Chapter III: Exploring in Time (2)

Obviously, however, we cannot carry out our calculations in this simple light-hearted way; it would be absurd to suppose that the sun’s last ton of substance will radiate energy at the same rate as his present stupendous mass of two thousand million million million million tons. A series of investigations which culminated in a paper published by Eddington in 1924, disclosed that, in a general sort of way, a star’s luminosity depends mainly on its weight. The dependence is not very precise, and neither is it universal, but when we are told a star’s weight we can say that its luminosity is likely, with a high degree of probability, to lie within certain fairly narrow limits. For instance most stars whose weight is nearly equal to the sun are found to have about the same luminosity as the sun. In general, as might be expected, stars of light weight radiate less than heavy stars, but also—and this could not have been foreseen—the differences in their radiations are far greater than the differences in their weights. The law which we have already noticed to hold for a few stars in the neighbourhood of the sun is true, although in a somewhat different sense, for the stars as a whole—the candle-power per ton is greatest in the heaviest stars. For example, the average star of half the weight of the sun does not radiate anything like half as much energy as the sun: the fraction is more like an eighth. This consideration extends the future life of the sun, and indeed of all the stars, almost indefinitely. A sort of parsimony seems to creep over the stars in their old age; so long as they have plenty of weight to squander, they squander it lavishly, but they contract their scale of expenditure when they have little left to spend. The sand runs slowly through the hour-glass when there is little left to run.

In the same way, the average star of double the sun’s weight does not merely radiate twice as much energy as the sun; it radiates about eight times as much. We must keep this in view in estimating the past life of the sun; it shortens the sun’s past life just as surely as the opposite effect lengthens its future life. Observation tells us at what rate the average star of any given weight spends its weight in the form of radiation, and, on the supposition that the sun has behaved like this typical average star at the corresponding stage of its own past history, we can draw up a table exhibiting its gradual change of weight as its life progressed. Selected entries from this table would read somewhat as follows:

2 × 10⁹ years ago, the sun had 1·00013 times its present weight
1,000 × 10⁹ ” ” 1·07 ” ”
2,000 × 10⁹ ” ” 1·16 ” ”
5,700 × 10⁹ ” ” double ”
7,100 × 10⁹ ” ” 4 times ”
7,400 × 10⁹ ” ” 8 ” ”
7,500 × 10⁹ ” ” 20 ” ”
7,600 × 10⁹ ” ” 100 ” ”

The first entry represents roughly the time since the earth was born. It shews that, during the whole existence of the earth, the sun’s weight has changed by only an inappreciable fraction of the whole. Consequently, it seems likely, although naturally we cannot be certain, that when the earth was born the sun was much the same as it now is, and that it has been the same, in all essential respects, throughout the whole life of the earth.

To come to appreciably different conditions we have to go back to remote aeons far beyond the time of the earth’s birth. We are free to do this, for we have seen that the earth’s whole life is only a moment in the lives of the stars. We have estimated the latter as being something of the order of 5 to 10 million million years, and it is only when we go back an appreciable fraction of these long periods that we find the sun’s weight differing appreciably from its present weight. We have, for instance, to go back more than 5 million million years to find the sun with double its present weight. When we go back much further than this a new phenomenon appears; the weight of our hypothetical past sun begins to go up by leaps and bounds. In time it begins to double and more than double every 100,000 million years, and we cannot go back as far as 8 million million years without postulating a sun of quite impossibly high weight. The sun must, then, have been born some time within the last 8 million million years.

The exact figures of our table may be open to suspicion, but as a general fact of observation there is no doubt that very massive stars radiate away their energy, and therefore also their weight, with extraordinary rapidity. Indeed the process is so rapid that we may disregard all that part of a star’s life in which it has more than about 10 times the weight of the sun—this is lived at lightning speed. Apart from all detailed calculations, this general principle fixes a definite limit to the ages, not only of the sun, but also of every other star. The upper limit to the age of the sun is certainly somewhere in the neighbourhood of 8 million million years.

This agrees well enough with the general age of from 5 to 10 million million years that other calculations have assigned to the stars in general. The calculations thus reinforce one another, and it looks as if at least two of the pieces of the puzzle were beginning to fit satisfactorily together. If all the stars in the sky were similar to the sun, we might feel a good deal of confidence in the conclusions we have reached.

Unfortunately, difficulties emerge as soon as we discuss the ages of stars which at present have many times the weight of the sun. The table on p. 164 shews that a class of stars (spectral type _A_ 0) of six times the weight of the sun have motions in space which conform well enough to the law of equipartition of energy. Unless this is a pure coincidence (and this is unlikely, in view of the fact that other groups of only slightly less weight conform equally well), we must assign an age of from 5 to 10 million million years to these very massive stars. Yet the average star of this weight is emitting about a hundred times as much radiation as the sun, which means that it is halving its weight every 150,000 million years. Clearly this process cannot have gone on for anything like 5 or 10 million million years.

Still more luminous stars present the problem in an even more acute form. The star _S_ Doradus in the Lesser Magellanic Cloud is at present emitting 300,000 times as much radiation as the sun. Whereas the sun is pouring its weight out into space at the rate of 650 Niagaras, _S_ Doradus is pouring it out at the rate of 200,000,000 Niagaras; every 50 million years it loses a weight equal to the total weight of the sun. It is obviously absurd to imagine that this star can have been losing weight at this rate for millions of millions of years.

For such a star as _S_ Doradus only two alternatives seem open. Either it was created quite recently (on the astronomical time-scale), and so is still at the very beginning of its prodigal youth, or else its loss of weight has in some way been inhibited through the greater part of its life. A good many arguments weigh against the hypothesis of recent creation. The star is a member of a star-cloud in which we should naturally expect all the members to be of approximately equal age. It is in a region of space in which there are no indications that stars are still being born. And, even if we accept the hypothesis of recent creation for this particular star, we are still at a loss to explain how the other massive stars, which figure in the table on p. 164, can be old enough for equipartition of energy to have become already established.

For many reasons it seems preferable, and indeed almost inevitable, to suppose that these highly luminous and very weighty stars have in some way been saved from energetic radiation, with its consequential rapid wasting of weight, throughout the greater part of their lives. In brief, we suppose that they are cases of arrested development, whose weight and general appearance equally belie their true ages. Later (p. 318) we shall come upon a physical mechanism which explains very simply and naturally how this could happen.

If this hypothesis can be accepted, it clears up the whole situation. As soon as we accept it, we become free to assign any age we please to the stars, and naturally select that indicated by the law of equipartition of energy, at any rate for those classes of stars which are found to conform to this law.

The exceptionally luminous stars which we have just had under discussion are comparatively rare objects in the sky. The vast majority of stars have luminosities and weights comparable with, or distinctly less than, those of the sun, and for these the difficulty does not exist. Indeed the hypothesis of arrested development would break down under its own weight if we had to invoke its help for many stars; it is tenable just because we seldom need to use it. We may accept the table on p. 180 as giving the past history of the sun with tolerable accuracy, thus fixing its age at something under 8 million million years, and a generally similar table would apply to most of the stars in the sky.

THE SOURCE OF STELLAR ENERGY

The ages of 5 million million years or more which we have been led to assign to the stars imply that at birth the sun must have had at least double, and more probably several times, its present weight. For every ton which existed in the sun at its birth only a few hundred-weight remain to-day. The rest of the ton has been transformed into radiation and, streaming away into space, has left the sun for ever.

In the preceding chapter, we had occasion to discuss the transformation of weight into radiation which accompanies the spontaneous disintegration of radio-active atoms. The most energetic instance of this phenomenon known on earth is the transformation of uranium into lead, in which about one part in 4000 of the total weight is transformed into radiation. In the sun, the corresponding fraction may be half, or nine-tenths, or even 99 per cent., but, whatever it is, it certainly exceeds one part in 4000. Thus the process by which the sun generates its light and heat must involve a far more energetic transformation of material weight into radiation than any process known on earth.

Perrin and Eddington at one time suggested that this process may be the building up of complex atomic nuclei out of protons and electrons. The simplest, and most favourable example of this, which was especially considered by Eddington, is to be found in the building up of the helium nucleus. The constituents of a helium atom are precisely identical with those of four hydrogen atoms, namely, four electrons and four protons. If these constituents could be rearranged without any transformation of material weight into radiation, the helium atom would have precisely four times the weight of the hydrogen atom. In actual fact Aston finds that the ratio of weights is only 3·970. The difference between this and 4·000 must represent the weight of the radiation which goes off when, if ever, the helium atom is built up by the coalescence of four hydrogen atoms. The loss of weight, one part in 130, is very much greater than occurs in radio-active transformations, but even so it does not provide adequate lives for the stars. The transformation of a sun which originally consisted of pure hydrogen into one consisting wholly of helium would only provide radiation at the sun’s present rate of radiation for about 100,000 million years, and the dynamical evidence of equipartition of energy, etc., as well as other evidence which we shall consider later (p. 188 below), demands far longer lives for the stars than this.

THE ANNIHILATION OF MATTER. Modern physics is only able to suggest one process capable of providing a sufficiently long life for a radiating star; it is the actual annihilation of matter. Various lines of evidence go to shew that the atoms in very massive stars are not, for the most part, fundamentally different from those in less massive stars. Thus the primary cause of the difference in weight between a heavy star and a light star is not a difference in the quality of the atoms; it is a difference in their number. A heavy star can only change into a light star through the actual disappearance of atoms; these must be annihilated, and their weight transformed into radiation.

I first drew attention in 1904 to the large amount of energy capable of being liberated by the annihilation of matter, positive and negative electric charges rushing together, annihilating one another and setting their energy loose in space as radiation. The next year Einstein’s theory of relativity provided a means for calculating the amount of energy which would be produced by the annihilation of a given amount of matter; it shewed that energy is set free at the rate of 9 × 10²⁰ ergs per gramme, regardless of the nature or condition of the substance which is annihilated. I subsequently calculated the length of lives which this source of energy permitted to the stars, but the calculated lives of millions of millions of years seemed greater than were needed by the astronomical evidence available at the time. Since then a continual accumulation of new evidence, particularly that discussed in the present chapter, has been seen to demand stellar lives of precisely these lengths, with the result that the majority of astronomers now regard annihilation of matter as the most probable source of stellar energy.

Other considerations in addition to those just mentioned point to the annihilation of matter as the fundamental process going on in the stars. If there were no annihilation of matter, a star could only change its weight by some small fraction of the whole, such, for example, as the one part in 4000 which accompanies radio-active disintegration, or as the one part in 130 which would result from the building up of helium atoms out of hydrogen. A star would retain its weight practically unaltered through its life. This would of course necessarily impose far shorter lives on the stars than we believe them to have had, for nothing can alter the fact that the sun loses 360,000 million tons of weight every day in radiation, so that if its weight cannot change much, it cannot have radiated for long.

We have seen that in the present universe, a star’s luminosity depends mainly on its weight. If we imagine that the same condition of things has always prevailed, then stars which retained the same weight throughout their lives would have to retain approximately the same luminosity, at any rate until their capacity for radiation became exhausted. Otherwise, contrary to observation, we should find stars with weights equal to the sun having all possible degrees of luminosity. Thus if we discard the hypothesis of the annihilation of matter, it becomes necessary to imagine some controlling mechanism, of a kind which would compel stars having the weight of the sun always to radiate at about the same rate as the sun, at least until sheer exhaustion prevents them from radiating any more, and similarly for stars of all other weights.

There does not seem to be any general objection against supposing such a controlling mechanism to exist, and indeed such mechanisms have been advocated by Russell and Eddington. But when we consider such a mechanism in detail, we encounter various objections which we shall consider in Chapter V (p. 294), the principal of which is that stars controlled by it would, so far as we can see, be in a highly explosive state. And immediately we abandon the hypothesis of such a controlling mechanism, the observed close dependence of luminosity on weight compels us to suppose that a star’s weight decreases as its luminosity diminishes, which leads us back immediately to the annihilation of matter.

A further consideration which points in the same direction may be mentioned here. We have seen how the “candle-power per ton of weight” is greatest in the heavier stars. As an immediate consequence the loss of weight per ton is greatest in the heaviest stars. In the time in which a massive star loses a hundred-weight per ton, a star of light weight may lose only a few pounds per ton. The consequence is that the passage of time tends to equalise the weights of the stars. This principle no doubt explains in large part why the present stars shew no very great range of weight. It also leads to interesting consequences when applied to the two components of a binary system. It shews that as a binary system ages, its two components ought continually to become more nearly equal in weight. Thus the two components ought to differ less in weight in old binaries than in young.

This last conclusion can be tested observationally. As regards spectroscopic binaries, Aitken finds that the ratio of weights of the two constituents of a binary increases from about 0·70 for young systems of large weight to 0·90 for older systems in which the constituents are about similar to the sun. The direction of change is that predicted by theory; the amount of change indicates a time-interval of the order of millions of millions of years between the two states concerned. Other astronomers have studied the corresponding problems presented by eclipsing and visual binaries, and have reached almost identical conclusions. The predictions of theory seem to be confirmed by each type of binary system separately.

On the whole, in whatever direction we try to escape from the hypothesis of annihilation of matter, the alternative hypothesis we set up to explain the facts seems to lead back in time to the annihilation of matter.

We must not overlook the revolutionary nature of the change which this hypothesis introduces into physical science. The two fundamental corner-stones of nineteenth-century physics, the conservation of matter and the conservation of energy, are both abolished, or rather are replaced by the conservation of a single entity which may be matter and energy in turn. Matter and energy cease to be indestructible and become interchangeable, according to the fixed rate of exchange of 9 × 10²⁰ ergs per gramme.

Yet, looked at from another angle, the hypothesis only carries physics one stage further along the road it has already trodden in the past. Heat, light, electricity have all in turn proved to be forms of energy; the annihilation hypothesis only proposes to add another to the list, so that matter itself also becomes a form of energy.

According to this hypothesis all the energy which makes life possible on earth, the light and heat which keep the earth warm and grow our food, and the stored up sunlight in the coal and wood we burn, if traced far enough back, are found to originate out of the annihilation of electrons and protons in the sun. The sun is destroying its substance in order that we may live, or, perhaps we should rather say, with the consequence that we are able to live. The atoms in the sun and stars are, in effect, bottles of energy, each capable of being broken and having its energy spilled throughout the universe in the form of light and heat. Most of the atoms with which the sun and stars started their lives have already met this fate; the remainder are doubtless destined to meet it in time. Scientific writers of half a century ago delighted in the picturesque description of coal as “bottled sunshine”; they asked us to think of the sunshine as being bottled-up as it fell on the vegetation of the primaeval jungle, and stored for use in our fireplaces after millions of years. On the modern view we must think of it as re-bottled sunshine, or rather re-bottled energy. The first bottling took place millions of millions of years ago, before either sun or earth was in being, when the energy was first penned up in protons and electrons. Instead of thinking prosaically of our sun as a mere collection of atoms, let us think of it for a moment as a vast storehouse of bottles of energy which have already lain in storage for millions of millions of years. So enormous is the sun’s supply of these bottles, and so great the amount of energy stored in each that, even after radiating light and heat for 7 or 8 million million years, it still has enough left to provide light and heat for millions of millions of years yet to come.

Two quantitative considerations may help to shew these processes in a clearer light. We have seen that the sun’s present store of atoms would, at the present rate of breakage, last for 15 million million years. This means that every year only one atom in 15 million million is broken, a fraction which may seem absurdly small to produce the sun’s vast continuous outpourings of energy. Let us, however, reflect that the energy which is continually pouring out of the sun’s surface at the rate of about 50 horse-power per square inch is generated throughout the vast interior of the sun’s body; the stream of energy which emerges from a square inch of surface is the concentration of all the energy generated in a cone of a square inch cross-section, but of 433,000 miles depth. Such a cone contains about 10³³ atoms, and although only one in 15 million million is broken each year, there are still about two million million atoms destroyed each second.

Even so, the amount of energy set free by the annihilation of matter is rather surprising; it is of an entirely different order of magnitude from that made available by any other treatment. The combustion of a ton of the best coal in pure oxygen liberates about 5 × 10¹⁶ ergs of energy; the annihilation of a ton of coal liberates 9 × 10²⁶ ergs, which is 18,000 million times as much. In the ordinary combustion of coal we are merely skimming off the topmost cream of the energy contained in the coal, with the consequence that 99·999999994 per cent. of the total weight remains behind in the form of smoke, cinders or ash. Annihilation leaves nothing behind; it is a combustion so complete that neither smoke, ash, nor cinders is left. If we on earth could burn our coal as completely as this, a single pound would keep the whole British nation going for a fortnight, domestic fires, factories, trains, power-stations, ships and all; a piece of coal smaller than a pea would take the _Mauretania_ across the Atlantic and back.

Purely astronomical evidence has led to the conclusion that atoms are continually being annihilated in the sun and stars. Here we have a piece of the puzzle which fits perfectly on to those we tentatively fitted together in the last chapter. As we there saw, recent investigations in mathematical physics suggest that the highly penetrating radiation received on earth has its origin in the annihilation of matter out in space. And the amount of this radiation received on earth is so great that we had to suppose the underlying annihilation of matter to be one of the fundamental processes of the universe; we now discover that it is in all probability the process which keeps the sun and stars shining and the universe alive.

PHYSICAL INTERPRETATION. It is perhaps worth trying to probe still one stage further into the physical nature of this process of annihilation of matter, although it must be premised that what follows is speculative in the sense that no direct observational confirmation is at present available.

We saw (p. 135) how the electrodynamical theory current in the last century required that the nucleus and electron of the hydrogen atom should approach ever closer and closer to one another with the mere passage of time, until finally they rushed together and coalesced. When this happened, the negative charge of the electron and the positive charge of the nucleus would neutralise one another and their energy would go off in a flash of radiation similar to the flash of lightning which indicates that the negative and positive charges in two opposing thunderclouds have met and neutralised one another.

The more recent quantum theory calls a halt to this motion as soon as the nucleus and electron have approached to within a distance of 0·53 × 10⁻⁸ centimetres of one another, and by so doing keeps the universe in being as a going concern (p. 135). Other halts are also established at 4, 9, 16, etc. times this distance, but here the prohibition on further progress is not absolute. At these longer distances the demand of the quantum theory “thus far shalt thou go and no further” seems to be replaced by “thou shalt go no further until after a long time.” And it now seems possible, on the astronomical evidence, that the prohibition at the shorter distance may not be absolute either. From the physical end nothing is known for certain, although here again it seems contrary to the newer conceptions of physics, as embodied in the wave-mechanics, that any such absolute prohibition should exist, either for the hydrogen atom or for other more complex atoms. Perhaps after waiting a long time in the orbit nearest to the nucleus, the electron is permitted, or even encouraged or compelled, to proceed; it merges itself into the nucleus and a flash of radiation is born in a star. This provides the most obvious mechanism for the annihilation of electrons and protons which the evidence of astronomy seems to demand. It will, however, be clearly understood that this is a purely conjectural conception of the mechanism; we shall return to a further consideration of this very intricate problem in Chapter V.

If this conjecture should prove to be sound, not only the atoms which provide stellar light and heat, but also every atom in the universe, are doomed to destruction, and must in time dissolve away in radiation. The solid earth and the eternal hills will melt away as surely, although not as rapidly, as the stars:

The cloud-capped towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve,
And ... leave not a rack behind.

And if the universe amounts to nothing more than this, shall we carry on the quotation:

We are such stuff
As dreams are made on; and our little life
Is rounded with a sleep,

—or shall we not?