# The Chandrasekhar-Eddington dispute

Astronomy and Mathematics have always had a close relationship: often one has influenced the development of the other. Sometimes, however, the discrepancies between mathematical results and astronomical intuition can generate controversy. The following is such an occasion.

In 1930, the Indian physics student Subrahmanyan Chandrasekhar obtained a scholarship to continue his studies in the University of Cambridge after graduating from Presidency College in Madras, India. On his long trip to England by ship, he pondered some questions regarding white dwarfs and arrived at an important result. Earlier, Chandrasekhar had applied Fermi-Dirac statistics to Ralph Fowler's work on white dwarfs and arrived at important results regarding their density. He had also shown that they had a stellar configuration of polytropic index 3 (a polytrope is stellar material in equilibrium under its own gravity, following an equation of state of the form $P = K\rho i^{1+1/n}$, where $n$ is the polytropic index). While on the ship, Chandrasekhar applied the principles of special relativity to his previous workings, and deduced that the stellar configuration was actually of index 3 and, furthermore, that a star could not become a white dwarf when its mass surpassed a certain limit. His original result for this limit was of 0.91 times the mass of the Sun. This was due to the mean molecular weight of stellar material being estimated to be 2.5 at the time. It was later revised to 2.0, so Chandrasekhar changed his result to 1.44 times the mass of the Sun.

Unbeknown to him, Edmund Stoner (of Leeds University), in collaboration with Wilhelm Anderson (of Tartu University, in Estonia), had arrived at similar results. Inspired by James Jeans' suggestion that the material in the core of a star might not follow Boyle's law, Stoner examined the effects of Fermi-Dirac degeneracy (a jamming of electrons in phase space) in the density of stellar material and derived a value for a limiting mass "above which the gravitational kinetic equilibrium considered will not occur." He published his results (his estimate for the limit was of 1.7 times the mass of the Sun) in: The limiting density in white dwarf stars, (Philosophical Magazine 7 (1929), 63-70); and The equilibrium of dense stars, (Philosophical Magazine 9 (1930), 944-963). However, Chandrasekhar's result was stronger because he used a more sophisticated model for a star, unlike Stoner and Anderson who had idealised a star as a sphere of uniform density. At the time, Chandrasekhar did not want to speculate on the fate of a star that surpassed that critical mass. Anderson also refrained from even mentioning it in his 1929 paper Über die Grenzdichte der Materie und der Energie, published in Zeitschrift für Astrophysik 56 (1929), 851-856.

Chandrasekhar arrived at Cambridge and presented his results to Ralph Fowler, who sent his paper to Arthur Milne, to ask for his opinion. Both of them were sceptical, because the existence of the limit brought to the forefront a question to which they did not have an answer: what happened to a star whose mass was over the limit? Nevertheless, Milne eventually got Chandrasekhar's paper The highly collapsed configurations of a stellar mass published in the March 1931 issue of the Monthly Notices of the Royal Astronomical Society. A shorter version of the paper The maximum mass of ideal white dwarfs, which Chandrasekhar had submitted in 1930, was published in volume 74 of the American Astrophysical Journal later in 1931. Chandrasekhar did not abandon the problem, however, and, in 1932, while at the University of Copenhagen, he published a new article Some remarks on the state of matter in the interior of stars in volume 5 of the German Zeitschrift für Astrophysik.

Once he finished his PhD, examined by Fowler and Eddington in an almost comic oral during which the two examiners spent much of the time arguing, and was elected a Fellow of Trinity College in 1933, Chandrasekhar visited Russia in 1934, where the enthusiasm of astronomers like Victor Ambartsumian and Lev Landau convinced him to resume his research on the topic. Although it was recorded that Chandrasekhar was irritated by Ambartsumian, nevertheless he greatly appreciated his outstanding feeling for astrophysics. During his years in Cambridge, Chandrasekhar had earned the friendship of some important astronomers like Edward Milne and Arthur Eddington. Eddington was particularly interested in Chandrasekhar's work on white dwarfs, as he thought it could help settle a long-standing dispute between him and Milne about Eddington's model of the stars.

Milne had proposed that every star must have a core composed of degenerate material, while Eddington's model considered the entirety of the star to behave like a perfect gas. Eddington had dismissed Milne's modifications in a meeting of the Royal Astronomical Society in 1929:-
I have not read professor Milne's paper, but I hardly think it is necessary, for it would be absurd for me to pretend that professor Milne has the remotest chance of being right.
Eddington wrote in a similar fashion about Milne's paper in a 1929 paper:-
It is difficult to discuss this paper. Professor Milne did not enter into detail as to why he arrives at results so widely different from my own; and my interest in the rest of the paper is dimmed because it would be absurd to pretend that I think there is the remotest chance of his being right.
To which Milne replied in The analysis of stellar structure (1930):-
Sir Arthur Eddington has dug a most valuable trench into unknown territory. But he has encountered a rocky obstacle which he cannot get around. If he would make the mental effort to scramble up the sides of the trench he would find the surrounding country totally different from what he had imagined and the obstacle entirely an underground one.
In the following months Eddington, Milne and James Jeans continued to have heated arguments both in private and at meetings of the Royal Astronomical Society about these matters.

Chandrasekhar's research implied that a star over the critical mass would act as a perfect gas, which gave Eddington the edge in the argument with Milne. However, Chandrasekhar's conclusion that not all stars would eventually become white dwarfs went directly against Eddington's model.

In 1934, he finished two papers on his theory of white dwarfs in which he had improved the results of his 1931 paper by obtaining an exact solution to the equation of state, which accounted for inhomogeneous polytropes, through extensive numerical analysis. In it he stated that, following a suggestion of Ambartsumian he had calculated the exact equation which governed the structure of spheres of gas consisting of both fast and slow moving electrons. He said that what he had done was so straightforward that:-
... it is surprising that it has not been isolated by anyone previously.
He submitted the paper to the Royal Astronomical Society and was invited to present his results during a meeting in January 1935. Chandrasekhar was not a stranger to these meetings, held on the second Friday of each calendar month, in London. Fowler had introduced him to the Society in 1930, and he had become a fellow in 1933. In previous meetings, he had talked about his ideas to other scientists, but this time he was bringing an exact solution. However, Eddington arranged it so that he would present his own paper straight after Chandrasekhar. In it, titled Relativistic Degeneracy, he directly attacked Chandrasekhar's arguments, not on their mathematical correctness, but on their physical meaning. He argued Chandrasekhar's work was based on a conceptual error:-
The formula is based on a combination of relativity mechanics and non-relativity quantum theory, and I do not regard the offspring of such a union as born in lawful wedlock.
He also appealed to preconceived notions of how the universe should be:-
I think there should be a law of Nature to prevent a star from behaving in this absurd way!
Chandrasekhar later said that this statement made him realise that:-
... despite this man's incredible physical insight, he has always operated with preconceived ideas.
Eddington did not stop here and reiterated his arguments in the General Assembly of the International Astronomical Union in Paris, in mid-July of that year. This was a large meeting attended by around 300 astronomers from 33 countries. When interviewed in 1977, Chandrasekhar remembered vividly how he had been made fun of by Eddington [30]:-
Eddington gave an hour's talk, criticising my work extensively and making it into a joke. I sent a note to Russell [Henry Norris Russell was presiding], telling him I would wish to reply. Russell sent back a note saying, "I prefer that you didn't." And so I had no chance even to reply; and accept the pitiful glances of the audience.
Eddington followed up his talk to the Royal Astronomical Society publishing a slightly edited version (On 'relativistic degeneracy', 1935 ) and with a couple of new papers years later (The hydrogen content of white dwarf stars in relation to stellar evolution, 1939, and The physics of white dwarf matter, 1940).

Eddington's arguments were successful at the time because, on top of Eddington's fame and charisma, his conclusions were more intuitively satisfying for many, despite the fact that his theoretical proceedings were hard to follow and involved discussion of relativity. He reinforced the established idea of a simple model for the evolution of the stars (in which they all eventually become white dwarfs). There was also no direct observation at the time that proved Eddington wrong or corroborated Chandrasekhar. William McCrea, who was present in the meeting, later said to K C Wali (in 1979), lamenting not having objected to Eddington's arguments at the time:-
... my instinct seemed to tell me that [Eddington] might be right. ... [His arguments] were superficially satisfying to me, and since they satisfied Eddington, I confess that I was content to let it go like that.
Since Chandrasekhar was not allowed to respond in any of the 1935 meetings where Eddington attacked his work, he decided to defend himself through other methods. He realised that at the centre of the issue was the confusion of a group of astronomers over a mainly physical problem. To solve the controversy then, he would need the help from experts in the field. In a letter to his friend Leon Rosenfeld soon after the January RAS meeting, he said:-
The result [of the controversy] is that there is going to be a long period of stress and confusion and if somebody like Bohr can authoritatively make a pronouncement in the matter it will be of the greatest value for further progress in the subject.
Rosenfeld was a Belgian physicist who was then working under Niels Bohr in Copenhagen. He brought the matter to Bohr's attention and reported back to Chandrasekhar that they were "absolutely unable to see any meaning in Eddington's statements" but that the question seemed to be "quite simple." In the hopes of getting them to settle the controversy, Chandrasekhar sent Eddington's manuscript to Rosenfeld and Bohr, who then in turn sent it to Wolfgang Pauli. However, even though they agreed with Chandrasekhar's papers, they were not willing to enter the discussion, claiming to be too busy at the time.

Chandrasekhar managed to respond to Eddington's arguments in 1935. In Relativistic Degeneracy, a paper jointly written with Danish physicist Christian Møller, he criticises Eddington's paper saying:-
... we [he and Møller] are quite unable to follow [Eddington's] arguments.
Then, they proceed to refute the arguments of his follow-up paper, showing that Eddington's own methods can be used to derive the very same equation that he claimed was the product of a misunderstanding. Chandrasekhar's call for help to physicists bore its fruits one year later, in Rudolf Peierls' Note on the derivation of the equation of state for a degenerate relativistic gas. Here, Peierls gave a proof of one of the physical facts that Eddington had doubted, that the pressure-density relation of a degenerate relativistic gas enclosed in a certain volume is independent of the shape of the volume. Peierls tackled the problem again in On Lorentz Invariance in the Quantum Theory (published 1942, submitted 1941), together with Paul Dirac and Maurice Pryce. The paper criticises Eddington's misunderstanding of the use of relativity, and hence the objections he had made to Chandrasekhar's work, through the lens of theoretical physics. It starts:-
In a recent paper, Eddington raises an objection against the customary use of the Lorentz transformation in quantum mechanics, ... when applied to ... the behaviour of a degenerate gas. This objection seems to us to be mainly based on a misunderstanding, and our purpose here is to show that the practice of theoretical physicists on this point is quite consistent.
Apart from these direct rebuttals, Eddington's objections did not attract much attention in published works, they were either omitted or relegated to footnotes, for example, in Ralph Fowler's Statistical Mechanics (1936) or George Gamow's The Birth and Death of the Sun (1940). This incident was certainly one of the low points of Eddington's scientific career and it is interesting to try to understand why he fought so strongly for his point of view. Looking for a reason behind this blunder, Werner Israel posits that it was due to an unorthodox definition of particle that Eddington had proposed in 1923's The Mathematical Theory of Relativity. This would have set him in opposition to the Stoner-Anderson formula and Chandrasekhar's limit even before he took into consideration their, at the time, unexplainable consequences. We offer another strong possibility.

Eddington was working on what he called his "fundamental theory" which was his attempt to unify general relativity, special relativity and quantum theory. His ideas involved the seven primitive constants of physics which Eddington sought to relate in simple numerical ways. Despite the mystical element to these ideas, he began to consider this work the crowning pinnacle of his scientific achievements. If he accepted Chandrasekhar's work, however, his "fundamental theory" was destroyed. This became clear in 1939 but first let us give one further example of Eddington attacking Chandrasekhar.

Eddington had been invited in 1936 to lecture at a conference to celebrate the tercentenary of Harvard University. He had used the occasion to attack Chandrasekhar who, he said in his lecture [14]:-
... put the stars back in precisely the same difficulty from which Fowler had rescued them. The small stars could cool down all right and end their days as dark stars in a reasonable way. But above a certain mass ... the star could never cool down, but must go on radiating and contracting until heaven knows hat becomes of it. That did not worry Chandrasekhar; he seemed to like the stars to behave that way, and believes that is what really happens. But I felt the same objections as earlier to this stellar buffoonery; at least it was sufficient that there must be something wrong with the physical formula used.
In 1937 Chandrasekhar took up a position in the United States. He met up again with Eddington in Cambridge, England, when he was on his way to the International Astrophysical Colloquium held in Paris in July 1939. At dinner at the high table Chandrasekhar, Eddington, Dirac and the young physicist Maurice Pryce (who had recently married Max Born's daughter) were seated together. In [3] Chandrasekhar's description of events is recorded:-
In 1939, Chandrasekhar published his finalised theory of white dwarfs in An Introduction to the Study of Stellar Structure. It was well received: for example Strömgren called it [27]:-
... a very valuable handbook for the astrophysicist.
Ledoux, in [20], said it deserved to be compared to the works of Eddington, Jeans and Rosseland. After that, he decided to move on to other fields of study.

In his 1935 papers, Chandrasekhar proposed that if a star is over its critical mass, it will lose mass until it is below the limit, and hence every star would eventually become a white dwarf:-
A possibility is that at some stage in the process of contraction ... a profuse ejection of matter will begin to take place. This ... will continue till the mass of the star becomes small enough for central degeneracy to be possible. ... Thus, whether the star is of large or small mass, the final stage in its evolution is always the white-dwarf stage.
This idea was purely speculation, but it took root over the next twenty years. For example Gamow and Schoenberg write in Neutrino Theory of Star Collapse (1941) [16]:-
On the other hand, the stars possessing a mass larger than the critical one will undergo a much more extensive collapse, ... [driving away] more and more material from the surface. The process will probably not stop until the expelled material brings the mass of the remaining star below the critical value.
The idea of ejection of mass was not critically challenged until the 1960s, when the study of stars with masses above the limit led to the development of the theory of black holes.

For more on this topic see our article on Black Holes at THIS LINK.

F Hoyle, W A Fowler, G R Burbridge and E M Burbridge write in On relativistic astrophysics (1963) [17]:-
The conventional view is that in all cases [where the mass is above Chandrasekhar's limit] a process of mass ejection takes place, conveniently reducing the final mass below one or other of the mass limits we mentioned. We wish to emphasise this view is no more than superstition.
... even though science deals with abstractions and grand cosmic issues that dwarf our small human lives, those who carry out scientific research and spend their lives absorbed in calculations and theories are human beings. They may be driven by irrational impulses, as was Eddington when he refused to believe a result which did not square with his view of the universe; and, as they go about their intensely solitary work, they may by haunted by passions, jealousies, fears, ambitions and disappointments. Chandrasekhar's eternal quest for personal peace could never be fulfilled.

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Written by J G Mena and T S C Peres, University of St Andrews
Last Update November 2018