Radiation of Mass

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

Atoms or systems into ruin hurl’d,

And now a bubble burst, and now a world.

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 thatthe 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. Thismaybe 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 thetwo 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 smallermass 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 wemay 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 onp. 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 sointelligently; 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 fartherand 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 mayhave 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 onlythe 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 theexhaustibility 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:[1]Fig. 1is from a photograph taken by Mr. Evershed at Kodaikanal Observatory, Madras.Fig. 2is from the Mount Wilson Observatory, California.[2]I am indebted to Professor C. T. R. Wilson for Figs.Fig. 3-Fig. 6.[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.[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.[5]The mean density of Capella is nearly the same as the density of the air.[6]Unless otherwise indicated ‘gaseous’ is intended to mean ‘composed ofperfectgas’.[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 onp. 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.[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 starof 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.[9]Rougher estimates were made much earlier.[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.[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’.[11]The mass-luminosity relation was not suspected at the time of which I am speaking.[13]My references to ‘perfect gasof the density of platinum’ and ‘material2,000 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.[14]Photographed by Dr. W. H. Wright at the Lick Observatory, California.[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.[16]It does not givebothtemperature 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.[17]Hydrogen (being element No. 1) has only one planet electron.[18]Fig. 9is a photograph of the ‘flash spectrum’ of the sun’s chromosphere taken by Mr. Davidson in Sumatra at the eclipse of 14 January 1926.[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.[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.[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.[22]Photograph taken by E. T. Cottingham and the author in Principe at the total eclipse of 29 May 1919.[23]We refer to calcium as it occurs in the chromosphere, i. e. with one electron missing.[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.[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.[26]From a photograph taken at the Royal Observatory, Cape of Good Hope.[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.[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.[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.[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).[31]You may wonder why, having said that the sun contains 2,000 quadrillion tons of energyat 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 ofsomething, 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.[32]Aston in his latest researches has been able to detect that the oxygen atom is just appreciably lighter than the four helium atoms.[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.[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.[35]The term ‘dwarf stars’ is not meant to includewhite dwarfs.[36]We can scarcely suppose that all stars after reaching the main series pass throughpreciselythe 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.[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.[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.

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

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

[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.

[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.

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

[6]Unless otherwise indicated ‘gaseous’ is intended to mean ‘composed ofperfectgas’.

[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 onp. 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.

[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 starof 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.

[9]Rougher estimates were made much earlier.

[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.

[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’.

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

[13]My references to ‘perfect gasof the density of platinum’ and ‘material2,000 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.

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

[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.

[16]It does not givebothtemperature 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.

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

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

[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.

[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.

[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.

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

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

[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.

[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.

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

[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.

[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.

[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.

[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).

[31]You may wonder why, having said that the sun contains 2,000 quadrillion tons of energyat 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 ofsomething, 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.

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

[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.

[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.

[35]The term ‘dwarf stars’ is not meant to includewhite dwarfs.

[36]We can scarcely suppose that all stars after reaching the main series pass throughpreciselythe 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.

[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.

[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.

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