INTERNAL STRUCTURE OF STARS1

A. Eddington.

On December 13, 1920, the angular diameter of a star was measured for the first time with the aid of an apparatus invented by Prof. A. A. Michelson.

Until then all stars had seemed only bright points, and there had been no way to distinguish a star from a geometrical point. But on that memorable evening the 20-foot interferometer, built at the Mount Wilson Observatory, was directed at the star Betelgeuse, and the measurement showed that this star has a disk with a diameter of \( \frac{1}{20} \) of an arc second—this corresponds to the apparent size of a two-kopeck coin placed at a distance of 80 km. We know the approximate distance of Betelgeuse (unfortunately, it cannot be determined with the same accuracy as the distances of many other stars), so that from this apparent size we can approximately calculate the true dimensions of the star. The diameter of Betelgeuse is no less than 300 million kilometers. The Earth’s orbit would fit entirely inside it.

Thus, among the stars there are not only individuals of comparatively small size, like the Sun—some stars are truly giants in comparison with the Sun. We may add one more step to the astronomical multiplication table—a million Earths make one Sun, ten million Suns make one Betelgeuse. This comparison refers to volumes, not to the quantity of matter. For the moment we leave open the question whether, in order to obtain such a giant, one must take the material of ten million Suns and unite it into one whole, or whether one must take the material of the Sun and divide it ten million times. Undoubtedly, the latter answer is closer to the truth. I allow that Betelgeuse contains more substance than the Sun (perhaps 50 times more), but its enormous dimensions are due chiefly to the rarefied state of

of this substance. It resembles an enormous balloon of low density, considerably less than the density of air, whereas the material of the Sun is compressed to a density greater than that of water.

Whether a star belongs to the number of these balloon-like bodies, or whether its density is like that of the Sun, depends on the stage of its development. It is natural to think that stars gradually condense out of rarefied matter, so that they become denser and denser as the history of their lives unfolds. At the present time we can observe in the sky examples of stars at any stage of development. Most of the stars visible to the naked eye are in an early, rarefied state; this is explained not by the fact that young stars are more numerous, but by the fact that their large dimensions make them brighter and more conspicuous. What I wish to tell about the interiors of stars concerns chiefly young, rarefied stars—the so-called giant stars. The reason for this is that the properties of matter in the state of an ideal gas are known to us much better than its properties in the condensed state. Although the difficulties encountered in considering dense stars like the Sun are not insurmountable, it is natural that the greatest results have been achieved in the easier problem of the giant stars.

Internal Temperatures.

We observe physical conditions only on the surfaces of stars, and at first sight it may seem impossible to learn anything about conditions inside them. Let us consider, for example, the question of temperature. The nature of the light we receive from Betelgeuse indicates a temperature of \(3{,}000^\circ\) C,—a temperature not excessively high even in comparison with our terrestrial experience. But this, of course, refers to the layer adjoining the surface, from which the light we observe proceeds; it is the temperature of the surface of a furnace, giving us no possibility of estimating the monstrous temperature inside it.

I shall not try to explain in detail how we manage to calculate the internal temperature, but I may perhaps be able to show that there is a path along which one can proceed with the aid of suitable mathematical methods.

Elasticity is a well-known property of gases—everyone is acquainted with it through its practical application in the pneumatic brake. The elasticity, or pressure, of a gas is caused by its heat, in other words by the energy of motion of its molecules, moving in all directions and ceaselessly tending to scatter.

The greater the heat, the greater the pressure. But at every point inside a star a certain state of equilibrium is reached; on the one hand, we have the weight of all the upper layers, which press downward and

seek to compress the gas inside the star; on the other hand, we have the elasticity of this internal gas, which tends to expand and push the upper layers outward. In view of the fact that neither one nor the other occurs, and the star remains practically unchanged for hundreds of years, we must conclude that these two tendencies precisely balance each other.

At every point the elasticity, and hence also the heat, must be exactly of the magnitude required to support the weight of the upper layers. This is the principal guiding thread that we use to determine the amount of heat at various depths inside the star.

The internal temperature varies among different stars, but usually at the center it reaches from 2 to 20 million degrees. Do not think that this is so high a temperature that ordinary conceptions are inapplicable to it. This temperature must be understood literally. Temperature is a way of describing the speed of motion of the smallest, elementary particles of matter. At ordinary temperatures the average speed of helium atoms is approximately \(1 \tfrac{1}{2}\) km per second; at 4 million degrees it is equal to 150 km per second. This speed is great, but not so great as to alarm us. Sir Ernest Rutherford describes helium atoms moving at a speed of 150,000 km per second. I cannot compete with him. My physicist colleagues are usually disappointed by the unhurried motion of atoms in the stars.

Material and Ether Heat.

We must imagine a typical giant star as a mass of matter whose mean density corresponds approximately to the density of air and whose dimensions are at least a thousand times greater than the Sun’s. The atoms of which it consists rush in all directions at speeds of up to 150 km per second, continually colliding and changing their paths. Each atom is continuously drawn downward by the attraction of the entire mass and just as continuously pushed back out by collisions with the atoms below. The energy of this atomic motion constitutes a large store of heat, but it is only part of the total store contained in the star. The star contains another store of heat of a different kind—ether heat, or ether waves, similar to those that carry solar heat across 150 million kilometers of empty space. These waves also rush in all directions inside the star. They are confined as though in a cage formed by matter, which allows them only slowly to seep out into outer space. An ether wave striving for freedom is caught and absorbed by an atom, then emitted by it in a new direction and passes from atom to atom. For hundreds of years it may wander in this labyrinth, until by chance it finds itself...

on the surface of the star. Once liberated, it may travel for an indefinitely long time through space, until at last it reaches some distant world, perhaps enters the eye of an astronomer, and tells him that the star is shining.

The possession of this double store of heat is a peculiarity not found in any hot body among those more nearly known to us. This is a new phase of matter, unattainable in laboratory experiments, although, fortunately, its theory is so simple that there can be no substantial doubt as to its properties. True, in addition to the heat consisting in the motion of molecules, a red-hot piece of iron also contains a little of this ethereal heat, but it amounts to less than a billionth part of the total quantity. The ethereal part becomes significant only in giant stars. Red-hot metal radiates ethereal heat, but does not contain any appreciable store of it; it converts material heat into it as required in this form of heat. The star dispenses with this mode of working without reserve; and, although it continuously converts stores of heat from one form into the other, it nevertheless always keeps ready a reserve for a thousand years and radiates only the leakage of this reserve of ethereal heat. Former theories did not take this circumstance into account; it was assumed that there exist continuous convection currents carrying hot matter from the inner regions, which replaces the radiated and cooled matter at the surface. At the present time it has become clear that the difficulty lies, perhaps, in another direction—by what means does the star retain its store of ether waves, why do they not slip out of it faster than we actually observe? This change of view has made it necessary to modify the earlier theories of Lane and others, and in general has considerably simplified the problem.

In hot bodies with which we deal in laboratories, heat is found almost wholly in material form, while its ethereal part is quite insignificant; in giant stars heat is divided between its two forms in approximately equal quantities.

Can we imagine a third state, in which the heat would be almost wholly ethereal, and its material part insignificant? We can imagine it, without doubt; but it is interesting, and I think noteworthy, that we do not encounter this state in nature.

Pressure of Light

You have heard of the pressure of light—of the fact that light really possesses mass and weight and momentum (quantity of motion), and exerts a certain very slight pressure upon any object that blocks its path. Rays of light, or waves of ether, are like the wind and yet usually

extremely weak wind; but the intense ether energy inside the star causes a strong wind. This wind inflates the star. It supports, to a certain degree, the weight of the upper layers and thereby reduces the share of gravity supported by the elasticity of the gases. Of course, this circumstance must be taken into account in our calculations of internal temperatures—which will lower them in comparison with the assumptions of the old theory. Just as ether and matter share thermal energy between them, so the ether wind and the elasticity of matter share the weight of the upper layers. We have the possibility of calculating in what ratio they distribute this weight between them. In a first approximation, one and the same ratio holds throughout the entire interior of the star and depends only on its total mass, but does not depend either on the density or even on the chemical composition of the substance. Moreover, in order to carry out this calculation, no astronomical knowledge is needed; all the constants entering into the formula have been determined by the physicist in his laboratory. We need to know the molecular weight of the substance, but I shall tell you later how it can be determined, despite our ignorance of the elements that may be in the interior of the star; this is one of the advantages of the fact that we are dealing with very high temperatures.

Let us imagine a physicist on a planet whose sky is covered with clouds, and let this physicist, who has never heard of the stars, begin to carry out these calculations for gas spheres of various sizes. Let him begin with a sphere containing \(10\) g, then \(100\) g, \(1{,}000\) g, and so on, so that the \(n\)-th sphere contains \(10^n\) g. They will rapidly increase in size: No. 1 has approximately the weight of a letter, No. 5—of a man, No. 8—of an airship, No. 10—of an ocean steamer; for the later numbers it is difficult to find comparisons. The following table gives part of our physicist’s results:

It is clear why I omit the remaining part of the table: it consists of long rows of zeros and nines. Only for the 33rd, 34th, and 35th sphere does the table become interesting; then it again begins to be filled with zeros and nines. The struggle between ether and matter for predominant influence upon the state of the spheres is too unequal to be interesting; only from number 33 to number 35 can anything more curious be expected.

Let us now draw aside the veil of clouds that enveloped our physicist, and allow him to look at the sky. He will find there thousands of millions of gaseous spheres, and the mass of each of them lies between the masses of the 33rd and 35th sphere of our table. The lightest of the stars known to us is slightly lighter than the 33rd sphere, the heaviest—slightly greater than the 35th. The overwhelming majority lie between numbers 33 and 35, precisely where ether pressure begins to become an important factor in the situation.

It is highly remarkable that the matter of the universe originally condensed into units of approximately constant mass. The stars differ very considerably from one another in their brightness, density, temperature, etc., but they all contain approximately the same quantity of matter. With few exceptions, their mass is contained between one-half and five times the mass of the Sun. I think that we can no longer seriously doubt the basic cause of this fact, although the details of the explanation may present difficulties.

Gravitation is a force that condenses matter; if it encountered no opposition, it would draw together ever greater and greater quantities of matter and would create spheres of monstrous dimensions. On the contrary, ether pressure is the principal separating force (it is assisted, of course, by the centrifugal force of stellar rotation); its function is to hinder the assemblage of large masses. But, as we have seen, the opposition becomes serious only when the mass has already reached the magnitude of the 33rd sphere. If this opposition really manifests itself, then it will put an end to any further increase of mass before it has become equal to the 35th sphere, for by that time ether pressure has in fact completely reduced to nothing its more passive partner (material pressure). We need not know exactly how great the opposition must be in order to prevent an increase of mass, for the opposition, once it has become perceptible, grows very rapidly and quickly reaches any required magnitude. Throughout the whole universe, the stars testify to us that the growth of masses caused by gravitation continues precisely up to the moment when it is halted by the appearance of sufficiently powerful opposing forces.

Stages of Increasing and Decreasing Temperature.

In 1870, Homer Lane showed that, as a gaseous star contracts, its temperature rises. Betelgeuse is a typical representative of the first stage, when the temperature has already risen just enough for the star to begin shining. The star will continue to contract and to become hot; its light will change from red to yellow, and then to white. But this, evidently, cannot continue without end. When condensation has advanced sufficiently far, the matter will become too dense to obey the laws of ideal gases. Then other laws begin to prevail. The rise in temperature becomes less rapid, stops, and, finally, the temperature falls. It can be calculated that the greatest temperature is reached at a density approximately equal to from \(1/4\) to \(1/3\) of the density of water. The Sun is denser than water; consequently, it has already passed the summit and is in the stage of falling temperature. So long as the fall of temperature continues, the brightness of the star changes hardly at all. Calculations show that the increase in the amount of light and heat radiated per square meter of surface and the decrease of the surface area compensate one another, so that the total radiation remains almost constant. But on the descending part of the path, both the fall of temperature and the decrease of surface diminish the amount of light, which rapidly decreases between the successive stages, or types, into which we divide the history of a star. This is entirely consistent with observations.

A star passes through any temperature level twice—once on the ascending path, and a second time on the descending path. We are accustomed to classifying stars chiefly according to their surface temperature, because the spectral character of the light, its color, and the chemical composition revealed in it depend predominantly upon that temperature. But this classification mixes together stars of an early, ascending stage with stars of a later, descending stage. For example, a star like Betelgeuse, only beginning its development, is assigned to the same class as a dense red star that has traversed its course and attained a second youth. Both are red stars of low temperature, and this was sufficient for the first attempts at classification. Sir Norman Lockyer always energetically maintained the existence of ascending and descending series; but in this he was almost alone among spectroscopists. He did not succeed in actually establishing the distinction between ascending and descending stars, although at times he was very close to the correct criterion. We owe this distinction to Russell and Hertzsprung. They discovered it not by means of spectroscopy, but thanks to measurements of the absolute brightness of stars: great brightness

ascending stars, due to their large sizes, sharply distinguishes them from descending stars, at least in the groups with low temperature. At the highest temperatures the two series merge with one another.

The distinction, the disentangling of these two series and the discovery of the true sequence of stellar evolution, is probably the most revolutionary and fraught with consequences among the new discoveries of stellar physics. It began to displace the former views around 1914. It is noteworthy that this discovery was made on the basis of observations belonging to the domain of classical astronomy, and not to what is usually called astrophysics. The data for it were parallaxes, proper motions, orbits of binary stars, etc.

The spectroscopists erred on the question of the sequence of evolution, and the indication of the correct path fell to the share of the branch of astronomy competing with them. But the spectroscopists did not remain embarrassed for long. Adams and Kohlschütter found an easy spectroscopic method for distinguishing ascending and descending stars. Although our principal present task consists in the investigation of the interior of a star, we may perhaps nevertheless rise for a moment to the surface, in order to consider that difference in surface conditions on rarefied and on condensed stars which enables the spectroscopist to distinguish them from one another.

Surface Conditions.

The state of the outer layers of a star is influenced, apparently, by only two factors: 1) the intensity of the stream of radiant energy crossing these layers, and 2) the intensity of the gravitational attraction holding them on the star.

The first factor is determined by the effective temperature, so that we have two variable factors—temperature and gravity. The spectrum probably changes in accordance with changes in the conditions determined by these factors. We must not expect that it will be possible to arrange the spectra exactly in a single sequence; they may vary in two directions.

The usual classification depends predominantly on the temperature factor; we may call it the longitudinal sequence. Adams’s new method aims chiefly to disentangle the transverse sequence, corresponding predominantly to the factor of gravity. It may be said that his method is, properly speaking, a means that makes it possible to determine the value of gravity at the surface of a star, although as yet there is no possibility of expressing these values in numbers. It is clear that, owing to the greater distance of the surfac-

distance from the center, gravity will be weaker in the rarefied stage than in the dense stage.

The decrease in gravity is associated with a decrease in density at the corresponding temperatures. This causes a considerable change in the state of the gas, namely its ionization. At moderately high temperatures atoms begin to lose one or several of their more weakly bound electrons; this process is called ionization. Low density facilitates ionization; high density hinders it. The theory of ionization in stellar atmospheres was developed chiefly by Megh Nad Saha, who obtained many interesting results. Here it is sufficient for us to note only that ionized atoms excite special spectra which for a long time were considered distinct from the spectra of neutral atoms. The low density of the atmosphere of rarefied stars should strengthen the “spark” (enhanced) lines belonging to ionized atoms in comparison with the “arc” lines caused by neutral atoms. Generally speaking, the difference is not especially great, but the atoms of certain elements, for which the conditions are of a critical character, are particularly sensitive to changes in density. This is precisely the criterion which Adams and Kohlschütter found empirically, and by means of which it is very easy to distinguish the ascending and descending series. To a limited extent it also makes it possible to distinguish the larger and smaller stars of one and the same series.

Although stars begin to shine after reaching a temperature of about 3,000°, and return to this temperature at the end of their luminous state, not all of them climb the ladder of temperatures to one and the same height. The more massive stars climb higher than the lighter ones. To a certain degree we can calculate the height to which they will rise; but I fear that at present the figures are very inaccurate, although there is hope of correcting them in the near future. The surface temperature of the Sun at present is about 5,900°; I do not think that it was ever hotter than 6,600°, for it does not possess sufficient mass to rise farther. Sirius, 2½ times more massive than the Sun, has reached 11,000° and at the present moment is practically at its maximum, having only just had time to turn back. Still hotter stars are known, such as the star Rigel—they must be still more massive.

At the other end of the scale, a star with a mass less than \(1/7\) of the Sun’s mass would not be able to reach 3,000° and would hardly be able to shine. In any case, such small masses must form very rarely, the reasons for which have already been indicated earlier in this lecture. The well-known fact that hot stars are on average more massive than colder stars is, we see, explained by the elimination of the smaller stars as the temperature level rises.

Atoms and Electrons.

Up to now we have described the interior of the star as a whirl of atoms and waves of ether. Now we must introduce yet a third kind of inhabitant joining in the dance. The star contains an enormous number of free electrons—unbound units of negative electricity. Electrons, more numerous than atoms, rush about with a speed a hundred times greater—corresponding to their small mass, which is only \(1/1850\) of the mass of the hydrogen atom. These electrons originate from atoms, being torn out of them at the high temperatures that occur here. The atom is compared with a miniature solar system: a complex central nucleus, bearing a positive charge, corresponds to the Sun, and around it, at comparatively large distances, a number of electrons, corresponding to planets, move in circular and elliptical orbits. For each element we know the number of satellite-electrons: sodium has 11, iron 26, tin 50, uranium 92. Our own solar system with 8 planets corresponds to an oxygen atom. The thermodynamic theory, for which we are chiefly indebted to Nernst, makes it possible to calculate approximately how many of these satellites are detached from the atom at a given temperature and density; it turns out that in a typical star a large fraction of them must be in the free state.

This condition brings with it the solution of our principal difficulty concerning the molecular weight of the substance of a star. In order to perform calculations concerning the state of a star, we must know this weight; but at first sight it may seem hopeless to determine the molecular weight without knowing which elements make up the principal mass of the substance. But suppose first that the temperature is so high that all the satellite-electrons have been torn out of the atom. A sodium atom would then decompose into 12 particles, namely into 11 electrons and 1 radiated atom; its atomic weight 23 would be divided among 12 independent particles, so that the mean atomic weight of each of them would be \(23/12 = 1.92\). Take next iron; its atomic weight 56 is divided among 27 particles, the mean being 2.07. For tin we shall have 119 divided by 51; the mean is 2.34. For uranium, 238 divided by 93 gives a mean of 2.56. The choice of element is immaterial; the mean weight of the final particles (which, of course, we take as the molecular weight) always lies near 2. If the stars were a little hotter than they actually are, our problem would be extraordinarily simplified. Unfortunately, they are not hot enough to cause complete splitting, and the stage of splitting depends on the actual temperature of the star, which introduces a considerable complication. Usually at least half of the electrons are detached, and the molecular weight must be considered to lie

between 3 and 4. I hope that the theory of this dissociation of electrons will be improved, for at present it is the chief obstacle to the rapid development of the theory of the structure of stars. It is a great relief to know that the molecular weight lies between 3 and 4; but we have reached a stage at which, for further progress, it is necessary to know the molecular weight for each star with much greater precision.

Brightness and Mass.

We have described a physicist on a cloud-veiled planet who, from laboratory data, was able to predict the magnitude of those masses into which the matter of the universe must gather. Let us now set him a more difficult task. We tell him that we have observed these gaseous masses and, choosing one of them, equal, let us suppose, to his 35th sphere, we ask him to predict how brightly it will shine. I have already mentioned that a star maintains a practically unchanged brightness throughout the time in which it is an ideal gas of increasing temperature; consequently, the physicist need be given no data apart from the exact mass. Resorting to the former method, we shall imagine a series of lamps with brightnesses of 10 candles, 100 candles, 1,000, and so on. The physicist’s task is to indicate that lamp in this series which approximately corresponds to the star. I believe that at present he has the possibility of accomplishing his task and indicating (correctly) the 31st lamp. But for this purpose it is not enough for him to know only everything concerning the heat stored in the interior of the star, for the brightness of a star depends on the rate of escape of the ether waves. This brings us to a new subject—the restraining force of the atoms of matter, which hold back the radiant flux.

Another name for this restraining force is opacity. A substance that strongly impedes the passage of light and heat waves is called opaque. The rise of temperature toward the center of a star causes heat to flow toward the surface, to lower temperature levels; the opacity of matter hinders this flow. The struggle of these two factors determines the amount of escaping light and heat. We have calculated the internal distribution of temperatures, so that we know everything concerning the first factor; since, in addition, we can determine the magnitude of the external flux, this should reveal the value of the second factor—opacity. The external flux can be observed because it is the light and heat radiated by the star.

One of the difficulties of astronomy is the circumstance that our information about the stars is of a very fragmentary character.

For one star we know its mass very accurately, but do not know its absolute brightness; for another we know the brightness, but do not know the mass; for a third we can determine the density accurately, but nothing more. For Sirius, Procyon, and α Centauri our information is fairly complete and accurate, but not one of them is a giant star in the state of an ideal gas, and therefore they are useless for our present considerations. However, during the past year we have been fortunate enough to obtain complete and very accurate information about one giant star, Capella. This is yet another result, beneficial to astronomy, of Prof. Michelson’s interferometric method. The mass of the brighter component of Capella (which is a binary star) is 4.2 times greater than the mass of the Sun; its brightness is 160 times greater than the solar one. These data we can use, in the way I have described, to determine the opacity, the coefficient of absorption of Capella; it turns out to be equal to 150 units in the C.G.S. system. To illustrate the significance of this figure, let us enter Capella and find a region in which the density is equal to the density of the terrestrial atmosphere familiar to us; a layer of this gas, only 6 inches thick, constitutes an almost opaque screen. Only \(1/20\) of the radiant energy incident on one of its sides will pass through to the opposite side; all the rest will be absorbed by the gas.

Absorption of X-rays in stars.

At first it seems astonishing that 6 inches of gas can so successfully bar the way to ether waves, but such a phenomenon could have been expected on the basis of the general data of physics. We give different names to ether waves according to the length of their wave. The longest of them are Hertzian waves, which are used in radiotelegraphy; then come the invisible heat waves, then light waves, then photographic, or ultraviolet, waves. After them come the Roentgen, or X-rays, and, finally, the shortest, \(\gamma\)-rays, emitted by radioactive substances.

Where in this series should the ether waves of the interior of a star be placed? This depends exclusively on the temperature, and at stellar temperatures ether waves are X-rays—more precisely, they are very “soft” X-rays. But X-rays, and especially soft X-rays, are strongly absorbed by all bodies. The opacity we have found for Capella is of the same order of magnitude as the laboratory-measured opacity of terrestrial bodies with respect to X-rays. The following table gives some of the laboratory results in comparison with the astronomical value for Capella:

THE INTERNAL STRUCTURE OF STARS

We investigated the absorption of X-rays in a star in parallel with the laboratory investigation of the same question. In one respect the physicist possesses a great advantage—he can change the material with which he experiments, whereas we must confine ourselves to the material composing the star, whatever it may be. But, as you see from the table, the physicist is also interested in how absorption changes for different wavelengths. In this question we can follow him and even outstrip him, for certain practical difficulties restrict the physicist to a narrow region of waves, whereas we can investigate a region in which the wavelength varies by at least a ratio of 10 to 1, using stars of different temperatures for this purpose. True, our results are still not very precise: we have only one star, Capella, for which a really good determination is possible, but for other stars approximate values can be determined. Terrestrial results indicate an extraordinarily rapid change of absorption with small changes of wavelength (as is evident from the table); on the contrary, astronomical results lead to an almost constant absorption coefficient. We cannot yet definitely establish whether it increases or decreases with increasing wavelength; in any case there is nothing here resembling the rapid changes given in the preceding table. This profound discrepancy between the astronomical and laboratory results compels us to investigate more attentively the theory of absorption in a star. We shall see that there is a fundamental reason for this discrepancy.

We have taken advantage, in comparison with our physicist, enveloped in clouds, of the fact that we have first taken a look at a real star. We are not going to allow him to do the same. To determine absorption he must not use astronomical observations; he must be able to predict the astronomical

value, whether proceeding from pure theory or from terrestrial experiments. This investigation is especially interesting because it at once brings us to those problems which at the present time most captivate physicists. We began with the investigation of the interior of a star; in the very near future we shall find ourselves in the interior of the atom.

At the present time it is recognized that, when ether waves fall upon an atom, they are not absorbed by it continuously. The atom calmly awaits a convenient opportunity and then suddenly swallows a whole portion at once. The waves are bound into packets, which are called quanta, and the atom is left to choose only between two possibilities—to absorb either the whole portion or nothing. Usually the portion is too large for the atom to be able to digest it, but the atom does not stop to discuss this question—it falls victim to its own greed; in short, it bursts. One of its satellite electrons leaps out with great speed, carrying with it the excess energy which the atom was unable to retain. Atoms could not go on bursting continuously for an indefinitely long time if the opposite process of restoration did not take place. The electrons they have cast out move about, encounter other atoms; after a while the burst atom meets, under suitable conditions, a free electron and compels it to remain and fill the gap. Now the atom is restored and ready for the next portion, as soon as the opportunity presents itself.

As a consequence of this, there is a great difference between the absorption of X-rays in the laboratory and in a star. In our laboratories the atoms are very meagerly fed; only in small quantities can we prepare beams of X-rays which constitute their food. Long before the atoms are given a second chance to eat, they have already been restored and are ready for it. But in the stars the intensity of the X-rays is monstrous; the atoms are overfed and cannot take advantage of the numerous convenient opportunities. The consumption of food by a hungry hunter is limited by his dexterity; the consumption of a well-to-do gentleman is limited by the capacity of his stomach. Laboratory experiments determine the atom’s dexterity in obtaining nourishment; stellar experiments determine the time needed by the atom to recover after taking food and to be ready for the next. That is why, in these two cases, absorption obeys different laws.

Electron Capture

In order to predict the stellar coefficient of absorption, we must focus our attention on the rate of restoration of the burst atoms. The atom wanders everywhere, announcing a free vacancy for an electron, while the numerous free electrons idly but ...

THE INTERNAL STRUCTURE OF STARS

are revolving around. Many of the electrons will draw near, inspect the position presenting itself, and move away again. How, then, can the atom entice an electron to itself? I shall indicate to you a solution of this question which I am inclined to regard as comparatively probable, although few agree with me. We may compare the electron with a wandering comet entering the solar system from outside; here, however, we must bear in mind that the planets (the satellite electrons) repel the newcomer, while the Sun (the positive nucleus) attracts it. Dynamics teaches us that, if no actual collision takes place, the newcomer is hardly ever captured; rather, after introducing a certain disorder, it again goes off to infinity. There are exceptions, for example when the Sun and Jupiter combine their forces in order to capture a comet; but these exceptions will be very rare under the conditions corresponding to the atom. In some cases the newcomer itself will carry off one of the planets, thus compensating itself for the cases when it is itself captured.

As regards the restoration of the atom, then, probably, on the average result the benefit from the invasion is equal to the harm.

Since the more delicate means of persuasion do not help, it would seem that nothing remains for the atom but to secure electrons for itself by means of brute force. Therefore I think that the capture of an electron usually occurs because the electron rushes into the positive nucleus of the atom. This nucleus has an exceedingly complex structure; thus, for example, iron consists of 86 separate charges, in some state of equilibrium. If an electron happens to fly at full speed into this close crowd, it will disturb it and thereby lose its energy; of course, it will rebound again, but with a lower velocity, insufficient for escaping from the atom’s sphere of attraction1.

We arrive at the conclusion that this is the only method, in accordance with the laws of dynamics, by means of which the atom can take possession of the electrons necessary for its restoration. Therefore I have come to the conclusion that the trap for electrons is in reality nothing other than the positive nucleus—the central region of the atom, which, as is known, has a radius of about \(10^{-12}\) cm. It must be remembered that the nucleus attracts electrons and will draw many into the trap

such electrons, which initially were not directed at all by its initial energy.

This theory has been criticized chiefly on the grounds that it is in full agreement with the laws of dynamics. At first glance this may not seem a serious objection, but we have become so accustomed to the fact that the behavior of atoms violates the classical laws that any theory which does not violate these laws may be met with distrust. While admitting that in the mysterious region inside the atom there is room for unknown possibilities, we must nevertheless note that the present problem belongs to that class of investigations in which physicists apply the ordinary laws of dynamics and, moreover, often with great success. It considers the motion of a free electron, not yet incorporated into any permanent quantized system. This problem occurs in the theory of the electrical conductivity of metals, in thermionic phenomena, and in the scattering of α- and β-particles. In these problems physicists usually assume (rightly or not) that the classical laws of dynamics are fulfilled, and we have merely followed their example (for better or worse). In particular, in Rutherford’s experiments on scattering, the classical laws of the action of forces proved valid almost right down to the very surface of the nucleus. Apparently there are sufficiently well-founded reasons in favor of approaching our stellar problem in the same way, although I agree that unforeseen circumstances may interfere with the matter1.

A substantial argument in our favor is the circumstance that this theory in fact gives for the absorption coefficient a value in agreement with astronomical observations. Thus, for Capella the calculated value is 110, whereas the observed value is 150. There are some doubtful factors which may alter the result by a factor of 2, and perhaps even by a factor of 3; we do not attach importance to exact agreement. But on the basis of this hypothesis it proves possible to predict, with an accuracy of one stellar magnitude, the brightness of those stars whose mass is known, such as, for example, Capella. This completely solves the problem posed to us by physics on a cloud-enveloped planet. It may be added that this theory also gives an explanation of the fact that in giant stars absorption is almost not

depends on the wavelength; but this is the result of something more elementary, which becomes obvious as soon as we realize that our problem is connected with the question of the rapidity with which atoms recover; many other theories of the conditions of recovery would also lead to the same conclusion.

The Source of Stellar Energy.

The store of ether heat and the store of material heat may be compared with the accumulators of a power station. Until we have discovered the dynamo machines, the accumulators would be exhausted in a few millennia, if they were not charged anew. Where is the source of energy that maintains (and, in the period of rising temperature, increases) this internal store? At present we consider this source to be subatomic energy. One theory holds that within a star, from simpler elements, more complex elements are gradually being created, and that this process is accompanied by the release of energy. A more radical view is that matter is entirely annihilated and thereby liberates all its structural energy. If we dwell on the first theory, then the most striking of the facts known to us is the formation of helium from hydrogen. We do not know how to manufacture helium from hydrogen, but we know that it is in fact manufactured; we also know that 0.8% of the mass disappears during this process—this must correspond to the mass of that energy—ether waves—which is released during the transformation. Ether waves are very light in weight, and the quantity of energy that can be obtained from this source is colossal. If 5% of a star consists of hydrogen which is transformed into helium at the first stage of the formation of light elements, then this provides an amount of energy sufficient for all reasonable demands.

It may be that we would have been inclined to expect that the youngest stars consist almost entirely of hydrogen, since it is unlikely that the evolution of heavy elements began before the interior became hot enough to stimulate this process. But here one difficulty arises. Astronomical considerations do not allow the possibility that even the earliest stars contained more hydrogen than a very moderate proportion. I pointed out the circumstance that our calculations in fact do not depend on the chemical composition of the star, but at the same time one limitation must be introduced—it is necessary that the star should not consist of hydrogen. Hydrogen leads to results that differ considerably from the case of all the other 91 elements.

If we suppose that hydrogen is a constituent part of a star, then in most cases this will destroy the general agreement of the theory with the observed—

observation; moreover, we can explain to ourselves the satisfactory nature of this agreement precisely by the fact that we have noted the disappearance of this agreement when ordinary elements are replaced by hydrogen. Therefore I believe that the process of creating elements from electrons and protons must have begun even before the stellar stage had been reached. This is one of the curious scattered facts that we encountered in investigating the interior of a star: we have the right to deny the possibility that a star consists predominantly of hydrogen, although any of the other 91 elements may be present in any quantity. Still more curious is that hydrogen is precisely the element from which we would be inclined to construct a star, so that, apparently, a chance denial hits the mark.

An admixture of hydrogen reduces the proportion of ether energy and ether pressure and thereby gives gravitation the possibility of gathering larger masses. The formation of the occasionally encountered stars of exceptionally great mass (from 20 to 80 times exceeding the mass of the Sun) may be due to the accidental predominance of hydrogen in the region where they arose—this means that the matter was at a more primitive stage in the evolution of the elements.

We should not attach great importance to the question of whether our attempts to investigate the interior of a star have led us to anything close to the ultimate truth. I think that we have succeeded in elucidating several basic factors of the problem and in convincing ourselves how many different tendencies are involved in it. The partial results already achieved agree sufficiently well with observations to inspire us with the cheerful thought that we have begun, from the right end, to untangle the knot of difficulties. Nowhere have we encountered difficulties that might seem insurmountable. Undoubtedly, from the point of view of mathematical physics, gaseous matter at very high temperatures is the simplest kind of substance. To understand everything that occurs, for example, in the material from which a table is made is a truly difficult problem, the solution of which perhaps exceeds the hopes of contemporary science; but it does not seem too optimistic for us to hope that in the not too distant future we shall be able to understand such a simple thing as a star.

Translated by Iv. Tamm.

1. Whereas rapidly moving particles undoubtedly penetrate into the atom by the path indicated by us, some believe that slow electrons (such as, for example, electrons in stars) are turned back already at the surface of the atom. This view apparently arose at a time when the positive charge of the atom was considered to be a large sphere of the same extent as the atom, and seems incompatible with modern views. It is unknown to present-day widespread theories of electrical conductivity. Even if one imagined that a neutral atom could so successfully shield itself from electrons, a strongly positive atom in a star is still unlikely to be able to avoid invasion. ↩↩↩