Introduction

Rapid expansion in nova, stellar atmospheres of supernovae and similar objects is established through several spectroscopic observations. In spectra of these objects, the absorption lines shift towards the violet side from the rest position indicating matter outflow in their atmospheres. These lines are accompanied by red shifted emission characteristics of P Cygni type as seen in figure 7.1 (see Beals (1950), Kuan and Kuhi (1975)). Beals (1929, 1931) interpreted the large widths in the lines of WR spectra to be due to the velocities of expansion of the order of 3000 km s-1 indicative of a rapid outflow of the matter in the outer layers of these stars. He suggested that this outflow of matter is influenced by the radiation pressure in the medium.

It is difficult to obtain the solution of the transfer equation in such spherical media. Beals (1929, 1930, 1931, 1934), Chandrasekhar (1934), Gerasimovič (1934) and Wilson (1934) investigated this problem assuming the medium to be optically thin, neglecting the transfer effects. Struve and Elvey (1934) found that the Doppler widths derived from the flat part of the growth curve were much larger than the thermal value, which they attributed to the ‘turbulent’ motion in the atmosphere which is non-thermal. Struve's observations (1946) showed large scale velocities through the fact that the line profile widths in certain stars were larger than the Doppler widths obtained from curve of growth analysis of their spectra.

Introduction

We have studied homogeneous, plane parallel scattering atmospheres in chapter 5, using the principles of invariance in semi-infinite and finite media. These problems are solvable by the standard techniques of differential equations and expressible in standard functions. The X- and Y-functions of Chandrasekhar are solutions of certain integral equations. These cannot be used in a non-homogeneous media unless one sacrifices the physical characteristics of the medium. These solutions have been tabulated and it is difficult to use them in practical problems. One has to make serious physical approximations or resort to a numerical approximation. The principles of invariance are essentially the statement of the conservation of energy. Conservation of energy in a finite region can be expressed by what is called the ‘interaction principle’. In the limit of vanishing thickness of the medium these principles lead to the integro-differential equations of radiative transfer. The principle of interaction (see Redheffer (1962), Preisendorfer (1965), Grant and Hunt (1969b)) generalizes the invariance principles particularly in a finite medium. The basic idea of the interaction principle is to specify the radiation field in terms of the transmitted and reflected radiation at any given point in the medium.

Carlson (1963) and Lathrop and Carlson (1967) used a numerical version of the discrete ordinate technique in neutron reactor calculations. By integrating the radiative transfer equation over a finite volume in space coordinates and using the mean value theorem for integrals, we can develop difference equations that conserve flux.

Introduction

So far we have seen the problems of radiative transfer mostly in one-dimensional plane parallel or spherically symmetric geometries in the context of astrophysical situations. The book by Sen and Wilson (1990) deals extensively with the basic techniques for solving radiative transfer problems in spherically and cylindrically symmetric media (see also Leong and Sen (1969, 1970, 1971a,b), Uesugi and Tsujita (1969), Kho and Sen (1972)). Taking into account the effects of geometrical convergence and oblique incidence arising out of the sphericity of the medium, Bellman, Kagiwada, Kalaba and Ueno (see the references given in Sen and Wilson (1990)) solved the problems of the diffuse transmission of light from a central point source through an inhomogeneous spherical shell medium. Tsujita used a corresponding method to solve transfer problems in infinite cylindrical media (see Sen and Wilson (1990)). Certain approximate techniques such as ray-by-ray methods may be useful in special circumstances, but we need to explore the solution in multi-dimensional geometries so that any I (X1, Y1, Z1; t1) can be correlated to any other I (X2, Y2, Z2; t2) exactly. This is essential especially in scattering media which generate diffuse radiation fields.

However, the developments in multi-dimensional radiative transfer are not as advanced as the one-dimensional (including curved geometries) case. The solution of multi-dimensional radiative transfer is most important and is needed in astrophysical problems. We shall sketch some of the available results in this chapter.

Astrophysicists analyse the light coming from stellar atmosphere-like objects with widely differing physical conditions using the solution of the equation of radiative transfer as a tool. A method of obtaining the solution of the transfer equation developed to suit a given physical condition need not necessarily be useful in a situation with different physical conditions. Furthermore, each individual has his/her preferences to a particular type of methodology. These factors necessitated the development of several widely differing methods of solving the transfer equation.

In the second half of the twentieth century several books were written on the subject of radiative transfer: one each by Chandrasekhar, Kourganoff and Sobolev, two books by Mihalas, two by Kalkofen and more recently two books by Sen and Wilson. These books, which describe the developments of the transfer theory, will remain milestones. They will be of great value to the researcher in this field. A beginner needs to understand the basic concepts and the initial development of the subject to proceed to use the latest advances. It is felt that it is necessary to have a book on radiative transfer which presents a comprehensive view of the subject as applied in astrophysics or more particularly in stellar atmospheres and objects with similar geometrical and physical conditions. This book serves such a purpose. Several methods are presented in the book so that the students of radiative transfer can familiarise themselves with the techniques old and new.

We shall study the state of polarization of the radiation field in this chapter. We need to study the scattering problems exactly since light is generally polarized on scattering. A good example of this is Rayleigh scattering. An initially unpolarized beam of radiation when scattered at an angle Θ to the direction of the incident beam becomes partially plane polarized in the ratio 1 : cos2 Θ in the directions perpendicular and parallel to the plane of scattering, which is also the plane of the direction of the incident and scattered of radiation. The diffuse radiation field arising out of the scattering of light in the atmosphere must therefore be partially polarized and we need to formulate the transfer equation correctly and conveniently, so that many important problems such as polarization in stellar (or solar) planetary atmospheres (including that of sunlit sky) are studied correctly.

There are several polarization observations of stars, for example the T Tauri stars which emit linearly and circularly polarized radiation (Bastian 1982, 1985, Nadeau and Bastian 1986). Magalhãs et al. (1986) obtained polarimetric observations of the semi-regular variable L2 Puppis.

Any radiation field is described by four parameters: (i) the intensity, (ii) the degree of polarization, (iii) the plane of polarization and (iv) the ellipticity of the radiation at each point and in any given direction. However, it is very difficult to include such diverse parameters – intensity, a ratio, an angle and a pure number – in any symmetrical way in formulating the transfer equation.

Introduction

In chapter 7, we studied the solution of the transfer equation in the rest frame of the observer. There are two difficulties in this way of treating the transfer equation: (1) the absorption and emission coefficients become angle dependent due to Doppler shifts in the frequency of the photon and hence become anisotropic and aberration of light is generated; and (2) the coupling between angle and frequency creates the practical problem of dealing with an unmanageably large mesh size for computation in scattering problems. This restricts the expansion velocities to a few times the mean thermal velocities. When the expansion velocities are very high one needs to use the comoving frame or the moving frame of the material. As the observer is with the moving frame, no Doppler shifts in the frequency of the photon occur and the opacity and emissivity are isotropic. One can use the redistribution functions for a static atmosphere. For problems involving scattering integrals, one can use a line profile with a band width which contains the full profile and which does not contain the velocity components. This makes the angle–frequency mesh small enough to contain the full profile of a static medium.

Lorentz transformations are used to describe the change (in the relevant physical variables) between the rest and comoving frames. Lorentz transforms are applicable when the relative velocity of the two frames is uniform and constant. In reality we find that the velocities are functions of not only radii but also time.

Introduction

The complete linearization method of Auer and Mihalas (1969) was a significant advance in solving complex problems of radiative transfer and was followed by the work of Rybicki (1971), Kalkofen (1974) and others. These are basically Newton–Raphson linearization methods which are highly efficient but are not favourably oriented towards computer time and storage. Certain problems such as those which involve radiation hydrodynamics require faster methods with sometimes a little loss of accuracy. Operator perturbation techniques were developed to meet the needs of these problems. An excellent survey of these methods is given in Kalkofen (1987).

Wu (1992) developed a method that can deal with complex models with a high rate of convergence in multi-level non-LTE line formation calculations. It essentially consists of linearization of the transfer equation and constraints, then solving them separately. It overcomes the disadvantage of requiring the simultaneous solution of the corresponding equations by the complete linearization method and the poor convergence rate. Hubený and Lanz (1992) suggested two approaches to accelerate the method of complete linearization. The first one is the so called Kantorovich variant of the Newton–Raphson method by which the Jacobi matrix of the system is fixed. This reduces the calculation of the number of matrix inversions considerably and retains them fixed during the subsequent computations. The second approach is the application of Ng acceleration. These approaches reduce the computer time by about 2–5 times.

Exact numerical methods become costly in terms of computer time when the radiation field is coupled with hydrodynamics. In such situations one needs methods which are fast and give insight into the physics of the problem in an easy and quick manner. Escape probability methods satisfy these requirements to a large extent and therefore became popular. There are ‘first order methods’ due to Biberman, Holstein, Sobolev and Zanstra, which are reviewed by Irons (1979a,b). The methods due to Athay (1972a,b), Rybicki (1972), Frisch and Frisch (1975), Canfield et al. (1981, 1984), Scharmer (1981, 1983, 1984) and others are the so called ‘second order methods’. We shall describe these and others methods in this chapter. These methods have been reviewed by Rybicki (1984).

Nordlund (1984) developed a method for obtaining an iterative solution of radiative transfer in a spherically symmetric atmosphere using a single ray approximation. The convergence is achieved in 2–3 iterations to give an accuracy better than 1% in the source function.

The Monte Carlo technique has been used by several authors (see, for example, Magnan (1970), Panagia and Ranieri (1973); Pozdnyakov et al. (1976)).

Surfaces of constant radial velocity

The geometrical region from which most of the observed emission at a given frequency x comes is likely to be a thin zone centred on a surface of constant radial velocity in such a way that υz = µυr = x, where the term radial velocity means the velocity along the line of sight, which is different from υr the velocity along the radius vector (see Mihalas (1978)) measured from the centre of the star.

Glass plates theory

The most fundamental characteristic of the radiation field in dispersive media such as stellar atmospheres, planetary atmospheres, planetary nebulae is the diffuse radiation which arises from multiple scattering of radiation by the media. This has been studied through an approach called the principle of invariance, or invariant imbedding, due to Ambarzumian (see books by Chandrasekhar (1960), Sobolev (1963), Kourganoff (1963), Wing (1962), Preisendorfer (1965)). Bellman and his collaborators have published several papers on this subject (see the bibliography at the end of the chapter). Before we study this principle, we shall see how the concept was developed by Sir George Stokes in his glass plate theory (1852, 1862). In remarkably simple papers, he derived the transmission and reflection factors when a ray of light passes through a system of glass plates. We shall see below how he obtained the principle of invariance of reflectance when several glass plates are arranged parallel to each other, one on top of the other.

He obtained difference equations for the reflection of radiation by a pile of identical glass plates and derived certain commutation relations for sets of glass plates. It is remarkable that he was able to obtain transmission and reflection factors which look similar to those obtained in more complicated media such as stellar atmospheres. We shall derive the transmission and reflection factors for the set of glass plates following the treatment given in Hottel and Sarofim (1967).