Rainer Wehrse

M dwarf spectra from 0.6 to 1.5 micron - A spectral sequence, model atmosphere fitting, and the temperature scale

Red/infrared (0.6-1.5 micron) spectra are presented for a sequence of well-studied M dwarfs ranging from M2 through M9. A variety of temperature-sensitive features useful for spectral classification are identified. Using these features, the spectral data are compared to recent theoretical models, from which a temperature scale is assigned. The red portion of the model spectra provide reasonably good fits for dwarfs earlier than M6. For layer types, the infrared region provides a more reliable fit to the observations. In each case, the wavelength region used includes the broad peak of the energy distribution. For a given spectral type, the derived temperature sequence assigns higher temperatures than have earlier studies - the difference becoming more pronounced at lower luminosities. The positions of M dwarfs on the H-R diagram are, as a result, in closer agreement with theoretical tracks of the lower main sequence.

An efficient method for the solution of 3-D radiative transfer problems

Abstract A novel efficient method for the solution of radiative transfer problems in media of two or three dimensions is presented. The approach is based on an implicit discretization of the transfer equation in Cartesian frames. The resulting linear system of equations is solved by a combination of recursion and iteration. Since the explicit handling of large matrices is avoided, the corresponding code is relatively simple and runs well on all types of modern computers. As examples of applications, we consider several spherical and non-spherical dust clouds heated by a central source and derive their temperature distributions and emergent spectra. Comparison of our spherical models with those of Rowan-Robinson shows good agreement.

Numerical methods in multidimensional radiative transfer

This paper presents a finite element method for solving the resonance line transfer problem in moving media. The algorithm is capable of dealing with three spatial dimensions, using hierarchically structured grids which are locally refined by means of duality-based a posteriori error estimates. Application of the method to coherent isotropic scattering and complete redistribution gives a result of matrix structure which is discussed in the paper. The solution is obtained by way of an iterative procedure, which solves a succession of quasi-monochromatic radiative transfer problems. It is therefore immediately evident that any simulation of the extended frequency-dependent model requires a solution strategy for the elementary monochromatic transfer problem, which is fast as well as accurate. The present implementation is applicable to arbitrary model configurations with optical depths up to 103–104. Additionally, a combination of a discontinuous finite element method with a superior preconditioning method is presented, which is designed to overcome the extremely poor convergence properties of the linear solver for optically thick and highly scattering media. The contents of this article is as follows: Introduction Overview: numerical methods Monochromatic 3D radiative transfer Polychromatic 3D line transfer Test calculations Applications Multi-model preconditioning Conclusion

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.

Analytical solution of the non-discretized radiative transfer equation for a slab of finite optical depth

Abstract An analytical solution of the radiative transfer equation for a plane parallel slab of finite depth is presented without requiring discretization in space or angles. The solution, expressed in terms of the matrix hyperbolic tangent function, avoids the problem of exponentially increasing terms so that it is also suitable for numerical calculations. By using the resolvent of the transfer equation we obtain simple series expansions for the matrix elements.

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.

Mathematical aspects of the plane-parallel transfer equation

Abstract The modelling of high-quality spectra from laboratory experiments or astronomical observations often requires the consideration, for example, of media with strong density inhomogeneities that can be treated only statistically, of complicated redistribution functions and/or of very many lines. Since the numerical methods available for solution of the corresponding radiative transfer equations are not efficient enough to deal with such situations, basically new methods have to be deviced. In order to provide a sound foundation for such algorithms, the relevant mathematical properties of the radiative transfer equation for static, plane-parallel media with coherent, isotropic scattering are investigated in this paper. The starting point is the solution in terms of the matrix tangent hyperbolic function. It is shown that this operator is the sum of a diagonal operator plus the difference between two positive operators of finite trace, which have the completely unexpected property that the highest eigenvalue practically coincides with the trace so that, in a zeroth approximation, the other eigenvalues can be neglected and the operators can be represented in the form λ max ¦V max 〉〈V max ¦ . This observation allows an explicit approximate solution. By means of a non-local representation of the transfer equation, we deduce that the equation has a variational principle and we prove that the outgoing intensities are positive whenever the ingoing intensities and the de-excitation parameter are positive, as is required by physics.

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.

Radiative Transfer with Many Spectral Lines

The radiative transfer equation, a partial integro-differential equation, is of particular interest for astronomers since it links the spectral properties of the light received (e.g. on Earth) with the properties of the matter from the place of origin (e.g. a star) to the place of the observer.