brought these conclusions into excellent order. It recognized a series
of _giant_ stars, comparatively diffuse stars with temperature
rising, and a series of _dwarf_ or dense stars with temperature
falling. The two series merged at the highest temperatures. An
individual star during its lifetime went up the giant series to its
highest temperature and then down the dwarf series. The brightness
remained fairly steady throughout the giant stage because the
continually increasing temperature counterbalanced the reduction of
the surface area of the star; in the dwarf stage the decreasing
temperature and the contraction of the surface caused a rapid decrease
of brightness as the star progressed down the series. This was in
accordance with observation. The theory has dominated most recent
astrophysical research and has been instrumental in bringing to light
many important facts. One example must suffice. Although we may have a
giant and a dwarf star with the same surface temperature, and therefore
showing very similar spectra, nevertheless a close examination of the
spectrum reveals tell-tale differences; and it is now quite easy to
ascertain from the spectrum whether the star is a diffuse giant or a
dense dwarf.
The attractive feature of the giant and dwarf theory was the simple
explanation given for the up-and-down progress of the temperature.
The passing over from the giant to the dwarf series was supposed to
occur when the density had reached such a value (about one-quarter the
density of water) that the deviation of the material from a perfect gas
began to be serious. It was shown by Lane fifty years ago that a globe
of perfect gas must rise in temperature as it contracts, his method of
finding the internal temperature being that considered on p. 12; thus
the rising temperature in the giant stage is predicted. But the rise
depends essentially on the easy compressibility of the gas; and when
the compressibility is lost at high density the rising temperature may
be expected to give place to falling temperature so that the star cools
as a solid or liquid would do. That was believed to account for the
dwarf stage.
I have been trying to recall ideas of twenty and ten years ago, and you
must not suppose that from the standpoint of present-day knowledge I
can endorse everything here stated. I have intentionally been vague
as to whether by the hotness of a star I mean the internal or the
surface temperature since ideas were formerly very loose on this point;
I have made no reference to white dwarfs, which are now thought to be
the densest and presumably the oldest stars of all. But it is the last
paragraph especially which conflicts with our latest conclusions, for
we no longer admit that stellar material will cease to behave as a
perfect gas at one-quarter the density of water. Our result that the
material in the dense dwarf stars is still perfect gas (p. 38) strikes
a fatal blow at this part of the giant and dwarf theory.
It would be difficult to say what is the accepted theory of stellar
evolution to-day. The theory is in the melting-pot and we are still
waiting for something satisfactory to emerge. The whole subject is in
doubt and we are prepared to reconsider almost anything. Provisionally,
however, I shall assume that the former theory was right in assuming
that the sequence of evolution is from the most diffuse to the densest
stars. Although I make this assumption I do not feel sure that it is
allowable. The former theory had strong reasons for making it which
no longer apply. So long as contraction was supposed to be the source
of a star’s heat, contraction and increasing density were essential
throughout its whole career; with the acceptance of subatomic energy
contraction ceases to play this fundamental role.
I propose to confine attention to the dwarf stars[35] because it is
among them that the upset has occurred. They form a well-defined series
stretching from high surface-temperature to low surface-temperature,
high luminosity to low luminosity, and the density increases steadily
along the series. We now call this the Main Series. It comprises the
great majority of the stars. To fix ideas let us take three typical
stars along the series--Algol near the top, the Sun near the middle,
and Krueger 60 near the bottom. The relevant information about them is
summarized below:
Mean Central
density temperature Surface
Mass (Water (million temperature Luminosity
Star. (Sun = 1). =1). deg.). (deg.) Colour. (Sun = 1).
Algol 4·3 0·15 40 12,000 white 150
Sun 1 1·4 40 6,000 yellow 1
Krueger
60 0·27 9·1 35 3,000 red 0·01
The idea of evolution is that these represent the stages passed
through in the life-history of an individual star.[36] The increasing
density in the third column should be noticed; according to our
accepted criterion it indicates that the order of development is
Algol→Sun→Krueger 60. A confusion between internal temperature and
surface temperature is responsible for some of the mistakes of the
older theories. To outward view the star cools from 12,000° to 3,000°
in passing down the series, but there is no such change in its internal
heat. The central temperature remains surprisingly steady. (No special
reliance can be placed on the slight falling off apparently shown by
Krueger 60.) It is very remarkable that all stars of the main series
have a central temperature of about 40 million degrees as nearly as we
can calculate. It is difficult to resist the impression that there is
some unusual property associated with this temperature, although all
our physical instincts warn us that the idea is absurd.
But the vital point is the decrease of mass shown in the second column.
_If an individual star is to progress any part of the way down the
main series it must lose mass_. We can put the same inference in a
more general way. Now that it has been found that luminosity depends
mainly on mass, there can be no important evolution of faint stars from
bright stars unless the stars lose a considerable part of their mass.
It is this result which has caused the hypothesis of annihilation of
matter to be seriously discussed. All progress in the theory of stellar
evolution is held up pending a decision on this hypothesis. If it is
accepted it provides an easy key to these changes. The star may (after
passing through the giant stage) reach the stage of Algol, and then by
the gradual annihilation of the matter in it pass down the main series
until when only one-sixteenth of the original mass remains it will be
a faint red star like Krueger 60. But if there is no annihilation of
matter, the star when once it has reached the dwarf stage seems to be
immovable; it has to stay at the point of the series corresponding to
its constant mass.
Let it be clearly understood what is the point at issue. The stars
lose mass by their radiation; there is no question about that. The
sun is losing 120 billion tons annually whether its radiation comes
from annihilation of matter or any other internal source. The question
is, How long can this loss continue? Unless there is annihilation of
matter, all the mass that can escape as radiation will have escaped in
a comparatively short time; the sun will then be extinct and there is
an end to the loss and to the evolution. But if there is annihilation
of matter the life of the sun and the loss of mass continue far longer,
and an extended track of evolution lies open before the sun; when it
has got rid of three-quarters of its present mass it will have become a
faint star like Krueger 60.
Our choice between the possible theories of subatomic energy only
affects stellar evolution in one point--but it is the vital point.
Unless we choose annihilation of matter, we cut the life of a star so
short that there is no time for any significant evolution at all.
I feel the same objection that every one must feel to building
extensively on a hypothetical process without any direct evidence that
the laws of Nature permit of its occurrence. But the alternative is to
leave the stars in sleepy uniformity with no prospect of development
or change until their lives come to an end. Something is needed to
galvanize the scene into that activity, whether of progress or decay,
in which we have so long believed. Rather desperately we seize on the
one visible chance. The petrified system wakes. The ultimate particles
one by one yield up their energy and pass out of existence. Their
sacrifice is the life-force of the stars which now progress on their
high adventure:
Atoms or systems into ruin hurl’d,
And now a bubble burst, and now a world.
_Radiation of Mass_
Our first evidence of the extent of the time-scale of stellar evolution
was afforded by the steadiness of condition of δ Cephei. This was
supplemented by evidence of the great extension of geological time on
the earth. We could not do more than set an upper limit to the rate of
progress of evolution and a lower limit to the age of the stars. But
this limit was sufficient to rule out the contraction hypothesis and
drive us to consider the store of subatomic energy.
We now make a new attack, which depends on the belief that _the rate
of evolution is determined by the rate at which a star can get rid of
its mass_. We are here considering only the evolution of faint stars
from bright stars, and there will remain scope for a certain amount of
development in the giant stage to which our arguments will not directly
apply. But to abandon all lines of evolution between bright stars and
faint stars would mean admitting that one star differs from another
star in brightness because it was different originally. This _may_
be true; but we ought not to surrender the main field of stellar
evolution without making a fight for it.
By the new line of attack we reach a definite determination of the
time-scale and not merely a lower limit. We know the rate at which
stars in each stage are losing mass by radiation; therefore we can find
the time taken to lose a given mass and thereby pass on to a stage of
smaller mass. Evolution from Algol to the Sun requires five billion
years; evolution from the Sun to Krueger 60 requires 500 billion years.
It is interesting to note that stars in the stage between the Sun and
Krueger 60 are much more abundant than those between Algol and the
Sun--a fact somewhat confirmatory of the calculated duration of the
two stages. The abundance of faint stars does not, however, increase so
rapidly as the calculated duration; perhaps the stellar universe has
not existed long enough for the old stars to be fully represented.
A star of greater mass than Algol squanders its mass very rapidly, so
that we do not increase the age of the Sun appreciably by supposing it
to have started with greater mass than Algol. The upper limit to the
present age of the Sun is 5·2 billion years however great its initial
mass.
But, it may be asked, cannot a star accelerate its progress by getting
rid of matter in some other way than by radiation? Cannot atoms escape
from its surface? If so the loss of mass and consequent evolution will
be speeded up, and the time required may perhaps even be brought within
range of the alternative theory of transmutation of the elements. But
it is fairly certain that the mass escaping in the form of material
atoms is negligible compared with that which imperceptibly glides
away in the form of radiation. You will perhaps be in doubt as to
whether the 120 billion tons per annum lost by the sun in radiation is
(astronomically regarded) a large quantity or a small quantity. From
certain aspects it is a large quantity. It is more than 100,000 times
the mass of the calcium chromosphere. The sun would have to blow off
its chromosphere and form an entirely fresh one every five minutes in
order to get rid of as much mass in this way as it loses by radiation.
It is obvious from solar observation that there is no such outrush
of material. To put it another way--in order to halve the time-scale
of evolution stated above it would be necessary that a billion atoms
should escape each second through each square centimetre of the sun’s
surface. I think we may conclude that there is no short cut to smaller
mass and that radiation is responsible for practically the whole loss.
We noticed earlier (p. 25) that Nature builds stars which are much
alike in mass, but allows herself some deviation from her pattern
amounting sometimes to a mistake of one 0. I think we may have done
her an injustice, and that she is more careful over her work than we
supposed. We ought to have examined coins fresh from her mint; it was
not fair to take coins promiscuously, including many that had been in
circulation for some hundreds of billions of years and had worn rather
thin. Taking the newly formed stars, i. e. the diffuse stars, we find
that 90 per cent. of them are between 2½ and 5½ times the mass of the
sun--showing that initially the stars are made nearly as closely to
pattern as human beings are. In this range radiation pressure increases
from 17 to 35 per cent, of the whole pressure; I think this would be
expected to be the crucial stage in its rise to importance. Our idea
is that the stellar masses initially have this rather close uniformity
(which does not exclude a small proportion of exceptional stars outside
the above limits); the smaller masses are evolved from these in course
of time by the radiation of mass.
For the time being the sun is comfortably settled in its present state,
the amount of energy radiated being just balanced by the subatomic
energy liberated inside it. Ultimately, however, it must move on.
The moving on, or evolution, is continuous, but for convenience of
explanation we shall speak of it as though it occurred in steps.
Two possible motives for change can be imagined, (1) the supply of
subatomic energy might fall off by exhaustion and no longer balance the
radiation, and (2) the sun is slowly becoming a star of smaller mass.
In former theories the first motive has generally been assumed, and we
may still regard it as effective during the giant stage of the stars;
but it is clear that the motive to move down the main series must be
loss of mass.[37] Apparently the distinction between giant and dwarf
stars, replacing the old distinction of perfect and imperfect gas, is
that the prolific and soon exhausted supplies of subatomic energy in
the giant stage disappear and leave a much steadier supply in the dwarf
stage.
When the sun has become a star of smaller mass it will need to resettle
its internal conditions. Suppose that at first it tries to retain its
present density. As explained on p. 12, we can calculate the internal
temperature, and we find that the reduced mass coupled with constant
density involves lower temperature. This will slightly turn off the
tap of subatomic energy, because there can be little doubt that the
release of subatomic energy is more rapid at higher temperature. The
reduced supply will no longer be sufficient to balance the radiation;
accordingly the star will contract just as it was supposed to do
on the old contraction hypothesis which corresponds to the tap of
subatomic energy being turned off altogether. The motive is loss of
mass; the first consequence is an increase of density which is another
characteristic of progress down the main series.
Tracing the consequences a little farther, the increasing density
causes a rising temperature which in turn reopens the tap of subatomic
energy. As soon as the tap is opened enough to balance the rate of
radiation of the star, the contraction stops and the star remains
settled in equilibrium at the smaller mass and higher density.
You will see that the laws of release of subatomic energy must be
invoked if we are to explain quantitatively why a particular density
corresponds to a particular mass in the progress down the main series.
The contraction has to proceed so far as to bring the internal
conditions to a state in which the release of energy is at the exact
rate required to balance the radiation.
I am afraid this all sounds very complicated, but my purpose is to
show that the adjustment of the star after an alteration of mass is
automatic. After a change of mass the star has to re-solve the problem
of the internal conditions necessary for its equilibrium. So far as
mechanical conditions are concerned (supporting the weight of the
upper layers) it can choose any one of a series of states of different
density provided it has the internal temperature appropriate to that
density. But such equilibrium is only temporary, and the star will not
really settle down until the tap of subatomic energy is opened to the
right extent to balance the rate of radiation which, as we have already
seen, is practically fixed by the mass. The star fiddles about with the
tap until it secures this balance.
One important conclusion has been pointed out by Professor
Russell. When the star is adjusting the tap it does not do so
_intelligently_; one trial must automatically lead to the next
trial, and it is all-important that the next trial should automatically
be nearer to and not farther from the right rate. The condition that it
shall be nearer to the right rate is that the liberation of subatomic
energy shall increase with temperature or density.[38] If it decreases,
or even if it is unaltered, the trials will be successively farther
and farther from the required rate, so that although a steady balance
is possible the star will never be able to find it. It is therefore
essential to admit as one of the laws of liberation of subatomic energy
that the rate increases with temperature or with density or with both;
otherwise subatomic energy will not fulfil the purpose for which it was
introduced, viz. to keep the star steady for a very long time.
The strange thing is that the condition of balance is reached when the
central temperature is near 40 million degrees--the same whether the
star is at the top, middle, or bottom of the main series. Stars at the
top release from each gramme of material 700 ergs of energy per second;
the sun releases 1 ergs per second; Krueger 60 releases 0·08 ergs per
second. It seems extraordinary that stars requiring such different
supplies should all have to ascend to the same temperature to procure
them. It looks as though at temperatures below this standard not even
0·08 ergs per second is available, but on reaching the standard the
supply is practically unlimited. We can scarcely believe that there is
a kind of boiling-point (independent of pressure) at which matter boils
off into energy. The whole phenomenon is most perplexing.
I may add that the giant stars have temperatures considerably below 40
million degrees. It would appear that they are tapping special supplies
of subatomic energy released at lower temperatures. After using up
these supplies the star passes on to the main series, and proceeds to
tap the main supply. It seems necessary to suppose further that the
main supply does not last indefinitely, so that ultimately the star (or
what is left of it) leaves the main series and passes on to the white
dwarf stage.
We are now in a position to deal with a question which you may
have wished to ask earlier. Why does δ Cephei pulsate? One possible
answer is that the oscillation was started off by some accident.
So far as we can calculate an oscillation, if once started, would
continue for something like 10,000 years before becoming damped down.
But 10,000 years is now deemed to be an insignificant period in the
life of a star, and, having regard to the abundance of Cepheids, the
explanation seems inadequate even if we could envisage the kind of
accident supposed. It is much more likely that the pulsation arises
spontaneously. Enormous supplies of heat energy are being released in
the star--far more than enough to start and maintain the pulsation--and
there are at least two alternative ways in which this heat can be
supposed to operate a mechanism of pulsation.
Here is one alternative. Suppose first that there is a very small
pulsation. When compressed the star has higher temperature and density
than usual and the tap of subatomic energy is opened more fully. The
star gains heat, and the expansive force of the extra heat assists
the rebound from compression. At greatest expansion the tap is turned
off a little and the loss of heat diminishes the resistance to the
ensuing compression. Thus the successive expansions and compressions
become more and more vigorous and a large pulsation grows out of an
infinitesimal beginning. It will be seen that the star works the tap of
subatomic energy just as an engine works the valve admitting heat into
its cylinder; so that the pulsations of a star are started up like the
pulsations of an engine.
The only objection that I can find to this explanation is that it
is too successful. It shows why a star may be expected to pulsate;
but the trouble is that stars in general do not pulsate--it is only
the rare exceptions that behave in this way. It is now so easy to
account for the Cepheids that we have to turn back and face the more
difficult problem of accounting for the normal steady stars. Whether
the pulsation will start up or not depends on whether the engine of
pulsation is sufficiently powerful to overcome the forces tending to
damp out and dissipate pulsations. We cannot predict the occurrence
or non-occurrence from any settled theory; we have rather to seek to
frame the laws of release of subatomic energy so as to conform to our
knowledge that the majority of the stars remain steady, but certain
conditions of mass and density give the pulsatory forces the upper hand.
Cepheid pulsation is a kind of distemper which happens to stars at a
certain youthful period; after passing through it they burn steadily.
There may be another attack of disease later in life when the star is
subject to those catastrophic outbursts which occasion the appearance
of ‘new stars’ or novae. But very little is known as to the conditions
for this, and it is not certain whether the outbreak is spontaneous or
provoked from outside.
So long as we stick to generalities the theory of subatomic energy
and especially the theory of annihilation of matter makes a fairly
promising opening. It is when we come to technical details that doubts
and perplexities arise. Difficulties appear in the simultaneous
presence of giant and dwarf stars in coeval clusters, notwithstanding
their widely different rates of evolution. There are difficulties in
devising laws of release of subatomic energy which will safeguard the
stability of the stars without setting every star into pulsation.
Difficulties arise from the fact that as a rule in the giant stage the
lower the temperature and density the more rapid the release of energy;
and although we account for this in a general way by considering the
exhaustibility of the more prolific sources of energy, the facts are
not all straightened out by such a scheme. Finally grave difficulties
arise in reconciling the laws of release inferred from astronomical
observation with any theoretical picture we can form of the process
of annihilation of matter by the interplay of atoms, electrons, and
radiation.
The subject is highly important, but we cannot very well pursue it
further in this lecture. When the guidance of theory is clear interest
centres round the broad principles; when the theory is rudimentary,
interest centres round technical details which are anxiously
scrutinized as they appear to favour now one view now another. I have
dealt mainly with two salient points--the problem of the source of a
star’s energy and the change of mass which must occur if there is any
evolution of faint stars from bright stars. I have shown how these
appear to meet in the hypothesis of annihilation of matter. I do not
hold this as a secure conclusion. I hesitate even to advocate it as
probable, because there are many details which seem to me to throw
considerable doubt on it, and I have formed a strong impression that
there must be some essential point which has not yet been grasped. I
simply tell it you as the clue which at the moment we are trying to
follow up--not knowing whether it is false scent or true.
I should have liked to have closed these lectures by leading up to
some great climax. But perhaps it is more in accordance with the
true conditions of scientific progress that they should fizzle out
with a glimpse of the obscurity which marks the frontiers of present
knowledge. I do not apologize for the lameness of the conclusion, for
it is not a conclusion. I wish I could feel confident that it is even a
beginning.
FOOTNOTES:
[Footnote 1: Fig. 1 is from a photograph taken by Mr. Evershed at
Kodaikanal Observatory, Madras. Fig. 2 is from the Mount Wilson
Observatory, California.]
[Footnote 2: I am indebted to Professor C. T. R. Wilson for Figs. 3-6.]