Why the Kelvin–Helmholtz timescale is not really their timescale

Available online at www.sciencedirect.com

New Astronomy Reviews 51 (2008) 803–813 www.elsevier.com/locate/newastrev

Why the Kelvin–Helmholtz timescale is not really their timescale Giora Shaviv Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel Available online 18 March 2008

Abstract We review the works of Helmholtz and Kelvin and their timescale and show that it is not what is called in classical text books the Kelvin–Helmholtz time scale. Next we review the work of Ritter and show that he was the first to calculate what is erroneously attributed today to Kelvin and Helmholtz. We conclude that the correct name is the Ritter–Kelvin–Helmholtz timescale. Ó 2008 Elsevier B.V. All rights reserved. Keyword: Kelvin–Helmholtz

1. Introduction A basic problem scientists faced in the mid of the 19th century was the energy source of the Sun. There were different ideas, but it was only after the discovery of the law of the conservation of energy could this problem be given a physical solution. Hence the discoverers of this conservation law, Mayer and Helmholtz, were the first to offer a solution for the energy of the Sun. However, due to a fundamental wrong assumption they could not get the right solution to the model they suggested. We start by a short review of the discovery of the conservation of energy and show what the wrong assumption was and who solved it correctly. 2. Energy conservation, Helmholtz and Mayer Herman von Helmholtz (1821–1894) was the first physicist to provide an apparent bona fide physical solution to the problem of the energy source of the Sun. Helmholtz was in a position to do so because he discovered the conservation law of energy, not only the mechanical one but all forms of energy were combined into one conservation law. The problem of extracting energy from the gravitational energy of the Sun required just such a conservation law (Helmholtz, 1856, 1862–1863). E-mail address: [email protected] 1387-6473/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2008.03.009

The idea of having a conservation law and the existence of conserved quantities was not new. Already in 1668 John Wallis (1616–1703) suggested that momentum is conserved. Gottfried (1646–1716) suggested the conservation law of mechanical energy (potential plus kinetic energy). Antoine Lavoisier (1743–1794), who is often referred to as the father of modern chemistry, was the first to formulate clearly and unambiguously the law of mass conservation. So the idea that certain quantities can be conserved was not new. However, combining such different entities like heat, mechanical work and radiations for example, into one conservation law was new. All forms of energy are conserved together even if the energy transforms from one form to another. Those were the days before Rudolf Clausius (1822–1888) had shown in 1850 that heat is associated with the kinetic motion of molecules and not that many years after when Count Rumford had shown in 1798 that the ‘supposed carrier of heat, the caloric’ cannot be conserved. We recall how in those days physicists and chemists invented ‘imponderable fluids’ to solve problems, the caloric for heat, the phlogiston for burning and the ether for light. The caloric for heat was killed by Rumford. The phlogiston for burning was killed by Lavoisier and the ether for light by the Michelson–Morely experiment and Albert Einstein (1879–1955). But despite Rumford’s proofs, there were scientists who kept thinking in terms of the caloric theory, most notable being Carnot (1796– 1832) (Carnot, 1872) who discovered the Carnot cycle

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and laid the grounds for the second law of thermodynamics when the first law (the conservation of energy) was not yet known. One can, however, ‘translate’ Carnot arguments into present day thermodynamics if one replaces ‘caloric’ with ‘entropy’, namely the entropy increases or does not change in any process. The entropy never decreases. (The idea of entropy the way we know it today is due to Clausius and came later, in 1865.) The belief in ‘ponderable fluids’ like caloric was the main barrier which made it difficult for scientists to accept the conservation of energy. What happens to the ‘caloric’, for example, was the question. As a matter of fact, Helmholtz was not the first to discover the law of energy conservation, the discoverer was Robert Mayer (1814–1878) who was a german surgeon on a Dutch vessel sailing in the tropics. It was during the therapeutic removal of blood in the tropics that he recognized that the venous blood of the Europeans was pale red and looked like the oxygenated arterial blood. It was a known phenomenon but never before explained. Mayer supposed that the oxygen is needed to power the muscles and keep the body warm. However, the hot weather of the tropics requires a lower metabolic rate to keep the body’s temperature, and hence demanding less oxygen than in a colder climate. He concluded, therefore, that the human body needs less oxygen in tropics than in cold temperate zones to maintain the body at a constant temperature. Moreover, he suggested that the heat produced during muscular effort must also be derived from the chemical energy stored in food; the input of energy is the food and the output is a ‘force’ (i.e., in those days what we call today energy was called force) that must be balanced. From this, he drew the surprising conclusion that motion and heat are ‘different manifestations of one and the same energy’. So, they ‘must permute and transform into one another’. This out of the blue conclusion was not easily accepted by the scientific establishment. Upon his return to Germany and setting a private practice, Mayer summarized his ideas in a paper and sent them to Johann Christian Poggendorff (1796–1877), the editor of the Annalen der Physik und Chemie. In this paper he postulated the conservation of force, which we call today the conservation law of energy. However, probably owing to Mayer’s lack of advanced training in physics, it contained some fundamental errors and the paper was rejected by Poggendorff. Mayer did not give up; he studied physics and argued with the Tu¨bingen physics professor Johann Gottlieb No¨rremberg, who naturally rejected his claim. In exchange Mayer got some ideas as to how to prove experimentally his point. So Mayer not only demonstrated how mechanical energy converts into heat but also measured the conversion factor of the transformation, namely, the mechanical equivalent of heat. The result of his investigations was published in 1842 in Justus von Liebig’s Annalen der Chemie und Pharmacie (Mayer, 1842). Liebig (1803–1883) was one of the greatest chemists of the 19th century and clearly recognized the importance of Mayer’s discovery and accepted it for publication. Three years later

Mayer published the book: ‘The Organic Movement in Connection with the Metabolism’ in which the numerical value of the conversion factor is given, a value which deviates only by about 10% from the present day value. A point of clarification. The time is before the second law of thermodynamics was known so that the conversion of heat to mechanical work and back was a problem and one had to show that energy was not lost despite the irreversible process. One of the most versatile scientists who ever lived, Hermann von Helmholtz, was born in Potsdam, Germany, to a high school teacher. Helmholtz was attracted to physics from young age, but his family could not afford the financial means needed to let him study physics. Instead, his father persuaded him to take up medicine, since the education of physicians was supported by the state provided they served for several years as doctors in the Prussian army. Helmholtz attended the Institute for Medicine and Surgery in Berlin from 1838 to 1842 and served as an army surgeon from 1843 to 1848. But his soul was always in research; even in the army barracks he set up a small laboratory for research in physiology and physics, the subjects he loved. On July 23, 1847, Helmholtz presented a paper on the conservation of energy at a meeting of the Berlin Physical Society. It was a talk at a reasonably high level of mathematical sophistication intended to convince physicists that energy is conserved in any closed physical system (strictly speaking he spoke on physical processes). The paper was submitted to Poggendorff, the editor of the Annalen der Physik, and was rejected as being too long and too mathematical for his readers. So Helmholtz published the results in a pamphlet that was soon recognized as one of the most important papers in physics. Poggendorff had the dubious honor to reject the first two, apparently independent, seminal papers with one of the greatest discoveries in physics, the discovery of energy conservation. The Helmholtz’s bold and path-breaking paper, written when he was only 26 years old, was his first and most fundamental statement of the principle of conservation of energy. In his 1847 book Helmholtz showed convincingly that the conservation of energy is indeed universally valid. Note that the full details of the energy conservation law were published by Helm¨ ber die Erhaltung der Kraft’ (on the holtz in the book ‘U conservation of force) 1847 and not in a refereed journal. Interesting, one of the most important laws in physics was discovered by two trained physicians and not physicists. Helmholtz showed that the assumption that work cannot be produced from nothing leads to the conservation of kinetic energy. He then applied this principle to a variety of different situations. He demonstrated that in various situations where energy appeared to be lost, it was in fact converted into heat. This happens in collisions, expanding gases, muscle contraction, and other situations. The book looked at a broad number of applications including electrostatics, galvanic phenomena and electrodynamics. In

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our days we can state that Helmholtz proved the non-existence of perpetuum mobile (of the first kind). During the same years that Mayer and Helmholtz were active in Germany, the English scientists were also busy understanding the connection between heat and mechanical energy. The dominant player was James Prescott Joule (1818–1889). In 1845 (Joule, 1845) he discovered what is known today as the Joule’s law, namely the connection between the electric current and the heat it produces. The discovery was presented to the Royal Society but was not highly regarded probably because in those days (and until 1854) Joule managed the brewery he inherited from his parents and did not have the auspice of a university or a research institute. Two years later he found the numerical equivalent between the electric current and the heat generated, and in 1845 he did away with the electric current and experimented the conversion of mechanical energy to heat and vice versa. In this way he discovered what is known today as the Joule constant, namely the conversion factor between mechanical energy and heat (Joule, 1843–1850). Objectors to the idea of irreversibility, which Joule’s experiments implied, brought the thermoelectric effect, the direct conversion of temperature differences to electric power by building a voltage difference as a counter example. Reversibility means that one type of energy can be converted to another type and back to the original form. Irreversibility means that if mechanical energy, for example, is converted into heat, the heat cannot be completely converted back into mechanical energy without losses. This is exactly what Carnot discovered. The thermoelectric effect is reversible while the Joule heating, the conversion of mechanical energy into heat is not. The current flow between the two edges of the metal placed at a temperature difference (Thomson heat) is indeed reversible and so is the heat released or absorbed when current flows. The reversibility is in the sense that when the current is reversed the effect remains the same but changes sign. On the other hand, heat conduction in wires and the heat dissipated in the resistance are not reversible. Also in 1847, one of Joule’s presentations at the British Association in Oxford was attended by distinguished figures like George Gabriel Stokes, Michael Faraday, and William Thomson. Stokes was inclined to believe Joule, Faraday was much struck with it though he had some doubts and Thomson, who later would be known as Kelvin, was intrigued but skeptical. Though the initial attitude of Thomson was that Joule’s results demanded theoretical explanation, he gradually became convinced that Joule might be right and in his 1848 paper, in which he established the existence of an absolute temperature and what is known today as the Kelvin scale, Thomson wrote on one hand that ‘. . .the conversion of heat (or caloric) into mechanical effect is probably impossible, certainly undiscovered’ and on the other hand in a footnote he raised his first doubts about the caloric theory, referring to Joule’s ‘very remarkable discoveries’. The use of the term ‘caloric’ may explain the difficulty Kel-

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vin had with the question what happens to the caloric upon conversion of heat into mechanical energy. Surprisingly, Thomson did not send Joule a copy of his paper but when Joule eventually read it he wrote to Thomson, claiming that his studies had demonstrated the conversion of heat into work and that he was planning further experiments. Thomson replied, revealing that he was planning his own experiments and hoping for a reconciliation of their two views. Though Thomson conducted no new experiments over the next 2 years he became increasingly dissatisfied with Carnot’s caloric theory and convinced of Joule’s claims. In his 1851 paper (Thomson, 1851) Thomson was willing to go no further than a compromise and declared ‘the whole theory of the motive power of heat is founded on . . . two . . . propositions, due respectively to Joule, and to Carnot and Clausius’. The correspondence between Joule and Thomson gave birth to a fruitful collaboration: Joule conducted the experiments and Thomson analyzed the results and suggested further experiments. The collaboration extended from 1852 to 1856 and culminated in the discovery of the Joule–Thomson effect. The collaboration with the esteemed Thomson helped to bring about general acceptance of Joule’s work and the kinetic theory of gases. 3. The source of solar energy The first attempt to explain the energy source of the Sun is due to Mayer who suggested in 1848 in a hardly cited or mentioned paper (Mayer, 1848b) that meteors falling on the Sun are the source of energy. Having in his hand what is called today the Joule constant, he was able even to calculate what we would call today the heating by accretion of matter coming mainly along the plane of the ecliptic. The experimental proof for the existence of such matter close to the Sun was the zodiac light. He was aware that as a consequence of such an accretion the mass and the radius of the (liquid) Sun would lead to observable results. To solve this problem he assumed that the Sun loses ether. The most popular solution to the problem of the source of energy of the Sun is due to Helmholtz, who on February 7, 1854 suggested in a popular address delivered at Ko¨nigsberg on the occasion of Kant’s commemoration that the Sun derives its energy from small rock-like pieces or even dust-like particles that were spread out in space and fall onto it. Helmholtz was influenced by the ‘nebular hypothesis’ advanced by Kant and Laplace. The nebular theory, which was very popular at the time, assumed that the planets formed from mergers of dust and gas rotating around the Sun. Laplace was very influenced by the shape of the ring nebulae. These are nebulas with a shape of a ring and were called planetary nebula because of the hypothesis advocated by Laplace that they are the progenitors of planetary systems. This was in complete contrast to our present day understanding of these objects. The chunks of matter – gas-meteors according to Helmholtz, fell inward to what is

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now the Sun’s position, releasing their huge gravitational energy upon colliding with the mass already present at the center, to form a very hot molten sphere of mass. The basic idea was, therefore, that matter fall onto the Sun. The Sun converts the gravitational energy into heat which is to be released later as the heat to space in the form of radiation. Helmholtz, as the discoverer of the conservation of energy, was able to realize that the process of mass accretion by the liquid Sun converts potential energy into kinetic energy and subsequently into heat. The process has two phases. In the first phase the Sun heats up when the rain of meteorites hits it continuously. In the second phase the hot Sun cools by radiation from the surface. Assuming that a heat capacity of the Sun is similar to that of water gives a temperature which when divided by the rate at which the Sun loses energy yields an age. Since the heat capacity was not known, the calculation had an uncertainty. The age Helmholtz got was few million years. The original idea of Helmholtz was that the Sun is under a heavy meteor shower. The mass of meteors that should fall on the Sun in a year in order to supply the energy radiated away is about 6  1025 g m or about 1% of the mass of the Earth per year. It is easy to realize that the accretion of such a mass by the Sun would affect the orbit of the Earth in a noticeable way and change the length of the year by about a second per year. This was in those days an easily detectable change. Clearly, if the meteors come from within the Earth orbit, no such change is expected and if they come from within the radius of Mercury, even Mercury would not feel the change in the gravitational force. But in the latter case the available energy would be much smaller. These consequences of the hypothesis were not observed and hence Helmholtz supposed that the Sun must be in the second phase. We end with an upsetting note: In 1848, Mayer learned about Joule’s papers and wrote to the French Acade´mie des Sciences to assert his priority. His letter was published in the Comptes Rendus (Mayer, 1848a) and Joule was quick to react. Thomson’s close relationship with Joule allowed him to be dragged into the controversy. The pair planned that Joule would admit Mayer’s priority for the idea of the mechanical equivalent but to claim that experimental verification rested with Joule. Some of the greatest names in English science were drafted to help Joule, like Rankine and Maxwell. But it did not help. On May 18, 1850, Mayer attempted suicide and we can guess that the controversy did not help his mental state. Several years later, in 1862, John Tyndall (1820–1893) who inherited the position of the great Faraday and was a great scientist on his own, continued Faraday’s tradition of public popular talks and argued in a lecture entitled ‘On Force’ at the Royal Institution that Mayer was to be credited with conceiving and measuring the mechanical equivalent of heat (Tyndall, 1873). Thomson and his followers lost their temper and started an ugly campaign on the pages of the Philosophical Magazine. Justice prevailed and Mayer’s priority

is now recognized. For more on the Mayer–Joule controversy see Lloyd (1970). In 1905 Carl Barus (1856–1935), a physicist, summarized for the Science Magazine Barus (1905), the major scientific achievements of the 19th century. The law of the conservation of energy was not mentioned at all. 4. Darwin By the time Darwin published his Earth shaking theory about the Origin of the Species by Means of Natural Selection in 1859 (Darwin, 1859) the proponents of biblical age of the Earth had disappeared only to surface again in the second half of the 19th century in the form of the creationist theory for life and the panspermia hypothesis. Is the Genesis-Geology debate of any relevance today? One might have hoped that it is part of the past, nonetheless there are some morals to learn even today. It is not rare to see today explanations of scientific evidence based on religious beliefs. It is interesting at this point to learn the attitude of one of the greatest Jewish rabbi Moshe Ben Maimon (1135–1204) known as Maimonides, who at the pre-scientific era claimed that it is wrong to read Genesis literally. He claimed that one has to understand the Bible in a way that is compatible with the findings of science. Indeed, Maimonides wrote that if science and the Bible are in conflict, it is either because science is not understood or the Bible is misinterpreted. Maimonides argued that if science proved a point, then the finding should be accepted and the Holy Scriptures should be interpreted accordingly. While the theory fermented in Darwin’s mind for a long time (he published the book when he was about 50 years old), it burst after his famous trip on the H.M.S. Beagel (1831–1836). During this trip, in which he collected data, Darwin studied Lyell’s book. Darwin’s theory is very strongly connected to Lyell’s claim that the Earth is very old. Moreover, Darwin applied the same principles that Lyell used in geology to biology, namely gradual evolution. The principle of natural selection or the survival of the fittest is an extremely simple principle though its manifestation can take many different forms as described in detail by Darwin in his book and are not the subject here. Our interest here is not in the biological theory but in the fact that it determines a time scale for the evolution. In those days, astronomy and astrophysics were unable to set the time scale for the evolution and resorted to biology for an estimate. Even Eddington years later, relied on biological timescales to refute the contraction hypothesis, namely Eddington argued the other way round! Darwin could not set a time scale for his process, but only observe its consequences. However, he realized that the time needed for it to take place must be very long, much longer than the inferred biblical age for the Earth. The interesting thing is that Darwin himself carried out the geological calculation to find the minimum age of the Earth, and it is this issue which interests us here. The particular example Darwin

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chose was the denudation of the Weald, a great valley that extends between the North and South Downs in the southern part of England. The data were taken from Ramsay but Darwin had to guess the critical number: the rate of denudation. Darwin guessed that the sea would erode into the 500 feet cliff at a rate of 1 inch per century and got that it would take about 306,662,400 years. While Darwin had good reasons for the estimated rate of erosion, still it was not a measured number. A heavy assumption in this calculation is the constancy of the process during such a long period. Said differently, the average rate over long periods was simply guessed. We know today that the surface of the Earth is relatively young compared to the age of the Earth. The surface of the Earth changes perpetually and the estimates Darwin (and geologists) made from erosion were, therefore, minimal and related to the relatively short time scale of continent motion, mountain formation and valleys erosions. Darwin must have been aware of the Helmholtz calculation of the age of the Sun but apparently decided to ignore the contradiction, in the direction that did not helped his case. Maybe because it came from Europe? But when Thomson pointed to the discrepancy, he could not ignore it any longer. The greatness and ingenuity of Darwin’s phenomenological investigation led to the hypothesis of evolution without providing a mechanism, without knowing anything about DNA, genes and mutations. It took science the needed time to establish the phenomenological findings on biochemical grounds and find the responsible mechanisms. The great hypothesis preceded the evidence for a mechanism. 5. First attempt to give numbers for the meteor theory Already in 1860 Waterston (Waterston, 1860) (1811– 1883) tried to use Joule’s new result and calculate the temperature of the Sun. His first attempt was to calculate the temperature that a meteor hitting the surface of the Sun would reach. Assuming that the meteor and the Sun behave like water he got incorrectly the fantastic central temperature of 55 million degrees (the correct number using his data is 16 million degrees.) He argued that if the Sun in made of iron, the temperature would have been nine times higher. Waterston simply converted the kinetic energy of the falling meteor to heat using Joule’s conversion factor. He continued and calculated the mass of the meteor that would supply the present power emitted by the Sun and claimed that if this meteor does not fall, the Sun should cool by the amount the meteor should supply. This strange argument led him to conclude that the Sun cools by 4.59 °C per year. Had he continued his reasoning, he would have reached the conclusion that the lifetime of the Sun is 22 million years, a conclusion that the old Darwin would not have appreciated. Interesting, Waterston calculated the effect mass accreted by the Sun would have on the length of the year of the Earth and rejected the possibility that masses as large as that of the Earth could fall off the

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Sun because he calculated the change in the length of the year to be 130 seconds per year, certainly noticeable by astronomers at that time. Kelvin was very impressed by Waterston results and hailed them publicly. 6. Kelvin A special role in the fight against the evolution theory of Darwin was played by Lord Kelvin (1824–1907). It is difficult to comprehend the enormous influence Kelvin had on science in the second half of the 19th century and the first part of the 20th century. William Thomson (later Lord Kelvin) was a professor of physics in Glasgow and had numerous discoveries in physics, the most important of which are in thermodynamics: the formulation of the second law of thermodynamics (Thomson, 1852). Kelvin carried out two independent calculations, one was the age of the Earth and the other was the age of the Sun. The calculation of the age of the Sun was carried out in 1862 (Thomson, 1862). It is interesting to study the content of this paper. First, Kelvin examined the idea that the heat generated in the solar atmosphere by falling meteors is sufficient to generate the entire power of the Sun (an idea sometimes later referred to Robert Mayer but no mention of him could be found in Kelvin’s paper). Kelvin compared this idea with the assumption that the Sun is an incandescent liquid mass losing the initial heat generated by falling meteors in the past. Kelvin rejected the first possibility (which was Helmholtz’s first version) by noting that the mass accreted by the Sun in 2000– 3000 years would be sufficient to change its mass in such a way as to affect the length of the year. The amount of mass needed to generate the Sun’s luminosity is 1/47 of the mass of the Earth which is 1/15,000,000 of the mass of the Sun and for this to be possible, one has to assume that the total mass of meteors in the plane of the planets is of the order of 1/5000 of the mass of the Sun. However, such a mass would affect the motion of the planets to a measurable extent, which was not noticed. We recall that already in 1840 (LeVerrier, 1840) Urbain LeVerrier (1811–1877) calculated the advance of the perihelion of Mercury due to perturbation from other planets. While there was a small discrepancy to be later explained by Einstein, it was too small to support a resistance by a swarm of meteors needed to explain the energy of the Sun. To avoid disturbances by the meteors to the motion of Mercury, the meteors had to be so close to the Sun as to reduce significantly the energy release by falling onto the Sun. Kelvin was, therefore, forced to assume that the Sun was heated at the beginning and since then radiates its energy, namely the Sun cools. Kelvin took from Herschel and Pouillet the total energy the Sun releases (6  1030 times the heat needed to raise the temperature of one pound of water by 1 °C). He then assumed that the solar material is similar to that of the Earth (and here he relied on spectroscopic observations of Kirchoff and Bunsen who

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discovered that elements of on the Sun are those found on the earth). Next, Kelvin assumed that the specific heat of the solar material resembles that of water so that he divided the above energy output by the mass of the Sun and the specific heat. In this way he obtained the cooling time of the Sun. As the Sun cools, it must contract like every material on the Earth which contracts upon cooling. Kelvin found that at the present rate of cooling the Sun should contract by about 1/120,000 of its diameter per every degree centigrade that it cools. If the specific heat is that of water, then the Sun must contract by 1% per 860 years, a change that must be noticed by astronomers. Since a contraction at such a rate posed an observational problem (it was not seen), Kelvin assumed that the material of the Sun must have a different heat capacity from that of the Earth. Here Kelvin committed the same misdeed for which he attacked Darwin. Namely, when the result of the calculation did not agree with the desired outcome, he changed the input parameters. Kelvin continued and realized that upon the contraction of the Sun, different parts of the Sun must do work which he cannot calculate because he does not know the run of the density inside the Sun. If the density is constant as Helmholtz assumed, then the energy of the Sun should suffice for 20,000 years. This number may increase if the density increases towards the center of the Sun. Next, argued Kelvin, ‘it is in the highest degree improbable that mechanical energy can in any case increase in a body contracting in virtue of cooling. It is certain that it really does diminish very notably in every case hitherto experimented on. It must be supposed, therefore, that the Sun always radiates away in heat something more than the Joule-equivalent of the work done on his contracting mass, by mutual gravitation of its parts. Hence, in contracting by one tenth per cent of his diameter, or three tenth per cent in his bulk, the Sun must give out something either more, or not greatly less, than 20,000 years’ heat. . .’ By this argument, in which the Sun contracts by 1% per 20,000 years, Kelvin solved the problem of the unobserved contraction of the Sun. However, to minimize the possible observation of the Sun’s contraction Kelvin proposed that the Sun may have a very large specific heat, as high as 10,000 times that of water. In this case the contraction upon cooling is reduced far below any observation limit. Of course, by this Kelvin donated the Sun a total energy 10,000 times higher. Kelvin ended this part by sneering at Darwin about his calculation of the ‘denudation of the Weald’. One strong storm he claimed would erode the cliff 1000 times more than the rate of 1 inch per century assumed by Darwin. This was of course a wild guess by Kelvin. After showing that the Sun cools, Kelvin compared the energy production per square foot on the surface of the Sun with the energy produced in the furnace of a locomotive and concludes of that they are similar. Since he assumed that the Sun is liquid, he did not calculate any heat transfer inside the Sun, as he assumed the heat to be carried by currents in the liquid Sun.

Finally came the question of the origin of the total amount of the Sun’s heat. Kelvin concluded that the theory of the present day meteoritic showers must be rejected in favor of the theory which hypothesized that all the solar heat was generated by past massive meteor showers, and here he returned to the hypothesis of Helmholtz. Finally, he concluded that the age of the Sun is not smaller than 10 million years if the density is constant and may be even 100 million years if the density increases inward. Kelvin’s calculation was wrong on many counts. To mention one, it was Eddington who years later showed that the effective specific heat of stars is negative, namely stars lose energy by radiation heat up, and not cool down! The calculation of the age of the Earth was more complicated than the calculation of the contraction of the Sun and significantly more uncertain. Kelvin knew about Fourier’s earlier work and even defined it as ‘a poem’ but he dismissed the conclusion of Fourier, namely that the available data then did not allow to calculate the rate of cooling. The first question to answer when attempting to calculate the age of the Earth was the initial state of the Earth and Kelvin assumed that the Earth was molten and hence an initial temperature must be assumed. Next, one has to assume how well the various layers in the Earth conduct heat. One needs to know the rise of temperature with depth, namely at what depth the temperature increases by one degree. Kelvin took yearly averaged values measured in mines. Finally, one has to assume a value for the heat capacity of the Earth’s material and the heat conductivity. The age obtained was few tens of thousand years with a large error because of inaccurate data. As we know today, the calculation was wrong because the radioactive heat source was ignored. It is amazing that Kelvin, the scientist, was not bothered by his own finding of a discrepancy between the ages of the Sun and the Earth. He was ready to accept that the Earth is younger than the Sun which he calculated to be younger than the age quoted by Darwin. The discrepancy could have been removed had Kelvin assumed a large heat capacity for the Earth’s interior, the physics of which was as unknown as that of the Sun. He probably did not do so because he was not interested in increasing the resulting Earth’s age. 7. The Darwin–Kelvin controversy The vocal arguments that the widely esteemed but also self-esteemed Kelvin had with the community of geologists occupy many pages and are not the subject here. He was wrong and they were right. Adding to his irritation was the infiltration of biology into his subject, personified as Charles Darwin and the Origin of Species to be followed by more books. As we mentioned above, in his 1862 paper on the age of the Sun, Thomson mocked at Darwin’s attempt to estimate the geological ages. Five years later Thomson’s friend Fleeming Jenkin wrote a long review of the Origin of Species

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and dismissed Darwin’s attempt at quantitative geology as a calculation of the kind engineers refer to as guess at the half and multiply by two. But of no avail, by that time Darwin’s book had already gained popularity and had gone through several editions. In the discussion part of the book, Darwin still wanted the evolution to take place for a long time, but admitted a defeat on this count. Kelvin’s scientific terror won: Darwin’s estimated age of the earth was removed from the coming editions of his book. Even by the mid-1860s, Thomson’s arguments about the age of the Earth were practically ignored by geologists, who were at that time not as quantitative scientists as today. It disturbed him that his ‘rock-solid thermodynamic arguments’ were rejected on the grounds that physics and geology should not be mixed. Physicists thought correctly that all systems must obey the laws of thermodynamics as they are so general. In 1867, when the British Association met at Dundee, Thomson argued with Andrew Ramsay, who was one of those who provided Darwin with his geological data. Ramsay was of the opinion that geological ages are very long, even as long as billion years. Thomson objected that the Sun, being a finite body, could not possibly shine for so long. Ramsay responded that this point of physics had nothing to do with him: I am as incapable of estimating and understanding the reasons which you physicists have for limiting geological time as you are incapable of understanding the geological reasons for our unlimited estimates. Thomson responded by saying that physics can be explained to everyone provided he is really willing to listen and understand. So far as he was concerned, Thomson was not telling geologists how to conduct their science, only that their theories could not disregard the universal laws of thermodynamics. Kelvin wanted the geologists to listen to him but refused to listen to them. Very little has changed in the data since Fourier arrived at his conclusion about the inaccuracy of the age calculation, but the arrogant self-assured Kelvin ignored what Fourier wrote 50 years earlier and continued with the calculation of the cooling of the Earth. Darwin could console himself that Kelvin argued not only with him but with many others on various topics. He argued at the beginning with Joule, then was convinced and collaborated with him. He argued with the entire community of geologists. He argued with Tyndall about magnetism but was found wrong. Tyndall fought Kelvin on many fronts. It is interesting to mention here that in 1874 Tyndall attacked Kelvin’s version of the source of energy of the Sun. He calculated a la` Kelvin the amount of energy that would be obtained if all planets fell onto the Sun (including the rotation energy) and found an age of 45,586 years. He then cited the original Helmholtz idea of the contraction of the Laplace original nebula and got a temperature of 28 million degrees and calculated that the cooling of the Sun to the temperature of the Earth would take 17 million years. The incessant propaganda by Kelvin spelled a disaster for Darwin who felt obliged to change some of his age of the Earth estimates (though he was right) and wrote bitter

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letters to Lyell (1868). Geologists like Huxley tried to debate Kelvin but of no avail. It was either Darwin or Kelvin. Both could not be right. Did Kelvin notice the discrepancies that his calculations led to? We do not know and in any case, it is not mentioned in his writings. But Kelvin was one of the greatest physicist and the establishment of the laws of thermodynamics was to his credit. So how to explain it? Kelvin was a very religious person. He could not live with Darwin’s evolution of life and human beings. While Darwin did not address the question of the origin of life, Kelvin did. In August 1871 he addressed the British Association for the Advancement of Science meeting held in Edinburgh and exposed his hypothesis for the origin of life. First he discussed the hypothesis that under meteorological conditions very different from the present, dead matter may have run together or crystallized or fermented into ‘germs of life’ or ‘organic cells’ or protoplasm and claimed that science brought evidence against this hypothesis though he did not provide any manifestation. He claimed that dead matter must be under the influence of life matter to become alive. He then suggested searching for spontaneous generation of life. Kelvin proceeded and asked ‘how life originate on Earth?’ He stated that ‘if probable solution, consistent with the ordinary course of nature, can be found, we must not invoke an abnormal act of Creative Power’. Kelvin then continued with the example of how wind carried vegetation onto the cooled lava of a volcano and soon reached the idea of panspermia. The essence of the theory is because we all confidently believe that there are at present, and have been from time immemorial, many worlds of life besides our own we must regard it as probable in the highest degree that there are countless seed-bearing meteoric stones moving about through space. Thus Kelvin claimed that The hypothesis that life originated on this Earth through moss-grown fragments from the ruins of another world may seem wild and visionary; all I maintain is that it is not unscientific. The amazing thing is that Kelvin realized that his idea required evolution and he indeed stated that all creatures now living on Earth have proceeded by orderly evolution from some such origin. Kelvin effectively believed in evolution and right after cited from ‘The Origin of Species’ by Darwin but he omitted two sentences as he explained: I have omitted two sentences . . . describing briefly the hypothesis of the ‘origin of species by natural selection’, because I have always felt that this hypothesis does not contain the true theory of evolution, if evolution there has been, in biology and here comes the crux of the matter I feel profoundly convinced that the argument of design has been greatly too much lost sight of in recent zoological speculations. This explains in my opinion why Kelvin was not bothered by the inconsistencies. Darwin died on April 19, 1882m, at 73 years of age, not knowing that he was right about the age of the Earth. Kelvin died on December 17, 1907, not before radioactivity was discovered and became the tool for dating, knowing

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that he was wrong on quite many issues, including the age of the Earth, new physics, ether and many more. The obituary the Times of London run on Lord Kelvin was 13 columns long and included a discussion, more than a column long, on the age of the Earth and the Sun and how Kelvin fought the geologists. Michael Faraday (1791–1867), James Clerk Maxwell and Lord Kevin were leading scientists in the second half of the nineteenth century with colossal contributions to science. Yet, their names are hardly known outside the narrow scientific circles, in dramatic contrast to the name of Darwin. Faraday, Maxwell and Kelvin were scientists and very religious. In particular the first two completely separated their religious beliefs from their scientific life, research and findings (and are not known to have expressed opinions in the age of the Earth and Sun controversy). Maxwell is known to have interest in astronomy as he solved the problem of the Saturn rings. Darwin on the other hand, was not an atheist. He described himself as an agnostic, and it is likely that he retained a belief in some kind of personal God, although not a deity who interferes continuously in the evolutionary process and in human affairs. Was Kelvin wrong? The calculations Kelvin did were correct. His great failure was to not recognize that something was missing in his assumption and hence that there was a fundamental flaw, something in the assumptions was wrong! This was his great fiasco. The problem was his loss of critical review when it came to matters that touch upon Holy writings. He did not have the hunch that something fundamental was missing, and this was the heating of the Earth by radioactivity. But was this accidental? Probably not. And so said Kelvin I cannot admit that, with regard to the origin of life, science neither affirms nor denies Creative Power. Science positively affirms Creative Power. It is not in dead matter that we live and move and have our being, but in the creating and directing Power which science compels us to accept as an article of belief. As an example of what the discrepancy should have stimulated it is interesting to see what T.C. Chamberlain, a leading geologist, said in 1899 in one of his addresses: ‘Is present knowledge relevant to the behavior of matter under such extraordinary conditions as obtain in the interior of the Sun sufficiently exhaustive to warrant the assertion that no unrecognized sources of heat reside there? What the internal composition of the atoms may be is as yet an open question. Is it not improbable that they are complex organizations and the seats of enormous energies? . . . No cautious chemist would probably venture to assert that the component atomecules, to use a convenient phrase, may not have the energies of rotation, revolution, position and be otherwise comparable to those of a planetary system. Nor would he probably be prepared to affirm or deny that the extraordinary conditions which reside in the center of the Sun may not set free a portion of this energy.’ In short, the discrepancy called for an examination of all assumptions, and the geologist was right.

8. The birth of the theory of stellar structure The only models of stars and the Sun at this time (the beginning of the second half of the 19th century) were models of liquid stars. The logic was simple. The mass and the radius of the Sun were known, so it was simple to calculate the mean specific density of the Sun and it is 1.45 g m/cc. As water is practically incompressible, the assumption about liquid stars was understood as incompressible stars. Many theorems were proved about such stars, researchers discussed problems like their stability, the possibility of fission when rotating fast or cooling as discussed by Kelvin and Helmholtz. On the other hand Kelvin and Helmholtz could not discuss gravitational energy derived from compression of the Sun, as the Sun in their view was incompressible. The configuration of rotating liquid masses acted on by self-gravity was a topic investigated by some of the greatest mathematicians like Colin Maclaurin (1698–1746) (after him the objects are named Maclaurin ellipsoids) Carl Jacobi (1804–1851) (after him the tri-axial objects are called Jacobian ellipsoids), Jules Henri Poincare (1854–1912) (who found that these objects can transform into pear-shaped objects) and George Darwin (the son). Ritter turned fascinating mathematical models into an exotic topic of no relevance to real stars by considering gaseous stars. The first to consider gaseous stars was Zoellner in 1871 (Zoellner, 1871). However, Zoellner assumed that the temperature is constant throughout the star. This assumption is problematic because it leads to an infinite mass, an unacceptable solution. August Ritter was a professor of mechanics at the polytechnical uniersity of Aachen. During the period 1878–1883 Ritter published a series of 18 papers in the Wiedemann’s Annalen with two basic assumptions, the first of which being that the stars, the Sun included, are gaseous. While the papers were crucial to stellar structure, they did not attract the proper attention of the scientific community and in 1898 the Editor, George Elery Hale,1 of the newly formed Astrophysical Journal, decide that Ritter’s series of papers was so important that he initiated the publication of the 16th paper in the Astrophysical Journal. The paper, ApJ, 8, 1898, p. 293, included a special preface by the Editor, a very unusual move. The ApJ paper contained also a complete reference of the entire series of papers. It is fantastic to see how in his 12th paper (Ritter, 1881) Ritter applied the newly discovered law of the emission of radiation by black body just discovered by Stefan Jozef Stefan (1835–1893) in 1879 (Stefan, 1879) to relate the emissivity of two stars. It was Ritter who discovered the fundamental formula L ¼ 4pRrT 4e . Moreover, Ritter succeeded to obtain for the first time a relation between the central temperature and the surface temperature. This was a major 1 The Astrophysical Journal was established in 1895 under the editorship of Hale George Ellery Hale (1868–1938) and James Edward Keeler (1857–1900). (Edwin Brant Frost, 1866–1935, served as an assistant editor from this date and as editor during 1902–1935.)

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breakthrough since the temperature at the center was found to be tens of millions degrees K and not few thousand degrees as essentially fixed by Kelvin. Ritter was the first to convert the Kelvin–Helmholtz hypothesis into what it is known today, namely the Sun has a fixed mass and is gaseous. The entire Sun contracts and not the accreted matter. Upon contraction/compression, the Sun sinks deeper into the gravitational potential well, heats up and radiates away only half of the energy extracted from the gravitational field. Ritter discovered that the Sun has a negative effective specific heat though he did not formulate it in these terms. If the temperature on the surface is few thousand degrees, then inside it is higher and Ritter assumed correctly that at such a temperature the matter must be in a gaseous form, vindicating a posteriori his assumption about the state of the matter in the Sun. Moreover, the gas was assumed to follow the law of ideal gas, the simplest known gas law on Earth. In the lack of knowledge of an energy source or how energy is transported from the core to the surface, Ritter assumed the second fundamental assumption, namely that the run of temperature and density throughout the Sun is adiabatic. Usually ‘adiabatic’ means that no heat enters or leaves a particular element of the gas. Here the meaning was a bit different. Take an element at a given point of the star. Clearly, it is under the force of gravity and pressure and when the forces balance each other the element is in equilibrium (at rest). Now move this element to another location in the star; as the pressure in the new location is different, the element say, contracts, raises its pressure and consequently its temperature so as to be at the same pressure as the pressure in the new location. If the movement of the element from one location to the other is carried out without adding/removing heat to/from the element, then we define the movement as adiabatic. So, if the displaced element reaches the new location with pressure equal to that in the new location but without any heat absorbed/released we call it an adiabatic lapse of temperature. The first to discover the adiabatic lapse was Kelvin when he analyzed the run of the temperature in the atmosphere of the Earth in 1861 Thomson (1911). So what Ritter did was simply to assume the same temperature lapse for the Sun (though Ritter actually cited Mohn and Guldberg 1878 (Mohn and Goldberg, 1878). Did Ritter ignore british papers? Once he made this assumption he could solve for the run of the temperature, pressure and density throughout the entire Sun. As soon as Ritter had the solution for the density throughout the Sun he could carry out the first correct calculation of the gravitational energy release by a contracting Sun without any additional assumptions about heat capacity, initial temperature, etc. Moreover, Ritter dispensed with the meteor shower, with the cooling of the Sun as well as with all the problems of observations of meteors, where is this mass, what is the maximum mass of meteors which can be hidden without disturbing the orbit of Mercury and so on. Ritter got the correct ‘clean’ relation between

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the rate of radius contraction and the rate of energy loss from the surface. It is Ritter who derived for the time that sKH ¼ GM 2 =RL. He could not calculate what the rate of energy output of the Sun should be; , this was done 50 years later by Eddington, but given the power output of the Sun, he calculated the rate of change of the radius of the Sun. Assuming that the Sun radiated at a constant rate all the time, Ritter found that the radius of the Sun must have been 215 times its present radius 5,509,864 years ago. As the radius of the Earth orbit around the Sun is 215 times the radius of the present day Sun, this result means that the maximum age of the Earth is about 5 million years. Ritter checked also what happened if he relaxed the assumption of constant power output by the Sun but found only small changes in the results. It is amusing, but Ritter noted correctly, that when the radius of the Sun was 215 times its present value, the Sun as seen from the Earth, occupied half the sky and in mid day, Sun filled all the sky! Ritter ended his calculation of the age of the Sun by writing that he could only give a maximum age of four million years as the original assumption must still be regarded as hypotheses imperfectly satisfied. It is astonishing to note that 25 years after Kelvin discovered the idea of adiabatic run of the temperature in the atmosphere and after Ritter had completed his series of papers on gaseous stars, Kelvin solved the problem of the equilibrium of a gas under its own gravitation only and derived independently most of the results of Thomson (1887). The 1887 paper promised a follow-up paper which appeared more than 20 years later, posthumously (Thomson, 1908) and contained a nice summary of the problem prior to the publication of the monumental work by Emden. The work of Ritter in Europe paralleled in time that of Lane in the US. Lane, Jonathan Homer, (1819–1880) was an American mathematician who served in the US coast survey and from 1869 till 1880 was associated with the bureau of weights and measures. He devoted considerable attention to astronomy and was sent, under the auspices of the coast survey, to survey solar eclipses in many places. As part of an attempt to determine the surface temperature of the Sun, Lane (Lane, 1869), wrote down for the first time the set of equations describing a gas sphere in hydrostatic equilibrium. Lane is frequently credited with constructing the first physical model of the solar interior in particular and stellar interior in general (he did not assume an adiabatic lapse). He supposed that the stars are in hydrostatic equilibrium and supplemented it with the assumption of a polytrope. Interestingly, although his model predicted the central temperature of the Sun reasonably well (when compared to present day known value), his predicted surface temperature of 30,000 K was well off the mark. This is because Lane’s work was carried without the aid of Stefan’s radiation law, which he was unaware of. Instead he relied on the earlier work of Dulong and Petit and of Hopkins for the rate of radiant energy from heated surfaces.

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Back to Europe, in 1902 Robert Emden (1862–1940) published a paper on the structure of the Sun and in 1907 he published the monograph Gaskugeln (Gaseous spheres) (Emden, 1907). Emden knew about Ritter’s and Lane’s papers. The book by Emden ended the era of stellar models without energy transfer. The most famous equation which governs the hydrostatic equilibrium of stars is called the Lane–Emden equation, and it served as the starting point of any theoretical work on stars until the early fifties of the last century when new methods and computers were introduced. Ritter discovered first a particular form of this equation when the temperature lapse is adiabatic. 9. Was solar contraction observed? Once Ritter assumed the Sun and the stars to be gaseous the idea of contraction and expansion of stars became plausible. In 1879 Ritter proposed that the source of the light variation in d Cepheid type stars is periodic changes in the radius. The idea had to wait almost 40 years for Eddington to provide a mechanism and derive the period-mean density law. Thus, if the variable stars derive their energy from contraction, the mean density must increase and the period decrease. So in 1917 (Eddington, 1917), Eddington examined 126 years of measurements of the period of d Cephei and found that the maximum change was found to be 0.106 ± 0.011 seconds per year! The contraction theory would demand a period change of 40 seconds per year. Thus, Eddington found a solid observational proof that the stellar contraction theory as a source of energy was in contradiction with the observations. The earliest documented search for changes in the size of the Sun is probably von Lindenau (von Lindenau, 1809) even before the idea that contraction supplies the energy. von Lindenau used a transit instrument to follow the Sun for over 2 years. The results indicated a periodic change in the diameter of the Sun, in agreement with similar observations carried out in Greenwich about 50 years earlier. On the basis of these results of Lindenau concluded that the Sun is an ellipsoid and hence must be rotating along its longer axis. The radius of the equator was by about 1/280 to 1/140 smaller than polar radius. The results and the conclusions were criticized right away by Bessel (1809) who claimed that the error resulted from periodic changes in the instruments. Piazzi (1831) and Bianchi (1831) repeated the measurements of von Lindenau and got the result that indeed the Sun is an ellipsoid, but the polar radius is smaller than the equatorial by about 1/249. The contradictory results apparently convinced astronomers that the Sun is round. Secchi was mainly interested in sun spots and prominences on the Sun. In 1871 he (Secchi, 1872) approached the problem of the measure of the Sun and discovered a correlation between the number of sun spots and the diameter of the Sun, namely the diameter was maximal when the number of sun spots was maximal. This result was confirmed by observation in Palermo and by Hilfiker (1878).

The results by Secchi, though based on a relatively small number of observations (187 in total and during one year), attracted attention as they came after Helmholtz and Kevin hypothesis. So Auwers (1873) examined thoroughly the observational evidence of Secchi as well as his predecessors, and concluded that the fluctuations in the solar diameter caused these observational errors and there was no foundation to the claim that the Sun’s diameter changes with time. A year later, Newcomb and Holden (1874) reached the conclusion that solar variability with a period of several days can be excluded but short time variability, like hours, cannot be ruled out. This new result activated Auwers once more (Auwers, 1886) and he decided to reduce all the data using an equation which allows for periodic variations as well as a gradual secular change in the radius. This time Auwers’ data reduction indicated that the variations in the number of sun spots was indeed correlated with changes in the solar radius. But Auwers was unhappy with the results and continued to accumulate data from various observers. He discovered that some of the results were periodic and some showed abrupt changes. Moreover, the results of Dunkin showed a secular contraction of 0.006 arc seconds per year while the results of Downing showed an expansion of the sun by 0.01 arc second per year which corresponds to a change of 6  106 in the radius per year. Auwers careful conclusion, after observing 26,000 observations, was that the Sun did not show any long period variation. All results can be attributed to the variations in the temperature of the instrument. The question whether the solar variations are real or not was taken up by Poor, who carried out a very thorough analysis of all possible ways to measure the changes in the Sun (Poor, 1908) and concluded that The exact shape of the Sun is not known. The generally accepted idea that the Sun is a sphere is at least open to question. Practically every series of measures heretofore made show departure from a spherical form; but these departures are extremely minute. There appear to be correlations with Sun’s spots. If the Sun extracts energy from meteors we should witness meteors falling on it. Moreover, it is natural to expect to see meteors in the vicinity of the Sun even if they do not fall on the Sun. Indeed, in 1879 Penrose (Penrose, 1879) in the wake of an unusual eclipse and corona observed reported about meteors moving through the solar corona appeared in the literature. However, the reports on dark objects falling on the surface of the Sun are very scant. Denning examined these reports in 1914 (Denning, 1914) and dismissed them all. In summary, there was no evidence whatsoever that any mass falls onto the Sun. 10. Conclusions In the present day language, Mayer, Helmholtz and Kelvin spoke on accretion onto a liquid sphere. Ritter was the first to consider the gravitational energy from a contracting sphere of gas. Lane also considered spheres of gas but he

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was less interested in the age of the Sun, energy, etc. Kelvin’s calculation was deliberately wrong. It seems, therefore, only fair to immortalize Ritter on the timescale he was the first to calculate correctly. For more expanded exposition (see Shaviv, 2008). Acknowledgements It is my pleasure to thank my collaborator on accretion disks, Jean-Pierre Lasota, for his help in finding practically inaccessible historical papers in French libraries. I wish him many more years of productive research and me more years of collaboration with him. References Auwers, A., 1873. On the alleged variability of the sun’s diameter. MNRAS 34, 22, This paper is a summary by Lynn of the original one published in Berlin: Ueber eine angebliche Vera¨n derlichkeit des Sonnendurchmessers, Monatsberichte of the Royal Academy of Sciences at Berlin, May, 1873. Auwers, A., Neue Untersuchungen u¨ber den Durchmesser der Sonner, I, II, Sitzungsberichte of the Berlin Academy, December 1886 and June 1887. Barus, Carl, 1905. The progress of physics in the nineteenth century. Science 22, 385. Bessel, Zach, 1809. Monatliche Correspondenz. Bianchi, G., 1831. AN, No. 213, 9, p. 365. Carnot, S., 1872. Annal. Sci. de l’Ecole Normale Superier, Ser. 2. Darwin, C., 1859. On the Origin of Species by Natural Selection. Appleton, The fifth edition was published in 1872. Denning, W.F., 1914. The Observatory 37 (10), 417. Eddington, A.S., 1917. The Observatory 40, 290. Emden, R., 1907. Gaskugeln. Teubner, Leipzig. Helmholtz, On the Conservation of Force, Introduction to a Series of Lectures Delivered at Karlsruhe in the Winter of 1862–1863, The Harvard Classics, 1909-14, translated by E. Atkinson. von Helmholtz, H., 1856. On the interaction of natural forces Ko¨nigsberg, February 7, 1954. Philos. Mag. 11 (4), 489. Hilfiker, J., 1878. Ueber die bestimmung der constante der sonnenparallaxe MIT besonderer berucksichtigung der oppositionsbeobachtungen. Buchdruckerei B.F. Haller, Bern. Joule, J.P., 1845. Philos. Mag. 27, 205. Joule, J.P., 1843–1850. On the mechanical equivalent of heat, Abstract of papers communicated to the Roy. Soc. London, 5, 839.

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Thomson, W., 1851. On the dynamical theory of heat, with numerical results deduced from Mr Joule’s equivalent of a thermal unit and M. Regnault’s observations of steam. Trans. Roy. Soc. Edin., and Philos. Mag., IV, 1852. Thomson, W., 1852. On the universal tendency in nature to the dissipation of mechanical energy. Philos. Mag. 4, 256. Thomson, W., 1862. (Lord Kelvin) On the age of the sun’s heat. Macmillan’s Mag. 5, 288, Transaction of the Royal Society of Edinburgh, April, 1864. On the Dynamical Theory of Heat, with numerical results deduced from Mr. Joules equivalent of a, Philos. Mag., 1853; On the secular cooling of the Earth, Philos. Mag., 1863; On the Reduction of Observations of Underground Temperature, Trans. Roy. Soc., Edinburgh, 1860. Thomson, W., 1911. (Lord Kelvin) Mathematical and Physical papers, vol. 3, Cambridge, p. 255. Thomson, W., 1887. Philos. Mag. 22, 287, The paper contains reference to Lane but not to Ritter. Thomson, W., 1908. The Problem of a Spherical Gaseous Nebula, Collected Papers, 5, 254. Lane, J.H., 1869. Am. J. Sci. 2nd Ser. 50, 57. LeVerrier, U.J.J., 1840. Sur les variation se´culaires des e´le´ments elliptiques des sept plan e`t principale:Mercure, Venus, la terre, Mars, Jupiter, Saturn et Uranus. J. Math. Pure. Appl. 4, 220. Ann. Obs. Imp. Paris, 1859. Lloyd, J.T., 1970. Notes and Records of the Royal Society of London, 25, 211. von Lindenau, F.B.A., 1809. In: Zach (Ed.), Monatliche Correspondenz. Mayer, R.J., 1842. Bemerkungen ueber die Kraefte der unbelebten Natur. Ann. Chem. Pharm. 43, 233. Mayer, R.J., 1848a. Comptes Rendus 27, 385. Mayer, R.J., 1848b. Beitrage zur Dynamik des Himmels. Popularer Darstallung, Heilbronn. Mohn, H., Goldberg, C.M., 1878. Uber die Temperatura¨nderung in ¨ sterr. Ges. f. verticaler Richtung in der Atmospha¨re. Ztschr. d.O Meteorologie 8. Newcomb, S., Holden, E.S., 1874. On the possible periodic changes in the sun’s apparent diameter. Am. J. Sci. Art. (October). Penrose, F.C., 1879. The Observatory 2, 302. Piazzi, G., 1831. Specola Astronomica di Palermo, LIV, VI. Poor, C.L., 1908. An investigation of the figure of the sun and of possible variations in its size and shape. Ann. N. Y. Acad. Sci. 18, 385. Ritter, A., 1881. Wiedemann Annalen 14, 610. Stefan, J., 1879. Sber. Math. Naturw. Classe K. Akad. Wiss Wien 79, 391. Secchi, A., 1872. MNRAS 32, 226. Also, Atti del’Accademia del Lincei, January 1872. Shaviv, G., 2008. The synthesis of the chemical elements – a touch of god? Magnes Pub., The Hebrew University Press. Tyndall, J., 1873. Heat a Mode of Motion. Appleton & Co., pp. 488–489. Waterston, J., 1860. MNRAS 20, 198. ¨ ber die stabilita¨t kosmischer Masses. Leipzig. Zoellner, E., 1871. U