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This review describes advances in radiative transfer theory since about 1985. We stress fundamental aspects and emphasize modern methods for the numerical solution of the transfer equation for spatially multidimensional problems, for both unpolarized and polarized radiation. We restrict the discussion to two-level atoms with noninverted populations for given temperature, density and velocity fields.

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. ([CITE]) is discussed.

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 use of the conventional opacity distribution function (ODF) to deal with very many spectral lines is restricted to static media. In this paper, its generalization to differentially moving media is derived from the analytical solution of the comoving-frame radiative transfer equation. This generalized ODF depends on only two parameters, on the wavelength position (as in the static case) and in addition on a wavelength interval Δ over which the line extinction is averaged. We present two methods for the calculation of the generalized ODF: (i) in analogy to the static case, it is derived from the mean values of the extinction coefficients over wavelength intervals Δ , (ii) it is calculated under the assumption that the lines follow a Poisson point process. Both approaches comprise the conventional static case as the limit of vanishing velocities, i.e. of $\Delta\to 0$. The averages of the extinction for all values of Δ contain the necessary information about the Doppler shifts and about the correlations between the extinction at different wavelengths. The flexible statistical approximation of the lines by a Poisson point process as an alternative to calculating the averages over all Δ from a deterministic "real"spectral line list, has the advantage that the number of parameters is reduced, that analytical expressions allow a better insight into the effects of the lines on the radiative transfer, and that the ODFs and their corresponding characteristic functions can be calculated efficiently by (fast) Fourier transforms. Numerical examples demonstrate the effects of the relevant parameters on the opacity distribution functions, in particular that with increasing line density and increasing Δ the ODF becomes narrower and its maximum is shifted to larger extinction values.

The analytical solution of the plane-parallel radiative transfer equation is obtained in the two-stream approximation for a large class of continuous distributions of the de-excitation coecient " (constant, linear, parabolic, with spikes etc.). We present also the method of the discrete space theory for obtaining solutions of the transfer equation in the media with strong density inhomogeneities. These sets of the analytical solutions can be used for the solution of the inverse problem. The deduction of the internal distribution of" from observational data is facilitated in the case of isothermal media, since the characteristic behavior of the solution refers to the certain behavior of ". As an example, we nd the corresponding parameters of the constant and linear distributions of " precisely.

ABSTRA C T We present calculations of the thermal structure of accretion funnels heated by hard X-ray emission from the accretion shock, soft X-ray reprocessed radiation from the white dwarf surface and magnetic heating in the matter‐field interaction region close to the orbital plane. The calculations determine self-consistently the thermal structure of the funnel, allowing for radiative transfer, electron scattering and the trapping of radiation within the funnel, and represent a substantial improvement on previous purely kinematical models. We show that, while models that allow only for X-ray heating can explain the observed intensities of the Balmer lines, they cannot, at the same time, explain the intensities of the He i and He ii lines in the optical spectra. These lines appear to be formed mainly in the magnetically heated transition region near the orbital plane, with this region playing a role similar to the hotspot in accretion discs. We show that, with the inclusion of this region, models can be constructed that are in close agreement with the optical line and continuum emission observed in AM Herculis systems: that is, they exhibit a flat or inverted Balmer decrement, He i lines, a strong He ii l 4686 line and complex emission-line profiles which vary dramatically in velocity and shape over the orbital period of the white dwarf. We also show that the continuum emission from the accretion funnel provides an important source of unpolarized background radiation, which reduces the degree of polarization of the cyclotron radiation from the accretion shocks, and produces the polarization standstills that are a well-known characteristic of these systems.

Significant progress has recently been achieved in the solution of the radiative transfer equation for various geometries, velocities, scattering modes etc. In particular, it has been become possible for the first time to derive an orthogonal system of functions that solves the transfer equation for plane media, to model numerically radiation fields in arbitrarily shaped 3D configurations with a prescribed accuracy, and to obtain general quadrature solutions for spheres with relativistic velocities. The latter are of special importance since they lead to accurate and convenient expressions for the diffusion limit in moving media. In this contribution the new techniques are reviewed and the consequences for our understanding of radiation fields briefly discussed.

Analytical solutions of the radiative transfer equation recently derived for differentially moving media allow the efficient inclusion of an arbitrarily large number of spectral lines both in a deterministic and in a stochastic way. For a deterministic line set and not too small velocities, a high computational speed-up is achieved if the solution is expressed not in terms of the extinction coefficient but by its much smoother wavelength integral. The assumption of a Poisson point process for the number of lines, combined with suitable distributions for the line positions and other line parameters, is a flexible presentation of the wavelength dependence of the line extinction coefficient which well describes deterministic “real” lines and requires only few parameters. The intricate connection between line properties and the emergent intensity can then be evaluated to a large part analytically. Assuming Lorentz profiles and a power law for the line strength distribution, we discuss the effect of many spectral lines on the radiation field of a typical AGN accretion disk, in particular, on the emerging radiation and on its diffusion limit holding in the interior.

For the modelling of the radiation fields in stellar winds and the resulting forces new efficient algorithms are presented. In the first one, the radiative transfer equation for moving 3D media is solved analytically with the assumption that the source function is known, eg. from the solution of the NLTE rate equations. For a wind with inhomogeneities an a-posteriori error controlled finite element code is described that takes scattering explicitly into account. Finally, we present possibilities for the accurate inclusion of an arbitrary number of spectral lines in a deterministic and in a stochastic way.

The structure of accretion discs around (super-)massive black holes is discussed in this contribution with special emphasis on the radiation fields. These are of crucial importance for the understanding of these objects since photons control in most cases not only the temperature distributions but also the pressures and shapes. Recent progress in the modelling of photon fields now provides the means for a much improved understanding of the consequences of the multidimensional structure of the discs as well as of the effects of the strong space time curvature and of the high velocities involved. However, the simultaneous inclusion of the NLTE level populations and of many spectral lines is still a major problem. It is also demonstrated that special and general relativity effects strongly distort the apparent brightness distributions and spectra such accretion discs so that a solution of the inverse problem will be very difficult.

We generalize the problem of the semi-gray model to cases in which a non-negligible fraction of the stellar radiation falls on the long-wavelength range, and/or that the planetary long-wavelength emission penetrates into the transparent short wavelength domain of the absorption. Second, applying the most general assumptions and independently of any particular properties of an absorber, we show that the greenhouse effect saturates and any Earth-like planet has a maximal temperature which depends on the type of and distance to its main-sequence star, its albedo and the primary atmospheric components which determine the cutoff frequency below which the atmosphere is optically thick. For example, a hypothetical convection-less planet similar to Venus, that is optically thin in the visible, could have at most a surface temperature of 1200-1300K irrespective of the nature of the greenhouse gas. We show that two primary mechanisms are responsible for the saturation of the runaway greenhouse effect, depending on the value of the wavelength above which the atmosphere becomes optically thick. Unless this wavelength is small and resides in the optical region, saturation is achieved by radiating the thermal flux of the planet through the short wavelength tail of the thermal distribution. This has the observational implication, the radiation from such a planet should be skewed towards the NIR. Otherwise, saturation takes place by radiating through windows in the FIR., 13 pages 14 figures

In this contribution we want to discuss M star atmospheres and their dependence on molecular opacities. A star belongs to the spectral class M if its optical and infrared spectrum shows strong bands of TiO and numerous strong metal lines so that for wavelengths < 4000 Å there is “hardly any flux left” (Jaschek & Jaschek 1987). M stars cover a very large range in luminosity: M dwarfs are the intrinsically faintest stars, whereas M giants and supergiants reach luminosities that are among the highest known. General properties of these objects are given in Table 1 (after Schmidt-Kaler 1982).

The photospheres of type II supernovae during the coasting phase ( ~ 1 week to 1 month after outburst) are characterized by moderate temperatures (Teff ~ 7...12 × 10^3K) , large radii (R ~ 10^13...10^15cm) , very low densities (p ~ 10^-13... 10^-16 gcm-3) and high expansions velocities (up to ~ 30000 km-1 in the outer layers). These properties lead to large geometrical extensions (i.e. the photons can travel a large fraction of the radius without being absorbed) and to electron scattering as the main source of opacity. Therefore these photospheres are very different from those of stars in hydrostatic equilibrium. In this review we first describe the input physics for models and essential numerical aspects of the model construction. In particular we stress the central role of an adequate treatment of the radiative transfer equation which should simultaneously take into account the sphericity, a high ratio of scattering to absorption, and relativistic effects. Subsequently we discuss details of the emergent spectra and their relevance for parameter determinations. Finally, model spectra are compared with observations of the bright supernovae SN 1979C and SN 1980 K, which seem to be more average examples of type II supernovae than SN 1987A, and possibilities for using these objects for extragalactic distance determinations are investigated.

For the modelling of the radiation fields in stellar winds and the resulting forces new efficient algorithms are presented. In the first one, the radiative transfer equation for moving 3D media is solved analytically with the assumption that the source function is known, eg. from the solution of the NLTE rate equations. For a wind with inhomogeneities an a-posteriori error controlled finite element code is described that takes scattering explicitly into account. Finally, we present possibilities for the accurate inclusion of an arbitrary number of spectral lines in a deterministic and in a stochastic way.

Continuum and line blanketing models for the photospheres of novae in the early stages of their outbursts are presented. The expanding envelopes are characterized by a very slow increase of density with decreasing radius which leads to very large geometrical extensions and large temperature differences between the inner and outer parts. The spectra show a large IR excess and a small Balmer jump which may be either in absorption or in emission. For the parameters considered (T eff = 104,1.5 x 104, 2 x 104K, R = 1011 cm, solar composition), most lines are in absorption. The effects of both modifications in the temperature structure (e.g. by heating from shock fronts) and changes in the abundances of the heavy elements on the emergent spectra are briefly discussed.

In recent years cool white dwarfs have been studied for various aspects ( see e.g. Wlnqet et ai.,1987 Wlnget and van Horn, 1987, Koester, 1987, Llebert, 1980) and much effort has been invested in attempts to interpret the energy distributions of these stars ( Greensteln, 1984, Zeldler-K.T. et al, 1986, Llebert et al., 1987, and others). However, it seems that ln spite of these efforts the spectra In particular of the very cool objects with effective temperatures below about 6000 K are not yet fully understood, since they are extremely dlverse and each objects needs special consideration. In addltlon, the analyses are extremely dlfflcult because the principal constluents of the atmospheres ( H, He ) and elements, which may donate the majority of electrons, are essentially invisible. Since usually only one lonlsatlon stage of an element is present, this implies that the gas pressure Pg is hlgh ( compared e.g. to the solar photosphere ), the accurate value of Pg , however, cannot be determined reliably.

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

We have calculated a large grid of hydrogen-rich supernova photospheres, in which radii, effective temperatures, density profiles, and expansion velocities have been varied. Spherical geometry, radiative equilibrium and LTE level populations are assumed. In the quasi-exact radiative transfer, the dilution of the radiation field, and scattering as well as absorption (by all relevant continuous processes and up to 150 000 lines in some models) are accurately considered. Good agreement can be obtained with the UV and IR spectra of supernovae 1979C, 1980K, and 1987A as observed during the coasting phase. Potential methods of parameter determinations for SN II are briefly discussed.

Problems of re-ionization of the Universe, in particular the source of the ionizing photons and the time evolution, are presently of great interest (cf. Peacock1, Liddle and Lyth2). The proper modeling of the phenomenon is a formidable task since it involves, among others, the formation of supermassive black holes, of galaxies and of stars, and the propagation of ionizing radiation in a medium of complex 3D geometry showing strong deviations from local thermodynamical equilibrium. In this paper we refer to the latter problem, i.e. we study the time development of the radiation field and the ionization degree of the intergalactic medium and some (proto-)galaxies when they are illuminated by a massive, just formed Pop III star. All hydrodynamics is neglected.

For the interpretation of long-term continuum variations of AGNs detailed non-grey models for stationary accretion disks are used to estimate the brightness and color variations as functions of the accretion rate. Possible consequences of multiple solutions of the structure equations are briefly discussed.

Model atmospheres for supernovae of type II have been calculated taking into account the effects of sphericity and velocity fields. We obtain a good fit for the energy distribution of the recent supernovae SN 1980 K and SN 1987 A.

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.

The M dwarfs GL 832 (spectral type M2), GL 887 (M1), and GL 1 (M2) have been observed with the ISOPHOT spectrophotometers in the range 2.5–12µm at high signal-to-noise ratios. Since optical and ground based near-infrared data are also available, more than 95 % of the total flux is now covered for these objects. It is seen that between 2.5 µm and about 9 µm the fluxes fall off faster than a Rayleigh-Jeans tail of a black body distribution and that for longer wavelenghts the decline is smaller. This indicates that the ISO data are probing the region around the temperature minimum and that our stars have some kind of chromosphere although Ha emission is not observed. Comparison with a model indicates that the features are considerably weaker than predicted.

We present non-LTE model atmosphere calculations for the LMC nova 1988 #1 during the optically thick phase shortly after the outburst. The models are able to reproduce the characteristic features of the observed spectra and will be used to determine the atmospheric parameters and chemical abundances of LMC novae.

We describe the construction and main properties of stationary accretion discs that are supposed to produce the high luminosities of Active Galactic Nuclei. It is found that for most parameter combinations the discs are geometrically thin everywhere. However, for high accretion rates and/or low local energy generation rates they may get thick close to the central object. The distributions of the temperature, the gas pressure and the vertical optical depth are characterized by steep gradients close to the surface and very flat stratifications in the inner parts. The ratio of the radiation pressure to the gas pressure varies over several orders of magnitude within a disc. The emergent fluxes can be well approximated by power laws in a wide wavelength range but turn over in the middle or extreme UV range. It is argued that this range is best suited for spectral diagnosis from the continuum.

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 Beer's 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 Milne's 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.

Using infrared and high-resolution optical observations of the M dwarf Gliese 1, we investigated the temperatures in the upper atmospheric layers of this star with low atmospheric activity. To fit the H and metal line profiles, the normal radiative equilibrium temperature decrease must be truncated at about log above which a steep chromospheric (T ≈ 4900 K) rise must be imposed. Unfortunately, the position of the onset of the chromosphere depends to some extent on the temperature distribution in the inner parts of the photosphere. The chromosphere is just not sufficiently optically thick to be seen in the infrared up to ≈ 30 μ m. The persistent strength of the TiO bands leads us to check for indications of surface inhomogeneities with negative results.

We consider a medium of non‐negligible optical depth in which high energy line photons are Compton scattered. For the calculation of the angle and energy dependent specific intensities we first cast the correlations between the angles and energies of the incoming and the outgoing photons (as determined by the Compton conditions) into the form of a redistribution function, which is weighted by the Klein‐Nishina cross‐section in order to account for the energy dependence. We then discretized the angle × energy space so that the radiative transfer equation transforms from an intergro‐differential equation into a system of ordinary differential equations subject to boundary conditions on both sides. This system is subsequently solved in a numerically stable way without any further approximation by means of the discrete‐ordinate‐matrix‐exponential (‘‘Dome’’) method so that all emergent intensities and the energy converted into heat are obtained.Since the redistribution function is very complex and involves sin...

The formation of chromospheres and coronae of accretion disks around white dwarfs is studied by means of stationary non-grey models with parameterized opacities. It is shown that very high temperatures in the outer layers occur if the pressure dependence of the opacity is moderate or strong (\(\frac{{d\ln \kappa }}{{d\ln P}}\) > 0.25) and the Shakura-Sunyaev parameter α is close to unity.

Analytical and numerical solutions of the radiative transfer equation recently obtained in Heidelberg are reviewed. In particular, a multidimensional finite-element algorithm for unstructured grids that can be adaptively refined is discussed in detail.

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.

In the continuum and line-blanketed models presented here, nova atmospheres are characterized by a very slow decrease of density with increasing radius. This feature leads to very large geometrical extensions so that there are large temperature differences between the inner and outer parts of the line-forming regions. The theoretical spectra show a large IR excess and a small Balmer jump which may be either in absorption or in emission. For the parameters considered (effective temperature of about 10 exp 4 K, L = 2 x 10 exp 4 solar luminosities, outer boundary density of about 3 x 10 exp -15 g cm exp -3, mass-loss rate of 10 exp -5 solar masses/yr), most lines are in absorption. The effects of changes in the abundances of the heavy elements on the emergent spectra are discussed. The strong unidentified features observed in ultraviolet spectra of novae are found in actuality to be regions of transparency within the Fe 'forest'. Ultraviolet spectra obtained from the IUE archives are displayed, and spectral synthesis of these spectra is done using the theoretical atmospheres.

In recent years cool white dwarfs have been studied for various aspects ( see e.g.Winget et al.,1987 Winget and van Horn, 1987, Koester, 1987, Llebert, 1980) and much effort has been Invested in attempts to interpret the energy distributions of these stars ( Greenstein, 1984, Zeldler-K.T. et al, 1986, Llebert et al., 1987, and others). However, it seems that in spite of these efforts the spectra in particular of the very cool objects with effective temperatures below about 6000 K are not yet fully under-stood, since they are extremely diverse and each objects needs special consideration. In addition, the analyses are extremely difficult because the principal constiuents of the atmospheres ( H, He ) and elements, which may donate the majority of electrons, are essentially invisible. Since usually only one ionlsatlon stage of an element is present, this implies that the gas pressure Pg is high ( compared e.g. to the solar photosphere ), the accurate value of Pg, however, cannot be determined reliably.

First reported at the IAU Colloquium No. 53 on White Dwarfs (McGraw et al. 1979), PG 1159-035 (GW Vir) is the prototype of a new class of very hot, pulsating, pre-white dwarf stars. It shows complicated, nonradial pulsation modes which have been studied exhaustively, both observationally and theoretically. The effective temperature has been crudely estimated as 100,000 K with log g ~ 7 (Wesemael, Green and Liebert 1985, hereafter WGL).

The main characteristic of white dwarf atmospheres is a pressure that is at least two orders of magnitude higher than in main sequence star atmospheres of the same effective temperature. This is due to the high gravity (log g ≈ 8), extreme metal under-abundances (Δlog εM ≳ 2.5), and (in many cases) the replacement of hydrogen by helium as the main constituent. As consequences the atmospheres are very thin (ΔR ≲ a few km) and the level populations of all species are given by the Boltzmann distribution (perhaps with the exception of the extreme outer layers, Greenstein 1973, Pilachowski 1984). Thus, models can be calculated under the assumption of plane parallel radiative transfer and local thermodynamic equilibrium, which facilitates the numerics very much;but special care has to be taken of pressure effects (e.g., broadening of spectral lines, quenching of levels, changes in the dissociation-ionization equilibria). In addition, the proper consideration of convection, which is very effective and may reach into the optically thin layers, makes the construction of model atmospheres for white dwarfs rather tedious and costly.

Accretion discs have come to be known as a frequent phenomenon in astrophysics (Frank, King and Raine 1985). While accretion discs are of various dimensions, scales, temperatures and so on, we limit the discussion here to accretion discs around white dwarfs. Extension of the present discussion to other conditions, in particular accretion discs in premain sequence stars and self gravitating accretion discs, will be made in subsequent communications.