Tag / white dwarf

by David D. Nolte

Orion’s Dog: The Serious Science of Sirius

The constellation Orion strides high across the heavens on cold crisp winter nights in the North, followed at his heel by his constant companion, Canis Major, the Great Dog.  Blazing blue from the Dog’s proud chest is the star Sirius, the Dog Star, the brightest star in the night sky.  Although it is only the seventh closest star system to our sun, the other six systems host dimmer dwarf stars. Sirius, on the other hand, is a young bright star burning blue in the night.  It is an infant star, really, only as old as 5% the age of our sun, coming into being when Dinosaurs walked our planet.

The Sirius star system is a microcosm of mankind’s struggle to understand the Universe.  Because it is close and bright, it has become the de facto bench-test for new theories of astrophysics as well as for new astronomical imaging technologies.  It has played this role from the earliest days of history, when it was an element of religion rather than of science, down to the modern age as it continues to test and challenge new ideas about quantum matter and extreme physics.

Sirius Through the Ages

To the ancient Egyptians, Sirius was the star Sopdet, the welcome herald of the flooding of the Nile when it rose in the early morning sky of autumn.  The star was associated with Isis of the cow constellation Hathor (Canis Major) following closely behind Osiris (Orion).  The importance of the annual floods for the well-being of the ancient culture cannot be underestimated, and entire religions full of symbolic significance revolved around the heliacal rising of Sirius.

Fig. Canis Major.

To the Greeks, Sirius was always Sirius, although no one even as far back as Hesiod in the 7th century BC could recall where it got its name.  It was the dog star, as it was also to the Persians and the Hindus who called it Tishtrya and Tishya, respectively.  The loss of the initial “T” of these related Indo-European languages is a historical sound shift in relation to “S”, indicating that the name of the star dates back at least as far as the divergence of the Indo-European languages around the fourth millennium BC.  (Even more intriguing is the same association of Sirius with  dogs and wolves by the ancient Chinese and by Alaskan Innuits, as well as by many American Indian tribes, suggesting that the cultural significance of the star, if not its name, may have propagated across Asia and the Bering Strait as far back as the end of the last Ice Age.)  As the brightest star of the sky, this speaks to an enduring significance for Sirius, dating back to the beginning of human awareness of our place in nature.  No culture was unaware of this astronomical companion to the Sun and Moon and Planets.

The Greeks, too, saw Sirius as a harbinger, not for life-giving floods, but rather of the sweltering heat of late summer.  Homer, in the Iliad, famously wrote:

And aging Priam was the first to see him

sparkling on the plain, bright as that star

in autumn rising, whose unclouded rays

shine out amid a throng of stars at dusk—

the one they call Orion's dog, most brilliant,

yes, but baleful as a sign: it brings

great fever to frail men. So pure and bright

the bronze gear blazed upon him as he ran.

The Romans expanded on this view, describing “the dog days of summer”, which is a phrase that echoes till today as we wait for the coming coolness of autumn days.

The Heavens Move

The irony of the Copernican system of the universe, when it was proposed in 1543 by Nicolaus Copernicus, is that it took stars that moved persistently through the heavens and fixed them in the sky, unmovable.  The “fixed stars” became the accepted norm for several centuries, until the peripatetic Edmund Halley (1656 – 1742) wondered if the stars really did not move.  From Newton’s new work on celestial dynamics (the famous Principia, which Halley generously paid out of his own pocket to have published not only because of his friendship with Newton, but because Halley believed it to be a monumental work that needed to be widely known), it was understood that gravitational effects would act on the stars and should cause them to move.

Fig. Halley’s Comet

In 1710 Halley began studying the accurate star-location records of Ptolemy from one and a half millennia earlier and compared  them with what he could see in the night sky.  He realized that the star Sirius had shifted in the sky by an angular distance equivalent to the diameter of the moon.  Other bright stars, like Arcturus and Procyon, also showed discrepancies from Ptolemy.  On the other hand, dimmer stars, that Halley reasoned were farther away, showed no discernible shifts in 1500 years.  At a time when stellar parallax, the apparent shift in star locations caused by the movement of the Earth, had not yet been detected, Halley had found an alternative way to get at least some ranked distances to the stars based on their proper motion through the universe.  Closer stars to the Earth would show larger angular displacements over 1500 years than stars farther away.  By being the closest bright star to Earth, Sirius had become a testbed for observations and theories of the motions of stars.  With the confidence of the confirmation of the nearness of Sirius to the Earth, Jacques Cassini claimed in 1714 to have measured the parallax of Sirius, but Halley refuted this claim in 1720.  Parallax would remain elusive for another hundred years to come.

The Sound of Sirius

Of all the discoveries that emerged from nineteenth century physics—Young’s fringes, Biot-Savart law, Fresnel lens, Carnot cycle, Faraday effect, Maxwell’s equations, Michelson interferometer—only one is heard daily—the Doppler effect [1].  Doppler’s name is invoked every time you turn on the evening news to watch Doppler weather radar.  Doppler’s effect is experienced as you wait by the side of the road for a car to pass by or a jet to fly overhead.  Einstein may have the most famous name in physics, but Doppler’s is certainly the most commonly used.   

Although experimental support for the acoustic Doppler effect accumulated quickly, corresponding demonstrations of the optical Doppler effect were slow to emerge.  The breakthrough in the optical Doppler effect was made by William Huggins (1824-1910).  Huggins was an early pioneer in astronomical spectroscopy and was famous for having discovered that some bright nebulae consist of atomic gases (planetary nebula in our own galaxy) while others (later recognized as distant galaxies) consist of unresolved emitting stars.  Huggins was intrigued by the possibility of using the optical Doppler effect to measure the speed of stars, and he corresponded with James Clerk Maxwell (1831-1879) to confirm the soundness of Doppler’s arguments, which Maxwell corroborated using his new electromagnetic theory.  With the resulting confidence, Huggins turned his attention to the brightest star in the heavens, Sirius, and on May 14, 1868, he read a paper to the Royal Society of London claiming an observation of Doppler shifts in the spectral lines of the star Sirius consistent with a speed of about 50 km/sec [2].

Fig. Doppler spectroscopy of stellar absorption lines caused by the relative motion of the star (in this illustration the orbiting exoplanet is causing the star to wobble.)

The importance of Huggins’ report on the Doppler effect from Sirius was more psychological than scientifically accurate, because it convinced the scientific community that the optical Doppler effect existed.  Around this time the German astronomer Hermann Carl Vogel (1841 – 1907) of the Potsdam Observatory began working with a new spectrograph designed by Johann Zöllner from Leipzig [3] to improve the measurements of the radial velocity of stars (the speed along the line of sight).  He was aware that the many values quoted by Huggins and others for stellar velocities were nearly the same as the uncertainties in their measurements.  Vogel installed photographic capabilities in the telescope and spectrograph at the Potsdam Observatory [4] in 1887 and began making observations of Doppler line shifts in stars through 1890.  He published an initial progress report in 1891, and then a definitive paper in 1892 that provided the first accurate stellar radial velocities [5].  Fifty years after Doppler read his paper to the Royal Bohemian Society of Science (in 1842 to a paltry crowd of only a few scientists), the Doppler effect had become an established workhorse of quantitative astrophysics. A laboratory demonstration of the optical Doppler effect was finally achieved in 1901 by Aristarkh Belopolsky (1854-1934), a Russian astronomer, by constructing a device with a narrow-linewidth light source and rapidly rotating mirrors [6].

White Dwarf

While measuring the position of Sirius to unprecedented precision, the German astronomer Friedrich Wilhelm Bessel (1784 – 1846) noticed a slow shift in its position.  (This is the same Bessel as “Bessel function” fame, although the functions were originally developed by Daniel Bernoulli and Bessel later generalized them.)  Bessel deduced that Sirius must have an unseen companion with an orbital of around 50 years.  This companion was discovered by accident in 1862 during a test run of a new lens manufactured by the Clark&Sons glass manufacturing company prior to delivery to Northwestern University in Chicago.  (The lens was originally ordered by the University of Mississippi in 1860, but after the Civil War broke out, the Massachusetts-based Clark company put it up for bid.  Harvard wanted it, but Northwestern got it.)  Sirius itself was redesignated Sirius A, while this new star was designated Sirius B (and sometimes called “The Pup”). 

Fig. White dwarf and planet.

The Pup’s spectrum was measured in 1915 by Walter Adams (1876 – 1956) which put it in the newly-formed class of “white dwarf” stars that were very small but, unlike other types of dwarf stars, they had very hot (white) spectra.  The deflection of the orbit of Sirius A allowed its mass to be estimated at about one solar mass, which was normal for a dwarf star.  Furthermore, its brightness and surface temperature allowed its density to be estimated, but here an incredible number came out: the density of Sirius B was about 30,000 times greater than the density of the sun!  Astronomers at the time thought that this was impossible, and Arthur Eddington, who was the expert in star formation, called it “nonsense”.  This nonsense withstood all attempts to explain it for over a decade.

In 1926, R. H. Fowler (1889 – 1944) at Cambridge University in England applied the newly-developed theory of quantum mechanics and the Pauli exclusion principle to the problem of such ultra-dense matter.  He found that the Fermi sea of electrons provided a type of pressure, called degeneracy pressure, that counteracted the gravitational pressure that threatened to collapse the star under its own weight.  Several years later, Subrahmanyan Chandrasekhar calculated the upper limit for white dwarfs using relativistic effects and accurate density profiles and found that a white dwarf with a mass greater than about 1.5 times the mass of the sun would no longer be supported by the electron degeneracy pressure and would suffer gravitational collapse.  At the time, the question of what it would collapse to was unknown, although it was later understood that it would collapse to a neutron star.  Sirius B, at about one solar mass, is well within the stable range of white dwarfs.

But this was not the end of the story for Sirius B [7].  At around the time that Adams was measuring the spectrum of the white dwarf, Einstein was predicting that light emerging from a dense star would have its wavelengths gravitationally redshifted relative to its usual wavelength.  This was one of the three classic tests he proposed for his new theory of General Relativity.  (1 – The precession of the perihelion of Mercury. 2 – The deflection of light by gravity.  3 – The gravitational redshift of photons rising out of a gravity well.)  Adams announced in 1925 (after the deflection of light by gravity had been confirmed by Eddington in 1919) that he had measured the gravitational redshift.  Unfortunately, it was later surmised that he had not measured the gravitational effect but had actually measured Doppler-shifted spectra because of the rotational motion of the star.  The true gravitational redshift of Sirius B was finally measured in 1971, although the redshift of another white dwarf, 40 Eridani B, had already been measured in 1954.

Static Interference

The quantum nature of light is an elusive quality that requires second-order experiments of intensity fluctuations to elucidate them, rather than using average values of intensity.  But even in second-order experiments, the manifestations of quantum phenomenon are still subtle, as evidenced by an intense controversy that was launched by optical experiments performed in the 1950’s by a radio astronomer, Robert Hanbury Brown (1916 – 2002).  (For the full story, see Chapter 4 in my book Interference from Oxford (2023) [8]).

Hanbury Brown (he never went by his first name) was born in Aruvankandu, India, the son of a British army officer.  He never seemed destined for great things, receiving an unremarkable education that led to a degree in radio engineering from a technical college in 1935.  He hoped to get a PhD in radio technology, and he even received a scholarship to study at Imperial College in London, when he was urged by the rector of the university, Sir Henry Tizard, to give up his plans and join an effort to develop defensive radar against a growing threat from Nazi Germany as it aggressively rearmed after abandoning the punitive Versailles Treaty.  Hanbury Brown began the most exciting and unnerving five years of his life, right in the middle of the early development of radar defense, leading up to the crucial role it played in the Battle of Britain in 1940 and the Blitz from 1940 to 1941.  Partly due to the success of radar, Hitler halted night-time raids in the Spring of 1941, and England escaped invasion.

In 1949, fourteen years after he had originally planned to start his PhD, Hanbury Brown enrolled at the relatively ripe age of 33 at the University of Manchester.  Because of his background in radar, his faculty advisor told him to look into the new field of radio astronomy that was just getting started, and Manchester was a major player because it administrated the Jodrell Bank Observatory, which was one of the first and largest radio astronomy observatories in the World.  Hanbury Brown was soon applying all he had learned about radar transmitters and receivers to the new field, focusing particularly on aspects of radio interferometry after Martin Ryle (1918 – 1984) at Cambridge with Derek Vonberg (1921 – 2015) developed the first radio interferometer to measure the angular size of the sun [9] and of radio sources on the Sun’s surface that were related to sunspots [10].  Despite the success of their measurements, their small interferometer was unable to measure the size of other astronomical sources.  From Michelson’s formula for stellar interferometry, longer baselines between two separated receivers would be required to measure smaller angular sizes.  For his PhD project, Hanbury Brown was given the task of designing a radio interferometer to resolve the two strongest radio sources in the sky, Cygnus A and Cassiopeia A, whose angular sizes were unknown.  As he started the project, he was confronted with the problem of distributing a stable reference signal to receivers that might be very far apart, maybe even thousands of kilometers, a problem that had no easy solution. 

After grappling with this technical problem for months without success, late one night in 1949 Hanbury Brown had an epiphany [11], wondering what would happen if the two separate radio antennas measured only intensities rather than fields.  The intensity in a radio telescope fluctuates in time like random noise.  If that random noise were measured at two separated receivers while trained on a common source, would those noise patterns look the same?  After a few days considering this question, he convinced himself that the noise would indeed share common features, and the degree to which the two noise traces were similar should depend on the size of the source and the distance between the two receivers, just like Michelson’s fringe visibility.  But his arguments were back-of-the-envelope, so he set out to find someone with the mathematical skills to do it more rigorously.  He found Richard Twiss.

Richard Quentin Twiss (1920 – 2005), like Hanbury Brown, was born in India to British parents but had followed a more prestigious educational path, taking the Mathematical Tripos exam at Cambridge in 1941 and receiving his PhD from MIT in the United States in 1949.  He had just returned to England, joining the research division of the armed services located north of London, when he received a call from Hanbury Brown at the Jodrell Bank radio astronomy laboratory in Manchester.  Twiss travelled to meet Hanbury Brown in Manchester, who put him up in his flat in the neighboring town of Wilmslow.  The two set up the mathematical assumptions behind the new “intensity interferometer” and worked late into the night. When Hanbury Brown finally went to bed, Twiss was still figuring the numbers.  The next morning, the tall and lanky Twiss appeared in his silk dressing gown in the kitchen and told Hanbury Brown, “This idea of yours is no good, it doesn’t work”[12]—it would never be strong enough to detect the intensity from stars.  However, after haggling over the details of some of the integrals, Hanbury Brown, and then finally Twiss, became convinced that the effect was real.  Rather than fringe visibility, it was the correlation coefficient between two noise signals that would depend on the joint sizes of the source and receiver in a way that captured the same information as Michelson’s first-order fringe visibility.  But because no coherent reference wave was needed for interferometric mixing, this new approach could be carried out across very large baseline distances.

After demonstrating the effect on astronomical radio sources, Hanbury Brown and Twiss took the next obvious step: optical stellar intensity interferometry.  Their work had shown that photon noise correlations were analogous to Michelson fringe visibility, so the stellar intensity interferometer was expected to work similarly to the Michelson stellar interferometer—but with better stability over much longer baselines because it did not need a reference.  An additional advantage was the simple light collecting requirements.  Rather than needing a pair of massively expensive telescopes for high-resolution imaging, the intensity interferometer only needed to point two simple light collectors in a common direction.  For this purpose, and to save money, Hanbury Brown selected two of the largest army-surplus anti-aircraft searchlights that he could find left over from the London Blitz.  The lamps were removed and replaced with high-performance photomultipliers, and the units were installed on two train cars that could run along a railroad siding that crossed the Jodrell Bank grounds.

Fig. Stellar Interferometers: (Left) Michelson Stellar Field Interferometer. (Right) Hanbury Brown Twiss Stellar Intensity Interferometer.

The target of the first test of the intensity interferometer was Sirius, the Dog Star.  Sirius was chosen because it is the brightest star in the night sky and was close to Earth at 8.6 light years and hence would be expected to have a relatively large angular size.  The observations began at the start of winter in 1955, but the legendary English weather proved an obstacle.  In addition to endless weeks of cloud cover, on many nights dew formed on the reflecting mirrors, making it necessary to install heaters.  It took more than three months to make 60 operational attempts to accumulate a mere 18 hours of observations [13].  But it worked!  The angular size of Sirius was measured for the first time. It subtended an angle of approximately 6 milliarcseconds (mas), which was well within the expected range for such a main sequence blue star.  This angle is equivalent to observing a house on the Moon from the Earth.  No single non-interferometric telescope on Earth, or in Earth orbit, has that kind of resolution, even today.  Once again, Sirus was the testbed of a new observational technology.  Hanbury Brown and Twiss went on the measure the diameters of dozens of stars.

Adaptive Optics

Any undergraduate optics student can tell you that bigger telescopes have higher spatial resolution.  But this is only true up to a point.  When telescope diameters become not much bigger than about 10 inches, the images they form start to dance, caused by thermal fluctuations in the atmosphere.  Large telescopes can still get “lucky” at moments when the atmosphere is quiet, but this usually only happens for a fraction of a second before the fluctuation set in again.  This is the primary reason that the Hubble Space Telescope was placed in Earth orbit above the atmosphere, and why the James Webb Space Telescope is flying a million miles away from the Earth.  But that is not the end of Earth-based large telescoped.  The Very Large Telescope (VLT) has a primary diameter of 8 meters, and the Extremely Large Telescope (ELT), coming online soon, has an even bigger diameter of 40 meters.  How do these work under the atmospheric blanket?  The answer is adaptive optics.

Adaptive optics uses active feedback to measure the dancing images caused by the atmosphere and uses the information to control a flexible array of mirror elements to exactly cancel out the effects of the atmospheric fluctuations.  In the early days of adaptive-optics development, the applications were more military than astronomic, but advances made in imaging enemy satellites soon was released to the astronomers.  The first civilian demonstrations of adaptive optics were performed in 1977 when researchers at Bell Labs [14] and at the Space Sciences Lab at UC Berkeley [15] each made astronomical demonstrations of improved seeing of the star Sirius using adaptive optics.  The field developed rapidly after that, but once again Sirius had led the way.

Star Travel

The day is fast approaching when humans will begin thinking seriously of visiting nearby stars—not in person at first, but with unmanned spacecraft that can telemeter information back to Earth.  Although Sirius is not the closest star to Earth—it is 8.6 lightyears away while Alpha Centauri is almost twice as close at only 4.2 lightyears away—it may be the best target for an unmanned spacecraft.  The reason is its brightness. 

Stardrive technology is still in its infancy—most of it is still on drawing boards.  Therefore, the only “mature” technology we have today is light pressure on solar sails.  Within the next 50 years or so we will have the technical ability to launch a solar sail towards a nearby star and accelerate it to a good fraction of the speed of light.  The problem is decelerating the spaceship when it arrives at its destination, otherwise it will go zipping by with only a few seconds to make measurements after its long trek there.

Fig. NASA’s solar sail demonstrator unit (artist’s rendering).

A better idea is to let the star light push against the solar sail to decelerate it to orbital speed by the time it arrives.  That way, the spaceship can orbit the target star for years.  This is a possibility with Sirius.  Because it is so bright, its light can decelerate the spaceship even when it is originally moving at relativistic speeds. By one calculation, the trip to Sirius, including the deceleration and orbital insertion, should only take about 69 years [16].  That’s just one lifetime.  Signals could be beaming back from Sirius by as early as 2100—within the lifetimes of today’s children.

Footnotes

[1] The section is excerpted from D. D. Nolte, The Fall and Rise of the Doppler Effect, Physics Today (2020)

[2] W. Huggins, “Further observations on the spectra of some of the stars and nebulae, with an attempt to determine therefrom whether these bodies are moving towards or from the earth, also observations on the spectra of the sun and of comet II,” Philos. Trans. R. Soc. London vol. 158, pp. 529-564, 1868. The correct value is -5.5 km/sec approaching Earth.  Huggins got the magnitude and even the sign wrong.

[3] in Hearnshaw, The Analysis of Starlight (Cambridge University Press, 2014), pg. 89

[4] The Potsdam Observatory was where the American Albert Michelson built his first interferometer while studying with Helmholtz in Berlin.

[5] Vogel, H. C. Publik. der astrophysik. Observ. Potsdam 1: 1. (1892)

[6] A. Belopolsky, “On an apparatus for the laboratory demonstration of the Doppler-Fizeau principle,” Astrophysical Journal, vol. 13, pp. 15-24, Jan 1901.

[7] https://adsabs.harvard.edu/full/1980QJRAS..21..246H

[8] D. D. Nolte, Interference: The History of Optical Interferometry and the Scientists who Tamed Light (Oxford University Press, 2023)

[9] M. Ryle and D. D. Vonberg, “Solar Radiation on 175 Mc/sec,” Nature, vol. 158 (1946): pp. 339-340.; K. I. Kellermann and J. M. Moran, “The development of high-resolution imaging in radio astronomy,” Annual Review of Astronomy and Astrophysics, vol. 39, (2001): pp. 457-509.

[10] M. Ryle, ” Solar radio emissions and sunspots,” Nature, vol. 161, no. 4082 (1948): pp. 136-136.

[11] R. H. Brown, The intensity interferometer; its application to astronomy (London, New York, Taylor & Francis; Halsted Press, 1974).

[12] R. H. Brown, Boffin : A personal story of the early days of radar and radio astronomy (Adam Hilger, 1991), p. 106.

[13] R. H. Brown and R. Q. Twiss. ” Test of a new type of stellar interferometer on Sirius.” Nature 178, no. 4541 (1956): pp. 1046-1048.

[14] S. L. McCall, T. R. Brown, and A. Passner, “IMPROVED OPTICAL STELLAR IMAGE USING A REAL-TIME PHASE-CORRECTION SYSTEM – INITIAL RESULTS,” Astrophysical Journal, vol. 211, no. 2, pp. 463-468, (1977)

[15] A. Buffington, F. S. Crawford, R. A. Muller, and C. D. Orth, “1ST OBSERVATORY RESULTS WITH AN IMAGE-SHARPENING TELESCOPE,” Journal of the Optical Society of America, vol. 67, no. 3, pp. 304-305, 1977 (1977)

[16] https://www.newscientist.com/article/2128443-quickest-we-could-visit-another-star-is-69-years-heres-how/

by David D. Nolte

Chandrasekhar’s Limit

Arthur Eddington was the complete package—an observationalist with the mathematical and theoretical skills to understand Einstein’s general theory, and the ability to construct the theory of the internal structure of stars.  He was Zeus in Olympus among astrophysicists.  He always had the last word, and he stood with Einstein firmly opposed to the Schwarzschild singularity.  In 1924 he published a theoretical paper in which he derived a new coordinate frame (now known as Eddington-Finkelstein coordinates) in which the singularity at the Schwarzschild radius is removed.  At the time, he took this to mean that the singularity did not exist and that gravitational cut off was not possible [1].  It would seem that the possibility of dark stars (black holes) had been put to rest.  Both Eddington and Einstein said so!  But just as they were writing the obituary of black holes, a strange new form of matter was emerging from astronomical observations that would challenge the views of these giants.

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.

White Dwarf

Binary star systems have always held a certain fascination for astronomers.  If your field of study is the (mostly) immutable stars, then the stars that do move provide some excitement.  The attraction of binaries is the same thing that makes them important astrophysically—they are dynamic.  While many double stars are observed in the night sky (a few had been noted by Galileo), some of these are just coincidental alignments of near and far stars.  However, William Herschel began cataloging binary stars in 1779 and became convinced in 1802 that at least some of them must be gravitationally bound to each other.  He carefully measured the positions of binary stars over many years and confirmed that these stars showed relative changes in position, proving that they were gravitational bound binary star systems [2].  The first orbit of a binary star was computed in 1827 by Félix Savary for the orbit of Xi Ursae Majoris.  Finding the orbit of a binary star system provides a treasure trove of useful information about the pair of stars.  Not only can the masses of the stars be determined, but their radii and densities also can be estimated.  Furthermore, by combining this information with the distance to the binaries, it was possible to develop a relationship between mass and luminosity for all stars, even single stars.  Therefore, binaries became a form of measuring stick for crucial stellar properties.

Comparison of Earth to a white dwarf star with a mass equal to the Sun. They have comparable radii but radically different densities.

One of the binary star systems that Hershel discovered was the pair known as 40 Eridani B/C, which he observed on January 31 in 1783.  Of this pair, 40 Eridani B was very dim compared to its companion.  More than a century later, in 1910 when spectrographs were first being used routinely on large telescopes, the spectrum of 40 Eridani B was found to be of an unusual white spectral class.  In the same year, the low luminosity companion of Sirius, known as Sirius B, which shared the same unusual white spectral class, was evaluated in terms of its size and mass and was found to be exceptionally small and dense [3].  In fact, it was too small and too dense to be believed at first, because the densities were beyond any known or even conceivable matter.  The mass of Sirius B is around the mass of the Sun, but its radius is comparable to the radius of the Earth, making the density of the white star about ten thousand times denser than the core of the Sun.  Eddington at first felt the same way about white dwarfs that he felt about black holes, but he was eventually swayed by the astrophysical evidence.  By 1922 many of these small white stars had been discovered, called white dwarfs, and their incredibly large densities had been firmly established.  In his famous book on stellar structure [4], he noted the strange paradox:  As a star cools, its pressure must decrease, as all gases must do as they cool, and the star would shrink, yet the pressure required to balance the force of gravity to stabilize the star against continued shrinkage must increase as the star gets smaller.  How can pressure decrease and yet increase at the same time?  In 1926, on the eve of the birth of quantum mechanics, Eddington could conceive of no mechanism that could resolve this paradox.  So he noted it as an open problem in his book and sent it to press.

Subrahmanyan Chandrasekhar

Three years after the publication of Eddington’s book, an eager and excited nineteen-year-old graduate of the University in Madras India boarded a steamer bound for England.  Subrahmanyan Chandrasekhar (1910—1995) had been accepted for graduate studies at Cambridge University.  The voyage in 1930 took eighteen days via the Suez Canal, and he needed something to do to pass the time.  He had with him Eddington’s book, which he carried like a bible, and he also had a copy of a breakthrough article written by R. H. Fowler that applied the new theory of quantum mechanics to the problem of dense matter composed of ions and electrons [5].  Fowler showed how the Pauli exclusion principle for electrons, that obeyed Fermi-Dirac statistics, created an energetic sea of electrons in their lowest energy state, called electron degeneracy.  This degeneracy was a fundamental quantum property of matter, and carried with it an intrinsic pressure unrelated to thermal properties.  Chandrasekhar realized that this was a pressure mechanism that could balance the force of gravity in a cooling star and might resolve Eddington’s paradox of the white dwarfs.  As the steamer moved ever closer to England, Chandrasekhar derived the new balance between gravitational pressure and electron degeneracy pressure and found the radius of the white dwarf as a function of its mass.  The critical step in Chandrasekhar’s theory, conceived alone on the steamer at sea with access to just a handful of books and papers, was the inclusion of special relativity with the quantum physics.  This was necessary, because the densities were so high and the electrons were so energetic, that they attained speeds approaching the speed of light. 

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.  He discovered that electron degeneracy pressure could balance the force of gravity if the mass of the white dwarf were smaller than about 1.4 times the mass of the Sun.  But if the dwarf was more massive than this, then even the electron degeneracy pressure would be insufficient to fight gravity, and the star would continue to collapse.  To what?  Schwarzschild’s singularity was one possibility.  Chandrasekhar wrote up two papers on his calculations, and when he arrived in England, he showed them to Fowler, who was to be his advisor at Cambridge.  Fowler was genuinely enthusiastic about  the first paper, on the derivation of the relativistic electron degeneracy pressure, and it was submitted for publication.  The second paper, on the maximum sustainable mass for a white dwarf, which reared the ugly head of Schwarzschild’s singularity, made Fowler uncomfortable, and he sat on the paper, unwilling to give his approval for publication in the leading British astrophysical journal.  Chandrasekhar grew annoyed, and in frustration sent it, without Fowler’s approval, to an American journal, where “The Maximum Mass of Ideal White Dwarfs” was published in 1931 [6].  This paper, written in eighteen days on a steamer at sea, established what became known as the Chandrasekhar limit, for which Chandrasekhar would win the 1983 Nobel Prize in Physics, but not before he was forced to fight major battles for its acceptance.

The Chandrasekhar limit expressed in terms of the Planck Mass and the mass of a proton. The limit is approximately 1.4 times the mass of the Sun. White dwarfs with masses larger than the limit cannot balance gravitational collapse by relativistic electron degeneracy.

Chandrasekhar versus Eddington

Initially there was almost no response to Chandrasekhar’s paper.  Frankly, few astronomers had the theoretical training needed to understand the physics.  Eddington was one exception, which was why he held such stature in the community.  The big question therefore was:  Was Chandrasekhar’s theory correct?  During the three years to obtain his PhD, Chandrasekhar met frequently with Eddington, who was also at Cambridge, and with colleagues outside the university, and they all encouraged Chandrasekhar to tackle the more difficult problem to combine internal stellar structure with his theory.  This could not be done with pen and paper, but required numerical calculation.  Eddington was in possession of an early electromagnetic calculator, and he loaned it to Chandrasekhar to do the calculations.  After many months of tedious work, Chandrasekhar was finally ready to confirm his theory at the 1934 meeting of the British Astrophysical Society. 

The young Chandrasekhar stood up and gave his results in an impeccable presentation before an auditorium crowded with his peers.  But as he left the stage, he was shocked when Eddington himself rose to give the next presentation.  Eddington proceeded to criticize and reject Chandrasekhar’s careful work, proposing instead a garbled mash-up of quantum theory and relativity that would eliminate Chandrasekhar’s limit and hence prevent collapse to the Schwarzschild singularity.  Chandrasekhar sat mortified in the audience.  After the session, many of his friends and colleagues came up to him to give their condolences—if Eddington, the leader of the field and one of the few astronomers who understood Einstein’s theories, said that Chandrasekhar was wrong, then that was that.  Badly wounded, Chandrasekhar was faced with a dire choice.  Should he fight against the reputation of Eddington, fight for the truth of his theory?  But he was at the beginning of his career and could ill afford to pit himself against the giant.  So he turned his back on the problem of stellar death, and applied his talents to the problem of stellar evolution. 

Chandrasekhar went on to have an illustrious career, spent mostly at the University of Chicago (far from Cambridge), and he did eventually return to his limit as it became clear that Eddington was wrong.  In fact, many at the time already suspected Eddington was wrong and were seeking for the answer to the next question: If white dwarfs cannot support themselves under gravity and must collapse, what do they collapse to?  In Pasadena at the California Institute of Technology, an astrophysicist named Fritz Zwicky thought he knew the answer.

Fritz Zwicky’s Neutron Star

Fritz Zwicky (1898—1874) was an irritating and badly flawed genius.  What made him so irritating was that he knew he was a genius and never let anyone forget it.  What made him badly flawed was that he never cared much for weight of evidence.  It was the ideas that mattered—let lesser minds do the tedious work of filling in the cracks.  And what made him a genius was that he was often right!  Zwicky pushed the envelope—he loved extremes.  The more extreme a theory was, the more likely he was to favor it—like his proposal for dark matter.  Most of his colleagues considered him to be a buffoon and borderline crackpot.  He was tolerated by no one—no one except his steadfast collaborator of many years Ernst Baade (until they nearly came to blows on the eve of World War II).  Baade was a German physicist trained at Göttingen and recently arrived at Cal Tech.  He was exceptionally well informed on the latest advances in a broad range of fields.  Where Zwicky made intuitive leaps, often unsupported by evidence, Baade would provide the context.  Baade was a walking Wikipedia for Zwicky, and together they changed the face of astrophysics.

Zwicky and Baade submitted an abstract to the American Physical Society Meeting in 1933, which Kip Thorne has called “…one of the most prescient documents in the history of physics and astronomy” [7].  In the abstract, Zwicky and Baade introduced, for the first time, the existence of supernovae as a separate class of nova and estimated the total energy output of these cataclysmic events, including the possibility that they are the source of some cosmic rays.  They introduced the idea of a neutron star, a star composed purely of neutrons, only a year after Chadwick discovered the neutron’s existence, and they strongly suggested that a supernova is produced by the transformation of a star into a neutron star.  A neutron star would have a mass similar to that of the Sun, but would have a radius of only tens of kilometers.  If the mass density of white dwarfs was hard to swallow, the density of a neutron star was billion times greater!  It would take nearly thirty years before each of the assertions made in this short abstract were proven true, but Zwicky certainly had a clear view, tempered by Baade, of where the field of astrophysics was headed.  But no one listened to Zwicky.  He was too aggressive and backed up his wild assertions with too little substance.  Therefore, neutron stars simmered on the back burner until more substantial physicists could address their properties more seriously.

Two substantial physicists who had the talent and skills that Zwicky lacked were Lev Landau in Moscow and Robert Oppenheimer at Berkeley.  Landau derived the properties of a neutron star in 1937 and published the results to great fanfare.  He was not aware of Zwicky’s work, and he called them neutron cores, because he hypothesized that they might reside at the core of ordinary stars like the Sun.  Oppenheimer, working with a Canadian graduate student George Volkoff at Berkeley, showed that Landau’s idea about stellar cores was not correct, but that the general idea of a neutron core, or rather neutron star, was correct [8].  Once Oppenheimer was interested in neutron stars, he kept going and asked the same question about neutron stars that Chandrasekhar had asked about white dwarfs:  Is there a maximum size for neutron stars beyond which they must collapse?  The answer to this question used the same quantum mechanical degeneracy pressure (now provided by neutrons rather than electrons) and gravitational compaction as the problem of white dwarfs, but it required detailed understanding of nuclear forces, which in 1938 were only beginning to be understood.  However, Oppenheimer knew enough to make a good estimate of the nuclear binding contribution to the total internal pressure and came to a similar conclusion for neutron stars as Chandrasekhar had made for white dwarfs.  There was indeed a maximum mass of a neutron star, a Chandrasekhar-type limit of about three solar masses.  Beyond this mass, even the degeneracy pressure of neutrons could not support gravitational pressure, and the neutron star must collapse.  In Oppenheimer’s mind it was clear what it must collapse to—a black hole (known as gravitational cut-off at that time). This was to lead Oppenheimer and John Wheeler to their famous confrontation over the existence of black holes, which Oppenheimer won, but Wheeler took possession of the battle field [9].

Derivation of the Relativistic Chandrasekhar Limit

White dwarfs are created from the balance between gravitational compression and the degeneracy pressure of electrons caused by the Pauli exclusion principle. When a star collapses gravitationally, the matter becomes so dense that the electrons begin to fill up quantum states until all the lowest-energy states are filled and no more electrons can be added. This results in a balance that stabilizes the gravitational collapse, and the result is a white dwarf with a mass density a million times larger than the Sun.

If the electrons remained non-relativistic, then there would be no upper limit for the size of a star that would form a white dwarf. However, because electrons become relativistic at high enough compaction, if the initial star is too massive, the electron degeneracy pressure becomes limited relativistically and cannot keep the matter from compacting more, and even the white dwarf will collapse (to a neutron star or a black hole). The largest mass that can be supported by a white dwarf is known as the Chandrasekhar limit.

A simplified derivation of the Chandrasekhar limit begins by defining the total energy as the kinetic energy of the degenerate Fermi electron gas plus the gravitational potential energy

The kinetic energy of the degenerate Fermi gas has the relativistic expression


where the Fermi k-vector can be expressed as a function of the radius of the white dwarf and the total number of electrons in the star, as

If the star is composed of pure hydrogen, then the mass of the star is expressed in terms of the total number of electrons and the mass of the proton

The total energy of the white dwarf is minimized by taking its derivative with respect to the radius of the star

When the derivative is set to zero, the term in brackets becomes

This is solved for the radius for which the electron degeneracy pressure stabilizes the gravitational pressure

This is the relativistic radius-mass expression for the size of the stabilized white dwarf as a function of the mass (or total number of electrons). One of the astonishing results of this calculation is the merging of astronomically large numbers (the mass of stars) with both relativity and quantum physics. The radius of the white dwarf is actually expressed as a multiple of the Compton wavelength of the electron!

The expression in the square root becomes smaller as the size of the star increases, and there is an upper bound to the mass of the star beyond which the argument in the square root goes negative. This upper bound is the Chandrasekhar limit defined when the argument equals zero

This gives the final expression for the Chandrasekhar limit (expressed in terms of the Planck mass)

This expression is only approximate, but it does contain the essential physics and magnitude. This limit is on the order of a solar mass. A more realistic numerical calculation yields a limiting mass of about 1.4 times the mass of the Sun. For white dwarfs larger than this value, the electron degeneracy is insufficient to support the gravitational pressure, and the star will collapse to a neutron star or a black hole.

By David D. Nolte, Jan. 7, 2019

[1] The fact that Eddington coordinates removed the singularity at the Schwarzschild radius was first pointed out by Lemaitre in 1933.  A local observer passing through the Schwarzschild radius would experience no divergence in local properties, even though a distant observer would see that in-falling observer becoming length contracted and time dilated. This point of view of an in-falling observer was explained in 1958 by Finkelstein, who also pointed out that the Schwarzschild radius is an event horizon.

[2] William Herschel (1803), Account of the Changes That Have Happened, during the Last Twenty-Five Years, in the Relative Situation of Double-Stars; With an Investigation of the Cause to Which They Are Owing, Philosophical Transactions of the Royal Society of London 93, pp. 339–382 (Motion of binary stars)

[3] Boss, L. (1910). Preliminary General Catalogue of 6188 stars for the epoch 1900. Carnegie Institution of Washington. (Mass and radius of Sirius B)

[4] Eddington, A. S. (1927). Stars and Atoms. Clarendon Press. LCCN 27015694.

[5] Fowler, R. H. (1926). “On dense matter”. Monthly Notices of the Royal Astronomical Society 87: 114. Bibcode:1926MNRAS..87..114F. (Quantum mechanics of degenerate matter).

[6] Chandrasekhar, S. (1931). “The Maximum Mass of Ideal White Dwarfs”. The Astrophysical Journal 74: 81. Bibcode:1931ApJ….74…81C. doi:10.1086/143324. (Mass limit of white dwarfs).

[7] Kip Thorne (1994) Black Holes & Time Warps: Einstein’s Outrageous Legacy (Norton). pg. 174

[8] Oppenheimer was aware of Zwicky’s proposal because he had a joint appointment between Berkeley and Cal Tech.

[9] See Chapter 7, “The Lens of Gravity” in Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press, 2018).