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.