Chandrasekhar Calculates the Upper Limit of a White Dwarf Star's Mass

Date 1931-1935

Subrahmanyan Chandrasekhar developed a mathematically rigorous theory of the structure of white dwarf stars that placed their maximum mass at 1.4 solar masses.

Locale India; England

Key Figures

• Subrahmanyan Chandrasekhar (1910-1995), Indian-born and British-trained theoretical astrophysicist
• Arthur Stanley Eddington (1882-1944), English astrophysicist
• Ralph H. Fowler (1889-1944), English astrophysicist
• Walter Sydney Adams (1876-1956), American astronomer

Summary of Event

White dwarf stars have challenged and perplexed astronomers since their accidental discovery in the mid-nineteenth century. The German astronomerFriedrich Wilhelm Bessel noted a wobble in the path of the star Sirius as it moved across the sky. After eliminating recognizable sources of error, in 1844 he concluded that a small companion star must be affecting the motion of the larger, brighter Sirius. From the wobble in the motion of the larger star, the mass of the smaller star was calculated to be that of the Sun.

In 1915, Walter Sydney Adams managed to channel the light from the companion star into a spectrograph. The light from the star, now called Sirius B, indicated that the surface of the star was almost as hot as Sirius. From the temperature and the brightness of Sirius B, astronomers calculated that Sirius B had a radius of about 24,000 kilometers (approximately 14,913 miles—about twice that of Earth). Packing a mass nearly that of the Sun into a volume fifty thousand times smaller yielded densities that were much larger than astronomers had ever known: One cubic centimeter (about 0.06 cubic inch) of the star—less than the size of a throat lozenge—would weigh 100 kilograms (about 220 pounds).

Sir Arthur Stanley Eddington, the foremost astrophysicist of his time, was not completely convinced that these very small but bright stars, later called white dwarfs, were indeed so dense. Many other skeptics, however, were convinced by the 1925 measurement of the “redshift” of Sirius B. Light trying to escape from a white dwarf is strongly affected by the extreme gravitational force arising from the large mass of the white dwarf. The photons of light lose energy as they struggle against the intense gravity. The frequency of the light is “shifted” toward the red end of the spectrum (reflecting the loss of energy) as the light struggles to escape. Albert Einstein’s general theory of relativity predicts that light will be affected in this manner by gravity. The amount of “shift” was equal to that predicted by Einstein’s theory.

Eddington’s influential The Internal Constitution of the Stars (1926) attempted to bring together fifty years of work involving the mechanical and physical conditions of stellar interiors. When it came to white dwarfs, his theory ran into problems. In his theory, most of a star’s lifetime was spent balancing the outward pressure of the escaping heat of nuclear reactions with the inward pressure of gravity. Eventually, the store of nuclear fuel would be depleted and the star would collapse into itself, becoming a white dwarf. The atomic nuclei, which make up the mass of the white dwarf, would then keep cooling and the electrons that had been ripped from the nuclei would be able to reattach themselves to the nuclei in the star. The problem was that the amount of energy required to re-form the atoms of the star would be more than that available in the star. In effect, the star would not have enough energy to cool down. This paradox puzzled Eddington.

Eddington believed that the pace of work in the field was quickening and that the newly developed field of quantum mechanics might be able to cast light on the theory of stellar interiors. He was correct on both counts. The paradox introduced by Eddington was resolved shortly after it was stated. Ralph H. Fowler resolved the paradox using the recently developed quantum mechanics, but he showed that white dwarf stars were even stranger than anticipated. The pressure that kept the star from contracting indefinitely was the result not of the temperature of the star but of “electron degeneracy.” In the intense heat and pressure of a star’s interior, electrons are torn away from nuclei and move about freely. In the classical theory, the electrons can move about unrestricted. According to quantum theory, however, the electrons are restricted to a discrete set of energies. In a normal star, electrons typically occupy many of the higher allowed energy levels.

In the interior of a white dwarf star, however, the electrons enter a special energy state. Electrons occupy all the lower energy levels. In this special case, the pressure exerted by the electrons becomes independent of the temperature. The star, according to Fowler, can no longer contract. The electrons cannot be forced into lower energy levels. The electrons are said to be “degenerate” because the electrons have become “neutralized”—they are no longer a factor in determining the resistance to gravitational collapse. Fowler resolved Eddington’s paradox by showing that a white dwarf can resist the force of gravity through electron degeneracy. The temperature of the star no longer matters. White dwarfs can live out their lives slowly cooling off.

Subrahmanyan Chandrasekhar followed the latest developments in astrophysics during his studies in theoretical physics in India. Upon graduation in 1930, he went to Trinity College, Cambridge, on a scholarship. He won a physics contest, for which he received a copy of Eddington’s The Internal Constitution of the Stars. He began to question Eddington’s conclusions concerning white dwarfs and Fowler’s calculations concerning electron degeneracy. He calculated that electrons in the dense core of a white dwarf would be moving at a velocity nearly that of light, so corrections must be made to the classical formulas describing the behavior of matter.

Chandrasekhar made the necessary corrections and realized that the effect was dramatic. For stars with a mass greater than about 1.4 times that of the Sun, the “pressure” exerted by electron degeneracy would not be enough to overcome the force of gravity. Instead of a long, slow cooling off, such stars would continue to contract, apparently indefinitely. Chandrasekhar did not speculate on the ultimate fate of stars of more than 1.4 solar masses. Calculations done years later by others showed that those stars form either neutron stars or black holes.

From 1931 to 1935, Chandrasekhar published a series of papers of his findings. During this time, he worked with Fowler and Eddington. By 1935, Chandrasekhar had developed a detailed, quantitative, mathematically rigorous theory of white dwarf stars, and he fully expected Eddington to accept his theory. Eddington gave no indication to Chandrasekhar that he had any doubts about the surprising results Chandrasekhar’s theory predicted. In 1935, Chandrasekhar was scheduled to present his results to the Royal Astronomical Society. Eddington also presented a paper, but to Chandrasekhar’s surprise it included an attack on Chandrasekhar’s theory.

However, work on white dwarfs continued, and further evidence was presented in support for his calculations. Chandrasekhar’s ideas gained gradual acceptance in the 1940’s and 1950’s as more white dwarfs were discovered and as spectrographic evidence mounted.

Significance

Chandrasekhar’s theory introduced the notion that not all stars behave as benignly in their old age as white dwarfs. He did not speculate what would happen to a star with a mass above the limit. For stars with masses below the limit, he devised a complete theory to account for their properties. He won the Nobel Prize in 1983 for his theoretical studies on the structure and evolution of stars.

Chandrasekhar’s limit is the dividing line between the strange but benign white dwarfs and the truly exotic black holes, pulsars, and neutron stars. It established the possibility that the strange behavior of stars nearing the end of their lives as white dwarfs could get stranger. Chandrasekhar’s legacy is the mathematical order that he brought to the theory of white dwarfs. He continued to bring mathematical order to other areas of astrophysics, including black holes.

Bibliography

Asimov, Isaac. The Collapsing Universe. 1977. Reprint. New York: Pocket Books, 1986. Engagingly written volume, intended for a wide audience, takes particular care to emphasize the immense range of stellar phenomena. Excellent introduction to astrophysics for the lay reader.

Chandrasekhar, Subrahmanyan. Eddington: The Most Distinguished Astrophysicist of His Time. Cambridge, England: Cambridge University Press, 1983. Slim volume presents two Sir Arthur Stanley Eddington Centenary Lectures, encapsulating both Chandrasekhar’s personality and his relationship with Eddington. Reviews Eddington’s contributions to astrophysics with grace and style and politely points out where Eddington was incorrect. Moderately technical.

Cooke, Donald A. The Life and Death of Stars. New York: Crown, 1985. Discusses the life history of stars in clear language, with minimal use of technical terms (which are carefully introduced in the early chapters). Profusely illustrated with both color and black-and-white photographs, charts, and diagrams. Highly recommended as a general introduction to stellar astronomy.

Cropper, William H. Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking. New York: Oxford University Press, 2001. Presents portraits of the lives and accomplishments of important physicists and shows how they influenced one another with their work. Chapter 28 is devoted to Subrahmanyan Chandrasekhar. Includes glossary and index.

Miller, Arthur I. Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes. Boston: Houghton Mifflin, 2005. Provides background on the history of the idea of black holes and describes the debate between Chandrasekhar and Eddington concerning the nature of black holes as well as the implications of that debate for twentieth century science.

Shipman, Harry L. Black Holes, Quasars, and the Universe. 2d ed. Boston: Houghton Mifflin, 1980. Written for nonastronomers and for use as a supplemental text in university courses for nonscientists. Includes discussion of black holes and white dwarfs. Provides summaries throughout.

Tierney, John. “Subrahmanyan Chandrasekhar: Quest for Order.” In A Passion to Know, edited by Allen L. Hammond. New York: Charles Scribner’s Sons, 1984. Interview with Chandrasekhar accompanied by some additional biographical information. Provides an interesting perspective on Chandrasekhar’s personality and work habits; gives minimal attention to the technical details of his theories.