Stars and Stuff (Part 2) ? the Indian Giant

In a previous post, we discussed the determination of the nature of white dwarf stars, such as Sirius B, which are very hot and extremely dense. Sir Arthur Eddington, a leading light of the Royal Astronomical Society (RAS) insisted that such stars, indeed all stars, would have a structure which obeyed the ideal gas law, regardless of the star’s mass. Professor Edward Milne disagreed with this view, citing recent theoretical developments on the subject of highly-compressed matter and there arose heated arguments between the two.

Concurrently in 1930, a young and brilliant physics graduate, Subrahmanyan Chandrasekhar (known as simply ‘Chandra’), had been awarded an Indian government scholarship at the age of 19 and was en voyage to England to read for a doctorate at Cambridge. His supervisor-to-be was a friend of Edward Milne’s, Ralph Fowler, who had already helped Chandra to publish a paper in the Monthly Notices of the RAS whilst he was an undergraduate.

Chandra, of course, was familiar with the works of Eddington and Milne and, whilst enjoying the cramped and rather spartan conditions of the long voyage to England (on a cargo vessel with limited passenger accommodation facilities), busied himself with reviewing Milne’s 1930 paper and examining the implications of the (Einstein’s) relativistic effects of electron degeneracy. Chandra realised that electron degeneracy motion becomes relativistic at very high densities; that is the electron’s momentum increases as a consequence of its relativistic velocity and the outward electron degeneracy pressure available to counter the inward pressure of gravity is at its limit once the speed of light is reached. He also further developed Milne’s models which considered stellar core densities separately from the outer shell densities.

A gifted mathematician, Chandra reworked and rearranged Milne’s equations to subsume effects such as temperature, relativity and other factors involved under relativistic conditions into terms of a constant K2, pressure p and density ? and examined how increasing a stellar core mass affected the relation between core density and radius. The results of his calculation of the limits of the derived relativistic equations of state led him to an astounding conclusion; one, upon arrival at Cambridge, he shared with Fowler who sent his paper to Milne; subsequently Milne had it published in the Monthly Notices of the RAS.

The conclusion was: Quote.

Thus the completely relativistic model considered as the limit of the composite series is a point mass with [core density] ?c = ?! [infinite] The theory gives this result because p = K2? (4/3) [the relativistic equation of state] allows any density provided the pressure [p] be sufficiently high. We are bound to assume therefore that a stage must come beyond which the equation of state p = K2?(4/3) is not valid, for otherwise we are led to the physically inconceivable result that for [0.92 of the solar mass] M = 0.92 ? (-3/2), [radius] r1 = 0, and [core density] ? = ?. End quote.

Chandrasekhar, S. (1931). The highly collapsed configurations of a stellar mass. Monthly Notices of the Royal Astronomical Society 91(5), 456-466.

This is known as the Chandrasekhar Limit for the core mass of a star beyond which, instead of forming a white dwarf, the core collapses to a point: a singularity of infinite density and zero radius. The limit was initially calculated to be 0.92 of the solar mass; that was later revised to 1.44 after the actual mean molecular weight of stellar material was determined to be 2, not 2.5.

This concept was so radical that Chandra covered himself with the phrase “physically inconceivable result” ? like Einstein before him, he could not quite believe the implications of his own calculations! Further on, he says of masses exceeding his calculated limit: Quote.

…the whole configuration has completely “collapsed” into one mass of incompressible matter at the highest density matter is capable of. We have in the limit, so to say, a “solid star.”

End quote.

Chandrasekhar, S. op. cit.


Eddington, when he realised how radically and comprehensively Chandra’s work had disproved the ideal gas model and had shown that there was a mass-dependent continuum of possible conditions ranging from ideal gas to point mass, set about denigrating Chandra. Eddington scoffed that there was “no such thing as relativistic degeneracy”. At a RAS meeting in 1935 he made Chandra the butt of a joke calling the calculation “almost a reductio ad absurdum of the relativistic degeneracy formula” and Chandra was denied an opportunity for a rebuttal. Despite Chandra being supported by many contemporary astrophysicists, Eddington could not be swayed even as the evidence against his ideal gas model mounted and he continued his relentless campaign against Chandra’s findings. For Eddington, the science was settled ? no naysayers were permitted in the field of astrophysics.

Despite Eddington’s fierce criticism of Chandra on this particular topic, he was in fact, along with Ralph Fowler, one of Chandra’s oral examiners; Chandra was awarded his PhD and became a fellow of Trinity College in 1933. In subsequent work, he further evaluated the intermediate conditions where the inexorable force of gravity crushes the electrons out of their orbits to combine with the protons to form a neutron star.

Chandra left England and took up a post at the University of Chicago where he became a professor and published extensively on astrophysics, stellar dynamics and, in his later years, on black holes (the “point mass with ?c = ? !”) and the theory of gravitational waves (of which, more in Part 3). In 1983, more than 50 years after his ground-breaking work ? a delay not in small measure caused by Eddington’s fame, influence and antagonistic intransigence ? Chandra and Fowler were jointly awarded the Nobel Prize for Physics.

Chandra died in 1995; remembered as “a classical applied mathematician whose research was primarily applied in astronomy and whose like will probably never be seen again.”

(to be continued)

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