B. Baschek

DIFFERENTIALLY MOVING MEDIA WITH MANY SPECTRAL LINES: STOCHASTIC APPROACH

Abstract Based upon the analytical solution of the radiative transfer equation for a given source function and a new approach to account for very many spectral lines contributing to the extinction, the connection between line properties and the emergent intensity is derived under the assumption that the wavelengths of the line centers follow a Poisson point process, whereas the other line parameters may have arbitrary distribution functions. A comparison with the widely used list of Kurucz shows that the Poisson distribution well describes deterministic “real” lines. The presentation by a Poisson point process requires only a modest number of parameters and is very flexible. It allows most operations to be carried out analytically and hence is very suitable to study the intricate influence of many lines on radiation fields in differentially moving media. We consider a simplified case of the solution of the radiative transfer equation in order to demonstrate the basic effects of the velocity field upon the emerging radiation field. Expressions for the expectation value of the intensity are derived, and examples are given for Lorentz line profiles and infinitely sharp lines, in particular as functions of the velocity gradient and the mean line density.

The diffusion of radiation in moving media - III. Stochastic representation of spectral lines

In this paper we present analytical expressions for the radiative flux, the effective extinction coefficient, and the radiative acceleration deep inside a differentially moving, very optically thick medium with many spectral lines. It is shown that the line contribution is essentially given by the characteristic function of wavelength averages of the extinction coefficient. It can be calculated either by means of the generalized opacity distribution function or by means of a Poisson point process model. Several examples are given to demonstrate the basic consequences of line densities, widths, and strengths, as well as velocity gradients on the diffusive flux. We demonstrate that for Poisson distributed lines the diffusive flux decreases with increasing absolute values of the velocity gradient (i.e. that the flux has a maximum for static media), and that such lines have hardly a direct influence on the radiative acceleration.

The diffusion of radiation in moving media - IV. Flux vector, effective opacity, and expansion opacity

For a given velocity and temperature field in a differentially moving 3D medium, the vector of the radiative flux is derived in the diffusion approximation. Due to the dependence of the velocity gradient on the direction, the associated effective opacity in general is a tensor. In the limit of small velocity gradients analytical expression are obtained which allow us to discuss the cases when the direction of the flux vector deviates from that of the temperature gradient. Furthermore the radiative flux is calculated for infinitely sharp, Poisson distributed spectral lines resulting in simple expressions that provide basic insight into the effect of the motions. In particular, it is shown how incomplete line lists affect the radiative flux as a function of the velocity gradient. Finally, the connection between our formalism and the concept of the expansion opacity introduced by Karp et al. (1977) is discussed.

Markov Chain Monte Carlo solutions for radiative transfer problems

Algorithms for the solution of the radiative transfer equation that are simultaneously accurate, fast, and easy to implement are still highly desirable, in particular for multidimensional media. We present a stochastic algorithm that solves the transfer equation by assuming that the transfer of radiation can be modelled as a Markov process. It is a generalization of the classical Monte Carlo method and can be applied to the solution of the Milne equation. known subject in the theory of Markov processes (see Meyer 1966). The classical potential given by the potential equation ΔΦ = −4πρ (cf. Eq. (6)) is the potential of the Brownian mo- tion. In Sect. 4, we show that Beers law and the radiative transfer equation can be derived by means of simple probabilistic argu- ments and the corresponding difference equation may be solved very simply by a Markov Chain Monte Carlo method. We also demonstrate how various forms of the transfer equation (differ- ential, difference, or stochastic equation) and the corresponding solutions transform into each other. In Sect. 5, the probabilis- tic interpretation of the discretized Feautrier equation is derived and discussed. To highlight the wide range of application of the Markov Chain Monte Carlo algorithm, the Milne equation is shown in Sect. 6 to be the potential equation of a Markov pro- cess. We note, that Milnes equation is defined in three dimen- sional space, so it is much easier to treat than the usual radiative transfer equation in five dimensional space (for fixed time and frequency variable). A general scheme for the numerical appli- cation is given. In Sect. 7, the numerical properties of MCMC are exemplified by means of several 1D and 3D configurations. We end with a discussion of the advantages and disadvantages of MCMC compared to the classical Monte Carlo approach and grid methods.