PART II ….

The British astronomer R. d'E. Atkinson was the first to suggest, in 1931, that the capture of a proton by an atom could liberate enough energy to light the sun. Eight years later, Hans Bethe and C. von Weizacker presented the same idea but marshalled better evidence for it in their study of thermonuclear fusion process known as the carbon-nitrogen-oxygen, or 'CNO cycle'. The CNO cycle was soon found to work well for stars like our own sun where internal temperatures had been estimated to be about 20 million degrees. Yet, the majority of the sars in the sky were less luminous than the sun. The red dwarf stars like Kruger 60A whose core temperature was only 16 million degrees was a case in point. That slight temperature difference translated into a 100-fold reduction in energy production and a predicted luminosity for Kruger 60A about 100 times fainter than it was known to be. So, what is it that powers stars cooler than the sun? The answer was provided by Hans Bethe who showed that a fusion reaction which converted hydrogen into helium, but not involving the CNO reaction, would work at these low temperatures. More advanced burning cycles have been studied since then which are capable not only of supplying even greater quantities of energy to support a star against gravitational collapse, but reactions capable of creating all of the known elements in the periodic table, in the cores of very massive stars.

The first stellar models that showed, in detail, how a star evolves from the hydrogen fusion phase called the 'main sequence', through the red giant phase did not become available until electronic computers were developed. Prior to the advent of computers, the computations had to be done by hand using desk calculators. This led to trade-offs between using a crude model of the star's interior and taking many steps in time, or using a moderately detailed model of the interior but taking only a handful of time steps.

In 1955, R. Harm and Martin Schwarzschild published 15 ‘models’; some calculated by hand, others by using the electronic computer at the Princeton Institute for Advanced Study. The models presented the star's interior in three zones: the core, the outer envelope and the intermediate zone where convection would likely occur. Radiation pressure was ignored, as were differences in chemical composition between the zones, and no internal energy source was treated in detail. It took one full year of laborious work on a desk calculator to construct the hand-calculated models that were computed for a total of 127 time steps. The models specified the changes in 18 quantities in each of the three zones. In contrast, the computer-generated model was followed for 37 time steps and required less than a day to compute. Continuing in a steady progression as faster computers were developed, present-day computers can calculate complete stellar models in less than one minute!

The computational extension of the models from the hydrogen burning phase to later stages began in earnest in 1961 with the appearance of several papers announcing detailed, independent studies of 5, 10 and 15 solar mass stars by Chushiro Hayashi, Robert Cameron and Emil Polak. They used IBM 650 and 7090 computers, splitting each star into a dozen or more internal shells. Their program followed the evolution of each star's structure, shell by shell, through the helium burning stage. For the most massive stars, the carbon and neon fusion stages were followed as well. They watched as the stars swelled to enormous dimensions and became red supergiants, as their cores collapsed switching first to helium burning, then to carbon and neon.

By 1964, the role of neutrinos in producing added pressure in the dense cores of more massive stars was discovered and incorporated into the models. John Cox and Edwin Salpeter also examined the evolution of stars where electron degeneracy pressure was important. A similar calculation for stars 4 to 8 times the mass of the sun done by David Arnett in 1969 showed that if the carbon burning cycle was triggered in a core that was degenerate, the entire star would blow up in a 'Carbon Detonation Supernova'. Whether anything was left behind other than an expanding cloud of gas seemed to depended very critically on the density of the star's core before the detonation, and just how much pressure the neutrinos escaping from the star's core produced in the overlaying matter. Depending on the core's density and mass, what would be left behind the star after this explosion would be: nothing, a white dwarf, or possibly a neutron star.

Since the 1960's, computer models have become more sophisticated. Periodic revisions have been made in the number of nuclear reactions that are considered, as well as updates in the reaction rates and energy yields based on more exact theoretical calculations supplemented by experimental results. The detailed role of convective mixing in transporting energy from place to place within the star and changing the composition of the star is also being studied, as are the roles of rotation and mass loss. As the models become more refined, they are used to an ever-increasing degree in explaining the observed details of known stars. Some stars show an overabundance of certain elements over others that cannot be entirely explained by temperature effects alone. This suggests that convective mixing seems to be the culprit, wherein the elements in deeper layers in the star are mixed with the visible surface layers. Then again, for the peculiar A-type stars, convection may be suppressed by strong magnetic fields that have been measured on the surfaces of these stars, so that atomic diffusion driven by radiation pressure may be a more important factor.

A related area of study concerns the evolution of binary stars. The presence of a nearby star can alter the evolution of both stars, especially if matter is being pulled from one companion and dumped onto the other. The gravitational stresses that result inside a star with a close companion can alter convection patterns and mix enriched hydrogen gas with hydrogen depleted material in the core, so that one star, essentially, gets to re-live its youthful, hydrogen burning phase all over again as though it had just been born.

The final stages in the evolution of stars are also of great theoretical interest. Exactly how do planetary nebulae form? How are neutron stars and black holes produced from supernova explosions? Do all supernovae produce identifiable remnants? Although we are tantalisingly close to answering these questions and can do so in general terms, the details are still a bit vague.

I have spoken about mathematical models for stars, but I have not really described for you what I mean by this terminology. How do you reduce a pinpoint of light in the sky into a collection of equations, and what would these equations look like? The basic equations defining the structure and evolution of a star have been known for nearly a century. They describe what determines whether a star is stable, or subject to gravitational collapse. They describe how energy is transported from the core of the star to its surface, and how the density and temperature of the gas varies from the core to the surface. This theoretical model must also describe how much energy is liberated by the various possible fusion reactions occurring in each gram of matter in the core. When we express all these relationships and interdependencies in symbolic form, we get the 'equations of stellar structure'.

But these equations are not enough because you also have to specify how the pressure inside a star, which supports it against gravitational collapse, is dependent on the values for other physical quantities like a star's chemical composition, temperature, and density, which may, in turn, change from place to place inside the star. The amount of energy released in the thermonuclear fires in the star's core, also depend on these quantities as does the stellar opacity. The equation linking the pressure to the other variables is called the 'Equation of State' by the astronomical cognoscenti, and its form can change as the star evolves or as you dissect the star and examine various layers within it. The pressure due to light radiation and high temperature gas is usually expressed by, For high gas densities near 10⁵ grams/cc, we also have to include electron degeneracy pressure, caused when electrons are squeezed together into a small volume. The opacity of a star determines how transparent it will be to its own emitted light radiation. Since radiation pressure is in many cases the most important internal support for a star, its accurate specification is crucial. Depending on the kind of interaction involved between matter and the light streaming out from the star's deep interior, the mathematical description of the transparency of the star's matter takes-on a variety of different forms. The sum total of these will determine how opaque the star is at a particular point in its interior, and how much radiation pressure will result. To write down all the different forms of the matter-radiation interaction that contribute to a star's opacity would easily fill a book of this size!

Although gravity is the ultimate source of energy for heating a star's interior, it is the nuclear reactions that provide the energy from which the star's internal pressure is ultimately derived. A complicated network of interdependent equations is required to account for the energy released by fusion reactions and how they change the internal element composition of a star. These describe how rapidly one element is converted into another by fusion or radioactive decay, and shows how the rate of energy release depends on the local temperature and density of the star. To assemble these equations, one must first write down all the important pathways by which the conversion from one element to another occurs, and the energy released at each step. For example, when the cores of stars more massive than the sun reach temperatures exceeding 100 million degrees, the so-called Triple Alpha reaction becomes important in supplying the thermal pressure needed to prevent further gravitational collapse. In this fusion reaction, two helium nuclei fuse into a single beryllium nucleus; then, after an additional helium nucleus fuses with the beryllium, one obtains a single carbon nucleus as nuclear 'ash'. The reaction also produces a considerable amount of energy.

At still higher temperatures appropriate to pre-supernova conditions where temperatures exceed 5 billion degrees, one encounters reactions that convert carbon into oxygen, oxygen into magnesium and silicon, and finally silicon into iron. All these reactions are very temperature sensitive. For instance, in Triple Alpha fusion, the reactions produce 10 times more energy at 105 million degrees than at 100 million degrees! Where does a star get the high energies and temperatures to allow these reactions to proceed? The answer is from the gravitational collapse of the core of the star under its own weight. Just as a rock gains speed and kinetic energy as it falls to the ground unsupported, the matter inside the core of a star, if unsupported by a counter-balancing pressure, will continue its fall towards the stellar core. In so doing, it gains kinetic energy that appears as an increase in temperature of the gas.

The change in the chemical composition of a star as it 'burns' one element and leaves behind another as a nuclear 'ash' can be represented by yet another set of equations. Modern nuclear reaction networks such as those used to study the last years of a star about to become a supernova, incorporate over 250 nuclear species and their isotopes, along with their highly interdependent equations of interconversion. Having considered the interior of the star and what goes into describing its inner workings, what of its outer layers?

How does a star look to a distant observer? All you see through the eyepiece of the most powerful telescope is the radiation emitted by the surface of the star. The interior is completely hidden from view. Not only that, but the light produced in the star's dense core requires millions of years to reach its surface, before it can start its journey to earth. There are models available for predicting the strengths and shapes of the atomic spectral lines emitted by the surface gases, but these models depend on the temperature, density, composition and surface gravity of the star. You can obtain predictions for these quantities at a particular instant in the life of a star using your stellar evolution model. These 'stellar atmosphere' models are very complicated; to merely write down the necessary equations would fill up several books this size. The most sophisticated model now in routine use is the one developed by Robert Kurutz at the Centre for Astrophysics in Cambridge, and his co-workers. His model contains 1,760,000 spectral lines for elements between hydrogen and nickel, and computes the expected spectrum shape and line intensities for most kinds of stars commonly studied in detail.

In addition to high temperature plasmas of charged atoms, stars are known to contain magnetic fields. A detailed study of the sun reveals a strong surface field of about 1 gauss, and sunspots where the fields are thousands of times stronger, along with a periodic 22-year cycle of magnetic polarity reversal, better known as the Sunspot Cycle. Other phenomena related to stellar magnetic fields include prominences, flares and coronal holes. Magnetic fields have been detected on nearly 100 stars, mostly of the peculiar A-type, which have surface fields 100 to 30,000 times stronger than the sun's. Sunspot cycles have also been observed on a number of nearby stars. Thanks to the rapid influx of data from satellite observations of the sun, and long-term studies from ground-based earth observatories, the detailed description of the role of magnetic fields in our sun has evolved rapidly from crude 'back of the envelope' calculations to highly sophisticated theoretical models. Presumably, the physics of the magnetic fields on more distant stars can also be described by this same theory, or simple modifications of it.

Solar Dynamo Theory provides a mathematical framework for understanding.......…