_Exploring in Time_
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 hour-glass; 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, although possibly of more complicated mathematics,
to estimate the time which has elapsed since the process first started.
THE AGE OF THE EARTH
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 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 landmarks 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[18] gives the
observed maximum thicknesses as follows:
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 ”
[18] In discussing the earth’s age, I have borrowed extensively from
Professor Holmes’ book, _The Age of the Earth_.
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 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:
_History of One ounce of Uranium_
Initially: 1 oz. uranium No lead
After 100 million years 0·985 oz. uranium 0·013 oz. lead
” 1000 ” 0·865 ” ” 0·116 ” ”
” 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 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 radio-active 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.
These two physical estimates of the time which has elapsed since the
earth solidified stand as follows:
_Age of the Earth by the Radio-active Clock_
1. From the lead-uranium }
ratio in radio-active } More than 1400 million years.
rocks }
2. From the relative abundance }
of uranium and } Less than 3400 million years.
actino-uranium }
Various astronomical methods are 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:
_Age of the Solar System by the Astronomical Clock_
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.
THE AGES OF THE STARS
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, 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 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 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 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 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 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 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—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.
_Equipartition of Energy in Stellar Motions_
+----------------+-----------+-----------+-----------+-------------+
| | Average | Average | Average |Corresponding|
| | weight | speed | energy | temperature |
| Type of star | _M_ | _C_ | ½_MC_² | (degrees) |
| | (grammes) |(cms./sec.)| (ergs) | |
+----------------+-----------+-----------+-----------+-------------+
| Spectral | | | | |
| type _B_ 3 |19·8 × 10³³|14·8 × 10⁵ |1·95 × 10⁴⁶|1·0 ×10⁶² |
| ” _B_ 8·5|12·9 |15·8 |1·62 |0·8 |
| ” _A_ 0 |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_ 0 | 5·0 |35·9 |3·24 |1·6 |
| ” _F_ 5 | 3·1 |47·9 |3·55 |1·7 |
| ” _G_ 0 | 2·0 |64·6 |4·07 |2·0 |
| ” _G_ 5 | 1·5 |77·6 |4·57 |2·2 |
| ” _K_ 0 | 1·4 |79·4 |4·27 |2·1 |
| ” _K_ 5 | 1·2 |74·1 |3·39 |1·7 |
| ” _M_ 0 | 1·2 |77·6 |3·55 |1·7 |
+----------------+-----------+-----------+-----------+-------------+
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 presents itself is whether this approximate
equality of energy can be attributed to any other cause than
long-continued 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.
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 years to meet. For how long must six
particles of dust, 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 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[19].
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.
[19] The eccentricity of orbit _e_ is distributed in such a way that
all values of _e_² from _e_² = 0 to _e_² = 1 are equally probable.
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 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 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.
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:
_The Approach to Equipartition of Energy in Binary Orbits_
+-------------+--------------+----------+---------------------+
| | Observed |Observed |Number to be expected|
|Eccentricity | number of |number of | theoretically |
| of Orbits |spectroscopic |visual | when the |
| | binaries |binaries | 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 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 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 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 Lemaître. 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 _à 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.
THE SUN’S RADIATION
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, 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 shooting-stars. 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 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 _rôles_ 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 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 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 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?