The Chandrasekhar Limit: A Point-Form Summary. This section of the course notes, and the associated PowerPoint presentation, makes the following critical points: small objects can sustain their structure because of the electrical bonds between molecules and atoms. A bigger object, like the Earth, can only take on a spherical shape because of its enormous self-gravity, but it does not need to be hot to maintain itself against the inward pull of gravity objects as massive as stars, however, cannot resist gravity unless they are hot -- at least according to 'classical physics.' In the 1930s, however, it was realised that the then-new physics of quantum mechanics provided an important new insight into this question it turns out that electrons resist being compressed tightly together, to an extent which has nothing to do with their electrical charges, in a way and to a degree which is completely unpredicted by classical physics. The phenomenon is known as 'electron degeneracy' this only becomes important at a density about a million times that of water, so is unknown in ordinary materials here on Earth. But this density is reached in the cores of sun-like stars as they convert helium to carbon and then contract the recognition of this effect was first accomplished by Chandrasekhar, a brilliant young student in Cambridge. His presentation of it, however, was shot down completely by Eddington, the dominant British astrophysicist of the time among other things, Chandra showed that there was a limit to the mass of a star which could be supported by electron degeneracy. The 'Chandrasekhar limit' is about 1.4 solar masses, and implies that more massive stars may be doomed to collapse under the influence of gravity. Eddington disliked this notion (which, however, could have led to predictions of black holes) Chandra regrouped to do other equally brilliant work, and was subsequently proven completely correct and awarded the Nobel Prize for his many contributions when sun-like stars become white dwarfs, they will be made almost entirely of carbon nuclei and a sea of electrons (which provide the sustaining pressure). Being carbon, they can be thought of (very loosely!) as 'diamonds in the sky', as in the old nursery rhyme. (But the density is unlike any real diamond!) we get to see the white dwarf remnant because the dying star 'puffs off' its outer parts, perhaps 20-50% of the original mass of the star, in the form of a shell expanding gently into space the shell is known as a 'planetary nebula', but this is a complete misnomer, an accident of history. It has absolutely nothing to do with planets the precise cause for the star to 'shed its skin' is not perfectly understood (there are a variety of plausible theories, and there may be several mechanisms at work) as the gas shell expands, we see the white dwarf as a hot 'cinder' in the centre of the nebula. It will radiate its heat away into space and cool and fade to a faint 'clinker' in space. Long before that happens, however, the gas will dissipate into space there are literally tens of millions of white dwarfs in our own galaxey

Associated Readings from the Text.

Please look at: Chapter 17, pages 552-558. Chapter 18, pages 575-576.

Your Expectations.

As we have seen, gravity seeks to pull a star in on itself. Although this tendency can be resisted by thermal pressure, the energy which a star can extract from nuclear reactions is a limited resource which seems to be able to stave off the inevitable collapse for only a limited time. For instance, when hydrogen is exhausted, leaving a helium ash, the stellar core loses some of its heat and pressure support, so gravity makes it shrink until it is hot enough to fuse helium into carbon. Thereafter, and indeed fairly promptly, the helium is all converted to carbon - an `ash' of a different sort - so you would logically expect another round of heat loss and contraction. In this way, one after another, you might expect all of the various elements to be formed and exhausted in turn as the process continues. What, then, could ever halt the collapse of a star into a final very dense state? Will gravity always win? Consider the following: Sufficiently small objects , like pencils or people, thumbtacks or streetcars, pretzels or asteroids, maintain their structural integrity because of the strong electrical bonds which hold their various constituent atoms in their relative positions. There is no need for a sustaining thermal pressure from within. A thumbtack will remain a thumbtack even if it is cooled down to absolute zero. Moderately large objects , like the Earth or moon, are so strongly self-gravitating that they cannot be scuplted into shapes very different from spherical. No Earth-sized planet could ever be shaped like a banana! The electrical bonds simply cannot `hold up' very elaborate structures or large-scale features. But the self-gravity of a planet is in fact of a middling size, and although it is strong enough to prevent the planet from maintaining arbitrary shapes, it is far too weak to cause a complete inward collapse even if the planet cools off completely. Let me reemphasise that the internal temperature of a planet is in fact irrelevant to its integrity. As we have seen, the very process of formation, as well as the later slow differentiation of its constituent material, leads to the internal heating of a planet; so too does the energy released by the slow decay of radioactive elements. Thereafter, the planet's proportions (the relatively small surface area) make it hard for the heat to escape. But the integrity of the planet does not depend on the internal temperature. Even when cold, the material within the Earth will successfully resist further compression despite the self-gravity. By contrast, consider the largest bodies - the stars. Classical physics, of the sort understood a hundred years ago, was faced with the difficulty that the slow loss of thermal energy seemed to imply that stars would inevitably have to succumb to gravity, and shrink catastrophically. No known material could resist the overwhelming gravitational force. The astronomers and physicists of those days were unaware of two important pieces of information: They did not know about nuclear reactions within stars. But, as we have seen, these do not solve the problem of an inevitable collapse -- they merely postpone it. But they also did not know about the quantum mechanical nature of matter, an aspect of modern physics which has the surprising consequence that for some stars, gravity does not win. Among the survivors will be the sun - although it will be in a very different form than it is today.

The Importance of Quantum Mechanical Effects.

Some lectures ago, I pointed out that the internal structure of ordinary stars is surprisingly simple to understand because the material obeys a law known as the Perfect Gas Law. This law states, in essence, that the pressure a gas exerts depends on the following: the speed with which the individual particles are moving around. Think of the air within a balloon. As each atom or molecule hits the skin of the balloon, it exerts an outward push. The higher the speed, the harder the push. The speed, of course, is a measure of the temperature of a gas, so this tells us that hot gases exert more pressure than cooler gases - which is why the hot gas in a hot-air balloon expands the skin of the balloon outwards. the number of particles per unit of volume of the gas. This also makes sense: the more particles you have whizzing around, the more collisions there are every second with the walls of the container (or the skin of the balloon). As a consequence, the pressure is higher, since each collision delivers a little push. In a star like the sun, this says it all. The core is hot and dense, so there are lots of particles moving around at very high speed. But at still higher densities, in conditions we do not encounter in day-to-day life, there is yet another sort of pressure, one which does not depend on temperature. This is the realm of quantum mechanics. We first encountered quantum mechanics in Physics 015 when we considered the origin of the absorption and emission lines in the spectra of stars and hot gases. In general, quantum mechanics is the science of the very small, the study of (among other things) the electrons and protons which constitute the atoms around us. One of the fascinating consequences of quantum mechanics is that, when ordinary matter is compressed to extremely high density, a surprising and completely unexpected source of pressure arises - a pressure which comes from what is known as electron degeneracy.

Electron Degeneracy: The Exclusion Principle.

As you know, electrons are sub-atomic particles. In Physics 015, when describing the nature of atomic spectra, I invited you to think of electrons as being rather like small planets orbiting a sun (representing the nucleus of an atom). Of course, I was quick to point out the inadequacy of that analogy, which breaks down when we realize that the orbits which the electrons can follow are quantized: an electron cannot simply take up any orbit it likes. (Indeed, it is not correct to think of the electrons as being in well-defined orbits anyway.) But the analogy proved useful in qualitative terms. The consequence of my analogy is probably that you visualise an electron as being like a tiny billiard ball, either orbiting an atomic nucleus like a tiny asteroid around a star or else, in an ionized gas, flying freely through space on its own, completely liberated. (This is qualitatively how I visualise them.) Strictly speaking, quantum mechanics requires you to think of electrons as being rather ``fuzzy'' in appearance: rather than being discrete point-like objects, they are more correctly thought of as regions in which there is a certain probability of finding a particular electron at a particular time. It is never possible to be absolutely precise about exactly where any particular electron is: we can only localise it to a certain extent. A remarkable consequence of this ``fuzziness,'' however, is that electrons suffer from a kind of claustrophobia: they do not like to be closely packed together. To understand this, visualise the following experiment: Imagine holding half a dozen billiard balls in your hands. (If you have never handled one, let me tell you that billiard balls are very hard indeed.) Let us imagine putting them into boxes of various sizes, to see how they fit. You could certainly put them into the big cardboard box in which your new refrigerator was recently delivered: indeed they would be lost inside. You could also fit them comfortably into a shoebox - they would rattle and roll around in there. But if you tried to pack them into a much smaller box - say, one the size of a carton of cigarettes - you might just be able to squeeze them in. Clearly there is no way that you would be able to squeeze them into a box the size of a pack of cards. The point, of course, is that the billiard balls have a certain size, and you lack the strength to crush them to any smaller volume, so what determines whether or not they will fit into a box is simply the physical volume they occupy. Imagine how surprised you would be, then, if you had trouble getting six billiard balls into the refrigerator-sized box! Imagine dropping the first few into the box, without problem, but then finding that the last one needed to be pushed like mad - and even then you could only force it into the box when one of the other billiard balls came bursting out through the cardboard side of the box. This is, in effect, how the electrons behave. If you do a simple-minded calculation of the `size' of an electron, you will find out that it is very small. For that reason, you might conclude that you could easily put a lot of electrons into some small box, only to discover that you can't. It is not simply that they are `too big', as in the problem of trying to squeeze billiard balls into a box the size of a deck of cards. It is rather that they seem to repel each other with extra vigour when pushed into close proximity. Please do not misunderstand me! This has nothing to do with the fact that the electrons carry electrical charges. It is true that their identical charges mean that they repel each other, but I am talking now of an additional resistance above and beyond that electrical repulsion. In short, when electrons get into moderately close confines, they simply refuse to allow other electrons near them. The physics described in this very qualitative discussion has its origin in something known as the Exclusion Principle, which states in essence that it is not possible to pack clouds of electrons arbitrarily close together - certainly not as closely as you might have thought from a naive calculation of the `size' of the electrons. The material is called degenerate, although the word is not being used in a sense which is meant to criticise its morals! -- it has an extra, different meaning in physics.

In the Core of the Sun.

As we have seen, the sun consumes its hydrogen to form helium ash. The core then contracts, and in fact the matter briefly becomes degenerate even before helium fusion starts. That is, the inward collapse of the core is slowed by the `unexpected' pressure of degenerate electrons. (For our purposes here, that does not matter, although it does affect the details of the way the sun consumes its helium while a red giant). What I do want to emphasise, however, is what happens next. After the helium in the core is burned up to produce carbon, and the energy supply is again exhausted, the core loses some of its thermal energy and contracts. This contraction is quickly stopped when the electrons within the core are jammed so closely together that the material becomes fully degenerate. No further collapse will take place: the core has reached a stable configuration, and will collapse no further, even if it now cools off forever. (For the moment, of course, it is still terribly hot, but that heat will eventually be radiated away without any damage being done to the star: no further collapse.) Given the need to squash the electrons into close proximity before these effects become important, you may not be surprised to learn that electron degeneracy occurs at densities which are about a million times that of water -- and thus in material which is utterly unlike any we encounter in day-to-day life. The white dwarfs are the stellar remnants which are supported in this way. They are the left-over cores of long-dead sunlike stars.

The Story of Chandrasekhar.

In class, while describing the discovery of the implications of quantum mechanics on star structure, I paid tribute to one of the greatest astrophysicists of this century - Subramanyan Chandrasekhar, an Indian-born scientist who made his career home the University of Chicago. He was a Nobel-prize winner, and is known to all astronomers simply as `Chandra.' His story is a gripping one, so I will repeat some of it here.

Chandra's Earliest Work: The Chandrasekhar Limit.

As a young man in the 1930's, Chandra won a fellowship to do graduate research at the University of Cambridge, and made his way there by boat from India. While on shipboard, and indeed while not yet having even begun his graduate studies, Chandra worked out most of what I have described to you about the importance of electron degeneracy in stellar structure. What I did not tell you above is that there are important considerations which arise from relativity theory as well, considerations which Chandra successfully and correctly incorporated. (Please remember that white dwarfs were already known by this time, so there were objects to which this reasoning would almost surely apply.) One of the implications of Chandra's work was that stars could be supported by electron degeneracy only up to a certain size - about one-and-a-half times the mass of the sun. In other words, more massive stars would still succumb to gravity . (We will see later that this statement needs some modification but is still roughly correct.) This limiting mass has come to be known as the Chandrasekhar Limit. In Cambridge, Chandra worked with Sir Arthur Eddington, then recognised as the world's greatest astrophysicist. As was customary at the time, Chandra prepared a talk for presentation at a meeting of the Royal Astronomical Society in London. (I gave one of these myself while a post-doctoral fellow in Cambridge, with considerably less impact than Chandra's presentation.) He discovered that Eddington was to speak after him, but Eddington would only tell him that he ``....had a surprise or two up his sleeve.'' The surprise was devastating: after Chandra's talk, Eddington got up and dismissed it all as complete rubbish. We now recognize that Eddington did not know what he was talking about, especially in the incorporation of the elements of relativity theory into the work, but only a few people in the audience were capable of judging that, and most of the people present felt that Chandra, then a very young and unknown scientist, had misfired badly. His later appeals to others to demonstrate public support for his reasoning had little effect. One reason that Eddington was so negative, by the way, was not so much that he disbelieved in the pressure support within the small stars. It was rather that he resisted the apparently inescapable conclusion that the bigger stars cannot avail themselves of this saving grace. Since there exist stars which are five, ten, twenty times the mass of the sun, it seems that they are doomed to collapse completely under the influence of gravity - which is where we now think black holes come from! Eddington found this notion repugnant. Remarkably, Chandra expressed no public bitterness over Eddington's mistaken public dismissal of his work, and was a lifelong supporter and admirer of Eddington. What he chose to do, instead of nursing a grudge, was develop contributions in other areas of astrophysics.

Chandra's Later Contributions.

Chandra's career, once he moved to America, can be roughly divided into decade-long contributions, each ending with the publication of an important and indeed definitive book on the subject. Chandra first published a book on Stellar Structure, then one on Stellar Dynamics (how stars move around under the influence of one another's gravity in a huge system like the Milky Way), and so on. His last book, published shortly before his death a few years ago, dealt with black holes in astrophysics, and when he died he was working on a complete re-examination of the scientific contributions of Isaac Newton. In addition to this, he served for decades as the Managing Editor of the Astrophysical Journal, and almost single-handedly made it the most important astronomical research journal in the world. I am very glad to be able to tell you that he was awarded the Nobel Prize about a decade ago for his cumulative contributions to astrophysics, and that the award made special mention of his very earliest work into star structure (work which had by then long since been recognized as completely correct.)

Twinkle, Twinkle.

Some of you may know a particular piece of music by Mozart, a set of piano variations on a simple theme which most of you would recognize as `Twinkle, Twinkle, Little Star'' (The theme was not known to Mozart by that name.) Although it may seem far-fetched, if you can remember my referring to this piece of music, it may help you to remember an important point about the way the sun will end up. After helium is fused into carbon in the core of the sun, no further thermonuclear reactions will take place. As we saw, this is because further reactions would require still higher temperatures. While such temperatures would result if the core were to contract much more, this will not happen in the sun thanks to the support provided by the pressure of degenerate electrons. Consequently, the small dense core of the sun, consisting almost entirely of nuclei of carbon and myriads of free electrons, will simply cool off as an unchanging cinder. Remembering the music should help you to keep this fact at your fingertips, because the familiar song goes on to say "...like a diamond in the sky.'' Diamonds are, of course, pure carbon! In a sense, the white dwarfs are indeed 'diamonds in the sky.' But please do not overinterpret what I am saying: as hard as diamonds are, they are merely crystalline arrangements of conventional carbon atoms at everyday densities. The carbon in the dying core of the sun, however, will be at a density a million times that of water, so it is certainly not like a diamond in any property other than its carbon composition.

The Diamond Exposed.

I have been describing what happens to the very core of a sun-like star, and have told you that it will end up as an extremely dense hot ball of carbon, supported by the pressure of degenerate electrons. This is in fact the description of a white dwarf, one example of which is the faint companion star to Sirius. But all of this happens deep within the heart of a red giant star with a thick envelope of cool hydrogen-rich gas. How do we ever get to see the white dwarf? There are two obvious possibilities. Perhaps the gas in the outer parts of the star falls back down and adds to the accumulated mass of the dense core, so that eventually all the material in the star is part of the white dwarf remnant. Alternatively, it could be that the gas actually leaves the star and spreads out into empty space, so that we can now see, for the first time, the exposed hot core. Which of these is correct? Once again, we do not just rely on our theoretical models to tell us! Instead, we look at the stars in the sky to see what happens. The answer, we discover, is that sun-like stars at this late stage of their life shed something like ten to twenty percent of their original mass into space, `puffing' a cloud of gas out into the void and leaving the naked white dwarf exposed.

Planetary Nebulae.

The evidence for this is found in the existence of the so-called planetary nebulae. (Beautiful examples of this are shown in your textbook on page 557.) First, I must give you an explanation and warning. The word nebula, the Latin word for cloud, was often used historically to refer to objects which looked fuzzy when seen through a telescope. But this particular sort of object was called a planetary nebula when first observed because to the eye, through a modest telescope, they look like faint round disks of light, just as a planet would. (A century or more ago, astronomers could not take the sorts of photographs which reveal the beautiful detail shown in the text.) Please remember, however, that these objects have nothing whatever to do with planets or planetary formation. The name is a complete misnomer, an accident of history. When we study the planetary nebulae, we find out the following: the gas we see is moving away from a central, very hot star. This expansion shows up in the Doppler shift of features in the spectrum of light emitted by the gas, but can also be seen in repeated photographs taken many years apart. The nebulae are getting bigger as the years pass. The Doppler shifts tell us that the gas is moving outwards at a speed of about 30 km/sec. This may sound fast, but by astronomical standards it is a modest pace, about as fast as the Earth moves in its orbit around the sun. This justifies my saying that the gas is gently `puffed' off the star. Although the gas looks like a ring (indeed, one of the nebulae is called the Ring Nebula), it is coming off as a shell, like an expanding bubble. The ring-like appearance can be attributed to the fact that there are more atoms contributing to the total light when you look along an edge of the nebula than when you look through the centre. The gas in the nebula is glowing because of a fluorescence mechanism. The central star is so hot that it emits lots of ultraviolet light. These ultraviolet photons are captured by atoms in the outward-moving gas cloud, and the atoms re-readiate much of the energy as visible photons. So the light coming from the gas cloud originated in the hot star and is, in a sense, merely 'repackaged' before being sent on its way. There are thousands of planetary nebulae in our galaxy, and it seems that on average each one of them lasts a few tens of thousands of years before the gas dissipates into space.

Why Does the Star Shed Its Skin?

What makes the star puff off its gas? The answer to this question is not perfectly understood - but we know it happens! There are various astrophysical explanations I will not describe in detail, some of which are more plausible than others. There may be shock waves passing through the outer parts of the star, or instabilities in the thermonuclear reactions which convert helium to carbon (just as a burning candle can flare up and down). Unfortunately, the behaviour of stars as they pass through the red giant phase to their final demise is rather complex. Indeed, one thing I have not told you is that the observed properties of stars at such times can change fairly quickly, back and forth. For a time, the surface may be somewhat hotter than usual, then somewhat cooler. The star can actually become variable, pulsating in and out like a beating heart and changing in brightness as it does so. (We will see a very interesting and more extreme example of this when we study the massive stars.) It is an over-simplification, therefore, to think of a sun-like star becoming a quiet, unchanging, stable red giant until it finally and simply puffs off a shell of gas to expose its white dwarf core. Some of these messy details are discussed on pages 554-558 of your text, and the predicted future of the sun is shown in a pair of figures on page 559. You will see that the sun will (at various times) expand so much that the Earth may be just within its outer parts, and its luminosity will grow to more than a thousand times its present value. But there will be various ups and downs in these properties as the sun goes through its death throes. This does not change the overall picture of progress through the red giant phase and mass ejection (as a planetary nebula) to the white dwarf cinder.

A White Dwarf Forever.

After the envelope of the star has been cast off, the hot white dwarf core merely cools off. There are no more thermonuclear reactions, and the heat within the star is simply that which is left over from its earlier existence. As this heat is radiated into space, the star will get cooler, and thus slowly go from blue to red in colour, eventually getting so cool that it gives off no visible light at all, at which time it will have become a `black dwarf.' The drop in temperature also means that every part of the surface of the star will give off less light than before, so the star will become very dim indeed. Newly-formed white dwarfs, therefore, are destined to follow a `cooling track' in the HR diagram, getting cooler and fainter as they age. Given enough time, a white dwarf will cool off until it is absolutely stone cold. As noted, however, even then the star can resist the inward pull of gravity thanks to the pressure of degenerate electrons. Where we once had a conspicuous sun-like star, we will now find a cold, faint `clinker' of essentially pure carbon at enormously high density. The Milky Way galaxy contains millions of these remnants. Previous chapter:Next chapter

0: Physics 016: The Course Notes, spring 2005. 1: The Properties of the Sun: 2: What Is The Sun Doing? 3: An Introduction to Thermonuclear Fusion. 4: Probing the Deep Interior of the Sun. 5: The Sun in More Detail. 6: An Introduction to the Stars. 7: Stars and Their Distances: 8: The HR Diagram: 9: Questions Arising from the HR Diagram: 10: The Importance of Binary Stars: 11: Implications from Stellar Masses: 12: Late in the Life of the Sun: 13: The Importance of Star Clusters in Understanding Stellar Evolution: 14: The Chandrasekhar Limit: 15: Supernovae: The Deaths of Massive Stars, 16: Pulsars: 17: Novae: 18: An Introduction to Black Holes: 19: Gravity as Geometry: 20: Finishing Off Black Holes: 21: Star Formation: 22: Dust in the Interstellar Medium: 23: Gas in the ISM: 24: The Size and Shape of Our Galaxy: 25: The Discovery of External Galaxies: 26: Galaxies of All Kinds: 27: The Expanding Universe: 28: Quasars and Active Galaxies: 29: The Hot Big Bang: 30: The Geometry of the Universe: 31: Closing Thoughts:

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