An exploding star, or
supernova,
produces during its short fireworks display roughly as much light as a whole galaxy, over a hundred billion normal stars. It is hard to come to grips with the sheer power released during such an event. Just seconds before the final explosion begins, the star is calmly burning the last available
bits of fuel, until the pressure generated by the star’s last gasp is no longer sufficient to hold up its incredibly dense iron core, containing as much mass as our sun but compressed into a region the size of Earth—a million times smaller. In less than one second this entire mass collapses inward, in the process releasing huge amounts of energy. This collapse continues until the entire core is contained in a ball of roughly a 10-kilometer (6-mile) radius—about the size of New Haven, Connecticut. At this point, the matter is so dense that a teaspoonful would weigh thousands of tons. More important, the extremely dense atomic nuclei of the iron begin to crowd together, “touching,” as it were. At this point, the material suddenly stiffens, and a whole new source of pressure, due to the interactions of these closely packed nuclei, becomes important. The collapse halts, the core “bounces,” and a shock wave is driven outward, through the core, thousands of miles to the outer shell of the star, which is literally blown off and is visible to us as a supernova.
This picture of the collapse of the core of an exploding star was built up over decades of painstaking analytical and numerical work by teams of researchers after the first proposal by S. Chandrasekhar in 1939 that such an unfathomable scenario could take place. It is all an outgrowth of the simple ideas of hydrostatic equilibrium, which we believe governs the sun’s structure. For fifty-odd years after first being proposed, the processes governing stellar collapse remained pure speculation. It had been centuries since the last supernova in our galaxy had been observed, and even then, all that could be seen were the outer fireworks, occurring far from where the real action was taking place, deep inside the star.
All of this changed on February 23, 1987. On that day a supernova was observed in the Large Magellanic Clouds, a small
satellite system at the outer edge of our galaxy, about 150,000 light years away. This was the closest supernova to have been observed in the last four hundred years. It turns out that the visual fireworks associated with a supernova form just the tip of the iceberg. More than a thousand times as much energy is emitted, not in light, but—you guessed it—in almost invisible neutrinos. I say
almost
invisible because even though almost every neutrino emitted by the supernova could go through the Earth undetected, the laws of probability tell us that, although rarely, a neutrino will have a measurable interaction in a detector of smaller dimensions. In fact, one can estimate that at the moment that “neutrino blast” from the distant supernova passed across the Earth, one of every million or so people who happened to have their eyes closed at just the right time might have seen a flash induced by light produced when a neutrino bounced off an atom in their eyes.
Thankfully, however, we didn’t have to depend on eyewitness accounts of this remarkable phenomenon. Two large detectors, each containing over 1,000 tons of water, located deep underground on opposite sides of the Earth had been outfitted with eyes for us. In each detector tank, thousands of photosensitive tubes lay ready in the darkness, and on February 23, in a ten-second period coincident in both detectors, nineteen separate neutrino-induced events were observed. Few as this may seem, it is almost precisely the number of events that one would predict to result from a supernova on the other side of our galaxy. Moreover, the timing and energy of the neutrinos were also in good agreement with predictions.
Whenever I think about this, I am still amazed. These neutrinos are emitted directly from the dense collapsing core, not from the surface of the star. They give us direct information about this
crucial period of seconds associated with the catastrophic collapse of the core. And they tell us that the theory of stellar collapse—worked out in the absence of direct empirical measurement over thirty-odd years, and based on extrapolating to its extreme limits the same physics of hydrostatic equilibrium responsible for determining the sun’s structure—is totally consistent with the data from the supernova. Confidence in our simple models led us to understand one of the most exotic processes in nature.
There is yet one more example of the remarkable power of approximating the sun as a sphere. Even as the solar neutrino problem was being resolved, another puzzle regarding stellar structure seemed to remain. If we assume the same solar modeling procedure that we use to understand the sun to predict how stars evolve, we can use a comparison of theory and observation to date not only our sun (about 4.55 billion years old), but also the oldest stars in our galaxy. When this procedure was applied to stars in some isolated systems on the edges of our galaxy called globular clusters, it was found that such systems were greater than 15 billion years old.
At the same time we can use the fact that our observed universe is expanding—and assuming this expansion is slowing down, as is natural given that gravity is an attractive force—to determine the age of our universe by measuring its expansion rate today. The argument is relatively simple: We observe how fast galaxies at a known distance from us today are moving away from us, and then assuming that they have been moving away from us at this speed or greater during the whole history of the universe, we can put an upper limit on how long it would have taken them to recede to their present distance since the big bang. After eighty years of trying, we were finally able to measure the expansion rate to about 10 percent accuracy, and we found that if the expansion
of the universe has been slowing down, then the universe had to be less than about 11 billion years old.
This presented a problem, as it suggested that the stars in our galaxy were older than the universe! This was not the first time stellar ages had caused such a problem, and each time resolving the problem shed new insights into the universe. For example, in the 1800s an estimate was made of the age of the sun, by assuming it was a large ball of carbon, burning like coal. Given the mass of the sun, one could determine how long it would take for it to fully use up all its fuel, and the answer was about 10,000 years. While this meshed nicely with literal interpretations from the Bible about the age of the universe, by this time the evidence of fossils and geological strata had revealed the Earth to be far older than this. Near the end of the nineteenth century it was then pointed out by two well-known physicists, Lord Kelvin in the UK and Helmoltz in Germany, that if the material in the sun was collapsing due to the large gravitational field of the sun, this could provide energy to power the sun, and they worked out that this process could persist for perhaps 100 million years before exhausting all of the available energy. While this was a vast improvement, by this time geology and evolutionary biology had made it clear that the Earth was billions of years old, again causing a problem since it was hard to imagine how the Earth could be much older than the sun.
In the 1920s this problem was so severe that the distinguished astrophysicist, Sir Arthur Stanley Eddington argued there had to be another, as of yet unknown, energy production mechanism that could keep the sun burning for billions of years. Many people were skeptical of this claim, as the 10 million degree temperature in the interior of the sun may seem very hot by terrestrial standards, it did not seem hot enough to allow for some new kind of
physics to intervene. In one of my favorite put-downs in science, Eddington urged all those who did not believe his claim to “go and find a hotter place!” As it turned out, Eddington was vindicated in the 1930s when the physicist Hans Bethe, whose name will reappear shortly, recognized that the newly discovered nuclear reactions that could be used to build a bomb on Earth could in fact power the sun, allowing it to survive for as long as 10 billion years in the form we now observe it. Bethe won the Nobel Prize for his calculations, which now form the basis of the Standard Solar Model.
To return to the question of whether Standard Solar Model approximations would break down when trying to determine the age of the oldest stars, with several colleagues, I was able to reexamine the age estimates of globular clusters; and we were able to explicitly incorporate the amount of uncertainty that was introduced into the results by the approximations used to derive them and show that the oldest globular clusters in our galaxy could be as young as 12 billion years, but not much younger, thus suggesting that the apparent conflict between their age and the age of the universe was a real one.
Partly motivated by this result, but also by other data that appeared to be inconsistent with a universe whose expansion was slowing, in 1995 a colleague from Chicago and I were driven to heretically (and perhaps also somewhat facetiously) suggest that perhaps the expansion of the universe was not slowing down but instead speeding up! This may sound implausible but in fact the exotic possibility of a repulsive form of gravity had in fact been proposed by Albert Einstein in 1916, shortly after he developed his general theory of relativity, to allow for a static universe—then thought to be the case—but was discarded when it was discovered that the universe was expanding. Moreover, our
suggestion was not as completely crazy as it may have sounded as it turns out that we now understand an extra “repulsive” form of gravity could naturally result if empty space possessed a new form of energy which, in the context of modern particle theory, is allowed.
Remarkably, in 1998 two teams—using observations of the apparent brightness of supernovae in distant galaxies to trace the expansion rate of the universe over time—independently discovered that the expansion of the universe
is
accelerating! This experimental discovery completely changed our picture of the expanding universe, so that trying to understand the nature of this dark energy is perhaps the most important outstanding problem in cosmology. Most interesting from the point of view of the present discussion, however, is the fact that once one incorporates this fact into a determination of the age of the universe, one gets an age estimate of approximately 13–14 billion years, which is in perfect agreement with the age determination of the oldest stars in our galaxy.
Thus, the simple approximation of stars as spheres has continued for over 200 years to lead us to discover new and profound aspects of the universe.
The previous examples demonstrate the great discovery power made possible by approximation in physics, but they should not distract us from the more fundamental fact that without approximation, we can do almost nothing. With it, we can make predictions that can be tested. When the predictions are wrong, we can then focus on the different aspects of the approximations we make, and in so doing we have learned almost everything we now know about the universe. In the words of James Clerk
Maxwell, the most famous and successful theoretical physicist of the nineteenth century: “The chief merit of a theory is that it shall guide experiment, without impeding the progress of the true theory when it appears.”
2
Sometimes physicists simplify the world on the basis of sound intuition, but most often they do it because they have no other choice. There is a well-known allegory that physicists like to tell: If you are walking at night on a poorly lit street and you notice that your car keys are not in your pocket, where is the first place you look? Under the nearest streetlight, of course. Why? Not because you expect that you would necessarily lose your keys there, but rather because that is the only place you are likely to find them! So, too, much of physics is guided by looking where the light is.
Nature has so often been kind to us that we have come to take it sort of for granted. New problems are usually first approached using established tools, whether or not it is clear that they are appropriate, because it is all we can do at the time. If we are lucky, we can hope that even in gross approximation, some element of the essential physics has been captured. Physics is full of examples where looking where the light is has revealed far more than we had any right to expect. One of them took place shortly after the end of World War II, in a chain of events that carried with it elements of high drama and at the same time heralded the dawn of a new era in physics. The final result was the picture we now have of how physical theory evolves as we explore the universe on ever smaller or larger scales. This idea, which I never see discussed in popular literature, is fundamental to the way modern physics is done.
The war was over, physicists were once again trying to explore fundamental questions after years of war-related work, and the
great revolutions of the twentieth century—relativity and quantum mechanics—had been completed. A new problem had arisen when physicists attempted to reconcile these two developments, both of which I shall describe in more detail later in the book. Quantum mechanics is based on the fact that at small scales, and for small times, not all quantities associated with the interactions of matter can be simultaneously measured. Thus, for example, the velocity of a particle and its position cannot be exactly determined at the same instant, no matter how good the measuring apparatus. Similarly, one cannot determine the energy of a particle exactly if one measures it over only a limited time interval. Relativity, on the other hand, stipulates that measurements of position, velocity, time, and energy are fundamentally tied together by new relationships that become more evident as the speed of light is approached. Deep inside of atoms, the motion of particles is sufficiently fast that the effects of relativity begin to show themselves, yet at the same time the scales are small enough so that the laws of quantum mechanics govern. The most remarkable consequence of the marriage of these two ideas is the prediction that for times that are sufficiently small so that it is impossible to measure accurately the energy contained in a certain volume, it is impossible to specify how many particles are moving around inside it. For example, consider the motion of an electron from the back of your TV tube to the front. (Electrons are microscopic charged particles, which, along with protons and neutrons, make up all atoms of ordinary matter. In metals, electrons can move about under the action of electric forces to produce currents. Such electrons are emitted by the metal tip of a heating element in the back of the TV and strike the screen at the front and cause it to shine, producing the picture you see.) The laws of quantum mechanics tell us that for any very short interval it is impossible
to specify exactly which trajectory the electron takes, while at the same time attempting to measure its velocity. In this case, when relativity is incorporated into the picture, it suggests that if this is the case, during this short interval one cannot claim with certainty that there is only
one
electron traveling along. It is possible that spontaneously both
another
electron and its antiparticle with opposite charge, called a positron, can appear out of empty space and travel along with the electron for a short time before these two extra particles annihilate each other and disappear, leaving nothing behind but the original electron! The extra energy required to produce these two particles out of nothing for a short time is allowed for because the energy of the original electron moving along cannot be accurately measured over such short times, according to the laws of quantum mechanics.