HOW THE SUN’S HEAT IS MAINTAINED.
The Contraction of a Body—Helmholtz Explained Sun-heat—Change of a
Mile every Eleven Years in the Sun’s Diameter—Effect of
Contraction on Temperature—The Solar Constant—Limits to the
Solar Shrinkage—Astronomers can Weigh the Sun—Density of the
Sun—Heat Developed by the Falling Together of the Solar
Materials—Contraction of Nebula to Form the Earth—Heat Produced
in the Earth’s Contraction—Similar Calculation about the
Sun—Earth and Sun Contrasted—Heat Produced in the Solar
Contraction from an indefinitely Great Nebula—The Coal-Unit
Employed—Calculation of the Heat given out by the Sun.
THE law which declares that a body which gives out heat must in general
submit to a corresponding diminution in volume appears, so far as we can
judge, to be one of those laws which have to be obeyed not alone by
bodies on which we can experiment, but by bodies throughout the extent
of the universe. The law which bids the mercury ascend the stem of the
thermometer when the temperature rises, and descend when the temperature
falls, affords the principle which explains some of the grandest
phenomena of the heavens. Applied to the solar system it declares that
as the sun, in dispensing its benefits to the earth day by day, has to
pour forth heat, so in like manner must it be diminishing in bulk.
Assuming that this principle extends sufficiently widely through time
and space, we shall venture to apply its consequences over the mighty
spaces and periods required for celestial evolution. We disdain to
notice the paltry centuries or mere thousands of years which include
that infinitesimal trifle known as human history. Our time conceptions
must undergo a vast extension.
It was Helmholtz who first explained by what agency the sun is able to
continue its wonderful radiation of heat, notwithstanding that it
receives no appreciable aid from chemical combination. Helmholtz pointed
out that inasmuch as the sun is pouring out heat it must, like every
other cooling body, contract. We ought not, indeed, to say every cooling
body; it would be more correct to say, every body which is giving out
heat, for the two things are not necessarily the same. Indeed, strange
as it may appear, it would be quite possible that a mass of gas should
be gaining in temperature even though it were losing heat all the time.
At first this seems a paradox, but the paradox will be explained if we
reflect upon the physical changes which the gas undergoes in consequence
of its contraction.
Let us dwell for a moment on the remarkable statement that the sun is
becoming gradually smaller. The reduction required to sustain the
radiation corresponds to a diminution of the diameter by about a mile
every eleven years. It may serve to impress upon us the fact of the
sun’s shrinkage if we will remember that on that auspicious day when
Queen Victoria came to the throne the sun had a diameter more than five
miles greater than it had at the time when her long and glorious career
was ended. The sun that shone on Palestine at the beginning of the
present era must have had a diameter about one hundred and seventy miles
greater than the sun which now shines on the Sea of Galilee. This
process of reduction has been going on for ages, which from the human
point of view we may practically describe as illimitable. The alteration
in the sun’s diameter within the period covered by the records of man’s
sway on this earth may be intrinsically large; it amounts no doubt to
several hundreds of miles. But in comparison with the vast bulk of the
sun this change in its magnitude is unimportant. A span of ten thousand
years will certainly include all human history. Let us take a period
which is four times as long. It is easy to calculate what the diameter
of the sun must have been forty thousand years ago, or what the diameter
of the sun is to become in the next forty thousand years. Calculated at
the rate we have given, the alteration in the sun’s diameter in this
period amounts to rather less than four thousand miles. This seems no
doubt a huge alteration in the dimensions of the orb of day. We must,
however, remember that at the present moment the diameter of the sun is
about 863,000 miles, and that a loss of four thousand miles, or
thereabouts, would still leave a sun with a diameter of 859,000 miles.
There would not be much recognisable difference between two suns of
these different dimensions. I think I may say that if we could imagine
two suns in the sky at the same moment, which differed only in the
circumstance that one had a diameter of 863,000 miles and the other a
diameter of 859,000 miles, it would not be possible without careful
telescopic measurement to tell which of the two was the larger.
After a contraction has taken place by loss of heat, the heat that still
remains in the body is contained within a smaller volume than it had
originally. The temperature depends not only on the actual quantity of
heat that the mass of gas contains, but also on the volume through which
that quantity of heat is diffused. If there be two equal weights of gas,
and if they each have the same absolute quantity of heat, but if one of
them occupies a larger volume than the other, then the temperature of
the gas in the large volume will not be so high as the temperature of
the gas in the smaller volume. This is indeed so much the case, that the
reduction of volume by the loss of heat may sometimes have a greater
effect in raising the temperature than the very loss of heat which
produced the contraction has in depressing it. On the whole, therefore,
a gain of temperature may be shown. This is what, indeed, happens not
unfrequently in celestial bodies. The contraction having taken place,
the lesser quantity of heat still shows to such advantage in the reduced
volume of the body, that no decline of temperature will be perceptible.
It may happen that simultaneously with the decrease of heat there is
even an increase of temperature.
The principle under consideration shows that, though the sun is now
giving out heat copiously, it does not necessarily follow that it must
at the same time be sinking in temperature. As a matter of fact,
physicists do not know what course the temperature of the sun is
actually taking at this moment. The sun may now be precisely at the same
temperature at which it stood a thousand years ago, or it may be cooler,
or it may be hotter. In any case it is certain that the change of
temperature per century is small, too small, in fact, to be decided in
the present state of our knowledge. We cannot observe any change, and to
estimate the change from mechanical principles would only be possible if
we knew much more about the interior of the sun than we know at present.
We are forced to the conclusion that the energy of the sun, by which we
mean either its actual heat or what is equivalent to heat, must be
continually wasting. A thousand years ago there was more heat, or its
equivalent, in the sun than there is at present. But the sun of a
thousand years ago was larger than the sun that we now have, and the
heat, or its equivalent, a thousand years ago may not have been so
effective in sustaining the temperature of the bigger sun as the lesser
quantity of heat is in sustaining the temperature of the sun at the
present day. It will be noticed that the argument depends essentially on
the alteration of the size of the sun. Of course if the orb of day had
been no greater a thousand years ago than it is now, then the sun of
those early days would not only have contained more heat than our
present sun, but it must have shown that it did contain more heat. In
other words, its temperature would then certainly have been greater than
it is at present.
Thus we see the importance—so far as radiation is concerned—of the
gradual shrinking of the sun. The great orb of day decreases, and its
decrease has been estimated numerically. We cannot, indeed, determine
the rate of decrease by actual telescopic measurement of the sun’s disc
with the micrometer; observations extending over a period of thousands
of years would be required for this purpose. But from knowing the daily
expenditure of heat from the sun it is possible to calculate the amount
by which it shrinks. We cannot conveniently explain the matter fully in
these pages. Those who desire to see the calculation will find it in the
Appendix. Suffice it to say here that the sun’s diameter diminishes
about sixteen inches in every twenty-four hours. This is an important
conclusion, for the rate of contraction of the solar diameter is one of
the most significant magnitudes relating to the solar system.
It was Helmholtz who showed that the contraction of the sun’s diameter
by sixteen inches a day is sufficient to account for the sustentation of
the solar radiation. For immense periods of time the heat may be
dispensed with practically unaltered liberality. The question then
arises as to what time-limit may be assigned to the efficiency of our
orb. Obviously the sun cannot go on contracting sixteen inches a day
indefinitely. If that were the case, a certain number of millions of
years would see it vanish altogether. The limit to the capacity of the
sun to act as a dispenser of light and heat can be easily indicated. At
present the sun, in its outer parts at all events, is strictly a
vaporous body. The telescope shows us nothing resembling a solid or a
liquid globe. The sun seems composed of gas in which clouds and vapours
are suspended. In the sun’s centre the temperature is probably very much
greater than any temperature which can be produced by artificial means;
it would doubtless be sufficient not only to melt, but even to drive
into vapour the most refractory materials. On the other hand, the
enormous condensing pressure to which those materials are submitted by
the stupendous mass of the sun will have the effect of keeping them
together and of compressing them to such an extent that the density of
the gas, if indeed we may call it gas, is probably as great as the
density of any known matter. The fact is that the terms liquids, gases,
and solids cease to retain intelligible distinctions when applied to
materials under such pressure as would be found in the interior of the
sun.
Astronomers can weigh the sun. It may well be imagined that this is a
delicate and difficult operation. It can, however, be effected with but
little margin of uncertainty, and the result is a striking one. It
serves no useful purpose to express the sun’s weight as so many myriads
of tons. It is more useful for our present purpose to set down the
density of the sun, that is to say, the ratio of the weight of the orb,
to that of a globe of water of the same size. This is the useful form in
which to consider the weight of the sun. Astronomers are accustomed to
think of the weight of our own earth in this same fashion, and the
result shows that the earth is rather more than five times as heavy as a
globe of water of the same size. We can best appreciate this by stating
that if the earth were made of granite, and had throughout the density
which we find granite to possess at the surface, our globe would be
about three times as heavy as a globe of water of the same size. If,
however, the earth had been entirely made of iron, it would be more than
seven times as heavy as a globe of water of the same size. As the earth
actually has a density of 5, it follows that our globe taken as a whole
is heavier than a globe of granite of the same size, though not so heavy
as a globe of iron.
In the matter of density there is a remarkable contrast between the sun
and the earth. The sun’s density is much less than that of the earth. Of
course it will be understood that the sun is actually very much heavier
than our globe; it is indeed more than three hundred thousand times
greater in weight. But the sun is about a million three hundred thousand
times as big as the earth, and it follows from these figures that its
density cannot be more than about a fourth of that of the earth. The
result is that, at present, the sun is nearly half as heavy again as a
globe of water the same size. We have used round numbers: the density of
the sun is actually 1.4.
[Illustration: Fig. 17.—ARGO AND THE SURROUNDING STARS AND NEBULOSITY.
(_Photographed by Sir David Gill, K.C.B._)]
In the following manner we explain how heat is evolved in the
contraction of the sun. In its early days the sun, or rather the
materials which in their aggregate form now constitute the sun, were
spread over an immense tract of space, millions of times greater than
the present bulk of the sun. We see nebulosities even now in the heavens
which may suggest what the primæval nebula may have been before the
evolution had made much progress. Look for instance at Sir David Gill’s
photograph of the Nebula in Argo in Fig. 17, or at the Trifid Nebula in
Fig. 18. We may, indeed, consider the primæval nebula to have been so
vast that particles from the outside falling into the position of the
present solar surface would acquire that velocity of three hundred and
ninety miles a second which we know the attraction of the sun is capable
of producing on an object which has fallen in from an indefinitely great
distance. As these parts are gradually falling together at the centre,
there will be an enormous quantity of heat developed from their
concurrence. Supposing, for instance, that the materials of the sun were
arranged in concentric spherical shells around the centre, and imagining
these shells to be separated by long intervals, so that the whole
material of the sun would be thus diffused over a vast extent, then
every pound weight in the outermost shell, by the very fact of its
sinking downwards to the present solar system, would acquire a speed of
390 miles a second, and this corresponds to as much energy as could be
produced by the burning of three tons of coal. But be the fall ever so
gentle, the great law of the conservation of energy tells us that for
the same descent, however performed, the same quantity of heat must be
given out. Each pound in the outer shell would therefore give out as
much heat as three tons of coal. Every pound in the other shells, by
gradual descent into the interior, would also render its corresponding
contribution. It then becomes easily intelligible how, in consequence of
the original diffusion of the materials of the sun over millions of
times its present volume, a vast quantity of energy was available. As
the sun contracted this energy was turned into radiant heat.
We may anticipate a future chapter so far as to assume that there was a
time when even this solid earth of ours was a nebulous mass diffused
through space. We are not concerned as to what the temperature of that
nebulous mass may have been. We may suppose it to be any temperature we
please. The point that we have now to consider is the quantity of heat
which is generated by the contraction of the nebula. That heat is
produced in the contraction will be plain from what has gone before. But
we may also demonstrate it in a slightly different way. Let us take any
two points in the nebula, P and Q. After the nebula has contracted the
points which were originally at P and Q will be found at two other
points, A and B. As the whole nebula in its original form was larger
than the nebula after it has undergone its contraction, the distance P Q
is generally greater than the distance A B. We may suppose the
contraction to proceed uniformly, so that the same will be true of the
distance between any other two particles. The distance between every
pair of particles in the contracted nebula will be less than the
distance between the same particles in the original nebula.
[Illustration: Fig. 18.—TRIFID NEBULA IN SAGITTARIUS
(Lick Observatory, California).
(_From the Royal Astronomical Society Series._)]
If two attracting bodies, A and B, are to be moved further apart than
they were originally, force must be applied and work must be done. We
may measure the amount of that work in foot-pounds, and then,
remembering that 772 foot-pounds of work are equivalent to the unit of
heat, we may express the energy necessary to force the two particles to
a greater distance asunder in the equivalent quantity of heat. If,
therefore, we had to restore the nebula from the contracted state to the
original state, this would involve a forcible enlargement of the
distance A B between every two particles to its original value, P Q.
Work would be required to do this in every case, and that work might, as
we have explained, be expressed in terms of its equivalent heat value.
Even though the temperature of the nebula is the same in its contracted
state as in its original state, we see that a quantity of heat might be
absorbed or rendered latent in forcing the nebula from one condition to
the other. In other words, keeping the temperature of the nebula always
constant, we should have to apply a large quantity of heat to change the
nebula from its contracted form to its expanded form.
It is equally true that when the nebula is contracting, and when the
distance between every two particles is lessening, the nebula must be
giving out energy, because the total energy in the contracted state is
less than it was in the expanded state. This energy is equivalent to
heat. We need not here pause to consider by what actual process the heat
is manifested; it suffices to say that the heat must, by one of the
general laws of Nature, be produced in some form.
We are now able to make a numerical estimate. We shall suppose that the
earth, or rather the materials which make the earth, existed originally
as a large nebula distributed through illimitable space. The
calculations show that the quantity of heat, generated by the
condensation of those materials from their nebulous form into the
condition which the earth now has, was enormously great. We need not
express this quantity of heat in ordinary units. The unit we shall take
is one more suited to the other dimensions involved. Let us suppose a
globe of water as heavy as the earth. This globe would have to be five
or six times as large as the earth. Next let us realise the quantity of
heat that would be required to raise that globe of water from freezing
point to boiling point. It can be proved that the heat, or its
equivalent, which would be generated merely by the contraction of the
nebula to form the earth, would be ninety times as great as the amount
of heat which would suffice to raise a mass of water equal in weight to
the earth from freezing point to boiling point.
We apply similar calculations to the case of the sun. Let us suppose
that the great luminary was once diffused as a nebula over an
exceedingly great area of space. It might at first be thought that the
figures we have just given would answer the question. We might perhaps
conjecture that the quantity of heat would be such as would raise a mass
of water equal to the sun’s mass from freezing to boiling point ninety
times over. But we should be very wrong in such a determination. The
heat that is given out by the sun’s contraction is enormously greater
than this estimate would represent, and we shall be prepared to admit
this if we reflect on the following circumstances. A stone falling from
an indefinitely great distance to the sun would acquire a speed of 390
miles a second by the time it reached the sun’s surface. A stone falling
from an indefinitely great distance in space to the earth’s surface
would, however, acquire a speed of not more than seven miles a second.
The speed acquired by a body falling into the sun by the gravitation of
the sun is, therefore, fifty-six times as great as the speed acquired by
a body falling from infinity to the earth by the gravitation of the
earth. As the energy of a moving body is proportional to the square of
its velocity, we see that the energy with which the falling body would
strike the sun, and the heat that it might consequently give forth,
would be about three thousand times as great as the heat which would be
the result of the fall of that body to the earth. We need not therefore
be surprised that the drawing together of the elements to form the sun
should be accompanied by the evolution of a quantity of heat which is
enormously greater than the mere ratio of the masses of the earth and
sun would have suggested.
There is another line of reasoning by which we may also illustrate the
same important principle. Owing to the immense attraction possessed by
the large mass of the sun, the weights of objects on that luminary would
be very much greater than the weights of corresponding objects here.
Indeed, a pound on the sun would be found by a spring-balance to weigh
as much as twenty-seven pounds here. If the materials of the sun had to
be distributed through space, each pound lifted a foot would require
twenty-seven times the amount of work which would be necessary to lift a
pound through a foot on the earth’s surface. It will thus be seen that
not only the quantity of material that would have to be displaced is
enormously greater in the sun than in the earth, but that the actual
energy that would have to be applied per unit of mass from the sun would
be many times as great as the quantity of energy that would have to be
applied per unit of mass from the earth to effect a displacement through
the same distance. To distribute the sun’s materials into a nebula we
should therefore require the expenditure of a quantity of work far more
than proportional to the mere mass of the sun. It follows that when the
sun is contracting the quantity of work that it will give out, or, what
comes to the same thing, the amount of heat that would be poured forth
in consequence of the contraction per unit of mass of the sun will
largely exceed the quantity of heat given out in the similar contraction
of the earth per unit of mass of the earth.
These considerations will prepare us to accept the result given by
accurate calculation. It has been shown that the heat which would be
generated by the condensation of the sun from a nebula filling all space
down to its present bulk is two hundred and seventy thousand times the
amount of heat which would be required to raise the temperature of a
mass of water equal to the sun from freezing point to boiling point.
This is a result of a most instructive character. The amount of heat
that would be required to raise a pound of water from freezing point to
boiling point would, speaking generally, be quite enough if applied to a
pound of stone or iron to raise either of these masses to a red heat.
If, therefore, we think of the sun as a mighty globe of stone or iron,
the amount of heat that would be produced by the contraction of the sun
from the primæval nebula would suffice to raise that globe of stone or
iron from freezing point up to a red heat 270,000 times. This will give
us some idea of the stupendous amount of heat which has been placed at
the disposal of the solar system by the process of contraction of the
sun. This contraction is still going on, and consequently the yield of
heat which is the consequence of this contraction is still in progress,
and the heat given out provides the annual supply necessary for the
sustenance of our solar system.
There is one point which should be specially mentioned in connection
with this argument. We have here supposed that the current supply of
radiant heat from the sun is entirely in virtue of the sun’s
contraction. That is to say, we suppose the sun’s temperature to be
remaining unaltered. This is perhaps not strictly the case. There may be
reason for believing that the temperature of the sun is increasing,
though not to an appreciable extent.
It will be convenient to introduce a unit that will be on a scale
adapted to our measurements. Let us think of a globe of coal as heavy as
the sun. Now suppose adequate oxygen were supplied to burn that coal, a
definite quantity of heat would be produced. There is no present
necessity to evaluate this in the lesser units adapted for other
purposes. In discussing the heat of the sun, we may use what we call the
_coal-unit_, by which is to be understood the total quantity of heat
that would be produced if a mass of coal equal to the sun in weight were
burned in oxygen. It can be shown by calculations, which will be found
in the Appendix, that in the shrinkage of the sun from an infinitely
great extension through space down to its present bulk the contraction
would develop the stupendous quantity of heat represented by 3,400
coal-units. It is also shown that one coal unit would be adequate to
supply the sun’s radiation at its present rate for 2,800 years.
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