Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  7.3.1 Review

  I will deal with topics in atomic and nuclear physics in later chapters, but a few points are needed here. An atom consists of a central nucleus about which orbits negatively charged electrons. The nucleus comprises positively charged protons (supplying the attractive force to keep the negatively charged electrons around it) and neutrons. The strong nuclear force holds the protons and neutrons together to form the nucleus. It is the production of nuclei which concerns us here since electrons are attracted later to form atoms.

  The simplest element is hydrogen, denoted by H1, which has one proton as its nucleus and one electron orbiting that proton. We can also add neutrons to produce heavy hydrogen: the deuteron D or H2 has one proton and one neutron in its nucleus; adding one more neutron gives the triton or H3.

  The elements build up, with more and more protons in the nucleus and with corresponding numbers of electrons in orbits around the nucleus. Thus we get the nuclei for

  helium: He3, two protons, one neutron; He4, two protons, two neutrons;

  lithium: Li5, three protons, two neutron; Li6, three protons, three neutrons;

  beryllium: Be7, four protons, three neutrons;

  and so on through the familiar periodic table of elements. Increasing the number of protons by one each time, we find the next in line to be boron, carbon, nitrogen, and oxygen.

  Notice that the number of neutrons may be varied, but in each case the number will affect the stability of the nucleus. (There is more on that topic in chapter 11.) Of great importance is the fact that the Li5 and beryllium nuclei are not stable, and after forming, they quickly split apart again.

  The light elements, hydrogen and helium, are most abundant in the universe with all the other elements occurring thousands or more times less often. Still, those more complex elements are needed to form us! Where they come from is a vital question for science to answer.

  7.3.2 The Fusion Process

  In the fusion process, two nuclei join together to form a new type of nucleus. (More details of this process will be given in sections 11.6.1 and 11.7.) This process occurs with the lighter nuclei up to iron, Fe55, which contains twenty-six protons. Energy is released during the fusion process, and I will return to the importance of this in chapter 11. (For elements beyond iron, it is the fission process that dominates nuclear physics.) Here is an example of a chain of fusion processes taken from the notebook of Fred Hoyle9 (more of whom in a moment):

  O16 + He4 → F19 + H1

  F19 + H1 → Ne20 + hν

  Ne20 + He4 → Na23 + H1

  Na23 + H1 → Mg24 + hν

  Energy released can be in the form of kinetic energy or photons (shown as hν in the above).

  It would appear that fusion can take us from the original few big bang constituents to the whole range of elements. Certainly helium was formed and is abundant in the universe. But there is an interaction problem when we consider the heavier elements. The nuclei are positively charged and repel each other with the electrical Coulomb force. One nucleus must get close to another in order for the very short-range, strong nuclear force to take over. In other words, nuclei must overcome the “Coulomb barrier” if fusion is to take place. This was extremely unlikely to happen in the conditions prevailing soon after the big bang.

  The solution to the interaction problem is found by looking at the interior of stars that are formed and collapse according to the action of the long-range gravitational force. Eventually the stellar interiors become incredibly hot and dense, with enormous pressures created to counter gravitational collapse. It is in these circumstances that nuclei can be propelled over or through the Coulomb barrier and fusion occurs. We have a process called stellar nucleosynthesis in which the heavier nuclei are built up from the lighter ones beginning with hydrogen and helium.

  A pioneering paper on the subject was published in 1954 by Fred Hoyle, and the great Synthesis of the Elements in Stars was published in 1957 by Margaret Burbidge, Geoffrey Burbidge, William Fowler, and Hoyle. These papers examined chains of reactions (like those set out in Hoyle's notebook) producing the elements and calculated how they could operate in conditions likely to be found in stellar interiors. These are truly landmark papers and calculations.

  A final step is needed: How do the elements created inside stars get to form us? The answer is that stars can explode, and material from them is scattered throughout the universe. That matter is then drawn into clumps that can form planets like Earth. Then there can be objects on Earth like us. “We are all made from stardust” is a wonderful way to say it. Surely that is one of science's most fantastic and beautiful stories!

  7.3.3 A Momentous Hiccup

  The whole building up process, starting with hydrogen H1 and helium He4, sounds brilliant, but there is an almost immediate problem: the beryllium nucleus Be8 needed to move up to carbon C12 is unstable. Unless something special occurs, no carbon can be produced. But carbon is produced (as our bodies testify!), and Hoyle found the mechanism. There can be a resonance situation in which, despite the rapid decay of beryllium, the mechanism still allows carbon to form. In his 1954 paper, Hoyle writes the process (with the alpha particle notation α for He4)

  α + α ↔ Be8 Be8 + α → C12 + γ

  The second reaction produces the gamma ray γ. But hidden in there is the vital step: the carbon nucleus must be able to assume a most particular state.

  An aside is required: The nucleus is a mixture of protons and neutrons held together by the strong nuclear force, which overcomes the protons’ electrical repulsion forces. According to the appropriate mechanics (that is quantum mechanics, to be introduced more fully in chapter 10), the protons and neutrons form a lowest energy configuration called the ground state. However, if energy is added, they may take up new configurations called excited states.

  Now, back to the fusion process that gives us carbon. Hoyle calculated that to form the resonance situation for the production of carbon, an excited state of the carbon nucleus must be involved. This excited state must have an energy above the ground state of about 7.65MeV. (MeV is a unit of energy in nuclear physics. Appendix 3 of Hoyle's 1975 textbook provides a very clear explanation of this work.) Since carbon does exist, Hoyle believed that this particular excited state of the carbon nucleus must also exist. He took his prediction to William Fowler and colleagues who were persuaded to try the necessary experiment. The excited state did indeed exist, just as Hoyle had predicted. This prediction must rank as one of the most remarkable and profound in all of physics.

  Solving the problem of carbon production was a brilliant achievement by a remarkable man. Fred Hoyle (1939–2001) was born in Yorkshire, England, and studied mathematics and physics at Cambridge University. He went on to become an outstanding astrophysicist, contributing to many areas of the subject. (For some unknown and shameful reason, he was never awarded the Nobel Prize, although his collaborators, like William Fowler, were.) Hoyle was a proponent of the steady state theory of the universe. He disliked the expanding universe model we accept today, and he scornfully called it the big bang model! Hoyle's sarcasm has been lost, and the mocking name he invented remains in use today. His autobiography, Home is Where the Wind Blows, is a captivating read; his version of the carbon story is given in chapter 18, “An Unknown Level in Carbon 12.”

  It is impossible not to put calculation 23, Hoyle makes carbon in my list of important calculations.

  7.3.4 Mankind and the Universe

  Recall that Copernicus moved the earth from the center of the universe and that gradually we have found that there is nothing special about our Earth or its position in the universe. We have been led to the cosmological principle as discussed in section 7.2.1. Nevertheless, it seems to be part of human nature to return to the so-called big questions like: How and why did it all begin? What is the point of life? What is the meaning of life? What is our place in the whole scheme of things, in the whole universe? While many scientists would say these are not questions for science to answe
r, in recent times, others have taken aspects of them more seriously. The result has been to add the anthropic principle to the cosmological principle introduced in section 7.2. (Anthropic from the Greek anthropos, human being.)

  Brandon Carter introduced the anthropic principle (AP) in 1974. In its weak form (WAP) it states that

  what we can expect to observe must be restricted by the conditions necessary for our presence as observers.

  This seems noncontroversial; since we exist, the conditions in the universe must allow for that. The existence of the necessary excited state in the carbon nucleus as predicted by Hoyle is often cited as an example of the WAP in action.

  The strong anthropic principle (SAP) states that

  the universe necessarily has the properties requisite for life—life that exists at some time in its history.

  That word necessarily makes this a much more controversial statement. Essentially we can read SAP as saying that the universe must have observers and so must have conditions suitable for their existence. In recent years, there has been much discussion about the “fine-tuning” of the physical parameters, like force strengths, which seem to be necessary for life (as we know it) to exist. It is sometimes claimed that the required coincidences are exquisitely precise and tuned to one another. Of course, it is not too big a step to invoke the idea of a god who carefully designed the universe, and the whole debate extends into a number of philosophical and religious issues.

  (The literature on this subject is large; the interested reader might start with Paul Davies's book The Goldilocks Enigma—Goldilocks, remember, wanted everything to be just right. For a brief introduction see chapter 8 in Harrison's Cosmology. An insightful and entertaining opinion of anthropic principles is given by mathematician-philosopher-writer Martin Gardner. He calls the final anthropic principle (FAP) the “completely ridiculous anthropic principle”—you can work out the abbreviation for yourself and get an idea of Gardner's take on the subject!)

  To conclude, it is appropriate to return to Hoyle. His work on carbon predates the anthropic principle, and certainly in his 1954 paper there is no mention of the necessity of carbon for life. (The paper “An Anthropic Myth” by Helge Kragh provides an extended discussion.) As well as using the excited state of carbon, Hoyle noted the precise energy level of a state in oxygen which this time acted in the opposite sense; if the state in oxygen was a little different, all the carbon would quickly convert to oxygen. He wrote:

  The positioning of these levels [in carbon and oxygen] depends on the electrical repulsion between protons, and the strength of the nuclear forces that bind protons and neutrons within the nuclei. Change those two opposing effects only slightly, and the levels in C12 and O16could be changed by an amount that would produce a world essentially without carbon, and hence without life as we know it.10

  Certainly Hoyle found a reason for some very careful tuning of physical parameters. In his autobiography he asks:

  Was the existence of life a result of a set of freakish coincidences in nuclear physics?…Or is the universe teleological, with the laws deliberately designed to permit the existence of life, the common religious position?11

  Debates related to calculation 23, Hoyle makes carbon are set to continue for a very long time.

  7.4 WHAT IS THE MATTER?

  In section 7.2.2, we saw that the average density of matter in the universe is the key parameter in determining its nature—whether it expands or contracts, for example. Hence discovering how much matter is distributed throughout the universe is a central problem of cosmology. We already saw that the strange dark energy contributes about three-quarters of the density to be used in equation (7.3), leaving only about 25 percent of “ordinary matter.” However, the peculiarities in the matter question do not stop there. As observational techniques were refined in the twentieth century, a new puzzle arose.

  7.4.1 Large-Scale Motion in Galaxies

  Gradually it became possible to observe galaxies, their motion, and the motion of their component stars and gases. These are large-scale gravitational phenomena, and so Newtonian theory can be applied. Some of the earliest studies by Jan Oort in 1932 and Fritz Zwicky in 1933 already showed that matching observations and dynamical theory was not straightforward. In his 1937 paper, Zwicky found that different methods failed to give consistent values for the masses of galaxies (nebulae).

  The work of astronomer Vera Rubin provides a good example of how the field has progressed. Vera Rubin's life and achievements illustrate the struggles women faced in order to succeed in the field of astronomy and how someone can be an important scientist as well as a wife and proud mother. (Her inspirational story is told in the biographical essay by Robert Irion.) In a (now) amusing sign of the times, the Washington Post headlined a story in 1950 about Rubin with “Young Mother Figures Center of Creation by Star Motion.” Remembering this, in 1993 when President Clinton recognized her brilliant career, some of Rubin's friends suggested the headline: “Old Grandmother Gets National Medal of Science”! The famous George Gamow supervised Rubin's PhD studies. When she visited him at the Applied Physics Laboratory at John Hopkins University, their meeting had to take place in the lobby because women were not allowed in the laboratories. Rubin was a pioneer; in 1965, she was the first woman allowed to observe at the Palomar Observatory in California.

  Starting in 1965, in collaboration with Kent Ford, Vera Rubin made extensive measurements of motion in the Andromeda, M31 galaxy. Later work extended to other galaxies. (I recommend Rubin's articles in Science, Scientific American, and Physics Today for an introduction to this work and the story of its progress and impact.) The result was observational data on the rotational velocity of stars at different distances from the galaxy center. Many results for different galaxies were accumulated, and figure 7.5 shows two examples taken from Rubin's 1983 Science paper.

  Figure 7.5. Rotation curves for components of the galaxies NGC 801 and UGC 2885 as a function of the component distance from the galaxy center. Reprinted with permission of AAAS, from Vera C. Rubin, “The Rotation of Spiral Galaxies,” Science 220, no. 4604 (1983).

  Notice that the velocity of the stars remains the same as we move away from the galaxy center. It is almost as if we had discovered that Jupiter, Saturn, and Uranus all travel with the same velocity. There is something extremely troubling about that result, as I explain below with the relevant calculations, and it led Rubin to give this remarkable summary for her paper:

  There is accumulating evidence that as much as 90 percent of the mass of the universe is non-luminous and is clumped, halo-like, around individual galaxies. The gravitational force of this dark matter is presumed to be responsible for the high rotational velocities of stars and gas in the discs of spiral galaxies.12

  (Of course, Rubin gave this 90 percent figure not knowing that a very large contribution to the universe was yet to be discovered in the form of dark energy as mentioned above in section 7.2.4.)

  There were also results for the motion of several galaxies, or clusters of galaxies, which gave strange results when the virial theorem was used to analyze them. To make clear the magnitude of the effects under discussion, consider this breathtaking start to the 1974 paper by Ostriker, Peebles, and Yahil:

  There are reasons, increasing in number and quality, to believe that the masses of ordinary galaxies may have been underestimated by a factor of 10 or more. Since the mean density of the universe is computed by multiplying the observed number density of galaxies by the typical mass per galaxy, the mean density of the universe would have been underestimated by the same factor.13

  The crucial parameter—the density of the universe—used so far might be wrong by a factor of ten!

  7.4.2 How It Works

  It is time for one of those calculations linking observations and theory.

  We can understand the source of the above conclusions by using two simple results from dynamics. First, we need to know the force exerted on a body of mass m in a spherically symmetr
ical mass distribution. In his Principia, Isaac Newton calculated that if the body is at a distance R from the center, then it experiences a gravitational force equivalent to a mass M(R) placed at that center, where M(R) is the sum of all the mass in the spherical distribution out to that radius R. A key point is that the mass outside the sphere of radius R exerts no force on the body—all the different contributions average out to zero. The force is given by equation (6.1) after substituting r = R and M = M(R).

  Second, we need to know the dynamical law that force equals mass times acceleration; for the body rotating around the mass distribution center with radius R that acceleration is V(R)2/R where V(R) is the body's velocity at radius R.

  Putting those two facts together gives

  Thus the observed velocities V(R) are directly related to the mass distribution M(R).

  One case is very simple. Suppose the galaxy mass distribution ends at a radius RG and M(RG) = MG, then for bodies at a distance R > RG equation (7.4) gives

  This means that for a galaxy of size RG and mass MG, the dust and few stars out at the edge of the galaxy should have velocities which decrease like the square root of their distance from the galaxy center. This is also the case for planets orbiting the sun; the outer planets are very slow compared with the inner ones as the 1/√R indicates. (In fact this is also what Kepler's third law tells us.) But Vera Rubin did not find that type of velocity decrease; she found that the velocities remain constant.

  Generally, equation (7.4) tells us that if V(R) is a constant, as the observations suggest, M(R) must be proportional to R. We cannot simply use the galaxy mass MG found by adding up the masses of its constituent stars; the mass continues to increase as R increases.

  Thus the galaxy masses required to fit the observed component velocity patterns do not correspond to the luminous matter. There must be some other “dark matter.” As Rubin said about her observation, “this dark matter is presumed to be responsible for the high rotational velocities.”