Author: Sir James Jeans, M. A., D. Sc., Sc. D., LL. D., F. R. S.
[p. 150] We have explored space to the furthest depths to which our telescopes can probe; we have explored into the intricacies of the minute structures we call atoms, of which the whole material universe is built; we now wish to go exploring in time. Man’s individual span of life, and indeed the whole span of time covered by our historical records — some few thousands of years at most — are both far too short to be of any service for our purpose. We must find far longer measuring rods with which to sound the depths of past time and to probe forward into the future.
Our general method will be one which the study of geology has already made familiar. Undeterred by the absence of direct historical evidence, the geologist insists that life has existed on earth for millions of years, because fossil remains of life are found to occur under deposits which, he estimates, must have taken millions of years to accumulate. As he digs down through different strata in succession, he is exploring in time just as truly as the geographer who travels over the surface of the earth is exploring in space. A similar method can be used by the astronomer. We find some astronomical effect, quality, or property, which exhibits a continual accumulation or decrease, like the sand in the bottom or top half of the hourglass; we estimate the rate at which this increase or decrease is occurring at the present moment, and also, if we can, the rate at which it must have occurred under the different conditions prevailing in the past. It then becomes a question, perhaps of mere arithmetic, [p. 151] although possibly of more complicated mathematics, to estimate the time which has elapsed since the process first started.
The method is well exemplified in the comparatively simple problem of the age of the earth.
The first scientific attempt to fix the age of the earth was made by Halley, the astronomer, in the year 1715. Each day the rivers carry a certain amount of water down to the sea, and this contains small amounts of salt in solution. The water evaporates and in due course returns to the rivers; the salt does not. As a consequence the amount of salt in the oceans goes on increasing ; each day they contain a little more salt than they did on the preceding day, and the present salinity of the oceans gives an indication of the length of time during which the salt has been accumulating. “We are thus furnished with an argument,” said Halley, somewhat optimistically, “for estimating the duration of all things.”
This line of argument does not lead to very precise estimates of the earth’s age, but calculations based on modern data suggest that it must be many hundreds of millions of years.
More valuable information can be obtained from the accumulation of sediment washed down by the rain. Every year that passes witnesses a levelling of the earth’s surface. Soil which was high up on the slopes of hills and mountains last year has by now been washed down to the bottoms of muddy rivers by the rain and is continually being carried out to sea. The Thames alone carries between one and two million tons of soil [p. 152] out to sea every year. For how long will England last at this rate, and for how long can it have already lasted? In our own lifetimes we have seen large masses of land round our coasts form landslides, and either fall wholly into the sea or slip down nearer to sea-level. Such conspicuous land-marks as the Needles, and indeed a large part of the southern coast of the Isle of Wight, are disappearing before our eyes. The geologist can form an estimate of the rapidity with which these and similar processes are happening, and so can estimate how long sedimentation has been in progress to produce the observed thickness of geological layers.
These thicknesses are very great; Professor Arthur Holmes[1] gives the observed maximum thicknesses as follows :
| Era | Thickness |
|---|---|
| Pre-Cambrian | at least 180,000 feet |
| Palaeozoic Era (Ancient life) | 185,000 „ |
| Mesozoic Era (Mediaeval life) | 91,000 „ |
| Cainozoic Era (Modern life) | 73,000 „ |
We can form a general idea of the rate at which these sediments have been deposited. Since Rameses II reigned in Egypt over 3000 years ago, sediment has been deposited at Memphis at the rate of a foot every 400 or 500 years; the excavator must dig down 6 or 7 feet to reach the surface of Egypt as it stood when Rameses II was king. The present rate of denudation in North America is estimated to be one foot in 8600 years; similar estimates for Great Britain indicate a rate of one foot in 3000 years. With geological strata deposited at an average rate of one foot per 1000 years, the total 529,000 feet of strata listed above would [p. 153] require over 500 million years for their deposition. At a rate of one foot per 4000 years, the time would be about 2100 million years.
This method of estimating geological time has been described as the “Geological hour-glass.” We see how much sand has already run, we notice how fast it is running now, and a calculation tells us how long it is since it first started to run. The method suffers from the usual defect of hour-glasses, that there is no guarantee that the sand has always run at a uniform rate. Geological methods suffice to shew that the earth must be hundreds of millions of years old, but to obtain more definite estimates of its age, the more precise methods of physics and astronomy must be called in. Fortunately the radio-active atoms discussed in the previous chapter provide a perfect system of clocks, whose rate so far as we know does not vary by a hair’s breadth from one age to another.
We have seen how, with the lapse of sufficient time, an ounce of uranium disintegrates into 0*865 ounce of lead and 0.135 ounce of helium. The process of disintegration is absolutely spontaneous; no physical agency known in the whole universe can either inhibit or expedite it in the tiniest degree. The following table shews the rate at which it progresses:
| Initially: | 1 oz. uranium | No lead |
|---|---|---|
| After 100 million years | 0.985 oz. uranium | 0.013 oz. lead |
| „ 1000 „ | 0.865 „ | „ 0116 „ „ |
| „ 2000 „ | 0.747 „ | „ 0.219 „ |
| „ 3000 „ | 0.646 „ | „ 0.306 „ |
and so on. Thus a small amount of uranium provides a perfect clock, provided we are able to measure the [p. 154] amount of lead it has formed, and also the amount of uranium still surviving, at any time we please. When the earth first solidified, many fragments of uranium were imprisoned in its rocks, and may now be used to disclose the age of the earth. We are not entitled to assume that all the lead which is found associated with uranium has been formed by radio-active integration. But, by a fortunate chance, lead which has been formed by the disintegration of uranium is just a bit different from ordinary lead; the latter has an atomic weight of 207.2, while the former is of atomic weight only 206.0. Thus a chemical analysis of any sample of radioactive rock shews exactly how much of the lead present is ordinary lead, and how much has been formed by radio-active disintegration. The proportion of the amount of lead of this latter kind to the amount of uranium still surviving tells us exactly for how long the process of disintegration has been going on.
In general all the samples of rock which are examined tell much the same story, and the radio-active clock is found to fix the time since the earth solidified at 1400 million years or more. The clock cannot tell us for how long before this the earth had existed in a plastic or fluid state, since in this earlier state the products of disintegration were liable to become separated from one another.
Aston has recently discovered a new isotope (see p. 118) of uranium, called actino-uranium. As uranium and its isotope have different periods of decay, the relative abundance of the two is continually changing. From the ratio of the amounts of these substances now surviving on earth, Rutherford has calculated that the age of the earth cannot exceed 3400 million years, and is probably substantially less.
[p. 155] These two physical estimates of the time which has elapsed since the earth solidified stand as follows :
| Origin | Age |
|---|---|
| 1) From the lead-uranium ratio in radio-active rocks |
More than 1400 million years |
| 2) From the relative abundance of uranium and actino-uranium |
Less than 3400 million years |
Various astronomical methods arc also available for determining the time since the solar system came into being. Here the “clocks” are provided by the shapes of the orbits of various planets and satellites. The orbits do not change at uniform rates, but their changes are determined by known laws, so that the mathematician can calculate the rates at which change occurred under past conditions, and hence, by totalling up, can deduce the time needed to establish present conditions. The following two estimates are both due to Dr H. Jeffreys :
| Method | Age |
|---|---|
| 1) From the orbit of Mercury | …From 1000 to 10,000 million years. |
| 2) „ „ the Moon | …Roughly about 4000 million years. |
While these various figures do not admit of any very exact estimate of the earth’s age, they all indicate that this must be measured in thousands of millions of years. If we wish to fix our thoughts on a round number, probably 2000 million years is the best to select.
We now turn to the far more difficult problem of determining the ages of the stars.
We shall not approach it by a direct frontal attack, [p. 156] but start far away from our real objective. Let us in fact start at the extreme other end of the universe, and delve a bit further into the properties of a gas.
Equipartition of Energy in a Gas. We have pictured a gas as an indiscriminate flight of molecule- bullets. These fly equally in all directions, occasionally crashing into one another, and in so doing, changing both their speeds and directions of flight. We have seen that the total energy of motion undergoes no decrease when such collisions occur. If one of the molecules taking part in a collision has its speed checked, the other has its speed increased by such an amount that the energy lost by one molecule is gained by the other. Total energy of motion is “conserved.”
Into this random hail of bullets, let us imagine that we project a far heavier projectile, which we may call a cannon-ball, with a speed equal to about the average speed of the bullets. The energies of the various projectiles are proportional jointly to their weights and to the squares of their speeds, so that in the present case, in which the speeds are all much the same, the big projectile has more energy than the bullets simply on account of its greater weight. If it weighs as much as a thousand bullets, it has a thousand times as much energy as each single bullet.
Yet the heavy projectile cannot for long continue swaggering through its lesser companions with a thousand times its fair share of energy. Its first experience is to encounter a hail of bullets on its chest. Very few bullets hit it in the back, for they are only moving at about its own speed, and so can hardly overtake it from behind. Moreover, even if they do, their blows on its back are very feeble because they are hardly moving faster than it. But the shower of blows [p. 157] on its chest is serious; every one of these tends to check its speed, and so to lessen its energy. And as the total energy of motion is conserved at every collision, it follows that, while the big projectile is losing energy all the time, the little ones must be gaining energy at its expense.
For how long will this interchange of energy go on? Will it, for instance, continue until the big projectile has lost all its energy, and been brought completely to rest? The problem is one for the mathematician, and it admits of a perfectly exact mathematical solution, which Maxwell gave as far back as 1859. The big projectile is not deprived of all its energy. As its speed gradually decreases, conditions change in all sorts of ways. When we allow for this change of conditions, we find that the energy of the big projectile goes on decreasing, not until it has lost all its energy, but until it has no more energy than the average bullet. When this stage is reached, the hits of the bullets are as likely on the average to increase the energy of the big projectile as to decrease it, so that this ends up by fluctuating around an amount equal to the average energy of the little projectiles.
Maxwell, and others after him, further shewed that no matter how many kinds of molecules there may be mixed together in a gas, and no matter how widely their weights may differ from one another, their repeated collisions must ultimately establish a state of things in which big molecules and little, light and heavy, all have the same average energy. This is known as the theorem of equipartition of energy. It does not mean that at any single instant all the molecules have precisely the same energy; obviously such a state of things could not continue for a moment, since the first [p. 158] collision between any pair of molecules would upset it immediately. But on averaging the energy of each molecule over a sufficiently long period of time — say a second, which is a very long time indeed in the life of a molecule, being the time in which at least a hundred million collisions occur — we shall find that the average energy of all the molecules is the same, regardless of their weights.
The same theorem can be stated in a slightly different form. Air consists of a mixture of molecules of different kinds and of different weights — molecules of helium which are very light, molecules of nitrogen which are far heavier, each weighing as much as seven molecules of helium, and the still heavier molecules of oxygen, each with the weight of eight molecules of helium. In its alternative form, the theorem tells us that at any instant the average energy of all the molecules of helium, in spite of their light weights, is exactly equal to the average energy of the molecules of nitrogen, and again each of these is exactly equal to the average energy of the molecules of oxygen. The lighter types of molecule make up for their small weights by their high speeds of motion. Similar statements are of course true for any other mixture of gases.
The truth of the theorem is confirmed observationally in a great variety of ways. In 1846, Graham measured the relative speeds with which the molecules of different kinds of gas moved, by observing the rates at which they streamed through an orifice into a vacuum; these proved to be such that the average energies of the various types of molecules were precisely equal to one another. Even earlier than this, Leslie and others had used this method to determine the [p. 159] relative weights of different molecules, although without fully understanding the underlying theory. Thus it may be accepted as a well-established law of nature that no molecule is allowed permanently to retain more energy than his fellows; in respect of their energies of motion, a gas forms a perfectly organised communistic state in which a law, which they cannot evade, compels the molecules to share their energies equally and fairly.
Subject to certain slight modifications, the same law applies also to liquids and solids. In liquids and gases, we can actually perform an experiment analogous to that of projecting our imaginary cannon-ball into the hail of molecule-bullets, and watch events. We may take a few grains of very fine powder, powdered gamboge or lycopodium seed, for instance, and let these play the part of super-molecules amongst the ordinary molecules of a gas or liquid. A powerful microscope shews that these super-molecules are not brought completely to rest, but retain a certain liveliness of movement, as they are continually hit about by the smaller and quite invisible true molecules. It looks for all the world as though they were affected by a chronic St Vitus’ dance, which shews no signs of diminishing as time goes on. These movements are called “Brownian movements”, after Robert Brown, the botanist, who first observed them in the sap of plants. Brown at first interpreted them as evidence of real life in the small particles affected by them, an interpretation which he had to abandon when he found that particles of wax shewed the same movements. In a series of experiments of amazing delicacy, Perrin not only observed, but also measured, the Brownian movements of small solid particles as they were hit about [p. 160] by the molecules of air and other gases, and deduced the weights of the molecules of these gases with great accuracy.
Stellar Equipartition Of Energy. We can now get back to the stars. The theorem of equipartition of energy is true not only of the molecules of a gas, and of a solid, and of a liquid ; it is true also of the stars of the sky. The processes of mathematics are applicable to the very great as well as to the very small, and a theorem which is proved true for the minutest of atoms is equally true for the most stupendous of stars, provided of course that the premisses on which it is based remain true, and do not suffer by transference from the small to the great end of the universe.
Now the conditions which are necessary for the theorem of equipartition of energy to be true happen to be amazingly simple; indeed it is difficult to believe that such wide consequences can follow from such simple conditions. They amount to practically nothing beyond a law of continuity and a law of causation ; in other words, that the state of the system at any instant shall follow inevitably from its state at the preceding instant, or if you like, that there shall be no free-will among the molecules or stars or other bodies whose motions are under discussion. In the present turmoil as to the fundamental laws of physics, we cannot be entirely certain as to how far these very simple conditions are fulfilled in the molecular problem, although abundant observational evidence makes it clear that the law of equipartition holds, at any rate to an exceedingly good approximation, in an ordinary gas.
On the other hand, there is not the slightest doubt as to what determines the motions of the stars; it is the law of gravitation, every star attracting every [p. 161] other star with a force which varies inversely as the square of their distance apart. This is Newton’s form of the law, but it is a matter of complete indifference for our present purpose whether we use the law in Newton’s or in Einstein’s form; for stellar problems the two are practically indistinguishable, and there is abundant evidence, particularly from the observed orbits of binary stars, in favour of either. The essential point is that, from the single supposition that the motions of the stars are governed by either of these laws of gravitation — or, for the matter of that, by any other not entirely dissimilar law — we can prove the theorem of equipartition of energy to be true for these motions. No subtle statement of exact conditions is required; the mere law of gravitation, together with the supposition that the stars cannot exercise free-will as to whether they obey it or not, is enough.
It is important to understand quite clearly what precisely the theorem asserts when applied to the stars. It does not of course assert that all the stars in the sky have equal energies. It does not even assert that on the average the heavy-weight stars in the sky have the same energy as the light-weight stars. What it asserts is that if we put any miscellaneous assortment of stars into space, then, after they have interacted with one another for a sufficient length of time (this is the essential point), those which started with more than their fair share of energy will have been compelled to hand over their excess to stars with lesser energy, so that the average energy of all the different types of stars must necessarily become reduced to equality in the long run.
In the molecular problem, the interaction between the molecules takes place through the medium of [p. 162] collisions, and equipartition of energy is established, to a very good approximation, after some eight or ten collisions have happened to each molecule. In ordinary air, this requires a period of only about a hundred- millionth part of a second.
In the stellar problem, we are dealing with very different lengths of time; collisions only occur at intervals of thousands of millions of millions of years. If the stars only redistributed their energy when actual collisions occurred, we might surmise that a close approximation to equipartition of energy would not be attained until after each star had experienced eight or ten collisions, and this would require a really stupendous length of time. Actually no such length of time is needed because the numerous gravitational pulls, even between stars which are at a considerable distance apart, equalise energy far more efficiently and expeditiously than the very rare direct hits. Every time that two stars happen to pass even fairly near to one another in their wanderings, each pulls the other a bit out of its course, and the directions and speeds of motion of both stars are changed — by much or little according as the stars pass quite close to one another or keep at a substantial distance apart. In brief, each approach of stars causes an interchange of energy, and after sufficient time, these repeated interchanges of energy result in the total energy being shared equally, on the average, between the stars, regardless of differences in their weights.
Now the crux of the situation, to which all this has been leading up, is that observation shews that stars of different weights are moving with different average speeds, these average speeds being such that equipartition of energy already prevails among the stars — [p. 163] not absolutely exactly, but to a tolerably good approximation.
The question of how long the stars must have interacted to reach such a condition now becomes one of absolutely fundamental importance, for the answer tells us the ages of the stars.
Stellar velocities. We have already seen (p. 48) how stars which form binary systems can be weighed, such weighings disclosing weights ranging from about a hundred times the weight of the sun to only a fifth of its weight. The speeds of motion of binary systems can be measured in precisely the same way as the speeds of single stars. As far back as 1911, Halm, with an accumulation of such measurements before him, pointed out that the heaviest stars moved the most slowly. He found that, on the average, the heaviest of known stars had approximately the same energy of motion as the lightest, the high speeds of the latter just about making up for the smallness of their weights, and so suggested that the velocities of the stars, like those of the molecules of a gas, might be found to conform to the law of equipartition of energy. It appeared to be a case of Brownian movements on a stupendous scale.
Since then a great deal more observational evidence has accumulated, and an exhaustive investigation made by Dr Seares of Mount Wilson in 1922 leaves very little room for doubt that the motions of the stars shew a real, and fairly close, approximation to equipartition of energy. The table overleaf shews the final result of Seares’ discussion.
The stars are first classified according to the different types of spectrum their light shews when analysed in a spectroscope.
| Type of star | Average weight M (grammes) |
Average speed C (cms.asec.) |
Average energy ½MC2 (ergs) |
Corresponding temperature (degrees) |
|---|---|---|---|---|
| Spectral type B 3 | 19.8 x l033 | 14.8 x l05 | 1.95 x l046 | 1.0 x l062 |
| „ B 8.5 | 12.9 | 15.8 | 1.62 | 0.8 |
| „ A O | 12.1 | 24.5 | 3.63 | 1.8 |
| „ A 2 | 10.0 | 27.2 | 3.72 | 1.8 |
| „ A 5 | 8.0 | 29.9 | 3.55 | 1.7 |
| „ F O | 5.0 | 35.9 | 3.24 | 1.6 |
| „ F 5 | 3.1 | 47.9 | 3.55 | 1.7 |
| „ G O | 2.0 | 64.6 | 4.07 | 2.0 |
| „ G 5 | 1.5 | 77.6 | 4.57 | 2.2 |
| „ K O | 1.4 | 79.4 | 4.27 | 2.1 |
| „ K 5 | 1.2 | 74.1 | 3.39 | 1.7 |
| „ M O | 1.2 | 77.6 | 3.55 | 1.7 |
[p. 164] These different types of stars have very different average weights ; the second column of the table shews that they exhibit a range of over 16 to 1. The third column, which gives the average speeds of these different types of stars, shews that the heaviest stars move the most slowly, and the lightest on the whole the most rapidly. The next column gives the average energy of motion of the different types of stars. This shews that the variation in speeds is just about that needed to make the average energies of all types of stars equal. An exception certainly occurs in the first two lines, which refer to the heaviest stars of all. Apart from these, the remaining ten lines shew a ratio of 10 to 1 in weight, whereas the average deviation of energy from the mean is only one of 9 per cent.
From this we see that the motions of the stars shew a real approach, and even a fairly close approach, to equipartition of energy. The question which naturally [p. 165] presents itself is whether this approximate equality of energy can be attributed to any other cause than longcontinued gravitational interaction between the stars. This latter agency could undoubtedly produce it, but could anything else produce a similar result? The last column of the table provides the answer. It shews the temperatures to which a gas would have to be raised, in order that each of its molecules should have the same energy as the different types of stars. This may well seem an absurd calculation. A star weighing millions of millions of millions of tons goes hurtling through space at a speed of about 1,000,000 miles an hour; are we seriously setting out to inquire how hot a gas must be for every single one of its tiny molecules to have the same energy of motion, the same power of doing damage — for that is what energy of motion really amounts to — as the star? The calculation is undoubtedly absurd, and it is meant to be, because it is leading up to a reductio ad absurdum. If the observed equipartition of energy were brought about by any physical agency, such as pressure of radiation, bombardment by molecules, by atoms or by high speed electrons, this agency would have to be at a temperature, or in equilibrium with matter at a temperature, of the order of those given in the last column. These are temperatures of the order of 1062 degrees. We can be pretty sure no such temperature exists in nature, whence the argument runs that the observed equipartition of energy cannot have been brought about by physical means, and so must be the result of gravitational interaction between the stars.
The age of the stars is, then, simply the length of time needed for gravitational forces to bring about as good an approximation to equipartition of energy as is observed.
[p. 166] The calculation of this length of time presents a complicated but by no means intractable problem. All the necessary data are available, and as the method of calculation is well understood from previous experience in the theory of gases, the mathematician may be trusted to supply a reliable and reasonably exact answer when we ask him, but even without his help we can see that the time must be very long indeed.
Leaving actual figures aside for the moment, we may find it easier to think in terms of the scale-model we constructed in the first chapter (p. 88). We took our scale so small that the stars were reduced to tiny specks of dust; we noticed that space is so little crowded with stars that in our model the specks of dust had to be placed over 200 yards apart ; to put it all in a concrete form, we found that Waterloo Station with only six specks of dust left in it is more crowded with dust than space is with stars. Now let the model come to life, so as to represent the motions of the stars. To keep the proportions right, the speed of the stars must of course be reduced in the same proportion as the linear dimensions of the model. In this the earth’s yearly journey round the sun of 600 million miles had become reduced to a pin-head a sixteenth of an inch in diameter, or, say, a fifth of an inch in circumference. As the stars move through space with roughly the same speed as the earth in its orbit, we may suppose the yearly journey of each speck of dust in our model also to be about a fifth of an inch. Thus each speck of dust will move about an inch in five years, roughly 16 feet in a thousand years — or say a ten-millionth part of a snail’s pace. Even if two specks started moving directly towards one another it would take them about 20,000 vears to meet. For how long must six particles of dust, [p. 167] floating blindly about in Waterloo Station, move at this pace before each has had enough close meetings with other specks of dust for their energy of motion to become thoroughly redistributed?
The mathematician, carrying out exact calculations with respect to the actual weights, speeds and distances of the stars, finds that the observed degree of approximation to equipartition of energy shews that gravitational interaction must have continued through millions of millions of years, most probably from 5 to 10 millions of millions of years. This, then, must be the length of life of the stars.
It is a stupendous length of time, and before finally accepting it we may well look for confirmation from other sources. In estimating the age of the earth we were able to invoke assistance from all kinds of clocks, astronomical, geological and physical ; happily they all told much the same story. In the present problem only astronomical clocks are available, but fortunately there are no fewer than three of these, and again they agree in saying much the same thing.
The orbits of binary systems. We have already seen (p. 47) how the two constituents of a binary system permanently describe closed elliptical orbits about one another, because neither can escape from the gravitational hold of its companion. Energy can reside in the orbital motion of these systems, as well as in their motion through space. And strict mathematical analysis shews that a long succession of gravitational pulls from passing stars must finally result in equipartition of energy, not only between the energies of motion of one system and another through space, but also between the various orbital motions of which each binary system is capable. When this final state of [p. 168] equipartition is ultimately reached, the orbits of the systems will not all be similar, but it can be shewn that their shapes will be distributed according to a quite simple statistical law[2]. As the orbits of actual systems are not found to conform to this law, it is clear that the stars have not yet lived long enough to attain equipartition of energy in respect of their orbital motions. It is impossible to discuss how far they have travelled along the road to equipartition without knowing the point, or points, from which they started.
The question of the origin of binary systems will be discussed more fully in the next chapter. For the moment it may be said that they appear to come into being in two distinct ways.
Practically all astronomical bodies are in a state of rotation about an axis. The earth rotates about its axis once every 24 hours, and Jupiter once every 10 hours, as is shewn by the motion of the red spot and other markings on its surface. The surface of the sun rotates every 26 days or so; we can follow its rotation by watching sun-spots, faculae and other features moving round and round its equator. There are theoretical grounds for supposing that the sun’s central core rotates considerably faster than this, most probably performing a complete rotation in comparatively few days. And it is likely that all the other stars in the sky are also in rotation, some fast and some slow. We shall see later how, with advancing age, a star is likely to shrink in size, and this shrinkage generally causes its speed of rotation to increase. Now mathematical theory shews that there is a critical speed of rotation which cannot be exceeded with safety. If the star rotates too fast [p. 169] for safety, it simply bursts into two, much as a rotating fly-wheel may burst if it is driven at too high a speed. It is in this way that one class of binary stars come into being. With a few exceptions this class is identical with the class of spectroscopic binaries described in Chapter I (p. 52); the two component stars are generally too close together to appear as distinct spots of light in the telescope, only spectroscopic evidence telling us that we are dealing with two distinct bodies.
Another class of binaries, the visual binaries, which appear quite definitely as pairs of spots of light in the telescope, probably have a different origin. We shall see later how the stars first come into being as condensations of nebulous gas, a whole shoal being born when a single great nebula breaks up. It must often happen that adjacent condensations are so near as to be unable to elude each other’s gravitational grip. In time these shrink down into normal stars, while the gravitational forces remain just as powerful as before, and we are left with a pair of stars which must permanently journey through space in double harness, because they have not energy of motion enough ever to get clear of one another’s gravitational hold. This mechanism produces a class of binaries which is precisely similar to that formed by the break-up of single stars, except for an enormous difference in scale. The distance between the two components of such a system must be comparable with the original distance between separate condensations in the primaeval nebula out of which the stars were born, and so is enormously greater than the corresponding distance in spectroscopic binaries, which is comparable only with the diameter of an ordinary star which has broken into pieces. This explains why visual binaries appear as distinct [p. 170] pairs of spots of light, while spectroscopic binaries do not.
In the final state of equipartition of energy, the shapes of the orbits will, as we have seen, be distributed according to a definite statistical law. This law of distribution is the same for all sizes of orbit. On the other hand, the time needed for equipartition of energy to bring this law about is not the same for all sizes of orbit ; it is far greater for the compact orbits of the spectroscopic binaries than for the more open orbits of the visual binaries. The reason for this is that changes in the shape of an orbit are caused merely by the difference of the gravitational pulls of a passing star on the two components of the binary. If the two components are very close together, the passing star exerts practically the same forces on both. These forces affect the motions of the two components in precisely the same way, with the result that the motion of the binary system as a whole through space is changed, but the shape of orbit remains unaltered. The passing star gets a grip on the motion of the binary as a whole, but none on the orbits of the components. On the other hand, when the components are far apart, the gravitational forces acting on the two may be widely different, so that a substantial change in the shape of the orbit may result, even if the encounter is not a very close one. In visual binaries, in which the components are usually hundreds of millions of miles apart, the time necessary to establish the final distribution of the “eccentricities,” by which the shapes of elliptical orbits are measured, is once again found to be of the order of millions of millions of years, but it is something like a hundred times as great as this for the far more compact spectroscopic binaries.
[p. 171] The following table, compiled from material given by Dr Aitken of Lick Observatory, shews the observed distribution of eccentricities in the orbits of those binaries for which accurate information is available :
| Eccentricity of Orbits | Observed number of spectroscopic binaries | Observed number of visual binaries | Number to be expected theoretically when the final state is attained |
|---|---|---|---|
| 0 to 0.2 | 78 | 7 | 6 |
| 0.2 „ 0.4 | 18 | 18 | 18 |
| 0.4 „ 0.6 | 16 | 28 | 30 |
| 0.6 „ 0.8 | 6 | 11 | 42 |
| 0.8 „ 1.0 | 1 | 4 | 54 |
Let us look first at the spectroscopic binaries. In the observed orbits, we see that low eccentricities predominate, no fewer than 78 out of 119 having an eccentricity of less than one-fifth. In other words, most spectroscopic binaries have nearly circular orbits. Both theory and observation shew that when a star first divides up into a spectroscopic binary, the orbits of the two components must be nearly circular, so that the table of observed orbits provides very little evidence of any progressive change of shape in the orbits as a whole. In contrast to this, the last column of the table shews the proportion of orbits of different eccentricities which is to be expected when, if ever, equipartition of energy is finally attained. Here high eccentricities, representing very elongated orbits, predominate ; only one orbit in twenty-five is so nearly circular as to have an eccentricity less than a fifth.
In general the observed numbers tabulated in the [p. 172] second column shew no resemblance at all to the theoretical numbers tabulated in the fourth column. In other words, the spectroscopic binaries shew no suggestion of any near approach to the final state, most of them retaining the low eccentricity of orbit with which they started life. We should naturally expect this, since we have seen that hundreds or even thousands of millions of millions of years would be needed for these orbits to attain a final state of equipartition, and the stars cannot be as old as this, for if they were, their motions through space ought to shew absolutely perfect equipartition, which they certainly do not.
Turning now to the third column, we see that the visual binaries shew a good approach to the theoretical final state up to an eccentricity of about 0.6, but not beyond. The deficiency of orbits of high eccentricity may mean that gravitational forces have not had sufficient time to produce the highest eccentricities of all, but part, and perhaps all, of it must be ascribed to the simple fact that orbits of high eccentricity are exceedingly difficult to detect observationally and to measure accurately.
Clearly, then, the study of orbital motions, like that of motions through space, points to gravitational action extending over millions of millions of years. In each case there is an exception to “prove the rule.” In the case we have just considered it is provided by the spectroscopic binaries, which are so compact that their constituents can defy the pulling-apart action of gravitation; in the former case it was provided by the B-type stars, which are so massive, possibly also so young, that the gravitational forces from less weighty stars have not yet greatly affected their motion.
When these two lines of evidence are discussed in [p. 173] detail, they agree in suggesting that the general age of the stars is about that already stated, namely, from five to ten millions of millions of years.
Moving clusters. A third line of evidence, which also tells much the same story, may be briefly mentioned. The conspicuous groups of bright stars in the sky, such as the Great Bear, the Pleiades and Orion’s Belt, consist for the most part of exceptionally massive stars which move in regular orderly formation through a jumble of slighter stars, like a flight of swans through a confused crowd of rooks and starlings. Swans continually adjust their flight so as to preserve their formation. The stars cannot, so that their orderly formation must in time be broken by the gravitational pull of other stars. The lighter stars are naturally knocked out of formation first, while the most massive stars retain their formation longest. Observation suggests that this is what actually happens to a moving star-cluster; at any rate the stars which remain in formation generally have weights far above the average. And, as we can calculate the time necessary to knock out the lighter stars, we can at once deduce the ages of those which are left in.
The result of the calculation confirms those already mentioned, so that we find that the three available astronomical clocks all tell much the same time. They agree in indicating an age of the order of five to ten millions of millions of years for the stars as a whole.
Another line of investigation, to be mentioned later (p. 188) again points to a similar age.
It is perhaps a little surprising that this age should prove to be so much longer than the age of the earth, although there is of course no positive reason why the earth should not have been born during the last few [p. 174] moments of the lives of the stars. It is perhaps also a little surprising that it should prove to be much longer than the age suggested, very vaguely it is true, by the cosmologies of de Sitter and Lemaitre. If we accept the apparent velocities of recession of the most distant nebulae as real, we find that some thousands of millions of years of motion at their present speeds would just about account for their present distances from us, so that a few thousands of millions of years ago, the nebulae must have been far more huddled together than they now are. This is of course very different from saying that the time which has elapsed since the creation of the nebulae can only be a few thousands of millions of years, yet we might reasonably have expected a priori that the two periods would be at least comparable.
To state the difficulty in a slightly different form, a period of two thousand million years seems to have made a great deal of difference to the earth, and if the apparent speeds of nebular motion are real, it has made a great deal of difference to the general arrangement in space of the great nebulae, so that it is odd that it should make so little difference to the stars that we need to postulate an age a thousand times as great before we can explain their present condition.
These considerations may seem to suggest that the estimate just made of stellar ages should be accepted with caution and perhaps even with suspicion. Yet if we reject it, so many facts of astronomy are left up in the air without any explanation, and so much of the fabric of astronomy is thrown into disorder (see p. 187, below), that we have little option but to accept it, and suppose that the stars have actually lived through times of the order of millions of millions of years.
[p. 175]
During the whole of some such vast period of time, the sun has in all probability been pouring out light and heat at least as profusely as at present. Indeed a mass of evidence, to which we shall return later, shews that young stars emit more radiation than older stars, so that during most of its long life the sun must have been pouring out energy even more lavishly than now.
If our ancestors thought about the matter at all, they probably saw nothing remarkable in this profuse outpouring of light and heat, particularly as they had no conception of the stupendous length of time during which it had lasted. It was only in the middle of last century, when the principle of conservation of energy first began to be clearly understood, that the source of the sun’s energy was seen to constitute a scientific puzzle of really first-class difficulty. The sun’s radiation obviously represented a loss of energy to the sun, and, as the principle of conservation shewed that energy could not originate out of nothing, this energy necessarily came from some source or store adequate to supply vast outpourings of energy over a very long period of time. Where was such a store to be found?
The sun at present pours out radiation at such a rate that if the necessary energy were generated in a power- station outside the sun, this station would have to burn coal at the rate of many thousands of millions of millions of tons a second. There is of course no such power-station. The sun is entirely dependent on its own resources; it is a ship on an empty ocean. And if, like such a ship, the sun carried its own store of coal, [p. 176] or if, as Kant imagined, its whole substance were its store of coal, so that its light and heat came from its own combustion, the whole would be burnt into ashes and cinders in a few thousand years at most.
The history of science records one solitary attempt to explain the sun’s energy as coming in from outside. We have seen how the energy of motion of a bullet is transformed into heat when the speed of the bullet is checked. An astronomical example of the same effect is provided by the familiar phenomenon of shootingstars. These are bullet-like bodies which fall into the earth’s atmosphere from outer space. So long as such a body is travelling through empty space, its fall towards the earth continually increases its speed, but, as it enters the earth’s atmosphere, its speed is checked by air-resistance, and the energy of its motion is gradually transformed into heat. The shooting-star becomes first hot and then incandescent, emitting the bright light by which we recognise it. Finally, the heat completely vaporises it, and it disappears from sight, leaving only a momentary trail of luminous gas behind. The original energy of motion of the shooting-star has been transformed into light and heat — the light by which we see it, and the heat by which it is ultimately vaporised.
In 1849, Robert Mayer suggested that the energy which the sun emitted as radiation might accrue to it from a continuous fall of shooting-stars or similar bodies into the solar atmosphere. The suggestion is untenable, because a simple calculation shews that a mass of such bodies equal to the weight of the whole earth would hardly maintain the sun’s radiation for a century, and that the infall needed to maintain the sun’s radiation for 30 million years would double its weight. As it is [p. 177] quite impossible to admit that the sun’s weight can be increasing at any such rate, Mayer’s hypothesis has to be abandoned.
In 1853 Helmholtz put forward a very similar theory, the famous “contraction-hypothesis,” according to which the sun’s own shrinkage sets free the energy which ultimately appears as radiation. If the sun’s radius shrinks by a mile, its outer atmosphere falls through a height of a mile and sets free as much energy in so doing as would be yielded up by an equal weight of shooting-stars falling through a mile and having their motion checked. On Helmholtz’s theory, the different parts of the sun’s own body performed the roles which Mayer had allotted to shooting-stars falling in from outside; they performed these same parts again and again, until ultimately the sun had shrunk so far that it could shrink no further. Yet Helmholtz’s theory, like that of Mayer, failed to survive the test of numerical computation. In 1862 Lord Kelvin calculated that the shrinkage of the sun to its present size could hardly have provided energy for more than about 50 million years of radiation in the past, whereas the geological evidence already noticed (p. 151) shews that the sun must have been shining for a period enormously longer than this.
To track down the actual source of the sun’s energy with any hope of success, we must give up guessing, and approach the problem from a new angle. We have seen (p. 120) how radiation carries weight about with it, so that any body which is emitting radiation is necessarily losing weight; the radiation emitted by a searchlight of 50 horse-power would, we saw, carry away weight at the rate of about a twentieth of an ounce a century. Now each square inch of the sun’s surface is [p. 178] in effect a searchlight of just about 50 horse-power, whence we conclude that weight is streaming away from every square inch of the sun’s surface at the rate of about a twentieth of an ounce a century. Such a loss of weight seems small enough, until we multiply it by the total number of square inches which constitute the whole surface of the sun. It then appears that the sun as a whole is losing weight at the rate of rather over 4 million tons a second, or about 250 million tons a minute — something like 650 times the rate at which water is streaming over Niagara.
The Past Histories of the Sun and Stars. Let us carry on the multiplication. Two hundred and fifty million tons a minute is 360,000 million tons a day. Thus the sun must have weighed 360,000 million tons more than now at this time yesterday, and will weigh 360,000 million tons less at this time to-morrow. And 360,000 million tons a day is 131 million million tons a year. We can dig as far into the past as we like in this way and can probe as far as we like into the future. But soon we encounter the usual trouble which besets all calculations of this kind — the sand does not always run through the hour-glass at the same rate. The rate at which the sun loses weight will not vary appreciably between to-day and to-morrow, or even over a century or a million years, but we must be on our guard against going too far. If the sun continued to radiate at precisely its present rate, a simple sum in division shews that it would last for just about 15 million million years, by which time its last ounce of weight would be disappearing. Incidentally this gives us a vivid conception of the enormous weight of the sun; it could go on pouring away its substance into space at 650 times the rate at which water is pouring [p. 179] over Niagara for 15 million million years before becoming exhausted.
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 [p. 180] left to spend. The sand runs slowly through the hourglass 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:
| Time | Weight |
|---|---|
| 2,000,000,000 years ago, the sun had | 1.00013 times its present weight |
| 1,000,000,000,000 „ „ | 1.07 |
| 2,000,000,000,000 „ „ | 1.16 |
| 5,700,000,000,000 „ „ | double „ |
| 7,100,000,000,000 „ „ | 4 times „ |
| 7,400,000,000,000 „ „ | 8 times „ |
| 7,500,000,000,000 „ „ | 20 „ |
| 7,600,000,000,000 „ „ | 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 [p. 181] 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 [p. 182] 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 [p. 183] 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 [p. 184] 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 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 hundredweight 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 radioactive 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 ninetenths, 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 [p. 185] 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 [p. 186] 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 x 10 20 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 [p. 187] 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 [p. 188] 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 [p. 189] 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 x 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 storedup 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 [p. 190] 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 [p. 191] 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 x 1016 ergs of energy; the annihilation of a ton of coal liberates 9 x 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 [p. 192] 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 x 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.” [p. 193] 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?