I should make it plain that the curve in Fig. 7 is traced by pure
theory or terrestrial experiment except for the one constant determined
by making it pass through Capella. We can imagine physicists working
on a cloud-bound planet such as Jupiter who have never seen the stars.
They should be able to deduce by the method explained on p. 25 that
if there is a universe existing beyond the clouds it is likely to
aggregate primarily into masses of the order a thousand quadrillion
tons. They could then predict that these aggregations will be globes
pouring out light and heat and that their brightness will depend on
the mass in the way given by the curve in Fig. 7. All the information
that we have used for the calculations would be accessible to them
beneath the clouds, except that we have stolen one advantage over
them in utilizing the bright component of Capella. Even without this
forbidden peep, present-day physical theory would enable them to assign
a brightness to the invisible stellar host which would not be absurdly
wrong. Unless they were wiser than us they would probably ascribe to
all the stars a brilliance about ten times too great[7]--not a bad
error for a first attempt at so transcendent a problem. We hope to
clear up the discrepant factor 10 with further knowledge of atomic
processes; meanwhile we shelve it by fixing the doubtful constant by
astronomical observation.
_Dense Stars_
The agreement of the observational points with the curve is remarkably
close, considering the rough nature of the observational measurements;
and it seems to afford a rather strong confirmation of the theory. But
there is one awful confession to make--_we have compared the theory
with the wrong stars_. At least when the comparison was first made
at the beginning of 1924 no one entertained any doubt that they were
the wrong stars.
We must recall that the theory was developed for stars in the condition
of a perfect gas. In the right half of Fig. 7 the stars represented
are all diffuse stars; Capella with a mean density about equal to that
of the air in this room may be taken as typical. Material of this
tenuity is evidently a true gas, and in so far as these stars agree
with the curve the theory is confirmed. But in the left half of the
diagram we have the Sun whose material is denser than water, Krueger
60 denser than iron, and many other stars of the density usually
associated with solid or liquid matter. What business have they on the
curve reserved for a perfect gas? When these stars were put into the
diagram it was not with any expectation that they would agree with
the curve; in fact, the agreement was most annoying. Something very
different was being sought for. The idea was that the theory might
perhaps be trusted on its own merits with such confirmation as the
diffuse stars had already afforded; then by measuring how far these
dense stars fell below the curve we should have definite information
as to how great a deviation from a perfect gas occurred at any given
density. According to current ideas it was expected that the sun would
fall three or four magnitudes below the curve, and the still denser
Krueger 60 should be nearly ten magnitudes below.[8] You see that the
expectation was entirely unfulfilled.
The shock was even greater than I can well indicate to you, because
the great drop in brightness when the star is too dense to behave
as a true gas was a fundamental tenet in our conception of stellar
evolution. On the strength of it the stars had been divided into two
groups known as giants and dwarfs, the former being the gaseous stars
and the latter the dense stars.
Two alternatives now lie before us. The first is to assume that
something must have gone wrong with our theory; that the true curve for
gaseous stars is not as we have drawn it, but runs high up on the left
of the diagram so that the Sun, Krueger 60, &c., are at the appropriate
distances below it. In short, our imaginary critic was right; Nature
had hidden something unexpected inside the star and so frustrated our
calculations. Well, if this were so, it would be something to have
found it out by our investigations.
The other alternative is to consider this question--Is it impossible
that a perfect gas should have the density of iron? The answer is
rather surprising. There is no earthly reason why a perfect gas should
not have a density far exceeding iron. Or it would be more accurate to
say, the reason why it should not is _earthly_ and does not apply
to the stars.
The sun’s material, in spite of being denser than water, really is
a perfect gas. It sounds incredible, but it must be so. The feature
of a true gas is that there is plenty of room between the separate
particles--a gas contains very little substance and lots of emptiness.
Consequently when you squeeze it you do not have to squeeze the
substance; you just squeeze out some of the waste space. But if
you go on squeezing, there comes a time when you have squeezed out
all the empty space; the atoms are then jammed in contact and any
further compression means squeezing the substance itself, which is
quite a different proposition. So as you approach that density the
compressibility characteristic of a gas is lost and the matter is no
longer a proper gas. In a liquid the atoms are nearly in contact;
that will give you an idea of the density at which the gas loses its
characteristic compressibility.
The big terrestrial atoms which begin to jam at a density near that
of the liquid state do not exist in the stars. The stellar atoms have
been trimmed down by the breaking off of all their outer electrons. The
lighter atoms are stripped to the bare nucleus--of quite insignificant
size. The heavier atoms retain a few of the closer electrons, but have
not much more than a hundredth of the diameter of a fully arrayed atom.
Consequently we can go on squeezing ever so much more before these
tiny atoms or ions jam in contact. At the density of water or even of
platinum there is still any amount of room between the trimmed atoms;
and waste space remains to be squeezed out as in a perfect gas.
Our mistake was that in estimating the congestion in the stellar
ball-room we had forgotten that crinolines are no longer in fashion.
It was, I suppose, very blind of us not to have foreseen this result,
considering how much attention we had been paying to the mutilation
of the atoms in other branches of the investigation. By a roundabout
route we have reached a conclusion which is really very obvious. And
so we conclude that the stars on the left of the diagram are after
all not the ‘wrong’ stars. The sun and other dense stars are on the
perfect gas curve because their material is perfect gas. Careful
investigation has shown that in the small stars on the extreme left of
Fig. 7 the electric charges of the atoms and electrons bring about a
slight deviation from the ordinary laws of a gas; it has been shown by
R. H. Fowler that the effect is to make the gas not _imperfect_
but _superperfect_--it is _more_ easily compressed than an
ordinary gas. You will notice that on the average the stars run a
little above the curve on the left of Fig. 7. It is probable that the
deviation is genuine and is partly due to superperfection of the gas;
we have already seen that imperfection would have brought them below
the curve.
Even at the density of platinum there is plenty of waste space, so
that in the stars we might go on squeezing stellar matter to a density
transcending anything known on the earth. But that’s another story--I
will tell it later on.
* * * * *
The general agreement between the observed and predicted brightness of
the stars of various masses is the main test of the correctness of our
theories of their internal constitution. The incidence of their masses
in a range which is especially critical for radiation pressure is also
valuable confirmation. It would be an exaggeration to claim that this
limited success is a proof that we have reached the truth about the
stellar interior. It is not a proof, but it is an encouragement to work
farther along the line of thought which we have been pursuing. The
tangle is beginning to loosen. The more optimistic may assume that it
is now straightened out; the more cautious will make ready for the next
knot. The one reason for thinking that the real truth cannot be so very
far away is that in the interior of a star, if anywhere, the problem
of matter is reduced to its utmost simplicity; and the astronomer is
engaged on what is essentially a less ambitious problem than that of
the terrestrial physicist to whom matter always appears in the guise
of electron systems of the most complex organization.
We have taken the present-day theories of physics and pressed them to
their remotest conclusions. There is no dogmatic intention in this;
it is the best means we have of testing them and revealing their
weaknesses if any.
In ancient days two aviators procured to themselves wings. Daedalus
flew safely through the middle air and was duly honoured on his
landing. Icarus soared upwards to the sun till the wax melted which
bound his wings and his flight ended in fiasco. In weighing their
achievements, there is something to be said for Icarus. The classical
authorities tell us that he was only ‘doing a stunt’, but I prefer to
think of him as the man who brought to light a serious constructional
defect in the flying-machines of his day. So, too, in Science. Cautious
Daedalus will apply his theories where he feels confident they will
safely go; but by his excess of caution their hidden weaknesses remain
undiscovered. Icarus will strain his theories to the breaking-point
till the weak joints gape. For the mere adventure? Perhaps partly; that
is human nature. But if he is destined not yet to reach the sun and
solve finally the riddle of its constitution, we may at least hope to
learn from his journey some hints to build a better machine.
LECTURE II
SOME RECENT INVESTIGATIONS
IT will help us to appreciate the astronomical significance of what we
have learnt in the previous lecture if we turn from the general to the
particular and see how it applies to individual stars. I will take two
stars round which centre stories of special interest, and relate the
history of our knowledge of them.
_The Story of Algol_
This is a detective story, which we might call ‘The Missing Word and
the False Clue’.
In astronomy, unlike many sciences, we cannot handle and probe the
objects of our study; we have to wait passively and receive and decode
the messages that they send to us. The whole of our information
about the stars comes to us along rays of light; we watch and try
to understand their signals. There are some stars which seem to be
sending us a regular series of dots and dashes--like the intermittent
light from a lighthouse. We cannot translate this as a morse code;
nevertheless, by careful measurement we disentangle a great deal of
information from the messages. The star Algol is the most famous of
these ‘variable stars’. We learn from the signals that it is really two
stars revolving round each other. Sometimes the brighter of the two
stars is hidden, giving a deep eclipse or ‘dash’; sometimes the faint
star is hidden, giving a ‘dot’. This recurs in a period of 2 days 21
hours--the period of revolution of the two stars.
There was a great deal more information in the message, but it was
rather tantalizing. There was, so to speak, just one word missing. If
we could supply that word the message would give full and accurate
particulars as to the size of the system--the diameters and masses of
the two components, their absolute brightness, the distance between
them, their distance from the sun. Lacking the word the message told us
nothing really definite about any of these things.
In these circumstances astronomers would scarcely have been human if
they had not tried to guess the missing word. The word should have told
us how much bigger the bright star was than the fainter, that is to
say, the ratio of the masses of the two stars. Some of the less famous
variable stars give us complete messages. (These could accordingly be
used for testing the relation of mass and absolute brightness, and are
represented by triangles in Fig. 7.) The difficulty about Algol arose
from the excessive brightness of the bright component which swamped and
made illegible the more delicate signals from the faint component. From
the other systems we could find the most usual value of the mass ratio,
and base on that a guess as to its probable value for Algol. Different
authorities preferred slightly different estimates, but the general
judgement was that in systems like Algol the bright component is twice
as massive as the faint component. And so the missing word was assumed
to be ‘two’; on this assumption the various dimensions of the system
were worked out and came to be generally accepted as near the truth.
That was sixteen years ago.[9]
In this way the sense of the message was made out to be that the
brighter star had a radius of 1,100,000 kilometres (one and a half
times the sun’s radius), that it had half the mass of the sun, and
thirty times the sun’s light-power, &c. It will be seen at once that
this will not fit our curve in Fig. 7; a star of half the sun’s mass
ought to be very much fainter than the sun. It was rather disconcerting
to find so famous a star protesting against the theory; but after
all the theory is to be tested by comparison with facts and not with
guesses, and the theory might well have a sounder basis than the
conjecture as to the missing word. Moreover, the spectral type of Algol
is one that is not usually associated with low mass, and this cast some
suspicion on the accepted results.
If we are willing to trust the theory given in the last lecture we
can do without the missing word. Or, to put it another way, we can
try in succession various guesses instead of ‘two’ until we reach one
that gives the bright component a mass and luminosity agreeing with
the curve in Fig. 7. The guess ‘two’ gives, as we have seen, a point
which falls a long way from the curve. Alter the guess to ‘three’
and recalculate the mass and brightness on this assumption; the
corresponding point is now somewhat nearer to the curve. Continue with
‘four’, ‘five’, &c.; if the point crosses the curve we know that we
have gone too far and must take an intermediate value in order to reach
the desired agreement. This was done in November 1925, and it appeared
that the missing word must be ‘five’, not ‘two’--a rather startling
change. And now the message ran--
Radius of bright component = 2,140,000 kilometres.
Mass of bright component = 4·3 x sun’s mass.
If you compare these with the original figures you will see that there
is a great alteration. The star is now assigned a large mass much
more appropriate to a B-type star. It also turns out that Algol is
more than a hundred times as bright as the sun; and its parallax is
0·028"--twice the distance previously supposed.
At the time there seemed little likelihood that these conclusions could
be tested. Possibly the prediction as to the parallax might be proved
or disproved by a trigonometrical determination; but it is so small as
to be almost out of range of reasonably accurate measurement. We could
only adopt a ‘take it or leave it’ attitude--‘If you accept the theory,
_this_ is what Algol is like; if you distrust the theory, these
results are of no interest to you.’
But meanwhile two astronomers at Ann Arbor Observatory had been making
a search for the missing word by a remarkable new method. They had in
fact found the word and published it a year before, but it had not
become widely known. If a star is rotating, one edge or ‘limb’ is
coming towards us and the other going away from us. We can measure
speeds towards us or away from us by means of the Doppler effect on
the spectrum, obtaining a definite result in miles per second. Thus
we can and do measure the equatorial speed of rotation of the sun
by observing first the east limb then the west limb and taking the
difference of velocity shown. That is all very well on the sun, where
you can cover up the disk except the special part that you want to
observe; but how can you cover up part of a star when a star is a mere
point of light? _You_ cannot; but in Algol the covering up is done
for you. The faint component is your screen. As it passes in front
of the bright star there is a moment when it leaves a thin crescent
showing on the east and another moment when a thin crescent on the west
is uncovered. Of course, the star is too far away for you actually to
see the crescent shape, but at these moments you receive light from
the crescents only, the rest of the disk being hidden. By seizing
these moments you can make the measurements just as though you had
manipulated the screen yourself. Fortunately the speed of rotation of
Algol is large and so can be measured with relatively small error. Now
multiply the equatorial velocity by the period of rotation;[10] that
will give you the circumference of Algol. Divide by 6·28, and you have
the radius.
That was the method developed by Rossiter and McLaughlin. The latter
who applied it to Algol found the radius of the bright component to be
2,180,000 kilometres.
So far as can be judged his result has considerable accuracy; indeed it
is probable that the radius is now better known than that of any other
star except the sun. If you will now turn back to p. 44 and compare
it with the value found from the theory you will see that there is
cause for satisfaction. McLaughlin evaluated the other constants and
dimensions of the system; these agree equally well, but that follows
automatically because there was only one missing word to be supplied.
In both determinations the missing word or mass ratio turned out to be
5·0.
This is not quite the end of the story. Why had the first guess at
the mass ratio gone so badly wrong? We understand by now that a
disparity in mass is closely associated with a disparity in brightness
of the two stars. The disparity in brightness was given in Algol’s
original message; it informed us that the faint component gives about
one-thirteenth of the light of the bright one. (At least that was how
we interpreted it.) According to our curve this corresponds to a mass
ratio 2½, which is not much improvement on the original guess 2. For a
mass ratio 5 the companion ought to have been much fainter--in fact its
light should have been undetectable. Although considerations like these
could not have had much influence on the original guess, they seemed at
first to reassure us that there was not very much wrong with it.
Let us call the bright component Algol A and the faint component Algol
B. Some years ago a new discovery was made, namely Algol C. It was
found that Algol A and B together travel in an orbit round a third star
in a period of just under two years--at least they are travelling round
in this period, and we must suppose that there is something present
for them to revolve around. Hitherto we had believed that when Algol
A was nearly hidden at the time of deepest eclipse all the remaining
light must come from Algol B; but now it is clear that it belongs to
Algol C, which is always shining without interference. Consequently the
mass ratio 2½ is that of Algol A to Algol C. The light from Algol B is
inappreciable as it should be for a mass ratio 5.[11]
The message from Algol A and B was confused, not only on account of
the missing word, but because a word or two of another message from
Algol C had got mixed up with it; so that even when the missing word
was found to be ‘five’ and confirmed in two ways, the message was
not quite coherent. In another place the message seemed to waver
and read ‘two-and-a-half’. The finishing step is the discovery that
‘two-and-a-half’ belongs to a different message from a previously
unsuspected star, Algol C. And so it all ends happily.
The best detective is not infallible. In this story our astronomical
detective made a reasonable but unsuccessful guess near the beginning
of the case. He might have seen his error earlier, only there was a
false clue dropped by a third party who happened to be present at the
crime, which seemed to confirm the guess. This was very unlucky. But it
makes all the better detective story of it.
_The Story of the Companion of Sirius_
The title of this detective story is ‘The Nonsensical Message’.
Sirius is the most conspicuous star in the sky. Naturally it was
observed very often in early days, and it was used by astronomers along
with other bright stars to determine time and set the clocks by. It
was a _clock star_, as we say. But it turned out that it was not
at all a good clock; it would gain steadily for some years, and then
lose. In 1844 Bessel found out the cause of this irregularity; Sirius
was describing an elliptic orbit. Obviously there must be something
for it to move around, and so it came to be recognized that there was
a dark star there which no one had ever seen. I doubt whether any one
expected it would ever be seen. The Companion of Sirius was, I believe,
the first invisible star to be regularly recognized. We ought not to
call such a star hypothetical. The mechanical properties of matter are
much more crucial than the accidental property of being visible; we
do not consider a transparent pane of glass ‘hypothetical’. There was
near Sirius something which exhibited the most universal mechanical
property of matter, namely, exerting force on neighbouring matter
according to the law of gravitation. That is better evidence of the
existence of a material mass than ocular evidence would be.
However, eighteen years later the Companion of Sirius was actually
seen by Alvan Clark. This discovery was unique in its way; Clark was
not looking at Sirius because he was interested in it, but because
Sirius was a nice bright point of light with which to test the optical
perfection of a large new object-glass that his firm had made. I dare
say that when he saw the little point of light close to Sirius he was
disappointed and tried to polish it away. However, it stayed, and
proved to be the already known but hitherto unseen Companion.
The big modern telescopes easily show the star and rather spoil the
romance; but as romance faded, knowledge grew, and we now know that the
Companion is a star not much less massive than the sun. It has ⅘ths of
the mass of the sun, but gives out only ¹⁄₃₆₀th of the sun’s light. The
faintness did not particularly surprise us;[11] presumably there should
be white-hot stars glowing very brightly and red-hot stars glowing
feebly, with all sorts of intermediate degrees of brightness. It was
assumed that the Companion was one of the feeble stars only just red
hot.
In 1914 Professor Adams at the Mount Wilson Observatory found that it
was not a red star. It was white--white hot. Why, then, was it not
shining brilliantly? Apparently the only answer was that it must be a
very small star. You see, the nature and colour of the light show that
its surface must be glowing more intensely than the sun’s; but the
total light is only ¹⁄₃₆₀th of the sun’s; therefore the surface must
be less than ¹⁄₃₆₀th of the sun’s. That makes the radius less than
¹⁄₁₉th of the sun’s radius, and brings the globe down to a size which
we ordinarily associate with a planet rather than with a star. Working
out the sum more accurately we find that the Companion of Sirius is a
globe intermediate in size between the earth and the next larger planet
Uranus. But if you are going to put a mass not much less than that of
the sun into a globe not very much larger than the earth, it will be
a tight squeeze. The actual density works out at 60,000 times that of
water--just about a ton to the cubic inch.
We learn about the stars by receiving and interpreting the messages
which their light brings to us. The message of the Companion of Sirius
when it was decoded ran: ‘I am composed of material 3,000 times denser
than anything you have ever come across; a ton of my material would
be a little nugget that you could put in a match-box.’ What reply can
one make to such a message? The reply which most of us made in 1914
was--‘Shut up. Don’t talk nonsense.’