively, described by the atomic and Thomson electron scattering redistribution functions, which in general depend on both the frequencies and directions of incident and scattered photons. In this paper, we consider the polarized spectral line formation in spherically symmetric extended and expanding media accounting for the angle-dependent partial frequency redistribution (AD-PRD) in scattering on both atoms and electrons. We solve this computationally demanding polarized transfer problem using an accelerated lambda iteration method and a method based on orders of scattering approach. In the case of expanding spherical medium, the concerned transfer problem is solved in the comoving frame. Because of the computational limitations, we consider optically thin isothermal spherically symmetric media of different extensions for the static case as well as when the velocity fields are present. For the considered model, we show that the AD-PRD effects on the linear polarization profiles are significant and have to be accounted for.

Atomic line radiative transfer with MCFOST

Aims. We present MCFOST-art, a new non-local thermodynamic equilibrium radiative transfer solver for multilevel atomic systems. The code is embedded in the 3D radiative transfer code MCFOST and is compatible with most of the MCFOST modules. The code is versatile and designed to model the close environment of stars in 3D. Methods. The code solves for the statistical equilibrium and radiative transfer equations using the Multilevel Accelerated Lambda Iteration method. We tested MCFOST-art on spherically symmetric models of stellar photospheres as well as on a standard model of the solar atmosphere. We computed atomic level populations and outgoing fluxes and compared these values with the results of the TURBOspectrum and RH codes. Calculations including expansion and rotation of the atmosphere were also performed. We tested both the pure local thermodynamic equilibrium and the out-of-equilibrium problems. Results. In all cases, the results from all codes agree within a few percent at all wavelengths and reach the sub-percent level between RH and MCFOST-art. We still note a few marginal discrepancies between MCFOST-art and TURBOspectrum as a result of different treatments of background opacities at some critical wavelength ranges.

LOC program for line radiative transfer

Context. Radiative transfer (RT) modelling is part of many astrophysical simulations. It is used to make synthetic observations and to assist the analysis of observations. We concentrate on modelling the radio lines emitted by the interstellar medium. In connection with high-resolution models, this can be a significant computationally challenge. Aims. Our aim is to provide a line RT program that makes good use of multi-core central processing units (CPUs) and graphics processing units (GPUs). Parallelisation is essential to speed up computations and to enable large modelling tasks with personal computers. Methods. The program LOC is based on ray-tracing (i.e. not Monte Carlo) and uses standard accelerated lambda iteration methods for faster convergence. The program works on 1D and 3D grids. The 1D version makes use of symmetries to speed up the RT calculations. The 3D version works with octree grids, and to enable calculations with large models, is optimised for low memory usage. Results. Tests show that LOC results agree with other RT codes to within ∼2%. This is typical of code-to-code differences, which are often related to different interpretations of the model set-up. LOC run times compare favourably especially with those of Monte Carlo codes. In 1D tests, LOC runs were faster by up to a factor ∼20 on a GPU than on a single CPU core. In spite of the complex path calculations, a speed-up of up to ∼10 was also observed for 3D models using octree discretisation. GPUs enable calculations of models with hundreds of millions of cells, as are encountered in the context of large-scale simulations of interstellar clouds. Conclusions. LOC shows good performance and accuracy and is able to handle many RT modelling tasks on personal computers. It is written in Python, with only the computing-intensive parts implemented as compiled OpenCL kernels. It can therefore also a serve as a platform for further experimentation with alternative RT implementation details.

Water isotopologues in the circumstellar envelopes of M-type AGB stars

AIM: In this study we examine rotational emission lines of two isotopologues of water: H$_2$$^{17}$O and H$_2$$^{18}$O. By determining the abundances of these molecules, we aim to use the derived isotopologue --- and hence oxygen isotope --- ratios to put constraints on the masses of a sample of M-type AGB stars that have not been classified as OH/IR stars. METHODS: We use detailed radiative transfer analysis based on the accelerated lambda iteration method to model the circumstellar molecular line emission of H$_2$$^{17}$O and H$_2$$^{18}$O for IK Tau, R Dor, W Hya, and R Cas. The emission lines used to constrain our models come from Herschel/HIFI and Herschel/PACS observations and are all optically thick, meaning that full radiative transfer analysis is the only viable method of estimating molecular abundance ratios. RESULTS: We find generally low values of the $^{17}$O/$^{18}$O ratio for our sample, ranging from 0.15 to 0.69. This correlates with relatively low initial masses, in the range $\sim1.0$ to 1.5 M$_\odot$ for each source, based on stellar evolutionary models. We also find ortho-to-para ratios close to 3, which are expected from warm formation predictions. CONCLUSIONS: The $^{17}$O/$^{18}$O ratios found for this sample are at the lower end of the range predicted by stellar evolutionary models, indicating that the sample chosen had relatively low initial masses.

POLARIZED LINE FORMATION IN NON-MONOTONIC VELOCITY FIELDS

ABSTRACT For a correct interpretation of the observed spectro-polarimetric data from astrophysical objects such as the Sun, it is necessary to solve the polarized line transfer problems taking into account a realistic temperature structure, the dynamical state of the atmosphere, a realistic scattering mechanism (namely, the partial frequency redistribution—PRD), and the magnetic fields. In a recent paper, we studied the effects of monotonic vertical velocity fields on linearly polarized line profiles formed in isothermal atmospheres with and without magnetic fields. However, in general the velocity fields that prevail in dynamical atmospheres of astrophysical objects are non-monotonic. Stellar atmospheres with shocks, multi-component supernova atmospheres, and various kinds of wave motions in solar and stellar atmospheres are examples of non-monotonic velocity fields. Here we present studies on the effect of non-relativistic non-monotonic vertical velocity fields on the linearly polarized line profiles formed in semi-empirical atmospheres. We consider a two-level atom model and PRD scattering mechanism. We solve the polarized transfer equation in the comoving frame (CMF) of the fluid using a polarized accelerated lambda iteration method that has been appropriately modified for the problem at hand. We present numerical tests to validate the CMF method and also discuss the accuracy and numerical instabilities associated with it.

Sulphur molecules in the circumstellar envelopes of M-type AGB stars

The sulphur compounds SO and SO$_2$ have not been widely studied in the circumstellar envelopes of asymptotic giant branch (AGB) stars. By presenting and modelling a large number of SO and SO$_2$ lines in the low mass-loss rate M-type AGB star R Dor, and modelling the available lines of those molecules in a further four M-type AGB stars, we aim to determine their circumstellar abundances and distributions. We use a detailed radiative transfer analysis based on the accelerated lambda iteration method to model circumstellar SO and SO$_2$ line emission and molecular data files for both SO and SO$_2$ that are more extensive than those previously available. Using 17 SO lines and 98 SO2 lines to constrain our models for R Dor, we find an SO abundance of 6.7x10$^{-6}$ and an SO$_2$ abundance of 5x10$^{-6}$ with both species having high abundances close to the star. We also modelled $^{34}$SO and found an abundance of 3.1x10$^{-7}$, giving an $^{32}$SO/$^{34}$SO ratio of 21.6. We derive similar results for the circumstellar SO and SO$_2$ abundances and their distributions for the low mass-loss rate object W Hya. For these stars, the circumstellar SO and SO$_2$ abundances are much higher than predicted by chemical models and these two species may account for all available sulphur. For the higher mass-loss rate stars, we find shell-like SO distributions with peak abundances that decrease and peak abundance radii that increase with increasing mass-loss rate. The positions of the peak SO abundance agree very well with the photodissociation radii of H$_2$O. We find evidence that SO is most likely through the photodissociation of H$_2$O and the subsequent reaction between S and OH. The S-bearing parent molecule appears not to be H$_2$S. The SO$_2$ models suggest an origin close to the star for this species, also disagreeing with current chemical models.

GAUSS-SEIDEL AND SUCCESSIVE OVERRELAXATION METHODS FOR RADIATIVE TRANSFER WITH PARTIAL FREQUENCY REDISTRIBUTION

The linearly-polarized solar limb spectrum that is produced by scattering processes contains a wealth of information on the physical conditions and magnetic fields of the solar outer atmosphere, but the modeling of many of its strongest spectral lines requires solving an involved non-LTE radiative transfer problem accounting for partial redistribution (PRD) effects. Fast radiative transfer methods for the numerical solution of PRD problems are also needed for a proper treatment of hydrogen lines when aiming at realistic time-dependent magnetohydrodynamic simulations of the solar chromosphere. Here we show how the two-level atom PRD problem with and without polarization can be solved accurately and efficiently via the application of highly convergent iterative schemes based on the Gauss-Seidel (GS) and Successive Overrelaxation (SOR) radiative transfer methods that had been previously developed for the complete redistribution (CRD) case. Of particular interest is the Symmetric SOR method, which allows us to reach the fully converged solution with an order of magnitude of improvement in the total computational time with respect to the Jacobi-based local ALI (Accelerated Lambda Iteration) method.

Radiative Transfer Methods: new exact results for testing the accuracy of the ALI numerical method for a stellar atmosphere

Tests are presented of the 1D Accelerated Lambda Iteration method, which is widely used for solving the radiative transfer equation for a stellar atmosphere. We use our ARTY code as a reference solution and tables for these tests are provided. We model a static idealized stellar atmosphere, which is illuminated on its inner face and where internal sources are distributed with weak or strong gradients. This is an extension of published tests for a slab without incident radiation and gradients. Typical physical conditions for the continuum radiation and spectral lines are used, as well as typical values for the numerical parameters in order to reach a 1% accuracy. It is shown that the method is able to reach such an accuracy for most cases but the spatial discretization has to be refined for strong gradients and spectral lines, beyond the scope of realistic stellar atmospheres models. Discussion is provided on faster methods.

A new type of photoionized code required for the new era of X-ray spectroscopy

With the advent of the present and future spatial X-ray missions, it becomes crucial to model correctly the line spectrum of X-ray emitting media. We have built a photoionization code, Titan, solving the transfer of a thousand lines and of the continuum with the Accelerated Lambda Iteration method, which is most reliable for line transfer. In all other photoionization codes the line intensities are computed with the so-called escape probability formalism, used in its simplest approximation. In a previous paper (Dumont et al. 2003), it was shown that this approximation leads to errors which can exceed one order of magnitude in the case a Thomson thickness of the order of unity. We show here that it also happens in the case of a Thomson thickness of 0.001 to 0.1. The errors on the line fluxes and line ratios are of the order of 30% for a column density of 10^{20} cm^{-2}, and a factor five for a column density of 10^{23} cm^{-2}, in conditions appropriate for Seyfert 2 and for the Warm Absorber of Seyfert 1.

Multilevel Line Transfer with the Implicit Integral Method

Once the need for an iterative procedure in order to solve the problem of the formation of spectral lines in the case of a model atom with many energy levels, the sequel is to seek for the most effective form of such an iterative scheme. It is an almost universal is assumed within all the iterative methods for the solution of those radiative transfer problems, in which the transfer equations are coupled to the state of the matter, to take as the input of each step of iterations the values of the opacity coefficients obtained as a result of the previous one. This is, for instance, the procedure used to correct the temperature in the computation of stellar atmosphere models, or that to build the Λ -operator (either the exact or the approximated one) within the Accelerated Lambda Iteration methods. Yet, if we assume, in order to solve the multilevel line transfer problem, that at each step of iterations the line opacities are known, we can express via the statistical equilibrium equations the populations of the energy levels - and consequently the source functions of the relevant spectral lines - as a linear function of the full set of the corresponding mean intensities of the radiation field. Once such linear forms for the source functions, we are able to solve without the need of any further approximation the radiative transfer equations for are obtained lines, now linearly coupled through the above linear forms of the statistical equilibrium equations. This is achieved by means of the Implicit Integral Method that we already presented in a series of previous papers.

Dust radiation fields and pumping of excited state OH masers in W3(OH)

I show that it is quite possible to obtain saturating amplification in the 13.44-GHz maser which is so far unique to W3(OH), using a combined radiative and collisional pumping scheme. The dominant radiative part of the pump involves far-infrared line overlap, and the far-infrared continuum is provided by dust, modelled as either a two-component mixture or composite grains. Transport of the far-infrared radiation is carried out via the accelerated lambda iteration method. The observational link between 13.44-GHz and 6035-MHz masers is reproduced by the model. Inadequate amplification in Sobolev models probably results from the optical depth limitations imposed by this approximation. I review the dust models used in far-infrared pumping models of OH masers, and conclude that the main consequence of moving from skewed blackbody functions to more sophisticated models is a selective pumping enhancement when ice mantles are included.

Atomic data for non-LTE model stellar atmospheres

We briefly review the state-of-the-art in non-LTE modeling of hot star atmospheres. During the last decade, the development of a new class of radiation transfer methods, based on the Accelerated Lambda Iteration method, have brought considerable progress in this field. Severe limitations of the size of model atoms used in non-LTE model atmospheres have been removed, so that realistic model atoms can now be considered. In recent years, the problem of non-LTE line blanketing, which aims to include the influence of thousands to millions of lines on the atmospheric structure, has been attacked, and first non-LTE line-blanketed models have been calculated. We illustrate the importance of non-LTE line blanketing with the example of a hot subdwarf, BD +75° 325.As a consequence of these developments, there is a need for extensive radiative and collisional atomic databases to build non-LTE models. Recent projects, like the Opacity Project, provide a substantial quantity of the necessary data. We discuss the current needs of atomic data for the purpose of building reliable non-LTE line-blanketed model atmospheres. A critical evaluation of the quality of such extensive sets of atomic data is essential. Finally, we present a new, IDL-based, graphic tool intended to facilitate the manipulation of large amounts of data needed to build the realistic model atoms that we use.

The solution of radiative transfer problems in molecular bands without the LTE assumption by accelerated lambda iteration methods

An iterative method based on the use of approximate transfer (or Λ) operators, which was designed initially to solve multilevel NLTE line formation problems in stellar atmospheres, is adapted and applied to the solution of the NLTE molecular band radiative transfer in planetary atmospheres. The matrices to be constructed and inverted are much smaller than those used in the traditional Curtis matrix technique, which makes possible the treatment of more realistic problems (including rotational NLTE, overlapping of lines in the bands and overlapping of bands with continuua) using relatively small computers. This technique converges much more rapidly than straightforward iteration between the transfer equation and the equations of statistical equilibrium (Λ-iteration). A test application of this new technique to the solution of NLTE radiative transfer problems for optically-thick and thin bands (the 4.3 μm CO 2 band in the Venusian atmosphere and the 4.7 and 2.3 μm CO bands in the Earth's atmosphere) is described. The necessity for careful treatment of rotational NLTE in weak lines of the 4.3 μm CO 2 fundamental band is shown. The possible influence of neglecting or treating approximately the CO 2 10 μm laser transitions and also effects of the overlap of the 4.3 μm CO 2 with the continuum are investigated. The rotational NLTE effects in the 4.7 μm CO band in the upper Earth's atmosphere are estimated.