Saturday, August 27, 2011

Chandrasekhar, Fermi-Dirac and White Dwarfs

My earlier blog post on the collaboration between India's Satyendranath Bose and the great Albert Einstein resulting in the historic Bose-Einstein Statistics in quantum physics has received appreciative reaction from many readers, particularly from those with a background of advanced Physics. Not unexpectedly, some of them have suggested that I extend the story of quantum statistical mechanics by describing the work of the great Indian-born theoretical Physicist Subrahmanian Chandrasekhar on the theory of white dwarf stars in which the other great edifice of quantum statistics, the Fermi-Dirac Statistics, plays a definitive role.   It would be quite logical to attempt this since most of the underlying concepts of quantum statistical physics have already been outlined in my earlier writing, though in a purely qualitative manner and without recourse to any mathematical formalism.  It is for this reason that I find great pleasure in presenting the following narrative featuring a towering astrophysicist and Nobel laureate of the twentieth century with strong Indian roots.  I had taught FD Statistics and Chandrasekhar's theory of white dwarf stars at the post-graduate level with as much relish as I did the work of Bose and Einstein.

I strongly suggest that readers without a background of collegiate Physics should read this only after reading my earlier post on Bose and Einstein.

Birth and Death of Stars

Stars go through a birth-death cycle just like humans and animals, but with an enormously larger life span, measured in billions of years.  The astrophysical evidence for this began to emerge only about a hundred years ago.  Let me first summarize the key steps in stellar evolution as we understand the process today.  What happens to a star towards the end of its life span depends crucially on how massive it was after its initial formation and how much of the initial mass is left over at the terminal stage of its life time. Since our Sun is a stable and typically average star in our galaxy, its known mass has long been employed as the standard unit of mass in Astrophysics.

Stars are born in diffuse interstellar clouds of gas (mostly hydrogen) and dust called nebulae.  The nebular matter coalesces into irregular clumps and eventually develops into dense 'protostars' under gravitational binding.  The surrounding diffuse material is also gravitationally pulled inward by the protostar, thereby increasing its density, as also the pressure and temperature, at the centre.  A protostar becomes a regular star when its core temperature rises to about ten million degrees on the Kelvin scale.  At such temperatures, the hydrogen nuclei fuse together to form helium nuclei, releasing a tremendous amount of energy in the process as envisaged by Einstein's Special Theory of Relativity.  It is this energy that the star emits outward in the form heat, light and other forms of electromagnetic radiation. In due course the outward radiation pressure balances the inward gravitational pressure and the star settles into a steady and stable state for most of its overall lifetime.  Fortunately for us, our Sun is in this phase of its evolution and is expected to remain so for several billion years more.  Therefore, our planet Earth is in no immediate danger from the Sun.  Whatever threat it faces today is almost entirely internal and self inflicted, through monumental levels of human immaturity, greed, indulgence, ignorance and stupidity!

Massive stars generally evolve into red giants when their hydrogen fuel begins to run out inside.  The star's core shrinks as the hydrogen gets exhausted, but its outer layers expand considerably and cool as the star begins to burn hydrogen in the layer surrounding the core.  A red giant can grow hundreds of times larger in diameter than the Sun, but its surface will be relatively cool, making it glow in the red color.  A super massive star may undergo a second expansion after exhausting its hydrogen and become a supergiant. This happens when helium nuclei in the core of massive red giants fuse to form carbon.  These thermonuclear reactions release a second wave of energy and a red giant expands into a supergiant as a result. A supergiant star may grow as large in relation to the Sun as the Sun itself is in relation to the Earth.

When a red giant that is 6-8 times as massive as the Sun runs out of fuel it may become unstable, collapse under its own weight, ejecting its outer layers into space to form a planetary nebula. The extremely dense and compact inner core will continue to glow because of energy trapped inside and become a white dwarf star.  White dwarfs can become as small as the earth, yet be nearly as massive as the Sun.  Given sufficient time, the white dwarf will transform itself into a black dwarf (not a black hole) after losing its energy steadily through radiation.  As we shall see, Chandrasekhar's path breaking work related to the theory of formation of such extraordinary stars, necessitating the application of both Quantum Statistical Mechanics and Relativity to the unique problem in the early thirties of the last century.

Super massive stars, typically more than eight times the mass of the Sun, may use up their nuclear fuel and collapse under their own weight.  The collapse leads to an incredibly gigantic explosion, sends out a shock wave through space followed by a shell of material ejected at great speeds from the star's atmosphere.  Called a supernova, this is a relatively rare event within any galaxy.  Only five such events have been observed within our own Milky Way galaxy in the last millennium (in 1006, 1054, 1181, 1572 and 1604).  The energy emitted in a supernova explosion can equal the brightness of an entire galaxy.  The supernova may completely destroy the star or may leave the core of the star intact.  Such a core may become a neutron star (or pulsar).  A neutron star may contain a mass that is equal to 1.4 to 3 solar masses, compressed into a volume about 20 km in diameter. The core of such an object is so dense that the inward gravitational pressure is sufficient to fuse the protons and electrons of normal matter into a 'sea' of neutrons.   

If the stars are even more massive than those that end up as neutron stars, the gravitational pressure inside the core may be so great as to cause even the neutron star to collapse further into what has come to be known as a black hole, the ultimate fate of a super massive star.  The gravitational force of a black hole is so strong as to hold back even light from escaping from inside. Black holes can become as small as a football, yet be as massive as the Sun.

Very little of these stellar evolutionary processes had been understood at the time of this narrative which brought Subrahmanian Chandrasekhar to the centre stage of Astrophysics in the early thirties.  Indeed, even the mechanism of production of the enormous amounts of energy emitted by stars had not been understood.  In a sense Astrophysics itself was going through its birth pangs.

Chandrasekhar – Madras to Cambridge

Born in 1910, Subrahmanian Chandrasekhar was the nephew (brother's son) of the great C V Raman whose famous discovery on the scattering of light in Calcutta in 1928 came to be called the Raman Effect and won him very quickly the 1930 unshared Nobel Prize for Physics (In sharp contrast, S Chandrasekhar had to wait half a century for his equally well deserved award!).  Unlike his uncle, the rather reticent S Chandrasekhar was interested primarily in mathematics (inspired by the genius of Srinivasa Ramanujan) and theoretical physics and didn't follow in the footsteps of the flamboyant and domineering Raman.  An exceptionally brilliant student in the Presidency College, Madras, Chandrasekhar came under the influence of the well known Cambridge astrophysicist Ralph Fowler and decided to seek his academic fortune in that great university city, whose most famous resident was the egregious Sir Isaac Newton in the seventeenth century.  

[Incidentally, C V Raman had another nephew, Sivaramakrishnan Chandrasekhar, who did follow in the footsteps of Raman and achieved considerable fame on his own for the study of liquid crystalline materials as head of a division in the Raman Research Institute founded by Raman in Bangalore after retiring from the Indian Institute of Science, Bangalore in 1948.  I had the great privilege of working under this S Chandrasekhar as a doctoral student during 1977-81; one of the tools of my study involved Laser Raman Scattering by aligned liquid crystalline molecules.  He was a Fellow of the Royal Society of England (FRS), one of very few Indian scientists to have achieved this distinction.  It is a curious coincidence that my interests also extend to Astrophysics, a discipline whose stature has been immensely enhanced by the contributions of the other S Chandrasekhar.]

In Madras, Chandrasekhar had the good fortune to meet and interact with Arnold Sommerfeld, one of the architects of quantum physics who was on a visit to India. He had studied Sommerfeld's latest works but discovered that their contents were already outdated by developments triggered by the new Quantum Mechanics.  More importantly, he had corresponded with the eminent astrophysicist R H Fowler of Cambridge and this proved very helpful in his quest for a placement in Cambridge under a government of India scholarship.  Not yet twenty years old and with two research papers already under his belt, he reached Cambridge with great expectations and with some rudimentary ideas for his seminal work on white dwarf stars to follow.  Amidst a galaxy of some of the greatest luminaries in contemporary science, he found himself rather out of place initially and was never at ease throughout his life at Cambridge.  Foremost among them was Sir Arthur Eddington, a towering personality and the acknowledged high priest of Astrophysics of that era, who often conducted himself like one.  There were also others like E A Milne, R H Fowler and the hugely talented but reclusive P A M Dirac, co-architect of Fermi-Dirac statistics upon which the theory of white dwarf stars was based.

Degenerate Stars

When a star has exhausted its source of energy inside its core and the outward radiation pressure cannot balance the inward gravitational pressure, the equilibrium is upset and the star should shrink under its own gravity.  If the star is sufficiently massive, the shrinkage should continue unabated and, in principle, the star should shrink into oblivion, giving rise to the mathematical physicist's nightmare of what is called a singularity when the star becomes infinitely dense.  Fortunately, the rules of quantum physics prevent this from happening as we shall now see.

At the temperatures prevailing inside the star's core, the electrons are stripped from the atoms and we have a sea of charged particles called plasma.  These are identical spin-half particles and obey the Pauli Exclusion Principle according to which no two such particles can occupy the same energy state.  Because of the very high density, the electrons lie very close together and constitute a degenerate electron gas, the term degenerate meaning that all the energy levels available are filled up with electrons.  Once a star becomes degenerate in this manner gravity cannot compress it anymore because there is no more space available to be taken up.  The inward pull of gravity is balanced by the outward degeneracy pressure exerted by the electrons distributed among all available energy levels. The contribution of the atomic nuclei in the star's core to this process is negligibly small.

Fermi-Dirac Statistics

Fermi–Dirac statistics describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle. It is named after Enrico Fermi and Paul Dirac, who discovered it independently of each other.

The Italian-born physicist Fermi was one of the most outstanding physicists of the twentieth century, equally adept at both theoretical and experimental physics, something of a rarity in the discipline.  Apart from the F-D statistics, Fermi is best known for his work on the theory of beta decay.  In the area of experimental physics, Fermi pioneered slow neutron induced radioactivity which ultimately led to the discovery of nuclear fission and the harnessing of atomic energy.  He was one of the key players associated with the development of the 'atom bomb' as part of the famous Manhattan Project during the Second World War.  Fermi was the recipient of the unshared Nobel Prize for Physics in 1938.

Paul Dirac, a close friend of S Chandrasekhar at Cambridge, was one of the most brilliant theoretical physicists of the twentieth century.  Apart from his work on the F-D statistics, Dirac played a stellar role in the development of the new quantum mechanics, relativistic quantum theory and quantum electrodynamics.  His work led to the eventual discovery of antiparticles. In 1932 he was appointed Lucasian Professor of Mathematics at Cambridge, the chair once held by the legendary Sir Isaac Newton.  In 1933 he shared the Nobel Prize for Physics with another pioneer of the new quantum mechanics, Erwin Schrodinger of Austria.

F–D statistics applies to identical particles with half-integer spin in a system in thermal equilibrium. The Fermi–Dirac distribution of particle energies over these states takes into account the condition that no two particles can occupy the same state, which has a considerable effect on the properties of the system. Since Fermi–Dirac statistics applies to particles with half-integer spin, they have come to be called fermions. It is most commonly applied to electrons, which are fermions with spin 1/2. Fermi–Dirac statistics is a part of the more general field of quantum statistical mechanics and uses the principles of quantum mechanics.

The energy distribution of electrons in the stellar core is governed by the Fermi-Dirac statistics. R H Fowler had first applied F-D statistics to describe the behavior of such a star, called a white dwarf star, and S Chandrasekhar was to carry it forward to its logical conclusion.

White Dwarf Stars

White dwarfs are old stars that have exhausted their available nuclear fuel and collapsed; yet continue to radiate light from thermal energy trapped in them during their collapse. Currently, there is observational evidence for the existence of about ten thousand white dwarf stars.

Interestingly, the first white dwarf to be recognized as such was the long known 'invisible' companion of Sirius, the brightest star in the night sky. It was originally detected by its gravitational attraction on the larger, brighter star and only later observed visually as a faint object, called Sirius B. It is about 10,000 times fainter than Sirius or about 500 times fainter than the Sun. Its mass is slightly less than that of the Sun, and its size a little less than that of Earth. Its surface temperature of about 25,000 K means that the energy emission per unit area from the surface must be vastly greater than that of the Sun. Because Sirius B is so faint, its surface area and thus its volume must be very small, and its average density is of the order of 100,000 times that of water.

Chandrasekhar Limit

Even while traveling to Cambridge from Madras, Chandrasekhar had developed an elementary theory of white dwarf stars and discussed it with others after settling down there.  Reservations about it expressed by both Milne and Fowler were both valid and convincing.  He had to look at the behavior of the degenerate electron gas on the basis of Einstein's theory of relativity and doing so was a big challenge. After considerable effort he successfully developed an exact theory from which it emerged that there was an upper limit of 1.44 solar mass for any star in its terminal phase to end up as a white dwarf.  Called the Chandrasekhar limit, this was to become one of the most significant discoveries in all of Astrophysics.  Observational evidence is fully consistent with this result. For stars more massive than 1.44 solar masses at the terminal phase there had to be other possibilities, waiting to be understood later.  As we now know, neutron stars and black holes are the prime candidates.

Chandrasekhar's study of the equilibrium of white dwarf stars led to the existence of a mass-radius relationship through which a unique radius could be assigned within the limit to a white dwarf star of a given mass; the larger the mass, the smaller the radius.

White dwarf stars will have exhausted all their nuclear fuel and so have no residual nuclear energy sources. Their compact structure also prevents further gravitational contraction. The energy radiated by them is provided by the residual thermal energy of the ions composing its core. That energy diffuses outward and the white dwarf cools down slowly. Following the complete exhaustion of this reservoir of thermal energy, a process that takes several more billions of years, the white dwarf stops radiating and has by then reached the final stage of its evolution, becoming a black dwarf.  

In 1983 Subrahmanian Chandrasekhar was awarded the Nobel Prize for Physics, half a century after he had done his pioneering work on the theory of white dwarf stars.  Numerous other distinctions came to him, including the editorship of the highly prestigious Astrophysical Journal for an unbroken period of twenty years. He authored some of the most famous technical monographs in astrophysics including An Introduction to the Study of Stellar Structure (1939), Principles of Stellar Dynamics (1942), Radiative Transfer (1950), and Hydrodynamic and Hydromagnetic Stability (1961) after a prolonged and exhaustive study of each of these fields. He also wrote Truth and Beauty: Aesthetics and Motivations in Science (1987) combining his views on science and philosophy. Despite these impressive accomplishments and an abiding liking for English literature, he remained very much Indian in outlook and habitsHe shared the Nobel Prize with American astrophysicist William Fowler who was rewarded for his contributions on the synthesis of elements inside stars.

The Eddington Episode

Chandrasekhar presented a summary of his findings in a symposium in Cambridge towards the end of 1934, attended by many physics luminaries of the day, including Eddington.  Before that, he had informally discussed the topic with several people, including Eddington with whom he had built up a good personal relationship. Before the symposium Eddington had dropped dark hints that he had something decisive to say about Chandrasekhar's findings.  After the presentation, Eddington who had listened to it in stony silence stood up and made some devastatingly sarcastic comments about Chandrasekhar's theory and dismissed the concept of degeneracy, central to the theory, saying that, "there should be a law of Nature to prevent a star from behaving in this absurd way".  He later called Chandrasekhar's description of stellar evolution as 'stellar buffoonery'.  Chandrasekhar's attempt to defend his ideas fell on deaf ears and he was left totally baffled and humiliated by the outcome.  What was worse, none of his associates and well wishers who had privately expressed their agreement with him had the courage to speak out against the high priest's diatribe.  Strangely, they included some of the greatest names in the world of physics in those days.

In all fairness it must be said that Eddington's reputation had been well earned.  He had made seminal contributions in the field of stellar structures and his theories appeared in book form in 1926 as The Internal Constitution of the Stars, which became an important source for training an entire generation of astrophysicists, including Chandrasekhar.  Equally importantly, Eddington had become the best known contemporary interpreter of Einstein's theories of relativity despite the negative influence of the First World War in keeping England and Germany apart even in the field of science.  He was almost wholly responsible for transforming Einstein into an international celebrity overnight after his successful total solar eclipse expedition to a remote island off the west coast of Africa in 1919 to verify a key prediction of general relativity theory.

One of Eddington's popular publications, The Nature of the Physical World, was a major milestone in serious scientific literature aimed at the intelligent lay reader and was one of the earliest such works I remember to have read with great interest.  It preceded my exposure to Einstein's work.

This Eddington-Chandrasekhar episode has been one of the most puzzling in the history of science.  There was no credible evidence to suggest that Eddington's attack was personally or racially motivated. Not only had he been in good terms with Chandrasekhar before it but also he continued to remain so long after it.  Indeed, he had actually helped Chandrasekhar professionally on several occasions, most notably when the latter unexpectedly got elected as a fellow of Cambridge's famous Trinity College. 

Whatever the reason may have been, the episode left a very bitter taste in Chandrasekhar's mouth; he never felt at home thereafter and was all too glad to quit England for America when the opportunity presented itself two years later.  Finding no immediate support for his theory of white dwarf stars, which he was convinced to be perfectly correct as also discovered by the scientific community much later, Chandrasekhar discontinued his hugely promising investigations of stellar evolution and embraced another field.  If he had continued in the field he might even have paved way for the discoveries of neutron stars and black holes that happened much later.

Science and Truth

Authority, dogma, faith, belief, superstition, etc., are human traits found all too commonly in almost all walks of everyday life.  However, one doesn't expect to encounter them in the world of science which is characterized precisely by their absence. But, since science cannot always be separated from what scientists do, it is not uncommon to find these traits in the day to day affairs and actions of scientists. The Eddington affair narrated here is a classic example of this.  The history of science is replete with such episodes that have had an immediate negative impact on the march of scientific progress*.  However, the damage has always been temporary; science in its relentless pursuit of truth has always bounced back and triumphed in the end.  Even Eddington's influence could not negate the white dwarf star theory for too long.  The scientist advocating the theory may have suffered, but not science itself.  This amazing proclivity is what sets science apart from other human pursuits.

[* This is far too important for just a casual reference like this and I intend to expand on the theme in a future blog post]


A V G Rao said...

Well written. Even a casual reader can understand this.

Surya.S.P. said...

Sir,the way u have built up the concept is impressive.It is simple and best in content. Final conclusion drawn is very appealing.

Ashwin Sasheendran MN said...

Sir, i would like to get the derivation of FD statistics from Pauli's Exclution Principle. Will you please help to get it.

Ashwin Sasheendran MN said...

Sir, i would like to get the derivation of FD statistics from Pauli's Exclution Principle. Will you please help to get it.

Dr S N Prasad said...

Ashwin, I shall be happy to meet your request if you provide me your email id. My own email id is: