Chapter IV: Part 4

Energy of 1·8. 10^{54} ergs has a mass 2. 10^{33} grammes which is the mass of the sun; consequently that is the sum total of the energy which the sun contains--the energy which has to last it all the rest of its life.[31] We do not know how much of this is capable of being converted into heat and radiation; if it is all convertible there is enough to maintain the sun’s radiation at the present rate for 15 billion years. To put the argument in another form, the heat emitted by the sun each year has a mass of 120 billion tons; and if this loss of mass continued there would be no mass left at the end of 15 billion years.

_Subatomic Energy_

This store of energy is, with insignificant exception, energy of constitution of atoms and electrons; that is to say, subatomic energy. Most of it is inherent in the constitution of the electrons and protons--the elementary negative and positive electric charges--out of which matter is built; so that it cannot be set free unless these are destroyed. The main store of energy in a star cannot be used for radiation unless the matter composing the star is being annihilated.

It is possible that the star may have a long enough life without raiding the main energy store. A small part of the store can be released by a process less drastic than annihilation of matter, and this might be sufficient to keep the sun burning for 10,000,000,000 years or so, which is perhaps as long as we can reasonably require. The less drastic process is transmutation of the elements. Thus we have reached a point where a choice lies open before us; we can either pin our faith to transmutation of the elements, contenting ourselves with a rather cramped time-scale, or we can assume the annihilation of matter, which gives a very ample time-scale. But at present I can see no possibility of a third choice. Let me run over the argument again. First we found that energy of contraction was hopelessly inadequate; then we found that the energy must be released in the interior of the star, so that it comes from an internal, not an external, source; now we take stock of the whole internal store of energy. No supply of any importance is found until we come to consider the electrons and atomic nuclei; here a reasonable amount can be released by regrouping the protons and electrons in the atomic nuclei (transmutation of elements), and a much greater amount by annihilating them.

Transmutation of the elements--so long the dream of the alchemist--is realized in the transformation of radio-active substances. Uranium turns slowly into a mixture of lead and helium. But none of the known radio-active processes liberate anything like enough energy to maintain the sun’s heat. The only important release of energy by transmutation occurs at the very beginning of the evolution of the elements.

We must start with hydrogen. The hydrogen atom consists simply of a positive and negative charge, a proton for the nucleus plus a planet electron. Let us call its mass 1. Four hydrogen atoms will make a helium atom. If the mass of the helium atom were exactly 4, that would show that all the energy of the hydrogen atoms remained in the helium atom. But actually the mass is 3·97; so that energy of mass a 0·03 must have escaped during the formation of helium from hydrogen. By annihilating 4 grammes of hydrogen we should have released 4 grammes of energy, but by transmuting it into helium we release 0·03 grammes of energy. Either process might be used to furnish the sun’s heat though, as we have already stated, the second gives a much smaller supply.

The release of energy occurs because in the helium atom only two of the four electrons remain as planet electrons, the other two being cemented with the four protons close together in the helium nucleus. In bringing positive and negative charges close together you cause a change of the energy of the electric field, and release electrical energy which spreads away as ether-waves. That is where the 0·03 grammes of energy has gone. The star can absorb these ether-waves and utilize them as heat.

We can go on from helium to higher elements, but we do not obtain much more release of energy. For example, an oxygen atom can be made from 16 hydrogen atoms or 4 helium atoms; but as nearly as we can tell it has just the weight of the 4 helium atoms, so that the release of energy is not appreciably greater when the hydrogen is transmuted into oxygen than when it is transmuted into helium.[32] This becomes clearer if we take the mass of a hydrogen atom to be 1·008, so that the mass of helium is exactly 4 and of oxygen 16; then it is known from Dr. Aston’s researches with the mass-spectrograph that the atoms of other elements have masses which are very closely whole numbers. The loss of 0·008 per hydrogen atom applies approximately whatever the element that is formed.

The view that the energy of a star is derived by the building up of other elements from hydrogen has the great advantage that there is no doubt about the possibility of the process; whereas we have no evidence that the annihilation of matter can occur in Nature. I am not referring to the alleged transmutation of hydrogen into helium in the laboratory; those whose authority I accept are not convinced by these experiments. To my mind the existence of helium is the best evidence we could desire of the possibility of the _formation_ of helium. The four protons and two electrons constituting its nucleus must have been assembled at some time and place; and why not in the stars? When they were assembled the surplus energy must have been released, providing a prolific supply of heat. Prima facie this suggests the interior of a star as a likely locality, since undoubtedly a prolific source of heat is there in operation. I am aware that many critics consider the conditions in the stars not sufficiently extreme to bring about the transmutation--the stars are not hot enough. The critics lay themselves open to an obvious retort; we tell them to go and find a _hotter place_.

But here the advantage seems to end. There are many astronomical indications that the hypothesis attributing the energy of the stars to the transmutation of hydrogen is unsatisfactory. It may perhaps be responsible for the rapid liberation of energy in the earliest (giant) stages when the star is a large diffuse body radiating heat abundantly; but the energy in later life seems to come from a source subject to different laws of emission. There is considerable evidence that as a star grows older it gets rid of a large fraction of the matter which originally constituted it, and apparently this can only be contrived by the annihilation of the matter. The evidence, however, is not very coherent, and I do not think we are in a position to come to a definite decision. On the whole the hypothesis of annihilation of matter seems the more promising; and I shall prefer it in the brief discussion of stellar evolution which I propose to give.

The phrase ‘annihilation of matter’ sounds like something supernatural. We do not yet know whether it can occur naturally or not, but there is no obvious obstacle. The ultimate constituents of matter are minute positive charges and negative charges which we may picture as centres of opposite kinds of strain in the ether. If these could be persuaded to run together they would cancel out, leaving nothing except a splash in the ether which spreads out as an electromagnetic wave carrying off the energy released by the undoing of the strain. The amount of this energy is amazingly large; by annihilating a single drop of water we should be supplied with 200 horsepower for a year. We turn covetous eyes on this store without, however, entertaining much hope of ever discovering the secret of releasing it. If it should prove that the stars have discovered the secret and are using this store to maintain their heat, our prospect of ultimate success would seem distinctly nearer.

I suppose that many physicists will regard the subject of subatomic energy as a field of airy speculation. That is not the way in which it presents itself to an astronomer. If it is granted that the stars evolve much more slowly than on the contraction-hypothesis, the measurement of the output of subatomic energy is one of the commonest astronomical measurements--the measurement of the heat or light of the stars.[33] The collection of observational data as to the activity of liberation of subatomic energy is part of the routine of practical astronomy; and we have to pursue the usual course of arranging the measurements into some kind of coherence, so as to find out how the output is related to the temperature, density, or age of the material supplying it--in short, to discover the laws of emission. From this point onwards the discussion may be more or less hypothetical according to the temperament of the investigator; and indeed it is likely that in this as in other branches of knowledge advances may come by a proper use of the scientific imagination. Vain speculation is to be condemned in this as in any other subject, and there is no need for it; the problem is one of induction from observation with due regard to our theoretical knowledge of the possibilities inherent in atomic structure.

I cannot pass from this subject without mentioning the penetrating radiation long known to exist in our atmosphere, which according to the researches of Kohlhörster and Millikan comes from outer space. Penetrating power is a sign of short wave-length and intense concentration of energy. Hitherto the greatest penetrating power has been displayed by Gamma rays originated by subatomic processes occurring in radio-active substances. The cosmic radiation is still more penetrating, and it seems reasonable to refer it to more energetic processes in the atom such as those suggested for the source of stellar energy. Careful measurements have been made by Millikan, and he concludes that the properties accord with those which should be possessed by radiation liberated in the transmutation of hydrogen; it is not penetrating enough to be attributed to a process so energetic as the annihilation of protons and electrons.

There seems to be no doubt that this radiation is travelling downwards from the sky. This is shown by measurements of its strength at different heights in the atmosphere and at different depths below the surface of mountain lakes; it is weakened according to the amount of air or water that it has had to traverse. Presumably its source must be extra-terrestrial. Its strength does not vary with the sun’s altitude, so it is not coming from the sun. There is some evidence that it varies according to the position of the Milky Way, most radiation being received when the greatest extension of the stellar system is overhead. It cannot come from the _interior_ of the stars, the penetrating power being too limited; all the hottest and densest matter in the universe is shut off from us by impenetrable walls. At the most it could come only from the outer rind of the stars where the temperature is moderate and the density is low; but it is more likely that its main source is in the diffuse nebulae or possibly in the matter forming the general cloud in space.[34]

We must await further developments before we can regard the supposed subatomic origin of this radiation as other than speculative; we mention it here only as a possible opening for progress. It will be of great interest if we can reach by this means a more direct acquaintance with the processes which we assume to be the source of stellar energy; and the messages borne to us by the cosmic rays which purport to relate to these processes deserve the closest attention. Our views of stellar energy are likely to be affected on one crucial point. Hitherto we have usually supposed that the very high temperature in the interior of a star is one of the essential conditions for liberation of subatomic energy, and that a reasonably high density is also important. Theoretically it would seem almost incredible that the building up of higher elements or the annihilation of protons and electrons could proceed with any degree of vigour in regions where encounters are rare and there is no high temperature or intense radiation to wake the atoms from apathy; but the more we face the difficulties of all theories of the release of subatomic energy the less inclined we are to condemn any evidence as incredible. The presence of sodium and calcium in the cosmical cloud, of helium and nebulium in the diffuse nebulae, of titanium and zirconium in large quantities in the atmospheres of the youngest stars, bears witness that the evolution of the elements is already far advanced during the diffuse prestellar stage--unless indeed our universe is built from the debris of a former creation. From this point of view it is fitting that we should discern symptoms of subatomic activity in open space. But the physicist may well shake his head over the problem. How are four protons and two electrons to gather together to form a helium nucleus in a medium so rare that the free path lasts for days? The only comfort is that the mode of this occurrence is (according to present knowledge) so inconceivable under any conditions of density and temperature that we may postulate it in the nebulae--on the principle that we may as well be hung for a sheep as for a lamb.

_Evolution of the Stars_

Twenty years ago stellar evolution seemed to be very simple. The stars begin by being very hot and gradually cool down until they go out.

On this view the temperature of a star indicated the stage of evolution that it had reached. The outline of the sequence was sufficiently indicated by the crude observation of colour--white-hot, yellow-hot, red-hot; a more detailed order of temperature was ascertained by examining the light with a spectroscope. The red stars naturally came last in the sequence; they were the oldest stars on the verge of extinction. Sir Norman Lockyer strongly opposed this scheme and to a considerable extent anticipated the more modern view; but most astronomers pinned their faith to it up to about 1913.

Ten years ago more knowledge had been gained of the densities of stars. It seemed likely that density would be a more direct criterion of evolutionary development than temperature. Granted that a star condenses out of nebulous material, it must in the youngest stage be very diffuse; from that stage it will contract and steadily increase in density.

But this necessitates an entire rearrangement of the scheme of evolution, because the order according to density is by no means the same as the order according to surface temperature. On the former view all the cool red stars were old and dying. But a large number of them are now found to be extremely diffuse--stars like Betelgeuse, for instance. These must be set down as the very youngest of the stars; after all it is not unnatural that a star just beginning to condense out of nebulous material should start at the lowest stage of temperature. Not all the red stars are diffuse; there are many like Krueger 60 which have high density, and these we leave undisturbed as representing the last stage of evolution. Both the first and last periods of a star’s life are characterized by low temperature; in between whiles the temperature must have risen to a maximum and fallen again.

The ‘giant and dwarf theory’ proposed by Hertzsprung and Russell brought these conclusions into excellent order. It recognized a series of _giant_ stars, comparatively diffuse stars with temperature rising, and a series of _dwarf_ or dense stars with temperature falling. The two series merged at the highest temperatures. An individual star during its lifetime went up the giant series to its highest temperature and then down the dwarf series. The brightness remained fairly steady throughout the giant stage because the continually increasing temperature counterbalanced the reduction of the surface area of the star; in the dwarf stage the decreasing temperature and the contraction of the surface caused a rapid decrease of brightness as the star progressed down the series. This was in accordance with observation. The theory has dominated most recent astrophysical research and has been instrumental in bringing to light many important facts. One example must suffice. Although we may have a giant and a dwarf star with the same surface temperature, and therefore showing very similar spectra, nevertheless a close examination of the spectrum reveals tell-tale differences; and it is now quite easy to ascertain from the spectrum whether the star is a diffuse giant or a dense dwarf.

The attractive feature of the giant and dwarf theory was the simple explanation given for the up-and-down progress of the temperature. The passing over from the giant to the dwarf series was supposed to occur when the density had reached such a value (about one-quarter the density of water) that the deviation of the material from a perfect gas began to be serious. It was shown by Lane fifty years ago that a globe of perfect gas must rise in temperature as it contracts, his method of finding the internal temperature being that considered on p. 12; thus the rising temperature in the giant stage is predicted. But the rise depends essentially on the easy compressibility of the gas; and when the compressibility is lost at high density the rising temperature may be expected to give place to falling temperature so that the star cools as a solid or liquid would do. That was believed to account for the dwarf stage.

I have been trying to recall ideas of twenty and ten years ago, and you must not suppose that from the standpoint of present-day knowledge I can endorse everything here stated. I have intentionally been vague as to whether by the hotness of a star I mean the internal or the surface temperature since ideas were formerly very loose on this point; I have made no reference to white dwarfs, which are now thought to be the densest and presumably the oldest stars of all. But it is the last paragraph especially which conflicts with our latest conclusions, for we no longer admit that stellar material will cease to behave as a perfect gas at one-quarter the density of water. Our result that the material in the dense dwarf stars is still perfect gas (p. 38) strikes a fatal blow at this part of the giant and dwarf theory.

It would be difficult to say what is the accepted theory of stellar evolution to-day. The theory is in the melting-pot and we are still waiting for something satisfactory to emerge. The whole subject is in doubt and we are prepared to reconsider almost anything. Provisionally, however, I shall assume that the former theory was right in assuming that the sequence of evolution is from the most diffuse to the densest stars. Although I make this assumption I do not feel sure that it is allowable. The former theory had strong reasons for making it which no longer apply. So long as contraction was supposed to be the source of a star’s heat, contraction and increasing density were essential throughout its whole career; with the acceptance of subatomic energy contraction ceases to play this fundamental role.

I propose to confine attention to the dwarf stars[35] because it is among them that the upset has occurred. They form a well-defined series stretching from high surface-temperature to low surface-temperature, high luminosity to low luminosity, and the density increases steadily along the series. We now call this the Main Series. It comprises the great majority of the stars. To fix ideas let us take three typical stars along the series--Algol near the top, the Sun near the middle, and Krueger 60 near the bottom. The relevant information about them is summarized below:

Mean Central density temperature Surface Mass (Water (million temperature Luminosity Star. (Sun = 1). =1). deg.). (deg.) Colour. (Sun = 1). Algol 4·3 0·15 40 12,000 white 150 Sun 1 1·4 40 6,000 yellow 1 Krueger 60 0·27 9·1 35 3,000 red 0·01

The idea of evolution is that these represent the stages passed through in the life-history of an individual star.[36] The increasing density in the third column should be noticed; according to our accepted criterion it indicates that the order of development is Algol→Sun→Krueger 60. A confusion between internal temperature and surface temperature is responsible for some of the mistakes of the older theories. To outward view the star cools from 12,000° to 3,000° in passing down the series, but there is no such change in its internal heat. The central temperature remains surprisingly steady. (No special reliance can be placed on the slight falling off apparently shown by Krueger 60.) It is very remarkable that all stars of the main series have a central temperature of about 40 million degrees as nearly as we can calculate. It is difficult to resist the impression that there is some unusual property associated with this temperature, although all our physical instincts warn us that the idea is absurd.

But the vital point is the decrease of mass shown in the second column. _If an individual star is to progress any part of the way down the main series it must lose mass_. We can put the same inference in a more general way. Now that it has been found that luminosity depends mainly on mass, there can be no important evolution of faint stars from bright stars unless the stars lose a considerable part of their mass.

It is this result which has caused the hypothesis of annihilation of matter to be seriously discussed. All progress in the theory of stellar evolution is held up pending a decision on this hypothesis. If it is accepted it provides an easy key to these changes. The star may (after passing through the giant stage) reach the stage of Algol, and then by the gradual annihilation of the matter in it pass down the main series until when only one-sixteenth of the original mass remains it will be a faint red star like Krueger 60. But if there is no annihilation of matter, the star when once it has reached the dwarf stage seems to be immovable; it has to stay at the point of the series corresponding to its constant mass.

Let it be clearly understood what is the point at issue. The stars lose mass by their radiation; there is no question about that. The sun is losing 120 billion tons annually whether its radiation comes from annihilation of matter or any other internal source. The question is, How long can this loss continue? Unless there is annihilation of matter, all the mass that can escape as radiation will have escaped in a comparatively short time; the sun will then be extinct and there is an end to the loss and to the evolution. But if there is annihilation of matter the life of the sun and the loss of mass continue far longer, and an extended track of evolution lies open before the sun; when it has got rid of three-quarters of its present mass it will have become a faint star like Krueger 60.

Our choice between the possible theories of subatomic energy only affects stellar evolution in one point--but it is the vital point. Unless we choose annihilation of matter, we cut the life of a star so short that there is no time for any significant evolution at all.

I feel the same objection that every one must feel to building extensively on a hypothetical process without any direct evidence that the laws of Nature permit of its occurrence. But the alternative is to leave the stars in sleepy uniformity with no prospect of development or change until their lives come to an end. Something is needed to galvanize the scene into that activity, whether of progress or decay, in which we have so long believed. Rather desperately we seize on the one visible chance. The petrified system wakes. The ultimate particles one by one yield up their energy and pass out of existence. Their sacrifice is the life-force of the stars which now progress on their high adventure:

Atoms or systems into ruin hurl’d,
And now a bubble burst, and now a world.

_Radiation of Mass_

Our first evidence of the extent of the time-scale of stellar evolution was afforded by the steadiness of condition of δ Cephei. This was supplemented by evidence of the great extension of geological time on the earth. We could not do more than set an upper limit to the rate of progress of evolution and a lower limit to the age of the stars. But this limit was sufficient to rule out the contraction hypothesis and drive us to consider the store of subatomic energy.

We now make a new attack, which depends on the belief that _the rate of evolution is determined by the rate at which a star can get rid of its mass_. We are here considering only the evolution of faint stars from bright stars, and there will remain scope for a certain amount of development in the giant stage to which our arguments will not directly apply. But to abandon all lines of evolution between bright stars and faint stars would mean admitting that one star differs from another star in brightness because it was different originally. This _may_ be true; but we ought not to surrender the main field of stellar evolution without making a fight for it.

By the new line of attack we reach a definite determination of the time-scale and not merely a lower limit. We know the rate at which stars in each stage are losing mass by radiation; therefore we can find the time taken to lose a given mass and thereby pass on to a stage of smaller mass. Evolution from Algol to the Sun requires five billion years; evolution from the Sun to Krueger 60 requires 500 billion years. It is interesting to note that stars in the stage between the Sun and Krueger 60 are much more abundant than those between Algol and the Sun--a fact somewhat confirmatory of the calculated duration of the two stages. The abundance of faint stars does not, however, increase so rapidly as the calculated duration; perhaps the stellar universe has not existed long enough for the old stars to be fully represented.

A star of greater mass than Algol squanders its mass very rapidly, so that we do not increase the age of the Sun appreciably by supposing it to have started with greater mass than Algol. The upper limit to the present age of the Sun is 5·2 billion years however great its initial mass.

But, it may be asked, cannot a star accelerate its progress by getting rid of matter in some other way than by radiation? Cannot atoms escape from its surface? If so the loss of mass and consequent evolution will be speeded up, and the time required may perhaps even be brought within range of the alternative theory of transmutation of the elements. But it is fairly certain that the mass escaping in the form of material atoms is negligible compared with that which imperceptibly glides away in the form of radiation. You will perhaps be in doubt as to whether the 120 billion tons per annum lost by the sun in radiation is (astronomically regarded) a large quantity or a small quantity. From certain aspects it is a large quantity. It is more than 100,000 times the mass of the calcium chromosphere. The sun would have to blow off its chromosphere and form an entirely fresh one every five minutes in order to get rid of as much mass in this way as it loses by radiation. It is obvious from solar observation that there is no such outrush of material. To put it another way--in order to halve the time-scale of evolution stated above it would be necessary that a billion atoms should escape each second through each square centimetre of the sun’s surface. I think we may conclude that there is no short cut to smaller mass and that radiation is responsible for practically the whole loss.

We noticed earlier (p. 25) that Nature builds stars which are much alike in mass, but allows herself some deviation from her pattern amounting sometimes to a mistake of one 0. I think we may have done her an injustice, and that she is more careful over her work than we supposed. We ought to have examined coins fresh from her mint; it was not fair to take coins promiscuously, including many that had been in circulation for some hundreds of billions of years and had worn rather thin. Taking the newly formed stars, i. e. the diffuse stars, we find that 90 per cent. of them are between 2½ and 5½ times the mass of the sun--showing that initially the stars are made nearly as closely to pattern as human beings are. In this range radiation pressure increases from 17 to 35 per cent, of the whole pressure; I think this would be expected to be the crucial stage in its rise to importance. Our idea is that the stellar masses initially have this rather close uniformity (which does not exclude a small proportion of exceptional stars outside the above limits); the smaller masses are evolved from these in course of time by the radiation of mass.

For the time being the sun is comfortably settled in its present state, the amount of energy radiated being just balanced by the subatomic energy liberated inside it. Ultimately, however, it must move on. The moving on, or evolution, is continuous, but for convenience of explanation we shall speak of it as though it occurred in steps. Two possible motives for change can be imagined, (1) the supply of subatomic energy might fall off by exhaustion and no longer balance the radiation, and (2) the sun is slowly becoming a star of smaller mass. In former theories the first motive has generally been assumed, and we may still regard it as effective during the giant stage of the stars; but it is clear that the motive to move down the main series must be loss of mass.[37] Apparently the distinction between giant and dwarf stars, replacing the old distinction of perfect and imperfect gas, is that the prolific and soon exhausted supplies of subatomic energy in the giant stage disappear and leave a much steadier supply in the dwarf stage.

When the sun has become a star of smaller mass it will need to resettle its internal conditions. Suppose that at first it tries to retain its present density. As explained on p. 12, we can calculate the internal temperature, and we find that the reduced mass coupled with constant density involves lower temperature. This will slightly turn off the tap of subatomic energy, because there can be little doubt that the release of subatomic energy is more rapid at higher temperature. The reduced supply will no longer be sufficient to balance the radiation; accordingly the star will contract just as it was supposed to do on the old contraction hypothesis which corresponds to the tap of subatomic energy being turned off altogether. The motive is loss of mass; the first consequence is an increase of density which is another characteristic of progress down the main series.

Tracing the consequences a little farther, the increasing density causes a rising temperature which in turn reopens the tap of subatomic energy. As soon as the tap is opened enough to balance the rate of radiation of the star, the contraction stops and the star remains settled in equilibrium at the smaller mass and higher density.

You will see that the laws of release of subatomic energy must be invoked if we are to explain quantitatively why a particular density corresponds to a particular mass in the progress down the main series. The contraction has to proceed so far as to bring the internal conditions to a state in which the release of energy is at the exact rate required to balance the radiation.

I am afraid this all sounds very complicated, but my purpose is to show that the adjustment of the star after an alteration of mass is automatic. After a change of mass the star has to re-solve the problem of the internal conditions necessary for its equilibrium. So far as mechanical conditions are concerned (supporting the weight of the upper layers) it can choose any one of a series of states of different density provided it has the internal temperature appropriate to that density. But such equilibrium is only temporary, and the star will not really settle down until the tap of subatomic energy is opened to the right extent to balance the rate of radiation which, as we have already seen, is practically fixed by the mass. The star fiddles about with the tap until it secures this balance.

One important conclusion has been pointed out by Professor Russell. When the star is adjusting the tap it does not do so _intelligently_; one trial must automatically lead to the next trial, and it is all-important that the next trial should automatically be nearer to and not farther from the right rate. The condition that it shall be nearer to the right rate is that the liberation of subatomic energy shall increase with temperature or density.[38] If it decreases, or even if it is unaltered, the trials will be successively farther and farther from the required rate, so that although a steady balance is possible the star will never be able to find it. It is therefore essential to admit as one of the laws of liberation of subatomic energy that the rate increases with temperature or with density or with both; otherwise subatomic energy will not fulfil the purpose for which it was introduced, viz. to keep the star steady for a very long time.

The strange thing is that the condition of balance is reached when the central temperature is near 40 million degrees--the same whether the star is at the top, middle, or bottom of the main series. Stars at the top release from each gramme of material 700 ergs of energy per second; the sun releases 1 ergs per second; Krueger 60 releases 0·08 ergs per second. It seems extraordinary that stars requiring such different supplies should all have to ascend to the same temperature to procure them. It looks as though at temperatures below this standard not even 0·08 ergs per second is available, but on reaching the standard the supply is practically unlimited. We can scarcely believe that there is a kind of boiling-point (independent of pressure) at which matter boils off into energy. The whole phenomenon is most perplexing.

I may add that the giant stars have temperatures considerably below 40 million degrees. It would appear that they are tapping special supplies of subatomic energy released at lower temperatures. After using up these supplies the star passes on to the main series, and proceeds to tap the main supply. It seems necessary to suppose further that the main supply does not last indefinitely, so that ultimately the star (or what is left of it) leaves the main series and passes on to the white dwarf stage.

We are now in a position to deal with a question which you may have wished to ask earlier. Why does δ Cephei pulsate? One possible answer is that the oscillation was started off by some accident. So far as we can calculate an oscillation, if once started, would continue for something like 10,000 years before becoming damped down. But 10,000 years is now deemed to be an insignificant period in the life of a star, and, having regard to the abundance of Cepheids, the explanation seems inadequate even if we could envisage the kind of accident supposed. It is much more likely that the pulsation arises spontaneously. Enormous supplies of heat energy are being released in the star--far more than enough to start and maintain the pulsation--and there are at least two alternative ways in which this heat can be supposed to operate a mechanism of pulsation.

Here is one alternative. Suppose first that there is a very small pulsation. When compressed the star has higher temperature and density than usual and the tap of subatomic energy is opened more fully. The star gains heat, and the expansive force of the extra heat assists the rebound from compression. At greatest expansion the tap is turned off a little and the loss of heat diminishes the resistance to the ensuing compression. Thus the successive expansions and compressions become more and more vigorous and a large pulsation grows out of an infinitesimal beginning. It will be seen that the star works the tap of subatomic energy just as an engine works the valve admitting heat into its cylinder; so that the pulsations of a star are started up like the pulsations of an engine.

The only objection that I can find to this explanation is that it is too successful. It shows why a star may be expected to pulsate; but the trouble is that stars in general do not pulsate--it is only the rare exceptions that behave in this way. It is now so easy to account for the Cepheids that we have to turn back and face the more difficult problem of accounting for the normal steady stars. Whether the pulsation will start up or not depends on whether the engine of pulsation is sufficiently powerful to overcome the forces tending to damp out and dissipate pulsations. We cannot predict the occurrence or non-occurrence from any settled theory; we have rather to seek to frame the laws of release of subatomic energy so as to conform to our knowledge that the majority of the stars remain steady, but certain conditions of mass and density give the pulsatory forces the upper hand.

Cepheid pulsation is a kind of distemper which happens to stars at a certain youthful period; after passing through it they burn steadily. There may be another attack of disease later in life when the star is subject to those catastrophic outbursts which occasion the appearance of ‘new stars’ or novae. But very little is known as to the conditions for this, and it is not certain whether the outbreak is spontaneous or provoked from outside.

So long as we stick to generalities the theory of subatomic energy and especially the theory of annihilation of matter makes a fairly promising opening. It is when we come to technical details that doubts and perplexities arise. Difficulties appear in the simultaneous presence of giant and dwarf stars in coeval clusters, notwithstanding their widely different rates of evolution. There are difficulties in devising laws of release of subatomic energy which will safeguard the stability of the stars without setting every star into pulsation. Difficulties arise from the fact that as a rule in the giant stage the lower the temperature and density the more rapid the release of energy; and although we account for this in a general way by considering the exhaustibility of the more prolific sources of energy, the facts are not all straightened out by such a scheme. Finally grave difficulties arise in reconciling the laws of release inferred from astronomical observation with any theoretical picture we can form of the process of annihilation of matter by the interplay of atoms, electrons, and radiation.

The subject is highly important, but we cannot very well pursue it further in this lecture. When the guidance of theory is clear interest centres round the broad principles; when the theory is rudimentary, interest centres round technical details which are anxiously scrutinized as they appear to favour now one view now another. I have dealt mainly with two salient points--the problem of the source of a star’s energy and the change of mass which must occur if there is any evolution of faint stars from bright stars. I have shown how these appear to meet in the hypothesis of annihilation of matter. I do not hold this as a secure conclusion. I hesitate even to advocate it as probable, because there are many details which seem to me to throw considerable doubt on it, and I have formed a strong impression that there must be some essential point which has not yet been grasped. I simply tell it you as the clue which at the moment we are trying to follow up--not knowing whether it is false scent or true.

I should have liked to have closed these lectures by leading up to some great climax. But perhaps it is more in accordance with the true conditions of scientific progress that they should fizzle out with a glimpse of the obscurity which marks the frontiers of present knowledge. I do not apologize for the lameness of the conclusion, for it is not a conclusion. I wish I could feel confident that it is even a beginning.

FOOTNOTES:

[Footnote 1: Fig. 1 is from a photograph taken by Mr. Evershed at Kodaikanal Observatory, Madras. Fig. 2 is from the Mount Wilson Observatory, California.]

[Footnote 2: I am indebted to Professor C. T. R. Wilson for Figs. 3-6.]

[Footnote 3: Primarily it is the electric charge and not the high speed of particles which determines their appearance in these photographs. But a high-speed particle leaves behind it a trail of electrically charged particles--the victims of its furious driving--so that it is shown indirectly by its line of victims.]

[Footnote 4: Other substitutions for silver do not as a rule cause greater change, and the differences are likely to be toned down by mixture of many elements. Excluding hydrogen, the most extreme change is from 48 particles for silver to 81 particles for an equal mass of helium. But for hydrogen the change is from 48 to 216, so that hydrogen gives widely different results from other elements.]

[Footnote 5: The mean density of Capella is nearly the same as the density of the air.]

[Footnote 6: Unless otherwise indicated ‘gaseous’ is intended to mean ‘composed of _perfect_ gas’.]

[Footnote 7: For this prediction it is unnecessary to know the chemical composition of the stars, provided that extreme cases (e. g. an excessive proportion of hydrogen) are excluded. For example, consider the hypotheses that Capella is made respectively of (_a_) iron, (_b_) gold. According to theory the opacity of a star made of the heavier element would be 2½ times the opacity of a star made of iron. This by itself would make the golden star a magnitude (= 2½ times) fainter. But the temperature is raised by the substitution; and although, as explained on p. 23, the change is not very great, it increases the outflow of heat approximately 2½ times. The resultant effect on the brightness is practically no change. Whilst this independence of chemical constitution is satisfactory in regard to definiteness of the results, it makes the discrepant factor 10 particularly difficult to explain.]

[Footnote 8: Observation shows that the sun is about 4 magnitudes fainter than the average diffuse star of the same spectral class, and Krueger 60 is 10 magnitudes fainter than diffuse stars of its class. The whole drop was generally assumed to be due to deviation from a perfect gas; but this made no allowance for a possible difference of mass. The comparison with the curve enables the dense star to be compared with a gaseous star _of its own mass_, and we see that the difference then disappears. So that (if there has been no mistake) the dense star is a gaseous star, and the differences above mentioned were due wholly to differences of mass.] [Footnote 9: Rougher estimates were made much earlier.]

[Footnote 10: The observed period of Algol is the period of revolution, not of rotation. But the two components are very close together, and there can be no doubt that owing to the large tidal forces they keep the same faces turned towards each other; that is to say, the periods of rotation and of revolution are equal.]

[Footnote 11: It may be of interest to add that although the proper light of Algol B is inappreciable, we can observe a reflection (or re-radiation) of the light of Algol A by it. This reflected light changes like moonlight according as Algol B is ‘new’ or ‘full’.]

[Footnote 11: The mass-luminosity relation was not suspected at the time of which I am speaking.]

[Footnote 13: My references to ‘_perfect gas_ of the density of platinum’ and ‘_material_ 2,000 times denser than platinum’ have often been run together by reporters into ‘perfect gas 2,000 times denser than platinum’. It is scarcely possible to calculate what is the condition of the material in the Companion of Sirius, but I do not expect it to be a perfect gas.]

[Footnote 14: Photographed by Dr. W. H. Wright at the Lick Observatory, California.]

[Footnote 15: Nos. 43, 61, 75 are recent discoveries and may require confirmation. There now remain only two gaps (85 and 87) apart from possible elements beyond uranium.]

[Footnote 16: It does not give _both_ temperature and pressure, but it gives one if the other is known. This is valuable information which may be pieced together with other knowledge of the conditions at the surface of the stars.]

[Footnote 17: Hydrogen (being element No. 1) has only one planet electron.]

[Footnote 18: Fig. 9 is a photograph of the ‘flash spectrum’ of the sun’s chromosphere taken by Mr. Davidson in Sumatra at the eclipse of 14 January 1926.]

[Footnote 19: The helium line in the Ring Nebula on which we have already commented is not a member of the Pickering Series, but it has had the same history. It was first supposed to be due to hydrogen, later (in 1912) reproduced by Fowler terrestrially in a mixture of helium and hydrogen, and finally discovered by Bohr to belong to helium.]

[Footnote 20: This, of course, is found from the other lines of the spectrum which genuinely belong to the star and shift to and fro as it describes its orbit.]

[Footnote 21: As the word temperature is sometimes used with new-fangled meanings, I may add that 15,000° is the temperature corresponding to the individual speeds of the atoms and electrons--the old-fashioned gas-temperature.]

[Footnote 22: Photograph taken by E. T. Cottingham and the author in Principe at the total eclipse of 29 May 1919.]

[Footnote 23: We refer to calcium as it occurs in the chromosphere, i. e. with one electron missing.]

[Footnote 24: There is an awkwardness in applying the term ‘apparent’ to something too small to be seen; but, remembering that we have armed ourselves with an imaginary telescope capable of showing the disk, the meaning will be clear.]

[Footnote 25: Densities below that of air have been found for some of the Algol variables by an entirely different kind of investigation, and also for some of the Cepheid variables by still another method. There are also many other examples of stars of bulk comparable with that of Betelgeuse.]

[Footnote 26: From a photograph taken at the Royal Observatory, Cape of Good Hope.]

[Footnote 27: For comparison, the nearest fixed star is distant 4 light years. Apart from clusters we rarely deal with distances above 2,000 light years.]

[Footnote 28: One cannot always be sure that what is true of the cluster stars will be true of stars in general; and our knowledge of the nearer stars, though lagging behind that of the stars in clusters, does not entirely agree with this association of colour and brightness.]

[Footnote 29: The term nebula covers a variety of objects, and it is only the nebulae classed as spirals that are likely to be outside our stellar system.]

[Footnote 30: This can be checked because uranium lead has a different atomic weight from lead not so derived. Ordinary lead is a mixture of several kinds of atoms (isotopes).]

[Footnote 31: You may wonder why, having said that the sun contains 2,000 quadrillion tons of energy _at the most_, I now assume that it contains just this amount. It is really only a verbal point depending on the scientific definition of energy. All mass is mass of _something_, and we now call that something ‘energy’ whether it is one of the familiar forms of energy or not. You will see in the next sentence that we do not assume that the energy is convertible into known forms, so that it is a terminology which commits us to nothing.]

[Footnote 32: Aston in his latest researches has been able to detect that the oxygen atom is just appreciably lighter than the four helium atoms.]

[Footnote 33: A measurement of the heat observed to flow from a continuous fountain of heat is a measurement of the output of the fountain, unless there is a storing of energy between the output and the outflow. The breakdown of the Kelvin time-scale indicates that the storing in the stars (positive or negative) and consequent expansion or contraction is negligible compared to the output or outflow.]

[Footnote 34: The stars all put together cover an area of the sky much less than the apparent disk of the sun, so that unless their surface-layers are generating this radiation very much more abundantly than the sun does, they cannot be responsible for it.]

[Footnote 35: The term ‘dwarf stars’ is not meant to include _white dwarfs_.]

[Footnote 36: We can scarcely suppose that all stars after reaching the main series pass through _precisely_ the same stages. For example, Algol, when it has become reduced to the mass of the Sun, may have slightly different density and temperature. But the observational evidence indicates that these individual differences are small. The main series is nearly a linear sequence; it must have some ‘breadth’ as well as ‘length’, but at present the scatter of the individual stars away from the central line of the sequence seems to be due chiefly to the probable errors of the observational data and the true breadth has not been determined.]

[Footnote 37: Exhaustion of supply without change of mass would cause the star to contract to higher density; it would thus have a combination of density and mass which (according to observation) is not found in any actual stars.]

[Footnote 38: This increase was assumed in our detailed description of the automatic adjustment of the star, and it will be seen that it was essential to assume it.]

APPENDIX

_Further Remarks on the Companion of Sirius_

I HAVE preferred not to complicate the Story of the Companion of Sirius with details of a technical kind; some further information may, therefore, be welcome to those readers who are curious to learn as much as possible about this remarkable star. I am also able to add a further instalment of the ‘detective story’ which has just come to hand, the sleuth this time being Mr. R. H. Fowler.

The star is between the eighth and ninth magnitude, so that it is not an excessively faint object. The difficulty in detecting it arises entirely from the overpowering light of its neighbour. At favourable epochs it has been seen easily with an 8-inch telescope. The period of revolution is 49 years.

The Companion is separated from Sirius by a distance nearly equal to the distance of Uranus from the Sun--or twenty times the earth’s distance from the sun. It has been suggested that the light might be reflected light from Sirius. This would account for its whiteness, but would not directly account for its spectrum, which differs appreciably from that of Sirius. To reflect ¹⁄₁₀₀₀₀th of the light of Sirius (its actual brightness) the Companion would have to be 74 million miles in diameter. The apparent diameter of its disk would be 0"·3, which, one would think, could scarcely escape notice in spite of unfavourable conditions of observation. But the strongest objection to this hypothesis of reflected light is that it applies only to this one star. The other two recognized white dwarfs have no brilliant star in their neighbourhood, so that they cannot be shining by reflected light. It is scarcely worth while to invent an elaborate explanation for one of these strange objects which does not cover the other two.