THE SOURCE OF STELLAR ENERGY1

A. S. Eddington.

It is usually considered that the gravitational energy released in the contraction of celestial bodies is insufficient to supply the large amount of heat that is given off by the stars in the form of radiation. The Helmholtz–Kelvin hypothesis of contraction leads to values that are too low for the age of the sun and the solar system. A revision of Kelvin’s calculations in the light of the most recent investigations has not introduced any significant change into the figures; according to the theory of contraction, the age of the sun is 46 million years, or 19 million years if reckoned from the time when the temperature of its photosphere reached 3000°. But even these numbers are too high, since in the calculation no account was taken of the energy retained in the sun in the form of ionization energy, which considerably lowers the amount of energy given off as radiation.

Physical and geological methods lead to the conclusion that the age of the earth is much greater—even if the reckoning is not begun from the very start of its existence as a planet. The age of ancient rocks is usually estimated from the ratio of the amounts of uranium and lead at 1200 million years; Prof. Joly gives lower values, but none of them is so small as to save the contraction hypothesis. The evolution of the earth–moon system likewise requires an enlargement of the Kelvin time scale. Astronomy gives us a method based on the study of the variable stars of the constellation Cepheus; if δ Cephei has no other source of energy besides contraction, then one can calculate the rate of change of its density and, consequently, the change in the period of its pulsations. This calculated value of the period has proved to be much smaller than the observed one; evolution, at least in the stage passed through by the variable stars of Cepheus, proceeds at a speed no greater than one hundredth of the speed calculated from the contraction hypothesis. This con—

This argument applies not only to the theory of pulsation, but also to the theories competing with it, which assume that the period of light is closely connected with the interior of the star, regardless of whether this is a period of oscillation or of rotation.

Obviously we need a time scale that would give an age for the Sun of at least \(10^{10}\) years; in any case we cannot lower our requirements below \(10^9\) years. Consequently, we must seek a more abundant source of energy, one that could maintain the temperature of the Sun and the stars over this prolonged period. We can at once narrow the field of our search: only those sources of energy are suitable which liberate heat deep within the star. The difficulty of the problem lies in the fact that we must provide a sufficient quantity of energy not only for radiation from the surface, but also for maintaining the internal heat of the star, which prevents the collapse of its gravitating mass. To maintain the Sun in its present rarefied state, it is necessary that within it there be maintained a temperature gradient from \(6000^\circ\) at the surface to \(40\,000\,000^\circ\) at the center; if this gradient is not maintained, the Sun will begin to contract and to evolve rapidly. It is clear that the temperature gradient cannot be maintained by supplying heat “from the lower end.” If this year the Sun were to encounter a swarm of meteorites that bombarded it with energy sufficient for a year’s radiation, this would not add even one year, nor even one day, to the life of the Sun; the internal regulation of temperature would continue without any change. The only result would be that during that year the Sun would give us twice its usual radiation.

Since we cannot imagine an external source of heat capable of manifesting itself at the center of a star, we must definitively reject the idea that a star draws energy from its path. It follows from this that stars contain within themselves a hidden energy, of which they must have enough for their entire life. But the energy cannot be completely hidden; it must reveal itself, since it possesses mass or itself constitutes mass. An energy of \(9\cdot10^{20}\) ergs has a mass of 1 gram, and this must constitute part of the mass of the star as determined by astronomical methods. An energy of \(1.8\cdot10^{54}\) ergs has a mass of \(2\cdot10^{33}\) g, which is equal to the mass of the Sun; consequently, this is the sum of the energies contained in the Sun. We do not know what part of this energy may be converted into radiation; but if this conversion could occur completely, the amount of energy would be sufficient to maintain the radiation of the Sun at its present rate for 15 trillion—\(15\cdot10^{12}\)—years. In other words, the heat annually radiated by the Sun possesses

with a mass of 120 trillion tons; if this loss of mass continued at the same rate, the entire mass would be expended in 15 trillion years.

SUBATOMIC ENERGY.

This store of energy is kept, with few exceptions, in the form of the structural energy of atoms and electrons, i.e. subatomic energy. If we accept that the Kelvin time scale is insufficient, that our rejection of external sources of energy is justified, and that the generally accepted view of the conservation of mass and energy is correct, then the source of energy must be of subatomic origin, since all the others are excluded. A large part of the store consists of the energy connected with the structure of electrons and protons; it cannot be liberated until they are destroyed. We must suppose that, when a proton collides with an electron, their electric charges mutually neutralize one another, and nothing remains except a flash in the ether, which propagates in the form of an electromagnetic wave carrying away energy. The possibility of such a case has long been discussed; I found the first indication of it in Larmor’s book (Larmor, “Aether and Matter,” 1900), where it is vividly described how a proton and an electron may be obtained if the ether is cut along a tube and the tube is twisted through a certain angle; there, too, the cautious suggestion is made that someday this tube may untwist back again. Another possibility is that a small, but perhaps sufficient, fraction of the store of energy may be liberated by a less catastrophic process—namely, by the transmutation of elements. In the formation of a helium atom from protons and electrons, 0.8% of the mass disappears and, consequently, 0.8% of the energy. If such an evolution of the elements takes place in the stars, the energy liberated in this process may be precisely the source we are seeking.

The conclusion that an appreciable part of the energy of protons and electrons can be liberated without their destruction we owe to Aston’s investigations with the mass spectrograph in 1920. He showed that the deviations of the chemical atomic weights from whole numbers are explained by the fact that the elements are mixtures of isotopes; the weights of individual atoms come very close to whole numbers if the atomic weight of oxygen is taken as \(O = 16\). Only for hydrogen was a small but very significant deviation confirmed: its atomic weight is 1.008, and not exactly unity. This shows that an individual proton in hydrogen has a mass of 1.008; but if it is brought into close proximity with electrons, as in the nucleus of helium or of the higher elements, its mass is reduced to 1.000. The difference undoubtedly represents the usual decrease of electrostatic energy when positive and negative charges are brought closer together; according to ordinary

according to the laws of electromagnetism the disappearing energy (and mass) is carried away in the form of radiation. If helium is formed in the stars, a quantity of radiant energy is released which may be sufficient to maintain their temperature; although mass is radiated away, the total number of protons and electrons may remain the same. In 1920 Prof. Perrin (J. Perrin) and I considered this hypothesis probable, but later reflections raised doubts as to its plausibility.

The further stages in the evolution of the higher elements release, proportionally, a much smaller amount of energy—no more than one percent of the total amount, and even that only if the original material is hydrogen. In most works on the radioactive theory of the origin of stellar energy there was no need to ascribe to the stars any definite chemical constitution, except for the condition that they should not contain hydrogen in an especially large proportion. Hydrogen gives results very different from all the other elements; consequently, strange as it may seem, the fragmentary information we possess concerning the chemical composition of the inner parts of the stars bears on this important question. I do not think that we can admit a content of more than 10% hydrogen even in the youngest stars; hence it follows that the evolution of the elements goes very far already in the pre-stellar stage of the nebula. In that case the transformation of the elements can release no more than one-thousandth part of all the subatomic energy, and the lifetime of the sun, past and future, is calculated at \(1.5 \cdot 10^{10}\) years, i.e. just barely enough. If we wish to obtain a greater duration for its life, we must resort to the hypothesis of the annihilation of protons and electrons. Which, then, of these two possibilities—the transformation or the annihilation of matter—provides the principal reserve of stellar energy?

Until recently the problem of the source of stellar energy had significance only in connection with the scale of time, and there was no pressing necessity to choose between scales of \(10^{10}\) and \(10^{13}\) years. In 1916, when I began the study of the questions set forth in the two preceding lectures, closer contact with this problem seemed inevitable, since it was necessary to make some assumption about the relative distribution of the source of energy in different parts of the star. This distribution could be calculated from Kelvin’s theory; but that theory was already obsolete, and the development of the problem on such a basis was of no interest. Progress became possible only when it was understood that the distribution of the source of energy has an insignificant influence on the results, an influence which can be taken into account only in the second approximation; for this we were not yet ready. At the beginning of 1924 the problem was sharply posed: in opposition to the theory of evolution then accepted, the conclu-

...view that dwarf stars are still in the state of an ideal gas. It became clear that any new theory of stellar evolution must be closely connected with the laws of subatomic energy.

The Problem of Evolution.

It is known that there exists a definite statistical distribution of stars—a curve in the diagram magnitude (luminosity)—spectral type of the star—around which all stars tend to group themselves. The merit of the theory of giant and dwarf stars was that it pointed to this curve as the path of evolution of the average star. Any arbitrary change in the rate of liberation of subatomic energy would cause the star to move to another point on the curve, and the laws of subatomic energy therefore determined only the rate of displacement, i.e. the time scale, but did not determine the path. The new conclusion makes this explanation of the statistical distribution unacceptable. An arbitrary change in the liberation of subatomic energy would force dwarf stars to leave the path of evolution; therefore such arbitrary changes must be excluded, i.e. if we wish to understand the statistical distribution of stars, we must reckon with the laws governing the liberation of subatomic energy.

In particular, the close dependence which we found between the luminosity and the mass of stars excludes the possibility of the evolution of faint stars from bright ones, unless we allow a significant change of mass during the life of a star. Not only in the theory of giant and dwarf stars, but also in the theory that preceded it, evolution from bright stars of types \(B\) and \(A\) to small stars of type \(M\) along the dwarf sequence was a fundamental assumption. To avoid the complete collapse of these ideas, we must allow a substantial decrease of mass and, consequently, the annihilation of matter. From this point of view, which has practical significance for astronomers, the hypothesis of the annihilation of matter is the most conservative; this does not mean that it is necessarily true, but it justifies a quite serious discussion of it.

It is possible that a star may change its mass not only as a result of radiation, but also in other ways; in any case these changes are, in all probability, much smaller. With a sufficient degree of certainty we can set an upper limit to the increase of a star’s mass from the attraction of diffuse matter out of space; it is much less than the loss by radiation. The loss of matter by stars must likewise be much smaller. According to my calculations the chromosphere of the Sun would have to move outward with a constant velocity of \(100\) km per sec in order to carry away as much mass as is carried away from the Sun by radiation. It goes without saying that such a material...

the flow could be detected by the Doppler effect. Therefore I think that the rate of mass loss by a star may be taken as equal to its radiation; consequently, this is a known quantity (independently of our views on subatomic energy), and the whole question is whether the lifetime of a star is long enough for this loss to be a determining factor in stellar evolution. If the lifetime of a star is taken to be the period during which the loss exceeds 1%, protons and electrons must disappear.

I think that most physicists regard the question of subatomic energy as a field for mere speculation. But to astronomers the matter appears otherwise. If we accept that a star evolves much more slowly than the contraction hypothesis assumes, so that the amount of energy emitted by the star is practically equal to the energy liberated within it, the measurement of the yield of subatomic energy is reduced to one of the ordinary astronomical measurements—the measurement of the temperature and brightness of a star. Of course, the astronomer is not satisfied with such measurements alone over an unlimited time; he tries to bring them into some definite relation, to find their dependence on the internal temperature and density, or on the age of the star. If a physicist had in his laboratory unknown sources of energy, the amount of which he could measure and the physical conditions of whose release he could determine, he would probably not show such indecision in discussing the causes and laws of these phenomena. The astronomical study of subatomic energy is no less direct than in this case; although our attempts to arrange the results of observation in a definite order have so far been unsuccessful, the problem itself is no more speculative than any induction from experimental data.

Exhaustion of the Source of Energy.

Let us now consider some difficulties that arise in comparing the results of observation.

The Sun liberates 2 ergs per gram per second, whereas the star Capella liberates 58 ergs per g sec. The density of the Sun is 620 times greater, and the internal temperature 3.7 times higher, than that of Capella.

If density and temperature in general have any effect on the liberation of subatomic energy, they must, of course, accelerate it. In particular, an increase in density brings closer together the participants in any process. Yet the Sun, despite its greater density and temperature, emits less energy than Capella. I think the only explanation of this lies in the fact that

that the Sun is beginning to wear out; its high density and temperature must support an insufficient influx of energy. The Sun is evidently an aging star. Thus we must reckon with an important factor—the source of the influx of energy.

Let us now consider both components of Capella: the star is divided into two parts; according to the law of the dependence of mass on brightness, the heavier component has radiated more ergs per gram than the lighter component. Consequently, it has spent the greater part of its store. However, the star with the greater mass, being more exhausted, at lower density and lower temperature, still releases more ergs per gram than the second star with the smaller mass.

It may be supposed, if the separation of the stars occurred during the last 100,000 years, that the components have not yet reached a stable state; in that case it would be unfair to equate the release of subatomic energy with the observed radiation. But the case of Capella is typical for spectroscopic binary stars1 in general, and they are so numerous that they cannot all be of such recent origin. However, in most eclipsing variable stars the faint component is colder, and by this one can explain their weaker release of energy. In these stars the components are at a very small distance from one another, sometimes almost touching, and they probably formed recently. Precisely those stars for which one could forgive their exceptional behavior (in view of their recent formation) do not require forgiveness.

Jeans (Jeans) suggested that, when a star divides, the heavier component appropriates the central part of the original star, where the heavy elements are concentrated, containing more active sources of energy. Two objections can be raised to this. First, insofar as this can be calculated, diffusion is incapable of producing any significant separation of light elements from heavy ones during the lifetime of a giant star. Second, the rotation of the star causes rotational currents, which sufficiently mix the material of the star. The latter conclusion is based on von Zeipel’s theorem (von Zeipel), which Jeans criticized; but I am convinced that von Zeipel is right. Therefore it is probable that, when a star divides, the initial structure of both components is the same.

As for the second objection, some doubts may arise concerning the adequacy of the mixing: it is possible that the rotational currents are distributed in layers. In a viscous fluid

vortical motion has a tendency to break up into layers, so that a star may divide into two or three spherical layers with good mixing within each layer and poor mixing of the material located in the different layers. Nevertheless I do not think that the general mixing has ceased completely. The diffusion with which it has to contend proceeds extremely slowly; no less than \(10^{13}\) years must pass before the heaviest elements separate from the lightest. The age of Capella at its division could not have been much greater than \(10^{11}\) years. Moreover, although the heavy elements tend to sink toward the center in small stars, in stars of great mass, such as Capella, there is evidently an opposite tendency. According to physical theory, the heaviest atoms have the largest absorption coefficients and experience the greatest radiation pressure. The interaction of the gravitational and electric fields with radiation pressure leads to interesting results: in Capella the heavy elements, together with hydrogen and helium, must flow toward the surface, while the remaining light elements flow toward the center.

The Main Sequence.

Let us now turn to the stars of the main sequence, extending from types \(O\) and \(B\) downward along the former dwarf series to type \(M\). The vast majority of stars belongs to this sequence, so that, in all probability, it corresponds to the greater part of a star’s life. The stage of a giant star is evidently a temporary stopping-place during which a very active but soon exhausted influx of energy is consumed; then the star passes onto the main sequence, where it remains until the chief store of its energy is exhausted. It is in this stage that there occurs (if it is at all possible) the liberation of energy as a consequence of the destruction of protons and electrons, since a star cannot move along the main sequence if it does not undergo a substantial loss of mass. One might have expected that stars would remain in this stage to the end of their lives, since no reason is apparent why protons and electrons should weary of mutual destruction; however, evidently in stars there is, or there forms with time, more resistant material, so that, at least in some stars, there remains an unburned residue, which is then subjected to a more severe trial in the stage of “white dwarf” stars.

In the main sequence we encounter such simplicity in the laws of subatomic energy that it may perhaps present even greater difficulties than the complications considered above. As Russell (H. N. Russell) has pointed out, in this stage all stars have practically the same internal temperature. I take it to be 40 million degrees; Russell adopts a somewhat smaller value.

THE SOURCE OF STELLAR ENERGY

The good constancy of the internal temperature is excellently illustrated by a diagram published by Russell (Nature, August 8, 1925, vol. 116, p. 209) for a considerable number of individual stars. To his remarks on this matter the following may be added. It is difficult to take account of the observational errors for each star and to establish what weight should be assigned to individual discrepancies. We therefore propose the inverse problem: accepting a constant internal temperature for main-sequence stars of \(40\,000\,000^\circ\), we find the relation between luminosity and spectral type, and compare it with the relation given by the general statistics of stars.

From the mass of the star, placed in the first column, and the temperature adopted for the interior of the star, one can derive, by the theory of radiation, the bolometric magnitude and the effective temperature1. In the last column the spectral types are indicated in accordance with the effective temperatures, which seems to me the most generally accepted criterion. Comparing in this table the third and fifth columns, we obtain a relation between magnitude and spectral type which is expressed by a curve coinciding, with great accuracy, with the central line of the main sequence constructed on the basis of statistical data. Without attaching too much significance to the exactness of the agreement, we must nevertheless conclude that there is a remarkable constancy of temperature in this phase of evolution.

Does a star require an inflow of energy of 680 ergs/g per sec, as in Puppis, of 2 ergs/g per sec, as in the Sun, or of 0.08 ergs/g per sec, as in Krueger 60? In order that a star should receive this energy, its internal temperature must be equal to \(40\,000\,000^\circ\). Obviously, at this critical temperature the inflow of energy occurs as needed, ad libitum. Can we suppose that at \(40\,000\,000^\circ\) energy is freely released from matter, just as at \(100^\circ\) steam is released from water? I think that physicists would find it very difficult to reconcile such extraordinary behavior with any generally accepted principle; nevertheless astronomical observations compel us to insist on this conception.

Super-stability.

Here we encounter a new difficulty, depending on “super-stability.” Let us imagine a vertical cylinder with a piston, containing heated gas. In order to increase the resemblance of this model to a star, we must suppose that heat is continuously supplied to the chamber of the cylinder, and that an equal quantity of it escapes outward through the walls. The piston will remain in stable equilibrium on the cushion of heated gas, and, if it is slightly displaced, it will return to its position of equilibrium after a few oscillations. Suppose now that the piston sets in motion a valve admitting an additional quantity of heat into the chamber when the piston moves downward, and cutting off the heat supply when the piston moves upward. If we disregard the heat escaping outward, we shall see that the chamber gains heat on compression and loses it on rarefaction, exactly as does the cylinder of any heat engine. Consequently, a slight displacement of the piston will set the machine in motion, and the piston will begin to move up and down with increasing amplitude. I call the state in which such a “steam engine” finds itself “super-stable.” This is not instability (a tendency to move away from the state of equilibrium), but a tendency to return to equilibrium, manifested with such vigor that the resulting oscillations are amplified instead of being damped. Now we shall see that, if the critical temperature is reached, the oscillations of a star set the valve in motion in exactly the same way: if a star contracts, it becomes hotter; the greater part of it is heated above the critical temperature, and more subatomic energy is released. If it expands, its central part cools below the critical temperature, and the inflow of heat ceases. Consequently, the star begins to behave like a steam engine. If the power of the machine is small, it cannot overcome the dissipative forces, and the star remains stable; but, at least, faint stars,

SOURCE OF STELLAR ENERGY

occupying normally a small region near the center above the critical temperature, will pass into a super-stable state. We know of stars—Cepheid variable stars—which for this reason, or for some other, are super-stable and as a result pulsate; but they are unusual, and they are all either giant stars, or are situated at the beginning of the main sequence; these are not at all the stars that ought to fit this theory.

I have already said that the difficulty presented by super-stability is an objection to too hasty an acceptance of the critical temperature at \(40\,000\,000^\circ\); but, in general, super-stability too is a difficulty. The strait between the Scylla of instability and the Charybdis of super-stability becomes so narrow that it is hard to indicate any law of subatomic energy that would safely pass between them. By way of an attempt, I am prepared to cut the knot of difficulties by introducing one more complication. I suppose that a change of temperature or density does not cause an immediate change in the rate of liberation of energy; there is a lag of several months, or perhaps hundreds of years, so that short-period changes have no effect at all. This could occur if temperature and density affected the rate of formation of self-disintegrating matter, which would then spontaneously begin to release its energy at a rate independent of temperature and density1.

As we have seen, we have already encountered quite a few difficulties. I am bold enough to suppose that we shall somehow extricate ourselves from them, but such a statement is not especially convincing. I think that a good theory ought at some point to have predicted the correct results from the very beginning, and not invariably to make mistakes and then offer its apologies by introducing new complications. The situation looks as though we do not yet have in our hands the right key to a general synthesis of the problem of subatomic energy, and I do not intend to defend any one theory in preference to the others. Here, however, we shall continue to collect the few facts that can be drawn from astronomical observations.

Dependence on Temperature and Density

In considering the stability of a star, we can show that the rate of liberation of energy increases with the growth of temperature or density, or of both simultaneously. Let the rate of liberation of subatomic energy be \(E\), and the rate of radiation from the star be \(L\). In a stationary state \(E=L\). Now suppose that \(E\) falls below \(L\); the star contracts (as also in Kelvin’s hypothesis, which took \(E=0\)). It is usually assumed that \(L\) increases as the radius diminishes, and this further increases the deficit. Further contraction follows, with a further increase of \(L\), and the star gradually collapses. To save the star, we must suppose that the rise of temperature and density during contraction causes an increase in \(E\); \(E\) becomes greater than \(L\), and again returns the star to a state of equilibrium. The threatening instability is not catastrophic, since the collapse of the star would take a time comparable with the Kelvin time scale; but we must, of course, take measures against such a collapse, since we have not accepted the Kelvin time scale.

In Jeans’s theories it is assumed that the liberation of subatomic energy does not depend on temperature and density (by analogy with ordinary radioactivity). This view was also expressed by Nernst. I was obliged to reject it from the very beginning of my investigations on radiative equilibrium, since it does not allow \(E\) and \(L\) to be brought into equilibrium for different stars. In order to change \(E\) by a factor of 3 or 4, colossal changes of density are required, and, as was indicated in the last paragraph, the adjustment of \(L\) is, in all probability, effected in the opposite direction. What is essential is that the adjustment should be produced by a change in \(E\). True, theoretical physics does not agree that stellar temperatures can noticeably alter subatomic processes; but the difficulties advanced by theoretical physics must evidently be overcome. Thus, for example, it was objected that the temperature of the stars is not high enough for the transformation of hydrogen into helium, and thereby a possible source of energy was rejected. But helium exists, and if the critics suppose that the stars are not hot enough for helium to have been able to form there, let them indicate a hotter place.

The assumption that \(E\) does not depend on temperature and density gives a rigid rule into which, apparently, the astronomical data cannot be fitted. It requires that the energy (per gram) liberated and radiated by any star should depend only on the age of the star. It is unclear from what zero point one should begin to reckon the age. In any case, the study of stellar groups of the same age, such as the Pleiades, Hyades, Praesepe, and others, does not confirm this hypothesis.

Evidence for a Decrease in Mass

Let us now consider in greater detail the theory which asserts that the mass of a star undergoes considerable changes in the course of its evolution. As we have already indicated, this theory is based on a hypothesis allowing for the annihilation of electrons and protons, since the transmutation of elements by itself is incapable of supplying an amount of energy sufficient for the accepted duration of a star’s life. Evolution downward along the main sequence becomes impossible unless the possibility of a change in mass is admitted; the star would have had to evolve very rapidly until it reached the main sequence, then remain unchanged for the greater part of its life, and afterward pass into the stage of a white dwarf. This means that evolution would cease to play a significant role in astrophysics. The fact that stars leaving the main sequence (white dwarfs) usually have a smaller mass than stars approaching the main sequence (giants) is an argument in favor of a considerable change of mass at this stage.

The statistics of giant stars also supports this view. If it is assumed that the most diffuse state of a star is the earliest one, then the average mass of a star at present, or soon after its birth, can be found from the statistics of giant stars of types $K$ and $M$. Determining the mass from the luminosity, we find that 90% of these young stars possess masses exceeding the mass of the Sun by 2.4 to 5.5 times; the average mass of stars in general, however, is less than the mass of the Sun. This compels us to suppose that the greater part of the stars have lost a considerable part of their initial mass. This result agrees well with the conclusion drawn in the first lecture, namely that stellar masses correspond to the critical state, where radiation pressure begins to prevail. If the average mass for all stars is taken to be from $1/3$ to one solar mass, we obtain that the radiation pressure ranges from 0.007 to 0.05 of the total pressure. This value is obviously too small. But we ought to have taken the initial masses before any losses occurred; if this is done, then masses exceeding the solar mass by 2.4–5.5 times lead to a relative radiation pressure of 0.17–0.35, which is quite appropriate in order of magnitude.

Hence I conclude that stars are ordinarily born with a mass exceeding the mass of the Sun by not less than a factor of 2. One might object to this that we do not observe diffuse stars of smaller mass, because they condense very rapidly. But why? They do not exhaust their store of energy so rapidly, and could evolve more slowly.

A. S. EDDINGTON

Stars of the Same Age in Star Clusters.

However, the opponents of our view have, perhaps, an equally strong argument, which we shall now try to explain. First of all let us give a table showing the time required for a star to change its mass or brightness by a definite amount. This can be done because \(dM/dt=-L/c^2\), where the mass of the emitted \(L\) ergs is equal to \(L/c^2\) grams. Since \(L\) can be represented as a function of \(M\), on the basis of the law connecting the mass with the brightness of a star, the equation can be integrated1, and \(M\) can be represented as a function of \(t\).

This table is applicable above all to the main sequence, where the internal temperature remains constant, but it may also be used, with sufficient approximation, for giant stars. We see that, however great the initial mass may have been, after a trillion years no mass greater than 2 (mass of the Sun \(=1\)) can remain. Consequently, if a star cluster contains stars with mass greater than 2, i.e. with brightness greater than \(+2^m\), the age of the star cluster cannot exceed a trillion years. Looking at the lower part of the table, however, we see that a trillion years is wholly insufficient for the evolution of the brighter stars. Therefore, if a star cluster contains, among others, stars of the 5th to 7th magnitude, the latter must have been born with the same mass and brightness as they possess at the present time. Many star clusters contain both bright and faint stars at the same time, and our reasoning shows that in this case the faint stars could not have undergone any appreciable evolution. If we must deny the evolution of dwarf stars in clusters, can we admit the evolution of dwarf stars in general? In any case

it gives the impression that we must abandon the thought that stars cannot be born with masses smaller than 2.

Nevertheless, although observations of star clusters seem to contradict our conclusion, we are somewhat encouraged by the fact that they do so, as it were, reluctantly. (Nature so rarely pays attention to our predictions concerning subatomic energy that even a reluctant denial on her part gives us grounds to take pride.) I learned from Hertzsprung that moving star clusters such as, for example, the Hyades, Pleiades and Praesepe contain far fewer faint dwarfs than they are supposed to. In them there is almost not a single star fainter than \(+7^m\). In globular clusters the ratio of the number of dwarfs to giants also seems to be much smaller than is usual in the system of the Milky Way. Therefore, perhaps, in our reasoning about the difficulty of the coexistence of bright and faint stars in systems of the same age there lies a grain of truth, although we have, of course, not yet found the proper formulation for it.

In the system of the Milky Way, where, it is assumed, we are dealing with stars of the most varied ages, the number of stars in each stage should be proportional to the duration of that stage. Consequently, the numbers in the last column of our table ought to have been proportional to the number of stars of the corresponding magnitude. However, this prediction is justified only for masses smaller than 2. The numbers corresponding to the upper part of the table must be considerably reduced, since the greater part of the stars has an initial mass between 2 and 5. The distribution of luminosities predicted in this way is well confirmed by the statistics of observations.

Another test, giving some support to the theory of evolution with loss of mass, was carried out by Vogt. From the table it is easy to see that if the initial masses of two stars are very different, then with the passage of time their masses approach equality. Therefore in binary stars we should expect a large discrepancy between the masses of the two components only in the early stages of evolution; in the later spectral types the mass ratio should approach unity. This seems to be confirmed.

One might perhaps have hoped that the assumption of loss of mass, affecting the dynamics of a binary-star system, would help to solve some hitherto unresolved problems concerning their separation and eccentricity. Such investigations were carried out by Jeans and Smart; the results proved unfavorable, and the new theory has shed no light at all on these stubborn problems. From all the other discussions of this question I also could not extract anything especially favorable or unfavorable for this theory.

A. S. EDDINGTON

The Compton Effect.

As an illustration of the close connection between the successes of pure physics and astronomy, let us note that the discovery of the Compton effect resolved one real difficulty concerning subatomic sources of stellar energy. According to the quantum theory, the radiation obtained in the formation of helium from hydrogen should have a wavelength of 0.00041 angstrom units, while the annihilation of a proton and an electron should give rise to radiation of \(0.000013\) Å. What is the mechanism by which this high-frequency radiation is transformed into the ordinary forms of the star’s thermal energy? The usual absorption coefficients fail, since they are proportional to the cube of the wavelength. The last of the quanta just mentioned would probably pass right through the star—and would go off to the end of the world, finding nothing that could absorb it and turn it into heat. It might be scattered many times, since the coefficient of scattering by electrons, although it decreases with decreasing wavelength, nevertheless does not fall so rapidly. However, so long as it was believed that the wavelength is not changed in scattering, this did not help to solve the difficulty. Now the theory of the Compton effect tells us that the wavelength increases by \(0.024(1-\cos \theta)\) Å in each act of scattering. Thus, however small the initial wavelength may be, the first scattering reduces the radiation to ordinary \(\gamma\)-radiation. A large part of the energy passes to the electron, which recoils with an enormous store of energy; one can without any difficulty imagine how the energy of the radiation degrades into ordinary molecular heat.

Penetrating Radiation.

The question of the origin of the penetrating radiation found in the earth’s atmosphere is of considerable interest in connection with subatomic energy. Several years ago an astronomical aspect was given to this problem by the experiments of Kolhörster, who tried to show that the radiation moves from above and therefore, evidently, comes into our atmosphere from without. He also found that the intensity of the radiation depends on the altitude of the Milky Way, so that the radiation reached a maximum when a large part of the stars was directly above us. The altitude of the sun did not change the radiation, which showed that the sun is not its source. The penetrating power was evidently much greater than that of all known \(\gamma\)-rays, and it seemed clear that this radiation might originate from some powerful subatomic source, such as the transformation of hydrogen or the annihilation of matter. Quite recently Millikan reported the results of an extensive and careful investigation of this radiation. He is firmly convinced that it comes from extraterrestrial sources; he finds that

its penetrating power corresponds to the quantum of energy released in the formation of helium from hydrogen.

Being unfamiliar with the technical details and with the difficulties of the experiments, I cannot take the side of Kolhörster and Millikan or of their opponents. Earlier it seemed to me that the penetrating power was so great that this made it possible to decide the question of the source of the radiation. It pointed to such a concentration of energy as could be obtained only in powerful subatomic processes; it seemed, at any rate, less sensational to admit them in the depths of space than on our planet. But I was somewhat shaken by Wilson’s calculations (C. T. R. Wilson), concerning electrons torn loose during a thunderstorm. He showed that these electrons can acquire energy sufficient to produce such radiation without any subatomic processes. In this way, however, it is difficult to explain the downward direction of the penetrating radiation.

If physicists in the end agree on the extra-terrestrial origin of this radiation, what significance can this have for the problems we are considering? The chief conclusion that can be drawn from this is that subatomic processes can occur at comparatively low temperatures, not reaching 100,000°. All truly hot matter in the universe is shut in behind walls thick enough to stop and scatter the most penetrating radiation; the source of the radiation cannot reach us. The choice must be made between the photosphere of stars and the matter of nebulae in space. The latter source is more probable. If stars are not more powerful sources of energy than the sun, penetrating radiation from the stars must stand in approximately the same relation to the penetrating radiation from the sun as starlight stands to sunlight; experiments show that no appreciable fraction of penetrating radiation comes from the sun, and still more negligible must be the radiation from the stars. Diffuse and dark nebulae, and matter scattered in interstellar space, must possess a mass approximately equal to the total mass of the stars; moreover, there is no screening here, so that all the radiation issuing from them is able to reach us.

If this is true, we must draw one more important conclusion: subatomic processes can occur at extremely low density. From the astronomer’s point of view, I must welcome these conclusions. It would be very difficult for us to explain the presence of helium and other elements in nebulae, and of heavier elements in the spectra of young stars, if the higher elements were not already formed at the pre-stellar stage, at low temperature and density. However, the physicist, at these words, will perhaps shake his head. How can protons and electrons meet and annihilate one another in a medium so...

is rarefied enough for the mean free path to take several years? How do 4 protons and 2 electrons combine and form a helium nucleus? It may well be better that the latter case is so improbable under all conditions of temperature and density that we are left to suppose its possibility in nebulae. Let us also recall that these conclusions are supported by observations (so difficult to bring into connection with theory), namely, that diffuse stars of low temperature are in general especially abundant in subatomic energy.

I should like to end this course of lectures by bringing them to some effective conclusion. But perhaps a more modest ending is more in keeping with the conditions of scientific progress—one that casts a faint ray into the darkness marking the limits of present knowledge. I do not apologize for the weakness of my conclusion, since it is not a conclusion. I should like to feel confident that it is at least a beginning.

1. Monthly Notices, Roy. Astr. Soc. 84, p. 328. ↩↩↩↩↩