_Stars_
The process of carving out the universe which we considered in the last
chapter ends normally with a simple star, although special accidents
may have other consequences. As the result of close approaches with
other stars, a tiny fraction of the total number of the stars, perhaps
about one star in 100,000 (p. 341 below), may be attended by a retinue
of planets. Another fraction, still small, although far greater than
the foregoing, appears to have broken up as the result of excessive
rotation, and formed binary or perchance multiple systems. But the
destiny of the majority of stars is to pursue their paths solitary
through space, neither breaking up of themselves nor being broken
up by other stars. The only contact such stars have with the outer
universe is that they are incessantly pouring away radiation into
space. This outpouring of radiation is almost entirely a one-way
process, any radiation a star may receive from other stars being quite
inappreciable in comparison with the amount it is itself emitting. The
radiation is accompanied by a loss of weight, and this again is all
give and no take, the weight of any stray matter the star may sweep
up out of space, like that of any radiation it receives, being quite
inappreciable in comparison with the weight it loses by radiation.
Without unduly straining the facts, the normal object in the sky may be
idealised as a solitary body, alone in endless space, which continually
pours out radiation and receives nothing in return.
In the present chapter we shall consider the sequence of changes which
such a star may be expected to experience during the course of its
life. Having already discussed the mechanical accidents to which stars
are liable, namely, fission through rotation and break-up through the
tidal action of a passing star, we now turn to consider the life of
a normal star which escapes all accidents until it finally becomes
extinct through mere old age.
It will be necessary in the first place to describe the physical states
of the various types of stars observed in the sky, and as a preliminary
to this we must explain how the observations of the astronomer are
translated into a form which gives us direct information as to the
condition of the star.
SURFACE-TEMPERATURE. In Chapter II (p. 140) we saw how each colour of
light or wave-length of radiation has a special temperature associated
with it, light of this colour predominating when a body is heated up
to the temperature in question. For instance, a body raised to what we
call a red-heat emits more red light than light of any other colour,
and so looks red to the eye.
Thus if a star looks red, it is legitimate to infer that its surface
is at the temperature we describe as a red-heat. If another star has
the colour of the carbon of an arc-light, we may conclude that its
surface is at about the same temperature as the arc. In this way we can
estimate the temperatures of the surfaces of the stars.
In practice the procedure is not so crude as the foregoing description
might seem to imply. The astronomer passes the light from a star
through a spectroscope, thus analysing it into its different colours.
By a process of exact measurement, he then determines the proportions
in which the different colours of light occur. This shews at once which
colour of light is most plentiful in the spectrum of the star. Either
from this or from the general distribution of colours, he can deduce
the temperature of the star’s surface.
[Illustration: Fig. 15. Distribution of radiation of different
wave-lengths at various temperatures.]
We have already seen (p. 123) how Planck discovered the law according
to which the radiation emitted by a full radiator is distributed
amongst the different colours or wave-lengths of the spectrum. The four
curves shewn in fig. 15 represent the theoretical distribution for the
radiation emitted by surfaces at the four temperatures 3000, 4000, 5000
and 6000 degrees respectively. The different wave-lengths of light are
represented by points on the horizontal axis, the marked wave-lengths
being measured in the unit of a hundred-millionth part of a centimetre,
which is usually called an Angstrom. The height of the curve above such
a point represents the abundance of radiation of the wave-length in
question.
The two methods of determining stellar temperature will be easily
understood by reference to these curves. The 6000 degrees curve reaches
its greatest height at a wave-length of 4800 Angstroms, so that if
light of wave-length 4800 Angstroms proves to be most abundant in the
spectrum of any star, we know that the star’s surface has a temperature
of 6000 degrees. The second method consists merely in examining to
which of the theoretical curves shewn in fig. 15 the observed curve can
be fitted most closely.
Either of these methods indicates that the temperature of the sun’s
surface is about 6000 degrees absolute, which is nearly twice the
temperature of the hottest part of the electric arc. The total amount
of light and heat received on earth from the sun shews that the
sun’s radiation must be very nearly, although not quite, the “full
temperature radiation” (p. 123) of a body at this temperature. This is
also shewn by the sun’s radiation being distributed among the various
colours in a way which conforms very closely to the theoretical curve
for a full radiator at 6000 degrees shewn in fig. 15.
The surface-temperature of a star can also be estimated from its
spectral type. Many of the lines in stellar spectra are emitted by
atoms from which one or more electrons have been torn off by the
heat of the star’s atmosphere. We know the temperatures at which the
electrons in question are first stripped off their atoms, and so can
deduce the star’s temperature.
The temperatures which correspond to the different types of stellar
spectra as shewn in Plate VIII (p. 51), are approximately as follows:
+-------------+-----------+
|Spectral type|Temperature|
+-------------+-----------+
| _B_ | 23,000 |
| _A_ | 11,000 |
| _F_ | 7,400 |
| _G_ | 6,000 |
| _K_ | 5,100 |
| _M_ | 3,400 |
+-------------+-----------+
The last three entries in the table refer only to normal stars having
diameters comparable with that of the sun. We shall find (p. 276) that
a second class of stars (giants) exist, whose diameters are enormously
greater than the sun’s. These have the substantially lower temperatures
shewn below:
+-------------+-----------+
|Spectral type|Temperature|
+-------------+-----------+
| _G_ | 5600 |
| _K_ | 4200 |
| _M_ | 3200 |
+-------------+-----------+
In studying stellar structure and mechanism, we are less concerned with
the heat of the star’s surface as measured by its temperature, than
with the amount of radiation it pours out per square inch.
This of course depends on the temperature; the hotter a surface,
the more radiation it emits. But the temperature does not measure
the quantity of radiation emitted. If we double the temperature
of a surface it emits 16 times, not twice, its previous amount of
radiation; the radiation from each square inch of surface varies
as the fourth power of the temperature. As a consequence, a star
with a surface-temperature of 3000 degrees, or half that of the sun,
emits only a sixteenth part as much radiation per square inch as the
sun[23]. The radiation of each star is a compound of light, heat and
ultra-violet radiation, and the proportions of these are not the
same in different stars; the cooler a star’s surface the greater the
fraction of its radiation which is emitted as heat. Thus the star at
3000 degrees will emit nothing like as much as a sixteenth of the sun’s
light per square inch, but will emit more than a sixteenth of the sun’s
heat.
[23] This is shewn in fig. 15, the area of the 3000 degree curve being
only a sixteenth of the area of the 6000 degree curve.
This shews that the total emission of radiation of a star cannot be
estimated from its visual brightness alone; a substantial allowance
must always be made for invisible radiations, both for the invisible
heat at the red end of the spectrum and for the invisible ultra-violet
radiation at the other end. The importance of these corrections is
shewn in fig. 16. The four thick curves are identical with those
already given in fig. 15, and shew how the radiation from a star
of given surface-temperature is distributed over the different
wave-lengths. The total radiation emitted at any temperature is of
course represented by the whole area enclosed between the corresponding
curve and the horizontal axis. The eye is only sensitive to radiation
of wave-lengths lying between 3750 and 7500 Angstroms, so that of all
this radiation only that part in the shaded strip is visible, all the
rest representing invisible radiation.
We see at once that a fair proportion of the radiation emitted by a
star at 6000 degrees comes within the range of visibility, but only
a small fraction of that emitted by a star at 3000 degrees. Taking
the stars as a whole, star-light forms only a small part of the total
radiation of the stars.
[Illustration: Fig. 16. Distribution of radiation into visible and
invisible.]
If our eyes were suddenly to become sensitive to all kinds of
radiation, and not to visual light alone, the appearance of the
sky would undergo a strange metamorphosis. The red stars Betelgeux
and Antares, which are at present only 12th and 16th in order of
brightness, would flash out as the two brightest stars in the sky,
while Sirius, at present the brightest of all, would sink to third
place. A star in the very undistinguished constellation of Hercules
would be seen as the sixth brightest star in the sky. It is the star
α Herculis, at present outshone by about 250 stars. As a consequence
of its extremely low temperature of 2650 degrees, this star emits its
radiation almost entirely in the form of invisible heat. For instance
it emits 60 times as much heat as the blue star η Aurigae, whose
temperature is about 20,000 degrees, but only four-fifths as much light.
Allowances for invisible radiation have been made in all the
calculations referred to in the present book, although it has not been
thought necessary continually to restate this.
STELLAR DIAMETERS. It is easy to measure the diameter of a planet,
because this appears in the telescope as a disc of finite size. But
the stars are too remote for their diameters to be measured in the
same way. No star appears larger in the sky than a pin-head held at a
distance of six miles, and no telescope yet built can shew an object of
this size as a disc. All stars, even the nearest and largest, appear as
mere points of light[24], so that their diameters can only be measured
by roundabout methods.
[24] The large round images of stars shewn in the frontispiece result
merely from over-exposure, and have nothing to do with the sizes of the
stars.
When a star’s distance is known, we can tell its luminosity from its
apparent brightness. From this, after allowing for invisible radiation,
we can deduce the star’s total outpouring of energy—so many million
million million million horse-power. We also know its outpouring
of energy per square inch of surface, because this depends only on
its surface-temperature which we deduce directly from spectroscopic
observation. Knowing these two data, it is a mere matter of simple
division to calculate the number of square inches which make up the
star’s surface, and this immediately tells us the diameter of the star.
The diameters of exceptionally large stars may be measured more
directly by an instrument known as the interferometer. When we focus a
telescope on a star we do not, strictly speaking, see only a point of
light, but a point of light surrounded by a rather elaborate system of
rings of alternating light and darkness, called a diffraction pattern.
It might be thought that the size of these rings would tell us the
size of the star, but the two have nothing to do with one another.
The rings represent a mere instrumental defect, their size depending
solely on the size and optical arrangement of the telescope. Following
a method suggested by Fizeau in 1868, Professor Michelson has shewn
how even this defect can be turned to useful ends, and by its aid has
produced what is perhaps the most ingenious and sensational instrument
in the service of modern astronomy—the interferometer. In effect, this
instrument superposes two separate diffraction patterns of the same
star, and sets one off against the other in such a way as to disclose
the size of star producing them. The diameters of a few of the largest
stars have been measured in this way, so that we may say that we
know their sizes from direct observation. In every case the directly
measured diameter agrees fairly well, although not perfectly, with that
calculated indirectly in the way already explained. The discrepancies,
which are not serious, appear to result from red stars not being
accurate “full radiators” in the sense explained on p. 123.
The interferometer method is only available for the largest stars, but
at the extreme other end of the scale the theory of relativity has
come to the rescue. Einstein shewed it to be a necessary consequence
of his theory of relativity that the spectrum of a star should be
shifted towards the red end by an amount depending on both the weight
and the diameter of the star. If, then, a star’s weight is known, the
observed spectral shift ought immediately to tell us its diameter. This
spectral shift has recently been observed in the light received from
the companion of Sirius, and measurements of its amount lead to a value
for the star’s diameter which agrees exactly with that calculated from
its luminosity. Thus at both ends of the scale, for the very largest as
well as for the very smallest of stars, direct observation confirms the
values calculated for the diameters of the stars.
We may accordingly feel every confidence in the calculated diameters
of all stars, even when these cannot be checked by direct measurement.
Indeed a discrepancy between the true and calculated diameters could
only arise in one way. The diameters are calculated on the assumption
that the stars emit their full temperature-radiation. If the stars had
been partially transparent like the nebulae, or solid bodies like the
moon, this assumption would have been false, and its falsity would
at once have been shewn by discordances between the calculated and
measured diameters of the stars. The fact that no large discordances
appear suggests that the stars emit nearly full temperature-radiation
throughout the whole range of size from the largest to the smallest.
THE VARIETY OF STARS
Observation shews that the physical characteristics of the stars
vary enormously, so that it is easy, as we shall soon see, to tell
a sensational story by contrasting extremes, setting the brightest
against the dimmest, the biggest against the smallest, and so on. This
would, however, give a very unfair impression of the inhabitants of the
sky; it would be like judging a nation from the giants and dwarfs, the
strong men and the fasting men, seen inside the showman’s tent.
We shall obtain a more balanced impression of the actual degree of
diversity shewn by the stars as a whole if we consider the physical
states of those stars which are nearest the sun. By taking these
precisely in the order in which they come, we avoid any suspicion of
going out of our way to introduce stars merely because they are bizarre
or exceptional. The small group of stars obtained in this way may be
expected to form a fair sample of the stars in the sky, although of
course it will not be a large enough sample to include extremes. We
need not discuss the sun itself in detail because it will figure as our
standard star, with reference to which all comparisons are made.
_The System of α Centauri._ This system consists of three constituent
stars, which are believed to be our three nearest neighbours in space.
The brightest, α Centauri _A_, is very similar to the sun. It is of
the same colour and spectral type, but weighs 14 per cent. more and is
about 12 per cent. more luminous. Being of the same colour as the sun,
it emits the same amount of radiation per square inch. Thus its 12 per
cent. greater luminosity shews that it must have a surface 12 per cent.
greater, and therefore a diameter 6 per cent. greater, than the sun.
The second constituent, α Centauri _B_, is considerably redder than
the sun, its surface-temperature being only about 4400 degrees against
the sun’s 6000 degrees or so. It has 97 per cent. of the sun’s weight,
but only about a third of its luminosity. Yet, as a consequence of
its low temperature, it needs 50 per cent. more area than the sun to
discharge a third of the sun’s radiation; this makes its diameter 22
per cent. greater than that of the sun. α Centauri _A_ and α Centauri
_B_ together form a visual binary, the two components revolving about
one another in a period of 79 years.
Neither of these two constituents is very dissimilar from the sun, but
the third star of the system, Proxima Centauri, is of an altogether
different type. It is red in colour, with a surface-temperature of
only about 3000 degrees. It is exceedingly dim, emitting only a
ten-thousandth part as much light as the sun, and has only a fourteenth
part of the sun’s diameter. Its weight is unknown.
[Illustration: Fig. 17. The System of α Centauri, with the Sun for
comparison.]
The sizes of the three stars of this system, with that of the sun for
comparison, are shewn in fig. 17.
_Munich 15040._ This is a single faint star about which little is
known. Its surface is red, with a temperature probably little above
2500 degrees, and it emits only ¹/₂₅₀₀th of the light of the sun.
_Wolf 359._ This is the faintest star yet discovered, but beyond this
very little is known about it. It is red in colour and emits only about
1/50,000th of the light of the sun.
_Lalande 21185._ Another faint red star, emitting ¹/₂₀₀th of the light
of the sun.
_The System of Sirius._ This consists of two very dissimilar stars,
there being some suspicion that a third also may exist.
The principal star, Sirius _A_, which appears as the brightest
star in the sky (the Dog-star), is white in colour and has a
surface-temperature of about 11,000 degrees. As this is nearly twice
the sun’s temperature, Sirius _A_ emits nearly 16 times as much
radiation per square inch as the sun. Its luminosity is about 26
times that of the sun, and this requires the star’s diameter to be 58
per cent. greater than that of the sun. It has nearly four times the
sun’s volume, but only 2·45 times its weight, so that matter is not
as closely packed in Sirius _A_ as in the sun. An average cubic metre
contains 1·42 ton in the sun, but only 0·93 ton in Sirius _A_.
[Illustration: Fig. 18. The System of Sirius, with the Sun for
comparison.]
The faint companion Sirius _B_ is one of the most interesting stars
in the sky. It is of nearly the same colour and spectral type as
Sirius _A_, but emits only a ten-thousandth part as much light. After
allowing for the slight difference in surface-temperature, we find
that its surface is only ¹/₂₅₀₀th, and its diameter ¹/₅₀th of that of
Sirius _A_. Yet Sirius _A_ weighs only three times as much as Sirius
_B_, although having 125,000 times its volume. It is not Sirius _A_
but Sirius _B_ that is remarkable; the average density of matter in
the latter is about 60,000 times that of water, the average cubic inch
containing nearly a ton of matter. Fig. 18 shews the sizes of the two
components of Sirius drawn to the same scale as fig. 17.
_B.D. 12° 4523 and Innes 11 h. 12 m., 57·2°._ Two stars, as to the
physical state of which nothing is known, except that they are very
faint, emitting ¹/₁₄₀₀th and 1/10,000th of the sun’s light respectively.
_Cordoba 5 h. 343 and τ Ceti._ Two faint stars, both of reddish colour,
emitting ¹/₆₀₀th and a third of the sun’s light respectively.
[Illustration: Fig. 19. The System of Procyon, with the Sun for
comparison.]
_The System of Procyon._ This is a binary system, similar in many
respects to Sirius. The main star, Procyon _A_, is of the same general
type as the sun, but weighs 24 per cent. more, and emits 5½ times as
much light. Its surface-temperature is about 7000 degrees, and its
diameter is 1·80 times that of the sun.
The faint companion, Procyon _B_, is so faint that nothing is known as
to its physical condition except that it emits only 1/30,000th of the
light of the sun. Its weight is 39 per cent. of the sun’s weight.
Fig. 19 shews the sizes of the two components of Procyon on the same
scale as before.
Next in order, as we recede from the sun, come eight very
undistinguished stars, every one of which is redder and fainter than
the sun, none of them having a surface-temperature higher than 5000
degrees, and none of them emitting more than a quarter of the sun’s
light. After these we come to:
_The System of Kruger 60._ This is a binary system in which both
components are small, red and dim.
[Illustration: Fig. 20. The System of Kruger 60, with the Sun for
comparison.]
The brighter component, Kruger 60 _A_, has a surface-temperature of
3200 degrees, and emits ¹/₄₀₀th of the light of the sun. Its diameter
is a third, and its weight a quarter of the sun’s; so that its
substance must be packed about 7 times as closely as that of the sun.
The fainter component, Kruger 60 _B_, has a similar surface-temperature
but emits only 1/14,000th of the sun’s light. Its diameter is a sixth,
and its weight a fifth of the sun’s; so that its substance must be
packed about 40 times as closely as that of the sun. The system is
illustrated in fig. 20.
_van Maanen’s star._ Another very faint star, which has the high
surface-temperature of 7000 degrees. Notwithstanding this, it only
emits ¹/₆₀₀₀th of the sun’s light. Consequently its diameter is only
about ¹/₁₁₀th of the sun’s, the star being if anything smaller than the
earth. Its weight is unknown, but its substance is in all probability
packed even more closely than in Sirius _B_.
The discussion of this sample of stars suggests that the majority of
stars in space are smaller, cooler and fainter than the sun. Stars
exist which are far brighter than the sun, but they are exceptional,
the average star in the sky being a small dull dim affair in comparison
with our sun.
With this sample of the average population of the sky before us, we may
proceed to discuss the various characteristics of stars in a systematic
way, without fearing to mention extremes. Let us begin with their
weights.
STELLAR WEIGHTS. The two stars of smallest known weight in the whole
sky are the faint constituent of Kruger 60, just discussed, and the
faintest constituent of the triple system ο₂ Eridani, each of which has
a fifth of the sun’s weight. But the stars whose weights are known are
so few that there can be no justification for supposing these to be the
smallest weights which occur in the whole universe of stars. A general
survey of the situation, on lines to be indicated later (p. 281),
suggests that there may be many stars of still smaller weight, but
that very few are likely to have weights which are enormously smaller.
Probably very few stars weigh as little as a tenth of the sun’s weight.
The vast majority of stars have weights intermediate between this
and ten times the sun’s weight. Stars which weigh even three times
as much as the sun are rare, those which weigh ten times as much are
very rare, probably only about one star in 100,000 having ten times
the weight of the sun. Even higher weights undoubtedly occur—we have
already mentioned Plaskett’s star, whose two constituents have more
than 75 and 63 times the sun’s weight respectively, and the quadruple
system 27 Canis Majoris which to all appearances weighs 940 times as
much as the sun—but such instances are very, very unusual. We may say
that as a general rule the weights of the stars lie within the range
of from a tenth to ten times the sun’s weight, and we shall find that
stars differ less in their weights than in most of their other physical
characteristics.
LUMINOSITY. A far greater range is shewn, for instance, in the
luminosities of the stars—in their candle-powers measured in terms of
the sun’s candle-power as unity. The most luminous star known is _S_
Doradus, already mentioned, with 300,000 times the luminosity of the
sun, while the least luminous is Wolf 359 with only a fifty-thousandth
part of the luminosity of the sun. The range of stellar luminosities,
as of stellar weights, extends about equally on the two sides of the
sun, so that the sun is rather a medium star in respect both of weight
and luminosity. It is medium in the sense of being about half-way
between extremes, but we have seen that there are many more stars below
than above it.
In comparison with the very moderate range of stellar weights, the
range of luminosity is enormous; _S_ Doradus is 15,000,000,000 times
as luminous as Wolf 359. If _S_ Doradus is a lighthouse, Wolf 359 is
something less than a firefly, the sun being an ordinary candle. If the
sun suddenly started to emit as much light and heat as _S_ Doradus, the
temperature of the earth and everything on it would run up to about
7000 degrees, so that both we and the solid earth would disappear into
a cloud of vapour. On the other hand, if the sun’s emission of light
and heat were suddenly to sink to that of Wolf 359, people at the
earth’s equator would find that their new sun only gave as much light
and heat at mid-day as a coal fire a mile away; we should all be frozen
solid, while the earth’s atmosphere would surround us in the form of
an ocean of liquid air. So far as we know, there is no possibility of
the sun suddenly beginning to behave like _S_ Doradus, but we shall
see later that the possibility of its behaving like Wolf 359 is not
altogether a visionary dream.
SURFACE-TEMPERATURE AND RADIATION. Sirius has been found to have the
highest surface-temperature of all the stars near the sun; it is about
11,000 degrees, or nearly double that of the sun. Going further afield,
we find many stars with far higher surface-temperatures. For instance,
Plaskett’s star is credited with a temperature of 28,000 degrees,
although it must be admitted that a substantial element of uncertainty
enters into all estimates of very high stellar temperatures.
At the other extreme, stellar temperatures ranging down to about 2500
degrees are comparatively common. The lowest temperatures of all
are confined to variable stars of a very special type (long period
variables) in which the light-variation is accompanied by, and indeed
mainly arises from, a variation in the temperature of the star’s
surface. The temperature of these stars when at the lowest, ranges
down to 1650 degrees, which is but little above the temperature of an
ordinary coal fire. In many of them, the temperature varies through a
large range, but it never sinks so low that the star becomes completely
invisible. Thus there is a range of temperature below about 2500
degrees which no star is known to occupy, except for the long-period
variables which only enter it at intervals. This would seem to suggest
that the number of absolutely dark stars in the sky is relatively
small. Other lines of evidence lead to the same conclusion. If a
star ceased to shine, its gravitational pull would still betray its
existence. Although we could not detect a single dark star in this way,
we could detect a multitude. If a great proportion of stars were dark,
we should probably suspect the existence of the dark stars from their
effects on the motions of the remainder, so that general gravitational
considerations preclude the possibility of there being a great number
of dark stars.
So far as our present knowledge goes, the temperature of stellar
surfaces ranges, in the main, from about 30,000 degrees down to about
2500, the lower limit being extended to about 1650 for long-period
variables at their lowest temperatures.
Apart from the long-period variables, this is only a 12 to 1 range,
so that the temperatures of the stars are more uniform than either
their luminosities or their weights. We must, however, remember that
a star’s radiation per square inch is far more fundamental than its
surface-temperature, and that a 12 to 1 range in the latter involves a
range of over 20,000 to 1 in the former. If we include the long-period
variables, there is a range of about 110,000 to 1 in the emission of
radiation per square inch.
In terms of horse-power, the sun emits energy at the rate of 50
horse-power per square inch, a star with a surface-temperature of 1650
degrees emits only 0·35 horse-power per square inch, while Plaskett’s
star, with a surface-temperature of 28,000 degrees, emits about 28,000
horse-power per square inch. In plain English, each square inch of this
last star pours out enough energy to keep an Atlantic liner going at
full speed, hour after hour, and century after century. And the energy
emitted per square inch by the surfaces of various stars covers the
whole range from the power of a liner to that of a man in a row-boat.
SIZE. The four stars of largest known diameter are the following:
+-------------------+------------+-----------+
| Star |Diameter in | Diameter |
| |terms of sun| in miles |
+-------------------+------------+-----------+
|Antares | 450 |390,000,000|
|α Herculis | about 400 |346,000,000|
|ο Ceti (at max.) | 300 |260,000,000|
|Betelgeux (average)| 250 |216,000,000|
+-------------------+------------+-----------+
All these diameters have been measured directly by the interferometer.
On the scale used in figs. 17 to 20, in which the sun is about the size
of a sixpence, the circle necessary to represent ο Ceti would be as
large as the floor of a good-sized room, while the second star of the
system (for ο Ceti is binary) would be the size of a grain of sand. We
may obtain some idea of the immense size of these stars by noticing
that every one of their diameters is larger than the diameter of the
earth’s orbit, so that if the sun were to expand to the size of any one
of them we should find ourselves inside it.
These stars must be exceedingly tenuous. Antares, for instance,
occupies 90,000,000 times as much space as the sun, so that if its
substance were as closely packed, it would weigh 90,000,000 times
as much as the sun. Yet, in actual fact, it probably has only about
40 or 50 times the sun’s weight, the difference between this number
and 90,000,000 arising from the difference between the densities of
Antares and the sun. On the average a ton of matter in the sun occupies
considerably less than a cubic yard; in Antares it occupies rather more
space than the interior of Saint Paul’s Cathedral. Yet a detailed study
of stellar interiors shews that we can attach but little meaning to an
average of this sort. It is quite likely that matter at the centre of
Antares is packed nearly (but not quite) as closely as matter at the
centre of the sun (p. 291 below). The huge size of Antares is probably
due mainly to an enormously extended atmosphere of very tenuous gas,
and there is not much point in striking an average between this and the
compact matter at the centre of the star.
The mysterious objects known as planetary nebulae, of which examples
are shewn in Plate II (p. 28), ought perhaps to be regarded as
stars of still larger diameter. At the centre of each of these the
telescope discloses a comparatively faint star with an extremely high
surface-temperature. Surrounding this is the nebulosity from which
these objects derive their somewhat unfortunate name. This is in all
probability merely an atmosphere of even greater extent than that
surrounding the four stars of our table. Van Maanen estimates the
diameter of the nebulosity of the Ring Nebula in Lyra (fig. 2 of Plate
II) to be 570 times that of the earth’s orbit, or about 106,000,000,000
miles. This nebulosity, however, differs from the atmosphere of an
ordinary star in being very nearly transparent; we can see through
106,000,000,000 miles of the Ring Nebula but can only see a few tens or
hundreds of miles into an ordinary star.
At the other extreme of size, the smallest known star, van Maanen’s
star (p. 268) is just about as large as the earth; over a million such
stars could be packed inside the sun and still leave room to spare. And
yet its weight is in all probability comparable, not with that of the
earth, but with that of the sun; at a guess it may have about a fifth
of the sun’s weight. To pack a fifth of the sun’s substance inside
a globe of the size of the earth, the average ton of matter must be
packed into a space of about the size of a small cherry—ten tons or so
to the cubic inch. The solidity of the earth suggests that its atoms
must be packed pretty closely together, but the atoms in van Maanen’s
star must be packed 66,000 times more closely.
How is it done? As we shall shortly see, there is only one possible
answer. The atom consists mostly of emptiness—we compared the carbon
atom to six wasps buzzing about in Waterloo Station. Let us break the
atom up into its constituent parts, pack these together as closely
as they will go, and we see the way in which matter is packed in
van Maanen’s star. Six wasps which can roam throughout the whole of
Waterloo Station can nevertheless be packed inside a very small box.
GIANTS AND DWARFS. There is a continuous series of stars between the
limits of weight we have mentioned, and the same is true of the limits
of temperature (and so also of colour) and of size.
[Illustration: Fig. 21. Stars of different colours arranged in order of
luminosity.]
Within these specified limits, I can find you a star of any weight or
of any colour or of any size you like. But this does not mean that you
may specify the weight _and_ colour _and_ size of the star you want,
and that I will undertake to find it for you; if the weight is right
the colour may be wrong, and so on. For instance, if you ask for a red
star I can find you a very heavy one or a very light one, but it is no
good your asking for one of intermediate weight. So far as we know, red
stars of intermediate weight simply do not exist. The same is true as
regards size—there are no red stars of intermediate size. Hertzsprung
noticed in 1905 that the red stars could be divided sharply into two
distinct classes characterised by large and small size—he called them
Giants and Dwarfs. Russell, studying the question further in 1913,
confirmed Hertzsprung’s earlier conclusions, and shewed that the
giant-dwarf division extended to stars of other colours than red.
Imagine that we have a series of coloured ladders, one for each colour
of star—red, orange, etc. Take all the red stars and stand them (in
imagination) on the different rungs of the red ladder. Do not merely
place them on at random; arrange them in order of their luminosities,
placing those of highest luminosity upper-most. Further let several
stars stand on the same rung if their luminosities are about equal. To
make the arrangement definite, let each rung of the ladder represent
5 times higher luminosity than the rung immediately below it, so that
each rung has a definite luminosity associated with it[25].
[25] For purely practical reasons the height is not taken proportional
to the luminosity but to its logarithm; without some such device
as this it would be impossible to represent the range of more than
1,000,000 to 1 in the observed luminosities of red stars.
With this agreement we are now ready to proceed. We take our red stars
and place each on the appropriate rung of the red ladder, and so on for
each other colour. The result is shewn diagrammatically in fig. 21, the
different stars being represented by crosses.
The red stars will be found to lie as on the right of the diagram,
Hertzsprung’s division into giants and dwarfs being very clearly
marked. The orange stars lie as on the next ladder to the left; as
Russell found, the division again appears, but is less marked.
THE RUSSELL DIAGRAM. Let us make ladder diagrams of this kind for each
colour of star, and put them side by side in their proper order, so as
to represent stars of all possible colours. We obtain a diagram of the
kind shewn in fig. 22. This type of diagram was introduced by Russell
in 1913, and is now generally known as a Russell diagram.
The letters at the top of the diagram represent spectral types
of stars, because these provide a better and more exact working
classification than the names of colours. The colours which
approximately correspond to the various spectral types are indicated at
the bottom of the diagram.
Only a very few sample stars are shewn, but all known stars are found
to be concentrated around the positions of these few typical stars.
Broadly speaking, there are two distinct and disconnected regions which
are occupied by stars. First, and most important, is a region shaped
rather like a reversed γ: the central line of this region is marked in
by a continuous thick line, following a determination of its position
by Redman. Second, there is a smaller region near the left-hand bottom
corner of the diagram. The stars which occupy this region are very
faint, and have far higher surface-temperatures than other stars of
similar luminosity.
We have already seen how a star’s diameter can be calculated from
its surface-temperature and luminosity. This amounts to the same
thing as saying that two stars which occupy the same position in the
Russell diagram must have the same diameter. Thus there is a definite
diameter associated with each point in the diagram, and we can map out
stellar diameters in the diagram, just as we can map out heights above
sea-level on a geographical map, by a system of “contour lines.” In
the present case the “contour lines” prove to be a system of almost
parallel curves. These lie roughly as shewn by the broken lines in fig.
22, all stars lying on any one line having the same diameter.
[Illustration: Fig. 22. The Russell diagram.]
This diagram throws a flood of light on the general question of stellar
diameters. We see at once that stars of the biggest diameters—100
times the sun’s diameter or more—must necessarily be red stars of high
luminosity. And in actual fact the stars of large diameter shewn in the
table on p. 272 are all red and have very high luminosities; they are
red giants.
The majority of the stars in the sky lie in the belt which runs across
the diagram of fig. 22 from top left-hand to bottom right-hand. This is
known as the “main-sequence.” The position of this band with reference
to the “contour lines” of diameters shews that main-sequence stars
are of moderate diameters. The brightest of all may have twenty times
the diameter of the sun, while the faintest may have only about a
twentieth of the sun’s diameter, but they all have diameters which are
at least comparable with that of the sun. The sample of stars from near
the sun, which we have already discussed, provides many instances of
main-sequence stars; we have, in order of decreasing luminosity:
+--------------+----------+-------------+
| Star |Luminosity|Diameter (in |
| | |terms of sun)|
+--------------+----------+-------------+
|Sirius _A_ | 26·3 | 1·58 |
|Procyon _A_ | 5·5 | 1·80 |
|α Centauri _A_| 1·12 | 1·07 |
|Sun | 1·00 | 1·00 |
|α Centauri _B_| 0·32 | 1·22 |
|τ Ceti | 0·32 | 0·95 |
|ε Indi | 0·15 | 0·82 |
|Kruger 60 _A_ | 0·0026 | 0·33 |
| ” _B_ | 0·0007 | 0·17 |
|Wolf 359 | 0·00002 | 0·03 |
+--------------+----------+-------------+
This table shews clearly how stellar luminosity and diameter decrease
together as we pass down the main-sequence.
The remaining group of stars in fig. 22, those in the bottom left-hand
corner, are generally known as “white dwarfs.” Their position in the
diagram shews that their diameters must be excessively small. The
vicinity of the sun provides three examples of this class of star, as
shewn in the following table:
+-------------------+-------------+-----------------+
| Star | Luminosity | Diameter (in |
| | | terms of sun) |
+-------------------+-------------+-----------------+
| Sirius _B_ | 0·0026 | 0·03 |
| ο₂ Eridani _B_ | 0·0031 | 0·018 |
| van Maanen’s star | 0·00016 | 0·009 |
+-------------------+-------------+-----------------+
In addition to these, the faint companion of ο Ceti is certainly
a white dwarf, while Procyon _B_ may be. These are the only known
examples of white dwarfs, but the extreme faintness of these stars
makes them very difficult of detection, so that it is quite likely that
they are fairly frequent objects in space.
In the table on p. 279, the main-sequence stars were intended to be
arranged in the order of luminosity, but this happens also to be the
order of weights. The weights of three of the stars are unknown; those
of the remainder are as follows:
+-------------------+-------------+-----------------+
| Star | Luminosity | Weight (in |
| | | terms of sun) |
+-------------------+-------------+-----------------+
| Sirius _A_ | 26·3 | 2·45 |
| Procyon _A_ | 5·5 | 1·24 |
| α Centauri _A_ | 1·12 | 1·14 |
| Sun | 1·00 | 1·00 |
| α Centauri _B_ | 0·32 | 0·97 |
| Kruger 60 _A_ | 0·0026 | 0·25 |
| ” _B_ | 0·0007 | 0·20 |
+-------------------+-------------+-----------------+
Like the luminosities, the weights fall off steadily as we pass down
the main-sequence, although, as already remarked, weight falls far less
rapidly than luminosity.
The only stars whose weights can be measured directly are the
components of binary systems, and these are relatively few in number.
Seares found, however, that the weights of binary systems conformed to
the law of equipartition of energy already explained in Chapter III,
so that it is highly probable that other stars which are not binary
also conform, for it is difficult to imagine any reason why binary
systems should attain to a state of equipartition sooner than other
stars. It will be remembered that this state is defined by a purely
statistical law connecting the weights and speeds of motion of stars,
so that the fact that a system of stars has attained this state can
give no information as to the weight of an individual star whose speed
is known, but it makes it possible to determine the average weight of
any group of stars in terms of their average speeds of motion. In this
way Seares has determined the average weights of stars of different
assigned luminosities and spectral types—in other words, the average
weights of the stars represented at the various points in the diagram
of fig. 22. The results he obtained are shewn by the thick curved lines
in fig. 23. The arrangement of these curves confirms the inference we
have drawn from a few selected stars; the weight of main-sequence stars
falls off steadily as we pass down the sequence from high luminosity to
low.
These curved lines specify the average weight of the stars represented
at each point in the Russell diagram, and the diameters are already
known from fig. 22. From these two data the mean density of the star
can of course be calculated. The mean densities as calculated by
Seares are shewn by the broken lines in fig. 23.
[Illustration: Fig. 23. Stellar weights and densities in the Russell
diagram, according to Seares.]
This completes our collection of observational material. We now
turn to the far more difficult problem of discussing what it all
means. Here we leave the firm ground of ascertained fact, to enter
the shadowy morasses of conjecture, hypothesis and speculation. The
questions we shall discuss are some of the most interesting in the
whole of astronomy, to which it must be admitted that science has so
far obtained only lamentably dusty answers. The reader who is hot for
certainties may prefer to read something other than the remainder of
the present chapter.
THE PHYSICAL CONDITION OF THE STARS
The foregoing collection of observational data has provided abundant
proof that stars to certain specifications do not exist at all. To put
the same thing in another way, there are certain regions in the Russell
diagram which are wholly unoccupied by stars.
To take the most conspicuous instance of all, there are no stars at
all to the left of the main-sequence in the Russell diagram (fig. 22),
until we come to the quite detached group of white dwarfs. Why are
there no stars in intermediate conditions? Why, to make the example
still more precise, does no star exist of the same colour as Sirius but
with only a tenth of its luminosity? Why do we have to go down to the
white dwarf ο₂ Eridani _B_, with a luminosity of only a ten-thousandth
of that of Sirius, before we can find a star to match Sirius in colour?
A hypothesis which occurs naturally to the mind is that the
main-sequence stars and the white dwarfs may form distinct groups
because they are of entirely different ages—they may represent distinct
creations. As stars age they decrease in weight and in luminosity, so
that it is natural to interpret the small weights and extremely low
luminosity of the white dwarfs as evidence of an age far greater than
the age of the normal main-sequence stars. Yet this hypothesis does not
appear to be tenable.
With the single exception of van Maanen’s star, every star which is
either known or suspected to be a white dwarf forms one component of a
binary system, and in every case its companion is a main-sequence star
or (in the case of ο Ceti) a red giant. We have already seen how rare
it is for two stars to approach near to one another in space. It must
be an almost inconceivably rare event for two stars, originally moving
as independent bodies, so to meet in their random wanderings, that the
big one “captures” the little one, and they henceforth journey together
through space. For it can be shewn that, for such an event to occur,
something more than a close approach is needed; a close approach must
take place in the presence of yet a third star, so that no fewer than
three stars must chance to come near one another simultaneously in
their wanderings through the vast emptinesses of space. It is almost
inconceivable that this should happen in a single instance, but it is
straining the probabilities too far to suppose that it has happened in
the case of every single known white dwarf but one. Thus we have to
suppose that the white dwarfs and their more normal companions have
been together since birth, and so were born at the same time out of the
same nebula.
The difference between white dwarfs and main-sequence stars cannot,
then, be a mere difference of age, and it would seem as though there
must be some physical reason militating against the existence of stars
in intermediate conditions. Taking a more general view of the question,
we are led to investigate whether the absence of stars built to certain
specifications can be attributed to such stars needing physical
properties which nature cannot provide. This leads directly into the
general question of the structure and mechanism of the stars.
THE INTERNAL CONSTITUTION OF THE STARS
Most investigations on the structure of the stars have proceeded on
the supposition that their interiors are gaseous throughout. Without
accepting this supposition as final truth, we may adopt it for the
moment, for the purely opportunist reason that it provides the most
convenient line of approach to an excessively difficult problem.
A mathematical theorem, generally known as Poincaré’s theorem, proves
to be of the utmost service in discussing the internal state of a
gaseous star. We have seen how Helmholtz thought that the energy
of the sun’s radiation might come from the sun’s contraction, each
layer falling in upon the next inner layer as the latter shrunk, and
transforming the energy set free by its fall into heat and light. It is
easy to estimate how much energy would be set free by a contraction of
this kind. For instance, Lord Kelvin calculated that the contraction
of the sun, as it shrunk from infinite size to its present diameter
of 865,000 miles, would liberate about as much energy as the sun now
radiates in 50 million years. In terms of ergs, the sun’s shrinkage
would liberate 6 × 10⁴⁸ ergs of energy.
Poincaré’s theorem states that the total energy of motion of all the
molecules in any gaseous star whatever is equal to precisely half the
total energy which the star would have liberated in shrinking down to
its present size. The theorem is true quite independently of whether
the star ever has so shrunk or not: nothing is involved but the present
state of the star.
One interesting consequence is that the further a gaseous star
shrinks, the hotter it becomes; if a star shrinks to half its present
size, the total energy set free by its shrinkage from infinite size
is doubled, so that the total energy of motion of its molecules is
doubled, and therefore its average temperature is doubled. This is a
special case of what is generally known as Lane’s law.
Let us go on with our calculation for the special case of the sun.
Poincaré’s theorem tells us that, if the sun is gaseous, the total
energy of motion of all its molecules is 3 × 10⁴⁸ ergs. The next thing
we want to know is how many molecules there are in the sun. The sun’s
weight is 2 × 10³³ grammes, but how many molecules are there to a
gramme? The answer of course depends on the type of molecule concerned;
there are 3 × 10²³ molecules in a gramme of hydrogen, 2 × 10²² in a
gramme of air and only 2·5 × 10²¹ in a gramme of uranium.
If we suppose the sun to be made of air, it must consist of 4 × 10⁵⁵
molecules, so that the average energy of motion of each molecule must
be 7·5 × 10⁻⁸ ergs, and this represents an _average temperature_,
for the sun’s interior, of 375 million degrees. In 1907 Emden, by a
different calculation, found that if the sun were made of air, the
_temperature at its centre_ would be 455 million degrees. Apart from
details, it is clear that the interior temperature of a sun made of air
would be one of hundreds of millions of degrees.
So far all study of stellar interiors had proceeded on the supposition
that the stars were formed of complete atoms or even molecules. In
1917, I made a simple calculation, of the type already explained on p.
141, and found that the quanta of radiation which fly about at such
temperatures would be energetic enough not merely to break up the
molecules of air into atoms, but also to strip all, or nearly all, of
the electrons from the atoms. At such temperatures each molecule of
air would break up into its constituent nuclei and electrons just as
surely as, on a hot day, a lump of ice breaks up into its constituent
molecules. The electric forces which, in quieter surroundings, would
unite the electrons and nuclei, first into atoms and then into complete
molecules, find themselves powerless against the incessant hail of
rapidly moving projectiles and the shattering blows of quanta of
high energy; it would be like trying to build a house of cards in
a hurricane. A sun consisting of molecules of air proves to be an
inconsistency, a contradiction; our hypothesis has defeated itself, and
we must start again from the beginning.
We may start wherever we like, but the conclusion which we must finally
reach is that, no matter what kind of molecules the sun consists of,
the heat at the sun’s centre will break them up, either completely or
nearly so, into their constituent nuclei and electrons. The same is
true for all other stars, and this introduces an extreme simplification
into the problem of the interior constitution of the stars. We cannot
say how many complete molecules there are to a gramme without knowing
the nature of the molecules, but once let these molecules be broken up
into their constituent parts, and we know at once the total number of
constituent parts, nuclei and electrons, which go to make up a gramme.
For, the atomic weights of all elements except hydrogen are nearly
double their atomic numbers (p. 110). Hence, as Eddington first pointed
out, the total number of nuclei and protons in a fully broken-up atom
of any substance except hydrogen must be equal to about half the
atomic weight of the atom. We may probably disregard the possibility
of a star consisting to any great extent of hydrogen. If so, the
number of constituent parts in a gramme of fully broken-up stellar
matter must be about 3 × 10²¹, regardless of the type of molecule from
which these parts originate. And when we know the total number of such
parts in any star, it becomes easy to calculate the temperature of the
star’s interior, either from the theorem of Poincaré just mentioned or
otherwise. The temperature will be the same as though the star were
made of unbroken molecules of hydrogen.
Emden calculated in 1907 that the central temperature of a sun of
this kind would be about 31,500,000 degrees. Later and more refined
calculations by Eddington led to an almost identical temperature, but
some still later calculations of my own give the substantially higher
figure of 55,000,000 degrees. There is no need for the moment to
discuss which of these figures is nearest the truth. Their diversity
will indicate what kind of degree of uncertainty attaches to all
calculations of this type.
It is easy to see the physical necessity for this high temperature.
The heat which flows away from the sun’s surface must first have been
brought there from its interior. Heat only flows from a hotter to
a cooler place, and a vigorous flow of heat is evidence of a steep
temperature gradient. The temperature must rise sharply as we pass
from the sun’s surface towards its centre, and this rise, continued
along the whole 433,000 miles to the centre, must result in a very high
temperature indeed being reached there.
The calculated central temperature of 30 to 60 million degrees so
far transcends our experience that it is difficult to realise what
it means. Let us, in imagination, keep a cubic millimetre of ordinary
matter—a piece the size of an ordinary pin-head—at a temperature of
50,000,000 degrees, the approximate temperature at the centre of the
sun. Incredible though it may seem, merely to maintain this pin-head
of matter at such a temperature—i.e. to replenish the energy it loses
by radiation from its six faces—will need all the energy generated by
an engine of three thousand million million horse-power; the pin-head
of matter would emit enough heat to kill anyone who ventured within a
thousand miles of it.
High though this temperature is, calculations shew that it would not
suffice to break up the stellar molecules completely. It would strip
the atoms of all their electrons down to the _K_-rings (p. 134), but
these would remain intact. It needs temperatures even higher than those
we are now considering to strip the _K_-ring electrons from the nucleus
of an atom. This result is true for the whole range, from about 30 to
60 million degrees, within which the temperature of the sun’s centre is
at all likely to lie, and it is true almost independently of the atomic
weight or atomic number of the atoms of which we suppose the sun to be
built.
Thus if the sun is wholly gaseous, its central parts must consist of a
collection of atoms stripped down to their _K_-rings, but not beyond,
flying about independently like the molecules of a gas, and with them,
also flying about like the molecules of a gas, all the stripped-off
electrons which originally formed the _L_-ring, the _M_-ring, etc., of
the atoms, the whole being at a temperature of somewhere between 30
and 60 million degrees. As we pass outwards towards the sun’s surface
we come to lower temperatures, at which the atoms are less completely
broken up. Finally, close to the sun’s surface, we may meet atoms which
are completely formed except perhaps for one or two of their outermost
electrons. In the surfaces of the coolest stars of all, we even find
complete molecules, as, for example, the molecules of titanium oxide
and magnesium hydride, which shew themselves in the spectra of the red
stars.
When the internal constitution of other stars is investigated in
the same way, all main-sequence stars are found to have about the
same central temperatures as the sun. Moreover, this is not the only
property which they have in common. Fig. 23, which exhibits Seares’
calculations of mean stellar densities, shews that the mean densities
of main-sequence stars are all approximately the same, except for
comparatively small deviations at the two extremities.
The mean density of the sun is 1·4, which means that the average cubic
metre in the sun contains 1·4 ton of matter. At the sun’s centre, the
density is about 100 times this, so that a cubic metre there contains
about 140 tons of matter. For comparison, a cubic metre of lead
contains only about 11 tons. If all stars were built on the same model
as the sun, any two stars which had the same mean density would also
have equal densities at their centres. But in stars having several
times the weight of the sun, a new factor comes into play, namely
pressure of radiation—the pressure which radiation exerts in virtue of
the weight it carries about with it. In most stars this pressure is
insignificant in comparison with the pressure produced by the impact
of material atoms and electrons, but in very massive stars it is large
enough to influence the structure of the star. It is to this that the
very massive stars whose diameters are tabulated on p. 272 owe their
abnormally large size. It is a general consequence of the disturbing
effects of radiation-pressure, that the weight of a very massive star
is far more concentrated in its central regions than that of a lighter
star, so that if a light and a massive star have the same average
density, the latter will have by far the higher density at its centre.
When this disturbing factor is allowed for, all stars in the upper part
of the main-sequence are found to have approximately the same densities
in their central regions, a density about equal to that at the centre
of the sun, which we may estimate at 140 tons to the cubic metre. And
we have already seen that the central regions of these stars have also
approximately the same temperatures as the centre of the sun, whence
it follows that their physical conditions are all substantially the
same. Thus the atoms in the central regions of all these stars must be
broken down to the same extent as the atoms in the central regions of
the sun. The _K_-rings of electrons survive intact, but the outer rings
are transformed into a hail of electrons flying about like independent
molecules.
With sufficient accuracy for our present purpose, all the stars on
the main-sequence, except perhaps those at its extreme lower end, may
be supposed to be in the same physical condition. On account of this
property, the main-sequence forms an admirable base-line from which to
carry out a survey of the Russell diagram in respect of the physical
conditions of stellar interiors.
Fig. 22 shews that a star to the right of the main-sequence has a
greater diameter than a main-sequence star of the same weight.
Consequently the energy it would emit in shrinking to its present
diameter is less, and hence its molecular energy of motion is less (by
Poincaré’s theorem). It follows that its internal temperatures are
lower, and its atoms are less completely broken up. Red giants such as
Antares are found only to have central temperatures of from one to five
million degrees, and their atoms probably retain intact not only their
_K_-rings of electrons, but also their _L_-rings and part at least of
their _M_-rings.
To the left of the main-sequence we come to a region in which stars,
if they occurred at all, would have shrunk further, and so would have
higher temperatures and more thoroughly broken atoms. Actually no
stars are encountered until we come to the white dwarfs. Calculation
shews that the central temperatures of these must be many hundreds of
millions of degrees at least, and that their atoms must be stripped
of electrons right down to the nuclei. Except for a small number of
atoms which may have escaped this general fate, the stellar matter must
consist of nuclei stripped absolutely bare, and of free electrons, all
flying independently through the star. The high densities of these
stars provide a convincing proof of the accuracy of this result. The
mean density of Sirius _B_ is certainly over 50,000, while that of van
Maanen’s star is probably over 300,000. There is no way in which matter
can be packed as closely as this, except that of stripping the atoms of
electrons right down to their bare nuclei.
The clearest general impression we can form of the Russell diagram in
terms of physical condition is probably obtained as follows:
We think first of two detached bands of stars, one, the white dwarf
group, formed by stars in which all the electrons are torn off the
atoms; and the other, the main-sequence, formed of stars in which the
atoms are still surrounded by their _K_-rings of electrons, while the
exterior rings have been torn off. Starting from about the middle of
the main-sequence is the spur branch leading up to the red giants, as
shewn in fig. 22. As we pass along this, the internal temperatures of
the stars decrease, so that the stellar atoms are less broken up than
in main-sequence stars. In the red giants at the extreme end, even
_M_-ring electrons may still remain.
RUSSELL’S HYPOTHESIS
Two entirely different explanations of this distribution of stars have
been suggested. In 1925 Russell put forward a theory which centred
primarily around the fact that the temperatures at the centres of the
main-sequence stars are all very nearly equal. Let us simplify the
situation for a moment by imagining it to be an ascertained fact that
the temperatures at the centres of _all_ stars are precisely the same,
say 32,000,000 degrees. If this were a sure fact, it would be natural
to conjecture that the stars had some sort of controlling mechanism
by which they continually adjusted their central temperatures to this
exact figure, so that if ever the temperature fell below 32,000,000
degrees the mechanism would come into play and raise the temperature
to precisely this amount, while if it increased to above this figure,
the mechanism would come into play and depress it. Such controlling
mechanisms are of course common in engineering practice; there are for
instance the safety-valves which keep the pressure in a boiler always
uniform, the Watts-governor which keeps an engine going always at the
same rate of speed, and the thermostat which keeps the temperature of a
room constant.
A mechanism is already known for raising the temperature at a star’s
centre. If a star is not generating any energy at all in its interior,
either by the annihilation of matter or otherwise, its emission of
radiation causes it to shrink, and this, as we have seen (p. 285),
causes its temperature to rise. Thus it is easy to keep a star’s
central temperature up to 32,000,000 degrees by arranging that no
energy shall be generated so long as the temperature at the centre is
below 32,000,000 degrees, and this is the main hypothesis on which
Russell’s theory is based. He supposes that no energy at all is
generated by matter at temperatures below 32,000,000 degrees, but that,
as soon as this temperature is reached, matter begins to annihilate
itself in sufficient quantity to provide for the radiation of a star.
The trouble with the theory is that it seems impossible to regulate the
temperature from the other end. A star whose central temperature is
below 32,000,000 degrees must be contracting without generating heat.
The contraction will not stop dead the moment the critical temperature
is attained; its momentum will carry it on until the central
temperature substantially exceeds 32,000,000 degrees. As soon as the
temperature seriously exceeds 32,000,000 degrees at the centre, that of
a substantial piece of the star will be 32,000,000 degrees or higher.
The annihilation of all this matter must produce a profusion of heat
which would raise the temperature of the star still further, resulting
in more and more annihilation of matter, until finally the whole
star disappeared in a flash of radiation. Indeed Russell’s theory
supposes that matter at 32,000,000 degrees is in a similar condition to
gunpowder at its flash point. Mathematical analysis then shews that a
star whose centre is at a temperature of 32,000,000 degrees would be in
the state of a keg of gunpowder with a spark at its centre, and—well,
“ohne hast, ohne rast” hardly describes the subsequent course of events.
Eddington has suggested that the stability of the stars might be
saved by imagining a time-lag between the instant at which matter
attained the critical temperature necessary for annihilation and the
instant at which this annihilation occurred. It has not yet been
proved that the proposed remedy could be made effective, but even if
it could, other difficulties remain. As the normal star inhabits the
main-sequence, Russell supposed it to be a property of normal matter
to annihilate itself at a temperature of about 32,000,000 degrees, the
supposedly uniform central temperature of all main-sequence stars.
It then became necessary to introduce further special assumptions to
explain the luminosity of white dwarfs and of stars on the red giant
spur line, whose centres are at temperatures very different from
32,000,000 degrees. He accordingly supposed that such stars contained
other types of matter which dissolved into radiation at temperatures
which were respectively higher and lower than 32,000,000 degrees. Even
if the stability difficulty could be overcome, this latter series
of assumptions seems to me to be so artificial as to compel the
abandonment of this interesting theory.
A discussion of the difficulties of Russell’s theory led me to
undertake a mathematical investigation of the stability of stars in
general, and this was found to provide a simple and somewhat unexpected
explanation of the otherwise incomprehensible distribution of stars in
the Russell diagram; it is in brief that the unoccupied regions of the
diagram represent stars in an unstable condition. I do not know what
proportion of astronomers accept this explanation; some, whose opinion
I value, do not. I do not think that much so far written in this book
would be seriously challenged by competent critics, but it is only fair
to say that at this point we are entering controversial ground.
THE HYPOTHESIS OF LIQUID STARS
Let us begin by imagining an enormous number of stars built on
all possible plans, out of all kinds of substances. Mathematical
investigation shews that some of these stars may be unable to shine
with a steady light for either or both of two reasons—they may explode,
like a heated keg of gunpowder, or they may have an inherent tendency
to contract or expand without limit. Whether a star escapes the first
pitfall or not depends mainly upon the properties of the substance of
which it is built; whether it escapes the second depends mainly upon
the way it is built. The two pitfalls are not altogether distinct, and
when we consider the stability of wholly gaseous stars of enormously
great weight, we find that the pits on the two sides of the path merge
into one, or at most only a narrow strip of safe ground is left between
them. Nevertheless stars of enormously great weight are known to exist,
and continue shining steadily. If then, these stars are wholly gaseous,
they must occupy the one safe spot of ground between the two pits,
and this informs us both as to the way they are built and as to the
properties of the substance of which they are built.
We find that such stars only escape both pitfalls if their substance
possesses properties which appear very improbable, and contrary to
anything of which we have any experience or knowledge in physics;
in brief, for such a star to remain stable, the annihilation of its
matter must proceed at a rate which depends on the temperature. Such
a property seems in every way contrary to the physical principles
explained in Chapter II, as it is to all our expectations of atomic
behaviour. The annihilation of matter is a far more violent change,
and involves quanta of far higher energy, than mere radio-active
disintegration, and as the latter process is not affected by
temperature changes, it hardly seems possible that the process of
annihilation should be, at any rate until we reach temperatures of the
order of the 2,200,000,000,000 degrees tabulated on p. 144[26].
[26] This provides a further objection to Russell’s hypothesis, which,
to avoid confusion, was not mentioned on p. 295.
We have, however, already found indications that the stars are not
purely gaseous, since purely gaseous masses could not form close binary
systems of the type observed in the spectroscopic binaries (p. 222).
Such systems can only be formed out of a mass which simulates the
properties of a liquid rather than those of a gas; the mass need not
be wholly liquid, but there must be a considerable divergence from the
state of a pure gas, at any rate in its central regions. Additional
evidence to the same effect will also emerge later (pp. 310, 311).
As soon as we admit that the interiors of the stars need not be in a
completely gaseous state, the whole situation changes, even a slight
departure from the gaseous state being found to impart a great deal of
additional stability to the star. If a star of great weight is purely
gaseous in its structure, the region of stability between the two
pitfalls is reduced to a narrow strip, and only by treading this can
the star escape the alternative fates of exploding or collapsing. But
if the star has a liquid, or partially liquid, centre, this strip of
safe land is so wide that, consistently with stability, the stellar
material may have exactly the property that we should _à priori_ expect
to find, namely, that its annihilation proceeds, like radio-active
disintegration, at the same rate at all temperatures. If the substance
of the star has this property, the star can be in no danger of
exploding, for a mass of uranium or radium does not explode whatever
we do to it. And mathematical analysis shews that if the centre of the
star is either liquid, or partially so, there is no danger of collapse;
the liquid centre provides so firm a basis for the star as to render a
collapse impossible.
These considerations suggest the two complementary hypotheses:
1. That the annihilation of stellar matter proceeds
spontaneously, not being affected by the temperature of the
star.
2. That the central regions of the stars are not in a
purely gaseous state; their atoms, nuclei and electrons are
so closely packed that they cannot move freely past one
another, as in a gas, but rather jostle one another about
like the molecules of a liquid.
If we have been right (p. 149) in attributing the observed highly
penetrating radiation in the earth’s atmosphere to the annihilation
of matter in distant astronomical bodies, then the first hypothesis
is confirmed. For the radiation could not retain its observed high
penetrating power if it had already penetrated any great thickness
of matter. The struggle of passing through matter lengthens the
wave-length of all kinds of radiation (the quanta get weaker all the
time), and so diminishes its penetrating power. Thus, wherever the
highly penetrating radiation originated, it must have got out into
empty space without much of a struggle, and this is the same thing as
saying that it must have originated in matter at a comparatively low
temperature. Thus the existence of the highly penetrating radiation
proves that matter can be annihilated in great quantities at quite
low temperatures; the high temperatures of stellar interiors are not
needed, as Russell’s theory asserts, for annihilation to occur.
A simple calculation shews that there can be no appreciable
annihilation of the earth’s substance. In the sun about one atom in
every 10¹⁹ is annihilated every minute; if even a ten-thousandth part
as many atoms as this were annihilated in the earth, its surface would
be too hot for human habitation. We can no longer explain this by
saying that the sun is hot and the earth cool, so that annihilation
goes on in the former but not in the latter. We must rather suppose
that the atoms in the sun are of a different type from those on earth.
Solar atoms spontaneously annihilate themselves, terrestrial atoms do
not, or at least do not to any appreciable extent.
The second of our hypotheses, that the central regions of the stars
are more like a liquid than a gas, is, as we have seen, a necessary
consequence of the first, but independent evidence in its favour is
found in the formation of binary systems by fission. For in Chapter IV
(p. 222) we saw that a star could only break up by fission if it had a
liquid, or a nearly liquid, centre.
THE STABILITY OF STELLAR STRUCTURES
For the present, let us tentatively accept the hypothesis that
the generation of stellar energy occurs spontaneously, like
the disintegration of radio-active atoms. The atoms which are
responsible for the light and heat of the stars may be regarded as
super-radio-active atoms which spontaneously annihilate themselves and
so change their substance into radiation.
We have already seen that, on this view of the mechanism of generation
of stellar energy, a star can only continue to shine steadily if its
central regions are not in a purely gaseous condition. A star built on
foundations of highly compressible gas meets the same fate as a house
built on sand: it collapses. A purely gaseous star is a dynamically
unstable structure, and must continually contract until the atoms in
its central regions are so closely packed that their state can no
longer be regarded as gaseous. Then, and then only, can the star exist
permanently as a stable structure. Thus the central regions of any
actual permanent star, the sun for instance, must be in a state which
for brevity we may describe as liquid.
Now let us imagine the sun to be expanded to ten times its present
diameter. This would diminish its density to a thousandth part of its
original value. The actual sun is 40 per cent. more dense than water,
but the expanded sun would only be as dense as ordinary atmospheric
air. The atoms and electrons, having moved ten times farther apart,
would be so distant from one another that the new sun might be regarded
as wholly gaseous. Thus it would be dynamically unstable and could not
remain in its wholly gaseous state.
Our imaginary expanded sun is of course no longer a main-sequence star
in the Russell diagram. In expanding the sun to ten times its present
size we move it off the main-sequence into a region entirely vacant of
stars—in fact, into the great gulf which lies between the red giants
and the red dwarfs (see fig. 22, p. 278). Thus, it appears that even if
we deliberately place a star in this region, it does not stay there but
immediately contracts until it gets on to the main-sequence. May not
this explain why the region in question is untenanted by stars?
Next let us imagine the sun contracted to a tenth of its present
diameter, so that its atoms and electrons move ten times nearer to one
another. Its mean density is thereby increased from 1·4 to 1400 times
that of water, and its central density from about 140 to 140,000.
You may check me here by pointing out that if the sun is already in
a liquid state it cannot be compressed to any such extent—a liquid
cannot usually have its density increased a thousand-fold. But we
have already noticed that halving a star’s diameter doubles its
temperature throughout. In the same way reducing a star’s diameter to
a tenth increases its temperature ten-fold, so that the sun’s central
temperature will be increased from, say, 50 million degrees to 500
million degrees. And at this latter temperature atoms hardly exist any
longer as such—the stellar matter consists almost entirely of free
electrons and nuclei. And these are so minute, that the increase of the
sun’s mean density from 1·4 to 1400 times the density of water is not
only possible, but leaves the sun’s substance in a state which may best
be described as gaseous. Once again, then, the new sun is dynamically
unstable. It would be represented by a point well to the left of the
main-sequence, near the middle of the unoccupied region between the
main-sequence and the white dwarfs, but as it is unstable it cannot
maintain its position here. Again we see that even if we place a star
in this region it cannot stay there. And, again—may it not be that the
reason why this region is unoccupied is that it represents unstable
stars?
Once more you may check me. If I have made my point, it has been by the
help of the rise of temperature which accompanies contraction. When
we imagined the sun to expand, ought we not to have considered the
fall of temperature which accompanies expansion? The answer is that we
ought, but it would have made no difference. Lowering the temperature
will cause a number of _L_-rings, and possibly also of _M_-rings, of
electrons to re-form, so that the new atoms will be of larger size,
but they will not lose their freedom of motion sufficiently to make
the sun stable. It would have been different if we had been discussing
a star of 10 or 50 times the sun’s weight; then it can be shewn that
the re-formation of _K_- and _L_-rings would have produced a series
of stable configurations. And the spur branch in the Russell diagram
exists to provide a home for just such stars.
The whole problem is too complicated to be discussed satisfactorily in
this fragmentary way; its proper discussion involves very complicated
mathematical analysis. Mathematical discussion shews that the Russell
diagram can be divided into regions representing stable and unstable
configurations in the manner shewn in fig. 24.
[Illustration: Fig. 24. Stable and unstable configurations in the
Russell diagram.]
The unstable areas are so marked; the remaining areas are stable. The
dots which form a sort of background to the diagram represent 2100
stars whose absolute magnitudes are known through their parallaxes
having been determined spectroscopically at Mount Wilson. The
observational material is not perfect, for considerable uncertainty
attaches to all spectroscopic parallaxes of _B_-type stars, and
_A_-type stars are almost unrepresented because it is practically
impossible to obtain their parallaxes by the spectroscopic method.
The theoretical curves are probably still more imperfect, yet, such
as they are, they seem to suggest very forcibly that the occupied
and unoccupied regions coincide with those representing stable and
unstable configurations; after making all possible allowances for the
imperfections both of theory and of observation, too much agreement
remains to be explained away as mere coincidence.
Thus the conclusion to which mathematical discussion seems to lead is
that the regions in the Russell diagram which are occupied, represent
stars whose central regions are in a liquid, or nearly liquid, state.
All other stars are unstable, so that the corresponding regions in the
Russell diagram are necessarily vacant. To put it in less technical
language, all the stars in the sky must have liquid, or nearly liquid,
centres.
Here we have a piece of the puzzle which seems to fit on to the piece
we unearthed in Chapter IV, where we found that a star could only break
up by fission if it had a liquid, or nearly liquid centre. Evidence
accumulates that the stars have liquid rather than gaseous centres.
Criticism of the foregoing hypothesis—generally described as the
“liquid-star” hypothesis—has mainly taken the form that the diameters
of the _K_-rings of atoms are so small that the _K_-ring atoms in
the sun’s central regions cannot possibly be packed closely enough
to involve any substantial departure from the gaseous state. It is
difficult to discuss, and still more to meet, this criticism without
knowing the precise diameters of these _K_-ring atoms. We of course
know the diameters assigned to the _K_-ring by Bohr’s theory (p. 129),
but no one any longer contends that this theory gives a true picture
of the atom. It provides a good working model within limits, but we do
not know where the limits end. The only practical experience we have
of _K_-ring atoms is with helium atoms; Bohr’s theory assigns to these
a diameter of 0·54 × 10⁻⁸ cms. Yet solid and liquid helium provide a
practical illustration of the closeness with which helium atoms can be
packed; in these each atom occupies a sphere of diameter 4 × 10⁻⁸ cms.,
or over 400 times the space allotted to it by Bohr’s theory. It looks
as though we are still far from definite knowledge of the dimensions of
_K_-rings of electrons.
The new wave-mechanics of Schrödinger draws a very different picture of
atomic interiors from the simpler theory of Bohr which it is rapidly
superseding. Even the electron is something very different from the
electron of Bohr’s old theory. It is the old-fashioned electron only
when it is at an infinite distance from the nucleus. As it gradually
approaches this nucleus, it undergoes a metamorphosis of a kind which
no one has yet succeeded in describing, and it is utterly impossible
to say what form it may have assumed by the time it is doing what we
call “describing a _K_-ring orbit.” All we know about the _K_-ring
orbit is its energy, and it seems impossible to predict the amount of
space occupied by such an orbit until we have a better knowledge of the
qualities of the article which is describing it.
We have of course to admit that the physical evidence, such as it is,
seems to point to _K_-ring atoms being substantially smaller than is
needed for the liquid-star hypothesis. But the astronomical evidence
seems to me stronger and more reliable, and to point in exactly the
opposite direction. And here we must leave the puzzle until further
pieces come to light.
Until we know the kind of atoms of which a particular star is composed,
we cannot calculate the extent to which they will be broken up by the
temperature of the star’s interior. As a consequence, the theoretical
curves of demarcation between stable and unstable configurations cannot
be calculated without assuming definite atomic numbers for the stellar
atoms.
The curves shewn in fig. 24 have been drawn for an atomic number of
about 95, this being slightly higher than the atomic number, 92, of
uranium. This atomic number was selected because it was found to
produce the best agreement between theory and observation, but we shall
see that other considerations justify our choice.
STELLAR STRUCTURE
A star, like a house or a pile of sand, is a structure which would
collapse under its own weight were it not that each layer is held up
against gravity by the pressure which the next inner layer of the star
exerts upon it. This pressure is not, like ordinary gas-pressure, the
result of the impacts of complete molecules. It is produced in part by
the impact of a certain number of atoms which have been stripped of
electrons almost or quite down to their nuclei, but to a far greater
extent by the impact of a hail of free electrons. In massive stars, an
additional pressure is produced by the impact of radiation which, as
we have seen, carries weight about with it, and so exerts pressure on
any obstacle it encounters. The combined impacts of free electrons, of
atoms (or bare nuclei), and of radiation prevent the star from falling
in under its own gravitational attraction.
This gives a reasonably good snapshot picture of a star’s structure.
The corresponding picture of its mechanism is obtained by thinking
of the nuclei as α-ray particles, of the free electrons as β-ray
particles, and of the radiation as γ-rays (although in most stars the
main bulk of the radiation has the wave-length of X-rays). All these
thread their way through the star, and, precisely as in laboratory
work, the β-rays are more penetrating than the α-rays, and the γ-rays
are more penetrating than either.
THE TRANSPORT OF ENERGY IN A STAR. We have seen how the heat of a gas
is merely the energy of its molecular motion. Conduction of heat in
a gas is usually studied by regarding each molecule as a carrier of
energy; when it collides with a second molecule the energy of the two
colliding molecules is redistributed between them, and in this way heat
is transported from hotter to cooler regions. Each molecule has a power
of transport which is jointly proportional to its energy of motion, its
speed of motion, and its “free-path”—the distance it travels between
successive collisions.
In the interior of a star, there are three distinct types of carrier in
action—atoms (or bare nuclei), free electrons, and radiation. We can
compare their relative capacities as carriers by multiplying up the
energy, speeds and free-paths of each. For this purpose the “free-path”
of radiation may be taken to be the distance the radiation travels
before 37 per cent. of it has been absorbed, since it can be shewn that
this is the average distance it carries its energy. On carrying out
the calculation, the carrying capacity of both nuclei and electrons is
found to be insignificant in comparison with that of the radiation.
The nuclei and electrons may have the greater amount of energy to
carry, but owing to their feebler penetrating powers, the distance
over which they carry it, their free-path, is far less than that of
the radiation. Their speed of transport is also less, since radiation
transports its energy with the velocity of light. In this way it comes
about that practically the whole transport of energy from the interior
of a star to its surface is by the vehicle of radiation.
This general principle was first clearly stated by Sampson in 1894.
He also shewed how the temperature of any small fragment of a star’s
interior must be determined by the condition that it receives just as
much radiation as it emits, but his detailed applications were vitiated
through his using an erroneous law of radiation. Twelve years later,
Schwarzschild independently advanced the same idea, and expressed it in
mathematical equations of “radiative equilibrium” which have formed the
basis of every subsequent discussion of the problem.
Just because radiation completely outstrips atoms and electrons in
carrying energy from a star’s interior to its surface, it follows that
the build of a star must be determined by the opacity of the matter in
its interior. If this is altered, the carrying power of the radiation
is altered, and this affects the whole structure of the star. A star
whose interior was entirely transparent could not retain any heat at
all; its whole interior would be at a very low temperature and the
star would be of enormous extent. On the other hand, in a very opaque
star, all energy would remain accumulated at the spot at which it was
generated, so that the interior temperature would become very high and
the star’s diameter would be correspondingly small. It is, of course,
the intermediate cases which are of practical interest, but the extreme
instances just mentioned shew how a star’s build depends on its opacity.
Unfortunately it is impossible to obtain any sort of direct measurement
of the opacity of stellar matter. We cannot even measure the opacity
of terrestrial matter under stellar conditions, since the interiors of
the stars are at incomparably higher temperatures than any available
in the laboratory. However, we know that the opacity of stellar matter
is due to the atoms, nuclei and free electrons of which it is composed
checking the onward journey of radiation, and although we cannot obtain
a sample of stellar matter, we know fairly definitely how many atoms,
nuclei and electrons such a sample would contain. Thus it becomes a
matter of theoretical calculation to determine its opacity.
Such a calculation was carried through by Dr Kramers of Copenhagen
in 1923, and his results gained general acceptance. Quite recently
Mr Gaunt of Cambridge has made a much more refined calculation, and
obtained results which agree very closely with those originally given
by Kramers. In so far as these results can be tested in the laboratory,
they agree well with observation. And, although there is a big gap
between laboratory conditions and stellar conditions, it is difficult
to see how Kramers’ formula could fail in the stars.
From this formula we can determine the build of the stars completely,
or, if the build of the star is supposed to be known, Kramers’ formula
tells us the rate at which energy flows to its surface (this depending
entirely on the opacity of the star’s substance), and this in turn
tells us at what rate energy must be generated inside the star for it
to be able to remain in equilibrium in the configuration in question.
As might be expected, configurations of different diameters are found
to require different rates of generation of energy. In nature, a star
must adjust its diameter to suit the rate at which it is generating
energy; in so doing it fixes not only its diameter but also its
surface-temperature, colour and spectral type. If a star’s rate of
generation of energy were suddenly to change, the star would expand or
contract until it had assumed the radius and temperature suited to its
new rate of generation of energy.
Detailed calculation shews that, for wholly gaseous stars, large
diameters correspond to feeble generation of energy and _vice versa_.
Thus, if the stars were wholly gaseous, red giants would be less
luminous than main-sequence stars of the same weight. Seares’ diagram,
reproduced on p. 282, shews that the reverse is actually the case, a
red giant emitting from 10 to 20 times as much total radiation as an
equally massive main-sequence star. This provides evidence against the
stars being wholly gaseous, but there is stronger evidence than this.
For wholly gaseous stars, the thick lines shewn in Seares’ diagram
would be straight slant lines, slanting upwards to the left. The wide
divergence between such a system of slant lines and the curves shewn in
fig. 23 gives some indication of the extent to which the condition of
stellar matter diverges from the purely gaseous state.
According to Kramers’ theory, the opacity of matter depends on the
atomic numbers and atomic weights of the atoms of which it is built,
a large clot of matter in the form of a massive atomic nucleus being
far more effective in absorbing radiation than a large number of
small clots of the same total weight. Everyday terrestrial experience
shews that this is so. It is for this reason that the physicist and
surgeon both select lead as the material with which to screen their
X-ray apparatus; they find that a ton of lead is far more effective
in stopping unwanted X-rays than a ton of wood or of iron. If we knew
the strength of an X-ray apparatus, and the total weight of shielding
material round it, we could form a very fair estimate of the atomic
weight of the shielding material by measuring the amount of X-radiation
which escaped through it.
A very similar method may be used to determine the atomic weights of
the atoms of which the stars are composed. A star is in effect nothing
but a huge X-ray apparatus. We know the weights of many of the stars,
and the rate at which they are generating X-rays is merely the rate at
which they are radiating energy away into space. If we could cut each
atomic nucleus in a star into two halves, we should halve the opacity
of the star, so that radiation would travel twice as far through the
star before being absorbed. If the star were wholly gaseous, this would
result in its expanding to four times its original diameter, and in its
surface-temperature being halved. It follows that we can calculate the
atomic weight of the atoms of which a star is composed from the weight,
luminosity and surface-temperature of the star.
The atomic weights of a number of stars, which I calculated on the
supposition that the stars were wholly gaseous, came out in practically
every case higher than that of uranium, which is the weightiest atom
known on earth. They not only proved to be higher, but enormously
higher; so high indeed, as to seem utterly improbable. Again the
explanation seems to be that the stars are not wholly gaseous. As
soon as stellar interiors are supposed to be partially liquid, the
calculated atomic weights are reduced enormously. They can no longer be
determined exactly, but the atomic number of about 95 to which we were
led from a consideration of the Russell diagram seems to be entirely
consistent with all the known facts.
Indeed other considerations seem to suggest that the atomic numbers
of stellar atoms must be higher than 92. _A priori_ stellar radiation
might either originate in types of matter known to us on earth or
else in other and unknown types. When once it is accepted that high
temperature and density can do nothing to accelerate the generation of
radiation by ordinary matter, it becomes clear that stellar radiation
cannot originate in types of matter known to us on earth. Other types
of matter must exist, and as, with two exceptions, all atomic numbers
up to 92 (uranium) are already occupied by terrestrial elements, it
seems probable that these other types must be elements of higher atomic
weight than uranium.
These super-heavy atoms must not be expected to disclose their presence
in stellar spectra, for these only inform us as to the constitution of
the atmospheres of the stars. And as the lighter atoms float to the
top it is these, in the main, which figure in stellar spectra. If the
sun’s atmosphere had contained any considerable number of super-heavy
atoms when the planets were born, some of them ought still to exist in
the earth. There cannot be any great number, or their high generation
of energy would betray them. The simplest view seems to be that the
heavier atoms sink to the centre in the stars, and that the earth was
formed mainly or solely out of the lighter atoms which had floated to
the sun’s surface.
STELLAR EVOLUTION
We have supposed that the stars were born initially as condensations in
the outer fringes of spiral nebulae. These condensations would, from
the mode of their formation, necessarily be of all sorts of sizes, and
subject only to the single restriction that none of them could be below
a certain limit of weight. Thus we should not expect the stars, either
at birth or subsequently, to be all of the same size or weight, or all
in the same physical condition. The stars would start their existences
at different points in the Russell diagram, but we may imagine that
their initial positions are limited to those parts of the diagram which
can be occupied by stars—either, as the liquid-star hypothesis would
suggest, because these are the only stable configurations, or else for
some other reason so far undiscovered. Each year a star loses a certain
amount of weight, and its rate of generating energy, and so also its
luminosity, are correspondingly reduced, with the result that it moves
to a new position in the diagram. Turning to Seares’ diagram of stellar
weights on p. 282, we may think of the curves of equal weight as a
flight of steps—very uneven steps, it is true—each representing a lower
weight than the one above. Whatever a star’s evolution may be, it is
essential that it should always be _down_ the steps: any upward step is
impossible.
We can trace out two possible paths of stellar evolution in the Russell
diagram which involve no entry into regions unoccupied by stars—two
roads along which the stellar army may march as they transform their
substance into radiation. The first is of course the “main-sequence,”
which a great number of considerations suggest to be the main line of
march of the stellar army. The branch which starts from the red giants
in the Russell diagram represents a second possible line of march, a
certain number of stars travelling along this branch until they reach
the main-sequence as blue or white stars, and then travelling down the
lower half of the main-sequence to end as faint red stars passing on to
ultimate extinction.
Progress along each of these roads is accompanied by a continuous
shrinkage in the size of the star, its diameter steadily decreasing.
This is not the same thing as saying that the star’s density
continually increases, for the star is continually diminishing in
weight, so that even if the star’s density remained the same, its
diameter would decrease. Nevertheless a study of Seares’ determinations
of mean densities, as shewn in the diagram on p. 282, suggests that
there is a continuous increase of density, although this becomes very
slight in the middle reaches of the main-sequence.
Practically every theory of stellar evolution which has ever been
propounded has imagined the march of the stellar army to be of the
same general type as that just described, although perhaps present-day
opinion is inclined to treat the main-sequence as the principal line of
march, whereas earlier theories supposed the youngest stars to march
solely along the red giant branch, only joining the main-sequence with
middle age. The first serious theory of all, that of Lockyer, was
expressed in terms of branches of ascending and descending temperature,
these together forming the last-mentioned line of march in the Russell
diagram. A theory which Russell propounded in 1913 again assigned
to the stars the lines of march just described. It also attempted a
physical explanation, since abandoned, as to why the stars followed
these particular paths rather than others. His more recent theory
of 1925 only differed from his earlier theory in giving the new
explanation, which we have already discussed (p. 293), as to why the
stars followed these particular paths.
At present, it is probably fair to say that nearly, and perhaps quite,
all astronomers are agreed that the evolutionary paths of the stars
are of the general type we have described. Some stars start as red
giants, some as blue, some possibly in intermediate conditions. As
they age, all move downwards in the Russell diagram, their various
paths converging to a point at the fork of the reversed γ shewn in fig.
22, and after passing this point they move down the main-sequence.
On the other hand, there is the widest difference of opinion as to
the physical interpretation which is to be assigned to these paths.
Most astronomers are probably suspending judgment until some definite
observational evidence is obtained to decide between conflicting
theories.
When the stars first came into being as flecks of fiery spray thrown
off by spinning nebulae, they would consist of mixtures of atoms of all
kinds, some perhaps being so short-lived as to transform themselves
almost at once into radiation, and others having such long lives
that they may properly be described as permanent. Except for a small
number of radio-active atoms, the earth must consist entirely of
atoms of this latter type. Calculation shews that terrestrial atoms
must have enormously longer lives than the average stellar atom,
otherwise their self-annihilation would make the earth too hot for
habitation. The permanent atoms in a star contribute almost nothing to
its energy-generating capacity, and so merely add to its weight. The
shortest lived atoms of all contribute greatly to the star’s generation
of energy while adding but little to its weight. In general the shorter
the life of any type of atom, the greater the proportion of its numbers
annihilated per year, and so the greater the amount of energy it
generates per ton of weight.
A star begins life with a large proportion of short-lived atoms, and
so at first generates energy furiously. As it ages, the shortest
lived atoms disappear first, and in so doing reduce the average
energy-generation of the star per ton so that, as a star’s weight
decreases, so also must its rate of generation of energy per ton.
Finally all the atoms with much energy-generating capacity have
disappeared, and the star is left, a shrunken and diminished mass of
atoms which have very little capacity for generating radiation.
To put the same thing in another way, the rate at which a star
generates energy per ton is proportional to the death-rate in its
population of atoms. To say that Sirius generates 16 times as much
energy per ton as the sun is only another way of saying that the
average atom in Sirius has only a sixteenth of the expectation of life
of the solar atoms; their death-rate is 16 times as high. As those
types of atoms which have the highest death-rate gradually die off in
any star, the average death-rate of the population decreases, or, in
other words, as a star ages its capacity for energy-generation per ton
decreases.
This agrees with the findings of observational astronomy. The most
massive stars not only generate more energy than less massive stars,
as is in any case to be expected; they also generate enormously
more energy per ton. This is illustrated by the following list of
main-sequence stars:
+-----------------+-----------------+--------------------+
| Star | Weight |Generation of energy|
| |(in terms of sun)| (ergs per gramme) |
+-----------------+-----------------+--------------------+
|Pearce’s Star _A_| 36·3 | 15,000 |
|_V_ Puppis _A_ | 19·2 | 1,000 |
|Sirius _A_ | 2·45 | 29 |
|Sun | 1·00 | 1·90 |
|ε Eridani | (0·45) | 0·26 |
|Kruger 60 _B_ | 0·20 | 0·021 |
+-----------------+-----------------+--------------------+
Seares’ diagram of stellar weights (p. 282) shews that this is a
general property of the stars. To repeat our former metaphor, the
stars squander their substance lavishly in their youth, while they
have plenty left to spend, but parsimony comes over them with old age.
Theoretical considerations have now given us an explanation of this
phenomenon.
The same diagram shews that two stars of the same weight do not usually
have the same luminosity. In general, giant stars on the spur branch
leading out to the red giants have substantially higher luminosities
than the main-sequence stars of equal weight. We have already noticed
how a red giant may emit as much as 10 or 20 times the radiation of an
equally massive main-sequence star. The same story is repeated when we
pass from the main-sequence stars to the white dwarfs. Main-sequence
stars emit enormously more radiation—anything up to 500 times more—than
white dwarfs of equal weight. This is illustrated by the three
following white dwarfs, which may be compared with the last three stars
of the preceding table:
+-----------------+-----------------+--------------------+
| Star | Weight |Generation of energy|
| |(in terms of sun)| (ergs per gramme) |
+-----------------+-----------------+--------------------+
|Sirius _B_ | 0·85 | 0·0027 |
|ο₂ Eridani _B_ | 0·44 | 0·002 |
|van Maanen’s star| (0·20) | (0·00055) |
+-----------------+-----------------+--------------------+
We have hitherto supposed generation of energy to be spontaneous and
so unaffected by changes of physical conditions. Yet the facts just
mentioned seem to suggest that this can hardly be the whole truth of
the matter. To state the objection in terms of a concrete instance,
Sirius _A_ and its white dwarf companion Sirius _B_ must in all
probability have been born at the same time out of the same nebula
(p. 284), yet the former generates 4000 times as much energy per ton
as the latter. It seems improbable that so great a difference can be
attributed to different types of atoms; the common origin of the two
stars almost precludes this. We know that the atoms are in different
physical conditions in the two stars; in Sirius _A_ they have retained
their _K_-rings intact, while in Sirius _B_, the white dwarf, they are
completely broken up into bare nuclei and free electrons. If the two
components of Sirius consist of essentially the same types of atoms,
as their common origin would lead us to expect, then the enormous
difference in the rates at which these atoms generate energy would seem
to depend on the different physical conditions of their atoms.
The considerations brought forward in Chapter III make it highly
probable that a star’s rate of generation of energy depends on the
physical condition of its atoms. We there supposed stellar energy to
be generated through electrons coalescing with protons; protons exist
only in atomic nuclei, and purely physical considerations led to the
conjecture that the only electrons which can coalesce with a particular
proton are those which are momentarily describing orbits around the
nucleus in which the proton resides. A study of stellar structure
supports this hypothesis, for if energy could be generated by free
electrons falling into nuclei, it can be shewn that the whole star
would be unstable and would explode in a flash of radiation. On this
hypothesis, a star in which only a few atoms have any electrons left
in orbital motion can of course generate but little energy. This at
once explains the feeble energy-generating powers of the white dwarfs,
and also gives an inkling as to why the red giants, in which _L_- and
_M_-rings of electrons survive, generate more energy than main-sequence
stars of equal weights.
As a star ages and its weight decreases, it continually has to pick
out new configurations such as make its emission of energy equal to
its internal rate of generation of energy. The same star may be a red
giant, a main-sequence star and a white dwarf in turn. Stripped of
technicalities this means that a star continually adjusts its diameter
to suit its varying rate of generation of energy.
On the hypothesis just considered, a star alters both its emission and
its generation of energy on changing its diameter. At every instant it
has to select a diameter for which the two exactly balance. The star
has so large a range of rates of generation, according as it has few
or many electrons left in orbital motion, that it is likely always to
be able to find a configuration of equilibrium. At any rate all the
stars in the sky appear to have done so, with the exception of the
long period variables which are continually expanding and contracting
as though they could not hit upon a diameter at which their income and
expenditure of energy would just balance.
This same hypothesis immediately makes it possible for all the great
variety of stars in the galactic system to be of approximately the
same age, and so to have been all born out of the same nebula. The
most luminous galactic stars can hardly have been generating energy
at their present rate for more than about 100,000 million years—any
longer age would require an impossibly high weight to start with. Yet
the motions of the stars indicate that even these highly luminous
stars must have been in existence for at least 50 times this period.
The apparent contradiction disappears if we admit that the extreme
luminosity of the very brightest stars may be a recent development, and
that for perhaps 98 per cent. of its life such a star was losing but
little energy because most of its atoms were stripped bare of electrons
and so were immune from annihilation. The requisite proportion of 98
per cent. may seem suspiciously large, but we have stated an extreme
case; we need only demand so large a proportion in the case of a very
rare type of star: probably not more than one star in ten million is of
such a type.
In their earlier dormant state, these stars, which are now so
luminous, would in effect have been white dwarfs of enormous weight.
Observational astronomy provides no evidence that any such stars exist,
but there is certainly no evidence that they do not exist. Very massive
stars are known to be very rare objects, so that in all probability we
should have to travel a long distance from the sun before finding one,
and then it might be so distant as to be invisible from the earth. In
any case, a very distant star of feeble luminosity would be exceedingly
likely to escape detection. We could not infer that no such stars
existed from the fact of none having yet been found.
Moreover, it is far from absolutely certain that such stars have
not been found. Very massive white dwarfs ought to have higher
surface-temperatures than either massive main-sequence stars or than
the known white dwarfs, all of which are of small weight. A whole group
of stars is known—the _O_-type stars—whose spectra indicate very high
temperatures indeed. These are usually interpreted as stars of enormous
luminosity at enormous distances, but it is possible that some at least
of them may be stars of feeble luminosity at moderate distances. In
particular the central stars of the planetary nebulae are of types _O_
and _B_. But whereas the normal main-sequence star of these spectral
types is generally about a thousand times as luminous as the sun, the
central stars of the planetary nebulae are found to be considerably
less luminous than the sun. Dr Gerasimovič has recently shewn that 52
of these stars have an average luminosity only about two-thirds of that
of the sun. Other observers have found even lower luminosities. As the
surface-temperatures of these stars must be of the order of 30,000
degrees, they must be minute in size; indeed a simple calculation shews
that they can hardly have more than about a tenth of the sun’s radius.
Not only are they small in size, but there are indications that they
are of great weight. Spectroscopic evidence shews that in many cases
the surrounding nebulae are in rotation, and, just as with the galaxy
(p. 69) and the huge extra-galactic nebulae (p. 71), their weights can
be calculated from their observed speeds of rotation. These seem to
indicate average weights of something like 40 or 50 times the weight of
the sun. Campbell and Moore, of Lick Observatory, have calculated that
the planetary nebula N.G.C. 7009 must have 162 times the weight of the
sun, but many factors combine to make this estimate rather uncertain.
All this goes to suggest that the central stars of the planetary
nebulae must, in all probability, be regarded as “white dwarfs” of
enormous weight. It seems possible that they are exactly the type of
star needed by our theory. The 52 planetary nebulae already mentioned
have the same average distance from the sun, and the same distribution
with respect to the galaxy, as the _O_-type stars, so that it is
possible that _O_-type stars and planetary nebulae are forms assumed
either successively or alternatively in the course of evolution of
similar bodies. It should be mentioned that the planetary nebulae
exhibit spectral displacements which, if interpreted in the ordinary
way, would indicate that they are moving with very high speeds, in
some cases reaching hundreds of miles a second. Such speeds would be
entirely inappropriate to stars of large weight. Yet this objection
rather defeats itself, for if the spectral shifts really arose from
high speeds of motion, the highest of the observed speeds would suffice
to carry the planetary nebulae possessing them clear of the galactic
system altogether; they would be random travellers through our system
of stars, whereas their orderly arrangement and concentration near the
galactic plane suggests very forcibly that they are permanent members
of it. Thus the observed large shifts of the spectral lines can hardly
indicate large speeds in space; some other interpretation must be
found. Professor Perrine considers that they arise in the main from
internal motions, contraction, expansion and rotation of the nebulae.
If so, the motions in space cannot be determined, but at least they no
longer present any difficulty. If these stars are, as our hypothesis
requires, white dwarfs of very great weight and small radius, their
spectra ought to exhibit a relativity displacement to the red of the
same type as that actually observed in Sirius _B_. It is possible that
this may be found to present a more serious difficulty, although it
is also possible that it again may prove to be merged in the large
observed displacements.
In view of all this, it seems quite possible that both the planetary
nebulae, and also other _O_- or _B_-type stars of feeble luminosity,
may be very massive stars in the dormant condition contemplated by
the hypothesis we have just been discussing. In brief, we imagine
that a massive star may have its weight conserved through existing as
a planetary nebula or a dwarf _O_-type star for millions of millions
of years, and then burst out as a highly luminous star with all the
appearance of extreme youth. But there is at present insufficient
observational evidence either for or against such a hypothesis. Some
pieces of the puzzle are missing, and we can only wait until they turn
up.
WHITE DWARFS. Apart from these hypothetical massive white dwarfs,
astronomers generally regard ordinary white dwarfs as the final stage
in stellar evolution. There is general agreement that they are stars
with central temperatures so high that their atoms are stripped bare of
electrons, but there is no general consensus of opinion as to why stars
shrink to this condition.
On the liquid-star hypothesis, the unoccupied regions in the Russell
diagram represent unstable configurations. Usually a slight loss of
weight by a star merely moves it to a new position in the diagram
contiguous to the old one. Sometimes, however, this slight move may
happen to carry the star into an unstable region of the diagram, in
which case it will hurriedly traverse this region, until finally it
ends up in some entirely different stable configuration.
The liquid-star hypothesis explains the white dwarf state quite simply
as the final state to which a star shrinks cataclysmically when its
generation of energy is no longer sufficient to entitle it to a place
in the main-sequence. In this state the star radiates so little energy
that annihilation and decay are almost entirely checked. We have seen
that if the sun went on radiating at its present rate for 15 million
million years, its whole weight would be transformed into radiation.
By contrast, van Maanen’s star can, and probably will, go on radiating
at its present rate for 15 million million years without losing more
than about a thousandth part of its present weight. We may think of the
white dwarf state as a final state from which change and decay have so
nearly disappeared that a star which shrinks to this state acquires a
new lease of life for a period of thousands of millions of millions of
years—we can only wonder to what purpose.