# Busbridge - Mathematics of Radiative Transfer

Ida Busbridge was a pure mathematician who applied her mathematical skills to the problems of solar physics. Her only book is

*The Mathematics of Radiative Transfer*published by*Cambridge University Press*in 1960. We give below (i) the Publisher's description, (ii) the Preface, (iii) an extract from the Introduction, (iv) an extract from a review, and (v) Walter Stibbs' comments on the book. We note that Walter Stibbs was an examiner of Busbridge's D.Sc. submission, the main part of which was*The Mathematics of Radiative Transfer*.**1. Publisher's description**

This tract gives a simple but rigorous treatment of some of the mathematical problems that arise in the theory of the transfer of radiation through the atmosphere of a star. Similar problems occur in the theory of the diffusion of neutrons and in the study of temperature-wave flow in solids; so the solutions found in one theory can often be applied in the others. Dr Busbridge's starting-point is the equation of transfer. The first section provides the auxiliary mathematics, and the second discusses the Milne equations. Some unsolved and incompletely solved problems are considered in an appendix. The language and notation of astrophysics is used throughout, for brevity and simplicity, but translation into other notations is usually fairly easy. Over the years the subject had grown considerably, and several outstanding problems have been solved, though the total amount of rigorous work is small. This tract will help to clear up confusions which exist and will provide an introduction to some of the more powerful mathematical techniques available.

**2. Preface**

E Hopf's tract on

*Mathematical Problems of Radiative Equilibrium*(now out of print) has, for twenty-six years, provided the basis for almost all rigorous work on transfer theory. Meanwhile the subject has grown enormously, and only a few authors have dealt with questions concerning the existence and uniqueness of solutions. The theory of finite atmospheres, in particular, has been in a confused state for some years.

In writing this tract, I have incorporated (with modifications) those parts of Hopf's tract which have proved most valuable in the development of the subject and those parts which are needed in the subsequent analysis. I have tried throughout to keep the treatment as simple as possible and to obtain results which will be of practical use. By simplifying the treatment it has been possible to consider. larger number of topics, but even so there are many important aspects of the subject which have only been touched upon, and others which have been ignored.

I should like to express my sincere thanks to Professor E C Titchmarsh, who read and criticised a large part of the book. Although I have not adopted all his suggestions, the final text owes a great deal to him. I should also like to thank Dr J B Sykes, who obtained for me copies of de-classified papers on neutron diffusion and who helped me with translations of Russian papers.

I.W.B.

St Hugh's College

Oxford

January 1960.

**3. Introduction**

In its early stages, transfer theory was concerned with the transfer of radiation through the atmosphere of a star. It gave rise to some interesting mathematical problems which were considered in detail by E Hopf in his tract

*Mathematical Problems of Radiative Equilibrium*. Hopf was mainly concerned with problems in which energy is conserved - the so-called conservative case - but the non-conservative case is of equal or even greater importance today. Moreover, many problems which were only partially solved when Hopf wrote his tract have now been solved completely.

Interest in transfer theory was stimulated when it was found that the theory of the diffusion of neutrons led to the same mathematical equations. When the transition rules are known, solutions found in one theory can often be applied in the other.

A more recent development has been the theory of temperature-wave flow in solids. This leads to mathematical equations of the same form, but real functions are replaced with complex ones.

It is the object of this tract to give a simple and rigorous account of some of the principal mathematical results and of the methods which have proved successful in the exact solution of problems. The language and notation of astrophysics is used. Moreover, in order to keep the treatment within bounds, the restrictions usual in astrophysics have been imposed. These do not apply to all the developments mentioned above, but complete generality would require a much larger volume and research workers should experience little difficulty in making the appropriate adjustments for themselves.

The starting point of this book is the equation of transfer, and no attempt has been made to deal rigorously with its derivation. Hopf wrote in [his tract]: "The entire subject ought to be treated anew and the fundamental equations be derived in a rigorous way, the main tool being the general theory of measure." This task, which remained neglected for twenty-three years, has at last ben attempted by R W Preisendorfer, but his treatment is far too long for inclusion in a work of this size.

The main mathematical methods in transfer theory have developed independently of one another, but precise results can be obtained more easily if methods are, to a certain extent, made interdependent. In particular, the important 'Ambartsumian technique' ... has been based on results in all the preceding chapters

...

Statistical concepts play a major role in the theory of scattering and absorption of radiation, and a strong case can therefore be made for treating the whole subject from a statistical viewpoint. This has been done by V V Sobolev and (following him) S Ueno. A stochastic integro-differential equation is derived for the probability that a photon, which is absorbed at a certain point, will be re-admitted in a certain direction. The same equation (with a different notation) is obtained in a solution using the Ambartsumian technique, and the subsequent analysis is identical in the two methods. For this reason no detailed consideration of the probabilistic treatment is included.

A subject which is still developing constantly presents new problems to the mathematician and some old ones are sometimes left unsolved by the way. Some notes on some of the incompletely solved problems of transfer theory have been added in an appendix.

**4. Review by J Howlett**

J Howlett reviewed

*The Mathematics of Radiative Transfer*in

*The Mathematical Gazette*

**46**(355) (1962), 84-85. The review begins:

Dr Busbridge's book can be regarded as a replacement and extension of Hopf's 1934 Tract - "

*Mathematical Problems of Radiative Equilibrium*" - which is now out of print. It is wholly mathematical with only the briefest mention of physical ideas; in effect it is a short treatise on a particular class of integral equations.

In the first chapter the general equation of radiation transfer in a scattering medium - conveniently called an atmosphere - is set up; this is a linear integrodifferential equation which can be reduced to a linear integral equation. Chapter 2 discusses what are called the

*H*-functions, defined by a non-linear integral equation and used later as a basis for expressing the solutions of the Milne and related equations, Chapter 3, the properties of two integral operators which are generalisations of that occurring in the Milne equation. The rest of the book (Chapters 4-10) deals with the solution of the Milne equation ...

**5. Walter Stibbs' comments**

Walter Stibbs was an examiner of Busbridge's D.Sc. submission, the main part of which was her Tract

*The Mathematics of Radiative Transfer*. He writes about the book in 'M G Adam and D W N Stibbs, Ida Winifred Busbridge, 10 February 1908-27 December 1988,

*Quarterly Journal of the Royal Astronomical Society*

**33**(4) (1992), 455-459'.

Detailed study of her Tract is a rewarding experience not only in the elegant use of rigorous mathematics, which she appears to have found easy, but also as a revelation of what she appears to have found difficult. It is evident from the Tract, as well as from some of her publications, that physical intuition for those gifted with it, but that it was necessary to verify mathematically many of the things that those so gifted took for granted. A case in point is the development of Principles of Invariance by Chandrasekhar on the basis of the early work by Ambartsumian who had used a physical approach to problems of diffuse reflection. At the beginning of Chapter 6 of her Tract, Miss Busbridge makes the following assertion: "The application of these principles is not easy, and until a precise statement is given of the physical conditions which are sufficient to ensure their truth, any solution based on them ought to be verified in another way." In one of her publication, in the Monthly Notices for 1955, Miss Busbridge carefully verifies mathematically the principle of invariance as applied to the case of completely non-coherent scattering and interlocked multiplet lines in the theory of line formation. Again, in the case of anisotropic scattering, which is treated in Chapter 10 of her Tract, she comments: "Chandrasekhar's method employs 'principles of invariance'. The equations for anisotropic scattering are derived and these are then reduced to H equations (if the atmosphere is semi-infinite) by an inspire insight denied to the reader." Towards the end of the same Chapter, she derives the law of diffuse reflection from the mathematical foundations of the subject in preference to it being regarded as a physical law to be called upon when required and, finally, she gives a mathematical derivation of one of the invariance equations, for an atmosphere with a constant net flux, otherwise known as an invariance arising from the asymptotic solution at infinity, which is perhaps one of the most difficult of the invariance equations to grasp. This leaves the reader with the impression that whereas Miss Busbridge might have been to some extent envious of those who had inspired physical insight, she nevertheless felt to be in some sense suspect and, in any case, unnecessary to use it when techniques are available to deduce the results with mathematical rigour from the fundamental equations of the problem.

Last Updated April 2020