Chapter I 1 2 3 4 5 6

Borderlands of Science

Copyright © 1999

by Charles Sheffield


3.1 Stars. Everything between atoms and stars, roughly speaking, belongs to chemistry. Although you and I are certainly subject to the laws of physics, we are chemical objects. Our metabolism and structure are controlled by the laws of chemistry. The same is largely true of planets. The shape of the Earth is defined by gravity, but most of the activities within it, or on its surface, or in its atmosphere, are decided by the laws of chemistry.

This is not true of stars. To understand how a star like the Sun can shine for billions of years, you need physics.

The modern view of stars, as giant globes of hot gas, began in 1609, when Galileo turned his home-made telescope upwards. Rather than a perfect sphere whose nature defied explanation, Galileo found that the Sun was a rotating object with lots of surface detail like sunspots and solar flares.

Over the next couple of hundred years, the size and the temperature of the sun were determined. It is a ball of gas, about a million miles across, with a surface at 6,000 degrees Celsius. What was not understood at all, even a hundred years ago, was the way that the sun stays hot.

Before 1800, that was not a worry. The universe was believed to be only a few thousand years old (Archbishop Ussher of Armagh, working through the genealogy of the Bible, in 1654 announced that the time of creation was 4,004 B.C., on October 26th. No messing about with uncertainty for him.)

In the eighteenth century, the scriptural time-scale prevented anyone worrying much about the age of the Sun. If it started out very hot in 4000 B.C., it hadn't had time to cool down yet. If it were made entirely of burning coal, it would have lasted long enough. A chemical explanation was adequate.

Around 1800, the geologists started to ruin things. In particular, James Hutton proposed his theory of geological uniformitarianism (Hutton, 1795).

Uniformitarianism, in spite of its ugly name, is a beautiful and simple idea. According to Hutton, the processes that built the world in the past are exactly those at work today: the uplift of mountains, the tides, the weathering effects of rain and air and water flow, these shape the surface of the Earth. This is in sharp distinction to the idea that the world was created just as it is now, except for occasional great catastrophic changes like the Biblical Flood.

The great virtue of Hutton's theory is that it removes the need for assumptions. Anything that shaped the past can be assessed by looking at its effectiveness today.

The great disadvantage of the theory, from the point of view of anyone pondering what keeps the Sun hot, is the amount of time is takes for all this to happen. We can no longer accept a universe only a few thousand years old. Mountain ranges could not form, seabeds be raised, chalk deposits laid down, and solid rocks erode to powder, in so short a time. Millions of years, at a minimum, are needed.

A Sun made of coal will not do. Nothing chemical will do. In the 1850's, Hermann von Helmholtz and Lord Kelvin finally proposed a solution, drawn from physics, that could give geology more time. They suggested that the source of the Sun's heat was gravitational contraction. If the material of the Sun were slowly falling inward on itself, that would release energy. The amount of energy produced by the Sun's contraction could be precisely calculated.

Unfortunately, it was still not enough. While Lord Kelvin was proposing an age for the Sun of 20 million years, the ungrateful geologists, and still more so the biologists, were asking considerably more. Charles Darwin's ORIGIN OF SPECIES came out in 1859, and evolution seemed to need much longer than mere tens of millions of years to do its work. The biologists wanted hundreds of millions at a minimum; they preferred a few billion.

No one could give it to them during the whole of the 19th century. Lord Kelvin, who no matter what he did could not come up with any age for the Sun greater than 100 million years and was in favor of a number far less, became an arch-enemy of the evolutionists. An "odious spectre" is what Darwin called him. But no one could refute his physical arguments. A scientific revolution was needed before an explanation was available for a multi-billion year age of the Sun.

That revolution began, as we saw, in the 1890's. The atom, previously thought indivisible, had an interior structure and could be broken into smaller pieces. By the 1920's it was realized that lightweight atoms could also combine, to form heavier atoms. In particular, four atoms of hydrogen could fuse together to form one atom of helium; and if that happened, huge amounts of energy could be produced.

Perhaps the first person to realize that nuclear fusion was the key to what makes the sun go on shining was Eddington. Certainly he was one of the first persons to develop the idea systematically, and equally certainly he believed that he was the first to think of it. There is a story of Eddington sitting out one balmy evening with a girl friend. She said, "Aren't the stars pretty." And he said, "Yes, and I'm the only person in the world who knows what makes them shine."

It's a nice story, but it's none too likely. Eddington was a lifelong bachelor, a Quaker, and a workaholic, too busy to have much time for idle philandering. Just as damning for the anecdote, Rudolf Kippenhahn, in his book 100 BILLION SUNS (Kippenhahn, 1979), tells exactly the same story — about Fritz Houtermans.

Even Eddington could not say how hydrogen fused to form helium. That insight came ten years later, with the work of Hans Bethe and Carl von Weizsäcker, who in 1938 discovered the "carbon cycle" for nuclear fusion.

However, Eddington didn't have to know how. He had all the information that he needed, because he knew how much energy would be released when four hydrogen nuclei changed to one helium nucleus. That came from the mass of hydrogen, the mass of helium, and Einstein's most famous formula, E = mc2.

Eddington worked out how much hydrogen would have to be converted to provide the Sun's known energy output. The answer is around 600 million tons a second. That sounds like a large amount, but the Sun is a huge object. To keep the Sun shining as brightly as it shines today for five billion years would require that less than eight percent of the Sun's hydrogen be converted to helium.

Why pick five billion years? Because other evidence suggests an age for the Earth of about 4.6 billion years. Nuclear fusion is all we need in the Sun to provide the right time-scale for geology and biology on Earth. More than that, the Sun can go on shining just as brightly for another five billion years, without depleting its source of energy.

But how typical a star is the Sun? It certainly occupies a unique place in our lives. All the evidence, however, suggests that the Sun is a rather normal star. There are stars scores of times as massive, and stars tens of times as small. The Sun sits comfortably in the middle range, designated by astronomers as a G2 type dwarf star, in what is known as the main sequence because most of the stars we see can be fitted into that sequence.

The life history of a star depends more than anything else on its mass. That story also started with Eddington, who in 1924 discovered the mass-luminosity law. The more massive a star, the more brightly it shines. This law does not merely restate the obvious, that more massive stars are bigger and so radiate more simply because they are of larger area. If that were true, because the mass of a star grows with the cube of its radius, and its surface area like the square of its radius, we might expect to find that brightness goes roughly like mass to the two-thirds power (Multiply the mass by eight, and expect the brightness to increase by a factor of four). In fact, the brightness goes up rather faster than the cube of the mass (Multiply the mass by eight, and the brightness increases by a factor of more than a thousand).

The implications of this for the evolution of a star are profound. Dwarf stars can go on steadily burning for a hundred billion years. Massive stars squander their energy at a huge rate, running out of available materials for fusion in just millions of years.

(A word of warning: Don't put into your stories a star that's a thousand times the mass of the Sun, or one-thousandth. The upper limit on size is set by stability, because a contracting ball of gas more than about 90 solar masses will oscillate wildly, until parts of it are blown off into space; what's left will be 90 solar masses or less. At the lower end, below maybe one-twelfth of the Sun's mass, a star-like object cannot generate enough internal pressure to initiate nuclear fusion and should not be called a "star" at all.)

The interesting question is, what happens to massive stars when their central regions no longer have hydrogen to convert to helium? Detailed models, beginning with Fred Hoyle and William Fowler's work on stellar nucleosynthesis in the 1940's, have allowed that question to be answered.

Like a compulsive gambler running out of chips, stars coming to the end of their supply of hydrogen seek other energy sources. At first they find it through other nuclear fusion processes. Helium in the central core "burns" (not chemical burning, but the burning of nuclear fusion) to form carbon, carbon burns to make oxygen and neon and magnesium. These processes call for higher and higher temperatures before they are significant. Carbon burning starts about 600 million degrees (as usual, we are talking degrees Celsius). Neon burning begins around a billion degrees. Such a temperature is available only in the cores of massive stars, so for a star less than nine solar masses that is the end of the road. Many such stars settle down to old age as cooling lumps of dense matter. Stars above nine solar masses can keep going, burning neon and then oxygen. Finally, above 3 billion degrees, silicon, which is produced in a process involving collisions of oxygen nuclei, begins to burn, and all the elements are produced up to and including iron. By the time that we reach iron, the different elements form spherical shells about the star's center, with the heaviest (iron) in the middle, surrounded by shells of successively lighter elements until we get to a hydrogen shell on the outside.

Now we come to a fact of great significance. No elements heavier than iron can be produced through this nuclear synthesis process in stars. Iron, element 26, is the place on the table of elements where nuclear binding energy is maximum. If you try to "burn" iron, fusing it to make heavier elements, you use energy, rather than producing it. Notice that this has nothing to do with the mass of the star. It is decided only by nuclear forces.

The massive star that began as mainly hydrogen has reached the end of the road. The final processes have proceeded faster and faster, and they are much less efficient at producing energy than the hydrogen-to-helium reaction. Hydrogen burning takes millions of years for a star of, say, a dozen solar masses. But carbon burning is all finished in a few thousand years, and the final stage of silicon burning lasts only a day or so.

What happens now? Does the star sink into quiet old age, like most small stars? Or does it find some new role?

And one more question. We can explain through stellar nucleosynthesis the creation of every element lighter than iron. But more than 60 elements heavier than iron are found on Earth. If they are not formed by nuclear fusion within stars, where did they come from?

2.2 Stellar endings. We have a star, of ten or more solar masses, running out of energy. The supply provided by the fusion at its center, of silicon into iron, is almost done. In the middle of the star is a sphere of iron "gas" about one and a half times the mass of the sun and at a temperature of a few billion degrees. It acts like a gas because all the iron nuclei and the electrons are buzzing around freely. However, the core density is millions of times that of the densest material found on Earth. Outside the central sphere, like layers of an onion, sit shells of silicon, oxygen and carbon, helium and neon and hydrogen, and smaller quantities of all the other elements lighter than iron.

When the source of fusion energy dries up, iron nuclei capture the free electrons in the iron gas. Protons and electrons combine. The energy that had kept the star inflated is sucked away. The core collapses to become a ball of neutrons.

The near-instantaneous gravitational collapse unleashes a huge amount of energy, enough to blow all the outer layers of the star clear away into space. What is left behind is a "neutron star" — a solid sphere of neutrons, spinning on its axis many times a second, only a few miles across but with a mass as much as the Sun's mass.

When such an object was observed, as a rapidly but regularly varying radio source, it seemed difficult to imagine anything in nature that could explain the signal. The team at Cambridge who discovered the first one in 1967 called it a pulsar. They wondered, even if they were reluctant to say so in public, if they had found signals from some alien civilization. When other pulsars were discovered and Thomas Gold proposed that the radio sources were provided by rotating neutron stars, astronomers realized that such a possibility had been pointed out long ago — in 1934, in a prophetic paper by Walter Baade and Fritz Zwicky. The most astonishing thing about the paper was that the neutron itself had been discovered only two years earlier, in 1932.

Could life ever exist on the surface of such a body, with its immense gravitational and magnetic field, and its extreme temperature and dizzying rotation? You might think not; but the novel DRAGON'S EGG (Forward, 1980) explores that wild possibility, as does FLUX (Baxter, 1993).

And how much is a "huge" amount of energy? When a star collapses and blows up like this, in what is known as a supernova, it shines for a time as brightly as a whole galaxy. Its luminosity can temporarily increase by a factor of one hundred billion. If that number doesn't tell you much, try it this way: if a candle in Chicago were to "go supernova," you would easily be able to read a newspaper by its light in Washington, D.C.

The explosion of the supernova also creates pressures and temperatures big enough to generate all the elements heavier than iron that could not be formed by standard nucleosynthesis in stars. So finally, after a long, complex process of stellar evolution, we have found a place where substances as "ordinary" as tin and lead, or as "precious" as silver, gold, and platinum, can be created.

For completeness, I will point out that there are actually two types of supernova, and that both can produce heavy elements. However, the second kind cannot happen to an isolated star. It occurs only in binaries, pairs of stars, close enough together that material from one of them can be stolen gravitationally by the other.

The star that does the stealing must be a small, dense star of the type known as a white dwarf, while its partner is usually a larger, diffuse, and swollen star known as a red giant. As more and more matter is stolen from the more massive partner, the white dwarf star shrinks in size, rather than growing. When its mass reaches 1.4 times the mass of the Sun (known as Chandrasekhar's limit) it collapses. The result is a huge explosion, with a neutron star left behind as a possible remnant. The outgoing shock wave creates heavy elements, and ejects them from the system along with the rest of the star's outer layers.

If you are thinking of using a supernova as part of a story, note that according to current theory the nearest binary star to us, Alpha Centauri A and B, is not a candidate. I am not discouraging the idea of using such a supernova, since I have just done it myself (AFTERMATH, 1998). The flux of radiation and high-energy particles from an Alpha Centauri supernova can do interesting things to Earth. But you'll need to do some ingenious talking if you want the idea to seem plausible.

Supernovas are rather like nuclear power stations. What they produce is important to us — it is the very stuff of which our own bodies and many of our most valued products are made. We prefer, however, not to have one in our own local neighborhood.

What is the final fate of a star that explodes and becomes a neutron star? That depends on the mass of the part that's left. One possibility is that it remains a neutron star to the end of its life. Another more exotic possibility is that it shrinks further and becomes a black hole. That intriguing option we will describe in the next section, after which we will expand the scale of our exploration.

3.3 Black holes. The story of black holes begins with Albert Einstein and the theory of general relativity.

In 1916, soon after the publication of the field equations in their final form, Karl Schwarzschild produced the first exact solution. Einstein was reportedly quite surprised, because of the complicated nature of the field equations - a set of ten coupled non-linear partial differential equations. As Einstein wrote to Max Born, twenty years later, "If only if were not so damnably difficult to find rigorous solutions."

The "Schwarzschild solution" gave the gravitational field for an isolated mass, which later became known as the Schwarzschild black hole. At the time, it was considered to be mathematically interesting, but of no physical significance. Soon after Schwarzschild's work, Reissner and Nordstrom solved the general relativity equations for a spherical mass that also carried a charge. It too was regarded with no special interest.

In 1939, Oppenheimer and Snyder studied the collapse of a star under gravitational forces — a situation which certainly does have physical significance, since it is a common stellar occurrence.

Two remarks from the summary of their paper are worth quoting: "Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of the sun, this contraction will continue indefinitely." In other words, not only can a star collapse, but if it is heavy enough there is no way that the collapse and contraction can be stopped. And "the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles." This is the first modern picture of a black hole, a body with a gravitational field so strong that light cannot escape from it. We say "modern picture" because John Michell in 1783, and Pierre Laplace in 1798, independently noted that a sufficiently massive body would have an escape velocity from its surface that exceeded the speed of light.

The idea of a "gravitational radius" came straight from the Schwarzschild solution. It is the distance from the center where the reddening of light becomes infinite, and it defines a sphere. Any light coming from inside that sphere can never be seen by an outside observer. Today the surface of the sphere has a variety of names, all defining the same thing: the surface of infinite red shift, the trapping surface, the one-way membrane, and the event horizon. Since the gravitational radius for the Sun is only three kilometers, if it were squeezed down to this size (which will never happen, fortunately, as a result of gravity) conditions inside the collapsed body would be difficult to imagine. The density of matter would be about twenty billions tons per cubic centimeter.

You might suppose that the Oppenheimer and Snyder paper, with its apparently bizarre conclusions, would have produced a sensation. In fact, it aroused little notice. It too was looked on as a mathematical oddity, a result that physicists did not need to take too seriously. The resurgence of interest in the solutions of the equations of general relativity did not take place until after Einstein's death in 1955, and it was one of the leaders of that renaissance, John Wheeler, who in 1958 provided the inspired name for the Schwarzschild solution at the gravitational radius: the black hole.

The object described by the Schwarzschild and Reissner/Nordstrom solutions could have a mass, and a charge, and that was all. The next development came in 1963, and it was a big surprise to everyone in the field.

Roy Kerr had been exploring a particular form of the Einstein field equations. The analysis was highly mathematical and seemed to be wholly abstract — until Kerr found that he could produce an exact form of solution. It included the Schwarzschild black hole as a special case, but there was more, another quantity that Kerr was able to associate with spin. For the first time, the possibility of a spinning black hole had appeared. It could also, as was shown a couple of years later by Ezra Newman and collaborators, have an associated charge.

From this point on, I am for convenience going to call the charged, spinning Kerr-Newman black hole a kernel. It has a number of fascinating properties useful to science fiction writers.

First, since it carries a charge a kernel can be moved from place to place using electric and magnetic fields. Second, the kernel has associated with it not the single characteristic surface of the Schwarzschild solution (the sphere defined by the gravitational radius), but two. In this case, the surface of infinite red shift is distinct from the event horizon.

To visualize the surfaces, take a hamburger bun and hollow out the inside, enough to let you put a round hamburger entirely within it. For a kernel, the outer surface of the bread (which is a sort of ellipsoid in shape) is the surface of infinite red shift, the "static limit" within which nothing can remain at rest, no matter how hard and efficiently its rocket engine works. Inside the bun, the surface of the meat patty forms a sphere, the "event horizon" from which no light or particle can ever escape to the outside. We can never find out anything about what goes on within the meat patty's surface, so its composition and nature, like that of many hamburgers, must remain a complete mystery. For a kernel, the bun and patty surfaces touch only at the north and south poles of the axis of rotation (the top and bottom centers of the bun). A really interesting region, however, lies between these two surfaces. It is called the ergosphere, and it has a most unusual property, pointed out in 1969 by Roger Penrose (yes, the same Penrose as in Chapter 2 — he is a highly versatile and creative individual, who has made major contributions to relativity theory and other fields).

Penrose showed that it is possible for a particle to dive in toward the kernel from outside, split in two when it is inside the ergosphere, and then have part of it ejected to the exterior in such a way that the piece has more total energy than the particle that went in. If we do this, we have extracted energy from the black hole.

Note that this must be a kernel, a spinning black hole, not a Schwarzschild black hole. The energy that we have gained comes from the rotational energy of the hole itself.

If the kernel starts out with only a little spin energy, we can use the energy-extraction process in reverse, to provide more rotational energy. We will refer to that as "spinning up" the kernel. "Spin down" is the opposite process, the one that extracts energy.

One other property of a kernel will prove useful later. Every kernel (but not a Schwarzschild black hole) possesses a "ring singularity." It appears possible to remain far enough from the singularity to avoid destruction by tidal forces, but close enough to take advantage of peculiar aspects of space-time there. This is discussed further in Chapter 9.

Since it can be proved that a black hole has as properties only mass, charge, spin, and magnetic moment, and the last one is fixed completely by the other three, that seems to say all that can be said about kernels. This result, that all black holes are completely defined by three constants, is a theorem that is often stated in the curious form, "A black hole has no hair."

That was the situation until 1974, when Stephen Hawking produced a result that shocked everyone. In perhaps the biggest surprise in all black hole history, he proved that black holes are not black.

This calls for some explanation. General relativity and quantum theory were both developed in this century, but they have never been combined in a satisfactory way. Physicists have known this and been uneasy about it for a long time. In attempting to move toward what John Wheeler referred to as "the fiery marriage of general relativity with quantum theory," Hawking studied quantum mechanical effects in the vicinity of a black hole. He found that particles and radiation can (and must) be emitted from the hole.

The smaller (and therefore less massive) the hole, the faster the rate of radiation. Hawking was able to relate the mass of the black hole to a temperature, and as one would expect, a "hotter" black hole pours out radiation and particles faster than a "cold" one. For a black hole the size of the Sun, the associated temperature is far lower than the background temperature of the Universe (the 2.7 Kelvins background radiation). Such a black hole receives more energy than it emits, so it will steadily increase in mass. However, there is no rule of nature that says a black hole has to be big and massive. For a black hole of a few billion tons (the mass of a small asteroid) the temperature is so high, at ten billion degrees, that the black hole will radiate itself away to nothing in a gigantic and rapid burst of radiation and particles. Furthermore, a spinning black hole will preferentially radiate particles that decrease its spin, while a charged black hole will prefer to radiate charged particles that reduce its overall charge.

These results are so strange that in 1972 and 1973 Hawking spent a lot of time trying to find the mistake in his own calculations. Only when he had performed every check that he could think of was he finally forced to accept the conclusion: black holes are not black after all; and the smallest black holes are the least black.

We have discussed the properties of kernels, without asking the crucial question: Do they exist?

For a while, it was thought that very small black holes, weighing only a hundredth of a milligram, might have been created in the Big Bang. The Hawking radiative process showed that any of those, if they ever existed, would have gone long since. Big black holes, however, seem not only possible, but inevitable. If the core of a collapsing star is massive enough (more than about three times the mass of our Sun), then after the star explodes to a supernova, Oppenheimer and Snyder's results apply. The remnant star is forced to collapse without limit, and no force in the universe is powerful enough to stop it.

Black holes, if they exist at all, ought therefore to be common throughout the universe, perhaps enough to make a sizeable contribution to the missing mass needed to close it (see Chapter 4). However, some people object to the very idea of black hole existence. Associated with them is a singularity — an infinity — that no one has been able to explain away, and singularities are generally regarded as evidence that a theory has something wrong with it. Einstein himself was reported to consider black holes as a "blemish to be removed from his theory by a better mathematical formulation."

Until that better mathematical formulation comes along, black holes are an acceptable part of theoretical physics; but what is the experimental evidence for them?

We have a problem. A black hole, unless it is small (the mass of, say a small asteroid) will not radiate measurable energy. Also, we know of no way that a black hole less than about three solar masses can form. Black holes are therefore, by definition, not directly visible. Their existence, like the existence of quarks, depends not on observing them, but on the role they play in simplifying and explaining other observations.

A black hole's presence must be detected by indirect effects. For example, matter falling into a black hole will be ripped apart and give off a powerful radiation signal; but so will matter that falls into a neutron star. Distinguishing between the two is a subtle and difficult problem. One of the early and best candidates for a solar-sized black hole is the source known as Cygnus X-1.

Very large black holes probably lie at the heart of many galaxies, and are the mechanism that powers quasars. It is also possible to regard the whole universe as a black hole, within which we happen to live; but these are conjectures, not established facts. In view of Einstein's comment, maybe any other possible explanation is to be preferred.

Despite lack of final proof of their existence, black holes form a valuable weapon in the writer's arsenal. In fact, they are so accepted a feature of the science fiction field that they can be introduced without further explanation, like an alien or a faster-than-light drive. Black holes, of various sizes and properties, can be found in hundreds of stories.

I will mention just a handful, so you will not be tempted to write a classic story that already exists: THE HOLE MAN, by Larry Niven (Niven, 1973); IMPERIAL EARTH, by Arthur Clarke (Clarke, 1975); BEYOND THE BLUE EVENT HORIZON, by Frederik Pohl (Pohl, 1980); and EARTH, by David Brin (Brin, 1990). All of these employ the Schwarzschild black hole. I like better the spinning, rotating black hole. With my preferred name for them, a kernel (for Kerr-Newman black hole), they are used in all the stories in ONE MAN'S UNIVERSE, and in PROTEUS UNBOUND (Sheffield, 1983, 1989).

Copyright © 1999 by Charles Sheffield
Chapter I 1 2 3 4 5 6

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Baen Books 02/02/03