J. Robert Oppenheimer's Neutron Core Trilogy

Bachelor’s Programme in Physics and Astronomy

J. Robert Oppenheimer’s Neutron Core Trilogy

A Search for Full Stellar Collapse in the Late Nineteen Thirties

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Abstract The theoretical physicist J. Robert Oppenheimer wrote three papers on the end state in the life of stars, of which the last described indeﬁnite gravitational contraction, nowadays known as a black hole. These papers concern physics, astronomy and general relativity: three concepts that were very separated from each other during the nineteen thirties. This thesis gives an overview of the theory of stellar evolution between 1910 and 1940, and the scientiﬁc debate of the relation between physics and astronomy, in order to get a deeper understanding of the three papers of interest. It will be shown that Oppenheimer’s turn to this more astronomical topic was not fully unexpected, since the path of his career between 1925 and 1930 and Richard C. Tolman played an important role in this. Furthermore this thesis will conclude that Oppenheimer’s results can be seen as an anomaly in the theoretical physics paradigm from the nineteen twenties until forties.

Report Bachelor Project (size 15 ECTS) Conducted between May 4th 2020 and October 6th 2020 Supervisor: prof. dr. J.A.E.F van Dongen Examiner: dr. M.L. Vonk Institute of Theoretical Physics Faculty of Science, University of Amsterdam Populaire samenvatting

De laatste jaren zijn zwarte gaten een aantal keer in het nieuws geweest; bijvoorbeeld in april 2019 toen er voor het eerst een foto was gemaakt van een zwart gat. Zwarte gaten, plekken in de ruimte waar massa naartoe wordt getrokken en waar licht niet uit kan ontsnappen, houden de natuurkunde al enkele tijd bezig. Sinds eind jaren vijftig van de vorige eeuw, is dit een bekend thema binnen de natuurkunde, maar eind jaren dertig van diezelfde eeuw is er ook al onderzoek naar gedaan. J. Robert Oppenheimer was een theoretisch natuurkundige, die vooral bekend is als de ’maker’ van de atoombom. Echter heeft hij zich ook bezig gehouden met andere onderwerpen in de natuurkunde, onder andere zwarte gaten. In 1938 en 1939 heeft hij drie artikelen geschreven over neutronensterren, waarbij het laatste artikel beschrijft wat er gebeurt als een ster volledig in elkaar stort; in de jaren zestig is hier de naam zwart gat aan gegeven. In deze scriptie worden deze drie artikelen bekeken, in de hoop deze artikelen beter te begrijpen. Ook wordt er gekeken naar de ontwikkeling van de theorie over sterevolutie tussen 1910 en 1940, om te kijken hoe de artikelen van Oppenheimer zich verhouden ten opzichte hiervan. Uiteindelijk zal blijken dat Oppenheimer zijn tijd vooruit was in het ontwikkelen van de theorie over volledige ineenstorting van neutronensterren. Echter zal ook duidelijk worden dat, kijkend naar zijn carri`erepad,het niet volledig onverwacht is dat hij deze theorie ontwikkelde.

2 Contents

1 Introduction4

2 Stellar Evolution7 2.1 The Rise of White Dwarfs...... 7 2.2 Energy Problem of White Dwarfs...... 8 2.3 Mass Problem of White Dwarfs...... 9 2.4 Final State of White Dwarfs...... 11

3 Oppenheimer’s Quest for Full Collapse 12 3.1 Prologue...... 12 3.2 Entering the Stellar Realm...... 13 3.3 Part One: Unstable Neutron Cores...... 14 3.4 Part Two: Massive Neutron Cores...... 15 3.5 Part Three: No More Neutron Cores...... 16

4 Conclusion 18

References 19

3 1

Introduction

J. Robert Oppenheimer is best known as the ’father of the atomic bomb’, but he was a lot more than that. As Bernstein (1996) states: ”Great science sometimes produces a legacy that outstrips not only the imagination of its practitioners, but also their intentions” and three papers Oppen- heimer wrote in 1938 and 1939 turn out to be a great example of this. As was typi- cal of Oppenheimer, he wrote the papers with three of his students: Robert Serber, George M. Volkoff and Hartland S. Snyder. Bernstein(2004) even calls the paper with Snyder one of the great papers in twentieth century physics: The paper is a pioneer in describing how a black hole can be formed. Figure 1: A picture of J. Robert Oppen- 1 This topic would become very popular in heimer. the late nineteen fifties–John A. Wheeler would introduce the term black hole in 1967–and with the first astronomical picture of a black hole fresh in our minds, it is safe to say that this phenomenon is still a very relevant topic in both physics and astronomy.

Oppenheimer was born in 1904, which means that his scientific career flourished during the same era as the development of quantum mechanics and Einstein’s theory of general relativity. Oppenheimer gained a prominent place in the scientific world: With Harvard as his alma mater and his doctorate from the University of Göttingen, he found himself in a perfect position to do physics. His time at Berkeley is even said to play an important role in making this university the centre of theoretical physics in the United States. Bird and Sherwin(2005) describe that, not long after Oppenheimer started teaching, the word started to spread around the country, that if one aspired a career in theoretical physics, Berkeley was the place to do this.

1Picture taken from https://alchetron.com/J-Robert-Oppenheimer on 10 September 2020.

4 As stated, only three papers of Oppenheimer’s oeuvre are on the topic of neutron cores and today physicists agree that this part of his career was groundbreaking. The last paper of this trilogy was published on the day Nazi Germany invaded Poland and drew little attention at the time: in fact the trilogy as a whole experienced a period of neglect after its publication.2 However, nowadays we know that Oppen- heimer’s three papers opened the door to twenty-ﬁrst century black hole physics (Bird and Sherwin, 2005, 89). If Oppenheimer was moved to work on this topic, why did he write only three papers about it? This question carries a lot of other questions with it. One can search for the answer by looking at Oppenheimer and the paradigm of theoretical physics during the nineteen twenties and thirties.3 One could also broaden the scope of research by looking at Oppenheimer, other physicists who worked on this topic and the physics paradigm of the nineteen twenties and thirties.

In this thesis we will focus on Oppenheimer and try to better understand the position of the trilogy on neutron cores in his scientiﬁc career. This will be done by looking at the development of his career and physics during the nineteen twenties and thirties, concerning the topic of neutron cores, up to and including the publication of Oppenheimer’s third article. In order to do this, section2 gives an overview of a history of stellar evolution in the making, focusing on what eventually let to the hypothesis of neutron cores. In the third section, we will look at Oppenheimer and the three papers of interest. We will place Oppenheimer within the historical context from section two, to get a deeper understanding of the three papers and try to place these within his career. By the end of this thesis we will compare the position of Oppenheimer’s neutron core trilogy in his career and the theoretical physics paradigm from the nineteen twenties and thirties, with the evolution of science as described by Thomas S. Kuhn.

Before we start our discussion about neutron cores and Oppenheimer, a few general remarks, should be made. To do historical research about the early twentieth century, it is important to know what the scientific world was like during this time. Since some call this era a golden age of physics, a lot can be said: we will try to point out the things, relevant for this thesis. First of all it is important to realise that the landscape of scientific disciplines was not like it is nowadays: fields as cosmology or astrophysics were not known in the way we know them now. A good anecdote on this is given by Bonolis(2017), who explains that the first Texas Symposium in 19634 was actually set up with the idea of merging general relativity and astrophysics. The organisers of the symposium thought this was about time, since the suspicion existed that relativity had something to do with quasars and they came up with the name relativistic astrophysics for this new discipline. An important note is that this event took place almost thirty years after Oppenheimer published his work on full stellar collapse. So a clear example of a deviation from physics today, is that physics and astronomy were a lot more separate

2See Hufbauer(2005) for an extensive reading on the possible reasons for this. 3Where paradigm is the common intellectual framework shared by members of a certain community for whom there are enough puzzles left to resolve (Kuhn, 1962, 10). 4It was called the International Symposium on Quasi-stellar Sources and Gravitational Collapse and chaired by Oppenheimer. The Texas Symposia evolved into a long series, from (Bonolis, 2017, 313).

5 from each other in the nineteen twenties and thirties. A second issue, which is connected to what is just said, is that Einstein’s theory of general relativity experienced a so-called low-watermark period from 1920 until 1955: During this period only some specialists worked on it, putting it in a state of little progress and deﬁnitely not in the position of a conventional theory (Blum et al., 2016, 344). The same Texas Symposium as just mentioned, can be seen as the moment when general relativity started to get more attention–after a period in which is was attributed to mathematicians–and in doing so blurred the lines between physics and astronomy (De Swart et al., 2017, 6). The renaissance of general relativity that followed, is closely connected to the position cosmology had: As the theory of general relativity got more attention, astronomy started to focus more on cosmology and by the nineteen sixties, cosmology had become a more respectable enterprise than it was during the nineteen twenties until forties (Smith, 2008, 91). What we learn from this, is that during the nineteen thirties the boundaries between physics and astronomy were still very sharp and that cosmology was not yet in a position, which was closely related to physics.

6 2

Stellar Evolution

To get a deeper understanding of the three papers written by Oppenheimer and coauthors Robert Serber, George M. Volkhoﬀ and Hartland S. Snyder in the years 1938 and 1939, it is necessary to know what moved Oppenheimer to write these papers. Since these papers concern the end state of stars, it is needed to follow the path physics and astronomy took concerning stellar evolution, which eventually led to what we today call a black hole. Oppenheimer wrote his papers in the late nineteen thirties so in this discussion, we will ﬁnd ourselves in the second, third and fourth decade of the twentieth century: An overview of the discovery of stellar evolution in that period of time, is given below.

2.1 The Rise of White Dwarfs

As a starting position, we look at the astronomer Walter S. Adams. In 1915 he succeeded in determining the spectrum of the companion of Sirius (Sirius B) and saw that this star had a great mass and low luminosity (Adams, 1915, 236). Fur- thermore he saw that the star was white, which was an exceptional observation at that time, since all known dwarfs were red (Bonolis, 2017, 316). Known dwarfs at that point, were mostly results from the research of Hertzsprung(1905) and Russell (1914), who independently of one another came up with a classiﬁcation of the stars, in which main-sequence stars–which Hertzsprung called dwarfs–and giants were dis- tinguished. We now know that Adams was the ﬁrst to observe a white dwarf : a relatively small star, with a high density in which fusion reactions do not take place anymore. However, Adams did not call Sirius B a white dwarf yet. Nine years later one of the leading astronomers of his time, Arthur S. Eddington– most famous for the relation between mass and luminosity of stars–publicised the observation of Adams. An important note is that it was commonly thought that equilibrium against the gravitational contraction was maintained in all stars by internal pressure of matter, being warmed into a gas (Bonolis, 2017, 316). This idea also plays a role in the star model Eddington developed: The Eddington model assumes that the pressure of a star is provided by a perfect–nowadays called ideal– gas and radiation, and that the ratio of gas pressure and gravitational pressure was preserved. Eddington(1924) stated that for main sequence stars 5 the total radiation

5In main sequence stars fusion of hydrogen takes place and thermal pressure compensates gravitational contraction (we call this hydrostatic balance). In 1924 the former was not known and the latter was assumed.

7 will be a function of mass, as long as the star can be taken as a perfect gas. He also explained that an ordinary gas will become incompressible at high density, but that very high temperatures within the star can break down this compression limit: If the temperature is high enough, the electrons will not be bound to the nuclei anymore and this state was called compressed or dense. In this case the density will only be limited by the sizes of the electrons and nuclei, making it possible for stars to consist of matter more dense than any known material at that time (id., 787). By this time the term white dwarf was adopted to describe a certain group of stars, and Eddington used it in his article (id., 786). Eddington stated that Sirius B could be an example of a dense star and in doing so advocated an addition of white dwarfs to the common stellar theory at that time, which was mostly focused on main sequence stars. He also said that his hypothesis about the star could be tested by measuring the gravitational redshift of the stars emitted light (id., 788). This redshift is a consequence of time dilation, since the frequency of light is connected to the passing of time at the position of emission (Thorne, 1995, 131). Subsequently Adams(1925) measured the redshift, confirmed Eddington’s finding about dense matter and executed a new test of Einstein’s theory of relativity (Bonolis, 2017, 317). For the first time general relativity was used to understand compact objects, which are light years away from earth (id., 318). At this point the existence of the white dwarf was confirmed and it was placed on the scientific map: The era of the white dwarf had begun.

2.2 Energy Problem of White Dwarfs

There was still a lot unknown about these white dwarfs and Eddington pointed out some of the main problems, starting with the energy source of the star. From Eddington’s point of view there is no way out of the compressed state: he did not see how a star could ever escape from it. This state was only possible under very high temperature and this means that, if the star cools down, the temperature reduction makes the compressed state unattainable. To preserve the equilibrium of the star when cooling down, something had to compensate for the gravity and Eddington (1926) stated that this would be expansion of the star. This is where the energy problem of white dwarfs arises. It was known that the energy fuel of white dwarfs had been exhausted and that it had no other resources, so the question was where the star would get energy from in order to expand (Bonolis, 2017, 318). One of Eddington’s colleagues in Cambridge, Ralph H. Fowler, picked up this question. Fowler made use of two new theories in nineteen twenties physics. Firstly, the Fermi-Dirac statistics, published by Fowler’s research student Dirac(1926) who followed the laws of quantum mechanics and Pauli’s exclusion principle from 1925, to give a statistical description of identical fermions. Secondly, Fowler’s own research in which he applied statistical mechanics to electrons and protons in order to get a deeper understanding of matter when it is under the temperature and density condition of stars (Fowler, 1926a). These two theories enabled Fowler to publish, only three months after Dirac’s publication, a paper in which he developed the quantum mechanics of identical particles to look at the relation between quantum mechanics and statistical mechanics (Fowler, 1926c). The insights from this paper made it possible for Fowler to respond to Eddington’s question concerning the energy source of white dwarfs. Fowler(1926b) came up with the idea of electron

8 degeneracy. He argued that Eddington was wrong in assuming a direct link between temperature and energy. Temperature is a way to measure radiation and radiation depends directly on temperature. But radiation depends only on energy as far as temperature determines the energy (id., 115). Fowler concluded that the problem raised by Eddington was not difficult to solve, if one would use quantum statistical mechanics instead of classical statistical mechanics. In this case, one would see that the limitation can be found when the energy is still large, but the temperature is zero. Then the temperature no longer plays a role and the matter is in its lowest possible quantum state: the degenerate state (ibid.). Furthermore, he stated that the pressure of this fully degenerate electron gas will balance the gravitational force so that collapse was avoided. The findings of Fowler marked a big step forward in the stellar evolution theory, as it was the first application of the new quantum statistics to a compact object and accounted for the earlier discovered characteristics of white dwarfs (Bonolis, 2017, 319). The white dwarf was now a testing ground for both quantum mechanics and general relativity.

2.3 Mass Problem of White Dwarfs

The first understanding of white dwarfs was given by Eddington and Fowler, but a lot of questions remained unanswered. A second major question concerned the mass and in particular if there was a limit to the mass of these stars. In the late nineteen twenties the German physicist Wilhelm R.K. Anderson entered the stage: he said that the highest possible density for matter, described by Fowler as the degenerate state, needed such a high pressure, that it should only occur in the central core of a star (Anderson, 1928). This meant that the degenerate state was not just possible for white dwarfs–for which the idea was initially developed–but also for other kinds of stars as long as their density was great enough (Bonolis, 2017, 322). Moreover he said that an upper mass limit was undeniable, since the density was limited too. Anderson was not the only one thinking about these aspects of white dwarfs: independent of him, Edmund C. Stoner worked on the mass limit of white dwarfs. Starting at Fowler’s idea of electron degeneracy, he discussed whether there would be a limiting density due to the compression of electrons. He stated that as the star shrinks, the density will increase and this process will reach a limit when the gravitational energy is great enough to press the electrons closer together (Stoner, 1929). In doing so, the idea of a superdense core in stars was initiated by Stoner, who came to the conclusion that white dwarfs contain a superdense core which approaches the limiting density and are therefore in an almost incompressible state (Bonolis, 2017, 324). Meanwhile, Edward A. Milne published his findings concerning the relations between luminosity, mass and temperature of stars: Milne said that his theory was made from a different standpoint than Eddington’s theory (Milne, 1929, 17). Milne stated that a star model in which both mass and energy are taken as a point source, would be more stable than the Eddington model. Furthermore he explained that Eddington’s theory–and theories that followed his–failed to account for white dwarfs, since they assumed stars to be masses of perfect gas, while such a star could impossibly be in a steady state (Milne, 1930, 4). Milne also criticised Eddington for not taking the generation of energy in stars into account and for the fact that he fully reasoned from an equilibrium perspective. Milne analysed the equilibrium state of

9 stars and tried to use this to understand the energy generation, so that eventually he could combine these two aspects in one theory. His conclusion was that stars should have a heavy but small core or must be very dense, correspondingly centrally- condensed or collapsed.6 Note that the Milne model, differs from Eddington’s model on several points: These models would be the topic of a scientific debate for quite some years. For a while it seemed that the white dwarf, presented by Adams and Eddington, had a dense core, where electron degeneracy appeared, explained by Anderson and Fowler respectively, and that this state was the final state in the life cycle of stars. Stoner added that the density was limited and Milne explained that the core was condensed. However, soon it became clear that the degenerate electrons should be treated relativistically. Stoner(1930) added relativity to the ideas of Anderson by calculating the effect of the relativistic change of mass and found a limiting mass above which the star would not find itself in gravitational kinetic equilibrium: gravity and pressure would not be able to balance each other out. He confirmed Anderson’s speculation of a limit mass for white dwarfs, but at this point both men did not say anything about stars, that are heavier than the mass limit (Bonolis, 2017, 325). This was around the same time that Subrahmanyan Chandrasekhar (Chandra) came all the way from India to work under Fowler in Cambridge. He started his time there with some papers on the new Fermi-Dirac statistics, but it did not take long until he found himself thinking about white dwarfs. As he used statistical mechanics to understand the dense cores of these stars, he found that some levels of the degenerate gas are relativistic, which Fowler did not take into account (id., 328). This meant that the internal pressure would not grow faster than the gravitational force, so that for stars heavier than the mass limit, the radius would go to zero, causing the star to collapse. Furthermore he calculated this mass and found the 7 value we now know as the Chandrasekhar limit, 1.44M (Chandrasekhar, 1931). This implied that for stars heavier than this mass, the white dwarf state would not exist in a stable way. However, at this point some physicists were convinced that white dwarfs represented the final state of all stars, so Chandra’s mass limit was not immediately accepted by everyone. One of the physicists that was interested in the same topic as Chandra, was Lev D. Landau at the university of Leningrad. He worked on the relativistic effects in the degenerate core of stars and explicitly pointed out the existence of Chan- dra’s mass limit. Landau(1932) worked on the possible existence of dense cores in stars–following Milne–and added that these might look like an enormous nucleus. Important here is to note that Landau was already a well-known physicist: he had worked in both Copenhagen and Zürich where he became acquainted with many prominent physicists (Bonolis, 2017, 331). His article attracted the attention of a lot of other physicists and transitioned the idea of dense stars from astronomy- to the world of physics (Bonolis, 2017, 336). About the fate of stars heavier than the mass limit, Landau said that, the current quantum theory did not prevent a system like that from collapsing to a point. To this he added that in reality such heavy masses

6Milne was here the ﬁrst to use the term collapsed in the ﬁeld of astrophysics to describe the state of a star (Bonolis, 2017, 326). 7 −2 The expression found by Chandra was 5.76µe M , where µe is the average molecular weight per electron, and in this case the expected value µe = 2 is used (Bonolis, 2017, 328).

10 did exist and that versions of these collapsed stars were never seen, which made him conclude that stars heavier than the mass limit, would have regions in which the laws of quantum mechanics would be violated, since these would not prevent the collapse. Besides the mass problem of white dwarfs, Landau also returned to the energy problem. He tried to avoid the collapse to a point, which was contradicting the conservation of energy. He prepared a new source of energy by stating that the radiation of stars might not be due to annihilation of protons and electrons, but by a violation of the law of energy, which he said to be no longer valid in relativistic quantum theory (Landau, 1932). Important to note is that Landau’s paper was published before Chadwick(1932) discovered the neutron and this discovery would play an important role in understanding Landau’s ﬁndings.

2.4 Final State of White Dwarfs

The discovery of the neutron started a new era in physics, particularly in the field of nuclear physics and for white dwarf theories: For the latter, the impact was huge, because it seemed that the neutron could play an important role in the density of the core. Being a quite small and neutral particle, the neutron could more easily be in a high density state and behave like a perfect gas (Bonolis, 2017, 336). The first to discuss the energy problem, including neutrons, was Theodor E. Sterne. Sterne(1933) described that neutrons had greater packing fractions than other kinds of nuclei and that in dense stars, the matter at low temperature would be compressed into neutrons (Bonolis, 2017, 340). The latter caught the attention of the astronomer Walter Baade and physicist Fritz Zwicky and they together came to a separation of the novae in ordinary novae and supernovae, which are very en- ergetic since mass is annihilated in large amounts. Furthermore they proposed that supernovae represented the transition from ordinary novae into neutron stars, which embodied extremely closely packed neutrons (Baade and Zwicky, 1933). These neutron stars may be very small, while possessing an extremely high density, since the gravitational packing can become very large in a cold neutron star. These cold neutron stars would represent the most stable configuration of this matter by its very nature (Baade and Zwicky, 1934, 263). Meanwhile Chandra was still working on his maximum mass for white dwarfs, focusing on stars whose mass exceeds the mass limit. Chandrasekhar(1932) stated that for these heavy stars, the material would not become degenerate. Moreover he stated that the only way to avoid the singularity–a full collapse–would be a maximum density. Eddington reacted to Chandra’s work by stating that it was a reductio ad absurdum to combine relativistic mechanics with non relativistic quantum theory (Bonolis, 2017, 347). However, more physicists started to realise Eddington’s statement was probably wrong and Chandra’s mass limit was right. Especially since Eddington himself already explained the relativistic effects of a powerful gravitational field exerted by a heavy star, ten years earlier (Eddington, 1926, 6). He said there that first, the gravitational force would be too great for light to escape from it. Second, that the redshift of the spectral lines would shift the spectrum out of existence and third, that the mass would curve spacetime, so that it would close up around the star, leaving us outside. Now that a history of stellar evolution is given, we will turn to our main character, J. Robert Oppenheimer.

11 3

Oppenheimer’s Quest for Full Collapse

Oppenheimer’s trilogy ends with the pioneering idea of full collapse for stars heavier than a certain limit. To better understand these papers we should know how and why Oppenheimer wrote these papers and to do this we shall look at a few aspects of his scientiﬁc career and life.

3.1 Prologue

How did Oppenheimer get involved in the topic of contraction, especially since it was not common for a theoretical physicist in the nineteen thirties to spend time on cosmology and astrophysics. The historian Karl Hufbauer(2005) gives several reasons for Oppenheimer’s acquaintance with the scientific debate about the relation between physics and astronomy in the years 1929 and 1930. First, Oppenheimer went to Cambridge in 1925–after graduating at Harvard in chemistry–for about a year to work as a postgraduate student under Fowler, who was working on the energy problem of white dwarfs during that time. Second, while staying in Zürich in 1929, he wrote a paper about characteristics of radiation from a free electron passing by an atom with relativistic velocity (Oppenheimer, 1929). He picked up this problem because it was relevant for the work of his colleague Wolfgang Pauli, who was trying to develop a relativistic quantum mechanics. Oppenheimer’s paper used theoretical physics and he decided to apply this to free electrons in the stellar interior. Furthermore a reference to Eddington can be found in it. Third, after Zürich Oppenheimer went back to the States to work at both Berkeley and Caltech. At the latter institute he spent time with Richard C. Tolman, a mathematical physicist who was then mostly working on general relativity. We know Oppenheimer was at least talking to Tolman about this, since Tolman thanked him for these talks in two papers he wrote (Tolman, 1930a, 895)(Tolman, 1930b, 919). Furthermore Fowler, Milne and others started criticising Eddington’s model around 1929 and the Eddington-Milne debate probably reached Oppenheimer (Hufbauer, 2005, 33,34). These three issues show that Oppenheimer was aware of the scientific debate concerning a combination of physics and astronomy and that he was sur- rounded by academics from both physics and astronomy around 1930. However, at this point we cannot say that he was expecting to enter the astronomy discussion in the future.

12 To see why he did end up in that debate, we need more insight in his scientiﬁc path as a whole. Oppenheimer was part of the group of young scientists in the nineteen twenties and thirties that entered the physics world right after the birth of quantum mechanics. He found himself in the quantum mechanics paradigm and this means that he was mostly working on combining this new theory with special relativity–another characteristic of the nineteen twenties and thirties theoretical physics paradigm–and using quantum mechanics to solve some routine problems.

3.2 Entering the Stellar Realm

From around 1930 until 1937 Oppenheimer’s research went well, but he was not yet living up to the potential the scientific community and himself saw in him in the early nineteen twenties (Hufbauer, 2005, 36). In 1933 Oppenheimer, on his own, showed interest in the topic of stellar theory for the first time by hosting, and speaking at, a talk on ”Stars and Nuclei”8 at Caltech. The focus of this talk was the growing interest in nuclear phenomena and the stellar energy problem. At this point there were two leading theories: Eddington’s and Milne’s as outlined in section2. We do not know of Oppenheimer’s interest in the physics of stars during the period 1933 until 1937. However, we do know that he spent time with George Gamow at symposia and at the Oppenheimer family ranch in New Mexico. Gamow was an advocate of Landau’s theory on condensed cores, so Oppenheimer might have been at least aware of that theory (Hufbauer, 2005, 37). In the next year Oppenheimer made the decision to work on stellar theory together with one of his graduate students, George M. Volkoff. After doing some research on the topic, Volkoff was ready to talk about ”The Source of Stellar En- ergy” at a Berkeley physics seminar.9 Since he did not have a new solution for the energy problem–which was a little outdated by now–it is likely that he talked about progress concerning both Eddington’s and Milne’s model, where Gamow and Sterne’s research–as pointed out in section2–were examples for the latter (Hufbauer, 2005, 38). A couple of months later, Oppenheimer, Fowler–meanwhile at Caltech– and Minkowski, arranged a meeting where Oppenheimer talked about ”The Physical Problem of Stellar Energy”.10 The topic of the talk was the theory of possible nuclear changes in lighter elements and their possible application in stars and he mentioned a star model with a high central concentration of neutrons (Hufbauer, 2005, 39). This shows his interest in stellar evolution, but we also recognise ideas from Landau and Sterne. His theoretical peers started to get interested in the topic as well: He learned that Hans A. Bethe was working on the stellar energy problem and Landau and Gamow were publishing new findings on the topic too. Correspondence between Bethe and Oppenheimer tells us that Oppenheimer was most impressed by Lan- dau’s theory and his idea about energy generation in stars as a result of accretion of stellar matter on their neutron cores:”that...one will be forced to [Landau type] core models...in spite of many difficulties with them”.11 Bethe responded to Op-

8Scheduled April 14th 1933 at Mt. Wilson-Caltech Astronomy and Physics Club, from (Hufbauer, 2005, 36). 9Scheduled on November 29 1937, from (Hufbauer, 2005, 38). 10June meeting of the AAAS in San Diego 1938, from (Hufbauer, 2005, 39). 11Oppenheimer to Bethe, 13 June 1938, from (Hufbauer, 2005, 39).

13 penheimer with an account of what would turn out as his carbon-nitrogen-oxygen cycle of energy generation within ordinary stars, where he was following Eddington’s model (Bethe, 1939). Oppenheimer reacted again to Bethe that he was not sure if the ”astronomers would admit a high enough concentration of nitrogen in the sun, or a low enough concentration of hydrogen in Capella, to make your explanation ten- able”.12 Note the subtle appearance of the division of physics and astronomy in the way they talk about the others: This shows us that these ﬁelds were still very much apart and that the academics from these ﬁeld were working isolated from each other instead of working together. Even though Oppenheimer was questioning Bethe’s theory, he also admitted that there were problems with the core model: Landau had underestimated the minimum stable core mass by a factor of about one thousand (Hufbauer, 2005, 40).

3.3 Part One: Unstable Neutron Cores

In the summer of 1938 Oppenheimer and his postdoc Robert Serber decided to look further into Bethe’s solution of the stellar energy problem and Landau’s suggestion of stellar neutron cores. In September Serber gave a talk on Bethe’s theory at the Berkeley physics seminar13 and afterwards Oppenheimer and Serber(1938) wrote a one page note on it. It starts with pointing out the hope of the scientific community that all stars have about the same structure and mechanism of energy generation. And they stated that the energy generation, as described by Bethe and Critchfield(1938), was applicable to main sequence stars when the Eddington model is used. However, for stronger radiating stars these reactions to describe energy generation would not suffice: Oppenheimer and Serber stated that to solve this problem other nuclear reactions should be assigned or one should expect severe deviations from the Ed- dington model. They tried to find a solution by considering a deviation from the Eddington model in the form of condensed neutron cores. They argued that for the discussion of the role of such cores in the stellar theory, it is essential to know the minimum mass of the core in order to be stable (Oppenheimer and Serber, 1938). This question is the main question of their paper. First they argued that, to be stable with respect to the most strongly bound nuclei, the neutron’s free energy in the core should be less than that in the nucleus 1 and keeping this in mind they found a minimum mass of 6 M . They assumed the core was uniformly dense, but said this assumption barely influenced the derived limit. The second half of the note is devoted to what consequences the existence of the mass limit has for the Eddington model: They explained that for a star with 1 a core above 6 M , the model would break down completely, since the degenerate region surrounding it, would use up the entire mass of the star. Then they said that, to fully determine the stability of neutron cores, one would need to know the contribution of nuclear forces to the core binding–in this case the binding between neutrons–which was still an open question in 1938. Nonetheless, Oppenheimer and Serber gave two possible outcomes of their research. Firstly, one could assume the forces between neutrons to be of the spin exchange

12Oppenheimer to Bethe, 24 June 1938, from (Hufbauer, 2005, 39). 13On 5 September 1938, from (Hufbauer, 2005, 40).

14 type and they argued that this would lower the mass limit to 0.1M . However, the result would still be the same: The degenerate area would have a mass of around the solar mass, so they argued that if such a star existed, the Eddington model would be completely wrong, since the degenerate area around the star would still be using up the mass of the star. They did state that the very massive stars might be an exception to this. Secondly, one could assume that the only relevant factor for the binding energy between neutrons, is the diﬀering kinetic energy in line with Pauli’s principle. In this case the limiting mass would be reduced considerably: Oppenheimer and Serber gave a value of 0.03M in order for a star with a neutron core to be stable. They concluded that no core will be formed unless nuclear sources are exhausted, at least in the star’s centre. But they did acknowledge that their argument cannot show that actual stars have cores: However, they stated that the forces of the spin exchange type exclude the possibility of a core for stars with a mass in the order of the sun (Oppenheimer and Serber, 1938).

3.4 Part Two: Massive Neutron Cores

Oppenheimer decided not to go on with the energy generation in stars but turn to a different direction or more precisely, look beyond the energy production: He started thinking about stars, that used up all their thermonuclear energy sources. This turn was a little out of character for Oppenheimer and Hufbauer(2005) gives two reasons for why he still did it. Firstly, his student Volkoff still had a great interest– and already worked on it for a year–in the topic and did not want to give up on the stellar theory. Secondly, Oppenheimer learned that Tolman was thinking about a new way to apply general relativity to the stars. Oppenheimer and Serber’s results relied on Newtonian gravity, just like Landau did, but Oppenheimer and Tolman soon agreed on how to switch to general relativity (id., 41). Oppenheimer and Volkoff started their article with the statement that it is important to know the distribution of energy sources and their dependence on physical conditions within the star, in order to solve the problem of stellar structure (Oppen- heimer and Volkoff, 1939, 374). These things were still unknown when Eddington worked on this problem, which is why different assumption were made about this, during that time. This variety of assumptions led to different star models, for in- stance the Eddington and Milne model as outlined in section2. Landau came up with the idea of first trying to understand a system in equilibrium– where no energy is generated–and then see if this gives any information about the energy production. Oppenheimer and Volkoff stated that this tactic works for white dwarfs, but fails for main sequence stars, for which the Eddington model told them that the material is non-degenerate so that both temperature and energy production play a role in the equilibrium conditions of the star (ib.). Furthermore Landau showed that for a cold non-degenerate Fermi gas no stable equilibrium exists for masses greater than a critical mass. Oppenheimer and Volkoff pointed out that this can be interesting: If thermonuclear reactions are the standard source of energy for main sequence stars, Landau’s case might be helpful for what happens when the thermonuclear sources are exhausted (ib.). One of the possibilities is that the star would form a neutron core, for which the minimal mass in order to be stable is 0.1M , as was examined by Oppenheimer and Serber. Oppenheimer and Volkoff focused on the upper mass limit for which this

15 neutron core would be stable. Landau said that this was 6M , but Oppenheimer and Volkoff argued against this value by pointing out that Landau used Newtonian gravitational theory and an inaccurate equation of state (id., 375). Oppenheimer and Volkoff sought to establish what the result would be if general relativity and a more exact equation of state were used. Note that Landau’s ideas played an important role in the motivation of Oppenheimer and Volkoff’s paper. Someone else who plays a role in their paper is Tolman: He was working on the same topic and the three men decided to publish the articles in parallel. In order to find the upper mass limit of a stable neutron core, Oppenheimer and Volkoff started off with a general relativistic treatment of the equilibrium of spheri- cally symmetric distributions of matter. After determining the matter distribution in general they used this for the special case of a cold neutron gas. They noted that the solution will be independent of the mass of the neutron (id., 378). After a quite lengthy calculation, they found the following results: There exist no static solutions 3 1 3 1 for m > 4 M , two solutions for 3 M < m < 4 M and one for all m < 3 M , where m is the mass of the star (ib.). The first result is the most exotic. It was already known that neutron cores lighter than 0.1M can hardly be stable because of beta decay, the disintegration of neutrons into electrons and nuclei. And Landau showed earlier that neutron cores will not be formed by collapse of ordinary matter 14 for masses under 1.5M . The combination of their own findings with Landau’s yields the final conclusion of their paper. Firstly, if stable neutron stars would ever form, their mass would 3 not succeed 4 M and secondly, only stars with a mass above 1.5M could collapse beyond the white dwarf stage. Ergo, they said it is very unlikely that static neutron cores play a great role in stellar evolution since ending up in this state is exceptional rather than usual. What exactly happens to stars–heavier than 1.5M –after the energy sources are exhausted, remained vague. They did present the, according to them, two possible answers about the final state of massive stars: The equation of state that they had used, failed to describe highly condensed matter or the star would continue to contract indefinitely and never reaches equilibrium (id., 381). However, they stated that if under high compression the repulsive force between the neutrons grows, this could prevent the collapse, but this would not enable static solutions for massive stars, since at low densities the neutrons cannot affect the equation of state so that the gravitational mass of the core will be finite (ib.). By the end of the article they said that, in order to get a deeper understanding of the final fate of heavy stars, one should consider the non-static solutions and hope to find a slowed contraction so that these states can be seen as quasi-static instead of equilibrium solutions. They finish the article stating that these solutions are being investigated.

3.5 Part Three: No More Neutron Cores

After the paper with Volkoff, Oppenheimer was not done with the topic. He and his graduate student Hartland S. Snyder continued where he and Volkoff left off, by poring over the static and non-static solutions for heavy stars that exhausted

14We now know this as the Tolman-Oppenheimer-Volkoﬀ (TOV) limit and the recent gravitational wave observations give us a value of m . 2.17M (Margalit and Metzger, 2017).

16 their thermonuclear sources. Oppenheimer himself described the results they found as ”very odd”15 but after both men spoke about the topic on several conferences16 Oppenheimer and Snyder(1939) published their paper. Oppenheimer and Snyder were interested in looking at the non-static solutions 3 of the field equations for stars heavier than 4 M , which have exhausted their thermonuclear energy sources. Stars like these will collapse and release four different kinds of energy. If the mass of the original star was small enough or one of the four sources of energy reduces the mass enough to end up below the limit, the star will end up in the white dwarf graveyard. Oppenheimer and Snyder looked at systems for which this state was out of reach. They said that the gravitational effect of escaping matter and deviations from spherical symmetry due to rotation can be neglected for stars in this collapsing state. For stars in this state, the boundary radius rb that indicates the stellar matter, would approach the gravitational radius r0–nowadays called the Schwarzschild radius–since the gravitational contraction is dominant with respect to the pressure of the matter (id., 456). Oppenheimer and Snyder explained that when looking at the star, the reference frame determines the impression one gets of the star, as follows from Einstein’s theory of relativity. This was both a crucial and a remarkable turn, since Oppenheimer and Snyder found themselves in the middle of the low-watermark period of Einstein’s theory of relativity. In order to investigate this more thoroughly they solved the Einstein equations with the pressure set to zero corresponding to free gravitational collapse (id., 457). They believed that this calculation would also be relevant for cases in which the pressure is not zero, as long as the mass is great enough for collapse. Just as in the article with Volkoff, there is a reference to earlier work of Tolman. They decided to follow Tolman(1934) in solving the field equations with the pressure set to zero, but used a different coordinate system, which was comoving with the matter (Oppenheimer and Snyder, 1939, 457). They argued that a local observer of the star would see matter falling inwards with a velocity close to that of light, while for a distant observer this movement would be slowed down by a factor (1 − r0/rb) (ib.). Furthermore all energy emitted outwards would be prevented from escaping by both the Doppler shift, relativistic gravitational red-shift due to the strong gravitational field around the star, and gravitational deflection of light, which would force the light to escape through a cone around the outward normal of the star. Ergo, they said, the star closes itself off from distant observers, leaving nothing but the gravitational field. And even though this will take infinite time for a distant observer, an observer comoving with the matter will see this happen within a finite time. They attached a statement on the assumptions of their study, saying that, although actual stars would collapse slower due to rotation, radiation and pressure of matter, the qualitative and quantitative arguments would remain. Therefore their final conclusion said that this behaviour can be expected for all stars which cannot end in a stable stationary state.

15In a letter to George Uhlenbeck, from (Hufbauer, 2005, 44). 16Oppenheimer spoke about ”Stellar Energies” at the Caltech Physics Research Conference in May 1939 and later that month Volkoﬀ talked about it at Caltech’s Seminar on Theoretical Physics, both from (Hufbauer, 2005, 44).

17 4

Conclusion

The aim of this thesis was to get a deeper understanding of the three papers Oppen- heimer and three of his students wrote on neutron cores in 1938 and 1939. To do this we started with the path stellar evolution took during the second until fourth decade of the nineteenth century. We have seen how, through the work of Adams and Eddington, the white dwarf gained a prominent place in astronomy. It is set out that Fowler added electron degeneracy to the debate and Anderson placed this degenerate matter in the star’s centre. Meanwhile both Stoner and Milne publicised the idea of very dense cores and the latter criticised the until then generally accepted Eddington star model. Chandra examined the mass limit of white dwarfs in order to be stable, and Landau was the one who made it known to physicists, together with his ideas of dense stars. We have seen that the debate changed after Chadwick’s discovery of the neutron: Suddenly there was a particle, which would be very suitable for the dense cores of stars. The section about stellar evolution ended with Bethe’s idea of energy production in stars, the acceptance of Chandra’s mass limit for white dwarfs and the open question what would happen to stars, whose mass exceeds this limit. In the third chapter we zoomed in on Oppenheimer. We have seen how Oppenheimer entered the debate concerning the relation between physics and astronomy through his acquaintance with Fowler in Cambridge, Pauli in Zürich and Tolman at Caltech. However, these three contacts were not enough to expect Oppenheimer to become a prominent figure in this debate in the future. In 1933 he showed interest in the topic for the first time by hosting a talk on it and in 1934 he and Volkoff started to work on the topic. During the same year, he showed his interest in stellar evolution and we recognised the ideas of Landau and Sterne in his talk. Meanwhile the theoretical physics community started to get more interested as well: Bethe and Oppenheimer talked about the stellar energy problem and Oppenheimer pleaded for Landau type core models and this is where we saw the beginning of Oppenheimer’s trilogy. His recognition of Landau’s underestimation of the minimum stable core mass, moved him to look deeper into it. Afterwards, in 1938, he and Serber wrote their one page note on the minimum mass for a neutron core to be stable. We have seen that they found a value of 1 6 M and said that Eddington’s star model would break down for stars with a core heavier than the limit. They ended the note stating that neutron cores will only be formed if nuclear sources are exhausted, at least in the star’s centre, but their research did not show that actual stars have neutron cores. After this Oppenheimer

18 and Volkoff wrote an article on stars, that used up all their thermonuclear sources and in this article Oppenheimer switched from Newtonian gravity–on which the results with Serber relied–to relativistic gravity. In this article the main focus is finding the upper mass limit for a neutron core to be stable. It was not immediately clear why Oppenheimer decided to turn to this topic, but we have seen that Tolman played an important role in this. He was working on applying general relativity to the stars in a new way and he, Oppenheimer and Volkoff ended up publishing two articles in parallel. Another important figure for this article is Landau, since Op- penheimer and Volkoff found inspiration in his statement about the impossibility of equilibrium for stars above a certain limit: They did argue against the upper mass limit Landau found, but his idea in general can be seen as part of their incentive to write the paper. The results of their paper said that the maximum mass for 3 stable neutron stars will be 4 M and only stars, whose mass exceeds 1.5M can collapse beyond the white dwarf stage. While the latter article only looked at the static solutions, the last article of Oppenheimer’s trilogy considered the non-static solutions of heavy stars, that exhausted their thermonuclear sources. In this article Oppenheimer and Snyder set out what happens in this case: They explained that, for a distant observer, the star seems to collapse fully and closes itself off, but this will take a infinite amount of time. However for an observer comoving with the star this will take a finite time. Tolman again played a role in the article, since Oppen- heimer and Snyder’s calculations to solve the field equations followed Tolman’s.

The analysis of Oppenheimer’s trilogy and the paradigm of theoretical physics in the nineteen twenties until forties, tell us a few things. First, we can say that it was unusual for Oppenheimer–a theoretical physicist–to turn to a stellar problem, since physics and astronomy were still quite separate scientific fields. We have seen that it was mostly Tolman from 1930 and the path of Oppenheimer’s career during 1925 and 1930, which informed him on a theoretical physics debate that was closer to astronomy. Second, the theory Oppenheimer used in the second and third paper–general relativity–was in its low-watermark period during the years 1938 and 1939, which means not a lot of scientists were actively doing research on it. Again, Tolman played a role in Oppenheimer’s choice for this, since Tolman was one of the few, who was working on general relativity. Third, we can say that Oppenheimers’s neutron core trilogy resulted in the statement about full stellar collapse. This idea can be seen as an anomaly in the theoretical physics paradigm of the nineteen twenties until forties, since it brought a result, which was not satisfactory in the paradigm. The fact that Oppenheimer’s results included a singularity in spacetime and an infinite timescale for the distant observer, made it difficult to integrate into physics at the time. The article did not get a lot of attention right away–it would take until the nineteen fifties before physics really started to investigate full stellar collapse and eventually black holes. A reason for this can be found in the low-watermark period: General relativity research was just not that popular at the time, as well as Oppenheimer’s neutron core trilogy.

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