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.
_Printed in England at the_ OXFORD UNIVERSITY PRESS
_by John Johnson Printer to the University_