Danksrede – Rom, 16.11.1994 (englisch)

Grossbritannien - USA/Deutschland

Fred Hoyle e Martin Schwarzschild

Balzan Preis 1994 für Astrophysik (Evolution der Sterne)

Für ihre bahnbrechenden Beiträge zur Theorie der Sternentwicklung, die als Bezugspunkt der modernen Entwicklung dieses Forschungsgebiets gelten.

Released on the occasion of the ceremony for the awarding of the 1994 Balzan Prize for astrophysics (evolution of stars)

FRED HOYLE

ADVANCES IN THE UNDERSTANDING OF THE EVOLUTION OF STARS
released on the occasion of the ceremony for the awarding of the 1994 Balzan Prize for astrophysics (evolution of stars)
lt is a matter of general interest to understand why science has developed with a startling exponential rapidity over the past few centuries. Why did it happen over the last millennium and not before, or later? Some exceptional combination of circumstances perhaps , of which I have often felt the building of the great cathedrals of medieval times was one. Large and splendid buildings had of course been constructed in classic al Athens and Rome, and even at earlier dates in other places. What characterized the medieval cathedrals , however, was the remarkable extent of their diffusion. Most towns of any consequence throughout western Europe had one , which meant that the pool of skilled workmen and the understanding of mechanical technology became very widely spread. With the consequence that in almost the first moment when an abstract scientific idea was conceived it was readily possible to give it a practical expression. lt became natural to turn abstract ideas into explicit experiment, permitting scientific knowledge to advance in known steps rather than in speculations piled on speculations, a process that in earlier times eventually collapsed under the weight of accumulating uncertainty. The study of the evolution of stars stands in a somewhat paradoxical light in relation to these considerations. On the one hand , it is not possible to experiment directly with stars, while on the other hand I can think of no part of science where the outcome of laboratory studies is of wider application than it is to the physics of stars. All the four forces of physics-gravitation, the electromagnetic interaction, the weak interaction and the strong interaction – are deployed to the very limit of what has become known about them from studies in the laboratory.
Although we cannot experiment on whole stars, experiments are carried on in immense profusion on the products of stars . Indeed , without the products of stars there could be no experiments at all, no technology, no Earth, no life. For with only three or four rare exceptions the chemical elements were produced in the stars. Without the stars there would be no chemistry, no molecules , no building of molecules into the materials of our everyday world.
One would think that from time immemorial people would have argued that stars are objects like the Sun, but at great distances away from us. And yet among the immensely intelligent writers of classical times I have never seen a clear-cut statement to this effect. Presumably the thought of just how far those distances would need to be was a seriously inhibiting factor. It certainly inhibited medieval authors. I have not been able to find mention of it even in Kepler’s brilliant and revolutionary book the New Astronomy published in 1609, not in its six hundred and more pages. Giordano Bruno is said, however, to have advanced this view at about the time of Kepler’s book, and by the end of the seventeenth century Isaac Newton had given a sound argument to show that stars must be distant bodies of the nature of the Sun.
There the matter largely rested until the end of the first third of the nineteenth century when the development of refracting telescopes in Germany at last permitted the distances of a few of the nearest stars to be measured. They were of course at the vast distances which had been anticipated, although the now certain knowledge that it was so still created a sensation. Although little had been done in the eighteenth century on stars as bodies to be studied individually, extensive pioneering work had gone ahead on their distribution in space. The first sky surveys were made and the observed star distribution of the Milky Way was tentatively identified as a flat disklike aggregate, which today we call our galaxy.
Steps towards the study of individual stars, and of the Sun especially, were taken in the last third of the nineteenth century, following the discovery of the spectroscope. It now became apparent that there were many different kinds of stars and emphasis passed for a while into classifying the various kinds, into the types and subtypes which have served as a useful framework of discussion throughout the present century. Also in the last third of the nineteenth century attempts were made to understand the physics of the interior of stars, with particular reference to the Sun, in terms of the newly discovered properties of gases. The enormous mass of the Sun was now known, and it was realised that the fact that the Sun is not in a state of collapse must mean that its interior pressure is extremely high, implying an exceedlingly high central temperature, with a corresponding temperature gradient necessarily occurring from the centre outwards and with energy necessarily flowing down this temperature gradient. lndeed, by assuming the interior of the Sun was in convective equilibrium the first stellar model was calculated by Homer Lane in 1869, a calculation which set a kind of starter’ s pistol for the race towards understanding the nature of stars that was to occupy much of the attention of astronomers over the ensuing century.
This first tentative exploration already turned up a deep physical problem. It was realized by Lord Kelvin in Scotland and von Elmholtz in Germany that the great heat present in the interior of the Sun could be understood in terms of the effects of gravitation. As a body shrinks it tends to become hot, and if the body is of great mass the amount of heat released is very large, adequate to supply the needed high pressure at the centre of the Sun. With the continuing escape of sunlight out into space, this process is required to persist. To maintain it, as Kelvin and Helmholtz saw the situation, the Sun must continue to shrink more and more as time goes on, in order to provide for the energy that is constantly being radiated out into space. This requirement raised two questions . Would the rate of shrinkage be sufficient to be observed? The answer was no, and it wasn’t. So no trouble there. Second question: Would the shrinkage be slow enough for the Sun to have been not unduly different from what it is now over the whole of geological time? To answer this question it was necessary to know the true extent of geological time and through the last quarter of the nineteenth century there was controversy over this issue, with Kelvin claiming not more than ten million years and geologists claiming more than a hundred million years. By the beginning of the present century the controversy had ended in a victory for the geologists. The gravitational energy released by the shrinkage of the Sun was not sufficient to account for the amount of energy which the Sun has radiated over the course of geological time. So what other source of energy could there be? The answer to this important question had to wait another forty years, to the beginning of my own research career. To anticipate the answer, the Sun and other stars are able to radiate prodigious quantities of energy, in the case of the Sun for almost five thousand million years, because they are nuclear reactors.
At the beginning of the present century physicists believed their science to be almost complete. There were only a few apparently awkward little things which had still to be understood , they thought. Actually, the “little things” were really indications of vast new territories of scientific thought which as yet, in 1900, were not even conceived of. The source of the Sun ‘s energy was one of those ‘ little things ‘.It foreshadowed the science of nuclear physics . lt is inter sting that fully a half of the ” little things ” of the year 1900 came from astronomy. Besides the energy of the Sun there were the dark lines discovered by Fraunhofer in the spectrum of the Sun, the interferometer measurements made by Michelson, and the excess rotation of the perihelion of Mercury discovered by Le Verrier. ln the laboratory there was the newly discovered radioactivity of Becquerel , the frequency distribution of black-body radiation, and not a great deal else.
The position today as we draw to the close of the twentieth century is similar to what it was at the end of the nineteenth , with physicists claiming that little remains to clear up except a few “little things”, the “little things” nowadays being mostly to do with cosmology . Perhaps one can be pardoned for suspecting that the eventual story will turn out as before, that the “little things” are really indications of vast new territories still to be discovered.
It was realised in the first decade of the present century that there could be equilibrium structures for a hot gaseous massive body that were not necessarily convective. It was also realised that energy would flow slowly out of such a body through radiation, and that this would go on inexorably , even if there was no convection, a problem that was solved in Germany by Karl Schwarzschild, the father of my fellow recipient of the Balzan Prize .
The data necessary for solving the energy problem of the Sun at last began to come along in the second decade of the century, with the determination of the masses of more than 200 nuclides, by Aston in Cambridge. It was then apparent that the aggregation of four hydrogen atoms into a helium atom would provide the richest energy source for stars, although there were many transmutations among atoms that would provide lesser sources . The concept of four hydrogen atoms fusing into helium did not have general approval, however, because the conditions inside stars did not at that time seem favourable to it. Nevertheless, there were a few who argued for this point of view, notably Perrin in France and Eddington in Cambridge.
A long step was taken by George Gamow in 1928-29 towards understanding atomic transmutations in stars, but the precise means whereby hydrogen forms into helium had to await another decade, until 1938 – 39 when the detailed nuclear reactions responsible for this key transformation were at last described by Hans Bethe.
Despite the lack of vital information on the energy sources of the stars, Eddington had shown in the early 1920 ‘ s. that from a knowledge of the mass and radius of a star and from the requirement that it be long-lasting, as the Sun has been long- lasting, it was possible to deduce how bright it must be, a result which became known as the mass-luminosity law. From this work a central temperature for the Sun of twenty million degrees was inferred and this provided a crucial datum for Bethe in his determination of the nuclear properties of the central regions of the Sun.
lt was at this point that my own interest in the field began. Eddington ‘s work and Bethe ‘s work had been separated both in logic and in time, a separation that was made possible by Eddington ‘s initial assumption of a star’s radius. It had to be possible to dispense with this assumption I realised. To rework Eddington ‘s calculation without assuming a star’s radius but now putting into the calculation the nuclear physics that Bethe had discovered. This project was carried through in the first months of the second world war, in collaboration with my colleague Lyttleton. It permitted a star’s radius to be calculated as well as its luminosity.
There was one interesting detail that needed to be put right before further advances could be made, however . It related to the chemical compositions of stars at their birth. In the 1920 ‘s, a high percentage of heavy elements such as iron had been assumed present inside stars like the Sun. This error had survived into the late nineteen thirties, despite a large observational preponderance of hydrogen in stellar atmospheres being known from 1926-28. Even despite an alternative possibility , with only two percent or so of heavy elements compared to hydrogen being shown to be feasible by the work of Strömgren in the early 1930’s. It was not until the immediate post-war years of 1946-47 that this alternative possibility, with some 98% of a star’s initial material (or even more in some cases) being hydrogen and helium in a proportion by numbers of atoms of about 12 to 1 became widely accepted. lt had the effect of dropping the central temperature of the Sun from about twenty million degrees to about fifteen million degrees, changing the balance of importance of the nuclear reactions that had been discussed by Bethe.
With this detail in place it then became possible to ask the crucial question which defines the problem of the evolution of the stars. As nuclear reactions change the initial composition of a star, how does its structure change? It would be a long time, extending beyond my own work in the subject, before a fully satisfactory answer to this question would be given. But some partial explorations could be done and these proved sufficiently interesting in their relations to observation to encourage further efforts. Indeed, this feature was to prove the touchstone of the problem, namely that at all stages as the complexities increased the output of worthwhile results proved sufficient to encourage the necessary further efforts to be made.
As the years advanced into the 1950’s it became clear that research which had begun with the purpose of understanding the nature of the stars had a second objective of as much importance as the first. For the nuclear reactions that were causing the stars to change , causing them to evolve , were also building more complex atoms from simpler ones. Could this be the origin of the chemical elements one was compelled to wonder? The origin of the materials of our everyday world? These questions had been asked already some years before 1950 but it was not until the early 1950′ s that their answer was seen to be affirmative. Stars were found possessing variable proportions of the different elements. which would not be the case if they all came from some initially invariable source . And radioactively unstable nuclides were observed spectroscopically to be present in some stars, which themselves had evidently to be the places of origin of the nuclides in question. Much of my own work in that period was concerned with these questions and with attempting to solve the further problem of how materials that are synthesised inside stars ultimately gets out of their places of origin , at first out into interstellar space and then into new stars and planets. This phase of the work culminated in 1957 in a paper written in collaboration with Geoffrey and Margaret Burbidge and with William Fowler.
I have already mentioned how astronomers were using spectroscopie methods to examine the stars even as early as the last third of the nineteenth century, work which continued at full pace into our own century. I also mentioned that many varieties of stars were round by these methods. ln an age when the internal processes within stars were at best only vaguely understood, these many varieties were thought to be distinct kinds of stars, independent of each other. The thought was that the distinctness now observed in the many varieties had somehow already existed at the birth of the stars. But with the developments I described above it became apparent that in its lifetime one and the same star plays many parts. The determination of these parts – a long-lived early placid period, than an enormous increase of outer volume to what is called a giant phase accompanied perhaps by atmospheric oscillations and short-lived internal explosions or ‘flashes’, loss of material from the extended atmosphere out into space leading to a shedding of the outer envelope, a luminous blue phase which again may be oscillating and a final demise into either a placid white dwarf or into the enormous explosion of a supernova, still to be followed by a dramatic finale as a rapidly rotating pulsar or (as some would have it) by a black hole. In truth an adventurous life.
Of course. stars differ one to another in some obvious respects. They differ in the amount of matter they contain in their masses . And they differ in their initial content of heavier elements, in particular in their content of metals such as iron. This is because iron is itself produced in stars and is thrown out explosively by supernovae into interstellar space, a process that is not spatially uniform. This leads to an irregularity in the initial metal contents of stars depending on where in relation to past supernovae they happen to be born. And there is a still larger variability depending on the time when stars were born. Those that formed early in the life time of our galaxy have less in the way of heavy elements than those born late , because for those born early there had not yet been adequate time for much in the way of heavy elements to have been produced.
The obvious difference in masses among the stars leads to differences in their evolutionary properties. But it soon became apparent that among stars of the same mass there were differences of evolution that were often large. The problem was that the other difference, that in heavy element content seemed at first to be far too minor to be of much consequence . Even for stars with the most heavy elements only about 2 percent of the mass was in heavy elements, the rest being hydrogen (about 73 percent) and helium (about 25 percent). Stars with small heavy element content could be a hundred, or even a thousand times, less than this. How could such small proportions be of any major consequence?
This question was answered collaboratively by Martin Schwarzschild and myself. We were able to show that initial differences of heavy element content lead in subtle ways to differences often very marked differences, in evolutionary behavior. thus opening out rich lines of stellar evolutionary development to be investigated.
When stars are found in clusters it can be taken that their ages and their initial compositions are all effectively the same. Then by studying evolutionary differences with respect to their individual masses, it is possible to determine the age of the cluster in question . For this observations of individual cluster stars are needed usually presented in what is called a colour-magnitude diagram. Already before 1960 this technique had resulted in a determination of the age of the stars of our own galactic disk at about 12 billion years. This estimate has hardly shifted ever since . It has been obtained repeatedly by investigators down the years. Star clusters outside the disk of our galaxy in what is called the halo of the galaxy turned out to be somewhat older at 14 – 15 billion years and this too is an estimate that has scarcely changed at all with the years.
The years have brought immense improvement of detail, made possible by the intensive use of digital computers. It is not unreasonable to say that the branch of astronomy to which a Balzan prize has been a warded this year, the evolution of stars, began the present century as vague and ill-understood , but that it will go out from the 20 th into the 21s t Century as one of the best understood parts of astronomy and astrophysics . Rather as the 18th and 19th centuries bequeathed celestial mechanics in an almost finished form to our present century, so we shall be bequeathing the twin subjects of stellar evolution and nucleosynthesis to the future. It has been an immense privilege to have been associated in some measure with this development.

MARTIN SCHWARZSCHILD

EARLY RESEARCH ON STELLAR EVOLUTION IN GLOBULAR CLUSTERS

The theory of stellar structure and evolution has had a history which was paced by the development of three broad fields of physics: classical hydrostatics, atomic physics and nuclear physics. Without the laws governing these branches of physics astronomers would have had no basis for the analysis o f stellar structure .
Hydrostatics was well developed before the turn of the century, and several mathematically inclined astronomers used the consequent physical Jaws to compute models for stars in hydrostatic equilibrium. This development was summarized in Emden’s book “Gaskugeln ” . The difficulty at that stage was that hydrostatics by itself does not give the relation between gas pressure and gas density. For this missing relation one had to substitute a guess, which was then put into the form of a simple power law. The resulting stellar models were called polytropes.
How did it come that these theoretical astronomers applied so much bright thought and numerical efforts – with nothing but logarithm tables! – to a problem which in its basis included a pure guess? Did they feel that at times a science gets into a state in which numerical exploration may be a major tool for subsequent advance? In any case, their efforts in studying the poly tropes were richly justified by the extensive use of their polytropes in the following phases of the theory of stellar structure.
Early in this century, atomic physics advanced far enough to provide quantitative laws for the interactions between light and matter. Specifically, it became possible to determine the rate at which light is absorbed by the hot gases in the stellar interior. Soon Eddington realized that this advance in physics made it possible to add the consideration of thermal equilibrium to that of hydrostatic equilibrium in the theory of the stellar interior. Inside a star thermal equilibrium takes the form not of isothermal conditions common in laboratory experiments, but of a steep temperature gradient from the hot interior to the cool surface of a star. The temperature gradient causes a transport mechanism to carry the energy produced in the stellar core outwards to finally leave the star by being radiated away from the star’s surface. In the deep interior of most stars the energy transport mechanism is provided by radiation for which the relevant physics had just been established.
Still, the theory of stellar structure was incomplete: the nuclear processes which provide the energy sources in the stellar core were as yet unknown. In spite of this incompleteness of the physical basis Eddington, in a brilliant turn, showed that with the physics at hand one could derive a relation between the mass of a star and its total luminosity. This relation became a keys tone for the theory of stellar structure. For certain nearby stars masses and luminosities had been observationally determined and thus a hypothetical development was turned into a theory with observational checks.
One other problem, that of the highly condensed white dwarfs, was solved in the period prior to the required advance in nuclear physics. This was accomplished by Chandrasekhar in a definitive sweep. Atomic physicists had established the pressure-density relation for dense, fully degenerate gases. This relation does not involve the temperature. Therefore, if combined with the laws of hydrostatics, it defines uniquely a sequence of models for the white dwarfs. Chandrasekhar showed that this sequence terminates in an upper mass limit – which in turn provides another contact with observations.
It would seem relevant to note the great differences in attitude towards computation as a scientific tool among those active in these early phases or the theory of stellar structure. To begin with , researchers like Emden employed computation vigorously. Perhaps they were still affected by the example set by the researchers in classical planetary orbit theory. Later and in stark contrast, Eddington and also Milne looked down on computation as an inferior tool. That attitude, however, did not prevent Eddington from choosing those approximations to his radiative thermal equilibrium equations that would cast them in the form of the polytrophic equations for which Emden had given the numerical solutions. Subsequent Eddington ‘s derivation of the mass-luminosity relation, he and Milne engaged in a protracted and acerbic dispute about a certain feature of that relation. This dispute could have been resolved by some explorative computations. But both disdained such exploration and died with the dispute unresolved.
Computation was reintroduced into stellar interiors work by a student of Eddington and Milne, Chandrasekhar. Even though he turned out to be the most gifted analyst of all involved, nevertheless he did not hesitate to carry through extensive computations, both for the white dwarf models and for rotational and tidal perturbation studies. Thus he set the style for all of us involved in the subsequent phases of the theory of stellar evolution – an essential turn in preparation for the a appearance or electronic computers.
It was 1938 when the breakthrough occurred that brought nuclear physics into the theory or the stellar interior. Von Weizsaecker in Europe and Bethe in America uncovered the specific nuclear processes which transmute hydrogen into helium in the cores or stars and by this fusion provide the stellar energy sources. They even provided the first estimates of the rates of these processes at the temperatures and densities characteristic for the stellar interior. Bethe immediately applied the new knowledge to main-sequence stars, the most common type in the solar neighborhood, and found that the observed data for these stars fitted well the high temperature sensitivity of the nuclear rates. Thus the scene was set for exciting research on stellar evolution.
I fear we should admit that most of us were slow in committing ourselves to this new challenging problem. We had excuses: the war occupied many of us for a number of years. But there was one astronomer, Oepik in Tartu , who had the insight to realize that the fusion process would decrease the hydrogen content in the core – while increasing the helium content – and thus produce an inhomogeneous composition in the star. He computed some sample models of such inhomogeneous stars and showed that these models seemed to fit the red giant stars. He published this work as early as 1938 and 1939 – and the rest of us completely ignored his papers. Why? It is true that Oepik tended to express himself exceedingly sharply about the research of others. That made studying his papers at times unpleasant. But did this human cause really suffice to make us fail to accept the big step Oepik had accomplished and thus to retard progress in our field by quite a few years? In retrospect, a depressing thought.
After the end of the war when we were free again to concentrate on stellar evolution. main-sequence stars were securely identified with the early phases of stellar evolution and red giant stars were suspected of representing later evolution phases. However, the transition between these two phases was not well documented by observations. This gap was convincingly filled by Sandage, then a student of Baade at Mt. Wilson. He succeeded in observing the color-magnitude (Hertzsprung-Russell) diagram for a nearby globular star cluster to such faint magnitudes that the main sequence and its turnoff to the subgiants were well documented. Thus, Sandage ‘s one diagram demanded a long program for research in stellar evolution. To push ahead that program, Sandage joined us in Princeton for a year.
From here on I will focus this essay narrowly on the study of evolution of globular cluster stars. In parallel with this particular study, research has been going forward on a broad front including the evolution of various star types, particularly more massive stars which encounter exciting new physical situations, but also stellar rotation, magnetic fields, pulsations, solar neutrinos and solar seismology. To do justice to ail this imaginative work would take a big book. Here then we restrain ourselves to the story of globular cluster stars – which at its start was fascinating indeed and still presents mysterious puzzles.
Even the turnoff from the initial main sequence phase in globular clusters presented new problems. In this phase the hydrogen fuel in the core is exhausted and the core heats by gravitational contraction until the temperature even at its edge, where hydrogen is still plentiful, gets hot enough for fusion. By this process , hydrogen fusion, which stops in the core, commences in a shell surrounding the core . We had been used to the idea that gravitational contraction is generally an ignorably small energy source in non degenerate stars compared to the nuclear sources. So it took us a while to realize that gravitational contraction nevertheless can and does play a key role in short transition phases, such as the turnoff from the main sequence . Furthermore , the appearance of a substantial gravitational energy source required a new numerical technique for its inclusion into our model building scheme – a technique that in due course was developed.
The work with Sandage had one immediate result of cosmological interest. Comparison of the theoretical stellar models representing the turnoff with the observed total luminosity of a star in the turnoff permitted the determination of the age of the globular cluster stars. The result was three billion years, which compared reasonably well with the age of the universe of two billion years as derived from Hubble ‘s rate of universal expansion. Only both were about a factor four wrong, in the same direction!
By the time Sandage had to return to Mt. Wilson, our model star had nicely passed through the subgiant sequence, but it refused to turn up the giant sequence. Instead, it disappeared into the infrared, quite in contradiction to the observed color – magnitude diagram. In due course, Fred Hoyle came to Princeton and two hectic but glorious spring semesters ensued. It is only slightly exaggerated to describe our daily procedure as follows. In the morning, Hoyle presents today’s new idea; all the rest of the day Richard Harm, my longtime research associate, and I strain ourselves to determine , by detailed computations, whether today’s idea is good or bad; great ups and downs, but over the weeks the good ideas push our work ahead.
As a first step in our investigation, we had to force our model star to turn from the subgiants up the observed giant branch . The preceding year, Osterbrock, at that time a Princeton postdoc, trained by Chandrasekhar and Stroemgren at Yerkes, following ideas developed by Biermann and Cowling , constructed models for lower main-sequence dwarfs that had convective envelopes. The latter had been computed for the first time on von Neumann’s electronic computer. Osterbrock’s models for faint dwarfs fitted the observations well. To apply the same ideas to the red giants, it was necessary to reformulate the surface boundary conditions, to construct deep convective envelopes and to fit them to radiative interior models. In carrying out this program we adopted some approximations to make the computational load tolerable. The equations we developed for this purpose were soon thereafter used with great insight by Hayashi for deriving what is now known as the Hayashi limit, a valuable general concept in the theory of stellar structure.
When our model star moved higher and higher along the giant and supergiant branch, its core compressed to the degenerate state and its evolution speeded up so as to heat the helium-rich core by gravitational contraction. This required another complication in the equations, the addition of a gravitational energy source, which we handled in rather rough approximation. We knew what would happen. Mestel had analyzed this type of a situation years earlier . When the stellar model reached the top of the supergiant branch, the central temperature attained the value for helium ignition, and a thermal runaway – not an explosion – ensued. The extra energy from the helium fusion heated the core and lifted it out of degeneracy. This process led to a more normal model, with helium burning in a nondegenerate core, though complicated by continuing hydrogen burning in a shell further out. We managed to construct two approximate samples of such models . They turned out to fit coarsely the ” horizontal branch”, known ever since Shapley’s observation of the brighter part of the color- magnitude diagram of globular clusters – an exciting identification indeed , though still quite insecure.
Hoyle and I wrote this work up in a long paper. We submitted it to the Supplements of the Astrophysical Journal, not to the Journal itself, in part because our paper was long and in part because we wanted to help Chandrasekhar in establishing the Supplements, which were new then.
Clearly , the evolution phases of globular cluster stars that Hoyle and I had covered in roughest approximation had to be redone with much greater certainty as well as higher accuracy, so that comparison with observations would become more instructive. Also, we still had not reached the end of the star’s life. But such improvements and extensions were a daunting task at the time. Electronic computers were just reaching tempting levels in speed and capacity. But our mathematical methods were still those developed in the pre-computer epoch.
The physics of stellar structure presents a mathematical problem with four first-order, highly non linear differential equations with four boundary conditions, two at the center of the star and two at its surface. We used to split this ” boundary-value” problem into two ” initial-value ” problems, one starting from the center and reaching halfway out in the star and the other starting at the surface and reaching inwards to the same halfway point. This method worked well for main-sequence models, but proved intolerably cumbersome for advanced giant models. The impasse was resolved by Henyey who, following ideas of von Neumann, devised a method in which the entire star is handled simultaneously by successive approximations . This scheme requires a large computer memory , which was just then becoming available . Thus, the jump was made from thinking in terms of hand computing to thin king in terms of modern computers.
Subsequently. the evolution phases previously covered in rough approximation were recomputed with much enhanced certainty, by a widening group of researchers. Luckily, the earlier work was confirmed in its broad features, but new useful details were uncovered, particularly some that permitted a more precise interpretation of the “horizontal” branch and its variations from cluster to cluster.
Next the evolution was followed beyond the end of the horizontal branch. At this phase, the star has exhausted the helium fuel in its core and helium burning moves out to a shell surrounding the carbon- rich core, while the hydrogen­ burning shell continues to simmer much further out. The star starts moving up the supergiant branch for a second time.
When this phase was reached, a nasty numerical instability was encountered. The Henyey method refused to converge . We spent much time experimenting with mathematical tricks, being convinced for quite a time that our numerical scheme must be at fault. By and by, however, it became apparent that convergence could be enforced by the adoption of unusually small time steps between successive models in an evolution sequence. The star was suffering from a thermal runaway in the helium burning shell, producing thermal flashes in regular cycles .
We had blindly stumbled in to a new thermal instability of a star. Exactly the same happened within a month to Kippenhahn and Weigert, who were following the evolution of a much more massive star. We all had been warned by Ledoux that he knew of no general proof of thermal stability in stellar structure , a warning that the rest of us had bliss fully ignored. After our computers had presented us with the helium-shell burning instability, it was easy to understand its physical cause . Thus, we experienced a neat example of von Neumann ‘s thesis that computers were good for exploration and nothing was lost if such exploration was followed by thinking leading to understanding.
Later work showed that the star, after some fifty helium-shell flash cycles, arrives again at the top of the supergiant branch. but now in a very insecure state. ln the deep interior the helium shell burns in violent flashes. ln the envelope a pulsational instability causes long- period variations, observationally well documented by the long-period variables which appear exactly at the top of the supergiant branch. These pulsations appear to reach amplitudes so large that they cause the ejection of gaseous shells which after much expansion can be observed as ” planetary nebulae” . Finally, there may even be a separate surface instability which causes a persistent stream of matter outward from the surface, observed as a ” stellar wind” .
In spite of the difficulties of following this dynamic phase in detail, it seems fairly sure that the star emerges in a contracted form as a ” planetary nucleus” and finally condenses into a white dwarf, with no nuclear energy sources, persistently cooling and fading.
Altogether then a fairly complete and detailed theory has by now been achieved covering the entire life of a star of the type characteristic for globular clusters. I would like to emphasize once more that this theory has two essential bases, the laws of physics which govern the stellar interior and the relevant astronomical observations, specifically here the color-magnitude diagram of globular clusters. Nobody would imagine that we could construct the story of a star’s life without the basis of the governing laws of physics. But I think the history here sketched also shows that we would not have gotten far without the guidance of the astronomical observations; we would have gone astray at every turn of the star’s life. It would seem incredibly lucky that on these two foundations this multifaceted stellar story could be constructed.

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