Unabridged response to letters from Michael Nauenberg and Eric Blackman
(Physics Today, July 2011, page 8)
by Kameshwar Wali
In the December 2010 issue of Physics Today, Freeman Dyson and I wrote articles of 4000 words in honor of Subrahmanyan Chandrasekhar’s centenary. We did not intend to review the complete history of the theory of white dwarfs. However, in view of the comments and questions raised by Michael Nauenberg and Eric Blackman, I am compelled to respond briefly.
Nauenberg contends that Chandra was aware of Wilhelm Anderson’s work related to relativistic effects on white dwarf stars and cites Chandra’s letter to his father on 30 August 1929. In it, Chandra indeed wrote,
As for my paper, as I had nearly completed, writing it out, a paper by a German—Wilhelm Anderson appeared discussing the same problem. Even mathematically his treatment was identical to mine. So the satisfaction is that I was able to do it independently. I do not intend sending it for publication.1
A careful look shows that the Anderson paper Chandra referred to could not have been the one about relativistic corrections to electron degeneracy2 that Nauenberg refers to. In his immediately previous letter to his father in early June 1929, Chandra listed five papers, four of which had been submitted for publication. About the fifth one, he wrote, “I may send a note to ‘Nature’ this week on ‘The Einstein’s method of deriving the Planck’s formula and the new statistics.’ ” It was this paper Chandra had in mind in the 30 August letter to his father, and it has nothing to do with the subject of white dwarfs.
Chandra had come across Ralph Fowler’s paper but did not work on the theory of white dwarfs until the early spring and summer of 1930. At the beginning of the year, he learned that he had been awarded a Government of India scholarship to go to England, and he opted to go to Cambridge University to work under Fowler. He had with him a paper he had completed before his departure from Madras. By combining Fowler’s ideas with Arthur Eddington’s polytropic considerations for a star, Chandra had been able to obtain a more detailed picture of a white dwarf. One conclusion in the paper, that the central density was six times the mean density, led him to think of the necessity of relativistic effects and the derivation of the critical mass limit. To the best of my knowledge, Chandra was not aware of either Anderson’s or Edmund Stoner’s work before he arrived in Cambridge.
Chandra was certain about his calculation of the maximum mass. But it is not surprising that he said, “At first I didn’t understand what this limit meant and I didn’t know how it would end.” Chandra’s conclusion that a star has a mass limit that can be calculated in terms of fundamental laboratory constants would cause anyone to be puzzled, especially coming as it did from one who had just completed his undergraduate degree and who admittedly had “knowledge of physics, astronomy, and mathematics rudimentary in the extreme at the time, the spring and early summer of 1930.”3
Chandra was confident that his result was correct in the extreme relativistic limit (the ideal white dwarf, as he said). In the following years, he did try to fit various models in the framework of Eddington’s standard model for a star and to incorporate some of Edward Arthur Milne’s ideas for collapsed configurations. Since stars much more massive than the critical mass existed in nature, it was not unreasonable to try various ideas. How is the idea that a star is a uniform, homogeneous sphere of constant density in Anderson’s and Stoner’s work any less ad hoc than what Chandra was trying to do?
Nauenberg contests Freeman Dyson’s account (Physics Today, December 2010, page 44) and mine (page 38) about reactions of Fowler and Milne to the two papers Chandra presented on his arrival in Cambridge.4 Here is what Chandra wrote:
Soon after arriving in England, I showed these results [drafts of the two papers] to R. H. Fowler. Fowler drew my attention to two papers by Stoner,[5] one of which had appeared earlier that summer. In these two papers Stoner had considered the energetics of homogeneous spheres on the assumption that the Fermi–Dirac distribution prevailed in them. While Stoner’s results gave some valid inequalities for the problem, he had not derived the structure of the equilibrium configurations in which all the governing equations are satisfied. Fowler, of course, appreciated this difference, and he was satisfied with my detailed results pertaining to the nonrelativistic configurations. But he appeared skeptical of my result on the critical mass, and so was E. A. Milne, to whom he had communicated it.6
Nauenberg is also mistaken with regard to Milne’s 2 November 1930 letter “promptly” responding to Chandra. Milne was referring to the paper titled “The highly collapsed configurations of a stellar mass”7 and not to the maximum-mass paper that Fowler had sent to him in September. Reference 7 is about the extension of Milne’s work on collapsed configurations, including a relativistically degenerate core. Chandra stated that clearly in both the abstract and the summary. Besides, why would Chandra send it to the Astrophysical Journal on 12 November 1930 if Milne had a positive response for its publication in the Monthly Notices of the Royal Astronomical Society on 2 November?
Milne remained skeptical about the maximum mass limit, since it contradicted his basic assumption that every star should have a degenerate core obeying Fermi–Dirac statistics.8
Finally, as for recognition and acclaim at the time (1934–35), it is a matter of perception. Eddington’s authority prevailed, and a cloud of suspicion, especially among astronomers, shadowed Chandra’s work.9
As regards Eric Blackman’s comment, appropriate references have been amply recorded in published papers. In an earlier article in Physics Today (October 1982, page 33), I noted, “Von Wilhelm Anderson and Edmund Clifton Stoner had independently considered relativistic effects on electron degeneracy. Their work implied the existence of a limiting mass.” In a comprehensive review, Werner Israel, after a detailed discussion of the works of Anderson and Stoner, writes,
In the calculations of Stoner and Anderson the astrophysical aspects were handled very schematically: the star was simply idealized as a sphere of uniform density. In view of the crudeness of this approxi-mation it must be considered fortuitous their estimates for the critical mass came so close to the correct value of 1.44 solar masses (for a helium star). The definitive value was obtained by the 19-year-old Subrahmanyan Chandrasekhar in July 1930 (without knowing at the time of Stoner’s work) during a sea voyage from India to England.10
As Blackman points out, Chandra also recorded appropriate references to the papers of Anderson and Stoner. In “The highly collapsed configurations of a stellar mass (Second paper)”11 Chandra includes the following bibliographic footnote referring to the equation of state of degenerate electrons,
The law P = K2(ρ)4/3 was first used by the author in his paper on “Highly Collapsed Configurations,”etc. (M.N., 91, 456, 1931). This law also has been derived by E. C. Stoner (M.N., 92, 444, 1932), T. E. Sterne (M.N., 93, 764, 1933), and is also implicitly contained in J. Frenkel (Z.f. Physik, 50, 234, 1928). The law has also been used by L. Landau (Physik. Zeits. d. Soviet Union, I, 285, 1932). It may also be pointed out that the law P = K2(ρ)4/3 is implicit in certain equations in a paper by F. Jutner (Z.f. Physik, 47, 542, 1928), equations in #13, 17; our equation (6) above is a limiting case of Jutner’s integral Q(α, γ; +I). This last work of Jutner is related to his earlier work on the relativistic theory of an ideal classical gas, for a convenient summary of which see W. Pauli, Relativitatstheorie (Leipzig, Teubner), #49.
I leave it to Blackman to prioritize and place Chandra in his proper place.
