Stars and Atoms

[Illustration]

[Illustration: Fig. 1. THE SUN. Hydrogen photograph]

                            STARS AND ATOMS




                                  BY

                            A. S. EDDINGTON

      M.A., D.Sc., LL.D., F.R.S., Plumian Professor of Astronomy
                    in the University of Cambridge




                   NEW HAVEN: YALE UNIVERSITY PRESS
           LONDON: HUMPHREY MILFORD, OXFORD UNIVERSITY PRESS
                                 1927

    Ich häufe ungeheure Zahlen,
    Gebürge Millionen auf,
    Ich setze Zeit auf Zeit und Welt auf Welt zu Hauf.

    A. VON HALLER.




                                PREFACE


‘STARS and Atoms’ was the title of an Evening Discourse given at
the meeting of the British Association in Oxford in August 1926. In
adapting it for publication the restrictions of a time limit are
removed, and accordingly it appears in this book as three lectures.
Earlier in the year I had given a course of three lectures in King’s
College, London, on the same topics; these have been combined with the
Oxford lecture and are the origin of most of the additions.

A full account of the subject, including the mathematical theory, is
given in my larger book, _The Internal Constitution of the Stars_
(Camb. Univ. Press, 1926). Here I only aim at exposition of some of the
leading ideas and results.

The advance in our knowledge of atoms and radiation has led to many
interesting developments in astronomy; and reciprocally the study of
matter in the extreme conditions prevailing in stars and nebulae has
played no mean part in the progress of atomic physics. This is the
general theme of the lectures. Selection has been made of the advances
and discoveries which admit of comparatively elementary exposition;
but it is often necessary to demand from the reader a concentration
of thought which, it is hoped, will be repaid by the fascination of
the subject. The treatment was meant to be discursive rather than
systematic; but habits of mind refuse to be suppressed entirely and
a certain amount of system has crept in. In these problems where our
thought fluctuates continually from the excessively great to the
excessively small, from the star to the atom and back to the star, the
story of progress is rich in variety; if it has not lost too much in
the telling, it should convey in full measure the delights--and the
troubles--of scientific investigation in all its phases.

Temperatures are expressed throughout in degrees Centigrade. The
English billion, trillion, &c. (10^{12}, 10^{18}, &c.) are used.

                                                               A. S. E.




CONTENTS

LECTURE I. THE INTERIOR OF A STAR        9

Temperature in the Interior                      11

Ionization of Atoms                              17

Radiation Pressure and Mass                      24

The Interior of a Star                           26

Opacity of Stellar Matter                        28

The Relation of Brightness to Mass               31

Dense Stars                                      36

LECTURE II. SOME RECENT INVESTIGATIONS  42

The Story of Algol                               42

The Story of the Companion of Sirius             48

Unknown Atoms and Interpretation of Spectra      53

Spectral Series                                  59

The Cloud in Space                               63

The Sun’s Chromosphere                           70

The Story of Betelgeuse                          76

LECTURE III. THE AGE OF THE STARS       85

Pulsating Stars                                  85

The Cepheid as a ‘Standard Candle’               90

The Contraction Hypothesis                       94

Subatomic Energy                                 99

Evolution of the Stars                          106

Radiation of Mass                               113

APPENDIX

Further Remarks on the Companion of Sirius      122




LIST OF ILLUSTRATIONS

FIG.

1. The Sun. Hydrogen Spectroheliogram.
(J. Evershed)

2. Solar Vortices. Hydrogen Spectroheliogram.
(Mount Wilson Observatory)

3. Tracks of Alpha Particles (helium atoms).
(C. T. R. Wilson)

4. Tracks of Beta Particles (electrons).
(C. T. R. Wilson)

5. Ionization by X-rays. (C. T. R. Wilson)

6. Ions produced by Collision of a Beta particle.
(C. T. R. Wilson)

7. The Mass-luminosity Curve.

8. The Ring Nebula in Lyra. Slitless Spectrogram.
(W. H. Wright)

9. Flash Spectrum of Chromosphere showing
Head of the Balmer Series. (British
Eclipse Expedition, 14 Jan. 1926)

10. Solar Prominence. (British Eclipse Expedition,
29 May 1919)

11. Star Cluster ω Centauri. (Cape Observatory)




                               LECTURE I

                        THE INTERIOR OF A STAR


THE sun belongs to a system containing some 3,000 million stars. The
stars are globes comparable in size with the sun, that is to say,
of the order of a million miles in diameter. The space for their
accommodation is on the most lavish scale. Imagine thirty cricket balls
roaming the whole interior of the earth; the stars roaming the heavens
are just as little crowded and run as little risk of collision as the
cricket balls. We marvel at the grandeur of the stellar system. But
this probably is not the limit. Evidence is growing that the spiral
nebulae are ‘island universes’ outside our own stellar system. It may
well be that our survey covers only one unit of a vaster organization.

A drop of water contains several thousand million million million
atoms. Each atom is about one hundred-millionth of an inch in diameter.
Here we marvel at the minute delicacy of the workmanship. But this is
not the limit. Within the atom are the much smaller electrons pursuing
orbits, like planets round the sun, in a space which relatively to
their size is no less roomy than the solar system.

Nearly midway in scale between the atom and the star there is another
structure no less marvellous--the human body. Man is slightly nearer to
the atom than to the star. About 10^{27} atoms build his body; about
10^{28} human bodies constitute enough material to build a star.

From his central position man can survey the grandest works of Nature
with the astronomer, or the minutest works with the physicist. To-night
I ask you to look both ways. For the road to a knowledge of the stars
leads through the atom; and important knowledge of the atom has been
reached through the stars.

       *       *       *       *       *

The star most familiar to us is the sun. Astronomically speaking, it is
close at hand. We can measure its size, weigh it, take its temperature,
and so on, more easily than the other stars. We can take photographs of
its surface, whereas the other stars are so distant that the largest
telescope in the world does not magnify them into anything more than
points of light. Figs. 1 and 2[1] show recent pictures of the sun’s
surface. No doubt the stars in general would show similar features if
they were near enough to be examined.

I must first explain that these are not the ordinary photographs.
Simple photographs show very well the dark blotches called sunspots,
but otherwise they are rather flat and uninteresting. The pictures
here shown were taken with a spectroheliograph, an instrument which
looks out for light of just one variety (wave-length) and ignores all
the rest. The ultimate effect of this selection is that the instrument
sorts out the different levels in the sun’s atmosphere and shows what
is going on at one level, instead of giving a single blurred impression
of all levels superposed. Fig. 2, which refers to a high level, gives a
wonderful picture of whirlwinds and commotion. I think that the solar
meteorologists would be likely to describe these vortices in terms not
unfamiliar to us--‘A deep depression with secondaries is approaching,
and a renewal of unsettled conditions is probable.’ However that may
be, there is always one safe weather forecast on the sun; cyclone or
anticyclone, the temperature will be _very warm_--about 6,000° in
fact.

[Illustration: Fig. 2. THE SUN. Hydrogen photograph]

But just now I do not wish to linger over the surface layers or
atmosphere of the sun. A great many new and interesting discoveries
have recently been made in this region, and much of the new knowledge
is very germane to my subject of ‘Stars and Atoms’. But personally I am
more at home underneath the surface, and I am in a hurry to dive below.
Therefore with this brief glance at the scenery that we pass we shall
plunge into the deep interior--where the eye cannot penetrate, but
where it is yet possible by scientific reasoning to learn a great deal
about the conditions.


                  _Temperature in the Interior_


By mathematical methods it is possible to work out how fast the
pressure increases as we go down into the sun, and how fast the
temperature must increase to withstand the pressure. The architect can
work out the stresses inside the piers of his building; he does not
need to bore holes in them. Likewise the astronomer can work out the
stress or pressure at points inside the sun without boring a hole.
Perhaps it is more surprising that the temperature can be found by pure
calculation. It is natural that you should feel rather sceptical about
our claim that we know how hot it is in the very middle of a star--and
you may be still more sceptical when I divulge the actual figures!
Therefore I had better describe the method as far as I can. I shall not
attempt to go into detail, but I hope to show you that there is a clue
which might be followed up by appropriate mathematical methods.

I must premise that the heat of a gas is chiefly the energy of motion
of its particles hastening in all directions and tending to scatter
apart. It is this which gives a gas its elasticity or expansive
force; the elasticity of a gas is well known to every one through its
practical application in a pneumatic tyre. Now imagine yourself at
some point deep down in the star where you can look upwards towards
the surface or downwards towards the centre. Wherever you are, a
certain condition of balance must be reached; on the one hand there
is the weight of all the layers above you pressing downwards and
trying to squeeze closer the gas beneath; on the other hand there is
the elasticity of the gas below you trying to expand and force the
superincumbent layers outwards. Since neither one thing nor the other
happens and the star remains practically unchanged for hundreds of
years, we must infer that the two tendencies just balance. At each
point the elasticity of the gas must be just enough to balance the
weight of the layers above; and since it is the heat which furnishes
the elasticity, this requirement settles how much heat the gas must
have. And so we find the degree of heat or temperature at each point.

The same thing can be expressed a little differently. As before, fix
attention on a certain point in a star and consider how the matter
above it is supported. If it were not supported it would fall to the
centre under the attractive force of gravitation. The support is given
by a succession of minute blows delivered by the particles underneath;
we have seen that their heat energy causes them to move in all
directions, and they keep on striking the matter above. Each blow gives
a slight boost upwards, and the whole succession of blows supports the
upper material in shuttlecock fashion. (This process is not confined
to the stars; for instance, it is in this way that a motor car is
supported by its tyres.) An increase of temperature would mean an
increase of activity of the particles, and therefore an increase in
the rapidity and strength of the blows. Evidently we have to assign a
temperature such that the sum total of the blows is neither too great
nor too small to keep the upper material steadily supported. That in
principle is our method of calculating the temperature.

One obvious difficulty arises, The whole supporting force will depend
not only on the activity of the particles (temperature) but also on
the number of them (density). Initially we do not know the density of
the matter at an arbitrary point deep within the sun. It is in this
connexion that the ingenuity of the mathematician is required. He has
a definite amount of matter to play with, viz. the known mass of the
sun; so the more he uses in one part of the globe the less he will have
to spare for other parts. He might say to himself, ‘I do not want to
exaggerate the temperature, so I will see if I can manage without going
beyond 10,000,000°.’ That sets a limit to the activity to be ascribed
to each particle; therefore when the mathematician reaches a great
depth in the sun and accordingly has a heavy weight of upper material
to sustain, his only resource is to use large numbers of particles to
give the required total impulse. He will then find that he has used up
all his particles too fast, and has nothing left to fill up the centre.
Of course his structure, supported on nothing, would come tumbling
down into the hollow. In that way we can prove that it is impossible
to build up a permanent star of the dimensions of the sun without
introducing an activity or temperature exceeding 10,000,000°. The
mathematician can go a step beyond this; instead of merely finding a
lower limit, he can ascertain what must be nearly the true temperature
distribution by taking into account the fact that the temperature
must not be ‘patchy’. Heat flows from one place to another, and any
patchiness would soon be evened out in an actual star. I will leave the
mathematician to deal more thoroughly with these considerations, which
belong to the following up of the clue; I am content if I have shown
you that there is an opening for an attack on the problem.

This kind of investigation was started more than fifty years ago. It
has been gradually developed and corrected, until now we believe that
the results must be nearly right--that we really know how hot it is
inside a star.

I mentioned just now a temperature of 6,000°; that was the temperature
near the surface--the region which we actually see. There is no serious
difficulty in determining this surface temperature by observation;
in fact the same method is often used commercially for finding the
temperature of a furnace from the outside. It is for the deep regions
out of sight that the highly theoretical method of calculation is
required. This 6,000° is only the marginal heat of the great solar
furnace giving no idea of the terrific intensity within. Going down
into the interior the temperature rises rapidly to above a million
degrees, and goes on increasing until at the sun’s centre it is about
40,000,000°.

Do not imagine that 40,000,000° is a degree of heat so extreme that
temperature has become meaningless. These stellar temperatures are to
be taken quite literally. Heat is the energy of motion of the atoms or
molecules of a substance, and temperature which indicates the degree
of heat is a way of stating how fast these atoms or molecules are
moving. For example, at the temperature of this room the molecules of
air are rushing about with an average speed of 500 yards a second; if
we heated it up to 40,000,000° the speed would be just over 100 miles
a second. That is nothing to be alarmed about; the astronomer is quite
accustomed to speeds like that. The velocities of the stars, or of the
meteors entering the earth’s atmosphere, are usually between 10 and 100
miles a second. The velocity of the earth travelling round the sun is
20 miles a second. So that for an astronomer this is the most ordinary
degree of speed that could be suggested, and he naturally considers
40,000,000° a very comfortable sort of condition to deal with. And if
the astronomer is not frightened by a speed of 100 miles a second, the
experimental physicist is quite contemptuous of it; for he is used
to handling atoms shot off from radium and similar substances with
speeds of 10,000 miles a second. Accustomed as he is to watching these
express atoms and testing what they are capable of doing, the physicist
considers the jog-trot atoms of the stars very commonplace.

Besides the atoms rushing to and fro in all directions we have in the
interior of a star great quantities of ether waves also rushing in
all directions. Ether waves are called by different names according
to their wave-length. The longest are the Hertzian waves used in
broadcasting; then come the infra-red heat waves; next come waves of
ordinary visible light; then ultra-violet photographic or chemical
rays; then X-rays; then Gamma rays emitted by radio-active substances.
Probably the shortest of all are the rays constituting the very
penetrating radiation found in our atmosphere, which according to
the investigations of Kohlhörster and Millikan are believed to reach
us from interstellar space. These are all fundamentally the same but
correspond to different octaves. The eye is attuned to only one octave,
so that most of them are invisible; but essentially they are of the
same nature as visible light.

The ether waves inside a star belong to the division called X-rays.
They are the same as the X-rays produced artificially in an X-ray tube.
On the average they are ‘softer’ (i.e. longer) than the X-rays used
in hospitals, but not softer than some of those used in laboratory
experiments. Thus we have in the interior of a star something familiar
and extensively studied in the laboratory.

Besides the atoms and ether waves there is a third population to join
in the dance. There are multitudes of free electrons. The electron is
the lightest thing known, weighing no more than ¹⁄₁₈₄₀ of the lightest
atom. It is simply a charge of negative electricity wandering about
alone. An atom consists of a heavy nucleus which is usually surrounded
by a girdle of electrons. It is often compared to a miniature solar
system, and the comparison gives a proper idea of the _emptiness_
of an atom. The nucleus is compared to the sun, and the electrons to
the planets. Each kind of atom--each chemical element--has a different
quorum of planet electrons. Our own solar system with eight planets
might be compared especially with the atom of oxygen which has eight
circulating electrons. In terrestrial physics we usually regard the
girdle or crinoline of electrons as an essential part of the atom
because we rarely meet with atoms incompletely dressed; when we do meet
with an atom which has lost one or two electrons from its system, we
call it an ‘ion’. But in the interior of a star, owing to the great
commotion going on, it would be absurd to exact such a meticulous
standard of attire. All our atoms have lost a considerable proportion
of their planet electrons and are therefore _ions_ according to
the strict nomenclature.


                     _Ionization  of Atoms_


At the high temperature inside a star the battering of the particles
by one another, and more especially the collision of the ether waves
(X-rays) with atoms, cause electrons to be broken off and set free.
These free electrons form the third population to which I have
referred. For each individual the freedom is only temporary, because it
will presently be captured by some other mutilated atom; but meanwhile
another electron will have been broken off somewhere else to take its
place in the free population. This breaking away of electrons from
atoms is called _ionization_, and as it is extremely important in
the study of the stars I will presently show you photographs of the
process.

My subject is ‘Stars and Atoms’; I have already shown you photographs
of a star, so I ought to show you a photograph of an atom. Nowadays
that is quite easy. Since there are some trillions of atoms present
in the tiniest piece of material it would be very confusing if
the photograph showed them all. Happily the photograph exercises
discrimination and shows only ‘express train’ atoms which flash past
like meteors, ignoring all the others. We can arrange a particle of
radium to shoot only a few express atoms across the field of the
camera, and so have a clear picture of each of them.

Fig. 3[2] is a photograph of three or four atoms which have flashed
across the field of view--giving the broad straight tracks. These are
atoms of helium discharged at high speed from a radio-active substance.

I wonder if there is an under-current of suspicion in your minds
that there must be something of a fake about this photograph. Are
these really the single atoms that are showing themselves--those
infinitesimal units which not many years ago seemed to be theoretical
concepts far outside any practical apprehension? I will answer that
question by asking you one. You see a dirty mark on the picture. Is
that somebody’s thumb? If you say Yes, then I assure you unhesitatingly
that these streaks are single atoms. But if you are hypercritical
and say ‘No. That is not anybody’s thumb, but it is a mark that
shows that somebody’s thumb has been there’, then I must be equally
cautious and say that the streak is a mark that shows where an atom
has been. The photograph instead of being the impression of an atom
is the impression of the impression of an atom, just as it is not the
impression of a thumb but the impression of the impression of a thumb.
I don’t see that it really matters that the impression is second-hand
instead of first-hand. I do not think we have been guilty of any more
faking than the criminologist who scatters powder over a finger-print
to make it visible, or a biologist who stains his preparations with
the same object. The atom in its passage leaves what we might call a
‘scent’ along its trail; and we owe to Professor C. T. R. Wilson a most
ingenious device for making the scent visible. Professor Wilson’s ‘pack
of hounds’ consists of water vapour which flocks to the trail and there
condenses into tiny drops.

You will next want to see a photograph of an electron. That also can be
managed. The broken wavy trail in Fig. 3 is an electron. Owing to its
small mass the electron is more easily turned aside in its course than
the heavy atom which rushes bull-headed through all obstacles. Fig. 4
shows numerous electrons, and it includes one of very high speed which
on that account was able to make a straight track. Incidentally it
gives away the device used for making the tracks visible, because you
can see the tiny drops of water separately.

[Illustration: Fig. 3]

[Illustration: Fig. 4. FAST-MOVING ATOMS AND ELECTRONS]

We have seen photographs of atoms and free electrons. Now we want a
photograph of X-rays to complete the stellar population. We cannot
quite manage that, but we can very nearly. Photographs _by_
X-rays are common enough; but a photograph _of_ X-rays is a
different matter. I have already said that electrons can be broken
away from atoms by X-rays colliding with them. When this happens the
free electron is usually shot off with high velocity so that it is one
of the express electrons which can be photographed. In Fig. 5 you see
four electrons shot off in this way. You notice that they all start
from points in the same line, and it does not require much imagination
to see in your mind a mysterious power travelling along this line
and creating the explosions. That power is the X-rays which were
directed in a narrow beam along the line (from right to left) when the
photograph was taken. Although the X-rays are left to your imagination,
the photograph at any rate shows the process of ionization which is so
important in the stellar interior--the freeing of electrons from the
atoms by the incidence of X-rays. You notice that it is just a chance
whether the X-ray ionizes an atom when it meets it. There are trillions
of atoms lying about (of which the photograph takes no notice); but,
nevertheless, the X-rays travel a long way before meeting the atom
which they choose to operate on.

Finally I can show you the other method of ionizing atoms by battering
of a more mechanical kind--in this case by the collision of a fast
electron. In Fig. 6 a fast electron was travelling nearly horizontally,
but the tiny water-drops that should mark its track are so spread out
that you do not at first trace the connexion. Notice that the drops
occur in pairs. This is because the fast electron battered some of the
atoms along its track, wrenching away an electron from each. You see
at intervals along the track a broken atom and a free electron lying
side by side, though you cannot tell which is which. Occasionally the
original fast electron was too vigorous and there is more of a mix up,
but usually you can see clearly the two fragments resulting from the
smash.[3]

A cynic might remark that the interior of a star is a very safe
subject to talk about because no one can go there and prove that you
are wrong. I would plead in reply that at least I am not abusing the
unlimited opportunity for imagination; I am only asking you to allow
in the interior of the star quite homely objects and processes which
can be photographed. Perhaps now you will turn round on me and say,
‘What right have you to suppose that Nature is as barren of imagination
as you are? Perhaps she has hidden in the star something novel which
will upset all your ideas.’ But I think that science would never have
achieved much progress if it had always imagined unknown obstacles
hidden round every corner. At least we may peer gingerly round the
corner, and perhaps we shall find there is nothing very formidable
after all. Our object in diving into the interior is not merely
to admire a fantastic world with conditions transcending ordinary
experience; it is to get at the inner mechanism which makes stars
behave as they do. If we are to understand the surface manifestations,
if we are to understand why ‘one star differeth from another star in
glory’, we must go below--to the _engine-room_--to trace the
beginning of the stream of heat and energy which pours out through the
surface. Finally, then, our theory will take us back to the surface
and we shall be able to test by comparison with observation whether we
have been badly misled. Meanwhile, although we naturally cannot prove a
general negative, there is no reason to anticipate anything which our
laboratory experience does not warn us of.

The X-rays in a star are the same as the X-rays experimented on in a
laboratory, but they are enormously more abundant in the star. We can
produce X-rays like the stellar X-rays, but we cannot produce them
in anything like stellar abundance. The photograph (Fig. 5) showed
a laboratory beam of X-rays which had wrenched away four electrons
from different atoms; these would be speedily recaptured. In the star
you must imagine the intensity multiplied many million-fold, so that
electrons are being wrenched away as fast as they settle and the atoms
are kept stripped almost bare. The nearly complete mutilation of the
atoms is important in the study of the stars for two main reasons.

The first is this. An architect before pronouncing an opinion on the
plans of a building will want to know whether the material shown in
the plans is to be wood or steel or tin or paper. Similarly it would
seem essential before working out details about the interior of a
star to know whether it is made of heavy stuff like lead or light
stuff like carbon. By means of the spectroscope we can find out a
great deal about the chemical composition of the sun’s atmosphere;
but it would not be fair to take this as a sample of the composition
of the sun as a whole. It would be very risky to make a guess at the
elements preponderating in the deep interior. Thus we seem to have
reached a deadlock. But now it turns out that when the atoms are
thoroughly smashed up, they all behave nearly alike--at any rate in
those properties with which we are concerned in astronomy. The high
temperature--which we were inclined to be afraid of at first--has
simplified things for us, because it has to a large extent eliminated
differences between different kinds of material. The structure
of a star is an unusually simple physical problem; it is at low
temperatures such as we experience on the earth that matter begins
to have troublesome and complicated properties. Stellar atoms are
nude savages innocent of the class distinctions of our fully arrayed
terrestrial atoms. We are thus able to make progress without guessing
at the chemical composition of the interior. It is necessary to make
one reservation, viz. that there is not an excessive proportion of
hydrogen. Hydrogen has its own way of behaving; but it makes very
little difference which of the other 91 elements predominate.

The other point is one about which I shall have more to say later.
It is that we must realize that the atoms in the stars are mutilated
fragments of the bulky atoms with extended electron systems familiar to
us on the earth; and therefore the behaviour of stellar and terrestrial
gases is by no means the same in regard to properties which concern the
size of the atoms.

To illustrate the effect of the chemical composition of a star, we
revert to the problem of the support of the upper layers by the
gas underneath. At a given temperature every independent particle
contributes the same amount of support no matter what its mass or
chemical nature; the lighter atoms make up for their lack of mass by
moving more actively. This is a well-known law originally found in
experimental chemistry, but now explained by the kinetic theory of
Maxwell and Boltzmann. Suppose we had originally assumed the sun to
be composed entirely of silver atoms and had made our calculations
of temperature accordingly; afterwards we change our minds and
substitute a lighter element, aluminium. A silver atom weighs just
four times as much as an aluminium atom; hence we must substitute
four aluminium atoms for every silver atom in order to keep the mass
of the sun unchanged. But now the supporting force will everywhere be
quadrupled, and all the mass will be heaved outwards by it if we make
no further change. In order to keep the balance, the activity of each
particle must be reduced in the ratio ¼; that means that we must assign
throughout the aluminium sun temperatures ¼ of those assigned to the
silver sun. Thus for unsmashed atoms a change in the assigned chemical
composition makes a big change in our inference as to the internal
temperature.

But if electrons are broken away from the atom these also become
independent particles rendering support to the upper layers. A
free electron gives just as much support as an atom does; it is of
much smaller mass, but it moves about a hundred times as fast. The
smashing of one silver atom provides 47 free electrons, making with
the residual nucleus of the atom 48 particles in all. The aluminium
atom gives 13 electrons or 14 particles in all; thus 4 aluminium atoms
give 56 independent particles. The change from smashed silver to an
equal mass of smashed aluminium only means a change from 48 to 56
particles, requiring a reduction of temperature by 14 per cent. We
can tolerate that degree of uncertainty in our estimates of internal
temperature;[4] it is a great improvement on the corresponding
calculation for unsmashed atoms which was uncertain by a factor 4.

Besides bringing closer together the results for different varieties
of chemical constitution, ionization by increasing the number of
supporting particles lowers the calculated temperatures considerably.
It is sometimes thought that the exceedingly high temperature assigned
to the interior of a star is a modern sensationalism. That is not so.
The early investigators, who neglected both ionization and radiation
pressure, assigned much higher temperatures than those now accepted.


                     _Radiation Pressure and Mass_


The stars differ from one another in mass, that is to say, in
the quantity of material gathered together to form them; but the
differences are not so large as we might have expected from the great
variety in brightness. We cannot always find out the mass of a star,
but there are a fair number of stars for which the mass has been
determined by astronomical measurements. The mass of the sun is--I will
write it on the blackboard--

2000000000000000000000000000 tons

I hope I have counted the 0’s rightly, though I dare say you would not
mind much if there were one or two too many or too few. But Nature
_does_ mind. When she made the stars she evidently attached great
importance to getting the number of 0’s right. She has an idea that
a star should contain a particular amount of material. Of course she
allows what the officials at the mint would call a ‘remedy’. She may
even pass a star with one 0 too many and give us an exceptionally
large star, or with one 0 too few, giving a very small star. But these
deviations are rare, and a mistake of two 0’s is almost unheard of.
Usually she adheres much more closely to her pattern.


[Illustration: Fig. 5. IONIZATION BY X-RAYS]


[Illustration: Fig. 6. IONIZATION BY COLLISION]


How does Nature keep count of the 0’s? It seems clear that there must
be something inside the star itself which keeps check and, so to speak,
makes a warning protest as soon as the right amount of material has
been gathered together. We think we know how it is done. You remember
the ether waves inside the star. These are trying to escape outwards
and they exert a pressure on the matter which is caging them in. This
outward force, if it is sufficiently powerful to be worth considering
in comparison with other forces, must be taken into account in any
study of the equilibrium or stability of the star. Now in all small
globes this force is quite trivial; but its importance increases with
the mass of the globe, and it is calculated that at just about the
above mass it reaches equal status with the other forces governing
the equilibrium of the star. If we had never seen the stars and were
simply considering as a curious problem how big a globe of matter
could possibly hold together, we could calculate that there would be
no difficulty up to about two thousand quadrillion tons; but beyond
that the conditions are entirely altered and this new force begins to
take control of the situation. Here, I am afraid, strict calculation
stops, and no one has yet been able to calculate what the new force
will do with the star when it does take control. But it can scarcely be
an accident that the stars are all so near to this critical mass; and
so I venture to conjecture the rest of the story. The new force does
not _prohibit_ larger mass, but it makes it risky. It may help a
moderate rotation about the axis to break up the star. Consequently
larger masses will survive only rarely; for the most part stars will be
kept down to the mass at which the new force first becomes a serious
menace. The force of gravitation collects together nebulous and chaotic
material; the force of radiation pressure chops it off into suitably
sized lumps.

This force of radiation pressure is better known to many people under
the name ‘pressure of light’. The term ‘radiation’ comprises all kinds
of ether-waves including light, so that the meaning is the same. It
was first shown theoretically and afterwards verified experimentally
that light exerts a minute pressure on any object on which it falls.
Theoretically it would be possible to knock a man over by turning a
searchlight on him--only the searchlight would have to be excessively
intense, and the man would probably be vaporized first. Pressure of
light probably plays a great part in many celestial phenomena. One of
the earliest suggestions was that the minute particles forming the
tail of a comet are driven outwards by the pressure of sunlight, thus
accounting for the fact that a comet’s tail points away from the sun.
But that particular application must be considered doubtful. Inside
the star the intense stream of light (or rather X-rays) is like a wind
rushing outwards and distending the star.


                       _The Interior of a Star_


We can now form some kind of a picture of the inside of a star--a
hurly-burly of atoms, electrons, and ether-waves. Dishevelled atoms
tear along at 100 miles a second, their normal array of electrons being
torn from them in the scrimmage. The lost electrons are speeding 100
times faster to find new resting places. Let us follow the progress of
one of them. There is almost a collision as an electron approaches an
atomic nucleus, but putting on speed it sweeps round in a sharp curve.
Sometimes there is a side-slip at the curve, but the electron goes on
with increased or reduced energy. After a thousand narrow shaves, all
happening within a thousand millionth of a second, the hectic career is
ended by a worse side-slip than usual. The electron is fairly caught,
and attached to an atom. But scarcely has it taken up its place when
an X-ray bursts into the atom. Sucking up the energy of the ray the
electron darts off again on its next adventure.

I am afraid the knockabout comedy of modern atomic physics is not very
tender towards our aesthetic ideals. The stately drama of stellar
evolution turns out to be more like the hair-breadth escapades on the
films. The music of the spheres has almost a suggestion of--_jazz_.

And what is the result of all this bustle? Very little. The atoms
and electrons for all their hurry never get anywhere; they only
change places. The ether-waves are the only part of the population
which accomplish anything permanent. Although apparently darting in
all directions indiscriminately, they do on the average make a slow
progress outwards. There is no outward progress of the atoms and
electrons; gravitation sees to that. But slowly the encaged ether-waves
leak outwards as through a sieve. An ether-wave hurries from one atom
to another, forwards, backwards, now absorbed, now flung out again
in a new direction, losing its identity, but living again in its
successor. With any luck it will in no unduly long time (ten thousand
to ten million years according to the mass of the star) find itself
near the boundary. It changes at the lower temperature from X-rays to
light-rays, being altered a little at each re-birth. At last it is so
near the boundary that it can dart outside and travel forward in peace
for a few hundred years. Perhaps it may in the end reach some distant
world where an astronomer lies in wait to trap it in his telescope and
extort from it the secrets of its birth-place.

It is the leakage that we particularly want to determine; and that
is why we have to study patiently what is going on in the turbulent
crowd. To put the problem in another form; the waves are urged to flow
out by the temperature gradient in the star, but are hindered and
turned back by their adventures with the atoms and electrons. It is
the task of mathematics, aided by the laws and theories developed from
a study of these same processes in the laboratory, to calculate the
two factors--the factor urging and the factor hindering the outward
flow--and hence to find the leakage. This calculated leakage should,
of course, agree with astronomical measurements of the energy of heat
and light pouring out of the star. And so finally we arrive at an
observational test of the theories.


                      _Opacity of Stellar Matter_


Let us consider the factor which hinders the leakage--the turning back
of the ether-waves by their encounters with atoms and electrons. If
we were dealing with light waves we should call this obstruction to
their passage ‘opacity’, and we may conveniently use the same term for
obstruction to X-rays.

We soon realize that the material of the star must be decidedly opaque.
The quantity of radiation in the interior is so great that unless it
were very severely confined the leakage would be much greater than the
amount which we observe coming out of the stars. The following is an
illustration of the typical degree of opacity required to agree with
the observed leakage. Let us enter the star Capella and find a region
where the density is the same as that of the atmosphere around us;[5]
a slab of the material only two inches thick would form a screen so
opaque that only one-third of the ether-waves falling on one side
would get through to the other side, the rest being absorbed in the
screen. A foot or two of the material would be practically a perfect
screen. If we are thinking of light-waves this seems an astonishing
opacity for material as tenuous as air; but we have to remember that
it is an opacity to X-rays, and the practical physicist knows well the
difficulty of getting the softer kinds of X-rays to pass through even a
few millimetres of air.

There is a gratifying accordance in general order of magnitude between
the opacity inside the star, determined from astronomical observation
of leakage, and the opacity of terrestrial substances to X-rays of more
or less the same wave-length. This gives us some assurance that our
theory is on the right track. But a careful comparison shows us that
there is some important difference between the stellar and terrestrial
opacity.

In the laboratory we find that the opacity increases very rapidly with
the wave-length of the X-rays that are used. We do not find anything
like the same difference in the stars although the X-rays in the cooler
stars must be of considerably greater wave-length than those in the
hotter stars. Also, taking care to make the comparison at the same
wave-length for both, we find that the stellar opacity is less than the
terrestrial opacity. We must follow up this divergence.

There is more than one way in which an atom can obstruct ether-waves,
but there seems to be no doubt that for X-rays both in the stars and in
the laboratory the main part of the opacity depends on the process of
ionization. The ether-wave falls on an atom and its energy is sucked up
by one of the planet electrons which uses it to escape from the atom
and travel away at high speed. The point to notice is that in the very
act of absorption the absorbing mechanism is broken, and it cannot
be used again until it has been repaired. To repair it the atom must
capture one of the free electrons wandering about, inducing it to take
the place of the lost electron.

In the laboratory we can only produce thin streams of X-rays so that
each wave-trap is only called upon to act occasionally. There is
plenty of time to repair it before the next time it has a chance of
catching anything; and there is practically no loss of efficiency
through the traps being out of order. But in the stars the stream of
X-rays is exceedingly intense. It is like an army of mice marching
through your larder springing the mouse-traps as fast as you can set
them. Here it is the time wasted in resetting the traps--by capturing
electrons--which counts, and the amount of the catch depends almost
entirely on this.

We have seen that the stellar atoms have lost most of their electrons;
that means that at any moment a large proportion of the absorption
traps are awaiting repair. For this reason we find a smaller opacity in
the stars than in terrestrial material. The lowered opacity is simply
the result of overworking the absorbing mechanisms--they have too
much radiation to deal with. We can also see why the laws of stellar
and terrestrial opacity are somewhat different. The rate of repair,
which is the main consideration in stellar opacity, is increased by
compressing the material, because then the atom will not have to wait
so long to meet and capture a free electron. Consequently the stellar
opacity will increase with the density. In terrestrial conditions there
is no advantage in accelerating the repair which will in any case be
completed in sufficient time; thus terrestrial opacity is independent
of the density.

The theory of stellar opacity thus reduces mainly to the theory of
the capture of electrons by ionized atoms; not that this process
is attended by absorption of X-rays--it is actually attended by
emission--but it is the necessary preliminary to absorption. The
physical theory of electron-capture is not yet fully definitive; but it
is sufficiently advanced for us to make use of it provisionally in our
calculations of the hindering factor in the leakage of radiation from
the stars.


                 _The Relation of Brightness to Mass_


We do not want to tackle too difficult a problem at first, and so we
shall deal with stars composed of perfect gas. If you do not like
the technical phrase ‘perfect gas’ you can call it simply ‘gas’,
because all the terrestrial gases that you are likely to think of are
without sensible imperfection. It is only under high compression that
terrestrial gases become imperfect. I should mention that there are
plenty of examples of gaseous[6] stars. In many stars the material
is so inflated that it is more tenuous than the air around us; for
example, if you were inside Capella you would not notice the material
of Capella any more than you notice the air in this room.

For gaseous stars, then, the investigation will give formulae by which,
given the mass of the star, we can calculate how much energy of heat
and light will leak out of it--in short, how bright it will be. In
Fig. 7 a curve is drawn giving this theoretical relation between the
brightness and mass of a star. Strictly speaking, there is another
factor besides the mass which affects the calculated brightness; you
can have two stars of the same mass, the one dense and the other puffed
out, and they will not have quite the same brightness. But it turns
out (rather unexpectedly) that this other factor, density, makes very
little difference to the brightness, always provided that the material
is not too dense to be a perfect gas. I shall therefore say no more
about density in this brief summary.

       *       *       *       *       *

Here are a few details about the scale of the diagram. The brightness
is measured in magnitudes, a rather technical unit. You have to
remember that stellar magnitude is like a golfer’s handicap--the bigger
the number, the worse the performance. The diagram includes practically
the whole range of stellar brightness; at the top -4 represents
almost the brightest stars known, and at the bottom 12 is nearly the
faintest limit. The difference from top to bottom is about the same as
the difference between an arc light and a glow worm. The sun is near
magnitude 5. These magnitudes refer, of course, to the true brightness,
not to the apparent brightness affected by distance; also, what is
represented here is the ‘heat brightness’ or heat intensity, which is
sometimes a little different from the light intensity. Astronomical
instruments have been made which measure directly the heat instead of
the light received from a star. These are quite successful; but there
are troublesome corrections on account of the large absorption of heat
in the earth’s atmosphere, and it is in most cases easier and more
accurate to infer the heat brightness from the light brightness, making
allowance for the colour of the star.

[Illustration: Fig. 7. The Mass-luminosity Curve.]

The horizontal scale refers to mass, but it is graduated according to
the logarithm of the mass. At the extreme left the mass is about ⅙ x
sun, and on the extreme right about 30 x sun; there are very few stars
with masses outside these limits. The sun’s mass corresponds to the
division labelled 0·0.

Having obtained our theoretical curve, the first thing to do is to test
it by observation. That is to say, we gather together as many stars as
we can lay hands on for which both the mass and absolute brightness
have been measured. We plot the corresponding points (opposite to the
appropriate horizontal and vertical graduations) and see whether they
fall on the curve, as they ought to do if the theory is right. There
are not many stellar masses determined with much precision. Everything
that is reasonably trustworthy has been included in Fig. 7. The
circles, crosses, squares, and triangles refer to different kinds of
data--some good, some bad, some very bad.

The circles are the most trustworthy. Let us run through them from
right to left. First comes the bright component of Capella, lying
beautifully on the curve--because I drew the curve through it. You
see, there was one numerical constant which in the present state of
our knowledge of atoms and ether-waves, &c., it was not possible to
determine with any confidence from pure theory. So the curve when it
was obtained was loose in one direction and could be raised or lowered.
It was anchored by making it pass through the bright component of
Capella which seemed the best star to trust to for this purpose. After
that there could be no further tampering with the curve. Continuing to
the left we have the fainter component of Capella; next Sirius; then,
in a bunch, two components of α Centauri (the nearest fixed star) with
the Sun between them, and--lying on the curve--a circle representing
the mean of six double stars in the Hyades. Finally, far on the left
there are two components of a well-known double star called Krueger 60.

The observational data for testing the curve are not so extensive and
not so trustworthy as we could wish; but still I think it is plain from
Fig. 7 that the theory is substantially confirmed, and it really does
enable us to predict the brightness of a star from its mass, or vice
versa. That is a useful result, because there are thousands of stars of
which we can measure the absolute brightness but not the mass, and we
can now infer their masses with some confidence.

Since I have not been able to give here the details of the calculation,
I should make it plain that the curve in Fig. 7 is traced by pure
theory or terrestrial experiment except for the one constant determined
by making it pass through Capella. We can imagine physicists working
on a cloud-bound planet such as Jupiter who have never seen the stars.
They should be able to deduce by the method explained on p. 25 that
if there is a universe existing beyond the clouds it is likely to
aggregate primarily into masses of the order a thousand quadrillion
tons. They could then predict that these aggregations will be globes
pouring out light and heat and that their brightness will depend on
the mass in the way given by the curve in Fig. 7. All the information
that we have used for the calculations would be accessible to them
beneath the clouds, except that we have stolen one advantage over
them in utilizing the bright component of Capella. Even without this
forbidden peep, present-day physical theory would enable them to assign
a brightness to the invisible stellar host which would not be absurdly
wrong. Unless they were wiser than us they would probably ascribe to
all the stars a brilliance about ten times too great[7]--not a bad
error for a first attempt at so transcendent a problem. We hope to
clear up the discrepant factor 10 with further knowledge of atomic
processes; meanwhile we shelve it by fixing the doubtful constant by
astronomical observation.


                             _Dense Stars_


The agreement of the observational points with the curve is remarkably
close, considering the rough nature of the observational measurements;
and it seems to afford a rather strong confirmation of the theory. But
there is one awful confession to make--_we have compared the theory
with the wrong stars_. At least when the comparison was first made
at the beginning of 1924 no one entertained any doubt that they were
the wrong stars.

We must recall that the theory was developed for stars in the condition
of a perfect gas. In the right half of Fig. 7 the stars represented
are all diffuse stars; Capella with a mean density about equal to that
of the air in this room may be taken as typical. Material of this
tenuity is evidently a true gas, and in so far as these stars agree
with the curve the theory is confirmed. But in the left half of the
diagram we have the Sun whose material is denser than water, Krueger
60 denser than iron, and many other stars of the density usually
associated with solid or liquid matter. What business have they on the
curve reserved for a perfect gas? When these stars were put into the
diagram it was not with any expectation that they would agree with
the curve; in fact, the agreement was most annoying. Something very
different was being sought for. The idea was that the theory might
perhaps be trusted on its own merits with such confirmation as the
diffuse stars had already afforded; then by measuring how far these
dense stars fell below the curve we should have definite information
as to how great a deviation from a perfect gas occurred at any given
density. According to current ideas it was expected that the sun would
fall three or four magnitudes below the curve, and the still denser
Krueger 60 should be nearly ten magnitudes below.[8] You see that the
expectation was entirely unfulfilled.

The shock was even greater than I can well indicate to you, because
the great drop in brightness when the star is too dense to behave
as a true gas was a fundamental tenet in our conception of stellar
evolution. On the strength of it the stars had been divided into two
groups known as giants and dwarfs, the former being the gaseous stars
and the latter the dense stars.

Two alternatives now lie before us. The first is to assume that
something must have gone wrong with our theory; that the true curve for
gaseous stars is not as we have drawn it, but runs high up on the left
of the diagram so that the Sun, Krueger 60, &c., are at the appropriate
distances below it. In short, our imaginary critic was right; Nature
had hidden something unexpected inside the star and so frustrated our
calculations. Well, if this were so, it would be something to have
found it out by our investigations.

The other alternative is to consider this question--Is it impossible
that a perfect gas should have the density of iron? The answer is
rather surprising. There is no earthly reason why a perfect gas should
not have a density far exceeding iron. Or it would be more accurate to
say, the reason why it should not is _earthly_ and does not apply
to the stars.

The sun’s material, in spite of being denser than water, really is
a perfect gas. It sounds incredible, but it must be so. The feature
of a true gas is that there is plenty of room between the separate
particles--a gas contains very little substance and lots of emptiness.
Consequently when you squeeze it you do not have to squeeze the
substance; you just squeeze out some of the waste space. But if
you go on squeezing, there comes a time when you have squeezed out
all the empty space; the atoms are then jammed in contact and any
further compression means squeezing the substance itself, which is
quite a different proposition. So as you approach that density the
compressibility characteristic of a gas is lost and the matter is no
longer a proper gas. In a liquid the atoms are nearly in contact;
that will give you an idea of the density at which the gas loses its
characteristic compressibility.

The big terrestrial atoms which begin to jam at a density near that
of the liquid state do not exist in the stars. The stellar atoms have
been trimmed down by the breaking off of all their outer electrons. The
lighter atoms are stripped to the bare nucleus--of quite insignificant
size. The heavier atoms retain a few of the closer electrons, but have
not much more than a hundredth of the diameter of a fully arrayed atom.
Consequently we can go on squeezing ever so much more before these
tiny atoms or ions jam in contact. At the density of water or even of
platinum there is still any amount of room between the trimmed atoms;
and waste space remains to be squeezed out as in a perfect gas.

Our mistake was that in estimating the congestion in the stellar
ball-room we had forgotten that crinolines are no longer in fashion.

It was, I suppose, very blind of us not to have foreseen this result,
considering how much attention we had been paying to the mutilation
of the atoms in other branches of the investigation. By a roundabout
route we have reached a conclusion which is really very obvious. And
so we conclude that the stars on the left of the diagram are after
all not the ‘wrong’ stars. The sun and other dense stars are on the
perfect gas curve because their material is perfect gas. Careful
investigation has shown that in the small stars on the extreme left of
Fig. 7 the electric charges of the atoms and electrons bring about a
slight deviation from the ordinary laws of a gas; it has been shown by
R. H. Fowler that the effect is to make the gas not _imperfect_
but _superperfect_--it is _more_ easily compressed than an
ordinary gas. You will notice that on the average the stars run a
little above the curve on the left of Fig. 7. It is probable that the
deviation is genuine and is partly due to superperfection of the gas;
we have already seen that imperfection would have brought them below
the curve.

Even at the density of platinum there is plenty of waste space, so
that in the stars we might go on squeezing stellar matter to a density
transcending anything known on the earth. But that’s another story--I
will tell it later on.

       *       *       *       *       *

The general agreement between the observed and predicted brightness of
the stars of various masses is the main test of the correctness of our
theories of their internal constitution. The incidence of their masses
in a range which is especially critical for radiation pressure is also
valuable confirmation. It would be an exaggeration to claim that this
limited success is a proof that we have reached the truth about the
stellar interior. It is not a proof, but it is an encouragement to work
farther along the line of thought which we have been pursuing. The
tangle is beginning to loosen. The more optimistic may assume that it
is now straightened out; the more cautious will make ready for the next
knot. The one reason for thinking that the real truth cannot be so very
far away is that in the interior of a star, if anywhere, the problem
of matter is reduced to its utmost simplicity; and the astronomer is
engaged on what is essentially a less ambitious problem than that of
the terrestrial physicist to whom matter always appears in the guise
of electron systems of the most complex organization.

We have taken the present-day theories of physics and pressed them to
their remotest conclusions. There is no dogmatic intention in this;
it is the best means we have of testing them and revealing their
weaknesses if any.

In ancient days two aviators procured to themselves wings. Daedalus
flew safely through the middle air and was duly honoured on his
landing. Icarus soared upwards to the sun till the wax melted which
bound his wings and his flight ended in fiasco. In weighing their
achievements, there is something to be said for Icarus. The classical
authorities tell us that he was only ‘doing a stunt’, but I prefer to
think of him as the man who brought to light a serious constructional
defect in the flying-machines of his day. So, too, in Science. Cautious
Daedalus will apply his theories where he feels confident they will
safely go; but by his excess of caution their hidden weaknesses remain
undiscovered. Icarus will strain his theories to the breaking-point
till the weak joints gape. For the mere adventure? Perhaps partly; that
is human nature. But if he is destined not yet to reach the sun and
solve finally the riddle of its constitution, we may at least hope to
learn from his journey some hints to build a better machine.




                              LECTURE II

                      SOME RECENT INVESTIGATIONS


IT will help us to appreciate the astronomical significance of what we
have learnt in the previous lecture if we turn from the general to the
particular and see how it applies to individual stars. I will take two
stars round which centre stories of special interest, and relate the
history of our knowledge of them.


                         _The Story of Algol_


This is a detective story, which we might call ‘The Missing Word and
the False Clue’.

In astronomy, unlike many sciences, we cannot handle and probe the
objects of our study; we have to wait passively and receive and decode
the messages that they send to us. The whole of our information
about the stars comes to us along rays of light; we watch and try
to understand their signals. There are some stars which seem to be
sending us a regular series of dots and dashes--like the intermittent
light from a lighthouse. We cannot translate this as a morse code;
nevertheless, by careful measurement we disentangle a great deal of
information from the messages. The star Algol is the most famous of
these ‘variable stars’. We learn from the signals that it is really two
stars revolving round each other. Sometimes the brighter of the two
stars is hidden, giving a deep eclipse or ‘dash’; sometimes the faint
star is hidden, giving a ‘dot’. This recurs in a period of 2 days 21
hours--the period of revolution of the two stars.

There was a great deal more information in the message, but it was
rather tantalizing. There was, so to speak, just one word missing. If
we could supply that word the message would give full and accurate
particulars as to the size of the system--the diameters and masses of
the two components, their absolute brightness, the distance between
them, their distance from the sun. Lacking the word the message told us
nothing really definite about any of these things.

In these circumstances astronomers would scarcely have been human if
they had not tried to guess the missing word. The word should have told
us how much bigger the bright star was than the fainter, that is to
say, the ratio of the masses of the two stars. Some of the less famous
variable stars give us complete messages. (These could accordingly be
used for testing the relation of mass and absolute brightness, and are
represented by triangles in Fig. 7.) The difficulty about Algol arose
from the excessive brightness of the bright component which swamped and
made illegible the more delicate signals from the faint component. From
the other systems we could find the most usual value of the mass ratio,
and base on that a guess as to its probable value for Algol. Different
authorities preferred slightly different estimates, but the general
judgement was that in systems like Algol the bright component is twice
as massive as the faint component. And so the missing word was assumed
to be ‘two’; on this assumption the various dimensions of the system
were worked out and came to be generally accepted as near the truth.
That was sixteen years ago.[9]

In this way the sense of the message was made out to be that the
brighter star had a radius of 1,100,000 kilometres (one and a half
times the sun’s radius), that it had half the mass of the sun, and
thirty times the sun’s light-power, &c. It will be seen at once that
this will not fit our curve in Fig. 7; a star of half the sun’s mass
ought to be very much fainter than the sun. It was rather disconcerting
to find so famous a star protesting against the theory; but after
all the theory is to be tested by comparison with facts and not with
guesses, and the theory might well have a sounder basis than the
conjecture as to the missing word. Moreover, the spectral type of Algol
is one that is not usually associated with low mass, and this cast some
suspicion on the accepted results.

If we are willing to trust the theory given in the last lecture we
can do without the missing word. Or, to put it another way, we can
try in succession various guesses instead of ‘two’ until we reach one
that gives the bright component a mass and luminosity agreeing with
the curve in Fig. 7. The guess ‘two’ gives, as we have seen, a point
which falls a long way from the curve. Alter the guess to ‘three’
and recalculate the mass and brightness on this assumption; the
corresponding point is now somewhat nearer to the curve. Continue with
‘four’, ‘five’, &c.; if the point crosses the curve we know that we
have gone too far and must take an intermediate value in order to reach
the desired agreement. This was done in November 1925, and it appeared
that the missing word must be ‘five’, not ‘two’--a rather startling
change. And now the message ran--

Radius of bright component = 2,140,000 kilometres.

Mass of bright component = 4·3 x sun’s mass.

If you compare these with the original figures you will see that there
is a great alteration. The star is now assigned a large mass much
more appropriate to a B-type star. It also turns out that Algol is
more than a hundred times as bright as the sun; and its parallax is
0·028"--twice the distance previously supposed.

At the time there seemed little likelihood that these conclusions could
be tested. Possibly the prediction as to the parallax might be proved
or disproved by a trigonometrical determination; but it is so small as
to be almost out of range of reasonably accurate measurement. We could
only adopt a ‘take it or leave it’ attitude--‘If you accept the theory,
_this_ is what Algol is like; if you distrust the theory, these
results are of no interest to you.’

But meanwhile two astronomers at Ann Arbor Observatory had been making
a search for the missing word by a remarkable new method. They had in
fact found the word and published it a year before, but it had not
become widely known. If a star is rotating, one edge or ‘limb’ is
coming towards us and the other going away from us. We can measure
speeds towards us or away from us by means of the Doppler effect on
the spectrum, obtaining a definite result in miles per second. Thus
we can and do measure the equatorial speed of rotation of the sun
by observing first the east limb then the west limb and taking the
difference of velocity shown. That is all very well on the sun, where
you can cover up the disk except the special part that you want to
observe; but how can you cover up part of a star when a star is a mere
point of light? _You_ cannot; but in Algol the covering up is done
for you. The faint component is your screen. As it passes in front
of the bright star there is a moment when it leaves a thin crescent
showing on the east and another moment when a thin crescent on the west
is uncovered. Of course, the star is too far away for you actually to
see the crescent shape, but at these moments you receive light from
the crescents only, the rest of the disk being hidden. By seizing
these moments you can make the measurements just as though you had
manipulated the screen yourself. Fortunately the speed of rotation of
Algol is large and so can be measured with relatively small error. Now
multiply the equatorial velocity by the period of rotation;[10] that
will give you the circumference of Algol. Divide by 6·28, and you have
the radius.

That was the method developed by Rossiter and McLaughlin. The latter
who applied it to Algol found the radius of the bright component to be

                         2,180,000 kilometres.

So far as can be judged his result has considerable accuracy; indeed it
is probable that the radius is now better known than that of any other
star except the sun. If you will now turn back to p. 44 and compare
it with the value found from the theory you will see that there is
cause for satisfaction. McLaughlin evaluated the other constants and
dimensions of the system; these agree equally well, but that follows
automatically because there was only one missing word to be supplied.
In both determinations the missing word or mass ratio turned out to be
5·0.

This is not quite the end of the story. Why had the first guess at
the mass ratio gone so badly wrong? We understand by now that a
disparity in mass is closely associated with a disparity in brightness
of the two stars. The disparity in brightness was given in Algol’s
original message; it informed us that the faint component gives about
one-thirteenth of the light of the bright one. (At least that was how
we interpreted it.) According to our curve this corresponds to a mass
ratio 2½, which is not much improvement on the original guess 2. For a
mass ratio 5 the companion ought to have been much fainter--in fact its
light should have been undetectable. Although considerations like these
could not have had much influence on the original guess, they seemed at
first to reassure us that there was not very much wrong with it.

Let us call the bright component Algol A and the faint component Algol
B. Some years ago a new discovery was made, namely Algol C. It was
found that Algol A and B together travel in an orbit round a third star
in a period of just under two years--at least they are travelling round
in this period, and we must suppose that there is something present
for them to revolve around. Hitherto we had believed that when Algol
A was nearly hidden at the time of deepest eclipse all the remaining
light must come from Algol B; but now it is clear that it belongs to
Algol C, which is always shining without interference. Consequently the
mass ratio 2½ is that of Algol A to Algol C. The light from Algol B is
inappreciable as it should be for a mass ratio 5.[11]

The message from Algol A and B was confused, not only on account of
the missing word, but because a word or two of another message from
Algol C had got mixed up with it; so that even when the missing word
was found to be ‘five’ and confirmed in two ways, the message was
not quite coherent. In another place the message seemed to waver
and read ‘two-and-a-half’. The finishing step is the discovery that
‘two-and-a-half’ belongs to a different message from a previously
unsuspected star, Algol C. And so it all ends happily.

The best detective is not infallible. In this story our astronomical
detective made a reasonable but unsuccessful guess near the beginning
of the case. He might have seen his error earlier, only there was a
false clue dropped by a third party who happened to be present at the
crime, which seemed to confirm the guess. This was very unlucky. But it
makes all the better detective story of it.


                _The Story of the Companion of Sirius_


The title of this detective story is ‘The Nonsensical Message’.

Sirius is the most conspicuous star in the sky. Naturally it was
observed very often in early days, and it was used by astronomers along
with other bright stars to determine time and set the clocks by. It
was a _clock star_, as we say. But it turned out that it was not
at all a good clock; it would gain steadily for some years, and then
lose. In 1844 Bessel found out the cause of this irregularity; Sirius
was describing an elliptic orbit. Obviously there must be something
for it to move around, and so it came to be recognized that there was
a dark star there which no one had ever seen. I doubt whether any one
expected it would ever be seen. The Companion of Sirius was, I believe,
the first invisible star to be regularly recognized. We ought not to
call such a star hypothetical. The mechanical properties of matter are
much more crucial than the accidental property of being visible; we
do not consider a transparent pane of glass ‘hypothetical’. There was
near Sirius something which exhibited the most universal mechanical
property of matter, namely, exerting force on neighbouring matter
according to the law of gravitation. That is better evidence of the
existence of a material mass than ocular evidence would be.

However, eighteen years later the Companion of Sirius was actually
seen by Alvan Clark. This discovery was unique in its way; Clark was
not looking at Sirius because he was interested in it, but because
Sirius was a nice bright point of light with which to test the optical
perfection of a large new object-glass that his firm had made. I dare
say that when he saw the little point of light close to Sirius he was
disappointed and tried to polish it away. However, it stayed, and
proved to be the already known but hitherto unseen Companion.

The big modern telescopes easily show the star and rather spoil the
romance; but as romance faded, knowledge grew, and we now know that the
Companion is a star not much less massive than the sun. It has ⅘ths of
the mass of the sun, but gives out only ¹⁄₃₆₀th of the sun’s light. The
faintness did not particularly surprise us;[11] presumably there should
be white-hot stars glowing very brightly and red-hot stars glowing
feebly, with all sorts of intermediate degrees of brightness. It was
assumed that the Companion was one of the feeble stars only just red
hot.

In 1914 Professor Adams at the Mount Wilson Observatory found that it
was not a red star. It was white--white hot. Why, then, was it not
shining brilliantly? Apparently the only answer was that it must be a
very small star. You see, the nature and colour of the light show that
its surface must be glowing more intensely than the sun’s; but the
total light is only ¹⁄₃₆₀th of the sun’s; therefore the surface must
be less than ¹⁄₃₆₀th of the sun’s. That makes the radius less than
¹⁄₁₉th of the sun’s radius, and brings the globe down to a size which
we ordinarily associate with a planet rather than with a star. Working
out the sum more accurately we find that the Companion of Sirius is a
globe intermediate in size between the earth and the next larger planet
Uranus. But if you are going to put a mass not much less than that of
the sun into a globe not very much larger than the earth, it will be
a tight squeeze. The actual density works out at 60,000 times that of
water--just about a ton to the cubic inch.

We learn about the stars by receiving and interpreting the messages
which their light brings to us. The message of the Companion of Sirius
when it was decoded ran: ‘I am composed of material 3,000 times denser
than anything you have ever come across; a ton of my material would
be a little nugget that you could put in a match-box.’ What reply can
one make to such a message? The reply which most of us made in 1914
was--‘Shut up. Don’t talk nonsense.’

But in 1924 the theory described in the last lecture had been
developed; and you will remember that at the end it pointed to the
possibility that matter in the stars might be compressed to a density
much transcending our terrestrial experience. This called back to mind
the strange message of the Companion of Sirius. It could no longer
be dismissed as obvious nonsense. That does not mean that we could
immediately assume it to be true; but it must be weighed and tested
with a caution which we should not care to waste over a mere nonsense
jingle.

It should be understood that it was very difficult to explain away
the original message as a mistake. As to the mass being ⅘ths of the
sun’s mass there can be no serious doubt at all. It is one of the very
best determinations of stellar mass. Moreover, it is obvious that the
mass must be large if it is to sway Sirius out of its course and upset
its punctuality as a clock. The determination of the radius is less
direct, but it is made by a method which has had conspicuous success
when applied to other stars. For example, the radius of the huge star
Betelgeuse was first calculated in this way; afterwards it was found
possible to measure directly the radius of Betelgeuse by means of
an interferometer devised by Michelson, and the direct measurement
confirmed the calculated value. Again the Companion of Sirius does not
stand alone in its peculiarity. At least two other stars have sent us
messages proclaiming incredibly high density; and considering our very
limited opportunities for detecting this condition, there can be little
doubt that these ‘white dwarfs’, as they are called, are comparatively
abundant in the stellar universe.

But we do not want to trust entirely to one clue lest it prove false
in some unsuspected way. Therefore in 1924 Professor Adams set to
work again to apply to the message a test which ought to be crucial.
Einstein’s theory of gravitation indicates that all the lines of the
spectrum of a star will be slightly displaced towards the red end of
the spectrum as compared with the corresponding terrestrial lines. On
the sun the effect is almost too small to be detected having regard to
the many causes of slight shift which have to be disentangled. To me
personally Einstein’s theory gives much stronger assurance of the real
existence of the effect than does the observational evidence available.
Still it is a striking fact that those who have made the investigation
are now unanimous in their judgement that the effect really occurs on
the sun, although some of them at first thought that they had evidence
against it. Hitherto Einstein’s theory has been chiefly regarded by
the practical astronomer as something he is asked to test; but now the
theory has a chance to show its mettle by helping us to test something
much more doubtful than itself. The Einstein effect is proportional to
the mass divided by the radius of the star; and since the radius of the
Companion of Sirius is very small (if the message is right) the effect
will be very large. It should in fact be thirty times as large as on
the sun. That lifts it much above all the secondary causes of shift of
the lines which made the test on the sun so uncertain.

The observation is very difficult because the Companion of Sirius
is faint for work of this kind, and scattered light from its
overpoweringly brilliant neighbour causes much trouble. However, after
a year’s effort Professor Adams made satisfactory measurements, and he
found a large shift as predicted. Expressing the results in the usual
unit of kilometres per second, the mean of his measurements came to 19,
whilst the predicted shift was 20.

Professor Adams has thus killed two birds with one stone. He has
carried out a new test of Einstein’s general theory of relativity, and
he has shown that matter at least 2,000 times denser than platinum is
not only possible but actually exists in the stellar universe.[13]
This is the best confirmation we could have for our view that the sun
with a density 1½ times that of water is still very far indeed from
the maximum density of stellar matter; and it is therefore entirely
reasonable that we should find it behaving like a perfect gas.

I have said that the observation was exceedingly difficult. However
experienced the observer, I do not think we ought to put implicit trust
in a result which strains his skill to the utmost until it has been
verified by others working independently. Therefore you should for the
present make the usual reservations in accepting these conclusions.
But science is not just a catalogue of ascertained facts about the
universe; it is a mode of progress, sometimes tortuous, sometimes
uncertain. And our interest in science is not merely a desire to hear
the latest facts added to the collection; we like to discuss our hopes
and fears, probabilities and expectations. I have told the detective
story so far as it has yet unrolled itself. I do not know whether we
have reached the last chapter.


             _Unknown Atoms and Interpretation of Spectra_


It should be understood that this matter of enormous density is
not supposed to be any strange substance--a new chemical element
or elements. It is just ordinary matter smashed about by the high
temperature and so capable of being packed more tightly--just as more
people could be squeezed into a room if a few bones were broken. It
is one of the features of astronomical physics that it shows us the
_ordinary_ elements of the earth in an _extraordinary_
state--smashed or ionized to a degree that has either not been
reproduced or has been reproduced with great difficulty in the
laboratory. It is not only in the inaccessible interior of the star
that we find matter in a state outside terrestrial experience.

Here is a picture of the Ring Nebula in Lyra (Fig. 8).[14] It is taken
through a prism so that we see not one ring but a number of rings
corresponding to different lines of the spectrum and representing the
different kinds of atoms which are at work producing the light of the
nebula. The smallest ring, which is rather faint (marked by an arrow),
consists of light produced by the helium atoms in the nebula--not
ordinary helium but smashed helium atoms. It was one of the great
laboratory achievements of recent times when Professor A. Fowler in
1912 succeeded in battering helium atoms in a vacuum tube sufficiently
to give this kind of light, already well known in the stars. Two other
rings are due to hydrogen. With these three exceptions none of the
rings have yet been imitated in the laboratory. For instance, we do not
know what elements are producing the two brightest rings on the extreme
right and left respectively.

We are sometimes asked whether any new elements show themselves in
the stars which are not present or are not yet discovered on the
earth. We can give fairly confidently the answer No. That, however,
is not because everything seen in the stars has been identified with
known terrestrial elements. The answer is in fact given not by the
astronomer but by the physicist. The latter has been able to make out
the orderly scheme of the elements; and it transpires that there are
no gaps left for fresh elements until we come to elements of very high
atomic weight, which would not be likely to rise into the atmosphere of
a star and show themselves in astronomical observation. Every element
carries a number, starting with hydrogen which is No. 1, and going up
to uranium which is No. 92.

[Illustration: Fig. 8. THE RING NEBULA IN LYRA]

[Illustration: Fig. 9. HYDROGEN--THE BALMER SERIES]

And what is more, the element carries its number-plate so conspicuously
that a physicist is able to read it. He can, for instance, see that
iron is No. 26 without having to count up how many known elements
precede it. The elements have been called over by their numbers, and up
to No. 84 they have all answered ‘Present’.[15]

The element helium (No. 2) was first discovered by Lockyer in the
sun, and not until many years later was it found on the earth.
Astrophysicists are not likely to repeat this achievement; they cannot
discover new elements if there aren’t any. The unknown source of the
two rings close together on the right of the photograph (a bright ring
and a fainter ring) has been called _nebulium_. But nebulium is
not a new element. It is some quite familiar element which we cannot
identify because it has lost several of its electrons. An atom which
has lost an electron is like a friend who has shaved off his moustache;
his old acquaintances do not recognize him. We shall recognize nebulium
some day. The theoretical physicists are at work trying to find laws
which will determine exactly the kind of light given off by atoms in
various stages of mutilation--so that it will be purely a matter of
calculation to infer the atom from the light it emits. The experimental
physicists are at work trying more and more powerful means of battering
atoms, so that one day a terrestrial atom will be stimulated to give
nebulium light. It is a great race; and I do not know which side to
back. The astronomer cannot do much to help the solution of the problem
he has set. I believe that if he would measure with the greatest care
the ratio of intensity of the two nebulium lines he would give the
physicists a useful hint. He also provides another clue—though it is
difficult to make anything of it--namely, the different sizes of the
rings in the photograph, showing a difference in the distribution of
the emitting atoms. Evidently nebulium has a fondness for the outer
parts of the nebula and helium for the centre; but it is not clear what
inference should be drawn from this difference in their habits.

The atoms of different elements, and atoms of the same element in
different states of ionization, all have distinctive sets of lines
which are shown when the light is examined through a spectroscope.
Under certain conditions (as in the nebulae) these appear as bright
lines; but more often they are imprinted as dark lines on a continuous
background. In either case the lines enable us to identify the element,
unless they happen to belong to an atom in a state of which we have
had no terrestrial experience. The rash prophecy that knowledge of the
composition of the heavenly bodies must be for ever beyond our reach
has long been disproved; and the familiar elements, hydrogen, carbon,
calcium, titanium, iron, and many others, can be recognized in the most
distant parts of the universe. The thrill of this early discovery has
now passed. But meanwhile stellar spectroscopy has greatly extended its
scope; it is no longer chemical analysis, but physical analysis. When
we meet an old acquaintance there is first the stage of recognition;
the next question is ‘How are you?’ After recognizing the stellar
atom we put this question, and the atom answers, ‘Quite sound’ or
‘Badly smashed’, as the case may be. Its answer conveys information
as to its environment--the severity of the treatment to which it is
being subjected--and hence leads to a knowledge of the conditions of
temperature and pressure in the object observed.

Surveying the series of stars from the coolest to the hottest, we can
trace how the calcium atoms are at first whole, then singly ionized,
then doubly ionized--a sign that the battering becomes more severe as
the heat becomes more intense. (The last stage is indicated by the
disappearance of all visible signs of calcium, because the ion with two
electrons missing has no lines in the observable part of the spectrum.)
The progressive change of other elements is shown in a similar way. A
great advance in this study was made in 1920 by Professor M. N. Saha,
who first applied the quantitative physical laws which determine the
degree of ionization at any given temperature and pressure. He thereby
struck out a new line in astrophysical research which has been widely
developed. Thus, if we note the place in the stellar sequence where
complete calcium atoms give place to atoms with one electron missing,
the physical theory is able to state the corresponding temperature or
pressure.[16] Saha’s methods have been improved by R. H. Fowler and
E. A. Milne. One important application was to determine the surface
temperatures of the hottest types of stars (12,000°—25,000°), since
alternative methods available for cooler stars are not satisfactory
at these high temperatures. Another rather striking result was the
discovery that the pressure in the star (at the level surveyed by the
spectroscope) is only ¹⁄₁₀₀₀₀th of an atmosphere; previously it had
been assumed on no very definite evidence to be about the same as that
of our own atmosphere.

We commonly use the method of spectrum analysis when we wish to
determine which elements are present in a given mineral on the earth.
It is equally trustworthy in examining the stars since it can make no
difference whether the light we are studying comes from a body close
at hand or has travelled to us for hundreds of years across space. But
one limitation in stellar work must always be remembered. When the
chemist is looking, say, for nitrogen in his mineral, he takes care to
provide the conditions which according to his experience are necessary
for the nitrogen spectrum to show itself. But in the stars we have
to take the conditions as we find them. If nitrogen does not appear,
that is no proof that nitrogen is absent; it is much more likely that
the stellar atmosphere does not hit off the right conditions for the
test. In the spectrum of Sirius the lines of hydrogen are exceedingly
prominent and overwhelm everything else. We do not infer that Sirius
is composed mainly of hydrogen; we infer instead that its surface is
at a temperature near 10,000°, because it can be calculated that that
is a temperature most favourable for a great development of these
hydrogen lines. In the sun the most prominent spectrum is iron. We do
not infer that the sun is unusually rich in iron; we infer that it
is at a comparatively low temperature near 6,000° favourable for the
production of the iron spectrum. At one time it was thought that the
prominence of hydrogen in Sirius and of metallic elements in the sun
indicated an evolution of the elements, hydrogen turning into heavier
elements as the star cools from the Sirian to the solar stage. There is
no ground for interpreting the observations in that way; the fading of
the hydrogen spectrum and the increase of the iron spectrum would occur
in any case as the result of the fall of temperature; and similar
spurious appearances of evolution of elements can be arranged in the
laboratory.

It is rather probable that the chemical elements have much the same
relative abundance in the stars that they have on the earth. All the
evidence is consistent with this view; and for a few of the commoner
elements there is some positive confirmation. But we are limited to the
outside of the star as we are limited to the outside of the earth in
computing the abundance of the elements, so that this very provisional
conclusion should not be pressed unduly.


                           _Spectral Series_


To illustrate further this kind of deduction, let us consider the
spectrum shown in Fig. 9 and see what may be learnt from it. With a
little trouble we can disentangle a beautifully regular series of
bright lines. The marks above will assist you to pick out the first few
lines of the series from the numerous other spectra mixed up with it.
Noticing the diminishing spacing from right to left, you will be able
to see that the series continues to the left for at least fifteen lines
beyond the last one marked, the lines ultimately drawing close together
and forming a ‘head’ to the series. This is the famous Balmer Series of
hydrogen, and having recognized it we identify hydrogen as one of the
elements present in the source of the light. But that is only the first
step, and we can proceed to further inferences.

Professor Bohr’s theory of the hydrogen atom teaches us that each line
of the series is emitted by an atom in a different state. These ‘states
of excitation’ can be numbered consecutively, starting from the normal
state of the hydrogen atom as No. 1. The light emitted in the first
few states comes into the part of the spectrum not reproduced here, and
the first line in our picture corresponds to state No. 8. Counting to
the left from this you will recognize the successive lines without much
difficulty up to state No. 30. Now the successive states correspond to
more and more swollen atoms, that is to say, the planet electron[17]
makes a wider and wider circuit. The radius (or more strictly the
semi-axis) of its orbit is proportional to the square of the number of
the state, so that the orbit for state No. 30 is 900 times larger than
the orbit for the normal atom No. 1. The diameter of the orbit in No.
30 is approximately a ten-thousandth of a millimetre. One inference can
be drawn immediately--the spectrum shown in Fig. 9 was not produced in
any terrestrial laboratory. In the highest vacuum that can be used in
terrestrial spectroscopy the atoms are still too crowded to leave room
for an orbit so large as this. The source must be matter so tenuous
that there is vacant space for the electron to make this wide circuit
without colliding with or suffering interference from other atoms.
Without entering into further detail we can conclude that Fig. 9 is a
spectrum of matter more rarefied than the highest vacuum known on the
earth.[18]

It is interesting to notice that, whereas throughout most of the
picture the lines are shown on a dark background, at the extreme left
the background is bright; the change occurs just at the point where the
Balmer Series comes to an end. This background of light is also due to
hydrogen and it is caused in the following way. The swollen atoms in
state No. 30 or thereabouts are perilously near the bursting-point,
so it is natural that along with them there should be atoms which
have overstepped the limit and burst. They have lost their planet
electrons and are occupied in catching new ones. Just as energy is
required in order to wrench away an electron from an atom, so there
will be superfluous energy to be got rid of when the atom tames a wild
electron. This superfluous energy is radiated and forms the bright
background referred to. Without entering into technicalities of the
theory, we can see that it is appropriate that this light from the
burst atoms should appear in the spectrum immediately beyond the lines
from the most swollen atoms, since bursting is a sequel to overswelling.

Whilst you have this photograph of the Balmer Series before you I
may take the opportunity of recounting the history of another famous
series. In some of the hottest stars a related series of lines known
as the Pickering Series was discovered in 1896. This is spaced on
precisely the same regular plan, but the lines fall half way between
the lines of the Balmer Series--not exactly half way because of the
gradually diminishing intervals from right to left, but just where
one would naturally interpolate lines in order to double their number
whilst keeping the spacing regular. Unlike the Balmer Series, the
Pickering Series had never been produced in any laboratory. What
element was causing it? The answer seemed obvious; surely these two
related series, one fitting half way between the other, must belong to
different modes of vibration of the same atom, hydrogen. That seemed to
be the only possible answer at the time; but we have learned more about
atoms since then. We may fairly argue that the ideal simplicity of
these two series indicates that they are produced by an atomic system
of the simplest possible type, viz. an atom with one planet electron;
but it must be remembered that this condition only tells us how the
atom is _clothed_, not what the atom is. The helium atom (or, for
that matter, the uranium atom) can on occasion masquerade in the scanty
attire of the hydrogen atom. Normal helium has two planet electrons;
but if one of these is lost, it becomes hydrogen-like and copies the
simple hydrogen system on a different scale. It is significant that the
Pickering Series appears only in the very hottest stars--in conditions
likely to cause loss of an electron. The difference between hydrogen
and hydrogen-like helium is firstly the difference of atomic weight;
the helium nucleus is four times as massive. But this scarcely affects
the spectrum because both nuclei are so massive that they remain almost
unshaken by the dancing electron. Secondly, the helium nucleus has
a double electric charge; this is equivalent to substituting in the
vibrating system a controlling spring of twice the strength. What can
be more natural than that the doubled force of the spring should double
the number of lines in the series without otherwise altering its plan?
In this way Professor Bohr discovered the real origin of the Pickering
Series; it is due to ionized helium, not to hydrogen.[19]

The heavy nucleus, whether of hydrogen or helium, remains almost
unshaken by the atomic vibration--almost, but not quite. At a later
date Professor A. Fowler succeeded in reproducing the Pickering
Series in the laboratory and was able to measure the lines with much
greater accuracy than could be achieved in stellar spectroscopy; he
was then able to show from his measures that the nucleus is not quite
irresponsive. It was a delicate double-star problem transferred to the
interior of the atom; or perhaps a closer analogy would be the mutual
influence of the sun and Jupiter, because Jupiter, having a thousandth
of the mass of the sun, disturbs it to about the same extent that the
light electron disturbs the hydrogen nucleus. Ionized helium is a
faithful copy of the hydrogen atom (on the altered scale) in everything
except the ‘shake’; the shake is less than in hydrogen because the
helium nucleus is still more massive and rock-like. The difference
of shake throws the Pickering Series of helium and the Balmer Series
of hydrogen slightly out of step with respect to one another; and by
measuring this misfit Professor Fowler was able to make a very accurate
determination of the shake and therefore of the mass of the electron.
In this way the mass of the electron is found to be ¹⁄₁₈₄₄th of the
mass of the hydrogen nucleus; this agrees well with the mass found
by other methods, and the determination is probably not inferior in
accuracy to any of them.

And so the clue first picked up in stars 300 light years away, followed
in turn by the theoretical and the experimental physicist, leads in the
end to the smallest of all things known.


                         _The Cloud in Space_


Having already considered the densest matter in the universe, we now
turn to consider the rarest.

In spite of great improvements in the art of exhausting vessels we
are still a long way from producing a _real_ vacuum. The atoms
in a vacuum tube before it is exhausted muster a formidable number
containing about twenty digits. High exhaustion means knocking off
five or six noughts at the end of that number; and the most strenuous
efforts to knock off one more nought seem ludicrously ineffective--a
mere nibbling at the huge number that must remain.

Some of the stars are extremely rarefied. Betelgeuse, for example, has
a density about a thousandth that of air. We should call it a vacuum
were it not contrasted with the much greater vacuosity of surrounding
space. Nowadays physicists have no difficulty in producing a better
vacuum than Betelgeuse; but in earlier times this star would have been
regarded as a very creditable attempt at a vacuum.

The outer parts of a star, and especially the light appendages such as
the solar chromosphere and corona, reach much lower densities. Also the
gaseous nebulae are, as their appearance suggests, extremely tenuous.
When there is space enough to put a pin’s head between adjacent atoms
we can begin to talk about a ‘real vacuum.’ At the centre of the Orion
nebula that degree of rarefaction is probably reached and surpassed.

A nebula has no definite boundary and the density gradually fades off.
There is reason to think that the fading off becomes slow at great
distances. Before we pass entirely out of the sphere of one nebula we
enter the sphere of another, so that there is always some residual
density in interstellar space.

I believe that, reasoning from the tailing off of the nebulae, we are
in a position to make an estimate of the amount of matter remaining
unaggregated in space. An ordinary region where there is no observable
nebulosity is the highest vacuum existing--within the limits of the
stellar system at least--but there still remains about _one atom in
every cubic inch_. It depends on our point of view whether we regard
this as an amazing fullness or an amazing emptiness of space. Perhaps
it is the fullness that impresses us most. The atom can find no place
of real solitude within the system of the stars; wherever it goes it
can nod to a colleague not more than an inch away.

Let us approach the same subject from a different angle.

In the ‘Story of Algol’ I referred to the way in which we measure the
velocity of rotation of the sun. We point the spectroscope first on one
limb of the sun and then on the other. Taking any one of the dark lines
of the spectrum, we find that it has shifted a little between the two
observations. This tells us that the material which imprinted the line
was moving towards or away from us with different velocities in the two
observations. That is what we expected to find; the rotation of the
sun makes solar material move towards us on one side of the disk and
away from us on the other side. But there are a few dark lines which
do not show this change. They are in just the same position whether
we observe them on the east or on the west of the sun. Clearly these
cannot originate on the sun. They have been imprinted on the light
after it left the sun and before it reached our telescope. We have
thus discovered a medium occurring somewhere between the sun and our
telescope; and as some of the lines are recognized as belonging to
oxygen, we can infer that it is a medium containing oxygen.

This seems to be the beginning of a great discovery, but it ends in a
bathos. It happens that we were already aware of a medium containing
oxygen lying somewhere between our telescope and the sun. It is a
medium essential to our existence. The terrestrial atmosphere is
responsible for the ‘fixed’ lines seen in the sun’s spectrum.

Just as the spectroscope can tell us that the sun is turning round (a
fact already familiar to us from watching the surface markings), so
it can tell us that certain stars are wandering round an orbit, and
therefore are under the influence of a second star which may or may
not be visible itself. But here again we sometimes find ‘fixed’ lines
which do not change with the others. Therefore somewhere between the
star and the telescope there exists a stationary medium which imprints
these lines on the light. This time it is not the earth’s atmosphere.
The lines belong to two elements, calcium and sodium, neither of which
occur in the atmosphere. Moreover, the calcium is in a smashed state,
having lost one of its electrons, and the conditions in our atmosphere
are not such as would cause this loss. There seems to be no doubt that
the medium containing the sodium and ionized calcium--and no doubt
many other elements which do not show themselves--is separate from the
earth and the star. It is the ‘fullness’ of interstellar space already
mentioned. Light has to pass one atom per cubic inch all the way from
the star to the earth, and it will pass quite enough atoms during its
journey of many hundred billion miles to imprint these dark lines on
its spectrum.

At first there was a rival interpretation. It was thought that the
lines were produced in a cloud attached to the star--forming a kind
of aureole round it. The two components travel in orbits round each
other, but their orbital motion need not disturb a diffuse medium
filling and surrounding the combined system. This was a very reasonable
suggestion, but it could be put to the test. The test was again
_velocity_. Although either component can move periodically to
and fro within the surrounding cloud of calcium and sodium, it is
clear that its average approach to us or recession from us taken over
a long time must agree with that of the calcium and sodium if the star
is not to leave its halo behind. Professor Plaskett with the 72-inch
reflector at the Dominion Observatory in British Columbia carried out
this test. He found that the secular or average rate of approach of
the star[20] was in general quite different from the rate shown by the
fixed calcium or sodium lines. Clearly the material responsible for
the fixed lines could not be an appendage of the star since it was not
keeping pace with it. Plaskett went farther and showed that whereas
the stars themselves had all sorts of individual velocities, the
material of the fixed lines had the same or nearly the same velocity
in all parts of the sky, as though it were one continuous medium
throughout interstellar space. I think there can be no doubt that this
research demonstrates the existence of a cosmic cloud pervading the
stellar system. The fullness of interstellar space becomes a fact of
observation and no longer a theoretical conjecture.

The system of the stars is floating in an ocean--not merely an ocean
of space, not merely an ocean of ether, but an ocean that is so far
material that one atom or thereabouts occurs in each cubic inch. It is
a placid ocean without much relative motion; currents probably exist,
but they are of a minor character and do not attain the high speeds
commonly possessed by the stars.

Many points of interest arise, but I will only touch on one or two.
Why are the calcium atoms ionized? In the calm of interstellar space
we seem to have passed away from the turmoil which smashed the calcium
atoms in the interior of a star; so at first it seems difficult to
understand why the atoms in the cloud should not be complete. However,
even in the depths of space the breaking-up of the atom continues;
because there is always starlight passing across space, and some of
the light-waves are quite powerful enough to wrench a first or second
electron away from the calcium atom. It is one of the most curious
discoveries of modern physics that when a light-wave is attenuated
by spreading, what it really suffers from is _laziness_ rather
than actual loss of power. What is weakened is not the power but the
probability that it will display the power. A light-wave capable of
bursting an atom still retains the power when it is attenuated a
million-fold by spreading; only it is a million times more sparing in
the exercise of the power. To put it another way, an atom exposed to
the attenuated waves will on the average have to wait a million times
longer before a wave chooses to explode it; but the explosion when it
does occur will be of precisely the same strength however great the
attenuation. This is entirely unlike the behaviour of water-waves; a
wave which is at first strong enough to capsize a boat will, after
spreading, become too weak. It is more like machine-gun fire which is
more likely to miss a given object at greater distance but is equally
destructive if it hits. The property here referred to (the quantum
property) is the deepest mystery of light.

Thus in interstellar space electrons are still being torn from calcium
atoms, only very infrequently. The other side of the question is the
rate of repair, and in this connexion the low density of the cosmic
cloud is the deciding factor. The atom has so few opportunities for
repair. Roving through space the atom meets an electron only about
once a month, and it by no means follows that it will capture the first
one it meets. Consequently very infrequent smashing will suffice to
keep the majority of the atoms ionized. The smashed state of the atoms
inside a star can be compared to the dilapidation of a house visited by
a tornado; the smashed state in interstellar space is a dilapidation
due to ordinary wear and tear coupled with excessive slackness in
making repairs.

A calculation indicates that most of the calcium atoms in interstellar
space have lost two electrons; these atoms do not interfere with the
light and give no visible spectrum. The ‘fixed lines’ are produced by
atoms temporarily in a better state of repair with only one electron
missing; they cannot amount at any moment to more than one-thousandth
of the whole number, but even so they will be sufficiently numerous to
produce the observed absorption.

We generally think of interstellar space as excessively cold. It is
quite true that any thermometer placed there would show a temperature
only about 3° above the absolute zero--if it were capable of
registering so low a reading. Compact matter such as a thermometer, or
even matter which from the ordinary standpoint is regarded as highly
diffuse, falls to this low temperature. But the rule does not apply
to matter as rarefied as the interstellar cloud. Its temperature is
governed by other considerations, and it will probably be not much
below the surface-temperature of the hottest stars, say 15,000°.
Interstellar space is at the same time excessively cold and decidedly
hot.[21]


                       _The Sun’s Chromosphere_


Once again we shift the scene, and now we are back in the outer parts
of the sun. Fig. 10[22] shows one of the huge prominence flames which
from time to time shoot out of the sun. The flame in this picture was
about 120,000 miles high. It went through great changes of form and
disappeared in not much more than twenty-four hours. This was rather an
exceptional specimen. Smaller flames occur commonly enough; it seems
that the curious black marks in Fig. 1, often looking like rifts,
are really prominences seen in projection against the still brighter
background of the sun. The flames consist of calcium, hydrogen, and
several other elements.

We are concerned not so much with the prominences as with the layer
from which they spring. The ordinary atmosphere of the sun terminates
rather abruptly, but above it there is a deep though very rarefied
layer called the chromosphere consisting of a few selected elements
which are able to float--float, not on the top of the sun’s atmosphere,
but on the _sunbeams_. The art of riding a sunbeam is evidently
rather difficult, because only a few of the elements have the necessary
skill. The most expert is calcium. The light and nimble hydrogen atom
is fairly good at it, but the ponderous calcium atom does it best.

The layer of calcium suspended on the sunlight is at least 5,000 miles
thick. We can observe it best when the main part of the sun is hidden
by the moon in an eclipse; but the spectroheliograph enables us to
study it to some extent without an eclipse.

[Illustration: Fig. 10. SOLAR PROMINENCE]

[Illustration: Fig. 11. STAR CLUSTER ω CENTAURI]

On the whole it is steady and quiescent, although, as the prominence
flames show, it is liable to be blown sky-high by violent outbursts.
The conclusions about the calcium chromosphere that I am going to
describe rest on a series of remarkable researches by Professor Milne.

How does an atom float on a sunbeam? The possibility depends on the
pressure of light to which we have already referred (p. 26). The
sunlight travelling outwards carries a certain outward momentum; if the
atom absorbs the light it absorbs also the momentum and so receives a
tiny impulse outwards. This impulse enables it to recover the ground it
was losing in falling towards the sun. The atoms in the chromosphere
are kept floating above the sun like tiny shuttlecocks, dropping a
little and then ascending again from the impulse of the light. Only
those atoms which can absorb large quantities of sunlight in proportion
to their weight will be able to float successfully. We must look rather
closely into the mechanism of absorption of the calcium atom if we are
to see why it excels the other elements.

The ordinary calcium atom has two rather loose electrons in its
attendant system; the chemists express this by saying that it is a
divalent element, the two loose electrons being especially important in
determining the chemical behaviour. Each of these electrons possesses
a mechanism for absorbing light. But under the conditions prevailing
in the chromosphere one of the electrons is broken away, and the
calcium atoms are in the same smashed state that gives rise to the
‘fixed lines’ in the interstellar cloud. The chromospheric calcium
thus supports itself on what sunlight it can gather in with the one
loose electron remaining. To part with this would be fatal; the atom
would no longer be able to absorb sunlight, and would drop like a
stone. It is true that after two electrons are lost there are still
eighteen remaining; but these are held so tightly that sunlight has no
effect on them and they can only absorb shorter waves which the sun
does not radiate in any quantity. The atom therefore could only save
itself if it restored its main absorbing mechanism by picking up a
passing electron; it has little chance of catching one in the rarefied
chromosphere, so it would probably fall all the way to the sun’s
surface.

There are two ways in which light can be absorbed. In one the atom
absorbs so greedily that it bursts, and the electron scurries off with
the surplus energy. That is the process of ionization which was shown
in Fig. 5. Clearly this cannot be the process of absorption in the
chromosphere because, as we have seen, the atom cannot afford to lose
the electron. In the other method of absorption the atom is not quite
so greedy. It does not burst, but it swells visibly. To accommodate the
extra energy the electron is tossed up into a higher orbit. This method
is called excitation (cf. p. 59). After remaining in the excited orbit
for a little while the electron comes down again spontaneously. The
process has to be repeated 20,000 times a second in order to keep the
atom balanced in the chromosphere.

The point we are leading up to is, Why should calcium be able to float
better than other elements? It has always seemed odd that a rather
heavy element (No. 20 in order of atomic weight) should be found in
these uppermost regions where one would expect only the lightest atoms.
We see now that the special skill demanded is to be able to toss up an
electron 20,000 times a second without ever making the fatal blunder
of dropping it. That is not easy even for an atom. Calcium[23] scores
because it possesses a possible orbit of excitation only a little way
above the normal orbit so that it can juggle the electron between
these two orbits without serious risk. With most other elements the
first available orbit is relatively much higher; the energy required
to reach this orbit is not so very much less than the energy required
to detach the electron altogether; so that we cannot very well have a
continuous source of light capable of causing the orbit-jumps without
sometimes overdoing it and causing loss of the electron. It is the
wide difference between the energy of excitation and the energy of
ionization of calcium which is so favourable; the sun is very rich in
ether-waves capable of causing the first, and is almost lacking in
ether-waves capable of causing the second.

The average time occupied by each performance is ¹⁄₂₀₀₀₀th of a second.
This is divided into two periods. There is a period during which the
atom is patiently waiting for a light-wave to run into it and throw
up the electron. There is another period during which the electron
revolves steadily in the higher orbit before deciding to come down
again. Professor Milne has shown how to calculate from observations of
the chromosphere the durations of both these periods. The first period
of waiting depends on the strength of the sun’s radiation. But we focus
attention especially on the second period, which is more interesting
because it is a definite property of the calcium atom, having nothing
to do with local circumstances. Although we measure it for ions in
the sun’s chromosphere, the same result must apply to calcium ions
anywhere. Milne’s result is that an electron tossed into the higher
orbit remains there for an average time of a hundred-millionth of a
second before it spontaneously drops back again. I may add that during
this brief time it makes something like a million revolutions in the
upper orbit.

Perhaps this is a piece of information that you were not particularly
burning to know. I do not think it can be called interesting except to
those who make a hobby of atoms. But it does seem to me interesting
that we should have to turn a telescope and spectroscope on the sun to
find out this homely property of a substance which we handle daily. It
is a kind of measurement of immense importance in physics. The theory
of these atomic jumps comes under the quantum theory which is still
the greatest puzzle of physical science; and it is greatly in need of
guidance from observation on just such a matter as this. We can imagine
what a sensation would be caused if, after a million revolutions round
the sun, a planet made a jump of this kind. How eagerly we should try
to determine the average interval at which such jumps occurred! The
atom is rather like a solar system, and it is not the less interesting
because it is on a smaller scale.

There is no prospect at present of measuring the time of relaxation
of the excited calcium atom in a different way. It has, however, been
found possible to determine the corresponding time for one or two other
kinds of atoms by laboratory experiments. It is not necessary that
the time should be at all closely the same for different elements;
but laboratory measurements for hydrogen also give the period as a
hundred-millionth of a second, so there is no fault to find with the
astronomical determination for calcium.

The excitation of the calcium atom is performed by light of two
particular wave-lengths, and the atoms in the chromosphere support
themselves by robbing sunlight of these two constituents. It is true
that after a hundred-millionth of a second a relapse comes and the
atom has to disgorge what it has appropriated; but in re-emitting
the light it is as likely to send it inwards as outwards, so that
the _outflowing_ sunlight suffers more loss than it recovers.
Consequently, when we view the sun through this mantle of calcium the
spectrum shows gaps or dark lines at the two wave-lengths concerned.
These lines are denoted by the letters H and K. They are not entirely
black, and it is important to measure the residual light at the centre
of the lines, because we know that it must have an intensity just
strong enough to keep calcium atoms floating under solar gravity; as
soon as the outflowing light is so weakened that it can support no
more atoms it can suffer no further depredations, and so it emerges
into outer space with this limiting intensity. The measurement gives
numerical data for working out the constants of the calcium atom
including the time of relaxation mentioned above.

The atoms at the top of the chromosphere rest on the weakened light
which has passed through the screen below; the full sunlight would blow
them away. Milne has deduced a consequence which may perhaps have a
practical application in the phenomena of explosion of ‘new stars’ or
novae, and in any case is curiously interesting. Owing to the Doppler
effect a moving atom absorbs a rather different wave-length from a
stationary atom; so that if for any cause an atom moves away from the
sun it will support itself on light which is a little to one side of
the deepest absorption. This light, being more intense than that which
provided a balance, will make the atom recede faster. The atom’s own
absorption will thus gradually draw clear of the absorption of the
screen below. Speaking rather metaphorically, the atom is balanced
precariously on the summit of the absorption line and it is liable to
topple off into the full sunlight on one side. Apparently the speed
of the atom should go on increasing until it has to climb an adjacent
absorption line (due perhaps to some other element); if the line is too
intense to be surmounted the atom will stick part-way up, the velocity
remaining fixed at a particular value. These later inferences may be
rather far-fetched, but at any rate the argument indicates that there
is likely to be an escape of calcium into outer space.

By Milne’s theory we can calculate the whole weight of the sun’s
calcium chromosphere. Its mass is about 300 million tons. One scarcely
expects to meet with such a trifling figure in astronomy. It is less
than the tonnage handled by our English railways each year. I think
that solar observers must feel rather hoaxed when they consider the
labour that they have been induced to spend on this airy nothing.
But science does not despise trifles. And astronomy can still be
instructive even when, for once in a way, it descends to commonplace
numbers.


                       _The Story of Betelgeuse_


This story has not much to do with atoms, and scarcely comes under
the title of these lectures; but we have had occasion to allude to
Betelgeuse as the famous example of a star of great size and low
density, and its history is closely associated with some of the
developments that we are studying.

No star has a disk large enough to be seen with our present telescopes.
We can calculate that a lens or mirror of about 20 feet aperture would
be needed to show traces even of the largest star disk. Imagine for a
moment that we have constructed an instrument of this order of size.
Which would be the most hopeful star to try it on?

Perhaps Sirius suggests itself first, since it is the brightest star in
the sky. But Sirius has a white-hot surface radiating very intensely,
so that it is not necessary that it should have a wide expanse.
Evidently we should prefer a star which, although bright, has its
surface in a feebly glowing condition; then the apparent brightness
must be due to large area. We need, then, a star which is both red
and bright. Betelgeuse seems best to satisfy this condition. It is
the brighter of the two shoulder-stars of Orion--the only conspicuous
red star in the constellation. There are one or two rivals, including
Antares, which might possibly be preferred; but we cannot go far wrong
in turning our new instrument on Betelgeuse in the hope of finding the
largest or nearly the largest star disk.

You may notice that I have paid no attention to the distances of these
stars. It happens that distance is not relevant. It would be relevant
if we were trying to find the star of greatest actual dimensions; but
here we are considering the star which presents the largest apparent
disk,[24] i.e. covers the largest area of the sky. If we were at twice
our present distance from the sun, we should receive only one-quarter
as much light; but the sun would look half its present size linearly,
and its apparent area would be one-quarter. Thus the light per unit
area of disk is unaltered by distance. Removing the sun to greater
and greater distance its disk will appear smaller but glowing not
less intensely, until it is so far away that the disk cannot be
discriminated.

By spectroscopic examination we know that Betelgeuse has a surface
temperature about 3,000°. A temperature of 3,000° is not unattainable
in the laboratory, and we know partly by experiment and partly by
theory what is the radiating power of a surface in this state. Thus it
is not difficult to compute how large an area of the sky Betelgeuse
must cover in order that the area multiplied by the radiating power
may give the observed brightness of Betelgeuse. The area turns out to
be very small. The apparent size of Betelgeuse is that of a half-penny
fifty miles away. Using a more scientific measure, the diameter of
Betelgeuse predicted by this calculation is 0·051 of a second of arc.

No existing telescope can show so small a disk. Let us consider briefly
how a telescope forms an image--in particular how it reproduces that
detail and contrast of light and darkness which betrays that we are
looking at a disk or a double star and not a blur emanating from a
single point. This optical performance is called resolving power; it is
not primarily a matter of magnification but of aperture, and the limit
of resolution is determined by the size of aperture of the telescope.

To create a sharply defined image the telescope must not only
bring light where there ought to be light, but it must also bring
darkness where there ought to be darkness. The latter task is the
more difficult. Light-waves tend to spread in all directions, and
the telescope cannot prevent individual wavelets from straying on to
parts of the picture where they have no business. But it has this one
remedy--for every trespassing wavelet it must send a second wavelet
by a slightly longer or shorter route so as to arrive in a phase
opposite to the first wavelet and cancel its effect. This is where the
utility of a wide aperture arises--by affording a wider difference of
route of the individual wavelets, so that those from one part of the
aperture may be retarded relatively to and interfere with those from
another part. A small object-glass can furnish light; it takes a big
object-glass to furnish darkness in the picture.

Now we may ask ourselves whether the ordinary circular aperture is
necessarily the most efficient for giving the wavelets the required
path-differences. Any deviation from a symmetrical shape is likely to
spoil the definition of the image--to produce wings and fringes. The
image will not so closely resemble the object viewed. But on the other
hand we may be able to sharpen up the tell-tale features. It does
not matter how widely the image-pattern may differ from the object,
provided that we can read the significance of the pattern. If we cannot
reproduce a star-disk, let us try whether we can reproduce something
distinctive of a star-disk.

A little reflection shows that we ought to improve matters by blocking
out the middle of the object-glass, and using only the extreme
regions on one side or the other. For these regions the difference of
light-path of the waves is greatest, and they are the most efficient in
furnishing the dark contrast needed to outline the image properly.

But if the middle of the object-glass is not going to be used, why go
to the expense of manufacturing it? We are led to the idea of using
two widely separated apertures, each involving a comparatively small
lens or mirror. We thus arrive at an instrument after the pattern of a
rangefinder.

This instrument will not show us the disk of a star. If we look through
it the main impression of the star image is very like what we should
have seen with either aperture singly--a ‘spurious disk’ surrounded by
diffraction rings. But looking attentively we see that this image is
crossed by dark and bright bands which are produced by interference
between the light-waves coming from the two apertures. At the centre
of the image the waves from the two apertures arrive crest on crest
since they have travelled symmetrically along equal paths; accordingly
there is a bright band. A very little to one side the asymmetry causes
the waves to arrive crest on trough, so that they cancel one another;
here there is a dark band. The width of the bands decreases as the
separation of the two apertures increases, and for any given separation
the actual width is easily calculated.

Each point of the star’s disk is giving rise to a diffraction image
with a system of bands of this kind, but so long as the disk is small
compared with the finest detail of the diffraction image there is no
appreciable blurring. If we continually increase the separation of the
two apertures and so make the bands narrower, there comes a time when
the bright bands for one part of the disk are falling on the dark bands
for another part of the disk. The band system then becomes indistinct.
It is a matter of mathematical calculation to determine the resultant
effect of summing the band systems for each point of the disk. It can
be shown that for a certain separation of the apertures the bands will
disappear altogether; and beyond this separation the system should
reappear though not attaining its original sharpness. The complete
disappearance occurs when the diameter of the star-disk is equal to
1⅕ times the width of the bands (from the centre of one bright band to
the next). As already stated, the bandwidth can be calculated from the
known separation of the apertures.

The observation consists in sliding apart the two apertures until the
bands disappear. The diameter of the disk is inferred at once from
their separation when the disappearance occurred. Although we measure
the size of the disk in this way we never _see_ the disk.

We can summarize the principle of the method in the following way. The
image of a point of light seen through a telescope is not a point but
a small diffraction pattern. Hence, if we look at an extended object,
say Mars, the diffraction pattern will blur the fine detail of the
marking on the planet. If, however, we are looking at a star which
is almost a point, it is simpler to invert the idea; the object, not
being an ideal point, will slightly blur the detail of the diffraction
pattern. We shall only perceive the blurring if the diffraction pattern
contains detail fine enough to suffer from it. Betelgeuse on account
of its finite size must theoretically blur a diffraction pattern; but
the ordinary diffraction disk and rings produced with the largest
telescope are too coarse to show this. We create a diffraction image
with finer detail by using two apertures. Theoretically we can make the
detail as fine as we please by increasing the separation of the two
apertures. The method accordingly consists in widening the separation
until the pattern becomes fine enough to be perceptibly blurred by
Betelgeuse. For a smaller star-disk the same effect of blurring would
not be apparent until the detail had been made still finer by further
separation of the apertures.

This method was devised long ago by Professor Michelson, but it was
only in 1920 that he tried it on a large scale with a great 20-foot
beam across the 100-inch reflector at Mount Wilson Observatory. After
many attempts Pease and Anderson were able to show that the bright
and dark bands for Betelgeuse disappeared when the apertures were
separated 10 feet. The deduced diameter is 0·045 a second of arc in
good enough agreement with the predicted value (p. 78). Only five or
six stars have disks large enough to be measured with this instrument.
It is understood that the construction of a 50-foot interferometer
is contemplated; but even this will be insufficient for the great
majority of the stars. We are fairly confident that the method of
calculation first described gives the correct diameters of the stars,
but confirmation by Michelson’s more direct method of measurement is
always desirable.

To infer the actual size of the star from its apparent diameter, we
must know the distance. Betelgeuse is rather a remote star and its
distance cannot be measured very accurately, but the uncertainty will
not change the general order of magnitude of the results. The diameter
is about 300 million miles. Betelgeuse is large enough to contain the
whole orbit of the earth inside it, perhaps even the orbit of Mars. Its
volume is about fifty million times the volume of the sun.

There is no direct way of learning the mass of Betelgeuse because
it has no companion near it whose motion it might influence. We
can, however, deduce a mass from the mass-brightness relation in
Fig. 7. This gives the mass equal to 35 x sun. If the result is
right, Betelgeuse is one of the most massive stars--but, of course,
not massive in proportion to its bulk. The mean density is about
one-millionth of the density of water, or not much more than
one-thousandth of the density of air.[25]

There is one way in which we might have inferred that Betelgeuse is
less dense than the sun, even if we had had no grounds of theory or
analogy for estimating its mass. According to the modern theory of
gravitation, a globe of the size of Betelgeuse and of the same mean
density as the sun would have some remarkable properties:

Firstly, owing to the great intensity of its gravitation, light would
be unable to escape; and any rays shot out would fall back again to the
star by their own weight.

Secondly, the Einstein shift (used to test the density of the Companion
of Sirius) would be so great that the spectrum would be shifted out of
existence.

Thirdly, mass produces a curvature of space, and in this case the
curvature would be so great that space would close up round the star,
leaving us outside--that is to say, _nowhere_.

Except for the last consideration, it seems rather a pity that the
density of Betelgeuse is so low.

       *       *       *       *       *

It is now well realized that the stars are a very important adjunct to
the physical laboratory--a sort of high-temperature annex where the
behaviour of matter can be studied under greatly extended conditions.
Being an astronomer, I naturally put the connexion somewhat differently
and regard the physical laboratory as a low-temperature station
attached to the stars. It is the laboratory conditions which should be
counted abnormal. Apart from the interstellar cloud which is at the
moderate temperature of about 15,000°, I suppose that nine-tenths of
the matter of the universe is above 1,000,000°. Under _ordinary_
conditions--you will understand my use of the word--matter has rather
simple properties. But there are in the universe exceptional regions
with temperature not far removed from the absolute zero, where the
physical properties of matter acquire great complexity; the ions
surround themselves with complete electron systems and become the atoms
of terrestrial experience. Our earth is one of these chilly places and
here the strangest complications can arise. Perhaps strangest of all,
some of these complications can meet together and speculate on the
significance of the whole scheme.




                              LECTURE III

                         THE AGE OF THE STARS


WE have seen that spatially the scale of man is about midway between
the atom and the star. I am tempted to make a similar comparison as
regards time. The span of the life of a man comes perhaps midway in
scale between the life of an excited atom (p. 74) and the life of a
star. For those who insist on greater accuracy--though I would not
like to claim accuracy for present estimates of the life of a star--I
will modify this a little. As regards mass, man is rather too near to
the atom and a stronger claimant for the midway position would be the
hippopotamus. As regards time, man’s three score years and ten is a
little too near to the stars and it would be better to substitute a
butterfly.

There is one serious moral in this fantasy. We shall have to consider
periods of time which appall our imagination. We fear to make such
drafts on eternity. And yet the vastness of the time-scale of stellar
evolution is _less_ remote from the scale of human experience than
is the minuteness of the time-scale of the processes studied in the
atom.

Our approach to the ‘age of the stars’ will be devious, and certain
incidental problems will detain us on the way.


                           _Pulsating Stars_


The star δ Cephei is one of the variable stars. Like Algol, its
fluctuating light sends us a message. But the message when it is
decoded is not in the least like the message from Algol.

Let me say at once that experts differ as to the interpretation of the
message of δ Cephei. This is not the place to argue the matter, or to
explain why I think that rival interpretations cannot be accepted. I
can only tell you what is to the best of my belief the correct story.
The interpretation which I follow was suggested by Plummer and Shapley.
The latter in particular made it very convincing, and subsequent
developments have, I think, tended to strengthen it. I would not,
however, claim that all doubt is banished.

Algol turned out to be a pair of stars very close together which from
time to time eclipse one another; δ Cephei is a single star which
pulsates. It is a globe which swells and contracts symmetrically with
a regular period of 5⅓ days. And as the globe swells and contracts
causing great changes of pressure and temperature in the interior, so
the issuing stream of light rises and falls in intensity and varies
also in quality or colour.

There is no question of eclipses; the light signals are not in the form
of ‘dots’ and ‘dashes’; and in any case the change of colour shows
that there is a real change in the physical condition of the source of
the light. But at first explanations always assumed that _two_
stars were concerned, and aimed at connecting the physical changes with
an orbital motion. For instance, it was suggested that the principal
star in going round its orbit brushed through a resisting medium which
heated its front surface; thus the light of the star varied according
as the heated front surface or cooler rear surface was presented
towards us. The orbital explanation has now collapsed because it is
found that there is literally no room for two stars. The supposed orbit
had been worked out in the usual way from spectroscopic measurements of
velocity of approach and recession; later we began to learn more about
the true size of stars, first by calculation, and afterwards (for a
few stars) by direct measurement. It turned out that the star was big
and the orbit small; and the second star if it existed would have to
be placed inside the principal star. This overlapping of the stars is
a _reductio ad absurdum_ of the binary hypothesis, and some other
explanation must be found.

What had been taken to be the approach and recession of the star as
a whole was really the approach and recession of the surface as it
heaved up and down with the pulsation. The stars which vary like δ
Cephei are diffuse stars enormously larger than the sun, and the total
displacement measured amounts to only a fraction of the star’s radius.
There is therefore no need to assume a bodily displacement of the star
(orbital motion); the measures follow the oscillation of that part of
the star’s surface presented towards us.

The decision that δ Cephei is a single star and not double has
one immediate consequence. It means that the period of 5⅓ days is
_intrinsic_ in the star and is therefore one of the clues to
its physical condition. It is a free period, not a forced period. It
is important to appreciate the significance of this. The number of
sunspots fluctuates from a maximum to minimum and back to maximum in
a period of about 11½ years; although we do not yet understand the
reason for this fluctuation, we realize that this period is something
characteristic of the sun in its present state and would change if any
notable change happened to the sun. At one time, however, there was
some speculation as to whether the fluctuation of the sunspots might
not be caused by the revolution of the planet Jupiter, which has a
period not so very different; if that explanation had been tenable
the 11½-year period would have been something forced on the sun from
without and would teach us nothing as to the properties of the sun
itself. Having convinced ourselves that the light-period of δ Cephei
is a free period of a single star, belonging to it in the same way
that a particular note belongs to a tuning-fork, we can accept it as
a valuable indicator of the constancy (or otherwise) of the star’s
physical condition.

In stellar astronomy we usually feel very happy if we can determine
our data--parallax, radius, mass, absolute brightness, &c.--to within
5 per cent.; but the measurement of a period offers chances of far
superior accuracy. I believe that the most accurately known quantity in
the whole of science (excluding pure mathematics) is the moon’s mean
period, which is commonly given to twelve significant figures. The
period of δ Cephei can be found to six significant figures at least.
By fastening an observable period to the intrinsic conditions of a
star we have secured an indicator sensitive enough to show extremely
small changes. You will now guess why I am approaching ‘the age of the
stars’ through the Cepheid variables. Up to the present they are the
only stars known to carry a sensitive indicator, by which we might hope
to test the rate of evolutionary change. We believe that δ Cephei like
other stars has condensed out of a nebula, and that the condensation
and contraction are still continuing. No one would expect to detect the
contraction by our rough determinations of the radius even if continued
for a hundred years; but the evolution must indeed be slow if an
intrinsic period measurable to 1 part in 10,000,000 shows no change in
a century.

It does not greatly matter whether or not we understand the nature of
this intrinsic period. If a star contracts, the period of pulsation,
the period of rotation, or any other free period associated with it,
will alter. If you prefer to follow any of the rival interpretations
of the message of δ Cephei, you can make the necessary alterations
in the wording of my argument, but the general verdict as to the
rate of progress of evolution will be unchanged. Only if you detach
the period from the star itself by going back to the old double star
interpretation will the argument collapse; but I do not think any of
the rival interpreters propose to do that.

It is not surprising that these pulsating stars should be regarded
with special interest. Ordinary stars must be viewed respectfully like
the objects in glass cases in museums; our fingers are itching to
pinch them and test their resilience. Pulsating stars are like those
fascinating models in the Science Museum provided with a button which
can be pressed to set the machinery in motion. To be able to see the
machinery of a star throbbing with activity is most instructive for the
development of our knowledge.

The theory of a steady star, which was described in the first lecture,
can be extended to pulsating stars; and we can calculate the free
period of pulsation for a star of assigned mass and density. You
will remember that we have already calculated the heat emission or
brightness and compared it with observation, obtaining one satisfactory
test of the truth of the theory; now we can calculate the period of
pulsation and by comparing it with observation obtain another test.
Owing to lack of information as to a certain constant of stellar
material there is an uncertainty in the calculation represented by a
factor of about 2; that is to say, we calculate two periods, one double
the other, between which with any reasonable luck the true period
ought to lie. The observational confirmation is very good. There are
sixteen Cepheid variables on which the test can be made; their periods
range from 13 hours to 35 days, and they all agree with the calculated
values to within the limits of accuracy expected. In a more indirect
way the same confirmation is shown in Fig. 7 by the close agreement of
the squares, representing Cepheid variables, with the theoretical curve.


                 _The Cepheid as a ‘Standard Candle’_


Cepheid variables of the same period are closely similar to one
another. A Cepheid of period 5⅓ days found in any part of the universe
will be practically a replica of δ Cephei; in particular it will be
a star of the same absolute brightness. This is a fact discovered
by observation, and is not predicted by any part of the theory yet
explored. The brightness, as we have seen, depends mainly on the mass;
the period, on the other hand, depends mainly on the density; so that
the observed relation between brightness and period involves a relation
between mass and density. Presumably this relation signifies that for a
given mass there is just one special density--one stage in the course
of condensation of the star--at which pulsations are liable to occur;
at other densities the star can only burn steadily.

This property renders the Cepheid extremely useful to astronomers. It
serves as a standard candle--a source of known light-power.

In an ordinary way you cannot tell the _real_ brightness of a
light merely by looking at it. If it appears dim, that may mean either
real faintness or great distance. At night time on the sea you observe
many lights whose distance and real brightness you cannot estimate;
your judgement of the real brightness may be wrong by a factor of a
quintillion if you happen to mistake Arcturus for a ship’s light. But
among them you may notice a light which goes through a regular series
of changes in a certain number of seconds; that tells you that it
is such-and-such a lighthouse, known to project a light of so many
thousand candlepower. You may now estimate with certainty how far off
it is--provided, of course, that there is no fog intervening.

So, too, when we look up at the sky, most of the lights that we see
might be at any distance and have any real brightness. Even the most
refined measurements of parallax only succeed in locating a few of the
nearer lights. But if we see a light winking in the Cepheid manner with
a period of 5⅓ days, we know that it is a replica of δ Cephei and is a
light of 700 sun-power. Or if the period is any other number of days
we can assign the proper sun-power for that period. From this we can
judge the distance. The apparent brightness, which is a combination
of distance and true brightness, is measured; then it is a simple
calculation to answer the question, At what distance must a light
of 700 sun-power be placed in order to give the apparent brightness
observed? How about interference by fog? Careful discussions have
been made, and it appears that notwithstanding the cosmical cloud in
interstellar space there is ordinarily no appreciable absorption or
scattering of the starlight on its way to us.

With the Cepheids serving as standard candles distances in the stellar
universe have been surveyed far exceeding those reached by previous
methods. If the distances were merely those of the Cepheid variables
themselves that would not be so important, but much more information is
yielded.

Fig. 11[26] shows a famous star-cluster called ω Centauri. Amongst the
thousands of stars in the cluster no less than 76 Cepheid variables
have been discovered. Each is a standard candle serving to measure the
distance primarily of itself but also incidentally of the great cluster
in which it lies. The 76 gauges agree wonderfully among themselves, the
average deviation being less than 5 per cent. By this means Shapley
found the distance of the cluster to be 20,000 light years. The light
messages which we receive to-day were sent from the cluster 20,000
years ago.[27]

The astronomer, more than other devotees of science, learns to
appreciate the advantage of not being too near the objects he is
studying. The nearer stars are all right in their way, but it is a
great nuisance being in the very midst of them. For each star has to
be treated singly and located at its proper distance by elaborate
measurements; progress is very laborious. But when we determine the
distance of this remote cluster, we secure at one scoop the distances
of many thousands of stars. The distance being known, the apparent
magnitudes can be turned into true magnitudes, and statistics and
correlations of absolute brightness and colour can be ascertained.
Even before the distance is discovered we can learn a great deal from
the stars in clusters which it is impracticable to find out from less
remote stars. We can see that the Cepheids are much above the average
brightness and are surpassed by relatively few stars. We can ascertain
that the brighter the Cepheid the longer is its period. We discover
that the brightest stars of all are red.[28] And so on. There is a
reverse side to the picture; the tiny points of light in the distant
cluster are not the most satisfactory objects to measure and analyse,
and we could ill spare the nearer stars; but the fact remains that
there are certain lines of stellar investigation in which remoteness
proves to be an actual advantage, and we turn from the nearer stars to
objects fifty thousand light years away.

About 80 globular clusters are known with distances ranging from
20,000 to 200,000 light years. Is there anything yet more remote? It
has long been suspected that the spiral nebulae,[29] which seem to be
exceedingly numerous, are outside our stellar system and form ‘island
universes’. The evidence for this has become gradually stronger, and
now is believed to be decisively confirmed. In 1924 Hubble discovered
a number of Cepheid variables in the great Andromeda nebula which is
the largest and presumably one of the nearest of the spirals. As soon
as their periods had been determined they were available as standard
candles to gauge the distance of the nebula. Their apparent magnitude
was much fainter than that of the corresponding Cepheids in globular
clusters, showing that they must be even more remote. Hubble has since
found the distance of one or two other spirals in the same way.

With the naked eye you can see the Andromeda nebula as a faint patch of
light. When you look at it you are looking back 900,000 years into the
past.


                     _The Contraction Hypothesis_


The problem of providing sufficient supplies of energy to maintain the
sun’s output of light and heat has often been debated by astronomers
and others. In the last century it was shown by Helmholtz and
Kelvin that the sun could maintain its heat for a very long time by
continually shrinking. Contraction involves an approach or fall of
the matter towards the centre; gravitational potential energy is thus
converted and made available as heat. It was assumed that this was
the sole resource since no other supply capable of yielding anything
like so large an amount was known. But the supply is not unlimited,
and on this hypothesis the birth of the sun must be dated not more
than 20,000,000 years ago. Even at the time of which I am speaking the
time-limit was found to be cramping; but Kelvin assured the geologists
and biologists that they must confine their outlines of terrestrial
history within this period.

About the beginning of the present century the contraction theory was
in the curious position of being generally accepted and generally
ignored. Whilst few ventured to dispute the hypothesis, no one seems to
have had any hesitation, if it suited him, in carrying back the history
of the earth or moon to a time long before the supposed era of the
formation of the solar system. Lord Kelvin’s date of the creation was
treated with no more respect than Archbishop Ussher’s.

The serious consequences of the hypothesis become particularly
prominent when we consider the diffuse stars of high luminosity; these
are prodigal of their energy and squander it a hundred or a thousand
times faster than the sun. The economical sun could have subsisted
on its contraction energy for 20,000,000 years, but for the high
luminosity stars the limit is cut down to 100,000 years. This includes
most of the naked-eye stars. Dare we believe that they were formed
within the last 100,000 years? Is the antiquity of man greater than
that of the stars now shining? Do stars in the Andromeda nebula run
their course in less time than their light takes to reach us?

It is one thing to feel a limitation of time-scale irksome, ruling out
ideas and explanations which are otherwise plausible and attractive; it
is another thing to produce definite evidence against the time-scale.
I do not think that astronomers had _in their own territory_ any
weapon for a direct attack on the Helmholtz-Kelvin hypothesis until
the Cepheid variables supplied one. To come to figures: δ Cephei emits
more than 700 times as much heat as the sun. We know its mass and
radius, and we can calculate without difficulty how fast the radius
must contract in order to provide this heat. The required rate is one
part in 40,000 per annum. Now δ Cephei was first observed carefully
in 1785, so that in the time it has been under observation the radius
must have changed by one part in 300 if the contraction hypothesis is
right. You remember that we have in δ Cephei a very sensitive indicator
of any changes occurring in it, viz. the period of pulsation; clearly
changes of the above magnitude could not occur without disturbing this
indicator. Does the period show any change? It is doubtful; there is
perhaps sufficient evidence for a slight change, but it is not more
than ¹⁄₂₀₀th of the change demanded by the contraction hypothesis.

Accepting the pulsation theory, the period should diminish 17 seconds
every year--a quantity easily detectable. The actual change is not
more than one-tenth of a second per year. At least during the Cepheid
stage the stars are drawing on some source of energy other than that
provided by contraction.

On such an important question we should not like to put implicit
trust in one argument alone, and we turn to the sister sciences for
other and perhaps more conclusive evidence. Physical and geological
investigations seem to decide definitely that the age of the
earth--reckoned from an epoch which by no means goes back to its
beginnings as a planet--is far greater than the Helmholtz-Kelvin
estimate of the age of the solar system. It is usual to lay most stress
on a determination of the age of the rocks from the uranium-lead ratio
of their contents. Uranium disintegrates into lead and helium at a
known rate. Since lead is unlike uranium in chemical properties the two
elements would not naturally be deposited together; so that the lead
found with uranium has presumably been formed by its decomposition.[30]
By measuring how much lead occurs with the uranium we can determine
how long ago the uranium was deposited. The age of the older rocks
is found to be about 1,200 million years; lower estimates have been
urged by some authorities, but none low enough to save the contraction
hypothesis. The sun, of course, must be very much older than the earth
and its rocks.

We seem to require a time-scale which will allow at least
10,000,000,000 years for the age of the sun; certainly we cannot
abate our demands below 1,000,000,000 years. It is necessary to look
for a more prolific source of energy to maintain the heat of the
sun and stars through this extended period. We can at once narrow
down the field of search. No source of energy is of any avail unless
it liberates heat in the deep interior of the star. The crux of the
problem is not merely the provision for radiation but the maintenance
of the internal heat which keeps the gravitating mass from collapsing.
You will remember how in the first lecture we had to assign a certain
amount of heat at each point in the stellar interior in order to keep
the star in balance. But the internal heat is continually running away
towards the cooler outside and then escaping into space as the star’s
radiation. This, or its equivalent, must be put back if the star is to
be kept steady--if it is not to contract and evolve at the rate of the
Kelvin time-scale. And it is no use to put it back at the surface of
the star--by bombarding the star with meteors, for example. It could
not flow up the temperature-gradient, and so it would simply take the
first opportunity of escaping as additional radiation. You cannot
maintain a temperature-gradient by supplying heat at the bottom end.
Heat must be poured in at the top end, i. e. in the deep interior of
the star.

Since we cannot well imagine an extraneous source of heat able to
release itself at the centre of a star, the idea of a star picking up
energy as it goes along seems to be definitely ruled out. _It follows
that the star contains hidden within it the energy which has to last
the rest of its life_.

Energy has mass. Many people would prefer to say--energy _is_
mass; but it is not necessary for us to discuss that. The essential
fact is that an erg of energy in any form has a mass of 1·1. 10^-21
grammes. The erg is the usual scientific unit of energy; but we can
measure energy also by the gramme or the ton as we measure anything
else which possesses mass. There is no real reason why you should not
buy a pound of light from an electric light company--except that it
is a larger quantity than you are likely to need and at current rates
would cost you something over £100,000,000. If you could keep all this
light (ether-waves) travelling to and fro between mirrors forming a
closed vessel, and then weigh the vessel, the observed weight would be
the ordinary weight of the vessel plus 1 lb. representing the weight of
the light. It is evident that an object weighing a ton cannot contain
more than a ton of energy; and the sun with a mass of 2.000 quadrillion
tons (p. 24) cannot contain more than 2.000 quadrillion tons of energy
at the most.

Energy of 1·8. 10^{54} ergs has a mass 2. 10^{33} grammes which is the
mass of the sun; consequently that is the sum total of the energy which
the sun contains--the energy which has to last it all the rest of its
life.[31] We do not know how much of this is capable of being converted
into heat and radiation; if it is all convertible there is enough to
maintain the sun’s radiation at the present rate for 15 billion years.
To put the argument in another form, the heat emitted by the sun each
year has a mass of 120 billion tons; and if this loss of mass continued
there would be no mass left at the end of 15 billion years.


                          _Subatomic Energy_


This store of energy is, with insignificant exception, energy of
constitution of atoms and electrons; that is to say, subatomic energy.
Most of it is inherent in the constitution of the electrons and
protons--the elementary negative and positive electric charges--out
of which matter is built; so that it cannot be set free unless these
are destroyed. The main store of energy in a star cannot be used for
radiation unless the matter composing the star is being annihilated.

It is possible that the star may have a long enough life without
raiding the main energy store. A small part of the store can be
released by a process less drastic than annihilation of matter, and
this might be sufficient to keep the sun burning for 10,000,000,000
years or so, which is perhaps as long as we can reasonably require.
The less drastic process is transmutation of the elements. Thus we
have reached a point where a choice lies open before us; we can either
pin our faith to transmutation of the elements, contenting ourselves
with a rather cramped time-scale, or we can assume the annihilation of
matter, which gives a very ample time-scale. But at present I can see
no possibility of a third choice. Let me run over the argument again.
First we found that energy of contraction was hopelessly inadequate;
then we found that the energy must be released in the interior of the
star, so that it comes from an internal, not an external, source; now
we take stock of the whole internal store of energy. No supply of
any importance is found until we come to consider the electrons and
atomic nuclei; here a reasonable amount can be released by regrouping
the protons and electrons in the atomic nuclei (transmutation of
elements), and a much greater amount by annihilating them.

Transmutation of the elements--so long the dream of the alchemist--is
realized in the transformation of radio-active substances. Uranium
turns slowly into a mixture of lead and helium. But none of the known
radio-active processes liberate anything like enough energy to maintain
the sun’s heat. The only important release of energy by transmutation
occurs at the very beginning of the evolution of the elements.

We must start with hydrogen. The hydrogen atom consists simply of a
positive and negative charge, a proton for the nucleus plus a planet
electron. Let us call its mass 1. Four hydrogen atoms will make a
helium atom. If the mass of the helium atom were exactly 4, that would
show that all the energy of the hydrogen atoms remained in the helium
atom. But actually the mass is 3·97; so that energy of mass a 0·03
must have escaped during the formation of helium from hydrogen. By
annihilating 4 grammes of hydrogen we should have released 4 grammes of
energy, but by transmuting it into helium we release 0·03 grammes of
energy. Either process might be used to furnish the sun’s heat though,
as we have already stated, the second gives a much smaller supply.

The release of energy occurs because in the helium atom only two of the
four electrons remain as planet electrons, the other two being cemented
with the four protons close together in the helium nucleus. In bringing
positive and negative charges close together you cause a change of
the energy of the electric field, and release electrical energy which
spreads away as ether-waves. That is where the 0·03 grammes of energy
has gone. The star can absorb these ether-waves and utilize them as
heat.

We can go on from helium to higher elements, but we do not obtain much
more release of energy. For example, an oxygen atom can be made from 16
hydrogen atoms or 4 helium atoms; but as nearly as we can tell it has
just the weight of the 4 helium atoms, so that the release of energy
is not appreciably greater when the hydrogen is transmuted into oxygen
than when it is transmuted into helium.[32] This becomes clearer if
we take the mass of a hydrogen atom to be 1·008, so that the mass of
helium is exactly 4 and of oxygen 16; then it is known from Dr. Aston’s
researches with the mass-spectrograph that the atoms of other elements
have masses which are very closely whole numbers. The loss of 0·008 per
hydrogen atom applies approximately whatever the element that is formed.

The view that the energy of a star is derived by the building up of
other elements from hydrogen has the great advantage that there is no
doubt about the possibility of the process; whereas we have no evidence
that the annihilation of matter can occur in Nature. I am not referring
to the alleged transmutation of hydrogen into helium in the laboratory;
those whose authority I accept are not convinced by these experiments.
To my mind the existence of helium is the best evidence we could desire
of the possibility of the _formation_ of helium. The four protons
and two electrons constituting its nucleus must have been assembled at
some time and place; and why not in the stars? When they were assembled
the surplus energy must have been released, providing a prolific supply
of heat. Prima facie this suggests the interior of a star as a likely
locality, since undoubtedly a prolific source of heat is there in
operation. I am aware that many critics consider the conditions in the
stars not sufficiently extreme to bring about the transmutation--the
stars are not hot enough. The critics lay themselves open to an obvious
retort; we tell them to go and find a _hotter place_.

But here the advantage seems to end. There are many astronomical
indications that the hypothesis attributing the energy of the stars
to the transmutation of hydrogen is unsatisfactory. It may perhaps be
responsible for the rapid liberation of energy in the earliest (giant)
stages when the star is a large diffuse body radiating heat abundantly;
but the energy in later life seems to come from a source subject to
different laws of emission. There is considerable evidence that as a
star grows older it gets rid of a large fraction of the matter which
originally constituted it, and apparently this can only be contrived
by the annihilation of the matter. The evidence, however, is not very
coherent, and I do not think we are in a position to come to a definite
decision. On the whole the hypothesis of annihilation of matter seems
the more promising; and I shall prefer it in the brief discussion of
stellar evolution which I propose to give.

The phrase ‘annihilation of matter’ sounds like something supernatural.
We do not yet know whether it can occur naturally or not, but there is
no obvious obstacle. The ultimate constituents of matter are minute
positive charges and negative charges which we may picture as centres
of opposite kinds of strain in the ether. If these could be persuaded
to run together they would cancel out, leaving nothing except a splash
in the ether which spreads out as an electromagnetic wave carrying off
the energy released by the undoing of the strain. The amount of this
energy is amazingly large; by annihilating a single drop of water we
should be supplied with 200 horsepower for a year. We turn covetous
eyes on this store without, however, entertaining much hope of ever
discovering the secret of releasing it. If it should prove that the
stars have discovered the secret and are using this store to maintain
their heat, our prospect of ultimate success would seem distinctly
nearer.

I suppose that many physicists will regard the subject of subatomic
energy as a field of airy speculation. That is not the way in which
it presents itself to an astronomer. If it is granted that the stars
evolve much more slowly than on the contraction-hypothesis, the
measurement of the output of subatomic energy is one of the commonest
astronomical measurements--the measurement of the heat or light of the
stars.[33] The collection of observational data as to the activity of
liberation of subatomic energy is part of the routine of practical
astronomy; and we have to pursue the usual course of arranging the
measurements into some kind of coherence, so as to find out how the
output is related to the temperature, density, or age of the material
supplying it--in short, to discover the laws of emission. From this
point onwards the discussion may be more or less hypothetical according
to the temperament of the investigator; and indeed it is likely that in
this as in other branches of knowledge advances may come by a proper
use of the scientific imagination. Vain speculation is to be condemned
in this as in any other subject, and there is no need for it; the
problem is one of induction from observation with due regard to our
theoretical knowledge of the possibilities inherent in atomic structure.

I cannot pass from this subject without mentioning the penetrating
radiation long known to exist in our atmosphere, which according
to the researches of Kohlhörster and Millikan comes from outer
space. Penetrating power is a sign of short wave-length and intense
concentration of energy. Hitherto the greatest penetrating power
has been displayed by Gamma rays originated by subatomic processes
occurring in radio-active substances. The cosmic radiation is still
more penetrating, and it seems reasonable to refer it to more energetic
processes in the atom such as those suggested for the source of
stellar energy. Careful measurements have been made by Millikan, and
he concludes that the properties accord with those which should be
possessed by radiation liberated in the transmutation of hydrogen; it
is not penetrating enough to be attributed to a process so energetic as
the annihilation of protons and electrons.

There seems to be no doubt that this radiation is travelling downwards
from the sky. This is shown by measurements of its strength at
different heights in the atmosphere and at different depths below the
surface of mountain lakes; it is weakened according to the amount of
air or water that it has had to traverse. Presumably its source must be
extra-terrestrial. Its strength does not vary with the sun’s altitude,
so it is not coming from the sun. There is some evidence that it varies
according to the position of the Milky Way, most radiation being
received when the greatest extension of the stellar system is overhead.
It cannot come from the _interior_ of the stars, the penetrating
power being too limited; all the hottest and densest matter in the
universe is shut off from us by impenetrable walls. At the most it
could come only from the outer rind of the stars where the temperature
is moderate and the density is low; but it is more likely that its main
source is in the diffuse nebulae or possibly in the matter forming the
general cloud in space.[34]

We must await further developments before we can regard the supposed
subatomic origin of this radiation as other than speculative; we
mention it here only as a possible opening for progress. It will be of
great interest if we can reach by this means a more direct acquaintance
with the processes which we assume to be the source of stellar energy;
and the messages borne to us by the cosmic rays which purport to
relate to these processes deserve the closest attention. Our views
of stellar energy are likely to be affected on one crucial point.
Hitherto we have usually supposed that the very high temperature in the
interior of a star is one of the essential conditions for liberation of
subatomic energy, and that a reasonably high density is also important.
Theoretically it would seem almost incredible that the building up of
higher elements or the annihilation of protons and electrons could
proceed with any degree of vigour in regions where encounters are rare
and there is no high temperature or intense radiation to wake the atoms
from apathy; but the more we face the difficulties of all theories of
the release of subatomic energy the less inclined we are to condemn
any evidence as incredible. The presence of sodium and calcium in
the cosmical cloud, of helium and nebulium in the diffuse nebulae,
of titanium and zirconium in large quantities in the atmospheres of
the youngest stars, bears witness that the evolution of the elements
is already far advanced during the diffuse prestellar stage--unless
indeed our universe is built from the debris of a former creation.
From this point of view it is fitting that we should discern symptoms
of subatomic activity in open space. But the physicist may well shake
his head over the problem. How are four protons and two electrons to
gather together to form a helium nucleus in a medium so rare that the
free path lasts for days? The only comfort is that the mode of this
occurrence is (according to present knowledge) so inconceivable under
any conditions of density and temperature that we may postulate it in
the nebulae--on the principle that we may as well be hung for a sheep
as for a lamb.


                       _Evolution of the Stars_


Twenty years ago stellar evolution seemed to be very simple. The stars
begin by being very hot and gradually cool down until they go out.

On this view the temperature of a star indicated the stage of evolution
that it had reached. The outline of the sequence was sufficiently
indicated by the crude observation of colour--white-hot, yellow-hot,
red-hot; a more detailed order of temperature was ascertained by
examining the light with a spectroscope. The red stars naturally
came last in the sequence; they were the oldest stars on the verge
of extinction. Sir Norman Lockyer strongly opposed this scheme and
to a considerable extent anticipated the more modern view; but most
astronomers pinned their faith to it up to about 1913.

Ten years ago more knowledge had been gained of the densities of
stars. It seemed likely that density would be a more direct criterion
of evolutionary development than temperature. Granted that a star
condenses out of nebulous material, it must in the youngest stage be
very diffuse; from that stage it will contract and steadily increase in
density.

But this necessitates an entire rearrangement of the scheme of
evolution, because the order according to density is by no means the
same as the order according to surface temperature. On the former
view all the cool red stars were old and dying. But a large number of
them are now found to be extremely diffuse--stars like Betelgeuse,
for instance. These must be set down as the very youngest of the
stars; after all it is not unnatural that a star just beginning to
condense out of nebulous material should start at the lowest stage of
temperature. Not all the red stars are diffuse; there are many like
Krueger 60 which have high density, and these we leave undisturbed
as representing the last stage of evolution. Both the first and last
periods of a star’s life are characterized by low temperature; in
between whiles the temperature must have risen to a maximum and fallen
again.

The ‘giant and dwarf theory’ proposed by Hertzsprung and Russell
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.]

[Footnote 3: Primarily it is the electric charge and not the high speed
of particles which determines their appearance in these photographs.
But a high-speed particle leaves behind it a trail of electrically
charged particles--the victims of its furious driving--so that it is
shown indirectly by its line of victims.]

[Footnote 4: Other substitutions for silver do not as a rule cause
greater change, and the differences are likely to be toned down by
mixture of many elements. Excluding hydrogen, the most extreme change
is from 48 particles for silver to 81 particles for an equal mass of
helium. But for hydrogen the change is from 48 to 216, so that hydrogen
gives widely different results from other elements.]

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

[Footnote 6: Unless otherwise indicated ‘gaseous’ is intended to mean
‘composed of _perfect_ gas’.]

[Footnote 7: For this prediction it is unnecessary to know the chemical
composition of the stars, provided that extreme cases (e. g. an
excessive proportion of hydrogen) are excluded. For example, consider
the hypotheses that Capella is made respectively of (_a_) iron,
(_b_) gold. According to theory the opacity of a star made of
the heavier element would be 2½ times the opacity of a star made of
iron. This by itself would make the golden star a magnitude (= 2½
times) fainter. But the temperature is raised by the substitution;
and although, as explained on p. 23, the change is not very great, it
increases the outflow of heat approximately 2½ times. The resultant
effect on the brightness is practically no change. Whilst this
independence of chemical constitution is satisfactory in regard
to definiteness of the results, it makes the discrepant factor 10
particularly difficult to explain.]

[Footnote 8: Observation shows that the sun is about 4 magnitudes
fainter than the average diffuse star of the same spectral class, and
Krueger 60 is 10 magnitudes fainter than diffuse stars of its class.
The whole drop was generally assumed to be due to deviation from a
perfect gas; but this made no allowance for a possible difference
of mass. The comparison with the curve enables the dense star to be
compared with a gaseous star _of its own mass_, and we see that
the difference then disappears. So that (if there has been no mistake)
the dense star is a gaseous star, and the differences above mentioned
were due wholly to differences of mass.] [Footnote 9: Rougher estimates
were made much earlier.]

[Footnote 10: The observed period of Algol is the period of revolution,
not of rotation. But the two components are very close together, and
there can be no doubt that owing to the large tidal forces they keep
the same faces turned towards each other; that is to say, the periods
of rotation and of revolution are equal.]

[Footnote 11: It may be of interest to add that although the proper
light of Algol B is inappreciable, we can observe a reflection (or
re-radiation) of the light of Algol A by it. This reflected light
changes like moonlight according as Algol B is ‘new’ or ‘full’.]

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

[Footnote 13: My references to ‘_perfect gas_ of the density of
platinum’ and ‘_material_ 2,000 times denser than platinum’ have
often been run together by reporters into ‘perfect gas 2,000 times
denser than platinum’. It is scarcely possible to calculate what is
the condition of the material in the Companion of Sirius, but I do not
expect it to be a perfect gas.]

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

[Footnote 15: Nos. 43, 61, 75 are recent discoveries and may require
confirmation. There now remain only two gaps (85 and 87) apart from
possible elements beyond uranium.]

[Footnote 16: It does not give _both_ temperature and pressure,
but it gives one if the other is known. This is valuable information
which may be pieced together with other knowledge of the conditions at
the surface of the stars.]

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

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

[Footnote 19: The helium line in the Ring Nebula on which we have
already commented is not a member of the Pickering Series, but it has
had the same history. It was first supposed to be due to hydrogen,
later (in 1912) reproduced by Fowler terrestrially in a mixture of
helium and hydrogen, and finally discovered by Bohr to belong to
helium.]

[Footnote 20: This, of course, is found from the other lines of the
spectrum which genuinely belong to the star and shift to and fro as it
describes its orbit.]

[Footnote 21: As the word temperature is sometimes used with
new-fangled meanings, I may add that 15,000° is the temperature
corresponding to the individual speeds of the atoms and electrons--the
old-fashioned gas-temperature.]

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

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

[Footnote 24: There is an awkwardness in applying the term ‘apparent’
to something too small to be seen; but, remembering that we have armed
ourselves with an imaginary telescope capable of showing the disk, the
meaning will be clear.]

[Footnote 25: Densities below that of air have been found for some of
the Algol variables by an entirely different kind of investigation, and
also for some of the Cepheid variables by still another method. There
are also many other examples of stars of bulk comparable with that of
Betelgeuse.]

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

[Footnote 27: For comparison, the nearest fixed star is distant 4 light
years. Apart from clusters we rarely deal with distances above 2,000
light years.]

[Footnote 28: One cannot always be sure that what is true of the
cluster stars will be true of stars in general; and our knowledge of
the nearer stars, though lagging behind that of the stars in clusters,
does not entirely agree with this association of colour and brightness.]

[Footnote 29: The term nebula covers a variety of objects, and it is
only the nebulae classed as spirals that are likely to be outside our
stellar system.]

[Footnote 30: This can be checked because uranium lead has a different
atomic weight from lead not so derived. Ordinary lead is a mixture of
several kinds of atoms (isotopes).]

[Footnote 31: You may wonder why, having said that the sun contains
2,000 quadrillion tons of energy _at the most_, I now assume
that it contains just this amount. It is really only a verbal point
depending on the scientific definition of energy. All mass is mass of
_something_, and we now call that something ‘energy’ whether it is
one of the familiar forms of energy or not. You will see in the next
sentence that we do not assume that the energy is convertible into
known forms, so that it is a terminology which commits us to nothing.]

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

[Footnote 33: A measurement of the heat observed to flow from a
continuous fountain of heat is a measurement of the output of the
fountain, unless there is a storing of energy between the output and
the outflow. The breakdown of the Kelvin time-scale indicates that the
storing in the stars (positive or negative) and consequent expansion or
contraction is negligible compared to the output or outflow.]

[Footnote 34: The stars all put together cover an area of the sky
much less than the apparent disk of the sun, so that unless their
surface-layers are generating this radiation very much more abundantly
than the sun does, they cannot be responsible for it.]

[Footnote 35: The term ‘dwarf stars’ is not meant to include _white
dwarfs_.]

[Footnote 36: We can scarcely suppose that all stars after reaching
the main series pass through _precisely_ the same stages. For
example, Algol, when it has become reduced to the mass of the Sun, may
have slightly different density and temperature. But the observational
evidence indicates that these individual differences are small. The
main series is nearly a linear sequence; it must have some ‘breadth’ as
well as ‘length’, but at present the scatter of the individual stars
away from the central line of the sequence seems to be due chiefly to
the probable errors of the observational data and the true breadth has
not been determined.]

[Footnote 37: Exhaustion of supply without change of mass would
cause the star to contract to higher density; it would thus have a
combination of density and mass which (according to observation) is not
found in any actual stars.]

[Footnote 38: This increase was assumed in our detailed description of
the automatic adjustment of the star, and it will be seen that it was
essential to assume it.]




                               APPENDIX

             _Further Remarks on the Companion of Sirius_


I HAVE preferred not to complicate the Story of the Companion of
Sirius with details of a technical kind; some further information may,
therefore, be welcome to those readers who are curious to learn as much
as possible about this remarkable star. I am also able to add a further
instalment of the ‘detective story’ which has just come to hand, the
sleuth this time being Mr. R. H. Fowler.

The star is between the eighth and ninth magnitude, so that it is not
an excessively faint object. The difficulty in detecting it arises
entirely from the overpowering light of its neighbour. At favourable
epochs it has been seen easily with an 8-inch telescope. The period of
revolution is 49 years.

The Companion is separated from Sirius by a distance nearly equal
to the distance of Uranus from the Sun--or twenty times the earth’s
distance from the sun. It has been suggested that the light might be
reflected light from Sirius. This would account for its whiteness, but
would not directly account for its spectrum, which differs appreciably
from that of Sirius. To reflect ¹⁄₁₀₀₀₀th of the light of Sirius (its
actual brightness) the Companion would have to be 74 million miles in
diameter. The apparent diameter of its disk would be 0"·3, which, one
would think, could scarcely escape notice in spite of unfavourable
conditions of observation. But the strongest objection to this
hypothesis of reflected light is that it applies only to this one star.
The other two recognized white dwarfs have no brilliant star in their
neighbourhood, so that they cannot be shining by reflected light. It
is scarcely worth while to invent an elaborate explanation for one of
these strange objects which does not cover the other two.

The Einstein effect, which is appealed to for confirmation of the high
density, is a lengthening of the wave-length and corresponding decrease
of the frequency of the light due to the intense gravitational field
through which the rays have to pass. Consequently the dark lines in the
spectrum appear at longer wave-lengths, i.e. displaced towards the red
as compared with the corresponding terrestrial lines. The effect can be
deduced either from the relativity theory of gravitation or from the
quantum theory; for those who have some acquaintance with the quantum
theory the following reasoning is probably the simplest. The stellar
atom emits the same quantum of energy hν as a terrestrial atom, but
this quantum has to use up some of its energy in order to escape from
the attraction of the star; the energy of escape is equal to the mass
hν/c^2 multiplied by the gravitational potential Φ at the surface of
the star. Accordingly the reduced energy after escape is hν(1 - Φ/c^2);
and since this must still form a quantum hν', the frequency has to
change to a value ν' = ν(1 - Φ/c^2). Thus the displacement ν' - ν is
proportional to Φ, i.e. to the mass divided by the radius of the star.

The effect on the spectrum resembles the Doppler effect of a velocity
of recession, and can therefore only be discriminated if we know
already the line-of-sight velocity. In the case of a double star the
velocity is known from observation of the other component of the
system, so that the part of the displacement attributable to Doppler
effect is known. Owing to orbital motion there is a difference of
velocity between Sirius and its Companion amounting at present to 43
km. per sec. and this has been duly taken into account; the observed
difference in position of the spectral lines of Sirius and its
Companion corresponds to a velocity of 23 km. per sec. of which 4 km.
per sec. is attributable to orbital motion, and the remaining 19 km.
per sec. must be interpreted as Einstein effect. The result rests
mainly on measurements of one spectral line H_β. The other
favourable lines are in the bluer part of the spectrum, and since
atmospheric scattering increases with blueness, the scattered light
of Sirius interferes. However, they afford some useful confirmatory
evidence.

Of the other white dwarfs ο^2 Eridani is a double star, its companion
being a red dwarf fainter than itself. The red shift of the spectrum
will be smaller than in the Companion of Sirius and it will not
be so easy to separate it from various possible sources of error.
Nevertheless the prospect is not hopeless. The other recognized white
dwarf is an unnamed star discovered by Van Maanen; it is a solitary
star, and consequently there is no means of distinguishing between
Einstein shift and Doppler shift. Various other stars have been
suspected of being in this condition, including the Companions of
Procyon, 85 Pegasi, and Mira Ceti.

If the Companion of Sirius were a perfect gas its central temperature
would be about 1,000,000,000°, and the central part of the star would
be a million times as dense as water. It is, however, unlikely that the
condition of a perfect gas continues to hold. It should be understood
that in any case the density will fall off towards the outside of the
star, and the regions which we _observe_ are entirely normal. The
dense material is tucked away under high pressure in the interior.

Perhaps the most puzzling feature that remains is the extraordinary
difference of development between Sirius and its Companion, which must
both have originated at the same time. Owing to the radiation of mass
the age of Sirius must be less than a billion years; an initial mass,
however large, would radiate itself down to less than the present mass
of Sirius within a billion years. But such a period is insignificant
in the evolution of a small star which radiates more slowly, and it
is difficult to see why the Companion should have already left the
main series and gone on to this (presumably) later stage. This is
akin to other difficulties in the problem of stellar evolution, and I
feel convinced that there is something of fundamental importance that
remains undiscovered.

Until recently I have felt that there was a serious (or, if you like,
a comic) difficulty about the ultimate fate of the white dwarfs.
Their high density is only possible because of the smashing of the
atoms, which in turn depends on the high temperature. It does not seem
permissible to suppose that the matter can remain in this compressed
state if the temperature falls. We may look forward to a time when the
supply of subatomic energy fails and there is nothing to maintain the
high temperature; then on cooling down, the material will return to
the normal density of terrestrial solids. The star must, therefore,
expand, and in order to regain a density a thousandfold less the radius
must expand tenfold. Energy will be required in order to force out
the material against gravity. Where is this energy to come from? An
ordinary star has not enough heat energy inside it to be able to expand
against gravitation to this extent; and the white dwarf can scarcely be
supposed to have had sufficient foresight to make special provision for
this remote demand. Thus the star may be in an awkward predicament--it
will be losing heat continually _but will not have enough energy to
cool down_.

One suggestion for avoiding this dilemma is like the device of a
novelist who brings his characters into such a mess that the only
solution is to kill them off. We might assume that subatomic energy
will never cease to be liberated until it has removed the whole
mass--or at least conducted the star out of the white dwarf condition.
But this scarcely meets the difficulty; the theory ought in some way to
guard automatically against an impossible predicament, and not to rely
on disconnected properties of matter to protect the actual stars from
trouble.

The whole difficulty seems, however, to have been removed in a recent
investigation by R. H. Fowler. He concludes unexpectedly that the
dense matter of the Companion of Sirius has an ample store of energy
to provide for the expansion. The interesting point is that his
solution invokes some of the most recent developments of the quantum
theory--the ‘new statistics’ of Einstein and Bose and the wave-theory
of Schrödinger. It is a curious coincidence that about the time that
this matter of transcendently high density was engaging the attention
of astronomers, the physicists were developing a new theory of matter
which specially concerns high density. According to this theory matter
has certain wave properties which barely come into play at terrestrial
densities; but they are of serious importance at densities such as that
of the Companion of Sirius. It was in considering these properties
that Fowler came upon the store of energy that solves our difficulty;
the classical theory of matter gives no indication of it. The white
dwarf appears to be a happy hunting ground for the most revolutionary
developments of theoretical physics.

To gain some idea of the new theory of dense matter we can begin by
referring to the photograph of the Balmer Series in Fig. 9. This shows
the light radiated by a large number of hydrogen atoms in all possible
states up to No. 30 in the proportions in which they occur naturally in
the sun’s chromosphere. The old-style electromagnetic theory predicted
that electrons moving in curved paths would radiate continuous
light; and the old-style statistical theory predicted the relative
abundance of orbits of different sizes, so that the distribution
of light along this continuous spectrum could be calculated. These
predictions are wrong and do not give the distribution of light shown
in the photograph; _but they become less glaringly wrong as we draw
near to the head of the series_. The later lines of the series
crowd together and presently become so close as to be practically
indistinguishable from continuous light. Thus the classical prediction
of continuous spectrum is becoming approximately true; simultaneously
the classical prediction of its intensity approaches the truth. There
is a famous Correspondence Principle enunciated by Bohr which asserts
that for states of very high number the new quantum laws merge into the
old classical laws. If we never have to consider states of low number
it is indifferent whether we calculate the radiation or statistics
according to the old laws or the new.

In high-numbered states the electron is for most of the time far
distant from the nucleus. Continuous proximity to the nucleus indicates
a low-numbered state. Must we not expect, then, that in extremely
dense matter the continuous proximity of the particles will give
rise to phenomena characteristic of low-numbered states? There is
no real discontinuity between the organization of the atom and the
organization of the star; the ties which bind the particles in the
atom, bind also more extended groups of particles and eventually the
whole star. So long as these ties are of high quantum number, the
alternative conception is sufficiently nearly valid which represents
the interactions by forces after the classical fashion and takes no
cognizance of ‘states’. For very high density there is no alternative
conception, and we must think not in terms of force, velocity, and
distribution of independent particles, but in terms of states.

The effect of this breakdown of the classical conception can best
be seen by passing at once to the final limit when the star becomes
a single system or molecule in state No. 1. Like an excited atom
collapsing with discontinuous jumps such as those which give the
Balmer Series, the star with a few last gasps of radiation will reach
the limiting state which has no state beyond. This does not mean that
further contraction is barred by the ultimate particles jamming in
contact, any more than collapse of the hydrogen atom is barred by
the electron jamming against the proton; progress is stopped because
the star has got back to the first of an integral series of possible
conditions of a material system. A hydrogen atom in state No. 1 cannot
radiate; nevertheless its electron is moving with high kinetic energy.
Similarly a star when it has reached state No. 1 no longer radiates;
nevertheless its particles are moving with extremely great energy.
What is its temperature? If you measure temperature by radiating power
its temperature is absolute zero, since the radiation is nil; if you
measure temperature by the average speed of molecules its temperature
is the highest attainable by matter. The final fate of the white dwarf
is to become at the same time the hottest and the coldest matter in
the universe. Our difficulty is doubly solved. Because the star is
intensely hot it has enough energy to cool down if it wants to; because
it is so intensely cold it has stopped radiating and no longer wants to
grow any colder.

We have described what is believed to be the final state of the white
dwarf and perhaps therefore of every star. The Companion of Sirius
has not yet reached this state, but it is so far on the way that the
classical treatment is already inadmissible. If any stars have reached
state No. 1 they are invisible; like atoms in the normal (lowest) state
they give no light. The binding of the atom which defies the classical
conception of forces has extended to cover the star. I little imagined
when this survey of Stars and Atoms was begun that it would end with a
glimpse of a Star-Atom.



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