LECTURE ITHE INTERIOR OF A STAR

THE sun belongs to a system containing some 3,000 million stars. The stars are globes comparable in size with the sun, that is to say, of the order of a million miles in diameter. The space for their accommodation is on the most lavish scale. Imagine thirty cricket balls roaming the whole interior of the earth; the stars roaming the heavens are just as little crowded and run as little risk of collision as the cricket balls. We marvel at the grandeur of the stellar system. But this probably is not the limit. Evidence is growing that the spiral nebulae are ‘island universes’ outside our own stellar system. It may well be that our survey covers only one unit of a vaster organization.

A drop of water contains several thousand million million million atoms. Each atom is about one hundred-millionth of an inch in diameter. Here we marvel at the minute delicacy of the workmanship. But this is not the limit. Within the atom are the much smaller electrons pursuing orbits, like planets round the sun, in a space which relatively to their size is no less roomy than the solar system.

Nearly midway in scale between the atom and the star there is another structure no less marvellous—the human body. Man is slightly nearer to the atom than to the star. About 1027atoms build his body; about 1028human bodies constitute enough material to build a star.

From his central position man can survey the grandest works of Nature with the astronomer, or the minutest works with the physicist. To-night I ask you to look both ways. For the road to a knowledge of the starsleads through the atom; and important knowledge of the atom has been reached through the stars.

The star most familiar to us is the sun. Astronomically speaking, it is close at hand. We can measure its size, weigh it, take its temperature, and so on, more easily than the other stars. We can take photographs of its surface, whereas the other stars are so distant that the largest telescope in the world does not magnify them into anything more than points of light. Figs.1and2[1]show recent pictures of the sun’s surface. No doubt the stars in general would show similar features if they were near enough to be examined.

I must first explain that these are not the ordinary photographs. Simple photographs show very well the dark blotches called sunspots, but otherwise they are rather flat and uninteresting. The pictures here shown were taken with a spectroheliograph, an instrument which looks out for light of just one variety (wave-length) and ignores all the rest. The ultimate effect of this selection is that the instrument sorts out the different levels in the sun’s atmosphere and shows what is going on at one level, instead of giving a single blurred impression of all levels superposed.Fig. 2, which refers to a high level, gives a wonderful picture of whirlwinds and commotion. I think that the solar meteorologists would be likely to describe these vortices in terms not unfamiliar to us—‘A deep depression with secondaries is approaching, and a renewal of unsettled conditions is probable.’ However that may be, there is always one safe weather forecast on the sun; cyclone or anticyclone, the temperature will bevery warm—about 6,000° in fact.

i002Fig. 2. THE SUN. Hydrogen photograph

Fig. 2. THE SUN. Hydrogen photograph

But just now I do not wish to linger over the surface layers or atmosphere of the sun. A great many new and interesting discoveries have recently been made in this region, and much of the new knowledge is very germane to my subject of ‘Stars and Atoms’. But personally I am more at home underneath the surface, and I am in a hurry to dive below. Therefore with this brief glance at the scenery that we pass we shall plunge into the deep interior—where the eye cannot penetrate, but where it is yet possible by scientific reasoning to learn a great deal about the conditions.

By mathematical methods it is possible to work out how fast the pressure increases as we go down into the sun, and how fast the temperature must increase to withstand the pressure. The architect can work out the stresses inside the piers of his building; he does not need to bore holes in them. Likewise the astronomer can work out the stress or pressure at points inside the sun without boring a hole. Perhaps it is more surprising that the temperature can be found by pure calculation. It is natural that you should feel rather sceptical about our claim that we know how hot it is in the very middle of a star—and you may be still more sceptical when I divulge the actual figures! Therefore I had better describe the method as far as I can. I shall not attempt to go into detail, but I hope to show you that there is a clue which might be followed up by appropriate mathematical methods.

I must premise that the heat of a gas is chiefly the energy of motionof its particles hastening in all directions and tending to scatter apart. It is this which gives a gas its elasticity or expansive force; the elasticity of a gas is well known to every one through its practical application in a pneumatic tyre. Now imagine yourself at some point deep down in the star where you can look upwards towards the surface or downwards towards the centre. Wherever you are, a certain condition of balance must be reached; on the one hand there is the weight of all the layers above you pressing downwards and trying to squeeze closer the gas beneath; on the other hand there is the elasticity of the gas below you trying to expand and force the superincumbent layers outwards. Since neither one thing nor the other happens and the star remains practically unchanged for hundreds of years, we must infer that the two tendencies just balance. At each point the elasticity of the gas must be just enough to balance the weight of the layers above; and since it is the heat which furnishes the elasticity, this requirement settles how much heat the gas must have. And so we find the degree of heat or temperature at each point.

The same thing can be expressed a little differently. As before, fix attention on a certain point in a star and consider how the matter above it is supported. If it were not supported it would fall to the centre under the attractive force of gravitation. The support is given by a succession of minute blows delivered by the particles underneath; we have seen that their heat energy causes them to move in all directions, and they keep on striking the matter above. Each blow gives a slight boost upwards, and the whole succession of blows supports the upper material in shuttlecock fashion. (This process is not confined to the stars; for instance, it is in this way that a motor car issupported by its tyres.) An increase of temperature would mean an increase of activity of the particles, and therefore an increase in the rapidity and strength of the blows. Evidently we have to assign a temperature such that the sum total of the blows is neither too great nor too small to keep the upper material steadily supported. That in principle is our method of calculating the temperature.

One obvious difficulty arises, The whole supporting force will depend not only on the activity of the particles (temperature) but also on the number of them (density). Initially we do not know the density of the matter at an arbitrary point deep within the sun. It is in this connexion that the ingenuity of the mathematician is required. He has a definite amount of matter to play with, viz. the known mass of the sun; so the more he uses in one part of the globe the less he will have to spare for other parts. He might say to himself, ‘I do not want to exaggerate the temperature, so I will see if I can manage without going beyond 10,000,000°.’ That sets a limit to the activity to be ascribed to each particle; therefore when the mathematician reaches a great depth in the sun and accordingly has a heavy weight of upper material to sustain, his only resource is to use large numbers of particles to give the required total impulse. He will then find that he has used up all his particles too fast, and has nothing left to fill up the centre. Of course his structure, supported on nothing, would come tumbling down into the hollow. In that way we can prove that it is impossible to build up a permanent star of the dimensions of the sun without introducing an activity or temperature exceeding 10,000,000°. The mathematician can go a step beyond this; instead of merely finding a lower limit, he can ascertain what must be nearly the true temperaturedistribution by taking into account the fact that the temperature must not be ‘patchy’. Heat flows from one place to another, and any patchiness would soon be evened out in an actual star. I will leave the mathematician to deal more thoroughly with these considerations, which belong to the following up of the clue; I am content if I have shown you that there is an opening for an attack on the problem.

This kind of investigation was started more than fifty years ago. It has been gradually developed and corrected, until now we believe that the results must be nearly right—that we really know how hot it is inside a star.

I mentioned just now a temperature of 6,000°; that was the temperature near the surface—the region which we actually see. There is no serious difficulty in determining this surface temperature by observation; in fact the same method is often used commercially for finding the temperature of a furnace from the outside. It is for the deep regions out of sight that the highly theoretical method of calculation is required. This 6,000° is only the marginal heat of the great solar furnace giving no idea of the terrific intensity within. Going down into the interior the temperature rises rapidly to above a million degrees, and goes on increasing until at the sun’s centre it is about 40,000,000°.

Do not imagine that 40,000,000° is a degree of heat so extreme that temperature has become meaningless. These stellar temperatures are to be taken quite literally. Heat is the energy of motion of the atoms or molecules of a substance, and temperature which indicates the degree of heat is a way of stating how fast these atoms or molecules are moving. For example, at the temperature of this room the molecules of air are rushing about with an average speed of 500 yards a second; ifwe heated it up to 40,000,000° the speed would be just over 100 miles a second. That is nothing to be alarmed about; the astronomer is quite accustomed to speeds like that. The velocities of the stars, or of the meteors entering the earth’s atmosphere, are usually between 10 and 100 miles a second. The velocity of the earth travelling round the sun is 20 miles a second. So that for an astronomer this is the most ordinary degree of speed that could be suggested, and he naturally considers 40,000,000° a very comfortable sort of condition to deal with. And if the astronomer is not frightened by a speed of 100 miles a second, the experimental physicist is quite contemptuous of it; for he is used to handling atoms shot off from radium and similar substances with speeds of 10,000 miles a second. Accustomed as he is to watching these express atoms and testing what they are capable of doing, the physicist considers the jog-trot atoms of the stars very commonplace.

Besides the atoms rushing to and fro in all directions we have in the interior of a star great quantities of ether waves also rushing in all directions. Ether waves are called by different names according to their wave-length. The longest are the Hertzian waves used in broadcasting; then come the infra-red heat waves; next come waves of ordinary visible light; then ultra-violet photographic or chemical rays; then X-rays; then Gamma rays emitted by radio-active substances. Probably the shortest of all are the rays constituting the very penetrating radiation found in our atmosphere, which according to the investigations of Kohlhörster and Millikan are believed to reach us from interstellar space. These are all fundamentally the same but correspond to different octaves. The eye is attuned to only one octave, so that most of them are invisible; but essentially they are of thesame nature as visible light.

The ether waves inside a star belong to the division called X-rays. They are the same as the X-rays produced artificially in an X-ray tube. On the average they are ‘softer’ (i.e. longer) than the X-rays used in hospitals, but not softer than some of those used in laboratory experiments. Thus we have in the interior of a star something familiar and extensively studied in the laboratory.

Besides the atoms and ether waves there is a third population to join in the dance. There are multitudes of free electrons. The electron is the lightest thing known, weighing no more than ¹⁄₁₈₄₀ of the lightest atom. It is simply a charge of negative electricity wandering about alone. An atom consists of a heavy nucleus which is usually surrounded by a girdle of electrons. It is often compared to a miniature solar system, and the comparison gives a proper idea of theemptinessof an atom. The nucleus is compared to the sun, and the electrons to the planets. Each kind of atom—each chemical element—has a different quorum of planet electrons. Our own solar system with eight planets might be compared especially with the atom of oxygen which has eight circulating electrons. In terrestrial physics we usually regard the girdle or crinoline of electrons as an essential part of the atom because we rarely meet with atoms incompletely dressed; when we do meet with an atom which has lost one or two electrons from its system, we call it an ‘ion’. But in the interior of a star, owing to the great commotion going on, it would be absurd to exact such a meticulous standard of attire. All our atoms have lost a considerable proportion of their planet electrons and are thereforeionsaccording to the strict nomenclature.

At the high temperature inside a star the battering of the particles by one another, and more especially the collision of the ether waves (X-rays) with atoms, cause electrons to be broken off and set free. These free electrons form the third population to which I have referred. For each individual the freedom is only temporary, because it will presently be captured by some other mutilated atom; but meanwhile another electron will have been broken off somewhere else to take its place in the free population. This breaking away of electrons from atoms is calledionization, and as it is extremely important in the study of the stars I will presently show you photographs of the process.

My subject is ‘Stars and Atoms’; I have already shown you photographs of a star, so I ought to show you a photograph of an atom. Nowadays that is quite easy. Since there are some trillions of atoms present in the tiniest piece of material it would be very confusing if the photograph showed them all. Happily the photograph exercises discrimination and shows only ‘express train’ atoms which flash past like meteors, ignoring all the others. We can arrange a particle of radium to shoot only a few express atoms across the field of the camera, and so have a clear picture of each of them.

Fig. 3[2]is a photograph of three or four atoms which have flashed across the field of view—giving the broad straight tracks. These are atoms of helium discharged at high speed from a radio-active substance.

I wonder if there is an under-current of suspicion in your minds that there must be something of a fake about this photograph. Arethese really the single atoms that are showing themselves—those infinitesimal units which not many years ago seemed to be theoretical concepts far outside any practical apprehension? I will answer that question by asking you one. You see a dirty mark on the picture. Is that somebody’s thumb? If you say Yes, then I assure you unhesitatingly that these streaks are single atoms. But if you are hypercritical and say ‘No. That is not anybody’s thumb, but it is a mark that shows that somebody’s thumb has been there’, then I must be equally cautious and say that the streak is a mark that shows where an atom has been. The photograph instead of being the impression of an atom is the impression of the impression of an atom, just as it is not the impression of a thumb but the impression of the impression of a thumb. I don’t see that it really matters that the impression is second-hand instead of first-hand. I do not think we have been guilty of any more faking than the criminologist who scatters powder over a finger-print to make it visible, or a biologist who stains his preparations with the same object. The atom in its passage leaves what we might call a ‘scent’ along its trail; and we owe to Professor C. T. R. Wilson a most ingenious device for making the scent visible. Professor Wilson’s ‘pack of hounds’ consists of water vapour which flocks to the trail and there condenses into tiny drops.

You will next want to see a photograph of an electron. That also can be managed. The broken wavy trail inFig. 3is an electron. Owing to its small mass the electron is more easily turned aside in its course than the heavy atom which rushes bull-headed through all obstacles.Fig. 4shows numerous electrons, and it includes one of very high speed which on that account was able to make a straight track. Incidentally it gives away the device used for making the tracks visible, because you can see the tiny drops of water separately.

i003Fig. 3

Fig. 3

i004Fig. 4. FAST-MOVING ATOMS AND ELECTRONS

Fig. 4. FAST-MOVING ATOMS AND ELECTRONS

We have seen photographs of atoms and free electrons. Now we want a photograph of X-rays to complete the stellar population. We cannot quite manage that, but we can very nearly. PhotographsbyX-rays are common enough; but a photographofX-rays is a different matter. I have already said that electrons can be broken away from atoms by X-rays colliding with them. When this happens the free electron is usually shot off with high velocity so that it is one of the express electrons which can be photographed. InFig. 5you see four electrons shot off in this way. You notice that they all start from points in the same line, and it does not require much imagination to see in your mind a mysterious power travelling along this line and creating the explosions. That power is the X-rays which were directed in a narrow beam along the line (from right to left) when the photograph was taken. Although the X-rays are left to your imagination, the photograph at any rate shows the process of ionization which is so important in the stellar interior—the freeing of electrons from the atoms by the incidence of X-rays. You notice that it is just a chance whether the X-ray ionizes an atom when it meets it. There are trillions of atoms lying about (of which the photograph takes no notice); but, nevertheless, the X-rays travel a long way before meeting the atom which they choose to operate on.

Finally I can show you the other method of ionizing atoms by battering of a more mechanical kind—in this case by the collision of a fast electron. InFig. 6a fast electron was travelling nearly horizontally, but the tiny water-drops that should mark its track are so spread outthat you do not at first trace the connexion. Notice that the drops occur in pairs. This is because the fast electron battered some of the atoms along its track, wrenching away an electron from each. You see at intervals along the track a broken atom and a free electron lying side by side, though you cannot tell which is which. Occasionally the original fast electron was too vigorous and there is more of a mix up, but usually you can see clearly the two fragments resulting from the smash.[3]

A cynic might remark that the interior of a star is a very safe subject to talk about because no one can go there and prove that you are wrong. I would plead in reply that at least I am not abusing the unlimited opportunity for imagination; I am only asking you to allow in the interior of the star quite homely objects and processes which can be photographed. Perhaps now you will turn round on me and say, ‘What right have you to suppose that Nature is as barren of imagination as you are? Perhaps she has hidden in the star something novel which will upset all your ideas.’ But I think that science would never have achieved much progress if it had always imagined unknown obstacles hidden round every corner. At least we may peer gingerly round the corner, and perhaps we shall find there is nothing very formidable after all. Our object in diving into the interior is not merely to admire a fantastic world with conditions transcending ordinary experience; it is to get at the inner mechanism which makes stars behave as they do. If we are to understand the surface manifestations,if we are to understand why ‘one star differeth from another star in glory’, we must go below—to theengine-room—to trace the beginning of the stream of heat and energy which pours out through the surface. Finally, then, our theory will take us back to the surface and we shall be able to test by comparison with observation whether we have been badly misled. Meanwhile, although we naturally cannot prove a general negative, there is no reason to anticipate anything which our laboratory experience does not warn us of.

The X-rays in a star are the same as the X-rays experimented on in a laboratory, but they are enormously more abundant in the star. We can produce X-rays like the stellar X-rays, but we cannot produce them in anything like stellar abundance. The photograph (Fig. 5) showed a laboratory beam of X-rays which had wrenched away four electrons from different atoms; these would be speedily recaptured. In the star you must imagine the intensity multiplied many million-fold, so that electrons are being wrenched away as fast as they settle and the atoms are kept stripped almost bare. The nearly complete mutilation of the atoms is important in the study of the stars for two main reasons.

The first is this. An architect before pronouncing an opinion on the plans of a building will want to know whether the material shown in the plans is to be wood or steel or tin or paper. Similarly it would seem essential before working out details about the interior of a star to know whether it is made of heavy stuff like lead or light stuff like carbon. By means of the spectroscope we can find out a great deal about the chemical composition of the sun’s atmosphere; but it would not be fair to take this as a sample of the compositionof the sun as a whole. It would be very risky to make a guess at the elements preponderating in the deep interior. Thus we seem to have reached a deadlock. But now it turns out that when the atoms are thoroughly smashed up, they all behave nearly alike—at any rate in those properties with which we are concerned in astronomy. The high temperature—which we were inclined to be afraid of at first—has simplified things for us, because it has to a large extent eliminated differences between different kinds of material. The structure of a star is an unusually simple physical problem; it is at low temperatures such as we experience on the earth that matter begins to have troublesome and complicated properties. Stellar atoms are nude savages innocent of the class distinctions of our fully arrayed terrestrial atoms. We are thus able to make progress without guessing at the chemical composition of the interior. It is necessary to make one reservation, viz. that there is not an excessive proportion of hydrogen. Hydrogen has its own way of behaving; but it makes very little difference which of the other 91 elements predominate.

The other point is one about which I shall have more to say later. It is that we must realize that the atoms in the stars are mutilated fragments of the bulky atoms with extended electron systems familiar to us on the earth; and therefore the behaviour of stellar and terrestrial gases is by no means the same in regard to properties which concern the size of the atoms.

To illustrate the effect of the chemical composition of a star, we revert to the problem of the support of the upper layers by the gas underneath. At a given temperature every independent particle contributes the same amount of support no matter what its mass or chemical nature; the lighter atoms make up for their lack of mass bymoving more actively. This is a well-known law originally found in experimental chemistry, but now explained by the kinetic theory of Maxwell and Boltzmann. Suppose we had originally assumed the sun to be composed entirely of silver atoms and had made our calculations of temperature accordingly; afterwards we change our minds and substitute a lighter element, aluminium. A silver atom weighs just four times as much as an aluminium atom; hence we must substitute four aluminium atoms for every silver atom in order to keep the mass of the sun unchanged. But now the supporting force will everywhere be quadrupled, and all the mass will be heaved outwards by it if we make no further change. In order to keep the balance, the activity of each particle must be reduced in the ratio ¼; that means that we must assign throughout the aluminium sun temperatures ¼ of those assigned to the silver sun. Thus for unsmashed atoms a change in the assigned chemical composition makes a big change in our inference as to the internal temperature.

But if electrons are broken away from the atom these also become independent particles rendering support to the upper layers. A free electron gives just as much support as an atom does; it is of much smaller mass, but it moves about a hundred times as fast. The smashing of one silver atom provides 47 free electrons, making with the residual nucleus of the atom 48 particles in all. The aluminium atom gives 13 electrons or 14 particles in all; thus 4 aluminium atoms give 56 independent particles. The change from smashed silver to an equal mass of smashed aluminium only means a change from 48 to 56 particles, requiring a reduction of temperature by 14 per cent. We can tolerate that degree of uncertainty in our estimates of internaltemperature;[4]it is a great improvement on the corresponding calculation for unsmashed atoms which was uncertain by a factor 4.

Besides bringing closer together the results for different varieties of chemical constitution, ionization by increasing the number of supporting particles lowers the calculated temperatures considerably. It is sometimes thought that the exceedingly high temperature assigned to the interior of a star is a modern sensationalism. That is not so. The early investigators, who neglected both ionization and radiation pressure, assigned much higher temperatures than those now accepted.

The stars differ from one another in mass, that is to say, in the quantity of material gathered together to form them; but the differences are not so large as we might have expected from the great variety in brightness. We cannot always find out the mass of a star, but there are a fair number of stars for which the mass has been determined by astronomical measurements. The mass of the sun is—I will write it on the blackboard—

2000000000000000000000000000 tons

I hope I have counted the 0’s rightly, though I dare say you would not mind much if there were one or two too many or too few. But Naturedoesmind. When she made the stars she evidently attached great importance to getting the number of 0’s right. She has an idea that a star should contain a particular amount of material. Of course she allows what the officials at the mint would call a ‘remedy’. She may even pass a star with one 0 too many and give us an exceptionally large star, or with one 0 too few, giving a very small star. But these deviations are rare, and a mistake of two 0’s is almost unheard of. Usually she adheres much more closely to her pattern.

i005Fig. 5. IONIZATION BY X-RAYS

Fig. 5. IONIZATION BY X-RAYS

i006Fig. 6. IONIZATION BY COLLISION

Fig. 6. IONIZATION BY COLLISION

How does Nature keep count of the 0’s? It seems clear that there must be something inside the star itself which keeps check and, so to speak, makes a warning protest as soon as the right amount of material has been gathered together. We think we know how it is done. You remember the ether waves inside the star. These are trying to escape outwards and they exert a pressure on the matter which is caging them in. This outward force, if it is sufficiently powerful to be worth considering in comparison with other forces, must be taken into account in any study of the equilibrium or stability of the star. Now in all small globes this force is quite trivial; but its importance increases with the mass of the globe, and it is calculated that at just about the above mass it reaches equal status with the other forces governing the equilibrium of the star. If we had never seen the stars and were simply considering as a curious problem how big a globe of matter could possibly hold together, we could calculate that there would be no difficulty up to about two thousand quadrillion tons; but beyond that the conditions are entirely altered and this new force begins to take control of the situation. Here, I am afraid, strict calculation stops, and no one has yet been able to calculate what the new force will do with the star when it does take control. But it can scarcely be an accident that the stars are all so near to this critical mass; and so I venture to conjecture the rest of the story. The new force doesnotprohibitlarger mass, but it makes it risky. It may help a moderate rotation about the axis to break up the star. Consequently larger masses will survive only rarely; for the most part stars will be kept down to the mass at which the new force first becomes a serious menace. The force of gravitation collects together nebulous and chaotic material; the force of radiation pressure chops it off into suitably sized lumps.

This force of radiation pressure is better known to many people under the name ‘pressure of light’. The term ‘radiation’ comprises all kinds of ether-waves including light, so that the meaning is the same. It was first shown theoretically and afterwards verified experimentally that light exerts a minute pressure on any object on which it falls. Theoretically it would be possible to knock a man over by turning a searchlight on him—only the searchlight would have to be excessively intense, and the man would probably be vaporized first. Pressure of light probably plays a great part in many celestial phenomena. One of the earliest suggestions was that the minute particles forming the tail of a comet are driven outwards by the pressure of sunlight, thus accounting for the fact that a comet’s tail points away from the sun. But that particular application must be considered doubtful. Inside the star the intense stream of light (or rather X-rays) is like a wind rushing outwards and distending the star.

We can now form some kind of a picture of the inside of a star—a hurly-burly of atoms, electrons, and ether-waves. Dishevelled atoms tear along at 100 miles a second, their normal array of electrons being torn from them in the scrimmage. The lost electrons are speeding 100times faster to find new resting places. Let us follow the progress of one of them. There is almost a collision as an electron approaches an atomic nucleus, but putting on speed it sweeps round in a sharp curve. Sometimes there is a side-slip at the curve, but the electron goes on with increased or reduced energy. After a thousand narrow shaves, all happening within a thousand millionth of a second, the hectic career is ended by a worse side-slip than usual. The electron is fairly caught, and attached to an atom. But scarcely has it taken up its place when an X-ray bursts into the atom. Sucking up the energy of the ray the electron darts off again on its next adventure.

I am afraid the knockabout comedy of modern atomic physics is not very tender towards our aesthetic ideals. The stately drama of stellar evolution turns out to be more like the hair-breadth escapades on the films. The music of the spheres has almost a suggestion of—jazz.

And what is the result of all this bustle? Very little. The atoms and electrons for all their hurry never get anywhere; they only change places. The ether-waves are the only part of the population which accomplish anything permanent. Although apparently darting in all directions indiscriminately, they do on the average make a slow progress outwards. There is no outward progress of the atoms and electrons; gravitation sees to that. But slowly the encaged ether-waves leak outwards as through a sieve. An ether-wave hurries from one atom to another, forwards, backwards, now absorbed, now flung out again in a new direction, losing its identity, but living again in its successor. With any luck it will in no unduly long time (ten thousand to ten million years according to the mass of the star) find itself near the boundary. It changes at the lower temperature from X-rays tolight-rays, being altered a little at each re-birth. At last it is so near the boundary that it can dart outside and travel forward in peace for a few hundred years. Perhaps it may in the end reach some distant world where an astronomer lies in wait to trap it in his telescope and extort from it the secrets of its birth-place.

It is the leakage that we particularly want to determine; and that is why we have to study patiently what is going on in the turbulent crowd. To put the problem in another form; the waves are urged to flow out by the temperature gradient in the star, but are hindered and turned back by their adventures with the atoms and electrons. It is the task of mathematics, aided by the laws and theories developed from a study of these same processes in the laboratory, to calculate the two factors—the factor urging and the factor hindering the outward flow—and hence to find the leakage. This calculated leakage should, of course, agree with astronomical measurements of the energy of heat and light pouring out of the star. And so finally we arrive at an observational test of the theories.

Let us consider the factor which hinders the leakage—the turning back of the ether-waves by their encounters with atoms and electrons. If we were dealing with light waves we should call this obstruction to their passage ‘opacity’, and we may conveniently use the same term for obstruction to X-rays.

We soon realize that the material of the star must be decidedly opaque. The quantity of radiation in the interior is so great that unless it were very severely confined the leakage would be much greater than the amount which we observe coming out of the stars. The following is anillustration of the typical degree of opacity required to agree with the observed leakage. Let us enter the star Capella and find a region where the density is the same as that of the atmosphere around us;[5]a slab of the material only two inches thick would form a screen so opaque that only one-third of the ether-waves falling on one side would get through to the other side, the rest being absorbed in the screen. A foot or two of the material would be practically a perfect screen. If we are thinking of light-waves this seems an astonishing opacity for material as tenuous as air; but we have to remember that it is an opacity to X-rays, and the practical physicist knows well the difficulty of getting the softer kinds of X-rays to pass through even a few millimetres of air.

There is a gratifying accordance in general order of magnitude between the opacity inside the star, determined from astronomical observation of leakage, and the opacity of terrestrial substances to X-rays of more or less the same wave-length. This gives us some assurance that our theory is on the right track. But a careful comparison shows us that there is some important difference between the stellar and terrestrial opacity.

In the laboratory we find that the opacity increases very rapidly with the wave-length of the X-rays that are used. We do not find anything like the same difference in the stars although the X-rays in the cooler stars must be of considerably greater wave-length than those in the hotter stars. Also, taking care to make the comparison at the same wave-length for both, we find that the stellar opacity is less than the terrestrial opacity. We must follow up this divergence.

There is more than one way in which an atom can obstruct ether-waves, but there seems to be no doubt that for X-rays both in the stars and in the laboratory the main part of the opacity depends on the process of ionization. The ether-wave falls on an atom and its energy is sucked up by one of the planet electrons which uses it to escape from the atom and travel away at high speed. The point to notice is that in the very act of absorption the absorbing mechanism is broken, and it cannot be used again until it has been repaired. To repair it the atom must capture one of the free electrons wandering about, inducing it to take the place of the lost electron.

In the laboratory we can only produce thin streams of X-rays so that each wave-trap is only called upon to act occasionally. There is plenty of time to repair it before the next time it has a chance of catching anything; and there is practically no loss of efficiency through the traps being out of order. But in the stars the stream of X-rays is exceedingly intense. It is like an army of mice marching through your larder springing the mouse-traps as fast as you can set them. Here it is the time wasted in resetting the traps—by capturing electrons—which counts, and the amount of the catch depends almost entirely on this.

We have seen that the stellar atoms have lost most of their electrons; that means that at any moment a large proportion of the absorption traps are awaiting repair. For this reason we find a smaller opacity in the stars than in terrestrial material. The lowered opacity is simply the result of overworking the absorbing mechanisms—they have too much radiation to deal with. We can also see why the laws of stellar and terrestrial opacity are somewhat different. The rate of repair, which is the main consideration in stellar opacity, is increased bycompressing the material, because then the atom will not have to wait so long to meet and capture a free electron. Consequently the stellar opacity will increase with the density. In terrestrial conditions there is no advantage in accelerating the repair which will in any case be completed in sufficient time; thus terrestrial opacity is independent of the density.

The theory of stellar opacity thus reduces mainly to the theory of the capture of electrons by ionized atoms; not that this process is attended by absorption of X-rays—it is actually attended by emission—but it is the necessary preliminary to absorption. The physical theory of electron-capture is not yet fully definitive; but it is sufficiently advanced for us to make use of it provisionally in our calculations of the hindering factor in the leakage of radiation from the stars.

We do not want to tackle too difficult a problem at first, and so we shall deal with stars composed of perfect gas. If you do not like the technical phrase ‘perfect gas’ you can call it simply ‘gas’, because all the terrestrial gases that you are likely to think of are without sensible imperfection. It is only under high compression that terrestrial gases become imperfect. I should mention that there are plenty of examples of gaseous[6]stars. In many stars the material is so inflated that it is more tenuous than the air around us; for example, if you were inside Capella you would not notice the material of Capella any more than you notice the air in this room.

For gaseous stars, then, the investigation will give formulae by which, given the mass of the star, we can calculate how much energy of heat and light will leak out of it—in short, how bright it will be. InFig. 7a curve is drawn giving this theoretical relation between the brightness and mass of a star. Strictly speaking, there is another factor besides the mass which affects the calculated brightness; you can have two stars of the same mass, the one dense and the other puffed out, and they will not have quite the same brightness. But it turns out (rather unexpectedly) that this other factor, density, makes very little difference to the brightness, always provided that the material is not too dense to be a perfect gas. I shall therefore say no more about density in this brief summary.

Here are a few details about the scale of the diagram. The brightness is measured in magnitudes, a rather technical unit. You have to remember that stellar magnitude is like a golfer’s handicap—the bigger the number, the worse the performance. The diagram includes practically the whole range of stellar brightness; at the top -4 represents almost the brightest stars known, and at the bottom 12 is nearly the faintest limit. The difference from top to bottom is about the same as the difference between an arc light and a glow worm. The sun is near magnitude 5. These magnitudes refer, of course, to the true brightness, not to the apparent brightness affected by distance; also, what is represented here is the ‘heat brightness’ or heat intensity, which is sometimes a little different from the light intensity. Astronomical instruments have been made which measure directly the heat instead of the light received from a star. These are quite successful; but there are troublesome corrections on account of the large absorption of heat in the earth’s atmosphere, and it is in most cases easier and more accurate to infer the heat brightness from the light brightness, making allowance for the colour of the star.

i007Fig. 7. The Mass-luminosity Curve.

Fig. 7. The Mass-luminosity Curve.

The horizontal scale refers to mass, but it is graduated according to the logarithm of the mass. At the extreme left the mass is about ⅙ x sun, and on the extreme right about 30 x sun; there are very few stars with masses outside these limits. The sun’s mass corresponds to the division labelled 0·0.

Having obtained our theoretical curve, the first thing to do is to test it by observation. That is to say, we gather together as many stars as we can lay hands on for which both the mass and absolute brightness have been measured. We plot the corresponding points (opposite to the appropriate horizontal and vertical graduations) and see whether they fall on the curve, as they ought to do if the theory is right. There are not many stellar masses determined with much precision. Everything that is reasonably trustworthy has been included inFig. 7. The circles, crosses, squares, and triangles refer to different kinds of data—some good, some bad, some very bad.

The circles are the most trustworthy. Let us run through them from right to left. First comes the bright component of Capella, lying beautifully on the curve—because I drew the curve through it. You see, there was one numerical constant which in the present state of our knowledge of atoms and ether-waves, &c., it was not possible to determine with any confidence from pure theory. So the curve when it was obtained was loose in one direction and could be raised or lowered. It was anchored by making it pass through the bright component of Capella which seemed the best star to trust to for this purpose. Afterthat there could be no further tampering with the curve. Continuing to the left we have the fainter component of Capella; next Sirius; then, in a bunch, two components of α Centauri (the nearest fixed star) with the Sun between them, and—lying on the curve—a circle representing the mean of six double stars in the Hyades. Finally, far on the left there are two components of a well-known double star called Krueger 60.

The observational data for testing the curve are not so extensive and not so trustworthy as we could wish; but still I think it is plain fromFig. 7that the theory is substantially confirmed, and it really does enable us to predict the brightness of a star from its mass, or vice versa. That is a useful result, because there are thousands of stars of which we can measure the absolute brightness but not the mass, and we can now infer their masses with some confidence.

Since I have not been able to give here the details of the calculation, I should make it plain that the curve inFig. 7is traced by pure theory or terrestrial experiment except for the one constant determined by making it pass through Capella. We can imagine physicists working on a cloud-bound planet such as Jupiter who have never seen the stars. They should be able to deduce by the method explained onp. 25that if there is a universe existing beyond the clouds it is likely to aggregate primarily into masses of the order a thousand quadrillion tons. They could then predict that these aggregations will be globes pouring out light and heat and that their brightness will depend on the mass in the way given by the curve inFig. 7. All the information that we have used for the calculations would be accessible to them beneath the clouds, except that we have stolen one advantage overthem in utilizing the bright component of Capella. Even without this forbidden peep, present-day physical theory would enable them to assign a brightness to the invisible stellar host which would not be absurdly wrong. Unless they were wiser than us they would probably ascribe to all the stars a brilliance about ten times too great[7]—not a bad error for a first attempt at so transcendent a problem. We hope to clear up the discrepant factor 10 with further knowledge of atomic processes; meanwhile we shelve it by fixing the doubtful constant by astronomical observation.

The agreement of the observational points with the curve is remarkably close, considering the rough nature of the observational measurements; and it seems to afford a rather strong confirmation of the theory. But there is one awful confession to make—we have compared the theory with the wrong stars. At least when the comparison was first made at the beginning of 1924 no one entertained any doubt that they were the wrong stars.

We must recall that the theory was developed for stars in the condition of a perfect gas. In the right half ofFig. 7the stars representedare all diffuse stars; Capella with a mean density about equal to that of the air in this room may be taken as typical. Material of this tenuity is evidently a true gas, and in so far as these stars agree with the curve the theory is confirmed. But in the left half of the diagram we have the Sun whose material is denser than water, Krueger 60 denser than iron, and many other stars of the density usually associated with solid or liquid matter. What business have they on the curve reserved for a perfect gas? When these stars were put into the diagram it was not with any expectation that they would agree with the curve; in fact, the agreement was most annoying. Something very different was being sought for. The idea was that the theory might perhaps be trusted on its own merits with such confirmation as the diffuse stars had already afforded; then by measuring how far these dense stars fell below the curve we should have definite information as to how great a deviation from a perfect gas occurred at any given density. According to current ideas it was expected that the sun would fall three or four magnitudes below the curve, and the still denser Krueger 60 should be nearly ten magnitudes below.[8]You see that the expectation was entirely unfulfilled.

The shock was even greater than I can well indicate to you, becausethe great drop in brightness when the star is too dense to behave as a true gas was a fundamental tenet in our conception of stellar evolution. On the strength of it the stars had been divided into two groups known as giants and dwarfs, the former being the gaseous stars and the latter the dense stars.

Two alternatives now lie before us. The first is to assume that something must have gone wrong with our theory; that the true curve for gaseous stars is not as we have drawn it, but runs high up on the left of the diagram so that the Sun, Krueger 60, &c., are at the appropriate distances below it. In short, our imaginary critic was right; Nature had hidden something unexpected inside the star and so frustrated our calculations. Well, if this were so, it would be something to have found it out by our investigations.

The other alternative is to consider this question—Is it impossible that a perfect gas should have the density of iron? The answer is rather surprising. There is no earthly reason why a perfect gas should not have a density far exceeding iron. Or it would be more accurate to say, the reason why it should not isearthlyand does not apply to the stars.

The sun’s material, in spite of being denser than water, really is a perfect gas. It sounds incredible, but it must be so. The feature of a true gas is that there is plenty of room between the separate particles—a gas contains very little substance and lots of emptiness. Consequently when you squeeze it you do not have to squeeze the substance; you just squeeze out some of the waste space. But if you go on squeezing, there comes a time when you have squeezed out all the empty space; the atoms are then jammed in contact and any further compression means squeezing the substance itself, which isquite a different proposition. So as you approach that density the compressibility characteristic of a gas is lost and the matter is no longer a proper gas. In a liquid the atoms are nearly in contact; that will give you an idea of the density at which the gas loses its characteristic compressibility.

The big terrestrial atoms which begin to jam at a density near that of the liquid state do not exist in the stars. The stellar atoms have been trimmed down by the breaking off of all their outer electrons. The lighter atoms are stripped to the bare nucleus—of quite insignificant size. The heavier atoms retain a few of the closer electrons, but have not much more than a hundredth of the diameter of a fully arrayed atom. Consequently we can go on squeezing ever so much more before these tiny atoms or ions jam in contact. At the density of water or even of platinum there is still any amount of room between the trimmed atoms; and waste space remains to be squeezed out as in a perfect gas.

Our mistake was that in estimating the congestion in the stellar ball-room we had forgotten that crinolines are no longer in fashion.

It was, I suppose, very blind of us not to have foreseen this result, considering how much attention we had been paying to the mutilation of the atoms in other branches of the investigation. By a roundabout route we have reached a conclusion which is really very obvious. And so we conclude that the stars on the left of the diagram are after all not the ‘wrong’ stars. The sun and other dense stars are on the perfect gas curve because their material is perfect gas. Careful investigation has shown that in the small stars on the extreme left ofFig. 7the electric charges of the atoms and electrons bring about aslight deviation from the ordinary laws of a gas; it has been shown by R. H. Fowler that the effect is to make the gas notimperfectbutsuperperfect—it ismoreeasily compressed than an ordinary gas. You will notice that on the average the stars run a little above the curve on the left ofFig. 7. It is probable that the deviation is genuine and is partly due to superperfection of the gas; we have already seen that imperfection would have brought them below the curve.

Even at the density of platinum there is plenty of waste space, so that in the stars we might go on squeezing stellar matter to a density transcending anything known on the earth. But that’s another story—I will tell it later on.

The general agreement between the observed and predicted brightness of the stars of various masses is the main test of the correctness of our theories of their internal constitution. The incidence of their masses in a range which is especially critical for radiation pressure is also valuable confirmation. It would be an exaggeration to claim that this limited success is a proof that we have reached the truth about the stellar interior. It is not a proof, but it is an encouragement to work farther along the line of thought which we have been pursuing. The tangle is beginning to loosen. The more optimistic may assume that it is now straightened out; the more cautious will make ready for the next knot. The one reason for thinking that the real truth cannot be so very far away is that in the interior of a star, if anywhere, the problem of matter is reduced to its utmost simplicity; and the astronomer is engaged on what is essentially a less ambitious problem than that of the terrestrial physicist to whom matter always appears in the guiseof electron systems of the most complex organization.

We have taken the present-day theories of physics and pressed them to their remotest conclusions. There is no dogmatic intention in this; it is the best means we have of testing them and revealing their weaknesses if any.

In ancient days two aviators procured to themselves wings. Daedalus flew safely through the middle air and was duly honoured on his landing. Icarus soared upwards to the sun till the wax melted which bound his wings and his flight ended in fiasco. In weighing their achievements, there is something to be said for Icarus. The classical authorities tell us that he was only ‘doing a stunt’, but I prefer to think of him as the man who brought to light a serious constructional defect in the flying-machines of his day. So, too, in Science. Cautious Daedalus will apply his theories where he feels confident they will safely go; but by his excess of caution their hidden weaknesses remain undiscovered. Icarus will strain his theories to the breaking-point till the weak joints gape. For the mere adventure? Perhaps partly; that is human nature. But if he is destined not yet to reach the sun and solve finally the riddle of its constitution, we may at least hope to learn from his journey some hints to build a better machine.

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