S. Chandrasekhar, Radiation Transfer, Oxford, At the Clarendon Press, 1950, p. 393.

The complex of questions connected with the propagation of radiation in media in the presence of appreciable secondary scattering has long attracted the attention of physicists and astrophysicists. This is due to the fact that the processes of light propagation in media largely determine the course of phenomena in the atmospheres of stars and planets, in surface marine layers, and in a number of other cases. Interest in these questions has grown especially in connection with the development of the theory of neutron diffusion, which proceeds from the same equations as the theory of radiation transfer. Until recently, the literature lacked general methods for considering radiation-transfer processes that were at the same time suitable for the effective solution of concrete problems.

The situation changed sharply in 1943–1944, when V. A. Ambartsumian formulated the so-called principle of invariance and, on its basis, obtained solutions to a number of very important problems. Naturally, Chandrasekhar’s reviewed book, which appeared in 1950, is devoted to a considerable extent to the exposition of Ambartsumian’s method and its applications.

In Chapter I the basic concepts are established and the fundamental equations describing the transfer of radiation in a scattering and absorbing medium are derived. In the same chapter the mathematical features of the various problems that will be considered in the following sections of the book are also discussed.

The best-known, but laborious, approximate method for solving these problems consists in passing from the integro-differential transfer equation to a system of differential equations of the first order. This passage can be carried out by means of any of the formulas of approximate integration. Chapter II sets forth this method and considers

various formulas of approximate integration are given. Of interest is the computation of the frequently occurring integral of the form

\[ \int_{0}^{\infty} I(t)\,E_{1}(|t-\tau|)\,d\tau, \]

where \(E_{1}(|t-\tau|)\) is the integral exponential.

The next four chapters are devoted to the transfer of radiation in a plane-parallel semi-infinite atmosphere. The main attention is concentrated on the study of two cases: 1) the atmosphere diffusely reflects the radiation incident upon it; 2) the atmosphere is subjected to external irradiation. The intensity of the flux propagating in it in the direction toward the boundary does not depend on depth.

In Chapter III, as if for contrast with what follows, the solution of these problems by the approximate method described above is presented. Despite the substantially simplifying assumption of isotropic scattering, this solution is cumbersome and laborious.

In Chapter IV the principle of invariance, formulated by V. A. Ambartsumian, is introduced. The application of this principle makes it possible to reduce both basic problems of the theory of radiative transfer to the solution of one and the same functional equation. Thereby a new effective method of solution is obtained and, at the same time, the internal connection between these problems is established.

The so-called \(H\)-function, which is the solution of Ambartsumian’s equation, plays such an important role in the modern theory of radiative transfer that Chandrasekhar considered it necessary to devote the entire fifth chapter to the analysis of its mathematical properties.

Chapter VI considers radiative transfer in a plane-parallel semi-infinite atmosphere in the presence of anisotropic scattering. The application of the principle of invariance makes it possible in this case as well, without particular difficulty, to carry the solution through to numerical values with the desired degree of accuracy.

V. A. Ambartsumian showed that the principle of invariance can be successfully applied also when considering radiative transfer in atmospheres of finite thickness. In this case the problem of transmission and reflection of radiation is reduced to finding the solution of a system of two functional equations.

In Chapters VII and VIII the derivation of these equations is given and the mathematical properties of the functions that are their solution are considered.

In Chapter IX this method is applied to concrete problems of the propagation of radiation in a semi-infinite atmosphere under various scattering laws.

Chapter X is undoubtedly of interest, in which the influence of polarization is taken into account. In the same chapter the problem of radiative transfer is made concrete for the case of a planetary atmosphere. In doing so, the reflecting properties of the planet’s surface are taken into account.

Chapters XI and XII touch upon certain astrophysical questions. In Chapter XI the methods presented earlier are applied to the problem of the distribution of energy in stellar spectra. Naturally, considerable attention is devoted to accounting for the absorption of radiation by negative hydrogen ions.

Chapter XII—the formation of absorption lines in stellar spectra—raises serious perplexities. Chandrasekhar, preserving the position of his earlier works, proceeds from the assumption of radiative equilibrium for any portion of a spectral line, without taking into account the change in photon frequency in the process of absorption and subsequent emission. Meanwhile, even the Doppler effect leads to a change in frequency of the order of the line width.

The sharp dependence of the absorption coefficient on frequency within a spectral line makes it necessary to take such processes into account. A theory that takes into account the peculiarities of radiation transfer in spectral lines was published in the Soviet physical literature in 1947. Chandrasekhar, however, considers it possible to ignore these processes and does not take the trouble to substantiate this assumption.

The last chapter, Chapter XIII, considers several questions related to the main subject matter of the book. A strange impression is produced by the paragraph devoted to the rate of emergence of “trapped” radiation. Chandrasekhar still ignores the peculiarities of radiation diffusion and notes only in a footnote that “a consideration of the same question, but under slightly different initial assumptions,” is given in the works of Kenty and Holstein.

It is appropriate to note that the treatment of the question in these “slightly different” works is completely and fundamentally different from Chandrasekhar’s presentation.

Kenty tried to take into account the shape of the spectral line while remaining within the framework of the usual diffusion theory, and arrived at an absurd result (an infinite value of the diffusion coefficient). Holstein, however, proceeded from the theory of radiation transfer developed in the Soviet Union, which takes into account changes of frequency within a spectral line. It can hardly be said that these works “differ slightly” from Chandrasekhar’s presentation.

Such disregard for the work of other authors is characteristic of the entire book. Chandrasekhar could not fail to note the role of V. A. Ambartsumian in the formulation of the invariance principle. However, he did not find it necessary to point out that Ambartsumian, using this method, also solved a number of fundamental problems in the theory of radiation transfer. There are also no references to very interesting works by other Soviet authors (V. V. Sobolev, E. S. Kuznetsov, E. R. Mustel, and others).

A major shortcoming of the book is its pronounced tendency toward a formally mathematical treatment of the problems.

The author is not interested in the physical conditions determining the formulation of a given problem and does not analyze the assumptions that are made in doing so. The book contains no references at all to experimental material or any comparison of the results of theory with experience.

Despite these shortcomings, Chandrasekhar’s book is of undoubted interest to a wide circle of physicists and astrophysicists, since it contains a systematic and sufficiently complete exposition of the mathematical methods of the theory of radiation transfer. The value of the book is also enhanced by the fact that it contains very detailed tables of functions important in the theory of radiation transfer, calculated by Chandrasekhar and his collaborators.

L. Biberman