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.