Edmund C. Stoner and the Discovery of the Maximum Mass of White Dwarfs

For example, see Dyson Freeman , “The death of a star”, Nature, ccccxxxviii (2005), 1086.

Bates L. F. , “Edmund Clifton Stoner, 1899–1968”, Biographical memoirs of Fellows of the Royal Society, xv (1969), 201–37.

Cantor G. , “The making of a British theoretical physicist: E. C. Stoner's early career”, The British journal for the history of science, xxvii (1994), 277–90.

Stoner E. C. , Magnetism (London, 1930). This was the first monograph containing the new quantum theory of magnetism.

Stoner E. C. , “The distribution of electrons among atomic levels”, Philosophical magazine, xlvii (1924), 719–36.

Fleming G. N. , “The evolution of Pauli's exclusion principle”, Studies in history and philosophy of modern physics, xxxviii (2007), 202–8. Fleming remarks that “in the paper [ref. 5] Stoner came within a hair's breadth of enunciating the electron Pauli exclusion principle himself”. For an historical description of the origin of the exclusion principle and Stoner's role in its formulation see Heilbron J. L. , “The origins of the exclusion principle”, Historical studies in the physical sciences, xiii (1982), 1982–310. Heilbron's perceptive comment, “of psychological interest is Pauli's continual misstatement of the key observations he took from Stoner”, strikes a chord here, because Chandrasekhar's continual neglect to acknowledge Stoner's priority in the discovery of the unusual properties of white dwarfs is also the main reasons why Stoner's contributions in this field have been forgotten.

Fowler R. H. , “On dense matter”, Monthly notices of the Royal Astronomical Society, lxxxvii (1926), 114–22. Fowler died at the age of fifty-two, and E. A. Milne wrote that “this was the most original paper of his lifetime”, Obituary Notices, ibid., cv (1945), 85. Fowler, and later Chandrasekhar, referred to the degeneracy pressure of electrons as due to Fermi-Dirac statistics, which is based on Pauli's exclusion principle. But white dwarf calculations were done at zero temperature, and in this case quantum statistics does not play any role, and only the exclusion principle is required. Stoner's work, op. cit. (ref. 5), which led to Pauli's formulation of the exclusion principle, indicates that R. H. Fowler's first encounter with this principle occurred through his contact with Stoner.

Actually, the density of the companion of Sirius, one of the only three white dwarfs that were known at the time, was underestimated by an order of magnitude. For the history of the early observation of a white dwarf, see Holberg J. B. Wesemael F. , “The discovery of the companion of Sirius and its aftermath”, Journal for the history of astronomy, xxxviii (2007), 162–74.

Eddington A. S. , The internal constitution of the stars (1st edn 1926; reprinted, New York, 1959), 82–83.

Thomas L. H. , “The calculation of atomic fields”, Proceedings of the Cambridge Philosophical Society, xxiii (1927), 542–8. Thomas had been a student at Trinity College, Cambridge, where Fowler had been appointed a College Lecturer in Mathematics in 1920, but at the time Thomas wrote his paper he was visiting Bohr in Copenhagen. In the case of an atom, the forces are electrostatic which are repulsive between electrons and attractive between an electron and the nucleus. Treating the electrons as a degenerate gas, Thomas arrived at an equation equivalent to the Lane-Emde equation for a polytrope of index 3/2, but with an opposite sign. A year later Enrico Fermi also derived the same equation which is now known as the Thomas-Fermi equation, see Fermi E. , “Über die Anwendung der statistischen Methode auf die Probleme des Atombaues”, Falkenhagen, Quantentheorie und Chemie, Leipziger Votraeger, 1928, 95–111. The application of the n = 3/2 polytrope to obtain the properties of white dwarfs for the non-relativistic equation of state for degenerate electrons was first carried out in 1930, by E. A. Milne (see ref. 21 below), who referred to Thomas's work, and a year later by Chandrasekhar (see ref. 18 below). At about the same time, Landau also derived the extension of the Thomas-Fermi equation for the extreme relativistic equation of state of a degenerate electron gas, corresponding to the Lane-Emden n = 3 polytrope, and obtained the white dwarf limit, op. cit. (ref. 29).

Stoner E. C. , “The limiting density in white dwarfs”, Philosophical magazine, vii (1929), 63–70.

Stoner's method for obtaining the properties of white dwarfs was based on the concept that at equilibrium, the sum of the internal energy and the gravitational energy of the star should be a minimum for a fixed mass of the star. Fowler had assumed that the atoms in a white dwarf were completely ionized, and that the internal energy and pressure was entirely due to a degenerate electron gas, while the ions mainly accounted for the mass of the star. Stoner understood that as the star contracts, the gravitational energy decreases, and since the density increases, the internal energy also increases. Hence, the total energy of the star either decreases or increases during the contraction of the star. By conservation of energy, when the total energy of the star decreases, radiation and/or other forms of energy must be emitted by the star. But without an external source of energy, the total energy of an isolated star cannot increase. Hence the contraction of the star must end if the total energy reaches a minimum, and then the star reaches an equilibrium.

About 35 years ago, without being aware of Stoner's seminal work, I applied the energy minimum principle to obtain the properties of white dwarfs in the uniform density approximation, with an approximate form of the relativistic equation of state similar to Stoner's. Now I find, as expected, that my results were similar to Stoner's. Compare, for example, the mass—radius relation shown here in Fig. 1 with the corresponding Fig. 1 in Nauenberg M. , “Analytic approximations for the mass—radius relation and energy of zero-temperature stars”, The astrophysical journal, clxxv (1972), 417–30. At the time, I sent a pre-print of my article to Chandrasekhar with a cover letter asking for his comments, but unfortunately I did not receive a response; such a response would have alerted me a long time ago about Stoner's work.

Anderson W. , “Über die Grenzdichte der Materie und der Energie”, Zeitschrift für Physik, liv (1929), 851–6. Even before the appearance of Stoner's paper (see below, ref. 16), Anderson attempted to introduce the effect of special relativity on the equation of state of a degenerate electron gas, and he also speculated that the necessary high density could occur in the interior of stars, apparently without being aware of Fowler's paper (op. cit. (ref. 7)), see Anderson W. , “Gewöhnliche Materie und stralende Energie als verschiedene ‘Phasen’ eines und desselben Grundstoffes”, Zeitschrift für Physik, liv (1929), 1929–44.

This relativistic equation of state for a degenerate electron gas is often called the Anderson-Stoner equation. But Anderson's relativistic analysis and his formulation of this equation given in opera cit. (ref. 14) is incorrect.

Stoner E. C. , “The equilibrium of white dwarfs”, Philosophical magazine, ix (1930), 944–63.

Stoner , op. cit. (ref. 11).

Chandrasekhar S. , “The density of white dwarfs”, Philosophical magazine, xi (1931), 592–7.

Eddington , op. cit. (ref. 9).

Stoner E. C. Tyler F. , “A note on condensed stars”, Philosophical magazine, xi (1931), 986–93. In this paper the authors did not obtain the energy minimum for the extreme relativistic equation state by taking density distribution for the n = 3 polytropic solution. This calculation leads to the same value for the critical mass obtained by Chandrasekhar. But Stoner's condition that the derivative of the energy with respect to the central density is zero, is satisfied because the energy itself also vanishes in this limit. These subtle mathematical issues may have been the reason why the authors did not attempt to do this calculation.

Milne E. A. , “The analysis of stellar structure”, Monthly notices of the Royal Astronomical Society, xci (1930), 4–55.

Chandrasekhar S. , “The Ritchmyer Memorial Lecture: Some historical notes”, American journal of physics, xxxvii (1969), 577–84. Chandrasekhar recalls: “Soon after arriving in England, I showed these results to R. H. Fowler. Fowler drew my attention to two papers by Stoner, 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 statistics prevailed in them. While Stoner's result 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 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 communicated it”. Two months after Chandrasekhar arrived in Cambridge, Milne sent him a letter concerning his paper on the maximum mass of a white dwarf which Fowler had forwarded to Milne. Contrary to Chandrasekhar's recollection that Milne appeared sceptical of his result, Milne wrote:. I have been interested in your paper, it seems to me very useful. I have long been aware that [the] relativistic form of the equation of state, under degenerate conditions would be required, but it is hopeless to construct a new theory unless one goes a step at a time and works out each case fully first. So I have deliberately carried through the complete theory of centrally-condensed and collapsed stars with p = Kρ^5/3 for an equation of state. When my paper appears you will be able to revise my estimate using methods similar to the ones I am developing. I think then your paper might well be accepted by R.A.S. for M.N. I believe there is already very great pressure in the November number, so your paper would have to wait a little in any case. (Milne's letter, dated 2 Nov. 1930, is in Chandrasekhar's archive at the University of Chicago.).

According to a widely-quoted account described in Wali's biography, Chandra (ref. 42), 76, during Chandrasekhar's August 1930 voyage from India to England, “it suddenly occurred to him to ask the question: If the central density is so high, will the relativistic effects be important?”. But in a letter to his father dated 30 August 1929, Chandrasekhar wrote: “As far my paper I had it nearly completed, writing it out, a paper by a German, Wilhelm Anderson, appeared discussing the same problem. Even mathematically this treatment was identically to mine. So the satisfaction is that I was able to do it independently. I do not intend sending it for publication.” (Wali, private communication.) In his papers, Anderson, op. cit. (ref. 14), attempted to introduce the effect of special relativity on the equation of state of a degenerate electron gas. Hence, already a year before arriving in Cambridge, Chandrasekhar had become aware that special relativity changed this equation of state; but Anderson's analysis was faulty and Chandrasekhar apparently made the same error in his treatment.

Chandrasekhar S. , “The maximum mass of ideal white dwarfs”, Astrophysical journal, lxxiv (1931), 81–82.

Fowler forwarded Chandrasekhar's result to Milne, who was an astrophysicist at Oxford University. Milne, while acknowledging that Chandrasekhar had worked out the relativistic degenerate star “most beautifully”, wrote to him that “the flaw in your reasoning is that you cannot prove that the solution appropriate to the outer parts of the relativistic degenerate core is Emden's solution, it may be one of the others” ( Wali , op. cit. (ref. 42), 121). For any polytropic solution, the density decreases uniformly from the centre of the star and vanishes at its boundary. Hence, for a sufficiently large central density the extreme relativistic equation of state is a valid approximation in the core, but it would fail near the boundary where the electrons become non-relativistic. Eventually, Chandrasekar was led to the conclusion that a consistent solution for the critical mass required that the central density become infinite, as Stoner had shown earlier in the uniform density approximation, because in this limit the envelope, where the electrons would be non-relativistic, vanishes.

Weart S. , Interview with Chandrasekhar, Niels Bohr Library, American Institute of Physics, 1977.

Chandrasekhar S. , “The highly collapsed configurations of a stellar mass”, Monthly notices of the Royal Astronomical Society, xci (1931), 456–66.

Livanova A. , Landau: A great physicist and teacher, transl. by Sykes J. B. (Oxford, 1980).

Landau L. D. , “On the theory of stars”, Physikalische Zeitschrift der Sowjetunion, i (1932), 285–8. Reprinted in the Collected papers of L. D. Landau, ed. by Haar D. T. (New York, 1965), 60–62. In this paper, Landau does not explain how he arrived at his equation for the Fermi energy, but from his later writings it is clear that his approach was similar to that of Fermi and Thomas, op. cit. (ref. 10), who argued that this energy plus the potential at a given radial distance is a constant.

Thomas , op. cit. (ref. 10).

Landau , op. cit. (ref. 29).

On the basis of the assumption that the molecular weight of white dwarfs is 2, Landau obtained 1.5 solar mass for the value of the critical mass of a white dwarf. This value differs from the accepted value of 1.4 solar mass only because Landau underestimated the mass of the Sun by 7%. Originally, Chandrasekhar's value for the critical mass was given as 0.91 solar mass, because he had taken for the molecular weight the value 2.5 assumed by astronomers at that time.

Landau's paper was submitted for publication on February 1931, before the appearance of Chandrasekhar's paper, which was published in the July 1931 issue of the Astrophysical journal (see ref. 24). But in his “Historical notes”, op. cit. (ref. 22), 583, Chandrasekhar claimed that “Landau isolated the critical mass apparently without knowledge of my results published two years earlier [my italics]”, giving 1933 as the publication date of Landau's article, although it had appeared a year earlier. In a footnote in his biography Chandra, op. cit. (ref. 42), 121, Wali “points out” that Landau's paper was published “a year later” than Chandrasekhar's, without clarifying that there was a year's delay due to the slow publication rate of the Soviet journal where Landau's paper appeared. Adding to this confusion, in his book Empire of the stars, A. Miller stated that Landau made his calculations after he had returned to the Soviet Union, and that “he was unaware of Chandra's earlier work because as a physicist Landau did not read the Astrophysical journal“, op. cit. (ref. 42), 158.

Chandrasekhar , op. cit. (ref. 24).

During a later visit with Bohr in Copenhagen, on the same day that the news of Chadwick's discovery of the neutron appeared, Landau suggested the possibility of formation of neutron stars, where the source of internal pressure was due to degenerate neutrons rather than to electrons. But he did not publish his idea until six years later, in Landau L. , “The origin of stellar energy”, Nature, cxli (1938), 333–4.

By applying minimum energy principle, Stoner obtained an analytic expression that gave the mass—density relation in parametric form, showing that the density is a function that increases monotonically, and more rapidly than the square of the star's mass. In particular, he obtained the fundamental result that the density approaches infinity for a finite mass. This is the limiting mass of white dwarfs, in which the mass scale is entirely determined by some of the fundamental constants of Nature.

Chandrasekhar S. , “The highly collapsed configurations of a stellar mass (second paper)”, Monthly notices of the Royal Astronomical Society, xcv (1935), 207–25. Chandrasekhar's numerical results are given in Table I of his paper which is reproduced as Table 25 in his book An introduction to the study of stellar structure (Chicago, 1939). The curve given here in Fig. 1, which is based on Stoner's 1930 analytic calculation (see op. cit. (ref. 16)), is nearly identical to Fig. 2 in Chandrasekhar's paper, reproduced as Fig. 31 in his book.

Milne , op. cit. (ref. 21).

Chandrasekhar , op. cit. (ref. 27).

Chandrasekhar S. , Eddington, the most distinguished astrophysicist of his time (Cambridge, 1983).

Stoner E. C. , “The minimum pressure of a degenerate electron gas”, Monthly notices of the Royal Astronomical Society, xcii (1932), 651–61, and “Upper limits for densities and temperatures in stars”, ibid., 662–76. In the last of his five papers on white dwarfs, Stoner followed Eddington's suggestion to apply his relativistic equation of state for a degenerate electron gas taking into account the effect of radiation pressure on the equilibrium state of dense stars.

Wali K. C. , Chandra: A biography of S. Chandrasekha (Chicago, 1991). Miller A. , Empire of the stars: Obsession, friendship, and betrayal in the quest for black holes (Boston, 2005).

Chandrasekhar S. , “On stars, their evolution and their stability”, Nobel lectures in physics 1981–1990 (Singapore, 1995), 142–64. This paper contains a complete list of Chandrasekhar's papers on white dwarfs.

At a meeting of an American Physical Society at Stanford University in December 1933, Walter Baade and Fritz Zwicky made the suggestion that the origin of supernova explosions was due to the collapse of massive stars into neutron stars. But apparently they were unaware of the existence of a white dwarf mass limit, because this limit was not mentioned in the abstract of their report which appeared as a letter to the editor, “Remarks on super-novae and cosmic rays”, Physical review, xlvi (1934), 76–77.

In 1939 Chandrasekhar met Eddington during dinner at high table in Cambridge and asked him: “How much of your fundamental theory depends on your ideas on relativistic degeneracy?” Eddington replied, “Why, all of it”. Miller , op. cit. (ref. 42), 131.

Eddington A. S. , “On relativistic degeneracy”, Monthly notices of the Royal Astronomical Society, xcv (1935), 194–206; “Note on relativistic degeneracy”, Monthly notices of the Royal Astronomical Society, xcvi (1935), 1935–21; “The pressure of a degenerate electron gas and related problems”, Proceedings of the Royal Society, A clii (1935), 253–72. Eddington's re-examination of relativistic degeneracy led him to some extreme statements such as: “The Stoner-Anderson modification is fallacious … a rigorous treatment leads to the original [Fowler] equation of state”, and “The Stoner-Anderson formula does not exist”, quoted in Mestel L. , “Arthur Stanley Eddington: Pioneer of stellar structure theory”, Journal of astronomical history and heritage, vii (2004), 2004–73.

A detailed critique of Eddington's objections to Stoner's relativistic equation of state for a degenerate electron gas can be found in Schatzman E. , White dwarfs (Amsterdam, 1958), 68–73.

Pauli, whose opinion was also requested, responded sarcastically that “Eddington did not understand physics”, Wali , op. cit. (ref. 42), 131.

On 23 January 1931, replying to the umpteenth letter from Chandrasekhar, Leon Rosenfeld wrote with respect to Eddington's objection to relativistic degenerary, “Wouldn't it be a good policy to leave him alone, instead of losing one's time and temper in fruitless arguments?”, Wali , op. cit. (ref. 42), 130.

Møller C. Chandrasekhar S. , “Relativistic degeneracy”, Monthly notices of the Royal Astronomical Society, xcv (1935), 673–6.

Peierls R. , “Note on the derivation of the equation of state for a degenerate relativistic gas”, Monthly notices of the Royal Astronomical Society, xcvi (1936), 780–4.

Chandrasekar , op. cit. (ref. 37).

Stoner E. C. , “The minimum pressure of a degenerate electron gas”, Monthly notices of the Royal Astronomical Society, xcii (1932), 651–61.

McDougall J. Stoner E. C. , “Computation of Fermi-Dirac Functions”, Philosophical transactions of the Royal Society of London , A ccxxxvii (1938), 67–104. During the discussion period after his presentation at the 1939 conference on astrophysics in Paris, Chandrasekhar mentioned that “one of his pupils and Dr. Stoner are working on the problem of what happens in the region in which the equation of state of degenerate matter approaches the equation of state of a perfect gas”, op. cit. (ref. 44, 2001), 106.

Lee T. D. , “Hydrogen content and energy-productivity mechanism of white dwarfs”, Astrophysical journal, cxi (1950), 625L.

See Chandrasekar , op. cit. (ref. 22), 583.

Chandrasekar S. , An introduction to the study of stellar structure (New York, 1967).

Chandrasekhar , op. cit. (ref. 57), 361.

Chandrasekhar , op. cit. (ref. 37).

Chandrasekhar , op. cit. (ref. 57), 421.

I have found several other occasions when Chandrasekhar used the word “isolate”, which may give a clue to its meaning in the present context. For example, in his book Eddington, the most distinguished astrophysicist of his time, Chandrasekhar stated that when Eddington calculated the relation between mass and pressure in a star, he did not “isolate” its dependence on natural constants, “a surprising omission in view of his later preoccupations with natural constants”, op. cit. (ref. 40), 14. Likewise, in his 1983 Nobel speech, Chandrasekhar remarked that his inequality Eq. (14) “has isolated the combination of natural constants of the dimension of mass”, op. cit. (ref. 43), 144. But it was Stoner who first isolated the natural constants that determine the limiting mass of a white dwarf ( Stoner , op. cit. (ref. 16)).

Stoner , op. cit. (ref. 16), 949–51.

Chandrasekhar , op. cit. (ref. 57), 451.

Stoner , opera cit. (refs 11 and 16).

Chandrasekhar , op. cit. (ref. 57), 451.

Wali , op. cit. (ref. 42).

Weart , op. cit. (ref. 26).

Miller , op. cit. (ref. 42), 14.

Miller , op. cit. (ref. 42), 133.

See, for example, opera cit. (refs 22, 26, 40 and 42).

Chandrasekhar , op. cit. (ref. 40), 50.

Chandrasekhar , op. cit. (ref. 22).

Milne , op. cit. (ref. 21).

By his own admission, initially Chandrasekhar was puzzled by his result, and he was not able to show until several months later that the critical mass was a maximum, and that in this limit the density was infinite. Moreover, he did not pursue the implications of this result, and for several years he assumed that at a certain value of the density, matter would become incompressible, an idea proposed earlier by Milne to avoid infinite density at the centre of his models of a star, op. cit. (ref. 21). Chandrasekhar formulated this idea as follows:

We are bound to assume therefore that a stage must come beyond which the equation of state p = Kρ^4/3 is not valid, for otherwise we are led to the physically inconceivable result that for M = .92M_s [M_s = solar mass and μ = 2.5], r₁ = 0, and ρ = ∞. As we do not know physically what the equation of state is that we are to take, we assume for definiteness the equation for the homogeneous material ρ = ρ_max, where ρ_max is the maximum density of which the material is capable… [op. cit. (ref. 37), 463].

For M > .92M_s Chandrasekhar assumed that there was a homogeneous core with ρ =ρ_max surrounded by a relativistic envelope. This required, however, an unrealistic model of the star, where the density must become discontinuous at an interface. It was not until 1934 that he dropped these crude models, after visiting Ambartsumian in Moscow, who suggested that he integrate directly the equations for gravitational equilibrium by applying the full relativistic equation of state for a degenerate electron gas at arbitrary densities; in other words, that he apply Stoner's equation of state.

Eddington , op. cit. (ref. 9), 114.

It is of interest to inquire what the relation is between the minimum energy principle use by Stoner, and the equation of gravitational equilibrium used by Chandrasekhar. Treating the minimum energy principle as a variational problem in which the total energy is a functional of the density, and this density is a function of the radial distance from the centre of the star, this variational approach leads to the equation of gravitational equilibrium. Its solution is the quantum mechanical ground state of an electron gas in the gravitational field of the ions, maintaining charge neutrality. This connection explains why Stoner and Chandrasekhar obtained the same relations for the density and mass of the star as functions of fundamental constants, but with somewhat different dimensionless quantities. I have not found any evidence that either Stoner or Chandrasekhar was aware of this connection, but it is implicit in Landau's work, op. cit. (ref. 29).

Landau, op. cit. (ref. 29). The mathematical details can be found at http://physics.ucsc.edu∼/michael.

Israel W. , “Dark stars: The evolution of an idea”, in 300 years of gravity , ed. by Hawking S. Israel W. (Cambridge, 1987), 199–276; and “Imploding stars, shifting continents, and the inconstancy of matter”, Foundations of physics, xxvi (1996), 1996–626. Also, A source book in astronomy and astrophysics 1900–1975, edited by Lang K. R. Gingerich O. (Cambridge, MA, 1979), 433–44. Fred Hoyle credited Stoner with the discovery of the white dwarf mass limit (Wali, private communication). His former student, Leon Mestel, also mentioned Stoner and Anderson in connection with this limit, see Mestel L. , “The theory of white dwarfs”, Monthly notices of the Royal Astronomical Society, cxii (1952), 1952–97.

“Discussion of papers 4 and 5 by A. S. Eddington and E. A. Milne”, The observatory, lviii (1935), 37–39.

Milne , op. cit. (ref. 21).

Chandrasekhar , op. cit. (ref. 27).

As late as 1934, Chandrasekhar still thought that the only “possible equations of state” for a degenerate electron gas were either the non-relativistic or the extreme relativistic forms of Stoner's exact equation of state, see Chandrasekhar S. , “The physical state of matter in the interior of stars”, The observatory, lvii (1934), 93–99.

Stoner , “The minimum pressure …” (ref. 41).

Stoner , “Upper limits …” (ref. 41).

Eddington A. S. , “Upper limits to the central temperature and density of a star”, Monthly notices of the Royal Astronomical Society, xci (1931), 444–6.

For a given star mass M and central density ρ_c, Eddington had shown that the central pressure P must be less than the central pressure of a star with uniform density ρ_c, see op. cit. (ref. 85). Eddington's theorem is the inequality.

P_c < ½(4π/3)^1/3GM^2/3ρ_c^4/3.

which is mentioned in his letter to Stoner (see Fig. 2), but with the first factor on the right hand side of this inequality given incorrectly as 1/3.